text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: How to pass XML request in RestAssured test case For my rest service(create student), I have to pass a XML request(request type is application/xml) like the following one:
<student>
<name>abc</name>
</age>
<addresses>
<add1>
<add_line1>pqr</add_line>
<city>sdf</city>
</add1>
<add2>
<add_line1>pqr</add_line>
<city>sdf</city>
</add2>
<addresses>
</student>
How can we pass the whole xml to the test case?
@Test
public final void testCreateStudent() {
string xmlstring=//whole xml in double quote
expect()
.statusCode(200)
.body(hasXPath("/Response/status[text()='true']")
.with()
.body(xmlstring)
.when()
.post("/service/student/create");
}
I am new in this field. can any one suggest me how to do it efficiently? Is the Xpath is correct for getting status?, if my response is like:
<response>
<id>123</id>
<status>true</status>
<response>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,925 |
Float is a side table originally inspired by a hat making class that I took. I was interested in how easily felt could be manipulated over a mold. I was immediately taken with the idea to create a table. When I came up with a form that I fell in love with, I made the decision to use fiberglass for its structural and economical benefits. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,067 |
Joey Bada$$ Drops Three-Track 'The Light Pack' EP
Joey Bada$$ drops a fresh EP with a guest appearance from Pusha T.
As the clock struck midnight on Friday, Joey Bada$$ gifted his fans a three-track EP titled The Light Pack. The project has just one feature–Pusha T on "No Explanation." Joey announced the project with lyrics from the project's intro, "The Light."
"This is mumble rap extermination," he wrote. "This is godly interpolation." According to hints left on his social media page, the three tracks symbolize the mind, body, and soul.
Joey recently spoke with Complex about his upcoming LP. The project will serve as his debut album at Columbia Records. "We're trying to come soon," he said. "It's almost ready. On Instagram, my fans were pressing me and I told them, 'Listen, it's not going to be a longer wait than six months.' That's the best I can say. We're really right there."
The Brooklyn MC has explored other ventures in recent years. In 2016, he modeled for Calvin Klein. That same year, he joined the cast of USA's Mr. Robot. The following year, he'd release his last album, All-Amerikkkan Badass.
Back in February, Joey Bada$$ shared a tribute to Pop Smoke on Instagram. "A lil something I wrote in light of all the fucked up shit going on," he wrote. "I need hip hop to have a hug."
Additionally, he's fresh off the first season of Hulu's Wu-Tang: An American Saga. Joey is playing the role of Inspectah Deck. This past May, Joey donated $25,000 to support homeless students in New York City during the COVID-19 pandemic.
Stream Joey Badass$$ – The Light Pack on all major digital streaming platforms.
KOTA the Friend, Joey Bada$$, and Bas Drop New Track "B.Q.E."
CJ Fly of Pro Era Drops Debut Album 'RUDEBWOY' Featuring Joey Bada$$, Conway the Machine
← New Video: Blu, Exile "Roots Of Blue"
T.I. – Top Back (Remix) Official Music Video (Dirty) [[HD]] w/ LYRICS → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,677 |
{"url":"https:\/\/blender.stackexchange.com\/questions\/55919\/how-do-you-find-the-nodes-youre-looking-for","text":"# How do you find the nodes you're looking for\n\nWhen following tutorials, I often come across one of the Math nodes (\"Greater than\", \"Power\", \"Modulo\"...), which, unless you use the Compositor for several hours every day, I don't think you can know by heart.\n\nOf course, I usually end up assuming that the nodes I'm looking for are inside the \"Math\" node - and usually that's exactly where they are.\n\nI just came across a tutorial requesting this node:\n\nSure enough, I thought I'd find it in the Math node. Not there.\n\nI searched again:\n\nStill nothing.\n\nI think it would be a great idea for the developers of Blender to show these words in search results even if the word is a part of the node (rather than the node's name itself), for example:\n\nMath > Modulo\n\ncould show up in the search results, thus informing the user where the node they're looking for is and what it's hidden inside.\n\nDoes anyone know of a way to find where to find certain functions of a node when it doesn't show up in the search results (or when it's a sub-menu of a node)?\n\nIdeally an offline way of doing this.\n\nGoing back to look for the Soft Light...\n\n\u2022 Soft Light is under color -> mixRgb \u2013\u00a0user2816 Jun 15 '16 at 5:29\n\u2022 You mean \"Color > Mix\"? \u2013\u00a0MicroMachine Jun 15 '16 at 5:41\n\u2022 mine is named mixRgb , but yes. then you can change it to soft light. \u2013\u00a0user2816 Jun 15 '16 at 5:43\n\u2022 Weird in 2.77 is says \"Mix\". Another idea would be that older node names show up in search results for a little while... :) Thanks root \u2013\u00a0MicroMachine Jun 15 '16 at 5:51\n\u2022 @root 'MixRGB' is used for Cycles nodes, probably to distinguish it from the 'Mix Shader' node. 'Mix' is used in the compositor. \u2013\u00a0Ray Mairlot Jun 15 '16 at 10:26\n\nI think it would be a great idea for the developers of Blender to show these words in search results even if the word is a part of the node (rather than the node's name itself).\n\nYou should suggest this to the developers, if enough people find it useful it will have a better chance to be implemented,there is a section in blender about small tasks so that new developers can pick a small task and try to code them to be implemented in blender , yours seem like a good task.\n\nDoes anyone know of a way to find where to find certain functions of a node when it doesn't show up in the search results\n\nThere aren't many nodes like that that have those kind of sub menus, if you use blender long enough and with practice you will know their location by heart.\n\nto ask for a feature to be implemented in blender , see : Best Place to put Feature Requests?\n\n\u2022 thanks! I created an account on developer.blender.org, but I can't understand how to create new tasks... do you know? @root \u2013\u00a0MicroMachine Jul 13 '16 at 21:38\n\u2022 I don't think users create tasks but blender developers, for example see how this bug report was turned into a task \u2013\u00a0user2816 Jul 13 '16 at 21:52\n\nOne way to learn more about the active node is to use the python console and type one of the following commands:\n\n# compositor nodes\nbpy.context.scene.node_tree.nodes.active.bl_idname\n# material nodes\nbpy.context.active_object.active_material.node_tree.nodes.active.bl_idname\n# texture nodes\nbpy.context.active_object.active_material.active_texture.node_tree.nodes.active.bl_idname\n\n\nKnowing the bl_idname helps you figure out which node type it lives in. Of course, if you already have the node in your tree, you can just duplicate it.\n\nWhen figuring it out from a tutorial, you're rather more boned.\n\nI hopped into my copy of the blender source and did a\n\n\\$ find . -name locale -prune -o -type f -print0 | xargs -0 grep -i \"soft light\" | less\n\n\nand the results included\n\n.\/intern\/cycles\/kernel\/shaders\/node_mix.osl: if (type == \"Soft Light\")\n.\/intern\/cycles\/render\/nodes.cpp: enm.insert(\"Soft Light\", NODE_MIX_SOFT);\n\n\nThe first one is about OSL, but mentions \"mix\". The second one required me to read the source, but it showed up in a function called mix_type_init(). I checked the Color>MixRGB node and it has a Soft Light option.\n\nClearly, this is way more of a pain than a regular user can be expected to go through (how many have a clone of the git repo lying around and are either running unix or have cygwin?)\n\nThis might actually be an opportunity for an addon which has specific operators for each variant of the node types which (when the addon is enabled) make all those text strings show up in the spacebar menu. Of course, that addon's utility would be related to how often it is updated. If blender invents new node variants and the addon is not updated, that node variant is harder to find.","date":"2020-07-06 09:50:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4172630310058594, \"perplexity\": 1489.4333346697033}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-29\/segments\/1593655890157.10\/warc\/CC-MAIN-20200706073443-20200706103443-00394.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/deviation-of-newtons-gravity-equations.71560\/","text":"# Deviation of Newton's Gravity Equations\n\n1. Apr 14, 2005\n\n### eNathan\n\n$$T = \\sqrt { \\frac{2d} {g} }$$\nTherefore (and I had to come up with this on my own because I could not find it on the internet :rofl:)\n$$D = \\frac{t^2 * g} {2}$$\n\nBut I cannot find the logic in why the time equals 2 times the distance over the acceleration of gravity on earth. Now, I do understand the units. The Time is in seconds, and the Distance is in meters, because the acceleration is in meters per second (any other units will work just fine).\n\nBut why is the distance multiplied by 2?\nAnd why is the square root function used?\n\n2. Apr 14, 2005\n\n### dextercioby\n\nDo you know calculus...?\n\nDaniel.\n\n3. Apr 14, 2005\n\n### eNathan\n\nhehe, I only wish I did. I am researching it in my spare time, and I plan to take classes on it but in shorts no I dont know calc.\n\nBut it wont hurt if you try to explained it to me anyway if that was your plan :)\n\n4. Apr 14, 2005\n\n### dextercioby\n\nTo integrate Newton's second law in scalar form\n\n$$\\frac{d^{2}y(t)}{dt^{2}}=g$$\n\nsubject to the initial conditions\n\n$$y\\left(t_{0}\\right)=y_{0}$$\n\n$$\\frac{dy}{dt}\\left(t_{0})=:v\\left(t_{0}\\right)=v_{0}$$\n\nDaniel.\n\n5. Apr 14, 2005\n\n### whozum\n\nMore simply, I'm sure you know the position equation for an object in constant acceleration:\n\n$$x = x_0 + v_0t + \\frac{gt^2}{2}$$\n\nNotice if $x_0 = v_0 = 0 [\/tex] then solving for t: $$t = \\sqrt{\\frac{2x}{g}}$$ Your equation is only valid when the initial velocity is zero. 6. Apr 15, 2005 ### eNathan $$x = x_0 + v_0t + \\frac{gt^2}{2}$$ Hmn, I think I get your point, but why do you square numbers all the time anyway? I mean, I see it everywhere. e=mc^2 ^2 ^2 blah every equation in physics involves $$x^2$$ :rofl: Maybe if you explain why everything need be squared, I can understand it Thx 7. Apr 15, 2005 ### arivero Hope it is a type, and you use meters per second squared for the acceleration. Also, this thread should be titled \"Deduction (or derivation) of Galilean law for free falling bodies\"; Newton has no role here except to confirm us that acceleration g is approximately constant at Earth surface. Indeed the original proof of the time square formula can be found in internet here: http:\/\/www.mpiwg-berlin.mpg.de\/Galileo_Prototype\/DHTML\/D202.HTM Last edited: Apr 15, 2005 8. Apr 15, 2005 ### arivero This requires a bit of history. In order to get students, Galileo did not publish the parabolic trajectory until Cavalieri did (rightly attributing it to master Galileo). Then he become furious for a moment but finally he thought better and he finished his Discorsi Concerning Two New Sciences.. There, Salviati reads to his friends some selected passages of a mysterious -actually, fictitious- latin book tittled De motu locali. Sort of Necronomicon of physics. In the above site you can see the quoted paragraphs and their relationship to drafts in a notebook of Galileo, mss 72, probably used during the composition of the book. Last edited: Apr 15, 2005 9. Apr 15, 2005 ### whozum The reason is calculus, and definitions of acceleration, velocity, and position. Acceleration is the rate of change of velocity: [itex] a = \\frac{dv}{dt} [\/tex] so then [itex] \\int{a}{dt} = \\int{dv} [\/tex] Integral of a constant acceleration \"a\" is just 'a' times the dependant variable, t. v = at+v_0. v_0 is the integration constant. [itex] v = at+v_0$\n\nVelocity is the rate of change of position:\n\n$v = at+v_0 = \\frac{dx}{dt}$ so then [itex] x = \\int{(at+v_0)}{dt} [\/tex]\n\nIntegral of a linear function of time 'at' is 'at^2\/2':\n\n$$x = \\int{at+v_0}{dt} = v_0t+\\frac{at^2}{2} + x_0$$","date":"2016-10-23 03:26:35","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\": 0.6485831141471863, \"perplexity\": 984.0009743974634}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988719139.8\/warc\/CC-MAIN-20161020183839-00121-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
Q: Probability of a sample mean I am working on the following problem:
The Department of Environmental Protection demands that your power plant show a mean NOx emission less than 5lb/MWh. Your plant's daily NOx emission fluctuates randomly with a mean of 4.9lb/MWh and a standard deviation of 0.5lb/MWh.
If the DEP randomly selects and tests emissions on 30 independent days, what is the probability that the sample mean will exceed the limit? (Hint: sampling distribution, use cdf functions to calculate the probability (normalcdf()).
I am not sure how to start this -- I thought I would just plug the information into normalcdf() in my calculator, but I think this doesn't take the 30 sample size into account. Any advice?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,847 |
Stop leaning your TV against the wall.
The Wali universal table top TV stand for 22-65 inch LCD TVs is down to $9 with code OIH7BUFC. This table top stand normally sells for around $20 and doesn't drop directly from that price except through codes like this. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,090 |
package gov.nih.nci.evs.restapi.util;
import gov.nih.nci.evs.restapi.bean.*;
import gov.nih.nci.evs.restapi.common.*;
import java.io.File;
import java.io.FileInputStream;
import java.io.InputStream;
import java.util.Arrays;
import java.io.*;
import java.util.*;
import java.util.Map.Entry;
public class ConceptDetailsGenerator {
HierarchyHelper hh = null;
Vector parent_child_vec = null;
String TREE_VARIABLE_NAME = "demoList";
String HYPERLINK = "https://nciterms65.nci.nih.gov/ncitbrowser/pages/concept_details.jsf?dictionary=NCI_Thesaurus&type=terminology&key=null&b=1&n=0&vse=null&code=";
HashMap propertyHashMap = null;
public ConceptDetailsGenerator() {
this.propertyHashMap = null;
}
public ConceptDetailsGenerator(HashMap propertyHashMap) {
this.propertyHashMap = propertyHashMap;
}
public void setHYPERLINK(String hyperlinkUrl) {
this.HYPERLINK = hyperlinkUrl;
}
public String getHyperLink(String code) {
if (HYPERLINK != null) {
return HYPERLINK + code;
} else {
return null;
}
}
public String encode(String t) {
if (t == null) return null;
StringBuffer buf = new StringBuffer();
for (int i=0; i<t.length(); i++) {
char c = t.charAt(i);
if (c > 126) {
buf.append(" ");
} else {
String s = "" + c;
if (s.compareTo("'") == 0) {
buf.append("\\'");
} else {
buf.append(s);
}
}
}
return buf.toString();
}
public void writeHeader(PrintWriter out, String title) {
out.println("<!doctype html>");
out.println("<html lang=\"en\">");
out.println("<head>");
out.println(" <meta charset=\"utf-8\">");
out.println(" <title>" + title + "</title>");
out.println(" <link rel=\"stylesheet\" href=\"tree.css\">");
writeFunction(out);
out.println("</head>");
}
public void writeFunction(PrintWriter out) {
out.println(" <script type=\"text/javascript\" src=\"http://ajax.googleapis.com/ajax/libs/jquery/1.4.2/jquery.min.js\"></script>");
out.println(" <script type=\"text/javascript\" src=\"js/jquery.sapling.min.js\"></script>");
out.println(" <link rel=\"stylesheet\" type=\"text/css\" href=\"css/styleSheet.css\">");
out.println(" <script type=\"text/javascript\">");
out.println(" $(document).ready(function() {");
out.println(" $('#demoList').sapling();");
out.println(" });");
out.println(" </script>");
}
public String getIndentation(int level) {
StringBuffer buf = new StringBuffer();
for (int i=0; i<level; i++) {
buf.append("\t");
}
return buf.toString();
}
public static Boolean isEven (Integer i) {
return (i % 2) == 0;
}
public String propertyHashMap2HTML(HashMap propertyHashMap) {
StringBuffer buf = new StringBuffer();
buf.append("<table class=\"datatable_960\" border=\"0\" width=\"100%\" >").append("\n");
buf.append("<tr>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Source Code</th>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Source Term</th>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Target Code</th>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Target Term</th>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Property</th>").append("\n");
buf.append(" <th class=\"dataCellText\" scope=\"col\" align=\"left\">Property Value</th>").append("\n");
buf.append("</tr>").append("\n");
buf.append(" <col width=\"10%\">").append("\n");
buf.append(" <col width=\"20%\">").append("\n");
buf.append(" <col width=\"10%\">").append("\n");
buf.append(" <col width=\"20%\">").append("\n");
buf.append(" <col width=\"10%\">").append("\n");
buf.append(" <col width=\"30%\">").append("\n");
Iterator it = propertyHashMap.keySet().iterator();
while (it.hasNext()) {
String key = (String) it.next();
String line = CSVFileReader.csv2Delimited(key, "|");
Vector u = StringUtils.parseData(line, '|');
String source_code = (String) u.elementAt(0);
String source_term = (String) u.elementAt(1);
String target_code = (String) u.elementAt(2);
String target_term = (String) u.elementAt(3);
StringBuffer property_buf = new StringBuffer();
HashMap hmap = (HashMap) propertyHashMap.get(key);
Vector properties = new Vector();
Iterator it2 = hmap.keySet().iterator();
while (it2.hasNext()) {
String key2 = (String) it2.next();
properties.add(key2);
}
properties = new gov.nih.nci.evs.restapi.util.SortUtils().quickSort(properties);
int line_num = 0;
for (int k=0; k<properties.size(); k++) {
StringBuffer line_buf = new StringBuffer();
String property = (String) properties.elementAt(k);
Vector values = (Vector) hmap.get(property);
values = new gov.nih.nci.evs.restapi.util.SortUtils().quickSort(values);
for (int k2=0; k2<values.size(); k2++) {
buf.append("<tr>").append("\n");
if (line_num == 0) {
line_buf.append("<td class=\"textbody\">").append(source_code).append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append(source_term).append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append("<a href='").append(getHyperLink(target_code)).append("'>").append(target_code).append("</a>").append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append(target_term).append("</td>").append("\n");
} else {
line_buf.append("<td class=\"textbody\">").append("").append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append("").append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append("").append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append("").append("</td>").append("\n");
}
String value = (String) values.elementAt(k2);
String col1 = property;
String col2 = value;
if (k2 != 0) {
col1 = "";
}
line_buf.append("<td class=\"textbody\">").append(col1).append("</td>").append("\n");
line_buf.append("<td class=\"textbody\">").append(col2).append("</td>").append("\n");
buf.append(line_buf.toString()).append("\n");
buf.append("</tr>").append("\n");
line_num++;
line_buf = new StringBuffer();
}
}
}
buf.append("</table>").append("\n");
return buf.toString();
}
public void writeBody(PrintWriter out, String title) {
out.println("<body>");
out.println(" <center><h3>" + title + "</h3></center>");
out.println(" <hr>");
String content = null;
try {
content = propertyHashMap2HTML(this.propertyHashMap);
} catch (Exception ex) {
ex.printStackTrace();
}
out.println(content);
out.println("</body>");
}
public void writeFooter(PrintWriter out) {
out.println("</html>");
}
public Vector sortByLabel(Vector codes) {
if (codes == null || codes.size()<=1) return codes;
Vector w = new Vector();
HashMap hmap = new HashMap();
for (int i=0; i<codes.size(); i++) {
String code = (String) codes.elementAt(i);
String label = hh.getLabel(code);
hmap.put(label, code);
w.add(label);
}
w = new SortUtils().quickSort(w);
Vector v = new Vector();
for (int i=0; i<w.size(); i++) {
String label = (String) w.elementAt(i);
String code = (String) hmap.get(label);
v.add(code);
}
return v;
}
public void generate(PrintWriter out, String title) {
writeHeader(out, title);
writeBody(out, title);
writeFooter(out);
}
public void generate(String outputfile, String title) {
long ms = System.currentTimeMillis();
PrintWriter pw = null;
try {
pw = new PrintWriter(outputfile, "UTF-8");
generate(pw, title);
} catch (Exception ex) {
} finally {
try {
pw.close();
System.out.println("Output file " + outputfile + " generated.");
} catch (Exception ex) {
ex.printStackTrace();
}
}
System.out.println("Total run time (ms): " + (System.currentTimeMillis() - ms));
}
public static Vector filterRoots(Vector parent_child_vec) {
Vector w = new Vector();
for (int i=0; i<parent_child_vec.size(); i++) {
String line = (String) parent_child_vec.elementAt(i);
if (line.indexOf("|<Root>|") == -1) {
w.add(line);
}
}
return w;
}
public HashMap appendPropertiesToMappingEntries(String serviceUrl, String namedGraph, Vector mapping_entries) {
HashMap propertyHashMap = new HashMap();
OWLSPARQLUtils owlSPARQLUtils = new OWLSPARQLUtils(serviceUrl);
ParserUtils parserUtils = new ParserUtils();
Vector v = mapping_entries;
for (int i=1; i<mapping_entries.size(); i++) {
String line = (String) mapping_entries.elementAt(i);
String line1 = line;
if (line.indexOf("|") == -1) {
line1 = CSVFileReader.csv2Delimited(line, "|");
}
Vector u = StringUtils.parseData(line1, '|');
String source_code = (String) u.elementAt(0);
String source_term = (String) u.elementAt(1);
String target_code = (String) u.elementAt(2);
if (target_code != null && target_code.length() > 0) {
String query = owlSPARQLUtils.construct_get_properties_by_code(namedGraph, target_code);
Vector w = owlSPARQLUtils.getPropertiesByCode(namedGraph, target_code);
if (w == null || w.size() == 0) {
//System.out.println("\tgetPropertiesByCode returns null???");
} else {
w = parserUtils.getResponseValues(w);
HashMap hmap = createPropertyHashMap(w);
propertyHashMap.put(line, hmap);
}
}
}
return propertyHashMap;
}
public void dumpPropertyHashMap(HashMap propertyHashMap) {
Iterator it = propertyHashMap.keySet().iterator();
while (it.hasNext()) {
String key = (String) it.next();
HashMap hmap = (HashMap) propertyHashMap.get(key);
System.out.println(key);
Vector properties = new Vector();
Iterator it2 = hmap.keySet().iterator();
while (it2.hasNext()) {
String key2 = (String) it2.next();
properties.add(key2);
}
properties = new gov.nih.nci.evs.restapi.util.SortUtils().quickSort(properties);
for (int k=0; k<properties.size(); k++) {
String property = (String) properties.elementAt(k);
System.out.println(property);
Vector values = (Vector) hmap.get(property);
values = new gov.nih.nci.evs.restapi.util.SortUtils().quickSort(values);
for (int k2=0; k2<values.size(); k2++) {
String value = (String) values.elementAt(k2);
System.out.println("\t" + value);
}
}
}
}
public HashMap createPropertyHashMap(Vector w) {
HashMap hmap = null;
if (w == null || w.size() == 0) {
return hmap;
}
hmap = new HashMap();
for (int j=0; j<w.size(); j++) {
String t = (String) w.elementAt(j);
Vector u = StringUtils.parseData(t, '|');
//Secondary Malignant Neoplasm|code|C4968
String property_name = (String) u.elementAt(1);
String property_value = (String) u.elementAt(2);
Vector v = new Vector();
if (hmap.containsKey(property_name)) {
v = (Vector) hmap.get(property_name);
}
v.add(property_value);
hmap.put(property_name, v);
}
return hmap;
}
public String toString() {
StringWriter out = new StringWriter();
PrintWriter writer = new PrintWriter(out);
String content = propertyHashMap2HTML(this.propertyHashMap);
writer.print(content);
return writer.toString();
}
public static void main(String[] args) {
String serviceUrl = args[0];
String namedGraph = args[1];
String mappingfile = args[2];
System.out.println("serviceUrl: " + serviceUrl);
System.out.println("namedGraph: " + namedGraph);
System.out.println("mappingfile: " + mappingfile);
System.out.println("Generating property hashmap...");
Vector mapping_entries = Utils.readFile(mappingfile);
HashMap propertyHashMap = new ConceptDetailsGenerator().appendPropertiesToMappingEntries(serviceUrl, namedGraph, mapping_entries);
ConceptDetailsGenerator generator = new ConceptDetailsGenerator(propertyHashMap);
String hyperlinkUrl = "https://ncimappingtool-dev.nci.nih.gov/ncimappingtool/pages/concept_details.jsf?ng=http://ncicb.nci.nih.gov/xml/owl/EVS/Thesaurus.owl&code=";
generator.setHYPERLINK(hyperlinkUrl);
String content = generator.propertyHashMap2HTML(propertyHashMap);
/*
String outputfile = "mapping.html";
String title = "Concept Details";
generator.generate(outputfile, title);
*/
System.out.println(content);
}
}
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"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/dcds.2014.34.4515","text":"# American Institute of Mathematical Sciences\n\nNovember\u00a0 2014,\u00a034(11):\u00a04515-4535. doi:\u00a010.3934\/dcds.2014.34.4515\n\n## Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation\n\n 1 Institut f\u00fcr Mathematik, Universit\u00e4t Klagenfurt, Universit\u00e4tsstra\u00dfe 65-67, 9020 Klagenfurt am W\u00f6rthersee, Austria, Austria\n\nReceived\u00a0 November 2013 Revised\u00a0 March 2014 Published\u00a0 May 2014\n\nWe consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.\nWe use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.\nMoreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.\nAdditionally, we provide results concerning exponential decay of solutions of the nonlinear equation.\nCitation: Rainer Brunnhuber, Barbara Kaltenbacher. Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4515-4535. doi: 10.3934\/dcds.2014.34.4515\n##### References:\n\nshow all references\n\n##### References:\n [1] Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934\/cpaa.2021057 [2] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934\/dcds.2020393 [3] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934\/dcds.2020382 [4] Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934\/dcdsb.2020227 [5] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934\/cpaa.2021006 [6] Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934\/dcds.2021001 [7] Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934\/nhm.2021005 [8] Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934\/dcds.2021022 [9] Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $n$-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934\/cpaa.2021018 [10] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934\/cpaa.2004.3.849 [11] Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934\/cpaa.2021017 [12] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934\/dcds.2017258 [13] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schr\u00f6dinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934\/dcdss.2020450 [14] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934\/dcdsb.2018055 [15] Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934\/dcdsb.2020233 [16] Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021\u00a0 doi: 10.3934\/dcdsb.2021091 [17] Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021\u00a0 doi: 10.3934\/dcdsb.2021103 [18] Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schr\u00f6dinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934\/cpaa.2021030 [19] Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$^+$-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete & Continuous Dynamical Systems - B, 2021\u00a0 doi: 10.3934\/dcdsb.2021090 [20] Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934\/cpaa.2021064\n\n2019\u00a0Impact Factor:\u00a01.338","date":"2021-04-16 15:23:05","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\": 0.5223907232284546, \"perplexity\": 4023.499188796518}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038066981.0\/warc\/CC-MAIN-20210416130611-20210416160611-00502.warc.gz\"}"} | null | null |
Q: List of data frames containing row and column indices to select data from a larger data frame I have a list of data frames. One of the columns contains the name of the data frame:
a <- data.frame(PIN = c(1:3), Item = c("a", "a", "a"))
b <- data.frame(PIN = c(4:6), Item = c("b", "b", "b"))
List <- list(a, b)
And a larger data frame I want to extract data from. The values in the first columns in my list of data frames correspond to the values in the first column or the larger data frame, and the names/values in the second columns of my list of data frames correspond to column names in the larger data frame:
DF <-data.frame(PIN = c(1:10), a = c(101:110), b = c(201:210), c = c(301:310))
I want to add columns to the data frames in my list extracting data from DF, using PIN as a row index and Item as a column index. So essentially, I will end up with:
a <- data.frame(PIN = c(1:3), Item = c("a", "a", "a"), a = c(101:103))
b <- data.frame(PIN = c(4:6), Item = c("b", "b", "b"), b = c(204:206))
List <- list(a, b)
How can I do this?
A: With tidyr, dplyr and purrr we could do:
DF <- tidyr::gather(DF, key = "Item", value, -PIN)
purrr::map(List, dplyr::left_join, DF, by = c("PIN", "Item"))
[[1]]
PIN Item value
1 1 a 101
2 2 a 102
3 3 a 103
[[2]]
PIN Item value
1 4 b 204
2 5 b 205
3 6 b 206
Note that you will get a warning message because the example data contains factors. Add stringsAsFactors = FALSE when constructing the data frames to avoid this warning message.
You can subsitute the purrr map function with base R's lapply using:
lapply(List, dplyr::left_join, DF, by = c("PIN", "Item"))
Additionally, one alternative to Uwe's solution could be:
library(tidyverse)
DF <- gather(DF, key = "Item", value, -PIN)
List %>%
map(mutate_if, is.factor, as.character) %>% # optional, but solves the warning message by converting factors to character
map_df(rbind, .id = "id") %>%
left_join(DF) %>%
split(.$id)
$`1`
id PIN Item value
1 1 1 a 101
2 1 2 a 102
3 1 3 a 103
$`2`
id PIN Item value
4 2 4 b 204
5 2 5 b 205
6 2 6 b 206
Data:
# Create list
List <- list(
data.frame(PIN = c(1:3), Item = c("a", "a", "a")),
data.frame(PIN = c(4:6), Item = c("b", "b", "b"))
)
# Create data frame
DF <- data.frame(
PIN = c(1:10),
a = c(101:110),
b = c(201:210),
c = c(301:310)
)
A: You may want to reconsider your data structures:
*
*A list of equally structured data.frames can be combined into one larger data object which avoids to iterate over the list elements.
*As the "payload" columns a, b, and c of DF all contain the same data type, DF can be reshaped from wide to long format. This converts the column names to data elements which can be more easily used for a lookup operation.
So, my suggestion is:
library(data.table)
keys <- rbindlist(List, idcol = TRUE)
DT <- setDT(melt(DF, id.vars = "PIN", variable.name = "Item"))
DT[keys, on = .(PIN, Item)]
PIN Item value .id
1: 1 a 101 1
2: 2 a 102 1
3: 3 a 103 1
4: 4 b 204 2
5: 5 b 205 2
6: 6 b 206 2
If you still need a list result:
split(DT[keys, on = .(PIN, Item)], by = ".id")
$`1`
PIN Item value .id
1: 1 a 101 1
2: 2 a 102 1
3: 3 a 103 1
$`2`
PIN Item value .id
1: 4 b 204 2
2: 5 b 205 2
3: 6 b 206 2
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← Weaver Watch Update
Season Scores May 23 →
Quarter Season All-Stars
As of today, teams have played 38 to 41 games, which means the 2011 baseball season is about one quarter over. Let's look at the players who have had the most success over the last six weeks. I'll build a 25-man roster in each league, with ten pitchers and fifteen hitters. There will be no rule about representing every team and, best of all, no fan voting, so players will only be selected on merit.
We'll start in the National League. We'll need two catchers, and we've got three to five solid contenders. I'll go with Arizona's Miguel Montero, whose .339 weighted on base average (wOBA) leads all NL catchers, and San Francisco's Buster Posey, whose .256/.347/.368 batting line may be nothing to write home about, but whose defense has been so good that fangraphs and baseball reference both have him leading all NL catchers in WAR. Apologies to Brian McCann of the Braves.
At first base, it has to be Cincinnati's Joey Votto, especially after last year's snub. Votto leads all National Leaguers with 2.7 fWAR and 2.6 bWAR, getting on base at an absurd .471 clip and slugging .566. Gaby Sanchez of the Marlins is hitting .331/.410/.538, good for 1.9 bWAR and a few at-bats off the bench in an All-Star game. Apologies to Prince Fielder and Ike Davis.
At second, Cincinnati's Brandon Phillips narrowly edges out Milwaukee's Rickie Weeks, whose identical .381 wOBA isn't enough to overcome Phillips' 12-point OBP advantage or his far superior defense. We'll keep Weeks on as a reserve.
Our starting third baseman is certainly Philadelphia's Placido Polanco, who is batting .347/.389/.447 with three stolen bases in three tries for the NL's best team. Apologies to Chipper Jones and Pablo Sandoval, who's been phenomenal in limited playing time.
At shortstop, Troy Tulowitzki has slowed down enough after his hot start to hand the starting gig over to Jose Reyes, who's passing his Giants audition with flying colors, leading all shortstops in runs created while stealing 12 bases in 15 attempts. Tulowitzki, probably the National League's best player since last summer, certainly warrants a spot on the bench. Apologies to Alex Gonzalez and Jimmy Rollins.
The NL's starting left fielder is Matt Holliday of the Cardinals. If not for Votto, Holliday would be the frontrunner for NL MVP with his .394 batting average and .614 slugging percentage. The Dodgers' Matt Kemp has been the best center fielder, combining power and speed with seven home runs and 12 stolen bases. In right, we'll take Kemp's teammate Andre Ethier, whose 30-game hitting streak boosted his batting average to .355 and his slugging percentage to an even .500. We'll take another outfielder, St. Louis's Lance Berkman, as our starting designated hitter, since Berkman has earned all of his 1.7 fWAR (or his 1.6 bWAR, if you'd prefer), and then some with his bat, as his defense has been well below replacement level. We've got room for two more position players on our team, so we'll take Philly's Shane Victorino, a defensive replacement who can hit a little (.503 slugging percentage and eight steals), and Milwaukee's Ryan Braun, who leads the league with 11 homers. Apologies to Drew Stubbs, Colby Rasmus, and Carlos Beltran.
On the mound, our starter has to be Florida's Josh Johnson. While Roy Halladay has accumulated more WAR according to both keepers of the stat, he's had two more starts in which to do so. Johnson, who leads the league with a 1.63 ERA and a 177 Season Score, has more bWAR than Halladay per start, despite a FIP more than a run higher. Halladay, whose 1.49 FIP suggests he'll pitch even better than his 2.05 ERA going forward, is certainly on our team, as is teammate Cole Hamels. I'll use Season Score to choose the rest of my starting pitchers. Tim Lincecum of the Giants actually leads Halladay with a 166 Season Score, 11 points behind Johnson. Cardinals Kyle Lohse and Jaime Garcia are fourth and fifth. Atlanta's Jair Jurrjens is a few spots down the list in 13th, but he missed more than two weeks with an injury to start the season and has returned 5-0 with an incredible 1.66 ERA. The Dodgers' Clayton Kershaw and the Diamondbacks' Ian Kennedy round out the starter portion of the staff. Apologies to Shaun Marcum and Jhoulys Chacin.
Since starters provide much more value than relievers, I'll stick with one reliever per league, and in the NL it has to be the Braves' Craig Kimbrel, who has a ridiculous 31 strikeouts in less than 20 innings. Apologies to Kimbrel's teammate Johnny Venters, as well as Ryan Madson of the Phillies.
In the American League, our starting catcher is Yankees reclamation project Russell Martin, the best catcher in the National League in 2007 and 2008. It appeared as though Martin may have been finished after two ugly seasons in LA, but he's hit more homers (seven) in his first 33 games at the Bandbox in the Bronx than he did in either of the last two seasons at Chavez Ravine. We'll take Baltimore's Matt Wieters as our other catcher. Wieters' .336 wOBA can't match Alex Avila's .367, but both keepers of WAR love Wieters's defense, each ranking him first among AL catchers.
At first base, Adrian Gonzalez, Miguel Cabrera, and Paul Konerko have all put up similarly impressive offensive numbers (all wOBAs between .407 and .412), but it's Gonzalez's defense that sets him apart (the five homers in his last five games don't hurt either). We'll keep Cabrera and his .980 OPS on as a designated hitter. Apologies to Konerko and Justin Smoak of the Mariners.
Either of the two best second basemen in the American League could play multiple positions or serve as a utilityman off the bench. We'll give the second base nod to Tampa's second baseman/outfielder Ben Zobrist and his .568 slugging percentage, while rewarding Angels first/second baseman Howie Kendrick's phenomenal offensive and defensive play (2.3 fWAR) with the utility spot.
At third, it's all Texas's Adrian Beltre, whose ten home runs and stellar defense cancel out his .304 on base percentage. Kevin Youkilis, our backup, has hit better than Beltre (.246/.385/.484), but has struggled defensively in his return to the hot corner. Apologies to Alex Rodriguez, who has cooled off after a hot start, and Jack Hannahan, who has helped buoy the Indians with strong offensive and defensive play.
We'll take another Angel who has played multiple positions as our shortstop. Maicer Izturis has hit .319/.366/.467 while playing second base, third base, and shortstop. We'll take a pure shortstop, Cleveland's Asdrubal Cabrera as a defensive replacement (though his .484 slugging percentage might warrant an at-bat or two as well). Apologies to Detroit's Jhonny Peralta.
Toronto's Jose Bautista, who leads all of baseball in virtually every meaningful category, most notably home runs (16) and OPS (1.388), probably deserves to play center field and shortstop and pitch a few innings, but he'll have to settle for the starting right field gig. In addition to his Bondsian hitting display, Bautista has fielded his position well, accumulating 3.9 WAR in 40 games, establishing a pace that could net him 15.8 WAR in a full season (Babe Ruth's 14.7 WAR in 1923 is the best season ever for a position player). New York's Curtis Granderson brings his 13 homers to our starting center field assignment, and we'll throw Tampa's Matt Joyce and his .615 slugging percentage in left. We've got two spots left, and one has to go to Kansas City's Jeff Francoeur and his .550 slugging percentage and excellent defense. Our defensive replacement will be Twins center fielder Denard Span. Apologies to the league's best true designated hitters, Jason Kubel, David Ortiz, and Travis Hafner, who haven't hit quite enough to be more valuable than any of these 15 guys.
Our American League starter could be any of five pitchers. LA's Jered Weaver made a strong case by winning his first six starts and striking out 61 batters to-date. Tampa's James Shields leads the league in average game score at 71.3 and has been a model of consistency. Boston's Josh Beckett is just a tick behind at 71 and leads the league with a 1.75 ERA. Oakland's Trevor Cahill leads the league with 2.5 bWAR, again delivering an ERA under 2 despite a FIP over 3. The best pick, though, is the Angels' Dan Haren, whose 186 Season Score is the best in the majors. Haren's 1.93 ERA is backed up by a 2.08 FIP, and his 7.63 strikeouts per walk are the best in the league. Texas's Alexi Ogando is another easy choice with his 0.869 WHIP, as is current strikeout leader (and co-leader in no-hitters) Justin Verlander of Detroit. Finally, Baltimore's Zach Britton may be succeeding with smoke and mirrors, carrying a 2.42 ERA despite striking out just over five batters per nine, but he's getting batters out, which is what counts (at least according to bWAR, where he's fifth in the AL). Our last pick could go several different ways, but we'll take Tampa's David Price, whose 2.89 FIP supports a 3.12 ERA and a 5-3 record. Apologies to Boston's Jon Lester and Cleveland's Josh Tomlin.
Our one relief pitching spot comes down to the classic Mariano Rivera-Jonathan Papelbon duel, so we'll pick… neither. While Rivera and his 1.45 ERA and Papelbon and his 11.34 K/9 lead AL relievers with .8 fWAR, the guy in third place, Chicago's Sergio Santos, has yet to surrender an earned run in 19 innings and carries a nifty 10.42 K/9 of his own.
Let's summarize by lining our teams up for battle:
1. Jose Reyes, ss, Mets
2. Matt Kemp, cf, Dodgers
3. Matt Holliday, lf, Cardinals
4. Joey Votto, 1b, Reds
5. Lance Berkman, dh, Cardinals
6. Andre Ethier, rf, Dodgers
7. Brandon Phillips, 2b, Reds
8. Placido Polanco, 3b, Phillies
9. Miguel Montero, c, Diamondbacks
p. Josh Johnson, Marlins
Ryan Braun, Brewers
Jaime Garcia, Cardinals
Roy Halladay, Phillies
Cole Hamels, Phillies
Jair Jurrjens, Braves
Ian Kennedy, Diamondbacks
Clayton Kershaw, Dodgers
Craig Kimbrel, Braves
Tim Lincecum, Giants
Kyle Lohse, Cardinals
Buster Posey, Giants
Gaby Sanchez, Marlins
Troy Tulowitzki, Rockies
Shane Victorino, Phillies
Rickie Weeks, Brewers
1. Curtis Granderson, cf, Yankees
2. Ben Zobrist, 2b, Rays
3. Adrian Gonzalez, 1b, Red Sox
4. Jose Bautista, rf, Blue Jays
5. Miguel Cabrera, dh, Tigers
6. Adrian Beltre, 3b, Rangers
7. Matt Joyce, lf, Rays
8. Russel Martin, c, Yankees
9. Maicer Izturis, ss, Angels
p. Dan Haren, Angels
Josh Beckett, Red Sox
Zack Britton, Orioles
Asdrubal Cabrera, Indians
Trevor Cahill, A's
Jeff Francoeur, Royals
Howie Kendrick, Angels
Alexi Ogando, Rangers
David Price, Rays
Sergio Santos, White Sox
James Shields, Rays
Denard Span, Twins
Justin Verlander, Tigers
Jered Weaver, Angels
Matt Wieters, Orioles
Kevin Youkilis, Red Sox
There are a lot of superstars among these 50 names- guys we would expect to make just about every quarter's All-Star team. There are a few budding stars- Jose Bautista is making a case as baseball's best player and Matt Wieters may finally be cashing in on all the promise he brought to Baltimore as a hot prospect a few years ago. But there are also some surprises here- Miguel Montero and Alexi Ogando would surprise more people by continuing to play this well than they would by fading into the background in the remaining three quarters.
We've got four Cardinals without Albert Pujols, four Phillies without Chase Utley, and four Rays without Evan Longoria. We've got more Angels (four) than Red Sox (three) or Yankees (two). Every AL team is represented except the Mariners, and Felix Hernandez and Michael Pineda were two of the last four or five pitchers out. In contrast, five NL teams- Washington, Chicago, Pittsburgh, Houston, and San Diego, don't have a player on the team nor one I found deserving of an apology (though fWAR loves the way Matt Garza got himself to 2-4, 4/17). I suspect the majority of these guys will make the actual All-Star team, at least after the injury replacements are named and the fans vote for the 45th man on each team. Those that don't will never forget these first 40 or so games, when they relived some old magic or started some water cooler talk about their future promise.
Now let's turn it over to the fans and watch Derek Jeter get elected.
This entry was posted in All-Star Game, Angels, Cardinals, Phillies, Rays. Bookmark the permalink.
6 Responses to Quarter Season All-Stars
I like the shot at Jeter at the end lol.
I agree with your Phillies picks. I'm a bit concerned that Halladay has thrown 2 consecutive complete games and lost. Their problems are bigger than not having Chase Utley. They're almost getting to the point that they may need to do to their outfielders what the Flyers did with their goalies, which as we saw, gets ugly.
Realistically, you could just use the Braves' bullpen for this game. Those guys are nasty. The Phillies are going to have to do a lot better against the rest of the league because I think the Braves will have their number on the head-to-head matchup for the rest of this season.
Nick, it's funny you should mention the Braves owning the Phillies. I was going to write a post about Braves-Phillies becoming the best rivalry in the game, as they're both loaded with pitching and have won something like 18 of the last 19 division titles between them. Then David Schoenfeld at ESPN wrote the same thing this morning about Cardinals-Reds. I'm not sure I agree with him that the players have to hate each other to make a great rivalry, but he's got a point about the drama in that rivalry. Still, the pitching matchups when the Braves and Phillies play practically guarantee three great games, and probably will for a few years, until the Phillies' veterans start to decompose.
Paul Lucero says:
Where is Justin Masterson? He is 5-1 with a 2.43 ERA. or at least Josh Tomlin. 5-1 with a 2.56 ERA. Indians have one of the better pitching staffs in the league. They should get some credit with 1-2 of their starters. Granted, the starters you named are having great starts to the season, but just because Verlander has thrown a no-no (almost 2) doesn't mean he should be an All-Star. Pre-no-no, 2-3 3.46 ERA. Just sayin' bud. Good teams though. I like your Asdrubal Cabrera choice, had me smiling.
Paul, thanks for commenting. The Indians have gotten some great pitching this year, but I have Masterson and Tomlin just outside the top nine. And Verlander leads the league in strikeouts, so I'm not sure he's the one you should be going after. To wit:
Verlander- 65 IP, 62 K, 21 BB, .94 WHIP
Britton- 52 IP, 29 K, 16 BB, 1.02 WHIP
Masterson 52 2/3 IP, 40 K, 17 BB, 1.27 WHIP
Tomlin- 52 2/3 IP, 27 K, 9 BB, 0.85 WHIP
Mike, Young's having a good year, but I don't see why it's a joke to exclude him from this team. Matt Joyce has a 1.030 OPS; Young's is .878. Joyce has created one more run than Young in three fewer games and plays defense every day. I was looking only at this season's numbers when I made these teams, but if I was considering past track records, Young would still be in line behind Hafner and Ortiz among true DHs, and Rodriguez and Konerko among offense-first position players. I'll reevaluate when Young has more WAR than Melky Cabrera.
where is Michael Young? Is this a joke? He is hitting over .500 w/ RISP. Best DH in the league. Sorry but you managed to put Joyce on ur list but not a guy w/ a track record like Young?
Pingback: An Early All-Star Ballot | Replacement Level Baseball Blog | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,259 |
Rob Kardashian and Blac Chyna Share First Photo of Baby Dream!
Rob Kardashian and Blac Chyna are proud parents to a beautiful baby girl!
Just a few hours after the engaged couple welcomed their first child together, Dream Renee Kardashian, on Thursday, the two released the first official photo of their little bundle of joy.
The pic shows baby Dream all cuddled up in a blue-and-pink striped blanket and matching beanie, adorably holding her daddy's pinky finger.
Others noted her resemblance to her grandfather, Robert Kardashian, while some Twitter users simply couldn't stop gushing over her "precious" features.
Shortly after, Chyna's BFF Amber Rose shared another pic of Dream, captioning it, "A Dream come true #DreamKardashian ?."
Chyna reportedly gave birth via C-section at Cedars-Sinai Medical Center in Los Angeles, with Dream weighing in at 7 pounds, 5 ounces.
"Everyone is doing great!" a source told ET at the time. "Rob and Chyna are so happy to finally meet Dream."
This is the first child for Rob, 29, and the second for Chyna. The 28-year-model is also mother to 4-year-old son, King Cairo, whom she shares with her ex, rapper Tyga.
Hear more on Dream's arrival in the video below. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,580 |
Perspective is a web app that amasses and organizes geo-location based communities around sensitive issues where individuals can be honest about their opinions, beliefs, and experiences while keeping their anonimity/privacy.Once an active group passes a certain threshold in either topic population or activity then a SEED of that topic will be sent to another FERTILE (active PERSPECTIVE community) somewhere else in the world.Communities around these sensitive issues will be exposed to experts, people that overcame topic related issues, and foundations that are attempting to solve the problem.Users can choose to either Donate to these foundations from anywhere in the world using Bitcoin or Choose to get Involved. Get a New Perspective ... Change The World...
## Copyright and License
Copyright 2013-2015 Iron Summit Media Strategies, LLC. Code released under the [Apache 2.0](https://github.com/IronSummitMedia/startbootstrap-grayscale/blob/gh-pages/LICENSE) license.
| {
"redpajama_set_name": "RedPajamaGithub"
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\section{Study rationale}
Testing represents 30-50\% of software development costs~\cite{biffl2006value} and about 50\% of development time~\cite{pan1999software}. There are many testing phases, the first round(s) usually utilizing alpha testers. Their number is limited by a company size, and even for big companies it is impossible to duplicate the myriad of possible hardware/software configurations. Whereas alpha testers are typically company employees, beta testers are the first product users outside the company. Their feedback can greatly influence the product before it is used by standard customers.
Thanks to the Internet, thousands of beta testers with different devices and practices can report their feedback on a product before its full launch. An additional benefit of beta testing lies in involving international aspects. Since beta test participants can come from different locations, localization issues (such as language, currency, culture or local standards) can be detected and reported~\cite{betatesting}. Furthermore, cultural background also affects the perceived usability~\cite{wallace2009effect}. Therefore, beta testers bring huge benefits by detecting potential hardware conflicts and performing usability checking.
Our work summarizes a large-scale case study on beta testers and standard users of an online security software. While many alpha or beta testing studies have been published, the idea of comparing beta testers and standard users has been rarely tackled before. Mantyla et al. investigated the related question \textit{Who tested my software?}~\cite{mantyla2012tested}, but their study was limited to three companies' employees. Other studies~\cite{kanij2015empirical, merkel2010does} provide some insights into the software tester population, yet are mainly based on specific sub-populations, such as people interested in testing, users of specialized forums and LinkedIn, or companies' employees, so a selection bias may occur.
We compare beta testers and standard users in a number of aspects.
Firstly, we focus on technology. Having similar devices with regard to the technical aspects (hardware, OS, etc.) is the basic requirement for successful software beta testing. Since physical environment is one aspect that influences usability testing~\cite{sauer2010influence}, the device used to test an application may influence its usability too. For example, security software running in background can decrease the perceived overall performance of the machine and thus the perceived usability. Therefore, participants with a low-end hardware may encounter different usability issues than those with a high-end one. Usually, beta testers are thought of as being the problem solvers, or early adopters~\cite{betatesterstype} with most recent and hi-tech hardware. Therefore, many issues could stay unnoticed during beta testing.
Secondly, we examine user demographics. Existing research shows that users' IT-related behavior is largely affected by their gender, age, education and cultural background. For example, a greater rate of computer use and online activities was associated with lower age, higher education and being male~\cite{2015world}. The differences in IT usage are also related to the country of origin~\cite{ono2007digital}. The countries differ in the development of information society, leading to varying access opportunities and creating digital disparities between nations~\cite{chinn2010ict,cuervo2006multivariate}. As a result, there could be nations with more computer savvy populations and/or populations more inclined to use free software (even though still in beta). For example, anecdotal evidence has it that the Japanese move towards emerging technologies more slowly than other nations~\cite{2011japan}.
Thirdly, the different patterns of Internet/computer usage are associated with other individual characteristics, such as users' computer self-efficacy and privacy perceptions. Computer self-efficacy~\cite{compeau1995computer} reflects the extent to which the user believes he/she is capable of working efficiently with a computer. Users with a higher computer self-efficacy tend to use the computer more~\cite{compeau1995computer}, adopt new technology faster~\cite{hill1986communicating,venkatesh2003user} and have better performance in computer-related tasks~\cite{downey2009accurately}. Regarding privacy perception, marketing research constantly shows how consumer online behavior (such as willingness to provide personal information or intention to use online services) are affected by their privacy concerns~\cite{malhotra2004internet,sheehan2002toward}. Since beta testing usually includes sharing one's system, localization or even personal information with a company, it may discourage users with higher privacy concerns or those who store more private data on their computers. However, this may be an important segment of end users population with distinct expectations from the final product.
\section{Methodology}
The study was conducted in cooperation with ESET, an online security software company with over 100 million users in more than 200 countries and territories\footnote{https://www.eset.com/int/about/}. Two samples were used for analyses: beta testers and standard users of ESET security software solution for Windows.
ESET Beta program allows anybody to download the product beta version and become a public beta tester. Despite the fact that users fill out a questionnaire before they can beta test the product, the company does not use any criteria to select beta testers. The main role of testers is to report bugs and/or suggest improvements. ESET's beta testers are motivated by the opportunity to use a beta product for free and possibility to use the product sooner than standard users.
The sample of beta testers ($N$ = 87\,896) was collected from June to December 2015, the sample of standard users ($N$ = 536\,275) from January to March 2016.
Firstly, we collected anonymized system parameters for each installation including processor configuration, RAM size, operating system, country, and time spent on each installation screen. Countries were identified by GeoIP2\footnote{https://www.maxmind.com/en/geoip2-databases}. One data record represents one installation of the software.
Secondly, a questionnaire was introduced to users at the end of the installation process. Filling out a questionnaire was voluntary and we used no incentives other than stating that completing it will help ESET improve their products. Of the beta testers sample, 6\,008 users filled out at least one questionnaire item (7.800\%) and the same applied to 27\,751 standard users (5.560\%). Since we collected the data from installations in English, the questionnaire was also presented in English. The questionnaire was a source for demographic data and privacy perceptions. No identification data were collected.
\subsection{Data cleaning}
During the data cleaning, we first removed installations coming from ESET's internal IP space (0.282\% of the sample) to exclude company alpha testers. Further, since each data entry reflected one installation, there might have been duplicated entries from the same device. To prevent possible biases, we identified cases with the same combination of hardware specification and IP address, randomly selected one and deleted the rest (this removed 7.429\% of the data).
As noted, the questionnaire was voluntary and only a sub-sample of users completed it. The whole questionnaire was presented on four screens. We used the time spent on each screen to clean the data -- those who spent less than 6 seconds on a screen with two items and those who spent less than 7 seconds on a screen with three items were considered as invalid and their data from respective screen were omitted from analyses ($N$ = 10\,151; 30.1\% of the questionnaire respondents).
The final cleaned sample for our study thus consists of 576\,170 installations on unique devices, including 29\,598 questionnaires with at least one answered item, see Table~\ref{tab:participants-numbers}.
\begin{table}[tbp]
\centering
\begin{tabular}{l||r|>{\raggedleft}p{1.3cm}|r}
& Unique devices & \multicolumn{2}{c}{Filled questionnaires} \\ \hline \hline
Beta testers & 77\,028 & 5\,514 & 7.158\% \\ \hline
Standard users & 499\,142 & 24\,084 & 4.825\% \\ \hline \hline
Total & 576\,170 & 29\,598 & 5.137\%
\end{tabular}
\caption{An overview of participant numbers (after data cleaning).}
\label{tab:participants-numbers}
\end{table}
\subsection{Analytical strategy}
To assess the differences between standard users and beta testers, we used the $\chi^2$ test (categorical data) and t-tests (interval data). Analyses on large samples typically show statistically significant results even for very small effects. When considering such results, it is important to interpret effect sizes rather than to rely on significance alone. Therefore, we calculated Cramer's~$V$ ($\varphi_c$) for categorical data and Cohen's~$d$ for interval data. For $\varphi_c$, the value of 0.1 is considered as small, 0.3 as medium and 0.5 as a large effect size, and for $d$, the respective values are 0.2, 0.5, and 0.8~\cite{cohen1988,field2002design}.
The fact that the questionnaire data came only from a sub-sample of users may imply a bias in the results (see the limitations). To gain an insight in the differences between the samples with and without the questionnaire, we compared these users with regard to the parameters available for all of them (platform information, CPU performance, RAM and OS version). The effect sizes of found differences were negligible ($\varphi_c$ < 0.034). Therefore we are confident that the questionnaire data are informative, despite being obtained from only a small subsample of users.
\section{Technology}
First we look at the technological aspects. We inspect the hardware platform information (32/64-bit), CPU model, RAM size and OS version.
\subsection{Hardware}
The platforms differ only slightly between the subsamples -- 35.3\% of beta testers use 32-bit systems while about 34.5\% of the standard users do so ($\chi^2$(1)= 20.998, $\varphi_c$ = -0.006, $p$ < 0.001, $N$ = 576\,170).
The CPU performance has been categorized into four groups (low-end, mid-low, mid-high, high-end) based on the PassMark \textit{CPU Mark} criterion~\cite{passmark}. The CPU name was matched against the PassMark online database. Since CPU names are not standardized, we were unable to assign the score in 3.040\% of the cases ($N_{noCpuMark}$ = 17\,514, proportionally distributed between beta testers and standard users).
Beta testers are more represented in the low-performance category and standard users in the mid-high category. The proportions are quite similar in mid-low and high-end categories (see Figure~\ref{fig:cpuram}). Although statistically significant, the effect size is small ($\chi^2$(3)= 1187.546, $\varphi_c$ = 0.045, $p$ < 0.001, $N$ = 576\,170).
RAM size was grouped into 4 categories (0-2\,GB, 2-4\,GB, 4-8\,GB, 8\,GB and higher). Standard users' proportion is higher in the `2-4\,GB' category, while beta testers dominate in the lowest `0-2\,GB' category. The proportions in two highest categories are similar, see Figure~\ref{fig:cpuram}. The small effect size suggests the differences are negligible, despite being significant ($\chi^2$(3)= 206.926, $\varphi_c$ = 0.019, $p$ < 0.001, $N$ = 576\,170).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/cpu.pdf}
\includegraphics[width=\columnwidth]{img/ram.pdf}
\caption{Basic hardware characteristics for beta testers and standard users (read more in Study Limitations, same for the following figures).}
\label{fig:cpuram}
\end{figure}
\subsection{Operating system}
Beta testers prevail in the two newest OS versions (Windows 8, Windows 10), while standard users dominate in Windows 7, and have nearly equal representation in Windows Vista and XP, see Figure~\ref{fig:winversions}. The effect size is again small ($\chi^2$(2)= 1\,925.745, $\varphi_c$ = 0.058, $p$ < 0.001, $N$ = 575\,979). Other Windows versions (Windows 98, Windows 2000, etc.) were also marginally present but were omitted due to the extremely low counts (<0.001\%, $N_{otherWinVersions}$ = 191). Note that the study targeted only users of Microsoft Windows software.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/os.pdf}
\caption{Comparison of operating system version between beta testers and standard users.}
\label{fig:winversions}
\end{figure}
Windows 10 was more often used by beta testers, even though their data were collected sooner -- i.e., standard users had more time to upgrade. This observation indicates that beta testers are often recruited from early adopters~\cite{betatesterstype}.
\pagebreak
\subsection{Specific configurations}
We were also interested in specific configurations of users' devices. We combined all four technological aspects together (OS platform, CPU performance, RAM size, OS). This led to the identification of 116 unique hardware+software combinations in the whole dataset with frequencies from 1 to 43\,519 (7.556\%). The sample of standard users included 114 combinations (i.e., there were two specific combinations found among beta testers' devices, which were not found among standard users), and the sample of beta testers included 102 combinations. However, the combinations not present in beta testers were also rather marginally present among standard users ($N_{onlyStandard}$ = 52, 0.010\%). We conclude that for almost every standard user in the sample, there was a beta tester in the sample with the same combination of examined parameters.
\section{Demography}
This section discusses participants' cultural and demographic profiles. We focus on the country of origin, gender, age and achieved education.
\subsection{Country of origin}
As noted in the Methodology section, the country was based on GeoIP2. This procedure failed to assign a country to 0.4\% cases ($N_{noCountry}$ = 2\,408). For a better overview, countries were grouped by continents and both subsamples were compared (see Figure~\ref{fig:continents}). We observed significant differences -- beta testers substantially dominate in South America and Europe, while standard users are more often based in Asia, Africa and Australia/Oceania ($\chi^2$(5)= 39\,049.72, $\varphi_c$ = 0.261, $p$ < 0.001, $N$ = 573\,538).
A detailed information regarding most represented countries can be seen in Table~\ref{tab:countries}. Only Iran, India, Egypt and the USA are among these most represented ones in both subsamples.
These issues are now investigated in the company w.r.t. product localization and usability, where country differences are quite likely to play a role.
\begin{table}
\centering
\begin{tabular}{|p{1.8cm}rr|p{1.8cm}rr|} \hline
\multicolumn{3}{|c|}{\bfseries Standard users} & \multicolumn{3}{c|}{\bfseries Beta testers} \\
\bfseries Country & \bfseries N & \bfseries \% & \bfseries Country & \bfseries N & \bfseries \% \\ \hline
Iran & 81\,035 & 16.2 & Mexico & 5\,662 & 7.4 \\ \hline
USA & 50\,220 & 10.1 & Indonesia & 5\,117 & 6.6 \\ \hline
India & 26\,532 & 5.3 & Brazil & 4\,251 & 5.5 \\ \hline
Indonesia & 25\,959 & 5.2 & China & 4\,132 & 5.4 \\ \hline
UK & 25\,173 & 5.0 & Peru & 3\,422 & 4.4 \\ \hline
Egypt & 21\,649 & 4.3 & Russia & 3\,348 & 4.3 \\ \hline
Romania & 16\,582 & 3.3 & Ukraine & 2\,979 & 3.9 \\ \hline
Pakistan & 15\,831 & 3.2 & Spain & 2\,513 & 3.3 \\ \hline
Peru & 15\,280 & 3.1 & Egypt & 2\,393 & 3.1 \\ \hline
Philippines & 14\,904 & 3.0 & <unknown> & 2\,314 & 3.0 \\ \hline
South Africa & 13\,951 & 2.8 & Iran & 2\,306 & 3.0 \\ \hline
UAE & 11\,584 & 2.3 & India & 1\,771 & 2.3 \\ \hline
Thailand & 10\,719 & 2.1 & Argentina & 1\,679 & 2.2 \\ \hline
Australia & 10\,621 & 2.1 & USA & 1\,560 & 2.0 \\ \hline
Germany & 8\,259 & 1.7 & Poland & 1\,543 & 2.0 \\ \hline
\end{tabular}
\caption{The most represented countries in the subsamples of beta testers and standard users.}
\label{tab:countries}
\end{table}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/continents.pdf}
\caption{The comparison of beta testers and standard users with respect to the continent they come from.}
\label{fig:continents}
\end{figure}
\subsection{Gender and age}
For basic information regarding demography, see Figure~\ref{fig:demographics}. In both subsamples, males represent a vast majority, however, there are more females among standard users ($\chi^2$(1)= 277.493, $\varphi_c$ = 0.099, $p$ < 0.001, $N$ = 28\,328).
Standard users are on average older than beta testers ($M_{beta}$ = 32.96, $SD$ = 12.974; $M_{standard}$ = 35.74, $SD$ = 16.327; $t$-test(25\,938) = 11.108; $p$ < 0.001; $d$ = 0.195, $N$ = 28\,940). Due to wide age range (11-80), we categorized the age into 7 groups to examine the differences in a more informative way (see Figure~\ref{fig:demographics}). Between the age of 21 and 50, there are significantly more beta testers than standard users, while the opposite applies to other categories ($\chi^2$(6)= 366.286, $\varphi_c$ = 0.119, $p$ < 0.001).
\begin{figure*}
\centering
\begin{minipage}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{img/gender.pdf}\\
\includegraphics[width=\textwidth]{img/education.pdf}
\end{minipage}
\includegraphics[width=0.48\textwidth]{img/age.pdf}
\caption{Demographic comparison of the subsamples based on the questionnaire data.}
\label{fig:demographics}
\end{figure*}
\subsection{Achieved education}
Education shows a consistent pattern in both subsamples, with college education being represented the most and primary school education the least. The pattern remains the same even when the youngest users (i.e., those who could not have reached higher education levels) are omitted. Beta testers are more represented in secondary education than standard users, but the effect size is small ($\chi^2$(2)= 237.085, $\varphi_c$ = 0.038, $p$ < 0.001, $N$ = 26\,354).
\subsection{Other demographic insights}
We combined the above-mentioned demographic information to examine whether beta testers represent various demographic segments of standard users well. By combining seven categories of age, gender, and education, we identified 42 unique combinations. Only two combinations were present in the sample of standard users, but not among beta testers. Both were females 71-80 years old, one with primary ($N_{standard}$ = 4) and the other with college education ($N_{standard}$ = 109). Remaining combinations were present in both subsamples with a fairly similar distribution. The highest difference was found among males aged 31-40 with college education, who were represented more often among beta testers (14.172\%) than standard users (9.539\%).
\section{Computer self-efficacy and privacy perception}
We assessed users' computer self-efficacy and privacy perception, using dedicated questions in an optional questionnaire. Furthermore, we inspected installation-related actions such as displaying the target installation folder.
The installation included an option to change the installation folder. In order to do this users had to click on the `change installation folder' link on one of the screens during the installation process to go to respective screen. This is also the only way the user could see the default installation folder, which is not displayed otherwise. Only a minority did this, with beta testers visiting the screen more than twice as much as standard users (1.1\% of standard users and 2.6\% of beta testers). The difference is statistically significant, however, the effect size is negligible ($\chi^2$(1)= 1215.180, $\varphi_c$ = 0.046, $p$ < 0.001, $N$ = 576\,170).
\subsection{Computer self-efficacy and digital skills}
Two questions assessed users digital skills:
\begin{itemize}
\item Do you consider yourself to be a skilled computer user? (Likert scale from (1) \textit{not at all skilled} to (6) \textit{extremely skilled}.)
\item Regarding this computer, are you IT technician? (Y/N)
\end{itemize}
Beta testers are more often IT technicians ($\chi^2$(1)= 285.988, $\varphi_c$ = 0.110, $p$ < 0.001, $N$ = 23\,607) and judge themselves as more skilled than standard users ($M_{beta}$ = 4.46, $SD$ = 1.313; $M_{standard}$ = 4.18; $SD$ = 1.473; $t$-test(22\,631) = -11.743; $p$ < 0.001; $d$ = 0.200, $N$ = 22\,633).
\subsection{Privacy perception}
The last part of our questionnaire involved questions about how private data are stored in users' computers, how sensitive the users are regarding their privacy, and their beliefs about the computer being generally a safe device. All items were measured on 6-point Likert scale ranging from (1) \textit{not at all} to (6) \textit{extremely (private/sensitive/safe)}:
\begin{itemize}
\item Do you consider the data in this computer private?
\item In general, are you sensitive about your privacy?
\item In general, do you consider computers to be safe devices against online attacks, e.g., viruses, hacking, phishing, etc.?
\end{itemize}
Both samples reported the same average level of private data in their computers ($M_{beta}$ = 4.678, $SD$ = 1.419; $M_{standard}$ = 4.690, $SD$ = 1.560; $t$-test(24\,323) = 0.504; $p$ = 0.614, $N$ = 24\,325) and both were quite similar in being privacy sensitive ($M_{beta}$ = 4.755, $SD$ = 1.376; $M_{standard}$ = 4.809; $SD$ = 1.492; $t$-test(23\,976) = 2.272; $p$ < 0.05; $d$ = 0.037, $N$ = 23\,978). We found a small difference in their evaluations of general computer safety: beta testers considered computers as slightly safer devices than standard users ($M_{beta}$ = 4.098, $SD$ = 1.712; $M_{standard}$ = 3.902; $SD$ = 1.819; $t$-test(23\,832) = -6.784; $p$ < 0.001; $d$ = 0.111, $N$ = 23\,834). We observed that beta testers consider themselves skilled IT users and they also consider the computer being a safer device than standard users do. This might suggest they are aware of security risks connected with computer usage and feel capable of preventing them.
\section{Study limitations}
There are limitations out of our control that may have influenced the presented results. Despite the careful cleaning process, we cannot be completely sure that each record corresponds to a unique participant/device. The OS version is based on the Windows system variable \textit{CurrentVersion} that does not differentiate end user and server products. However, we presume the amount of servers in the study is negligible, as the installed ESET product is designed for end user devices. We also lacked details of devices' technological aspects, which might have shown more nuanced configuration discrepancies.
The small ratio of users filling in the questionnaire may pose other limitations.
Firstly, self-selection and non-response bias might have skewed our results. For instance, the majority of users have college education -- these may recognize the value of user feedback better and be more willing to complete a product-related questionnaire. However, they did not differ in terms of hardware nor software from those skipping the questionnaire.
Secondly, there were only limited options to validate users' answers. Despite the thorough cleaning, some flawed questionnaire answers may have remained.
Thirdly, the questionnaire was distributed in English, which might have discouraged users not proficient in the language.
The datasets have different numbers of participants and come from different times. This may have influenced, for example, the number of people using Windows 10 as the study was conducted during the free upgrade period. Furthermore, the research was based only on the English mutation of the software, missing the customers preferring other languages.
\section{Conclusions}
We have cooperated with the software security company ESET in a large-scale comparison between beta testers and standard users of their main product. We focused on technological aspects, demographics and computer self-efficacy of nearly 600\,000 users.
Beta testers were early adopters of newer operating systems -- their distribution was significantly skewed towards newest versions (despite having less time for Windows 10 migration). They also tend to be younger, more often males, and perceive themselves as more skilled with their computers and also are more often IT technicians, supporting the `beta testers as geeks' picture. However, their hardware (platform, CPU performance and RAM size) was very similar to that of standard users, somewhat contradicting this popular image.
A striking difference was found in the countries of origin. From the top ten most represented countries only three were represented in both subsamples.
Overall, beta testers in our case study represented the population of standard users reasonably well: we have not observed any standard user segment that would be largely underrepresented in the sample of beta testers. ESET approach not to filter beta testers in any way and go with `the more testers the better', followed by analyses of selected observed differences, seems sufficient. For large international companies who are able to attract large numbers of beta testers, this may be the most efficient approach. However, for smaller, local, or not so well-established companies, this approach would probably not yield representative outcomes and may even shift the development focus in a wrong direction~\cite{dolan1993maximizing}.
\newpage
Additional resources, including the article presentation video, are available
at http://crcs.cz/papers/cacm2018.
\subsection*{Actionable takeaways}
\begin{itemize}
\item Use data you can collect to find out who your users and beta testers are. Consider the country of origin, software \& hardware configuration and basic demographics you know.
\item The fewer testers you have, the pickier you should be about their selection.
\item When testing international products, ensure beta testers are culturally representative of standard users to identify localization and cultural usability issues.
\item Testers should be representative of standard users. Keep checking that this is the case -- or remediate with additional analyses and more carefully reached conclusions.
\end{itemize}
\subsection*{Acknowledgement}
We thank Masaryk University (MUNI/M/1052/2013) and Miroslav Bartosek for support, and to anonymous reviewers and Vit Bukac for valuable feedback.
\bibliographystyle{abbrv}
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?> | {
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} | 1,807 |
\section{Introduction}\label{sec:int}
General Relativity breaks down at high enough energies. At some time in the early universe, it reached an energy scale which cannot be described by Einstein's theory. At those regimes we need a quantum theory of gravity. Attempts for that are found in the literature as Loop Quantum Gravity (\cite{ashtekar/2004,ashtekar/2011,rovelli/1990,rovelli/1998}) and M-theory (\cite{banks/1997,berman/2011,dasgupta/1999}).
Plenty of efforts have been made in trying to describe the observable universe as a $3-$brane embedded in a $5$D space-time named {\it bulk}. Some of these models rise as a special case of 11-dimensional M-theory. In Randall-Sundrum (RS) I braneworld model (\cite{randall/1999}), such a universe set up, with two branes - the weak brane, where we live in, and the gravity brane - was presented as an optimistic alternative to solve the hierarchy problem. Differently from the other fundamental forces, gravity would not be stuck on the weak brane. It would always ``realize" the effects of an extra compactified dimension. Note that according to Einstein, gravity is connected with the geometry of the (entire) space-time; in this way, it is expected to act in each dimension.
In RSII braneworld model (\cite{randall/1999b}), the weak brane is ``sent to infinite" and we and all the standard model particles are found at the gravity brane. Note that despite gravity ``leaks" through the extra dimension, it is concentrated near the brane, i.e., the graviton probability function has a maximum at the brane and exponentially falls as one moves away from the brane through the fifth dimension, which differently from RSI case, can be large. Indeed, this happens because the extra dimension in RSII model is considered to be warped. Moreover, RSII braneworld is capable of recovering, in a certain energy regime, the newtonian gravitational potential, as carefully discussed in (\cite{randall/1999b}).
A lot of cosmological models have been derived from RSII model (check, for instance, (\cite{binetruy/2000,flanagan/2000,kim/2000,apostolopoulos/2006,campos/2003}) and references therein, and also (\cite{brax/2004}) for a seminal braneworld cosmology review).
RSII model has also been considered as the universe set up for generalized models of gravity, such as $f(R)$ theories (\cite{nojiri/2003,shamir/2010,masoudi/2013,hussain/2011}), as one can see, for instance, in (\cite{balcerzak/2011,bazeia/2013,xu/2015}).
Recently, a more general theory of gravity was proposed by T. Harko and collaborators, for which, besides the general dependence on the Ricci scalar $R$ as in $f(R)$ theories, the gravitational part of the action also depends on a generic function of $T$, the trace of the energy-momentum tensor, namely the $f(R,T)$ gravity (\cite{harko/2011}). Although plenty of extradimensional cosmological models have been derived from $f(R,T)$ theories (e.g. (\cite{moraes/2015,moraes/2014,ram/2013,reddy/2013,rao/2015,reddy/2012,samanta/2013})), none of them has considered braneworld scenarios so far. Therefore, it seems reasonable and promising to consider $f(R,T)$ gravity in the scope of braneworld scenarios, more specifically, the RSII model, which is the aim of the present work.
In fact, $f(R,T)$ gravity has already been considered in braneworld models (\cite{bazeia/2015,cm/2015}). However, such applications were made in thick (non-RS) brane models and no cosmological scenarios have been derived.
Note that the $f(R,T)$ theories predict a coupling between matter, through the dependence on $T$, and geometry, through the dependence on $R$. The $T$-dependence could be interpreted as inducted from the existence of quantum effects (\cite{harko/2011}). In fact, this dependence links with known illustrious proposals such as geometrical curvature inducing matter and geometrical origin of matter content in the universe (\cite{shabani/2013,farhoudi/2005}). Meanwhile, the RSII bulk contains only gravity (geometry), and the brane (matter) arises as a geometrical manifestation of such an empty 5D space-time. The match between these features justify and make promising the consideration of $f(R,T)$ theories in braneworlds.
As it will be shown below, the braneworld approach raises some new quantities, not found in standard cosmology, like the 5D bulk cosmological constant $\Lambda^{(5)}$ and the brane tension $\sigma$. In (\cite{mm/2014}) it was presented a pioneer form for constraining the values that the brane tension cosmological parameter can assume, from the study of the sign of gravitational waves emitted by a binary system of neutron stars.
It is worth mentioning that recently, in the literature, it has been common to see efforts focused on probing the existence of a braneworld extra dimension. Since the photons are confined to the brane, the extra dimension cannot be observed and its probing is made indirectly, as one can check in (\cite{simonetti/2011,yagi/2011,johannsen/2009,farajollah/2013,rahimov/2011}), for instance.
\section{The $f(R,T)$ gravity}\label{sec:frt}
Recently proposed by T. Harko et al., the $f(R,T)$ gravity considers the gravitational part of the action as being dependent on a function of $R$, the Ricci scalar, and $T$, the trace of the energy-momentum tensor $T_{\mu\nu}$ (\cite{harko/2011}). The dependence on $T$ is justified by the consideration of quantum effects. In this way, the gravitational part of the action reads
\begin{equation}\label{frt1}
S_G=\frac{1}{16\pi}\int d^4x\sqrt{-g}f(R,T),
\end{equation}
with $f(R,T)$ being the function of $R$ and $T$, $g$ the determinant of the metric and we will work with units such that $c=G=1$.
By varying Eq.(\ref{frt1}) with respect to the metric, one obtains the following field equations (FEs):
\begin{align}\label{frt2}
&f_R(R,T)R_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}+(g_{\mu\nu}\Box-\nabla_\mu\nabla_\nu)\nonumber \\
&f_R(R,T)=8\pi T_{\mu\nu}-f_T(R,T)T_{\mu\nu}-f_T(R,T)\Theta_{\mu\nu}.
\end{align}
In (\ref{frt2}), as usually, $R_{\mu\nu}$ stands for the Ricci tensor, $T_{\mu\nu}=g_{\mu\nu}\mathcal{L}_m-2\partial\mathcal{L}_m/\partial g^{\mu\nu}$ is the energy-momentum tensor, with $\mathcal{L}_m$ being the matter lagrangian, $\nabla_\mu$ is the covariant derivative with respect to the symmetric connection associated to $g_{\mu\nu}$ with $\mu,\nu$ running from $0$ to $3$, as originally proposed by the authors, $f_R(R,T)\equiv\partial f(R,T)/\partial R$, $f_T(R,T)\equiv\partial f(R,T)/\partial T$, $\Box\equiv\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu)/\sqrt{-g}$ and $\Theta_{\mu\nu}\equiv g^{\alpha\beta}\delta T_{\alpha\beta}/\delta g^{\mu\nu}$.
It can be seen, from several cosmological models obtained via $f(R,T)$ gravity, such as (\cite{rao/2015,baffou/2015,shabani/2014,moraes/2015b}), that the extra terms\footnote{``Extra terms" when compared to standard gravity field equations.} in the FEs (\ref{frt2}) may be the responsible for the present cosmic acceleration the universe is passing through (\cite{riess/1998,perlmutter/1999}). Such a cosmological feature in standard cosmology is justified by the existence of the cosmological constant $\Lambda$, which yields a sort of ``anti"-gravitational effect on the Einstein's FEs. In this way, $f(R,T)$ gravity is able to predict the cosmic acceleration without the necessity of invoking the cosmological constant. Consequently, such a scenario is free of the cosmological constant problem (\cite{weinberg/1989,peebles/2003,luongo/2012}).
\section{The field equations of the $f(R,T)$ braneworld model}\label{sec:fe}
The total action in RSII model consists of the bulk and brane actions, as (\cite{brax/2004})
\begin{equation}\label{rs1}
S_{bulk}=-\int d^{5}x\sqrt{-g^{(5)}}\left[\frac{R^{(5)}}{16\pi}+\Lambda^{(5)}\right],
\end{equation}
\begin{equation}\label{rs2}
S_{brane}=\int d^{4}x\sqrt{-g}(-\sigma).
\end{equation}
In the equations above, $g^{(5)}$ is the determinant of the 5D metric, $R^{(5)}$ is the 5D Ricci scalar, $\Lambda^{(5)}$ is the bulk cosmological constant, $g$ the determinant of the 4D metric and $\sigma$ is the brane tension.
Let me assume, as in (\cite{brax/2004}), that the RSII bulk metric reads
\begin{equation}\label{m1}
ds^{2}=a^{2}(t,y)b^{2}(t,y)(dt^{2}-dy^{2})-a^{2}(t,y)\eta_{ij}dx^{i}dx^{j}.
\end{equation}
In (\ref{m1}), $a$ and $b$ are the scale factors and $\eta_{ij}$ is the Minkowski metric with $i,j=1,2,3$, so that (\ref{m1}) is consistent with a homogeneous and isotropic brane located at $y=0$, with $y$ being the extra space-like dimension.
The non-null components of the Einstein tensor in the bulk for (\ref{m1}) are
\begin{equation}\label{m2}
G_0^{0}=3\left[2\left(\frac{\dot{a}}{a}\right)^{2}+\frac{\dot{a}}{a}\frac{\dot{b}}{b}-\frac{a''}{a}+\frac{a'}{a}\frac{b'}{b}+\tilde{\sigma} b^{2}\right],
\end{equation}
\begin{equation}\label{m3}
G_1^{1}=3\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}-\left(\frac{\dot{b}}{b}\right)^{2}-3\frac{a''}{a}-\frac{b''}{b}+\left(\frac{b'}{b}\right)^{2}+\tilde{\sigma} b^{2},
\end{equation}
\begin{equation}\label{m4}
G_2^{2}=G_3^{3}=G_1^{1},
\end{equation}
\begin{equation}\label{m5}
G_0^{4}=3\left(-\frac{\dot{a}'}{a}+2\frac{\dot{a}a'}{a^{2}}+\frac{\dot{a}}{a}\frac{b'}{b}+\frac{a'}{a}\frac{\dot{b}}{b}\right),
\end{equation}
\begin{equation}\label{m6}
G_4^{4}=3\left[\frac{\ddot{a}}{a}-\frac{\dot{a}}{a}\frac{\dot{b}}{b}-2\left(\frac{a'}{a}\right)^{2}-\frac{a'}{a}\frac{b'}{b}+\tilde{\sigma} b^{2}\right],
\end{equation}
with dots and primes standing for derivations with respect to time and extra dimension, respectively, and $\tilde{\sigma}\equiv 8\pi\sigma/\sqrt{6}$.
Naturally, those non-null components of the Einstein tensor must be related to the matter-energy content of the universe. In order to construct such relations, let me develop Eq.(\ref{frt2}) in $5$D, by assuming $f(R,T)=R+2\lambda T$, with $\lambda$ a constant. Such an $f(R,T)$ functional form has been widely explored in cosmology, as it can be seen, for instance, in (\cite{harko/2011,moraes/2015,moraes/2015b,moraes/2014,kumar/2015,mahanta/2014}). In 5D, one has, from (\ref{frt2}):
\begin{equation}\label{frtx}
G_{AB}=8\pi T_{AB}+\lambda Tg_{AB}+2\lambda(T_{AB}+Pg_{AB}),
\end{equation}
for which $A,B$ run from $0$ to $4$ and $P$ is the pressure of the universe.
It should be stressed that for the energy-momentum tensor of a perfect fluid on the rhs of Eq.(\ref{frtx}), one must consider the contributions from both the bulk and the brane. For instance, in the above equation, $P=p_{bulk}+p$, with $p$ being the brane pressure. In the same way, below, $\rho$ will stand for the brane density. Therefore, with the help of Eqs.(\ref{m2})-(\ref{m6}), the development of (\ref{frtx}) yields
\begin{align}\label{m7}
&3\left[2\left(\frac{\dot{a}}{a}\right)^{2}+\frac{\dot{a}}{a}\frac{\dot{b}}{b}-\frac{a''}{a}+\frac{a'}{a}\frac{b'}{b}+ \tilde{\sigma} b^{2}\right]=4(2\pi+\lambda)\Lambda^{(5)}\nonumber \\
&+(8\pi+3\lambda)\rho-\lambda p,
\end{align}
\begin{align}\label{m8}
&3\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}-\left(\frac{\dot{b}}{b}\right)^{2}-3\frac{a''}{a}-\frac{b''}{b}+\left(\frac{b'}{b}\right)^{2}+\tilde{\sigma} b^{2}=8\pi\Lambda^{(5)}\nonumber\\
&+\lambda\rho-(8\pi+\lambda)p,
\end{align}
along with the following constraint relations
\begin{equation}\label{m9}
\dot{a}'=2\dot{a}a'+\dot{a}+a'\frac{b'}{b},
\end{equation}
\begin{equation}\label{m10}
\sigma=\frac{\sqrt{6}}{8\pi b^{2}}\left[-\frac{\ddot{a}}{a}+\frac{\dot{a}}{a}\frac{\dot{b}}{b}+2\left(\frac{a'}{a}\right)^{2}+\frac{a'}{a}\frac{b'}{b}\right].
\end{equation}
It should be noted that the equations above stand for the brane location. Moreover, I have used the relation $\rho_{bulk}=-p_{bulk}=\Lambda^{(5)}$, which is assumed in RS models (\cite{brax/2004,binetruy/2000}), and both $T_4^{0}$ and $T_4^{4}$ are null, the latter because, as mentioned in Section \ref{sec:int}, the extra dimension contains only gravity.
\section{A matter-dominated universe in $f(R,T)$ braneworld scenario}\label{sec:mdu}
When deriving the cosmological solutions for the $f(R,T)$ brane model, I will assume the universe is dominated by matter, with equation of state (EoS) $p=0$ to be substituted in (\ref{m7})-(\ref{m8}). Such a consideration along with Eq.(\ref{m10}) make us able to write
\begin{equation}\label{m14}
\frac{5}{2}\left(\frac{\ddot{a}}{a}-\frac{a''}{a}\right)-\left(\frac{\dot{a}}{a}\right)^{2}+\left(\frac{a'}{a}\right)^{2}=-\tilde{\Lambda}^{(5)},
\end{equation}
with $\tilde{\Lambda}^{(5)}\equiv(-44\pi/9)\Lambda^{(5)}$ and $\lambda=8\pi/3$. In (\ref{m14}), as in (\cite{binetruy/2000}), the fifth dimension was considered to be static.
In order to solve Eq.(\ref{m14}) we will use the method of separation of variables. By taking $a(t,y)=\theta(t)\varepsilon(y)$, with $\theta(t)$ and $\varepsilon(y)$ being functions of $t$ and $y$ only, respectively, we obtain from (\ref{m14}):
\begin{equation}\label{m15}
\frac{5}{2}\frac{\ddot{\theta}}{\theta}-\left(\frac{\dot{\theta}}{\theta}\right)^{2}=0,
\end{equation}
\begin{equation}\label{m16}
\frac{5}{2}\frac{\varepsilon''}{\varepsilon}-\left(\frac{\varepsilon'}{\varepsilon}\right)^{2}=\tilde{\Lambda}^{(5)},
\end{equation}
for which we have taken the constant of separation to be $-\tilde{\Lambda}^{(5)}$.
The solution of Eq.(\ref{m15}) is
\begin{equation}\label{m17}
\theta (t)=C_{1}(3t+{C}_{2})^{5/3},
\end{equation}
with $C_1$ and $C_2$ being constants.
On the other hand, the solution of (\ref{m16}) will depend on the sign of the constant $\tilde{\Lambda}^{(5)}$. By choosing positive values for $\tilde{\Lambda}^{(5)}$, the solutions will be exponentials of $y$, while when $\tilde{\Lambda}^{(5)}$ is negative, an oscillatory dependence for $y$ in the scale factor is obtained as $\varepsilon\sim\cos y$. An oscillatory dependence for such a scale factor is non-physical, so we shall discard the case $\tilde{\Lambda}^{(5)}<0$.
If we choose to work with the gauge $\tilde{\Lambda}^{(5)}=1$, we have
\begin{equation}\label{m18}
\varepsilon (y)=C_3(C_4e^{\alpha y}+1)^{5}e^{-\beta y},
\end{equation}
with $C_3$ and $C_4$ being constants, $\alpha =2\sqrt{6}/5$ and $\beta =\sqrt{6}/3$.
Now, from Eqs.(\ref{m17})-(\ref{m18}), we can write the scalar factor $a(t,y)$ as
\begin{equation}
a(t,y)=C_0(3t+C_2)^{5/3}(C_4e^{\alpha y}+1)^{5}e^{-\beta y}, \label{ad3}
\end{equation}
with $C_0\equiv C_1C_3$. Here, it is important to remark that the solutions of Eqs.(\ref{m15}) and (\ref{m16}), given by Eqs.(\ref{m17}) and (\ref{m18}), respectively, contain four arbitrary integration constants, which are determined by initial conditions. By taking $a(0,y)=0$ yields $C_0\neq 0$ and $C_2=0$. Moreover, as we are interested in determining the Hubble
factor $H=\dot{a}/a$, it is not necessary to find the values of $C_0$ and
$C_4$, since such a ratio will remove the dependence on these constants.
From Eq.(\ref{ad3}), one is able to plot the Hubble parameter $H=\dot{a}/a$ of the $f(R,T)$ braneworld model. Such a cosmological parameter is depicted below.
\begin{figure}[h!]
\vspace{1cm}
\includegraphics[{height=05cm,angle=00}]{hubble2}
\vspace{0.3cm}
\caption{Plot of the Hubble parameter.}
\label{FIG1}
\end{figure}
Furthermore, in possession of Eq.(\ref{ad3}), one is also able to calculate the deceleration parameter $q=-a\ddot{a}/\dot{a}^{2}$, so that the universe expansion is accelerating when $q<0$ and decelerating when $q>0$. From Eq.(\ref{ad3}), one obtains $q=-0.4$, in accordance with the acceleration of the universe expansion.
\section{Conclusions}\label{sec:dis}
In this work, we have presented a novel cosmological scenario, which is derived from the consideration of the $f(R,T)$ theory of gravity in the RSII braneworld. Although plenty of extradimensional cosmological models have been derived in $f(R,T)$ gravity (check, for instance, (\cite{moraes/2015,moraes/2014,ram/2013})), those were not obtained from the braneworld scenario consideration. Rather, they were obtained from Kaluza-Klein models (\cite{overduin/1997,moraes/2016}). Although the unification of $f(R,T)$ gravity with braneworld scenarios was made (\cite{bazeia/2015,cm/2015}), in such works, the braneworld set up was not RS like, and last, but not least, no cosmological solutions were obtained.
In Section \ref{sec:mdu}, the solutions of Eq.(\ref{m14}) favoured a negative bulk cosmological constant $\Lambda^{(5)}$ (remind that $\Lambda^{(5)}>0$ yields a non-physical scale factor). It is well known that the bulk cosmological constant must, indeed, be negative (\cite{brax/2004,randall/1999,randall/1999b,campos/2003,germani/2001}). In fact, a positive bulk cosmological constant would accelerate the ``leaking" of gravity through the extra dimension, making it harder to locate such a force on the brane. While a negative bulk cosmological constant is sometimes imposed to braneworld cosmological models, in the present work, this feature has emerged naturally.
Still in Section \ref{sec:mdu} we have derived the resultant cosmological parameters of the model. Fig.\ref{FIG1} reveals a well-behaved evolution for the Hubble parameter. Firstly note that, from standard cosmology, $H\sim t_H^{-1}$, with $t_H$ being the Hubble time. It can be seen from Fig.\ref{FIG1} that such a feature is conserved in the $f(R,T)$ brane model. Furthermore, $H$ is restricted to positive values, assuring the universe expansion.
The deceleration parameter is defined in such a way that if $q<0$, the universe expansion is accelerating. Type Ia Supernovae (\cite{riess/1998,perlmutter/1999}) and temperature fluctuations on the cosmic microwave background radiation observations (\cite{hinshaw/2013}) confirm our universe is undergoing a phase of accelerated expansion. In Section \ref{sec:mdu}, the result predicted by the model for the present value of the deceleration parameter was $q=-0.4$, in accordance with an accelerated expansion.
Moreover, $q=-0.4$ is in the range of accepted values for $q$ at present as one can see in (\cite{hinshaw/2013,giostri/2012}), in which for the latter, the authors have combined Baryon Acoustic Oscillations observations with Supernovae Ia data to constrain the values of the deceleration parameter. Here, it is worth mentioning that the values of the constants $C_0$ and $C_4$ have no effect on the numerical value of $q$ in the present model. In fact, there was no necessity of assuming any values for them.
A value for $q$ which is also in accordance with (\cite{giostri/2012}) was already found by considering $f(R,T)$ gravity in another extradimensional model, the Kaluza-Klein gravity (\cite{overduin/1997}). In that approach, in order to obtain a negative accepted value for $q$ it was necessary to assume the EoS $p=-\rho$ in the model FEs (\cite{moraes/2014}). Such an EoS is in agreement with recent Wilkinson Mapping Anisotropy Probe observations, which predict $\omega=-1.073^{+0.090}_{-0.089}$ at present for a cosmological constant dominated universe (\cite{hinshaw/2013}). On the other hand, in the present $f(R,T)$ braneworld model, it was not necessary to assume the EoS above. Recall that in Section \ref{sec:mdu}, I have assumed $p=0$. Such an EoS stands for a matter dominated universe, i.e., there was no need of assuming any kind of exotic fluid for the universe composition and nevertheless, an accelerated expansion was obtained.
Furthermore, it is worth stressing that some extradimensional $f(R,T)$ models found in the literature could not be able to generate a negative deceleration parameter, predicting, this way, a decelerating expansion (\cite{reddy/2012,reddy/2013}). On the other hand, from a generic functional form for $f(R,T)$ and without the necessity of invoking an exotic EoS for the universe, the present model was able to predict the cosmic acceleration.
\acknowledgments
PHRSM would like to thank S\~ao Paulo Research Foundation (FAPESP), grant 2015/08476-0, for financial support. PHRSM is also thankful to the anonymous referee, for his/her comments, which made the mathematics of the paper more robust, yielding a more realistic cosmological model. RACC thanks to UFABC and CAPES for financial support
\bibliographystyle{spr-mp-nameyear-cnd}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,730 |
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package org.apache.fop.render.pdf;
import java.awt.geom.Rectangle2D;
import java.io.File;
import java.io.IOException;
import java.util.HashMap;
import java.util.List;
import org.junit.Test;
import org.apache.pdfbox.pdmodel.PDDocument;
import org.apache.pdfbox.pdmodel.PDPage;
import org.apache.fop.pdf.PDFArray;
import org.apache.fop.pdf.PDFDictionary;
import org.apache.fop.pdf.PDFDocument;
import org.apache.fop.pdf.PDFName;
import org.apache.fop.pdf.PDFNumber;
import org.apache.fop.pdf.PDFObject;
import org.apache.fop.pdf.PDFPage;
import org.apache.fop.pdf.PDFResources;
import org.apache.fop.pdf.PDFStructElem;
import org.apache.fop.render.pdf.pdfbox.PDFBoxAdapter;
import org.apache.fop.render.pdf.pdfbox.TaggedPDFConductor;
import junit.framework.Assert;
public class TaggedPDFConductorTestCase {
private static final String LINK = "/linkTagged.pdf";
private static final String HELLO = "/helloWorld.pdf";
private static final String TABLE = "/emptyRowTable.pdf";
private static final String OTF = "/otf.pdf";
private static final String IMAGE = "/hello2.pdf";
private static final String NOPARENTTREE = "/NoParentTree.pdf";
private PDFPage pdfPage;
private PDFDocument pdfDoc;
@Test
public void testHandleLogicalStructure() throws IOException {
PDFStructElem elem = new PDFStructElem();
runConductor(LINK, elem);
checkStructure(elem, 0);
elem = new PDFStructElem();
runConductor(HELLO, elem);
PDFNumber mcid = (PDFNumber)elem.getKids().get(0);
int test = mcid.getNumber().intValue();
Assert.assertEquals(test, 0);
elem = new PDFStructElem();
runConductor(TABLE, elem);
Assert.assertEquals(print(elem), "/Div/Part/Sect/Table/TBody/TR/TD/P/TD/P/TR/TD/TD");
elem = new PDFStructElem();
elem.put("Alt", "alt-text");
runConductor(OTF, elem);
Assert.assertEquals(print(elem), "/Div/Part/Art/P/Span");
Assert.assertNull(elem.get("Alt"));
}
private String print(PDFStructElem x) throws IOException {
StringBuilder sb = new StringBuilder(x.get("S").toString());
if (x.getKids() != null) {
for (PDFObject k : x.getKids()) {
if (k instanceof PDFStructElem) {
sb.append(print((PDFStructElem) k));
}
}
return sb.toString();
}
return "";
}
private void runConductor(String pdf, PDFStructElem elem) throws IOException {
setUp();
PDDocument doc = PDDocument.load(new File(getClass().getResource(pdf).getFile()));
PDPage srcPage = doc.getPage(0);
elem.setObjectNumber(2);
PDFBoxAdapter adapter = new PDFBoxAdapter(
pdfPage, new HashMap(), new HashMap<Integer, PDFArray>());
PDFLogicalStructureHandler handler = setUpPDFLogicalStructureHandler();
new TaggedPDFConductor(elem, handler, srcPage, adapter).handleLogicalStructure(doc);
}
private void setUp() {
Rectangle2D r = new Rectangle2D.Double();
pdfPage = new PDFPage(new PDFResources(pdfDoc), 0, r, r, r, r);
pdfDoc = new PDFDocument(" ");
pdfDoc.makeStructTreeRoot(null);
pdfPage.setObjectNumber(1);
pdfPage.setDocument(pdfDoc);
}
private PDFLogicalStructureHandler setUpPDFLogicalStructureHandler() {
PDFLogicalStructureHandler handler = new PDFLogicalStructureHandler(pdfDoc);
handler.getParentTree().setDocument(pdfDoc);
handler.startPage(pdfPage);
return handler;
}
private void checkStructure(PDFStructElem elem, int index) {
String [] types = {"Part", "Sect", "P"};
List<PDFObject> list = elem.getKids();
if (index != 3) {
PDFStructElem kid = (PDFStructElem)list.get(0);
String test = ((PDFName)kid.get("S")).getName();
String expected = types[index];
Assert.assertEquals(test, expected);
index++;
checkStructure(kid, index);
} else {
PDFDictionary firstKid = (PDFDictionary) list.get(0);
int test = ((PDFNumber)firstKid.get("MCID")).getNumber().intValue();
int expected = 0;
Assert.assertEquals(test, expected);
PDFDictionary firstKidSibling = (PDFDictionary) list.get(2);
test = ((PDFNumber)firstKidSibling.get("MCID")).getNumber().intValue();
expected = 2;
Assert.assertEquals(test, expected);
PDFStructElem second = (PDFStructElem)list.get(1);
List secondKids = second.getKids();
PDFStructElem secKid = (PDFStructElem) secondKids.get(0);
List secondKidKids = secKid.getKids();
PDFDictionary leafElem = (PDFDictionary)secondKidKids.get(0);
test = ((PDFNumber)leafElem.get("MCID")).getNumber().intValue();
expected = 1;
Assert.assertEquals(test, expected);
}
}
@Test
public void testTaggedImagePDF() throws IOException {
PDFStructElem elem = new PDFStructElem();
runConductor(IMAGE, elem);
Assert.assertEquals(print(elem), "/Div/Part/Sect/P/Image");
}
@Test
public void testCreateDirectDescendants() throws IOException {
PDFStructElem elem = new PDFStructElem();
runConductor(NOPARENTTREE, elem);
Assert.assertEquals(print(elem), "/Div/Document");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,384 |
Scottish Football Monitor | Asking the questions the media won't ask.
Asking the questions the media won't ask.
The server is currently down and we have technicians working on it. It may take some time, so in the interim we can use the old home as a base whilst we get things fixed.
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The global discourse about educational policy and change has narrowed considerably because of a preoccupation with the high performing systems, as defined by large-scale international assessments, and the factors that contribute to their success. Building on Alma Harris and Michelle Jones' book, Leading Futures: Global Perspectives on Educational Leadership, the Leading Futures series is premised on the contention that more contextual and culturally sensitive accounts of educational change are needed in order to consider broader attributions and explanations of educational performance.
The Leading Futures series provides a platform for sharing different views on the process and practice of changing education systems for the better. Its intention is to open up the contemporary debate on school and system performance through critical policy analysis, empirical enquiry and contextualized accounts of system performance.
This post by Alma Harris, Michelle Jones, Jan Heijmans and Job Christians is the first in the Leading Futures series.
The Dutch Way: Is the Netherlands a best kept educational secret?
Alma Harris, Michelle Jones, Jan Heijmans and Job Christians.
Unlike many other education systems, the Netherlands appears to be delivering both educational quality and equity. So why does the Dutch system do so well? To attribute its success to a handful of structural features or to certain strategies is one way to go. However, accurately identifying causal attributions for better system performance is far from straightforward or fool proof. In complex education systems there are often multiple reasons for better outcomes that interact and intersect.
In this post, we argue that the Dutch system provides an example of "principled educational performance," combining a focus on democratic values with an approach to policymaking that relies on both collaboration and autonomy.
The global interest in the high performing education systems shows no signs of slowing down. The interest in borrowing from the best has placed the international spotlight on a select group of education systems and not others. Earlier this year, the OECD published "Supporting Teacher Professionalism," drawing upon the 2013 TALIS survey in order to explore teachers' and principals' perceived professionalism. Thirty-four countries were scored on three measures: teachers' professional knowledge, work autonomy, and access to peer networks. Of all the education systems that scored highest on the index of professionalism, seven were in Europe and the Netherlands placed fourth in this group.
The Dutch education system is not necessarily on the radar of policy makers in search of better performance but a quick look at the Dutch system makes interesting reading. The evidence shows that Dutch students perform very well in international student assessments and as a country, the Netherlands has remained just outside the PISA top ten, for successive rounds. At the primary level, results from both the 2011 TIMSS and PIRLS assessments indicate an exceptionally good performance for Dutch students aged nine to ten. Among all participating countries, in these international assessments, the Netherlands was only outperformed by seven countries in mathematics and science, and by nine countries in reading.
Turning next to the all important PISA scores. In 2012, 15-year-olds in the Netherlands achieved results significantly above the OECD average in the 3 areas tested (mathematics, reading and science). Only two other OECD countries achieved significantly higher performance levels in mathematics. In 2011, the Netherlands had the lowest rate of 15-29 year-olds not in employment, education or training across all OECD countries: 7% compared to an OECD average of 16%. While there are some who argue that above average is not good enough, from different vantage points and using different indicators it would appear that Dutch education system is performing well.
Yet, the Dutch seem to be remarkably quiet about their educational successes and accomplishments. Possibly this is because unlike some of their near European neighbours, they are not among the big hitters in PISA. Yet, they have a track record in educational equity that should be the envy of many countries in Europe and beyond. Take for example the fact that the Netherlands has fewer low performers and more high performers than the OECD average. Significantly fewer Dutch 15-years-olds scored below the PISA performance level 2, which is believed to mark the basic competency which enables active participation in a society. The impact of student socioeconomic background on performance in mathematics was less pronounced in the Netherlands than at the OECD average. The Netherlands also has an above average proportion of resilient students i.e. students who manage to overcome difficult socio-economic circumstances and exceed expectations, when compared to students in other countries.
It is no accident that the Netherlands is one of the OECD's most devolved education systems, with schools enjoying a high degree of autonomy. This particular brand of autonomy however is not to be confused with increased privatization of schooling or the erosion of local control of schooling. Rather, this particular brand of localalized empowerment is based upon the principle of freedom of education where public and private schools are on an equal footing and all schools receive public funding, provided that they meet the requirements for schools in their sector. In the Netherlands, all teachers receive high quality teacher training at bachelors and masters level plus there is a great emphasis on teacher autonomy and professionalism. The Education Cooperative, which involves over 200,000 teachers, is run by teachers for teachers with the chief aim of safeguarding the quality of the profession.
Before concluding that the Netherlands is some educational utopia where schools and teachers are blissfully free from any interference, think again. The central government sets learning objectives and quality standards that apply to both public and private schools. The Inspectorate of Education monitors school quality and compliance with central rules and regulations. Unlike many other education systems however the Dutch system balances support and pressure in a positive way. While there is a framework of standards, with broadly formulated goals, there are also additional resources and teaching support in schools that need it the most. If schools improve, they are rewarded with more autonomy and freedom to innovate, if they are considered high performing they can apply for Excellent School status.
Of particular note is the fact that the Dutch education system is not overly encumbered with regulation, prescription and standardisation. There is no national curriculum in the Netherlands, however certain learning objectives are stipulated by the Ministry and are expected to be met at the end of primary and lower secondary education. There is testing in the Netherlands and notably, the system stands out internationally for its high-quality standardised assessments. While the issue of testing remains for some Dutch educators somewhat controversial, on balance, the pressure to compete and perform is not as acute as in many other countries. The norms of the Dutch society are collaborative and this threads its way through the very fabric of schooling. Competition hardly plays a role in Dutch educational culture; students are seldom graded against each other or expected to compete against one another.
In terms of equity, the Netherlands is a particularly strong system example. It is the only country participating in PIRLS where all students achieved, at least, the low international benchmark of performance in reading. In addition, 99% of the Dutch students achieved at least the low international benchmark in mathematics and science in TIMSS. Young people in the Netherlands, up to age of 18, must attend school until they attain a basic qualification and there is a strong policy on truancy and absenteeism. The Ministry has signed performance agreements on student dropout with municipalities and schools in 39 regions, which ensures that the most vulnerable young people are supported. In 2006, the government introduced a successful program (Aanval op de uitval) with a regional approach to promote school success and to avoid early school drop outs. A recent OECD report shows that in terms of low-performing students, the Netherlands is far below the OECD average. In the Netherlands, students from low socio-economic backgrounds are 1.72 times more likely to be low performers than their peers with high socio-economic status which is below the OECD average (2.37 times). A higher proportion of Dutch disadvantaged students attend schools with students from better-off backgrounds than the OECD average.
In summary, the Netherlands demonstates a strong comitment to collective and equitable development. As Professor Wilma Vollebergh, University of Utrecht and Netherlands Institute for Social Research reports, it has a social culture and Dutch educational policy-making reflects power-sharing and consenses in decision-making. Such strong cultural norms and values are at the heart of educational practice and largely explain the performance of its education system. The national belief in fairness, equity and justice not only drives the education system but also, at a practical level, translates into a collective effort to ensure success for every child in every setting. A recent study of 200,000 students from 42 countries concluded that Dutch students are happy and have high levels of well-being.
What can we take away from the Dutch approach?
So what can we take away from the Dutch education system? Essentially, there are three things. First, that the Netherlands does not rely on school competition or market forces to secure better educational performance. Conversely, it relies on strong collaboration between teachers and schools to raise achievement and attainment. Second, it does not exclude students from its education system who are disadvantaged, marginalised or are refugees from another country. Instead, it makes every effort to ensure that young people, from all backgrounds, do not leave school early and that they enter the workforce qualified to participate.Third, the Dutch system shows that it is perfectly possible to combine educational equity and quality. While some may argue that there is more work to be done, compared to many other countries the Dutch education system is undoubtedly moving in the right direction.
For those interested in navigating the slopes of quick-fix, high performance, the Netherlands is categorically off-piste. The Dutch way is epitomized by a long history and a proud tradition of building civic society around democratic values that continue to define both an education system and a country. In years to come, when the high-octane remedies for better educational performance have been over-sold to the point where they have lost their lustre and attraction to policy makers, Dutch educators will still be striving, in their quiet but determined way, for educational excellence through equity. With hindsight, it might indeed be the case, that one of our most principled educational performers was there all along.
Dr. Alma Harris is Professor of Educational Leadership and Director of the Institute of Educational Leadership at the University of Malaya.
Dr. J. Heijmans is Chair of the Executive Board KPZ (teacher training Center Zwolle) in the Netherlands.
Job Christians is a former teacher and founder/director of Onderwijs Maak Je Samen (organization for professional development) in the Netherlands.
The Department of Education is providing scholarships to primary and secondary school teachers to undertake an undergraduate or graduate degree. Teachers can apply for a scholarship of €7,000/year to improve their level of education. The Ministry of Education believes this will also improve the quality of teaching in the Netherlands.
One percent of Dutch primary schools have a status of "academic schools." These schools have contracted with research departments to establish research activities in their schools which are cooperatively implemented by primary school teachers and researchers. The results of this research are expected to directly benefit the quality of teaching and learning in these schools, as the schools enrolled in the program share data with one another. The Department of Education sponsors these activities as a means to improve the quality of elementary education.
Segregation of students with different ethnic backgrounds in schools (leading to what is called "white" and "black" schools) is a problem in a number of Dutch cities; segregation is hard to tackle as parents are free to choose a school for their children. The Dutch Department of Education has implemented a number of pilot projects to increase cooperation between these schools and to have students with different backgrounds work together. | {
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{"url":"https:\/\/crypto.stackexchange.com\/questions\/62673\/if-the-riemann-hypothesis-solved","text":"If the Riemann hypothesis solved\n\n\"Sir Michael Atiyah claims to have demonstrated a simple solution to the Riemann hypothesis\".\n\nIf solved, what results we will have in Cryptography?\n\n\u2022 Nothing, because people have quite often assumed RH to hold anyways. \u2013\u00a0SEJPM Sep 27 '18 at 9:05\n\u2022 This is what in my mind, when I see the new. \u2013\u00a0kelalaka Sep 27 '18 at 9:06","date":"2019-02-18 16:48:56","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\": 0.8887492418289185, \"perplexity\": 999.5003060124366}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247487595.4\/warc\/CC-MAIN-20190218155520-20190218181520-00616.warc.gz\"}"} | null | null |
Q: How can I separate JSON object elements into separate arrays in Python? {'labels': 'Middle East', 'default': 193}
{'labels': 'Coronavirus pandemic', 'default': 116}
{'labels': 'Europe', 'default': 90}
{'labels': 'United States', 'default': 67}
{'labels': 'Health', 'default': 63}
I have JSON objects like above.
I am trying to make below type array in python. plz help
{'labels': ['Middle East', 'Coronavirus pandemic', 'Europe', 'United States', 'Health'], 'default': [193, 116, 90, 67, 63]}
A: You can use list comprehension, assuming you have all those dictionaries inside an array:
dictionaries=[
{'labels': 'Middle East', 'default': 193},
{'labels': 'Coronavirus pandemic', 'default': 116},
{'labels': 'Europe', 'default': 90},
{'labels': 'United States', 'default': 67},
{'labels': 'Health', 'default': 63}]
labels=[d['labels'] for d in dictionaries]
default=[d['default'] for d in dictionaries]
result={'labels' : labels, 'default': default}
print(result)
Output:
{'labels': ['Middle East', 'Coronavirus pandemic', 'Europe', 'United States', 'Health'], 'default': [193, 116, 90, 67, 63]}
A: you can create a function merge that can take multiple dictionaries and then merge those dictionaries
def merge(*dicts):
res = {}
for d in dicts:
for key in d.keys():
if key in res:
res[key].append(d[key])
else:
res[key] = []
res[key].append(d[key])
return res
a = {'labels': 'Middle East', 'default': 193}
b = {'labels': 'Coronavirus pandemic', 'default': 116}
c = {'labels': 'Europe', 'default': 90}
d = {'labels': 'United States', 'default': 67}
e = {'labels': 'Health', 'default': 63}
c = merge(a, b, c, d, e)
print(c)
output:
{'labels': ['Middle East', 'Coronavirus pandemic', 'Europe', 'United States', 'Health'], 'default': [193, 116, 90, 67, 63]}
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Q: Spanned images on Android ListView I'm trying to convert an HTML text with images using Html.fromHtml and set it inside a ListView, but images appears as squares! I have implemented an imageGetter and also I set
StrictMode.ThreadPolicy policy = new StrictMode.ThreadPolicy.Builder().permitAll().build();
StrictMode.setThreadPolicy(policy);
to avoid android.os.NetworkOnMainThreadException
Here's my code:
private class MyTask extends AsyncTask<Void, Void, Void>{
@Override
protected Void doInBackground(Void... arg0) {
try {
URL rssUrl = new URL(link);
SAXParserFactory mySAXParserFactory = SAXParserFactory.newInstance();
SAXParser mySAXParser = mySAXParserFactory.newSAXParser();
XMLReader myXMLReader = mySAXParser.getXMLReader();
RSSHandler myRSSHandler = new RSSHandler();
myXMLReader.setContentHandler(myRSSHandler);
InputSource myInputSource = new InputSource(rssUrl.openStream());
myXMLReader.parse(myInputSource);
myRssFeed = myRSSHandler.getFeed();
} catch (MalformedURLException e) {
e.printStackTrace();
} catch (ParserConfigurationException e) {
e.printStackTrace();
} catch (SAXException e) {
e.printStackTrace();
} catch (IOException e) {
e.printStackTrace();
}
return null;
}
@Override
protected void onPostExecute(Void result) {
if (myRssFeed!=null)
{
List<Map<String, String>> data = new ArrayList<Map<String, String>>();
for (RSSItem item : myRssFeed.getList()) {
Spanned span = Html.fromHtml(item.getDescription().toString(), getImageHTML(), null);
Map<String, String> datum = new HashMap<String, String>(0);
datum.put("title", item.getTitle());
datum.put("desc", span.toString());
data.add(datum);
}
SimpleAdapter adapter2 = new SimpleAdapter(getApplicationContext(), data, R.layout.listview, new String[] {"title", "desc"}, new int[] {R.id.text1,
R.id.text2});
setListAdapter(adapter2);
And here's the ImageGetter:
public ImageGetter getImageHTML(){
ImageGetter ig = new ImageGetter(){
public Drawable getDrawable(String source) {
StrictMode.ThreadPolicy policy = new StrictMode.ThreadPolicy.Builder().permitAll().build(); StrictMode.setThreadPolicy(policy);
try{
Drawable d = Drawable.createFromStream(new URL(source).openStream(), "src name");
d.setBounds(0, 0, d.getIntrinsicWidth(),d.getIntrinsicHeight());
return d;
}catch(IOException e){
Log.v("IOException",e.getMessage());
return null;
}
}
};
return ig;
}
Can someone help me?
A: I'm not sure weather myRssFeed.getList() or item.getDescription() performes any network operation (Just guessing only), Just try adding your for loop logic inside doInBackground method.
List<Map<String, String>> data = new ArrayList<Map<String, String>>();
for (RSSItem item : myRssFeed.getList()) {
Spanned span = Html.fromHtml(item.getDescription().toString(), getImageHTML(), null);
Map<String, String> datum = new HashMap<String, String>(0);
datum.put("title", item.getTitle());
datum.put("desc", span.toString());
data.add(datum);
}
adapter2 = new SimpleAdapter(getApplicationContext(), data, R.layout.listview, new String[] {"title", "desc"}, new int[] {R.id.text1,
R.id.text2});
Declare SimpleAdapter adapter2 as a global variable in MyTask and move above code snippet to doInBackground method
| {
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require 'pp'
require 'optparse'
require 'singleton'
require 'opscode/expander/flattener'
require 'opscode/expander/loggable'
require 'opscode/expander/version'
module Opscode
module Expander
def self.config
@config ||= Configuration::Base.new
end
def self.init_config(argv)
config.apply_defaults
remaining_opts_after_parse = Configuration::CLI.parse_options(argv)
# Need to be able to override the default config file location on the command line
config_file_to_use = Configuration::CLI.config.config_file || config.config_file
config.merge_config(Configuration::Base.from_chef_compat_config(config_file_to_use))
# But for all other config options, the CLI config should win over config file
config.merge_config(Configuration::CLI.config)
remaining_opts_after_parse
end
class ChefCompatibleConfig
attr_reader :config_hash
def initialize
@config_hash = {}
end
def load(file)
file = File.expand_path(file)
instance_eval(IO.read(file), file, 1)
end
def method_missing(method_name, *args, &block)
if args.size == 1
@config_hash[method_name] = args.first
elsif args.empty?
