text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
PREFIX="${ATS_BUILD_BASEDIR}/install"
REMAP="${PREFIX}/etc/trafficserver/remap.config"
RECORDS="${PREFIX}/etc/trafficserver/records.config"
TWEAK=""
[ "1" == "$enable_tweak" ] && TWEAK="-tweak"
# Change to the build area (this is previously setup in extract.sh)
set +x
cd "${ATS_BUILD_BASEDIR}/build"
./configure \
--prefix=${PREFIX} \
--with-user=jenkins \
--enable-ccache
# Not great, but these can fail on the "docs' builds for older versions, sigh
${ATS_MAKE} -i ${ATS_MAKE_FLAGS} V=1 Q=
${ATS_MAKE} -i install
[ -x ${PREFIX}/bin/traffic_server ] || exit 1
ldd ${PREFIX}/bin/traffic_server
# Get NPM v12
source /opt/rh/rh-nodejs12/enable
# Setup and start ATS with the required remap rule
echo "map http://127.0.0.1:8080 http://192.168.3.1:8000" >> $REMAP
${PREFIX}/bin/trafficserver start
set -x
cd /home/jenkins/cache-tests
npm run --silent cli --base=http://127.0.0.1:8080 > /CA/cache-tests/${ATS_BRANCH}.json
cat /CA/cache-tests/${ATS_BRANCH}.json
${PREFIX}/bin/trafficserver stop
# Now run it again, maybe, with the tweaked configs
if [ "" != "$TWEAK" ]; then
# echo "CONFIG proxy.config.http.cache.required_headers INT 1" >> $RECORDS
echo "CONFIG proxy.config.http.negative_caching_enabled INT 1" >> $RECORDS
${PREFIX}/bin/trafficserver start
cd /home/jenkins/cache-tests
npm run --silent cli --base=http://127.0.0.1:8080 > /CA/cache-tests/${ATS_BRANCH}${TWEAK}.json
echo "TWEAKED RESULTS"
cat /CA/cache-tests/${ATS_BRANCH}${TWEAK}.json
${PREFIX}/bin/trafficserver stop
fi
# We should check the .json file here, but not yet
exit 0
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,884 |
Crenidens crenidens, the karanteen seabream or karanteen, is a species of ray-finned fish from the sea bream family Sparidae which was described by the Swedish zoologist Peter Forsskål in 1775. It is native to the western Indian Ocean but has colonised the eastern Mediterranean Sea since 1970. It is one of only three species in genus Crenidens, the others being the little known Crenidens macracanthus and the partially sympatric C. indicus.
Description
Crenidens crenidens has an oblong to ovoid shaped body which is slightly compressed. Smaller adults, less than 19.7 cm in length, show a nose like bump in front of the eye. The mouth extends back to the anterior nostril and there are three rows of teeth in each jaw, the upper front jaw has 8-9 brown-tipped incisor like teeth, each bearing five denticulations which give the edge of the teeth a wavy appearance. The inner rows have a few teeth of similar form but the other teeth are granular. The scales finely ctenoid, the scaly cheeks contrasting with the scaleless interorbital region. The dorsal fin has eleven each of spiny and soft rays, while the anal fin has three spiny and ten soft rays. The caudal fin is forked. There are 52-60 scales making up the lateral line.
C. crenidens is silvery greenish-blue or olive in colour with darker narrow longitudinal stripes created by dark spots on the scales which are above the level of pectoral fins; the fins are coloured dull yellowish or olive with a dark margin on the dorsal fin and infrequently the axil pectoral fin shows darkish axils. They can grow to 30 cm in length in their native range, although the maximum in the Mediterranean is 20 cm, the more usual measurement is between 10 and 16 cm in length.
Distribution
Crenidens crenidens is indigenous to the western Indian Ocean from the Red Sea south along the coast of eastern Africa South Africa and has been also reported from southern Madagascar. In 1970 it was recorded for the first time in the Mediterranean in the Bardawil Lagoon (northern Egypt) and has since spread to Israel and to Libya. The most likely route for this species to have followed to colonise the Mediterranean is through the Suez Canal.
Biology
Crenidens crenidens is found in shallow coastal water, over sandy substrates which are frequently covered with sea grass. Its main food is algae but it also feeds on smaller invertebrates, such as crustaceans and worms. The eggs and larvae are planktonic. Off the Libyan coast C. crenidens has developed a distinct breeding season from November to February and the fish move away from the coast to spawn elsewhere in March and April. The males attain sexual maturity at 14 cm length and some females reach maturity at around 13 cm to 13.9 cm but for 50% of females maturity occurs at around 15.4 cm length. Fecundity is dependent on the weight of the female and varies from 678 eggs in smaller females to 9,888 eggs.
Taxonomy
There were two generally recognised subspecies:
C. c. crenidens (Forsskål, 1775) which occurs in the Red Sea and southwards along the eastern African coast;
C. c. indicus Day 1873 which extends from the Red Sea to the Persian Gulf and Nicobar Islands.
The two taxa occur sympatrically in the Red Sea. Recent work has supported the raising of C.c. indicus to species level once more and that a third species which is currently regarded as a junior synonym of C.c. indicus, Crenidens macracanthus, known from only two specimens from the Arabian Sea, is a separate valid species.
Uses
Crenidens crenidens is caught all year round in the northern Indian Ocean using trammel nets and beach seines and is consumed fresh, however, in the southern part of that Ocean it is fished mainly for bait.
References
Sparidae
Fish described in 1775
Taxa named by Peter Forsskål | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,496 |
Skin Revive Exfoliant is your multi-tasking skin cleanser and exfoliant in one, in a generous 75ml size! Our unique combination of natural microbeads and cleanser enhances the exfoliating process, with no harsh abrasive action. The cleansing properties of our hero ingredient Kumerahou gently loosen the cement (oil, dirt and debris) around dead skin cells, allowing them to be more effectively removed by the Jojoba and Candelilla wax beads without micro tears or damage to the skin. Skin is revived by revealing the younger cells beneath and stimulating healthy cell renewal, with a fresh, herby mint fragrance to awaken and refresh.
Aqua (water), Glycerin (vegetable glycerin)**, Decyl Glucoside, Glucose, Mel (manuka honey), Pomaderris kumerahou flowers/leaf extract, Euphrasia officinalis (eyebright), Phormium tenax leaf juice (NZ flax gel), Galactoarabinan, Hamamelis virginiana (witchhazel) extract, Xanthan gum, Glucose oxidase, Lactoperoxidase. ** Certified organic. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,094 |
Learn how to display and style a cluster of related content using Tabbed or Accordion panels in Adobe Muse.
Open the Widgets Library panel (Window > Widgets Library). Expand Panels, and select the Tabbed Panels widget. Drag the widget from the Widgets Library to your Design panel.
The Tabbed Panel widget displays the default formatting when you drag it onto a page.
Click the top tab on the right side three times. The first click selects the entire Tabbed Panel, the second click selects the Tab Container, and the third click selects the Tab on the right side.
Check the Selection Indicator to see which sub-element of the widget is currently selected. Press Delete (Mac) or Backspace (Windows) to delete the third tab. After making this change, there are only two tabs remaining. Click away, anywhere else on the page.
Click the widget once again, to select the entire Tabbed Panel. Drag the handles to resize the entire widget to a width of 840 and a height of 645. As you drag the handles, you can refer to the measurements that appear as you drag.
You can also check the Transform panel to see the values in the W (width) and H (height) fields.
Review the width and height fields to check the dimensions of the resized widget.
Use the Selection tool to position the Tabbed Panel widget on the page, near the top, and centered vertically.
Click the top center position of the Pin interface in the Control panel, to pin the Tabbed Panel widget in place. To learn more about pinning objects in position so that they don't scroll, see this link.
While the Tabbed Panel widget is still selected, click the left tab to select the Tab Container (the element that holds both top tabs). Use the Selection tool to drag the center handle down, until the height is approximately 98 pixels.
Set the spacing values for the Tab Container in the Spacing panel.
After making these changes, use the Selection tool to drag the entire Tabbed Panel widget up to the very top of the page, so that the top of the Tabbed Panel is butted up against the bottom of the header rectangle with no vertical space between the two.
If necessary, you can zoom in to get a closer look of how the elements are aligned. Set the magnification back to 100% after you are finished.
Now that you've added the Tabbed Panel widget, the next part involves updating the styles to design the two tabs.
Click the left tab three times, to select the left Tab. Open the States panel and select the Normal state. Use the Fill menu to set the fill color. Set the stroke width to 0.
Select the Rollover state in the States panel. Set the fill color of the Rollover state of the tab to 571E00. Notice that when you set the fill color for the Rollover state, the Mouse Down state also updates automatically.
Select the Active state and also apply the same fill color to the tab while in the Active state. Now the Normal state is filled with the fill color you chose in the previous step, and the other three states are filled with 571E00.
Style the label text by updating the settings in the Text panel.
While the Label is still selected, click the New Style button at the bottom of the Paragraph Styles panel. To reapply this formatting whenever you want, with a single click, rename this new paragraph style. For example, head-tabs.
When you look in the States panel, the formatting you applied to the Label while the Normal state is selected is also applied to both the Rollover and Mouse Down states. Select the Active state and then click the head-tabs paragraph style again, to apply the formatting to the Active state. Now all the states in both the tabs have been styled.
Select the Normal state again.
Use the Text tool to select the label text in all the tabs, and enter appropriate names for your labels.
If you select the left Tab and look at the States panel, you'll see the final formatting applied to each tab and label. Click the right tab and notice that the states look the same, except for the text content of each tab.
Review the formatting of each tab in the States panel.
Take a moment to click back and forth between the two different tabs, that have different labels. The placeholder text for both the tabs are different, so you can tell that one container is displayed as the other is hidden.
Click the right tab in your widget. When the right tab is selected, the corresponding content area is displayed below the tab. This is the same behavior that occurs both in Design view when editing the page and on the live website. This is what makes Tabbed Panel widgets very helpful, because you can efficiently display much more content within a single page.
Select the Content Area for your tabs. The Selection Indicator displays the words Content Area when it is selected. Select the placeholder image file and press Delete (Mac) or Backspace (Windows) to delete it.
Use the Text tool to select the existing placeholder text header. Select the bold header text, Mauris sit amet, and delete it.
Temporarily switch from Adobe Muse to your desktop. Open the sample files folder and locate the file named text-about-ourstory.txt. Double-click the file to open it in a text editor. Copy the first portion of the page, up until the line that reads: ... crafted through years of baking for friends and family.
Return to Muse. Use the Text tool to select the existing placeholder text. Paste the text content you copied in the text frame. Leave one extra line and return to the top of the text frame.
While the text is still selected, click the New Style button at the bottom of the Paragraph Styles panel. Double-click the new paragraph style and rename it. For example, body. This will make it easier to reapply the same formatting to other text content.
In the Swatches panel, double-click the color value #222222 and rename this color. For example, katieblack.
In the Control panel, set the Stroke color of the text field to #222222 (katieblack) and set the Stroke Width to 5 pixels.
Use the Fill menu to set the fill color of the text field to a light beige color (#F8F3E2). Click the folder next to the Image section and browse to select the file in the sample files folder named bg-texture.png. Set the Fitting menu to: Tile.
In the Spacing panel, set the Left and Right spacing. For example, you can set the Left spacing to 24 and the Right spacing to 15.
Set the spacing values to create visual space around the pasted text copy.
Next, you'll add content to each of the content area containers that correspond to each of the tabs.
Use the Selection tool to resize the text field. Drag it to the left side of the container, and then use the handles to expand its width to approximately half of the available content area space (approximately 390 pixels wide by 381 pixels high).
Next, you'll duplicate the existing text frame, to create a second text frame that fills the right side of the content area.
Use the Selection tool to select the text frame. Press and hold Option (Mac) or Alt (Windows) as you drag a duplicate copy of the text frame to the right side of the content area (see the image below).
Use the alignment guides that appear to ensure that the right text frame is aligned horizontally with the left text frame.
Duplicate the existing text frame, to create a second column on the right side.
Temporarily switch from Muse to the text-editing program that contains the content for your Tabbed Panel. Copy the content that you want to paste in your duplicate tab.
Switch back to Muse. Delete all the existing text in the duplicated right text frame. Paste the new text content that you copied from the clipboard.
As needed, use the Selection tool to position both text frames so that they are aligned and centered within the content area at the desired location.
While the text frame is still selected, press Escape once to select the Content Area of the duplicated tab. Set the stroke width to 0, and set the fill color to None.
Copying text frames is almost complete. The last part involves adding a small image that wraps inside the text of the left text frame.
Accordion Panel widgets are helpful when you want to fit a lot of content into a smaller area of screen real estate. The collapsing and expanding behavior allows visitors to click a label panel and see the corresponding content area appear. In addition to desktop sites, Accordion Panel widgets are often used to display content in mobile layouts.
In this section, you'll learn how to add content to the individual pages in your site. You'll work with another type of widget, called an Accordion widget. The Panel widgets (both Accordion and Tabbed Panel widgets) are helpful because they make it possible to display more content on a page within a smaller area of screen real estate.
Open your project in the Design view and begin editing it.
Choose File > Place. In the Import dialog box, navigate to the folder where you have your assets, and select the image file you need. Click Select, and then click once at the top of the page to place the image at its original size on the page as the header text. You'll add the Accordion widget below this header.
Click Panels to expand the list of Panel widgets. Select the Accordion widget. Drag the Accordion widget onto the page and position it in the upper left side of the page content region.
Drag and place the Accordion widget on your page.
With the Selection tool, select the various components of the widget. As you select the entire widget, the Selection Indicator in the top left corner of the Control panel displays the word "Widget." As you click again, notice how the Selection Indicator updates to tell you that you have selected a container, or a text frame. The Selection Indicator tells you which part of the widget is currently selected.
To deselect that element of a widget, either click the Escape key to back out of the current selection one level, or press it repeatedly to back up out of the nested elements. Alternatively, you can simply click away from the widget and onto another part of the page.
By default, the Accordion widget has two panels. The top panel displays the placeholder tab name Lorem 1 and the bottom panel displays the placeholder tab name Ipsum 2.
Open the Widget Options dialog box by clicking the round blue arrow icon. Ensure that the option Edit Together is checked. This option ensures that any changes you make to one tab name in the Accordion will be applied to all tabs.
Ensure Edit Together is checked in the Option panel in Accordion widgets.
With the Selection tool, click the widget once to select it, and then drag the right handle to the right to expand the width of the widget, making it wide enough to cover two columns as shown in the image below.
While the widget is selected, drag the right handle to expand it to cover two columns.
Click the plus (+) sign at the bottom of the widget, below Ipsum 2, to add a third panel. The third panel displays the placeholder tab name Ipsum 3.
In the next section, you'll learn how to place text content into a panel of the Accordion widget to populate it. But first, you'll update the top tab name that appears above the first panel in the Accordion widget.
Use the Selection tool to select the top tab's text frame, currently named Lorem 1. Click "Lorem 1" once to select the entire widget and expand the container that corresponds to the top tab, and then click in the "Lorem 1" text to select the text frame. The words "Text Frame" are displayed in the Selection Indicator when the text frame is selected.
Double-click the placeholder text "Lorem 1" to edit it and then type the text that you want.
Select the light gray larger container beneath the first tab. Use the Text tool to drag open a text frame in the larger container.
In the new text frame, type your header text.
Select the top tab of the Accordion widget that displays the header.
Click the round blue arrow button to open the Options menu. Deselect the option Edit Together. Click away from the Options menu to close it.
Click the top tab twice until the Selection Indicator displays the words "Text Frame." With the top tab's text frame selected, click the Fill link in the Control panel to open the Fill menu. Click the color picker and set the background fill color to none, rather than the default dark gray color.
Click the folder icon in the Image section to open the Import dialog box. Navigate to your assets folder, and select the image that you want to add.
Open the Spacing panel. Press and hold the up arrow in the Padding: L (Left) section to increase the space before the header so that the text is centered on the background image for the tab.
Use the Spacing panel to increase the left padding, and center text in the accordion.
The first tab of the Accordion widget is complete.
If you deselected the Edit Together setting when you edited the first tab, the changes do not appear for the other tabs. You have to seperately style the other tabs in your Accordion Panel.
By default, Accordion Panel widgets are set up to display the label panels on top and the content areas below, in a vertical orientation.
When you add an Accordion Panel widget to a page, it is displayed vertically.
In Design view, open the Widgets Library and expand the Panels section.
Select the Accordion option from the Panels section and drag it to the page.
Using the Selection tool, click once to select the entire widget. The Selection Indicator in the top left corner of the Control panel displays the word: Accordion.
Holding the Shift key while rotating constrains the proportions so you can move the widget in 45-degree increments. A tooltip appears to display the current rotate value as you are rotating the widget.
Rotate the widget to -90° or +90°, depending which way you choose to rotate the accordion.
After rotating the widget, click the Preview link or preview the page in a browser to see that the Accordion still expands and collapses the panels as expected. Check whether the label containers are scripted to display the corresponding content area when clicked.
Now all that remains is to populate the labels and content area containers with content that is oriented "face up" instead of displaying sideways.
Follow these steps to display text content within the content areas. You'll use the same technique described above to rotate the text field in a content area container.
Click the default text in the content area twice. The first time selects the entire Accordion widget and the second time selects the content area container. While the content area is selected, use the Selection tool to expand the size of the content area, by clicking and dragging its resize handles.
While the content area is still selected, click once again to select the text frame inside the content area. Hover your cursor near one of the text frame corners until you see the rotation cursor displayed. Press and hold the Shift key while rotating the text frame to the same 90-degree rotation as the outer widget.
Rotate the existing placeholder text content if you want to display text in the content area.
After rotating the text frame, use the Selection tool to reposition it within the content area. Use the Text tool to replace the placeholder text with the actual text you want to display and then use the text formatting options in the Control panel and the Text panel to style the text (or apply paragraph styles to update its appearance).
Place images, draw rectangles, or embed HTML to populate the content area containers as desired. You don't have to rotate this content because it is placed upright within the rotated container automatically.
To update the text in the label containers, you have several options.
If the labels are short and you want the text to remain "sideways" (to simulate books on a bookshelf, for example) you can use the Text tool to select the existing placeholder text and type the desired labels (see the image below).
Edit the label text using the Text tool for Accordion widgets.
If you do not want to use text labels, you can simply delete the existing placeholder text in the label containers and then use the formatting options in the Control panel or Fill panel (along with the States panel) to apply a different color to each label (see the image below). To update the appearance of each label container individually, click the blue arrow button to access the Options menu and disable the Edit Together option.
Use the Fill menu and text formatting options to color code the label containers and the corresponding text content.
Another option is to use an image-editing program to create images that are inserted in the label panels. Place the images on the page, cut them, select the label text with the text tool, and then paste the images into the label containers. If desired, the images can contain vertically oriented text (see the image below).
Insert images with vertically oriented text in the label containers.
Select the Accordion widget where you want to add the anchor link. Click the tab where you want to add the anchor link.
Click the Anchor tool to load an anchor link in the Place Gun.
Select the Anchor tool from the Tools panel.
Place the anchor link in the Accordion widget.
A pop-up to rename the anchor is displayed. Type in a suitable name for your anchor, and click OK.
You can now link this anchor to any page element from any page in your website.
To link a page element to this anchor, select the page element that you want to link. From the Hyperlinks drop-down on top, choose the anchor that you created.
In your live site, whenever you click the page element, the page displays the Accordion tab where you had placed the anchor link. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,143 |
Q: Multiple font colours in a single row in a JList I am creating an application in which we show a couple hundred records in a JList.
Occasionally, we would want to highlight certain values in each record with a different font colour.
I have tried using HTML which works perfectly, although this would go horribly wrong if any of these records contained any sort of html tags, as they would be rendered.
Unforunately you can't have:
normaltext<html><font color="red">redtext</font></html>
as it seems the HTML has to be at the beginning.
I have tried overriding the getListCellRendererComponent, where I can create a JPanel, and have a JLabel for each part of the record with a different foreground colour, but this is inefficient for large records.
I'd rather not use any third party solutions due to licensing issues.
Does anyone have any solutions for this scenario?
Thanks
A:
I have tried using HTML which works perfectly, although this would go
horribly wrong if any of these records contained any sort of html
tags, as they would be rendered.
Actually if the text contains html tags you can escape the tags and wrap it in <html> escaped text of record</html> marking text wth desired colors
UPDATE working example
public static void main(String[] args) {
JFrame f=new JFrame();
String text="<html>An example of tag <html> with <font color='red'><input></font> tag colored</html>";
f.add(new JLabel(text));
f.pack();
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.setVisible(true);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 447 |
In der Liste der Kulturdenkmale in Kellinghusen sind alle Kulturdenkmale der schleswig-holsteinischen Stadt Kellinghusen (Kreis Steinburg) aufgelistet (Stand: 6. Januar 2020).
Legende
Sachgesamtheiten, Bauliche Anlagen und Gründenkmale
|}
Weblinks
Quelle
Kellinghusen
! | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,696 |
Author: Quote Banq
A vault of quotes that contains enough wisdom to last an eternity.
Self- Esteem
"Poor self esteem places us in an adversarial relationship with our own well being." "Self Esteem is the reputation we acquire with ourselves." -Nathaniel Branden https://www.youtube.com/watch?v=8UGJ5efn_kU https://www.youtube.com/watch?v=mfFUVnwCNVY
Do your work then step back…
"When starting a new task, project or anything of importance, let us remember to quote the Tao Te Ching, "The only path to serenity is to do your work and…
That Inner Spark
"How many of us have had a spark of insight that could change our lives? How often have we let the forces of doubt and fear put that spark out?"…
The Average Citizen
"It is interesting how the average citizen cares little about politics, yet politics cares greatly about the average citizen." -R.B. Royal
12 Feb 2018 15 Apr 2018
"The greater the number of owners, the less the respect for common property. People are much more careful of their personal possessions than of those owned communally; they exercise care…
Guiding Forces
"The economy and progress are driven by businesses, individuals and governments and guided by emotions and irrationality. Though it should be intellect and logic with the reigns." -R.B. Royal
"Most people will come up with ten reasons why they can't do something rather than come up with one good reason why they can and should. Be the kind who…
"A great idea without commitment and execution will leave you discouraged, doubtful and negative. Don't believe the excuses you tell yourself" -R.B. Royal
"If voting mattered they wouldn't let you do it." -George Carlin
Lets Fix Society
8 Feb 2018 15 Apr 2018
"Want to know what is wrong with a child? Look at the parents. Want to know whats wrong with the parents? Look at society and culture. Look at its values.… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,566 |
Q: iOS: Timer animation on separate thread I have a simple animation to make a label look like its counting upwards, which I put in a timer.
NSTimer *timer = [NSTimer scheduledTimerWithTimeInterval:0.01f target:self selector:@selector(animateScore:) userInfo:nil repeats:YES];
[[NSRunLoop mainRunLoop] addTimer:timer forMode:NSRunLoopCommonModes];
The animate score selector increments the labels text until it gets to the desired integer then invalidates the timer.
Right now, it stalls while other UI stuff is going such as reloading table view sections.
I tried running this code, but it didn't work:
dispatch_async(dispatch_get_global_queue(DISPATCH_QUEUE_PRIORITY_DEFAULT, 0), ^{
NSTimer *timer = [NSTimer scheduledTimerWithTimeInterval:0.01f target:weakSelf selector:@selector(animateScore:) userInfo:nil repeats:YES];
[[NSRunLoop mainRunLoop] addTimer:timer forMode:NSRunLoopCommonModes];
});
A: The preferred way to do this is using a special timer class CADisplayLink, which fires every time the device's screen is updated. Something like:
@property (nonatomic, strong) CADisplayLink *displayLink;
-(void)viewDidLoad {
// …
self.displayLink = [CADisplayLink displayLinkWithTarget:self selector:@selector(displayLinkDidFire:)];
[self.displayLink addToRunLoop:[NSRunLoop mainRunLoop] forMode:NSRunLoopCommonModes];
// …
}
-(void)displayLinkDidFire:(CADisplayLink *)displayLink {
// Update label text
}
A: scheduledTimerWithTimeInterval:target:selector:userInfo:repeats: already adds the timer to the current run loop (which may not be the main run loop since you execute this on a background queue).
Use timerWithTimeInterval:target:selector:userInfo:repeats: instead. This creates the timer but does not add it to the current run loop.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,692 |
Q: Showing 1st 25 record or 1st 50 record on Kendo Grid I have the codes below:
$("#grid").kendoGrid({
dataSource: {
data: setData(),
pageable: false
},
scrollable: true,
sortable: true,
selectable: true,
columns: [
{ field: "Name", title: "Name", width: 230" },
{ field: "Sex", title: "Sex", width: 50},
{ field: "Ca", title: "C.A." , width: 55},
{ field: "TotalScore", title: "Total Score", width: 100},
{ field: "Rank", title: "Rank", width: 60 }
]
});
$("#button1").on("click", function() {
// show 1st 25 records
});
$("#button2").on("click", function() {
// show 1st 50 records
});
Now what I want, when the I click a button "button1," it will only show the 1st 25 records or when i click "button2" it will only show the 1st 50 records.
Is that possible?
Thanks
A: *
*create a function named bindGrid:
function bindGrid(data) {
if ($("#grid").data("kendoGrid"))
$("#grid").data("kendoGrid").destory()
$("#grid").empty();
$("#grid").kendoGrid({
dataSource: {
data: data,
pageable: false
},
scrollable: true,
sortable: true,
selectable: true,
columns: [
{ field: "Name", title: "Name", width: 230 },
{ field: "Sex", title: "Sex", width: 50 },
{ field: "Ca", title: "C.A.", width: 55 },
{ field: "TotalScore", title: "Total Score", width: 100 },
{ field: "Rank", title: "Rank", width: 60 }
]
});
}
*store grid data in a global variable and call bindGrid function:
var gridData=setData();
bindGrid(gridData);
*Now handle button clicks and rebind grids
$("#button1").on("click", function () {
// show 1st 25 records
var newDataSource = [];
$.each(gridData, function (i, item) {
if (i < 25) {
newDataSource.push(item);
}
});
//rebind grid
bindGrid(newDataSource); });
$("#button2").on("click", function () {
// show 1st 50 records
var newDataSource = [];
$.each(gridData, function (i, item) {
if (i < 50) {
newDataSource.push(item);
}
});
//rebind grid
bindGrid(newDataSource); });
A: $("#grid").kendoGrid({
dataSource: {
data: setData(),
pageable: false
},
scrollable: true,
sortable: true,
selectable: true,
columns: [
{ field: "Name", title: "Name", width: 230" },
{ field: "Sex", title: "Sex", width: 50},
{ field: "Ca", title: "C.A." , width: 55},
{ field: "TotalScore", title: "Total Score", width: 100},
{ field: "Rank", title: "Rank", width: 60 }
]
});
$("#button1").on("click", function() {
$("#grid").data("kendoGrid").dataSource.pageSize(25);
});
$("#button2").on("click", function() {
$("#grid").data("kendoGrid").dataSource.pageSize(50);
});
Example here: http://dojo.telerik.com/aQID
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,635 |
{"url":"https:\/\/www.techwhiff.com\/learn\/prelab-1-in-your-lab-notebook-draw-the-schematic\/407847","text":"# Prelab 1. In your lab notebook draw the schematic diagram for a three-phase wye-connected induction motor...\n\n###### Question:\n\nPrelab 1. In your lab notebook draw the schematic diagram for a three-phase wye-connected induction motor with its three-phase stator winding connected to the three-phase power supply through 25-\u03a9 resistors. Connect two er analyzers in the circuit (between the 25-\u03a9 resistors and the motor) to read the total power absorbed by the induction motor. What is the function of the resistors inserted in the circuit? 2. After reading this lab handout, draw the circuit diagram for the induction motor-synchronous generator set as described.\n\n#### Similar Solved Questions\n\n##### Question 19 4 pts A survey of 64 families yields the following data for the number...\nQuestion 19 4 pts A survey of 64 families yields the following data for the number of children per family. Number of Children Number of Families (Frequencies) 8 1 11 2 18 11 3 4 6 5 4 2 6 7 1 8 2 1 9 What proportion of the families in the survey had more than 4 children? Express your answer as a dec...\n##### Describe the receptor for glucagon or ephinephrine, and the steps leading to the activation of Protein...\nDescribe the receptor for glucagon or ephinephrine, and the steps leading to the activation of Protein Kinase A....\n##### The following two assets and payout data are given below. Asset A: Pays a return of...\nThe following two assets and payout data are given below. Asset A: Pays a return of $2,000 20% of the time and$500 80% of the time. Asset B: Pays a return of $1,000 50% of the time and$600 50% of the time If both assets can be acquired for the same price, as a risk-averse investor, you would prefe...\n##### Explain the process called electrophoresis to distinguish hidden genetic variation in the form of isozymes.\nExplain the process called electrophoresis to distinguish hidden genetic variation in the form of isozymes....\n##### The reader understands derivatives, and knows the definition of instantaneous velocity (dx\/dt), and knows how to...\nThe reader understands derivatives, and knows the definition of instantaneous velocity (dx\/dt), and knows how to calculate integrals but is struggling to understand them. Use students\u2019 prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of v...\n##### Taking exam, please help! calculate ending inventory and cogs using the following inventory costing methods. THANKYOU...\nTaking exam, please help! calculate ending inventory and cogs using the following inventory costing methods. THANKYOU 10.00 points Consider the following information for Maynor Company, which uses a periodic inventory system Transaction Units Unit Cost Total Cost January 1 Beginning Inventory 31 Mar...\n##### Java coding question: Write a complete program that contains a method named repeat that takes a...\njava coding question: Write a complete program that contains a method named repeat that takes a String as a parameter and returns nothing The method will print (use System.out.println) the input string 5 times. The main method will call the method repeat with the value \"Java\"....\n##### QUESTION 11 a value represents the 95th percentile, this means that: 0 95% of all values...\nQUESTION 11 a value represents the 95th percentile, this means that: 0 95% of all values are below this value 0 95% of all values are above this value 0 95% of the time you will observe this value O there is a 5% chance that this value is incorrect O there is a 95% chance that this value is correct ...","date":"2023-02-07 05:57:01","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\": 0.33853477239608765, \"perplexity\": 1314.2902991575575}, \"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-2023-06\/segments\/1674764500384.17\/warc\/CC-MAIN-20230207035749-20230207065749-00186.warc.gz\"}"} | null | null |
using System;
using System.Collections.Generic;
using System.ComponentModel.DataAnnotations;
using System.Text;
using CSC.CSClassroom.Model.Classrooms;
using CSC.CSClassroom.Model.Users;
namespace CSC.CSClassroom.Model.Users
{
/// <summary>
/// Represents a class admin that has chosen to receive messages
/// from students in a particular section.
/// </summary>
public class SectionRecipient
{
/// <summary>
/// The ID of the section message recipient.
/// </summary>
public int Id { get; set; }
/// <summary>
/// The recipient.
/// </summary>
[Display(Name = "Admin")]
public int ClassroomMembershipId { get; set; }
public ClassroomMembership ClassroomMembership { get; set; }
/// <summary>
/// The section for which the user will receive messages.
/// </summary>
public int SectionId { get; set; }
public Section Section { get; set; }
/// <summary>
/// Whether or not to view announcements to this section.
/// </summary>
[Display(Name = "View Announcements")]
public bool ViewAnnouncements { get; set; }
/// <summary>
/// Whether or not to receive e-mails for announcements to this section.
/// </summary>
[Display(Name = "E-mail Announcements")]
public bool EmailAnnouncements { get; set; }
/// <summary>
/// Whether or not to view messages from students in this section.
/// </summary>
[Display(Name = "View Messages")]
public bool ViewMessages { get; set; }
/// <summary>
/// Whether or not to receive e-mails for messages from students in this section.
/// </summary>
[Display(Name = "E-mail Messages")]
public bool EmailMessages { get; set; }
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,030 |
Come along to our FREE mini me art sessions and have some quality time with your little ones. Our free activity is running from 10.30am – 12.30pm on the following dates with a different theme each time.
Just turn up and enjoy the fun!
Entry tickets must be purchased or a valid Annual pass must be held for you to take part in this activity.
Godstone Farm reserve the right to change or cancel these activities at any time.
Godstone Farm reserve the right to change the terms and conditions at any time.
This activity is subject to availability, it is on a first come first serve basis.
Godstone Farm will allocate the final location of the craft workshop on the farm, on the day.
Introducing our beautiful miniature Shetland foal, Xyla! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,466 |
Q: I'm passing a list of JSON file is not getting display in the Android application I was Passing a this JSON http://www.mocky.io/v2/5cacde192f000078003a93bb
i was trying to print a just a category_name
I'm not able to get the data list , when i pass the object with out the data list just like http://www.mocky.io/v2/5cb859344c0000092ed3d4df
private Category_name category_name;
public Category_name getCategoryName() {
return category_name;
}
}
public class Category_name {
@SerializedName("category_name")
public String name;
public String getName() {
return name;
}
}````
i can access that through the NewAdapter.java
with the following code
@Override
public void onBindViewHolder(NewsViewHolder holder, int position) {
Log.e("Its coming","NewAdapter");
ApiObject apiObject = apiObjectList.get(position);
holder.title.setText(apiObject.getCategoryName().getName());
}
with the same code I'm not able to get the data list
@SerializedName("data")
public List<Data> data;
public List<Data> getData() {
return data;
}
public class Data {
@SerializedName("details")
private Category_name category_name;
public Category_name getCategoryName() {
return category_name;
}
}
public class Category_name {
@SerializedName("category_name")
public String name;
public String getName() {
return name;
}
}
@Override
public void onBindViewHolder(NewsViewHolder holder, int position) {
Log.e("Its coming","NewAdapter");
ApiObject apiObject = apiObjectList.get(position);
holder.title.setText(apiObject.getData().getCategoryName().getName());
}
I'm not able to access the getCategoryName();
Please help thanks in advance
A: use json 2 pojo conversion to create proper model class of json data
http://www.jsonschema2pojo.org/
pass whole example object to adapter constructer.
A: I think you need to follow these way of POJO parsing according to your JSON response.
public class Data{
@Serialization("status")
@Expose
private String status;
@Serialization("data")
@Expose
private List<MyData> data;
Then
public class MyData{
@Serialization("details")
@Expose
private List<Details> getDetails();
@Serialization("product_count")
@Expose
private String Product_count;
@Serialization("products")
@Expose
private List<Products> getProducts();
//setter and getters
}
Details POJO
Public class Details{
@Serialization("category_id")
@Expose
private String category_id;
@Serialization("category_name")
@Expose
private String category_name;
@Serialization("category_icon")
@Expose
private String category_icon;
//setter and getters
}
Products POJO
Public class Products{
@Serialization("product_id")
@Expose
private String product_id;
@Serialization("product_name")
@Expose
private String product_name;
@Serialization("product_image")
@Expose
private String product_icon;
etc
//setter and getters
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
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This month is all about IPL. Hoardings, radio ads, tv ads, newspapers. The IPL Fever is on everywhere and Hyderabad has not been spared. Our home team – The Sunrisers Hyderabad is in full form and this is evident with them being on the top in the IPL Scoreboard. Their previous home match played against KXIP or the debut match against comeback team Rajasthan Royals, were all a worthwhile Experience with the orange team holding 3 victories in the home ground.
The Ease of Booking:Unlike earlier you don't have to stand in a crowded line and shout which seat you want while the crowd moves you like a pendulum and you get to smell sweaty underarms. Say goodbye and thank God! You can simply book tickets for all SRH home matches at your own ease, while you swirl the chair; at EventsNow.com. You can select the stand you want, the number of tickets and simply book tickets using various payment modes offered. All you pay is a few bucks as convenience fee. But, convenience this kind, is worth it.
Redemption:Wish you could skip the line to redeem the tickets you purchased online? Yay! Looks like your prayers have been answered. EventsNow has set up 'just for you' redemption centres at 3 different locations including Gymkhana Grounds in Secunderabad, LB Stadium in Basheerbagh, and Saroor Nagar Indoor Stadium. So, all you must do is visit the one of the redemption centres, show the OTP and your booking ID to get the tickets.
Retail Counter:You don't have to be worried about booking tickets online only. EventsNow.com allows customers to book tickets via various methods and retail counters are one of them. All you need to do is decide the match you wish to watch, the stand you wish to book for and leave the rest to the ticket booking team. Not sure about using card? Don't worry, you can pay for the tickets using Cash.
Seating:To offer a great experience, the seating for all home matches has been divided into 4 parts. While North and South blocks offer behind the pitch view, East and West offer side view of the pitch. There are 3 levels of seating Ground floor, First Floor and Terrace.
Pricing:Who said good things come at a great cost? To let you enjoy the match live amidst crazy cheering and great excitement, the prices for tickets begin from Rs.500 onwards.
Special Corporate Team:EventsNow has a specialised Corporate team which can help you book bulk and corporate tickets at one go without worrying about availability. You can also book an enclosed airconditioned box with full hospitality to enjoy matches like a VIP.
Parking:Worried about parking? Sit back and relax! Because each stand has its own dedicated parking. Simply drive, park, lock and go to watch the match hassle free without worrying about your vehicle.
Food & Beverages: Love to eat while you enjoy the match? You don't need to carry food from home. Various food stalls will be setup at the stadium to let you scintillate your taste buds.
Customer Service:Along with on ground Staff, a team of customer service team will be available to answer all your queries and payment related worries. Wondering where? They are just a call away.
With so many facilities and many more, cheering for the SRH team can never be boring. Clear your throats to shout loud for the team as you watch Bhuvaneshwar Kumar pick wickets and Shikhar Dhawan once again play in the ground as the opening batsman again. But, before it all visit https://www.eventsnow.com/ to book your tickets because entry without tickets is strictly prohibited.
See you at the next match in the Stadium! | {
"redpajama_set_name": "RedPajamaC4"
} | 259 |
Bathroom Vanity Double Sink Interior Architecture | Partitions Net bathroom vanity double sink 48. bathroom vanity double sink top. bathroom vanity double sink 48 inches.
Elegant Bathroom Vanity Double Sink In Vanities Furniture. Glamorous Bathroom Vanity Double Sink On Adorna 92 Inch Transitional White Finish. Picturesque Bathroom Vanity Double Sink At Best Costco Epic 75 On. Vanity Bathroom Double Sink In Buy Vincent 72 Inch Solid Wood Charcoal. Mesmerizing Bathroom Vanity Double Sink On Great 49 Photos HTSREC COM. Miraculous Bathroom Vanity Double Sink On Home Decorators Collection Aberdeen 60 In W X 22 D Bath. Likeable Bathroom Vanity Double Sink On Luxury 60 Inch. Miraculous Bathroom Vanity Double Sink At Awesome TEDx Design. Awesome Bathroom Vanity Double Sink At Bliss 60 Gray Oak Mount Modern. Entranching Bathroom Vanity Double Sink Of Startling Costco Vanities Elegant Sinks. Awesome Bathroom Vanity Double Sink On Bathrooms Design 60 2. Captivating Bathroom Vanity Double Sink On Unique 55 Inch Shower Room Idea. Wonderful Bathroom Vanity Double Sink Of With Center Tower Cabinet Linen. Unique Bathroom Vanity Double Sink At 16 Large Single Com. Spacious Bathroom Vanity Double Sink In 67 Inch With Travertine Counter Top. Exquisite Bathroom Vanity Double Sink On 60 Silo Christmas Tree Farm. Vanity Bathroom Double Sink On Brilliant Sofa Stunning Ideas 30 With In. Amusing Bathroom Vanity Double Sink Of Adorna 88 Inch Set With Trough Style Sinks. Romantic Bathroom Vanity Double Sink Of Bath Photo Gallery Dakota Kitchen Sioux Falls SD. Glamorous Bathroom Vanity Double Sink At Vanities Lowe S Canada. Entranching Bathroom Vanity Double Sink In 72 Perfecta PA 5126 Cabinet Dark. Glamorous Bathroom Vanity Double Sink Of Luxury 60 Inch. Minimalist Bathroom Vanity Double Sink On Home Decorators Collection Hamilton 61 In W X 22 D Bath. Entranching Bathroom Vanity Double Sink At Amazing Top Notch Wall Mounted Floating. Elegant Bathroom Vanity Double Sink Of Shop Ancerre Designs Audrey Sapphire Gray With White Natural. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,246 |
{"url":"https:\/\/dataspace.princeton.edu\/handle\/88435\/dsp010c483n01s","text":"Please use this identifier to cite or link to this item: http:\/\/arks.princeton.edu\/ark:\/88435\/dsp010c483n01s\n Title: A Scholzian Approach to the Local Langlands Correspondence for $$\\mathrm{GL}_n$$ over function fields Authors: Li, Daniel Advisors: Morel, Sophie M. Contributors: Taylor, Richard L. Department: Mathematics Class Year: 2017 Abstract: Let $$F$$ is a local field of characteristic $$p$$. Inspired by work of Scholze, we construct a map $$\\pi\\mapsto\\sigma(\\pi)$$ from irreducible smooth representations of $$\\mathrm{GL}_n(F)$$ to $$n$$-dimensional Weil representations of $$F$$. We prove that this map uniquely satisfies a purely local compatibility condition on traces of a test function $$f_{\\tau,h}$$, and we also prove that this map is compatible with parabolically inducing tensor products. It is expected that $$\\pi\\mapsto\\sigma(\\pi)$$ equals the local Langlands correspondence for $$\\mathrm{GL}_n$$ over $$F$$, up to Frobenius semisimplification. URI: http:\/\/arks.princeton.edu\/ark:\/88435\/dsp010c483n01s Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Mathematics, 1934-2020","date":"2020-10-26 14:11:48","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\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8385453820228577, \"perplexity\": 1034.2236314288011}, \"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-2020-45\/segments\/1603107891228.40\/warc\/CC-MAIN-20201026115814-20201026145814-00639.warc.gz\"}"} | null | null |
Iran rushing toward 30 percent inflation
Home/Iran/Iran rushing toward 30 percent inflation
A currency exchange office in the Iranian capital, Tehran, August 2018. (ILNA)
By Ali Ranjipour
A government-run statistics unit has published its monthly report on Iran's consumer price index, revealing a new hike in consumer goods — a rise in prices that is hitting lower-income families and rural communities hardest.
On Thursday, November 22, the Statistical Center of Iran published its monthly report. According to the study, the general price for goods and services has increased by a significant 34.9 percent since the same month one year ago. The report looks specifically at the cost of food, and reveals that inflation was actually higher for rural households — an abnormal situation that could have devastating consequences.
What Costs were Most Affected?
According to statistics published by the center, the highest increase in prices for consumer items related to food and beverages, which rose by 4.3 percent for the month and by a shocking 48 percent for the year. The lowest rate of inflation was seen in the housing and fuel sector, which rose by a little over 20 percent.
It is worth noting that housing and fuel costs make up the biggest share — almost one-third — of expenses for an average Iranian household. With this in mind, if the increase in the prices of housing and fuel equaled those in other sectors, this would amount to a much higher inflation rate in consumer items than the current 34.9 percent.
When it comes to food, prices for all items except bread, cereal and sugar rose by more than 43 percent for the year. Iranian households purchase bread and cereals more than any other food items. This means that if the inflation rate for these items had been similar to other items, the rate for food and consumer prices in general would have been much higher.
Rural vs Urban Areas
The statistics show that the inflation for the previous month, as with the month before it, was higher in rural areas than in urban areas. Rural households spent 37 percent more on consumer goods compared to the same month last year, whereas in rural areas the spending was around 34 percent more.
A high inflation rate in rural areas is not a normal phenomenon, and it can have devastating consequences. Usually, the cost of living and inflation linked to it is lower in rural areas than in the cities, but in recent months the reverse has happened, making the lives of Iranian villagers even more difficult.
For the last 50 years the increased migration from rural to urban areas has brought about one of the biggest social problems in Iran, and has led to social chaos and the expansion of margin-dwelling in cities. Up until recently, the main motivation for migrants has been perceived or real opportunities in the cities and the hope of having a greater income, but a higher cost of living in rural areas compared to cities is bound to make margin-dwelling even worse.
The Effects on Lower-Income Families
Considering that the increase in food prices plays an important role in the current structure of the country's inflation, it is middle class and lower-income communities that suffer more; the impact on more affluent households is obviously less dramatic.
The above chart illustrates the increase in the cost of household expenditures for the Iranian calendar month of Aban 1397 (October 23-November 22, 2018). As can be seen, the inflation for the second to seventh deciles of the population are almost the same. Then it follows a downward path from the seventh to the tenth deciles, which refers the most affluent households. The first decile, the poorest one-tenth of the population's households, stands out. The inflation for this decile of households is lower probably because they are forced to buy less food, and specifically less meat, fruit and fresh vegetables. They also spend comparatively less on bread and cereals.
The Inflation Crystal Ball
Not long ago, the website Rouhani Meter, which keeps scores on President Rouhani's promises, estimated that if the monthly inflation rate remains at 2.5 percent, the annual rate of inflation, calculated by the Central Bank and the Statistical Center of Iran, would reach to between 27 to 30 percent. Five and a half years ago, when Rouhani was first elected to the presidency, he promised to bring inflation down to 25 percent. And during his campaign for the 2017 presidential election, he promised to keep the inflation rate to single digits. Neither promises have been fulfilled and, under current conditions, it is questionable whether they will be fulfilled anytime soon.
Iran Wire
About Track Persia
Track Persia is a Platform run by dedicated analysts who spend much of their time researching the Middle East, in due process we fall upon many indications of growing expansionary ambitions on the part of Iran in the MENA region and the wider Islamic world. These ambitions commonly increase tensions and undermine stability.
View all posts by Track Persia
By Track Persia|2018-11-28T18:21:43+00:00November 29th, 2018|Iran|
Iranians enthralled by Rouhani's hasty departure from Khamenei's Friday prayer
Corruption in Iran's Red Crescent
Iran banned from hosting international football
Khamenei's sermon: The Supreme Leader vs the people
Iranian Shahnaz Akmali: "I go to prison as a mother with hope for better days ahead"
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Copyright 2016 Track Persia | All Rights Reserved | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,428 |
Q: How do I mock express.Application with Jest? Some class that takes express as an argument (basic DI):
class App {
constructor(express: express.Application) { /* ... */ }
}
My test:
// this doesn't work ("argument not assignable"):
//const expressMock = jest.mock("express");
//let app = new App(expressMock);
// so how do I mock it?
const expressMock = ???
let app = new App(expressMock);
How do I mock express.Application with Jest? The whole thing, not just a request, or route, etc.
A: Technically, express.Application cannot be mocked, it's an interface. It's an object that it represents that should be mocked.
As jest.mock documentation states, it
Returns the jest object for chaining.
So this is not a correct way to retrieve mocked object:
const expressMock = jest.mock("express");
A correct one would be
jest.mock("express");
...
const expressMock = require("express");
Moreover, App accepts not express factory function but application object, which is a result of express() call.
jest.mock without factory function results in auto-mocked express factory which won't produce proper application object.
Since expressMock is passed to App directly in tests, there's no necessity to mock Express module. A mock that contains bare minimum implementation can be passed instead:
const expressMock = {
use: jest.fn(),
...
} as any as express.Application;
let app = new App(expressMock);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,371 |
\section{Introduction}
It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose-Einstein condensation. A brief introduction into these areas is given in the respective sections; for recent monographs on these topics see \cite{bord01-353-1,dalv99-71-463,eliz95b,eliz94b,kirs02b,milt01b,milt04-37-209,peth02b,pita03b}. We will consider exclusively spectral zeta functions, that is zeta functions arising from the eigenvalue spectrum of suitable differential operators. Applications like those in number theory \cite{apos76b,apos90b,dave67b,titc51b} will not be considered in this contribution.
There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every spectral zeta function. Those tools are introduced in Section 2 using the well-studied examples of the Hurwitz \cite{hurw82-27-86}, Epstein \cite{epst03-56-615,epst07-63-205} and Barnes zeta function \cite{barn03-19-426,barn03-19-374}. In Section 3 it is explained how these different examples can all be thought of as being generated by the same mechanism, namely they all result from eigenvalues of suitable (partial) differential operators. It is this relation with partial differential operators that provides the motivation for analyzing the zeta functions considered in these lectures. Motivations come for example from the questions "Can one hear the shape of a drum?" and "What does the Casimir effect know about a boundary?". Finally "What does a Bose gas know about its container?" The first two questions are considered in detail in Section 4. The last question is examined in Section 5 where we will see how zeta functions can be used to analyze the phenomenon of Bose-Einstein condensation. The Conclusions will point towards recent developments for the analysis of spectral zeta functions and their applications.
\section{Some basic zeta functions}
In this section we will construct analytical continuations of basic zeta functions. From these we will determine the meromorphic structure, residues at singular points and special function values.
\subsection{Hurwitz zeta function}
We start by considering a generalization of the Riemann zeta function \beq \zeta_R (s) = \sum_{n=1}^\infty \frac 1 {n^s} . \label{zetarie}\eeq
\begin{definition}
Let $s\in \C$ and $0<a<1$. Then for $\Re s >1 $ the Hurwitz zeta function is defined by \beq \zeta_H (s,a) = \sum_{n=0}^\infty \frac 1 {(n+a)^s}.\nn\eeq
\end{definition}
Clearly, we have that $\zeta_H (s,1) = \zeta _R (s)$. Results for $a=1+b>1$ follow by observing \beq \zeta_H (s,1+b) = \sumnn \frac 1 {(n+1+b)^s} = \zeta _H (s,b) - \frac 1 {b^s}.\nn\eeq
In order to determine properties of the Hurwitz zeta function, one strategy is to express it in term of 'known' zeta functions like the Riemann zeta function.
\begin{theorem}
For $0<a<1$ we have \beq \zeta_H (s,a) = \frac 1 {a^s} + \sumkn (-1)^k \frac{\Gamma (s+k)} {\Gamma (s) k!} a^k \zeta_R (s+k).\nn\eeq
\end{theorem}
\begin{proof}
Note that for $|z| <1$ we have the binomial expansion \beq (1-z)^{-s} = \sumkn \frac{\Gamma (s+k)} {\Gamma (s) k!} z^k.\nn\eeq
So for $\Re s>1$ we compute
\beq \zeta_H (s,a) &=& \frac 1 {a^s} + \sumne \frac 1 {n^s} \frac 1 {\left( 1 + \frac a n \right)^s} \nn\\
&=& \frac 1 {a^s} + \sumne \frac 1 {n^s} \sumkn (-1)^k \frac{\Gamma (s+k)} {\Gamma (s) k!} \left( \frac a n \right)^k \nn\\
&=& \frac 1 {a^s} + \sumkn (-1)^k \frac{\Gamma (s+k)} {\Gamma (s) k!} a^k \sumne \frac 1 {n^{s+k}} \nn\\
&=& \frac 1 {a^s} + \sumkn (-1)^k \frac{\Gamma (s+k)} {\Gamma (s) k!} a^k \zeta _R (s+k),\nn\eeq
which is the assertion.
\end{proof}
From here it is seen that $s=1$ is the only pole of $\zeta_H (s,a)$ with $\res \zeta_H (1,a) =1$.
In determining certain function values of $\zeta_H (s,a)$ the following polynomials will turn out to be useful.
\begin{definition}
For $x\in \C$ we define the Bernoulli polynomials $B_n (x)$ by the equation
\beq \frac{ z e^{xz}}{e^z-1} = \sumnn \frac{B_n (x)} {n!} z^n, \quad \quad \mbox{where } |z|<2\pi.\label{bernoullipol}\eeq
\end{definition}
Examples are $B_0 (x) =1$ and $B_1 (x) = x-1/2$. The numbers $B_n (0)$ are called Bernoulli numbers and are denoted by $B_n$. Thus \beq \frac z {e^z-1} = \sum_{n=0}^\infty \frac{B_n }{n!} z^n, \quad \quad \mbox{where } |z|<2\pi.\label{bernoullinum}\eeq
\begin{lemma}\label{lem2.2}
The Bernoulli polynomials satisfy
\begin{enumerate}
\item
\beq B_n (x) = \sum_{k=0}^n { n \choose k} B_k x^{n-k} ,\nn\eeq
\item
\beq B_n (x+1) - B_n (x) = n x^{n-1} \quad \quad \mbox{if }n\geq 1 ,\nn\eeq
\item
\beq (-1)^n B_n (-x) = B_n (x) + n x^{n-1}, \nn\eeq
\item
\beq
B_n (1-x) = (-1)^n B_n (x) .\nn\eeq
\end{enumerate}
\end{lemma}
\begin{exercise} Use relations (\ref{bernoullipol}) and (\ref{bernoullinum}) to show the assertions of Lemma \ref{lem2.2}.\end{exercise}
We now establish elementary properties of $\zeta_H (s,a)$.
\begin{theorem}\label{the2.3}
For $\Re s >1$ we have
\beq \zeta_H (s,a) = \frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \frac{e^{-at}}{1-e^{-t}} dt.\label{zetahurint}\eeq
Furthermore, for $k\in\N_0$ we have \beq \zeta_H (-k , a) = - \frac{B_{k+1}(a)} {k+1} .\nn\eeq
\end{theorem}
\begin{proof}
We use the definition of the Gamma-function and have
\beq \Gamma (s) = \intl _0^\infty u^{s-1} e^{-u} du = \lambda^s \intl_0^\infty t^{s-1} e^{-t \lambda} dt .\label{Gamma}\eeq
This shows the first part of the Theorem,
\beq \zeta_H (s,a) &=& \sumnn \frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} e^{-t (n+a)} dt
= \frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \sumnn e^{-t (n+a)} dt\nn\\
&=&\frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \frac{e^{-at}}{1-e^{-t}}dt.\nn\eeq
Furthermore we have
\beq \zeta_H (s,a) &=&
= \frac 1 {\Gamma (s)} \intl _0^\infty t^{s-2} \frac{ t e^{-ta}}{1-e^{-t}} dt \nn\\
&=& \frac 1 {\Gamma (s)} \intl_0^1 t^{s-2} \frac{ (-t) e^{-ta}} {e^{-t} -1} dt +
\frac 1 {\Gamma (s)} \intl_1^\infty t^{s-2} \frac{ (-t) e^{-ta}} {e^{-t} -1} dt .\nn\eeq
The integral in the second term is an entire function of $s$. Given the Gamma-function has singularities at $s=-k$, $k\in \N_0$, only the first term can possibly contribute to the properties $\zeta_H (-k,a)$ considered. We continue and write
\beq \frac 1 {\Gamma (s) } \intl_0^1 t^{s-2} \frac{ (-t) e^{-ta}}{e^{-t} -1} dt &=& \frac 1 {\Gamma (s)} \intl_0^1 t^{s-2} \sumnn \frac{ B_n (a)} {n!}(-t)^n dt \nn\\
&=& \frac 1 {\Gamma (s) } \sumnn \frac{B_n (a)} {n!} \frac {(-1)^n}{s+n-1} ,\nn\eeq
which provides the analytical continuation of the integral to the complex plane.
From here we observe again
\beq \res \zeta_H (1,a) &=& B_0 (a) =1 , \nn\eeq
and the second part of the Theorem
\beq
\zeta_H (-k , a) &=& \lim_{\ep\to 0} \frac 1 {\Gamma (-k+\ep )} \frac{B_{k+1} (a) } { (k+1)! } \frac{ (-1)^{k+1}} \ep \nn\\
&=& \lim_{\ep\to 0} (-1)^k k! \ep \frac{B_{k+1} (a) } { (k+1)! } \frac{ (-1)^{k+1}} \ep = - \frac{ B_{k+1} (a)} {k+1} \nn\eeq
follows.
\end{proof}
The disadvantage of the representation (\ref{zetahurint}) is that it is valid only for $\Re s >1$. This can be improved by using a complex contour integral representation. Starting point is the following representation for the Gamma-function \cite{grad65b}.
\begin{lemma}\label{lem2.4}
For $z\notin\Z$ we have
\beq \Gamma (z) = - \frac 1 {2i\sin (\pi z)} \intl _{\Cc} (-t)^{z-1} e^{-t} dt ,\nn\eeq
where the anticlockwise contour $\Cc$ consists of a circle $\Cc_3$ of radius $\epsilon < 2\pi$ and straight lines $\Cc_1$, respectively $\Cc_2$, just above, respectively just below, the $x$-axis; see Figure \ref{fig1}.
\end{lemma}
\begin{figure}[ht]
\setlength{\unitlength}{1cm}
\begin{center}
\begin{picture}(10,6.5)
\thicklines
\put(0,3){\vector(1,0){10}} \put(5.0,0){\vector(0,1){6}}
\put(8.0,5.5){{\bf $t$-plane}}
\qbezier(5.3,3.01)(8.0,3.01)(9.50,3.01) \qbezier(5.3,2.99)(8.0,2.99)(9.50,2.99)
\put(5.0,3.0){\circle{0.6}}
\put(5.0,3.0){\circle{0.59}}
\put(5.0,3.0){\circle{0.61}}
\put(5.0,3.0){\circle{0.58}}
\put(5.0,3.0){\circle{0.62}}
\put(9.30,3.2){$\Cc_1$}
\put(9.30,2.6){$\Cc_2$}
\put(4.3,3.2){$\Cc_3$}
\put(8.9,3.735){\vector(-1,0){1.7}}
\put(7.2,2.265){\vector(1,0){1.6}}
\put(7.2,3.9){$-t = e^{-i \pi} u$}
\put(7.2,1.85){$-t = e^{i \pi} u$}
\end{picture}
\caption{Contour $\Cc$ in Lemma \ref{lem2.4}.}\label{fig1}
\end{center}
\end{figure}
\begin{proof}
Assume $\Re z >1$. As the integrand remains bounded along ${\Cc}_3$, no contributions will result as $\ep \to 0$. Along ${\Cc}_1$ and ${\Cc}_2$ we parameterize as given in Figure \ref{fig1} and thus for $\Re z>1$
\beq \lim_{\epsilon\to 0}\,\,\intl_{\Cc} (-t)^{z-1} e^{-t} dt &=& \intl_\infty^0 e^{-i \pi (z-1)} u^{z-1} e^{-u} du
+ \intl_0^\infty e^{i\pi (z-1)} u^{z-1} e^{-u} du \nn\\
&=& - \intl_0^\infty u^{z-1} e^{-u} \left( e^{i\pi z} - e^{-i\pi z} \right) du\nn\\
&=& - 2i\sin (\pi z) \intl_0^\infty u^{z-1} e^{-u} du ,\nn\eeq
which implies the assertion by analytical continuation.
\end{proof}
This representation for the Gamma-function can be used to show the following result for the Hurwitz zeta function.
\begin{theorem}
For $ s \in \C $, $s\notin \N$, we have \beq \zeta_H (s,a) = - \frac{\Gamma (1-s)}{2\pi i} \intl_{\Cc} \frac{(-t)^{s-1} e^{-ta}}{1-e^{-t}} dt,\nn\eeq
with the contour $\Cc$ given in Figure \ref{fig1}.
\end{theorem}
\begin{proof}
We follow the previous calculation to note
\beq \intl_{\Cc} \frac{(-t)^{s-1} e^{-ta}}{1-e^{-t}} dt = - 2i\sin (\pi s ) \intl_0^\infty t^{s-1} \frac{e^{-ta}}{1-e^{-t}} dt,\nn\eeq
and we use \cite{grad65b} $$\sin (\pi s) \Gamma (s) = \frac \pi {\Gamma (1-s)}$$
to conclude the assertion.
\end{proof}
From here, properties previously given can be easily derived. For $s\in \Z$ the integrand does not have a branch cut and the integral can easily be evaluated using the residue theorem. The only possible singularity enclosed is at $t=0$ and to read off the residue we use the expansion
$$- (-t)^{s-2} \frac{(-t) e^{-ta}}{e^{-t}-1} = - (-t)^{s-2} \sumnn \frac{B_n (a)} {n!} (-t)^n .$$
\subsection{Barnes zeta function}\label{secbarn}
The Barnes zeta function is a multidimensional generalization of the Hurwitz zeta function.
\begin{definition}\label{defbarn}
Let $s\in \C$ with $\Re s >d$ and $c\in \R_+$, $\vec r \in \R_+^d$. The Barnes zeta function is defined as \beq \zeta_\Bb (s,c|\vec r) = \sum_{\vec m \in \N_0^d} \frac 1 {(c+\vec m \cdot \vec r)^s}.\label{eqbarn}\eeq If $c=0$ it is understood that the summation ranges over $\vec m \neq \vec 0$.
\end{definition}
For $\vec r = \vec 1_d:=(1,1,...,1,1)$, the Barnes zeta function can be expanded in terms of the Hurwitz zeta function.
\begin{example}\label{ex1}
Let us consider $d=2$ and $\vec r = (1,1)$. Then
\beq\zeta_\Bb (s,c|\vec 1_2) &=& \sum_{\vec m \in \N_0^2} \frac 1 {(c+m_1+m_2)^s}= \sumkn \frac {k+1} {(c+k)^s} = \sumkn \frac{k+c+1-c}{(c+k)^s} \nn\\
&=& \zeta_H (s-1,c) + (1-c) \zeta_H (s,c).\nn\eeq
\end{example}
\begin{example}\label{ex2}
Let $e_k^{(d)}$ be the number of possibilities to write an integer $k$ as a sum over $d$ non-negative integers. We then can write
\beq \zeta_\Bb (s,c| \vec 1_d) = \sum_{\vec m\in \N_0^d} \frac 1 {(c+m_1+...+m_d)^s} = \sumkn e_k^{(d)} \frac 1 {(c+k)^s} .\nn\eeq
The coefficient $e_k^{(d)}$ can be determined for example as follows. Consider \beq \frac 1 {(1-x)^d} &=& \frac 1 {1-x} \cdot \cdot \cdot \frac 1 {1-x} = \left( \sum_{l_1=0}^\infty x^{l_1} \right) \cdot \cdot \cdot \left( \sum_{l_d=0}^\infty x^{l_d}\right) \nn\\
&=& \sum_{l_1=0}^\infty \cdot \cdot \cdot \sum_{l_d=0}^\infty x^{l_1+...+l_d} = \sumkn e_k^{(d)} x^k.\nn\eeq
On the other side, using the binomial expansion \beq \frac 1 {(1-x)^d} &=& \sumkn \frac{\Gamma (d+k)} {\Gamma (d) k!} x^k = \sumkn \frac{(d+k-1)!}{(d-1)! k!} x^k \nn\\
&=& \sumkn {d+k-1 \choose d-1 } x^k.\nn\eeq
This shows \beq \zeta_{\mathcal B} (s,c| \vec 1_d) = \sumkn {d+k-1 \choose d-1} \frac 1 {(c+k)^s} ,\nn\eeq
which, once the dimension $d$ is specified, allows to write the Barnes zeta function as a sum of Hurwitz zeta functions along the lines in Example \ref{ex1}.
\end{example}
It is possible to obtain similar formulas for $r_i$ rational numbers \cite{dowk94-162-633,dowk94-35-4989}.
For some properties of the Barnes zeta function the use of complex contour integral representations turns out to be the best strategy.
\begin{theorem}\label{barnrep}
We have the following representations:
\beq \zeta_\Bb (s,c| \vec r) &=& \frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \frac{e^{-ct}}{\prod_{j=1}^d \left(1-e^{-r_j t}\right)} dt \nn\\
&=& - \frac {\Gamma (1-s)} {2\pi i} \intl_\Cc (-t)^{s-1} \frac{e^{-ct}}{\prod_{j=1}^d \left(1-e^{-r_j t}\right)} dt,\nn\eeq
with the contour $\Cc$ given in Figure \ref{fig1}.
\end{theorem}
\begin{exercise}
Use equation (\ref{Gamma}), respectively Lemma \ref{lem2.4}, to proof Theorem \ref{barnrep}.
\end{exercise}
The residues of the Barnes zeta function and its values at non-positive integers are best described using generalized Bernoulli polynomials \cite{norl22-43-121}.
\begin{definition}\label{defber}
We define the generalized Bernoulli polynomials $B_n^{(d)} (x|\vec r)$ by the equation \beq \frac{e^{-xt}}{\prod_{j=1}^d \left( 1-e^{-r_j t}\right)} = \frac{ (-1)^d}{\prod_{j=1}^d r_j} \sumnn \frac{(-t)^{n-d} }{n!} B_n^{(d)} (x|\vec r).\nn\eeq
\end{definition}
Using Definition \ref{defber} in Theorem \ref{barnrep} one immediately obtains the following properties of the Barnes zeta function.
\begin{theorem}\label{polebarn}
We have \begin{enumerate}
\item
\beq \res \zeta_\Bb (z,c|\vec r) = \frac{ (-1)^{d+z}}{(z-1)! (d-z)! \prod_{j=1}^d r_j} B_{d-z}^{(d)} (c|\vec r ), \quad \quad z=1,2,...,d, \nn\eeq
\item
\beq \zeta_\Bb (-n , c| \vec r ) = \frac{ (-1)^d n!} {(d+n)! \prod_{j=1}^d r_j } B_{d+n} ^{(d)} (c| \vec r ) .\nn\eeq
\end{enumerate}
\end{theorem}
\begin{exercise}
Use the first representation of $\zeta_\Bb (s,c|\vec r)$ in Theorem \ref{barnrep} together with Definition \ref{defber} to show Theorem \ref{polebarn}. Follow the steps of the proof in Theorem \ref{the2.3}.
\end{exercise}
\begin{exercise}
Use the second representation of $\zeta_\Bb (s,c| \vec r)$ in Theorem \ref{barnrep} together with Definition \ref{defber} and the residue theorem to show Theorem \ref{polebarn}.
\end{exercise}
\subsection{Epstein zeta function}
We now consider zeta functions associated with sums of squares of integers \cite{epst03-56-615,epst07-63-205}.
\begin{definition}\label{epdef}
Let $s\in \C$ with $\Re s > d/2$ and $c\in \R_+$, $\vec r \in \R_+^d.$ The Epstein zeta function is defined as
\beq \zeta_\Ee (s,c| \vec r ) = \sum_{\vec m \in \Z ^d} \frac 1 {(c+r_1 m_1^2 + r_2 m_2^2+...+r_d m_d^2)^s}.\nn\eeq
If $c=0$ it is understood that the summation ranges over $\vec m \neq \vec 0$.
\end{definition}
\begin{lemma}
\label{mellinep}
For $\Re s > d/2$, we have \beq \zeta _\Ee (s,c| \vec r) = \frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \sum_{\vec m \in \Z ^d} e^{-t (r_1 m_1^2 + ... + r_d m_d^2 + c)} dt .\nn\eeq
\end{lemma}
\begin{proof} This follows as before from property (\ref{Gamma}) of the Gamma-function.\end{proof}
As we have noted in the proof of Theorem \ref{the2.3}, it is the small-$t$ behavior of the integrand that determines residues of the zeta function and special function values. The way the integrand is written in Lemma \ref{mellinep} this $t\to 0$ behavior is not easily read off. A suitable representation is obtained by using the Poisson resummation \cite{hill62b}.
\begin{lemma}
\label{poisson}
Let $r \in \C$ with $\Re r >0$ and $t\in \R_+$, then
\beq \sum_{l=-\infty} ^\infty e^{-trl^2} = \sqrt{\frac \pi {tr} } \sumlm e^{-\frac{\pi ^2} {rt} l^2} .\nn\eeq
\end{lemma}
\begin{exercise}\label{expoi}
If $F(x)$ is continuous such that
$$\int\limits_{-\infty}^\infty \,\, | F(x) | dx < \infty,$$ then we
define its Fourier transform by
$$\hat F (u) = \int\limits_{-\infty}^\infty \,\, F(x) e^{-2\pi i x
u} \,\, dx.$$ If $$\int\limits_{-\infty}^\infty \,\, | \hat F (u)
| \,\, du < \infty , $$ then we have the Fourier inversion formula
$$F(x) = \int\limits_{-\infty} ^\infty \,\, \hat F (u) \,\,
e^{2\pi i x u} \,\, du.$$ Show the following Theorem: Let $F\in
L^1 (\R)$. Suppose that the series $$\sum_{n\in\Z} \,\,
F(n+v)$$ converges absolutely and uniformly in $v$, and that
$$\sum_{m\in\Z} \,\, | \hat F (m) | < \infty.$$ Then
$$\sum_{n\in\Z} \,\, F(n+v) = \sum_{n\in\Z} \,\, \hat F
(n) e^{2\pi i nv}.$$ Hint: Note that $$G(v) =
\sum_{n\in\Z}\,\, F(n+v)$$ is a function of $v$ of period
$1$. \end{exercise}
\begin{exercise}
Apply Exercise \ref{expoi} with a suitable function $F(x)$ to show the Poisson resummation formula Lemma \ref{poisson}.
\end{exercise}
In Lemma \ref{poisson} it is clearly seen that the only term on the right hand side that is not exponentially damped as $t\to 0$ comes from the $l=0$ term. Using the resummation formula for all $d$ sums
in Lemma \ref{mellinep}, after resumming the $\vec m = \vec 0$ term contributes
\beq \zeta_\Ee ^{\vec 0} (s,c| \vec r ) &=&
\frac 1 {\Gamma (s)} \intl_0^\infty t^{s-1} \frac{ \pi^{d/2}}{t^{d/2} \sqrt{ r_1 \cdot \cdot \cdot r_d}} e^{-ct} dt \nn\\
&=&\frac{\pi^{d/2}}{\sqrt{r_1 \cdot \cdot \cdot r_d } \,\,\Gamma (s)} \intl_0^\infty t^{s-d/2-1} e^{-ct} dt \nn\\
&=& \frac{\pi ^{d/2}}{\sqrt{r_1 \cdot \cdot \cdot r_d} } \frac{\Gamma \left( s-\frac d 2 \right)}{\Gamma (s) c^{s-d/2}} .\nn\eeq
All other contributions after resummation are exponentially damped as $t\to 0$ and can be given in terms of modified Bessel functions \cite{grad65b}.
\begin{definition}
\label{bessel}
Let $\Re z^2 >0$, then we define the modified Bessel function $K_\nu (z)$ as
\beq K_\nu (z) = \frac 1 2 \left( \frac z 2 \right)^\nu \intl_0^\infty e^{-t - \frac{z^2} {4t}}\,\, t^{-\nu -1} dt.\nn\eeq
\end{definition}
Performing the resummation in Lemma \ref{mellinep} according to Lemma \ref{poisson}, with Definition \ref{bessel} one obtains the following representation of the Epstein zeta function valid in the whole complex plane \cite{eliz94b,terr73-183-477}.
\begin{theorem}
\label{bessel1}
We have
\beq \zeta_\Ee (s,c| \vec r) &=& \frac{ \pi^{d/2}}{\sqrt{r_1 \cdot \cdot \cdot r_d}} \frac{\Gamma \left( s- \frac d 2 \right)}{\Gamma (s)} c^{\frac d 2 -s} + \frac{2 \pi ^s c^{\frac{d-2s} 4}} {\Gamma (s) \sqrt{ r_1 \cdot \cdot \cdot r_d}} \nn\\
& &\hspace{-1.0cm}\times\sum_{\vec n \in \Z ^d / \{\vec 0 \}} \left[ \frac{ n_1^2} {r_1} + ... + \frac{n_d^2} {r_d} \right] ^{\frac 1 2 \left( s- \frac d 2 \right) } K_{\frac d 2 -s} \left( 2 \pi \sqrt c \left( \frac{n_1^2} {r_1} + ... + \frac{n_d^2} {r_d} \right)^{1/2} \right) .\nn\eeq
\end{theorem}
\begin{exercise}
Show Theorem \ref{bessel1} along the lines indicated.
\end{exercise}
From Definition \ref{bessel} it is clear that the Bessel function is exponentially damped for large $\Re z^2$. As a result the above representation is numerically very effective as long as the argument of $K_{d/2-s}$ is large. The terms involving the Bessel functions are analytic for all values of $s$, the first term contains poles. As an immediate consequence of the properties of the Gamma-function one can show the following properties of the Epstein zeta function.
\begin{theorem}\label{the2.11}
For $d$ even, $\zeta_\Ee (s,c| \vec r)$ has poles at $s=\frac d 2, \frac d 2 -1, ..., 1$, whereas for $d$ odd they are located at $s= \frac d 2, \frac d 2 -1, ..., \frac 1 2, - \frac{2l+1} 2$, $l\in \N_0$. Furthermore,
\beq \res \zeta _\Ee (j,c| \vec r ) &=& \frac{ (-1)^{\frac d 2 +j} \pi ^{\frac j 2} c^{\frac d 2 -j} } {\sqrt{r_1 \cdot \cdot \cdot r_d}\,\, \Gamma (j) \Gamma \left( \frac d 2 -j +1\right)} , \nn\\
\zeta_\Ee (-p,c| \vec r ) &=& \left\{ \begin{array}{ll}
0 & \mbox{for $d$ odd}\\
\frac{ (-1)^{\frac d 2 } p! \pi ^{\frac d 2} c^{\frac d 2 +p} } {\sqrt{r_1 \cdot \cdot \cdot r_d} \,\,\Gamma \left( \frac d 2 +p +1\right)} & \mbox{for $d$ even}.\end{array} \right. \nn\eeq
\end{theorem}
\begin{exercise}
Use Theorem \ref{bessel1} and properties of the Gamma-function to show Theorem \ref{the2.11}.
\end{exercise}
This concludes the list of examples for zeta functions to be considered in what follows. A natural question is what the motivations are to consider these zeta functions. Before we describe a few aspects relating to this question let us mention how all these zeta functions, and many others, result from a common principle.
\section{Boundary value problems and associated zeta functions}
In this section we explain how the considered zeta functions, and others, are all associated with eigenvalue problems of (partial) differential operators.
\begin{example}
Let $M=[0,L]$ be some interval and consider the {\it Dirichlet} boundary value problem.
\beq P \phi_n (x) := - \frac{\partial^2}{\partial x^2} \phi _n (x) = \lambda_n \phi_n (x), \quad \quad \phi_n (0) = \phi_n (L) =0 .\nn\eeq
The solutions to the boundary value problem have the general form
\beq \phi _n (x) = A \sin (\sqrt {\lambda_n} x ) + B \cos ( \sqrt{\lambda_n} x) .\nn\eeq
Imposing the Dirichlet boundary condition shows we need
\beq \phi _n (0) = B=0, \quad \quad \phi_n (L) = A \sin (L \sqrt{\lambda_n}) = 0, \nn\eeq
which implies $$\lambda_n = \frac{n^2 \pi^2} {L^2} , \quad \quad n\in\N.$$
We only need to consider $n\in\N$ because non-positive integers lead to linearly dependent eigenfunctions. The zeta function $\zeta_P (s)$ associated with this boundary value problem is defined to be the sum over all eigenvalues raised to the power $(-s)$, namely
\beq \zeta_P (s) = \sumne \lambda_n^{-s} ,\quad \quad \Re s > \frac 1 2.\nn\eeq
So here the associated zeta function is a multiple of the zeta function of Riemann,
\beq \zeta_P (s) = \sumne \left( \frac{n \pi} L \right)^{-2s} = \left( \frac L \pi \right)^{2s} \zeta_R (2s) .\nn\eeq
\end{example}
\begin{example}
The previous example can be easily generalized to higher dimensions. We consider explicitly two dimensions; for the higher dimensional situation
see \cite{ambj83-147-1}. Let $M=\{ (x,y) | x\in [0,L_1], y\in [0,L_2]\}.$ We consider the boundary value problem with Dirichlet boundary conditions on $M$, that is
\beq P \phi_{n,m} (x,y) &=& \left(- \frac{\partial^2}{\partial x^2} - \frac{\partial ^2} {\partial y^2} + c \right) \phi _{n,m} (x,y) = \lambda_{n,m} \phi_{n,m} (x,y) , \nn\\
\phi_{n,m} (0,y) &=& \phi_{n,m} (L_1,y)= \phi_{n,m} (x,0) = \phi _{n,m} (x,L_2) =0 .\nn\eeq
Using the process of separation of variables, eigenfunctions are seen to be
\beq \phi_{n,m} (x,y) = A \sin \left(\frac{n\pi x}{L_1}\right) \sin \left(\frac{m\pi y}{L_2}\right) , \nn\eeq
with the eigenvalues $$\lambda_{n,m} = \left( \frac{ n\pi} {L_1} \right)^2 + \left( \frac{m\pi} {L_2}\right)^2 + c, \quad \quad n,m\in\N.$$
The associated zeta function therefore is
\beq \zeta_P (s) = \sumne \summe \left[ \left( \frac{n \pi } {L_1}\right)^2 + \left( \frac{ m \pi} {L_2} \right)^2 +c \right] ^{-s} , \nn\eeq
which can be expressed in terms of the Epstein zeta function given in Definition \ref{epdef} as follows,
\beq \zeta_P (s) &=& \frac 1 4 \zeta_\Ee \left( s,c\left| \left( \left( \frac \pi {L_1} \right)^2,\left( \frac \pi {L_2} \right)^2\right)\right) \right. \nn\\
& & - \frac 1 4 \zeta_\Ee \left( s,c\left| \left( \frac \pi {L_1} \right)^2 \right)\right. -
\frac 1 4 \zeta_\Ee \left( s,c\left| \left( \frac \pi {L_2} \right)^2 \right)\right. + \frac 1 4 c^{-s}. \eeq
\end{example}
\begin{example}
Similarly one can consider periodic boundary conditions instead of Dirichlet boundary conditions, this means the manifold $M$ is given by $M=S^1\times S^1$. In this case the eigenfunctions have to satisfy
\beq \phi_{n,m} (0,y) &=& \phi_{n,m} (L_1,y) , \quad \quad \frac \partial {\partial x} \phi_{n,m} (0,y) = \frac \partial {\partial x} \phi_{n,m} (L_1,y) , \nn\\
\phi_{n,m} (x,0) &=& \phi_{n,m} (x, L_2) , \quad \quad \frac \partial {\partial y} \phi_{n,m} (x,0) = \frac \partial {\partial y} \phi_{n,m} (x,L_2) .\nn\eeq
This shows \beq \phi_{n,m} (x,y) = A e^{i \frac{2 \pi n} {L_1} x} \,\, e^{i \frac{2\pi m} {L_2} y} , \nn\eeq
which implies for the eigenvalues $$ \lambda_{n,m} = \left( \frac{ 2\pi n} {L_1}\right)^2 + \left( \frac{ 2\pi m} {L_2} \right)^2 + c, \quad \quad (n,m) \in \Z^2.$$ The associated zeta function therefore is
\beq \zeta_P (s) = \zeta _\Ee \left(s,c| \vec r \right), \quad \quad \vec r = \left( \left(\frac{2\pi} {L_1}\right)^2, \left(\frac{2\pi} {L_2}\right)^2\right).\nn\eeq
Clearly, in $d$ dimensions one finds \beq \zeta_P (s) = \zeta_\Ee \left( s,c| \vec r \right), \quad \quad \vec r = \left( \left( \frac{2\pi} {L_1} \right)^2 , ..., \left( \frac{2\pi } {L_d} \right)^2 \right).\nn\eeq
\end{example}
\begin{example}\label{exHO}
As a final example we consider the Schr{\"o}dinger equation of atoms in a harmonic oscillator potential. In this case $M=\R^3$, and the eigenvalue equation reads
\beq \left\{ - \frac {\hbar^2} {2m} \Delta + \frac m 2 \left( \omega_1 x^2+\omega_2 y^2 + \omega _3 z^2\right) \right\} \phi_{n_1,n_2,n_3} (x,y,z) = \lambda_{n_1,n_2,n_3} \phi_{n_1,n_2,n_3} (x,y,z).\nn\eeq
This differential equation is augmented by the condition that eigenfunctions must be square integrable, $\phi_{n_1,n_2,n_3} (x,y,z) \in \mathcal{L}^2 (\R^3).$ As is well known, this gives the eigenvalues $$\lambda_{n_1,n_2,n_3} = \hbar \omega_1 \left( n_1 + \frac 1 2 \right) + \hbar \omega_2 \left( n_2 + \frac 1 2 \right) + \hbar \omega_3 \left( n_3 + \frac 1 2 \right) , \quad (n_1,n_2,n_3)\in\N_0^3.$$
This clearly leads to the Barnes zeta function
\beq \zeta_P (s) = \zeta_\Bb (s,c| \vec r),\nn\eeq
where \beq c&=& \frac 1 2 \hbar (\omega_1 + \omega_2 + \omega_3), \quad \quad \vec r = \hbar \left(\omega_1,\omega_2,\omega_3\right).\nn\eeq
If $M=\R$ is chosen the Hurwitz zeta function results.
\end{example}
The above examples illustrate how the zeta functions considered in Section 2 are all related in a natural way to eigenvalues of specific boundary value problems. In fact, zeta functions in a much more general context are studied in great detail. For our purposes the relevant setting is the setting of Laplace-type operators on a Riemannian manifold $M$, possibly with a boundary $\partial M$. Laplace-type means the operator $P$ can be written as $$P = - g^{jk} \nabla _j^V \nabla_k^V - E,$$ where $g^{jk}$ is the metric of $M$, $\nabla^V$ is the connection on $M$ acting on a smooth vector bundle $V$ over $M$, and where $E$ is an endomorphism of $V$. Imposing suitable boundary conditions, eigenvalues $\lambda_n$ and eigenfunctions $\phi_n$ do exist, $$P \phi_n (x) = \lambda_n \phi_n (x), $$ and assuming $\lambda_n>0$ the zeta function is defined to be $$\zeta _P (s) = \sum_{n=1}^\infty \lambda_n ^{-s}$$ for $\Re s$ sufficiently large. If there are modes with $\lambda_n=0$ those have to be excluded from the sum. Also, if finitely many eigenvalues are negative the zeta function can be defined by choosing nonstandard definitions of the principal value for the argument of complex numbers, but we will not need to consider those cases.
\section{(Some) Motivations to consider zeta functions}
There are many situations where properties of zeta functions in the above context of Laplace-type operators are needed. In the following we present a few of them, but many more can be found for example in the context of number theory \cite{apos76b,apos90b,dave67b,titc51b} and quantum field theory \cite{bord01-353-1,buch92b,byts03b,byts96-266-1,dett92-377-252,dunn08-41-304006,dunn05-94-072001,eliz95b,espo94b,espo98b,kirs02b,sach92-65-652}.
\subsection{Can one hear the shape of a drum?}
Let $M$ be a two-dimensional membrane representing a drum with boundary $\partial M$. The drum is fixed along its boundary. Then possible vibrations of the drum and its fundamental tones are described by the eigenvalue problem \beq \left( - \frac{\partial^2}{\partial x^2} - \frac {\partial^2} {\partial y^2} \right) \phi_n (x,y) = \lambda_n \phi_n (x,y) , \quad \quad \left.\phi_n (x,y) \right|_{(x,y) \in \partial M} =0.\nn\eeq
Here, $(x,y)$ denotes the variables in the plane, the eigenfunctions $\phi_n (x,y)$ describe the amplitude of the
vibrations and $\lambda_n$ its fundamental tones. In 1966 Kac \cite{kac66-73-1} asked if just
by listening with a perfect ear, so by knowing all the fundamental tones
$\lambda_n$, it is possible to hear the shape of the drum. One problem in answering this question is, of course, that in
general it will be impossible to write down the eigenvalues $\lambda_n$
in a closed form and to read off relations with the shape of the drum
directly. Instead one has to organize the spectrum intelligently in
form of a spectral function to reveal relationships between the eigenvalues
and the shape of the drum. In this context a particularly fruitful spectral
function is the heat kernel
$$
K(t) = \sum_{n=1}^\infty e^{-\lambda_n t},
$$
which as $t$ tends to zero clearly diverges. Given that some relations between the fundamental tones and properties of the drum are hidden in the $t\to 0$ behavior let us consider this asymptotic behavior very closely. Before we come back to the setting of the drum, let us use a few examples to get an idea what the structure of the $t\to 0$ behavior of the heat kernel is expected to be.
\begin{example}\label{heatcircle} Let $M=S^1$ be the circle with circumference $L$ and let $P = - \partial^2/\partial x^2$. Imposing periodic boundary conditions eigenvalues are $$\lambda_k = \left( \frac{2\pi k} L \right)^2 , \quad \quad k\in\Z ,$$ and the heat kernel reads
\beq K_{S^1} (t) = \sum_{k=-\infty}^\infty e^{- \left( \frac{2\pi k} L \right)^2t} .\nn\eeq
From Lemma \ref{poisson} we find the $t\to 0$ behavior $$K_{S^1} (t) = \frac 1 {\sqrt{4\pi t}} L +
\left( \mbox{exponentially damped terms}\right). $$ Note that with obvious notation this could be written as
$$K_{S^1} (t) =\frac 1 {\sqrt{4\pi t}} \mbox{vol} (M) + \left( \mbox{exponentially damped terms}\right). $$
\end{example}
\begin{example} \label{heattorus} The heat kernel for the $d$-dimensional manifold $M=S^1\times \cdot \cdot \cdot \times S^1$ with $P=-\Delta$ clearly gives a product of the above and thus $$K_M (t) = K_{S^1} (t) \times \cdot \cdot \cdot \times K_{S^1} (t) = \frac 1 {(4\pi t)^{d/2}} \mbox{vol} (M) +
\left( \mbox{exponentially damped terms}\right). $$
\end{example}
\begin{example} \label{massivetorus} To avoid the impression that there is always just one term that is not exponentially damped consider $M$ as above but $P=-\Delta + m^2$. Then
\beq K(t) &=& e^{-m^2 t} K_M (t) = e^{-m^2 t} \left( \frac 1 {(4\pi t)^{d/2}} \mbox{vol} (M) +
\mbox{exponentially damped terms} \right)\nn\\
&=& \frac 1 {(4\pi)^{d/2}} \mbox{vol} (M) \sum_{\ell =0}^\infty \frac{(-1)^\ell} {\ell !} m^{2\ell } t^{\ell - \frac d 2}
+ \left(\mbox{exponentially damped terms}\right) .\nn\eeq
\end{example}
In fact, the structure of the heat kernel observed in this last example is the structure observed for the general class of Laplace-type operators.
\begin{theorem}\label{heatnobound}
Let $M$ be a $d$-dimensional smooth compact Riemannian manifold without boundary and let $$P= - g^{jk} \nabla_j^V \nabla_k^V - E ,$$ where $g^{jk}$ is the metric of $M$, $\nabla^V$ is the connection on $M$ acting on a smooth vector bundle $V$ over $M$, and where $E$ is an endomorphism of $V$.
Then as $t\to 0$, $$K (t) \sim \sum_{k=0}^\infty a_k \,\,t^{k-d/2}$$ with the so-called heat kernel coefficients $a_k$.
\end{theorem}
\begin{proof} See, e.g., \cite{gilk95b}. \end{proof}
In Example \ref{massivetorus} one sees that $$a_k = \frac 1 {(4\pi)^{d/2}} \frac{(-1)^k}{k!} m^{2k} \mbox{vol}(M).$$ In general, the heat kernel coefficients are significantly more complicated and they depend upon the geometry of the manifold $M$ and the endomorphism $E$ \cite{gilk95b}.
Up to this point we have only considered manifolds without boundary. In order to consider in more detail questions relating to the drum, let us now see what relevant changes in the structure of the small-$t$ heat kernel expansion occur if boundaries are present.
\begin{example}\label{heatinterval}
Let $M=[0,L]$ and $P=-\partial^2/\partial x^2$ with Dirichlet boundary conditions imposed. Normalized eigenfunctions are then given by $$\varphi_\ell (x) = \sqrt{\frac 2 L} \sin\left( \frac{\pi \ell x} L \right)$$ and the associated eigenvalues are $$\lambda_\ell = \left( \frac{\pi \ell} L \right)^2, \quad \quad \ell \in \N.$$ Using Lemma \ref{poisson} this time we obtain \beq K(t) = \frac 1 {\sqrt{4\pi t}} \mbox{vol}(M) - \frac 1 2 + (\mbox{exponentially damped terms}).\label{intbounddir}\eeq
Notice that in contrast to previous results we have integer and half-integer powers in $t$ occurring.
\end{example}
\begin{exercise}
There is a more general version of the Poisson resummation formula than the one given in Lemma \ref{poisson}, namely
\beq \sum_{\ell =-\infty} ^\infty e^{-t (\ell +c)^2} = \sqrt{\frac \pi t } \sum_{\ell =-\infty}^\infty e^{-\frac{\pi^2} t \ell^2 - 2\pi i \ell c} .\label{genpoisson}\eeq Apply Exercise \ref{expoi} with a suitable function $F(x)$ to show equation (\ref{genpoisson}).
\end{exercise}
\begin{exercise}\label{localheat}
Consider the setting described in Example \ref{heatinterval}. The local heat kernel is defined as the solution of the equation
\beq \left( \frac \partial {\partial t} - \frac{\partial^2}{\partial x^2}\right) K(t,x,y) =0 \nn\eeq
with the initial condition
\beq \lim_{t\to 0} K(t,x,y) = \delta (x,y) .\nn\eeq
In terms of the quantities introduced in Example \ref{heatinterval} it can be written as \beq K(t,x,y) = \sum_{\ell =1}^\infty \varphi _\ell (x) \varphi_\ell (y) e^{-\lambda_\ell t} .\nn\eeq
Use the resummation (\ref{genpoisson}) for $K(t,x,y)$ and the fact that $$K(t) = \intl _0^L K(t,x,x) dx$$ to rediscover the above result (\ref{intbounddir}).
\end{exercise}
\begin{exercise} \label{heatboundint} Let $M=[0,L]$ and $$ P= - \frac{\partial ^2} {\partial x^2} + m^2$$ with Dirichlet boundary conditions imposed. Find the small-$t$ asymptotics of the heat kernel.\end{exercise}
\begin{exercise} \label{highdbound} Let $M=[0,L] \times S^1\times\cdot \cdot \cdot \times S^1$ be a $d$-dimensional manifold and $$P=-\frac{\partial^2}{\partial x^2} + m^2.$$ Impose Dirichlet boundary conditions on $[0,L]$ and periodic boundary conditions on the circle factors. Find the small-$t$ asymptotics of the heat kernel.\end{exercise}
As the above examples and exercises suggest, one has the following result.
\begin{theorem}\label{heatbound} Let $M$ be a $d$-dimensional smooth compact Riemannian manifold with smooth boundary and
let $$P= - g^{jk} \nabla_j^V \nabla_k^V - E ,$$ where $g^{jk}$ is the metric of $M$, $\nabla^V$ is the connection on $M$ acting on a smooth vector bundle $V$ over $M$, and where $E$ is an endomorphism of $V$. We impose Dirichlet boundary conditions.
Then as $t\to 0$, $$K (t) \sim \sum_{k=0,1/2,1,...}^\infty a_k \,\, t^{k-d/2}$$ with the heat kernel coefficients $a_k$.
\end{theorem}
\begin{proof} See, e.g., \cite{gilk95b}. \end{proof}
As for the manifold without boundary case, Theorem \ref{heatnobound}, the heat kernel coefficients depend upon the geometry of the manifold $M$ and the endomorphism $E$, and in addition on the geometry of the boundary. Note, however, that in contrast to Theorem \ref{heatnobound} the small-$t$ expansion contains integer and half-integer powers in $t$.
The same structure of the small-$t$ asymptotics is found for other boundary conditions like Neumann or Robin, see \cite{gilk95b}, and the coefficients then also depend on the boundary condition chosen.
In particular, for Dirichlet boundary conditions one can show the identities
\beq a_0 = (4 \pi)^{-d/2} \mbox{vol}(M), \quad \quad a_{1/2} = (4\pi )^{-(d-1)/2} \left( - \frac 1 4 \right) \mbox{vol}(\partial M),\label{leadcoef}\eeq
a result going back to McKean and Singer \cite{mcke67-1-43}. In the context of the drum, what the formula shows is that
by listening with a perfect ear one can indeed hear certain properties
like the area of the drum and the circumference of its boundary. But as has been
shown by Gordon, Webb and Wolpert \cite{gord92-27-134}, one cannot hear all details of the
shape.
\begin{exercise}
Use Exercise \ref{highdbound} to verify the general formulas (\ref{leadcoef}) for the heat kernel coefficients.
\end{exercise}
Instead of using the heat kernel coefficients to make the above statements, one could equally well have used zeta function properties for equivalent statements. Consider the setting of Theorem \ref{heatbound}.
The associated zeta function is given by $$ \zeta_P (s) = \sum_{n=1}^\infty \lambda_n^{-s},$$ where it follows from Weyl's law \cite{weyl12-71-441,weyl15-39-1} that this series is convergent for $\Re s>d/2$. The zeta function is related with the heat kernel by \beq \zeta_P (s) = \frac 1 {\Gamma (s)} \intl _0^\infty t^{s-1} K(t) dt , \label{zetaheat}\eeq
where equation (\ref{Gamma}) has been used. This equation allows us to relate residues and function values at certain points with the small-$t$ behavior of the heat kernel. In detail,\beq \mbox{Res } \zeta_P (z) &=& \frac{a_{\frac d 2 -z}} {\Gamma (z)} , \quad \quad z=\frac d 2 , \frac{d-1} 2 , ..., \frac 1 2, - \frac{2n+1} 2 , n \in\N_0, \label{zeheres}\\
\zeta_P (-q) &=& (-1)^q q! a_{\frac d 2 +q} , \quad \quad q\in\N_0. \label{zehefun}\eeq
Keeping in mind the vanishing of the heat kernel coefficients $a_k$ with half-integer index for $\partial M=\emptyset$, see Theorem \ref{heatnobound}, this means for $d$ even the poles are actually located only at $z=d/2,d/2-1,...,1$. In addition, for $d$ odd we get $\zeta_P (-q)=0$ for $q\in\N_0$.
\begin{exercise} Use Theorem \ref{heatbound} and proceed along the lines indicated in the proof of Theorem \ref{the2.3} to show equations (\ref{zeheres}) and (\ref{zehefun}). \end{exercise}
Going back to the setting of the drum properties of the zeta function relate with the geometry of the surface. In particular, from (\ref{leadcoef}) and (\ref{zeheres}) one can show the identities
\beq \res \zeta_P (1) = \frac{\mbox{vol}(M)} {4\pi}, \quad \res \zeta _P \left( \frac 1 2 \right) = - \frac{\mbox{vol}(\partial M)} {2\pi}, \nn\eeq
and the remarks below equation (\ref{leadcoef}) could be repeated.
\subsection{What does the Casimir effect know about a boundary?}
We next consider an application in the context of quantum field theory in finite systems. The importance of this topic lies in the fact that
in recent years, progress in many fields has been triggered by the
continuing miniaturization of all kinds of technical devices. As
the separation between components of various systems tends towards
the nanometer range, there is a growing need to understand every
possible detail of quantum effects due to the small sizes involved.
Very generally speaking, effects resulting from the finite
extension of systems and from their precise form are known as the
Casimir effect. In modern technical devices this effect is
responsible for up to 10$\%$ of the forces encountered in
microelectromechanical systems
\cite{chan01-87-211801,chan01-291-1941}. Casimir forces are of
direct practical relevance in nanotechnology where, e.g., sticking
of mobile components in micromachines might be caused by them
\cite{serr95-4-193}. Instead of fighting the occurrence of the
effect in technological devices, the tendency is now to try and
take technological advantage of the effect.
Experimental progress in recent years has been impressive and for
some configurations allows for a detailed comparison with
theoretical predictions. The best tested situations are those of
parallel plates \cite{bres02-88-041804} and of a plate and a
sphere
\cite{chan01-291-1941,chen06-74-022103,lamo97-78-5,lamo00-84-5673,mohi98-81-4549};
recently also a plate and a cylinder has been considered
\cite{brow05-72-052102,emig06-96-080403}. Experimental data and theoretical
predictions are in excellent agreement, see, e.g.,
\cite{bord01-353-1,decc05-318-37,klim06-39-6485,lamo05-68-201}.
This interplay between theory and experiments, and the intriguing
technological applications possible, are the main reasons for the
heightened interest in this effect in recent years.
In its original form, the effect refers to the situation of two
uncharged, parallel, perfectly conducting plates. As predicted by
Casimir \cite{casi48-51-793}, the plates should attract with a
force per unit area, $F(a)\sim 1/a^4$, where $a$ is the distance
between the plates. Two decades later Boyer \cite{boye68-174-1764} found
a repulsive pressure of magnitude $F (R) \sim 1/R^4$ for a
perfectly conducting spherical shell of radius $R$. Up to this day an intuitive understanding of the opposite signs found is lacking.
One of the main questions in the context of the Casimir effect therefore is how the occurring forces depend on the geometrical properties of the system considered. Said differently, the question is "What does the Casimir effect know about a boundary?" In the absence of general answers one approach consists in accumulating further knowledge by adding bits of understanding based on specific calculations for specific configurations. Several examples will be provided in this section and we will see the dominant role the zeta functions introduced play. However, before we come to specific settings let us briefly introduce the zeta function regularization of the Casimir energy and force that we will use later on.
We will consider the Casimir effect in a quantum field theory of a non-interacting scalar field under {\it external} conditions. The action in this case is \cite{itzy80b}
\beq S[\Phi ] = - \frac 1 2 \intl_M \Phi (x) \left( \Delta - V(x) \right) \Phi (x) \,\, dx\label{lec29a}\eeq
describing a scalar field $\Phi (x)$ in the background potential $V(x)$. We assume the Riemannian manifold $M$ to be of the form $M=S^1 \times M_s$, where the circle $S^1$ of radius $\beta$ is used to describe finite temperature $T=1/\beta$ and $M_s$, in general, is a $d$-dimensional Riemannian manifold with boundary. For the action (\ref{lec29a}) the corresponding field equations are \beq (\Delta -V(x)) \Phi (x) =0 . \label{lec29b}\eeq
If $M_s$ has a boundary $\partial M_s$, these equations of motion have to be supplemented by boundary conditions on $\partial M_s$. Along the circle, for a scalar field, periodic boundary conditions are imposed.
Physical properties like the Casimir energy of the system are conveniently described by means of the path-integral functionals \beq Z [V] = \int e^{-S [\Phi ] } \,\, D\Phi , \label{lec29c}\eeq
where we have neglected an infinite normalization constant, and the functional integral is to be taken over all fields satisfying the boundary conditions. Formally, equation (\ref{lec29c}) is easily evaluated to be
\beq \Gamma [V] = - \ln Z[V] = \frac 1 2 \ln \det \left[ ( - \Delta + V(x) )/\mu^2\right] , \label{lec29d}\eeq
where $\mu$ is an arbitrary parameter with dimension of a mass to adjust the dimension of the arguments of the logarithm.
\begin{exercise}
In order to motivate equation (\ref{lec29d}) show that for $P$ a positive definite Hermitian $(N\times N)$-matrix one has
\beq \intl_{\R^n} e^{-\frac 1 2 (x,Px)} (dx) = (\det P)^{-1/2},\nn\eeq where $(dx) = d^nx (2\pi)^{-n/2}$. For $P=-\Delta + V(x)$ and interpreting the scalar product $(x,Px)$ as an $L^2 (M)$-product, one is led to (\ref{lec29d}) by identifying $D\Phi$ with $(dx)$.
\end{exercise} Clearly equation (\ref{lec29d}) is purely formal because the eigenvalues $\lambda_n$ of $-\Delta + V(x)$ grow without bound for $n\to\infty$ and thus expression (\ref{lec29d}) needs further explanations.
In order to motivate the basic definition let $P$ be a Hermitian $(N\times N)$-matrix with positive eigenvalues $\lambda_n$. Clearly $$\left.\ln \det P = \sum_{n=1}^N \ln \lambda_n = - \frac d {ds} \sum_{n=1}^N \lambda_n^{-s} \right|_{s=0} = \left. -\frac d {ds} \zeta_P (s) \right|_{s=0} , $$ and the determinant of $P$ can be expressed in terms of the zeta function associated with $P$. This very same definition, namely \beq \ln \det P = - \zeta_P ' (0) \label{lec29e}\eeq with \beq \zeta_P (s) = \sum_{n=1}^\infty \lambda_n^{-s}\label{lec29ea}\eeq is now applied to differential operators as in (\ref{lec29d}). Here, the series representation is valid for $\Re s$ large enough, and in (\ref{lec29e}) the unique analytical continuation of the series to a neighborhood about $s=0$ is used.
This definition was first used by the mathematicians Ray and Singer \cite{ray71-7-145} to give a definition of the Reidemeister-Franz torsion. In physics, this regularization scheme took its origin in ambiguities of dimensional regularization when applied to quantum field theory in curved spacetime \cite{dowk76-13-3224,hawk77-55-133}. For applications beyond the ones presented here see, e.g., \cite{buch92b,byts03b,dett92-377-252,dunn08-41-304006,dunn05-94-072001,espo94b,espo98b,sach92-65-652}.
The quantity $\Gamma [V]$ is called the effective action and the argument $V$ indicates the dependence of the effective action on the external fields. The Casimir energy is obtained from the effective action via \beq E=\frac \partial {\partial \beta} \Gamma [V] = - \frac 1 2 \frac \partial {\partial \beta} \zeta_{P/\mu^2} ' (0) . \label{lec29f}\eeq
Here, we will only consider the zero temperature Casimir energy \beq E_{Cas} = \lim_{\beta \to \infty} E \label{lec29fa}\eeq and we will next derive a suitable representation for $E_{Cas}$. We want to concentrate on the influence of boundary conditions and therefore we set $V(x) =0$. The relevant operator to be considered therefore is $$P = - \frac{\partial^2}{\partial \tau^2} - \Delta_s$$ where $\tau \in S^1$ is the imaginary time and $\Delta_s$ is the Laplace operator on $M_s$. In order to analyze the zeta function associated with $P$ we note that eigenfunctions, respectively eigenvalues, are of the form
\beq \phi_{n,j} (\tau , y) &=& \frac 1 \beta e^{\frac{2\pi in} \beta \tau } \varphi_j (y) , \nn\\
\lambda_{n,j} &=& \left( \frac{ 2\pi n} \beta \right)^2 + E_j ^2, \quad \quad n\in\Z,\nn\eeq
with $$-\Delta_s \varphi _j (y) = E_j ^2 \varphi_j (y),$$
where $y\in M_s$.
For the non-selfinteracting case considered here, $E_j$ are the one-particle energy eigenvalues of the system. The relevant zeta function therefore has the structure \beq \zeta_P (s) = \sum_{n=-\infty}^\infty \sum_{j=1}^\infty \left( \left( \frac{2\pi n} \beta \right)^2 + E_j^2\right)^{-s} . \label{lec29g}\eeq
We repeat the analysis outlined previously, namely we use equation (\ref{Gamma}) and we apply Lemma \ref{poisson} to the $n$-summation. In this process the zeta function $$\zeta_{P_s} (s) = \sum_{j=1}^\infty E_j^{-2s}$$ and the heat kernel $$K_{P_s} (t) = \sum_{j=1}^\infty e^{-E_j^2 t} \sim \sum_{k=0,1/2,1,...}^\infty a_k \,\, t^{k-\frac d 2}$$ of the spatial section are the most natural quantities to represent the answer,
\beq
\zeta_P (s) &=& \frac 1 {\G (s)} \snuu \intl_0^\infty t^{s-1}
e^{-\left(\frac{2\pi n } \beta \right)^2 t } K_{P_s} (t)\,dt \nn\\
&=& \frac \beta {\sqrt{4\pi}} \frac{\G (s-1/2)}{\G (s)} \zeta_{P_s} (s-1/2)
\nn\\
& &
+ \frac \beta {\sqrt{\pi}\,\,\G (s)} \sneu \intl_0^\infty
t^{s-3/2} e^{-\frac{n^2\beta^2}{4t}} K_{P_s} (t)\,dt .\nn
\eeq
For the Casimir energy we need ($D=d+1$)
\beq
\zeta ' _{P / \mu^2} (0) &=& \zeta_P ' (0) +\zeta_P (0) \ln \mu^2 \nn\\
&= &-\beta \left( FP\,\,\zeta_{P_s} (-1/2) +
2(1-\ln 2) \Res \zeta_{P_s} (-1/2) \right.\nn\\
& &\left. -\frac 1 \beta \zeta_P (0)
\ln \mu^2 \right)
+ \frac \beta {\sqrt{\pi} } \sneu \intl_0^\infty t^{-3/2}
e^{-\left(\frac{n^2\beta^2}{4t}\right)}K_{P_s} (t) \,dt\nn\\
&=& -\beta \left(FP\,\,\zeta_{P_s} (-1/2)
- \frac 1 {\sqrt{4\pi}} a_{D/2} \left[(\ln\mu^2) +
2 (1-\ln 2 ) \right]\right) \nn\\
& & +
\frac \beta {\sqrt{\pi} } \sneu \intl_0^\infty t^{-3/2}
e^{-\left(\frac{n^2\beta^2}{4t}\right)}K_{P_s} (t)\, dt,
\label{lec29h}
\eeq
with the finite part $FP$ of the zeta function and where equations (\ref{zeheres}) and (\ref{zehefun}) together with the fact that $$K_M (t) = K_{S^1} (t) \,\, K_{P_s} (t)$$ have been used, in particular
\beq \mbox{Res }\zeta_{P_s} \left( - \frac 1 2 \right) = - \frac{ a_{D/2}} {2\sqrt \pi} , \quad \quad \zeta_P (0) = \frac \beta {\sqrt{4\pi}} a_{D/2} . \label{relzetcoe}\eeq
At $T=0$ we obtain for the Casimir energy, see equations (\ref{lec29f}) and (\ref{lec29fa}),
\beq
E_{Cas} = \lim_{\beta \to\infty} E = \frac 1 2 FP\,\,\zeta_{P_s} (-1/2)
-\frac 1 {2\sqrt{4\pi}} a_{D/2}
\ln \tilde \mu^2 , \label{lec29i}
\eeq
with the scale $\tilde \mu = (\mu e / 2)$. Equation (\ref{lec29i}) implies that as long as $a_{D/2} \neq 0$ the Casimir energy contains a finite ambiguity and renormalization issues need to be discussed. Note from (\ref{relzetcoe}) that whenever $\zeta_{P_s} (-1/2)$ is finite no ambiguity exists because $a_{D/2}=0$. In the specific examples chosen later we will make sure that these ambiguities are absent and therefore a discussion of renormalization will be unnecessary.
In a purely formal calculation one essentially is also led to equation (\ref{lec29i}). As mentioned, in the quantum field theory of a free scalar field the eigenvalues of a Laplacian are the square of the energies of the quantum fluctuations.
Writing the Casimir energy as (one-half) the sum over the energy of all quantum fluctuations one has
\beq E_{Cas} = \frac 1 2 \sum_{k=0}^\infty \lambda_k^{1/2}, \label{lec30}\eeq
and a formal identification 'shows'
\beq E_{Cas} = \frac 1 2 \zeta_{P_s} \left( - \frac 1 2 \right).\label{lec31}\eeq
Clearly, the expression (\ref{lec30}) is purely formal as the series diverges. However, when $\zeta_{P_s} (-1/2)$ turns out to be finite this formal identification yields the correct result. Otherwise, the ambiguities given in (\ref{lec29i}) remain as discussed above.
An alternative discussion leading to definition (\ref{lec29i}) can be found in \cite{blau88-310-163}.
As a first example let us consider the configuration of two parallel plates a distance $a$ apart analyzed originally by Casimir \cite{casi48-51-793}. For simplicity we concentrate on a scalar field instead of the electromagnetic field and we impose Dirichlet boundary conditions on the plates. The boundary value problem to be solved therefore is $$-\Delta u_k (x,y,z) = \lambda_k u_k (x,y,z) $$ with $$u_k(0,y,z) = u_k(a,y,z) =0.$$ For the time being, we compactify the $(y,z)$-directions to a torus with perimeter length $R$ and impose periodic boundary conditions in these directions. Later on, the limit $R\to \infty$ is performed to recover the parallel plate configuration. Using separation of variables one obtains normalized eigenfunctions in the form
$$u_{\ell_1 \ell_2 \ell} (x,y,z) = \sqrt{\frac 2 {aR^2}} \sin \left( \frac{\pi \ell} a x\right) e^{i \frac {2\pi\ell_1 y} R } e^{i \frac {2\pi\ell_2 z} R } $$ with eigenvalues $$\lambda_{\ell _1 \ell _2 \ell } = \left( \frac{ 2\pi \ell _1} R \right)^2+\left( \frac{ 2\pi \ell _2} R \right)^2+
\left( \frac{ \pi \ell } a \right)^2 , \quad (\ell _1, \ell _2) \in \Z^2, \quad \ell \in \N.$$
This means we have to study the zeta function
\beq \zeta (s) = \sum_{(\ell _1, \ell _2) \in \Z^2} \,\,\sum_{\ell =1}^\infty \left[ \left( \frac{2\pi \ell_1} R \right)^2 + \left( \frac{2\pi \ell_2} R \right)^2 + \left( \frac{\pi \ell} a \right)^2 \right]^{-s} . \label{lec32}\eeq
As $R\to\infty$ the Riemann sum turns into an integral and we compute using polar coordinates in the $(y,z)$-plane \beq \zeta (s) &=& \left( \frac R {2\pi}\right)^2 \sum_{\ell =1}^\infty \,\,\int\limits_{-\infty} ^\infty \,\, \int\limits_{-\infty} ^\infty \left[ k_1^2+k_2^2 + \left(\frac{\pi \ell} a \right)^2 \right]^{-s} \,\,dk_2 \,\, dk_1\nn\\
&=& \left( \frac R {2\pi}\right)^2 \sum_{\ell =1}^\infty 2\pi \int\limits_0^\infty k \left[ k^2 + \left( \frac{\pi \ell} a \right)^2 \right]^{-s} \,\,dk \nn\\
&=& \left.\frac{R^2} {2\pi} \frac 1 {2 (1-s)} \sum_{\ell =1}^\infty \left[ k^2 + \left( \frac{\pi \ell} a \right)^2 \right]^{-s+1} \right|_0^\infty \nn\\
&=& - \frac{R^2} {4\pi (1-s)} \sum_{\ell =1}^\infty \left( \frac{\pi \ell} a \right)^{2 (-s+1)} \nn\\
&=& - \frac{R^2}{4\pi (1-s)} \left( \frac \pi a \right)^{2-2s} \zeta_R (2s-2).\nn\eeq
Setting $s=-1/2$ as needed for the Casimir energy we obtain \beq \zeta \left( - \frac 1 2 \right) = - \frac{R^2} {4\pi } \,\, \frac 2 3 \,\, \left( \frac \pi a \right)^3 \zeta_R (-3) = - \frac{ R^2 \pi^2} { 720 a^3} . \label{lec33} \eeq
The resulting Casimir force {\it per area} is \beq F_{Cas} = -\frac \partial {\partial a} \frac{E_{Cas}}{R^2} = -\frac{\pi^2} {480 a^4} . \label{lec34}\eeq
Note, that this computation takes into account only those quantum fluctuations from between the plates. But in order to find the force acting on the, say, right plate the contribution from the right to this plate also has to be counted. To find this part we place another plate at the position $x=L$ where at the end we take $L\to\infty$. Following the above calculation, we simply have to replace $a$ by $L-a$ to see that the associated zeta function produces $$ \zeta \left( - \frac 1 2 \right) = - \frac{R^2 \pi^2}{720 (L-a)^3} $$ and the contribution to the force on the plate at $x=a$ reads $$F_{Cas} = \frac{\pi^2} {480 (L-a)^4}.$$
This shows the plate at $x=a$ is always attracted to the closer plate. As $L\to\infty$ it is seen that equation (\ref{lec34}) also describes the total force on the plate at $x=a$ for the parallel plate configuration.
\begin{exercise}
Consider the Casimir energy that results in the previous discussion when the compactification length $R$ is kept finite. Use
Lemma \ref{bessel1} to give closed answers for the energy and the resulting force. Can the force change sign depending on $a$ and $R$?
\end{exercise}
More realistically plates will have a finite extension. An interesting setting that we are able to analyze with the tools provided are pistons. These have received an increasing amount of interest because they allow the unambiguous prediction of forces \cite{cava04-69-065015,hert05-95-250402,kirs09-79-065019,mara07-75-085019,teo09-672-190}.
Instead of having parallel plates let us consider a box with side lengths $L_1,L_2$ and $L_3$. Although it is possible to find the Casimir force acting on the plate at $x=L_1$ resulting from the interior of the box, the exterior problem has remained unsolved until today. No analytical procedure is known that allows to obtain the Casimir energy or force for the outside of the box. This problem is avoided by adding on another box with side lengths $L-L_1,L_2$ and $L_3$ such that the wall at $x=L_1$ subdivides the bigger box into two chambers. The wall at $x=L_1$ is assumed to be movable and is called the piston. Each chamber can be dealt with separately and total energies and forces are obtained by adding up the two contributions. Assuming again Dirichlet boundary conditions and starting with the left chamber, the relevant spectrum reads
\beq \lambda_{\ell _1 \ell _2 \ell _3} = \left( \frac{\pi \ell _1} {L_1}\right)^2+\left( \frac{\pi \ell _2} {L_2}\right)^2+\left( \frac{\pi \ell _3} {L_3}\right)^2, \quad \quad \ell_1, \ell _2,\ell _3 \in \N, \label{lec34a}\eeq and the associated zeta function is \beq \zeta (s) = \sum_{\ell _1,\ell _2,\ell _3 \in \N} \left[\left( \frac{\pi \ell _1} {L_1}\right)^2+\left( \frac{\pi \ell _2} {L_2}\right)^2+\left( \frac{\pi \ell _3} {L_3}\right)^2\right]^{-s}. \label{lec35}\eeq
One way to proceed is to rewrite (\ref{lec35}) in terms of the Epstein zeta function in Definition \ref{epdef}.
\begin{exercise}
Use Lemma \ref{bessel1} in order to find the Casimir energy for the inside of the box with side lengths $L_1,L_2$ and $L_3$ and with Dirichlet boundary conditions imposed.
\end{exercise}
Instead of using Lemma \ref{bessel1} we proceed as follows. We write first \beq \zeta (s) &=& \frac 1 2 \sum_{\ell _1 = - \infty}^\infty \,\,\sum_{\ell _2, \ell_3 =1}^\infty \left[ \left( \frac{\pi \ell _1} {L_1}\right)^2 + \left( \frac{ \pi \ell _2} {L_2} \right)^2 + \left( \frac{\pi \ell_3} {L_3} \right)^2 \right] ^{-s} \nn\\
& -& \frac 1 2 \sum_{\ell _2, \ell _3 =1}^\infty \left[ \left( \frac{\pi \ell_2} {L_2} \right)^2 + \left( \frac{\pi \ell _3}{L_3}\right)^2 \right]^{-s} . \label{lec36} \eeq
This shows that it is convenient to introduce \beq \zeta _{\mathcal C}(s) = \sum_{\ell _2, \ell _3 =1}^\infty \left[ \left( \frac{\pi \ell_2} {L_2}\right)^2 + \left( \frac{\pi \ell _3}{L_3}\right)^2 \right]^{-s} . \label{lec37}\eeq
We note that this could be expressed in terms of the Epstein zeta function given in Definition \ref{epdef}. However, it will turn out that this is unnecessary.
Also, to simplify the notation let us introduce $$\mu_{\ell_2 \ell_3} ^2 = \left( \frac{ \pi \ell_2}{L_2}\right)^2 + \left( \frac{ \pi \ell_3}{L_3}\right)^2 . $$ Using equation (\ref{Gamma}) for the first line in (\ref{lec36}) we continue \beq \zeta (s) &=& \frac 1 {2 \Gamma (s)} \sum_{\ell_1=-\infty}^\infty \,\,\sum_{\ell_2, \ell_3 =1}^\infty \int\limits_0^\infty t^{s-1} \exp \left\{ - t \left[ \left( \frac{\pi \ell_1}{L_1}\right)^2 + \mu_{\ell_2\ell_3}^2 \right] \right\} dt \nn\\
&- &\frac 1 2 \zeta_{\mathcal C} (s).\nn\eeq
We now apply the Poisson resummation in Lemma \ref{poisson} to the $\ell_1$-summation and therefore we get
\beq \zeta (s) &=& \frac {L_1} {2\sqrt \pi \,\,\Gamma (s)} \sum_{\ell_1 =-\infty} ^\infty \,\,\sum_{\ell_2, \ell_3=1}^\infty \int\limits_0^\infty t^{s-\frac 3 2 } \exp \left\{ - \frac{L_1^2 \ell_1^2} t - t \mu_{\ell_2\ell_3}^2\right\} dt\nn\\
& -& \frac 1 2 \zeta_{\mathcal C} (s).\label{lec37a}\eeq
The $\ell_1 =0$ term gives a $\zeta _{\mathcal C}$-term, the $\ell_1\neq 0$ terms are rewritten using (\ref{bessel}). The outcome reads \beq \zeta (s) &=& \frac{L_1 \Gamma \left( s-\frac 1 2 \right)}{2 \sqrt \pi \,\,\Gamma (s)} \zeta _{\mathcal C} \left( s- \frac 1 2 \right) - \frac 1 2 \zeta _{\mathcal C} (s) \label{lec38} \\
&+& \frac{2 L_1 ^{s+ \frac 1 2 }}{\sqrt \pi \,\,\Gamma (s)} \sum_{\ell_1, \ell _2, \ell _3 =1}^\infty \left(\frac{\ell_1^2}{\mu_{\ell_2 \ell_3}^2} \right)^{\frac 1 2 \left( s- \frac 1 2\right)} K_{\frac 1 2 -s } \left( 2 L_1 \ell_1 \mu_{\ell_2 \ell_3} \right).\nn\eeq We need the zeta function about $s=-1/2$ in order to find the Casimir energy and Casimir force.
Let $s=-1/2+\epsilon$. In order to expand equation (\ref{lec38}) about $\epsilon =0$ we need to know the pole structure of $\zeta_{\mathcal C} (s)$. From equation (\ref{bessel1}) it is expected that $\zeta_{\mathcal C} (s)$ has at most a first order pole at $s=-1/2$ and that it is analytic about $s=-1$. So for now let us simply assume the structure \beq \zeta_{\mathcal C} \left( - \frac 1 2 + \epsilon \right) &=& \frac 1 \epsilon \mbox{Res } \zeta_{\mathcal C} \left( - \frac 1 2 \right) + \mbox{FP } \zeta_{\mathcal C} \left( - \frac 1 2 \right) + {\mathcal O} (\epsilon ), \nn\\
\zeta _{\mathcal C} (-1+\epsilon ) &=& \zeta_{\mathcal C} (-1) + \epsilon \zeta_{\mathcal C} ' (-1) + {\mathcal O} (\epsilon ^2), \nn\eeq
where $\mbox{Res } \zeta_{\mathcal C} (-1/2)$ and $\mbox{FP }\zeta_{\mathcal C} (-1/2)$ will be determined later.
With this structure assumed, we find \beq \zeta \left( - \frac 1 2 + \epsilon \right) &=& \frac 1 \epsilon \left( \frac{L_1}{4\pi} \zeta_{\mathcal C} (-1) - \frac 1 2 \mbox{Res } \zeta_{\mathcal C} \left( - \frac 1 2 \right) \right)\nn\\
& & + \frac{L_1} {4\pi} \left( \zeta_{\mathcal C} ' (-1)
+ \zeta _{\mathcal C} (-1) ( \ln 4 -1)\right) - \frac 1 2 \mbox{FP } \zeta_{\mathcal C} \left( - \frac 1 2 \right) \label{lec39} \\
& &-\frac 1 \pi \sum_{\ell_1, \ell_2, \ell_3 =1}^\infty \left| \frac{\mu_{\ell_2 \ell_3}}{\ell_1} \right| K_1 \left( 2 L_1 \ell_1 \mu_{\ell_2 \ell_3}\right) .\nn\eeq
This shows that the Casimir energy for this setting is unambiguously defined only if
$\zeta_{\mathcal C} (-1) =0$ and $\mbox{Res }\zeta_{\mathcal C} (-1/2)=0$.
\begin{exercise} \label{exep2} Show the following analytical continuation for $\zeta_{\mathcal C} (s)$:
\beq \zeta_{\mathcal C} (s) &=& - \frac 1 2 \left( \frac{L_3} \pi \right)^{2s} \zeta_R (2s) + \frac{L_2 \Gamma \left( s - \frac 1 2 \right)}
{2 \sqrt \pi \,\,\Gamma (s)} \left( \frac{L_3} \pi \right)^{2s-1} \zeta_R (2s-1) \label{lec40} \\
& &+ \frac{2 L_2 ^{s+1/2}}{\sqrt \pi \,\,\Gamma (s)} \sum_{\ell_2=1}^\infty \sum_{\ell _3 =1}^\infty \left( \frac{\ell_2 L_3}{\pi \ell_3}\right)^{s-1/2} K_{\frac 1 2 -s} \left( \frac{2\pi L_2 \ell_2 \ell_3} {L_3} \right) \nn.\eeq
Read off that $\zeta_{\mathcal C} (-1) = \mbox{Res } \zeta _{\mathcal C} (-1/2) =0$. \end{exercise}
Using the results from Exercise \ref{exep2} the Casimir energy, from equation (\ref{lec39}), can be expressed as
\beq E_{Cas} &=& \frac{L_1}{8\pi} \zeta_{\mathcal C} ' (-1) - \frac 1 4 \mbox{FP } \zeta_{\mathcal C} \left( - \frac 1 2 \right) \label{lec41}\\
&-& \frac 1 {2\pi} \sum_{\ell_1, \ell _2, \ell_3 =1}^\infty \left| \frac{\mu_{\ell_2 \ell_3}}{\ell_1} \right| K_1 (2L_1 \ell_1 \mu_{\ell _2 \ell_3}) . \nn\eeq
\begin{exercise} \label{exenergy} Use representation (\ref{lec40}) to give an explicit representation of the Casimir energy (\ref{lec41}).\end{exercise}
For the force this shows \beq F_{Cas} = - \frac 1 {8\pi} \zeta_{\mathcal C} ' (-1) + \frac 1 {2\pi } \sum_{\ell_1 , \ell_2, \ell _3 =1}^\infty \left| \frac{ \mu_{\ell_2 \ell_3}}{\ell_1} \right| \frac \partial {\partial L_1} K_1 (2 L_1 \ell_1 \mu_{\ell_2 \ell_3} ) .\label{lec42}\eeq
\begin{exercise} \label{exkelvin} Use Definition \ref{bessel} to show that $K_\nu (x)$ is a monotonically decreasing function for $x\in\R_+$.\end{exercise}
\begin{exercise} Determine the sign of $\zeta_{\mathcal C} ' (-1)$. What is the sign of the Casimir force as $L_1 \to \infty$? What about $L_1 \to 0$?\end{exercise}
Remember that the results given describe the contributions from the interior of the box only. The contributions from the right chamber are obtained by replacing $L_1$ with $L-L_1$. This shows for the right chamber
\beq E_{Cas} &=& \frac{L-L_1}{8\pi} \zeta _{\mathcal C} ' (-1) - \frac 1 4 \mbox{FP } \zeta _{\mathcal C} \left( - \frac 1 2 \right) \nn\\
& -& \frac 1 {2\pi} \sum_{\ell_1, \ell_2, \ell_3=1}^\infty \left| \frac{ \mu_{\ell _2 \ell_3}}{\ell_1} \right| K_1 (2 (L-L_1) \ell_1 \mu_{\ell_2 \ell_3} ) ,\nn\\
F_{Cas} &=& \frac 1 {8\pi} \zeta_{\mathcal C} ' (-1) + \frac 1 {2\pi} \sum_{\ell_1 , \ell_2, \ell_3 =1}^\infty \left| \frac{\mu_{\ell_2 \ell_3}}{\ell_1} \right| \frac{\partial}{\partial L_1} K_1 (2 (L-L_1) \ell_1 \mu_{\ell_2 \ell_3}) .\nn\eeq
Adding up, the total force on the piston is
\beq F_{Cas}^{tot} &=& \frac 1 {2\pi } \sum_{\ell_1 , \ell_2, \ell _3 =1}^\infty \left| \frac{ \mu_{\ell_2 \ell_3}}{\ell_1} \right|
\frac \partial {\partial L_1}K_1 (2 L_1 \ell_1 \mu_{\ell_2 \ell_3} )\nn\\&+& \frac 1 {2\pi} \sum_{\ell_1 , \ell_2, \ell_3 =1}^\infty \left| \frac{\mu_{\ell_2 \ell_3}}{\ell_1} \right| \frac{\partial}{\partial L_1} K_1 (2 (L-L_1) \ell_1 \mu_{\ell_2 \ell_3}) . \label{lec43}\eeq
This shows, using the results of Exercise \ref{exkelvin}, that the piston is always attracted to the closer wall.
Although we have presented the analysis for a piston with rectangular cross-section, our result in fact holds in much greater generality. The fact that we analyzed a rectangular cross-section manifests itself in the spectrum (\ref{lec34a}), namely the part $$\left( \frac{\pi \ell_2} {L_2} \right)^3 + \left( \frac{\pi \ell_3}{L_3} \right)^2 $$ is a direct consequence of it. If instead we had considered an arbitrary cross-section ${\mathcal C}$, the relevant spectrum had the form $$\lambda_{\ell _1 i} = \left( \frac{\pi \ell_1} {L_1} \right)^2 + \mu_i^2,$$ where, assuming still Dirichlet boundary conditions on the boundary of the cross-section ${\mathcal C}$, $\mu_i^2$ is determined from $$ \left.\left( - \frac{ \partial ^2}{\partial y^2} - \frac{\partial ^2}{\partial z^2} \right) \phi_i (y,z) = \mu_i^2 \phi_i (y,z) , \quad \quad \phi_i (y,z) \right|_{(y,z) \in \partial {\mathcal C}} =0.$$ Proceeding in the same way as before, replacing $\mu_{\ell_2 \ell _3}$ with $\mu_i$ and introducing $\zeta_{\mathcal C} (s)$ as the zeta function for the cross-section, $$\zeta_{\mathcal C} (s) = \sum_{i=1}^\infty \mu_i^{-2s},$$ equation (\ref{lec37a}) remains valid, as well as equations (\ref{lec38}) and (\ref{lec39}). So also for an arbitrary cross-section the total force on the piston is described by equation (\ref{lec43}) with the replacements given and the piston is attracted to the closest wall.
\begin{exercise} In going from equation (\ref{lec37a}) to (\ref{lec38}) the fact that $\mu_{\ell_2 \ell_3 }^2 > 0$ is used. Above we used $\mu_i^2>0$
which is true because we imposed Dirichlet boundary conditions.
Modify the calculation if boundary conditions are chosen (like Neumann boundary conditions) that allow for $d_0$ zero modes $\mu_i^2 =0$ \cite{kirs09-79-065019}.\end{exercise}
We have presented the piston set-up for three spatial dimensions, but a similar analysis can be performed in the presence of extra dimensions \cite{kirs09-79-065019}. Once this kind of calculation is fully understood for the electromagnetic field it is hoped that future high-precision measurements of Casimir forces for simple configurations such as parallel plates can serve as a window into properties of the dimensions of the universe that are somewhat hidden from direct observations.
As we have seen for the example of the piston, there are cases where an unambiguous prediction of Casimir forces is possible. Of course the set-up we have chosen was relatively simple and for many other configurations even the sign of Casimir forces is unknown. This is a very active field of research; some references are \cite{bord01-353-1,emig07-99-170403,full09-42-155402,milt01b,milt04-37-209,scha09-102-060402}. Further discussion is provided in the Conclusions.
\section{Bose-Einstein condensation of Bose gases in traps}
We now turn to applications in statistical mechanics. We have chosen to apply the techniques in a quantum mechanical system described by the Schr{\"o}dinger equation \beq\left( - \frac{\hbar^2}{2m} \Delta + V(x,y,z)\right) \phi_k (x,y,z) = \lambda_k \phi_k (x,y,z), \label{lec50a}\eeq
that is we consider a gas of quantum particles of mass $m$ under the influence of the potential $V(x,y,z)$. Specifically, later we will consider in detail the harmonic oscillator potential $$V(x,y,z) = \frac m 2 \left( \omega_1 x^2 + \omega_2 y^2 + \omega _3 z^2\right) $$ briefly mentioned in Example \ref{exHO}, as well as a gas confined in a finite cavity.
Thermodynamical properties of a {\it bose} gas, which is what we shall consider in the following, are described by the (grand canonical) partition sum \beq q = - \sum_{k=0}^\infty \ln \left( 1-e^{-\beta (\lambda_k -\mu) } \right) , \label{lec50}\eeq where $\beta$ is the inverse temperature and $\mu$ is the chemical potential. We assume the index $k=0$ labels the unique ground state, that is, the state with smallest energy eigenvalue $\lambda_0$. From this partition sum all thermodynamical properties are obtained. For example the particle number is \beq N= \left.\frac 1 \beta \frac{\partial q}{\partial \mu} \right|_{T,V} = \sum_{k=0}^\infty \frac 1 {e^{\beta (\lambda_k - \mu)} -1} , \label{lec51}\eeq
where the notation $(\partial q/\partial \mu|_{T,V})$ indicates that the derivative has to be taken with temperature $T$ and volume $V$ kept fixed.
The particle number is the most important quantity for the phenomenon of Bose-Einstein condensation.
Although this phenomenon was predicted more than 80 years ago \cite{bose24-26-178,eins24-22-261}
it was only relatively recently experimentally verified \cite{ande95-269-198,brad95-75-1687,davi95-75-3969}.
Bose-Einstein condensation is one of the most interesting properties of a system of bosons. Namely, under certain conditions it is possible to have a phase transition at a critical value of the temperature in which all of the bosons can condense into the ground state. In order to understand at which temperature the phenomenon occurs a detailed study of $N$, or alternatively $q$, is warranted. This is the subject of this section.
We first note that from the fact that the particle number in each state has to be non-negative it is clear that $\mu<\lambda_0$
has to be imposed.
It is seen in (\ref{lec50}) that as $\beta \to 0$ (high temperature limit) the behavior of $q$ cannot be easily understood. But contour integral techniques together with the zeta function information provided makes the analysis feasible and it will allow for the determination of the critical temperature of the bose gas.
Let us start by noting that from $$\ln (1-x) = - \sum_{n=1}^\infty \frac{x^n} n , \quad \quad \mbox{for }|x|<1,$$ the partition sum can be rewritten as \beq q = \sum_{n=1}^\infty \sum_{k=0}^\infty \frac 1 n e^{-\beta (\lambda_k - \mu ) n } . \label{lec52}\eeq
The $\beta \to 0$ behavior is best found using the following representation of the exponential.
\begin{exercise} \label{expcon} Given that $$\lim_{|y| \to \infty} | \Gamma (x+iy)|
\,\,e^{\frac \pi 2 |y|}\,\, |y|^{\frac 1 2 -x} = \sqrt{2\pi} ,
\quad x,y\in\R ,$$ and $$\Gamma (z) = \sqrt{ 2\pi} e^{\left(
z-\frac 1 2 \right) \log z \,\, - z } \left( 1+ o (1)\right),$$ as
$|z| \to \infty$, show that
\beq e^{-a} = \frac 1 {2\pi i} \int\limits_{\sigma-i\infty} ^{\sigma
+ i\infty} \,\, a^{-t}\,\, \Gamma (t) \,\,dt,\label{lec53}\eeq valid for $\sigma >
0$, $|\mbox{arg}\,\, a| < \frac \pi 2 - \delta $, $0<\delta \leq
\pi /2$. \end{exercise}
Before we apply this result to the partition sum (\ref{lec52}) let us use a simple example to show how this formula allows us to determine asymptotic behavior of certain series in a relatively straightforward fashion. From Lemma \ref{poisson} we know that
\beq \sum_{\ell =1}^\infty e^{-\beta \ell^2} &=& \frac 1 2 \sum_{\ell =-\infty}^\infty e^{-\beta \ell^2} - \frac 1 2 \nn\\
&=& \frac 1 2 \sqrt{\frac \pi \beta }- \frac 1 2 + \sqrt{\frac \pi \beta} \sum_{\ell =1}^\infty e^{-\frac{\pi^2} \beta \ell^2} .\label{lec52c}\eeq
As $\beta \to 0$ it is clear that the series on the left diverges and Lemma \ref{poisson} shows that the leading behavior is described by a $1/\sqrt{\beta}$ term, followed by a constant term, followed by exponentially damped corrections. Let us see how we can easily find the polynomial behavior as $\beta \to 0$ from (\ref{lec53}). We first write \beq \sum_{\ell =1}^\infty e^{-\beta \ell ^2} = \sum_{\ell =1}^\infty \frac 1 {2\pi i} \intl _{\sigma - i\infty} ^{\sigma + i\infty} (\beta \ell ^2)^{-t} \Gamma (t)dt. \nn\eeq
Here, $\sigma > 0$ is assumed by Exercise \ref{expcon}. However, in order to be allowed to interchange summation and integration we need to impose $\sigma >1/2$ and find
\beq \sum_{\ell =1}^\infty e^{-\beta \ell ^2} &=& \frac 1 {2\pi i} \intl_{\sigma-i\infty} ^{\sigma+i\infty} \beta^{-t} \Gamma (t) \sum_{\ell =1}^\infty \ell ^{-2t}dt \nn\\
&=& \frac 1 {2\pi i} \intl_{\sigma - i\infty} ^{\sigma + i \infty} \beta^{-t} \Gamma (t) \zeta_R (2t) dt.\nn\eeq
In order to find the small-$\beta$ behavior, the strategy now is to shift the contour to the left. In doing so we cross over poles of the integrand generating polynomial contributions in $\beta$. For this example, the right most pole is at $t=1/2$ (pole of the zeta function of Riemann) and the next pole is at $t=0$ (from the Gamma-function). Those are all singularities present as $\zeta_R (-2n)=0$ for $n\in\N$.
Therefore, \beq \sum_{\ell =1}^\infty e^{-\beta \ell ^2} &=& \beta^{-\frac 1 2} \,\,\Gamma \left( \frac 1 2 \right) \frac 1 2 + \beta^0 \cdot 1 \cdot \zeta_R (0) +\frac 1 {2\pi i} \intl_{\tilde \sigma -i\infty}^{\tilde \sigma + i\infty} \beta^{-t} \Gamma (t) \zeta_R (2t)dt \nn\\
&=& \frac 1 2 \sqrt{\frac \pi \beta} - \frac 1 2 + \frac 1 {2\pi i}\intl_{\tilde \sigma -i\infty}^{\tilde \sigma + i\infty} \beta^{-t} \Gamma (t) \zeta_R (2t) dt,\nn\eeq where $\tilde \sigma <0$ and where contributions from the horizontal lines between $\tilde \sigma \pm i\infty$ and $\sigma\pm i\infty$ are neglected. For the remaining contour integral plus the neglected horizontal lines one can actually show that they will produce the exponentially damped terms as given in (\ref{lec52c}). How exactly this actually happens has been described in detail in \cite{eliz89-40-436}.
\begin{exercise}
Argue how $\sum_{n=1}^\infty \,\, e^{-\beta n^\alpha},$
$\beta
>0, \alpha >0$, behaves as $\beta \to 0$ by using the above procedure. Determine the leading three terms in the expansion assuming that the contributions from the contour at infinity can be neglected.
\end{exercise}
\begin{exercise}
Find the leading three terms of the small-$\beta$ behavior of
$$\sum_{n=1}^\infty \log \left( 1-e^{-\beta n}\right)$$
assuming that the contributions from the contour at infinity can be neglected.
\end{exercise}
Let us next apply the above ideas to the partition sum (\ref{lec52}). As a further warmup, for simplicity, let us first set $\mu =0$. Not specifying $\lambda_k$ for now and using $$\zeta (s) = \sum_{k=0}^\infty \lambda_k^{-s}$$
for $\Re s >M$ large enough to make this series convergent,
we write
\beq q &=& \sum_{n=1}^\infty \sum_{k=0}^\infty \frac 1 n e^{-\beta \lambda_k n} \nn\\
&=& \sum_{n=1}^\infty \sum_{k=0}^\infty \frac 1 n \frac 1 {2\pi i} \intl_{\sigma - i\infty}^{\sigma+i\infty} (\beta \lambda_k n)^{-t} \Gamma (t) dt \nn\\
&=& \frac 1 {2\pi i} \intl_{\sigma - i\infty}^{\sigma + i \infty} \beta^{-t} \Gamma (t) \left( \sum_{n=1}^\infty n^{-t-1}\right) \left( \sum_{k=0}^\infty \lambda_k ^{-t} \right) dt\nn\\
&=& \frac 1 {2\pi i} \intl _{\sigma -i\infty}^{\sigma + i\infty} \beta^{-t} \Gamma (t) \zeta_R (t+1) \zeta (t)dt .\nn\eeq
Here $\sigma >M$ is needed for the algebraic manipulations to be allowed.
It is clearly seen that the integrand has a double pole at $t=0$. The right most pole (at $M$) therefore comes from $\zeta (t)$, and the location of this pole determines the leading $\beta \to 0$ behavior of the partition sum.
For the harmonic oscillator potential, in the notation of Example \ref{exHO}, the Barnes zeta function occurs and we have
\beq q = \frac 1 {2\pi i} \intl_{\sigma-i\infty}^{\sigma + i\infty}\beta^{-t} \Gamma (t) \zeta_R (t+1) \zeta_{\mathcal B} (t,c|\vec r)dt .\label{lec60}\eeq
The location of the poles and its residues are known for the Barnes zeta function, see Definition \ref{defber} and Theorem \ref{polebarn}, in particular one has $$\mbox{Res } \zeta_{\mathcal B} (3,c| \vec r) = \frac 1 {2 \hbar^3\Omega^3}, $$ where, as is common, the geometric mean of the oscillator frequencies $$\Omega = (\omega_1 \omega_2 \omega_3)^{1/3}$$ has been used. The leading order of the partition sum therefore is
\beq q = \frac{\pi^4}{90} \frac 1 {(\beta \hbar \Omega )^3} + {\mathcal O} (\beta^{-2}).\nn\eeq
\begin{exercise}
Use Definition \ref{defber} and Theorem \ref{polebarn} to find the subleading order of the small-$\beta$ expansion of the partition sum $q$.
\end{exercise}
\begin{exercise}
Consider the harmonic oscillator potential in $d$ dimensions and find the leading and subleading order of the small-$\beta$ expansion of the partition sum $q$.
\end{exercise}
If instead of considering a bose gas in a trap we consider the gas in a finite three-dimensional cavity $M$ with boundary $\partial M$ we have to augment the Schr{\"o}dinger equation (\ref{lec50a}) by boundary conditions. We choose Dirichlet boundary conditions and thus the results for the heat kernel coefficients (\ref{leadcoef}) are valid.
From equation (\ref{zeheres}) we conclude furthermore that the rightmost pole of $\zeta (s)$ is located at $s=3/2$ and that $$\res \zeta \left( \frac 3 2 \right) = \frac{a_0}{\Gamma \left( \frac 3 2 \right) } = \frac{\mbox{vol}(M)}{4\pi^2},$$ furthermore the next pole is located at $s=1$. For this case, the leading order of the partition sum therefore is $$q=\frac 1 {(4\pi\beta)^{3/2}} \zeta_R \left( \frac 5 2 \right) \mbox{vol}(M) + {\mathcal O} (\beta^{-1}).$$ One way to read this result is that the bose gas does know the volume of its container because it can be found from the partition sum. This is completely analogous to the statement for the drum where we used the heat kernel instead of the partition sum.
Subleading orders of the partition sum reveal more information about the cavity, see the following exercise. But as for the drums, the gas does not know all the details of the shape of the cavity because there are different cavities leading to the same eigenvalue spectrum \cite{gord92-27-134}. Those cavities cannot be distinguished by the above analysis.
\begin{exercise}
Consider a bose gas in a $d$-dimensional cavity $M$ with boundary $\partial M$. Use (\ref{leadcoef}) and (\ref{zeheres}) to find the leading and subleading order of the small-$\beta$ expansion of the partition sum $q$. What does the bose gas know about its container, meaning what information about the container can be read of from the high-temperature behavior of the partition sum?
