So, you think you'd like to participate in discussions on the MinGW-Dvlpr mailing list? This presumably implies that you have an interest in MinGW development issues, and would be interested in joining the MinGW Development Team.
In fact, MinGW-Dvlpr is a private mailing list, exclusively for the use of MinGW Project Administrators and members of the MinGW Development Team, and specifically as a forum where they can discuss project development issues. Subscription is strictly restricted to the project administrators, and to bona-fide Mingw Developers. If you are interested in actively participating in MinGW development, then please read on; if, on the other hand, you are mostly interested in just learning the Developers' perspective on some MinGW or MSYS related topic, then please subscribe to the MinGW-Users List or the MinGW-MSYS List, and ask your questions there.
So, still interested in joining? Please do not attempt to subscribe to the MinGW-Dvlpr list directly; your subscription request will surely be refused! Instead, you should apply by email, to mingw-dvlpr-owner@lists.sourceforge.net. You will need to have a SourceForge user account, for which you must state your user name, give a brief resume of your experience,and explain how you would like to contribute to MinGW development; the Project Administrator will then automatically subscribe you to the MinGW-Dvlpr list, on your successful enrolment as a MinGW Developer.
Please note that once you have joined as a developer you are expected to remain subscribed to the developer mailing list, (which should not present you with any problem, because it is very low volume). If you still wish to use an alternate method to access developer email (e.g. Gmane) then please let us know.
Do note that you may contribute to MinGW development, simply by submitting patches; you do not need to be formally enrolled as a MinGW Developer to do this. However, if you adopt this approach, and feel that you would like to be recognised as involved with the development on a more formal basis, then by all means please ask to be considered for enrolment.


text: pip install /cache/user-job-dir/mindspore_open_src/mindspore_ascend-0.3.1-cp37-cp37m-linux_x86_64.whl --no-deps --ignore-installed

redpajama_set_name: RedPajamaGithub


text:
History Hits
Ink: A History of Tattooing
Matt Lodder is the world's leading expert on the history of tattoos. He has found evidence of people using ink or charcoal on their bodies stretching back thousands of years. He explodes myths at every turn. Tattoos were common long before Captain Cook allegedly imported them back from the Pacific in the 18th century, and he demonstrates that they were never the exclusive preserve of marginalised subcultures, but actually adorned the bodies of royalty.
Up Next in History Hits
Margaret Beaufort: Mother of the Tudors
Nicola Tallis comes on the show to talk about the extraordinary Margaret Beaufort: 'Mother of the Tudors' and the ancestor of all subsequent royals.
Modern Spain
Sir Paul Preston CBE comes on the show to discuss Spain's modern history and explain how the stripping away of its empire in the early 19th century had such a catastrophic impact on the country.
Pandemics: Science and History
I was thrilled to be joined by the legendary Peter Frankopan, Professor of Global History at Oxford University and bestselling author of 'The Silk Roads: A New History of the World'. In this podcast we discussed the current crisis in a wider historical context, and Peter gave some fascinating ins...

redpajama_set_name: RedPajamaCommonCrawl


The Energy Check, founded in 2014 are an independent energy consultancy supporting businesses globally to secure energy contracts, reduce energy costs and carbon footprints while promoting the uptake of energy efficiency measures & renewable energy technology.
The Energy Check offers their customers a wide range of services including LED Lighting, Commercial Battery Storage, Solar PV, Online Home Energy Switching and more.
Over the past 3 years, The Energy Check have saved their customers over £3.5 million pounds on their energy bills, over 80,000t on their CO2 emissions and over 8GWh of energy.
Contact The Energy Check today by calling 0191-691-2096 or visit their website at www.theenergycheck.co.uk. Be sure to also give them a like on Facebook by clicking HERE and a follow on Twitter by clicking HERE.


text:
package com.gemstone.gemfire.internal.tools.gfsh.app.command.task;

import java.io.DataInput;
import java.io.DataOutput;
import java.io.IOException;

import com.gemstone.gemfire.cache.Cache;
import com.gemstone.gemfire.cache.CacheFactory;
import com.gemstone.gemfire.cache.Region;
import com.gemstone.gemfire.internal.tools.gfsh.command.CommandResults;
import com.gemstone.gemfire.internal.tools.gfsh.command.CommandTask;
import com.gemstone.gemfire.internal.tools.gfsh.util.RegionUtil;

/**
 * RegionPathTask retrieves an entire list of region paths in the connected
 * server or in the entire distributed system in which the connected server
 * belongs.
 */
public class RegionPathTask implements CommandTask
{
	private static final long serialVersionUID = 1L;

	private boolean regionsInDistributedSystem = false;
	private boolean recursive = true;
	private String parentRegionPath = null;

	/**
	 * Returns all region paths in the entire distributed system. This
	 * constructor call is equivalent to new RegonPathTask(true, true, null);
	 */
	public RegionPathTask()
	{
	}

	/**
	 * Returns all region paths starting from the top level.
	 *
	 * @param regionsInDistributedSystem
	 *            if false, returns all region paths found in the cache. If
	 *            true, returns all region paths found in the entire distributed
	 *            system.
	 * @param recursive
	 *            if true, returns all nested region paths, otherwise, returns
	 *            the top-level region paths
	 */
	public RegionPathTask(boolean regionsInDistributedSystem, boolean recursive)
	{
		this(regionsInDistributedSystem, recursive, null);
	}

	/**
	 * @param regionsInDistributedSystem
	 *            if false, returns all region paths found in the cache. If
	 *            true, returns all region paths found in the entire distributed
	 *            system.
	 * @param recursive
	 *            if true, returns all nested region paths, otherwise, returns
	 *            the top-level region paths
	 * @param parentRegionPath
	 *            the parent region path
	 */
	public RegionPathTask(boolean regionsInDistributedSystem, boolean recursive, String parentRegionPath)
	{
		this.regionsInDistributedSystem = regionsInDistributedSystem;
		this.recursive = recursive;
		this.parentRegionPath = parentRegionPath;
	}

	public CommandResults runTask(Object userData)
	{
		String[] regionPaths = null;
		Cache cache = CacheFactory.getAnyInstance();
		if (regionsInDistributedSystem) {

			// get region paths defined in this cache only

			if (parentRegionPath == null) {
				regionPaths = RegionUtil.getAllRegionPaths(CacheFactory.getAnyInstance(), recursive);
			} else {
				Region region = cache.getRegion(parentRegionPath);
				if (region != null) {
					regionPaths = RegionUtil.getAllRegionPaths(region, recursive);
				}
			}

		} else {

			// get region paths defined in all of the caches in the distributed
			// system

			if (parentRegionPath == null) {
				regionPaths = RegionUtil.getAllRegionPathsInDistributedSystem(cache.getDistributedSystem(), recursive);
			} else {
				Region region = cache.getRegion(parentRegionPath);
				if (region != null) {
					regionPaths = RegionUtil.getAllRegionPaths(region, recursive);
				}
			}

		}
		CommandResults results = new CommandResults(regionPaths);
		return results;
	}

	public boolean isRegionsInDistributedSystem()
	{
		return regionsInDistributedSystem;
	}

	public void setRegionsInDistributedSystem(boolean regionsInDistributedSystem)
	{
		this.regionsInDistributedSystem = regionsInDistributedSystem;
	}

	public boolean isRecursive()
	{
		return recursive;
	}

	public void setRecursive(boolean recursive)
	{
		this.recursive = recursive;
	}

	public String getParentRegionPath()
	{
		return parentRegionPath;
	}

	public void setParentRegionPath(String parentRegionPath)
	{
		this.parentRegionPath = parentRegionPath;
	}

	public void fromData(DataInput input) throws IOException, ClassNotFoundException
	{
		regionsInDistributedSystem = input.readBoolean();
		recursive = input.readBoolean();
		parentRegionPath = input.readUTF();
		if (parentRegionPath.equals("\0")) {
			parentRegionPath = null;
		}
	}

	public void toData(DataOutput output) throws IOException
	{
		output.writeBoolean(regionsInDistributedSystem);
		output.writeBoolean(recursive);
		if (parentRegionPath == null) {
			output.writeUTF("\0");
		} else {
			output.writeUTF(parentRegionPath);
		}
	}

}


In The Incredibles 2, Helen is called on to lead a campaign to bring Supers back, while Bob navigates the day-to-day heroics of "normal" life at home with Violet, Dash and baby Jack-Jack -- whose superpowers are about to be discovered.
Their mission is derailed, however, when a new villain emerges with a brilliant and dangerous plot that threatens everything. But the Parrs don't shy away from a challenge, especially with Frozone by their side.
That's what makes this family so INCREDIBLE.
Join your favourite monster family as they embark on a vacation adventure on a luxury monster cruise ship so Drac can finally take a break from handling everyone else's vacation at the hotel.
But they hit rocky waters when Mavis realizes that Drac has fallen for the mysterious captain of the ship, Ericka, who hides a dangerous secret that could sink their plans for a dream vacation.
It seems to the Teens that all the major superheroes out there are starring in their own movies—everyone but the Teen Titans, that is! But de facto leader Robin is determined to remedy the situation, and be seen as a star instead of a sidekick. If only they could get the hottest Hollywood film director to notice them.
With a few madcap ideas and a song in their heart, the Teen Titans head to Tinsel Town, certain to pull off their dream. But when the group is radically misdirected by a serious Super-Villain and his maniacal plan to take over the Earth, things really go awry. The team finds their friendship and their fighting spirit failing, putting the very fate of the Teen Titans themselves on the line!
Four teenagers in detention discover an old video game console with a game they've never heard of. When they decide to play, they are immediately sucked into the jungle world of Jumanji in the bodies of their gaming avatars (Dwayne Johnson, Jack Black, Kevin Hart, and Karen Gillan).
They'll have to complete the adventure of their lives filled with fun, thrills and danger or be stuck in the game forever!
Surly Squirrel and the gang are back, returning to Liberty Park after they are forced to leave their easy life at the nut store.
Desperate to prove his talent, Miguel finds himself in the stunning and colourful Land of the Dead following a mysterious chain of events. Along the way, he meets charming trickster Hector, and together, they set off on an extraordinary journey to unlock the real story behind Miguel's family history.
With the war to end all wars looming, a young army "doughboy" Robert Conroy has his life forever changed when a little dog with a stubby tail wanders into camp as the men of the 102nd Infantry Regiment train on the grounds of Yale University. Conroy gives his new friend a name, a family, and a chance to embark on the adventure that would define a century.
Despite lacking formal military working dog training, Stubby the dog and his human companions find themselves in the trenches of France and on the path to history. French Infantryman Gaston Baptiste befriends the duo and accompanies them along their epic journey through harsh conditions and incredible acts of courage.
For his valorous actions, Stubby is recognized as the first canine ever promoted to the rank of Sergeant in U.S. Army history.
A.X.L. is a top-secret, robotic dog created by the military to help protect tomorrow's soldiers. Code named by the scientists who created him, A.X.L. stands for Attack, Exploration, Logistics, and embodies the most advanced, next-generation artificial intelligence.
After an experiment gone wrong, A.X.L. is discovered hiding alone in the desert by a kind-hearted outsider named Miles, who finds a way to connect with him after activating his owner-pairing technology. Together, the two develop a special friendship based on trust, loyalty and compassion. Helping Miles gain the confidence he's been lacking, A.X.L. will go to any length to protect his new companion, including facing off against the scientists who created him and who will do anything to get him back.
Knowing what is at stake if A.X.L. is captured, Miles teams up with a smart, resourceful ally named Sara to protect his new best friend on a timeless, epic adventure for the whole family.
In 2045, the world is on the brink of chaos and collapse. But the people have found salvation in the OASIS, an expansive virtual reality universe created by the brilliant and eccentric James Halliday.
When Halliday dies, he leaves his immense fortune to the first person to find a digital Easter egg he has hidden somewhere in the OASIS, sparking a contest that grips the entire world. When an unlikely young hero named Wade Watts decides to join the contest, he is hurled into a breakneck, reality-bending treasure hunt through a fantastical universe of mystery, discovery and danger.
The beloved garden gnomes Gnomeo and Juliet are back for a whole new adventure.
When they first arrive in the city with their friends and family, the biggest concern is getting their new garden ready for spring. However, they soon discover that someone is kidnapping garden gnomes all over London. When everyone in their garden goes missing – there's only one gnome to call… SHERLOCK GNOMES.