@config_hash[method_name] or super
else
super
end
end
end
module Configuration
class InvalidConfiguration < StandardError
end
class Base
include Loggable
def self.from_chef_compat_config(file)
config = ChefCompatibleConfig.new
config.load(file)
from_hash(config.config_hash)
end
def self.from_hash(config_hash)
config = new
config_hash.each do |setting, value|
setter = "#{setting}=".to_sym
if config.respond_to?(setter)
config.send(setter, value)
end
end
config
end
def self.configurables
@configurables ||= []
end
def self.validations
@validations ||= []
end
def self.defaults
@defaults ||= {}
end
def self.configurable(setting, default=nil, &validation)
attr_accessor(setting)
configurables << setting
defaults[setting] = default
validations << validation if block_given?
setting
end
configurable :config_file, File.expand_path(File.dirname(__FILE__) + '/../../../conf/opscode-expander.rb')
configurable :index do
invalid("You must specify this node's position in the ring as an integer") unless index.kind_of?(Integer)
invalid("The index cannot be larger than the cluster size (node-count)") unless (index <= node_count.to_i)
end
configurable :node_count do
pp :self => self
invalid("You must specify the cluster size as an integer") unless node_count.kind_of?(Integer)
invalid("The cluster size (node-count) cannot be smaller than the index") unless node_count >= index.to_i
end
configurable :ps_tag, ""
configurable :solr_url, "http://localhost:8983"
configurable :amqp_host, '0.0.0.0'
configurable :amqp_port, '5672'
configurable :amqp_user, 'chef'
configurable :amqp_pass, 'testing'
configurable :amqp_vhost, '/chef'
configurable :log_level, :info
# override the setter for log_level to also actually set the level
def log_level=(level)
if level #don't accept nil for an answer
level = level.to_sym
Loggable::LOGGER.level = level
@log_level = log_level
end
level
end
def initialize
reset!
end
def reset!(stdout=nil)
self.class.configurables.each do |setting|
send("#{setting}=".to_sym, nil)
end
@stdout = stdout || STDOUT
end
def apply_defaults
self.class.defaults.each do |setting, value|
self.send("#{setting}=".to_sym, value)
end
end
def merge_config(other)
self.class.configurables.each do |setting|
value = other.send(setting)
self.send("#{setting}=".to_sym, value) if value
end
end
def fail_if_invalid
validate!
rescue InvalidConfiguration => e
@stdout.puts("Invalid configuration: #{e.message}")
exit(1)
end
def invalid(message)
raise InvalidConfiguration, message
end
def validate!
self.class.validations.each do |validation_proc|
instance_eval(&validation_proc)
end
end
def vnode_numbers
vnodes_per_node = VNODES / node_count
lower_bound = (index - 1) * vnodes_per_node
upper_bound = lower_bound + vnodes_per_node
upper_bound += VNODES % vnodes_per_node if index == node_count
(lower_bound...upper_bound).to_a
end
def amqp_config
{:host => amqp_host, :port => amqp_port, :user => amqp_user, :pass => amqp_pass, :vhost => amqp_vhost}
end
end
module CLI
@config = Configuration::Base.new
@option_parser = OptionParser.new do |o|
o.banner = "Usage: opscode-expander [options]"
o.on('-c', '--config CONFIG_FILE', 'a configuration file to use') do |conf|
@config.config_file = File.expand_path(conf)
end
o.on('-i', '--index INDEX', 'the slot this node will occupy in the ring') do |i|
@config.index = i.to_i
end
o.on('-n', '--node-count NUMBER', 'the number of nodes in the ring') do |n|
@config.node_count = n.to_i
end
o.on('-l', '--log-level LOG_LEVEL', 'set the log level') do |l|
@config.log_level = l
end
o.on_tail('-h', '--help', 'show this message') do
puts "opscode-expander #{VERSION}"
puts ''
puts o
exit 1
end
o.on_tail('-v', '--version', 'show the version and exit') do
puts "opscode-expander #{VERSION}"
exit 0
end
end
def self.parse_options(argv)
@option_parser.parse!(argv.dup)
end
def self.config
@config
end
end
end
end
end
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Homepage / Community / Hill Therapy – Who Says you Have to Sit to Meditate
Hill Therapy – Who Says you Have to Sit to Meditate
By: Ashley Hunter Arnold
Yesterday morning I was sitting with Austin and a few friends in the bus as rain pelted the metal roof. We were talking about the breakfast I was making, music we listened to the night before and the transcending meditative state that movement–particularly dance–can provide under certain circumstances, after a certain length of time, under a certain mindset. We called it nutrition. Full-body-soul-connecting nutrition. A different type of nutrition than what comes up almost daily in my conversations since I am studying nutritional therapy. But an important nutrition just the same.
Having been a dancer most of my life, I'm intimately acquainted with this type of nutrition. But dancing, I have discovered, isn't the only place to soak it in. Running, it turns out, is an equally powerful source … if you allow it to be one. It's not as easy as it sounds. It often takes literally stepping back from everything that you think running is—removing all labels of what constitutes a run, what it means to be a runner … all the way down to even removing the act of running as a form of exercise. And simply letting running be movement. In the same way that dance is movement. In fact, let it be dance. Or, more simple still, let it be.
I know there are all kinds of "running as meditation" practices out there. I've never read much about any of them. Call me stubborn or anti-establishment if you want (it's my Aquarius nature), but the truth is, I thrive on discovering things organically in a trial-by-fire fashion that sometimes results in chaos. But thankfully, many times, in bliss. As was the case when I discovered what I call Hill Therapy.
OK, I need to back up a minute because you may be wondering why I would even think to relate meditation to hills, those painful calf-crushing, earth-coming-at-your-face monsters that leave your lungs begging for more air and your legs screaming with lactic acid. But see, I love hills. I love them so much that I would prefer to run uphill continuously even if that meant I'd receive no downhill running reward. I know a lot of people accept hills—even grow to love them—partly because of the view (if there even is one at the top) and also (perhaps more so) for the downhill that follows.
But me? I couldn't give a cat's lick about the downhill. It's cool, I guess. But what I really love about hills is actually that very earth-coming-at-your-face feeling. It's the feeling in your legs, it's the burning in your lungs. And then, it's the stillness at the top. It's the view (if there is one), it's the melding of self and outside of self that makes it all worth it. It is, I suppose, the lesson.
OK, so Hill Therapy. …
There's this hill next to the property where we parked our tiny-house bus just outside of Asheville, North Carolina. It's only .57 miles. But it is between probably a 12- and 15-percent grade the whole way. The first time I walked up it, I was tired. It's paved and treelined and relatively free of traffic save the occasional pick up truck and stray dog. At the top, for probably the last .05 miles, it turns to gravel under towering trees alongside a nondescript waterfall, sort of tucked away in a fold on the hill.
The first time I ran it, I was panting and ecstatic. I stood gaping at the water pouring over worn down rocks. I took a few minutes to "soak it in" before turning around and, reluctantly, trotting back the way I came.
The next day, I did it again. I did it twice actually. A little over two miles. At the top on the second lap I stopped a little longer. I held my right hand over my heart, my left hand over my belly and closed my eyes. I took big deep-belly breaths. I listened to the water. To the breeze rustling the leaves. To the far-off sound of dogs barking. I opened my arms and my eyes and made large scooping motions. Scooping toward my center. I scooped in the trees, the water, the folded hillside, the gravel, even the barking dogs. I scooped it all in and breathed.
The third day I wanted to run it five times. I was getting excited. I was letting the runner in me that loves hills and loves to push more and more and more and farther and farther and farther take over. But I stopped myself. Only two, I said. In fact, I said it out loud. This hill wasn't about a run. I didn't know what it was at the time or why I felt so compelled to go up it over and over. But I just let it be. I let myself, as they say, sit with it.
The fourth day I was having a rough morning. I'd received some not-so-great news the night before, I'd had an argument, I hadn't slept well and I felt unsettled. I decided not to do the hill. I reasoned that it was a waste of time, that it was boring and I didn't like it anyway. I was leaving to meet a friend for lunch in 45 minutes anyway.
But still, I was so worked up that I felt off-center. I felt lost in the air space above my head. I'd left intuition and reasoning behind. I was in a spiral of self-destructing chaos. I went outside and stood there facing the sun for a minute (something I love to do almost every morning) …. maybe hoping it would just give me the answers. … I went back inside and put on my running clothes.
Up the hill I went.
I pushed my anger into the hill. I pushed my frustration into each step. I let it all come bubbling up over and over and I threw it onto the pavement as I pulled up and up and up along an imaginary rope. Higher and higher. The more upset I felt, the more the earth pushed back. The more in-my-face the steep felt, the more I had to soften and relax if I wanted to get up the hill at all. … The more the hill forced me to let go.
When I got to the top, I walked. Slow and deliberate steps as I worked to calm my breathing. I was back on the gravel patch beneath green canopy and alongside the flowing water. The sky was visible, but only in patches between the leaves. I felt completely blanketed within the Earth. Grounded. I scooped my arms again. Two, five, 10 times. I pulled it into my center and breathed the Earth into me. Nothing I felt 10 minutes ago existed. I felt simultaneously completely free and also totally connected.
After several minutes I went back down. I walked some but mostly just let gravity pull me. With each downward pull I focussed on returning to the earth, on grounding, on sinking and rooting into it. With each step, I focussed on stillness. With each step I imagined myself melding further and further into the landscape around me until I felt like I wasn't even me running down the hill at all.
This time, when I reached the bottom, I turned to the hill and with my hands in prayer, I said thank you.
Ashley Hunter Arnold
Ashley Hunter Arnold is a writer and filmmaker currently living and running in Asheville, NC. Her favorite foods are kale and cake with lots of icing. You can connect with her on Instagram @ashleyharnold.
Exertional Heat Stroke and Lessons Learned
Chrysta Archer January 19, 2022
Bring on Your Best: 2022
Lizi Bolanos-Nauth January 18, 2022
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,910 |
Q: Bootstrap 4 nav toggle button not working? I am trying to build the responsive navbar using the bootstrap 4, where the navbar has display the toggle button if the size of screen reduces by certain pixels width, I am able to display the toggle button but i am not able to display the link items when i am clicking on toggle button i don't know why is this happening also i have tried the jquery script first followed by the bootstrap link, popper script and at last the bootstrap script but still it is not displaying nav items after clicking on toggle button can anyone help me with this.
here is the code i have added in header
<head>
<meta charset="utf-8"/>
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<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css">
<title>Nav Bar</title>
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here is the code i have added in my body
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A: You need to check that you are reading the documentation for the version that you are using. If you look here https://getbootstrap.com/docs/4.5/components/navbar/ (note the version in the URL) you will see the data attribute names did not include the 'bs-' portion, that was added in v5.
Change this:
<button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbarNav" aria-controls="navbarNav" aria-expanded="false" aria-label="Toggle navigation">
To:
<button class="navbar-toggler" type="button" data-toggle="collapse" data-target="#navbarNav" aria-controls="navbarNav" aria-expanded="false" aria-label="Toggle navigation">
A: In your <button class="navbar-toggler"> change the toggle attribute from data-bs-toggle to data-toggle.
Do the same with your target attribute and remove bs part from your data-bs-target
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A:
<!doctype html>
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/bootstrap@4.6.0/dist/css/bootstrap.min.css" integrity="sha384-B0vP5xmATw1+K9KRQjQERJvTumQW0nPEzvF6L/Z6nronJ3oUOFUFpCjEUQouq2+l" crossorigin="anonymous">
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<nav class="navbar navbar-expand-lg navbar-dark bg-dark">
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<a class="navbar-brand" href="#">TarotByPoonam</a>
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<li class="nav-item">
<a class="nav-link active" aria-current="page" href="#">Home</a>
</li>
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| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,689 |
As its wounded, defiant provocation of a title suggests, "Gook" looks head-on at racial animosity, American-style. Set on the first day of the 1992 Los Angeles riots, Justin Chon's drama is uneven but bristling with life, and it offers a new perspective on a calamitous moment, one whose 25th anniversary has been commemorated in recent months with a slew of potent documentaries.
The writer-director's fictional spin mixes street-level grit, melodrama and deadpan humor, placing a young black girl — memorably played by newcomer Simone Baker — at the hopeful, precarious center of a story that unfolds from the point of view of working-class Korean Americans.
"Do the Right Thing" is an obvious inspiration (Chon has also cited the hard-hitting French film "La Haine" and, on the opposite end of the spectrum, he throws in a blatant nod to "Clerks"), and as in Spike Lee's landmark feature, the action revolves around a neighborhood store. In this case it's a shop offering steeply discounted women's footwear, some of it acquired through such back channels as the back of a truck. Chon, an actor whose credits include "Twilight," stars as Eli, who owns and runs the shop with his younger brother, Daniel (stand-up comic David So, affecting in his first film role).
A standalone storefront behind a chain-link fence in Paramount — a few miles from the conflagrations that will erupt in South Los Angeles as the day proceeds — it's a place that Westside shoppers might at first mistake for an abandoned property. But enough customers flock to it that the brothers' unofficial shop assistant, 11-year-old Kamilla (Baker), keeps busy. She's a sunburst of a girl, sporting tomboy cutoffs and a low-key riff on a Billie Holiday gardenia, but her open face sometimes crumples into an old soul's scowl of worry.
Kamilla's sister, Regina (Omono Okojie), and tightly wound brother, Keith (Curtiss Cook Jr.), who are raising her, regularly warn her not to go the store. Their reasons, and the exuberant girl's connection to the place, are gradually revealed but never explicitly spelled out. Chon's judicious handling of backstory, through shards of individual memories but without clunky flashbacks, delivers depth charges of emotion.
At times, though, his assured grasp of the naturalistic milieu can give way to stilted or showy exchanges. As L.A. burns and the story moves toward its climax, it can feel overdetermined, especially regarding Kamilla's purpose in the drama. But the low-budget movie, shot in artful black-and-white by Ante Cheng, pulses with yearning and sorrow and love for its characters. Its brightening touches of underplayed humor strengthen and comment on the main action.
In one sly, matter-of-fact detail, the "push" and "pull" signs on the store's front door are reversed; anyone wishing to open the door should heed each sign as an instruction to do the opposite of what it says. It's a nifty joke given the brothers' conflicted feelings toward the business they inherited. Neither young man's heart is really in it, although as the movie opens, on the day of the verdicts in the Rodney King case, taskmaster Eli hasn't come to terms with his own dissatisfaction. Daniel, on the other hand, softer than his wiry sibling both physically and temperamentally, secretly has his sights set on becoming a professional singer.
Chon makes the economic realities and paradoxes of their situation fully felt. Eli and Daniel are behind on the store's rent and live in far-from-glamorous circumstances, yet as business owners they face the resentment of some of their predominantly black clientele, who view them as exploiters. At the same time they're subject to random beatings by Latino gangbangers. There are also more targeted attacks by Keith, with Cook, who made an impression in the New York coming-of-age indie "Naz & Maalik," stirring up compellingly tangled depths in what might have been a walking cliché.
Within his stripped-down locales, Chon taps into an atmosphere of hair-trigger tensions and free-floating hatred, heightened by the events of that April 29 but hardly new. There are obvious intimations of Latasha Harlins' shooting death in Kamilla's interactions with a belligerent Korean liquor store owner. He's played by a very good Sang Chon, the director's father (who was a child actor in his native South Korea and whose Paramount store was looted during the '92 riots). Through the unpredictable old man, the movie delves into a Korean American generational divide, to both comical and poignant effect.
"Gook" might not achieve everything it aims for, but it's the work of someone reaching high. Viewing a pivotal event through a personal lens, Chon brings the futility of bigotry into vivid, aching focus. As Eli and Kamilla, two parentless Angelenos, watch smoke rise above the skyline, there's no question that they are family. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,618 |
Haus der Geschichte steht für
Haus der Europäischen Geschichte
Haus der Geschichte der Bundesrepublik Deutschland, Bonn
Haus der Geschichte Baden-Württemberg, Stuttgart
Haus der Bayerischen Geschichte
Haus der Geschichte Nordrhein-Westfalen, Düsseldorf?, geplant
Haus der Geschichte des Ruhrgebiets, Bochum
Haus der Geschichte Wittenberg
Haus der Geschichte, Darmstadt, siehe Hessisches Staatsarchiv Darmstadt#Haus der Geschichte
Haus der Geschichte Österreich, eröffnet am 10. November 2018
Haus der Geschichte Niederösterreich, eröffnet am 9. September 2017
Siehe auch:
Haus der Stadtgeschichte
Haus der Essener Geschichte / Stadtarchiv
Gothaer Haus der Versicherungsgeschichte | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,716 |
\section{Introduction}
Many complex problems require planning many steps ahead. People often struggle with such problems because their capacity for planning is limited \citep{prystawski2022resource}. To overcome this challenge, individuals, managers, and educators often break complex problems down into a series of subgoals \citep{simon1975functional,catrambone1998subgoal,drucker2012practice}.
In principle, it should be possible to use models of human planning \citep[e.g.,][]{callaway2022rational}, goal-pursuit\citep[e.g.,][]{prystawski2022resource}, and problem-solving \citep[e.g.,]{newell1972human,anderson2013architecture} to predict which subgoals are most beneficial for people. This creates an untapped opportunity to derive practical recommendations for which subgoals managers and individuals should set from cognitive models of bounded rationality \citep{lieder2020resource,lewis2014computational,gershman2015computational}.
Here, we formulate a mathematically precise normative theory of (self-)management by goal-setting that makes it possible to derive optimal subgoals from computational models of bounded rationality \citep{lieder2020resource}. The basic idea is that goal-setting serves to decompose a complex problem into a series of simpler problems that require less planning. To apply this theory, we combined it with a computational model of human goal-pursuit and an optimization algorithm.
This approach allowed us to improve the problem-solving performance of bounded agents and crowd workers in simulations and an online experiment, respectively. These findings suggest that it might be possible to ground prescriptive theories of managerial and personal goal-setting in computational models of bounded rationality.
The outline of this paper is as follows: We first introduce the relevant theoretical background. We then propose our normative theory of goal-setting for (self-)management. The following two sections evaluate our theory in simulations and an online experiment. We close by discussing our results, their implications, limitations, and directions for future work.
\section{Background}
\subsection{Computational models of bounded rationality}
Human cognition is constrained by bounded cognitive resources and having to solve complex problems in a limited amount of time \citep{lieder2020resource}. To deal with this, the brain uses heuristic strategies that can perform reasonably well with limited resources. The framework of resource rationality \citep{lieder2020resource} formalizes this intuition and seeks to understand human behavior as the optimal use of limited cognitive resources and information. Formally, the resource-rational heuristic that people should use in an environment $E$ is
\begin{equation*}
h^\star = \argmax_{h \in H_B} \mathbb{E} [\RR(h, E, B)]
\end{equation*}
where $H_B$ is the set of all heuristics that can be implemented by brain $B$ and the resource rationality $\RR$ of a heuristic $h$ implemented by brain $B$ in an environment $E$ is
\begin{multline*}
\RR(h, E, B) = \mathbb{E}_{P(\textrm{result} | s_0, h, E, B)}[u(\textrm{result})] \\ - \mathbb{E} [\textrm{cost}(t_h, \rho) | h, s_0, B, E]
\end{multline*}
where $u(\textrm{result})$ is the person's subjective utility of the result obtained by using the heuristic $h$ in situation $s_0$ and $\textrm{cost}(t_h, \rho)$ is the total opportunity cost of investing resources $\rho$ used by heuristic $h$ for time $t_h$.
Resource-rational models take into account cognitive limitations and perform as well as possible under those constraints. This framework has been successfully applied to various cognitive processes such as planning \citep{callaway2022rational}, goal pursuit \citep{prystawski2022resource}, and decision-making \citep{bhui2021resource}.
\subsection{Simulated Microworlds}
Simulated Microworlds (SMWs) are used to study problem-solving in naturalistic, dynamic environments \citep{funke1993microworlds}. They consist of various variables that interact according to a dynamical system with discrete time steps. The agent can directly manipulate a subset of the variables while only indirectly influencing the values of other variables. For instance, the ultimate aim of the owner of a bakery is to maximize profits. However, she cannot directly control her profit, nor can she influence other factors like demand and sales. Instead, she has to decide how much to spend on advertising, rent, wages, raw materials, etc. in a way that would lead to maximum profit. This makes SMWs a suitable paradigm to study problem-solving in the real world.
In this study, we adopt the SMW introduced by \citet{mohnert2019testing}, in which participants manage a farm. Their goal is to bring various crops (also referred to as `states') close to specified target values by adjusting how much fertilizer and pesticides (also referred to as `actions') are deployed on the fields (Fig.~\ref{fig:no_subgoal_sc}). In this SMW, participants have complete knowledge of the relationships between all variables, and suboptimality in their behavior can therefore be attributed to their limited cognitive resources. The dynamics of the system are described by the following equation:
\begin{equation*}
\textbf{s}_{t+1} = \textbf{f}(\textbf{s}_t, \textbf{a}_t) = \textbf{A}\textbf{s}_t + \textbf{B}\textbf{a}_t
\end{equation*}
where $\textbf{s}_t \in \mathbb{R}^N$ is the \textit{state} containing values of various crops at time $t$, $\textbf{a}_t \in \mathbb{R}^M$ is the \textit{action} consisting of the amounts of various resources used at time $t$, $\textbf{A} \in \mathbb{R}^{N \times N}$ determines how various crops affect themselves and each other from time $t$ to $t+1$, and $\textbf{B} \in \mathbb{R}^{N \times M}$ determines how resources affect crops.
\subsection{Resource Rational Models of Goal Pursuit}
In simulated micro-worlds, participants often struggle to achieve the task's ultimate goal \citep{prystawski2022resource, funke1993microworlds} because doing so requires planning many steps ahead and taking various factors into account, which can be cognitively demanding. To effectively model how people pursue goals in SMWs, it is important to consider these cognitive limitations. To this end, \citet{prystawski2022resource} developed resource-rational models of goal pursuit, which accommodate limits on people's attention and how many steps they can plan ahead. They found that the model that best explained the problem-solving behavior of the largest proportion of participants in the SMW described above was a hill-climbing model inspired by Newell and Simon's foundational research on human problem-solving \citep{newell1972human}.
\subsubsection{The Hill-Climbing model}
This model assumes that the agent's limited computational resources prevent it from planning more than one step into the future. The agent, therefore, chooses the combination of inputs (actions) that maximally reduces the distance to the goal in the very next round, while also minimizing the cost of those inputs. Any input can be described in terms of the direction of the change in the system's state and the distance between the previous state and the next state. Since the goal of the agent is to minimize its distance from the goal, the agent moves in the direction opposite to the gradient of this distance. Concretely, the action chosen by the agent is:
\begin{equation*}
\textbf{a}_t = - \lambda \cdot \lambda_{opt} \cdot \nabla_{\textbf{a}} || \textbf{f}(\textbf{s}_t, \textbf{a}) - \textbf{g} ||_2
\end{equation*}
where $\lambda_{opt}$ is the optimal step size\footnote{The optimal step size is the step size that minimizes the distance from the goal immediately after a single step in the optimal direction. It is obtained by setting the derivative of the distance from the goal equal to zero while keeping the direction fixed.} in the direction of the negative gradient, $\textbf{g}$ is the goal, $\textbf{f}(\textbf{s}_t, \textbf{a})$ is the state of the agent after taking action $\textbf{a}$ in the state $\textbf{s}_t$. The gradient is evaluated at $\textbf{a} = 0$. $\lambda$ is a free parameter that captures people's tendency to take steps that are systematically smaller or larger than the optimal step size. Stochasticity in people's actions is captured via noise added to the distance and the direction of the agent's actions (see \citet{prystawski2022resource} for details).
\subsection{The resource-rational perspective on goal-setting}
It has been proposed that the function of goals is to reduce the amount of planning that is necessary to reach good decisions \citep{lieder2020resource}. According to a recent refinement of this perspective \citep{correa2020resource,correa2022humans}, an optimal sequence of subgoals should minimize the sum of the costs of the actions the person will take and the mental effort they have to invest into planning to select those actions.
\subsection{Goal-setting in management and self-regulation}
A common approach to management is \textit{management by objectives} \citep{drucker2012practice}. In this approach, the manager's first responsibility is to break down a complex problem into a series of subgoals (\textit{objectives}) that their subordinates can achieve efficiently. The manager assigns their subordinates one subgoal at a time. The employees then work towards the assigned subgoal, and once they accomplish it, the manager assigns them the next subgoal. The purpose of this management practice is to enable teams to achieve challenging long-term goals whose achievement requires considerable amounts of planning and problem-solving. Numerous studies have consistently found that organizational productivity benefits from management by objectives in general \citep{rodgers1991impact} and its goal-setting component in particular \citep[e.g.,][]{mento1987meta}.
Moreover, people can also use goal-setting to manage themselves, improve their own performance, and help themselves achieve their long-term goals \citep{carver2001self,latham1991self,zimmerman2012goal}.
\section{A normative theory of goal-setting for (self-)management}
The finding that people routinely improve the performance of others (or themselves) through goal-setting raises several interesting, interrelated questions about which subgoals are most effective, what makes them so effective, and how supervisors managing teams (and individuals managing themselves) can generate them. Here, we approach this question from the perspective of rational analysis \citep{anderson2013adaptive} by formalizing the problem people managing others or themselves solve by setting objectives. For the ease of reading and understanding, we describe the theory for the case of managerial goal-setting. But, in principle, the theory also applies when the person being managed is the manager herself.
Based on the research on goal pursuit summarized in the previous section \citep{prystawski2022resource}, a key problem that managers have to solve is that the path to achieving the organization's ultimate goals may be very long and complex relative to individual employees' capacity and/or propensity for planning. From this perspective, managerial goal-setting serves to reduce the amount of planning that is required for effective goal pursuit \citep[cf. ][]{correa2020resource,correa2022humans}. This, in turn, increases employees' performance on problems that are beyond the cognitive capacity of any single individual. From this perspective, the function of managerial goal-setting is to translate a long-term goal into a sequence of subgoals such that when employees devote their limited cognitive resources to the pursuit of those subgoals they will make more progress towards the long-term goal than they would if they pursued the long-term goal directly.
To formalize this idea, we model employees' decisions using the recently developed model of boundedly rational goal pursuit introduced above \citep{prystawski2022resource}. Concretely, we model the employee as an agent $\alpha$ that interacts with an environment $E$ by taking action $\textbf{a}_t \in \mathbb{R}^M = (a_t^0, a_t^1, \ldots, a_t^M)$ at time $t$ based on the environment's state $\textbf{s}_t \in \mathbb{R}^N = (s_t^0, s_t^1, \ldots, s_t^N)$. A goal $g$ is characterized by its target values $\textbf{s}_g \in \mathbb{R}^N$, threshold $\delta_g \in \mathbb{R}$, and scale $\gamma_g \in \mathbb{R}^{N \times N}$. The scale captures the possibility that it is more important to bring some state variables closer to their target values than others, and the threshold specifies how close the agent needs to be to the target values to successfully reach the goal. The agent is considered to have achieved the goal if
$
\sqrt{(\textbf{s} - \textbf{s}_g)^\intercal \gamma_g (\textbf{s} - \textbf{s}_g)} \leq \delta_g,
$
where $\gamma_g$ is a diagonal matrix with its $i$\textsuperscript{th} value being equal to the scale of the $i^{th}$ state variable. Because values of the scale parameter can be hard to interpret for humans, we converted them to tolerance values for human participants in our experiment. The tolerance $\theta_i$ for the $i^{th}$ state variable is computed as
$\theta_i = \frac{\delta_g}{\sqrt{N \cdot (\gamma_g)_i}} \refstepcounter{equation}(\theequation) \label{eq:scale2tolerance}$,
where $(\gamma_g)_i$ is the $i^{th}$ diagonal element of $\gamma_g$.
Subgoals are defined similarly, but for a subset of state variables. Concretely, a subgoal $\epsilon$ can be defined for a set of state variables $D_{\epsilon} = (i_1, i_2, ..., i_d)$ with target values $\textbf{s}_{\epsilon}$, scale $\gamma_{\epsilon}$, and threshold $\delta_{\epsilon}$. The condition for successfully achieving a subgoal is
$
\sqrt{(\textbf{s}_r - \textbf{s}_{\epsilon})^\intercal \gamma_{\epsilon} (\textbf{s}_r - \textbf{s}_{\epsilon})} \leq \delta_{\epsilon},
$
where $\textbf{s}_r$ is the \textit{reduced state} which is obtained by considering the state variables of $\textbf{s}$ that are included in $D_{\epsilon}$, i.e., $\textbf{s}_r = (s^i | i \in D_{\epsilon})$. Scales for subgoals can be converted to tolerances as done in Eq.~\ref{eq:scale2tolerance} by replacing $N$ with the size of $D_\epsilon$, $\delta_g$ with $\delta_\epsilon$, and $\gamma_g$ with $\gamma_\epsilon$.
The agent starts at $t = 0$ from state $\textbf{s}_0$ and tries to achieve a series of subgoals $\epsilon_1, \epsilon_2, \ldots, \epsilon_k$ and subsequently the final goal $g$. Let $G = (\epsilon_1, \epsilon_2, \ldots, \epsilon_k, g)$. The trajectory $\tau$ of the agent is $\tau = (\textbf{s}_0, \textbf{a}_0, \textbf{s}_1, \textbf{a}_1, \ldots, \textbf{s}_{T-1}, \textbf{a}_{T-1}, \textbf{s}_T)$, where $T$ is the duration of the trial. The expected performance of the agent $\alpha$ is the expected value of the quality $\phi(\tau, g, E)$ of potential trajectories $\tau$ across all trajectories that might occur, that is
\begin{equation}
\label{eq:performance}
\phi_{\textbf{s}_0}^{G}(\alpha) = \mathbb{E}_{P(\tau | \textbf{s}_0, G, \alpha, E)} \left[\phi(\tau, g, E)\right].
\end{equation}
In principle, more subgoals would lead to better performance. But, in practice, the desired number of subgoals ($k$) is limited by the manager's time, the frequency of the manager's communication with the employees, and the employee's need for autonomy \citep{ryan2006self}.
With all of these definitions in place, we can now define the optimal solution to the problem of managerial goal-setting as selecting the sequence of subgoals $\epsilon_1^\star, \epsilon_2^\star, \ldots, \epsilon_k^\star$ that maximizes the employee's expected performance given uncertain knowledge about its capacities
($P(\alpha)$) as
\begin{equation}
\label{eq:optimalsubgoals}
\epsilon_1^\star, \epsilon_2^\star, \ldots, \epsilon_k^\star = \argmax_{\epsilon_1, \epsilon_2, \ldots, \epsilon_k} \mathbb{E}_{P(\alpha)}[\phi_{\textbf{s}_0}^{G}(\alpha)].
\end{equation}
If the normative theory of managerial goal-setting formalized in Equation~\ref{eq:optimalsubgoals} is potentially useful, then subgoals derived from this theory should improve worker's performance. In the following two sections, we test this prediction with simulated and real workers, respectively.
\section{Improving the performance of bounded agents}
We test our normative theory of managerial goal-setting in the SMW described in the following paragraph. In this section, we apply the theory to compute one optimal subgoal for a resource-rational model of goal-pursuit and check if it improves the model's problem-solving performance.
\paragraph{The management problem: maximizing the productivity of a farm}
The simulated micro-world used in the present study simulates the problem of managing a farm (see Fig.~\ref{fig:no_subgoal_sc}). Starting from the state $s_0$, the goal of the agents is to bring the values of certain crops (corresponding to state, $\textbf{s}_t$; shown on the right in Fig.~\ref{fig:no_subgoal_sc}) close to the target values specified by the manager. Agents can do so over the course of $T = 20$ steps by using various costly resources (corresponding to action, $\textbf{a}_t$; shown on the left in Fig.~\ref{fig:no_subgoal_sc}). Further, agents have complete information about how crops are influenced by resources and each other, as shown by the weighted edges in Fig.~\ref{fig:no_subgoal_sc}. In particular, we studied the setting where agents start from $\textbf{s}_0 = [80, 20, 90, 10, 70]^\intercal$ and pursue the goal with $\textbf{s}_g = [0, 0, 0, 0, 0]^\intercal, \gamma_g = \textbf{I}_5$ (i.e., identity matrix of size 5), $\delta_g = 50$, and $\delta_{\epsilon} = 1$.
The performance of the agent is higher the larger the number of time steps for which it achieves the goal. Since resources are costly, using more resources leads to lower performance. We formalize this using the \textit{goal-achievement score} (GAS). If the agent achieves the goal for $x$ out of $T$ time steps, and $y = \sum_t ||\textbf{a}_t||_1$, then GAS is defined as
\begin{equation}
\textrm{GAS}(\tau, g, E) = \textrm{max}(0, w_1 + w_2 \cdot x - w_3 \cdot y) \label{eq:GAS}
\end{equation}
where $w_1$, $w_2$, and $w_3$ capture the starting endowment, the reward for achieving the goal, and the cost of resources, respectively. Here, we used $w_1 = 0.2, w_2 = 0.3, w_3 = 0.005$ to capture that achieving the goal is most important. The aim of the agent is to maximize its GAS, while the aim of the manager is to provide a sequence of subgoals to the agent such that pursuing them helps it maximize its GAS.
An important feature of this environment is that the state variable Crowding has an edge with weight = +1.5 starting and ending in it, which creates a positive feedback loop. Without intervention, this feedback loop would cause the value of Crowding to increase exponentially over time. To prevent this, the agent has to bring the value of Crowding close to 0. We therefore predicted that a good subgoal should include Crowding = 0.
\paragraph{Computing an (approximately) optimal subgoal for boundedly rational employees}
To approximate the optimal goal defined in Equation~\ref{eq:optimalsubgoals}, we approximated the expectation in Eq.~\ref{eq:performance} by running $\eta$ noisy simulations of the given agent. We approximated the expectation in Equation~\ref{eq:optimalsubgoals} by averaging the performances ($\phi_{\textbf{s}_0}^{G}(\alpha)$ in Eq.~\ref{eq:optimalsubgoals}) over a population ($\omega$) of hill-climbing agents with different values of step size $\lambda$ that covered the behavior of the largest proportion of participants in \citet{prystawski2022resource}. To derive $\omega$, we first selected participants from \citet{prystawski2022resource} which were best explained using the hill-climbing model, arranged their step sizes (the only free parameter) in ascending order, and selected 30 equally spaced step sizes to cover the entire range of participants. For each agent, we measured the quality of its trajectories by the goal achievement score defined in Eq.~\ref{eq:GAS}.
To further simplify the computational problem, we considered only one 2-dimensional subgoal (i.e., $k = 1$) and used $\eta = 1$. The distance and angular noise in the actions of the hill-climbing agent were drawn from an exponential distribution (with intensity parameter, $\nu = 0.1$) and a von Mises distribution (centered at $0^\circ$ with concentration parameter $\kappa = 40$), respectively.
We then approximated the optimal subgoal defined in Eq.~\ref{eq:optimalsubgoals}, by maximizing the simulated performance using Cross-Entropy (CE) Optimization \citep{de2005tutorial}. We ran a separate Cross-Entropy procedure for every possible pair of state variables to optimize for $\textbf{s}_{\epsilon}$ and $\gamma_{\epsilon}$, and then chose the state variables with the highest performance. We ran the CE procedure for 10 iterations with 1000 candidate subgoals in every iteration and selected the top 20\% subgoals in every iteration to refine the distribution of potential subgoals.
\paragraph{Results}
The best subgoal according to our subgoal discovery procedure was Crowding = 0 and SpaceWorms = 4 with the $2 \times 2$ diagonal matrix having elements 0.121 and 0.012 respectively as the scale parameter $\gamma_\epsilon$. This scale parameter translates to tolerance values of ±2 and ±6 for the two subgoal variables, respectively. This is consistent with our prediction that a good subgoal should include Crowding = 0. Simulations with $\eta=100$ showed that the goal-achievement scores of agents are higher in the presence of subgoals vs. without them according to a Mann-Whitney U-test (0.696 vs. 0.069, $U = 5.9 \times 10^6, p < 0.001$). This highlights the efficacy of the subgoal and our procedure. The standard deviations of the two target values in $\textbf{s}_\epsilon$ across 5 runs of the subgoal discovery procedure were 0.71 and 1.41, respectively, which shows that the subgoals discovered by our method are reproducible.
\section{Improving the performance of crowdworkers}
We performed a pre-registered experiment to test if the subgoals generated by our method help people in problem-solving. To do so, we tested if providing the subgoal computed in the previous section improves people's performance in the simulated micro-world described above (Figure~\ref{fig:no_subgoal_sc}). Participants in the subgoal condition were asked to pursue the subgoal computed by our method before pursuing the final goal (Figure~\ref{fig:subgoal_sc}), whereas participants in the control condition were directly asked to pursue the final goal (Figure~\ref{fig:no_subgoal_sc}). The pre-registration is available at \href{https://aspredicted.org/5W2_GTV}{\texttt{https://aspredicted.org/5W2\_GTV}.}
\subsection{Methods}
\paragraph{Participants}
We recruited 441 crowd workers from the online study platform \textit{Positly}, out of which 234 identified as male, 190 identified as female, and 17 chose not to disclose their gender. The minimum and maximum ages reported were 21 and 76, respectively, with the average age being 40.28 ($SD = 12.05$). Participants spent an average of 39.7 minutes in the experiment. They earned a base pay of \$3 for completing the training. After this, they participated in a practice trial consisting of six rounds ($T=6$). Participants who achieved the specified goal in this trial were paid \$0.15 and were invited to participate in the main experiment. 302 people participated in the main experiment and received a performance-based bonus, with the average value of the bonus being \$1.28.
\paragraph{Procedure}
At the beginning of the experiment, participants were shown three instruction videos and were given a chance to participate in three practice trials. Then, they had to take a quiz testing their attentiveness and understanding of SMWs. Following this, they were invited to participate in another practice trial of six rounds ($T=6$). Participants who achieved the specified goal in this trial were invited to take part in the main experiment. We only analyzed data from the main experiment, which was completed by 302 participants.
We randomly assigned each participant to one of two conditions: the subgoal condition ($n=150$) and the no subgoal condition ($n=152$). Before starting the main experiment, participants in the subgoal condition were informed that they would receive a subgoal that would help them achieve the final goal. Additionally, they were instructed that subgoals can be defined for a subset of crops, and that they should bring the values of these subgoal measures within the specified tolerances of the target values. During the experiment, only the subgoal crops had target values and tolerances next to them (Fig. \ref{fig:subgoal_sc}). Participants were given a message upon successfully achieving the subgoal, following which the subgoal disappeared and the final goal was displayed.
In the main experiment, which consisted of one trial of 20 rounds (i.e., $T = 20$), participants earned a bonus payment that was equal to their goal-achievement score in USD. Participants in both conditions were informed that they would earn a higher bonus by bringing all crops within their target ranges while using as few resources as possible. They were also informed that negative values of resources have the same cost as positive values. The current value of the bonus was displayed on the screen throughout the trial. In addition, the total distance from the subgoal/final goal was also displayed on the screen.
\paragraph{Materials}
The experiment involved managing a farm on an alien planet (Fig.~\ref{fig:no_subgoal_sc}) with the goal of bringing the values of certain crops within the specified tolerances of their target values. The target value, tolerance, and current value of each farming measure were shown alongside it on the screen. Participants could influence the values of crops using various costly resources. They could select the amount (positive or negative) for each resource by either typing in the desired value in the corresponding box or by using the up/down arrow keys. The causal relationships between variables were shown via weighted edges. To reduce cluttering, self-connections were only shown when a variable amplified its value over time (Crowding in Fig. \ref{fig:subgoal_sc}).
The starting position and final goal in both conditions were equal to those used previously to compute optimal subgoals. All participants in the subgoal condition were given the subgoal computed by our automatic method: Crowding = 0±2 and SpaceWorms = 4±6. For the practice trials, we employed an SMW which was different from the one used in the main experiment but followed the same rules.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{no_subgoal.png}
\end{center}
\caption{Screenshot of the Simulated Microworld shown to participants in the experiment.}
\label{fig:no_subgoal_sc}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{subgoal.png}
\end{center}
\caption{Screenshot of the Simulated Microworld showing the subgoal discovered by our method. Note that crops not in the subgoal do not have corresponding target and tolerance values because the subgoal has not been achieved yet.}
\label{fig:subgoal_sc}
\end{figure}
\subsection{Results}
The average goal-achievement scores were 1.33 and 1.23 in the subgoal and no subgoal conditions, respectively. This difference was not statistically significant according to a Mann-Whitney U-test ($U=10595$, $p = 0.11$). However, the proportion of participants with a positive score was significantly higher in the subgoal condition than in the control condition according to a two-proportions z-test (43.33\% vs. 31.6\%, $z= -2.11$, $p=0.035$).
The amount of resources used by a participant was computed as $\sum_t ||\textbf{a}_t||_1$. We found that participants in the subgoal condition used substantially fewer resources than participants in the control condition (494.45 vs. 859.5, $U= 13092$, $p = 0.012$).
Additionally, we also computed participants' distance score as
$
\textrm{DS} = \sqrt{||\textbf{s}_T - \textbf{s}_g||_2^2 + c \cdot \sum_{t=0}^{T-1} ||\textbf{a}_t||_2^2}.
$
The distance score captures how close an agent gets to the final goal at the end of the trial while penalizing it for using more resources. Smaller values of DS are better. Following \citet{prystawski2022resource}, we used $c = 0.01$. In the subgoal condition, the median value of the distance score was numerically lower than in the control condition (492 vs. 988). However, because of the high variance, this difference was not statistically significant according to a one-sided Mann-Whitney U-test ($U= 12466$, $p = 0.08$).
Taken together with the significant reduction in the amount of resources used, this result indicates that subgoals can help people achieve their goals more efficiently.
In summary, the subgoal condition performed numerically better than the control condition on all outcome measures, was significantly more likely to achieve a positive goal-achievement score, and used the farm's resources significantly more efficiently. These findings are inconclusive, but broadly consistent with the interpretation that the automatically derived subgoal had a small positive effect on people's performance in the problem-solving task.
\section{Discussion}
Goal-setting is commonly used to improve people's ability to solve complex problems \citep{drucker2012practice,catrambone1998subgoal,locke2002building}. One of the reasons why goal-setting is effective is that it reduces the amount of planning that is necessary for goal achievement. It should therefore be possible to leverage resource-rational models of planning and goal-pursuit \citep[e.g.,][]{callaway2022rational,prystawski2022resource} to improve the theory and practice of improving performance through goal-setting \citep{locke2002building}.
To explore this approach, we have proposed a normative theory for (self-)management by goal-setting. We have applied this theory to computationally derive subgoals from a resource-rational model of goal pursuit.
Our proof-of-concept simulations and experiment suggest that it might be possible to derive helpful goal suggestions from resource-rational models of goal pursuit. This illustrates that it is, at least in principle, possible to ground recommendations for goal-setting in the theory of resource-rationality.
Our work could therefore be considered a first step towards establishing empirically supported computational models of bounded rationality as a micro-foundation for prescriptive theories of (self-)management.
The main limitation of the present work is that empirical evidence for our method's ability to improve human problem-solving was mixed. To explain the mixed results, it is worth noting that the self-amplifying dynamics of the simulated micro-world we used in this experiment made participants' scores extremely variable. This variability, in turn, reduced the power of our statistical tests. As a consequence, even large numerical difference were not always statistically significant. Concretely, the high variability of the scores resulted from the presence of a positive feedback loop that caused Crowding to increase exponentially over time. Once Crowding exceeded a certain value, the exponential growth became unstoppable and participants could no longer control the system. This made the task very challenging for participants, even in the presence of the optimal subgoal.
Future work should investigate why our participants benefited less from the provided subgoal than our simulations had predicted, and improve the model (and the resulting subgoal) accordingly.
Follow-up experiments should also investigate the moderating role of individual differences in cognitive ability and motivation.
Moreover, our assessment of the method was limited to a single problem in just one simulated micro-world. Future experiments should assess the generalizability of our findings to other problems in other environments. Such studies could jointly identify under which conditions the subgoals recommended by our method are most beneficial and who benefits the most.
The work presented in this article builds on, extends, and applies the resource-rational perspective on goals and goal-setting \citep{lieder2020resource}. While previous work explored this idea in planning tasks with discrete states \citep{correa2020resource,correa2022humans}, we have applied the resource-rational perspective on goal-setting to complex problem-solving in dynamic environments with continuous states and inputs. Moreover, while previous work made the unrealistic assumption that the goal is always achieved, our normative theory of goal-setting takes into account that goal-achievement is the exception rather than the norm and that maintaining the desired state is also an important part of the problem. Another innovation of our theory is that it draws on an evidence-based process model of human goal-pursuit, instead of assuming that people use search algorithms that were developed for computers.
Moreover, while \citet{correa2020resource} and \citet{correa2022humans} studied how people do and machines should decompose tasks into subtasks, our goal was to formulate a normative theory of (self-)management.
As far as we know, previous research on improving managerial goal-setting did not explicitly engage with mechanistic models of goal pursuit and bounded rationality and did not develop computational methods for computing optimal subgoals.
The work we have presented in this short article can be extended in several directions. One direction is to improve the current method for computing optimal subgoals. Possible improvements include generating a series of multiple subgoals, improving the accuracy and/or speed of the optimization algorithm, considering higher-dimensional subgoals, incorporating the mental effort of planning and goal-pursuit into the objective function \citep[cf.][]{correa2020resource,correa2022humans}.
Further, future work can also use our normative theory of goal-setting as the starting point for resource-rational analyses of how people set goals for others and themselves, respectively. In addition, our work can be extended to make the subgoals more adaptive by learning from the behavior of the employee and adjusting the model of goal pursuit and the subsequent subgoals accordingly.
Finally, we hope that in the long run, the research begun in this project will improve the theory and practice of individual and organizational goal-setting. As a step in this direction, future experiments should compare the effectiveness of the subgoals generated by our method against the effectiveness of subgoals derived from previously proposed heuristics and subgoals chosen by participants. Another step in this direction could be to use an improved version of our theory to generate optimal subgoals for a wide range of complex problems, and then characterize what features useful subgoals have in common.
Although research on grounding prescriptive theories in computational models of bounded rationality is still in its infancy, it is at least beginning to suggest that this is possible to leverage the rigorous methods of computational cognitive science to generate practical knowledge and useful technologies for helping people and organizations become more effective \citep{Lieder2022Interdisciplinary,LiederPrentice_LIS}. We hope that future work in this direction will establish a solid cognitive science foundation for helping people, teams, and organizations set better goals.
\printbibliography
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,020 |
Q: Why is desktop.ini "Ready to Be Written to the Disc" of any CD or DVD inserted under Windows 7? I have a netbook with an external CD/DVD burner running Windows 7 Starter. Whenever I put any CD or DVD (doesn't have to be blank) in the drive and navigate to its path in Windows Explorer, it tells me under "Files Currently on the Disc":
Files Ready to Be Written to the Disc
(1) desktop.ini 23/03/2011
I can't recall ever doing anything that I would've expected would send any files to be written the CD/DVD.
The contents of the .ini file doesn't look suspicious:
[.ShellClassInfo]
LocalizedResourceName=@%SystemRoot%\system32\shell32.dll,-21815
Is this some Windows 7 feature I didn't know about? A known glitch? Likely the result of some accidental and unnoticed drag and drop? How do I undo it? Is it safe to just delete the desktop.ini via Windows Explorer?
A: The post Desktop.ini file keeps showing up as a file to be written to a blank CD says it is normal behavior. One person suggests hiding system files so you don't see it.
A: Here is what happened:
*
*In the process of a burning a CD/DVD or preparing to burn a CD/DVD, you entered the special "Burn DVD" folder Windows uses as a staging area and changed the sort order, arrangement, or other layout/view option to be different than the default view.
*A hidden "desktop.ini" file is created automatically to hold this new layout information, so the next time you view the folder it will remember your preference. This is normal behavior for any folder on your file system, and if you go looking for it you can find this file all over the place.
*After burning the CD, this file was cleared out along with everything else. However, your layout was still non-standard, and so it is recreated when the folder is closed.
*You now have a file sitting in the special staging folder, and that trips up the "File waiting to be burned to CD" notice.
A: Ignore it, it's just a file similar to the thumbs.db file you see. It can also be considered the same as the .DS_STORE file found in Macs. It just contains certain configuration for the current directory.
A: Right click on the following folder:
C:\Users\Administrator\AppData\Local\Microsoft\Windows\Burn
or
C:\Users*[your user name]*\AppData\Local\Microsoft\Windows\Burn
Select Properties > Security TAB > Edit > Click on any user name > CHECK: Deny (to the right of Full Control) > OK > OK
Desktop.ini no longer shows up, I don't know what other consequences this will have other than maybe not directly burning to the disc, but most of us use third party burning software anyway.
A: I was having the same issue reading a DVD with files burned from another computer.
Viewing the DVD would only display a Desktop.ini file with no actual data files.
"Ready to Be Written to the Disc" of any CD or DVD inserted under Windows 7.
Solution - Since the system was a laptop, I was able to remove the DVD drive from the bay. After a few seconds, pop it back in which Win7 re-discovered the drive.
Result, the files now appear normally without displaying the Desktop.ini file.
A: I had the same problem, and when I looked under burn options, the option had switched from Audio to DVD or data. Switched it back to Audio, and the annoying 'waiting to burn desktop' etc. didn't show. Before this change, WMP burned onto an audio CD but it was unplayable in CD players because of that annoying add-on.
A: *
*Set these folder view options:
*
*[Enable] Show hidden files, folders, and drives
*[Disable] Hide protected operating system files [1]
*Delete any desktop.ini file in these paths:
*
*C:\Users\Administrator\AppData\Local\Microsoft\Windows\Burn\Temporary Burn Folder
*C:\Users\you\AppData\Local\Microsoft\Windows\Burn\Temporary Burn Folder
[1] The .ini extension is recognized as a "protected operating system file", so it remains hidden until this option is disabled.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,215 |
This Week at Fifty & Five 9.17.13
Twitter Files Request To Go Public As you've probably heard, Twitter tweeted last Thursday that it had filed a confidential request for going public. This is huge news, as it is the biggest IPO since Facebook's $16 billion offering in May 2012, and many have predicted that it will be the biggest tech IPO of […]
Twitter Files Request To Go Public
As you've probably heard, Twitter tweeted last Thursday that it had filed a confidential request for going public. This is huge news, as it is the biggest IPO since Facebook's $16 billion offering in May 2012, and many have predicted that it will be the biggest tech IPO of the year at an estimated $15 to $20 billion valuation. Since Twitter has only started to really focus on revenue as of recently, it was able to "confidentially" submit its S-1 to the Securities and Exchange Commission, which requires that current revenue be lower than $1 billion.
Industry experts expect Twitter's road to the stock exchange to be much smoother (and potentially more lucrative for investors) than Facebook's. When Facebook went public in 2012, it set its price too high, issued too many shares, and IPO'd too late in the company's life cycle. By learning from these missteps, Twitter can hopefully have a more successful debut. Although we don't know exactly when Twitter will IPO, it plans to float its shares on the NYSE, not the NASDAQ, which seems like a smart move considering the technical difficulties that Facebook faced on the NASDAQ. We have high hopes for Twitter and are excited to see how big the valuation is when it comes out.
Facebook Boosts Image Size For Link Share Ads
Facebook has boosted the image size for their link share ads and organic posts, challenging marketers to take a more visual approach. Now when a page or person shares a link, the image takes up a large amount of real estate within News Feed. According to the Facebook & Journalists page, the ideal image size for these posts is now 1200 pixels x 627 pixels, with the minimum being 560 x 292.
Facebook has been constantly pushing for a focus on big, bold images. When the company released the newest version of News Feed, the photos were the most striking feature, pressuring marketers to give more thought to how images are featured. Many pages have started posting a photo with a link to the website within the text prompt. The update Facebook announced Tuesday fixes this. Now, when a user clicks on either the image or the text on a link post, they will be taken to the website. This will lead to a higher click-through rate and more exposure for a page's website through these posts.
Pinterest Hires First International Employees
Pinterest is hiring abroad for the first time, adding country managers in France and the UK to help grow the site's user base in Europe, according to a company spokesperson. The new managers will build out the local Pinterest communities overseas and connect with brands and partners. Pinterest is also looking to hire marketing managers in London and Paris, and may open international offices after building a larger workforce outside the United States.
Google+ Adds Photo Editing Tools
Google+ has just given its members a variety of web-based photo editing tools powered by Snapseed. The tools include a number of filters, sliders for adjusting specific elements, and features to let you tweak only a selected part of the image. The one catch is that users have to use Chrome, Android, or iOS to access the photos they want to edit. The updates will roll out gradually, and many people are interested to see if Facebook responds with photo editing of its own.
Facebook Tests Video Autoplay
Facebook will begin testing a new mobile feature that plays videos automatically as users scroll through their News Feeds. The videos will begin playing as they come into view on the screen, and users can preview them within the News Feed without clicking on or opening them. Each video will play silently and can be viewed with sound upon clicking. The new capability will only be rolled out a small group of random users, and videos will only play automatically if they are uploaded directly to Facebook – not embedded from other sites like YouTube. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,128 |
Yes indeed the "change agents" have socially engineered us into this mess.
in response to reader comment: The Change Agents Have Done Their Work Well!
Submitted by Homefront (Australia), May 7, 2008 at 21:38
Duliece,
I agree with you totally. Society has been pregnant with the new cohort of neomarxist nonsense since the 1960-70s and now we are about to reap the fruit of the seeds the socialists planted in our education system. Broadly speaking these are the death of free and rational thought and the collapse of the western world.
The latter is almost inevitable and perhaps the only thing that will be able to rid us of this communalistic nonsense of Multiculturalist cultural relativism. In fact many in the left know they are pushing us towards a collapse - many of the older theorists wanted to bring such a collapse about as a necessary step to fashion their utopia.
I think that a more likely result of left wing bottom feeder dominance is a dystopia like the one fleshed out in the novel "The Hatred of Angels" or Mark Steyn's "America Alone". All of the ways to avoid the looming apocalypse like recapturing our institutions (suggested by Steyn) are almost impossible to achieve.
Universities and Schools are left wing ideological fortresses - they're expecting an assault. The other course of action that is equally implausible is establishing parallel institutions and fortifying them against the left. Who would we staff them with? Taking journalism as an example there are only around 4 neoconservative journalists in the entire of Australia (Andrew Bolt and Janet Albrechtson are the only two that come immediately to mind).
It is almost impossible to see how we can recover from our present predicament. We have an entire generation who are incapable of critical thought and will only transmit their ignorance to the next generation (to the extent that they are prepared to even breed). In Australia the dominant educational paradigm which has driven this catastrophe is called "student centred learning" which is essentially designed around elevating the bottom rung students at the expense of almost everyone else. It is an exemplar of the maxim "a system designed around the worst qualities of humankind will tend to reflect them".
A female University Academic who migrated to Australia from China and lived under Mao's regime said that she thought neoconservative assertions that the educational system in Australia was Maoist were unfair to Mao. The basis for her assertion was the fact that she found that most of the Australian undergraduates she had to deal with were both ideologically indoctrinated and lacking the most basic academic skills. At least the idealogues churned out of the Maoist system had basic Academic skills. The best students academically she lamented were often the most closed minded - apparently this is the price of an A in the Australian high school system.
The end product is a cohort of unshakable, unthinking Rudd supporters in Australia. And in America perhaps an Obama presidency. Ever wonder why the Republicans look like they're getting older every year? It's because the University and School system in America years ago discontinued the conservative product line.
Homefront.
Mark my comment as a response to Yes indeed the "change agents" have socially engineered us into this mess. by Homefront | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,111 |
\section{Introduction}\label{sec:introduction}
Primitive Equations constitute a fundamental model in geophysical fluid dynamics.
The present work is devoted to the study of Gaussian invariant measures in the stochastically forced 2-dimensional case:
the model under analysis is thus a stochastic PDE of the form:
\begin{equation}\label{eq:generalmodel}
\begin{cases}
\partial_t v + v \partial_x v + w \partial_z v + \partial_x p = \mathcal{D}(\Delta)v + \eta, \\
\partial_z p = 0, \\
\partial_x v + \partial_z w = 0,
\end{cases}
\end{equation}
where $(x,z)$ are coordinates of the bounded domain $D=[0,2\pi]^2$
on which suitable boundary conditions are imposed,
$(v,w)$ are the components of the \emph{velocity} vector field,
$p$ is the \emph{pressure},
the term $\mathcal{D}(\Delta)$ describes a dissipation mechanism
and $\eta$ is a Gaussian stochastic process.
It is in fact the case with $\mathcal{D}(\Delta)=\nu\Delta$, and $\eta=0$ to be usually
referred to as \emph{2-dimensional Primitive Equations} (2dPE), together with its variants including effects
such as density and temperature variations, and other geophysical effects.
When those physical phenomena are neglected,
equations \eqref{eq:generalmodel} have many aspects in common with the 2-dimensional Navier-Stokes equations:
\begin{equation*}
\begin{cases}
\partial_t u + (u\cdot \nabla)u+\nabla p=\nu \Delta u,\\
\div u=0,
\end{cases}
\end{equation*}
describing the evolution of the velocity field $u$ of an incompressible viscous fluid.
This familiarity naturally leads to look for applications of concepts and techniques developed
in the extensive theory of Navier-Stokes equations, especially in the 2-dimensional setting.
However, the nonlinearity of 2-dimensional Primitive Equations is in fact harder to treat.
When considering the stochastically forced case, it is well-known that 2-dimensional
stochastic Navier-Stokes equations (SNS) in their vorticity form:
\begin{equation*}
\begin{cases}
\partial_t \omega + (u\cdot \nabla)\omega=\nu \Delta \omega+\sqrt{2\nu} \xi,\\
\div u=0, \, \omega=\curl u,
\end{cases}
\end{equation*}
$\omega$ being the scalar \emph{vorticity} field, preserve the so-called \emph{enstrophy measure}
when driven by space-time white noise $\xi$.
The enstrophy measure is a Gaussian random distribution, corresponding to space white noise
at the level of $\omega$: its name is due to the fact that its covariance is the quadratic form associated
to the quadratic observable $\norm{\omega}^2_{L^2}$, known as \emph{enstrophy}.
Indeed, enstrophy is a first integral of motion in the case $\nu=0$, the 2-dimensional Euler equations,
and enstrophy measure is the unique, ergodic invariant measure of the linear part of the dynamics
when $\nu>0$.
Notwithstanding the low space regularity under enstrophy measure,
existence and pathwise uniqueness of stationary solutions for SNS in this setting
are by now classical results due to \cite{DPDe02,AlFe04}.
There exists of course another quadratic invariant for Euler equations:
the energy $\norm{u}^2_{L^2}$. When SNS is driven by space-time white noise at the level of velocity,
the \emph{energy measure}, a white noise at the level of $u$, is \emph{formally} preserved.
The cursive is here in order, because the energy measure regime is so singular that
no solution theory is yet available in this case. Nonetheless, the existing stochastic analysis techniques
allow to deal with such regime in \emph{hyperviscous} cases, that is replacing the viscous term
$\Delta u$ with $-(-\Delta)^\theta u$, $\theta>1$. Indeed,
a procedure known as \emph{It\=o trick} in the literature related to regularisation by noise
is employed in \cite{GuJa13} to give meaning and solve SNS under energy measure
with sufficiently strong hyperviscosity. We also mention the recent development \cite{GuTu19},
in which Kolmogorov equations are solved by means of Gaussian analysis tools,
broadening the result to solutions absolutely continuous with respect to energy measure.
The analogue of vorticity field for 2-dimensional Primitive Equations is $\omega=\partial_z v$,
as the quadratic observable $\norm{\partial_z v}^2_{L^2}$ is a first integral
of the 2-dimensional \emph{hydrostatic Euler equations}:
\begin{equation}\label{eq:2dhe}
\begin{cases}
\partial_t v + v \partial_x v + w \partial_z v + \partial_x p =0, \\
\partial_z p = 0, \\
\partial_x v + \partial_z w = 0.
\end{cases}
\end{equation}
Prescribing the correct additive Gaussian noise $\eta$, the linear part of \eqref{eq:generalmodel}
with $\mathcal{D}(\Delta)=\Delta$ preserves the Gibbsian measure associated to $\norm{\partial_z v}^2_{L^2}=\norm{\omega}^2_{L^2}$,
formally defined by
\begin{equation*}
d\mu(\omega)=\frac1Z e^{-\frac12 \norm{\omega}^2_{L^2}} d\omega,
\end{equation*}
that is, white noise distribution for $\omega$.
However, as we will detail below,
the stationary regime with $\mu$-distributed marginals for \eqref{eq:generalmodel} is
not comparable to the enstrophy measure stationary regime of SNS, because of the
more singular nonlinearity. Indeed, unlike in \cite{DPDe02,AlFe04},
the nonlinear terms of \eqref{eq:generalmodel} can not be defined as distributions when $\partial_z v$ has law $\mu$.
Still, as in the case of energy measure SNS, hyperviscosity allows to apply the techniques of \cite{GuJa13}.
In this work we present a solution theory of 2-dimensional Primitive Equations in the hyperviscous setting
$\mathcal{D}(\Delta)=-(-\Delta)^\theta$, for large enough $\theta$ and a suitable stochastic forcing.
The regularising effect of hyperviscosity for Navier-Stokes and Primitive Equations
is well-understood in the deterministic setting, and it is often used in numerical simulations \cite{LJTN11};
we refer to \cite{La58,Li59,LiTeSh92a} and, more recently, \cite{Hu20} for a thorough discussion.
The main contribution of the present paper is thus to introduce a Gaussian invariant measure
in the context of 2-dimensional Primitive Equations, and then to exploit the techniques
of \cite{GuJa13} to provide a first well-posedness result for this singular SPDE in a hyperviscous setting.
Although stochastic versions of Primitive Equations both in two and three dimensions
have already been considered, to the best of our knowledge the existing literature is limited
to regimes more regular than ours. To mention a few relevant previous works,
in \cite{GHZi08,GHTe11,TM10}, 2-dimensional Primitive Equations
are considered with a multiplicative noise taking values in function spaces,
the same is done in the 3-dimensional case in \cite{DeGHTeZi12,GaSu16},
and in \cite{GHKuViZi14} the authors prove the existence of an invariant measure in this setting.
In the 2-dimensional cases, large deviation principles are studied in \cite{GaSu12,SuGaLi18}.
Let us also mention the works \cite{PeTeWi04,BrKaLe05} on deterministic 2-dimensional
Primitive Equations, whose study began with \cite{LiTeSh92a,LiTeSh92b,LiTeSh93},
and \cite{MaWo12} on their inviscid version, by which the vorticity formulation
we present below for our model is inspired.
The paper is structured as follows: in \autoref{sec:vortform} we rigorously introduce a
stochastic version of 2-dimensional Primitive Equations in terms of vorticity $\omega=\partial_z v$
and a Gaussian measure formally preserved by the dynamics. We then summarize how the
theory of \cite{GuJa13} applies and state a well-posedness result for martingale solutions
in sufficiently hyperviscous cases. In \autoref{sec:details} we then collect
details and computations completing the proof of our main results.
\renewcommand{\abstractname}{Acknowledgements}
\begin{abstract}
The authors would like to thank Franco Flandoli and Massimiliano Gubinelli
for many useful suggestions and relevant conversations,
and Martin Saal for introducing them to the literature on Primitive Equations.
\end{abstract}
\section{Vorticity Formulation and Conservation Laws}\label{sec:vortform}
The model under consideration in the remainder of the paper is the following stochastic
PDE in the space domain $D=[0,2\pi]^2\ni(x,z)$,
\begin{equation}\label{eq:2dspe}
\begin{cases}
\partial_t v + v \partial_x v + w \partial_z v + \partial_x p = -(-\Delta)^\theta v + \eta, \\
\partial_z p = 0, \\
\partial_x v + \partial_z w = 0.
\end{cases}
\end{equation}
Here, $v=v(t,x,z)$ is the horizontal velocity, $w=w(t,x,z)$ is the vertical velocity,
$p=p(t,x)$ is the pressure. The parameter $\theta$ and the additive Gaussian noise $\eta$
will be specified below. The unknown fields $v,w$ are subject to the following boundary conditions:
\begin{equation}\label{eq:bc}
\begin{cases}
w=0, & \mbox{ if } z=0,2\pi,\\\
v=0, & \mbox{ if } x=0,2\pi,\\
\partial_z v=0, & \mbox{ if } z=0,2\pi.
\end{cases}
\end{equation}
The first two lines impose \emph{impermeability} of the boundary;
the third one is called a \emph{free boundary condition} for the surface and the bottom of $D$.
Before moving on, we discuss another possible choice in the next paragraph.
\subsection{On Physically Realistic Boundary Conditions}\label{ssec:physrel}
While free boundary conditions are suited to describe interfaces between fluids
such as the ocean surface, they can not be used to model a solid boundary such as the ocean bottom.
Instead, one should consider a no-slip boundary condition, leading to a different set of conditions:
\begin{equation}\label{eq:physbc}
\begin{cases}
w=0, & \mbox{ if } z=0,2\pi,\\\
v=0, & \mbox{ if } x=0,2\pi,\\
v=0, & \mbox{ if } z=0,\\
\partial_z v=0, & \mbox{ if } z=2\pi.\\
\end{cases}
\end{equation}
In other words, we are assuming that the full velocity field $(v,w)$ vanishes on the bottom side.
We prefer the choice \eqref{eq:bc} since
Laplace operator can be diagonalised on functions satisfying that set of boundary conditions.
This is not true when we consider Dirichlet boundary at the bottom, since
the eigenvalue problem is overdetermined.
In that case, Fourier analysis can still be carried through with an orthonormal basis
differing from usual trigonometric functions, see \cite[Section 6]{BrKaLe05}.
We also refer to \cite{GHTe11} for further discussion on boundary condition,
and conclude the paragraph observing that one can reduce conditions \eqref{eq:physbc} to the ones \eqref{eq:bc}.
Assume that $(v,w)$ is a smooth solution of \eqref{eq:2dspe} on $D$ satisfying \eqref{eq:physbc},
for simplicity in the case $\eta=0$. Then, if we extend the solution to the doubled domain
$\tilde D=[0,2\pi]\times [-2\pi,2\pi]$ so that $v,w$ are odd functions in the $z$ direction,
we have obtained a solution of \eqref{eq:2dspe} on $\tilde D$ satisfying \eqref{eq:bc}.
The size and aspect ratio of the domain is in fact irrelevant in our discussion.
\subsection{Vorticity Formulation}\label{ssec:vortform}
Let us first assume to be dealing with smooth solutions of \eqref{eq:2dspe},
driven by a smooth deterministic $\eta$.
The aim is to derive an equivalent formulation of the model
in terms of the only scalar field {vorticity} $\omega=\partial_z v$,
on which we will focus the remainder of our discussion.
First of all, let us notice that $v$ must always have zero average in the $z$ direction,
since the incompressibility equation $\partial_x v+\partial_z w=0$ and boundary conditions imply, for all $x\in[0,2\pi]$:
\begin{equation*}
\partial_x \int_0^{2\pi} v(x,z')dz'=\int_0^{2\pi} \partial_x v(x,z')dz'=-\int_0^{2\pi} \partial_z w(x,z')dz'=0,
\end{equation*}
from which it follows
\begin{equation}\label{eq:vzeroaverage}
\int_0^{2\pi} v(x,z')dz'=\int_0^{2\pi} v(0,z')dz'=0.
\end{equation}
Because of this, the solution $A(v)$ of the linear problem
\begin{equation*}
\begin{cases}
-\partial_z^2 A(v)(x,z)=v(x,z), &(x,z)\in [0,2\pi]\times (0,2\pi),\\
A(v)(x,z)=0, &z=0,2\pi,
\end{cases}
\end{equation*}
is well defined for all $v$ satisfying our hypothesis.
Another property of solutions $(v,w)$ holding independently of time is that
$w$ is a \emph{diagnostic variable}, \emph{i.e.} it is completely determined by $v$:
\begin{equation*}
w (x,z) = w(x,0) -\int_0^z \partial_x v(x,z') dz'=\partial_x\partial_z A(v)(x,z).
\end{equation*}
Neglecting for a moment boundary conditions,
equations \eqref{eq:2dspe} can thus be rewritten in terms of only $v,p$ by
\begin{equation*}
\begin{cases}
\partial_t v + v \partial_x v +\partial_x\partial_z A(v) \partial_z v + \partial_x p =-(-\Delta)^\theta v + \eta,\\
\partial_z p = 0.
\end{cases}
\end{equation*}
The system is then further simplified by considering the equation for vorticity $\omega=\partial_z v$,
which does not involve the pressure $p$:
\begin{equation}\label{eq:vorticityformulation}
\partial_t \omega+ \nabla^\perp A(\omega)\cdot \nabla\omega=-(-\Delta)^\theta \omega + \partial_z \eta,
\end{equation}
where $\nabla^\perp=(-\partial_z,\partial_x)$.
Notice that $v$ is completely determined by its partial derivative $\partial_z v$ and
the zero average condition \eqref{eq:vzeroaverage}, so \eqref{eq:vorticityformulation}
is equivalent to \eqref{eq:2dspe}. Let us also remark that $A(\omega)$ is well-defined
since $\omega$ has zero average in the $z$ direction, and that $A$ --to be rigorously
defined below as an operator on function spaces-- commutes with derivatives.
Let us briefly discuss boundary conditions for $\omega$. Conditions on $v$ immediately prescribe
$\omega(x,0)=\omega(x,2\pi)=0$ for $x\in[0,2\pi]$ and,
moreover, since $v$ is constant along the $z$ direction at $x=0,2\pi$,
we also have $\partial_z v(0,z)=\partial_z v(2\pi,z)=0$ for all $z\in [0,2\pi]$:
overall $\omega$ must vanish on $\partial D$.
The condition $w=0$ on $z=0,2\pi$ is not as easy to translate into a condition for $\omega$,
but we will bypass the issue with our Fourier series approach below.
It is worth noticing, however, that it is because of the boundary condition on $w$
that $A(\omega)$ is well defined.
\begin{rmk}
The relation between boundary conditions for $(v,w)$ and $\omega$ is thoroughly discussed
in \cite{BrKaLe05} in the setting of \autoref{ssec:physrel}.
\end{rmk}
To conclude the paragraph, let us observe that thanks to the driving vector field $\nabla^\perp A(\omega)$
being Hamiltonian, smooth solutions of the hydrostatic Euler equation \eqref{eq:2dhe} in vorticity form,
\begin{equation}\label{eq:2dhevort}
\partial_t \omega+ \nabla^\perp A(\omega)\cdot \nabla\omega=0,
\end{equation}
with $(v,w)=\nabla^\perp A(\omega)$ satisfying boundary conditions, preserve the quadratic observable $\int_D \omega^2 dxdz$. Quite remarkably, this feature is peculiar to the two-dimensional case, since the quantity $\omega$ does not seem to have a counterpart in higher dimensions.
\subsection{A Rigorous Functional Analytic Setting}\label{ssec:analfun}
As we described above, we are not interested in regular solutions of \eqref{eq:2dspe},
but rather to singular, distributional regimes. It is thus convenient to
encode in Fourier series the boundary conditions, and then set up
our results in distribution spaces defined by means of Fourier expansions.
The general Fourier series expansion of a smooth function $\omega$ on $D$
such that $A(\omega)$ is well-defined and $(v,w)=\nabla^\perp A(\omega)$ satisfy boundary conditions \eqref{eq:bc} is
\begin{equation*}
\omega(x,z)=\sum_{k\in\mathbb{N}^2_0} \hat \omega_k e_k(x,z),
\quad e_k(x,z)=\frac1\pi \sin(k_1 x)\sin(k_2 z),
\end{equation*}
where the $e_k$'s form an orthonormal set in $L^2(D)$, $\hat \omega_k$ are the Fourier coefficients of $\omega$
and $k=(k_1,k_2)\in \mathbb{N}^2_0=(\mathbb{N}\setminus\set{0})^2$.
We will denote
\begin{equation*}
\S=\set{\omega=\sum_{k\in\mathbb{N}^2_0} \hat \omega_k e_k: \forall p\in\mathbb{R}\, \sum_{k \in \mathbb{N}^2_0} |k|^p \abs{\hat\omega_k}<\infty}.
\end{equation*}
Equivalently, $\S$ is the space of smooth functions $\omega$ on $D$ belonging to the domain of $A$
and such that $(v,w)=\nabla^\perp A(\omega)$ satisfies the boundary conditions \eqref{eq:bc}.
We then denote by $\S'$ its dual space,
represented by Fourier series whose coefficients grow at most polynomially.
Brackets $\brak{\cdot,\cdot}$ will denote duality couplings between functions and distributions
\begin{equation*}
\brak{f,g}=\sum_{k\in\mathbb{N}^2_0} \hat f_k \hat g_k,
\end{equation*}
defined whenever the right-hand side converges. Let us also introduce, for $m\in \mathbb{N}$,
the projection onto the linear space of functions generated by $e_k$ with $|k|^2=k_1^2+k_2^2\leq m^2$,
\begin{equation*}
\pi_n: \S'\to \S, \quad \omega\mapsto \pi_m\omega=\sum_{\substack{k \in \mathbb{N}^2_0, \\|k|\leq m}} \hat \omega_k e_k.
\end{equation*}
Following \cite{GuJa13}, we set up our analysis on the Banach spaces
\begin{equation*}
\mathcal{F} L^{p,\alpha}=\set{\omega\in\S':
\norm{\omega}^p_{\mathcal{F} L^{p,\alpha}}=\sum_{k\in\mathbb{N}^2_0} |k|^{\alpha p}|\hat\omega_k|^p<\infty},
\quad \alpha\in\mathbb{R},p\geq 1,
\end{equation*}
and their $p=\infty$ version with $\norm{\omega}_{\mathcal{F} L^{\infty,\alpha}}=\sup_{k\in\mathbb{N}^2_0} |k|^\alpha |\hat\omega_k|$.
Moving to the Fourier expression of the dynamics \eqref{eq:2dspe}, the crux is clearly the nonlinear term,
whose Fourier expansion is given by
\begin{equation}\label{eq:fourierb}
\nabla^\perp A(\omega)\cdot\nabla\omega=B(\omega)=\sum_{k\in\mathbb{N}^2_0} B_k(\omega)e_k,
\quad
B_k (\omega) =\sum_{h\in\mathbb{Z}^2_0} \hat\omega_h \hat\omega_{k- h} \frac{k \cdot h^{\perp}}{h_2^2},
\end{equation}
where $\mathbb{Z}_0^2 = (\mathbb{Z}\setminus\{0\})^2$ and, for $h=(h_1,h_2) \in \mathbb{Z}_0^2$, $\hat\omega_h = \text{sign}(h_1 h_2)\hat\omega_{(|h_1|,|h_2|)}$.
With vorticity formulation at hand, the difficulty inherent to the nonlinear term
is now apparent: looking at the $z$ component of the divergence-less vector field
$\nabla^\perp A(\omega)$, the loss of one $\partial_x$ derivative is not compensated
by the gain of one $\partial_z$ derivative. Indeed, such unbalance marks the difference
between \eqref{eq:2dspe} and 2-dimensional SNS, which is especially evident in the
Fourier series expansion \eqref{eq:fourierb}.
\subsection{Gaussian Invariant Measures and Driving Noise}
Referring to \cite{DPZa14}, we now introduce the stochastic analytic tools we will employ below.
Invariance of $S(\omega)=\frac12\int_D \omega(x,z)^2 dxdz$ for \eqref{eq:2dhe}
suggests that existence of an invariant \emph{Gibbs measure} formally defined by
\begin{equation}\label{eq:enstrophymeasure}
d\mu(\omega)=\frac1Z e^{- S(\omega)}d\omega.
\end{equation}
Since $S$ is quadratic, \eqref{eq:enstrophymeasure} can be understood as a Gaussian measure on $\S'$
with covariance operator $\id$, a multiple of \emph{space white noise} on $D$.
In other words, $\mu$ is the law of the centred Gaussian process $\chi$ indexed by $\mathcal{F} L^{2,0}$ with covariance
\begin{equation*}
\expt{\chi(f)\chi(g)}= \brak{f,g}, \quad f,g\in \mathcal{F} L^{2,0}.
\end{equation*}
Such $\mu$ can be interpreted as the law of a random distribution
supported on all $\mathcal{F} L^{2,\alpha}$ with $\alpha<-1$, the spaces into which the reproducing kernel
Hilbert space $\mathcal{F} L^{2,0}$ has Hilbert-Schmidt embedding.
Although a fixed realisation of the random field $\chi$
is only a distribution, couplings $\brak{f,\chi}=\chi(f)$ for $f\in \mathcal{F} L^{2,0}$
are defined as random variables in $L^2(\mu)$ (It\=o integrals).
Another equivalent formulation is in terms of infinite products: formally expanding $S$ by
Parseval formula, we can write
\begin{equation*}
d\mu(\omega)=\prod_{k\in\mathbb{N}^2_0} \pa{\frac1{\sqrt{2\pi}}e^{-\frac12|\hat\omega_k|^2}d\hat\omega_k},
\end{equation*}
that is, under $\mu$ the Fourier coefficients $\hat \omega_k$ are independent identically distributed
standard Gaussian variables.
As a consequence, for all $\alpha<0$, $\mu$ is supported by $\mathcal{F} L^{\infty,\alpha}$.
Looking at the laws of Fourier components under $\mu$ it is also clear why
under this measure equations \eqref{eq:2dhe} and \eqref{eq:maineq} are \emph{singular}:
the series defining a single coefficient $B_k(\omega)$ of the vector field
diverges almost surely under $\mu$.
On the other hand, the expected value under $\mu$ of each summand in the series defining $B_k(\omega)$
vanishes, which is a formal but suggestive argument supporting the invariance of $\mu$.
In fact, the argument becomes rigorous when considering Galerkin truncations of $B$,
and we will make essential use of this in the following.
The \emph{space-time} analogue of $\mu$, which we will use to define the stochastic forcing
for \eqref{eq:2dspe}, can be defined in two equivalent ways.
First, we can consider the centred Gaussian field $\xi$ indexed by $L^2([0,T], \mathcal{F} L^{2,0})$,
with $T\in [0,\infty]$, whose covariance is given by
\begin{equation*}
\expt{\xi(\Phi)\xi(\Phi')}=\brak{\Phi,\Phi'}_{L^2([0,T], \mathcal{F} L^{2,0})}.
\end{equation*}
When coupled with test functions of the form $\bm{1}_{[0,t]}(s)\phi(x,z)$, $\phi\in \S$,
$\xi$ can be regarded as the cylindrical Wiener process $W_t$ on $\mathcal{F} L^{2,0}$:
\begin{equation*}
\brak{\xi, \bm{1}_{[0,t]}\phi}=\brak{W_t,\phi}=\sum_{k\in\mathbb{N}^2_0} \hat\phi_k \beta^k_t,
\end{equation*}
the latter part being the usual Karhunen-Lo\`eve decomposition with
$(\beta^k_t)_{k\in \mathbb{N}^2_0}$ independent standard Wiener processes.
For all $\theta>0,\nu>0$, the Gaussian measure $\mu$ is the unique,
ergodic invariant measure of the infinite-dimensional Langevin's dynamics
\begin{equation}\label{eq:langevin}
\partial_t X=-\nu (-\Delta)^\theta X + \sqrt{2\nu}(-\Delta)^{\theta/2}\xi,
\end{equation}
which can be interpreted, by means of Fourier decomposition, as the system of independent one-dimensional SDEs
\begin{equation*}
d\hat X_k=-\nu |k|^{2\theta} \hat X_k dt + \sqrt{2\nu}|k|^{\theta} d\beta^k_t, \quad k \in \mathbb{N}_0^2.
\end{equation*}
For the sake of simplicity, and without loss of generality, we will set $\nu=1$ in the following.
Let us conveniently introduce a symbol for the Generator of the dynamics \eqref{eq:langevin}:
first we define cylinder functionals on $\S'$ by
\begin{equation*}
\mathcal{C} = \set{F\in L^2(\mu): \, F(\omega)=f(\hat\omega_{k_1},\dots \hat\omega_{k_r}),
\, f\in C^\infty(\mathbb{R}^r), \,k_1,\dots k_r\in \mathbb{N}^2_0, r\in\mathbb{N}},
\end{equation*}
and for $F\in\mathcal{C}$ we denote
\begin{equation}\label{eq:ougenerator}
\L_\theta F(\omega)= \sum_{i=1}^r |k_i|^{2\theta} \pa{-\hat\omega_{k_i} \partial_i f+\partial_i^2 f}.
\end{equation}
Let us also introduce the \emph{carr\'e du champ} of the diffusion operator $\L_\theta$:
for $F,G\in\mathcal{C}$,
\begin{equation}\label{eq:carreduchamp}
\mathcal{E}_\theta(F,G)(\omega)=\sum_{i=1}^n |k_i|^{2\theta} \partial_i f\partial_i g,
\end{equation}
which satisfies the Gaussian integration by parts formula
\begin{equation*}
\expt[\mu]{F \L_\theta G}=-\expt[\mu]{\mathcal{E}_\theta(F,G)}.
\end{equation*}
The above arguments finally lead us to consider the combination of dynamics \eqref{eq:2dhe} and \eqref{eq:langevin}
as a SPDE preserving $\mu$:
\begin{equation}\label{eq:maineq}
\partial_t \omega+ \nabla^\perp A(\omega)\cdot \nabla\omega
=-(-\Delta)^\theta \omega + \sqrt{2}(-\Delta)^{\theta/2}\xi.
\end{equation}
As already noticed, the nonlinear part of the dynamics
is not well defined for functions $\omega$ in the regularity regime dictated by $\mu$,
or rather, it can be given a rigorous meaning only by exploiting cancellations
due to the structure of the stochastic equation as a whole.
\begin{rmk}
In terms of $v$, the latter equation reads
\begin{equation*}
\begin{cases}
\partial_t v + v \partial_x v +\partial_x\partial_y A(v) \partial_y v + \partial_x p
=- (-\Delta)^\theta v +\partial_z \sqrt{2} (-\Delta)^{\theta/2}\xi, \\
\partial_z p = 0.
\end{cases}
\end{equation*}
The forcing term should have white noise regularity in $x$, and Brownian regularity in $y$,
although the covariance structure is a nontrivial copula of the two.
\end{rmk}
\begin{rmk}
Just as in the case of 2D Euler or stochastic Navier-Stokes equations,
the invariant measure associated to enstrophy is not able to describe
peculiar features of the fluid-dynamic model, such as turbulence phenomena.
In fact, such measures are preserved
by any flow of measure-preserving diffeomorphisms of the domain,
among which the Euler flow is a very distinguished case.
Energy ensembles should be in fact more relevant, but they are supported
on quite larger distribution spaces.
\end{rmk}
\section{Regularisation by Noise in Hyperviscous Regimes}
In this section we outline how the solution theory of \cite{GuJa13}
(known as \emph{Energy Solutions} theory in the context of stochastic Burgers and KPZ equations)
applies to our model in a sufficiently hyperviscous regime.
Computations differ from that work only by small details: we collect them in
the last section for the sake of completeness, and in the present one we only
recall the core ideas.
\subsection{Controlled Processes and Martingale Solutions}
We recall the notion of controlled process from \cite{GuJa13}.
\begin{definition}\label{def:controlledprocess}
For $\theta\geq 0$ and $T> 0$ we define the space $\mathcal{R}_{\theta,T}$ of stochastic processes
with trajectories of class $C([0,T],\S')$ such that any $\omega\in \mathcal{R}_{\theta,T}$ satisfies:
\begin{enumerate}
\item $\omega$ is stationary and for any $t\in [0,T]$, $\omega_t\sim \eta$;
\item there exists a stochastic process $\mathcal{A}$ with trajectories $C([0,T],\S')$
starting from $\mathcal{A}_0=0$ and with null quadratic variation such that,
for any $\phi\in\S$,
\begin{equation*}
\brak{\phi,\omega_t}-\brak{\phi,\omega_0}
+\int_0^t \brak{(-\Delta)^\theta\phi,\omega_s}ds-\brak{\phi,\mathcal{A}_t}=M_t(\phi)
\end{equation*}
is a martingale with respect to the filtration of $\omega$, and it has quadratic variation
$\bra{M(\phi)}_t=2t \norm{(-\Delta)^{\theta/2}\phi}^2_{\mathcal{F} L^{2,0}}$;
\item the reversed process $\tilde \omega_t=\omega_{T-t}$ satisfies condition (2)
with $\tilde \mathcal{A}_t=-\mathcal{A}_{T-t}$.
\end{enumerate}
\end{definition}
Notice that in fact elements of $\mathcal{R}_{\theta,T}$ are the couples $(\omega,\mathcal{A})$.
The forward and backward martingale equations defining the class $\mathcal{R}_{\theta,T}$
allow to obtain good \emph{a priori} estimates for nonlinear functionals of controlled process,
in a procedure by now commonly known as \emph{It\=o trick}, especially in literature
related to regularisation by noise techniques, see \cite{Fl10,FlGuPr10,BeFlGuMa19}.
In the next paragraph we detail how the It\=o trick produces good estimates on Galerkin approximations of \eqref{eq:maineq}:
the idea behind \autoref{def:controlledprocess} is to collect the features of those approximants
allowing such estimates, to form a class of processes on which the nonlinear term of \eqref{eq:maineq} is defined.
In \autoref{sec:details} we will prove the following:
\begin{lemma} \label{lem:nonlinearity}
Let $\theta > 2$, $T> 0$ and $\omega\in \mathcal{R}_{\theta,T}$. Then for every $\zeta < -1$
\begin{equation*}
\lim_{m\to \infty} \int_0^t B(\pi_m \omega_s)ds
\end{equation*}
exists as a limit in $C([0,T],\mathcal{F} L^{\infty,\zeta})$.
We denote by $\int_0^t B(\omega_s)ds$ the limiting process.
\end{lemma}
The latter lemma shows that the nonlinear functional $B(\omega)$ can be defined for $\omega\in\mathcal{R}_{\theta,T}$
as a distribution in both space \emph{and} time.
Let us observe that Fourier truncation $\pi_m$ in \autoref{lem:nonlinearity} can in fact be replaced
with a large class of mollifiers, the limit being independent of such choice:
for the sake of keeping the exposition simple, we refrain from going into details.
We can now give a notion of martingale solution to \eqref{eq:maineq}.
\begin{definition}\label{def:martsolution}
Let $\theta> 2$, $T> 0$ and $\omega\in \mathcal{R}_{\theta,T}$. We say that $\omega$
is a \emph{martingale solution} to \eqref{eq:maineq} if it holds almost surely, for any $t\in[0,T]$,
\begin{equation*}
\mathcal{A}_t=\int_0^t B(\omega_s)ds.
\end{equation*}
The solution is \emph{pathwise unique} if, for any two controlled processes
$\omega,\tilde\omega\in\mathcal{R}_{\theta,T}$ defined on the same probability space,
satisfying conditions (2) and (3) of \autoref{def:martsolution} with the same martingales and
with $\omega_0=\tilde\omega_0$ almost surely, then almost surely, for all $t\in [0,T]$, $\omega_t=\tilde\omega_t$.
\end{definition}
The following is the main result of the paper: its proof will be given in \autoref{sec:details}.
\begin{thm} \label{thm:main}
Let $T>0$. For any $\theta>2$ there exists a solution to \eqref{eq:maineq} in the sense of \autoref{def:martsolution}.
Moreover, for $\theta>3$ the solution is pathwise unique.
\end{thm}
\subsection{Galerkin Approximation and the It\=o Trick}
Let us introduce approximating processes $(\omega^m)_{m\in\mathbb{N}}$ by their Fourier coefficients dynamics:
for $k\in\mathbb{N}^2_0$,
\begin{align}\label{eq:galerkin}
d \hat\omega_k^m &= B^m_k(\omega^m) dt - |k|^{2\theta} \hat\omega^m_k dt + \sqrt{2}|k|^\theta d\beta^k_t,
\end{align}
where $B^m(\omega)=\pi_m B(\pi_m\omega)$, and $\omega^m_0\sim \mu$.
The vector field $B^m$ satisfies
\begin{align}\label{eq:divergencefree}
\div_\mu B^m(\omega)
&= \div_\mu \sum_{\substack{k \in \mathbb{N}^2_0, \\|k|\leq m}}\sum_{\substack{h \in \mathbb{Z}^2_0, \\|h|\leq m}}
\hat\omega_h \hat\omega_{k- h} \frac{k \cdot h^{\perp}}{h_2^2} e_k\\ \nonumber
&= \sum_{\substack{k \in \mathbb{N}^2_0, \\|k|\leq m}}\sum_{\substack{h \in \mathbb{Z}^2_0, \\|h|\leq m}} \pa{\partial_{\hat\omega_k}\pa{\hat\omega_h \hat\omega_{k- h}}
-\hat\omega_h \hat\omega_{k- h} \hat\omega_k} \frac{k \cdot h^{\perp}}{h_2^2}=0.
\end{align}
As a consequence, \eqref{eq:galerkin} has a unique, (probabilistically) strong, global in time solution since
$\mu$ is preserved by the linear part of the dynamics, and thus \cite[Theorem 3.2]{Cr83} applies.
In the following, we denote by $\mathbb{P}^m_\mu$ the law of $\omega^m$ in $C(\mathbb{R}_+,\S')$.
By It\=o formula, for any cylinder function $F\in\mathcal{C}$,
$F(\omega) = f(\hat\omega_{k_1},\dots, \hat \omega_{k_n})$, it holds
\begin{equation*}
d F(\omega^m)=\L_\theta F(\omega^m)dt+ \mathcal{G}^m F(\omega^m)dt
+ \sum_{i=1}^n \partial_i f(\hat\omega_{k_1}^m,\dots,\hat \omega_{k_n}^m) \sqrt{2}|k_i|^\theta d\beta^{k_i}_t,
\end{equation*}
where $\L_\theta$ is defined in \eqref{eq:langevin} and
\begin{equation*}
\mathcal{G}^m F(\omega)=\sum_{i=1}^n \partial_i f(\hat\omega_{k_1},\dots, \hat\omega_{k_n}) B^m_{k_i}(\omega) dt.
\end{equation*}
In other words, the process
\begin{equation}\label{eq:mart1}
M^{F,m}_t= F(\omega_t^m) - F(\omega_0^m) - \int_0^t \L_\theta F (\omega^m_s) ds
- \int_0^t \mathcal{G}^m F (\omega^m_s) ds
\end{equation}
is a martingale with quadratic variation
\begin{equation*}
[M^{F,m}]_t = 2 \int_0^t \sum_{i=1}^n |k_i|^{2\theta} \pa{\partial_i f(\hat\omega_{k_1}^m,\dots,\hat \omega_{k_n}^m)}^2 ds
= 2 \int_0^t \mathcal{E}_\theta(F)(\omega_s^m) ds.
\end{equation*}
Let us point out that, thanks to the hydrodynamic form of the nonlinearity,
$\mathcal{G}^m$ is a skew-symmetric operator with respect to $\mu$:
indeed, since
\begin{equation*}
\brak{\omega,B^m(\omega)}=\brak{\omega, \pi_m(\nabla^\perp A(\pi_m \omega)\cdot \nabla\pi_m\omega)}=0,
\end{equation*}
Gaussian integration shows that
\begin{equation*}
\expt[\mu]{F\mathcal{G}^m G}=-\expt[\mu]{G\mathcal{G}^m F}, \quad \forall F,G\in\mathcal{C}.
\end{equation*}
Let us then consider the reversed process $\tilde \omega^m_t=\omega^m_{T-t}$, for a fixed time horizon $T>0$:
$\tilde \omega^m$ is a Markov process whose generator is the adjoint of the one of $\omega^m$,
that is $\L_\theta-\mathcal{G}^m$. The process
\begin{equation}\label{eq:mart2}
\tilde M^{F,m}_t= F(\tilde \omega_t^m) - F(\tilde \omega_0^m) - \int_0^t \L_\theta F (\tilde \omega^m_s) ds
- \int_0^t \mathcal{G}^m F (\tilde \omega^m_s) ds
\end{equation}
is thus another martingale with quadratic variation $2 \int_0^t \mathcal{E}_\theta(F)(\omega_s^m) ds$.
To sum up, we have shown that $\omega^m$ is a controlled process in the sense of \autoref{def:controlledprocess}.
The trick is now to sum the martingale identities \eqref{eq:mart1}, \eqref{eq:mart2}
for $\omega^m$ and $\tilde{\omega}^m$: in doing so the nonlinear skew symmetric part,
together with boundary terms, is canceled,
leaving us with martingales term and the symmetric Ornstein-Uhlenbeck generator,
\begin{align*}
\tilde{M}^{F,m}_{T-t} - \tilde{M}^{F,m}_T - M^{F,m}_t = 2 \int_0^t \L_\theta F (\omega^m_s) ds.
\end{align*}
Burkholder-Davis-Gundy inequality thus provides, together with stationarity of $\omega^m$,
the following powerful estimate: for $p\geq 1$ there exists a constant $C_p>0$ only depending on $p$ such that
for all $F\in\mathcal{C}$
\begin{equation} \label{eq:itotrick}
\expt[\mathbb{P}^m_\mu]{\sup_{t \in [0,T]} \abs{\int_0^t \L_\theta F (\omega_s^m) ds}^p}
\leq C_p \sqrt T \expt[\mu]{\abs{\mathcal{E}_\theta F}^{p/2}}.
\end{equation}
\begin{rmk}
As already observed, \autoref{def:controlledprocess} actually collects the elements we used to
establish \eqref{eq:itotrick}; indeed, the latter holds more generally for any controlled process $\omega\in\mathcal{R}_{\theta,T}$.
\end{rmk}
Inequality \eqref{eq:itotrick} provides good estimates on time integrals of observables for $\omega^m$,
provided that we are able to solve a Poisson equation in Gaussian space.
The main aim are clearly bounds to establish the limit in \autoref{lem:nonlinearity},
which can be obtained by means of \eqref{eq:itotrick} by solving
\begin{equation*}
\L_\theta H^m_k(\omega)=B^m_k(\omega).
\end{equation*}
Since $B^m_k(\omega)$ belongs to the second chaos in the Wiener chaos decomposition of $L^2(\mu)$,
and since $\L_\theta$ is diagonalised by such decomposition, it is easy to obtain the explicit solution
\begin{equation}\label{eq:soluzh}
H_k^m (\omega) = - \chi_{\{|k| \leq m\}} \sum_{\substack{h,\ell \in \mathbb{Z}^2_0, \\ h + \ell = k \\ |h|,|\ell|\leq m}}
\hat\omega_h \hat\omega_{\ell} \frac{\ell \cdot h^{\perp}}{h_2^2 (|h|^2 +|\ell|^2)^{\theta}}.
\end{equation}
The computation is completely analogous to \cite[Section 3]{GuJa13}, to which we refer.
With the latter expression at hand, one only needs to estimate moments of $\mathcal{E}_\theta(H_k^m)$:
we report such computation in the next section, together with some variants from which
\autoref{thm:main} follows.
\section{Proof of Main Result} \label{sec:details}
We complete in this Section the proof of \autoref{thm:main}.
First, by means of the It\=o trick estimate \eqref{eq:itotrick}
we obtain bounds on Galerkin approximations: the last two paragraphs are then
devoted to existence and uniqueness of martingale solutions.
In this section, the symbol $\lesssim$ denotes inequality up to a positive multiplicative
constant uniform in the involved parameters.
\subsection{Controlling the Nonlinear Term} \label{subsec:apriori}
We start from the expression \eqref{eq:soluzh} for $H^m_k$
to obtain estimates on the nonlinear term in \eqref{eq:maineq}.
By definition of $\mathcal{E}_\theta$, \eqref{eq:carreduchamp}, one has
\begin{equation*}
\mathcal{E}_\theta(H^m_k)(\omega)=\chi_{\{|k| \leq m\}} \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} |h|^{2\theta} \abs{\frac{2(k-h) \cdot h^{\perp}}{h_2^2(|k-h|^2+|h|^2)^\theta} \hat\omega_{k-h}}^2,
\end{equation*}
therefore, taking expectation with respect to $\mu$, for $|k|\leq m$
\begin{align*}
\mathbb{E}_\mu \left[\mathcal{E}_\theta(H^m_k) \right]
&= \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} |h|^{2\theta} \abs{\frac{2(k-h) \cdot h^{\perp}}{h_2^2(|k-h|^2+|h|^2)^\theta}}^2 \\
&\lesssim \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} \frac{|k|^2 |h|^{2+2\theta}}{|k-h|^{4\theta}+|h|^{4\theta}}
\lesssim \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} \frac{|k|^2 |h|^{2}}{|k-h|^{2\theta}+|h|^{2\theta}}.
\end{align*}
Now we use the fact that, for $\theta > 2$,
\begin{equation*}
\sum_{h \in \mathbb{Z}^2_0} \frac{| h |^2}{| k - h |^{2 \theta}+| h |^{2 \theta}} \lesssim
| k |^{4 - 2 \theta}
\end{equation*}
(see \cite[Lemma 16]{GuTu19}) to deduce the following estimate uniformly in $m$:
\begin{equation*}
\mathbb{E}_\mu \left[\mathcal{E}_\theta(H^m_k) \right] \lesssim | k |^{6 - 2 \theta}.
\end{equation*}
Similarly, increments are controlled by
\begin{equation*}
\sup_{n>m} \mathbb{E}_\mu [ \mathcal{E}_{\theta} (H_k^n-H_k^m)]
\lesssim | k |^2 m^{ 4-2\theta} .
\end{equation*}
With these estimates at hand, by means of \eqref{eq:itotrick} and Gaussian hypercontractivity,
one can prove the following estimates on the nonlinear term of \eqref{eq:maineq}.
\begin{lemma} \label{lem:apriori1}
Let $G^m_t \coloneqq \int_0^t B(\pi_m \omega_s)ds$ and
${\mathbb{P}^m_\mu}$ be the distribution of the stationary solution of \eqref{eq:galerkin} described above.
For any $n>m$ we have the following estimates:
\begin{gather}
\left\| \sup_{t \in [0,T]} \left( G^{m}_t\right)_k \right\|_{L^p({\mathbb{P}^m_\mu})} \lesssim |k|^{3 - \theta} T^{1/2}, \label{eq:apriori2}\\
\left\| \sup_{t \in [0,T]} \left( G^{n}_t\right)_k - \left( G^{m}_t\right)_k \right\|_{L^p({\mathbb{P}^m_\mu})} \lesssim |k| T^{1/2} m^{2-\theta}. \label{eq:apriori3}
\end{gather}
\end{lemma}
\begin{lemma} \label{lem:apriori2}
Let $\tilde{G}^m_t \coloneqq \int_0^t e^{-(t-s)(-\Delta)^\theta} B(\pi_m \omega_s)ds$ and
${\mathbb{P}^m_\mu}$ as above.
For any $m$ fixed, $n>m$, $s,t \in [0,T]$, $s<t$, we have the following estimates:
\begin{gather}
\left\| \sup_{t \in [0,T]} \left( \tilde{G}^{m}_t\right)_k \right\|_{L^p({\mathbb{P}^m_\mu})} \lesssim |k|^{3 - 2\theta}, \label{eq:apriori5}\\
\left\| \sup_{t \in [0,T]} \left( \tilde{G}^{n}_t\right)_k - \left( \tilde{G}^{m}_t\right)_k \right\|_{L^p({\mathbb{P}^m_\mu})} \lesssim |k|^{-1} m^{4-2\theta}, \label{eq:apriori6}\\
\sup_{m}\left\| \left( \tilde{G}^{m}_t\right)_k - \left( \tilde{G}^{m}_s\right)_k \right\|_{L^p({\mathbb{P}^m_\mu})} \lesssim |k|^{3-2\theta+2\varepsilon\theta} (t-s)^{\varepsilon},\label{eq:aprioricont}
\end{gather}
where the last inequality is meant to hold for $\varepsilon>0$ small enough.
\end{lemma}
Proofs of the previous estimates follow along the lines of Lemma 5,
Lemma 6 and Corollary 1 of \cite{GuJa13}, so we refrain from repeating them here.
\subsection{Existence for $\theta>2$}
We first prove \autoref{lem:nonlinearity}, which gives a meaning to the nonlinear term of \eqref{eq:maineq}.
The result easily follows from \autoref{lem:apriori1}.
\begin{proof}[Proof of \autoref{lem:nonlinearity}]
Let $G^m_t \coloneqq \int_0^t B(\pi_m \omega_s)ds$.
It is clear that $G^m$ is a random process with values in $C([0,T],\mathcal{F} L^{\infty,\zeta})$ for every $m$ and $\zeta \in \mathbb{R}$.
Since $\theta>2$, \eqref{eq:apriori3} gives for any $p$ and $n>m$
\begin{equation*}
\mathbb{E}_{\mathbb{P}^m_\mu} \left[ \sum_k |k|^{\zeta p} \left| \sup_{t \in [0,T]} \left( G^{n}_t\right)_k - \left( G^{m}_t\right)_k \right|^p \right] \to 0
\end{equation*}
as $m \to \infty$ whenever $\zeta <-2/p -1$.
Taking $p$ sufficiently large, for any $\zeta < -1$ we obtain the almost sure uniform
convergence of $G^m$ in the space $C([0,T],\mathcal{F} L^{\infty,\zeta})$.
\end{proof}
We are now ready to prove the first part of \autoref{thm:main}.
The proof relies on \autoref{subsec:apriori} and Skorokhod Theorem.
\begin{proof}[Proof of \autoref{thm:main}, existence]
Let us consider the mild formulation of \eqref{eq:galerkin}:
\begin{align} \label{eq:Galerkin}
\omega^m_t &= e^{-t(-\Delta)^\theta} \omega_0 + \int_0^t e^{-(t-s)(-\Delta)^\theta} B^m(\omega^m_s) ds \\ \nonumber
&\quad + \sqrt{2}(-\Delta)^{\theta/2} \int_0^t e^{-(t-s)(-\Delta)^\theta} d\beta_s,
\end{align}
where $\omega_0 \sim \mu$ and $\beta$ is a cylindrical Wiener process on $\mathcal{F} L^{2,0}$. Define
\begin{equation*}
\mathcal{A}^m_t \coloneqq \int_0^t B^m(\omega^m_s) ds , \quad \tilde{\mathcal{A}}^m_t \coloneqq \int_0^t e^{-(t-s)(-\Delta)^\theta} B^m(\omega^m_s) ds.
\end{equation*}
We prove that, for $\varepsilon>0$ sufficiently small and $\zeta < -1$, the laws of the processes $\left( \omega^m , \mathcal{A}^m, \tilde{\mathcal{A}}^m, \beta \right)_m$ are tight in $C([0,T],\mathcal{X})$, where
\begin{equation*}
\mathcal{X} \coloneqq \mathcal{F} L^{\infty,\zeta} \times \mathcal{F} L^{\infty,\theta - 3 - \varepsilon} \times \mathcal{F} L^{\infty,2\theta - 3 -\varepsilon} \times\mathcal{F} L^{\infty,-\varepsilon}.
\end{equation*}
By Borel-Cantelli theorem applied to Fourier expansions, the law $\mu$ is concentrated on $\mathcal{F} L^{\infty,-\varepsilon}$,
and the stochastic convolution takes values in $C([0,T],\mathcal{F} L^{\infty,\theta -\varepsilon})$ for every $\varepsilon>0$.
Tightness in this space is given by Fernique Theorem.
Tightness of $(\tilde{\mathcal{A}}^m)_m$ descends from \autoref{eq:aprioricont}
and tightness of $({\mathcal{A}}^m)_m$ descends from \autoref{eq:apriori2}.
Hence, by a standard application of Prokhorov Theorem and Skorokhod Theorem,
we deduce the a.s. convergence, up to a subsequence and a change of the underlying probability space,
of $\left( \omega^m , \mathcal{A}^m, \tilde{\mathcal{A}}^m, \beta \right)$ towards some random variable
$\left( \omega , \mathcal{A}, \tilde{\mathcal{A}}, \beta \right)$ in $C([0,T],\mathcal{X})$ which satifies
\begin{align*}
\omega_t &= e^{-t(-\Delta)^\theta} \omega_0 + \tilde{\mathcal{A}}_t + \sqrt{2}(-\Delta)^{\theta/2} \int_0^t e^{-(t-s)(-\Delta)^\theta} d\beta_s \\
&= \omega_0 + \int_0^t (-\Delta)^\theta \omega_s ds + \mathcal{A}_t + \sqrt{2}(-\Delta)^{\theta/2} \beta_t.
\end{align*}
Now it is easy to check that $\omega \in \mathcal{R}_{\theta,T}$, see \cite{GuJa13} for details.
\end{proof}
\subsection{Uniqueness for $\theta>3$}
\begin{proof}[Proof of \autoref{thm:main}, uniqueness]
We have constructed a sequence of $\omega^m$ converging a.s. to a solution $\omega$ as random variables in $C([0,T], \mathcal{F} L^{\infty,\zeta})$ for every $\zeta < -1$. Here we prove uniqueness, which comes from an estimate on the quantity $\pi_m(\omega^m - \omega)$ in a suitable space, where $\omega \in \mathcal{R}_{\theta,T}$ is a controlled solution to \eqref{eq:maineq} and $\omega^m$ is its Galerkin approximation defined by \eqref{eq:Galerkin}. In particular, we prove that $\pi_m(\omega^m - \omega)$ converges a.s. to zero in the space $C([0,T], \mathcal{F} L^{\infty,\xi})$, for suitable $\xi > \zeta$. This would conclude the proof by uniqueness at the level of the Galerkin truncations.
It is easy to see that
\begin{align*}
\pi_m(\omega^m_t - \omega_t) \coloneqq \delta_t^m &= \int_0^t e^{-(t-s)(-\Delta)^{\theta}} (B^m(\omega^m_s) - \pi_m B(\omega_s)) ds \\
&= \int_0^t e^{-(t-s)(-\Delta)^{\theta}} (B^m(\omega^m_s) - B^m(\omega_s)) ds \\
&+ \int_0^t e^{-(t-s)(-\Delta)^{\theta}} (B^m(\omega_s) - \pi_m B(\omega_s)) ds \\
&= \alpha^m_t + \gamma^m_t.
\end{align*}
for every $m$, and thus for every $\xi$
\begin{align*}
\sup_k \sup_{t \in [0,T]} |k|^\xi |(\delta^m_t)_k| \leq \sup_k \sup_{t \in [0,T]} |k|^\xi |(\alpha^m_t)_k| + \sup_k \sup_{t \in [0,T]} |k|^\xi |(\gamma^m_t)_k|.
\end{align*}
By \eqref{eq:apriori5}, \eqref{eq:apriori6} and interpolation, $\gamma^m$ satisfies
\begin{equation*}
\left\| \sup_{t \in [0,T]} \left| (\gamma^m_t)_k \right|\right\|_{L^{p}({\mathbb{P}^m_\mu})} \lesssim |k|^{3-2\theta+\varepsilon} m^{-\varepsilon},
\end{equation*}
and therefore for every $\xi< 2\theta - 3$ we have
\begin{equation*}
\sup_k \sup_{t \in [0,T]} |k|^\xi |(\gamma^m_t)_k| \to 0 \mbox{ a.s. for } m \to \infty.
\end{equation*}
On the other hand, since
\begin{equation*}
\left| B^m(\omega^m_s) - B^m(\omega_s) \right| \lesssim \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} |k||h||(\omega^m_s + \omega_s)_h| |(\omega^m_s - \omega_s)_{k-h}|,
\end{equation*}
we obtain the following bound on $\alpha^m$:
\begin{align*}
\sup_{t \in [0,T]} |k|^\xi |(\alpha^m_t)_k| \lesssim &|k|^\xi \sup_h \sup_{t \in [0,T]} |h|^\xi |(\delta^m_t)_h| \\
&\times \sup_{t \in [0,T]} \left| \int_0^t e^{-(t-s)|k|^{2\theta}} \sum_{\substack{h\in \mathbb{Z}^2_0,\\|h|\leq m}} |h|^{1-\xi}|k||(\omega^m_s + \omega_s)_{k-h}| ds\right|.
\end{align*}
If $\xi>3$ the series $\sum_{h \in \mathbb{Z}^2_0}|h|^{1-\xi}$ converges, therefore by H\"older inequality
\begin{gather*}
\left| \int_0^t e^{-(t-s)|k|^{2\theta}} \sum_{|h| \leq m} |h|^{1-\xi}|(\omega^m_s + \omega_s)_{k-h}| ds\right| \\
\lesssim \left| \int_0^t\sum_{|h| \leq m} |h|^{1-\xi} e^{-p'(t-s)|k|^{2\theta}} ds\right|^{1/p'}
\left| \int_0^t \sum_{|h| \leq m} |h|^{1-\xi}|(\omega^m_s + \omega_s)_{k-h}|^p ds\right|^{1/p} \\
\lesssim |k|^{-2\theta/p'} \left| \int_0^t \sum_{|h| \leq m} |h|^{1-\xi}|(\omega^m_s + \omega_s)_{k-h}|^p ds\right|^{1/p}.
\end{gather*}
Taking $p'\to 1$ such that $\xi + 1 -2\theta/p' < 0$ and using the fact that $\omega^m$ and $\omega$ have marginals $\sim \mu$, we finally get
\begin{equation*}
\sup_k \sup_{t \in [0,T]} |k|^\xi |(\delta^m_t)_k| \to 0 \mbox{ a.s. for } m \to \infty,
\end{equation*}
for every $3<\xi<2\theta - 3$, which corresponds to the additional contraint $\theta>3$. The proof is complete.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,161 |
Q: Kubernetes Worker Node in Status NotReady I have been trying to setup a K8s cluster on a set of Raspberry Pi's. Here is a link to my GitHub page that describes the whole set up:
https://github.com/joesan/plant-infra/blob/master/pi/README.md
I'm now stuck with the last step where I join my worker nodes with the master. I did issue the join command on the worker node, but after that I check the nodes in the master and I get to see the following:
pi@k8s-master-01:~ $ kubectl get nodes
NAME STATUS ROLES AGE VERSION
k8s-master-01 Ready master 56m v1.9.6
k8s-worker-01 NotReady <none> 26m v1.9.6
k8s-worker-02 NotReady <none> 6m v1.9.6
The question is, do I need to install the container network like weave also on the worker nodes?
Here is the log file from the worker node:
pi@k8s-worker-02:~ $ journalctl -u kubelet
-- Logs begin at Thu 2016-11-03 17:16:42 UTC, end at Tue 2018-05-01 11:35:54 UTC. --
May 01 11:27:28 k8s-worker-02 systemd[1]: Started kubelet: The Kubernetes Node Agent.
May 01 11:27:30 k8s-worker-02 kubelet[334]: I0501 11:27:30.995549 334 feature_gate.go:226] feature gates: &{{} map[]}
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.005491 334 controller.go:114] kubelet config controller: starting controller
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.005584 334 controller.go:118] kubelet config controller: validating combination of defaults and flags
May 01 11:27:31 k8s-worker-02 kubelet[334]: W0501 11:27:31.052134 334 cni.go:171] Unable to update cni config: No networks found in /etc/cni/net.d
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.084480 334 server.go:182] Version: v1.9.6
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.085670 334 feature_gate.go:226] feature gates: &{{} map[]}
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.092807 334 plugins.go:101] No cloud provider specified.