\end{exercise}
In order to examine the phenomenon of Bose-Einstein condensation we have to consider non-vanishing chemical potential. Close to the phase transition, as we will see,
more and more particles have to reside in the ground state and the value of the chemical potential will be close to the smallest eigenvalue, which is the 'critical' value for the chemical potential,
$\mu_c = \lambda_0$. Near the phase transition, for the expansion to be established, it will turn out advantageous to rewrite $\lambda_k-\mu$ such that the small quantity $\mu_c-\mu$ appears, $$\lambda_k-\mu = \lambda_k - \mu_c + \mu_c - \mu = \lambda_k - \lambda_0 + \mu_c - \mu.$$
Given the special role of the ground state, we separate off its contribution and write
\beq q = q_0 +\sum_{n=1}^\infty {\sum_{k=1}^{\,\,\,\infty }} \,\,\frac 1 n \,e^{-\beta n (\lambda_k -\lambda_0 )} \,\, e^{-\beta n (\mu_c -\mu)} .\nn\eeq
Note that the $k$-sum starts with $k=1$, which means that the ground state is not included in this summation. Employing the representation (\ref{lec53}) only to the first exponential factor and proceeding as before we obtain \beq q = q_0 + \frac 1 {2\pi i} \intl_{\sigma - i\infty} ^{\sigma + i \infty} \beta^{-t} \Gamma (t) \mbox{Li}_{1+t} \left( e^{-\beta (\mu_c - \mu)}\right) \zeta_0 (t) dt, \label{lec61}\eeq
with the polylogarithm \beq \mbox{Li}_n (x) = \sum_{\ell =1}^\infty \frac{x^\ell} {\ell^n}, \label{lec62}\eeq
and the spectral zeta function \beq \zeta_0 (s) = {\sum_{k=1}^{\,\,\,\infty}}(\lambda_k - \lambda_0)^{-s}.\nn\eeq
In order to determine the small-$\beta$ behavior of expression (\ref{lec61}) let us discuss the pole structure of the integrand.
Given $\mu_c - \mu >0$, the polylogarithm $\mbox{Li}_{1+t} (e^{-\beta (\mu_c -\mu )})$ does not generate any poles. Concentrating on the harmonic oscillator, we find
\beq \mbox{Res } \zeta_0 (3) &=& \frac 1 {2(\hbar \Omega)^3}, \nn\\
\mbox{Res } \zeta_0 (2) &=& \frac 1 {2\hbar^2} \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right).\nn\eeq
Note that $\zeta_0 (s)$ is the Barnes zeta function as given in Definition \ref{defbarn} with $c=0$ where we have to exclude $\vec m= \vec 0$ from the summation. However, clearly the residues at $s=3$ and $s=2$ can still be obtained from Theorem \ref{polebarn} with $c\to 0$ taken.
Shifting the contour to the left we now find
\beq q &=& q_0 + \frac 1 {(\beta \hbar \Omega)^3} \mbox{Li} _4 \left( e^{-\beta (\mu_c -\mu)}\right)\nn\\
& &+ \frac 1 {2(\beta \hbar)^2} \mbox{Li}_3 \left( e^{-\beta (\mu_c -\mu)}\right) \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right) + ...\nn\eeq
In order to find the particle number $N$ we need the relation for the polylogarithm \beq \frac{\partial \mbox{Li}_n (x)} {\partial x} = \frac 1 x \mbox{Li}_{n-1} (x),\nn\eeq which follows from (\ref{lec62}). So
\beq N &=& N_0 +\frac 1 {(\beta \hbar \Omega)^3} \mbox{Li} _3 \left( e^{-\beta (\mu_c -\mu)}\right)\nn\\
& &+ \frac 1 {2(\beta \hbar)^2} \mbox{Li}_2 \left( e^{-\beta (\mu_c -\mu)}\right) \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right) + ...\nn\eeq
\begin{exercise} \label{propol} Use (\ref{lec53}) and (\ref{lec62}) to show $$ \mbox{Li}_n \left( e^{-x}\right) = \zeta_R (n) - x \zeta_R (n-1) + ...$$ valid for $n>2$. What does the subleading term look like for $n=2$?
\end{exercise}
As the critical temperature is approached $\mu\to\mu_c$ and with Exercise \ref{propol} the particle number close to the transition temperature becomes \beq N &=& N_0 +\frac {\zeta_R (3)} {(\beta \hbar \Omega)^3}
+ \frac {\zeta_R (2)} {2(\beta \hbar)^2} \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right) + ...\label{lec63}\eeq
The second and third term give the number of particles in the {\it excited} levels (at high temperature close to the phase transition).
The critical temperature is defined as the temperature where all excited levels are completely filled such that lowering the temperature the ground state population will start to build up. This means the defining equation for the critical temperature $T_c = 1/\beta _c$ in the approximation considered is
\beq N = \frac 1 {(\beta_c \hbar \Omega)^3} \zeta_R (3) + \frac 1 {2(\beta_c\hbar)^2} \zeta_R (2) \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right).\eeq
Solving for $\beta_c$ one finds \beq T_c = T_0 \left\{ 1-\frac{\zeta_R (2)} {3 \zeta_R (3)^{2/3}} \,\,\delta \,\, N^{-1/3}\right\}.\nn\eeq
Here, $T_0$ is the critical temperature in the bulk limit ($N\to\infty$)
\beq T_0 = \hbar \Omega \left( \frac N {\zeta_R (3)} \right)^{1/3}\nn\eeq
and \beq \delta = \frac 1 2 \Omega^{2/3} \left( \frac 1 {\omega_1 \omega_2} + \frac 1 {\omega_1 \omega_3} + \frac 1 {\omega_2 \omega_3} \right).\nn\eeq
Different approaches can be used to obtain the same answers \cite{gros95-50-921,gros95-208-188,haug97-225-18,haug97-55-2922}.
If only a few thousand particles are used in the experiment the finite-$N$ correction is actually quite important. For example the first successful experiments on Bose-Einstein condensates were done with rubidium \cite{ande95-269-198} at frequencies $\omega_1 = \omega_2 = 240 \pi / \sqrt 8$ s$^{-1}$ and $\omega_3 = 240 \pi$s$^{-1}$. With $N=2000$ one finds $T_c \sim 31.9$nK$=0.93\,\, T_0$ \cite{kirs96-54-4188}, a significant correction compared to the thermodynamical limit.
\begin{exercise}
Consider the bose gas in a $d$-dimensional cavity. Find the particle number and the critical temperature along the lines described for the harmonic oscillator. What is the correction to the critical temperature caused by the finite size of the cavity? (For a solution to this problem see \cite{kirs99-59-158}.)
\end{exercise}
\section{Conclusions}
In these lectures some basic zeta functions are introduced and used to analyze the Casimir effect and Bose-Einstein condensation for particular situations. The basic zeta functions considered are the Hurwitz, the Barnes and the Epstein zeta function. Although these zeta functions differ from each other they have one property in common: they are based upon a sequence of numbers that is explicitly known and given in closed form. The analysis of these zeta functions and of the indicated applications in physics is heavily based on this explicit knowledge in that well-known summation formulas are used.
In most cases, however, an explicit knowledge of the eigenvalues of, say, a Laplacian will not be available and an analysis of the associated zeta
functions will be more complicated. In recent years a new class of examples where eigenvalues are defined implicitly as solutions to transcendental equations has become accessible. In some detail let us assume that eigenvalues are determined by equations of the form
\beq F_\ell ( \lambda_{\ell ,n}) =0\label{implieigen}\eeq with $\ell ,n$ suitable indices. For example when trying to find eigenvalues and eigenfunctions of the Laplacian whenever possible one resorts to separation of variables and $\ell$ and $n$ would be suitable 'quantum numbers' labeling eigenfunctions. To be specific consider a scalar field in a three dimensional ball of radius $R$ with
Dirichlet boundary conditions. The eigenvalues $\lambda_k$ for
this situation, with $k$ as a multiindex, are thus determined
through
$$-\Delta \phi_k (x) = \lambda_k \phi _k (x), \quad \quad \phi (x)
|_{|x| = R} =0 .$$ In terms of spherical coordinates $(r,\Omega)$,
a complete set of eigenfunctions may be given in the form
$$\phi_{l,m,n} (r, \Omega ) = r^{-1/2} J_{l+1/2}
(\sqrt{\lambda_{l,n} }r ) Y _{l,m} (\Omega) ,$$ where $Y_{l,m}
(\Omega )$ are spherical surface harmonics \cite{erde55b}, and
$J_\nu$ are Bessel functions of the first kind \cite{grad65b}.
Eigenvalues of the Laplacian are determined as zeroes of Bessel
functions. In particular, for a given angular momentum quantum
number $l$, imposing Dirichlet boundary conditions, eigenvalues
$\lambda_{l,n}$ are determined by \beq J_{l+1/2} \left(
\sqrt{\lambda_{l,n} }R\right) =0 .\label{1}\eeq
Although some properties of the zeroes of Bessel functions are well understood \cite{grad65b},
there is no closed form for them available and
we encounter the situation described by (\ref{implieigen}). In order to find properties of the zeta function associated with this kind of boundary value problems the idea is to use the argument principle or Cauchy's residue theorem. For the situation of the ball one writes the zeta function in the form \beq \zeta (s) = \sum_{l=0}^\infty (2l+1) \frac 1 {2\pi i}
\int\limits_\gamma k^{-2s} \frac
\partial {\partial k} \ln J_{l+1/2} (kR) dk,\label{2}\eeq
where the contour $\gamma$ runs counterclockwise and must enclose
all solutions of (\ref{1}). The factor $(2l+1)$ represents the
degeneracy for each angular momentum $l$ and the summation is over
all angular momenta. The integrand has singularities exactly at the eigenvalues and one can show that the residues are one such that the definition of the zeta function is recovered. More generally, in other coordinate systems, one would have, somewhat symbolically,
\beq\zeta (s) = \sum_j d_j \frac 1 {2\pi i}
\int\limits_\gamma k^{-2s} \frac \partial
{\partial k} \ln F_j (k) dk, \eeq
the task being to
construct the analytical continuation of this object.
The details of the procedure will depend very much
on the properties of the special function $F_j$ that enters, but often all the information needed can be found \cite{kirs02b}.
Nevertheless, for many separable coordinate systems this program has not been performed but efforts are being made in order to obtain yet unknown precise values for the Casimir energy for various geometries.
\section*{Acknowledgements}
This work is supported by the National Science Foundation Grant
PHY-0757791.
Part of the work was done while the author
enjoyed the hospitality and partial support of the
Department of Physics and Astronomy of
the University of Oklahoma. Thanks go in particular to Kimball Milton and his group who
made this very pleasant and exciting visit possible.
\def\cprime{$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
| {
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Is Mesut Ozil's Kicker interview the first shot fired in battle for Arsenal's future?
January 9, 2017 by blogonyougunners, posted in Arsenal, premier-league
Mesut Ozil has set an army of tongues wagging and hares running in the last 24 hours after an interview with German publication Kicker was published in which he appeared to link his future with that of Arsene Wenger.
It's no secret that Mesut is in talks with Arsenal about extending his stay in North London and many believe money will be at the heart of whether the German opts to sign or not.
But, in his interview, which was published on Sunday, Mesut was quoted as saying he was awaiting assurances over Wenger's future before talks could progress.
He is reported* to have said: "To be completely honest: in my career, it's never been about money. It is about fun, about trust. The club knows that I am here because of Arsène Wenger above all else, he brought me here and I relish his faith. The club also knows that I want to know what the manager is going to do."
If that's not a bombshell statement – at least in relative terms – then I'm not sure what is.
One thing is certain, it is very much a public show of faith and support in Wenger continuing as Arsenal manager. The implication – although not explicit – is that if Wenger leaves in the summer, when his contract expires, Mesut will likely follow him out of the door.
It's a rare occurrence in modern football to see a player show such steadfastly loyalty to his manager and should, in its simplest form, be applauded. Loyalty is a commodity that I thought went out with East 17 at the end of the 90s but it appears a few last vestiges remain.
I do think, however, it is also a show of faith that will put the German at odds with a significant number of Arsenal fans, who are vociferously against Wenger continuing at the club after his contract expires in the summer. While I don't think these fans will turn their backs on Mesut, I think it will colour their opinion of him from this moment on. Some will see it as holding the club to ransom, against the will of fans.
Others, who are just as vociferously for Wenger remaining as Arsenal manager, will see it as succour to their cause, 'if we lose Arsene, we lose one of our world class talents' they will argue. They wouldn't be wrong to do so.
In the middle of it all is the club who, if they weren't walking a tightrope before, most certainly are now. I don't know whether they have opened talks with Wenger about a new contract, or whether they even plan to, but this is a significant new factor in the decision that is looming large like a storm cloud over the Emirates.
If Mesut and Wenger leave, will Alexis Sanchez follow suit? Will there be such disruption in the make-up of the team that we risk a period of turbulence?
Or will a new manager come in, bringing fresh ideas and new players, while transforming the squad into a well-organised, title-winning machine?
These are all things the board will have to consider and, whatever they decide, it is certain to upset a large portion of fans.
Of course, Mesut can't be blamed for that, he is simply repaying the faith shown in him by Wenger right back to his days at Werder Bremen. Good on him.
But I get the feeling this interview is the first battle line drawn in what is destined to be a rocky period ahead, particularly if things deteriorate on the pitch.
I have no doubt the comments will be put to Arsene in his weekly press conference and it will be fascinating to see where we go from here.
*The full interview is surprisingly frank and can be found here: http://news.arseblog.com/2017/01/mesut-ozil-kicker-interview-full-transcript/
Tagged Arsene Wenger, Mesut Ozil
Previous postArsenal show that doing things by half is the key to success
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"redpajama_set_name": "RedPajamaCommonCrawl"
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If I work, I'll wake up at 400am to leave by 430am for my 12.5 hour day shift. I get off at 630pm and drive an hour (traffic-willing) to Santa Rosa to catch the last 30-45 min of training. When I'm not working I try to hit up double days. Either class in the morning and again at night or cardio in the morning and class at night. I drive to Concord and San Francisco to cross train at Combat Fitness and El Nino as well, which also takes up some time. The driving can wear you down more than training but I love my sister schools and the work they put into me when I train with them.
I want to test myself…to constantly push myself to be better than I was yesterday. Nothing will show you what is in you more than facing your fear and dominating it, than thinking you've reached your threshold yet pushing beyond it, than overcoming the valleys when you lose but learning from it, and experiencing the highs of winning and going farther than you've ever thought you'd go. With fighting, the joy is in the journey.
U.S. Retired Navy Seal Jocko Willink. I've read his book Extreme Ownership and I listen to his podcast. It's been on my bucket list to go down to his school in San Diego and train with him. I admire what he and our military have done for this country and I am motivated by his "get after it" mentality. There's no room for feeling sorry for oneself and the only option is taking ownership of your life and dominating.
IF YOU COULD FIGHT ANYONE, LIVING OR DEAD, WHO WOULD YOU FIGHT AND WHY?
I don't really have a specific person in mind. Other than fighting for sport what really makes me want to fight someone is when they are taking advantage of or abusing someone unable to defend themselves. Bullying makes me pretty mad too. I tend to want to stand up to that. It took me a little while to figure out how to stand up for myself, so I'm pretty protective now. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,031 |
"Thirteen Days in September" & "The Art of Adapting"
IN THE WORKS FOR FALL 2015:
Written by acclaimed sports author and oral historian Harvey Frommer, with an intro by pro football Hall of Famer Frank Gifford, When It Was Just a Game tells the fascinating story of the ground-breaking AFL–NFL World Championship Football game played on January 15, 1967: Packers vs. Chiefs. Filled with new insights, containing commentary from the unpublished memoir of Kansas City Chiefs coach Hank Stram, featuring oral history from many who were at the game—media, players, coaches, fans—the book is mainly in the words of those who lived it and saw it go on to become the Super Bowl, the greatest sports attraction the world has ever known. Archival photographs and drawings help bring the event to life.
Two books as different from each other as can be are the focus of this review. One deals with a fateful meeting among leaders that changed the course of history. The other is a novel about a recently separated woman and her challenges.
"Thirteen Days in September" by Lawrence Wright (Knopf, $27.95, 245 pages) is a masterpiece. Compelling, informative, as timely as today's headlines, its focus is on the meeting at Camp David of President Carter, Israel's Prime Minister Menachem Begin and Egypt's President Anwar Sadat.
Wright, the Pulitzer Prize winning author of The Looming Tower, is at the very top of his game in his newest effort as takes us day-by-day into the interplay of the three world leaders at that 1978 conference. The book includes a brief background history of Egyptian-Israeli relations that goes back to the Exodus The story of three powerful men representing three major religions gives the book texture and nuance. HIGHLY RECOMMENDED
"The Art of Adapting" by Cassandra Dunn (Touchstone, $25.00, 354 pages) mixes together four voices Lana, Matt, Byron and Abby in a dramatic story of how and why families look out for and help each other. Brother Matt has Asperger's Syndrome but provides a very unexpected anchor for sister Lana, the newly separated mother of teenagers.
Dunn's writing is moving, at times inspiring. "The Art of Adapting" is about coping, surviving and prevailing. There is much of all of us in this graceful book. It is a NOTABLE ACHIEVEMENT.
You can reach Harvey Frommer at:
Harvey Frommer is in his 38th year of writing books. A noted oral historian and sports journalist, the author of 42 sports books including the classics: "New York City Baseball,1947-1957" and "Shoeless Joe and Ragtime Baseball," his acclaimed REMEMBERING YANKEE STADIUM was published in 2008 and his REMEMBERING FENWAY PARK: AN ORAL AND NARRATIVE HISTORY OF THE HOME OF RED SOX NATION was published to acclaim in 2011. The prolific Frommer is at work on When It Was Just a Game, An Oral History on Super Bowel One.
His work has appeared in the New York Times, Los Angeles Times, Washington Post, New York Daily News, Newsday, USA Today, Men's Heath, The Sporting News, among other publications.
FROMMER SPORTSNET (syndicated) reaches a readership in the millions and is housed on Internet search engines for extended periods of time.
on Twitter: http://twitter.com/south2nd
on Linked In: http://www.linkedin.com/profile/edit?locale=en_US
on the Web: http://www.dartmouth.edu/~frommer
Dr. Frommer is the Official Book Reviewer of Travel-Watch.
*Autographed copies of Frommer books are available .
Other Frommer sports related articles can be found at:
http://www.baseballlibrary.com/baseballlibrary/excerpts/growing_up_baseball.stm
http://www.baseballsavvy.com/archive/w_pumpsie.html
http://discuss.washingtonpost.com/wp-srv/zforum/02/sports_frommer083002.htm
http://www.redsoxnation.net/forums/index.php?showtopic=15388&s=93d3ba714519f785d290b4305f62c42d
Harvey Frommer along with his wife, Myrna Katz Frommer are the authors of five critically acclaimed oral/cultural histories, professors at Dartmouth College, and travel writers who specialize in cultural history, food, wine, and Jewish history and heritage in the United States, Europe, and the Caribbean.
This Article is Copyright © 1995 - 2014 by Harvey Frommer. All rights reserved worldwide. | {
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} | 8,045 |
Tag: horror
Kate's Review: "Secret Santa"
Book: "Secret Santa" by Andrew Shaffer
Publishing Info: Quirk Books, November 2020
Where Did I Get This Book: I received an eARC from the publisher via NetGalley.
Book Description: The Office meets Stephen King, dressed up in holiday tinsel, in this fun, festive, and frightening horror-comedy set during the horror publishing boom of the '80s, by New York Times best-selling satirist Andrew Shaffer.
Out of work for months, Lussi Meyer is desperate to work anywhere in publishing. Prestigious Blackwood-Patterson isn't the perfect fit, but a bizarre set of circumstances leads to her hire and a firm mandate: Lussi must find the next horror superstar to compete with Stephen King, Anne Rice, and Peter Straub. It's the '80s, after all, and horror is the hottest genre.
But as soon as she arrives, Lussi finds herself the target of her co-workers' mean-spirited pranks. The hazing reaches its peak during the company's annual Secret Santa gift exchange, when Lussi receives a demonic-looking object that she recognizes but doesn't understand. Suddenly, her coworkers begin falling victim to a series of horrific accidents akin to a George Romero movie, and Lussi suspects that her gift is involved. With the help of her former author, the flamboyant Fabien Nightingale, Lussi must track down her anonymous Secret Santa and figure out the true meaning of the cursed object in her possession before it destroys the company—and her soul.
Review: Thank you to Quirk Books and NetGalley for providing me with an eARC of this novel!
Happy Holidays, everyone! I know that it's kind of a weird holiday season for SO MANY reasons, but I hope that everyone is making do with the circumstances and being safe as well as finding joy and togetherness. Even if that means doing it via FaceTime. In our house we're wrapping up Hanukkah and getting ready to have a solitary Christmas, which means I'm digging into books when I'm not eating all the latkes. If you're like me and like it when the horror genre and Yuletide combine, enjoying movies like "Gremlins", or "Black Christmas" (more the old one. The new one was cathartic, but also SUPER hamfisted), then "Secret Santa" by Andrew Shaffer may be the kind of book you want with your holiday cheer.
Is this a scene from "Gremlins", or is this me stuffing my face with Christmas cookies? (source)
Shaffer is known for his satire and cheeky humor, so it's safe to say that "Secret Santa" isn't the scariest book you could read this time of year. Luckily, I wasn't expecting it to be terrifying, so that worked for me, for the most part. I liked Lussi, our ambitious protagonist, as she fits the 'ambitious woman in a man's world' mold in a way that adds to the story. You understand her wants and her determination to succeed in the publishing industry, especially as a young woman in the 1980s. I liked her sarcasm and her wit, and I felt that her characterization fit into the story as a whole, reflecting a snide and cutthroat time and place. The mystery as to what is going on at Blackwood-Patterson when things start to go awry is a slow build, and it reads less like a horror novel building up the dread and more like a strange whodunnit. By the time we circle back to the actual origin on what is going on (I don't want to spoil TOO much, but do know that occultism and Nazis do enter into it. Take that as you will), the lack of scares was a little frustrating. That isn't to say that there aren't creepy elements involved with this tale. Let's just say that if you don't like dolls, you will find a lot to be scared about. But overall, the scary elements are very obviously harkening to a very specific time in horror publishing, when pulp paperbacks were the rage and strange concepts weren't hard to come by (I seriously suggest looking into "Paperbacks from Hell" by Grady Hendrix if you want more information on this). This will work for some people, but it may leave others in the cold.
But what worked the most was that this book is clearly a love letter to 1980s horror fiction, be it paperback pulp horror novels or films that involve tiny beings that wreak havoc and gaslight those around them into thinking they are losing their minds. You can tell that Shaffer really loves the horror of this era, and the winks and nods to the genre are fun for someone like me who has an affection for it. Sometimes the 80s references in general got a little heavy handed, but you feel like you're in on the joke, so I was able to deal with it with minimal eye rolling. This book is very clearly a love letter to a very specific kind of fiction, and I, for one, really loved seeing it all unfold. You can just feel the fun he was having writing this book, and frankly, that's charming as hell.
"Secret Santa" is a tongue in cheek ode to horror paperbacks with a festive holiday bow placed right on top. If you're looking for some holiday creeps, it could be the right book to have by the fire with a glass of eggnog.
Rating 7: Entertaining and sardonic, "Secret Santa" has some Christmas fun as well as some creepy moments if you don't like dolls. It's not terribly scary, but it has more than enough 80s horror nostalgia to make up for it.
"Secret Santa" isn't included on any Goodreads lists as of yet, but it would fit in on "Creepy Christmas" to be sure!
Find "Secret Santa" at your library using WorldCat, or a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on December 17, 2020 October 8, 2020 Categories Kate's ReviewsTags 7 Rating, horrorLeave a comment on Kate's Review: "Secret Santa"
Kate's Review: "This Is Not a Ghost Story"
Book: "This Is Not a Ghost Story" by Andrea Portes
Publishing Info: HarperTeen, November 2020
Book Description: I am not welcome. Somehow I know that. Something doesn't want me here.
Daffodil Franklin has plans for a quiet summer before her freshman year at college, and luckily, she's found the job that can give her just that: housesitting a mansionfor a wealthy couple.
But as the summer progresses and shadows lengthen, Daffodil comes to realize the house is more than it appears. The spacious home seems to close in on her, and as she takes the long road into town, she feels eyes on her the entire way, and something tugging her back.
What Daffodil doesn't yet realize is that her job comes with a steep price. The house has a long-ago grudge it needs to settle . . . and Daffodil is the key to settling it.
I love haunting and eerie ghost stories of the Gothic variety, though I will admit that I sometimes find them to be predictable. The genre itself has such specific building blocks to it that a little bit of a road map is in place from the get go, which is perfectly okay. Because of this, I thought that I knew where "This Is Not a Ghost Story" by Andrea Portes was going to go when I picked it up. I was still excited to read it, mind you, but my overconfident ass thought that it wouldn't have much new to say. But lo and behold, this story took me by surprise in all the best ways possible.
While the set up definitely seems run of the mill (money strapped college student agrees to house sit an isolated home, strange things start happening), Portes has created a Gothic ghost story that feels unique and fresh specifically because of how she has chosen to tell it. Daffodil, our first person protagonist, has a stream of consciousness and anxious voice in her narration, and as she tells the reader what is happening to her in this house and in the town around it, we have a slow build up of dread in the way that Daffodil would be experiencing it. As she tries to write off strange occurrences as they happen, we see the panic rise and rise until she is unable to deny that something very bad is happening, which I REALLY liked. From things that could very easily be explainable to the absolutely disturbing, Daffodil's stream of consciousness builds the tension and also becomes VERY relatable as the story goes on. And all along she is cracking wise, making funny observations, and generally cracking me up, which helped cut the tension but didn't ruin it. But along with the present bad things happening inside the house and on the grounds, we also get a slow unfolding of Daffodil's past, and why her anxiety and unease is ever present. Most of this involves a love story with her high school boyfriend Zander, someone that she had always felt was out of her league but who loved her very much. Their high school romance felt real, and as it runs its course in her memories and unfolds in tragic ways, you see a whole other side to Daffodil that makes her all the more endearing.
In terms of scares and plotting, "This Is Not A Ghost Story" has some elements that are old hat, and some reveals that didn't quite catch me by surprise in the way that they were probably supposed to do. But Portes still manages to write these elements and reveals in a way that made them enjoyable, and they still felt pretty fresh and convincing. I worry that if I say too much we'll start getting into spoiler territory, but I do want to mention two aspects that worked really well for me. The first is the uncanny creepiness, of things going missing or ending up in places that don't make sense. The other is the slow building unease with the people on and around the property, from a construction worker on the guest house named Mike to a nosy and stuffy neighbor named Penelope. We think one thing about them at first, but Portes picks away at our perceptions of them and makes them suspenseful in their own way (though I will say that Daffodil does the thing that MANY women do when it comes to men who may be threatening: she second guesses herself and her instincts. THIS was so well done that it felt like a GREAT way to show the target audience that no, your instincts should probably be listened to).
So while it may not have shocked me or really scared me too much, "This Is Not A Ghost Story" was an enjoyable and poignant ghostly tale of trauma, forgiveness, and the things that haunt us. Fans of Gothic horror should definitely check it out!
Rating 8: A haunting, bittersweet, and sardonically funny Gothic tale, "This Is Not a Ghost Story" will keep you guessing, and will stay with you.
"This Is Not a Ghost Story" is new and not on many specific Goodreads lists, but I think it would fit in on "Modern Gothic", and "Not the Normal Paranormal".
Find "This Is Not a Ghost Story" at your library using WorldCat, or a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on December 10, 2020 November 10, 2020 Categories Kate's ReviewsTags 8 Rating, horror, Young AdultLeave a comment on Kate's Review: "This Is Not a Ghost Story"
Kate's Review: "The Easton Falls Massacre: Bigfoot's Revenge"
Book: "The Easton Falls Massacre: Bigfoot's Revenge" by Holly Rae Garcia and Ryan Prentice Garcia
Publishing Info: Close to the Bone, October 2020
Where Did I Get This Book: I received an eARC from the authors.
Book Description: US Army Veteran Henry Miller embarks on a hunt at the edge of the Black Forest, but strays from the path and finds himself too close to the East Cascade Mountain Range.
Something lurks in the forest on the other side of those mountains. An ancient race of Bigfoot that have kept to themselves for centuries, until one of them defies the warnings and roams too far from the safety of their home.
When these two intersect, alliances are broken and events set in motion that will leave residents of the town of Easton Falls, Washington, fighting for their lives.
Review: Thank you to Holly Rae Garcia and Ryan Prentice Garcia for sending me an eARC of this novella!
Back when I was a kiddo, along with my supernatural and ghost obsession I was also very into cryptids and cryptozoology. I would check out books from my school library about The Loch Ness Monster, The Abominable Snowman, and, of course, Bigfoot. As time went on my fascination with such things waned, but I'm still game to talk about weird cryptic stories if anyone else is (especially if we are talking about my boi Mothman!). I haven't really dabbled into much creature feature horror in my book repertoire. Enter Holly Rae Garcia reaching out to me asking if I would be interested in reading and reviewing the novella "The Easton Falls Massacre: Bigfoot's Revenge", which she wrote with her husband, Ryan Prentice Garcia. I was taken with the description, and said yes, yes I would. It's been awhile since I last did a stint with some Bigfoot lore.
Nuff said. (source)
"The Easton Falls Massacre: Bigfoot's Revenge" is a bloody, tense, fun horror novella in which humans have to contend with the wrath of Bigfoot, and let me tell you, I had a blast reading it. The authors do a fantastic job of fitting in relationship angst, small town drama, and a sense of foreboding and isolation, all while building up to a gory and unrepentant gore-a-thon where a couple Bigfoots enter a small town's city limits and fuck shit up. There is a little background given to the area (being the Pacific Northwest, Bigfoot Central U.S.A.), as well as hints dropped about how the Indigenous people who had been there before connect to the Bigfoot lore. While I'm always a bit skittish when it comes to Indigenous belief systems and mythology being used in fantasy and/or horror media, I will say that in this book it wasn't trotted out repeatedly or focused upon too much (that said, as a white woman, I can't speak for Indigenous People). Along with a solid setting, we have some pretty solid characters too. Our protagonist, Henry, has a tough backstory which gives him a pall of sadness, and there are enough fraught and messy aspects to him and his relationship to his lover Kate that make you connect and feel for both of them. You also get a good sense about the town and how the people function within it, and how their relationships grow, change, and sometimes turn toxic. All of this is accomplished in a short novella, and I was impressed that so much was explored in the number of pages we had to work with.
And now the Bigfoot stuff. Fun as hell. I don't want to give many spoilers, of course, but just know that the reasons behind the 'revenge' aspect that is promised in the title is pretty understandable. While I could sympathize with Henry, and certainly the townspeople that we encountered, ultimately I was here to see Bigfoots take out a bunch of humans, because humans are the WORST. And this book certainly delivers that. The descriptions of the various death scenes, and the aftermaths, are gruesome and over the top, and absolutely feel like they could be those you'd find in a B-schlock horror creature feature from Troma. Which makes the read super entertaining.
Halloween may be just behind us, but if you're like me and always looking to extend the season by a hair, you should definitely consider picking up "The Easton Falls Massacre: Bigfoot's Revenge". And hey, if you live in the Pacific Northwest, just be careful when you wander into the woods. You never know what you could find!
Rating 8: A quick bit of creature feature horror for cryptid enthusiasts especially, "The Easton Falls Massacre: Bigfoot's Revenge" reads like a cheesy horror movie in all the best ways possible!
"The Easton Falls Massacre: Bigfoot's Revenge" isn't featured on any Goodreads lists as of now, but I think it would fit in on "Sasquatch Books", and "Cryptids".
Find "The Easton Falls Massacre: Bigfoot's Revenge" at the authors' website!
Author thelibraryladiesmnPosted on November 5, 2020 October 7, 2020 Categories Kate's ReviewsTags 8 Rating, horror, Short storiesLeave a comment on Kate's Review: "The Easton Falls Massacre: Bigfoot's Revenge"
Kate's Review: "The Haunting of Beatrix Greene"
Book: "The Haunting of Beatrix Greene" by Rachel Hawkins, Ash Parsons, and Vicky Alvear Shecter.
Publishing Info: Serial Box, October 2020/January 2021 (this is expanded upon in the review)
Book Description: Beatrix Greene has made a name for herself in Victorian England as a reputable spiritual medium, but she's a fraud: even she knows ghosts aren't real. But when she's offered a lucrative job by James Walker—a scientist notorious for discrediting pretenders like her—Beatrix takes the risk of a lifetime. If her séance at the infamously haunted Ashbury Manor fools him, she will finally have true financial freedom. If she fails, her secret will become her public shame.
But James has his own dark secrets, and he believes only a true medium can put them to rest. When Beatrix's séance awakens her real gift—and with it, a vengeful spirit—James finds that the answers he seeks are more dangerous than he could have imagined. Together, with a group of supernatural sleuths, Beatrix and James race to settle the ghost's unrest before it strikes— or else they might not make it out of the haunted manor alive.
New York Times bestselling author Rachel Hawkins, along with Ash Parsons and Vicky Alvear Shecter, weaves darkness, death, and a hint of desire into this suspenseful mystery for fans of Sherlock Holmes and Crimson Peak.
We are wrapping up our Horrorpalooza reads, and Halloween is this weekend. First and foremost, Happy Halloween everyone! What better way to end the spooky reading season than with a good old fashioned haunted house story? "The Haunting of Beatrix Greene" is that, but with some modern lens tweaks and a unique storytelling style that I'm still kind of trying to wrap my head around. But if a fraudulent medium and an old manor on the English moors are involved, I'm going to be on board regardless of stylistic choices.
Would I do that? Can't be sure. Would I READ about it? Every time. (source)
Since I've been noting the format, that's the aspect of the story I'll address first. "The Haunting of Beatrix Greene" is published by Serial Box, an organization that releases books and audiobooks in weekly episodes, each episode written by different authors. Our authors for this book are Rachel Hawkins, Ash Parsons, and Vicky Alvear Shecter. It feels like it's a Round Robin writing exercise, which is definitely unique and not something that I've really encountered outside of fan fiction. I think that when you are experiencing it in this way, that is in weekly episodes like a TV show or podcast, that is a pretty cool thing. But in this format where it's just a book that collects them all together but still calls them episodes as opposed to chapters, it feels a little strange. That is a bit exacerbated by the fact that the actual complete book isn't going to be coming out until January, but the episodes have started dropping on Serial Box now, something that I wasn't totally aware of when I requested this book. I think that this is confusing, frankly, and the 'one chapter a week' format may not appeal to all. If you want to do the whole book in one go, January will be when your time comes, according to Amazon.
But, there was a lot that I liked about this story in terms of the bare bones of the haunted house theme. The biggest stand out for me is Beatrix herself, a woman who is making a life as a medium during the time in England when Spiritualism was having its first big boom. Beatrix doesn't actually believe in ghosts, and uses the kinds of tricks and strategies that many of those charlatan spiritualists used, like cold reading and ringers. But we also get to see that Beatrix isn't doing this because she's conniving or sociopathic. Rather, she's trying to survive as a single woman during a time where options are limited. When she is invited by skeptical scientist (and charlatan exposer) James Walker to conduct a seance at an old manor called Ashbury Hall, she feels a need to prove herself to a seemingly arrogant scientist, and to protect her reputation so she can keep making a living. I loved Beatrix, and felt that she was nuanced and complicated. James, too, had some complexities and nuance to his character, and didn't just serve as an antagonist foil who is ultimately going to be a love interest to Beatrix. He has his own personal stake in wanting to have her come to Ashbury Manor.
And yes, there is a romance between them, and yes, it feels a little unrealistic given that this story takes place in such a short time AND they find themselves in a very haunted and dangerous house. But the chemistry and banter between Beatrix and James sizzles, so I was very easy to forgive it. Along with the romance, of course, is a ghost story, and I thought that that part of it was also pretty well done. We have some fun nods to the genre, with believers and unbelievers getting in way too deep, and a house with a tragic history that goes back far beyond the time that the first brick was laid. The horror aspects have some moments of genuine scares and a little bit of gore, but I would also say that this is a friendlier read for horror lite people who may not want to be SUPER scared. A lot is crammed into this short tale (clocking in at less than two hundred pages), but I feel like Hawkins, Parsons, and Shecter are able to pull it all together and never make it feel rushed or haphazard. And going back to the format for just a moment, even though the chapters alternate between different authors, their styles meld together well enough that it always felt like a unified narrative, which isn't always easy to do.
"The Haunting of Beatrix Greene" is fun and just a little bit spooky, and a nice addition to the many other ghostly Gothic tales that came before it.
And that wraps up Horrorpalooza 2020! I hope that you all have a safe and happy Halloween!
Rating 7: A spooky and entertaining Gothic tale of (semi)terror, "The Haunting of Beatrix Greene" has some good scares and some good characters, but the format seems unnecessary and the way it's released may be confusing to some people.
"The Haunting of Beatrix Greene" is new and not included on any Goodreads lists yet, but it would fit in on "Haunted Houses", and "Historical Ghost Fiction".
Find "The Haunting of Beatrix Greene" at Serial Box. In January, find it at your local library using WorldCat, or a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 29, 2020 September 30, 2020 Categories Kate's ReviewsTags 7 Rating, horror, RomanceLeave a comment on Kate's Review: "The Haunting of Beatrix Greene"
Kate's Review: "Night of the Mannequins"
Book: "Night of the Mannequins" by Stephen Graham Jones
Publishing Info: Tor.Com, September 2020
Book Description: Stephen Graham Jones returns with Night of the Mannequins, a contemporary horror story where a teen prank goes very wrong and all hell breaks loose: is there a supernatural cause, a psychopath on the loose, or both?
Review: Thank you to NetGalley for providing me with an eARC of this novella!
While the "Scary Stories To Tell in the Dark" books had many stories that messed me up, the one that scarred me the most was that of "Harold", in which two farmers create a scarecrow to be a joke of a friend, which then comes to life and wreaks havoc. The idea that an inanimate but human looking object could come to life and kill you really scared me. So doing some research into "Night of the Mannequins" by Stephen Graham Jones (beyond the appropriately vague description) got me pretty hyped for the idea of a mannequin coming to life and killing teens in a friend group. After all, mannequins are a bit creepy enough on their own, right?
And we can buiiiiild this thing together, staaanding strong forever… (source)
Now, it is admittedly going to be hard to talk about this novella in detail without potentially treading towards spoiler territory, and I REALLY don't want to spoil anything for those who don't want to be spoiled. So just be warned…. there may be hints of spoilers in this review.
Our protagonist/first person narrator is Sawyer, a teenage boy in a group of friends who like to pull pranks on each other, and who at one time found a department store mannequin that they decided to make into their mascot. They called him Manny, and brought him along on all kinds of adventures. As they grew up, Manny was left behind, but as they are nearing the end of high school Sawyer thinks that one more prank with Manny could be fun. And it is… until Sawyer sees Manny stand up and walk away. What comes next is a story that reads like a slasher movie, with a lot of weird deranged action, a very funny narrative voice, and a lot of ambiguity as to what exactly is happening to Sawyer and his friends, and whether or not a mannequin has come to life with a taste for revenge. There isn't much dread to be found here, but what you do have is a lot of splatterpunk gore descriptions, action that reads like a movie, and a twisted up perception of what is real and what isn't.
Sawyer is both incredibly funny to follow as well as authentic in his frenzied teenage voice, his ruminations and planning clearly leaving some logic out of his plans in his hopes to save people from Manny the Mannequin. I found myself laughing out loud, even at moments where it probably wasn't appropriate to be doing so, but like in a slasher film, part of the entertainment is seeing the crazed and over the top kill scenes. Jones sprinkles a little bit of interesting pathos in every once in awhile, be it hints as to Sawyer's family life or the lives of his friends, as well the fear of losing your childhood and what comes next. I also have to say that Jones does a really good job of making the reader question almost everything in terms of reliability and reality. By the time I got towards the end I thought that I had everything clear in my mind, but then Jones managed to pull the rug out from under me again! His stories have a bit of a brutality to them, but there is always a bit of wryness to go with it, and I really like that.
"Night of the Mannequins" is strange and filled with splatterpunk themes, but it definitely has some inner machinations that are intriguing to find and explore. Plus, it's a quick read, the perfect one for a season-appropriate afternoon of horror leisure reading. Discover Stephen Graham Jones if you haven't, and you could totally start here.
Rating 8: A weird and disturbing (but also fun) slasher kinda story. It's a hoot as well as a trip, and it's exactly the kind of entertainment a slasher kinda story should be!
"Night of the Mannequins" isn't on any Goodreads lists yet, but I think that it would fit in on "Haunted Dolls", and "Indigenous Fiction Books".
Find "Night of the Mannequins" at your library using WorldCat, or at a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 27, 2020 August 27, 2020 Categories Kate's ReviewsTags 8 Rating, horror, Novella1 Comment on Kate's Review: "Night of the Mannequins"
Kate's Review: "The Silence of the Lambs"
Book: "The Silence of the Lambs" by Thomas Harris
Publishing Info: St. Martin's Press, July 1988
Book Description: A serial murderer known only by a grotesquely apt nickname—Buffalo Bill—is stalking women. He has a purpose, but no one can fathom it, for the bodies are discovered in different states. Clarice Starling, a young trainee at the FBI Academy, is surprised to be summoned by Jack Crawford, chief of the Bureau's Behavioral Science section. Her assignment: to interview Dr. Hannibal Lecter—Hannibal the Cannibal—who is kept under close watch in the Baltimore State Hospital for the Criminally Insane.
Dr. Lecter is a former psychiatrist with a grisly history, unusual tastes, and an intense curiosity about the darker corners of the mind. His intimate understanding of the killer and of Clarice herself form the core of The Silence of the Lambs—an ingenious, masterfully written book and an unforgettable classic of suspense fiction.
Review: I first read "The Silence of the Lambs" when I was a freshman in high school. My mom and I were at a local drug store and they had the mass market paperback for sale, and she was kind enough to purchase it for me because she never censored what I wanted to read (even if she probably sighed to herself about her daughter's morbid curiosities). I read it very quickly, completely immersed in the story. I saw the film shortly thereafter, and both are now very high on my lists in terms of favorite books and films. There has been a debate lately between film fans on Twitter as to whether "The Silence of the Lambs" is horror or not. Given that I watch it ever Halloween Season and my friends and I did a Netflix Party of it on one of our weekly Terror Tuesdays, I can see the argument for it being within the horror genre (though I myself flip flop between yes and no). Because of this, I decided that it was time to revisit the story in book form, and that I would include it in this year's Horrorpalooza. And picking it up again felt like I was visiting an old friend.
But not one that I'm planning on having for dinner or anything… (source)
This book is still so good. While I think that I PROBABLY like the movie better, that is only because the movie is so perfect at bringing all of these three dimensional and amazing characters to life. Hannibal Lecter is a literary villain who stands above so many, but this book is 100% Clarice Starling's. Harris created a 'badass female protagonist' who feels so real, so relatable, and so nuanced that I'm continually shocked that a man wrote her (given that sometimes male authors can miss the mark when it comes to writing lady characters). You feel Clarice's ambition, her frustration, her smarts and her anxiety and her need to solve the Buffalo Bill case, and you completely understand why she would go to the lengths she does… like getting close to Hannibal, even though he is incredibly manipulative and dangerous. I also really appreciated the moments of misogyny and sexism that she has to endure, as for 1988 for a guy to put those in, and to make them sting and hurt without feeling overdone or corny, that's impressive. Clarice is such an important and formative feminist icon for me, and I was worried that revisiting her might not hold up as well. But it did. Hannibal, too, is a fascinating character, and while he doesn't have the same amount of page time as Clarice (which is just fine), his insidiousness and his charm makes him very creepy, as well as vastly entertaining. But for me, it's all about Clarice.
I had also forgotten how well Harris slowly builds to the Buffalo Bill mystery that is the true heart of "The Silence of the Lambs". You get small references to it here and there, but it takes awhile to realize that this story is the one that Clarice is going head first into. Seeing her slowly gather her evidence, be it through talking with Lecter or going into a storage container to find evidence, or going to an autopsy and finding a bug, we get to go along with Clarice, see the pathology unfold, and then see Bill in action. And Harris really knows how to write a suspenseful scene. Even though I have read the book before and seen the movie countless times, I found myself getting nervous and anxious during some of the action moments. Especially during the Buffalo Bill kidnapping we get to witness on the page.
I will say that Buffalo Bill, while a really well done villain (and completely under appreciated in the movie. Ted Levine is GOD TIER and gets overshadowed by Hopkins. I get why, but my GOD, every time I watch Levine just blows me away), feels problematic now given that Bill is clearly dealing with some kind of gender dysphoria. I do know that Bill is based on a whole smorgasbord of serial killers, and that Jerry Brudos is almost assuredly the one who manifests in Bill's obsession with womanhood (as during one of his attacks he was dressed like a woman, would dress up in his victims clothing, and had a huge thing about womens shoes). But while it's stated that Bill isn't 'actually transsexual' (paraphrasing from the text here) in the book, it still feels like there are shades of transphobia with the character. I think it says more about the time it was written than much else, but it's definitely something to think about, and stands out for all the wrong reasons today.
Overall, "The Silence of the Lambs" is still a gripping, scary, and masterful classic that blurs the lines between thriller and horror. Re-reading was a joy, and I am glad I jumped back into it.
Rating 10: An enduring thriller classic that touches on real life horrors and (mostly) holds up.
"The Silence of the Lambs" is included on the Goodreads lists "I Like Serial Killers", and "Best Female Lead Characters".
Find "The Silence of the Lambs" at your library using WorldCat, or a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 22, 2020 October 16, 2020 Categories Kate's ReviewsTags 10 Rating, horror1 Comment on Kate's Review: "The Silence of the Lambs"
Kate's Review: "Plain Bad Heroines"
Book: "Plain Bad Heroines" by Emily M. Danforth
Publishing Info: William Morrow, October 2020
Book Description: The award-winning author of The Miseducation of Cameron Post makes her adult debut with this highly imaginative and original horror-comedy centered around a cursed New England boarding school for girls—a wickedly whimsical celebration of the art of storytelling, sapphic love, and the rebellious female spirit.
Our story begins in 1902, at The Brookhants School for Girls. Flo and Clara, two impressionable students, are obsessed with each other and with a daring young writer named Mary MacLane, the author of a scandalous bestselling memoir. To show their devotion to Mary, the girls establish their own private club and call it The Plain Bad Heroine Society. They meet in secret in a nearby apple orchard, the setting of their wildest happiness and, ultimately, of their macabre deaths. This is where their bodies are later discovered with a copy of Mary's book splayed beside them, the victims of a swarm of stinging, angry yellow jackets. Less than five years later, The Brookhants School for Girls closes its doors forever—but not before three more people mysteriously die on the property, each in a most troubling way.
Over a century later, the now abandoned and crumbling Brookhants is back in the news when wunderkind writer, Merritt Emmons, publishes a breakout book celebrating the queer, feminist history surrounding the "haunted and cursed" Gilded-Age institution. Her bestselling book inspires a controversial horror film adaptation starring celebrity actor and lesbian it girl Harper Harper playing the ill-fated heroine Flo, opposite B-list actress and former child star Audrey Wells as Clara. But as Brookhants opens its gates once again, and our three modern heroines arrive on set to begin filming, past and present become grimly entangled—or perhaps just grimly exploited—and soon it's impossible to tell where the curse leaves off and Hollywood begins.
A story within a story within a story and featuring black-and-white period illustrations, Plain Bad Heroines is a devilishly haunting, modern masterwork of metafiction that manages to combine the ghostly sensibility of Sarah Waters with the dark imagination of Marisha Pessl and the sharp humor and incisive social commentary of Curtis Sittenfeld into one laugh-out-loud funny, spellbinding, and wonderfully luxuriant read.
A few years ago I picked up the YA book "The Miseducation of Cameron Post" by Emily M. Danforth, as it had been put on a Pride display at the library I was working at at the time. I read it and liked it, and saw the movie and liked that as well. I told myself that I would be on the lookout for more books by Danforth, but admittedly didn't really pay too close attention to her publications. When I saw the book "Plain Bad Heroines" on NetGalley and read the description, it caught my eye enough that I requested it and got a copy… and then when I put two and two together that this, too, was by Danforth, I was even more excited to read it!
"Plain Bad Heroines" is a mixed bag of genres, perspectives, themes, and narratives. It definitely has a horror story within its pages, but it also has some romance, some historical high strangeness, and some cheeky tongue in cheek humor, with a number of wry citations thrown in by a humorous narrator. The crux of this story is a former school estate called Brookhants, where at the turn of the 20th century a number of gruesome deaths occurred. We get to see the timeline of these deaths, and see the mysteries surrounding them, but we also get to see a modern narrative involving a movie set and crew that is trying to make a horror movie based upon a book written about the mysterious deaths and the supposedly haunted and/or cursed grounds. The past story has a focus on Libbie and Alex, two lovers who are running the school where a number of girls, who were also involved in various sapphic relationships and were obsessed with a book with lesbian themes, died in horrific ways. The modern tale focuses on three women involved in the production of a new horror movie about the school: Merritt, who wrote the book about Brookhants and framed it as a queer feminist history; Harper Harper, the superstar actress who champions her own queerness; and Audrey, a former child actress who is hoping to reinvent herself. The two timelines are interspersed together and unfold with tragedy, humor, longing, and Gothic horror.
But even with suspense, romantic drama, Hollywood nonsense, and some actual horror moments that set me on edge, "Plain Bad Heroines" is also a very earnest, charming, and funny tale. The narrative jumps between timelines but connects with a humorous and ever nudging Narrator, with citations, side comments, and the occasional period appropriate illustration ever at hand. It works so well, and while I was worried that it may take away from the ghost story (and the body horror elements, SO MANY BODY HORROR ELEMENTS), it never did. While I mostly liked the modern story more, I did like getting the background and context of the haunted school and seeing how the curse and its fallout was affecting Harper, Audrey, and Merritt in the modern day. Fair warning: if you have a thing about yellow jackets, content warnings ABOUND in this book. Danforth hits many a horror moment, which was great to see and something I didn't necessarily expect from her given her other book. Yet she does it with ease, and pulls off lots of unsettling moments.
But it's really the characters that propelled the story for me, both the ones from the past and in the present. Libbie, Alex, and the other characters in the past storyline were well described and grounded in historical truths that are very sad when it comes to lesbian relationships during that time period. You know that societal constraints are driving many things out of their control, and the sadness of the complications, and the doom you know is coming, made these characters very sympathetic, even when they were making decisions and choices that may have been damaging and hurtful. But (once again) it's the modern women who really stood out, all of their complexities and nuances on display and perfectly drawn out. While Harper and Merritt have a lot of great moments of goodness and badness, it was Audrey who really captivated me, her desperation to move on from her old life and to find something new incredibly palpable. I loved watching all of them interact, and how Danforth put the power of womens' relationships, be they romantic or platonic, at the forefront.
I really enjoyed "Plain Bad Heroines". Danforth is such a dynamic writer, and if you want something spooky this season that isn't too scary, this will surely captivate you as it did me.
Rating 9: A complex, wry, and genuinely creepy book about Gothic mysteries, untimely deaths, sapphic romance, and a whole lot of yellow jackets, "Plain Bad Heroines" is a pleasant surprise from Danforth and a fun Halloween read!
"Plain Bad Heroines" isn't included on many Goodreads lists as of yet, but I think it would fit in on "Sapphic Book Lists".
Find "Plain Bad Heroines" at your library using WorldCat, or at a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 20, 2020 September 14, 2020 Categories Kate's ReviewsTags 9 Rating, horror1 Comment on Kate's Review: "Plain Bad Heroines"
Kate's Review: "Clown in a Cornfield"
Book: "Clown in a Cornfield" by Adam Cesare
Publishing Info: HarperTeen, August 2020
Book Description: Quinn Maybrook just wants to make it until graduation. She might not make it to morning.
Quinn and her father moved to tiny, boring Kettle Springs to find a fresh start. But ever since the Baypen Corn Syrup Factory shut down, Kettle Springs has cracked in half. On one side are the adults, who are desperate to make Kettle Springs great again, and on the other are the kids, who want to have fun, make prank videos, and get out of Kettle Springs as quick as they can.
Kettle Springs is caught in a battle between old and new, tradition and progress. It's a fight that looks like it will destroy the town. Until Frendo, the Baypen mascot, a creepy clown in a pork-pie hat, goes homicidal and decides that the only way for Kettle Springs to grow back is to cull the rotten crop of kids who live there now.
Review: I am not afraid of clowns. I have friends who are, but I myself don't really have much beef with them outside of sometimes finding them a little pointless. Even the likes of Pennywise of John Wayne Gacy's Pogo just don't really make me tap into my inner coulrophobic. But I do like a book that reads like a slasher story, and reading the description of "Clown in a Cornfield" by Adam Cesare felt like exactly that. Throw in some Millennial resentment towards older generations that don't quite get the road we've had to travel, and I was eager to dive in and see what Cesare was going to do with all of it.
"Clown in a Cornfield" is a bit of a slasher story, a bit of small town secrets story, and some 'okay, Boomer' memes all mixed together to create a YA horror tale. On a few levels, this works out pretty well and makes for fun reading. The very concept of a bunch of teens being slaughtered by someone wearing a clown mask is great horror fodder, but "Clown in a Cornfield" takes it a few steps further than that and works through some generational angst that is playing out in the real world. The town of Kettle Springs, the setting of this book, is having a bit of a reckoning when it comes to the older people in town versus the teenagers. The older people want Kettle Springs to stay the same, living off of good family values, hard work, and the corn syrup factory that has given the town jobs and prosperity, until recently, that is. The younger generation, specifically the teens, just want to live their lives and then move on. Cesare takes a pretty realistic conflict and pumps it full of blood and guts, and it works pretty well, with those with traditional values blaming inevitable changes in values for all the ills within the town. It could have been heavy handed, but Cesare keeps his tongue planted in cheek firmly enough that it's a rather effective satire. I also liked a few of our main characters, namely Quinn, the new girl in town who is trying to fit in. Quinn has enough tragic backstory to give her a little bit of pathos, but also stands on her own two feet well enough that she is likable and endearing.
But that said, some of the executions of the plot points didn't work as well for me. Besides Quinn and a couple other characters, we don't really get to know enough about a number of the people we're following so that it doesn't feel like the stakes are too high when the clown Frendo ("No Country for Old Men" reference?) comes a knocking with weaponry and murderous intent. I don't really care too much when a slasher film just has a bunch of stereotypes to act as machete fodder for a masked killer, but I think that on the page you have a little more wiggle room to give us some insight into your characters, even if it's just a little bit. Along with that, the pacing was a little off at times, feeling a bit rushed in some places but kind of draggy in others. I bought the plot overall, as it really is just a slasher story and I know what I'm getting into there. But I think that had there been a little more focus on fleshing out some other characters and less on making super cool kills happen, it probably would have worked a little better. Especially since the satire was pretty well thought out.
Inevitable progress to traditionalists everywhere. (source)
"Clown in a Cornfield" is a pretty fun read. I think that it would have worked better as a gory limited series, but Cesare left room for a sequel, and it was good enough that I would definitely read it. If you don't like clowns, maybe skip it? But if you're like me, this could be a fun read for this time of year.
Rating 6: A sly premise and some fun characters keep this story afloat, though the plot is a little hasty at times and the scares feel like they'd work better on screen than on the page.
"Clown in a Cornfield" isn't included on many relevant Goodreads lists, but it would fit in on "Clown Horror". Obviously.
Find "Clown in a Cornfield" at your library using WorldCat, or a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 15, 2020 September 27, 2020 Categories Kate's ReviewsTags 6 Rating, horror, Young Adult2 Comments on Kate's Review: "Clown in a Cornfield"
Kate's Review: "Ring Shout"
Book: "Ring Shout" by P. Djèlí Clark
Publishing Info: Tor.com, October 2020
Book Description: Nebula, Locus, and Alex Award-winner P. Djèlí Clark returns with Ring Shout, a dark fantasy historical novella that gives a supernatural twist to the Ku Klux Klan's reign of terror.
D. W. Griffith is a sorcerer, and The Birth of a Nation is a spell that drew upon the darkest thoughts and wishes from the heart of America. Now, rising in power and prominence, the Klan has a plot to unleash Hell on Earth.
Luckily, Maryse Boudreaux has a magic sword and a head full of tales. When she's not running bootleg whiskey through Prohibition Georgia, she's fighting monsters she calls "Ku Kluxes." She's damn good at it, too. But to confront this ongoing evil, she must journey between worlds to face nightmares made flesh–and her own demons. Together with a foul-mouthed sharpshooter and a Harlem Hellfighter, Maryse sets out to save a world from the hate that would consume it.
Review: Thanks to NetGalley for providing me with an eARC of this novella!
I have said it many a time, the horror genre can be so effective at symbolism and social commentary wrapped up in a creepy and spooky tale. You've seen it with such films as "Night of the Living Dead", "Get Out", "Candyman", "Dawn of the Dead", the list goes on and on and on. Books, too, do this very well, with recent titles like "Lovecraft Country" and "The Devil in Silver" being two that come to mind for me. I always love some horror that has more to say about society than just ghosts and ghouls, and "Ring Shout" by P. Djèlí Clark is a new title that scratches that itch. Sure, it looks like it's a book about demons, demon hunters, and black magic. But it's also a story of the very real horrors of American Racism.
I am usually nervous when I'm about to start a novella that seems like it has a lot of distance to cover and a lot of complicated themes, as it's hard to fit all of that into limited pages. But even though "Ring Shout" clocks in at less than two hundred pages, Clark does a great job of developing his characters, building an alternative American history with mystical themes, and hitting the metaphors out of the park with biting satire and, in some cases, dark humor. We follow Marys, Sadie, and Chef, three Black women who, in post WWI America, are fighting against demons summoned by dark sorcerer D.W. Griffith that have fed upon white people's racism and led to a resurgence of the KKK. There are Klans, who are white racists whose hate for Black (and other non-white groups) has been amplified, and Ku Kluxes, actual demons disguised under the robes. Maryse, Sadie, and Chef, with the help from a Gulluh woman with magic and insight and other freedom fighters, are hoping to stop the end of the world fueled by racism, and Maryse may hold the key to it all. What I liked best about this story was that the three main characters are Black women, and they are given a whole lot of agency, motivation, and unique characterizations that make them all very enjoyable and fun. Maryse especially has a lot of complexity, her anger and determination pushing her forward. Clark gives all of them unique voices, but Maryse's in particular stands out.
The social commentary has a lot to work with her. While it could have been easy to just say 'demons are the problems behind the racism in the Jim Crow South' (something you kind of saw in "Abraham Lincoln: Vampire Hunter" with vampires and chattel slavery), Clark doesn't let racist white people off the hook here. While it's true that the demons are part of it, those who are influenced already had hate in their hearts. That hate was just used to help the demons gain power. There is also a less obvious but just as powerful metaphor in this story through the magical system regarding these demons that Clark puts in. The only people who can see the demons in their true forms are people from marginalized groups (this is mostly Black people, but there is a Jewish character in the group fighting against the Ku Kluxes that can see them as well); white people cannot. It's a clever way to call out white people's 'I don't see color' hypocrisy, as well as a metaphor for the microaggressions that have a blind eye turned to them even when marginalized groups who are affected by them say that they are, indeed, there. I greatly enjoyed that part of the mythos.
And Clark pulls all of this together in a cohesive and engaging story in less than two hundred pages! With some pretty gnarly Lovecraftian imagery and body horror thrown in for good measure (there's also a nod to this extremely problematic horror icon in this story, which was super fun to see). It made for a bite sized horror treat that I was able to read and enjoy in one sitting, the perfect quick tale for this Halloween season!
"Ring Shout" is a new social commentary horror classic in the making! Treat yourself to something a little more complex this ghostly season, you won't be disappointed.
Rating 8: Steeped in sharp and biting satire and commentary, "Ring Shout" is a story of demons, heroines, and American racism.
"Ring Shout" is included on the Goodreads lists "Beyond Butler: Spec Fic by Authors of Color", and "ATY 2020 – About Racism and Race Relations".
Find "Ring Shout" at your library using WorldCat, or at a local independent bookstore using IndieBound!
Author thelibraryladiesmnPosted on October 13, 2020 August 20, 2020 Categories Kate's ReviewsTags 8 Rating, horror, Short storiesLeave a comment on Kate's Review: "Ring Shout" | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,660 |
(taskwarrior time-of-day shell script)
is a taskwarrior extension designed to filter-out tasks that are not appropriate for the current (now) time of day, for 2 selected reports. Using the extension is as easy as adding +morn +aft +day +eve +night tags to any tasks that would be best performed at those times.
Status: core functions working well
See also: tod.txt for more requirements/ install/ config/ using ToD info
Caveat: Perpetual Proof-of-Concept until reviewed and tested by "real programmers". Lock up your daughters and backup your data!
### Features
- easily set preferred time-of-day for a task by adding a +morn +aft +day +eve +night tag
- easily re-set time-of-day with 'ToD [morn|aft|day|eve|night|none]'
- does not use or interfere with task context
- designed to work with tw-needs-hook (another persistent filter tw-extension)
- view help with 'ToD help'
#### Planned
- a cron-job that runs ToD.sh, changes rc.tod and applies tod filters.
- more easily customized times and hours
- applicable to any number of reports, not just 2 (ready and ls by default)
- on-add and on-modify hooks to act on any date-change, adding "+8hrs" (or whatever) to any task with a +morn tag (for example), if no hours were specified.
### Advantages
If you use taskwarrior, you could easily filter for all "evening tasks" with 'task +eve ready' but that over-simplification would hide any task without an +eve tag. That's potentially a problem, because many of the hidden tasks might well be actionable any time, including the evening, unless you were _meticulous_ with your tags.
Instead, what this extension does, is to filter selected reports with the _inverse_ of the rc.tod setting. When rc.tod=eve, ( -morn -aft -day -night ) ia added to the "ready" and "ls" reports, concealing tasks specified for some other time-of-day.
### Benefits
De-selecting reports that are not actionable "now" can reduce a large list by half, dramatically improving the signal-to-noise ratio, and making it clearer which tasks can (and should) be done immediately.
### Issues
There are always bugs, and there could always be improvements. Contributions, ideas and comments are welcome at https://github.com/linuxcaffe/tw-tod-sh
| {
"redpajama_set_name": "RedPajamaGithub"
} | 252 |
@extends('api.master')
@section('title', 'Suyabay: Registration of a new App')
@endsection
@section('content')
@include('api.includes.contents.newapp')
@endsection | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,587 |
\subsection*{Abstract}
For $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator
\[\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(\delta^2), \quad (\varphi_{I\times J}) \mapsto \sum \varphi_{I\times J}f_{I \times J}h_{I \times J}.
\]
Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(\delta^2)$
\[|f|=|x|^{1-\theta}|y|^{\theta}\quad\quad \text{ and }\quad\quad \|x\|_{X_0}^{1-\theta}\|y\|^{\theta}_{H^2(\delta^2)}\leq C\|f\|_{H^p(\delta^2)},
\] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(\delta^2)$ and $H^2(\delta^2)$.
Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_f$ involves the Haar coefficients of $f$ and its atomic decomposition. We treated the one-parameter $H^p$-spaces in [{\em Houston Journal Math.}~2015].
\section{Introduction}
Let $Y_0,Y$ be Banach spaces. An operator $T\in L(Y_0,Y)$ is called \textit{2-summing} if there is a constant $C$ such that for every choice of finite sequences $(\varphi_i)$ in $Y_0$, we have
\begin{equation}
\label{eq:summing}
\bigg(\sum_{i=1}^n{\norm{T\varphi_i}^2}\bigg)^{\frac{1}{2}}\leq C\,\sup{\bigg\{\Big(\sum_{i=1}^n{\abs{\varphi^*(\varphi_i)}}^2\Big)^{\frac{1}{2}}:\,\varphi^* \in B_{Y_0^*}\leq 1\bigg\}}.
\end{equation}
In the early 70's the concepts of type and cotype were mainly developed by\\ J.~Hoffmann-J{\o}rgensen, S.~ Kwapien, B.~Maurey and G.~Pisier, see \cite{MR0356155,MR0341039,MR0346510,MR0308834,MR0331017,MR0443015,MR0333673}.
A Banach space $Y$ is called of \textit{cotype 2} if there is a constant $C$ such that for all finite sequences $(y_i)$ in $Y$
\begin{equation*}
\label{eq:cotype}
\bigg(\sum_{i=1}^n\norm{y_i}^2\bigg)^{\frac{1}{2}}\leq C \bigg(\int_0^1{\biggnorm{\sum_{i=1}^n{r_i(t)y_i}}_X^2dt}\bigg)^{\frac{1}{2}},
\end{equation*}where $(r_i)_{i \in \mathbb{N}}$ denotes the independent Rademacher system.
One famous theorem due to Maurey (\cite{MR0399818, MR0344931}, see also \cite{MR0487403}) combining absolutely summing operators and the concept of cotype states that every bounded operator
\[T\colon\ell^{\infty}\rightarrow Y\] is $2$-summing, whenever $Y$ is of cotype $2$.
In particular, if
\[\norm{T\varphi}_Y\leq \sup_{i \in \mathbb{N}}\abs{\varphi_i},\]
then $T$ satisfies \eqref{eq:summing}
and by Pietsch's factorization theorem (cf.~\cite{MR1144277}) there exists a constant $C$ such that
\begin{equation}
\label{eq:pietsch}
\norm{T\varphi}_Y\leq C \left(\int_{\Omega}\abs{\varphi}^2d\mu\right)^{\frac{1}{2}},
\end{equation}
where $\mu$ is a Borel probability measure on $\Omega=B_{(\ell^{\infty})^*}$, called Pietsch measure.
Another concept going back to the 70's are Hardy spaces of martingales and their atomic decomposition, cf.~\cite{MR0447953,MR0312166,MR539351,MR602392,MR590628,MR584078}.
In our recent paper \cite{MulPent} we exhibited a connection between these two concepts.
In the present work we further extend and exploit these newly found connections. We consider operators from $\ell^{\infty}$ into bi-parameter dyadic Hardy spaces $H^p(\delta^2)$ that act as multipliers on the Haar system. By the above, these multiplication operators are $2$-summing and satisfy therefore \eqref{eq:pietsch}.
In our main result (Theorem \ref{th:main1}) we determine explicit formulae for the Pietsch measure of these multiplication operators. We recall that for general absolutely summing operators the existence of a Pietsch measure is given by a Hahn-Banach argument and is therefore not constructive.
Let ${\mathcal D}$ be the set of dyadic intervals.
Let $(f_{I \times J})_{I \times J\in {\mathcal D}\times {\mathcal D}}$ be a real sequence indexed by the dyadic rectangles ${\mathcal D}\times {\mathcal D}$. The space $H^p(\delta^2)$ consists of all functions
\begin{equation*}
\label{eq:func}
f=\sum_{I \in {\mathcal D}}\sum_{J \in {\mathcal D}}f_{I \times J}h_{I \times J},
\end{equation*} where $h_{I\times J}=h_I \otimes h_J$, which satisfy
\begin{equation*}
\norm{f}_{H^p(\delta^2)}=\left(\int_{[0,1]^2}\Big(\sum_{I\in {\mathcal D}}\sum_{J\in {\mathcal D}}f^2_{I\times J}1_{I\times J}\Big)^{\frac{p}{2}}\,dm\right)^{\frac{1}{p}} <\infty,
\end{equation*}where $m$ denotes the Lebesgue measure on $[0,1]^2$. Every $f \in H^p(\delta^2)$ defines a multiplication operator of the form
\begin{equation}
\label{eq:mult1}
\begin{split}
{\mathcal M}_f\colon\ell^{\infty}({\mathcal D} \times {\mathcal D}) &\rightarrow H^p(\delta^2)\\
(\varphi_{I \times J}) &\mapsto \sum_{I}\sum_{J}\varphi_{I \times J}f_{I \times J} h_{I \times J}.
\end{split}
\end{equation}
For $1\leq p \leq 2$ the Hardy spaces $H^p(\delta^2)$ are of cotype $2$ and therefore the multiplication operators ${\mathcal M}_f$ are $2$-summing and have Pietsch measures.
In our main theorem (Theorem \ref{th:main1}) we use the atomic decomposition of $f \in H^p(\delta^2)$ to give explicit formulae for these Pietsch measures. In particular, we determine $\omega=(\omega_{I \times J})_{I \times J \in {\mathcal D} \times {\mathcal D}}$ with $\omega_{I \times J}\geq 0$ and $\sum \omega_{I \times J}\leq 1$ such that
for all $\varphi \in \ell^{\infty}({\mathcal D}\times {\mathcal D})$ the following holds
\begin{equation}
\label{eq:pietsch2}
\norm{{\mathcal M}_f(\varphi)}_{H^p(\delta^2)} \leq C \norm{f}_{H^p(\delta^2)}\bigg(\sum_{I \in {\mathcal D}}\sum_{J \in {\mathcal D}}\abs{\varphi_{I\times J}}^2\omega_{I\times J}\bigg)^{\frac{1}{2}}.
\end{equation}
The explicit formulae for $\omega$ are given by equation \eqref{eq:omega} in Section \ref{sec:mainth}.
Multiplication operators such as given in \eqref{eq:mult1} played an important role in the development of Banach space theory. See for instance the proof by Lindenstrauss and Pe{\l}czy{\'n}ski on the uniqueness of the unconditional basis in $\ell^1$ (\cite{MR0500056,MR0231188}).
Bi-parameter Hardy spaces $H^p(\delta^2)$ may be regarded as vector-valued Hardy spaces $H^p_X$, where $X=H^p$. In \cite[Theorem 3.3, (3.19)]{MulPent} we obtained partially constructive formulae for the Pietsch measures of Haar multipliers on $\ell^\infty$ into the vector-valued Hardy spaces $H^p_X$. In the scalar-valued case, i.e.~$X=\mathbb{R}$, we obtained fully constructive formulae for the Pietsch measures of the multiplication operators, see \cite[Theorem 3.1.]{MulPent}.
With this in mind, our present theorem (Theorem \ref{th:main1}) gives fully constructive results for a special class of vector-valued Hardy spaces and simultaneously we extend in a non-trivial way the scalar-valued one-parameter case to the bi-parameter case.
\smallskip
\noindent
\textit{Application.} The Banach spaces $H^p(\delta^2)$ form Banach lattices whose lattice structure is induced by their unconditional basis $(h_{I \times J})$ and they are related through Calder{\'o}n's product formula
\begin{equation}
\label{eq:calderon}
H^p(\delta^2)=\left(H^1(\delta^2)\right)^{1-\theta}\left(H^2(\delta^2)\right)^{\theta}, \quad 0<\theta<1,\,\, \frac{1}{p}=1-\theta+\frac{\theta}{2}.
\end{equation}
This follows by combining the one-parameter identities (cf.~\cite[Theorem 8.2.]{MR1070037}) with Calder{\'o}n's theorem (\cite[Paragraph 13.6]{MR0167830}).
Therefore, Pisier's extrapolation statement (\cite[Theorem 2.10]{MR557371}) can be adapted to the family of $H^p(\delta^2)$ spaces and reads in this setting as follows
\begin{equation*}
\label{eq:pisier}
H^p(\delta^2)=(X_0)^{1-\theta}(H^2(\delta^2))^{\theta}, \quad \theta=2-\frac{2}{p}.
\end{equation*}
Here $X_0$ is the Banach lattice of all elements $x=\sum_{I}\sum_Jx_{I \times J}h_{I \times J}$ for which
\begin{equation}
\label{eq:X0norm}
\norm{x}_{X_0}=\sup{\left\{\biggnorm{\sum_I\sum_J\abs{x_{I \times J}}^{1-\theta}\abs{y_{I \times J}}^{\theta}h_{I \times J}}_{H^p(\delta^2)}\right\}}<\infty,
\end{equation}
where the supremum is taken over all $y=\sum_{I,J}y_{I \times J}h_{I\times J}$ with $\norm{y}_{H^2(\delta^2)}\leq 1$.
Specifically, \eqref{eq:pisier} asserts that given $f \in H^p(\delta^2)$ there is $x \in X_0$ and $y \in H^2(\delta^2)$ such that
\begin{equation}
\label{eq:pisier2}
\abs{f}=\abs{x}^{1-\theta}\abs{y}^{\theta} \quad \text{and} \quad \norm{x}_{X_0}^{1-\theta}\norm{y}_{H^2(\delta^2)}^{\theta}\leq C\norm{f}_{H^p(\delta^2)}.
\end{equation}
Pisier shows in his proof that the weight $\omega=(\omega_{I \times J})$ given by equation \eqref{eq:pietsch2} yields factors for $f$. Hence, our explicit formulae for $\omega=(\omega_{I \times J})$ determined in Theorem \ref{th:main1} allow us to give constructive factors of $f$ satisfying \eqref{eq:pisier2}.
\section{Preliminaries}
\label{sec:prelim}
\subsection{Bi-parameter Hardy spaces $H^p(\delta^2)$}
The \textit{dyadic intervals} ${\mathcal D}$ on the unit interval are given by
\begin{equation*}
{\mathcal D}=\left\{\left[2^{-n}(k-1),\,2^{-n}k\right[:\, n,k \in \mathbb{N}_0,\, 0\leq k < 2^n\right\}
\end{equation*}
and the \textit{dyadic rectangles $\mathcal{R}$} on the unit square are given by $\mathcal{R}={\mathcal D}\times {\mathcal D}.$
Let ${\mathcal C} \subseteq \mathcal{R}$ be a collection of dyadic rectangles. Then we denote by ${\mathcal C}^*$ the \textit{pointset} covered by the union of all dyadic rectangles in the collection ${\mathcal C}$.
The space $\ell^{\infty}(\mathcal{R})$ is the space of all sequences $\varphi=(\varphi_{IJ})_{I \times J \in \mathcal{R}}$, indexed by the dyadic rectangles, with $\norm{\varphi}_{\infty}=\sup_{I\times J\in \mathcal{R}}\abs{\varphi_{IJ}}<\infty.$
For every $I \in {\mathcal D}$ we define the $L^{\infty}$- normalised \textit{Haar function} $h_I$ to be $+1$ on the left half of $I$, $-1$ on the right half of $I$ and zero on $[0,1]\setminus I$.
The an-isotropic \textit{2D Haar system} $(h_{I\times J})_{I \times J \in \mathcal{R}}$ indexed by the dyadic rectangles is defined as follows
\[h_{I \times J}(s,t):=h_I(s)h_J(t), \quad I,J \in {\mathcal D}, \, (s,t)\in [0,1]^2.\]
Let $(f_{IJ})_{I\times J \in \mathcal{R}}$ be a real sequence and $f=(f_{IJ})_{I\times J \in \mathcal{R}}$ the real vector indexed by the dyadic rectangles. The \textit{square function of $f$} is defined as follows
\begin{equation*}
S(f)(s,t)=\bigg(\sum_{I\times J \in \mathcal{R}}f_{IJ}^21_{I\times J}(s,t)\bigg)^{\frac{1}{2}},\quad (s,t) \in [0,1]^2.
\end{equation*}
The \textit{bi-parameter dyadic Hardy space $H^p(\delta^2)$}, $0<p\leq 2$, consists of vectors $f=(f_{IJ})_{I\times J \in \mathcal{R}}$ for which
\begin{equation*}
\norm{f}_{H^p(\delta^2)}=\left(\int_{[0,1]^2}S^p(f)(s,t)\, dm(s,t)\right)^{\frac{1}{p}} <\infty,
\end{equation*}where $m$ is the Lebesgue measure on $[0,1]^2$.
Systematically we use the notation $\norm{f}_2=\norm{f}_{H^2(\delta^2)}$.
For convenience we identify $f=(f_{IJ})_{I \times J \in \mathcal{R}}\in H^p(\delta^2)$ with its formal Haar series
\begin{equation}
\label{eq:haarseries}
f=\sum_{I \times J \in \mathcal{R}}f_{IJ}h_{I \times J}.
\end{equation}
\subsection{Atomic decomposition}
\label{sec:atdec}
Let $0<p\leq 2$ and $f \in H^p(\delta^2)$ with Haar expansion \eqref{eq:haarseries}.
For every $n\in \mathbb{Z}$ we define the set
\[F_n=\left\{(s,t) \in [0,1]^2:\, S(f)(s,t)>2^n\right\}\]
and the collection of dyadic rectangles
\[\mathcal{R}_n=\left\{I\times J \in \mathcal{R}:\,\abs{I\times J \cap F_n}>\frac{\abs{I\times J}}{2},\, \abs{I\times J \cap F_{n+1}}\leq \frac{\abs{I\times J}}{2}\right\}.\]
Then $f=\sum_{n \in \mathbb{Z}}f_n$, where
\[f_n=\sum_{I \times J \in \mathcal{R}_n}f_{IJ}h_{I\times J}\]
and the following inequalities hold
\begin{align}
&\label{eq:atdec}
\norm{f}^p_{H^p(\delta^2)} \leq \sum_{n \in \mathbb{Z}}{\norm{f_n}^p_{H^p(\delta^2)}}\leq \sum_{n \in \mathbb{Z}}{\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}\norm{f_n}^p_{2}}\leq A_p \norm{f}^p_{H^p(\delta^2)}.
\end{align}
The family $(f_n, \mathcal{R}_n)_{n \in \mathbb{Z}}$ is called the \textit{atomic decomposition} of $u \in H^p(\delta^2)$.
This decomposition originates from \cite{MR0312166,MR539351,MR590628,MR584078}.
Note that the right-hand side inequality in \eqref{eq:atdec} results from the following
\begin{equation}
\label{eq:l2at}
\begin{split}
\norm{f_n}_{2}^2&=\int_{[0,1]^2} S^2(f_n)\,dm \leq 2\int_{[0,1]^2} S^2(f_n)1_{F_{n+1}^c} \, dm\leq 2\cdot 2^{2(n+1)}\abs{\mathcal{R}_n^*}\\
&\leq 8\cdot 2^{2n} \Big|\Big\{M_S(1_{F_n})>\frac{1}{2}\Big\}\Big|\leq C \,2^{2n}\abs{F_n}.
\end{split}
\end{equation}
Here $M_S$ is the strong maximal operator (cf.~\cite{Jessen1935},\cite{MR1315539}) in $[0,1]^2$ given by
\[M_S(1_{F_n})(s,t)=\sup_{R \ni (s,t)}\frac{1}{\abs{R}}\int_R 1_{F_n}\,dm,\]
where the supremum is taken over all rectangles $R$ in $[0,1]^2$ with side length parallel to the axes.
Boundedness estimates for the strong maximal operator (cf.~\cite{Jessen1935}) give rise to bi-parameter Fefferman-Stein strong maximal operator estimates (cf.~\cite[Theorem 1.]{MR0284802}). We exploit these Fefferman-Stein inequalities in the following form.
\begin{lemma}
\label{le:bownik}
Fix $\varepsilon >0$. Suppose that for each $I \times J \in \mathcal{R}$ the subset $E_{I \times J} \subseteq I \times J$ is a measurable set with $\frac{\abs{E_{I\times J}}}{\abs{I\times J}}>\varepsilon$. Then for any $f \in H^p(\delta^2)$, $0<p<\infty$, with Haar expansion
$f=\sum_{I \times J \in \mathcal{R}}f_{IJ}h_{I\times J}$,
the following holds
\begin{equation*}
\norm{f}_{H^p(\delta^2)}\leq C_p(\varepsilon)\, \biggnorm{\Big(\sum_{I \times J \in \mathcal{R}}\abs{f_{IJ}}^21_{E_{I\times J}}\Big)^{\frac{1}{2}}}_{L^p}.
\end{equation*}
\end{lemma}
Frazier and Jawerth (\cite[Theorem 2.7.]{MR1070037}) give a proof for the one-parameter version of this lemma. Their proof can be adapted to the setting above.
\subsection{Modified H\"older inequality}
See \cite[p.~61 (65.)]{MR0046395}.
Let $(\Omega, \Sigma, \mu)$ be a measure space and $r>1$ or $r<0$. Then for all measurable functions $f,g$ on $\Omega$
\begin{equation}
\label{eq:modholder}
\int_{\Omega} f^rg^{1-r}d\mu \geq \left(\int_{\Omega}f d\mu\right)^r \left(\int_{\Omega}g \,d\mu\right)^{1-r}.
\end{equation}
\medskip
\section{The main Theorem}\label{sec:mainth}
Let $0<p\leq 2$.
Every $f \in H^p(\delta^2)$ defines a multiplication operator of the form
\begin{equation*}
\label{eq:mult2}
{\mathcal M}_f\colon \ell^{\infty}(\mathcal{R})\rightarrow H^p(\delta^2),\,\, (\varphi_{IJ})\mapsto \sum_{I \times J \in \mathcal{R}} \varphi_{IJ}f_{IJ}h_{I \times J}
\end{equation*}
and clearly we have
\begin{equation*}
\norm{{\mathcal M}_f\colon\ell^{\infty}(\mathcal{R})\rightarrow H^p(\delta^2)}\leq \norm{f}_{H^p(\delta^2)}.
\end{equation*}
Banach space theory as described in the introduction guarantees that these multiplication operators are $2$-summing and satisfy \eqref{eq:pietsch2}.
In Theorem \ref{th:main1} we determine explicit formulae for the weights $\omega=(\omega_{I \times J})$ given in \eqref{eq:pietsch2}.
Every multiplication operator ${\mathcal M}_f$ is induced by a function $f \in H^p(\delta^2)$. These functions admit an atomic decomposition $(f_n,\mathcal{R}_n)_{n \in \mathbb{N}}$ satisfying the equations in \eqref{eq:atdec}. This is the input for our construction and the output is equation \eqref{eq:omega} determining $\omega$ explicitly.
\begin{theorem}
\label{th:main1}
Let $0<p\leq 2$ and $f \in H^p(\delta^2)$ with Haar expansion
\[f=\sum_{I\times J \in \mathcal{R}}{f_{IJ}h_{I\times J}}
\]and atomic decomposition $(f_n, \mathcal{R}_n)_{n \in \mathbb{Z}}$. Then the sequence $\left(\omega_{IJ}\right)_{I\times J \in \mathcal{R}}$, defined by
\begin{equation}
\label{eq:omega}
\omega_{IJ}=\frac{1}{A_p\norm{f}^{p}_{H^p(\delta^2)} }\frac{\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}f_{IJ}^2\abs{I}\abs{J}}{\norm{f_n}^{2-p}_{2}}, \hspace{0.5 cm} I \times J \in \mathcal{R}_n,
\end{equation}satisfies
\begin{equation*}
\sum_{I\times J \in \mathcal{R}}{\omega_{IJ}}\leq 1
\end{equation*}and for each $\varphi \in \ell^{\infty}(\mathcal{R})$ the following inequality holds
\begin{equation*}
\label{eq:mainth}
\norm{{\mathcal M}_f(\varphi)}_{H^p(\delta^2)}\leq A_p \|f\|_{H^p(\delta^2)}\Big(\sum_{I\times J \in \mathcal{R}}{\abs{\varphi_{IJ}}^2 \omega_{IJ}}\Big)^{\frac{1}{2}}.
\end{equation*}
\end{theorem}
\begin{proof}
Note that the left-hand side inequality of (\ref{eq:atdec}) depends only on the fact that $(\mathcal{R}_n)_{n\in \mathbb{Z}}$ forms a partition of $\mathcal{R}$. Hence, for all $\varphi=(\varphi_{IJ}) \in\ell^{\infty}(\mathcal{R})$ the following estimate holds
\begin{equation*}
\begin{split}
\biggnorm{\sum_{I\times J \in \mathcal{R}}{\varphi_{IJ} f_{IJ} h_{I\times J}}}^p_{H^p(\delta^2)} &=\biggnorm{\sum_{n \in\mathbb{Z}}\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ} f_{IJ} h_{I \times J}}}^p_{H^p(\delta^2)}\\
& \leq \sum_{n \in\mathbb{Z}}{\biggnorm{\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ} f_{IJ} h_{I \times J}}}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\\
&= \sum_{n \in\mathbb{Z}}{\biggnorm{\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ} \frac{f_{IJ}}{\norm{f_n}_{2}} h_{I \times J}}}^p_{2}\norm{f_n}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}.
\end{split}
\end{equation*}
With
\begin{align*}
\biggnorm{\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ} \frac{f_{IJ}}{\norm{f_n}_{2}} h_{I \times J}}}^p_{2} &= \Big(\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}_{2}^2}\abs{I}\abs{J}}\Big)^{\frac{p}{2}}
\end{align*}
and H\"older's inequality we get
\begin{equation*}
\begin{split}
&\biggnorm{\sum_{I \times J \in \mathcal{R}}{\varphi_{IJ} f_{IJ} h_{I \times J}}}^p_{H^p(\delta^2)} \leq \sum_{n \in\mathbb{Z}}{\Big(\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}_{2}^2}\abs{I}\abs{J}}\Big)^{\frac{p}{2}}\norm{f_n}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\\
&\qquad\quad\,\,\,=\sum_{n \in\mathbb{Z}}{\Big(\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}_{2}^2}\abs{I}\abs{J}\norm{f_n}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\Big)^{\frac{p}{2}}\Big(\norm{f_n}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}\Big)^{1-\frac{p}{2}}}\\
&\qquad\quad\,\,\,\leq \Big(\sum_{n \in\mathbb{Z}}\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}^{2-p}_{2}}\abs{I}\abs{J}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\Big)^{\frac{p}{2}}\Big(\sum_{n \in\mathbb{Z}}\norm{f_n}^p_{2}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}\Big)^{1-\frac{p}{2}}.
\end{split}
\end{equation*}
\begin{align*}
&\biggnorm{\sum_{I \times J \in \mathcal{R}}{\varphi_{IJ} f_{IJ} h_{I \times J}}}_{H^p(\delta^2)}^p\\
&\qquad\qquad\leq A_p^{1-\frac{p}{2}}\norm{f}^{p(1-\frac{p}{2})}_{H^p(\delta^2)}\Big(\sum_{n\in \mathbb{Z}}\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}^{2-p}_{2}}\abs{I}\abs{J}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\Big)^{\frac{p}{2}}\\
&\qquad\qquad= A_p\norm{f}^p_{H^p(\delta^2)}\Big(\sum_{n \in\mathbb{Z}}\sum_{I \times J \in \mathcal{R}_n}{\varphi_{IJ}^2\frac{f_{IJ}^2}{\norm{f_n}^{2-p}_{2}\norm{f}^{p}_{H^p(\delta^2)}A_p}\abs{I}\abs{J}\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}}\Big)^{\frac{p}{2}}.
\end{align*}
Recall that
\begin{equation}
\label{eq:pr3}
\norm{f_n}_{2}^2=\sum_{I \times J \in \mathcal{R}_n}{f_{IJ}^2\abs{I}\abs{J}}.
\end{equation}
By the right-hand side inequality in equation (\ref{eq:atdec}) and by equation (\ref{eq:pr3}) we obtain for the sequence $\left(\omega_{IJ}\right)_{I \times J \in \mathcal{R}}$, defined by
\[\omega_{IJ}=\frac{1}{A_p\norm{f}^{p}_{H^p(\delta^2)} }\frac{\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}f_{IJ}^2\abs{I}\abs{J}}{\norm{f_n}^{2-p}_{2}}, \hspace{0.5 cm} I \times J \in \mathcal{R}_n,
\]the following estimate
\begin{align*}
\sum_{I \times J \in \mathcal{R}}{\omega_{IJ}}&=\frac{1}{A_p\norm{f}^p_{H^p(\delta^2)}}\sum_{n \in\mathbb{Z}}\sum_{I \times J \in \mathcal{R}_n}{\frac{\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}f_{IJ}^2\abs{I}\abs{J}}{\norm{f_n}^{2-p}_{2}}}\\
&=\frac{1}{A_p\norm{f}^p_{H^p(\delta^2)}}\sum_{n\in\mathbb{Z}}{\abs{\mathcal{R}_n^*}^{1-\frac{p}{2}}\norm{f_n}^p_{2}}\,\leq 1.
\end{align*}
\end{proof}
\section{Another application of the atomic decomposition}
Pisier's extrapolation lattice $X_0$ defined in \eqref{eq:X0norm} is known to coincide with $H^1(\delta^2)$. This follows by a specialisation of a general theorem of Cwikel and Nilsson (see \cite{MR1996919}).
Their extrapolation method is applicable, since $H^p(\delta^2)$ spaces are related through Calderon's product formula (cf.~equation \eqref{eq:calderon}).
The space $X_0$ is of particular importance to our work in this paper. Hence, we take the opportunity to complement the work of \cite{MR1996919,MR1070037} with a direct argument based on the atomic decomposition of $H^p(\delta^2)$. We build our strategy by exploiting the formulae used by \cite{MR2157745, MR2183484, zbMATH06162617} for similar purposes. In particular, we refer to Bownik's paper \cite{zbMATH06162617} for the formula \eqref{eq:gfunc} and the idea of using Lemma \ref{le:bownik} in the proof of the following theorem.
\begin{theorem}
\label{th:int}
Let $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{IJ} h_{I \times J}$. Then for $0<\theta<1$ and $q$ given by
\begin{equation}
\label{eq:intq}
\frac{1}{q}=\frac{1-\theta}{p}+\frac{\theta}{2},
\end{equation}
the following holds:
\begin{equation*}
c_p \norm{f}_{H^p(\delta^2)}^{1-\theta}\leq \sup{\left\{\biggnorm{\sum_{I\times J \in \mathcal{R}}\abs{f_{IJ}}^{1-\theta}\abs{g_{IJ}}^{\theta}h_{I \times J}}_{H^q(\delta^2)}\right\}}\leq \norm{f}_{H^p(\delta^2)}^{1-\theta},
\end{equation*}where the supremum is taken over all functions $g=\sum g_{IJ}h_{I\times J}$ with $\norm{g}_{2}\leq 1$.
\end{theorem}
\begin{proof}
We start with the proof of the right-hand side inequality. Let
\[h=\sum_{I \times J \in \mathcal{R}}\abs{f_{IJ}}^{1-\theta}\abs{g_{IJ}}^{\theta}h_{I\times J}.\]
Then, by applying H\"older's inequality for sequence spaces with $1-\theta+\theta=1$, we obtain the following inequality for the square functions
\begin{align*}
S^q(h)\leq S^{q(1-\theta)}(f)\,S^{q\theta}(g).
\end{align*}
Integrating over $[0,1]^2$ and applying H\"older's inequality with $\frac{q(1-\theta)}{p}+\frac{q\theta}{2}=1$ yields
\begin{equation*}
\norm{h}_{H^q(\delta^2)}\leq \norm{f}_{H^p(\delta^2)}^{1-\theta}\,\,\norm{g}_{2}^{\theta}.
\end{equation*}
For the left-hand side inequality we show that for every $f \in H^p(\delta^2)$ there exists a function $g \in H^2(\delta^2)$ such that $\norm{g}_{2}^2\leq c_p\norm{f}^p_{H^p(\delta^2)}$ and
\begin{align}
\label{eq:bow0}
\biggnorm{\sum_{I\times J \in \mathcal{R}}\abs{f_{IJ}}^{1-\theta}\abs{g_{IJ}}^{\theta}h_{I \times J}}_{H^q(\delta^2)}^q&\geq C_p\, \norm{f}^p_{H^p(\delta^2)}.
\end{align}
Let $(f_n,\mathcal{R}_n)_{n \in \mathbb{Z}}$ be the atomic decomposition of $f \in H^p(\delta^2)$.
Let $g=\sum g_{IJ}h_{I \times J}$, where
\begin{equation}
\label{eq:gfunc}
\abs{g_{IJ}}=2^{-\frac{n}{2}(2-p)}\abs{f_{IJ}}, \quad I \times J \in \mathcal{R}_n.
\end{equation} Then, by equation \eqref{eq:l2at}, we have
\begin{equation}
\label{eq:lhs}
\begin{split}
\norm{g}^2_{2}&=\sum_{n \in \mathbb{Z}}2^{-n(2-p)}\norm{f_n}_2^2\leq C\sum_{n \in \mathbb{Z}}2^{-n(2-p)}2^{2n}\abs{F_n}=C\sum_{n \in \mathbb{Z}}2^{np}\abs{F_n}\\
&\leq c_p\norm{f}_{H^p(\delta^2)}^p.
\end{split}
\end{equation}
To prove equation \eqref{eq:bow0} we use Lemma \ref{le:bownik} with sets $E_{I \times J}=I\times J \cap F_n$, for $I \times J \in \mathcal{R}_n$ and obtain
\begin{equation}
\label{eq:bow3}
\begin{split}
\norm{f}_{H^p(\delta^2)}^p&=\left(\int_{[0,1]^2}S^p(f)\,dm\right)^{\frac{q}{p}}\left(\int_{[0,1]^2} S^p(f)\,dm\right)^{1-\frac{q}{p}}\\
&\leq C_p^q\left(\int_{[0,1]^2}\Big(\sum_{I\times J}\abs{f_{IJ}}^21_{E_{I\times J}}\Big)^{\frac{p}{2}}\,dm\right)^{\frac{q}{p}}\left(\int_{[0,1]^2}S^p(f)\,dm\right)^{1-\frac{q}{p}}.
\end{split}
\end{equation}
Let $h=\left(\sum_{I\times J \in \mathcal{R}}\abs{f_{IJ}}^21_{E_{I\times J}}\right)^{\frac{1}{2}}$.
Note that by the modified H\"older inequality (cf.~equation \eqref{eq:modholder}) we have
\begin{equation}
\label{eq:revholder}
\left(\int_{[0,1]^2} h^p\,dm\right)^{\frac{q}{p}}\left(\int_{[0,1]^2} S^p(f)\,dm\right)^{1-\frac{q}{p}}\leq \int_{[0,1]^2} h^qS^{p-q}(f)\,dm.
\end{equation}
Combining equation \eqref{eq:bow3} and \eqref{eq:revholder} yields
\begin{equation}
\label{eq:bow2}
\begin{split}
\norm{f}^p_{H^p(\delta^2)}&\leq C_p^q\int_{[0,1]^2}\Big(\sum_{I\times J \in \mathcal{R}}\abs{f_{IJ}}^21_{E_{I\times J}}\Big)^{\frac{q}{2}}S^{p-q}(f)\, dm\\
&=C_p^q\int_{[0,1]^2}\Big(\sum_{n\in \mathbb{Z}}\sum_{I\times J \in \mathcal{R}_n}\abs{f_{IJ}}^21_{I\times J}1_{F_n}\Big)^{\frac{q}{2}}S^{p-q}(f)\, dm.
\end{split}
\end{equation}
We know that $S(f)1_{F_n}>2^{n}1_{F_n}$. Since $q>p$, it follows that
\begin{equation}
\label{eq:S2}
S(f)^{p-q}1_{F_n}<2^{-n(q-p)}1_{F_n}.
\end{equation}
Equation \eqref{eq:intq} gives the identity $q-p=\frac{q\theta}{2}(2-p)$. Hence, putting equation \eqref{eq:S2} into equation \eqref{eq:bow2} yields
\begin{equation}
\label{eq:rhs}
\begin{split}
\norm{f}_{H^p(\delta^2)}^p&\leq C_p^q\int_{[0,1]^2}\Big(\sum_{n\in \mathbb{Z}}2^{-n\theta(2-p)}\sum_{I\times J \in \mathcal{R}_n}\abs{f_{IJ}}^21_{I\times J}1_{F_n}\Big)^{\frac{q}{2}}\, dm\\
&\leq C_p^q\int_{[0,1]^2}\Big(\sum_{n\in \mathbb{Z}}2^{-n\theta(2-p)}\sum_{I\times J \in \mathcal{R}_n}\abs{f_{IJ}}^21_{I \times J}\Big)^{\frac{q}{2}}\, dm\\
&=C_p^q\int_{[0,1]^2}\Big(\sum_{I\times J \in \mathcal{R}}\abs{f_{IJ}}^{2(1-\theta)}\abs{g_{IJ}}^{2\theta}1_{I \times J}\Big)^{\frac{q}{2}}\, dm\\
&=C_p^q\biggnorm{\sum_{I \times J \in \mathcal{R}}\abs{f_{IJ}}^{1-\theta}\abs{g_{IJ}}^{\theta}h_{I \times J}}_{H^q(\delta^2)}^q.
\end{split}
\end{equation}
Summarizing equations \eqref{eq:lhs} and \eqref{eq:rhs} yields
\begin{align*}
\norm{f}_{H^p(\delta^2)}^{1-\theta}\norm{g}_{2}^{\theta}\leq c_{p}^{\frac{\theta}{2}}\norm{f}_{H^p(\delta)^2}^{\frac{p}{q}}\leq C_{p} \,\biggnorm{\sum_{I \times J \in \mathcal{R}}\abs{f_{IJ}}^{1-\theta}\abs{g_{IJ}}^{\theta}h_{I \times J}}_{H^q(\delta^2)}.
\end{align*}
\end{proof}
\subsection*{Acknowledgements}
We would like to thank M.~Bownik for helpful discussions during the preparation of this paper.
This research has been supported by the Austrian Science foundation (FWF) Pr.Nr.P22549, Pr.Nr.P23987 and Pr.Nr.P28352.
\bibliographystyle{alpha}
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Chinonele alcătuiesc o clasă de compuși organici derivați de la compusul tipic 1,4-benzochinonă (denumită și chinonă, de unde vine denumirea clasei de substanțe) , care sunt componenți de bază în sinteza unor coloranți. Cei mai importanți sunt benzochinona, naftachinona și 9,10-antrachinona.
Exemple
Referințe
Surse externe
Despre chinone la US National Library of Medicine Medical Subject Headings (MeSH)
Vezi și
Hidrochinonă
Coloranți sintetici | {
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\section{Introduction}
\subsection{Background}
\myparagraph{Active learning.}
Active learning is a subfield of machine learning, in which at any
time, the learning algorithm is able to query an oracle for the label
of a particular data point. One model for active learning is the
membership query synthesis model \cite{a-qcl-87}. Here, the learner
wants to minimize the number of oracle queries, as such queries are
expensive---they usually correspond to either consulting with a
specialist, or performing an expensive computation. In this setting,
the learning algorithm is allowed to query the oracle for the label of
any data point in the instance space. See \cite{s-alls-09} for a more
in-depth survey on the various active learning models.
\myparagraph{PAC learning.}
A classical approach for learning is using random sampling, where one
gets labels for the samples (i.e., in the above setting, the oracle is
asked for the labels of all items in the random sample). PAC learning
studies the size of the sample needed. For example, consider the
problem of learning a halfplane for $n$ points $\PS \subset \Re^2$,
given a parameter $\eps \in (0,1)$. The first stage is to take a
labeled random sample $\RSet \subseteq \PS$. The algorithm computes
any halfplane that classifies the sample correctly (i.e., the
hypothesis). The misclassified points lie in the symmetric difference
between the learned halfplane, and the (unknown) true halfplane, see
\figref{active:learning}. In this case, the error region is a double
wedge, and it is well known that its VC dimension \cite{vc-ucrfe-71}
is a constant (at most eight). As such, by the $\eps$-net Theorem
\cite{hw-ensrq-87}, a sample of size $O(\eps^{-1} \log\eps^{-1})$ is
an $\eps$-net for double wedges, which implies that this random
sampling algorithm has at most $\eps n$ error.
\setlength{\savedparindent}{\parindent}%
\medskip%
\noindent%
\begin{minipage}{0.82\linewidth}
\setlength{\parindent}{\savedparindent}%
A classical example of a hypothesis class that cannot be learned
is the set of convex regions (even in the plane). Indeed, given a
set of points $\PS$ in the plane, any sample $\RSet \subseteq \PS$
cannot distinguish between the true region being $\CHX{\RSet}$ or
$\CHX{\PS}$. Intuitively, this is because the hypothesis space in
this case grows exponentially in the size of the sample (instead
of polynomially).
\end{minipage}\hfill
\begin{minipage}{0.17\linewidth}
\hfill%
\includegraphics{figs/convex}
\end{minipage}%
\myparagraph{Weak $\eps$-nets.}
Because $\eps$-nets for convex ranges do not exist, an interesting
direction to overcome this problem is to define weak $\eps$-nets
\cite{hw-ensrq-87}. A set of points $\RSet$ in the plane, not
necessarily a subset of $\PS$, is a \emph{weak $\eps$-net} for $\PS$
if for any convex body $\body$ containing at least $\eps n$ points of
$\PS$, it also contains a point of $\RSet$. \Matousek and Wagner
\cite{mw-ncwe-03} gave a weak $\eps$-net construction of size
$O(\eps^{-d}(\log \eps^{-1})^{O(d^2 \log d)})$, which is doubly
exponential in the dimension. The state of the art is the recent
result of Rubin \cite{r-ibwen-18}, that shows a weak $\eps$-net
construction in the plane of size (roughly) $O(1/\eps^{3/2})$.
However, these weak $\eps$-nets cannot be used for learning such
concepts. Indeed, the analysis above required an $\eps$-net for the
symmetric difference of two convex bodies of finite complexity, see
\figref{active:learning}.
\subsection{Problem and motivation}
\myparagraph{The problem.}
In this paper, we consider a variation on the active learning problem,
in the membership query synthesis model. Suppose that the learner is
trying to learn an unknown convex body $\body$ in $\Re^d$.
Specifically, the learner is provided with a set $\PS$ of $n$
unlabelled points in $\Re^d$, and the task is to label each point as
either inside or outside $\body$, see \figref{classify}. For a query
$\query \in \Re^d$, the oracle either reports that $\query \in \body$,
or returns a hyperplane separating $\query$ and $\body$ (as a proof
that $\query \not\in \body$). Note that if the query is outside the
body, the oracle answer is significantly more informative than just
the label of the point. The problem is to minimize the overall number
of queries performed.
\begin{figure}[t]
\phantom{}%
\hfill%
{\includegraphics[page=4,scale=0.4]{figs/linear_classifier_net}}%
\hfill%
{\includegraphics[page=4,scale=0.4]{figs/convex_body_net}}
\hfill\phantom{}%
\captionof{figure}{The shaded region shows the symmetric
difference between the hypothesis and true classifier.
(I) Learning halfspaces. (II) Learning arbitrary convex regions.}%
\figlab{active:learning}
\end{figure}
\begin{figure}[t]
\phantom{}%
\hfill%
{\includegraphics[page=1,width=0.25\linewidth]{figs/classify}}
\hfill
{\includegraphics[page=2,width=0.25\linewidth]{figs/classify}}
\hfill
{\includegraphics[page=3,width=0.25\linewidth]{figs/classify}}%
\hfill\phantom{}%
\captionof{figure}{(I) A set of points $\PS$. (II) The unknown
convex body $\body$. (III) Classifying all points of $\PS$ as
either inside or outside $\body$.}%
\figlab{classify}
\end{figure}
\myparagraph{Hard and easy instances.}
Note that in the worst case, an algorithm may have to query the oracle for
all input points---such a scenario happens when the input points are
in convex position, and any possible subset of the points can be the
points in the (appropriate) convex body. As such, the purpose here is
to develop algorithms that are \emph{instance sensitive}---if the
given instance is easy, they work well. If the given instance is hard,
they might deteriorate to the naive algorithm that queries all points.
\setlength{\savedparindent}{\parindent}%
\medskip%
\noindent%
\begin{minipage}{0.8\linewidth}
\setlength{\parindent}{\savedparindent}%
Natural inputs where one can hope to do better, are when
relatively few points are in convex position. Such inputs are grid
points, or random point sets, among others. However, there are
natural instances of the problem that are easy, despite the input
having many points in convex position. For example, consider when
the convex body is a triangle, with the input point set being
$n/2$ points spread uniformly on a tiny circle centered at the
origin, while the remaining $n/2$ points are outside the convex
body, spread uniformly on a circle of radius $10$ centered at the
origin. Clearly, such a point set can be classified using a
constant number of oracle queries (in the best case).
See \figref{easy:not:easy} for some related examples.%
\end{minipage}\hfill
\begin{minipage}{0.18\linewidth}
\hfill%
\centerline{\includegraphics{figs/easy}}%
\end{minipage}%
\myparagraph{Additional motivation \& some previous work.}
\begin{compactenumA}
\item \emph{Separation oracles.}
The use of separation oracles is a common tool in optimization
(e.g., solving exponentially large linear programs) and operations
research. It is natural to ask what other problems can be solved
efficiently when given access to this specific type of oracle.
\medskip%
\item \emph{Other types of oracles.} Various models of computation
utilizing oracles have been previously studied within the community.
Examples of other models include nearest-neighbor oracles (i.e.,
black-box access to nearest neighbor queries over a point set
$\PS$) \cite{hkmr-sevps-16}, and proximity probes (which given a
convex polygon $\body$ and a query $\query$, returns the distance
from $\query$ to $\body$) \cite{pavg-eppam-13}. It is reasonable to
ask what classification-type problems can be solved with few oracle
queries when using separation oracles.
Furthermore, other types of active learning models
(in addition membership query model) have also been studied within
the learning community, see, for example, \cite{a-qcl-87}.
\medskip%
\item \emph{Active learning.} As discussed, the problem at hand can
be interpreted as active learning a convex body in relation to a set
of points $\PS$ that need to be classified (as either inside or
outside the body), where the queries are via a separation oracle. We
are unaware of any work directly on this problem in the theory
community, while there is some work in the machine learning
community that studies related active learning classification
problems \cite{cal-igwal-94, gg-oalumi-07, s-alls-09}.
Specifically, Cohn et~al.\xspace\cite{cal-igwal-94} propose a similar
problem to ours. For example, when the unknown body to be learned is
an axis-parallel rectangle in the plane, they use neural networks to
both learn and decide how to best query the oracle.
\end{compactenumA}
\subsection{Our results}
\begin{compactenumA}
\item We develop a greedy algorithm, for points in the plane,
which solves the problem using $O( \indexX{\PS} \log n)$ oracle
queries, where $\indexX{\PS}$ is the largest subset of points of
$\PS$ in convex position. See \thmref{greedy-method}. It is known
that for a random set of $n$ points in the unit square,
$\Ex{\indexX{\PS}} = \Theta( n^{1/3})$ \cite{ab-lcc-09}, which
readily implies that classifying these points can be solved using
$O( n^{1/3} \log n)$ oracle queries. A similar bound holds for the
$\sqrt{n} \times \sqrt{n}$ grid. An animation of this algorithm is
on YouTube \cite{j-agca-18}. We also show that this algorithm
can be implemented efficiently, using dynamic segment trees,
see \lemref{impl-greedy}.
\medskip%
\item The above algorithm naturally extends to three dimensions,
also using $O(\indexX{\PS} \log n)$ oracle queries. While the
proof idea is similar to that of the algorithm in 2D, we believe
the analysis in three dimensions is also technically interesting.
See \thmref{greedy-method-3d}.
\medskip%
\item For a given point set $\PS$ and convex body $\body$, we
define the separation price $\priceY{\PS}{\body}$ of an instance
$(\PS, \body)$, and show that any algorithm classifying the points
of $\PS$ in relation to $\body$ must make at least
$\priceY{\PS}{\body}$ oracle queries (\lemref{lower:bound}).
As an aside, we show that when $\PS$ is a set of $n$ points
chosen uniformly at random from the unit square and $\body$
is a (fixed) smooth convex body,
$\Ex{\priceY{\PS}{\body}} = O(n^{1/3})$, and this bound is
tight when $\body$ is a disk (our result also
generalizes to higher dimensions, see \lemref{ex:sep:pr}).
For randomly chosen points, the separation price is related
to the expected size of the convex hull of $\PS \cap \body$,
which is also known to be $\Theta(n^{1/3})$ \cite{b-rpcba-07}.
We believe this result may be of independent interest, see
\apndref{ex:sep:pr}.
\medskip%
\item In \secref{improved:2d} we present an improved algorithm
for the 2D case, and show that the number of queries made is
$O(\priceY{\PS}{\body} \log^2 n)$. This result is $O(\log^2 n)$
approximation to the optimal solution, see
\thmref{improved:alg:2d}.
\medskip%
\item We consider the extreme scenarios of the problem: Verifying
that all points are either inside or outside of $\body$. For each
problem we present a $O(\log n)$ approximation algorithm to the
optimal strategy. The results are presented in
\secref{emptiness:2d}, see \lemref{greedy-method-empty} and
\lemref{reverse:emptiness}.
\medskip%
\item \secref{applications} presents an application of the above
results, we consider the problem of minimizing a \emph{convex} function
$\Mh{f} : \Re^3 \to \Re$ over a point set $\PS$. Specifically, the
goal is to compute $\arg\min_{\pnt \in \PS} \Mh{f}(\pnt)$. If $\Mh{f}$
and its derivative can be efficiently evaluated at a given a query
point, then $\Mh{f}$ can be minimized over $\PS$ using
$O(\indexX{\PS} \log^2 n)$ queries to $\Mh{f}$ (or its derivative) in
expectation. We refer the reader to \lemref{discrete:min}.
Given a set of $n$ points $\PS$ in $\Re^d$, the discrete geometric
median of $\PS$ is a point $\pnt \in \PS$ minimizing the function
$\sum_{\Mh{q} \in \PS} \normX{\pnt - \Mh{q}}$. As a corollary of
\lemref{discrete:min}, we obtain an algorithm for computing the
discrete geometric median for $n$ points in the plane. The algorithm
runs in $O(n\log^2 n \cdot (\log n \log\log n + \indexX{\PS}))$
expected time. See \lemref{discrete:med}. In particular, if $\PS$
is a set of $n$ points chosen uniformly at random from the unit
square, it is known $\Ex{\indexX{\PS}} = \Theta(n^{1/3})$
\cite{ab-lcc-09}---the discrete geometric median can be computed
in $O(n^{4/3} \log^2 n)$ expected time.
While the discrete median is easy to approximate, we are unaware
of any sub-quadratic algorithm for the discrete case even in the
plane.
\end{compactenumA}
\section{The greedy algorithm in two and three dimensions}
\seclab{greedy-2d-3d}
\subsection{Preliminaries}
For a set of points $\PS \subseteq \Re^2$, let $\CHX{\PS}$ denote
the convex hull of $\PS$. Given a convex body
$\body \subseteq \Re^d$, two points
$\pnt, \pntx \in \Re^d \setminus \intX{\body}$ are \emphi{mutually visible},
if the segment $\pnt \pntx$ does not intersect $\intX{\body}$, where
$\intX{\body}$ is the interior of $\body$.
We also use the notation
$\PS \cap \body = \Set{\pnt \in \PS}{\pnt \in \body}$.
For a point set $\PS \subseteq \Re^d$, a \emphi{centerpoint} of
$\PS$ is a point $\cpnt \in \Re^d$, such that for any closed
halfspace $h^+$ containing $\cpnt$, we have
$\cardin{h^+ \cap \PS} \geq \cardin{\PS}/(d+1)$. A centerpoint
always exists, and it can be computed exactly in
$O(n^{d-1} + n \log n)$ time \cite{c-oramt-04}.
%
\subsection{The greedy algorithm in 2D}
\subsubsection{Operations}
\seclab{operations}
Initially, the algorithm copies $\PS$ into a set $\USet$ of
unclassified points. The algorithm is going to maintain an inner
approximation $\Mh{B} \subseteq \body$. There are two types of
updates (\figref{operations} illustrates the two operations):
\begin{compactenumA}
\smallskip%
\item \ExpandOp{}$(\pnt)$: Given a point
$\pnt \in \body \setminus \Mh{B}$, the algorithm is going to:
\begin{compactenumi}
\item Update the inner approximation:
$\Mh{B} \leftarrow \CHX{\Mh{B} \cup \brc{\pnt}}$.
\item Remove (and mark) newly covered points:
$\USet \leftarrow \USet \setminus \Mh{B}$.
\end{compactenumi}
\medskip%
\item \RemoveOp{}$(\Line)$: Given a closed halfplane $\Line^+$
such that $\intX{\body} \cap \Line^+ = \emptyset$, the algorithm
marks all the points of $\USet_\Line = \USet \cap \intX{\Line^+}$
as being outside $\body$, and sets
$\USet \leftarrow \USet \setminus \USet_\Line$.
\end{compactenumA}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\phantom{}\hfill%
{$\vcenter{\hbox{\includegraphics[scale=0.5]{figs/operations_expand}}}$}
\hfill%
{$\vcenter{\hbox{\includegraphics[scale=0.5]{figs/operations_remove}}}$}
\hfill\phantom{}
\end{minipage}
\caption{(I) Performing $\ExpandOp{}(\pnt)$, and marking points
inside $\body$. (II) Performing $\RemoveOp{}(\Line)$, and marking
points outside $\body$.}
\figlab{operations}
\end{figure}
\subsubsection{The algorithm}
\seclab{round}%
The algorithm repeatedly performs rounds, as described next, until the
set of unclassified points is empty.
At every round, if the inner approximation $\Mh{B}$ is empty, then the
algorithm sets $\USet^+ = \USet$. Otherwise, the algorithm picks a
line $\Line$ that is tangent to $\Mh{B}$ with the largest number of
points of $\USet$ on the other side of $\Line$ than $\Mh{B}$. Let
$\Line^-$ and $\Line^+$ be the two closed halfspace bounded by
$\Line$, where $\Mh{B} \subseteq \Line^-$. The algorithm computes the
point set $\USet^+ = \USet \cap \Line^+$. We have two cases:
\medskip%
\begin{compactenumA}[label=\Alph*.]
\item If $\cardin{\USet^+} = O(1)$, then the algorithm queries the
oracle for the status of each of these points. For every point
$\pnt \in \USet^+$, such that $\pnt \in \body$, the algorithm
performs \ExpandOp{}$(\pnt)$. Otherwise, the oracle returned a
separating line $\Line$, and the algorithm calls
\RemoveOp{}$(\Line^+)$.
\medskip%
\item Otherwise, the algorithm computes a centerpoint
$\cpnt \in \Re^2$ for $\USet^+$, and asks the oracle for the
status of $\cpnt$. There are two possibilities:
\begin{compactenumA}[label*=\Roman*.]
\item If $\cpnt \in \body$, then the algorithm performs
\ExpandOp{}$(\cpnt)$.
\item If $\cpnt \notin \body$, then the oracle returned a
separating line $\LineA$, and the algorithm performs
\RemoveOp{}$(\LineA)$.
\end{compactenumA}
\end{compactenumA}
\subsubsection{Analysis}
Let $\Mh{B}_i$ be the inner approximation at the start of the $i$\th
iteration, and let $\iEmpty$ be the first index where $\Mh{B}_\iEmpty$
is not an empty set. Similarly, let $\USet_i$ be the set of
unclassified points at the start of the $i$\th iteration, where
initially $\USet_1 = \USet$.
\begin{lemma}
\lemlab{iterations:empty:inapx}%
%
The number of (initial) iterations in which the inner
approximation is empty is $\iEmpty = O(\log n)$.
\end{lemma}
\begin{proof}
As soon as the oracle returns a point that is in $\body$, the
inner approximation is no longer empty. As such, we need to bound
the initial number of iterations where the oracle returns that the
query point is outside $\body$. Let $f_i = \cardin{\USet_i}$, and
note that $\USet_1 = \PS$ and $f_1 = \cardin{\PS} = n$. Let
$\cpnt_i$ be the centerpoint of $\USet_i$, which is the query point
in the $i$\th iteration ($\cpnt_i$ is outside $\body$). As such,
the line separating $\cpnt_i$ from $\body$, returned by the
oracle, has at least $f_i/3$ points of $\USet_i$ on the same side
as $\cpnt_i$, by the centerpoint property. All of these points get
labeled in this iteration, and it follows that
$f_{i+1} \leq (2/3) f_i$, which readily implies the claim, since
$f_{\iEmpty} < 1$, for $\iEmpty = \ceil{\log_{3/2} n} +1$.
\end{proof}
\begin{defn}[Visibility graph]
\deflab{visi:graph} Consider the graph $\Graph_i$ over $\USet_i$,
where two points $\pnt, \Mh{r} \in \USet_i$ are connected $\iff$
the segment $\pnt \Mh{r}$ does not intersect the interior of
$\Mh{B}_i$.
\end{defn}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\phantom{}\hfill%
{$\vcenter{\hbox{\includegraphics[page=1,scale=0.8]{figs/intervals}}}$}
\hfill%
{$\vcenter{\hbox{\includegraphics[page=2,scale=0.8]{figs/intervals}}}$}
\hfill%
{$\vcenter{\hbox{\includegraphics[page=3,scale=0.8]{figs/intervals}}}$}
\hfill\phantom{}%
\end{minipage}
\caption{}
\figlab{intervals}
\end{figure}
\myparagraph{The visibility graph as an interval graph.}
For a point $\pnt \in \USet_i$, let $\IY{i}{\pnt}$ be the set of all
directions $v$ (i.e., vectors of length $1$) such that there is a line
perpendicular to $v$ that separates $\pnt$ from $\Mh{B}_i$. Formally,
a line $\Line$ separates $\pnt$ from $\Mh{B}_i$, if the interior of
$\Mh{B}_i$ is on one side of $\Line$ and $\pnt$ is on the (closed)
other side of $\Line$ (if $\pnt \in \Line$, the line is still
considered to separate the two). Clearly, $\IY{i}{\pnt}$ is a circular
interval on the unit circle. See \figref{intervals}.
The resulting set of intervals is
$\IntSet_i = \Set{\IY{i}{\pnt}}{\pnt \in \USet_i}$. It is easy to
verify that the intersection graph of $\IntSet_i$ is $\Graph_i$.
Throughout the execution of the algorithm, the inner approximation
$\Mh{B}_i$ grows monotonically, this in turn implies that the
visibility intervals shrinks over time; that is,
$\IY{i}{\pnt} \subseteq \IY{i-1}{\pnt}$, for all $\pnt \in \PS$ and
$i$. Intuitively, in each round, either many edges from $\Graph_i$
are removed (because intervals had shrunk and they no longer
intersect), or many vertices are removed (i.e., the associated points
are classified).
\begin{defn}
Given a set $\IntSet$ of objects (e.g., intervals) in a domain
$\Domain$ (e.g., unit circle), the \emphi{depth} of a point
$\pnt \in D$, is the number of objects in $\IntSet$ that contain
$\pnt$. Let $\depthX{\IntSet}$ be the maximum depth of any point
in $\Domain$.
\end{defn}
When it is clear, we use $\depthX{\Graph}$ to denote
$\depthX{\IntSet}$, where $\Graph = (\IntSet, \Edges)$ is the
intersection graph in \defref{visi:graph}.
First, we bound the number of edges in this visibility graph
$\Graph$ and then argue that in each iteration, either many edges
of $\Graph$ are discarded or vertices are removed (as they are
classified).
\begin{lemma}
\lemlab{int-graphs}%
%
Let $\IntSet$ be a set of $n$ intervals on the unit circle, and
let $\Graph = (\IntSet, \Edges)$ be the associated intersection
graph. Then $\cardin{\Edges} =O(\indep \clique^2)$, where
$\clique= \depthX{\IntSet}$ and $\indep = \indep(\Graph)$ is the
size of the largest independent set in $\Graph$. Furthermore,
the upper bound on $\cardin{\Edges}$ is tight.
\end{lemma}
\begin{proof}
Let $J$ be the largest independent set of intervals in
$\Graph$. The intervals of $J$ divide the circle into
$2\cardin{J}$ (atomic) circular arcs. Consider such an arc
$\arc$, and let $K(\arc)$ be the set of all intervals of $\IntSet$
that are fully contained in $\arc$. All the intervals of $K(\arc)$
are pairwise intersecting, as otherwise one could increase the
size of the independent set. As such, all the intervals of
$K(\arc)$ must contain a common intersection point. It follows
that $\cardin{K(\arc)} \leq \clique$.
Let $K'(\arc)$ be the set of all intervals intersecting
$\arc$. This set might contain up to $2\clique$ additional
intervals (that are not contained in $\arc$), as each such
additional interval must contain at least one of the endpoints of
$\arc$. Namely, $\cardin{K'(\arc)} \leq 3 \clique$. In particular,
any two intervals intersecting inside $\arc$ both belong to
$K'(\arc)$. As such, the total number of edges contributed by
$K'(\arc)$ to $\Graph$ is at most
$\binom{3\clique}{2} = O(\clique^2)$. Since there are
$\leq 2 \indep$ arcs under consideration, the total number of
edges in $\Graph$ is bounded by $O(\indep \clique^2)$, which
implies the claim.
The lower bound is easy to see by taking an independent set of
intervals of size $\indep$, and replicating every interval
$\clique$ times.
\end{proof}
\begin{lemma}
\lemlab{segments:intersect}%
%
Let $\PS$ be a set of $n$ points in the plane lying above the
$x$-axis, $\cpnt$ be a centerpoint of $\PS$, and
$\SegSet = \binom{\PS}{2}$ be set of all segments induced by
$\PS$. Next, consider any point $\Mh{r}$ on the $x$-axis. Then,
the segment $\cpnt \Mh{r}$ intersects at least $n^2/36$ segments of
$\SegSet$.
\end{lemma}
\begin{proof}
If the segment $\cpnt \Mh{r}$ intersects the segment
$\pnt_1 \pnt_2$, for $\pnt_1, \pnt_2 \in \PS$, then we consider
$\pnt_1$ and $\pnt_2$ to no longer be mutually visible.
It suffices to lower bound the number of pairs of points
which lose mutual visibility of each other.
\medskip
\centerline{\includegraphics{figs/center_point}}
\medskip
Consider a line $\Line$ passing through the point $\cpnt$. Let
$\Line^+$ be the closed halfspace bounded by $\Line$ containing
$\Mh{r}$. Note that $\cardin{\PS \cap \Line^+} \geq n/3$, since
$\cpnt$ is a centerpoint of $\PS$, and $\cpnt \in \Line$. Rotate
$\Line$ around $\cpnt$ until there are $\geq n/6$ points on each
side of $\Mh{r}\cpnt$ in the halfspace $\Line^+$. To see why this
rotation of $\Line$ exists, observe that the two halfspaces
bounded by the line spanning $\Mh{r}\cpnt$, have zero points on one
side, and at least $n/3$ points on the other side --- a continuous
rotation of $\Line$ between these two extremes, implies the
desired property.
Observe that points in $\Line^+$ and on opposite sides of the
segment $\cpnt\Mh{r}$ cannot see each other, as the segment
connecting them must intersect $\cpnt\Mh{r}$. Consequently, the
number of induced segments that $\cpnt\Mh{r}$ intersects is at
least $n^2/36$.
\end{proof}
\begin{lemma}
\lemlab{depth:reduce}%
%
Let $\Graph_i$ be the intersection graph, in the beginning of the
$i$\th iteration, and let
$\nEdgesX{i} = \cardin{\EdgesX{\Graph_{i}}}$. After the $i$\th
iteration of the greedy algorithm, we have
$\nEdgesX{i+1} \leq \nEdgesX{i} - \clique^2/36$, where
$\clique = \depthX{\Graph_i}$.
\end{lemma}
\begin{proof}
Recall that in the algorithm $\USet^+ = \USet_i \cap \Line^+$ is
the current set of unclassified points and $\Line$ is the line
tangent to $\Mh{B}_i$, where $\Line^+$ is the closed halfspace
that avoids the interior of $\Mh{B}_i$ and contains the largest
number of unlabeled points of $\USet_i$. We have that
$\clique = \cardin{\USet^+}$.
If a \RemoveOp{} operation was performed in the $i$\th iteration,
then the number of points of $\USet^+$ which are discarded is at
least $\clique/3$. In this case, the oracle returned a separating
line $\LineA$ between a centerpoint $\cpnt$ of $\USet^+$ and the
inner approximation. For the halfspace $\LineA^+$ containing
$\cpnt$, we have
$t_i = \cardin{\USet^+ \cap \LineA^+} \geq \cardin{\USet^+}/3 \geq
\clique/3$. Furthermore, all the points of $\USet^+$ are pairwise
mutually visible (in relation to the inner approximation
$\Mh{B}_i$). Namely,
\begin{math}
\nEdgesX{i+1}%
=%
\cardin{\EdgesX{\Graph_{i} - (\USet^+ \cap \LineA^+)}}%
\leq%
\nEdgesX{i} - \binom{t_i}{2}%
\leq%
\nEdgesX{i} - \clique^2 /36.
\end{math}
If an \ExpandOp{} operation was performed, the centerpoint $\cpnt$
of $\USet^+$ is added to the current inner approximation
$\Mh{B}_i$. Let $\Mh{r}$ be a point in $\Line \cap \Mh{B}_i$, and
let $\cpnt_i$ be the center point of $\USet_i$ computed by the
algorithm. By \lemref{segments:intersect} applied to
$\Mh{r}, \cpnt$ and $\USet^+$, we have that at least $\clique^2/36$
pairs of points of $\USet^+$ are no longer mutually visible to each
other in relation to $\Mh{B}_{i+1}$. We conclude, that at least
$\clique^2/36$ edges of $\Graph_i$ are no longer present in
$\Graph_{i+1}$.
\end{proof}
\begin{defn}
\deflab{index}%
%
A subset of points $X \subseteq \PS \subseteq \Re^2$ are in
\emphi{convex position}, if all the points of $X$ are vertices of
$\CHX{X}$ (note that a point in the middle of an edge is not
considered to be a vertex). The \emphi{index} of $\PS$, denoted
by $\indexX{\PS}$, is the cardinality of the largest subset of
$\PS$ of points which are in convex position.
\end{defn}
\begin{theorem}
\thmlab{greedy-method}%
%
Let $\body$ be a convex body provided via a separation oracle, and
let $\PS$ be a set of $n$ points in the plane. The greedy
classification algorithm performs
$O\bigl((\indexX{\PS}+1) \log n\bigr)$ oracle queries. The
algorithm correctly identifies all points in $\PS \cap \body$
and $\PS \setminus \body$.
\end{theorem}
\begin{proof}
By \lemref{iterations:empty:inapx}, the number of iterations (and
also queries) in which the inner approximation is empty is
$O(\log n)$, and let $\iEmpty = O(\log n)$ be the first iteration
such that the inner approximation is not empty. It suffices to
bound the number of queries made by the algorithm after the inner
approximation becomes non-empty.
For $i \geq \iEmpty$, let $\Graph_i = (\USet_i, \Edges_i)$ denote
the visibility graph of the remaining unclassified points
$\USet_i$ in the beginning of the $i$\th iteration. Any
independent set in $\Graph_i$ corresponds to a set of points
$X \subseteq \PS$ that do not see each other due to the presence
of the inner approximation $\Mh{B}_i$. That is, $X$ is in
convex position, and furthermore $\cardin{X} \leq \indexX{\PS}$.
For $0 \leq t \leq n$, let $\startX{t}$ be the first iteration
$i$, such that $\depthX{ \Graph_i} \leq t$. Since the depth of
$\Graph_i$ is a monotone decreasing function, this quantity is
well defined. An \emphi{epoch} is a range of iterations between
$s(t)$ and $s(t/2)$, for any parameter $t$. We claim that an epoch
lasts $O( \indexX{ \PS})$ iterations (and every iteration issues
only one oracle query). Since there are only $O( \log n)$
(non-overlapping) epochs till the algorithm terminates, as the
depth becomes zero, this implies the claim.
So consider such an epoch starting at $i = \startX{t}$. We have
$m = \nEdgesX{i} = \cardin{\EdgesX{\Graph_i}} = O( \indexX{\PS}
t^2)$, by \lemref{int-graphs}, since $\indexX{\PS}$ is an
upper bound on the size of the largest independent set in
$\Graph_i$. By \lemref{depth:reduce}, as long as the depth of the
intervals is at least $t/2$, the number of edges removed from the
graph at each iteration, during this epoch, is at least
$\Omega(t^2)$. As such, the algorithm performs at most
$O(m_i/t^2) = O( \indexX{\PS} )$ iterations in this epoch, till
the maximum depth drops to $t/2$.
\end{proof}
\subsubsection{Implementing the greedy algorithm}
With the use of dynamic segment trees \cite{mn-dfc-90} we show
that the greedy classification algorithm can be implemented
efficiently.
\begin{lemma}
\lemlab{impl-greedy}%
%
Let $\body$ be a convex body provided via a separation oracle, and
let $\PS$ be a set of $n$ points in the plane. If an oracle query
costs time $T$, then the greedy algorithm can be implemented in
$O\bigl(n\log^2 n\log\log n + T\cdot\indexX{\PS} \log n\bigr)$
expected time.
\end{lemma}
\begin{proof}
The algorithm follows the proof of \thmref{greedy-method}. We
focus on efficiently implementing the algorithm once inner
approximation is no longer empty. Let $\USet \subseteq \PS$ be
the subset of unclassified points. By binary searching on the
vertices of the inner approximation $\Mh{B}$, we can compute the
collection of visibility intervals $\IntSet$ for all points in
$\USet$ in $O(\cardin{\USet}\log m) = O(n\log n)$ time (recall
that $\IntSet$ is a collection of circular intervals on the unit
circle). We store these intervals in a dynamic segment tree
$\mathcal{T}$ with the modification that each node $v$ in $\mathcal{T}$ stores
the maximum depth over all intervals contained in the subtree
rooted at $v$. Note that $\mathcal{T}$ can be made fully dynamic to
support updates in $O(\log n \log \log n)$ time \cite{mn-dfc-90}.
An iteration of the greedy algorithm proceeds as follows. Start by
collecting all points $\USet^+ \subseteq \USet$ realizing the
maximum depth using $\mathcal{T}$. When $t = \cardin{\USet^+}$, this
step can be done in $O(\log n + t)$ time by traversing $\mathcal{T}$.
We compute the centerpoint of $\USet^+$ in $O(t \log t)$ expected
time \cite{c-oramt-04} and query the oracle using this
centerpoint. Either points of $\USet$ are classified (and we
delete their associated intervals from $\mathcal{T}$) or we improve the
inner approximation. The inner approximation (which is the convex
hull of query points inside the convex body $\body$) can be
maintained in an online fashion with insert time $O(\log n)$
\cite[Chapter 3]{ps-cg-85}. When the inner approximation expands,
the points of $\USet^+$ have their intervals shrink. As such, we
recompute $\IX{\pnt}$ for each $\pnt \in \USet^+$ and reinsert
$\IX{\pnt}$ into $\mathcal{T}$.
As defined in the proof of \thmref{greedy-method}, an epoch is the
subset of iterations in which the maximum depth is in the range
$[t/2, t]$, for some integer $t$. During such an epoch, we make
two claims:
\smallskip%
\begin{compactenumi}
\item there are $\sigma = O(n)$ updates to $\mathcal{T}$, and
\item the greedy algorithm performs $O(n/t)$
centerpoint calculations on sets of size $O(t)$.
\end{compactenumi}
\smallskip%
Both of these claims imply that a single epoch of the greedy
algorithm can be implemented in expected time
$O(\sigma \log n \log\log n + n\log n + T\cdot \indexX{\PS})$.
As there are $O(\log n)$ epochs, the algorithm can be
implemented in expected time
$O(n \log^2 n \log\log n + T\cdot \indexX{\PS}\log n)$.
We now prove the first claim. Recall that we have a collection of
intervals $\IntSet$ lying on the circle of directions. Partition
the circle into $k$ atomic arcs, where each arc contains $t/10$
endpoints of intervals in $\IntSet$. Note that $k = 20n/t =
O(n/t)$. For each circular arc $\arc$, let
$\IntSet_\arc \subseteq \IntSet$ be the set of intervals
intersecting $\arc$. As the maximum depth is bounded by $t$, we
have that $\cardin{\IntSet_\arc} \leq t + t/10 = 1.1t$. In
particular, if $\Graph[\IntSet_\arc]$ is the induced subgraph of
the intersection graph $\Graph$, then $\Graph[\IntSet_\arc]$ has
at most $\binom{\cardin{\IntSet_\arc}}{2} = O(t^2)$ edges.
In each iteration, the greedy algorithm chooses a point in an
arc $\arc$ (we say that $\arc$ is \emph{hit}) and edges
are only deleted from $\Graph[\IntSet_\arc]$.
The key observation is that an arc $\arc$ can only be hit
$O(1)$ times before all points of $\arc$ have depth below
$t/2$, implying that it will not be hit again until the next
epoch. Indeed, each time $\arc$ is hit, the number of edges
in the induced subgraph $\Graph[\IntSet_\arc]$ drops
by a constant factor (\lemref{depth:reduce}). Additionally,
when $\Graph[\IntSet_\arc]$ has less than
$\binom{t/2}{2}$ edges then any point on $\arc$ has
depth less than $t/2$. These two facts imply that an arc
is hit $O(1)$ times.
When an arc is hit, we must reinsert
$\cardin{\IntSet_\arc} = O(t)$ intervals into $\mathcal{T}$. In
particular, over a single epoch, the total number of hits
over all arcs is bounded by $O(k)$. As such,
$\sigma = O(kt) = O(n)$.
For the second claim, each time an arc is hit, a single
centerpoint calculation is performed. Since each arc
has depth at most $t$ and is hit a constant number
of times, there are $O(k) = O(n/t)$ such
centerpoint calculations in a single epoch, each costing
expected time $O(t\log t)$.
\end{proof}
\subsection{The greedy algorithm in 3D}
\seclab{greedy:3d}
Consider the 3D variant of the 2D problem: Given a set of points
$\PS$ in $\Re^3$ and a convex body $\body$ specified via a
separation oracle, the task at hand is to classify, for all the points
of $\PS$, whether or not they are in $\body$, using the fewest
oracle queries possible.
The greedy algorithm naturally extends, where at each iteration $i$ a
plane $\LineB_i$ is chosen that is tangent to the current inner
approximation $\Mh{B}_i$, such that it's closed halfspace (which
avoids the interior of $\Mh{B}_i$) contains the largest number of
unclassified points from the set $\USet_i$. If the queried centerpoint
is outside, the oracle returns a separating plane and as such points
can be discarded by the \RemoveOp{} operation. Similarly, if the
centerpoint is reported inside, then the algorithm calls the
\ExpandOp{} and updates the 3D inner approximation $\Mh{B}_i$. %
\subsubsection{Analysis}
Following the analysis of the greedy algorithm in 2D, we
(conceptually) maintain the following set of objects: For a point
$\pnt \in \USet_i$, let $\pdisk_i(\pnt)$ be the set of all unit length
directions $v \in \Re^3$ such that a plane perpendicular to $v$
separates $\pnt$ from $\Mh{B}_i$. Let
$\PDSet_i = \Set{\pdisk_i(\pnt)}{\pnt \in \USet_i}$. A set of objects
form a collection of \emphi{pseudo-disks} if the boundary of every
pair of them intersect at most twice. The following claim shows that
$\PDSet_i$ is a collection of pseudo-disks on $\sphereC$, where
$\sphereC$ is the sphere of radius one centered at the origin.
\begin{lemma}
The set
$\PDSet_i = \Set{\pdisk_i(\pnt) \subseteq \sphereC}{\pnt \in
\USet_i}$ is a collection of pseudo-disks.
\end{lemma}
\begin{proof}
Fix two points $\pnt, \Mh{r} \in \USet_i$ such that the boundaries
of $\pdisk_i(\pnt)$ and $\pdisk_i(\Mh{r})$ intersect on
$\sphereC$. Let $\Line$ be the line in $\Re^3$ passing through
$\pnt$ and $\Mh{r}$. Consider any plane $\LineB$ such that $\Line$
lies on $\LineB$. Since $\Line$ is fixed, $\LineB$ has one degree
of freedom. Conceptually rotate $\LineB$ until becomes tangent to
$\Mh{B}_i$ at point $\Mh{u}'$. The direction of the normal to this
tangent plane, is a point in
$X = \partial\pdisk_i(\pnt) \cap \partial\pdisk_i(\Mh{r})$. Note
that this works also in the other direction --- any point in $X$
corresponds to a tangent plane passing through $\Line$. The
family of planes passing through $\Line$ has only two tangent
planes to $\body$. It follows that $\cardin{X}=2$. As such, any
two regions in $\PDSet_i$ intersect as pseudo-disks.
\end{proof}
We need the following two classical results that follows from the
Clarkson-Shor \cite{cs-arscg-89} technique.
\SaveContent{\LemmaNumVerticesDepthBody}%
{%
Let $\PDSet$ be a collection of $n$ pseudo-disks, and let
$\vDY{\depthk}{\Arr}$ be the set of all vertices of depth at most
$\depthk$ in the arrangement $\Arr = \ArrX{\PDSet}$. Then
$\cardin{\vDY{\depthk}{\Arr}} = O(n\depthk)$.%
}%
\begin{lemma}[{{\normalfont Proof in \apndref{num:v:depth}}}]
\lemlab{num:vertices:depth}%
%
\LemmaNumVerticesDepthBody{}
\end{lemma}
\SaveContent{\LemmaNumEdgesDepthStatement}%
{%
Let $\PDSet$ be a collection of $n$ pseudo-disks. For two integers
$0 < t \leq k$, a subset $X \subseteq \PDSet$ is a
\emphi{$( t,k)$-tuple} if
\begin{compactenumi*}
\item $\cardin{X} \leq t$,
\item $\exists \pnt \in \cap_{\pdisk \in X} \pdisk$, and
\item $\depthY{\pnt}{\PDSet} \leq \depthk$.
\end{compactenumi*}
Let $\tuplesZ{t}{\depthk}{n}$ be the set of all $(\leq t,k)$-tuples
of $\PDSet$. Then
$\cardin{\tuplesZ{t}{\depthk}{n}} = O(n t k^{t-1})$. %
}
\begin{lemma}[{{\normalfont Proof \apndref{proof:num:edges:depth}}}]
\lemlab{num:edges:depth}%
%
\LemmaNumEdgesDepthStatement{}
\end{lemma}
\begin{lemma}
\lemlab{num:edges:pdisks}%
%
Let $\Graph_i = (\PDSet_i, E_i)$ be the intersection graph of the
pseudo-disks of $\PDSet_i$ (in the $i$\th iteration). If
$\ArrX{\PDSet_i}$ has maximum depth $\depthk$, then
$\cardin{E_i} = O(n\depthk)$. Furthermore,
$\indep(\Graph_i) = \Omega(n/\depthk)$, where $\indep(\Graph_i)$
denotes the size of the largest independent set in $\Graph_i$.
\end{lemma}
\begin{proof}
The first claim readily follows from \lemref{num:edges:depth} ---
indeed, $\cardin{E_i} = \tuplesZ{2}{\depthk}{n} = O(n\depthk)$ ---
since every intersecting pair of pseudo-disks induces a
corresponding $(2,\depthk)$-tuple.
For the second part, Tur\'an\xspace's Theorem states that any graph has an
independent set of size at least $n/\pth{\avgdeg(\Graph_i) + 1}$,
where $\avgdeg(\Graph_i) = 2\cardin{E_i}/n \leq c \depthk$ is the
average degree of $\Graph_i$ and $c$ is some constant. It follows
that $\indep(\Graph_i) \geq n/(c\depthk + 1) = \Omega(n/\depthk)$.
\end{proof}
The challenge in analyzing the greedy algorithm in 3D is that
mutual visibility between pairs of points is not necessarily lost as
the inner approximation grows. As an alternative, consider the
\emph{hypergraph} $\Mh{H}_i = (\PDSet_i, \Mh{\mathcal{E}}_i)$, where a triple
of pseudo-disks $\pdisk_1, \pdisk_2, \pdisk_3 \in \PDSet_i$ form a
hyperedge $\brc{\pdisk_1,\pdisk_2,\pdisk_3} \in \Mh{\mathcal{E}}_i$ $\iff$
$\pdisk_1 \cap \pdisk_2 \cap \pdisk_3 \neq \varnothing$ (this is
equivalent to the condition that the corresponding triple of points
span a triangle which does not intersect $\Mh{B}_i$).
As in the analysis of the algorithm in 2D, we first bound the
number of edges in $\Mh{H}_i$ and then argue that enough progress
is made in each iteration.
\begin{lemma}
\lemlab{num:triples:pdisks}%
%
Let $\Mh{H}_i = (\PDSet_i, \Mh{\mathcal{E}}_i)$ be the hypergraph in
iteration $i$, and let $\Graph_i$ be the corresponding
intersection graph of $\PDSet_i$. If $\ArrX{\PDSet_i}$ has
maximum depth $\depthk$, then
$\cardin{\Mh{\mathcal{E}}_i} = O(\indep(\Graph_i) \depthk^3)$.
\end{lemma}
\begin{proof}
\lemref{num:edges:pdisks} implies that $\Graph_i$ has an
independent set of size $\Omega(f_i/k)$, where
$f_i = \cardin{\PDSet_i}$. \lemref{num:edges:depth} implies that
$\cardin{\Mh{\mathcal{E}}_i} \leq \cardin{\tuplesZ{3}{\depthk}{f_i}} =
O(f_i\depthk^2) = O(\indep(\Graph_i) \depthk^3)$.
\end{proof}
The following is a consequence of the Colorful Carath\'eodory\xspace Theorem
\cite{b-gct-82}, see Theorem 9.1.1 in \cite{m-ldg-02}.
\begin{theorem}
\thmlab{cpnt:many:simplices}%
%
Let $\PS$ be a set of $n$ points in $\Re^d$ and $\cpnt$ be the
centerpoint of $\PS$. Let $\SSet = \binom{\PS}{d+1}$ be the
set of all $d+1$ simplices induced by $\PS$. Then for
sufficiently large $n$, the number of simplices in $\SSet$ that
contain $\cpnt$ in their interior is at least $c_d n^{d+1}$, where
$c_d$ is a constant depending only on $d$.