With the 2019 approaching fast , all preparations and the initial stockings have been completed. Many thanks to all those who gave up their time attending work parties..
no bales will be placed in the Loch in 2019 - this is being done to determine if the bales are effective in reducing algae blooms in summer and autumn.
Existing members must pay, in full, for their 2019 season ticket prior to the start of the 2019 season.
Never bring the fish into the boat - keep the fish in the net and the net in the water.
Fish should be kept in the net in the water , giving them enough time to recover , before letting the fish swim out of the net.


text:
Q: How to get Generalised touch points in android i am working on an application where on touch of an imageview i have to add/place another image.
I am able to do so using below code:
            int[] viewCoords = new int[2];
            imageview.getLocationOnScreen(viewCoords);

            int touchX = (int) event.getX();
            int touchY = (int) event.getY();

            int imageX = touchX - viewCoords[0]; // viewCoords[0] is the X coordinate
            int imageY = touchY - viewCoords[1];

            RelativeLayout.LayoutParams lp = new RelativeLayout.LayoutParams(
                    RelativeLayout.LayoutParams.WRAP_CONTENT,
                    RelativeLayout.LayoutParams.WRAP_CONTENT);
            ImageView iv = new ImageView(getApplicationContext());
            lp.setMargins(imageX, imageY, 0, 0);
            iv.setLayoutParams(lp);
            iv.setImageDrawable(getResources().getDrawable(
                    R.mipmap.circ));
            layout.addView(iv);

This works only on one device.
I am Storing all the touched points to the server. Multiple users can see the edited image.
However if i try to plot this co-ordinate/points it gets placed at different position.
Tried searching but dint find any generalised way of getting X and Y values which are independent of the screen size and density.
Any help will be Appreciated.

redpajama_set_name: RedPajamaStackExchange


text:
Our Expertise | Audit Advisory
Deep technical bench and trusted insights into the regulators' perspectives.
For US and foreign network audit firms, adherence to relevant regulatory requirements and the dynamic evolution of professional standards covering auditing, independence, accounting, and financial reporting requires continuous attention and diligence. Our diverse team of professionals includes several former senior officials from the Public Company Accounting Oversight Board (PCAOB), including the Divisions of Enforcement and Investigation (DEI) and Inspections and Registration (IR), as well as a Board Special Advisor. In addition, we have former senior officers from the Securities and Exchange Commission (SEC), Division of Enforcement, certified public accountants, including many former audit firm partners, standard-setters, and other regulatory officials from the PCAOB, the SEC, and the Financial Accounting Standards Board (FASB). Our team provides insights into the regulators' perspectives to assist them in understanding and responding to regulatory inquiries. In addition to assisting clients in navigating the complexities of the regulatory environment, we also assist clients in navigating the complexities of the regulatory environment and judicial process, from case assessment through expert testimony, as well as compliance verification services following resolution of the matter.
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We have firsthand experience as standard-setters and regulators to assist firms with assessments of their systems of quality control in advance of regulatory inspections.
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Our technical experts provide a truly independent source in disputed or contentious issues.
Our seasoned experts include former standard-setters and rule-makers who have provided expert testimony in a broad array of lawsuits, administrative proceedings, and mediation capacities.
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We bring regulatory credibility and deep experience to monitoring or independent consultant situations.
Our Solutions Leverage the Expertise of:
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We provide independent and objective analysis of accounting disputes, white paper preparation, and resolution of registrant issues with regulators.
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A credible investigation can greatly protect an organization by not allowing potentially isolated issues from becoming more systemic, while also minimizing disruption to the company's reputation and enterprise value.
Regulatory & Technical Accounting Advisory
We offer a wealth of assistance with independent and objective analysis of accounting disputes, white paper preparation, and resolution of registrant issues with regulators.
For audit advisory matters, our visual communications consultants work with our team of professionals to understand the significant business issues and develop the solutions necessary to make complicated information easy to understand. We develop the most effective presentation solutions because we understand how critical communication is to our clients in resolving these significant matters.
White Collar & Securities
Our professionals have deep fraud and regulatory expertise necessary to offer valuable insights with regulatory issues.
Steven E. Richards
Mitchell Friedman
Kevin Stephens
Gene L. Deetz


text:
using JsTestAdapter.TestServerClient;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace KarmaTestAdapterTests.Expectations
{
    public static class SpecExtensions
    {
        private static string FormatNull()
        {
            return FormatValue(null);
        }

        private static string FormatValue(this object value)
        {
            if (value == null)
            {
                return "null";
            }
            if (value is string)
            {
                return string.Format(@"""{0}""", value);
            }
            if (value is char)
            {
                return string.Format(@"'{0}'", value);
            }
            return string.Format(@"{0}", value);
        }

        public static StringBuilder AppendLine(this StringBuilder sb, string format, params object[] args)
        {
            return sb.AppendFormat(format, args).AppendLine();
        }

        public static string Format(this IEnumerable<Spec> specs)
        {
            return string.Join("", specs.Select(s => s.Format()));
        }

        public static string Format(this ExpectedSpec spec)
        {
            var text = new StringBuilder();

            if (spec != null)
            {
                text.AppendLine("=====================================================================");
                text.AppendLine("Expected spec");
                text.AppendLine("---------------------------------------------------------------------");
                text.AppendLine("FullyQualifiedName: {0}", spec.FullyQualifiedName.FormatValue());
                text.AppendLine("DisplayName:        {0}", spec.DisplayName.FormatValue());
                text.AppendLine("Description:        {0}", spec.Description.FormatValue());
                text.AppendLine("FileName:           {0}", spec.FileName.FormatValue());
                text.AppendLine("LineNumberFrom:     {0}", spec.LineNumberFrom.FormatValue());
                text.AppendLine("LineNumberTo:       {0}", spec.LineNumberTo.FormatValue());
                text.AppendLine("Success:            {0}", spec.Success.FormatValue());
                text.AppendLine("Skipped:            {0}", spec.Skipped.FormatValue());
                text.AppendLine("Output:             {0}", spec.Output.FormatValue());
                text.AppendLine("StackTrace:         {0}", spec.StackTrace.FormatValue());
                text.AppendLine("IsValid:            {0}", spec.IsValid.FormatValue());
                text.AppendLine("InvalidReason:      {0}", spec.InvalidReason.FormatValue());
                text.AppendLine("=====================================================================");
                text.AppendLine();
            }

            return text.ToString();
        }

        public static string Format(this Spec spec)
        {
            var text = new StringBuilder();

            if (spec != null)
            {
                text.AppendLine("=====================================================================");
                text.AppendLine("Karma spec");
                text.AppendLine("---------------------------------------------------------------------");
                text.AppendLine("FullyQualifiedName: {0}", spec.FullyQualifiedName.FormatValue());
                text.AppendLine("DisplayName:        {0}", spec.DisplayName.FormatValue());
                text.AppendLine("Id:                 {0}", spec.Id.FormatValue());
                text.AppendLine("Description:        {0}", spec.Description.FormatValue());
                text.AppendLine("Suite:              {0}", spec.Suite.FormatSuite());
                text.AppendLine("Source:             {0}", spec.Source.Format());
                if (spec.Results != null)
                {
                    foreach (var result in spec.Results)
                    {
                        result.Format(text);
                    }
                }
                text.AppendLine("=====================================================================");
                text.AppendLine();
            }

            return text.ToString();
        }

        public static string FormatSuite(this IEnumerable<string> suite)
        {
            return suite != null ? string.Format("[ {0} ]", string.Join(", ", suite.Select(s => s.FormatValue()))) : FormatNull();
        }

        public static string Format(this Source source)
        {
            if (source == null)
            {
                return FormatNull();
            }
            var result = source.FileName;
            if (source.LineNumber.HasValue)
            {
                result += string.Format(" Line {0}", source.LineNumber);
            }
            if (source.ColumnNumber.HasValue)
            {
                result += string.Format(" Column {0}", source.ColumnNumber);
            }
            return result.FormatValue();
        }

        public static void Format(this SpecResult result, StringBuilder text)
        {
            text.AppendLine("---------------------------------------------------------------------");
            text.AppendLine("Result:             {0}", result.Name.FormatValue());
            text.AppendLine("Success:            {0}", result.Success.FormatValue());
            text.AppendLine("Skipped:            {0}", result.Skipped.FormatValue());
            text.AppendLine("Output:             {0}", result.Output.FormatValue());
            text.AppendLine("Time:               {0}", result.Time.FormatValue());
        }

        public static string GetOutcome(this SpecResult result)
        {
            if (result.Skipped)
            {
                return "Skipped";
            }
            return result.Success ? "Passed" : "Failed";
        }
    }
}