May 01 11:27:31 k8s-worker-02 kubelet[334]: I0501 11:27:31.110132 334 certificate_store.go:130] Loading cert/key pair from ("/var/lib/kubelet/pki/kubelet-client.crt", "/var/lib/
May 01 11:27:39 k8s-worker-02 kubelet[334]: E0501 11:27:39.905417 334 machine.go:194] failed to get cache information for node 0: open /sys/devices/system/cpu/cpu0/cache: no suc
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.911993 334 server.go:428] --cgroups-per-qos enabled, but --cgroup-root was not specified. defaulting to /
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.914203 334 container_manager_linux.go:242] container manager verified user specified cgroup-root exists: /
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.914272 334 container_manager_linux.go:247] Creating Container Manager object based on Node Config: {RuntimeCgroupsName
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.914895 334 container_manager_linux.go:266] Creating device plugin manager: false
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.919031 334 kubelet.go:291] Adding manifest path: /etc/kubernetes/manifests
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.919197 334 kubelet.go:316] Watching apiserver
May 01 11:27:39 k8s-worker-02 kubelet[334]: E0501 11:27:39.935754 334 reflector.go:205] k8s.io/kubernetes/pkg/kubelet/config/apiserver.go:47: Failed to list *v1.Pod: Get https:/
May 01 11:27:39 k8s-worker-02 kubelet[334]: E0501 11:27:39.937449 334 reflector.go:205] k8s.io/kubernetes/pkg/kubelet/kubelet.go:480: Failed to list *v1.Node: Get https://192.16
May 01 11:27:39 k8s-worker-02 kubelet[334]: E0501 11:27:39.937492 334 reflector.go:205] k8s.io/kubernetes/pkg/kubelet/kubelet.go:471: Failed to list *v1.Service: Get https://192
May 01 11:27:39 k8s-worker-02 kubelet[334]: W0501 11:27:39.948764 334 kubelet_network.go:139] Hairpin mode set to "promiscuous-bridge" but kubenet is not enabled, falling back t
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.949871 334 kubelet.go:577] Hairpin mode set to "hairpin-veth"
May 01 11:27:39 k8s-worker-02 kubelet[334]: W0501 11:27:39.951008 334 cni.go:171] Unable to update cni config: No networks found in /etc/cni/net.d
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.952122 334 client.go:80] Connecting to docker on unix:///var/run/docker.sock
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.952976 334 client.go:109] Start docker client with request timeout=2m0s
May 01 11:27:39 k8s-worker-02 kubelet[334]: W0501 11:27:39.959045 334 cni.go:171] Unable to update cni config: No networks found in /etc/cni/net.d
May 01 11:27:39 k8s-worker-02 kubelet[334]: W0501 11:27:39.971616 334 cni.go:171] Unable to update cni config: No networks found in /etc/cni/net.d
May 01 11:27:39 k8s-worker-02 kubelet[334]: I0501 11:27:39.971765 334 docker_service.go:232] Docker cri networking managed by cni
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.002411 334 docker_service.go:237] Docker Info: &{ID:25GN:65LU:UXAR:CUUY:DOQH:ST4A:IQOE:PIDR:BKYC:UVJH:LI5H:HQSG Contai
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.002766 334 docker_service.go:250] Setting cgroupDriver to cgroupfs
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.058142 334 remote_runtime.go:43] Connecting to runtime service unix:///var/run/dockershim.sock
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.098202 334 kuberuntime_manager.go:186] Container runtime docker initialized, version: 18.04.0-ce, apiVersion: 1.37.0
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.110512 334 server.go:755] Started kubelet
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.112242 334 kubelet_node_status.go:273] Setting node annotation to enable volume controller attach/detach
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.114014 334 server.go:129] Starting to listen on 0.0.0.0:10250
May 01 11:27:40 k8s-worker-02 kubelet[334]: E0501 11:27:40.114962 334 kubelet.go:1281] Image garbage collection failed once. Stats initialization may not have completed yet: fai
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.133665 334 server.go:299] Adding debug handlers to kubelet server.
May 01 11:27:40 k8s-worker-02 kubelet[334]: E0501 11:27:40.141790 334 event.go:209] Unable to write event: 'Post https://192.168.0.101:6443/api/v1/namespaces/default/events: dia
May 01 11:27:40 k8s-worker-02 kubelet[334]: E0501 11:27:40.175654 334 container_manager_linux.go:583] [ContainerManager]: Fail to get rootfs information unable to find data for
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.175765 334 fs_resource_analyzer.go:66] Starting FS ResourceAnalyzer
May 01 11:27:40 k8s-worker-02 kubelet[334]: I0501 11:27:40.176241 334 volume_manager.go:247] Starting Kubelet Volume Manager
lines 1-41
Any idea as to why my worker nodes show up as NotReady?
EDIT: I traced the error with the kubectl describe nodes command:
Name: k8s-worker-02
Roles: <none>
Labels: beta.kubernetes.io/arch=arm
beta.kubernetes.io/os=linux
kubernetes.io/hostname=k8s-worker-02
Annotations: node.alpha.kubernetes.io/ttl=0
volumes.kubernetes.io/controller-managed-attach-detach=true
Taints: <none>
CreationTimestamp: Tue, 01 May 2018 11:26:50 +0000
Conditions:
Type Status LastHeartbeatTime LastTransitionTime Reason Message
---- ------ ----------------- ------------------ ------ -------
OutOfDisk False Tue, 01 May 2018 11:40:17 +0000 Tue, 01 May 2018 11:26:43 +0000 KubeletHasSufficientDisk kubelet has sufficient disk space available
MemoryPressure False Tue, 01 May 2018 11:40:17 +0000 Tue, 01 May 2018 11:26:43 +0000 KubeletHasSufficientMemory kubelet has sufficient memory available
DiskPressure False Tue, 01 May 2018 11:40:17 +0000 Tue, 01 May 2018 11:26:43 +0000 KubeletHasNoDiskPressure kubelet has no disk pressure
Ready False Tue, 01 May 2018 11:40:17 +0000 Tue, 01 May 2018 11:26:43 +0000 KubeletNotReady runtime network not ready: NetworkReady=false reason:NetworkPluginNotReady message:docker: network plugin is not ready: cni config uninitialized. WARNING: CPU hardcapping unsupported
How can I solve this?
A: *
*Go to your master node in master node go to /etc/cni/net.d
*In that folder you will find a cni config file
*Copy that file and upload the same file in /etc/cni/net.d of your worker node
*your worker node will be ready in 1 to 2 mins
*if this is not working add a comment
A: You need to try the below solutions and see if any one of the them would be helpful
*
*Check your firewalld status.If it is Running Stop it.
*Check your Kube-dns status.Sometimes it may be down or throwing some error
*Try to reload and Restart your Kubelet
A: I had the same issue and like some I have the kiss of death when it comes to installs on standard perfectly normal equipment so NONE of the items anywhere helped until I rejoined the worker nodes to the master.
My install is on three physical machines. One master and two workers. All needed reboots.
I wasn't expecting it to work but it did. Probably won't work for you but if nothing else works, maybe give this a shot. you will need your join token, which you probably don't have, but I'll show you how to get it so you DON'T have to go to another set of pages and search for it:
sudo kubeadm token list
copy the TOKEN field data, the output looks like this (no, that's not my real one):
TOKEN
ow3v08ddddgmgzfkdkdkd7 18h 2018-07-30T12:39:53-05:00 authentication,signing The default bootstrap token generated by 'kubeadm init'. system:bootstrappers:kubeadm:default-node-token
THEN join the cluster here. Master node IP is the real IP address of your machine:
sudo kubeadm join --token <YOUR TOKEN HASH> <MASTER_NODE_IP>:6443 --discovery-token-unsafe-skip-ca-verification
A: I managed to fix this! This is how I did it:
$sudo nano /etc/systemd/system/kubelet.service.d/10-kubeadm.conf
After that I just commented the line that contained the KUBELET_NETWORK_ARGS
After that I just rebooted the system and I can see the node being in Ready status!
A: You can also install Flannel CNI plugin as below
git clone https://github.com/containernetworking/cni
cd cni
git checkout v0.5.2
./build.sh
cp bin/* /opt/cni/bin
mkdir -p /etc/cni/net.d
Download kube-flannel.yml and kube-flannel-rbac.yml from here
then try this and your node should be Ready
kubectl apply -f kube-flannel.yml -f kube-flannel-rbac.yml
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,649 |
The Arawak Cay port is projecting to generate between $4.918 million to $5.761 million in operating income between 2013-2015, according to local newspaper The Tribune. Unveiling the $10 million initial public offering for the Nassau Container Port's holding company, Arawak Port Development (APD) chief executive, Michael Maura Jr, said the company's annual gross revenues could fluctuate between $25-27 million over the next three years.
Dec. 19/11 (NATARIO McKENZIE) THE ARAWAK CAY port is projecting to generate between $4.918 million to $5.761 million in operating income between 2013-2015, its chief executive saying it expected to enjoy a significant boost in container throughput volumes from the $2.6 billion Baha Mar project.
Unveiling the $10 million initial public offering (IPO) for the Nassau Container Port's holding company, Arawak Port Development (APD) chief executive, Michael Maura Jr, said the company's annual revenues could fluctuate between $25 and 27 million over the next three years.
Mr Maura said the Government, in the Memorandum of Understanding (MoU) signed with the 19 private sector shipping industry companies, had sought to strike a balance between investor returns and minimising shipping costs and associated tariffs for the Bahamian business community and general public. The MoU, he said, permits APD to earn an investment return rate of between 10 to 12 per cent.
"Income streams are driven by the number of containers landing, the amount of bulk material coming from cement vessels, the break bulk cargo and the number of cars coming to Nassau," Mr Maura said.
"We, at the end of the day, had to come up with a revenue number which, multiplied by cargo volumes, allowed us to cover our operating costs and provide the public and our owners essentially with a 10 per cent return. Our gross revenues will fluctuate between $25 million, going into $27 million, over next three years. Our operating income is in the range of $5 to $6 million.
He added: "We are actually using just a little shy of 60,0000 TEUs (Twenty-foot Equivalent Unit or a 20-foot shipping container) as our first full year of operations.
"I believe that when Baha Mar kicks in and it really starts moving, we are going to see our volumes exceed that. The Government, through the MOU, has permitted the company to earn a rate of return between 10 and 12 per cent. That rate over return is based on traffic," Mr Maura explained.
"In good times, what is likely to happen is we see a spike in volume. When they increase, it means we are generating more revenue. In those periods of time, what our traffic amendment model does is it looks to bring our returns back down to a level of 10 to 12 per cent, so the company does not generate super profits.
Mr Maura told Tribune Business that discussions on how best to make 20 per cent of the company available to Bahamian institutional and retail investors led to the delay of the $10 million IPO's launch.
APD and its financial advisers/placement agents, CFAL and Providence Advisors, had initially hoped to launch the IPO in late September/early October. But Mr Maura told Tribune Business that discussions between the Government and private sector on how to structure the IPO had delayed it.
"Basically what was happening was that the Government and the private sector shareholders were having discussions on how to make the 20 per cent of the company available," Mr Maura said.
Mr Maura noted that when APD was incorporated two years ago, it had five million in authorised shares, but only 4 million was issued, leaving one million shares unused.
"What ended up happening is the company has issued that million and when people buy those shares it represents 20 per cent equity in the company," Mr Maura said.
When asked as to how he felt investor appetite would be towards the offering, Mr Maura said that while many persons may have already decided on how to spend their cash this Christmas, APD had decided to leave the offering open until the end of Januaryn 2012.
"There are some who have already figured how to spend their money this Christmas," he conceded.
In terms of the company's proposed private placement, scheduled for next year, Mr Maura told Tribune Business: "Based on the financial structure of the company, we have $40 million in equity and a $43 million bridge loan facility makes up an $83 million budget.
The $43 million will be targeted at specific investors only. It will replace the $43 million bank loan intended to finance the Arawak Cay port's construction.
This is an excerpt from The Tribune as it appeared on December 20, 2011. For updates or to read the current version of this post in its entirety, please click here.
Wells Fargo & Company (NYSE: WFC) and EverKey Global Partners (EverKey) announced last week that they have signed a definitive agreement for Wells Fargo to acquire EverKey, an investment boutique that offers global equity strategies to institutional clients, reports insurancenewsnet.com. EverKey has offices in The Bahamas and New York and manages global equity strategies for institutional clients in Japan and North America.
The Arawak Cay port will add an extra $800 million to the gross domestic product, and more than 1,200 full-time jobs, over an initial 20-year period, by freeing up 20 acres of prime land in downtown Nassau for redevelopment, reports local paper The Tribune. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,532 |
{"url":"https:\/\/solvedlib.com\/statement-of-cash-flowsindirect-method-the,46394","text":"# Statement of Cash Flows\u2014Indirect Method The comparative balance sheet of Merrick Equipment Co. for December 31,...\n\n###### Question:\n\nStatement of Cash Flows\u2014Indirect Method\n\nThe comparative balance sheet of Merrick Equipment Co. for December 31, 20Y9 and 20Y8, is as follows:\n\n Dec. 31, 20Y9 Dec. 31, 20Y8 Assets Cash $269,720$253,210 Accounts receivable (net) 97,710 90,940 Inventories 275,830 269,260 Investments 0 104,320 Land 141,470 0 Equipment 304,320 238,050 Accumulated depreciation\u2014equipment (71,250) (64,190) Total assets $1,017,800$891,590 Liabilities and Stockholders' Equity Accounts payable $184,220$175,640 Accrued expenses payable 18,320 23,180 Dividends payable 10,180 8,020 Common stock, $10 par 54,960 43,690 Paid-in capital: Excess of issue price over par-common stock 206,610 121,260 Retained earnings 543,510 519,800 Total liabilities and stockholders\u2019 equity$1,017,800 $891,590 Additional data obtained from an examination of the accounts in the ledger for 20Y9 are as follows: 1. Equipment and land were acquired for cash. 2. There were no disposals of equipment during the year. 3. The investments were sold for$93,890 cash.\n4. The common stock was issued for cash.\n5. There was a $65,610 credit to Retained Earnings for net income. 6. There was a$41,900 debit to Retained Earnings for cash dividends declared.\n\nRequired:\n\nPrepare a statement of cash flows, using the indirect method of presenting cash flows from operating activities. Use the minus sign to indicate cash outflows, cash payments, decreases in cash, or any negative adjustments.\n\n Merrick Equipment Co. Statement of Cash Flows For the Year Ended December 31, 20Y9 Cash flows from operating activities: $Adjustments to reconcile net income to net cash flow from operating activities: Changes in current operating assets and liabilities: Net cash flow from operating activities$ Cash flows from (used for) investing activities: $Net cash flow used for investing activities Cash flows from (used for) financing activities: Net cash flow from financing activities$ Cash at the beginning of the year Cash at the end of the year $## Answers #### Similar Solved Questions 1 answer ##### Determine the xx and yy components of the reaction force on the rocker arm ABCABC at... Determine the xx and yy components of the reaction force on the rocker arm ABCABC at the pin for equilibrium. Determine the magnitude of the force along the spring DFDF for equilibrium. 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At I (b) What the maximum value of the integral? maximum value... 5 answers ##### SONYOUR WORLJadc32,38,39,42,45Mean: 32+38+39+42+45-196196\/5-39.2Standard Deviation:Disrussion; What do these values mzan forJack?SHoWYOUL WORK Juan; 22, 2930 30 31Meen\"Stancero Deviazion;Ciurton: Wha: d3 nese valuemean ior Juan?Iiof 2F73 wod:D;Focus SONYOUR WORL Jadc32,38,39,42,45 Mean: 32+38+39+42+45-196 196\/5-39.2 Standard Deviation: Disrussion; What do these values mzan forJack? SHoWYOUL WORK Juan; 22, 2930 30 31 Meen\" Stancero Deviazion; Ciurton: Wha: d3 nese valuemean ior Juan? Iiof 2 F73 wod: D;Focus... 1 answer ##### 2. Define the hydrologic cycle component and indicate how they are different from each other for... 2. 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Let be a map Define the map prove or disprove 2) for all 3) for all A B We were unable to transcribe this imagef(and) = f(c) n (D) CD CA f-1( EF) = f-1(E)f-1(F) We were unable to transcribe this image... 1 answer ##### Solve the given equation. $\\cos ^{2} \\theta-\\cos \\theta-6=0$ Solve the given equation. $\\cos ^{2} \\theta-\\cos \\theta-6=0$... 1 answer ##### Values of Trigonometric Functions Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example$1 .$$\\sin 15^{\\circ}$ Values of Trigonometric Functions Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example$1 .$\\sin 15^{\\circ}$... 5 answers ##### What is the surface area of an object generated when the graph of f (x) = 3x on the interval [2,5] is revolved about the x axis? 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Do the monatomic vs. diatomic gases look right? 1. Problem 1.44. For example: Do simple solids like aluminum; carbon, or copper have the Cp values you expect? Why do the more complicated solid compounds have generally larger Cp values? Do the monatomic vs. diatomic gases look right?... 1 answer ##### From The Law of Healthcare Administration 8th Edition (2012) by Showalter - Chapter 10. 7. How... From The Law of Healthcare Administration 8th Edition (2012) by Showalter - Chapter 10. 7. How does informed consent apply to someone who had not signed an advance directive? To a newborn? To a mature minor? 8. Under what circumstances may consent be refused for the artificial administration of nutr... 1 answer ##### Help needed ASAP, really confused with multiple choice got (a) Two highways intersect as shown in Fig. 4-55. At the instant shown, a police car P is 800 m from the intersection and moving at 40 km\/h. 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Use a while loop.(BLOCK DIAGRAM REQUIRED)... 1 answer ##### Constants 1 Periodic Table \u25bc Part A A rigid container holds hydrogen gas at a pressure... Constants 1 Periodic Table \u25bc Part A A rigid container holds hydrogen gas at a pressure of 3.5 atm and a temperature of 40 \u00b0C. What will the pressure be if the temperature is lowered to -40 \u00b0C? Express your answer to two significant figures and include the appropriate units. PValue Unit... 1 answer ##### & 2x-5y = 8 y = 3x+1 & 2x-5y = 8 y = 3x+1... 5 answers ##### Given the following probabilities for an event E, find the odds for and against E.50.23 (D) 0.62For all answers_ type the odds as fraction. Simplify your answers:(A) Odds for =Odds against = Given the following probabilities for an event E, find the odds for and against E. 5 0.23 (D) 0.62 For all answers_ type the odds as fraction. Simplify your answers: (A) Odds for = Odds against =... 1 answer ##### Material science Questions: In the given crystal structure ? (1) Write the postions for the four... material science Questions: In the given crystal structure ? (1) Write the postions for the four dots. (II) Write the direction for the arrow.... 5 answers ##### The power factor of an AC circuit having resistance(R)$and inductance$(L)$connected in series and an angular velocity$omega$is(A)$R \/ omega L$(B)$R \/left(R^{2}+omega^{2} L^{2}ight)^{1 \/ 2}$(C)$omega L \/ R$(D)$R \/left(R^{2}-omega^{2} L^{2}ight)^{1 \/ 2}$The power factor of an AC circuit having resistance$(R)$and inductance$(L)$connected in series and an angular velocity$omega$is (A)$R \/ omega L$(B)$R \/left(R^{2}+omega^{2} L^{2} ight)^{1 \/ 2}$(C)$omega L \/ R$(D)$R \/left(R^{2}-omega^{2} L^{2} ight)^{1 \/ 2}$... 5 answers ##### Attempt$Considcr the ncutralization rcaction2HNOs (a9) Ba(OH)z (aq)2H,9 O(I) Ba(NOz), (uq) A0.115 L sample of an unknown HNO, solution requircd 39.5 mL. of 0.100 M Ba(OH)z for complete neutralization: What is thc concentration of the HNO, solution?concentrution:TOOLS X10\nAttempt \\$ Considcr the ncutralization rcaction 2HNOs (a9) Ba(OH)z (aq) 2H,9 O(I) Ba(NOz), (uq) A0.115 L sample of an unknown HNO, solution requircd 39.5 mL. of 0.100 M Ba(OH)z for complete neutralization: What is thc concentration of the HNO, solution? concentrution: TOOLS X10...\n##### Consider the following time series data.Week123456Value171415111813Using the naive method (most recent value) as the forecast forthe next week, compute the following measures of forecastaccuracy.(a) mean absolute errorMAE = (b)mean squared errorMSE = (c)mean absolute percentage error (Round your answer to twodecimal places.)MAPE = %(d)What is the forecast for week 7?\nConsider the following time series data. Week 1 2 3 4 5 6 Value 17 14 15 11 18 13 Using the naive method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy. (a) mean absolute error MAE = (b)mean squared error MSE = (c)mean absolute percentage...\nProton motive force Multiple Choice used to synthesize ATP used tO produce NADH: used to synthesize ADP uses ADP and ATP is used to produce FADH2:...\n##### Please show how to do doing excel\nPlease show how to do doing excel...\n##### Choose which of the following reactions would NOT proceed as written: \u041026 + CH3S \u041e\u043d OH...\nChoose which of the following reactions would NOT proceed as written: \u041026 + CH3S \u041e\u043d OH SCH3 1 \u0437 + PBr3 3 + H3PO3 \u041e\u041d Br \u0428 HI + MeOH \u041e\u041d CN CI NaCN IV NaCl \u041e\u041d (A) I only (B) I and Il only (C) Il only (D) IV only (E) Il and IIl only...\n##### For fle) = * * * * 7, waluto av \u2022 3 \u2013 Aas, samples For...\nfor fle) = * * * * 7, waluto av \u2022 3 \u2013 Aas, samples For f(x) = x2 + 4x + 7, evaluate f(x + 2) - f(x). Simplify. 192 Tutorial Let f(x) = 9x - 36. 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\section{Introduction} \label{sec:intro}
The fact that the cosmic-ray (CR) spectrum extends up to extremely high energies, as well as the difficulties encountered by standard acceleration mechanisms to energize particles up to such energies, has led to a wide search for ways to boost the maximum energy and/or alternative acceleration mechanisms. While we can safely say that CR acceleration at the shocks associated with supernova (SN) explosions is now confirmed experimentally, it is also true that convincing observational proof that SN remnants (SNRs) can accelerate protons to energies in excess of $\sim 100$ TeV is, thus far, missing. Even from the theoretical point of view, SNRs associated with Type Ia and ordinary core-collapse SNe may be connected to CR acceleration up to $\sim 100$ TeV, while higher energies require much more extreme conditions, perhaps to be found in rare and very energetic SN explosions \cite[]{cristofari2020}.
While standard acceleration processes, such as the second-order Fermi process \cite[]{Fermi49} and diffusive shock acceleration \cite[]{1977ICRC11.132A,1977DoSSR2341306K,1978ApJ221L29B}, have received a lot of attention throughout the years, more recently, the acceleration of charged particles in realistic MHD turbulence has been attracting an increasing level of attention \citep[e.g.,][and references therein]{LazarianEA20}, with special emphasis on magnetic reconnection. Early on, attention focused on contributions to energization by MHD activity in special regions within a dynamic plasma \citep{Giovanelli47,dungey1953lxxvi}. Examination of diverse scenarios for energization has over time become increasingly varied and complex \citep{AmbrosianoEA88, DmitrukEA04-tp, DrakeEA06,DrakeEA09,KowalEA12-ptcles}, suggesting that a more general perspective may be led by a simpler view of the physical principles at work.
In 2D configurations, there exist simple conservation laws that define and constrain particle orbits \citep{sonnerup1971adiabatic,MattEA84} so that when turbulent fluctuations are present, charged test particles can be confined and accelerated \citep{AmbrosianoEA88,DrakeEA06} in secondary magnetic flux structures or ``islands''. Initially applied to smaller-gyroradius particles such as electrons in the context of magnetic reconnection geometries,
this idea was further developed by \citet{OkaEA10}, who noted that coalescence in a multiple-island context can efficiently accelerate electrons due to ``anti-reconnection'' electric fields associated with such mergers. Furthermore, when applied to ``pickup'' of protons in reconnection jets \citep{DrakeEA09} that subsequently feed into a Fermi process as multiple magnetic islands contract and merge, a more complex proton energization process may be developed involving multiple islands \citep{DrakeEA10}. More recently, \citet{trotta2020fast} showed that transrelativistic electrons can be significantly accelerated while trapped in turbulent structures and experiencing curvature drift. Such a behavior, obtained in a 2D configuration where the background turbulent plasma is modeled with a hybrid particle-in-cell method while energetic electrons are treated as test particles, has also been verified at different intensities of the turbulent fluctuations. The basic physical elements of these models were formalized in transport theories \citep{Zank2014transport,leRouxEA15,leroux2018self,leroux2019modeling} that facilitate applications (see also the recent reviews by \citet{khabarova2021currentsheets, pezzi2021currentsheets}).
Complementary to these developments, there has been a parallel line of studies that begin with the premise that charged particle energization might be treated as being due to plasma dynamics that from the onset is complex and, in fact, turbulent. Early efforts demonstrated the feasibility of efficient turbulent acceleration \citep{DmitrukEA03-testpart} in three dimensions, although numerical limitations such as small system size and lack of resonant power at numerical grid scales (however see \citep{LeheEA09}) cast doubt on conclusions concerning scaling laws.
Numerical experiments have improved ever since \citep{DmitrukEA04-tp,KowalEA12-ptcles,DalenaEA14}, including turbulence effects associated with current sheets and reconnection, contracting islands, and proliferation of plasmoids and/or secondary islands \citep{KowalEA11,KowalEA12-ptcles}. Similar effects have been characterized in relativistic plasmas \citep{hoshino2012stochastic, zhdankin2017kinetic,zhdankin2019electron,ComissoSironi18,GuoEA20,KilianEA20}.
Most important, it has been understood that the initial efforts to characterize acceleration in or around reconnection regions focused on the physical processes responsible for extracting particles from the thermal background and injecting them into some energization process at work on larger scales. Yet, the particle velocity, the thermal velocity, and the Alfv\'en speed were of the same order of magnitude, making it very difficult to identify the nature of the acceleration mechanism (first or second order) and the scaling laws that could be used for higher-energy particles.
Injecting test particles in a 2D or 3D simulation box of MHD turbulence has been an independent method to investigate both the transport of such particles and their energization, although in some of the previous articles (e.g. \citep{KowalEA11,KowalEA12-ptcles}) the Larmor radius of the particles at the beginning of the simulation was chosen to be much smaller than the grid spatial spacing, which makes transport unrealistic, due to the lack of resonance with the turbulence and thereby unreliable diffusive transport.
When self-consistent broadband turbulence is involved, particle energization becomes complex due to interactions with the internal structure of large- and small-scale flux tubes, as well as the current sheets, vortices, and reconnection sites that typify flux-tube boundaries and their mutual interactions. Common to a number of these treatments of energization including turbulence is the role of direct acceleration for particles of smaller Larmor radii, transitioning to the involvement of perpendicular acceleration at larger energy \citep{DmitrukEA04-tp,DalenaEA14,ComissoSironi18,comisso2021pitch,trotta2020fast}.
Although some works indicated that high energies can be also attained by second-order processes (e.g., \citet{ArznerEA06,SioulasEA20}), the role of temporary trapping has been highlighted in various contexts as it can dramatically influence both transport and acceleration \citep{KowalEA11,KowalEA12-ptcles,TooprakaiEA16,DalenaEA14}. Indeed, the ubiquity of turbulent coherent structures and islands/plasmoids/flux ropes produced by magnetic reconnection makes the potential of the trapping mechanism significant for different systems. In the solar and stellar coronae, magnetic loops may provide sufficient conditions to entrap particles \citep{vlahos2018particle}. In the interplanetary medium, there are observations supporting the idea that particles are locally accelerated when trapped in merging or coalescing islands \citep{khabarova2017energetic, MalandrakiEA19}. Particle trapping may also provide a source for particle reacceleration downstream of shocks \citep{zank2015diffusive,nakanotani2021interaction}. Other systems, e.g. the intracluster medium, could also encompass such a mechanism although observations are usually explained in terms of second-order processes \citep{vazza2017turbulence, brunetti2020second}. The importance of trapping for accelerating particles has been also recently highlighted by \citet{lemoine2021particle} in the context of strong and intermittent turbulence by relating the energization process and the gradients of the bulk velocity and magnetic fields.
Intuitively, trapping can enhance acceleration when appropriate electric fields are encountered, but it can also inhibit stochastic acceleration when the required transport is thwarted. The full range of possibilities for such effects remains to be exhaustively explored. The study of \citet[][see also \citet{KowalEA12-ptcles}]{KowalEA11} is instructive on several salient features. In these numerical experiments, the authors observe two small magnetic islands merging with a central elongated island. A selected particle gains considerable energy after entering the system by following field lines through one small island, eventually becoming trapped within the large island, circuiting it numerous times, and gaining energy exponentially and mainly when passing near the region of merger with the small island.
These conclusions were illustrated in detail only for 2D turbulence in a configuration that was optimized to create reconnecting islands. However, the test particles injected in a snapshot of the simulation were initially subgrid, which means that their transport could not be described in a realistic way. Eventually, the energy of the particles, after an exponential increase, reaches the regime in which resonances could in principle be relevant for particle transport, but this phenomenon was not discussed. Despite these shortcomings,
this approach demonstrates clearly the complex interplay between the transport effects that entrain the particle within the island and near the acceleration region, along with the special circumstances that support the electric field responsible for the energization itself.
In the present paper, we advance such a scenario for strong turbulence and particle energization in the highly relativistic particle regime, having in mind the implications that the process may have for the acceleration of very-high-energy CRs. In particular, we continue the examination of these complex interactions of charged particles with turbulence by performing relativistic test-particle simulations with a turbulent electromagnetic field produced by means of 3D MHD simulations, not specifically devised to produce reconnection regions. Our focus is on clarifying of the nature of the trapping or entrainment that leads particles to rapid acceleration to the highest energies.
We find that the bulk of the test particles injected in the simulation box went through a secular-second order acceleration due to the random plasma motions, which in turn induced random electric fields. The interaction of the particles with these plasma motions leads to stochastic energy change, which we characterize in terms of plasma properties and manage to associate with a diffusive motion in momentum space. The transport of the same particles in physical space is also found to be well described through a diffusive motion. The latter is in a range of scales where the anisotropic cascade of the turbulence with respect to the local magnetic field does not seem to have a visible effect as yet.
In addition to this acceleration process that is clearly at the second order in the quantity $v_A/c\ll 1$, we also identify a small fraction of particles that manage to get trapped in selected regions associated with the interaction between flux tubes. These particles all have pitch-angle cosine very close to zero, a necessary condition for trapping, and go through an exponential phase of energy increase. We provide a characterization of these regions in terms of physical observables that could be measured in the simulation. We also build a simple model of the region where this phenomenon occurs and manage to reproduce the main properties of the acceleration process and the time scales involved. The acceleration process is similar to a first-order mechanism in which the particle trajectory in the plane perpendicular to the local magnetic field encounters a gradient of plasma velocity (i.e. of the induced electric field).
The paper is structured as follows. In Sect. \ref{sec:background} we present the MHD simulations adopted in the present work, while in Sect. \ref{sec:testparticle} we describe the test-particle code and the first results obtained in terms of physical space transport. Then, in Sect. \ref{sec:results} we focus on the main numerical outcomes of the work concerning particle energization. Sect. \ref{sec:discussion} discusses a simple model of the acceleration region and derives the main properties of the acceleration mechanism and the main time scales involved. Moreover, we discuss the implication of our findings for astrophysical systems. Finally, in Sect. \ref{sec:conclusions} we conclude by summarizing our results and illustrating future developments.
\section{MHD Simulation Background} \label{sec:background}
In order to study the transport and acceleration of charged test particles, we follow particle evolution
in electromagnetic fields obtained through incompressible three-dimensional MHD simulations. These simulations solve the following set of equations:
\begin{eqnarray}
\frac{\partial {\bm u}}{\partial t} + \left({\bm u} \cdot \nabla\right) {\bm u} &&= - \frac{1}{\rho}\nabla P + \frac{1}{\rho}{\bm j} \times {\bm B} + \nu \nabla^2 {\bm u} \label{eq:MHD1}\\
\frac{\partial {\bm B}}{\partial t} + \left({\bm u} \cdot \nabla\right) {\bm B} &&= \left({\bm B} \cdot \nabla\right) {\bm u} + \eta \nabla^2 {\bm B} \\
\nabla \cdot {\bm u} &&= \nabla \cdot {\bm B} = 0 \label{eq:MHD2}
\end{eqnarray}
where ${\bm u}({\bm r},t)$ is the magnetofluid speed composed only of its fluctuating part, and ${\bm B}({\bm r},t)$ is the magnetic field that is decomposed into a uniform mean $B_0$ and a zero-mean fluctuation $\bm b$, ${\bm B}({\bm r},t) = \bm{B_0} + \bm{b}({\bm r},t)=B_0 {\bm e}_z + \bm{b}({\bm r},t)$. Furthermore, $P$ is the thermal pressure, and $\rho$ is the magnetofluid density. The current density is ${\bm j}=\nabla \times {\bm B}$, while $\nu$ and $\eta$ are the viscosity and resistivity, respectively. The flow is incompressible $\nabla \cdot {\bm u} =0$, and
the density is uniform $\rho={\rm const}$.
Lengths, time, and velocities in Equations (\ref{eq:MHD1}--\ref{eq:MHD2}) are respectively normalized to a typical length $L_A$, time $t_A$ and to the Alfv\'en speed $v_A=L_A/t_A=\bar{B}/\sqrt{4\pi m_p\bar{n}}$, where $\bar{B}$ and $\bar{n}$ are reference values for the magnetic field and for the background number density. We here adopt $L_A=81.5\, {\rm pc}$, corresponding to $L_{\rm box}=512 \, {\rm pc}$, while ${\bar B}=1\, \mu G$ and ${\bar n}=1\, {\rm cm}^{-3}$. Unless specified, hereafter we assume normalized variables.
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{j2_B0=0.png}
\caption{Rendering of the
current density $j^2({\bm r})$ shows a plethora of intermittent coherent structures. Such structures form a template for the possibility of rare acceleration events.}
\label{fig:j2B0=0}
\end{figure}
Equations (\ref{eq:MHD1}--\ref{eq:MHD2}) are solved in a 3D Cartesian periodic box of size $L_{\rm box}=2\pi$, with spatial
resolution $N_x=N_y=N_z=1024$ adopting a pseudo-spectral method in a Fourier basis. The time advancement is performed with a second-order Runge-Kutta scheme and the 2/3 rule for spatial dealiasing is chosen \citep{PattersonOrszag71}.
Small values of resistivity and viscosity $\eta=\nu=2\times 10^{-4}$
are introduced to define the well-resolved spectral domain. The dissipative wavenumber $k_{\rm diss}$
(the reciprocal of the Kolmogorov length scale) for the considered runs is always smaller by a factor 2 than the maximum resolved wavenumber $k_{\rm max}$ \citep[for further details, see][]{BandyopadhyayEA18-prx}.
Large-scale uncorrelated fluctuations of ${\bm u}$ and ${\bm b}$ are introduced at $t=0$ and turbulence develops, producing small-scale fluctuations. We focused here on the case with $u_{\rm rms}=b_{\rm rms}=1$ and $B_0=0$. The role of a finite background magnetic field and compressibility will be discussed in a separate forthcoming work.
We then selected the time instant at which the turbulent activity is strongest (i.e. highest dissipation). The complex and highly structured pattern of the turbulence is displayed in Figure \ref{fig:j2B0=0}, showing the contour plot of ${\bm j}$ in the 3D domain. Vortices and magnetic islands, as well as intense current sheets where magnetic reconnection may be at work, naturally emerge as elementary structures of the turbulent flow. The omnidirectional spectrum of magnetic energy (Figure \ref{fig:MagneticSpectrum}) indicates that an inertial range, whose length is about a decade in wavenumber space, develops before dissipative effects steepen the spectrum at higher wavenumber $k$.
In the inertial range, the slope is rather compatible with either the
Kolmogorov or Kraichnan predictions;
these are respectively displayed in green and orange dashed lines in Figure \ref{fig:MagneticSpectrum} (see also the inset in the same figure). By numerical evaluating the correlation length $l_c$ of the magnetic field, we find $l_c=0.218$ ($l_c=17.7\, {\rm pc}$ in physical units), corresponding to protons with energy $E\sim 16\, {\rm PeV}$ in the typical field ${\bar B}$.
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_MagneticOmnidirSpectrum.pdf}
\caption{Omnidirectional spectrum of the magnetic energy. The green dashed (orange dotted-dashed) line shows the Kolmogorov (Kraichnan) prediction. The small inset displays the magnetic energy spectrum compensated by the Kolmogorov (green dashed) and the Kraichnan (orange dotted-dashed) slope. The gray dashed vertical lines indicate the wavenumber associated with the initial particle gyroradius. }
\label{fig:MagneticSpectrum}
\end{figure}
\section{Methods: test-particle propagation details}\label{sec:testparticle}
We numerically integrate the motion equations of $N_p=10^{5}$ relativistic test particles of positive charge $e$ and mass $m_p$ moving in the turbulent electromagnetic field obtained by means of the incompressible MHD simulations described above. The normalized particle equations of motion are
\begin{eqnarray}
&&\frac{d \bm{x}}{dt} = {\bm v} \label{eq:rsc}\\
&&\frac{d {\bm p}}{dt} = \alpha \left( {\bm E} + {\bm v} \times {\bm B} \right) \label{eq:usc}
\end{eqnarray}
where ${\bm x}=(x,y,z)$, ${\bm v}$, and ${\bm p}=\gamma {\bm v}$ are the particle position, velocity, and momentum, while ${\bm E}$ and ${\bm B}$ are the electric and magnetic fields. Equations (\ref{eq:rsc}-\ref{eq:usc}) are scaled analogously to MHD simulations. In normalized units, the Lorentz factor reads $\gamma = 1/\sqrt{1 - (\beta_A v)^2} = \sqrt{1 + (\beta_A p)^2}$, where
$\beta_A = v_A/c$. The electric field in Equation (\ref{eq:usc}) is derived through Ohm's law: ${\bm E}= -{\bm u}\times{\bm B}+\eta {\bm j}$.
The parameter $\alpha = t_A \Omega_{0}$, where $\Omega_0=e \bar{B}/m_p c$ is the proton cyclotron frequency, can be easily rewritten as $\alpha=L_A/d_p$, with $d_p$ the proton skin depth of the background plasma. $\alpha$ is thus connected to the extension of the inertial range of the turbulence with respect to kinetic, dissipative scales \citep{DmitrukEA04-tp,GonzalezEA16}. In a $\beta_p\sim 1$ plasma (with $\beta_p$ the thermal to magnetic pressure ratio), the parameter $\alpha$ corresponds to the inverse of the normalized gyroradius of nonrelativistic
particles moving with speed $\sim v_A$. Previous works considering the injection of thermal particles into the acceleration region were hence forced to reduce $\alpha$ to much smaller and computationally feasible values.
Such a requirement provides particles with a gyroradius at least larger than the grid size, so that resonant scattering might be properly taken into account. On the other hand, relativistic particles moving at the speed of light have a much larger gyroradius because $\gamma\gg 1$, thus removing the constraint on the value of $\alpha$. For the
parameters described above, $\alpha\sim10^{12}$ and $\beta_A\sim10^{-5}$.
To save computational resources, we only artificially increase $\beta_A=5\times 10^{-2}$.
Because we are interested in the energization of relativistic particles moving in a nonrelativistic environment, we assume stationary electromagnetic fields, i.e. $\partial {\bm B}/\partial t = \partial {\bm E}/\partial t =0$ (magnetostatic approximation), and we consider a static snapshot of these fields when turbulence is fully developed.
Eqs. (\ref{eq:rsc}--\ref{eq:usc}) are integrated by adopting the relativistic Boris method \citep{ripperda2018comprehensive,dundovic2020novel}. The electric and magnetic fields are interpolated at the particle position through a trilinear interpolation method \citep{birdsall2004plasma}. We verified that the results presented here are not affected by adopting a more accurate yet significantly slower 3D cubic spline method (not shown here). Particles are injected homogeneously throughout the computational box at a given energy and with isotropic velocity direction on the unit 3D sphere. The time step is set to $1/50$ of the initial gyroperiod.
Most of the results here adopt the initial gyroradius to be $r_{g,0}\simeq 0.1 l_c=0.02$, corresponding to $E_0\simeq 1.6\, {\rm PeV}$. The resonant wavenumber $k_{r_g}=1/r_g = 50$, reported in Figure \ref{fig:MagneticSpectrum} with a vertical dashed gray line, resides in the inertial range of turbulence, and it is also quite far from the dissipative scales where the resistive electric field is expected to become important. This ensures that the acceleration process studied here is mainly driven by the inductive electric field, this being the most relevant term for analyzing the energization of relativistic particles whose gyroradius is much larger than the typical length where dissipative and resistive effects are expected to steepen the magnetic spectrum. To double-check, we also verified
that our results are not affected by the resistive field, in that if we exclude the resistive component from the computation of the electric field, the energization process is basically unchanged. This shows that for the high-energy particles we are interested in, namely when the Larmor radius exceeds the thickness of the reconnection regions, the energization is not due to the resistive fields but rather to the induced electric fields due to the plasma motion.
Although here we are most interested in how particles react to electric fields in the simulation box, it is first worth studying how particles move in the magnetic field, especially to confirm that we find diffusive motion and to identify possible differences with respect to cases where turbulence is synthetic rather being the result of an MHD simulation.
In order to study particle transport in physical space, we performed a subset of test-particle simulations by excluding the electric field. A diffusive regime after a ballistic transient is always recovered. When reaching the diffusive plateau, the isotropic diffusion coefficient is computed as $D_{\rm iso}=(D_{xx}+D_{yy}+D_{zz})/3$ with
\begin{equation}
D_{xx}(\Delta t) = \frac{\langle (\Delta x(\Delta t))^2 \rangle}{2 \Delta t}\, .
\label{eq:D_run}
\end{equation}
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_Dsat.pdf}
\caption{Mean free path $\lambda_{\rm iso}$ as a function of the particle gyroradius. The vertical dashed and dotted-dashed gray lines correspond to the particle gyroradius adopted here and to the gyroradius corresponding to the dissipative scale, respectively. The blue dotted-dashed line refers to the QLT prediction $D_{\rm iso} \sim r_g^2$ obtained for $r_g\gtrsim l_c$. The red dashed line reports the prediction of \citet{SubediEA17} (Equation (53) of \citet{dundovic2020novel}).}
\label{fig:Diso-sat}
\end{figure}
Figure \ref{fig:Diso-sat} shows the isotropic mean free path $\lambda_{\rm iso}=3 D_{\rm iso}/c$ as a function of the particle gyroradius $r_g$, normalized in the usual way to the correlation scale $l_c$. The typical behavior of the path length as a function of energy is the same as that found in synthetic turbulence, with a low-energy trend that reflects the shape expected from a given isotropic power spectrum \citep{SubediEA17,dundovic2020novel}. In particular, the dotted-dashed red line in Figure
\ref{fig:Diso-sat} implements Equation (53) of \citet{dundovic2020novel} for the Kolmogorov case, where $l_{\rm iso}$ is the bend-over scale of the synthetic model in \citet{dundovic2020novel}, being $l_{\rm iso}\sim 2 l_c$. At variance with synthetic models of the turbulent field, the slope of the isotropic power spectrum is not very well defined here because of the limited dynamical range (see Figure \ref{fig:MagneticSpectrum}).
At high energies, when the gyroradius satisfies the condition $r_g>l_c$, the diffusion coefficient becomes weakly dependent upon the power spectrum, $D_{\rm iso}\sim r_g^2$ (blue dotted-dashed line). For gyroradii $r_g/l_c<0.01$, several numerical effects, especially dissipation, limit the validity of the approach by reducing the power available in the form of modes that the particle gyration can resonate with. As a result, the numerically computed path length departs from the dotted-dashed red line at such low energies. The vertical dashed gray line identifies the energy of the particles used below for our investigation of energization.
This correspondence to the isotropic spatial diffusion theory is itself a result of some significance: on one hand, it validates the approach of \citet{SubediEA17} and \citet{dundovic2020novel}
with a ``realistic'' turbulent and intermittent magnetic field obtained from the numerical evolution of MHD equations. On the other hand, this may be considered somewhat surprising because the anisotropic cascade development with respect to the local magnetic field, expected in MHD \citep{GS1994}, seems to have little effect on the diffusion properties. It is likely that the effects of the anisotropic cascade are not fully developed as yet, due to the limited dynamical range imposed by the numerical constraints. These results seem to be in good agreement with those of \cite{Cohet2016}.
\section{Results on particle energization}\label{sec:results}
The turbulent motion of the plasma in the simulation box leads to the unavoidable creation of inductive electric fields, which are expected to have random orientations. As such, their presence is expected to lead to changes in the energy of the particles, due to the presence of such inductive electric fields in the Lorentz force. This effect is expected to cause the energy of a particle to increase or decrease depending on the relative orientation of the particle momentum and the local electric field, namely a typical second-order phenomenon. On the other hand, the complex structure of MHD turbulence is known to trigger additional phenomena that may lead to more rapid energization of the particles (see for instance \citet{KowalEA11}). Here we investigate all these phenomena in great detail, stressing that the simulation was not carried out to maximize the formation of reconnection regions or other peculiar structures. The phenomena we see are, in this sense, very generic.
We find that in, addition to an overall second-order stochastic acceleration mechanism, a first-order process is at work as well, due to the temporary trapping of particles in coherent structures.
\subsection{Stochastic energization}
Figure \ref{fig:DE-run} displays the running energy diffusion coefficient as a function of energy, assuming that in fact the motion of the particles can be described in terms of a random walk in momentum space:
\begin{equation}
D_{EE} = \frac{\langle (\Delta E(\Delta t))^2\rangle}{2 \Delta t}.
\label{eq:DEE}
\end{equation}
It is evident that after a transient, the energy diffusion coefficient saturates at a roughly constant value. This implies the presence of diffusion in energy space, thus revealing the typical nature of a second-order process \citep{Ostrowski97}. The energy diffusion coefficient $D_{EE}\simeq 0.01 v_A E_0^2/l_c$ implies a characteristic time for the energy diffusion process $\tau_{\rm diff,E}=E^2/D_{EE}\sim 10^2 l_c/v_A$ that is consistent with the large time scale for growth of the average energy that occurs at late times in our simulations (not shown here). The slight increase recovered in $D_{EE}$ for very large $\Delta t$ may be due to the fact that the average gyroradius starts to increase on this timescale due to other processes (see below), thus making $D_{EE}$ move away from the plateau.
\begin{figure}[!thb]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_DERun.pdf}
\caption{Running energy diffusion coefficient $D_{EE}$ as a function of $\Delta t$.}
\label{fig:DE-run}
\end{figure}
Another signature of an active stochastic energization mechanism comes by looking at the probability density functions (PDFs) of the relative energy gain $\Delta E(\Delta t)/E$, displayed in Figure \ref{fig:dEtoE-dist} for $\Delta t= 0.05 l_c/v_A$ (blue) and $\Delta t=0.5 l_c/v_A$ (black). The PDFs have been computed averaging on the initial time instants $t$ up to $t=t_{\rm max}\simeq 22 l_c/v_A$. Particles are likely to undergo both increases and decreases in energy. Although the distribution is peaked at small values, larger changes of energy up to $\gtrsim \beta_A$ (dashed gray lines in Figure \ref{fig:dEtoE-dist}) are allowed. The distribution functions are skewed toward the positive value of energy changes because the standardized skewness is $\tilde{s}=0.20$ for $\Delta t= 0.05 l_c/v_A$ and $\tilde{s}=0.12$ for $\Delta t= 0.5 l_c/v_A$, where $\tilde{s}=s/\sigma^3$. Here,
$\sigma$ and $s$ are, respectively, the standard deviation and the skewness (third-order moment) of the distribution function. The distribution function of the relative energy gains is manifestly non-Maxwellian for small $\Delta t$ and tends to recover the Maxwellian shape for larger $\Delta t$. Indeed, the kurtosis $\kappa$---defined as the fourth-order moment of the PDF normalized by $\sigma^4$--- is $\kappa = 4.84$ and $\kappa=3.69$ for $\Delta t= 0.05 l_c/v_A$ and $\Delta t= 0.5 l_c/v_A$, respectively.
\begin{figure}[thb!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_dEtoE-dist_avert.pdf}
\caption{Probability density functions (PDFs) of $\Delta E/E$ with $\Delta t=0.05 l_c/v_A$ (blue) and $\Delta t=0.5 l_c/v_A$ (black). The red and orange dashed curves correspond to the associated Gaussian distributions. Dotted-dashed and dashed gray lines indicate the values of $\beta_A^2$ and $\beta_A$, respectively. }
\label{fig:dEtoE-dist}
\end{figure}
The presence of positively skewed PDFs indicates that energy increase is more favorable than energy decrease. This provides the secular direction of the process, leading to a net energy gain.
Clearly, all acceleration processes are at work simultaneously, and it is not trivial to discriminate among them by looking at a collection of particles: What we can say is that basically all test particles launched in the simulation box suffer from the second-order process illustrated above. As we discuss below, a small fraction of particles happen to be trapped in selected structures and get energized through a first-order process. It is not clear to what extent these few particles can affect the shape of the high-energy-gain tail of the PDF shown in Figure \ref{fig:dEtoE-dist}. As a consequence of the fact that only a few particles experience trapping, it is in general rather difficult to evaluate statistical properties of the population of trapped particles, such as the transport coefficients. In future work, we plan to explore the potential of novel methods, for instance, those commonly adopted in biophysics and based on single-particle trajectories \citep{golding2004RNA,saxton2012wanted,trotta2020particle}.
\subsection{Particle trapping in coherent structures: First-order acceleration}
In Figure \ref{fig:ParticleEnergySpectra} we show the spectra of particles in the simulation box, after a time $t$ indicated in the figure. A few comments are in order: (1) Particles are injected at energy $E_0$ which, for the natural units adopted here, corresponds to $E_0=1.6\, {\rm PeV}$. (2) As time evolves, the second-order process leads to a broadening of the distribution function, namely there are both particles losing energy and particles gaining energy. On average, however, the particle energy increases as one can see by noticing that the peak of the distribution moves toward higher energies. (3) Contemporaneously an approximate power law is created at high energies that eventually extends to particles with energies such that $r_g\simeq l_c$. This typically happens at times $t\gtrsim 10l_c/v_A$.
\begin{figure}[hbt!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_csi=0.02_beta=0.05_g-dist_early.pdf}
\caption{Particle PDFs at different time instants showing the energization process. The vertical dotted-dashed gray line highlights $r_g=l_c$, while the orange dotted-dashed line displays the ${-5/2}$ power-law slope.}
\label{fig:ParticleEnergySpectra}
\end{figure}
For the sake of comparison, we also report in Figure \ref{fig:ParticleEnergySpectra} a line indicating the spectrum $\propto E^{-5/2}$, which was predicted and observed by different groups, although for very different systems and with different qualitative premises.
The pioneering numerical simulations of \citet{AmbrosianoEA88} found evidence for accelerated particles with a power-law tail with a slope compatible with $-5/2$ using two-dimensional simulations. As we discuss below, the dimensionality of the problem is very important in assessing the efficiency of trapping processes that are responsible for particle energization. Moreover, \citet{AmbrosianoEA88} focused on the extraction of particles from the thermal bath, for which the particle velocity remains close to that of the background thermal particles. As we discuss below, the acceleration, the trapping, and escape from the acceleration region work in somewhat different ways for relativistic particles. Moreover, as it is well known, the slope in energy should not be the same in any case for relativistic and nonrelativistic particles: This is so even for particles accelerated at a strong shock, for which the spectrum of accelerated particles in momentum is $f(p)\propto p^{-4}$, but when expressed in terms of energy, it is $N(E)\propto E^{-2}$ for relativistic particles and $\propto E^{-3/2}$ for nonrelativistic particles.
The slope $-5/2$ was also predicted by \citet{degouveiadalpino2005production} and \citet{delvalle2016properties}, where a shock-like toy model was introduced to describe a reconnection region: The particles would be advected into the reconnection region with the inflowing plasma and would be expelled (the analog of escape to downstream in the case of a shock) at the speed of the reconnection exhaust. This determines a sort of universal spectrum for the accelerated particles. Such universality was later criticized by \citet{drury2012firstorder}. In fact, for particles that are being energized, it is unlikely that the velocity of the exhaust plays any role in the shaping of the spectrum of accelerated particles because the particles' Larmor radius becomes quickly larger than the thickness of the current sheet. As we mentioned above, the spectrum $E^{-5/2}$ is shown in Figure \ref{fig:ParticleEnergySpectra} only as a reference, while it appears to be asymptotically reached in our simulations only for exceedingly long times compared with the dynamical time of the MHD turbulence.
In the perspective of understanding the nature of the acceleration processes at work, in the top panel of Figure \ref{fig:Expgrowth}, we show the temporal evolution of the gyroradius averaged on the full particle ensemble (red dashed line), while the red shadowed area corresponds to the standard deviation of the averaged gyroradius. We clearly see that there is a secular increase in the particles' energy, which we attribute to the random interaction with the inductive electric fields in the simulation box. The black curve shows the temporal evolution of the particle gyroradius that, after a time $T\sim 22 l_c/v_A$, turns out to be the most energetic particle in the simulation. It is worth noticing that for early times, namely on the left side of the first vertical dashed line ($t\approx 8 l_c/v_A$), the fluctuations in the particle gyroradius (or its energy) are compatible with the fluctuations expected based on the bulk of the particles in the simulation (shaded area). Moreover, the particle energy is clearly carrying out a random walk, in that it increases and decreases, a typical feature of a second-order process.
At a time $t\approx 8 l_c/v_A$ an exponential increase of the particle energy starts (the plot is in lin-log scale) and lasts for about $\sim 10 l_c/v_A$. The end of this period is marked by the rightmost vertical dashed line ($t\approx 17.5 l_c/v_A$). This period of rapid energization suggests that a small number of particles experience some new phenomenon. This number must be small because the energy gained by such particles is visibly larger than the typical deviation from the mean (shadowed area).
\begin{figure}[htp!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_ExpGrowth.pdf}
\caption{Typical behavior of a trapped particle showing exponential growth of energy that occurs within the two vertical green dashed lines. The particle gyroradius for the trapped particles increases exponentially over a time-scale $\tau \gtrsim 10 l_c/v_A$. This growth is much faster than the growth of the averaged gyroradius, where the average is performed on the full ensemble of test particles (red dashed line). The dashed red area represents the standard deviation of the averaged gyroradius. The bottom panel shows the particle trajectory illustrating that the particle is trapped. }
\label{fig:Expgrowth}
\end{figure}
We notice that, superposed on the main regular energy growth, there remain visible smaller-scale oscillations of the particle energy (see inset in Figure \ref{fig:Expgrowth}). In particular, we visually identify a smaller-scale oscillation whose period is $T \simeq 5\times 10^{-2} l_c/v_A$, which corresponds to the particle gyromotion $\tau_g=2\pi/\Omega_g=2\pi r_g/c$. A larger-scale modulation with period $T \simeq 5 \times 10^{-1}l_c/v_A$ is also observed, and this may be correlated with fluctuations of the magnetic field intensity on this timescale. This is suggestive of the simultaneous presence of additional processes, such as mixing of second-/first-order processes and the role of mirroring or drifts. The multiscale complexity of the overall energization dynamics is evidenced by the appearance of at least four timescales in Figure \ref{fig:Expgrowth} ---the exponential time scale, the gyromotion, the modulation seen in the inset, and the second-order energy gain seen to the left and right sides of the exponential phase.
\begin{figure*}[!thb]
\centering
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\columnwidth]{3D_Bfield.png}
\end{minipage}
\begin{minipage}{0.48\textwidth}
\includegraphics[width=0.96\columnwidth]{3D_Bfield_INSET.png}
\end{minipage}
\caption{Particle trajectory in the 3D domain, with the points colored with the particle energy, where the color scale goes from blue to red as the particle energy increases. Magnetic field lines, colored with the magnitude of the magnetic field itself (again from blue to red as the magnetic field magnitude increases), indicate that the particle is trapped in a flux tube and that it is accelerated when the flux tube is feeling the gradients associated with the interaction with another large-scale structure. The right panel shows an inset of the left plot zoomed in the trapping region and limited in time to a few particle gyrations. The green line in each panel corresponds to the correlation length $l_c$.}
\label{fig:3Dplots}
\end{figure*}
The peculiar behavior of the particles during the exponential phase is best illustrated in the bottom panel of Figure \ref{fig:Expgrowth}, where we show the particle's trajectory. One can see that, during the stage of exponential energy growth, the particle is trapped in a small region of the computational domain of size $\sim 0.5 l_c$. In fact, the spatial excursion per unit $l_c/v_A$ is about 10 times smaller between the dashed lines than outside that region. The particle escapes from the trapping region when its gyroradius becomes comparable with the island size $l_{\rm isl}\sim l_c$.
The phenomenology of this trapping can be also appreciated by looking at the particle trajectory in the 3D domain. Figure \ref{fig:3Dplots} shows the particle trajectory as dots colored with the particle energy, where the color scale goes from blue to red as particle energy increases. Magnetic field lines near the trapping region are also displayed, colored with the amplitude of the field itself (again going from blue to red as the magnetic field amplitude increases). When the particle is not trapped, it carries out an erratic motion in the whole computational domain, akin to an unconstrained random walk. The trapping is associated with a spherical-like motion constrained within a flux-tube-like structure. The energization occurs when the flux tube is perturbed by another large-scale structure, more easily appreciated in the right panel of Figure \ref{fig:3Dplots}. This confirms the scenario that an intense acceleration can occur when magnetic islands and, more in general, large-scale plasma structures, are interacting (collapsing, merging, etc) with other similar structures \citep{DrakeEA06,KowalEA11}, leading to a locally strong magnetic field gradient.
It is important to point out here that there is no evident association of the structure responsible for the exponential growth of the particle energy with the process of magnetic reconnection. Magnetic reconnection is a sufficient condition for generating large-scale islands where particles can be trapped. Indeed, it can be expected that when the magnetic field reconnects in a turbulent environment, the magnetic islands produced by reconnection interact, thus allowing an intense and fast energization process. However, it is apparently not necessary that reconnection be present during the energization process itself. Other configurations without the explicit invocation of magnetic reconnection, such as the interaction of two large-scale turbulent structures (e.g. flux ropes, as recently reported in recent Parker Solar Probe observations by \citet{pecora2021parker}), may provide a similar behavior, provided that the magnetic geometry of the interaction region favors particle trapping. We remark that the direct acceleration due to the electric field at the reconnection site is negligible for the relativistic particles considered in the present work, given that such particles have a gyroradius much larger than the typical width of current sheets.
\subsection{Characterization of trapping and concomitant energy gain}
\label{sec:character}
In order to characterize the coherent structure that entraps and gives a significant boost to the particle energy, we calculate the current density ${\bm j}=\nabla \times {\bm B}$ and the normalized magnetic helicity $h_m={\bm a}\cdot{\bm B}/(|{\bm a}||{\bm B}|)$, with ${\bm B}=\nabla \times {\bm a}$, interpolated at the particle position. The current density is a direct proxy of the small-scale gradients of the magnetic field, and an intense current density is expected to highlight small-scale structures and current sheets where magnetic reconnection and, in general, dissipative processes may occur \citep[see][and references therein]{pezzi2021dissipation}. On the other hand, magnetic helicity measures the topology of the magnetic field: In particular, a nonnull magnetic helicity indicates twisted, helical magnetic structures.
The values of these variables at the particle position are displayed in Figure \ref{fig:Structure}. The exponential phase is limited by the green dashed vertical lines. The structure responsible for the exponential growth of the particle energy is a relatively quiet region in which the current density is relatively smooth. In comparison, the current outside the structure
easily reaches intense values $j\gtrsim 4 j_{\rm rms}$, but such intense peaks are not evident within the structure. The magnetic fluctuations are also less intense within the structure, as the rms value of magnetic fluctuations is reduced there by a factor of $3-4$ with respect to the global value.
\begin{figure}[htp!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_Structure.pdf}
\caption{Current density ${\bm j}=\nabla \times {\bm B}$ (top), scaled to its rms value, and normalized magnetic helicity $h_m={\bm a}\cdot{\bm B}/(|{\bm a}||{\bm B}|)$ (bottom), computed at the particle position, as a function of time. The red dashed line corresponds to a large-scale current average performed over $\Delta t\simeq0.5 l_c/v_A$. }
\label{fig:Structure}
\end{figure}
The structure is furthermore characterized by a finite magnetic helicity, suggesting a flux-tube and/or plasmoid-like shape where magnetic field lines wrap helically on themselves. A finite magnetic helicity also suggests that the structure tends to be force free as ${\bm a}|| {\bm b} \rightarrow {\bm j}|| {\bm b}$, i.e. it may be a large-scale quasi-equilibrium structure typical of intermittent plasma turbulence, where nonlinearities are depleted \citep{MatthaeusEA15}.
The properties of the particle trapped in the accelerating coherent structure are also remarkable. The top panel of Figure \ref{fig:muloc} illustrates the pitch-angle cosine of the particle, here defined as
\begin{equation}
\mu_{\rm loc} = \frac{{\bm B}\cdot {\bm v}}{|{\bm B}||{\bm v}|} = \cos \theta_{{\bm v}{\bm B}},
\end{equation}
because the regular field is absent.
Concurrently with the period when the particle is trapped, its pitch-angle cosine displays reduced oscillations around the mean zero value. In fact, $\mu_{\rm loc}$ oscillations, estimated as $(\Delta \mu_{\rm loc})_{\rm rms}$, are weaker by a factor of $3-4$ inside the structure with respect to outside.
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_B0=0_muloc.pdf}
\caption{Top: time evolution of the pitch-angle cosine $\mu_{\rm loc}$. The time window associated with trapping is limited by the vertical green dashed lines. The red dashed line is the large-scale current average performed over $\Delta t\simeq0.5 l_c/v_A$. Bottom: time spectrum of the $\mu_{\rm loc}$, when the particle is trapped. The yellow shaded area corresponds to the gyroradius range experienced by the particle during this time window. }
\label{fig:muloc}
\end{figure}
The trapped particle has a peculiar motion configuration, with a pitch angle almost perpendicular to the local field. This indicates that the particle is trapped within an elongated 2D-like flux tube and, in particular, it moves in the plane perpendicular to the tube axis. As a consequence, within this period, the particle mainly experiences mainly a perpendicular energization. The evolution of the particle magnetic moment (not shown here) also supports this view because it shows a secular growth on the same timescale as the exponential energy growth of the particle, while the magnetic field is roughly constant within the same time window. Similar observations have been pointed out in the different contexts of the so-called ``second stage'' of acceleration of nonrelativistic particles \citep{DalenaEA14} and, more recently, of electron acceleration in nonrelativistic plasma turbulence \citep{trotta2020fast} and particle acceleration in relativistic plasma turbulence including radiative losses \citep{comisso2021pitch}.
To get insights into the nature of $\mu_{\rm loc}$ oscillations in the time period corresponding to trapping, the bottom panel of Figure \ref{fig:muloc} displays the Fourier time spectrum of $\mu_{\rm loc}$, performed in such a window. The dashed yellow area corresponds to the frequencies associated with the particle gyroradius, which changes within the period due to the particle energization. High-frequency $\mu_{\rm loc}$ oscillations, associated with the particle gyromotion, are combined with lower-frequency fluctuations, possibly related to smaller-scale turbulent fluctuations or other effects, such as mirroring and drifts.
To demonstrate that the most intense energization is statistically associated with a small pitch-angle cosine, Figure \ref{fig:PDFmu-Econd} displays the PDFs of the pitch-angle cosine conditioned to the particle energy. In particular, we computed the PDFs by considering the full ensemble of particles up to the time $T\simeq 22 l_c/v_A$ and by setting the threshold $E_{\rm thr}=70\% E_{\rm max}$ (blue) and $E_{\rm thr}=95\% E_{\rm max}$ (red), where
$E_{\rm max}=24.4\, {\rm PeV}$ (i.e. $r_{g,{\rm max}}/l_c=2.2$). Particles below the threshold show a distribution compatible with isotropy, with a mean pitch-angle value of $1/2$, reported in Figure \ref{fig:PDFmu-Econd} with a gray dashed line. On the other hand, the most energetic particles display a strongly anisotropic distribution, peaked at small $\mu_{\rm loc}$, becoming more evident for larger thresholds.
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{figMHD-ApJ_PDFmu_ECond.pdf}
\caption{PDFs of $\mu_{\rm loc}$ conditioned with the particle energy for the top $30\%$ of the most energetic particles (blue) and for the top $5\%$ (red), with the maximum energy being $E_{\rm max}=24.4\, {\rm PeV}$ ($r_{g,{\rm max}}/l_c=2.2$). The horizontal dashed gray line displays the $1/2$ value, corresponding to isotropic distribution. The exponential acceleration responsible for the high-energy tail is associated with a small pitch-angle cosine $\mu$. Because we are not removing particles once they enter the exponential phase, they consequently undergo the standard pitch-angle process, thus producing the observed spreading in the $\mu_{\rm loc}$ distribution.
}
\label{fig:PDFmu-Econd}
\end{figure}
This confirms that the most-energetic particles preferentially have a small local pitch-angle cosine, i.e. that they move perpendicular to the local magnetic field. As we discuss in \S \ref{sec:discussion}, this may be the very cause of the trapping: The velocity of the particles parallel to the local field is such that the particle stays in the structure long enough to be energized (while possibly drifting) by gradients in the magnetic field. During this process, the particle pitch angle changes gradually, as would be expected due to the quasi-conservation of the adiabatic invariant: Because the field changes along the trajectory (at most by the $\delta B/B_{\rm rms}$ calculated at the scale of the Larmor radius of the particles) and $v_\perp p_\perp/B$ must stay constant, then $p_\perp$ must change, hence the pitch angle changes \cite[]{volk1975} by about $\delta B/B_{\rm rms}$. This leads to a spread in the PDF of the pitch-angle distribution of the most energetic particles.
\section{Discussion of the physical processes at work}
\label{sec:discussion}
Here we discuss our results and their implications for astrophysical systems. It has been found that relativistic particles moving in stationary turbulent electric and magnetic fields, constituted by a snapshot in time of the incompressible MHD simulations, experience a complex energization process owing to the turbulent inductive electric field. The character of the process reveals at least a twofold nature. The second-order, stochastic acceleration affecting the whole ensemble of particles goes hand in hand with a first-order mechanism, impacting only a few particles that are trapped in large-scale coherent structures of turbulence. These trapped particles experience exponential energy growth until the time when the gyroradius becomes comparable with the transverse size of the structure.
The phase of exponential increase in particle energy is of special interest for its potential implications for astrophysical systems. As discussed in previous sections, this stage is rather complex: It requires that a particle enter a region where two large-scale structures seem to be interacting, thereby leading to gradients in the induced electric field and most likely in the magnetic field as well. The number of particles showing evidence of this exponential energy increase is very small, only a few out of $10^5$ ($0.001\%$). All these particles seem to have a very small pitch-angle cosine, namely $\mu\sim 0$. They also exhibit a similar phenomenology in terms of characteristic times: The trapping time is quite large ($\sim 5-10 l_c/v_A$), and various oscillations, as described in the previous section, are recovered. The formation of such structure in a generic turbulence box is difficult to characterize in a quantitative way, also due to intermittency: The standard Kolmogorov scalings could be locally violated due to spatial inhomogeneities \citep{MatthaeusEA15}. In particular, magnetic fluctuations could become weaker in large-scale coherent structures with respect to the globally averaged values because nonlinearities could be partially suppressed within large-scale structures, and perhaps elevated near flux-tube boundaries. In fact, as discussed earlier in this article, the structure in which the exponential acceleration takes place does not show evidence for special activity: it might have originated from a reconnection event, but it is not a site of reconnection. Indeed, reconnection plays no role in causing the rapid energy increase that we observe in the simulations.
Figure \ref{fig:3Dplots} illustrates very clearly how complex the region is where the particle energy is seen to increase exponentially. The only properties that we can confidently associate to this region are (1) the presence of a rather organized large-scale flux tube that seems to be interacting with another structure, and (2) a gradient in the local electric field, due to plasma motion. Moreover, it is reasonable to speculate that there may be a gradient in the magnetic field due to the interaction between structures.
Given the complexity of the situation, it may be helpful to use a toy model to illustrate the different effects and check whether the qualitative picture is reproduced. The toy model gives us the opportunity to comment on the different physical processes at work.
We show the geometry of our toy model in Figure \ref{fig:toy}: The magnetic field ${\bm B}$ in that region is assumed to be oriented in the $\hat z$ direction, but it is assumed to have a gradient in the direction of the center of the tube, within a ring (green region) of size $L_{\rm grad}$. We also assume that at least on a fraction of the surface of the tube, the plasma velocity has a gradient along the $\hat x$ direction (see zoom-in in the lower part of Figure \ref{fig:toy}). In the figure, this gradient is shown in the form of a sign reversal of the velocity but this is not required. We will comment below on the implications of a fluid velocity crossing the value $u=0$.
Let us start by discussing the role of a gradient in the magnetic field, although the existence of a strong gradient does not emerge in an evident way from the simulations: A magnetic field, with or without gradients, cannot make work on charged particles. Hence, the exponential energy increase is certainly not related to such a gradient. On the other hand, the gradient introduces a drift, whose direction depends on the direction of the gradient. With reference to the situation illustrated in the lower part of Figure \ref{fig:toy}, this grad-B drift leads the particle to move in the $y$ direction, namely to stay in the green-colored region. The drift velocity is proportional to the gradient:
\begin{equation}
{\bm v}_D\simeq\frac{pvc}{2q}\frac{{\bm B} \times {\nabla}B}{B^3}\approx
\frac{1}{2}\frac{r_L(p)v}{L_{\rm grad}} \hat y,
\label{eq:vD}
\end{equation}
namely, the lower the gradient, the lower the drift velocity, so that the particle can stay longer in the island. Note that Equation (\ref{eq:vD}) includes the grad-B drift, while the curvature drift is subdominant because we showed that the motion occurs at $\mu \sim 0$, i.e. $p_{\rm perp}\simeq p$.
Inspired by the results of our simulations, we first assume that the particle has a very small pitch-angle cosine $\mu$, namely, its motion is constrained to be in the $x-y$ plane (blue curve in Figure \ref{fig:toy}). A small value of $\mu$ means that the particle can travel in the $\hat z$ direction with velocity $\sim c\mu$ (all of the particles in the simulation are relativistic) and will eventually leave the island in a time $\sim L_{\rm isl}/c\mu$. For $\mu\sim 1$ (red curve in Figure \ref{fig:toy}), the escape time would be exceedingly fast and no appreciable acceleration can take place (see below).
At the same time that the particle rotates in the $x-y$ plane, it is advected with the local plasma, which is expected to move at speeds close to the local Alfv\'en speed $v_A$. The region of the gradient in the plasma speed (which corresponds to the gradient in the induced electric field) can then be crossed in a time $L_{\rm grad}/v_A\sim \eta L_{\rm isl}/v_A$, where we assumed that the gradient develops on a fraction $\eta$ of the size of the island, $L_{\rm grad}\sim \eta L_{\rm isl}$. The time to escape the island along $\hat z$ exceeds the time scale of advection if $\mu\eta \ll \frac{v_A}{c}$. This apparently simple and rather constraining conclusion on the pitch angle of the particles in the acceleration region is in fact affected by several phenomena that can possibly enhance the trapping: one such phenomenon is diffusion, but it is hard to imagine that it may be effective. In fact, the particles that we are considering have an initial Larmor radius of only one order of magnitude smaller than the coherence scale, which in turn is of the same of order as the size of the island, $L_{\rm isl}$. Hence, the escape time is a fraction of $l_{c}^2/(c l_c/3)\sim 3l_c/c$, where we assumed that the diffusion coefficient is not too far from that expected at $r_L\sim l_c$ (see Figure \ref{fig:Diso-sat}). The time estimated in this way is very close to the ballistic time scale (within an order of magnitude), too short to be of any relevance for CR trapping. Moreover, the fact that the pitch angle is close to zero makes diffusion ineffective, because the resonance condition would require modes with a very large wavenumber $k$, which is not accessible in our simulations, and not abundant in the standard turbulence spectrum anyway, especially after accounting for anisotropy in the cascade.
A second phenomenon that is instead expected to be more effective is associated with the existence of perturbations in the magnetic field along the $\hat z$ direction: Because on the scale of a few gyrations it is a good approximation to assume that the quantity $v_\perp p_\perp/B$ is conserved, if there are fluctuations of order $\Delta=\delta B(1/r_L)/\delta B(1/L_{\rm isl})$ in the local field, the conservation of this quantity also implies that $p_\perp$ must be changing, namely $\mu$ must be changing. Oscillations in $B$ imply oscillations in the pitch angle of magnitude $\propto \Delta^{1/2}$ \cite[]{volk1975}. Unfortunately, for the parameters of our simulation, the quantities $\Delta^{1/2}$ and $v_A/c$ are too close to identify this effect unambiguously. However, for more realistic values of the ratio $v_A/c$, the effect of oscillations associated with longitudinal gradients in the magnetic field should dominate and force particle trapping provided that $\mu$ is smaller than $\Delta$.
We finally come to the effect of the gradient in the plasma velocity. If there were a homogeneous plasma velocity $u$ in the $\hat x$ direction, this would result in an induced electric field $E_y=\frac{u(x)}{c}B$, directed in the $\hat y$ direction. Notice that as long as the plasma speed is spatially constant, one can always move into a reference frame in which this velocity is absent: Hence, the presence of this induced electric field only introduces a drift velocity in the $\hat x$ direction and no net increase in the energy of the particles can exist. In other words this drift is only the ${\bm E}\times {\bm B}$ drift and leads to the particle advection with the background plasma in the $\hat x$ direction. The case where the velocity of the plasma is not constant in the $x$ direction is more interesting. Notice that the simulations used here are incompressible, hence $\nabla\cdot {\bm u}=0$. This does not contradict the assumption that a gradient $du/dx\neq 0$ exists, for instance, if two structures are moving against each other at $u\sim v_A$.
Below we show that the existence of this drift is the very source of particle acceleration. We will reach this conclusion in two independent ways.
The evolution in time of the particle distribution function $f$ of the particles, under the effect of the $du/dx$ gradient alone, can be written as:
\begin{equation}
\frac{\partial f}{\partial t}-\frac{1}{3}\frac{du}{dx}\frac{\partial f}{\partial \xi}=0,
\label{eq:transport}
\end{equation}
where $\xi=\ln(p)$, and we used $du/dx\approx v_A/L_{\rm grad}$. If the two structures are moving against each other, a more likely estimate for the gradient would be $2v_A/L_{\rm grad}$, but the difference is only quantitative, not qualitative. Using the method of characteristics, one gets
\begin{equation}
\frac{dp}{p}=\frac{1}{3}\frac{du}{dx}dt \to p(t)\propto \exp\left[ \frac{v_A t}{3 L_{\rm grad}}\right].
\label{eq:exp1}
\end{equation}
The particle momentum is expected to grow exponentially due to the presence of a difference in velocity felt by particles during gyration around the magnetic field. One can look at this phenomenon in at least two independent ways: One way is to imagine that in each particle orbit the electric field in one half is not exactly compensated by the second half, hence there is a net electric field that can energize the particle. The second way to look at this as a first-order Fermi process, in which the particle bounces (each half-orbit) on a fluid moving with a different speed, qualitatively similar to what happens close to a shock front, where however the gradient is much larger. The acceleration has to come to an end when the gyroradius becomes comparable with the size of the trapping region unless more constraining phenomena occur on shorter time scales.
Notice that the factor of $1/3$ in Equation (\ref{eq:transport}) is derived under the assumption of particle isotropy, which clearly does not apply here. We expect that the time scale of acceleration of particles with $\mu\sim 0$ must be somewhat shorter than the $3L_{\rm grad}/v_A$ suggested by Equation (\ref{eq:exp1}).
\begin{figure}[htb!]
\centering
\includegraphics[width=\columnwidth]{trapping.pdf}
\caption{Sketch of the geometry of the flux tube interacting with another structure (top part). At the bottom is shown a zoom-in into the region where the gradient of plasma velocity is present.}
\label{fig:toy}
\end{figure}
In order to prove that our conclusion is solid, we also derive an analogous result in a more formal way, starting from the equation of motion of a particle moving in a setup as in Figure \ref{fig:toy}. In this derivation, we ignore the effect of particle drift due to the gradient of the magnetic field, hence
\begin{eqnarray}
\frac{d p_x}{dt}=\frac{q}{c} v_y B \label{eq:motionx} \\
\frac{d p_y}{dt}=-\frac{q}{c} v_x B +q \frac{u(x)}{c} B,
\label{eq:motiony}
\end{eqnarray}
where we used the expression for the local induced electric field derived above. Recalling now that ${\bm p} = m \gamma {\bm v}$, where $\gamma$ is the Lorentz factor of the particle, and multiplying Equation (\ref{eq:motionx}) by $v_x$ and Equation (\ref{eq:motiony}) by $v_y$, we can deduce that
\begin{eqnarray}
\frac{1}{2}m \gamma \frac{d v_x^2}{dt}+ m v_x^2 \frac{d\gamma}{dt}=\frac{q}{c} v_x v_y B \\
\frac{1}{2}m \gamma \frac{d v_y^2}{dt}+ m v_y^2 \frac{d\gamma}{dt}=- \frac{q}{c} v_x v_y B +\frac{q}{c}u(x)v_y B.
\label{eq:motion1}
\end{eqnarray}
Summing all terms and recalling some basic relations of relativistic kinematics, one obtains
\begin{equation}
\frac{dp}{dt}=\frac{q}{c} B v_y u(x).
\label{eq:motion2}
\end{equation}
At this point we assume that the particle velocity is already relativistic, $v\simeq c$, and that the particle trajectory is weakly modified by the gradient in $u(x)$. This means that the gradient is assumed to be weak on the scale of the Larmor gyration of the particles. In this case, Equation \ref{eq:motion2} can be averaged over many gyrations and recalling that $x(t)\propto sin(\Omega t)$ and $v_y(t)\propto sin(\Omega t)$, the average leads to
\begin{equation}
\left\langle \frac{dp}{dt}\right\rangle = \frac{1}{2} a p,
\end{equation}
where $a\approx v_A/L_{\rm grad}$ is the gradient of $u(x)$. It follows that
\begin{equation}
\langle \frac{dp}{p}\rangle = \frac{1}{2} \frac{v_A}{L_{\rm grad}} dt \to p(t)\propto \exp\left[ \frac{1}{2} \frac{v_A}{L_{\rm grad}} t \right].
\end{equation}
This result is formally the same as in Equation (\ref{eq:exp1}), but it correctly shows that the time for exponential increase of the momentum is slightly shorter than expected for an isotropic particle distribution.
In the absence of escape mechanisms, the trapping is expected to cease when the particle gyroradius becomes comparable with the island size, $r_L\sim L_{\rm isl}$. This provides a characteristic time for the duration of the energization process,
\begin{equation}
t \approx \frac{2 L_{\rm grad}}{v_A} \Lambda,
\label{eq:Texp}
\end{equation}
where
$r_{L,0}=p_0 c/e B$ is the initial particle Larmor radius and $\Lambda=\ln{\left(\frac{L_{\rm isl}}{r_{L,0}}\right)}$.
In order to have effective acceleration, the acceleration time must be shorter than all the time scales of escape parallel to $B$ and due to drifts.
By imposing that the time of escape along $\hat z$, $\tau_{{\rm cross},||}=L_{\rm isl}/c\mu$, be longer than the acceleration time, we obtain the constraint
\begin{equation}
\mu \lesssim \frac{v_A}{c} \frac{L_{\rm isl}}{2 L_{\rm grad}} \frac{1}{\Lambda}.
\label{eq:mulim}
\end{equation}
This result provides us with a simple explanation of the reason why only particles with a small value of $\mu$ actually show evidence of exponential momentum increase.
The issue of drifts is more subtle: the ${\bm E}\times {\bm B}$ drift, as discussed above, simply leads to particle advection along the ${\bm x}$ direction. The region of the gradient is then crossed in a time of order $\tau_{{\rm esc},x}\sim L_{\rm grad}/v_A$, comparable to, though shorter than, the acceleration time. However, if the two islands move against each other with roughly the same speed, $\sim v_A$, the guiding center of the particle may be advected at speeds much smaller than $v_A$, as a result of the fact that $u(x)$ crosses zero.
The drift along the $\hat y$ direction is due to the gradient in the magnetic field and in principle can be very fast because the drift velocity is given by Equation (\ref{eq:vD}), so that the time to drift out of the region where the gradient exists is of order
\begin{equation}
\tau_{{\rm esc},y}\simeq \frac{L_{\rm grad}^2}{\frac{1}{3}r_L v}\simeq
\frac{3\eta^2L_{\rm isl}}{0.1 c},
\end{equation}
where we assumed that $r_L\sim 0.1 L_{\rm isl}$. For the parameters adopted in the simulation, this time is $\tau_{{\rm esc},y}\sim 0.3 L_{\rm isl}/c$, much shorter than all other times scales. In the presence of such an effect, no appreciable acceleration should be expected. The evidence of an exponential increase in particle energy suggests that either the gradient in the magnetic field is a small fraction of $B/L_{\rm grad}$, or that it is present all along the surface of the flux tube (green region in the top part of Figure \ref{fig:toy}), so that the particle drifts around the tube while retaining its pitch angle, as discussed above.
It would be very useful to further investigate the phenomena discussed above, but because in our simulations $v_A=c/20$, such time scales are too close to each other to reach a definitive conclusion on the hierarchy of drift timescales.
We conclude our in-depth discussion of the physical processes at work in the acceleration region by commenting on the role of reconnection. As we discussed in \S \ref{sec:character}, our simulations suggest that the region where particle energization is fast is not very active: It may be the result of a reconnection event, but it is not a reconnection region. This is also to be expected: The current sheet is very thin compared with the Larmor radius of the relativistic particles considered here, hence the resistive electric fields cannot be responsible for particle acceleration. In this sense, a model like the one of \cite{degouveiadalpino2005production}, in which the escape from the region is regulated by the speed of the exhaust of the reconnection phenomenon, appears to be not well justified. On the other hand, \cite{KowalEA11} correctly pointed out that interacting islands away from reconnection events can energize particles, even exponentially. This result was, however, obtained in simulations that were optimized to create reconnection regions. We show that even in a generic 3D MHD simulation box there are structures that lead to the same physical phenomenon, and we provide a physical explanation of the phenomenon and a recipe of the conditions required for the acceleration to take place. It is also worth noticing that a first-order particle acceleration in converging islands in relativistic turbulence was also found by \cite{ComissoSironi18}. The case of relativistic particles moving in a relativistic plasma $v_A\sim c$ may be considerably different from the one described above.
The final part of this section is devoted to a discussion of the possible relevance of these phenomena for astrophysical turbulence, for instance, in the Galaxy as a whole.
In principle the exponential increase associated with phenomenon of particle trapping in interacting islands may be of great importance: For instance, if we take the turbulence in the Galaxy as our laboratory, one would expect $v_A\sim 10\, {\rm km}\, {\rm s}^{-1}$ and $L_{\rm isl}\sim\,$few tens of parsec, with a typical magnetic field of $3\,\mu$G. If a CR particle enters one such structure and suffers an exponential increase in energy up to the point where the Larmor radius equals $L_{\rm isl}$, a maximum energy of $\sim 20$ PeV would be reached, tantalizingly close to the energy of the knee. This simple numerical estimate stimulates some additional questions: What is the time scale for such acceleration? And what is the probability that a CR particle may encounter such a peculiar structure before escaping the Galaxy?
The time scale for exponential increase is as in Equation (\ref{eq:Texp}): One can see that for realistic values of the parameters, the acceleration time is $\sim (1-10) l_c/v_A$, where we assumed that $L_{\rm isl}\sim l_c$. This seems to be in accordance with the numerical results shown in Figure \ref{fig:Expgrowth} (top panel). The fact that this time is comparable to or exceeds the eddy turnaround time $l_c/v_A$ is a source of concern because both the simulations and the toy model discussed above assume that the turbulence is static (the propagation of test particles was carried out in a snapshot at a given time of the MHD simulation).
For turbulence in the Galaxy, the acceleration time estimated above is of order $\sim 1$ million years. For particles at the knee, the escape time from the Galaxy can be deduced from an extrapolation to high energies of the low-energy ($\lesssim 1$ TeV) confinement time as inferred from secondary/primary ratios \cite[]{Evoli2019} and from the $Be/B$ ratio, as discussed recently by \cite{Evoli2020}, using AMS-02 data. A reasonable estimate for such escape time at $E\sim\, {\rm PeV}$ is of $\sim 0.5$ Myr, comparable with the acceleration time. The situation can be considered somewhat more promising if one thinks that lower-energy particles (with longer confinement time) are the ones required to be trapped in the interacting islands and eventually getting energized.
Assuming that in a time of the order of the confinement time a CR particle can probe the statistical properties of the turbulence, the question arises of how many particles can potentially interact with a region where trapping occurs and particle energy increases exponentially. In the simulations we ran, only a few particles out of the $10^5$ ($0.001\%$) experienced exponential energy increase. The probability of order $10^{-5}\div 10^{-4}$ that one of the particles, while carrying out a diffusive motion, may encounter a region of size $\sim l_c$ in which there are the right conditions for trapping to occur is hard to quantify in that it is the convolution of turbulence properties (volume filling factor of islands that interact in the right way, in the presence of intermittency) and properties of the particle trajectory at the time of entering the island: As we discussed above, only particles that approach the region with very small $\mu$ can be trapped in the region for long enough to experience the exponential energy increase.
Because diffusion isotropizes the particle distribution function (outside the island), the fraction of particles that at any point in the box have a pitch-angle cosine $\mu\sim v_A/c$ is $\sim \frac{1}{2}v_A/c = 0.025$, where the numerical value refers to the conditions of the simulation, while in the Galaxy that number would be $\sim 1.7\times 10^{-5}$. The fact that only a few particles out of $10^5$ suffer exponential energy increase implies that, in the simulation, the filling factor of the island that allows such phenomenon is very small, of the order of $\sim 10^{-3}$.
A dedicated investigation of these issues would be most important as a future development of the concepts discussed in the present article and is crucial to assess the importance of these phenomena for particle acceleration in nature.
\section{Conclusions} \label{sec:conclusions}
The use of high-resolution magnetohydrodynamic simulation data in conjunction with orbit calculations of a large number of charged test particles enables detailed examination of both
spatial transport and energization in a generic 3D incompressible turbulence simulation not specifically devised to study reconnection.
The limited number of scales that can be simulated in 3D at this time makes it difficult to go much beyond the state of the art in terms of investigating particle transport: Nevertheless we determined the diffusion coefficient of relativistic particles in a range of scales where resonances are numerically accessible in the simulation. This bounds us to about one decade in energy below the energy for which the Larmor radius equals the correlation length $l_c$ of the turbulence. At such energies, it is difficult to spot the effect of anisotropic cascade with respect to the local magnetic field, typical of MHD, although such anisotropy is certainly visible when studied in terms of statistical indicators \citep{matthaeus2012local}. In this sense, our results on the diffusion coefficient are compatible with those of \cite{Cohet2016} but do not add to it. They are also compatible with the results previously obtained by \citet{SubediEA17} and \citet{dundovic2020novel} using synthetic isotropic turbulence, rather than MHD turbulence. This latter result confirms that the effects of anisotropic cascades are not yet visible on the scales accessible to particles.
Our results on particle energization are much more interesting: We find that the bulk of the test particles simulated here are subject to a secular, second-order acceleration process, due to the interaction of the particles with the random electric fields induced through plasma motion in the simulation. A few out of the $10^5$ test particles for which we simulate the trajectories happen to suffer very fast acceleration---in fact exponential in time. This process is seen to end when the Larmor radius becomes comparable with the size of the magnetic structure in which the particles reside.
The second-order stochastic acceleration process has been analyzed by employing standard techniques including computation of the running diffusion coefficient in energy space. We find empirical values for energy diffusion $D_{EE}\simeq 0.01 v_A E_0^2/l_c$, and for the associated characteristic time $\tau_{\rm diff,E} \sim 10^2 l_c/v_A$. These quantities are in qualitative agreement with naive expectations: The energy of a particle is expected to change in a random way (increase or decrease) due to the interaction with induced electric fields, so that their energy changes as $\left(\frac{\xi v_A}{c}\right)^2 E$, with $\xi\lesssim 1$, in a time that is approximately $\sim l_c/c$ (see Figure \ref{fig:Diso-sat}). It follows that the diffusion coefficient in energy can be estimated as $D_{EE}=\left(\frac{v_A}{c}\right)^2 E \frac{c}{v_A}\frac{v_A}{\xi l_c}$. For the $v_A/c=1/20$ adopted in our calculations $D_{EE}\approx \xi^2 0.05 E^2/l_c$.
We characterize further the second-order acceleration process by evaluating the time-dependent probability distribution functions, for both particle energy and time increments of particle energy. These additional tests show the second-order nature of the phenomenon.
As mentioned above, a few of the test particles in the simulation appear to have a quite different behavior, in that in addition to the slow second-order process, they happen to encounter special locations in the box where the energy is seen to increase exponentially, for times exceeding or about 10 Alfv\'en crossing times of the magnetic coherence length.
Closer examination reveals that these particles maintain a pitch angle not far from 90$^\circ$ during this period, while their spatial trajectory is highly confined and differs greatly from its more typical random walk nature. In essence, these particles have become temporarily trapped within particular turbulence structures that are characterized by strong gradients. The structures are not directly associated with regions of activity in the plasma, and in fact, if any, they show less than normal activity, although visual inspection of these regions suggests the existence of extended interaction areas with at least another structure.
We built a toy model that includes in a simplified way the main drifts and gradients in the local electric fields. The toy model shows that the main reason for rapid particle energization is a sort of first-order Fermi process in which the energy of the particles grows due to a local gradient in the plasma velocity field. Both the temporal evolution of the energization process and the typical time scales for particle acceleration, as well as the maximum energy, are qualitatively reproduced in a correct way.
\section{Acknowledgements}
P.B. and O.P. are grateful to G. Kowal, L. Sironi, and S. Servidio for very useful conversations on topics related to this article. O.P. also appreciated friendly yet significant discussions with D. Trotta. We also thank Riddhi Bandyopadhyay for his support in making use of the MHD simulations and the anonymous reviewer whose comments definitely improved our manuscript. Test-particle simulations here presented have been performed on the Newton cluster at the University of Calabria (Italy). W.H.M.'s research is partially supported by NSF-DOE Grant PHY2108834 and by the Parker Solar Probe ISOIS project under subcontract SUB0000165 from Princeton. Last but not least, O.P. thanks Maria Pezzi for her precious helpfulness.
\bibliographystyle{plainnat}
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\newcommand{\stereo} {\emph{{S}{T}{E}{R}{E}{O}}}
\newcommand{\au} {{A}{U}\ }
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\newcommand{\MHD} {{M}{H}{D}\ }
\newcommand{\mhd} {{M}{H}{D}\ }
\newcommand{\RMHD} {{R}{M}{H}{D}\ }
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\newcommand{\alfven} {{A}lfv{\'e}n\ }
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| {
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{"url":"https:\/\/anhngq.wordpress.com\/2011\/07\/11\/energy-functionals-associate-to-integrals-of-exponential-type\/","text":"# Ng\u00f4 Qu\u1ed1c Anh\n\n## July 11, 2011\n\n### Energy functionals associate to integrals of exponential\u00a0type\n\nFiled under: PDEs \u2014 Ng\u00f4 Qu\u1ed1c Anh @ 15:05\n\nThe purpose of this note is to derive some integral functionals $\\mathcal F$ associated to the following\n\n$\\displaystyle R(x)e^u$\n\nin the weak form in the sense that each critical point of $\\mathcal F$ is a weak solution for equation\n\n$\\displaystyle R(x)e^u = 0.$\n\nFor simplicitly, we denote\n\n$\\displaystyle \\mathcal I(u) = \\int_M R(x)e^udv_g$\n\nwhere $M$ is a Riemannian manifold with metric $g$ and $u$ a function sitting in an appropriate Sobolev space. To be exact, we shall find a functional $\\mathcal F$ so that its first variation, denoted by $\\delta\\mathcal F$, equals $\\mathcal I$.\n\nType 1. We shall find $\\mathcal F$ of the following form\n\n$\\displaystyle \\mathcal F(u) = C\\int_M R(x)e^udv_g$\n\nfor some constant $C$ to be determined later. To this purpose, we try to calculate the first variation of $\\mathcal F$ in the direction $v$. Indeed,\n\n$\\displaystyle\\delta \\mathcal{F}(u)(v) = C\\frac{d}{{dt}}\\int_M R (x){e^{u + tv}}d{v_g}{\\bigg|_{t = 0}} = C\\int_M R (x){e^u}vd{v_g}.$\n\nTherefore, we may choose $C=1$.\n\nType 2. We shall find $\\mathcal F$ of the following form\n\n$\\displaystyle \\mathcal F(u) = C\\ln\\left(\\int_M R(x)e^udv_g\\right)$\n\nfor some constant $C$ to be determined later. The first variation of $\\mathcal F$ in the direction $v$ can be computed as below\n\n$\\displaystyle\\begin{gathered} \\delta \\mathcal{F}(u)(v) = C\\frac{d}{{dt}}\\ln \\left( {\\int_M R (x){e^{u + tv}}d{v_g}} \\right){\\bigg|_{t = 0}} \\hfill \\\\ \\qquad= C\\dfrac{{\\displaystyle\\int_M R (x){e^{u + tv}}vd{v_g}}}{{\\displaystyle\\int_M R (x){e^{u + tv}}d{v_g}}}{\\bigg|_{t = 0}} \\hfill \\\\ \\qquad= \\dfrac{C}{{\\displaystyle\\int_M R (x){e^u}d{v_g}}}\\int_M R (x){e^u}vd{v_g}. \\hfill \\\\ \\end{gathered}$\n\nThus, in this case, we have to choose\n\n$\\displaystyle C = \\int_M R (x){e^u}d{v_g}.$","date":"2017-08-20 00:10:30","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\": 25, \"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\": 0.8355520367622375, \"perplexity\": 185.5302992996814}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886105955.66\/warc\/CC-MAIN-20170819235943-20170820015943-00082.warc.gz\"}"} | null | null |
The Diet passed on Friday a sweeping reform of nuclear inspections to allow regulators to conduct unannounced inspections of nuclear plants and give them unlimited access to needed data.
The enactment of the revised nuclear reactor regulation law comes after the International Atomic Energy Agency suggested Japan, which has been holding periodic inspections using checklists, needs a more flexible system.
The new inspection system, based on the U.S. system, will be implemented from fiscal 2020 after the Nuclear Regulation Authority sets specific rules. | {
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Azuz. Thank you for watching this D-Day.
the Soviets pushing in from the East. Less than a year later, Germany surrendered, World War II in Europe was over.
million GM vehicles had an ignition switch floor. It could shut off the car and disable airbags, power steering and any lock brakes. Wild driving.
Reserve? You`ve got three seconds, go!
On February 3rd, 2014, Janet Yellen became the first woman to lead the Fed, the Federal Reserve. That`s your answer and that`s your shoutout.
was one of the major stories we covered this year. Here`s a look back at some others.
Shutdown. At 12:01 a.m. on Tuesday, the possibility became a reality. And the sides in this U.S. government face-off were blaming each other.
major parts of the government. All because they didn`t` like one law.
conference, he`s not going to go to the constitutional event where we are supposed to come together and compromise.
UNIDENTIFIED FEMALE: It`s safe to say the rollout of Obamacare and Healthcare.gov was less than perfect.
demolished. Tacloban`s mayor said everyone he talked to had lost someone.
Chicago, this is the fifth snowiest winter ever recorded with almost 67 inches of snow so far.
snow and we are only in February.
end of apartheid and earned him the Nobel Peace Prize.
host country has done that since 1952.
It was a tumultuous weekend in Ukraine. We`ve told you about protests in the capital Kiev that led to the ouster of Ukraine`s president last month.
Most of those protesters want their country to have closer ties with Europe.
But many Ukrainians like their ousted president, want closer ties with Russia. And one region where a support for Russia is strong is Crimea.
survivors aboard Malaysia`s Airline Flight 370, which vanished more than two weeks ago.
information from a British satellite company and accident investigators indicates that plane crashed somewhere in the southern Indian Ocean.
home of the Lakers, hello to everyone watching in Nixon.
are happy to see the thunderbirds today. They are online at Connetquot High School.
sporting event. It runs from June 12 through July 13. And we`ll be - what happened there when we return.
Jumping back to the United States, there is an election this fall. Every seat in the U.S. House of Representatives and about a third of the U.S.
social studies. And you can count on CNN STUDENT NEWS for your mid-term election coverage.
almost all we have for meow, I`m Carl Azuz. It`s been a pleasure, a privilege and a blessing hosting this show for the best audience we know.
We`ll leave you with a reflection on World News from last August through now. | {
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{"url":"https:\/\/nigerianscholars.com\/past-questions\/mathematics\/question\/365934\/","text":"Home \u00bb \u00bb Find the value of x in the equation: 9x = 729\n\n# Find the value of x in the equation: 9x = 729\n\n### Question\n\nFind the value of x in the equation: 9x = 729\n\nA) 81\n\nB) 27\n\nC) 18\n\nD) 9\n\nE) 3","date":"2021-10-19 06:25:38","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8254581093788147, \"perplexity\": 410.72370358318443}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585242.44\/warc\/CC-MAIN-20211019043325-20211019073325-00338.warc.gz\"}"} | null | null |
{"url":"https:\/\/zbmath.org\/?q=an:1307.60050","text":"## Local probabilities for random walks with negative drift conditioned to stay nonnegative.(English)Zbl\u00a01307.60050\n\nSummary: Let $$\\{S_n: n\\geq 0\\}$$ with $$S_0=0$$ be a random walk with negative drift and let $$\\tau_x=\\min\\{k>0: S_k<-x\\}$$, $$x\\geq 0$$. Assuming that the distribution of i.i.d. increments of the random walk is absolutely continuous with subexponential density we find the asymptotic behavior, as $$n\\to \\infty$$, of the probabilities $$\\operatorname{P}(\\tau_x=n)$$ and $$\\operatorname{P}(S_n\\in [y,y+\\Delta), \\tau_x>n)$$ for fixed $$x$$ and various ranges of $$y$$. The case of lattice distribution of increments is considered as well.\n\n### MSC:\n\n 60G50 Sums of independent random variables; random walks 60F10 Large deviations\nFull Text:","date":"2023-01-27 01:50:27","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\": 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\": 0.8206398487091064, \"perplexity\": 256.54006850014235}, \"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\/1674764494852.95\/warc\/CC-MAIN-20230127001911-20230127031911-00455.warc.gz\"}"} | null | null |
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Health and social care, Public safety and emergencies, Coronavirus in Scotland
Addressing the impact of COVID-19.
Extra support for people affected by alcohol and drug use during the coronavirus (COVID-19) pandemic has been announced by the Minister for Public Health Joe FitzPatrick.
Funding of £166,000 from the Scottish Government will support:
• a programme to increase access to naloxone in new settings, including investment in kits to add to the existing outlets for naloxone supply
• increased capacity within the Know the Score helpline and webchat service
• a COVID-19 publicity campaign signposting support for alcohol and drug users during the pandemic
• funding to link people with others in recovery through the Scottish Recovery Consortium
Help will also be available for those affected by another's alcohol and drug use through the Supporting Families Fund. This includes a discretionary emergency fund for families, online wellbeing activities, family meals, wellbeing packs and counselling support.
Mr FitzPatrick said:
"Responding to the coronavirus pandemic is one of the biggest challenges of our lifetime and it will disrupt so many lives like never before. Protecting and supporting everyone during these unprecedented times is our priority.
"We've worked closely with organisations on the frontline to create this package of measures to ensure we continue to support those affected by alcohol and drug use.
"By working together and observing the latest guidance, we can all play our part in keeping everyone safe."
The Scottish Government worked alongside national commissioned organisations (NCOs) including Scottish Drugs Forum, Scottish Recovery Consortium, Scottish Families Affected by Alcohol and Drugs, Crew, Alcohol Focus Scotland, We Are With You and Scottish Health Action on Alcohol Problems as well as Public Health Scotland on key initiatives to address the impact of COVID-19.
The funding will come from the overall total of £93.8m which is being invested in 2020/21 to reduce harm from drug and alcohol use. This includes new funding of £20m in the Scottish Budget to ensure the Drug Deaths Taskforce and Alcohol and Drug Partnerships can respond to the needs of those who are most at risk.
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All content is available under the Open Government Licence v3.0, except for graphic assets and where otherwise stated | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,412 |
Hesperisternia vibex is een slakkensoort uit de familie van de Buccinidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1833 door Broderip.
Buccinidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,322 |
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Congress strikers dig in
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<P><font face="Times New Roman, Times, serif" size="4">Solidarity rally shows support for Chicago's HERE Local 1</font><br>
<font face="Times New Roman, Times, serif" size="5"><b>Congress strikers dig in</b></font></P>
<P><font face="Times New Roman, Times, serif" size="2"><b>By Lauren Fleer and Lee Sustar</b></font><font face="Arial, Helvetica, sans-serif" size="2"> | July 18, 2003 | Page 11</font></P>
<P><font face="Times New Roman, Times, serif" size="3">CHICAGO--More than 200 people gathered outside the Congress Hotel July 12 as the strike by Hotel Employees and Restaurant Employees Local 1 members entered its fourth week. Community and religious organizations sponsored this protest, which took its theme from the biblical story of the fall of Jericho. </P>
<P>Contingents from Jobs with Justice, Service Employees International Union Local 880, the Student-Labor Action Project and the ISO all turned out in good numbers. In a lively march, participants circled the building seven times toting signs and noisemakers, demanding that hotel management give a just contract to the 130 workers whose wages were slashed by 7 percent May 19.</P>
<P>The July 12 protest followed an earlier solidarity rally July 2 with 500 people from organized labor in attendance. Forcing low wages on workers is an old tactic by Congress Hotel owner Albert Nasser, whose Gelmart Industries runs garment sweatshops in Asia, Africa and Latin America.</P>
<P>But if Nasser's reach is international, so is solidarity for the Congress Hotel strikers. Gelmart workers in the Philippines held a two-hour rally July 2 in solidarity with the picket-line protest in Chicago.</P>
<P>Local 1 President Henry Tamarin--who was arrested on the picket line at the outset of the strike--has told workers that they face a long struggle. Neither side has initiated contact since the June 15 walkout.</P>
<P>But the workers' 24-hour picket line has impeded the hotel's ability to do business. Hotel management has stated that occupancy is averaging between only 20 and 30 percent, and this week the hotel was forced to shut down its restaurant for lunch and dinner, presumably due to an inability to staff it. This is an important victory for the picketers, who have played a direct role in ushering customers to other hotels, as it illustrates a weakness and vulnerability on the side of management.</P>
<P>Nevertheless, more is needed to win. One important way to bring pressure on management is to organize solidarity action by the unionized building maintenance staff, who are members of the International Union of Operating Engineers (IUOE) Local 399, and the security guards, who are members of the Service Employees International Union. Members of both unions are crossing the picket line, which weakens the strike.</P>
<P>So far, Local 1 hasn't asked these unions to honor their picket lines. But while management would try to target workers who take action, support from the Chicago Federation of Labor (CFL), could keep those workers from being isolated. In fact, CFL President Dennis Gannon is a member of another IUOE local in the city.</P>
<P>While Congress Hotel workers are determined to fight as long as it takes, management hopes that the low-wage workers won't be able to afford a long battle. That's why it's important to step up the solidarity action now.</P>
<P>A union victory at the Congress Hotel would provide a recent example to all Chicago workers that organizing and solidarity can beat the bosses' stranglehold. A win at the Congress would be a win for all of us, and workers from every industry have good reason to support this strike.</P>
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"redpajama_set_name": "RedPajamaGithub"
} | 1,453 |
Jocelyne ist ein Vorname.
Herkunft und Bedeutung
Der Name ist die französische weibliche Variante von Joscelin, siehe Jocelyn.
Varianten sind unter anderem Joselyn, Joslyn, Jocelin, Josceline, Josslyn, Joss (englisch), Josselin, Joceline, Josseline (französisch).
Bekannte Namensträgerinnen (Auswahl)
Jocelyne Boisseau (* 1953), französische Schauspielerin
Jocelyne Gout (* 1968), französische Fußballspielerin
Jocelyne Lamoureux (* 1989) US-amerikanische Eishockeyspielerin und -trainerin
Jocelyne Saucier (* 1948), ist eine kanadische Schriftstellerin
Jocelyne Villeton (* 1954), französische Langstreckenläuferin
Einzelnachweise
Weiblicher Vorname | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 36 |
Q: GraphQL file uploads and cURL: json.decoder.JSONDecodeError: Expecting ',' delimiter: line 1 column 188 (char 187) I have a graphql-server that is running at localhost:3001 I want to send files to a python flask server together with some variables in a cURL request. Here is what i'm sending
curl http://localhost:3001/graphql/ -F operations='{ "query": "mutation ConvertDocToPDF($input: ConvertWordDocToPDFInputType!){ convertDocToPDF(input: $input){ success } } ", "variables": { "input" : { "file": null, "saveName": "hello" } ] } }' -F map='{ "0": ["variables.input.file"] }' -F 0=@README.md
Surprisingly I'm getting an error which says:
json.decoder.JSONDecodeError: Expecting ',' delimiter: line 1 column 188 (char 187)
[11/Jan/2023 09:38:31] "POST /graphql/ HTTP/1.1" 500 107387
What may be the problem in this request. Any help input will be appreciated
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,833 |
'use strict';
const assert = require('./../../assert');
const common = require('./../../common');
let battle;
describe("Screen Cleaner", function () {
afterEach(function () {
battle.destroy();
});
it("should remove screens from both sides when sent out", function () {
battle = common.createBattle();
battle.setPlayer('p1', {team: [
{species: 'Mew', ability: 'synchronize', moves: ['reflect']},
{species: 'Mr. Mime-Galar', ability: 'screencleaner', moves: ['psychic']},
]});
battle.setPlayer('p2', {team: [
{species: 'Mew', ability: 'synchronize', moves: ['lightscreen', 'reflecttype']},
]});
battle.makeChoices('move reflect', 'move lightscreen');
battle.makeChoices('switch 2', 'move reflecttype');
assert(!battle.p1.sideConditions.reflect);
assert(!battle.p2.sideConditions.lightscreen);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,202 |
Spoiler Alert — today's post is slightly off-topic….
Like all of my fellow law bloggers–or blawgers but I refuse to use that term–I am not only the author of a lawyer blog, I am a lawyer blog fan. I have therefore followed with interest the blogs discussing law schools, new law school graduates, the employment prospects of new lawyers, the growth and contraction of the mega-firm world, and general discontent of new lawyers with life and (seemingly) everything else. The question has been asked repeatedly on some of these blogs if law school is "worth it," and various economic tests have been devised to find an answer.
Some commentators have spent a great deal of time discussing the miseries and woes of newly-minted lawyers; indeed, AboveTheLaw.com, seems to have been founded by young lawyers specifically to discuss the woes of newly minted lawyers. Law professors–always in search of something to write about–have wandered into the field, and have done some very sobering analyses of the facts of life for newly minted lawyers who want to buy a house as just one example.
When law professors and folks publishing on-line lawyer magazines and blogs talk about trouble in the legal profession that is one thing. As Michael Gambon says to Daniel Craig in the movie Layer Cake–when Craig's character learns that his idol in the criminal underworld has for many years been a police informant: How did you think these people make a living?
But when noteworthy law bloggers like Steven Harper, who have real legal careers start retiring from big law firms to write full-time about lawyer reform, well, that's another thing entirely. I have touched upon topics like law firm economics and qui tam history before in this blog, but today's post is a little different.
Some of the criticism offered by the above sources (and by other sources) is on-point. Some of it is not. All of it is worth reading, for reasons I may or may not get around to explaining in later posts. But what I want to talk about today–because I am afraid at least some members of Generations X and Y may not know this–is the fact that it always has been, and always will be, damn hard to make a living as a lawyer.
Lets start with that most important of all figures to legal education–OWHJ, or Oliver Wendell Holmes Jr. Most of us heard about him while we were applying to law school, and 100% of us heard his name on the first day of law school. OWHJ had by any measure a towering intellect. He used a word where the rest of us use a sentence. He thought in paragraphs. His judicial writings rise above the normal fodder of lawyers and are in fact nothing less than English prose. His public speeches on everything from the Civil War to the meaning of life are worth reading and re-reading.
But he was a miserable failure in the practice of law; and in fact he failed to make enough money to support himself.
From 1866 (when he was admitted to the Massachusetts Bar) to 1882 (when he accepted a position at Harvard Law School) Holmes struggled to make a living. Law Professor James Chen made headlines with his formula for calculating how much money new lawyers would need to make to payoff their loans and buy a house, but home ownership didn't become difficult for young lawyers recently, because OWHJ lived with his parents until they died.
And yes, he was married, to one Fanny Bowditch Dixwell. And yes, Mrs. OWHJ was very unhappy that her husband didn't make enough to buy thier own house. But don't take my word for it, just read Yankee from Olympus. That book is also important for another reason relevant to this post. Yankee from Olympus captures–and very well in my opinion–a snapshot of the Holmes family in all of their historical grandeur. You see, OWHJ wasn't just a great legal mind, he was a bona fide product of the Boston upper classes.
When OWHJ was a small child Henry Wadsworth Longfellow read to him; at age seven he had a first-row seat at the funeral of John Quincy Adams, the Sixth President of the United States. All three of his names–that is, Oliver, Wendell and Holmes–were given to him to make sure that everyone would know that he was part of not one but three privileged families.
In short, the family history of Oliver Wendell Holmes was nothing short of the history of Massachusetts.
So yes, to say he was well-connected doesn't quite do it–this guy knew everyone who was anyone. So if he couldn't bring in enough clients to pay the bills, how are any of the rest of us supposed to make it?
That's right, Winfield Scott–whose impression on the United States Army can still be seen to this day, and is rivaled only by George Marshall–started out as a lawyer. When he couldn't make a living in private practice, he changed careers and joined the United States Army.
And these are just two of the noteworthy people who were unable to make a living as lawyers–how many non-noteworthy lawyers failed to cut it in the practice of law in history? More than a few I assure you. How many political careers were launched by the inability to make a living as a lawyer? More than a few.
In those days, of course, you didn't need a six-figure debt to get a law license, you didn't even need an apprenticeship, although it was better to have one under your belt. Certainly the need for an expensive and lengthy education serves as a "barrier to entry" to the practice of law, and that of course is by design. You see, back in the 19th century–when the practice of law was still open to any snake-oil salesman with a distaste for physical labor–the market for legal services wasn't less crowded that it is today, it was more crowded.
So lawyers did what every other group of tradesmen did in the 19th century when their livelihood was threatened–they closed ranks and organized to protect their own. This is according to Lawrence Freidman.
The moral to the story is this–and it is sometimes said to be the oldest wisdom in the book. There ain't no such thing as a free lunch. Are there too many law schools? Yes, there certainly are, but I am betting we will see as many as one-third fewer law schools ten years from now. Do law schools charge too much for tuition nowadays? Yes, they certainly do. But they charge what the market will support; otherwise they would be out of business, and therefore the question becomes a more fundamental question about the nature of our economy and about the fundamental human desire to make money and become famous.
Simply put, I think that much of the discontent among young lawyers is because they think it is an automatic given that everything should be handed to them. They want to dial it in, so to speak, and make tons of money. They just don't understand that to make a good living in a profession, a person has to become the right blend of artist and businessperson. Too much of one or the other and you are in trouble. And you most definitely can't dial it in–it takes constant effort and constant attention.
I offer no opinion on whether a "young person" should go into law as a career, but I can say this–I am glad I chose to become a lawyer. Moreover, what is happening with the legal profession is not limited to the legal profession, as many commentators from other sectors of the economy point out. If anyone really, really wants to get at the bottom of what is going in the legal profession–and elsewhere in our economy–I think a must-read book is Tyler Cowen's The Great Stagnation. It's a short book but an important one for those of us interested in where things are headed. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,084 |
Q: CSS class will be applied different to div element and different to button element I am working on a project and I have a page with 3 buttons.
The first button is a button tag and the other 2 are div tags.
I have used a class to these 2 divs to make them look like my button tag.
The problem is that I have written the same CSS rules for the button tag and the div with the class but as you can see from the photo below they are not the same.
Here is my code:
HTML:
<!DOCTYPE html>
<html>
<head>
<title>Δομημένος προγραμματισμός</title>
<meta charset="utf-8"/>
</head>
<body>
<button onclick="exercise()">Click me</button>
<div class="button" onclick="klimakoti()">Klimakoti xreosi</div>
<div class="button" onclick="hw()">Height and weight</div>
</body>
</html>
CSS:
button{
border: 0px;
padding: 15px;
font-size: 18px;
color: green;
font-style: italic;
border-radius: 6px;
border-left:4px dashed #f22aa9;
}
.button{
border: 0px;
padding: 15px;
font-size: 18px;
color: green;
font-style: italic;
border-radius: 6px;
border-left:4px dashed #f22aa9;
}
Please help me!!!
A: You have to apply a background color to override the browser default:
button, .button {
border: 0px;
padding: 15px;
font-size: 18px;
color: green;
font-style: italic;
border-radius: 6px;
border-left:4px dashed #f22aa9;
background-color: transparent;
}
edit — if you want the <div> elements to look like the button, rather than the opposite, you have to do a few other things:
button, .button {
border: 0px;
padding: 15px;
font-size: 18px;
color: green;
font-style: italic;
font-family: sans-serif;
border-radius: 6px;
border-left:4px dashed #f22aa9;
background-color: #ddd;
margin: 0;
display: inline-block;
}
I've made sure that the font family is the same, and that they're displayed as inline elements.
A: You have to add a background color and set the display type to inline-block(, wich i believe is the default button display).
button, .button {
border: 0px;
padding: 15px;
font-size: 18px;
font-family:arial;
color: green;
font-style: italic;
border-radius: 6px;
border-left:4px dashed #f22aa9;
background-color: grey;
display:inline-block;
}
Here is the final result: http://jsfiddle.net/h25h5oz5/
A: You have to make sure that you have added a background color and added display:inline-bl
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,973 |
Carex oblanceolata är en halvgräsart som beskrevs av Tetsuo Michael Koyama. Carex oblanceolata ingår i släktet starrar, och familjen halvgräs. Inga underarter finns listade i Catalogue of Life.