\end{theorem}
Next, we argue that in each iteration of the greedy algorithm, a
constant fraction of the edges in $\Mh{H}_i$ are removed. The
following is the higher dimensional version of
\lemref{segments:intersect}.
\begin{lemma}
\lemlab{cpnt:many:simplex:faces}%
%
Let $\PS$ be a set of $n$ points in $\Re^3$ lying above the
$xy$-plane, $\cpnt$ be the centerpoint of $\PS$ and
$T = \binom{P}{3}$ be the set of all triangles induced by
$\PS$. Next, consider any point $\Mh{r}$ on the $xy$-plane. Then
the segment $\cpnt\Mh{r}$ intersects at least $\Omega(n^3)$
triangles of $T$.
\end{lemma}
\begin{proof}
Let $\SSet = \binom{\PS}{d+1}$ be the set of all simplices
induced by $\PS$. \thmref{cpnt:many:simplices} implies that the
centerpoint $\cpnt$ is contained in $n^4/\constA$ simplices of
$\SSet$ for some constant $\constA > 1$. Let $K$ be a
simplex that contains $\cpnt$ and observe the segment $\cpnt\Mh{r}$
must intersect at least one of the triangular faces $\tau$ of
$K$. As $K \in \SSet$, charge this simplex
$K$ to the triangular face $\tau$. Applying this counting
to all the simplices containing $\cpnt$, implies that at least
$n^4/\constA$ charges are made. On the other hand, a triangle
$\tau$ can be charged at most $n-3$ times (because a simplex can
be formed from $\tau$ and one other additional point of
$\PS$). It follows that $\cpnt\Mh{r}$ intersects at least
$(n^4/\constA)/ (n-3) = \Omega(n^3)$ triangles of $T$.
\end{proof}
\begin{lemma}
\lemlab{depth:reduce:3d}%
%
In each iteration of the greedy algorithm, the number of edges in
the hypergraph $\Mh{H}_i = (\PDSet_i, \Mh{\mathcal{E}}_i)$ decreases by at
least $\Omega(\depthk^3)$, where $\depthk$ is the maximum depth of
any point in $\ArrX{\PDSet_i}$.
\end{lemma}
\begin{proof}
Recall that $\USet^+ = \USet_i \cap \LineB^+$ is the current set
of unclassified points and $\LineB$ is the plane tangent to
$\Mh{B}_i$, where $\LineB^+$ is the closed halfspace that avoids
the interior of $\Mh{B}_i$ and contains the largest number of
unlabeled points. Note that $|\USet^+| \geq \depthk$.
In a \RemoveOp{} operation, arguing as in \lemref{depth:reduce},
implies that the number of points of $\USet^+$ which are discarded
is at least $\depthk/4$. Since all of the discarded points are in
a halfspace avoiding $\Mh{B}_i$, it follows that all the triples
they induce are in $\Mh{H}_i$. Namely, at least
$\binom{\depthk/4}{3} = \Omega(k^3)$ hyperedges get discarded.
In an \ExpandOp{} operation, the centerpoint $\cpnt$ of $\USet^+$
is added to the current inner approximation $\Mh{B}_i$. Since all
of the points of $\USet^+$ lie above the plane $\LineB$, applying
\lemref{cpnt:many:simplex:faces} on $\USet^+$ with the centerpoint
$\cpnt$ and a point lying on the plane $\LineB$ inside the
(updated) inner approximation, we deduce that at least
$\Omega(\depthk^3)$ hyperedges are removed.
\end{proof}
\begin{theorem}
\thmlab{greedy-method-3d}%
%
Let $\body \subseteq \Re^3$ be a convex body provided via a
separation oracle, and let $\PS$ be a set of $n$ points in
$\Re^3$. The greedy classification algorithm performs
${O\bigl((\indexX{\PS}+1) \log n\bigr)}$ oracle queries. The
algorithm correctly identifies all points in $\PS \cap \body$
and $\PS \setminus \body$.
\end{theorem}
\begin{proof}
The proof is essentially the same as \thmref{greedy-method}.
Arguing as in \lemref{iterations:empty:inapx} implies that there
are at most $O(\log n)$ iterations (and thus also oracle queries)
in which the inner approximation is empty.
Now consider the hypergraph $\Mh{H}_1 = (\PDSet_1, \Mh{\mathcal{E}}_1)$ at
the start of the algorithm execution. As the algorithm
progresses, both vertices and hyperedges are removed from the
hypergraph. Let $\Mh{H}_i = (\PDSet_i, \Mh{\mathcal{E}}_i)$ denote the
hypergraph in the $i$\th iteration of the algorithm. Recall that
$\PDSet_i$ is a set of pseudo-disks associated with each of the
points yet to be classified. Observe that any independent set of
pseudo-disks in the corresponding {intersection graph} $\Graph_i$
corresponds to an independent set of points with respect to the
inner approximation $\Mh{B}_i$, and as such is a subset of points
in convex position. Therefore, the size of any such independent
set is bounded by $\indexX{\PS}$.
Let $\depthk_i$ denote the maximum depth of any vertex in the
arrangement $\ArrX{\PDSet_i}$. \lemref{num:triples:pdisks}
implies that
$\cardin{\Mh{\mathcal{E}}_i} = O\pth{\indexX{\PS} \depthk_i^3}$.
\lemref{depth:reduce:3d} implies that the number of hyperedges in
the $i$\th iteration decreases by at least
$\Omega(\depthk_i^3)$. Namely, after $O( \indexX{\PS})$
iterations, the maximum depth is halved. It follows that after
$O( \indexX{\PS} \log n)$ iterations, the maximum depth is zero,
which implies that all the points are classified. Since the
algorithm performs one query per iteration, the claim follows.
\end{proof}
\section{An instance-optimal approximation %
in two dimensions}
\seclab{improved:2d}
Before discussing the improved algorithm, we present a lower bound on
the number of oracle queries performed by any algorithm that
classifies all the given points. We then present the the improved
algorithm, which matches the lower bound up to a factor of $O(\log^2n)$.
\subsection{A lower bound}
\seclab{lower:bound}
Given a set $\PS$ of points in the plane, and a convex body $\body$,
the \emphi{outer fence} of $\PS$ is a closed convex polygon $\Fout$
with minimum number of vertices, such that $\body \subseteq \Fout$ and
$\body \cap \PS = \Fout \cap \PS$. Similarly, the \emphi{inner
fence} is a closed convex polygon $\Fin$ with minimum number of
vertices, such that $\Fin \subseteq \body$ and
$\body \cap \PS = \Fin \cap \PS$. Intuitively, the outer fence
separates $\PS \setminus \body$ from $\partial \body$, while the
inner fence separates $\PS \cap \body$ from $\partial \body$. The
\emphi{separation price} of $\PS$ and $\body$ is
\begin{equation*}
\priceY{\PS}{\body} = \nVX{ \Fin} + \nVX{ \Fout},
\end{equation*}
where $\nVX{F}$ denotes the number of vertices of a polygon $F$. See
\figref{easy:not:easy} for an example.
\begin{figure}[t]
\centerline{\hfill%
\includegraphics[page=1]{figs/splitting}%
\hfill%
\includegraphics[page=2]{figs/splitting}%
\hfill%
\includegraphics[page=3]{figs/splitting} \hfill%
\phantom}
\caption{The separation price, for the same point set, is
different depending on how ``tight'' the body is in relation to
the inner and outer point set.}
\figlab{easy:not:easy}
\end{figure}
\begin{lemma}
\lemlab{lower:bound} Given a point set $\PS$ and a convex body
$\body$ in the plane, any algorithm that classifies the points of
$\PS$ in relation to $\body$, must perform at least
$\priceY{\PS}{\body}$ separation oracle queries.
\end{lemma}
\begin{proof}
Consider the set $Q$ of queries performed by the optimal algorithm
(for this input), and split it, into the points inside and outside
$\body$. The set of points inside, $\Qin = Q \cap \body$
has the property that $\Qin \subseteq \body$, and furthermore
$\CHX{\Qin} \cap \PS = \body \cap \PS$ --- otherwise, there
would be a point of $\body \cap \PS$ that is not
classified. Namely, the vertices of $\CHX{\Qin}$ are vertices of a
fence that separates the points of $\PS$ inside $\body$ from the
boundary of $\body$. As such, we have that
$\cardin{\Qin} \geq \nVX{\CHX{\Qin}} \geq \nVX{\Fin}$.
Similarly, each query in $\Qout = Q \setminus \Qin$ gives rise to
a separating halfplane. The intersection of the corresponding
halfplanes is a convex polygon $H$ which contains $\body$, and
furthermore contains no point of $\PS \setminus \body$. Namely,
the boundary of $H$ behaves like an outer fence. As such, we have
$\cardin{\Qout} \geq \nVX{H} \geq \nVX{\Fout}$.
Combining, we have that
$\cardin{Q} = \cardin{\Qin} + \cardin{\Qout} \geq \nVX{\Fin} +
\nVX{\Fout} = \priceY{\PS}{\body}$, as claimed.
\end{proof}
In \apndref{ex:sep:pr}, we show that when $\PS$ is a set of $n$ points
chosen uniformly at random from a square and $\body$ is a smooth convex
body, $\Ex{\priceY{\PS}{\body}} = O(n^{1/3})$. Thus, when the
points are randomly chosen, one can think of $\priceY{\PS}{\body}$ as
growing sublinearly in $n$. Of course, for much more contrived
instances, one would expect $\priceY{\PS}{\body}$ to be much smaller
than $\indexX{\PS}$.
\subsection{Useful operations}
\seclab{useful}
We start by presenting some basic operations that the new algorithm
will use.
\subsubsection{A directional climb}
Given a direction $v$, a \emphi{directional climb} is a sequence of
iterations, where in each iteration, the algorithm finds the extreme
line perpendicular to $v$, that is tangent to the inner approximation
$\Mh{B}$. The algorithm then performs an iteration with this line, as
described in \secref{round}. See \figref{directional:climb} for an
illustration. The directional climb ends when the outer halfspace
induced by this line contains no unclassified point.
\begin{figure}[t]
\phantom{}\hfill%
\includegraphics{figs/directional_climb}
\hfill\phantom{}
\caption{A directional climb. An iteration is done using the line
$\Line$. After updating $\Mh{B}$ to include the query $q$, the
algorithm chooses a new extreme line $\LineA$ tangent to $\Mh{B}$ in
the direction of $v$.}
\figlab{directional:climb}
\end{figure}
\begin{claim}
A directional climb requires $O( \log n)$ oracle queries.
\end{claim}
\begin{proof}
Consider the tangent to $\Mh{B}$ in the direction of $v$. At each
iteration, we claim the number of points in this halfplane is
reduced by a factor of $1/3$. Indeed, if the query (i.e.,
centerpoint) is outside $\body$ then at least a third of these
points got classified as being outside. Alternatively, the tangent
halfplanes moves in the direction of $v$, since the query point is
inside $\body$. But then the new halfspace contains at most $2/3$
fraction of the previous point set --- again, by the centerpoint
property.
\end{proof}
\subsubsection{Line cleaning}
\begin{figure}[t]
\centerline{%
\hfill%
\includegraphics[page=1,scale=0.7]{figs/pocket}%
\hfill%
\includegraphics[page=2,scale=0.7]{figs/pocket}%
\hfill\phantom{}%
}
\caption{Unclassified points and their pockets.}
\figlab{pockets}%
\end{figure}
A \emphi{pocket} is a connected region of
$\CHX{\USet \cup \Mh{B}} \setminus \Mh{B}$, see \figref{pockets}. For
the set $\PS$ of input points, consider the set of all lines
\begin{equation}
\LinesX{\PS}%
=%
\Set{\mathrm{line}(\pnt, \Mh{r})}{\pnt,\Mh{r} \in \PS}%
\eqlab{lines:x}%
\end{equation}
they span.
Let $\Line$ be a line that splits a pocket $\Pocket$ into two regions,
and furthermore, it intersects $\Mh{B}$. Let
$\Interval = \Line \cap \Pocket$, and consider all the intersection
points of interest along $\Interval$ in this pocket. That is,
\begin{equation*}
\IPSetZ{\Pocket}{\Line}{\PS}%
=%
\Interval \cap \LinesX{\PS}%
=%
\Set{\bigl. (\Pocket \cap \Line) \cap \LineA }{
\LineA \in \LinesX{\PS}}.
\end{equation*}
In words, we take all the pairs of points of $\PS$ (each such pair
induces a line) and we compute the intersection points of these lines
with the interval $\Interval$ of interest. Ordering the points of
this set along $\Line$, a prefix of them is in $\body$, while the
corresponding suffix are all outside $\body$. One can easily compute
this prefix/suffix by doing a binary search, using the separation
oracle for $\body$ --- see the lemma below for details. Each answer
received from the oracle is used to update the point set, using
\ExpandOp{} or \RemoveOp{} operations, as described in
\secref{operations}. We refer to this operation along $\Line$ as
\emphi{cleaning} the line $\Line$. See \figref{clean}.
\begin{figure}[h]
\centerline{%
\hfill%
\begin{minipage}{0.3\linewidth}
\includegraphics[page=1,width=0.99\linewidth]{figs/ps_2}%
\end{minipage}
\hfill%
\begin{minipage}{0.3\linewidth}
\includegraphics[page=3,width=0.99\linewidth]{figs/ps_2}%
\end{minipage}
\hfill%
\begin{minipage}{0.3\linewidth}
\includegraphics[page=4,width=0.92\linewidth]{figs/ps_2}\\%
\includegraphics[page=5,width=0.92\linewidth]{figs/ps_2}%
\end{minipage}
\hfill\phantom{}%
} \vspace*{-0.25cm}%
\caption{Line cleaning. All the intersection points of interest
along $\Line$ are classified. The binary search results in the
oracle returning a line $\LineA$ that separates the points
outside from the points inside. }
\figlab{clean}%
\end{figure}
\begin{lemma}
Given a pocket $\Pocket$, and a splitting line $\Line$, one can
clean the line $\Line$ --- that is, classify all the points of
$\IPSet = \IPSetZ{\Pocket}{\Line}{\PS}$ using
$O\pth{ \log n \bigr.}$ oracle queries. By the end of this
process, $\Pocket$ is replaced by two pockets, $\Pocket_1$ and
$\Pocket_2$ that do not intersect $\Line$. The pockets $\Pocket_1$
or $\Pocket_2$ may be empty sets.
\end{lemma}
\begin{proof}
First, we describe the line cleaning procedure in more detail.
The algorithm maintains, in the beginning of the $i$\th iteration,
an interval $\IntervalA_i$ on the line $\Line$ containing all the
points of $\IPSet$ that are not classified yet. Initially,
$\IntervalA_1 = \Pocket \cap \Line$. One endpoint, say
$\pnt_i \in \IntervalA_i$ is on $\partial \Mh{B}_i$, and the
other, say $\pnt_i'$, is outside $\body$, where $\Mh{B}_i$ is the
inner approximation in the beginning of the $i$\th iteration.
In the $i$\th iteration, the algorithm computes the set
$\IPSet_i = \IntervalA_i \cap \IPSet$. If this set is empty, then
the algorithm is done. Otherwise, it picks the median point
$\Mh{u}_i$, in the order along $\Line$ in $\IPSet_i$, and queries
the oracle with $\Mh{u}_i$. There are two possibilities: %
\medskip%
\begin{compactenumA}
\item If $\Mh{u}_i \in \body$ then the algorithm sets
$\IPSet_{i+1} = \IPSet_i \setminus [\pnt_i, \Mh{u}_i)$, and
$\IntervalA_{i+1} = \IntervalA_i \setminus [\pnt_i,\Mh{u}_i)$.
\smallskip%
\item If $\Mh{u}_i \notin \body$, then the oracle provided a
closed halfspace $h^+$ that contains $\body$. Let $h^-$ be the
complement open halfspace that contains $\Mh{u}_i$. The
algorithm sets $\IPSet_{i+1} = \IPSet_{i} \setminus h^-$ and
$\IntervalA_{i+1} = \IntervalA_i \cap h^+$.
\end{compactenumA}
\medskip%
This resolves the status of at least half the points in
$\IPSet_i$, and shrinks the active interval. The algorithm repeats
this till $\IPSet_i$ becomes empty. Since
$\cardin{\IPSet} = O(n^2)$, this readily implies that the
algorithm performs $O( \log n)$ iterations.
We now argue that the pocket is split --- that is, $\Pocket_1$ and
$\Pocket_2$ do not intersect $\Line$. Assume that it is false, and
let $\Mh{B}'$ be the inner approximation after this procedure is
done. Let $L$ (resp.~$R$) be the points of
$\USet_\Pocket = \USet \cap \Pocket$
that are unclassified on one side (resp.~other side) of
$\Line$. If the pocket is not split, then there are two points
$\pnt \in L$ and $\Mh{r} \in R$, such that
$\pnt\Mh{r} \cap \Mh{B}' = \emptyset$, and
$\partial \CHX{\Mh{B}' \cup L \cup R}$ intersects $\Line$ at the
point $\Mh{u} = \pnt \Mh{r} \cap \Line$. However, by construction,
the point $\Mh{u} \in \IPSet$. As such, the point $\Mh{u}$ is now
classified as either being inside or outside $\body$, as it is a
point in $\IPSet$. If $\Mh{u}$ is outside, then the halfplane $h^-$
that classified it as such, must had classified either $\pnt$ or
$\Mh{r}$ as being outside $\body$, which is a contradiction. The
other option, is that $\Mh{u}$ is classified as being inside, but
then, it is in $\Mh{B}'$, which is again a contradiction, as it
implies that $\Mh{B}'$ intersects the segment $\pnt \Mh{r}$.
\end{proof}
\subsubsection{Vertical pocket splitting}
\seclab{pocket:split}%
Consider a pocket $\Pocket$ such that all of its points lie
vertically above $\Mh{B}$, and the bottom of $\Pocket$ is part of a
segment of $\partial \Mh{B}$, see \figref{v:pocket}. Such a pocket can
be viewed as being defined by an interval on the $x$-axis
corresponding to its two vertical walls. Let $\USet_\Pocket$ be the
set of unclassified points in this pocket. In each iteration, the
algorithm computes the center point of $\USet_\Pocket$, and queries
the separation oracle. As long as the query point is outside $\body$,
the algorithm performs a \RemoveOp{} operation using the returned
separating line.
When the oracle returns that the query point $\query$ is inside
$\body$, the algorithm computes the vertical line $\Line_\query$
through $\query$. The algorithm now performs line cleaning on this
vertical line. This operation splits $\Pocket$ into two sub-pockets.
Crucially, since $\query$ was a centerpoint for $\USet_\Pocket$,
the number of points in each of the two sub-pockets is at most
$2\cardin{\USet_\Pocket}/3$. See \figref{v:pocket}.
\begin{figure}[t]
\centerline{%
\hfill%
\includegraphics[page=1]{figs/vertical_pocket}%
\hfill%
\includegraphics[page=2]{figs/vertical_pocket}%
\hfill%
\phantom{}}%
%
\caption{Vertical pocket splitting. The figure on the right is
somewhat misleading --- none of the unclassified points in the
new pockets are mutually visible to each other after the line
cleaning operation was done on the separating line.}%
\figlab{v:pocket}%
\end{figure}
\subsection{The algorithm}
\seclab{improved:alg}
The algorithm starts in the same way as the greedy algorithm of
\secref{round}, until we obtain a non-empty inner approximation.
The algorithm also maintains the convex hull of the unclassified
points together with the inner approximation.
Next, the algorithm performs two directional climbs in the positive
and negative directions of the $x$-axis. This uses $O( \log n)$ oracle
queries and results in a computed segment
$\Mh{v} \Mh{v}' \subseteq \body$, where $\Mh{v}, \Mh{v}'$ are vertices of
the inner approximation $\Mh{B}$, such that all unclassified points
lie in the vertical strip induced by these two points.
The algorithm now handles all points of $\USet$ lying above
$\Mh{v} \Mh{v}'$ (the points below the line are handled in a
similar fashion). Let $\Mh{B}^+$ be the set of vertices of $\Mh{B}$ in
the top chain. Note that $\Mh{B}^+$ consists of at most $O(\log n)$
vertices. For each vertex $v$ of $\Mh{B}^+$, the algorithm performs
line cleaning on the vertical line going through $v$. This results in
$O(\log n)$ vertical pockets, where all vertical lines passing
originally through $\Mh{B}^+$ are now clean.
The algorithm repeatedly picks a vertical pocket. If the pocket
contains less three points the algorithm queries the oracle for the
classification of these points, and continues to the next pocket.
Otherwise, the algorithm performs a vertical pocket splitting
operation, as described in \secref{pocket:split}. The algorithm stops
when there are no longer any pockets (i.e., all the points above the
segment $\Mh{v} \Mh{v}'$ are classified). The algorithm then runs the
symmetric procedure below this segment $\Mh{v} \Mh{v}'$.
\subsection{Analysis}
\begin{figure}[t]
\centerline{%
\hfill%
\begin{minipage}{0.4\linewidth}
\centering
\includegraphics[page=1,scale=1.2]{figs/inner_fence}%
\end{minipage}
\hfill%
\begin{minipage}{0.4\linewidth}
\centering
\includegraphics[page=2]{figs/inner_fence}\\%
\includegraphics[page=3]{figs/inner_fence}%
\end{minipage}
\hfill%
\phantom{}
}
\caption{Constructing the polygon $\PolygonA$ from an inner
fence $\Polygon$.}%
\figlab{inner:fence}%
\end{figure}
\begin{lemma}
\lemlab{separate:but}%
%
Given a point set $\PS$, and a convex polygon $\Polygon$ that is
an inner fence for $\PS \cap \body$; that is,
$\PS \cap \body \subseteq \Polygon \subseteq \body$. Then, there
is a convex polygon $\PolygonA$, such that
\begin{compactenumA}
\item
$\PS \cap \body \subseteq \PolygonA \subseteq \Polygon$.
\item $\nVX{\PolygonA} \leq 2\nVX{\Polygon}$ (where $\nVX{Q}$
denotes the number of vertices of the polygon $Q$).
\item Every edge of $\PolygonA$ lies on a line of
$\LinesX{\PS}$, see \Eqref{lines:x}.
\end{compactenumA}
\end{lemma}
\begin{proof}
Any edge $\edge$ of $\Polygon$ that does not contain any point of
$\PS$ on it can be moved parallel to itself into the polygon
until it passes through a point of $\PS$. Next, split the edges
that contain only a single point of $\PS$, by adding this point
as a vertex.
Consider a vertex $v$ of the polygon that is not in $\PS$ ---
and consider the two adjacent vertices $u,w$, which must be in
$\PS$. If $\triangle uvw \setminus uw$ contains no point of $\PS$,
then we delete $v$ from the polygon and replace it by the edge
$uw$. Otherwise, move $v$ towards $u$, until the edge $vw$ hits a
point of $\PS$. Next, move $v$ towards $w$, till the edge $vu$
hits a point of $\PS$. See \figref{inner:fence}.
Repeating this process so that all edges contain two points of
$\PS$ means that properties (A) and (C) are met.
Additionally, the number of edges of the new polygon $\PolygonA$
is at most twice the number of edges of $\Polygon$,
implying property (B).
\end{proof}
Consider the inner and outer fences $\Fin$ and
$\Fout$ of $\PS$ in relation to $\body$. Applying
\lemref{separate:but} to $\Fin$, results in a convex polygon
$\PolygonA$ that separates $\PS \cap \body$ from
$\partial \body$, that has at most $2 \nVX{\Fin}$ vertices. Let
$\VSet$ be the set of all vertices of the polygons $\Fin, \Fout$
and $\PolygonA$.
The following two Lemmas state that if a vertical pocket $\Pocket$
containing no vertex of $\VSet$, then all points in $\Pocket$ can be
classified using $O(\log n)$ oracle queries. Finally, we analyze the
scenario when $\Pocket$ contains at least one vertex of $\VSet$.
\begin{lemma}
\lemlab{no:vertex:outside}
Let $\Pocket$ be a vertical pocket created during the algorithm with
current inner approximation $\Mh{B}$. Suppose that
$\VSet \cap \Pocket = \varnothing$, then all points in
$\PS \cap \Pocket$ are outside $\body$.
\end{lemma}
\begin{proof}
Assume without loss of generality that $\Pocket$ lies above $\Mh{B}$.
Let $\USet = \PS \cap \Pocket$ be the set of unclassified points in
the pocket. Note that $\Pocket$ is bounded by two vertical lines that
were previously cleaned.
By assumption, $\Pocket$ does not contain any vertex of $\PolygonA$.
It follows that there is a single edge of $\PolygonA$ which
intersects the two vertical lines bounding $\Pocket$. Let
$\Mh{u}_L, \Mh{u}_R$ be these two intersection points, one lying on
each line. By definition, we have $\Mh{u}_L, \Mh{u}_R \in \body$.
Furthermore, $\Mh{u}_L, \Mh{u}_R$ lie on lines of $\LinesX{\PS}$ by
construction of $\PolygonA$. Since both vertical lines bounding
$\Pocket$ were cleaned, it must be that the segment
$\Mh{u}_L \Mh{u}_R \subseteq \Mh{B}$. Since all points of $\USet$ are
above $\Mh{B}$, this implies that $\USet$ lies above
$\Mh{u}_L \Mh{u}_R$ and thus above $\PolygonA$. Namely, all points of
$\USet$ are outside $\body$.
\end{proof}
\begin{lemma}
\lemlab{no:vertex:classify} Let $\Pocket$ be a vertical pocket with
$\VSet \cap \Pocket = \varnothing$. Then during the vertical pocket
splitting operation of \secref{pocket:split} applied to $\Pocket$,
all oracle queries are outside $\body$. In particular, all points of
$\PS \cap \Pocket$ are classified after $O(\log n)$ oracle queries.
\end{lemma}
\begin{proof}
Let $\USet = \PS \cap \Pocket$. By \lemref{no:vertex:outside}, all
points of $\USet$ lie outside $\body$. Assume that the first
statement of the Lemma is false, and let $\USet' \subseteq \USet$ be
the set of unclassified points such that $\query$ was the centerpoint
for $\USet'$ and $\query \in \body$. Now $\query$ is inside a
triangle induced by three points of $\USet'$. Namely, there are (at
least) two points outside $\body$ in this pocket that are not
mutually visible to each other with respect to $\body$. But this implies
that $\Fout$ must have a vertex somewhere inside the vertical pocket
$\Pocket$, which is a contradiction.
Hence, all oracle queries made by the algorithm are outside $\body$.
Each such query results in a constant reduction in the size of
$\USet$, since the query point is a centerpoint of the unclassified
points. It follows that after $O(\log\cardin{\USet}) = O(\log n)$
queries, all points in $\Pocket$ are classified.
\end{proof}
\begin{theorem}
\thmlab{improved:alg:2d}
Let $\body$ be a convex body provided via a separation oracle, and
let $\PS$ be a set of $n$ points in the plane. The improved
classification algorithm performs
\begin{math}
O\pth{ \bigl[ 1 +\priceY{\PS}{\body}\bigr] \log^2 n}
\end{math}
oracle queries. The algorithm correctly identifies all points
in $\PS \cap \body$ and $\PS \setminus \body$.
\end{theorem}
\begin{proof}
The initial stage involves two directional climbs and $O( \log n)$
line cleaning operations, and thus requires $O( \log^2 n)$
queries.
A vertical pocket that contains a vertex of $\VSet$ is charged
arbitrarily to any such vertex. Since the number of points in a
pocket reduces by at least a factor of $1/3$ during a split
operation, this means that a vertex of $\VSet$ is charged at most
$O(\log n)$ times. Each time a vertex gets charged, it has to pay
for the $O(\log n)$ oracle queries that were issued in the
process of creating this pocket, and later on for the price of
splitting it. Thus, we only have to account for queries performed
in vertical pockets that do not contain a vertex of $\VSet$.
By \lemref{no:vertex:classify}, such a pocket will have all
points inside it classified after $O(\log n)$ oracle queries.
However, the above implies that there are at most
$O([1+\priceY{\PS}{\body}] \log n)$ vertical pockets with no
vertex of $\VSet$ throughout the algorithm execution.
Since handling such a pocket requires $O( \log n)$ queries, the
bound follows.
\end{proof}
\section{On emptiness variants in two dimensions}
\seclab{emptiness:2d}
Here, we present two instance-optimal approximation algorithms for
solving the following two variants:
\begin{compactenumA}
\smallskip%
\item Emptiness: Find a point $\pnt \in \PS \cap \body$, or
using as few queries as possible, verify that
$\PS \cap \body = \varnothing$.
\smallskip%
\item Reverse emptiness: Find a point
$\pnt \in \PS \setminus (\PS \cap \body)$, or using as few
queries as possible, verify that $\PS \cap \body = \PS$.
\end{compactenumA}
\smallskip%
For both variants we present $O( \log n )$ approximation
(the algorithm for emptiness is randomized), improving
over the general approximation algorithm of \secref{improved:2d}
which provides a $O( \log^2 n)$ approximation.
\subsection{Emptiness: Are all the points outside?}
Here we consider the problem of verifying that all the given points
are outside the convex body.
\myparagraph{Algorithm.}
The algorithm is a slight modification of the algorithm of
\secref{round}. At each iteration the point set $\USet^+$ is the
largest set of currently unclassified points in $\PS$ contained in
some halfspace tangent to the current inner approximation
$\Mh{B}$. Let $\clique = \cardin{\USet^+}$. We make the following
changes: If $\clique = O(1)$, test the membership of each point
individually. Otherwise, choose a random point $\query \in
\USet^+$. If $\query$ is found to be inside $\body$, we are done, as
$\query$ is our witness. Otherwise $\query$ is outside, and a
\RemoveOp{} operation is performed. The algorithm then performs
a regular iteration on $\USet^+$, as described in \secref{round}.
\myparagraph{Analysis.}
Let $\Graph_i$ be the intersection graph (see \defref{visi:graph})
over the points outside $\body$ in the beginning of the $i$\th
iteration. We need the following technical Lemma.
\SaveContent{\LemmaIndepFin}%
{%
Suppose $\PS \cap \body = \varnothing$. Then at any iteration
$i$, the largest independent set in the visibility graph $\Graph_i$
is at most $\cardin{\Fout}$. }
\begin{lemma}[{{\normalfont Proof in \apndref{proof:indep:fin}}}]
\lemlab{indep:fin}%
%
\LemmaIndepFin{}
\end{lemma}
\begin{lemma}
\lemlab{greedy-method-empty}%
%
Let $\body$ be a convex body provided via a separation oracle, and
let $\PS$ be a set of $n$ points in the plane. The randomized greedy
classification algorithm for emptiness performs
${O\bigl((\cardin{\Fout}+1) \log n\bigr)}$ oracle queries
with high probability. The algorithm always correctly
verifies that $\PS \cap \body = \varnothing$ or
finds a witness point of $\PS$ inside $\body$.
\end{lemma}
\begin{proof}
Suppose $\PS \cap \body = \varnothing$. Then \lemref{indep:fin}
along with the proof of \thmref{greedy-method} implies the result,
by replacing the quantity $\indexX{\PS}$ with $\cardin{\Fout}$.
If $\PS \cap \body \neq \varnothing$, let $\USet^+$ be a set of
points in the current iteration,
$\USetin{+} = \USet^+ \cap \body$, and
$\USetout{+} = \USetout{+} \setminus \USetin{+}$. Observe that
$\USetin{+}$ remains the same throughout the algorithm execution,
while $\USetout{+}$ shrinks. If
$\cardin{\USetout{+}} > \cardin{\USet^+}/2$, then by
\lemref{depth:reduce} the number of edges removed from $\Graph_i$
is $\Omega\pth{\cardin{\USetout{+}}^2}$ (though the hidden
constants will be smaller). Thus, after at most
$O\bigl((\cardin{\Fout}+1)\log n \bigr)$ iterations, we must
encounter an iteration in which there is a set of points
$\USet^+$ with $\cardin{\USetout{+}} < \cardin{\USet^+}/2$. Now
the probability that our randomly sampled point lies in
$\USetin{+}$ is at least 1/2. In particular, after an additional
$O(\log n)$ iterations, the probability that we fail to find a
witness point is at most $1/n^{\Omega(1)}$, thus implying the bound
on the number of queries.
\end{proof}
\subsection{Reverse emptiness: Are all the points inside?}
Here we consider the problem of verifying that all the given points
are inside the convex body.
\subsubsection{Algorithm}
\myparagraph{Initialization.}
Let $\mathcal{D} = \CHX{\PS}$. Define $\Mh{v}, \Mh{v}' \in \PS$ to be the
extreme left and right vertices of $\mathcal{D}$. Let $\Mh{v}_1$ and $\Mh{v}_2$
be the vertices adjacent to $\Mh{v}$ on $\mathcal{D}$. Similarly define
$\Mh{v}'_1$ and $\Mh{v}'_2$ for $\Mh{v}'$. The algorithm asks the oracle
for the status of $\Mh{v}$, $\Mh{v}_1$, $\Mh{v}_2$, $\Mh{v}'$, $\Mh{v}'_1$,
and $\Mh{v}'_2$. If any of them are outside, the algorithm halts
and reports the witness found. Otherwise, all points must lie either
above or below the horizontal segment $\Mh{v}\pntD'$. We now describe
how to handle the points above $\Mh{v}\pntD'$ (the below case is
handled similarly).
Let $\Mh{\ch^+}$ be the polygonal chain which is $\mathcal{D}$ clipped inside
region bounded by the segment $vv'$ and two vertical lines
passing through $\Mh{v}$ and $\Mh{v}'$.
Label the edges along $\mathcal{D}^+$ by $\edgeB_1, \ldots, \edgeB_k$
clockwise from $\Mh{v}$ to $\Mh{v}'$. For $1 \leq i < j \leq k$, let
$\ChRangeY{i}{j}$ be the polygonal chain consisting of the consecutive
edges $\edgeB_i, \ldots, \edgeB_j$. The algorithm now invokes the
following recursive procedure.
\myparagraph{Recursive procedure.}
A recursive call is described by two indices $(i,j)$, the goal is
to verify that all the points of $\PS$ lying below $\ChRangeY{i}{j}$
are inside $\body$.
For a given recursive instance $(i,j)$, the algorithm proceeds as
follows. Begin by computing the lines $\Line_i$ and $\Line_j$ through
the edges $\edgeB_i$ and $\edgeB_j$ respectively. Let
$\query = \Line_i \cap \Line_j$ be the point of intersection. The
algorithm asks the oracle for the status of $\query$. If $\query$ is
inside, then all points below $\ChRangeY{i}{j}$ must also be in
$\body$. The algorithm classifies the appropriate points and returns.
Otherwise $\query$ is outside, and generates two recursive calls. Let
$\ell = \floor{(i + j)/2}$ and $\edgeB_\ell = (x,y)$ be the middle edge
in the chain $\ChRangeY{i}{j}$. The algorithm queries the oracle with
$x$ and $y$. If either $x$ or $y$ is outside, the algorithm returns
the appropriate witness found. Otherwise $x$ and $y$ are both
inside. The algorithm recurses on the instances $(i, \ell)$ and
$(\ell, j)$.
\subsubsection{Analysis.}
The analysis will use the polygon $\PolygonA$, as defined in
\lemref{separate:but}, applied to $\Fin$. Specifically, it is an inner
fence where $\cardin{\PolygonA} = O(\cardin{\Fin})$ and every edge of
$\PolygonA$ lies on a line of $\LinesX{\PS}$, see \Eqref{lines:x}. Note
that $\mathcal{D} \subseteq \PolygonA$ and every edge of $\mathcal{D}$ lies on a line
of $\LinesX{\PS}$. For each edge $\edge$ of $\PolygonA$, let
$\Line_\edge \in \LinesX{\PS}$ be the line containing $\edge$. We can
match every edge $\edge$ of $\PolygonA$ with the edge $\edgeB(\edge)$
of $\mathcal{D}$ which lies on $\Line_\edge$. If an edge $\edgeB$ of $\mathcal{D}$ is
matched to some edge of $\PolygonA$, we say that $\edgeB$ is
\emphi{active}. A recursive call $(i,j)$ is \emphi{alive} if the query
$\query = \Line_i \cap \Line_j$ generated is outside $\body$.
\begin{lemma}
\lemlab{num:recursive:calls}%
%
The number of alive recursive calls at the same recursive depth
is at most $\cardin{\PolygonA} = O(\cardin{\Fin})$.
\end{lemma}
\begin{proof}
Fix an alive recursive call $(i,j)$ with edges
$\edgeB_i, \ldots, \edgeB_j$ of $\mathcal{D}$. Suppose that none of these
edges are active. Because $\PolygonA$ is an inner fence for $\PS$
and $\body$, there must be a vertex $\Mh{v}$ of $\PolygonA$ lying on
or above the chain $\ChRangeY{i}{j}$. Let $\edge_1$ and $\edge_2$
be the edges adjacent to $v$ in $\PolygonA$. For $\ell = 1, 2$,
consider $\edgeB(\edge_\ell)$, the edge of $\mathcal{D}$ matched to
$e_\ell$. Since there are no active edges in $\ChRangeY{i}{j}$, we
have $\edgeB(\edge_\ell) \not\in \{\edgeB_i, \ldots, \edgeB_j\}$
for $\ell = 1, 2$. This readily implies that all vertices in the
polygonal chain $\ChRangeY{i}{j}$ are contained in the wedge formed
by $\Mh{v}$ and the two edges $\edge_1$ and $\edge_2$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{figs/active_alive_instance}
\end{figure}
In particular, the query $\query$ generated is inside $\PolygonA$
and thus $\body$. Contradicting that the recursive call was alive.
It follows that each alive recursive call must contain at least
one active edge. The number of active edges is bounded by
$\cardin{\PolygonA}$, implying the result.
\end{proof}
\begin{lemma}
\lemlab{reverse:emptiness}%
%
Let $\body$ be a convex body provided via a separation oracle, and
let $\PS$ be a set of $n$ points in the plane. The
classification algorithm for reverse emptiness performs
$O\bigl(\cardin{\Fin} \log n\bigr)$ oracle queries.
The algorithm correctly verifies that $\PS \cap \body = \PS$
or finds a witness point of $\PS$ outside $\body$.
\end{lemma}
\begin{proof}
Suppose all points of $\PS$ are inside $\body$. By
\lemref{num:recursive:calls}, there are at most $O(\cardin{\Fin})$
alive recursive calls at each level of the recursion tree. Since
the depth of the recursion tree is $O(\log n)$, the number of
total alive recursive calls throughout the algorithm is
$O(\cardin{\Fin} \log n)$. At each alive recursive call of the
above algorithm, $O(1)$ queries are made. This implies the result.
Otherwise not all points of $\PS$ are inside $\body$. At least
one such point outside of $\body$ must be a vertex on the convex
hull $\mathcal{D}$. Hence after at most $O(\cardin{\Fin} \log n)$ oracle
queries, this vertex will be queried and found to be outside
$\body$.
\end{proof}
\section{Application: Minimizing a convex function}
\seclab{applications}
Suppose we are given a set of $n$ points $\PS$ in the plane and a convex
function $\Mh{f} : \Re^2 \to \Re$. Our goal is to compute the point in $\PS$
minimizing $\min_{\pnt \in \PS} \Mh{f}(\pnt)$. Given a point
$\pnt \in \Re^2$, assuming that we can evaluate $\Mh{f}$ and the derivative
of $\Mh{f}$ at $\pnt$ efficiently, we show that the point in $\PS$ minimizing
$\Mh{f}$ can be computed using $O(\indexX{\PS} \log^2 n)$ evaluations
to $\Mh{f}$ or its derivative.
\begin{defn}
Let $\Mh{f} : \Re^d \to \Re$ be a convex function. For a number
$c \in \Re$, define the \emphi{level set of $\Mh{f}$} as
$\LvSetY{\Mh{f}}{c} = \Set{\pnt \in \Re^d}{\Mh{f}(\pnt) \leq c}$. If
$\Mh{f}$ is a convex function, then $\LvSetY{\Mh{f}}{c}$ is a convex set
for all $c \in \Re$.
\end{defn}
\begin{defn}
Let $\Mh{f} : \Re^d \to \Re$ be a convex (and possibly
non-differentiable) function. For a point $\pnt \in \Re^d$, a
vector $v \in \Re^d$ is a \emphi{subgradient} of $\Mh{f}$ at $\pnt$
if for all $\Mh{q} \in \Re^d$,
$\Mh{f}(\Mh{q}) \geq \Mh{f}(\pnt) + \DotProdY{v}{\Mh{q} - \pnt}$. The
\emphi{subdifferential} of $\Mh{f}$ at $\pnt \in \Re^d$, denoted by
$\partial \Mh{f}(\pnt)$, is the set of all subgradients $v \in \Re^d$
of $\Mh{f}$ at $\pnt$.
\end{defn}
It is well known that when the domain of $\Mh{f}$ is $\Re^d$ and $\Mh{f}$ is
a convex function, then $\partial \Mh{f}(\pnt)$ is a non-empty set of all
$\pnt \in \Re^d$ (for example, see \cite[Chapter 3]{f-ina-13}).
Let $\alpha = \min_{p \in \PS} \Mh{f}(p)$. We have that
$\LvSetY{\Mh{f}}{\alpha} \cap \PS = \Set{p \in \PS}{\Mh{f}(p) = \alpha}$
and $\LvSetY{\Mh{f}}{\alpha'} \cap \PS = \varnothing$ for all $\alpha' < \alpha$.
Hence, the problem is reduced to determining the smallest value $r$
such that $\LvSetY{\Mh{f}}{r} \cap \PS$ is non-empty.
\begin{lemma}
\lemlab{decision}
Let $\PS$ be a collection of $n$ points in the plane.
For a given value $r$, let $\body_r = \LvSetY{\Mh{f}}{r}$.
The set $\body_r \cap \PS$ can be computed using
$O(\indexX{\PS} \log n)$ evaluations to $\Mh{f}$ or
its derivative. If $T$ is the time needed to evaluate $\Mh{f}$
or its derivative, the algorithm can be implemented in
$O(n\log^2 n\log\log n + T\cdot \indexX{\PS} \log n)$ expected
time.
\end{lemma}
\begin{proof}
The Lemma follows by applying \thmref{greedy-method}. Indeed, let
$\body_r = \LvSetY{\Mh{f}}{r}$ be the convex body of interest. It remains
to design a separation oracle for $\body_r$.
Given a query point $\query \in \Re^2$, first compute
$c = \Mh{f}(\query)$. If $c \leq r$, then report that
$\query \in \body_r$. Otherwise, $c > r$. In this case, compute
some gradient vector $v$ in $\partial \Mh{f}(\query)$. Using the vector
$v$, we can obtain a line $\Line$ tangent to the boundary of
$\LvSetY{\Mh{f}}{c}$ at $\query$. As
$\LvSetY{\Mh{f}}{r} \subseteq \LvSetY{\Mh{f}}{c}$, $\Line$ is a separating
line for $\query$ and $\body_r$, as desired.\footnote{Note that $\query$
lies on $\Line$. If we require that $\query$ lies in the interior
of one of the halfspaces bounded by $\Line$, we can shift
$\Line$ infinitesimally to properly separate $\query$ and $\body_r$.}
As such, the number of separation oracle queries needed to determine
$\body_r \cap \PS$ is bounded by $O(\indexX{\PS} \log n)$
by \thmref{greedy-method}.
The implementation details of \thmref{greedy-method} are given in
\lemref{impl-greedy}.
\end{proof}
\myparagraph{The algorithm.}
Let $\alpha = \min_{p \in \PS} \Mh{f}(p)$.
For a given number $r \geq 0$, set $\PS_r = \LvSetY{\Mh{f}}{r} \cap \PS$.
We develop a randomized algorithm to compute $\alpha$.
Set $\PS_0 = \PS$. In the $i$\th iteration, the algorithm chooses a
random point $p_i \in \PS_{i-1}$ and computes $r_i = \Mh{f}(p_i)$.
Next, we determine $\PS_{r_i}$ using \lemref{decision}. In doing so,
we modify the separation oracle of \lemref{decision} to store the
collection of queries $S_i \subseteq \PS$ which satisfy $\Mh{f}(s) = r_i$
for all $s \in S_i$. We set $\PS_{i+1} = \PS_{r_i} \setminus S_i$.
Observe that all points $p \in \PS_{i+1}$ have $\Mh{f}(p) < r_i$.
The algorithm continues in this fashion until we reach an iteration
$j$ in which $\cardin{\PS_{j+1}} \leq 1$. If $\PS_{j+1} = \{q\}$ for
some $q \in \PS$, output $q$ as the desired point minimizing the
geometric median. Otherwise $\PS_{j+1} = \varnothing$, implying that
$\PS_{r_j} = S_j$, and the algorithm outputs any point in the set
$S_j$.
\myparagraph{Analysis.}
We analyze the running time of the algorithm. To do so, we argue
that the algorithm invokes the algorithm in \lemref{decision} only
a logarithmic number of times.
\begin{lemma}
\lemlab{jumps}
In expectation, the above algorithm terminates after $O(\log n)$
iterations.
\end{lemma}
\begin{proof}
Let $V = \Set{\Mh{f}(p)}{p \in \PS}$ and $N = \cardin{V}$. For a
number $r$, define $V_r = \Set{i \in V}{i \leq r}$. Notice that we
can reinterpret the algorithm described above as the following
random process. Initially set $r_0 = \max_{i \in V} i$. In the
$i$\th iteration, choose a random number $r_i \in
V_{r_{i-1}}$. This process continues until we reach an iteration
$j$ in which $\cardin{V_{r_j}} \leq 1$.
We can assume without loss of generality that
$V = \{1, 2, \ldots, N\}$. For an integer $i \leq N$,
let $T(i)$ be the expected number of iterations needed for the
random process to terminate on the set $\{1, \ldots, i\}$. We have
that $T(i) = 1 + \frac{1}{i-1} \sum_{j=1}^{i-1} T(i-j)$, with
$T(1) = 0$. This recurrence solves to $T(i) = O(\log i)$.
As such, the algorithm repeats this random process
$O(\log N) = O(\log n)$ times in expectation.
\end{proof}
\begin{lemma}
\lemlab{discrete:min} Let $\PS$ be a set of $n$ points in $\Re^2$
and let $\Mh{f} : \Re^2 \to \Re$ be a convex function. The point in
$\PS$ minimizing $\Mh{f}$ can be computed using
$O(\indexX{\PS} \log^2 n)$ evaluations to $\Mh{f}$ or its
derivative. The bound on the number of evaluations holds in
expectation. If $T$ is the time needed to evaluate $\Mh{f}$ or its
derivative, the algorithm can be implemented in
$O(n\log^3 n\log\log n + T\cdot \indexX{\PS} \log^2 n)$ expected
time.
\end{lemma}
\begin{proof}
The result follows by combining \lemref{decision} and
\lemref{jumps}.
\end{proof}
\subsection{The discrete geometric median}
Let $\PS$ be a set of $n$ points in $\Re^d$. For all $x \in \Re^d$, define
the function $\Mh{f}(x) = \sum_{q \in \PS - x} \normX{x - q}$.
The \emphi{discrete geometric median} is defined as the point in $\PS$
minimizing the quantity $\min_{p \in \PS} \Mh{f}(p)$.
Note that $\Mh{f}$ is convex, as it is the sum of convex functions.
Furthermore, given a point $\pnt$, we can compute $\Mh{f}(\pnt)$ and the
derivative of $\Mh{f}$ at $\pnt$ in $O(n)$ time. As such, by
\lemref{discrete:min}, we obtain the following.
\begin{lemma}
\lemlab{discrete:med}
Let $\PS$ be a set of points in $\Re^2$. Then
the discrete geometric median of $\PS$ can be computed in
$O(n\log^2 n \cdot (\log n \log\log n + \indexX{\PS}))$
expected time.
\end{lemma}
\begin{remark}
For a set of $n$ points $\PS$ chosen uniformly at random from the unit
square, it is known that in expectation $\indexX{\PS} = \Theta(n^{1/3})$
\cite{ab-lcc-09}.
As such, the discrete geometric median for such a random set $\PS$
can be computed in $O(n^{4/3} \log^2 n)$ expected time.
\end{remark}
\BibTexMode{%
\SoCGVer{%
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,669 |
Q: Mount user-specific folder on logon There is a small network with an active directory, and there is one machine with RAID-1 storage. How can I mount a user-specific shared folder, for example:
\\NetworkStorage\Storage\%USERNAME%
when %USERNAME% performs logon?
A: you can use the NET command to mount a drive:
NET USE F: \\NetworkStorage\Storage\%USERNAME%
(Note that you need to use back slashes, not forward slashes)
This could be added to your logon script.
ALSO, you can specify a "home" drive for each user in his or her active directory profile. This can be mounted automatically, or with the following command:
NET USE F: /HOME
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,793 |
Q: How do I dynamically assign the ImageUrl property of an image, using data binding? I've got a SQL Server database with a table in it, which lists the file names of images. It's my intention to assign the ImageUrl of an Image control on the page, from the data in the table. I've placed a SQLDataSource control on the page, and then tried putting a FormView control there, and an Image control within that. But I don't see how I can assign the value to the ImageUrl property via data binding.
A: inside whatever type of control your using to return the data you would do something like...
<asp:imagebutton id="btnId" runat="server" ImageUrl='<%# Bind("ImgUrl") %>' />
Play with that and give it a try... most of what i do is in datagridviews or repeaters... but that doesnt really matter much its the bind or eval that matters.
A bit more information for you Rod.
<asp:SqlDataSource
id="SqlDataSource1"
runat="server"
DataSourceMode="DataReader"
ConnectionString="<%$ ConnectionStrings:MyNorthwind%>"
SelectCommand="SELECT LastName FROM Employees">
</asp:SqlDataSource>
<asp:SqlDataSource
id="SqlDataSource2"
runat="server"
DataSourceMode="DataReader"
ConnectionString="<%$ ConnectionStrings:MyNorthwind%>"
SelectCommand="SELECT FirstName FROM Employees">
</asp:SqlDataSource>
<asp:ListBox
id="ListBox1"
runat="server"
DataTextField="LastName"
DataSourceID="SqlDataSource1">
</asp:ListBox>
<asp:ListBox
id="ListBox2"
runat="server"
DataTextField="FirstName"
DataSourceID="SqlDataSource2">
</asp:ListBox>
Let me know if that helps
A: A cleaner way of doing this sort of binding, well cleaner to me at least, would be to handle the binding in the ItemDataBound event.
So you would do something like:
Image imageToBind = e.Item.FindControl("imgTest") as Image;
image.ImageUrl = (string)DataBinder.Eval(e.Item.DataItem, "ColumnName");
I just find that to be more elegant than doing it in the actual markup.
A: .aspx.cs
public string GetImage(string status)
{
if (status=="Active")
return "~/images/green_acti.png";
else
return "~/images/red_acti.png";
}
.aspx
<asp:TemplateField HeaderText="|| Status ||">
<ItemTemplate>
<asp:Image ID="imgGreenAct" ImageUrl='<%# GetImage(Convert.ToString(DataBinder.Eval(Container.DataItem, "truck_status")))%>' AlternateText='<%# Bind("truck_status") %>' runat="server" />
</ItemTemplate>
</asp:TemplateField>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 921 |
George is a Debt Recovery Executive within the Dispute Resolution team, having joined Morrisons Solicitors in early 2016 he deals primarily with money claims and the enforcement of Judgment Debts.
He has specialised in Debt Recovery for four years, joining Morrisons from a boutique Financial Services and Litigation Support company, where he had been managing the Debt Recovery Department.
George is particularly involved in the obtaining of Charging Orders as a method of enforcing money Judgments, and dealt with numerous applications during the transition of the application process from local Courts to the County Court Money Claims Centre in April 2016. During this period he liaised frequently with the Court to provide feedback on this process.
His other primary focus is providing assistance to solicitors in the Dispute Resolution team in the obtaining of money Judgments, in matters involving both commercial and private debtors.
In another life George was the singer and guitarist in a relatively successful band who achieved minor acclaim, and one day intend to obtain an injunction to prevent the publication of any photographs or recordings from this period of time. He also has a profound dislike of moths. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,720 |
Q: unable to get deployment statu During VMware vCenter 7 deployment on the second stage of installation, after a reboot, when trying to log in https://:5480/ redirects to https://:5480/configurev2/#/
"unable to get deployment status"
at the same time, the center itself allows entering normally through 443
This is already the 10th attempt to install. Checked all the articles about FQDN, /etc/resolve and hostname issues. The hostname both A and PTR registered and everything resolves normally and there is plenty of space (over 1 TB)
I deploy the vCenter on OVH dedicated servers. In order to map a failover IP you need to register the MAC address on OVH first then manually enter newly generated MAC address to vm.
The vCenter appliance vm is created by the installer and MAC is generated automatically. After the first stage is completed, I have to manually shut down the vm and enter a new MAC address manually
Everything starts up, I complete the installation, but after the reboot I see the message
"unable to get deployment status"
when I installed 6.7 in the same way, there were no such problems
Has anyone experienced something similar?
There is no error in the logs that could lead to a clue
The only thing that comes to my mind is to deploy pfsense and distribute the local network
vCenter version
7.0.3.01000
A: I managed to find a solution to this one.
it's impossible to use a failover ip for vCenter applience , becase of MAC address generation.
so the only option is to use pfsanse or vrack and generate a local network
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 291 |
\section{Introduction}
\label{section:Introduction}
\subsection{Quantum Phase Transitions}
\label{subsection:QPTs}
Unlike ordinary phase transitions, quantum phase transitions
(QPTs)~\cite{Vojta} take place at a temperature of absolute zero
and are consequently driven entirely by quantum fluctuations,
rather than thermal ones. Moreover, the long-range, algebraically
decaying correlations which are characteristic of critical
many-body ground states are due entirely to entanglement. As such,
one would expect entanglement measures (see
Ref.~\cite{EntMeasures} for an overview) to provide further
insight into the fundamental physical underpinnings of QPTs,
possibly complementing the conventional condensed-matter approach,
which relies mostly on two-point correlation functions.
Indeed, Osterloh and co-workers~\cite{OsterlohEtAl} demonstrated
singular and scaling behaviour of a bipartite entanglement measure
in the vicinity of the critical point in the one-dimensional XY
spin model. On the other hand, this behaviour could be understood
entirely from that of the relevant two-point correlation functions
as these are sufficient to determine the two-particle reduced
density matrices. Furthermore, the entanglement between just two
sites is not particularly well suited to capturing the large-scale
behaviour of correlations, which becomes all the more relevant in
the critical regime. This was recognized in
Ref.~\cite{BlockEntIC}, where the scaling behaviour of the
entanglement between a contiguous \emph{block} of $L$ sites with
the rest of the lattice was first considered in lattice field
theories. It was found that the entanglement entropy of
\emph{noncritical} chains with short-ranged interactions reaches a
saturation level, while the entanglement entropy diverges
logarithmically with $L$ in the field limit, i.e. in the case of
\emph{critical} chains. This approach was subsequently pursued
numerically~\cite{VidalLatorre} in the particular context of the
one-dimensional XY model and, in more generality, in an analytical
treatment based on random matrix theory~\cite{Keating}, applicable
to any spin chain Hamiltonian that can be cast into a quadratic
form of fermionic operators.
At the same time, progress has been hampered by the realization
that \emph{bi-partite} entanglement, whether of blocks of spins or
otherwise, at best only gives us an incomplete, local picture of
the entanglement exhibited by generic many-body ground states. It
is something of a truism, of course, to note that a truly global,
multi-partite approach is indispensable if one wishes to capture
the entanglement that pervades critical many-body ground states
simultaneously at all length scales (for recent work on
multi-partite entanglement in quantum spin chains see for example
Refs.~\cite{Bruss,Costantini}).
Unfortunately, the theory of multi-partite entanglement is still
very much in its infancy. It is true that many of the entanglement
measures used for bi-partite states carry straightforward
generalizations to the multi-partite setting. This is particularly
true for distance-based entanglement measures, such as the
relative entropy of entanglement~\cite{RelEnt} or the closely
related geometric entanglement~\cite{Geometric}: the entanglement
of a state is then simply quantified in terms of the minimum
distance of that state from the set of all multi-partite separable
states, rather than the set of all bi-partite separable states.
However, it is one thing to generalize the axiomatic definitions
of entanglement measures to include the multi-partite case, but
quite another to still be able to compute these in practice.
Moreover, the theory of multi-partite entanglement is still
plagued by a host of different candidates for suitable
entanglement measures, even for the case of pure states (see
Ref.~\cite{EntMeasures} for more details).
In view of these difficulties, attention has shifted to include
other, potentially related, means of characterizing
QPTs~\cite{Zanardi}. One such approach centres around the notion
of \emph{geometric
phase}~\cite{Berry,Resta,Carollo1,Carollo2,Hamma}, and it is this
approach which we wish to pursue in the remainder of this work.
\subsection{Geometric Phase}
\label{subsection:GeometricPhase}
In his seminal paper~\cite{Berry}, Berry investigated the phase
picked up by an eigenstate of a parameter-dependent Hamiltonian
when transported adiabatically around a closed trajectory in
parameter space. It turns out that in addition to the well-known
dynamical phase there is also a geometric component to the phase,
which is observable, at least in principle. While the dynamical
phase provides a measure of the duration of the Hamiltonian's
evolution and is independent of the geometry of the trajectory
followed, conversely, the geometric phase is independent of the
rate at which the state is transported around the loop (as long as
this is slow enough for the adiabatic theorem to rule out
transitions to neighbouring, orthogonal states) and depends solely
upon the geometry of the trajectory. The geometric phase had
hitherto been widely overlooked as just another unphysical phase
factor, and certainly had not been granted the level of
recognition it enjoys nowadays.
Before going on to explain how Berry's phase may be used to probe
for QPTs, it is useful here to first restate a simple example
given in Berry's original paper, which serves to illustrate the
concept of geometric phase. To that end, consider a single
spin-$1/2$ particle which is coupled to an external magnetic
field, and suppose the spin is initially aligned with the magnetic
field, i.e. the particle is in an eigenstate of the system's
Hamiltonian. Then the spin will remain aligned with the magnetic
field vector (the particle remains in the instantaneous
eigenstate) when the magnetic field vector is made to rotate
adiabatically. Upon completion of some closed trajectory, the
final state of the particle differs from its initial one by an
overall phase factor, which is part dynamical, part geometric in
origin. Now, it turns out that the geometric phase is in fact
equal in size to precisely half the solid angle subtended at the
origin by the magnetic field vector's trajectory in parameter
space. This result not only serves to highlight in a particularly
acute way the geometric character of Berry's phase, but it also
gives an indication of the origin of Berry's phase: the spin's
state is degenerate when the magnetic field is switched off, and
the geometric phase can thus be interpreted as providing a measure
of the view of the circuit as seen from that point of degeneracy
at the origin of the parameter space.
In fact, in any general setting, the emergence of geometric phases
can always be traced back to the presence of isolated
singularities in parameter space. Formally, Berry's phase can be
related to the curvature of the Hilbert space bundle over the base
space of parameters~\cite{Simon}, and it is this interpretation of
Berry's phase which renders its potential usefulness as a
diagnostic tool for criticality plausible: points of degeneracy
are associated with a greater curvature of the associated Hilbert
space bundle, and therefore it might be reasonable to expect
Berry's phase to act as a signature for quantum critical points in
interacting spin systems.
Of course, the geometric phase of $N$ non-interacting spin-$1/2$
particles with respect to a particular circuit is simply $N$ times
the geometric phase of a single particle, which in turn can be
related to the circuit's solid angle, as explained above. However,
as we switch on the spin-spin interactions, this simple
interpretation of the geometric phase in terms of solid angles
starts to break down. It is of great interest to know how the
geometric phase of a set of interacting spins with respect to a
given circuit changes as a function of the coupling parameter, and
whether the geometric phase is able to signal the presence of
critical points in the system. An important point to note here is
that the circuit in parameter space need only pass \emph{near} the
critical point for the Berry phase to register it. In other words,
the system need not actually undergo the quantum phase transition
for the Berry phase to pinpoint its presence and location in
parameter space, a consideration which assumes particular
importance in the light of the difficulties that may be associated
with physically implementing actual QPTs.
\subsection{Previous Work}
\label{subsection:PreviousWork}
A relationship between the geometric phase and criticality in spin
chains was examined by Carollo and
Pachos~\cite{Carollo1,Carollo2}. Specifically, the authors
analyzed the Berry phase of the ground state of a one-dimensional
spin-$1/2$ $XY$ model
\begin{equation}
H = \sum_{l=-M}^{M} \left( \frac{1 + \gamma}{2}\sigma_l^x
\sigma_{l+1}^x + \frac{1 - \gamma}{2}\sigma_l^y \sigma_{l+1}^y +
\lambda \sigma_l^z \right),
\end{equation}
where $M = (N-1)/2$ for an odd number of spins $N$. The Berry
phase, as computed with respect to a rotation of the complete
Hamiltonian around the $z$-axis, was shown to exhibit the
following property in the thermodynamical limit
$N\rightarrow\infty$:
\begin{equation}
\label{eq:PT}
\lim_{\gamma\rightarrow 0}(\varphi/M) =
\begin{cases}
0 \,(\text{mod }2\pi), & \text{for} |\lambda| > 1 \nonumber \\
\text{finite}, & \text{for} |\lambda| < 1. \nonumber
\end{cases}
\end{equation}
As the $XY$ model is critical for $|\lambda| < 1$ (and
non-critical otherwise), the quantity given above may be
interpreted as a signature of criticality for this model.
Unfortunately, the Berry phase \emph{per spin}, $\varphi/M$, is
not a physical quantity that is accessible via experiments. We can
only measure the \emph{total} Berry phase $\varphi$, which
vanishes for all values of $\lambda$: i.e.
$(1/M)\lim_{\gamma\rightarrow 0}\varphi = 0 \,(\text{mod } 2\pi),
\forall\lambda$. This is in agreement with Hamma~\cite{Hamma}, who
also showed, via a different line of argument, that for a finite
number of spins, the Berry phase acquired by the ground state is
always trivial ($0$ or $2\pi$). As an alternative, it was
suggested that the same quantity would be non-trivial in the
thermodynamic limit: $\lim_{\gamma\rightarrow 0}\lim_{M
\rightarrow \infty}(\varphi/M) \neq 0$. However, for any physical,
finite system, no matter how large, one is not able to detect
criticality in this manner. In other words, any finite-size
scaling behaviour is completely lacking, i.e. $\lim_{M \rightarrow
\infty}\lim_{\gamma\rightarrow 0}(\varphi/M) = 0\,\mbox{(mod
2$\pi$)}$; it is only in the thermodynamic limit itself that this
method represents a signature of criticality, i.e. the limiting
case is reached discontinuously. As such, the result's worth is
perhaps more of an abstract, mathematical nature, than of any
real, physically measurable consequence.
In the present work, in contrast, we employ a technique based on
the geometric phase that is able to detect critical points without
the need to contract the circuit's radius to zero or to extend the
number of particles to infinity. We do so by considering
discretized circuits that pass directly through the critical
point, rather than circumnavigating it. This procedure and the
obtained results are outlined in the following.
\section{Procedure and Outline}
\label{section:Outline}
We start by introducing the concept of a Bargmann
invariant~\cite{Bargmann} and its associated phase, which may be
regarded as a generalized Berry phase. It will be seen that
Bargmann invariants are ideally suited for \emph{numerically}
analyzing the gauge-invariant phase induced by evolving states
along a \emph{discretized} circuit in parameter space. We then
introduce the model of interacting spin-$1/2$ particles with which
we will be concerned for the rest of this paper and discuss some
of its more salient features. The Bargmann invariant and
associated phase is computed numerically for a chosen discretized
circuit and plotted as a function of the spin coupling strength
$J$. Of note will be the `speed' (to be defined) with which the
Bargmann invariant changes as a function of $J$ and the behaviour
of the magnitude of the Bargmann invariant; we will find that both
of these quantities can act as adequate signatures of criticality.
At this stage, it may be helpful to depict the procedure
schematically:
\begin{figure}[htp]
\label{fig:schematic}
\centering
\psfrag{A}[tc][tc]{$J$}
\psfrag{B}[tc][tc]{$J_c$}
\psfrag{Sn}[bl][bl]{$\underline{\alpha}_{\mathcal{N}} = \underline{\alpha}_{0}$}
\psfrag{SS}[rc][rc]{$\underline{\alpha}_{s+1}$}
\psfrag{SS1}[rc][rc]{$\underline{\alpha}_{s}$}
\includegraphics[width = .38\textwidth]{schematic}
\caption{
A schematic representation of the procedure used to compute the
Bargmann invariant with respect to some chosen circuit
$\{\underline{\alpha}_s\}$ in parameter space, as a function of the
spin coupling strength $J$.
}
\end{figure}
Let us fix the coupling parameter to some value $J_0$ and choose a
particular circuit composed of $\mathcal{N}$ vertices
$\underline{\alpha}_s$, each representing the set of parameters
needed (together with $J_0$) in order to completely describe the
Hamiltonian of the system at that point. At each individual point
on the circuit we numerically compute the ground state of the
system. The Bargmann invariant associated with the circuit at
$J_0$ is then obtained by cyclically multiplying successive
overlaps of these ground state wavefunctions. This will be
explained in more detail in section~\ref{section:Bargmann}, which
is devoted entirely to the subject of Bargmann invariants; for
now, we merely note that, in order to be sure to acquire a
gauge-invariant quantity in this way, the wavefunctions at the
start and end points of the circle need to be taken as
\emph{identical}. By repeating this procedure for a range of
different coupling strengths (generating a complex number, the
Bargmann invariant, for each value), we start to build up a
picture of the general trend. We present our results for varying
lengths of the spin chain, contrast the case of spin-$1/2$
particles with that of spin-$1$ particles, and discuss the effect
of increasing the number $\mathcal{N}$ of constituent vertices of
the circuit. Of particular interest, of course, is what, if
anything, may be said about the Bargmann invariant as the coupling
parameter crosses its critical point $J_c$. The report concludes
with a summary and brief discussion of our work.
\section{Bargmann Invariant and Phase}
\label{section:Bargmann}
In quite general terms one may define the phase difference $\chi$
between two non-orthogonal state vectors, $\ket{\Psi_1}$ and
$\ket{\Psi_2}$, by the relation
\begin{equation*}
\exp{(i \chi)} \equiv
\braket{\Psi_1}{\Psi_2}/|\braket{\Psi_1}{\Psi_2}|,
\end{equation*}
so that $\chi = \text{arg}\braket{\Psi_1}{\Psi_2}$. Naturally,
individual state vectors are only defined up to arbitrary phase
factors; $\chi$ is thus gauge-dependent and its value may not be
assigned any direct physical meaning. On the other hand, any
\emph{cyclic} combination of state vectors, such as
$\text{arg}(\braket{\Psi_1}{\Psi_2}\braket{\Psi_2}{\Psi_3}\braket{\Psi_3}{\Psi_1})$
is manifestly gauge-invariant and may therefore potentially
represent a physically relevant, measurable quantity. This
observation features strongly in the early works of
Pancharatnam~\cite{Pancharatnam} and it also appears in the form of
a remark in the celebrated proof of Wigner's theorem by
Bargmann~\cite{Bargmann}.
Introducing some notation, we define the \emph{Bargmann invariant}
with respect to a general $\mathcal{N}$-vertex circuit in parameter
space by
\begin{equation}
\label{eq:phasedefn2}
\mathcal{C} =
\prod_{s=0}^{\mathcal{N}-1}\braket{\psi_s}{\psi_{s+1}} =
\mbox{tr}{\prod_{s=0}^{\mathcal{N}-1}\proj{\psi_s}},
\quad
\psi_{\mathcal{N}} = \psi_{0}.
\end{equation}
The associated \emph{Bargmann phase} is denoted by $\varphi =
\text{arg}(\mathcal{C})$. Modulo $2\pi$, the global Bargmann phase
is just the sum of the individual phases, i.e. $\varphi =
\sum_{s=0}^{\mathcal{N}-1}\text{arg}\braket{\psi_s}{\psi_{s+1}}$.
Note that we require $\mathcal{N} \geqslant 3$ for non-trivial
Bargmann phases.
Suppose now, for argument's sake, that we are dealing with state
vectors $\ket{\psi(\underline{\alpha}_s)}$ that represent the
instantaneous ground states of a family of parameter-dependent
Hamiltonians $H(\{\underline{\alpha}\})$. Then, in the continuum
limit where the sum above is replaced by a contour integral over an
infinitesimal phase along a (smooth) closed circuit in parameter
space, the Bargmann invariant reduces to the usual Berry phase. In
this sense Bargmann invariants may be regarded as generalized Berry
phases. The key advantage afforded by the generalized formulation
lies in its computational ease: Bargmann invariants lend themselves
directly to numerical computation, and thus, crucially, to scenarios
where parts of the circuit under consideration represent
non-integrable Hamiltonians, as is the case in the present work. An
obvious potential drawback is that the discretization procedure may
lack the simple interpretational underpinnings of the Berry phase in
terms of physical adiabatic processes.
\section{The Spin Model}
\label{section:SpinModel}
Consider a spin chain with nearest-neighbour $XX$ interactions in
the presence of an external magnetic field, which is aligned along
an arbitrary direction $\overrightarrow{n}$. The system is
described by the following Hamiltonian:
\begin{equation}
\label{eq:generalham}
H_{\overrightarrow{n}} = J \sum_{k=1}^{N}\sigma^{x}_{k}\sigma^{x}_{k+1}
+ B \sum_{k=1}^{N}\sigma^{\overrightarrow{n}}_{k},
\quad N \geqslant 2,
\end{equation}
where, $\overrightarrow{n} = (n_x,n_y,n_z)$ denotes a vector of
unit length, so that $\sigma^{\overrightarrow{n}} = \sum_{\alpha}
n_{\alpha}\sigma^{\alpha}$ with $\sum_{\alpha} n_{\alpha}^2 = 1$,
$\alpha \in \{x,y,z\}$. Note that we impose periodic boundary
conditions, so that $\sigma^{x}_{N+1} \equiv \sigma^{x}_1$.
As already outlined in the previous two sections, our aim is now
to analyze the Bargmann invariant induced by the ground states of
a family of Hamiltonians (\ref{eq:generalham}) that is
characterized by a set of $\mathcal{N}$ unit vectors
$\{\overrightarrow{n}\}$. Typically, we will imagine the magnetic
field vectors of that family of Hamiltonians to be arranged on the
vertices of a regular polygon residing on the surface of the Bloch
sphere (i.e. $B = 1$); an example of such a circuit is depicted in
Fig.~\ref{fig:Bloch2}. For a given coupling strength, $J_0$, the
Bargmann invariant can now be obtained by numerically computing
the ground states at each of the circuit's vertices and applying
Eq.~(\ref{eq:phasedefn2}). Having chosen our circuit in parameter
space, we would then like to investigate the manner in which the
Bargmann invariant depends on the choice of spin coupling
strength, $J$. Of particular interest are circuits that slice
through any critical points of the system. Does the Bargmann
invariant undergo a marked shift when the parameters pass through
their critical values? Of further interest is the dependence on
the number $\mathcal{N}$ of vertices of the regular polygon
circuit, as well as the dependence on the number $N$ of spins in
the chain.
Leaving the choice of circuit in parameter space open for now, it
is evident from the form of Hamiltonian (\ref{eq:generalham}) that
there are two `special' directions in which the magnetic field
vector may point, namely the $x$-direction and any direction in
the $y-z$ plane. The former is essentially a classical model while
the latter corresponds to the quantum Ising model. In both cases,
the Hamiltonian is analytically soluble. Of course, it is well
known that the Ising model exhibits criticality when $B = J$,
provided we find ourselves in the thermodynamic limit of an
infinite spin chain. Aside from the two special cases outlined,
however, the Hamiltonian will in general be non-integrable at any
point along the circuit; this forces us to resort to numerical
simulations of a limited number of spins, which moves the Ising
model's critical point outside our region of accessibility. On the
other hand, when the magnetic field vector points in the
$x$-direction, the Hamiltonian does possess a critical point, even
for a \emph{finite} number of spins, as outlined in the following.
\section{The Critical Point} \label{section:CriticalPoint}
A special case of the class of Hamiltonians (\ref{eq:generalham})
occurs when the magnetic field vector points in the $x$-direction:
\begin{equation}
\label{eq:specialham}
H_x =
J \sum_{k=1}^{N}\sigma^{x}_{k}\sigma^{x}_{k+1}
+ B \sum_{k=1}^{N}\sigma^{x}_{k}, \quad N \geqslant 2.
\end{equation}
This Hamiltonian is purely classical: its two constituent terms
commute and are diagonal in the \{\ket{\pm}\} eigenbasis. In the
following we will demonstrate that $H_x$ possesses a critical
point~\cite{footnote1} at $J=J_c \equiv |B|/2$.
It is not difficult to find the ground state of this model. When
$J \leqslant J_c$, it is simply
\begin{equation}
\label{eq:PT}
\ket{E^{(0)}} =
\begin{cases}
\ket{-}^{\otimes N}, \text{ for }B \geqslant 0 \, \& \, J \leqslant J_c,& \nonumber \\
\ket{+}^{\otimes N}, \text{ for }B \leqslant 0 \, \& \, J \leqslant J_c,& \nonumber
\end{cases}
\end{equation}
with the corresponding (non-degenerate) ground state energy given
by $E^{(0)} = N(J-2J_c)$.
In the regime $J \geqslant J_c$, the situation is only slightly
more involved as the parity of $N$ now becomes important. For an
\emph{even} number of spins, the (doubly degenerate) ground state
energy is given by $E^{(0)} = -NJ$, the corresponding eigenstates
being $\ket{+-}^{\otimes N/2}$ and $\ket{-+}^{\otimes N/2}$. For
an \emph{odd} number of spins, the ground state energy is $N$-fold
degenerate and reads $E^{(0)} = -NJ + 2(J-J_c)$. The ground state
is given by
\begin{equation}
\label{eq:PT}
\ket{E^{(0)}} =
\begin{cases}
\ket{-+-+- \cdots -+-}, \text{ for }B \geqslant 0 \, \& \, J \geqslant J_c,& \nonumber \\
\ket{+-+-+ \cdots +-+}, \text{ for }B \leqslant 0 \, \& \, J \geqslant J_c,& \nonumber
\end{cases}
\end{equation}
and all $N$ translations thereof. In order to convince oneself
that this is indeed the correct ground state, one can easily show
that no individual spin flip is capable of lowering the energy any
further.
From the analysis above it follows that the ground state energy
$E^{(0)}$ can always be written as a function of $|J-J_c|$,
demonstrating non-analyticity at the critical point $J_c$.
Moreover, the energy gap vanishes in the region $J \geqslant J_c$.
\section{The Circuit} \label{section:CriticalPoint}
In light of the previous discussion, the parameter circuit which
suggests itself is the following: a regular polygon residing on
the Bloch sphere of which one edge is bisected by the $x$-axis, as
illustrated in Fig.~\ref{fig:Bloch2}.
\begin{figure}[htp]
\centering
\includegraphics[width = .348\textwidth]{BlochSphere2}
\caption{
A schematic representation of the circuit's topology. The critical
$x$-axis slices mid-way through an edge that connects two adjacent
vertices of the polygon.
}
\label{fig:Bloch2}
\end{figure}
As we move along the vertices of the circuit, the critical $x$-axis
is crossed in the direction of positive $z$-axis.
\section{Results for the Spin-$1/2$ Chain}
\label{section:Results}
We now turn to the results of our simulation of the Bargmann
invariant for the spin-$1/2$ chain, which are summarized in
Fig.~\ref{fig:spinonehalf}. The circuit used in the simulation is
the regular polygon described in the previous section, but
consists of $\mathcal{N}=100$ vertices.
\begin{figure}[htp]
\centering
\psfrag{Xlabel}[tc][tc]{}%
\psfrag{Ylabel}[bc][bc]{}%
\psfrag{replaceX}[tc][tc]{}%
\psfrag{replaceY}[bc][bc]{}%
\hspace{-0.1cm}%
\subfigure{\label{subfig:3cnum}
\includegraphics[width = .22\textwidth]{N3cnum100clipped}
}\hspace{0.2cm}%
\subfigure{\label{subfig:3phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{N3BargPhaseclipped}
}\\[-0.2cm]%
\hspace{-0.1cm}%
\subfigure{\label{subfig:5cnum}
\includegraphics[width = .22\textwidth]{N5cnum100clipped}
}\hspace{0.2cm}%
\subfigure{\label{subfig:5phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{N5BargPhaseclipped}
}\\[-0.2cm]%
\hspace{-0.1cm}%
\subfigure{\label{subfig:7cnum}
\includegraphics[width = .22\textwidth]{N7cnum100clipped}
}\hspace{0.2cm}%
\subfigure{\label{subfig:7phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{N7BargPhaseclipped}
}\\[-0.2cm]%
\hspace{-0.1cm}%
\subfigure{\label{subfig:9cnum}
\includegraphics[width = .22\textwidth]{N9cnumclipped}
}\hspace{0.2cm}%
\subfigure{\label{subfig:9phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{N9BargPhaseclipped}
}\\[-0.2cm]%
\hspace{-0.1cm}%
\subfigure{\label{subfig:11cnum}
\includegraphics[width = .22\textwidth]{11N100cnumclipped}
}\hspace{0.2cm}%
\subfigure{\label{subfig:11phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{11N100BargPhaseclipped}
}\\[-0.2cm]%
\caption{
Bargmann invariants and their phases. The rows correspond to the
number of spin-$\frac{1}{2}$ particles $N$, increasing from the top
as $3,5,7,9,11$. The left column depicts the Bargmann invariants
$\mathcal{C}$ on the unit disk in the complex plane, each point of
the graph stemming from a different value of the coupling parameter
$J$. The right-hand column shows the corresponding Bargmann phase
$\varphi/\pi$ as a function of the coupling parameter $(J-J_c)/r$.
See the text for a detailed description of observations.
}\label{fig:spinonehalf}
\end{figure}
The Bargmann invariant and its phase were computed for different
spin chain lengths, ranging in odd numbers from $3$ to $11$, and
shown in Fig.~\ref{fig:spinonehalf} as increasing from the top.
Note that the results for even numbers of spins have been omitted
as they showed no qualitative difference. This may be due to
finite-size effects of the kind which have been observed in
similar studies elsewhere, see for example Ref.~\cite{Kay}. The
left-hand column of the figure depicts the Bargmann invariants
$\mathcal{C}$ on the unit disk in the complex plane. Each data
point of the graph stems from a different value of the coupling
parameter $J$. The right-hand column shows the corresponding
Bargmann phase $\varphi/\pi$, as a function of the coupling
parameter $(J-J_c)/r$, where $r$ refers to the radius of the
circle within which the $\mathcal{N}$-polygon circuit is
inscribed. Note that in this case the radius was chosen as
$10^{-5}$, but the graphs, as they are plotted, are qualitatively
independent of the size of this radius.
We would like to point out several features that are apparent from
Fig.~\ref{fig:spinonehalf}. First and foremost, the Bargmann
invariant $\mathcal{C}$ is observed to always pass through the
origin of the complex plane just when the spin coupling strength
attains its critical value of $J_c = 1/2$. In other words, the
magnitude $|\mathcal{C}|$ vanishes at the critical point. This is
one of the features which we would like to propose as a signature
of criticality, and is discussed further in
section~\ref{subsection:magsignature}.
Another trend that is immediately apparent from
Fig.~\ref{fig:spinonehalf} is that with increasing spin chain
length the Bargamann invariants `wind ever more loops' around the
origin. This statement can be made a little more precise by
referring to the corresponding Bargmann phases. These graphs are
characterized by discontinuities at the critical point: `jumps' by
$\pi$ alternating with `jumps' by $2\pi$. However, on joining up
those separate pieces in the graphs end-to-end, one notices that
the `image' covered by the resulting graphs grows with $N$ as $\pi
(N-1)/2$. This result may be explained by way of referring to the
$J=0$ limit of non-interacting spins for which the Berry phase
grows linearly with the number of spins. Also note that for
spin-$1$ chains, the `joined-up' Bargmann phase covers twice as
much ground for any given number of particles (see
Fig.~\ref{fig:spinone}).
In addition, we should note that we obtain Berry's solid angle
result for the case of non-interacting spins, $J=0$. Approximating
the solid angle of our regular polygon circuit as that of the
enveloping cone, the Bargmann phase $\varphi$ asymptotically
approaches Berry's solid angle result as the number of vertices of
the polygon is increased.
Of course, in the present work only spin chains of up to $11$
particles were simulated, and the exponential growth in the size of
the Hilbert space with increasing particle number precludes one from
simulating chains that are much longer than that~\cite{footnote2}.
However, from the preceding discussion the trend for Bargmann
invariants and phases of longer spin chains should by now be fairly
obvious. In particular, it has become clear that, perhaps
surprisingly, precious little may be deduced about criticality from
the Bargmann phase. Instead, a much clearer indication of the
existence of a critical point is the vanishing magnitude of the
Bargmann invariant itself.
\section{Signatures of Criticality}
\subsection{Magnitude of the Bargmann Invariant}
\label{subsection:magsignature}
It was already apparent from Fig.~\ref{fig:spinonehalf} that the
magnitude of the Bargmann invariant $\mathcal{C}$ is able to
assume the role of a diagnostic tool for critical points in the
finite-sized interacting spin chain considered. For clarity, the
magnitude of the graphs of $\mathcal{C}$ in
Fig.~\ref{fig:spinonehalf} are plotted separately in
Fig.~\ref{fig:varyspinnumber}.
\begin{figure}[htp]
\centering
\psfrag{Xlabel}[tc][tc]{$(J-J_c)/r$}
\psfrag{Ylabel}[bc][bc]{$|\mathcal{C}|$}
\includegraphics[width=.45\textwidth, height=.3\textwidth]{abscnum100all}
\caption{
Plots of $|\mathcal{C}|$ for varying lengths of the spin chain,
ranging from $N=3$ to $N=11$ (odd numbers only). The longer the
spin chain, the more sharply peaked is the graph. The number of
vertices on the circuit is kept constant, at $\mathcal{N} = 100$.
}\label{fig:varyspinnumber}
\end{figure}
An analogous plot is shown in Fig.~\ref{fig:varypointnumber}, but
this time for a fixed number of spins $N$ and a varying number of
circuit vertices $\mathcal{N}$.
\begin{figure}[htp]
\centering
\psfrag{Xlabel}[tc][tc]{$(J-J_c)/r$}
\psfrag{Ylabel}[bc][bc]{$|\mathcal{C}|$}
\psfrag{a}[tl][tl]{$\mathcal{N} = 300$}
\psfrag{b}[tl][tl]{$\mathcal{N} = 100$}
\includegraphics[width=.45\textwidth, height=.3\textwidth]{N3allabscnumA}
\caption{
Plots of $|\mathcal{C}|$ for varying numbers of vertices on the
circuit, ranging from $\mathcal{N}=100$ to $\mathcal{N}=300$ (in
steps of $50$). The more vertices there are on the circuit, the
more sharply peaked is the graph. The spin chain length is kept
constant, at just $N = 3$. Note that qualitatively nothing changes
as we increase the chain length (tested up to $N=13$).
}\label{fig:varypointnumber}
\end{figure}
Of course, there is a perfectly plausible explanation for these
results: as the circuit passes over the critical point the ground
state changes abruptly, so that the overlap of the ground states
on either side of the critical point will be rather small in
magnitude. This in turn forces the magnitude of the product of
overlaps along the circuit in Eq.~\ref{eq:phasedefn2} to be small.
In fact, similar results were observed in a work by Zanardi and
Paunkovi$\acute{\text{c}}$~\cite{Zanardi}, who concluded that the
ground state overlap function is itself a good characterization of
QPTs.
\subsection{`Speed' of the Bargmann Invariant}
Another characteristic that is sensitive to the phase transition
is the `speed' with which the Bargmann invariant changes as a
function of the coupling parameter, which we define by $v_s :=
|\mathcal{C}_{s+1}-\mathcal{C}_s|/(J_{s+1} - J_s)$. As is evident
from Fig.~\ref{fig:speed}, the speed picks up markedly in the
vicinity of the critical point. Again, this phenomenon is a direct
result of the rapid change of the ground state near a critical
point.
\begin{figure}[htp]
\centering
\psfrag{Xlabel}[tc][tc]{$(J-J_c)/r$}
\psfrag{Ylabel}[bc][bc]{\text{`speed'} $v$}
\includegraphics[width=.45\textwidth, height=.3\textwidth]{3N100speed}
\caption{
Plot of the `speed', $v_s :=
|\mathcal{C}_{s+1}-\mathcal{C}_s|/(J_{s+1} - J_s)$, as a function
of the spin coupling strength. The spin chain length is $N=3$ and
the circuit is composed of $\mathcal{N} = 100$ vertices.
}\label{fig:speed}
\end{figure}
\section{Results for the Spin-$1$ Chain}
\label{section:spinone}
The results presented in Fig.~\ref{fig:spinone} derive from
simulations that are entirely analogous to those that gave rise to
Fig.~\ref{fig:spinonehalf}, but they refer to the spin-$1$ chain
model, rather than the spin-$1/2$ model, Eq.~\ref{eq:generalham}.
\begin{figure}[htb]
\centering
\subfigure{\label{subfig:5cnum}%
\includegraphics[width = .19\textwidth]{spinone5N100cnumclipped}
}\hspace{.25cm}%
\subfigure{\label{subfig:7cnum}
\includegraphics[width = .19\textwidth]{spinone7N100cnumclipped}
}\\[0.2cm]%
\psfrag{Xlabel}[tc][tc]{$(J-J_c)/r$}
\psfrag{Ylabel}[bc][bc]{$\varphi/\pi$}
\subfigure{\label{subfig:5phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{spinone5N100ppclipped}
}\hspace{.5cm}%
\subfigure{\label{subfig:7phase}
\includegraphics[width=.1815\textwidth,height=.1727\textwidth]
{spinone7N100ppclipped}
}\\[0.2cm]%
\subfigure{\label{subfig:5joinedphase}
\includegraphics[width=.2\textwidth,height=.18\textwidth]
{spinone5N100ppcontclipped}
}\hspace{.25cm}%
\subfigure{\label{subfig:7joinedphase}
\includegraphics[width=.2\textwidth,height=.18\textwidth]
{spinone7N100ppcontclipped}
}\\[0.2cm]%
\psfrag{Ylabel}[bc][bc]{$|\mathcal{C}|$}
\subfigure{\label{subfig:5abscnum}
\includegraphics[width=.18\textwidth,height=.17\textwidth]
{spinone5N100abscnumclipped}
}\hspace{.7cm}%
\subfigure{\label{subfig:7abscnum}
\includegraphics[width=.18\textwidth,height=.17\textwidth]
{spinone7N100abscnumclipped}
}\\[0.2cm]%
\caption{
These plots depict the Bargmann invariants for spin-$1$ chains of
lengths $N=5$ (left-hand column) and $N=7$ (right-hand column),
and allow for direct comparison and contrast with the spin-$1/2$
case of Fig.~\ref{fig:spinonehalf}. For a detailed description of
the results the reader is kindly referred to the text in
section~\ref{section:spinone}.
}
\label{fig:spinone}
\end{figure}
The Bargmann invariants are shown for spin-$1$ chains of lengths
$N=5$ (left-hand column) and $N=7$ (right-hand column), and allow
for direct comparison and contrast with the spin-$1/2$ case of
Fig.~\ref{fig:spinonehalf}. The circuit is composed of
$\mathcal{N} = 100$ vertices. Again, the uppermost row shows the
trajectory traced out in the complex plane by the Bargmann
invariant as the spin coupling strength is varied from $J_c - 2r$
to $J_c + 2r$. The second row from the top shows the corresponding
Bargmann phase; notice that points in the immediate vicinity of
the critical point have been disregarded as the phase of a complex
number becomes increasingly prone to error as the origin of the
complex plane is approached. The next row pictures the overall
`extent' of the phase, with the individual pieces of the graph
having been joined up end-to-end, as described in
section~\ref{section:Results}. Finally, the magnitude of the
Bargmann invariant is shown, and is clearly seen to vanish in the
vicinity of the critical point.
Qualitatively, the results for spin-$1$ chains are very similar to
those of spin-$1/2$ chains. In particular, the magnitude of the
Bargmann invariant still represents a signature for the critical
point. Of note is also the fact that the `joined-up' Bargmann
phase extends twice as high for spin-$1$ chains as it does for
spin-$1/2$ chains, for the same number of particles. Again, this
can be understood from Berry's solid angle result for
non-interacting particles, where the geometric phase is
proportional to the spin dimension.
\section{Summary and Discussion}
\label{section:Summary}
In summary, we have shown how the magnitude and `speed' of the
Bargmann invariant can act as a signature of the critical point for
the finite one-dimensional spin chain model considered. This stands
in stark contrast to the Bargmann phase, which is oblivious to the
phase transition as long as the spin chain length remains finite.
It is conjectured that the signatures presented here in the
context of a specific spin chain model will uphold their validity
also in a wider, more general setting. It would be worthwhile to
conduct experimental investigations with a view to testing this
conjecture; similarly, any theoretical proof of the proposed
conjecture would be of considerable interest.
Finally, it is worth stressing that the observed signatures of
criticality are very likely a direct consequence of the finite
`speed' with which the circuit is traversed in parameter space,
i.e. a result of the discrete nature of the circuit. No matter how
finely spaced are the neighbouring vertices on the circuit, the
adiabatic approximation will always break down in the immediate
vicinity of the critical point, causing Landau-Zener tunnelling
effects~\cite{LandauZener} to take centre stage. In the limit of a
smooth, continuous circuit, the observed signatures of criticality
would likely disappear altogether.
\begin{acknowledgments}
The authors would like to acknowledge discussions with K.
Audenaert at early stages of this project, as well as helpful
comments by D. Gross and K. Kieling. We thank L. Reuter for her
invaluable help and patience in creating Fig.~\ref{fig:Bloch2} of
this paper.
This work was funded in part by Hewlett-Packard Ltd. via an EPSRC
CASE award, by QIP IRC with support from the EPSRC (GR/S82176/0),
by the EU Integrated Project Qubit Applications (QAP), which is
funded by the IST directorate as contract no. 015848, by the
Alexander von Humboldt Foundation and the The Royal Society.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,480 |
#ifndef HW_S390_SCLP_H
#define HW_S390_SCLP_H
#include <hw/sysbus.h>
#include <hw/qdev.h>
#define SCLP_CMD_CODE_MASK 0xffff00ff
/* SCLP command codes */
#define SCLP_CMDW_READ_SCP_INFO 0x00020001
#define SCLP_CMDW_READ_SCP_INFO_FORCED 0x00120001
#define SCLP_READ_STORAGE_ELEMENT_INFO 0x00040001
#define SCLP_ATTACH_STORAGE_ELEMENT 0x00080001
#define SCLP_ASSIGN_STORAGE 0x000D0001
#define SCLP_UNASSIGN_STORAGE 0x000C0001
#define SCLP_CMD_READ_EVENT_DATA 0x00770005
#define SCLP_CMD_WRITE_EVENT_DATA 0x00760005
#define SCLP_CMD_WRITE_EVENT_MASK 0x00780005
/* SCLP Memory hotplug codes */
#define SCLP_FC_ASSIGN_ATTACH_READ_STOR 0xE00000000000ULL
#define SCLP_STARTING_SUBINCREMENT_ID 0x10001
#define SCLP_INCREMENT_UNIT 0x10000
#define MAX_AVAIL_SLOTS 32
#define MAX_STORAGE_INCREMENTS 1020
/* CPU hotplug SCLP codes */
#define SCLP_HAS_CPU_INFO 0x0C00000000000000ULL
#define SCLP_CMDW_READ_CPU_INFO 0x00010001
#define SCLP_CMDW_CONFIGURE_CPU 0x00110001
#define SCLP_CMDW_DECONFIGURE_CPU 0x00100001
/* SCLP response codes */
#define SCLP_RC_NORMAL_READ_COMPLETION 0x0010
#define SCLP_RC_NORMAL_COMPLETION 0x0020
#define SCLP_RC_SCCB_BOUNDARY_VIOLATION 0x0100
#define SCLP_RC_INVALID_SCLP_COMMAND 0x01f0
#define SCLP_RC_CONTAINED_EQUIPMENT_CHECK 0x0340
#define SCLP_RC_INSUFFICIENT_SCCB_LENGTH 0x0300
#define SCLP_RC_STANDBY_READ_COMPLETION 0x0410
#define SCLP_RC_INVALID_FUNCTION 0x40f0
#define SCLP_RC_NO_EVENT_BUFFERS_STORED 0x60f0
#define SCLP_RC_INVALID_SELECTION_MASK 0x70f0
#define SCLP_RC_INCONSISTENT_LENGTHS 0x72f0
#define SCLP_RC_EVENT_BUFFER_SYNTAX_ERROR 0x73f0
#define SCLP_RC_INVALID_MASK_LENGTH 0x74f0
/* Service Call Control Block (SCCB) and its elements */
#define SCCB_SIZE 4096
#define SCLP_VARIABLE_LENGTH_RESPONSE 0x80
#define SCLP_EVENT_BUFFER_ACCEPTED 0x80
#define SCLP_FC_NORMAL_WRITE 0
/*
* Normally packed structures are not the right thing to do, since all code
* must take care of endianness. We cannot use ldl_phys and friends for two
* reasons, though:
* - some of the embedded structures below the SCCB can appear multiple times
* at different locations, so there is no fixed offset
* - we work on a private copy of the SCCB, since there are several length
* fields, that would cause a security nightmare if we allow the guest to
* alter the structure while we parse it. We cannot use ldl_p and friends
* either without doing pointer arithmetics
* So we have to double check that all users of sclp data structures use the
* right endianness wrappers.
*/
typedef struct SCCBHeader {
uint16_t length;
uint8_t function_code;
uint8_t control_mask[3];
uint16_t response_code;
} QEMU_PACKED SCCBHeader;
#define SCCB_DATA_LEN (SCCB_SIZE - sizeof(SCCBHeader))
/* CPU information */
typedef struct CPUEntry {
uint8_t address;
uint8_t reserved0[13];
uint8_t type;
uint8_t reserved1;
} QEMU_PACKED CPUEntry;
typedef struct ReadInfo {
SCCBHeader h;
uint16_t rnmax;
uint8_t rnsize;
uint8_t _reserved1[16 - 11]; /* 11-15 */
uint16_t entries_cpu; /* 16-17 */
uint16_t offset_cpu; /* 18-19 */
uint8_t _reserved2[24 - 20]; /* 20-23 */
uint8_t loadparm[8]; /* 24-31 */
uint8_t _reserved3[48 - 32]; /* 32-47 */
uint64_t facilities; /* 48-55 */
uint8_t _reserved0[100 - 56];
uint32_t rnsize2;
uint64_t rnmax2;
uint8_t _reserved4[120-112]; /* 112-119 */
uint16_t highest_cpu;
uint8_t _reserved5[128 - 122]; /* 122-127 */
struct CPUEntry entries[0];
} QEMU_PACKED ReadInfo;
typedef struct ReadCpuInfo {
SCCBHeader h;
uint16_t nr_configured; /* 8-9 */
uint16_t offset_configured; /* 10-11 */
uint16_t nr_standby; /* 12-13 */
uint16_t offset_standby; /* 14-15 */
uint8_t reserved0[24-16]; /* 16-23 */
struct CPUEntry entries[0];
} QEMU_PACKED ReadCpuInfo;
typedef struct ReadStorageElementInfo {
SCCBHeader h;
uint16_t max_id;
uint16_t assigned;
uint16_t standby;
uint8_t _reserved0[16 - 14]; /* 14-15 */
uint32_t entries[0];
} QEMU_PACKED ReadStorageElementInfo;
typedef struct AttachStorageElement {
SCCBHeader h;
uint8_t _reserved0[10 - 8]; /* 8-9 */
uint16_t assigned;
uint8_t _reserved1[16 - 12]; /* 12-15 */
uint32_t entries[0];
} QEMU_PACKED AttachStorageElement;
typedef struct AssignStorage {
SCCBHeader h;
uint16_t rn;
} QEMU_PACKED AssignStorage;
typedef struct SCCB {
SCCBHeader h;
char data[SCCB_DATA_LEN];
} QEMU_PACKED SCCB;
typedef struct sclpMemoryHotplugDev sclpMemoryHotplugDev;
#define TYPE_SCLP_MEMORY_HOTPLUG_DEV "sclp-memory-hotplug-dev"
#define SCLP_MEMORY_HOTPLUG_DEV(obj) \
OBJECT_CHECK(sclpMemoryHotplugDev, (obj), TYPE_SCLP_MEMORY_HOTPLUG_DEV)
struct sclpMemoryHotplugDev {
SysBusDevice parent;
ram_addr_t standby_mem_size;
ram_addr_t padded_ram_size;
ram_addr_t pad_size;
ram_addr_t standby_subregion_size;
ram_addr_t rzm;
int increment_size;
char *standby_state_map;
};
static inline int sccb_data_len(SCCB *sccb)
{
return be16_to_cpu(sccb->h.length) - sizeof(sccb->h);
}
void s390_sclp_init(void);
sclpMemoryHotplugDev *init_sclp_memory_hotplug_dev(void);
sclpMemoryHotplugDev *get_sclp_memory_hotplug_dev(void);
void sclp_service_interrupt(uint32_t sccb);
void raise_irq_cpu_hotplug(void);
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,355 |
Камень — потухший стратовулкан, расположенный в Восточном вулканическом поясе полуострова Камчатка.
Общая информация
Находится в центральной части полуострова Камчатка, рядом с Ключевским вулканом. Около вулкана расположены ледник Шмидта и ледник Богдановича. Входит в восточный вулканический пояс. Расположен в Ключевской группе вулканов.
При высоте — 4585 м над уровнем моря является вторым по высоте вулканом на Камчатке.
Дата последнего извержения неизвестна, вулканологи предполагают, что вулкан потух в конце плейстоцена. После завершения вулканической деятельности склоны вулкана были сильно разрушены многочисленными обвалами. Породы слагающие его, представлены переслаивающимися лавами андезито-базальтового состава с вкрапленниками пироксена и плагиоклаза.
Восхождения на вершину вулкана Камень совершаются с западной стороны и являются в силу крутизны склонов исключительно альпинистским мероприятием.
Примечания
Ссылки
о вулкане в каталоге метаданных ИВиС ДВО РАН
Вулкан Камень на краеведческом сайте о Камчатке.
Видеонаблюдение за вулканом.
Топографическая карта
Вулканы Камчатского края
Стратовулканы
Вулканы-четырёхтысячники | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,990 |
\section{Introduction}
In spite of intense theoretical and experimental exploration over
the past years, many features of one-dimensional strongly interacting systems have
not yet been clarified. For example, charge fractionalization, which has been observed in fractional quantum Hall effect(FQHE) edge states\cite{glattli}, has not been observed directly in carbon nanotubes. Furthermore, though it is believed
\cite{shotproposal,safiprl01,eunah,chetan} that the statistics of LL quasiparticles can be
measured in various setups, actual experiments addressing this
issue are much fewer \cite{statexp}.
Shot noise is a powerful tool to extract information about the charge and statistics of the elementary excitations of a system. In particular, it is believed that high frequency noise contains important information about the statistics of quasiparticles \cite{eunah,chetan}, as well as about the charge fractionalization in systems such as carbon nanotubes, where charge fractionalization is masked at zero frequency by the metallic leads \cite{dolcini,ines}.
While some high frequency noise measurements have been performed for diffusive conductors \cite{sch}, the relevant range of frequencies was until recently too high and out of the experimental reach for one-dimensional systems; with new experiments being developed \cite{deblock,glattli-new}, this range of frequencies will likely become experimentally accessible in the near future. Moreover, these groups will have access to the non-symmetrized noise, i.e. both the emission and the absorption portions of the noise. The emission noise (the non-symmetrized noise at positive frequencies) quantifies the amount of photons of specific frequencies which are emitted by the fluctuating system, while the absorption noise (the non-symmetrized noise at negative frequencies) measures the amount of photons emitted by an active detector that can be absorbed by the fluctuating system. In general the emission noise is non-zero only when energy is put into the system (e.g. by applying an external voltage, or working at finite temperature), and only for frequencies smaller than the applied voltage/temperature, consistent with energy conservation; however there is no such constraint on the absorption noise.
High-frequency symmetrized noise at zero temperature has been addressed theoretically \cite{chamon1,saleur,ffnoise}. Similarly, some aspects of the non-symmetrized noise have been addressed theoretically for non-interacting systems \cite{deblock} and for situations in which the effects of interactions can be taken into account perturbatively \cite{hekking}. However, non-symmetrized noise in one-dimensional interacting systems has not been previously analyzed theoretically.
Here we compute the high-frequency finite-temperature non-symmetrized noise in a FQHE sample at filling factors of a simple fraction $\nu=1/2n+1$ form, and we compare the results to the symmetrized noise. Most of our analysis is done for the weak backscattering limit, but we also compute and discuss briefly the opposite strong-backscattering limit.
The symmetrized noise is proportional to the Fourier transform of the expectation value of the anticommutator of two current fluctuation operators $\langle \{\Delta j(y,0),\Delta j(x,t)\} \rangle$; the non-symmetrized noise is proportional to the Fourier transform of $\langle \Delta j(y,0) \Delta j(x,t) \rangle$.
We should stress that we can obtain access to both the quantum
regime when the frequency is much larger than the temperature, and the classical regime, in which the frequency is much smaller than the temperature. We focus on \\
$\bullet$ the auto-correlations (noise) in the total current flowing trough the system,
\\
$\bullet$ auto-correlations and cross-correlations of individual currents flowing trough the four terminals of a FQHE sample (see Fig.~\ref{sample}).
The auto-correlations of the total current, as well as of individual branches contain two types of noise:
\\
$\bullet$ noise in the absence of backscattering (which is also independent of voltage).
\\
$\bullet$ backscattering-induced noise (which is the difference between the noise in
the presence and in the absence of backscattering).
The cross-correlations between the outgoing right-movers and the
outgoing left-movers are solely backscattering-induced. In our
analysis we will focus mainly on the backscattering-induced
component of the noise, as it probes the very nature of
quasiparticles.
When we study backscattering-induced noise we also need to
distinguish between
\\
$\bullet$ zero voltage noise,
\\
$\bullet$ the excess noise, which is the difference between the noise at a finite voltage and the noise at zero voltage. Since the noise in the absence of backscattering is independent of voltage, this is the same as the difference between the backscattering-induced noise at a finite voltage and the backscattering-induced noise at zero voltage.
We focus first on the auto-correlations of chiral outgoing
branches, and the cross-correlations between outgoing branches
with opposite chiralities. We find that these are entirely
even in frequency, and also do not depend on the distance
between the point where the current is measured and the
backscattering site. The most striking feature observed is a
singularity at the Josephson frequency (JF) $\omega_0=\nu e V/\hbar$
(proportional to the applied voltage $V$, and to
the filling factor $\nu$ equal to the fractional charge $g$). This feature appears as a
cusp in a non-interacting system ($\nu=1$) such that the noise
decreases linearly from $2 g e I_B$ at zero frequency to zero at
the JF. For interacting systems with $g<1/2$, for example for
$g=\nu=1/3$ the noise exhibits an inverse power-law divergence at
the JF, and decays to zero for frequencies larger than the JF. At
finite temperature the singularity is rounded-off, and the noise
exhibits a peak slightly below the JF. The position and the width
of the peak, as well as the manner in which the noise decays to
zero above the JF depend on temperature: at low
temperature the peak is very sharp and close to the JF, but when
the temperature is increased the peak moves to lower frequencies,
widens and disappears completely for temperatures comparable to
the applied voltage. The zero temperature limit of our results is consistent
with the behavior of the symmetrized cross-correlations obtained in Ref. \cite{chamon1}.
Next we analyze the dependence on frequency of the non-symmetrized
noise in the total current. We focus first on the
situation in which the distance between the point where the
current is measured and the backscattering site is much smaller than
$v_F /\nu \omega$, where $v_F$ is the Fermi velocity, for all
frequencies $\omega$ probed experimentally. We find that the
emission noise is roughly equal to the noise in the outgoing
chiral branches, however the absorption noise exhibits a positive
peak slightly below the JF, and a negative dip slightly above the
JF. The average of the two, which is the symmetrized noise, is
similar in structure to the absorption noise.
Another important quantity that we study is the non-symmetrized excess noise (the difference between the non-symmetrized noise at finite and zero voltage). Consistent with previous findings \cite{deblock} we show that this is even for non-interacting electrons. However we find that it becomes non-symmetric in the presence of interactions. This is a signature of Luttinger liquid physics which will probably exist also in other one-dimensional systems such as carbon nanotubes even in the presence of metallic leads.
If the distance $x$ between the backscattering site and the measuring point
is significant, oscillations with a
period $2 \pi v_F/ \nu x$ in the noise dependence on frequency also appear. Such oscillations do
not occur in the auto-correlations of individual branches or in
cross-correlations of outgoing branches, but are manifest in the
absorption noise and in the symmetrized noise of the total current. If an
average over the position of the measuring point is performed,
both the non-symmetrized and the symmetrized noise are reduced to
the form of the individual branch correlations described
previously.
We treat the edges of the FQHE as infinite chiral Luttinger liquids, and we use a perturbative Schwinger-Keldysh formalism \cite{Keldysh} to compute the backscattered current and the noise up to second order in the backscattering amplitude.
In section 2 we present the basics of the mathematical formalism used to calculate the noise. Some of the details of the calculation are outlined in Appendices A and B. In Section 3 we present our results for the non-symmetrized noise and a comparison with the symmetrized noise. In section 4 we discuss our results in comparison with the Landauer-B\"uttiker approach, we discuss also the correspondence with the generalized Kubo formula and with the fluctuation-dissipation theorem, as well as the results for the high-frequency non-symmetrized noise in the tunneling limit. We conclude in section 5.
\section{Formalism}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{sample.eps}
\vspace{0.15in} \caption{\small Schematics of the sample}
\label{sample}
\end{center}
\end{figure}
We model the edges of the FQHE as an infinite Luttinger liquid described by the Hamiltonian
\begin{equation}
{\mathcal{H}} ={\mathcal{H}}_{0} \, + \, {\mathcal{H}}_{B}, \label{L}
\end{equation}
where ${\mathcal{H}}_{0}$ describes the interacting one-dimensional system, and ${\mathcal{H}}_{B}$ accounts for the backscattering.
Explicitly, we have
\begin{eqnarray}
{\mathcal{H}}_0 &=&\frac{\hbar v_F}{2} \int_{-\infty}^{\infty}
dx \left[ \Pi^2 + \frac{1}{g^2}
(\partial _x\Phi )^2\right] \\
{\mathcal{H}}_B &=& \Gamma e^{i \sqrt{4 \pi} \Phi(x_0,t)+2 i k_F
x_0}+ h.c.
\label{l0}
\end{eqnarray}
Here, $\Phi(x,t)=\Phi_R(x,t)+\Phi_L(x,t)$ is the standard Bose field operator in
bosonization and $\Pi(x,t)=\partial_t \Phi(x,t)/v_F$ is its conjugate momentum density.
The parameter $g$ is the value of the fractional charge of the free excitations of the model; for the FQHE it has been shown \cite{fqhe} that $g$ is equal to the filling factor $\nu$. Also $k_F$ is the Fermi wavelength, and $x_0$ is the position of the backscattering site; for simplicity we will set $x_0$ to zero in our calculations.
In bosonization, the current operators are related to the bosonic
chiral fields $\Phi_{R/L}$ through
\begin{equation}
j_{R/L} (x,t) = \frac{e}{\sqrt{\pi}} \partial_t
\Phi_{R/L}(x,t) \; \label{current} \\
\end{equation}
and
\begin{equation}
j(x,t)=j_R(x,t)+j_L(x,t).
\end{equation}
The presence of an applied voltage can be taken into account by the shift in the tunneling operator such that $\Gamma\rightarrow\Gamma e^{i \omega_0 t}$, where $\omega_0=g e V/\hbar$ is the Josephson frequency associated with the applied voltage.
The finite frequency non-symmetrized noise $S^1_{a b}(x,y,\omega)$ is defined as
\begin{eqnarray}
S^1_{a b}(x,y,\omega) \equiv 2 \int_{-\infty}^{\infty} dt~e^{i\omega t}
\left\langle \Delta j_a (y,0) \Delta j_b(x,t)
\right\rangle \; , \label{noise}
\end{eqnarray}
where $a,b$ stand for the right/left moving indices, and $\Delta
j_a(x,t) = j_a(x,t) - \langle j_a(x,t) \rangle$ is the current
fluctuation operator at position $x$ and time $t$.
For positive frequencies, Eq. (\ref{noise}) corresponds to emission noise, and for negative frequencies it corresponds to absorption noise. The noise in the total current is given by
\begin{equation}
S^1(x,y,\omega)= S^1_{RR}(x,y,\omega)+S^1_{LL}(x,y,\omega)+S^1_{LR}(x,y,\omega)+S^1_{RL}(x,y,\omega).
\end{equation}
The symmetrized noise can be obtained by symmetrizing the above with respect to frequency, i.e.
\begin{equation}
S^{0}_{a b}(x,y,\omega)=[S^1_{a b}(x,y,\omega) +S^1_{ba}(y,x,-\omega)]/2,
\end{equation}
and similarly for the noise in the total current.
The calculation of the symmetrized and of the non-symmetrized noise is performed using the non-equilibrium Schwinger-Keldysh formalism\cite{Keldysh}; the details are presented in Appendices A and B.
\section{Results}
We find the noise in the absence of backscattering to be
\begin{eqnarray}
s^{1}_{a b}(x,y,\omega)&=&
\delta_{a b}\frac{2 e^2 \omega^2}{\pi}\tilde{\cal C}^{+-}_{a}(x,y,\omega)\label{s01}
\\
s^{0}_{a b}(x,y,\omega)&=&
\delta_{a b}\frac{2 e^2 \omega^2}{\pi}\tilde{\cal C}^{\cal K}_{a}(x,y,\omega)
\end{eqnarray}
or equivalently, using the Keldysh transformations presented in the Appendices,
\begin{eqnarray}
s^{\alpha}_{a b}(x,y,\omega)
&=&\delta_{a b}\frac{e^2 \omega^2}{\pi} \{\tilde{\cal C}^{\cal K}_{a}(x,y,\omega)+\alpha [\tilde{\cal C}^{\cal A}_a(x,y,\omega)-
\tilde{\cal C}_a^{\cal R}(x,y,\omega)]\}
\label{seq}
\end{eqnarray}
Since our formulas will apply both for symmetrized and non-symmetrized noise, we will use the index $\alpha$ to distinguish between them, such that $\alpha=0$ for
the symmetrized noise and $\alpha=1$ for the non-symmetrized noise. Also $\tilde{\cal C}^{+-}$ is the Fourier transform of the two point correlator which is given by ${\cal C}^{+-}_{a b}(x,0;y,t)= \langle \Phi_a(y,t) \Phi_b(x,0) \rangle$. We can see that Eq.(\ref{s01}) is consistent with
the definitions from Eq.(\ref{current}).
The $\tilde{\cal C}^{{\cal R},{\cal A},{\cal K}}$ are the corresponding Fourier transforms of the retarded, advanced and symmetric Green's functions of the system in the absence of backscattering. They are presented in detail in Appendix B, and we can write
\begin{eqnarray}
&&\tilde{\mathcal{C}}^{\cal A}_{L/R}(x,y,\omega)=-\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \theta[\pm (x-y)] \nonumber \\&&
\tilde{\mathcal{C}}^{\cal R}_{L/R}(x,y,\omega)=\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \theta[\mp (x-y)] \nonumber \\&&
\tilde{\mathcal{C}}^{\cal K}_{L/R}(x,y,\omega)=\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big).
\label{gf}
\end{eqnarray}
For simplicity we present two situations, $x=y>0$ (the currents
are measured at the same position), and $x=-y>0$ (the currents are
measured at equal and opposite distances from the backscattering site). More general
situations can be easily obtained from the formalism presented in
the Appendices.
For $x=y>0$ we can write for the noise in the absence of backscattering:
\begin{eqnarray}
&&s^{\alpha}_{R R}(x,x,\omega)=s^{\alpha}_{L
L}(x,x,\omega)=\frac{g \omega e^2}{2 \pi}
\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] \end{eqnarray}
for the chiral branches, with the cross-correlations
between the right and the left-movers vanishing in the absence of backscattering: $s^{\alpha}_{R
L}(x,-x,\omega)=0$. For the total current we find \begin{eqnarray}
&&s^{\alpha}(x,x,\omega)=\frac{g \omega e^2}{\pi}
\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big].
\label{seqt} \end{eqnarray}
As also described in the Introduction, the total
noise is the sum of the noise in the absence of backscattering and of
the backscattering-induced noise: $S^{\alpha}(x,y,\omega)=s^{\alpha}(x,y,\omega)+\delta
S^{\alpha}(x,y,\omega)$.
Following the treatment described in Appendix A, using the same notations and following similar lines with Ref. \cite{dolcini},
we find that
the backscattering-induced components of the correlations between
chiral currents can be written as:
\begin{equation}
\delta S^{\alpha}_{a b}
(x,y,\omega)=S^A_{ a b}(x,y,\omega)+S^C_{a b}(x,y,\omega)+\alpha S^N_{a b}(x,y,\omega),
\label{qw}
\end{equation}
where again $\alpha=0$ for the symmetrized noise and $\alpha=1$ for the non-symmetrized noise.
Here we have
\begin{eqnarray}
S^A_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi} \tilde{\cal C}^{\cal R}_{a}(x,0,\omega)
\tilde{\cal C}^{\cal R}_{b}(y,0,-\omega)f_A(\omega),\label{n1} \\
S^C_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi}[ \tilde{\cal C}^{\cal K}_{a}
(x,0,\omega)\tilde{\cal C}^{\cal R}_{b}(y,0,-\omega)f_C(-\omega)
\nonumber \\&&
+\tilde{\cal C}^{\cal R}_{a}(x,0,\omega)\tilde{\cal C}^{\cal K}_{b}
(y,0,-\omega)f_C(\omega)],\label{n2}\\
S^N_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi}[\tilde{\cal C}^{\cal A}_{a}(x,0,\omega)
\tilde{\cal C}^{\cal R}_b(y,0,-\omega)f_C(-\omega)
\label{n3} \\&& -
\tilde{\cal C}^{\cal R}_a(x,0,\omega)\tilde{\cal C}^{\cal A}_b(y,0,-\omega)f_C(\omega)].
\nonumber
\end{eqnarray}
The functions $f_A(\omega)$ and $f_C(\omega)$ are given by
\begin{equation}
f_A(\omega) = \int_{-\infty}^\infty dt \, e^{i \omega t}
\left\langle \left\{ \Delta j_B(t), \Delta j_B(0) \right\}
\right\rangle , \label{s_A}
\end{equation}
where $\Delta j_B(t) = j_B(t) - \langle j_B(t)
\rangle$. The expectation values are performed with respect to the full action.
Also, $j_B(t)$ is the backscattering
current operator at the backscattering site
\begin{equation}
j_{B}(t) = - \frac{e}{\hbar} \frac{\delta
\mathcal{H}_B(\Phi)}{\delta \Phi(0,t)} \; \label{ib_def} .
\end{equation}
Similarly
\begin{equation}
f_C(\omega)= \int_0^\infty dt \left( e^{i \omega t}-1 \right)
\left\langle \left[ j_B(t),j_B(0) \right]
\right\rangle \; . \label{s_C}
\end{equation}
In this notation, the emission noise is the non-symmetrized noise taken at positive
frequencies, while the absorption noise is the non-symmetrized
noise at negative frequencies.
We note that by summing up the chiral components of the noise to obtain the noise in the total current,
we recover the same structure for the symmetrized noise as the one presented in Ref.\cite{dolcini}.
Up to this point the calculation is non-perturbative and we can analyze a few {\it non-perturbative}
features of our results.
For example we note that if one studies the case $x=y>0$,
the fluctuations in the chiral currents will be independent of position. Similarly
for the situation $x=-y>0$, the
cross-correlations between the outgoing right-movers and the
outgoing left-movers are spatially independent. This is because the noise terms described above depend on position only through the free propagation of a chiral
moving mode from the first measuring point to the backscattering
site $\tilde{\mathcal{C}}_{L/R}(x,0,\omega)$, and through the free
propagation of another chiral mode from the backscattering site
to the second measuring point $\tilde{\mathcal{C}}_{L/R}(y,0,-\omega)$.
Thus the total phase accumulated during the propagation cancels
for $\delta S_{RR/LL}(x,x,\omega)$ and for $\delta S_{RL}(x,-x,\omega)$ which are position
independent. However other quantities, such as
the cross-correlations between the incoming and outgoing right-movers $\delta S_{RR}(x,-x,\omega)$,
are affected by interference effects, and are hence position dependent.
We also note that, as it can be seen from the above
non-perturbative formulas and from Eqs.(\ref{gf}), the chiral $\delta S^1_{RR}(x,x,\omega)$, and $\delta
S^1_{RL}(x,-x,\omega)$ current correlations are equal to each other, and also even in frequency
(independent of $\alpha$ - the emission
noise is equal to the absorption noise). However $\delta
S^1_{RR}(x,-x,\omega)$ and the backscattering noise in the total
current $\delta S^1(x,x,\omega)$ are non-symmetric (dependent on $\alpha$ - the emission
noise is different from the absorption noise).
\subsection{Perturbative results}
Using Eq.(\ref{qw}) we can now evaluate our results perturbatively up to second order in $\Gamma$ to find
\begin{eqnarray} &&\delta S^{\alpha}_{R
R}(x,x,\omega)= \delta S^{\alpha}_{R L}(x,-x,\omega) \nonumber
\\&& =g e \sum_{m=\pm 1}\Big\{\coth\Big[\frac{\hbar (\omega+m
\omega_0)}{2 k_B T}\Big]-\coth\Big(\frac{\hbar \omega}{2 k_B
T}\Big)\Big\} I_B(\omega+m \omega_0). \label{crosseq1} \end{eqnarray}
Similarly
\begin{equation} \delta S^{\alpha}_{R R/L L}(x,-x,\omega)= \mp
\frac{g^2}{4 \pi}\Big[\coth\Big(\frac{\hbar \omega}{2 k_B
T}\Big)-\alpha\Big]f_C(\pm \omega) e^{\pm 2 i g \omega x/v_F}
\label{crosseq11} \end{equation}
The other cross-correlators are evaluated in Appendix A in Eq.(\ref{qet}). Using the above formulas and
Eq.(\ref{qet}), we find the noise in the total current to be:
\begin{eqnarray}
&&\delta S^{\alpha}(x,x,\omega)=g e \sum_{m=\pm
1}\Big\{\coth\Big[\frac{\hbar (\omega+m \omega_0)}{2 k_B
T}\Big]-\coth\Big(\frac{\hbar \omega}{2 k_B
T}\Big)\Big\}I_B(\omega+m \omega_0)
\\&&
-\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]\Big\{
\sum_{m=\pm 1}I_B(\omega+m \omega_0)\cos\Big( \frac{2 g \omega x}{v_F}\Big)
+ i \frac{g^2}{4 \pi} \sin\Big(\frac{2 g \omega x}{v_F}\Big)[f_C(\omega)+f_C(-\omega)]\Big\} \nonumber
\label{crosseq}
\end{eqnarray}
where we can evaluate $f_C$ (presented in Eq.(\ref{s_C})) perturbatively:
\begin{eqnarray}
f_C(\omega)=4 \pi i \Big(\frac{e}{\hbar}\Big)^2 |\Gamma|^2 \int_0^\infty dt (e^{i \omega t}-1)
\cos(\omega_0 t) \Big[\frac{\pi t k_B T}{\hbar \sinh(\pi t k_B T/\hbar)}\Big]^{2 g} {\rm{Im}}\big[(1+i t \epsilon_h/\hbar)^{-2 g}\big].
\end{eqnarray}
Also
\begin{equation}
I_B(\Omega)= - \frac{g e}{\hbar^2} |\Gamma|^2 {\cal{F}}_g(\Omega)
\end{equation}
is the value of the backscattered current for an applied voltage equal to $\Omega$, with
\begin{eqnarray}
&&{\cal F}_g(\omega)=\int_0^{\infty} \sin(\omega t)\Big[\frac{\pi t k_B T}{\hbar \sinh(\pi t k_B T/\hbar)}\Big]^{2 g} {\rm{Im}}\big[(1+i t \epsilon_h/\hbar)^{-2 g}\big] dt.
\label{calf}
\end{eqnarray}
When taking into account $\hbar\omega\ll\epsilon_h$ this becomes
\begin{eqnarray}
&&{\cal F}_g(\omega) \approx -\sin(\pi g)\Big(\frac{\pi k_B T}{\epsilon_h}\Big)^{2 g} 2^{2 g} \int_0^{\infty} \sin(\omega t) \sinh^{-2 g}(\pi t k_B T/\hbar) dt
\nonumber \\
&&=i \sin(\pi g) \Big(\frac{\pi k_B T}{\epsilon_h}\Big)^{2 g} 2^{2 g-2} \Gamma(1-2 g)\Big[\frac{\Gamma(g-i \tilde{\omega})}{\Gamma(1-g-i \tilde{\omega})}-\frac{\Gamma(g+i \tilde{\omega})}{\Gamma(1-g+i \tilde{\omega})}\Big]
\end{eqnarray}
where $\tilde{\omega}=\hbar \omega/2 \pi k_B T$.
For $g=1/2$
\begin{equation}
{\cal F}_{1/2}(\omega)=-\frac{\pi \hbar}{2 \epsilon_h} \tanh\Big(\frac{\hbar \omega}{2 k_B T}\Big)
\end{equation}
while for $g=1$(non-interacting system),
\begin{equation}
{\cal F}_1(\omega)=-\frac{\pi \hbar^2}{2 \epsilon_h^2} \omega
\end{equation}
We can note directly from Eq. (\ref{crosseq}) that for frequencies much larger than the temperature, i.e in the quantum regime, the emission noise ($\omega >0$) is independent of position, and it decays to zero for frequencies slightly larger than the Josephson frequency.
We evaluate some simple limits ($g=1$, and $g=1/2$) to obtain:
\begin{eqnarray}
\frac{\delta S^{\alpha}}{e I_B}&=&\frac{1}{\omega_0}\Big\{\sum_{m=\pm1}(\omega+m \omega_0)\coth\Big(\frac{\hbar \omega+m \hbar \omega_0}{2 k_B T}\Big)
-2\omega \Big[2\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]\Big\}
\label{ni}
\end{eqnarray}
for $g=1$, where $I_B$ is the value of the backscattering current. The result for the symmetrized noise ($\alpha=0$) is in agreement with the Landauer-B\"uttiker formalism \cite{buttiker} which
finds that the noise dependence on frequency for an arbitrary amount of backscattering is:
\begin{equation}
S^0(\omega)=\frac{e^2}{2 \pi \hbar} \Big[{\cal T}(1-{\cal T})\sum_{m=\pm1}(\hbar \omega+m e V)\coth\Big(\frac{\hbar \omega+m e V}{2 k_B T}\Big)+2 {\cal T}\hbar \omega \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big) \Big],
\label{lb}
\end{equation}
where $\cal T$ is the transmission of the barrier.
We see that by taking the limit ${\cal T}\rightarrow1$ (weak backscattering), and expanding this result in $1-{\cal T}\propto |\Gamma|^2$
we retrieve the same behavior as that presented in Eq.(\ref{ni}).
For $g=1/2$, we obtain
\begin{equation}
\frac{\delta S^{\alpha}}{g e I_B}=\Big\{2-\Big[2 \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] \sum_{m=\pm1}\tanh \Big(\frac{\hbar \omega+m\hbar \omega_0}{2 k_B T}\Big)\Big\}\coth\Big(\frac{\hbar \omega_0}{2 k_B T}\Big)
\end{equation}
which for $\alpha=0$ (symmetrized noise) is consistent with the small $\Gamma$ limit of the exact calculation presented in Ref. \cite{chamon1} performed using a scattering approach.
We also note that the expansion of $I_B(x)$ around $x=0$ is of the form of $a x+ b x^3$.
Similarly, $I_B(x)\approx x^{2 g-1}$ for $x \gg 1$. This is consistent with the standard Luttinger liquid theory, where the current is linear with voltage for voltages much smaller than the temperature, and has a power-law dependence on voltage for voltages much larger than the temperature. Also, this implies that the zero-temperature inverse power-law divergences in the first term in $\delta S$ at the JF are rounded off quadratically at finite temperature. The second term in $\delta S$ is linear in the vicinity of the JF.
Before plotting our results we will also say a few words about the range of validity of the perturbative expansion. To ensure that the perturbative approach is valid we need to check that the backscattering-induced noise is much smaller than
the noise in the absence of backscattering. For frequencies far from the Josephson frequencies this translates into the criterion: $[max(e V, k_B T, \omega)/\epsilon_h]^{2-2 g} \gg |\Gamma|^2/\epsilon_h$, where $\epsilon_h$ is the high energy cutoff of the problem. For the regime when $\omega$ is comparable to $\omega_0$, the criterion is harder to write down, and we have to check that the height of the peaks near the Josephson frequency is sufficiently limited by the temperature.
In Fig.~\ref{sym} we plot the non-symmetrized
backscattering-induced correlations between the chiral currents
$\delta S^1_{R R}(x,x,\omega)=\delta S^1_{R L}(x,-x,\omega)$.
These correlations are $even$, and independent of position.
They also exhibit a singularity at the JF $\omega_0=g e V/\hbar$.
This is cusp-like for a non-interacting system ($\nu=1$). For
$g=1/2$ the noise has a step-like transition, while for $g=\nu=1/3$ the
noise exhibits a peak slightly below the JF, and falls to zero for
frequencies larger than the JF.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{crossc5.eps}
\caption{\small The backscattering-induced {\it non-symmetrized} cross-correlations between the outgoing right-movers and the outgoing left-movers
renormalized by $ g e I_B$ plotted as a function of frequency (in
units of $\omega_0$) for $g=\nu=1/3$ (full line), $g=\nu=1$
(dashed line), and $g=1/2$ (dotted line). The $g=1/2$ situation
does not correspond to any FQHE state, but is drawn here only for
comparison. We set $k_B T/\hbar \omega_0=0.03$. Note that the cross-correlations are even (the emission component is equal to the absorption component). Note also that this plot describes also the excess cross-correlation noise (the zero voltage cross-correlations vanish), and that in the absence of backscattering the cross-correlations are zero.} \label{sym}
\end{center}
\end{figure}
From Eq.(\ref{crosseq1}) we can see that in the absence of an applied voltage the correlators described above vanish (the backscattering-induced noise is equal to the excess noise). Also, while the fluctuations in the current of outgoing right-movers are non-zero in the absence of backscattering, there is no such component for the cross-correlations between the outgoing right-movers and the outgoing left-movers. Thus the cross-correlations are of great experimental relevance when we are interested in isolating the backscattering-induced noise; while for some of the noise quantities that can be measured in a FQHE four-terminal setup the contributions in the absence of backscattering (of order $1$) may mask the backscattering-induced noise (of order $|\Gamma|^2 \ll 1$), this is not the case for the cross-correlations.
We now present our results for the non-symmetrized noise in the total current.
For simplicity we start with the limit $g \omega x/v_F\ll1$ which is presented in Fig.~\ref{temp}. Consistent with Eq. (\ref{crosseq1}), we see that the emission noise is similar to the chiral current correlations presented above. At low temperature the emission noise vanishes for frequencies larger than the JF, consistent with the intuitive picture that no photon with an energy larger that the applied voltage can be emitted by the fluctuating system. While the emission noise behaves similar to the chiral current correlators, the absorption noise exhibits a positive peak slightly below the JF, and a negative dip slightly above the JF. The function connecting the negative and the positive resonances has a linear dependence of frequency when it passes through the JF. As depicted in Fig.~\ref{temp}, with increasing the temperature the peaks broaden and move away from the JF, to disappear for temperatures of the same order of magnitude
as the applied voltage.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{temp.eps}
\caption{\small The backscattering non-symmetrized noise in the total current $\delta S^1(x,x,\omega)$ renormalized
by $ g e I_B$ plotted as a function of frequency in units of
$\omega_0$ for $g=\nu=1/3$ and different temperatures $k_B T/\hbar \omega_0=0.01$(full line), $k_B T/\hbar \omega_0=0.1$ (dotted line), $k_B T/\hbar \omega_0=0.3$ (dashed line), and $k_B T/\hbar \omega_0=1$ (dashed-dotted line). We consider $g \omega x/v_F \ll 1$ and for simplicity we denote $\delta S^1(x,x,\omega)\equiv \delta S^1(\omega)$.}
\label{temp}
\end{center}
\end{figure}
The corresponding excess noise is depicted in Fig.~\ref{excessn}. Indeed, as described in Ref. \cite{deblock}, the emission and absorption excess noises are equal for non-interacting electrons. However, they are {\it different} in the presence of interactions.
The asymmetry between the emission and absorption high-frequency excess noises is an important signature of Luttinger liquid physics which should be looked for also in carbon nanotubes where the presence of metallic leads masks the physics of charge fractionalization in the zero frequency noise.
Also, we note that the absorption excess noise can even
become negative in some regions of frequency for $g<1/2$.
A similar behavior was also found
for the symmetrized noise of one-dimensional interacting systems connected to metallic leads \cite{dolcini}, as well as for the case of a ballistic single-channel
quantum wire, capacitively coupled to a gate \cite{negexcess}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{exnoisen.eps}
\caption{\small The excess noise in the total current $S^1_V(x,x,\omega)-S^1_{V=0}(x,x,\omega)$ (in units of
$g e I_B$), as a function of frequency (in units of $\omega_0$) for $\nu=1$ (dashed line), $\nu=1/3$ (full line) and $\nu=1/2$ (dotted line). We set $k_B T/\hbar \omega_0=0.03$ and
$g \omega x/v_F\ll 1$. For simplicity we denote $S^1(x,x,\omega)\equiv S^1(\omega)$.}
\label{excessn}
\end{center}
\end{figure}
The dependence of the absorption noise on frequency in the limit in which $g \omega x/v_F$ is of order $1$ is presented in Fig.~\ref{posn}. For the temperature considered here,
$k_B T/\hbar \omega_0=0.03$, the emission noise is basically unchanged and is still characterized by the behavior presented in Fig.~\ref{temp}; note that oscillations with a period of $\pi v_F/g x$ appear in the absorption part of the noise. By averaging over the position of the measuring point, the absorption noise in the total current actually becomes equal to the emission noise.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{nsnoisep1.eps}
\vspace{0.15in} \caption{\small The backscattering
absorption noise in the total current $\delta S^1(x,x,\omega<0)$ (renormalized by $ g e I_B$) as a function of frequency (in units of $\omega_0$) for $g=\nu=1/3$ (full line),
$g=\nu=1$ (dashed line). We set $k_B T/\hbar \omega_0=0.03$. For positive frequencies (emission noise), we retrieve the same behavior as the one described in Fig.~\ref{temp}.} \label{posn}
\end{center}
\end{figure}
The total non-symmetrized noise (including the noise in the absence of backscattering) described by Eq.(\ref{seq}) is depicted in Fig.~\ref{nstot}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{tnoisen.eps}
\caption{\small The total non-symmetrized noise $S^1(x,x,\omega)=s^1(x,x,\omega)+\delta S^1(x,x,\omega)$ in the total current (including the noise in the absence of backscattering) for $g=1$ (dashed line) and $g=1/3$ (full line) (in arbitrary units), as a function of frequency (in units of $\omega_0$), for
$g \omega x/v_F\ll 1$. We set $k_B T/\hbar \omega_0=0.03$. For simplicity we denote
$S^1(x,x,\omega)\equiv S^1(\omega)$.}
\label{nstot}
\end{center}
\end{figure}
We see that the emission spectrum is solely backscattering-induced,
consistent with the fact that at small temperature the amount of
photons that can be emitted by the system is very small if no
energy is transferred to the system (e.g. through applying a
voltage).
Next we focus on the symmetrized noise. We note that in general it is not identical, but it has a similar behavior to the absorption noise. In Fig.~\ref{excess} we plot the
symmetrized excess noise in the total current as a
function of frequency for a few values of the parameter $g$, when
$g \omega x/v_F \ll 1$. For a strongly interacting one-dimensional system such as the one described by $g=1/3$, the excess symmetrized noise has regions in which it becomes negative, similar to the absorption excess noise. Experimentally this may be used as a the signature of Luttinger liquid physics in the presence of strong interactions between electrons.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{exnoises.eps}
\caption{\small The symmetrized excess noise in the total current $S^0_V(x,x,\omega)-S^0_{V=0}(x,x,\omega)$ in units of $g e I_B$, as a
function of frequency in units of $\omega_0$ for $g=\nu=1/3$ (full
line), $g=\nu=1$ (dashed line), and $g=1/2$ (dotted line). We set $k_B T/\hbar \omega_0=0.03$ and $g \omega x/v_F \ll 1$. For simplicity we denote
$S^0(x,x,\omega)\equiv S^0(\omega)$.}
\label{excess}
\end{center}
\end{figure}
When $g \omega x/v_F$ is of order $1$, the dependence of the symmetrized noise on frequency is depicted in Fig.~\ref{pos}. We note again the presence of oscillations with period $\pi v_F/g x$, similar to the case of the absorption noise. If an average with respect to the measuring point is performed, the symmetrized noise also reduces to the behavior of the cross-correlations presented in Fig.~\ref{sym}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.5in]{snoisep.eps}
\caption{\small The symmetrized backscattering-induced noise in the total current $\delta S^0(x,x,\omega)$ (renormalized by $g e I_B$) as a function of frequency (in units of $\omega_0$) for
$\nu=g=1/3$ (full line), and $\nu=g=1$ (dashed line). The ratio
$\omega_0/\omega_L$ is set to $10$, where the energy scale
associated to the length $\omega_L= v_F/g x$.
We set $k_B T/\hbar \omega_0=0.03$.} \label{pos}
\end{center}
\end{figure}
For the case of the total symmetrized noise the noise in the absence of backscattering (of order $1$) gives rise to a linear background with a slope proportional to $g$; backscattering (of order $|\Gamma|^2$) creates a small cusp-like feature at $\omega_0$ for $g=1$ and a small ``bump-like" feature for $g=1/3$. This is also similar to the absorption noise in Fig.~\ref{nstot}.
We would like to mention here that this behavior if similar to the behavior of a different physical system; as it turns out, one can map the impurity problem in a
Luttinger liquid to the problem of a coherent one-dimensional conductor embedded in an ohmic environment with an arbitrary resistance \cite{safisaleur}. For the latter, the high frequency symmetrized noise was computed
\cite{ffnoise} by combining a scattering matrix approach with a real time effective action formalism, and a similar behavior was found, with the difference that the Josephson singularity appears in that situation at a value proportional to the value of the voltage across the sample $U$ and not to the value of the voltage source $V$. We will examine the consequences of this mapping for the high-frequency non-symmetrized noise of a coherent conductor embedded in an ohmic environment in a separate publication.
\section{Discussion}
\subsection{Comparison with the Landauer-B\"uttiker approach for the symmetrized noise}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{sample.eps}
\vspace{0.15in} \caption{\small Schematics of the sample}
\label{sample1}
\end{center}
\end{figure}
We can compare our results to those obtained using a standard Landauer-B\"uttiker (LB)
approach \cite{buttiker} for non-interacting electrons.
In the limit when the distance from the backscattering site to the
measuring point is very small, $\omega x/v_F \ll 1$, we can write the symmetrized noise
in the total current \begin{equation} S^0(t)=\langle
\{I_{R_{out}}(t)+I_{L_{in}}(t),I_{R_{out}}(0)+I_{L_{in}}(0)\}\rangle
\end{equation} where $I_{R/L_{in/out}}$ are the currents of incoming/outgoing
right/left movers with the convention that all currents are
positive if flowing from left to right in Fig.~\ref{sample1}. The
tunneling current between the two edges is denoted $I_B$. Noting
that $I_{R_{in}}-I_{R_{out}}=I_{L_{in}}-I_{L_{out}}=I_B$, we can
rewrite the total noise (up to terms of the form
$\langle I_{R/L_{in}} I_{R/L_{in}}\rangle$) as \begin{eqnarray} S^0(t)&=&\langle
\{I_B(t),I_B(0)\} \rangle+ \langle \{I_{R_{out}} (t),
I_{R_{in}}(0)\} \rangle + \langle \{I_{L_{out}} (t) ,I_{L_{in}}(0)
\}\rangle \nonumber \\&& +\langle \{I_{R_{in}} (t),
I_{R_{out}}(0)\} \rangle + \langle \{I_{L_{in}} (t)
,I_{L_{out}}(0) \}\rangle \end{eqnarray}
We can connect this relation to our results presented in
Eqs.(\ref{n1},\ref{n2}). We can identify the fluctuations in the
tunneling current between the edges $\langle \{I_B(t),I_B(0)\}
\rangle$ with the term $S_A$ in Eq.(\ref{n1}). As described in Eq.(\ref{lb}), for non-interacting
systems this term will be given by ${\cal T}(1-{\cal T})\sum_{m=\pm1}(\hbar \omega+m e V)\coth[(\hbar \omega+m e V)/2 k_B T]$. Perturbatively
we found this term to be $\sum_{m=\pm1}\coth[(\omega+m \omega_0)/2k_B T]$ $I_B(\omega+m
\omega_0)$. In the case
of small backscattering $1-{\cal T}\approx |\Gamma|^2$, and the two expressions are consistent in the non-interacting limit for which $I_B(\omega) \propto \omega$.
Similarly we see that we can identify the Fourier transform of $\langle \{I_{R_{out}} (t), I_{R_{in}}(0)\} \rangle$ \\
$+ \langle \{I_{L_{out}} (t), I_{L_{in}}(0) \}\rangle$ $+\langle \{I_{R_{in}} (t), I_{R_{out}}(0)\} \rangle $ $+ \langle \{I_{L_{in}} (t) ,I_{L_{out}}(0) \}\rangle$ with the term denoted $S_C$ in Eq.(\ref{n2}).
We note that this term is due to correlations between currents in the same reservoir.
In the LB formalism in the absence of interactions (see Eq.(\ref{lb})) this term is proportional to ${\cal T}\hbar \omega$ $\coth(\hbar \omega/2 k_B T)$, while perturbatively
we found it to be proportional to $-\coth(\omega/2k_B T)$ $\sum_{m=\pm1}$ $I_B(\omega+m \omega_0)$.
We see that again the two expressions agree in the non-interacting limit for which $I_B(\omega) \propto \omega$, in the case of small backscattering.
In a four-terminal setup such as a FQHE bar, one has experimental access to the chiral current correlations as well as to the noise in the total current. We can see from the formulas presented in Eqs.(\ref{crosseq1},\ref{crosseq}) that in this situation one can also $measure$ the two terms described above separately, for example by combining the symmetrized cross-correlations and the symmetrized noise in the total current: $S_A=2\delta S^0_{RL}(x,-x,\omega)-\delta S^0(x,x,\omega)$ and
$S_C=\delta S^0(x,x,\omega)-\delta S^0_{RL}(x,-x,\omega)$. Similar combinations of the emission and the absorption noise could also be used to separate the two terms. The individual measurements of $S_A$ and $S_C$ may allow one to extract important information about the system; for example we note that for non-interacting systems $S_C$ is voltage-independent; the fact that
$S_C$ depends on voltage only in the presence of interactions could provide a direct way to reveal
them in a FQHE system.
\subsection{Generalized Kubo formula}
It has been shown that one can also write a generalization of the Kubo formula for non-linear systems in the linear regime corresponding to small values of the applied voltage. In this regime the antisymmetric part of the current noise can be related to the differential AC conductivity such that
\begin{equation}
S(-\omega)-S(\omega)=2 \hbar \omega G_d(\omega)
\end{equation}
where $G_d(\omega)$ is the differential conductivity of the system.
We can derive a more general relation for our system by analyzing the antisymmetric part of the noise.
We start from the general position-dependent non-perturbative results presented in Eqs.(\ref{seq},\ref{n1},\ref{n2},\ref{n3}), and we analyze the connection between the noise and the Green's functions of the system modified to take into account the effect of local backscattering. Thus, we can rewrite Eqs.(\ref{seq},\ref{n1},\ref{n2},\ref{n3}) such that
the total symmetrized current fluctuations are related to the
Keldysh Green's function $\tilde{\bf C}^{\cal K}$ in the presence of backscattering:
\begin{equation}
S^0(x,y,\omega)=\frac{e^2\omega^2}\pi \tilde{\bf C}^{\cal K}(x,y,\omega)
\label{se}
\end{equation}
while the non-symmetrized noise is related to $\tilde{\bf C}^{+-}$:
\begin{equation}
S^1(x,y,\omega)=\frac{e^2\omega^2}\pi \tilde{\bf C}^{+-}(x,y,\omega)
\label{sn}
\end{equation}
where consistent with our previous notations, the indices $0$ and $1$ denote the symmetrized/non-symmetrized noises respectively.