text:
Home > Anime > News > Netflix's The Witcher Nightmare of the Wolf Review: Stream it or Skip it?
Netflix's The Witcher Nightmare of the Wolf Review: Stream it or Skip it?
There are a lot of solid reasons why Geralt of Rivia doesn't behead a vampire and then say, "Fangs for the memories," in The Witcher.
Friday, 27 August 2021, 01:11 IST
There are a lot of solid reasons why Geralt of Rivia doesn't behead a vampire and then say, "Fangs for the memories," in The Witcher. Unlike the first season of the show, this new film is animated. The Witcher: Nightmare of the Wolf serves as a prologue film introducing Vesemir, Geralt's instructor.
In the first sequence of Netflix's animated spinoff, The Witcher: Nightmare of the Wolf, his mentor Vesemir battles the Leshen, and his sense of humor is more caustic and blunter. That disparity might be explained in part by the film's finale, which revealed a number of worldbuilding aspects that carry over into the main series.
Vesemir is a young man who quit his job as a servant to pursue his dream of becoming a witch. "The Witcher" transforms him into one of the finest witchers in the world after going through gruesome preliminaries, analyses, and painful testing. Years later, the adjacent country Kaedwen hires Vesemir to battle a mysterious beast in the forest that is killing citizens. Tetra Gilcrest leads them to a mystically contaminated kitsu, where they locate witches that need to be eliminated.
The kitsu-mahr has been kidnapping female elves and experimenting on them to produce hybrids like herself. Still, she has only succeeded in creating one girl, with the others dying horribly. Finally, tetra persuades their monarch to execute genocide on the whole village after discovering proof that the witcher mages use mutagenic alchemy to create new monsters, effectively providing more employment for witchers. Tetra fabricates the death of the kitsu's "daughter" by claiming Vesimir killed her, and the kitsu joins the humans in killing the witchers.
Only Vesemir and the youngsters who had recently passed the witcher trials are still alive. Geralt obtains his favorite piece of jewelry, the witcher wolf necklace, as a youngster. It's also a symbol of Vesemir's personal transformation, as the Witcher realizes that a family is all he wants. He's perplexed by the novices, which stand in for the vast number of children he'll never have owing to his Witcher transformation.
Nonetheless, at the end of the story, Vesemir appears to have found peace with his predicament in everyday life and instead accepts Geralt and his companions as his new wards, thereby making them the family he has been searching for all this time. Thus, it serves the film well that The Witcher: Nightmare of The Wolf manages to make this narrative element more of an effective method to link the histories of two key characters, rather than undermining Vesemir's one moment in the limelight, by presenting Geralt at any point in time.
With the franchise's popularity rising, there should be more opportunities for sequels like The Witcher: Nightmare of The Wolf, which expand on The Continent's environment and the characters that inhabit it outside of the main plot.
How many episodes of spy x family are out
BAKI: where to watch Baki, release date ,cast and plot


text:
class CreateReleasesLinkTable < ActiveRecord::Migration[5.0]
  include Gitlab::Database::MigrationHelpers

  DOWNTIME = false

  def change
    create_table :release_links, id: :bigserial do |t|
      t.references :release, null: false, index: false, foreign_key: { on_delete: :cascade }
      t.string :url, null: false
      t.string :name, null: false
      t.timestamps_with_timezone null: false

      t.index [:release_id, :url], unique: true
      t.index [:release_id, :name], unique: true
    end
  end
end


Summary The pavilion in Silver Plume, Colorado has a hipped roof, arched entry and windows, and one smoking chimney. One couple is holding hands and looking right as they approach another group of three people. Other people are in the background. A steep cliff band is behind the pavilion and borders Clear Creek. The pavilion was associated with the Argentine Central Railroad that was built to transport ore, but later became a popular tourist attraction.
Subject Silver Plume (Colo.); Pavilions--Colorado--Silver Plume--1900-1920; Railroads--Colorado--Silver Plume--1900-1920; Argentine Central Railway Company.


Christianity … is always in need of re-simplifying, going back to its origins, ridding itself of the excessive superstructure it has acquired through history.
In my kitchen there are six LAYERS of wallpaper. When it gets muggy, you can see the walls sag a bit as the weight of all the layers pull at each other. In some parts of the house, you can see the original wallpaper, just scratches here and there. To expose the original wallpaper (or even the paint), we'd have to tear the pieces off slowly and steadily.
In a simple phrase from Fr. Comblin, that encapsulates one mission of this website.
That's what the writings on Wikipedia are all about: removing the layers of bureaucracy to expose Christian opportunities and lay empowerment.
That what Hacking Christianity is all about: peeling back the layers and removing the wallpaper that covers up the truth and life of the Christian faith.
And if we have to tear down the walls to get at the heart of the Christian faith…we will.
such a great quote. i love it. i saw that yesterday and wanted to tweet it, but alas, it was more than 140 characters.


text:
Rosetta@home has been updated to version 5.43
Message boards : News : Rosetta@home has been updated to version 5.43
Rosetta@home has been updated to version 5.43. The new version has simplified the graphics in an effort to reduce graphics-related crashes for a small fraction of your users. Display of sidechains has been turned off temporarily as well as mouse rotation of the models. You can expect these features to return soon! For details, see this thread.


Local geology can influence how an area will respond to an earthquake. For example, on the east coast, the velocity contrast between the top two sediment layers dominates the amplification of ground motion. In this work, we characterize shear wave velocities of the two layers with cubic splines that approximate dispersion curves created by plotting phase velocity vs. frequency data from a seismic survey. We used more than 5,000 shots obtained through 14 km of seismic data collected near an M~7 earthquake in 1886 that devastated Charleston, South Carolina. The data was very noisy; therefore we simultaneously smoothed the data while fitting it to splines that could potentially estimate slopes and intercepts of the dispersion curves. We analyzed cubic splines and the smoothing technique used.
Clizzie, Nicole, "Using Cubic Splines to Evaluate Shear Wave Velocities of the 1886 Charleston Earthquake Region" (2019). 2019 Undergraduate Research and Scholarship Conference. 33.


Gorboduc, play by Thomas Norton and Thomas Sackville that takes as its subject Gorboduc, a mythical king of ancient Britain. First performed in 1561, it is the earliest English tragic play in blank verse.
Norton and Sackville's play is derived from Geoffrey of Monmouth's Historia regum Britanniae (1135–38; History of the Kings of Britain), which relates the dispute between Gorboduc's two sons, Ferrex and Porrex, over who would succeed him as king. Norton and Sackville depict Gorboduc as a good ruler who gives his kingdom away during his lifetime to his sons. The sons quarrel, and Porrex, the younger, kills Ferrex. Gorboduc's queen, Videna, avenges the death of her more-beloved older son by murdering Porrex. Gorboduc and his queen are, in turn, murdered by their horrified former subjects.
One of the first English tragedies to take Senecan tragedy as its model, Gorboduc is a blend of English and Classical elements. It ignores the unities of time and place and adds non-Classical dumb shows before each act, but it employs Classic formalities such as chorus and messenger. Gorboduc premiered before Queen Elizabeth I on Jan. 18, 1561. It was published in 1565 with printing errors and in better form in 1570 as The Tragedy of Ferrex and Porrex.


Can I edit in a 1080p timeline and then export to 4K with proper rescale for all items?
"In Premiere you can actually edit in a 1080p timeline and then change the resolution on export to 4K."
Read that, was wondering if I can do it in blender. Atm whenever I change the resolution of the workspace all my edits are not scaled 4x as well, they stay the old size. By edits I mean image overlays, text, stuff not self contained in the original video file.
So atm I'm basically rendering out at a higher format so all items are selfcontained in a video/imageformat and then re-rendering everything at 4k. I want to skip the extra step of needing to render inbetween though.
Browse other questions tagged rendering video-sequence-editor or ask your own question.
VSE: how to zoom in on image without pixel blur?


text:
Grotz: Rent is due on Eagles' mysterious…
Grotz: Rent is due on Eagles' mysterious 2021 preseason plan
Eagles head coach Nick Sirianni looks on during a preseason game against the New York Jets Aug. 27. Few regulars saw the field in that one, leaving open the question of just what the Eagles will look like, besides well-rested, in Sunday's opener in Atlanta. (AP Photo/Adam Hunger)
By Bob Grotz | rgrotz@delcotimes.com | Reading Eagle
PUBLISHED: September 9, 2021 at 2:34 a.m. | UPDATED: September 9, 2021 at 4:48 a.m.
PHILADELPHIA — Throughout training camp and preseason, Nick Sirianni and the Eagles have denied us answers to several basic questions fans have taken for granted over the years.
Real fundamental stuff like, what kind of offense are you going to run?
Suppressing that is supposed to give them a competitive advantage. It's the old "trust us, what you don't know, can't hurt you" strategy.
But like Jalen Hurts would say, the rent is due Sunday. When the Eagles open the season on the road against the favored Atlanta Falcons, we get to see if the secrecy is worth the lengths the Eagles have gone to preserve it.
This is the most mysterious Eagles rollout since the days of Chip Kelly's up-tempo offense. Excuse me for not expecting the same sort of fireworks as that Monday night eight years ago when the Eagles rolled into FedEx Field and Michael Vick led the Eagles to a 26-7 lead at the intermission in what would be a triumphant debut for the Chipster.
"When you're a new coach, everybody is still trying to figure you out," Eagles center Jason Kelce said Wednesday. "I still remember Chip Kelly when we were out there and we put up I think 50-some points against Washington in his debut. We had over 50-some plays in the first half with that tempo. Obviously, a different offense now but I'm excited to go out there and do this the first time with Nick and with this team."
It only seemed like 50-some points, Kelly and the Eagles hanging on to secure a 33-27 victory. The Eagles ran 53 plays in the first half. In the second half, the Philly defense was on the field twice as long as the offense. A roller coaster, for sure. Quality entertainment as well.
What we know about Sirianni and the Eagles is they're advocates of the sports science plan to get to the season healthy. A handful of regulars were limited Wednesday, which certainly beats the lengthy casualty list (Lane Johnson, Javon Hargrave, Miles Sanders, Derek Barnett, Brandon Brooks) towed into the opener against Washington last year.
How did Sirianni and the sports science staff do it? They barely practiced. A long training camp session was 90 minutes. Several practices were 75 minutes. The Eagles didn't have a two-hour practice until Bill Belichick and the New England Patriots came here for joint sessions. You have to wonder what Belichick was thinking when the Eagles intimated that 90 minutes would be fine.
Then there is the preseason, in which Hurts and most of the regulars played about 10 snaps. Quarterback Matt Ryan didn't play in any of the Falcons' three preseason games, but he has 201 more starts and 112 more victories than Hurts.
Former Eagles linebacker Seth Joyner, a longtime TV and radio analyst, appreciates the concern over injuries yet is puzzled that the Eagles have bombarded players with don't-get-hurt messaging.
"The question is how do you bring it all together with a new coaching staff when you don't play these guys in the preseason trying to protect them from getting hurt?." Joyner said on WBCB 1490 radio's Pro Football Report with Merrill Reese. "That's my major issue. Now they can get to the regular season and everybody's healthy but they can start dropping like flies in Week 1, and all of that protection that you've created is for naught.
"I'm not a big proponent of this culture of fear. In one of the most violent sports in the world, the NFL is now conforming. Because at the end of the day what are they saying? These are our starters. These are the guys that we believe in but we need to put them in bubble wrap to make sure we get to Week 1 and they're 100 percent healthy."
Sanders and Kelce, not surprisingly, feel fantastic. And they should. Sanders played one series in the preseason, Kelce two.
"I feel amazing, actually," Sanders said. "I took a different approach on my offseason workouts, too. Definitely with the way training camp is going on, how they do it now to keep us fresh I feel amazing. And I think everybody else feels good, too."
That alternative approach was to stay in shape, rather than get in shape. Kelce, who turns 34 in November, had an extra veteran's day off per week.
"I feel good," Kelce said. "I think we had a really good training camp. The coaches all did a phenomenal job, and the players. The strength and conditioning and the sports science really put together a plan to try to keep us healthy and do some things that have plagued us the last couple years. We're pretty healthy going into the first week. I feel really good about that."
Sirianni gave such a long-winded answer when asked if he and his team were ready, that I'll condense it for you. They are right where he wants them to be on a Wednesday before meeting another new coaching staff and culture.
"I definitely feel like we put ourselves in a position to be ready, to be here right now," Sirianni said. "The hay is not in the barn yet. There is still a lot of work to be done."
A word of warning Nick: The rent is due Sunday.
Bob Grotz | Reporter
More in Philadelphia Eagles
After rough finish, Jalen Hurts ready for next Eagles act
Grotz: Despite Rocky prep, Eagles get flattened by knock-out Bucs
Eagles Notebook: Hobbled, humbled Jalen Hurts 'not defined' by Bucs loss
McCaffery: Struggles against contenders dulled Eagles' season