Källor
Starrar
oblanceolata | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,785 |
using System;
namespace Visitor
{
public class VisitorA : Visitor
{
public VisitorA()
{
}
public override void VisitConcreteElementA(ElementA elementA)
{
Console.WriteLine("{0} visited by {1}", elementA.GetType().Name, this.GetType().Name);
}
public override void VisitConcreteElementB(ElementB elementB)
{
Console.WriteLine("{0} visited by {1}", elementB.GetType().Name, this.GetType().Name);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,913 |
\section{Introduction}\label{sec:intro}
Solar flares strongly influence space weather and their prediction using photospheric magnetic field observations has been studied extensively in recent years (e.g. \cite{abramenko05,mcateer05,schrijver07,leka07,georgoulis07,qahwaji08,colak08,barnesleka08,colak09,mason10,yu10,yang13, alghraibah15, boucheron15,bobra2015, liu17,daei17, gheibi17, liudeng17, raboonik17, nishizuka2017, nishizuka18, huang18})
In this paper we analysed a dataset of magnetic feature (MF) properties generated by Solar Monitor Active Region Tracker (SMART) \citep{higgins11}, an automated system for detecting and tracking active regions (AR) from SOHO Michelson Doppler Interferometer (MDI) magnetograms. SMART determines MF properties such as region size, total flux, flux imbalance, flux emergence rate, Schrijver's R-value and Falconer's measurement of non-potentiality. Each MF detection was classed as flaring or non-flaring if it produced a C-class or above flare within the 24 hours following the observation.
This dataset was previously analysed by \cite{ahmed11} and in this paper one of the aims was to improve on their results. We considered a number of classification approaches: binary logistic regression (LR) \citep{cox58}, which is a linear classifier, as well as the classifiers that allow for nonlinear classification rules, namely: Random Forests (RF) \citep{breiman01}, support vector machines (SVMs) \citep{vapnik98} and a set of Deep Feedforward Neural Network (DNN) architectures. We considered feature selection for the linear model: the LR classifier was applied to a small subset of MF properties selected by a marginal relevance (MR) criteria \citep{dudoit02} and the full feature set. We also used Lasso \citep{tibshirani96}, a model related to LR that simultaneously performs classification and feature selection.
To assess how the results of a predictive model will generalize to an independent dataset we used cross-validation where the training of algorithms and feature selection are carried out on the training set and the presented results are shown for the test set. True Skill Scores (TSS), Heidke Skill Score (HSS), Receiver Operating Characteristic (ROC) curves and Area Under ROC curve (AUC) were used as measures of classifier performance.
For the dataset analysed by \cite{ahmed11} we found that the linear classifier using only the top three features selected by MR yielded good classification rates with the highest TSS of 0.84, sensitivity (recall) of $95\%$ and specificity of $89\%$. This is a significant improvement on the previous analysis of this data. None of the other approaches that we considered exceeded this performance.
SMART detects MFs automatically and independently from NOAA active regions. A large number of detections are small magnetic flux regions that have no associated sunspot structure and do not possess many of the properties that SMART calculates, yielding values close to zero for some of the features. These detections never flare and it is relatively easy for a forecasting system to get them correct. In order to compare our results to a broader section of the flare forecasting literature, we analysed a second set of results that correspond to SMART detections which are NOAA active regions by initially filtering the SMART dataset. For this reduced dataset, the same linear classifier with the top three features selected by MR yielded TSS of 0.67 with corresponding sensitivity and specificity of $87\%$ and $80\%$. None of the other models, including more comprehensive searches of the feature space and nonlinear classifiers, were able to improve on this performance.
Based on the classification results as well as the visualization of the data we show that there is no advantage in including a larger number of features or fitting more complex, non-linear models for these datasets.
For comparison with \cite{ahmed11} we used the same split of the data into training and testing sets. For the full dataset of all SMART detection, the training set is large, comprising of 330,000 instances and as such puts constraints on the choice of classifier methodology. For example, kernel based classifiers such as SVM require the computation of an $n \times n$ dimensional kernel matrix and do not scale well to data where $n$, the number of instances is large. To evade this issue we took the approach of subsampling from the full training set to construct 50 smaller training sets of 400 instances each. The SVMs were trained on these small training sets whereas DNNs involve highly parameterized models and were trained on the full training set. LR and RF were trained on the small subsampled training sets and the full training set. We found that using the entire training set to train the algorithms gives no improvement in test classification rates.
The analysis of the data was done in R, a free software environment for statistical computing and graphics \citep{R17}. The graphical displays were produced using ggplot2 \citep{wickham} and plot3D \citep{soetaert17} packages. The code to reproduce the analysis and graphics can be accessed at \url{https://github.com/domijan/Sola}.
The paper is organized as follows: in Section~\ref{sec:data} we describe the dataset, in Section~\ref{sec:methods} we briefly outline the method used for feature selection; the classification algorithms; the cross-validation settings for assessing classifier performance and the forecast performance measures. Section~\ref{sec:results} presents the results and in Section~\ref{sec:discussion} we make some concluding comments.
\section{Data}\label{sec:data}
Data are line-of-sight magnetograms from \emph{SOHO}/MDI. Magnetic feature properties were extracted by Solar Monitor Active Region Tracking algorithm \citep{higgins11}. Flares are from \emph{Geostationary Operational Environmental Satellite (GOES)} soft X-ray (1--8\,\AA) flare lists provided by NOAA/SWPC.
SMART detects MFs automatically and independently from NOAA active regions. Following \citep{ahmed11} we defined an ``MF detection'' as an individual SMART MF detected in one MDI magnetogram. Each MF detection was classified as flaring or non-flaring if it produced a C-class or above flare within the 24 hours following the observation. In order to minimize the error caused by projection effects, only MF detections located within 45 deg from solar disc center were considered. The dataset comprises of MF detections generated by SMART from April 1996 - Dec 2010. A list of SMART MF features used in this analysis with descriptions is given in Table~\ref{table:features}.
In this paper we study two datasets: the ``full SMART dataset'' of all MF detections generated by SMART from 1996 April 1 to 2010 December 31 and a ``NOAA AR dataset'' containing of only those SMART MF detections that can be associated with NOAA ARs. The second dataset is derived by retaining only those SMART MF detections whose boundaries encompass the coordinates of one or more NOAA ARs after these were time-rotated to the MDI observation times used by SMART.
\section{Methods}\label{sec:methods}
\subsection{Classification algorithms}\label{sec:class}
\textbf{Logistic regression} is a well established framework for modelling and prediction of data where the response variable of interest is binary. It is a subset of the Generalized Linear Models (GLMs)\citep{NeldWedd72} that are widely used across a range of scientific disciplines and are available in almost all statistical software packages.
For each MF detection in a training dataset, we have a feature vector $\bm{X}_{i}$ and an observed class label $Y_i \in \{0,1\}$, denoting if it produced a flare. The distribution of of $Y_i$ is modeled by a Bernoulli ($p_i$) distribution, where $p_i = P(Y_i = 1 |\bm{X}_{i}, \bm{\beta})$ denotes the probability of flare and $\bm{\beta}$ is the parameter vector. In this model we use the logit link, where $p_i$ is the logistic function of a linear combination of the explanatory features:
$$p_i = \frac{1}{1+e^{\beta_0+ \bm{\beta} \bm{X}_{i}}}.$$
The model coefficients $\bm{\beta}$ are estimated using maximum likelihood and are used to estimate $p_i$. The class of a MF detection $i$ can be predicted by thresholding the estimated $p_i$ at a particular value therefore giving a linear classification algorithm. As such, LR is a related model to Fisher's linear discriminant analysis (LDA) \citep{fisher36} and SVMs with linear kernels. Other classification methods that we considered were chosen because they take very different approaches to inducing nonlinearity, feature selection and model-fitting.
LR is an example of a feedforward neural network architecture with a single neuronal unit, single layer, and sigmoid activation function.
By adding units and layers, the neural networks extend the LR model to complex models with non-linear classification boundaries. The architectures with two or more hidden layers are generally called \textbf{deep neural networks} (DNNs) \citep{geron18}. There are many types of DNN architectures and these models that have been successfully applied in the domains of text and image analysis. In this paper we employed a multi-layer perceptron (MLP) which consists of a sequence of densly connected layers of neurons. This is the classic architecture for the data where the feature vector does not have a hierarchical structure, as is the case for image or text data.
In this paper we considered a range of fully connected layers architectures: two hidden layers with 8 and 4 units (DNN\_8\_4), two hidden layers with 16 units each (DNN\_16\_16), two hidden layers with 256 and 32 units (DNN\_256\_32) and three hidden layers with 13, 6 and 6 units each (DNN\_13\_6\_6). We chose {\em tanh} activations for the hidden units. The output layer for each of the networks is a single sigmoid unit and the loss function is set as the binary cross-entropy. The networks have been trained over 200 epochs with a mini-batch size of 1024 and using the ADAM optimization strategy.
We did consider deeper architectures, but they were overfitting the data and we also considered different activation functions but there was no difference in the algorithm performance.
A \textbf{support vector machine} (SVM) is a kernel extension of a binary linear classifier that constructs a hyperplane to separate two classes. The hyperplane is chosen so that the smallest perpendicular distance of the training data to the hyperplane (margin) is maximized. A tuning parameter (cost) controls the number of observations that are allowed to violate the margin or the hyperplane. Kernel trick is a general technique that can be applied to any optimization problem which can be rewritten so that it takes the inner products between pairs of the training observations as opposed to the observations themselves. When the inner product is replaced with a more general kernel, the observations are implicitly mapped to a higher dimensional feature space where the optimization takes place. For linear classifiers, this has the effect of fitting nonlinear decision boundaries in the original feature space. The shape of the boundary is determined by the choice of kernel and its parameterization.
\textbf{Random Forests} (RF) provide a different approach to the classification problem and to feature selection. They grow a number of decision trees on bootstrapped samples of the training set. Each tree recursively partitions the feature space into rectangular subregions where the predicted class is the most common occurring class. At each iteration, a tree algorithm searches through all the possible split-points along a randomly selected subset of features to find a partition which minimizes the region impurity, measured by the Gini index. For a binary problem, the Gini index is given by: $2\hat{p}_m(1-\hat{p}_m)$, where $\hat{p}_m$ is the proportion of flaring observations in region $m$. A single consensus prediction is obtained from all the trees using majority vote which allows for very complex and nonlinear decision boundaries.
The total decrease in the Gini index from splitting on a feature, averaged over all trees can be taken as an estimate of that feature's importance.
\subsection{Feature selection}
For this study, we use the Marginal Relevance (MR) score to rank the features in order of their capability to discriminate between the two classes (flare/non-flare). The MR score for each feature is the ratio of the between-class to within-class sum of squares. This idea underpins many statistical methodologies and is frequently used in genetics to screen out a large number of spurious features, see, for example, \cite{dudoit02}.
The approach to feature selection using MR screens out the unnecessary features before applying logistic regression. MR considers the information in each feature independently so the highest ranked features can be correlated and do not necessarily form the optimal subset for the purposes of classification.
For these datasets we also fitted Lasso \citep{tibshirani96}, a model related to LR, but where the coefficients $\beta$ are simultaneously shrunk to zero using a penalty which is controlled by a tuning parameter. Lasso provides a more sophisticated approach to feature selection than MR and simultaneously reduces dimensionality of the feature space and performs classification. Lasso is implemented in R package glmnet \citep{friedman09}.
All features were used to train the nonlinear classifiers. SVMs combine all the feature information into a distance matrix (kernel) and can cope with correlated inputs and a small number of spurious features. In RF all the features are used to grow the trees and at each iteration randomly selected subsets are jointly considered for subdividing the feature space.
\subsection{Cross-validation}\label{sec:CV}
For consistency and comparison with \cite{ahmed11} we use the MF detections from April 1996 - December 2000 and January 2003 - December 2008 to train the classification algorithms and the MF detections from January 2001 - December 2002 and January 2009 to December 2010 comprise the test set.
The number of flaring/non-flaring SMART detections in the training and testing sets for both the full and NOAA AR data are shown in Table~\ref{table:detections}.
The training set is further subsampled by randomly drawing 200 instances of flares and 200 instances of non-flares to form 50 smaller training sets.
The full SMART dataset contains 490,997 non-flaring and 27,244 (5.4\%) flaring instances and therefore exhibits a large class imbalance. In the NOAA AR dataset 18.6\% of detections were classed as flares. Class imbalance is a common problem and has received a great deal of attention in classification literature, see, for example \cite{chawla04}. In construction of the subsampled training sets we uniformly sampled instances of flares and non-flares but adjusted the mixture of the classes, an approach known as case-control sampling. Logistic regression models fitted to subsamples can be converted to a valid model using a simple adjustment to the intercept, see \cite{fithian14}.
\begin{table}[ht]
\caption{The number of flaring/non-flaring SMART detections in the full and reduced datasets.}\label{table:detections}
\begin{tabular}{lllll}
\hline
& \multicolumn{2}{l}{Full SMART dataset}& \multicolumn{2}{l}{NOAA AR dataset}\\
\hline
& Training set & Testing set & Training set & Testing set \\
\hline
flare & 16673 &10571 &1137 &707 \\
non-flare& 313617 & 177380&5272 & 2789 \\
\hline
\end{tabular}
\end{table}
\subsection{Forecast performance measures}\label{sec:forecast}
In a binary classification problem we can designate one outcome as positive (flare) and the other as negative (no flare). For algorithms with probabilistic outputs, binary forecasts are obtained by thresholding $p$, e.g. predicting a flare if the estimated $p>0.5$. A confusion matrix is constructed by cross-tabulating the predicted with the observed classes. This presents the number of true positives TP (flare predicted and observed), false positives FP (flare predicted but not observed), true negatives TN (no flare predicted and none observed) and false negatives FN (no flare predicted but observed).
The true positive rate (TPR), or sensitivity, is the proportion of correctly classified flares out of all the flares observed in the sample TPR = TP/ (TP+FN). The true negative rate (TNR), or specificity, is the proportion of true negatives out of all the non-flaring instances.
The false positive rate is FPR = 1 - TNR and the false negative rate is FNR = 1 - TPR.
A classifier that performs well will give a high TPR and TNR and, consequently, low FPR and FNR. For classifiers that give probabilistic outputs, sensitivity (TPR) can be increased by lowering the threshold of $p$, but this automatically increases the FPR. An optimal choice of the threshold is context dependent: the cost of FNs might be higher than FPs.
For a ($0,1$) range of thresholds, receiver operating characteristic (ROC) curve plots the TPR vs FPR. ROC curve and the corresponding area under the ROC curve (AUC) are used for comparing the performance of algorithms over the entire range of thresholds. The ideal ROC curve is in the top left corner, giving high TPR and a low FPR, and the maximum possible value for AUC is 1.
For a single threshold or for classifiers with non-probabilistic outputs, the elements of the confusion matrix can be combined in a number of ways to obtain a single measure of the performance of a given method.
Accuracy (ACC) gives the proportion of correctly classified observations over both classes.
True skill statistic (TSS), \citep{youden50, hanssen65} combines the sensitivity and specificity by taking TSS = TPR + TNR - 1.
Heidke skill score (HSS) \citep{heidke26} measures the fraction of correct predictions after adjusting for the predictions which would be correct due to random chance.
For more details on forecast performance measures used in solar flare literature see \cite{bloomfield12, barnesleka08, allClear16}.
\section{Results}\label{sec:results}
LR with top three features selected by MR (LR) and SVMs were trained on 50 subsampled training sets. The results for the SVM were obtained from R package e1071 \citep{e1071}, with a Gaussian kernel with the bandwidth parameter set to $0.03$ for the full SMART dataset and $0.01$ for the NOAA AR dataset. The cost of constraints violation was set to $1$.
LR with top three features selected by MR (LR3), the full set of features (LR13), RF and DNN were trained on the full training set. The DNN architectures were: two layers with 8 and 4 units (DNN\_8\_4), two layers with 16 units each (DNN\_16\_16), two layers, 256 and 32 units (DNN\_256\_32) and three layers 13,6 and 6 units each (DNN\_13\_6\_6).
For RF, $500$ trees were grown, where at each iteration three variables were randomly sampled as candidates for each split. The tuning parameters of RF and SVMs were tuned over grids using cross-validation of the training set. For support vector machines (SVMs) we tried out Gaussian, Anova and Laplacian kernels and Bayesian Kernel Projection Classifier (BKPC) \citep{domijan09}, a sparse Bayesian variant using lower dimensional projections of the data in the feature space. All three kernels and BKPC performed equally well in the training set at the optimal values of their kernel parameters so, in the paper we present the results of the Gaussian kernel as it is best known.
RF is implemented in R library randomForest \citep{liaw02}. MR algorithm is implemented in R library BKPC \citep{domijan16}. DNNs were fitted using Keras library in R \citep{keras17}.
For the purposes of analysis, some of the features were log transformed (high-gradient neutral-line length in the region, neutral-line length in the region, Falconer's $WL_{SG}$ value, Schrijver's $R$ value, total un-signed magnetic flux and flux emergence rate). Same transformations were found to be adequate for both the full SMART dataset and the NOAA AR dataset. For both datasets, the features in the training sets were scaled to have zero mean and unit variance and the same scaling was then applied to the test sets.
The forecast performance measures (TPR, TNR, TSS, ACC and HSS), described in Section~\ref{sec:forecast}, were calculated for the test set at a range of thresholds of $p$.
For the classifiers trained on 50 subsampled training sets, 50 classification rules were obtained and consequently the median values, $2.5$th and $97.5$th percentiles of the resulting forecast performance measures are reported. This can be used to assess the sensitivity of the algorithms to the choice of the training sets.
\subsection{Full SMART dataset}\label{sec:resultsFULL}
\begin{figure}[t!]
\centerline{
\hspace*{0.015\textwidth}
\includegraphics[width=7.5cm, height=5cm]{ROCcompareV3.pdf}
\hspace*{-0.03\textwidth}
\includegraphics[width=7.5cm, height=5cm]{ROCcompareRDV3.pdf}
}
\vspace{-0.42\textwidth}
\centerline{\Large \bf
\hspace{-0.088 \textwidth} \small{(a)}
\hspace{0.545\textwidth} \small{(b)}
\hfill}
\vspace{0.38\textwidth}
\caption{ROC curves, calculated for the training set for LR (LR with 3 features trained on subsamples), LR3 (LR with 3 features trained on the full training set), LR13 (LR with all features trained on the full training set), RF, SVM and the DNN architectures. (a) Full SMART dataset, (b) NOAA ARs dataset. }\label{fig:roccompare}
\end{figure}
Figure~\ref{fig:roccompare}(a) shows the ROC curves plotted for all the classifiers for the full SMART dataset. For the algorithms trained on 50 subsampled training sets (LR and SVM), the ROC curve is obtained from the median TPR and FPR over the $(0,1)$ range of thresholds. The ROC curves are very close and show that after careful tuning, all models perform equally well and converge to the same results in terms of the performance measures.
\begin{table}[t]
\caption{AUC, the highest TSS and HSS for all the classifiers. The first two (LR and SVM) were fitted to subsampled training sets. The rest were trained on the full training set. }\label{table:TopTSS}
\begin{tabular}{llll}
\hline
Classifier & TSS& HSS & AUC\\
\hline
LR & 0.84 (0.83, 0.84)& 0.63 (0.60, 0.64)& 0.966 (0.962, 0.968)\\
SVM & 0.83 (0.83, 0.84)& 0.56 (0.51, 0.60)& 0.949 (0.942, 0.956) \\
LR3 & 0.84 & 0.64& 0.967\\
LR13 & 0.84 & 0.64&0.967\\
RF & 0.83 & 0.63&0.964\\
DNN\_8\_4 & 0.83 & 0.63& 0.966\\
DNN\_16\_16 & 0.83 & 0.63&0.966\\
DNN\_256\_32 & 0.83 & 0.63&0.966\\
DNN\_13\_6\_6 & 0.83 & 0.64&0.965\\
\hline
\end{tabular}
\end{table}
AUC, the highest TSS and HSS for all classifiers for the full SMART data are given in Table~\ref{table:TopTSS}. For algorithms trained on 50 subsamples, the reported value is the median with $2.5$th and $97.5$th percentiles given in brackets. AUC ranged from 0.949 to 0.967, TSS ranged from 0.83 to 0.84 and HSS ranged from 0.56 to 0.64. This again shows that no model convincingly outperformed the others in terms of predictive ability for this dataset. The linear model with only three features works as well as the more complex models that allow for nonlinear classification boundaries. Likewise, including extra features in the linear model did not improve performance. The results for LR with three features are the same for the algorithm trained on the subsampled training sets and the full training set (LR and LR3), showing that small datasets of 400 instances are sufficient to train this model. Narrow confidence bands for LR and SVM indicate that the classification results are consistent across the subsampled training sets.
For the logistic regression algorithm with the three input features trained on the subsampled training sets (LR) the median values for TSS, TPR, TNR, ACC and HSS at each threshold are presented in Table~\ref{table:test} in the Appendix. The 2.5th and 97.5th percentiles for TSS and HSS are given in brackets. For comparison, Table~\ref{table:testLRfull} presents the results for the same algorithm trained on the full training set (LR3).
For LR, the highest TSS of 0.84 is obtained at the thresholds between 0.04 to 0.08 which give the TPR in the range of 0.96 and 0.92 and TNR of 0.88 and 0.91 respectively. The results from \cite{ahmed11} give a TPR of 0.523, TNR of 0.989 with HSS of 0.595 for the machine learning algorithm and TPR of 0.814 and HSS of 0.512 (TNR is not reported) for the automated solar activity prediction (ASAP).
Area under ROC curve (not reported by \cite{ahmed11}) was calculated for the 50 curves and the median AUC value for LR is 0.966 with 2.5th and 97.5th percentiles of 0.962, 0.968. For the same model trained on the full training set using three features (LR3), the AUC was 0.967. Using all features in the model (LR13) did not increase AUC from 0.967.
\subsection{Choice of skill scores and threshold}
The LR model fits a sigmoid surface over the range of $\bm{X}$ and the decision boundary separating the two predicted classes at any threshold is linear.
Note that in this dataset a single MF is tracked though time and will be recorded multiple times throughout its lifetime. For the purpose of this analysis, all MF detections are treated as individual measurements. This can pose a problem for interpreting the probabilistic inference of LR which is underpinned by the independence assumption: the standard error estimates for the coefficients are no longer reliable and $p$ cannot be interpreted as probability. However, in this paper, we do not make use of probabilistic inference and we treat the logistic regression model as a deterministic linear classifier, where the choice of thresholding value of estimated $p$ is based on context requirements: comparing the acceptable levels of true positive and true negative rates for different thresholds. Furthermore, by training algorithms on very small subsets of the original data (100 instances of flares randomly drawn from 16,673 and 300 non-flares from 313,617) one is unlikely to get many detections of the same MF in the same sub-sample, which helps evade this problem. Alternatively, one could enforce the randomly drawn detections to have large enough time cadences between them (e.g. more than two weeks) in order to ensure that the same MF is not recorded in the same training set multiple times throughout its lifetime.
Figure~\ref{fig:SENSSPECfull} shows the calculated sensitivity (TPR), specificity (TNR), ACC, TSS and HSS over a range of probability thresholds (0.01 to 0.99 in steps of 0.01), where $\hat{p}$ was estimated from the LR3 model. The proportion of flaring instances in the full training set is $0.05$, shown as the vertical line on the graph. This figure shows increase in TNR at the expense of TPR with increase of the threshold. At the lowest threshold $p = 0.01$, the $82\%$ of non-flares are correctly identified and this increases to $89\%$ at 0.05. Likewise, $98\%$ of flares are correctly predicted at the threshold of 0.01 and $95\%$ are correctly identified at the threshold of 0.05. The maximum estimated TSS = TNR + TPR - 1 is 0.84. Accuracy, a measure of overall error rate is maximized at the threshold of $p=0.5$, which gives TNR of $99\%$, but misclassifies $50\%$ of the flares.
This illustrates how ACC is a very poor choice of metric for data with a large class imbalance. At threshold of 1, classifying all observations as negative, one will still get a $95\%$ accuracy score and choosing a threshold of $p=0.5$ will misclassify over half the flare detections. Given that the assumptions for probabilistic inference in LR are met, the $p$ estimates the likelihood that an observation is going to flare given its feature information. Before fitting the model and utilizing the feature information, the probability that a randomly selected detection will flare is 0.05. Unlike LDA, the prior information about flare prevalence is not incorporated in the model. Thus it is sensible to take this prior as a threshold as opposed to $0.5$. For this threshold, the algorithm will classify a detection as flare if the estimated $p$ is greater than the probability we would assign to a randomly selected detection. Figure~\ref{fig:SENSSPECfull} shows that this threshold strikes a sensible balance between TPR and TNR and indicates that TSS is a better choice than HSS for imbalanced data. HSS is maximized at a higher threshold of $0.3$ with TPR of 0.72.
ROC curves allow for comparison of classifiers with probabilistic outputs over the range of thresholds. AUC summarises the forecast performance in a single score, however, one could argue that comparing algorithm performance at an `optimal' threshold is more useful than over the entire range. When comparing classifier performance using skill scores, we argue that it is useful to plot skill score curves over the threshold range.
\begin{figure}[t!]
\centerline{
\includegraphics[width=10cm, height=7cm]{ChoosePV3.pdf}
}
\caption{Sensitivity (TPR), specificity (TNR), accuracy, TSS and HSS over a range of probability thresholds. Estimated probability was obtained from the logistic regression with three features on the full SMART dataset. The proportion of flares in the training dataset is 0.05 (vertical line). }\label{fig:SENSSPECfull}
\end{figure}
Figure~\ref{fig:tsscompare} shows the TSS curves for different classifiers (for LR the curve is median TSS with $2.5$th and $97.5$th percentile band). Figure~\ref{fig:tsscompare}(a) shows the TSS curves from the algorithms trained and tested on the full SMART dataset. The maximum TSS values obtained from all classifiers are very close (ranging from 0.82-0.84) but are obtained at different thresholds for $p$ since the TSS curves differ in shape. This is due to the fact that the probability of flare is estimated differently in these models. For the RF the estimate of $p$ is non-parametric. By default SVM produces categorical outputs, however probabilistic extensions exist and the R implementation fits a probabilistic regression model that assumes (zero-mean) Laplace-distributed errors for the predictions. For DNNs, in order to balance the classes, the class weight of 20 was used for the flare class labels in the computation of the loss function, which is equivalent to upsampling to match the majority class. Therefore, for the DNN models, TSS is optimised at a threshold of 0.5.
\begin{figure}[t!]
\centerline{
\hspace*{0.015\textwidth}
\includegraphics[width=7.5cm, height=5cm]{TSScompareFullV3.pdf}
\hspace*{-0.03\textwidth}
\includegraphics[width=7.5cm, height=5cm]{TSScompareRDV3.pdf}
}
\vspace{-0.42\textwidth}
\centerline{\Large \bf
\hspace{-0.095 \textwidth} \small{(a)}
\hspace{0.545\textwidth} \small{(b)}
\hfill}
\vspace{0.38\textwidth}
\caption{TSS or median TSS curve for all the classifiers. LR (LR with 3 features trained on subsamples), LR3 (LR with 3 features trained on the full training set), LR13 (LR with all features trained on the full training set), RF, SVM and four DNN architectures. TSS curve for LR (full line) has $2.5$th and $97.5$th percentile band. (a) Full SMART dataset, (b) NOAA AR dataset.}\label{fig:tsscompare}
\end{figure}
\subsection{NOAA AR dataset}\label{sec:resultsRED}
Figure~\ref{fig:roccompare}(b) shows the ROC curves plotted for all the classifiers for the NOAA AR data. TSS curves are given in Figure~\ref{fig:tsscompare}(b). For LR the curve is median TSS with $2.5$th and $97.5$th percentile band. Compared to the results of the algorithms trained and tested on the full SMART data, the performance of all algorithms is significantly weaker if trained and tested on the NOAA AR dataset, however this is still competitive with the results reported elsewhere in the solar flare forecasting literature, for example see \cite{allClear16}.
\begin{table}[t]
\caption{NOAA AR data: AUC, the highest TSS and HSS for all the classifiers. The first two (LR and SVM) were fitted to subsampled training sets. The rest were trained on the full training set. }\label{table:TopTSSRD}
\begin{tabular}{llll}
\hline
Classifier & TSS& HSS & AUC\\
\hline
LR & 0.66 (0.658, 0.664)& 0.59 (0.57, 0.59)& 0.90 (0.90, 0.90)\\
SVM & 0.64 (0.63, 0.66)& 0.57 (0.55, 0.58)& 0.90 (0.89, 0.90) \\
LR3 & 0.66 & 0.59& 0.91\\
LR13 & 0.67 & 0.58& 0.91\\
RF & 0.64 & 0.57&0.90\\
DNN\_8\_4 & 0.66 & 0.57& 0.90\\
DNN\_16\_16 & 0.67 & 0.59&0.90\\
DNN\_256\_32 & 0.66 & 0.59&0.90\\
DNN\_13\_6\_6 & 0.66 & 0.58&0.90\\
\hline
\end{tabular}
\end{table}
For the NOAA AR dataset, AUC, the highest TSS and HSS for all classifiers are given in Table~\ref{table:TopTSSRD}. AUC ranged from 0.9 to 0.91, TSS ranged from 0.64 to 0.67 and HSS ranged from 0.57 to 0.59. For logistic regression algorithms (LR, LR3, LR13) and RF, the TSS is optimised at the threshold of $p \approx 0.18$ which is the prevalence of flares in the training set of this data. For DNN architectures the class weight of 5 was used for the flare class labels in the computation of the loss function. For full output of LR and LR13 results, see Table~\ref{table:testNOAA} and Table~\ref{table:testDNNfull} in the Appendix).
The results show that the algorithms with top three features (LR and LR3) perform as well as the linear model with all features (LR13) and all the nonlinear algorithms (DNNs, RF and SVM). In addition, small datasets of 400 instances are sufficient to train the linear model. Narrow confidence bands for LR and SVM indicate that the classification results are consistent across the subsampled training sets.
\subsection{Feature analysis and selection}
The marginal relevance score for each feature was derived from the data used to train the classification algorithm (detections recorded from April 1996 - December 2000 and January 2003 - December 2008). Features in order of their marginal relevance derived from the full and the NOAA AR dataset are given in Table~\ref{table:features}. The third and fourth columns give the importance order of the top six features obtained from the Random Forest in both datasets. The top features selected by Lasso are given in columns five and six. Column titles with (R) denotes the importance measures were derived for the reduced NOAA AR dataset.
MR selects high-gradient neutral-line length in the region (LsgMm), maximum gradient along polarity inversion line (MxGradGpMm) and neutral-line length in the region (LnlMm) as the top three features for both full SMART and NOAA AR dataset. For both datasets, the best performing and most parsimonious Lasso model had three features, but selected area of the region (AreaMmsq) or total un-signed magnetic flux (BfluxMx) instead of maximum gradient along PIL (MxGradGpMm). These algorithms had the same classification performance as LR3 and LR13. In addition to neutral-line length in the region (LnlMm), features with highest importance selected by RF were Schrijver's $R$ value (RvalMx), total un-signed magnetic flux (BfluxMx) and Falconer's $WL_{SG}$ value (WLsgGpMm).
The forecast performance measures for the Lasso models, RFs and LR with three or more features were similar in both datasets. Many other approaches to feature selection exist, but this indicates that there is little scope for improvement with more thorough exploration of the feature space.
\begin{sidewaystable}[ht]
\caption{SMART magnetic features in order of their marginal relevance obtained from the training dataset for the dataset of all SMART detections.
The second column MR(R) is the MR feature ranking obtained from the training set of the NOAA AR detections only. Columns RF and RF(R) present the variable importance order obtained from running the Random Forest on the training set of the full and NOAA AR (R) datasets. Columns Lasso and Lasso (R) present the features selected in the four sparsest models fitted to the training set of the full and NOAA AR (R) datasets.}\label{table:features}
\begin{center}
\small{
\begin{tabular}{rrrrrrll}
\hline
MR & MR(R) & RF& RF(R)&Lasso&Lasso(R)& Feature & Description\\
\hline
1 &1 &5 &5 & 1 &1& LsgMm &High-gradient neutral-line length in the region\\
2 &3 &$-$&6 & $-$&$-$& MxGradGpMm & Maximum gradient along polarity inversion line \\
3 &2 &1 &1 & 1 &1& LnlMm & Neutral-line length in the region\\
4 &5 &4 &3 &$-$&$-$& WLsgGpMm & Falconer's $WL_{SG}$ value\\
5 &6 &6 &$-$&1& $-$&AreaMmsq & Area of the region\\
6 &7 &2 &2&$-$&$-$& RvalMx & Schrijver's $R$ value\\
7 &9 &$-$&$-$&$-$&$-$& B & Largest magnetic field value \\
8 &8 &$-$&$-$&$-$& $-$&HGlonwdth & Heliographic longitudinal extent\\
9 &4 &3 &4&$-$& 1&BfluxMx & Total un-signed magnetic flux\\
10&10 &$-$&$-$&$-$&$-$& HGlatwdth & Heliographic latitudinal extent\\
11&11 &$-$&$-$&$-$& $-$&MednGrad & Median gradient along the neutral line \\
12&12 &$-$&$-$ &$-$& $-$&Bfluximb & Flux imbalance fraction in the region\\
13&13 &$-$&$-$&$-$& $-$&DBfluxDtMx & Flux emergence rate\\
\hline
\end{tabular}
}
\end{center}
\end{sidewaystable}
Figure~\ref{fig:topfeats} and \ref{fig:topfeats2} in the Appendix plot the marginal densities from the training set of the full SMART dataset and NOAA AR dataset of some of the top features selected by MR, Lasso and RF. Large peaks close to 0 in the non-flare distribution show that the full SMART dataset is dominated by the small magnetic flux regions that do possess many of the properties that SMART calculates and never flare.
All the displayed features contain information on whether an observation is likely to be a flare, however, all contain a significant amount of overlap in the distributions of flaring and non-flaring regions. Schrijver's $R$ and Falconer's $WL_{SG}$ values show a clear separation between two clusters, one of which has a very low proportion of flares, whereas the density plots for most other features indicate a steady increase in the proportion of flares, which gives an insight into why splitting on these variables would lead to a decrease in the Gini index.
Figure~\ref{fig:train3d} shows one subsampled training set and test set of the full dataset of all SMART detections in three dimensions corresponding to features with the highest marginal relevance. The detections are colored by their class (flare/non-flare). In the test set, a large proportion of detections is located around 0, but due to over-plotting, it is harder to see how these observations dominate the dataset compared to density plots. These plots show that whereas the features contain information about classes, there in an overlap between them in this feature space - the classes are not perfectly separable. The shape of the classification boundary will not change this and the scope for improvement when using more complex algorithms in this feature space is limited.
\begin{figure}[t!]
\centerline{
\hspace*{0.015\textwidth}
\includegraphics[width=8cm,height=8cm]{3dTrain.png}
\hspace*{-0.03\textwidth}
\includegraphics[width=8cm,height=8cm]{3dTest.png}
}
\caption{MF detections in the (a) one training dataset and (b) testing dataset, in 3 dimensions (3 features with the highest marginal relevance), red = flare, black = no flare.}\label{fig:train3d}
\end{figure}
\section{Discussion}\label{sec:discussion}
In order to classify MF detections we used a number of classification approaches including binary logistic regression, SVMs, RFs and a set of DNN architectures.
Categorical forecasts were obtained by thresholding the estimated probability from these models. Skill scores, curves of TSS, ROC curves and area under the ROC were used to compare the performance of the classification approaches. We discussed the choice of skill scores and optimal thresholds for various model settings.
The flare prediction results that we obtained from the linear classifier with a very sparse subset of features compare favorably to those found elsewhere in the literature and show a significant improvement on the results of the previous analysis of this data. We found that, in terms of classification performance, there was no benefit in using more features or more flexible models that allow for nonlinear classification boundaries as all approaches converged to the same result. Furthermore, we found no decrease in performance when training the algorithms on very small subsampled training sets.
By plotting the data in the selected dimensions we see that the classes are not perfectly separable in the space of SMART features and that there is limited scope for improvement in using more complex algorithms on this dataset.
A better performance, however, might be obtained by using the deep learning networks to learn the forecasting patterns directly from magnetograms of solar active regions as opposed to using the features computed from the magnetograms. Some work on DNNs for solar flare prediction has been done by \cite{nishizuka18} and \cite{huang18}.
Direct comparisons with other published methods are difficult because of differences in data sets, the definition of an event, and evaluation and reporting of classification results \cite{allClear16}. It would be of interest to carry out a comparative study of classification algorithms, such as presented here, to the Space-weather HMI Active Region Patch (SHARP) \citep{bobra2014} data from the \emph{Solar Dynamics Observatory's}(SDO) Helioseismic and Magnetic Imager (HMI). The data have previously been analysed by \cite{bobra2015} and \cite{liudeng17} who used SVMs and Random Forests.
The work presented in this paper is fully reproducible with code for variable selection, sub-sampling and classification available via GitHub.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,977 |
Q: Single page: set elements at bottom of page in any browser or device I am developing a website where I have a top menu and a background image with some text on the home page. This covers the whole screen. BUT at the bottom of the screen there have to be 3 blocks who show up at the bottom of any screen or device. I am able to to this for Chrome but for Firefox for example the blocks are way far down at IE the same... What I want:
I am currently doing applying following code on the div containing the 3 blocks:
position: relative;
top: -150px;
But as mentioned above in another browser this div shows up to low so the user won't see it until he scrolls...
Any solutions here?
A: Simple change:
position: fixed;
bottom: 0;
Better idea to wrap it inside another <div class="bottom-stuff"> and give the rules to .bottom-stuff.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,891 |
import '../../utils/dismiss-local-backup';
import {
login,
createPost,
createPostAndExit,
updateExistingPostAndExit,
exitEditor,
goToWorkflow,
goToCollections,
updateWorkflowStatus,
publishWorkflowEntry,
assertWorkflowStatusInEditor,
assertPublishedEntry,
deleteEntryInEditor,
assertOnCollectionsPage,
assertEntryDeleted,
assertWorkflowStatus,
updateWorkflowStatusInEditor,
} from '../../utils/steps';
import { workflowStatus, editorStatus } from '../../utils/constants';
export default function({ entries, getUser }) {
it('successfully loads', () => {
login(getUser());
});
it('can create an entry', () => {
login(getUser());
createPostAndExit(entries[0]);
});
it('can update an entry', () => {
login(getUser());
createPostAndExit(entries[0]);
updateExistingPostAndExit(entries[0], entries[1]);
});
it('can publish an editorial workflow entry', () => {
login(getUser());
createPostAndExit(entries[0]);
goToWorkflow();
updateWorkflowStatus(entries[0], workflowStatus.draft, workflowStatus.ready);
publishWorkflowEntry(entries[0]);
});
it('can change workflow status', () => {
login(getUser());
createPostAndExit(entries[0]);
goToWorkflow();
updateWorkflowStatus(entries[0], workflowStatus.draft, workflowStatus.review);
updateWorkflowStatus(entries[0], workflowStatus.review, workflowStatus.ready);
updateWorkflowStatus(entries[0], workflowStatus.ready, workflowStatus.review);
updateWorkflowStatus(entries[0], workflowStatus.review, workflowStatus.draft);
updateWorkflowStatus(entries[0], workflowStatus.draft, workflowStatus.ready);
});
it('can change status on and publish multiple entries', () => {
login(getUser());
createPostAndExit(entries[0]);
createPostAndExit(entries[1]);
createPostAndExit(entries[2]);
goToWorkflow();
updateWorkflowStatus(entries[2], workflowStatus.draft, workflowStatus.ready);
updateWorkflowStatus(entries[1], workflowStatus.draft, workflowStatus.ready);
updateWorkflowStatus(entries[0], workflowStatus.draft, workflowStatus.ready);
publishWorkflowEntry(entries[2]);
publishWorkflowEntry(entries[1]);
publishWorkflowEntry(entries[0]);
goToCollections();
assertPublishedEntry([entries[2], entries[1], entries[0]]);
});
it('can delete an entry', () => {
login(getUser());
createPost(entries[0]);
deleteEntryInEditor();
assertOnCollectionsPage();
assertEntryDeleted(entries[0]);
});
it('can update workflow status from within the editor', () => {
login(getUser());
createPost(entries[0]);
assertWorkflowStatusInEditor(editorStatus.draft);
updateWorkflowStatusInEditor(editorStatus.review);
assertWorkflowStatusInEditor(editorStatus.review);
updateWorkflowStatusInEditor(editorStatus.ready);
assertWorkflowStatusInEditor(editorStatus.ready);
exitEditor();
goToWorkflow();
assertWorkflowStatus(entries[0], workflowStatus.ready);
});
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,130 |
Q: Make Strobe Light from LED light on button click i have made a flashlight app and i want to try to make add-on to allow a strobe light feature.
I want to set it up on a different button thought, not the same one. i think i need to use a timer , but i have never used a timer because im new to java.
here is my code for the flashlight:
public class FlashLightActivity extends Activity {
private boolean isLighOn = false;
private Camera camera;
private Button button;
@Override
protected void onStop() {
super.onStop();
if (camera != null) {
camera.release();
}
}
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
button = (Button) findViewById(R.id.buttonFlashlight);
Context context = this;
PackageManager pm = context.getPackageManager();
if (!pm.hasSystemFeature(PackageManager.FEATURE_CAMERA)) {
Log.e("err", "Device has no camera!");
return;
}
camera = Camera.open();
final Parameters p = camera.getParameters();
button.setOnClickListener(new OnClickListener() {
public void onClick(View arg0) {
if (isLighOn) {
Log.i("info", "torch is turn off!");
p.setFlashMode(Parameters.FLASH_MODE_OFF);
camera.setParameters(p);
camera.stopPreview();
isLighOn = false;
} else {
Log.i("info", "torch is turn on!");
p.setFlashMode(Parameters.FLASH_MODE_TORCH);
camera.setParameters(p);
camera.startPreview();
isLighOn = true;
}
}
});
}
}
A: You could use a Handler
public class Strobe extends Activity {
private LinearLayout mLinearLayout;
private Handler mHander = new Handler();
private boolean mActive = false;
private boolean mSwap = true;
private final Runnable mRunnable = new Runnable() {
public void run() {
if (mActive) {
if (mSwap) {
mLinearLayout.setBackgroundColor(Color.WHITE);
mSwap = false;
mHander.postDelayed(mRunnable, 20);
} else {
mLinearLayout.setBackgroundColor(Color.BLACK);
mSwap = true;
mHander.postDelayed(mRunnable, 100);
}
}
}
};
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
mLinearLayout = (LinearLayout) findViewById(R.id.strobe);
startStrobe();
}
private void startStrobe() {
mActive = true;
mHander.post(mRunnable);
}
}
Set a Theme to the Activity to make it full screen.
android:theme="@android:style/Theme.NoTitleBar.Fullscreen"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,700 |
\section{INTRODUCTION}
Not long after the discovery of pulsars---whose characteristic signal was linked to magnetic fields \citep{hewish_etal_1968}---the potential role of magnetic fields in the core-collapse supernova (CCSN) explosion mechanism began to be investigated \citep[e.g.,][]{leblanc_wilson_1970,bisnovatyi-kogan_etal_1976,meier_etal_1976,symbalisty_1984}.
In principle, a differentially rotating proto-neutron star (PNS) could both amplify magnetic fields and serve as an energy reservoir available to be tapped by those fields, giving rise to magnetically powered explosions.
An early conclusion, however, was that both unrealistically rapid rotation \emph{and} unrealistically strong magnetic fields would be needed at the pre-collapse stage for magnetic fields to play a principal role in the explosion dynamics \citep{leblanc_wilson_1970,symbalisty_1984}.
In more recent years interest in strong magnetic fields has returned in connection with a number of observables related to core-collapse supernovae, including asymmetries in the explosion ejecta \citep{wheeler_etal_2002}, natal neutron star kick velocities \citep{laiQian_1998}, and especially the high-energy electromagnetic activity connected to some neutron stars known as magnetars, or Anomalous X-ray Pulsars (AXPs) and Soft Gamma Repeaters (SGRs) \citep[e.g.,][]{duncanThompson_1992,thompsonDuncan_2001,hurley_etal_2005,woodsThompson_2006}.
AXPs and SGRs are characterized by quiescent X-ray luminosities as high as $10^{35}$~erg s$^{-1}$, with sporadic outbursts releasing up to $10^{41}$~erg per event.
Gamma-ray outbursts from SGRs are even more energetic, an extreme example being the giant flare from SGR 1806-20, which released an estimated $10^{46}$~erg over 380~s \citep{hurley_etal_2005}.
Furthermore, AXPs and SGRs are neutron stars characterized by relatively long rotation periods ($P\gtrsim1$~s) and high spin-down rates ($\dot{P}\gtrsim10^{-12}$~ss$^{-1}$) \citep[e.g.,][]{lorimerKramer_2005}.
As their rotational energy cannot account for the electromagnetic emission, and because of the strong magnetic torques implied by high spin-down rates, they are believed to be young neutron stars powered by dissipation of extremely strong surface magnetic fields \citep[$10^{14}$-$10^{15}$~G,][]{duncanThompson_1996,thompsonDuncan_2001}.
On the theoretical side, the discovery of the magneto-rotational instability (MRI) by \citet{balbusHawley_1991} and its application to CCSNe \citep[initiated by][]{akiyama_etal_2003} relaxed the requirement of strong pre-collapse $B$-fields,
renewing interest in magnetic fields as a possible key ingredient in the explosion mechanism of some supernovae \citep[i.e., those from rapidly rotating progenitor cores; e.g.,][]{wheeler_etal_2002,obergaulinger_etal_2005,moiseenko_etal_2006,burrows_etal_2007,takiwaki_etal_2009}.
(The MRI results in exponential growth of the magnetic energy on the rotation timescale.)
However, the rotational energy falls off quadratically with increasing rotation period, and is about $5\times10^{49}$~erg for a 20~ms period PNS---much less than the characteristic CCSN explosion energy of $\sim10^{51}~\mbox{erg}\equiv1~\mbox{Bethe (B)}$.
Thus any magneto-rotationally driven supernovae likely would be peculiar events, since magnetic progenitor cores tend to rotate slowly at the pre-collapse stage \citep{heger_etal_2005}.
Leaving aside the explosion mechanism, the relationship between the formation of neutron star magnetic fields and CCSNe is still an open and interesting question, particularly in the case of magnetars (AXPs and SGRs) \citep{lorimerKramer_2005}.
\citet{thompsonDuncan_1993} argued that such strong fields must be generated during the neutrino cooling epoch after the collapse of the progenitor's iron core, and possibly before the explosion is initiated ($\sim1$~s after core collapse).
Their model remains one of the prevailing theories for magnetar formation, and includes a convective $\alpha-\Omega$ dynamo, which operates when the rotation period is comparable to the turnover time of entropy-driven convection ($\lesssim3$~ms) near the surface of the PNS.
The rapid turnover time may suggest that magnetars are formed in the magnetically-driven explosion of collapsed, rapidly rotating progenitors, whose remnant is spun down by MHD processes at later times.
\citet{bonanno_etal_2003,bonanno_etal_2005} found that neutron finger instabilities \citep[e.g.,][]{bruennDineva_1996} may also result in dynamo action in PNSs with rotation periods as long as 1~s.
In this scenario, the formation of neutron star magnetic fields may be slow (compared to the explosion time scale), and their creation is not necessarily tied to dynamics in the supernova explosion.
The MRI may also operate near the surface of the PNS, and contribute to neutron star magnetization.
The lack of sufficient rotational energy in magnetized pre-collapse progenitor cores, as predicted by stellar evolution models \citep{heger_etal_2005}, has sparked some recent interest in MHD processes in non-rotating CCSN environments \citep{endeve_etal_2010,guilet_etal_2011,obergaulingerJanka_2011}.
These studies investigate field amplification mechanisms and the possible role of amplified $B$-fields on the dynamics of slowly or non-rotating collapsed progenitors, in which rotational MHD processes are insignificant.
In particular, \citet[][hereafter \citetalias{endeve_etal_2010}]{endeve_etal_2010} studied magnetic field amplification by the stationary accretion shock instability \citep[SASI,][]{blondin_etal_2003}.
The SASI is central to the theory of CCSNe: recent simulations lead to the conclusion that it likely plays an important role in neutrino-powered explosions \citep{bruenn_etal_2006,buras_etal_2006,mezzacappa_etal_2007,scheck_etal_2008,marekJanka_2009,suwa_etal_2010,muller_etal_2012}, and may also explain certain observables of pulsars, including their proper motion \citep{scheck_etal_2004} and spin \citep{blondinMezzacappa_2007}.
Thus, magnetic fields may be an important part of a supernova model if the SASI is found to be sensitive to their presence.
In \citetalias{endeve_etal_2010} we adopted the idealized model of \citet{blondin_etal_2003} and \citet{blondinMezzacappa_2007}, and added a weak radial (split monopole) magnetic field.
We presented results from 2D (axisymmetric) and 3D MHD simulations of the SASI, and found that SASI-driven flows may result in significant magnetic field amplification.
Magnetic field evolution in axisymmetric simulations was found to be geometrically constrained.
Moreover, the non-axisymmetric spiral SASI mode \citep{blondinMezzacappa_2007} dominates the post-shock flows in 3D simulations at late times.
The nonlinear evolution of the spiral mode drives vigorous turbulence below the shock, which results in exponential amplification of $B$-fields due to ``stretching" \citep[e.g.,][]{ott_1998}, and the magnetic energy becomes concentrated in intense, intermittent magnetic flux ropes.
We presented results from models with non-rotating and weakly rotating initial conditions, and weak ($10^{10}$~G) and stronger ($10^{12}$~G) initial magnetic fields.
The magnetic fields were not found to reach dynamically significant levels (i.e., components of the Maxwell stress tensor did not contribute significantly to the total stress), and hence no impact of magnetic fields on local or global dynamics was demonstrated.
However, we found that SASI-induced turbulent magnetic field amplification is very sensitive to the spatial resolution adopted in the numerical simulations.
Most of the 3D models presented in \citetalias{endeve_etal_2010} were performed at ``medium" spatial resolution (grid cells with sides $\Delta l=1.56$~km), while one model was performed with ``high" spatial resolution ($\Delta l=1.17$~km).
The thickness of magnetic flux ropes was found to decrease in proportion to $\Delta l$.
We did not observe convergence of $B$-field amplification with increasing spatial resolution.
Nevertheless, the simulations implied neutron star magnetization as a result of SASI-induced magnetic field amplification.
This paper continues and extends the investigations initiated in \citetalias{endeve_etal_2010}.
It improves on our previous study in several important ways, including (1) coverage of a larger parameter space, (2) higher spatial resolution (up to $1280^{3}$ zones), and (3) computation of kinetic and magnetic energy spectra.
With the new set of simulations we investigate in some detail the nature of SASI-driven turbulence, and the growth and impact of magnetic fields during operation of the SASI.
We investigate the saturation level of magnetic energy in our simulations, the (kinetic) energy reservoir available for magnetic field amplification, and the factors determining the magnetic energy growth rate.
We also consider as in \citetalias{endeve_etal_2010} the impact of amplified magnetic fields on global shock dynamics, in particular any impact they may have on the SASI.
Finally, we attempt to quantify the levels of neutron star magnetization that may be expected from SASI dynamics.
We find that the SASI-driven turbulence shares several similarities with non-helical turbulence \citep[e.g.,][]{brandenburg_etal_1996,haugen_etal_2004}, and results in an efficient small-scale dynamo.
Magnetic fields grow exponentially in the turbulent flows driven by the SASI as long as the ``kinematic regime'' remains valid.
The kinematic regime ends when the magnetic energy becomes comparable (locally) to the kinetic energy of the turbulent flows---the magnetic energy source.
From the computed energy spectra we estimate the ``turbulent" kinetic energy, $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$, available for magnetic field amplification, and, in our idealized model, $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$ constitutes about $10\%$ of the total kinetic energy below the shock ($E_{\mbox{\tiny kin}}\sim5\times10^{49}$~erg).
The total magnetic energy saturates at about $E_{\mbox{\tiny mag}}\sim5\times10^{47}$~erg.
The presence of amplified magnetic fields results in less kinetic energy on small spatial scales, but we find no impact of magnetic fields on global shock dynamics, which is consistent with considerations of the energetics.
However, magnetic field evolution remains sensitive to numerical resolution, and magnetic fields are subject to significant numerical dissipation during the saturated state, and our ability to quantify fully the impact of magnetic fields in a more realistic situation is therefore limited.
The magnetic energy growth time decreases with increasing resolution, and, based on the turnover time of the SASI-driven turbulence, is estimated to be a few milliseconds.
We argue that the MHD processes studied in this paper may contribute significantly to strong, small-scale neutron star magnetic fields, and provide a connection between the magnetic fields of neutron stars at birth and supernova dynamics.
The saturation energies may be sufficient to power flaring activity of AXPs, and possibly SGRs.
Moreover, their formation does not require progenitor rotation.
\section{SETUP OF NUMERICAL SIMULATIONS}
We employ the same numerical code and three-dimensional initial conditions we used in \citetalias{endeve_etal_2010}, which follow closely the adiabatic setup described in \citet{blondin_etal_2003} and \citet{blondinMezzacappa_2007}: a stationary accretion shock is placed at a radius $r=R_{\mbox{\tiny{Sh}}}=200$~km, and a highly supersonic flow is nearly free-falling towards the shock for $r>R_{\mbox{\tiny{Sh}}}$ with $\edens{kin}+\edens{grav}\approx0$.
Between the shock and the PNS the flow settles subsonically---obeying the Bernoulli equation $\edens{kin}+\edens{int}+P+\edens{grav}=0$---and is nearly in hydrostatic equilibrium.
Matter is allowed to flow through an inner boundary placed at $r=R_{\mbox{\tiny PNS}}=40$~km.
The mass density and pressure just inside $R_{\mbox{\tiny PNS}}$ are determined from values just outside $R_{\mbox{\tiny PNS}}$ using power-law extrapolations: $\rho\propto r^{-3}$ and $P\propto r^{-4}$, respectively (a procedure that proved necessary in order to maintain the steady state of the unperturbed initial condition).
\begin{figure}
\epsscale{1.0}
\plotone{./figure1.png}
\caption{Plot of the initial condition for the non-rotating weak-field model with $B_{0}=10^{10}$~G ($\model{10}{0.0}{00}$): internal energy density ($\edens{int}$, solid line), magnitude of gravitational potential energy density ($|\edens{grav}|$, dash-dot line), kinetic energy density ($\edens{kin}$, dotted line), and magnetic energy density ($\edens{mag}$, dashed line) versus radial distance from the center of the PNS. The surface of the PNS (our inner boundary) is fixed at $r=R_{\mbox{\tiny PNS}}=40$~km and the shock is initially located at $r=R_{\mbox{\tiny Sh}}=200$~km. Inside the shock $|\edens{grav}|$, $\edens{int}$ and $\edens{mag}$ follow roughly the same power-law ($\propto r^{-4}$), while $\edens{kin}\propto r^{-1}$. The flow is in steady state free-fall outside $R_{\mbox{\tiny{Sh}}}$, with $\edens{kin}$ and $|\edens{grav}|$ proportional to $r^{-2.5}$, and $\edens{int}\propto r^{-2}$. $\edens{mag}$ has been multiplied by $10^{6}$ to become visible on the plot. (The dashed line is also identical to $\edens{mag}$ in the strong-field model ($\model{13}{0.0}{00}$), cf. Table \ref{tab:computedModels}.) \label{fig:initialCondition}}
\end{figure}
Figure \ref{fig:initialCondition} displays the initial configuration of a spherically symmetric, non-rotating stationary accretion shock with a weak radial magnetic field ($B_{0}=1\times10^{10}$~G; the initial magnetic fields in our simulations are discussed in further detail below).
We plot internal energy density $\edens{int}=P/(\gamma-1)$, kinetic energy density
$\edens{kin}= \rho\vect{u}\cdot\vect{u}/2$,
magnetic energy density
$\edens{mag}=\vect{B}\cdot\vect{B}/(2\mu_{0})$,
and the magnitude of the gravitational potential energy density $\edens{grav}=\rho\Phi$ versus radial distance from the center of the star.
Here $\rho$, $\vect{u}$, $P$, $\vect{B}$, and $\Phi$ are the mass density, fluid velocity, fluid pressure, magnetic flux density (magnetic field), and gravitational potential, respectively.
The vacuum permeability is denoted $\mu_{0}$.
We adopt the ideal gas equation of state, with the ratio of specific heats set to $\gamma=4/3$.
The time-independent point-mass gravitational potential is $\Phi=-GM/r$, where $G$ is Newton's constant and $M=1.2~M_{\odot}$ is the mass of the central object.
The accretion rate ahead of the shock is $\dot{M}=0.36~M_{\odot}\mbox{ s}^{-1}$, which is held fixed during the simulations.
Our numerical simulation code, GenASiS, solves the adiabatic, non-relativistic, ideal MHD equations including gravity \citepalias[cf. Eqs. (1)-(4) in][]{endeve_etal_2010}.
Starting from the semi-analytic initial condition, balance equations for mass density $\rho$, momentum density $\vect{S}=\rho\vect{u}$, and magneto-fluid energy density $\edens{fluid}=\edens{int}+\edens{kin}+\edens{mag}$ are evolved with a second-order HLL-type ideal MHD scheme in a manner that ensures conservation of mass and energy (i.e., volume integrals of $\rho$ and $\edens{fluid}+\edens{grav}$) to numerical precision.
The magnetic induction equation is evolved in a divergence-free manner via the method of constrained transport \citep{evansHawley_1988}. \citepalias[See][and the references therein for further details. See also Appendix \ref{app:numericalDissipation} in this paper.]{endeve_etal_2010}
Without initial perturbations the initial configuration in Figure \ref{fig:initialCondition} remains stationary.
In order to initiate the SASI we perturb the initial condition by adding small ($\sim1\%$) random perturbations to the initial pressure profile in the region $r\in[R_{\mbox{\tiny PNS}},R_{\mbox{\tiny{Sh}}}]$.
These perturbations initiate the SASI and allow us to study the evolution of magnetic fields in SASI-driven flows.
The topology, strength and distribution of magnetic fields in core-collapse supernova progenitors are highly uncertain.
A similar uncertainty applies to our knowledge of the angular momentum distribution in the progenitor core.
These uncertainties then apply directly to the initial conditions of simulations aimed at studying the evolution and impact of magnetic fields in core-collapse supernovae.
Rotation and magnetic fields in stellar interiors are intimately coupled in a complex multidimensional interplay.
Stellar core rotation can drive the evolution of magnetic fields, while the magnetic fields can play an important role in distributing the core's angular momentum \citep[e.g.,][]{spruit_2002}.
Three-dimensional stellar evolution models (even without magnetic fields) extending all the way to iron core collapse are currently not available.
However, some insight into the issue of core magnetic fields (and rotation) is provided by recent stellar evolution calculations \citep[e.g.,][]{heger_etal_2005,meynet_etal_2011}.
In particular, \citet{heger_etal_2005} included magnetic fields in their calculations, and found that magnetic torques can significantly reduce the rotation rate of the pre-collapse iron core.
The resulting magnetic fields were dominated by a toroidal component $B_{\varphi}$ ($B_{\varphi}/B_{r}=10^{3}-10^{4}$, where $B_{r}$ is the poloidal (radial) component of the magnetic field).
They also reported that the core rotation rate and magnetic field strength at the pre-supernova stage is an increasing function of progenitor mass.
In the iron core of their 15~$M_{\odot}$ model, the toroidal and poloidal magnetic fields are $B_{\varphi}\approx5\times10^{9}$~G and $B_{r}\approx8\times10^{5}$~G, respectively, while in their 35~$M_{\odot}$ model, the toroidal and poloidal magnetic fields are $B_{\varphi}\approx1\times10^{10}$~G and $B_{r}\approx1\times10^{7}$~G, respectively.
Accounting for the three orders of magnitude increase attained during core-collapse, the \citet[][]{heger_etal_2005} models predict the post-bounce toroidal and poloidal magnetic fields to be in the range of $10^{12}-10^{13}$~G and $10^{9}-10^{10}$~G, respectively.
This is in the range of `common pulsars', inferred from observations of pulsar spin periods and corresponding spin-down rates \citep{lorimerKramer_2005}, but significantly lower than that of magnetars \citep[][]{duncanThompson_1992}.
\begin{table}
\begin{center}
\caption{Tabular overview of computed models. \label{tab:computedModels}}
\begin{tabular}{cccc}
Model & $B_{0}$ (G) & $l_{0}$ (cm$^{2}$ s$^{-1}$) & $t_{\mbox{\tiny end}}$ (ms) \\
\tableline
\tableline
\model{10}{0.0}{00} & $1\times10^{10}$ & 0.0 & 1100 \\
\model{10}{1.5}{15} & $1\times10^{10}$ & $1.5\times10^{15}$ & 878 \\
\model{10}{4.0}{15} & $1\times10^{10}$ & $4.0\times10^{15}$ & 678 \\
\tableline
\model{12}{0.0}{00}\tablenotemark{a} & $1\times10^{12}$ & 0.0 & 1126 \\
\model{12}{1.5}{15} & $1\times10^{12}$ & $1.5\times10^{15}$ & 1000 \\
\model{12}{4.0}{15} & $1\times10^{12}$ & $4.0\times10^{15}$ & 644 \\
\tableline
\model{13}{0.0}{00} & $1\times10^{13}$ & 0.0 & 1100 \\
\tableline
\tableline
\end{tabular}
\tablecomments{$^{\rm a}$ Model computed with multiple grid resolutions.}
\end{center}
\end{table}
Investigating the role of initial $B$-field topology in our simulations is beyond the scope of this study, which is restricted to an initially radial (split monopole) magnetic field configuration; $B_{r}=\mbox{sign}(\cos\vartheta)\times B_{0}(R_{\mbox{\tiny PNS}}/r)^{2}$, where $\vartheta$ is the polar angle.
We only vary the strength of the initial magnetic field $B_{0}$ at the surface of the PNS ($r=R_{\mbox{\tiny PNS}}$).
In particular, we vary $B_{0}$ in the range from $1\times10^{10}$~G to $1\times10^{13}$~G (cf. Table \ref{tab:computedModels}).
The initial magnetic energy density profile for the model with $B_{0}=1\times10^{10}$~G is represented by the dashed line in Figure \ref{fig:initialCondition}, where it has been boosted by a factor of $10^{6}$ to become visible on the plot.
(The corresponding profile for the model with $B_{0}=1\times10^{13}$~G is identical to the dashed line in Figure \ref{fig:initialCondition}.)
Clearly, when comparing the magnetic energy density to $\edens{kin}$ and $\edens{int}$, all our models are initiated with weak magnetic fields.
From the perspective of the \citet{heger_etal_2005} models, our initial magnetic fields are purely poloidal and stronger than their predicted poloidal fields, but comparable (in magnitude) to their predicted toroidal magnetic fields.
Based on the expected multidimensional character of the post-bounce supernova dynamics (e.g., convection and the SASI) and the strength of the magnetic fields ($\edens{mag}$ is small relative to $\edens{kin}$ and $\edens{int}$) we do not expect the magnetic fields to retain the anisotropic ($B_{\varphi}/B_{r}\gg 1$) configuration predicted by the stellar evolution calculations of \citet{heger_etal_2005}.
Progenitors from multidimensional stellar evolution calculations may deviate significantly from their spherically symmetric counterparts \citep[][]{arnettMeakin2011}.
We believe the initial magnetic field configuration we have chosen has only (at best) a secondary impact on our results, and that initial insight into the MHD evolution in core-collapse supernovae can be obtained from the simulations presented here.
For comparison, the non-rotating simulations recently presented by \citet{obergaulingerJanka_2011} with weak initial magnetic fields (models s15-B10 and s15-B11 in that study) start with purely poloidal pre-collapse core magnetic fields of $1\times10^{10}$~G and $1\times10^{11}$~G, respectively.
After core-collapse and shock stagnation, the strength of the magnetic field in the stable layer separating the PNS convection zone and the gain region is about $4\times10^{12}$~G and $3\times10^{13}$~G, respectively (cf. Table 2 in \citet{obergaulingerJanka_2011}).
This layer coincides roughly with our inner boundary at $r=R_{\mbox{\tiny PNS}}$.
Thus, the magnetic field strength in their collapsed weak-field models is comparable (initially) to that of our strongest-field model.
Our rotating models are initiated by setting the pre-shock gas into rotation about the $z$-axis by specifying the azimuthal velocity according to $u_{\varphi}=l_{0}\sin\vartheta/r$, where $l_{0}$ is the (constant) specific angular momentum.
We have computed rotating models where $l_{0}$ has been set to $1.5\times10^{15}$~cm$^{2}$ s$^{-1}$ and $4.0\times10^{15}$~cm$^{2}$ s$^{-1}$.
The discretized ideal MHD equations are solved in a cubic computational domain with sides $L$ and volume $V_{L}=L^{3}$.
Cartesian coordinates are employed.
The computational domain is divided into $N$ zones in each spatial dimension, resulting in $N^{3}$ cubic zones with sides $\Delta l=L/N$.
To conserve computational resources we start our simulations in a relatively small computational domain with $L=L_{\mbox{\tiny min}}=600$~km and $N=512$ (resulting in $\Delta l\approx1.17$~km).
The time-step in our simulations (limited by the Courant-Friedrichs-Lewy condition) is about 5-10~$\mu$s, depending on the stage of the particular run.
The runs are evolved to a physical time of about 1~s, which results in about $10^{5}$ time-steps taken per simulation.
The MHD solver is parallelized using the Message Passing Interface (MPI), and the computational domain is subdivided into blocks containing an equal number of zones, which are distributed among MPI processes.
During the simulations we keep the number of zones per block (MPI process) fixed to $32^{3}$.
Once the SASI evolves into the nonlinear regime the volume encompassed by the shock grows, and the shock eventually interacts with the boundary of the computational domain.
When this happens, we expand the computational domain by adding a layer of $32^{3}$-zones blocks (i.e., we add 64 zones in each coordinate direction) and restart the simulation from the last checkpoint written before the shock interacted with the boundary of the computational domain.
We repeat this process, and run our simulations until the shock interacts with the boundary of the largest computational box $L=L_{\mbox{\tiny max}}=1500$~km, or the simulation time reaches $t=1100$~ms, whichever occurs first.
Since we keep $\Delta l$ fixed during the simulations, the largest computational domain is covered by 1280 zones in each spatial dimension.
During a run, we write simulation output for analysis and visualization every 2~ms of physical time, resulting in tens of Terabytes of data from each model.
\section{SIMULATION RESULTS}
We focus on the magnetic field evolution during the nonlinear phase of the SASI, during which magnetic fields are amplified most effectively and the potential for back-reaction of the induced fields on the fluid flow is greatest.
We do not apply any rigorous criterion for the onset of nonlinearity; we simply note when the accretion shock deviates noticeably from its spherically symmetric initial shape, and the post-shock velocity field has developed a significant non-radial component.
(The upper panels of Figure \ref{fig:machNumberAndVorticity} are representative of the early nonlinear phase.)
\begin{figure*}
\epsscale{1.0}
\plotone{./figure2.png}
\caption{Nonlinear operation of the spiral SASI mode in model $\model{13}{0.0}{00}$: flow Mach number $|\vect{u}|/c_{S}$ (left panels) and magnitude of fluid vorticity $|\boldsymbol\omega|(\equiv|\curl{\vect{u}}|)$ (right panels). The adiabatic sound speed is $c_{S}=\sqrt{\gamma P/\rho}$. Snapshots are taken at $t=720$~ms (upper panels) and $t=820$~ms (lower panels). To highlight the spiral mode pattern in each panel, the viewing normal is aligned with the total angular momentum in $V_{\mbox{\tiny{Sh}}}$. The shock surface is traced out by the white contour. Velocity vectors where $|\vect{u}|\ge10^{4}$~km s$^{-1}$ are shown in the left panels. \label{fig:machNumberAndVorticity}}
\end{figure*}
Our simulations vary in initial magnetic field strength and spatial resolution, and feature both non-rotating and rotating configurations (see Table \ref{tab:computedModels}).
We focus first---and predominantly---on non-rotating models, often referring to model $\model{10}{0.0}{00}$ (non-rotating model with $B_{0}=1\times10^{10}$~G) as the ``weak-field model,'' and to model $\model{13}{0.0}{00}$ (non-rotating model with $B_{0}=1\times10^{13}$~G) as the ``strong-field model.''
The rotating models are also briefly discussed, but we find that the turbulent magnetic field amplification is mostly unaffected by rotation.
We initiate the SASI with random pressure perturbations in the post-shock flow in order to avoid biased excitation of particular modes (i.e. ``sloshing'' vs. ``spiral''), and find that all our simulations exhibit flows typical of the spiral mode. This is consistent with our results in \citetalias{endeve_etal_2010}, and with \citet{blondinMezzacappa_2007}, who found the spiral mode to dominate the late-time evolution independent of the initial perturbation. It is also consistent with the conclusions of \cite{fernandez_2010}, who demonstrated that the spiral modes of the SASI can be viewed as a superposition of sloshing modes out of phase, and that any superposition of sloshing modes with non-zero relative phase leads to spiral modes and angular momentum redistribution in the post-shock flow, which potentially spins up the underlying PNS \citep[see also][]{blondinShaw_2007}.
The development of spiral SASI modes thus seems to be a general outcome (in 3D) of perturbing the (convectively stable) spherically symmetric initial condition.
Moreover, a recent laboratory experiment---a shallow water analogue to a shock instability \citep[SWASI,][]{foglizzo_etal_2012}---found spiral modes to emerge favorably from the nonlinear phase.
\subsection{Turbulence from Spiral SASI Modes}
Figure \ref{fig:machNumberAndVorticity} illustrates the flows that develop from the nonlinear spiral SASI mode.
The renderings are created from the strong-field model, but the hydrodynamic developments exhibited by this model and highlighted in Figure \ref{fig:machNumberAndVorticity} are typical of all our non-rotating models.
The flow Mach number is shown in the left panels, and in the right panels the magnitude of fluid vorticity is displayed.
The upper panels ($t=720$~ms) depict the early development of the nonlinear spiral SASI mode.
The shock surface is still quasi-spherical, but significant angular momentum redistribution has occurred in the post-shock flow, and the presence of strong counterrotating flows is apparent (cf. velocity vectors).
The shock triple-point \citep[cf.][]{blondinMezzacappa_2007}, positioned to the lower left ($\sim$seven o'clock position), has just formed, and is visible as the kink in the shock surface (cf. white contour).
The shock triple-point (a line segment extending across the shock surface) connects the pre-shock accretion flow and the two counterrotating flows in the post-shock gas.
It moves on the shock surface in the counterclockwise direction in Figure \ref{fig:machNumberAndVorticity}.
A layer of strongly sheared flows extends from the triple-point, downstream from the shock.
This is clearly seen in both plots of fluid vorticity.
This shear flow is one site of post-shock vorticity generation in our simulations.
The post-shock flows are still subsonic for $t=720$~ms.
In the lower panels ($t=820$~ms) the shock triple-point has completed about one and a half revolutions along the shock surface and is now positioned to the upper right ($\sim$two o'clock position).
The shock volume has grown by more than a factor of three compared to the upper panels, and the shape of the accretion shock and the mass distribution in the shocked cavity are even more aspherical.
The supersonic pre-shock accretion flow impinges on the shock at an oblique angle due to the aspherical shock and its off-center position.
The significant tangential velocity component (relative to the shock surface), which is preserved across the shock, leads to supersonic post-shock flows ahead of (and directed towards) the triple-point.
These supersonic flows, which strengthen the shear flow discussed above, are directed down towards the PNS and result in further vorticity generation as they decelerate up the density gradient or impinge on the PNS.
The inviscid vorticity equation is obtained by taking the curl of Euler's equation \citep[e.g.,][]{landauLifshitz_1959}
\begin{equation}
\pderiv{\boldsymbol\omega}{t}
=\curl{(\vect{u}\times\boldsymbol\omega)}
+\f{1}{\rho^{2}}\gradient{\rho}\times\gradient{P},
\label{eq:vorticityEquation}
\end{equation}
where the first term on the right-hand side describes changes in vorticity due to fluid motions, and the second term is the baroclinic vector.
(Magnetic fields are neglected in Eq. (\ref{eq:vorticityEquation}), but it remains valid for weak magnetic fields.)
In particular, vorticity may be generated in regions where isosurfaces of density and pressure intersect.
Figure \ref{fig:vorticityProduction} displays the polytropic constant $\kappa=P/\rho^{\gamma}$ (a proxy for entropy) in a slice through model $\model{13}{0.0}{00}$ at $t=820$~ms with focus on the shear flows emanating from the shock triple-point.
Contours of constant density (dashed) and pressure (solid) are also plotted.
The density and pressure contours are mostly parallel, but diverge strongly in the shear layer, indicating intersecting density and pressure isosurfaces and vorticity generation through the baroclinic term in Eq. (\ref{eq:vorticityEquation}).
(Also, significant vorticity amplification occurs in the shear layer and elsewhere through the ``advection" term in Eq. (\ref{eq:vorticityEquation}).)
Vorticity is generated, amplified, and distributed in a large fraction of the post-shock volume during operation of the SASI.
(Movie 1 in the online material shows the generation and evolution of vorticity in the time interval from $t=720$~ms to $t=820$~ms.)
The vorticity field exhibits strong intermittency in the late stages of SASI evolution.
We also note the development of vorticity tube structures (vortex tubes) during the operation of the spiral mode.
(See also Movie 2, which shows a full revolution of a vorticity still-frame at $t=820$~ms.)
\citet{meeBrandenburg_2006} pointed out that the presence of vorticity may be helpful for turbulent magnetic field amplification.
\begin{figure}
\epsscale{1.0}
\plotone{./figure3.png}
\caption{Slice through model \model{13}{0.0}{00} at $t=820$~ms showing the distribution of the polytropic constant $\kappa=P/\rho^{\gamma}$ around the shear layer associated with the shock triple-point. Contours of constant density ($\rho=6\times10^{8}$~g~cm$^{-3}$ and $\rho=3\times10^{8}$~g~cm$^{-3}$; dashed black and gray, respectively) and pressure ($P=1.7\times10^{27}$~erg~cm$^{-3}$ and $P=6.5\times10^{26}$~erg~cm$^{-3}$; solid black and gray, respectively) are also plotted. \label{fig:vorticityProduction}}
\end{figure}
Strongly forced accretion-driven turbulence develops as a result of the SASI, and the post-shock flow becomes roughly divided into a supersonic (driving) component and a subsonic (volume-filling) turbulent component (cf. lower left panel in Figure \ref{fig:machNumberAndVorticity}).
This is also reflected in the probability density function (PDF) of the velocity field below the shock.
In the left panel in Figure \ref{fig:pdfVelocityAndVorticity} we plot normalized PDFs of the $x$-component of the velocity.
We plot the \emph{total} PDF (solid black line), associated with the subsonic \emph{and} supersonic flow, and the PDF associated with the subsonic flow only (dashed black line).
The supersonic flows contribute only to the tails of the distribution.
The center of the distribution moves in response to the triple-point's movement along the shock surface (cf. gray curves), but when averaged over one revolution about the PNS, the PDF is practically centered on zero: we find $\left[\xi=u_{x}/u_{\mbox{\tiny rms}}\right]_{\mbox{\tiny PDF}}=\int_{-\infty}^{\infty}\xi\,\mbox{PDF}(\xi)\,d\xi\approx0.019$.
In the right panel of Figure \ref{fig:pdfVelocityAndVorticity} we plot the PDF of the $x$-component of the vorticity.
The vorticity PDF is more peaked, has extended exponential tails, and is also centered about zero; $\left[\omega_{x}/\omega_{\mbox{\tiny rms}}\right]_{\mbox{\tiny PDF}}\approx0.002$.
Similar vorticity distributions were found in simulations of convectively driven turbulence by \cite{brandenburg_etal_1996} and attributed to intermittency in hydrodynamic turbulence \citep[see also][]{kraichnan_1990,ishihara_etal_2009}.
\citet{brandenburg_etal_1996} characterized the intermittency of a variable $f$ by the kurtosis of its PDF
\begin{equation}
\mbox{Kurt}(f)=\left[f^{4}\right]_{\mbox{\tiny PDF}}/\left[f^{2}\right]_{\mbox{\tiny PDF}}^{2},
\end{equation}
where $\left[f^{n}\right]_{\mbox{\tiny PDF}}=\int_{-\infty}^{\infty}\,f^{n}\,\mbox{PDF}(f)\,df$ is the $n$-th moment of the PDF (assuming zero mean).
For the $x$-component of the velocity below the shock we find $\mbox{Kurt}(u_{x}/u_{\mbox{\tiny rms}})\approx4.6$, and for the $x$-component of the vorticity we find $\mbox{Kurt}(\omega_{x}/\omega_{\mbox{\tiny rms}})\approx26.7$ \citep[i.e., similar to][]{brandenburg_etal_1996}.
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure4a.png}
{./figure4b.png}
\caption{Normalized probability density functions (PDFs) of the $x$-component of velocity (left) and the $x$-component of vorticity (right). The PDFs are constructed from the post-shock flows in model \model{13}{0.0}{00} during the nonlinear operation of the spiral SASI mode (from $t=810$~ms to $t=922$~ms, which corresponds to roughly one full revolution of the shock triple-point about the PNS). The rms values of velocity and vorticity below the shock are $u_{\mbox{\tiny rms}}=\sqrt{2E_{\mbox{\tiny kin}}/M_{\mbox{\tiny{Sh}}}}$ and $\omega_{\mbox{\tiny rms}}=\sqrt{2\Omega/V_{\mbox{\tiny{Sh}}}}$, respectively, and $E_{\mbox{\tiny kin}}$, $M_{\mbox{\tiny{Sh}}}$, and $\Omega$ are the kinetic energy, mass, and enstrophy in $V_{\mbox{\tiny{Sh}}}$, the volume bounded by the shock surface and the surface of the PNS. We show PDFs for individual time states in gray and the average over all the time states in black. In the left panel, the total PDFs are represented by the solid lines, while the dashed lines are the PDFs constructed from subsonic flows only ($|\vect{u}|/c_{S}<1$). The (averaged) PDF associated with the subsonic flow fits well with the Gaussian $0.01975\times\exp{[-3.1\times(u_{x}/u_{\mbox{\tiny rms}})^{2}]}$. (The PDFs constructed from the other velocity and vorticity components look similar.) \label{fig:pdfVelocityAndVorticity}}
\end{figure*}
\subsection{Amplification of Weak Magnetic Fields from Turbulence: Elementary Concepts}
The SASI-driven hydrodynamic developments result in turbulent amplification of initially weak magnetic fields, which is the focus of this study \citepalias[see also][]{endeve_etal_2010}.
Here we very briefly review some elementary concepts pertaining to such magnetic field amplification \citep[for details, see for example reviews by][]{ott_1998,brandenburgSubramanian_2005}.
Stellar interiors are extremely good electrical conductors.
In a perfectly conducting fluid the electric field is $\vect{E}=-(\vect{u}\times\vect{B})$, and Faraday's law (the induction equation), which governs the evolution of the magnetic field, becomes
\begin{equation}
\pderiv{\vect{B}}{t}=\curl{(\vect{u}\times\vect{B})},
\label{eq:inductionEquation}
\end{equation}
where the right-hand side (the induction term) describes changes to the magnetic field due to fluid motions.
We note that, modulo the baroclinic vector, Eqs. (\ref{eq:vorticityEquation}) and (\ref{eq:inductionEquation}) have identical form, suggesting a possible analogy between $\boldsymbol\omega$ and $\vect{B}$ \citep{batchelor_1950}.
Moreover, \citet{batchelor_1950} argued that the distribution of $\boldsymbol\omega$ and $\vect{B}$ will be similar in fully developed turbulence.
An important difference, however, is that the vorticity equation is nonlinear, while the induction equation is linear for a specified velocity field.
Nevertheless, similarities between vorticity and magnetic field are observed in our simulations.
Equation (\ref{eq:inductionEquation}) can be combined with the mass conservation equation to form \citep[e.g.,][]{landauLifshitz_1960}
\begin{equation}
\dderiv{}{t}\left(\f{\vect{B}}{\rho}\right)=\left(\f{\vect{B}}{\rho}\cdot\nabla\right)\vect{u},
\label{eq:inductionEquationStretching}
\end{equation}
where $d/dt=\partial/\partial t+(\vect{u}\cdot\nabla)$, and $\vect{B}/\rho$ changes due to gradients in the velocity field.
Equation (\ref{eq:inductionEquationStretching}) has an important physical interpretation.
It has the exact same form as the evolution equation for an infinitesimal ``fluid line" connecting fluid elements and moving with the flow.
Thus, two infinitely near fluid elements initially connected by a magnetic field line remain on that magnetic field line, and the value of $\vect{B}/\rho$ varies in proportion to the distance between the fluid elements \citep{landauLifshitz_1960}.
Thus, the magnetic field is ``frozen" in a perfectly conducting fluid.
In an approximately incompressible fluid, the magnetic field grows in direct proportion to the separation between fluid elements.
The interpretation of Eq. (\ref{eq:inductionEquationStretching}) is equivalent to the following simple consideration of a magnetic flux tube, with strength $b$, length $l$, and cross-section $a$, which permeates (and is frozen in) a fluid element with density $\rho$:
let the fluid element be ``stretched" by the flow to a new state (characterized by $b'$, $l'$, $a'$, and $\rho'$).
Then, mass conservation ($\rho'l'a'=\rho la$) and magnetic flux conservation ($b'a'=ba$) gives
\begin{equation}
\f{b'}{\rho'}=\f{b}{\rho}\times(l'/l).
\end{equation}
In the incompressible limit, the field is amplified in direct proportion to the stretching of the tube.
At the same time, the flux tube undergoes a decrease in the scale perpendicular to the stretching ($a'=a\times(l/l')<a$).
The decrease in flux tube cross-section proceeds until (1) the field becomes strong enough to react back on the fluid, preventing further stretching, or (2) resistive (non-ideal) effects become important (Section \ref{sec:magneticEnergyGrowthRates}), or a combination of (1) and (2).
Stretching is a very useful concept for turbulent $B$-field amplification.
The frozen-in condition can result in rapid magnetic field amplification in a turbulent flow.
Initially adjacent fluid elements separate rapidly, perhaps exponentially, in the chaotic flows that characterize turbulence \citep{ott_1998}.
Thus, an initially weak magnetic field (i.e., $\vect{u}$ is independent of $\vect{B}$) may amplify exponentially by stretching, and the growth rate is roughly given by the inverse turnover time of turbulent eddies \citep[e.g.,][]{kulsrudAnderson_1992}.
This is also apparent from a simplistic consideration of Eq. (\ref{eq:inductionEquationStretching}): the turbulent velocity varies significantly $\sim\mathcal{O}(u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$) over a turbulent eddy of size $\lambda_{\mbox{\tiny eddy}}$, hence $B^{-1}(dB/dt)\sim u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}/\lambda_{\mbox{\tiny eddy}}$.
Exponential amplification of weak magnetic fields is commonly seen in MHD turbulence simulations \citep[e.g.,][]{choVishniac_2000,brandenburg_2001,haugen_etal_2004}.
Exponential growth ceases when the magnetic field becomes strong enough to cause a back-reaction on the fluid (i.e., $\vect{u}$ becomes dependent on $\vect{B}$).
Amplification of weak magnetic fields through turbulent stretching is initially a kinematic mechanism (i.e., described by Eq. (\ref{eq:inductionEquation}) for a specified velocity field).
As such, it differs from magnetic field amplification by the MRI in a fundamental way.
The MRI is a dynamic mechanism, described by the full MHD system of equations, and requires the Lorentz force to be included in Euler's equation.
(Also, the MRI requires differential rotation in the PNS to operate, while a turbulent dynamo can operate without PNS rotation.)
However, for weak progenitor $B$-fields \citep{heger_etal_2005}, both mechanisms require high spatial resolution for simulation \citepalias[e.g.,][]{obergaulinger_etal_2009,endeve_etal_2010}, and may ultimately be computationally prohibitive to capture properly in large-scale multi-physics simulations.
We comment here on the amount of kinetic helicity $\boldsymbol{\omega}\cdot\vect{u}$ in our simulations of SASI-driven turbulence.
Kinetic helicity is a measure of ``handedness" (or lack of mirror symmetry) in the turbulent flows, and is an important quantity in dynamo theory \citep[e.g.,][and reference therein]{brandenburgSubramanian_2005}.
Turbulent flows with kinetic helicity can support a so-called inverse cascade and produce large scale magnetic fields (i.e., larger than the turbulent forcing scale) in the nonlinear, saturated state \citep[e.g.,][]{meneguzzi_etal_1981,brandenburg_2001}.
Non-helical turbulence results in mostly small-scale magnetic fields \citep[e.g.,][]{brandenburg_etal_1996,haugen_etal_2004}.
We have constructed PDFs of the relative kinetic helicity $h_{\mbox{\tiny kin}}=(\boldsymbol\omega\cdot\vect{u})/(\omega_{\mbox{\tiny rms}}u_{\mbox{\tiny rms}})$ in the post-shock flow.
The kinetic helicity distributions are similar to the vorticity distributions (strongly peaked with exponential tails), with $\left[h_{\mbox{\tiny kin}}\right]_{\mbox{\tiny PDF}}\approx-8.3\times10^{-4}$ and $\mbox{Kurt}(h_{\mbox{\tiny kin}})\approx28.4$.
Despite the apparent handedness associated with the spiral SASI mode, the resulting turbulence is essentially non-helical.
(This may change, however, if a rapidly---and differentially---rotating PNS is included in the model.)
Thus, we expect that SASI-driven turbulence in our simulations results in magnetic field amplification due to a non-helical small-scale dynamo.
\subsection{Time Evolution of Global Quantities}
\label{sec:timeGlobal}
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure5a.png}
{./figure5b.png}
\plottwo{./figure5c.png}
{./figure5d.png}
\caption{Time-evolution of global quantities integrated over the shock volume $V_{\mbox{\tiny{Sh}}}$, bounded by the surface of the PNS, $\partial V_{\mbox{\tiny PNS}}$, and the surface of the shock, $\partial V_{\mbox{\tiny{Sh}}}$, in non-rotating models in which the initial magnetic field has been varied from $1\times10^{10}$~G to $1\times10^{13}$~G. Plotted are kinetic energy $E_{\mbox{\tiny kin}}$ (top left), magnetic energy change (relative to the initial) $\Delta E_{\mbox{\tiny mag}}/E_{\mbox{\tiny mag},0}$ (top right), angular momentum $|\vect{L}|$ (bottom left), and average shock radius $\bar{R}_{\mbox{\tiny Sh}}=(3V_{\mbox{\tiny{Sh}}}/(4\pi))^{1/3}$. Models $\model{10}{0.0}{00}$, $\model{12}{0.0}{00}$, and $\model{13}{0.0}{00}$ are represented by solid, dashed, and dotted lines, respectively. The initial magnetic energy $E_{\mbox{\tiny mag},0}$ in these models is $2.3\times10^{-12}$~B, $2.3\times10^{-8}$~B, and $2.3\times10^{-6}$~B, respectively. The dash-dotted lines in the top panels are proportional to $\exp{(t/\tau)}$, where the growth times are $\tau=85$~ms and $\tau=66$~ms in the top left and top right panels, respectively. \label{fig:overviewNonRotating}}
\end{figure*}
An overview of the simulations with non-rotating initial conditions is given in Figure \ref{fig:overviewNonRotating}, in which we plot the time-evolution of selected globally integrated quantities for models $\model{10}{0.0}{00}$ (solid lines), $\model{12}{0.0}{00}$ (dashed lines), and $\model{13}{0.0}{00}$ (dotted lines): kinetic energy (upper left), relative magnetic energy change (upper right), angular momentum (lower left), and average shock radius (lower right).
All quantities are obtained from integration over the shock volume $V_{\mbox{\tiny{Sh}}}$, bounded by the surface of the PNS $\partial V_{\mbox{\tiny PNS}}$ and the surface of the shock $\partial V_{\mbox{\tiny{Sh}}}$.
The magnetic energy change is scaled with the initial magnetic energy for easy comparison across the models ($(E_{\mbox{\tiny mag}}-E_{\mbox{\tiny mag,0}})/E_{\mbox{\tiny mag,0}}$ is plotted).
The kinetic energy of the settling flow beneath the shock is initially about $2\times10^{-3}$~B. It begins to grow rapidly during the initial ramp-up phase of the SASI, which starts around 400~ms in all models.
In particular, for the weak-field model, the post-shock kinetic energy grows exponentially with a nearly constant growth rate over the time period extending from $t\approx510$~ms to $t\approx720$~ms.
The growth time during this epoch is about $\tau\approx85$~ms (cf. dash-dotted line in the top left panel in Figure \ref{fig:overviewNonRotating}).
The kinetic energy in the models with a stronger initial magnetic field grows somewhat slower initially ($t\lesssim660$~ms), and then at a rate similar to that of the weak-field model.
The growth slows down considerably for $t\gtrsim800$~ms, but the kinetic energy continues to grow throughout the nonlinear phase and reaches similar levels in all three models, with variability on a shorter timescale superimposed.
When averaged over the time interval extending from 900~ms to 1100~ms we find\footnote{The temporal average of a variable $X$, over the interval $T=t_{2}-t_{1}$, is denoted $\langle X\rangle_{t_{1}}^{t_{2}}=\f{1}{T}\int_{t_{1}}^{t_{2}}X\,dt$.} $\timeAverage{E_{\mbox{\tiny kin}}}{0.9}{1.1}=$ 0.051~B, 0.048~B, and 0.044~B for models $\model{10}{0.0}{00}$, $\model{12}{0.0}{00}$, and $\model{13}{0.0}{00}$, respectively.
While these time-averaged post-shock kinetic energies are slightly smaller in the models with a stronger initial magnetic field, we have not found convincing evidence that this slight decrease in global kinetic energy is a result of a stronger magnetic field.
We find, however, that strong magnetic fields affect flows on small spatial scales (cf. Section \ref{sec:spectralAnalysis}).
In terms of spherical harmonics, the SASI is characterized by exponentially growing power in low-order modes \citep{blondinMezzacappa_2006}.
As a result of this, the shock surface deviates exponentially from its initially spherical shape.
The obliquity of the shock front relative to the pre-shock accretion flow causes the non-radial post-shock kinetic energy to grow (also exponentially) at the expense of thermal energy \citep{blondin_etal_2003}.
We have decomposed the post-shock kinetic energy into radial and non-radial components; $E_{\mbox{\tiny kin},\parallel}=\f{1}{2}\int_{V_{\mbox{\tiny{Sh}}}}\rho u_{r}^{2}\,dV$ and $E_{\mbox{\tiny kin},\perp}=\f{1}{2}\int_{V_{\mbox{\tiny{Sh}}}}\rho(u_{\vartheta}^{2}+u_{\varphi}^{2})\,dV$, respectively.
The non-radial component grows much faster ($\tau\lesssim50$~ms) than the radial component, and the growth seen in Figure \ref{fig:overviewNonRotating} is due to a combination of the two.
(The kinetic energy associated with the three velocity components become similar in the saturated state, and $E_{\mbox{\tiny kin},\perp}\approx2E_{\mbox{\tiny kin},\parallel}$.)
Saturation of the post-shock kinetic energy may be due to the development of turbulence via secondary instabilities \citep[e.g.,][and Section \ref{sec:spectralAnalysis} in this paper]{guilet_etal_2010}.
The exact details that determine the growth rate of the post-shock kinetic energy are tied directly to the physical origin of the SASI, which is not the focus of this paper.
We focus primarily on magnetic field amplification in the flows that result from SASI activity.
The magnetic energy grows at the expense of the turbulent kinetic energy below the shock (cf. Section \ref{sec:spectralAnalysis}).
After an initial spurt, all the models shown in Figure \ref{fig:overviewNonRotating} experience an early period of exponential magnetic energy growth with essentially the same growth rate (cf. the temporal window from $t=650$~ms to $t=780$~ms).
Such evolution is expected in a kinematic growth regime, in which the magnetic field's back-reaction on the fluid is negligible.
The magnetic energy in the weak-field model ($\model{10}{0.0}{00}$, $B_{0}=1\times10^{10}$~G) grows exponentially at a nearly constant rate, with growth time $\tau\approx66$~ms, until the end of the simulation ($t=1100$~ms), and receives a total boost of about five orders of magnitude.
The magnetic energy growth time is significantly shorter ($\sim25\%$) than the total kinetic energy growth time during the overlapping period of exponential growth.
In the model with $B_{0}=1\times10^{12}$~G ($\model{12}{0.0}{00}$) we also find that $E_{\mbox{\tiny mag}}$ grows steadily until the end of the run ($t=1126$~ms).
The magnetic energy in this model initially grows at the same rate as the weak-field model, but its growth rate clearly tapers off at later times ($t\gtrsim900$~ms).
The strong-field model ($\model{13}{0.0}{00}$, $B_{0}=1\times10^{13}$~G) also exhibits exponential magnetic energy growth ($\tau\approx66$~ms) early on.
Then, around $t\approx780$~ms, its growth rate drops almost discontinuously, and $E_{\mbox{\tiny mag}}$ grows by only about $50\%$ for the remainder of the simulation (until $t=1100$~ms).
Model $\model{13}{0.0}{00}$ receives a total boost in magnetic energy of about a factor of 300.
The abrupt change in the magnetic energy growth rate observed in the strong-field model occurs when magnetic fields become dynamically important in localized regions below the shock (cf. Section \ref{sec:radialProfiles}).
At the end of the simulations the magnetic energy in models $\model{10}{0.0}{00}$, $\model{12}{0.0}{00}$, and $\model{13}{0.0}{00}$ has reached about $1.8\times10^{-7}$~B, $2.3\times10^{-4}$~B, and $8.9\times10^{-4}$~B, respectively.
The magnetic energy in the weak-field model is many ($\sim$ five) orders of magnitude below the post-shock kinetic energy at this point.
In the strong-field model it saturates below $10^{-3}$~B, which is also significantly less than the total kinetic energy in the post-shock flow.
Also, the boost in magnetic energy in the strong-field model is almost an order of magnitude lower than the difference in average post-shock kinetic energy between the strong-field and the weak-field models, which is about $7\times10^{-3}$~B.
At the end of the simulations the magnetic energy in the strong-field model is less than a factor of four larger than in model $\model{12}{0.0}{00}$, while it initially was a factor $10^{2}$ larger.
We point out that the magnetic energies listed above are values recorded when the simulations were stopped after roughly one explosion time ($t\sim1$~s).
(SASI-induced magnetic field amplification ceases once the explosion is initiated.)
However, the listed values should \emph{not} be interpreted as upper limits on the magnetic energy for the different initial magnetic fields.
The magnetic energy growth rate is (for reasons we detail in this paper) underestimated by the numerical simulations, and we suspect that all models---independent of the initial magnetic field---will reach a saturated state, similar to the strong-field model, well within an explosion time.
The issue of magnetic energy growth, saturation, and its effect on the post-shock flow will be discussed in later sections.
The induced magnetic fields do not notably affect the global characteristics of the shock evolution.
The plots of total angular momentum $|\vect{L}|$ in $V_{\mbox{\tiny{Sh}}}$ and the average shock radius $\bar{R}_{\mbox{\tiny Sh}}$ show that these quantities reach similar values in all the non-rotating models.
The angular momentum reaches a few $\times10^{47}$~g cm$^{2}$ s$^{-1}$, consistent with the comparable models of \citet{blondinMezzacappa_2007} and \citet{fernandez_2010}.
Moreover, during the nonlinear evolution, after the period of exponential growth of the angular momentum in $V_{\mbox{\tiny{Sh}}}$, we find $|\vect{L}|\lesssim f\dot{M}\bar{R}_{\mbox{\tiny Sh}}^{2}$, with $f\approx 0.25$ \citep[cf.][]{fernandez_2010}.
The average shock radius exhibits significant variability, and briefly exceeds 500~km in some of the models.
In particular, we find $\timeAverage{\bar{R}_{\mbox{\tiny Sh}}}{0.9}{1.1}=$ 484~km, 466~km, and 438~km, for models $\model{10}{0.0}{00}$, $\model{12}{0.0}{00}$, and $\model{13}{0.0}{00}$, respectively.
The larger-amplitude oscillations in kinetic energy and the smaller average shock radius exhibited by the strong-field model ($\model{13}{0.0}{00}$) may be attributed to this model's somewhat different nonlinear evolution, which is due to the stochastic nature of the nonlinear SASI rather than to the stronger initial field.
Model $\model{13}{0.0}{00}$ first evolves into a typical spiral mode pattern (cf. Figure \ref{fig:machNumberAndVorticity}), but later develops a flow pattern reminiscent of the sloshing mode, with two oppositely directed high-speed streams, emanating from opposite sides of the shock, terminating on opposite sides of the PNS, or colliding head-on deep in the shocked cavity.
The appearance of this flow pattern coincides with the turnover in the angular momentum seen in the lower left panel ($t\approx950$~ms).
Note that there is little or no response in the magnetic energy evolution due to these rearrangements in the large scale flow.
This is consistent with the magnetic field being amplified by small-scale rather than large-scale flows.
Comparing Figure \ref{fig:overviewNonRotating} with Figure 12 of \citetalias{endeve_etal_2010}, we note that spatial resolution ($\Delta l=1.17$~km in this paper versus $\Delta l=1.56$~km in \citetalias{endeve_etal_2010}) affects global magnetic quantities much more than global fluid quantities.
In particular, models 3DB10Rm and 3DB12Rm in \citetalias{endeve_etal_2010} correspond to models $\model{10}{0.0}{00}$ and $\model{12}{0.0}{00}$ respectively.
The increased spatial resolution has no significant impact on the post-shock kinetic energy, total angular momentum, average shock radius, or exponential growth time for the kinetic energy ($\tau\approx85$~ms).
The magnetic energy in model 3DB12Rm is boosted by a factor of $2\times10^{3}$, while model 3DB10Rm received a boost of less than a factor of $10^{3}$.
We also measured an exponential growth time for the magnetic energy of $\tau\approx71$~ms over several hundred milliseconds in \citetalias{endeve_etal_2010}.
These results are somewhat different than the results presented in Figure \ref{fig:overviewNonRotating}, which show that the increased resolution results in a larger boost in the magnetic energy and shorter exponential growth time ($\tau \approx 66$~ms), and that the magnetic energy growth at late times depends on the initial magnetic field strength.
The sensitivity to spatial resolution was also pointed out in our previous study, and it will be further discussed later in this paper.
\subsection{Evolution of Spherically Averaged Radial Profiles and Saturation of Magnetic Energy}
\label{sec:radialProfiles}
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure6a.png}
{./figure6b.png}
\plottwo{./figure6c.png}
{./figure6d.png}
\caption{Spherically averaged radial profiles at selected times for the non-rotating weak-field (left panels) and the strong-field (right panel) models. Upper panels: rms magnetic field strength (solid black) and maximum magnetic field strength (dashed red). For reference, the initial magnetic field profile is plotted in each panel (dash dot). We also plot reference lines proportional to $\exp{(-r/L_{B})}$ (dash-dotted) with $L_{B}=120$~km (weak-field case) and $L_{B}=100$~km (strong-field case). The magnetic field strengths have been normalized to the initial value $B_{0}$ at $r=R_{\mbox{\tiny PNS}}$. Lower panels: magnetic energy density (solid black), kinetic energy density (dotted blue), and thermal pressure (dashed red). For model $\model{10}{0.0}{00}$ profiles are plotted for 666~ms, 792~ms, 918~ms, and 1100~ms (marked by diamonds in the upper right panel of Figure \ref{fig:overviewNonRotating}), while for model $\model{13}{0.0}{00}$ we plot radial profiles for times 720~ms, 820~ms, 922~ms, and 1100~ms (marked by squares in the upper right panel of Figure \ref{fig:overviewNonRotating}). Thicker lines indicate a more advanced time state. Note that the magnetic energy densities in the weak-field model (lower left panel) has been multiplied by a factor of $10^{4}$. \label{fig:sphericalProfilesNonRotating}}
\end{figure*}
The magnetic energy in the strong-field model reaches saturation relatively early in the nonlinear evolution ($t\approx780$~ms).
To help elucidate the physical conditions under which the magnetic energy growth in our simulations is quenched, we plot, in Figure \ref{fig:sphericalProfilesNonRotating}, spherically averaged radial profiles from the evolution of the weak-field model ($\model{10}{0.0}{00}$, left panels) and the strong-field model ($\model{13}{0.0}{00}$, right panels).
In the upper panels we plot the rms magnetic field strength $B_{\mbox{\tiny rms}}$ (solid lines) and the maximum magnetic field strength $B_{\mbox{\tiny max}}$ (dashed red lines) versus radial distance from the center of the PNS.
Values are computed in spherical shells centered on the origin of the computational domain $r=\sqrt{x^{2}+y^{2}+z^{2}}=0$, with thickness $\delta L=20$~km and volume $\delta V_{i}$, which includes all zones with radial coordinate $r\in[r_{i}-\delta L/2,r_{i}+\delta L/2)$, with $r_{i}=50$~km, $70$~km,$\ldots$,$(L-\delta L)/2$.
The rms magnetic field is computed from the shell-volume-averaged\footnote{$\volumeAverage{X}{V}=\f{1}{V}\int_{V}X\,dV$ denotes the volume average of $X$ over the volume $V$.} magnetic energy density $B_{\mbox{\tiny rms}}=\sqrt{2\mu_{0}\volumeAverage{\edens{mag}}{\delta V_{i}}}$, while $B_{\mbox{\tiny max}}$ is simply the maximum magnetic field over all zones in each shell.
In the lower panels we plot the shell-volume-averaged magnetic energy density $\volumeAverage{\edens{mag}}{\delta V_{i}}$ (solid lines), kinetic energy density $\volumeAverage{\edens{kin}}{\delta V_{i}}$ (dotted blue lines), and fluid pressure $\volumeAverage{P}{\delta V_{i}}$ (dashed red lines) versus radial distance.
For each model we plot four profiles (time states) of each variable.
The time states, which are also indicated with diamonds ($\model{10}{0.0}{00}$) and squares ($\model{13}{0.0}{00}$) in the upper right panel of Figure \ref{fig:overviewNonRotating}, are chosen to emphasize the temporal magnetic field evolution in each of the models and to contrast the two models.
The spherically averaged radial profiles further illustrate the differences in magnetic field evolution of the weak-field and strong-field models.
The magnetic field in model $\model{10}{0.0}{00}$ intensifies steadily throughout the nonlinear evolution, at all radii below the shock, and $B_{\mbox{\tiny rms}}$ evolves self-similarly during the later stages.
Toward the end of the weak-field run, the rms magnetic field has received a boost of about two orders of magnitude near $R_{\mbox{\tiny PNS}}$, and stays above $B_{0}$ for $r\lesssim500$~km.
The maximum magnetic field is typically an order of magnitude above the rms magnetic field, which is an indication of strong spatial intermittency in the magnetic field (see also Figures \ref{fig:magneticFieldStrength} and \ref{fig:pdfBFieldAndVorticityDotBField} below).
In the lower left panel of Figure \ref{fig:sphericalProfilesNonRotating} we see that the magnetic energy density is many orders of magnitude smaller than the kinetic energy density and pressure for the shown times, consistent with kinematic magnetic field growth.
Not even in localized regions does the magnetic energy density become comparable to the kinetic energy density or the pressure.
At the end of the simulation, there are only a few zones in which the ratio of magnetic-to-kinetic energy exceeds $10^{-2}$.
The strong-field model's magnetic energy evolution is not governed by kinematic growth (except for during the initial boost received at early times), but rather by dynamic interactions with the fluid in a saturated state (and numerical dissipation; see Sections \ref{sec:spectralAnalysis} and \ref{sec:magneticEnergyGrowthRates}).
The magnetic energy in this model falls off the exponential growth curve around $t=780$~ms.
The thin solid curve in the upper right panel in Figure \ref{fig:sphericalProfilesNonRotating} ($t=720$~ms) represents the transition from the initial magnetic field profile (dash-dot curve) to the saturated state.
Although the post-shock flow is governed by vigorous turbulence at later times, there are only minor changes to the rms and maximum magnetic field profiles.
The relative boost of $B_{\mbox{\tiny rms}}$ and $B_{\mbox{\tiny max}}$ in model $\model{13}{0.0}{00}$ is about an order of magnitude less than what is observed in model $\model{10}{0.0}{00}$.
At the end of the strong-field model, the rms magnetic field exceeds $10^{13}$~G for $r\lesssim225$~km, while the maximum magnetic field exceeds $10^{14}$~G out to $r\approx200$~km.
At the end of both runs the rms magnetic field follows an exponential decrease with radius, $B_{\mbox{\tiny rms}}$ is proportional to $\exp{(-r/L_{B})}$, where the characteristic length scale $L_{B}$ is about 120~km and 100~km for model $\model{10}{0.0}{00}$ and model $\model{13}{0.0}{00}$, respectively (cf. dash-dotted lines).
From the upper panels in Figure \ref{fig:sphericalProfilesNonRotating} it is apparent that the exponential decrease in $B_{\mbox{\tiny rms}}$ with radius holds reasonably well in both models throughout the nonlinear evolution.
Moreover, the averaged kinetic energy density below the shock has increased significantly compared to the initial condition (Figure \ref{fig:initialCondition}) and roughly follows a power-law in radius $r^{\alpha}$, with the exponent $\alpha$ varying between $-2.7$ and $-2.3$.
The decrease in kinetic energy density with radius is mostly due to the decrease in mass density: the mass density falls off as $r^{-3}$ inside $r=150$~km, and almost as $r^{-2}$ outside $r=150$~km.
During the runs, the spherically averaged pressure remains relatively quiescent inside $r=150$~km, where it falls off as $r^{-4}$.
As noted above, the decrease in kinetic energy density (the source of magnetic energy) follows a power law with radius, while the magnetic energy density (and $B_{\mbox{\tiny rms}}$) decreases exponentially with radius.
On the other hand, we find that the enstrophy $\f{1}{2}\volumeAverage{\omega^{2}}{\delta V_{i}}$ also decreases exponentially with radius, with a length scale comparable to (but somewhat shorter than) that of $\volumeAverage{\edens{mag}}{V_{i}}$.
Moreover, by comparing the lower panels of Figure 6 in \citetalias{endeve_etal_2010} (showing $|\vect{B}|$) with Figure 11 in \citetalias{endeve_etal_2010} (showing $|\boldsymbol\omega|$) we see that the spatial distribution of magnetic field and vorticity is very similar.
These observations support a similarity between vorticity and magnetic field \citep{batchelor_1950}.
(We have not investigated the physical reasons for the particular spatial distribution of vorticity and magnetic field in further detail, but we plan to do so in a future study.)
The relative boost in $B_{\mbox{\tiny rms}}$ decreases monotonically with the initial field strength in our models.
The results from model $\model{12}{0.0}{00}$ (not shown) confirm this trend.
This is because the models with stronger initial fields reach saturation during the simulation.
(The growth rate is the same in all models during the kinematic regime.)
Saturation occurs when the magnetic energy becomes comparable (locally) to the kinetic energy.
In particular, we find that the kinematic growth regime ends when $B_{\mbox{\tiny max}}^{2}/(2\mu_{0})\lesssim\volumeAverage{\edens{kin}}{\delta V_{i}}$.
For model $\model{12}{0.0}{00}$ we find $B_{\mbox{\tiny max}}^{2}/(2\mu_{0})\approx 0.1\times\volumeAverage{\edens{kin}}{\delta V_{i}}$ and $B_{\mbox{\tiny max}}^{2}/(2\mu_{0})\approx 0.3\times\volumeAverage{\edens{kin}}{\delta V_{i}}$ for $t=966$~ms and $t=1126$~ms, respectively, which represent time states after the magnetic energy growth has fallen off the exponential curve with growth time $\tau=66$~ms (Figure \ref{fig:overviewNonRotating}).
In model $\model{13}{0.0}{00}$, the ratio $B_{\mbox{\tiny max}}^{2}/(2\mu_{0}\volumeAverage{\edens{kin}}{\delta V_{i}})$ stays above $0.3$ for the three most advanced time states shown in Figure \ref{fig:sphericalProfilesNonRotating}, and hovers around unity at the end of the simulation.
In both models the ratio stays remarkably constant with distance from the PNS, varying by less than a factor of two, out beyond $r=300$~km (although exact details vary by model).
Thus, turbulence-induced magnetic fields may impact dynamics in localized regions throughout the shock volume.
\subsection{Full Spatial Distributions, Intermittency, and Saturation of Magnetic Energy}
\label{sec:spatialDistributions}
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure7a.png}
{./figure7b.png}
\caption{Magnetic field magnitude $|\vect{B}|$ near the end ($t=1068$~ms) of the weak-field simulation ($\model{10}{0.0}{00}$). The left panel shows a global view of the magnetic fields below the shock (traced out with a white contour). The right panel shows the magnetic field in a $200$~km$^{3}$ volume near the PNS. \label{fig:magneticFieldStrength}}
\end{figure*}
Figure \ref{fig:magneticFieldStrength} shows volume renderings of the magnetic field magnitude in model $\model{10}{0.0}{00}$ near the end of the simulation ($t=1068$~ms).
The left panel shows a global view of the amplified magnetic fields below the shock (white contour), and the right panel shows a zoomed in view in a $200$~km$^{3}$ volume near the PNS.
These renderings illustrate the complicated, highly intermittent magnetic fields that develop from SASI-induced turbulent flows.
The magnetic energy is concentrated in thin, folded, intense flux ropes.
Notice in the left panel that amplified magnetic fields do not extend all the way to the shock, but are confined to a smaller volume, which is characterized by highly turbulent flows (see also the distribution of vorticity for model $\model{13}{0.0}{00}$ in the volume renderings in Figure \ref{fig:machNumberAndVorticity}, and the associated movies).
The spatial distribution of the magnetic field (in particular the intermittency) in the strong-field model is similar to that of the weak-field model.
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure8a.png}
{./figure8b.png}
\caption{Normalized PDFs of the magnetic field components (left panel) and the cosine of the angle between vorticity and magnetic field (right panel) in model \model{10}{0.0}{00}. The PDFs are constructed by averaging over the time period from $804$~ms to $918$~ms. The PDFs of the $x$-, $y$-, and $z$-components of the magnetic field (solid, dotted, and dashed lines, respectively) are practically indistinguishable. \label{fig:pdfBFieldAndVorticityDotBField}}
\end{figure*}
Normalized PDFs of individual components of the magnetic field from model \model{10}{0.0}{00} are plotted in the left panel of Figure \ref{fig:pdfBFieldAndVorticityDotBField}.
The shape of the distributions is strongly peaked with extended exponential tails (similar to the vorticity distributions in Figure \ref{fig:pdfVelocityAndVorticity}), and the PDFs of the different magnetic field components are practically indistinguishable.
The intermittency is high, $\mbox{Kurt}(B_{x}/B_{\mbox{\tiny rms}})\approx\mbox{Kurt}(B_{y}/B_{\mbox{\tiny rms}})\approx\mbox{Kurt}(B_{z}/B_{\mbox{\tiny rms}})\approx32.5$ (somewhat larger than, but comparable to, the intermittency of the vorticity field), which is consistent with the visual impression given in Figure \ref{fig:magneticFieldStrength}.
We note that the PDFs are highly symmetric, which implies small overall polarity of the field; the magnitude of the mean values $\left[B_{x}/B_{\mbox{\tiny rms}}\right]_{\mbox{\tiny PDF}}$, $\left[B_{y}/B_{\mbox{\tiny rms}}\right]_{\mbox{\tiny PDF}}$, and $\left[B_{z}/B_{\mbox{\tiny rms}}\right]_{\mbox{\tiny PDF}}$ are all less than $0.01$.
The PDF of the cosine of the angle between $\boldsymbol\omega$ and $\vect{B}$ is plotted in the right panel of Figure \ref{fig:pdfBFieldAndVorticityDotBField}, which shows that the vorticity and magnetic field tend to be aligned or anti-aligned, and gives further support to the similarity between the magnetic and vorticity fields.
Similar distributions were also reported by \citet{brandenburg_etal_1996}.
The ratio $B_{\mbox{\tiny max}}^{2}/(2\mu_{0}\volumeAverage{\edens{kin}}{\delta V_{i}})$ (used in Section \ref{sec:radialProfiles} to characterize saturation of the magnetic energy) is still only an approximate measure of the relative strength of the magnetic field since the kinetic energy density is averaged over the shell.
The highly intermittent magnetic fields created by turbulence have been ``expelled" from the fluid \citep{thompsonDuncan_1993}.