Here we can see that Eqs.(\ref{seq},\ref{n1},\ref{n2},\ref{n3}) imply that the Green's functions $\tilde{\bf C}^{\cal K}$ and $\tilde{\bf C}^{{\cal R},{\cal A}}$ obey Dyson-type equations with the roles of self-energy being played by the
functions $f_A(\omega)$ and $f_C(\omega)$.
\begin{eqnarray}
e^2\tilde{\bf C}^{\cal R}(x,y,\omega)&=&e^2 \tilde{\cal C}^{\cal R}(x,y,\omega)-\tilde{\cal C}^{\cal R}(x,0,\omega) f_c(\omega)
\tilde{\cal C}^{\cal R}(0,y,\omega)\\
\tilde{\bf C}^{\cal A}(x,y,\omega)&=&\tilde{\bf C}^{\cal R}(y,x,-\omega)\\
e^2\tilde{\bf C}^{\cal K}(x,y,\omega)&=& e^2\tilde{\cal C}^{\cal K}(x,y,\omega)-\tilde{\cal C}^{\cal R}(x,0,\omega)f_A(\omega)
\tilde{\cal C}^{\cal A}(0,y,\omega)
\nonumber \\
&&-\tilde{\cal C}^{\cal R}(x,0,\omega) f_C(\omega) \tilde{\cal C}^{\cal K}(0,y,\omega)
-\tilde{\cal C}^{\cal K}(x,0,\omega) f_C(-\omega) \tilde{\cal C}^{\cal A}(0,y,\omega)
\end{eqnarray}
Noting that by the definition of the Keldysh transformation
\begin{equation}
2 \tilde{\bf C}^{+-}=\tilde{\bf C}^{\cal K}+\tilde{\bf C}^{\cal A}-
\tilde{\bf C}^{\cal R}
\end{equation}
we can see that the difference between the symmetrized and the non-symmetrized noises
comes from $\tilde{\bf C}^{\cal A}-
\tilde{\bf C}^{\cal R}$. Notice that a similar behavior will be observed if one works with
the chiral Green's function instead of the total Green's functions.
Thus for the fluctuations of the current evaluated at the same position in space we can write:
\begin{eqnarray}
&&S^1(x,x,\omega)-S^1(x,x,-\omega)=\frac{e^2 \omega^2}{\pi}[
\tilde{\bf C}^{+-}(x,x,\omega)-\tilde{\bf C}^{+-}(x,x,-\omega)]
\nonumber \\&&
=\tilde{\bf C}^{\cal A}(x,x,\omega)-\tilde{\bf C}^{\cal A}(x,x,-\omega)-\tilde{\bf C}^{\cal R}(x,x,\omega)+\tilde{\bf C}^{\cal R}(x,x,-\omega)
\end{eqnarray}
We can also show that
$\tilde{\bf C}^A(x,x,-\omega)$ $=\tilde{\bf C}^R(x,x,\omega)$ $=-\tilde{\bf C}^R(x,x,-\omega)^*$, and we can rewrite the above
$exact$ expression as
\begin{eqnarray}
S^1(x,x,-\omega)-S^1(x,x,\omega)=\frac{4e^2\omega^2}{\pi} {\rm{Re}}[
\tilde{\bf C}^{\cal R}(x,x,\omega)].
\end{eqnarray}
If we use the definition of the retarded Green's function presented in Eq.(\ref{cret}), and the definition of the current in Eq.(\ref{current}), we find $ \omega^2 \tilde{\bf C}^{\cal R}(x,x,\omega)$ to be proportional to the Fourier transform of the non-linear response function of the system $\theta(t-t')
\langle [j_a(\mathbf{x,t}), j_a(\mathbf{x,t'}) ] \rangle$.
In the limit $e V\ll \hbar \omega$ we can relate the non-local AC conductivity and the retarded Green's function by
\begin{equation}
\sigma(x,y,\omega)=\frac{2e^2\omega}{h}\tilde{\bf C}^{\cal R}(x,y,\omega)
\label{sig}
\end{equation}
and we find that
\begin{equation}
S^1(x,x,-\omega)-S^1(x,x,\omega)=4 \hbar \omega {\rm{Re}}[
\sigma(x,x,\omega)].
\end{equation}
This extends the generalized Kubo formula presented in \cite{kubo}. The extra factor of $2$ is due to our definition of noise.
We can now also verify explicitly how our perturbative
results satisfy this condition. For clarity we focus on the case $\omega x\ll 1$ and we drop all spatial indices.
We find:
\begin{eqnarray}
S^1(-\omega)-S^1(\omega)&=&s^1(-\omega)-s^1(\omega)
+\delta S^1(-\omega)-\delta S^1(\omega)
\nonumber \\
&=&\frac{2 g e^2 \omega}{\pi}-2 g e \sum_{m=\pm1} I_B(\omega+m \omega_0)
\end{eqnarray}
We note that for small applied voltages this can be expanded as
\begin{equation}
S^1(-\omega)-S^1(\omega)=4 \hbar \omega \sigma_0(\omega)-4 g e I_B(\omega)
\end{equation}
where $\sigma_0(\omega)=g e^2/h$ is the unperturbed conductivity of the system.
Following Ref. \cite{dolcini} we define the conductivity of the system to be
\begin{equation}
\sigma(x,y,\omega)=\sigma_0(x,y,\omega)+ \sigma_{\rm
BS}(x,y,\omega) \label{sig12}
\end{equation}
where $\sigma_0(x,y,\omega)=g e^2/h$ is the conductivity of the
system in the absence of the backscattering, and
\begin{eqnarray}\nonumber
\sigma_{\rm BS}(x,y,\omega)= - \frac{2}{\hbar \omega} \left(
\frac{\pi \lambda}{e}\right)^2 \sigma_0(x,0,\omega)
\sigma_0(0,y,\omega) \\
\label{sigbs} \times \int_0^\infty dt \, (e^{i \omega t}-1) \left(
\sum_{s=\pm} s \, e^{4 \pi {\cal C}(0,s t;0,0)} \right)
\label{sg}
\end{eqnarray}
is the contribution to the conductivity due to backscattering. Here $\cal C$
is the generalized two-point function for an infinite Luttinger liquid in the absence of backscattering given in Eq.(\ref{Creg_def}) of Appendix A. The backscattered current can be expressed in terms of this conductivity as
\begin{equation}
{\rm{Re}}[\sigma_{\rm BS}(\omega)]=-\frac{ge}{\hbar \omega} I_B(\omega)
\end{equation}
Thus we can write
\begin{equation}
S^1(-\omega)-S^1(\omega)=4 \hbar \omega \{\sigma_0+{\rm{Re}}[\sigma_{\rm BS}(\omega)]\}
\end{equation}
which is consistent with the generalized Kubo formula presented above.
\subsection{Fluctuation dissipation theorem}
We check that the noise at zero voltage can indeed be related by the fluctuation-dissipation theorem to the real part of the conductivity.
The noise is described by Eqs.(\ref{se},\ref{sn}) presented in the previous subsection. At zero voltage we expect that
\begin{equation}
\tilde{\bf C}^{\cal K}(x,y,\omega)=2 \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)
{\rm{Re}}[\tilde{\bf C}^{\cal R}(x,y,\omega)]
\end{equation}
In the absence of backscattering, we see from Eq.(\ref{gf}) that this is indeed the case. Using the relation between the retarded Green's function and the conductivity in Eq.(\ref{sig}), we then expect
\begin{equation}
S^{\alpha}(x,x,\omega)=2 \hbar \omega \Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] {\rm Re}[\sigma(x,x,\omega)].
\end{equation}
We can check how this relation is satisfied by our perturbative results. We found that the noise in the absence of an applied voltage is:
\begin{equation}
S^{\alpha}(x,x,\omega)=\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]\Big[\frac{g \omega e^2}{\pi} + 2 g e I_B(\omega)\Big]
\end{equation}
which can be rewritten as
\begin{equation}
S^{\alpha}(x,x,\omega)=2 \hbar \omega \Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] \{\sigma_0(x,x,\omega)+{\rm Re}[\sigma_{\rm BS}(x,x,\omega)]\}.
\end{equation}
This is indeed the fluctuation-dissipation theorem for the symmetrized and for the non-symmetrized noise.
\subsection{Tunneling limit}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{samplet.eps}
\vspace{0.15in} \caption{\small Schematics of the sample for the tunneling regime}
\label{samplet}
\end{center}
\end{figure}
For completion we can also analyze the noise in the limit when the magnitude of the backscattering is very large. In this limit we can treat the system as being disconnected into two subsystems by the scattering site, with only a small amount of tunneling between the two subsystems. This limit is not so relevant for measuring the fractional charge of the quasiparticles, as in this situation the shot noise contains primarily information about the electrons tunneling between the two subsystems. We thus expect the ratio of the noise at zero frequency to the tunneling current to be proportional to $e$ instead of $g e$ \cite{kf}. Also, the Josephson frequency is shifted to $e V/\hbar$ instead of $g e V\hbar$. The total noise in this limit is given solely by a single term describing the fluctuations in the tunneling current between the two edges. For simplicity, we restrict ourselves to the limit $\omega x/v_F \ll1 1$ and we find the noise to be:
\begin{eqnarray}
S_t^{\alpha}(\omega)&=&\frac{1}{4 \pi}\tilde{f}_A(\omega)-\alpha \frac{1}{4 \pi} [\tilde{f}_C(\omega)-\tilde{f}_C(-\omega)]
\nonumber \\
&=& e \sum_{m=\pm 1}\coth\Big[\frac{\hbar (\omega+m \tilde{\omega}_0)}{2 k_B T}-\alpha\Big]I_t(\omega+m \tilde{\omega}_0)
\end{eqnarray}
with $\tilde{\omega}_0=e V/\hbar$, and
\begin{equation}
\tilde{f}_A(\omega) = \int_{-\infty}^\infty dt \, e^{i \omega t}
\left\langle \left\{ \hat{I}_t(t), \hat{I}_t(0) \right\}
\right\rangle
\end{equation}
where $\hat{I}_t$ is the tunneling current operator between the two edges.
Similarly
\begin{equation}
\tilde{f}_C(\omega) = \int_0^\infty dt e^{i \omega t}
\left\langle \left[ \hat{I}_t(t),\hat{I}_t(0) \right]
\right\rangle
\end{equation}
Also
\begin{equation}
I_t(\omega)= - \frac{e}{\hbar^2} |\Gamma|^2 {\cal{F}}_{1/g}(\omega)
\end{equation}
is the value of the tunneling current for an applied voltage equal to $\omega$, with
\begin{eqnarray}
{\cal F}_{1/g}(\omega)=i \sin\Big(\frac{\pi}{g}\Big) \Big(\frac{\pi k_B T}{\epsilon_h}\Big)^{2/g} 2^{2/g-2} \Gamma(1-2/g)\Big[\frac{\Gamma(1/g-i \tilde{\omega})}{\Gamma(1-1/g-i \tilde{\omega})}-\frac{\Gamma(1/g+i \tilde{\omega})}{\Gamma(1-1/g+i \tilde{\omega})}\Big]
\end{eqnarray}
where $\tilde{\omega}=\hbar \omega/2 \pi k_B T$.
We see that indeed at zero frequency, the symmetrized noise $S^0(\omega \rightarrow 0)=2 e I_t(\tilde{\omega}_0)$, while at high frequency the noise has a singularity at a Josephson frequency $\tilde{\omega}_0$ given by $e V/\hbar$. Also, information about the Luttinger liquid characteristics of the FQHE edges is contained in the $1/g$ exponent, which is the dual of the $g$ exponent in the case of small backscattering between the two edges.
When we plot the non-symmetrized noise as a function of frequency for a few values of $g$ (see Fig.~\ref{tunn}), we find that indeed the case of $g=1$ is consistent with the known results for non-interacting fermions. The case of small $g$ shows that indeed the noise has a power-law dependence $\omega^{2/g-1}$ on frequency; however the singularities at the Josephson frequency $\tilde{\omega}_0$ are very soft in this situation, for example the singularity in the emission noise is of the type $\theta(\tilde{\omega}_0-\omega)|\omega-\tilde{\omega}_0|^5$ for $g=1/3$ which is very weak and becomes basically impossible to observe. A similar situation happens for the absorption noise, as can be seen from Fig.~\ref{tunn}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{tunneling.eps}
\vspace{0.15in} \caption{\small The non-symmetrized noise in the tunneling current $S_t(\omega)$ as a function of frequency (in units of $\omega_0=e V/\hbar$) for $\nu=1/3$ (full line), and $\nu=1$ (dashed line). The noise $S_t(\omega)$ is given in units of $e I_t$ for the case of $g=1$, while for $g=1/3$ the units are arbitrary to facilitate the plot of the two curves on the same figure without a loss of information at large negative frequencies.}
\label{tunn}
\end{center}
\end{figure}
\section{Conclusion}
We computed the high frequency non-symmetrized noise for a FQHE sample with small backscattering. At zero frequency we retrieved the classical result, $S=2 g e I_B$. At finite frequency and zero temperature the most striking feature we observe is a singularity at the Josephson frequency $\omega_0=g e V/\hbar$. This singularity is rounded off by temperature. For a non-interacting system ($\nu=1$) this feature is cusp-like, while it has power-law resonant characteristics (a peak, or a ``positive peak - negative dip'' structure depending on the quantity measured) in a strongly interacting system, e.g. ($\nu=1/3$).
Other important aspects that we observed were that the
non-symmetrized correlations of chiral outgoing branches are
entirely even in frequency. However, the backscattering-induced
noise in the total current is not even. If one looks instead
at the excess noise we see that this is even in the absence
of interactions, but it becomes asymmetric if the
electron-electron interactions are taken into account. Thus, the
asymmetry with respect to positive and negative frequencies
in the excess noise can be used to assess the Luttinger
liquid character of a system. Also we noted that the
excess noise (both symmetrized and non-symmetrized) can become
negative in some regions of frequencies, and for
very strong electron-electron interactions; this may provide a
signature of Luttinger liquid physics.
If the distance between the scattering site and the measuring point is significant, oscillations of the noise with respect to frequency will also appear. The period of these oscillations is proportional to the fractional charge $g$. Such oscillations do not occur in correlations of chiral outgoing branches, but are manifest in the absorption part of the non-symmetrized noise of the total current, as well as in the symmetrized noise of the total current.
We have also analyzed also the limit when the strength of backscattering is very large and the system
is disconnected in two subsystems, in this situation the noise
is dominated by electrons tunneling between the two subsystems.
We should note that, while not done here, for the case of weak backscattering it would also be interesting to compute the higher order corrections in the backscattering amplitude $\Gamma$ for the emission and absorption noises at high frequencies.
As shown in \cite{chetan}, the fourth order contributions to the high frequency symmetrized noise in a Laughlin state as well as in a non-Abelian Pfaffian state contain information about fractional and non-Abelian statistics; it would also be interesting to see what are the effects of the higher order corrections for the non-symmetrized noise.
\vspace{.2in}
\noindent{\bf\large Acknowledgements:}~
We would like to thank Richard Deblock, Christian Glattli, Frank Hekking, Fabien Portier, Patrice Roche, Bertrand Reulet and Hubert Saleur for interesting discussions. CB acknowledges the support of a Marie Curie Intra-European Fellowship.
\vspace{.4in}
{\noindent{\Large{\bf Appendix A: The Keldysh formalism used to calculate the non-symmetrized noise}}}
\vspace{.1in}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{pfaf2.eps}
\vspace{0.15in} \caption{\small The two branches of the Keldysh contour}
\label{pfaf}
\end{center}
\end{figure}
As depicted in Fig.\ref{pfaf}, we introduce the standard Keldysh time contour \cite{Keldysh} and we denote by $\Phi^+_a$ and $\Phi^-_a$ ($a=R/L$ denotes the chirality) the complex fields on
the upper and lower time branch of the contour. We
introduce the generating functional
\begin{eqnarray}
&&Z[J]= \frac{1}{\cal{N}_Z} \int {\mathcal D}
\Phi^{\pm}_{R/L} \exp \Big\{-\frac{1}{2} \int d\mathbf{r'}
d\mathbf{r''} \sum_{\eta,\eta'=\pm} \sum_{a=R/L}\Phi^{\eta}_a(\mathbf{r'})
({\mathcal C}^{-1})^{\eta,\eta'}_{a}
(\mathbf{r'},\mathbf{r''})
\Phi^{\eta'}_a(\mathbf{r''}) \Big\} \nonumber
\\ && \times \exp\Big\{\sum_{\eta=\pm} \Big( -\frac{i}{\hbar} \eta
\int_{-\infty}^{+\infty} d t' \mathcal{H}_B[\Phi^{\eta}]\Big) +\sum_{\eta=\pm,a=R/L}i \int
d\mathbf{x} J_a^{\eta}(\mathbf{x})\Phi_a^{\eta}(\mathbf{x})\Big)\Big\}
\label{gen-fun-1}
\end{eqnarray}
where the vector label $\mathbf{r'}$ stands for
$\mathbf{r'}=(x',t')$, $\int d\mathbf{r'}=\int_{-\infty}^{+\infty}
dt' \int_{-\infty}^{+\infty} dx'$, and $\cal{N}_Z$ is a
normalization factor, which assures that $Z[0]=1$. In
Eq.~(\ref{gen-fun-1}), ${\mathcal
C}^{-1}(\mathbf{r'},\mathbf{r''})$ is the inverse of a $4 \times
4$ matrix defined by the four free correlators
\begin{equation}
{\mathcal C}^{\eta,\eta'}_{a a}(\mathbf{r'};\mathbf{r''})={\mathcal C}^{\eta,\eta'}_{a}(\mathbf{r'};\mathbf{r''}) = \langle
\Phi^\eta_a(\mathbf{r'}) \Phi^{\eta'}_a(\mathbf{r''}) \rangle_0
\end{equation}
and
\begin{equation}
{\mathcal C}^{\eta,\eta'}_{a b}(\mathbf{r'};\mathbf{r''})=0
\end{equation}
if $a \ne b$.
Also, $\langle \ldots \rangle_0 $ indicates the average performed
with respect to the free Hamiltonian (\ref{l0}) along the Keldysh
contour.
For a Luttinger liquid with a backscattering site and in presence of an applied voltage,
we have
\begin{equation}
\langle \Phi_a(\mathbf{x}) \rangle = \frac{1}{2} \sum_{\eta=\pm}
\langle \Phi^\eta_a(\mathbf{x}) \rangle =-\frac{i}{2} \sum_{\eta=\pm}\left.
\frac{\delta Z[J]}{\delta J_a^{\eta}(\mathbf{x})} \right|_{J=0}\;.
\label{Phi-VM1}
\end{equation}
One can simplify the notation by introducing infinite-dimensional
vectors and matrices where {\it both} $\mathbf{r}$, $\eta$ and $a$ are
component labels. Defining
\begin{equation}
\mathbf{\Phi}= \left(
\begin{array}{c} \Phi^{+}_R(\mathbf{r}) \\ \Phi^{-}_R(\mathbf{r})\\ \Phi^{+}_L(\mathbf{r}) \\\Phi^{-}_L(\mathbf{r}) \end{array}
\right),
\end{equation}
\begin{equation}
\mathbf{J}= \left(
\begin{array}{r}
J_R^+(\mathbf{r})\\
J_R^-(\mathbf{r})\\J_L^+(\mathbf{r})\\
J_L^-(\mathbf{r})
\end{array}
\right) ,
\end{equation}
and
\begin{equation}
\mathsf{C}= \left(
\begin{array}{cccr}
\mathcal{C}^{++}_{R}(\mathbf{r},\mathbf{r'}) &
\mathcal{C}^{+-}_{R}(\mathbf{r},
\mathbf{r'})&0&0 \\
\mathcal{C}^{-+}_{R}(\mathbf{r},\mathbf{r'}) &
\mathcal{C}^{--}_{R}(\mathbf{r},
\mathbf{r'}) &0&0\\
0&0&\mathcal{C}^{++}_{L}(\mathbf{r},\mathbf{r'}) &
\mathcal{C}^{+-}_{L}(\mathbf{r},
\mathbf{r'})\\
0&0&\mathcal{C}^{-+}_{L}(\mathbf{r},\mathbf{r'}) &
\mathcal{C}^{--}_{L}(\mathbf{r},
\mathbf{r'})\\
\end{array}
\right) ,
\end{equation}
one can rewrite the generating functional (\ref{gen-fun-1}) as
\begin{eqnarray}
\displaystyle Z[J]= \frac{1}{\cal{N}_Z} \int {\mathcal D}
\mathbf{\Phi} \, e^{-\frac{1}{2} \Big(\mathbf{\Phi}^T {\mathsf
C}^{-1} \mathbf{\Phi} -2 i \mathbf{J}^T
\mathbf{\Phi} \Big)}
\exp{\left\{ - \frac{i}{\hbar} \sum_{\eta=\pm} \eta
\int_{-\infty}^{+\infty} d t' \mathcal{H}_B[\Phi^{\eta}]
\right\}}\, ,
\end{eqnarray}
where the superscript ${}^T$ indicates the transpose. Shifting the
fields
\begin{equation}
\mathbf{\Phi} \rightarrow \mathbf{\Phi}+{\mathbf{A}}\, ,
\hspace{1cm} {\mathbf{A}}=i{\mathbf{\mathsf{C}}}
\mathbf{J} \, , \label{shift-in-field}
\end{equation}
the generating functional can be factorized into
\begin{equation}
Z[J]=Z_0[J] Z_B[J] \; , \label{fun-gen-fact}
\end{equation}
where $Z_0$ and $Z_B$ are given below. In particular, $Z_0$ is the
generating functional in the absence of a backscatterer and reads
\begin{equation}
Z_0[J]= e^{-\frac{1}{2} \mathbf{J}^T {\mathsf C}
\mathbf{J}} \label{Z0}
\end{equation}
The second factor $Z_B$ in (\ref{fun-gen-fact}) is the generating
functional
\begin{eqnarray}\label{ZB}
Z_B[J(\mathbf{r})]
= \left \langle \exp{\left( -
\frac{i}{\hbar} \sum_{\eta=\pm} \eta \int_{-\infty}^{+\infty}
\mathcal{H}_B[\Phi^{\eta}_R+\Phi^{\eta}_L+A^{\eta}_R+A^{\eta}_L] \, d t' \right)}
\right\rangle_0,
\end{eqnarray}
which weighs the backscattering term, and where the dependence
on the source field $J(\mathbf{x})$ is contained in the shift
${\mathbf{A}}$ defined in Eq.~(\ref{shift-in-field}). In
components, the latter reads explicitly
\begin{eqnarray}
&&{\mathbf{A}}_{a}^{\eta}=\int d \mathbf{x}
i \sum_{\eta'=\pm} C^{\eta \eta'}_{a}(\mathbf{r};\mathbf{x}) J_a^{\eta'}(\mathbf{x})
\end{eqnarray}
The Keldysh Green's functions can be related to the regular Green's function using the
relations:
\begin{equation}
C_a^{\eta_1 \eta_2}=\frac{\eta_1 C_a^{\cal A}+\eta_2 C_a^{\cal R}+C_a^{\cal K}}{2}
\end{equation}
Here $C_a^{{\cal R}/{\cal A}}$ are the retarded and advanced Green's function respectively, $\eta_{1,2}=\pm$, and $C_a^{{\cal K}}$ is defined in Appendix B; for an infinite Luttinger liquid we evaluate it explicitly in Appendix B.
The finite frequency non-symmetrized noise $S_{a b}(x,y,\omega)$ is defined as
\begin{eqnarray}
S_{a b}(x,y,\omega) = \int_{-\infty}^{\infty} dt e^{i\omega t}
2 \left\langle \Delta j_a (y,0) \Delta j_b(x,t)
\right\rangle \; , \label{noiseeq}
\end{eqnarray}
where $a,b$ stand for the right/left moving indices, and $\Delta
j_a(x,t) = j_a(x,t) - \langle j_a(x,t) \rangle$ is the current
fluctuation operator, with $j_a(x,t)=e \partial_t \Phi_a(x,t)/\sqrt{\pi}$.
Thus, using the Keldysh formalism we can express the non-symmetrized noise as
\begin{eqnarray}
S_{a b}(x,y,\omega)&=&\int_{-\infty}^{\infty} dt_x e^{i \omega t_x}\frac{2 e^2}{\pi} \partial_{t_x} \partial_{t_y} \langle \Phi_a(\mathbf{y})\Phi_b(\mathbf{x})\rangle |_{t_y=0}\\ &=&
-\frac{2 e^2}{\pi} \int_{-\infty}^{\infty} dt_x e^{i \omega t_x}\Big\{\partial_{t_x} \partial_{t_y}
\frac{\partial^2 Z}{\partial J_a^+(\mathbf{x})\partial J_a^-(\mathbf{y})}\Big\}\Big|_{t_y=0}.
\end{eqnarray}
The noise in the total current is obtained by summing up $S_{RR}+S_{LL}+S_{RL}+S_{LR}$.
The procedure of taking the functional derivatives is lengthy but straightforward and follows closely the procedure described in Ref. \cite{dolcini}. We obtain
\begin{equation}
S^{\alpha}_{a b}
(x,y,\omega)=s^{\alpha}_{a b}(x,y,\omega)+
S^A_{ a b}(x,y,\omega)+S^C_{a b}(x,y,\omega)+\alpha S^N_{a b}(x,y,\omega),
\end{equation}
where $\alpha=0$ for the symmetrized noise and $\alpha=1$ for the non-symmetrized noise.
Here we have
\begin{eqnarray}
s^{\alpha}_{a b}(x,y,\omega)&=&\delta_{a b}\frac{e^2 \omega^2}{\pi} \{\tilde{\cal C}^{\cal K}_{a}(x,y,\omega)+\alpha [\tilde{\cal C}^{\cal A}_a(x,y,\omega)-
\tilde{\cal C}_a^{\cal R}(x,y,\omega)]\},\\
S^A_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi} \tilde{\cal C}^{\cal R}_{a}(x,0,\omega)
\tilde{\cal C}^{\cal R}_{b}(y,0,-\omega)f_A(\omega),\nonumber \\
S^C_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi}[ \tilde{\cal C}^{\cal K}_{a}
(x,0,\omega)\tilde{\cal C}^{\cal R}_{b}(y,0,-\omega)f_C(-\omega)
\nonumber \\&&
+\tilde{\cal C}^{\cal R}_{a}(x,0,\omega)\tilde{\cal C}^{\cal K}_{b}
(y,0,-\omega)f_C(\omega)],\nonumber \\
S^N_{a b}(x,y,\omega)&=&-\frac{\omega^2}{\pi}[\tilde{\cal C}^{\cal A}_{a}(x,0,\omega)
\tilde{\cal C}^{\cal R}_b(y,0,-\omega)f_C(-\omega)\nonumber \\&&-
\tilde{\cal C}^{\cal R}_a(x,0,\omega)\tilde{\cal C}^{\cal A}_b(y,0,-\omega)f_C(\omega)]
\nonumber
\end{eqnarray}
with
\begin{equation}
\tilde{\mathcal{C}}^m(x,y,\omega)=\int_{-\infty}^{\infty} \,
e^{i \omega t} \, {\mathcal{C}}^m(x,t;y,0) \, dt \; ,
\end{equation}
for $m={\cal A,R,K}$, and $x_0$ is the backscattering position
which we will set to zero.
The functions $f_A(\omega)$ and $f_C(\omega)$ are given by
\begin{equation}
f_A(\omega) = \int_{-\infty}^\infty dt \, e^{i \omega t}
\left\langle \left\{ \Delta j_B(t), \Delta j_B(0) \right\}
\right\rangle , \label{s1_A}
\end{equation}
where $\Delta j_B(t) = j_B(t) - \langle j_B(t)
\rangle$. The expectation values are performed with respect to the full action.
Also, $j_B(t)$ is the backscattering
current operator at the backscattering site
\begin{equation}
j_{B}(t) = - \frac{e}{\hbar} \frac{\delta
\mathcal{H}_B(\Phi)}{\delta \Phi(0,t)} \; \label{ib1_def} .
\end{equation}
Similarly
\begin{equation}
f_C(\omega) = \int_0^\infty dt \left( e^{i \omega t}-1 \right)
\left\langle \left[ j_B(t),j_B(0) \right]
\right\rangle \; . \label{s1_C}
\end{equation}
We can evaluate $f_A$ and $f_C$ perturbatively up to second order in $\Gamma$ to find:
\begin{eqnarray}
f_A(\omega)&=&
4 \pi \Big(\frac{e}{\hbar}\Big)^2 |\Gamma|^2 \int_0^\infty dt \cos(\omega t)
\cos(\omega_0 t) \sum_{s=\pm} e^{4 \pi {\cal C}(0,s t;0,0)} \\
&=& 2 \pi i \Big(\frac{e}{\hbar}\Big)^2 |\Gamma|^2 \sum_{m=\pm 1} \coth \Big[ \frac{\hbar(\omega+m \omega_0)}{2 k_B T}\Big]\int_0^\infty dt \sin[(\omega+m \omega_0) t] \sum_{s=\pm}s e^{4 \pi {\cal C}(0,s t;0,0)} \nonumber
\end{eqnarray}
and
\begin{eqnarray}
f_C(\omega)=2 \pi \Big(\frac{e}{\hbar}\Big)^2 |\Gamma|^2 \int_0^\infty dt (e^{i \omega t}-1)
\cos(\omega_0 t) \sum_{s=\pm}s e^{4 \pi {\cal C}(0,s t;0,0)}
\end{eqnarray}
where $\omega_0=g e V/\hbar$, and ${\cal C}$ is the generalized
correlation function
\begin{eqnarray}
{\cal C}(x,t;y,0)= \left\langle \, \Phi(x,t) \Phi(y,0) - \frac{\Phi^2(x,t) +
\Phi^2(y,0)}{2} \, \right\rangle_0 \label{Creg_def}.
\end{eqnarray}
with $\Phi=\Phi_R+\Phi_L$.
The Green's functions for an infinite Luttinger liquid system are presented in Appendix B. Using their specific forms we find that for $x=y>0$, the noise in the absence of backscattering is
\begin{eqnarray}
&&s^{\alpha}_{RR}(x,x,\omega)=s^{\alpha}_{L L}(x,x,\omega)=\frac{g \omega e^2}{2 \pi} \Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] \nonumber \\
&&s^{\alpha}(x,x,\omega)=\frac{g \omega e^2}{\pi}
\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big] \end{eqnarray}
with the cross correlations between the right and the left-movers
vanishing. The backscattering induced noise is \begin{eqnarray}
&&\delta S^{\alpha}_{R R}(x,x,\omega)=\frac{g^2}{4\pi} f_A(\omega)- \frac{g^2}{4 \pi} \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)[f_C(\omega)-f_C(-\omega)]\nonumber \\
&&\delta S^{\alpha}_{LL}(x,x,\omega)=0\nonumber \\
&&\delta S^{\alpha}_{RL}(x,x,\omega)+\delta S^{\alpha}_{LR}(x,x,\omega)=
- \frac{g^2}{4 \pi} \Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]\times
\nonumber \\&&
\times \Big\{\exp\Big(\frac{2 i g \omega |x|}{v_F}\Big) f_C(\omega)
- \exp\Big(-\frac{2 i g \omega |x|}{v_F}\Big) f_C(-\omega)\Big\}
\label{qet}
\end{eqnarray}
Here, as before, we denote the symmetrized noise by $\alpha=0$ and the non-symmetrized noise by $\alpha=1$.
The noise in the total current is
\begin{eqnarray}
&&\delta S^{\alpha}(x,x,\omega)=
\frac{g^2}{4\pi} f_A(\omega)- \frac{g^2}{4 \pi} \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)[f_C(\omega)-f_C(-\omega)]
- \frac{g^2}{4 \pi} \Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]
\times
\nonumber \\&& \times \Big\{\exp\Big( \frac{2 i g \omega |x|}{v_F}\Big) f_C(\omega)
- \exp\Big(-\frac{2 i g \omega |x|}{v_F}\Big) f_C(-\omega)\Big\}
\end{eqnarray}
Similarly, we find:
\begin{eqnarray}
&&\delta S^{\alpha}_{RL}(x,-x,\omega)=\frac{g^2}{4\pi} f_A(\omega)- \frac{g^2}{4 \pi} \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)[f_C(\omega)-f_C(-\omega)]
\nonumber \\&&
\delta S^{\alpha}_{R R/L L}(x,-x,\omega)= \mp \frac{g^2}{4 \pi}\Big[\coth\Big(\frac{\hbar \omega}{2 k_B T}\Big)-\alpha\Big]f_C(\pm \omega) e^{\pm 2 i g \omega x/v_F}
\end{eqnarray}
Noting that
\begin{equation}
f_A(\omega)=- 4 \pi \frac{e^2}{\hbar^2} |\Gamma|^2 \sum_{m=\pm 1}\coth\Big[\frac{\hbar (\omega+m \omega_0)}{2 k_B T}\Big]{\cal F}_g(\omega+m \omega_0)
\end{equation}
and
\begin{equation}
f_C(\omega)-f_C(-\omega)=- 4 \pi \frac{e^2}{\hbar^2} |\Gamma|^2 \sum_{m=\pm 1}{\cal F}_g(\omega+m \omega_0)
\end{equation}
where ${\cal F}_g(\omega)$ was defined in Eq. (\ref{calf}), we retrieve the equations presented in section 3.
\vspace{.2in}
{\noindent{\Large{\bf Appendix B - Chiral Green's functions for an infinite Luttinger liquid}}}
\vspace{.1in}
The correlations for two chiral operators in a Luttinger liquid can be computed from the fundamental correlations:
\begin{eqnarray}
{\cal C}_{a}(x,t;y,0)= \left\langle \, \Phi_a(x,t) \Phi_a(y,0) - \frac{\Phi^2_a(x,t) +
\Phi^2_a(y,0)}{2} \, \right\rangle_0 \label{Creg_def2}
\end{eqnarray}
We find that
\begin{equation}
{\cal C}_{R/L}(x,t;y,0)=-\frac{g}{4 \pi} \log \Big\{\frac{1+i[t\mp g(x-y)/v_F]\omega_h}{1\mp i g(x-y)/v_F \omega_h}\Big\}+\frac{g}{4 \pi} \log
\Big\{\frac{\pi T[t \mp g(x-y)/v_F]}{\sinh \pi T [t \mp g(x-y)/v_F]}\Big\}
\end{equation}
with the two-point function between a right mover and a left mover being zero. The total two-point function ${\cal C}$ is obtained by summing the two components.
By definition:
\begin{eqnarray}
{\mathcal C}_{a}^{\cal A}(\mathbf{r};\mathbf{r'}) &=& - \theta(t'-t)
\langle [ \Phi_a(\mathbf{r}), \Phi_a(\mathbf{r'}) ] \rangle_0 \; , \label{cadv} \\
{\mathcal C}_{a}^{\cal R}(\mathbf{r};\mathbf{r'}) &=& \theta(t-t')
\langle [ \Phi_a(\mathbf{r}), \Phi_a(\mathbf{r'}) ] \rangle_0 \; ,
\label{cret}
\\ {\mathcal C}_{a}^{\cal K}(\mathbf{r};\mathbf{r'}) &=& \langle \{
\Phi_a(\mathbf{r}),\Phi_a(\mathbf{r'})\} \rangle_0 \; \label{ckel}
\end{eqnarray}
with the corresponding Fourier transforms:
\begin{equation}
\tilde{\mathcal{C}}^m(x,y,\omega)=\int_{-\infty}^{\infty} \,
e^{i \omega t} \, {\mathcal{C}}^m(x,t;y,0) \, dt \; ,
\end{equation}
for $m={\cal A,R,K}$.
We obtain
\begin{eqnarray}
&&\tilde{\mathcal{C}}^{\cal A}_{L/R}(x,y,\omega)=-\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \theta[\pm (x-y)] \nonumber \\&&
\tilde{\mathcal{C}}^{\cal A}(x,y,\omega)=-\frac{g}{2 \omega}e^{-i g \omega |x-y|/v_F}
\end{eqnarray}
and
\begin{eqnarray}
&&\tilde{\mathcal{C}}^{\cal R}_{L/R}(x,y,\omega)=\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \theta[\mp (x-y)] \nonumber \\&&
\tilde{\mathcal{C}}^{\cal R}(x,y,\omega)=\frac{g}{2 \omega}e^{i g \omega |x-y|/v_F}
\end{eqnarray}
while, up to some constants
\begin{eqnarray}
&&\tilde{\mathcal{C}}^{\cal K}_{L/R}(x,y,\omega)=\frac{g}{2 \omega}e^{\mp i g \omega (x-y)/v_F} \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big) \nonumber \\&&
\tilde{\mathcal{C}}^{\cal K}(x,y,\omega)=\frac{g}{\omega} \cos[\omega(x-y)/v_F] \coth\Big(\frac{\hbar \omega}{2 k_B T}\Big).
\end{eqnarray}
We observe that the total $\cal C$ satisfies the fluctuation-dissipation theorem, while this is not the case for the individual chiral Green's functions.
| {
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Nienstedt, King and Associates was formed by the Realtor power duo of Amy Nienstedt and Racheal King. Some people call them a "match made in Heaven" due to their combined 20+ years of real estate experience, as well as their no-nonsense, get it done attitudes. Today's world has become pretty impersonal; automated tellers, voice mail systems, warehouse stores. Convenient, yes, but sometimes don't you wish you could find people who care?
Amy and Racheal know that when it comes to people, no two individual's needs or goals are exactly the same. That's why they never try to fit their clients into a mold or use a cookie-cutter approach. Amy and Racheal know each person needs individual attention. They believe that each client has their own unique mark and make great strides to help that individual meet his particular needs in real estate transactions.
Founding partner of Nienstedt, King and Associates, Amy is a 10 year residential real estate industry veteran and has represented hundreds of buyers and sellers in a wide array of real estate transactions. Her reputation for protecting her clients' interests, savvy negotiation skills, uncompromising integrity and "outside the box" marketing strategies are the hallmarks of Amy's service. Exceptionally loyal clientele, personal referrals, and repeat business have formed the foundation of Amy's career. So much so, that she was recently named "Summerville's Favorite Real Estate Agent" by the readers of the Summerville Journal Scene.
A Charleston resident since 2004, Amy is an enthusiastic member of the community and enjoys gardening, baking, fishing the waters of the lowcountry, and showing clients the best that Charleston has to offer!
Those who work with Racheal will tell you it's a spark of creativity that fuels her. They know what was effective in the industry yesterday is old news today. Racheal knows that today's home buyers and sellers are more educated and sophisticated than in years past. She drives herself to provide the level of incomparable service today's clients demand. Racheal's desire to stay ahead of this constantly changing industry finds her persistently searching for better, faster and smarter techniques to get the job done. She has that unique talent to turn every challenge into an opportunity and each potential failure into a success by doing the right thing in the right way at the right time.
Lakefront Home in The Coves at Cane Bay!
Lovely Courtyard Home – Price Drastically Reduced!
©2017 Nienstedt, King, & Associates | Website by Charleston Tech Support, Inc. | {
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Muzeum vévody Antona Ulricha (německy Herzog Anton Ulrich-Museum, zkratkou HAUM) v Brunšviku je muzeum výtvarného umění, založené již roku 1754 a disponující jednou z nejvýznamnějších sbírek starého umění v Německu. Pojmenované je podle zakladatele sbírky, místního vévody Antona Ulricha (1633–1714), jednoho z typických představitelů osvíceného absolutismu; sbírku pak otevřel veřejnosti jeho nástupce Karel I. Brunšvicko-Wolfenbüttelský.
Galerie
Externí odkazy
Muzea v Německu
Vzniklo 1754 | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/support.bioconductor.org\/p\/111667\/","text":"Question: DESeq2 baseMean from DiffBind summarizedExperiment\n0\n8 months ago by\nrbronste60\nrbronste60 wrote:\n\nSo I am using DiffBind to create a count matrix as summarizedExperiment and then running this through DESeq2 and I was wondering about ways to filter for specific baseMean ranges within this data? My script looks like the following:\n\nlibrary(\"DESeq2\")\nlibrary(\"ggplot2\")\nlibrary(\"BiocParallel\")\nlibrary(\"DiffBind\")\n\nparallel=TRUE\nBPPARAM=MulticoreParam(4)\n\n# Create a dba.count object, important that dba.count be unnormalized (DBA_SCORE_READS).\n\n# Retrieve counts from dba.count\n\nnrows <- 1095036\nncols <- 12\ncounts <- matrix(runif(nrows * ncols, 1, 1e5), nrows)\nrowRanges<-GRanges(rangedCounts)\n\nsampleNames<-c(\"MMV1\", \"MMV2\", \"MMV3\", \"FMV1\", \"FMV2\", \"FMV3\", \"MME7\", \"MME8\", \"MME9\", \"FME1\", \"FME2\", \"FME3\")\nsampleSex<-c(\"male\", \"male\", \"male\", \"female\", \"female\", \"female\", \"male\", \"male\", \"male\", \"female\", \"female\", \"female\")\ncolData<-data.frame(sampleName=sampleNames, sex=sampleSex)\n\n# Retrieve counts from dba.count without the site interval ranges\n\ncounts <- as.matrix(mcols(rangedCounts))\n\n# Construct a SummarizedExperiment\n\nse<-SummarizedExperiment(assays=list(counts=counts),rowRanges=rowRanges, colData=colData)\n\n# DESeq2 analysis\n\ntable(sampleSex)\n\ndds<-DESeqDataSet(se, design= ~sex)\n\ndds$group <- factor(paste0(dds$sex))\n\ndesign(dds) <- ~0 + group\n\ndds <- dds[ rowSums(counts(dds)) > 1, ]\n\ndds <- DESeq(dds, betaPrior = FALSE)\nwritten 8 months ago by rbronste60\nAnswer: C: DESeq2 baseMean from DiffBind summarizedExperiment\n1\n8 months ago by\nMichael Love23k\nUnited States\nMichael Love23k wrote:\n\nI don\u2019t follow the question. You want to pre-filter the dds based on the mean of counts? You can pre-filter the object using dds[keep,] where keep may be defined as rows with mean of normalized count greater than some value or rows where x or more samples have a count greater than or equal to y. This is up to you.\n\nYes actually that answers part of it, however an even deeper and perhaps slightly naive question is how the baseMean called by DESeq2 relates to the original counts and how does that change with normalization within DESeq2 - trying to determine if the problem is with the stringency of called peaks by MACS2 or something happening to during the differential peak calling. An example being that a lot of shared peaks among samples may have a high baseMean such as those at promoters with the differential peaks having low baseMeans relatively - therefore the overall change in the differential peaks is small. Thanks again!\n\n1\n\nTake a look here:\n\nhttps:\/\/bioconductor.org\/packages\/release\/bioc\/vignettes\/DESeq2\/inst\/doc\/DESeq2.html#more-information-on-results-columns\n\nYes this is quite helpful however I guess I am still a bit confused as what baseMean represents in an RNA-seq experiment vs the experiment I am indicating with the code above? The binding matrix in question above is made from ATAC-seq data. Thanks.\n\n1\n\nThe column that DESeq2 creates called \u201cbaseMean\u201d is rowMeans(counts(dds, normalized=TRUE)) regardless of the type of input data.\n\nSure of course, I just mean that baseMean in gene expression data represents to some approximation the transcript abundance however in ChIP or ATAC data merely represents the fragment pileup at a specific site, unless I am incorrect?\n\n1\n\nI'll leave it to the DiffBind package authors I suppose\n\n1\n\nIn this case, where the count score is DBA_SCORE_READS, the binding matrix does not have pileup scores, but rather the total number of reads that align overlapping each (consensus) peak. This is similar to what you get in RNA data, where the expression matrix has the raw number of reads that overlap the gene (exons), only in the ATAC case, instead of transcripts, they are overlapping open chromatin regions.","date":"2019-04-22 06:50:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.46221187710762024, \"perplexity\": 7055.0437822579015}, \"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-2019-18\/segments\/1555578544449.50\/warc\/CC-MAIN-20190422055611-20190422081611-00559.warc.gz\"}"} | null | null |
Director David Nutter discusses Game of Thrones Season 8. Says security on the set was like the Gestapo
Abhinav Pathak
Game of Thrones is finally coming to an end with the final season airing in April 2019 and we've all been waiting for the first episode to air. As the time comes closer more teasers, images, and interviews start surfacing on the internet but this isn't enough to ease the cravings for Season 8 whose trailer would soon drop – HBO's #ForTheThrone campaign is the biggest proof! Recently, director David Nutter, who is directing three episodes this season appeared in an interview on the Huffington Post.
Previously, Nutter also addressed the Game of Thrones fans via Reddit where he each episode of the new season to be over one hour long. But, in this interview, he talks about the Game of Thrones security, fake scenes shot for this season, premiere episodes, and the ending.
Nutter had been missing from the series for complete two seasons (6 and 7) and now he returns for the final season again. Upon being asked what it was like to him, he replied:
"After 2015, after the fifth season, I had a couple back surgeries and I missed Season 6 and 7. And it broke my heart to do so. You go through physical therapy and all the things you go through for back surgery. There are times when I'd be in excruciating physical therapy sessions, and I'd be wondering if I'd ever have a chance to direct again ― let alone direct "Game of Thrones."
Fortunately, I came back. I was all repaired. They welcomed me back with open arms, and not only welcomed me back with open arms but welcomed me back to actually have me involved in and direct three of the six episodes [of Season 8]."
The revelation of Jon Snow A.K.A. Aegon Targaryen's heritage is a big game changer for the upcoming season, which was revealed while Nutter was totally absent from the series. On directing this in the new season he says,
"I'm just so honoured to be involved in the material because it's so wonderfully written. It's so wonderfully performed. It's so tremendously executed. Just to be a part of that world is a dream come true… [Showrunners David Benioff and Dan Weiss] are wonderful to work with. They're such great collaborators as well as teachers and mentors. In many respects for me, it's just getting a chance to be involved in what they're doing."
I remember Kit Harington talking about filming a few fake scenes last season on Jimmy Kimmel Live. David Nutter reveals the same scenario even this season by saying they need to maintain some amount of secrecy seeing paparazzi and seeing drone cameras hovering around. He reveals,
"Well, sometimes there were paparazzi in amazing places ― on construction cranes and all kinds of crazy places, to try to get a point of view of things. They were all over, everywhere, trying to get in on what was happening. But it was definitely a situation where there was no paper on the set, [that] type of thing."
"[The production team] wanted to make sure nobody knew what was happening, and they went to the nth degree as they do on the show in general. They basically take it to the point where it's like the Gestapo. It's tough to get answers."
Finally talking about his take and what he can tell us about the ending of Game of Thrones, Nutter revealed:
"All I know is that David and Dan spent a lot of time to tell the story in a proper fashion, and the audience will be completely satisfied. Not everybody will be satisfied, but I feel the audience will be satisfied with the direction the series goes. It lives up to all the building it's coming to, I promise you that."
Read the full interview on the Huffington Post here. What was your take on David Nutter's latest interview? Tell us in the comments section down below.
Related Topics:Dan WeissDavid Benioffdavid nutterend-gamegameofthronesseason8
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Saurav Jha
A show like Game of Thrones doesn't simply reach the stature it did without everything being high class. An area where the show truly held its own was costume designing. Designers, Michele Clapton and April Ferry really pushed the envelope with their work on the show and now they have already begun spreading their wings elsewhere. Read on!
If you can still recall some of the costumes worn by the characters of Cersei Lannister and Daenerys Targaryen on the show, you can count yourself in the Michele Clapton fan club as well. Now, as per the hbothenevers.com, the makers have roped in Emmy award-winning Michele Clapton as the costume designer for the upcoming HBO show The Nevers.
Clapton did some truly world-class work for Thrones and won many awards for her work as well. For Game of Thrones fans, there is more to look forward to in The Nevers, Stunt coordinator Rowley Irlam, production designer Gemma Jackson and producer Duncan Muggoch are going to be part of the new show.
With so many great names associated with the show, The Nevers already looks like a winner. The show's release date isn't out yet but it is expected to premiere in 2021.
Let us know your thoughts on the news!
Game of Thrones stars have continued on their merry ways and have continued to do some great stuff. While some have gone on to do award-winning roles like Richard Madden, some have gone on to portray roles on the big screen like Emilia Clarke. Another popular cast member who has quietly gone on to open her production house is Natalie Dormer a.k.a. Margaery Tyrell of Thrones. Read on to find out more!
Natalie Dormer has recently launched a production house called Dog Rose Productions. As per The Wrap, Dormer has decided to produce a series on female aviators in World War 2 and it has been titled as Spitfire Sisters. The show will be written by playwright and screenwriter Morgan Lloyd Malcolm.
Regarding the story Dormer said:
"This is a barely known story, a thrilling story; an important story ready to be told by such a passionate voice as Morgan's."
This is going to be the second series under Dormer's newly launched production house. The first announced project was the drama Vivling based on the life of British actress Vivien Leigh.
Natalie Dormer really got the limelight through her portrayal of Margaery Tyrell on Game of Thrones. Although her character was killed off in Season 6, fans still remember her character. In fact, the treatment given to her on-screen family had pissed her off!
Isn't it great to see former Game of Thrones stars doing their own things and making some great stuff of their own? We hope Spitfire Sisters goes to achieve great things for Dormer and her production house.
Let us know your views on this announcement in the comments below!
Game of Thrones star Richard Madden has been cast alongside Bollywood star Priyanka Chopra Jonas for a lead role in an upcoming Amazon series. The series is titled Citadel and is said to be an action-packed spy series. Read on for more details!
As per Variety, Citadel is going to be Avengers: Endgame directors Anthony and Joe Russo's next offering. Amazon announced that the show is going to have multiple versions. Chopra and Madden are going to star in the U.S. "mothership" edition of the series. The Indian series will be developed by Raj Nidimoru and Krishna D.K. of The Family Man fame.
Madden is coming off his award-winning role in Bodyguard. He, of course, played the role of Robb Stark in Game of Thrones as well and became a popular figure. Madden is currently filming for a Marvel Cinematic Universe movie Eternals slated for November 2020 release.
Amazon announced Citadel back in July 2018 and have lots of hope from the series. The studio stated:
"All of the local series are meant to enhance the entire entertainment experience and will be available for the viewer to deep dive into an imagined layered world."
The studio hasn't yet announced the rest of the star cast and the exact release date of the show.
It is going to be exciting to see how the series turns out. One thing is for sure, it is going to be another feather in the cap for Richard Madden. Let us know your views on the news!
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Game of Thrones becomes the most torrented TV show for the last time | {
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\section{Introduction}
Densifying the deployments of access points (APs) is one of the most significant drivers for increasing the throughput of wireless network \cite{gupta2000capacity,tse2005fundamentals}. As the radio resources can be reused across smaller spatial scales, the network throughput is expected to increase linearly with the AP density \cite{tse2005fundamentals}.
However, the sustainability of such a linear throughput gain with the AP densification has been challenged in the recent studies of 5G mmWave networks \cite{andrews2017modeling,bai2015coverage,jia2016impact,bai2014analysis,saha2018integrated,alammouri2018unified}, mainly due to the inherently high density of mmWave LAN infrastructure \cite{jia2016impact,bai2015coverage,alammouri2018sinr,rappaport2013broadband,rappaport2013millimeter,rappaport2015wideband}.
Specifically, the mmWave network is most likely to be operated in the interference-limited regime and thus the throughput will reach the plateau when the aggregate interference from the APs becomes overwhelming \cite{bai2015coverage,andrews2017modeling}.
To suppress the interference, the hybrid precoding is performed at the mmWave device, where the hybrid precoding technique is the combination of analog beamforming and spatial multiplexing \cite{andrews2017modeling,bai2015coverage,alkhateeb2015limited,sun2014mimo}. However, it has been proved in \cite{alammouri2020escaping} that the densification plateau cannot be avoided with the finite analog beamforming gain. In this paper, we propose to overcome the throughput densification plateau by deploying the spatial multiplexing at mmWave APs. With that goal, we provide the comprehensive performance analysis for the downlink mmWave networks.
In the context of dense APs and strong interference, we use the stochastic geometry to facilitate the performance analysis for the mmWave network, where the APs and end-users are modeled as two independent Poisson Point processes (PPPs).
The use of stochastic geometry in modeling the communication system has been widely accepted due to its accuracy and tractability in analyzing the signal-to-interference-plus-noise ratio (SINR) distribution for the wireless network \cite{andrews2011tractable,haenggi2012stochastic}.
In \cite{andrews2017modeling, bai2015coverage, di2015stochastic, singh2015tractable}, the stochastic geometry has been extended to study the SINR distribution for mmWave networks by incorporating the distinguishing features of the mmWave system.
For example, to model the high penetration loss of mmWave signal, the study in \cite{di2015stochastic, bai2015coverage, andrews2017modeling} introduces the mmWave link state, namely the line-of-sight (LOS) and the non-line-of-sight (NLOS) state.
In \cite{alkhateeb2015limited, bai2015coverage, di2015stochastic, kulkarni2017performance,saha2018integrated}, the mmWave link is modeled with the sectorized beam patterns, which characterizes the analog beamforming gain of the hybrid precoding performed at the mmWave device.
In addition to the analog beamforming gain, we introduce the spatial multiplexing gain to capture the impact of multiple RF chains on the mmWave hybrid precoding \cite{jia2016impact}.
To take account of the blockage effects on the mmWave network, we use the relative density to quantify the number of APs throughout this paper. The relative density is defined as the average number of LOS APs that an end-user can observe, which was first introduced in \cite{bai2015coverage}.
In Section II, we propose to model the mmWave network by incorporating the hybrid precoding architecture in \cite{alkhateeb2015limited}, where both the analog beamforming gain and the spatial multiplexing gain are considered.
In Section III, we derive the analytical expressions of the coverage probability, the throughput of the fixed-rate coding scheme, and the throughput upper bound for mmWave networks.
The analysis in Section III is then used to investigate the effect of AP densification on mmWave networks in Section IV. Also, the results of Section III provide the basis for the performance evaluation of the mmWave network throughput in Section V.
Our results provide two key system design insights for the mmWave network. In Section IV, we analyze how the throughput of a mmWave network is affected by the AP density, where we assume that the fixed-rate coding scheme is employed. Moreover, we introduce a key performance indicator, termed densification gain, to capture the throughput scaling law as the AP density increases.
Our analysis shows that without the spatial multiplexing at the APs, the throughput of a mmWave network will reach the plateau when the AP density grows very large. However, by employing spatial multiplexing, the mmWave network can continuously harvest the throughput gain from the AP densification.
In Section V, we numerically quantify the throughput of different coding schemes for the mmWave network.
The fixed-rate coding scheme with the optimal rate threshold is used to provide the tightest throughput lower bound, while the multi-rate coding scheme is developed for further throughput improvements. We also numerically compare these two schemes to the throughput upper bound as the spatial multiplexing gain increases.
Our results suggest that for the mmWave network, the performance of the fixed-rate coding scheme is less satisfactory with the larger spatial multiplexing gain, where the multi-rate coding scheme is required to explore the potential throughput gain.
\section{Modeling mmWave Network}
We consider the downlink mmWave network, where the end-users and APs are distributed in the area according to independent PPPs $\Phi_{\text{U}}$ and $\Phi_{\text{A}}$ with intensities $\lambda_{\text{U}}$ and $\lambda_{\text{A}}$, respectively.
Each mmWave AP is assumed to be equipped with $k$ RF chains. The mmWave end-user is assumed to be equipped with one RF chain due to its small form factor.
\subsection{Hybrid Precoding}
The mmWave hybrid precoding consists of the analog beamforming at the antenna arrays and spatial multiplexing in the baseband \cite{andrews2017modeling,bai2015coverage}.
Assume that the mmWave AP is equipped with $k>1$ RF chains, then the AP can simultaneously transmit independent data streams to multiple end-users.
In the following, we introduce the spatial multiplexing gain to denote the number of data streams transmitted by the mmWave AP.
Note that the spatial multiplexing gain captures the impact of multiple RF chains on the hybrid precoding.
\begin{defn}
The spatial multiplexing gain for a mmWave AP is defined as the number of independent data streams transmitted by the AP.
\label{def: spatial multiplexing gain}
\end{defn}
Note that the spatial multiplexing gain of a mmWave AP is limited by the number of RF chains equipped by the AP. Assume that all the RF chains are activated, then the spatial multiplexing gain at the mmWave AP equals to $k$.
For the mmWave end-user, the spatial multiplexing gain is always one since the end-user has only one RF chain due to the small form factor.
It follows that an AP can simultaneously transmit to $k$ end-users.
For the mmWave device, the analog beamforming can be characterized by the main lobe gain, the main lobe beamwidth and the side lobe gain \cite{andrews2017modeling}. We denote the main lobe gain of the mmWave AP as $G_{\text{A}}$, while the side lobe gain is denoted by $g_{\text{A}}$, $g_{\text{A}} \ll G_{\text{A}}$. Let $\theta_{\text{A}}$ denote the main lobe width of the mmWave AP, where we assume $\theta_{\text{A}} \leq \frac{2\pi}{k}$.
For the mmWave end-user, we denote $\theta_{\text{U}}$ as the main lobe width, where the analog beamforming gain of the main lobe and the side lobe are denoted by $G_{\text{U}}$ and $g_{\text{U}}$, respectively.
Note that the hybrid precoding is the combination of the spatial multiplexing and analog beamforming \cite{bai2015coverage,andrews2017modeling}.
In the implementation of hybrid precoding, the area centered at a mmWave AP is equally partitioned into $k$ spatially orthogonal sectors with $\frac{2\pi}{k}$ angular width, as shown in Fig.\ref{fig:spatial multiplexing gain}.
We assume that all mmWave APs have full buffers, meaning that there is at least one user at each sector of the AP.
Due to the lack of scattering, the mmWave channel is almost deterministic, which means that the channels of nearby end-users are highly correlated \cite{bai2015coverage,saha2018integrated}.
To minimize the inter-beam interference in such a scenario, we assume that a mmWave AP assigns one analog beam to each sector, where the end-users within the same sector are scheduled via the orthogonal time-division and frequency-division techniques.
We remark that the end-users located in different sectors can reuse the same time-frequency resources at the AP, which has been referred to as the sectorization gain \cite{yang2000sectorization}.
We assume that the analog beam of an end-user and its intended AP are aligned. However, the analog beams of interfering APs are considered to be randomly oriented for the end-user. Denote the hybrid precoding gain of the mmWave AP as $\mathbb{G}_\text{A}$. For an unintended mmWave link, the hybrid precoding gain of the AP can then be written as $\mathbb{P}(\mathbb{G}_\text{A} = G_\text{A}) = \frac{k\theta_{\text{A}}}{2\pi}$ and $\mathbb{P}(\mathbb{G}_\text{A} = g_\text{A}) = 1 - \frac{k\theta_{\text{A}}}{2\pi}$, while the hybrid precoding gain of the end-user is denoted by $\mathbb{G}_\text{U}$ with $\mathbb{P}(\mathbb{G}_\text{U} = G_\text{U}) = \frac{\theta_{\text{U}}}{2\pi}$ and $\mathbb{P}(\mathbb{G}_\text{U} = g_\text{U}) = 1 - \frac{\theta_{\text{U}}}{2\pi}$.
In this paper, we focus on the effect of the spatial multiplexing gain $k$ on the performance of mmWave network. Thus, the analog beamforming gains are assumed to be finite constants.
\begin{figure}
\vspace*{-10pt}
\centering
\subfloat[{\scriptsize AP with $k=3$ RF chains i.e. spatial multiplexing gain $k = 3$.} \label{Figure: AP topology with k = 3}]{%
\includegraphics[width=0.48\linewidth]{Figures/Spatial_Multiplexing_Gain3.pdf}}
\hfill
\subfloat[{\scriptsize AP with $k=6$ RF chains i.e. spatial multiplexing gain $k = 6$.}\label{Figure: AP topology with k = 6}]{%
\includegraphics[width=0.48\linewidth]{Figures/Spatial_Multiplexing_Gain6.pdf}}
\caption{\textcolor{black}{Illustration of hybrid precoding at mmWave AP with different number of RF chains}}
\label{fig:spatial multiplexing gain}
\vspace*{-20pt}
\end{figure}
\subsection{Relative Density}
The mmWave system experiences a much higher penetration loss than the sub-6GHz system, which leads to dramatically different path loss for the line-of-sight (LOS) and non-line-of-sight (NLOS) mmWave links.
Denote $\ell(r)$ as the path loss of a mmWave link with length $r$.
To capture the blockage effects in the mmWave network, the path loss model $\ell(r)$ has two states, namely a LOS state and a NLOS state \cite{singh2015tractable,andrews2017modeling}. If the mmWave link is LOS, then the path loss $\ell(r)$ is proportional to $r^{\alpha_{\text{L}}}$ \cite{singh2015tractable}. Such a model can be applied to a NLOS link by replacing the path loss exponent $\alpha_\text{L}$ with $\alpha_{\text{N}}$ \cite{singh2015tractable,andrews2017modeling,rappaport2013millimeter}.
For the mmWave network with dense deployments of APs, it is shown in \cite{bai2015coverage,andrews2017modeling} that the path loss model can be further simplified to an equivalent LOS-ball of radius $R_\text{B}$ with the inverse
\begin{equation}
\ell(r)^{-1} =
\left\lbrace
\begin{split}
& r^{-\alpha_\text{L}}, \; \text{if}\;r\leq R_\text{B}\\
& 0, \qquad \text{if}\;r> R_\text{B}
\end{split},
\right.
\label{LOS-ball}
\end{equation}
with the path loss exponent $\alpha_\text{L}\leq 2$.
\textcolor{black}{Moreover, the mmWave link is deterministic if the density of APs is large, which means that the mmWave channel fading is negligible \cite{bai2015coverage}.
By ignoring the channel fading, the coverage of a mmWave AP is equivalent to the intersection of the Voronoi cell and the disk of radius $R_\text{B}$, as shown in Fig.\ref{fig:voronoi_diagram}.
For the mmWave end-user, we then introduce the relative density to characterize the number of mmWave APs within its LOS region.}
\begin{prop}
In a mmWave network, the relative density $\psi$ is defined as the average number of LOS APs that an end-user can observe. Given the intensity $\lambda_{\text{A}}$ of $\Phi_{{\text{A}}}$, the relative density can then be given as
\begin{eqnarray}
\psi = \pi \lambda_{\text{A}} R_\text{B}^2,
\label{eq: relative density}
\end{eqnarray}
where $R_\text{B}$ is the radius of LOS-ball in (\ref{LOS-ball}).
\label{prop: relative density}
\end{prop}
\vspace{-10pt}
\begin{proof}
Note that $\Phi_{\text{A}}$ follows a PPP, which is translation-invariant \cite{haenggi2012stochastic}. Thus, the average number of LOS APs is identical for each mmWave end-user. The result then follows the Campbell's Theorem for sums on a PPP, as shown in \cite[Theorem 4.1]{haenggi2012stochastic}.
\end{proof}
It can be observed from (\ref{eq: relative density}) that with a fixed $R_\text{B}$, the relative density $\psi$ is proportional to the intensity of mmWave APs $\lambda_{\text{A}}$, where $R_\text{B}$ is determined by the propagation environment \cite{bai2015coverage}. In our analysis, $R_\text{B}$ is assumed to be a constant. It follows that the increase in the relative density $\psi$ in (\ref{eq: relative density}) is equivalent to the increase in the AP density $\lambda_{\text{A}}$. Thus, we use $\psi$ and $\lambda_{\text{A}}$ alternatively in the sequel.
\begin{figure}[!t]
\vspace{-0.1cm}
\centering
\includegraphics[width=2.5in]{Figures/Voronoi_Diagram.pdf}
\caption{Coverage of mmWave APs. Black dots represent the mmWave APs. For the end-user in the green area, it will be served by the AP located in the same Voronoi cell. The end-user in the white area is out of coverage.}
\label{fig:voronoi_diagram}
\vspace{-20pt}
\end{figure}
\subsection{SINR and Coverage Probability}
In an interference-limited network, the received SINR is considered as one of the most important performance indicators \cite{haenggi2012stochastic,andrews2011tractable}.
Given the SINR threshold $\tau$, the coverage probability of mmWave network with the relative density $\psi$ is then defined as the probability that a randomly chosen end-user can achieve a SINR larger than $\tau$ i.e.
\begin{equation}
\mathcal{C}_\psi(\tau,k) \triangleq \mathbb{P}\left(\text{SINR}>\tau \left| \mathbb{P}(\mathbb{G}_\text{A} = G_\text{A}) = k\cdot\frac{\theta_{\text{A}}}{2\pi} \right. \right).
\label{def: P_cov}
\end{equation}
Obviously, the coverage probability $\mathcal{C}_\psi(\tau,k)$ increases as the SINR threshold $\tau$ decreases.
Note that the distribution of mmWave APs $\Phi_{\text{A}}$ follows a PPP, which is translation-invariant \cite{haenggi2012stochastic}. Thus the SINR distribution of each end-user is identical. By randomly selecting an end-user from $\Phi_{\text{U}}$, we then shift the origin of our coordinate system to the location of that end-user.
Assume that the end-user at the origin is associated with the mmWave AP located at $x_0$.
To maximize the signal strength at the origin, the location $x_0$ is selected such that $x_0 = \text{arg}\max\limits_{x\in\Phi_{{\text{A}}}}\ell(|x|)^{-1}$. Such a strategy is widely adopted in the implementation of wireless network, which is known as the minimum path loss association rule \cite{andrews2011tractable,haenggi2012stochastic,bai2015coverage,di2015stochastic,andrews2017modeling,saha2018integrated}.
By following $x_0 = \text{arg}\max\limits_{x\in\Phi_{{\text{A}}}}\ell(|x|)^{-1}$, all the interfering APs are farther than distance $|x_0|$.
Given $x_0$, the received power at the origin resulted by the interfering APs can then be written as
\begin{eqnarray}
\mathcal{I}&=& \sum_{x \in \Phi_{\text{A}} \cap \mathcal{B}(|x_0|,R_\text{B})} \mathbb{G}_\text{A}\mathbb{G}_\text{U}|x|^{-\alpha_{\text{L}}},
\label{eq: interference of LOS}
\end{eqnarray}
where $\mathcal{B}(|x_0|,R_\text{B})$ denotes the ring centered at the origin and of the radius ranging from $|x_0|$ to $R_\text{B}$; $\mathbb{G}_\text{A}, \mathbb{G}_\text{U}$ represent the hybrid precoding gain of the mmWave AP and end-user, respectively.
Conditioning on $x_0$, the coverage probability $\mathcal{C}_\psi(\tau,k)$ in (\ref{def: P_cov}) can then be written as
\begin{equation}
\mathcal{C}_\psi(\tau,k)
= \mathbb{P}\left(G_{\text{A}}G_{\text{U}} |x_0|^{-\alpha_{\text{L}}} > {\tau \mathcal{I}} \right).
\label{eq: LOS coverage probability}
\end{equation}
Here, the thermal noise is ignored \cite{bai2015coverage}. Due to $G_\text{A} \gg g_\text{A}$, the signal from the main lobe of $x\in\Phi_{\text{A}}\backslash\{x_0\}$ is considered as the interference by the end-user at the origin, while the signal from side lobes is treated as the background noise.
Next, we derive the coverage probability in (\ref{eq: LOS coverage probability}) based on Alzer's Lemma in \cite{alzer1997some}.
\begin{thm}
For the mmWave network of relative density $\psi$ and spatial multiplexing gain $k$, the coverage probability $\mathcal{C}_\psi(\tau,k)$ at the SINR threshold $\tau$ is given as
\begin{eqnarray}
\mathcal{C}_{\psi}(\tau,k)
&=&
\psi e^{-\psi}{\sum_{j =1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\int_{0}^{1} \exp
\left\lbrace
\psi
\mathbb{E}
\left[
\Lambda_j
\left(\tau,r,\frac{\mathbb{G}_\text{A} \mathbb{G}_\text{U}}{G_\text{A}G_\text{U}} \right)
\right]
\right\rbrace
\text{d} r,
\label{eq: simplified coverage probability}
\end{eqnarray}
where $\Lambda_j(\tau,r,\cdot)$ is the incomplete gamma function defined in (\ref{eq: taylor parameter}); $\mu$ is the shape parameter of small scale fading channel; $\mathbb{G}_\text{A}, \mathbb{G}_\text{U}$ denote the hybrid precoding gain of mmWave AP and mmWave end-user, respectively; the thermal noise is ignored.
\label{thm: coverage probability for dense mmWave network}
\end{thm}
\begin{proof}
Refer to Appendix \ref{proof: coverage probability of LOS ball}.
\end{proof}
To calculate the coverage probability $\mathcal{C}_\psi(\tau,k)$, we choose the shape parameter as $\mu = 10$, which has been validated for modeling the deterministic mmWave channel \cite{bai2015coverage}. Note that the incomplete gamma function $\Lambda_j(\tau,r,\cdot)$ of the coverage probability $\mathcal{C}_\psi(\tau,k)$ in (\ref{eq: simplified coverage probability}) can be fast evaluated by using most numerical tools. In other words, $\mathcal{C}_\psi(\tau,k)$ in Theorem \ref{thm: coverage probability for dense mmWave network} can be efficiently computed with a simple numerical integral over a finite interval. In the following, we continue to show that the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ is a monotonically decreasing function of $k$.
\begin{cor}
For the mmWave network of relative density $\psi$, the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ at the SINR threshold $\tau$ decreases with the spatial multiplexing gain $k$, $1 \leq k\leq \frac{\theta_{\text{A}}}{2\pi}$.
\label{cor: bound of coverage probability}
\end{cor}
\begin{proof}
Note that the interference $\mathcal{I}$ in (\ref{eq: interference of LOS}) monotonically increases with the hybrid precoding gain $\mathbb{G}_\text{A}$, where $\mathbb{G}_\text{A}$ increases with $k$. The Corollary immediately follows.
\end{proof}
In the prior work \cite{bai2015coverage,di2015stochastic}, the coverage probability of mmWave network is derived without considering the spatial multiplexing at APs, which is equivalent to setting the spatial multiplexing gain as $k=1$. Therefore, the results in \cite{bai2015coverage,di2015stochastic} correspond to the special case of $\mathcal{C}_\psi(\tau,1)$ in (\ref{eq: simplified coverage probability}).
\subsection{Coding Scheme and Throughput}
Note that the SINR threshold $\tau$ of the coverage probability $\mathcal{C}_\psi(\tau,k)$ in (\ref{def: P_cov}) is determined by the coding scheme applied to the mmWave network.
Assume that the mmWave network employs the fix-rate coding scheme at the target rate threshold $\rho$. Then the transmission is considered successful only if the maximum achievable rate of the end-user exceeds $\rho$, i.e., $W \log_2(1+\text{SINR}) > \rho$, where $W$ denotes the bandwidth assigned to the end-user.
It follows that with the fixed-rate coding scheme of $\rho$, an end-user can achieve a rate
\begin{equation}
\mathcal{R} =
\left\lbrace
\begin{split}
& \rho, \;\text{if}\;\;W \log_2(1+\text{SINR}) > \rho\\
& 0, \;\text{if}\;\;W \log_2(1+\text{SINR}) \leq \rho
\end{split}.
\right.
\label{eq: data rate of end-user}
\end{equation}
Note that the downlink throughput of a mmWave network is defined as the aggregate rate of all end-users.
By implementing the fixed-rate coding scheme with the rate threshold $\rho$, it follows from (\ref{eq: data rate of end-user}) that the throughput can be written as
\begin{eqnarray}
\mathcal{T}(\rho; \psi,k)
&\triangleq& \lambda_{\text{U}}\mathcal{R} = \lambda_{\text{U}}\rho\mathbb{P}\left(W \log_2(1+\text{SINR})>\rho\right) \nonumber\\
&=& \lambda_{\text{U}} \rho \mathbb{P}\left(\text{SINR}>2^{\rho/W}-1\right)
= \lambda_{\text{U}} \rho \mathcal{C}_\psi(2^{\rho/W}-1, k).
\label{eq: throughput of fixed-rate coding scheme}
\end{eqnarray}
Obviously, $\lambda_{\text{U}} \rho$ is the upper bound for the throughput $\mathcal{T}(\rho; \psi,k)$ in (\ref{eq: throughput of fixed-rate coding scheme}), which is achieved only if the interference at end-users is completely canceled. However, with the finite analog beamforming gain and dense APs, the interference of the mmWave network is strong and thus significantly affects the throughput, which implies $\mathcal{T}(\rho; \psi,k) \ll \lambda_{\text{U}} \rho$.
A natural extension of the fixed-rate coding scheme is the multi-rate coding scheme. Instead of choosing one rate threshold $\rho$, the multi-rate coding scheme selects a finite set of rate thresholds $\{\rho_i\}_{i=1}^M,M>1$. By applying the multi-rate coding scheme to the mmWave network, the rate of an end-user can then be written as
\begin{equation}
\hat{\mathcal{R} }=
\left\lbrace
\begin{split}
& \rho_M, \;\text{if}\;\;W \log_2(1+\text{SINR}) >\rho_M\\
& \rho_i, \;\text{if}\;\; \rho_i < W \log_2(1+\text{SINR}) \leq \rho_{i+1}, \text{for} \;1 \leq i \leq M-1\\
& 0, \;\text{if}\;\;W \log_2(1+\text{SINR}) \leq \rho_1
\end{split}.
\right.
\label{eq: data rate of end-user - multi-rate coding scheme}
\end{equation}
Thus, the throughput of the mmWave network with the multi-rate coding scheme is given as
\begin{eqnarray}
&&\mathcal{T}\left( \{\rho_i\}_{i=1}^{M} \,;\, \psi,k \right)
\triangleq \lambda_{\text{U}}\hat{\mathcal{R}} \nonumber\\
&&= \lambda_{\text{U}} \sum_{i=1}^{M-1} \rho_i
\left[\mathcal{C}_\psi(2^{\rho_i/W}-1, k) - \mathcal{C}_\psi(2^{\rho_{i+1}/W}-1, k) \right] + \lambda_{\text{U}}\rho_M\mathcal{C}_\psi(2^{\rho_M/W}-1, k),
\label{eq: throughput of multi-rate coding scheme}
\end{eqnarray}
which is bounded by
\begin{equation}
\mathcal{T}(\rho_1; \psi, k) \leq \mathcal{T}\left( \{\rho_i\}_{i=1}^{M} \,;\, \psi,k \right) < \lambda_{\text{U}} \rho_M \mathcal{C}_\psi(2^{\rho_1/W}-1, k).
\label{eq: bound on throughput of multi-rate coding scheme}
\end{equation}
Note that the throughput $\mathcal{T}\left( \{\rho_i\}_{i=1}^{M} \,;\, \psi,k \right) $ can be obtained by evaluating the coverage probability in (\ref{eq: simplified coverage probability}) at each SINR threshold $\tau_i = 2^{\rho_i/W}-1, i \in [1,M]$. Hence, there is no difference in analyzing the throughput for the fixed-rate coding scheme and multi-rate coding scheme. Without loss of generality, we will focus on the analysis of fixed-rate coding scheme in the sequel.
For a mmWave network, the upper bound of the throughput is achieved if each end-user can instantaneously reach the maximum achievable rate $W\log_2(1+\text{SINR})$. Therefore, the throughput upper bound for the mmWave network of $\psi$ and $k$ is defined as
\begin{equation}
\overline{\mathcal{T}}\left(\psi,k \right) = \lambda_{\text{U}}\mathbb{E}\left[ W\log_2(1+\text{SINR}) \right].
\label{eq: throughput upper bound definition}
\end{equation}
To achieve the upper bound in (\ref{eq: throughput upper bound definition}), the mmWave network needs to work with any arbitrarily small SINR, which may not be feasible in practice.
However, we can evaluate the performance of a coding scheme by comparing the corresponding throughput with the upper bound $\overline{\mathcal{T}}\left(\psi,k \right)$.
The rest of the paper focuses on the impact of relative density $\psi$ and spatial multiplexing gain $k$ on the throughput of mmWave networks. Other system parameters are assumed to be constants as follows.
The analog beamforming gain $G_{\text{A}} = 20\,\text{dB}, G_{\text{U}} = g_{\text{A}} = 0\,\text{dB}, g_{\text{U}} = -10\,\text{dB}, \theta_{\text{A}} = 30^{\circ}$ and $\theta_{\text{U}} = 90^{\circ}$ are used in all the numerical results \cite{bai2015coverage,rappaport2013millimeter}. The radius of the LOS-ball in (\ref{LOS-ball}) is assumed to be $R_\text{B} = 200$m and $\alpha_\text{L} = 2$ is used in the numerical results \cite{bai2015coverage,andrews2017modeling}. The total available bandwidth of $2$GHz and the intensity of mmWave end-users $\lambda_{\text{U}} = 10^4\text{km}^{-2}$ are used in all numerical calculations.
\section{Performance Analysis in mmWave Network}
In this section, we derive the analytical expressions for the coverage probability, the throughput corresponding to the fixed-rate coding scheme and the throughput upper bound of mmWave networks.
Note that all the performance metrics are characterized as the polynomials of spatial multiplexing gain $k$. Moreover, the derived polynomial formulas separate the positive and negative effects of the AP densification on the performance of mmWave networks.
\subsection{Coverage Probability Analysis}
For the mmWave network of relative density $\psi$, the coverage probability at SINR threshold $\tau$ is derived in Theorem \ref{thm: coverage probability for dense mmWave network}, where the expression of $\mathcal{C}_\psi(\tau,k)$ is given in (\ref{eq: simplified coverage probability}). However, it is still obscure how the coverage probability is affected by the relative density $\psi$.
To provide more insights, we derive the polynomial formula for the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ as follows.
\begin{thm}
For a mmWave network of relative density $\psi$, the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ at the SINR threshold $\tau$ can be written as a polynomial of the spatial multiplexing gain $k$ as follows:
\begin{eqnarray}
\mathcal{C}_{\psi}(\tau,k)
&=&
c_0(\tau,\psi) +
\sum\limits_{l=1}^{\infty} c_l (\tau,\psi) k^l,
\label{eq: SINR interms of polynomial}
\end{eqnarray}
where the expression of coefficient $c_0(\tau,\psi)$ is given in (\ref{eq: coefficient c_0}); $c_l(\tau,\psi)$ for $l\in\mathbb{N}^+$ is given in (\ref{eq: coefficient c_l}).
The coverage probability $\mathcal{C}_{\psi}(\tau,k)$ can be further approximated by its $L$ first terms, where
\begin{eqnarray}
\mathcal{C}_{\psi}(\tau,k) \approx c_0(\tau,\psi) + \sum\limits_{l=1}^{L} c_l (\tau,\psi) k^l
\label{eq: coverage probability with finite terms}
\end{eqnarray}
with the approximation error
\begin{equation}
\left| \mathcal{C}_{\psi}(\tau, k) - \sum\limits_{l=0}^{L} c_l (\tau,\psi) k^l \right|
<\frac{(2^{\mu}-1) e^\psi \psi^{L+2}}{(L+2)!}.
\label{eq: approximation error of coverage probability}
\end{equation}
\label{thm: taylor expansion of coverage probability}
\end{thm}
\vspace*{-20pt}
\begin{proof}
Refer to Appendix \ref{proof: taylor expansion of coverage probability}.
\end{proof}
Note that the coefficients $c_0(\tau,\psi), c_l(\tau,\psi)$ are independent of the spatial multiplexing gain $k$. It implies that $k$ is decoupled from $\psi$ and $\tau$ in (\ref{eq: SINR interms of polynomial}).
\textcolor{black}{
For the special case of the spatial multiplexing gain $k=1$, it follows from Corollary \ref{cor: bound of coverage probability} that $\mathcal{C}_{\psi}(\tau, 1)$ provides the upper bound for $\mathcal{C}_{\psi}(\tau, k)$. Thus, we have $\mathcal{C}_{\psi}(\tau,k) \leq \mathcal{C}_{\psi}(\tau, 1) = \sum\limits_{l=0}^{\infty} c_l (\tau,\psi)$.
}
In Theorem \ref{thm: taylor expansion of coverage probability}, the coverage probability of mmWave network is written as a sum of $c_0(\tau,\psi)$ and $\sum\limits_{l=1}^{\infty} c_l (\tau,\psi) k^l$.
Following the proof of Theorem \ref{thm: taylor expansion of coverage probability}, the term $c_0(\tau,\psi)$ in (\ref{eq: coefficient c_0}) is equivalent to the probability $\mathbb{P}(\text{SNR}>\tau)$, which is monotonically increasing with the relative density $\psi$. That is to say, $c_0(\tau,\psi)$ represents the gain in the coverage probability with the AP density. Moreover, $c_0(\tau,\psi)$ is independent of $k$ since the signal strength at the end-users is not affected by the spatial multiplexing gain of mmWave APs.
The second term $\sum\limits_{l=1}^{\infty} c_l (\tau,\psi) k^l$ in (\ref{eq: SINR interms of polynomial}) captures the loss in the coverage probability due to the aggregate interference of the APs. It follows that the detrimental effect of the AP densification on the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ can be expressed as a polynomial of the spatial multiplexing gain $k$.
\subsection{Throughput Analysis for Fixed-Rate Coding Scheme}
Assume that the fixed-rate coding scheme with the rate threshold $\rho$ is implemented. Then the throughput of mmWave network $\mathcal{T}(\rho; \psi,k)$ is given in (\ref{eq: throughput of fixed-rate coding scheme}).
Note that the coverage probability $\mathcal{C}_\psi(\tau,k)$ is derived in Theorem \ref{thm: taylor expansion of coverage probability}. To obtain the throughput, we then need to derive the bandwidth $W$ for mmWave end-users.
For analytical tractability, a common assumption is that the mmWave AP deploys the round-robin scheduling to allocate the time-frequency resources.
It follows that the end-user at the origin is assigned with the bandwidth $W = B/\Psi$, where $B$ denotes the total bandwidth available at the mmWave AP at $x_0$ and $\Psi$ represents the total number of end-users that share the radio resources with the end-user at the origin \cite{singh2013offloading,saha2019unified}.
It follows from the minimum path loss association rule that a mmWave AP covers the area within its LOS-ball of radius $R_\text{B}$ and belonging to its Voronoi cell, as shown in Fig.\ref{fig:voronoi_diagram}.
We remark that the LOS-ball model limits the coverage of a single mmWave AP, which differentiates the AP coverage in mmWave bands from the typical Voronoi cell in \cite{singh2013offloading}.
Given the spatial multiplexing gain $k$, the coverage area of the mmWave AP is then divided into $k$ sectors, as illustrated in Fig.\ref{fig:spatial multiplexing gain}.
It follows that $\Psi$ refers to the number of end-users which are covered by the mmWave AP at $x_0$ and within the same sector of $\frac{2\pi}{k}$ angle width as the origin.
Next, we present the main result on the bandwidth distribution $W = B/\Psi$ for the end-user at the origin.
\begin{thm}
Consider a mmWave network of relative density $\psi$, where the spatial multiplexing gain of APs is $k$. Then the bandwidth assigned to the end-user at the origin equals to $W = B/\Psi$, where $\Psi$ has the probability mass function (PMF)
\begin{equation}
\mathbb{P}\left(\Psi = n\right) =\mathcal{K}_\text{T}(n,k; \psi,\lambda_{\text{U}}), \; n \in\mathbb{N}^+,
\label{eq: PMF tagged AP sector}
\end{equation}
and the expression of $\mathcal{K}_\text{T}(\cdot)$ is shown in (\ref{eq: tagged end-user load}).
Since $\psi \geq 1$, the PMF of $\Psi$ can be further simplified as the expression in (\ref{eq: simplied tagged load}).
\label{thm: PMF for tagged AP sector}
\end{thm}
\begin{proof}
Refer to Appendix \ref{proof: bandwidth distribution}.
\end{proof}
Following Theorem \ref{thm: PMF for tagged AP sector}, the average bandwidth $\overline{W}$ for the mmWave end-user at the origin is derived as follows.
\begin{cor}
In the mmWave network of relative density $\psi$ and spatial multiplexing gain $k$, the average bandwidth assigned to the end-user at the origin is given as
\begin{equation}
\overline{W} = \frac{B}{\mathbb{E}[\Psi]} = \frac{B}{1 + \xi\frac{\pi R_\text{B}^2 \lambda_{\text{U}}}{k \psi}},
\label{eq: average bandwidth}
\end{equation}
where $B$ is the total bandwidth available at the mmWave AP; $\xi = 1.28$ is the bias factor for the end-user at the origin.
\label{cor: average bandwidth}
\end{cor}
\begin{proof}
Following the similar steps in \cite{singh2013offloading}, we have ${\mathbb{E}[\Psi]} = 1 + \xi\frac{\pi R_\text{B}^2 \lambda_{\text{U}}}{k \psi}$ with the bias factor $\xi$, where $\xi = 1.28$ is the bias factor for the Voronoi cell containing the origin. For the Voronoi cell that does not contain the origin, we have the bias factor $\xi = 1$.
\end{proof}
Since we have $\xi\frac{\pi R_\text{B}^2 \lambda_{\text{U}}}{k \psi} \gg 1$, Corollary \ref{cor: average bandwidth} then implies that the average bandwidth $\overline{W}$ of the mmWave end-user at the origin scales linearly with the relative density $\psi$.
It implies that as the density of APs increases, the bandwidth gain of end-users is the key enabler for increasing the throughput of mmWave network.
Based on the coverage probability derived in Theorem \ref{thm: taylor expansion of coverage probability} and the bandwidth distribution derived in Theorem \ref{thm: PMF for tagged AP sector}, we then present the main result on the throughput of mmWave networks.
\begin{thm}
Consider the mmWave network of relative density $\psi$, where the spatial multiplexing gain of the mmWave AP equals to $k$. Then the throughput achieved by the fixed-rate coding scheme with the rate threshold $\rho$ can be given as
\begin{equation}
\mathcal{T}(\rho; \psi,k)
= \lambda_{\text{U}} \rho \mathbb{E}_{\Psi}\left[ c_0 \left(2^{\rho \Psi/ B} - 1,\psi \right) \right] + \lambda_{\text{U}} \rho \sum_{l=1}^{\infty} k^l\mathbb{E}_{\Psi}\left[ c_l \left(2^{\rho \Psi/ B} - 1,\psi \right) \right],
\label{eq: rate coverage}
\end{equation}
where $B$ is the total bandwidth available at the AP;
the PMF of $\Psi$ is derived in (\ref{eq: PMF tagged AP sector});
the expression of $c_l(\tau,\psi)$ for $l\in\mathbb{N}$ is given in Theorem \ref{thm: taylor expansion of coverage probability}.
\label{thm: rate coverage}
\end{thm}
\begin{proof}
Given the bandwidth distribution $W = B/\Psi$ in Theorem \ref{thm: PMF for tagged AP sector}, the throughput under the fixed-rate coding scheme in (\ref{eq: throughput of fixed-rate coding scheme}) can then be written as
\begin{eqnarray}
\mathcal{T}(\rho; \psi,k) &=& \lambda_{\text{U}} \rho \mathbb{P}\left(\frac{B}{\Psi}\log_2(1+\text{SINR})>\rho\right) \nonumber\\
&=& \lambda_{\text{U}} \rho \mathbb{E}_{\Psi}\left[ \mathcal{C}_{\psi}(2^{\rho \Psi/B} - 1,k) \right]
\label{eq: rate coverage in coverage probability},
\end{eqnarray}
where $\mathcal{C}_{\psi}(\tau, k)$ is the coverage probability of the mmWave network at the SINR threshold $\tau = 2^{\rho n/B } - 1$ for $\Psi = n$.
The throughput in (\ref{eq: rate coverage}) can be immediately obtained from the coverage probability in (\ref{eq: SINR interms of polynomial}) and the PMF of $\Psi$ in (\ref{eq: PMF tagged AP sector}).
\end{proof}
It follows from (\ref{eq: rate coverage}) that the throughput of mmWave network can be written as the summation of two terms.
As proved in Theorem \ref{thm: taylor expansion of coverage probability}, $c_0(\tau,\psi)$ in the first term denotes the probability $\mathbb{P}(\text{SNR} > \tau)$ and is a monotonically increasing function of $\psi$, since the desired signal strength increases with the AP density.
The second term of (\ref{eq: rate coverage}) corresponds to the contribution of the aggregate interference received by the end-user, which captures the detrimental effect of the AP densification on the throughput of mmWave network.
It follows from Theorem \ref{thm: PMF for tagged AP sector} that the bandwidth for the end-user, $B/\Psi$, tends to increase with $\psi$. Thus the densification of APs can lower the SINR threshold $\tau = 2^{\rho \Psi/ B} - 1$ in (\ref{eq: rate coverage}). In other words, the variable $\tau = 2^{\rho \Psi/ B} - 1$ in (\ref{eq: rate coverage}) characterizes the throughput gain provided by the increase in the end-user's bandwidth as the AP density increases.
\subsection{Throughput Upper Bound Characterization}
While the throughput achieved by the fixed-rate coding scheme will vary with different choices of rate threshold $\rho$, the throughput upper bound is irrelevant to $\rho$. Given the instantaneous SINR, the maximum achievable rate for the end-user is set by the Shannon bound, which equals to $W\log_2(1 +\text{SINR})$. It follows that the upper bound of the network throughput is achieved if each end-user can reach its instantaneously maximum achievable rate \cite{alammouri2018sinr}. In the following Theorem, we derive the upper bound of the throughput for the mmWave network.
\begin{thm}
Consider the mmWave network of relative density $\psi$, where the spatial multiplexing gain of APs equals to $k$. By assuming that the interference is treated as the noise, the upper bound of the throughput is given as
\begin{equation}
\overline{\mathcal{T}}\left(\psi,k \right)
=
\lambda_{\text{U}}\mathbb{E}_{\Psi}\left[\frac{B}{\Psi}\right]
\log_2(e)
\left(
\int_{0}^{\infty} \frac{c_0 (\tau,\psi)}{1+\tau} \text{d}\,\tau
+
\sum\limits_{l=1}^{\infty} k^{l}
\int_{0}^{\infty} \frac{c_l (\tau,\psi)}{1+\tau} \text{d}\,\tau
\right) .
\label{eq: bounds on throughput}
\end{equation}
Here,
the PMF of $\Psi$ is derived in Theorem \ref{thm: PMF for tagged AP sector}; $c_l(\tau,\psi)$ is given in Theorem \ref{thm: taylor expansion of coverage probability}.
\label{thm: network throughput}
\end{thm}
\begin{proof}
See Appendix \ref{proof: throughput upper bound}.
\end{proof}
We remark that the coverage probability in (\ref{eq: SINR interms of polynomial}), the throughput in (\ref{eq: rate coverage}), and the throughput upper bound in (\ref{eq: bounds on throughput}) decouple the improvement and degradation of the network performance as the mmWave AP density increases. Before delving into the asymptotic trends in the dense AP deployments, we want to emphasize that the full buffer is assumed for all the performance analysis in this section. It implicitly means that to apply the results in (\ref{eq: SINR interms of polynomial}), (\ref{eq: rate coverage}) and (\ref{eq: bounds on throughput}), the density of mmWave end-users $\lambda_{\text{U}}$ needs to be sufficiently large, which can guarantee that no AP sectors are idle.
\section{Effect of AP Densification on mmWave Network}
We now investigate how the throughput of a mmWave network scales with the AP density, where the implementation of the fixed-rate coding scheme is assumed throughout this section.
Note that the scaling law of throughput is different in the power-limited and interference-limited scenarios.
Therefore, we first derive the range of relative density threshold $\psi^*$, below which the mmWave network is power-limited and beyond which is interference-limited.
Then, we introduce the concept of densification gain, which captures the throughput gain as the density of APs increases.
Our goal here is to illustrate the presence of the densification plateau for the mmWave network without spatial multiplexing. Moreover, we demonstrate that such a densification plateau can be overcome by employing the spatial multiplexing at the mmWave APs.
\subsection{Relative Density Threshold for mmWave Network}
For the mmWave network, let $\psi^*(\tau,k)$ denote the relative density that maximizes the coverage probability i.e. $\psi^*(\tau,k) = \text{arg}\max\limits_{\psi\in\mathbb{R}^+} \mathcal{C}_\psi(\tau, k)$, where $\psi^*(\tau,k)$ is defined as the relative density threshold.
Intuitively, the mmWave network with the relative density $\psi < \psi^*(\tau,k) $ is in the power-limited regime, where the coverage probability is limited by the signal strength due to the lack of AP coverage. It follows that the coverage probability will benefit from increasing the AP density, or equivalently, increasing $\psi$.
However, as $\psi$ increases, the mmWave network will transit from the power-limited regime into the interference-limited regime, where the coverage probability starts to decay with $\psi$.
In \cite{bai2015coverage}, $\psi^* = 4$ is shown to be the empirically relative density threshold.
Next, we analytically derive the range of the relative density threshold.
\begin{lem}
For a mmWave network, the range of relative density threshold $\psi^*(\tau,k)$ satisfies $2 \leq \psi^*(\tau,k) \leq 4$ for all the SINR threshold $\tau \leq 15$dB and spatial multiplexing gain $k\leq \frac{2\pi}{\theta_{\text{A}}}$.
\label{lem: optimal relative density}
\end{lem}
\begin{proof}
To prove the range of $\psi^*(\tau,k)$, we need to show that $\mathcal{C}_{\psi}(\tau,k)$ increases with the relative density $\psi$ for $\psi<2$, while decreases with $\psi$ for $\psi>4$.
For the mmWave network with the relative density $\psi < 2$, the fact $|\sum\limits_{l=1}^{\infty} c_l (\tau,\psi) k^l |<0.1, \forall \tau\leq15\text{dB, }\, \forall k \leq \frac{2\pi}{\theta_{\text{A}}},$ demonstrates that the interference has a negligible impact on the coverage probability $\mathcal{C}_{\psi}(\tau,k)$ in (\ref{eq: SINR interms of polynomial}) within that range of relative density. It also indicates SINR $\approx$ SNR when the mmWave network has a relative density $\psi < 2$. As the desired signal strength increases with the AP density, $\mathcal{C}_{\psi}(\tau,k)$ increases with $\psi$ if $\psi<2$.
However, if $\psi > 4$, we have $c_0(\tau,\psi) > 0.95, \forall \tau\leq15\text{dB}, \forall k \leq \frac{2\pi}{\theta_{\text{A}}}$, which implies that only marginal gain in SNR can be obtained beyond the relative density $\psi = 4$ and thus the interference determines the coverage probability. It follows that the mmWave network is in the interference-limited region and $\mathcal{C}_{\psi}(\tau,k)$ decreases with $\psi$ if $\psi > 4$.
\end{proof}
\begin{figure}[!t]
\vspace{-0.1cm}
\centering
\includegraphics[width=0.5\textwidth]{Figures/optimal_density_marked.pdf}
\caption{The variation of coverage probability with relative density $\psi$ for mmWave network. For illustration purpose, the SINR threshold is selected as $\tau=10$dB and $\tau = 15$dB, which satisfies the SINR threshold condition i.e. $\tau \leq 15$dB in Lemma \ref{lem: optimal relative density}. It can be observed that $\mathcal{C}_\psi(\tau,k)$ increases with $\psi$ in the power-limited regime i.e. $\psi < 2$ and decay with $\psi$ in the interference-limited regime i.e. $\psi > 4$.}
\label{fig:optimal_density}
\vspace{-20pt}
\end{figure}
For a mmWave network, the relative density threshold $\psi^*(\tau,k)$ corresponds to the AP density that is sufficiently large in terms of the desired signal strength, yet not over-densifying in terms of the aggregate interference.
In Fig.\ref{fig:optimal_density}, we numerically demonstrate the range of $\psi^*(\tau,k)$ by plotting the variation of coverage probability $\mathcal{C}_{\psi}(\tau,k)$ with the relative density $\psi$. Here, the SINR thresholds are chosen as $\tau = 10$dB and $\tau = 15$dB for illustration purposes.
It is observed that for the relative density $\psi < 2$, the coverage probability will always benefit from increasing $\psi$. This is because the coverage probability $\mathcal{C}_\psi(\tau,k)$ with the relative density $\psi < 2$ is limited by the desired signal strength for all the SINR threshold $\tau < 15$dB. It follows that the desired signal strength increases with the AP density, which results in a significant improvement in the coverage probability.
However, for the mmWave network of relative density $\psi\geq4$, the interference dominates the coverage probability. Therefore, increasing $\psi$ will harm the coverage probability due to the increase in the interfering APs.
Fig.\ref{fig:optimal_density} also numerically demonstrates that with the SINR threshold $\tau \leq 15\text{dB}$, the relative density threshold $\psi^*(\tau,k)$ ranges from 2 to 4, which agrees with the statement in Lemma \ref{lem: optimal relative density}.
\subsection{Densification Gain}
Next, we introduce the densification gain to study the effect of AP densification on the throughput of mmWave networks.
For a mmWave network of relative density $\psi$, the densification gain is defined as the following ratio,
\begin{eqnarray}
\gamma(\psi, k) \triangleq \frac{\mathcal{T}(\rho(\psi); \psi, k)}{\mathcal{T}(\rho_0; 1, k)}
= \frac{\mathcal{T}(\psi\rho_0; \psi, k)}{\mathcal{T}(\rho_0; 1, k)},
\label{def: densification gain}
\end{eqnarray}
where $\rho_0$ denotes the rate threshold when the mmWave network has the relative density $\psi = 1$; the throughput $\mathcal{T}(\rho(\psi); \psi, k)$ with the rate threshold $\rho(\psi)$ is given in (\ref{eq: rate coverage}). Note that we choose the rate threshold $\rho(\psi) = \psi\rho_0$. Correspondingly, by assuming that the interference can be completely eliminated, the throughput of mmWave network $\lambda_{\text{U}}\rho(\psi)$ can scale linearly with $\psi$.
Note that the value of densification gain $\gamma(\psi, k)$ characterizes the relationship between the AP density and the throughput of mmWave network as follows. If the densification gain $\gamma(\psi,k)$ equals to $\psi$, then it follows from (\ref{def: densification gain}) that $\mathcal{T}(\psi\rho_0; \psi, k) = \psi \mathcal{T}(\rho_0; 1, k)$, which means that the throughput scales linearly with the relative density $\psi$. For the case that $\gamma(\psi,k)$ is upped bounded by a finite constant denoted by $a$, the throughput $\mathcal{T}(\psi\rho_0; \psi, k)$ converges to a finite number $a \mathcal{T}(\rho_0; 1, k)$, which illustrates the existence of densification plateau for mmWave networks.
Next, we derive the densification gain $\gamma(\psi, k)$ for different relative densities $\psi$. Following Lemma \ref{lem: optimal relative density}, the relative density $\psi < \psi^*(\tau,k)$ and $\psi > \psi^*(\tau,k)$ correspond to the power-limited regime and the interference-limited regime, respectively. We start to derive the densification gain $\gamma(\psi,k)$ for the power-limited mmWave network i.e. $\psi < 2$.
\begin{lem}
For the mmWave network of the relative density $\psi < 2$, the densification gain $\gamma(\psi,k)$ satisfies the inequality $\gamma(\psi,k) \geq \psi$.
\label{lem: densification gain - power-limited regime}
\end{lem}
\begin{proof}
It is equivalently to prove $\mathcal{T}(\psi \rho_0; \psi, k) \geq \psi\mathcal{T}(\rho_0; 1, k)$. By approximating the bandwidth $W$ by the average bandwidth in (\ref{eq: average bandwidth}), it follows from Lemma \ref{lem: optimal relative density} that the throughput $\mathcal{T}(\psi \rho_0; \psi, k) = \lambda_\text{U}\psi\rho_0\mathcal{C}_\psi(2^{\rho_0/kW_0} - 1, k) \geq \psi \lambda_{\text{U}}\rho_0\mathcal{C}_1(2^{\rho_0/kW_0} - 1, k) = \psi\mathcal{T}(\rho_0; 1,k)$ with $W_0 = \frac{\xi\pi R_\text{B}^2\lambda_{\text{U}}}{B}$ for all $\psi < 2$ if the SINR threshold $\tau = 2^{\rho_0/kW_0} - 1 < 2^{\rho_0/W_0} - 1 < 15$dB. Note that for the mmWave network with the large bandwidth available, $\tau < 15$dB is considered to be a valid assumption \cite{bai2015coverage,rappaport2013broadband,rappaport2015wideband}. The Lemma then follows.
\end{proof}
Lemma \ref{lem: densification gain - power-limited regime} implies that for the power-limited mmWave network, the throughput increases at least linearly with the AP density.
Next, we move to the interference-limited mmWave network when the relative density $\psi$ becomes large. We first obtain the asymptotic result on the densification gain for the mmWave network without spatial multiplexing i.e. $k=1$.
\begin{lem}
For the mmWave network with the spatial multiplexing gain $k=1$, the densification gain $\gamma(\psi,1)$ is upper bounded by a finite constant as the relative density $\psi \rightarrow +\infty$.
\label{lem: plateau for k=1}
\end{lem}
\begin{proof}
See Appendix \ref{proof: lem - densification plateau}.
\end{proof}
Since the densification gain $\gamma(\psi,1)$ is saturated to a finite constant as $\psi$ increases, the existence of densification plateau for $k=1$ is then proved in Lemma \ref{lem: plateau for k=1}.
In the following, we continue to show that the densification gain $\gamma(\psi,k)$ is proportional to $\psi$ if the spatial multiplexing gain $k$ increases linearly with $\psi$.
\begin{lem}
For the interference-limited mmWave network, the densification gain $\gamma(\psi,k)$ is proportional to the relative density $\psi$ if the spatial multiplexing gain $k$ increases linearly with $\psi$ as $\psi \rightarrow +\infty$.
\label{lem: densification gain with increased k}
\end{lem}
\begin{proof}
Following the similar steps in (\ref{proofeq: limitation of SIR - upper}) of Appendix \ref{proof: lem - densification plateau}, we then have
\begin{eqnarray}
\lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \lambda_{\text{A}} \text{SINR}
&\geq&
\frac{ 1 }{2 \pi \sigma R_\text{B}^{\alpha_{\text{L}}}},
\label{proofeq: limitation of SIR - lower}
\end{eqnarray}
where $\sigma = \int_{0}^{R_\text{B}} r^{-\alpha_\text{L}+ 1} \text{d}r$ is a non-trivial finite constant.
It follows from Lemma \ref{lem: densification gain - power-limited regime} that $\mathcal{T}(\psi\rho_0; \psi, k) \approx \lambda_{\text{U}}\psi\rho_0\mathcal{C}_\psi(\tau,k)$ at $\tau = 2^{\frac{1}{k}\cdot \frac{\rho_0\xi\pi R_\text{B}^2\lambda_{\text{U}}}{B}}-1$. By letting $\tau_0 = \frac{1}{2\lambda_{\text{A}}} \cdot \frac{1}{2 \pi \sigma R_\text{B}^{\alpha_{\text{L}}}}$, we can conclude from (\ref{proofeq: limitation of SIR - lower}) that $\lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \text{SINR} \geq \lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \frac{1}{\lambda_{\text{A}}} \cdot \frac{1}{2 \pi \sigma R_\text{B}^{\alpha_{\text{L}}}} > \tau_0$. Note that $ \psi = \pi R_\text{B}^2 \lambda_\text{A}$ and $R_\text{B}$ is unchanged, thus $\lim\limits_{\psi \rightarrow +\infty}\text{SINR} = \lim\limits_{\lambda_{\text{A}} \rightarrow +\infty}\text{SINR}> \tau_0$ a.s.. It means that $\mathcal{C}_\psi(\tau_0,k) = \mathbb{P}(\text{SINR} > \tau_0)$ is a finite non-trial number if $\psi$ is sufficiently large. It follows that if $\lim\limits_{\psi \rightarrow +\infty}\tau= \tau_0$, then we have $\lim\limits_{\psi \rightarrow +\infty} \mathcal{C}_\psi(\tau,k) = \mathcal{C}_\psi(\tau_0,k)$. Note that $\lim\limits_{\psi \rightarrow +\infty}\tau= \lim\limits_{\psi \rightarrow +\infty} \; 2^{\frac{1}{k}\cdot \frac{\rho_0\xi\pi R_\text{B}^2\lambda_{\text{U}}}{B}}-1$ and $\tau_0 = \frac{1}{\psi} \cdot \frac{R_\text{B}^{-\alpha_{\text{L}} + 2}}{4\sigma}$. By using the fact that $\log_2(1 + 1/\psi) \approx 1/\psi$ as $\psi \rightarrow +\infty$, the proof then completes.
\end{proof}
Lemma \ref{lem: densification gain with increased k} implies that given the rate threshold $\rho$, the increased spatial multiplexing gain $k$ improves the coverage probability $\mathcal{C}_\psi(\tau,k)$ by lowering the SINR threshold $\tau$, which consequently drives the gain in the throughput $\mathcal{T}(\rho; \psi, k)$.
Note that an implicit assumption on the mmWave AP is that the spatial multiplexing gain satisfies $k\leq \frac{2\pi}{\theta_{\text{A}}}$, where $\theta_{\text{A}}$ is the main lobe width of the AP. Therefore, as $k$ increases with $\psi$, $\theta_{\text{A}}$ also needs to decrease with $\psi$.
\begin{figure}[!t]
\vspace{-0.1cm}
\begin{minipage}[t]{0.47\textwidth}
\centering
\includegraphics[width=3in]{Figures/Achievabel_Throughput_vs_psi_marked.pdf}
\caption{The throughput of mmWave network (normalized by $\lambda_{\text{U}}$), i.e., $\mathcal{T}(\rho(\psi); \psi,k)/\lambda_{\text{U}}$, with spatial multiplexing gain $k=1$ and $k=12$. The rate threshold of fixed-rate coding scheme is chosen as $\rho(\psi) = \psi\rho_0$ with $\rho_0 = 80$Mbps.}
\label{fig:achievable_throughput}
\end{minipage} %
\hfill
\begin{minipage}[t]{0.47\textwidth}
\centering
\includegraphics[width=3in]{Figures/Densification_Gain_vs_k_marked.pdf}
\caption{Densification gain $\gamma(\psi,k)$ vs. spatial multiplexing gains $k$ ($\rho_0 = $80Mz). Without spatial multiplexing i.e. $k=1$, the mmWave network reaches the densification plateau at $\psi = 4$ since $\gamma(\psi,1) \leq 4$. With $k\geq 8$, $\gamma(\psi,k) \geq \psi$ means the throughput scales at least linearly with $\psi$.}
\label{fig:densification_gain}
\end{minipage}
\vspace{-20pt}
\end{figure}
In Fig.\ref{fig:achievable_throughput}, we compare the throughput normalized by $\lambda_{\text{U}}$, i.e., $\mathcal{T}(\rho(\psi); \psi, k)/\lambda_{\text{U}}$ for the mmWave network with different relative densities $\psi$.
It can be seen from Fig.\ref{fig:achievable_throughput} that for the spatial multiplexing gain $k=1$, the AP densification beyond $\psi = 4$ does not provide any further improvement for the throughput of mmWave network, which illustrates that the mmWave network with $k=1$ reaches the densification plateau at $\psi = 4$.
In sharp contrast to $k=1$, the throughput of the mmWave network with $k=12$ can be improved by increasing the AP density for $\psi \leq 10$, which implies that the deployment of spatial multiplexing at mmWave APs overcomes the densification plateau for the mmWave network.
In Fig.\ref{fig:densification_gain} we show the variation of densification gain $\gamma(\psi,k)$ with $k$ for different relative densities $\psi$.
From Fig.\ref{fig:densification_gain}, it is clear that for the relative density $\psi = 2$, the densification gain $\gamma(\psi,k) \geq \psi$ holds for all $k$, as stated in Lemma \ref{lem: densification gain - power-limited regime}.
As the relative density $\psi$ increases from $2$ to $4$, the mmWave network transits from the power-limited region to the interference region.
Then the densification gain $\gamma(\psi,k)$ for $k>1$ diverges from $\gamma(\psi,1)$.
Fig. \ref{fig:densification_gain} numerically demonstrates Lemma \ref{lem: plateau for k=1} and \ref{lem: densification gain with increased k}.
It can be observed that with the spatial multiplexing gain $k=1$, the densification gain $\gamma(\psi,1)$ for different relative densities $\psi$ are clustered to a constant $a\leq 4$. It indicates that with $k=1$, only marginal throughput gain can be observed beyond $\psi = 4$. In other words, the mmWave network with $k=1$ reaches the densification plateau at the relative density $\psi = 4$.
In contrast to $k=1$, for $k>1$, the densification gain $\gamma(\psi,k)$ scales with $\psi$, which illustrates that the throughput can consistently increase with the AP density. It is equivalent to say that the densification plateau of the mmWave network can be avoided by deploying the spatial multiplexing gain $k\propto \psi$ at the APs.
\section{Numerical Results on mmWave Network Throughput}
This section shows the numerical results on throughput gains as the spatial multiplexing gain at mmWave APs increases.
Note that by choosing the optimal rate threshold $\rho^*$, the fixed-rate coding scheme provides the tightest lower bound for the throughput of mmWave networks, while the throughput upper bound is derived in (\ref{eq: bounds on throughput}).
Moreover, we compare the multi-rate coding scheme with the throughput upper and lower bounds, which motivates a performance-complexity trade-off for coding schemes in the mmWave network.
\subsection{Throughput Gain via Spatial Multiplexing}
We start to provide an intuitive explanation on how the spatial multiplexing gain affects the throughput of mmWave network.
Corollary \ref{cor: average bandwidth} shows that the average bandwidth of end-user scales linearly with $k$.
Consequently, increasing $k$ equivalently lowers the SINR threshold $\tau = 2^{\rho \Psi/B} - 1$ for the coverage probability $\mathcal{C}_\psi(\tau,k)$ in (\ref{eq: rate coverage in coverage probability}) and thus improves the network throughput $\mathcal{T}(\rho;\psi,k)$.
For completeness, the minor drawback of increasing $k$ is shown in Corollary \ref{cor: bound of coverage probability} as the coverage probability $\mathcal{C}_\psi(\tau,k)$ decreasing with $k$.
From the analytical viewpoint, by fixing the relative density $\psi$, the throughput in (\ref{eq: rate coverage}) is a polynomial of the spatial multiplexing gain $k$, while the polynomial's coefficient $\mathbb{E}_{\Psi}\left[ c_l \left(2^{\rho \Psi/ B} - 1,\psi \right)\right]$ and $k$ are also correlated due to the dependency of $\Psi$ on $k$, as shown in Theorem \ref{thm: PMF for tagged AP sector}.
To numerically demonstrate the effect of spatial multiplexing on the throughput of mmWave networks, $\mathcal{T}(\rho; \psi,k)/\lambda_{\text{U}}$ in (\ref{eq: rate coverage}) is plotted against $\rho$ for different spatial multiplexing gains $k$ in Fig.\ref{fig: throughput vs. rate -fixed rate coding scheme}. Here, the relative density of mmWave network is chosen as $\psi = 4$ and $\psi = 10$.
From Fig.\ref{fig: throughput vs. rate -fixed rate coding scheme}, it can be clearly seen that the throughput $\mathcal{T}(\rho; \psi, k)$ always increases with the spatial multiplexing gain $k$ regardless of the chosen $\psi$ and $\rho$. Hence, we can conclude that the deployment of spatial multiplexing at mmWave APs is always beneficial for the throughput of mmWave networks.
\begin{figure}[!t]
\vspace{-0.1cm}
\centering
\includegraphics[width=0.5\textwidth]{Figures/NT_optimal_rate_marked.pdf}
\caption{Comparison of the throughput $\mathcal{T}(\rho; \psi,k)/\lambda_{\text{U}}$ for the spatial multiplexing gain $k=1$ and $k=12$. Here, relative density $\psi = 4$ (in red) and $\psi = 10$ (in blue) represent the mmWave network with dense APs and ultra-dense APs, respectively. By deploying the spatial multiplexing gain $k=12$ at mmWave APs, the improvement in the throughput is clearly seen. Note that the fixed-rate coding scheme with the optimal $\rho^*(\psi,k)$ is used to provide the lower bound for mmWave network throughput.}
\label{fig: throughput vs. rate -fixed rate coding scheme}
\vspace{-20pt}
\end{figure}
\subsection{Throughput Upper and Lower Bounds}
By deploying the fixed-rate coding scheme, the throughput $\mathcal{T}(\rho; \psi, k)$ of the mmWave network achieves its maximum value when the rate threshold $\rho$ is chosen such that $\rho^*(\psi,k) = \text{arg}\max\limits_{\rho\in\mathbb{R}^+}\,\mathcal{T}(\rho; \psi, k)$.
In Fig.\ref{fig: throughput vs. rate -fixed rate coding scheme}, we show the variation of throughput normalized by $\lambda_{\text{U}}$, i.e., $\mathcal{T}(\rho; \psi, k)/\lambda_{\text{U}}$ with the rate threshold $\rho$. It can be seen that with the same relative density $\psi$, the optimal rate threshold $\rho^*(\psi,k)$ increases as the spatial multiplexing gain $k$ increases. If $k$ is fixed, then the mmWave network with higher relative density $\psi$ requires a larger rate threshold $\rho^*(\psi,k)$ to maximize the throughput $\mathcal{T}(\rho; \psi, k)$.
By choosing the optimal rate threshold $\rho^*(\psi,k)$, the fixed-rate coding scheme then provides the tightest lower bound for the throughput of mmWave network.
In Fig.\ref{fig:NT_vs_k}, we plot the throughput upper bound $\overline{\mathcal{T}}(\psi, k)/\lambda_\text{U}$ in (\ref{eq: bounds on throughput}) and the tightest lower bound $\mathcal{T}(\rho^*; \psi, k)/\lambda_{\text{U}}$ for the mmWave network with different spatial multiplexing gains. Here, we choose the relative densities as $\psi = 4$ and $\psi = 10$, which correspond to the dense and ultra-dense deployments of mmWave APs, respectively.
It can be seen in Fig.\ref{fig:NT_vs_k} that both the upper bound and the tightest lower bound of the network throughput increases as the spatial multiplexing gain $k$ increases. It illustrates that the overall performance of the mmWave network is improved by increasing $k$.
By comparing the upper bound and the tightest lower bound in Fig.\ref{fig:NT_vs_k}, it can be observed that the gap between the two bounds grows when the spatial multiplexing gain $k$ increases. Consequently, we can conclude that for the mmWave network, the performance of the fixed-rate coding scheme becomes less satisfactory as the spatial multiplexing gain $k$ increases.
\begin{figure}
\vspace*{-0pt}
\centering
\subfloat[{\scriptsize Relative density $\psi = 4$ represents the dense deployments of mmWave APs.} \label{fig:NT_vs_k_psi4}]{%
\includegraphics[width=0.48\linewidth]{Figures/NT_vs_k_psi4.pdf}}
\hfill
\subfloat[{\scriptsize Relative density $\psi = 10$ represents the ultra-dense deployments of mmWave APs.}\label{fig:NT_vs_k_psi10}]{%
\includegraphics[width=0.48\linewidth]{Figures/NT_vs_k_psi10.pdf}}
\caption{\textcolor{black}{Comparison of the throughput upper and lower bounds for the mmWave network, where the throughput achieved by the multi-rate coding scheme lies in between.}}
\label{fig:NT_vs_k}
\vspace*{-20pt}
\end{figure}
\subsection{Throughput of Multi-Rate Coding Scheme}
To achieve the throughput upper bound $\overline{\mathcal{T}}(\psi, k)$, each mmWave AP is required to deploy the coding scheme that can be instantaneously updated according to the SINR value of each end-user. However, the operational complexity of such a strategy is prohibitively high, which renders it impractical to be implemented in the mmWave network.
In sharp contrast to the upper bound, the implementation of the fixed-rate coding scheme is simple.
However, by fixing the rate threshold, any received rate higher than $\rho$ cannot be explored. Moreover, the received rate smaller than $\rho$ is always considered as an outage for the mmWave end-user. Therefore, the fixed-rate coding scheme results in a sub-optimal network throughput $\mathcal{T}(\rho; \psi, k)$.
Note that the multi-rate coding scheme defined in (\ref{eq: data rate of end-user - multi-rate coding scheme}) is a compromise between the operational complexity and the throughput performance \cite{alammouri2018unified}. It follows from (\ref{eq: throughput of multi-rate coding scheme}) that the multi-rate coding scheme is expected to achieve a throughput in between the upper and lower bounds. While the comprehensive study of a multi-rate coding scheme is out of the scope, in Fig.\ref{fig:NT_vs_k} we numerically show the throughput corresponding to the rate thresholds $\{\rho_i\}_{i=1}^{10}$ with $\rho_i = 1+30(i-1)$ in units of MHz.
As can be seen, by employing such a multi-rate coding scheme, the mmWave network can achieve almost the throughput upper bound when the spatial multiplexing gain is small. Moreover, the throughputs provided by the multi-rate coding scheme increase by more than two times compared to the tightest lower bounds. Hence, we can conclude that the multi-rate coding scheme is necessary to guarantee the performance of mmWave network, especially when the spatial multiplexing gain at APs is large.
\section{Conclusion}
We proposed to model the downlink mmWave network incorporating the hybrid precoding gain, where both the analog beamforming gain and the spatial multiplexing gain $k$ were considered.
We then derived the coverage probability, the throughput of the fixed-rate coding scheme, and the throughput upper bound. With these analytical expressions, we separated the positive and negative effects of AP densification on mmWave networks.
Our analysis then proved that without the spatial multiplexing, over-densifying APs leads to the densification plateau for the network throughput. To overcome such a performance bottleneck, the mmWave APs need to deploy the spatial multiplexing, primarily because the spatial multiplexing gain improves the throughput by lowering the SINR threshold for mmWave end-users.
We optimized the fixed-rate coding scheme to provide the throughput lower bound, while the throughput achieved by the multi-rate coding scheme was also evaluated for different $k$. For comparison, we also quantified the throughput upper bound. Our numerical results verified that all these throughputs increase as the spatial multiplexing gain $k$ increases. However, the gap between the throughput upper and lower bounds increases rapidly with $k$, which implied that the fixed-rate coding scheme is inefficient in exploiting the potential throughput gain for mmWave networks with large $k$. Further, the multi-rate coding scheme showed a satisfactory performance when $k$ grows, which triggered a potential study on the complexity-performance trade-off for the coding-scheme of mmWave networks.
\appendices
\section{\label{proof: coverage probability of LOS ball}}
With the LOS-ball model in (\ref{LOS-ball}), the coverage probability in (\ref{eq: LOS coverage probability}) can be written as (\ref{eq: simplified coverage probability}) with
\begin{eqnarray}
\Lambda_j(\tau,r,G)
&\triangleq&
\frac{2r}{\alpha_{\text{L}}} \left( j \eta \tau G \right)^{\frac{2}{\alpha_{\text{L}}}}
\Gamma\left( -\frac{2}{ \alpha_{\text{L}}}; j \eta \tau G , j \eta \tau G r^{\frac{\alpha_{\text{L}}}{2}}\right)
\label{eq: SINR parameter definition}\\
&=&
\frac{2r}{\alpha_{\text{L}}} \left( j \eta \tau G \right)^{\frac{2}{\alpha_{\text{L}}}}
\int
_{j \eta \tau G r^{\frac{\alpha_{\text{L}}}{2}}}
^{j \eta \tau G}
t^{-\frac{2}{\alpha_{\text{L}}} - 1} e^{-t}
\;\text{d}t.
\label{eq: taylor parameter}
\end{eqnarray}
Note that $ \Lambda_j(\tau,r,G)$ is in the form of incomplete gamma function.
Here, we show the sketch of deriving the coverage probability of mmWave network with LOS-ball of radius $R_\text{B}$.
We start to derive the PDF of the distance from a end-user to its nearest AP in $\Phi_{\text{A}}$, which is denoted by $r$. Without loss of generality, consider the end-user at the origin. Following the network model, the APs which are LOS to the user form an inhomogeneous PPP with density $\lambda_{\text{A}}$ for $r\leq R_\text{B}$.
It follows from \cite{andrews2011tractable} that the PDF and CDF of $r$ can be derived as
\begin{eqnarray}
f_{\text{L}}(r) = 2 \pi r \lambda_\text{A} e^{-2 \pi \lambda_\text{A} \int_{0}^{r}xdx}, \nonumber\\
F_{\text{L}}(r) = 1-e^{-2\pi \lambda_{\text{A}} \int_{0}^{r} {x \text{d}x} }.
\label{eq: f_l and F_l}
\end{eqnarray}
for $r\leq R_\text{B}$.
For the end-user at the origin, the coverage probability can be written as,
\begin{equation}
\mathcal{C}_\psi(\tau,k) \triangleq \mathbb{P}\left( \text{SINR}>\tau \left| \mathbb{P}(\mathbb{G}_\text{A} = G_\text{A}) = k\cdot\frac{\theta_{\text{A}}}{2\pi} \right. \right) = \mathbb{E}_{r}\left[\mathcal{C}_\psi (\tau,k,r)\right], \nonumber
\end{equation}
where $\mathcal{C}_\psi (\tau,k,r) \triangleq \mathbb{P}( \text{SINR}>\tau \;|\; |x_0| = r, \mathbb{P}(\mathbb{G}_\text{A} = G_\text{A}) = k\cdot\frac{\theta_\text{A}}{2\pi} )$ is the coverage probability conditioned on the distance $r$ from the origin to the AP at $x_0$.
To derive $\mathcal{C}_\psi (\tau,k,r)$, we need to first derive the Laplace transform of the interference $\mathcal{I}$ in (\ref{eq: interference of LOS}) given that $r = |x_0|$, as shown in \cite[]{andrews2011tractable}.
It follows from \cite[Theorem 4.9]{haenggi2012stochastic} that the Laplace transform of $\mathcal{I}$ evaluated at $s$, denoted by $\mathcal{L}_{\mathcal{I}}(s)$, can be given as
\begin{eqnarray}
\mathcal{L}_{\mathcal{I}}(s) \triangleq \mathbb{E} \left[ e^{-s \mathcal{I}} \right]
= \int_{r}^{R_\text{B}} \left(
1-\mathbb{E}
\left[
e^{- s h \mathbb{G}_\text{A} \mathbb{G}_\text{U} x^{-\alpha_{\text{L}}}}
\right]
\right) x \; \text{d}x ,
\label{eq: Laplace Transform derivation}
\end{eqnarray}
where $h = 1$ means no fading in deterministic channel. As shown in \cite[Appendix D]{bai2015coverage}, $h$ can be considered as a gamma random variable with the shape parameter goes to infinity.
By using Alzer's Lemma in \cite{alzer1997some}, the conditional coverage probability can be written as follows:
\begin{eqnarray}
\mathcal{C}_{\psi} (\tau,k,r)
&\approx&
{\sum_{j=1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\mathcal{L}_{\mathcal{I}}
\left( \tau \Omega j \right) ,
\label{eq: conditional SINR Nakagami LOS}
\end{eqnarray}
where $\mu$ is the shape parameter of channel fading $h$ and $\mu \geq 10$ is a good estimation on the deterministic channel \cite{bai2015coverage}; and $\Omega = \frac{\eta r^{\alpha_\text{L}} }{G_{\text{A}}G_{\text{U}}}$ with $\eta = \mu (\mu !)^{-\frac{1}{\mu}}$.
The rest of proof follows the same steps in \cite[Theorem 3]{bai2015coverage}.
\section{\label{proof: taylor expansion of coverage probability}}
The coefficients $c_0(\tau)$ and $c_l(\tau,\psi)$ in (\ref{eq: SINR interms of polynomial}) are defined as
\begin{eqnarray}
c_0 (\tau,\psi) = \psi e^{-\psi}
{\sum_{j=1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\int_{0}^{1}
\exp
\left\lbrace
\psi
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r,\frac{g_\text{A}}{G_\text{A}} \right)
\right.\nonumber\\
+
\left.
\psi
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
\right\rbrace
\text{d} r,
\label{eq: coefficient c_0}
\end{eqnarray}
\begin{eqnarray}
c_l (\tau,\psi) &=& \frac{\psi e^{-\psi}}{l!}
{\sum_{j=1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\int_{0}^{1}
e^
{
\frac{\psi\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r,\frac{g_\text{A}}{G_\text{A}} \right)
+
\psi\left(1- \frac{\theta_\text{U}}{2\pi} \right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
}
\nonumber\\
&&
\times
\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l
\left[
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r,1\right)
\right.
\left.
+
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{U}}{G_\text{U}}\right)
\right.
\nonumber\\
&&
\left.
-
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r, \frac{g_\text{A}}{G_\text{A}}\right)
-
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
\right]^l
\text{d} r, \;\;\; l\in\mathbb{N}^+.
\label{eq: coefficient c_l}
\end{eqnarray}
Here, $\Lambda_j(\tau,r,G)$ is given in (\ref{eq: taylor parameter}).
We start to prove that $c_0 = \mathbb{P}(\text{SNR} > \tau)$. By ignoring the thermal noise \cite{bai2015coverage}, the background noise is solely resulted by the side lobes of interfering APs. It is equivalent to letting the hybrid precoding gain $\mathbb{G}_\text{A} = g_\text{A}$ for all $x\in \Phi_{\text{A}}\backslash \{x_0\}$. By plugging $\mathbb{G}_\text{A} = g_\text{A}$ into the coverage probability in (\ref{eq: SINR interms of polynomial}), the expression of $c_0(\tau,\psi)$ immediately follows.
Next, we derive the Taylor series expansion for the expression in (\ref{eq: simplified coverage probability}).
By denoting $
\mathcal{F}_{\psi}(k,\tau,j)
=
\int_{0}^{1} \exp
\left\lbrace
\psi
\mathbb{E}
\left[
\Lambda_j
\left(\tau,r,\frac{\mathbb{G}_\text{A}\mathbb{G}_\text{U}}{G_\text{A}G_\text{U}} \right)
\right]
\right\rbrace
\text{d} r$,
the coverage probability in (\ref{eq: simplified coverage probability}) can then be written as
\begin{equation}
\mathcal{C}_{\psi}(\tau,k)
=
\psi e^{-\psi}{\sum_{j =1}^{\mu}(-1)^{j+1}{\mu\choose j}} \mathcal{F}_{\psi}(k,\tau,j).
\label{proofeq: coverage probability with F}
\end{equation}
By plugging in the hybrid precoding gain $\mathbb{G}_\text{A}$ and $\mathbb{G}_\text{U}$, the term $\mathcal{F}_{\psi}(k,\tau,j)$ can be further expanded as follows:
\begin{eqnarray}
\mathcal{F}_{\psi}(k,\tau,j)
&=&
\int_{0}^{1}
\exp
\left\lbrace
\psi
\left[
k
\frac{\theta_{\text{A}}\theta_{\text{U}}}{4\pi^2}
\Lambda_j\left( \tau,r,1\right)
+
k
\frac{\theta_{\text{A}}}{2\pi}
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{U}}{G_\text{U}}\right)
\right.
\right.
\nonumber\\
&&
\left.
\left.
+
\left(
1-\frac{k\theta_{\text{A}}}{2\pi}
\right)
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r, \frac{g_\text{A}}{G_\text{A}}\right)
+
\left(
1-\frac{k\theta_{\text{A}}}{2\pi}
\right)
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
\right]
\right\rbrace
\text{d} r.
\nonumber
\end{eqnarray}
It follows that the $l^\text{th}$ derivative of $\mathcal{F}_{\psi}(k,\tau,j)$ can be written as
\begin{eqnarray}
\mathcal{F}_{\psi}^{(l)}(k,\tau,j)
&=&
\int_{0}^{1}
\frac{\partial^l}{\partial k^l}
\exp
\left\lbrace
\psi
\left[
k
\frac{\theta_{\text{A}}\theta_{\text{U}}}{4\pi^2}
\Lambda_j\left( \tau,r,1\right)
+
k
\frac{\theta_{\text{A}}}{2\pi}
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{U}}{G_\text{U}}\right)
\right.
\right.
\nonumber\\
&&
\left.
\left.
+
\left(
1-\frac{k\theta_{\text{A}}}{2\pi}
\right)
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r, \frac{g_\text{A}}{G_\text{A}}\right)
+
\left(
1-\frac{k\theta_{\text{A}}}{2\pi}
\right)
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
\right]
\right\rbrace
\text{d} r,
\nonumber
\end{eqnarray}
which follows from the Leibniz integral rule.
Thus, the Taylor series expansion of $\mathcal{F}_{\psi}(k,\tau, j)$ at $k = 0$ can be written as
\begin{eqnarray}
&&\mathcal{F}_{\psi}(k,\tau, j) = \sum\limits_{l=0}^{\infty}
\frac{\mathcal{F}_{\psi}^{(l)}(0,\tau,j)}{l!} k^l \nonumber\\
&=&
\sum_{l=0}^{\infty}
\frac{k^l}{l!}
\int_{0}^{1}
e^
{
\frac{\psi\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r,\frac{g_\text{A}}{G_\text{A}} \right)
+
\left(\psi- \frac{ \psi \theta_\text{U}}{2\pi} \right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
}
\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l
\left[
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r,1\right)
+
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{U}}{G_\text{U}}\right)
\right.
\nonumber\\
&&
\left.
-
\frac{\theta_{\text{U}}}{2\pi}
\Lambda_j\left( \tau,r, \frac{g_\text{A}}{G_\text{A}}\right)
-
\left(
1-\frac{\theta_{\text{U}}}{2\pi}
\right)
\Lambda_j\left(\tau,r,\frac{g_\text{A}g_\text{U}}{G_\text{A}G_\text{U}}\right)
\right]^l
\text{d} r.
\label{proofeq: Taylor expansion}
\end{eqnarray}
Now we prove that the Taylor series in (\ref{proofeq: Taylor expansion}) converge to $\mathcal{F}_{\psi}(k,\tau, j)$ in the region $k \in [0,+\infty)$. Following the definition of $\Lambda_j\left( \tau,r,G\right)$ in (\ref{eq: taylor parameter}), we have
\begin{eqnarray}
0 < \Lambda_j\left( \tau,r,G\right)
\stackrel{(a)}{<}\frac{2r}{\alpha_{\text{L}}} \left( j \eta \tau G \right)^{\frac{2}{\alpha_{\text{L}}}}
\int
_{j \eta \tau G r^{\frac{\alpha_{\text{L}}}{2}}}
^{j \eta \tau G}
t^{-\frac{2}{\alpha_{\text{L}}} - 1}
\;\text{d}t = 1-r, \quad\forall G>0,\; r\in(0,1],
\label{proofeq: Taylor parameter upper bound}
\end{eqnarray}
where (a) follows from $ e^{-t}< 1$ for $t > 0$.
Then, the absolute value of $\mathcal{F}_{\psi}^{(l)}(0,\tau,j)$ can be upper bounded by
\begin{eqnarray}
|\mathcal{F}_{\psi}^{(l)}(0,\tau,j)|
<
\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l
\int_{0}^{1}
e^
{ \psi(1-r) }
(1-r)^l
\text{d} r
< \frac{e^\psi}{l+1}\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l.
\label{proofeq: Taylor expansion upper bound}
\end{eqnarray}
Therefore, $\mathcal{F}_{\psi}(k,\tau, j)$ is real analytic on $k \in [0,+\infty)$.
Define
\begin{equation}
c_l (\tau, \psi) \triangleq \frac{\psi e^{-\psi}}{l!}
{\sum_{j=1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\mathcal{F}_{\psi}^{(l)}(0,\tau,j).
\label{proofeq: c_l in F}
\end{equation}
The expression of $c_l(\tau,\psi)$ for $l \in \mathbb{N}^+$ in (\ref{eq: coefficient c_l}) then follows.
Define
$c_l (\tau, \psi, k) \triangleq \frac{\psi e^{-\psi}}{l!}
{\sum_{j=1}^{\mu}(-1)^{j+1}{\mu\choose j}}
\mathcal{F}_{\psi}^{(l)}(k,\tau,j)$, where $k \in [1,K]$ and $K= \frac{2\pi}{\theta_{\text{A}}}$.
Next, we start to derive the bound for $\mathcal{F}_{\psi}^{(l)}(k,\tau,j)$ i.e. the $l^\text{th}$ derivative of $\mathcal{F}_{\psi}(k,\tau,j)$. By following the similar steps in the derivation of (\ref{proofeq: Taylor expansion upper bound}), we have
\begin{eqnarray}
|\mathcal{F}_{\psi}^{(l)}(k,\tau,j)|
< \frac{e^{2\psi}}{l+1}\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l.
\label{proofeq: Parameter F bound}
\end{eqnarray}
Given the degree of polynomial $L$, the approximation error is upper bounded by
\begin{eqnarray}
& \max\limits_{k\in[1,K],\tau\in\mathbb{R}^+} &\left| \mathcal{C}_{\psi}(\tau,k) - \sum\limits_{l=0}^{L} c_l (\tau,\psi) k^l \right|
= \left| \sum\limits_{l=L+1}^{\infty} c_l (\tau,\psi) K^l \right|
\nonumber\\
&\stackrel{(a)}{=}&
K^{L+1}
| c_{L+1}(\tau, \psi, k^*) |
\qquad
\text{for some}\; k^* \in [1,K] \nonumber \\
&\stackrel{(b)}{<}&
\frac{ e^\psi \psi^{L+2} }{(L+2)!}
{\sum_{j=1}^{ \mu}
{\mu\choose j }}
\left(
\frac{K \theta_{\text{A}}}{2\pi}
\right)^{L+1} \nonumber
\stackrel{(c)}{=}
\frac{(2^{\mu}-1) e^\psi \psi^{L+2}}{(L+2)!},
\nonumber
\end{eqnarray}
where (a) follows the Lagrange's formula; (b) follows the inequality (\ref{proofeq: Parameter F bound}); (c) follows $K = \frac{2\pi}{\theta_{\text{A}}}$.
\section{\label{proof: bandwidth distribution}}
Consider the mmWave network, where the channel follows the path loss model $r^{-\alpha_{\text{L}}}$ with no fading. According to the minimum path loss association rule, the serving area of each AP can be modeled by a Voronoi cell \cite{andrews2011tractable}. For $\Phi_{\text{A}}$ of intensity $\lambda_{\text{A}}$, the size distribution of a Voronoi cell is derived in \cite{ferenc2007size},
with the PDF
\begin{equation}
f(y) \propto y^{2.5}\exp{(-3.5{\lambda_{\text{A}}}y)}, \;\;\forall\; y>0.
\label{proofeq: Voronoi cell}
\end{equation}
Based on the PDF of Voronoi cell in (\ref{proofeq: Voronoi cell}), we then derive the size distribution of an AP sector. Let a randomly selected AP be the origin. It follows from the isotropic property of the PPP $\Phi_\text{U}$ that the end-users are isotropically distributed around the origin. Thus, the serving region of the selected AP is isotropically distributed around the origin. The area distribution of an AP sector can then be written as
\begin{equation}
f_{\mathcal{A}}(y) =
\left\lbrace
\begin{split}
&\frac{3.5^{3.5}{k\lambda_{\text{A}}}}{\Gamma(3.5, \frac{\psi}{3.5k^2\lambda_{\text{A}}^2})}
({k\lambda_{\text{A}}}y)^{2.5}\exp{(-3.5{k\lambda_{\text{A}}}y)},
\; \text{for}\; y \leq \frac{\psi}{k\lambda_{\text{A}}}\\
&0, \; \text{for} \; y > \frac{\psi}{k\lambda_{\text{A}}}
\end{split}.
\right.
\label{eq: the area distribution of AP sector}
\end{equation}
Here, $\Gamma(s,x) = \int_{0}^{x} t^{s-1}e^{-t} \text{d}t$ is the incomplete gamma function.
We remark that the serving area of a mmWave AP is restricted to the LOS-ball with radius $R_\text{B}$. Therefore, for the mmWave AP sector, $f_{\mathcal{A}}(y) = 0$ when $y>\frac{1}{k} \pi R_\text{B}^2 = \frac{\psi}{k\lambda_{\text{A}}}$.
The end-user located at the origin is served by the tagged AP. We use $\mathcal{A}'$ to denote the serving area within a sector of the tagged AP. It can be proved by a minor modification of \cite[lemma2]{yu2013downlink} that the distributions of $\mathcal{A}'$ and $\mathcal{A}$ in (\ref{eq: the area distribution of AP sector}) are related as follows:
\begin{eqnarray}
f_{\mathcal{A}'}(y)& \propto& y f_{\mathcal{A}}(y) \label{biased sector} \nonumber\\
&=& \frac{3.5^{4.5}{k\lambda_\text{A}}}{\Gamma(4.5, \frac{\psi}{3.5k^2\lambda_{\text{A}}^2})}({k\lambda_\text{A}}y)^{3.5}\exp{(-3.5{k\lambda_\text{A}}y)}, y \leq \frac{\psi}{\lambda_{\text{A}}},
\label{proofeq: biased sector 2}
\end{eqnarray}
where $\Gamma(s,x)$ is the incomplete gamma function $\Gamma(s,x) = \int_{0}^{x} t^{s-1}e^{-t} \text{d}t$. It follows from (\ref{proofeq: biased sector 2}) that the area distribution of the tagged AP sector is biased since the end-user at the origin is more likely to lie in the larger area.
Note that $\Psi$ represents the number of end-users that share the same pool of radio resources as the origin.
By following \cite{singh2013offloading}, the PMF of $\Psi$ can be obtained as follows:
\begin{eqnarray}
\mathbb{P}({\Psi}= n)
&=& \mathcal{K}_\text{T}(n,k; \psi,\lambda_{\text{U}})
= \frac{\left. \left(\int_{0}^{\psi/\lambda_{\text{A}}} {\exp{(\lambda_\text{U}y(z-1))}} f_{\mathcal{A}'}(y) \, \text{d}y\right)^{(n-1)} \right|_{z=0}}{(n-1)!} \nonumber\\
&=&
\frac{3.5^{4.5} \; \Gamma\left(n+3.5, \frac{\psi}{k\lambda_{\text{A}}(\lambda_{\text{U}}+3.5k\lambda_\text{A})}\right)}
{(n-1)! \; \Gamma\left(4.5,\frac{\psi}{3.5k^2\lambda_{\text{A}}^2}\right)}
\left(\frac{\lambda_\text{U}}{k\lambda_\text{A}}\right)^{n-1}
\left(3.5+\frac{\lambda_\text{U}}{k\lambda_\text{A}}\right)^{-n-3.5}, n\geq 1,
\label{eq: tagged end-user load}
\end{eqnarray}
where $\lambda_{\text{A}} = \psi/\pi R_\text{B}^2$ with $R_\text{B}$ being the radius of LOS ball; $\Gamma(s,x)$ is the incomplete gamma function $\Gamma(s,x) = \int_{0}^{x} t^{s-1}e^{-t} \text{d}t$.
Note that $n \geq 1$ since $\Psi$ always contains the end-user at the origin.
By assuming the relative density $\psi \geq 1$, the end-user can observe at least one LOS AP \cite{bai2015coverage}, which results in
$\Gamma\left(n+3.5, \frac{\psi}{\lambda_{\text{A}}(\lambda_{\text{U}}+3.5k\lambda_\text{A})}\right) \approx \Gamma(n+ 3.5)$ and
$\Gamma\left(4.5,\frac{\psi}{3.5k\lambda_{\text{A}}^2}\right) \approx \Gamma(4.5)$. It follows that
\begin{equation}
\mathbb{P}\left(\Psi= n\right) =\frac{3.5^{3.5} \; \Gamma\left( n+3.5 \right)}
{(n-1)! \; \Gamma\left( 3.5 \right)}
\left(\frac{\pi R_\text{B}^2 \lambda_\text{U}}{k \psi}\right)^{n-1}
\left(3.5+\frac{\pi R_\text{B}^2 \lambda_\text{U}}{k \psi}\right)^{-n-3.5}, n\geq 1,
\label{eq: simplied tagged load}
\end{equation}
where $\Gamma(s) = \int_{0}^{+\infty} t^{s-1}e^{-t} \text{d}t$ is the gamma function; $R_\text{B}$ is the radius of the LOS-ball.