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The word Babri was literally used to define something abnormal, out of sense or mad. Three instances mentioned in Babar-Nama support this. Babar's journal is replete with precise detail with a telling image or idiom as "a bud resembling a sheep's heart", "fell like water on fire" which lingers in the reader's mind long after the event or anecdote has receded. Few of the phrases and words in the Babar Nama are now part of everyday language in India and Pakistan as "Namak Haram" means lack of trustworthiness, "hamesha" means always, "bakhshish" means gift, "maidan" means plain area, "julab" means laxative and the most important, which is of our use here, is "Babri/Baburi /Bavala" means related to unhealthy mental state or mad or abandoned or one who is abnormal. Babar writes about his infatuation, after his marriage in March AD 1500, for a boy as, "In those days I discovered myself a strange inclination - no, a mad infatuation-for a boy in the camp's bazaar, his name was Babri/Baburi being apposite. Until then I had no inclination of love and ... a couplet of Muhammad Salih came to my mind: When I see my friend I am abashed with shame; My companions look at me, I look away sans aim. This couplet suited my state of mind perfectly. In that maelstrom of desire and passion, and under the stress of youthful folly, I used to wander, bareheaded and barefoot, through streets and lanes, orchards and vineyards. I showed civility neither to friends nor to strangers, took no care of myself or others." Babar clearly stated that guy's name was Bavara as he was of raging and flickering nature and Babar himself became Bavara or crazy for him to attain his sexual proximity.... That is why the term "Babri Mosque" is specially used only for the construction that was done according to Mughal architecture at Ramjanmabhoomi because it was made for Hindus not for Muslims. Babri Mosque means Mosques of infidels-insane Hindus.


text:
Contact: Dr. Gary J. Bennett, Quality Milk Production Services, 877-645-5523, 315-379-3930
Jpgs:
KlebsiellaLabWorkbw.jpg - At the QMPS laboratory i Canton, Dr. Marcos Munoz from Chile examines one of many test samples collected for the Klebsiella mastitis prevention and controls research project funded by the Northern New York Agricultural Development Program. Photo: Cornell University
KlebsiellaSamplingbw.jpg - Dr. Ruth N. Zadoks collects a test sample from a dairy barn alleyway. The sample was evaluated for Klebsiella bacteria as part of the Northern New York Agricultural Development Program project. Photo: Cornell University
New Klebsiella Mastitis Fact Sheet Now Online at www.nnyagdev.org:
Knowledge Offers New Tool for Preventing Infection
Klebsiella bacteria cause mastitis in dairy cows. reduce milk production, do not respond to treatment of the udder with antibiotics and come in so many different strains that North Country farmers are frustrated in their battle to reduce the infection. If not caught early, Klebsiella causes cows to be permanently removed from milk production, and the infection is often fatal to the animals. Northern New York Agricultural Development Program-funded Klebsiella research involving North Country dairy herds is providing new knowledge and tools for preventing the infection. The Klebsiella Mastitis Prevention and Control fact sheet is now online in the Dairy information section of the regional program�s website at www.nnyagdev.org.
New Fact Sheet Equips Farmers to Battle Infection
Doug Shelmidine of Sheland Farms in Belleville, NY, is among the Northern New York Agricultural Development Program farmer-advisors and was a leading proponent for the Klebsiella project that collected and analyzed milk and manure samples from cows and from their farm environments.
Shelmidine says, �Klebsiella has become an issue for more and more Northern New York farms. This project gives us a look at outcomes from several herds so each dairy can weigh the factors that influence the rate of infection and make changes for our own operations. There may not be a cure for Klebsiella and there are no shortcuts to cleanliness and proper milking procedures, but the more we can learn about reducing exposure and increasing prevention measures, the better.�
Identifying Critical Control Points
To identify critical control points to combat Klebsiella, the farmer-driven Northern New York Agricultural Development Program (NNYADP) enlisted the aid of regional dairy farmers and Cornell University researchers and veterinarians Dr. Ruth Zadoks and Dr. Gary J. Bennett with Quality Milk Production Services (QMPS). Milk, manure, feed, water and bedding samples taken from NNY dairy herds were analyzed for Klebsiella by the QMPS laboratories in Canton and Ithaca, NY. Several DNA fingerprinting methods were used to identify and type hundreds of different strains.
Zadoks says, �If you have cows, you have Klebsiella. We have found various strains of the Klebsiella family of bacteria on farms in manure, bedding, water, feed, and milking equipment.�
Bennett says approximately 80 percent of the cows tested in more than 10 herds over the summer of 2005 were positive for some strain of Klebsiella in their manure. The average for Northern New York herds in the summer of 2007 was around 60 percent. Each cow sheds multiple strains of Klebsiella.
�On rare occasions we have seen multiple cows in the same herd with the same strain of Klebsiella in their udders. To ensure that cows do not infect each other, it is important to separate animals with mastitis from the rest of the herd. Cows excrete Klebsiella in their feces and via the manure it spreads everywhere. We think that transmission can take place via the milking machine and via milk that leaks into the cow stalls,� Zadoks says.
Research Shows Keeping Alleys Clean is Key
�Klebsiella does not just come from bedding. This research showed us how big a role the cleanliness of the alleyways and of the cows plays in this infection and provides us with new tools to prevent Klebsiella mastitis,� Zadoks says. �Alleyway hygiene is an important and often overlooked component of Klebsiella control.�
When cows walk through manure in alleyways, they contaminate their udders, teats and legs. Then they lie down with their teats on top of their manure-soiled legs creating further exposure to the Klebsiella. Two out of three cows tested in Northern New York had Klebsiella on her legs and the skin of her teats.
�We are directing farmer attention to alleyways, including the exit from the milking parlor, and holding pens, where Klebsiella is extremely common,� Zadoks says.
Developing vaccines to fight the many strains of Klebsiella is a long-term prospect. Vaccination strategies that work against the less aggressive E. coli bacteria do not work with Klebsiella.
What can farmers do while waiting for a vaccine to fight Klebsiella, which can be signaled by a rise in somatic cell count?
Diligence to Proper Milking Practices Reduces Risk
Dairyman Andrew Bilow milks 1,000 cows at the farm he co-owns with his father and cousin in Malone, NY. The farm has been named a Dairy of Distinction noted for its cleanliness for the past 10 years.
�Our somatic cell count was climbing and we could not find a reason why. After participating in this Klebsiella project, we changed our pre-dip and post-dip treatments and our counts fell to consistently-acceptable levels within six months,� Bilow says.
At the W.H. Miner Agricultural Research Institute in Chazy, NY, Herdsperson Anna Pape helps oversee a milking herd of 280 cows.
�This project confirmed that we needed to remain diligent to good milking practices. We are paying particular attention to cleaning teat ends to reduce the opportunity for exposure when they open to let the milk out,� Pape says.
�And now that we understand the true lack of sensitivity to antibiotics by Klebsiella, we are better at treating infected cows with hydration and pain management, reducing their exposure to secondary infections, and feeding alfalfa meal and electrolytes to maintain their appetite,� Pape adds.
Bedding and Stall Changes Can Help
Zadoks says that testing of various types of dairy bedding shows that switching from sawdust and other bedding with high organic matter to sand bedding can decrease Klebsiella exposure. She cautions, however, that once sand bedding that tests Klebsiella-free prior to use is put into the barn and becomes contaminated with manure, it can become a source of the Klebsiella infection.
Bilow and Pape both say their farms will make adjustments in the barn to reduce exposure. Bilow Dairy will convert to using sand bedding in 2008. The Miner Institute dairy barn is not set up to use sand bedding and has larger-than-industry-standard size stalls to promote cow comfort. Pape says, �We are looking at adjusting the brisket boards to position the cows farther back in the stall so more manure will fall into the alley and not in the stall.� At any time of year, �stand, alley and barn hygiene is critical for controlling exposure to Klebsiella. Using bedding with the least amount of organic matter reduces exposure. Milking cows you know to be infected last and disinfecting equipment after milking cows with mastitis are also good practices,� Zadoks says. �Each farm must develop a plan for its unique conditions.�
Some good news for farmers in the colder climate of Northern New York is that the rate of Klebsiella-based mastitis tends to be lower in winter months.
The Klebsiella Mastitis Prevention and Control fact sheet can be downloaded from the Northern New York Agricultural Development Program website at www.nnyagdev.org.
The Klebsiella Mastitis Prevention and Control fact sheet on the sources, transmission and control methods for Klebsiella-based mastitis will be sent to producers whenever the QMPS labs identify Klebsiella in one of that producer�s farm samples. Veterinarians and Cornell Cooperative Extension educators will be distributing the Klebsiella fact sheet to dairy producers in Clinton, Essex, Franklin, Jefferson, Lewis and St. Lawrence counties.
A committee of dairy, fruit, livestock, maple and vegetable producers guides the Northern New York Agricultural Development Program that funds research and educational outreach with practical results for NNY farmers. Dairymen Jon Greenwood of Canton and Joe Giroux of Plattsburgh co-chair the program. #