As evolution proceeds, an increasing percentage of the total magnetic energy resides in regions where the ratio of magnetic-to-kinetic energy $\beta_{\mbox{\tiny kin}}^{-1}(=v_{\mbox{\tiny A}}^{2}/|\vect{u}|^{2})$ exceeds $10^{-2}$, $10^{-1}$, and $1$: $55\%$, $10\%$, and $0.5\%$, respectively, for model $\model{12}{0.0}{00}$ at $t=966$~ms, while for $t=1126$ the respective percentages have increased to $72\%$, $20\%$, and $1.5\%$.
(The percentages quoted for $t=1126$~ms are very similar to those quoted in \citetalias{endeve_etal_2010} for model 3DB12Ah, which was computed with the same resolution and initial condition as model $\model{12}{0.0}{00}$, but with a different initial perturbation.)
The fraction of the total magnetic energy concentrated in regions with high $\beta_{\mbox{\tiny kin}}^{-1}$ stays relatively constant during the saturated state of the strong-field model ($t\ge780$~ms): we find $\sim90\%$ ($\beta_{\mbox{\tiny kin}}^{-1}\ge10^{-2}$), $\sim50\%$ ($\beta_{\mbox{\tiny kin}}^{-1}\ge10^{-1}$), and $\sim7\%$ ($\beta_{\mbox{\tiny kin}}^{-1}\ge1$).
Volume renderings in Figure \ref{fig:inverseKineticBeta} show the spatial distribution of $\beta_{\mbox{\tiny kin}}^{-1}$ late in the evolution of the strong-field model, illustrating the spatial \emph{and} temporal intermittency of turbulence-induced strong magnetic fields.
The snapshots are temporally separated by $10$~ms, which is longer than the turbulent turnover time (cf. Section \ref{sec:spectralAnalysis}), but is significantly shorter than both the advection time and the Alfv{\'e}n crossing time (defined loosely as $\bar{R}_{\mbox{\tiny{Sh}}}/|\vect{u}|$ and $\bar{R}_{\mbox{\tiny{Sh}}}/v_{A}$ respectively).
Concentrations of high $\beta_{\mbox{\tiny kin}}^{-1}$, which can briefly exceed unity in localized regions, are scattered throughout the shock volume.
As noted above, about $50\%$ of the total magnetic energy resides in regions where $\beta_{\mbox{\tiny kin}}^{-1}\ge10^{-1}$, and these magnetic fields occupy less than $10\%$ of the total shock volume.
(Movie 3 and Movie 4 in the online material show the time-evolution of $\beta_{\mbox{\tiny kin}}^{-1}$ from $t=1050$~ms to $t=1100$~ms and a full revolution of $\beta_{\mbox{\tiny kin}}^{-1}$ for $t=1100$~ms, respectively.)
Alfv{\'e}n waves are likely excited by the SASI activity discussed in this paper.
\citet{suzuki_etal_2008} performed simulations in spherical symmetry and investigated the role of Alfv{\'e}n waves on the core-collapse supernova explosion mechanism.
They found that---for sufficiently strong magnetic fields ($\gtrsim2\times10^{15}$~G)---heating associated with Alfv{\'e}n wave energy dissipation may revive the stalled shock.
For weaker magnetic fields no shock revival was observed.
We have not attempted to identify Alfv{\'e}n waves, or energy dissipation due to Alfv{\'e}n waves, in our simulations (the highly dynamic nature of the SASI-driven flows makes this a nontrivial task).
However, the magnetic fields attained in our strong-field model are significantly weaker than $10^{15}$~G and we do not expect Alfv{\'e}n wave heating due to the mechanism studied by \citet{suzuki_etal_2008} to result in significant energy deposition near the shock (or elsewhere) in our simulations.
\citet{guilet_etal_2011} recently suggested a mechanism for magnetic field amplification in the vicinity of an Alfv{\'e}n surface (i.e., where $v_{\mbox{\tiny A}}=|\vect{u}|$).
In their model, Alfv{\'e}n waves, excited for example by the SASI, may amplify near the Alfv{\'e}n surface and create a dynamic back-reaction.
However, the turbulent nature of the hydromagnetic evolution in our simulations may result in unfavorable conditions for this mechanism to operate.
In particular, regions where $v_{\mbox{\tiny A}}=|\vect{u}|$ appear and disappear in a highly intermittent manner (cf. Figure \ref{fig:inverseKineticBeta}, and associated movies).
\begin{figure*}
\epsscale{1.0}
\plotone{./figure9.png}
\caption{Select snapshots of the logarithm of the magnetic-to-kinetic energy ratio $\beta_{\mbox{\tiny kin}}^{-1}=v_{\mbox{\tiny A}}^{2}/|\vect{u}|^{2}$ late in the highly nonlinear magnetically saturated phase of the strong-field model ($\model{13}{0.0}{00}$). The Alfv{\'e}n speed is $v_{\mbox{\tiny A}}=|\vect{B}|/\sqrt{\mu_{0}\rho}$. The snapshots are separated by 10~ms, taken at $t=1090$~ms (left) and $t=1100$~ms (right). \label{fig:inverseKineticBeta}}
\end{figure*}
\subsection{Spectral Analysis}
\label{sec:spectralAnalysis}
Further important insight into the numerical simulations can be gained from a Fourier decomposition of the magnetic and kinetic energy.
In particular, our analysis presented in \citetalias{endeve_etal_2010} lacked the ability to quantify the amount of turbulent kinetic energy available to amplify the magnetic field as well as the magnetic field's impact on the evolution of the small-scale flows.
We seek to address these questions in this section.
Following \citet{ryu_etal_2000} we compute Fourier amplitudes from components of the velocity and magnetic fields in the computational domain\footnote{The Fourier transforms are computed using the FFTW library documented at http://www.fftw.org.}
\begin{equation}
\widehat{X}(\vect{k})
=\f{1}{V_{\mbox{\tiny L}}}\int_{V_{\mbox{\tiny L}}}X(\vect{x})\times\exp\left( i \vect{k}\cdot\vect{x}\right)\,dV,
\label{eq:fourierTransform}
\end{equation}
where $X(\vect{x})$ represents $\sqrt{\rho}u_{j}$ or $B_{j}$, with $j\in\{x,y,z\}$.
We then compute the kinetic and magnetic spectral energy density on a $k$-space shell
\begin{equation}
\widehat{e}_{\mbox{\tiny kin}}(k)
=\f{1}{2}\int_{k\mbox{-shell}}
\sum_{j}|\widehat{\sqrt{\rho}u_{j}}|^{2}
k^{2}\,d\Omega_{k}
\end{equation}
and
\begin{equation}
\widehat{e}_{\mbox{\tiny mag}}(k)
=\f{1}{2\mu_{0}}\int_{k\mbox{-shell}}
\sum_{j}|\widehat{B_{j}}|^{2}
k^{2}\,d\Omega_{k},
\end{equation}
respectively.
The magnitude of the wave vector (wavenumber) is $k=|\vect{k}|=(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})^{1/2}$ and $d\Omega_{k}$ is a solid angle element in Fourier space.
Proper normalization of the Fourier transform ensures that integration of the spectral energy densities over $k$-space equals real-space integrals of the corresponding energy densities; i.e.,
\begin{equation}
\int_{k_{\mbox{\tiny min}}}^{k_{\mbox{\tiny max}}} \widehat{e}_{\mbox{\tiny kin}}\,dk
=\int_{V_{L}}e_{\mbox{\tiny kin}}\,dV
\equiv \widehat{E}_{\mbox{\tiny kin}} \label{eq:parsevalKin}
\end{equation}
and
\begin{equation}
\int_{k_{\mbox{\tiny min}}}^{k_{\mbox{\tiny max}}} \widehat{e}_{\mbox{\tiny mag}}\,dk
=\int_{V_{L}}\edens{mag}\,dV
\equiv \widehat{E}_{\mbox{\tiny mag}}, \label{eq:parsevalMag}
\end{equation}
where the integrals over $k$ extend from $k_{\mbox{\tiny min}}=2\pi/L$ (defined by the spatial scale of the computational box $L$) to $k_{\mbox{\tiny max}}=2\pi/\Delta l$ (limited by finite grid resolution).
The real-space integrals extend over the volume of the computational domain $V_{L}$.
Note that $\widehat{E}_{\mbox{\tiny mag}}$, the total magnetic energy in the computational domain, practically equals the magnetic energy below the shock wave $E_{\mbox{\tiny mag}}$, while a similar equality does not hold for the kinetic energy ($\widehat{E}_{\mbox{\tiny kin}}>E_{\mbox{\tiny kin}}$) because the kinetic energy of the supersonic accretion flow above the shock (included in the Fourier transform) is substantial, but contributes mostly to the spectrum at small $k$ (see Figure \ref{fig:compensatedKineticEnergySpectra}).
\subsubsection{Varying the Initial Magnetic Field Strength}
The evolution of magnetic energy in Fourier space is shown in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating}, in which we plot magnetic energy spectra during the nonlinear SASI for the weak-field model (left panel) and the strong-field model (right panel), for the same times used for the spherically averaged radial profiles displayed in Figure \ref{fig:sphericalProfilesNonRotating}.
Initially (not shown), the magnetic energy spectrum decreases monotonically with increasing $k$.
Then, most of the magnetic energy resides on relatively large scales ($k\lesssim0.1$), and $\edensk{mag}$ is roughly proportional to $k^{-2}$ for larger $k$-values.
The energy spectra in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating} exhibit features typical of MHD turbulence simulations \citep[see][for a recent comprehensive review]{brandenburgSubramanian_2005}.
The spectral magnetic energy density increases with wavenumber, roughly as $\edensk{mag}\propto k^{3/2}$, for small $k$-values \citep[cf.][]{kazantsev_1968,brandenburgSubramanian_2005}. It reaches a maximum around 0.2-0.3~km$^{-1}$, where the turnover is due to numerical diffusivity (Figure \ref{fig:energySpectraResolutionStudy}), beyond which it decreases rapidly with increasing $k$.
The spectral magnetic energy density remains basically self-similar over the time-intervals displayed in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating}.
The weak-field model exhibits the exponential growth on all scales typical of a kinematic small-scale dynamo \citep{brandenburgSubramanian_2005}, the peak value increasing by almost three orders of magnitude (from $\sim6\times10^{-10}$~B~km to $\sim4\times10^{-7}$~B~km), as seen in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating}. In contrast the peak value saturates in the strong-field model, increasing by only a factor of about four (to $\sim2\times10^{-3}$~B~km). However the spectral shape stays relatively unchanged in both cases, with the full width at half maximum being roughly constant over time ($\sim0.40$~km$^{-1}$ and $\sim0.38$~km$^{-1}$ for the weak-field model and the strong-field model, respectively).
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure10a.png}
{./figure10b.png}
\caption{Temporal evolution of the spectral magnetic energy density $\edensk{mag}$ for the weak-field (left) and the strong-field (right) models, respectively. The spectral distributions are plotted at 666~ms, 792~ms, 918~ms, and 1100~ms (model $\model{10}{0.0}{00}$) and 720~ms, 820~ms, 922~ms, and 1100~ms (model $\model{13}{0.0}{00}$), respectively (i.e., the same times for the respective models as those displayed in Figure \ref{fig:sphericalProfilesNonRotating}). The dotted vertical reference lines denote spatial scales (from left to right) of 300~km, $20\times\Delta l$, and $10\times\Delta l$, where $\Delta l=1.17$~km is the size of a computational cell. The dash-dot line in each panel is proportional to $k^{3/2}$. The mean magnetic wavenumber $\bar{k}_{\mbox{\tiny mag}}$ (Eq. (\ref{eq:meanMagneticWaveNumber})) is indicated by diamonds. \label{fig:spectralMagneticEnergyDensityNonRotating}}
\end{figure*}
The spectral shape in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating} appears unchanged across the two models, even though the strong-field model saturates and the weak-field model does not.
There are, however, small differences.
At the end of the simulations ($t=1100$~ms), the normalized spectra $\edensk{mag}/\max(\edensk{mag})$ of models $\model{10}{0.0}{00}$ and $\model{12}{0.0}{00}$ lie practically on top of each other for all $k$.
The corresponing spectrum of the strong-field model follows those of the weaker-field models for large $k$ (although the peak is slightly shifted to the left), but has ``excess" power for $k\lesssim0.1$~km$^{-1}$: the integral $\f{1}{\max(\edensk{mag})}\int_{k_{\mbox{\tiny min}}}^{0.1}\edensk{mag}\,dk$ is about $65\%$ larger in the strong-field model than in the weaker-field models.
In the simulations of non-helical MHD turbulence by \citet{haugen_etal_2004}, the magnetic energy spectra grow self-similarly until saturation ($\edensk{mag}\sim\edensk{kin}$), which occurs first on smaller spatial scales, and on larger scales later (i.e., the smallest wavenumber where $\edensk{mag}\sim\edensk{kin}$ moves to even smaller $k$), and the magnetic energy spectrum appears to align itself with the kinetic energy spectrum with $\edensk{mag}\gtrsim\edensk{kin}$, almost up to the forcing scale.
This may suggest that the shape of the saturated and unsaturated spectra should differ more than displayed in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating}.
Despite the differences (saturated or not), we have not been able to find a reasonable explanation for why the spectral shape in the two models remain so similar.
From the spectral magnetic energy distribution we obtain the mean magnetic wavenumber
\begin{equation}
\bar{k}_{\mbox{\tiny mag}}=\f{1}{\widehat{E}_{\mbox{\tiny mag}}}\int_{k_{\mbox{\tiny min}}}^{k_{\mbox{\tiny max}}}k\edensk{mag}\,dk,
\label{eq:meanMagneticWaveNumber}
\end{equation}
and the characteristic spatial scale of the magnetic field $\bar{\lambda}_{\mbox{\tiny mag}}=2\pi/\bar{k}_{\mbox{\tiny mag}}$.
(The mean magnetic wavenumber is indicated by a diamond on each of the energy spectra in Figure \ref{fig:spectralMagneticEnergyDensityNonRotating}.)
During the initial ramp-up to nonlinear SASI evolution we find that $\bar{\lambda}_{\mbox{\tiny mag}}$ decreases rapidly with time, from $\bar{\lambda}_{\mbox{\tiny mag}}\approx60$~km initially to $\bar{\lambda}_{\mbox{\tiny mag}}\approx20$~km around $t=650$~ms, and stays relatively constant thereafter in all the non-rotating models.
Magnetic field amplification in our simulations is caused by turbulent stretching of flux tubes.
In a kinematic dynamo the characteristic scale of the magnetic field decreases exponentially.
If the kinematic approximation remains valid, the decrease is halted by resistive dissipation when the spatial dimension of the field (the flux tube thickness) approaches the resistive scale \citep{shekochihin_etal_2002}.
The kinematic approximation remains valid throughout the evolution of the weak-field model.
The temporal constancy of $\bar{\lambda}_{\mbox{\tiny mag}}$ for $t\ge650$~ms in the weak-field model is a strong indication that numerical diffusion plays an important role in our simulations.
In our numerical scheme, we have adopted the HLL Riemann solver \citep{harten_etal_1983}, which approximates the MHD Riemann problem by only considering the left- and right-propagating fast magnetosonic waves.
This approximation results in diffusive evolution of intermediate waves (e.g., slow magnetosonic, Alfv{\'e}n, and entropy waves), and is the main source of dissipation in our simulations.
(No other form of dissipation, physical or numerical, has been explicitly included in our simulations.)
The inherent diffusivity of schemes based on the HLL Riemann solver also affects the evolution of small-scale structures (e.g., turbulence induced magnetic fields.
See Appendix \ref{app:numericalDissipation} for further details on the source and nature of numerical dissipation of the magnetic energy in our simulations.)
Moreover, in the strongly nonlinear regime of the SASI at $t\gtrsim750$~ms we find that $\bar{\lambda}_{\mbox{\tiny mag}}$ is somewhat larger ($\sim10\%$) in models with a stronger initial magnetic field.
Specifically, we find $\timeAverage{\bar{\lambda}_{\mbox{\tiny mag}}}{0.8}{1.1}\approx18$~km for the weak-field model and $\timeAverage{\bar{\lambda}_{\mbox{\tiny mag}}}{0.8}{1.1}\approx20$~km for the strong-field model.
This trend is consistent with the magnetic field becoming strong enough to cause a back-reaction on the fluid through the magnetic tension force and thereby limit the extent to which magnetic flux tubes are stretched and bent by the chaotic flow induced by the SASI.
\citet{shekochihin_etal_2001} observed a strong anti-correlation between the magnetic field strength and the curvature of magnetic flux tubes in their small-scale dynamo simulations; i.e., that the strongest magnetic fields are less curved.
(This effect could potentially be much stronger in simulations similar to ours, but performed at significantly higher spectral resolution, where the magnetic diffusion scale would move to larger $k$-values.)
We find that the magnetic curvature radius $\lambda_{\mbox{\tiny c}}$ and the magnetic rms scale $\lambda_{\mbox{\tiny rms}}$ \citepalias[cf. Eqs. (16) and (17), respectively, in][]{endeve_etal_2010} evolve similarly to $\bar{\lambda}_{\mbox{\tiny mag}}$.
In particular, for the weak-field model we find $\timeAverage{\lambda_{\mbox{\tiny c}}}{0.8}{1.1}\approx9.5$~km and $\timeAverage{\lambda_{\mbox{\tiny rms}}}{0.8}{1.1}\approx3.7$~km.
The corresponding values for the strong-field model are about $10\%$ larger.
Note that $\lambda_{\mbox{\tiny c}}$ and $\lambda_{\mbox{\tiny rms}}$ combined characterize the structure of the magnetic field.
They measure respectively how sharply magnetic flux tubes are bent and how thinly they are stretched.
Such information is not contained in $\bar{\lambda}_{\mbox{\tiny mag}}$ alone.
Spectral kinetic energy distributions from the non-rotating models with different initial field strengths are shown in Figure \ref{fig:spectralKineticEnergyDensityNonRotating}.
The stochastic nature of the SASI and the turbulent flows necessitates the use of temporally averaged spectra when cross-comparing the models; in particular,
the kinetic energy spectra shown in Figure \ref{fig:spectralKineticEnergyDensityNonRotating} are averaged over the time period extending from 800~ms to 1100~ms (that is, we plot $\timeAverage{\edensk{kin}}{0.8}{1.1}$ versus $k$).
The spectra show that the majority of the kinetic energy resides on relatively large spatial scales (small $k$; see the dotted vertical reference lines).
The spectral kinetic energy density is roughly proportional to $k^{-4/3}$ for small $k$-values ($k\lesssim0.2$~km$^{-1}$), while for larger $k$-values ($k\gtrsim0.6$~km$^{-1}$) the flow is heavily influenced by numerical dissipation and the kinetic energy decreases rapidly with increasing wavenumber ($\edensk{kin}\propto k^{-9/2}$).
Magnetic field amplification is driven by the turbulent flows, and it is the kinetic energy of the small-scale motions that is available to be tapped by the magnetic fields.
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure11a.png}
{./figure11b.png}
\caption{Left panel: (time-averaged) kinetic energy spectra $\edensk{kin}$ from non-rotating models $\model{10}{0.0}{00}$ (solid line), $\model{12}{0.0}{00}$ (dotted line), and $\model{13}{0.0}{00}$ (dotted line). Right panel: difference in spectral kinetic energy relative to the weak-field reference model ($1-\edensk{kin}/\widehat{e}_{\mbox{\tiny kin}}^{\,\mbox{\tiny ref}}$), where $\widehat{e}_{\mbox{\tiny kin}}^{\,\mbox{\tiny ref}}$ is the spectral kinetic energy density of the weak-field model ($\model{10}{0.0}{00}$). In both panels we include vertical reference lines to indicate spatial scales of 300~km, $20\times\Delta l$, and $10\times\Delta l$. In the left panel we also include reference lines proportional to power-laws in $k$: $k^{-5/3}$ (long-dashed), $k^{-4/3}$ (dash-dotted), and $k^{-9/2}$ (dash-dot). \label{fig:spectralKineticEnergyDensityNonRotating}}
\end{figure*}
When comparing the non-rotating models in the left panel of Figure \ref{fig:spectralKineticEnergyDensityNonRotating}, we see a decreasing trend in the spectral kinetic energy density for larger wavenumbers ($k\gtrsim0.2$) in models with a stronger initial magnetic field.
(The decrease in kinetic energy on small scales is balanced by a corresponding increase in magnetic energy.)
We emphasize this difference further in the right panel, where we plot the difference in $\edensk{kin}$ for the stronger field models relative to the weak-field model, $1-\edensk{kin}/\widehat{e}_{\mbox{\tiny kin}}^{\,\mbox{\tiny ref}}$.
(The spectral kinetic energy density of the weak-field reference model is here denoted $\widehat{e}_{\mbox{\tiny kin}}^{\,\mbox{\tiny ref}}$.)
The spectral kinetic energy density in model $\model{12}{0.0}{00}$ is reduced by up to $\sim6\%$, while in the strong-field model it is reduced by a maximum of $\sim15\%$ relative to the weak-field model.
The largest difference is seen in the strongly diffusive regime around $k=0.6$~km$^{-1}$.
These results demonstrate that even relatively weak initial magnetic fields can be amplified and impact the flow, although only on small spatial scales.
For larger spatial scales ($k\lesssim0.06$) the differences are caused by differences in the pre-shock kinetic energy (due to differences in $\bar{R}_{\mbox{\tiny Sh}}$ and box size $L$).
The kinetic energy of the pre-shock flows, density stratification (resulting in the real-space power-law (in radius) in the spherically averaged kinetic energy density (Section \ref{sec:radialProfiles})), and compressibility (due to the presence of supersonic flows below the shock) contribute to the kinetic energy spectra in Figure \ref{fig:spectralKineticEnergyDensityNonRotating}.
To further investigate details about the shape of the kinetic energy spectra due to these factors, in particular the $-4/3$ slope (as opposed to the $-5/3$ slope of Kolmogorov turbulence), we have computed (1) kinetic energy spectra with the pre-shock flow velocity set to zero $\edensk{kin}^{\,\mbox{\tiny I}}(k)$, (2) kinetic energy spectra with the pre-shock flow velocity set to zero \emph{and} corrected for the radial density stratification $\edensk{kin}^{\,\mbox{\tiny II}}(k)$ (i.e., we use $X=\sqrt{\volumeAverage{\rho}{V_{\mbox{\tiny L}}}}u_{j}$ with $j\in\{x,y,z\}$ in Eq. (\ref{eq:fourierTransform})), and (3) kinetic energy spectra with the pre-shock flow velocity set to zero \emph{and} corrected for radial density stratification, \emph{but} with local compressibility retained $\edensk{kin}^{\,\mbox{\tiny III}}(k)$ (i.e., we use $X=\sqrt{\bar{\rho}}u_{j}$, where $\bar{\rho}=\volumeAverage{\rho}{V_{\mbox{\tiny L}}}\times(\rho/\volumeAverage{\rho}{\delta V_{i}})$, in Eq. (\ref{eq:fourierTransform})).
Results from these calculations are plotted in Figure \ref{fig:compensatedKineticEnergySpectra}, where we plot compensated kinetic energy spectra from the weak-field model (averaged over the time interval from $804$~ms to $918$~ms): $\edensk{kin}\times k^{5/3}$ (solid line), $\edensk{kin}^{\,\mbox{\tiny I}}\times k^{5/3}$ (dotted line), $\edensk{kin}^{\,\mbox{\tiny II}}\times k^{5/3}$ (dashed line), and $\edensk{kin}^{\,\mbox{\tiny III}}\times k^{5/3}$ (dash-dot line).
(In Figure \ref{fig:compensatedKineticEnergySpectra}, $\edensk{kin}^{\,\mbox{\tiny II}}$ and $\edensk{kin}^{\,\mbox{\tiny III}}$ have been multiplied by a factor of three for convenient comparison with $\edensk{kin}$ and $\edensk{kin}^{\,\mbox{\tiny I}}$.)
\begin{figure}
\epsscale{1.0}
\plotone{./figure12.png}
\caption{Compensated kinetic energy spectra from model \model{10}{0.0}{00}. (The spectra are computed by averaging over the time period from $804$~ms to $918$~ms.) The solid line corresponds to the kinetic energy spectrum shown as the solid line in left panel in Figure \ref{fig:spectralKineticEnergyDensityNonRotating}. The kinetic energy spectrum obtained when setting the pre-shock flow velocity to zero $\edensk{kin}^{\,\mbox{\tiny I}}$ is represented by the dotted line. Kinetic energy spectra with corrections for density stratification (also with the pre-shock flow excluded), $\edensk{kin}^{\,\mbox{\tiny II}}$ and $\edensk{kin}^{\,\mbox{\tiny III}}$, are represented by the dashed and dash-dot lines, respectively (see text for details). Note the narrow inertial range in the stratification-corrected spectra ($\edensk{kin}^{\,\mbox{\tiny II}},\edensk{kin}^{\,\mbox{\tiny III}}\propto k^{-5/3}$) in $k\in[0.04,0.1]$~km$^{-1}$ (i.e., spatial scales from $\sim160$~km to $\sim60$~km). \label{fig:compensatedKineticEnergySpectra}}
\end{figure}
When comparing $\edensk{kin}$ and $\edensk{kin}^{\,\mbox{\tiny I}}$ in Figure \ref{fig:compensatedKineticEnergySpectra} we see that the supersonic pre-shock accretion flow contributes to the energy spectrum, mostly for small $k$-values, but the two spectra remain similar in shape.
(By integrating the two spectra over all $k$ we find $\widehat{E}_{\mbox{\tiny kin}}\approx0.053$~B and $\widehat{E}_{\mbox{\tiny kin}}^{\,\mbox{\tiny I}}\approx0.040$~B.)
On the other hand, the kinetic energy spectra change markedly when the density stratification is excluded from its computation.
The $-4/3$ spectral slope seen in the left panel of Figure \ref{fig:spectralKineticEnergyDensityNonRotating} is due to density stratification from $R_{\mbox{\tiny PNS}}$ to $R_{\mbox{\tiny{Sh}}}$.
Effects due to compressibility are subdominant and do not change the shape of the spectrum in any significant way.
(\citet{kritsuk_etal_2007} scaled the velocity with $\rho^{1/3}$ to recover Kolmogorov $-5/3$ scaling in spectra from simulations of supersonic isothermal turbulence.)
Moreover, we observe a narrow inertial range in $k\in[0.04,0.1]$~km$^{-1}$ (i.e., spatial scales from $\sim160$~km to $\sim60$~km) where $\edensk{kin}^{\,\mbox{\tiny II}},\edensk{kin}^{\,\mbox{\tiny III}}\propto k^{-5/3}$.
For larger $k$-values (around $k=0.2$~km$^{-1}$) we observe a bump in the $\edensk{kin}^{\,\mbox{\tiny II}}$ and $\edensk{kin}^{\,\mbox{\tiny III}}$ spectra \citep[e.g.,][and references therein]{ishihara_etal_2009}, which is less pronounced for $\edensk{kin}^{\,\mbox{\tiny III}}$.
We note that the simulations by \citet{haugen_etal_2003,haugen_etal_2004} argue in favor of a $k^{-5/3}$ spectrum for non-helical MHD turbulence.
(Given infinite spectral resolution, the $\edensk{kin}$ spectrum could possibly also follow $-5/3$ scaling for larger $k$, where the spectrum presumably would be less influenced by density stratification.)
The identification of Kolmogorov-like spectra in our simulations helps us associate post-shock flows with turbulence.
In Figure \ref{fig:compensatedKineticEnergySpectra}, the peak in the $\edensk{kin}^{\,\mbox{\tiny II}}$ and $\edensk{kin}^{\,\mbox{\tiny III}}$ spectra for smaller $k$-values ($k\approx0.02$~km$^{-1}$; i.e., spatial scales around 300~km) is associated with the large scale SASI flows (cf. Figure \ref{fig:machNumberAndVorticity}), which drive post-shock turbulence.
The peak is located around $k=0.05$~km$^{-1}$ (i.e., spatial scales around 125~km) in the earlier stages of SASI-development, when the spectrum also has a steeper-than-$-5/3$ slope for larger $k$.
When the SASI develops nonlinearly, and the average shock radius begins to increase, the peak moves to smaller $k$-values, and the $\edensk{kin}^{\,\mbox{\tiny II}}$ and $\edensk{kin}^{\,\mbox{\tiny III}}$ spectra develop Kolmogorov slopes.
Thus, the power in the large scale flows cascades to smaller-scale flows.
In particular, integrating the $\edensk{kin}^{\,\mbox{\tiny I}}$ spectrum in Figure \ref{fig:compensatedKineticEnergySpectra} over $k$, from $k=0.04$ to $k_{\mbox{\tiny max}}$, gives $0.022$~B.
Thus, a large fraction (up to $\sim50\%$) of the kinetic energy below the shock may be associated with turbulence.
These observations suggest that the SASI saturates due to the development of turbulence via secondary instabilities (e.g., the Kelvin-Helmholtz instability), which feed on the power in the low-order SASI modes \citep[e.g.,][]{guilet_etal_2010}.
The turbulent energy is either dissipated via viscous heating, or converted into magnetic energy and dissipated via Joule heating.
However, we find that significantly less than $50\%$ of the post-shock kinetic energy is accessed for magnetic field amplification (cf. Figure \ref{fig:turbulentKineticEnergy}).
We use the unmodified kinetic energy spectrum $\edensk{kin}$ in our further analysis since it is related to the total kinetic energy via Eq. (\ref{eq:parsevalKin}), and therefore most useful for extracting quantitative information from our simulations.
From the spectral kinetic energy density we obtain the turbulent kinetic energy in our simulations
\begin{equation}
E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}
=\int_{k_{\mbox{\tiny tur}}}^{k_{\mbox{\tiny max}}}\edensk{kin}\,dk.
\label{eq:turbulentKineticEnergy}
\end{equation}
For the purpose of studying magnetic field amplification, we have chosen to define turbulent flows to include flows residing on scales with $k\gek_{\mbox{\tiny tur}}=2\pi/\lambda_{\mbox{\tiny tur}}$, where the turbulent spatial scale covers 25 grid cells $\lambda_{\mbox{\tiny tur}}=25\times\Delta l\approx30$~km (for $\Delta l=1.17$~km).
For reference, $\lambda_{\mbox{\tiny tur}}$ is more than an order of magnitude smaller than the average shock radius, which again is comparable to the forcing scale of the turbulent flows (i.e., the scale of the supersonic downdrafts from the shock triple-point), but comparable to $R_{\mbox{\tiny PNS}}$ ($\sim25\%$ smaller).
This particular choice for $k_{\mbox{\tiny tur}}$ is motivated by several factors, including (1) most of the magnetic field amplification occurs on spatial scales with $k>k_{\mbox{\tiny tur}}$ (Figures \ref{fig:spectralMagneticEnergyDensityNonRotating} and \ref{fig:energySpectraResolutionStudy}), (2) any dynamic effect of the magnetic field is seen on scales with $k\gtrsimk_{\mbox{\tiny tur}}$ (Figure \ref{fig:spectralKineticEnergyDensityNonRotating}), and (3) the flow Taylor microscale, $\lambda_{\mbox{\tiny T}}=\sqrt{\volumeAverage{u^{2}}{V_{\mbox{\tiny{Sh}}}}/\volumeAverage{|\curl{\vect{u}}|^{2}}{V_{\mbox{\tiny{Sh}}}}}$, which measures the average size of turbulent eddies \citep[e.g.,][]{ryu_etal_2000}, is comparable to $\lambda_{\mbox{\tiny tur}}$ (about a factor of two smaller).
\begin{figure}
\epsscale{1.0}
\plotone{./figure13.png}
\caption{Turbulent kinetic energy (Eq. (\ref{eq:turbulentKineticEnergy}), black lines) and total magnetic energy (red lines) versus time in non-rotating models in which the initial magnetic field strength is varied: $\model{10}{0.0}{00}$ (solid), $\model{12}{0.0}{00}$ (dashed), and $\model{13}{0.0}{00}$ (dotted). The dash-dotted line is proportional to $\exp{(t/\tau)}$, with $\tau=100$~ms. The long-dashed horizontal line indicates the upper limit for turbulent kinetic energy, assuming Kolmogorov scaling applies for $k\gek_{\mbox{\tiny tur}}$. ($10^{-2}~\mbox{B}=10^{49}~\mbox{erg}$.) \label{fig:turbulentKineticEnergy}}
\end{figure}
In Figure \ref{fig:turbulentKineticEnergy} we plot the time evolution of the turbulent kinetic energy in the non-rotating models.
The turbulent kinetic energy evolves similarly to the total kinetic energy below the shock (cf. top left panel in Figure \ref{fig:overviewNonRotating}). It grows exponentially during the ramp-up of the SASI and reaches a saturation level, where the intermittent time variability is superimposed on a barely noticeable overall growth.
During the exponential growth phase, the growth rate is somewhat lower and the saturation level about an order of magnitude below the total kinetic energy beneath the shock.
The exponential growth time in the weak-field model is $\tau\approx100$~ms.
The time-averaged saturation levels for the turbulent kinetic energy in the respective models are found to be $\timeAverage{E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}}{0.8}{1.1}=5.90\times10^{-3}$~B ($\model{10}{0.0}{00}$), $5.48\times10^{-3}$~B ($\model{12}{0.0}{00}$), and $5.11\times10^{-3}$~B ($\model{13}{0.0}{00}$), which implies $\sim7\%$ ($4.2\times10^{-4}$~B) and $\sim13\%$ ($7.9\times10^{-4}$~B) reductions with respect to the weak-field model for model $\model{12}{0.0}{00}$ and model $\model{13}{0.0}{00}$, respectively.
These reductions in turbulent energy are comparable to the increase in magnetic energy in the respective models.
Thus the magnetic energy grows at the expense of the turbulent kinetic energy.
The saturation level for the turbulent kinetic energy in our models is only about a factor of two below what is obtained by (hypothetically) assuming Kolmogorov scaling ($\edensk{kin}\propto k^{-5/3}$) for $k\gek_{\mbox{\tiny tur}}$ (indicated by the long-dashed line in Figure \ref{fig:turbulentKineticEnergy}). Thus, $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}\approx 10^{-2}$~B may serve as an upper limit for turbulent kinetic energy in our models, and therefore also as a reasonable upper limit on the magnetic energy attainable in these simulations.
(The turbulent kinetic energy may depend on the accretion rate ahead of the shock, which is held fixed in our simulations.
Thus, the upper limit on $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$ is only approximate.)
We find that the magnetic energy in model $\model{13}{0.0}{00}$ saturates at about $10\%$ of the turbulent kinetic energy.
One must keep in mind that the magnetic energy is also heavily influenced by numerical dissipation during the saturated phase.
It is entirely possible that the magnetic energy can grow beyond the levels seen in our simulations, but probably not much above the long-dashed horizontal line in Figure \ref{fig:turbulentKineticEnergy}.
Small-scale dynamo simulations commonly show that the magnetic energy spectrum lies slightly above the kinetic energy spectrum on the smallest scales during saturation \citep[e.g.,][]{brandenburgSubramanian_2005}.
The magnetic energy does not exceed the kinetic energy in any part of the spectrum in our simulations.
This may be due to finite resolution and numerical dissipation on the smallest scales.
\subsubsection{Varying the Spatial Resolution}
In \citetalias{endeve_etal_2010} we found that magnetic field amplification from SASI-induced turbulent flows is very sensitive to the spatial resolution adopted in the numerical simulations.
(In general, increased spatial resolution results in stronger magnetic fields and improves the conditions for a dynamical influence of magnetic fields.)
With the energy spectra we continue to study the effect of resolution in this section.
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure14a.png}
{./figure14b.png}
\caption{Energy spectra from simulations in which the spatial resolution has been varied. Kinetic energy spectra are plotted in the left panel, and the fractional magnetic energy enclosed by $k$-space shell with radius $k$, $f_{\mbox{\tiny mag}}(k)$ (cf. Eq. (\ref{eq:eMagEnclosed})), is plotted in the right panel. Results are plotted for the non-rotating model with $B_{0}=1\times10^{12}$ ($\model{12}{0.0}{00}$). The spatial resolution in these runs has been set to $\Delta l=2.34$~km (dotted lines), $1.56$~km (dashed lines), and $1.17$~km (solid lines). The kinetic energy spectra are averaged over the time period extending from $800$~ms to $1100$~ms. For each model, magnetic energy spectra are plotted at $t=800$~ms, $900$~ms, $1000$~ms, and $1100$~ms (thicker lines represent more advanced time states). The mean magnetic wavenumber (cf. Eq. (\ref{eq:meanMagneticWaveNumber})) is denoted with a diamond on each spectrum in the right panel. In both panels we include vertical reference lines indicating the spatial scale of 300~km (dash-dot line), and $10\times\Delta l$ (with line styles matching each of the models). \label{fig:energySpectraResolutionStudy}}
\end{figure*}
Energy spectra from simulations of the non-rotating model with $B_{0}=1\times10^{12}$~G for various spatial resolutions are plotted in Figure \ref{fig:energySpectraResolutionStudy}.
In the left panel we plot the spectral kinetic energy density.
In the right panel we plot the fractional magnetic energy enclosed by the $k$-space shell with radius $k$
\begin{equation}
f_{\mbox{\tiny mag}}(k)=
\f{1}{\widehat{E}_{\mbox{\tiny mag}}}
\int_{k_{\mbox{\tiny min}}}^{k}\edensk{mag}(k')\,dk',
\label{eq:eMagEnclosed}
\end{equation}
where $f_{\mbox{\tiny mag}}(k)$ is normalized to the total magnetic energy $\widehat{E}_{\mbox{\tiny mag}}$ so that $f_{\mbox{\tiny mag}}(k_{\mbox{\tiny max}})=1$.
Results from three simulations are presented, and the grid size has been varied by a factor of two: $\Delta l=2.34$~km (low resolution; dotted lines), $\Delta l=1.56$~km (medium resolution; dashed lines), and $\Delta l=1.17$~km (high resolution; solid lines).
Kinetic energy spectra are averaged over a time period extending from $800$~ms to $1100$~ms, while magnetic energy spectra are plotted for $t=800$~ms, $900$~ms, $1000$~ms, and $1100$~ms.
The kinetic energy spectra are very similar and follow each other closely for small wavenumbers ($k\lesssim0.2$~km$^{-1}$).
Numerical dissipation influences the kinetic energy for larger $k$-values, and $\edensk{kin}$ begins to fall off more rapidly with increasing $k$.
The fall-off starts at smaller $k$-values for the lower resolution runs: $\edensk{kin}$ falls below $10^{-3}$~B km$^{-1}$ around $k=0.5$~km$^{-1}$ in the low resolution run, and around $k=0.8$~km$^{-1}$ in the high resolution run.
Since most of the kinetic energy below the shock resides on large scales, and includes the flows associated with the supersonic stream ahead of the shock triple-point (Figure \ref{fig:machNumberAndVorticity}), the total kinetic energy below the shock, $\eShock{kin}$, is insensitive to the spatial resolution.
During the highly nonlinear operation of the SASI spiral mode we find $\timeAverage{\eShock{kin}}{0.8}{1.1}=0.045$~B, $0.043$~B, and $0.044$~B, for the low, medium, and high resolution model, respectively.
Our definition of $k_{\mbox{\tiny tur}}$, used in Eq. (\ref{eq:turbulentKineticEnergy}), is not optimal when comparing simulations computed with different spatial resolutions, since it results in smaller $k_{\mbox{\tiny tur}}$ and more turbulent kinetic energy in models with larger $\Delta l$ ($\timeAverage{E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}}{0.8}{1.1}=8.6\times10^{-3}$~B, $6.3\times10^{-3}$~B, and $5.5\times10^{-3}$~B, for low, medium, and high resolution, respectively).
For the purpose of quantifying the increase in kinetic energy on small scales due to higher resolution, we fix $k_{\mbox{\tiny tur}}=0.2$~km$^{-1}$ and find that $\timeAverage{E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}}{0.8}{1.1}$ increases (linearly) by a factor of two when $\Delta l$ is decreased by a corresponding factor of two ($\timeAverage{E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}}{0.8}{1.1}=2.7\times10^{-3}$~B, $4.5\times10^{-3}$~B, and $5.5\times10^{-3}$~B for low, medium, and high resolution, respectively).
The self-similar evolution of the magnetic energy spectra is clearly displayed in the right panel of Figure \ref{fig:energySpectraResolutionStudy}.
The different models are well separated, while each model's temporally separated spectra fall practically on top of each other.
The spectra are shifted to larger $k$-values when the resolution is increased.
The shift to the right in the spectrum is a direct consequence of a corresponding shift of the spatial scale where numerical diffusion dominates.
The characteristic scale of the magnetic field is roughly constant with time over the time period displayed, but decreases linearly with increasing spatial resolution.
In particular, we find $\timeAverage{\bar{\lambda}_{\mbox{\tiny mag}}}{0.8}{1.1}\approx32$~km, $23$~km, and $18$~km, for the low, medium, and high resolution runs, respectively.
Similarly, the magnetic rms scale $\lambda_{\mbox{\tiny rms}}$ (the average flux tube thickness) decreases by almost a factor of two when $\Delta l$ is reduced by a factor of two (from $7.0$~km to $3.8$~km).
The shift to smaller spatial scales---in particular the decrease in the flux tube thickness---afforded by higher spatial resolution results in stronger magnetic fields and an increase in the magnetic energy.
The integrated magnetic energy below the shock in the low resolution model reaches saturation for $t<800$~ms, and does not grow much beyond $3\times10^{-6}$~B.
(Saturation of magnetic energy in this model is solely due to numerical dissipation.
The influence of magnetic fields on small scale flows emphasized in the right panel of Figure \ref{fig:spectralKineticEnergyDensityNonRotating} is not observed in lower-resolution models.)
In the high resolution model the magnetic energy grows throughout the run, and reaches $2\times10^{-4}$~B near the end (dashed red line in Figure \ref{fig:turbulentKineticEnergy}).
It is interesting to note that during the time span from 800~ms to 1100~ms, between $68\%$ and $77\%$ of the total magnetic energy resides on scales smaller than $\lambda_{\mbox{\tiny tur}}$, nearly independent of spatial resolution.
(There is a weak decrease in the percentage with increasing resolution).
The corresponding percentage for spatial scales smaller than $10\times\Delta l$ varies between $16\%$ and $21\%$.
We expect the magnetic energy spectra will continue to move to higher wavenumbers when the resolution is increased beyond that of our simulations.
The shift to smaller spatial scales (smaller flux tube cross-section) is accompanied by stronger magnetic fields, and we expect the flux tube cross-section to decrease until the magnetic fields become strong enough to cause a back-reaction on the fluid through the Lorentz force.
(\citet{haugen_etal_2003} presented converged magnetic energy spectra in their simulations of non-helical MHD turbulence.
In their converged spectra, most of the magnetic energy resides at a wavenumber $\sim5$ times the minimum wavenumber in the computational domain.)
\subsection{Magnetic Energy Growth Rates}
\label{sec:magneticEnergyGrowthRates}
In this section we focus on the relative importance of mechanisms that control the exponential growth rate of magnetic energy when the magnetic field is weak and the kinematic approximation remains valid.
We also consider the impact of finite numerical resolution on the growth rate in our simulations.
An eddy turnover time $\tau_{\mbox{\tiny eddy}}=\bar{\lambda}_{\mbox{\tiny mag}}/u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$ is commonly invoked as the characteristic exponential growth time of magnetic fields in a turbulent small-scale dynamo \citep[e.g.,][]{kulsrudAnderson_1992}. Here $\bar{\lambda}_{\mbox{\tiny mag}}$ is the characteristic spatial scale of the magnetic field defined below Eq. (\ref{eq:meanMagneticWaveNumber}). The turbulent rms velocity is
\begin{equation}
u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}=\left(\f{2E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}}{M_{\mbox{\tiny{Sh}}}}\right)^{1/2},
\label{eq:rmsVelocity}
\end{equation}
where $M_{\mbox{\tiny{Sh}}}$ is the mass in $V_{\mbox{\tiny{Sh}}}$.
The use of $M_{\mbox{\tiny{Sh}}}$ in Eq. (\ref{eq:rmsVelocity}), instead of only the mass of the flow included in $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$, results in an underestimate of $u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$.
On the other hand, $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$ (and therefore $u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$) is also sensitive to the definition of $k_{\mbox{\tiny tur}}$, which may be larger than the value we use in Eq. (\ref{eq:turbulentKineticEnergy}) and result in smaller $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$.
Nevertheless, Eq. (\ref{eq:rmsVelocity}) provides a reasonable order-of-magnitude estimate of $u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$.
We find that $u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$ grows rapidly during the initial ramp up of the SASI and then levels off at later times.
In the non-rotating models we find $\timeAverage{u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}}{0.8}{1.1}\approx4000$~km s$^{-1}$.
A turbulent rms velocity of several $\times10^{3}$~km s$^{-1}$ is consistent with an inspection of the subsonic flows below the shock: the average velocity among the zones with $|\vect{u}|/c_{S}\le 1$ is about $7000$~km s$^{-1}$.
For the non-rotating models $\tau_{\mbox{\tiny eddy}}$ is about $5$~ms during the highly nonlinear stage of strong SASI activity.
(Another commonly used expression for the eddy turnover time, $\volumeAverage{|\curl{\vect{u}}|^{2}}{V_{\mbox{\tiny{Sh}}}}^{-1/2}$, gives a similar result.)
We now investigate the individual magnetic energy growth rates relevant to our simulations.
Assuming a non-ideal electric field $-(\vect{u}\times\vect{B})+\eta\vect{J}$ with scalar resistivity $\eta$, the evolution equation for the magnetic energy density is easily obtained by dotting the Maxwell-Faraday (induction) equation with $\vect{B}/\mu_{0}$:
\begin{equation}
\pderiv{\edens{mag}}{t}
+\divergence{\vect{P}}
=-\vect{u}\cdot\left(\vect{J}\times\vect{B}\right)
-\f{1}{\mu_{0}}\vect{B}\cdot\curl{\left(\eta\vect{J}\right)},
\label{eq:magneticEnergyEquation}
\end{equation}
where $\vect{P}=[\vect{u}(\vect{B}\cdot\vect{B})-\vect{B}(\vect{B}\cdot\vect{u})]/\mu_{0}$ and $\vect{J}=\left(\curl{\vect{B}}\right)/\mu_{0}$. (See also Eq. (10) as well as the discussion in Section 3.3 in \citetalias{endeve_etal_2010}.) The first and second terms on the right-hand-side of Eq. (\ref{eq:magneticEnergyEquation}) represent work done against the Lorentz force ($W_{\mbox{\tiny L}}$) and magnetic energy decay due to resistive (Joule) dissipation ($-Q_{\mbox{\tiny J}}$), respectively. Kinetic energy of the flow is converted into magnetic energy if $W_{\mbox{\tiny L}}>0$.
It is apparent from Eq. (\ref{eq:magneticEnergyEquation}) that the total magnetic energy growth rate $\tau_{\mbox{\tiny tot}}^{-1}=\langle\edens{mag}\rangle^{-1}\langle\partial\edens{mag}/\partial t\rangle$ equals the sum $\tau_{\vect{J}\times\vect{B}}^{-1}+\tau_{\vect{P}}^{-1}+\tau_{\mbox{\tiny J}}^{-1}$ of individual rates due to work done against the Lorentz force, accretion of magnetic energy (Poynting flux) through $\partial V_{\mbox{\tiny PNS}}$, and resistive energy dissipation.
(The angle brackets in the total rate imply an integral over a volume bounded by the surface $\partial V_{\mbox{\tiny PNS}}$ and a spherical surface enclosing the accretion shock.)
The Poynting flux through the spherical surface enclosing the accretion shock vanishes because $\vect{u}\parallel\vect{B}$ ahead of the shock .
The Poynting flux through $\partial V_{\mbox{\tiny PNS}}$ and resistive dissipation generally result in decay of the magnetic energy in the computational domain.
The decay must be overcome by the Lorentz work term in order for the magnetic energy to increase.
In \citetalias{endeve_etal_2010} we found flux tube stretching by turbulent flows driven by the spiral SASI mode to be the dominant mechanism for magnetic field amplification (see also Figure \ref{fig:magneticEnergyGrowthRatesB0_1E10L0_0_0E00} below).
The magnetic energy growth rate due to work done against the Lorentz force is
\begin{eqnarray}
\tau_{\vect{J}\times\vect{B}}^{-1}
&=&
\f{1}{E_{\mbox{\tiny mag}}}\int_{V}\vect{u}\cdot\left(\vect{J}\times\vect{B}\right)\,dV \nonumber \\
&\approx&
2u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}/\bar{\lambda}_{\mbox{\tiny mag}} = 2 \tau_{\mbox{\tiny eddy}}^{-1},
\label{eq:growthRateLorentzWork}
\end{eqnarray}
where the turbulent rms velocity $u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}$ and the characteristic scale of the magnetic field $\bar{\lambda}_{\mbox{\tiny mag}}$ have been used.
(The factor two in the second part of Eq. (\ref{eq:growthRateLorentzWork}) stems from the factor of one half in the definition of magnetic energy, but is probably not important for this rough estimate.)
The corresponding growth time is then approximately
\begin{equation}
\tau_{\vect{J}\times\vect{B}}
\approx
2.5~\mbox{ms}
\left(\f{\bar{\lambda}_{\mbox{\tiny mag}}}{20~\mbox{km}}\right)
\left(\f{u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}}{4000~\mbox{km s}^{-1}}\right)^{-1}.
\label{eq:growthTimeLorentzWork}
\end{equation}
\begin{figure}
\epsscale{1.0}
\plotone{./figure15.png}
\caption{Magnetic energy growth rates versus time for the non-rotating weak-field model ($\model{10}{0.0}{00}$).
The growth rates are based on terms appearing in Eq. (\ref{eq:magneticEnergyEquation}) and are due to work done against the Lorentz force (solid line) and Poynting flux losses due to accretion through the spherical surface with $r=R_{\mbox{\tiny PNS}}$ (dash-dot line).
We also plot the growth rates due to compression (dotted line) and stretching (dashed line) (Eqs. (13) and (11) in \citetalias{endeve_etal_2010}, respectively). \label{fig:magneticEnergyGrowthRatesB0_1E10L0_0_0E00}}
\end{figure}
The volume occupied by the PNS is excluded from our simulations, and the magnetic energy in $V_{\mbox{\tiny{Sh}}}$ is also affected by accretion of magnetized matter through $\partial V_{\mbox{\tiny PNS}}$.
The decay rate due to this process is
\begin{equation}
\tau_{\vect{P}}^{-1}
=
\f{1}{E_{\mbox{\tiny mag}}}\oint_{\partial V_{\mbox{\tiny PNS}}}\vect{P}\cdot d\vect{S}
\approx
\f{3\dot{M}}{2\pi\rho_{0}L_{B}^{3}},
\label{eq:growthRatePoynting}
\end{equation}
where in the rightmost estimate we have adopted the exponential decrease of magnetic field with radius over a characteristic length scale $L_{B}$ (cf. Figure \ref{fig:sphericalProfilesNonRotating}) to relate the magnetic energy in $V_{\mbox{\tiny{Sh}}}$ to the field strength at the surface of the PNS: $E_{\mbox{\tiny mag}}\approx\f{B_{0}^{2}}{2\mu_{0}}\f{4\pi}{3}L_{B}^{3}$.
The decay time due to accretion through the inner boundary is then approximately
\begin{eqnarray}
\tau_{\vect{P}}
&\approx&
90~\mbox{ms}
\left(\f{\rho_{0}}{3\times10^{10}~\mbox{g cm}^{-3}}\right)\times \nonumber \\
&&\times\left(\f{L_{B}}{100~\mbox{km}}\right)^{3}
\left(\f{\dot{M}}{0.36~M_{\odot}\mbox{ s}^{-1}}\right)^{-1}
\label{eq:growthTimePoynting}
\end{eqnarray}
The average mass density around $r=R_{\mbox{\tiny PNS}}$, denoted $\rho_{0}$, stays fairly constant throughout the simulations.
In Figure \ref{fig:magneticEnergyGrowthRatesB0_1E10L0_0_0E00} we plot the growth rates $\tau_{\vect{J}\times\vect{B}}^{-1}$ (solid line) and $\tau_{\vect{P}}^{-1}$ (dash-dot line) versus time for model $\model{10}{0.0}{00}$.
This model exhibits exponential magnetic energy growth throughout with a growth time of about 66~ms.
The growth rates plotted in Figure \ref{fig:magneticEnergyGrowthRatesB0_1E10L0_0_0E00} are computed from numerical approximations (second-order finite differences) to the integral expressions, and not the approximations provided by the rightmost expressions in Eqs. $(\ref{eq:growthTimeLorentzWork})$ and $(\ref{eq:growthTimePoynting})$.
We also include the growth rates due to stretching $\tau_{\gradient{\vect{u}}}^{-1}$ and compression $\tau_{\divergence{\vect{u}}}^{-1}$ (Eqs. (11) and (13) in \citetalias{endeve_etal_2010}, respectively), and the plot shows that stretching dominates over compression.
The rates remain quasi-steady for $t\gtrsim750$~ms, and in particular we find $\timeAverage{\tau_{\vect{J}\times\vect{B}}^{-1}}{0.9}{1.1}\approx480$~s$^{-1}$ and $\timeAverage{\tau_{\vect{P}}^{-1}}{0.9}{1.1}\approx9$~s$^{-1}$. (We also find $\timeAverage{\tau_{\gradient{\vect{u}}}^{-1}}{0.9}{1.1}\approx515$~s$^{-1}$, and $\timeAverage{\tau_{\divergence{\vect{u}}}^{-1}}{0.9}{1.1}\approx76$~s$^{-1}$.)
We note that there is good agreement between the numerically computed growth rates and the growth rates predicted by the estimates provided by the rightmost expressions in Eqs. $(\ref{eq:growthTimeLorentzWork})$ and $(\ref{eq:growthTimePoynting})$.
Furthermore, the relative importance of these rates in determining the total magnetic energy growth rate becomes clear: since $\tau_{\vect{J}\times\vect{B}}\ll\tau_{\vect{P}}$, accretion of magnetic energy through $\partial V_{\mbox{\tiny PNS}}$ has virtually no effect on the growth of magnetic energy in $V_{\mbox{\tiny{Sh}}}$.
The discrepancy between the millisecond growth time predicted by Eq. (\ref{eq:growthRateLorentzWork}) and the numerically measured growth time ($\tau\approx66$~ms; Figure \ref{fig:overviewNonRotating}) suggests that numerical dissipation plays an important role in controlling the growth time for magnetic energy in our simulations.
This is further supported by the results presented in Section \ref{sec:spectralAnalysis}, which show that the magnetic energy develops on spatial scales that are strongly affected by numerical dissipation (see also Appendix \ref{app:numericalDissipation}).
If not suppressing field growth entirely, numerical dissipation tends to increase the magnetic energy growth time.
The characteristic decay rate due to resistive dissipation of magnetic energy is
\begin{equation}
\tau_{\mbox{\tiny J}}^{-1}
=
\f{1}{E_{\mbox{\tiny mag}}}\int_{V}\f{1}{\mu_{0}}\vect{B}\cdot\curl{\left(\eta\vect{J}\right)}\,dV
\approx
\f{2\eta}{\lambda_{\mbox{\tiny d}}^{2}},
\label{eq:growthRateDissipation}
\end{equation}
where we have introduced the dissipation scale $\lambda_{\mbox{\tiny d}}$.
The decay time due to resistive dissipation is then
\begin{equation}
\tau_{\mbox{\tiny J}}
\approx
R_{\mbox{\tiny m}}\left(\f{\lambda_{\mbox{\tiny d}}}{\bar{\lambda}_{\mbox{\tiny mag}}}\right)^{2}\tau_{\vect{J}\times\vect{B}},
\label{eq:growthTimeDissipation}
\end{equation}
where the magnetic Reynolds number is defined as $R_{\mbox{\tiny m}}=u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}\bar{\lambda}_{\mbox{\tiny mag}}/\eta$.
The magnetic Reynolds number in the supernova environment is expected to be extremely large; on the order $10^{17}$ in the PNS \citep[][]{thompsonDuncan_1993}.
As far as the magnetic energy growth rate is concerned, resistive effects are only relevant on very small scales, and the growth is most likely curbed by dynamical interactions with the fluid through magnetic tension forces before the magnetic field develops to resistive scales due to turbulent stretching of flux ropes \citep{thompsonDuncan_1993}.
\citet{thompsonDuncan_1993} list (in their Table 1) the resistivity in the PNS convection zone ($\eta=1\times10^{-4}$~cm$^{2}$~s$^{-1}$).
Adopting this value, the resistive decay time for a magnetic field varying on a spatial scale of, for example, $1$~m (i.e., much smaller than any scale resolved by our simulations) becomes very long ($\tau_{\mbox{\tiny J}}=5\times10^{7}$~s) compared to the explosion time ($\sim1$~s).
Resistive effects are, however, important to consider in numerical MHD simulations of astrophysical systems.
We do not explicitly include resistivity in our simulations, but the numerical scheme incorporates an effective numerical resistivity in the induction equation in order to stabilize the solution when discontinuities or underresolved gradients appear in the flow (see Appendix \ref{app:numericalDissipation} for further details).
An approximation of the total growth rate can be obtained by combining Eqs. (\ref{eq:growthRateLorentzWork}) and (\ref{eq:growthRateDissipation}):
\begin{equation}
\tau_{\mbox{\tiny tot}}^{-1}
\approx
\tau_{\vect{J}\times\vect{B}}^{-1}
\left[1-\f{1}{R_{\mbox{\tiny m}}}\left(\f{\bar{\lambda}_{\mbox{\tiny mag}}}{\lambda_{\mbox{\tiny d}}}\right)^{2}\right].
\label{eq:totalGrowthRate}
\end{equation}
In our simulations we have $\tau_{\mbox{\tiny tot}}\gg\tau_{\vect{J}\times\vect{B}}$ and $\lambda_{\mbox{\tiny d}}\lesssim\bar{\lambda}_{\mbox{\tiny mag}}$.
Defined this way, the magnetic Reynolds number in our simulations is therefore somewhat larger than, but still close to, unity ($R_{\mbox{\tiny m}}\gtrsim 1$).
This conclusion is consistent with the observations from the energy spectrum plots above, which show that a sizable fraction of the magnetic energy resides on spatial scales where numerical diffusion is significant.
Our simulations are therefore likely to grossly underestimate the magnetic energy growth rates that can be expected under more realistic physical conditions (i.e., where $R_{\mbox{\tiny m}}\gg1$).
We point out here that the saturation of magnetic energy observed in model $\model{13}{0.0}{00}$ does \emph{not} mean that $\tau_{\vect{J}\times\vect{B}}^{-1}\approx0$~s$^{-1}$ for this model.
We find that the amplified magnetic fields in model $\model{13}{0.0}{00}$ result in about a $10\%$ reduction in $\tau_{\vect{J}\times\vect{B}}^{-1}$ relative to the weak-field model, which, because of numerical dissipation, results in a significant reduction in the total growth rate $\tau_{\mbox{\tiny tot}}^{-1}$ (we expect $\tau_{\mbox{\tiny J}}^{-1}$ to be the same in both models).
In Appendix \ref{app:numericalDissipation} we measure numerically the magnetic energy decay rate due to resistive dissipation in one of our simulations.
We find $\tau_{\mbox{\tiny J}}^{-1}\approx380$~s$^{-1}$, which is comparable to, but still somewhat smaller than $\tau_{\vect{J}\times\vect{B}}^{-1}$.
The decay rates $\tau_{\mbox{\tiny J}}^{-1}$, along with $\lambda_{\mbox{\tiny d}}(\approx\bar{\lambda}_{\mbox{\tiny mag}})$ and $\eta_{\mbox{\tiny num}}$ (Appendix \ref{app:numericalDissipation}), and $\tau_{\vect{J}\times\vect{B}}^{-1}$ do not vary significantly with time during the highly nonlinear stage of the SASI.
Thus, the numerically measured growth time ($\tau_{\mbox{\tiny tot}}^{-1}\approx66$~ms; Figure \ref{fig:overviewNonRotating}) is mostly the result of two large and competing processes: growth due to $\tau_{\vect{J}\times\vect{B}}^{-1}$ and decay due to $\tau_{\mbox{\tiny J}}^{-1}$.
For increasing spatial resolution (i.e., increasing $R_{\mbox{\tiny m}}$) we expect $\tau_{\mbox{\tiny tot}}^{-1}\to\tau_{\vect{J}\times\vect{B}}^{-1}$ (cf. Eq. (\ref{eq:totalGrowthRate})).
Indeed, we have carried out simulations with different spatial resolutions and measured the magnetic energy growth rate when an epoch of exponential growth can be identified.
We find that the growth rate increases with increasing resolution: for the lowest resolution model ($\Delta l=2.34$~km) the exponential growth time is about 150~ms, while in a simulation with $\Delta l=0.78$~km the exponential growth time decreases to about 50~ms\footnote{To conserve computational resources, this model was not run to completion but until the computational domain consisted of $1280^{3}$ zones ($t\approx880$~ms). At this time the SASI is still ramping up and the magnetic energy growing rapidly.}.
In fact, a divergent increase in the magnetic energy growth rate with increasing magnetic Reynolds number (i.e., resolution) has been reported in direct numerical simulations of MHD turbulence \citep[e.g.,][]{haugen_etal_2004} and recently in simulations of turbulent star formation using adaptive mesh refinement \citep[e.g.,][]{syr_etal_2010,federrath_etal_2011}.
These authors show results from simulations in which the resolution has been doubled several times, and they find that the magnetic energy growth rate increases as a power law with increasing magnetic Reynolds number.
The sensitivity of the magnetic field evolution to numerical resolution does raise concerns about what aspects of our simulations are relevant to core-collapse supernovae.
Dissipation due to finite grid resolution tends to suppress magnetic energy growth.
At face value, our simulations (falsely) result a negative assessment on the efficiency of SASI-induced magnetic field amplification.
However, as the resolution is increased the growth rate increases and the resulting magnetic fields become stronger.
Further analysis suggests that the simulations grossly underestimate the growth rates and fields that may obtain in the supernova environment.
The SASI-induced turbulent magnetic field amplification mechanism is a robust result from our simulations.
Only the growth rate, saturation amplitude, and the dynamical impact of the amplified magnetic fields remain uncertain.
An important consequence of the implied millisecond growth time is that any weak seed magnetic fields may be amplified to saturation levels ($|\vect{B}|\approx\sqrt{\mu_{0}\rho}|\vect{u}|$) in a core-collapse supernova if the SASI operates and drives vigorous turbulent flows below the shock.
The kinetic energy available to amplify the magnetic energy (some fraction of $E_{\mbox{\tiny kin}}^{\mbox{\tiny tur}}$) is not sufficient for magnetic fields generated in this way to play a principal role in the explosion dynamics.
We cannot, however, completely rule out the possibility that SASI-generated magnetic fields play a secondary role in the dynamics leading to core-collapse supernovae.
\begin{figure*}
\epsscale{1.0}
\plottwo{./figure16a.png}
{./figure16b.png}
\caption{Kinetic energy (left) and magnetic energy (right) versus time from simulations with varying degree of initial rotation. The specific angular momentum in the pre-shock flow has been set to $l=0.0$~cm$^{2}$~s$^{-1}$ (solid; $\model{10}{0.0}{00}$), $1.5\times10^{15}$~cm$^{2}$~s$^{-1}$ (dashed; $\model{10}{1.5}{15}$), and $4.0\times10^{15}$~cm$^{2}$~s$^{-1}$ (dotted; $\model{10}{4.0}{15}$). The magnetic field strength at $r=R_{\mbox{\tiny PNS}}$ is initially $B_{0}=1\times10^{10}$~G in all the models. \label{fig:kineticAndMagneticEnergy_rotating}}
\end{figure*}
\subsection{Simulations with Initial Rotation}
\label{sec:rotatingModels}
Results from rotating models are shown in Figure \ref{fig:kineticAndMagneticEnergy_rotating}, in which we plot kinetic energy (left panel) and magnetic energy (right panel) versus time.
Rotating models with $B_{0}=1\times10^{10}$~G, and $l_{0}=1.5\times10^{15}$~cm$^{2}$~s$^{-1}$ ($\model{10}{1.5}{15}$; dashed lines) and $l_{0}=4.0\times10^{15}$~cm$^{2}$~s$^{-1}$ ($\model{10}{4.0}{15}$; dotted lines) are compared with the corresponding non-rotating model ($\model{10}{0.0}{00}$; solid lines).
The most notable difference between these models is the earlier onset of the SASI observed in the rotating models.
The post-shock flow is set into rotation about the $z$-axis as the pre-shock material with angular momentum advects downstream.
The kinetic energy in the post-shock flow in model $\model{10}{1.5}{15}$ increases initially by $\sim50\%$, and settles momentarily into a quiescent state, which lasts for about 200~ms.
Then, for $t\gtrsim300$~ms, the nonlinear phase of the SASI sets in, and the kinetic energy begins to grow exponentially with a growth time $\tau\approx55$~ms, which is notably faster than in the non-rotating model.
Model $\model{10}{4.0}{15}$ receives a stronger initial perturbation due to more angular momentum ahead of the shock, and the kinetic energy in this model grows rapidly by a factor of $\sim5$ before settling into a short, quasi-steady state with $E_{\mbox{\tiny kin}}\sim10^{49}$~erg.
The kinetic energy begins to grow again for $t\gtrsim200$~ms.
The kinetic energy in all models eventually reaches similar levels in the strongly nonlinear phase of the SASI; when averaged over the last $200$~ms of each run we find $\timeAverage{E_{\mbox{\tiny kin}}}{0.68}{0.88}=0.052$~B and $\timeAverage{E_{\mbox{\tiny kin}}}{0.48}{0.68}=0.050$~Bs for models $\model{10}{1.5}{15}$ and $\model{10}{4.0}{15}$ respectively.
(We reported $\timeAverage{E_{\mbox{\tiny kin}}}{0.9}{1.1}=0.051$~B for model $\model{10}{0.0}{00}$ in Section \ref{sec:timeGlobal}.)
The earlier onset of the nonlinear SASI in the rotating models is consistent with \citet{blondinMezzacappa_2007} and \citet{yamasakiFoglizzo_2008}.
However, model $\model{10}{4.0}{15}$ is perturbed relatively hard when the rotating pre-shock material advects downstream, and the model does not settle into a quiescent state, as is observed in $\model{10}{1.5}{15}$.
We think it is very likely that the early SASI development in model $\model{10}{4.0}{15}$ is partially a result of our method of initiating the rotating models.
Nevertheless, the purpose of these simulations is to study the effect of rotation on turbulent magnetic field amplification during the non-linear phase, and our rotating models are suitable for this purpose.
The evolution of the magnetic energy below the shock during nonlinear SASI-operation (right panel in Figure \ref{fig:kineticAndMagneticEnergy_rotating}) in the rotating models is not significantly different from model $\model{10}{0.0}{00}$.
All models exhibit exponential magnetic energy growth during the late stages.
The magnetic energy in model $\model{10}{1.5}{15}$ grows exponentially with a growth time $\tau\approx44$~ms during the early stages (from $t\approx340$~ms to $t\approx550$~ms), and grows at a rate similar to model $\model{10}{0.0}{00}$ later on ($t\gtrsim600$~ms).
The magnetic energy in model $\model{10}{4.0}{15}$ grows exponentially at a somewhat slower rate than the other models ($\tau\approx85$~ms).
However, all models have reached similar levels at the end of the respective runs.
In particular, we find $E_{\mbox{\tiny mag}}\approx1.2\times10^{-7}$~B ($t=878$~ms) for $\model{10}{1.5}{15}$ and $E_{\mbox{\tiny mag}}\approx3.4\times10^{-8}$~B ($t=678$~ms) for $\model{10}{4.0}{15}$.
\section{MAGNETIZATION OF PROTONEUTRON STARS}
\label{sec:pnsMagnetization}
In \citetalias{endeve_etal_2010} we pointed out that the underlying PNS may be significantly magnetized due to SASI-induced magnetic field amplification.
In this section we estimate in a similar manner the degree of PNS magnetization predicted by the current set of simulations.
Adopting Eq. (\ref{eq:magneticEnergyEquation}), the magnetic energy in the volume occupied by the PNS $V_{\mbox{\tiny PNS}}$ at some time $t>t_{0}$ is
\begin{eqnarray}
E_{\mbox{\tiny mag}}(t)
&=&E_{\mbox{\tiny mag}}(t_{0}) \nonumber \\
& &
+\int_{t_{0}}^{t}\,dt'
\left(
\int_{V_{\mbox{\tiny PNS}}}\left(W_{\mbox{\tiny L}}-Q_{\mbox{\tiny J}}\right)\,dV
\right. \nonumber \\
& &
\left.
\hspace{1.5cm}
-\int_{\partial V_{\mbox{\tiny PNS}}}\vect{P}\cdot d\vect{S}
\right).
\label{eq:pnsMagneticEnergy}
\end{eqnarray}
Here $\vect{P}$ is the Poynting flux through the surface of the PNS,
and $W_{\mbox{\tiny L}}$ and $Q_{\mbox{\tiny J}}$ are obtained from $\vect{u}$ and $\vect{B}$, which must be computed with an appropriate physical model of the PNS.
The dissipative term $Q_{\mbox{\tiny J}}$ also involves the resistivity $\eta$.
Resistive dissipation is not likely to suppress field amplification in the PNS \citep{thompsonDuncan_1993}, but may be important to the long-term evolution of the neutron star magnetic field (strength and topology).
Evaluation of the volume integral on the right-hand-side of Eq. (\ref{eq:pnsMagneticEnergy}) involves numerical simulations of the hydro-magnetic evolution inside the PNS during the explosion phase of core-collapse supernovae and subsequent PNS cooling, and includes neutrino radiation-magnetohydrodynamic simulations of dense nuclear matter.
Such calculations are well beyond the scope of this study.
Earlier works have suggested numerous mechanisms for field amplification in the PNS, including winding by differential rotation \citep[e.g.,][]{wheeler_etal_2002}; the magneto-rotational instability \citep{akiyama_etal_2003}; and convective dynamo action, driven by entropy gradients, lepton gradients, or both \citep[e.g.,][]{thompsonDuncan_1993,bonanno_etal_2003,bonanno_etal_2005}.
All these mechanisms operate inside or on the surface of the PNS and rely on rotation.
We exclude the PNS from our simulations and do not address field amplification mechanisms in its interior.
Our simulations, however, focus on field amplification by the SASI exterior to the PNS, which is often ignored in models addressing the origin of pulsar magnetism.
From our simulations we compute the increase in magnetic energy in the volume occupied by the PNS due to the Poynting flux through the surface bounding it,
\begin{equation}
E_{\mbox{\tiny mag},\vect{P}}(t)
=-\int_{t_{0}}^{t}\,dt'\int_{\partial V_{\mbox{\tiny PNS}}}\vect{P}\cdot d\vect{S}.
\label{eq:pnsMagneticEnergyPoyntingFlux}
\end{equation}
We then estimate the PNS magnetic field due to SASI activity $\langle B_{\mbox{\tiny PNS},\vect{P}}\rangle=(2\mu_{0}E_{\mbox{\tiny mag},\vect{P}}/V_{\mbox{\tiny PNS}})^{1/2}$ \citepalias[cf. Eq. (18) in][]{endeve_etal_2010}. Results from these estimates for the rotating and non-rotating models with varying initial magnetic field strengths are listed in Table \ref{tab:pnsMagnetization}.
\begin{table}
\begin{center}
\caption{PNS Magnetic field estimates. \label{tab:pnsMagnetization}}
\begin{tabular}{cccc}
Model & $t_{\mbox{\tiny end}}$ (ms) & $E_{\mbox{\tiny mag},\vect{P}}$ (Erg) & $\langle B_{\mbox{\tiny PNS},\vect{P}}\rangle$ (G) \\
\tableline
\tableline
\model{10}{0.0}{00} & 1100 & $1.14\times10^{44}$ & $3.3\times10^{12}$ \\
\model{10}{1.5}{15} & 878 & $6.70\times10^{43}$ & $2.5\times10^{12}$ \\
\model{10}{4.0}{15} & 678 & $2.52\times10^{43}$ & $1.5\times10^{12}$ \\
\tableline
\model{12}{0.0}{00} & 1126 & $3.16\times10^{47}$ & $1.7\times10^{14}$ \\
\model{12}{1.5}{15} & 1000 & $1.15\times10^{48}$ & $3.3\times10^{14}$ \\
\model{12}{4.0}{15} & 644 & $1.74\times10^{47}$ & $1.3\times10^{14}$ \\
\tableline
\model{13}{0.0}{00} & 1100 & $4.49\times10^{48}$ & $6.5\times10^{14}$ \\
\tableline
\tableline
\end{tabular}
\tablecomments{Magnetic energy accumulated on the proto-neutron star in computed models. The inferred magnetic field strength resulting from SASI-induced magnetic field amplification is also listed.}
\end{center}
\end{table}
Our results show that the magnetic energy generated by SASI activity may result in significant magnetization of the PNS.
The magnetic energies generated in some of the models meet the energy requirements to power the total flare energy released per SGR \emph{and} the persistent X-ray emission \citep{thompsonDuncan_2001}.
The models with the weakest initial magnetic field predict field strengths in the range of ordinary pulsars (a few $\times10^{12}$~G), while the models with stronger initial magnetic fields predict fields in the magnetar range (exceeding $10^{14}$~G).
The magnetic field in the strong-field model ($\model{13}{0.0}{00}$) saturates dynamically, and this model may represent an upper limit to the fields attainable from this process.
On the other hand, the weak-field models do not reach saturation.
The magnetic energy in these models continues to grow at an underestimated rate, and the maximum attainable field strength/energy is also limited by finite grid resolution.
The PNS field strengths predicted by these models are therefore artificially low.
It then seems likely, given infinite grid resolution, that PNS magnetic fields can exceed $10^{14}$~G due to the SASI alone, independent of the initial magnetic field strength.
Moreover, as finite grid resolution severely limits the exponential growth rate of magnetic energy, the duration of SASI operation may be less critical.
The amount of initial rotation in the models does not seem to affect the degree of PNS magnetization.
The field strengths listed in Table \ref{tab:pnsMagnetization} should also be corrected for additional magnetic field amplification as the PNS cools and contracts.
From conservation of magnetic flux through the PNS surface, contraction from a 40~km radius to a radius of about 15~km boosts the surface field by a factor of $\sim7$.
We point out that the PNS magnetic fields resulting from the turbulent flows driven by the SASI are likely small-scale and disordered.
A connection to the dipolar magnetic field structure inferred for neutron stars is currently missing, and, of course, the SASI alone cannot fully explain the origin of pulsar magnetism.
However, pulsar magnetic fields are thought to consist of a global dipole field superimposed with higher order multipole (small-scale) components, and pulsar magnetism is most likely a result of the combined action of multiple amplification mechanisms.
While the inferences we can make are limited by resolution (which affects the magnetic growth rate) and the absence of important physics (which determines the time to explosion), our simulations suggest that the SASI could in principle make a nontrivial contribution.
\section{SUMMARY, DISCUSSION, AND CONCLUSIONS}
\label{sec:discussionConclusions}
We present results from three-dimensional MHD simulations of the SASI.
The simulations are initiated from a configuration that resembles the early stalled shock phase in a core-collapse supernova, albeit with simplified physics that excludes critical components of a supernova model (e.g., neutrino transport, self-gravity, and the PNS itself).
On the other hand, our simulations are computed with a spatial resolution that is currently inaccessible to state-of-the-art supernova models in three spatial dimensions, and they may therefore provide valuable insight into MHD developments in core-collapse supernovae.
In particular we study the evolution and amplification of magnetic fields in SASI-driven flows in order to assess the effects of the amplified magnetic fields on supernova dynamics, and the possible role of the SASI in magnetizing the PNS.
This paper is a continuation and extension of the study initiated in \citetalias{endeve_etal_2010}.
The simulations reported here were performed with higher spatial resolution (up to $1280^{3}$ grid cells) and cover a broader parameter range than the 3D simulations presented in \citetalias{endeve_etal_2010}: we have varied the strength of the initial magnetic field and the degree of rotation in the flow ahead of the shock (including no rotation).
We have also varied the spatial resolution in some of the simulations, and extended the analysis from \citetalias{endeve_etal_2010} to include a Fourier decomposition of the kinetic energy and magnetic energy in the simulations.
Our main findings are
\begin{itemize}
\item[1.] The SASI-driven turbulence that develops is essentially non-helical, and shares similarities with convectively driven MHD turbulence \citep[e.g.,][]{brandenburg_etal_1996}.
(See also ``box turbulence" simulations by \cite{haugen_etal_2004}.)
When corrected for density stratification, the kinetic energy spectra associated with the post-shock flow develop Kolmogorov-like $-5/3$ scaling (i.e., $\edensk{kin}^{\,\mbox{\tiny II}},\edensk{kin}^{\,\mbox{\tiny III}}\propto k^{-5/3}$; Section \ref{sec:spectralAnalysis}) in a narrow wavenumber range.
Moreover, inspection of the time evolution of the kinetic energy spectra reveals that the power in low-order SASI modes (i.e., large scale flows) cascades to higher-order modes (i.e., smaller-scale flows), and that a significant fraction (up to $\sim50\%$) of the post-shock kinetic energy can be associated with turbulence (although a smaller fraction is involved in magnetic field amplification).
This further suggests that the non-linear SASI saturates due to the development of turbulence via secondary instabilities \citep[e.g.,][]{guilet_etal_2010}.
\item[2.] The magnetic energy grows exponentially with time in turbulent flows driven by the SASI, as long as the kinematic regime obtains.
Our simulations develop flows characteristic of the SASI spiral mode \citep[e.g.,][]{blondinMezzacappa_2007}.
These flows drive vigorous turbulence below the shock ($u_{\mbox{\tiny rms}}^{\mbox{\tiny tur}}\sim4000$~km~s$^{-1}$), which amplifies magnetic fields by stretching.
The resulting magnetic field is highly intermittent and consists of thin, intense magnetic flux ropes.
\item[3.] Simulations initiated with weak or moderate rotation evolve similarly to non-rotating models as far as the magnetic field amplification mechanism is concerned.
However, models with initial rotation develop the nonlinear spiral SASI flows earlier, and exponential magnetic energy growth sets in sooner.
The earlier onset of the SASI in models with initial rotation is consistent with the results of \citet{blondinMezzacappa_2007} and \citet{yamasakiFoglizzo_2008}.
\item[4.] The magnetic energy grows at the expense of the kinetic energy available in the turbulent flows driven by the SASI.
Our simulations show that strong magnetic fields emerge on small (turbulent) spatial scales, and reduce the turbulent kinetic energy on those scales.
For our reference spatial resolution, magnetic fields impact flows on scales with wavenumber $k>k_{\mbox{\tiny dyn}}\approx0.1-0.2$~km$^{-1}$ ($\lambda_{\mbox{\tiny dyn}}=2\pi/k_{\mbox{\tiny dyn}}\lesssim30-60$~km) and peak around $k=0.6$~km$^{-1}$ ($\sim10$~km) (Figure \ref{fig:spectralKineticEnergyDensityNonRotating}).
That is, magnetic fields do not affect the portion of the kinetic energy spectrum with $k\lesssimk_{\mbox{\tiny dyn}}$.
The turbulent kinetic energy (that is, the kinetic energy on spatial scales below some specified cutoff) in models with larger magnetic fields is reduced compared to models initiated with weaker magnetic fields, indicating a dynamical impact of the amplified magnetic field.
\item[5.] The magnetic field evolution in our simulations remains very sensitive to the spatial resolution.
Key parameters extracted from simulations performed with increasing spatial resolution do not converge in the range covered in this study.
Both the final magnetic energy attained and the rate at which the magnetic energy grows increase with increasing grid resolution.
In particular, estimates using data extracted from our simulations suggest that the magnetic energy may grow exponentially on a millisecond timescale under physically realistic conditions, with very large magnetic Reynolds numbers, as opposed to the $\sim50$-$60$~ms timescale measured directly in our runs.
\item[6.] The magnetic energy saturates when the magnetic energy density becomes comparable to the kinetic energy density (i.e., $|\vect{B}|/\sqrt{\mu_{0}\rho}\gtrsim|\vect{u}|$) in localized regions of the flow.
Only our ``strong-field'' model (with the largest initial magnetic field) reaches this saturated state.
The subsequent magnetic field evolution remains highly dynamic: strong fields are advected through the flow, are temporarily weakened, and then reemerge in a seemingly stochastic manner.
\item[7.] The magnetic fields amplified by the SASI are not likely to play an important role in the explosion dynamics (but see further discussion below).
The presence of amplified magnetic fields does not result in noticeable effects on the global shock dynamics in our simulations, and this can be understood as a matter of simple energetics.
Magnetic energy grows at the expense of kinetic energy, and the kinetic energy content in the post-shock flow during vigorous SASI activity ($\sim5\times10^{-2}$~B) is not enough for magnetic fields to become energetically significant to the explosion ($\sim1$~B).
This was also pointed out in \citetalias{endeve_etal_2010}.
Moreover, the turbulent kinetic energy---which powers SASI-driven field amplification---accessible for magnetic field amplification only amounts to about $10\%$ of the total kinetic energy below the shock.
We further point out that our estimate for turbulent kinetic energy is \emph{not} critically sensitive to the numerical resolution (Section \ref{sec:spectralAnalysis} and Figure \ref{fig:turbulentKineticEnergy}).
A rapidly rotating (millisecond period) PNS would provide an energy reservoir large enough to power magnetically-driven explosions \citep[e.g.,][]{burrows_etal_2007}, but it is not likely that rotation would be this strong in most supernova progenitors \citep[e.g.,][]{heger_etal_2005}.
These observations suggest a rather passive role of magnetic fields in the overall dynamics of at least most supernovae.
\item[8.] Our simulations suggest that SASI-induced magnetic field amplification may play an important role in determining the strength of the magnetic field in proto-neutron stars and young pulsars.
Upon integrating the Poynting flux through the surface encompassing the PNS, we estimate that the magnetic energy accumulated on the PNS may account for magnetic field strengths exceeding $10^{14}$~G.
This is stronger than the canonical dipole field inferred for typical pulsars, and in this connection two points must be emphasized.
First, SASI-driven amplification is expected to cease when the explosion takes off, so that different delay times to explosion (which may for example be a function of progenitor mass) may result in different degrees of PNS magnetization.
Second, the SASI-amplified portion of the field accumulated by the PNS will at least initially be disordered and not of the large-scale, dipolar character of the fields inferred from pulsar spindown.
\end{itemize}
Despite the pessimism of point 7 above regarding the relevance of SASI-amplified magnetic fields to the explosion mechanism, we caution that the sensitivity of magnetic field amplification and evolution to numerical resolution prevents us from completely dismissing magnetic fields as unimportant to supernova dynamics in weakly rotating progenitors.
Certainly, our simulations cannot accurately describe the dynamical interaction between the magnetic field and the fluid on small scales.
An initially weak magnetic field is amplified exponentially in turbulent flows when the flux tubes are stretched and their cross sectional area decreases.
In a realistic post-shock supernova environment, where $R_{\mbox{\tiny m}}$ is extremely large, field amplification is likely quenched by dynamic back-reaction on the fluid before the flux tube thickness reaches the resistive scale \citep{thompsonDuncan_1993}.
The resistive decay time then remains much longer than the dynamical timescale of hydro-magnetic interactions.
But in numerical simulations the flux tube cross section inevitably approaches the grid scale, and numerical dissipation sets in and prevents further strengthening of the magnetic field.
This occurs in all our simulations.
(The strong-field model ($\model{13}{0.0}{00}$) develops dynamically relevant magnetic fields, but is also strongly affected by numerical dissipation.)
Our simulations suggest that magnetic fields become dynamically relevant on spatial scales smaller than $\lambda_{\mbox{\tiny dyn}}\sim30$~km (Figure \ref{fig:spectralKineticEnergyDensityNonRotating}).
The global shock dynamics remains unaffected by the presence of magnetic fields (e.g., Figure \ref{fig:overviewNonRotating}).
However, we cannot rule out the possibility that flows on scales larger than $\lambda_{\mbox{\tiny dyn}}$ could ultimately be affected by hydro-magnetic interactions emerging from small-scale turbulent flows.
Simulations of non-helical MHD turbulence \citep[e.g.,][]{haugen_etal_2004} show that the magnetic energy grows exponentially on the turnover time, on all spatial scales during the kinematic regime.
(We also observe exponential growth on all scales in our runs during this regime.)
The kinematic regime ends when the magnetic energy becomes comparable to the kinetic energy.
This occurs on a scale by scale basis.
Magnetic energy growth slows down considerably after this equipartition, which occurs first on the smallest spatial scales, and the magnetic energy spectrum settles somewhat above the kinetic energy spectrum.
(We also observe that magnetic energy growth is quenched when $\edens{mag}\sim\edens{kin}$, but the magnetic energy spectrum stays below the kinetic energy spectrum for all $k$ in our simulations.)
At later times in MHD turbulence simulations, the largest spatial scale at which $\edensk{mag}\gtrsim\edensk{kin}$ (i.e., $\lambda_{\mbox{\tiny dyn}}$) increases, and may approach the driving scale of the turbulent forcing.
For helical MHD turbulence, which may be more relevant when a rapidly rotating PNS is included in the model, $\lambda_{\mbox{\tiny dyn}}$ can even grow beyond the forcing scale \citep[e.g.,][]{meneguzzi_etal_1981,brandenburg_2001}.
However, the timescale for this process is relatively slow, and increases with $R_{\mbox{\tiny m}}$ \citep{brandenburg_2001}.
Nevertheless, it would be desirable to determine the largest scale at which the magnetic energy equilibrates with the kinetic energy in SASI-driven flows.
The lack of sufficient spectral resolution in our simulations prevents us from determining whether magnetic fields can become strong on large enough scales to alter post-shock flows in a significant way.
The SASI may play an important role in improving the conditions for successful neutrino-driven explosions \citep[e.g.,][]{bruenn_etal_2006,buras_etal_2006,mezzacappa_etal_2007,marekJanka_2009,suwa_etal_2010,muller_etal_2012}.
If amplified magnetic fields can alter the evolution of the SASI and change the conditions (making them more, or less, favorable) for energy deposition by neutrinos, then magnetic fields may play a secondary but relevant role in the dynamics of a broader range of core-collapse supernovae (i.e. not just those arising from rapidly rotating progenitor stars).
This point was also argued by \citet{obergaulingerJanka_2011}, who studied magnetic field amplification in non-rotating collapsed stellar cores using axisymmetric simulations that included the PNS and neutrino transport.
They found that the SASI and convection contribute to magnetic field amplification, and observed the most pronounced shock expansion in the model where the magnetic field was strong enough to alter the post-shock flow topology.
(This model was initiated with a strong pre-collapse magnetic field.)
However, axial symmetry severely constrains magnetic field evolution driven by the SASI \citepalias[see][]{endeve_etal_2010} and, most likely, also convectively driven field amplification.
Simulations similar to those of \citet{obergaulingerJanka_2011} in full 3D, where the SASI spiral mode can develop and drive turbulent field amplification, are therefore highly desired.
Such simulations will improve on our simulations in (at least) two important ways:
\begin{itemize}
\item[1.] A significant amount of magnetic energy (comparable to that in $V_{\mbox{\tiny{Sh}}}$) is lost through the boundary at $r=R_{\mbox{\tiny PNS}}$ in our models, and not accounted for in the subsequent dynamics.
3D simulations with the PNS included do not suffer from this artificial limitation, and will allow us to better assess the role of SASI-induced magnetic fields.
\item[2.] Simulations that include neutrino transport develop neutrino-driven convection, both in the PNS and in the shocked mantle. This convective activity will impact the evolution of magnetic fields, and possibly also the SASI. We will then be able to study magnetic field evolution in a much more physically realistic supernova environment.
Moreover, with neutrino transport included, we will be able to directly address the role of magnetic fields on neutrino-powered explosions.
\end{itemize}
The constraint on numerical resolution in order to properly describe turbulent flows may still be computationally prohibitive, especially when additional (necessary) physics components are added to the models.
This may be partially circumvented with the use of adaptive mesh refinement techniques and improved numerical algorithms.
Local (or semi-global) simulations \citep[e.g.,][]{obergaulinger_etal_2009}, adopting physical conditions and forcing functions relevant to the supernova environment (i.e., derived from global multi-physics simulations), may also be necessary to study turbulent magnetic field evolution and its impact on supernova dynamics in more detail.
More investigations, using both local and global simulations, are needed to better understand the role of magnetic fields in core-collapse supernovae.
In summary, we conclude from our simulations that magnetic fields in core-collapse supernovae may be amplified exponentially by turbulence on a millisecond timescale; i.e., much shorter than the time between core bounce/shock formation and initiation of the explosion.
Details of the impact on explosion dynamics by SASI-amplified magnetic fields remain unclear, but on energetic grounds alone the role of these magnetic fields is likely sub-dominant.
The simulations further suggest that small-scale neutron star magnetic fields in the $10^{14}-10^{15}$~G range may be formed, which may be sufficient to power some of the energetic activity that define AXPs and SGRs.
\acknowledgments
This research was supported by the Office of Advanced Scientific Computing Research and the Office of Nuclear Physics, U.S. Department of Energy.
This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory provided through the INCITE program.
We are grateful for support from members of the National Center for Computational Sciences during the execution and analysis of the simulations, especially Bronson Messer.
We also thank an anonymous referee for comments that helped us improve the manuscript.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,846 |
Hound Dog est une chanson de rock 'n' roll écrite par Jerry Leiber & Mike Stoller, dont l'interprétation la plus célèbre est celle d'Elvis Presley, éditée par RCA en 1956.
Premiers enregistrements
La chanson, écrite en 1952 en collaboration avec Johnny Otis, est d'abord enregistrée en mars 1953 sous forme de rhythm and blues par la chanteuse afro-américaine « Big Mama » Thornton (dont le vrai nom était Willie Mae Thornton) (Peacock 1612). . Elle se classe à la première place des charts R&B pendant sept semaines. Rufus Thomas la parodie en enregistrant Bear Cat sur Sun Records, qui sera n° 3. D'après le magazine américain Billboard, il s'agit de « la réponse chantée la plus rapide jamais mise sur le marché ».
Cinq versions country sont enregistrées simultanément en 1953 par Billy Starr, Tommy Duncan, Eddie Hazelwood, Jack Turner et Cleve Jackson. Little Esther la chante également cette année-là.
Elle est reprise en 1954 par Freddie Bell & The Bellboys dans une version rock 'n' roll. Sortie en 1955 (Teen 101), les paroles sont légèrement différentes.
Elvis Presley
Elvis, qui entend Freddie Bell interpréter ce morceau à Las Vegas en , s'en inspire. Il la chante une première fois à la télévision le dans le Milton Berle Show. Cette diffusion, qui dépasse tous les records d'audience mesurée jusqu'alors, fait scandale. Elvis l'interprète une seconde fois mémorable au Tonight Show de Steve Allen le juillet, devant un basset impassible, affublé d'un smoking et cadré uniquement au-dessus de la ceinture. Il l'enregistre le dans les studios RCA de New York, avec les Jordanaires dans les chœurs, D.J. Fontana à la batterie, Bill Black à la basse et Scotty Moore à la guitare.
Le 45 tours sort le avec Don't Be Cruel sur l'autre face. Le nom du cocompositeur Johnny Otis a alors disparu de la liste des auteurs de cette chanson. Les deux titres atteignent la première place des hit-parades américain au mois d'août. Hound Dog et à la fois n° 1 dans les charts pop, rhythm & blues et country & western du magazine Billboard. Elvis Presley reçoit un disque d'or pour la vente de de copies de ce single en six mois. Au total, la chanson s'est vendue à plus de 10 millions d'exemplaires. Ce succès phénoménal incitera le « King » à demander d'autres compositions au tandem Leiber/Stoller.
On peut entendre cette version dans de nombreux films comme American Graffiti, Forrest Gump, Lilo et Stitch, Des hommes d'honneur. La chanson introduit même le film Indiana Jones et le Royaume du crâne de cristal.
Scotty Moore, reprend ce morceau sur son album The Guitar That Changed the World! en 1964.
Reconnaissance
En 2004, la version d'Elvis est classée plus grande chanson de tous les temps par le magazine Rolling Stone. Celle de Big Mama Thornton est intronisée au Blues Hall of Fame de la Blues Foundation en 2006 dans la catégorie « Enregistrement classique du Blues — Single ».
Autres reprises
Cette chanson est reprise également par de nombreux autres artistes, parmi lesquels :
Chubby Checker, sur l'album For Twisters Only (1961)
Johnny Hallyday, sur l'album Sings America's Rockin' Hits (1962)
Pat Boone, sur Pat Boone Sings Guess Who?, un album de reprises du King (1963)
Little Richard, sur Little Richard is Back! (1964)
The Everly Brothers, sur Rock'n Soul (1965)
The Easybeats, sur Good Friday (1967)
Jimi Hendrix, sur Radio One (1988, session radio de 1967)
Albert King, sur King Does the King's Thing, autre album de reprises d'Elvis (1970)
John Lennon, sur Live in New York City (concert enregistré en 1972, paru en 1984)
John Entwistle, sur Rigor Mortis Sets In (1973)
The Rolling Stones, sur Handsome Girls (live in Memphis, Tennessee, )
Scorpions, sur Tokyo Tapes (1978)
Eric Clapton, sur Journeyman (1989)
The Residents, sur The King & Eye, encore un album de reprises d'Elvis (1989)
Frank Zappa, sur Beat the Boots I (1991)
Bryan Adams, sur It's Now or Never - The Tribute to Elvis (1994)
Status Quo, sur Famous in the Last Century (2000)
Etta James, sur Matriarch of the Blues (2000)
Willy DeVille, sur In Berlin (2003)
Albert Lee et Scotty Moore sur le DVD Hommage au roi du rock 'n' roll - Live at Abbey Road (2005)
James Taylor, sur Covers (2008)
Nadja, sur Nadja (2009)
Handcrafted Soul sur Life Sounds Better (2014)
et aussi Jerry Lee Lewis, Vigon (1966), Billy "Crash" Craddock, Shakin' Stevens (1983), Dread Zeppelin (1990), Rockapella (1992), Marva Wright (1993), Robert Palmer (2003), Kate Nash, etc.
Elle est interprétée par le groupe Sha Na Na dans la bande originale du film Grease avec John Travolta et Olivia Newton-John (1978), et par Jeff Beck et Jed Leiber pour la musique du film Lune de miel à Las Vegas'' (1992).
Notes et références
Liens externes
Le choc provoqué par la retransmission du Hound Dog d'Elvis à la télé américaine
Chanson de 1953
Chanson de 1956
Chanson interprétée par Elvis Presley
Single numéro un dans le Hot R&B/Hip-Hop Songs
Grammy Hall of Fame Award
Chanson interprétée par Jerry Lee Lewis
Chanson écrite par Jerry Leiber et Mike Stoller
Chanson interprétée par Etta James
Single certifié argent au Royaume-Uni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,860 |
#region Header
// --------------------------------------------------------------------------
// Tethys.Logging
// ==========================================================================
//
// A (portable) logging library for .NET Framework 4.5, Silverlight 4 and
// higher, Windows Phone 7 and higher and .NET for Windows Store apps.
//
// ===========================================================================
//
// <copyright file="Program.cs" company="Tethys">
// Copyright 2009-2015 by Thomas Graf
// All rights reserved.
// Licensed under the Apache License, Version 2.0.
// Unless required by applicable law or agreed to in writing,
// software distributed under the License is distributed on an
// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND,
// either express or implied.
// </copyright>
//
// System ... Portable Library
// Tools .... Microsoft Visual Studio 2013
//
// ---------------------------------------------------------------------------
#endregion
namespace TestApp.Common.Logging231
{
using System;
using System.Windows.Forms;
/// <summary>
/// Program main class.
/// </summary>
public static class Program
{
/// <summary>
/// The main entry point for the application.
/// </summary>
[STAThread]
public static void Main()
{
Application.EnableVisualStyles();
Application.SetCompatibleTextRenderingDefault(false);
Application.Run(new MainForm());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,279 |
\section{Introduction}
In the past two decades, tandem mass spectrometry (\emph{MS/MS}) has
become an indispensable tool for identifying the proteins present in a
complex biological sample. At the output of a typical MS/MS
experiment, a collection of spectra is produced on the
order of tens-to-hundreds of thousands, each of which is
representative of a protein subsequence, called a \emph{peptide},
present in the original complex sample. The main challenge in MS/MS
is accurately identifying the peptides responsible for generating each
output spectrum.
The most accurate identification methods search a database of peptides
to first score peptides, then rank and return the top-ranking such
peptide. The pair consisting of a scored candidate
peptide and observed spectrum is typically referred to as a
\emph{peptide-spectrum match} (PSM). However, PSM scores returned
by such database-search methods are often difficult to compare across
different spectra (i.e., they are poorly calibrated), limiting the
number of spectra identified per search~\cite{keich2014importance}.
To combat such poor calibration, post-processors are typically used to
recalibrate PSM scores~\cite{kall:semi-supervised,
spivak:improvements, spivak:direct}.
Recent work has attempted to exploit generative scoring functions for
use with discriminative classifiers to better recalibrate PSM scores;
by parsing a DBN's \emph{Viterbi path} (i.e., the most probable
sequence of random variables), heuristically derived features were
shown to improve discriminative recalibration using support vector
machines (SVMs). Rather than relying on heuristics, we look towards
the more principled approach of a Fisher
kernel~\cite{jaakkolaFisherKernelNips1998}. Fisher kernels
allow one to exploit the sequential-modeling strengths of generative
models such as DBNs, which offer vast design flexibility for
representing data sequences of varying length, for use with
discriminative classifiers such as SVMs, which offer superior accuracy
but often require feature vectors of fixed length. Although the
number of variables in a DBN may vary given different observed
sequences, a Fisher kernel utilizes the fixed-length gradient of the
log-likelihood (i.e., the \emph{Fisher score}) in the feature-space of
a kernel-based classifier. Deriving the Fisher scores of a DBN for
Rapid Identification of Peptides (DRIP)~\cite{halloran2014uai-drip},
we show that the DRIP Fisher kernel greatly improves upon the previous
heuristic approach; at a strict FDR of
$1\%$ for the presented datasets, the heuristically derived DRIP
features improve accuracy over the base feature set by an average
$6.1\%$, while the DRIP Fisher kernel raises this average improvement to
$11.7\%$ (Table~\ref{table:datasets} in
Appendix~\ref{appendix:impact}), thus
nearly doubling the total accuracy of DRIP post-processing.
Motivated by improvements offered by the DRIP Fisher kernel, we look
to extend this to other models by defining a generative model
representative of the large class of existing scoring
functions~\cite{craig:tandem, eng:approach, eng:comet, kim:msgfPlus,
howbert:computing, wenger2013proteomics, mcilwain:crux}. In
particular, we define a DBN (called \emph{Theseus}\footnote{In honor of
Shannon's magnetic mouse, which could learn to traverse a small
maze.})
which, given an observed spectrum,
evaluates the universe of all possible PSM scores. In this work, we
use Theseus to model PSM score distributions with respect to the
widely used XCorr scoring function~\cite{eng:approach}. The resulting
Fisher kernel once again improves discriminative
post-processing accuracy. Furthermore, with the generative model in
place, we explore inferring parameters of the modeled scoring function
using max-product inference and gradient-based learning. The
resulting coordinate ascent learning algorithm outperforms standard
maximum-likelihood learning. Most importantly, this overall learning
algorithm is unsupervised which, to the authors' knowledge, is the
first MS/MS scoring function parameter estimation procedure not to
rely on any supervision. We note that this overall training
procedure may be adapted by the many MS/MS search algorithms whose
scoring functions lie in the class modeled by Theseus.
The paper is organized as follows. We discuss background information in
Section~\ref{section:background}, including the process by which MS/MS
spectra are produced, the means by which spectra are identified, and
related previous work. In Section~\ref{section:dripFisherScores}, we
extensively discuss the log-likelihood of the DRIP model and derive
its Fisher scores. In Section~\ref{section:theseus}, we introduce
Theseus and derive gradients of its log-likelihood. We then discuss
gradient-based unsupervised learning of Theseus parameters and present
an efficient, monotonically convergent coordinate ascent algorithm.
Finally, in Section~\ref{section:results},
we show that DRIP and Theseus Fisher kernels substantially improve
spectrum identification accuracy and that Theseus' coordinate ascent
learning algorithm provides accurate unsupervised parameter
estimation.
\section{Background}
\label{section:background}
\begin{figure*}
\centering
\includegraphics[trim=0.9in 0.0in 1.0in 0.4in,clip=true,
width=0.48\textwidth]{target.pdf}
\caption{{\small Example tandem mass spectrum with precursor charge
$c(s)=2$ and generating peptide $x =
\mbox{LWEPLLDVLVQTK}$. Plotted in red and blue are, respectively,
b- and y-ion peaks (discussed in
Section~\ref{section:theoreticalSpectra}), while spurious observed
peaks (called \emph{insertions}) are colored gray. Note $y_1,
b_1,b_4,$ and $b_{12}$ are absent fragment ions (called
\emph{deletions}).}}
\label{fig:exampleSpectrum}
\end{figure*}
A typical tandem mass
spectrometry experiment begins by cleaving proteins into peptides
using a digesting enzyme. The resulting peptides are then separated
via liquid chromatography and subjected to two
rounds of mass spectrometry. The first round measures the mass and
charge of the intact peptide, called the \emph{precursor mass} and
\emph{precursor charge}, respectively. Peptides are then fragmented
and the fragments undergo a second round of mass spectrometry,
the output of which is an observed spectrum indicative of the
fragmented peptide. The x-axis of this observed spectrum denotes
\emph{mass-to-charge} (\emph{m/z}), measured in thomsons (\ensuremath{\mathsf{Th}}),
and the y-axis is a unitless intensity measure, roughly proportional
to the abundance of a single fragment ion with a given m/z value. A
sample such observed spectrum is illustrated in
Figure~\ref{fig:exampleSpectrum}.
\subsection{MS/MS Database Search}\label{section:databaseSearch}
Let $s$ be an observed spectrum with
precursor mass $m(s)$ and precursor charge $c(s)$. In order to
identify $s$, we search a database of peptides, as follows.
Let $\mathcal{P}$ be the set
of all possible peptide sequences. Each peptide
$x \in \mathcal{P}$ is a string $x = x_1x_2 \dots x_{n}$ comprised of
characters, called \emph{amino acids}. Given a
peptide database $\mathcal{D} \subseteq \mathcal{P}$, we wish to find the
peptide $x \in \mathcal{D}$ responsible for generating $s$.
Using the precursor mass and charge, the set of peptides to be scored
is constrained by setting a mass tolerance threshold, $w$, such
that we score the set of \emph{candidate peptides}
$D(s, \mathcal{D}, w)= \left\{x: x
\in \mathcal{D} ,\, \left| \frac{m(x)}{c(s)}-m(s) \right| \leq
w\right\}$, where $m(x)$ denotes the mass of peptide $x$.
Note that we've overloaded $m(\cdot)$ to return either a peptide's or
observed spectrum's precursor mass; we similarly overload $c(\cdot)$.
Given $s$ and denoting an arbitrary scoring function as
$\psi(x,s)$, the output of a search algorithm is thus $x^* =
\argmax_{x \in D(m(s), c(s),\mathcal{D},
w)} \psi(x,s)$, the top-scoring PSM.
\subsubsection{Theoretical Spectra}\label{section:theoreticalSpectra}
In order to score a candidate peptide $x$, fragment ions corresponding
to suffix masses (called \emph{b-ions}) and prefix masses (called
\emph{y-ions}) are collected into a \emph{theoretical spectrum}. The
annotated b- and y-ions of the generating peptide for an observed
spectrum are illustrated in Figure~\ref{fig:exampleSpectrum}.
Varying based on the
value of $c(s)$, the $k$th respective b- and y-ion
pair of $x$ are
\begin{equation*}
b(x,c_b, k) = \frac{\sum_{i = 1}^k m(x_i) + c_b}{c_b},
\;\;\;\; y(x,c_y, k) =
\frac{\sum_{i = n-k}^n m(x_i) + 18+ c_y}{c_y},
\end{equation*}
where $c_b$ is the charge of the b-ion and $c_y$ is the charge of the
y-ion. For $c(s) = 1$, we have $c_b = c_y = 1$, since these are the only
possible, detectable fragment ions. For higher observed charge states
$ c(s) \geq 2$, it is unlikely for a single fragment ion to consume
the entire charge, so that we have $c_b + c_y = c(s)$, where $c_b,
c_y \in [1, c(s)-1]$. The b-ion offset corresponds to the mass of a $c_b$
charged hydrogen atom, while the y-ion offset corresponds to the mass of a water
molecule plus a $c_y$ charged hydrogen atom.
Further fragment ions
may occur, each corresponding to the loss of a molecular group off a
b- or y-ion. Called \emph{neutral losses}, these correspond to a loss
of either water, ammonia, or carbon monoxide. These fragment ions are
commonly collected into a vector $v$, whose elements are weighted
based on their corresponding fragment ion. For instance,
XCorr~\cite{eng:approach} assigns all b- and y-ions a weight of 50 and
all neutral losses a weight of 10.
\subsection{Previous Work}
Many scoring functions have been proposed for use in search
algorithms. They range from simple dot-product scoring functions
(X!Tandem~\cite{craig:tandem}, Morpheus~\cite{wenger2013proteomics}),
to cross-correlation based scoring functions
(XCorr~\cite{eng:approach}), to exact $p$-values over linear scoring
functions calculated using dynamic programming
(MS-GF+~\cite{kim:msgfPlus} and XCorr
$p$-values~\cite{howbert:computing}). The recently introduced
DRIP~\cite{halloran2014uai-drip} scores candidate peptides without
quantization of m/z measurements and allows learning the
expected locations of theoretical peaks given high quality, labeled
training data. In order to avoid quantization of the m/z
axis, DRIP employs a dynamic alignment strategy wherein two types of
prevalent phenomena are explicitly modeled:
spurious observed peaks, called \emph{insertions}, and absent
theoretical peaks, called \emph{deletions} (examples of both are
displayed in Figure~\ref{fig:exampleSpectrum}). DRIP then uses
max-product inference to calculate the most probable sequences of
insertions and deletions to score candidate peptides, and was shown to
achieve state-of-the-art performance on a variety of datasets.
In practice, scoring functions are often \emph{poorly calibrated}
(i.e., PSM scores from different spectra are difficult to compare to
one another), leading to potentially identified spectra left on the
table during statistical analysis.
In order to properly recalibrate such PSM scores, several
semi-supervised post-processing methods have been
proposed~\cite{kall:semi-supervised,
spivak:improvements,spivak:direct}. The most popular such method is
Percolator~\cite{kall:semi-supervised}, which, given the output target
and decoy PSMs (discussed in Section~\ref{section:results}) of a
scoring algorithm and features detailing
each PSM, utilizes an SVM to learn a discriminative classifier between
target PSMs and decoy PSMs. PSM scores are then recalibrated using
the learned decision boundary.
Recent work has attempted to leverage the generative nature of the DRIP
model for discriminative use by Percolator~\cite{halloran2016dynamic}.
As earlier mentioned, the output of DRIP is the most probable sequence
of insertions and deletions, i.e., the Viterbi path. However,
DRIP's observations are the sequences of observed spectrum
m/z and intensity values, so that the lengths of PSM's Viterbi paths
vary depending on the number of observed spectrum peaks. In order to
exploit DRIP's output in the feature-space of a discriminative
classifier, PSM Viterbi paths were heuristically mapped to a
fixed-length vector of features. The resulting heuristic features
were shown to dramatically improve Percolator's ability to
discriminate between PSMs.
\subsection{Fisher Kernels}
Using generative models to extract features for discriminative
classifiers has been used to great effect in many problem
domains by using Fisher
kernels~\cite{jaakkolaFisherKernelNips1998, jaakkola1999using,
elkan2005deriving}. Assuming a
generative model with a set of parameters $\theta$ and
likelihood $p(O | \theta) = \sum_{H}p(O, H | \theta)$, where $O$ is
a sequence of observations and $H$ is the set of hidden variables, the
\emph{Fisher score} is then $U_{o} =
\nabla_{\theta} \log p(O | \theta)$. Given observations
$O_i$ and $O_j$ of differing length (and, thus, different
underlying graphs in the case of dynamic graphical models), a
kernel-based classifier over these instances is trained using
$U_{O_i}$ and $U_{O_j}$ in the feature-space. Thus, a similarity
measure is learned in the gradient space, under the intuition that
objects which induce similar likelihoods will induce similar
gradients.
\section{DRIP Fisher Scores}\label{section:dripFisherScores}
\begin{figure}[htbp!]
\begin{center}
\includegraphics[page=8,trim=1.75in 1.2in 0.6in 0.0in,clip=true,
width=0.5\linewidth]{drip_publication_tierLabelled}
\end{center}
\caption{{\small Graph of DRIP, the frames (i.e., time instances) of which correspond to
observed spectrum peaks. Shaded nodes represent observed variables
and unshaded nodes represent hidden variables. Given an observed
spectrum, the middle frame (the chunk) dynamically expands to
represent the second observed peak to the penultimate observed peak.}
}
\label{fig:dripGraph}
\end{figure}
We first define, in detail, DRIP's log-likelihood, followed by the
Fisher score derivation for DRIP's learned parameters. For discussion
of the DRIP model outside the scope of this work, readers are directed
to~\cite{halloran2014uai-drip, halloran2016dynamic}.
Denoting an observed peak as a pair $(O^{\mbox{mz}}, O^{\mbox{in}})$ consisting of an
m/z measurement and intensity measurement, respectively, let $s =
(O^{\mbox{mz}}_1, O^{\mbox{in}}_1), (O^{\mbox{mz}}_2, O^{\mbox{in}}_2), \dots, (O^{\mbox{mz}}_{T},
O^{\mbox{in}}_{T})$ be an MS/MS spectrum of $T$ peaks and $x$ be a
candidate (which, given $s$, we'd like to score). We denote the
theoretical spectrum of
$x$, consisting of its unique b- and y-ions sorted in ascending
order, as the length-$d$ vector $v$. The graph of DRIP is
displayed in Figure~\ref{fig:dripGraph}, where variables which control
the traversal of the theoretical spectrum are highlighted in blue and
variables which control the scoring of observed peak measurements are
highlighted in red. Groups of variables are collected into time
instances called \emph{frames}. The frames of DRIP correspond to the
observed peak m/z and intensity observations, so that there are $T$
frames in the model.
Unless otherwise specified, let $t$ be an arbitrary frame $1 \leq t
\leq T$. $\delta_t$ is a multinomial random variable which dictates
the number of theoretical peaks traversed in a frame. The random
variable $K_t$, which denotes the index of the current theoretical
peak index, is a deterministic function of its parents, such that
$p(K_t = K_{t-1} + \delta_{t} | K_{t-1}, \delta_t) = 1$. Thus,
$\delta_t>1$ corresponds to the deletion of $\delta_t - 1$ theoretical
peaks. The parents of $\delta_t$ ensure that DRIP does not attempt to
increment past the last theoretical peak, i.e., $p(\delta_t > d -
K_{t-1}| d, K_{t-1}, i_{t-1}) = 0$. Subsequently, the theoretical peak value
$v(K_t)$ is used to access a Gaussian from a collection (the mean of
each Gaussian corresponds to a position along the m/z axis, learned
using the EM algorithm~\cite{dempster:maximum}) with which
to score observations. Hence, the state-space of the model is all
possible traversals, from left to right, of the theoretical spectrum,
accounting for all possible deletions.
When scoring observed peak measurements, the Bernoulli random variable
$i_t$ denotes whether a peak is scored
using learned Gaussians (when $i_t = 0$) or considered an insertion
and scored using an insertion
penalty (when $i_t = 1$). When scoring m/z observations, we thus have $p(O^{\mbox{mz}}_t | v(K_t),
i_t = 0) = f(O^{\mbox{mz}}_t | \mu^{\mbox{mz}}(v(K_t)), \sigma^2)$ and $p(O^{\mbox{mz}}_t |
v(K_t), i_t = 1) = a_{\mbox{mz}}$, where $\mu^{\mbox{mz}}$ is a vector of Gaussian
means and $\sigma^2$ the m/z Gaussian variance. Similarly, when scoring intensity
observations, we have $p(O^{\mbox{in}}_t | i_t = 0) = f(O^{\mbox{in}}_t | \mu^{\mbox{in}},
\bar{\sigma}^2)$ and $p(O^{\mbox{in}}_t | i_t = 1) = a_{\mbox{in}}$, where
$\mu^{\mbox{in}}$ and $\bar{\sigma}^2$ are the intensity Gaussian mean and variance,
respectively. Let $i_0 = K_0 = \emptyset$ and $\mathbf{1}_{ \{ \cdot
\}}$ denote the indicator function.