\section{\label{proof: throughput upper bound}}
To derive the analytical expression of the throughput upper bound, we notice that
\begin{eqnarray}
\mathbb{E} [ \log_2 (1 + \text{SINR})]
&\stackrel{(a)}{=}& \log_2(e) \int_{0}^{\infty} {\mathbb{P}(\ln (1 + \text{SINR}) > t)} \; \text{d}t \nonumber \\
&\stackrel{(b)}{=}& \log_2(e) \int_{0}^{\infty} {\frac{\mathbb{P}(\text{SINR} > \tau)}{1 + \tau}} \; \text{d}\tau
= \log_2(e) \int_{0}^{\infty} {\frac{\mathcal{C}_{\psi}(\tau,k)}{1 + \tau}} \; \text{d}\tau,
\label{proofeq: upper bound with P_cov}
\end{eqnarray}
where (a) follows from that for a positive random variable $X$, $\mathbb{E}(X) = \int_{t>0} \; \mathbb{P}(X > t) \; \text{d}t$; (b) is derived by the change of random variable $t = \ln(\tau + 1)$. It follows from Theorem \ref{thm: taylor expansion of coverage probability} that
\begin{eqnarray}
\mathbb{E} [ \log_2 (1 + \text{SINR})]= \log_2(e)
\sum\limits_{l=0}^{\infty} k^l
\int_{0}^{\infty} \frac{c_l (\tau,\psi)}{1+\tau} \text{d}\,\tau ,
\label{eq: spectral efficiency upper bound}
\end{eqnarray}
We then complete the proof of (\ref{eq: spectral efficiency upper bound}) by showing $\int_{0}^{\infty} \frac{c_l (\tau,\psi)}{1+\tau} \text{d}\,\tau < +\infty$. To that end, we first derive the bound for $\Lambda_j\left( \tau,r,G\right)$ in (\ref{eq: taylor parameter}), where
\begin{eqnarray}
0 < \Lambda_j\left( \tau,r,G\right)
< e^{-j\eta\tau G r^{\alpha_{\text{L}}/2}}(1-r), \quad\forall G>0,\; r\in(0,1].
\label{proofeq: tighter bound on Taylor parameter}
\end{eqnarray}
It follows from (\ref{proofeq: Taylor parameter upper bound}) that
$\mathcal{F}_{\psi}^{(l)}(0,\tau,j)$ in Appendix \ref{proof: taylor expansion of coverage probability} can be upper bounded by
\begin{eqnarray}
|\mathcal{F}_{\psi}^{(l)}(0,\tau,j)|
&<&
\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l
\left[
\int_{0}^{\frac{1}{(1+\tau)^{1/\alpha_{\text{L}}}}}
e^
{ \psi(1-r) }
(1-r)^l
\text{d} r
+
\int_{\frac{1}{(1+\tau)^{1/\alpha_{\text{L}}}}}^{1}
e^
{ \psi(1-r)-j\eta G l \tau r^{\alpha_\text{L}/2} }
(1-r)^l
\text{d} r
\right] \nonumber\\
&<&
e^\psi
\left(
\frac{\psi\theta_{\text{A}}}{2\pi}
\right)^l
\left(
\frac{1}{(1+\tau)^{\frac{1}{\alpha_{\text{L}}}}} + e^{-j\eta G l\frac{\tau}{\sqrt{1+\tau}}}
\right),
\end{eqnarray}
which illustrates $\int_{0}^{\infty} \frac{|\mathcal{F}_{\psi}^{(l)}(0,\tau,j)|}{1+\tau} \text{d}\,\tau < +\infty$. Further, $c_l(\tau,\psi)$ in (\ref{proofeq: c_l in F}) is a finite sum of $\mathcal{F}_{\psi}^{(l)}(0,\tau,j)$ and thus $\int_{0}^{\infty} \frac{c_l (\tau,\psi)}{1+\tau} \text{d}\,\tau < +\infty$.
The throughput upper bound in (\ref{eq: throughput upper bound definition}) can then be given as
\begin{eqnarray}
\overline{\mathcal{T}}(\psi, k)
=
\lambda_\text{U} \mathbb{E}\left[\frac{B}{\Psi}\log_2(1 + \text{SINR})\right]
&\stackrel{(a)}{=}& \lambda_\text{U} \mathbb{E}\left[\frac{B}{\Psi}\right] \mathbb{E}\left[ \log_2(1 + \text{SINR}) \right],
\label{eq: upper bound proof 2}
\end{eqnarray}
where (a) follows from that the SINR distribution is independent of the bandwidth distribution.
By plugging (\ref{eq: spectral efficiency upper bound}) into (\ref{eq: upper bound proof 2}), the result immediately follows.
\section{\label{proof: lem - densification plateau}}
First, we need to derive the asymptotic SINR distribution of mmWave network as the relative density $\psi \rightarrow +\infty$.
Following (\ref{eq: interference of LOS}) and (\ref{eq: LOS coverage probability}), the coverage probability at SINR threshold $\tau$ is lower and upper bounded by assuming all interfering hybrid precoding gain in (\ref{eq: interference of LOS}) as $\mathbb{G}_\text{A}\mathbb{G}_\text{U} = G_\text{A}G_\text{U}$ and $\mathbb{G}_\text{A}\mathbb{G}_\text{U} = g_\text{A}g_\text{U}$, respectively. It follows that the coverage probability in (\ref{eq: LOS coverage probability}) is bounded by
\begin{equation}
\mathbb{P} \left(
\frac{|x_0|^{-\alpha_{\text{L}}} }{\sum_{x \in \Phi_{\text{A}} \backslash \mathcal{B}(0,|x_0|)} |x|^{-\alpha_{\text{L}}}}> \tau \right)
<
\mathcal{C}_\psi(\tau, k)
<
\mathbb{P} \left(
\frac{|x_0|^{-\alpha_{\text{L}}} }{\sum_{x \in \Phi_{\text{A}} \backslash \mathcal{B}(0,|x_0|)} |x|^{-\alpha_{\text{L}}}}> \hat{\tau} \right) ,
\label{proofeq: bounds of SIR in the asymptotic region}
\end{equation}
where $x_0$ is the location of the AP serving the origin; $\hat{\tau} = \tau \frac{ g_\text{A} g_\text{U} }{ G_\text{A} G_\text{U} }$; the thermal noise is ignored since the signal from the side lobe of interfering APs is the major component of background noise when $\psi$ is large.
Note that we assume that $R_\text{B}$ is fixed for the mmWave network and thus the relative density $\psi$ increases with the AP density $\lambda_{\text{A}}$. For a given $x_0 \in \mathbb{R}^2$ and assume $\ell(|x_0|) = 1$ for $|x_0| < 1$,
\begin{eqnarray}
\lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \lambda_{\text{A}} \text{SINR}
&=&
\lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \lambda_{\text{A}} \frac{|x_0|^{-\alpha_{\text{L}}} }{\sum_{x \in \Phi_{\text{A}} \backslash \mathcal{B}(0,|x_0|)} |x|^{-\alpha_{\text{L}}}} \nonumber\\
&\stackrel{(a)}{=}& \lim\limits_{\lambda_{\text{A}} \rightarrow +\infty} \frac{\lambda_{\text{A}} |x_0|^{-\alpha_\text{L}} }{2 \pi \lambda_{\text{A}} \int_{0}^{R_\text{B}} r^{-\alpha_\text{L}+ 1} \text{d}r}
\stackrel{(b)}{=}
\frac{ 1}{2 \pi \sigma },
\label{proofeq: limitation of SIR - upper}
\end{eqnarray}
where $\sigma = \int_{0}^{R_\text{B}} r^{-\alpha_\text{L}+ 1} \text{d}r$ is a finite number; (a) follows from \cite[Lemma 1]{alammouri2018unified}; (b) follows that $\ell(|x_0|) \geq 1$.
When the spatial multiplexing gain $k=1$ and $\psi$ increases, we then have the densification gain $\gamma(\psi,1)$ in (\ref{def: densification gain}) as
\begin{eqnarray}
\lim\limits_{\psi \rightarrow +\infty} \gamma(\psi,1)
&=& \frac{1}{\mathcal{T}(\rho_0;1,1)} \lim\limits_{\psi \rightarrow +\infty} \mathcal{T}(\psi\rho_0; \psi, 1) \nonumber \\
&\stackrel{(a)}{=}& \frac{\lambda_\text{U}\rho_0}{\mathcal{T}(\rho_0;1,1)} \lim\limits_{\psi \rightarrow +\infty} \psi\mathcal{C}_\psi(2^{\rho_0/W_0}-1,1) \nonumber\\
&\stackrel{(b)}{\leq}& \frac{\lambda_\text{U}\rho_0}{\pi R_\text{B}^2 \mathcal{T}(\rho_0;1,1)} \lim\limits_{\lambda_\text{A} \rightarrow +\infty} \lambda_{\text{A}} \mathbb{P}\left(\text{SINR} > \hat{\tau}\right) \nonumber\\
&=& \frac{\lambda_\text{U}\rho_0}{\pi R_\text{B}^2 \mathcal{T}(\rho_0;1,1)} \lim\limits_{\lambda_\text{A} \rightarrow +\infty} \lambda_{\text{A}} \mathbb{E}\left( \mathbbm{1} (\text{SINR} > \hat{\tau}) \right) \nonumber \\
& = & \frac{\lambda_\text{U}\rho_0}{\pi R_\text{B}^2 \mathcal{T}(\rho_0;1,1)} \mathbb{E}\left( \lim\limits_{\lambda_\text{A} \rightarrow +\infty}\lambda_{\text{A}}\mathbbm{1} \left(\frac{\text{SINR}}{\hat{\tau}} >1 \right) \right) \nonumber\\
&\leq & \frac{\lambda_\text{U}\rho_0}{\pi R_\text{B}^2 \mathcal{T}(\rho_0;1,1)} \mathbb{E}\left(
\lim\limits_{\lambda_\text{A} \rightarrow +\infty}\lambda_{\text{A}}\text{SINR} / \hat{\tau} \right) \nonumber\\
&\stackrel{(c)}{\leq}& \frac{\lambda_\text{U}\rho_0}{2\pi^2 R_\text{B}^2 \sigma\hat{\tau} \mathcal{T}(\rho_0;1,1)},
\label{proofeq: limitation of densification gain with k=1}
\end{eqnarray}
where $\hat{\tau} = (2^{\rho_0/W_0}-1) \frac{ g_\text{A} g_\text{U} }{ G_\text{A} G_\text{U} }$; (a) is obtained by plugging (\ref{eq: average bandwidth}) into (\ref{eq: throughput of fixed-rate coding scheme}); (b) follows from (\ref{proofeq: bounds of SIR in the asymptotic region}); (c) follows from (\ref{proofeq: limitation of SIR - upper}). It follows from (\ref{proofeq: limitation of densification gain with k=1}) that $\gamma(\psi,1)$ is upper bounded by a finite constant.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\small
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,741 |
\section{Virtual alphabets in combinatorial Hopf algebras}
The notion of {\em virtual alphabet} provides a powerful symbolic notation
for dealing with endomorphisms of combinatorial Hopf algebras, at least
for those which can be realized in terms of (noncommutative) polynomials.
Examples include ${\bf Sym}$ (noncommutative symmetric
functions,~\cite{NCSF1,NCSF2}), ${\bf FQSym}$ (Free quasi-symmetric
functions~\cite{NCSF6,NCSF7}, which realize the Malvenuto-Reutenauer algebra
\cite{MR}), ${\bf WQSym}$ (Word quasi-symmetric functions~\cite{Hiv}), ${\bf PQSym}$
(Parking quasi-symmetric functions~\cite{NTp2}), and their subalgebras as long
as they are stable under the internal product.
These algebras can be regarded as generalizations of the Hopf algebra of
symmetric functions, for which this formalism was essentially promoted
by A. Lascoux~\cite{LS,Las}.
The algebra of symmetric functions is denoted by $Sym$. Apart from this
detail, our notation for symmetric functions follows Macdonald's
book~\cite{Mcd}.
\subsection{Virtual alphabets for symmetric functions}
It is convenient to use the generic term \emph{alphabet} to designate the
argument of a symmetric function. Indeed, a symmetric function $f$ is
characterized by its expression $f(x_1,x_2,\ldots)$ in terms of monomials in
an infinite sequence of independent indeterminates $x_i$, to which various
sets or multisets of algebraic expressions (numbers, monomials, cohomology
classes, vector bundles, \emph{etc}.) can be substituted.
The diversity of possible interpretations suggests to treat as far as possible
these arguments as formal symbols (or \emph{letters}, whence the term
\emph{alphabet}), and the possible occurence of multiple arguments suggests to
extend the usual meaning of this term and to understand it as a
\emph{multiset} of symbols.
Actually, a multiset $A=\{a,c,c,c,f,f\}$ is nothing but a formal linear
combination of symbols with nonnegative integer coefficients, and it is more
convenient to represent it as $A=a+3c+2f$. With this notation, the union of
multisets becomes just a sum $A+B$, and if we have sums, we can also have
differences $A-B$, at least when $B$ is contained in $A$.
This leads us to the point of this paragraph. When $B$ is not contained
in $A$, let us call the formal combination $A-B$ a \emph{virtual alphabet}.
It is easy to define the value of a symmetric function on a virtual alphabet.
Indeed, it is well known that any symmetric function $f$ can be expressed in
terms of the elementary symmetric functions $e_k$. The elementary functions
$e_k(A)$ of a genuine alphabet $A$ are the coefficients of the product
\begin{equation}
\lambda_t(A):=\prod_{a\in A}(1+ta)=\sum_k e_k(A)t^k
\end{equation}
and when $B$ is contained in $A$, those of the alphabet $C=A-B$ are
obtained by division of the generating functions:
\begin{equation}
\lambda_t(A-B)=\lambda_t(A)/\lambda_t(B)\,.
\end{equation}
When $B$ is not contained in $A$, this still defines coefficients
$e_k(A-B)$, which are, by definition, the elementary symmetric functions
of the virtual alphabet $A-B$. This defines all symmetric functions
of $A-B$, and one has for example the simple expression
$p_n(A-B)=p_n(A)-p_n(B)$ for the power sums.
More generally, a virtual alphabet can be defined by \emph{any} sequence
$(e_n(A))_{n\ge 1}$, interpreted as its elementary symmetric functions
(or as any sequence of independent generators of the algebra of
symmetric functions, such as power sums $p_n$ or complete homogeneous
functions $h_n$).
For example, although the exponential function $e^t$ has no zeros in the
complex plane, we can introduce a virtual alphabet ${\sym E}$ such that
\begin{equation}
e_n({\sym E}) =\frac1{n!},\,\qquad\qquad \lambda_t({\sym E})=e^t\,.
\end{equation}
This is a useful trick, allowing to understand a lot of formulas in
combinatorics or analysis as specializations of simple identities on symmetric
functions. For example, the exponential generating function of the derangement
numbers $d_n$ is just
\begin{equation}
\sigma_t(1-{\sym E}):=\lambda_{-t}(1-{\sym E})^{-1}=\frac{e^{-t}}{1-t}\,.
\end{equation}
Also, for a partition $\lambda$ of $n$, $n!s_\lambda({\sym E})=f_\lambda$, the
number of standard tableaux of shape $\lambda$.
Another virtual alphabet of interest is $1-q$, where $p_n(1-q)=1-q^n$. It
plays an essential role in the theory of Hall-Littlewood
functions~\cite{Mcd}.
Our expression of alphabets as formal sums or differences also
allows the consideration of products. For genuine alphabets $A$, $B$,
\begin{equation}
AB = \{a\,b\,\,|\,\, a\in A,\ b\in B\} =\sum_a a\sum_b b
\end{equation}
and for virtual alphabets, the symmetric functions of $AB$
are defined by any of the formulas like $p_n(AB)=p_n(A)p_n(B)$, or
\begin{equation}
h_n(AB)=\sum_{\lambda\vdash n}s_\lambda(A)s_\lambda(B)\,,
\end{equation}
the famous Cauchy identity (the $s_\lambda$ are the Schur functions).
More generally, for any function $f$, we have: $f(AX)=F(X)*\sigma_1(AX)$,
where $*$ denotes the internal product, and $\sigma_t(X)=\sum t^n h_n(X)$ is
the generating series of homogeneous complete functions.
Hence, with this formalism, we can consider \emph{transformations of
alphabets} on symmetric functions, which are ring homomorphisms mapping any
$f(X)$ to $f({\sym E} X)$, $f((1-{\sym E})X)$, $f((1-q)X)$, $f(X/(1-q))$, and so on.
Remark that in this setting, the virtual alphabet $1-q$ is the inverse of the
genuine alphabet $\{1,q,q^2\ldots\}\equiv 1+q+q^2+\cdots$.
Actually, $Sym$ is a $\lambda$-ring~\cite{Las,LS}, and the notion of plethysm
allows much more complicated transformations, but in the present paper, we
concentrate on the above mentioned ones since these can be defined only in
terms of the (combinatorial) Hopf algebra structure.
\subsection{Extension to combinatorial Hopf algebras}
Let $X$ and $Y$ be two infinite alphabets. Identifying expressions like
$f(X)g(Y)$ with the tensor product $f\otimes g$, we can regard the
transformations
\begin{equation}
\Delta:\ f(X)\longmapsto f(X+Y)
\end{equation}
and
\begin{equation}
\delta:\ f(X)\longmapsto f(XY)
\end{equation}
as linear maps $Sym\rightarrow Sym\otimes Sym$, that is, as comultiplications.
Moreover, both are (obviously) algebra morphisms, and the first one
being graded in the sense that
\begin{equation}
\Delta:\ Sym_n\longrightarrow \bigoplus_{i+j=n}Sym_i\otimes Sym_j
\end{equation}
defines actually a Hopf algebra structure, whose antipode is simply
$f(X)\mapsto f(-X)$.
Clearly, the transformations $f({\sym E} X)$, $f((1-{\sym E})X)$, $f((1-q)X$ and so on can
be defined only in terms of these coproducts, of the antipode, and of the
internal product. It turns out that most combinatorial Hopf algebras can be
realized in terms of polynomials in some infinite and \emph{totally ordered
alphabet}, denoted by $A=\{a_n|n\ge 1\}$ in the case of noncommuting letters,
and by $X=\{x_n|n\ge 1\}$ in the case of commuting letters. The basic example
is the pair Noncommutative Symmetric Functions -- Quasi-symmetric functions
$({\bf Sym},QSym)$ of mutually dual combinatorial Hopf algebras. Sums and
differences of alphabets are defined on both sides, and it is possible to
make sense of the product $XA$ (see~\cite{NCSF2}).
Hence, our transformations are defined in this case.
The subject of this article is the study of the $(1-{\sym E})$-transform in the
pair $({\bf Sym},QSym)$, and its extension to other combinatorial Hopf algebras.
\section{The $(1-{\sym E})$-transform in ${\bf Sym}$}
\subsection{Background}
Our notations for the Hopf algebra of noncommutative symmetric functions are
as in~\cite{NCSF1,NCSF2}. This Hopf algebra is denoted by ${\bf Sym}$, or by
${\bf Sym}(A)$ if we consider the realization in terms of an auxiliary alphabet.
Bases of its homogeneous component ${\bf Sym}_n$ are labelled by compositions
$I=(i_1,\ldots,i_r)$ of $n$. The noncommutative complete and elementary
functions are denoted by $S_n$ and $\Lambda_n$, and the notation $S^I$ means
$S_{i_1}\cdots S_{i_r}$. The ribbon basis is denoted by $R_I$.
The notation $I\vDash n$ means that $I$ is a composition of $n$.
The conjugate composition is denoted by $I^\sim$, the mirror image composition
by $\overline{I}$. The \emph{descent set} of $I$ is
${\rm Des\,}(I) = \{ i_1,\ i_1+i_2, \ldots , i_1+\dots+i_{r-1}\}$.
The graded dual of ${\bf Sym}$ is $QSym$ (quasi-symmetric functions).
The dual basis of $(S^I)$ is $(M_I)$ (monomial), and that of $(R_I)$
is $(F_I)$.
The Hopf structures on ${\bf Sym}$ and $QSym$ allows one to partially extend the
$\lambda$-ring notation of ordinary symmetric functions (see~\cite{NCSF2},
and~\cite{Las} for background on the original commutative version).
If $A$ and $X$ represent totally ordered sets of noncommuting and commuting
variables respectively, the noncommutative symmetric functions of $XA$ are
defined by
\begin{equation}
\sigma_t(XA)=\sum_{n\ge 0}t^n S_n(XA) =
\prod_{x\in X}^{\rightarrow}\sigma_{tx}(A)= \sum_I t^{|I|}M_I(X)S^I(A)\,.
\end{equation}
Now, $X$ can be a virtual alphabet, defined by an arbitrary specialization of
an independ set of generators of $QSym$.
An alternative way to express the transformation of alphabets defined by $X$
is~\cite{NCSF2}
\begin{equation}
F(XA)=F(A)*\sigma_1(XA)\,,
\end{equation}
where $*$ is the internal product. Since $\sigma_1(XA)$ is grouplike,
the $X$-transform is a bialgebra morphism, thanks to the splitting
formula
\begin{equation}
(F_1F_2\cdots F_r)*G = \mu_r \left[ (F_1\otimes\cdots\otimes F_r)* \Delta^r G
\right]
\end{equation}
where in the right-hand side, $\mu_r$ denotes the $r$-fold ordinary
multiplication and $*$ stands for the operation induced on ${\bf Sym}^{\otimes n}$
by $*$.
Thanks to the commutative image homomorphism ${\bf Sym}\rightarrow Sym$,
noncommutative symmetric functions can be evaluated on any element $x$ of a
$\lambda$-ring, $S_n(x)$ being $S^n(x)$, the $n$-th symmetric power. Recall
that $x$ is said \emph{of rank one} (resp. \emph{binomial}) if
$\sigma_t(x)=(1-tx)^{-1}$ (resp. $\sigma_t(x)=(1-t)^{-x}$).
The scalar $x=1$ is the only element having both properties. We usually
consider that our auxiliary variable $t$ is of rank one, so that
$\sigma_t(A)=\sigma_1(tA)$.
The argument $A$ of a noncommutative symmetric function can be a
\emph{virtual alphabet}. This means that, being algebraically independent,
the $S_n$ can be specialized to any sequence $\alpha_n\in{\mathcal A}$ of
elements of any associative algebra ${\mathcal A}$. Writing $\alpha_n=S_n(A)$
defines all the symmetric functions of $A$. Quasi-symmetric functions
of a virtual alphabet $X$ can be defined by a specialization of the
algebraic generators of $QSym$, which is more easily done by expressing
the noncommutative symmetric functions of $XA$ in terms of $A$.
The specializations $X={\sym E}$, defined by
\begin{equation}
S_n({\sym E} A)=\frac{1}{n!}S_1(A)^n
\end{equation}
(so that $F_I({\sym E})=\frac1{n!}$) and $X=\frac{1}{1-q}$, for which
\begin{equation}
S_n\left(\frac{A}{1-q}\right)=\frac{1}{(q)_n}\sum_{I\vDash n}q^{{\rm maj\,}(I)}R_I(A)
\end{equation}
are of special importance. The second one can be used to define the
peak algebras and simplifies considerably their
investigation~\cite{BHT,ANT,NST}.
Here, we will study the $(1-{\sym E})$- transform by the methods developed in these
references.
\subsection{Noncommutative derangement numbers}
A possible motivation for the $(1-{\sym E})$-transforms is the combinatorics of
derangements. Indeed, as already mentioned, the generating function of the
derangement numbers
\begin{equation}
D(t)=\sum_{n\ge 0} d_n\frac{t^n}{n!}=\frac{e^{-t}}{1-t}
\end{equation}
can be expressed as
\begin{equation}
D(t)=\sigma_1((1-{\sym E})t)\,.
\end{equation}
The specializations of the various bases of symmetric functions
at $1-{\sym E}$ are obtained by expanding the Cauchy kernel $\sigma_1((1-{\sym E})X)$,
and, analogously, its quasi-symmetric functions can be defined as the
coefficients of the expansion of the non-commutative Cauchy kernel
\begin{equation}
\sigma_1((1-{\sym E})A)=e^{-S_1(A)}\sigma_1(A)
\end{equation}
on any basis of noncommutative symmetric functions.
\bigskip
{\footnotesize
For example, since the quasi-monomial basis $M_I$ is dual to $S^I$,
we have
\begin{equation}
\sigma_1((1-{\sym E})A)=\sum_{n\geq0} S_n((1-{\sym E})A)= \sum_{I} M_I(1-{\sym E}) S^I(A),
\end{equation}
so that
\begin{equation}
S_1((1-{\sym E})A)= 0,\ S_2((1-{\sym E})A)= S_2(A) - S^{11}(A)/2,\
\end{equation}
\begin{equation}
S_3((1-{\sym E})A)= S_3(A) - S^{21}(A) + S_{111}(A)/3,
\end{equation}
hence impliying
\begin{equation}
M_{1}(1-{\sym E})=0,\ M_{2}(1-{\sym E})=1,\ M_{11}(1-{\sym E})=-1/2,
\end{equation}
and
\begin{equation}
M_{3}(1\!-\!{\sym E})=1,\ M_{21}(1\!-\!{\sym E})=-1,\
M_{12}(1\!-\!{\sym E})=0,\ M_{111}(1\!-\!{\sym E})=1/3.
\end{equation}
}
\bigskip
Since for $A=1$, $S_n((1-{\sym E})A)= d_n/n!$, these noncommutative symmetric
functions might be called noncommutative derangement numbers
(see~\cite{NT3} for other examples of
\emph{noncommutative combinatorial numbers}).
\bigskip
{\footnotesize
Another natural noncommutative analogue of $d_n$ is given by
\begin{equation}\label{desar}
\lambda_{-t}(A)(1-tS_1(A))^{-1}
\end{equation}
which gives back the $d_n$ for $A={\sym E}$, and $d_n(q)$ for $A=\frac1{1-q}$.
Its expansion of the ribbon basis is easily seen to be
\begin{equation}
\sum_{n\ge 0}t^n\sum_{i=1}^{[n/2]}\sum_{|J|=n-2i}R_{1^{2i}\triangleright J}
\end{equation}
(ribbons whose first column is of even height, or permutations whose
position of the first local minimum is even; the observation that these
permutations are counted by $d_n$ is due to D\'esarm\'enien, and an explicit
bijection has been given by D\'esarm\'enien and Wachs~\cite{DW1,DW2}).
Similarly,
\begin{equation}
\lambda_{u-t}(A)(1-tS_1(A))^{-1}
\end{equation}
gives a natural noncommutative analog of the generating series
$D(t,u)=e^{u-t}/(1-t)$ for permutations by number of fixed points,
and the expansion in ${\bf FQSym}$ of
\begin{equation}
\sigma_t(A) (1-t\,S_1(A))^{-1}
\end{equation}
yields a class of permutations in bijection with arrangements: for
$A={\sym E}$, this series is $\frac{e^t}{1-t}$.
}
\bigskip
Denote for short by a $\sharp$ the $(1-{\sym E})$ transform, \emph{i.e.},
\begin{equation}
F^\sharp := F((1-{\sym E})A)=F*\sigma_1^\sharp\,
\end{equation}
where
\begin{equation}
\label{eq-sigundiese}
\sigma_1^\sharp = \sigma_1((1-{\sym E})A)=e^{-S_1}\sigma_1,
\end{equation}
that is,
\begin{equation}
\label{eq-sndiese}
S_n^\sharp = \sum_{i=0}^n (-1)^i \frac{S_1^i}{i!} S_{n-i}.
\end{equation}
\begin{lemma}
\label{lem-proj}
The $\sharp$-transform is a projector, \emph{i.e.},
\begin{equation}
\sigma_1^\sharp*\sigma_1^\sharp=\sigma_1^\sharp\,,
\end{equation}
or, equivalently,
\begin{equation}
S_n^\sharp * S_n^\sharp = S_n^\sharp \text{\rm \ \ for all $n$}.
\end{equation}
\end{lemma}
\noindent \it Proof -- \rm
Since the $\sharp$ transform is a homomorphism,
\begin{equation}
\begin{split}
\sigma_1^\sharp*\sigma_1^\sharp
&= (e^{-S_1}\sigma_1)*\sigma_1^\sharp
= (e^{-S_1}*\sigma_1^\sharp)\, (\sigma_1*\sigma_1^\sharp) \\
&= e^{-S_1^\sharp}\, \sigma_1^\sharp = \sigma_1^\sharp.\\
\end{split}
\end{equation}
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\bigskip
Hence, $S_n^\sharp$ corresponds to an idempotent $\delta_n$ of the descent
algebra $\Sigma_n$. We shall see later that the dimension of the
representation ${\sym C}{\mathfrak S}_n\delta_n$ is the derangement number $d_n$ and that
$\delta_n$ coincides with Schocker's derangement idempotent \cite{Sc}, whence
the name \emph{derangement algebra} below for the corresponding ideal of the
descent algebra.
\subsection{The small derangement algebra ${\mathcal D}^{(0)}$}
\subsubsection{Definition of ${\mathcal D}^{(0)}$}
Imitating the construction of the peak algebra in \cite{BHT}, we define the \emph{small
derangement algebra}, or the \emph{derangement ideal} as
\begin{equation}
{\mathcal D}^{(0)} := {\bf Sym}^\sharp = {\bf Sym}(A)*\sigma_1^\sharp\,.
\end{equation}
By definition of the transformation of alphabet, ${\mathcal D}^{(0)}$ is a Hopf
subalgebra of ${\bf Sym}$, and thanks to the second equality, each homogenous
component ${\mathcal D}^{(0)}_n$ is a left ideal of ${\bf Sym}_n$ for the internal product,
explaing the two names for ${\mathcal D}^{(0)}$, depending on which property
we want to emphasize.
\subsubsection{Dimension and bases of ${\mathcal D}_n^{(0)}$}
We have $S_1^\sharp=0$, and the other $S_n^\sharp$ are clearly
algebraically independent. Hence, the dimension $d_n^{(0)}$ of ${\mathcal D}_n^{(0)}$
is given by the number of compositions of $n$ with no part equal to $1$. These
elements satisfy the induction
\begin{equation}
d_0^{(0)}=1,\quad d_1^{(0)}=0,\quad
d_{n}^{(0)} = d_{n-1}^{(0)}+d_{n-2}^{(0)} \text{\ for $n\geq2$},
\end{equation}
hence are shifted Fibonacci numbers.
Let us say that a composition is \emph{non-unitary} if it does not contain the
part $1$. Then, the $S^{I\sharp}$ with $I$ non-unitary form a basis of
${\mathcal D}_n^{(0)}$.
Since the coproducts
\begin{equation}
\Delta S_n^\sharp=\sum_{i+j=n}S_i^\sharp\otimes S_j^\sharp
\end{equation}
are given by the same formula as those of the $S_n$, we see that
${\mathcal D}_n^{(0)}$ is isomorphic to the quotient ${\bf Sym}'$ of ${\bf Sym}$ by the
two-sided ideal generated by $S_1$. Hence, its (graded) dual is isomorphic
to the subalgebra $QSym'$ of $QSym$ spanned by the $M_I$ for $I$ non-unitary.
Since $\sigma_1^\sharp$ is a projector, we have
\begin{equation}
f^\sharp*g^\sharp=(f*g)^\sharp\,.
\end{equation}
${\bf Sym}'$ being stable under the internal product, ${\mathcal D}_n^{(0)}$
is also $*$-isomorphic to ${\bf Sym}_n'$.
Recall that the ribbon basis of ${\bf Sym}$ is given by
\begin{equation}
R_I=\sum_{J\le I} S^J
\end{equation}
where $\le$ is the reverse refinement order. If $I$ is non-unitary,
so are all $J\le I$, thus
\begin{equation}
Q_I := R_I^\sharp\qquad (1\not \in I)
\end{equation}
is a basis of ${\mathcal D}_n^{(0)}$.
\subsection{Algebraic structure of $({\mathcal D}_n^{(0)},*)$}
Since the construction of ${\mathcal D}^{(0)}$ mimicks the one of the
peak ideal, obtained in \cite{BHT} as the image of the
$(1-q)$-tranform at $q=-1$,
one may expect that ${\mathcal D}^{(0)}$ shares many properties with the peak
ideal. There are some differences however.
Whilst the peak ideal has no unity for the internal product $*$, we have:
\begin{proposition}
\label{d0-unit}
For all $n$, $({\mathcal D}^{(0)}_n,*)$ is a unital algebra with $S_n^\sharp$
as neutral element.
\end{proposition}
\noindent \it Proof -- \rm
From Lemma \ref{lem-proj}, we already know that $\sigma_1^\sharp$ is neutral
on the right.
To prove that it is neutral on the left, let us consider its action
on the generating series of a basis of ${\mathcal D}^{(0)}$
\begin{equation}
\sigma_1(X\cdot(1-{\sym E})A)=\sum_I M_I(X){S^I}^\sharp(A)\,.
\end{equation}
We have
\begin{equation}
\sigma_1^\sharp*\sigma_1(X\cdot(1-{\sym E})A)
=\sigma_1^\sharp*\sigma_1(XA)*\sigma_1^\sharp
\end{equation}
and
\begin{equation}
\begin{split}
\sigma_1^\sharp*\sigma_1(XA)
&= \left( e^{-S_1(A)}\sigma_1(A \right)) * \sigma_1(XA) \\
&= \mu\left[ (e^{-S_1(A)}\otimes \sigma_1(A))
*_2 (\sigma_1(XA)\otimes\sigma_1(XA))\right] \\
&= (e^{-S_1(A)}*\sigma_1(XA))\,(\sigma_1(A) * \sigma_1(XA)) \\
&= e^{-S_1(XA)}\sigma_1(XA)
\end{split}
\end{equation}
so that
\begin{equation}
\begin{split}
\sigma_1^\sharp * \sigma_1(X\cdot(1-{\sym E})A)
&= \left[ e^{-S_1(XA)}\sigma_1(XA) \right] * \sigma_1^\sharp \\
&= e^{-S_1(X\cdot(1-{\sym E})A)}\,\sigma_1(X\cdot(1-{\sym E})A) \\
&= \sigma_1(X\cdot(1-{\sym E})A) \\
\end{split}
\end{equation}
since $S_1(X\cdot(1-{\sym E})A)=0$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\subsection{Representation theory of ${\mathcal D}_n^{(0)}$}
Now that we know that ${\mathcal D}_n^{(0)}$ is unital, we can investigate its
representation theory. We first look at the idempotents.
\subsubsection{Idempotents in ${\mathcal D}_n^{(0)}$}
Recall that the \emph{(right) Zassenhaus idempotents} $\zeta_n$ are
defined as the homogeneous elements of degree $n$ (that is,
$\zeta_n\in{\bf Sym}_n$) satisfying (see~\cite{NCSF2}):
\begin{equation}
\sigma_1 =: \prod_{k\geq1} e^{\zeta_k}
= e^{\zeta_1} e^{\zeta_2} e^{\zeta_3} \dots.
\end{equation}
Obviously, $\zeta_1=S_1$ so that $\zeta_1^\sharp=0$. Since the
$\sharp$-transform is multiplicative,
\begin{equation}
\sigma_1^\sharp= e^{\zeta_2^\sharp} e^{\zeta_3^\sharp} \dots
\end{equation}
but also
\begin{equation}
\sigma_1^\sharp= e^{-\zeta_1}\sigma_1=e^{\zeta_2} e^{\zeta_3} \dots
\end{equation}
so that
$\zeta_i^\sharp = \zeta_i$.
Extracting the term of degree $n$, we have
\begin{equation}
S_n^\sharp = \sum_{\gf{|\lambda|=n}{1\not\in\lambda}}
\frac{\zeta^\lambda}{m_\lambda},
\end{equation}
where for a partition $\lambda=(\lambda_1\ge\lambda_2\ge\dots)$,
$\zeta^\lambda:=\zeta_{\lambda_1}\cdots \zeta_{\lambda_r}$,
and
$m_\lambda:=\prod_{i\geq1}m_i(\lambda)!$, where $m_i(\lambda)$ the
multiplicity of $i$ in $\lambda$.
We shall make use of the notation
$e_\lambda := \frac{\zeta^\lambda}{m_\lambda}$.
For a composition $I$, we set $\zeta^I=\zeta_{i_1}\dots\zeta_{i_r}$ and
$m_I=m_\lambda$ if $I\!\downarrow=\lambda$.
\subsubsection{A basis of idempotents}
The $\zeta_n$ are primitive elements with commutative image $p_n/n$,
hence are Lie idempotents~\cite{NCSF2}.
As with any sequence of Lie idempotents, we can construct an idempotent
basis of ${\bf Sym}_n$ from the $\zeta_n$.
We need the following lemma from~\cite{NST}, easily derived from the splitting
formula (compare~\cite[Lemma 3.10]{NCSF2}).
Recall that the radical of $({\bf Sym}_n,*)$ is
${\mathcal R}_n = {\mathcal R}\cap{\bf Sym}_n$, where ${\mathcal R}$ is the
kernel of the commutative image ${\bf Sym}\rightarrow Sym$.
\begin{lemma}
\label{lem-NST}
Denote by ${\mathfrak S}(J)$ the set of distinct rearrangements of a composition $J$.
Let $I=(i_1,\ldots,i_r)$ and $J=(j_1,\ldots,j_s)$ be two compositions
of $n$. Then,
\smallskip (i) if $\ell(J)<\ell(I)$ then $\zeta^I*\zeta^J=0$.
\smallskip (ii) if $\ell(J)>\ell(I)$ then
$\zeta^I*\zeta^J \in {\rm Vect\,}\<\zeta^K\, : \, K\in{\mathfrak S} (J)\>\cap{\mathcal R}$.
More precisely,
\begin{equation}
\zeta^I*\zeta^J
= \sum_{\gf{\scriptstyle J_1,\ldots J_r}{\scriptstyle |J_k|=i_k}}
\< J\, ,\, J_1\, \shuffl \, \cdots \, \shuffl \, J_r \>\Gamma_{J_1}\cdots\Gamma_{J_r}
\end{equation}
where for a composition $K$ of $k$,
$\Gamma_K:=\zeta_k*\zeta^K$, and $\, \shuffl \,$ denotes the shuffle products
of compositions regarded as words over the positive integers.
\smallskip (iii) if $\ell(J)=\ell(I)$, then $\zeta^I*\zeta^J\not= 0$
only for $J\in{\mathfrak S} (I)$, in which case $\zeta^I*\zeta^J=m_I\, \zeta^I$.
\end{lemma}
\hfill \hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\begin{corollary}
\label{cor-NST}
(i) The elements
\begin{equation}
e_I=\frac1{m_I}\zeta^I\,,\quad I\vDash n\,,
\end{equation}
are all idempotents and form a basis of ${\bf Sym}_n$.
This basis contains in particular a complete set of minimal orthogonal
idempotents, $e_\lambda$ of ${\bf Sym}_n$.
(ii) The $e_I$ such that $I$ does not have a part equal to $1$ form a basis of
${\mathcal D}_n^{(0)}$.
(iii) The $e_\lambda$ with no part equal to $1$ in $\lambda$ form a complete
set of minimal orthogonal idempotents of ${\mathcal D}^{(0)}_n$. \hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{corollary}
\subsection{Cartan invariants of ${\mathcal D}^{(0)}_n$}
By (iii) of lemma \ref{lem-NST}, the indecomposable projective module
$P_\lambda={\mathcal D}^{(0)}_n*e_\lambda$ contains the $e_I$ for $I\in{\mathfrak S}(\lambda)$.
For $I\not\in{\mathfrak S}(\lambda)$, (i) and (ii) imply that $e_I*e_\lambda$ is in
${\rm Vect\,}\<\zeta^K\, : \, K\in{\mathfrak S} (\lambda)\>$. Hence, this space coincides with
$P_\lambda$. So, we get immediately an explicit decomposition
\begin{equation}
{\mathcal D}^{(0)}_n=\bigoplus_{\lambda\vdash n,\ 1\not\in\lambda}P_\lambda\,,\qquad
P_\lambda=\bigoplus_{I\in{\mathfrak S}(\lambda)}{\sym C} e_I\,.
\end{equation}
The Cartan invariants
\begin{equation}
c_{\lambda,\mu}={\rm dim\,}\,( e_\mu*{\mathcal D}^{(0)}_n*e_\lambda)
\end{equation}
are now easily obtained. The above space is spanned by the
\begin{equation}
e_\mu*e_I* e_\lambda = e_\mu* e_I\,,\quad I\in{\mathfrak S}(\lambda)\,.
\end{equation}
\noindent
From (ii) of Lemma~\ref{lem-NST}, this space has the dimension of the space
$[S^\mu(L)]_\lambda$, spanned by all symmetrized products of Lie polynomials
of degrees $\mu_1,\mu_2,\ldots$ formed from $\zeta_{i_1},\zeta_{i_2},\ldots$,
as in the classical result of Garsia-Reutenauer for the descent
algebra~\cite{GaR}.
\bigskip
In the following examples, partitions are ordered by reverse lexicographic
order.
For $n\leq4$, the Cartan matrix of ${\mathcal D}_n^{0}$ is trivial: it is the identity
matrix of size $d_n^{(0)}$, since there is at most one value in a partition
of $n$ with no part one.
For $n$ up to $7$, the Cartan invariants are given by the matrices
(\ref{qCar-dn0-567}) and~(\ref{qCar-dn0-89}) at $q=1$. Indeed, the
$q$-analogues defined from the Loewy series can be explicitely calculated.
\subsection{Quiver and $q$-Cartan invariants (Loewy series)}
We shall use the following modified refinement order on partitions: we write
$\lambda\prec_{p}\mu$, and say that $\lambda$ is $p$-finer than $\mu$, if each
part of $\mu$ is either a part of $\lambda$ or a sum of distinct parts of
$\lambda$. Hence, $\lambda$ covers $\mu$ iff $\mu$ is obtained from $\lambda$
by merging two distinct parts.
Still relying upon point (ii) of the lemma, we see that $c_{\lambda,\mu}=0$ if
$\lambda$ is not $p$-finer than (or equal to) $\mu$, and that if $\mu$ is
obtained from $\lambda$ by adding up two parts $\lambda_i,\lambda_j$, $e_\mu*
e_I=0$ if $\lambda_i=\lambda_j$ and is a nonzero element of the radical
otherwise.
The $q$-analogues of the Cartan invariants
\begin{equation}
c_{\lambda,\mu}
:= \sum_{k} q^k
{\rm dim\,}\left[ (e_\mu*{\rm rad\,}^k{\mathcal D}_n^{(0)}*e_\lambda)
/ {\rm rad\,}^{k+1}{\mathcal D}_n^{(0)} \right]
\end{equation}
can now be obtained from Proposition~\ref{d0-unit} and the following lemma.
\begin{lemma}
\label{BAe}
Let $A$ be an associative algebra. If $e$ is an idempotent of $A$ such that
$e$ is neutral in $B=Ae$, then
\begin{equation}
{\rm rad\,}^k B = e ({\rm rad\,}^k A) e.
\end{equation}
\end{lemma}
\noindent \it Proof -- \rm
Let $x\in {\rm rad\,} B$. There exists an integer $n$ such that
$(xB)^n=0$. Then,
\begin{equation}
(x eAe)^n = (xA)^n e = 0,
\end{equation}
so that, as well
\begin{equation}
(xA)^n exA = (xA)^n xA = (xA)^{n+1} =0.
\end{equation}
Thus, the right ideal $xA$ is nilpotent, which proves that $x\in{\rm rad\,} A$.
Since $x=exe$, $x\in e\,{\rm rad\,} A e$ and we have shown that
${\rm rad\,} B\subseteq e\,{\rm rad\,} A e$.
Conversely, if $x\in{\rm rad\,} A$, so that $x^n=0$ for a certain $n$, then
$(exe)^n=x^ne=0$, whence $exe\in{\rm rad\,} B$, which proves the claim for $k=1$.
Now, if $x\in{\rm rad\,}^k B$, $x=x_1\dots x_k$ with $x_i=ey_ie$, for some
$y_i\in{\rm rad\,} A$. Hence
\begin{equation}
x = ey_1e\dots ey_ke = e y_1y_2\dots y_r e \in e\,{\rm rad\,}^k Ae.
\end{equation}
Conversely, any $x$ of the form $ey_1\dots y_re$ with $y_i\in{\rm rad\,} A$ can be
rewritten as $x=ey_1e\dots ey_ke\in {\rm rad\,}^k B$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\medskip
Applying this to $A={\bf Sym}_n$, and $e=S_n^\sharp$, we obtain from the known
description of the $q$-Cartan matrices of ${\bf Sym}_n$~\cite{Sal}:
\begin{theorem}
(i) In the quiver of ${\mathcal D}^{(0)}_n$, there is an arrow $\lambda\rightarrow\mu$
iff $\mu$ is obtained from $\lambda$ by adding two distinct parts,
\noindent
(ii) The $q$-Cartan invariants of ${\mathcal D}^{(0)}_n$ are given by
\begin{equation}
c_{\lambda,\mu}(q)= c_{\lambda,\mu}\, q^{\ell(\lambda)-\ell(\mu)}.
\end{equation}
if $\lambda$ is finer than or equal to $\mu$, and $c_{\lambda,\mu}(q)=0$
otherwise.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{theorem}
The result can also be derived as follows:
In~\cite{BL}, it is shown that the powers of the radical of ${\bf Sym}_n$ for the
internal product coincide with the homogeneous component of degree $n$ of the
lower central series of ${\bf Sym}$ for the external product:
\begin{equation}
{\mathcal R}^{*j} =\gamma^j({\bf Sym})
\end{equation}
where $\gamma^j({\bf Sym})$ is the ideal of ${\bf Sym}$ generated by the commutators
$[{\bf Sym},\gamma^{j-1}({\bf Sym})]$.
Since ${\mathcal D}^{(0)}$ is a free associative algebra over a sequence of primitive
elements $(\zeta_k)_{k\ge 2}$ with the same internal product as in ${\bf Sym}$, the
argument of \cite{BL} can be reproduced \emph{verbatim}, and we see that
\begin{equation}
({\rm rad\,}{\mathcal D}^{(0)})^{*j} =\gamma^j({\mathcal D}^{(0)})=
{\mathcal R}^{*j}\cap {\mathcal D}^{(0)}\,.
\end{equation}
This shows that, in ${\mathcal D}^{(0)}$ as well as in ${\bf Sym}$, for $\lambda$ finer than
$\mu$, $e_\mu* e_I$ is nonzero modulo ${\rm rad }^{*2}$ iff $\mu$ is obtained
from $\lambda$ by summing two distinct parts.
And more generally, $e_\mu* e_I$ is in ${\rm rad}^{*k}$ and nonzero modulo
${\rm rad}^{*k+1}$ iff $\ell(\lambda)-\ell(\mu)=k$.
\subsection{Tables of the $q$-Cartan invariants of ${\mathcal D}_n^{(0)}$}
The labels for row and columns of the $q$-Cartan matrices, namely partitions
with no part one, are in reverse lexicographic order.
In the following matrices, the zero entries are represented by dots to enhance
readability.
For $n\leq4$, the $q$-Cartan matrix of ${\mathcal D}_n^{0}$ is the identity matrix of
size $d_n^{(0)}$.
The first non-trivial example arises for $n=5$.
The $q$-Cartan matrices of ${\mathcal D}_5^{(0)}$, ${\mathcal D}_6^{(0)}$, and ${\mathcal D}_7^{(0)}$
are respectively
{
\begin{equation}
\label{qCar-dn0-567}
\left(
\begin{array}{cc}
1 & q \\
. & 1 \\
\end{array}
\right)
\qquad
\left(
\begin{array}{cccc}
1 & q & . & . \\
. & 1 & . & . \\
. & . & 1 & . \\
. & . & . & 1 \\
\end{array}
\right)
\qquad
\left(
\begin{array}{ccccccccccccccc}
1 & q & q &q^2 \\
. & 1 & . & q \\
. & . & 1 & . \\
. & . & . & 1 \\
\end{array}
\right)
\end{equation}
}
and those of ${\mathcal D}_8^{(0)}$ and ${\mathcal D}_9^{(0)}$ are
{
\begin{equation}
\label{qCar-dn0-89}
\left(
\begin{array}{ccccccc}
1 & q & q & . &q^2&q^2& . \\
. & 1 & . & . & q & . & . \\
. & . & 1 & . & . & q & . \\
. & . & . & 1 & . & . & . \\
. & . & . & . & 1 & . & . \\
. & . & . & . & . & 1 & . \\
. & . & . & . & . & . & 1 \\
\end{array}
\right)
\qquad
\left(
\begin{array}{cccccccc}
1 & q & q & q &q^2&2q^2& . &q^3\\
. & 1 & . & . & q & q & . &q^2\\
. & . & 1 & . & . & q & . & . \\
. & . & . & 1 & . & q & . & . \\
. & . & . & . & 1 & . & . & q \\
. & . & . & . & . & 1 & . & . \\
. & . & . & . & . & . & 1 & . \\
. & . & . & . & . & . & . & 1 \\
\end{array}
\right)
\end{equation}
}
On these matrices, one can read the quiver of ${\mathcal D}^{(0)}_n$. Note
that it is a subquiver of the quiver of ${\bf Sym}_n$ (see \cite{Sal}), since
one cannot create parts $1$ by merging parts of non-unitary partitions.
\subsection{The (large) derangement algebra ${\mathcal D}={\mathcal D}^{(\infty)}$}
Pursuing the analogy with the peak algebra, let us define
\begin{equation}
{\mathcal D} := \bigoplus_{k\ge 0}S_k{\mathcal D}^{(0)}\,.
\end{equation}
Note already that ${\mathcal D}$ is not a subalgebra of ${\bf Sym}$. It is only a
sub-coalgebra. Moreover, we have:
\begin{theorem}
Each homogeneous component ${\mathcal D}_n$ of ${\mathcal D}$ is stable under $*$. It is a
unital algebra, since it contains $S_n$, the neutral element of $*$ in
${\bf Sym}$.
\end{theorem}
We shall prove a slightly more general result, interpolating between
${\mathcal D}^{(0)}$ and ${\mathcal D}$.
\subsection{A filtration of ${\mathcal D}$}
Define ${\mathcal D}_n^{(k)}$ by
\begin{equation}
\label{filt}
{\mathcal D}_n^{(k)} := \bigoplus_{j=0}^k S_j {\mathcal D}^{(0)}_{n-j}.
\end{equation}
For $k=0$, this is ${\mathcal D}_n^{(0)}$ and for
$k\geq n$, one recovers ${\mathcal D}_n$.
The following alternative definition of the filtration
will be useful in the sequel:
\begin{lemma}
\label{altfilt}
\begin{equation}
{\mathcal D}_n^{(k)} = \bigoplus_{j=0}^{\min(n,k)} S_1^j {\mathcal D}_{n-j}^{(0)}.
\end{equation}
\end{lemma}
\noindent \it Proof -- \rm
Expanding $\sigma_1=e^{S_1}\sigma_1^\sharp$, we see that
\begin{equation}
S_k\equiv \frac{S_1^k}{k!} \pmod{\bigoplus_{j<k} S_j {\mathcal D}_{n-j}^{(0)} }.
\end{equation}
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\subsection{Dimensions of the ${\mathcal D}_n^{(k)}$}
From~(\ref{filt}), we see that the dimension $d_n^{(k)}$ of ${\mathcal D}_n^{(k)}$ is
\begin{equation}
d_n^{(k)} = \sum_{i\leq k} d_{n-i}^{(0)}.
\end{equation}
For $k=\infty$, these are the usual Fibonacci numbers.
The first values are given in the following table:
\begin{equation}
\begin{array}{c|c|c|c|c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
d_n^{(0)} & 1 & 0 & 1 & 1 & 2 & 3 & 5 & 8 & 13& 21\\
\hline
d_n^{(1)} & 1 & 1 & 1 & 2 & 3 & 5 & 8 & 13& 21& 34\\
\hline
d_n^{(2)} & 1 & 1 & 2 & 2 & 4 & 6 & 10 & 16& 26& 42\\
\hline
d_n^{(3)} & 1 & 1 & 2 & 3 & 4 & 7 & 11 & 18& 29& 47\\
\hline
d_n^{(\infty)} & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21& 34& 55\\
\hline
\end{array}
\end{equation}
\subsection{The complete picture}
\begin{theorem}
For all $n$ and $k$, each homogeneous component ${\mathcal D}^{(k)}_n$ of ${\mathcal D}^{(k)}$
is stable under $*$.
It is a unital algebra and the neutral element is
\begin{equation}
\label{defPnk}
P_{n}^{(k)} := \sum_{i=0}^{\min(k,n)} \frac{S_1^i}{i!} S_{n-i}^\sharp.
\end{equation}
\end{theorem}
The theorem is a consequence of the following lemma:
\begin{lemma}
\label{lem-exp}
Let $f$ and $g$ be in ${\bf Sym}$. Then
\begin{equation}
\left(\frac{S_1^m}{m!}f^\sharp\right)
* \left(\frac{S_1^n}{n!} g^\sharp\right) =
\left\{
\begin{array}{cc}
0 & \text{if $m\not=n$}, \\
\frac{S_1^n}{n!} ( f^\sharp * g^\sharp) & \text{otherwise}.
\end{array}
\right.
\end{equation}
\end{lemma}
\noindent \it Proof -- \rm
Replacing the right factor by its generating series, we have
\begin{equation}
\begin{split}
(S_1^m f^\sharp) * (e^{S_1} g^\sharp)
&= \sum_{(g)} (S_1^m * (e^{S_1}g_{(1)}^\sharp))\,
(f^\sharp * (e^{S_1}g_{(2)}^\sharp)) \\
&= \sum_{(g)} (e^{S_1}g_{(1)}^\sharp * S_1^m)\,
(f^\sharp * (e^{S_1}g_{(2)}^\sharp)), \\
\end{split}
\end{equation}
since $S_1^m$ is central for $*$.
Now, $(e^{S_1}{S^I}^\sharp * S_1^m)=0$ if $I$ is not empty since
$S_n^\sharp * S_1^n=S_1^n * S_n^\sharp=0$ for $n\geq1$ as
$e^{-S_1} * \sigma_1^\sharp=1$.
So the whole sum reduces to
\begin{equation}
\begin{split}
S_1^m (f^\sharp * (e^{S_1}g^\sharp)) \\
\end{split}
\end{equation}
Thus,
\begin{equation}
f^\sharp*\left(e^{S_1}g^\sharp\right)
=f*\sigma_1^\sharp*\left(e^{S_1}g^\sharp\right)
\end{equation}
Consider now the generic case $g=\sigma_1(XA)$:
\begin{equation}
\begin{split}
\sigma_1^\sharp * \left(e^{S_1}\sigma_1(XA)^\sharp\right)
&= \left(e^{-S_1} \sigma_1\right) *\left(e^{S_1}\sigma_1(XA)^\sharp\right) \\
&= \left(e^{-S_1}*e^{S_1}\sigma_1(XA)^\sharp\right)
\left(e^{S_1}\sigma_1(XA)^\sharp\right) \\
&=
\left(e^{S_1}*e^{-S_1}\right)
\left(\sigma_1(XA)^\sharp*e^{-S_1}\right)
\left(e^{S_1}\sigma_1(XA)^\sharp\right) \\
&=e^{-S_1}
\left(e^{-S_1}*\sigma_1(XA)^\sharp\right)
\left(e^{S_1}\sigma_1(XA)^\sharp\right) \\
&=e^{-S_1}
e^{-S_1(XA)^\sharp}
e^{S_1}\sigma_1(XA)^\sharp \\
&=e^{-S_1} e^{S_1}\sigma_1(XA)^\sharp = \sigma_1(XA)^\sharp \\
\end{split}
\end{equation}
where the third and fourth equalities come from the fact that
$e^{-S_1}$ is central for $*$ and idempotent.
Multiplying by $f$ on the left yields
\begin{equation}
f^\sharp * \left(e^{S_1} g^\sharp\right) = f * g^\sharp,
\end{equation}
whence the statement.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\noindent \it Proof -- \rm[of the theorem]
From Lemma \ref{altfilt}, we have $P_{n}^{(k)}\in{\mathcal D}^{(k)}_n$.
Moreover, since the $S_1^k {S^I}^\sharp$ with $I$ non-unitary form a basis
of ${\mathcal D}_n^{(k)}$ adapted to the direct sum decomposition, that $P_{n}^{(k)}$
is neutral on both sides is equivalent to the already known fact that
$S_n^\sharp$ is neutral in ${\mathcal D}_n^{(0)}$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\begin{corollary}
\label{cor-morph}
The map $\phi_{m} : \ {\mathcal D}^{(0)}_k\rightarrow {\mathcal D}^{(m)}_{k+m}$ defined by
\begin{equation}
\phi_m(f) = \frac{S_1^{m}}{m!}f
\end{equation}
is a (non-unital) monomorphism of algebras.
\end{corollary}
\begin{corollary}
As an \emph{algebra}, ${\mathcal D}_n$ is isomorphic to the direct sum
\begin{equation}
{\mathcal D}_n \sim \bigoplus_{k=0}^n {\mathcal D}_{n-k}^{(0)}.
\end{equation}
\end{corollary}
\noindent \it Proof -- \rm
As a vector space, ${\mathcal D}_n$ is the direct sum of the spaces
$V_n^{(k)}=S_1^k{\mathcal D}_{n-k}^{(0)}$.
By Lemma~\ref{lem-exp}, $V_n^{(k)}*V_n^{(\ell)}=0$ for $k\not=\ell$, and by
Corollary~\ref{cor-morph}, each $V_n^{(k)}$ is a subalgebra isomorphic to
$D_{n-k}^{(0)}$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\subsection{Representation theory of ${\mathcal D}_n^{(k)}$}
We are now in a position to deduce the representation theory of the
${\mathcal D}^{(k)}_n$ from that of ${\mathcal D}^{(0)}_n$.
We can extend Corollary~\ref{cor-NST} to all values of $k$, as a direct
consequence of Lemma~\ref{lem-NST}.
\begin{corollary}
Let $e_I$ be the idempotents of ${\bf Sym}$ defined in Corollary~\ref{cor-NST}.
\noindent
(i) The $e_J$ such that $J=(1^j,I)$ with $j\le k$ and $I$ does not contain a
part $1$ form a basis of ${\mathcal D}_n^{(k)}$.
\noindent
(ii) The principal idempotents of ${\mathcal D}^{(k)}_n$ are the $e_\lambda$
such that $m_1(\lambda)\le k$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{corollary}
We can now state the general result on the representation theory of
${\mathcal D}_n^{(k)}$:
\begin{corollary}
The irreducible representations of ${\mathcal D}_n^{(k)}$ are of dimension $1$ and
parametrized by partitions of $n$ with at most $k$ parts equal to $1$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{corollary}
Note that the principal idempotents of ${\mathcal D}_n$ are also a complete
set of minimal orthogonal idempotents of ${\bf Sym}_n$.
\subsection{$q$-Cartan matrix of ${\mathcal D}_n^{(k)}$}
We order the labels of the rows and columns of the $q$-Cartan matrices, namely
partitions with at most $k$ ones, first by their number of ones and then in
reverse lexicographic order.
So, for example, with $n=5$ and $k=3$, the order is
\begin{equation}
[ 5,\ 32,\ 41,\ 221,\ 311,\ 2111 ].
\end{equation}
With this convention, the $q$-Cartan matrix of ${\mathcal D}_n^{(k)}$ is the
block-diagonal matrix obtained by putting on the diagonal the $q$-Cartan
matrices of ${\mathcal D}_{n-i}^{(0)}$ for $i\leq\min(k,n)$.
Indeed, from the previous results, one easily sees that
\begin{lemma}
\begin{equation}
{\mathcal D}_n^{(k)} = P_n^{(k)} * {\bf Sym}_n * P_n^{(k)}.
\end{equation}
\end{lemma}
Since each $P_n^{(k)}$ is a sum of orthogonal idempotents (the
$S_1^iS_{n-i}^\sharp/i!$), this proves that the $q$-Cartan matrix of
${\mathcal D}_n^{(\infty)}$ is deduced from the $q$-Cartan matrix of ${\bf Sym}_n$ by
putting to $0$ the entries whose row and column do not have the same number of
ones.
\subsection{Block projectors}
We have seen that each space $S_1^j {\mathcal D}_{n-j}^{(0)}$ has as unit a part of
$P_n^{(k)}$, namely
\begin{equation}
\label{dnk}
D_{n,k} = \frac{S_1^k}{k!} S_{n-k}^\sharp
\end{equation}
Their double generating series is
\begin{equation}
D(t,u;A):= \sum_{n\ge 0}\sum_{k=0}^n t^{n-k}u^k D_{n,k}(A)
= e^{(u-t)S_1(A)}\sigma_t(A),
\end{equation}
that is, the noncommutative analog of the generating function of $d_{n,k}$,
the number of permutations in ${\mathfrak S}_n$ with exactly $k$ fixed points:
\begin{equation}
D(t,u)=\sum_{n\ge 0}\frac{1}{n!}\sum_{k=0}^n d_{n,k} {u^k}t^{n-k}
=\frac{e^{u-t}}{1-t}.
\end{equation}
\subsection{$q$-dimension polynomials}
If one sums up the entries of the $q$-Cartan matrix of ${\mathcal D}_n^{(\infty)}$, one
gets the following polynomials in $q$, refining the Fibonacci numbers:
\begin{equation}
1,\ 2,\ 3,\ 5,\ q+7,\ 2\,q+11,\ q^2+5\,q+15,\
3\,q^2+9\,q+22, \dots
\end{equation}
better represented in the following triangle:
\begin{equation}
\begin{array}{cccccccccc}
1 \\
2 \\
3 \\
5 \\
7 & 1 \\
11 & 2 \\
15 & 5 & 1 \\
22 & 9 & 3 \\
30 & 17 & 7 & 1 \\
42 & 28 & 16 & 3 \\
56 & 47 & 31 & 9 & 1 \\
77 & 73 & 58 & 21 & 4 \\
101 & 114 & 102 & 47 & 12 & 1 \\
135 & 170 & 175 & 94 & 32 & 4 \\
176 & 253 & 286 & 183 & 74 & 14 & 1 \\
231 & 365 & 461 & 333 & 162 & 40 & 5 \\
\end{array}
\end{equation}
The first column (constant terms of the polynomials)
corresponds to the size of the matrix, hence the number of partitions of $n$.
The second column is sequence~A000097 of~\cite{Slo}:
from the characterization of the quiver of ${\bf Sym}_n$, we also have that the second column
gives to the number of ways of selecting two different parts different from
$1$, in all partitions of $n$.
Finally, it is also equal to the number of ways of selecting two different
parts, in all partitions of $n-2$, hence justifying that the number of
arrows in the quiver of ${\mathcal D}_n^{(\infty)}$ is equal to the number of arrows in
the quiver of ${\bf Sym}_{n-2}$.
\section{The $(1-{\sym E})$ transform in ${\bf WQSym}^*$}
\subsection{Word quasi-symmetric functions}
A word $u$ over ${\sym N}^*$ is said to be \emph{packed} if the set of letters
occuring in $u$ is an interval of ${\sym N}^*$ containing $1$.
The algebra ${\bf WQSym}(A)$ (Word Quasi-Symmetric functions) is defined as the
subalgebra of ${\sym K}\<A\>$ based on \emph{packed words} and spanned by the
elements
\begin{equation}
{\bf M}_u(A) := \sum_{{\rm pack\,}(w)=u} w,
\end{equation}
where ${\rm pack\,}(w)$ is the \emph{packed word} of $w$, that is, the word obtained
by replacing all occurrences of the $k$-th smallest letter of $w$ by $k$.
For example,
\begin{equation}
{\rm pack\,}(871883319) = 431442215.
\end{equation}
This is the invariant algebra of the quasi-symmetrizing action of ${\mathfrak S}(A)$ on
${\sym K}\<A\>$ \cite{NCSF7}.
Packed words can be identified with set compositions in an obvious way, and
geometrically, they can be interpreted as facets of the permutohedron:
a packed word $w = w_1\dots w_n$ with largest entry $\ell$ can be identified
with the set composition $[P_1, \dots, P_\ell]$ where
$P_j = \{ i\leq n \ |\ w_i = j\}$.
For example, $431442215$ corresponds to
$[\{3,8\},\{6,7\},\{2\},\{1,4,5\},\{9\}]$.
\medskip
Let ${\bf N}_u={\bf M}_u^*$ be the dual basis of $({\bf M}_u)$.
It is known that ${\bf WQSym}$ is a self-dual Hopf algebra~\cite{Hiv,NT06} and that
on the graded dual ${\bf WQSym}^*$, an internal product $*$ may be defined by
\begin{equation}
\label{intWQ}
{\bf N}_u * {\bf N}_v = {\bf N}_{{\rm pack\,}(u,v)},
\end{equation}
where the packing of biwords is defined with respect to the lexicographic
order on biletters, so that, for example,
\begin{equation}
{\rm pack\,}\left(\gf{42412253}{53154323}\right)
= 62513274.
\end{equation}
This product is induced from the internal product of parking
functions~\cite{NTpark,NT1,NTp2} and allows one to identify the homogeneous
components ${\bf WQSym}_n$ with the (opposite) Solomon-Tits algebras, in the sense
of~\cite{Patras}.
The (opposite) Solomon descent algebra, realized as ${\bf Sym}_n$, is embedded in
the (opposite) Solomon-Tits algebra realized as ${\bf WQSym}^*_n$ by
\begin{equation}
\label{S2NW}
S^I = \sum_{{\rm ev\,}(u)=I} {\bf N}_u,
\end{equation}
where ${\rm ev\,}(u)$ is the evaluation of $u$.
From now on, we shall denote ${\bf WQSym}^*$ by ${\mathcal W}$.
\subsection{Idempotents of ${\mathcal W}$}
\subsubsection{The semi-simple quotient}
It is known that the radical of ${\mathcal W}_n$ is spanned by the differences
\begin{equation}
{\bf N}_u - {\bf N}_v,
\end{equation}
where $v=\sigma(u)$ for some permutation $\sigma$ of the support of $u$,
${\rm supp}(u)=\{i\ |\ |u|_i\not=0\}$.
This is easily seen: Equation~(\ref{intWQ}) implies that the ${\bf N}_u-{\bf N}_v$ are
nilpotent of order $2$ if $v=\sigma(u)$, and that their span is an ideal
${\mathcal R}_n$. Moreover, any product of $n$ such factors
${\bf N}_{u_i}-{\bf N}_{v_i}$ vanishes since the product of two such factors is either
strictly finer than $u_i$ or zero, so that ${\mathcal R}_n$ is nilpotent.
The quotient $W_n/{\mathcal R}_n$ is semi-simple. It can be
identified with the (commutative) algebra of set partitions with $\wedge$ (the
inf for the refinement order on set partitions) as product.
Indeed, packed words encode set compositions, $u=u_1\dots u_n$ corresponding
to the set composition of $[n]$ in which $i$ belongs to the block $u_i$,
\emph{e.g.},
\begin{equation}
u = 21231 \Longleftrightarrow [\{2,5\},\{1,3\},\{4\}].
\end{equation}
and the left action of permutations amounts to permuting the blocks,
\emph{e.g.}, with $\sigma=231$,
\begin{equation}
\sigma(21231)=32312 \Longleftrightarrow [\{4\},\{2,5\},\{1,3\}].
\end{equation}
Hence, the idempotents of a complete family of ${\mathcal W}_n$ are parametrized by set
partitions of $[n]$.
\subsubsection{The idempotents of Saliola}
In~\cite{Sal}, Saliola has given a general recipe for constructing such
complete sets.
Given a packed word $u$, denote by $\Pi(u)$ the set partition obtained by
forgetting the order among the blocks of the set compositions encoded by $u$.
For each set partition $\pi$ of $[n]$, choose a linear combination
\begin{equation}
l_\pi=\sum_{\Pi(u)=\pi} c_u {\bf N}_u,
\end{equation}
where the coefficient $c_u$ depends only on the evaluation ${\rm ev\,}(u)$ of $u$, and
\begin{equation}
\sum_{\Pi(u)=\pi} c_u = 1.
\end{equation}
Start with the initial condition
\begin{equation}
e_{\{1\},\{2\},\dots,\{n\}}
= \frac{1}{n!} \sum_{\sigma\in{\mathfrak S}_n} {\bf N}_\sigma,
\end{equation}
hence equal to $S_1^n/n!$ in~${\bf Sym}$,
and define by the induction
\begin{equation}
\label{recFS}
e_\pi = l_\pi * ({\bf N}_{1^n} - \sum_{\pi'>\pi} e_{\pi'})
\end{equation}
where ${\bf N}_{1^n}=S_n$ is the identity of $*$, and $\geq$ is the refinement
order. Then, the $e_\pi$ form a complete set of orthogonal idempotents of
${\mathcal W}_n$.
\subsubsection{A non-recursive construction}
Families of Saliola idempotents can be computed for all ${\mathcal W}_n$
simultaneously, in a non-recursive way from families of idempotents of the
descent algebra ${\bf Sym}_n$ constructed by the method developed in~\cite{NCSF2}.
Recall that the starting point of this construction is a sequence of Lie
idempotents $\gamma_n\in{\bf Sym}_n$, that is, an arbitrary sequence of primitive
elements whose commutative image in $Sym$ is $p_n/n$.
Then, if we decompose the identity $S_n$ of ${\bf Sym}_n$ as
\begin{equation}
S_n = \sum_{I\vDash n} c_I \gamma^I,
\end{equation}
the elements
\begin{equation}
e_\lambda := \sum_{I\downarrow n} c_I \gamma^I,
\end{equation}
with $\lambda$ a partition of $n$ form a complete family of orthogonal
idempotents of ${\bf Sym}_n$.
Let us fix such a family, and define for each set partition $\pi$ of $[n]$
\begin{equation}
l_\pi = \sum_{\Pi(u)=\pi} c_{{\rm ev\,}(u)} {\bf N}_u.
\end{equation}
These elements satisfy Saliola's conditions: obviously, $c_{{\rm ev\,}(u)}$ depends
only on ${\rm ev\,}(u)$, and
\begin{equation}
\sum_{\Pi(u)=\pi} c_{{\rm ev\,}(u)} =
\prod_i m_i(\lambda)! \sum_{I\downarrow\lambda} c_I
\end{equation}
is the coefficient of $p_\lambda / z_\lambda$ in the commutative image of
$S_n$, which is $h_n$, so it is $1$.
Hence, the sequence $(\gamma_n)$ determines idempotents $e_\pi$ of the
${\mathcal W}_n$ by the recursion~(\ref{recFS}).
But we can also compute these directly as follows:
\begin{theorem}
The idempotents $e_\pi$ are given by the internal products
\begin{equation}
e_\pi = l_\pi * e_\lambda.
\end{equation}
\end{theorem}
\noindent \it Proof -- \rm
Let $\tilde{e}_\pi = l_\pi * e_\lambda$.
For $\pi=\{\{1\},\dots,\{n\}\}$,
we have $l_\pi = S_1^n/n!$, $e_\lambda= S_1^n/n!$, so that
$\tilde{e}_\pi = S_1^n/n!$.
Let $l_\lambda = \sum_{\Lambda(\pi)=\lambda} l_\pi$, where $\Lambda(\pi)$ is
the integer partition recording the block lengths of $\pi$.
We have
\begin{equation}
l_\lambda = \sum_{I\downarrow \lambda} c_I S^I
\equiv e_\lambda \mod\bigoplus_{l(I)>l(\lambda)} {\sym K} \gamma^I,
\end{equation}
so that $e_\lambda = l_\lambda * e_\lambda$ in ${\bf Sym}$.
We want to show that $\tilde{e}_\pi = e_\pi$.
For that, recall from~\cite{Sal} that, if $\Pi(u)\not\leq\pi$,
${\bf N}_u*e_\pi=0$,
so that
$l_\pi*e_{\pi'}=0$ if $\pi\not>\pi'$, and
$\Lambda(\pi')\not=\Lambda(\pi)$.
This implies in particular that
$e_\lambda = \sum_{\Lambda(\pi)=\lambda} e_\pi$.
Indeed, this is true for $\lambda=1^n$, and, by induction,
$e_\pi = l_\pi* ( {\bf N}_{1^n} - \sum_{l(\pi')>l(\pi)} e_{\pi'})$,
since $\pi'>\pi$ implies $l(\pi')>l(\pi)$.
Hence
\begin{equation}
e_\pi = l_\pi* ( {\bf N}_{1^n} - \sum_{l(\lambda')>l(\lambda)} e_{\lambda'})
= l_\pi* \sum_{l(\lambda')\leq l(\lambda)} e_{\lambda'}.
\end{equation}
Summing over $\pi$, we get
\begin{equation}
\sum_{\Lambda(\pi)=\lambda} e_\pi
= l_\lambda * \sum_{l(\lambda')\leq l(\lambda)} e_{\lambda'}
= e_\lambda.
\end{equation}
Now,
\begin{equation}
\begin{split}
e_\pi
&= l_\pi * ({\bf N}_{1^n} - \sum_{\pi'>\pi} e_{\pi'})
= l_\pi * \sum_{\pi'\not>\pi} e_{\pi'}
\\
&= l_\pi * (e_\lambda + \sum_{\pi'\not>\pi ; \Lambda(\pi')\not=\Lambda(\pi)}
e_{\pi'})
= l_\pi * e_\lambda = \tilde{e}_\pi.
\end{split}
\end{equation}
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\subsection{The $(1-{\sym E})$-transform in ${\mathcal W}$}
The embedding~(\ref{S2NW}) of ${\bf Sym}$ in ${\mathcal W}$ can be defined on the
generators as
\begin{equation}
S_n\longmapsto {\bf N}_{1^n}\,.
\end{equation}
It is clearly a bialgebra morphism.
The element
\begin{equation}
\sigma_1^\sharp = e^{-{\bf N}_1}\sum_{n\ge 0}{\bf N}_{1^n}
\end{equation}
is well-defined in ${\mathcal W}$, and so is the $\sharp$-transform
\begin{equation}
F^\sharp := F * \sigma_1^\sharp\,.
\end{equation}
\subsection{Bases of ${\mathcal W}^\sharp$}
Let us say that a packed word $u$ is {\em non-unitary} (and \emph{unitary}
otherwise) if no letter occurs exactly once in $u$. These words correspond to
set compositions without singletons.
\begin{proposition}
The ${\bf N}_u^\sharp$ with u non-unitary form a basis of ${\mathcal W}^\sharp$.
\end{proposition}
\noindent \it Proof -- \rm
Let us say that $v$ is finer than $u$ (and write $v>u$) if the set composition
encoded by $v$ is finer than the set composition encoding $v$.
Then,
\begin{equation}
{\bf N}_u * \sigma_1^\sharp = {\bf N}_u + \sum_{v} {c_{uv}{\bf N}_v},
\end{equation}
where $v>u$ or $v$ is unitary. Hence, the ${\bf N}_v^\sharp$ with $u$ non-unitary
are linearly independent.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\subsection{Algebraic structure of ${\mathcal W}^\sharp$}
Let ${\mathcal J}$ be the two-sided ideal of ${\mathcal W}$ generated by the ${\bf N}_u$
such that $u$ has at least a letter occuring exactly once.
The product rule (\ref{intWQ}) shows that ${\mathcal J}$ is an ideal for the
internal product as well. Hence, the projection
\begin{equation}
\pi:\ {\mathcal W}\longrightarrow {\mathcal W}/{\mathcal J}
\end{equation}
is a morphism for $*$. Its restriction to ${\mathcal W}^\sharp$ is then an isomorphism,
and clearly,
\begin{equation}
\pi(\sigma_1^\sharp)=\sigma_1\,.
\end{equation}
Since $\sigma_1$ is neutral in ${\mathcal W}/{\mathcal J}$, we have:
\begin{proposition}
$\sigma_1^\sharp$ is neutral in ${\mathcal W}^\sharp$.\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{proposition}
\noindent
Note that this proof would apply to ${\bf Sym}$ as well.
To summarize,
\begin{proposition}
${\mathcal W}^\sharp$ is isomorphic to ${\mathcal W}/{\mathcal J}$ as a Hopf algebra, and each
${\mathcal W}_n^\sharp$ is $*$-isomorphic to ${\mathcal W}_n/{\mathcal J}_n$, with
$D_n=S_n^\sharp={\bf N}_{1^n}^\sharp$ as neutral element.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{proposition}
\subsection{Representation theory of ${\mathcal W}^\sharp$}
We can now apply Lemma \ref{BAe} with
$A={\mathcal W}_n$,
$B={\mathcal W}_n^\sharp$ and $e={\bf S}_n^\sharp$.
Thus
\begin{equation}
{\rm rad\,}^k {\mathcal W}_n^\sharp = D_n * ({\rm rad\,}^k {\mathcal W}_n) * D_n.
\end{equation}
The irreducible representations of ${\mathcal W}_n$, which are one-dimensional
are parametrized by set partitions of $[n]$.
The $q$-Cartan matrices and quiver of ${\mathcal W}_n$ have been determined
in~\cite{Sc}:
\begin{equation}
\label{cabq}
c_{\alpha,\beta}(q) = c_{\alpha,\beta} q^{l(\alpha)-l(\beta)},
\end{equation}
where $l(\pi)$ is the number of blocks of a set partition $\pi$, and the
Cartan invariant $c_{\alpha,\beta}$ is $0$ if $\alpha$ is not finer than
$\beta$, and otherwise
\begin{equation}
c_{\alpha,\beta} = \prod_i (m_i-1)!,
\end{equation}
where for each block $B_i$ of $\beta$, $m_i$ is the number of blocks of
$\alpha$ into which $B_i$ has been split.
For example, with $\alpha = 12|3|4|56|7$ and $\beta=1234|567$,
we have: $c_{\alpha,\beta} = (3-1)! (2-1)! = 2$, $l(\alpha)=5$,
$l(\beta)=2$, so that
$c_{\alpha,\beta}(q) = 2q^3$.
\begin{theorem}
The $q$-Cartan matrix of ${\mathcal W}_n^\sharp$ is the restriction to rows and columns
indexed by non-unitary set partitions of $[n]$ of~(\ref{cabq}).
In particular, the vertices of the quiver of ${\mathcal W}_n^\sharp$ are the
non-unitary set partitions, and there is an arrow $\alpha\to\beta$ whenever
$\beta$ is obtained from $\alpha$ by merging two blocks.
\end{theorem}
\subsection{Analogue of ${\mathcal D}$ in ${\mathcal W}$}
Let ${\mathcal V}_n^{(0)}={\mathcal W}_n^\sharp$ and
\begin{equation}
{\mathcal V}_n^{(k)} = \bigoplus_{k=0}^n D_{n,k} * {\mathcal W}_n * D_{n,k}\,.
\end{equation}
Then, as in the case of ${\bf Sym}$, each ${\mathcal V}_n^{(k)}$ is a unital
subalgebra of ${\mathcal W}_n^\sharp$.
\section{The $(1-{\sym E})$-transform in ${\bf FQSym}$}
\subsection{Definition}
Recall that ${\bf FQSym}$ is based on permutations, that in the mutually dual
bases ${\bf F}_\sigma={\bf G}_{\sigma^{-1}}$, the internal product is defined by
\begin{equation}
{\bf F}_\sigma*{\bf F}_\tau={\bf F}_{\sigma\tau}\quad\text{or equivalently}\quad
{\bf G}_\sigma*{\bf G}_\tau={\bf G}_{\tau\sigma}\,,
\end{equation}
and that ${\bf Sym}$ is embedded into ${\bf FQSym}$ by $S_n={\bf G}_{12\dots n}$.
The transformation can therefore be defined by
\begin{equation}
{\bf F}_\sigma^\sharp := {\bf F}_\sigma*\sigma_1^\sharp\,.
\end{equation}
Since the splitting formula remains valid in ${\bf FQSym}$ when the right factor
of the internal product is in ${\bf Sym}$ \cite{NCSF7}, this is again a Hopf
algebra morphism.
As we shall see below, in ${\bf FQSym}$, the idempotent $D_n=S_n^\sharp$ as well
as the other $D_{n,k}$ defined in~(\ref{dnk}) admit an interesting
interpretation.
\subsection{The Tsetlin library (uniform case)}
The $(1-{\sym E})$-transform in ${\bf FQSym}$ is related to a classical problem in
probability theory known as the \emph{Tsetlin library} (see {\it e.g.,}
\cite{BHR}).
This is a Markov chain on ${\mathfrak S}_n$, defined by a shelf of $n$ books, which are
randomly picked by users and them put back at the left of the shelf after use.
In the uniform case (when all books are picked with the same probability), the
determination of the stationary distribution amounts to the diagonalisation of
the linear operator on ${\sym C}{\mathfrak S}_n$
\begin{equation}
t_n(f)=f\tau_n
\end{equation}
where
\begin{equation}
\tau_n = 12\ldots n+ 2134\dots n + 3124\dots n +\cdots + n12\dots n-1
\in {\sym C}{\mathfrak S}_n\,.
\end{equation}
This problem is also an ingredient of the proof of Hivert's conjecture by
Garsia and Wallach~\cite{GW}. It can be solved in many different ways.
The following one is quite natural in the context on Noncommutative
Symmetric Functions.
We start with the observation that $\tau_n$ is in the descent algebra
$\Sigma_n$. Indeed, $\tau_n=D_{\subseteq \{1\}}$ (the sum of permutations
having at most a descent at the first position), so that its representation
as a noncommutative symmetric function is $S^{1,n-1}$, a rather
well-understood element.
From this remark, we obtain immediately the eigenvalues of $t_n$. Indeed,
according to Proposition 3.12 of \cite{NCSF2}, these are the scalar products
$\langle h_{n-1}h_1,p_\lambda\rangle$ of ordinary symmetric functions.
Clearly, the scalar product evaluates to $m_1(\lambda)$, so that the spectrum
is $0,1,2,\ldots,n-2,n$.
Let us now construct the spectral projectors. To this aim, we shall need to
evaluate some polynomials in $t_n$.
Let us set $T_n=S^{1,n-1}$ and consider the generating function
\begin{equation}
T=\sum_{n\ge 0}T_n = S_1\sigma_1\,.
\end{equation}
Since the internal products $T_n*T_m$ are (by definition) $0$ for
$m\not=n$, we have
\begin{equation}
\sum_n T_n^{*r} = T^{*r}\,
\end{equation}
and using iteratively the splitting formula (\cite{NCSF2}, Prop. 2.1)
\begin{equation}
T*T^{*(r-1)} = (S_1\sigma_1)*T^{*(r-1)}=\mu[(S_1\otimes\sigma_1) *_2
\Delta(T^{*(r-1)})],
\end{equation}
we get the expression
\begin{equation}
T^{*r} = B_r(S_1)\sigma_1,
\end{equation}
where $B_r(x)$ are the Bell polynomials (this is the obvious noncommutative
analogue of the classical formula for the Kronecker powers of the
representation of ${\mathfrak S}_n$ by permutation matrices).
Using the fact that the coefficients of $B_n$ are the Stirling numbers of the
second kind $S(n,k)$, we obtain in ${\bf Sym}$
\begin{equation}
T*(T-1)*(T-2)*\cdots * (T-k+1)= S_1^k\sigma_1\,,
\end{equation}
and in particular, in degree $n$,
\begin{equation}
T_n*(T_n-1)*(T_n-2)*\cdots * (T_n-n+1) = S_1^n
\end{equation}
and as well
\begin{equation}
T_n*(T_n-1)*(T_n-2)*\cdots * (T_n-n+2) =S_1^n\,,
\end{equation}
and since it is plain that $S_1^n*T_n=nS_1^n$, so that $S_1^n*(T_n-n)=0$,
the minimum polynomial of $T_n$ is
\begin{equation}
P_n(x)=(x-n)\prod_{k=0}^{n-2}(x-k)\,.
\end{equation}
This shows that $T_n$ is semisimple, and allows an easy construction
of the spectral projectors.
Let us start with the kernel. The projector is given by $f\mapsto f*D_n$
where
\begin{equation}
D_n=\frac{(T_n-1)*(T_n-2)*\cdots*(T_n-n+2)*(T_n-n)}{(-1)(-2)\cdots(2-n)(-n)}
\end{equation}
but since $T_n-n+1$ is invertible, one can take as well
\begin{equation}
D_n=\frac{(-1)^n}{n!}(T_n-1)_{*n}
\end{equation}
where $(x)_n=x(x-1)\cdots (x-n+1)$, and the star means evaluation with the
internal product. This is a better choice, since we have now a simple
generating series for all these projectors,
\begin{equation}
\sum_{n\ge 0}D_nx^n=e^{-xS_1}\cdot\sigma_x= \sigma_x((1-{\sym E})A)\,.
\end{equation}
Indeed, we have $(x-1)_n=(x)_n-n(x-1)_{n-1}$, so that
\begin{equation}
\frac{(-1)^n}{n!}(T-1)_{*n} = \frac{(-S_1)^n}{n!}\sigma_1-
\frac{(-1)^{n-1}}{(n-1)!}(T-1)_{*\
n-1}=\sum_{k=0}^n\frac{(-S_1)^k}{k!}\sigma_1\,.
\end{equation}
The same reasoning shows that the projectors $D_{n,k}$ on the eigenspaces of
$k$ are given by the generating series
\begin{equation}
D(t,u)=\sum_{n\ge 0}\sum_{k=0}^nt^{n-k}u^k D_{n,k} = e^{(u-t)S_1}\sigma_1 \,,
\end{equation}
which is $\sigma_1(tA-(t-u){\sym E} A)$, so that these elements coincide with those
defined by~(\ref{dnk}).
\subsection{Characters of the associated modules}
The Frobenius characteristic of the left ideal of ${\sym C}{\mathfrak S}_n$ generated by
the idempotents $\delta_{n,k}$ corresponding to $D_{n,k}$ via the
identification $\sigma\leftrightarrow {\bf F}_\sigma$ can now be calculated
as follows (compare \cite[Cor. 4.2]{Sc}).
Since $\delta_{n,k}$ is an idempotent, its characteristic ${\rm ch\,}(\delta_{n,k})$
coincides with its cycle index $Z(\delta_{n,k})$. By the Gessel-Reutenauer
formula \cite{GR}, the coefficient of $p_\mu$ in $Z(\delta_{n,k})$ is equal
to $\< \underline{D_{n,k}}, L_\mu\>$, where $\underline{F}$ means the
commutative image of the noncommutative symmetric function $F$,
and for $\nu=1^{n_1}2^{n_2}\ldots$,
\begin{equation}
L_\nu=h_{n_1}[\ell_1]h_{n_2}[\ell_2]\cdots\,,\quad
\ell_n=\frac1n\sum_{d|n}\mu(d)p_{n/d}^d\,,
\end{equation}
$\mu$ denoting here the Moebius function.
Hence, the generating function of all the cycle indexes is
\begin{equation}
Z_Y(t,u)=\<D(t,u;X), L(X,Y) \>_X
=\<D(t,u;X),L(Y,X)\>_X
\end{equation}
by the symmetry formula \cite{ST}
\begin{equation}
L(X,Y)=L(Y,X)=\sum_\mu L_\mu(X)p_\mu(Y) = \prod_{n\ge
1}\sigma_{p_n(X)}[\ell_n(Y)]\,.
\end{equation}
Plugging this last expression into the scalar product and dualizing, we
obtain
\begin{equation}
\prod_{n\ge 1}\sigma_{p_n(t+(u-t){\sym E})}[\ell_n(Y)]
= \frac{\sigma_1((u-t)Y)}{1-tp_1(Y)}\,.
\end{equation}
In particular, specializing at $Y={\sym E}$ gives that the dimension
of ${\sym C}{\mathfrak S}_n\delta_{n,k}$ is $d_{n,k}$, the number of permutations in
${\mathfrak S}_n$ with exactly $k$ fixed points.
It is also easy to obtain the expansion of $Z_Y(t,u)$ as a combination
of the $L_\mu(Y)$. Indeed, writing $D(t,u;x)=\sigma_1((u-t){\sym E} X +tX)$,
we have $\<D(t,u),p_\mu\>=u^{m_1}\prod_{i\ge 2}t^{m_i}$, where $m_i$
is the multiplicity of the part $i$ in $\mu$. Hence,
\begin{equation}
Z(\delta_{n,k})=\sum_{m_1(\mu)=k}L_\mu
\end{equation}
We note that this is the quasi-symmetric generating function of the
permutations with exactly $k$ fixed points.
Note that $Z(D(t)$ is the commutative image of the generating series
of desarrangements (\ref{desar}).
\subsection{$q$-derangement numbers}
From the above considerations, one can easily derive a (known) closed formula
for the $q$-derangement numbers (compare \cite[Theorem 4.5]{Sc})
\begin{equation}
d_n(q):=\sum_{\sigma\in D_n}q^{{\rm maj\,} \sigma}\,.
\end{equation}
Indeed,
\begin{equation}
\begin{split}
d_n(q)&=\left\langle \sum_{\sigma\in D_n}{\bf F}_\sigma,
\sum_{\tau\in{\mathfrak S}_n}q^{{\rm maj\,}(\tau)}{\bf G}_\tau\right\rangle_{{\bf FQSym}}\\
&=\sum_{\sigma\in D_n}\sum_{|I|=n}q^{{\rm maj\,}(I)}\<F_{C(\sigma)},R_I\>
=\left\langle
\sum_{m_1(\mu)=0}L_\mu, K_n(q)\right\rangle\,.
\end{split}
\end{equation}
Hence,
\begin{equation}
\begin{split}
\sum_{n\ge 0} d_n(q)\frac{x^n}{(q)_n}
&=\left\langle \sigma_1\left[\sum_{n\ge 2}x^nl_n\right] ,
\sigma_1\left(\frac{X}{1-q}\right)\right\rangle\\
&=\left\langle \frac{\lambda_{-x}}{1-xp_1},
\sigma_1\left(\frac{X}{1\!-\!q}\right)
\right\rangle
=\lambda_{-x}\left(\frac{1}{1\!-\!q}\right)
\left(1-\frac{x}{1\!-\!q}\right)^{-1}\\
&=\frac{1-q}{1-x-q}\prod_{n\ge 0}(1-xq^n)
\end{split}
\end{equation}
so that finally~\cite{Wa}
\begin{equation}
d_n(q)=[n]!\sum_{k=0}^n\frac{(-1)^k}{[k]!}q^{\binom{k}{2}}\,.
\end{equation}
\subsection{Characters from Lie idempotents}
The expression
\begin{equation}
\label{charact}
{\rm ch}(\delta_{n,k})=\sum_{m_1(\lambda)=k}L_\lambda\,,
\end{equation}
is also a consequence of the following type of expressions
\begin{equation}
\label{decD}
D_{n,k}=\sum_{m_1(\lambda)=k}E_\lambda(\pi)
\end{equation}
in the notation of \cite[Theorem 3.16]{NCSF2}, for some sequence $\pi_n$ of
Lie idempotents in descent algebras. Indeed, \cite[Theorem 3.21]{NCSF2},
implies then that the character is given by (\ref{charact}). We have already
seen one such expression with $\pi_n=\zeta_n$, the Zassenhaus idempotents.
We can also write one involving the Hausdorff series. Writing as usual
\begin{equation}
\Phi =\sum_{n\ge 1}\phi_n=\log \sigma_1
\end{equation}
(the Solomon idempotents), we have
\begin{equation}
\sum_{n\ge 0}D_n= e^{-\phi_1} e^\Phi=e^{H(-\phi_1,\Phi)}
\end{equation}
where $H$ is the Hausdorff series. Taking $\pi_1=\phi_1$ and
$\pi_n=H_n(-\phi_1,\Phi)$ for $n\ge 2$, we obtain a sequence
of Lie idempotents (see, \emph{e.g.}, \cite{NCSF2}, Theorem 3.1),
from which it is easy to build a decomposition of the identity
\begin{equation}
\sigma_1 = e^{\pi_1}\exp\left\{\sum_{n\ge 2}\pi_n\right\}\,,
\end{equation}
and more explicitely,
\begin{equation}
\label{decId}
S_n=\sum_{r+s=n}\frac{1}{r!s!}\sum_{\ell(J)=r,|J|=n-s,1\not\in J}\pi^{1^s,J}.
\end{equation}
This gives in particular the decomposition (\ref{decD}),
with, for a partition $\lambda$ such that $m_1(\lambda)=0$,
\begin{equation}\label{projpi}
E_\lambda(\pi) =\frac{1}{\ell(\lambda)!}\sum_{I\downarrow \lambda}\pi^I\,,
\end{equation}
where $I\downarrow \lambda$ means that the nondecreasing rearrangement
of the composition $I$ is the partition $\lambda$.
\subsection{Eigenbases of $t_n$}
From Proposition 7.4 of \cite{NCSF3}, we know the image a projector
of the type (\ref{projpi}).
It is formed of weighted symmetrizations of Lie elements.
With the above $\pi_n$, the distribution is uniform, so that the kernel
consists in ordinary symmetrized products of Lie elements. Concretely, a basis
of ${\rm Ker\,} t_n$ in ${\sym C}{\mathfrak S}_n$ is for example
\begin{equation}
(\gamma_1\theta_{\lambda_1},\gamma_2\theta_{\lambda_2},\cdots,
\gamma_r\theta_{\lambda_r})
\end{equation}
where $(a,b,c)=abc+acb+bac+bca+cab+cba$, and so on (symmetrized products),
the $\gamma_k$ are the minimal representatives of the cycles of a
derangement, $\theta_n=[[\cdots[1,2],3],\cdots n]$ is a Dynkin element,
and $\lambda$ runs over partitions without part 1.
For example, a basis of ${\rm Ker\,} t_4$ of dimension $d_4=9$ is given by the
elements $[[[1,a],b]c]$ with $abc$ running over permutations of $234$, for
$L_4$, and by the three symmetrized products $([1,2],[3,4])$, $([1,3],[2,4])$,
and $([1,4],[2,3])$ for $L_{22}$.
Bases of the other eigenspaces are obtained by the same process, using
weighted symmetrizations as indicated in~\cite{NCSF3}.
Indeed, Equation~(\ref{decId}) shows that a basis of the eigenspace with
eigenvalue $s$ is given by
\begin{equation}
(\gamma_1\theta_{\lambda_1},\gamma_2\theta_{\lambda_2},\cdots,
\gamma_r\theta_{\lambda_r})\cdot (j_1\, \shuffl \, j_2\, \shuffl \,\cdots\, \shuffl \, j_s)
\end{equation}
where $\gamma_k$ are the minimal representatives of the cycles of length at
least $2$ a permutation of cycle type $(\lambda,1^s)$ having $s$ fixed points
$j_1,j_2,\ldots,j_s$.
To continue with $n=4$, a basis of the $1$ eigenspace is $[[i,j],k]\cdot l$
($i<j,k$, $ijkl$ a permutation of $1234$), dimension $8$, and a basis of the
$2$-eigenspace is given by $$[i,j]\cdot(kl+lk)\,\ i<j,\ k<l$$ where $ijkl$ is
a permutation of $1234$ (dimension $6$).
Finally, the $4$-eigenspace is one dimensional and generated by the full
symmetrizer.
Using the $\zeta_n$ instead of the $\pi_n$, we can replace symmetrized
products by ordinary products of homogeneous Lie polynomials taken
in nondecreassing order of the degrees.
The idempotents $\delta_{n,k}$ have been first studied by M.
Schocker~\cite{Sc} (apparently unaware of previous works on the subject and of
their relation with the Tsetlin library).
\subsection{A basis of ${\bf FQSym}^\sharp$}
We have seen that the $(1-{\sym E})$-transform is a bialgebra morphism in ${\bf FQSym}$.
Hence, its image ${\bf FQSym}^\sharp$ is a Hopf subalgebra.
The ${\mathfrak S}_n$-module $\Delta_{n,k}$ can be identified with
\begin{equation}
{\bf FQSym}_n^\sharp = {\bf FQSym}_n * D_{n,k},
\end{equation}
so that
\begin{equation}
{\rm dim\,} {\bf FQSym}_n^\sharp = d_n.
\end{equation}
It is therefore desirable to find a basis of ${\bf FQSym}^\sharp$ labeled by
derangements, or some other set of permutations naturally in bijection with
these. As we shall see, the natural transformation involved here is simply a
version of Foata's first fundamental transformation~\cite{Loth}.
Let $\gamma_n$ be the cycle
\begin{equation}
\gamma_n := n\,1\,2\dots n-1,
\end{equation}
so that
\begin{equation}
{\bf S}^{\gamma_n} = T_n=S_1 S_{n-1} = R_n + R_{1,n-1} =
\sum_{\sigma\in 1\, \shuffl \, 23\dots n} {\bf F}_\sigma.
\end{equation}
Since the $\sharp$-transform is a morphism for the product of ${\bf FQSym}$,
\begin{equation}
{\bf S}^{\gamma_n} * \sigma_1^\sharp = S_1^\sharp S_{n-1}^\sharp =0,
\end{equation}
and, for any permutation $\sigma\in{\mathfrak S}_n$,
\begin{equation}
({\bf F}_\sigma * {\bf S}^{\gamma_n})^\sharp
= {\bf F}_\sigma * {\bf S}^{\gamma_n} * \sigma^\sharp
= 0.
\end{equation}
Recall that $i$ is a \emph{left-right minimum} of $\sigma$ if
\begin{equation}
\sigma_j>\sigma_i \text{\ for all $j<i$}.
\end{equation}
Let $X_n$ be the set of permutations of ${\mathfrak S}_n$ such that $\sigma\cdot 0$ does
not have two consecutive left-right minima (that is, $\sigma$ does not end by
$1$ and does not have two consecutive LR-minima),
and let $Y_n = {\mathfrak S}_n \backslash X_n$.
\begin{lemma}
\label{FY}
For $\sigma\in Y_n$, write
\begin{equation}
\sigma\cdot 0 = \cdots \sigma_i \sigma_{i+1} \cdots
\end{equation}
where $i$ and $i+1$ is the first pair of consecutive LR-minima, and let
\begin{equation}
\sigma' = \sigma_i \cdot \sigma_1 \cdots \hat{\sigma_i} \cdots \sigma_n
\end{equation}
be the permutation obtained by moving $\sigma_i$ at the first position,
leaving the remaining letters unchanged, and removing the zero in the end.
Then
\begin{equation}
{\bf F}_{\sigma'} * {\bf S}^{\gamma_n} = {\bf F}_\sigma + \sum_{\tau\in T} {\bf F}_\tau,
\end{equation}
where the permutations of $T$ are lexicographically smaller than $\sigma$.
\end{lemma}
\noindent \it Proof -- \rm
The expression
\begin{equation}
{\bf F}_{\sigma'} * {\bf S}^{\gamma_n} =
\sum_{\tau \in \sigma_i\, \shuffl \, \sigma_1 \dots \hat{\sigma_i} \dots \sigma_n}
{\bf F}_\tau
\end{equation}
contains ${\bf F}_\sigma$ and, since $\sigma_i$ is a LR-minimum, $\sigma$ is the
maximal element of the previous sum.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\medskip
For a permutation $\sigma$ with LR-minima $i_1,\dots,i_p$, let
\begin{equation}
\phi(\sigma) = (\sigma_1\dots\sigma_{i_1-1})
(\sigma_{i_1}\dots\sigma_{i_2-1})
\dots
(\sigma_{i_p}\dots\sigma_{n}),
\end{equation}
where each parenthesis represents a cycle.
For example, with $\sigma = 62781453$,
\begin{equation}
\phi(\sigma) = (6)(278)(1453) = 47153682.
\end{equation}
This is Foata's first fundamental transformation (up to reversing the order on
the integers), hence a bijection.
Clearly, $\phi(\sigma)$ has fixed points whenever $\sigma\in Y_n$, so
$\phi$ induces a bijection between $X_n$ and derangements of ${\mathfrak S}_n$.
From Lemma~\ref{FY}, we see that the elements
\begin{equation}
({\bf F}_\sigma^\sharp)_{\sigma\in X_n}
\end{equation}
span ${\bf FQSym}_n^\sharp$. Since $|X_n|= d_n = {\rm dim\,} {\bf FQSym}_n^\sharp$, we have
finally
\begin{theorem}
The ${\bf F}_\sigma^\sharp$ for $\sigma\in X_n$ form a basis of ${\bf FQSym}^\sharp$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\end{theorem}
The sets $X_n$ have an interesting structure.
\begin{theorem}
The set $X_n$ is an ideal of the left weak order on ${\mathfrak S}_n$.
Its maximal elements are the left-shifted concatenations
\begin{equation}
w_I := w_{i_1} \blacktriangleright \dots \blacktriangleright w_{i_r},
\end{equation}
where $w_i:=1\,i\,i\!-\!1,\dots,2$, composition $I$ has no part $1$, and
$\alpha\blacktriangleright\beta = \alpha[\ell]\cdot \beta$ if $\beta\in{\mathfrak S}_\ell$.
\end{theorem}
\noindent \it Proof -- \rm
To show that $X_n$ is an ideal, we will prove that if $s_i$ denotes the
elementary transposition $(i,i+1)$, then $\sigma\in Y_n$ and
${\rm inv\,}(s_i\sigma)={\rm inv\,}(\sigma)+1$, implies $s_i\sigma\in Y_n$.
If $\sigma_k=r > s=\sigma_{k+1}$ are consecutive LR-minima of $\sigma$,
they will remain so for $s_i\sigma$, unless $i=r-1$, $r$, $s-1$, or $s$.
Since $s_i\sigma$ has one inversion more than $\sigma$, we can exclude the
case $i=r-1$: $r$ being a LR-minimum, $r-1$ cannot be to the left of $r$ in
$\sigma$. We can also exclude $i=s-1$ for the same reason.
If $i=r$, then $r$ is exchanged with $r+1$, which has to be to its right in
$\sigma$, so that again $\sigma_k$ and $\sigma_{k+1}$ are consecutive
LR-minima in $s_i\sigma$.
The same reasoning applies with $i=s$.
Hence $Y_n$ is a coideal, and consequently $X_n$ is an ideal.
Now, the elements $w_I$ are clearly in $X_n$ when $I$ has no part $1$, and any
exchange of consecutive values creating an inversion in such a $w_I$ would
create a pair of consecutive LR-minima. So these $w_I$ are maximal elements of
the ideal $X_n$.
Conversely, consider $\sigma\in X_n$ maximal.
Then consider the suffix $s$ of $\sigma$ beginning with $1$. The
maximality condition of $\sigma$ implies that if $t$ belongs to that suffix,
then $t-1$ also belongs to it. So this prefix is a permutation of an
${\mathfrak S}_{|s|}$, then should be $1|s|\dots 2$. The same now works by induction on
the permutation $\tau$ defined by $\sigma = \tau\blacktriangleright s$.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\medskip
For example, with $n=5$, we get the following three maximal elements of $X_n$:
\begin{equation}
15432,\ 35412,\ 45132.
\end{equation}
\medskip
The same proof can be adapted to the case of permutations that are images by
$\phi$ of permutations with at most $k$ fixed points.
Let $X_n^{(k)}$ be the image by $\phi$ of permutations with at most $k$ fixed
points. Then $X_n^{(k)}$ is the set of permutations with at most $k$
consecutive LR-minima.
\begin{theorem}
The set $X_n^{(k)}$ is a ideal of the left weak order on ${\mathfrak S}_n$.
Its maximal elements are the $w_I$ where $I$ runs over compositions with
\begin{itemize}
\item either $k-1$ ones and the remaining parts equal to $2$,
\item or exactly $k$ ones.
\end{itemize}
\end{theorem}
\noindent \it Proof -- \rm
The fact that $X_n^{(k)}$ is an ideal comes from the same idea as before: all
permutations greater than a given permutation $\sigma$ for the left weak order
have LR-minima at the same position.
By the same argument as in the previous theorem, the maximal elements must be
some $w_I$, where $I$ has at most $k$ ones. Now, it is clear that
\begin{equation}
w_I < w_J,
\end{equation}
in the left weak order iff $I$ can be obtained from $J$ by gluing parts equal
to $1$ with their next part. So the compositions described in the statement
are definitely maximal elements. And since all compositions with at most $k$
ones can be obtained from these ones by the gluing process, this ends the
proof.
\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}
\bigskip
Here is a table of the number of maximal elements of $X_n^{(k)}$
\begin{equation}
\begin{array}{|c||c|c|c|c|c|c|c|c|c|}
\hline
n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\hline
1 & 0 & 1 & & & & & & & \\
\hline
2 & 1 & 1 & 1 & & & & & & \\
\hline
3 & 1 & 2 & 2 & 1 & & & & & \\
\hline
4 & 2 & 3 & 3 & 3 & 1 & & & & \\
\hline
5 & 3 & 5 & 6 & 4 & 4 & 1 & & & \\
\hline
6 & 5 & 9 & 9 &10 & 5 & 5 & 1 & & \\
\hline
7 & 8 &15 &19 &14 &15 & 6 & 6 & 1 & \\
\hline
8 &13 &27 &31 &34 &20 &21 & 7 & 7 & 1 \\
\hline
\end{array}
\end{equation}
Note that the first column is obviously given by Fibonacci numbers since these
indeed count the number of compositions of $n$ in parts at least $2$.
The other columns are not known to~\cite{Slo} and neither is the sequence of
row sums.
But there exists a simple formula giving the number of maximal elements,
coming directly from their characterization:
\begin{equation}
m_{n,k} := \binom{(n+k-1)/2}{k-1} +
\sum_{\ell=0}^{\lfloor[\frac{n-k}{2}\rfloor]}
\binom{\ell+k}{k} \binom{n-k-\ell-1}{\ell-1},
\end{equation}
with the convention that a binomial coefficient with entries not in the
natural numbers is zero.
\subsection{Other bases of ${\bf FQSym}^\sharp$}
\begin{conjecture}
Let $<'$ be the order on permutations defined by
\begin{equation}
\sigma <' \tau \Longleftrightarrow \phi(\sigma) <_{\rm lex} \phi(\tau).
\end{equation}
Then the matrix of ${{\bf S}^\sigma}^\sharp$ of the ${\bf S}$ basis is triangular.
Moreover, the diagonal values are $1$ for the elements of $X_n$ and $0$ for
$Y_n$.
\end{conjecture}
\bigskip
For example, here are the matrices for $n=2$, $3$, and $4$ (Figure 1) where the zero
entries have been represented by dots to enhance readability.
The permutations are ordered as follows:
\begin{equation}
[21,\ 12],\qquad\qquad [321,\ 312,\ 231,\ 123,\ 132,\ 213].
\end{equation}
\begin{equation}
\begin{split}
[& 4321,\ 4312,\ 4231,\ 4123,\ 4132,\ 4213,\ 3421,\ 3412,\\
& 2341,\ 1234,\ 1243,\ 2314,\ 2431,\ 1423,\ 3241,\ 2134,\\
& 3142,\ 1324,\ 1432,\ 2413,\ 2143,\ 3214,\ 1342,\ 3124].
\end{split}
\end{equation}
\begin{equation}
\left(\begin{array}{cc}
. & -1/2 \\
. & 1 \\
\end{array}
\right)
\qquad
\left(\begin{array}{cccccc}
. & . & . & 1/3 & -1/3 & 2/3 \\
. & . & . & -1 & . & -1 \\
. & . & . & . & . & . \\
. & . & . & 1 & . & 1 \\
. & . & . & . & 1 & -1 \\
. & . & . & . & . & . \\
\end{array}
\right)
\end{equation}
\begin{figure}[ht]
{\tiny
\rotateleft{
$
\left(\begin{array}{cccccccccccccccccccccccccccccc}
.& .& .& .& .& .& .& 1/4& .& -1/8& 1/4& -3/8& .& 1/8& .& -1/4& 3/8&
-1/4& -1/4& 3/8& 1/2& -3/4& 1/8& -3/8 \\
.& .& .& .& .& .& .& -1/2& .& 1/2& .& 1/2& .& -1/2& .& 1& .&
1/2& .& -1/2& .& 1& .& 1 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& -1/2& . \\
.& .& .& .& .& .& .& .& .& -1& .& -1& .& .& .& -1& .&
-1/2& .& .& .& -1& .& -1 \\
.& .& .& .& .& .& .& .& .& .& -1& 1& .& .& .& .& -1&
1/2& .& .& -1& 1& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
-1/2& .& .& .& .& .& . \\
.& .& .& .& .& .& .& -1/2& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& 1& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& 1/2&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& 1& .& 1& .& .& .& 1& .&
.& .& .& .& 1& .& 1/2 \\
.& .& .& .& .& .& .& .& .& .& 1& -1& .& .& .& .& .&
.& .& .& 1& -1& .& -1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& 1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& -1/2&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& 1& .& -1& .&
.& .& 1& .& -1& .& -1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& -1/2&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& -1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& 1&
.& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
1& .& .& .& .& .& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& 1& -1& -1& 1& .& 1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& -1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& 1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& -1/2 \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& 1& . \\
.& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .& .&
.& .& .& .& .& .& 1 \\
\end{array}
\right)
$
}
}
\caption{The matrix of $\sharp$ in the ${\bf S}$ basis of ${\bf FQSym}$}
\end{figure}
\section{Other combinatorial Hopf algebras}
\subsection{The algebras ${\bf PQSym}$ and ${\bf CQSym}$}
There is an internal product on ${\bf PQSym}$ extending that of
${\bf WQSym}^*$~\cite{NTpark}. The $\sharp$-transform is defined in ${\bf PQSym}$
(it contains ${\bf Sym}$ as a subalgebra), but
${\bf PQSym}*{\bf WQSym}^* \subseteq {\bf WQSym}^*$, so that ${\bf PQSym}^\sharp = {\mathcal W}^\sharp$,
and we get nothing new.
\medskip
Similarly, the Catalan algebra ${\bf CQSym}$~\cite{NTp2}, we have
\begin{equation}
{\bf CQSym} * {\bf Sym} \subseteq {\bf Sym},
\end{equation}
so that
\begin{equation}
{\bf CQSym}^\sharp = {\bf Sym}^\sharp.
\end{equation}
\subsection{The algebra of planar binary trees ${\bf PBT}$}
The Loday-Ronco algebra of planar binary tree is not stable by the
$\sharp$-transform. Since ${\bf PBT}$ is the subalgebra of ${\bf FQSym}$ generated by
the $S^{\sigma}$ where $\sigma$ avoids the pattern $132$ (see~\cite{HNT}), we
have, for example :
\begin{equation}
{S^{213}}^\sharp = S^{123} - S^{132} - S^{312} + \frac{2}{3} S^{321}
\not\in {\bf PBT}.
\end{equation}
However, ${\bf PBT}^\sharp$ is a well-defined Hopf subalgebra of ${\bf FQSym}$.
\begin{conjecture}
The algebra ${\bf PBT}^\sharp$ is free over the set ${\bf P}_T^\sharp$, where $T$ runs
over trees with at least two nodes, and such that the right subtree of the root
is empty.
\end{conjecture}
In particular, the conjecture implies that the dimension of the homogeneous
components ${\bf PBT}_n^\sharp$ are given by the Fine numbers~\cite{DS,Slo},
sequence A000957:
\begin{equation}
1,\,0\, 1\, 2\, 6\, 18\, 57\, 186,\, 622 \dots
\end{equation}
\newpage
\footnotesize
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 360 |
premake.tree = { }
local tree = premake.tree
--
-- Create a new tree.
--
-- @param n
-- The name of the tree, applied to the root node (optional).
--
function tree.new(n)
local t = {
name = n,
children = { }
}
return t
end
--
-- Add a new node to the tree, or returns the current node if it already exists.
--
-- @param tr
-- The tree to contain the new node.
-- @param p
-- The path of the new node.
-- @param onaddfunc
-- A function to call when a new node is added to the tree. Receives the
-- new node as an argument.
-- @returns
-- The new tree node.
--
function tree.add(tr, p, onaddfunc)
-- Special case "." refers to the current node
if p == "." or p == "/" then
return tr
end
-- Look for the immediate parent for this new node, creating it if necessary.
-- Recurses to create as much of the tree as necessary.
local parentnode = tree.add(tr, path.getdirectory(p), onaddfunc)
-- Create the child if necessary
local childname = path.getname(p)
local childnode = parentnode.children[childname]
if not childnode or childnode.path ~= p then
childnode = tree.insert(parentnode, tree.new(childname))
childnode.path = p
if onaddfunc then
onaddfunc(childnode)
end
end
return childnode
end
--
-- Insert one tree into another.
--
-- @param parent
-- The parent tree, to contain the child.
-- @param child
-- The child tree, to be inserted.
--
function tree.insert(parent, child)
table.insert(parent.children, child)
if child.name then
parent.children[child.name] = child
end
child.parent = parent
return child
end
--
-- Gets the node's relative path from it's parent. If the parent does not have
-- a path set (it is the root or other container node) returns the full node path.
--
-- @param node
-- The node to query.
--
function tree.getlocalpath(node)
if node.parent.path then
return node.name
elseif node.cfg then
return node.cfg.name
else
return node.path
end
end
--
-- Remove a node from a tree.
--
-- @param node
-- The node to remove.
--
function tree.remove(node)
local children = node.parent.children
for i = 1, #children do
if children[i] == node then
table.remove(children, i)
end
end
node.children = {}
end
--
-- Sort the nodes of a tree in-place.
--
-- @param tr
-- The tree to sort.
--
function tree.sort(tr)
tree.traverse(tr, {
onnode = function(node)
table.sort(node.children, function(a,b)
return a.name < b.name
end)
end
}, true)
end
--
-- Traverse a tree.
--
-- @param t
-- The tree to traverse.
-- @param fn
-- A collection of callback functions, which may contain any or all of the
-- following entries. Entries are called in this order.
--
-- onnode - called on each node encountered
-- onbranchenter - called on branches, before processing children
-- onbranch - called only on branch nodes
-- onleaf - called only on leaf nodes
-- onbranchexit - called on branches, after processing children
--
-- Callbacks receive two arguments: the node being processed, and the
-- current traversal depth.
--
-- @param includeroot
-- True to include the root node in the traversal, otherwise it will be skipped.
-- @param initialdepth
-- An optional starting value for the traversal depth; defaults to zero.
--
function tree.traverse(t, fn, includeroot, initialdepth)
-- forward declare my handlers, which call each other
local donode, dochildren
-- process an individual node
donode = function(node, fn, depth)
if node.isremoved then
return
end
if fn.onnode then
fn.onnode(node, depth)
end
if #node.children > 0 then
if fn.onbranchenter then
fn.onbranchenter(node, depth)
end
if fn.onbranch then
fn.onbranch(node, depth)
end
dochildren(node, fn, depth + 1)
if fn.onbranchexit then
fn.onbranchexit(node, depth)
end
else
if fn.onleaf then
fn.onleaf(node, depth)
end
end
end
-- this goofy iterator allows nodes to be removed during the traversal
dochildren = function(parent, fn, depth)
local i = 1
while i <= #parent.children do
local node = parent.children[i]
donode(node, fn, depth)
if node == parent.children[i] then
i = i + 1
end
end
end
-- set a default initial traversal depth, if one wasn't set
if not initialdepth then
initialdepth = 0
end
if includeroot then
donode(t, fn, initialdepth)
else
dochildren(t, fn, initialdepth)
end
end
--
-- Starting at the top of the tree, remove nodes that contain only a single
-- item until I hit a node that has multiple items. This is used to remove
-- superfluous folders from the top of the source tree.
--
function tree.trimroot(tr)
local trimmed
-- start by removing single-children folders from the top of the tree
while #tr.children == 1 do
local node = tr.children[1]
-- if this node has no children (it is the last node in the tree) I'm done
if #node.children == 0 then
break
end
-- remove this node from the tree, and move its children up a level
trimmed = true
local numChildren = #node.children
for i = 1, numChildren do
local child = node.children[i]
child.parent = node.parent
tr.children[i] = child
end
end
-- found the top, now remove any single-children ".." folders from here
local dotdot
local count = #tr.children
repeat
dotdot = false
for i = 1, count do
local node = tr.children[i]
if node.name == ".." and #node.children == 1 then
local child = node.children[1]
child.parent = node.parent
tr.children[i] = child
trimmed = true
dotdot = true
end
end
until not dotdot
-- if nodes were removed, adjust the paths on all remaining nodes
if trimmed then
tree.traverse(tr, {
onnode = function(node)
if node.parent.path then
node.path = path.join(node.parent.path, node.name)
else
node.path = node.name
end
end
}, false)
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,517 |
Es consideren països catòlics o de tradició catòlica els països que històricament i tradicional han tingut una quantitat predominant de seguidors de l'Església Catòlica Romana en el conjunt de la població. Itàlia és potser el més conegut, particularment des que es troba el Vaticà a la ciutat de Roma. D'altres països coneguts per la predominant població catòlica a Europa són França, Espanya, Portugal, Àustria, Eslovàquia, Lituània, Polònia, Irlanda, Croàcia, Eslovènia, Suïssa i Malta. A altres parts del món també hi ha altres països de majoria catòlica, com les Filipines a Àsia, o els països de l'Amèrica del Sud. A Àfrica aquesta religió també ha començat a formar part de la nova identitat d'alguns països com Angola, Nigèria, Uganda i Guinea Equatorial.
La proporció referent a la població catòlica de cada país del món s'ha agafat de l'Informe Internacional de Llibertat Religiosa del 2004 realitzat pel Departament d'Estat dels Estats Units. D'altres fonts emprades inclouen el catholic-hierarchy.org i el The World Factbook. La població total de cada país va prendre's de l'Oficina del Cens dels Estats Units (estimacions del 2005). S'ha de prendre en consideració que alguns dels percentatges expressats en aquest article només fan referència a la població adulta del país, i són emprades en la població total del país. També s'ha de tenir en compte que les xifres indiquen el nombre de persones batejades, per això pot incloure tant catòlics no practicants com no seguidors. Així com els seguidors no registrats no són comptats.
Percentatges per països
Pitjant les fletxes que hi ha al costat dels noms de les capçaleres es poden ordenar les columnes,
Percentatges per regions
A Àfrica
A Àsia
A Europa
A Amèrica
A Oceania
Referències | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,884 |
Comment to The Line, Dec 3, 2021
Well, at least I know not to trust that "gut-level, feels right" intuition of Matt Gurney.
I can tell he's too young to remember that post-Vietnam era, with drastic limitations on everything because the price of oil had just quintupled, bringing recession. That's when there was a SERIOUS feeling of "we've hit our peak and this brief historical Good Times era is gone". I was told that, despite the embargo and new high prices and new tiny cars (everybody was going to have to buy tiny cars from then on, SUVs were not a dream) the oil would run out by 2000. Not told that by a sign-waving ecologist, but a UofC Engineering professor. Which made it funny that it was in the actual year 2000, we all had to listen to the "Peak Oil" theory and told the future was, again, a Mad Max apocalypse. (Fracking became cost-positive in 1999 and took off; no more "Peak Oil".)
I admit to frustration, talking to people whose parents didn't give them the Depresssion Training I got; and my brother's "Polio Summer of 1956" memories allowed me to mock those (young) journalists who said that this event was "not in living memory": http://brander.ca/c19/#polio
Sort of a side note, but it's an eye-roll when a conservative who's absorbed too much American militarism describes Canada has having passed from one "umbrella of protection", to another. The only country that's ever thought of attacking us was America, 1812. Our military only serves other's imperial adventures, from my Grandfather helping imperial Britain pacify invaded Boer farmers in 1901-1902, to helping America spend 1 year trying to arrest Osama bin Laden, and 19 trying to dominate Afghanistan for its own sake. Russia developed nukes in defensive terror that they'd be invaded again by the West, right after the Nazis tried; they never had a plan to nuke us into submission. "Protection", indeed.
But the main point is that conservatives have, all my life, been telling me that our civilization is decadent and weak and unable to rise to a challenge. That started in the same early 1970s, with the pampered, rich (by comparison to Depression-raised parents) spoiled Boomers, who had never known war. (That's why they had lost Vietnam, raised on candy and TV and comic books, not enough spanking because of Dr. Spock.)
It is frustrating how deliberately-oblivious people can be. Half of journalism seems to be about sequel stories to messes that were buried so we could *remain* oblivious after having the problem shaken in our face: the RCMP. The military. Indigenous water. Care homes. The pointless impossibility of progress in Afghanistan. Income inequality. And, above all, ecological stress and change.
But we're also totally amazing at what we come up with when we DO get going. Not just fracking, but mRNA! It's global warming, where my optimism is highest. There will be this bad century to come, a lot more bad weather, but we'll get through it. The world added a shocking 290 GW of renewables generation last year, 50% more than 2019, which was itself a record. Despite the demand, EV batteries dropped another 6% in price, with lithium sources popping up all over because now people are looking hard for it. (So much for peak lithium.)
There was, once upon a time, this generation that had not seen a large war in three generations, one of the longest peaces in European history. It was also a century of dramatic technological improvement, vast new wealth pouring into Britain from a colonized world. Then they had to fight WW1. AND that giant pandemic. They responded magnificently.
Have a little faith, Matt. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,335 |
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Home General Hierarchy of Courts and Justice System in Sri Lanka: Complete Overview
Hierarchy of Courts and Justice System in Sri Lanka: Complete Overview
Nikieta Aggarwal
This article is written by Ajay Jose, pursuing M.A. Business Laws from NUJS, Kolkata. The article describes the Hierarchy of Courts & Justice System in Sri Lanka.
Law is a set of rules to regulate behaviour of people in a given society at a given time. According to Hobbes[1]- law is an obligatory rule of conduct. The commands of him or them that have coercive power.
Salmond [2] states that law is the body of principals recognised and applied by the state in the administration of justice. There could be legal rules and moral rules. Legal rules are called laws and non-legal rules are called moral rules/customs/ethics. Legal rules cannot be breached and it is punishable. E.g. Cheating, robbery, rape, murder etc. Non legal rules are not punishable. E.g. Drinking alcohol on pooja days, not giving room in a bus to an elderly, unmarried couple living together.
Who has Created Laws?
It has evolved in the society by Custom, Convention, Religion, Beliefs etc. It is made by a supreme leader (a King) or a body of people (a parliament) in a society, almost all the areas in human activity are controlled or influenced by law.
Law has become a regulatory device in controlling the conflicting social interest.
Law must be able to cater to the needs of the society and influence behavior of people. Businesses are also part of the society and their behavior is also governed or influenced by law. There are so many laws that govern businesses in any society.
Laws of Sri Lanka
Sri Lanka [3] has been home for many communities for a long period of time. Sinhala, Tamil, Muslim are the dominant Ethnic groups. Buddhism, Hinduism, Islam and Christianity are the dominant Religions.
Further some geographical areas are dominated by certain communities whilst other areas have mixed communities.
The local laws were influenced by this multi-ethnic and multi-religious characteristics. Therefore, the local laws – Kandyan, Thesawalamai, Islamic laws have evolved based on such characteristics.
Our motherland has been invaded and conquered by Portuguese, Dutch and British[4]. Their laws and legal systems also have been introduced to our country in a substantial manner.
Therefore, the laws and legal systems can be broadly divided into two periods –
Laws before the foreign invasion.
Laws after the foreign invasion.
Sinhala law (now remain as Kandyan law)
There had been a comprehensive legal system in Sri Lanka. If not, the country would not have developed and governed in the past.
It would have been influenced by the laws and customs of indigenous people by the time the early settlers who migrated to Sri Lanka from Indian and the customs and laws that they brought into Sri Lanka by then. Now it is applicable as a personal law for people who get designated as Kandyans for the purposes of personal laws such as marriage, property rights etc.
Thesawalamai Law
It was in operation in the North Kingdom by the time Portuguese came to Sri Lanka. The sources of this law trace to customs and laws used by Dravidians in Malabar Coast and Coromandel Coast. It is still applicable in areas like Jaffna.
The meaning of Thesawalamai is something like "Law of the Land". So, it is applicable for people who live in a specific area.
Muslim Law
Although Muslim did not have a separate Kingdom, the settlers of Muslims had their own laws applicable to them but there is no hard evidence to show that Sinhalese Tribunals administered Muslim Law separately in their legal system. [5] Portuguese and Dutch recognized it as a separate law. Still, it is applicable as a personal law for Muslims especially in marriages, divorces etc.
The Influence of Western Laws and Legal Systems to Sri Lanka
Maritime Provinces were acquired by Portuguese in 1505. By that time, a well established legal system was in operation. [6] They agreed to uphold the system. But they could not comprehend the system and the substance of the local laws. When they administered, with or without their knowledge, their own systems, thinking creped into the system.
Dutch Influence 1656 to 1796
The laws that they were having by that time in their own country was the Dutch law influenced by the Roman laws. Therefore, it was called Roman-Dutch Law.
They applied their laws to Dutch settlers, their servants, locals who lived in their Forts and those who practiced Christianity.
The Roman Dutch was a comprehensive law and when they thought that local laws were unsuitable or silent, they used their R&D for locals too. Therefore it became a Residual law.
Both the Roman-Dutch and Portuguese Laws were not applied to Kandyans as such areas were never controlled by them.They establish an elaborate court system in Sri Lanka[7].
The British Influence 1796 to 1948
During 1796 Maritime Provinces surrendered to the British and formally ceded in 1802 and the Kandyan kingdom was also captured by the British in 1815. However, British continued to apply local laws and R &D law subject to changes done if any by introducing statutes of their own. The introduction of English Law and the judicial system greatly changed the laws and legal system of Sri Lanka.
Currently, the general law applicable to Sri Lanka is English Laws, R & D law and the statute law created by our own parliaments, much of them based on those ideas.
Property laws are still influenced by R &D and Commercial and Criminal Laws and procedures are mainly influenced by English Laws.
In contract Law, English law of Delicts alone with R &D are applicable. [8]
Kandyan law is applicable only to Kandyan Sinhalese, Thesawalamai is applicable to Tamils living in Jaffna and Muslim law for all Muslims living in Sri Lanka.
These special laws are restricted to their personal laws such as marriage, divorce, inheritance and land matters.
Classification of Laws
Punishment by imprisonment, penalty or death. Sate will prosecute, the alleged person named as accused. Courts and Procedures are different. It should prove beyond reasonable doubt.
E.g. Robbery, hurt, rape, death, fraud etc.
The offense should be stated in a written law such as the Penal Code, Exchange control Act, Customs Ordinance etc as an offense punishable.
Criminal Courts such as Magistrates Court and High Court will hear cases and Procedure is enumerated in Acts such as Criminal Procedure Code.
Damages or other relief, person sustained damage or breach has to institute legal proceedings, called plaintiff and the other party is called the defendant.
They are designed to protect private rights and state will not prosecute. Proof, balance of probabilities.
Civil courts will hear the cases and procedure is indicated in the civil procedure Code.
Roman Dutch Law
Court System of Sri Lanka
The court structure consists of –
A Supreme Court
A Court of Appeal
Primary Courts [9]
Additionally, there are numerous tribunals etc. In cases involving criminal law, a Magistrate's Court or a High Court is the only court with primary jurisdiction, the respective legal domains of each are provided in the Code of Criminal Procedure.
The preponderant majority of criminal law cases are initiated at a Magistrate's Court. These cases may be initiated by any police officer, or public servant, with a written or oral complaint to the magistrate (see section on Magistrate's Court).
Murder trials and various offenses against the State originate in a High Court (see section on High Courts).
Original jurisdiction over most civil matters lies with the relevant District Court (see section on District Courts).
Until 1972, The Judicial Committee of the Privy Council in Britain was the final court of appeal for Sri Lanka. The right of appeal to the Privy Council "was abolished…as there were concerns that any attempt to discard the existing Constitution in 1972 might be adjudged unconstitutional."
At that time, "Parliamentarians constituted themselves as members of what was termed the 'Constituent Assembly' to draft and adopt a new Constitution," which became effective on May 22, 1972. It has been suggested that the concerns of minority communities in Sri Lanka were not adequately considered in the drafting of that Constitution.
On Aug 31, 1978, another Constitution replaced the 1972 Constitution. Under this Constitution, for the first time in Sri Lanka an Executive President, elected by the entire country, became the leader of the country. Under the earlier Constitution, the government was headed by a Prime Minister, who, as a Member of Parliament, would have been elected by just one electorate in the country.
As a consequence of major civil strife that erupted in 1983, now efforts are being made to replace the 1978 Constitution with a new Constitution in order to grant greater political autonomy to the different regions in the country. These efforts, at present, seem stalled in an All Party Conference constituted by the current President, MahindaRajapakse. Furthermore, not all political parties in the country agreed to participate in the Conference.
There are also other courts such as the Kathi Courts that handle matrimonial disputes among Muslims, and numerous tribunals (see section on Other Courts).
The constitution of Si Lanka provides specific provisions as to the powers, duties, jurisdiction and the procedure of the Supreme Court and of the Court of Appeal.
The enshrining of the provisions of SC and Court of Appeal (CA) in the Constitution of SL as regards the highest courts in Sri Lanka is to guarantee the independence of the judiciary and also to safeguard the tenure of judges of the highest courts. [10]
Supreme Court is the Highest Court in Sri Lanka and it has the final appellate jurisdiction, matters relating constitutional affairs, Fundamental rights, Consultative jurisdiction, Election Petitions, hear any breach of parliamentary privileges, any matter that parliament may refer (Refer Article 118 to 136 of Sri Lanka Constitution) [11]
Appeal Court
The Next highest Court created by the constitution. The Appeal court has the jurisdiction for correction of any error in fact or law committed by any court of the first instance, tribunal or other judicial institution.
It has the power to affirm, reverse, correct or modify any order, judgment or decree or sentence given by above institutions. [12]
It also has the power to grant injunctions and writs etc. (Refer article 137 to 147 of Sri Lankan Constitution [13])
Courts of First Instances
The High Court
High Court has two judicial powers now. High Courts exercising criminal jurisdiction as well as Commercial jurisdictions. High Court has the power to conduct-
High Court has the power to conduct trials on original criminal jurisdiction.
High Court can conduct a trial by jury or trial at bar.
High court can impose any sentence or penalty prescribed by law.
Under the 13th amendment to the Sri Lankan Constitution, it has the appellate and revisionary jurisdiction by way of Provincial High Court.
Provincial High Court has been vested with appellate and revisionary jurisdiction in respect of orders and judgments of the magistrates Court, Primary Courts, labor Tribunals, Agrarian Services Commissioners Tribunals within the province.
Commercial High Court was established under the High Court of the Provinces act of 1996. It has jurisdiction to hear civil actions where the cause of action has arisen out of commercial transactions in which the debt, damage or demand exceeds One Million.
District Courts are empowered with unlimited original civil jurisdiction. This court deals in money, revenue, trust, insolvency, commercial matters, family matters such as divorce, adoption, custody of children etc. If aggrieved y a decision of DC, an appeal could be made to the Court of Appeal.
Magistrates Court (MC) – Criminal jurisdiction
This court exercises basic criminal summary jurisdiction and inquiries into the commission of offenses. Summary jurisdiction is applied where the Magistrate could read charges against an accused and asked whether he or she is guilty or not guilty.
If pleads guilty, the verdict of guilt will be recorded and punishment will be imposed.
If doesn't plead guilty, a trial will be fixed and witnesses will be called and cross-examined and a decision will be made.
Magistrate Courts have the power to imprison or impose penalties as prescribed by law. When alleged offense seems not be a one triable summarily by MC, the Magistrate will make a Preliminary Inquiry to find out whether there is enough evidence to forward it to High Court.
The charge will be read but accused will not be asked to plead guilty or not. Witnesses are produced and cross-examined to find out the sufficiency of the evidence to proceed to High Court. Aggrieved by the decisions of an MC, an appeal could be made to Provincial High Court of the relevant province.
Primary Court (Civil jurisdiction)
The judicature Act provides for primary court adjudication where the debt, damage, demand or claim does not exceed Rs.1500. Further Primary Court has very important jurisdictions such as – actions for the enforcement of local authority by-laws and recovery of revenue due to such authority.
Land disputes which are likely to cause the breach of the peace.
It could decide who is in the possession of the land but it does not decide on the ownership of the land which is a matter for District Court to decide.
Alternate Dispute Settlements
There are many steps taken by the judicial System to sort out problems out of the mainstream court system. The intention was to make things easy and settle disputes early, reduce the workload of courts etc.
Mediation Boards
The mediation Boards Act of 1995 and subsequent amendments govern these mediation aspects. The Minister has the power to set up Mediation Boards and set the areas that will come under such mediation boards. Commercial mediations are reality now in Sri Lanka.
Arbitration is also another form of dispute resolution. Arbitration Act of 1995 and subsequent amendments govern the procedure. Tribunals- Labour Tribunal established under industrial Disputes Act provides provisions for the employees to institute the action against employers.
Agricultural Tribunals established under the Agrarian Services Act of 1979 enables settling disputes as to cultivation and related matters[14].
Sri Lankan legal system was originally a combination of Roman-Dutch and English Law. The present system of judicial administration and organization is based on the current Constitution introduced in 1978 apart from the personal laws in existence.
No changes to the judicial system were made under the recent changes to the Constitution with the changes of the government towards Good Governance popularly known as "Yahapapalaya".
The Judiciary comprises of the Supreme Court, Court of Appeal, Provincial High Courts, District Courts, Magistrates Courts and Primary Courts. [15]
The Supreme Court is the highest court headed by the Chief Justice of and other judges. Infringement of any fundamental rights declared and recognized by Chapter 3 or 4 will have a remedy where Article 126 of the Constitution has exclusive jurisdiction to hear and determine any questions relating to infringement or imminent infringement by an Executive or administrative action.
That's all about Sri Lankan Justice System. What are your views about the justice system of Sri Lanka? Is it Justice Driven? Comment below & Don't forget to Share.
The Constitution: Sri Lanka. Chapters XV (Judiciary) and XVI (Superior Courts), in the Official Website of the Government of Sri Lanka.
Cooray, Anton. "Oriental and Occidental Laws in Harmonious Co-existence: The Case of Trusts in Sri Lanka." Electronic Journal of Comparative Law. v.12.1 (May 2008)
Courts of Law. The Sri Lankan "Ministry of Justice and Law Reforms,"
Goonesekera, Savitri. "The Roman Dutch Law in the Plural Legal System of Sri Lanka." The Colombo Law Review. Faculty of Law, University of Colombo, Sri Lanka. v.9, (1998) pp.1-36.
Jayasuriya, Dayanath C. "Sri Lanka," in Legal Systems of the World: A Political, Social, and Cultural Encyclopedia. Vol. IV, S-Z. ed. Herbert M. Kritzer. ABC.CLIO: Santa Barbara, CA (2002). pp. 1526-1531.
"Muslim Personal Law and Women," by Cat's Eye, in the Island newspaper, Sri Lanka, July 9, 2003. Access this URL and select the "Midweek Review" link on the left panel (Accessed Feb. 2007).
Rajepakse, Ruana. An Introduction to Law in Sri Lanka. Aitken Spence Printing, Pte, Ltd., 315 Vauxhall Street, Colombo 2, Sri Lanka 1989.
Reynolds, Thomas H., and Arturo A. Flores. Eds."Sri Lanka." (June 2004 release), in Foreign Law: Current Sources of Codes and Legislation in Jurisdictions of the World. William S. Hein &Co.Inc. Buffalo, New York. v.3 (2003) pp. Sri Lanka I- Sri Lanka 32.
[1] The 17th Century English philosopher Thomas Hobbes is now widely regarded as one of a handful of truly great political philosophers, whose masterwork Leviathan rivals in significance the political writings of Plato, Aristotle, Locke, Rousseau, Kant, and Rawls. Hobbes is famous for his early and elaborate development of what has come to be known as "social contract theory", the method of justifying political principles or arrangements by appeal to the agreement that would be made among suitably situated rational, free, and equal persons.
[2] Sir John William Salmond KC (3 December 1862 – 19 September 1924) was a legal scholar, public servant and judge in New Zealand.
[3] Sri Lanka, officially the Democratic Socialist Republic of Sri Lanka (formerly known as Ceylon), is an island country in South Asia near south-east India. Sri Lanka has maritime borders with India to the northwest and the Maldives to the southwest
[4]Goonesekera, Savitri. "The Roman Dutch Law in the Plural Legal System of Sri Lanka." The Colombo Law Review. Faculty of Law, University of Colombo, Sri Lanka. v.9, (1998) pp.1-36
[5] Muslim Personal Law and Women," by Cat's Eye, in the Island newspaper, Sri Lanka, July 9, 2003. Access this URL and select the "Midweek Review" link on the left panel (Accessed Nov 7)
[6]Jayasuriya, Dayanath C. "Sri Lanka," in Legal Systems of the World: A Political, Social, and Cultural Encyclopedia. Vol. IV, S-Z. ed. Herbert M. Kritzer. ABC.CLIO: Santa Barbara, CA (2002). pp. 1526-1531
[7]Rajepakse, Ruana. An Introduction to Law in Sri Lanka. Aitken Spence Printing, Pte, Ltd., 315 Vauxhall Street, Colombo 2, Sri Lanka.1989
[8]Muttettuwegama, Ramani. "'But I am both.' Equality in the Context of Women Living under Parallel Legal Systems: The Problem in Sri Lanka." A Briefing Document. (Accessed Feb. 2007)
[9] Courts of Law. The Sri Lankan "Ministry of Justice and Law Reforms," (Accessed Nov 6)
[10] [The] Constitution: Sri Lanka. Chapters XV (Judiciary) and XVI (Superior Courts), in the Official Website of the Government of Sri Lanka.
(Accessed Nov 3)
[11] Art 118. General jurisdiction of Supreme Courts, Art 119. Constitution of Supreme Court, Art 120. Constitutional jurisdiction of the Supreme Court, Art 121. Ordinary exercise of constitutional jurisdiction in respect of Bills, Art 122. Special exercise of constitutional jurisdiction in respect of urgent Bills, Art 123. Determination of Supreme Court in respect of Bills, Art 124. Validity of Bills and legislative process not to be questioned, Art 125. Constitutional jurisdiction in the interpretation of the Constitution, 126. Fundamental rights jurisdiction and its exercise, Art 127. Appellate Jurisdiction, Art 128. Right of appeal, Art 129. Consultative jurisdiction, Art 130. Jurisdiction in election and referendum petitions, Art 131. Jurisdiction in respect of the breaches of Parliamentary privileges, Art 132. Sitting of the Supreme Court, Art 133. Appointment of ad hoc Judges, Art 134. Right to be heard by the Supreme Court, Art 135. Registry of the Supreme Court and office of Registrar, Art 136. Rules of the Supreme Court
[12]Cooray, Anton. "Oriental and Occidental Laws in Harmonious Co-existence: The Case of Trusts in Sri Lanka." Electronic Journal of Comparative Law. v.12.1 (May 2008)
pp. 1-17. (Accessed Nov 5)
[13] Art 137. The Court of Appeal, Art 138. Jurisdiction of the Court of Appeal, Art 139. Powers in appeal, Art 140. Power to issue writs, other than writs of habeas corpus, Art 141. Power to issue writs of habeas corpus, Art 142. Power to bring up and remove prisoners, Art 143. Power to grant injunctions, Art 144. Parliamentary election petitions, Art 145. Inspection of records, Art 146. Sittings of the Court of Appeal, Art 147. Registry of the Court of Appeal and office of Registrar
[14] Reynolds, Thomas H., and Arturo A. Flores. Eds."Sri Lanka." (June 2004 release), in Foreign Law: Current Sources of Codes and Legislation in Jurisdictions of the World. William S. Hein &Co.Inc. Buffalo, New York. v.3 (2003) pp. Sri Lanka I- Sri Lanka 32
[15]Tambiah. H.W. "Sri Lanka," in Encyclopedia of Comparative Law: National Reports. ed. Victor Knapp. MartinusNijhoff Publishers: Dordrecht, Boston, Lancaster (1987) pp. S-125/S-136
Alternate Dispute Settlement
Commercial High Court
Dutch Influence
Law in Sri Lanka
Megistrates Court
Primary Court
Provincial High Court
Sinhala law
Summary Jurisdiction
The British Influence
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Genderless goes mainstream in luxury fashion sector | {
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Crisis Text Line shares data with a for-profit spinoff to make customer service software
The nonprofit said the data is anonymized, but privacy experts say that's not enough.
Crisis Text Line shares anonymized data with Loris.ai, a customer service software company.
Photo by Adrian Swancar/Unsplash
Crisis Text Line, a nonprofit that uses text messaging to help people struggling with suicide and other mental health issues, has been sharing data collected during those text conversations with a for-profit spinoff. The spinoff, Loris.ai, uses that data to create customer service software and has a revenue-sharing agreement with Crisis Text Line, according to a new POLITICO investigation.
Crisis Text Line told POLITICO that all of the data it shares with Loris.ai is anonymized and that the organization describes data usage in its terms of service. "Crisis Text Line obtains informed consent from each of its texters," the organization's general counsel told POLITICO. "The organization's data sharing practices are clearly stated in the Terms of Service & Privacy Policy to which all texters consent in order to be paired with a volunteer crisis counselor."
But privacy experts say anonymized data can sometimes be reverse-engineered, and people rarely read terms of service — perhaps even less so when they're reaching out for help in the midst of a mental health crisis. "The fact that the data is transferred to a for-profit company makes this much more troubling and could give the FTC an angle for asserting jurisdiction," Jessica Rich, former director of the Federal Trade Commission's Bureau of Consumer Protection, told POLITICO.
A former volunteer for Crisis Text Line has been leading an advocacy campaign calling for changes to the organization's data practices, including through a Change.org petition.
If you or a loved one needs help:
Call the National Suicide Prevention Lifeline at 800-273-8255 to reach a counselor at a locally-operated crisis center 24 hours a day for free. Here is their privacy policy.