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Fugitive Ex-Beauvoir Teacher Indicted on Child Porn Charge
Teacher accused of taking inappropriate pictures of boy
Published December 1, 2008 • Updated on December 1, 2008 at 8:33 pm
GREENBELT, Md. -- A federal grand jury indicted a former Beauvoir Elementary School teacher Monday on a charge of production of child pornography.
Eric Justin Toth, 26, remains a fugitive. He was featured on "America's Most Wanted."
In early June, Toth, who taught third-graders at Beauvoir Elementary School, was kicked off the campus at the National Cathedral in northwest Washington. On June 12, the school notified parents in a letter that a teacher was found in possession of a school-owned camera with inappropriate pictures of a boy. Sources told News4 that pictures of at least three other boys being touched in inappropriate ways were on a thumb drive.
In August, Toth's car was found in a parking garage at Minneapolis-St. Paul International Airport. A note in the car indicated that Toth was contemplating suicide and that his body would be found in a nearby lake, sources said. Investigators believe that may be a ruse, police said.
Toth had worked at Beauvoir for three years. Before the end of the 2007-08 school year, he resigned and told school officials he wouldn't be returning this school year.
It was common for Toth, formerly of Arlington, Va., to tutor and baby sit Beauvoir students, sources said. At least one of the boys photographed is believed to be a Beauvoir student, sources said.
If convicted, Toth faces a minimum mandatory sentence of 15 years in prison and a maximum of 30 years in prison.
5 Hurt, Including Children, in Deadly Shooting, Crash in Baltimore: Police


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text:
NVIDIA RTX 4070 Ti leaks reveal specs and potential price
The company, which is expected to unveil the GPU at CES, accidentally published the specs on its website.
Kris Holt
·Contributing Reporter
December 30, 2022, 4:03 PM ·1 min read
REUTERS/Dado Ruvic/Illustration
NVIDIA is expected to reveal its GeForce RTX 4070 Ti graphics card at CES next week, but it preemptively leaked the specs. Thanks to new rumors, we have a sense of the GPU's likely price too.
The RTX 4070 Ti is slated to have 12GB of GDDR6X memory with 7,680 Cuda cores that can be boosted to 2.61GHz, as Tom's Hardware notes. NVIDIA seemingly expects the card to deliver 4K gameplay at up to 240Hz, or 8K visuals at 60Hz with DSC and HDR enabled. The company claimed the RTX 4070 Ti will deliver around 3.5 times better performance than the 12GB RTX 3080 in Cyberpunk 2077 when the new RT Overdrive mode is enabled.
It has been widely believed that the latest card would essentially be a rebranded version of the 12GB RTX 4080. In October, NVIDIA reversed plans to release that model and suggested it would rebadge the GPU.
Meanwhile, rumors indicate NVIDIA will sell the RTX 4070 Ti for $799. It was previously expected that the price would be $899, but NVIDIA may have lowered it after the US delayed tariffs on GPUs that were set to resume on January 1st. Based on the RTX 4070 Ti's expected performance, Wccftech ran the numbers and found that, on a teraflop-to-dollar ratio, the GPU will offer 97 percent of the value proposition of the $1,599 RTX 4090.
We should find out official details about the RTX 4070 Ti, perhaps including the release date, very soon. NVIDIA has scheduled a CES edition of its GeForce Beyond event for January 3rd at 11AM ET.
Virginia vs Syracuse College Basketball Prediction, Game Preview
Virginia vs Syracuse preview, prediction, and breakdown for the college basketball game on Monday, January 30
Lansing State Journal
MSU hockey swept by No. 2 Minnesota, as big home series vs. Notre Dame looms
The Spartans fell 6-3 on Saturday night, dropping to No. 16 in the all-important Pairwise rankings.
Biden, McCarthy to discuss debt limit in talks on Wednesday
House Speaker Kevin McCarthy said Sunday he is looking forward to discussing with President Joe Biden a "reasonable and responsible way that we can lift the debt ceiling " when the two meet Wednesday for their first sit-down at the White House since McCarthy was elected to the post. McCarthy, R-Calif., said he wants to address spending cuts along with raising the debt limit, even though the White House has ruled out linking those two issues together as the government tries to avoid a potentially devastating financial default. "I know the president said he didn't want to have any discussion (on cuts), but I think it's very important that our whole government is designed to find compromise," McCarthy told CBS' "Face the Nation."
Lady Susan Hussey accompanies the King to church in return to royal fold
Lady Susan Hussey has joined the King and the Princess Royal at a church service at Sandringham in an apparent vote of confidence in the former royal aide.
Hosts South Africa won the toss and elected to bowl first, but after a slow start England accelerated superbly to post a formidable 342 for seven in their 50 overs. The home side reached their target with five balls remaining for a record chase at the Mangaung Oval, smashing the previous best mark of 274 by South Africa against Australia in 2020.
Michigan basketball game vs. Penn State Nittany Lions: Live game updates
Michigan Wolverines game time, TV channel info, radio, score, live updates and analysis as they play the Penn State Nittany Lions
Why this week's Fed vs. markets showdown could decide fate of early 2023 stock rally
The Federal Reserve and investors appear to be locked in a stare-down. What Fed Chair Jerome Powell says Wednesday could determine the winner.
No Charter School Applications in Louisville Despite Legislature's 'Urban Academy' Mandate
No one applied to open a charter school in Louisville, despite last year's mandate by the legislature that Jefferson County Public Schools approve a pilot charter by July 1 of this year. Monday was the deadline for charter school applicants in Jefferson County. JCPS spokesman Mark Hebert confirmed that no sponsor stepped up to open […]
DeTomaso Pantera Wears A Widebody Kit
Now this is how you do a widebody, Instagram artists!
Colts Wire
Colts will hold second head coach interview with Shane Steichen
The Colts are bringing in Eagles OC Shane Steichen for a second head coach interview.
The health conditions that warrant a trip to A&E and the ones that don't
When A&E doctor Mark Harmon's elderly mother slipped on the ice recently, he knew that hospital was the last place that would be good for her.
Pac-12 bubble watch: USC now leading the pack after Utah and Arizona State lose big
UCLA and Arizona are obviously the two best teams in the Pac-12. Who's No. 3? It's a fluid race, but USC is definitely leading in the push for an NCAA tourney bid.


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text:
\section{Introduction}

The quantum Hall effect continuous to intrigue both experimentalists
and theorists not only because of the beautifully rich patterns visible
in the data but also because of the fascinating physics involved in
the collective phenomena of strongly interacting systems.
The first suggestion of a connection between the quantum Hall effect
and the modular group appeared in \cite{Shapere+Wilczek}, although
these authors focused on a sub-group of the full modular group that did not
turn out to have any direct relevance to the current experimental data.
Subsequent papers on symmetries
of the phase diagram of the quantum Hall effect
\cite{KLZ,Lutken+Ross} appeared almost simultaneously from two very different
directions and laid the foundations for the application of
modular symmetry to the quantum Hall effect.  Although reference \cite{KLZ}
did not use the mathematical language of modular symmetry the \lq\lq Law
of Corresponding States'' put forward in that reference is in fact equivalent
to the assumption of modular symmetry \cite{BDa}.

Modular symmetry gives predictions \cite{BDc} for the manner in which
the conductivity of a two-dimensional quantum Hall sample flows,
as the temperature is varied keeping the magnetic field fixed ---
predictions which have already received strong experimental
support \cite{Murzin1,Murzin2,Taiwan1,Taiwan2,Taiwan3}.

While the flow diagram presented in \cite{BDc}
was for spin-split samples the experimental data
presented in \cite{Murzin1} is for spin-degenerate samples.
Zeeman splitting in the context of modular symmetry was analysed
in \cite{BDspin}
and a flow diagram, deformed by interactions between adjacent Landau levels,
was given but
the fully spin-degenerate case has not yet been treated using modular symmetry

In this paper restrictions on the temperature flow due to modular symmetry are
combined with a Zeeman splitting analysis
to determine the way in which
temperature flow changes as the Zeeman splitting is smoothly varied
between the two extremes of samples with well-split spins and fully spin-degenerate
samples.  The analysis is restricted to situations with particle-hole symmetry,
as modular symmetry is particularly powerful in this case but does not give
strong predictions otherwise.
A central prediction of the analysis is that pairs of
critical points of the quantum Hall phase transitions
between adjacent plateaux in spin-split samples must merge as the Zeeman splitting
is reduced, as shown in figures 12 and 13.

\section{The modular group}

The law of corresponding states \cite{KLZ,ZHK}, for isotropic
quantum Hall samples with
spins well split, can be written in terms of the complex
conductivity
\[ \sigma = \sigma_{xy} + i\sigma_{xx}\]
(for isotropic samples $\sigma_{xx}=\sigma_{yy}$).
A general map between two quantum Hall states can be constructed by iterating
two generating maps, \cite{Lutken+Ross,BDa,BL}:
the Landau level addition transformation, ${\bf L}$,
\[ \label{Complexsigma}
\sigma\rightarrow \sigma + 1,
\]
and the flux attachment transformation, ${\bf F}^2$,
\[
\sigma\rightarrow \frac{\sigma}{2\sigma+1},
\]
which attaches two units of statistical gauge field flux to each
electron (we use units with $\frac{e^2}{h}=1$).

In samples which enjoy particle-hole symmetry there is a third
map, the particle-hole transformation, ${\bf P}_1$,
$$
\sigma\rightarrow 1 - \overline\sigma.
$$
These maps generalise Jain's transformations on ground state wavefunctions,
\cite{Jain}, to include non-zero Ohmic conductivity.
Repeated iteration of ${\bf F}^2$ and ${\bf L}$ generate an infinite discrete
group which we shall denote $\Gamma_0(2)$.