Denoting DRIP's Gaussian parameters as $\theta$, the likelihood is
thus
{\small
\begin{align*}
p(s | x,\theta) &= \prod_{t = 1}^{T} p(\delta_{t} | K_{t-1}, d, i_{t-1})
\mathbf{1}_{\{ K_t = K_{t-1} + \delta_t \}}p(O^{\mbox{mz}}_t | K_t)p(O^{\mbox{in}}_t)\\
&= \prod_{t = 1}^{T} p(\delta_{t} | K_{t-1}, d, i_{t-1})
\mathbf{1}_{\{ K_t = K_{t-1} + \delta_t \}}(\sum_{i_t =
0}^{1}p(i_t)p(O^{\mbox{mz}}_t | K_t, i_t))(\sum_{i_t =
0}^{1}p(i_t)p(O^{\mbox{in}}_t | i_t))\\
&= \prod_{t = 1}^{T} \phi (\delta_t, K_{t-1},
i_t, i_{t-1}).
\end{align*}}
The only stochastic variables in the model are $i_t$ and $\delta_t$, where
all other random variables are either observed or
deterministic given the sequences $i_{1:T}$ and $\delta_{1:T}$. Thus,
we may equivalently write $p(s | x,\theta) =
p(i_{1:T}, \delta_{1:T} | \theta)$. The Fisher score of the
$k$th m/z mean is thus $\frac{\partial}{\partial \mu^{\mbox{mz}}(k)} \log p(s |
x,\theta) = \frac{1}{p(s | x,\theta)} \frac{\partial}{\partial
\mu^{\mbox{mz}}(k)} p(s |x, \theta)$, and we have (please see
Appendix~\ref{appendix:dripFisherKernelDerivation} for the full derivation)
{\small
\begin{align}
\frac{\partial}{\partial
\mu^{\mbox{mz}}(k)} p(s |x, \theta)=&
\frac{\partial}{\partial \mu^{\mbox{mz}}(k)}
\sum_{i_{1:T}, \delta_{1:T}} p(i_{1:T}, \delta_{1:T} | \theta)
=
\sum_{i_{1:T}, \delta_{1:T} : K_t = k, 1 \leq
t \leq T} \frac{\partial}{\partial \mu^{\mbox{mz}}(k)}p(i_{1:T}, \delta_{1:T}
| \theta) \nonumber \\
=&
\sum_{i_{1:T}, \delta_{1:T}} \mathbf{1}_{\{ K_t = k \}}
p(s| x,\theta)
\left ( \prod_{t: K_t = k}\frac{1}{p(O^{\mbox{mz}}_t | K_t)} \right )
\left (\frac{\partial}{\partial \mu^{\mbox{mz}}(k)}
\prod_{t: K_t = k}p(O^{\mbox{mz}}_t | K_t)
\right ). \nonumber
\end{align}
}
{\small
\begin{align}
\Rightarrow
\frac{\partial}{\partial \mu^{\mbox{mz}}(k)} \log p(s | x, \theta)
=& \sum_{t = 1}^T
p(i_{t}, K_{t} = k | s, \theta)
p(i_t = 0 | K_t, O^{\mbox{mz}}_t)\frac{(O^{\mbox{mz}}_t -
\mu^{\mbox{mz}}(k))}{\sigma^2}\label{eqn:fisherScoreMzMean}.
\end{align}}
Note that the posterior in
Equation~\ref{eqn:fisherScoreMzMean}, and thus the Fisher score, may
be efficiently computed using sum-product inference. Through similar
steps, we have
{\small
\begin{align}
\frac{\partial}{\partial \sigma^2(k)} \log p(s | x,\theta) =&
\sum_{t}
p(i_{t}, K_{t} = k | s, \theta)
p(i_t = 0 | K_t, O^{\mbox{mz}}_t)(\frac{(O^{\mbox{mz}}_t -
\mu^{\mbox{mz}}(k))}{2 \sigma^4} - \frac{1}{2\sigma^2})
\label{eqn:fisherMzVariance}\\
\frac{\partial}{\partial \mu^{\mbox{in}}} \log p(s | x,\theta) =&
\sum_{t}
p(i_{t}, K_{t} | s, \theta)
p(i_t = 0 | O^{\mbox{in}}_t)\frac{ (O^{\mbox{in}}_t -
\mu^{\mbox{in}})}{\bar{\sigma}^2}
\label{eqn:fisherIntensityMean}\\
\frac{\partial}{\partial \bar{\sigma}^2} \log p(s | x,\theta) =&
\sum_{t}
p(i_{t}, K_{t} | s, \theta)
p(i_t = 0 | O^{\mbox{in}}_t)
(\frac{(O^{\mbox{in}}_t -
\mu^{\mbox{in}})}{2 \bar{\sigma}^4} - \frac{1}{2\bar{\sigma}^2}),
\label{eqn:fisherIntensityCovariance}
\end{align}
}
where $\sigma^2(k)$ denotes the partial derivative of the variance for
the $k$th m/z Gaussian with mean $\mu^{\mbox{mz}}(k)$.
Let $U_{\mu} =
\nabla_{\mu^{\mbox{mz}}} \log p(s,x | \theta)$ and $U_{\sigma^2} =
\nabla_{\sigma^2(k)} \log p(s,x | \theta)$. $U_{\mu}$ and $U_{\sigma^2}$ are length-$d$
vectors corresponding to the mapping of a peptide's sequence of b- and
y-ions into $r$-dimensional space (i.e., dimension equal to an
m/z-discretized observed spectrum). Let $\mathbbm{1}$ be the
length-$r$ vector of ones. Defining $z^{\mbox{mz}}, z^{\mbox{i}} \in
\mathbb{R}^r$, the elements of which are the
quantized observed spectrum m/z and intensity values, respectively, we
use the following DRIP gradient-based features for SVM training in
Section~\ref{section:results}:
$|U_{\mu}|_1$, $|U_{\sigma^2}|_1$, $U_{\mu}^Tz^{\mbox{mz}}$,
$U_{\sigma^2}^Tz^{\mbox{i}}$, $U_{\mu}^T\mathbbm{1}$,
$U_{\sigma^2}^T\mathbbm{1}$, $\frac{\partial}{\partial \mu^{\mbox{in}}} \log
p(s,x | \theta)$, and $\frac{\partial}{\partial \bar{\sigma}^2} \log p(s,x |
\theta)$.
\section{Theseus}
\label{section:theseus}
Given an observed spectrum $s$, we focus on representing the universe
of linear PSM scores using a DBN. Let $z$ denote the
vector resulting from preprocessing the observed
spectrum, $s$. As a modeling example, we look to represent the
popular XCorr scoring function. Using subscript $\tau$ to denote a
vector whose elements are shifted $\tau$ units, XCorr's scoring
function is defined as
{\small
\begin{align*}
\mbox{XCorr}(s,x) &= v^T z - \sum_{\tau = -75}^{75} v^T z_{\tau}= v^T
(z - \sum_{\tau = -75}^{75} z_{\tau}) = v^T z',
\end{align*}}
where $z' = z - \sum_{\tau = -75}^{75} z_{\tau}$. Let $\theta \in
\mathbb{R}^l$ be a vector of XCorr weights for the various types of
possible fragment ions (described in
Section~\ref{section:theoreticalSpectra}). As described in
\cite{howbert:computing}, given $c(s)$, we
reparameterize $z'$ into a vector $z_{\theta}$ such that
$\mbox{XCorr}(x,s)$ is rendered as a dot-product between $z_{\theta}$
and a boolean vector $u$ in the reparameterized space. This
reparameterization readily applies to any linear MS/MS scoring
function. The $i$th element of $z_{\theta}$ is $z_{\theta}(i) = \sum_{j =
1}^{l} \theta(j) z_j(i)$, where $z_j$ is a vector whose
element $z_j(i)$ is the sum of all higher charged fragment ions
added into the singly-charged fragment ions for the $j$th fragment ion
type. The nonzero elements of $u$ correspond to the singly-charged
b-ions of $x$
and we have $u^Tz_{\theta} = \sum_{i =
1}^nz_{\theta}(m(x_i) + 1) = \sum_{i = 1}^n \sum_{j =
1}^l\theta(j) z_j(m(x_i) + 1) = v^Tz' = \mbox{XCorr}(s,x)$.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[page=19,trim=2.3in 1.95in 2.55in 0.9in,clip=true,
width=0.5\textwidth]{linear_decoder}
\end{center}
\caption{{\small
Graph of Theseus. Shaded nodes are observed
random variables and unshaded nodes are hidden (i.e., stochastic).
The model is unrolled for $n+1$ frames, including $B_0$ in frame zero.
Plate notation denotes $M$ repetitions of the model, where $M$ is the
number of discrete precursor masses allowed by the precursor-mass
tolerance threshold, $w$.}}
\label{fig:generativeModel}
\end{figure}
Our generative model is illustrated in
Figure~\ref{fig:generativeModel}.
$n$ is the maximum possible peptide length and $m$ is one of $M$
discrete precursor masses (dictated by the precursor-mass tolerance
threshold, $w$, and $m(s)$). A
\emph{hypothesis} is an instantiation of random variables across all
frames in the model, i.e., for the set of all possible
sequences of $X_i$ random variables, $X_{1:n} = X_1, X_2, \dots, X_n$,
a hypothesis is $x_{1:n} \in X_{1:n}$. In our case, each hypothesis
corresponds to a peptide and the corresponding log-likelihood its
XCorr score. Each frame after the first contains an amino acid random
variable so that we accumulate b-ions in successive frames and access
the score contribution for each such ion.
For frame $i$, $X_i$ is a random amino acid and $B_i$ the
accumulated mass up to the current frame. $B_0$ and $B_n$ are
observed to zero and $m$, respectively, enforcing the boundary
conditions that all length-$n$ PSMs considered begin with mass zero
and end at a particular precursor mass. For $i > 0$, $B_i$ is a
deterministic function of its parents, $p(B_i | B_{i-1}, X_{i}) =
p(B_i = B_{i-1} +m(X_i)) = 1$. Thus, hypotheses which do not respect
these mass constraints receive probability zero, i.e., $p(B_n \neq m |
B_{n-1}, X_n) = 0$. $m$ is observed to the value of
the current precursor mass being considered.
Let $\mathcal{A}$ be the set of amino acids, where $|\mathcal{A}| = 20$. Given
$B_i$ and $m$, the conditional distribution of $X_i$ changes such that
$p(X_i \in \mathcal{A} | B_{i-1} < m) = \alpha \mathcal{U}\{
\mathcal{A} \}, p(X_i = \kappa | B_{i-1} \geq m) = 1$, where
$\mathcal{U}\{ \cdot \}$ is the uniform distribution over the input
set and $\kappa \notin \mathcal{A}$, $m(\kappa) = 0$. Thus, when the
accumulated mass is less than $m$, $X_i$ is a random amino acid and,
otherwise, $X_i$ deterministically takes on a value with zero mass.
To recreate XCorr scores, $\alpha = 1 / |\mathcal{A}|$, though, in
general, any desired mass function may be used for $p(X_i
\in \mathcal{A} | B_{i-1} < m)$.
$S_i$ is a \emph{virtual evidence
child}~\cite{pearl:probabilistic}, i.e., a leaf node
whose conditional distribution need not be normalized to compute
probabilistic quantities of interest in the DBN. For our model, we
have $p(S_i | B_i < m, \theta) = \exp(z_{\theta}(B_i)) = \exp(\sum_{i =
1}^{|\theta|}\theta_i z_{i}(B_i))$ and $p(S_i | B_i \geq m, \theta)
= 1$.
Let $t'$ denote the first frame in which $m(X_{1:n}) \geq m$. The
log-likelihood is then $\log p(s, X_{1:n} | \theta)$
{\small
\begin{align*}
&= \log p(X_{1:n},
B_{0:n}, S_{1:{n-1}})\\
& = \log( \mathbf{1}_{ \{B_0 = 0
\}} ( \prod_{i=1}^{n-1} p(X_i | m, B_{i-1}) p(B_i = B_{i-1} +
m(X_i)) p(S_i| m, B_i, \theta) ) \mathbf{1}_{ \{B_{n-1} + m(X_n)
= m\}})\\
& = \log \mathbf{1}_{ \{B_0 = 0 \; \wedge m(X_{1:n}) = m \}} +
\log (\prod_{i=t'+1}^{n} p(X_i | m, B_{i-1}) p(B_i = B_{i-1}
+ m(X_i)) p(S_i | m, B_i, \theta)) + \\
& \;\;\;\; \log (
\prod_{i=1}^{t'} p(X_i | m, B_{i-1}) p(B_i = B_{i-1}
+ m(X_i)) p(S_i| m, B_i, \theta) )\\
& = \log \mathbf{1}_{ \{ m(X_{1:n})
= m \}} +
\log 1 + \log (
\prod_{i=1}^{t'} \exp(z_{\theta}(B_i)) )\\
& = \log \mathbf{1}_{ \{m(X_{1:n})
= m \}} + \sum_{i = 1}^{t'}z_{\theta}(B_i) = \log \mathbf{1}_{ \{B_0 = 0
\; \wedge \; m(X_{1:n}) = m \}} + \text{XCorr}(X_{1:t'}, s)
\end{align*}
}
The $i$th element of Theseus' Fisher score is thus
{\small
\begin{align}
\frac{\partial}{\partial \theta(i)} \log p(s | \theta) &=
\frac{\partial}{\partial \theta(i)} \log
\sum_{x_{1:n}}p(s, x_{1:n} | \theta)= \frac{1}{p(s
| \theta)} \frac{\partial}{\partial \theta(i)}
\sum_{x_{1:n}}p(s, x_{1:n} | \theta) \nonumber\\
&= \frac{1}{p(s | \theta)}
\sum_{x_{1:n}} \mathbf{1}_{ \{b_0 = 0 \; \wedge \;
m(x_{1:n}) = m \} } (\sum_{j=1}^{t'}z_i(b_j))\prod_{j=1}^{t'}
\exp(z_{\theta}(b_j)) \label{equation:theseusFisher}
,
\end{align}
}
While Equation~\ref{equation:theseusFisher} is generally difficult to
compute, we calculate it efficiently using sum-product inference.
Note that when the peptide sequence is observed,
i.e., $X_{1:n} = \hat{x}$, we have $\frac{\partial}{\partial
\theta(i)} \log p(s, \hat{x} | \theta) =
\sum_{j}z(m(\hat{x}_{1:j}))$.
Using the model's Fisher scores,
Theseus' parameters $\theta$ may be learned via maximum likelihood
estimation. Given a dataset of spectra $s^1, s^2, \dots, s^n$, we
present an alternate learning algorithm
(Algorithm~\ref{algorithm:coordinateAscent}) which
converges monotonically to a local optimum (proven in
Appendix~\ref{appendix:monotonicProof}). Within each iteration,
Algorithm~\ref{algorithm:coordinateAscent} uses max-product inference
to efficiently infer the most probable PSMs per iteration,
mitigating the need for training labels. $\theta$ is maximized in
each iteration using gradient ascent.
{\small
\begin{algorithm}
\caption{Theseus Unsupervised Learning Algorithm}\label{algorithm:coordinateAscent}
\begin{algorithmic}[1]
\While{not converged}
\For{$i = 1, \dots, n$}
\State $\hat{x}^i \leftarrow \argmax_{x^i \in \mathcal{P}} \log p(s^i, x^i | \theta)$
\EndFor
\State $\theta \leftarrow \argmax_{\theta} \sum_{i=1}^n \log p(s^i, \hat{x}^i |
\theta)$
\EndWhile
\end{algorithmic}
\end{algorithm}
}
\section{Results}
\label{section:results}
Measuring peptide identification performance is complicated by the
fact that ground-truth is unavailable for real-world data. Thus, in
practice,
it is most common to estimate the \emph{false discovery rate}
(\emph{FDR})~\cite{benjamini:controlling} by searching a decoy
database of peptides which
are unlikely to occur in nature, typically generated by
shuffling entries in the
target database~\cite{keich2015improved}. For a particular
score threshold, $t$, FDR is then calculated as the proportion of
decoys scoring better than $t$ to the number of targets scoring better
than $t$. Once the target and decoy PSMs are calculated, a
curve displaying the FDR threshold vs. the number of correctly
identified targets at each given threshold may be calculated. In
place of FDR along the x-axis, we use the
\emph{q-value}~\cite{keich2015improved}, defined to be the minimum
FDR threshold at which a
given score is deemed to be significant. As many applications
require a search algorithm perform well at low thresholds, we only
plot $q \in [0, 0.1]$.
The same datasets and search settings used to evaluate DRIP's
heuristically derived features in~\cite{halloran2016dynamic} are
adapted in this work. MS-GF+ (one of the most accurate search
algorithms in wide use, plotted for reference) was run using version
9980, with PSMs ranked by E-value and Percolator features calculated
using \texttt{msgf2pin}. All database
searches were run using a $\pm 3.0 \ensuremath{\mathsf{Th}}$ mass tolerance, XCorr
flanking peaks not allowed in Crux searches, and all search
algorithm settings otherwise left to their defaults. Peptides were
derived from the protein databases using trypsin cleavage rules
without suppression of proline and a single fixed carbamidomethyl
modification was included.
Gradient-based feature representations derived from DRIP and
XCorr were used to train an SVM classifier~\cite{kall:semi-supervised} and recalibrate PSM
scores. Theseus training and computation of XCorr Fisher scores were
performed using a customized version of Crux
v2.1.17060~\cite{mcilwain:crux}. For an XCorr PSM, a feature
representation is derived directly using both $\nabla_{\theta} \log
p(s | \theta)$ and $\nabla_{\theta} \log p(s,x | \theta)$ as defined
in Section~\ref{section:theseus}, representing gradient information
for both the distribution of PSM scores and the
individual PSM score, respectively.
DRIP gradient-based
features, as defined in Section~\ref{section:dripFisherScores}, were
derived using a customized version of the DRIP
Toolkit~\cite{halloran2016dynamic}
Figure~\ref{fig:absRanking}
displays the resulting search accuracy for
four worm and yeast dataset
. For the uncalibrated search
results in Figure~\ref{fig:theseusUnsupervisedLearning}, we show that
XCorr parameters may be learned without supervision using Theseus, and
that the presented coordinate descent algorithm (which estimates the
most probable PSMs to take a step in the objective space) converges to
a much better local optimum than maximum likelihood estimation.
\begin{figure}[htbp!]
\centering
\subfigure{\raisebox{9.0mm}{\includegraphics[trim=3.4in 1.0in 0.3in 1.7in,
clip=true,scale=0.485]{dripTideLegend.pdf}}}
\subfigure[Worm-1]{\includegraphics[trim=0.0in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-worm-01-ch123.pdf}}
\subfigure[Worm-2]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-worm-02-ch123.pdf}}
\subfigure[Worm-3]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-worm-03-ch123.pdf}}
\subfigure[Worm-4]{\includegraphics[trim=0.0in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-worm-04-ch123.pdf}}
\subfigure[Yeast-1]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-yeast-01-ch123.pdf}}
\subfigure[Yeast-2]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-yeast-02-ch123.pdf}}
\subfigure[Yeast-3]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-yeast-03-ch123.pdf}}
\subfigure[Yeast-4]{\includegraphics[trim=0.0in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{dripTide-dripKernel-yeast-04-ch123.pdf}}
\caption{Search accuracy plots measured by
$q$-value versus number of spectra identified for worm
(\emph{C. elegans}) and yeast (\emph{Saccharomyces cerevisiae})
datasets. All methods are post-processed using the
Percolator SVM classifier~\cite{kall:semi-supervised}. ``DRIP''
augments the standard set of DRIP features
with DRIP-Viterbi-path parsed PSM features (described
in~\cite{halloran2016dynamic}) and ``DRIP Fisher''
augments the heuristic set with gradient-based DRIP
features. ``XCorr,'' ``XCorr $p$-value,'' and ``MS-GF+'' use
their standard sets of Percolator features (described
in~\cite{halloran2016dynamic}), while ``XCorr $p$-value Fisher''
and ``XCorr Fisher'' augment the standard XCorr feature sets with
gradient-based Theseus features.
}
\label{fig:absRanking}
\end{figure}
\begin{figure}[htbp!]
\centering
\subfigure{\raisebox{9.0mm}{\includegraphics[trim=2.0in 1.0in 0.3in 1.7in,
clip=true,scale=0.45]{trainLegend}}}
\subfigure[Yeast-1]{\includegraphics[trim=0.0in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{theseusTrain-yeast-01}}
\subfigure[Yeast-2]{\includegraphics[trim=0.45in 0.0in 0.0in 0.05in,
clip=true,scale=0.3]{theseusTrain-yeast-02}}
\caption{Search accuracy of Theseus' learned scoring function
parameters. Coordinate ascent parameters are learned using
Algorithm~\ref{algorithm:coordinateAscent} and MLE parameters are
learned using gradient ascent.}
\label{fig:theseusUnsupervisedLearning}
\end{figure}
\subsection{Discussion}
DRIP gradient-based
post-processing improves upon the heuristically derived features in
all cases, and does so substantially on a majority of datasets. In
the case of the yeast datasets, this distinguishes DRIP
post-processing performance from all competitors and leads to
state-of-the-art identification accuracy. Furthermore, we note that
both XCorr and XCorr $p$-value post-processing performance are greatly
improved using the gradient-based features derived using Theseus,
raising performance above the highly similar MS-GF+ in several cases.
Particularly noteworthy is the substantial improvement in XCorr
accuracy which, using gradient-based information, is nearly
competitive with its $p$-value counterpart.
Considering the respective runtimes of the underlying search
algorithms, this thus presents a tradeoff for a researcher considering
search time and accuracy. In
practice, the DRIP and XCorr $p$-value
computations are at least an order of magnitude slower than XCorr
computation in Crux~\cite{wang2016faster}. Thus, the
presented work not only improves state-of-the-art accuracy, but also
improves the accuracy of simpler, yet significantly faster, search
algorithms.
Owing to max-product inference in graphical models, we also show that
Theseus may be used to effectively learn XCorr model
parameters (Figure~\ref{fig:theseusUnsupervisedLearning}) without
supervision. Furthermore, we show that XCorr $p$-values are also made
more accurate by training the underlying scoring function for which
$p$-values are computed. This marks a novel step towards unsupervised
training of uncalibrated scoring functions, as unsupervised learning
has been extensively explored for post-processor recalibration, but
has remained an open problem for MS/MS database-search scoring
functions.
The presented learning framework, as well as the
presented XCorr gradient-based feature representation, may be adapted
by many of the widely scoring functions represented by
Theseus~\cite{craig:tandem, eng:approach, eng:comet, kim:msgfPlus,
howbert:computing, wenger2013proteomics,
mcilwain:crux}.
Many exciting avenues are open for future work. Leveraging the large
breadth of graphical models research, we plan to explore other learning paradigms using
Theseus (for instance, estimating other PSMs using $k$-best Viterbi in
order to discriminatively learn parameters using algorithms such as
max-margin learning). Perhaps most exciting, we plan to further
investigate the peptide-to-observed-spectrum mapping derived from DRIP
Fisher scores. Under this mapping, we plan to explore learning distance
metrics between PSMs in order to identify proteins from peptides.
\noindent {\bf Acknowledgments}: This work was supported by the
National Center for Advancing Translational Sciences (NCATS), National
Institutes of Health, through grant UL1 TR001860.
\bibliographystyle{plain}
\setcitestyle{numbers, open={[}, close={]}}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,618 |
package org.hillview.remoting;
import org.hillview.dataset.api.IMap;
import java.io.Serializable;
import java.util.List;
/**
* Wrap an IMap object to be sent to a remote node for a flatMap operation.
* @param <T> Input type of the map function
* @param <S> Output type of the map function
*/
public class FlatMapOperation<T, S> extends RemoteOperation implements Serializable {
public final IMap<T, List<S>> mapper;
public FlatMapOperation(final IMap<T, List<S>> mapper) {
this.mapper = mapper;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,115 |
\section*{Introduction}
Matrix models have been very useful to generate and study random geometries in two dimensions. At large matrix size $N$, the $1/N$ expansion is a topological expansion, labeled by the genus of the random discrete surfaces. In the large $N$ limit, only planar maps on the sphere survive. These maps encode discrete geometries of fluctuating surfaces, making them very important in physics. A famous application is two-dimensional gravity coupled to conformal matter (central charge $c<1$) \cite{mm-review-difrancesco}.
Tensor models allow to extend those ideas to random geometries with more than two dimensions \cite{ambjorn-3d-tensors, sasakura-tensors, gross-tensors}. Their Feynman expansion is a sum over discretized (pseudo-)manifolds in dimension $d$ and it possesses a $1/N$ expansion \cite{Gur4, uncoloring}. A continuum limit exists, first found in \cite{critical-colored}, which can be coupled to (non-unitary) critical matter \cite{harold-hard-dimers, multicritical-dimers}, leading to different universality classes.
The progress obtained in the past few years on tensor models are due to the discovery that tensor models with a $U(N)^d$ symmetry naturally generate regular, edge-colored graphs (dual to triangulations of pseudo-manifolds) \cite{uncoloring}. Those graphs, in contrast with the stranded graphs initially considered in \cite{ambjorn-3d-tensors}, are amenable to analytical investigation. A combinatorial classification has been recently obtained, \cite{GurauSchaeffer}. In the same time, tensor models with quartic interactions have been re-formulated as matrix models, \cite{DSQuartic, GenericQuartic}. Both approaches have led to a double-scaling limit and more generally to a good understanding of the singularities at fixed order in the $1/N$ expansion. The double scaling limit has been extended to models beyond the quartic interactions in \cite{DSSD} using a typical tool of matrix models, the loop equations.
It thus appears that matrix model techniques can be useful in tensor models. Formulating tensor models as matrix models also opens the possibility of using the combinatorial techniques (or even maybe already existing results) on maps. However a precise study of the relationships between tensor and matrix models has not appeared yet. This is the program we start in the present article.
It was not obvious at first that matrix models techniques would be of any use. In particular, diagonalization and eigenvalues (together with the saddle point method or orthogonal polynomials) are among the most effective tools in random matrix models and they are not available for random tensors. Also the fact that $U(N)^d$-invariant random tensors become Gaussian at large $N$ \cite{universality}, and are thus very different from large $N$ matrix models, tends to establishing a clear distinction between matrices and tensors.
However those arguments are no longer relevant thanks to the intermediate field method which turns quartic models into multi-matrix models. In addition, there are two simple ideas which establish a direct connection between matrices and tensors, which we present and exploit in this article. They enable to understand the position of tensor models with respect to matrix models. Following those two ideas one after the other, we offer a novel presentation of random tensor models, in which results from tensor models are applied to matrix models and the other way around.
The first part \ref{sec:loops} is based on the observation that a collection of matrices $M_1,\dotsc, M_\tau$ may be packaged into a tensor of rank three and size $N\times N\times \tau$, whose first two indices are matrix indices while the third one is the label of the matrix. When the joint probability distribution on the matrices is of the form $e^{-V}$ for a polynomial potential $V$ that is $U(\tau)$-invariant, then we have a tensor model in disguise. We therefore introduce a family of $U(\tau)$ models which is shown to generate random surfaces dressed with configurations of oriented loops. We describe the bijection between the observables and the Feynman graphs of those $U(\tau)$ models and their corresponding tensor models.
All known $1/N$ expansions in tensor models rely on the \emph{degree} of the Feynman graphs dual to the triangulations. It was originally introduced in \cite{Gur3} to exhibit a $1/N$ expansion for tensor models for the first time. The degree was defined as a sum of genera of ribbon sub-graphs which are generated by matrix models embedded in the tensor theory \cite{JacketsMatrixModels}. It controls the balance between the number of faces and the number of vertices and reduces to the genus in two dimensions. The dominant triangulations of tensor models at large $N$ are those with vanishing degree and are known as \emph{melonic} triangulations \cite{critical-colored, uncoloring}, which have a specific, highly curved, geometry. They have been recently matched to random branched polymers \cite{melons-BP}, meaning that the continuous geometry is that of the continuous random tree. Melonic graphs are the ones which maximize the number of faces at fixed number of vertices \cite{critical-colored}.
In the $U(\tau)$ models, it turns out that the number of loops at fixed genus of the random surfaces, fixed numbers of edges and vertices, is counted by the degree of the 4-colored graph representative of the Feynman graph. This provides a new combinatorial interpretation of the degree. In particular, melonic graphs are those which maximize the number of loops. It also makes clear how the large $N$, melonic behavior of tensor models arise from a matrix model when $\tau\to\infty$. We then apply the classification of edge-colored graphs from \cite{GurauSchaeffer} to the quartic $U(\tau)$ model to get a classification of its loop configurations. Finally, the double scaling limit of tensor models is found to resum consistently the most critical loop configurations.
The intermediate field transformation is also performed on the quartic $U(\tau)$ models, leading to a two-matrix model. To our knowledge, this two-matrix model has never been studied in the matrix model literature and we do not even know its large $N$ free energy. It generates graphs formed by two maps glued together at their vertices (at least at one of them for the whole graph to be connected). Those graphs, also called \emph{nodal surfaces} do have already appeared in the literature \cite{EynardBookMulticut} and they may be well suited for a combinatorial analysis.
In a companion paper \cite{AngularIntegralsGaussian}, another simple relationship between matrix and tensor models is studied. It relies on re-packaging the set of $d$ indices into two disjoint sets which are interpreted as indices of range $N^p$ and $N^{d-p}$ so that $T$ becomes a (typically rectangular) matrix. The singular value decomposition then enables to perform partial integration over the angular degrees of freedom. The results in a notion of effective observables which actually allows the calculation of new expectations in the Gaussian distribution.
In addition to exploring relationships between random tensors and matrices, those approaches clarify the difficulties faced by random tensor theory in the light of familiar matrix models. We also hope that it sets a frame in which those challenges may be dealt with.
Finally, the section \ref{sec:interpolating} in appendix investigates the possibility of interpolating matrix and tensor models, a question often asked, or more generally interpolating various tensor models. We use for instance a tensor of size $N\times N \times \dotsb \times N^\beta$ where $\beta$ runs in $[0,1]$. This completes the analysis of \cite{new1/N} of tensor models with distinct index ranges. It is found that there are only two large $N$ behaviors in those models, $\beta=0$ and $\beta\in (0,1]$. The reason is that for $\beta=0$ we have a tensor model of rank $d-1$ but as soon as $\beta>0$ each face of colors $(0,d)$ contributes to the large $N$ scaling. However, the $1/N$ corrections are typically found to depend on $\beta$, but we do not know if this affects the continuum limits.
\section{The degree expansion in completely packed loop models on random surfaces} \label{sec:loops}
\subsection{Loop model on random surfaces}
Matrix models are known to generate discretized random 2D surfaces. Each term of the action has the form $\tr (AA^\dagger)^n$, where $A$ is a complex matrix, and creates ribbon vertices of degree $2n$. A matrix model generates random surfaces through the Wick theorem which connects these ribbon vertices together via ribbon lines. Following the recipe of \cite{difrancesco-folding}, the random surfaces can be decorated with oriented loops in the following way: let $\{A_i, A_i^\dagger, i = 1, \dotsc, \tau\}$ be a set of decorated matrices, and rewrite the terms $\tr (AA^\dagger)^n$ with various matrix labelings. We allow terms of the form
\begin{equation}
\label{eq:TraceInvariant}
V_{n,\sigma}(\{A_i,A_i^\dagger\}) = \sum_{\substack{\alpha_1, \dotsc\alpha_n\\\beta_1,..,\beta_n}} \tr\left( A_{\alpha_1}A^\dagger_{\beta_1} A_{\alpha_2} A^\dagger_{\beta_2}\,\dotsm\,A_{\alpha_n}A^\dagger_{\beta_n}\right) \prod_{k=1}^n \delta_{\alpha_k,\beta_{\sigma(k)}},
\end{equation}
where $\sigma$ is a permutation of $\{1,\dotsc,n\}$ (there are obviously redundancies in this parametrization).
Such terms can be interpreted as $n$ lines meeting, and possibly crossing, at a $2n$-valent ribbon vertex. The incoming line in position $k$ (corresponding to $A_{\alpha_k}$) crosses the vertex and go out in position $\sigma(k)$ (corresponding to $A^\dagger_{\alpha_{\sigma(k)}}$). This is pictured in figure \ref{fig:InterpretationAsLoops}. We call the drawing associated to $\sigma$ the \emph{link pattern labeled by $\sigma$}.
\begin{figure}
\center
\subfigure[Interpretation as crossing loops on the ribbon vertex generated by the term $\tr A_a A_b^\dagger A_b A_c^\dagger A_d A_a^\dagger A_c A_d^\dagger$]{\makebox[6cm]{\includegraphics[width=4cm]{CrossingLoops.pdf}}}
\hspace{1cm}
\subfigure[Interpretation as non-crossing loops of the term $\tr A_a A_b^\dagger A_b A_a^\dagger A_c A_c^\dagger A_d A_d^\dagger$\label{subfig:NonCrossingLoop}]{\makebox[6cm]{\includegraphics[width=4cm]{NonCrossingLoops.pdf}}}
\caption{Interpretation in terms of loops on ribbon vertices of the labeled matrix model. The loops are naturally oriented from $A_\alpha$ towards $A_\alpha^\dagger$. \label{fig:InterpretationAsLoops}}
\end{figure}
In this model, the most general action reads
\begin{equation} \label{matrixaction}
S(\{A_i,A_i^\dagger\}) = \sum_{i=1}^\tau \tr A_i A_i^\dagger + \sum_{(n,\sigma)} V_{n,\sigma}(\{A_i,A_i^\dagger\}),
\end{equation}
where the sums typically run over a finite set of terms only. In the Feynman expansion, propagators connect ribbon vertices so as to form random (orientable) surfaces, as usual in matrix models. Moreover, each half-line of a ribbon vertex carries a (incoming or outgoing) line with index $i=1,\dotsc,\tau$, and these half-lines are connected by propagators to create loops. Each propagator between two vertices identifies their label $i=1,\dotsc,\tau$. As a result, there is a free sum per loop, giving rise to a factor $\tau$, hence a factor $\tau^L$ for the whole ribbon graph, $L$ being its number of loops.
The free energy of the model admits the following expansion,
\begin{equation} \label{matrixF}
F = N^2f = -\ln \int \prod_{i=1}^{\tau}\diff A_i\diff A_i^\dagger \exp\left(-\frac{N}{\lambda}S(\{A_i,A_i^\dagger\})\right) = \sum_{\substack{\text{connected}\\\text{ribbon graphs G}}} \frac{1}{s(G)}N^{2-2g(G)}\lambda^{E-V}\tau^{L},
\end{equation}
where $s(G)$ is a symmetry factor, $E$ is the number of edges, $V$ of vertices, $F$ of faces and $L$ of loops. The $1/N$ expansion of the free energy is, as usual, the genus expansion, where the genus $g$ is
\begin{equation}
\label{eq:Genus}
2 - 2g(G) = F - E + V.
\end{equation}
It is worth noting that two kinds of configurations may happen:
\begin{itemize}
\item CPL configurations, where all loops are self and mutually avoiding. The name `CPL' comes from the Completely Packed Loop model. In what follows, we will see that these CPL configurations have a dominant role. They are generated by gluings of link patterns with no crossing, like on figure \ref{subfig:NonCrossingLoop}, i.e. \emph{planar} patterns up to rotations and reflection,
\item Configurations with crossings, where at least one loop crosses itself or another loop.
\end{itemize}
\subsection{Mapping to colored graphs and the degree expansion of tensor models}
We will map the Feynman graphs of our matrix model to a family of edge-colored graphs which we now introduce.
\subsubsection{Colored graphs and their degree}
\begin{definition}
A $\Delta$-colored graph is a regular bipartite graph (say, with black and white vertices) where each edge carries a color from the set $\{1,\dotsc,\Delta\}$ and such that the vertices have degree $\Delta$ and the edges incidnet to a vertex all have distinct colors.
\end{definition}
Some graphs are given in figure \ref{fig:bubbles}. If $2p$ denotes the number of vertices in such a graph, the total number of edges is $\Delta p$, and the number of edges of any given color is simply $p$. Furthermore, coloring gives an additional structure, which provides in particular a natural notion of faces. A \emph{face with colors} $(a,b) \in \{1,\dotsc,\Delta\}$ is a closed path with alternating colors $a$ and $b$. The total number of faces of a graph $G$ is $F(G) = \sum_{a<b} F_{ab}$, where $F_{ab}$ is the number of faces with colors $a,b$.
\begin{figure}
\includegraphics[scale=.5]{tensobs.pdf}
\caption {\label{fig:bubbles} Graphs on six vertices with 3 colors.}
\end{figure}
\begin{definition}
Let $\Delta\geq 3$ be an integer, $G$ be a connected, $\Delta$-colored graph with $2p$ vertices and $\sigma$ be a cycle on $\{1,\dotsc,\Delta\}$. The jacket $J$ associated to $\sigma$ is the connected ribbon graph which contains all the faces of colors $(\sigma^q(1),\sigma^{q+1}(1))$ for $q=0,\dots,\Delta-1$ in $G$. Therefore the number of faces in $J$ is given by $f_J = 2-2g_J + \Delta p -2p$, where $g_J$ is the genus of $J$. We define the degree $\omega(G)\in{\mathbb N}$ of $G$ as the sum of the genera of the jackets.
\end{definition}
One gets an (over-)counting of faces by summing the formulas of the genus over all jackets, leading to the following theorem.
\begin{theorem} \label{thm:degree}
Let $G$ be a $\Delta$-colored graph with $2p$ vertices. The number of faces and vertices are related to the degree as follows,
\begin{equation} \label{degree formula}
F - \frac{(\Delta-1)(\Delta-2)}{2}\,p = \Delta-1 -\frac2{(\Delta-2)!}\,\omega(G).
\end{equation}
\end{theorem}
For a 3-colored graph, $2-2\omega(G) = F-p = F - 3p +2p$, where $3p$ is the total number of lines. Therefore the degree then reduces to the well-known formula of the genus. The degree was introduced for 4-colored graphs in \cite{Gur3}, and generalized in \cite{Gur4}.
The colored graphs generated by a model of a single random tensor of rank $d$, $T_{a_1 \dotsb a_d}$, are obtained from the following Feynman rules. A \emph{bubble} is a connected $d$-colored graph, with colors $1,\dotsc,d$, like in figure \ref{fig:bubbles}. It generalizes the notion of ribbon vertex used in two dimensions \cite{uncoloring}. Propagators then create edges which connect black vertices to white vertices. We assign the color $0$ to these edges. Each vertex thus receives an edge of color 0 in addition to the $d$ other edges of its bubble. Therefore the Feynman graphs are $(d+1)$-colored graphs with colors $\{0,\dotsc,d\}$ built by gluing some bubbles, as in figure \ref{fig:feynmangraph}.
\begin{figure}
\includegraphics[scale=.5]{tensobsgraph.pdf}
\caption{\label{fig:feynmangraph} This is a $(3+1)$-colored graphs, obtained by connecting bubbles (in solid lines) via propagators (dashed lines, with the color 0).}
\end{figure}
Bubbles are colored graphs and therefore have a degree. Applying the degree formula \eqref{degree formula} to a Feynman graph $G$ with $d+1$ colors and to all its bubbles $\{B_\rho\}$, it comes
\begin{equation} \label{graph degree}
\sum_{a=1}^d F_{0a} - (d-1)(p-b) = d - 2\left[ \frac{1}{(d-1)!}\,\omega(G) - \frac{1}{(d-2)!}\sum_\rho \omega(B_\rho)\right].
\end{equation}
The quantity into square brackets is a positive integer, which is zero if and only if $\omega(B_\rho)=0$ for each bubble together with $\omega(G)=0$.
The large $N$ limit of tensor models is dominated by graphs which maximize the number of faces at fixed number of vertices and bubbles. They are therefore the graphs whose degrees vanish as well as the degrees of their bubbles. Such bubbles and Feynman graphs are called \emph{melonic}.
\begin{definition}
A closed melonic graph with $\Delta$ colors is built by recursive insertions of $(\Delta-1)$-dipoles, i.e. two vertices connected by $\Delta-1$ lines inserted on any line, starting from the closed graph on two vertices. The $(\Delta-1)$-dipole is represented in figure \ref{fig:dipole}, as well as a melonic graph.
\end{definition}
\begin{theorem}
The colored graphs of degree $\omega=0$ are the melonic graphs.
\end{theorem}
This theorem was proved in \cite{critical-colored}.
\begin{figure}
\includegraphics[scale=0.5]{D-1Dipole.pdf}
\hspace{3cm}
\includegraphics[scale=.35]{Melonic2pt.pdf}
\caption{\label{fig:dipole} On the left is a $(D-1)$-dipole with external color $c$. A 2-point (i.e. with two open half-edges) melonic graph on 4 colors is represented on the right. It is built by recursive insertions of 3-dipoles. A closed graph is obtained by connecting the two external open half-edges.}
\end{figure}
We say that a melonic graph has melons only on the colors $a_1,\dotsc, a_k$ if it can be constructed by dipole insertions on edges of colors $a_1,\dotsc,a_k$ only.
Edge-colored graphs have been recently classified according to their degree in \cite{GurauSchaeffer}.
\subsubsection{The corresponding tensor model}
In this section, we explain why our matrix model can be written as a tensor model. Note that not all multi-matrix models are tensor models in disguise, since the action of a tensor model for a tensor of size $N_1\times \dotsm \times N_d$ is required to be $U(N_1)\times \dotsm U(N_d)$-invariant, as we explain.
Let $T_{a_1 \dotsb a_d}$ be the entries of a tensor $T$, with $a_i = 1,\dotsc,N_i$ for $i=1,\dotsc,d$, and $\overline{T}_{a_1\dotsb a_d}$ for its complex conjugate. The algebra of polynomials in the entries of $T$ and $\overline{T}$ which are invariant under the fundamental action of $U(N_1)\times \dotsm \times U(N_d)$, that is
\begin{equation} \label{U(N)Transfo}
T_{a_1\dotsb a_d} \mapsto \sum_{b_1, \dotsc, b_d} U^{(1)}_{a_1 b_1}\,\dotsm\, U^{(d)}_{a_d b_d}\ T_{b_1\dotsb b_d},
\end{equation}
where $U^{(1)},\dotsc,U^{(d)}$ are $d$ independent unitary matrices (of different sizes), and similarly for the complex conjugate $\overline{T}$, is generated by a set of polynomials labeled by bubbles. Recall that bubbles here are connected, edge-colored graphs with $d$ colors. The correspondence between invariant polynomials and bubbles works as follows. Let $B$ be a bubble. To each white (respectively black) vertex of $B$ we associate a $T$ (respectively $\overline{T}$). An edge of color $c\in\{1,\dotsc,d\}$ between a white vertex and a black vertex means that the indices in the position $c$ of the corresponding $T$ and $\overline{T}$ are identified and summed over (from $1$ to $N_c$). The polynomial labeled by $B$ can be written explicitly. Let $\mathcal{W}$ be the set of white vertices, $\mathcal{B}$ the set of black vertices and $\mathcal{E}$ the set of edges. We identify an edge $e\in\mathcal{E}$ via the black vertex and the white vertex it connects and its color, thus $e = (w,b,c)$ with $w\in\mathcal{W}, b\in\mathcal{B}$. Then the polynomial, denoted $B(T,\overline{T})$, is
\begin{equation}
B(T, \overline{T}) = \prod_{k\in\mathcal{W}} \sum_{i_1^k, \dotsc, i_d^k}\ \prod_{l\in\mathcal{B}} \sum_{j_1^l, \dotsc, j_d^l}T_{i_1^k \dotsb i_d^k}\ \overline{T}_{j_1^l \dotsb j_d^l}\ \prod_{e=(w,b,c)\in\mathcal{E}}{\delta_{i_c^w,j_c^b}}.
\end{equation}
In order to write the matrix action \eqref{matrixaction}, defined from the set of matrices $\{A_i,A^\dagger_i\}_{i=1,\dotsc,\tau}$, as a tensor action, we make the quite obvious ansatz,
\begin{equation}
T_{a_1 a_2 a_3} = \left(A_{a_3}\right)_{a_1 a_2},\qquad \overline{T}_{a_1 a_2 a_3} = \left(A^\dagger_{a_3}\right)_{a_2 a_1}.
\end{equation}
It remains to check that the matrix potential has the required invariance, that is that each $V_{n,\sigma}(\{A_i,A_i^\dagger\})$ defined in \eqref{eq:TraceInvariant}, is actually an invariant polynomial labeled by a 3-colored bubble. The above definitions of $T$ and $\overline{T}$ shows that the matrix element of a product $(A_a A_b^\dagger)_{a_1 b_1}$ is exactly $\sum_{a_2}T_{a_1 a_2 a} \overline{T}_{b_1 a_2 b}$, i.e. a contraction along the second index. Similarly, $(A_b^\dagger A_a)_{b_2 a_2}=\sum_{a_1} \overline{T}_{a_1 b_2 b} T_{a_1 a_2 a}$ is a contraction on the first index. Finally, $\sum_{i=1}^\tau (A_i)_{a_1 a_2} (A_i^\dagger)_{b_2 b_1} = \sum_{a_3} T_{a_1 a_2 a_3} \overline{T}_{b_1 b_2 a_3}$ creates the contraction along the third index.
The action \eqref{matrixaction} is thus really a sum over invariant polynomials labeled by bubbles, and the quadratic part is obviously $\sum_{i=1}^\tau \tr A_iA_i^\dagger = \sum_{a_1,a_2,a_3} T_{a_1 a_2 a_3} \overline{T}_{a_1 a_2 a_3}$.
\begin{figure}
\center
\includegraphics[height=4cm]{LoopToTensor1.pdf}
\hspace{0.5cm}
\raisebox{2cm}{$\to$}
\hspace{0.5cm}
\includegraphics[height=4cm]{LoopToTensor2.pdf}
\hspace{0.5cm}
\raisebox{2cm}{$\to$}
\hspace{0.5cm}
\includegraphics[height=4cm]{LoopToTensor3.pdf}
\caption{The map from ribbon vertices with loop lines to 3-colored bubbles. \label{fig:LoopToTensor}}
\end{figure}
It is also interesting to proceed graphically. There is a straightforward mapping between the link patterns, i.e. the ribbon vertices of the matrix model, and bubbles with 3 colors and a single face of colors $(1,2)$, as shown in figure \ref{fig:LoopToTensor}. One draws an (unknotted) circle around the ribbon vertex, such that the intersections between the circle and the loop lines (labeled $1,\dotsc, \tau$) give rise to the vertices of the bubble (say an outgoing line gives a white vertex, and an incoming line gives a black vertex). The segments on the circle are given alternating colors 1 and 2. There are two possible choices to do that, and we choose the color 1 when going clockwise from a black to a white vertex. The loop lines which cross the ribbon vertex are then given the color 3, and they indeed connect white to black bubble vertices.
Notice that our ribbon vertices do not generate all 3-colored bubbles, but only those with a single face of colors $(1,2)$. This is because they come from single trace invariants in the matrix model.
Conversely, given a 3-colored bubble with a single face of colors $(1,2)$, one gets a unique ribbon vertex with loop lines. The ribbon vertex is determined by the face with colors $(1,2)$: there is one open ribbon line per vertex of the bubble. Each line of color 3 connects a black and a white bubble vertex and corresponds to an oriented loop line going through the ribbon vertex.
The planar patterns, used to build CPL configurations, are exactly the bubbles which are melonic with melonic insertions on the colors 1 and 2 only. This set of bubbles has been studied in \cite{SDEs} where it is shown to be in one-to-one correspondence with non-crossing partitions of $\{1,\dotsc,p\}$ up to rotations and reflections.
As an example, those on 4 vertices correspond to the terms $V_{n,\sigma}$ with $n=2$. There are only two permutations on two elements, hence two 3-colored graphs with a single face of colors $(1,2)$ which we denote $B_1$ and $B_2$, and corresponding to the identity $\sigma=(1)(2)$ and the transposition $\sigma=(12)$ (in cycle notation). Graphically, we have
\begin{equation} \label{QuarticBubbles}
\begin{aligned}
&V_{n=2,\sigma=(1)(2)} = \sum_{i,j=1}^\tau \tr( A_i A_i^\dagger A_j A_j^\dagger) = \begin{array}{c}\includegraphics[scale=.25]{4PointLinkPattern1.pdf}\end{array} = \begin{array}{c}\includegraphics[scale=.45]{4PointBubbleColor1.pdf}\end{array},\\
&V_{n=2,\sigma=(12)} = \sum_{i,j=1}^\tau \tr( A_i A_j^\dagger A_j A_i^\dagger) = \begin{array}{c}\includegraphics[scale=.25]{4PointLinkPattern2.pdf}\end{array} = \begin{array}{c}\includegraphics[scale=.45]{4PointBubbleColor2.pdf}\end{array}.
\end{aligned}
\end{equation}
Therefore, we get a tensor model whose free energy expansion is \eqref{matrixF}. This implies that if we could solve exactly the matrix models defined by the potentials \eqref{eq:TraceInvariant}, we would in fact get the exact solution of some rank-three tensor models, with tensors of size $N\times N\times \tau$ (for which the $1/N$ expansion is organized according to the genus of subgraphs with colors 0,1,2 while the degree partially controls the $1/\tau$ expansion, as we are going to see)
\subsubsection{The mapping}
The fact that the initial matrix model fits in the frame of tensor models suggests the existence of a bijection between the random surfaces decorated with loops and the $(3+1)$-colored Feynman graphs of the tensor model. From the correspondence between the ribbon vertices $V_{n,\sigma}$ and the bubbles established above, this bijection is quite trivial: only propagators have to be added. The edges of the ribbon graphs are simply mapped to edges of color 0, which indeed connect black to white vertices.
This allows to evaluate the number of faces, edges, vertices and loops of the matrix Feynman graphs in terms of faces, vertices and bubbles of the corresponding 4-colored graphs, as summarized in the Table \ref{tab:MatrixToTensorCorrespondence}.
\begin{table}
\begin{center}
\begin{tabular}{|c c c|}
\hline
Loops model graphs & \vline & 4-colored graphs \\
\hline
Faces: $F$ & $\to$ & Faces of colors (0,1), (0,2): $F_{01}+F_{02}$\\
Edges: $E$ & $\to$ & Vertexes: $p$\\
Vertexes: $V$ & $\to$ & Bubbles of color (1,2,3): $b$\\
Loops: $L$ & $\to$ & Faces of colors (0,3): $F_{03}$\\
\hline
\end{tabular}
\end{center}
\caption{The correspondence between the characteristics of the random surfaces with loops and the characteristics of the corresponding colored graphs. \label{tab:MatrixToTensorCorrespondence}}
\end{table}
The slightly non-trivial is to find a relation between the genus of the random surfaces and the degree of the colored graphs. Being 3-colored, the bubbles themselves can be interpreted as ribbon graphs. This is done by clockwise-ordering the edges of colors 1,2,3 around each white vertex, and counterclockwise-ordering them around each black vertex (then thickening the edges if desired). The degree of a bubble is then the genus of that discrete surface, $\omega(B) = g(B)$, and it is a measure of the amount of crossing of the loop lines at a given ribbon vertex. Obviously, the melonic bubbles are the planar ones.
Using this correspondence, the degree of a $(3+1)$-colored graph, equation \eqref{graph degree}, reads
\begin{equation}
F + L - 2(E - V) = 3 - \omega(G) +2\sum_\rho g(B_\rho),
\end{equation}
Since the genus of the ribbon graph fixes the number of faces at fixed number of edges and vertices, through equation \eqref{eq:Genus}, the number of loops $L$ can be extracted as a function of the degree, of the genus of the subgraph of colors (0,1,2), the genera of the bubbles, and of the numbers of edges and vertices
\begin{equation}
\label{eq:LoopCounting}
L = E - V + 1 + 2g(G) - \omega(G) + 2\sum_\rho g(B_\rho).
\end{equation}
This \emph{loop counting formula} is the main outcome of the mapping. To complete the loop counting, we prove that the quantity $\omega(G) - 2\sum_\rho g(B_\rho) - 2g(G) \geq 0$, and identify the configurations for which it vanishes. We use the obvious bound $L<F$, which together with \eqref{eq:LoopCounting} implies that
\begin{equation}
\omega(G) - 2\sum_\rho g(B_\rho) - 2g(G) > -1+2g(G).
\end{equation}
This means that if $g(G)\geq 1$, then the left hand side is strictly positive. Only the case $g(G)=0$ remains. Then we find
\begin{equation}
\omega(G) - 2\sum_\rho g(B_\rho) = \frac13\,\omega(G) +\frac23 \bigl(\omega(G) - 3\sum_\rho g(B_\rho)\bigr).
\end{equation}
In addition to $\omega(G)\geq 0$ as part of the Theorem \ref{thm:degree}, it can be proved that $\omega \geq 3\sum_\rho g(B_\rho)$, which is a particular case of the Lemma 7 in \cite{tree-algebra}.
We summarize the consequences of this analysis in the following proposition.
\begin{proposition} \label{prop:LoopCounting}
The number of loops on a random discrete surface $G$ decorated with oriented loops visiting all edges once (and such that the orientations on the edges around each vertex alternate), made of the gluing of $V$ link patterns $\{B_\rho\}_\rho$ via $E$ edges satisfies
\begin{equation*}
L = 1 + E - V + 2g(G) - \omega(G) + 2\sum_\rho g(B_\rho),
\end{equation*}
where $\omega(G)$ is the degree of the corresponding $(3+1)$-colored graph. Furthermore
\begin{itemize}
\item The graphs of degree zero are those which maximize the number of loops at fixed number of edges and vertices (and the link pattern at each vertex is planar).
\item They are planar, $\omega(G)=0\ \Rightarrow\ g(G)=0$.
\item At fixed genus, fixed numbers of edges and vertices of each allowed type, the degree measures how far $G$ is from the configuration which maximizes the number of loops.
\end{itemize}
\end{proposition}
We also note that the formula \eqref{eq:LoopCounting} can be used to get a bound on the maximal degree of the colored graphs built from 3-colored bubbles with a single face of colors 1,2. Since $L\geq1$, it comes (using the notation of colored graphs)
\begin{equation}
\omega(G)\leq p-b +2g(G) + 2\sum_\rho g(B_\rho).
\end{equation}
Let us illustrate a bit the melonic sector in the special case where only the terms with $n=2$ are kept in the action \eqref{matrixaction}. This leaves only the two link patterns, or the two 3-colored graphs (both planar), of the equation \eqref{QuarticBubbles}. In this model, all melonic insertions come from inserting appropriately one of these two bubbles on any edge of color 0. Assuming the edge of color 0 correponds to a ribbon edge with a loop going up north, the two possibilities are
\begin{equation}
\begin{array}{c}\includegraphics[scale=.5]{MelonicInsertionBubble1.pdf} \end{array} = \begin{array}{c}\includegraphics[scale=.25]{MelonicInsertion1.pdf} \end{array}\qquad \text{and}\qquad
\begin{array}{c}\includegraphics[scale=.5]{MelonicInsertionBubble2.pdf} \end{array} = \begin{array}{c}\includegraphics[scale=.25]{MelonicInsertion2.pdf} \end{array}.
\end{equation}
We see that a melonic insertion adds one loop, one ribbon vertex and one ribbon edge (and does not change the genus). By contrast, a non-melonic insertion on an edge of color 0 would be
\begin{equation}
\begin{array}{c}\includegraphics[scale=.5]{NonMelonicInsertionBubble2.pdf} \end{array} = \begin{array}{c}\includegraphics[scale=.25]{NonMelonicInsertion2.pdf} \end{array}
\end{equation}
which would not create a new loop.
In the section which follows, we solve explicitly the melonic sector ($\omega=0$) and further use the recent classification of colored graphs \cite{GurauSchaeffer} to organize the $1/\tau$ expansion.
\subsection{Scaling limits} \label{sec:ScalingLimits}
Using the counting of loops obtained in equation \eqref{eq:LoopCounting}, the free energy writes
\begin{equation}
N^2 f = \sum_{\substack{\text{connected}\\ \text{ribbon graphs}}} \left(\frac{N}{\tau}\right)^{2-2g(G)}\ \left(\lambda \tau\right)^{E-V}\ \tau^{3-\omega(G)}\ \frac1{s(G)}.
\end{equation}
This shows three contributions to the exponent of $\tau$. Those with the genus and with $E-V$ are not relevant since these quantities are controled by $N$ and $\lambda$. Consequently, the degree $\omega$ labels the expansion in the number of loops.
\subsubsection{Large $\tau$ limit}
Furthermore, it is possible to build a scaling limit which projects the loop model onto the melonic sector. To project onto the melonic family, the limit $\tau \to \infty$ is required. To ensure the limit is well-defined, we must scale $\lambda$ with $\tau$ as follows: $\lambda\tau = \tilde{\lambda}$, where $\tilde{\lambda}$ is kept finite. We also scale $N$ with $\tau$, and for convenience set their ratio to 1, $\tau = N$. The rescaled free energy $\tilde{f} = \frac{f}{\tau} = \frac{f}{N}$ then reads
\begin{equation} \label{FreeEnergyTensor}
N^3 \tilde{f} = N^2 f = \sum_{\substack{\text{4-colored}\\\text{connected graphs}}} N^{3-\omega(G)} \tilde{\lambda}^{E-V} \frac1{s(G)},
\end{equation}
It is finite in the large $N$, large $\tau$ limit, and its leading order in the $1/N$ expansion consists of melonic graphs.
It is interesting to perform the rescaling directly in the matrix integral (and setting $\tau$ to $N$ everywhere),
\begin{equation} \label{tensorscaling}
N^3 \tilde{f} = -\ln \int \prod_{i=1}^N \diff A_i \diff A_i^\dagger \exp\left( -\frac{N^2}{\tilde{\lambda}} S(\{A_i,A_i^\dagger\})\right)
\end{equation}
The factor $N^2$ in front of the action is exactly the standard scaling for a random tensor of rank-three and size $N^3$. This is natural in this scaling limit, since there are $\tau=N$ matrices, each of size $N\times N$.
We can write the solution quite explicitly in the large $N$ limit \cite{universality, uncoloring}. Indeed, large random tensors in a unitary-invariant distribution (invariant under \eqref{U(N)Transfo}) are subjected to a universality theorem, stating that all large $N$ expectations are Gaussian, with the covariance being the large $N$ 2-point function. For an invariant polynomial $B(T,\overline{T})$ of degree $p_B$ in $T$, this gives
\begin{equation}
\frac1N\,\langle B(T,\overline{T})\rangle = N^{-\omega^*(B)} \Bigl(C_B\ G_2^{p_B}+\mathcal{O}(1/N)\Bigr),
\end{equation}
where $G_2 = \lim_{N\to\infty} \langle T\cdot\overline{T}\rangle/N = \lim_{N\to\infty} \langle \sum_{i=1}^N \tr (A_iA_i^\dagger)\rangle/N$. $\omega^*(B) \geq 0$ and vanishes if and only if $B$ is melonic, meaning that melonic bubbles of the action are the only relevant ones at large $N$. Therefore, only the terms of the type $V_{n,\sigma}$ in \eqref{matrixaction} with $\sigma$ corresponding to a planar link pattern survive. Moreover, $C_B$ is the leading order number of Wick contractions and for a melonic bubble turns out to be 1 only. This way,
\begin{equation}
\frac1N\,\langle V_{n,\sigma \rm{ planar}}(\{A_i,A_i^\dagger\}) \rangle = G_2^n.
\end{equation}
All large $N$ calculations thus boil down to the leading order 2-point function. It is found thanks to the Schwinger-Dyson equation
\begin{equation}
\sum_{a_1,a_2,i=1,\dotsc,N} \int \prod_i dA_i\,dA_i^\dagger\ \frac{\partial}{\partial (A_i)_{a_1 a_2}} \Bigl((A_i)_{a_1 a_2}\ e^{-N^2 S(\{A_i,A_i^\dagger\})/\tilde{\lambda}}\Bigr) = 0,
\end{equation}
which after making the derivatives explicit and using the universality to close the system leads to the equation
\begin{equation}
\tilde{\lambda} - G_2 - \sum_{n, \text{planar }\sigma} n\,G_2^n = 0,
\end{equation}
which is polynomial as long as the action contains a finite collection of planar link patterns. It is a standard result that one then gets a square-root singularity for $G_2$ when approaching the critical value of $\tilde{\lambda}$, i.e. $G_2 \sim (\tilde{\lambda}_c - \tilde{\lambda})^{1/2}$. Therefore the singular part of the free energy behaves as $\tilde{f} \sim (\tilde{\lambda}_c - \tilde{\lambda})^{2-\gamma}$ with $\gamma=1/2$. The regime where $\tilde{\lambda}$ is close to $\tilde{\lambda}_c$ is called the \emph{continuum limit}.
\subsubsection{The $1/\tau$ expansion}
Thanks to the $1/N$ expansion, we can work at fixed genus. We can then take advantage of the recent classification of edge-colored graphs according to their degree \cite{GurauSchaeffer} to organize the the $1/\tau$ expansion.
This classification relies on the fact that only 2-point subgraphs and 4-point subgraphs can generate infinite family of graphs of constant degree\footnote{We remind the reader that tensor model at large $N$ are dominated by Gaussian contributions, i.e. 2-point functions, while the first $1/N$ correction only involves 2-point and 4-point functions, \cite{NLO, DSSD}.}. Once replaced by ``reduced'' 2-point and 4-point functions, there exists only a finite number of graphs of given degree. Those reduced graphs are called \emph{schemes} in \cite{GurauSchaeffer}.
In the following, we restrict the potential to $n=2$, leaving only room for the bubbles $B_1, B_2$ introduced in \eqref{QuarticBubbles}. (This reduces the source of 4-point functions; otherwise we would have for instance a 6-point bubble with an arbitrary 2-point function between two of its vertices also play the role of an effective 4-point bubble, and so on\footnote{The reference \cite{GurauSchaeffer} studies the whole set of colored graphs, which is somewhat simpler than focusing on the set of graphs generated by a given but arbitrary set of bubbles, except if this set is simple enough. This is the case for graphs built from quartic interactions ($n=2$ here), and this is the choice made in \cite{DSQuartic}.}). We recall the loop counting formula specialized to this case (i.e. with $E=2V$ and planar link patterns),
\begin{equation}
L = V +(3-\omega(G))-(2-2g(G)).
\end{equation}
Therefore the degree measures how far a loop configuration is from the one which maximizes the number of loops for a fixed number of vertices and a fixed genus.
The Schwinger-Dyson equation on $G_2$ simply reads $\lambda - G_2 - 4 G_2^2=0$, hence
\begin{equation}
G_2 = \frac{\sqrt{1+16\lambda}-1}{8},
\end{equation}
with $G_2\sim \lambda$ for small $\lambda$. The critical point which defines the continuum limit is $\lambda_c=-1/16$.
Given an arbitrary $(3+1)$-colored graph built from the bubbles of type $B_1, B_2$ glued along edges of color 0, one first reduces the purely melonic 2-point subgraphs, as the one in figure \ref{fig:dipole} (this is done recursively by identifying 2-cut edges; the order does not matter, as proved in \cite{GurauSchaeffer}). Arbitrary melonic insertions do not change the degree, and so does this reduction. This way, we have to consider only \emph{melon-free} graphs, while all melonic insertions are completely accounted for by simply using $G_2(\lambda)$ as the new propagator, i.e. $G_2$ becomes the weight associated to edges of color 0.
Second, one identifies \emph{chains}. In our model, chains are simply sequences of quartic bubbles glued in a chain-like manner,
\begin{equation*}
\begin{array}{c}\includegraphics[scale=.45]{GenericChain.pdf}\end{array}
\end{equation*}
where $c,c'$ are 1 and/or 2. Those chains have to be maximal, so they have 4 half-edges of color 0 as external edges. There are two types of chains.
\begin{itemize}
\item Those built from a sequence of a single bubble, either $B_1$ or $B_2$, and called \emph{unbroken chains}. In terms of ribbon graphs and loops, an unbroken chain takes the form
\begin{equation*}
\begin{array}{c}\includegraphics[scale=.4]{UnbrokenRibbonChain.pdf}\end{array}
\end{equation*}
with two possible orientations. It is clearly planar.
The generating function of unbroken chains with a weight $-2/\lambda$ on each bubble and $G_2(\lambda)$ on each edge of color 0 is
\begin{equation}
C_u(\lambda) = \sum_{n\geq 1} \frac{(-2)^n\,(G_2(\lambda))^{2(n-1)}}{\lambda^n} = -\frac{2}{\lambda + 2(G_2(\lambda))^2}.
\end{equation}
\item Those which contain both bubbles $B_1, B_2$ and called \emph{broken chains}. The generating function of broken chains is found by considering the one of arbitrary chains, with arbitrary 4-point bubbles (bubbles hence receiving the weight $-2\times 2/\lambda$, to account for the two possible types of bubbles at each time), and substracting the generating functions of the two unbroken chains,
\begin{equation}
\begin{aligned}
C_b(\lambda) &= \sum_{n\geq 1} \frac{(-4)^n\,(G_2(\lambda))^{2(n-1)}}{\lambda^n} -2C_u(\lambda) = -\frac{4}{\lambda + 4(G_2(\lambda))^2} + \frac{4}{\lambda + 2(G_2(\lambda))^2} \\
&= \frac{8(G_2(\lambda))^2}{\lambda + 2(G_2(\lambda))^2}\ \frac1{\lambda + 4 (G_2(\lambda))^2}.
\end{aligned}
\end{equation}
\end{itemize}
Chains can be arbitrarily long with no change in the degree of the melon-free graphs. We have to make sure that does not change the genus of the random surface neither. It is clear for the unbroken chains. As for the broken ones, if a bubble $B_i$ is inserted somewhere in an unbroken subchain of type $i$, including at an end, this does not change anything. So we are left with the case where a bubble $B_1$, for instance, is inserted in a subchain of bubbles $B_2$. Because the full chain is broken, there is another bubble $B_1$ somewhere, say on the right of the chain,
\begin{equation}
\begin{array}{c}\includegraphics[scale=.5]{BrokenChain.pdf}\end{array}
\end{equation}
where we have marked the added bubble in a bounding box. We have to evaluate the variation of the genus of the subgraph with colors 0,1,2 between before and after the insertion. Clearly, the number of ribbon edges changes by 2 and the number of ribbon vertices (i.e. bubbles) by 1. Therefore $\Delta (E-V) = 1$. To find the variation of the number of faces, it is more convenient to use the representation as an edge-colored graph rather than as a ribbon graph. The number of faces of the surface is $F=F_{01}+F_{02}$. The face of colors $(0,2)$ which arrives from the top left leaves on the bottom left, and that was already the case before the insertion because the chain is broken. There is however a new face of colors $(0,2)$, which goes around the bubble of type $B_2$ on the right of the bounding box. Therefore $\Delta F_{02}=1$. Moreover, there is no new face of colors $(0,1)$, so $\Delta F_{01}=0$. The variation of the genus is thus $-2\Delta g= \Delta (F-E+V) = 0$, meaning that the genus is independent of the length of the chain.
As a consequence, it is safe to simply contract chains into ``boxes'' called broken or unbroken chain-vertices with two incident edges of color 0 on one side of the box and two on the opposite side, and weight them with the generating functions $C_b(\lambda)$ or $C_u(\lambda)$ respectively. We obtain this way the set of \emph{schemes}, i.e. melon-free graphs with chain-vertices representing arbitrarily long chains. The key result of \cite{GurauSchaeffer} is then the finiteness of the number of schemes at any fixed degree.
Let $s$ be a scheme with $p\geq2$ black vertices, $\alpha$ unbroken chain-vertices and $\beta$ broken chain-vertices. Then the generating function of colored graphs, rooted on an edge of color 0, with scheme $s$ is
\begin{equation}
G_s(\lambda) = (G_2(\lambda))^{p}\,(C_u(\lambda))^\alpha\,(C_b(\lambda))^\beta.
\end{equation}
To get the free energy of the model (or rather its 2-point function), one substitutes $\lambda \tau$ instead of $\lambda$. Then the 2-point function at genus $g$ has the expansion
\begin{equation}
G_2^{(g)}(\lambda) = G_2(\lambda\tau)\,\delta_{g,0} + \sum_{\omega\geq1} \tau^{3-\omega} \sum_{\substack{\text{schemes $s$} \\ \omega(s)=\omega, g(s)=g}} (-2)^\alpha 8^\beta \frac{(G_2(\lambda\tau))^{p+2\beta+1}}{(\lambda\tau + 2(G_2(\lambda\tau))^2)^{\alpha+\beta}\ (\lambda\tau+4(G_2(\lambda\tau))^2)^\beta},
\end{equation}
where the sum over schemes at fixed genus and degree is finite. Here we have isolated the purely melonic part, which corresponds to the empty scheme with no vertices.
Of course, to complete the analysis, it is necessary to know how the degree of a scheme behaves as a function of the number of chains. Again, this was done in \cite{GurauSchaeffer}. If a chain-vertex is separating, i.e. if after its removal and after connecting the half-edges of color 0 together on each side of the chain vertex we get two connected components, then the degree of the graph is simply the sum of the degrees of both connected components. For an non-separating, unbroken chain of type $i$, such that there are two different faces of colors $(0,i)$ going through the chain, the degree is the degree of the graph with the chain removed plus one, meaning such a chain contribute to a factor $1/\tau$. In all other situations, a chain brings in a factor $1/\tau^3$.
A corollary of this analysis is the double scaling regime. First notice that the critical point defining the continuum limit is $\lambda_\tau = -1/(16\tau)$. The generating function $C_u(\lambda\tau)$ of unbroken chains is finite at criticality. However, the generating function of broken chains is singular. Indeed, its denominator contains
\begin{equation}
\lambda\tau + 4 (G_2(\lambda\tau))^2 = -\lambda\tau\ G_2(\lambda\tau)\ \sqrt{1-\lambda/\lambda_\tau}.
\end{equation}
Therefore a scheme with $\beta$ broken chains diverges as $(1-\lambda/\lambda_\tau)^{\beta/2}$ at criticality. The idea of the double scaling limit is to pick up the terms of arbitrary degree which maximize the divergence.
The answers provided in \cite{GurauSchaeffer} for generic edge-colored graphs and in \cite{DSQuartic} in the special case of quartic melonic interactions coincide. We consider a rooted, binary tree with a single loop attached to every leaf. For each such tree, we get an edge-colored graph of the model by replacing the edges with broken chains which are glued together at the vertices in the obvious way, while the loops on the leaves represent unbroken chains. Those loops break melonicity and were called cherries in \cite{DSQuartic}. Since all broken chains are separating, the degree is simply the number of unbroken ones, $\omega=n$. Moreover, the binary-tree structure of broken chains is the way to maximize the divergence at fixed number of cherries. The number of broken chains grows linearly as two times the number of cherries, so each such graph receives a factor $\tau^{-n}/\sqrt{1-\lambda/\lambda_\tau}^{2n}$. This shows that the optimal balance is reached by introducing $x = \tau (1-\lambda/\lambda_\tau)$ and sending $\tau\to\infty, \lambda\to\lambda_\tau$ while $x$ is kept fixed.
Using the same technique as in \cite{GurauSchaeffer} to extract the behavior of the degree as a function of the chain-vertices, the graphs of the double-scaling regime can be shown to be planar. Indeed, one breaks up the cherry trees into isolated vertices, edges which represent broken chains, and loops attached to the leaves and analyze the genus of each piece (found to vanish in all cases).
The resummation of this family is quite simple to perform\footnote{Compared to \cite{DSQuartic}, one has to set $D=3$ when $D$ enters the degree, but $D=2$ in the equations for criticality since we have only two quartic bubbles and not three.}. It has a square-root singularity in $x$ that is likely to lead to a branched polymer phase.
\subsection{Another bijection and the intermediate field method} \label{sec:Bimaps}
We have shown a bijection between the ribbon graphs with oriented loops generated by the matrix model \eqref{matrixF} and the edge-colored graphs of tensor models whose interactions are labeled by bubbles with a single cycle of colors $(1,2)$. In the case the potential is restricted to the two quartic terms in equation \eqref{QuarticBubbles}, there is a bijection between the graphs of the tensor model and a family of maps. It was first observed in \cite{BeyondPert} in quartic melonic models, and generalized to tensor models with arbitrary quartic interactions in \cite{GenericQuartic}. Algebraically, this bijection corresponds to the intermediate field method. Here, we first present the bijection, then the corresponding intermediate field theory.
Notice that the bubbles used in the quartic case, equation \eqref{QuarticBubbles}, have four vertices with a canonical partition in pairs. A canonical pair of vertices consists of those connected by a multiple edge (here two edges including the one of color 3). The two pairs are connected by two edges of color $i$ (here 1 or 2) and we have labeled the bubble by that color ($B_1$ and $B_2$). We are now going to represent $B_i$ as an edge of color $i$, as if the canonical pairs of vertices were contracted to single points.
Furthermore, each white/black vertex has an incident edge of color 0. This means that every edge of color 0 belongs to a single closed cycle made of alternating edges of color 0 and multiple edges. We map those cycles to vertices, while preserving the cyclic ordering of the bubbles. In our model, those vertices correspond to the faces of color $(0,3)$, i.e. the loops. Through this process, we represent every colored graph as a map, since the ordering around each vertex matters, with edges of colors 1 or 2.
For such a map, there are two canonical submaps, ${\cal M}_1$ and ${\cal M}_2$, which respectively correspond to the submaps containing only the edges of color 1 and 2. The faces of colors $(0,i)$ in the Feynman graph of the tensor model are mapped to the faces of the map ${\cal M}_i$. Moreover, the bubbles are mapped to edges and the loops to vertices.
Remarkably, many problems in tensor models become quite simple when formulated in this way. For instance, the dominance of the melonic sector: the question is how to maximize the number of loops at fixed number of ribbon vertices. After the mapping, it becomes how to maximize the number of vertices at fixed number of edges; the answer clearly being trees, which indeed are the representatives of the melonic edge-colored graphs. Further, the double scaling regime presented in the previous section is dominated by Motzkin trees (i.e. trees whose nodes can have zero, one or two children), such that there always is at least one change of color between two vertices of degree three, and with loops of arbitrary length and of a fixed color attached to the leaves.
Since we exhibited a bijection to maps, there may be a matrix model which generates them with the correct amplitudes. This works through the intermediate field method which transforms the initial matrix model \eqref{matrixF} with $n=2$ into a two-Hermitian-matrix model. Here it is useful to introduce independent coupling constants $\lambda_1, \lambda_2$ and consider
\begin{equation}
Z_{N,\tau}(\lambda_1,\lambda_2) = \int \prod_{i=1}^\tau dA_i\,dA_i^\dagger\ e^{-N\left(\sum_i \tr A_iA_i^\dagger + \lambda_1 \tr \sum_{i,j} A_i A_i^\dagger A_j A_j^\dagger + \lambda_2 \tr \sum_{i,j} A_i A_j^\dagger A_j A_i^\dagger\right)}.
\end{equation}
We can re-write each quartic term via a Gaussian integral over an auxiliary, Hermitian matrix,
\begin{equation}
e^{-N\lambda_1\tr \sum_{i,j} A_i A_i^\dagger A_j A_j^\dagger} = \int dM_1\ e^{-N \tr M_1^2 -2iN\sqrt{\lambda_1} \tr \sum_i M_1 A_i A_i^\dagger},
\end{equation}
up to irrelevant constants, and similarly for the other quartic term. The partition function is then
\begin{equation}
Z_{N,\tau}(\lambda_1,\lambda_2) = \int dM_1\,dM_2\,\prod_{i=1}^\tau dA_i\,dA_i^\dagger\ e^{-N\tr(M_1^2+M_2^2+\sum_i A_iA_i^\dagger) - 2iN \sqrt{\lambda_1} \tr M_1 \sum_i A_i A_i^\dagger - 2iN\sqrt{\lambda_2} \tr M_2 \sum_i A_i^\dagger A_i}.
\end{equation}
Performing the Gaussian integral on the $\tau$ matrices $A_i$, one gets,
\begin{equation} \label{TwoMatrixModel}
Z_{N,\tau}(\lambda_1,\lambda_2) = \int dM_1\,dM_2\ e^{-N\tr(M_1^2+M_2^2) - \tau \tr \ln \left(\mathbb{I}\otimes \mathbb{I} - 2i\sqrt{\lambda_1} M_1\otimes \mathbb{I} -2i\sqrt{\lambda_2}\mathbb{I}\otimes M_2\right)}.
\end{equation}
If the logarithm is expanded onto powers of $M_1, M_2$, it is clear that we have a generating function for the maps described above.
We are not aware of a solution of this model for arbitrary $\lambda_1,\lambda_2,\tau$ in the literature. Nevertheless, setting $\lambda_2=0$, one gets
\begin{equation} \label{SingleColorQuartic}
Z_{N,\tau}(\lambda_1,0) = \int \prod_{i=1}^\tau dA_i\,dA_i^\dagger\ e^{-N\left(\sum_i \tr A_iA_i^\dagger + \lambda_1 \tr \sum_{i,j} A_i A_i^\dagger A_j A_j^\dagger\right)} = \int dM\ e^{-N\tr M^2 - N\tau \tr \ln \left(\mathbb{I} - 2i\sqrt{\lambda_1} M_1\right)}.
\end{equation}
One recognizes here the generalized Penner model with a quadratic potential. We refer to \cite{PennerAllGenera} for an analysis with an arbitrary polynomial, at all genera, using the loop equations. In the case of the quadratic potential, the Penner model with coupling $\tau$ on the logarithmic part is equivalent to the quartic matrix model with $\tau$ matrices, which can actually be solved directly. For instance, a rectangular matrix of size $N\times \tau N$ can be formed, $C_{a_1 \alpha} = (A_i)_{a_1 a_2}$ with the ``fat'' index $\alpha=(a_2,i)$. Then the action is simply $\tr CC^\dagger + \lambda_1 \tr (CC^\dagger)^2$, and the partition function can be evaluated using techniques developed for rectangular matrix models, like the orthogonal polynomials in \cite{MyersTripleScaling} (and see \cite{toy-doublescaling} for an application to tensor models).
As far as we know, the quartic case with $\lambda_2=0$ is the only situation where a model of the generic class we have introduced has been solved. However, it should be emphasized that already the quartic model with $\lambda_2\neq 0$ is very different. In particular, for $\lambda_2=0$, the distinction between broken and unbroken chains disappears and all chains become singular at the critical point.
\section*{Conclusion}
The motivation of this article is to connect tensor models and its challenges to the more familiar framework of matrix models
With this in mind, the present article has been devoted to a novel presentation of random tensor models, from the view of matrix models. It is based on a really simple observations: that a tensor of size $N\times N \times \tau$ can be seen as a set of $\tau$ matrices.
In section \ref{sec:loops}, this observation allows to interpret models for tensors of size $N\times N\times \tau$ whose interactions have a single cycle of colors $(1,2)$ as $U(\tau)$-invariant matrix models. We describe this correspondence through a bijection between edge-colored graphs and random surfaces decorated with oriented loops and show that the degree, which organizes the $1/N$ expansion of tensor models, here organizes the expansion with respect to the number of loops on the random surfaces, via the equation \eqref{eq:LoopCounting}. That provides a new, combinatorial interpretation of the degree.
We have taken this as an opportunity to review the most recent results on tensor models applied in the context of the loop models. This approach also unravels the challenges faced by random tensor theory. It is emphasized that to our knowledge there is no known solution to those models (e.g. for the large $N$ free energy at finite $\tau$), beyond a very particular case which corresponds to a Penner model. Beyond this case, the most generic and explicit result is the classification of edge-colored graphs according to their degree, due to Gurau and Schaeffer \cite{GurauSchaeffer} which as we have explained in section \ref{sec:ScalingLimits} classifies the loop configurations at fixed genus and number of edges according to the number of loops. There is moreover a double-scaling limit which sums consistenly the most singular (at criticality) loop configurations. We hope that the relationship we have established between tensor models and loop models can lead to fruitful cross-fertilization.
While we have focused in section \ref{sec:ScalingLimits} on the scaling limits, further connections between matrix and tensor models have been reviewed in section \ref{sec:Bimaps}, based on the Hubbard-Stratanovich (intermediate field) transformation. It reveals that melonic quartic tensor models generate maps formed by maps with different edge colors glued together at vertices (or by duality, at the center of their faces), \cite{DSQuartic} (see also \cite{BeyondPert} for a constructive analysis (Borel summability) of this model and \cite{GenericQuartic} for an extension of those ideas to arbitrary quartic models). In the case of two edge colors, those maps have already appeared under the name of \emph{nodal surfaces} in \cite{EynardBookMulticut} as multicut solutions of the one-matrix model\footnote{Obviously the multicut solution satisfies the loop equations of the 1-matrix model and is not a solution of quartic tensor models which correspond to a different evaluation of the generating function of nodal surfaces.}. It has been further observed that such maps can be generated in a Givental-like fashion\footnote{We are indebted to Bertrand Eynard for pointing this out and we would like to thank St\'ephane Dartois for sharing his progress on such a re-formulation of tensor models.} \cite{GiventalDartois}.
We believe that viewing tensor models as matrix models constitutes an interesting research road and places tensor models in a frame where powerful tools are available. In particular, the intermediate field method turns quartic tensor models into matrix models which generate generalizations of nodal surfaces. Either techniques developed for matrix models, such as the topological recursion \cite{TopRec}, or combinatorial approaches, could lead to new results. Among the combinatorial approaches, bijective methods akin to Schaeffer's bijection for planar quadrangulations could be useful to solve the large $N$ limit (i.e. the planar sector) of the two-matrix model \eqref{TwoMatrixModel}, while algebraic methods have proved helpful to probe maps at arbitrary genus \cite{KPGouldenJackson, CarrellChapuyRecursion}. Preliminary calculations suggest that the large $N$ limit of \eqref{TwoMatrixModel} is a generalization of the $O(\tau)$ model (where the eigenvalues of $M_1$ are attracted to the mirror image of those of $M_2$) \cite{ExactO(n)Eynard, O(n>2)Eynard}.
\section*{Acknowledgements}
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,856 |
When teenagers Danielle and Gracie Sullivan (from Back of Beyond) take a clandestine car trip to visit their friend in Montana, little do they know it's the last time anyone will ever hear from them again. The girls and their car simply vanish. Cody Hoyt, who's just lost his job and has fallen off the wagon after a long stretch of sobriety, is in no condition to investigate. But his son Justin, whom the girls were going to visit, and his former partner, Cassie Dewell, convince him to drive south to their last known location. As Cody makes his way to the piece of highway where the girls went missing, Cassie discovers that there have been scores of disappearances in the state. There's a serial killer out there roaming the interstate, and Cody and Cassie must find him before he takes more lives.
Drawing on characters introduced in Back of Beyond, Box's stand-alone weaves together subplots into a nonstop, action-filled race against time. Rolling down the superhighway of suspense, this thriller will leave readers breathless.
Filled with believable characters and hard, realistic dialogue, Edgar-winner Box's perfectly paced novel offers a suspenseful story laced with more than a few shockingly unexpected plot twists.
As the film Duel demonstrated, the mere sight of an enormous truck speeding up behind you on a long, empty stretch of highway is never comforting; and Box works that inherent fear masterfully.
The result is a violent, tension-packed, well-written thriller spiced with Box's vivid portrayal of the Western landscape that he loves. Along the way, Box also drops in surprising insights about the itinerant lives of long-haul truckers.
Box's previous thrillers, most of them featuring Wyoming game warden Joe Pickett, have all featured strong, tough-talking male protagonists, but the heroes of this one are the underestimated Cassie, who proves to be both tough and resourceful, and the courageous Gracie, who keeps her head when her flighty sister falls apart.
The Highway is the second new thriller this year from the prolific Box, who seems to get better with every book.
Be warned: This is one scary novel. Fans of C.J. Box's excellent Joe Pickett series know that his characters get under your skin. This new series is even better than Pickett, but don't read this book in the dark of the night.
With The Highway, his latest contemporary Western suspense novel, C.J. Box is taking risks and flouting rules.
Through 13 novels featuring Wyoming game warden Joe Pickett and three standalone thrillers, the Cheyenne, Wyo., author has earned a reputation as one of better mystery writers working today.
But until The Highway, Box mostly played within the established one-good-guy-gets-one-bad-guy sandbox of the genre.
Not this time. It's clear that he feels he's built some capital with his readers — some measure of willingness to follow wherever he goes — and with The Highway, he's clearly cashing in and careening into new territory. And Box, a canny and consummate craftsman, gets away with it.
One of Box's bigger gifts is his ability to create sumptuous characters from spare, surgical strokes of prose. A favorite: "He drove his truck for eleven months and hunted elk with the other, and he could quote Shakespeare and Paul Harvey without missing a beat."
Want to share the literary road less traveled with people like that? Get off the genre interstate and take The Highway.
Box's ability to create vivid characters is one of the things that makes reading his books so enjoyable. He does an amazing job of not only creating the characters who are on the side of good, but the evil ones, too. His evil characters are even that much more stunning for their depiction and the author's ability to produce chills down the reader's spine.
I originally thought of this book as a race against time, but have since amended my description to a fast moving, edge of your seat page-turner. I really couldn't put it down. Box tells his story as it needs to be told, and is not afraid to let the action go where it may and let the bodies fall. Things can't always be neat and tidy and summed up in the drawing room by a fire. But when you think about it, when are things really that neat and tidy in real life?
C. J. Box's ability to explore the mind of a serial killer and in doing so create a three-dimensional character is frightening. It would make one wonder about his dark side, if he didn't also create such wonderfully well-rounded characters whose goodness overcomes their flaws.
In The Highway the reader is taken for a terrifying ride with killers and can only hope rescuers arrive before the journey ends in a bloody torture chamber. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,635 |
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DomiNations assaults Clash of Clans with a Civilization-like mobile strategy game
Dean Takahashi@deantak April 2, 2015 8:01 AM
DomiNations has army on army combat.
Image Credit: Big Huge Games
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Somebody is going to knock Clash of Clans out of the No. 1 spot on the top-grossing mobile apps charts one of these days. The latest to attempt this feat is DomiNations, the new Civilization-like mobile game from Big Huge Games and Asian gaming giant Nexon. DomiNations launches today on both Apple's iTunes App Store, Google Play, and Amazon.
If DomiNations succeeds, Big Huge Games can steal a chunk of revenue from Supercell, which generated $1.7 billion in 2014 from just three games, including Clash of Clans, which has been at the top for a couple of years of the mobile market — a sector that could reach $30 billion this year, according to research firm Newzoo.
The base-building strategy release taps Big Huge Games' heritage in human history games and applies it to the simpler touch interface of mobile. Its multiplayer version is akin to Supercell's Clash of Clans, but it allows for more interactive control of a battle, where you attack your enemy's stronghold. It is the latest title from veterans Brian Reynolds and Tim Train, who have been making strategy games like Alpha Centauri and Rise of Nations for 24 years.
"The common thread of successes has been history games," said Train, in an interview with GamesBeat. "What our team does well is make history fun and accessible to a broad audience.
"We also played a ton of mobile strategy games and were excited about doing a world history game set in all of human history, set on mobile. In the old days, if we sold 3 million units of Civilization II, we were excited. In this era, you can put a game out that has 100 million players. We felt genetically born and bred to do this game. We think history is one of the best intellectual properties you can have because everyone knows something about history."
Going off the strategy track
Above: Tim Train of Big Huge Games shows DomiNations.
Image Credit: Dean Takahashi
Reynolds and Train veered off the track of history games for a time. They most recently worked in game design roles at the Baltimore-based Zynga East studio, until the publisher shut it down in 2013. In 2014, they created a startup studio and bought their former name, Big Huge Games — formerly a PC game studio that made real-time strategy games Rise of Nations and Rise of Legends — after the state of Rhode Island auctioned it off last fall in the wake of the collapse of Curt Schilling's company, 38 Studios. Big Huge Games made Kingdoms of Amalur: Reckoning, for 38 Studios. In re-acquiring the Big Huge Games brand, Reynolds and Train tapped into the nostalgia for those strategy games of yore.
Reynolds and Train rebuilt their studio in Baltimore, and they signed up Nexon M, a mobile-focused division of Tokyo-based Nexon, as a publisher. Reynolds and Train got their start working for game design wizard Sid Meier. After a while, Reynolds branched out more and more to do his own thing, including leaving Meier's company, Firaxis, and starting Big Huge Games in 2000. For his whole career, Reynolds focused on PC strategy games. But at Zynga, he veered off into casual games such as FrontierVille and CityVille 2. Now he and Train are back doing strategy.
The new mobile strategy game lets players lead their own civilization from the Stone Age to the Space Age. They can evolve their civilization, build a single fortress, and conquer others in single-player campaigns. They can join clans and fight others in asynchronous multiplayer, where one player attacks another's fortress while the other one simply sets up defenses and watches the battle unfold. That part of the game is a lot like Clash of Clans.
Building something unique
Above: DomiNations has pretty 2.5D art.
Image Credit: Nexon
Train demoed the game to me yesterday. It covers the whole span of human history, and it has a cartoon art style, which is more accessible to wider audiences. But it also has a lot of strategic depth that hardcore gamers will like.
With DomiNations, you have more tactical options than in Clash of Clans.
"We feel it is absolutely different from Clash of Clans," Train said. "We looked at Age of Empires for inspiration. And, of course, Age of Empires is a very different game from Warcraft."
You can choose among different civilizations, including the British, Romans, Chinese, Germans, French, Japanese, and Greeks. Each has its own unique abilities. You have to manage your food and economy while training troops and hiring mercenaries to colonize neighboring civilizations.
"The British are known for their history of successful imperialism, so they get more loot from their successful attacks, and they're known for the longbowmen," Train said. "The Chinese are known for their numbers, so they get an advantage in adding citizens."
You can progress through different ages, including Dawn Age, Stone Age, Bronze, Iron, Classical, Medieval, and Enlightenment. After launch, Big Huge Games will release the Industrial Age, where you get access to weapons like tanks and planes.
You can get a rare resource, Diamonds, only by attacking other players. To win a Diamond, you have to defeat the enemy within a first minute of play. (Battles last a mere 3 minutes, 30 seconds). That means you have to destroy the enemy's Town Center. If an enemy has a particularly deadly building, you can lock it down for a time with Sabotage.
Your clan can help you out by lending you soldiers from different civilizations. That enables you to combine Alexander the Great's heavy cavalry with the Japanese samurai. Such combinations could help you defend your base or assault your enemy's fortress. Altogether, there are a couple of hundred types of units and more than 100 buildings. It has some cool units, like rampaging war elephants.
In contrast to Clash of Clans, DomiNations has more army on army combat, he said. It isn't a two-way real-time battle, with synchronous live gameplay. But it is much more interactive than other games.
Defending your base
Above: Serious combat in DomiNations
While you spend a lot of time attacking players, you also have to set up your base. The layout of your guns, walls, spawn points, and buildings matters a lot, Train said.
The player's defense is automated, and it revolves around where the player puts defensive guns and spawn points for soldiers. A lot of the strategy comes in deciding what assets to deploy first — cannon, cavalry, or infantry — in the hopes of drawing the enemy out. Once you dispatch the enemy's ground pounders, you can deploy your cannon. Then you can take out the walls and start knocking out enemy buildings. You can't control individual units, but you can set rally points every 30 seconds or so. When you do, your troops will converge on the rally point and attack whatever is there. Periodically, the defenders spawn soldiers, until you destroy the spawn points.
If you succeed in conquering an enemy's base, you can get more resources to build troops and fortify your own base. The single-player campaign leads players from the Fertile Crescent to the Napoleonic wars. You can activate one general at a time, but you can attain up to four. You get more over time, and those will influence battles in different ways.
"We've built DomiNations from the ground up to combine the in-depth strategy gameplay we've been creating for over 20 years with a fast, fun, and tactical experience customized for mobile devices," said Brian Reynolds, creative director for DomiNations, in a statement.
You get a gold bonus if you connect your buildings to roads. That helps your economy. But it makes your cities harder to defend because it expands the territory that must be defended. You can create Wonders of the World, like the Hanging Gardens, which gives a bonus for surrounding fruit trees.
A big bet for Nexon
Above: A Japanese faction base in DomiNations
DomiNations is an important game for Nexon, which has commissioned a lot of Western mobile and PC games from studios such as Shiver Entertainment, Envision Entertainment, and Turbo Studios. Nexon wants to become a global gaming giant, which means it has to expand from Asia to the West. If it succeeds, Nexon and Big Huge Games will win at their own version of DomiNations in the strategy battles of the game industry.
"Big Huge Games is ushering in the next generation of mobile strategy games with DomiNations, offering players an unrivaled amount of strategic choices and gameplay depth," said John Robinson, the general manager of Nexon M.
Nexon has been particularly good at monetization, and DomiNations uses some familiar monetization features. It's free-to-play, but you can purchase virtual currency, or crowns, with real money to speed up the development of buildings or other items.
DomiNations is slimmed down from strategy games like Civilization. You won't fight enemies across a giant map. You select enemies to attack from a kind of big soup, rather than going after your neighbors as you do in titles such as Boom Beach. It has no cooperative play or user-generated content. And right now, you only get one city.
Of course, this is a mobile game, and Big Huge Games can always expand it in the future if it's a hit. The game has been available in some territories for a few months, and the retention of users has been good, Train said.
Like other things Reynolds and Train have done, DomiNations is certainly ambitious. It remains to be seen if Big Huge Games has cracked the code on mobile, and broken the grip of Clash of Clans.
Register for GamesBeat's upcoming event: Driving Game Growth & Into the Metaverse | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,137 |
Q: All emoticons have the same value in java public static void main(String[] args) {
String bear1 = "";
String multiple = "\uD83D\uDC5A\uD83D\uDC5B\uD83D\uDC5C\uD83D\uDC55\uD83D\uDC5C\uD83D\uDC59\uD83D\uDC5E\uD83D\uDC61\uD83E\uDD84\uD83E\uDD93\uD83E\uDD84\uD83D\uDC34";
String bear2 = "\ud83d\udc3b";
String alien = "\uD83D\uDC7D";
System.out.println("bear 1: " + bear1+ " multiple:" + multiple + " bear2: " + bear2 + " alien: " + alien);
byte[] bear1B = bear1.getBytes();
byte[] multipleB = multiple.getBytes();
byte[] bear2B = bear2.getBytes();
byte[] alienB = alien.getBytes();
System.out.println("bear1B: " + Arrays.toString(bear1B));
System.out.println("multiple: " + Arrays.toString(multipleB));
System.out.println("bear2B : " + Arrays.toString(bear2B));
System.out.println("alienB : " + Arrays.toString(alienB));
}
Output:
bear 1: ? multiple:????????????? bear2: ? alien: ?
bear1B: [63]
multiple: [63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63]
bear2B : [63]
alienB : [63]
It does not seem like a failure of console, becouse the bytes are all 63 although they are different emoticons in "multiple" variable. I don't know if I'm doing something wrong but it seems like java interprets all emoticons as a single value (63).
A: I suspect this is a charset issue. This will probably be fixed if you specify a charset like this:
byte[] bear1B = bear1.getBytes(StandardCharsets.UTF_8);
with StandardCharsets being java.nio.charset.StandardCharsets.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,688 |
Тихонова Ніна Олександрівна (* 20 грудня 1923) — радянський і український художник по гриму.
Працювала на Київській кіностудії з 1937 р.
Була членом Національної Спілки кінематографістів України.
Фільмографія
Брала участь у створенні фільмів:
«Андрієш» (1954),
«Педагогічна поема» (1955),
«Над Черемошем» (1954),
«Партизанська іскра» (1958),
«Військова таємниця» (1959),
«Фортеця на колесах»,
«Кров людська — не водиця» (1960),
«Рибки захотілось» (1963),
«Лушка» (1964),
«До уваги громадян та організацій» (1966),
«Всюди є небо» (1967),
«Втікач з «Янтарного»» (1968),
«Чи вмієте ви жити?» (1970),
«Інспектор карного розшуку» (1971),
«Нічний мотоцикліст» (1972),
«Щовечора після роботи» (1973),
«Марина» (1975),
«Мустанг-іноходець» (1975),
«Припустимо — ти капітан...» (1976),
«Еквілібрист» (1976),
«Якщо ти підеш» (1978),
«Дамський кравець» (1990) тощо.
Посилання
Література
Самохвалова Л. Зачем красить негра? // ТУ парк. 1998. № 31. — С.6.
Персоналії за алфавітом
Українські художники кіно
Художники кіно СРСР
Українські художники по гриму
Художники кіно XX століття
Народились 1923
Народились 20 грудня
Члени Національної спілки кінематографістів України
Працівники Кіностудії ім. Довженка | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,492 |
- [Installation](#installation)
- [Configuration](#configuration)
- [Scanner](#scanner)
- [Fail-On: Failing on vulnerabilities](#fail-on-failing-on-vulnerabilities)
- [Valid Repos](#valid-repos)
- [Trusted Builder Identities and Trusted Builder Projects](#trusted-builder-identities-and-trusted-builder-projects)
- [Repository Checks](#repository-checks)
- [Repository Groups](#repository-groups)
- [Repository Authentication](#repository-authentication)
- [Organization Check](#organization-check)
- [Enabling Checks](#enabling-checks)
- [Checks Groups](#check-groups)
- [Signing Keys](#signing-keys)
- [OpenPGP Keys](#openpgp-keys)
- [Google KMS Keys](#google-kms-keys)
- [Usage](#usage)
## Installation
Install `voucher_server` by running:
```shell
$ go get -u github.com/grafeas/voucher/cmd/voucher_server
```
This will download and install the voucher server binary into `$GOPATH/bin` directory.
## Configuration
See the [Tutorial](TUTORIAL.md) for more thorough setup instructions.
An example configuration file can be found in the [config directory](../../config/config.toml).
The configuration can be written as a toml, json, or yaml file, and you can specify the path to the configuration file using "-c".
Below are the configuration options for Voucher Server:
| Group | Key | Description |
| :------------- | :--------------------------- | :---------------------------------------------------------------------------------------------------- |
| | `dryrun` | When set, don't create attestations. |
| | `scanner` | The vulnerability scanner to use ("clair" or "gca"). |
| | `failon` | The minimum vulnerability to fail on. Discussed below. |
| | `valid_repos` | A list of repos that are owned by your team/organization. |
| | `trusted_builder_identities` | A list of email addresses. Owners of these emails are considered "trusted" (and will pass Provenance) |
| | `trusted_projects` | A list of projects that are considered "trusted" (and will pass Provenance) |
| | `image_project` | The project in the metadata server that image information is stored. |
| | `binauth_project` | The project in the metadata server that the binauth information is stored. |
| `checks` | (test name here) | A test that is active when running "all" tests. |
| `server` | `port` | The port that the server can be reached on. |
| `server` | `timeout` | The number of seconds to spend checking an image, before failing. |
| `server` | `require_auth` | Require the use of Basic Auth, with the username and password from the configuration. |
| `server` | `username` | The username that Voucher server users must use. |
| `server` | `password` | A password hashed with the bcrypt algorithm, for use with the username. |
| `ejson` | `dir` | The path to the ejson keys directory. |
| `ejson` | `secrets` | The path to the ejson secrets. |
| `clair` | `address` | The hostname that Clair exists at. If "http://" or "https://" is omitted, this will default to HTTPS. |
| `repository.[alias]` | `org-url` | The URL used to determine if a repository is owned by an organization. |
| `required.[env]` | (test name here) | A test that is active when running "env" tests. |
Configuration options can be overridden at runtime by setting the appropriate flag. For example, if you set the "port" flag when running `voucher_server`, that value will override whatever is in the configuration.
Note that Repositories can be set multiple times. This is discussed futher below.
### Scanner
The `scanner` option in the configuration is used to select the Vulnerability scanner.
This option supports two values:
- `c` or `clair` to use an instance of CoreOS's Clair.
- `g` or `gca` to use Google Container Analysis.
If you decide to use Clair, you will need to update the clair configuration block to specify the correct address for the server.
### Fail-On: Failing on vulnerabilities
The `failon` option allows you to set the minimum vulnerability to consider an image insecure.
This option supports the following:
- "negligible"
- "low"
- "medium"
- "unknown"
- "high"
- "critical"
For example, if you set `failon` to "high", only "high" and "critical" vulnerabilities will prevent the image from being attested. A value of "low" will cause "low", "medium", "unknown", "high", and "critical" vulnerabilities to prevent the image from being attested failure.
### Valid Repos
The `valid_repos` option in the configuration is used to limit which repositories images must be from to pass the DIY check.
This option takes a list of repos, which are compared against the repos that images live in. An image will pass if it starts with any of the items in the list.
For example:
```toml
valid_repos = [
"gcr.io/team-images/",
"gcr.io/external-images/specific-project"
]
```
Will allow images that start with `gcr.io/team-images/` and `gcr.io/external-images/specific-project/` to pass the DIY check, while blocking other `gcr.io/external-images/`.
### Trusted Builder Identities and Trusted Builder Projects
The provenance check works by obtaining the build metadata for an image from the metadata service, and verifying that it both comes from a trusted project and was built by a trusted builder.
You can use this to ensure that only images built by your continuous integration pipeline can be deployed to the cloud, with exceptions for images built by trusted administrators.
For example:
```toml
trusted_builder_identities = [
"catherine@example.com",
"idcloudbuild.gserviceaccount.com"
]
trusted_projects = [
"team-images"
]
```
This would require that images be built in the `team-images` project, by either `catherine@example.com` or `idcloudbuild.gserviceaccount.com`. Images not built in the `team-images` project will fail Provenance regardless of who built them.
### Repository Checks
#### Repository Groups
Repository Groups are used to determine which Repository client should be used
to connect to a Repository.
They are defined as an alias (usually matching the name of the organization in the repository system) and a URL. Note that
the alias can contain only lower cases letters, dashes, and underscores (`[a-z_-]`).
The URL is used to determine if a repository is owned by a repository group.
For example, a repository owned by `Shopify`, it's URL should contain `github.com/Shopify`. A repository owned by `Grafeas`, should contain `github.com/grafeas`, and so on.
Repository groups are required to use the Repository checks.
You can define repository groups as follows:
```toml
[repository.shopify]
org-url = "https://github.com/Shopify"
[repository.grafeas]
org-url = "https://github.com/grafeas"
```
#### Repository Authentication
For the repository groups to authenticate with a repository server, you will
need to create secrets matching the alias name in your `ejson` file.
For example, for the `repository.shopify` block, you would need a `repositories`
block in your `ejson` with a `shopify` block inside of it:
```json
{
"_public_key" : "cb0849626842a90427a026dc78ab39f8ded6f3180477c848ed3fe7c9c85da93d",
"repositories": {
"shopify": {
"token": "EJ[1:vt978NEKynsJhZlCvg5XdSE2S2PM1JBPK0tlC05cQAc=:4mDZCykfieedtHGoM0UT+Wr6zPO9J6XO:/AuGJ3I2QVnk52qOLo0sQ+EzEAk=]"
}
}
}
```
Note that like aliases, the name of the block must only contain lower cases
letters, dashes, and underscores (`[a-z_-]`).
If your repository server supports personal access token, you can specify it
by setting the `token` key (as in the example above).
If your repository server supports app based authentication (as Github does),
you can specify the App ID, Installation ID, and the private key as follows:
```json
{
"_public_key" : "cb0849626842a90427a026dc78ab39f8ded6f3180477c848ed3fe7c9c85da93d",
"repositories": {
"shopify": {
"_id": "1234",
"_installation_id": "123456789",
"private_key": "EJ[1:vt978NEKynsJhZlCvg5XdSE2S2PM1JBPK0tlC05cQAc=:4mDZCykfieedtHGoM0UT+Wr6zPO9J6XO:/AuGJ3I2QVnk52qOLo0sQ+EzEAk=]"
}
}
}
```
#### Organization Check
The organization check is a dynamic check which uses the name of an organization to determine if code came from that organization.
For example, if you have defined an organization as follows:
```toml
[[repositories]]
alias = "Shopify"
org-url = "https://github.com/Shopify"
```
You can enable a check that verifies that an image came from that organization:
```toml
[checks]
is_shopify = true
```
The name of the check is `is_` followed by the name of the organization, converted to lowercase.
This check works by:
- looking up the build metadata for an image from your metadata service
- getting the repository information from that metadata
- connecting to the API of that repository (in this example, Github)
- verifying that the source code is associated with the organization that it says it is
### Enabling Checks
You can enable certain checks for the "all" option by updating the `checks` block in the configuration.
For example:
```toml
[checks]
diy = true
nobody = true
provenance = false
snakeoil = true
is_shopify = true
```
With this configuration, the `diy`, `nobody`, `snakeoil`, and `is_shopify` checks would run when running `all` checks. The `provenance` check will be ignored unless called directly.
### Check Groups
You can configure named groups of checks identically to how you would define an [enable
checks](#enabling-checks) block, by replacing the block heading with `required.[env]`
where `[env]` is a name of your choosing.
For example:
```toml
[required.myenv]
diy = true
nobody = true
provenance = false
snakeoil = true
is_shopify = true
```
With this configuration, the `diy`, `nobody`, `snakeoil`, and `is_shopify`
checks would run when running `myenv` checks. The `provenance` check will be
ignored unless called directly.
### Signing Keys
#### OpenPGP Keys
You can use OpenPGP keys by encrypting them in your ejson secrets file.
First, ensure the signer is set to `pgp` in the configuration:
```toml
signer = "pgp"
```
Then add keys to your ejson secrets file. [This is documented more completely
in the Tutorial](TUTORIAL.md#generating-keys-for-attestation).
#### Google KMS Keys
You can use Google KMS signing keys by switching the signer in the
configuration:
```toml
signer = "kms"
```
Then you would specify the KMS keys in the configuration file, by adding
`[[kms_keys]]` blocks.
For example, to have the DIY Check use a KMS key in the project `binauth` with
the name `diy-attestor`, you would use the following:
```toml
[[kms_keys]]
check = "diy"
path = "projects/binauth-staging/locations/global/keyRings/binauthz-keys/cryptoKeys/diy-attestor/cryptoKeyVersions/1"
algo = "SHA512"
```
## Usage
### Using Voucher Server to check an image
You can run Voucher in server mode by launching `voucher_server`, using the following syntax:
```shell
$ voucher_server [--port <port number>]
```
`voucher_server` supports the following flags:
| Flag | Short Flag | Description |
| :-------- | :--------------- | :------------------------------------------------------------------------- |
| `--config` | `-c` | The path to a configuration file that should be used. |
| `--port` | `-p` | Set the port to listen on. |
| `--timeout` | | The number of seconds to spend checking an image, before failing. |
For example:
```shell
$ voucher_server --port 8000
```
This would launch the server, utilizing port 8000.
You can connect to Voucher over http.
For example, using `curl`:
```shell
$ curl -X POST -H "Content-Type: application/json" -d "{\"image_url\": \"gcr.io/path/to/image@sha256:ab7524b7375fbf09b3784f0bbd9cb2505700dd05e03ce5f5e6d262bf2f5ac51c\"}" http://localhost:8000/all
```
The response will be something along the following lines:
```json
{
"image": "gcr.io/path/to/image@sha256:ab7524b7375fbf09b3784f0bbd9cb2505700dd05e03ce5f5e6d262bf2f5ac51c",
"success": false,
"results": [
{
"name": "provenance",
"error": "no occurrences returned for image",
"success": false,
"attested": false
},
{
"name": "snakeoil",
"success": true,
"attested": true
},
{
"name": "diy",
"success": true,
"attested": true
},
{
"name": "nobody",
"success": true,
"attested": true
}
]
}
```
More details about Voucher server can be read in the [API documentation](../../server/README.md).
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,189 |
{"url":"https:\/\/help.altair.com\/hwsolvers\/ms\/topics\/solvers\/ms\/sfsf_api.htm","text":"# Sfsf\n\nModel ElementSfsf defines a higher pair constraint. The constraint consists of a surface on one body rolling and sliding on a surface on a second body. The surfaces are required to have a unique contact point.\n\nSfsf\n\n## Attribute Summary\n\nName Property Modifiable by command? Designable?\nid Int ()\nlabel Str () Yes\nisurface Reference (\"Surface\") Yes Yes\njsurface Reference (\"Surface\") Yes Yes\nirm Reference (\"Marker\") Yes Yes\njrm Reference (\"Marker\") Yes Yes\nidisp Double ([0,0,0], count=3) \u00a0 FD Only\njdisp Double ([0,0,0], count=3) \u00a0 FD Only\nactive Bool () Yes\n\n## Usage\n\nSfsf (isurface=objSurface, irm=objMarker, jsurface=objSurface, jrm=objMarker, optional_attributes)\n\n## Attributes\n\nisurface\nReference to an existing surface.\nSpecifies the surface that is fixed to the first body.\nThe attribute isurface is mandatory.\nirm\nReference to an existing Marker.\nSpecifies a fixed marker that defines the coordinate system in which the isurface points are defined. It also implicitly defines the body on which the surface is \"etched\". The surface moves with the body.\nThe attribute irm is mandatory.\njsurface\nReference to an existing surface.\nSpecifies the surface that is fixed to the second body.\nThe attribute jsurface is mandatory.\njrm\nReference to an existing Marker.\nSpecifies a fixed marker that defines the coordinate system in which the ISURFACE points are defined. It also implicitly defines the body on which the surface is \"etched\". The surface moves with the body.\nThe attribute jrm is mandatory.\nid\nInteger\nSpecifies the element identification number. This number must be unique among all the Sfsf objects in the model.\nThis attribute is optional. MotionSolve will automatically create an ID when one is not specified.\nRange of values: id > 0\nlabel\nString\nSpecifies the name of the Sfsf object.\nThis attribute is optional. When not specified, MotionSolve will create a label for you.\nidisp\nList of three doubles.\nThese three values specify the location of the contact point on isurface at the input configuration as measured in the irm coordinate system.\nThe three values come as a set. All three must be specified or none may be defined.\nThis attribute is optional.\njdisp\nList of three doubles\nThese three parameters specify the location of the contact point on jsurface at the input configuration as measured in the jrm coordinate system. The three parameters come as a set. All three must be specified or none may be defined.\nThis attribute is optional.\nactive\nBool\nSelect one from True or False.\n\u2022 True indicates that the element is active in the model and it affects the behavior of the system\n\u2022 False indicates that the element is inactive in the model and it does not affect the behavior of the system. It is almost as if the entity was removed from the model, of course with the exception that can be turned \"ON\" when desirable.\nThe attribute active is optional. When not specified, active defaults to True\n\n## Example\n\nCreate two surfaces and define an Sfsf constraint between them.\n# Create the first surface object\nsurf1 = Surface (function=\"user (10)\", routine=Surface1, uclosed=True,\nvclosed=False,\nminpar=[0, -150], maxpar=[2*pi, 150], label=\"Surf-1\")\n\n# Create the second surface object\nsurf2 = Surface (function=\"user (10)\", routine= Surface2, uclosed=True,\nvclosed=False,\nminpar=[0, -1], maxpar=[2, 15], label=\"Surf-2\")\n\nic1 =[1.466,5.66,0.1]\nic2 = [-0.522,-0.852,-0.453]\n\nsfsf1 = Sfsf (isurface=surf1, irm=m1, jsurface=surf2, jrm=m2, idisp=ic1, jdisp=ic2)","date":"2023-03-26 15:12:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4197479486465454, \"perplexity\": 5510.260515091898}, \"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-2023-14\/segments\/1679296945473.69\/warc\/CC-MAIN-20230326142035-20230326172035-00176.warc.gz\"}"} | null | null |
Gonzalo Higuain scored the only goal, as Argentina beat Belgium 1-0 to progress into the semi-final stage of the World Cup.
Both teams came into the game, looking to book a place against either Holland or Costa Rica in the last four. It took the Napoli striker only eight minutes to fire in the winner.
Lionel Messi embarked on a brilliant solo run and found Di Maria's whose pass for deflected into Higuain's path. The striker struck it first time into the bottom corner past a stunned Thibaut Courtois.
The rest of the game panned out without any major event, except for the last moments when Messi broke free, but was denied by Courtois. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,299 |
Q: How can I instantiate a class in a C++ dll, so that it maintains memory inbetween function calls? I'm trying to write a dll that will store some values passed to it in between calls.
Currently, I have:
MyDll.h:
namespace MyDllNamespace
{
class MyClass
{
public:
static int getValue(int &a);
};
}
MyDll.cpp:
#include "ALERTDataAnalysisLibrary.h"
#include <stdexcept>
using namespace std;
namespace MyDllNamespace
{
int storedValue=0;
int MyClass::getValue(int &a)
{
a=a+storedValue;
}
}
MyDll.def
LIBRARY MyDllLibrary
EXPORTS
getValue @1
The idea is that the first time it is called, it returns the passed value, and the second time it's called it returns the sum of the passed value, and the previously passed value. (This is a bit cut down, the final version will have to work on arrays, and with more values, but one step at a time!)