Issie Lapowsky ( @issielapowsky) is Protocol's chief correspondent, covering the intersection of technology, politics, and national affairs. She also oversees Protocol's fellowship program. Previously, she was a senior writer at Wired, where she covered the 2016 election and the Facebook beat in its aftermath. Prior to that, Issie worked as a staff writer for Inc. magazine, writing about small business and entrepreneurship. She has also worked as an on-air contributor for CBS News and taught a graduate-level course at New York University's Center for Publishing on how tech giants have affected publishing.
crisis text line privacy data | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,284 |
{"url":"http:\/\/mathhelpforum.com\/calculus\/31894-integrating-factors.html","text":"# Math Help - integrating factors\n\n1. ## integrating factors\n\nHi i was wondering if anyone could help me with integrating factors.\ny'+1\/x(y)=x is the question\nHence the intergrating factor is e^the integral of 1\/x. this gives e^lnx +A. which leaves the integratig factor as x. Hence x(y'+1\/x(y))=x^2\nthen xy'+y=x^2. However after this point i am not sure where to go. the example tells me to go to dy\/dx(xy)=x^2. But i am not sure why to do that. If anybody could help it would be appreciated and if anybody has any more tips on integrating factors that would be nice! thanks\n\n2. Originally Posted by studentsteve1202\nHi i was wondering if anyone could help me with integrating factors.\ny'+1\/x(y)=x is the question\nHence the intergrating factor is e^the integral of 1\/x. this gives e^lnx +A. which leaves the integratig factor as x. Hence x(y'+1\/x(y))=x^2\nthen xy'+y=x^2. However after this point i am not sure where to go. the example tells me to go to dy\/dx(xy)=x^2. But i am not sure why to do that. If anybody could help it would be appreciated and if anybody has any more tips on integrating factors that would be nice! thanks\nthe whole point of using integrating factors is to turn the left hand side into the result of a product rule derivative. you will notice that xy' + y is the result you get if you differentiate the product xy implicitly. this is the idea. you can therefore go backwards and contract the expression on the right to the original form before it was differentiated and just take the anti-derivative of both sides.\n\nsee post #21 here\n\n3. Hello, studentsteve1202!\n\n$y'+\\frac{1}{x}\\,y\\:=\\:x$\n\nHence, the intergrating factor is: $e^{\\int\\frac{dx}{x}}$\nThis gives: $e^{\\ln x} \\:=\\:x$\n\nHence: . $xy' +y\\:=\\:x^2$\n\nHowever, after this point i am not sure where to go.\n\nWhen you multiplied through by the integrating factor,\n. . the left side became: . $xy' + y$\nwhich is the derivative of a product . . . namely, the derivative of $xy$\n. .\n(Check it out for yourself!)\n\nThe left side is always the derivative of: . $\\text{(Integrating factor)} \\times y$\n\nAnd that's why we can write: . $\\frac{d}{dx}(xy) \\:=\\:x^2\\quad\\Rightarrow\\quad d(xy) \\:=\\:x^2\\,dx$\n\nNow integrate both sides: . $\\int d(xy)\\;=\\;\\int x^2\\,dx$\n\n. . . . . . . . . .and we get: . . . $xy \\;\\;\\;=\\;\\;\\;\\frac{1}{3}x^3 + C$\n\nTherefore: . $y \\;=\\;\\frac{1}{3}x^2 + Cx^{-1}$","date":"2016-07-24 14:18:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 11, \"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\": 0.9433774352073669, \"perplexity\": 504.75186294094937}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2016-30\/segments\/1469257824037.46\/warc\/CC-MAIN-20160723071024-00067-ip-10-185-27-174.ec2.internal.warc.gz\"}"} | null | null |
\chapter{The Black Hole Information Paradox}
\author{%
Martin B EINHORN\\
{\it Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93106-4030\\
meinhorn@kitp.ucsb.edu}}
\AuthorContents{M.~B.~Einhorn}
\AuthorIndex{Einhorn}{M.~B.}
\section*{Abstract}
After a brief reminscence about work with K. Sato 25 years ago, a discussion is given of the black hole information paradox. It is argued that, quite generally, it should be anticipated that the states behind a horizon should be correlated with states outside the horizon, and that this quantum mechanical entanglement is the key to understanding unitarity in this context. This should be equally true of cosmologies with horizons, such as de~Sitter space, or of eternal black holes, or of black holes formed by gravitational collapse.
\section{Reminiscence on a collaboration.}
Several people have asked me how Katsu and I came to collaborate\cite{Einhorn:1980ik}. As this conference is an opportunity to pay tribute to Katsu on his 60th birthday, perhaps I may be permitted some reflections on our truly serendipitous meeting. First, a little background: thinking back to 1979, magnetic monopoles were a hot topic because it had been realized that every grand unified theory (GUT,) such as SU(5), contained monopoles which would be essentially stable at scales below the unification scale. The question arose about what their density should be today as relics of their production during the early universe\cite{Preskill:1979zi}. For this, an estimate of their production during the early universe was required. Stein, Toussaint, and I\cite{Einhorn:1980ym} had come to the conclusion that, if the universe underwent a second-order phase transition as the temperature cooled below the GUT scale, there would very likely be far too many relic monopoles around today. Guth and Tye\cite{Guth:1979bh} had reached the same conclusion independently. (As an aside, the two groups had submitted the papers to PRL with a request that they be published back-to-back. Both were rejected! How they came to be published separately is yet another interesting tale. Some of this story is told in Guth's popular account\cite{Guth1}, but you may come away with the impression that, thereafter, I dropped off the edge of the earth. Actually, I went to Copenhagen to spend a year visiting NORDITA and, in those pre-internet days, Alan and I lost touch for a while.
Shortly after arriving in January, 1980, I gave a seminar summarizing the results of this work and concluded by stating that I wanted next to understand was what happened if the phase transition were strongly first-order. After the seminar concluded, Katsu came up and showed me a preprint that he had just written entitled ``First Order Phase Transition Of A Vacuum And Expansion Of The Universe." (Although it was submitted in February, 1980, it was only published the following September.) Katsu was half-way through spending a year in Copenhagen at the invitation of Chris Pethick, who had been impressed by Katsu's earlier work, discussed elsewhere in these proceedings.
You can imagine my surprise upon reading this; it contained much of what I had set out to learn about general relativity in this context. This was wonderful, because he came at this from an astrophysics/cosmology background, whereas I came from a particle physics background and was still learning cosmology. Sato's analysis had been made generically, motivated by a first-order phase transition in the Weinberg-Salam model. He wasn't so familiar as I with Grand Unified Theories and not at all with the monopole puzzle, but he didn't need much convincing to become persuaded that this was the more natural context for applying this cosmology.
This was easier said than done, and it wasn't clear how that was to be married to the scenario of a GUT broken as the universe cooled and, in particular, what effects this might have on the monopole density and how the universe might appear after the phase transition. Guth and Tye were also led in the same direction, and, I later learned, had also realized that an exponential expansion of the universe would occur that might dilute the monopole density significantly greater than in the second-order case.
By mid-March, when I traveled to Erice to lecture on these things at the Europhysics Conference on Unification of the Fundamental Interactions\cite{Einhorn:1980rt},
we had developed a fairly clear picture that the exponential expansion would dilute the monopole density sufficiently, but it was much more difficult to see whether the universe would be left sufficiently homogeneous. I recall when I gave my talk, Lenny Susskind, who had just come from Stanford, volunteered that I had just described ``Guth's Cosmology." No surprise, Alan was at SLAC and had been led down the same path. The pictures that we had independently arrived at came to be called ``old inflation," and we all concluded that it left the universe too grainy to work. To his enduring credit, Alan realized that the exponential expansion might solve some other cosmological issues, and, as they say, the rest is history. Nevertheless, the inflationary universe emerged from the monopole problem, not from those other issues.
Unfortunately, Katsu had to return to Japan at the end of June, and in those days, collaboration at a distance was much harder than now, so we weren't able to continue working together. Sato and I submitted our work in July. Others, more stubborn than we, persisted in developing alternate models of inflation. I was then as now (and like many others) terribly bothered by the fine-tuning of the cosmological constant inherent in all such models and felt that, to make progress, we needed to understand naturalness. Remarkably, it seems that Nature doesn't care!
It is a great pleasure to be here to celebrate Katsu's 60th birthday and the 25th anniversary of our collaboration. I have always regarded our meeting as a most fortuitous happenstance. It is my pleasure to wish you ``otanjoubi omedetou gozaimasu."
\section{Introduction to the Black Hole Information Paradox
I now turn to the black hole (BH) information paradox.
In 1975, Hawking showed that, because of quantum effects, it appears to a stationary observer that radiation is emitted from a region near the horizon of a black hole.\cite{Hawking:1974sw} With the approximations he used, it seemed that the radiation was purely thermal. Shortly thereafter, he realized that, if a BH eventually evaporated due to this mechanism of energy loss, it would challenge the basic tenets of quantum mechanics. In the case of gravitational collapse, one may suppose that matter starts in a pure state, collapses to form a BH, which eventually evaporates leaving the universe in a thermal mixed state. However, unitary evolution in quantum mechanics implies that pure states evolve to pure states, so that that somewhere in this process, there must be a breakdown of unitarity.\cite{HawkingBHI}. This observation led to many attempts to circumnavigate this problem or to improve upon his approximations. (For a brief review with many citations to the literature, see ref.~\cite{Page:2004xp}.) This paradox is closely connected with Beckenstein's conjecture\cite{Bekenstein:1973ur} that the entropy of a BH is proportional to the area of its horizon rather than its volume, as might naively be expected in quantum field theory (QFT.) This, together with corresponding laws of BH thermodynamics, have generally become accepted, at least at the semiclassical level. Eventually, it suggests that, at a fundamental level when gravity is involved, QFT is a redundant description and that dynamics may be describable in terms of many fewer degrees of freedom than had been previously thought, a property that has been called the holographic principle\cite{'tHooft:1993gx, Susskind:1994vu}.
Motivated by Maldacena's treatment of a BH in AdS\cite{Maldacena:2001kr}, Hawking recently changed his mind and is now of the opinion that information is not lost via BH evaporation\cite{HawkingDublin, Hawking:2005kf}. The AdS/CFT correspondence does carry with it the strong suggestion that the evolution of a BH is unitary. The key element in Maldacena's view, which Hawking had overlooked, is the role of different topological configurations, but Barbon and Rabinovici\cite{Barbon:2003aq} showed that the situation is necessarily more complicated than Maldacena suggested. Moreover, Hawking's argument that, at the end of the day, gravitational collapse to a BH does not contribute to the S-matrix at infinity seems to throw the baby out with the bathwater, so his widely publicized ``concession" remains quite controversial. Simply regarding a BH like another resonance in scattering amplitudes does seem to beg the question. Thus, there remains much else to be understood. Precisely, how can a black hole appear at one and the same time to be black body and yet be part of a pure state? And how is that to be reconciled with the successful accounting in string theory of the BH microstates\cite{Strominger:1996sh,Maldacena:1997de,Gubser:1996de}?
The information paradox is sharpest in the case of gravitational collapse from a pure state having no horizon to a BH followed by its eventual evaporation leaving an apparently thermal state. The question is whether this process is like the burning of a book, in which there is no doubt that the information is encoded in the radiation even though it would be impossibly difficult to recover. Thus, the issue is a matter of principle rather than of observation. Unfortunately, the dynamics of the collapse followed by radiation are not well understood in a detailed way. What has been better understood is the quantum character of stationary BHs, for which there is no immediate paradox. To arrange for such a situation, one must artificially place sources at infinity so that the incoming energy precisely balances the radiation, leaving a stationary horizon. This allows one to consider a quantum field in a fixed background. We shall review this situation first and subsequently reflect on the case of gravitational collapse.
\section{Entanglement Entropy
Our proposal is that BH entropy can always be identified with entanglement entropy. (For the case of eternal BH's, this point of view is elaborated in ref.~\cite{BEY}.) I would like to review entanglement entropy in detail, but, given space limitations, I must refer you to ref.~\cite{BEY} or to Feynman's lectures\cite{Feynmanstatmech}, whose notation I will closely follow. The main point is as follows: Suppose a system, which might be the entire universe, is in a pure state $\ket{\psi}.$ An observer who is confronted with a horizon cannot make measurements on the system beyond that. Therefore, it is natural to express the state in a basis of states describing states within her causal sector $\ket{\phi_i}$ and states outside
$\ket{\theta_r}$. These span Hilbert spaces that we will call ${\cal{H}}_1$ and ${\cal{H}}_2,$ respectively, and the full space of states is the Hilbert space ${\cal{H}}= {\cal{H}}_1\otimes {\cal{H}}_2.$ The general state is then written as
\begin{equation}\label{purestate}
|\psi\rangle =\sum_{i,} C_{ir} |\phi_i \rangle_1 |\theta_r\rangle_2,
\end{equation}
or, as a density matrix,
$\rho=|\psi\rangle \langle\psi|.$ (Through this discussion, the indices may actually be vector indices, representing all the quantum numbers necessary to uniquely specify the basis states.)
Since $\rho$ corresponds to a pure state, $\rho^2=\rho.$
An observable for the observer outside the horizon corresponds to a Hermitian operator $A$ that acts within ${\cal{H}}_1,$ that is,
whose domain and range are both in the subspace ${\cal{H}}_1.$ Then the expectation value of $A$ is
\begin{eqnarray}
\langle\psi|A|\psi\rangle&=&\sum C_{js}^*C_{ir}
\langle\theta_s|\langle\phi_j|A|\phi_i\rangle|\theta_r\rangle\\
&=&\sum(C C^\dagger)_{ij}\langle\phi_j|A|\phi_i\rangle\\
&=&{\rm Tr_1}\left[A\rho_1\right],\ {\rm where\ \ }\rho_1\equiv {\rm Tr_2}[\rho]=C C^\dagger.
\end{eqnarray}
In physical terms, an observer constrained to subspace ${\cal{H}}_1$ appears to be in a mixed state described by the density matrix $\rho_1.$ Diagonalizing $\rho_1,$ it can be expressed in a new basis $|i\rangle_1$ for ${\cal{H}}_1$ as
\begin{equation}
\rho_1=\sum w_i|i\rangle_1{}_1\!\langle i|,\ {\rm with~eigenvalues\ } 0\le w_i\le1.
\end{equation}
The mixed state arises because of the correlations between ${\cal{H}}_1$ and ${\cal{H}}_2$ implied by the fact that, globally, the universe is in a pure state. Even if the universe were in its vacuum state; the observer would appear to observe a distribution of excited states. This is one of many such concepts that takes getting used to.
The entropy associated with the density matrix $\rho_1,$ called entanglement entropy, is defined as usual as
\begin{equation}
S_1=-\sum w_i \ln w_i = - {\rm Tr_1}[\rho_1\ln(\rho_1)].
\end{equation}
Similarly, an observer confined to ${\cal{H}}_2$ sees a density matrix
\begin{equation}
\rho_2\equiv {\rm Tr_2}[\rho]= C^\dagger C.
\end{equation}
Now it is easy to show that the nonzero eigenvalues of $\rho_1$ and $\rho_2$ are the {\bf same} (even if they have different dimensions.) In particular, the entanglement entropies of the two subspaces must be equal:
\begin{equation}
S_1=S_2.
\end{equation}
This simple observation has far-reaching consequences.
Finally, if there is equal likelihood to be in each state $|i\rangle_1,$ then $w_i=1/N_1,$ where $N_1$ is the dimension of ${\cal{H}}_1$. Then one sees that $S_1=\ln N_1.$ In that case, the entanglement entropy coincides with the log of the number of microstates, which is the common definition of entropy for a thermal ensemble.
\section{Quantum Field Theory in Curved Backgrounds
I cannot review here the nature of QFT in curved spacetime\cite{Birrell, Brout:1995rd}. Let me recall a few salient facts:
The Hilbert space of states is in general coordinate dependent, and there are alternative, frame-dependent definitions of the no-particle state. Often there is no conserved Hamiltonian, so there is no analogue of the lowest-energy state or ground state naturally associated with the vacuum for QFT in Minkowski space. Gravitational collapse is hard to discuss, but the entropy of eternal black holes or stationary spacetimes is somewhat easier. Classic discussions compare QFT in an inertial frame in flat space (Minkowski) versus a constantly accelerating frame (Rindler), or the nature of a Schwarzschild BH in Schwarzschild coordinates as compared with Kruskal-Szekeres coordinates.
Similarly, there are cosmologies such as de~Sitter spacetime, in which observers would observe radiation from the horizon, as well as more complicated cases, such as the Schwarzschild BH in asymptotically AdS space\cite{Maldacena:2001kr}.
All these situations have in common that there exists a stationary coordinate system with timelike Killing vector in which the spacetime admits a bifurcating Killing horizon. (See figure and references in \cite{BEY}.) The Hilbert space may be built up as a product $\cal{H}_{\rm L}\times \cal{H}_{\rm R}$ of Hilbert spaces representing states associated with the ``Left" or ``Right" portion of a fixed time slice, with Hamiltonians $\rm{H_L}$ or $\rm{H_R}$ for each. The sense of the time is reversed across the horizon, so the full Hamiltonian is $\rm{H=H_R-H_L}.$ A natural choice of the global vacuum state in such cases, consistent with the stationary metric, is the Hartle-Hawking state. Typically, in these cases, the vacuum takes the form\cite{Israel:1976ur}
\begin{equation}
\ket{0}=\frac{1}{\sqrt{Z}} \sum_i e^{-\frac{\beta E_i}{2}} \ket{E_i}_R\ket{E_i}_L,
\end{equation}
where $Z$ is the partition function, $E_i$ is the eigenvalue of $H_R,$ and $\beta$ is the inverse temperature associated with the Hawking radiation. For example, for the Schwarzschild BH, $\beta=8\pi G_NM_{BH},$ where $G_N$ is Newton's constant and $M_{BH}$ is the mass of the black hole. This gives a thermal density matrix for $\rho_{\rm R}$ and an entropy equal to the Beckenstein-Hawking value, $S_{BH}=A/4G_N,$ where $A$ is the area of the horizon.
Unfortunately, the case of eternal BH's, while illustrating the role of entanglement in BH entropy, does not shed much light on the nature of the information paradox. It may help to have a more microscopic picture of the mechanism of radiation and the associated back reaction on the background metric. While Hawking radiation is sometimes referred to as a tunneling process, the only quantitative development of this idea that I've seen is one by Parikh and Wilczek\cite{Parikh:1999mf}. A BH that radiates is analogous to the decay of an unstable particle, $M\rightarrow M'+\gamma,$
in which a particle of mass $M$ decays into another particle of mass $M'$ plus a photon. These authors show that the transition can be calculated semiclassically in a particularly nice, stationary coordinate system that is nonsingular at the horizon (while remaining asymptotically flat.) Energy is conserved however, so that $M=M'+\omega_\gamma,$ Thus, the black hole horizon correspondingly shrinks, in the Schwarzschild case, from $R=2G_NM$ to $R'=2G_NM'=2G_N(M-\omega_\gamma).$ They calculate the transition amplitude in WKB approximation. Interestingly, they do not obtain a perfectly thermal spectrum except for sufficiently small $\omega_\gamma.$ Their microscopic description suggests that the continual decrease of entropy during radiation is perfectly consistent with unitarity when the backreaction on the background is taken into account.
\section{Gravitational Collapse
There have been precious few calculations of gravitational collapse that bear on the issue of the information paradox. However, if one believes in quantum mechanics and unitarity, then it is easy to argue that the entropy of a BH so formed will be entanglement entropy. Imagine starting in the distant past with a pure state of matter in a background having no horizon and only infalling matter, with none outgoing $\ket{\psi, g_{\mu\nu}}.$ The matter, together with a self-consistently determined metric, evolves unitarily in time $U(t)\ket{\psi, g_{\mu\nu}}.$ At some point, the background develops a horizon with respect to a stationary observer outside. Eventually, all of the original matter will have fallen inside the horizon although there will remain outgoing radiation outside. Classically, the two regions are causally disconnected. Since measurements are classical, no observer subsequently can determine all the properties of the system, so it is natural to describe the Hilbert space as a product of states inside and outside the horizon. Globally, since the system is in a pure state, it will take the form of eq.~(\ref{purestate}). This must remain true as a function of time, even as the collapse continues while radiation is emitted from the neighborhood of the horizon. An exterior measurement during this period would necessarily involve a density matrix associated with the causal region of a given observer. But the concept of a measurement ``during this time" is already problematic, since one can only determine energy with limited accuracy in a finite time. (Basically, the approximation is that the metric changes slowly compared with the duration of the measurement to some accuracy.) We think eventually the BH so formed would evaporate. Thus, the BH is like a resonance in particle physics, in which an initial state of matter forms a long-lived configuration that eventually decays back to ordinary matter. Asymptotically, one only measures stable particles, ``from dust we were made, and to dust we shall return."\cite{Genesis3:19}
Perhaps one could describe the evolution of the metric quasi-statically and adiabatically. At first, that sounds contradictory, since that suggests no entropy production. However, globally, since the system remains in a pure state, there is nothing in principle to obstruct such an argument.
Recall that the entanglement entropy of each subsystem is always equal, so the question of the number of states inside and outside the BH seems not so relevant. From this point of view, it is puzzling that string theory calculations, counting black hole microstates, work\cite{Strominger:1996sh,Maldacena:1997de,Gubser:1996de}.
Based on the observation in Section~1 (below eq.~(8)), the suggestion would be that, by the time string theory is relevant, we are dealing with strongly interacting gravity, in which all states in the available phase space are equally likely to be populated. Some sort of ergodic theorem in which the time-averaged properties of such a system resemble ensemble averages would then be needed to reconcile these two points of view. This is how statistical mechanics normally reconciles the behavior of a unique system with that of a collection of similar systems. The brilliance of the argument in ref.~\cite{Strominger:1996sh} was the realization that, owing to supersymmetric nonrenormalization theorems, the counting could be done in the weak coupling regime.
\section{Summary and Conclusions
In the light of our previous discussion, it is worth reflecting on various other attempts to reconcile the information paradox with the usual laws of physics. One focus has been on the BH final state, whether there is a remnant left containing all the information that has fallen into the BH, or whether it completely evaporates. This line of development is misdirected, because Hawking radiation appears to come from the horizon. As the mass of the BH decreases, its horizon shrinks and its entropy diminishes. So one must understand whether or not this process involves information loss. The issue is not what happens at the very end.
Similarly, a great deal of attention has been devoted to the question of the singularity inside. This IS a very interesting question, and if it were resolved by a wormhole tunneling to a different classical spacetime, then certainly, not only information but also energy might disappear from our universe. The AdS/CFT correspondence provides hope that this is not the outcome and suggests that, in the end, all will be well. Regardless, this does not address the decrease of entropy in the initial stages discussed above. It may however effect things near the very end.
What happens in the final stage of evaporation is anybody's guess, so here is mine. I suspect that it is quite uneventful, since as the BH radiates, the mass decreases along with the entropy. Eventually, it becomes very small with very few states left to be entangled. (Recall $S_1=S_2$ always.) The final stage ends with a whimper, not a bang.
If the viewpoint elaborated here is correct, the BH information paradox will eventually be seen as similar to the EPR paradox\cite{Einstein:1935rr}. On the one hand, the situation seems to violate the laws of physics. On the other hand, it clearly does not, but, because it is counterintuitive, it still is fascinating. Thus, rather than a physics problem, it becomes a psychological problem, a bit like an optical illusion.
\section{Acknowledgement}
I would like to thank my coauthors of ref.~\cite{BEY} for extensive discussions about the topics surveyed herein. I also wish to thank the conference organizers for their hospitality and for the opportunity to return to Japan on this happy occasion.
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"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/formulative.io\/help\/Tutorials\/gettingStartedLanguage.html","text":"# Getting Started With the Language\n\n## Prerequisites\n\nThis guide assumes that you know how to create a project, edit and evaluate your formulas. If not, you can browse through it here Getting Started. We will be referring to these features throughout this document.\n\nFormulative is a functional language with its expressions closely linked to Mathematics. This means that problems modelled using mathematical constructs can - in most cases - be translated to Formulative using the same mathematical notation. Model and implementation can in many cases stay very close and the resulting implementation is terse.\n\nFormulative supports Numerical problems and up to now we have not implemented Symbolic computations.\n\nNot all models can be expressed with pure mathematics (nor is it always desirable). Therefore Formulative has generic functional constructs like map, reduce or fold to solve generic problems.\n\nThe expressions are evaluated lazily: when you ask it to compute some variable, it will build up a path of expressions in your model to get that result. You cannot control the procedural logic, how results are calculated, i.e. Formulative is not a procedural language.\n\nFormulative has some extras to help implement Calculations like \u201cmath-style\u201d functions, collections (sets, vectors and matrices) and operators to work with them, advanced date-time manipulation, etc.\n\n## Variables\n\nVariable identifiers follow math conventions, therefore it is important that two variables which render almost similarly might be in fact two different variables, e.g. $\\mathit{x}$ and $\\mathbit{x}$ are two distinct Variables.\n\nNote\n\nIdentifier embellishments are \u201cdecorators\u201d and create distinct variables\n\nFormulative also has some common built in identifiers, e.g.: $e,$i, $\\pi Find more details on variables here: Identifiers. ## Numerical expressions Create a simple expression that defines a decimal literal constant: x = 123.2 Note You can copy & paste formulas like the above (x = 123.2) into the formula editor Embellishments are significant, so now enter this formula: x' = 232.1 and check the Evaluation variable list, it will show you the two variables with their respective values: To use some built in constants and a numeric operator, create a literal constant with its value equal to pi square (${\\mathrm{\\pi }}^{\\mathrm{2}}$). The expression is: a =$\\pi^2\n\n\nOnce you entered the formula you can Evaluate the result:\n\nCheckout the reference for more on Numbers (literals, arithmetic, relational operators).\n\n## String expressions\n\nCreate the following expressions to do some String literal manipulation:\n\n\u2022 expression defining first name: n' = \"John\"\n\u2022 expression defining last name: n'' = \"Doe\"\n\u2022 expression to concatenate them: n''' =\u00a0 n' *** \" \" *** n''\n\nTo get a substring of the above variable ${\\mathit{n}}^{\u2034}$, add the following expression:\n\nn_'substr = (n''')_{1,...,4}\n\n\nThis will select the first 4 characters from the string:\n\nNote that character indexing starts with 1 (instead of 0 as in many other languages).\n\nCheckout the reference for more on Strings\n\n## Keywords\n\nKeywords are specific strings and are defined with an apostrophe, e.g. suppose you have payment frequencies, then you could use 'monthly as a specific value in the list of payment frequency values. You often use keywords in relational table resources to represent a strict set of values in some of the columns. More on this subject in the Tables section.\n\n## Date \/ Time & Durations\n\nFormulative has built in support for Date and Time manipulation. Start with a simple date literal:\n\n!d_'lit = '2018-10-23\n\n\nNext create a time literal:\n\n!t_'lit = '2018-10-23T01:02:03\n\n\nContinue with a duration adding 2 years and 3 months:\n\ndur_'lit = 2 $!year + 3$!month\n\n\nNow, let\u2019s add this to the !d_lit variable by:\n\n!d'_lit' = !d_'lit + dur_'lit\n\n\nYou can also convert between data and date-time literals.\n\nCreate a date-time from a date value (by extending it with 00:00:00):\n\n!t_'extend = $time(!d_'lit) Do it the other way, create a date by extracting the date part from a date-time: \u2022 create a time value: !t_'lit = '2018-10-23T01:02:03 \u2022 extract the date part: !d_'extract =$date(!t_'lit)\n\n\nUsing the built in functions $today and $now provide the current date and time respectively.\n\nThe following expression constructs a date from parts (Y, M, D):\n\n!d_'comp = $date(2018, 12, 6) You can use the$time(Y, M, D, h, m, s) similarly to construct a time value:\n\n!t_'comp = $time(2018, 12, 6, 12, 5, 0) Subtracting parts from a date or time is done using subscripts, e.g. to get the years part: n_'years = (!!d_'comp)_'year Checkout the reference for more on Date \/ Time & Durations ## Ranges, repeats, tuple A simple use of the range construct is to create some collection from it. For example to create a tuple with numbers from 1 to 10, you could write this: a = (1,...,10) Now that we have the numbers in a collection we could sum up say the first 5 numbers like this: b = {+}__{i=1}^^5 (a)_i What is going on in this formula? \u2022 b = is an assignment of a new variable \u2022 {+} is the summation operator that will be invoked on our collection of integers \u2022 _{i=1} the subscript notation is used on the operator to tell the index to begin from \u2022 ^5 the superscript notation tells the upper index to use \u2022 (a)_i is the element at position i in collection a You can use expressions in both the subscript and superscript, so if you wanted you could iterate over a previously unknown length by using the$len function like this:\n\nc = {+}__{i=1}^^{$len a} (a)_i With ranges and repeats you can also create iterator bindings that will bind a variable to a range of values and let you work with these values one by one in some expression. For example to create the same range of numbers and bind them to a variable used to create the Tuple, you could write this: d = (i | i = 1,...,10) What is going on in this formula? \u2022 d = is an assignment of a new variable \u2022 i = 1,...,10 binds i to a list of integers (given as a range here) \u2022 i the variable in the binding is a local variable in the expression which takes the listed values \u2022 | maps the expression on the left to the binding on the right, using iteration \u2022 the braces ( ) are important, they are not used for grouping but for collection construction: constructing a tuple. Note Had we used {{ }} instead of ( ) we would have created a set instead of a tuple. Creating a Tuple with this approach is more complex but has some additional features. For example, if we wanted to add up only even numbers between 1 and 10 we could add a condition to the range construct as follows: e = (i | i = 1,...,10 \/\\ i$mod 2 = 0)\n\n\nand to sum these up:\n\nf = {+}__{i=1}^^{$len e} (e)_i This is what your formulas look like (source and rendered): See this for more on lists and ranges: Lists See this for more on Tuples ## Finite Sets Constructing a finite set manually, by listing its elements: s_'fromLiteral = {{1,2,3,4,4,4,4,4}} Since we are creating a set there will be only 4 elements. Building on the range example above, we can create a set with the even numbers from 1 to 10 like this: S = {{i | i = 1,...,10 \/\\ i$mod 2 = 0}}\n\n\nThen adding the Set\u2019s values is very concise:\n\nsum_'S = {+}_{x (- S} x\n\n\nNote\n\nFor this sum operation {+} to work you need to have a Set.\n\nThere are other predefined operators on sets like multiplication or set algebra.\n\nCreate two sets of keywords (we will use letters for keywords), then we will see how the set operations work:\n\nP={{'a,'b,'c,'d}}\nT={{'a,'b,'e,'f}}\n\n\nSome of the Set Operations and their respective results using the above sets:\n\nSet operations and results\nSource Result\nP_'unionT = P (_) T ${\\mathit{P}}_{\\mathsf{unionT}}=\\left\\{\\mathsf{a},\\mathsf{b},\\mathsf{c},\\mathsf{d},\\mathsf{e},\\mathsf{f}\\right\\}$\nP_'intsT = P (~) T ${\\mathit{P}}_{\\mathsf{intsT}}=\\left\\{\\mathsf{a},\\mathsf{b}\\right\\}$\nP_'diffT = P \\\\ T ${\\mathit{P}}_{\\mathsf{diffT}}=\\left\\{\\mathsf{c},\\mathsf{d}\\right\\}$\n\nSee this for more on Finite Sets\n\n## Associations (maps) and Relational Tables\n\nAssociations are key-value maps. Keywords come handy when building keys of associations. If you wanted to represent attributes of some object you can use Associations. As a simple example, here is an association that holds the name, age and gender attributes of a person:\n\nJohn = {{'name |-> \"John\", 'age |-> 28, 'gender|->'male}}\n\n\nNote the following:\n\n\u2022 The mathematical model of an association is simply a (finite) function: it maps elements from a set to another set\n\u2022 Elements of associations are actually (key, value) pairs\n\u2022 The maplet operator |-> is syntactic sugar, i.e. key |-> value = (key, value)\n\u2022 However, when using the maplet operator |-> to construct an association the keys are checked for uniqueness\n\u2022 We were using keywords, e.g: 'name for the keys of the associations\n\u2022 We were using strings where there is no closed value set, e.g. the name attribute\n\u2022 We were using keywords again when the value is from a closed set like genders\n\nThen create a second person:\n\nThomas = {{'name |-> \"Thomas\", 'age |-> 32, 'gender |-> 'male}}\n\n\nTo create a relational table of the people\/characters in our example we can combine these into a Relational Table, which is just a plain set of associations:\n\nP = {{John, Thomas,\n{{'name |-> \"John\", 'age |-> 31, 'gender |-> 'male}},\n{{'name |-> \"Winona\", 'age |-> 13, 'gender |-> 'female}},\n{{'name |-> \"Lige\", 'age |-> 35, 'gender |-> 'male}},\n{{'name |-> \"Starling\", 'age |-> 29, 'gender |-> 'male}},\n{{'name |-> \"Rosalee\", 'age |-> 24, 'gender |-> 'female}}\n}}\n\n\nNotice that:\n\n\u2022 we have used John and Thomas variables in the table construction, from their previous declaration\n\u2022 we have added other associations by constructing them literally in place\n\nNow we can do some relational table manipulation:\n\nSelect Thomas\u2019s age with the following expression:\n\nAge_'T = {{(r)_'age | r (- P \/\\ (r)_'name = \"Thomas\"}}\n\n\nSome explanation of the above expression:\n\n\u2022 we are constructing the result as a set by the set construction operator: {{...}}\n\u2022 (r)_'age will select the age attribute (with the subscript) of the current element bound to the local variable r\n\u2022 | r (- P here we bind r, a local variable, to elements of the set P\n\u2022 \/\\ (r)_'name = \"Thomas\" is a condition that r has to satisfy. The subscript selects the attribute (name)\n\nSelect all female characters:\n\nC_'f = {{(r)_'name | r (- P \/\\ (r)_'gender = 'female}}\n\n\nWe could have used a simpler form of selection to create a new relational table (sub-select) with the sigma operator:\n\nC'_f = $\\sigma_{'gender|->'female} P Select all male characters older then 30: C_'gt30 = {{(r)_'name | r (- P \/\\ (r)_'gender = 'male \/\\ (r)_'age >= 30}} Some explanation of the above expression: \u2022 here we used more than one condition on the current element r See this for more on Associations (Maps) & Sets of Associations (Relational Tables) ## Parameter Tables Parameter tables you define (or upload) are Relational Tables, too. Make sure you create the scoring parameter table in this getting started tutorial: Add interest rate calculation based on Scoring To get the the annual rate for a customer with a score of 720 write an expression similar to how selection was done from the above relational table: S_'customer = (r)_'APR | r (- @Scores : (r)_'From <= 720 \/\\ (r)_'To >= 720 Notice the following: \u2022 we used @Scores to refer to the Parameter table (as a relational table) \u2022 after we bound r to elements of the Scores table, we used : to describe selection criteria, this is an alternative to using \/\\, as we did before. ## Sequences Start by creating a simple sequence literal: S = {<1,2,3,4,5>} Sequences are more complex than Sets, therefore the following expression that sums the above sequence will require some explanation: S_'sum = {+}_{(i,e) (- S} e Notice the following: \u2022 we are using the {+} operator similar to the example with Sets \u2022 the subscript _{(i,e) (- S} binds to the elements of Sequence $\\mathit{S}$ \u2022 the (i,e) part, however, is not as simple as in the case of sets, where we bound a single variable to the elements of the Set. \u2022 here we destructure each element of S into two distinct variables \u2022 i- the index of the element, and \u2022 e- the actual member @ index i, i.e. the value \u2022 then we use e to sum up the elements of the sequence The mathematical model of sequences is an association that has numeric keys in the range 1,...,n, for some n. Sequences can also be described as finite functions whose domain is an initial segment of the positive integers. All this means that sequences are functions, associations, and, as such, are sets as well. Therfore, if we iterate elements of a Sequence with the (- (element of) operator, we get (i,e) (index, member) pairs as elements. Note Sets have elements. Sequences have members. However, as sequences are also sets, their elements are pairs consisting of an index and the corresponding member. Therefore the following are all the same sequence: {< a, b, c >}, {{ 1 |-> a, 2 |-> b, 3 |-> c }}, {{ (1,a), (2,b), (3,c) }}. Create a subset of the above sequence with only the first two elements: S_'sub = {< e | i = 1,...,$len S; e = S(i) \/\\ i < 3 >}\n\n\nNotice the following:\n\n\u2022 {<...>} is the sequence constructor operator, so the result will be a sequence\n\u2022 e will refer to the value of each sequence element (index\/value) pair\n\u2022 i is bound to a range of indexes from 1 to the length of S, $len S \u2022 e = S(i) extracts the value from the sequence at position i \u2022 \/\\ i < 3 is a condition that tells to select only elements with indexes below 3 (indexes start at 1) \u2022 this example shows that several simple bindings can be joined with ;. If you wanted the square of all members of S you could write this: S_'subsq = {< e^2 | i = 1,...,$len S; e = S(i) \/\\ i < 3 >}\n\n\nNote\n\nNotice above that the expression e^2 is mapped to the binding on the right. This notation makes it possible to construct any sequence whose elements can be written as an expression of the index.\n\nNow that you have seen the hard way to create a subset with the first two elements here is a simpler but less generic way to do the same:\n\nS_'sub2 = ( S )_{< 1,...,2 >}\n\n\n## Functions\n\nBefore going further with map\/reduce\/fold\/sort functions we need to make a little detour into functions. Other than expressions assigning values to variables you can also define functions. Let us start with a simple one calculating the Area of a circle:\n\nf_'AofCircle: %R --> %R, r |--> r^2 $\\pi There are more than one way to define a function, let\u2019s dissect this one: \u2022 f_'AofCircle is the name of the function \u2022 %R --> %R describes that the function will have one argument (a real number, denoted by %R) and will return a real value (%R to the right of the arrow) \u2022 %R is a set that Formulative calls a \u201cdomain\u201d, you can find out more about domains here: Operations on Numeric Sequences \u2022 r names our single argument (which we previously described to be a real) \u2022 if we had more arguments we would have put brackets around like: (x,y,z) \u2022 to the right of the |--> literal we define the body of the function. To call our function and assign a value to a variable you could write: A = f_'AofCircle(x) Where x is an unbound variable which you can set when you evaluate your formula. Let\u2019s do something with a sequence of real numbers like finding out the average: avg:$Seq(%R) --> %R, S |--> 1\/($len S) * ({+}_{(i,e) (- S} e) Invoke our new avg function on the first 100 numbers wth this formula: x^^ = avg({<1,...,100>}) What happens when you have multiple arguments ? Here is a function that calculates the area of a rectangle: f_'AofRect: %R**%R -) (a,b) |--> a b (- %R Some explanation: \u2022 this notation is an alternative form of f_'AofRect: %R**%R --> %R, (a,b) |--> a b \u2022 we used the cartesian product of reals, %R**%R, to define the input parameter domain \u2022 the -) (\u201ccontains as element\u201d) operator described the argument list (a,b) \u2022 we used the invisible multiplier operator to do the multiplication \u2022 (- %R describes that the function returns a real number. Similarly to the invocation of the circle area function use this with unbound variables a and b: A_'rect = f_'AofRect(a,b) Then evaluate the function if you want to check the result, specify a and b as input variables and A_'rect as your output variable. There are alternative notations to declare functions which you can find here: Functions (Morphisms) ## Sequence operations Now that Functions were introduced we can get back and do some more operations with Sequences. ### Sorting To begin with let us create a sequence of unordered numbers: N_'us = {<33,55,22,11,5,324,10>} Sorting requires a comparator function that takes two parameters (a, b) and returns a %R (real). If the returned value is is less than 0, sort a to an index lower than b, i.e. a comes first. Our number comparator function is the following: numCmp: %Z**%Z --> %Z, (a,b) |--> a - b Since the sequence contains integers, we used the %Z (integers) domain. This expression will then do the sorting: N_'s =$sort_numCmp (N_'us)\n\n\nNote\n\nYou could have also sorted this sequence using a default sorter for integers by using this N_'sdef = $sort_%Z (N_'us) The expression defines the N_'s variable that will hold the sorted sequence. The subscript _numCmp tells the $sort function which comparator function to use for sorting.\n\n### Sorting Sequences of Associations\n\nLet us do something more complex, sort the People sequence according to age. Assuming the S_'people sequence is defined, before moving on, define the domain (or schema) of your Relational Table:\n\nPersonD = $Asc( {{'name}} --> %S, {{'age}} --> %N, {{'gender}} --> %K ) This declares that the PersonD domain is the set of all associations made up of: \u2022 a name keyword that maps to a String value %S \u2022 an age keyword that maps to a Number value %N \u2022 a gender keyword that maps to keywords %K With the domain declared we can write a comparator function: ageCmp: PersonD**PersonD --> %R, (e_1, e_2) |--> (e_1)_'age - (e_2)_'age Recall from the function introduction that following the function name declaration the Parameter Domains have to be specified. The comparator function used by the sort function is given two elements that it has to compare. The elements will be from the sequence therefore we can reuse the PersonD domain declaration to declare the domains of parameters e_1 and e_2. Note Instead of DomainD**DomainD we could have used a shorter form DomainD^2. The following expression will finally create the sequence containing the people, sorted by age: S_'peopleSorted =$sort_ageCmp(S_'people)\n\n\n### Reduce \/ Fold\n\nFormulative has support for map\/reduce\/fold algorithms that work on sequences. Working with sequences has show how mapping works (on any collection). Reduce and Fold however requires a collection where the order of elements is guaranteed. Sequences provide this guarantee, therefore to Reduce and Fold operate only on Sequences.\n\nAs a first example let\u2019s see how Reduce works. Create a reduce function that will add the square of the elements in an integer sequence:\n\nsqrRe: %Z**%Z --> %Z, (acc, item)\n|--> acc + item^2\n\n\nReduce functions take two arguments:\n\n\u2022 the accumulator acc (initially 0 because we are using the %Z (integer) domain)\n\u2022 the current item item\n\nInvoke the reduce function as shown below:\n\nRe_'sqr = $reduce_sqrRe(S) Note If you wanted to specify an initial accumulator value e.g. start with -10, you could specify it in the subscript along with the reducer function name: Re_'sqr =$reduce_{sqrRe, -10}(S)\n\nA Fold function will be used to count the even numbers in a sequence. The fold function is the following:\n\ncntFld : %R^2 --> %R, (a,e) |--> a + [ e $mod 2 = 0 ] The full signature of the fold function (a,e,i,s) is more complex than above (and the reduce callback function\u2019s): \u2022 a : the accumulator value (as with reduce) \u2022 e : the current element in the sequence (as with reduce) \u2022 i : the optional index value of the e -s index in the input sequence \u2022 s : the optional input sequence In order to do the counting evaluate this expression: Fld_'cnt =$fold_{ cntFld, 0 } ({<1,...,50>})\n\n\n## Matrix and Vector operations\n\nTo demonstrate the use of Matrices we are going to solve a simple linear regression, line-fitting problem.\n\nThe problem is of the form: $\\mathit{Y}=\\mathit{A}+\\mathit{b}\\mathit{X}$\n\nWhere Y is the criterion variable, X is the predictor variable. Variable A is the intercept and b is the slope of the curve.\n\nThe Example data is the following:\n\nLinear Regression Example data\nPredictor (X) Criterion (Y)\n1.0 1.0\n2.0 2.0\n3.0 1.3\n4.0 3.75\n5.0 2.25\n\nIn order to find out the best fitting line we will use linear transformations. The formula we are going to use is the normal equation formula: $\\mathit{\\varphi }={\\left({\\mathit{X}}^{\\mathit{T}}\\mathit{X}\\right)}^{-\\mathrm{1}}\\cdot {\\mathit{X}}^{\\mathit{T}}\\mathit{y}$\n\nIn order to implement out problem, let\u2019s create the design matrix X (with ${\\mathit{x}}_{\\mathrm{0}}=\\mathrm{1}$):\n\nX = [ 1 # 1 ##\n1 # 2 ##\n1 # 3 ##\n1 # 4 ##\n1 # 5]\n\n\nThen the vector for Y the criterion variable:\n\nY = [1, 2, 1.3, 3.75, 2.25]\n\n\nThe formula below implements the pseudo inverse calculation ${\\left({\\mathit{X}}^{\\mathit{T}}\\mathit{X}\\right)}^{-\\mathrm{1}}\\cdot {\\mathit{X}}^{\\mathit{T}}$\n\nX_'pinv= ($transpose(X) * X)^{-1} *$transpose(X)\n\n\nFinally we get the parameter vector, by this multiplication (we could have kept the two last formulas together, separated to make the calculation of the pseudo inverse more explicit):\n\n\\varphi = X_'pinv * Y\n\n\nWhen you evaluate the expressions you should get the parameter vector $\\mathit{\\varphi }=\\left[\\mathrm{0.785},\\mathrm{0.425}\\right]$ (cut precision in display)\n\nTherefore filling the parameters into the equation, we got to this solution:\n\n${\\mathit{Y}}^{\\prime }=\\mathrm{0.785}+\\mathrm{0.425}\\mathit{X}$\n\n### Calculating the standard error\n\nFinish this part by computing the standard error of the estimate based on errors of prediction.\n\nExample data with Prediction Errors\nPredictor (X) Criterion (Y) Prediction (Y\u2019) Error (Y-Y\u2019) Error Sqr (Y - Y\u2019)^2\n1.00 1.00 1.210 -0.210 0.044\n2.00 2.00 1.635 0.365 0.133\n3.00 1.30 2.060 -0.760 0.578\n4.00 3.75 2.485 1.265 1.600\n5.00 2.25 2.910 -0.660 0.436\nSum 15.00 10.30 10.30 0.000 2.791\n\nThe standard error of the estimate is a measure of the accuracy of predictions. Our regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error). The standard error of the estimate is defined below:\n\n${\\mathit{\\sigma }}_{\\mathsf{pred}}=\\sqrt{\\frac{\\left(\\sum _{\\mathit{i}=\\mathrm{1}}^{\\mathit{N}}{\\mathit{Err}}_{\\mathit{i}}\\right)}{\\mathit{N}}}$ where ${\\mathit{Err}}_{\\mathit{i}}$ is ${\\left({\\mathit{Y}}_{\\mathit{i}}-{\\mathit{Y}}_{\\mathit{i}}^{\\prime }\\right)}^{\\mathrm{2}}$ .\n\nTo perform this calculation we need to implement the followings:\n\n\u2022 implement the linear function formula to call to get the predicted results\n\u2022 create a sequence holding tuples of reals, one for X, Y, Y\u2019, Y - Y\u2019 and (Y - Y\u2019)^2\n\u2022 do a fold on the above sequence to sum the (Y - Y\u2019)^2 values\n\u2022 calculate the standard error according to the above formula.\n\nThe linear function implementation requires the following 3 expressions:\n\nA = (\\varphi)_1\nb = (\\varphi)_2\nf_'lin : %R --> %R, x |--> A + b x\n\n\nNote\n\nWe could have used (\\varphi)_1 and (\\varphi)_2 in place the linear function - ${\\mathit{f}}_{\\mathsf{lin}}:\\mathbb{R}\u27f6\\mathbb{R},\\mathit{x}\u27fc{\\left(\\mathit{\\varphi }\\right)}_{\\mathrm{1}}+{\\left(\\mathit{\\varphi }\\right)}_{\\mathrm{2}}\\cdot \\mathit{x}$ - as well.\n\nThe sequence holding tuples of X, Y, Y\u2019, Y - Y\u2019 and (Y - Y\u2019)^2:\n\nPred = {< (x, y, y', y - y', (y - y')^2) | i = 1, ..., $size Y; x = ( X )_{i,2}; y = (Y)_i; y' = f_'lin(x) >} Some explanation: \u2022 the (x, y, y', y - y', (y - y')^2) part will create a tuple with the values calculated in the binding to the righ of | \u2022 i = 1, ...,$size Y creates an iterator over the indexes of the Y vector (size)\n\u2022 x = ( X )_{i,2} gets the second element of the X (design) matrix, the predictor variable at index i\n\u2022 y = (Y)_i the criterion value at index i\n\u2022 y' = f_'lin(x) calls our linear function with the predictor variable to get the predicted value Y\u2019\n\nThe fold function that will summarize (Y - Y\u2019)^2 values (the 5th element in the tuples)\n\nfldPredErr: %R ** %R^5 --> %R, (acc, item)\n|--> acc + (item)_5\n\n\nThe calculation of the standard error given the above constructs:\n\n\\sigma_'pred = \\\/~{$fold_{fldPredErr, 0} (Pred) \/$size Y}\n\n\nThe above formula will do the fold on the Pred sequence variable and calculate the square root of the mean of squared errors. Rendered like this:\n\n${\\mathit{\\sigma }}_{\\mathsf{pred}}=\\sqrt{\\frac{{\\mathrm{fold}}_{\\mathit{fldPredErr},\\mathrm{0}}\\left(\\mathit{Pred}\\right)}{\\mathrm{size}\\mathit{Y}}}$\n\nFinally here is the original article from onlinestatbook","date":"2019-12-09 02:47:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 19, \"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\": 0.7010605335235596, \"perplexity\": 2899.2653186002376}, \"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-2019-51\/segments\/1575540517156.63\/warc\/CC-MAIN-20191209013904-20191209041904-00049.warc.gz\"}"} | null | null |
As Christmas draws to a close and the house is quiet for the first time today---I have stopped to remember the empty space in our lives that most of us have today---once filled by another. I have stopped and prayed many times today for my sweet friends that are acutely aware of the emptiness and asked that God would fill that void with sweet memories of the years that space was filled with love and joy.
One of the facts of life is that we all will lose someone dear to us---if we stay around long enough--and the longer you are here in this world--the greater the chance that you will lose those that are dear to you. I have no answer except the promise of eternity.
I have decided that it is better to have loved and lost---than to have never have loved. Grief is painful---but the joy of having loved ones in our lives is well worth the pain we must endure when they leave.
SO ---there are empty spaces in my life---quite a few--but I am who I am because of the love I have shared with those that are now gone. As the wind moans mournfully outside my window---I look to God and thank Him for the gift of loving others--those present and those that have left. He is teaching me---giving me a glimpse of how important I am to Him and the promise that there will never be an empty space with Him--He will be with me always.
A Final Merry Christmas & Happy Christmas Memories of Today and Yesterday! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
package org.safehaus.jettyjam.utils;
import javax.ws.rs.core.MediaType;
import org.junit.Rule;
import org.junit.Test;
import static junit.framework.TestCase.assertEquals;
/**
* Tests the Hello World Servlet in embedded mode.
*/
public class ServletTest {
@JettyContext(
servletMappings = {
@ServletMapping( servlet = TestServlet.class, spec = "/*" )
}
)
@Rule
public JettyResource service = new JettyUnitResource( this );
@Test
public void testHelloWorld() {
String result = service.newTestParams()
.setEndpoint( "/" )
.newWebResource().accept( MediaType.TEXT_PLAIN )
.get( String.class );
assertEquals( TestServlet.MESSAGE, result );
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,384 |
An amazing landscape just waiting to be explored with some of the best trekking possibilities worldwide. I knew that this was to be a favourite landscaping destination.
In contrast to Venice or Amsterdam Bruges is an underestimated gem. The Medieval city has some beautiful architecture & a visually stunning canal system that meanders throughout the ancient city. The streets weren't so overcrowded making a tour through the cobbled streets along the tree lined canals a relaxed affair.
No trip to Belgium is complete so I have been told without visiting the town of Bruges. Along with Amsterdam Bruges is also a city that bears the name as "Venice of the North". Then of course there is the film starring Colin Farrell, at this point it should become apparent that Belgium has more to offer than chocolate, waffles & beer. Indeed this is the Belgium that I hoped to see & enjoy. Beautiful, clean, cobbled streets adorned with very well kept old houses. Romantic looking side streets with cafes & restaurants, plus a market square that leaves one breathless with admiration. It is of no surprise that the city is a World UNESCO Heritage site since 2000 & has a very important economic position in Belgium due to its port.
Where the Koningstraat crosses with the Spiegelrei I spotted this oldtimer on the bridge with a nice backdrop of the Jan van Eyck Square.
As the evening draws in and the shadows get longer, the streets start to empty Bruges is at its best.
Romantic canals lined with opulent architecture & crossed by numerous stone bridges is a dream.
Just before the sun disappears it sheds its golden light on many of the old houses leaving others in dark shadow.
A chronicle of places I have visited & documented with word & picture in recent years.
Me, my gear, inspiration & answers to frequent questions. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,485 |
\section{Introduction}
Spectral graph theory deals with the relation between the structure of a graph and the eigenvalues (spectrum) of an associated matrix,
such as the adjacency matrix $A$ and the Laplacian matrix $L$.
Important types of relations are the spectral characterization.
These are conditions in terms of the spectrum of $A$ or $L$, which are necessary and sufficient for certain graph properties.
Two famous examples are: (i) a graph is bipartite if and only if the spectrum of $A$ is invariant under multiplication by $-1$, and
(ii) the number of connected components of a graph is equal to the multiplicity of the eigenvalue $0$ of $L$.
Properties that are characterized by the spectrum for $A$ as well as for $L$
are the number of vertices, the number of edges, and regularity.
If a graph is regular, the spectrum of $A$ follows from the spectrum of $L$, and vice versa.
This implies that for both $A$ and $L$ the properties of being regular and bipartite,
and being regular and connected are characterized by the spectrum.
If a property is not characterized by the spectrum, then there exist a pair of cospectral graphs
where one has the property and the other one not.
For many graph properties and several types of associated matrices, such pairs are not hard to find.
However, if we restrict to regular graphs it becomes harder and more interesting,
because a pair of regular cospectral graphs where one has a given property and the other one not
is a counter examples for a spectral characterization with respect to $A$, $L$, and several other types of matrices.
Such a pair of regular cospectral graphs
has been found for a number of properties, for example
for being distance-regular~\cite{H}, having a given diameter~\cite{HS}, and admitting a perfect matching~\cite{BCH}.
The {\em vertex-connectivity}~$\kappa(\G)$ of a graph $\G$ is the minimum number of vertices one has to delete from $\G$ such that the graph
becomes disconnected.
The {\em edge-connectivity}~$\kappa'(\G)$ is the minimum number of edges one has to delete from $\G$ to make the graph disconnected.
One easily has that $\kappa(\G)\leq\kappa'(\G)\leq\delta(\G)$ where $\delta(\G)$ is the minimal degree of $\G$.
Clearly $\kappa(\G)=0$ as well as $\kappa'(\G)=0$ just means that $\G$ is disconnected,
therefore these two properties are characterized by the spectrum when $\G$ is regular.
Fiedler~\cite{F} showed that the second smallest eigenvalue of the Laplacian matrix $L$ (called the {\em algebraic connectivity})
is a lower bound for the vertex- (and edge-) connectivity.
For a regular graph $\G$ there exist stronger spectral bounds for $\kappa(\G)$ (see \cite{A}) and $\kappa'(\G)$ (see~\cite{C}).
Here we show that for the vertex- and for the edge-connectivity in a connected regular graph there is in general no spectral characterization.
For $k \geq 2$ we present a pair of regular cospectral graphs $\G$ and $\G'$ of degree $2k$ and order $6k$,
where $\kappa(\G)=2k$ and $\kappa(\G')=k+1$.
The edge-connectivity turned out to be much harder.
Nevertheless, for every even $k\geq 4$ we found a pair of regular cospectral graphs $\G$ and $\G'$ of degree
$3k-5$, where $\kappa'(\G)=3k-5$ and $\kappa'(\G')=3k-6$.
The main tool is the following result of Godsil and McKay~\cite{GM}.
\begin{thm}\label{gm}
Let $\G$ be a graph, and let $X_1,\ldots, X_m,Y$ be a partition of the vertex set of $\G$
into $m+1$ classes, such that the following holds:\\
(i) For $1\leq i,j\leq m$, each vertex $x\in X_i$ has the same numbers of neighbors in $X_j$.
\\
(ii) For $1\leq i\leq m$ each vertex $y\in Y$ is adjacent to $0$, $\frac{1}{2}|X_i|$, or all vertices of $X_i$.
\\
Make a graph $\G'$ as follows:
For $0\leq i\leq m$ and each vertex $y\in Y$ with $\frac{1}{2}|X_i|$ neighbors in $X_i$,
delete the $\frac{1}{2}|X_i|$ edges between $y$ and $X_i$, and insert $\frac{1}{2}|X_i|$
edges between $y$ and the other vertices of $X_i$.
Then the graph $\G'$ thus obtained is cospectral with $\G$ with respect to the adjacency matrix.
\end{thm}
The corresponding operation is called {\em Godsil-McKay switching}.
In many applications $m=1$.
Then $X_1$ is called a {\em GM-switching set}, and condition (i) just means that the subgraph of $\G$ induced by $X_1$ is regular.
We assume familiarity with basic results from linear algebra and graph spectra; see for example \cite{BH}, or~\cite{CRS}.
As usual, $J_{m,n}$ (or just $J$) denotes the $m\times n$ all-ones matrix, $O_{m,n}$ (or just $O$) is the $m\times n$ all-zeros matrix,
and $I_n$ (or just $I$) denotes the identity matrix of order~$n$.
\section{Vertex-connectivity}
Suppose $k\geq 2$.
We define a $k$-regular graph $G$ on the integers modulo $3k-1$ as follows:
for $i=0,\ldots,2k-1$ vertex $i$ is adjacent to $\{k+i,k+i+1,\ldots,2k+i-1\}$ (mod~$3k-1$).
Then $G$ has no triangles and vertex-connectivity $k$ (indeed, between any pair of vertices there exists
$k$ vertex-disjoint paths).
Next we partition the vertex set $V$ of $G$ into four classes $V_0,\ldots,V_3$ as follows:
\[
V_0=\{0\},\ V_1=\{k,k+1,\ldots,2k-1\},\ V_2=\{1,2,\ldots,k-1\},\ V_3=\{2k,2k+1,\ldots,3k-2\}.
\]
So $V_1$ consists of the neighbors of vertex~$0$.
Note that $G$ contains a matching of size $k-1$ that matches vertices of $V_2$ with $V_3$.
Let $B$ be the corresponding partitioned adjacency matrix of $G$.
Then
\[
\ B=\left[
\begin{array}{cccc}
0 & J_{1,k} & O_{1,k-1} & O_{1,k-1} \\
J_{k,1} & O_{k,k} & B_{1,2} & B_{1,3} \\
O_{k-1,1} & B_{1,2}^\top & O_{k-1,k-1} & B_{2,3} \\
O_{k-1,1} & B_{1,3}^\top & B_{2,3}^\top & O_{k-1,k-1}
\end{array}
\right].
\]
Next we define
\[
\ N=\left[
\begin{array}{c}
I_{k+1} \\
J_{k-1,k+1}
\end{array}
\right],
\ K =
\left[
\begin{array}{c}
O_{k,k-1} \\
J_{k,k-1}
\end{array}
\right],
\ M =
\left[\, J\!-\!N\ \ K\ \ J\!-\!K\, \right].
\]
Finally, we define $\G$ to be the graph with adjacency matrix
\[
A=\left[
\begin{array}{ccc}
O_{2k,2k} & N & M \\
N^\top & \ J\!-\!I_{k+1}\ & O_{k+1,3k-1} \\
M^\top & O_{3k-1,k+1} & B
\end {array}
\right].
\]
Let $\{X,U,V\}$ be the corresponding partition of the vertex set of $\G$.
It follows that $\G$ is regular of degree $2k$, and that $X$ is a GM-switching set of $\G$.
In terms of $A$, Godsil-McKay switching replaces $N$ by $J-N$ and $M$ by $J-M$.
The graph $\G'$ thus obtained is cospectral with $\G$, and becomes disconnected if we delete the first $k+1$ vertices.
So the vertex-connectivity of $\G'$ is at most $k+1$.
To verify that the vertex-connectivity of $\G$ equals $2k$ we have to find $2k$
vertex-disjoint paths between any two distinct nonadjacent vertices $x$ and $y$ of $\G$ (see for example \cite{S}, Theorem~15.1).
This is a straightforward (and time consuming) activity, for which we have to distinguish several cases,
depending on the partition classes to which $x$ and $y$ belong.
Suppose $x,y\in V$. Then, because $G$ has vertex-connectivity $k$, there exist $k$ vertex-disjoint paths in $G$ between $x$ and $y$.
Let $X_x$ and $X_y$ be the sets of neighbors in $X$ of $x$ an $y$, respectively.
Then there exist $\ell = |X_x\cap X_y|$ vertex-disjoint paths between $x$ and $y$ of length $2$,
and $k-\ell$ vertex-disjoint paths between $X_x\setminus X_y$ and $X_y\setminus X_x$ via $U$ of length $3$.
Suppose $x,y\in X$.
If $x$ and $y$ are both adjacent to all vertices of $U$, then $x$ and $y$ are also adjacent to all vertices of $V_3$,
so there are $2k$ vertex-disjoint paths of length $2$ between $x$ and $y$.
If $x$ and $y$ are both adjacent to all vertices of $V_2$, then $x$ and $y$ have $2k-2$ common neighbors in $V$, and there is a
path between $x$ and $y$ of length $3$ via two vertices of $U$, and a path of length $4$ via a vertex of $X$.
If $x$ is adjacent to all vertices of $U\setunion V_3$, and $y$ is adjacent to all vertices of $V_2$,
then there exist $k-1$ vertex-disjoint paths of length $4$ using the matching between $V_2$ and $V_3$,
$x$ and $y$ have one common neighbor $z\in U$ and $k$ vertex-disjoint paths of length $5$ via $V_0\setunion V_1$, $X$ and $U\setminus\{z\}$.
Next suppose $x$ is the unique vertex in $X$ adjacent to all vertices of $V_3$ and just one vertex of $U$.
If $y$ is adjacent to all vertices of $U\setunion V_3$, then $x$ and $y$ have one common neighbor in $U$ and $k-1$ common neighbors in $V_3$,
furthermore there are $k$ vertex-disjoint paths of length $4$ via $V_1$, $X$ and $U$.
If $y$ is adjacent to all vertices of $V_2$, then $x$ and $y$ have $k-1$ vertex-disjoint paths of length $2$ via $V_1$,
$k-1$ disjoint paths of length $3$ via $V_3$ and $V_2$, one path of length $3$ via $V_0$ and $V_1$, and one path of length $3$ via an edge of $U$.
The remaining cases: $(x\in X,~y\in U)$, $(x\in X,~y\in V)$ and $(x\in U,~ y\in V)$ are left as an exercise.
In a similar way one can verify that $\G'$ has vertex connectivity $k+1$.
So we can conclude:
\begin{thm}
For every $k\geq 2$ there exists a pair of $2k$-regular cospectral graphs,
where one has vertex-connectivity $2k$ and the other one has vertex-connectivity $k+1$.
\end{thm}
The smallest cases ($k=2,3,4$) have been double checked by computer using the package newGRAPH~\cite{SBCS}.
\section{Edge-connectivity}\label{ec}
The construction of cospectral pairs of regular graphs with different edge-connectivity turned out to be much harder then
for the case of vertex-connectivity.
Again Godsil-McKay switching is the main tool, but now we apply Theorem~\ref{gm} with $m=2$.
(All attempts to find an example with $m=1$ failed; the difficulty is caused by the regularity requirement.)
For every even integer $k\geq 6$ we define a graph $\G$ for which the vertex set is partitioned into three classes, $X_1$, $X_2$ and $Y$,
and assume the corresponding partitioned adjacency matrix $A$ has the following structure.
\[
A=\left[
\begin{array}{ccc}
A_1 & L & M_1\\
L^\top & A_2 & M_2\\
M_1^\top & M_2^\top & B
\end{array}
\right],
\]
such that $A_1$, $A_2$ and $B$ are the adjacency matrices of the subgraphs $G_1$, $G_2$ and $H$ induced by $X_1$, $X_2$,
and $Y$, respectively.
Let $G$ be the graph induced by $X=X_1\cup X_2$, defined by
\[
L = \left[\begin{array}{cc} J_{k-1,k-1} & O \\ O & J_{k+1,k+1}-{C} \end{array}\right], \mbox{ and }
A_1=A_2=L-I_{2k},
\]
where $C$ is the adjacency matrix of the $(k+1)$-cycle.
Moreover, $H$ is a disconnected graph with adjacency matrix
\[
B = \left[\begin{array}{cccc}
B_1 & O & J_{2k-3,k-1} & O \\
O & B_2 & O & J_{2k-5,k+1} \\
J_{k-1,2k-3} & O & J-I_{k-1} & O\\
O & J_{k+1,2k-5} & O & J-I_{k+1}
\end{array}\right],
\]
where $B_1$ is the adjacency matrix of a $(k-4)$-regular graph $H_1$ of order $2k-3$,
and $B_2$ is the adjacency matrix of a $(k-6)$-regular graph $H_2$ of order $2k-5$
(here we use that $k\geq 6$ and even).
Finally we define
\[
\left[\begin{array}{c}
M_1 \\ M_2
\end{array}\right] =
\left[\begin{array}{cccccc}
O & J_{k,k-2} & O & O & O_{k,2k} \\
O & O & J_{k,k-2} & O & O_{k,2k} \\
J_{k,k-2} & O & O & O & O_{k,2k} \\
O & O & O & J_{k,k-2} & O_{k,2k}
\end{array}\right].
\]
A vertex from $H$, which is not a vertex of $H_1$ or $H_2$ is adjacent to no vertex of $X$.
A vertex from $H_1$ or $H_2$ is adjacent to exactly half or none of the vertices of $X_1$ and $X_2$.
It is easily checked that $G_1$, $G_2$ and $G$ are regular,
thus we can conclude that the given partition of $\G$ satisfies the conditions for GM-switching
given in Theorem~\ref{gm}.
To obtain the adjacency matrix $A'$ of the switched graph $\G'$ we have to replace $M_1$ and $M_2$ by
\[
\left[\begin{array}{c}
M'_1 \\ M'_2
\end{array}\right] \mbox{ by }
\left[\begin{array}{cccccc}
O & O & J_{k,k-2} & O & O_{k,2k} \\
O & J_{k,k-2} & O & O & O_{k,2k} \\
O & O & O & J_{k,k-2} & O_{k,2k} \\
J_{k,k-2} & O & O & O & O_{k,2k}
\end{array}\right].
\]
We know that $\G$ and $\G'$ are cospectral, and we easily have that $\G$ and $\G'$ are ($3k-5$)-regular of order $10k-8$.
To work out the edge-connectivity of $\G$ and $\G' $, we first observe that the graph $G_1$ (resp. $G_2$) has two
connected components $G_{1,1}$ and $G_{1,2}$ (resp. $G_{2,1}$ and $G_{2,2}$),
and we define $x_1$ (resp. $x_2$) to be the first vertex (ordered as in $A$) of the larger component $G_{1,2}$ (resp. $G_{2,2}$).
Let $y$ be the last vertex of $H_1$.
For $i,j=1,2$ let $X_{i,j}$ be the vertex set of $G_{i,j}$.
Then the adjacencies between $X$ and $Y$ are as follows.
\\
(i) the first $k-2$ vertices of $B_1$ are adjacent to $X_{2,1}\setunion\{x_2\}$ in $\G$, and to $X_{2,2}\setminus\{x_2\}$ in $\G'$;
\\
(ii) the second $k-2$ vertices of $B_1$ are adjacent to $X_{1,1}\setunion \{x_1\}$ in $\G$, and to $X_{1,2}\setminus\{x_1\}$ in $\G'$;
\\
(iii) the vertex $y$ of $B_1$ is adjacent to $X_{1,2}\setminus\{x_1\}$ in $\G$, and to $X_{1,1}\setunion\{x_1\}$ in $\G'$;
\\
(iv) the first $k-1$ vertices of $B_2$ are adjacent to $X_{1,2}\setminus\{x_1\}$ in $\G$, and to $X_{1,1}\setunion\{x_1\}$ in $\G'$;
\\
(v) the other $k-2$ vertices of $B_2$ are adjacent to $X_{2,2}\setminus\{x_2\}$ in $\G$, and to $X_{2,1}\setunion\{x_2\}$ in $\G'$.
From (i) to (v) we see that $\G$ and $\G'$ become disconnected if we delete the vertices $x_1$, $x_2$ and $y$
($\kappa(\G)=\kappa(\G')=3$).
We claim that there is no disconnecting set of edges in $\G$, which is smaller than the degree $3k-5$.
The only candidates for such a set are subsets of the edges between $X$ and $Y$.
The smallest disconnecting set of edges between $X$ and $Y$ has $3k-4$ edges and consists of the $2k-4$ edges
between $\{x_1,x_2\}$ and $H_1$, together with the $k$ edges between $y$ and $X_{1,2}$.
Therefore $\G$ has edge-connectivity $3k-5$.
However, after switching, we find a disconnecting edge set in $\G'$ consisting of the $2k-5$ edges between $\{x_1,x_2\}$ and $H_2$,
and the $k-1$ edges between $y$ and $X_{1,1}$ (indeed, we don't need the edge $\{y,x_1\}$).
Therefore the edge-connectivity of $\G'$ equals $3k-6$.
Thus we have:
\begin{thm}
For every $k\geq 6$ there exists a pair of $(3k-5)$-regular cospectral graphs, where one has edge-connectivity $3k-5$
and the other one has edge-connectivity $3k-6$.
\end{thm}
The smallest pair has degree 13, order 52 and edge-connectivities 13 and 12.
There is some variation possible in the above construction, which can lead to examples with other
degrees, orders and edge-connectivities.
For example, we can obtain a pair of 7-regular graphs of order 36 with edge connectivities 7 and 6 by the above construction with $k=4$
when we replace the component of $H$ containing $H_2$ by a graph with adjacency matrix
\[
\left[
\begin{array}{cccc}
O_{3,3} & I_3 & I_3 & I_3 \\
I_3 & J-I_3 & J-I_3 & J-I_3 \\
I_3 & J-I_3 & J-I_3 & J-I_3 \\
I_3 & J-I_3 & J-I_3 & J-I_3
\end{array}
\right].
\]
But we found no pair of cospectral regular graphs where one has edge-connectivity smaller than 6.
The cases $k=4$ and $k=6$ have been double checked by computer using the package newGRAPH~\cite{SBCS}.
\section{Final remarks}
If $\G$ and $\G'$ are regular cospectral graphs, then also the line graphs $L(\G)$ and $L(\G')$
are regular and cospectral (see for example~\cite{BH}, Section~1.4.5).
Moreover, for every graph $\G$, $\kappa(L(\G)) = \kappa'(\G)$.
So, the line graphs of the graphs described in Section~\ref{ec} give another infinite family of
cospectral pairs of regular graphs with different vertex-connectivity.
We can conclude that in general the property of being regular with a given vertex- or edge-connectivity
is not characterized by the spectrum.
However, for vertex-connectivity at most 2, and edge-connectivity at most 5 this may still be the case.
Especially interesting is the question if being regular with vertex- or edge-connectivity 1 is characterized
by the spectrum.
\\[10pt]
{\bf Acknowledgment}
I thank the referees for pushing me to be more explicit in the construction of Section~2;
it made the construction better understandable.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,954 |
Biografia
Nipote dello scrittore e critico musicale Bruno Barilli, esordì nel cinema interpretando piccoli ruoli in film musicali. Assieme al collega Paolo Bonacelli (con cui ebbe il figlio Leone) fondò la Compagnia del Porcospino con la quale, a Roma, mise in scena numerosi spettacoli teatrali di successo.
Filmografia
I ragazzi del juke-box, regia di Lucio Fulci (1959)
Tutti innamorati, regia di Giuseppe Orlandini (1959)
Urlatori alla sbarra, regia di Lucio Fulci (1960)
La commare secca, regia di Bernardo Bertolucci (1962)
I Giacobini, regia di Edmo Fenoglio - Sceneggiato televisivo (1962)
Il comandante, regia di Paolo Heusch (1963)
C'era una volta, regia di Francesco Rosi (1967)
Novecento, regia di Bernardo Bertolucci (1976)
Prosa teatrale
Un marziano a Roma di Ennio Flaiano, regia di Vittorio Gassman. Prima al Teatro Lirico di Milano il 23 novembre 1960.
Prosa televisiva Rai
Le colonne della società di Henrik Ibsen, regia di Mario Missiroli, trasmessa il 18 febbraio 1972.
Don Chisciotte di Miguel de Cervantes, regia di Maurizio Scaparro, trasmessa il 30 dicembre 1985.
Prosa radiofonica Rai
Orestiade di Eschilo, regia di Vittorio Gassman e Luciano Lucignani, trasmessa il 15 giugno 1960.
Adelchi di Alessandro Manzoni, regia di Vittorio Gassman, trasmessa il 31 gennaio 1961.
Varietà televisivi Rai
Moderato sprint, programma musicale con Fausto Papetti e Michelino, presenta Carlotta Barilli, regia di Vladi Orengo, 8 settembre 1962.
Musica in pochi, programma con Mario Pezzotta e Carlo Pes presentano Franca Aldrovandi con Carlotta Barilli, regia di Lino Procacci, 18 giugno 1963.
Note
Altri progetti
Collegamenti esterni
Attori teatrali italiani
Conduttori televisivi di Rai 2 degli anni 1960 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,859 |
\section{Introduction}
Sterile neutrinos with masses at the keV scale are a popular warm dark matter (WDM) candidate \cite{Olive:1982cb,Dodelson:1994sn,Shi:1998km,Fuller:2001sn,Dolgov:2000ew,Biermann:2006bu,Boyarsky:2006fg,Boyanovsky:2006it,Asaka:2007ls, Sierra:2008wj,Laine:2008pg,Wu:2009yr,Gelmini:2009xd,Boyarsky:2009ix,Kusenko:2009up,deVega:2009ku,deVega:2010yk,deVega:2011si, Araki:2011zg,Chen:2011ai,Merle:2011yv,Geng:2012jm}, that may potentially account for small-scale structure formation (see e.g \cite{Bode:2000gq,Zavala:2009ms, deVega:2010yk}) and possibly explain large pulsar kick velocities \cite{Kusenko:1997sp,Kusenko:2009up}. Sterile neutrino WDM can be produced non-thermally via (non)-resonant oscillations from the active neutrinos \cite{Dodelson:1994sn,Shi:1998km,Fuller:2001sn,Boyarsky:2007ay,Boyarsky:2008xj,Wu:2009yr,Kusenko:2009up, Boyarsky:2009ix,Das:2010ts,Watson:2011dw}, by decays from the inflaton \cite{Shaposhnikov:2006xi,Anisimov:2008qs}, or thermally with subsequent entropy dilution (see e.g. \cite{Lindner:2010ks,Liao:2010yx}). Typically, the parameter space spanned by the mass (hereafter $m_d$) and active-sterile mixing angle (hereafter $\theta_d$) for sterile neutrino WDM is most tightly constrained by Lyman-$\alpha$ \cite{Boyarsky:2008xj,Lindner:2010ks} and x-ray flux \cite{Boyarsky:2006ag,Boyarsky:2006rp,Boyarsky:2007ay,deVega:2011si,Watson:2011dw} bounds, along with free-streaming, Tremaine-Gunn and big-bang nucleosynthesis bounds, too (see e.g. \cite{Kusenko:2009up, Das:2010ts}). The aggregate effect of these bounds depends on the production mechanism of the sterile neutrino WDM. In particular, at present purely non-resonant production is disfavored, while windows exist for resonant production, production from inflaton decay, or from entropy-diluted thermal freeze out \cite{Boyarsky:2007ay,Boyarsky:2008xj,Boyarsky:2009ix,Kusenko:2009up}.
In this Note, we show that elementary keV \emph{Dirac} sterile neutrinos can be a natural feature of the composite neutrino scenario \cite{ArkaniHamedGrossman:1999,Okui:2004xn,GrossmanTsai:2008,McDonald:2010jm,Duerr:2011ks}, in the same way that the light fermions of the standard model (SM) can arise naturally in the extended technicolor framework \cite{Farhi:1981tc}. Briefly, the composite neutrino scenario is a class of theories in which the right-handed neutrinos are composite bound states of a confining hidden sector (CHS).
The possibility of such keV sterile neutrinos was first mentioned briefly in Ref. \cite{Grossman:2010iq}, and some of its x-ray flux bounds were investigated in \cite{Hundi:2011et}. In this Note, we present a more generalized discussion of this mechanism that is independent of the precise details of the confining sector, and then proceed to investigate the possible cosmological histories for this WDM candidate. We show certain classes of CHS's can naturally produce keV sterile neutrinos with active-sterile mixing angle in the resonant production window, and a freeze out temperature $\gtrsim$ TeV. Provided the post-inflation reheating temperature is below the TeV scale, then these keV sterile neutrinos could be WDM produced non-thermally via the usual resonant production mechanism \cite{Dodelson:1994sn,Shi:1998km,Fuller:2001sn,Boyarsky:2007ay,Boyarsky:2008xj,Wu:2009yr,Kusenko:2009up, Boyarsky:2009ix,Das:2010ts,Watson:2011dw}, or by a combination of inflaton decay and subsequent non-resonant production \cite{Shaposhnikov:2006xi,Anisimov:2008qs}.
As mentioned above, an alternative to non-thermal WDM production is ultra-relativistic thermal production followed by entropy dilution (see e.g. \cite{Lindner:2010ks}). This has the advantage of producing colder WDM than resonant production and can better evade the Lyman-$\alpha$ bounds. Usually the diluting entropy is produced by the out-of-equilibrium decay of a sufficiently long-lived heavy particle. In this Note we examine another compelling possibility: The first-order phase transition induced by the confinement of the hidden sector can also produce significant entropy if there is sufficient supercooling. This results in thermal keV WDM. We will discuss the details of this mechanism.
\section{The Composite Dirac Neutrino Model}
\subsection{Setup}
The generic theory of interest is a low-energy effective field theory below a scale $M$. Its group structure is $G_{\textrm{c}}\otimesG_{\textrm{F}}\otimesG_{\textrm{SM}}$, with $G_{\textrm{c}}$ a confining group called $\nu$-color, $G_{\textrm{SM}}$ the SM gauge groups (or a UV extension), and $G_{\textrm{F}}$ a global (or weakly gauged) hidden flavor group. The theory consists of three sectors
\begin{equation}
\chi \sim G_{\textrm{c}}\otimesG_{\textrm{F}}~,\qquad \xi \sim G_{\textrm{F}}~,\qquad q \sim G_{\textrm{SM}}\otimesG_{\textrm{F}}~,
\end{equation}
and which interact only via $M$-scale irrelevant operators. We call $\chi$ `preons' and say they belong to the CHS. Here $q$ denote the SM fields extended to also carry hidden flavor $G_{\textrm{F}}$, and we say $\xi$ comprise the `extended hidden sector' (EHS). We assume that the $\chi$ and $\xi$ are purely chiral fermions, but we emphasise that like the SM sector, the $\chi$ and $\xi$ may consist of various different irreps.
The $\nu$-color group confines at a confinement scale $\Lambda \ll M$. Necessarily $M \gg v$, the electroweak scale, so it is convenient to define two parameters
\begin{equation}
\epsilon \equiv \Lambda/M \ll 1~,\qquad \theta \equiv v/M \ll 1~.
\end{equation}
Confinement of the CHS produces preonic bound states, which we shall crudely denote as $\chi^p$: The superscript denotes the number of preons participating in the bound state. Formation of a scalar condensate $\chi^{m}$ with $\langle \chi^{m}\rangle \not= 0$ generically induces a spontaneous breaking of the hidden flavor group $G_{\textrm{F}} \to G_{\textrm{F}}^\prime\subset G_{\textrm{F}}$. This produces a new sub-$\Lambda$ effective field theory, which consists of: preonic bound states; $\xi$ and $q$ decomposed into $G_{\textrm{F}}^\prime$ irreps; and also light `hidden pions'. There are three crucial ideas:
(i) If the CHS has non-trivial $G_{\textrm{F}}^\prime$ anomalies, then anomaly matching of the CHS to its confined phase, with $\xi$ and $q$ acting as chiral spectators, implies that there are \emph{massless} fermionic bound states after confinement. The remaining bound states generically have masses $\sim \Lambda$, except for the hidden pions, which can be massless or have arbitrarily small masses, depending on the nature of the $G_{\textrm{F}}$ symmetry breaking. We assume the pion masses are sufficiently small that they make negligible contributions to the DM energy fraction.
Hereafter we shall assume $G_{\textrm{F}}^\prime = \textrm{U}(1)_{\textrm{F}}$, and that there are precisely three massless bound states all with the same $\textrm{U}(1)_{\textrm{F}}$ charge \footnote{In this case decomposition of $q$ under $G_{\textrm{F}} \to \textrm{U}(1)_{\textrm{F}}$ could result in multiple copies of SM irreps, also with the same $\textrm{U}(1)_{\textrm{F}}$ charges, which could be the source of flavor.}. For simplicity we assume the massless bound states have the same number of preons, hereafter denoted $n$, necessarily an odd integer. We shall suggestively denote these bound states as $n_R^i$, $i=1,2,3$ with $\textrm{U}(1)_{\textrm{F}}$ charge $F(n_R) = +1$. Explicit examples of preonic theories capable of producing such spectra are presented in Ref. \cite{Grossman:2010iq}. The corresponding sub-$\Lambda$ EFT that we shall consider hereafter is shown in Table \ref{tab:SLEFT}. In producing this EFT, we require that the mechanisms of $G_{\textrm{F}} \to \textrm{U}(1)_{\textrm{F}}$ breaking and electroweak symmetry breaking are independent, at least to a good approximation.
\begin{table}[h]
\begin{center}
\begin{tabular*}{0.5\textwidth}{@{\extracolsep{\fill}}c|ccccccc}
\hline
\\[-7pt]
$\quad$ & $\phi$ & $L^c_L$ & $E_R$ & $Q_L^c$ & $U_R$ & $D_R$ & $n_R$ \\[2pt]
\hline\hline
\\[-6pt]
F & $+1$ & 0 & $-1$ & 0 & $1$ & $-1$ & $+1$\\[2pt]
\hline
\end{tabular*}
\end{center}
\caption{ $\textrm{U}(1)_{\textrm{F}}$ charges assignments to the massless bound states $n_R$ and the SM fields $q = \{\phi, Q, U ,D,L,E\}$, which also have the usual SM charges (not shown). The $n_R$ are SM sterile by construction.}
\label{tab:SLEFT}
\end{table}
One can check $2Y-F = B-L$, so $\textrm{U}(1)_{\textrm{F}}$ is nonanomalous, and the electroweak symmetry breaking (EWSB) pattern is
\begin{equation}
\label{eqn:SBP}
\mbox{SU(2)}_{\rm L}\otimes \textrm{U}(1)_Y \otimes \textrm{U}(1)_{\textrm{F}} \to \mbox{U(1)}_{\rm EM}\otimes\mbox{U(1)}_{B-L}~.
\end{equation}
That is, one obtains Dirac neutrinos, with the $n_R$ acting as right-handed neutrinos. Note $\textrm{U}(1)_{\textrm{F}}$ may be gauged, but we assume its gauge coupling and kinetic mixing with the photon are sufficiently small that they can be neglected.
(ii) For the sub-$\Lambda$ EFT in Table \ref{tab:SLEFT}, there exist irrelevant operators that couple the preons of the massless $n_R$ -- i.e. the $G_{\textrm{c}}$ singlets $\chi^n$ -- to the SM singlet $\bar{L}_L\tilde{\phi}$. Such an operator is generically of form
\begin{equation}
\label{eqn:SYM}
\frac{1}{M^{3(n-1)/2}}\bar{L}_L\tilde{\phi}\chi^n \to \epsilon^{3(n-1)/2}\bar{L}_L\tilde{\phi}n_R~,
\end{equation}
after confinement. That is, this operator produces a suppressed Yukawa in the sub-$\Lambda$ EFT. Since $n_R$ are massless and there is $B-L$ symmetry (\ref{eqn:SBP}), this operator leads to light Dirac neutrino masses after EWSB, compared to the electroweak scale.
There may also be other vector-like right-handed fermionic bound states $N_R$ and $N^c_L$, with $F(N_{R,L}) = +1$ We shall again assume for simplicity they contain $n$ preons. Such bound states must form Dirac fermions with $\Lambda$ scale masses, and the $N_R$ will generically also have operators of form (\ref{eqn:SYM}). $N_{R,L}$ are therefore $\Lambda$-scale sterile Dirac neutrinos.
(iii) Under decomposition into $\textrm{U}(1)_{\textrm{F}}$ irreps, the chiral EHS fields $\xi$ may form real $\textrm{U}(1)_{\textrm{F}}$ representations and acquire masses. However, because the EHS couples only irrelevantly to the condensate vev $\langle \chi^m \rangle$ responsible for $G_{\textrm{F}} \to \textrm{U}(1)_{\textrm{F}}$, the mass terms must be suppressed. This is the same mechanism which suppresses the quark and lepton masses in Extended Technicolor theories \cite{Farhi:1981tc}. Explicitly, for a Dirac fermion $\xi_{R,L}$, such mass terms arise from operators of the form
\begin{equation}
\label{eqn:EMT}
\frac{1}{M^{(3m-2)/2}}\xi \chi^m \xi \to \Lambda\epsilon^{(3m-2)/2}\bar{\xi}_L\xi_R~,
\end{equation}
after confinement \footnote{There may also be mass cross terms involving $\xi_LN_R$, for example. However, we assume that such cross-terms, i.e involving composite and elementary states, are suppressed by the details of the UV theory above $M$. An analogous assumption must also be made for the proton decay operator $uude/M^2$.}. If also $F(\xi_{R,L}) = +1$, then there may exist irrelevant operators that couple the corresponding $G_{\textrm{c}}$ singlet $\chi^m\xi$ to $\bar{L}_L\tilde{\phi}$, noting any renormalizable coupling of $\xi$ directly to $\bar{L}_L\tilde{\phi}$ is forbidden by the $G_{\textrm{F}}$ chiral structure. That is, we could have
\begin{equation}
\label{eqn:ESM}
\frac{1}{M^{3m/2}}\bar{L}_L\tilde{\phi}\chi^m\xi \to \epsilon^{3m/2} \bar{L}_L\tilde{\phi}\xi_R~.
\end{equation}
Consequently, such a $\xi_{R,L}$ forms an \emph{elementary} sterile Dirac neutrino with naturally suppressed mass term $\sim \Lambda \epsilon^{(3m-2)/2}$ and coupling to the active sector $\sim \epsilon^{3m/2}$. In principle, there may be several species of such a Dirac neutrino, as well as other EHS fermions with $F \not= \pm 1$ that acquire Dirac or even Majorana masses of the same size.
\subsection{Spectrum}
We may classify the sub-$\Lambda$ EFT by a tuple $(n,m)$, where $n$ (odd $\ge 3$) is the number of preons in the sterile neutrino bound states, and $m$ (even $\ge 2$) is the number of preons in the symmetry breaking condensate. After EWSB, from eqs. (\ref{eqn:SYM})--(\ref{eqn:ESM}) a $(n,m)$ theory has neutrino mass term,
\begin{equation}
\Lambda \begin{pmatrix} \nu_L \\ \xi_L \\ N_L \end{pmatrix}^T \begin{pmatrix} \theta \epsilon^{\frac{3n -5}{2}} & \theta \epsilon^{\frac{3m -2}{2}} & \theta \epsilon^{\frac{3n -5}{2}}\\ 0 &\epsilon^{\frac{3m -2}{2}} & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}n_R \\ \xi_R \\ N_R\end{pmatrix}~,
\end{equation}
where $\nu_L$ is the SM active neutrino. Each entry of this mass matrix denotes the prefactor of an $\mathcal{O}(1)$ sub-block, whose dimensions depends on the number of species of each type of sterile neutrino. For example, the upper left entry must be $3 \times 3$.
For $m \le n-1$, the mass spectrum can be determined by expansions in $\epsilon$ and $\theta$. One obtains at leading order
\begin{equation}
\label{eqn:MS}
m_l \sim v\epsilon^{\frac{3(n-1)}{2}}~,\qquad m_d \sim \Lambda \epsilon^{\frac{3m-2}{2}}~, \qquad m_h \sim \Lambda~.
\end{equation}
Here the superscripts $l$, $d$ and $h$ denote `light', `dark' and `heavy'. The left-handed mass basis is, at leading order in $\epsilon$ and $\theta$,
\begin{equation}
\label{eqn:LHMB}
\begin{pmatrix} \nu^l_L \\ \nu^d_L \\ \nu^h_L \end{pmatrix}\sim \begin{pmatrix} 1 & \theta & \theta\epsilon^{\frac{3n-5}{2}} \\ \theta & 1 & \theta^2 \epsilon^{\frac{3n + 6m -9}{2}} \\ \theta\epsilon^{\frac{3n-5}{2}} & \theta^2\epsilon^{\frac{3n -5}{2}} & 1 \end{pmatrix}\begin{pmatrix} \nu_L \\ \xi_L \\ N_L \end{pmatrix}~,
\end{equation}
and the right-handed mass basis is
\begin{equation}
\label{eqn:RHMB}
\begin{pmatrix} \nu^l_R \\ \nu^d_R \\ \nu^h_R \end{pmatrix} \sim \begin{pmatrix} 1 & \theta^2\epsilon^{\frac{3(n-m-1)}{2}} & \theta^2\epsilon^{3n-5} \\ \theta^2\epsilon^{\frac{3(n-m-1)}{2}} & 1 & \theta^2 \epsilon^{\frac{3n + 3m -7}{2}} \\ \theta^2\epsilon^{3n-5} & \theta^2 \epsilon^{\frac{3n + 3m -7}{2}}& 1 \end{pmatrix}\begin{pmatrix} n_R \\ \xi_R \\ N_R \end{pmatrix}.
\end{equation}
We emphasise that eqs. (\ref{eqn:LHMB}) and (\ref{eqn:RHMB}) denote only sub-block prefactors; the entries of the sub-blocks themselves are generically $\mathcal{O}(1)$ numbers multiplied by the appropriate prefactor.
It is clear from eq. (\ref{eqn:LHMB}) that the dark-active mixing angle $\theta_d \sim \theta$. One can then rearrange eqs. (\ref{eqn:MS}) and (\ref{eqn:LHMB}) into
\begin{equation}
m_d\theta_d \sim v\bigg(\frac{m_l}{v}\bigg)^{\frac{m}{n-1}}~,\quad \frac{\Lambda}{m_d} \sim \bigg(\frac{m_l}{v}\bigg)^{\frac{2-3m}{3n-3}}~,
\end{equation}
in which the right-hand sides are fully specified by $(n,m)$ and the requirement that $m_l \sim 0.05$ eV, $v \simeq 174$ GeV. Figure \ref{fig:MTL} shows $\sin^2(2\theta_d)$ up to $\mathcal{O}(1)$ uncertainty as a function of $m_d$, with $m = n-1$. Theories with $m < n-1$ have much larger mixing angles, and are therefore ruled out by x-ray flux constraints, so we consider only $(n,n-1)$ theories henceforth. For such theories $M \sim 2\times 10^4(m_d/5~\mbox{keV})$ TeV, and we provide the corresponding $\Lambda$ and $\epsilon$ in Table \ref{tab:CE}.
\begin{figure}[t]
\begin{center}
\includegraphics{MassThetaL.pdf}
\end{center}
\caption{Mixing angle $\sin^2(2\theta_d)$ up to $\mathcal{O}(1)$ uncertainty (light gray) as a function of $m_d$, for $(n,n-1)$ theories. Also shown: Non-resonant production contours (dashed lines), labelled by the ratio of $\nu^d$ and DM energy fractions, $\Omega_d/\Omega_{\rm DM}$ \cite{Asaka:2007ls,Boyarsky:2009ix,Kusenko:2009up}; resonant total DM production contours (dash-dotted lines) for lepton asymmetries $Y_{\Delta L} = 8,~12,~16,~25 \times 10^{-6}$ (resp. top to bottom), and their corresponding Lyman-$\alpha$ lower bounds on the WDM mass (black dots) \cite{Boyarsky:2009ix}; the Lyman-$\alpha$ exclusion for thermally produced WDM with subsequent entropy dilution (hatched region, see e.g. \cite{Boyarsky:2008xj,Kusenko:2009up} and eq. (\ref{eqn:LAB}) below) assuming 100\% $\nu^d$ WDM; the x-ray flux exclusion for 100\% $\nu^d$ WDM fitted from most stringent archival data (heavy black line, see e.g. \cite{Boyarsky:2007ay,Boyarsky:2009ix}) and from the most recent observations of dwarf spheriodal galaxies \cite{Loewenstein:2012px} (heavy broken line).}
\label{fig:MTL}
\end{figure}
\begin{table}[h]
\begin{center}
\begin{tabular*}{0.5\textwidth}{@{\extracolsep{\fill}}c|ccc}
\hline
\\[-6pt]
$(n,m)$ & $\Lambda\times (5~ \mbox{keV}/m_d)$ (TeV)& $\epsilon\times(5~ \mbox{keV}/m_d)$ \\
\hline\hline
\\[-6pt]
$(3,2)$ & $1$ & $7\times10^{-5}$\\[2pt]
$(5,4)$ & $10^2$ & $8\times 10^{-3}$\\[2pt]
$(7,6)$ & $7\times 10^3$ & $9\times 10^{-2}$\\[2pt]
\hline
\end{tabular*}
\end{center}
\caption{Confinement scale $\Lambda$ and $\epsilon$ for $(n,n-1)$ theories. Such theories with $n>7$ have $\epsilon \not\ll 1$, and are not considered further.}
\label{tab:CE}
\end{table}
It is amusing to note that for the $(n,n-1)$ theories $m_d \sim 5$ keV implies $\sin^2(2\theta_d) \sim 3\times10^{-10}$, which matches the (as yet unconfirmed) \emph{Chandra} results in the Willman I dwarf galaxy \cite{Loewenstein:2009cm}.
\subsection{Dirac vs Majorana}
The keV sterile neutrinos in this Note are Dirac, in contrast with the Majorana sterile neutrinos often considered in other WDM scenarios. The WDM production mechanisms that we consider below produce dominantly symmetric DM -- the resonant production mechanism requires an asymmetry in the proper number density $(n_\nu-n_{\bar{\nu}})/n_\nu < 10^{-2}$ \cite{Fuller:2001sn,Wu:2009yr} -- so that the DM particles and antiparticles are present in the same abundances to a very good approximation. The x-ray flux bounds due to sterile neutrinos are therefore insensitive to the mass structure, since decay modes to the active neutrino and antineutrino are present in both cases: I.e, the x-ray flux is due to either $N \to \nu \gamma$ and $N\to \nu^c \gamma$ for a Majorana neutrino $N$, or $\nu^d \to \nu \gamma$ and $\bar{\nu}^d \to \bar{\nu} \gamma$ for the present scenario. Similarly, (non)-resonant production by conversion from the left-handed active neutrinos will produce the same sterile neutrino energy fraction, $\Omega_d$, regardless of the Dirac or Majorana nature of the masses. In Fig. \ref{fig:MTL} we therefore use the existing results for both the x-ray bounds and production processes, without any alteration for the Dirac mass structure.
The x-ray bounds could also be altered by exotic $\nu^d \to X\gamma $ decay channels, that might arise from $M$-scale irrelevant operators. We emphasize that the chiral and composite structure of the composite neutrino framework ensures any such operators are of sufficiently high dimension that the corresponding decay rates are negligible. For example, in the $(3,2)$ theory $\nu^d \to \gamma \nu^l$ or $\nu^d \to \gamma \Pi \nu^l$ could also arise from $\chi^3 \chi^2 F_{\mu\nu}\sigma^{\mu\nu}\xi/M^7$ which confines to $(\Lambda^5/M^7)n_R(\Lambda + \Pi) F_{\mu\nu}\sigma^{\mu\nu}\xi_L$. This respectively produces decay rates $ \sim \epsilon^{12}m_d^3/M^2$ or $ \sim \epsilon^{10}m_d^5/M^4$, that are negligible compared to the decay through mixing with the active neutrinos.
\subsection{Decoupling}
Our knowledge of the generic structure of the non-renormalizable operators permits us to consider the cosmological histories of the CHS and EHS, and therefore determine whether the $\nu^d$ sterile neutrinos can be a WDM candidate: satisfying the $(m_d,\theta_d)$ bounds is necessary but not sufficient for this. For the $(n,n-1)$ theories, we now enumerate various important processes and their freeze out temperatures, $T_{\rm fr}$. We assume the effective degrees of freedom at the TeV scale $g_* \sim 10^2$.
(i) $\bar{X}X \leftrightarrow \bar{Y}Y$, where $X,Y \in \{q,\xi,\chi \}$. These processes couple the SM, CHS and EHS. The dimension-5 operator $\phi^\dagger \phi \bar{X}X$ is heavily suppressed, since $X$ are all chiral. The leading operators are then the dimension-6
\begin{equation}
\frac{1}{M^2}\bar{X}\gamma^\mu X \bar{Y} \gamma_\mu Y~;~T_{\rm fr} \sim \bigg[\frac{g_*^{\frac{1}{2}}M^4}{M_{\rm pl}}\bigg]^{1/3} \sim~\mbox{TeV}~,
\end{equation}
and similarly for $\phi^\dagger \partial_\mu \phi\bar{X}\gamma^\mu X/M^2$. Note that the current collider constraint on the dark matter - quark interaction is insensitive to the coupling due to the large mediator mass, $M$ \cite{Rajaraman:2011uq,Fox:2011fk}.
(ii) $\bar{\xi}_R\xi_L \leftrightarrow 2\Pi$, where $\Pi$ denotes the hidden pions. This process is generated by the non-linear sigma operator
\begin{equation}
\label{eqn:XXPP}
m_d\bar{\xi}_R\xi_L e^{i\Pi/\Lambda}~;~T_{\rm fr} \sim \bigg[\frac{g_*^{\frac{1}{2}}\Lambda^4}{(m_d)^2M_{\rm pl}}\bigg] \sim~\mbox{TeV}~,
\end{equation}
for the $(3,2)$ theory, and much larger for $(5,4)$ and $(7,6)$.
(iii) $\bar{\nu}^d_L\nu_L^d \leftrightarrow \bar{q}q$. This can occur also through $W$ and $Z$ exchange, and must freeze out before BBN. The pertinent operators are
\begin{equation*}
\frac{g(\theta_{d})^2}{2c_W}\bar{\nu}^d_L\slashed{Z}\nu^d_L~,~\frac{g\theta_d}{\sqrt{2}}\bar{\nu}^d_L\slashed{W}\ell_L~;~T_{\rm fr} \!\sim \!\bigg[\frac{g_*^{\frac{1}{2}}m_W^4}{(\theta_{d})^4M_{\rm pl}}\bigg]^{1/3} \!\!\!\!\!\! \sim~\mbox{TeV}~.
\end{equation*}
(iv) $\bar{\nu}^l_L\nu^{l}_R \leftrightarrow 2\Pi$. This must also freeze out before the BBN epoch. The non-linear sigma coupling of $\nu^l_{L,R}$ to the hidden pions is suppressed by both the left and right mixing between active and sterile sectors. From eqs. (\ref{eqn:LHMB}) and (\ref{eqn:RHMB}) this leads to an extra prefactor of $(\theta_d)^3$ for the non-linear sigma operator in eq. (\ref{eqn:XXPP}), and therefore a decoupling much larger than the TeV scale.
(v) $2h \leftrightarrow 2\Pi$. This is generated by the operator $\phi^\dagger\phi (\chi^m)^\dagger \chi^m/M^{3m-2}$ which confines to the dimension-4 operator $\epsilon^{3m-2}\phi^\dagger\phi \Pi\Pi$. This becomes efficient only \emph{below}
\begin{equation}
T_{\rm fr} \sim \epsilon^{6m-4}M_{\rm pl}/ g_*^{\frac{1}{2}} \lesssim 10^{-7}~\mbox{eV}~,
\end{equation}
for $(n,n-1)$ theories, and therefore does not produce significant recoupling.
\section{Warm Dark Matter}
\subsection{Non-Thermal WDM}
The moral of the above analysis is that approximately below the TeV scale, the SM, CHS and EHS are decoupled. From Table. \ref{tab:CE}, confinement of the CHS also occurs at latest at the TeV scale. As a result, we may imagine a scenario in which the post-inflation reheating temperature $T_{\rm rh} <$ TeV. In this case, the sterile Dirac neutrinos $\nu^d$ might never be in thermal contact with the SM plasma, and therefore be produced non-thermally through the (non)-resonant production mechanism \cite{Dodelson:1994sn,Shi:1998km,Fuller:2001sn,Wu:2009yr}, forming the WDM.
As can be seen in Fig. \ref{fig:MTL}, the predicted $(m_d,\theta_d)$ values fall outside the $\Omega_d > \Omega_{\textrm{DM}}/2$ non-resonant production region, which itself is ruled out by the combination of Lyman-$\alpha$ \cite{Seljak:2006qw,Boyarsky:2008xj} and x-ray flux bounds \cite{Boyarsky:2007ay, Boyarsky:2009ix}. However the $(m_d,\theta_d)$ ranges still overlap an allowed window for full WDM resonant production if there is a sufficiently large lepton asymmetry \cite{Fuller:2001sn,Laine:2008pg,Wu:2009yr,Kusenko:2009up,Boyarsky:2009ix}. Alternatively, in this low reheat scenario, coupling of the sterile neutrinos to the inflaton -- an SM singlet -- could result in significant non-thermal WDM production from its decay \cite{Shaposhnikov:2006xi,Anisimov:2008qs}, with the remaining fraction (if any) produced by non-resonant production.
Just as for exotic x-ray decay channels, the chiral structure of the SM, CHS and EHS generically suppresses the operators that may produce non-thermal sterile neutrino WDM from SM decays. For example, from eqs. (\ref{eqn:ESM}) and (\ref{eqn:MS}) it is clear that Higgs to sterile neutrino decay rate is suppressed by $\epsilon^{3m} \sim (m_l/v)^2$ for $(n,n-1)$ theories, so there is no significant production from the Higgs decay channel. Along similar lines to the inflaton scenario, one might putatively extend the Higgs sector with a SM singlet that can decay to the EHS without such suppression (see e.g. \cite{Kusenko:2006rh,Petraki:2007gq}), however we do not make any such assumptions about the SM Higgs sector here.
One might also consider production via lepton or hadron decays such as $\tau \to e\xi\xi$ or $B \to K\xi\xi$ respectively. The chiral structure ensures such processes can only be mediated by operators of the form
\begin{equation}
\frac{\lambda_{ij}}{M^2}\bar{q}^i\gamma^\mu q^j\bar{\xi}\gamma_\mu\xi~.
\end{equation}
This type of operator necessarily produces FCNCs, too, but the large mediator scale $M$ easily evades the present bounds for quark FCNCs \cite{pdg:2012}. One finds for the dominant top decay process $\Gamma/H(m_t) \lesssim 10^{-4}$. For semi-relativistic tops in thermal equilibrium, this produces a sterile neutrino energy fraction $\Omega_d \sim 1\% ~ \Omega_{\textrm{DM}}$, so that this production channel can be neglected. Similarly, production from spin-1 bound state decays like $\rho_0 \to \xi\xi$ is negligible due to suppression of the rate by a $(\Lambda_{\textrm{qcd}}/M)^4$ factor.
\subsection{Thermal WDM}
The $(3,2)$ theory exhibits the interesting feature that the decoupling temperature of the EHS, $T_d$, the confinement temperature of the CHS, $T_c \sim \Lambda$, and decoupling of temperature the CHS, $T_\chi$, all occur at the TeV scale. In contrast to the non-thermal resonant scenario, for a $(3,2)$ theory one may plausibly consider a scenario in which all three sectors are initially in thermodynamic equilibrium, the lepton asymmetry is small, and
\begin{equation}
\label{eqn:OS}
T_d > T_c > T_\chi~.
\end{equation}
In this scenario, the EHS fermions $\xi$ freeze-out ultra-relativisitically before confinement, and there is no subsequent resonant production: from Fig. \ref{fig:MTL} we see that fractional non-resonant production at the $10\%~\Omega_{\textrm{DM}}$ level may still occur, but we shall neglect this henceforth as it is a subdominant contribution. Defining $Y \equiv n/s$ -- the ratio of the comoving number density and entropy density -- then for each \emph{Dirac} $\xi$ species
\begin{equation}
Y_{\xi} = \frac{135\zeta(3)}{2\pi^4}\frac{1}{g_{*_{\rm S}}^d}~,
\end{equation}
where $g_{*_{\rm S}}^d$ is entropic effective equilibrium number of degrees of freedom at freeze-out.
Even if only one species of $\xi$ -- the Dirac $\xi_{R,L}$ -- obtains a mass $m_d$, which we assume henceforth, such a $Y_\xi$ leads to over-closure unless $g_{*_{\rm S}}^d \sim 10^4$. This is unnaturally large since $g_{*_{\rm S}} \sim 10^2$ for the SM at this scale. However, if after freeze-out the entropy increases by a factor $\gamma$, then the frozen out species are diluted, $Y_{\xi} \to Y_\xi/\gamma$. The present-day energy fraction for the Dirac $\nu^d$, which are an admixture dominantly composed of $\xi_{R,L}$, is then
\begin{equation}
\label{eqn:PDEF}
\frac{\Omega_{d}}{\Omega_{\rm DM}} \simeq \frac{Y_\xi m_d s_0}{\rho_c\Omega_{\rm DM}} = \frac{1.1\times 10^4}{g_{*_{\rm S}}^d\gamma}\bigg(\frac{m_d}{5~\mbox{keV}}\bigg)~,
\end{equation}
in which we used $s_0 \simeq 2.89\times10^3$ cm$^{-3}$, $\rho_c \simeq 10.5 h^{2}$cm$^{-3}$keV, and $\Omega_{\rm DM} = 0.105h^{-2}$. It is clear that we need $g_{*_{\rm S}}^d\gamma \gtrsim 10^4$ for a DM candidate.
\subsection{Supercooled Confinement}
The ordering (\ref{eqn:OS}) permits us to consider the confinement of the CHS as the source of entropy that dilutes $Y_\xi$ after freeze-out. The entropy production from a confinement-induced first-order phase transition can be significant if it occurs suddenly after supercooling \cite{DeGrand:1984sc,Csorgo:1994dd}. That is, if the confinement phase transition (CPT) begins at a cooler temperature $T_i < T_c$, and the duration of the transition $\tau_c \ll 1/H(T_i)$, the Hubble time at temperature $T_i$.
Before confinement -- at temperature $T_i$ -- and after confinement -- at temperature $T_f>T_\chi$ --, we suppose that we have equilibrium plasmas. By construction
\begin{align}
g_{*_{\rm S}}(T_i) & \equiv g_{*_{\rm S}}^i = g_{*_{\rm S}}^{\rm SM} + g_{*_{\rm S}}^{\rm c} \simeq 2\times10^2~,\notag\\
g_{*_{\rm S}}(T_f) & \equiv g_{*_{\rm S}}^f \equiv g_{*_{\rm S}}^{\rm SM} + g_{*_{\rm S}}^{\rm bs} \simeq 10^2~.
\end{align}
Here $g_{*_{\rm S}}^{\rm SM}$, $g_{*_{\rm S}}^{\rm c}$ and $g_{*_{\rm S}}^{\rm bs}$ denote the effective equilibrium relativistic degrees of freedom in the SM, CHS and the bound states. By construction, for three $n_R$ we have $g_{*_{\rm S}}^{\rm bs}=2\cdot3\cdot(7/8)+N_{\Pi}$ with $N_{\Pi}$ the number of hidden pions. We have assumed $g_{*_{\rm S}}^{\rm bs} \sim 10$ and $g_{*_{\rm S}}^{\rm SM},~g_{*_{\rm S}}^{\rm c} \simeq 10^2$. Note that since the frozen out $\xi_{L,R}$ have only four degrees of freedom, then $g_{*_{\rm S}}^d \simeq g_{*_{\rm S}}^i$.
Since $T_f>T_\chi$, then such entropy production leads to reheating of \emph{both} the CHS and SM, because they only decouple later at $T_\chi$. This mutual reheating means the present DM temperature, $T^0_{d}$, compared to that of the active neutrinos, $T^0_\nu$, is just
\begin{equation}
\label{eqn:TRDM}
\frac{T^0_{d}}{T^0_\nu} = \bigg(\frac{g_{*_{\rm S}}^f}{\gammag_{*_{\rm S}}^d}\frac{g_{*_{\rm S}}^\nu}{g_{*_{\rm S}}^{\rm SM}}\bigg)^{1/3}\simeq \bigg(\frac{10.75}{1.1\times10^4(m_d/5~\mbox{keV})}\bigg)^{1/3}~,
\end{equation}
from eq. (\ref{eqn:PDEF}) and since $g_{*_{\rm S}}^{f} \simeq g_{*_{\rm S}}^{\rm SM}$. Equation (\ref{eqn:TRDM}) implies the entropy-diluted thermal WDM is red-shifted compared to the active neutrino plasma. The Lyman-$\alpha$ bounds \cite{Seljak:2006qw,Boyarsky:2008xj,Lindner:2010ks} require non-resonantly produced WDM -- at present temperature $T^0_\nu$ -- to satisfy $m_\textrm{nrp} > 10$ keV. Since the free-streaming length $\lambda_{\textrm{FS}} \propto T/m$ (see e.g. \cite{Kusenko:2009up}), this Lyman-$\alpha$ bound translates to $m_d > 10(T^0_d/T^0_\nu)$ keV. Together with eq. (\ref{eqn:TRDM}) we find that thermally produced $\nu^d$ may safely avoid the Lyman-$\alpha$ bound, provided
\begin{equation}
\label{eqn:LAB}
m_d>1.5~ \mbox{keV}~.
\end{equation}
This is the Lyman-$\alpha$ bound displayed in Fig. \ref{fig:MTL}.
Note also that the $n_R$ and hidden pion contribution to the effective number of neutrino degrees of freedom, $\delta N^{\rm eff}_\nu$, at the big-bang nucleosynthesis (BBN) epoch is
\begin{equation}
\delta N^{\rm eff}_\nu = (8/14)g_{*_{\rm S}}^{\rm bs}\big(g_{*_{\rm S}}^\nu/g_{*_{\rm S}}^{\rm SM}\big)^{4/3} \lesssim 0.26(g_{*_{\rm S}}^{\rm bs}/10)~.
\end{equation}
It is amusing to note that the right-handed neutrinos together with the hidden pions can supply sufficient effective degrees of freedom at the BBN epoch to significantly contribute to the observed $\delta N^{\rm eff}_\nu \sim 1$ excess (see e.g \cite{Komatsu:2010fb,Benson:2011ut}). In contrast, this is difficult to achieve with seesaw models, or even ad hoc Dirac neutrino models.
\subsection{Entropy Production Estimate}
The massive bound states typically have masses $x\Lambda$, with $x\gtrsim 1$, so they are non-relativistic. Their corresponding widths are generically also $\Gamma \sim\Lambda$. This leads to $\Gamma/H(T_i) \sim M_{\rm pl}\Lambda /T_i^2 \ggg 1$. In contrast, the longest-lived heavy bound state we could contemplate decays only via exchange of an $M$-scale boson, like the electroweak decay of the $\Lambda^0$ baryon of QCD. In this case, the decay rate is $\Gamma \sim \Lambda x^5 \epsilon^4$. For the $(3,2)$ theory $\epsilon \sim 10^{-4}$, so that $\Gamma/H(T_i) \gtrsim x^5 \epsilon^4 M_{\rm pl}\Lambda/T_i^2 \gg 1$. This means that even for a sudden CPT, the heavy bounds states all decay within $\tau_c$ and generically, predominantly produce hidden pions and $n_R$ with energies $\sim T_c$. It seems reasonable, then, to treat the CPT as a quasiequilibrium process, in which the non-relativistic heavy bound states have exponentially suppressed number and energy densities, while pions and $n_R$ are thermal with temperature $T_c$.
With this in mind, one can estimate the amount of entropy production by treating the CPT as a first-order phase transition in $g_{*_{\rm S}}$, as a function of $\zeta \equiv (RT)^3$. Here $R$ is the universe scale factor and $T$ the equilibrium temperature. The picture is that confinement begins at supercooled plasma temperature $T_i$, and suddenly produces the relativistic pions and $n_R$ at temperature $T_c$, so that $g_{*_{\rm S}}$ undergoes a jump at $\zeta_i = (R_iT_i)^3$ from $g_{*_{\rm S}}^i$ to
\begin{equation}
\label{eqn:GSF}
g_{*_{\rm S}}^{f\prime} = g_{*_{\rm S}}^{\rm SM} + g_{*_{\rm S}}^{\rm \rm bs}\big(T_c/T_i\big)^3~.
\end{equation}
This expression for $g_{*_{\rm S}}^{f\prime}$ follows just from the definition $g_{*_{\rm S}}(T) \equiv \sum_\alpha g_{*_{\rm S}}^\alpha(T_\alpha/T)^3$, a sum over species at different temperatures. After the phase transition, the plasma undergoes an adiabatic thermalization until $g_{*_{\rm S}} = g_{*_{\rm S}}^f$ and $T = T_f$. SM-CHS decoupling at $T_\chi$ follows thereafter. Figure \ref{fig:CH} shows this history.
\begin{figure}[t]
\begin{center}
\includegraphics{CosmoH.pdf}
\end{center}
\caption{A sketch of the thermal history. Species freeze-out ($a$-$b$) along the $S_i$ adiabat (lower dashed), is followed by the CPT ($b$-$c$), which is a first-order $g_{*_{\rm S}}$ phase transition in $\zeta$. The CPT is followed by thermalization (c-d) along the $S_f$ adiabat (upper dashed) until $g_{*_{\rm S}} = g_{*_{\rm S}}^f$ at which $T= T_f$. Once $T = T_\chi$, the CHS and SM decouple.}
\label{fig:CH}
\end{figure}
Provided $(T_c/T_i)^3 \gg g_{*_{\rm S}}^{\rm SM}/g_{*_{\rm S}}^{\rm bs} \sim 10$, the entropy production estimate from eq. (\ref{eqn:GSF}) is then
\begin{equation}
\gamma \equiv \frac{S_f}{S_i} = \frac{g_{*_{\rm S}}^{f\prime}\zeta}{g_{*_{\rm S}}^i\zeta} \simeq \frac{g_{*_{\rm S}}^{\rm \rm bs}}{g_{*_{\rm S}}^i}\bigg(\frac{T_c}{T_i}\bigg)^3~.
\end{equation}
The important feature of this na\"\i ve estimate is the $(T_c/T_i)^3$ dependence of the entropy production. A more careful treatment in Ref. \cite{DeGrand:1984sc} produces the result
\begin{equation}
\gamma \simeq \frac{1}{r}\bigg(\frac{r-1}{3}\bigg)^{3/4} \bigg(\frac{T_c}{T_i}\bigg)^3~,\qquad r \equiv \frac{g_{*_{\rm S}}^i}{g_{*_{\rm S}}^f}~.
\end{equation}
One also finds $T_f = [(r-1)/3]^{1/4}T_c$. Using this result and eq. (\ref{eqn:PDEF}), and fixing $r=2$, it follows that for $\Omega_d \le \Omega_{\rm DM}$ (i.e. $\gamma g_{*_{\rm S}}^d \ge1.1\times10^4 m^d/5~\mbox{keV}$) we require
\begin{equation}
\label{eqn:TCTI}
\frac{T_c}{T_i} \ge 6.3 \bigg(\frac{2\times10^2}{g_{*_{\rm S}}^{d}}\bigg)^{1/3}\bigg(\frac{m_d}{5~\mbox{keV}}\bigg)^{1/3}~.
\end{equation}
Note $T_f = 0.76T_c$ here, so it is plausible that $T_f > T_\chi$. By comparison to eq. (\ref{eqn:TCTI}), the QCD maximal supercooling is $T_c/T_i \simeq 1.7$ \cite{DeGrand:1984sc}. However, given that this upper bound will be sensistive e.g. to the tunneling probabilities between the metastable ($G_{\textrm{F}}$ symmetric) and stable ($G_{\textrm{F}}^\prime$ symmetric) vacua, the degree of supercooling required in this estimate is not implausible.
\section{Conclusions}
Within the composite neutrino framework, we have shown in this Note that keV sterile Dirac neutrinos can be naturally produced with mixing angles appropriate for non-thermal resonant production, provided the composite neutrinos are all comprised of $n$ preons and the scalar condensate vev has $n-1$ of them. Alternatively, for a $(3,2)$ theory, a single keV sterile Dirac neutrino species could be WDM produced by entropy-diluted ultrarelativistic freeze-out. In this latter case the entropy can be provided by a supercooled confinement-induced phase transition.
\acknowledgments
The authors thank Kfir Blum, Yuval Grossman, Roni Harnik, Bibhushan Shakya and Tomer Volansky for helpful discussions. This work is supported by the U.S. National Science Foundation through grant PHY-0757868.
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Australia Orders Review of TikTok's Data Collection
September 6, 2022 TruthUSA 0 Comments
Australia's Home Affairs Minister has ordered cybersecurity authorities to investigate social media giant TikTok's data collection security.
Clare O'Neil, who is also the cybersecurity minister, called on Australians who use TikTok to be cautious of data collection by the app.
"I'd say to Australians: if you're using TikTok, think about what data of yours might be being collected, and know that we're not always 100 percent confident of how that data's being used," O'Neil said on a TV program of Australian Broadcasting Corporation (ABC) on Sept. 5.
"We do need to take precautions in this digital age."
According to Digital 2022, a report released by Internet data research company WE ARE SOCIAL, the international version of TikTok has 7.38 million adult users in Australia, second only to Facebook, Facebook Messenger, and Instagram.
Clare Ellen O'Neil, Minister for Home Affairs and Minister for Cyber Security, at Parliament House on September 1, 2022, in Canberra, Australia. (Martin Ollman/Getty Images)
The move comes after TikTok Australia admitted in July that its employees in China had access to Australian users' data.
"Our security teams minimize the number of people who have access to data and limit it only to people who need that access in order to do their jobs." Brent Thomas, TikTok's Australian director of public policy, wrote in reply to James Paterson, the shadow minister for cybersecurity and countering foreign interference.
"We have policies and procedures that limit internal access to Australian user data by our employees, wherever they're based, based on need."
Concern About Technology in 'Authoritarian Countries'
The minister said this is more than TikTok's data collection.
"This is a much bigger issue about how, as Australians. We're going to engage with technology that's invented in authoritarian countries," O'Neill said on ABC.
TikTok is a hugely popular short-format video platform that allows users to create, share, and view 15-second videos, often featuring singing, dancing, or comedy.
Started in China as "Douyin" in September 2016, the app attracted 100 million Chinese users within one year and was relaunched as TikTok internationally in September 2017, attracting dozens of A-list celebrity users and partnerships with the NBA, NFL, and Comedy Central.
By 2020, TikTok reported nearly a billion active users worldwide—less than four years after its launch.
People walk past a logo of Bytedance, the China-based company which owns the short video app TikTok, or Douyin, at its office in Beijing, China, on July 7, 2020. (Thomas Suen/File Photo/Reuters)
However, the China-based social media has come under scrutiny because of censorship, its ownership by the Chinese company ByteDance, and the reported link to the Chinese Communist Party, which can make a direct request for access to the user data under the 2017 National Intelligence Law.
O'Neill said she had ordered the Home Affairs Department to begin a review on TikTok.
"But I would just say to Australians: This is a really hard and very new problem. There is no country in the world that has quite nailed this," she said.
"I talked to my counterparts in the United States, in Canada, in Britain, in other friendly countries, and we're all kind of trying to find our way through what is a set of very modern problems."
Shadow Cybersecurity Minister: Don't Rule out Banning TikTok
Liberal Senator James Paterson, shadow minister for cybersecurity and countering foreign interference, called on the government not to rule out banning TikTok.
Paterson said the Coalition would support "any appropriate initiatives the government proposes" after the review but suggested the government must consider all options, including a ban.
"I am concerned that prior to receiving any advice from the Department of Home Affairs, the minister has pre-emptively ruled out banning any social media apps which pose an unacceptable national security risk," Paterson told Sydney Morning Herald.
"It's possible that some of the privacy and cybersecurity risks with some of these apps can be successfully mitigated by regulation, but it's also foreseeable that tougher measures including banning are the only satisfactory solution with others."
Liberal Senator James Paterson in the Senate at Parliament House in Canberra, Australia, on Nov. 21, 2016. (AAP Image/Mick Tsikas)
Patterson said options for handling social media platforms could include strengthening data security regulations, such as requiring Australian users' data to be stored in Australia and blocking access from authoritarian countries such as PRC.
He said the government could also consider foreign interference disclosure measures, such as requiring accounts to be identified if they are associated with a foreign government.
The review will also look at other social media, including WeChat, another Chinese app popular among Chinese Australians, Sydney Morning Herald reported.
"It's not just about TikTok," O'Neil told the news outlet.
"We've got this basic problem here where we've got technology companies that are based in countries with a more authoritarian approach to the private sector, and this is a relatively new problem."
The Epoch Times reached out to Home Affairs Department to confirm if the review will include WeChat but has not received a response by publication time.
Daniel Teng contributed to this report.
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Foreship technical review leads ship battery power inland
The use of batteries as a main source of propulsion power for inland and coastal ships has passed a significant milestone, after Foreship delivered its Ship Applicability Type Technical Report to partners of the EU-funded research and innovation Current Direct project.
The naval architecture and marine engineering firm is developing specifications and requirements for the waterborne system and optimised ship design for the project whose 13 partners aim initially to demonstrate and commercialise a swappable 'e-house' containing one or more batteries to power inland vessels. A second study for the project, covering coastal ships applicability, is already under consideration.
The 'Horizon 2020' Current Direct project has estimated that the use of swappable batteries and Energy-as-Service platform could cut as much as 482,000 metric tonnes of CO2 equivalents from water transport emissions in Europe each year. With the average age of vessels operating on the Rhine standing at 50 years, the new Foreship analysis identifies a market size of up to 6,000 inland vessels as eligible for a battery-power electric solution.
Critically, Foreship is developing a framework for standardized interfaces to support the mechanical and electrical infrastructure that will be needed to enable such ambitions. In March 2022, this part of the Current Direct** project was cited as providing the basis for guidelines covering interfaces for both swappable and fixed battery containers under development by the influential Maritime Battery Forum*.
Foreship was also instrumental in the core development of the Current Direct Voyage Energy Planner Tool, which provides vessel owners with a battery swapping energy plan for inland waterway vessels based on a limited number of vessel and journey parameters.
Foreship Chief Technology Officer, Jan-Erik Räsänen, said that – with the EU considering a marine tax on residual fuel oil from 2023 – promising cases for low carbon liquid alternatives were being compromised by lack of availability, uncompetitive pricing, or both.
"Outside HFO, MGO and LNG, availability will be an issue for all other options until around 2035, whereas we have to do something now to meet decarbonisation goal," said Räsänen. "Zero-emission battery power is already viable and increasingly scalable."
The Current Direct swappable battery solution envisages one or more batteries installed in an e-house, offering approximately 3MWh of energy to support long distance zero-emission voyages. Specially engineered for waterborne transport, the lithium-ion batteries will reduce GHG emissions while eliminating shore charging points by swapping out the containerised solution in transit will control costs. Räsänen said that work on a prototype 'e-house' was underway, with a demonstrator vessel in the Port of Rotterdam expected to start operating in 2023, creating a "pathway" towards commercialisation. In parallel, liaison with waterway authorities on the regulatory and classification requirements for battery-powered vessels to operate along Europe's inland network is being handled by LR.
"The standardisation we develop in the Current Direct project will be critical for the contribution battery power can make to GHG goals on a commercial basis," he added. "As well as a standard footprint for the e-house and its foundations, we are developing the common power connections, standardised communications and fire precautions needed to deliver safe, flexible and green operations."
Foreship
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HIGH OUTPUT CAPACITY GUARANTEED PERFORMANCE MULTIPLE OUTPUT CAPACITIES SIMPLE OPERATION PRINCIPLE DESIGNED FOR EASY MAINTENANCE WIDE RANGE OF OPERATING PRESSURES 10-4 Series pumps have a 4" diameter air piston and a 1 �" stroke. Nine models are available with pressures up to 22,000 psig. When operating from 0 to rated hydraulic pressure, air consumption will be approximately 14 scfm of free air at 100 psi output. At lower air pressures and higher hydraulic pressures, air consumption will be reduced proportionately to flow rates indicated. | {
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Mobivity Selected to Power SMS Mobile Marketing Solutions for SUBWAY® Restaurant Chain in U.S.
Download as PDF June 30, 2015
PHOENIX, AZ -- (Marketwired) -- 06/30/15 -- Mobivity Holdings Corp. (OTCQB: MFON), an award-winning provider of proprietary and patented mobile marketing technologies and solutions, today announced it has been selected to power SMS mobile marketing solutions for the SUBWAY® restaurant chain. The SUBWAY® chain will use Mobivity's patented SMS text messaging technologies to operate local and national SMS marketing programs from a single platform.
The SUBWAY® brand, already leveraging Mobivity's SmartReceipt technology in a significant number of domestic locations, will engage local customers by promoting SMS text marketing programs through calls-to-action promoted in-location, including SmartReceipts, and via other forms of local and national messaging. By texting in keywords, such as "SOCALOFFERS", to "SUBWAY" (782929), SUBWAY® customers can elect to receive localized offers specific to their location. SUBWAY® plans to operate SMS programs across all of its more than 27,000 domestic locations.
"Our goal is to utilize opt-in SMS marketing programs as a personalized, direct connection to our customers and give them access to Subway with the right message, at the right time," stated Joost Zimmerman, Director of Digital Marketing at SUBWAY®. "Mobivity's unique combination of SMS text messaging and receipt-based targeting and display technology is a big driver to fulfilling our vision."
Early studies have shown that SmartReceipt driven SMS calls-to-action can improve SMS subscriber rates by as much as ten times. Additionally, SmartReceipt enables the unique ability to combine real-time sales data with SMS promotions to reveal performance insights and improve the efficacy of SMS marketing programs over time. SUBWAY® plans to leverage both technologies to achieve higher performing mobile marketing programs and drive greater value to the consumer.
"We're elated to partner with an iconic brand such as Subway and pioneer the evolution of mobile marketing together," said Dennis Becker, CEO at Mobivity. "With the expansive presence of Subway in thousands of locations worldwide, coupled with the ubiquity of mobile devices, we believe Subway is uniquely positioned to drive great value to both their customers and franchise operators, and we're very excited to be a part of that success."
SUBWAY® launched its inaugural SMS campaign with Mobivity on June 22nd, leveraging Mobivity's patent-pending SmartSMS technology, offering millions of customers in the Southern California area a free 6" sub with the purchase of a 30 oz drink.
SUBWAY® restaurant Development Agent Bob Grewal, who leads the brand's largest market in the Los Angeles area, commented, "SMS text messaging provides the broadest reach resulting in immediate results to consumers via the mobile channel. At the same time, it's imperative that consumers get the right message at the right time. Mobivity's SmartSMS and SmartReceipt technologies provide a unique combination of geographic and redemption control, with rich data tracking tools, that ensure consumers and Subway operators achieve these goals."
"We are thrilled to be working with Subway and launching our patent-pending SmartSMS technology in the LA market for our first campaign with this incredible brand," said Bill Van Epps, Executive Chairman of Mobivity and QSR Franchise Veteran. "Having worked in the restaurant industry for many years and understanding the business at the operational level, I know how important it is for operators to be able to use this highly effective mobile medium to drive sales and transactions, while still being able to control redemptions, keep offers time sensitive, location-specific, and tie everything back to clear attribution to prove ROI. Mobivity's SmartSMS solution was developed to specifically address these needs, and we are looking forward to seeing our technology answer such a highly anticipated industry need."
The campaign is being promoted via free-standing inserts, in-store marketing materials, and radio ads to market the launch of a subscription-based SMS text messaging program aimed at delighting customers while improving business performance for over 1,000 SUBWAY® locations in and around the Los Angeles metro area.
Mobivity is an award-winning provider of a suite of patented mobile marketing technologies designed to drive sales, enhance customer engagement, and reward customer loyalty for local businesses and national brands. Its solutions enable businesses across North America to drive incremental sales and profitability by quickly and effectively communicating to their existing customers to drive engagement, frequency, and loyalty. Included are SmartReceipt, compatible with nearly all POS systems, which transforms traditional retail transaction receipts into engaging "smart" receipts; an industry-leading text messaging product; and an innovative Stampt™ mobile loyalty application. Additionally, Mobivity offers a unique, high definition graphical system platform that allows its clients to enhance customer or fan experience by interacting with their mobile phones and video boards or screens in real time. Mobivity's clients include national brands such as CNN, Disney, the NFL, Sony Pictures, AT&T, Chick-fil-A, NBC Universal, Subway, Baskin Robbins, Jamba Juice, Sonic, U-Swirl, numerous professional sports teams, as well as thousands of small, local businesses across the U.S. For more information, visit www.mobivity.com.
About SUBWAY® Restaurants
Headquartered in Milford, Connecticut, and with regional offices in Amsterdam, Beirut, Brisbane, Miami and Singapore, the SUBWAY® chain was co-founded by Fred DeLuca and Dr. Peter Buck in 1965. Their partnership, which continues today, marked the beginning of a remarkable journey -- one that has made it possible for thousands of individuals to build and succeed in their own business.
For more information about the SUBWAY® chain, visit www.subway.com
Find us on Facebook: Facebook.com/subway
Follow us on Twitter: twitter.com/subwayfreshbuzz
SUBWAY® is a registered trademark of Doctor's Associates Inc.
Robert B. Prag
The Del Mar Consulting Group, Inc.