The infinite discrete group generated by repeated applications of
${\bf L}$ and ${\bf F}$ is called the modular group and is usually
denoted by $\Gamma(1)$ in the mathematical literature \cite{Koblitz}.
A general element $\gamma\in\Gamma(1)$
sends
\begin{equation}
\label{gammadef}
\sigma\ \rightarrow\ \gamma(\sigma):=\frac{a\sigma +b}{c\sigma +d},
\end{equation}
where $a,b,c$ and $d$ are any four integers satisfying $ad-bc =1$.
Group multiplication can be realised in terms of the $2\times 2$ matrix
\begin{equation}\label{gammaM}
\gamma=\left(
\begin{array}{cc}a&b\\ c&d\\ \end{array}
\right)\end{equation}
and demanding that $\det\gamma=1$. It is easy to check from the definition
(\ref{gammadef}) that, for any three
such matrices satisfying $\gamma_1\gamma_2=\gamma_3$,
we have $\gamma_1(\gamma_2(\sigma))=\gamma_3(\sigma)$. Thus the group multiplication
law is given by matrix multiplication.

The full modular group $\Gamma(1)$ is not a symmetry of quantum Hall effect, for example
the element $\left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\right)$
sends $\sigma \rightarrow -1/\sigma$ which has a fixed point for $\sigma=i$,
{\it i.e.} $\sigma_{xx}=1$, $\sigma_{xy}=0$.
There is no indication in the experimental data on the quantum
Hall effect that this point has any special significance (though it is
important in the insulator --- superconductor phase transition \cite{MFisher,Gammatheta}).

The group $\Gamma_0(2)$ is a sub-group of the modular group, it is represented
by matrices of the from (\ref{gammaM}) with the extra condition that $c$ be even.
In matrix notation Landau level addition and Flux attachment are represented
by
${\bf L}=
\left(
\begin{array}{cc} 1&1\\ 0&1 \\ \end{array}
\right)$ and ${\bf F}^2=
\left(\begin{array}{cc} 1&0\\ 2&1\\ \end{array}
\right).$

If the electron spins are not split then the story is different.
For simplicity first consider the situation if Zeeman splitting is completely
absent and states corresponding to two different electron spins
are completely degenerate.  Then Landau level addition sends
\[ \sigma\rightarrow \sigma + 2, \]
or $\frac\sigma 2 \rightarrow \frac \sigma 2 +1$,
which is ${\bf L}^2=\left(\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ \end{array}\right)$.
Attaching two units of flux to each electron sends
\[
\frac{\sigma} {2} \rightarrow
\frac {\left(\frac{\sigma} {2}\right)} {2\left(\frac{\sigma} {2}\right) + 1}
\qquad \Rightarrow \qquad \sigma \rightarrow \frac {\sigma}{\sigma + 1},\]
which is ${\bf F}=\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array}\right)$ acting
on $\sigma$.  Thus spin generate Landau levels give rise to an infinite discrete
group generate by ${\bf L}^2$ and ${\bf F}$.  A general element of this group
is of the form $\gamma=\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right)$
with $a$, $b$, $c$ and $d$ integers satisfying $ad-bc=1$,
but with $b$ restricted to be even.
We denote this group by $\Gamma^0(2)$.  In fact
that $\Gamma^0(2)$  acting on $\sigma$ is the same as $\Gamma_0(2)$
acting on $\frac \sigma 2$.  Particle-hole symmetry for degenerate
spins is also modified to ${\bf P}_2=\sigma\rightarrow 2-\overline\sigma$,
so ${\bf P}_2={\bf P}_1+1$.

For spins that have a slight splitting, so that the Landau levels
corresponding to opposite spins are not
completely degenerate but are still close enough for there to be some
mixing between Landau level wave-functions, one expects ${\bf L}^2$ to be a symmetry rather
than ${\bf L}$.  At the same time adding two units of statistical flux to
each electron in the individual levels is described by ${\bf F}^2$.
For this intermediate case therefore the group is generated by
${\bf L}^2$ and ${\bf F}^2$ and a general element is represented by
$\gamma=\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right)$
with $ad-bc=1$ and {\it both} $b$ and $c$ even integers.
This group is denoted by $\Gamma(2)$.
Clearly $\Gamma(2)\subset \Gamma_0(2)$ and
$\Gamma(2)\subset \Gamma^0(2)$.
We are led to suggest the following sequence of symmetries as the Zeeman
splitting relative to the cyclotron energy is varied from large to small:
\[\Gamma_0(2)\quad\longrightarrow\quad\Gamma(2)\quad\longrightarrow\quad
\Gamma^0(2),\]
thus the symmetry first decreases and then increases again
as the Zeeman splitting is varied.

Since the modular group has an infinite number of elements the law
of corresponding states maps between an infinite number of quantum Hall
phases, which is clearly a mathematical idealisation which is never
realised in any physical system.  Obviously the symmetry breaks down
in various limits, such as weak magnetic field, when the Landau level
splitting becomes comparable with thermal energies, or very strong
magnetic fields, when a Wigner crystal is expected to form.
The range of validity is discussed in \cite{BDb}.

It was pointed out in \cite{Sachdev} that the law of corresponding states
applies to the AC conductivity in the limit of infinite frequency
rather than the DC conductivity.  This is because the conductivity is a
a function of frequency over temperature,
$\sigma_{xx}\left(\frac \omega T\right)$,
and the derivation of the law of corresponding states in \cite{KLZ}
takes the limit $T\rightarrow 0$ before $\omega\rightarrow 0$
and this does
not in general commute with the limit required to extract the DC conductivity,
namely $\omega\rightarrow 0$ before $T\rightarrow 0$.
The precise relationship between these two
limits can only be explored in the context of a specific microscopic
theory for the conductivity and is not accessible solely through infra-red
effective action techniques such as the law of corresponding states.
Nevertheless the law of corresponding states
has been applied to DC conductivities, for example with regard to
temperature flows, and the experimental data are in
remarkable agreement with the predictions
\cite{Murzin1,Murzin2,Taiwan1,Taiwan2,Taiwan3,BDb}.
While some microscopic models may display a
symmetry which makes the order in which the limits are taken
irrelevant  \cite{Sachdev}
this is not generic, but since we do not commit ourselves to a specific
model here we cannot address this question directly.  Rather our philosophy will
be to develop the predictions of the law of corresponding states, see
where they lead, and future experiments will test their validity.

\section{Temperature flow}

Scaling arguments \cite{Wei} suggest that, at low temperatures, the
DC conductivity should be a function of a single variable,
$\sigma\left(\frac{\Delta B} {T^\kappa}\right)$,
rather than of the temperature $T$ and magnetic field $B$
separately.
Here $\Delta B= B- B_c$ is the deviation of the magnetic field
from the critical value $B_c$ separating two quantum Hall
phases and $\kappa$ is a scaling exponent,
experimentally $\kappa\approx 0.44$.
The conductivity depends on
the electron scattering length $l$ and we define the scaling
function
\[\Sigma_l(\sigma,\overline\sigma)=-l \frac {d\sigma}{dl}.\]
Assuming $l$ is a strictly monotonic function of temperature, increasing as
$T$ decreases, the temperature flow described by the scaling function
\[\Sigma_T(\sigma,\overline\sigma)= T \frac {d\sigma}{d T}\]
will have the same topology as the flow described by $\Sigma_l$ ---
they will have the same fixed points, $\Sigma_T=0$ if and only if
$\Sigma_l=0$.  We do not need to determine either $\Sigma_l$
or $\Sigma_T$ exactly, in fact if $s(T)$ is {\it any} monotonic
function of $T$, decreasing as $T$ decreases, then
the flow described by
\[ \Sigma_s(\sigma,\overline \sigma)=  s \frac {d\sigma}{d s} \]
will have the same topology as both that of $\Sigma_T$ and of $\Sigma_l$.

If the law of corresponding states
correctly describes the low temperature
physics then the scaling flow commutes with the
law of corresponding states map.  From this it can be concluded that
any value of the complex conductivity $\sigma_*$
that is a fixed point of the modular group
(in the sense that there exists an element $\gamma$ of the modular group
such that $\gamma(\sigma_*)=\sigma_*$)
must also be a fixed point of the scaling flow, \cite{BDa,BL}.
This implies that
\[\Sigma_s(\sigma_*,\overline \sigma_*)=0.\]
This follows because the assumption $\gamma(\sigma_*)=\sigma_*$ requires
\begin{equation}\label{SigmaFlow}
\Sigma_s(\sigma_*,\overline \sigma_*)=\Sigma_s\bigl(\gamma(\sigma_*),\gamma(\overline\sigma_*)\bigr)=s\left.\frac {d \gamma(\sigma)}{d s}\right|_{\sigma_*}
=\frac 1 {(c\sigma_* + d)^2}\Sigma_s(\sigma_*,\overline \sigma_*)\end{equation}
which is only possible if $(c\sigma_* +d)^2=1$ or
if $\Sigma_s(\sigma_*,\overline \sigma_*)=0$ or $\infty$.
If $(c\sigma_* + d)^2=1$ then
$a\sigma_* +b =\pm \sigma_*$ and, excluding the trivial case $a=\pm 1$, $b=0$,
this is
not possible if the Ohmic conductivity $(\sigma_*)_{xx}>0$ at the fixed point.
Assuming $\Sigma_s(\sigma_*,\overline\sigma_*)$ is not infinite
we conclude that
$\Sigma_s(\sigma_*,\overline \sigma_*)=0$
and $\sigma_*$ is a fixed point of the flow.
The fixed points with  $Im(\sigma_*)>0$ are isolated and easy to enumerate,
since
\[\gamma(\sigma_*)=\sigma_* \qquad \Rightarrow
\qquad \sigma_*=\frac{a-d \pm\sqrt{(a+d)^2-4}} {2c}.\]
Now $ad-bc=1$, with $bc$ even for all three groups $\Gamma_0(2)$, $\Gamma^0(2)$
and $\Gamma(2)$, hence $ad$ must be odd so both $a$ and $d$
must be odd.  Demanding $Im(\sigma_*)>0$ then requires that $-2<a+d<2$.
Hence $a+d=0$, $\pm 1$, but $a$ and $d$ are both odd
so $\pm 1$ is ruled out and we can conclude that $d=-a$.
Hence
\begin{equation}\label{sigmastar}
\sigma_*=\frac{a + i} {c}\end{equation}
as the Ohmic conductivity, $Im(\sigma_*)$, cannot be negative.
A matrix $\gamma$ that leaves $\sigma_*$ fixed must now be of the form
\[\gamma=\left(\begin{array}{cc} a & b \\ c & -a \\ \end{array}\right)\]
with $bc=-(1+a^2)$ with $a$ odd.  Let $a=2p+1$ for some integer $p$,
then $bc=-4p(p+1)-2$ and $bc=2$ mod $4$.  In particular $b$ and $c$ cannot
both be even so $\Gamma(2)$ has {\it no} fixed points with
$Im(\sigma)>0$.