I think what I want to do is stop getValue being static. Which would mean instantiating a class within the DLL (perhaps when it's loaded), and storing data in that. Then exposing a function of that class. I've no idea if that's the way to do it, I've not found anything like it after much searching.
A: You could export the whole class (and optionally use the pimpl idiom, but that's not mandatory):
In your DLL header file "MyDll.h", write the following:
#ifdef MYDLL_EXPORTS
#define MYDLL_API __declspec(dllexport)
#else
#define MYDLL_API __declspec(dllimport)
#endif
class MYDLL_API MyClass
{
...
};
Then, make sure that MYDLL_EXPORTS is defined in your DLL project only, so when you compile your DLL, your class is "dllexported", while when included in your main application, it's "dllimported".
Also, you won't need the .def file anymore.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,254 |
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,509 |
\section{Introduction}
\label{intro}
In the early stages of a design process towards third-generation gravitational wave detectors we can
indulge in the creative activity of inventing completely new instruments in various
forms or shapes.
However, third-generation detectors would be very large (several kilometers long) instruments and
we can expect that in case of a successful realisation
the final form and location of the detectors will have been strongly influenced by less scientific constraints
such as costs and politics. The most exciting period for many instrumental designers lies in the period
when out of the many possibilities we have to identify and optimise the best instrument
conceivable within an optimistic but realistic outlook on the available resources and technologies.
\section{The footprint of future interferometric detectors}
\label{sec:adf}
This section briefly reviews the reasoning behind the shape of current gravitational wave detectors and then
discusses alternative geometries which can be of interest for third-generation detectors. We
will use the terminology introduced in the review of a triangular configuration~\cite{Freise09} and
discriminate between the
\emph{geometry}, \emph{topology} and \emph{configuration} of a detector as follows:
\begin{itemize}
\item The \emph{geometry} describes the position information of one or several interferometers,
defined by the number of interferometers, their location and relative orientation.
\item The \emph{topology} describes the optical system formed by its core elements.
The most common examples are the Michelson, Sagnac and Mach--Zehnder topologies.
\item Finally the \emph{configuration} describes the detail of the optical layout and the set of parameters that
can be changed for a given topology, ranging from the specifications of the optical core
elements to the control systems, including the operation point of the main interferometer.\footnote{Note that the
addition of optical components to a given topology is often referred to as a change in configuration.}
\end{itemize}
\subsection{The L-shape}
Current gravitational wave detectors represent the most precise instruments for measuring length changes.
They are laser interferometers with km-long arms and are operated differently from many precision
instruments built for measuring an absolute length. Viewed from above
they resemble an L-shape with equal arm length.
This geometric form follows directly from the nature of gravitational
waves: gravitational waves are transverse, quadrupole waves, thus a length change measured along any axis
occurs with opposite sign along the axis orthogonal to the previous and the direction of propagation.
This key feature allows to make a differential measurement between two orthogonal interferometer
arms, yielding twice the amplitude of a single arm. More importantly a differential measurement allows us to
potentially discriminate between gravitational wave signals and those types of noise common to both arms, such as,
for example, laser amplitude noise. To achieve this the interferometer arms generally have to have approximately
the same length. The most simple L-shaped interferometer allowing to do this type
of measurement is the symmetric Michelson interferometer, on whose topology all current interferometric detectors
are based.
The long arm length of the detectors represents the simplest way to increase the signal-to-noise ratio
in the detector because the `tidal' effect of the gravitational wave increases with the base length over which the
measurement is taken, while the fundamental noises are connected to the interaction
of light with the optical components or the photo detection and thus do not scale with the length of the interferometer
arms.
We can summarise, provided specifications of the vacuum system housing the interferometer and
the performance of mirror position control systems are good enough, an increase in arm length will increase the
sensitivity of the detector proportionally.
Using the framework developed in~\cite{Jaranowski98} we can compute the sensitivity of a laser interferometer
with two arms to gravitational waves, taking into account the geometry of the detector, the location of the source and the
changes of both over time. The equations show directly that the arms of the detector do not
have to be perpendicular\footnote{The GEO detector for example features an opening angle of approximately $94^\circ$
in order to make the best use of the available site.}, the right angle, however, provides the maximum
response of an ideal detector to gravitational waves, which more generally can be written as
\begin{equation}
h(t)=F_{+}(t)h_{+}(t)+F_{\times}(t)h_{\times}(t)=\sin\zeta\,f(t,\psi, \dots)
\end{equation}
with $\zeta$ the opening angle of the interferometer arms, $F_{+}$ and $F_{\times}$ the beam pattern functions
and $f(t,\psi, \dots)$ a functions of the remaining parameters describing the geometry (the location of the detector and
of the source in space and time and the wave polarisation angle).
In summary we can say that for a gravitational wave of given direction and polarisation, a properly aligned symmetric L-shape
is an ideal optical layout for an interferometric
detector; the arms should be as long as possible and the sensitivity is maximised for an opening angle of $90^\circ$.
It should be noted that this does not put severe constraints on the type of interferometer topology used. In fact,
most common interferometer types can be used in a form that features two large symmetric arms in an L-shape
while potential other interferometer arms or sections are shortened such that they can be considered as part of one
corner of the detector.
\subsection{The triangle}
At any given moment an L-shaped detector can only detect one linear combination of polarisations of a
gravitational wave. However, for
estimation of source parameters from the measured signal, the full polarisation information can be essential
(see next section).
Thus it is of considerable interest to design a detector able to detect both polarisations (and thus the full content)
of a gravitational wave at all times. This can be achieved by combining two co-located
L-shaped detectors which are positioned at $45^\circ$ to each other.
Already more than 20 years ago it was recognised that a triangular geometry would provide the same
sensitivity to both polarisations as detectors at $45^\circ$ while requiring less enclosed space and fewer
end stations~\cite{MPQ-talk}. In particular, the sensitivity of the two geometries shown
in Figure~\ref{fig:triangle} differs only by $6\%$~\cite{Freise09}. The difference in the sensitivity to
different polarisations between a single L-shape and a triangular geometry can be best illustrated
with a plot of the so-called antenna pattern as shown in Figure~\ref{fig:triangleAP}.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.7\textwidth,keepaspectratio]{pic/TRIMI03.eps}
\caption{{\bf a)} Triangle geometry: three L-shaped detectors with 10\, km arm length
are positioned in a equilateral triangle.
{\bf b)} Four L-shaped detectors at $0^\circ$ and $45^\circ$. The integrated length of all
interferometer arms in both configurations is 60\,km and two interferometer arms can share
the same structure. Note that for avoiding noise correlations between two detectors the
neighbouring interferometer arms would probably be housed in a separate vacuum tubes.}
\label{fig:triangle}
\end{center}
\end{figure}
Using co-located detectors yields another advantage. Both layouts shown in Figure~\ref{fig:triangle} represent
detectors with redundancy. Redundancy here can be understood in relation to the continuous operation of the
detector as an observatory, or as a feature of the data streams generated by the full system. Redundancy in operation
is achieved by having multiple detectors which generate an equal or similar response to gravitational waves.
This is desirable in observatories which are expected to produce a quasi-continuous stream of astrophysical meaningful
data over an substantial amount of time. Typically laser interferometers cannot produce science data during
upgrades and maintenance work. Thus only alternate upgrading and data taking of redundant detectors can
avoid long down-times, for example during detector upgrades.
Such redundancy is obviously provided in the case of the 4 L-shaped detectors, where two detectors are
always identical but can be operated independently. However, one can easily show that the triangular geometry
provides exactly the same redundancy~\cite{Freise09}. For example, for three equal L-shaped interferometers
oriented at $0^\circ$, $120^\circ$ and $240^\circ$, one obtains:
\begin{equation}
-h_{0^\circ}= h_{240^\circ}+h_{120^\circ}\,,
\end{equation}
where the sign of the operation is defined by which ports of the interferometers
are used to inject the laser light. Thus the two interferometers at $120^\circ$ and $240^\circ$
create exactly the same response as the one at $0^\circ$. This allows to construct
so-called null-streams (or null-data streams)~\cite{nullstream}. Null-streams are a powerful
data analysis method that allows to identify noise which is uncorrelated between the
detectors. Even though this does not increase the sensitivity of a detector,
it can add significantly to the robustness of the data processing pipelines and thus lead, for example, to shorter
delays between an event and the generation of a trigger for follow-up searches with optical telescopes.
The triangular geometry represents the minimal setup in one plane that can resolve both polarisations
and provides redundancy for the generation of null-streams.
The idea of using a triangular geometry is considered with strong interest within the context of
the design study for a third-generation detector \emph{Einstein gravitational wave Telescope}~\cite{ET}.
Therefore we will in the following use the term \emph{ET-class} to refer to three L-shaped detectors in a
triangular geometry.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.25\textwidth]{pic/pyramid02.eps}
\caption{The minimal three-dimensional geometry from L-shaped detectors
is an equilateral pyramid in which each edge houses one interferometer arm. A redundant set
which extends the features of the triangular shape requires six L-shaped detectors
such that each edge of the pyramid houses two detector arms.}
\label{fig:pyramid}
\end{center}
\end{figure}
\subsection{The pyramid}
When all co-located interferometers of one detector are confined to one plane they provide maximum sensitivity for
gravitational wave travelling perpendicular to this plane but are not very efficient for gravitational waves
travelling parallel to the plane. In particular, if the polarisation of the wave is not aligned with the
plane of the detectors the sensitivity can drop to zero. This, for example, is illustrated in the center
antenna pattern in Figure~\ref{fig:triangleAP}: in the central region (with detector normal orthogonal to
the z-axis) the sensitivity oscillates between zero and half of the maximum sensitivity as a function of
the angle of the detector normal to the polarisation axis.
In order to maximise the detector response regardless of the source location the collocated
detectors must form a three-dimensional structure. Ideally an redundant set of six detectors should
be used to preserve the features discusses for the triangular geometry. Such a detector would exhibit
an almost spherical antenna pattern as a whole and can provide unique source location information by comparing
the signals of the co-located detectors. The minimal approach to a full three-dimensional structure
requires three L-shaped detectors. Such a geometry is shown in Figure~\ref{fig:pyramid} and the
corresponding antenna pattern is depicted in the right plot of Figure~\ref{fig:triangleAP}.
While the pyramid can be considered the ideal geometry maximising the detected signal independent of the
source location or polarisation, this geometry has not been studied in detail. The reasons for this
are probably practical problems related to the non-horizontal light beams as well as the non-vertical
mirrors. The former at least would increase construction costs of the detector disproportionally
and the latter would even require a completely different suspension concept than employed by
current detectors.
\section{The ideal microphone}
\label{sec:andrea}
The third-generation gravitational wave detectors will be charged
with the mission of opening the field of precision physics in general
relativity~\cite{will:2006}, and provide data to the astrophysics community
allowing to complement information obtained by electro-magnetic and neutrino
detectors~\cite{gwic:2009}.
The details of the mission may depend on the results obtained with the second-detector generation, the so-called \emph{advanced detectors}, currently under
construction~\cite{vicere:2005}.
In fact, we can expect that advanced detectors will have not only carried out
the first detection, but collected a sufficient number of events to allow
identifying the most promising classes of sources, thus possibly suggesting
directions for the optimization of the design.
\subsection{The sources of interest}
For the third-generation ground based detectors to really allow precision
measurements on all the sources of interest, several requirements
with impact on the observatory geometry and topology need to be considered.
It seems appropriate then to recapitulate the characteristics of the different
classes of sources (see \cite{sathya:schutz:2009} for an ample and recent
review).
\paragraph{Transient sources}
Under this name we group all those sources emitting signals which last from
tens of ms up to a few minutes in the detector observation band; for ground
based detectors, these include for instance the coalescing binaries of low and
intermediate mass~\cite{grishchuk:2001}, and collapsing massive stars~\cite{fryer:new:2003}.
The main requirement on third-generation GW observatories is to allow the
simultaneous reconstruction of the sky location and of the source
polarization; for instance, these informations allow to estimate the source
distance in coalescing binary events, and together with an optical counterpart
this would allow a direct measurement of the Hubble constant~\cite{schutz:1986}.
The sky location in turn needs to be accurate enough to optimize the
association with electromagnetic or neutrino transients, for instance to allow
an efficient follow up in the optical band.
\paragraph{Continuous, deterministic sources}
We group here all those point sources emitting continuous wave signals, like
spinning neutron stars~\cite{bonazzola:1997}; the signals are expected to be
almost monochromatic, like those emitted by the known pulsars, but may also
display a significant linewidth due, for instance, to the mechanism of mass
accretion, like in the low-mass X-ray binaries~\cite{ushomirsky:2000}.
In all cases, for such signals the observation time is of the order of several
months to years, and the relative motion of any individual detector and the
source allows to access also the polarization information, with an accuracy
inversely proportional to the signal to noise ratio (SNR) available.
\paragraph{Stochastic sources}
Here we consider both the stochastic background of cosmological origin, which
is generally expected to be modeled as a Gaussian, unpolarized noise, received
incoherently from all sky directions~\cite{lsc:2005}; and the background due
to a large number of astrophysical, unresolved sources~\cite{regimbau:2007}.
The noise nature of the signal requires to correlate the output of different
detectors, having uncorrelated noises, in order to estimate a background due
to gravitational waves~\cite{allen:romano:1999}; therefore, the sensitivity to
GW background scales with the product of each detector sensitivity.
\subsection{Requirements on topologies}
It seems effective to consider a few different and plausible scenarios:
a) single, ET-class observatory; b) an ET-class detector in collaboration with
several advanced detectors such as Advanced LIGO or Advanced VIRGO or
c) several ET-class detectors distributed over the world.
\subsubsection{Requirements on a single, ET-class observatory}
\label{subsub:singleET}
By an ET-class observatory we mean, as in Figure~\ref{fig:triangle} an ensemble of
instruments located at the same place on Earth, capable of reconstructing
both polarizations, with partial or full redundancy.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth,viewport=50 0 840 245,clip]{pic/ap_mi_pyramid.eps}
\caption{
The response of a detector to a linear polarised gravitational wave
as a function of the detector orientation. All three plots show the normalised sensitivity
to a wave travelling along the z-axis. Each data point represents the sensitivity of the
detector for a specific detector orientation defined by the detector normal
passing the respective data point and the origin. The colour
of the data point as well as its distance from the origin indicate the magnitude of the
sensitivity. The left plot depicts the response of a single Michelson, while the center plot
gives the response of a set of three interferometers in a `flat' triangular geometry and the
right plot refers to three detectors in a pyramid geometry.
}
\label{fig:triangleAP}
\end{center}
\end{figure}
The localization of transient sources is the main weakness of a single
observatory; the antenna patterns shown in Figure~\ref{fig:triangleAP} indicate
that all possible detector geometries feature relatively broad patterns and
offer very limited possibility of source localization.
For some sources, like coalescing binaries at a sufficient SNR,
a single observatory may still allow an approximate reconstruction of the sky
location, by exploiting the amplitude information carried by the different
harmonics of the signal;
however the angular accuracy
attainable is expected to be order(s) of magnitude smaller than what is
possible with a network of detectors, because of the lack of the
time-of-flight information.
For continuous, deterministic signals instead, a single ET-class detector
is sufficient to completely characterize the gravitational waves
received, and the null-stream information can be fully exploited.
For stochastic background signals, the main requirement on the topology is
that the detector antenna patterns, for each given polarization, display a
good overlap for a large fraction of sky directions.
In this case, having several detectors at the same location minimizes the loss
of signal coherency due to the finite correlation length of the stochastic
background signal, as measured by the so-called overlap reduction function
$\gamma(r f)$, where $r$ is the distance of the two detectors and $f$ represents
the signal frequency.
To provide a quantitative measure, recall that the stochastic background is
measured by the
quantity $\Omega_{GW}(f)$, the density of the
gravitational waves energy $\rho_{GW}$ per unit logarithmic frequency
\begin{equation}
\Omega_{GW}(f)\equiv\frac{1}{\rho_c}\frac{d\rho_{GW}}{d\log f} ,
\end{equation}
normalized to the critical energy $\rho_c$ for the Universe closure.
The sensitivity to $\Omega_{GW}$ scales as the product
$\tilde{h}_1(f)\times\tilde{h}_2(f)$ of the individual detector sensitivities;
the two Advanced LIGO detectors~\cite{advancedligo:2007} at the Hanford site
will be able to bound $\Omega_{GW}\le 10^{-9}$, and an ET detector will
improve this limit by about two orders of magnitude.
We stress again that a severe limit to the performance of co-located
instruments could result from disturbances that induce correlated noises.
The optical topology should be designed to reduce some of these effects,
for instance by requiring that different beams do not share any optical
element, or even that they are placed in different vacuum systems.
However, such solutions will decouple only some noise sources, like glitches
induced by dust crossing the beam or extra noise induced by the control system
steering the optical elements. Several other noise sources, for instance of
electromagnetic or seismic origin, including in the latter the gravity
gradient noise, will require solutions that are beyond the scope of this paper.
We should add that for continuos, narrow band gravitational waves emitted by
point sources these correlated noises are expected not to be a severe limit,
since the Doppler effect signature induced in the GW signals by the Earth
rotation and revolution should allow discriminating them.
\subsubsection{An ET-class observatory operated along with several advanced
detectors}
In addition to the capabilities expressed by an ET-class observatory, the
coincident operation with advanced detectors, despite their inferior
sensitivity, could bring advantages.
For point sources, a signal displaying a large SNR, say about 50, in ET, might
still be above detection threshold on one or two of the advanced detectors; in
such a case, the time-of-flight measurement would allow to constrain the
source position, better than what is possible in the ET observatory alone.
We recall that, as a rule of thumb, the solid angle in which a source can be
constrained by triangulation in a three-detectors network is of the order of
\begin{equation}
\delta\theta^2 \propto \frac{c^2}{A}\delta t^2
\end{equation}
where $\delta t$ is the relative timing error, depending on the kind of
signal and the SNR, and $A$ is the area of the network.
For $A\sim 10^7$ km$^2$, and a $\delta t \sim 0.5\cdot 10^{-3}$\,s, typical for
coalescing binaries signals~\cite{acernese:2007} having SNR~$\sim 10$, this
leads to an estimated error box about 0.02\% of the solid angle,
This source localization would improve significantly the amplitude and
polarization reconstruction carried out on the basis of the signal available
in the ET observatory; however, selecting events with sufficient SNR in the
advanced detectors, 10 times less sensitive than ET, could limit the event
rate by factors as large as 1000.
For continuous sources instead the sensitivity gap cannot be filled, and
advanced detectors would be completely useless in the network.
For stochastic background, the correlation of data from an ET-class and an
advanced detector would lead to better upper limits by one order
of magnitude, when comparing with what is possible with a pair of not
co-located advanced detectors.
However, this estimate would be at least one order of magnitude worse than
what could be done by the ET-class observatory exploiting its own multiple
detectors. We recall, moreover, that the signal correlation will be attenuated
by the overlap reduction function $\gamma(r f)$; an example is shown in Figure~\ref{fig:orf}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth,clip]{pic/orf.eps}
\caption{An example of overlap reduction function, showing that detector sites
need to be closer than $\sim 300$ km in order to collect, at 300\,Hz, O(50\%)
of the signal available to co-located detectors.}
\label{fig:orf}
\end{center}
\end{figure}
To limit the attenuation to O(50\%) at 300\,Hz, where ET displays its best
sensitivity, one should require the new detector to be built within about 300
km from one of the existing advanced detectors; this requirement, though, would
impair severely the source location capability of the mixed network, because
of the reduced effectiveness of the triangulation method.
\subsubsection{Several ET-class observatories}
This is actually not a single scenario, but could be split in two more cases:
a) each ET-class observatory hosts only one detector and
b) each ET-class observatory hosts more than one detector at the same
location.
\paragraph{Single detector ET-class observatories}
The first case could be represented by several L-shaped detectors located
around the globe. In such a scenario, we do not have really requirements on
the optical topology, but rather we need to optimize the distance and the
relative orientation so as to maximize the scientific return.
To improve sky location reconstruction, maximizing all the distances among the
sites is the main requirement, while the relative orientation needs to find a
compromise between sky coverage and the possibility to observe the same source
with several detectors.
The distance requirement collides of course with the needs of investigations
about the stochastic background, as already discussed.
Upper limits would still be improved, with respect to non co-located advanced
detectors, by about two orders of magnitude, but could not be compared with
the sensitivity of co-located ET-class detectors.
A compromise could result by locating two of the detectors of the network at a
close distance, say up to 300km as mentioned in section~\ref{subsub:singleET},
while building the others as distant as possible on the Earth
surface. We underline that in addition to the two ``close'' detectors, we
would need at least two more ``far'' detectors for optimal source localization,
\paragraph{Multiple detector ET-class observatories}
Such a scenario could consist of two or more sites, each hosting multiple
detectors, i.e. ET-class observatories; as such, it would inherit the benefits
discussed in the previous sections, while removing most or all of the
shortcomings.
In particular, already a two-site scenario would improve dramatically the sky
localization; the time of flight measurement would allow to limit the source
position to an annulus in the sky, and additional amplitude and polarization
information could allow to further constrain it.
The measurement of the stochastic background could be carried out
independently at each site, with maximum sensitivity provided that the
correlated noise issues mentioned in section~\ref{subsub:singleET} can be
coped with; therefore the sites could be placed just as far as possible so as
to optimize their triangulation capability.
In addition, having several sites with ET-class detectors would
allow to carry out sensitive radiometry measurements~\cite{mitra:2008},
thus mapping the background on the sky.
The large redundancy would provide, in addition to two coherent signal
channels for the gravitational wave polarizations, also a large number of
null-streams. These could be exploited, as already discussed, to reject the
noise background, but also to constrain alternative GW theories which foresee
other polarization components.
The constraints about the optical topology are derived almost directly from
what has been discussed in the previous sections, in particular it remains of
paramount importance to limit as much as possible the noise correlation among
detectors at the same site.
\begin{figure}[p]
\begin{center}
\includegraphics[width=\textwidth, viewport=0 0 680 460, clip]{pic/ET1.eps}
\includegraphics[width=\textwidth, viewport=0 0 680 460, clip]{pic/ET12.eps}
\caption{{\bf Upper plot:} Fundamental noise contributions to the sensitivity of a potential
advanced detector (blue solid line). The solid
red line is a rough approximation of the initial ET design target.
{\bf Lower plot:} Noise contributions and
resulting sensitivity of a potential ET configuration (black solid line) as presented in
\cite{ET-B}. (This sensitivity option is sometimes also referred to as \emph{ET-B}
and can be downloaded from \cite{ET}, sensitivity section).
} \label{fig:noise_budget}
\end{center}
\end{figure}
\section{The brute force approach}
\label{sec:stefan}
In this section we describe, based on \cite{ET-B}, how the ambitious sensitivity goal
for third-generation GW detectors such as the Einstein Telescope (ET)~\cite{ET}
could potentially be achieved. This approach makes use of rather conservative
technology concepts, but pushes some parameters significantly beyond what is used
in current detectors. In addition, it needs to be noted that the brute force approach
assumes that some mayor technical problems, we currently encounter in the first generation
GW detectors (such as for instance thermal induced mirror distortions or the abundance of
non-gaussian noise contributions) will be solved during the commissioning of the advanced
detectors.
We start from a standard detector topology
similar to a second-generation detector, i.e. a single Michelson interferometer
with arm cavities
featuring power and signal recycling.\footnote{Several of such interferometers
could then be combined to the various detector geometries as described
in section \ref{sec:adf}.} In addition, we restrict ourselves in this section to
the application of `conventional' technology. We use the term conventional in
the sense that we believe the noise
reduction techniques described in this section to be a rather conservative
approach.
Of course, a lot of more innovative technologies and
configurations have been suggested for application in ET.
As we will show in later sections of this article, some of these innovative
technologies, such as xylophone configurations (see section \ref{sec:ken}) and
advanced quantum noise reduction topologies (see section \ref{sec:QNR}),
might turn out to be more elegant and
less cost-intensive than the brute force approach presented in this section.
However, as these innovative
techniques often involve a lot of new physics and have not yet been fully
tested in prototypes, we find it useful to construct a first reference design from
tested technologies.
The upper plot of Figure \ref{fig:noise_budget} shows exemplary
the individual noise contributions (dashed traces) of an advanced
detector (second generation) together with the resulting sensitivity (blue
solid trace) and a rough estimate of the targeted ET sensitivity (red
solid trace). Compared to the sensitivity of second-generation detectors
the ET sensitivity target requires a sensitivity improvement of a factor
between 10 and 50 for all frequencies above 10\,Hz. Even more
ambitious is the targeted sensitivity improvement of up to several
orders of magnitude at very low frequencies (below 10\,Hz). It
is also worth noting that every noise contribution shown in this
plot is at least at some frequencies above the targeted ET sensitivity,
thus, when going from second to third generation, we have to improve
every single noise contribution. One rather simple but cost intensive
way to decrease all of these noise contributions by a factor of about
3 is to increase the arm length of ET to 10\,km. Though this helps
already a lot we have to further improve all noise contributions, as
we will discuss in more detail in the following sections.
\subsection{High-frequency noise contributions}
\label{sec:stefan:HF}
The high frequency sensitivity (above a few 100\,Hz) of advanced
detectors will be entirely
limited by photon shot noise which is the high frequency component of
quantum noise \cite{Buonanno01}. We assumed that the required shot noise
reduction of a factor of about 50, can be achieved by a combination
of the following changes. The shot noise contribution scales
inverse with the arm length of the interferometer, inverse with
the square root of the optical power stored in the arm cavities and
directly inverse with any applied quantum noise suppression, for instance from
the injection of squeezed light \cite{Caves81}. In addition, we can vary the two
signal recycling parameters \cite{Meers88} (signal recycling tuning phase and
reflectivity of the signal recycling mirror) to optimise the shape
of the quantum noise.
Detuned signal recycling allows to increase the peak sensitivity,
but this comes at the cost of significantly reduced high-frequency performance.
Moreover, detuned signal recycling would make the application of
squeezed light much more hardware intensive; in this case km-long,
low-loss filter cavities would be required \cite{Kimble02}.
Therefore, it seems likely that any broad-band ET interferometer
would feature tuned signal recycling.\footnote{As we will show in
section \ref{sec:ken}, detuned signal recycling
might be an interesting option for any low-frequency interferometer of
an ET xylophone configuration.}
In summary, in \cite{ET-B} it was assumed that the shot noise contribution will be improved
firstly by a factor 2 from increasing the circulating optical power by a factor 4, from
0.75\,MW to 3\,MW, secondly by a factor of 3.3 due to the increased arm length,
thirdly by about a factor of 3 from the application of 10\,dB of broad-band
squeezing and finally by another factor of a few from changing to tuned
signal recycling.
Please note that these individual improvement factors have been chosen
in a rather arbitrary way and only indicate a single of many potential
possibilities to reach the
required shot noise reduction for ET.\footnote{For instance the experience we
gain from commissioning of the second-generation detectors will
tell us whether a further factor 4 of power increase can realistically be
achieved. In case we find any power level above 1\,MW to be impractical,
we might have to increase the targeted quantum noise suppression through
the application of squeezed light.}
\subsection{Mid-frequency noise contributions}
Apart from the already discussed photon shot noise, for the mid-frequency
range of any potential ET observatory we have to improve the noise contribution
from residual gas pressure as well as all thermal noise contributions associated
with the test masses themselves. Overcoming the coating Brownian noise \cite{Harry07}
clearly imposes the biggest challenge, as it needs to be reduced by about a
factor of 20 to be compliant with the targeted ET sensitivity.
Generally the coating Brownian noise is inverse proportional to the beam spot size, inverse
proportional to the square root of the coating temperature and finally also depends
on the material parameters like loss angles and Young's modulus.
Improving the mechanical properties of the coating layers, i.e. searching
for alternative materials featuring lower mechanical losses, is
a field of intense ongoing research. However,
we have considered here the conservative approach of reducing the coating
Brownian noise only by means of increasing the arm length, increasing the
spot size and cooling the test masses.
The maximal beam size on the test masses is determined on one hand by
the maximal available size of mirror bulk material and on the hand by the
achievable polishing accuracy for the surface curvature. The model described
in \cite{ET-B} assumes an increase of the beam radius by a factor of 2 from 6\,cm
to 12\,cm, which corresponds for an arm length of 10\,km to a radius
of curvature of 5070\,m. One of the main consequences of such a large
beam size is that the mirrors have to be at least 60\,cm
in diameter to keep the clipping losses within an acceptable range.
As a nice side-effect increasing the spot size on the test masses also
reduces substrate Brownian noise and coating thermo-optic noise and
slightly reduces the contribution from residual gas pressure noise (due
to the larger volume of the beam).
It seems likely that the ET test masses have to be cooled to
cryogenic temperatures to make them compliant with the envisaged
sensitivity. Assuming that future research will find cryogenic coatings
with the same mechanical properties as the best currently available
room-temperature coatings, reducing the temperature of the mirrors from 290\,K
to 20\,K will further reduce coating Brownian noise by a factor of 3 to 4.
Please note that for various reasons fused silica, though standard
for room temperature interferometers, seems to be disadvantageous
at cryogenic temperatures. Sapphire and silicon are generally
considered as more likely candidates for cryogenic mirrors. An exemplary
analysis of the individual thermal noise contributions of a cryogenic
silicon test mass can be found in \cite{ET-C}.
Another way to reduce coating Brownian noise which will be discussed
in more detail in section \ref{sec:beams} is to sense the
test masses with non-conventional beam shapes.
\subsection{Low-frequency noise contributions}
\label{sec:stefan:LF}
As shown in Figure \ref{fig:noise_budget} the low frequency sensitivity
of second-generation GW observatories will be limited by a mixture of
four different fundamental noise sources: Photon radiation pressure
noise (the low-frequency component of quantum noise), seismic noise,
gravity gradient noise and suspension thermal noise. All four of these
noise contributions have to be reduced by vast amounts to reach
the ET sensitivity target.
The photon radiation pressure noise contribution scales proportional
to the square root of the circulating optical power, inverse
proportional to the arm length and inverse proportional to the mirror
mass. This means that in addition to the required improvement of a factor 20
shown in the upper graph of Figure \ref{fig:noise_budget}, we also have
to recover an additional factor of 2 originating from the factor 4 power
increase described in section \ref{sec:stefan:HF}. This total factor of
40 of required radiation pressure noise reduction can be achieved by
increasing the arm length by a factor 3.3, increasing the mirror mass
by a factor 3 from 42 to 126\,kg,\footnote{As it was recently shown
the coating Brownian noise of a mirror also depends
on its ratio of radius and thickness \cite{Somiya09}. It turns out that
the mirror thickness should be roughly similar to the mirror radius
in order to achieve reasonable coating Brownian noise. In our case with
60\,cm mirror diameter, this seems to make an even higher mirror mass
of about 200\,kg appearing more realistic \cite{ET-C}.}
a factor 3.2 from the above mentioned
10\,dB of quantum noise suppression\footnote{A simultaneous
reduction of the shot noise and radiation pressure noise by
squeezed vacuum in a tuned RSE configuration requires one filter
cavity, see section~\ref{sec:Filtercavities} for more details.}
and a final factor of about 1.3 gained from changing to tuned
signal recycling combined with a signal recycling mirror
transmittance of 10\,\%.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
& advanced detector & potential ET design \\
\hline
Arm length & 3\,km & 10\,km \\
SR-phase & detuned (0.15\,rad) & tuned (0.0\,rad)\\
SR transmittance & 11\,\% & 10\,\% \\
Input power (after IMC) & 125\,W & 500\,W \\
Arm power & ~0.75\,MW & ~3\,MW\\
Quantum noise suppression & none & 10\,dB \\
Beam radius & ~6\,cm & 12\,cm \\
Temperature & 290\,K & 20\,K \\
Suspension & Superattenuator & 5 stages of each 10\,m length\\
Seismic (for $f>1$ \,Hz) & $1\cdot 10^{-7}\,{\rm m}/f^2$ (surface) & $5\cdot 10^{-9}\,{\rm m}/f^2$ (underground) \\
Gravity gradient reduction & none & factor 50 required \\
Mirror masses & 42\,kg & 126\,kg \\
BNS range & ~150\,Mpc & ~ 2650\,Mpc\\
BBH range & ~ 800\,Mpc & ~ 25000\,Mpc \\
\hline
\end{tabular}
\caption{Summary of the parameter changes necessary to go from an
advanced detector sensitivity to the ET design target using the approach
described in \cite{ET-B}. The second and third
columns correspond to the upper and lower graph of Figure
\ref{fig:noise_budget}, respectively. \label{tab:summary}}
\end{center}
\end{table}
The seismic noise contribution can in general be reduced by either
reducing the ambient seismic level itself or by reducing the coupling
of the ambient seismic to the test masses, i.e. improving the
performance of the seismic isolation systems. Starting from an
ambient seismic (for frequencies above 1\,Hz)
of about $1\cdot 10^{-7}\,{\rm m}/f^2$
we assume that the seismic level can be reduced by a factor of 20
to about $5\cdot 10^{-9}\,{\rm m}/f^2$ by building ET in an underground
location
In a second, more
vigorous step we have to strongly push the seismic wall to lower
frequencies. Using passive pendulum systems this can only be achieved
by reducing the resonance frequencies of the suspensions. It has been
estimated that five pendulum stages of each 10\,m length, yielding a resonance
frequency of about 160\,mHz, together with a quiet underground
location can achieve the seismic noise suppression required for ET.
Similar to the seismic noise the gravity gradient noise contribution is
determined by the level of the ambient seismic and the corresponding
coupling transfer function to the test masses. However, the big difference
is that we have no means of influencing the
magnitude of coupling, because we cannot shield
gravity. As shown in Figure \ref{fig:noise_budget} we need to reduce
the gravity gradient noise contribution by a challenging factor of about
3000, when progressing from second-generation detectors to
ET. We considered a factor of 3 reduction from increasing the arm length,
a factor 20 from going to an underground location (see above) and optimistically
assumed that the remaining reduction factor of 50 can be achieved
by subtracting the gravity gradient noise from the GW channel by
making use of coherent signals from an array of seismometers. For a
more comprehensive discussion of gravity gradient noise, see \cite{Cella09}.
Note that at the current stage of investigations no comprehensive
analysis of the suspension thermal noise of a cryogenic ET suspension
exists. Therefore, we omitted the suspension thermal noise trace from the
lower graph of Figure \ref{fig:noise_budget}.
\subsection{Discussion of the described sensitivity option for a single
broad-band ET observatory}
A comparison of the most important interferometer parameters
of a typical second-generation detector and a potential ET
configuration, combining all noise reduction efforts discussed in
this section, is shown in Table \ref{tab:summary}.
Figure \ref{fig:noise_budget} shows clearly that
the presented ET sensitivity is limited by quantum noise at
nearly all frequencies. As a consequence (and considering
the levels and slopes of gravity gradient noise and coating Brownian
noise), using heavier mirrors would
immediately allow to improve the sensitivity between 3 and
30\,Hz significantly beyond the envisaged ET target. On the other
hand our efforts have not yet reached the originally targeted sensitivity in
the band between 100 and 300\,Hz. Improving the quantum noise in this
region by means of further increasing the optical power would simultaneously
reduce the low-frequency sensitivity. Similarly, detuned signal recycling
which offers the potential of reaching the targeted sensitivity around
200\,Hz, would worsen the sensitivity at the lower and higher frequencies.
This restriction of peak-sensitivity versus bandwidth can potentially be
overcome by so-called xylophone configurations \cite{ET-C}, i.e. by building several
narrow-band interferometers each optimised to give optimal performance
in a certain frequency band.
\section{Building a multi-band antenna}
\label{sec:ken}
As the targeted sensitivity and bandwidth for future gravitational wave detectors
is pushed
higher, eventually it must be true that it becomes better to employ multiple
interferometers rather than a single one. There are two main reasons for this,
and both of them, as discussed below, lead to an approach by which two or
more interferometers of rather narrow bandwidth can be combined to
better deliver the required broad-band sensitivity than a single interferometer.
The first concept, the traditional multi-interferometer approach
described in section~\ref{sec:xylo:trad}, was originally proposed
to increase the high frequency sensitivity of shot noise limited interferometers
featuring signal recycling. However, more recently it was realised that
the real benefit of xylophone interferometers is the potential to
significantly improve the low frequency sensitivity by resolving incompatibilities
interferometer parameters and noise reduction schemes. This highly
profitable approach will be described in detail in section~\ref{sec:xylo:LF}.
\subsection{The traditional multi-interferometer approach}
\label{sec:xylo:trad}
The original proposal for multiple interferometers, often called a `xylophone'
by analogy to the percussion instrument, arose in the context of the shot noise
limit of relatively narrow-band signal recycled interferometers operating at high
frequency. In this case a classical picture taking into account only shot noise
and coating thermal noise suffices to describe the situation and explain the approach.
The best way to characterise the performance of such a detector is by the spectral
density of the noise and the bandwidth taken together.
Here we assume
that the
performance limit arises mainly from shot noise, and that, at least at frequencies above
a few $100\,$Hz, thermal noise can be neglected.\footnote{This is always true at sufficiently
high frequency, as shot noise increases in cavity-based interferometers, and thermal
noise decreases, with increasing frequency (ignoring resonant modes in both
the optical and mechanical cases).}
The sensitivity is limited by shot noise because there is a limit to the optical energy
that may be stored in the interferometer. Setting aside the technique of resonant
sideband extraction (RSE)\footnote{RSE, a technique proposed specifically to allow
the photon storage time to be
increased without (to first order) affecting the signal bandwidth provides only a
modest degree of improvement as losses in the RSE system limit the extent
to which the storage time can be increased.}
the storage time for the photons in the arms of the
interferometer is capped by the desired bandwidth of the signal response. In this
case the energy limit is essentially a power limit -- set by the tolerable heating of
the cavity mirrors that form the arms of the interferometers. In the first and second-generation
detectors the heating leads to unwanted and eventually uncorrectable
distortion of the optical mode through the change of shape or change of refractive
index of the optic. In the advanced detectors, for example, it is intended to push
the circulating power to $\sim 1\,$MW, and that requires state of the art low
absorption optical materials and aggressive thermal compensation.\footnote{In
proposed cryogenic detectors the limiting light power is not yet known, but given
the difficulty of extracting heat at very low temperatures, it is unlikely to be much
greater than in the advanced detectors.}
Initial xylophone ideas considered the option of spatially overlapping interferometers
sharing the same mirrors, but employing different wavelengths \cite{phd.Freise}. Such multi-colour
detectors while potentially attractive in terms of cost, do not solve the heating problem.
Thus it is necessary to consider multiple interferometers each with their own optics,
though possibly sharing the same site, and even vacuum envelope. At higher
frequencies the mirrors can be relatively light as radiation pressure noise is not significant.
As vibration isolation is also simple above $\sim 100\,$Hz the isolation and suspension
systems which otherwise constitute a significant fraction of the complexity and cost of
a detector can be simple and compact. Thus it is not unrealistic to consider several
interferometers covering the band among them.
Figure~\ref{fig:MultipleSignalRecycledInterferometers} illustrates the basic approach with 5
signal recycled interferometers, combined to a xylophone detector.
As this example shows such a configuration leads to only moderately improved
sensitivity over a limited bandwidth. Thus from the point of third generation
observatories the prospects of this traditional xylophone approach are rather limited.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth,viewport=0 0 410 210, keepaspectratio]{./pic/xylo1.eps}
\caption{Shot noise limited sensitivity of a xylophone interferometer \cite{phd.Freise}
compared to an equivalent Dual-Recycled interferometer.
} \label{fig:MultipleSignalRecycledInterferometers}
\end{center}
\end{figure}
\subsection{Multi-interferometer solutions for low frequencies}
\label{sec:xylo:LF}
More recently, in the development of techniques for second and third-generation detectors,
it has been realised that multiple interferometers could become a key element
of future systems. A more complete description of the interferometer is required
to explain this approach.
There are several noise contributions that, broadly speaking, rise steeply
towards low frequency, including radiation pressure noise ($f^{-2}$ in non-QND
systems), suspension thermal noise (broadly $f^{-2.5}$ except near suspension
resonances) and gravity-gradient noise, which has a complicated spectrum.
The first two of these also scale with the mass $m$ of the mirror: directly in the
case of radiation pressure noise and as $m^{1/4}$ in the case of suspension
thermal noise. In addition, seismic noise, after filtering by the isolation and
suspension systems has a much steeper gradient, but in that case it is more
convenient to represent performance by a seismic wall. To push the wall down
to lower frequency requires larger and/or more sophisticated approaches to
isolation (see section \ref{sec:stefan:LF}).
These facts are the key ingredients in the design of xylophones for
application at low frequency (here taken to be below about 100\,Hz).
The changing mix of noise contributions as a function of frequency suggests
that it may well be worth considering interferometers designed to work over
a relatively narrow band -- perhaps an octave or two -- where one or two
noise contributions dominate. The advantage in this approach comes whenever
techniques are applied to reduce two noise sources that dominate in distinct bands
are mutually exclusive. The most prominent example for such an antagonism
is the scaling of quantum noise with the optical power. To improve the high
frequency sensitivity by means of reducing the photon shot noise contribution it is important
to increase the circulating light power. However, improving the low frequency
sensitivity by means of reducing photon radiation pressure noise requires
to decrease the optical power in the system.
Another example is the potential conflict of very high power
interferometers (to reduce shot noise) and cryogenics (to reduce suspension
thermal noise), originating from residual light absorption in the mirrors and their coatings.
Such an xylophone approach vastly increases the space of available designs, at the cost of
system complexity. Even with two interferometers covering the desired band
there are many aspects that can then be individually optimised for both bands:
mirror mass, size and material; beam size and mode type; operating temperature;
suspension type and isolation
cutoff frequency; interferometer topology including layout,
mirror transmittances, laser power, and readout scheme. Note that many of these
are independent so the number of combinations to be considered is large.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth,keepaspectratio]{./pic/ET-C.eps}
\caption{Strain sensitivity of a potential third-generation GW
observatory as described in \cite{ET-C}. The low frequency interferometer
(ET-LF) features low optical power, cryogenic silicon optics and very tall suspensions,
while the high frequency counterpart (ET-HF) makes use of very high
optical power, silica optics at room temperature and standard suspensions.
The resulting xylophone sensitivity is compared to the single broad-band
detector described in \cite{ET-B}. (This xylophone sensitivity is sometimes also
referred to as \emph{ET-C}
and can be downloaded from \cite{ET}, sensitivity section).
} \label{fig:ET-C}
\end{center}
\end{figure}
One approach that has been proposed is to keep the interferometric sensing as simple
as possible in the low frequency band, i.e.~without the use of a QND measurement.
In that case the balance between shot noise and radiation pressure is simple and indeed,
as the point is to use low light power in the lowest frequency bands, only shot noise,
with a white spectrum, need be considered. This approach leads to a cascade of
interferometers with massive mirrors, cryogenic suspensions, aggressive vibration
isolation, and modest light power at the low end, while at the high end the mirrors may
be smaller, the suspensions need not be cryogenic, vibration isolation is simply
achieved, but the light power must be high (perhaps also with squeezing or QND
techniques). The bands would possibly be approximately logarithmic each covering
somewhere between an octave and a decade in frequency.
A different approach has been taken in work towards the design of the Einstein
Telescope \cite{ET-C}. In this case the idea was to split the whole detection band, from below
10\,Hz to above 1\,kHz in two. The two interferometers both employ mirrors
weighing about 200\,kg, with the interferometers basically
configured in the traditional dual recycled Fabry-Perot Michelson
proposed for the second-generation detectors although enhanced by
including systems to suppress quantum noise by 10\,dB in both
cases\footnote{To achieve a broadband quantum noise suppression by
e.g. a factor of 10\,dB using squeezed vacuum, the low frequency
detector requires two filter cavities due to its detuned RSE
configuration, whereas the high frequency detector requires only
one filter cavity, see section~\ref{sec:Filtercavities} for more
details.}, but while the high frequency (HF) detector uses fused
silica at room temperature,
the LF detector mirrors are single crystal silicon for good performance at low
temperature (10\,K). The low frequency (LF) detector has 50\,m tall suspensions while those
of the HF detector are a few meters tall; the interferometry for the LF system uses
just 18\,kW of circulating power, for low heating with a conventional Gaussian beam
shape and has a response intended to give maximum sensitivity from about
$7\sim30\,$Hz, while the HF interferometer has 3\,MW of circulating power and employs
Laguerre-Gaussian 3,3 mode beams to better average over thermal noise
in the mirror coatings. Another difference is that the LF interferometer uses an
optical wavelength of 1550\,nm at which silicon is transparent, unlike the 1064\,nm
suited to the silica mirrors of the HF interferometer. The strain sensitivities
of the LF and HF interferometer as well as the resulting xylophone
sensitivity are shown in Figure~\ref{fig:ET-C}. Although this system is
proposed as a straw-man and the detail is subject to change, it provides a
good illustration of the complexity and variety of choice of parameters for
even a two band detector.
\section{Quantum noise reduction}
\label{sec:QNR}
We have shown that the target sensitivity of a detector as envisaged in the
ET design study could potentially be achieved by an
interferometer in a {\it conventional} configuration with ultimately
good parameters that we can expect in the future. The xylophone option
will ease the use of some incompatible parameters. On the other hand,
we also know that there are number of configurations theoretically
developed to be used for a future detector. These {\it advanced}
configurations would let us achieve the same sensitivity increase with easier
parameters.
Most of what we shall show in the following sections has not been
experimentally demonstrated but is expected to be in time for the
construction of third-generation detectors. In this section, we
focus on the reduction of quantum noise. At first we present in
Sec.~\ref{sec:Filtercavities} how filter cavities can be used to
transform a conventional interferometer using squeezed vacuum into
either a \textit{frequency dependent squeezed-input
interferometer} or a \textit{variational-output interferometer} to
reduce the quantum. In Sec.~\ref{sec:speedmeter} we will
introduce the concept of the {\it speedmeter} which can help
reducing the quantum radiation-pressure noise whereas in
Sec.~\ref{sec:opticalbar}, we introduce a {\it optical-bar}
regime, which is another concept for reducing quantum noise. More detailed
explanation of these scheme is given in a separate article in this
issue by Danilishin {\it et al}~\cite{GRG-QND}. Further
information can be found in a direct comparison of several quantum
noise reduction schemes which has been performed as part of the ET
design study~\cite{et_mueller_ebhard09}. In
Sec.~\ref{sec:kentaro}, we will introduce some more advanced
configurations for the reduction of classical noise.
\subsection{Filter cavity}\label{sec:Filtercavities}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=1\textwidth]{RSE_with_Filtercavities_unlinked.eps}
\caption{{\it Left panel}: Tuned RSE interferometer using one FC
to create a frequency dependent squeezed input achieving a
broadband quantum noise reduction. {\it Middle panel}: Detuned RSE
interferometer using two FCs to create a frequency dependent
squeezed input achieving a broadband quantum noise reduction. {\it
Right panel}: RSE interferometer using a pair of filter cavities
thereby changing the readout configuration to variational output.
FC: filter cavity; PD: photo diode; PRM: power-recycling mirror;
SRM: signal-recycling mirror.}
\label{fig:RSE-with-FC}
\end{center}
\end{figure}
In general, filter cavities can be used to turn a conventional
interferometer either into a \textit{frequency dependent
squeezed-input interferometer} where an initial frequency
independent squeezed vacuum is made frequency dependent before it
is injected into the interferometer or into a
\textit{variational-output interferometer} where the readout
quadrature of output light is made frequency dependent to maximise
the SNR of the detector. The performance of these two
approaches is different. Where the frequency dependent
squeezed-input interferometer achieves maximally a broadband
reduction of the initial quantum noise by the squeezing strength,
the variational-output interferometer additionally allows a
complete cancelation of the radiation pressure noise so that only
the shot noise component of the quantum noise remains in the
signal output. Layouts of both schemes for the RSE topology are
shown in Figure~\ref{fig:RSE-with-FC}. For more information about
frequency dependent squeezed-input and variational-output the
reader is referred to \cite{Kimble02,Harms03,Buonanno04}. In both
cases the light is reflected at one or several filter cavities to
imprint a certain frequency dependent phase shift. The
configuration of the interferometer determines hereby the number
of filter cavities required as well as their parameters to achieve
a certain quantum noise reduction. In the case of a single band
antenna approach as presented in section~\ref{sec:stefan} a tuned
RSE configuration is used. As a consequence a direct injection of
phase quadrature squeezed vacuum achieves a broadband reduction of
the quantum noise only in the shot noise limited regime. To
achieve a full broadband reduction in the whole frequency range of
the detector, a single filter cavity is necessary to transform the
initial phase squeezed vacuum into amplitude squeezed
vacuum to reduce the quantum noise in the radiation pressure
dominated frequency range simultaneously to the shot noise
dominated regime. The same topology but using a detuned RSE
configuration, is used in the low frequency detector of the
multi-band antenna approach in section~\ref{sec:ken}. It requires
two filter cavities to achieve the same broadband improvement in
the whole frequency range.
Both filter cavities together
compensate the dispersion introduced by the detuned signal
recycling cavity and the resulting non-symmetric phase delay
applied to the two signal sidebands of sideband frequencies
$\pm\Omega$ with respect to the carrier at frequency $\omega_0$.
In both RSE configuration cases, tuned or detuned, and assuming no
losses the initial quantum noise curve can be shifted downwards at
all frequencies by the squeezing factor as stated in
Table~\ref{tab:summary} resulting in the new squeezed quantum
noise. The concept of the filter cavity has already been
demonstrated in a table top experiment in 2005 \cite{Chelkowski05}
shortly followed by the demonstration of a broadband improvement
of the quantum noise by squeezed vacuum in a detuned signal
recycled configuration \cite{Vahlbruch05}.
\subsection{Speedmeter}\label{sec:speedmeter}
A so-called {\it speedmeter} is another way to suppress radiation
pressure noise in broadband. There are several ways to realize a
speedmeter, one of which is the well-known Sagnac configuration as is
shown in the left panel of Figure~\ref{fig:Sag-OB}. The incident light
is split into two and probes the motion of the mirrors in each arm
with a certain time delay. The vacuum field from the signal extraction
port is split into two and imposes radiation pressure noise with the
time delay. Thus the $a_1(\Omega)$ component is partially cancelled
($\Omega$ is multiplied). One difference is that, unlike the
variational readout, this cancellation is made before the
photo-detection, so the quantum inefficiency, supposedly the largest
part of the optical losses, is not a big problem for the
speedmeter. Although the optical loss of the mirrors limits the
ability of the speedmeter, the estimated noise curves with reasonable
amount of losses for ET, the speedmeter turns out to be the best
candidate for the low frequency measurement~\cite{GRG-QND}.
\subsection{Optical bars}\label{sec:opticalbar}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=.5\textwidth]{Sagnac.eps}
\hfill
\includegraphics*[width=.4\textwidth]{OpticalBar.eps}
\caption{{\it Left panel}: Resonant Sagnac interferometer. {\it Right panel}: Optical bar.\label{fig:Sag-OB}}
\end{center}
\end{figure}
The sensitivity of ET at high frequencies is limited by shot
noise. The noise level can be reduced by injecting high power beam which
causes various problems related to heating optical components through
light absorption. Injecting squeezed vacuum from
the signal extraction port helps reducing shot noise, but it is
limited by the optical losses. Signal recycling tuned at high
frequencies can reduce shot noise but it increases the optical losses
and ruins the effect of squeezing. After all it is not easy to improve
the sensitivity at high frequencies by conventional means. One way to circumvent the problem
is to use a so-called {\it optical-bar} scheme (right panel,
Figure~\ref{fig:Sag-OB}). In this scheme, a long arm cavity is locked
by an optical spring with the resonant frequency higher than the
observation band; the masses should be small and the circulating power
should be high. The motion of the rigidly locked cavity is measured by
another interferometer that is much shorter than the locked arm
cavity.
The arm cavity is as long as a conventional detector so that the strain caused
by gravitational waves is the
same, while the sensing part is short so that the cavity pole is pushed to a higher frequency.
A detuned signal-recycled interferometer, which is the configuration
of the second-generation gravitational-wave detectors, has the feature
of the optical bar scheme. In fact, the optical spring frequency is
close to the lower end of the observation band, so the signal
enhancement at the spring frequency is the main purpose to employ the
detuned configuration. It is, however, pointed out that the
sensitivity can be improved by an auxiliary probe light to measure the
mirror motion imposed by gravitational waves at frequencies lower than
the optical-spring resonance; this is called a {\it local readout
scheme}~\cite{LocalReadout}.
\section{Reducing thermal effects with exotic beam shapes}\label{sec:beams}
As stated earlier overcoming the coating Brownian thermal noise
\cite{Gorodetsky08} imposes one of the biggest challenges for
third generation gravitational wave detectors. This section reviews another
interferometric concept (which thus has some impact on the topology and
configuration of future detectors) which can be used to reduce said noise.
The Brownian thermal noise in general is a consequence of the non-zero thermal
energy causing random motions of e.g. the mirrors' reflecting
surface \cite{Vinet09}. Currently there exist several techniques
to reduce the thermal noise content sensed by a gravitational wave
detector:
\begin{itemize}
\item {Cooling of mirror test masses:} This technique directly
reduces the thermal energy $k_BT $ stored in the mirror test mass substrates of
the interferometer.
A clever choice of
the substrate material
allows the simultaneous reduction of Brownian thermal noise and thermo-elastic noise \cite{Rowan05}.
\item Use of {high-reflective mirrors with less coating layers:}
Several research efforts are under way to reduce the coating Brownian noise
by essentially reducing the coating thickness needed for creating the required
high reflectivity. One such technique, the \emph{end-mirror cavity} is briefly
explained in section~\ref{sec:endcavity}. Another approach
is to use mirrors with a single waveguide coating layer to achieve a high reflectivity \cite{Gossler07}.
Both techniques are in the early prototype stages but are considered very promising.
\item {Change the mode shape of the laser:} A change of
the laser beams mode shape used in gravitational wave detectors
can lower several thermal noise contributions simultaneously. \cite{Vinet09}.
\end{itemize}
This section focuses on the method using alternative beam shapes.
While this technique is relatively new, its technical readiness is more advanced such that
is currently considered for potential upgrades of advanced detectors.
\subsection{Prospects of alternative beam shapes}
Various alternative beam shapes have been proposed to be used in
gravitational wave detectors to reduce the measured thermal noise
as well as to lower thermal effects in general. In the following
we will concentrate on two beam shapes;
the so-called Mesa beams \cite{DAmbrosio04} and higher
order Laguerre-Gauss (LG) modes \cite{Vinet05}. In these beam shapes
that laser power is distributed more widely compared to the
currently used TEM$_{00}$ mode. Hence, the fluctuations e.g.
induced by the Brownian motion of the mirrors reflective surface
are canceled significantly by averaging over a larger region of
the mirror with the cross section of the beam used.
At first glance a perfectly flat beam would appear to be the best candidate in terms
of thermal noise for an alternative beam shape, because of its
uniformly distributed intensity profile. But such a rectangular
shaped beam intensity will spread out very quickly while
propagating, due to diffraction. This renders flat beams
impractical to be used in the long interferometer arms of a
gravitational wave detector. To circumvent this problem the Mesa
beams have been introduced. Their intensity profile has much
smoother edges compared to the flat beam thereby reducing the beam
spread to an acceptable level. The phase fronts of these Mesa
beams require mirrors that have a \textit{Mexican Hat} profile.
The fabrication of these mirrors is more complex compared to the
currently used spherical mirrors but first test experiment with a
mesa beam based cavity has been performed at Caltech
\cite{agresti06b}.
However, the use of non-spherical mirrors is new and much more
work is required to reach the level of accuracy which can currently be
achieved in interferometers with spherical mirrors. Therefor the
use of higher order Laguerre Gauss modes has been proposed.
While LG modes can also feature a wide intensity distribution
their phase fronts are spherical and they can be used with spherical
mirrors..
An excellent overview about the expected thermal noise of all the
alternative beam shapes mentioned above is given by Vinet
\cite{Vinet09}.
We find that the
linear spectral density of the coating Brownian thermal noise of
the mesa beam is a factor of $\sim\!1.5$ lower compared to the
TEM$_{00}$ mode. The LG$_{33}$ mode and LG$_{55}$
perform even better with reduction factors of $\sim\!2.2$ and
$\sim\!2.3$ respectively. For the substrate thermal noise the
results are similar; both LG modes outperform the Mesa beam
\cite{Mours06}. Nevertheless, the calculation of the thermo-elastic
noise for the same 1\,ppm clipping loss modes only shows a
performance increase for the Mesa beam with a reduction factor of
$\sim\!1.8$ compared to the TEM$_{00}$ mode. The two LG modes
produce a higher thermoelastic noise which is larger by a factor
of $\sim\!1.7$ in case of the LG$_{33}$ mode and larger by a
factor of $\sim\!2.5$ in case of the LG$_{55}$ mode with respect
to the TEM$_{00}$ mode. In the current ET scenario the
thermo-elastic noise is far away from being the limiting thermal
noise, however, this relies on the parameters used and in general a careful trade-off in
terms of beam size and clipping loss has to be made for optimising the thermal noise
contribution using LG modes.
\subsection{Impact of alternative beam shapes on detector configuration}
LG modes are expected to be fully compatible with the noise requirements
of high-precision measurements done by gravitational wave detectors~\cite{Chelkowski09}.
Experimental verification of such theoretical studies are in
progress. Of particular interest is the impact of the mode degeneracy in
optical cavities which are resonant for a higher-order Gaussian mode.
For any alternative beam shape considered, the focusing telescopes
and all the input/output optics component should be adapted to the new
beam shape. Also, the compatibility between alternative beam shape and
squeezing techniques should be studied.
As already pointed out, the use of Mesa beam requires particular
non-spherical mirrors, and create matching losses between the
input-output beams and the cavity beam. An complete analysis of a
\textit{mesa beam interferometer } (i.e. with power and signal
recycling) has not been performed.
Higher-order helical Laguerre-Gaussian beams are
not compatible with the use of triangular cavities, largely used in
interferometric detectors as pre-mode cleaners, pre-stabilization
cavities, input and output mode-cleaner. The reason is the 180 degrees
phase shift for the field distribution in the plane of the cavity due
to three mirror reflections. Mode-cleaner cavities with an even number
of mirrors should be used, or in alternatively sinusoidal
Laguerre-Gauss modes.
\subsection{Generation of alternative beam shapes}
One of the challenges in using alternative beam shapes is their
generation at high power and with low amplitude- and frequency noise
as well as the required stability in beam shape and position. Typically we
expect to be able to use laser amplifiers and mode-cleaning
cavities similarly as for the TEM$_{00}$ beams. However, dedicated
research for created such light sources for gravitational wave
detectors has just begun.
One well-known method of converting a Gaussian beam into an
alternative beam shape is to use an optical cavity which is resonant only (or
dominantly) for the required beam shape.
Experiments using this method in the context of
Mesa beams have been performed in Stanford
\cite{beyesdorf2006} and Caltech \cite{tarallo2007}.
If the parameters of the Gaussian beam are
optimized to have the highest overlap with the Mesa beam, the power
coupled can theoretically be more than 90\%~\cite{DAmbrosio03}. The
matching can be further increased to shaping the input beam, for
example using deformable mirrors~\cite{avino2006}.
Higher-order Laguerre-Gauss modes have been widely studied for their
property to carry orbital angular momentum~\cite{allen92}, with
several applications in cold atom physics, quantum physics and quantum
communications~\cite{heckenberg,mair}. Because of this
applications, most of their production techniques were mostly focused
on doughnut-like mode. Among
these techniques there are diffractive optics, phase plates coupled
with astigmatic mode converters, computer generated holograms and
optical fibers coupled with long period gratings.
For example, LG$_{21}$ modes have been produced using two diffractive optical
elements (DOE), starting from a pure Gaussian beam, with a conversion
efficiency of the order of 60\%~\cite{kennedy2002}. An incoming
Gaussian mode diffracts on the micro-structures etched on the DOEs and
the Laguerre-Gauss mode results from the sum of the diffracted
waves. The limitation on the conversion efficiency and on the mode
purity comes by the discreteness of the micro-structures. This
technique is commonly used to shape beams for industrial applications,
but the goal in that case is only to shape the power profile of the
beam and not to control the wavefront.
Another common technique to generate LG modes is to use an astigmatic
mode converter~\cite{beijersbergen1993}. The principle of an astigmatic mode converter relies on the Gouy phase
shift introduced by a pair of cylindrical lenses separated by a
suitable distance, which is a function of the focal length.
This setup transforms, in principle without any losses, an Hermite-Gaussian mode in a Laguerre-Gaussian mode. The
Hermite-Gaussian mode can be obtained passing a Gaussian beam through
a phase plate, or forcing the laser cavity to emit directly the HG
using a spatial filter. The potential drawback of the astigmatic
converter is due to the high order of the Hermite-Gaussian mode
needed: for example, to create a LG$_{33}$, a HG$_{63}$ is needed. A
doughnut-like (LG$_{10}$) mode has been recently created using this
technique, with a purity higher than 99\%~\cite{chu2008}; in this
case the Hermite-Gauss was created inside the laser cavity, by use of
a phase plate.
Computer generated holograms, obtained by computing the
theoretical interference between a Laguerre-Gauss mode and a plane
wave, have been used mostly to produce doughnut-like
modes, but the production of multi-ringed modes have been also
demonstrated \cite{arlt1998}. It should be remarked that this
technique does not produce pure Laguerre-Gauss modes, but a
superposition of LG modes with different quantum numbers, and this can
limit the conversion efficiency.
All the methods of
producing LG modes for quantum optics application are promising but
past research rarely include the modes of interest for GW detection, not did it investigate the excess noise
of the new light source regarding amplitude, frequency and beam jitter.
In general, the purity of the Laguerre-Gauss mode can be
increased injecting the beam in a mode-cleaner Fabry-Perot cavity. If
the cavity is kept resonant on the desired Laguerre-Gauss mode, the
unwanted spatial impurities appears under the form of non-resonant
optical modes, and they are filtered by the the cavity, proportionally
to its finesse. Furthermore, because the very high power needed in
future gravitational wave interferometers, the compatibility with high
power of the mode transformer should be addressed.
\section{Further techniques to reduce classical noise}
\label{sec:kentaro}
In this section, we introduce two examples of other advanced configurations which
have been proposed for reducing classical noise. Both of them will work as an auxiliary system
that can be added to the basic interferometer. We can regard these techniques as alternatives to realize the ET sensitivity with easier
parameters or to upgrade the sensitivity.
\subsection{Suspension point interferometry}
The ET design study envisages to be built a detector deep underground in a hard-rock mountain so that both
seismic noise and gravity-gradient noise are one or two orders smaller than the
first or second-generation gravitational-wave detectors. The mirrors will be
suspended by multiple pendulums, each of which can be 10\,m tall.
Further improvement can be made with a so-called suspension-point
interferometer (SPI) that was introduced by Drever many years
ago~\cite{Drever_SPI}. While setting up an interferometer on the
suspension platform to reduce rms motion of mirrors in the main
interferometer is an idea to be used in second-generation detectors, a
true SPI is the one with an interferometer locking the upper test
masses (left panel, Figure~\ref{fig:SPI-ETMC}). Remaining noise that
appears as the differential motion of the two masses suspended from
the rigidly locked masses is mainly the common motion of the upper
stage coupled through the imbalance of the suspension systems.
An experimental demonstration has been made at a prototype
interferometer~\cite{Aso}. One option with the SPI is to use it as a
part of the xylophone. The lower-stage interferometer would be for the
low-frequency measurement with low seismic noise, and the higher-stage
interferometer would be for the high-frequency or middle-frequency
measurement with higher power.
Alternative to the long suspension will be magnetic levitation, which
is also introduced by Drever~\cite{Drever_Maglev}. Attaching magnets
to upper and lower test masses and controlling the distance of two
magnets in the vertical direction, we can trap the lower test mass,
for the horizontal directions, in a very shallow potential similar to that of
an extremely long pendulum. The magnetic levitation might be also
a solution to reduce suspension thermal noise. It has been proposed to
use electrostatic force instead of the magnetic force so as to avoid
possible thermal noise in the magnetic system~\cite{ESD}.
Combination of the SPI and the magnetic levitation will let us tune up
the balance of the suspension systems so that the common-mode
rejection rate of the SPI can be much closer to unity. In fact, as the
purpose of the magnetic force here is to balance the suspension
systems, we can just add small magnets on test masses suspended by a
conventional pendulum.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=.35\textwidth]{SPI.eps}
\hfill
\includegraphics*[width=.6\textwidth]{ETMC.eps}
\caption{{\it Left panel}: The suspension-point interferometer cancels out seismic noise by a factor depending on the imbalance of the suspensions. {\it Right panel}: The end-mirror cavity (and the front-mirror cavity) realizes the mechanical separation of coating layers and also let us control and cancel out a big fraction of coating thermal noise.\label{fig:SPI-ETMC}}
\end{center}
\end{figure}
\subsection{End-mirror cavity}\label{sec:endcavity}
One of the limiting noise sources for a second-generation detector is Brownian
motion of the coatings.
Coating Brownian thermal noise strongly depends
on the numbers of coating layers used to build the reflective
coating layer of the mirror. The higher the reflectivity of a
mirror, the higher the number of coating layers needed and the
higher the coating Brownian thermal noise.
Reduction of coating thermal noise by mechanical separation of the
first few layers and the rest was proposed by
Khalili~\cite{Khalili}. A conventional mirror is replaced by an
anti-resonant cavity that has a few coatings on the first mirror and
more coatings on the second mirror (right panel,
Figure~\ref{fig:SPI-ETMC}). The cavity is locked anti-resonant so that
temperature fluctuation in the substrate of the first mirror or
coating thermal noise of the second mirror does not matter as far as
the reflectivity of the first mirror is reasonably high.
In fact, rigid control of the end-mirror cavity allows us to further reduce the reflectivity of
the first mirror~\cite{SomiyaETMC}. Total fluctuation of the
end-mirror cavity is measured by an auxiliary beam and is subtracted
from the output of the main beam in a proper way. Shot noise of the
auxiliary beam will be imposed instead, so the power of the auxiliary
beam should be high. While radiation pressure noise in the end-mirror
cavity is not a problem, radiation pressure noise of the auxiliary
beam in the main cavity cannot be suppressed by the control. As the
number of coating layers could be zero, coating thermal noise would
be totally replaced by quantum noise of the auxiliary beam. We could
use one of those quantum non-demolition techniques to overcome the
quantum limit and increase the power infinitely so that finally
coating thermal noise could be removed without excess control noise.
It should be noted, however, that a rigid control of such a short, essentially
marginally stable cavity is known to be very difficult and in all
likelihood the noise of the alignment control system will be a
limit to the performance of such a system.
\section{Summary and Outlook} \label{sec:summary}
Third-generation gravitational wave detectors are envisaged to be observatories providing
a continuous stream of astrophysical meaningful data. In order to achieve this goal these detectors
need to be designed and planned to be robust machines with a sensitivity ten times better than
the advanced detectors currently under construction. In this article we have discussed how these
goals affect the design of the core instrument, in other words the optical layout of the laser interferometer(s)
of the detector. From the nature of the gravitational radiation follows that long-baseline,
L-shaped interferometers represent the ideal instruments. In order to maximise the data quality
of a single detector or of a detector integrated in a network, we found that the use of co-located
interferometers are essential. A triangular geometry of three co-located interferometers is
currently under investigation within the Einstein Telescope design study. Further constraints of the
interferometer topology and configuration are not obvious from general principles but are
expected to follow from the technical detailed of optical schemes for reducing the quantum noise and
the thermal noise in the detectors.
We have reviewed the preliminary design work on the Einstein Telescope and showed that the envisaged
sensitivity can possibly be reached
by an moderately optimistic scaling of current technologies. In particular, we have reviewed the
noise contributions in a potential ET detector and have shown that a design featuring several
small-band interferometers might be superior over a single broadband design.
In addition, several new, advanced techniques can be used to provide a margin for further sensitivity
improvements. We have presented several such techniques, concentrating mainly on those whose
technical readiness is sufficient for a potential integration in future large scale projects over the next
decade.
Several research programs within the world-wide gravitational wave collaborations are dedicated
to the experimental realisation and testing of advanced techniques for third-generation detectors.
We thus expect that we will be able to significantly beat the sensitivity of the advanced detectors
which are currently under construction. This is an exciting outlook for the time when said advanced detectors
have made the first detections of gravitational waves and increasing the signal to noise ratio would enable
us to tap the full potential of the new field of gravitational wave astronomy.
\section{Acknowledgements}
This work has been supported by the Science and Technology
Facilities Council (STFC), the European Gravitational Observatory (EGO),
the United States National
Science Foundation (NSF) and the Seventh Framework Programme (Grant
Agreement 211743) of the European Commission. K.~Somiya is supported
by Japan Society for the Promotion of Science (JSPS).
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Der Rotermichbach (früher: Kreuzbach) ist ein etwa zwei Kilometer langer und linker Zufluss des Westerbachs im Landkreis Aschaffenburg im bayerischen Spessart.
Geographie
Verlauf
Der Rotermichbach entspringt auf einer Höhe von etwa dem Zieglersbrunnen an der Heiligkreuzkapelle, am Fuße des Habersberges (428 m), in der Nähe der Kreuzziegelhütte, zwischen Oberwestern und Großkahl.
Er fließt in südwestliche Richtung durch den Rotermichgrund und mündet schließlich an der Klotzenmühle südlich von Unterwestern auf einer Höhe von ungefähr von links in den aus dem Norden heranziehenden Westerbach.
Der etwa 2 km lange Lauf des Rotermichbachs endet ungefähr 68 Höhenmeter unterhalb seiner Quelle, er hat somit ein mittleres Sohlgefälle von circa 34 ‰.
Einzugsgebiet
Das etwa 1,4 km² große Einzugsgebiet des Rotermichbachs liegt im Spessart und wird durch ihn über den Westerbach, die Kahl, dem Main und den Rhein zur Nordsee entwässert.
Es grenzt
im Osten und Süden an das Einzugsgebiet der Kahl
im Westen an das des Westerbachs
und im Norden an das Herzbachs, der in den Westerbach mündet.
Flusssystem Kahl
Liste der Fließgewässer im Flusssystem Kahl
Siehe auch
Liste der Flüsse im Spessart
Weblinks
Einzelnachweise
Fließgewässer im Landkreis Aschaffenburg
Geographie (Westerngrund)
Fluss im Spessart
Geographie (Kleinkahl) | {
"redpajama_set_name": "RedPajamaWikipedia"
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export PATH=/opt/gcc-4.8.4/bin:/opt/gcc_infrastructure/bin:$PATH
export LD_LIBRARY_PATH=/opt/gcc_infrastructure/lib:/opt/gcc-4.8.4/lib64:/opt/gcc-4.8.4/lib:$LD_LIBRARY_PATH
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,831 |
In Texas, as in other states, ownership of the surface of land is often separated from ownership of substances that lie beneath it. For years, it has been common for sellers to reserve all or part of the "oil, gas and other minerals" that lie under land they are selling, thus retaining a mineral estate that is distinct from the surface estate being sold. Countless deeds in Texas contain similar reservation language. Although the concept of reserving the ownership of "oil, gas and other minerals" seems straightforward, from a mining perspective in Texas it unfortunately is not.
To put it simply, Texas courts have had a very difficult time establishing the meaning of the words "other minerals". The difficulty has stemmed from attempts by courts to balance the positions of the owner of the surface, and the owner of a subterranean asset who may be required to destroy the surface to enjoy it. The Texas Supreme Court has issued a number of opinions over the past forty years regarding the meaning of "other minerals" but has failed to arrive at a consistent definitional framework that produces predictable results. One Justice has described the law in this area as a "quagmirish fog".
Although it may not be possible to say exactly what the words "other minerals" mean in Texas, it is possible to say that certain materials that are extracted by surface mining methods have consistently been held not to be "other minerals" and thus are not reserved by the phrase "oil, gas and other minerals". Limestone, caliche, and sand and gravel are within this category. The Supreme Court repeatedly has held that these substances belong to the surface owner, unless it is clear from the wording of the reservation that a different result was intended by the original parties.
The Supreme Court has also ruled, and reaffirmed in later cases, that coal and iron ore which are located near the surface are likewise not "other minerals" and are owned by the surface owner unless it is clear from the wording of the reservation that a different result was originally intended. It should be mentioned that the decisions regarding coal and iron ore have been criticized by some legal scholars but it seems fairly doubtful that the Court will reverse its course in the future and treat surface coal and iron ore differently than it has treated stone, sand and gravel.
Uranium, however, is treated differently. The Supreme Court has ruled that uranium falls within the "ordinary and natural" meaning of the word "minerals" and is retained by the mineral owner in all reservations of "oil, gas and other minerals" occurring after June, 1983. Unfortunately, while the Court resolved the status of uranium, its reliance on the meaning of the word "minerals" provides little guidance with respect to a number of other materials. The word has an amorphous definition and it is impossible to predict whether the Court will rule that a particular material falls within the meaning of "minerals" or not.
The difficulty with the Court's approach is seen in a 2001 Court of Appeals holding that granite is a "mineral" and supporting that finding with evidence that when the granite in question was severed from the surface, in the 1890s, it had a special economic value due to the construction of the State Capitol; indicating that if it had not had such value, it may not be a "mineral". With deference to the Court, when reading its opinion one is reminded of Humpty Dumpty as he spoke to Alice in Through the Looking Glass: "When I use a word," he said, "it means just what I choose it to mean—neither more nor less."
Because the Texas Supreme Court has failed to establish a predictable framework to determine the types of materials that will, or will not, be included within the reservation of "oil, gas and other minerals", the owners of surface and mineral estates in many cases can only guess at the ownership of materials that have not yet been the subject of a Supreme Court ruling. With that fact in mind, parties who are contemplating the sale or lease of mining lands in the face of such a reservation should proceed with the utmost caution, obtaining the advice of a knowledgeable professional if there is any question regarding their legal positions. | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/2544042\/could-collatz-rules-3xb-hold-every-possible-song-melody-in-existance","text":"# Could Collatz rules $3x+b$ hold every possible song melody in existance?\n\nSince Modulo seems to have a significant role in the Collatz Conjecture for one reason or another, I wondered what would happen if the Conjecture was put to music (I was inspired by this interesting YouTube video). Shuffling about the internet, I came across this web tool called Piano Pencil Code Gym that would play any sequence of letters.\n\nI wrote some not very impressive code (and for 3x+5 defaulted to using an Excel spreadsheet with the =Mod function) to translate the trajectories into mod 7 numbers and then lazily used Word's replace feature to convert those into letters. I then copied and pasted the whole thing into the web tool and listened to the result. After playing around with some 3x+1 and 3x+5 trajectories, the immediate thought came about:\n\nIs it possible to find any music melody somewhere in any given 3x+b rule, where b is an odd integer?\n\nThe tunes can be repetitive (For $3x+1$, it seems to alternate between ...D B A D G G G B A whatever... and ...E F E F E F C C E F something...), but different rules such as 3x+5 may mix it up a bit. Since the scale is really basic, I suspected the possibility of every song melody appearing by coincidence was possible. I have no idea where to start with trying to answer this question other than playing some different Collatz Rules and hearing what happens.\n\nHere are some sequences I already generated. Feel free to copy and paste them into the web tool.\n\nJust as a heads up: I assigned A=1 and G=0. If you re-assign the mod numbers so C=1 for example, the music sounds different.\n\n$3x+1$ trajectories of 1-39 (odd seeds only)\n\np=new Piano\np.lt 90\n\n#1#\np.play \"D3 B2 A2\"\n\n#3#\np.play \"C C E B A D B A\"\n\n#5#\np.play \"E B A D B A\"\n\n#7#\np.play \"G A D F C C E F E F C E B A D B A\"\n\n#9#\np.play \"B G G G A D F C C E F E F C E B A D B A\"\n\n#11#\np.play \"D F C C E F E F C E B A D B A\"\n\n#13#\np.play \"F E F C E B A D B A\"\n\n#15#\np.play \"A D B G G A D F C E F C E B A D B A\"\n\n#17#\np.play \"C C E F E F C E B A D B A\"\n\n#19#\np.play \"E B A D B A D F C C E F E F C E B A D B A\"\n\n#21#\np.play \"G A D B A D B A\"\n\n#23#\np.play \"B G G A D F C E F C E B A D B A\"\n\n#25#\np.play \"D F C E B A D B A D F C C E F E F C E B A D2 B2 A3\"\n\n#27#\n\np.play \"F E F E F C C E B A D B G G A D B G G G A D F C E B A D B G G G A D F C C E B A D F C E F E F E F E F C C E B A D F C C E B A D B G G A D B A D B G G A D F C E F C C E F E F C C E F C E B A D B G G A D F C E F C E B A D B A\"\n\n#29#\n\np.play \"A D B A D F C C E F E F C E B A D B A\"\n\n#31#\n\np.play \"C C E B A D B G G A D B G G G A D F C E B A D B G G G A D F C C E B A D F C E F E F E F E F C C E B A D F C C E B A D B G G A D B A D B G G A D F C E F C C E F E F C C E F C E B A D B G G A D F C E F C E B A D B A\"\n\n#33#\n\np.play \"E B A D F C E B A D B A D F C C E F E F C E B A D B A\"\n\n#35#\n\np.play \"G A D F C E F C E B A D B A\"\n\n#37#\n\np.play \"B G G G G G A D F C C E F E F C E B A D B A\"\n\n#39#\n\np.play \"D F C C E B A D F C C E F C E B A D B A D F C C E F E F C E B A D2 B2 A3\"\n\n\n$230631$, A number with a really long Collatz trajectory. I found in the 3x+1 Completeness and Gamma Records Table from Eric Roosendaal's page. (http:\/\/www.ericr.nl\/wondrous\/comprecs.html)\n\nIt is a proud 442 steps long:\n\np.play \"B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 F1\/2 C1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 F1\/2 E1\/2 F1\/2 C1\/2 C1\/2 E1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 G1\/2 G1\/2 A1\/2 D1\/2 F1\/2 C1\/2 E1\/2 F1\/2 C1\/2 E1\/2 B1\/2 A1\/2 D1\/2 B1\/2 A1\/2\"\n\n\nWarning: This trajectory is 4 times longer than 27's. If you delete all of the 1\/2's, you are in for 5 whole minutes of Collatz Music.\n\nHere are some $3x+5$ trajectories (odd seeds only):\n\n#1#\n\np.play \"A A D B A\"\n\n#3#\n\np.play \"C G G E F B A D C E F C G G E F C E\"\n\n#5#\n\np.play \"E F C E\"\n\n#7#\n\np.play \"G E F B A D C E F C G G E F C E\"\n\n#9#\n\np.play \"B D B A D B A\"\n\n#some random numbers...#\n\n#75#\n\np.play \"E F C G G E F B A D B A D C E F C E F C E\"\n\n#12347#\n\np.play \"F B A A D C E F C G G E F C E F C E F C E F C G G E F B A D B A D B D B D B D B D B D B D B A D C E F C G G G E F C E F C G G G G G G E F C G G G E F C G G E F B A A D C E F C E F B A D C E F C E F B A D C E F C G G E F C E\"\n\n\nIt contains all binary strings as prefixes, (taking $1$ as $(3n + 1)\/2$ and $0$ as $n\/2$) I suppose, since if you take $n = m + 2 ^ k$ then $n$ has the exactly same prefix as $m$ and it repeats at least every $2 ^ k$ and the $n$ sequence has to be different from $m$ at some point.\n[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1], [0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1], [0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1]\nIf you see that every $k$ bit prefix repeat only and exactly after $2 ^ k$","date":"2021-09-22 12:28:40","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\": 0.36232200264930725, \"perplexity\": 425.2732980273716}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, 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actions :create
attribute :ruby_string, :kind_of => String, :name_attribute => true
attribute :user, :kind_of => String
attribute :patch, :kind_of => String
def initialize(*args)
super
@action = :create
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| {
"redpajama_set_name": "RedPajamaGithub"
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2008/02: "Music Lover Shows Legal Twist," Law Institute Journal (Law Institute of Victoria, Melbourne, Australia), January/February 2008, at 87:
Announcing the winner of LIJ's holiday song title competition (competition was set forth in its December 2007 issue), with the prize being "a copy of US attorney Lawrence Savell's fourth album The Lawtunes: Live at Blackacre." In its January/February 2008 issue, LIJ ran a story announcing the winner, showing him, a managing partner, proudly holding the CD, noting that the winner had "also won the Lawtunes prize in 2003" (an earlier CD in a prior LIJ competition).
2008/Winter: "David Winder Teacher, Director, Friend," Poly Prep Magazine, Winter 2008, at 12-13, online at http://www.polyprep.org/ftpimages/71/download/download_group1039_id303596.pdf:
Reminiscence quoted in tribute to beloved high school teacher:
I have nothing but very positive memories of David, be they of him as a dedicated and motivating teacher or as an encouraging director. But what I remember most was his response to a story I wrote for him for Latin class in 1971, shortly after my father passed away. The story involved a character who suffered just such a loss (perhaps a bit of self-therapy by a troubled 13-yearold). For that submission, David gave me two grades: one for Latin usage (a fully justified C+) and one for content. The latter grade he assigned to that otherwise pedestrian narrative was an A+, which communicated to me, in the profound poetry of his unexpressed explanation, his tremendous compassion and understanding.
— Lawrence Savell '75
2008/04/10: "Speakers," Cardozo Law School, Squadron Program in Law, Media and Society, published April 10, 2008, http://www.cardozo.yu.edu/MemberContentDisplay.aspx?ccmd=ContentDisplay&ucmd=UserDisplay&userid=10708&contentid=6000:
"In addition to conferences and symposia, the Squadron Program organizes monthly lunchtime talks with some of the leading practitioners of the media law community in New York. Guests speak on a wide range of issues from libel to artist rights. Practitioners also offer practical advice for Cardozo's aspiring media and communications lawyers. Speakers have included:
"Lawrence Savell- Mr. Savell is a media lawyer from Chadbourne & Parke LLP. His discussion was entitled The Media Lawyer in the Bullpen: Developing a Media Law Practice as an Addition to Another Primary Practice Area."
2008/07/00: "Teaching Bibliography," New England School of Law Library Website, http://www.nesl.edu/library/TeachingBib/Search_.cfm:
Included in "Resources and Services for New England School of Law Faculty: Teaching Bibliography" is:
Savell, Lawrence, Dear Professor Rosenstein: a short story, 83 Michigan Bar Journal, 5 50(3) (2004)
2008/07/00: "Juris Rocker," Cornell Alumni News, July/August 2008, at 85:
"Lawrence Savell '79
"The guitar came first—an acoustic six-string, circa 1976, that Lawrence Savell bought with a scholarship check that was supposed to go toward his Cornell tuition. The law career that inspired his music came a few years later. Now, Savell is a lawyer by day and a musician by night. He's often up late, writing and recording original songs that satirize his job as a litigator for Chadbourne & Parke in New York City. Most of his songs are inspired by everyday tasks—sifting through piles to find the one document necessary to win a case, ordering take-out at his desk, reading legal citation manuals. 'If I stopped coming to work,' he says, 'I'd run out of material.'
"Savell is still using the guitar he bought in Ithaca; his only formal training was a couple of group lessons in Willard Straight Hall. So far, he's sold more than 1,000 copies of his albums, which feature tunes that he describes as 'simple' and 'hummable.' (They include a love song entitled 'She's an Electronic Discovery' and an ode to Perry Mason's secretary, Della Street.) He makes most of his sales online, through his website (lawrencesavell.com) or amazon.com, or at his performances at law firm parties and legal conferences. But for Savell, selling albums isn't the point. He says lawyers often get a bad rap, and he wants to dispel the myth that people in his field are consumed by their careers. Says Savell: 'We're not as stuffy and as distant as we might be perceived.'"
2008/09/00: "Justice, Distractions and the Occasional Anecdote," Hearsay (Pace University School of Law student newspaper, "Adver[seriously]" column), September 2008, at 5, www.pace.edu/lawschool/files/hearsay/issues/Hearsay_2008_09.pdf:
"For a nice reprieve from studies and stress, it's important to have healthy distractions. I have found one such distraction in the form of a hot new band. That's right folks, you heard it here first. Their name: The LawTunes. The Album: Live at Blackacre. This album features such chart toppers as, '(She's an) Electronic Discovery,' 'Lawyers' Blood is Typo,' and 'Little Bluebook.' This beauty can be sampled and, should you feel so inspired, purchased at www.lawhumor.com. Enjoy!"
2008/10/01: "Lawyer2Lawyer - The Song!!," Robert Ambrogi's LawSites (blog), October 1, 2008, http://www.legaline.com/2008/10/lawyer2lawyer-song.html:
"Today we recorded the 150th episode of our weekly legal-affairs podcast Lawyer2Lawyer. The show is not yet ready to post, but as we were recording, our favorite lawyer/recording artist Larry Savell of LawTunes surprised and honored us with our very own Lawyer2Lawyer song that he composed and recorded to mark our third anniversary. It is very good, if I may say so myself, and VERY funny. He has posted it at this Lawyer2Lawyer song page in a choice of high- and low-fidelity MP3 and Windows Media files."
2008/10/01: "LawTunes' Larry Savell Serenades Lawyer2Lawyer On Our Third Anniversary," May It Please The Court (blog), October 1, 2008, http://www.mayitpleasethecourt.com/journal.asp?blogid=1905:
"This week marks our 150th podcast on Lawyer2Lawyer, and in honor of our third anniversary, Bob Ambrogi and I had Larry Savell of LawTunes back on the show, and he did us the great honor of composing and singing a song for the show. You can click on the link below to listen to the song, and here are the lyrics:
"Thanks to Larry Savell for permission to post the song and the links. You can visit Larry at LawTunes and buy his other music, including his latest release, Live at Blackacre."
2008/10/02: "Music & Law: The Final Frontier," Advocate's Studio (blog), October 2, 2008, http://advocatesstudio.wordpress.com/2008/10/02/music-law-the-final-frontier/:
"If you could actually hear me right now, you would be hearing 'oh my god, oh my god, oh my god,…' etc. This has the be the ultimate marriage of law and music. I must have been living under some kind of boulder, but apparently, in this age of extreme specialization, there actually is something known as 'The LawTunes' - legal humor lawyer gift music CDs by and for lawyers. With the tagline 'musical scales of justice', this unique musical group (or should I say 'firm'), has produced such gems as '(She's an) Electronic Discovery', 'Little Bluebook', 'Another Billable Christmas' and 'Rainmaker Reindeer.' You will be treated to clips of these songs when you enter the site. I still can't believe I heard the word 'certiorari' in a song and probably never will again. There are four CDs available, which can be had for a bargain price when you buy all four. The latest is the pictured album 'Live At Blackacre.' The band members are listed as follows, but I think these are stage names:
Micah ("Courthouse Steps") Jagger - Plaintiff's Counsel, Oral Argument, Strutting Around, Shirt Removal, Lead Vocals
Nicholas ("Prosser on") Tortolli - In-House Counsel, Bass, Alto Fax, Photocopier, Invoice Review, Vocals
Maxwell ("Max") Billings - External Defense Counsel, Drafting, Electric Guitar, Electronic Discovery, Wardrobe, Vocals
Smash ("Bones") Drummond - Prosecutor's Office, Drums/Percussion and Other Auditory Damage, Cellphone, Vocals
Malcolm ("Mal") Practisse - Solo Practitioner, Keyboards, Shredder, Excuses, Professional Liability Insurance, Vocals
"My hat is off to Larry Savell and The Lawtunes for embarking on such an artistically intellectual and uniquely melodic course! This is must-see web-surfing at its finest! My work here is done."
[Follow-up note: http://twitter.com/advocatesstudio/statuses/942699577] "advocatesstudio: I blogged about LawTunes tonight http://tinyurl.com/3eprnt and w/in 15 min. Larry Savell emails me to thank me. The WWW is a wondrous place. October 2, 2008"
2008/10/20: "'Tis the season[:] The right gifts for referrals can bring holiday cheer," LawyersUSA, October 20, 2008, http://www.lawyersusaonline.com/index.cfm/archive/view/id/432066:
"From a marketing standpoint, it's crucial for lawyers to think about what gifts to give to clients and referral sources. . . . [L]awyers need to have a little fun and be creative. Here are a few suggestions:
"Make it humorous.
At LawTunes (www.lawtunes.com), you can buy an album of hit pop songs rewritten with lawyer terms, featuring such hits as "She's an Electronic Discovery" and "Orderin' In."
2008/11/24: "Loose Ends, 11-24-08," Bitter Lawyer, November 24, 2008, http://www.bitterlawyer.com/index.php/columns_detail_comment/loose_ends_11_24_08/?cat_id=6:
"Deck the halls—the holiday season is officially here. And since lawyers don't know how to emotionally connect with anything unless it directly speaks to their profession, LawTunes.com is spreading Christmas cheer to the stunted law masses with a new law-related holiday album that's sure to please the gentile esquire in all of us. Since you want everything to remind you about your job (afterall, it's the only thing that defines you as a person), pump up the volume and put 'It's Gonna Be A Billable Christmas' on repeat. No word on a Hanukah CD counterpart, which is a bummer because 'Maccabean Rainmaker' and 'So Many HLS Loans, Not Enough Gelt' would be h-o-t tracks. [PRWeb.com] "
2008/12/00: "The Blawg 100," ABA Journal, December 2008, in print and at http://www.abajournal.com/magazine/blawg_100_2008:
Cover story on the top 100 legal blogs notes in its write-up of the podcast "Lawyer2Lawyer" that "the catchy Lawyer2Lawyer theme brags that listening is 'the most fun you can have while wearing a vest.'" Theme composed and performed by Lawrence Savell.
See follow-up in January 2009
2008/12/12: "Holiday Shopping Guide for Gift-Giving Lawyers," Law.com/The Legal Intelligencer, December 12, 2008, http://www.law.com/jsp/law/careercenter/lawArticleCareerCenter.jsp?id=1202426677720:
"If someone asks you for your own list, you can always ask for a copy of the new CD, 'The Lawtunes: Live at Blackacre,' described as 'lawyer-created law-related legal humor rock-and-roll.' Songs include 'It's Gonna Be a Billable Christmas,' 'Hey, Santa, I Appeal,' 'I Got A Footnote in My Stocking,' and 'Rainmaker Reindeer.' | {
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Q: The folders I deleted in android studio come back I don't know what file they are but i think they are related to android studio they come back when i delete accidentally i installed the .android folder directly in the directory i gave it another way from sdk manager but how do i solve this problem? I try to what ı know but i cant
*
*I try to change system environments path
*Restart my pc
UPDATE
The problem wasn't caused by android studio
UPDATE
Folder Content . All folders like that
A: It possible looks like the .git folder of a git repository or some kind other cache / object storage.
The easiest attempt to resolve this and to check if it's android studio causing this is to uninstall Android Studio. Remove all files. Remove the .android folder in your home directory. Remove (if present) the android sdk from %localappdata%\Android and the .AndroidStudio* folder from your user directory.
Then reinstall Android Studio
You can and should of course take backups from the settings if you want to.
| {
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In this post, we have shared some of the most useful Chrome tips which you will make your life a lot easier.
Google Chrome is the most widely used browser around the world no doubts. You and I daily spent hours doing stuff, doing our office work, college work and so on.
There are a couple of things which can boost your working performance with Chrome. It's those small things that can make the process a little fast and all those are inbuilt into the chrome which most of the users don't realize it's there.
Do let us know if you find at least one thing out of this article that is actually useful for your day to day to life or work or you consider to be cool.
Why not try it out to scroll through this article?
Instead, Press Ctrl+P on your keyboard or Right click mouse button and print.
Most of you might already be knowing this, but it's always good to get reminded of some of the things to start using them more often.
If you are trying to log in on some website and you are continuously getting some issues, the web page is not displaying properly, you just want to clear everything cache, cookies.
Incognito mode is quite useful and you should use it more often to be on the safer side.
Use it as long as you want and just close it, no need to worry about log out.
Sometimes you will come across really cool chrome extension but they aren't available on chrome web store which you need to install manually by downloading and loading from local storage.
You will see three more options after that Load Extension, Pack Extension and Update.
It's like a VPN which is always inbuilt into your chrome browser but can't hide your identity.
Once you are in developer tools it can look complicated as it's not meant for the normal user to do something.
Under Geo-location, you can select a different available location or manually enter longitude & latitude to fake your location for web pages.
If you want to check go to Google Maps and change your location and refresh the page to see your new location every time.
No need to copy text, new tab and paste for searching.
You might have too many passwords saved in your chrome browser locally or synced to google account.
Sometimes you forget password but it is saved in your browser but you can't see your password directly.
What you can do is visit this URL in your browser chrome://settings/passwords or go to settings search for manage passwords and you can see each and every password you have saved in your browser you can even export all passwords.
If you are away from your desktop or laptop but want to check all your passwords then you can visit passwords.google.com and login with the same account which is used to login in google chrome browser.
It might look scary as all your passwords are connected with google account, that's why always setup two factor authentication and other backup options for your Google account.
We suggest you to use apps like Last Pass or KeePass a password manager.
Click the star icon and bookmark it on the bookmark bar to quickly access your inbuilt notepad, make sure to rename bookmark to notepad.
It will start playing automatically with all options for control.
You can use this method to cast your local media files on chrome cast through cast this tab.
Google Cache comes very handy if a web page you are trying is open is offline and Chrome makes it really easy for you to access the cached version of any page. While you are on a page, click the address bar, type cache: before the URL and hit Enter. You'll be taken straight to the Google Cache if there's one.
When your connection is slow and webpage loading is terrible on the desktop, then try to access the mobile site of the website.
Usually, websites don't use too much data on mobile pages but there is one problem some website will redirect you back to there desktop website if you try to open a mobile website.
What you can do is open Developer tools (Shortcut F12) and select the mobile device from the dropdown list to pretend like you are using a mobile device.
Alternatively, Press Ctrl+H to see all your recently visited websites.
When chrome starts to lag a lot and not working responsively then open up the Chrome task manager settings → more tools → task manager (shortcut Shift + Esc) and check which web page or extension is using most of the resources.
You can use it to search directly on too many websites, for example, shopping website, food ordering website, you will get a hint from chrome if it's possible on current website URL, you can do this on Youtube, Google, Flipkart, Snapdeal, Facebook, Twitter and many more.
When you bookmark any website on your bookmark bar it shows an icon then title.
But why keep the title when there is an icon.
Always organize bookmarks in the folder but keeping a few URLs in the bookmarks bar with the icon is quite a good way.
For example, I have Play Console, Youtube, Reddit, Quora as an icon in my bookmarks bar.
Take your cursor to any other inactive tab and while holding Ctrl button click and if you press again you will go back to the tab on which you were previously.
Useful when you have too many tabs and you want to check a specific tab and then go back to the current tab, just hold Ctrl click on any other tab and click on the same tab again.
Clicking on the star icon at the end of the URL bar instantly bookmarks and gives you the option to select the folder.
But if you have tons of folder and subfolder it's not fast enough.
Instead, whenever you need to bookmark any specific page or website click on the URL bar it will select the whole URL then drag it to the folder or subfolder in your bookmarks bar and release.
It's quite fast for people with too many folders.
You will be using the same chrome but with different extensions and bookmark that will be synced to your account. you can even create separate launch shortcut to directly launch your personal chrome user account without switching between accounts.
believe me, it's as simple as it looks.
or bored with work, open this URL in your new tab and play the inbuilt t-rex game which you can only play while your internet connection isn't working.
You can activate extensions on specific keyboard combination and if extension allows multiple things that can be done using the extension you can set and manage those keyboard shortcuts of all extensions from one screen.
Just open extension page chrome://extensions/ then open side menu by pressing hamburger menu and select keyboard shortcuts or directly open chrome://extensions/shortcuts.
Keep in mind extension activate shortcut will act if an extension is clicked so keep in mind all extension don't work on just one click.
When you want to just focus on work and don't want any disturbance when you are doing your work on any website you can press F11 and keep yourself focused on single website opened.
In general use case, this is helpful when you want to keep yourself focused on a single tab, reading long articles, playing browser-based games, using web apps and learning from any website.
There is no need to always remove any extension when you just don't want to use it for short period.
You can just disable the extension from chrome://extensions/ and re-enable when you want to use it.
You can also disable the extra extension for performance purpose.
Often times you start browsing from one article to another and another.
You visit multiple pages on the same tab but if you need to quickly go back multiple pages back you need press back button too many times.
Next time when you need to go back multiple pages back in the same tab just hold the right button of the mouse on the back arrow near reload icon.
And you will see a list of pages just like below image.
Chrome can act as a basic file explorer for your computer. On a Windows machine, try typing C:\ into the Omnibox to start navigating around.
Chrome has its own basic tool for sharing your screen with other computers: All you need is Google's official Chrome Remote Desktop extension.
It allows you to establish secure connections for remote support or for your own remote access to apps and files.
Every single day you need to open a few fixed website when you start working.
Bookmark all those pages under one folder, just right click on the folder and select open all.
And all the website's are open, Isn't cool?
Open the website, Goto Three dots at the top right corner>> more tools>>Create a shortcut and keep the mark checked below it which says open as a window.
Shortcut feature is most useful for websites which are kind of web apps.
While there might be many more things you can do to save more time, work fast and increase productivity while working on Google Chrome, these were the most common that will be helping everyone.