Scott Wilfong
Alex Partners, LLC
Mobivity Contact:
Dennis Becker
Mobivity | {
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{"url":"http:\/\/yourchemistrymaster.blogspot.com\/2009\/11\/2-unit-6-page-2.html","text":"### +2 UNIT 6 PAGE- 2\n\nMOLECULARITY AND MECHANISM OF REACTION\nAccording to collision theory a chemical reaction takes place due to collision between the particles of the reactants. The number of reacting species(atoms, molecules or ions) which must collide simultaneously in order to bring about the chemical reaction is called the molecularity of the reaction. The molecularity of the reaction can be 1, 2 or 3 . For example, the decomposition of ammonium nitrite is a unimolecular reaction.\nNH4NO2 * N2 + 2 H2O\nSimilarly, the reaction involving simultaneous collision between two species is a bimolecular reaction.\n2 H I (g) * H2 (g)+ I2(g)\nIn the same way, the reaction between NO and O2 is trimolecular reaction.\n2 NO + O2 * 2 NO2\nIn most of the reactions, the molecularity does not exceed three. It is because the chances of simultaneous collisions between three or more particles are rare. In general , for elementary reactions, i.e., single step reactions, the molecularity of the reaction can be obtained from balanced chemical equation. However, in many reactions molecularity of the reaction as obtained from the balanced chemical equation may come out to be more than three. For example, in the reaction between HBr and O2 , the molecularity is five as given by balanced equation.\n4 HBr + O2 * 2 H2O + 2 Br2\nSince molecularity greater than three is not possible, therefore , the above reaction does not involve the simultaneous collision of all the reacting species in a single step. In fact, such chemical reactions proceed through a sequence of steps. Each step is an elementary step and involves the simultaneous collision of two or three species only. Such chemical reactions proceed through more than one steps are termed as complex reactions.\nThe detailed description of various steps (the stepwise reactions are not necessarily observable; they are merely proposed pathways from reactants to products which logically fit into all the available experimental data.) of the chemical reaction is called mechanism of reaction. For example, the above reaction occurs by the following steps :\nHBr + O2 * HOOBr\nHOOBr + HBr * 2 HOBr\n[HOBr + HBr * H2O + Br2 ] x 2\n------------------------------------------\n4 HBr + O2 * 2 H2O + 2 Br2\nIt is clear that from the above discussions that a complex reaction occurs in two or more steps. The term molecularity of overall reaction has no significance in this case.\nRATE LAW OR RATE EQUATION\nIn complex reactions, the rate expression written on the basis of the overall balanced equation has no significance at all. The true rate expression for such complex reactions can be evaluated on the basis of the experimental data only. For example, in the reaction between NO2 and F2 to yield nitryl fluoride, NO2F,\n2 NO2 (g) + F2 (g) * 2 NO2F(g)\nthe expected rate expression is :\nRate = k [NO2]2 [F2]\nBut it is experimentally found that the rate of this reaction is proportional to the product of single concentration term of NO2 and and F2. Thus the experimental rate of the reaction is given as :\nRate = k [NO2] [F2]\nSuch a mathematical expression that gives the true rate of a reaction in terms of concentration of the reactants , which actually influence the rate ,is called rate law.\nIn general, for any hypothetical reaction:\na A + b B * c C + d D\nThe rate law expression may be written as :\nRate = k [A]m [B]n\nIn the above rate law expression, the numerical value of m and n may or may not be same as a and b . The constant k in the rate law expression is called rate constant, velocity constant or specific reaction rate. Now, if the concentration of each of the reactants involved in the reaction is unity, i.e.,\n[ A] = [B] = 1 mol L\uf02d1 , then\nrate = k x 1 x 1 = k\nThus, the rate constant of a reaction at given temperature may be defined as the rate of the reaction when the concentration of each of the reacting species is unity.\n\nCharacteristics of rate constants\ni) The value of k is different for different reactions.\nii) It is a measure of intrinsic rate of reaction. This means larger the value of k, faster will be the reaction. Similarly, small value of k reflects slower reaction.\niii) At a fixed temperature, the value of k is a constant and characteristic of a reaction.\niv) For a particular reaction, k is independent of concentration but depends on temperature.\nThe important differences between the rate of the reaction and rate constant of the reaction are given in the following TABLE.\nTABLE\nRate of reaction Rate constant of reaction\n1. It is the speed at which the reactants are converted into the products.\n2. It depends upon the concentration of reactant species at that moment.\n\n3. It decreases with the\nprogress of reaction\ngenerally. 1. It is a constant of proportionality in the rate law expression.\n\n2. It refers to the rate of reaction at specific point when concentration of every reacting species is unity.\n3. It is a constant and does\nnot depend on the\nprogress of the reaction.\n\nRATE CONTROLLING STEP\nA complex reaction proceeds through more than one steps. Out of the various steps of the reaction, the slowest step will decide the rate of overall reaction because the reaction cannot take place faster than the slowest step. The slowest step of the complex reaction is called the rate controlling step or rate determining step.\nThe rate law expression of a complex reaction gives an indication about the slowest step in the mechanism of a reaction. For example, the decomposition of nitrous oxide:\n2 N2O * 2 N2 + O2\nthe rate law is found to be :\nRate = k [N2O]\nSince the rate of the reaction depends only upon the single power of N2O, it indicates that only one molecule of N2O is involved in the slow step. Thus, with this knowledge and chemical intuition a possible mechanism of the reaction may be proposed as :\n\nThe above postulated mechanism is consistent with the rate law expression.\nNote: It must be noted that there is no way to verify the absolute validity of the proposed mechanism. All we can do is to postulate a mechanism that is consistent with experimental data. If further investigations reveal information that is not consistent with postulated mechanism, it is to be modified accordingly so as to explain all the observations.\n\nProblems\n10. For the reaction:\n2 NO2 + F2 * 2 NO2F\nThe experimental rate law :\nRate = k [NO2] [F2]\nPropose the mechanism of the reaction.\n11. For the reaction :\n2 NO + Br2 * 2 NOBr\nif proceeds by the following mechanism:\n\nSuggest a rate law consistent with the mechanism.\nORDER OF A REACTION\nIt is an important parameter for every chemical reaction. It is always determined experimentally and cannot be written from the balanced chemical equation. It refers to the number of reacting particles whose concentration terms determine the reaction rate.\nThe Order of a reaction may be defined as the sum of the powers to which concentration terms in the rate law are raised to express the observed rate of a reaction.\nFor example, for a hypothetical reaction :\na A + b B + c C * Products\nif the rate law expression for this reaction is :\nRate = k [A]m [B]n [C]p\nThen the order of the reaction is equal to ( m + n + p ). These powers or exponents i.e., m, n and p have no relation to the stoichiometric coefficients a, b and c. Order of the reaction with respect to A is m, with respect to B is n and with respect to C is p. If the sum of the powers is equal to one, the reaction is first order reaction. If the sum of the powers is two or three, the reaction is second order or third order reaction respectively. The order of the reaction can also be zero or fractional.\nSome examples of reactions of different orders\na) Reactions of first order\ni) Decomposition of Nitrogen pentoxide.\nN2O5(g) * 2 NO2(g) + (1\/2) O2(g)\nRate = k [N2O5] ; Order = 1\nii) Decomposition of Hydrogen peroxide.\nH2O2 * H2O + (1\/2)O2\nRate = k[H2O2] ; Order = 1\niii) Dehydrohalogenation of Chloroethane.\nC2H5Cl * C2H4 + HCl\nRate = k [C2H5Cl] ; Order = 1\niv) Decomposition SO2Cl2.\nSO2Cl2 * SO2 + Cl2\nRate = k [SO2Cl2] ; Order = 1\nb) Reactions of second order\ni) Decomposition of nitrogen peroxide :\n2 NO2 * 2 NO + O2\nRate = k [NO2]2 ; Order = 2\nii) Reaction between hydrogen and iodine to give HI.\nH2 + I2 * 2 HI : Rate = k [H2] [I2] ; Order = 2\nc) Reactions of Third order :\ni) The reaction between nitric oxide and oxygen .\n2 NO(g) + O2(g) * 2 NO2(g)\nRate = k [NO]2 [O2] ; Order = 3\nii) Reaction between nitric oxide and chlorine :\n2NO + Cl2 **2 NOCl Rate = k [NO]2 [Cl2] ; Order = 3\nIt may be noted that the order of a reaction may or may not be a whole number. It can have zero or fractional values also.\nd. Reactions of fractional order\ni) Combination of CO and Cl2.\nCO + Cl2 * COCl2\nRate = k [CO]2 [Cl2]1\/2 ; Order = 2.5\nii) Decomposition of carbonyl chloride.\nCOCl2 * CO + Cl2\nRate = k [COCl2]3\/2 ; Order = 3\/2\niii) Decomposition of acetaldehyde.\nCH3CHO * CH4 + CO\nRate = k [CH3CHO] 3\/2 ; Order = 3\/2\ne. Zero order reaction : A number of zero order reactions are known in which the rate of reaction is independent of the concentration of the reactants. For example, the decomposition of ammonia at the surface of metals like gold, platinum etc. is a zero order reaction.\n\nIt has been observed that the rate of reaction is independent of the concentration of ammonia, i.e.,\nRate = k [NH3]0\n\uf05c Order of the reaction = zero.\nUnits of rate constants\nThe rate is the change in concentration with time. Therefore, the rate of reaction is expressed by concentration units divided by time units. If the concentrations are expressed in mol L\uf02d1 and time in seconds, then the units for rate of a reaction are\nmol L\uf02d1 s\uf02d1 as :\n\nThe units of rate constants of different orders are different.\nFor an nth order reaction:\n\nIf concentration is expressed in mol L\uf02d1 and time in seconds,\n\nFor First order reaction , the unit of rate constant is s\uf02d1 .\nThe unit of second order rate constant is L mol \uf02d1 s\uf02d1\nThe unit of third order rate constant is L2 mol\uf02d2 s\uf02d1\nFor gaseous reactions of nth order , k has units of ;\n(atm) 1\uf02d n s\uf02d1\nINTEGRATED RATE EXPRESSION\nThis is the most common method for studying the kinetics of chemical reactions. The instantaneous rate of a reaction is given by differential rate law equations. For example, for a general reaction :\na A + b B \uf0ae Products\nThe differential form of the rate law is transformed to integrated form of rate law by simple mathematics (calculus). The integrated form is more convenient because it helps to understand the variation of concentrations of the reactants with time. The integrated rate equations for different orders can be derived.\nZero order reaction\nA reaction is said to be of zero order, if its rate is independent of the concentration of the reactants. Consider a general reaction\nA \uf0ae Product\nLet [A] be the concentration of the reactant A and ko is the rate constant for the zero order reaction. For the zero order reaction, the rate of reaction is independent of the concentration of A.\n\nThus,\n\nThis form of the rate law is known as differential rate equation. Rearranging the above equation,\n\uf02d d [A] = k0 d t\nItegrating the above equation, we get,\n\n\uf02d [A] = k0 t + I \u2026\u2026\u2026\u2026(2)\nwhere I is the constant of integration. The value of I can be calculated from the initial concentration . For example, the initial concentration of A be [A]0 at t = 0 . Then equation becomes\n\uf02d [A]o = k0 x 0 + I\nI = \uf02d [A]0\nSubstituting the value of I in (2) we get,\n\uf02d [A] = k0 t \uf02d [A]0\nk0 t = [A]0 \uf02d [A]\n\nwhere [A]o is the initial concentration of A and [A] is the concentration at time t.\nAlternately, if the initial concentration of A is \u2018a\u2019 moles per litre and x moles of reactants gets changed to products in time t. Then concentration of A left after t be (a \uf02d x).\nA \uf0ae Product\nInitial concentration : a 0\nConcentration at time t : (a \uf02d x) x\n\uf05c [A]o = a , [A] = a \uf02d x so that\n[A]o \uf02d [A] = a \uf02d (a \uf02d x) = x\nThus equation (3) becomes,\n\nThe above equation is general equation for zero order reaction. The amount of the substance reacted is proportional to time.\nFirst order reaction\nConsider the general first order reaction\nA \uf0ae Products.\nThe differential rate law equation is \uf02d d[A] = k [A]\ndt\nThe differential form of rate law is transformed to integrated form of rate law by simple calculus. This equation is used to calculate the rate constant k.\nLet us illustrate this method by applying it to a first order reaction. Consider a first order reaction. For the first order reaction, the rate of this reaction is directly proportional to the concentration of the reactant A. Thus :\nRate = \uf02d d[A] = k1 [A] \u2026\u2026\u2026.(5)\ndt\nThis form of rate law is known as differential rate equation. Rearranging the above equation :\n\uf02d d[A] = k1dt \u2026\u2026\u2026.(6)\n[A]\nIntegrating the above equation we get:\n\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\uf020\n\uf02d ln [A] = k1 t + I \u2026\u2026(7)\nwhere I is the constant of integration. The value of I can be calculated from the initial concentrations. For example, the initial concentration of A can be [A]0 at t = 0. Then Equ.(7) becomes :\n\uf02d ln [A]0 = k 1 x 0 + I\n\uf05c I = \uf02d ln [A]0\nSubstituting the value of I in Eq.(7) , we get :\n\uf02d ln [A] = k1 t \uf02d ln [A]0\nRearranging ,\nln [A]0 \uf02d ln [A] = k 1 t\nor ln [A]0 = k 1 t\n[A]\nor k 1 = 1 ln [A]0 \u2026\u2026..(8)\nt [A]\nChanging the above expression to log base 10\n( ln x = 2.303 log x), we get:\n\nThe Equ .(9 ) is integrated rate expression for first order reaction.\nThe above equation may also be written in an alternate form. Let the initial concentration of A is 'a' moles per litre. Suppose at time t , x moles of reactants get changed into products. Then the concentration of A left after time t is (a \uf02d x). Therefore, [A]0= a and [A] = (a \uf02d x ).\nThus Equ. (9) becomes :\n\nSignificance of integrated Rate equation\nThe integrated Rate equation can be used in the following ways :\ni) Order of reaction : All reactions of first order obey the following equation :\n\nStarting with the known concentration of A, the concentration of the reactant [A], at different intervals of time (t) may be measured. The value of [A] at different time intervals are substituted in the above equation. If the value of k comes out to be a constant, then the reaction is of first order.\n(ii) Calculation of Rate constant : The integrated rate equation can be used to calculate the rate constant of a reaction. If the reaction is known to be first order , then by substituting initial concentration of reactant [A]0 and concentrations at different times (t) in the above equation , the value of k can be calculated.\nThe rate constants can be calculated graphically . The Equation (9) may be written as :\n\nThis equation is comparable to the equation of a straight line ( y = m x + c ). When a graph is plotted between log[A] against corresponding time interval we get a straight line. This is shown in Fig.\n\nPlot of t versus log[A] to calculate rate\nconstant for First Order Reaction.\n\nThe slope of this line is equal to :\nSlope = \uf02d k 1\n2.303\nFrom this the rate constant can be calculated.\nk 1 = \uf02d 2.303 (slope)\nEXPERIMENTAL DETERMINATION OF ORDER OF A REACTION\nThe following methods employed to determine the rate law , the rate constant and order of a reaction.\n1. GRAPHICAL METHOD : This method is used to determine the rate law of the reaction which involves only one reactant species. The various steps involved are :\ni) The concentrations of reacting substances are determined at different intervals by suitable method.\nii) A graph is plotted between concentration and time.\niii) From the plot of concentration vs time , the instantaneous rates corresponding to different concentrations determined by drawing tangents to the curve and subsequently calculating their slopes.\niv) The rate of reaction (calculated above step 3) is plotted versus concentration [A] or (concentration)2 , [A]2 and so on.\n(a) If rate of reaction rate remains constant in the rate versus concentration graph, it means that the rate is independent of the concentration of the reactant, i.e.,\nRate = k [A]0 = k\nTherefore, the reaction is of zero order.\n(b) If a straight line is obtained in rate versus concentration graph, it means that the rate is directly proportional to the concentration of the reactant, i.e.,\nRate = k [A]\nTherefore, the reaction is of first order.\n(c) If a straight line is obtained in the rate versus (concentration)2 graph, it means that\nRate = k [A]2\nTherefore, the order of the reaction is 2.\n(d) Similarly , if we get straight line in the rate versus (concentration)3 graph, then\nRate = k [A]3\nAnd the order of the reaction is 3.\nIn general, , if we get straight line by plotting graph of the rate versus (concentration)n, where n = 1,2,3\u2026 so on, then\nRate = k [A]n\nAnd order of the reaction is n.\n\nThese graphs are shown below.\n\nIllustration\nConsider the decomposition of nitrogen pentoxide\n2 N2O5(g) \uf0ae 4 NO2(g) + O2(g)\nA convenient method to follow the reaction is to measure at different times , the increase in pressure accompanying the reaction. From the measured values it is possible to first calculate the partial pressure of N2O5 and then the concentration in moles per litre of N2O5. The molar concentrations so obtained for different times are plotted in the form of curve(Fig 5). The values of slopes(or reaction rate) progressively decreases as the concentration of N2O5 decreases. This shows that the reaction depends on its concentration. On plotting rate against concentration, i.e., the rate against [N2O5], a straight line is obtained (Fig 6).\n\nIt means that the rate of reaction is proportional to the concentration of N2O5 or :\nRate \uf061 [N2O5]\nwhich in turn means that the rate law is :\nRate = k [N2O5]\nHowever, we do not get straight line by plotting rate of a reaction against [N2O5]2. This means that the reaction is not of second order.\nThus the correct rate law for this reaction is :\n\nand the order of the reaction is one.\nCalculation of Rate constant\nThe rate constant can be calculated by substituting the values of rate and concentration of [N2O5] in the rate expression :\n\nNote : From the above discussion, we notice that the coefficient of N2O5 in the balanced equation is 2 while the exponent in the rate law is 1. Thus the order of the reaction is not same as the coefficient in the balanced chemical equation.\nGraphical method for integrated rate equation\nGraphical method can also be applied for integrated rate equation. In this method, appropriate function of concentration is plotted against time. The resulting curve will be straight line only for the case in which the appropriate integrated equations has been used. The integrated rate equations for zero, first and second order reactions are given below :\n\nAs shown below, straight lines are obtained for a plot of [A] versus t for a zero order reaction, of log [A] versus t for a first order reaction and 1\/[A] versus t for second order reaction as shown below.\n\nThe curves also help to calculate the value of k from the slope of the straight line.\n2. INITIAL RATE METHOD\nThe graphical method cannot be applied for the reactions which involve more than one reactants. The rates of such reactions can be determined by initial rate method. This method involves the following steps:\n(a) The initial rate of the reaction, i.e., the rate at the beginning of the reaction is measured. This may be taken as the rate over an initial time interval that is short enough so that concentrations of the reactants do not change appreciably from their initial values. This corresponds to slope of the tangent to the concentration versus time graph at t = 0.\n(b) The initial concentration of only one reactant is changed(keeping other concentration constant) and the rate is determined again. From this the order with respect to that particular reactant is calculated.\n(c) The procedure is repeated with respect to each reactant until the overall rate law is fully determined.\n(d) The sum total of the individual orders with respect to each reactant gives the order of the reaction.\nIllustration\nThe method is illustrated by taking hypothetical reaction :\n2 A + 2 B \uf0ae Products.\nThe experimental data for this reaction is given below:\nExperiment Concentration\n[A] [B] Rate of reaction\nI 0.01 0.01 0.005\nII 0.02 0.01 0.020\nIII 0.02 0.03 0.060\nThe general form of the rate law may be written as :\nRate = k [A]p[B]q\nThen, the expression for initial rate is :\n(Rate)0 = k [A]op[B]oq\nwhere subscript zero denotes initial values. The problem involves the determination of p and q.\nConsider the experiments I and II and substituting the values we get,\n(Rate)1 = k (0.01)p(0.01)q = 0.005 \u2026\u2026.(1)\n(Rate)2 = k (0.02)p(0.01)q = 0.020 \u2026\u2026..(2)\nDividing Eq(2) by Eq.(1):\n\n(2)p = 4 or (2)p = 22 \uf05c p = 2\nSimilarly, comparing experiments II and III,\n(Rate)2 = k ( 0.02)p(0.01)q = 0.020 \u2026\u2026\u2026(3)\n(Rate)3 = k (0.02)p (0.03)q = 0.060 \u2026\u2026\u2026(4)\nDividing equation (4) by (3), we get :\n\n3q = 3 or q = 1\nTherefore , the order with respect to A is 2 and the order with respect to B is 1.\nThus, the rate law may be written as :\nRate = k [A]2 [B]\n3. Ostwald Isolation method\nThis method was introduced by Ostwald in 1902 and is used to find the order of a reaction with respect to one reactant at a time. The total order of the reaction is equal to the sum of the orders of reaction for individual reactants. This method is based on the principle that if the concentrations of all but one reactant are taken in excess, then during the course of the reaction, the concentration of those reactants taken in excess will remain almost constant and hence variation in rate will correspond to the concentration of that reactant whose concentration is small. This process is repeated with other reactants and order with respect to each reactant is determined. The overall order will be the sum of all these orders. For example, consider the general reaction :\na A + b B + c C \uf0ae Products\nSuppose we isolate A by taking B and C in large excess and get the order of the reaction with respect to A ( say p) . Similarly, we isolate B by taking A and C in excess and isolate C by taking A and B in excess and get order with respect to B and (say q) and C (say r) respectively.\nOverall order of the reaction n = p + q + r\nProblems\n12. Three experimental runs were carried out for the reaction between Cl2 and NO.\nCl2 + 2 NO(g) \uf0ae 2 NOCl(g).\nThe following rate law are determined.\nRun Initial concentration (mol\/L)\n[Cl2] [NO] Initial rate\nmol \/ L \/ s\n1 0.020 0.010 2.4 x 10\uf02d4\n2 0.020 0.030 2.16 x 10\uf02d3\n3 0.040 0.030 4.32 x 10\uf02d3\nDetermine :\ni) The order with respect to Cl2 and NO.\nii) The rate law. (iii) The rate constant.\n13. The decomposition of N2O5 in carbon tetrachloride solution has been investigated.\nN2O5(solution) \uf0ae2 NO2(solution) + (1\/2)O2(g)\nThe reaction has been found to be first order with\nfirst order rate constant 6.2 x 10\uf02d4 s\uf02d1. Calculate the\nrate of reaction when :\n(a) [N2O5] = 1.25 mol L-1 and (b) [N2O5] = 0.25 mol L\uf02d1\n(c) What concentration of N2O5 would give a rate 2.4 x 10\uf02d3 s\uf02d1 ?\n14. With the help of the following rate expressions of the reaction,\nfind out the overall order of the reactions and order with\nrespect to each reaction ?\n2 NO(g) + O2(g) \uf0ae 2 NO2(g) ; Rate = k[NO]2[O2]\n2 N2O(g) \uf0ae 2 N2 + O2(g) ; Rate = k [N2O]\n2 NO(g) + 2 H2(g) \uf0ae N2(g)+2H2O(g) ; Rate = k[NO]2[H2]\n15. Identify the reaction order from the following rate constants ;\ni) k = 2.3 x 105 L mol\uf02d1 s\uf02d1\nii) k = 2.3 x 10\uf02d1 s\uf02d1\n16. What is the rate of reaction and order of the reaction , if the mechanism is :\n2 NO + H2 \uf0ae N2 + H2O2 (slow)\nH2O2 + H2 \uf0ae 2 H2O (fast)\n17. The experimental data for the reaction :\n2 A + B2 \uf0ae 2 AB is :\nExperiment [A] [B2] Rate\n(mol L\uf02d1 s\uf02d1)\n1 0.50 0. 50 1.6 x 10\uf02d4\n2 0.50 1.00 3.2 x 10\uf02d4\n3 1.00 1.00 3.2 x 10\uf02d4\nWrite the most probable rate equation for the reaction\n18. For the following reactions, state the order with respect to each reactant and overall order :\na) 3 NO(g) \uf0ae N2O(g) + NO2(g)\nRate = k [NO]2\nb) H2O2 + 3 I\uf02d + 2 H+ \uf0ae 2 H2O + I3\uf02d(aq)\nRate = k [H2O2][ I\uf02d]\nc) CH3CHO(g) \uf0ae CH4(g) + CO(g)\nRate = k [CH3CHO]3\/ 2\nd) CHCl3(g) + Cl2(g) \uf0ae CCl4(g) + HCl(g)\nRate = k [CHCl3][ Cl2]1\/2\nWhat are the dimensions of the rate constants in each case.\n19. For the following reaction : 2 A + B + C \uf0ae A2B + C\nthe rate law has been determined to be :\nRate = k[A] [B]2 : k = 2.0 x 10\uf02d6 mol\uf02d2L\uf032\uf020s\uf02d1\n(a) For the reaction, determine the initial rate of reaction with [A] = 0.1 mol L\uf02d1;\n[B] = 0.2 mol \uf02d1; [C] = 0.8 mol L\uf02d1.\n(b) Determine the rate after 0.04 mol L\uf02d1 of A has reacted.\n20. The rates of reaction starting with initial concentrations 2 x 10\uf02d3 M and 1 x 10\uf02d3 M are equal to 2.4 x 10\uf02d4 M s\uf02d1 and 0.60 x 10\uf02d4 M s\uf02d1 respectively. Calculate the order of the reaction with respect to the reactant and also the rate constant.\n21. The initial rate of the reaction : A + 5 B + 6 C \uf0ae 3 L + 3 M\nhas been determined by measuring the rate of disapperance of A under the following conditions :\nExpt No [A]o M [B]o M [C]o M Initial rate M min\uf02d1\n1 0.02 0.02 0.02 2.08 x 10\uf02d3\n2 0.01 0.02 0.02 1.04 x 10\uf02d3\n3 0.02 0.04 -.02 4.16 x 10\uf02d3\n4 0.02 0.02 0.04 8.32 x 10\uf02d3\nDetermine the order of the reaction with respect to each reactant and overall order of the reaction. What is the rate constant ? Calculate the initial rate of the reaction when the concentration of all reactants is 0.01 M. Calculate the initial rate of change in concentration of B and L.\n22. The pressure of a gas decomposing at the surface of a solid catalyst has been measured at different times and results are given below;\nt (s) 0 100 200 300\nP (Pa) 4.00 x 103 3.50 x 103 3.00 x 103 2.5 x 103\nDetermine the order of the reaction, its rate constant and half life period.\n23. The rate of decomposition of N2O5 in CCl4 solution has been studied at 318 K and the following results are obtained :\nt min 0 135 339 683 1680\nC M 2.08 1.91 1.68 1,35 0. 57\nFind the order of the reaction and calculate its rate constant. What is the half-life period ?\n4. Half Life Period Method\nThe order of a reaction can also be determined by another method known as half life period method. This is discussed below.\nHALF LIFE PERIOD OF A REACTION\nHalf life of a reaction is defined as the time during which the concentration of the reactants is reduced to half of the initial concentration or it is the time required for the completion of half of the reaction. It is denoted by t1\/2 . Let us calculate the half life of a First Order reaction.\nFor a First Order reaction, we know that :\n\nIn general for an nth order reaction :\n\nThus it is clear from the above expression , that the half-life period or half-change time for first order reaction does not depend upon the initial concentration of the reactants. Similarly, the time required to reduce the concentration of reactant to any fraction of initial concentration for this type of reactions is also independent of the initial concentration.\nIn addition to calculation of t1\/2, the time required to complete different fractions of a first order reaction can also be calculated.\nFor example, the time required to complete 1\/3 rd of the reaction will be given as :\n\nIn general the time required to complete any fraction of a first order reaction is given as :\n\nwhere n = 2 to 9.\n\nHalf life period of zero order and second order reactions\nFor zero order reaction,\nk t = [A]o \uf02d [A]\nFor half life period, t\u00bd ,\n\nSimilarly for second order reactions.\n\nThese are summed up below:\n\nThese results give very interesting observation that only the half lives of first order reactions are independent of the concentrations. The half-lives versus concentration [A]o or 1\/[A]o plots are shown below.\n\nDEPENDENCE ON REACTION RATE ON TEMPERATURE\nThe rate of a chemical reaction is significantly affected by a change in temperature. For most of the chemical reactions, the rate increases with rise in temperature. The rate usually becomes doubled to trebled for each 10\uf0b0 rise in temperature.\n\nTemperature coefficient\nThe increase in the rate of reaction with rise in temperature is usually expressed in terms of a quantity known as temperature coefficient. It is defined as the ratio of rate constant of a reaction at two different temperatures separated by 10\uf0b0C. The two temperatures generally taken are 35\uf0b0C (308 K) and 25\uf0b0C (298 K).\n\nwhere k toC is the rate constant for the reaction at t\uf0b0C and k t + 10OC is the rate constant for the same reaction at t + 10\uf0b0C. For most homogeneous reactions the value of temperature constant lies between 2 and 3. This means that the rate constant and hence rate of reaction increases two- to- three fold for every 10\uf0b0 rise in temperature. For example, temperature coefficient of the reaction 2 HI \uf0ae H2 + I2 is 1.7. In some cases the value of temperature coefficient is found even greater than three. For example temperature coefficient for the decomposition of nitrogen pentoxide (N2O5) is found to be equal to 3.8.\n\n## QUESTIONS\n\nAtoms and Molecules\n1.","date":"2015-05-24 06:57:36","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\": 0.8348979949951172, \"perplexity\": 1263.5257489384328}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2015-22\/segments\/1432207927844.14\/warc\/CC-MAIN-20150521113207-00148-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
Q: Триггеры SQL, AFTER INSERT Есть таблица timetable с 3мя полями (id, time, user)
CREATE TRIGGER addTime
AFTER INSERT ON timetable
FOR EACH ROW
BEGIN
SET NEW.time = now();
END
Как должно работать: Вставляем запись со значениями для колонок id, user... time вставляется после действия INSERT.
Но это не работает т.к программа ругается на AFTER INSERT
A: Ключевые слова NEW и OLD можно применять только в BEFORE-триггере. Корректно триггер может выглядеть следующим образом:
DELIMITER //
CREATE TRIGGER addTime
BEFORE INSERT ON timetable
FOR EACH ROW BEGIN
SET NEW.time = NOW();
END//
Вы можете вызывать в AFTER-триггере операторы UPDATE для обновления данных, но только не в отношении текущей таблицы - MySQL не даст изменять ее. Однако, если ваша задача изменить время только одного вставляемого столбца, нет необходимости задействовать триггер, достаточно задать условие ON UPDATE для столбца time при определении таблицы timetable
CREATE TABLE timetable (
id int(11) NOT NULL,
...
`time` datetime NOT NULL DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP
);
Такой столбец можно не заполнять - ему автоматически будет назначаться текущее время.
| {
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Calera de León är en kommunhuvudort i Spanien. Den ligger i provinsen Provincia de Badajoz och regionen Extremadura, i den sydvästra delen av landet, km sydväst om huvudstaden Madrid. Calera de León ligger meter över havet och antalet invånare är .
Terrängen runt Calera de León är kuperad åt sydväst, men åt nordost är den platt. Terrängen runt Calera de León sluttar norrut. Den högsta punkten i närheten är Cerro de Tentudia, meter över havet, km söder om Calera de León. Runt Calera de León är det glesbefolkat, med invånare per kvadratkilometer. Närmaste större samhälle är Monesterio, km öster om Calera de León. I omgivningarna runt Calera de León
Medelhavsklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är januari, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är november, med i genomsnitt mm nederbörd, och den torraste är juli, med mm nederbörd.
Kommentarer
Källor
Externa länkar
Orter i Extremadura | {
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Mint has revamped its website, which is great, but it has really messed up the URLs of my columns. Now all my previous columns come in a "page not found" section.
Rather than going back and correcting every link, I am taking down my Mint "category" in the blog.
The latest one about Chinese martial arts is available here. | {
"redpajama_set_name": "RedPajamaC4"
} | 321 |
Glasses are an essential bit of kit for most of us on almost every ride, and the Julbo Aero sunnies are a quality performing pair, although there's just a single very lightly changeable photochromic lens included in the price.
To be honest, before these glasses arrived for test I had never heard of Julbo. A little internet research tells me it has been around since 1888, based in the Jura region of France, bordering Switzerland, making eyewear initially for the mountaineering sector then branching into other sports throughout the company's history – it knows its stuff that's for sure.
The Aero glasses are a multi-sport design, for running and cycling. The frame is constructed from plastic, and our pair came with Julbo's in-house unbreakable polycarbonate Zebra Light lens, a very light photochromic lens with minimal change in the tint (other lenses are available).
Also notable are the rubber nose section wings and lower arm sections, both of which help to form-fit all nose shapes and grip the head without discomfort. Included in the package is a soft case with a hard plastic lens protector and a soft bag which can be used to clean the glasses. For prescription lens users, the glasses include an Optical Clip system that accepts the prescription lens inserts. All of Julbo's glasses are covered by a lifetime guarantee, too, which adds peace of mind.
First impressions on a local ride were good. They fitted well, were comfortable from the off, and the panoramic lens was perfectly clear and free of blemishes or distortion. The venting kept any misting at bay, too. Since then I have ridden in a variety of conditions – early morning, late afternoon into the evening, and on the Continent in the kind of super-bright sunshine we Brits (normally!) only dream of. In all those conditions they worked really well, although in the sunshine abroad I would have liked a little more tint as the glare could get a bit much.
Only having the one lens option – albeit lightly photochromic – does limit them a little, with no clear or heavily tinted extras included, as you might find with other manufacturers (though certainly not all). However, for average UK conditions they were absolutely fine.
Comfort is another area they perform well in, the rubber nose section forming to the shape of your nose without causing any pressure spots, and the likewise-equipped lower arm sections gripping the side of your head gently but firmly, again with no pressure issues. Julbo also claims some shock absorbing qualities here, but I can't say I noticed any difference (other than the grip) against my all-plastic-armed alternatives.
Although they don't seem to be anti-scratch or aqua-repellent, the lens stayed relatively clean for a long time – other glasses I have used seem to attract dust and show up sweat marks a lot but these didn't, another bonus. A friend was keen to try them while out on a ride and he was suitably impressed with them too, commenting on the clarity and comfort.
While I never had an issue in choosing to wear them for any ride, I did have a slight issue with positioning and the height of the frame. When on the hoods or bar tops they were fine, but I did find that when riding on the drops or tucked down low for a more aero position the top of the frame was in my line of sight quite considerably, leading me to crane my neck uncomfortably further up, or look over the glasses and eventually suffer with runny eyes. I suspect it's because they were designed more with mountain bikers (and runners) in mind, who would generally stay in a more upright position.
Comparing them to other pairs of glasses it looks as though the lens and frame aren't much different in height, but that the nosepiece is deeper, so they sit slightly lower on your face. Not a major issue and it probably won't affect all people as I have a slim nose, but it's worth checking if you can try them on.
In terms of value, at £135 they aren't cheap, but the construction is top notch and the company does have a great reputation for its eyewear and lenses, particularly within running and mountaineering circles. Compare them with Oakley's top end Jawbreaker Prizm, with an RRP of £175, and they don't look quite so bad. If you'd like three lenses you can swap around, Salice's cheaper 016 RW are worth considering, as are Tifosi's Pro Escalate glasses, at £4.99 more than the Julbos.
To sum up, there are many, many options for sports glasses on the market and a wide variety of prices. The Julbo Aeros sit in the mid-range, but the quality of the lens is high-end stuff. My personal preference is for a photochromic lens with a bigger tint range, but saying that these did work well in most conditions, and are definitely worth considering.
The Julbo Aero frames are a multi-sport orientated pair of glasses, with a choice of lenses for different conditions, aimed at runners and cyclists, although marketed at the mountain bike community.
Julbo says, "The Aero has been designed with the help of world-class ultrarunners and mountain bikers. Its super lightweight (32g) frame offers our new Air link - extra slim & cushion - dampening temple system, 3D fit nose piece, a wide field of vision, and snug but comfortable fit. Its sleek design optimizes ventilation and air flow and three lens options cover every light condition."
The frame is made from high impact plastic, with the patent Zebra Light lens offering a very light photochromic change, 1-3 on Julbo's scale, suitable for running and cycling where crisp vision is required with no polarising. The wide panorama lens is suspended on the frame giving maximum ventilation all around. The nose incorporates a 3D fit system, where the soft rubber wings form to your nose shape, while the arms have the same rubberised material on the lower edge to improve comfort and provide grip to avoid the glasses slipping away. For prescription lens wearers, the glasses include an Optical Clip system for the prescription insert to connect to.
The glasses are really well constructed, with high quality materials used. The lens is suspended from the frame from only three anchor points, yet never felt flimsy or liable to break.
The lens is optically clear, the subtle coloured coating providing a crystal clear enhanced view ahead whatever the weather, while the rubber nosepiece and arm sections keep them comfortable and securely in place.
They have endured many rides and trips stuck in the helmet vents and jersey pockets without incident. I haven't dropped them at any point, but I'm sure they would survive intact, although the lens isn't anti-scratch coated it is supposedly unbreakable.
I don't think many pairs of modern plastic based glasses weigh too much and these don't either, with no noticeable pressure when wearing them or additional weight noticed when stashed.
I found these to be extremely comfortable glasses even on all-day rides thanks in part to the rubber nose wings. I have had issues in the past with hard nosepieces digging in after a while and becoming irritating, not so with these Julbos. The arms aren't set too tight by default, and I have a larger rather than smaller head and didn't suffer any discomfort at all; the rubber arm sections helped here too.
Retailing at £135, they aren't cheap, but the construction is top notch and the company does have a great reputation for its eyewear and lenses, particularly within running and mountaineering circles.
The glasses performed well in all light conditions, managing well in bright sunlight and only becoming difficult to see through when it was almost dark – even at dusk my vision was clear enough to not to need to remove them. There were no any issues with them slipping down the nose when on the move and turning the head, so no having to constantly push them back into place.
The optically clear panoramic lens is a highlight, and the lack of fogging due to the way it's suspended was a big help on cooler days when stopped.
I only had one gripe with the glasses, and that was that the top of the frame seemed lower than other pairs I have used. It didn't cause an issue in normal use but when on the drops and looking up the road, I found the top bar was right in my line of sight. I suspect this is because they were designed with runners and mountain bikers in mind who would generally stay in an upright position. This won't affect everyone, but it might be worth checking a tucked pose if you can try them on.
A very well engineered pair of glasses offering great performance for sport, but less versatile than some with a wider photochromic tint range, and the lack of polarisation means they aren't as suitable for off the bike activities where glare reduction would be a benefit. | {
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Сексът (още сексуален акт, полов акт, коитус или сношение) е сексуално проникване, между две живи същества – хора или животни. В по-широкото си значение думата означава не само съвкуплението (акт с проникване, копулация, при животните – съешаване), но и всякакъв интимен физически контакт с цел удовлетворяване на сексуалното желание. За повечето нехуманоидни животни сексуалният акт се извършва единствено с репродуктивна цел чрез инсеминация и последваща фертилизация. Все пак при делфините например е известно, че двойките се събират за сексуални взаимодействия дори когато женската е извън фертилен период на репродуктивния цикъл, както и има секс между партньори от един и същи пол. За разлика от тях, в повечето случаи хората правят секс за удоволствие.
Видове секс
В зависимост от пола на партньорите сексуалният контакт може да бъде разнополов и еднополов.
Секс между повече от двама души се нарича групов секс (оргия).
Сексуален контакт с цел финансова или материална изгода, а не (само) от желание за сексуално удовлетворение, се нарича проституция.
Прякото, буквално и в някои случаи вулгарно възпроизвеждане на полов акт чрез литературни, изобразителни и аудио-визуални средства се нарича порнография, а по-изтънченото и естетическо представяне се нарича еротика. Границите между двете понятия обаче понякога са твърде размити.
Сексуални действия
Действията, насочени към стимулиране на половите органи и целящи постигане на сексуално удовлетворение, се наричат сексуални действия. Те могат да бъдат:
анално проникване
аналингус
вагинално проникване
кунилингус
мастурбация
орален секс
петинг
псевдопроникване
трибадизъм
фелацио
фингъринг
фистинг
фрот
целувки
Религия и секс
Религиозните виждания по отношение на сексуалния акт варират силно между отделните религии, между отделните секти или разклонения на една религия, както и между различните вярващи.
В християнството днес на секса се гледа балансирано – от една страна, той е Божие благословение спрямо човека (в Стария Завет Бог благославя сексуалните връзки: "Плодете се и множете се"), а от друга – следва да бъде практикуван единствено в рамките на осветения от Църквата брачен съюз.
В Римокатолическата църква сексът е позволен само ако се използва за възпроизвеждане, което означава, че в брака е разрешен само вагиналният секс. Според католическата църква сексът не е за удоволствие, а за възпроизвеждане. Заради тази причина контрацепцията е забранена за католиците.
В исляма сексът е позволен само в брака със съпруга. Например позволени са вагиналният и оралният секс. Аналният секс е забранен в исляма.
Сексът в речта
Разпространени синоними на секс в българския език, смятани за вулгаризми, са еба̀не, чукане и други. Евфемизми са любя се, правя любов, спя с и други.
Вижте също
Сексология
Любов
Оргазъм
Според сексуалната ориентация и партньорите в секса
Разнополов секс
Еднополов секс
Хетеросексуалност
Хомосексуалност
Бисексуалност
Тематизиране на секса
Еротика
Порнография
Правни
Проституция
Изнасилване
Сексуален тормоз
Други
Генофобия
Източници
Външни препратки
Международна енциклопедия върху сексуалността
Правни съвети към прокурорите във Великобритания по въпроси касаещи сексуалния акт | {
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Whataburger Secret Menu
Whataburger Secret Menu Menu With Price 2021 – Whataburger Secret Menu Menu Price Near Me
Whataburger Secret Menu Menu Price
Check Whataburger Secret Menu Latest Menu Prices For January 2022
Whataburger Secret Menu With Prices 2022
Whataburger provides larger-than-usual patties as well as buns. Many connect this fast food restaurant with Texas (it began at Corpus Christi, after all) However, the company now has more than 760 locations spread across 1o state and expanding.
Many surveys place It at the top of the list when it is rated as the top hamburgers available in the United States. You've probably tried all the classic Whataburger dishes but have you ever tried the menu items from the menu hidden away?
The Whataburger Menu That Is Secret is quite extensive, and this is due to the fact that staff is generally willing to accommodate your requests. That's basically what is on the menu in secret. They're variants that customers have requested in the past and some were requested so many times that they became viral.
Whataburger Secret Menu Items
1). Honey BBQ Chicken Strip Sandwich
It's probably the most talked-about item on the menu that is hidden. It's so well-loved that there's a campaign on Facebook by enthusiastic fans who wish to keep this as an element on the menu.
If you encounter the Whataburger employee who isn't sure what this is, explain it as a standard chicken strip sandwich, but it's made equipped with BBQ sauce.
The reason this is part of the secret menu unique is it isn't possible to be certain that the restaurant has BBQ sauce available. If they don't have BBQ sauce, you could ask for the seasonal ingredients they have in stock instead.
2). The Whataburger Hulk
If Honey BBQ Chicken Strip Sandwich is an opponent in terms of the market, it's this drink. There is also a Facebook fan page for it. The green color of the product explains its name. It can turn your tongue green, too.
Making the Hulk does not require anything that is exclusive to Whataburger. The thing that makes it what it is Whataburger Hulk is that just Whataburger people are willing to create this for the customers of their company.
It's made by filling a glass of ice around 1/8 - 1/4 full of Powerade. After that, you make sure to add the Vault soda till the glass is completely filled.
3). Grilled Cheese
Sometimes, you and your buddies are at a restaurant and nobody needs a burger. In that situation, you can enjoy the toast of two pieces stuffed with melty, gooey American cheese. It's that easy and easy, it's really inexpensive. However, the taste!
4). Double-Double
The chain In-N-Out is the only restaurant to trademark a Double-Double name however it doesn't mean that Whataburger cannot offer the Double-Double version it has created.
This is especially relevant when it's listed on the menu that is secret. You can ask for it by name. If employees don't know about it, you can request two patties of beef, layers of cheese over the patties, as well as any vegetables you'd like.
5). Veggie Burger
If the thought to eat "all the veggies you want" is what gets you excited Then you should take a look at this veggie burger that is on the menu that is secret. Whereas other places remove the beef patties, Whataburger replaces the beef Patty with potatoes hashed brown potato patty.
6). Chicken and Pancakes
It's the Whataburger's take on waffles and chicken, but it's not something you can request in one item.
It is important to choose the best moment to order it and this is during breakfast time between 11 pm and 11 am. At this time you can also order pancakes at a cost. Also, you can purchase the biscuit sandwich, which includes chicken strips.
You can either skip the biscuit or take it out later and you'll have chicken served with pancakes. It's all you need with only an entree and the option of a la carte items!
7). Ranchero BOB
The most popular is Breakfast on a Bun with either sausage or bacon. The Ranchero version contains jalapenos inside the scrambled eggs, and they add green or red (whichever you'd like) Picante Sauce.
8). Whataburger with Whatever
This is the hidden menu item that truly lengthens this list if you're looking to be precise about the names of each item.
It's because this isn't an individual burger however, it is a category for burgers. It's the Whataburger that comes with any combination of usual toppings (like mustard, ketchup cheese tomato, lettuce onions as well as pickles.)
Since Whataburgers generally are very accommodating to requests, they allow you to choose your own burger.
Excellent taste, an extensive menu that is secret, and friendly staff. It's not surprising that Whataburger is fast being regarded as one of America's most-loved fast food restaurants!
What to order:
Whataburger and two Grilled kinds of cheese. When you receive your order, take the liberty of substituting the buns with a sandwich grilled cheese.
The menu for Whataburger is gorgeous and delicious, with no modifications, However, ordering a secret menu item could take things to a whole new level. When you next visit Whataburger, you should explore their menu that is secret. You might discover your new favorite food!
Whataburger Secret Menu, The details of the price list is a general overview of the menu that is available at the restaurant. So if you ask for Italian food at a Mexican restaurant might get wrong. So its good have a look at the menu and price list before ordering the food and is it available or not at the restaurant can be known. Thank u for visiting, you may share this with your friends and families too, even you may suggest to others about the specialties of the restaurant. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,737 |
This is something that has come up time and time again now, and with Amazing Radio going online-only with nothing but a single Shoutcast server streaming one format at one bitrate, now seemed like a good time to write about online streaming of radio stations.
Let's start by briefly looking at real broadcast operations – on FM and AM we try and maximise coverage (within our license), maximise compatibility, and of course we want to add as much value as we can with metadata like RDS (and now things like RadioDNS). We're trying to reach as many people as possible, with as little fuss as possible, and trying to give people the best possible service. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,000 |
\section{Introduction}
The current ``concordance'' cosmological model assumes a flat $\Lambda$CDM
universe composed of baryons, cold dark matter, and ``dark
energy'' that accelerates the expansion of the universe \citep{Spergel:03}.
The location of the first peak in the angular power spectrum of the
cosmic microwave background (CMB) radiation determines the
angular-size distance to the surface of last scattering. This
distance depends on the amount of dark energy and the geometry
and current expansion rate of the universe, \ifmmode {H_0} \else {$H_0$} \fi. If one does
not assume that the universe is flat, the CMB data alone are consistent
with a wide range of values of \ifmmode {H_0} \else {$H_0$} \fi. Thus, independent measurements
of \ifmmode {H_0} \else {$H_0$} \fi\ are needed to justify the flatness assumption and to determine
whether dark energy is the cosmological constant, $\Lambda$, of General
Relativity, a variable ``quintessence" \citep{Wetterich:88,Ratra:88},
or possibly something else. \citet{Hu:05} concludes that the most
important single complement to CMB data would be a precise
(e.g.~\ $\sim1$\% uncertainty) measurement of \ifmmode {H_0} \else {$H_0$} \fi.
The current ``best value'' for the Hubble Constant,
$\ifmmode {H_0} \else {$H_0$} \fi = 72 \pm 7~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi~{\rm Mpc}^{-1}$ from the HST
Key Project \citep{Freedman:01}, is based on luminosity distance
measurements to extragalactic Cepheid variables treated as
``standard candles.'' The 10\% uncertainty in \ifmmode {H_0} \else {$H_0$} \fi\ is dominated by
systematic errors that cannot easily be reduced by observations of
more galaxies.
Very Long Baseline Array (VLBA) observations of the H$_2$O\ megamaser
in the nearby Seyfert 2 galaxy NGC~4258\ have provided an accurate,
angular-diameter distance to the galaxy \citep{Herrnstein:99},
bypassing the problems of standard candles. The H$_2$O\ masers in NGC~4258\
arise in a thin (annular) disk viewed nearly edge-on \citep{Greenhill:95a}
and appear at galactocentric radii $R \approx 0.14~{\rm to}~0.28$~pc.
Maser lines near the systemic velocity of the galaxy come from gas moving
across the sky on the near side of the disk, and ``high-velocity lines,''
with relative velocities of up to $V \approx \pm1100$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi, come from gas
moving along the line of sight at the disk tangent points.
The high-velocity lines display a Keplerian rotation curve, implying a
central mass of $\approx4 \times 10^7$~\ifmmode {M_\odot} \else {$M_\odot$} \fi, presumably in the
form of a supermassive black hole (SMBH) \citep{Miyoshi:95}.
For NGC~4258, the velocities of individual systemic features are
observed to increase by $\approx 9~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi~{\rm yr}^{-1}$
\citep{Haschick:94,Greenhill:95b},
allowing a direct measurement of the centripetal acceleration
($a = V^2/R$) of clouds moving across our line of sight near the
nucleus \citep{Watson:94}.
Conceptually, the angular-diameter distance, $D_\theta$, to NGC~4258\
can be determined geometrically by dividing the {\it linear} radius of masers,
measured from Doppler shifts and accelerations ($R \approx V^2/a$),
by their {\it angular} radius, measured from a Very Long Baseline
Interferometric (VLBI) image ($\theta_R$).
Maser proper motions can be also be used to measure distance, but
generally yield less accurate distances than using accelerations.
Observations with the
VLBA of the H$_2$O\ masers in NGC~4258\ have been carefully modeled,
yielding the most accurate distance ($D_\theta = 7.2 \pm 0.5$ Mpc) to date
for a galaxy \citep{Herrnstein:99}.
Unfortunately, NGC~4258\ is too close to determine \ifmmode {H_0} \else {$H_0$} \fi\ directly
(i.e.~ by dividing its recessional velocity of 475~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ by its distance),
since the galaxy's deviation from the Hubble-flow could be a significant
fraction of its recessional velocity. Instead, the measured distance to
NGC~4258\ has been used to anchor the zero point of the Cepheid
period-luminosity relation \citep{Newman:01,Macri:06,Argon:07,Humphreys:08}.
However, galaxies with edge-on, disk-like H$_2$O\ masers, similar to
those in NGC~4258, that are distant enough to be in the Hubble flow
($D > 30$~ Mpc) could be used to measure \ifmmode {H_0} \else {$H_0$} \fi\ directly \citep{Greenhill:04}.
Surveys of galaxies for nuclear H$_2$O\ masers have been quite successful
and have identified more than 100 extragalactic nuclear H$_2$O\ masers
\citep{Claussen:86,Braatz:96,Greenhill:02,Henkel:02,Greenhill:03,
Braatz:04,Kondratko:06a,Kondratko:06b,Braatz:08a}.
In order to coordinate
efforts to find and image new sources of nuclear H$_2$O\ masers, we formed
a team of scientists active in this area of research
from the Harvard-Smithsonian Center for Astrophysics, the National Radio
Astronomy Observatory (NRAO), and the Max-Planck-Institut f\"ur
Radioastronomie (MPIfR). This effort, called the Megamaser Cosmology
Project (MCP), is aimed at measuring \ifmmode {H_0} \else {$H_0$} \fi\ directly, with $\approx 3$\%
accuracy, using a combination of VLBI imaging and single-dish monitoring
of nuclear H$_2$O\ masers toward $\sim10$ galaxies.
Recently \citet{Braatz:08a} discovered a relatively
strong H$_2$O\ maser ($S_\nu \approx 0.1$~Jy) toward the Seyfert 2
nucleus of UGC~3789. The H$_2$O\ maser spectrum has the characteristics of an
edge-on disk similar to NGC~4258. The UGC~3789\ masers span $\approx1500$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\
in Doppler shift and the systemic masers were observed to accelerate
by up to 8.1~\kmsperyr, suggesting an origin in a sub-pc
disk about a $\sim10^7$~\ifmmode {M_\odot} \else {$M_\odot$} \fi\ black hole.
In this paper, and in \citet{Braatz:08b} (hereafter Paper II), we report
results leading to the first MCP measurement of \ifmmode {H_0} \else {$H_0$} \fi.
Sensitive VLBI observations and
images of the nuclear H$_2$O\ masers toward UGC~3789\ are presented in this
paper. In Paper II, we present monitoring observations with
large single-dish telescopes,
which yield accelerations of H$_2$O\ masers.
The combination of the VLBI imaging and the single-dish acceleration data
may yield a measurement of \ifmmode {H_0} \else {$H_0$} \fi\ with an accuracy comparable to that of
the Hubble Key Project.
\section{Observations, Calibration, and Imaging}
We observed UGC~3789\ on 2006 December 10 for a total of 12 hours, with
the 10 NRAO
\footnote{The National Radio Astronomy Observatory is operated by
Associated Universities, Inc., under a cooperative agreement with
the National Science Foundation.}
VLBA antennas (under program BB227A), augmented by the Green Bank
Telescope (GBT) and the Effelsberg 100-m telescope
\footnote{The Effelsberg 100-m telescope is a facility of the
Max-Planck-Institut f\"ur Radioastronomie}.
The coordinates of the sources observed are listed in
Table~\ref{table:positions}.
We alternated between two observing modes: (1) a 60-min block
of continuous tracking of UGC~3789\ (self-calibration mode) and (2)
a 45-min block of rapid switching between UGC~3789\ and a nearby compact
continuum source J0728+5907 (phase-referencing mode).
The phase-referencing blocks were a ``back-up'' in the event that the
UGC~3789\ maser signal was not strong enough for self-calibration.
Both observing modes were successful. However, since the
self-calibration mode produced a much higher on-source duty cycle
and better phase calibration than the phase-referencing mode,
we only report results from the total of $\approx5$~hr
of self-calibration mode observations.
\begin{deluxetable}{lll}
\tablecolumns{3} \tablewidth{0pc}
\tablecaption{Source Positions}
\tablehead {
\colhead{Source} & \colhead{R.A. (J2000)} & \colhead{Dec. (J2000)}
\\
\colhead{} & \colhead{(h~~m~~s)} & \colhead{(d~~'~~'')}
}
\startdata
UGC~3789\ ........... &07 19 30.9566 &59 21 18.330 \\
J0728+5907 .........&07 28 47.2170 &59 07 34.128 \\
J0753+5352 .........&07 53 01.3846 &53 52 59.637 \\
\enddata
\tablecomments {Positions used for data correlation.
Imaging the UGC~3789\ maser spot at $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi=2689$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi, by phase-referencing
to J0728+5907, we found the maser spot
offset by ($-1,-15$)~mas (east,north) from the correlation position.}
\label{table:positions}
\end{deluxetable}
With a maximum recording rate of 512~Mbits~s$^{-1}$, we could
cover the entire range of detectable UGC~3789\ H$_2$O\ maser emission,
but not with dual-polarization for all frequency bands.
We centered 16-MHz bands at LSR velocities (optical definition)
as follows: left circularly polarized (LCP) bands at
3880, 3710, 3265, 2670 and 2500~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi and
right circularly polarized (RCP) at 3880, 3265 and 2670~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi.
The signals were sampled at the Nyquist rate (32~Mbits~s$^{-1}$) and
with 2 bits per sample.
We placed ``geodetic'' blocks at the start and end of our observations,
in order to solve for atmospheric and clock delay residuals for each
antenna \citep{RB04}. In these
blocks we rapidly cycled among 14~compact radio sources that
spanned a wide range of zenith angles at all antennas.
These data were taken in left circular polarization
with eight 16-MHz bands that spanned 492 MHz of bandwidth between
22.00 and 22.49 GHz; the bands were spaced in a
``minimum redundancy'' manner to sample,
as uniformly as possible, all frequency differences. The data were
correlated, corrected for ionospheric delays using total electron
content measurements \citep{walker-chatterjee:00}, and residual
multi-band delays and fringe rates were determined for all sources.
The multi-band delays and fringe rates were modeled as owing
to a vertical atmospheric delay and delay-rate, as well as a clock
offset and clock drift rate, at each antenna.
Using a least-squares fitting program, we estimated zenith atmospheric
path-delays and clock errors with accuracies typically $\approx0.5$~cm
and $\approx0.03$~nsec, respectively.
We observed the strong continuum source, J0753+5352,
hourly in order to monitor delay and electronic phase differences
among and across the IF bands. Generally, variations of phase across
the VLBA bandpasses are small ($<5^\circ$) across the central 90\% of
the band and thus we needed no bandpass corrections. We tested the
effect of bandpass corrections, using the J0753+5352 data,
and found position differences of $\approx0.002$~mas for maser
features mid-way between the band center and band edge.
The raw data recorded at each antenna were
cross-correlated with an integration time of 1.05~sec at
the VLBA correlation facility in Socorro, NM. For this short
integration time we had to correlate the data in two passes in
order to achieve sufficient spectral resolution (128 spectral
channels for each IF band) without exceeding the maximum correlator
output rate. Before calibration, the two correlation data sets
were ``glued'' together.
We calibrated the data using the NRAO Astronomical Image
Processing System (AIPS).
First, we corrected interferometer delays and phases for
the effects of diurnal feed rotation (parallactic angle) and
for small errors in the values of the Earth's orientation parameters
used at the time of correlation. By analyzing the data taken in
phase-referencing mode, we determined that the strong maser
feature at $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi=2689$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi, which we later used as the phase
reference for the self-calibration mode data, was offset
from the position of UGC~3789\ used in the VLBA correlator
by ($-1,-15$)~mas toward (east,north), respectively, relative
to J0728+5907.
Since the VLBA correlator model includes no ionospheric delays, we used
global total electron content models to remove ionospheric effects.
We then corrected the data for residual zenith atmospheric
delays and clock drifts, as determined from the geodetic block data.
While we obtained good atmospheric/clock corrections for most
antennas, insufficient data were obtained for the Effelsberg (EB) and
Mauna Kea (MK) antennas for this task. Thus, we later used
the data from the hourly observations of J0753+5352 to determine
final delay corrections for the UGC~3789\ data.
We corrected the interferometer visibility amplitudes for
the few percent effects of biases in the threshold levels of the data
samplers at each antenna. We also entered system temperature and
antenna gain curve information into calibration tables.
These tables were used later to convert correlation coefficients
to flux densities.
Next, we performed a ``manual phase-calibration'' to remove
delay and phase differences among all bands. This was
accomplished with data from one scan on a strong calibrator, 4C~39.25.
We did not shift the frequency axes of the maser interferometer spectra
to compensate for the Doppler shift changes during the $\pm5$~hr
UGC~3789\ observing track, as these effects were less than our velocity
resolution of 1.7~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi.
The final calibration involved selecting a maser feature
as the interferometer phase-reference. The strongest maser
feature in the spectrum peaked at $\approx0.07$~Jy and was fairly
broad. We found that using 5 channels spanning an LSR velocity
range of 2685 to 2692~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ (i.e.~\ channels 52 to 56 from the blue-shifted
high velocity band centered at $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi=2670$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi), adding
together both polarizations, and fitting fringes over a 1~min period
gave optimum results. The St. Croix (SC) antenna failed to produce
phase-reference solutions and data from that antenna were discarded.
For most antennas at most times the phases could be easily interpolated
between solutions. However, when the differences
between adjacent reference phases exceeded $60^\circ$,
the data between those times were discarded. This editing
was done on baseline (not antenna) data, since correlated phases
between antennas do not affect interferometer coherence.
After calibration, we Fourier transformed the gridded ({\it u,v})-data\
to make images of the maser emission in all spectral
channels for each of the five IF bands. The point-source response
function had FWHM of $0.35\times0.22$~mas elongated along a position
angle of $-17^\circ$ east of north. The images were
deconvolved with the point-source response using the CLEAN algorithm
and restored with a circular Gaussian beam with a 0.30~mas FWHM.
All images appeared to contain single, point-like maser spots.
We then fitted each spectral channel image with an elliptical Gaussian
brightness distribution in order to obtain positions and flux densities.
\section {Results \& Discussion}
Channel maps typically had rms noise levels of $\approx0.9$~mJy for the
dual-polarized IF bands and $\approx1.2$~mJy for the single-polarization
IF bands. The flux densities from the Gaussian fits for all spectral
channels in all IF bands were used to generate the interferometer spectrum
shown in Fig.~\ref{fig:spectrum}. When little signal was detected in a
spectral channel, as evidenced by a failed fit or a spot size greater
than 1~mas, we assigned that channel zero flux density.
\begin{figure}
\epsscale{0.9}
\plotone{f1.ps}
\caption{Interferometer spectrum of the 22~GHz H$_2$O\ masers toward
UGC~3789\ constructed from VLBI data using the VLBA, the GBT and the
Effelsberg antennas. The systemic velocity of the galaxy of
$3325\pm24$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi, as determined from HI observations,
is within the systemic velocity components shown in {\it green} colors.
High velocity components, shifted by up to $800$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ from the
systemic velocity, are shown in {\it blue} and {\it red} colors.
Broad-band spectra taken with the GBT before the VLBI observations
showed almost no detectable maser
features outside our observing bands, indicated by horizontal
lines below the spectrum.
\label{fig:spectrum}
}
\end{figure}
The flux densities and positions determined by Gaussian fitting each
spectral channel image are reported in Tables~\ref{table:reds},
\ref{table:systemics} and \ref{table:blues} for maser spots stronger
than 10~mJy. The positions of these spots are plotted in
Fig.~\ref{fig:spots}.
The nearly linear arrangement of the maser spots on the sky is striking.
The red- and blue-shifted high-velocity spots straddle the
systemic emission complex. This spatial-velocity arrangement
is characteristic of a nearly edge-on disk, as is well documented
for NGC~4258\ \citep{Herrnstein:05}.
\begin{figure}
\epsscale{0.9}
\plotone{f2.ps}
\caption{Map of the relative positions of individual maser spots toward UGC~3789.
High velocity blue-shifted ({\it blue}) and red-shifted ({\it red}) masers
straddle the systemic masers ({\it green} and expanded view {\it inset})
and the linear
arrangement of spots suggests that we are viewing a nearly edge-on
rotating disk, similar to that seen in NGC~4258. The $\approx2$~mas extent
of the maser spots in UGC~3789\ is approximately seven times smaller than for
NGC~4258, which is consistent with UGC~3789\ being at approximately seven times
greater distance. Formal fitting
uncertainties are given in Tables~\ref{table:reds}, \ref{table:systemics}
and \ref{table:blues} and are typically $<0.010$~mas.
\label{fig:spots}
}
\end{figure}
We calculated the position along the spot distribution (i.e.~ an impact
parameter along position angle of $41^\circ$ east of north)
and show a position-velocity plot in Fig.~\ref{fig:pv}.
The high-velocity masers display a Keplerian velocity ($V\propto1/\sqrt{R}$)
versus impact parameter (or radius), suggesting that the gravitational
potential is dominated by a SMBH. The Keplerian velocity pattern is
centered at $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi \approx 3265$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. This is slightly offset
from the central velocity of HI emission from the galaxy at
$\ifmmode {V_{\rm Helio}} \else {$V_{\rm Helio}$} \fi \approx 3325\pm24$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ \citep{Theureau:98}.
(Note: $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi - V_{\rm Helio} = 0.3$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ for UGC~3789.)
Correcting the maser velocity to the CMB reference frame (i.e.~
$V_{\rm CMB} \approx \ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi + 60~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi$), yields a recessional velocity
of 3325~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. Thus, for $\ifmmode {H_0} \else {$H_0$} \fi=72$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi~Mpc$^{-1}$,
UGC~3789's distance would be expected to be $\approx46$ Mpc.
\begin{figure}
\epsscale{0.9}
\plotone{f3.ps}
\caption{Position-Velocity plot of the maser spots toward UGC~3789.
The high velocity blue-shifted ({\it blue}) and red-shifted
({\it red}) spots display a Keplerian $1/\sqrt{R}$ rotation
curve, indicated by the {\it curved dotted lines}.
The systemic masers ({\it green}) are consistent with projected positions
and velocities for gas in Keplerian orbit at $R\approx0.43$~mas,
indicated by the {\it straight dotted line},
but small deviations from a linear distribution are apparent.
Impact parameter is defined as distance along a position angle of
$41^\circ$ east of north from an (east,north) offset of ($-0.4,-0.5$)~mas;
the plus sign ($+$) at ($-0.03$ mas, $3265$ \ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi) indicates the assumed
center of the SMBH.
\label{fig:pv}
}
\end{figure}
The detected blue-shifted high velocity masers sample disk radii
between 0.35 and 0.70~mas (0.08 to 0.16~pc) and achieve
rotation speeds as high as 792~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi, with respect to a central systemic
velocity of 3265~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. The detected red-shifted masers sample radii of
0.50 to 1.33~mas (0.11 to 0.30~pc) and achieve rotation speeds up to
647~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. Also shown by the straight dotted lines in Fig.~\ref{fig:pv}
is the position-velocity distribution expected for systemic maser spots
that lie at a radius of 0.43~mas from the central mass, whose assumed
location is indicated by the plus-sign ($+$) in the figure.
These spatial-kinematic parameters are comparable to those of the
H$_2$O\ masers in NGC~4258, which sample radii of $\approx0.14$ to $0.28$~pc
and achieve rotation speeds of $\approx1100$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. The moderately lower
rotation speed at a slightly smaller radius suggests that the SMBH
at the center of UGC~3789\ is less massive than the $3.9\times10^7$~\ifmmode {M_\odot} \else {$M_\odot$} \fi\
SMBH in NGC~4258. At a distance of 46~Mpc, the high-velocity
data for UGC~3789\ can be well fit by gas in circular orbit about a central
mass of $1.1\times10^7$~\ifmmode {M_\odot} \else {$M_\odot$} \fi, as shown by the blue and red dotted lines in
Fig.~\ref{fig:pv}.
As can be seen in Fig.~\ref{fig:spots}, the systemic features
lie between the high velocity features but are distributed along
a position angle of roughly $10^\circ$ (east of north). Thus, they are
misaligned by approximately $30^\circ$ with respect to the $41^\circ$
position angle of the disk, obtained by drawing a line through
the high velocity masers. This suggests that the UGC~3789\ disk may be
slightly inclined and warped and/or that the systemic masers
are not all at the same radius. (Note that the NGC~4258\ disk is
both inclined by $8^\circ$ with respect to our line of sight and
warped \citep{Herrnstein:05}.) Deviation from a perfectly flat,
edge-on disk for UGC~3789\ can also be seen in the position-velocity
plot (Fig.~\ref{fig:pv}) as a slight bending of the systemic maser
spots and in the variation in accelerations seen by \citet{Braatz:08a}.
Modeling of the disk will need to accommodate these complications.
We searched for continuum emission from the vicinity of the SMBH
(i.e.~\ near the position of the systemic velocity masers) by summing
channels 5 through 120 of the (red shifted) dual-circularly polarized
band centered at $\ifmmode {V_{\rm LSR}} \else {$V_{\rm LSR}$} \fi=3380$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi. We maximized the detection
sensitivity by natural weighting of the data when imaging.
The masers were detected at the expected position, offset by
$\approx1$~mas from the position of the SMBH, but no continuum
emission was detected at a $2\sigma$ limit of $<0.14$~mJy.
\section {Conclusions}
The discovery of H$_2$O\ masers emanating from a sub-pc disk
in the Seyfert 2 galaxy NGC~4258\ more than two decades
ago has led to detailed imaging of an AGN accretion disk.
Geometric modeling of the Keplerian orbits of the masers
yielded the most accurate distance to any galaxy, allowing
recalibration of the extragalactic distance scale. Now,
the recent discovery by \citet{Braatz:08a} of H$_2$O\ masers in
UGC~3789\ offers the opportunity to extend this technique to a
galaxy seven times more distant.
In this paper we presented VLBI images of the UGC~3789\ H$_2$O\ masers,
which showed that these masers are remarkably similar to those in NGC~4258.
In both sources, the spatial distribution is nearly linear, with high
velocity masers on both sides (both spatially and spectrally) of
systemic velocity masers. The masers trace gas in Keplerian orbits
with rotation speeds of $\sim1000$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ at radii of $\sim0.1$~pc,
presumably moving under the influence of a $\sim10^7$~\ifmmode {M_\odot} \else {$M_\odot$} \fi\ SMBH.
UGC~3789\ has a recessional velocity of $\approx 3325$~\ifmmode {{\rm km~s}^{-1}} \else {km~s$^{-1}$} \fi\ and is well
into the Hubble flow. The VLBI results presented in this paper will
be followed by detailed spectral monitoring data and disk modeling
in Paper II to determine the distance to UGC~3789. This angular-diameter
distance, when combined with its recessional velocity,
should yield a direct and accurate estimate of \ifmmode {H_0} \else {$H_0$} \fi.
\vskip 0.5truecm
{\it Facilities:} \facility{VLBA, GBT, Effelsberg}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,485 |
\section{#1} \label{#1}}
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\title{\bf Prime spectra of derived quiver representations}
\author{{\small \textsc{ Yu-Han Liu and Susan J. Sierra}}}
\begin{document}
\date{\today}
\maketitle
\setcounter{tocdepth}{2}
\setcounter{section}{0}
\setcounter{subsection}{1}
\mysec{Introduction}
\mysubsec{Introduction}
\p{} For every essentially small tensor triangulated category $T$, Balmer \cite{balmer_spectrumtt} defined functorially a ringed space $\mathrm{Spec}(T)$, called its prime spectrum. It has been shown in some cases that the construction $T \mapsto \mathrm{Spec}(T)$ does not lose information in the sense that the category $T$ can be reconstructed from $\mathrm{Spec}(T)$. For example, when $T_X=D(X)_\mathrm{parf}$ is the tensor triangulated category of perfect complexes on a scheme $X$, there is a natural morphism \[X\longrightarrow \mathrm{Spec}(T_X)\] of ringed spaces. This comparison map is an isomorphism when $X$ is \emph{topologically noetherian}; see \cite[Theorem~6.3]{balmer_spectrumtt}.
\p{} Not surprisingly, Balmer's construction loses a great deal of information when applied to noncommutative rings, as we show in this note. We give an alternate construction, showing how to construct an algebra from a tensor triangulated category. In some cases, our construction recovers much more information than Balmer's.
\p{} Fix a field $k$. Let $Q$ or $(Q,R)$ be a quiver \emph{with relations}; for legibility we will omit the relations $R$ from the notation $(Q,R)$ whenever convenient. Let $Q\mathrm{-Rep}_k$ be its abelian category of finite dimensional representations over $k$. Under suitable conditions on $R$ (see \rf{par:unital}), this category is equipped with a natural \emph{vertex-wise} tensor product which is an exact functor in each factor. Its bounded derived category $D(Q):=D^b(Q\mathrm{-Rep})$ is then a tensor triangulated category when the relations satisfy \rf{par:unital}. We will describe $\mathrm{Spec}(D(Q))$ in the case when $Q$ is finite and ordered (that is, without non-trivial oriented cycles).
It turns out that the ringed space $\mathrm{Spec}(D(Q))$ is not enough to recover $Q$, even in the case of quivers without relations; see \rf{par:examples}. Moreover, $\mathrm{Spec}(D(Q))$ is not (yet) a quiver, hence we do not expect a comparison map \[Q\longrightarrow \mathrm{Spec}(D(Q)).\]
We show, however, that one can still recover $Q$ (or, more accurately, its path algebra) from the tensor triangulated category $D(Q)$, at least in the case that $Q$ is finite and ordered.
\p{par:Pm} These results were motivated in part by the fact that the derived categories of some projective varieties are equivalent to derived categories of quiver representations. Consider the triangulated category $D(\mathbb P^m)=D^b(\mathrm{Coh}(\mathbb P^m))$. It is known to be equivalent to the category $D(S_m)$, where $S_m$ is the quiver described in \cite[Example~5.3 and Example~6.4]{bondal_repassalgandcoh}: it has $m+1$ ordered vertices and $m+1$ arrows between consecutive vertices, with commutativity relations (see \rf{par:unital}). Hence we have two different tensor products on the same triangulated category $T=D(\mathbb P^m)\cong D(S_m)$: one from the sheaf tensor product on $\mathbb P^m$ and the other the quiver tensor product. We will see that the two tensor products give very different spectra.
{\bf Acknowledgements.}\; The authors thank Ryan Kinser and Michael Wemyss for their helpful comments. The second author was partially supported by an NSF Postdoctoral Research Fellowship, grant DMS-0802935.
\mysubsec{Conventions and preliminary facts about quivers}
\p{} For a field $k$, $\mathbf{TT}_k$ denotes the category of (essentially small) $k$-linear tensor triangulated categories. Morphisms in this category are exact functors preserving the tensor products and unit objects.
\p{} If the base field $k$ is fixed, we denote $\mathrm{Vect}_k$ simply by $\mathrm{Vect}$, and similarly $\mathbf{TT}_k$ by $\mathbf{TT}$ and $Q\mathrm{-Rep}_k$ by $Q\mathrm{-Rep}$.
\p{} For any collection $M$ of objects in a tensor triangulated category $T$, $\langle M\rangle$ denotes the smallest \emph{tensor ideal} containing $S$. (For us, a tensor ideal is a full, thick, triangulated subcategory $J$ of $T$ that satisfies $V \otimes W \in J$ for any $V \in J$ and $W \in T$.) A tensor ideal $P$ is {\em prime} if $V \otimes W \in P$ implies that either $V \in P$ or $W \in P$.
\p{par:quivers} For definitions concerning quivers, see \cite{auslander_repartinalg}. Let $v \in Q_0$. For any quiver representation $V$ we denote by $V_v$ the vector space associated to the vertex $v$. We take however the following convention that is different from \cite{auslander_repartinalg}: We identify $Q\mathrm{-Rep}$ with the category of \emph{right} modules over the path algebra $kQ$, and take the corresponding convention on multiplication in $kQ$; for example, $e_m p = p$ for any path \emph{beginning} at the vertex $m$.
We consider mainly \emph{finite ordered quivers}, which means there are only finitely many vertices and arrows, and there are no oriented cycles. In this case we denote the vertices by integers $n=1, 2, \ldots$, ordered in such a way that there are non-trivial paths from $n$ to $m$ only when $n<m$.
\p{par:unital} We impose the condition on the relations $R$ of $(Q,R)$ that the the subcategory $(Q,R)\mathrm{-Rep}$ is a monoidal subcategory of $(Q,\emptyset)\mathrm{-Rep}$. More explicitly, we require that the unit object $U$ in $(Q,\emptyset)\mathrm{-Rep}$ (over any field) lies in the subcategory $(Q,R)\mathrm{-Rep}$, and $(Q,R)\mathrm{-Rep}$ is closed under the quiver tensor product.
We call relations involving only non-trivial paths satisfying this condition \emph{tensor relations}.
For example, relations generated by ``commutativity relations'' are tensor. This means that there are relations $p_i-q_i\in R$, where $p_i$ and $q_i$ are non-trivial paths with the same source and target, such that every relation in $R$ is of the form $r=\sum \lambda_i(p_i-q_i)$ with $\lambda_i\in k$.
\p{par:full} For any quiver $Q$, a \emph{full subquiver} $Q'$ is a quiver with $Q'_0 \subseteq Q_0$ and $Q'_1 \subseteq Q_1$, that further inherits all arrows between vertices: that is, for any arrow $a \in Q_1$ between vertices $n, m \in Q'_0$, we have $a \in Q'_1$.
\p{par:subq} Let $Q$ be a quiver and $R$ a set of (not necessarily tensor) relations, which we assume to be an ideal in the path algebra $kQ$; let $Q'$ be a full subquiver of $Q$. Denote by $f: Q'\rightarrow Q$ the inclusion map. Then there is an exact \emph{restriction functor} \[f^*:Q\mbox{-Rep}\rightarrow Q'\mbox{-Rep}.\]
Denote by $R\cap Q'$ the subset of $R$ consisting of relations all of whose paths are in $Q'$; we view $R\cap Q'$ as a set of relations on $Q'$. Then the image under $f^*$ of the subcategory $(Q,R)$-Rep is contained in $(Q',R\cap Q')$-Rep, and we have an induced exact functor \[f^*:(Q,R)\mbox{-Rep}\longrightarrow (Q',R\cap Q')\mbox{-Rep}.\]
We will need conditions under which its derived functor is full and essentially surjective. To this end it is convenient to find a right-inverse. Consider the exact \emph{extension by zero} functor \[f_*:Q'\mbox{-Rep}\rightarrow Q\mbox{-Rep}.\] Observe that this functor respects the vertex-wise tensor product, but does not preserve the unit object in general.
Denote by $\bar R$ the set of relations on $Q'$ obtained by setting every arrow occurring in $R$ but not in $Q'$ to be zero. Then the image under $f_*$ of $(Q',R\cap Q')$-Rep is contained in $(Q,R)$-Rep whenever $R\cap Q'=\bar R$. (Notice that the containment $\subseteq$ always holds.)
We say that $R$ and $Q'$ are \emph{compatible} if the equality $R\cap Q'=\bar R$ holds (equivalently, $R \cap Q \supseteq \bar R$), in which case we have
\centerline{\xymatrix{(Q,R)\mbox{-Rep}\ar@/^/[r]^-{f^*} & (Q',R')\mbox{-Rep},\ar@/^/[l]^-{f_*} }}
where $R':=R\cap Q'=\bar R$. These functors satisfy $f^*f_*=\mathrm{Id}$ on $(Q',R')$-Rep.
\p{par:qex} For example if $Q$ is the quiver
$\xymatrix@R=10pt{& 2\ar[dr]^{b} & \\ 1\ar[ur]^{a}\ar[dr]_{c} && 4, \\ & 3\ar[ur]_{d}}$
$R=(ab-cd)$, and $Q'$ is the full subquiver $1\stackrel{a}{\longrightarrow} 2 \stackrel{b}{\longrightarrow} 4$, then $\bar R=\{ab\}$ and $R\cap Q'=\emptyset$. Hence $R$ and $Q'$ in this example are not compatible.
On the other hand, notice that if $R$ has only non-trivial paths and $Q'$ consists of only one vertex then $R$ and $Q'$ are always compatible.
\p{par:tensorr} Let $(Q,R)$ be a quiver with tensor relations, and $Q'$ a full subquiver compatible with $R$; let $R':=R\cap Q'=\bar R$. We claim that $R'$ also consists of tensor relations. First, $f^*$ sends the unit object $U$ in $Q$-Rep to the unit object $U'$ in $Q'$-Rep. By assumption $U\in (Q,R)$-Rep, and so $U' = f^*(U)\in (Q',R')$-Rep.
Next, let $V',W'\in (Q',R')$-Rep; we need to show that $V'\otimes W'\in (Q',R')$-Rep too. Since $f^*$ respects tensor products, we have \[V'\otimes W'=f^*f_*(V')\otimes f^*f_*(W')=f^*(f_*(V')\otimes f_*(W')).\] But $f_*(V'), f_*(W')\in (Q,R)$-Rep, and so by assumption $f_*(V')\otimes f_*(W')\in (Q,R)$-Rep, whose image $V'\otimes W'$ under $f^*$ then lies in $(Q',R')$-Rep, as required.
\p{par:qwr} Note that the full abelian subcategory $(Q,R)$-Rep of $Q$-Rep is in general not closed under extensions. In particular there is an inclusion functor $D(Q,R)\rightarrow D(Q)$, but $D(Q,R)$ is not a triangulated subcategory of $D(Q)$. This inclusion is an exact tensor functor (under the condition in \rf{par:unital}) that is injective on objects, but is not full in general: For example, consider the quiver $Q$ with two vertices and two arrows $a$ and $b$ both going from one vertex to the other. Let $R:=(a-b)$. Then both simple objects in $Q$-Rep lie in $(Q,R)$-Rep, but most of their extensions do not lie in $(Q,R)$-Rep.
In general, all the simple objects of $Q$-Rep lie in $(Q,R)$-Rep, since all arrows act as the zero map in a simple representation.
\p{par:filtration} Let $(Q,R)$ be a finite ordered quiver with tensor relations (in particular, by the definition given in \rf{par:unital}, $R$ involves only non-trivial paths); let $q$ be the number of its vertices. Denote its vertices by $1, 2, \ldots, q$ so that there exists a non-trivial path from $n$ to $m$ only when $n<m$. (There is in general more than one such way to label the vertices.)
Let $Q_\ell$ be the full subquiver with vertices $\ell, \ell+1, \dots, q$, for every $\ell\leq q$.
\begin{lemma}\label{lem:comp} $Q_\ell$ and $R$ are compatible for every $\ell$. \end{lemma}
\begin{proof} With $\ell$ fixed, let $\bar R$ be as defined in \rf{par:subq}, then we need to show $\bar R\subseteq R\cap Q_\ell$. For any relation $r\in R$ denote by $\bar r\in \bar R$ the relation obtained by setting every arrow not in $Q_\ell$ to be zero.
The ideal of relations $R$ is generated by relations in $R$ of the form $r=\sum p_j$ with each $p_j$ a path between two fixed vertices, say from $n$ to $m$; in particular every arrow occurring in $r$ is between vertices $v,w$ with $n \leq v < w \leq m$.
If $n \geq \ell$ then $r=\bar r$ is contained in $Q_\ell$, since in this case $Q_\ell$ contains every arrow occurring in $r$.
If $n<\ell$ then at least one arrow in each path occurring in $r$ does not lie in $Q_\ell$, namely the arrow with source $n$. In this case $\bar r=0\in R\cap Q_\ell$ as well. \end{proof}
With the ordering of the vertices fixed, we have a filtration $...\subset K_{\ell+1}\subset K_\ell \subset K_{\ell-1}\subset \ldots U$ of the unit object $U\in Q$-Rep, so that the quotient $K_\ell/K_{\ell+1}$ is the simple object $U(\ell)$ supported at the vertex $\ell$.
\begin{cor}\label{cor:filtration} If $(Q,R)$ is a finite ordered quiver with tensor relations, then every $K_\ell$, $\ell=1, 2, \ldots, q$, lies in the subcategory $(Q,R)$-Rep.\end{cor}
\begin{proof} Notice that if $f$ denotes the inclusion of $Q_\ell$ in $Q$, then $K_\ell$ is the image under $f_{*}$ of the unit object $U_\ell$ in $Q_\ell$-Rep. Let $R_\ell:=R\cap Q_\ell$. Then by \rf{lem:comp} and \rf{par:tensorr} we know that $(Q_\ell,R_\ell)$ is a finite ordered quiver with tensor relations; in particular $U_\ell\in (Q_\ell,R_\ell)$-Rep. By \rf{lem:comp} and \rf{par:subq} we have $K_\ell=f_*U_\ell\in (Q,R)$-Rep. \end{proof}
\mysec{Balmer's constructions and their applications to quivers}
In this section, we recall briefly the constructions in \cite[Definition~2.1 and Definition~6.1]{balmer_spectrumtt}, and apply them to quiver representations. We see that a great deal of information is lost.
\mysubsec{Spectrum of a tensor triangulated category}
\p{} Let $T$ be a tensor triangulated category. Here we recall Balmer's definition of the topological space $\Spc T$.
\begin{defn}\label{def:spec}
The space $\Spc T$ is the set of prime $\otimes$-ideals of $T$. It is given the {\em Zariski topology}: a closed set is of the form
\[ \mathbf{Z}(S) := \{ P \in \Spc T \vert S \cap P = \emptyset \},\]
where $S$ is a set of objects in $T$.
\end{defn}
\p{} If $X$ is a (topologically) noetherian scheme, by \cite[Corollary~5.6]{balmer_spectrumtt}, the topological space $\Spc(D(X)_\mathrm{parf})$ is homeomorphic to $X$.
\p{} For any quiver $(Q,R)$ with tensor relations (see \rf{par:unital}) let $D(Q,R)$, and often just $D(Q)$, be the bounded derived category of $(Q,R)$-Rep, equipped with the vertex-wise tensor product.
We now calculate $\Spc D(Q)$ for a finite ordered quiver $Q$ with tensor relations, and show that the functor $Q \mapsto \Spc D(Q)$ loses a great deal of information; in fact, it retains merely the number of vertices of $Q$!
\p{} Recall that every object $V$ in $D(Q)$ can be represented by a bounded complex of objects in $Q\mathrm{-Rep}$. In particular its cohomology $H(V)=\bigoplus H^i(V)$ is again an object in $Q\mathrm{-Rep}$. We call the graded vector space $H(V)_n$ \emph{the cohomology of $V$ at the vertex $n$}.
\begin{lemma}\label{lem:main} Let $Q$ be a finite ordered quiver with tensor relations. For any object $V$ in $D(Q)$, we have $\langle V\rangle=\langle U(n)\,|\, H(V)_n\neq 0\rangle$, where $U(n)$ is the simple representation corresponding to the vertex $n$.\end{lemma}
\begin{proof} The unit object is given by the representation $U$ with $U_n\cong k$ at each vertex $n$ and identity maps between them for all arrows. Since $Q$ is finite and ordered, $U$ admits a filtration whose successive quotients are the simple representations $U(n)$; see \rf{par:filtration}.
Tensoring this filtration with $V$ we see that $\langle V\rangle$ contains the object $V(n)=V\otimes U(n)$ with $V(n)_n=V_n$ and $V(n)_m=0$ for $m\neq n$; all the arrows in the representation $V(n)$ are the zero map. On the other hand, $V$ is an extension of the $V(n)$'s; thus $V \in \ang{V(n)}$. We have obtained that $\langle V\rangle=\langle V(n)\,|\, n=1, 2,\ldots\rangle$.
The object $V(n)\in \langle V\rangle$ is a complex of vector spaces, and the construction in \cite[Chapter~III, \S~1.4 Proposition]{gelfand_manin_homologicalalgebra} shows that it is isomorphic in $D(Q)$ to the complex $H(V)_n$ of vector spaces with zero differentials. Since $\langle V\rangle$ is a thick triangulated subcategory, it contains the simple representation $U(n)$ whenever $H(V)_n\neq 0$.
\end{proof}
\begin{cor}\label{cor:1} If $V$ can be represented by a complex with non-zero cohomology at every vertex, then $\langle V\rangle=D(Q)$ is the unit ideal.\qed \end{cor}
\p{H_n} Consider the evaluation functor $V\mapsto V(n)=V\otimes U(n)$ from $Q\mathrm{-Rep}$ to the category $\mathrm{Vect}$. This functor is exact and preserves the tensor products as well as the unit object. Its derived functor from $D(Q)$ to $D^b(\mathrm{Vect})\cong \oplus \mathrm{Vect}[j]$ then sends $V$ to its cohomology $H(V)_n$ at $n$. The kernel of this derived functor is a tensor ideal, which we denote by $P_n$. It is the full subcategory of $D(Q)$ consisting of objects $V$ with $H(V)_n=0$.
\begin{theorem}\label{thm:main} Let $Q$ be a finite ordered quiver with tensor relations. Then $\Spc D(Q)$ is the discrete space $\{ P_n \vert \, n \in Q_0\}$. \end{theorem}
\begin{proof} Clearly $P_n \supseteq \langle U(m) \vert m \neq n \rangle$. Suppose that $V \not \in P_n$.
Then $H(V)_n\neq 0$, hence $\displaystyle V\oplus\bigoplus_{m\neq n}U(m)$ is an object in $\langle P_n, V\rangle$ which has non-zero cohomology at every vertex. By \rf{cor:1} we have $\langle P_n, V\rangle=D(Q)$. Thus $P_n$ is maximal, and by \cite[Proposition~2.3(c)]{balmer_spectrumtt} $P_n$ is prime.
Let $I$ be an ideal that is not contained in any $P_n$. Then for every $n$ we can find $V^n\in I$ with $H(V^n)_n\neq 0$. This implies that $\oplus V^n\in I$ has non-zero cohomology at every vertex, and $I$ must then be the unit ideal. Thus the $P_n$ are precisely the maximal ideals of $D(Q)$.
Let $P$ be a prime ideal in $D(Q)$. By the previous paragraph it is contained in $P_n$ for some $n$, in particular $U(n)\notin P$. But $U(m)\otimes U(n)=0\in P$ whenever $m\neq n$, hence $U(m)\in P$ for every $m\neq n$ since $P$ is prime. That is, $P=P_n$.\end{proof}
\p{} Let $T$ be the triangulated category $D^b(\mathbb P^m)$ which by \rf{par:Pm} is equivalent to $D^b(S_m)$. By \cite[Corollary~5.6]{balmer_spectrumtt} the spectrum $\mathrm{Spc}(T,\otimes_{\mathbb P^m})$ under the sheaf tensor product is homeomorphic to $\mathbb P^m$, while by \rf{thm:main} above $\mathrm{Spc}(T,\otimes_{S_m})$ is $m+1$ discrete points. The same phenomenon happens for example for Grassmannians by \cite{kapranov_dercatcoh}, and more generally for varieties admitting a full exceptional set of objects.
\mysubsec{The structure sheaf}
\p{support} Let $T$ be a triangulated tensor category. According to \cite[Definition~2.1]{balmer_spectrumtt} the space $\mathrm{Spc}(T)$ comes with a map $\mathrm{supp}(-)$ from the objects of $T$ to closed subsets of $\mathrm{Spc}(T)$ associating to every object $V$ in $T$ the set of prime ideals {\em not} containing $V$.
By comparing this with \rf{thm:main} we see that for any object $V$ of $D(Q)$ we have \[\mathrm{supp}(V)=\{P_n\,|\,H(V)_n\neq 0\}.\]
\p{} Recall from \cite[Definition~6.1]{balmer_spectrumtt} the definition of the structure sheaf on $\mathrm{Spc}(T)$: For any open subset $W\subseteq \mathrm{Spc}(T)$, denote by $Z$ its complement, and $T_Z$ the full triangulated subcategory of $T$ consisting of objects $a$ with $\mathrm{supp}(a)$ contained in $Z$. Then the localization functor \cite{verdier_derivedcategories} $T\rightarrow T/T_Z$ is a tensor functor between tensor triangulated categories. Denote still by $U$ the image of the unit object $U\in T$ in $T/T_Z$, then $\mathcal O_{\mathrm{Spec}(T)}$ is by definition the sheafification of the presheaf of rings
\begin{equation}\label{presheaf}
W\mapsto \mathrm{End}_{T/T_Z}(U).
\end{equation}
The {\em spectrum} of a tensor triangulated category $T$ is the pair $\Spec T := (\Spc T, \mc{O}_{\Spec(T)})$.
\p{} If $X$ is a topologically noetherian scheme, then $\Spec(D(X)_\mathrm{parf}) \cong X$ as locally ringed spaces, by \cite[Theorem~6.3]{balmer_spectrumtt}.
\p{} We now consider the same construction for $T = D(Q)$.
\begin{theorem}\label{thm:structure sheaf quiver}
Let $Q$ be a finite ordered quiver with tensor relations. Then $\mc{O}_Q := \mc{O}_{\Spec(D(Q))}$ is the constant sheaf of algebras $k$. That is, for any $W \subseteq \Spc(D(Q))= Q_0$, we have $\mc{O}_{Q}(W) \cong k^{\oplus W}$.
\end{theorem}
\begin{proof} Since $\mathrm{Spc}(D(Q))$ is a discrete topological space, it suffices to show that $\mathcal O_Q(\{v\})\cong k$ on the open set $\{v\}$ consisting of one point. Let $Z:=\mathrm{Spc}(D(Q))-\{v\}$ be the complement of $\{v\}$ and $Q'$ the full subquiver with only one vertex $v$. Denote by $f: Q'\rightarrow Q$ the inclusion. By identifying $D(Q')$ with $D^b(\mathrm{Vect})$ we see that $f^*$ is exactly the functor $V\mapsto H(V)_v$ on $D(Q)$. In particular we have $T_Z=\ker(f^*)$ and an induced functor \[\bar f^*: D(Q)/T_Z= D(Q)/\ker(f^*)\longrightarrow D(Q').\]
Since $Q'$ is compatible with tensor relations (see \rf{par:qex}), we may apply the next proposition to conclude the proof. \end{proof}
\begin{prop} Let $(Q,R)$ be a finite ordered quiver with relations. Let $f:Q'\rightarrow Q$ be the inclusion of a full subquiver compatible with the relations; let $R':=R\cap Q'$ as in \rf{par:subq}. Then the derived restriction functor $f^*:D(Q)\rightarrow D(Q')$ induces an equivalence \[\bar f^*: \overline T:=D(Q)/\ker(f^*)\longrightarrow D(Q').\]
\end{prop}
\begin{proof} We claim that $f^*$ is essentially surjective and full: Indeed, the derived functor of the extension by zero exact functor \[f_*:Q'\mathrm{-Rep}\rightarrow Q\mathrm{-Rep}\] is a right inverse of $f^*$ on both objects and morphisms (see \rf{par:subq}). It follows that $\bar f^*$ is also essentially surjective and full, and so it remains to show that it is faithful.
Let $X, Y \in \Ob(D(Q)) = \Ob(\overline T)$. An element of $\Hom_{\overline T}(X,Y)$ is represented by a diagram $X\stackrel{s}{\leftarrow}V\stackrel{g}{\rightarrow}Y$ in $T$,
where $s$ is such that $f^*s$ is an isomorphism. This is thought of as a ``morphism'' $g s^{-1}: X \to Y$.
Now let $X\stackrel{s}{\leftarrow}V\stackrel{g}{\rightarrow}Y$ represent a morphism in $\overline T$ that maps to zero under $\bar f^*$.
In particular $g:V\rightarrow Y$ is a morphism in $D(Q)$ such that $f^*g=0$. By applying $f^*$ to the distinguished triangle \[V\stackrel{g}{\longrightarrow} Y\stackrel{h}{\longrightarrow} \mathrm{cone}(g)\] we see that this implies $f^*h$ has a left inverse $m:f^*\mathrm{cone}(g)\rightarrow f^*Y$ in $D(Q')$.
Let $\tilde m: \mathrm{cone}(g)\rightarrow Y$ be in $D(Q)$ such that $f^*\tilde m$ is equal to $m$; such an $\tilde m$ exists since $f^*$ is full, as observed above.
Now $f^*(\tilde m\circ h)=m\circ f^*h$ is an isomorphism in $D(Q')$. Therefore $\mathrm{cone}(\tilde m\circ h)$ lies in $\ker(f^*)$ and this means that $\tilde m \circ h$ also becomes an isomorphism in $\overline T$. Hence $g$ maps to zero in $\overline T$ since $(\tilde m\circ h)\circ g$ maps to zero in $\overline T$. So $X\stackrel{s}{\leftarrow}V\stackrel{g}{\rightarrow}Y$ represents the zero morphism in $\overline T$. \end{proof}
\p{} The presheaf \rf{presheaf} formally contains more information than its sheafification $\mc{O}_{\Spec(T)}$. In the case of a quiver, this difference is significant. Let $v, w$ be vertices in a quiver $Q$ without relations, and let $W := \{ v, w\}$ considered as an open subset of $\mathrm{Spec}(D(Q))$; denote by $Q_W$ the full subquiver of $Q$ consisting of vertices $v$ and $w$.
Then $\End_{D(Q_W)}(U)$ is either $k$ or $k \oplus k$, depending on whether there are arrows between $v$ and $w$. Thus the presheaf \rf{presheaf} recovers the underlying graph of the quiver, while the structure sheaf recovers only the number of vertices.
\p{prime ideals} \emph{Prime ideals in the path algebra:} By \cite[page~53]{auslander_repartinalg} we know that the path algebra $kQ$ modulo its radical (which is the ideal generated by all the non-trivial paths) is isomorphic to the product of $\#Q_0$ copies of $k$. In particular we see that the two-sided prime ideals in $kQ$ are naturally in bijection with prime ideals in $D(Q)$. That is, the global sections of $\mc{O}_{\mathrm{Spec}(D(Q))}$ are naturally isomorphic to $k Q/J(kQ)$, and, since the map $kQ \to kQ/J(kQ)$ has a right inverse, to a subalgebra of $kQ$.
\p{par:examples} Let $Q_i$, $i\geq 1$, be the $i$-Kronecker quiver: the quiver (without relations) with two vertices and $i$ arrows all going from one vertex to the other. There are obvious morphisms $Q_i\rightarrow Q_j$ with $i<j$, inducing functors $D(Q_j)\rightarrow D(Q_i)$ of tensor triangulated categories. These in turn induce morphisms \[\mathrm{Spec}(D(Q_i))\longrightarrow \mathrm{Spec}(D(Q_j))\] of ringed spaces. From \rf{thm:main} we see easily that these are isomorphisms, in particular we cannot recover quivers from their prime spectra. Note that the presheaves \rf{presheaf} are isomorphic for all $Q_i$ as well.
\mysec{Functors of points}
\setcounter{subsection}{1}
\setcounter{subsubsection}{0}
\p{} Let $T, S \in \mathbf{TT}$. Define \[T(S):=(\mathrm{Hom}_{\mathbf {TT}}(T,S)/\cong\,)=(\mathrm{Fun}^{\otimes,1}(T,S)/\cong).\]
Elements in $T(S)$ will be called \emph{$S$-valued points in $T$}. We then have a set-valued covariant functor $T(-)$ on $\mathbf{TT}$ for every $T\in\mathbf{TT}$.
\p{} If $R$ is a commutative $k$-algebra, we define $T(R):= T(D^b(R-{\rm Mod}))$. Elements in $T(R)$ will be called \emph{$R$-rational points in $T$}.
\p{} Let $Q$ be a finite ordered quiver with tensor relations \rf{par:unital}; we will compute $D(Q)(k)$. We identify the tensor triangulated category $D^b(k) = D^b(\mathrm{Vect})$ with $\oplus \mathrm{Vect}[j]$ by taking cohomology of complexes of vector spaces.
The functors $V\mapsto H(V)_n$ in \rf{H_n} are in $D(Q)(k)$, and conversely:
\begin{prop}\label{prop: points of quiver} $D(Q)(k)=\{(V\mapsto H(V)_n)\,|\, n\in Q_0\}$.
\end{prop}
\begin{proof} Let $F\in D(Q)(k)$. Consider the filtration $\ldots \subset K_{n+1}\subset K_{n}\subset \ldots\subset U$ of the unit object in $D(Q)$ such that the quotient $K_n/K_{n+1}$ is the simple object $U(n)$ supported at the vertex $n$. By \rf{cor:filtration}, each $K_n$ lies in $D(Q)=D(Q,R)$.
Since $F$ preserves the unit objects, $F(U)$ is isomorphic to $k$ viewed as a graded vector space placed at degree $0$. Hence on applying $F$ to the distinguished triangle \[K_{n+1}\stackrel{f_{n+1}}{\longrightarrow} U\stackrel{g_{n+1}}{\longrightarrow} U/K_{n+1},\] we see that for any fixed vertex $n$ exactly one of $F(f_{n+1})$ and $F(g_{n+1})$ is equal to $0$, while the other one admits a one-sided inverse.
Note that for $n$ sufficiently large we have $F(f_{n+1})=0$.
Moreover, if $F(f_{n+1})=0$ then $F(f_{n+2})=0$: Indeed, suppose on the contrary that $F(f_{n+1})=0$ but $F(f_{n+2})\neq 0$, then we get a contradiction by applying $F$ to the following commutative diagram:
\centerline{\xymatrix{K_{n+2} \ar[r]^-{f_{n+2}} \ar[d] & U \ar@{=}[d]\\ K_{n+1}\ar[r]^-{f_{n+1}}& U.}}
So from now on let $n$ be minimal so that $F(f_{n+1})=0$. Consider the following diagram of distinguished triangles, where by assumption $F(f_n)$ is non-zero and admits a right inverse $m$:
\[
\xymatrix{K_{n+1} \ar[r]^-{f_{n+1}}\ar[d] & U \ar[r]\ar@{=}[d] & U/K_{n+1}\ar[d] &&& F(K_{n+1}) \ar[r]^-{0 }\ar[d] & F(U) \ar[r]\ar@{=}[d] & F( U/K_{n+1})\ar[d]
\\ K_n \ar[r]^-{f_n}\ar[d]_-q & U \ar[r]\ar[d] & U/K_n \ar[d] & \ar @/^/ @{..>} [r]^{F} && F(K_n) \ar[r]^{F(f_n)}\ar[d]_{F(q)} & F(U) \ar@/^1pc/[l]^{m} \ar[r]\ar[d] & F(U/K_n) \ar[d] \\
U(n) \ar[r] & 0 \ar[r] & U(n)[1] &&& F(U(n)) \ar[r] & 0 \ar[r] & F(U(n))[1].}\]
The map $F(q) \circ m: F(U)\rightarrow F(U(n))$ must be non-zero, since otherwise $m$ lifts to a non-zero map from $F(U)$ to $F(K_{n+1})$, giving a right inverse to $F(f_{n+1})$, which is zero by assumption.
This says that $F(U(n))$ contains $F(U)=k$ as a direct summand, and when viewed as an object in $\oplus \mathrm{Vect}[j]$, $F(U(n))$ must be isomorphic to $k$ since $U(n)\otimes U(n)=U(n)$.
The two functors $V\mapsto F(V)=F(U\otimes V)$ and $V\mapsto F(U(n)\otimes V)=F(V_n)$ are then isomorphic:
\[ F(V) = F(U \otimes V) = k \otimes F(V) = F(U(n))\otimes F(V) = F(U(n)\otimes V) = F(V_n).\]
Now the full subcategory of $D(Q)$ consisting of objects of the form $U(n)\otimes V$ is isomorphic to $D^b(k)$, and any exact functor preserving tensor products and the unit objects from $D^b(k)$ to itself is the identity. Hence on identifying $D^b(k)$ with $\oplus \mathrm{Vect}[j]$ we see that the functor $F$ is isomorphic to $V\mapsto H(V)_n$. \end{proof}
\p{par:sum} To summarize, we have established bijections for any finite ordered quiver $Q$ with tensor relations:
\hskip 1.5in\xymatrix{Q_0\ar[r] & D(Q)(k)\ar[r] & \mathrm{Spc}(D(Q)) \\ n \ar@{|->}[r] & (V\mapsto H(V)_n) \\ & F \ar@{|->}[r] & \ker F.}
\p{par:F_n} Let $F_n \in D(Q)(k)$ be the functor $V\mapsto H(V)_n$ corresponding to the vertex $n$. We now give a useful formula for the functors $F_n$. Denote by $M_n$ the indecomposable projective module of the path algebra $kQ$ of $Q$ with simple quotient $U(n)$, namely, it is the \emph{right} submodule of $kQ$ generated by the trivial path $e_n$ at the vertex $n$: $M_n=e_n(kQ)$ is the $k$-vector space spanned by all paths starting at the vertex $n$; see \cite[Page~59]{auslander_repartinalg}. Then \[kQ\cong \bigoplus_n M_n.\] Under the usual identification of $Q\mathrm{-Rep}$ with $\mathrm{mod-}kQ$, we may view $M_n$ as an object in the former abelian category.
\begin{lemma}\label{lem:F_n} We have $F_n(-)=R\mathrm{Hom}(M_n,-)$ on $D(Q)$.
\end{lemma}
\begin{proof} \begin{comment}Recall that $F_n$ may be given by the derived functor on $D(Q)$ of the \emph{exact} functor $-\otimes U(n)$ on $Q\mathrm{-Rep}$. Since $\mathrm{Hom}(M_n,-)$ is also exact, it suffices to show that $\Hom(M_n, \mbox{$\underline{\makebox[10pt]{}}$}) = F_n(\mbox{$\underline{\makebox[10pt]{}}$})$ on $Q\mathrm{-Rep}$.
Let $V$ be an object in $Q\mathrm{-Rep}$, and $\tilde V$ the $kQ$-module corresponding to it. Then $V\otimes U(n)=V_n=\tilde Ve_n$. On the other hand, the evaluation map from $\mathrm{Hom}(M_n,\tilde V)$ to $\tilde V$, sending $f\mapsto f(e_n)$, is clearly an injection since $M_n=e_n(kQ)$ and $f$ is a right $kQ$-module homomorphism.
The image of this evaluation map is exactly $\tilde Ve_n$, since $f(e_n)=f(e_n^2)=f(e_n)e_n$, and for any $xe_n\in \tilde Ve_n$, the assignment $f:e_n\mapsto xe_n$ extends to an element in $\mathrm{Hom}(M_n,\tilde V)$. \end{comment}
This is well-known, and follows for example from \cite[Exercise~III.10]{auslander_repartinalg}. \end{proof}
\mysec{Algebras associated to a tensor triangulated category}
\mysubsec{Defining the algebra}
\p{} Let $S, T\in\mathbf{TT}$. For any $F,G\in T(S)$ denote by $\mathrm{Hom}(F,G)$ the set of natural transformations. Since $T$ is a $k$-linear category, this is a $k$-vector space.
Let $U$ be the unit object of $T$.
On the $k$-vector space \[A(T,S):=\prod_{F,G\in T(S)}\mathrm{Hom}(F,G)\] we then have a partially defined product by composition, which gives an associative, in general non-commutative algebra by defining the product between elements which are not composable to be zero. We are most interested in the special case:
\[ A(T) := A(T, D^b(k)) := \prod_{F,G\in T(k)}\mathrm{Hom}(F,G).\]
\p{} The functor $A(\mbox{$\underline{\makebox[10pt]{}}$}, \mbox{$\underline{\makebox[10pt]{}}$})$ is contravariant in its first argument and covariant in its second.
\mysubsec{Recovering finite ordered quivers}
\p{} We now show that a finite ordered quiver with tensor relations \rf{par:unital} can be essentially recovered from its tensor triangulated category of representations. Recall that for a quiver $Q$ we denote by $D(Q)$ the bounded derived tensor category of finite dimensional representations of $Q$ over the fixed field $k$.
\begin{thm}\label{thm:recovery} Let $(Q, R)$ be a finite ordered quiver with tensor relations. The algebra $k Q/(R)$ of $(Q, R)$ over $k$ is naturally isomorphic to $A(D(Q))$.
\end{thm}
\begin{proof} Recall that we have a bijection (see \rf{par:sum}) from the set of vertices $Q_0$ to $T(k)$ given by $n\mapsto (F_n:V\mapsto H(V)_n)$.
Let $\Lambda:= kQ/(R)$.
The set of paths from vertex $n$ to $m$ is $e_n\Lambda e_m$ (where $e_n$ is the trivial path at $n$), and we first define a map \[\phi:e_n\Lambda e_m \longrightarrow \mathrm{Hom}(F_n,F_m)\] for every pair of vertices $n,m$. Let $p\in e_n\Lambda e_m$ be a path from $n$ to $m$, and $V\in D(Q)$, then we define \[\phi(p)_V: F_n(V)\rightarrow F_m(V)\] to be the morphism $H(V)(p): H(V)_n\rightarrow H(V)_m$ (where the cohomology $H(V)$ is a complex of $Q$-representations). This is well-defined since $H(V)(r)=0$ for every relation $r$. Note that this is defined without assumptions on $Q$ or $R$.
Now by the Yoneda lemma on the category $D(Q)$, remembering \rf{lem:F_n} that $F_n$ is represented by $M_n=e_n\Lambda $, we have \[\mathrm{Hom}(F_n,F_m)\cong F_m(M_n)\cong \mathrm{Hom}_{\Lambda }(M_m, M_n),\]sending $\rho: F_n\rightarrow F_m$ to $\rho_{M_n}(e_n)$, where $\rho_{M_n}:(F_n(M_n)=ke_n)\rightarrow F_m(M_n)$.
On the other hand we have a natural isomorphism \[\psi:\mathrm{Hom}_\Lambda (M_m,M_n)\longrightarrow e_n\Lambda e_m,\] sending $f: M_m\rightarrow M_n$ to $f(e_m)\in M_n$; see \rf{lem:F_n}. It is now easy to show that $\psi$ is the inverse of $\phi$:
\centerline{\xymatrix{e_n\Lambda e_m \ar[r]^-\phi & \mathrm{Hom}(F_n, F_m) \ar[r] & F_m(M_n) \ar[r] & \mathrm{Hom}(M_m, M_n) \ar[r]^-{\psi} & e_n\Lambda e_m \\ p \ar@{|->}[r] & (V\mapsto \phi(p)_V) \ar@{|->}[r]& \phi(p)_{M_n}(\mathrm{Id}_{M_n} ) \ar@{|->}[r] & (e_mq\mapsto pe_mq) \ar@{|->}[r] & pe_m=p.} }
The naturality in the statement means the following. Let $(Q, R)$ and $(Q', R')$ be two quivers with tensor relations; let $\Lambda:= kQ/(R)$ and let $\Lambda':= kQ'/(R')$. Suppose there is a $k$-algebra homomorphism $\Lambda \to \Lambda'$. We obtain an induced restriction functor $D(Q', R') \to D(Q, R)$ and a homomorphism $A(D(Q, R)) \to A(D(Q', R'))$.
Then these homomorphisms form a commutative square with the isomorphisms $\Lambda \cong A(D(Q))$ and $\Lambda'\cong A(D(Q'))$ above. We leave the details to the reader. \end{proof}
Thus a finite, ordered quiver $Q$ with tensor relations (or at least its path algebra) can be recovered from its tensor triangulated category $D(Q)$ of representations. By \cite[Chapter~III, Theorem~1.9(c)(d)]{auslander_repartinalg}, the quiver $Q$ can be recovered from its path algebra if the ideal generated by its relations $R$ lies between $J^2$ and $J^t$ for some integer $t$, where $J$ is the radical of $kQ$.
\mysubsec{Comparing the two constructions}
\p{} For a tensor triangulated category $T$, the structure sheaf $\mc{O}_{\Spec T}$ and the algebra $A(T)$ are naturally related. This follows from:
\begin{proposition}\label{prop:algebra}
Let $T, S$ be tensor triangulated categories, and let $U$ be the unit object of $T$. Then $A(T, S)$ is naturally an $\End_T(U)$-algebra.
\end{proposition}
\begin{proof}
Let $\phi \in \End_T(U)$. Then $\phi$ induces a natural transformation $\Phi: \Id_T \to \Id_T$, where $\Phi_M: M \to M$ is given by
\[ \xymatrix{ M \ar[r]^{\cong} & M \otimes U \ar[r]^{1 \otimes \phi} & M \otimes U \ar[r]^{\cong}& M.}\]
For every $F \in T(k)$, this then induces a natural transformation $F(\Phi): F\to F$, defined by
\[ F(\Phi)_M = F(\Phi_M): FM \to FM.\]
The map
\begin{align*}
z: \End_T(U) & \to A(T, S) \\
\phi & \mapsto (F(\Phi))_F
\end{align*}
is easily seen to be a ring homomorphism.
Let $F, G \in T(S)$ and let $\beta \in \Hom(F, G)$. Let $M \in T$. Then by naturality of $\beta$, the diagram
\[ \xymatrix{
FM \ar[r]^{F(\Phi_M)} \ar[d]_{\beta_M} & FM \ar[d]^{\beta_M} \\
GM \ar[r]_{G(\Phi_M)} & GM }
\]
commutes. That is, $G(\Phi) \circ \beta = \beta \circ F(\Phi)$ in $A(T, S)$, and $z(\phi) \in Z(A(T))$.
\end{proof}
\p{} From the previous result, we see that for any tensor triangulated category $T$, there are ring homomorphisms
\[ \xymatrix{
\End_T(U) \ar[r]^{\alpha} \ar[rd]_{z} & \Gamma(\mc{O}_{\Spec T}) \\
& A(T), }
\]
where $\alpha$ is induced from the canonical map from a presheaf to the associated sheaf.
In general, there does not seem to be any reason why there should be a vertical map (in either direction) completing the triangle. However, we have:
\begin{proposition}\label{prop:quiver maps} Let $Q$ be a finite ordered quiver with tensor relations and $T = D(Q)$. Then there are vertical maps so that the diagram
\[ \xymatrix{
\End_T(U) \ar[r]^{\alpha} \ar[rd]_{z} & \Gamma(\mc{O}_{\Spec T}) \ar@<1ex>[d] \\
& A(T) \ar@<1ex>[u],}
\]
commutes; further $\End_T(U) \cong Z(A(T))$.
\end{proposition}
\begin{proof}
It is easy to see that $\End_T(U) \cong k^{\pi_0(Q)}$ is the center of the path algebra $kQ$; here $\pi_0$ denotes the set of equivalence classes of vertices in $Q$ generated by the existence of arrows between them. The vertical maps come from \rf{prime ideals}.
\end{proof}
\p{} We note that for tensor triangulated categories $T = D^b(X)$ induced from a scheme $X$, the algebra $A(T)$ can be quite unpleasant, and does not necessarily recover $X$. For example, let $k$ be an algebraically closed field, and let $T := D^b(\mathbb P^1_k)$. Then $T(k) = \mathbb P^1(k)$; that is, the only tensor functors from $T$ to $D^b(k)$ are given by restriction to a $k$-point. The algebra $A(T)$ is then easily seen to be
\[ A(T) = \prod_{p \in \mathbb P^1(k)} k_p,\]
the direct product of the skyscraper sheaves. Indeed, it suffices to show that if $x,y$ are in $\mathbb P^1(k)$, then there are no non-zero natural transformations between the corresponding functors $x^*$ and $y^*$.
Suppose on the contrary that $\phi$ is a non-zero natural transformation from $x^*$ to $y^*$. Then there is a coherent sheaf $\mathcal F$ such that both fibres $x^*\mathcal F$ and $y^*\mathcal F$ are non-zero, and $\phi_\mathcal F$ is a non-zero map between them. In fact since $D^b(\mathbb P^1)$ is generated by $\mathcal O$ and $\mathcal O(1)$, we may assume that $\mathcal F$ is a line bundle. In this case the non-zero map $\phi_\mathcal F$ must be an isomorphism between the fibres $x^*\mathcal F$ and $y^*\mathcal F$.
Now let $\mathcal F':=y_*y^*\mathcal F$. Then we have $x^*\mathcal F'=0$ but $y^*\mathcal F\rightarrow y^*\mathcal F'= y^*y_*y^*\mathcal F$ is a non-zero map. Hence we obtain a commutative diagram, by the naturality of $\phi$:
\hskip 2.4in\xymatrix{x^*\mathcal F \ar[r]\ar[d]_-{\phi_\mathcal F}^-\cong & x^*\mathcal F'=0 \ar[d]_-{\phi_{\mathcal F'}}\\ y^*\mathcal F\ar[r]^-{\neq 0} & y^*\mathcal F'.}
But this implies that $\phi_\mathcal F=0$, a contradiction.
Thus for triangulated tensor categories coming from algebraic geometry, Balmer's construction is much better than ours.
It would be interesting to find a functorial construction that combines the good features of both and to prove a reconstruction theorem that generalizes simultaneously \rf{thm:recovery} and \cite{balmer_spectrumtt}.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,006 |
[Senate Report 117-10]
[From the U.S. Government Publishing Office]
Calendar No. 38
117th Congress } { Report
1st Session } { 117-10
TO AUTHORIZE THE SEMINOLE TRIBE OF FLORIDA TO LEASE OR TRANSFER CERTAIN
LAND, AND FOR OTHER PURPOSES
April 14, 2021.--Ordered to be printed
Mr. Schatz, from the Committee on Indian Affairs,
submitted the following
[To accompany S. 108]
[Including cost estimate of the Congressional Budget Office]
The Committee on Indian Affairs, to which was referred the
bill (S. 108) to authorize the Seminole Tribe of Florida to
lease or transfer certain land, and for other purposes, having
considered the same, reports favorably thereon without
amendment and recommends that the bill do pass.
The purpose of the bill is to authorize the Seminole Tribe
of Florida (Tribe) to convey or otherwise transfer land and
interests in land owned in fee by the Tribe without regard to
language of the Indian Non-Intercourse Act, codified at 25
U.S.C. 177.
BACKGROUND AND NEED FOR LEGISLATION
Seeking to diversify and strengthen its economic
development enterprises that support essential services for
Tribal members, the Seminole Tribe of Florida established a
real estate investment fund to invest in commercial properties.
However, due to restrictions set out in the Indian Non-
Intercourse Act, the Tribe is unable to secure a mortgage
through its investment fund for fee property without
Congressional approval while hindering the Tribe's ability to
manage its asset in a timely manner.
Originally enacted in 1790, the Indian Non-Intercourse Act
reserved the exclusive right to acquire Indian lands to the
federal government. The Act prevents the transfer, sale, lease,
or other land conveyances owned by an Indian Tribe without
Congressional approval. This prohibition applies to both trust
and fee lands, regardless of how the land was obtained, and was
originally intended to protect Indian Tribes by preventing the
loss of their lands.
On January 28, 2021, Senators Rubio and Rick Scott
introduced S. 108, a bill to authorize the Seminole Tribe of
Florida to lease or transfer certain land, and for other
purposes. The Senate referred the bill to the Committee on the
same day. The Committee held a duly called business meeting to
consider nine bills, including S. 108, on March 10, 2021. No
amendments were filed to S. 108. The Committee passed the bill
with eight other bills by voice vote and ordered it to be
favorably reported.
On January 4, 2021, Representatives Soto, Deutch, Hastings,
Crist, Wasserman Schultz, and Mast introduced a house companion
bill, H.R. 164, a bill to authorize the Seminole Tribe of
purposes, in the House of Representatives. The House of
Representatives referred the bill to the Committee on Natural
Resources on the same day. On February 18, 2021, the bill was
referred to the House Natural Resources Subcommittee for
Indigenous Peoples of the United States. No further action has
been taken on H.R. 164.
On June 25, 2020, Senators Rubio and Rick Scott introduced
S. 4079, a bill to authorize the Seminole Tribe of Florida to
lease or transfer certain land, and for other purposes, which
was referred to the Committee. On September 23, 2020, the
Committee held a legislative hearing on S. 4079. At the
hearing, Seminole Tribe of Florida Chairman, Marcellus Osceola
Jr., testified and the Bureau of Indian Affairs, U.S.
Department of the Interior, provided written testimony. On
November 18, 2020, the Committee held a duly called business
meeting at which S. 4079 was considered with ten other bills.
The Committee ordered the bill, without amendment, to be
reported favorably to the Senate by voice vote. S. 4079 without
amendment passed the Senate by unanimous consent on December
18, 2020. The bill was received in the House of Representatives
on December 21, 2020 and held at the desk. No further action
was taken on S. 4079.
A House companion bill, H.R. 7565, a bill to authorize the
Seminole Tribe of Florida to lease or transfer certain land,
and for other purposes, was introduced by Representatives Soto,
Crist, Mast, Wasserman Schultz, Deutch, Hastings, and Diaz-
Balart, on July 9, 2020, and referred to the House Committee on
Natural Resources. On August 5, 2020, the bill was referred to
the House Natural Resources Subcommittee for Indigenous Peoples
of the United States. Representative Frankel was added as a
cosponsor on September 16, 2020. On September 24, 2020, the
House Natural Resources Subcommittee for Indigenous Peoples of
the United States held a legislative hearing on H.R. 7565. No
further action was taken on H.R. 7565.
Section 1--Approval not required to validate certain land transactions
of the Seminole Tribe of Florida
Section 1(a) would allow for the Seminole Tribe of Florida
to transfer all or any part of its interest in real property
that is not held in trust by the United States but land the
Tribe owns in fee without further Congressional approval.
Section 1(b) makes clear that nothing in this section
authorizes the Seminole Tribe of Florida to transfer all or any
part of an interest in real property that is held in trust by
the United States for the benefit of the Seminole Tribe of
Florida; or affects the operation of any law governing leasing,
selling, conveying, warranting, or otherwise transferring any
interest in any real property that is held in trust by the
United States for the benefit of the Seminole Tribe of Florida.
COST AND BUDGETARY CONSIDERATIONS
U.S. Congress,
Congressional Budget Office,
Washington, DC, April 6, 2021.
Hon. Brian Schatz,
Chairman, Committee on Indian Affairs,
U.S. Senate, Washington, DC.
Dear Mr. Chairman: The Congressional Budget Office has
prepared the enclosed cost estimate for S. 108, a bill to
authorize the Seminole Tribe of Florida to lease or transfer
certain land, and for other purposes.
If you wish further details on this estimate, we will be
pleased to provide them. The CBO staff contact is Jon Speri.
Phillip L. Swagel,
S. 108 would allow the Seminole Tribe of Florida to sell,
lease, or otherwise transfer any property owned by the tribe
that is not held in trust by the United States. Under current
law, the tribe must receive Congressional approval before such
a transfer. Compensation for transfers would be paid directly
to the Seminole Tribe and such transactions would not affect
the federal budget.
The CBO staff contact for this estimate is Jon Sperl. The
estimate was reviewed by H. Samuel Papenfuss, Deputy Director
of Budget Analysis.
REGULATORY AND PAPERWORK IMPACT STATEMENT
Paragraph 11(b) of rule XXVI of the Standing Rules of the
Senate requires each report accompanying a bill to evaluate the
regulatory and paperwork impact that would be incurred in
carrying out the bill. The Committee believes that S. 108 will
have a minimal impact on regulatory or paperwork requirements.
EXECUTIVE COMMUNICATIONS
The Committee has received no communication from the
Executive Branch regarding S. 108.
CHANGES IN EXISTING LAW
On February 11, 2021, the Committee unanimously approved a
motion to waive subsection 12 of rule XXVI of the Standing
Rules of the Senate. In the opinion of the Committee, it is
necessary to dispense with subsection 12 of rule XXVI of the
Standing Rules of the Senate to expedite the business of the | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,492 |
{"url":"https:\/\/mathoverflow.net\/questions\/163849\/factor-of-2-in-the-definition-of-metric-contact-structure","text":"# Factor of 2 In the Definition of Metric Contact Structure\n\nIn Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation $$d\\kappa \\left( {X,Y} \\right) = g\\left( {X,\\Phi Y} \\right)$$ as well as other standard relation like ${\\Phi ^2} = - 1 + R \\otimes \\kappa$, $\\Phi R = 0$, etc.\n\nOn the other hand, in Tanno's papers and a few others literatures, the definitions involves $$d\\kappa \\left( {X,Y} \\right) = 2 g\\left( {X,\\Phi Y} \\right)$$ differing from Blair's by a factor of $2$.\n\nSo I try to figure out which is \"more correct\", and then comes one important equation that really confuses me, in Blair's book (Lemma 6.2): $${\\nabla _X}R = - \\Phi X - \\Phi hX, \\;\\;\\;\\;h \\propto \\mathcal{L}_R \\Phi\\;\\;\\;\\;\\;\\;\\;\\;(*)$$\n\nNow, let us consider the contact metric structure to be K-contact, and therefore $h = 0$, ${\\mathcal{L}_R}g = 0$, and also equation $(*)$ should still hold, hence (in components) $${\\nabla _X}R = - \\Phi X \\Leftrightarrow {\\nabla _m}{R^n} = - {\\Phi ^n}_m \\Rightarrow {\\nabla _m}{R_n} = - {g_{nk}}{\\Phi ^k}_m\\;\\;\\;\\;\\;\\;\\;\\;(**)$$\n\nOn the other hand, since $\\nabla$ is Levi-civita, we have $${d\\kappa } \\left( {X,Y} \\right) = \\left( {{\\nabla _m}{R_n} - {\\nabla _n}{R_m}} \\right){X^m}{Y^n} = g\\left( {X,\\Phi Y} \\right) = {g_{mk}}{\\Phi ^k}_n{X^m}{Y^n}$$ and therefore ${\\nabla _m}{R_n} - {\\nabla _n}{R_m} = {g_{mk}}{\\Phi ^k}_n$. Now comes the final point: using the Killing vector equation ${\\nabla _m}{R_n} = - {\\nabla _n}{R_m}$, one obtains $$- 2{\\nabla _m}{R_n} = 2{\\nabla _n}{R_m} = - {g_{mk}}{\\Phi ^k}_n$$ which directly contradict the equation $(**)$\n\nSo my question is\n\nIs Blair's definition or Tanno's \"more correct\"? Is Blair's definition incompatible with the Lemma 6.2? Note that Tannos definition has the correct factor of 2, so $(*)$ is compatible with the definition.\n\nThe discrepancy comes from which coefficient convention is being used for the covariant derivative d. Blair uses the convention that for any 1-form $\\eta$ $$d\\eta(X,Y) = \\frac{1}{2}\\left( X\\eta(Y) - Y\\eta(X) - \\eta[X,Y]\\right)$$ whereas Tanno is most likely using $$d\\eta(X,Y) = X\\eta(Y) - Y\\eta(X) - \\eta[X,Y].$$\nThe difference between these is conceptually nil, but computationally important. And, in fact, it is derived from the wedge product convention for 1-forms: $$\\eta \\wedge \\zeta = \\frac{1}{2}(\\eta \\otimes \\zeta - \\zeta \\otimes \\eta)$$ versus $$\\eta \\wedge \\zeta = \\eta \\otimes \\zeta - \\zeta \\otimes \\eta.$$\n\u2022 I agree with Brendan on the two definitions of $d$: I found at the bottom of page 62, Blair wrote down $d\\eta$ formula. I feel that a clear list of convention should be provided at the very beginning of the book. Thanks! \u2013\u00a0Lelouch May 4 '14 at 3:33","date":"2020-10-29 08:49:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 6, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9444091320037842, \"perplexity\": 414.5266207214143}, \"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-2020-45\/segments\/1603107903419.77\/warc\/CC-MAIN-20201029065424-20201029095424-00302.warc.gz\"}"} | null | null |
require 'spec_helper'
class InstrumentationExampleClient < ActiveRestClient::Base
base_url "http://www.example.com"
get :fake, "/fake", fake:"{\"result\":true, \"list\":[1,2,3,{\"test\":true}], \"child\":{\"grandchild\":{\"test\":true}}}"
get :real, "/real"
end
describe ActiveRestClient::Instrumentation do
it "should save a load hook to include the instrumentation" do
hook_tester = double("HookTester")
expect(hook_tester).to receive(:include).with(ActiveRestClient::ControllerInstrumentation)
ActiveSupport.run_load_hooks(:action_controller, hook_tester)
end
it "should call ActiveSupport::Notifications.instrument when making any request" do
expect(ActiveSupport::Notifications).to receive(:instrument).with("request_call.active_rest_client", {:name=>"InstrumentationExampleClient#fake"})
InstrumentationExampleClient.fake
end
it "should call ActiveSupport::Notifications#request_call when making any request" do
expect_any_instance_of(ActiveRestClient::Instrumentation).to receive(:request_call).with(an_instance_of(ActiveSupport::Notifications::Event))
InstrumentationExampleClient.fake
end
it "should log time spent in each API call" do
expect_any_instance_of(ActiveRestClient::Connection).
to receive(:get).
with("/real", an_instance_of(Hash)).
and_return(::FaradayResponseMock.new(OpenStruct.new(body:"{\"first_name\":\"John\", \"id\":1234}", response_headers:{}, status:200)))
expect(ActiveRestClient::Logger).to receive(:debug).with(/ActiveRestClient.*ms\)/)
expect(ActiveRestClient::Logger).to receive(:debug).at_least(:once).with(any_args)
InstrumentationExampleClient.real
end
it "should report the total time spent" do
# Create a couple of classes to fake being part of ActionController (that would normally call this method)
class InstrumentationTimeSpentExampleClientParent
def append_info_to_payload(payload) ; {} ; end
def self.log_process_action(payload) ; [] ; end
end
class InstrumentationTimeSpentExampleClient < InstrumentationTimeSpentExampleClientParent
include ActiveRestClient::ControllerInstrumentation
def test
payload = {}
append_info_to_payload(payload)
self.class.log_process_action(payload)
end
end
messages = InstrumentationTimeSpentExampleClient.new.test
expect(messages.first).to match(/ActiveRestClient.*ms.*call/)
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 532 |
Selfies, sunshine and gur cake at Dublin's Road to the Rising
The real star of the day is the sun, making its first lasting impression since the end of last summer
Mon, Apr 6, 2015, 15:00 Updated: Mon, Apr 6, 2015, 15:03
Conor Pope
Conor Pope takes a tour of O'Connell Street which was transformed as part of RTE's 'Road to the Rising'. Costumed volunteers, historical walking tours and restored artefacts helped evoke an Edwardian atmosphere in the capital. Video: Enda O'Dowd
Emma Leung (9) from Saggart takes part in RTÉ's Road to the Rising on O'Connell Street on Easter Monday 2015. Photograph: Gareth Chaney Collins
There is a Massy Brothers coffin-filled hearse parked at the top of O'Connell St just under the shadow of Parnell. It's surrounded by people laughing and taking selfies in the sun.
"I hear there's a body in there already. He's ready for take off," a man says. The crowd surrounding this most sombre of carriages laughs.
A woman looks on at all the mirth mournfully. "That's Sean Patrick O'Neill," she tells a confused looking boy from Brazil who has just happened upon the scene. "He was my cousin and he died on this street after he was hit by a tram. He's left behind 10 children. It is so, so sad."
She shakes her head and looks close to tears.
Hildo Marques Psousa's confusion grows. He moved here from Brazil a couple of weeks back and hasn't a notion what is going on in Dublin right now. Eventually he interrupts her.
"I don't think this is true," he says to her earnestly. "I think maybe this happens but not maybe to you. Not the way you are smiling."
It is Sean Patrick O'Neill's cousin's turn to looks confused but she continues to mourn his passing.
Eventually Hildo wanders on and it is only then that Rachel Fayne comes out of character. She is one of the 250 volunteers playing an acting role in RTÉ's Road to the Rising, the wonderful celebration of Dublin at Easter 1915 when it was unknowingly readying itself for an insurrection that would shape the next 100 years.
While the 21st Century Volunteers are sprinkled among the tens of thousands of people who have come out to play, the real star of the day is the sun, making its first lasting impression the city since the end of last summer.
Music fills the air and actors and members of the public mingle among the many stalls which have lined the city's main street. Unusually, perhaps, for a high-profile city centre event, there isn't a drunk teenager skulling cheap lager to be seen anywhere.
Near the James Larkin statue a large crowd has gathered. A small child is hoisted on to his dad's shoulders to see what is happening. "Ah it's just a man talkin, Da," he says. He sounds most unimpressed.
The hundred or so people who have formed a huddle mass around John Gibney as his historical walking tour of the area gets underway are a whole lot more impressed.
"I'd no idea we'd get such a great crowd", he says as he leads his group from Big Jim Larkin to the junction of Henry St and O'Connell St where the second stage of his tour starts. "It's making it difficult to navigate but sure we'll see how we get on."
As he picks his way gingerly through the crowd, his audience pepper him with questions about the Rising. He answers them all easily enough as he rubs his reddening neck. "I don't have any sunscreen on. I didn't expect it to be so sunny. I reckon I'll do one more tour and then retreat to the shadows like a mushroom," he says before leading his tour part up Moore St while a nearby choir belts out a melodic version of Kodaline's High Hopes.
The Waldorf Barber Shop has taken a stall near the spire and barber Christian Hoey is inviting passers by to sit in an antique barber's chair and pose for a picture as he tends to them.
In the quiet moments, he tends to his own perfectly sculpted handlebar moustache. Beside him is a big poster alerting would-be customers to the various beard styles that would have been popular in the year before the Rising.
"We had them before the hipsters," he says. "Before they were cool."
There is something of a commotion at the Manning's Bakery stall next door. Four women in early 20th century clothes are handing out free gur cake. Some people eye the dark brown paste sandwiched between slices of pale pastry suspiciously.
There are no suspicious eyes to be found among the Cully Sisters. They are delighted to have stumbled upon the bakery. Four of the five sisters worked in the bakery on Talbot St going back almost 50 years.
"I started there nearly 44 years ago when I was 14," Denise says. "The Gur Cake we used to sell was only gorgeous. Four of us worked there. Mind you Olive only lasted one day. She wouldn't wash the tables," she continues. Olive, standing at her shoulder nods in agreement.
There is a lot of interest in the free food, the queue for the carousel stretches further than those any of the other attractions. "It's longer than the queues at Disneyland," one punter says.
No-one seems to mind the wait, mind you. Sunshine will do that to people.
Mary Enright from Drumcondra is dressed as Helena Moloney, the prominent Republican, feminist and labour activist. Mary is having a blast.
"A lot of attention will be focused on the 100th anniversary next year but today we are celebrating the 99th and isn't that just as important. The city has been crying out for something like this for a long time," she says.
Just behind Mary/Helena are a couple of suffragettes. It is no accident that Evelyn and Greta Quinlan have come dressed in honour of those who fought for women's right to vote.
"We believe in equal rights," Evelyn says. "And considering the referendum on marriage equality that is coming up we are in a very political space right now so we had to dress up in costumes that would promote equality for all people."
She breaks off to pose for a picture. There is a lot of posing for pictures. For one day the city's main street has turned into Instagram paradise.
On a stage near the spire the Postman's Choir are finishing up a rousing chorus of Danny Boy - a tune that was enormously popular in 1915. It still has resonance today judging from the cheers and cries of encore when the postmen have delivered their last song.
Teresa Gogarty and three generations of her family are here for the day. They have only just arrived but are very impressed by what they have seen so far. "We have a 10-year-old who is learning about 1916 in school so it is nice for her to see it brought to life."
Her father-in-law Joe breaks in to the conversation. "We have a son-in-law whose grandfather took part in the Rising," he says enthusiastically. "He was a runner in Clanwilliam House. We have two candle sticks he took from there as the place crumbled."
The family wander off in search of diversion. It's not hard to find.
Evelyn Quinlan
Helena Moloney
James Larkin
Sean Patrick O Neill
Miriam Lord: 1916 celebrations a masterpiece of solemn dignity
Foreign diplomats join Minister at Glasnevin Easter ceremony
Commemorating 1916: We are all equally entitled to call ourselves republican
Time for a new Proclamation?
Gallery: Road to the Rising
Thinking Anew– Compassion in action
Adoption reforms: Why a referendum may be on the cards
Katherine Zappone is working on amendments to enable disclosure of information for adoptees
'It's a disgrace': Anger in Thurles as Traveller homes dispute goes to court
'Top of the range' houses built for Traveller families remain vacant due to long-running standoff
Presiding at the celebration of the Eucharist felt like the greatest honour
Thinking Anew: We only gather together to share the bread and wine because of Jesus | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,963 |
Le Veau d'or du meilleur montage (en ) est une récompense de l'industrie du film néerlandais récompensant le meilleur monteur.
La récompense est attribuée depuis 2003 dans le cadre de la cérémonie annuelle du Veau d'or.
Lauréats
2003 : Peter Alderliesten pour Phileine zegt sorry
2004 : Mario Steenbergen pour The Last Victory
2005 : Sander Vos pour Paradise Now
2006 : Menno Boerema, Albert Elings, Eugenie Jansen et Chris van Oers pour Jungle Rudy, Kroniek van een familie
2007 : Herman P. Koerts pour Kruistocht in Spijkerbroek
2008 : Robert Jan Westdijk pour Het echte leven
2009 : Esther Rots pour Kan door huid heen
2010 : Job ter Burg pour Tirza
2011 : Sander Vos pour Ingrid Jonker (Black Butterflies)
2012 : JP Luijsterburg pour De Heineken Ontvoering
2013 : Katherina Wartena pour Boven is het stil
2014 : Boudewijn Koole pour Happily Ever After
2015 : Mieneke Kramer pour Prins
2016 : Sander Vos pour Full Contact
2017 : Sander Vos pour Tonio
2018 : Wouter van Luijn pour Wij
2019 : Menno Boerema pour Het Wonder van Le Petit Prince
2020 : Ruben van der Hammen pour Ze noemen me Baboe
2021 : Marc Bechtold pour De Slag om de Schelde
Récompense de cinéma pour le meilleur montage
Veau d'or | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,113 |
\section{Introduction}
\label{sec:sec1}
Correlations between final-state particles in high energy collisions have
been extensively studied during the last decades. They can be due to phase space, energy-momentum
conservation, resonance production, hadronisation mechanisms, or be
dynamical in nature.
In the particular case of identical bosons the correlations
are enhanced by the Bose-Einstein effect~\cite{gold1,gold2}.
These Bose-Einstein correlations (BEC) are a consequence of quantum
statistics. The net result is that multiplets of
identical bosons are produced with smaller energy-momentum differences than
non-identical ones.
Several aspects of BEC have been measured in hadronic Z decays and are
well understood~\cite{LEPBECZ0}. It is natural to expect the
same behaviour in the hadronic decay of a single W. It is, however, not clear
how BEC manifest themselves in a system of two hadronically decaying W's, in
particular between bosons coming from different W's (inter-W BEC).
The separation between two W's before their decay is of the order of 0.1 fm,
compared to a typical hadronisation scale of
several fm. Therefore, due to the large overlap between the two hadron
sources, inter-W BEC cannot be a priori excluded.
However, it is unclear whether these are of the same
type as BEC measured inside a single decaying W, where they are, in contrast to the
traditional Hanbury-Brown and Twiss~\cite{hbt1} picture, not related to the total hadronisation volume.
Together with colour reconnection~\cite{skmodel,arcr}, the poor understanding of the
inter-W BEC effect introduces a large systematic uncertainty in the measurement
of the W mass in the channel $\rm{e^+e^- \to W^+W^- \to
q_1\overline{q_2}q_3\overline{q_4}}$~\cite{Lonnblad:1995mr,appref}.
The current statistical uncertainty of the combined LEP measurement in this
channel amounts to 35 MeV~\cite{lepewnote}, to be compared with the total systematic
uncertainty in this channel of 107 MeV, which is, however, expected to decrease with
improved measurements of colour reconnection. The effect of possible
inter-W BEC amounts to 35 MeV~\cite{lepewnote}. It is thus clear that a better
understanding of the phenomenon would help in reducing this
uncertainty.
Measuring inter-W BEC is challenging in practice because of a low
sensitivity to the effect. This is mainly due to the small fraction of
relevant particle pairs coming from different W's. Moreover, its isolation
from BEC inside a single W requires careful attention and needs to be
as model-independent as possible.
The scope of this paper is the model-independent analysis of the
correlations
of like-sign hadron pairs in $\rm{e^+e^- \to W^+W^-}$, where both W's
decay into hadrons, with the aim of determining the presence and size of
inter-W BEC.
The outline of the paper is as follows: In section~\ref{sec:sec2} the
mathematical
formalism applied throughout this analysis is specified and a brief
overview
of the analysis is given. In section~\ref{sec:sec3} experimental details,
such as the detector setup and WW event selection, are presented. Section~\ref{sec:sec4}
focuses on the mixing procedure employed in order to construct an inter-W
BEC-free reference sample from events where one W decays leptonically.
Section~\ref{sec:sec5} clarifies the details of the Monte Carlo models for
comparison to the data.
In section~\ref{sec:sec6} a detailed
overview
of the numerical analysis of the measured correlation functions is given
including a construction of weights applied in order to increase the
sensitivity of the analysis, the subtraction of background and the
determination of statistical errors and correlations of the bins.
Moreover the parametrisation of the correlation function is discussed.
In section~\ref{sec:sec7} results are presented and in section~\ref{sec:sec8} the
systematic
uncertainties are discussed. Finally, sections~\ref{sec:sec9}
and~\ref{sec:sec10} discuss the results and conclusions are given.
\section{Analysis method}
\label{sec:sec2}
The mathematical method used to extract a possible
inter-W BEC signal is largely based on
\cite{Chekanov:1998hi} and \cite{DeWolf:2001wz}. In
the case of two stochastically independent hadronically decaying W's,
the single and two-particle inclusive densities obey the following relations:
\begin{align}
\rho^{\rm WW}(1) &= \rho^{\rm W^+}(1)+\rho^{\rm W^-}(1), \\
\rho^{\rm WW}(1,2) &= \rho^{\rm W^+}(1,2)+\rho^{\rm W^-}(1,2)+
\rho^{\rm W^+}(1)\rho^{\rm W^-}(2)
+\rho^{\rm W^+}(2)\rho^{\rm W^-}(1),
\label{densities}
\end{align}
\noindent
where $\rho^{\rm W}(1)$ denotes the inclusive single particle density of one W and $\rho^{\rm W}(1,2)$
the inclusive two-particle density of one W. The densities $\rho^{\rm WW}(1)$ and
$\rho^{\rm WW}(1,2)$ then correspond to the single and two-particle inclusive
densities of a fully-hadronic WW event.
Assuming that the densities for the $\rm{\rm W^+}$ and the $\rm{\rm W^-}$ are the same,
which is correct if one does not look at the absolute sign of the particles'
charges, equation~(\ref{densities}) can be re-written as
\begin{equation}
\rho^{\rm WW}(1,2) = 2\rho^{\rm W}(1,2)+2\rho^{\rm W}(1)\rho^{\rm W}(2).
\label{simpler}
\end{equation}
The terms $\rho^{\rm WW}(1,2)$ and $\rho^{\rm W}(1,2)$ can be measured in
fully-hadronic and semi-leptonic WW decays respectively.
A pair or two-particle density $\rho^{\rm WW}(1,2)$ is trivial to construct.
The correlation measurement is made difficult by the fact that
only the correlations between particles coming from different W's
are of interest and there is no way of determining where the particles
originated from. Finally, in order to obtain a correlation function, it is necessary
to construct a reference sample of events without BEC between particles
coming from different W bosons. This sample corresponds, in our case, to the product of the
single particle densities $\rho^{\rm W}(1)\rho^{\rm W}(2)$.
Events where only one of the W's decays hadronically can be used
to address these challenges.
Taking two of these independent
hadronically decaying W's and mixing them to form one event allows an
emulation of a fully-hadronic WW event, having BEC inside each of the W's.
By construction these events
will have no correlations in pairs from different W's and the
measurement becomes a direct comparison between two event samples,
without any model dependence. The event mixing should follow closely the electroweak production of WW
events. Possible biases of the mixing procedure can be estimated by
applying the same procedure to large samples of simulated events.
Hence, the term $\rho^{\rm W}(1)\rho^{\rm W}(2)$ in equation~(\ref{simpler})
replaced by a
two-particle density $\rho^{\rm WW}_{\rm mix}$, obtained by combining particles
from two hadronic W decays taken from different semi-leptonic events.
The details of this ``mixing'' procedure are explained in
section~\ref{sec:sec4}.
Expressed in the variable $Q=\sqrt{-(p_1-p_2)^2}$, where $p_{1,2}$ stands for
the four-momentum of particles $1$ and $2$, equation~(\ref{simpler}) can be
re-written as
\begin{equation}
\rho^{\rm WW}(Q) = 2\rho^{\rm W}(Q)+2\rho^{\rm WW}_{\rm mix}(Q).
\label{msimpler}
\end{equation}
Keeping in mind that equation~(\ref{densities}) was formulated for
independent W decays, test observables can be constructed to search for deviations
from this assumption. Such deviations will indicate that particles from different W decays do
correlate. The observables considered are:
\begin{align}
\Delta \rho (Q) &= \rho^{\rm WW}(Q) - 2\rho^{\rm W}(Q) - 2\rho^{\rm WW}_{\rm mix}(Q), \label{drho}\\
D(Q) &= \frac{\rho^{\rm WW}(Q)}{2\rho^{\rm W}(Q) + 2\rho^{\rm WW}_{\rm
mix}(Q)}. \label{d}\\
\intertext{Given the definition of the genuine inter-W correlation function
$\delta_I(Q)$~\cite{DeWolf:2001wz}, it can be shown that}
\delta_I(Q) &= \frac{\Delta \rho (Q)}{2\rho^{\rm WW}_{\rm mix}(Q)}. \label{deli}
\end{align}
If no inter-W correlations exist, the variables $\Delta \rho (Q)$ and $\delta_I(Q)$ will be zero
for all values of $Q$, while $D(Q)$ will be equal to one.
Inter-W BEC will lead to an excess at small values of $Q$.
The selection of particles and pairs is straightforward, with
the strongest requirement that they should originate from the primary interaction.
Moreover, the selected WW candidates have a significant background which
must be subtracted using a model dependent procedure.
BEC in Z decays have been extensively measured and constitute a
natural basis to compare with inter-W BEC. The correlation functions
measured in Z events use simulated events without BEC as reference
samples. They are therefore close to being genuine correlation
functions but with large model-dependent systematic errors and
some dilution due to particles which are either not pions or which
do no originate from the primary interaction. When the inter-W
correlations are measured it is natural to compare to the Z and
single-W data using the same fitting functions. Since the inter-W
measurement uses data as reference, the model dependence
is no longer present.
The mixing procedure, which allows events to be mixed more than once,
leads to a rather involved description of the statistical properties
of the correlation function.
However, the same mixing can be used to investigate the sensitivity
to the inter-W BEC effect.
The applied mixing reuses semi-leptonic events up to 20 times,
which affects the precision depending on whether pairs
are constructed by mixing or come from inside single W's.
Finally, a pair-weighting technique was devised which improved the
sensitivity and is described in section~\ref{sec:sec6.1}.
For this purpose the mixed reference sample was
used to determine statistically whether particles come
from the same or different W's.
\section{Experimental details}
\label{sec:sec3}
\subsection{The DELPHI detector}
The DELPHI detector configuration for the LEP2 running evolved
compared to the one at LEP1~\cite{Aarnio:1991vx,Abreu:1996uz}.
The main changes relevant to the analysis described in this paper
were the extension of both the vertex and
the inner detectors. This ensured a very good track quality
also in the forward region down to small polar angles.
During the operation of the detector in the latter part of the year 2000 one sector of the TPC
malfunctioned and the data from this period are excluded from the
results.
In order to verify that a track originates from the primary
interaction it was required that the TPC participated in the measurement of
the track. This effectively required
the track to be within the polar angle region
$20$$^\circ$\xspace $< \theta < 160$$^\circ$\xspace.
The reconstructed charged particles were required
to fulfill the following criteria on the momentum, $p$, the momentum error,
$\Delta p / p$, and the impact parameters with respect to the event vertex in
the plane transverse to the beam, $\epsilon_{\perp}$, or parallel to the
beam, $\epsilon_{\parallel}$:
\begin{itemize}
\item 0.2 GeV $< p < p_{\rm beam}$;
\item $\Delta p / p < 1$;
\item $\epsilon_{\perp} < 0.4$ cm;
\item $\epsilon_{\parallel} < 1.0$ cm/sin$\theta$.
\end{itemize}
The two track reconstruction efficiency in DELPHI drops
for opening angles below 2.5$^{\circ}$. Since the mixing procedure does not
necessarily reproduce this drop in efficiency all particle pairs having an
opening angle below 2.5$^{\circ}$ were omitted in all two-particle density distributions.
These requirements lead to a typical efficiency of about
85\%
and reduce the total fraction of secondary tracks to about 5\%.
Secondary tracks are typically tracks from
secondary decays ($K^0$, $\Lambda^0$, etc.) or from secondary interactions
in the beam pipe and with detector material.
Particles not coming from the primary interaction or not being pions will
dilute the observed correlation.
The combined effect was estimated to reduce the measured BEC to about 70\% of
the nominal one. This dilution was not corrected for due to model dependence
and affects all pair densities in nearly the same way. When results from
different experiments are combined it will be necessary to apply such
corrections in order to get comparable results.
\subsection{Selection of WW events}
\begin{table}[!t]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
Year & 1998 & \multicolumn{4}{|c|}{1999}
&2000
\cr
\hline\hline
$\sqrt{s}$ (GeV)& 189 & 192 &196 &200 &202 & 204-209 \cr
$\mathcal{L}$ (pb$^{-1}$) & 158.0 & 25.9 & 76.9 & 84.3 & 41.1 & 163.4 \cr
\hline
\end{tabular}
\caption{The integrated luminosities, $\mathcal{L}$, for the various years of
LEP2 data-taking, expressed in units of pb$^{-1}$. The corresponding
centre-of-mass energies are also given.}
\label{lumitab}
\end{center}
\end{table}
The total analysed dataset amounts to an integrated luminosity of 549.6 pb$^{-1}$,
collected with the DELPHI detector during the years 1998--2000.
A summary of the integrated luminosity per energy point is
given in table~\ref{lumitab}.
The samples of fully-hadronic and semi-leptonic events required for
the WW BEC
analysis were selected using neural networks, developed in~\cite{crosspaper}
and~\cite{chhiggs}.
For the {\bf fully-hadronic} event selection, it was demanded that the events fulfill the
following requirements:
a large enough charge multiplicity,
a large effective centre-of-mass energy, large visible energy and
four or more jets.
The final selection was performed using a neural network trained on thirteen
event variables.
The dominant background contribution
comes from
the $\rm{q\overline{q}(\gamma)}$ events. All other backgrounds are
negligible.
Hadronically decaying ZZ events, which constitute 5\% of the selected
sample, were treated as signal as they, except for events where at least
one Z decays into b-quarks, will have similar space-time kinematics.
A comparison between data and simulated events of the neural network output
for the fully-hadronic selection is shown in figure~\ref{nnoutfh}.
By requiring a neural network output larger than a given value, a desired
purity or efficiency can be reached.
The whole analysis was repeated for several cuts on the neural network output,
selecting samples with an increasing purity, ranging from 83\% to 97\%.
This allowed the choice of an optimal working point, minimising the
sum of the statistical and background uncertainty, corresponding to a
selection efficiency and purity of 63\% and 92\% respectively, with 3252
events selected in total.
\begin{figure}[!tb]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./nnet4q_tot.eps}
&
\epsfxsize=7.5 cm
\epsfbox{./legend2.eps}
\end{tabular}
\caption{The neural network output variable for the fully-hadronic
event selection. The light shaded histogram are the signal,
while the dark shaded histograms correspond to the background
processes. The optimised selection cut is indicated by the arrow.}
\label{nnoutfh}
\end{figure}
\begin{figure}[!bth]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./nnetmu_tot.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./nnetel_tot.eps}
\\
\epsfxsize=8.5 cm
\epsfbox{./nnettau_tot.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./legend.eps}
\end{tabular}
\caption{The neural network outputs for the three semi-leptonic
event selections. The muon channel (a) and electron channel (b)
have small background
contaminations, due to the clear identification of the isolated
lepton. (c) The taus are more difficult to identify, resulting in a
higher background rate. All signal events are shown by the light
shaded histograms, the background events correspond to the dark
shaded histograms. The selection cuts are indicated by the arrows.}
\label{nnoutsl}
\end{figure}
The {\bf semi-leptonic} events were selected by requiring two hadronic jets, a well-isolated
identified muon or electron or (for tau candidates) a well-isolated particle
associated with missing momentum possibly from the neutrino.
The missing momentum direction was required to point away from the beam pipe.
Dedicated neural network trainings were used for all lepton flavours.
Combining all three lepton flavours, an overall efficiency and purity of respectively 58\% and 96\%
was reached, corresponding to 2567 selected events. The three neural network
outputs, corresponding to the three lepton flavours are shown in
figure~\ref{nnoutsl} for data and simulated events.
The WPHACT~\cite{wphact} generator with
the JETSET~\cite{Sjostrand:1995iq} hadronisation model was used for the
simulation of all signal and
four-fermion background events. The $\rm{q\overline{q}(\gamma)}$ background was
simulated using the KK2F~\cite{kk2f} generator and also hadronized with
JETSET.
\section{Mixing procedure}
\label{sec:sec4}
\begin{figure}[!t]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./mompaperf.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./evshap1.eps}
\\
\epsfxsize=8.5 cm
\epsfbox{./evshap2.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./evshap4.eps}
\end{tabular}
\caption{Comparison between simulated fully-hadronic events and simulated mixed events for
(a) the charged particle momenta, (b) the charge multiplicity, (c) the
missing momentum and (d) the reconstructed W mass. In all plots the WPHACT
generator using the BEI model was used.}
\label{mixqual}
\end{figure}
The mixed two-particle density, $\rho^{\rm WW}_{\rm mix}$, was constructed by combining the hadronically-
decaying W's from pairs of different semi-leptonic WW events,
$\rm{q\overline{q}l\overline{\nu_l}}$, from which the lepton was removed and
irrespective of the charge of the W's.
The momentum of each hadronic W can be constructed as the
visible W momentum or the W momentum after a constrained fit
imposing energy- and momentum-conservation and constraining the
two W masses in the event to be equal to 80.35 GeV.
After mixing the W's, the first method gives mixed events which
have smaller missing momenta than the fully-hadronic events while the
second gives larger missing momenta.