To summarise: $\Gamma_0(2)$ has fixed points above the real axis
of the form (\ref{sigmastar}) with $a$ odd and $c$ even;
$\Gamma^0(2)$ has fixed points above the real axis
of the form (\ref{sigmastar}) with $a$ and $c$ both odd;
$\Gamma(2)$ has no fixed points above the real axis.

Although any fixed point of the modular group with $Im(\sigma_*)>0$ must be
a fixed point of the flow the converse does not necessarily hold,
there could be fixed points of the flow that are not fixed points of
the modular group.  Any such point would have an infinite number of images
under the group action.  However there is no
sign any such extra fixed points in the experimental data for spin split
samples so we shall assume that there are no fixed points
for $\Gamma_0(2)$, other than those required by the symmetry.
For  brevity in the following this will be referred as the
{\it minimalist assumption}.
If we assume that the topology of the flow varies smoothly as the
Zeeman energy is varied then the $\Gamma_0(2)$ fixed points cannot
suddenly disappear when $\Gamma_0(2)$ is broken to the smaller group
$\Gamma(2)$, they must move down towards the real axis.
We shall likewise assume that the only fixed points of the flow for
degenerate spins are those of $\Gamma^0(2)$ and these move smoothly down
towards the real axis as the degeneracy is lifted.

The topology of the flow is determined by the fixed points and
some other rather mild assumptions \cite{BDc},
such as decreasing Ohmic conductivity when $\sigma_{xx}>>\sigma_{xy}$ as the temperature is reduced, as in a semi-conductor, and attractive fixed
points at integer quantum Hall plateaux.
We can plot the flow by changing variables from $\sigma$
to $\lambda(\sigma)$ where $\lambda$ is invariant under $\Gamma(2)$, {\it i.e.}
$\lambda\bigl(\gamma(\sigma)\bigr)=\lambda(\sigma)$, with $\gamma\in\Gamma(2)$,
\cite{BDspin,SemiCircle}.  Since $\Sigma_s(\sigma,\overline\sigma)$ represents a vector flow in
a two-dimensional space this is just a change of co-ordinates in that
space.  In the new parametrisation the flow is given by
\[
\Sigma_\lambda(\lambda,\overline\lambda)=s\frac{d \lambda}{d s}
=\Sigma_s(\sigma,\overline\sigma)\lambda'
\qquad \Rightarrow \qquad \Sigma_s(\sigma,\overline\sigma)=\frac{\Sigma_\lambda}{\lambda'} ,\]
where $\lambda'=\frac{d\lambda}{d\sigma}$.

The invariant function $\lambda$ that has the smallest number of poles
and zeros in the complex plane is unique, up to
a constant rescaling and addition of a constant, \cite{WW}.
It is most easily expressed in terms of Jacobi
$\vartheta$-functions:
\[\lambda=\frac{\vartheta_2^4}{\vartheta_3^4}\]
where
\begin{equation} \vartheta_3(\sigma)=\sum_{n=-\infty}^\infty e^{i\pi n^2\sigma}\qquad
\hbox{and}\qquad
\vartheta_2=2\sum_{n=0}^\infty e^{i\pi(n+\frac{1}{2})^2\sigma}.\end{equation}
It is also useful to define
\begin{equation} \label{theta-identity}
\vartheta^4_4(\sigma)=\sum_{n=-\infty}^\infty (-1)^n e^{i\pi n^2\sigma}
=\vartheta^4_3(\sigma)-\vartheta^4_2(\sigma)
.\end{equation}
Then these functions have the following transformation properties
under ${\bf L}$ and ${\bf F}^2$:
\begin{eqnarray}
\vartheta_2(\sigma+1)=e^{\frac{i\pi} {4}}\vartheta_2(\sigma) ,\kern -12pt
&\qquad & \vartheta_2\left(\frac{\sigma}{2\sigma+1}\right)=\sqrt{2\sigma+1}\;\vartheta_2(\sigma),\nonumber \\
\vartheta_3(\sigma+1)=\vartheta_4(\sigma),
\;&\quad\qquad & \vartheta_3\left(\frac{\sigma}{2\sigma+1}\right)=\sqrt{2\sigma+1}\;\vartheta_3(\sigma),\nonumber \\
\vartheta_4(\sigma+1)=\vartheta_3(\sigma),
\;&\quad\qquad & \vartheta_4\left(\frac{\sigma}{2\sigma+1}\right)=
\sqrt{2\sigma+1}\;e^{-\frac{i\pi}{2}}\vartheta_4(\sigma),\nonumber
\end{eqnarray}
(we use the notation of \cite{WW}).

Furthermore under ${\bf P}_2$, $\vartheta_i \rightarrow \overline\vartheta_i$
for $i=2,3,4$.  Hence particle-hole interchange swaps $\lambda\leftrightarrow
\overline\lambda$ and assuming particle-hole symmetry
has the important consequence for the scaling function that
\begin{equation}\label{PHinterchange}
\Sigma_\lambda(\lambda,\overline\lambda)=
\overline{\Sigma_\lambda(\lambda,\overline\lambda)}
=\Sigma_\lambda(\overline\lambda,\lambda).\end{equation}
This then implies that an expansion of $\Sigma$ in powers of $\lambda$
and $\overline\lambda$ has only real co-efficients and we draw
the important conclusion that, starting from
any point for which $\lambda$ is real, the flow
can never generate an imaginary part for $\lambda$.  In other words
any curve on which $\lambda$ is real is in integral curve of the
flow \cite{SemiCircle}: this is a key observation in creating the
flow diagrams below.  In particular $\lambda$ is real on vertical
lines above the integer points on the real line and one the semi-circles
of radius $1/2$ joining the integers (see figure 1).

While $\lambda$ is not invariant under the larger groups $\Gamma_0(2)$ and
$\Gamma^0(2)$ the following functions of $\lambda$ are:
\begin{itemize}
\item
$\mu=\frac{\lambda-1}{\lambda^2}=
-\frac{\vartheta_3^4\vartheta_4^4}{\vartheta_2^8}$ is invariant under $\Gamma_0(2)$;\hfill\break
\item
$\rho=\frac{\lambda}{(1-\lambda)^2}=
\frac{\vartheta_2^4\vartheta_3^4}{\vartheta_4^8}$ is invariant under $\Gamma^0(2)$.
\end{itemize}

The functions $\mu$ and $\rho$ can be used to determine the topology
of the flow for $\Gamma_0(2)$ and $\Gamma^0(2)$ respectively.
Fixed points of $\Gamma_0(2)$, respectively $\Gamma^0(2)$,
must be fixed points of the flow, and these are enumerated in equation
(\ref{sigmastar}).
Next, as above, we argue that demanding particle-hole symmetry
implies that curves on which $\mu$, respectively $\rho$, is real
will be integral curves of the flow.
The flow can then be modelled qualitatively
by taking $\Sigma_s$ to be a meromorphic function of $\sigma$, $\Sigma_s(\sigma)$.  Although
there is no physical argument for meromorphicity by plotting the flow
in this case we can get a picture of what it looks like, the inclusion
of $\overline\sigma$ dependence can  only result in a smooth deformation
of the meromorphic flow which leaves the fixed points invariant.
Meromorphic functions satisfying (\ref{SigmaFlow}) are called modular
forms in the mathematical literature \cite{Koblitz} and their properties
are well known.
For $\Gamma_0(2)$ it
was shown in \cite{BDc} that the minimalist assumption leads to
\begin{equation}\label{split-flow}
\Sigma_s(\sigma)=-\frac{\mu}{\mu'}=
\frac{\lambda(1-\lambda)}{\lambda'(2-\lambda)}=
\frac{1}{i\pi(2\vartheta_3^4 - \vartheta_2^4)},\end{equation}
where $\mu'=\frac{d\mu}{d\sigma}$ (the second and third forms of this
equation rely on equation (\ref{theta-identity}) and the fact that
\begin{equation} \label{lambdaprime}\frac{\lambda(1-\lambda)}{\lambda'}= \frac {1} {i\pi\vartheta_3^4},\end{equation}
a result that can be derived using the techniques in \cite{BDc,BDspin}).  The integral curves of the
flow (\ref{split-flow}) are plotted in figure 3.

It should be borne in mind
that (\ref{split-flow}) does not give a quantitative description
of the temperature flow, because the function $s(T)$ is undetermined.
By changing $s(T)$, $\Sigma_s$ can be multiplied by a function of $T$,
but as long as $s(T)$ is strictly monotonic and real this will not
change the fixed points nor will it change the fact that curves on which
$\mu$ is real are integral curves of the flow --- it will merely change
the {\it rate} at which the flow lines in figure 3 are traversed
as the temperature is changed, and this rate is not evident in the
figure.  Also physical quantum Hall samples cannot be expected to
give meromorphic flow, in a real sample $\Sigma(\sigma,\overline\sigma)$
will depend on $\overline\sigma$ and $\sigma$ independently, but
similar arguments apply: $\overline\sigma$ dependence can only distort the
flow smoothly from the meromorphic flow shown  leaving the fixed points
invariant and, again assuming particle-hole symmetry, the vertical lines
above the integers and the semi-circles, together with their images
under $\Gamma_0(2)$, will not be affected (equation (\ref{PHinterchange})
does not require meromorphicity).  Figure 3 should be compared
to the experimental data in \cite{Murzin2,Taiwan1,Taiwan2} for
spin split samples.

For $\Gamma^0(2)$ similar arguments applied to $\rho$ lead to
\begin{equation} \label{degenerate-flow}
\Sigma_s(\sigma)= \frac{\rho}{\rho'}=\frac{\lambda(1-\lambda)}{\lambda'(1+\lambda)}
=\frac{1}{i\pi (\vartheta_3^4 + \vartheta_2^4)}\end{equation}
and this flow is plotted in figure 11.
This should be compared to the experimental flow for the
spin degenerate sample in \cite{Murzin1} -- the agreement is remarkable.