Which one is the most useful one for your day to day usage? or If you have more suggestions which everyone can use do let us know in the comments section, we will love to hear from you. | {
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{"url":"https:\/\/discourse.pymc.io\/t\/modelling-a-timeseries-of-cumulative-maximum\/7829","text":"# Modelling a timeseries of cumulative maximum\n\nI am trying to model a process where I only get to see the historical maximums at any point in time.\n\nA simple generative setup in numpy for the data is as follows:\n\nimport numpy as np\nimport math\n\n# Model parameters\nmu = 10.\nsigma = 1.\n\n# Sample generation\nts = np.arange(100)\nobss = []\nobs = -math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\nimport matplotlib.pyplot as plt\nplt.plot(ts, obss);\n\n\nNow I want to code this as as PyMC3 model, fit the data and forecast the extrapolated value 10 timesteps after the last observation. I tried the following:\n\nt_forecast = 110\n\nwith pm.Model() as model:\nmu = pm.Normal('mu', mu=10, sigma=5)\nsigma = pm.Exponential('sigma', lam=1)\nrecords = []\n\nrecord = -pm.distributions.Constant('INITIAL FAKE RECORD', -10000)\n# Introduce observations\nfor (t, obs) in zip(ts,obss):\nnode_name = f'step {t\/\/STEP} record'\nrecord = pm.math.maximum(record, pm.Normal(node_name, mu, sigma),\nobserved = observation) # This doesn't work!\nrecords.append(record)\n\n# Introduce forecasted variable\nwhile t < x_forecast:\nt+= STEP\nrecord = pm.math.maximum(record, pm.Normal(f'step {t\/\/STEP} record', mu, sigma))\n\n# Sample posterior\ntrace = pm.sample(1000, step = [pm.Metropolis(), pm.NUTS()])\npm.traceplot(trace, ['mu', 'beta', f'step {t} record'])\n\n\nHowever this doesn\u2019t quite work - one cannot feed the observations like that.\nAccording to this previous answer I would need then to compute the joint distribution of all the timesteps.\n\nI tried working this out in a more general setting. Let\u2019s have X_n be some iid variables with cdf F_X, and let\u2019s define Y_n = \\max_{i\\le n} X_i.\n\nThen we have that\n\nP(Y_1\\le o_1, Y_2\\le o_2, ..., Y_n \\le o_n) = \\\\ P(Y_1\\le o_1) P(Y_2\\le o_2 | Y_1\\le o_1) \\cdots P(Y_n\\le o_n | Y_1\\le o_1, \\dots, Y_{n-1}\\le o_{n-1})\n\nExpanding the first few terms gives us\n\nP(Y_1\\le o_1) = P(X_1 \\le o_1) = F_X(o_1)\nP(Y_2\\le o_2 | Y_1\\le o_1) = P(\\max(X_1, X_2) \\le o_2 | X_1 \\le o_1) = \\\\ P(X_2 \\le o_2) P(X_1 \\le o_2 | X_1 \\le o_1) = F_X(o_2) \\frac{F_X(\\min\\{o_1, o_2\\})}{F_X(o_1)}\nP(Y_3\\le o_3 | Y_1\\le o_1, Y_2\\le o_2) = P(\\max(X_1, X_2, X_3) \\le o_3 | X_1\\le o_1, \\min(X_1, X_2)\\le o_2) = \\\\ P(X_3 \\le o_3) P(X_2 \\le o_3 | X_2 \\le o_2) P(X_1 \\le o_3 | X_1 \\le \\min(o_1, o_2)) = \\\\ F_X(o_3) \\frac{F_X(\\min\\{o_2, o_3\\})}{F_X(o_2)}\\frac{F_X(\\min\\{o_1, o_2, o_3\\})}{F_X(\\min\\{o_1, o_2\\})}\n\nThe expansion of the n th term of the product is then\n\nP(Y_n\\le o_n | Y_1\\le o_1, \\dots, Y_{n-1}\\le o_{n-1}) = \\prod_{1 \\le i\\le n} \\frac{F_X(\\min\\{o_n, o_{n-1}, o_{n - i + 1}\\})}{F_X(\\min\\{o_{n-1}, o_{n - i + 1}\\})}\n\nso the joint cdf of Y_1, \\dots, Y_n is then:\n\nP(Y_1\\le o_1, Y_2\\le o_2, ..., Y_n \\le o_n) = \\\\ P(Y_1\\le o_1) P(Y_2\\le o_2 | Y_1\\le o_1) \\cdots P(Y_n\\le o_n | Y_1\\le o_1, \\dots, Y_{n-1}\\le o_{n-1}) = \\\\ [F_X(o_1)] [F_X(o_2) \\frac{F_X(\\min\\{o_1, o_2\\})}{F_X(o_1)}] \\cdots [\\prod_{1 \\le i\\le n} \\frac{F_X(\\min\\{o_n, o_{n-1}, o_{n - i + 1}\\})}{F_X(\\min\\{o_{n-1}, o_{n - i + 1}\\})}] = \\\\\n\ndenominators cancel out previous terms in the product, and we end with:\n\n= \\prod_{1 \\le i\\le n} F_X(\\min\\{o_n, o_{n-1}, o_{n - i + 1}\\})) = \\\\ F_X(o_n) F_X(\\min\\{o_n, o_{n-1}\\})\\cdots F_X(\\min\\{o_n, o_{n-1}, \\dots, o_1\\})\n\nMy questions now are:\n\n1. Is my math correct?\n2. How can I implement a distribution with the cdf in the last equation in pymc3?\n2 Likes\n\nThere seem to be some issues with the math related to latex syntax or definition of latex blocks\n\nInteresting! It renders correctly for me\n\nFWIW the first time I loaded the page I got a similar error, but after I reloaded it once it never happened again. Very mysterious!\n\nI also noticed the math processing error went away when I refreshed.\n\n1. Is my math correct?\n\nNo, I don\u2019t think it is. So the expression of the density of Y_n = \\max \\{ X_1, \\ldots, X_n \\} where the X_i are i.i.d. and X_1 has a pdf f_X and a cdf F_X is:\n\nf_Y(y) = n[F_X(y)]^{n-1} f_X(y).\n\nIn order to derive this expression, you begin as you did where you write things out in terms of the CDF and do a bunch of manipulations. However, this only works in the 1D case from what I can tell. When you move to the joint CDF, things get complicated and it\u2019s unclear to me how to proceed. Particularly, the hard part seems to be conditioning Y_n on Y_{n-1}. I eventually tried taking a different approach which I\u2019ll detail below.\n\nDisclosure: I\u2019m no expert on order statistics so I may be entirely wrong\n\nI think this problem is much mathematically harder than you may have anticipated. From properties of maximums we know that Y_{n+1} = \\max \\{ X_1, \\ldots, X_n, X_{n+1} \\} = \\max \\{Y_n, X_{n+1} \\}. So, this would mean that P(Y_{n+1} \\leq y | Y_1, \\ldots, Y_n) = P(Y_{n+1} \\leq y | Y_n), i.e. that this sequence Y_i is Markovian in some sense. The distribution of Y_{n+1} only depends on the previous Y_{n} and the value of X_{n+1}. Let Y_{1:i} denote the collection \\{ Y_1, \\ldots, Y_n \\}. Then the likelihood for your time series of n observations becomes\n\np_{Y_{1:n}}(y_1, \\ldots, y_n | \\mu, \\sigma) = \\prod_{i=1}^{n-1} p_{ Y_{i+1}| Y_{1:i}}(y_{i+1} | y_1, \\ldots, y_{i}, \\mu, \\sigma) \\\\ = \\prod_{i=1}^{n-1} p_{ Y_{i+1}| Y_{i}}(y_{i+1} | y_{i}, \\mu, \\sigma).\n\nOk, so it looks like all we need is an expression for the density f_{Y_{n+1}| Y_{n}} of Y_{n+1} conditioned on an observation of Y_{n}.\n\nNow we\u2019re going to try a different way to figure out the cdf by reasoning about how could draw samples of the random variable Y_{n+1}|Y_{n} = y_n. When we go to simulate Y_{n+1}, we first draw a sample for X_{n+1}. Let\u2019s call this sample x_{n+1}. Then, we compare whether the sample is bigger or less than our current maximum y_n. If it\u2019s smaller, we then set y_{n+1} = y_{n}; otherwise we set y_{n+1} = x_{n+1}. In math notation this process can be written as:\n\n1. Flip a biased coin that yields heads with probability F_X (y_n) and tails with probability 1 - F_X(y_n).\n2. If the coin is heads, then set y_{n+1} = y_n. If the coin is tails, draw y_n \\sim Z where Z is a random variable that admits a density\nf_Z(z) = \\frac{f_X(z)}{1 - F_X(y_n)} \\mathbb{\\chi}_{z > y_n}.\n1. Return the value for y_{n+1} as your sample of Y_{n+1}.\n\nIntuitively, what we\u2019ve done is make a new CDF F_{Y_{n+1}} where we take all of the probability mass (under F_{X}) for values less than y_n and concentrate it at the single point y_n. Then, the CDF continues as it would for F_X except scaled so that F_{Y_{n+1}}(\\infty) = 1.\n\nHere\u2019s the cdf of your true RV X (normal with \\mu = 10, \\sigma = 1):\n\nHere\u2019s the cdf for Y_{n+1}, conditioned on an observation Y_{n} = 10:\n\nSo the random variable Y_{n+1} is a mixed random variable; it has both a continuous part and a discrete part. Unfortunately, this is where your approach breaks down: I think this breaks the statistical theory when trying to use the MCMC methods implemented in PyMC3 (Metropolis, NUTS, etc.). You could write out a density for the conditioned Y_{n+1} but this would involve Dirac delta functions.\n\nIMO, if my reasoning above is correct then I think the best you could do using PyMC3 is to ignore the sequence of maximums and consider only the last one, Y_{N}. This throws out information but then all you need is a density for Y_{N} which you already know (see the formula above). You could then take the logarithm of this density and write a custom PyMC3 distribution that has this log density under its logpdf method. As long as you code this log density using functions in pm.math or other theano code then you get the gradient of the log density for free, then you\u2019re all good to go to use NUTS.\n\nYour final code would look like:\n\nwith pm.Model() as model:\nmu = pm.Normal('mu', mu=10, sigma=5)\nsigma = pm.Exponential('sigma', lam=1)\n\n# This shows the model only the last observation. N is the number of X_i that\n# you've drawn leading up to Y_n, which will show up in your custom distribution definition\nlikelihood = YourCustomDistribution(\"y_N\", mu=mu, sigma=sigma, N=N, observed=obss[-1])\n\n\nThis is unsatisfying because an entire time series then only gives you a single observation. You could perhaps feed in the final values Y_N of an ensemble of time series generated the same way, but depending on your application you may not have access to this.\n\nTL;DR I don\u2019t think it is possible to approach this problem (in a satisfactory way) in PyMC3 since you end up doing MCMC on mixed continuous\/discrete distributions.\n\nSince I\u2019m curious, where is the data you\u2019re trying to model coming from?\n\n1 Like\n\nThank you again so much!\n\nLet me see if I understand everything correctly, please bear with all my questions!\n\nThe marginal pdf\n\nThe pdf you provided for Y_n makes sense to me.\n\nOne easy way to see it is that\n\nF_{Y_n}(o) = P(\\max X_{1:n} \\le o) = \\prod_{1\\le i \\le n} P(X_i = n) = [F_X(o)]^{n}\n\nhence the cdf is f_ {Y_n}(o) = F'_{Y_n}(o) = n [F_X(o)]^{n-1} f_X(o)\n\nThe joint pdf\n\nI am a bit confused because I am not sure whether you meant if my math is wrong because there is a specific mistake or because what we need is the pdf instead of the cdf.\n\nIf the latter, can\u2019t we derive the joint pdf by taking the partial derivatives of the cdf as usual?\n\nMixing random and continuous variables\n\nAccording to this NIPS paper NUTS cannot be applied to mixed random variables.\nAbout Metropolis I am confused, though this post here says it should work.\n\nI am actually very confused on what different methods are implemented in PyMC3 and what am I doing when calling eg pm.sample(1000, step = [pm.Metropolis(), pm.NUTS()]). Is that sampling first with Metropolis and then with NUTS or doing a combination of both?\n\nI am also confused about whether the distribution we want to model is mixed. Clearly the conditional distirbution Y_n | Y_{n-1} is, as you show in your answer. But what about the joint distribution of Y_{1:n}? And if it isn\u2019t and we can find the pdf or approximate it numerically, could we get it working in PyMC3?\n\nThe code\n\nI tried implementing your suggestion as follows:\n\nimport pymc3 as pm\n\ndist = pm.Normal.dist\nN = len(obss)\nlogp = lambda mu, sigma, value : N + (N-1) * dist(mu, sigma).logcdf(value).sum() + dist(mu, sigma).logp(value)\n\nwith pm.Model() as model:\nmu = pm.Normal('mu', mu=10, sigma=5)\nsigma = pm.Exponential('sigma', lam=1)\n\n# This shows the model only the last observation. N is the number of X_i that\n# you've drawn leading up to Y_n, which will show up in your custom distribution definition\nYn = pm.DensityDist(\"Yn\", logp, observed= {'mu':mu, 'sigma': sigma, 'value':obss[-1]})\n\ntrace = pm.sample(1000, step = [pm.Metropolis(), pm.NUTS()])\npm.traceplot(trace, ['mu', 'sigma', 'Yn'])\n\n\nHowever I am getting an error MissingInputError: Undeclared input. What am I doing wrong?\n\nAssuming fake independence\nI wonder whether we could get sensible-enough results by pretending that each Y_i is independent and feeding that into the model. This would introduce a bias upwards in the parameters, but maybe it is maneageble. What are your thoughts?\n\nThe data\nMy eventual goal is to model the World Records on speedrunning competitions (where people attempt to beat a videogame as fast as possible) The code to scrape the data is here, in case you are curious.\n\nThe exercise in the OP was a warmup before I think about what distribution to use for each attempt, what priors to use and how to model a minimum instead of a maximum.\n\nApologies in advance for this wall of text but I think it\u2019s a pretty nice solution to an awesome problem . The first section responds to your questions, the second section shows my proposed solution.\n\n# Q&A\n\nI am a bit confused because I am not sure whether you meant if my math is wrong because there is a specific mistake or because what we need is the pdf instead of the cdf.\n\nI took a second look and I stand corrected, I actually think your math is correct (although I only checked the expression for P(Y_2 \\leq o_2 | Y_1 \\leq o_1) ).\n\nIf the latter, can\u2019t we derive the joint pdf by taking the partial derivatives of the cdf as usual?\n\nI think this is where the problem comes up. You end up taking derivatives of min\u2019s\/max\u2019s and the result gets very complicated. For a custom distribution, you ultimately need the log pdf and not the CDF, so given your formula is correct I\u2019m not sure how you would proceed.\n\nAn additional comment is that while it might be possible that you could somehow figure out a formula for the joint CDF then take partial derivatives, you don\u2019t actually need the full joint CDF. But one of the observations I made was that you don\u2019t need the full joint PDF\/CDF, since\n\nY_{n+1} | \\big( Y_{n} = y_n, Y_{n-1} = y_{n_1}, \\ldots, Y_1 = y_1 \\big) = Y_{n+1} | Y_n = y_n.\n\nBecause of conditional independences amongst the sequence you only need to know the PDF\u2019s\/CDF\u2019s for the n conditional distributions f_{Y_{n+1}|Y_n} corresponding to Y_{n+1}| Y_n = y_n instead of the full pdf f_{Y_{1:n+1}}. This greatly simplifies things as opposed to trying to figure out the full joint pdf\/cdf.\n\nAccording to this NIPS paper NUTS cannot be applied to mixed random variables.\n\nI actually found this same paper when I was looking into this. I think there\u2019s a bit of terminology overloading going on, however. In this paper by \u201cmixed\u201d they are referring to random vectors where some components are discrete and some components are continuous, and not \u201cmixed\u201d as in a single component is both discrete & continuous. You could potentially have a vector of mixed random variables, where each component was both continuous and discrete.\n\nI am actually very confused on what different methods are implemented in PyMC3 and what am I doing when calling eg pm.sample(1000, step = [pm.Metropolis(), pm.NUTS()]) . Is that sampling first with Metropolis and then with NUTS or doing a combination of both?\n\nFrom briefly looking at the source, I think this is trying to assign a different step method for each variable of the posterior, in the order that you provide in the list. If you don\u2019t provide enough step methods as there are random variables, then it picks the best one according to whether it\u2019s discrete or continuous. I might be wrong about what precisely is going on here but my advice is to not mess with this.\n\nAssuming fake independence\n\nI don\u2019t think this is a good idea but you could try it.\n\n# My Solution\n\nI\u2019m not entirely convinced that the approach I\u2019m about to detail is correct, but I think it\u2019s within an epsilon-ball of being correct. So I\u2019ll go ahead and explain.\n\nThis post on stats StackExchange gives an excellent response to someone asking about the likelihood function of a mixed random variable. Translating this into our problem, we can express the CDF of the conditioned random variable Y_{n+1} | (Y_n = y_n, \\theta) as\n\nF_{Y_{n+1}|Y_n = y_n, \\theta} = F_{a, \\theta}(x) + F_{d, \\theta}(x) = \\lambda(\\theta) \\int_{\\infty}^x f_a(t; \\theta) \\, dt + (1 - \\lambda(\\theta)) \\sum_{t \\leq x} f_d(t; \\theta).\n\nHere \\theta is referring to the collection [\\mu, \\sigma] of the parameters of the underlying Normal distribution for each X_n. The first term in the sum above contains the contribution to the CDF from the continuous portion of the data, while the second term contains the contribution from the discrete part. The term \\lambda(\\theta) is the mixing probability, which you interpret as the probability that an observation came from the continuous part as opposed to the discrete part.\n\nIn the above, the density f_a(t;\\theta) corresponds to just the density I mentioned in my initial initial answer:\n\nf_a(t; \\theta) = f_Z(z; \\theta) = \\frac{f_X(z; \\theta)}{1 - F_X(y_n; \\theta)}\\chi_{z > y_n}\n\nwhere \\chi is just the indicator function. The discrete probability mass function f_d(t; \\theta), is just the \u201cdumb\u201d random variable that puts all of its mass on y_n. The \\lambda(\\theta) parameter is actually just\n\n\\lambda(y_{n-1}, \\theta) = 1 - F_X(y_{n-1}; \\theta)\n\nwhich was the probability of the biased coin landing on tails from my response.\n\nNow, that StackExchange answer proposed that the likelihood function could be computed as\n\n\\mathcal{L}(\\mathcal{Y}; \\theta) = \\prod_{i \\in C} \\lambda(\\theta) f_a(y_i; \\theta) \\,\\, \\prod_{i \\in D} (1 - \\lambda(\\theta)) f_d(y_i; \\theta)\n\nwhere we split the data \\mathcal{Y} = {y_1, \\ldots, y_n} into two sets, where C is the set of indices for which the observation y_i, i \\in C does not equal y_{i-1} (we flipped a tails and drew from the continuous distribution) and D is the set of indices for which the observation y_i, i \\in D where y_i = y_{i-1} (we flipped a heads and stuck with y_n). Note that you always know whether y_i came from the discrete\/continuous draw just by looking if the running maximum changed\/didn\u2019t change. The explicit steps to do this are:\n\n1. Write a function for the logarithm of the density f_Z(z; \\theta) I gave above.\n2. Assemble the list of \\lambda(y_{n-1}, \\theta)'s and (1 -\\lambda(y_{n-1}, \\theta))'s. These will be lists of two different lengths whose total length sums to the number of observations. Finally, take the logarithms of both lists, then sum them together and call it log_lambdas_sum.\n3. Write a master logp function that computes the logarithm of the likelihood I gave above. The log probability should end up being log_lambdas_sum + the sum of the \\log f_a(y_i; \\theta) for each i \\in C. We end up ignoring the parts for the discrete pmf since at all of the i \\in D the pmf f_d(y_i; \\theta) is just 1 so they drop out.\n\n# Summary\n\nI was probably too lazy and garbled some of the notation above, but hopefully you can get a sense of how you can calculate the likelihood. It turns out, we can decompose the likelihood function for the mixed random variable into its discrete and continuous parts, and then evaluate to total likelihood by evaluating the individual likelihoods on two partitioned sets of the data and weighting by the \\lambda factors. I am working on a notebook that implements this but can\u2019t promise when I\u2019ll get around to finishing it, so if you can see the light at the end of the tunnel I encourage you to try this yourself!\n\n1 Like\n\nThe wall of text is very much appreciated!\n\nLet me see if I understand the proposed solution. We have that we can decompose the conditional cdf as:\n\nF_{Y_{n+1} | Y_n=o_n}(o_{n+1}) = P(Y_{n+1} \\le o_{n+1} | Y_n=o_n) = \\\\ = P(X \\le o_{n+1}) I(o_n \\le o_{n+1}) = F_X(o_n) I(o_n \\le o_{n+1}) + I(o_n \\le o_{n+1}) \\int_{o_n}^{o_{n+1}}f_X = \\\\ = F_X(o_n) I(o_n \\le o_{n+1}) + \\frac{(1-F_X(o_n))}{(1-F_X(o_n))}I(o_n \\le o_{n+1}) \\int_{o_n}^{o_{n+1}}f_X = \\\\ = F_X(o_n) I(o_n \\le o_{n+1}) + (1-F_X(o_n))I(o_n \\le o_{n+1}) \\frac{\\int_{o_n}^{o_{n+1}}f_X}{(1-F_X(o_n))} = \\\\ = F_X(o_n) I(o_n \\le o_{n+1}) + (1-F_X(o_n))I(o_n \\le o_{n+1}) \\frac{\\int_{o_n}^{o_{n+1}}f_X}{\\int_{o_{n}}^{\\infty} f_X} = \\\\ \\lambda_{n+1} P(A_{n+1} \\le o_{n+1}) + (1-\\lambda_{n+1}) P(X_{n+1} \\le o_{n+1} | X_{n+1} > o_n)\n\nWhere we have that \\lambda_{n+1} = F_X(o_n) and A_{n+1} is a constant random variable with value o_{n}.\n\nHence we have expressed the conditional random variable Y_{n+1} | Y_n=y_n as a mixture of A_{n+1} and B_{n+1} = X_{n+1} | X_{n+1} > o_n, with mixing probability \\lambda_{n+1} = F_X(o_n).\n\nUsing the trick in the link you sent, we can then write the conditional likelihood for o_{n+1} given o_n as\n\n\\mathcal{L}_{n+1} (o_{n+1} | o_n) = \\lim_{\\epsilon\\to 0} \\frac{F_{Y_{n+1} | Y_n=o_n}(o_{n+1} + \\epsilon) - F_{Y_{n+1} | Y_n=o_n}(o_{n+1} - \\epsilon)}{2\\epsilon} \\approx \\\\ = \\left.\\begin{cases} \\lambda_{n+1} F_{A_{n+1}}(o_{n+1}), & \\text{if } o_{n} = o_{n+1}\\\\ (1 - \\lambda_{n+1}) f_{B_{n+1}}(o_{n+1}), & \\text{if } o_{n} < o_{n+1}\\\\ \\end{cases}\\right\\} = \\\\ = \\left.\\begin{cases} F_X(o_n) , & \\text{if } o_{n} = o_{n+1}\\\\ (1 - F_X(o_n)) \\frac{f_X(o_{n+1})}{\\int_{o_n}^\\infty f_X}, & \\text{if } o_{n} < o_{n+1}\\\\ \\end{cases}\\right\\} = \\\\ = \\left.\\begin{cases} F_X(o_n) , & \\text{if } o_{n} = o_{n+1}\\\\ f_X(o_{n+1}), & \\text{if } o_{n} < o_{n+1}\\\\ \\end{cases}\\right\\}\n\ndenoting as D the set of indices i>1 where o_i = o_{i-1} and C its complement, we can then denote the total likelihood as:\n\n\\mathcal{L}(\\mathbf o) = f_X(o_1)\\prod_{i\\in D} F_X(o_i) \\prod_{i\\in C} f_X(o_{i})\n\nTo make this work in pymc3, we have to code the logarithm of this expression as a python function, and implement it as a joint distribution object (which I am still confused about how to do)\n\nSo far what I could implement is the code for the log likelihood part.\nIf we assume that the underlying distribution of X_i is normal then it would be:\n\nimport numpy as np\nimport pymc3 as pm\n\ndef logp(os, mu, sigma, epsilon = 0.00001):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\nlog_likelihood = 0\no_n_minus_1 = -np.math.inf\nfor o_n in os:\nif np.abs(o_n - o_n_minus_1) < epsilon: # if the value is equal to the last one\nlog_likelihood += x_dist.logp(o_n).eval()\nelif o_n_minus_1 - o_n: #if the value is superior than last one\nlog_likelihood += x_dist.logcdf(o_n).eval()\nelse: # if the value is inferior than last one\nlog_likelihood += -np.math.inf\no_n_minus_1 = o_n\nreturn log_likelihood\n\n\nWe can test this log likelihood function on some sampled data:\n\n# TEST THE LOGP FUNCTION\n\n# Model parameters\nN = 20\nmu = 10.\nsigma = 0.1\n\n# Sample generation\nts = np.arange(N)\nobss = []\nobs = -math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\nmus = np.linspace(0., 20.)\nlog_likelihoods = np.vectorize(lambda mu : logp(obss, mu=mu, sigma=1.))(mus)\n\nimport matplotlib.pyplot as plt\nplt.plot(mus, log_likelihoods);\n\n\nThe peak of the loglikelihood corresponds to the value of \\mu we used to generate the data, so it seems to be working!\n\nI tried implementing this as a DensityDistribution in PyMC3 as this:\n\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp, observed = {'os':obss, 'mu': mu, 'sigma': sigma})\ntrace = pm.sample(1000)\npm.traceplot(trace, ['mu', 'sigma'])\n\n\nBut PyMC3 didn\u2019t like my logp; I get TypeError: Variables do not support boolean operations.\nI guess I needed to implement the function as you said, with different distirbution depending on whether the observations are the same or different.\n\n1 Like\n\nI will take a closer look later but the issue is that you need to use Theano\u2019s functions for Boolean comparison instead of just vanilla python if\/else statements. Check out theano.tensor.switch. You will also want to use theano.scan for your for loop, but maybe you can get by without it.\n\n1 Like\n\nI see!\n\n# Sample parameters\nN = 20\nmu = 10.\nsigma = 1.0\n\n# Sample generation\nts = np.arange(N)\nobss = []\nobs = -math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\n# PyMC3 model\ndef logp(os, mu, sigma, epsilon = 0.00001):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\nlog_likelihood= pm.math.constant(0.)\no_n_minus_1 = pm.math.constant(0.)\neps = pm.math.constant(epsilon)\nfor o_n in os:\nlog_likelihood += \\\npm.math.switch(pm.math.lt(pm.math.abs_(o_n - o_n_minus_1), eps),\nx_dist.logp(o_n),\nx_dist.logcdf(o_n)\n)\no_n_minus_1 = o_n\n\nreturn log_likelihood\n\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp, observed = {'os':obss, 'mu': mu, 'sigma': sigma})\ntrace = pm.sample(5000, return_inferencedata=True, idata_kwargs={\"density_dist_obs\": False})\npm.plot_trace(trace, ['mu', 'sigma'])\n\n\nI think it works!\n\n1 Like\n\nOk, maybe I spoke too fast.\n\nExecuting the previous script with \\mu = 10, \\sigma=1 and n=100 seems to result in a biased posterior (?)\n\nYes, I think we are close to celebrating but not quite. I kicked the number of observations N up to 200 and drew 300K samples x 4 chains. This was the result:\n\nThere is clearly some bias. I will take a look at this but I think you don\u2019t have the likelihood correct (not the implementation, I don\u2019t think the formula in your last post you gave is correct).\n\nWait! I mixed up the clauses in the switch.\n\nHere is the fixed code (plus some other small bug fixes)\n\n# Sample parameters\nN = 100\nmu = 10.\nsigma = 1.0\n\n# Sample generation\nts = np.arange(N)\nobss = []\nobs = -math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\n# PyMC3 model\ndef logp(os, mu, sigma, epsilon = 0.00001):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\nlog_likelihood= pm.math.constant(0.)\no_n_minus_1 = pm.math.constant(-np.math.inf)\neps = pm.math.constant(epsilon)\nfor o_n in os:\nlog_likelihood += \\\npm.math.switch(pm.math.lt(pm.math.abs_(o_n - o_n_minus_1), eps),\nx_dist.logcdf(o_n),\nx_dist.logp(o_n),\n)\no_n_minus_1 = o_n\n\nreturn log_likelihood\n\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp, observed = {'os':obss, 'mu': mu, 'sigma': sigma})\ntrace = pm.sample(5000, return_inferencedata=True, idata_kwargs={\"density_dist_obs\": False})\npm.plot_trace(trace, ['mu', 'sigma'])\n\n\nMuch better!\n\nWhat do you think?\n\n1 Like\n\nI think we can celebrate now.\n\nSo I initially thought that your likelihood you wrote down in an earlier post was wrong and that your code looked too simple, but I did the math and some terms actually cancel out. That being said, you had one small error. When you calculated the contribution from the likelihood for the data where the maximum didn\u2019t change, you need to use x_dist.logcdf(o_n_minus_1) rather than x_dist.logp(o_n). Can you see why? Here\u2019s what I get when I make this change.\n\nOk, so a little less biased perhaps. But it still seems like something weird is going on.\n\nOne thing you may have noticed is that if you increase the number of observations in your model, it takes a very long time to compile the model (yet sampling is very fast). This is because you are using a for loop in your code. I mentioned earlier that you should use theano.scan, but actually we can just vectorized the whole likelihood calculation like this:\n\n# Sample parameters\nN = 1000\nmu = 10.\nsigma = 1.0\n\n# Sample generation\nts = np.arange(N)\nobss = []\nobs = -math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\n### First, filter for the data where there were jumps in the running max\n\njump_idxs = np.where(np.diff(obss) > 0)[0] + 1 # This gives you the indices of the array where the max changed (except for the first element)\njump_idxs = np.insert(jump_idxs, 0, 0) # Include the first element as a jump\njump_idxs_right_before = jump_idxs[1:] - 1 # This gives you the indices right before the change\n\njump_data = obss[jump_idxs] # Slice out the data for the actual jumps. Serves as our o_n for the jumps\njump_data_right_before = obss[jump_idxs_right_before] # Same but for the right before the jumps (we'll use these as o_n_minus_1)\njump_data_right_before = np.insert(jump_data_right_before, 0, -np.inf) # Correct for first element\n\n### Second, filter for the data where the running max didn't change\n\nflat_idxs = np.setdiff1d(np.arange(len(obss)), jump_idxs)\nflat_idxs_right_before = flat_idxs - 1\n\nflat_data = obss[flat_idxs] # Serves as our o_n for the data with no_jumps\nflat_data_right_before = obss[flat_idxs_right_before]\n\n# PyMC3 model\ndef logp(jump_data, flat_data_right_before, mu, sigma):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\nlog_likelihood = pm.math.constant(0.)\n\n# Add likelihood contribution from the jump data\nlog_likelihood += pm.math.sum(x_dist.logp(jump_data))\n\n# Add likelihood contribution from the flat data\nlog_likelihood += pm.math.sum(x_dist.logcdf(flat_data_right_before))\n\nreturn log_likelihood\n\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.1, 10.0)\nYs = pm.DensityDist('Ys', logp, observed = {'jump_data':jump_data, 'flat_data_right_before':flat_data_right_before, 'mu': mu, 'sigma': sigma})\ntrace = pm.sample(20000, chains=4, return_inferencedata=True, idata_kwargs={\"density_dist_obs\": False})\n\n\nBasically, I just used the factorization of the likelihood to vectorized the contributions from both the data where max changed\/didn\u2019t change. You\u2019ll find that this code runs significantly faster, especially when you increase your number of observations N. I\u2019m not sure if there was something wrong with the code from earlier, but in this new code, the posterior on sigma actually looks a lot different\/better. This is with N = 100:\n\nI\u2019ve found that sometimes you will get chains with a huge number of divergences, whereas sometimes you will get chains with none at all. I\u2019m not sure what might be causing this, but I bet it has something to do with the discrete\/continuous nature of this problem. Perhaps we violate an assumption for using NUTS and something like Metropolis is better to use in this case. I\u2019ll have to think more about this.\n\n1 Like\n\nNeat implementation!\n\nI am confused.\n\nFirst, I think there must be a typo, since we want to use the CDF when dealing with flat data, right?\nGiven that and your code, I am reading that we should use \\mathcal{L}(o_{n+1} | o_n) = F_A(o_{n}) instead of \\mathcal{L}(o_{n+1} | o_n) = F_A(o_{n+1}) when o_n = o_{n+1}.\nBut if o_n = o_{n+1} then F_A(o_{n+1}) = F_A(o_{n}), so I must be misunderstanding something here\n\nWhen you calculated the contribution from the likelihood for the data where the maximum didn\u2019t change, you need to use x_dist.logcdf(o_n_minus_1) rather than x_dist.logp(o_n) .\n\nSorry, I meant to say:\n\nWhen you calculated the contribution from the likelihood for the data where the maximum didn\u2019t change, you need to use x_dist.logcdf(o_n_minus_1) rather than x_dist.logcdf(o_n).\n\nAnd thanks for pointing that out, yes you are correct. In my head I was thinking that it doesn\u2019t make sense to evaluate the likelihood on the same value that is already being plugged into the definition of the likelihood, but in the case o_n = o_{n+1} this doesn\u2019t matter. It shouldn\u2019t matter which you use, and I guess you don\u2019t even need to keep track of the _right_before arrays at all. You can just give the DensityDist flat_data itself rather than flat_data_right_before.\n\nI see!\n\nSo to synthesize, the code would look like:\n\n# Sample parameters\nN = 10\nmu = 10.\nsigma = 1.0\n\nimport numpy as np\nimport pymc3 as pm\n\n# Sample generation\nts = np.arange(N)\nobss = []\nobs = -np.math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\n### Separate the data where there were jumps in the running max\n### and the data where the running max stayed constant\njump_data = obss[jump_mask] # Slice out the data for the actual jumps. Serves as our o_n for the jumps\nflat_data = obss[~jump_mask] # Serves as our o_n for the data with no_jumps\n\n# PyMC3 model\ndef logp(jump_data, flat_data, mu, sigma):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\n# Add likelihood contribution from the jump data\nlog_likelihood = pm.math.sum(x_dist.logp(jump_data))\n\n# Add likelihood contribution from the flat data\nlog_likelihood += pm.math.sum(x_dist.logcdf(flat_data))\n\nreturn log_likelihood\n\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp, observed = {'jump_data':jump_data, 'flat_data':flat_data, 'mu': mu, 'sigma': sigma})\ntrace = pm.sample(2000, chains=4, return_inferencedata=True, idata_kwargs={\"density_dist_obs\": False})\npm.plot_trace(trace, ['mu', 'sigma'])\n\n\n2 Likes\n\nI\u2019m back with more questions!\n\nI want to make two plots for the model.\n\n1. A posterior predictive plot, showing a high density interval of the model conditional on the posterior for \\mu, \\sigma on top of the plot of actual observations to see whether the model is doing a good job at predicting those\n2. An extrapolation plot,\n\nFor 1) I tried following the example here.\n\nMy code is:\n\n# Sample parameters\nN = 100\nmu = 10.0\nsigma = 1.0\n\nimport numpy as np\nimport pymc3 as pm\nimport arviz as az\n\n# Random sampling\ndef random(point=None, size=None):\n# Retrive parameters\nmu = point['mu']\nsigma = point['sigma']\n\n# Generate sample\nts = np.arange(N)\nobss = []\nobs = -np.math.inf\nfor t in ts:\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\nreturn obss\n\nobss = random({'mu': mu, 'sigma':sigma})\n\n### Separate the data where there were jumps in the running max\n### and the data where the running max stayed constant\njump_data = obss[jump_mask] # Slice out the data where a new record is set\nflat_data = obss[~jump_mask] # Slice the data where the record is maintained\n\n# Logarithmic likelihood of the joint distribution of data\ndef logp(jump_data, flat_data, mu, sigma):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\n# Add likelihood contribution from the jump data\nlog_likelihood = pm.math.sum(x_dist.logp(jump_data))\n\n# Add likelihood contribution from the flat data\nlog_likelihood += pm.math.sum(x_dist.logcdf(flat_data))\n\nreturn log_likelihood\n\n# PyMC3 model definition\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp, random=random,\nobserved = {'jump_data':jump_data,\n'flat_data':flat_data,\n'mu': mu,\n'sigma': sigma}\n)\ntrace = pm.sample(1000, chains=2, tune=1000,\nreturn_inferencedata=True,\nidata_kwargs={\"density_dist_obs\": False})\n\npost_pred = pm.sample_posterior_predictive(trace, var_names=[\"Ys\"])\nidata = az.concat(trace, az.from_pymc3(posterior_predictive=post_pred))\naz.plot_ppc(idata)\n\n\nWhen I execute the above snippet, I am getting an internal Theano error MissingInputError: Input 0 of the graph (indices start from 0), used to compute sigmoid(mu_interval__), was not provided and not given a value..\n\nFor 2\u2026 I don\u2019t even know how to start.\n\nPS: Still very interested in your thoughts about the divergence and what parameters to use for the NUTS sampling if you think more about that!\n\nUpdate: I kind of gave up on Arviz and used Altair to plot the posterior predictive distribution:\n\n# Generate predictive posterior\nwith model:\npost_pred = pm.sample_posterior_predictive(trace, var_names=[\"Ys\"])\n\n# Preprocess predictive posterior\npost_pred_obss = post_pred['Ys']\nlower_bound = np.quantile(post_pred_obss, q=0.05, axis=0)\nmedian = np.quantile(post_pred_obss, q=0.5, axis=0)\nupper_bound = np.quantile(post_pred_obss, q=0.95, axis=0)\n\n# Encode as pandas DataFrame\nimport pandas as pd\ndata_df = pd.DataFrame({'t': ts, 'obs' : obss})\nhdi_df = pd.DataFrame({\n't': ts,\n'lower': lower_bound,\n'median' : median,\n'upper': upper_bound,\n})\n\n# Plot observations and predictive posterior with Altair\nimport altair as alt\ndata_chart = alt.Chart(data_df)\\\n.mark_line(\n# size=10,\ncolor='purple',\n)\\\n.encode(\nx=alt.X('t:Q', title='Timestep'),\ny=alt.Y('obs:Q', title='Cumulative maximum'),\n)\n\n# https:\/\/stackoverflow.com\/questions\/60649486\/line-chart-with-custom-confidence-interval-in-altair\nline_chart = alt.Chart(hdi_df).mark_line().encode(\nx='t:Q',\ny='median:Q'\n)\n\nband_chart = alt.Chart(hdi_df).mark_area(\nopacity=0.5\n).encode(\nx='t:Q',\ny='lower:Q',\ny2='upper:Q'\n)\n\n(data_chart + line_chart + band_chart)\\\n.properties(\nwidth=800,\nheight=400\n)\\\n.configure_axis(\nlabelFontSize=20,\ntitleFontSize=30\n)\n\n\nI am a bit nervous because I expected the uncertainty to grow wider with time, but we see it is a band of constant width. Is my intuition off or is this symptom of a problem?\n\nFor 2 I am still stuck. I tried cheating my way out of the problem by defining a chain of variables with the conditional marginal likelihood we previously derived. But I don\u2019t seem to be able to instruct PyMC to feed to the variable the last variable of the joint distribution we defined\n\n#@title PyMC model\n\n# Sample parameters\nN = 100\nM = 5\nmu = 10.0\nsigma = 1.0\n\nimport numpy as np\nimport pymc3 as pm\n\n# Random sampling function\ndef random(point=None, size=None, N=N):\n\n# Retrive parameters\nmu = point['mu']\nsigma = point['sigma']\n\n# Generate sample\nobss = []\nobs = -np.math.inf\nfor t in range(N):\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\nreturn obss\n\n# Generate fake observations\nts = np.arange(N)\nobss = random({'mu': mu, 'sigma':sigma})\n\n### Separate the data where there were jumps in the running max\n### and the data where the running max stayed constant\njump_data = obss[jump_mask] # Slice out the data where a new record is set\nflat_data = obss[~jump_mask] # Slice the data where the record is maintained\n\n# Logarithmic likelihood of the joint distribution of data\ndef logp(jump_data, flat_data, mu, sigma):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\n\n# Add likelihood contribution from the jump data\nlog_likelihood = pm.math.sum(x_dist.logp(jump_data))\n\n# Add likelihood contribution from the flat data\nlog_likelihood += pm.math.sum(x_dist.logcdf(flat_data))\n\nreturn log_likelihood\n\ndef next_logp(o_n, o_n_minus_1, mu, sigma):\nx_dist = pm.Normal.dist(mu=mu, sigma=sigma)\nlog_likelihood = \\\npm.math.switch(\npm.math.eq(o_n, o_n_minus_1),\nx_dist.logp(o_n),\nx_dist.logcdf(o_n)\n)\nreturn log_likelihood\n\n# PyMC3 model definition\nwith pm.Model() as model:\nmu = pm.Uniform('mu', 0., 20.)\nsigma = pm.Uniform('sigma', 0.5, 1.5)\nYs = pm.DensityDist('Ys', logp,\nshape = (N,),\nrandom=random,\nobserved = {'jump_data':jump_data,\n'flat_data':flat_data,\n'mu': mu,\n'sigma': sigma}\n)\n\nprevious = Ys[-1] # This does not work\nfor i in range(M):\nZi = pm.DensityDist(f'Z{i}', next_logp,\nobserved = {\n'previous' : previous,\n'mu': mu,\n'sigma': sigma,\n})\nprevious = Zi\n\n# Sample\ntrace = pm.sample(1000, chains=2, tune=1000,\nreturn_inferencedata=True,\nidata_kwargs={\"density_dist_obs\": False})\npm.plot_trace(trace, ['mu', 'sigma'])\n\n\nEg with the snippet above I get the error TypeError: 'MultiObservedRV' object is not subscriptable. How can I access one dimension of a joint distribution?\n\nUpdate: I could not figure out how to extend the model to include extra variables, so I just wrote my own sampler, that takes values from the trace and does the whole sampling procedure, the using Altair for plotting:\n\ndef random(point=None, size=None, N=N, initial_obs = -np.math.inf):\n\n# Retrive parameters\nmu = point['mu']\nsigma = point['sigma']\n\n# Generate sample\nobss = []\nobs = initial_obs\nfor t in range(N):\naux = mu + sigma*np.random.randn()\nobs = np.max([obs, aux])\nobss.append(obs)\nobss = np.array(obss)\n\nreturn obss\n\n#@title Extrapolate to future timesteps\n\nM = 100\n\nimport pandas as pd\n\n# Generate samples from the posterior distribution of future data\n\nmu_chains = trace['posterior']['mu'].data\nsigma_chains = trace['posterior']['sigma'].data\n\nextrapolation_data = []\nfor chain_number, (mu_chain, sigma_chain) in enumerate(zip(mu_chains, sigma_chains)):\nfor mu, sigma in zip(mu_chain, sigma_chain):\nextrapolated_obss = random({'mu':mu, 'sigma':sigma}, N=M, initial_obs=obss[-1])\nextrapolation_data += [extrapolated_obss]\nextrapolation_data = np.array(extrapolation_data)\n\nassert extrapolation_data.shape[1] == M\n\n# Preprocess extrapolated posterior\nlower_bound = np.quantile(extrapolation_data, q=0.05, axis=0)\nmedian = np.quantile(extrapolation_data, q=0.5, axis=0)\nupper_bound = np.quantile(extrapolation_data, q=0.95, axis=0)\n\nassert median.shape == (M,)\n\n# Print guesses for last observation\nprint(f\"The median guess for the record at time {N+M} is {median[-1]:.2f}\")\nprint(f\"The 90% confidence interval for the record at time {N+M} is {lower_bound[-1]:.2f} to {upper_bound[-1]:.2f}\")\n\n# Encode as pandas DataFrame\nimport pandas as pd\nex_ts = np.array(range(N, N+M))\next_df = pd.DataFrame({\n't': ex_ts,\n'lower': lower_bound,\n'median' : median,\n'upper': upper_bound,\n})\n\n# Plot the extrapolated data\n# https:\/\/stackoverflow.com\/questions\/60649486\/line-chart-with-custom-confidence-interval-in-altair\nimport altair as alt\nline_chart = alt.Chart(ext_df).mark_line().encode(\nx=alt.X('t:Q', title='Timestep'),\ny=alt.Y('median:Q', title='Cumulative maximum'),\n)\n\nband_chart = alt.Chart(ext_df).mark_area(\nopacity=0.5\n).encode(\nx='t:Q',\ny='lower:Q',\ny2='upper:Q'\n)\n\n(line_chart + band_chart)\\\n.properties(\nwidth=800,\nheight=400\n)\\\n.configure_axis(\nlabelFontSize=20,\ntitleFontSize=30\n)\n\n\nThere might be some bugs but I think the approach should work!\n\nExcellent!\n\nI am a bit nervous because I expected the uncertainty to grow wider with time, but we see it is a band of constant width. Is my intuition off or is this symptom of a problem?\n\nActually, your result is exactly what you should expect. As you march forward in time, there is an ever-smaller chance that an attempt will succeed the previous maximum. At some point, the maximum virtually never changes. The width of your posterior predictive on the maximum at this point is then entirely driven by the parametric uncertainty of the underlying normal random variable. The maximum, in the long run, settles down to about the 99.9% quantile of the underlying Gaussian if that makes sense. So the uncertainty band you get should have constant width, the lower bound corresponding to the posterior \\mu and \\sigma that yields the smallest 99.9% quantile compared to all other posterior samples, and similarly for the upper bound. Of course there is some adjustment to be made since you are really instead taking the lower 5% \/ upper 95% quantile across all of these quantiles.\n\n1 Like","date":"2022-07-07 01:59:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9110015034675598, \"perplexity\": 3140.4415204894503}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104683020.92\/warc\/CC-MAIN-20220707002618-20220707032618-00128.warc.gz\"}"} | null | null |
W:m Crichton & C:o Ab is a former engineering and shipbuilding company that operated in Turku, Grand Duchy of Finland in 1842–1913. The company also had another shipyard in Okhta, Saint Petersburg.
The company was founded as Cowie & Eriksson. At the beginning it produced steam engines, boilers and other engineering products. William Crichton became owner in 1862 and the company was named W:m Crichton & C:o The first shipbuilding slipway was constructed in 1864. The company became the biggest employer of Turku after acquiring the nearby yard Åbo Skeppswarf in 1883.
Crichton died in 1889, after which the operations were continued by investors. In 1896 the company started a new yard in Okhta, Saint Petersburg, to build ships for the local market. The operations were poorly organised, and both yards lost so much money that the entire company went bankrupt in 1913.
A new company, Ab Crichton, was established in 1914 to continue shipbuilding in Turku.
Name typography and spelling
The contemporary sources use alternative spellings Wm Crichton & Co, W:m Crichton & C:o and W:m Crichton & Co. for which the corresponding Russian transliteration was В:мъ Крейтонъ и Ко. ("V:m Kreiton i Ko.").
Origins
The company was started in 1842, when Swedish chief engineer Anders Thalus Eriksson and Scottish David Cowie got a permit to start a foundry and engineering works in Turku. The men had worked before in Stockholm for Samuel Owen who had built the first steam engines of Sweden. The location for the premises was selected an empty lot on east bank of river Aura, in address Itäinen Rantakatu 56–58. The start was funded by a loan of 15 000 silver rubles given by the senate. The operations started in 1844 and the portfolio consisted of steam engines and other machinery. In 1844 the company employed 86 people. While starting an engineering company from scratch included a high risk, there was need for steam engines in industrialising Finland. The products were steam engines and various castings and forgings, and the main customer became the Turku Old Shipyard located at the other side of the river.
First steam engines
The first known large Turku-built ship propelled by Cowie & Eriksson engine was the 1850 built Furst Menschikoff II.
Cowie & Eriksson got an order for an engine to power steam frigate Rurik, which was the largest ship built in Finland until then. In 1850 the company hired Scottish engineer William Crichton to accomplish installation and commissioning of the 300-hp engine; the project took three years in total. The ship was launched at the Turku Old Shipyard in 1851 and handed over in 1853, after which Crichton left the company.
The company suffered of lack of orders during the Crimean War. Eriksson decided to give up the business and sold his share of the company to local investor Erik Julin in 1857. After this the company operated under name D. Cowie & C:o. Julin started to seek for a partner who had experience about marine industry and business knowledge. He wrote a letter to William Crichton, who lived then in Kolpino, close to Saint Petersburg, and suggested him to buy Cowie's share of the company. Crichton took the offer and moved in Turku in 1862, after which the company was named W:m Crichton & C:o. Crichton was up to date about the latest development of naval engineering: propellers were replacing paddle wheels, wood gave way to steel and compound technology was entering in steam engines; this was enabled by the recently invented fire-tube boilers which withstood higher pressure.
Start of shipbuilding
Crichton wanted to extend the company portfolio to complete vessels. The company's first ships were not built in Turku but delivered in parts to Vähä-Äiniö yard at lake Päijänne, 230 km away from Turku. The ships were paddle steamers built by local craftsmen under supervision of specialists sent by Crichton. The first one, named Ensimmäinen ("first") was powered by a small, 12-hp engine and handed over in 1863. The subsequent Toinen ("second") was delivered in the following year and had a 25-hp engine. Both ships were intended for log tugging. Three more ships were built at Vähä-Äiniö yard: passenger ships Seura, which was a 28.8 metres long, and 33.7-metre Lahtis were both paddle steamers and handed over in 1864. In 1866 the Imperial Russian Army ordered Knyas Aleksei, a propeller equipped ship for Päijänne transportation.
In the meantime, a new slipway was taken into use in Turku in 1864; it could be applied both on newbuilding and repairs. A mast crane was built to lift engines and boilers into hulls and to unload steel sheets from barges. The first ships built on the new slipway were a tugboat and a couple of barges delivered to a Saint Petersburg customer in about 1865. It is likely that also few other vessels were built before 1867, when steam cruiser Suomi was built, but the original order documents have not survived.
Crichton built and installed small auxiliary steam engines on sailing ships in 1864–1873. The stern was modified to allow fitting of boiler, engine and propeller. The complete number of conversion projects remains unknown but the existing documents mention few barques and schooners with 30–40-hp engines. The concept was unsuccessful and in many cases the engines were removed after.
In 1868 Crichton delivered for Russian steam frigate Knyas Pozyarsk, a 35-feet longboat powered by a 5-hp steam engine. The boat type became one of the company's specialties; by 1905 Crichton had delivered at least 50 longboats. The hull was typically composite construction with steel ribs and wooden planking, sometimes completely made from steel. Engine output was generally under 10 hp, the most powerful one being 68 hp.
Growth
The 1867 Suomi, delivered to the Finnish Customs, was Crichton's largest ship produced by then. It was 110' 6" long, 18' wide, the draught was 7' and displacement 107 tonnes. The engine output was 320 ihp. Although most of the subsequent orders consisted of relatively small vessels, such as longboats, tug boats and small sailboats, Crichton predicted that demand for larger ships would grow and he wanted to follow the trend. Despite of the famine years in the 1860s, the yard sales grew steadily. In about 1870 Crichton bought more land from Itäinen Rantakatu 60–62 and built a larger, in particular wider slipway diagonally next to the river.
In March 1872 W:m Crichton & C:o received the largest order until then. The customer was Ångfartygsaktiebolaget Åbo, which ordered a passenger and freight ship for the Stockholm – Turku – Helsinki – Saint Petersburg route. The draught of the ship was 437 tonnes and the price was 200 000 marks. The delivery was scheduled for spring 1873. Although the size of the ship did not cause problems, the short lead time did – the ship, named Åbo ("Turku"), was launched at the end of May 1873 and it was ready for sea trial only at the end of September. The interior outfitting work was outsourced to the Turku Old Shipyard. Although Crichton presumably paid high penalties for the delay, the customer was satisfied with the result and the performance of the 100-hp compound engine which provided maximum cruise speed of 10 knots. In 1875 Turun Höyrylaiva Oy ordered a passenger steamer. Its size was 175×26×10 feet and net tonnage 357. Her 125-hp compound engine gave a maximum speed of 11 knots. S/S Finland was handed over in June 1876, now in due time.
In the early 1870s W:m Crichton & C:o employed 400 people. Most of the sales came from steam engines and other machinery, and just 30% of turnover came from ships. The company produced most of the needed tools, even drilling and turning machines, in its own workshops. Rapid growth of the company and the large orders required investments and various financial arrangements, creating large risks to its owners. At about the same time with making quotation for S/S Finland in 1874, the owners converted W:m Crichton & C:o a limited company in order to assure their personal property. The capital was 1 500 000 marks which was divided into 30 shares, each of 50 000 marks value; 20 shares were owned by William Crichton and 10 shares by trading house Julin. Erik Julin died already in the same year, after which the trading house was led by his son John Julin.
Due to the growing demand of larger ships Crichton became interested at the Turku Old Shipyard company located at the opposite side of river Aura. The yard had capacity for building large hulls. During the 1870s William Crichton grew his ownership in the Turku Old Shipyard and by the end of the decade he became majority shareholder and was appointed head of the company in 1878. Straight after this Crichton started investments at the old yard to enable building of large steel hulls; a workshop was built for bending steel ribs and after few years there were sheet steel forming and riveting facilities. When Crichton received two large orders with two steel ships in each in 1881, the hull production of Solovetski, Murmanski, Zurukan and Spasiteliy was subcontracted from the Old Shipyard. During that year Crichton handed over total 12 vessels, including customs boat Wiikinki, paddle steamer Krasnovodsk and six smaller vessels. Crichton became sole owner in 1883 and merged the yard to his own company. Subsequently, W:m Crichton & C:o became by far the largest company of Turku; the headcount was 936 and turnover 1 589 000 marks.
In 1879 Englishman John Eager became the company head designer. He was specialised on fast boats and designed a number of new vessel types, such as torpedo boats, gunboats and tanker ships.
When the Imperial Russian Navy started an extensive investment programme 1881, which included substantial extension of the Pacific fleet, the company received many beneficial orders. Crichton and Eager had created good contacts with the Russian authorities who decided about state purchases. The company gained good reputation in Moscow industrial fair in 1882, when W:m Crichton & C:o won a significant award, the Empire Seal. Despite of the positive reputation, getting orders from the state required a complex process, including formal and informal negotiations, official and unofficial money transfers. The company had a dedicated contact person in Saint Petersburg, commercial councellor F.W. Martinson, who handled the negotiations with the Russian officials.
In 1882 Crichton received orders for four tankers. 980-tonne Lastoska was powered by a 380-hp engine and delivered to an Astrakhan based customer. 670-tonne sisters Obiyt, Armenia and Atmosfer featured 80-hp engines and had length of 152' and width of 23'. The customer was Naphta Transport Company in Baku. It is possible that the ships were built by using drawings from Swedish Motala Works, which had built the world's first tanker ship just a few years earlier.
Years 1881–1882 were the record-breaking years for Crichton; the good order level reached until 1887. During these years W:m Crichton & C:o got 113 orders. Most of the vessels were smaller, such as longboats, small motorboats and sailing boats. Only three ships were large, over 1 000 tonnes in size. Notable passenger ships built in the 1880s were the 1887 delivered 345-tonne Ebba Munck and 1888 delivered 183-tonne Heben.
The premises were extended again in 1887, when two large sheds with slipways were built on the west bank.
John Eager's era
Changes in ownership
The company faced a major loss in April 1889 when William Crichton passed away just at the age of 62 years. John Eager took his post as company manager. None of Crichton's 12 children wanted to take the business. Shortly after four shares were sold to Eager, one to the major funding bank Nordiska Aktiebanken för Handel och Industrie. In the 1890s 13 of the remaining 15 shares were sold. Count Anatol Orloff-Davidoff from Saint Petersburg bought 12 shares and another nobleman from the same city, Ludwig von Havemann, bought one share. Supposedly, the Russia-based owners contributed the rapid growth of the company after the mid-1890s, but speculatively they also initiated the disastrous decision of starting operations in Saint Petersburg.
Trading house John Julin held ten shares; after death of John Julin the shares went to his two daughters, who were married to prosperous brothers Ernst and Magnus Dahlström. In 1898 the shares were transferred to trading house C. M. Dahlström and both brothers represented five shares each.
Count Orloff-Davidoff gave his shares to his sons Alexander and Alexei in 1906. Other new owners then were Russian German businessman Alfred Meiser and colonel Karl Björksten.
Ships for the imperial family
W:m Crichton & C:o won again the Empire Seal in industrial fair in Nizhny Novgorod in 1896; it also had an own exhibition stand with range of small steam engines in Paris World Expo in year 1900. The company won one gold medal but never managed to sell products to Central and Western Europe.
Grand duke Alexander Mikhailovich ordered a yacht from W:m Crichton & C:o in 1895. The 1897 built over 1 000-tonne steam yacht Tamara was the most luxurious ship built by Crichton. No records about the interior have survived but it was most likely ordered from Saint Petersburg. The length of the ship was 79.2 metres and the two 600-hp triple expansion steam engines provided maximum speed of 14 knots.
Another notable vessel delivered in 1897 was not significant in size, but the customer and structure makes it noteworthy. It was small steamship Peterhof, ordered by the Russian czar Nikolai II, who used it for travelling between the capital and Peterhof Palace. The ship was almost entirely built from brass alloy called delta metal, anti-magnetic material, which would not activate potential naval mines set by terrorists. The structures on the deck were decorated by gold plated mouldings. The ship was 61' long, 10.5* wide and its draught was 5'2". The 185-hp steam engine gave speed of 13 knots.
In 1903 the company delivered two steam cruisers, Kondor and Berkut, to the Russian Customs. Both of them had 900-hp engines and maximum speed of 14.5 knots.
The last luxurious vessel built by W:m Crichton & C:o was the 1905 delivered steam yacht Neva, which was built to the Navy Minister Aleksei Birilev. The ship was exceptionally paddle steamer. The engines generated output of 1 100 horsepower which gave her maximum speed of 15 knots, which is high for a such type of vessel.
At the turn of the century the company was still healthy, although delays in payments from Russia caused occasional shortages in cash reserves. During those years the company could not pay dividend. Due to unprofitable deals the company made loss by 153 000 marks in 1904 and 335 000 marks in 1905 although the yard was fully employed. The steam yacht Neva had been sold for 1 068 000 marks with a small profit, but eventually the project caused loss of 226 655 marks; the last part of the payment came from the customer as late as in 1909.
John Eager retired in summer 1906 and the new manager became Karl Björksten.
Okhta shipyard
In 1896 W:m Crichton & C:o leased premises in Okhta, Saint Petersburg, for building vessels and their engines for the Russian Admiralty. In the agreement the company committed to invest on the buildings and machinery, but despite of large sums spent on the facilities, a single steam engine was never built. Since the very beginning operations were poorly organised; schedules were not kept and raw materials were used wastefully. Until 1907 the losses at Okhta were caused by penalty payments and investments but after that some bad deals made earlier started to cause massive losses, most significant ones being four Kaiman-class submarine hulls, two minelayers and two icebreakers. In 1907 Björksten demanded that the technical management at Okhta yard must be changed. This happened in 1908. In 1909 the situation looked still hopeful, but the interest charges caused by the earlier losses made the situation unbearable.
Navy vessels
W:m Crichton & C:o produced its first torpedo boats for the Imperial Russian Navy in 1877–1878. These early models were small and featured steel hull and one or two torpedo launchers. The engines were 220-hp high pressure steam engines. The drawings for the boats were made by Admiralty Technical Department. Most of the boats were produced at Saint Petersburg yards. Crichton produced six units from the total 100 boats ordered by the Baltic Fleet.
In 1885 W:m Crichton & C:o produced its largest vessel by then, gunboat Bobr ("Beaver") with displacement of 1 187 tonnes. She featured two engines with total output of 1 140 hp, which gave her maximum speed of 11 knots. The vessel was heavily armed; two 9" cannons on the bow and a 6" cannon on the stern and additional smaller six 42-mm and four 37-mm guns. The gunboat served in the Russian Pacific fleet until it sank due to Japanese fire in Port Arthur bombing in 1904.
The Imperial Russian Navy kept modernising its fleet; in 1887 Crichton received and order for eight new generation torpedo boats. Four of them, Borgo, Ekenes, Kotka and Dagö had engine power of 1 000 hp. The four bigger ones Gogland, Nargen, Polanger and Pakerort had two engines, each of them with 1 000 hp output. The maximum speed was between 18–21 knots.
In 1893 two torpedo cruisers Vsadnik and Gajdamak were launched. The armour consisted of six 47-mm and three 37-mm guns and one torpedo launcher placed on the deck. The 3 300-hp engine output gave maximum speed of 21 knots. Both ships sunk in Port Arthur in December 1904 but Japanese lifted and repaired them, after which they served in the Imperial Japanese Navy until 1914.
An even bigger torpedo cruiser, 500-tonne torpedo cruiser Abrek was handed over in 1897. The value of the order was 1 340 000 marks. The 4 500 hp engine power enable maximum speed of 21 knots. Abrek featured two 75-mm and 47-mm guns and two torpedo launchers placed on the deck. The ship grounded at the front of Tallinn in 1906 and was removed from craft due to the damages.
In 1898 new Sokol type of torpedo boats, Kretyet and Korsyun, were launched. They were based on drawings made by Scottish Yarrow & Co after assignment of the Russian Navy. The total number of the boats was 26 and all the others were built at yards in Saint Petersburg area. Two high-power steam engines gave output of total 3 800 hp and maximum speed of 28 knots; there were two torpedo launchers and four guns size between 47 mm and 75 mm.
The Russian Navy ordered two Cyklon type torpedo boats which were designed by French Normándie shipyard. The engine output was the same 3 800 hp as in Sokol type but due to lighter structure and smaller fuel capacity they reached 29 knots' maximum speed. The ships were handed over in 1904.
After suffering heavy losses in the Russo-Japanese War, the Russian Navy placed orders for new craft: total 16 units of 650-tonne torpedo destroyers with engine power of 7 300 hp were ordered from various yards, two of them from W:m Crichton & C:o. The 75-metre vessels were designed by Stettiner Vulkan in Germany and featured two 100-mm and on 37-mm guns, four machine guns and three torpedo launchers for 450-mm torpedoes. The top speed of Ochotnik and Pogranityniy reached 25 knots. The vessels were handed over in 1906 and the deal of total 4 340 352 marks was the biggest the yard ever got. These were the last torpedo boats ever built in Turku. The orders of the Russian Navy ceased in 1906 due to political tensions between the Empire and Grand Duchy.
Combustion engines
The decision of starting combustion engine production originated most likely from Karl Björksten. As the company did not have the needed expertise, Björksten bought production licence for an engine type developed by Swedish engineer H.A. Bertheau. The first series of ten units was produced in 1906 and delivered to Okhta shipyard, to be mounted on cutters built there. The engine construction turned out to be technically failed; the engine problems delayed the handover and the company had to pay significant penalties. By 1907 total 24 engines were produced, of which 16 in Turku and 8 in Okhta, when the project was filed.
In 1909 W:m Crichton & C:o started producing Loke engines, which were possibly also of Swedish design. The first two units were built in Okhta and installed to power a generator of a lightvessel. Total 73 units were produced until 1912; 16 in Turku and 57 in Okhta.
Crichton's engines lost in properties to its neighbour and main competitor, Vulcan. Also another nearby company, Andrée & Rosenqvist, started engine production in 1912 and became successful in the field. Both of these companies had designed their own engines and had a lot of knowledge about engine production, but Crichton had chosen to work with bought drawings without understanding about the product.
1906–1913: decline and bankruptcy
Unprofitable deals, such as steam yacht Neva built in Turku in 1904 and four submarine hulls built in Okhta and handed over in 1905 caused massive losses to W:m Crichton & C:o. The failed combustion engines caused delays and warranty costs. Moreover, operations had relied on prepayments of new orders and therefore decline of order intake in 1906 led quickly to cash shortage. Ship order intake dropped drastically in 1906 leading to dismissal of hundreds of employees at the Turku yard. By spring 1907 the company was in a serious crisis. In addition to cash shortage the company's machinery and facilities were in poor shape and old fashioned, causing low productivity. The facilities required immediate investments, for which the company had no funds.
While steam engine production was still profitable and they were produced for the ships built at Okhta yard and sold also for some other yards in Saint Petersburg, only few boilers were built due to old and shattered boiler shop in Turku. Therefore, most of the Okhta built ships featured locally built boilers. All the engines were compound type with output range of 80–120 ihp, featuring 9" and 17" bores and 10" long stroke, or 9½" and 19" bores with stroke of 12".
In 1908 W:m Crichton & C:o made loss by nearly 1.4 million marks and the outstanding debt reached 3.7 million marks. The main creditors were banks, Orloff-Davidoff brothers and C. M. Dahlström company. The share capital was doubled from 1.5 million marks to 3 million in 1909, and in 1910–1911 the losses dropped to 290 000 marks. In the meantime the sales still declined and debt increased: at the end of 1911 the company was loaded by total 8.4 million marks' debt burden. Despite of the efforts, W:m Crichton & C:o was eventually filed for bankruptcy on 24 April 1913.
Re-establishment
Okhta yard was closed and the moveables were sold in the following year. C. M. Dahlström company held claims of over 4.5 million marks to the bankrupt estate. The Dahlström brothers took immediate actions to restart the operations in Turku, in order to secure their property. Therefore, only such property was sold that was not necessary for production. New company, Aktiebolaget Crichton, was founded in June 1914 under leadership of the Dahlström brothers.
Some of the longest-time served vessels
The hull of the 1860s built paddle steamer Lahtis has survived. The ship is currently undergoing a thorough rebuild.
Many of the W:m Crichton & C:o's vessels built for the public authorities served for decades, some of them surviving over 100 years of age. The main reason for this was the Finnish government's purchasing policy; as long as it was cheaper in a short term to operate old ships rather than invest on new craft, the old vessels were kept in service. The 1867 built steam cruiser Suomi, that served as customs control vessel, was in use until 1935. Hydrographic survey ship Sekstant, later Åland, operated from 1872 until 1959. The 1878 built light vessel Taipaleenluoto served on lake Ladoga until Winter War and the subsequent Moscow Peace Treaty in 1940, after which it stayed in the territory ceded to Soviet Union; the fate of the ship is unknown. Customs control ship Nordvakten built in the same year was broken up in 1959. The
Light vessel Relandersgrund built in 1888 served until 1977, when it was decommissioned under name Vuolle. The badly decayed ship was salvaged from a shipbreaker in 1991, after which she was thoroughly renovated and nowadays she serves as a restaurant ship in Helsinki. Relandersgrund is the oldest surviving Finnish light vessel.
In 1892 the company built two light ships, which were sisters: Nahkiainen was used until 1976 and Ärangsgrund was later converted into cargo ship that operated until the 1980s. The 1893 built pilot ship Saimaa has remained nearly original and she still operates in summertime by her Crichton-made steam engine.
References
Sources
Engine manufacturers of Finland
Crichton-Vulcan
1842 establishments in Finland | {
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Q: Mootools and Win8 Store app development Question 1:
Can we use Mixin pattern of Mootools in the development of Windows 8 Metro style app? In other words, can we override/substitute/extend WinJS.Class with Mootool's class?
Question 2:
For example, in Mootools if we have a base class Human:
var Human = new Class({
initialize: function(name, age) {
this.name = name;
this.age = age;
},
isAlive: true,
energy: 1,
eat: function() {
this.energy = this.energy + 1; //same as this.energy++
}
});
(using Mixin) an interface class Warrior:
var Warrior = new Class({
energy: 100,
kills: 0,
attack: function(target) {
target.isAlive = false;
this.energy = this.energy - 5;
this.kills++;
}
});
and a derived / concrete class Ninja:
var Ninja = new Class({
Extends: Human,
Implements: Warrior,
initialize: function(name, age, side) {
this.side = side;
this.parent(name, age);
}
});
How would we say this in WinJS accent using WinJS.Class.define, WinJS.Class.derive and WinJS.Class.mix?
A: If you want to use MooTools in your WinJS app, it should just work for the most part. There may be some warnings at startup, but as long as it doesn't violate the security framework for dynamically generated content, MooTools itself should just work. I wouldn't try to splice MooTools' code into WinJS, just use it as is.
WinJS.Class methods are, just like MooTools, defining JavaScript prototypes under the hood. "Types" you define should work together regardless of if you used MooTools or WinJS.
As far as your second question, I think you can do everything you need with just WinJS, the syntax is just different.
Defining your "human" constructor is straightforward:
var Human = WinJS.Class.define(
function(name, age) {
this.name = name;
this.age = age;
},
{
isAlive: true,
energy: 1,
eat: function() {
this.energy = this.energy + 1;
}
}
);
A mixin is defined simply as an object:
var Warrior = {
energy: 100,
kills: 0,
attack: function(target) {
target.isAlive = false;
this.energy = this.energy - 5;
this.kills++;
}
};
To do a derivation, you can use WinJS.Class.derive. This only gives you the inheritance part:
var Ninja = WinJS.Class.derive(Human,
function(name, age, side) {
this.side = side;
Human.call(this, name, age);
}
);
Then you do the mixin via WinJS.Class.mix:
WinJS.Class.mix(Ninja, Warrior);
And you should be all set to do:
var clyde = new Ninja("Clyde", "12", "confused");
| {
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} | 8,574 |
Current Positive Discipline Whole School Implementation Research Report
Research Supporting Positive Discipline
The Effectiveness of Adlerian Parent and Teacher Study Groups
Efficacy of Class Meetings
Learning Through Positive Discipline
The Positive Discipline School
The Effects of Class Meetings on Fifth-Grade Student Behavior
Authoritative Parenting and Outcomes of Positive Discipline Parent Training
Adlerian Resources
Adler Journals
Featured Dissertation
Holliday, M. (2014). Authoritative parenting and outcomes of Positive Discipline parent training: Parenting style and perceived efficacy. (Doctoral dissertation). Available from ProQuest Dissertations and Theses database.
The Adler School of Professional Psychology
This study was designed to investigate the impact of Adlerian parent trainings on parenting style and perceived competence, in order to determine if Positive Discipline parent training courses promote the authoritative parenting style. It was hypothesized that an Adlerian parent training would both promote the authoritative style and reduce authoritarian style and permissive style. It was also hypothesized that after attending a parent training, parents would note an increase in their sense of competence as parents. The central constructs were assessed through an online survey that included a measure of parenting style (Parenting Styles and Dimensions Questionnaire; PSDQ) and a measure of parenting competence (Parent Sense of Competence; PSOC). The sample consisted of 101 parents who attended one of 26 distinct Positive Discipline parent training group classes offered in cities across the United States. For the study, parents were assessed for parenting style and competence before the start of the course, after they completed the course, and at a 3-month follow-up period. Results indicated that parents experienced significant increases in both authoritativeness and sense of competence from pre-test to post-test. There was an even stronger significance associated with the increase in authoritativeness from pre-test to the three-month follow-up. Results also confirmed the hypothesis that attendance at the parent training would lead to reduced levels of authoritarian and permissive parenting styles. The study provides emperical support for the theoretical link between the Adlerian parenting model and the authoritative parenting model.
The Journal of Individual Psychology
Positive Discipline uses the theory of Individual Psychology to teach parents and educators effective methods for helping children become responsible, respectful, and resourceful. Positive Discipline is based on the teachings of Alfred Adler and Rudolph Dreikurs. The Journal of Individual Psychology provides a forum for dialogue pertaining to Adlerian practices, principles, and theoretical development. Articles relate to theoretical and research issues as well as concerns of practice and application of Individual Psychology. Below is a sample of articles published in the Journal of Individual Psychology specifically related to Positive Discipline or Adlerian practices incorporated by Positive Discipline. These journal publications show the rich history and foundation on which Positive Discipline is based. The editors of the Journal place great emphasis on maintaining high overall quality of research and scholarly contributions. - See more at: http://utpress.utexas.edu/index.php/journals/journal-of-individual-psychology
Gfroerer, K., Nelsen, J., & Kern, R. (2013). Positive Discipline: Helping children develop belonging and coping resources using Individual Psychology.
The Journal of Individual Psychology. 69, 294-304.
Abstract: The authors discuss the Positive Discipline program, which is based on Adlerian theory, with particular emphasis on fulfilling the individual's need to belong. The Positive Discipline program is available through a variety of media, including books, a website, audio and video recordings, and apps. Specific Positive Discipline tools that support the development of a child's sense of belonging and the development of life skills are discussed. These Adlerian methods can be practically applied in a variety of settings: schools, clinical practice, and through parent education.
McVittie, J, & Best, A. (2009). The impact of Adlerian-Based parenting classes on self-reported parental behavior.
The Journal of Individual Psychology, 65, 264-285.
Abstract: Public health research shows that the authoritative parenting style protects youth from risky and dangerous behavior. The purpose of this study was to evaluate whether Adlerian-based parent education classes influence parental behavior in the direction of being more authoritative. To this end, over 1,250 participants completed assessments at the end of 110 Adlerian parenting classes in the United States and Canada. Parent-guardians reported statistically significant changes in behavior: setting clearer limits, increasing their sense of positive connection, and decreasing harshness. These changes toward a more authoritative parenting style (based on parent report of their own behaviors) were statistically significant. The greatest changes were in younger parents, women, those with the lowest income, and those with fewer children. Results were also useful for the parent educators in assessing their teaching.
Nelsen, J. (1985). The three R's of logical consequences, the three R's of punishment, and the six steps for winning children over.
Individual Psychology: The Journal of Adlerian Theory, Research & Practice, 41, 161-165.
Abstract: Focuses on the concept of logical consequences, punishment and steps in the winning of children. Enumeration of the R's of logical consequences; Purpose of the R's in punishment; Significance of the winning techniques to children after cooling-off period.
Copyright of Journal of Individual Psychology is the property of University of Texas Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) The information above was shared with permission from University of Texas Press. | {
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L'Agorà dei giovani italiani è stata un'iniziativa pastorale della Chiesa cattolica italiana dedicata ai giovani.
Tale attività di evangelizzazione, indirizzata alla promozione del coinvolgimento dei giovani italiani nelle attività della Chiesa cattolica, è stata proposta nella sessione di marzo 2006 della Conferenza Episcopale Italiana e si è articolata in un percorso triennale (2006-2009). Ognuno dei tre anni pastorali è caratterizzato da:
un tema e dei contenuti;
una modalità di esercizio della testimonianza;
l'ascolto
la relazione
la cultura
la proposta di iniziative e progetti a livello diocesano;
un evento nazionale conclusivo dell'anno.
Il programma dell'Agorà
Anno pastorale 2006-2007
Il tema del primo anno è "Come io vi ho amato" (Gv 15,9): questo anno, dedicato all'ascolto del mondo giovanile, ha avuto come obiettivo il portare la Chiesa a contatto con il mondo esterno, ovvero fuori dagli spazi che le sono propri, per incontrare i giovani dove essi si trovano,
A livello diocesano dovevano instaurarsi iniziative di contatto e ascolto tra gruppi, comunità, diocesi.
Culmine del primo anno è l'incontro nazionale di Loreto: papa Benedetto XVI ha incontrato i giovani italiani vicino al Santuario della Santa Casa nei giorni 1º e 2 settembre 2007.
Anno Pastorale 2007-2008
Il tema del secondo anno è "Mi sarete testimoni" (At 1,8). L'anno è
A livello diocesano si sono sperimentate iniziative di "missioni feriali".
Il secondo anno è culminato con la Giornata Mondiale della Gioventù 2008, che si è tenuta a Sydney, in Australia, dal 15 al 20 luglio 2008.
Anno Pastorale 2008-2009
Il tema del terzo anno è "Fino ai confini della terra" (At 1,8). Questo anno è dedicato alla dimensione culturale e sociale dell'evangelizzazione.
A livello diocesano vi erano iniziative di cultura, comunicazione, carità e politica, che ha lanciato un "progetto culturale dei giovani italiani".
Il percorso triennale si è concluso con un evento che si è tenuto simultaneamente in ogni diocesi italiana, non solo nelle chiese ma anche in qualche "nuovo santuario" del nostro tempo (centri commerciali, stazioni, cinema, piazze, stadi, luoghi dell'emarginazione).
Voci correlate
Giornata Mondiale della Gioventù
Conferenza Episcopale Italiana
Prelatura territoriale di Loreto
Collegamenti esterni
Chiesa cattolica in Italia | {
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{"url":"https:\/\/imathworks.com\/tex\/tex-latex-change-color-in-beamer-theme\/","text":"# [Tex\/LaTex] Change color in beamer theme\n\nbeamercolor\n\nI'm preparing a presentation using the following beamer theme:\n\n\\usetheme{AnnArbor}\n\\usecolortheme{beaver}\n\n\nIs there any easy way to change the yellow color to cyan each time it is used in the theme? I have used Latex for some time but I have absolutely no idea how beamer works.\n\nEdit: Here is a MWE. When I compile this, the title box is yellow, and that is the color I wanted to change to blue.\n\n\\documentclass{beamer}\n\\usepackage[spanish,activeacute]{babel}\n\\usetheme{AnnArbor}\n\\usecolortheme{beaver}\n\\begin{document}\n\\title{My title}\n\\author{My name}\n\\date{\\today}\n\\begin{frame}\n\\titlepage\n\\end{frame}\n\\end{document}\n\n\nAnnArbor defines the background color for titles (and similar) with:\n\n\\setbeamercolor{titlelike}{parent=structure,bg=yellow!85!orange}\n\n\n(see beamerthemeAnnArbor.sty).\n\nTo change it into a kind of blue you can use something like:\n\n\\documentclass{beamer}\n\\usepackage[spanish,activeacute]{babel}\n\\usetheme{AnnArbor}\n\\usecolortheme{beaver}\n\\setbeamercolor{titlelike}{parent=structure,bg=cyan}\n\\begin{document}\n\\title{My title}\n\\author{My name}\n\\date{\\today}\n\\begin{frame}\n\\titlepage\n\\end{frame}\n\\end{document}\n\n\nThe result is:\n\nChanging the foreground color is as easy, e.g., use\n\n\\setbeamercolor{titlelike}{parent=structure,fg=yellow,bg=cyan}\n\n\nto get\n\nSee the beamer manual for more information about \\setbeamercolor and keys like fg and bg.","date":"2023-03-29 09:58:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.859399139881134, \"perplexity\": 2728.9302258956636}, \"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\/1679296948965.80\/warc\/CC-MAIN-20230329085436-20230329115436-00210.warc.gz\"}"} | null | null |
Q: Using common/default properties with multi-environment resource filtering? I have a Maven project with the following properties file structure:
./clientA/
dev.properties
staging.properties
production.properties
./clientB/
dev.properties
staging.properties
production.properties
and so forth and so forth.
90% of the properties are all the same across the clients within the same environment (e.g. dev props on clientA and clientB are usually all the same). The properties that are the same are not, however, guaranteed to be the same across every client.
What I would like to do is create a common.[dev|staging|production].properties files at the root level, move all common properties there, then, as needed, override the common properties at the client properties files, but that doesn't appear to be supported by Maven, e.g. I can't define the same property in two properties files used for filtering, because Maven uses the value only on the first properties file processed.
How would I achieve the same functionality in Maven? Is there another way to solving this issue? Or is it not possible?
Here's what I would like to do:
common.dev.properties:
foo=bar
clientA/dev.properties
foo=snafu
custom.prop=clientA Value
clientB/dev.properties
custom.prop=clientB Value
Filtered values should be:
clientA:
foo=snafu
custom.prop=clientA Value
clientB:
foo=bar
custom.prop=clientB Value
Instead of that what happens currently is:
clientA:
foo=bar (i.e. the common.dev.properties foo property value can not be overriden)
custom.prop=clientA Value
clientB:
foo=bar
custom.prop=clientB Value
Resources:
src/main/resources/test.properties
foo=${foo}
custom.prop=${custom.prop}
Resources-plugin config
<build>
<filters>
<filter>${basedir}/env/common.properties</filter>
<filter>${basedir}/env/${client.name}/${env}.properties</filter>
</filters>
<plugins>
<plugin>
<groupId>org.apache.maven.plugins</groupId>
<artifactId>maven-resources-plugin</artifactId>
<version>2.5</version>
<configuration>
<encoding>UTF-8</encoding>
</configuration>
<executions>
<execution>
<id>copy-resources</id>
<phase>process-resources</phase>
<goals>
<goal>copy-resources</goal>
</goals>
<configuration>
<outputDirectory>${basedir}/target/classes</outputDirectory>
<resources>
<resource>
<filtering>true</filtering>
<directory>src/main/resources</directory>
</resource>
</resources>
</configuration>
</execution>
</executions>
</plugin>
</plugins>
</build>
| {
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With its patented digital monitoring system, the Cadillock 4000 Bike constantly checks the minimum voltage level that the vehicle battery must maintain to be able to start the motorcycle. As soon as the voltage has fallen short of 11.9 V for a period of more than 60 seconds, the Cadillock 4000 Bike automatically interrupts the vehicle power supply.
Thus the Cadillock 4000 Bike protects your vehicle's battery from falling to a dangerously low state of charge or even being completely discharged, and reserves the necessary battery energy for an engine emergency start. Thus the Cadillock 4000 Bike prevents your vehicle not being able to be started due to a flat battery.
By interrupting the power supply by the Cadillock 4000 Bike the status of the motorcycle battery is very similar to the state that would be achieved by disconnecting the motorcycle battery. The electrical current is shut off.
The anti-theft device is activated by remote control. The Cadillock 4000 Bike with its patented circuit configuration separates the battery power supply, thus making it impossible to start the vehicle.
There is no interference with the vehicle's electronics during the installation of the Cadillock 4000 Bike.
The Cadillock 4000 Bike can be installed in just three steps and is particularly easy and cost-effective. Only a wrench and a few minutes' time are needed.
Connect the positive cable of the vehicle to the pole of the Cadillock 4000 Bike.
Connect the battery clamp of the Cadillock 4000 Bike to the positive pole of the motorcycle battery.
Connect the ground cable of the Cadillock 4000 Bike to the negative pole of the motorcycle battery. | {
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Following the red stripe you are lead down, deep into the bowels of the complex. Gradually the bright lighting of the entrance dims down to a more subdued level. Cables snake along the wall and an industrial air creeps in.
Finally the corridor opens out into a viewing gallery that overlooks a gallery of strange equipment. Some of it moving, some of it standing - staidly changing lights in complex patterns across itself. Signs indicate sub sections of equipment, most of them you can't quite make out. But the ones that you can read include 'Autonomic Responses', 'Image Analysis' & 'Syntax Analyser'. Some of the machinery looks complex, some simple and yet others look deceptively simple.
A locked door leads out to the left down metal stairs to the gallery floor. Mounted at strategic locations around the window railing are touch screens with brief informational blurbs and logos on them. On closer inspection the logos reveal themselves to be small diagrams outlining the machines associated with the text of the display. | {
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'use strict';
const fs = require('fs-extra');
let root = process.cwd();
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process.chdir(root);
return fs.remove(path).then(function () {
fs.mkdirsSync(path);
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process.chdir(root);
return fs.remove(path);
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\section{Introduction}
\label{sec:intro}
Emulsion films have recently been considered as possible position detectors for low-energy antimatter studies. These studies include the AEgIS (AD6) experiment at CERN~\cite{aegisprop,AEgIS_emul1,AEgIS_emul2,nature}, whose goal is the measurement of the Earth's gravitational acceleration on antihydrogen atoms. Another collaboration proposed emulsions for their studies on positrons, as described in~\cite{quplas}. In particular, in the case of the AEgIS experiment, the position-sensitive detector must have a micrometer-level resolution to allow the required sensitivity of $\sim$1\% for the gravitational acceleration measurement.
Spatial resolutions of $\sim$1-2~$\mu$m can be achieved with emulsion films~\cite{ereditato}, and they have been exploited before for the reconstruction of antihydrogen impact points from annihilation products~\cite{AEgIS_emul2}. Films with this resolution, combined with a time of flight detector, could allow the experimental goal to be achieved. In the same paper, a preliminary study of antiproton-nuclei annihilations was also reported.
That study assessed particle multiplicities resulting from antiproton annihilations on emulsion films and aluminium. Recently, again within the context of the AEgIS experiment, similar measurements were performed by means of a silicon detector also acting as an annihilation target~\cite{silicon}.
Apart from the obvious applications in nuclear physics, measuring the decay products of low-energy antiproton annihilation in different materials provides a useful check of the ability of standard Monte Carlo packages to reproduce fragment multiplicities, type and energy distributions stemming from antiproton (or antineutron) annihilations on nuclei at rest. Although measurements of the multiplicities of pions and other charged particles with energies higher than $\sim$ 50 MeV are available in the literature~\cite{ref1994,ref1995}, the production of highly ionizing nuclear fragments with short range has not been studied sufficiently. A measurement of the multiplicities of charged products in antiproton-aluminium annihilations was reported in~\cite{AEgIS_emul2}, although only 43 events were analysed in this study, and the tracking efficiency of the detector was limited to 80\%.
In this paper, we present the results of a study of the multiplicities of charged annihilation products on different target materials, namely copper, silver and gold, using emulsion detectors at the Antiproton Decelerator (AD~\cite{AD}) at CERN.
\section{Experimental setup}
Emulsion detectors were used to study the antiproton annihilation products generated in different materials. Before reaching the targets, the 5.3 MeV antiprotons from the AD ($3 \times 10^7$ $\overline{p}$/shot every 100 s) were slowed down using several different titanium and aluminium foils with variable thicknesses. Finally, the beam was collimated in a vacuum test chamber after crossing a titanium vacuum separation window with a thickness of 12~$\mu$m.
The emulsion detector was situated at the downstream end of the vacuum chamber ($\sim$1~m in length), where it could be reached by a defocused beam of low-energy antiprotons ($\sim$100~keV). This distance from the degrading layer was necessary to reduce the background due to annihilations taking place at the moderator.
A sketch of the experimental assembly is shown in Figure~\ref{fig:setup}. The emulsion detector was operated under ordinary vacuum conditions ($10^{-5}-10^{-6}$ mbar). The antiproton intensity measured by the detector was approximately 150/cm$^2$ per shot.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{setup_3.png}
\caption{Schematic setup of the experiment. An enlarged view of the target region is shown on the right.
} \label{fig:setup}
\end{figure}
For this study emulsion detectors were produced at the Laboratory for High Energy Physics (LHEP) of the University of Bern by pouring the emulsion gel with a thickness of $\sim$100~$\mu$m, provided by Nagoya University (Japan), on a glass plate (for a review on the emulsion technology see ~\cite{ereditato}). Glycerin was added to so that the emulsion could operate in vacuum~\cite{AEgIS_emul2}. This emulsion features a very low background with approximately 1-2 thermally induced grains per 1000 $\mu$m$^{3}$~\cite{quplas}.
\begin{figure}
\begin{center}
\begin{minipage}{0.6\textwidth}
\includegraphics[width=1\textwidth]{frame.png}
\end{minipage}
\begin{minipage}{0.35\textwidth}
\includegraphics[width=1\textwidth]{bare_emul.png}
\end{minipage}
\caption{Left and middle: Target arrangement in the two assemblies fixed to the emulsion films. Right: Antiproton annihilations on the bare emulsion surface.
} \label{fig:frame_and_bare_emul}
\end{center}
\end{figure}
Foils of copper, silver and gold, each having a thickness of 10~$\mu$m, were placed as targets at the end of the vacuum chamber, in front of the emulsion detectors. Figure~\ref{fig:frame_and_bare_emul} shows the targets ($2 \times 2$ cm$^{2}$ each) fixed to the emulsion film and an example of antiproton annihilating on the emulsion surface. At the antiproton energies obtained after degrading, all annihilations are expected to take place within a few $\mu$m of the target surface. During data taking, we collected approximately 1500 antiprotons per cm$^2$ in about 10 AD shots.
\section{Data analysis and results}
Data recorded by the emulsion detectors were scanned by an automatic optical microscope and then analysed by exploiting a recently developed fast 3D tracking algorithm~\cite{atmic}. The measured tracking efficiency of our detector was approximately 99\% for minimally ionizing particles over a wide angular range, as reported in~\cite{atmic}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{trackdensity.pdf}
\caption{
Profile of detected tracks ($xy$ positions of tracks at the emulsion layer) in one of the assemblies. The top right part was not included in our study.
} \label{fig:trackdensity}
\end{figure}
Figure~\ref{fig:trackdensity} shows the profile of detected tracks ($xy$ positions of tracks at the emulsion layer) in one of the assemblies.
Among the reconstructed tracks we only considered those that were longer than 30 $\mu$m to avoid considering tracks that were due to the background.
An angular cut of 0.4$<$tan$\theta$$<$2.0 (22$^\circ$$<$$\theta$$<$63$^\circ$) was applied, where $\theta$ is the track angle with respect to the beam direction.
To reconstruct a vertex, at least two three-dimensional tracks were required. The efficiency of vertex reconstruction was estimated by applying the criteria given above to the output of the FLUKA simulation. It was found to be 22\% for copper, 24\% for silver, and 18\% for gold.
Figure~\ref{fig:vz} (left) shows the distribution of the reconstructed vertex position perpendicular to the film surface ($z$-direction) in a subarea of the copper target. The peak in $z$ is a measure of the target foil position (the gap between the emulsion surface and the target foil). This measurement was performed by segmenting the analysed area into smaller areas since the target foils were neither flat nor in contact with the film surface.
The surface topography obtained from the reconstructed vertices is shown in Figure~\ref{fig:topo} for copper, silver and gold targets.
In our analysis, we only considered regions of the emulsion film surface with the $z$ value smaller than 100~$\mu$m because the vertex reconstruction efficiency was uniform within a few percent in this region.
The analysed fiducial area was 1.68 cm$^2$ for the copper target, 1.96 cm$^2$ for silver and 0.80 cm$^2$ for gold.
The fraction of signal vertices became dominant by requiring vertex reconstruction at the position of the target foil.
The nearly flat distribution in Figure~\ref{fig:vz} (right) is due to combinatorial background in the vertex reconstruction. The main source of the background was due to accidental combinations of tracks coming from annihilations taking place upstream in the apparatus, which were not completely excluded by the angular cut due to the broad angular distribution. The fraction of tracks from signal vertices to all detected tracks was estimated using the vertex finding efficiency described above and found to be 9\% for copper, 9\% for silver, and 6\% for gold.
The number of background vertices was estimated using all the detected tracks in the analysed area by subtracting the above signal track fraction, randomizing positions and slopes of the remaining tracks, reconstructing the vertices and counting the number of vertices that mimicked annihilations in the target.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{vz_Cu1_2000x7000.pdf}
\caption{Left: Distribution of the reconstructed vertex position perpendicular to the film surface (z-direction). Right: Estimated combinatorial background.
} \label{fig:vz}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{topo_3.png}
\caption{Surface topography for copper, silver and gold targets obtained from the reconstructed vertices. The vertical scale refers to the distance from the emulsion film.
} \label{fig:topo}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth]{ntot_29.pdf}
\\
Copper
\\
\includegraphics[width=0.65\textwidth]{ntot_47.pdf}
\\
Silver
\\
\includegraphics[width=0.65\textwidth]{ntot_79.pdf}
\\
Gold
\caption{
Left: Background contributions to the total multiplicities for the different targets. Right: Multiplicity distributions after subtraction of the background. The colored histograms show the Monte Carlo predictions.
}\label{fig:ntot}
\end{figure}
The background estimated from measured data, which depends on the number of tracks in the event, is shown in the left panes of Figure~\ref{fig:ntot}, while the right panes show the multiplicity distributions after subtraction of the background, compared with the Monte Carlo predictions based on the CHIPS \cite{chips} and FTFP (FTFP\_BERT\_TRV) \cite{ftfp} models in the GEANT4 (4.9.5.p02) and FLUKA (2011.2c) \cite{fluka} frameworks.
A total of 654 signal annihilation vertices were reconstructed for copper, 941 for silver and 233 for gold.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{cos_sd.pdf}
\caption{Signal density (S.D.) distribution as a function of track angle with respect to the beam direction. Tracks below the black line are defined as being minimally ionizing.
} \label{fig:cos_sd}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{nsnh_29_3.pdf}
\\
Copper
\\
\includegraphics[width=0.7\textwidth]{nsnh_47_3.pdf}
\\
Silver
\\
\includegraphics[width=0.7\textwidth]{nsnh_79_3.pdf}
\\
Gold
\caption{
Reconstructed multiplicity distributions for annihilations in copper, silver and gold foils for MIPs (left) and HIPs (right). The colored histograms show the Monte Carlo predictions by CHIPS, FTFP and FLUKA. The error bars of the histograms account for uncertainties in the $dE/dx$ classification.
}\label{fig:nsnh}
\end{figure}
\renewcommand{\arraystretch}{1.2}
\begin{table}[hbtb]
\footnotesize
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|l|}{} & \multicolumn{4}{c|}{Average multiplicity for MIPs} & \multicolumn{4}{c|}{Average multiplicity for HIPs} \\
\hline
& Data & CHIPS & FTFP & FLUKA & Data & CHIPS & FTFP & FLUKA \\
\hline
Copper & 1.07 $\pm$ 0.07 & 0.98$^{+0.07}_{-0.09}$ & 1.59$^{+0.09}_{-0.14}$ & 0.83$^{+0.08}_{-0.11}$ & 1.54 $\pm$ 0.07 & 1.46$^{+0.09}_{-0.07}$ & 0.60$^{+0.14}_{-0.10}$ & 1.68$^{+0.11}_{-0.08}$ \\
\hline
Silver & 1.02 $\pm$ 0.06 & 1.04$^{+0.08}_{-0.09}$ & 1.64$^{+0.09}_{-0.13}$ & 0.73$^{+0.07}_{-0.09}$ & 1.71 $\pm$ 0.07 & 1.33$^{+0.09}_{-0.08}$ & 0.51$^{+0.14}_{-0.09}$ & 1.87$^{+0.09}_{-0.07}$ \\
\hline
Gold & 0.92 $\pm$ 0.09 & 1.21$^{+0.10}_{-0.11}$ & 1.75$^{+0.09}_{-0.13}$ & 0.81$^{+0.07}_{-0.11}$ & 1.60 $\pm$ 0.09 & 1.04$^{+0.11}_{-0.09}$ & 0.39$^{+0.13}_{-0.09}$ & 1.60$^{+0.11}_{-0.06}$ \\
\hline
\end{tabular}
\caption{Measured average multiplicity and Monte Carlo predictions by CHIPS, FTFP (GEANT4) and FLUKA. Statistical errors and uncertainties in the background estimation are combined and reported for each set of data. The errors in the predictions account for uncertainties in $dE/dx$ classification.
}
\label{tb:ave}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
We were also able to discriminate between heavily ionizing particles (HIPs) such as protons and nuclear fragments and minimally ionizing particles (MIPs), namely pions.
Continuous dense tracks correspond to HIPs, while faint tracks are produced by MIPs, since the aligned grains of these last tracks are separated.
The local energy deposition ($dE/dx$) of each track can then be assessed in terms of signal density (S.D.) along the reconstructed tracks, using
\[
S.D. = \sum_{x,y,z\in C}S_{xyz}/L.
\]
Here, $x$, $y$ and $z$ are the coordinates of voxels in the 3D image data. $C$ is a group of voxels in a cylinder along the track, and $S_{xyz}$ is an 8-bit grey-scale signal of the voxel. $L$ is the length of a reconstructed track in the 3D image data. The S.D. is proportional to the $dE/dx$ of the particle and does not depend on the angle. However, there is a saturation effect for higher values of $dE/dx$.
Figure~\ref{fig:cos_sd} shows that the S.D. distribution of tracks revealed a peak at 3000 for MIPs.
As the simulated $dE/dx$ distribution of MIPs peaked at 1.2 MeV$\cdot$g$\cdot$cm$^{-2}$, we define particles with $dE/dx$ smaller than 2.4 MeV$\cdot$g$\cdot$cm$^{-2}$, corresponding to an S.D. below 6000 $\mu$m$^{-1}$, as being MIPs. The complementary particles are defined as HIPs.
Figure~\ref{fig:nsnh} shows the track multiplicity distributions for MIPs and HIPs. The errors on the data are statistical.
The histograms represent the Monte Carlo predictions by CHIPS, FTFP and FLUKA. The error bars of the Monte Carlo predictions account for uncertainties in the $dE/dx$ classification. These uncertainties were estimated using simulations, which smeared the threshold for assigning tracks to either class of ionizing particles by 20\% and checked for effects on the multiplicity distributions for MIPs and HIPs.
The statistical errors for the simulations were 0.01-0.02, which are significantly smaller than the errors reported above.
The average measured multiplicities are summarized in Table~\ref{tb:ave}.
Both CHIPS and FLUKA are in good agreement for copper, particularly in the case of MIPs. Neither CHIPS nor FTFP accurately describe particle multiplicities for annihilations on silver and gold nuclei, while FLUKA more closely reproduce the data than the other models.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth]{A_dep_mip.pdf}
\\
\includegraphics[width=0.65\textwidth]{A_dep_hip.pdf}
\caption{Particle multiplicity from antiproton annihilations as a function of atomic number for MIPs (top) and HIPs (bottom).
} \label{fig:A_dep}
\end{figure}
The mean values of particle multiplicities measured for the three target materials are shown in Figure~\ref{fig:A_dep} as a function of atomic number along with the simulation outcome. Results obtained for MIPs with the FTFP model do not agree with our experimental data for any material, while those obtained with both CHIPS and FLUKA are in fair agreement as far as copper is concerned, although only FLUKA reproduces the higher atomic number behavior. Good agreement with CHIPS was also found for annihilation on bare emulsions and for aluminium~\cite{AEgIS_emul2}. Multiplicities related to HIPs are well described by the FLUKA simulation, while the CHIPS and FTFP models clearly underestimate the number of particles produced by antiproton annihilation.
\section{Conclusions}
The goal of the study presented in this paper was to measure the products of low-energy antiproton annihilation on different materials, utilizing a secondary beam line of CERN's Antiproton Decelerator in the AEgIS experimental area. The characteristics (e.g. hadronization and fragmentation multiplicities) of low-energy antiprotons annihilating on nuclei are not well known, and experimental data are needed to validate models used by simulation packages such as GEANT4 and FLUKA.
We exposed several thin targets (Cu, Ag and Au) to the antiproton beam and measured fragment tracks using emulsion detectors with a vertex position resolution at the level of a few micrometers, which allowed the separation between minimally and highly ionizing particles. The fragment multiplicities we measured were not well reproduced by the different models used in Monte Carlo simulation with the exception of FLUKA, which is in good agreement with the particle multiplicities for both minimally and heavily ionizing particles. Future measurements with more materials are needed to gain a better understanding of antinucleon annihilations also on low-$Z$ materials, and to obtain a full description in terms of particle types, multiplicities, as well as energy, for a more complete benchmarking of Monte Carlo simulations.\\
\acknowledgments
This work was supported by the Swiss National Science Foundation Ambizione grant PZ00P2\_154833; Istituto Nazionale di Fisica Nucleare; a Deutsche Forschungsgemeinschaft research grant; an excellence initiative of Heidelberg University; European Research Council under the European Unions Seventh Framework Program FP7/2007-2013 (Grants No. 291242 and No. 277762); Austrian Ministry for Science, Research, and Economy; Research Council of Norway; Bergen Research Foundation; John Templeton Foundation; Ministry of Education and Science of the Russian Federation and Russian Academy of Sciences; and the European Social Fund within the framework of realizing the project, in support of intersectoral mobility and quality enhancement of research teams at Czech Technical University in Prague (Grant No. CZ.1.07/2.3.00/30.0034). The authors would like to acknowledge the contributions by the mechanical workshop at LHEP.
| {
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\section{Introduction}
\vspace{-0.3cm}
Since is launch in 2002 the {\it INTEGRAL} (Winkler et al. 2003) satellite is performing four main scientific aims: (i) a deep
exposure of the Galactic central radian, (ii) regular scans of the Galactic Plane, (iii) pointed observations of the Vela region
and (iv) Target of Opportunity follow-ups. IBIS (Ubertini et al. 2003) is a hard X-ray imaging instrument onboard {\it INTEGRAL}
with a large field of view (30$^{\circ}$), and it is the basis of several {\it INTEGRAL} surveys. Through this unique
capabilities, IBIS
permits the detection of sources at the mCrab level with a typical localization accuracy of 2-3 arcmin above 20 keV (Gros et al.
2003). From the first to the fourth IBIS survey catalog, both sensitivity and sky coverage improved substantially, enabling the
increase of the number of detected hard X-ray sources from 123 in the 1$^{\rm st}$ catalog to 723 in the 4$^{\rm th}$ one. A
fraction of these objects ($\sim$30\% in all catalogs) had no known or evident counterpart at other wavelengths and therefore
could not be associated with any known class of high-energy emitting sources. For this reason, since 2004 our group has been
performing an observational campaign employing telescopes located in the northern and the southern hemispheres to obtain optical
spectroscopy of the putative counterparts of these hard X-ray emitting objects in order to determine their actual nature.
Here we want to briefly illustrate the method we use to associate the optical counterpart to the corresponding unidentified
hard X-ray source and the progress of our identifications from the 1$^{\rm st}$ to the 4$^{\rm th}$ IBIS survey catalog by
pointing out the contribution of our group to this identification work.
\vspace{-0.3cm}
\section{The IBIS/{\it INTEGRAL} soft gamma-ray surveys}
\vspace{-0.3cm}
The 1$^{\rm st}$ IBIS soft gamma-ray survey catalog was performed in the first year of satellite operations, the regular scans
of the Galactic Plane yielding a survey with a sensitivity down to $\sim$1 mCrab (Bird et al. 2004). This allowed detecting
more than 120 sources (22\% of which were unidentified), many of them detected for the first time above 20 keV.
The second IBIS survey catalog (Bird et al. 2006) increased the sensitivity unveiling more than 200 sources with $\sim$27\%
of them unidentified.
Within the third IBIS survey catalog (Bird et al. 2007) 421 hard X-ray sources were detected, and $\sim$28\% had no
classification, while the fourth catalog (Bird et al. 2010) contains 723 hard X-ray emitting objects, with as much as $\sim$
29\% of unidentified sources.
\begin{figure*}[th!]
\begin{center}
\includegraphics[width=12cm]{survey.eps}
\caption{Evolution of the percentages of the detected X-ray sources (divided according to their classification) from the
1$^{st}$ to the 4$^{th}$ IBIS survey catalog.
}\label{sur}
\end{center}
\end{figure*}
As one can see in Fig. \ref{sur}, the majority of the identified sources in the first three catalogs was made of Galactic
objects, while in the 4$^{\rm th}$ one extragalactic sources constitute the largest group. Moreover, a marked evolution in
the percentages of Galactic sources is apparent: Low Mass X-ray Binaries (LMXBs) fell from 43\% of the
1$^{\rm st}$ survey to 13\% in the 4$^{\rm th}$ one; a similar, albeit less sharp, reduction holds for High Mass X-ray Binaries
(HMXBs). The number of detected Cataclysmic Variables (CVs) kept instead stable across the four catalogs.
As already mentioned, the detection of extragalactic sources, that is Active Galactic Nuclei (AGNs), skyrocketed from 4\% to
35\%. Besides,
while the number of detected sources increased dramatically (by a factor of $\sim$6) across the four surveys (of course due to
the larger and larger instrument exposure available), the percentage of the unknown objects remained almost constant. As we
will show below, these two results are a clear demonstration that our observational multisite campaign for the search of
optical counterparts of {\it INTEGRAL} hard X-ray sources is highly effective.
\vspace{-0.3cm}
\section{The identification method}
\vspace{-0.3cm}\
The first step for the determination of the nature of unidentified sources belonging to the IBIS catalogs is to search for
counterparts at other wavelengths. This, albeit the {\it INTEGRAL} error boxes are definitely smaller and more easily explorable
with respect to those afforded by past hard X-ray missions, is still quite complicated due to the fact that the uncertainty
circle area (which is of the
order of $\sim$10 arcmin$^2$) contains a large number of objects, especially at optical wavelengths. One therefore needs to
reduce the search area down to a size of less than a few arcseconds.
Cross correlations with catalogues in other wavebands can then be used to reduce the positional uncertainty to facilitate the
identification process. Stephen et al. (2006), by cross-correlating the 2$^{\rm nd}$ IBIS and the {\it ROSAT} catalogs,
demonstrated that when a bright, single soft X-ray object is found within the IBIS error circle, it is almost certainly the
lower-energy counterpart of the {\it INTEGRAL} source. Thus, the presence of a catalogued or archival {\it Swift}, {\it ROSAT},
{\it Chandra} and/or {\it XMM-Newton} source within the IBIS error circle of a hard X-ray object marks the position of its
longer wavelength counterpart with a precision of a few arcseconds or better. Thereby reducing the search area of a factor of 10$^{4}$.
Similarly, when no soft X-ray information is available, far-infrared or radio catalogs can be used for this task.
In this case, however, the reliability of the correlation is less strong (see e.g. Masetti et al. 2008a and references
therein), and one always needs confirmation by means of a clear soft X-ray detection from the putative counterpart.
Likewise, the presence of a bright and peculiar optical object (e.g. an unidentified galaxy, or a poorly studied
emission-line star) within the IBIS error circle can be a hint for it to be the counterpart of the IBIS source; again, however,
one has to wait for confirmation of emission at soft X-rays from the optical object to definitely prove any association.
Once this smaller, arcsec-sized position is available, a search for optical counterparts within its area is made with the use
of archival images (e.g., those of the DSS II-red survey\footnote{{\tt http://archive.eso.org/dss/dss}}). Then, optical
spectroscopy is performed on the optical object(s) within this smaller error box: the source which presents remarkable spectral
features (typically strong Balmer, Helium or forbidden emission lines) can definitely be identified as the optical counterpart
of the IBIS hard X-ray emitter. The characteristics of the optical spectrum eventually allow us to determine distance,
reddening, chemical composition, and in the very end the nature, of the considered {\it INTEGRAL} unidentified source.
\vspace{-0.3cm}
\section{Telescopes involved in this activity}
\vspace{-0.3cm}
Starting with the very first pilot project performed at the Loiano Telescope (Masetti et al. 2004),
during the more than 5-year long hunt for the identification of {\it INTEGRAL} sources set on by our group, optical
spectroscopy useful for our identification was performed at various telescopes worldwide. Below we present the list of all
ground-based facilities used until now within this project:
\begin{itemize}
\vspace{-0.1cm}
\item 1.5m telescope at the Cerro Tololo Interamerican Observatory, Chile;
\vspace{-0.3cm}
\item 1.52m ``Cassini'' telescope of the Astronomical Observatory of Bologna, in Loiano, Italy;
\vspace{-0.3cm}
\item 1.8m ``Copernicus'' telescope at the Astrophysical Observatory of Asiago, in Asiago, Italy;
\vspace{-0.3cm}
\item 1.9m ``Radcliffe'' telescope at the South African Astronomical Observatory, in Sutherland, South Africa;
\vspace{-0.3cm}
\item 2.1m telescope of the Observatorio Astr\'onomico Nacional in San Pedro M\'artir, Mexico;
\vspace{-0.3cm}
\item 2.15m ``Jorge Sahade'' telescope at the Complejo Astron\'omico el Leoncito, Argentina;
\vspace{-0.3cm}
\item 3.58m telescope ``Telescopio Nazionale Galileo'' at the Observatorio of the la Roque de Los Muchachos
in Canary Islands, Spain;
\vspace{-0.3cm}
\item 3.58m NTT of the ESO Observatory in La Silla, Chile;
\vspace{-0.3cm}
\item 3.6m telescope of the ESO Observatory in La Silla, Chile;
\vspace{-0.3cm}
\item 4.2m ``William Herschel Telescope'' at the Observatory of Roque the Los Muchachos in Canary Islands, Spain;
\vspace{-0.3cm}
\end{itemize}
plus archival spectra from 6dF\footnote{{\tt http://www.aao.gov.au/local/www/6dF}} and SDSS\footnote{{\tt http://www.sdss.org}}.
\section{Results}
Up to the time of this conference (October 2009), within our identification program we produced 9 refereed papers
(Masetti et al. 2004, 2006a,b,c,d, 2007, 2008a,b, 2009) plus a number of conference proceedings and of short
communications. Going into details, our work allowed us to spectroscopically determine or confirm the nature of 104
unidentified {\it INTEGRAL} sources, which can be divided into several subclasses as follows (see also Fig. \ref{res}):
\begin{itemize}
\vspace{-0.3cm}
\item 4 (persistent) LMXBs;
\vspace{-0.3cm}
\item 14 Be/X HMXBs (often with a highly reddened optical counterpart);
\vspace{-0.3cm}
\item 5 HMXBs with supergiant companion (often fast X-ray transients);
\vspace{-0.3cm}
\item 57 nearby AGNs (30 Seyfert 1 and 27 Seyfert 2) with redshift between 0.011 and 0.422;
\vspace{-0.3cm}
\item 3 X-ray Bright, Optically Normal Galaxies (XBONGs);
\vspace{-0.3cm}
\item 2 high-$z$ blazars (with redshift $\geq$1);
\vspace{-0.3cm}
\item 2 BL Lacs;
\vspace{-0.3cm}
\item 12 magnetic CVs;
\vspace{-0.3cm}
\item 4 Symbiotic stars;
\vspace{-0.3cm}
\item 1 Active star.
\vspace{-0.3cm}
\end{itemize}
Among these, we would like to mention a few outstanding objects, such as the Symbiotic X-ray Binary IGR J16194$-$2810
(Masetti et al. 2007) and the high-redshift ($z$ = 2.40) blazar Swift J1656.3$-$3302 (Masetti et al. 2008a).
We also stress that this work allowed the detection of a large number of new AGNs, especially in the so-called `Zone of
Avoidance', i.e. along the Galactic Plane, where the presence of Galactic dust and neutral hydrogen severely hampered
past studies of AGNs at both optical and soft X-ray wavelengths. It also gave us the possibility of detecting a
substantial number of new, possibly magnetic, CVs (e.g. Landi et al. 2009; Scaringi et al. 2009).
It is moreover remarked that, despite recent claims (Cerutti et al. 2009), the present program halved the number of
unidentified sources detected in the 3$^{\rm rd}$ IBIS catalog, and the same bright goal is expected for the 4$^{\rm th}$
survey (see next Section).
Moreover, as a service to the community, we regularly maintain a web archive reporting the main properties of each
{\it INTEGRAL} source identified through optical and near-infrared spectroscopy. This archive can be found at the URL:
\begin{figure*}[th!]
\begin{center}
\includegraphics[width=5cm]{results.eps}
\caption{The percentages of the various unidentified sources identified by our group up to October 2009, divided
according to their classification.}
\label{res}
\end{center}
\end{figure*}
\noindent
{\tt http://www.iasfbo.inaf.it/extras/IGR/main.html} .
\vspace{-0.3cm}
\section{Outlook}
\vspace{-0.3cm}
As already remarked, the 4$^{\rm th}$ IBIS survey has about 29\% of sources which lack an obvious counterpart,
which means that it hosts 208 sources of unidentified nature.
As for the past catalogs, we already started the identification work for these sources by means of optical spectroscopy,
and we alredy selected a sample of 25 sources for which a classification could be achieved using the approach
illustrated above {(Masetti et al. 2010)}. The majority of these newly-identified sources are AGNs (68\%), followed
by CVs and X-ray binaries (both 16\%). This new lot of identified sources already reduced by $\sim$ 12\% the whole amount
of unidentified sources in the 4$^{\rm th}$ IBIS catalog.
In conclusion, we point out that we are running a similar project in relation with the {\it Swift/BAT} sources belonging to
different catalogs (see Landi et al. 2007 and Parisi et al. 2009): we could identify or better classify 28 hard
X-ray emitting AGNs. Likewise, we are also performing a similar work on the unidentified sources detected with the
{\it Fermi} satellite: this allowed us to classify the GeV source 0FGL J2001.0+4352 as a BL Lac object (Bassani et al.
2009).
\vspace{-0.3cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 316 |
Middle Tennessee volleyball marched to Richmond, Virginia, this weekend to take part in the Virginia Commonwealth University Invitational. The Blue Raiders challenged Morgan State, VCU and Liberty University but were only able to come home with one victory.
The Blue Raiders faced the Morgan State Bears to kick off the round-robin tournament, whom they beat handily. The Blue Raiders took the first two sets, 25-21 and 25-13, respectively. The Bears would surge back and take the third set 25-23, but Middle Tennessee answered, taking the fourth set 26-24 and winning the match 3-1. Freshman Dora Peonia finished with a career-high 15 kills. Regan Coughlin added eight kills with 18 digs.
The Blue Raiders took on the host, VCU, in their second game of the invitational. After coming off a win earlier in the day, MTSU would fall to the Rams in three straight sets, 25-20, 25-19 and 28-26.
MTSU would seek to come away with a winning record under their belt, but the Liberty Flames thought otherwise. The Blue Raiders would ultimately lose in three straight sets — 25-23, 25-18 and 25-23 — to end the tournament.
With a 1-2 record in the VCU Invitational, the Blue Raiders now drop to 3-10 on the season. MTSU looks to open up conference play next Friday when they travel to UAB to take take on the Blazers (7-8). | {
"redpajama_set_name": "RedPajamaC4"
} | 2,954 |
Daewoo is ex-South-Korean second largest industrial concern, which became bankrupt in 1999 after a huge Asian financial collapse. Later 20 some subdivisions were sold or continue to work separately. General Motors acquired Daewoo Auto and Technology subdivision in the beginning of 21st century. After merging the company was called GM Daewoo. The list of car models that were produced by Daewoo included such as Matiz, Winstorm, Alpheon, Lacetti, Tosca and other. Some of vehicles that were produced by Daewoo were sold under Chevrolet brand name in Australia and a few Asian countries. Since Daewoo was a part of General Motor concern some of the concepts that are known under divergent names were also developed on the foundation of Daewoo. For instance, Chevrolet Aveo initially was produced by the ex-Korean subsidiary and it was called Daewoo Kalos. Nowadays Daewoo specifications of this model is rather different from what it originally was, surely it changed to better and now it has 1.8L engine with 136 horsepower, 5 manual or 6 automatic transmission and more.
Anyways, General Motors made a decision to shut down the production of cars under Daewoo brand-name and the subdivision itself was abolished due to doubtful reputation of concern in South Korea which was a trail of events that took place back in the end of 90-s. So if you are a votary of Daewoo vehicle series then you should trace the development of the model you like.
Car purchase is not like going out to get groceries, here you have to look through all the available Daewoo car models, take a good look at Daewoo technical specifications , compare prices and models of various brands and then draw a conclusion basing on which you are going to buy a car. So, if you are looking for a SUV then you should go through pictures of all Daewoo SUVs available because if you don't like the way car looks like then it's pointless to figure out what is inside of it. Since you're done with that we can move forward. If you don't have that much money then you might want to consider a possibility of acquiring a used SUV model of brand you like. In case you care about ecology a lot then you might ant to take a look at hybrid SUV models of Toyota or other brands you like. Look through pictures of all Daewoo SUVs you chose once again, sort it out once again – now you are ready to look through Daewoo technical specs and trims of each model to make the final decision. It's fun and usually people like to choose cars themselves unless you are a billionaire and you don't care if a car you bought doesn't meet some of your requirements so you can go back and buy a sedan model you like or switch it to Bentley convertible model right away. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,310 |
Q: Removing newline after tags? I am having a problem with removing linebreaks after the <h1> tag, as everytime it prints, it adds a line break straight after it, so something like <h1>Hello World!</h1> <h2>Hello Again World!</h2> prints out like this:
Hello World!
Hello Again World!
I am unsure on what tags I need to change in CSS, but I expect it's something to do with the padding or margins
I also want to keep the vertical padding if at all possible.
A: I just solved this problem by setting h1 margin value to minus in html style section. It works perfecly for my needs.
<style>
h1 {
display: block;
font-size: 0.9em;
margin-top: -1.91em;
margin-bottom: -1.91em;
margin-left: 0;
margin-right: 0;
font-weight: bold;
}
</style>
<h1 style="text-align:center"> Headline </h1>
A: Sounds like you want to format them as inline. By default, h1 and h2 are block-level elements which span the entire width of the line. You can change them to inline with css like this:
h1, h2 {
display: inline;
}
Here's an article that explains the difference between block and inline in more detail: http://www.webdesignfromscratch.com/html-css/css-block-and-inline/
To maintain vertical padding, use inline-block, like this:
h1, h2 {
display: inline-block;
}
A: <h1> tags have {display: block} set. They are block-level elements. To turn this off:
{display: inline}
A:
<style>
h1 {
padding: 0px;
margin: 0px;
}
</style>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
#include "winuwp_compat_internal.h"
#ifdef WINUWP
#ifdef main
#undef main
#endif /* main */
#include <Windows.h>
#ifdef __cplusplus
extern "C" {
#endif /* __cplusplus */
#ifdef __cplusplus
}
#endif /* __cplusplus */
int winuwpMainCC(int argc, char *argv[]);
int main(Platform::Array<Platform::String^>^ args)
{
WinUWP::MainConvertToUTF8 conv(args);
return winuwpMainCC(conv.argc(), conv.argv());
}
#endif /* WINUWP */
void winuwp_compat_main_utf8_cc() {}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,365 |
Q: How to return bool value with navigator.pop? I'm trying to return a bool value from the dialogbox but I do not understand why the value does not return as need. I have tried returning as a future value and returning the values together with context after popping the dialogbox.
final bool delete = await _showDialog();
print(delete);
Future<bool> _showDialog() {
bool result;
showDialog(
context: context,
builder: (BuildContext context) {
return AlertDialog(
title: Text('Delete Appointment'),
content: Text(
'Are you sure? \nThis action cannot undo.',
style: TextStyle(
color: Colors.red,
fontSize: 20
),
),
actions: <Widget>[
FlatButton(
color: Colors.blue,
child: Text(
'CANCEL',
style: TextStyle(
color: Colors.white,
fontSize: 20
),
),
onPressed: () {
setState(() => result = false);
//print(result);
Navigator.pop(context, result);
return Future.value(result);
},
),
SizedBox(
width: 50,
),
FlatButton(
color: Colors.red,
child: Text(
'CONFIRM',
style: TextStyle(
color: Colors.white,
fontSize: 20
),
),
onPressed: () {
setState(() => result = true);
//print(result);
Navigator.pop(context, result);
return Future.value(result);
},
)
],
);
}
);
}
A: Here's a working example based on your code:
@override
Widget build(BuildContext context) {
return Container(
child: RaisedButton(onPressed: () async {
bool delete = await _showDialog(context);
print(delete);
})
);
}
Future<bool> _showDialog(context) {
return showDialog(
context: context,
builder: (BuildContext context) {
return AlertDialog(
title: Text('Delete Appointment'),
content: Text(
'Are you sure? \nThis action cannot undo.',
style: TextStyle(
color: Colors.red,
fontSize: 20
),
),
actions: <Widget>[
FlatButton(
color: Colors.blue,
child: Text(
'CANCEL',
style: TextStyle(
color: Colors.white,
fontSize: 20
),
),
onPressed: () {
Navigator.pop(context, false);
},
),
SizedBox(
width: 50,
),
FlatButton(
color: Colors.red,
child: Text(
'CONFIRM',
style: TextStyle(
color: Colors.white,
fontSize: 20
),
),
onPressed: () {
Navigator.pop(context, true);
},
)
],
);
}
);
}
A: For Return True
Navigator.of(context).pop(true);
For Return False
Navigator.of(context).pop(false);
A: You pass the value you want to return to Navigator.pop and it will be returned by showDialog:
final result = await showDialog(
...
// At some point, Navigator.pop is called
Navigator.pop(context, true);
);
print(result); // Prints: true
A: check out this full woking code. you have to pass value which you want to return.
import 'package:flutter/material.dart';
class Delete extends StatefulWidget {
Delete({Key key}) : super(key: key);
@override
_DeleteState createState() => _DeleteState();
}
class _DeleteState extends State<Delete> with SingleTickerProviderStateMixin {
int index = 0;
@override
Widget build(BuildContext context) {
return Scaffold(
body: Center(
child: RaisedButton(
onPressed: () {
_showDialog().then((value) {
print(value.toString());
});
},
child: Text("press ME!"),
),
),
);
}
_showDialog() {
return showDialog(
context: context,
builder: (BuildContext context) {
return AlertDialog(
title: Text('Delete Appointment'),
content: Text(
'Are you sure? \nThis action cannot undo.',
style: TextStyle(color: Colors.red, fontSize: 20),
),
actions: <Widget>[
FlatButton(
color: Colors.blue,
child: Text(
'CANCEL',
style: TextStyle(color: Colors.white, fontSize: 20),
),
onPressed: () {
Navigator.pop(context, false);
},
),
SizedBox(
width: 50,
),
FlatButton(
color: Colors.red,
child: Text(
'CONFIRM',
style: TextStyle(color: Colors.white, fontSize: 20),
),
onPressed: () {
Navigator.pop(context, true);
},
)
],
);
});
}
}
A: onPressed:() {
result = false;
//print(result);
setState(() {});
Navigator.pop(context, result);
}
or
onPressed:() {
setState(() {
result = false;
//print(result);
Navigator.pop(context, result);
});
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,197 |
In Polen wurden Comics bereits vor dem Zweiten Weltkrieg in Tageszeitungen abgedruckt. Der allererste eigene polnische Comic-Held war der 1933 zum Leben erwachte Ziegenbock Koziołek Matołek von Zeichner Marian Walentynowicz und Texter Kornel Makuszyński.
Geschichte
Im großen Stil wurden Comics in Polen allerdings erstmals in den späten 1940er Jahren in Form von Heften oder Taschenbüchern veröffentlicht. Optischen und literarischen Einfluss auf die frühen Publikationen hatten dabei vor allem die US-amerikanischen Werke der 1930er und 1940er Jahre. Lizenzveröffentlichungen gab es kaum, meist waren veröffentlichte Comics ab den 1940ern eigene Produktionen polnischer Zeichner und Autoren. Der restliche europäische Comic erlangte erst Jahre später Einfluss auf die Comicschaffenden in Polen.
Zu den Comics der Zeit von 1948 bis zur Wende 1989 gehörten in erster Linie besonders idealistische und politisch engagierte Titel aus eigener Produktion, wie die heutige Kultserie Kapitan Żbik und eher bildende Serien, wie Historia Państwa Polskiego (dt. Die Geschichte des polnischen Staates). Wichtigster Vertreter von anspruchsvolleren Comics mit einem surrealistischen Humor war zu der Zeit Tadeusz Baranowski. Doch auch andere Comics, die eher auf Unterhaltung hinaus zielten, besaßen ihren festen Platz. Zu diesen zählten u. a. die Comicserien Tytus, Romek i A'Tomek von Papcio Chmiel für jüngere Leser (entfernt vergleichbar mit den Abrafaxen aus der DDR), Kapitan Kloss, eine Serie über einen polnischen Geheimagenten, der als Wehrmachtsoffizier Aufträge für die Sowjetische Armee im Dritten Reich erfüllt, die Funnyserien Kajko i Kokosz und Kajtek i Koko w Kosmosie (dt. Kajtek und Koko im Weltraum) von Janusz Christa sowie die Science-Fiction-Reihe Funky Koval von Jacek Rodek, die bedeutend und wegweisend für die Entwicklung des Mediums in Polen werden sollten.
Neben zahlreichen Comics wurde auch eine Reihe richtiger Fachmagazine und anderer Sekundärliteratur veröffentlicht. Zu einem der wichtigsten der kommunistischen Ära zählt das Magazin Relax, welches beispielsweise mit dem deutschen Comicmagazin Zack vergleichbar ist.
In Relax wurden neben Fachartikeln und Rezensionen auch zahlreiche Comicstrips von heute namhaften Künstlern veröffentlicht und teilweise auch speziell dafür produziert. So machten z. B. Autoren und Zeichner, wie Grzegorz Rosiński, der später durch die frankobelgische Fantasyserie Thorgal weltweites Ansehen erlangte, und Bogusław Polch, dessen Werk Bogowie z Kosmosu (dt. unter dem Titel Das Ende der Götzen erschienen) in mittlerweile mehr als fünfzehn Sprachen übersetzt wurde, über dieses Magazin Karriere.
Neben Relax befanden sich jede Menge anderer Zeitschriften auf dem Markt, die entweder in Kiosks, am Bahnhof oder im Buchhandel erhältlich waren, wie auch das schlicht benannte Magazin Komiks, das neben heimischen Comicstrips auch erstmals ausländische Titel herausbrachte.
Nach der Wende 1989 entwickelte sich ein plötzlicher Comicboom in Polen. Innerhalb der ersten drei Jahre bildeten sich Dutzende von kleinen und größeren Verlagen, die entweder eigene Reihen produzierten oder lizenzierte Serien aus dem Ausland veröffentlichten. Erstmals wurden auch verstärkt US-amerikanische und zahlreiche frankobelgische Comics veröffentlicht, die zuvor nur vereinzelt anzutreffen waren.
Seit Mitte der 1990er Jahre ist der Comic ein fester Bestandteil der polnischen Literaturszene. Junge Nachwuchskünstler werden durch verschiedene private und auch staatliche Aktionen gefördert. In Krakau und Warschau haben sich feste Comicfestivals etabliert. Zudem finden in Łódź und Warschau alljährlich Messen, wie das Internationale Festival des Comics statt, bei dem bereits internationale Größen, wie Moebius oder Milo Manara gastierten. Zahlreiche Fanzines halten die unabhängige Comicszenen am Leben. Wichtigste Vertreter sind AQQ und Ziniol, die am Kiosk und im Buchhandel verkauft werden. Letzteres veröffentlicht zudem regelmäßig Comics aus deutscher Produktion.
Heutige Situation
Gesellschaftlich war und ist der Comic nicht als Schundliteratur gebrandmarkt und hatte auch nicht die Schwierigkeiten sich durchzusetzen, wie im Rest des ehemaligen Ostblocks oder gar einigen Ländern Westeuropas. Die Funktionäre der kommunistischen Regierungspartei PZPR nutzten sogar das Medium zu Lern- und Propagandazwecken.
Derzeit spielen vor allem die einheimischen Comicschaffenden eine wichtige Rolle. Sie veröffentlichen innovative und teils sehr künstlerische Comics, was ihnen besonders durch moderne Wege zum Kunden ermöglicht wird. So findet sich in jeder Stadt mindestens ein Multimediaanbieter, meist Buchhandlungen der Kette Empik, in der es Comics sämtlicher Verlage zu kaufen gibt. Die wichtigsten dieser Verlagshäuser sind der in Breslau gegründete Verlag Mandragora, der auch in Deutschland bekannte Warschauer Verlag Kultura Gniewu, der bereits einige deutsche Comicproduktionen in Polen veröffentlicht hat und die polnische Tochterfirma des dänischen Verlagskonzerns Egmont Ehapa. Auch Comicadaptionen von polnischen Literaturklassikern, wie beispielsweise von Nobelpreisträger Henryk Sienkiewicz, sowie Comics zu historischen und aktuellen Ereignissen, wie zum Jubiläum der Solidarność-Bewegung, erfreuen sich großer Beliebtheit. Auf letztere hat sich besonders der Verlag Zin Zin Press spezialisiert, der auch international mit tschechischen und baltischen Comickünstlern zusammenarbeitet.
Einige der gegenwärtig beliebtesten Comicserien sind Jeż Jerzy von Rafał Skarżycki und Tomasz Lew Leśniak, die für das Kino als Animationsfilm verfilmt wurde, sowie Pierwsza Brygada von Tobiasz Piątkowski und Krzysztof Janicz. Populär und wegweisend ist zudem die Arbeit von Comickünstlerin Agata Nowicka, die Comics und Illustrationen mit dem Programm Paint entwickelt. Seit Ende der 1990er kann man zudem auch verstärkt japanische Mangas und koreanische Manhwas in Polen vorfinden. Wie in Deutschland, haben sich auch in Polen einige speziell dafür gegründete Verlage auf dem Markt gefunden, die lediglich Veröffentlichungen aus Asien anbieten.
Weblinks
Wrak – Polnische Presseagentur für Comics (poln.)
Gildia Komiksu – Polnisches Onlinemagazin (poln.)
Aleja Komiksu – Polnisches Onlinemagazin (poln.)
Komiks Polter – Polnisches Onlinemagazin (poln.)
Großes Archiv polnischsprachiger Comicpublikationen (poln.)
Comic in Polen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,601 |
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