It was therefore decided to use the average of the visible W momentum
with a weight of 0.4 and the fitted W momentum
with a weight of 0.6 to obtain the final W momentum. This procedure
gave the best agreement with respect to the missing momenta, and it was
cross checked that the mixing quality does not depend significantly
on the used weights.
In WW events the W's are nearly back-to-back due to momentum
conservation. In mixed events this was accomplished by requiring that
one W polar angle lay within 10$^{\circ}$ opposite to the polar angle of the
other W.
Pairings where the two W polar angles were within
10$^{\circ}$ after inverting the $z$-component
of one W were also accepted. The momenta of the W's were then approximatively
balanced by rotating one W around the beam axis so that the
W's became back-to-back in the plane transverse to the beam axis.
The above transformations reflect the azimuthal and forward-backward symmetry
of the DELPHI tracking detectors.
\begin{figure}[!t]
\begin{center}
\mbox{\epsfig{file=./r2q_long.eps,height=6.5cm}}
\caption{The two-particle correlation function for a single W decay, obtained
from semi-leptonic WW events in data and the BE$_{32}$ Monte Carlo model. The
model was tuned using Z data. Fits with
equation~(\ref{laguerre}) are superimposed.}
\label{r2q}
\end{center}
\end{figure}
\noindent
All mixed events were subjected to the same event selection as the
fully-hadronic events. The agreement between fully-hadronic events and mixed events was verified for several
event variables and single-particle distributions. Small differences in the
distributions are taken into account in the estimation of the systematic errors.
Examples are shown in figure~\ref{mixqual}, comparing the particle momenta,
charge multiplicity, total missing momentum and reconstructed W mass between
simulated fully-hadronic events and simulated mixed events.
These events were generated with the BEI model described in section~\ref{sec:sec5}.
\section{Monte Carlo models}
\label{sec:sec5}
All Monte Carlo generated events were hadronized using the JETSET algorithm
unless stated otherwise. BEC were included using the local reweighting
algorithm LUBOEI~\cite{Lonnblad:1998kk,Lonnblad:1995mr}.
It takes as starting point the hadrons produced by the string
fragmentation in JETSET, where no Bose-Einstein effects are present. Then
the momenta of identical bosons are shifted such that the inclusive distribution
of the relative separation $Q$ of identical pairs is enhanced by a factor
$f_2(Q) \geq 1$, parametrised with the phenomenological form
\begin{equation}
f_2(Q)=1+\lambda \exp(-Q^2R^2),
\end{equation}
where $Q$ is the difference in four-momentum of the pair, $\lambda$ and $R$
are free parameters related to the correlation strength and
the spatial scale of the source of the correlations.
The corresponding change in the momentum of the particles is not unique. In addition, energy and
momentum cannot be simultaneously conserved. In the model, the momentum
is always conserved and afterwards all three-momenta are rescaled by a
constant factor, close to unity, in order to restore energy conservation.
Even when BEC is only allowed for pairs coming from the same W (BEI), this global
rescaling introduces unreasonable negative shifts in the reconstructed W mass.
The BE$_{32}$ variant of LUBOEI overcomes this problem by including extra momentum
shifts to restore total energy conservation, instead of a rescaling of the
momenta.
The BE$_{32}$ model was tuned to hadronic Z decays, keeping all fragmentation
parameters fixed, giving a satisfactory result for all hadronic Z events and
hadronic Z events with
reduced b-content. The resulting LUBOEI
parameters for the correlation strength, $\lambda$, and the correlation
length scale, $R$, are
PARJ(92)=1.35 and PARJ(93)=0.34 GeV$^{-1}$ (= 0.6 fm), respectively.
Monte Carlo sets exceeding ten times the size of the WW data set were simulated at
each centre-of-mass energy.
The tuned model gives a good description of DELPHI's Z data (see
figure~\ref{bgtune}(a)) and the hadronic
decay of single W's. The latter is illustrated in figure~\ref{r2q}, where the
two-particle correlation function defined as
\begin{equation}
R_2(Q)-1=\frac{\rho^{\rm W}(Q)_{\rm data}}{\rho^{\rm W}(Q)_{\rm MC no BE}}-1,
\end{equation}
is shown for selected semi-leptonic W decays in data and MC simulation.
The dip around $Q=0.5-1.0$ GeV in the BE$_{32}$ curve in figure~\ref{r2q} is
understood to come from the conservation of the total multiplicity
in the model and is taken into account by the fit in equation
(\ref{laguerre}).
The signal at low Q values is naturally compensated by
a depletion at higher Q values.
In this paper the label BEI is used for the LUBOEI model
including only BEC between particles from the same W, BEA is used when
all particles are subjected to BEC and BE0 when all BEC are switched off.
\begin{figure}[!t]
\begin{center}
\begin{tabular}{cc}
\hspace{-1.cm}
\mbox{\epsfig{file=./fofq_bea.eps,height=5.5cm}} &
\mbox{\epsfig{file=./purity.eps,height=5.5cm}}\\
\end{tabular}
\caption{a):The fraction of pairs coming from different $W$'s, $F(Q)$, obtained
for a BEI MC sample without pair weights (dashed line) and using pair
weights (full line). b):The estimated pair purity, $p(Q,\gamma,\theta^*)$,
obtained from mixed semi-leptonic events.}
\label{fofq}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\begin{tabular}{cc}
\hspace{-1.cm}
\mbox{\epsfig{file=./qplots_wn.eps,height=9.cm}} &
\mbox{\epsfig{file=./qplots_wy.eps,height=9.cm}}\\
\end{tabular}
\caption{The two-particle densities, $\rho^{WW}(Q)$, $\rho^{W}(Q)$ and
$\rho^{WW}_{mix}(Q)$, for like-sign and unlike-sign particle pairs,
with no pair weights applied (a-c), and including pair weights (d-f).
The background contribution from $\rm{Z(\gamma)\to q\overline{q}(\gamma)}$ events in the $\rho^{WW}(Q)$ distribution
is also shown.}
\label{qplots}
\end{center}
\end{figure}
\section{Numerical analysis}
\label{sec:sec6}
The numerical analysis of the BEC measurement is complicated by
statistical correlations introduced by the methodology. Each charged particle
occurs in several pairs and each semi-leptonic event is
used to produce several different mixed events. All these
statistical correlations were included in the numerical analysis and
the performance evaluated using resampling techniques.
\subsection{Pair weights}
\label{sec:sec6.1}
The sensitivity of any inter-W BEC measurement is limited by the small
fraction of particle pairs coming from different W's, resulting in a small
$Q$ value. This is illustrated in figure~\ref{fofq}(a), where the fraction of
pairs from different W's, denoted as $F(Q)$, is shown.
It drops to around 15\% at very low $Q$ values.
In addition to $Q$, the Lorentz factor, $\gamma$, and the decay angle
of a particle pair, $\theta^*$, are sensitive to whether two particles come
from the same or from different W's. The decay angle, $\theta^*$, is defined
as the angle between the momentum vector of one of the particles in the
two-particle rest frame and the vector sum of the two particle momenta in the
lab frame. As such, each individual pair of tracks can be estimated to have a
probability $p(Q,\gamma,\theta^*)$ to come from different W's.
$p(Q,\gamma,\theta^*)$ was parametrised for this analysis
using large samples of simulated mixed semi-leptonic W decays.
Figure~\ref{fofq}(b) illustrates the distribution of $p(Q,\gamma,\theta^*)$ for pairs with $Q<0.5$ GeV.
The BEI model is shown for the mixed and same W's and compared to the
data results.
A particle is combined with the other particles in one event
when constructing $\rho^{\rm W}(Q)$. It is combined with
the other particles in many other events when constructing
$\rho^{\rm WW}_{\rm mix}(Q)$. Therefore, $\rho^{\rm WW}_{\rm mix}(Q)$
has smaller local fluctuations than $\rho^{\rm W}(Q)$ even though they are
constructed using the same particles.
Using a detailed error analysis it was determined that
pairs from the same W contribute a factor $1+c$ more to the
final variance of the BEC measurements at low $Q$-values than
pairs from different W's. For this analysis a value of $c=1.9$
was determined and used in the following.
The contribution to the statistical variance was estimated for samples of
pairs with a given purity and was found
to be proportional to $1 + c \cdot (1-p(Q,\gamma,\theta^*))$.
Finally, all pairs were weighted by $p(Q,\gamma,\theta^*)$
divided by the above variance factor.
Using these weights, the improvement in the statistical error
on the final measurement was 9\%. This is the reason for choosing the analysis
which weights all particle pairs with their information content
for the final result. The above procedure not only reduces the
statistical error but makes the analysis more sensitive to
pairs which originate from different W's.
The two-particle densities in $Q$ for the combined data set are
shown in figure~\ref{qplots} for both like-sign particle pairs and
unlike-sign particle pairs, with and without pair weights.
In both the fully-hadronic and semi-leptonic samples, the number of
unlike-sign pairs is higher than the number of like-sign pairs at $Q$ values
below 2 GeV. This is due to the large number of resonance decays
with masses in this range.
The region around $Q=0.7$ GeV is dominated by $\rm{\pi^+\pi^-}$ pairs
coming from the $\rho$ resonance which is abundantly present in hadronic
decays of the W. Reflections of three-body decays are also present in the
like-sign distributions. The two-particle densities of like-sign and
unlike-sign pairs for the mixed events
coincide, the reason for this being that all pairs in this
distribution contain particles from different events.
When pair weights are applied, contributions from resonance decays are down-weighted,
therefore the like-sign and unlike-sign spectra become more similar.
\subsection{Background subtraction}
The histograms in figure~\ref{qplots} show the contribution
from $\rm{q\overline{q}(\gamma)}$ background events as they are simulated
with the BE$_{32}$ model.
The density $\rho^{\rm WW}(Q)$ is, consequently,
corrected for this background using the expression
\begin{equation}
\rho^{\rm WW}(Q) = \frac{1}{N_{\rm tot}-N_{\rm q\overline{q}}}\left(\frac{dn_{\rm tot}}{dQ}-\frac{
dn_{\rm q\overline{q}}}{dQ}\right),
\end{equation}
\noindent
where $N_{\rm tot}$ and $N_{\rm q\overline{q}}$ are the total number of
selected events and the number of selected background events,
respectively, and
$n_{\rm tot}$ and $n_{\rm q\overline{q}}$ the respective number of particle
pairs from these events.
The correlation strength parameter, PARJ(92), was re-tuned to a value of
0.9 in order to get a better description of four-jet Z events,
which are more similar to the selected background than inclusive Z decays.
This re-tuning was only used for the background events.
Details on the background tuning are discussed in section~\ref{sec:sec8}.
\subsection{Fit parametrisation}
\begin{table}[!t]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|} \hline
{\footnotesize sample/parameter}&
{\footnotesize $\Lambda_{R_2}$}&
{\footnotesize $R(fm)$}&
{\footnotesize $\epsilon_d$}&
{\footnotesize $\delta_N$} &
{\footnotesize $\chi^2$/ndf}\\ \hline\hline
{\footnotesize BE32}&
{\footnotesize 0.77 $\pm$\xspace 0.02} &
{\footnotesize 0.59 $\pm$\xspace 0.01} &
{\footnotesize -0.78 $\pm$\xspace 0.02} &
{\footnotesize 0.013 $\pm$\xspace 0.003} &
{\footnotesize 141.8/96}\\\hline
{\footnotesize data}&
{\footnotesize 0.64 $\pm$\xspace 0.07} &
{\footnotesize 0.59 } &
{\footnotesize -0.78 } &
{\footnotesize 0.018 $\pm$\xspace 0.017} &
{\footnotesize 114.7/98}\\
\hline
\end{tabular}
\caption{Results of the fit, using equation~(\ref{laguerre}), to $R_2(Q)-1$,
obtained from semi-leptonic WW decays, both for data and the BE$_{32}$ model.}
\label{r2results}
\end{center}
\end{table}
In order to quantify an excess at small $Q$ values, fits were
performed to the inter-W correlation function $\delta_{\rm I}(Q)$.
The choice of fitting function is motivated by the shape of
the $R_2(Q)$ measurements. $\delta_I(Q)$ and $R_2(Q)$ have nearly
the same physical meaning but different systematics since $R_2(Q)$
is dependent on the fragmentation model used.
This means that the optimal fitting functions are not necessarily
the same and that comparisons between $R_2(Q)$ and $\delta_I(Q)$ results must
be done carefully.
It is known from BEC measurements of the hadronic final state
of a Z or a single W
that two particle correlation functions are reasonably well described by
either a Gaussian or an exponential parametrisation~\cite{LEPBECZ0,LEPBECWW}.
However, the BE$_{32}$ Monte Carlo samples show a more detailed
structure in the $Q$ range between 0.5 and 1.5 GeV, as can be seen
from figure~\ref{r2q}.
The slope observed around 1 GeV flattens out above
$Q=2$ GeV. Therefore in the plots of the correlation functions we restrict
to the $Q$ range 0-2 GeV. The fits are, however, performed in the
$Q$ range 0-4 GeV.
The dip around $Q=0.7$ GeV and the following slope of the correlation function
were treated as integral parts of the BE correlation function and as
integral parts of the BEC effect.
Therefore, all $R_2(Q)$ distributions are fitted with the parametrisation
\begin{equation}
R_2(Q)-1=\Lambda_{R_2} e^{-RQ}(1+\epsilon_d R Q)+\delta_N, \label{laguerre}
\end{equation}
where $\Lambda_{R_2}$ denotes the correlation strength, $R$ is related to the
source size, and the term with $\epsilon_d$ accommodates the dip around
$0.6 < Q < 1.0$ GeV.
The auxiliary term $\delta_N$ accounts for small differences in the charged
multiplicity of the signal and reference samples leading to a potential
bias in the normalisation.
The results of the correlation function, $R_2(Q)-1$, are summarised in
table~\ref{r2results} and shown in figure~\ref{r2q}.
In order to compare the measured correlation strength,
$\Lambda_{R_2}$, between data and the BE$_{32}$ model, the $R$ and $\epsilon_d$
parameters were fixed to the values obtained from the model.
The measured correlation strength in data was found to be slightly smaller
than in the model.
The significance of this difference is 1.8 standard deviations,
but it does not include systematic errors. Fitting measurements of
$R_2(Q)$ must typically include an additional slope parameter and implies
that it
takes into account the uncertainty due to fragmentation.
This leads to a large correlation with the $\epsilon_d$ parameter and
means that it is no longer possible to extract meaningful results
on $\epsilon_d$, since it is not known whether to attribute the dip to
the signal or the fragmentation of the reference sample. The $R$ parameter
will also be affected by this but to a lesser degree and finally
the $\Lambda_{R_2}$ parameter is the most stable. No reliable way has been
found to quantify the systematic errors on the $R_2(Q)$ measurements,
and therefore only a qualitative agreement between BE$_{32}$ and the
semi-leptonic data can be demonstrated.
\begin{figure}[!t]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./deltarho_1_wy.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./d_1_wy.eps}
\\
\epsfxsize=8.5 cm
\epsfbox{./deltarho_2_wy.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./d_2_wy.eps}
\end{tabular}
\caption{The $\Delta \rho (Q)$ (a,c) and $D(Q)$ (b,d) distributions for like-sign
and unlike-sign particle pairs, respectively. The BE32 Monte Carlo models
including BEC between all particle pairs (BEA), pairs coming from the
same W (BEI) are superimposed on the plots.}
\label{delta}
\end{figure}
When fitting $\delta_I(Q)$ the normalization, $\delta_N$, can no longer be
described as a simple additive parameter, and is included in the fits as:
\begin{equation}
\delta_I(Q) = \Lambda_I e^{-RQ} (1+\epsilon_dRQ) +
\delta_N (1+\frac{\rho^{\rm W}(Q)}{\rho_{\rm mix}^{\rm WW}(Q)} ).
\label{newdifit}
\end{equation}
The comparison of $\delta_I(Q)$ and $R_2(Q)$ results is limited by
the systematics of the $R_2(Q)$ measurements, as described above.
Througout the paper, for reasons of clarity, the symbols $\Lambda_{R_2}$ and $\Lambda_{I}$ wil be
used when referring to fits of the $R_2(Q)$ and $\delta_I(Q)$ distributions respectively.
\subsection{Statistical errors and correlations}
\label{sect:sect6.3}
The values of the two-particle density distributions for different bins are
statistically correlated, since in general a particle occurs in several pairs and
because of non-Poissonian fluctuations
in the overall particle multiplicity~\cite{alessandro}.
The covariance matrix elements for the histogrammed spectra are given by
$V_{i,j} = \braket{h_i h_j}-\braket{h_i}\braket{h_j}$,
where $h_i$ and $h_j$ are the contents of bins $i$ and $j$.
These covariances are
propagated in the computation of the errors related to the distributions in
equations~(\ref{drho})--(\ref{deli}). The use of pair weights
does not pose any problems within this approach.
The statistical errors shown in the figures in this paper were computed only
from the diagonal elements of the covariance matrices.
Because of the limited precision of $V$, the covariance matrix was
treated carefully from a numerical point of view in the subsequent fits.
The results and the covariance matrix were obtained from
the same data and therefore correlated. This correlation
is effectively enhanced by the substantial correlations
in $V$ and is a cause for biases in the fit results.
By choosing suitable transformations of the fitting functions
this bias can often be reduced. However, for the $\delta_I(Q)$ observable
the value of $\Lambda$ is already nearly unbiased when no transformation is applied.
All these biases were found to be inversely proportional to the number
of fitted events and the effect could therefore be estimated
by comparing the fit results on large samples to the fit
results on data-sized samples.
The final biases on the parameters $\Lambda$ and $\delta_N$ (see section
6.4) which were obtained for data-sized samples,
were estimated to be $0.040+0.031\cdot\Lambda$ and $-$0.0121, respectively.
The biases on $\epsilon_d$ and $R$ were found to be
completely negligible.
All the final fit results were corrected for these biases.
The statistical errors were verified using resampling techniques
and were found
to be unbiased within a relative precision of 2\%.
\section{Results}
\label{sec:sec7}
\begin{figure}[!t]
\begin{center}
\begin{tabular}{cc}
\hspace{-1.cm}
\mbox{\epsfig{file=./delifits_l.eps,height=10.5cm}}
&
\hspace{-.5cm}
\mbox{\epsfig{file=./delifits_u.eps,height=10.5cm}}\\
\end{tabular}
\caption{The $\delta_I(Q)$ distribution for like-sign (a) and unlike-sign (b)
particle pairs. The BE$_{32}$ models including BEC between all particle pairs
and pairs coming from the same W only are superimposed on the plots, together with the fit results
using equation~(\ref{newdifit}). The fit results are with fixed $R$ and
$\epsilon_d$ and are corrected for the bias mentioned in section~\ref{sect:sect6.3}.}
\label{delifits}
\end{center}
\end{figure}
\begin{table}[!t]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|} \hline
{\footnotesize sample/parameter}& {\footnotesize $\Lambda_I$}&{\footnotesize
$R(fm)$}&{\footnotesize $\epsilon_d$}&{\footnotesize $\delta_N$} &{\footnotesize
$\chi^2$/ndf}\\
\hline\hline
\multicolumn{6}{|c|}{{\footnotesize $R$ free and $\epsilon_d$ fixed to BEA values}}\\\hline
{\footnotesize BEA ($\pm$\xspace,$\pm$\xspace)} &
{\footnotesize 1.50 $\pm$\xspace 0.06} &
{\footnotesize 0.72 $\pm$\xspace 0.02} &
{\footnotesize -0.50 $\pm$\xspace 0.03} &
{\footnotesize -0.010 $\pm$\xspace 0.002} &
{\footnotesize 116.4/96} \\\hline
{\footnotesize Data ($\pm$\xspace,$\pm$\xspace)} &
{\footnotesize 1.42 $\pm$\xspace 0.63}&
{\footnotesize 1.14 $\pm$\xspace 0.33}&
{\footnotesize -0.50} &
{\footnotesize -0.002 $\pm$\xspace 0.020}&
{\footnotesize 88.3/97} \\\hline
{\footnotesize BEA (+,--\xspace)} &
{\footnotesize 0.30 $\pm$\xspace 0.03} &
{\footnotesize 0.41 $\pm$\xspace 0.03} &
{\footnotesize -0.60 $\pm$\xspace 0.09} &
{\footnotesize -0.010 $\pm$\xspace 0.002} &
{\footnotesize 110.8/96} \\\hline
{\footnotesize Data (+,--\xspace)} &
{\footnotesize 0.43 $\pm$\xspace 0.22} &
{\footnotesize 0.45 $\pm$\xspace 0.15} &
{\footnotesize -0.60}&
{\footnotesize 0.000 $\pm$\xspace 0.020}&
{\footnotesize 93.2/97}\\\hline
\multicolumn{6}{|c|}{{\footnotesize $R$ and $\epsilon_d$ fixed to BEA values}}\\\hline
{\footnotesize Data ($\pm$\xspace,$\pm$\xspace)} &
{\footnotesize 0.82 $\pm$\xspace 0.29}&
{\footnotesize 0.72}&
{\footnotesize -0.50}&
{\footnotesize -0.005 $\pm$\xspace 0.020}&
{\footnotesize 89.9/98}\\\hline
{\footnotesize BEI ($\pm$\xspace,$\pm$\xspace)} &
{\footnotesize 0.10 $\pm$\xspace 0.05} &
{\footnotesize 0.72} &
{\footnotesize -0.50}&
{\footnotesize -0.009 $\pm$\xspace 0.004}&
{\footnotesize 99.3/98}\\\hline
{\footnotesize BE0 ($\pm$\xspace,$\pm$\xspace)} &
{\footnotesize 0.02 $\pm$\xspace 0.02} &
{\footnotesize 0.72} &
{\footnotesize -0.50}&
{\footnotesize -0.010 $\pm$\xspace 0.002}&
{\footnotesize 98.0/98} \\\hline
{\footnotesize Data (+,--\xspace)} &
{\footnotesize 0.40 $\pm$\xspace 0.18}&
{\footnotesize 0.41} &
{\footnotesize -0.60} &
{\footnotesize -0.001 $\pm$\xspace 0.020}&
{\footnotesize 93.2/98}\\\hline
{\footnotesize BEI (+,--\xspace)} &
{\footnotesize 0.01 $\pm$\xspace 0.03}&
{\footnotesize 0.41} &
{\footnotesize -0.60} &
{\footnotesize -0.005 $\pm$\xspace 0.003}&
{\footnotesize 137.5/98} \\\hline
{\footnotesize BE0 (+,--\xspace)} &
{\footnotesize -0.04 $\pm$\xspace 0.02}&
{\footnotesize 0.41} &
{\footnotesize -0.60} &
{\footnotesize -0.009 $\pm$\xspace 0.002}&
{\footnotesize 138.0/98}\\\hline
\end{tabular}
\caption{Fit results to like-sign and unlike-sign $\delta_I(Q)$ with $R$ free and $R$
fixed. Only statistical errors are shown.}
\label{results}
\end{center}
\end{table}
Inter-W BE correlations were studied as function of the observables
$\Delta \rho (Q)$, $D(Q)$ and $\delta_I(Q)$ as defined in equations~(\ref{drho})--(\ref{deli}). These results
are shown as function of $Q$ in figures~\ref{delta},\ref{delifits} and compared to predictions of
the LUBOEI model.
In the like-sign distributions, an excess in data at low $Q$ values can be
observed. Numerical results were extracted from the $\delta_I(Q)$
distribution, shown in figure~\ref{delifits}. This choice was made since the
$\delta_I(Q)$ is a genuine inter-W correlation function having a clear
physical interpretation. Other results are given for comparison.
Note that none of the results are corrected for pion purity or secondary
tracks, as was mentioned in section~\ref{sec:sec3}.
The $\delta_I(Q)$ distribution for like-sign pairs was fitted
using equation~(\ref{newdifit}).
The fits to the BEA model were performed with all four fit parameters left free.
The correlation strengths obtained from
the BEI and BE0 models were used to estimate a possible bias for the
measurement which is treated as systematic error (see section~\ref{sec:sec8}).
The result where both $\Lambda_I$ and $R$ were left free in the fit provides
the most model-independent result. However, fixing the values of $R$ and $\epsilon_d$ to the ones predicted by LUBOEI
BE$_{32}$ tuned to DELPHI inclusive Z data, makes model comparisons easier
and contains the same statistical significance.
The results on the data and models are summarised in table~\ref{results}.
The main result of this paper is then the observed
correlation strength for like-sign pairs:
\begin{equation}
\Lambda_I (\pm,\pm) =0.82 \pm 0.29 {\rm (stat)} \pm 0.17 {\rm (syst)},
\end{equation}
with $R$ fixed to 0.72 fm. The first error is statistical and the second
error is systematic.
The evaluation of the systematic uncertainty of this result is discussed in section~\ref{sec:sec8}.
The expectation from the BEA model yields
\begin{equation}
\Lambda_{I_{BEA}} (\pm,\pm) =1.50 \pm 0.06 {\rm (stat)}.
\end{equation}
\begin{figure}[!t]
\hspace{-1.cm}
\mbox{\epsfig{file=./cont1.eps,height=7.cm}}
\caption{The 2-d $\Delta \chi^2$ curves are shown for $\Lambda_I$ and $R$.
(a) shows the results for like-sign pairs, while
in (b) the unlike-sign pair result is displayed.
The three contours correspond to a $\Delta \chi^2$ of
1, 4, and 9 respectively. The crosses show the prediction from LUBOEI
BE$_{32}$. The errors corresponding to the LUBOEI predictions are multiplied
by a factor 5 for clarity.}
\label{fig:2d}
\end{figure}
The correlations in unlike-sign pairs were also measured
using the same procedure as for the like-sign pairs.
The result is shown in figure~\ref{delifits}~(b) and summarised in
table~\ref{results}.
The distribution shows some enhancements at low $Q$ when compared to the prediction from
BEI. Fitting with the same expression as for the like-sign pairs, but
with $R$ and $\epsilon$ fixed to the unlike-sign BEA prediction, yields:
\begin{equation}
\Lambda_I (+,-) = 0.40 \pm 0.18 {\rm (stat)} \pm 0.22 {\rm (syst)}
\end{equation}
in agreement with the expectation from BEA:
\begin{equation}
\Lambda_{I_{BEA}} (+,-) = 0.30 \pm 0.03{\rm (stat)}.
\end{equation}
The LUBOEI model prediction for unlike-sign pairs has not been
tuned on Z events as it was the case for the like-sign pairs.
Therefore this prediction should be treated more carefully
and an interpretation will be made in the discussion section.
The numerical results of the simultaneous fit of $R$ and $\Lambda_I$ are shown
in table~\ref{results}.
Since the two parameters are strongly correlated the results
are presented as $\Delta \chi^2$ curves in 2-d plots.
Figure~\ref{fig:2d} shows the fit results for both the like-
and unlike-sign pairs.
In these fits, the value of $\epsilon_d$ is still fixed since the data
do not contain enough information about this parameter.
The position of the dip in the correlation function is hence
forced to scale like $R$.
The systematic error is of the same size as the measurement with $R$ fixed
and not included in the fits shown in table~\ref{results} and figure~\ref{fig:2d}.
\section{Systematic uncertainties}
\label{sec:sec8}
The model-independent analysis of inter-W correlations
performed in this paper avoids
potential biases by the use of the event mixing and a
direct data-to-data comparison. Therefore only few sources of systematic
uncertainty remain to be considered; these are the subtraction of the $\rm{Z \rightarrow q\overline{q}}$
background, the quality of the mixing of W final states
from
semi-leptonic events and the possible influence of colour reconnection.
\label{sect:bgsb}
\begin{figure}[!t]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./bgtune_1.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./bgtune_2.eps}
\\
\epsfxsize=8.5 cm
\epsfbox{./bgtune_3.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./bgtune_4.eps}
\end{tabular}
\caption{Top plots: The ratio of the two-particle densities of
like-sign particle pairs for Z events in data and in
Monte Carlo events using the BE$_{32}$ model with parameters PARJ(92)=1.35 and
PARJ(93)=0.34~GeV$^{-1}$, a) for the inclusive sample, b) for a four-jet sample with
$d_{34}>4.0$~GeV. The two bottom plots show the same comparison
but with a different BE$_{32}$ input parameter PARJ(92)=0.9.}
\label{bgtune}
\end{figure}
The background was subtracted using BE$_{32}$ simulation where the
correlation strength can be varied via the parameter PARJ(92).
Although the standard BE$_{32}$ tuning, using PARJ(92)=1.35, gives a good
description of inclusive
Z-events, it was found that an input parameter strength of PARJ(92)=0.9
gives a better description of Z-events having a clear four-jet
topology. This can be seen in figure~\ref{bgtune} where both
tunings are compared with inclusive Z-decays and four-jet like events.
The four-jet events were selected with the
LUCLUS~\cite{Sjostrand:1995iq} clustering algorithm with
$d_{34}>4.0$~GeV.
Both background samples were subtracted from the data and
half of the absolute difference in the final
result of $\Lambda_I$, 0.075, was taken as a systematic uncertainty due to the lack of
knowledge about BEC in $\rm{q\overline{q}(\gamma)}$ events.
In addition, data with a lower purity than at the working point were
used to estimate the correlations in the background itself.
Four different purity bins were chosen and fits performed in each bin
to obtain the dependency of $\Lambda_I$ on the event purity.
The measured correlation strength depends non-linearly on the correlation
strength parameter PARJ(92) used in the LUBOEI simulation of the background.
A linear interpolation between the model predictions for the two
parameter choices of the following form has therefore been used:
Model($b$) = $b\cdot$ Model(PARJ(92)=1.35) + $(1-b)\cdot$
Model(PARJ(92)=0.90).
The extrapolation was found also to apply satisfactory
outside the range $0<b<1$.
In each purity bin and for several different background subtractions
(as determined by different values of $b$)
the data were then fitted in order to obtain the corresponding
values of $\Lambda_I$ and $\partial\Lambda_I/\partial b$.
Only the uncorrelated errors were used in these fits, so that
the results are independent except for the semi-leptonic
events which are identical for each bin.
The results of these fits are shown in table~\ref{tab:back}.
The value of $b$ was then varied to obtain the background subtraction,
for which the $\Lambda_I$ values are independent of the purity.
The result of this variation gave the result: $b = -0.75 \pm 0.65$,
with $\chi^2=4.1$ for 3 degrees of freedom.
This value of $b$ is consistent with the value used, $b = 0.0 \pm 0.5$,
and is an additional strong indication that the correlations
in background 4-jet events are smaller than for the inclusive
sample ($b=+1$). The data are shown in figure~\ref{fig:back},
where the background with $b=-0.75$ is subtracted.
The amount of subtracted background was varied by 10\% (relative), which
changed $\Lambda_I$ by 0.007, which is negligible.
\begin{table}[!t]
\begin{center}
\begin{tabular}{|l|c|c|c|c|} \hline
Purity bin & a & b & c & d \\ \hline
Purity & 0.60 & 0.77 & 0.89 & 0.97 \\
$\Lambda_I$ $(b=0)$ & 0.39 $\pm$ 0.36 & 1.28 $\pm$ 0.35 & 0.74
$\pm$ 0.23 & 1.33 $\pm$ 0.33 \\
$\frac{\partial \Lambda_I}{\partial b}$ & -0.69 & -0.48 & -0.17 & -0.08 \\
$\Lambda_I$ $(b=-0.75)$ & 0.91 $\pm$ 0.40 & 1.64 $\pm$ 0.36 & 0.87
$\pm$ 0.24 & 1.39 $\pm$ 0.33 \\ \hline
\end{tabular}
\caption{Results for fits of individual bins in purity.
Only the uncorrelated errors are shown.}
\label{tab:back}
\end{center}
\end{table}
In the semi-leptonic events the background is a factor 2 smaller
than in the fully-hadronic events.
It was verified that the topologies of these backgrounds are
identical to those of the signal, i.e.\ mostly 2-fermion decays into
hadrons. These events behave with respect to BEC in the same way as inclusive
Z-events and therefore the BEC in these events are not expected to be
significantly different than in the signal W-events. Finally, these
background events do not suffer from the large extrapolation
uncertainties of the 4-jet background in the fully-hadronic channel.
The estimated systematic errors are shown in table~\ref{tabsyst}.
\begin{figure}[!t]
\hspace{-13. mm}
\begin{tabular}{cc}
\epsfxsize=8.5 cm
\epsfbox{./nielsbg1.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./nielsbg2.eps}
\\
\epsfxsize=8.5 cm
\epsfbox{./nielsbg3.eps}
&
\epsfxsize=8.5 cm
\epsfbox{./nielsbg4.eps}
\end{tabular}
\caption{$\delta_{\rm I}$ is plotted for the 4 different purity bins of table~\ref{tab:back}
(a-d).
The background is subtracted using $b=-0.75$.
The results are strongly correlated between the purity bins.}
\label{fig:back}
\end{figure}
\label{sect:bgre}
The selection of the data and the construction of the
mixed reference sample may introduce distortions in the
two-particle densities and therefore may lead to a non-zero value of
$\Lambda_I$, measured in Monte Carlo samples without inter-W BEC.
The fragmentation models used at LEP do not give a perfect
description of all the details of the soft fragmentation and
correlations. However, they constitute a reliable test for the
magnitude of variation which can be expected for such effects.
The largest absolute value of the measured $\Lambda_I$, 0.125, for these
models (PYTHIA~\cite{Sjostrand:1995iq}, HERWIG~\cite{HERWIG}, ARIADNE~\cite{ariadne}) was taken as a measure for the influence of selection
procedures and mixing method on the measurement.
The above method sums over many possible problems of the mixing.
In order to verify that the sum is not small due to an accidental
cancellation of large effects a weighting procedure was applied for
several event variables and single particle distributions as described
in section~\ref{sec:sec4}.
BEI events were weighted so that the mixed and fully-hadronic events
were in perfect agreement and the weighted events were refitted.
The maximum shift in $\Lambda_I$ was found to be 0.045 which is
easily covered by the inclusive estimation.
Detector effects are very small due to the analysis method.
Any simple variation of detector performance was found to give
zero shift in $\Lambda_I$. Higher order effects are still possible,
but these are also correlated with fragmentation properties
and it was therefore assumed that they are covered
by the previous estimate.
A systematic uncertainty was attributed to the Colour
Reconnection (CR) effect. This effect could have, in addition to
inter-W BEC drastic consequences for the W mass
measurement in the fully-hadronic channel~\cite{skmodel,arcr}.
As for BEC, it violates the
assumption that the two produced W bosons decay independently of each
other.
Colour reconnection occurs when independent colour singlets interact strongly
before hadron formation.
In fully-hadronic W decays it recombines partons from different parton
showers. After fragmentation, the resulting hadrons carry therefore a
mixture of energy-momentum of both original W's.
The CR effect has been modelled in various
ways~\cite{skmodel,arcr,Lep2yellow}. Only the extreme
models~\cite{skmodel}, where reconnection
occurs in all events, have been ruled out by the LEP
experiments~\cite{lepewnote}; however the absence of CR
is also disfavoured by the same information.
For this reason, three possible models of CR as
implemented in JETSET, ARIADNE and HERWIG, were used to estimate their
influence on this measurement.
The maximum difference in $\Lambda_I$ between the CR samples and
their equivalent models without CR implementation, 0.07, was found with the
HERWIG implementation and was conservatively taken as systematic uncertainty due to
the CR effect.
Finally, half of the bias correction (0.033) described in section~\ref{sect:sect6.3}
was conservatively attributed as a systematic error due to fit biases.
The total systematic uncertainty on the measured $\Lambda_I$ value is
the sum in quadrature of the contributions listed in
table~\ref{tabsyst} for both like-sign and unlike-sign particle pairs.
\begin{table}[!t]
\begin{center}
\begin{tabular}{|c|c|c|} \hline
syst source & contribution to $\Lambda_I (\pm,\pm)$ & contribution to $\Lambda_I (+,-)$\\
\hline\hline
background BE model & 0.07 & 0.02\\\hline
semi-leptonic Bg. & 0.01 & 0.01 \\\hline
cuts \& mixing & 0.12 & 0.04\\\hline
Colour Reconnection & 0.07 & 0.22\\\hline
Bias Correction & 0.03 & 0.03 \\\hline
Total syst. & 0.17 & 0.22\\\hline\hline
\end{tabular}
\caption{A breakdown of the systematic errors for the $\Lambda_I$
measurement with fixed $R$, for like-sign and unlike-sign particle pairs.}
\label{tabsyst}
\end{center}
\end{table}
\section{Discussion}
\label{sec:sec9}
The result, $\Lambda_I=0.82\pm 0.29\pm 0.17$, presented in section~\ref{sec:sec7}
shows an indication for BEC in like-sign pairs between two hadronically
decaying W bosons at the level of 2.4 $\sigma$ (standard deviations).
The effect is 2.2 $\sigma$ below the prediction of BE$_{32}$, assuming that
the systematic uncertainties due to cuts and mixing are 100\% correlated. The
spatial scale, $R$, of the correlations is higher at the level of 1.3 $\sigma$,
based on statistical uncertainties only.
The QCD background to the WW signal is by itself of interest to study,
and the data show that the BEC are weaker in these events than in
inclusive hadronic Z events.
The measurement where both $R$ and $\Lambda_I$ are left free
and measured from the data contains the most model-independent
information, while the result with fixed $R$ is optimal for
testing the excess in the direction predicted by BE$_{32}$.
The results for unlike-sign pairs are difficult to interpret
due to a large model dependency and systematic errors.
The unlike-sign pairs show a smaller excess at low Q values. The excess
is at the 1.4 $\sigma$ level and in some agreement with the BE$_{32}$
prediction, but a bit too large if the unlike-sign effect is scaled
to follow the like-sign.
The unlike-sign pairs are statistically
correlated with the like-sign pairs at the level of 60\%.
Therefore it is not possible to rule out a large statistical
component of these effects.
The tuning of the BE$_{32}$ model was verified using
the semi-leptonic WW events.
The overall agreement including the
description of the dip around $Q=0.7$ GeV was found to be excellent,
except that the data
show a bit less correlation than the prediction tuned on Z's.
To this 1.8 $\sigma$ statistical difference a systematic
uncertainty which comes from extrapolating from Z
data using the model has to be added. This uncertainty which is
difficult to control is also the reason why the main result of this
paper does not rely on BE$_{32}$ simulation.
Inside the LUBOEI models the input parameter PARJ(92) determines the
strength of the correlations. However, due to non-linear effects
in the implementation, there is no one-to-one correspondence
between the generated strength and the measured strength, $\Lambda_{R_2, I}$.
Even with a fixed value of PARJ(92) the observed $\Lambda_{R_2}$ does
depend on the multiplicity of the event. The used value of
PARJ(92)=1.35 leads to a measured value of $\Lambda_{R_2}=0.77 \pm 0.02$
for the semi-leptonic BE$_{32}$ events while the BEA result is $\Lambda_I=1.50 \pm 0.06$.
The two BE$_{32}$ results come from fits to $R_2$ and $\delta_I$
respectively. The systematics are expected to be small on these
numbers because they compare events which use the same fragmentation.
The above observation can be combined with the information that the
improvement in the statistical sensitivity to BE$_{32}$ was smaller
(9\%) than the improvement (17\%) expected from pure statistics.
This indicates that the BEA not only adds correlations between
particles from different W's compared to BEI, but increases
the strength of all correlations significantly. This enhancement
scenario is not supported by the 4-jet Z-data where
the BE$_{32}$ model clearly overestimates the correlations
when tuned on inclusive events.
Both $\Lambda_{R_2, I}$ and $R$ are subject to additional
corrections if their values are to be combined with other experiments.
$\Lambda_{R_2, I}$ is diluted due to pair impurities to a level of
about 70\%, while $R$ is only comparable to either
other data or models.
In conclusion this paper presents a model-independent measurement
of inter-W BEC.
The measurement does assume that intra-W BEC are
the same in 2-jet and 4-jet WW decays.
The direction in which the signal is looked for
is motivated by the measured BEC in semi-leptonic WW events and
implemented via the BE$_{32}$ model tuned to Z data.
The measurement is firmly based on the mixing of
the
hadronic final state of independent W bosons from semi-leptonic WW
events.
This technique yields small systematic uncertainties in spite of the
small fraction of particle pairs from different W's. A weighting
technique
allowed the effective purity to be raised to about 20\%.
The treatment of statistical errors and bin correlations of
the
correlation function allows for a reliable specification of the
statistical error of the quoted correlation strength. The remaining
systematic uncertainties and their dependence on model parameters
were carefully studied.
\section{Conclusion}
\label{sec:sec10}
Overall, there is an indication of correlations in like-sign pairs
from different W's with a significance of 2.4 standard deviations.
The results are 2.2 standard deviations lower than the expectation
of the BE$_{32}$ model which was tuned to DELPHI Z data and verified
on semi-leptonic WW events.
The spatial scale of the correlations is larger but consistent
with BE$_{32}$. The results from the unlike-sign pairs are inconclusive,
and their interpretation is model dependent.
The prediction of the LUBOEI model is disfavoured not only
by the behaviour of the WW data but also through the
investigations of the background. It is clear
that developments in the theoretical side of fragmentation
models are needed before the BEC results presented in this
paper can be fully understood.
These final results of the DELPHI experiment have a limited
statistical
accuracy. The precision would be significantly increased by a combination with the
results of the LEP experiments which can be found in~\cite{LEPBECWW}.
\subsection*{Acknowledgements}
\vskip 3 mm
We are greatly indebted to our technical
collaborators, to the members of the CERN-SL Division for the excellent
performance of the LEP collider, and to the funding agencies for their
support in building and operating the DELPHI detector.\\
We acknowledge in particular the support of \\
Austrian Federal Ministry of Education, Science and Culture,
GZ 616.364/2-III/2a/98, \\
FNRS--FWO, Flanders Institute to encourage scientific and technological
research in the industry (IWT), Belgium, \\
FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil, \\
Czech Ministry of Industry and Trade, GA CR 202/99/1362,\\
Commission of the European Communities (DG XII), \\
Direction des Sciences de la Mati$\grave{\mbox{\rm e}}$re, CEA, France, \\
Bundesministerium f$\ddot{\mbox{\rm u}}$r Bildung, Wissenschaft, Forschung
und Technologie, Germany,\\
General Secretariat for Research and Technology, Greece, \\
National Science Foundation (NWO) and Foundation for Research on Matter (FOM),
The Netherlands, \\
Norwegian Research Council, \\
State Committee for Scientific Research, Poland, SPUB-M/CERN/PO3/DZ296/2000,
SPUB-M/CERN/PO3/DZ297/2000, 2P03B 104 19 and 2P03B 69 23(2002-2004)\\
FCT - Funda\c{c}\~ao para a Ci\^encia e Tecnologia, Portugal, \\
Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134, \\
Ministry of Science and Technology of the Republic of Slovenia, \\
CICYT, Spain, AEN99-0950 and AEN99-0761, \\
The Swedish Research Council, \\
Particle Physics and Astronomy Research Council, UK, \\
Department of Energy, USA, DE-FG02-01ER41155. \\
EEC RTN contract HPRN-CT-00292-2002. \\
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,355 |
Workers' compensation is a specialized type of insurance for people who are injured in work-related accidents. California requires that all employers carry workers' compensation insurance for their employees. Benefits under workers compensation include medical costs and wage replacement. In some cases, workers' compensation might include death benefits for surviving family members of employees. The main benefit of workers' compensation for employers is that their employees, by accepting workers' comp benefits, are no longer allowed to file lawsuits against the company. In turn, employees get the benefit of having their work-related injuries paid for by their employer's insurance.
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Our goal is your recovery, both physical and financial. Don't hesitate to contact our firm today! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,099 |
Q: How to quality check this data in R? I have a df1 with 820 rows:
ID College Score Score3
EI01 0 1 2
EI01 0 1 6
EI08,EI07 1 4 4
EI08,EI07 1 4 8
EI02 0 0 9
EI05 1 2 2
EI06 1 10 12
I have a df2 with 713 rows:
ID Points
EI01 20
EI08,EI07 12
EI02 30
EI04 10
I have tried merging these two df by "ID" with all=FALSE but my new merged df is 864 rows, when I would only like it to maximally be 820 rows given the # in df1. I would like to know what I am doing wrong. Additionally, I would like to make a df3 that includes all of the IDs in df1 that are not present in df2.
A: If you want to join data and have an automatic QC check, you could consider using the tidylog package, which replaces many dplyr functions with more verbose ones.
Here's an example of tidylog::left_join() with your sample data:
> left_join(df1,df2)
Joining, by = "ID"
left_join: added one column (Points)
> rows only in x 2
> rows only in y (1)
> matched rows 5
> ===
> rows total 7
ID College Score Score3 Points
1: EI01 0 1 2 20
2: EI01 0 1 6 20
3: EI08,EI07 1 4 4 12
4: EI08,EI07 1 4 8 12
5: EI02 0 0 9 30
6: EI05 1 2 2 NA
7: EI06 1 10 12 NA
To find IDs that are in df1 and not in df2 you can use:
df1 %>%
filter(!ID %in% df2$ID) %>%
pull(ID)
[1] "EI05" "EI06"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,822 |
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The company serves in particular the market regions of German-speaking Switzerland, Ticino, southern Germany and Austria. For special products, it cooperates closely with other Model companies. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,606 |
import { each } from './utils'
/**
* Returns a mapping of the getDataBinding keys to
* the reactor values
*/
function getState(reactor, data) {
let state = {}
each(data, (value, key) => {
state[key] = reactor.evaluate(value)
})
return state
}
/**
* @param {Reactor} reactor
*/
export default function(reactor) {
return {
getInitialState() {
return getState(reactor, this.getDataBindings())
},
componentDidMount() {
this.__unwatchFns = []
each(this.getDataBindings(), (getter, key) => {
const unwatchFn = reactor.observe(getter, (val) => {
this.setState({
[key]: val
})
})
this.__unwatchFns.push(unwatchFn)
})
},
componentWillUnmount() {
while (this.__unwatchFns.length) {
this.__unwatchFns.shift()()
}
},
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,382 |
Getting Quiet… and Waiting
Commisions and Custom Art
Meditation Circles
Collectors' Blog
Understanding Clouds
Basic Class
Logo Design – The Lighting Connection
by Rita | Comments Off
You know the old saying, "Jack of all trades, master of none." Well, I beg to differ.
I ended 2017 doing some new logo designs, starting off with The Lighting Connection in Denver. A wonderful group of people, ready to let the design process unfold and a pleasure to work with. (See their info below.) Although I don't often do logo design, they had faith in my visual and imaginative skills to gave me the opportunity to blend some old-school ways with today's easy access to fonts and graphics.
From 1986 – 1991 I spent all day, every day doing design work — paste-up, hand lettering, illustration, darkroom work, color separations for film — for a printing company outside Kansas City which had national clients. As it turns out, six years of total immersion will bring you to that level of mastery enough to call it up whenever you need.
If you're contractor in the Denver area, contact this fantastic company!
the-lighting-connection.com
The Lighting Connection – Our Story
The Lighting Connection started operations in 1984 and has consistently been providing excellent lighting solutions and service since inception. Our business is unique in the Denver market as we do not offer a showroom nor do we have walk-in traffic. Our sole focus is you, the Colorado builder. We work closely with builders and their teams — including general contractors, electricians, superintendents, and purchasing agents to provide comprehensive design and delivery of complete lighting packages (residential, multi-family and commercial) across Colorado.
Our team has more than 75 combined years in the lighting industry, and we have been providing lighting solutions to many of the largest home builders (including national, regional and local builders) here in Colorado. We have the industry and logistics experience to deliver on all your lighting needs. Our team of professionals currently delivers lighting packages to more than 100 sites per week.
To deliver competitively-priced lighting fixtures with exceptional customer service. The Lighting Connection has a legacy of reliability in the lighting industry.
Our company will not waver from its mission, which means competitively-priced lighting fixtures and round-the-clock customer support. We will provide continued stellar builder support and comprehensive lighting packages for all building needs.
Also posted in Creative Process, Featured, What's New?
Green Turtle Cay, Bahamas
Bita Beach, Green Turtle Cay, Bahamas 9″ x 12″ Oil on Gesso Board
I spent January exploring the most fabulous 1.5 square miles, walking beaches instead of shoveling record snowfall in Colorado.
Green Turtle Cay is one of the barrier islands off mainland Great Abaco The Bahamas. It is considered part of the "Abaco Out Islands" and is 3 miles long and ½ mile wide. It was named after the once abundant green turtles that inhabited the area. Wikipedia
A wonderfully generous painting student has a house on White Sound. She and her husband offered me the use of their house for a month after taking my course. (Understanding Clouds) That course is no longer offered through Artist's Network University, so I'll be making it available through my own website later this year. Yes, this was an online course, so I had never met my hosts in person until two planes and a ferry brought me to their dock. It pays to have a good sense of people and I feel that this whole adventure is a natural outcome of teaching with a desire to inspire creativity and take artistic risks. Learning to paint better is a byproduct.
While on the island, I used the solitude to work on a first draft of a new screenplay. ( ritadoyleroberts.com ) Now that the story is nearly ready to send out, I am happy to be painting again…. but I do miss those beautiful beaches.
Also posted in Blog, Creative Process, Featured, Uncategorized, Writings
New Print Available – Mount Blanca
"Dusting of Snow on Mount Blanca"
12″ x 14″ Giclee Print on Canvas, Stretched
Sides printed black and ready to hang.
I was recently contacted by a someone who grew up in the San Luis Valley but now lives in Utah. He misses the landscape of his home and nothing says SLV more than Mount Blanca. He found me and this painting in a Google search. This piece is in the permanent collection of Trinidad State Junior College – Alamosa Campus so I had this print made for him. Now it's available to all.
History of Mount Blanca
(from Wikipedia)
Blanca Peak is known to the Navajo people as the Sacred Mountain of the East: Sisnaajiní[8] (or Tsisnaasjiní[9]), the Dawn or White Shell Mountain. The mountain is considered to be the eastern boundary of the Dinetah, the traditional Navajo homeland. It is associated with the color white, and is said to be covered in daylight and dawn and fastened to the ground with lightning. It is gendered male.[8]
Summitpost notes that "the first recorded ascent of Blanca by the Wheeler Survey was recorded on August 14, 1874, but to their surprise they found evidence of a stone structure possibly built by Ute Indians or wandering Spaniards."[10]
Also posted in Blog, Collectors, Creative Process, Featured, What's New?
Open Studio 2016
Tuesday through Saturday
Or call for appointment: 719-852-6976
The studio is loaded with original art plus signed, giclee prints and greeting cards. I'll be doing small watercolor paintings and/or oil pastel drawings each day of the open studio. These new pieces will be available immediately to the first bidder.
Much of my year has been focused on writing. I've revised a children's book manuscript and now I am writing a screenplay for the animated film version of the same story. It's a different brain process, making vivid pictures with words. The result has been that my artwork has become less literal, less representational. I've been exploring color, texture and unconscious prompts. This image is a detail of one of those explorations.
I still do representational work and I'll show that here too, during Open Studio. I'd love for any of these paintings to find their permanent homes while the Studio is open for visitors. This is your annual opportunity to take home affordable art work, directly from the artist, or just to see what's going on.
Please share this event with your friends.
Daily posts will be made at these links:
https://www.facebook.com/RitaRoberts.Artist/
https://www.facebook.com/RitaRoberts.Prints/
https://www.facebook.com/rita.roberts.7982
https://twitter.com/RitaDRoberts
https://www.linkedin.com/in/ritadroberts
Also posted in Blog, Creative Process, Featured, Uncategorized, What's New?
Colorado Supports the Arts!
The Colorado Creative Industries Division of the Office of Economic Development and International Trade (CCI), has awarded me with a Career Advancement grant. The grant is to be used toward developing, publishing and marketing my Children's Picture Book, Augustina, as well as a screenplay for animated film to accompany the book.
She's making progress toward being available in bookstores!
Augustina, is a humorous story about a lovable, barnyard oddball who dances her way into the carnival spotlight, and discovers her true identity — she is a heifer-potamus!
Educators, day care providers and social workers enthusiastically support this story:
"I see all kinds of application potential, everything from children who are adopted to helping children gain ego strength." — Carol C., Clinical Social Worker
Please contact me for more information about Augustina.
To capitalize on attending conferences and industry workshops I am simultaneously developing an additional story for both book and film. Hint: the characters are perfect for cute Halloween costumes in coming years. Stay tuned!
Also posted in Featured, What's New?, Writings
Art for the Environment
"The Fire Below" 14″ x 14″ Oil on Canvas
Art for the Endangered Landscape Show and Sale
This painting sold at an exhibit sponsored by Community Partnerships Gallery at Adams State University, Alamosa, Colorado last December. Proceeds go to help us keep Wolf Creek Wild.
Wolf Creek Pass and its heavily used highway corridor hold a critical place in the ongoing struggle to balance natural systems with human disruption. The pass bisects some of the wildest remaining primitive country in the southern Rocky Mountains. To the north of the pass is half-million acre Weminuche Wilderness and to the south is the South San Juan Wilderness holding 160,000 acres. The boundaries of these two wilderness areas come to within 6 miles of each other at their closest proximity, but those are treacherous miles for wildlife and plant populations to negotiate.
Wolf Creek was also the area selected to release the reintroduced Canada lynx, an endangered species throughout its historic range.
The most controversial endangerment to consider is a 10,000- person resort complex proposed by developers on a piece of private land adjacent to Wolf Creek Ski Area.
From its origination as a questionable land exchange in 1986 to its current incarnation of transfering yet again more public land, this proposal has galvanized opposing factions. For more in-depth information on this aspect go to Wolf Creek Developments.
The Art for the Endangered Landscape project strives to shed a different light on development issues from the aspect of loss of visual beauty. This art celebration also honors what we have now and what we have to lose in a tangible and visceral manner. I am happy to support this conservation effort!
Also posted in Featured, What's New?
Painting Acquired by Trinidad State Junior College
"Early Snow on Mt. Blanca" 14 " x 16″ – Oil on Linen
Colorado Creative Industries has acquired public art through direct purchase of existing artwork for Trinidad State Junior College's (TSJC) Valley Campus, located in Alamosa, Colorado, in conjunction with the project to update their main campus building. Colorado Art in Public Places Program requires one-percent of capital construction funds for new or renovated state buildings be set aside for the acquisition of works of art at the project site.
About the Painting
"Early Snow on Mt. Blanca" is one of the pieces acquired by this program. This painting depicts a familiar sign of winter in the San Luis Valley and is one that is close to my heart as an all-time favorite piece. The sun sets in the west while Mt. Blanca is briefly coated in pink to red hues. It is the prominent view walking or driving toward my home. I am happy that this painting will remain in the San Luis Valley.
Mount Blanca (Sisnaajini) Navajo Sacred Mountain
The mountain is considered to be the eastern boundary of the Dinetah, the traditional Navajo homeland. Blanca Peak should be thought of as the 'north arrow' on a map, which determines the orientation of a person's mind and physical presence on earth. (http://navajopeople.org/blog/mount-blanca-sisnaajini-navajo-sacred-mountain/)
About Trinidad State Junior College – Valley Campus:
The Valley Campus of Trinidad State is located in downtown Alamosa. This branch of the school is a commuter campus of approximately 600 students, that features friendly staff and small class sizes. Hands-on programs include Nursing, Massage, Auto and Diesel Mechanics and Welding. This campus also has an excellent Aquaculture program where students can learn how to run a fish hatchery. The Law Enforcement Academy and Auto Mechanics programs operate training sites at off campus locations. Students can also take a variety of courses offered in traditional classrooms including Business, Chemistry and Math. The Valley campus is unique in that the average age of students is late 20s. This non-traditional student population consists of students who are "starting over" and re-defining their career. At 7,600 feet in elevation, Alamosa sits in the middle of the San Luis Valley, the highest and largest mountain desert in the world. To the west are the majestic San Juan Mountains and to the east rises the rugged Sangre de Cristo Range. Great Sand Dunes National Park is located 35 miles to the north east and features the tallest dunes in North America. (From the announcement on callforentry.org)
My thanks to Trinidad State Junior College and the Colorado Creative Industries!
Also posted in What's New?
Art for the Endangered Landscape Honoring Wolf Creek
"The Fire Below" and "Sunset Clouds" 14″ x 14″ Oil on Canvas, $1,960 each. Click on image for a larger view.
Paintings, Sculpture, Photography, Jewelry by 40 Regional Artists. Proceeds go to help us keep Wolf Creek Wild.
Opening reception:
at 4:00pm – 7:00pm
For more information go to www.slvec.org
You may visit the show Monday through Friday from 9am to 5 pm from December 7 to December 19, 2015
Community Partnerships Gallery at Adams State University, Alamosa, Colorado.
Wolf Creek Pass and its heavily used highway corridor hold a critical place in the ongoing struggle to balance natural systems with human disruption. The pass bisects some of the wildest remaining primitive country in the southern Rocky Mountains. To the north of the pass is half-million acre Weminuche Wildernss and to the south is the South San Juan Wilderness holding 160,000 acres. The boundaries of these two wilderness areas come to within 6 miles of each other at their closest proximity, but those are treacherous miles for wildlife and plant populations to negotiate.
The Art for the Endangered Landscape project strives to shed a different light on development issues from the aspect of loss of visual beauty. This art celebration also honors what we have now and what we have to lose in a tangible and visceral manner.
Also posted in Blog, Collaborative Works, Featured, Uncategorized, What's New?
Cover Art for Colorado Central Magazine
Colorado Central Magazine
cozine.com
About the painting: This oil painting depicts fall cottonwoods on the historic Garcia Ranch. Reyes Garcia is now the steward of this ranch and allowed me to take a walk and paint this beautiful piece of property. As a retired professor of philosophy, environmental and indigenous studies, Reyes is deeply attuned to the legacy of his family's land and the way of life it has provided for generations. With the Garcia family having originally settled in Conejos County in the 1850's, he has a long history rooted in the special area between the Conejos and San Antonio Rivers in the southern part of the San Luis Valley.
Conserving the land and water is a way "to make my own small contribution to preserving the family legacy of ranching and the land-based culture of the ranchero tradition," Garcia writes. "… I came to understand this tradition includes putting into practice ecological values by virtue of an instinctual love of the land that engenders good stewardship and a deep respect for all life forms, the seasonal rotation of livestock and their humane treatment, the acequia irrigation system especially, the transmission of skills which make self-reliance possible…"
in 2013, the Rio Grande Headwaters Land Trust worked with Reyes to complete a voluntary conservation easement on the spectacular Garcia Ranch, to insure that this working ranch will remain intact with its senior water rights in perpetuity. Learn more about RiGHT's ongoing conservation work and the ranch at www.riograndelandtrust.org
Also posted in Blog, Collaborative Works, Collectors, Creative Process, Featured, Uncategorized, What's New?, Writings
Paint-Out to Celebrate the Wilderness Act
The Wilderness Act is well known for its succinct and poetic definition of wilderness:
"A wilderness, in contrast with those areas where man and his own works dominate the landscape, is hereby recognized as an area where the earth and its community of life are untrammeled by man, where man himself is a visitor who does not remain."
"Mountain Buddies" 10″ x 10″, Oil on Canvas. Available after the Paint Out at Frame Shop Creede.
The Wilderness Act of 1964 (Pub.L. 88–577) was written by Howard Zahniser of The Wilderness Society. It created the legal definition of wilderness in the United States, and protected 9.1 million acres (36,000 km²) of federal land. The result of a long effort to protect federal wilderness and to create a formal mechanism for designating wilderness, the Wilderness Act was signed into law by President Lyndon B. Johnson on September 3, 1964 after over sixty drafts and eight years of work. — from Wikipedia
The two equine friends are regulars to the Park Corrals. Friendly, always ready for a pet or a treat. Carrots are on my list of supplies to take some carrots to the paint out.
Also posted in Blog, Featured, What's New?
Coming Clean
Embracing the Snail
It's All Perfect
Pikes Peak Watercolor Society – Demo
© Rita Roberts - Web Design by Autumnbridge Media LLC | {
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"Read My Mind" is the first single by Sweetbox from the album Jade, with Jade Villalon as a frontwoman. "Read My Mind" samples Cavalleria Rusticana from Pietro Mascagni.
An acoustic version and the music video of the song can be found on the album Jade (Silver Edition). A live version of the song can be found on the album Live.
The song was composed by Geoman and Villalon, and the music video for the song was directed by Marc Schoelermann.
Track listing
Credits
The 'EP Edition' was remixed by Satoshi Hidaka.
The song was covered by Taiwan band S.H.E. releasing it as Always on My Mind.
References
Sweetbox songs
2002 singles
Songs written by Jade Villalon
2002 songs
Warner Music Group singles | {
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{"url":"https:\/\/docs.heavenland.io\/ecosystem\/heavenmarket\/staking\/mechanism","text":"# Mechanism\n\nHeavenland's staking differs from other staking solutions users are used to. This section describes details of Heavenland's staking mechanism.\nHeavenland's staking is fully managed by Staking Program on the Solana blockchain, which has been developed by the Heavenland team. To enable staking on an account, the account must be first initialized to reserve some space on the Solana blockchain to store account staking data. Any account can create a fixed limit of 100 stakes. The initialization is a one-time payment of approx. 0.05 SOL.\nThe Staking Program is composed of 2 parts - one that holds the staked assets (HTO and NFTs) and that's been already audited, and the part that handles the rest of the staking logic, which is, at the time of writing, under security audit.\n\n## HTO Release Rate [RR]\n\nThe total staking reward distributed among all active stakers comes from the in-game economy and pre-mine, which will be distributed in the first five years according to Tokenomics. The pre-mine source should incentivize early staking, while the in-game source will dominate the later stages. In the first months, the pre-mine source is derived from the expected number of HTO in circulation to give early stakers up to a 10% monthly reward.\nRR defines how many HTOs should be distributed among stakers every hour. RR will be changed once per 30 days, and its change will be announced a minimum 5 weeks ahead.\n\n## Heavenland Constant [$c_{HL}$]\n\nHeavenland defines\n$c_{HL}$\n, the Heavenland factor, that is common to all stakable assets. The primary purpose of\n$c_{HL}$\nis to limit the maximum number of HTOs that can be staked in total.\n$c_{HL}$\nwill be derived from the circulating HTO as Heavenland wants to prohibit staking all HTO in the early stages on just a few selected NFTs and will increase from the initial value of\n$c_{HL} = 30$\nas the HTO circulating supply increases.\nAll the changes to\n$c_{HL}$\nwill be announced at least 2 weeks in advance.\n\n## NFT Constant [$c_{NFT}$]\n\nTo define the staking limit for each NFT, Heavenland introduces\n$c_{NFT}$\n, a constant derived from NFT attributes.\n$c_{NFT}$\ntogether with cHL defines the total number of HTOs that can be staked on a given asset as\n$c_{HL} \\cdot c_{NFT}$\nUnlike\n$c_{HL}$\n, which will increase over time,\n$c_{NFT}$\nwon't change at all. Details for registered collections are described in the Stakable NFTs section.\n\n## Staking Multiplier [F]\n\nThe stake allows for periods of different duration, multiples of 30 days, with the shortest period being 30 days and the longest period being 180 days. The period defines your staking multiplier F.\nFor the shortest period of 30 days, F = 1. Every 30 days added to the staking period increases F by 0.2, so F = 2 for the longest period of 180 days.\n\n## Weighted HTO [wHTO]\n\nThe number of your staked HTO, together with the staking multiplier F, defines weighted HTO, wHTO, which is crucial for calculating the staking reward and voting power in DAO.\n$wHTO = F \\cdot HTO.$\nFor further calculations, it's helpful to introduce\n$\\Sigma wHTO$\n, which is the sum of wHTO of all stakes.\n\n## Reward [R]\n\nLet's assume a stake with wHTO runs from\n$t_{start}$\nto\n$t_{end}$\n(interval between these times will be 30, 60, \u2026 or 180 days), and let\n$t_{claim}$\nbe the time of the last claim (before the first claim,\n$t_{claim} = t_{start}$\n). In time\n$t_{now} = \\min(t_{now}, t_{end})$\n, staker executes claim function. Let's denote T as the interval (in hours) between\n$t_{claim}$\nand\n$t_{now}$\n. His reward R will be\n$R = wHTO \\cdot RR \/ \\Sigma wHTO \\cdot T.$","date":"2023-03-21 15:03:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 22, \"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\": 0.42853525280952454, \"perplexity\": 4650.356394467849}, \"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\/1679296943698.79\/warc\/CC-MAIN-20230321131205-20230321161205-00540.warc.gz\"}"} | null | null |
Best Crime Movies of 2005
Driven by tragedy, billionaire Bruce Wayne dedicates his life to uncovering and defeating the corruption that plagues his home, Gotham City. Unable to work within the system, he instead creates a new identity, a symbol of fear for the criminal underworld - The Batman.
Starring: Christian Bale, Michael Caine, Liam Neeson, Katie Holmes, Gary Oldman, Cillian Murphy, Tom Wilkinson, Rutger Hauer, Ken Watanabe, Morgan Freeman ..
Directed by: Christopher Nolan
Welcome to Sin City. This town beckons to the tough, the corrupt, the brokenhearted. Some call it dark… Hard-boiled. Then there are those who call it home — Crooked cops, sexy dames, desperate vigilantes. Some are seeking revenge, others lust after redemption, and then there are those hoping for a little of both. A universe of unlikely and reluctant heroes still trying to do the right thing in a city that refuses to care.
Starring: Bruce Willis, Mickey Rourke, Jessica Alba, Clive Owen, Devon Aoki, Rosario Dawson, Benicio del Toro, Michael Clarke Duncan, Alexis Bledel, Carla Gugino ..
Directed by: Frank Miller, Robert Rodriguez, Quentin Tarantino
Match Point is Woody Allen's satire of the British High Society and the ambition of a young tennis instructor to enter into it. Yet when he must decide between two women - one assuring him his place in high society, and the other that would take him far from it - palms start to sweat and a dark psychological match in his head begins.
Starring: Jonathan Rhys Meyers, Scarlett Johansson, Emily Mortimer, Brian Cox, Penelope Wilton, James Nesbitt, Ewen Bremner, Miranda Raison, Margaret Tyzack, Rupert Penry-Jones ..
Directed by: Woody Allen
Yuri Orlov is a globetrotting arms dealer and, through some of the deadliest war zones, he struggles to stay one step ahead of a relentless Interpol agent, his business rivals and even some of his customers who include many of the world's most notorious dictators. Finally, he must also face his own conscience.
Starring: Nicolas Cage, Bridget Moynahan, Jared Leto, Ethan Hawke, Eamonn Walker, Ian Holm, Sammi Rotibi, Tanit Phoenix, Shake Tukhmanyan, Jared Burke ..
Directed by: Andrew Niccol
Kiss Kiss Bang Bang
A petty thief posing as an actor is brought to Los Angeles for an unlikely audition and finds himself in the middle of a murder investigation along with his high school dream girl and a detective who's been training him for his upcoming role...
Starring: Robert Downey Jr., Val Kilmer, Michelle Monaghan, Corbin Bernsen, Dash Mihok, Larry Miller, Rockmond Dunbar, Shannyn Sossamon, Angela Lindvall, Indio Falconer Downey ..
Green Street Hooligans
After being wrongfully expelled from Harvard University, American Matt Buckner flees to his sister's home in England. Once there, he is befriended by her charming and dangerous brother-in-law, Pete Dunham, and introduced to the underworld of British football hooliganism. Matt learns to stand his ground through a friendship that develops against the backdrop of this secret and often violent world. 'Green Street Hooligans' is a story of loyalty, trust and the sometimes brutal consequences of living close to the edge.
Starring: Elijah Wood, Claire Forlani, Charlie Hunnam, Ross McCall, Leo Gregory, Marc Warren, Rafe Spall, Kieran Bew, Geoff Bell, Henry Goodman ..
Directed by: Lexi Alexander
An average family is thrust into the spotlight after the father commits a seemingly self-defense murder at his diner.
Starring: Viggo Mortensen, Maria Bello, Heidi Hayes, Ashton Holmes, William Hurt, Ed Harris, Peter MacNeill, Stephen McHattie, Greg Bryk, Kyle Schmid ..
Directed by: David Cronenberg
Raised as a slave, Danny is used to fighting for his survival. In fact, his "master," Bart, thinks of him as a pet and goes as far as leashing him with a collar so they can make money in fight clubs, where Danny is the main contender. When Bart's crew is in a car accident, Danny escapes and meets a blind, kindhearted piano tuner who takes him in and uses music to free the fighter's long-buried heart.
Starring: Jet Li, Morgan Freeman, Bob Hoskins, Vincent Regan, Dylan Brown, Tamer Hassan, Phyllida Law, Michael Jenn, Carole Ann Wilson, Kerry Condon ..
Directed by: Louis Leterrier
Four Brothers
Four adopted brothers return to their Detroit hometown when their mother is murdered and vow to exact revenge on the killers.
Starring: Mark Wahlberg, Tyrese Gibson, André Benjamin, Garrett Hedlund, Terrence Howard, Josh Charles, Sofía Vergara, Fionnula Flanagan, Chiwetel Ejiofor, Taraji P. Henson ..
Directed by: John Singleton
The Exorcism of Emily Rose
When a younger girl called Emily Rose dies, everyone puts blame on the exorcism which was performed on her by Father Moore prior to her death. The priest is arrested on suspicion of murder. The trial begins with lawyer Erin Bruner representing Moore, but it is not going to be easy, as no one wants to believe what Father Moore says is true.
Starring: Laura Linney, Tom Wilkinson, Campbell Scott, Jennifer Carpenter, Kenneth Welsh, Mary Beth Hurt, Colm Feore, Henry Czerny, Shohreh Aghdashloo, Duncan Fraser ..
Directed by: Scott Derrickson
Also check Best crime movies of 2004.
Check out our top containing the Best Crime Movies of 2005 - PickTheMovie.com. This top was obtained with our unique algorithm ordered by our unique ranking system. | {
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Posts Tagged 'perugia'
Amanda Knox demands $10,000 to talk to law students
Amanda Knox after the dramatic annulment in 2015
Acquitted ex- murder defendant hires herself out for talks
Row over Knox charging to speak to students
A media storm erupted when it transpired that the acquitted murder defendant was paid 'up to $10,000' to speak in front of law students at Roanoke College last week
Amanda Knox is reported as having registered as a speaker with an entertainments agency. Her entry shows she expects between US$5,000 – US$10,000 plus expenses for an appearance.
Kercher Attorney slams the enterprise as 'inappropriate'
The Meredith Kercher attorney, Fransesco Maresca is quoted deploring her tactlessness towards the victim's family.
ANSA, the Italian News Agency, reports him as saying:
"I hope I can convey how inappropriate this behavior is and how the family of Meredith Kercher can be adversely affected."
'Demonized'
The gist of Knox' talks are that she is a victim of 'demonization' by the Italian prosecutor, Giuliano Mignini and the press. In particular, Nick Pisa of the DAILY MAIL, has been hammered in the Netflix documentary, 'Amanda Knox', as writing salacious reports during the trial. The Netflix film was also shown at Roanoke College the day before as the background to her talk, which was on the theme of 'Truth Matters'.
Attendees at the event report Knox spoke 'emotionally'. Her book Waiting to be Heard was also on sale at a discounted price.
Knox maintains she is the victim of misogyny and a belief the aggravated murder was part of some kind of satanist rite by the Roman Catholic prosecutor. Court records do not support her claim that this was the grounds for her prosecution, although the fact of Halloween – the murder took place 1 Nov 2007, the day after – as was her boyfriend, Raffaele Sollecito's penchant for self-proclaimed satanist Marilyn Manson, his expressed wish on FACEBOOK for 'extreme experiences' and the violent manga material found at his apartment, were observations brought up by Mignini at the initial remand trial before Judge Matteini. Matteini concluded the crime was so serious and the likelihood of Knox absconding to the US was high. Thus she remanded the pair in custody.
Convicted, then unexpectedly freed
After the pair were convicted in 2010, after a trial, the convictions were upheld by the Appeal Court. The case was taken to the Supreme Court, who annulled the conviction in 2015 on the grounds of a 'flawed investigation' and 'undue press influence'. The judges, Marasca and Bruno, did however, remark in their written reasons that it was a judicial fact she was certainly present at the cottage during the murder, did wash off Kercher's blood from her hands and did cover up for Rudy Guede, also convicted.
'The burglary scene was staged", the courts ruled
The final Supreme Court ruled that the burglary was staged. In the Netflix film, Knox claims, 'Guede was the local burglar and he burgled my house.'
The pair were freed for the legal reason of 'Not Guilty due to insufficient evidence'. The words, 'innocent' and 'exonerated' do not appear anywhere in the judgment. In addition, the conviction for falsely accusing Patrick Lumumba of the assault and murder was upheld, for which Knox served three years, in addition to one year in remand.
Raffaele Sollecito bid for compensation rejected
Knox' co-defendant Sollecito failed in his attempt to win €500,000 in compensation last year, as it was deemed he lied time and again to the police, thus excluding himself from any because of misconduct during the investigation.
'I was wrongfully imprisoned for four years' Knox tells audiences
In her latest move, Knox is touring America demanding up to $10,000 per event claiming she has been declared innocent and exonerated. She tells audiences that she was 'wrongfully imprisoned' for four years. However, that conviction, for Calunnia (=US equivalent Obstruction of Justice) against Patrick Lumumba has never been overturned, and remains on her record.
Amanda Knox has an application to the European Court of Human Rights outstanding, since 2013, claiming a breach of Article 6 (right to a fair trial) and Article 3 (torture).
Tags:amanda knox, ANSA, bruno-marasca, Fransesco Maresca, general, italy, matteini, meredith kercher, Mignini, netflix, news, perugia, raffaele sollecito, roanoke college, rudy guede, supreme court
What Marasca-Bruno Supreme Court said about Raffaele Sollecito
Bongiorno, Sollecito, Maori legal team
Compensation claim by Raffaele Sollecito
BREAKING: Claim thrown out! 'ANSA) – PERUGIA, FEBRUARY 11 – Rejected by the Court of Appeal of Florence, the claim for wrongful imprisonment advanced by Raffaele Sollecito, finally acquitted of the charge of having participated in the murder of Meredith Kercher. He asked over 500 thousand euro for almost four years in jail before being released from prison. As learned by ANSA Tuscan courts have held contradictory his statements in the initial survey. ' – Too many lies in the early stages.
Motivation Report of the Florence Compensation Claim Dismissal now available:
This translation was done by a group of unpaid volunteers who are regular posters on the Perugiamurderfile.org message board devoted to discussing the murder of Meredith Kercher in Perugia, Italy, in November of 2007. The translation and editorial team was international in its make-up.
It was completed in February 2017, having been undertaken for the sole purpose of promoting a better understanding of this complex case, and to ensure that the facts are readily available to the English-speaking world without selective emphasis, misstatement or bias.
It has been translated on a "best efforts" basis, and has gone through multiple rounds of proofreading and editing, both to ensure its accuracy and to harmonize the language insofar as possible. Persons fluent in both Italian and English are invited and encouraged to contact PMF if they find any material errors that influence the meaning or intention of the judges. All such corrections will be investigated, made as required and brought to the attention of the public. The original Italian document is twelve pages long.
As with any translation, some terminology in Italian has no direct equivalent in English. Explanations have been provided where relevant. Similarly, readers are encouraged to submit any questions about legal or other concepts that may arise as they peruse the report. Our goal is to make the report as clear and as accurate as possible; to this end, it will be amended whenever doing so promotes this goal.
As the report was written and published in Italian, that language prevails in the event of a dispute over interpretation. This English-language version is provided for readers' convenience only; accordingly, it is a free translation and has no legal authority or status.
This translation may be freely copied or otherwise reproduced and transmitted in the unedited pdf format, provided that the translation or any excerpt therefrom is accompanied by the following attribution: "From the translation prepared by unpaid volunteers from http://www.perugiamurderfile.org to promote a better understanding of the circumstances surrounding the death of Meredith Kercher and the case against Amanda Knox and Raffaele Sollecito in the English-speaking world".
The compensation claim
Raffaele Sollecito, represented by his attorneys throughout the process, Avvocato Giulia Bongiorno and Luca Maori, is currently claiming compensation for 'wrongful imprisonment' in respect of the four years he served of a sentence of 25 years handed down for the Aggravated Murder of Meredith Kercher, 1 Nov 2007. The conviction was controversially overturned by the final Italian Supreme Court in March 2015, and its Motivational Report published – some three months late – in September 2015. It was only then Sollecito was able to commence compensation proceedings, as the Italian Penal Code provides for this, given, its long-winded legal process whereby defendants accused of serious crimes (i.e., one with a sentence of over three years custody) can be held on remand whilst awaiting trial. In theory, this should only be for up to one year.
The issues with the Marasca ruling
The Marasca verdict is considered controversial because Sollecito and his co-defendant, Amanda Knox had been found guilty at the first instance trial court (merits), which was upheld on appeal. It is unusual for the Supreme Court to have not remitted the case back to the Appeal (second instance) court as the Penal Code – as is standard in the UK and the USA – does not allow the Supreme Court to assess facts found at trial. The correct procedure is to send the disputed evidence back to the court which in the opinion of the Supreme court erred. Marasca did not rule a Section 530,1 'Not Guilty' acquittal, but a Section 530, 2 'Not Guilty' 'insufficient evidence', which some say is similar to Scottish Law, 'Not Proven'.However, the wording used, proscioglimento indicates a pre-trial 'charges dropped', rather than 'acquittal' (assoluzione).
Sollecito and Knox made several applications against being held in custody whilst awaiting trial and were turned down at every stage, including appeals and an application for 'house arrest' in lieu.
The prosecution opposed the application on the grounds of the seriousness of the crime, and in Knox' case, the standard ground that she might flee the country, as a foreigner to Italy. In addition, the prosecution had used special preventative powers to isolate the defendants (Knox, Sollecito and Guede) to prevent tampering with witnesses, a power which had been added to the Penal Code to assist in the fight against mafia gangs who did intimidate witnesses, often through their lawyers. Therefore the law allowed the prosecutors to deny the defendants an attorney until just before their remand hearings.
Sollecito's challenges
However, the award of compensation for having (a) been held in remand, and (b) serving a sentence until such time the conviction was overturned, is not automatic. The applicant has to show that they are factually 'not guilty', i.e., cannot possibly have committed the crime, perhaps because the 'real perpetrator' has come to light, or 'new evidence' presented. Neither of these scenarios apply in Sollecito's case. Whilst a defendant is allowed to 'lie' and indeed, does not need to swear an oath in testifying, this only holds true if they are guilty. Marasca did not find Sollecito or Knox, 'Not Gulty' as per Article 530,1, the common or garden 'Not Guilty' verdict.
Further, Sollecito refused to testify at his own trial, and made various misrepresentations and lies to the police. He argues in current tv and radio show rounds – for example, in the recent Victoria Derbyshire BBC morning show – that as he was a 'collector of knives' and had always carried a knife around since age thirteen, 'To carve on tables and trees', he explains, and thus argues, the police should not have viewed this with suspicion when he attended the questura carrying one in the days after the murder.
Sollecito's other difficulty is that Marasca, whilst criticising the investigation as 'flawed', and this being the main reason for acquittal, it nonetheless cuts Sollecito little slack.
How Marasca cuts Sollecito little slack
From the Marasca Supreme Court Motivational Report, Sept 2015:
It remains anyway strong the suspicion that he [Sollecito] was actually in the Via della Pergola house the night of the murder, in a moment that, however, it was impossible to determine. On the other hand, since the presence of Ms. Knox inside the house is sure, it is hardly credible that he was not with her.
And even following one of the versions released by the woman, that is the one in accord to which, returning home in the morning of November 2. after a night spent at her boyfriend's place, she reports of having immediately noticed that something strange had happened (open door, blood traces everywhere); or even the other one, that she reports in her memorial, in accord to which she was present in the house at the time of the murder, but in a different room, not the one in which the violent aggression on Ms. Kercher was being committed, it is very strange that she did not call her boyfriend, since there is no record about a phone call from her, based on the phone records within the file.
Even more if we consider that having being in Italy for a short time, she would be presumably uninformed about what to do in such emergency cases, therefore the first and maybe only person whom she could ask for help would have been her boyfriend himself, who lived only a few hundred meters away from her house.
Not doing this signifies Sollecito was with her, unaffected, obviously, the procedural relevance of his mere presence in that house, in the absence of certain proof of his causal contribution to the murderous action.
The defensive argument extending the computer interaction up to the visualization of a cartoon, downloaded from the internet, in a time that they claim compatible with the time of death of Ms. Kercher, is certainly not sufficient to dispel such strong suspicions. In fact, even following the reconstruction claimed by the defence and even if we assume as certain that the interaction was by Mr. Sollecito himself and that he watched the whole clip, still the time of ending of his computer activity wouldn't be incompatible with his subsequent presence in Ms. Kercher's house, given the short distance between the two houses, walkable in about ten [sic] minutes.
An element of strong suspicion, also, derives from his confirmation, during spontaneous declarations, the alibi presented by Ms. Knox about the presence of both inside the house of the current appellant the night of the murder, a theory that is denied by the statements of Curatolo, who declared of having witnessed the two together from 21:30 until 24:00 in piazza Grimana; and by Quintavalle on the presence of a young woman, later identified as Ms. Knox, when he opened his store in the morning of November 2.
An umpteenth element of suspicion is the basic failure of the alibi linked to other, claimed human interactions in the computer of his belongings, albeit if we can't talk about false alibi, since it's more appropriate to speak about unsuccessful alibi.
Sollecito in his police interview of the 5 Nov 2007, shortly after which he was arrested, withdrew his alibi from Amanda Knox. During the Nencini appeal phase, he and his advocate, Bongiorno, called a press conference to underline that Sollecito 'could not vouch for Knox' whereabouts between 8:45 pm and 1:00 am on the night of the murder. Sollecito has never once retracted this withdrawal of an alibi for Amanda.
Further, Marasca states:
The defensive argument extending the computer interaction up to the visualization of a cartoon, downloaded from the internet, in a time that they claim compatible with the time of death of Ms. Kercher, is certainly not sufficient to dispel such strong suspicions.
In fact, even following the reconstruction claimed by the defence and even if we assume as certain that the interaction was by Mr. Sollecito himself and that he watched the whole clip, still the time of ending of his computer activity wouldn't be incompatible with his subsequent presence in Ms. Kercher's house, given the short distance between the two houses, walkable in about ten [sic] minutes.
Sollecito had claimed he was surfing the internet until 3:00 am in one statement and claimed to have watched Naruto cartoon until 9:45 pm on the murder night. It winds up:
The technical tests requested by the defence cannot grant any contribution of clarity, not only because a long time has passed, but also because they regard aspects of problematic examination (such as the possibility of selective cleaning) or of manifest irrelevance (technical analysis on Sollecito's computer) given that is was possible, as said, for him to go to Kercher's house whatever the length of his interaction with the computer (even if one concedes that such interaction exists), or they are manifestly unnecessary, given that some unexceptionable technical analysis carried out are exhaustive (such are for example the cadaver inspection and the following medico-legal examinations).
Leading to the verdict:
Following the considerations above, it is obvious that a remand [rinvio] would be useless, hence the declaration of annulment without remand, based on art. 620 L) of the procedure code, thus we apply an acquittal [proscioglimento *] formula [see note just below] which a further judge on remand would be anyway compelled to apply, to abide to the principles of law established in this current sentence.
*[Translator's note: The Italian word for "acquittal" is actually "assoluzione"; while the term "proscioglimento" instead, in the Italian Procedure Code, actually refers only to non-definitive preliminary judgments during investigation phase, and it could be translated as "dropping of charges". Note: as for investigation phase "proscioglimento" is normally meant as a non-binding decision, not subjected to double jeopardy, since it is not considered a judgment nor a court's decision.] http://themurderofmeredithkercher.com/The_Marasca-Bruno_Report_(English)
The Issues Facing the Florence Appeal Court
Sollecito has clearly passed the first hurdle of being eligible to have a hearing for compensation. His legal team have asked for the maximum €516,000. A claimant who can successfully plead 'wrongful imprisonment' can claim €500, per diem imprisonment, up to a cap of €516,000.
Sollecito's legal team have referred to Marasca's criticism of the investigation as grounds for the full compensation, claiming Sollecito's "innocence and loss of youthful endeavours" because of the 'flaws'. Problem is, the issue of investigative flaws was never pleaded at trial, or at least, not upheld, by either the trial or appeal court judge. Marasca never really explains in which way this was a proven fact.
The Prosecutor's Office based at Florence is opposing the application. I would expect they will be relying on Matteini's remand hearing and Gemmelli's written reasons rejecting Sollecito's appeal against being kept in custody until the hearing.
The three judges who on 27 January 2017 in a hearing listed for five days announced they would issue their verdict 'within five days', as of 7 Feb 2017, some seven working days later, have yet to make a decision. Alternatively, the decision has been made, but the public and press have not yet accessed it. It could be Sollecito's legal team have yet to call a press conference, whilst they study the findings.
The panel will decide:
is Sollecito entitled to compensation?
if so, how much?
did he lie to police or mislead them?
if so, to what extent was he contributory to his being remanded?
to what extent the 'flawed investigation' a factor in his 'wrongful imprisonment'?
should Sollecito receive compensation for the one year remand in custody leading up to the trial?
should he be compensated for the three further years of a sentence served as a convicted prisoner, six months of it in solitary confinement?
should this be for both of the above, either of the above, or neither of them?
Sollecito has made noises that he plans further legal action against the prosecutor, based on Marasca's criticisms in its Motivational Report.
Sources: The Murder of Meredith Kercher com True Justice for Meredith Kercher
Tags:amanda knox, compensation, firenze, florence appeal court, gemmeli, giulia bongiorno, italy, marasca, marasca-bruno, matteini, meredith kercher, nencini, news, perugia, raffaele sollecito
Posted in Justice for Meredith Kercher | 3 Comments » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Skoki narciarskie na Mistrzostwach Świata w Narciarstwie Klasycznym 2007 – konkursy skoków narciarskich, przeprowadzone między 24 lutego a 3 marca 2007 roku w ramach Mistrzostw Świata w Narciarstwie Klasycznym 2007 w Sapporo.
W kalendarzu mistrzostw znalazły się trzy konkurencje w skokach narciarskich – dwie indywidualne (na skoczni normalnej i dużej) i jedna drużynowa. Po raz 33. skoczkowie narciarscy walczyli o medale mistrzostw świata.
W indywidualnym konkursie na skoczni dużej złoty medal zdobył Simon Ammann, srebro wywalczył Harri Olli, a brąz Roar Ljøkelsøy. W konkursie na skoczni normalnej najlepszy okazał się Adam Małysz, który wyprzedził Simona Ammanna i Thomasa Morgensterna. Drużynowe mistrzostwo zdobyła reprezentacja Austrii w składzie: Wolfgang Loitzl, Gregor Schlierenzauer, Andreas Kofler, Thomas Morgenstern. Srebrny medal w konkursie drużynowym uzyskali skoczkowie norwescy, a brązowy – reprezentanci Japonii.
Do zawodów zostało zgłoszonych 80 zawodników z 21 narodowych reprezentacji.
Przed mistrzostwami
W sezonie 2006/2007 przed rozpoczęciem mistrzostw rozegrano siedemnaście konkursów zaliczanych do klasyfikacji Pucharu Świata. Pięć spośród nich wygrał Gregor Schlierenzauer, cztery razy zwyciężał Anders Jacobsen, a trzy razy Adam Małysz. Po jednym wygranym konkursie na swoim koncie mieli również: Arttu Lappi, Simon Ammann, Andreas Küttel, Rok Urbanc i Michael Uhrmann. Na pierwszym miejscu w klasyfikacji generalnej plasował się Jacobsen przed Schlierenzauerem i Małyszem. Ci trzej zawodnicy zdaniem bukmacherów mieli największe szanse na zdobycie złotego medalu mistrzostw świata w Sapporo. Wśród faworytów wymieniano także takie nazwiska jak: Simon Ammann, Andreas Küttel, Roar Ljøkelsøy, Thomas Morgenstern, Dmitrij Wasiljew, Janne Ahonen, Andreas Kofler, Wolfgang Loitzl, Martin Koch, Matti Hautamäki i Arttu Lappi.
Jednym z faworytów był także Michael Uhrmann, który wygrał jeden konkurs Pucharu Świata w sezonie, jednak w trakcie serii treningowej na Ōkurayamie nie ustał skoku, w wyniku czego odniósł kontuzję kości śródstopia w prawej nodze i trafił do szpitala. Uhrmann nie wystartował w żadnej z konkurencji.
Jedyny konkurs drużynowy przed mistrzostwami, który odbył się 11 lutego 2007 w Willingen wygrała drużyna Austrii, która wyprzedziła ekipę Norwegii i Niemiec.
Indywidualnych tytułów mistrza świata z 2005 roku bronili: Rok Benkovič na skoczni normalnej oraz Janne Ahonen na skoczni dużej. Drużynowego mistrzostwa broniła reprezentacja Austrii.
Skocznie
Dwie spośród trzech konkurencji mistrzostw świata w skokach narciarskich w 2007 roku zostały przeprowadzone na obiekcie dużym w Sapporo. Rozmiar skoczni Ōkurayama jest równy 134 metry, a punkt konstrukcyjny umieszczony jest na 120. metrze. Trzeci konkurs skoków rozegrano na skoczni normalnej, Miyanomori, której wielkość wynosi 100 metrów, a punkt K znajduje się dziesięć metrów bliżej.
Jury
Dyrektorem konkursów w skokach narciarskich na mistrzostwach świata w Sapporo był Kyoichi Omori oraz z ramienia Międzynarodowej Federacji Narciarskiej, dyrektor zawodów Pucharu Świata, Walter Hofer. Asystentem Waltera Hofera był, podobnie jak w innych oficjalnych zawodach organizowanych przez FIS, Miran Tepeš. Sędzią technicznym był Torgeir Nordby, a jego asystentem - Reed Zuehlke.
Skład sędziowski poszczególnych konkursów przedstawia poniższa tabela.
Przebieg zawodów
Pierwszym konkursem skoków narciarskich podczas Mistrzostw Świata w Narciarstwie Klasycznym 2007 były zawody o tytuł indywidualnego mistrza świata na dużej skoczni. Zmagania rozpoczęły się 24 lutego 2007 punktualnie o godzinie 18:00 czasu lokalnego. Jury konkursu postanowiło, że zawody zostaną przeprowadzone z czternastej belki startowej. Pierwszym zawodnikiem w konkursie był Tobias Bogner, który oddał skok na odległość 93 metrów. W pierwszej serii konkursowej tylko siedmiu skoczków pokonało w swych próbach punkt konstrukcyjny skoczni Ōkurayama. Pierwszym, który tego dokonał, był Harri Olli. Startujący z dwudziestym siódmym numerem Fin dzięki uzyskanym 124 metrom zajmował po pierwszej kolejce drugą pozycję. Metr bliżej wylądował Roar Ljøkelsøy, co pozwoliło mu uplasować się na trzecim miejscu. Najdłuższy skok, na 125 metr, oddał reprezentant Szwajcarii, Simon Ammann i prowadził po pierwszej serii z przewagą 2,8 punktu nad Ollim. Zwycięzca kwalifikacji, Adam Małysz, po próbie na 123 metrów zajmował 4. pozycję. Słabsze skoki oddali lider i wicelider klasyfikacji generalnej Pucharu Świata. Anders Jacobsen za sprawą 114,5 metrów zajmował szesnaste, zaś Gregor Schlierenzauer, po próbie na 115 metrów, piętnaste miejsce. Szósty po pierwszej serii był obrońca tytułu, Janne Ahonen, który skoczył 123 metrów. W drugiej próbie skoczył siedem metrów dalej i utrzymał swoją pozycję. Piąte miejsce zajął Thomas Morgenstern po skokach na 122 i 131,5 metrów. Adam Małysz mimo odległości 133 metrów zajął miejsce tuż za podium. Brązowy medal po skoku na 135 metrów zapewnił sobie Roar Ljøkelsøy, zaś wicemistrzem świata z wynikiem 136,5 metrów został Harri Olli. Dwa metry bliżej skoczył Simon Ammann, jednak dzięki przewadze punktowej uzyskanej w pierwszej serii o 0,2 punktu został mistrzem świata.
Dzień później na tej samej skoczni rozegrano konkurs drużynowy. Tytuł mistrza świata z 2005 roku obronili Austriacy, którzy w składzie: Wolfgang Loitzl, Gregor Schlierenzauer, Andreas Kofler i Thomas Morgenstern wyprzedzili o 46,9 punktu drugich Norwegów. Podopieczni Miki Kojonkoskiego: Tom Hilde, Anders Bardal, Anders Jacobsen i Roar Ljøkelsøy mieli z kolei 47,4 punktu przewagi nad drużyną gospodarzy. Japończycy w składzie Shōhei Tochimoto, Takanobu Okabe, Daiki Itō i Noriaki Kasai wyprzedziła sklasyfikowaną na czwartej pozycji reprezentację Finlandii i zdobyła brązowy medal. Piątą pozycję zajęli Polacy, w skład której weszli: Kamil Stoch, Piotr Żyła, Robert Mateja i Adam Małysz. Po pierwszej serii drużyna polska plasowała się na czwartym miejscu ze stratą 3 punktów do Japończyków, lecz za sprawą skoku Roberta Matei w drugiej serii na 94,5 metra spadła na niższe miejsce. Najdłuższe skoki na odległości: 136 i 135 metrów oddał indywidualny mistrz świata, Simon Ammann, dzięki czemu ekipa Szwajcarii zajęła 7. pozycję.
Zmagania skoczków na mistrzostwach kończył konkurs indywidualny na normalnej skoczni. Zawody odbyły się 3 marca 2007 na skoczni Miyanomori. Konkurs otworzył skokiem na 87 metrów Robert Mateja. Jako pierwszy punkt K umieszczony na 90 metrze przekroczył Radik Żaparow, który uzyskał 90,5 metra. Obrońca tytułu ze skoczni normalnej w Oberstdorfie, Rok Benkovič z wynikiem 89,5 metra zajmował po pierwszej serii 26. miejsce. Najlepszy zawodnik z Austrii na dużej skoczni, Thomas Morgenstern po skoku na 95 metrów zajmował ex aequo 3. pozycję z Simonem Ammannem. Mistrz świata sprzed tygodnia skoczył półtora metra dalej, jednak otrzymał niższe noty za styl. Drugi był rodak Ammanna, Andreas Küttel (95,5 m), który miał nad mistrzami olimpijskimi przewagę 0,5 punktu. Po pierwszej kolejce prowadził Adam Małysz. Mistrz świata na normalnej skoczni z 2001 i dwukrotny mistrz świata z 2003 roku, po skoku na 102 metry miał 12,5 punktową przewagę nad Küttelem. Lider Pucharu Świata, Anders Jacobsen po skoku na 91,5 metra plasował się na czternastej pozycji. W drugiej próbie Norweg uzyskał rezultat o 2,5 metra lepszy, dzięki czemu awansował na 7. miejsce. Jako pierwszy z dwóch wspólnie zajmujących trzecią pozycję zawodników, z racji niższego numeru startowego, swój skok oddał Thomas Morgenstern i uzyskał 95 metrów. Simon Ammann wylądował metr dalej, dzięki czemu zapewnił sobie medal. Andreas Küttel po próbie na 92 metry stracił szanse na medal mistrzostw świata i ostatecznie został sklasyfikowany na 5. miejscu. Prowadzący po pierwszej kolejce Adam Małysz ponownie uzyskał najlepszy wynik serii - 99,5 metra i zdobył czwarty indywidualny tytuł mistrza świata w skokach narciarskich. Polak uzyskał 21,5 punktu przewagi nad srebrnym medalistą, Ammannem. Brązowy medal wywalczył Morgenstern.
Złoty medal Adama Małysza
Nie wliczając tzw. zasady podwójnego mistrzostwa, według której mistrzowie olimpijscy z lat 1924-1984 automatycznie zostawali mistrzami świata, Adam Małysz jako pierwszy zawodnik w historii zdobył cztery złote medale mistrzostw świata w konkurencjach indywidualnych w skokach narciarskich.
Medaliści
Konkurs indywidualny na skoczni HS 134 (24.02.2007)
Konkurs drużynowy na skoczni HS 134 (25.02.2007)
Konkurs indywidualny na skoczni HS 100 (03.03.2007)
Klasyfikacja medalowa
Wyniki
Kwalifikacje do konkursu indywidualnego na skoczni HS 134 (23.02.2007)
Legenda:
pq - zawodnik ma zapewnioną kwalifikację dzięki pozycji w pierwszej "15" klasyfikacji Pucharu Świata
Q - zawodnik zakwalifikował się do konkursu głównego
nq - zawodnik odpadł w kwalifikacjach
Konkurs indywidualny na skoczni HS 134 (24.02.2007)
Konkurs drużynowy na skoczni HS 134 (25.02.2007)
Kwalifikacje do konkursu indywidualnego na skoczni HS 100 (02.03.2007)
Legenda:
pq - zawodnik ma zapewnioną kwalifikację dzięki pozycji w pierwszej "15" klasyfikacji Pucharu Świata
Q - zawodnik zakwalifikował się do konkursu głównego
nq - zawodnik odpadł w kwalifikacjach
DNS - zawodnik nie wystartował
DSQ - dyskwalifikacja
Konkurs indywidualny na skoczni HS 100 (03.03.2007)
Składy reprezentacji
W poniższej tabeli przedstawione zostały składy wszystkich państw, które były reprezentowane przez przynajmniej jednego skoczka narciarskiego na mistrzostwach w 2007 roku. Mimo wcześniejszego powołania do reprezentacji, w żadnej z konkurencji nie uczestniczyli: Mario Innauer, Tami Kiuru, Fumihisa Yumoto, Kevin Horlacher, Michael Uhrmann, Stefan Hula, Ildar Fatkullin i Robert Kranjec. Wspomniany Uhrmann nie wystartował, ponieważ w wyniku kontuzji, której doznał w trakcie jednej z serii treningowych, zmuszony był zrezygnować ze startu w mistrzostwach świata.
Z uwagi na fakt, że tytułu mistrza świata na skoczni dużej z 2005 roku bronił Janne Ahonen, reprezentacja Finlandii wystawiła pięciu skoczków do kwalifikacji przed pierwszym konkursem indywidualnym na tym obiekcie. Analogicznie, za sprawą obrońcy tytułu ze skoczni normalnej – Roka Benkoviča, w kwalifikacjach do zawodów na skoczni K-90 wystartowało pięciu skoczków słoweńskich.
Legenda:
q – zawodnik nie zakwalifikował się do konkursu głównego;
DNS – zawodnik nie wystartował w konkursie głównym;
- – zawodnik nie został zgłoszony do kwalifikacji.
Upadki
W trakcie serii konkursowych, kwalifikacyjnych i treningowych rozegranych w ramach mistrzostw świata w Sapporo odnotowano dwa upadki skoczków narciarskich. Pierwszy z nich miał miejsce 21 lutego 2007 podczas drugiej serii treningowej przed kwalifikacjami do konkursu indywidualnego na skoczni dużej. Zawodnikiem, który nie ustał swojej próby był Michael Uhrmann. W wyniku upadku niemiecki skoczek doznał kontuzji kości śródstopia w prawej nodze, co uniemożliwiło mu start w mistrzostwach świata.
Podczas pierwszej serii konkursowej zawodów na skoczni Miyanomori upadł Harri Olli. Reprezentant Finlandii uzyskał 93,5 metra i mimo upadku, dzięki temu, że jego odległość stanowiła ponad 90% najdłuższego wyniku pierwszej serii, awansował do serii finałowej jako lucky loser.
W pozostałych startach nie odnotowano żadnego upadku.
Uwagi
Przypisy
Bibliografia
Skoki narciarskie na mistrzostwach świata w narciarstwie klasycznym
Mistrzostwa Świata w Narciarstwie Klasycznym 2007
sl:Svetovno prvenstvo v nordijskem smučanju 2007#Smučarski skoki | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 80 |
$LOAD_PATH.unshift 'lib'
require 'textpow'
syntax = Textpow.syntax('javascript')
text = File.read('examples/jquery.js')
processor = Textpow::RecordingProcessor.new
start = Time.now.to_f
(ARGV[0] || 1).to_i.times{ syntax.parse(text, processor) }
puts Time.now.to_f - start
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,418 |
Le phare de Ragged Point (en ), était un phare offshore de type screw-pile lighthouse situé sur la rivière Potomac dans le Comté de Saint Mary, dans le Maryland. Il fut le dernier construit en baie de Chesapeake et remplacé, en 1962, par une balise moderne.
Historique
La première demande d'éclairage à Ragged Point date de 1896. Les fonds ne furent cependant alloués qu'en 1906 et la construction ne commença que lorsque 5.000 dollars supplémentaires furent alloués. La construction a finalement commencé en 1910 et la lumière a été mise en service en mars de la même année. C'était le dernier phare érigé dans le Maryland.
Au début des années 1960, la lumière a été neutralisée par des avions participant à une mission d'entraînement de la Naval Air Station Patuxent River. Les gardiens ont pu faire signe aux pilotes qui avaient pensé que la lumière était vacante. En 1962, la maison a été démantelée et une tour a été construite sur l'ancienne fondation.
Description
Le phare actuel est un tourelle métallique à claire-voie, avec une balise, montée sur l'ancienne plateforme. Il porte une marque de jour. Il émet, à une hauteur focale de , un éclat blanc toutes les 6 secondes. Sa portée n'est pas connue.
Identifiant : ARLHS : USA-685 ; USCG : 2-16940 ; Admiralty : J1824 .
Voir aussi
Références
Lien connexe
Liste des phares du Maryland
Liens externes
Maryland Lighthouses
Lighthouses of the United States : Maryland Eastern Shore
Lighthouses of the United States : Maryland Western Shore
Maryland - ARLHS World List of Lights (WLOL)
Maryland - Online list of lights
Ragged Point Light - Lighthouse Explorer
Phare dans le comté de Saint Mary
Phare du XXe siècle | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,889 |
Former Chelsea and Tottenham midfielder Gus Poyet believes Manchester City midfielder Kevin De Bruyne could be as good as the legendary Zinedine Zidane if he keeps up his form.
Speaking to the Daily Star, Poyet, most recently seen on these shores managing Sunderland, showered praise on the 26-year-old who continues to deliver as a key man in Pep Guardiola's City side.
"Kevin De Bruyne [has been the standout player this season]," Poyet told the Daily Star.
"He's a player that is showing the world he knows what to do every time. If he can maintain this level, he's going to have to be up there with Zinedine Zidane.
"When he needed to accelerate the team, Zidane would accelerate the team. When he needed to stop and make an extra second he would. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,698 |
{"url":"http:\/\/physics.qandaexchange.com\/?qa=1866\/how-many-waves-have-superposed","text":"# How many waves have superposed?\n\n1 vote\n76 views\n\nThe displacement of particle executing periodic motion is given as :\n$y= 4 cos^2(t\/2)sin(1000t)$\nThe above expression may be considered to be result of superposition of :\n\n1. 2 waves\n2. 3 waves\n3. 4 waves\n4. 5 waves\n\nI tried this by changing $cos$ function,\nas $4 [(1+cost)\/2 ] sin1000t$\nbut now after I think its superposition of 2 waves but given answer is 3 waves , then how should Interpret the number waves ?\n\n$$4[(1+cost)\/2]sin1000t$$$$2\\sin 1000t + 2\\cos t \\sin 1000t$$\nUsing $2\\sin a \\cos b =\\sin (a+b)\/2 + \\sin(a-b)\/2$\n$$2\\sin 1000t + \\sin 1001t\/2 +\\sin 999t\/2$$","date":"2022-11-26 15:54:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6220183372497559, \"perplexity\": 2298.4481894938144}, \"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-2022-49\/segments\/1669446708010.98\/warc\/CC-MAIN-20221126144448-20221126174448-00269.warc.gz\"}"} | null | null |
Cottage Ranch - Complete Cad, Inc.Complete Cad, Inc.
With room to grow into, this semi-large cottage ranch has some great features. Cottage design, open concept floor plan, 9′ ceilings, large bedrooms, bonus room over the garage and full finished walk-out basement are just a few. The master suite is large and has a luxurious walk in bathroom with walk-in shower, private toilet area and double vanity sink. The closets in this home are generously sized walk in style with swing doors instead of bi-folds. The living room with cathedral ceiling has a fireplace and room for built-ins on both sides and large windows overlooking the back yard. The kitchen is spacious and features a peninsula with eating counter and a walk in pantry. The finished basement is massive and has endless design possibilities. Currently it's shown with an large recreation room, two large bedrooms with walk in closets and a full bathroom, as well as another multipurpose room and separate mechanical / storage area. | {
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
\label{sec-1}
The magnetism of antiferromagnetically coupled and geometrically
frustrated spin systems is a fascinating subject due to the
richness of phenomena that are observed
\cite{Ram:ARMS94,Gre:JMC01}. Realizations of such systems exist
in one, two, and three dimensions; the most famous being the
two-dimensional kagome lattice
\cite{Gre:JMC01,Diep94,NKH:EPL04,Zhi:PRL02,SHS:PRL02,Atw:NM02}
and the three-dimensional pyrochlore antiferromagnet
\cite{PhysRevLett.88.067203,Moe:CJP01,BrG:Science01,BAA:PRL03,PSS:PRL04,CML:PRL05,Hen:PRL06,SRM:JPA06,ZhT:PRB07}.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=35mm]{cubo-fig-1a.eps}
\quad
\includegraphics[clip,width=35mm]{cubo-fig-1b.eps}
\quad
\includegraphics[clip,width=45mm]{cubo-fig-1c.eps}
\caption{Cuboctahedron, icosidodecahedron, and (part of the) kagome lattice}
\label{F-1}
\end{figure}
It is very interesting and from the point of theoretical
modeling appealing that similar but zero-dimensional spin
systems -- in the form of magnetic molecules
\cite{BGG:JCSDT97,MSS:ACIE99,MTS:AC05,TMB:ACIE07,PLK:CC07} --
exist that potentially could show many of the special features
of geometrically frustrated antiferromagnets. Figure~\xref{F-1}
displays the zero-dimensional ``little brothers" of the kagome
antiferromagnet: the cuboctahedron which consists of squares
surrounded by triangles and the icosidodecahedron which consists
of pentagons surrounded by triangles. Such finite size
antiferromagnets offer the possibility to discover and
understand properties that are shared by the infinitely extended
lattices. An example is the discovery of localized independent
magnons \cite{SHS:PRL02,SSR:EPJB01}, which explain the unusual
magnetization jump at the saturation field. Also the plateau at
$1/3$ of the saturation magnetization that appears in systems
built of corner sharing triangles could be more deeply
investigated by looking at the cuboctahedron and the
icosidodecahedron \cite{SNS:PRL05,RLM:PRB08}.
In this article we continue investigations along this line. We
focus on two points. First we discuss the physics of the regular
cuboctahedron as a function of the single spin quantum number
$s=1/2, 1, 3/2$. For these cases all energy eigenvalues could be
obtained with the help of Irreducible Tensor Operator (ITO)
techniques \cite{GaP:GCI93,BCC:IC99,Wal:PRB00} and by
application of point group symmetries. As a second point we
investigate irregular cuboctahedra. This study is motivated by
recent magnetization measurements of the icosidodecahedral
molecules \mofe\ \cite{MSS:ACIE99} and \mocr\ \cite{TMB:ACIE07}
published in Ref.~\cite{SPK:PRB08} which could successfully be
interpreted by a \emph{classical} Heisenberg model with random
antiferromagnetic exchange couplings between the paramagnetic
ions.
\section{Theoretical model}
\label{sec-2}
The physics of many of the mentioned spin systems can be well
understood with the help of the isotropic Heisenberg model,
\begin{eqnarray}
\label{E-2-1}
\op{H}
&=&
- 2\,
\sum_{u<v}\;
J_{uv}\,
\op{\vec{s}}(u) \cdot \op{\vec{s}}(v)
\ .
\end{eqnarray}
Here the sum runs over pairs of spins given by spin operators
$\op{\vec{s}}$ at sites $u$ and $v$. A negative value of the
exchange interaction $J_{uv}$ corresponds to antiferromagnetic
coupling. We refer to a regular body, e.g. cuboctahedron, if
there are only nearest neighbor couplings of constant size
$J$. In the case of an irregular coupling the nearest-neighbor
couplings can assume values according to the chosen
distribution.
Since the Hamiltonian commutes with the total spin, we can find
a common eigenbasis $\{\ket{\nu} \}$ of $\op{H}$, $\op{S}^2$,
and $\op{S}_z$ and denote the related eigenvalues by $E_{\nu}$,
$S_{\nu}$, and $M_{\nu}$, respectively. The eigenvalues of
\fmref{E-2-1} are evaluated in mutually orthogonal subspaces
${\mathcal H}(S,M)$ of total spin $S$ and total magnetic quantum
number $M$ using Irreducible Tensor Operator (ITO) techniques
\cite{GaP:GCI93,BCC:IC99,Wal:PRB00}. In addition point group
symmetries have been applied for the regular cuboctahedron.
\section{The regular cuboctahedron}
\label{sec-3}
The regular cuboctahedron belongs to the class of geometrically
frustrated antiferromagnets built of corner-sharing triangles.
Such systems possess an extended magnetization plateau at $1/3$
of the saturation magnetization ${\mathcal M}_{\text{sat}}$ caused by dominant
up-up-down contributions \cite{SNS:PRL05,RLM:PRB08}, an
unusually high jump of the magnetization at the saturation field
due to independent magnons \cite{SSR:EPJB01,SHS:PRL02} as well
as low-lying singlets below the first triplet level
\cite{SSR:JMMM05,SSR:PRB07,RLM:PRB08}. These features are shared
for instance by the icosidodecahedron and by the kagome lattice.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=45mm]{cubo-fig-2a.eps}
\includegraphics[clip,width=45mm]{cubo-fig-2b.eps}
\includegraphics[clip,width=45mm]{cubo-fig-2c.eps}
\caption{Magnetization as a function of applied field at $T=0$
for the regular cuboctahedron with $s=1/2$, $s=1$, and
$s=3/2$. The extended plateau at ${\mathcal M}_{\text{sat}}/3$ is clearly
visible.}
\label{F-2}
\end{figure}
Figure~\xref{F-2} shows the magnetization curves at $T=0$ for
the regular cuboctahedron with $s=1/2$, $s=1$, and $s=3/2$. These
curves show besides the plateau at ${\mathcal M}_{\text{sat}}/3$ a jump to
saturation of height $\Delta M = 2$. Both features are reflected
by the differential susceptibility function which is displayed in
Fig.~\xref{F-3}. Each step in Fig.~\xref{F-2} corresponds to a
peak in Fig.~\xref{F-3}. One notices that the peaks are washed
out for higher temperatures, but that the minimum that
corresponds to the plateau at ${\mathcal M}_{\text{sat}}/3$ persists up to
temperatures of the order of the exchange coupling.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=45mm]{cubo-fig-3a.eps}
\includegraphics[clip,width=45mm]{cubo-fig-3b.eps}
\includegraphics[clip,width=45mm]{cubo-fig-3c.eps}
\caption{Differential susceptibility as a function of applied
field at $k_BT/|J| = 0.25, 0.5, 0.75, 1.0$
for the regular cuboctahedron with $s=1/2$, $s=1$, and
$s=3/2$. The smoother the curve, the higher the temperature.}
\label{F-3}
\end{figure}
As a function of the intrinsic spin $s$ the differential
susceptibility $\text{d}{\mathcal M}/\text{d} B$ exhibits two
properties. With increasing spin quantum number $s$ the
individual peaks oscillate more and more with smaller relative
amplitude, but the minimum at $1/3$ is actually sharpened. It
is known that in the classical limit, i.e. for
$s\rightarrow\infty$, the differential susceptibility is
practically flat below the saturation field except for the dip
at $1/3$ \cite{SNS:PRL05}.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=45mm]{cubo-fig-4a.eps}
\includegraphics[clip,width=45mm]{cubo-fig-4b.eps}
\includegraphics[clip,width=45mm]{cubo-fig-4c.eps}
\caption{Low-lying energy levels for the regular cuboctahedron
with $s=1/2$, $s=1$, and $s=3/2$. Numbers attached to selected
levels denote their multiplicities $d_S$; unlabeled levels
below the highest labeled level have $d_S=1$.}
\label{F-5}
\end{figure}
For zero field \figref{F-5} shows the low-lying energy
levels. In the case of $s=1/2$ (l.h.s. of \figref{F-5}) one
notices the low-lying singlets below the first triplet. These
states are a cornerstone of geometric frustration and as well
present in the kagome lattice and the icosidodecahedron with
$s=1/2$ \cite{SSR:JMMM05}. It is interesting to note that with
increasing $s$, i.e. towards a more classical behavior, the
number of these states decreases. For $s=1$ (middle of
\figref{F-5}) the first excited singlet level is already
(slightly) above the lowest triplet level. For $s=3/2$
(r.h.s. of \figref{F-5}) a doubly degenerate excited singlet
level remains below the lowest triplet, the others have
disappeared. This behavior, i.e. no excited singlets below the
lowest triplet for integer spins and a doubly degenerate excited
singlet below the lowest triplet, does not change anymore for
higher spin quantum numbers as can be checked e.g. by Lanczos
methods.
The rather high symmetry of the cuboctahedron leads to many
degenerate energy levels. As examples we label some low-lying
energy levels in \figref{F-5} by their multiplicity $d_S$,
i.e. by the degeneracy of the whole multiplet. The full
degeneracy including the multiplicity of the magnetic sublevels
$d_M$ is then $d=d_S\times d_M$. Clearly, such high
multiplicities have an important impact on the magnetocaloric
behavior since they increase the entropy for low temperatures
\cite{SSR:PRB07,HoZ:JPCS08}. In the following we would like to
discuss the
impact of low-lying singlets below the first triplet which in
the case of extended lattices are supposed to condense in
infinite number onto the ground state.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=45mm]{cubo-fig-5a.eps}
\quad
\includegraphics[clip,width=45mm]{cubo-fig-5b.eps}
\caption{Heat capacity (l.h.s.) and zero-field susceptibility
(r.h.s.) for the regular cuboctahedron with $s=1/2$, $s=1$,
and $s=3/2$.}
\label{F-4}
\end{figure}
Figure \xref{F-4} compares the heat capacity (l.h.s.) and the
zero-field susceptibility (r.h.s.) for the regular cuboctahedron
with $s=1/2$, $s=1$, and $s=3/2$. The heat capacity shows a
pronounced double peak structure for $s=1/2$ and $s=1$ which
dissolves into a broad peak with increasing spin quantum
number. The broad peak also moves to higher temperatures with
increasing $s$. The reason for the first sharp peak is
twofold. Since there are several gaps between the low-lying
levels the density of states has a very discontinuous structure
which results in the double peak structure. For $s=1/2$ the
low-lying singlets provide a very low-lying non-magnetic density
of states which is responsible for the fact that the first sharp
peak is at such low temperatures. For $s=1$ the first sharp peak
results from both excited singlet as well as lowest triplet
levels. For $s=3/2$ a remnant of the first sharp peak is still
visible; it is given by the low-lying singlets, but since they
are so few, also influenced by the lowest triplet levels.
The behavior of the heat capacity is contrasted by the
susceptibility on the r.h.s. of \figref{F-4} which reflects
mostly the density of states of magnetic levels and is only
weakly influenced by low-lying singlets. Therefore, the first
sharp peak, or any other structure at very low temperatures, is
absent .
\section{The irregular cuboctahedron}
\label{sec-4}
In this section we investigate how the magnetic properties of
the cuboctahedron change if random variations of the exchange
coupling parameters are introduced. This study is motivated by
recent magnetization measurements of the icosidodecahedral
molecules \mofe\ \cite{MSS:ACIE99} and \mocr\ \cite{TMB:ACIE07}
published in Ref.~\cite{SPK:PRB08}, which were interpreted by
assuming random distributions of exchange parameters in a
classical Heisenberg model description.
\begin{figure}[ht!]
\centering
\includegraphics[clip,width=45mm]{cubo-fig-6a.eps}
\includegraphics[clip,width=45mm]{cubo-fig-6b.eps}
\includegraphics[clip,width=45mm]{cubo-fig-6c.eps}
\caption{Differential susceptibility as a function of applied
field for the irregular cuboctahedron with $s=1/2$. L.h.s.:
dependence on the width $\Delta$ of the random distribution.
Middle: dependence on the temperature $k_BT/|J| = 0.25, 0.5,
0.75, 1.0$ for $\Delta=1.0 |J|$. R.h.s.: same as middle for
$\Delta=2.0 |J|$.
}
\label{F-6}
\end{figure}
We introduce variations of the exchange parameters of the
Hamiltonian \fmref{E-2-1} by replacing the common nearest
neighbor exchange parameter $J_{uv}=J$ with values of a flat
random distribution $J-0.5 \Delta \leq J_{uv} \leq J+0.5
\Delta$. Thus the mean exchange parameter is kept to be $J$. In
order to gain sufficient statistical certainty we use ensembles
of 10,000 spectra for realizations of the irregular
cuboctahedron with $s=1/2$; the results do not deviate from
those for 1,000 realizations. For larger $s$ the production of
sufficiently large ensembles is hindered by prohibitively many
diagonalizations of larger matrices.
Figure \xref{F-6} shows the differential susceptibility that
results from averages using distributions with various
$\Delta$. The figure on the l.h.s. compares $\text{d}{\mathcal
M}/\text{d} B$ at the rather low temperature of $k_BT=0.1|J|$ for
$\Delta=0$, i.e. the regular cuboctahedron, with $\Delta=0.5
|J|$, $\Delta=1.0 |J|$, and $\Delta=2.0 |J|$. One clearly sees
that the pattern which mainly originates from ground state level
crossings does not change much for $\Delta=0.5 |J|$ and
$\Delta=1.0 |J|$. It needs a variation as large as $\Delta=2.0
|J|$, i.e. ferromagnetic interactions occur, to qualitatively
change the differential susceptibility function. The reason is
that smaller variations do no alter the structure of low-lying
energy gaps. The singlet-triplet gap, which is approximately
$0.765 |J|$, does not vary very much for the ensembles with
smaller $\Delta$, and so does the singlet-triplet crossing which
is determined by the singlet-triplet gap. It needs an
appreciable variance of the exchange parameter distribution in
order to impose large variations of the level crossing fields.
The middle and the r.h.s. of \figref{F-6} display
$\text{d}{\mathcal M}/\text{d} B$ for temperatures $k_BT/|J| = 0.25,
0.5, 0.75, 1.0$ and $\Delta=1.0 |J|$ and $\Delta=2.0 |J|$,
respectively. As already explained, there is only very little
difference between the behavior of an irregular cuboctahedron
with $\Delta=1.0 |J|$ (middle) and the regular one. For
$\Delta=2.0 |J|$ (r.h.s.) the differential susceptibility is
much more smeared out which includes an appreciable broadening
at the saturation field. Considering the irregular
cuboctahedron we can conclude that the magnetic properties are
rather stable against random fluctuations of the exchange
parameters. This means that the striking behavior especially of
the experimental differential susceptibility of \mofe\ and
\mocr\ which shows no signs of level crossings at all
\cite{SPK:PRB08} needs further theoretical exploration of the
microscopic origin.
\section*{Acknowledgment}
Computing time at the Leibniz Computer Center in Garching is
greatly acknowledged as well as helpful openMP advices by Dieter
an Mey and Christian Terboven of High Performance Computing,
RWTH Aachen University. We thank Boris Tsukerblat for motivating
discussions about the Irreducible Tensor Operator technique.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,791 |
SNMP OID for SNMP monitors.
List of Acct-Application-Id attribute value pairs (AVPs) for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers. A maximum of eight of these AVPs are supported in a monitoring message.
Action to perform when the response to an inline monitor (a monitor of type HTTP-INLINE) indicates that the service is down. A service monitored by an inline monitor is considered DOWN if the response code is not one of the codes that have been specified for the Response Code parameter.
* NONE - Do not take any action. However, the show service command and the show lb monitor command indicate the total number of responses that were checked and the number of consecutive error responses received after the last successful probe.
* LOG - Log the event in NSLOG or SYSLOG.
* DOWN - Mark the service as being down, and then do not direct any traffic to the service until the configured down time has expired. Persistent connections to the service are terminated as soon as the service is marked as DOWN. Also, log the event in NSLOG or SYSLOG.
Number of consecutive probe failures after which the appliance generates an SNMP trap called monProbeFailed.
Name of the application used to determine the state of the service. Applicable to monitors of type CITRIX-XML-SERVICE.
Attribute to evaluate when the LDAP server responds to the query. Success or failure of the monitoring probe depends on whether the attribute exists in the response. Optional.
List of Auth-Application-Id attribute value pairs (AVPs) for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers. A maximum of eight of these AVPs are supported in a monitoring CER message.
The base distinguished name of the LDAP service, from where the LDAP server can begin the search for the attributes in the monitoring query. Required for LDAP service monitoring.
The distinguished name with which an LDAP monitor can perform the Bind operation on the LDAP server. Optional. Applicable to LDAP monitors.
Custom header string to include in the monitoring probes.
Name of the database to connect to during authentication.
IP address of the service to which to send probes. If the parameter is set to 0, the IP address of the server to which the monitor is bound is considered the destination IP address.
TCP or UDP port to which to send the probe. If the parameter is set to 0, the port number of the service to which the monitor is bound is considered the destination port. For a monitor of type USER, however, the destination port is the port number that is included in the HTTP request sent to the dispatcher. Does not apply to monitors of type PING.
Time value added to the learned average response time in dynamic response time monitoring (DRTM). When a deviation is specified, the appliance learns the average response time of bound services and adds the deviation to the average. The final value is then continually adjusted to accommodate response time variations over time. Specified in milliseconds, seconds, or minutes.
IP address of the dispatcher to which to send the probe.
Port number on which the dispatcher listens for the monitoring probe.
Domain in which the XenDesktop Desktop Delivery Controller (DDC) servers or Web Interface servers are present. Required by CITRIX-XD-DDC and CITRIX-WI-EXTENDED monitors for logging on to the DDC servers and Web Interface servers, respectively.
Time duration for which to wait before probing a service that has been marked as DOWN. Expressed in milliseconds, seconds, or minutes.
Default syntax expression that evaluates the database server's response to a MYSQL-ECV or MSSQL-ECV monitoring query. Must produce a Boolean result. The result determines the state of the server. If the expression returns TRUE, the probe succeeds.
For example, if you want the appliance to evaluate the error message to determine the state of the server, use the rule MYSQL.RES.ROW(10 .TEXT_ELE2.EQ("MySQL")).
Number of retries that must fail, out of the number specified for the Retries parameter, for a service to be marked as DOWN. For example, if the Retries parameter is set to 10 and the Failure Retries parameter is set to 6, out of the ten probes sent, at least six probes must fail if the service is to be marked as DOWN. The default value of 0 means that all the retries must fail if the service is to be marked as DOWN.
Name of a file on the FTP server. The appliance monitors the FTP service by periodically checking the existence of the file on the server. Applicable to FTP-EXTENDED monitors.
Filter criteria for the LDAP query. Optional.
Firmware-Revision value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
Name of a newsgroup available on the NNTP service that is to be monitored. The appliance periodically generates an NNTP query for the name of the newsgroup and evaluates the response. If the newsgroup is found on the server, the service is marked as UP. If the newsgroup does not exist or if the search fails, the service is marked as DOWN. Applicable to NNTP monitors.
Host-IP-Address value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers. If Host-IP-Address is not specified, the appliance inserts the mapped IP (MIP) address or subnet IP (SNIP) address from which the CER request (the monitoring probe) is sent.
Hostname in the FQDN format (Example: porche.cars.org). Applicable to STOREFRONT monitors.
HTTP request to send to the server (for example, "HEAD /file.html").
Inband-Security-Id for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
Time interval between two successive probes. Must be greater than the value of Response Time-out.
Set of IP addresses expected in the monitoring response from the DNS server, if the record type is A or AAAA. Applicable to DNS monitors.
Send the monitoring probe to the service through an IP tunnel. A destination IP address must be specified.
KCD Account used by MSSQL monitor.
Version number of the Citrix Advanced Access Control Logon Agent. Required by the CITRIX-AAC-LAS monitor.
Name of the logon point that is configured for the Citrix Access Gateway Advanced Access Control software. Required if you want to monitor the associated login page or Logon Agent. Applicable to CITRIX-AAC-LAS and CITRIX-AAC-LOGINPAGE monitors.
Calculate the least response times for bound services. If this parameter is not enabled, the appliance does not learn the response times of the bound services. Also used for LRTM load balancing.
Maximum number of hops that the SIP request used for monitoring can traverse to reach the server. Applicable only to monitors of type SIP-UDP.
Metric table to which to bind metrics.
Name for the monitor. Must begin with an ASCII alphanumeric or underscore _ character, and must contain only ASCII alphanumeric, underscore, hash #, period ., space , colon :, at @, equals =, and hyphen - characters.
Version of MSSQL server that is to be monitored.
Name of the network profile.
Name of the service identifier that is used to connect to the Oracle database during authentication.
Origin-Host value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
Origin-Realm value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
Password that is required for logging on to the RADIUS, NNTP, FTP, FTP-EXTENDED, MYSQL, MSSQL, POP3, CITRIX-AG, CITRIX-XD-DDC, CITRIX-WI-EXTENDED, CITRIX-XNC-ECV or CITRIX-XDM server. Used in conjunction with the user name specified for the username parameter.
Product-Name value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
Domain name to resolve as part of monitoring the DNS service (for example, example.com).
Type of DNS record for which to send monitoring queries. Set to Address for querying A records, AAAA for querying AAAA records, and Zone for querying the SOA record.
Account Session ID to be used in Account Request Packet. Applicable to monitors of type RADIUS_ACCOUNTING.
Account Type to be used in Account Request Packet. Applicable to monitors of type RADIUS_ACCOUNTING.
Called Station Id to be used in Account Request Packet. Applicable to monitors of type RADIUS_ACCOUNTING.
Source ip with which the packet will go out . Applicable to monitors of type RADIUS_ACCOUNTING.
Authentication key (shared secret text string) for RADIUS clients and servers to exchange. Applicable to monitors of type RADIUS and RADIUS_ACCOUNTING.
Calling Stations Id to be used in Account Request Packet. Applicable to monitors of type RADIUS_ACCOUNTING.
NAS-Identifier to send in the Access-Request packet. Applicable to monitors of type RADIUS.
Network Access Server (NAS) IP address to use as the source IP address when monitoring a RADIUS server. Applicable to monitors of type RADIUS and RADIUS_ACCOUNTING.
String expected from the server for the service to be marked as UP. Applicable to TCP-ECV, HTTP-ECV, and UDP-ECV monitors.
Response codes for which to mark the service as UP. For any other response code, the action performed depends on the monitor type. HTTP monitors and RADIUS monitors mark the service as DOWN, while HTTP-INLINE monitors perform the action indicated by the Action parameter.
Amount of time for which the appliance must wait before it marks a probe as FAILED. Must be less than the value specified for the Interval parameter.
Note: For UDP-ECV monitors for which a receive string is not configured, response timeout does not apply. For UDP-ECV monitors with no receive string, probe failure is indicated by an ICMP port unreachable error received from the service.
Response time threshold, specified as a percentage of the Response Time-out parameter. If the response to a monitor probe has not arrived when the threshold is reached, the appliance generates an SNMP trap called monRespTimeoutAboveThresh. After the response time returns to a value below the threshold, the appliance generates a monRespTimeoutBelowThresh SNMP trap. For the traps to be generated, the "MONITOR-RTO-THRESHOLD" alarm must also be enabled.
Maximum number of probes to send to establish the state of a service for which a monitoring probe failed.
Mark a service as DOWN, instead of UP, when probe criteria are satisfied, and as UP instead of DOWN when probe criteria are not satisfied.
RTSP request to send to the server (for example, "OPTIONS *").
String of arguments for the script. The string is copied verbatim into the request.
Path and name of the script to execute. The script must be available on the NetScaler appliance, in the /nsconfig/monitors/ directory.
Secondary password that users might have to provide to log on to the Access Gateway server. Applicable to CITRIX-AG monitors.
Use a secure SSL connection when monitoring a service. Applicable only to TCP based monitors. The secure option cannot be used with a CITRIX-AG monitor, because a CITRIX-AG monitor uses a secure connection by default.
String to send to the service. Applicable to TCP-ECV, HTTP-ECV, and UDP-ECV monitors.
SIP method to use for the query. Applicable only to monitors of type SIP-UDP.
SIP user to be registered. Applicable only if the monitor is of type SIP-UDP and the SIP Method parameter is set to REGISTER.
SIP URI string to send to the service (for example, sip:sip.test). Applicable only to monitors of type SIP-UDP.
URL of the logon page. For monitors of type CITRIX-WEB-INTERFACE, to monitor a dynamic page under the site path, terminate the site path with a slash /. Applicable to CITRIX-WEB-INTERFACE, CITRIX-WI-EXTENDED and CITRIX-XDM monitors.
Community name for SNMP monitors.
SNMP version to be used for SNMP monitors.
SQL query for a MYSQL-ECV or MSSQL-ECV monitor. Sent to the database server after the server authenticates the connection.
State of the monitor. The disabled setting disables not only the monitor being configured, but all monitors of the same type, until the parameter is set to enabled. If the monitor is bound to a service, the state of the monitor is not taken into account when the state of the service is determined.
Store the database list populated with the responses to monitor probes. Used in database specific load balancing if MSSQL-ECV/MYSQL-ECV monitor is configured.
Enable/Disable probing for Account Service. Applicable only to Store Front monitors. For multi-tenancy configuration users my skip account service.
This option will enable monitoring of services running on storefront server. Storefront services are monitored by probing to a Windows service that runs on the Storefront server and exposes details of which storefront services are running.
Store Name. For monitors of type STOREFRONT, storename is an optional argument defining storefront service store name. Applicable to STOREFRONT monitors.
Number of consecutive successful probes required to transition a service's state from DOWN to UP.
List of Supported-Vendor-Id attribute value pairs (AVPs) for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers. A maximum eight of these AVPs are supported in a monitoring message.
Probe the service by encoding the destination IP address in the IP TOS (6) bits.
The TOS ID of the specified destination IP. Applicable only when the TOS parameter is set.
The monitor is bound to a transparent device such as a firewall or router. The state of a transparent device depends on the responsiveness of the services behind it. If a transparent device is being monitored, a destination IP address must be specified. The probe is sent to the specified IP address by using the MAC address of the transparent device.
Code expected when the server is under maintenance.
String expected from the server for the service to be marked as trofs. Applicable to HTTP-ECV/TCP-ECV monitors.
Type of monitor that you want to create.
Unit of measurement for the Deviation parameter. Cannot be changed after the monitor is created.
Unit of measurement for the Down Time parameter. Cannot be changed after the monitor is created.
User name with which to probe the RADIUS, NNTP, FTP, FTP-EXTENDED, MYSQL, MSSQL, POP3, CITRIX-AG, CITRIX-XD-DDC, CITRIX-WI-EXTENDED, CITRIX-XNC or CITRIX-XDM server.
Validate the credentials of the Xen Desktop DDC server user. Applicable to monitors of type CITRIX-XD-DDC.
Vendor-Id value for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers.
List of Vendor-Specific-Acct-Application-Id attribute value pairs (AVPs) to use for monitoring Diameter servers. A maximum of eight of these AVPs are supported in a monitoring message. The specified value is combined with the value of vendorSpecificVendorId to obtain the Vendor-Specific-Application-Id AVP in the CER monitoring message.
List of Vendor-Specific-Auth-Application-Id attribute value pairs (AVPs) for the Capabilities-Exchange-Request (CER) message to use for monitoring Diameter servers. A maximum of eight of these AVPs are supported in a monitoring message. The specified value is combined with the value of vendorSpecificVendorId to obtain the Vendor-Specific-Application-Id AVP in the CER monitoring message.
Vendor-Id to use in the Vendor-Specific-Application-Id grouped attribute-value pair (AVP) in the monitoring CER message. To specify Auth-Application-Id or Acct-Application-Id in Vendor-Specific-Application-Id, use vendorSpecificAuthApplicationIds or vendorSpecificAcctApplicationIds, respectively. Only one Vendor-Id is supported for all the Vendor-Specific-Application-Id AVPs in a CER monitoring message. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,372 |
In regards to the white towels draped from the handles of the coffin, they are given to the pall bearers after the grave site burial so that the pall bearers have a clean surface to dry their hands after the traditional washing of the hands and before the luncheon that follows the funeral service. Do you know the significance of the oil and vinegar? When viewing the body, which is optional but often expected, approach and pause briefly in front of the casket. In some cases, widows may avoid social events for a full year. Hope you find this helpful. It is appropriate to briefly visit the bereaved at home after the funeral.
It shows that person is in your thoughts at this time. If you are feeling like there is an emptiness, you might speak with your priest about it. Boxes with images of saints for your , jewelry or anything precious. It is done to preserve space. Actually, this was done in the Victorian era by a number of people, religious or not. The gift of a rosary demonstrates that the child will be grounded in prayer as she matures.
In Greek, the eggs are called Kokkina-pasxalina avga. Giving Money Money is always very much appreciated as a Christening gift. The event often includes a celebration with friends and family, many of whom bring special gifts to commemorate the occasion. They introduce the recipe this way: Koliva is a traditional memorial dish in the Orthodox Church. Categorized in: This post was written by GreekBoston. My 16 year old daughter wants us to attend an Icarian convention. The paschal candles are also bought on this day, ready to be decorated before gifting.
Before we had professionals doing the actual burial, the pall bearers and other attendees would have helped fill in the grave. We will be in Greece for five weeks but not with the family all that time. Money Although it may not be traditional, the most common baptism gift among Greek Orthodox followers may be a check or money order. We really love these people and want to be prepared to participate as appropriate and still figure out how to have a vacation without offending. Common things to write include the name of the person being baptized, the date of the baptism, the names of the godparents, and the names of the parents.
If you decide to personalize an icon, be sure to write on the back, not the front, so that you can keep the image free of clutter. Really need some feedback,she has less than a few months left and I want answers so I can plan accordingly. This is why traditional Jews and Greek Orthodox insist on burying the body, and reject cremation. Greek Orthodox traditionally bow before the casket and kiss an icon or a cross placed on the chest of the deceased. Gail Rubin There is no reason to stay away from church, and often families find that being closer to the church and communing more frequently during this time of grief is helpful. Any baby-related item is appropriate here. High Fiber, Lactose Free, Soy Free, Vegan.
Please let me know what they tell you! It was common for families to carry the relics of their ancestors around with them. Is this what the Greek Orthodox believe? Baptism is one of the sacraments in the Greek Orthodox Church, according to the Roy Rosenzweig Center for History and New Media at George Mason University. So Jews and Christians of ancient times exhumed the body after a period of time, washed the bones, and placed them in an urn. The price's are very affordable as I have found the same icon's on other online store's with an increased price. Jews felt the human body should be respected as the image of God. I seem to recall someone keeping the box of bones under their bed; is that considered normal or odd? It is one of the best aspects of the Greek traditional customs.
There are also traditional baptism gifts that are given by the family and guests. I was informed by a lady who also lost her mother that a 2 year memorial is not as important as a 3 year memorial. Here are some ideas as to the best religious gifts you can give. Is there any advice you can give me while I bring along the kids to prepare them for the funeral and burial of their great grandmother? Through baptism a person dies and is buried with Christ by threefold emersion in the waters of the baptismal font. Or would you prefer to? The shipment was wonderfully packed a box within a box, since the shipment was porcelain.
The candles are used at the Easter Midnight service. Traditional Food Gifts Since much of Holy Week is spent fasting, by the time Easter Sunday comes around, food gifts are much appreciated. I am so sorry to hear about this experience with the Church. After the soul leaves the body, it keeps returning to its body for up to 30 days before finally releasing and moving on to the spirit world. Godparents participate in a baby's baptism in the Greek Orthodox Church. She is Greek Orthodox,I am wiccan.
Also, check if the couple registered anywhere and aim to get them a gift from their wish list. Maybe you could choose a deep blue, if you feel better in a color. My Mother has brain cancer and is in hospice care. It is related to passages recited regarding going to sleep in the earth. We were at a loss. The truth of the victory of salvation which was won on the cross was proven three days later when Christ rose from the dead and raised with him all of the righteous ones who had gone before and all of the righteous who would come after. | {
"redpajama_set_name": "RedPajamaC4"
} | 321 |
For correct tires for your vintage British car visit Universal Vintage Tire For quality US made custom fit car covers for your classic or modern British car visit carCoversdirect. Ads run until sold! Also instrument repair and restoration. Mail order all over the world.
After writing about the cars in my life Auto-Biography , I moved on to searching for examples of all the significant cars in my life through Curbside Classic. Now lest you get the idea from the headline that the treasure hunt is over; far from it. But this car is the old flame I most wanted to find. A lot of memories were made in our Stephanie drove a wagon, which is so different from the sedan that it will get its own CC, when I find one.
I would also hasten to add that what follows are only my way of doing things. The only thing that I would want to emphasize is to get a decent helmet, keep everything spotlessly clean and grease free — and to keep those needles pointy! The examples I have chosen to show are primarily to do with welding sheet between 0. For us, though, a typical task might be to weld a 0. So this is what we start with, a 5mm sheet, cut to size and with a sizeable hole in it, in which the sink eventually fits the smaller holes are for the taps etc.
A San Francisco cable car: The power to move the cable was normally provided at a "powerhouse" site a distance away from the actual vehicle. The London and Blackwall Railway , which opened for passengers in east London, England, in used such a system.
Me encanta tus tetas .y el color de tu pelo. Como tienes el coño?
You have now four men here in Wisconsin commenting to you and I know none of us would not have an issue with all of us giving you big dicks to pleasure you, you need to get us together for an all Wisconsin gangbang.
because of. Cuba at center of has been lobbying for The Caribbean to be switched out as part of North America into geographically/(financially CeAm. Which would be a crime and financial disaster for everyone. Obama and Michelle were laying out investment plans/offering up investors in that swtich-a-roo during his baseball trip to Cuba. Many Caribbean countries were already transf'd into CeAm as continental/port control (CIA even switched their notations). | {
"redpajama_set_name": "RedPajamaC4"
} | 4,622 |
Publisher: Broadview Press Inc
Mark Twain and Stephen Railton
From the Back Cover From its first appearance onward, Adventures of Huckleberry Finn has been both praised and condemned, enshrined as one of the world's great novels and banned from libraries and classrooms. This new edition is designed to enable modern readers to explore the sources of its greatness, and also to take a fresh, open-minded look at the source of the current controversy about its place in the canon: its representation of race and slavery. Based on the first American edition of 1885, this Broadview Edition includes all 174 original illustrations by E.W. Kemble. Appendices include contemporary reviews, passages deleted from the original manuscript, advertisements for the book, and a range of materials, from newspaper articles to minstrel show scripts to contemporary fiction, showing how race and slavery were depicted in the larger culture at the time. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,348 |
Q: Need help building Multiple range sliders on the same page that will display different images depending on the value of the slider I am building a web page I would like to have multiple range sliders on the same page that will display different images depending on the value of the slider . I can it do one slider but the second or third sliders dont work and also dont display the value of the slider.The images use the number of the slider value eg [0.jpg], [1,jpg], [2,jpg]
I must be doing something wrong
<style>
.slidecontainer {
width: 100%;
}
.slider {
-webkit-appearance: none;
width: 100%;
height: 40px;
border-radius: 5px;
background: #458525
outline: none;
opacity: 0.7;
-webkit-transition: .2s;
transition: opacity .2s;
}
.slider::-webkit-slider-thumb {
-webkit-appearance: none;
appearance: none;
width: 50px;
height: 50px;
border: 0;
background: url('./images/Golf-Ball1.png');
cursor: pointer;
}
.slider::-moz-range-thumb {
width: 25px;
height: 25px;
border-radius: 50%;
background: url('./images/Golf-Ball1.png');
cursor: pointer;
}
</style>
</head>
<body>
<div class="slidecontainer">
<img src="./images/0.jpg" alt="" id="img">
<input type="range" min="0" max="10" value="5" class="slider" id="myRange1">
<p>Value: <span id="value"></span></p>
<img src="./images/0.jpg" alt="" id="img">
<input type="range" min="0" max="10" value="5" class="slider" id="myRange2">
<p>Value: <span id="value"></span>
<?php echo "aaa"; ?><br/>
</div>
<script type="text/javascript">
var slider = document.getElementById("myRange");
var output = document.getElementById("value");
output.innerHTML = slider.value;
slider.oninput = function() {
output.innerHTML = this.value;
var img = document.getElementById("img");
img.setAttribute("src", "./images/" + this.value + ".jpg");
}
</script>
Any advice will be valuable to a newbe
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,596 |
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