For samples intermediate between degenerate and well split spins
$\Gamma(2)$ symmetry is not as powerful as there are no fixed points
with $\sigma_{xx}>0$.  Nevertheless we would expect there to be fixed
points of the flow, the fixed points of $\Gamma_0(2)$ for spin split
samples and of $\Gamma^0(2)$ for spin degenerate samples can hardly just
disappear when the Zeeman splitting is smoothly varied.
The minimalist assumption cannot be used for $\Gamma(2)$.
We can however assume that
the fixed points of $\Gamma_0(2)$ and $\Gamma^0(2)$
persist when the Zeeman splitting is varied, but
their position is no longer dictated by modular symmetry.
We seek a smooth deformation from figure 3 to figure 11,
equation (\ref{split-flow}) to equation (\ref{degenerate-flow}),
as the Zeeman splitting is increased, a deformation which is compatible with
particle-hole symmetry,
$\overline{\Sigma_\lambda(\lambda)}=\Sigma_\lambda(\overline\lambda)$.
In order to avoid creating new spurious fixed points we keep the
order of the polynomials in $\lambda$ fixed in the numerators and denominators
of (\ref{split-flow}) and (\ref{degenerate-flow}), and this
dictates that the interpolating  flow must be of the form
\begin{equation}\label{general-flowA}
\Sigma_s(\sigma)=
\frac{\lambda(A+B\lambda)}{\lambda'(C+D\lambda)}
,\end{equation}
where $A$, $B$, $C$ and $D$ are constants.
Particle-hole symmetry requires $\overline {\Sigma_s(\sigma)}
= \Sigma_s(\overline{\sigma})$, which
dictates that $A$, $B$, $C$ and $D$ be real.
Equation (\ref{lambdaprime}) shows that a factor $\frac{(A+B\lambda)}{\vartheta^4_3(1-\lambda)}$
will appear in $\Sigma_s(\sigma)$, unless $A=-B$, and this would
introduce a new zero when $\lambda=-\frac A B$ that is not there in
either (\ref{split-flow}) or (\ref{degenerate-flow}).
To avoid this we set $A=-B$ and the only possible deformation
that is compatible with our assumptions is, up to an overall constant factor,
\begin{equation}\label{general-flow}
\Sigma_s(\sigma)=
z\frac{\lambda(1-\lambda)}{\lambda'(\lambda+z)}=\frac{1}{i\pi}
\frac {z} {(\vartheta_2^4 + z \vartheta_3^4)},\end{equation}
with $z$ independent of $\sigma$ and real.  The free parameter $z$
varies from $z=- 2 $ for
$\Gamma_0(2)$ to $z=1$ for $\Gamma^0(2)$ (an overall factor of 2 multiplying
(\ref{split-flow}) does not change figure 3, indeed
(\ref{split-flow}) is only derived in \cite{BDc}
up to an overall positive constant).

Figure 2 shows how the fixed point $\sigma_*=\frac{1+i}{2}$ of $\Gamma_0(2)$
moves as $z$ is varied: particle-hole symmetry constrains it
to keep to the real curve of $\lambda$
shown in figure 1.   The point associated with $z=1$ can be gained from
from $z=-2$ either by going anti-clockwise or clockwise.
Suppose first that $z$ increases monotonically between $-2$ and $1$ as
the Zeeman splitting is varied smoothly from well-split spins, $\Gamma_0(2)$,
to degenerate spins, $\Gamma^0(2)$.
Figure 2 shows that, as $z$ is increased from $-2$ to $-1$, the
fixed point at $\sigma_*=\frac {1+i} {2}$ of the $\Gamma_0(2)$ flow (figure 3)
follows the semi-circular arc of radius $\frac 1 2$, moving down to the left
until it hits the origin in the $\sigma$-plane, $\sigma_*=0$ when $z=-1$.
It then continues up the imaginary axis, through $\sigma_*=i$
for $z=-\frac 1 2$,  to $\sigma_*=i\infty$ for $z=0$ (near $z=0$ equation (\ref{general-flow}) can be replaced with
\begin{equation}
\Sigma_s(\sigma)=\frac{1}{i\pi}
\frac {\tilde z} {(\tilde z \vartheta_2^4 + \vartheta_3^4)},\end{equation}
where $\tilde z = 1/z$, multiplying $\Sigma_s$ by a constant does not change the
topology of the flow).
The fixed point subsequently
moves down from $\sigma_*=1+i\infty$ to $\sigma_*=1+i$ (a $\Gamma^0(2)$ fixed point,
figure 11) as $z$ increases from
 $0$ to $1$.  While experimental data to date do show a fixed point
on the semi-circle spanning $1$ to $0$ that is to the left of $\frac{1+i}{2}$, \cite{Taiwan2,Taiwan3,Hilkeetal}, it does not seem likely that this sequence of flows can be the correct
one.  A fixed point at $\sigma_*=i$ when $z=-\frac 1 2 $ has never been seen in any Hall sample.
Indeed such fixed points were  identified in
\cite{Gammatheta} as being associated with bosonic pseudo-particles excitations
(as in the superconductor-insulator transition of \cite{MFisher} for example),
rather the fermionic pseudo-particle excitations of the quantum Hall effect.
Since no quantum Hall
sample to date has ever exhibited a critical point at $\sigma=i$
we exclude this possibility.

An alternative possibility is that the flow morphs from figure 3 to figure 11
by decreasing $z$ from $-2 $ going through $-\infty$ to
$+\infty$ to continue down to $+1$.
The nine plots in figures 3 to 11 show the series of flows for
\[z=-2,-10,-100,\pm\infty,100,20,10,2 \ \hbox{and}\ 1.\]
The fixed point at $\frac{3+i}{2}$ for samples with well-split
spins moves left and down as the Zeeman splitting is decreased, along
the semi-circle of radius $\frac 1 2$ centred
$\sigma=\frac 3 2$,
until it hits the real axis at $\sigma=1$ (for $|z|=\infty$),
where it merges with the incoming fixed point coming from
$\frac{1+i}{2}$ on the left.
It then moves vertically upwards to the point $1+i$ when $z=1$,
which is the fixed point for degenerate samples with symmetry $\Gamma^0(2)$.
Every flow in the sequence has $\Gamma(2)$ symmetry, which is enhanced
to $\Gamma_0(2)$ or $\Gamma^0(2)$ at the extreme values $z=-2$ and
$z=1$ respectively.

In real samples particle-hole symmetry is hardly likely to be an exact symmetry
of the system, there will be deviations from this picture.
But any deviations will
be small if particle-hole interchange is a reasonably good symmetry:
for example if $\frac{m_p - m_h}{m_p+m_h}$ is small, where $m_p$ is the
particle mass and $m_h$ the hole mass.
In particular the collision of the critical points at $\sigma=1$ ($|z|=\infty$)
seems likely to be an artifact of the mathematical idealisation of
exact particle-hole symmetry, as there is no obvious physical mechanism
governing the merging of two critical points as the Zeeman energy is reduced.
The most plausible scenario here is that the merge is postponed in a real
sample until the Zeeman splitting is reduced to very small values and
the proposed trajectory of a real sample, in which particle-hole interchange
is a good but not exact symmetry, is shown in figure 12.
This is
a small perturbation of the mathematically idealised flows shown in figures 3
to 11 which is still compatible with the proposed symmetries.

\section{Conclusions}

The topology of the temperature flow of conductivities
in quantum Hall samples is tightly constrained by the law
of corresponding states, expressed in terms of modular transformations
on the complex conductivity (\ref{Complexsigma}).
For the extreme cases of well-split spins and degenerate spins
the critical points in the complex plane are determined by the symmetry.
For intermediate values of the Zeeman splitting modular symmetry does
not determine the position of the critical points but one
can assume that they move around the complex plane in a continuous
manner as the Zeeman splitting is varied.

In the case of samples exhibiting particle-hole symmetry
modular symmetry is particularly powerful, leading to the
statement that the curves in figure 1, and their images under $\Gamma(2)$
modular transformations, will be trajectories of the conductivity
flow as the temperature is varied keeping the magnetic field fixed.
This statement should be true
for  any Zeeman splitting.
Zeeman energies which are large enough to give well-split spins
result in critical points at $\sigma_{xy}+i\sigma_{xx}=n+\frac{1 +i}{2}$
between integer Hall plateaux
$\sigma_{xy}=n$ and $n+1$ (figure 3).
Zeeman energies which are so small that the spins are degenerate
give critical points at $\sigma_{xy}+i\sigma_{xx}=2n+1+i$
between integer Hall plateaux $\sigma_{xy}=2n$ and $2n+2$ (figure 11).

The form of the flow as the
Zeeman splitting
is varied from the spin-split to the spin degenerate case is shown
in figures 3 to 11.
Figure 12 shows
proposed trajectories of two critical points as the Zeeman
energy is reduced in a real sample exhibiting symmetry under
particle-interchange which is good but not exact.

The analysis here has assumed that the parameter $z$ varies monotonically
as the Zeeman energy is varied, implying that the critical point at $\sigma=\frac{1+i}{2}$
in the transition between $\sigma=1$ and $\sigma=0$
in spin-split samples moves to the right, down towards to $\sigma=1$, as the Zeeman
splitting is reduced.  There is as yet no experimental evidence for such
behaviour:
indeed in \cite{Taiwan2,Taiwan3,Hilkeetal} a critical point is found on the semi-circle spanning
0 to 1 in the complex conductivity plane which is to the {\it left} of $\sigma=\frac{1+i}{2}$.
This could be a consequence of a constant re-scaling of $\sigma_{xx}$, \cite{BDspin},
or, perhaps more likely, it may indicate that $z$ is not monotonic.
It could be that the critical point at $\sigma=\frac{1+i}{2}$ ($z=-2$ in
equation (\ref{general-flow}))
in spin-split samples first starts to move down and to the left ($z>-2$)
and then reverses to retrace its steps back to $z=-2$ before starting to
travel down to the right towards $\sigma=1$ as $z$ decreases below $-2$.
A possible trajectory is shown in figure 13.
While this would be compatible with the experimental data to date
any physical explanation of such a trajectory, which
is certainly allowed by the law of corresponding states combined with
particle-hole symmetry, would go beyond the general predictions
following from these assumptions and would probably require a more specific
microscopic model.
More experimental data
would be welcome in order to determine the true behaviour of the critical
points.

B.P.D. acknowledges the warm hospitality of Pei-Ming Ho and Chi-Te
Liang during his visit funded by the National Science Council, Taiwan,
R.O.C. (NSC 098-2912-I-002-097). It is a pleasure to thank Pei-Ming Ho
and Chi-Te Liang for discussions.

\vfill\eject

redpajama_set_name: RedPajamaArXiv