text:
Global carbon trade collapses into hot air
Photo: redd-monitor.org
If there is one thing melting away faster than the Arctic ice cap it's the credibility of the global carbon trading system set up under the Kyoto treaty to address climate change. A United Nations sponsored panel said in a September 10 report that "global carbon markets ... are collapsing with potentially devastating consequences".
Panel member Joan MacNaughton told the Guardian: "The carbon market is profoundly weak." The Economist said it is "a shambles".
The report said the price of carbon offset credits issued under the UN's Clean Development Mechanism (CDM) had fallen 70% in the past year. The UN-backed carbon offsets now sell for less than $3: down from about $20 in 2008.
The CDM market allows firms in developed countries to buy offset credits from carbon-reduction projects in the global South. The UN oversees the process, giving out CDM credits for each tonne of carbon emissions it says has been saved. Businesses in rich countries that take part in the scheme can buy the CDM offsets to meet their emission cut targets – in effect, swapping emission cuts in the South for cash from the North.
But the market is falling over because there are far more CDM offsets for sale than there are buyers. European firms have been the biggest buyers of CDM offsets, but Europe's emissions trading scheme has hit a wall due its huge over-allocation of free permits to industry. That, combined with Europe's economic woes, means few of Europe's big polluters need to buy more offset credits to meet their emission cuts targets.
The rock bottom international offset price has implications for Australia's carbon price scheme, which recently joined with the European system. Australia's scheme gives the country's biggest polluters billions in compensation payouts. The government pays compensation at $23 a tonne of carbon, but these firms can meet 12.5% of their required cuts with international offsets that cost just $3 a tonne. Don't expect them to give the extra cash back to taxpayers.
Oversupply is also dogging another UN-backed market-based climate scheme. The Kyoto protocol allows countries that emit less than their target to sell their surplus emissions on the carbon market. But these surplus credits – which the UN calls Assigned Amount Units (AAU) – are in such oversupply they now sell for less than €1 a tonne.
A recent report by Point Carbon said a worldwide AAU surplus of 13.1 billion tonnes was generated from 2008 to 2012. The surplus could rise above 17 billion tonnes by 2020. CDM-Watch's Anja Kollmuss told the Inter Press Service: "The supply is three magnitudes bigger than the demand."
Most of these surplus credits are bogus – often known as "hot air" credits. Russia, Ukraine and Poland hold the most hot air – the de-industrialisation of these countries after the collapse of communism means their emissions are lower than in 1990, which the Kyoto protocol treats as its baseline year. The global economic crisis, which led to a fall in Europe's emissions, has added even more hot air to the scheme.
The oversupply crisis means Kyoto's carbon trading scheme now ensures it is cheap for companies to increase their pollution – hardly an incentive to switch to zero emissions technologies.
The UN's carbon markets might be a "shambles", but the UN's top climate change official Christina Figueres seems determined to pretend all is well. She issued a press release on September 7 to welcome the 1 billionth carbon offset credit issued under the CDM since 2005. "This exciting milestone is testament to the expanding use of the CDM," she said.
Of course, the "expanding use" of CDM offsets has led to its oversupply crisis. But the size of the CDM market also says nothing about its integrity. The truth is that most of the billion CDM offsets approved by the UN are fakes and do not represent emissions cuts.
The August 9 New York Times said about 46% of CDM credits have come from just 19 companies that have destroyed an industrial gas called HFC 23. The gas, which is a waste byproduct created during the manufacture of coolant gases for refrigerators, has a global warming potential 11,000 times that of carbon dioxide.
Realising they stood to make millions, these companies have sharply increased production of coolant gas (itself a powerful greenhouse gas) simply so they can be paid to destroy the byproduct HFC 23.
This CDM loophole has not just given an incentive to these companies to increase pollution, but it has done so at a remarkably high price. In 2008, Stanford University's Ken Michael Wara and David Victor estimated that by this year destroying the gas would cost about $100 million, but would create $4.7 billion in CDM credits.
The European Union says it will ban industrial gas offsets from its scheme from next year and Australia's carbon price scheme does not allow them. But the UN is likely to keep accrediting the bogus offsets. Chinese manufacturers have threatened to release the gas into the atmosphere if they are no longer paid to destroy it.
But the CDM's integrity problem extends beyond the waste gas issue. A September 10 investigation by India's Daily News & Analysis said: "Several projects in the country that are earning huge sums through the sale of carbon credits are, in fact, ineligible for the CDM scheme. Promoters of these projects have manipulated and backdated documents to meet requirements as prescribed by the CDM executive board."
It said the Indian body responsible for approving CDM proposals and submitting them for international accreditation routinely "approves CDM proposals without conducting field inspections to verify whether the project fulfils the eligibility criteria".
In theory, to be eligible for CDM accreditation a project must a) prove it cuts emissions and b) prove that the emissions cuts are additional to what would have happened anyway.
A July 2008 cable from the US consulate in Mumbai released by WikiLeaks said "executives of major Indian companies ... conceded that no Indian project could meet the 'additionality in investment criteria' to be eligible for [CDM] carbon credits". But the UN's CDM Executive Board had still approved 346 projects in 2008. Today, there are 827 CDM-approved projects in India.
The cable said the then-chairperson of India's CDM authority, R K Sethi, had "publically admitted that the National CDM Authority takes the 'project developer at his word' for clearing the additionality barriers".
Soumitra Ghosh, from India's North Eastern Society for Preservation of Nature and Wildlife, told Daily News & Analysis: "Of the 60 CDM projects that I have evaluated, there appeared not to be one that actually reduced emissions."
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The critically acclaimed and family loved Lego Movie has grossed over $425 million at the box-office and reintroduced the timeless toys to a lot people. If you have want to show off your love for Legos and try something new and fun for Easter then try this DIY Project – Lego Easter Eggs by ItsAlwaysAutumn.com.
Print the sheet on sticker photo paper or preferably Tattoo Paper .
Trim around your design, leaving a small border around it as shown.
Peel off the clear plastic portion. As you peel, you should see adhesive on your printed design.
Place the design where you want it on your egg.
Cover the tattoo with a damp cloth, gently pressing down to form the tattoo paper to the curve of the egg. Wait 10-15 seconds.
Carefully remove the paper, leaving your design on the egg.


A smiling chef putting the finishing touches on a plated seafood meal. Shallow dof, focus on the food and the garnishing fingers.
Whether you are seeking a private chef for your Cabo villa or you require staff for a corporate event, having the best efficient concierge services insures you get the most from your holiday and special event.
Finding a private chef can be a great way to have authentic Mexican cuisine served in the comforts and privacy of your vacation home.
Specialty diets, preferred meal choices, and other personal options are easily fulfilled with a private chef to cater your needs.
With so many qualified chefs in Cabo, you will be able to find the perfect one for your event. Let the local professional experts work for you!
Brown's is a private catering company that caters towards people staying in villas.
Chef in Cabo is a private catering company that caters towards people staying in villas.
Denni´s Catering has had experience catering in Los Cabos for over 10 years. Chef Denis Gabriel specializes in great food and quality staff services, butlers, cleaning crews, and even entertainment options. All of the assorted menu directorys are of international flair. Whether you have 2,000 or 2, Denni's catering can help you craft the most amazing meal.
El Matador Catering was founded in Los Cabos by Pablo Velez, a bull fighter with a passion for cooking. Whether your private special event is for work or pleasure, El Matador Catering can make sure your food is delicious. With extensive catering experience in birthdays, business conventions, weddings and other events, El Matador can craft an amazing experience.
Many times guests staying in private villas in Los Cabos hire private chefs to cook breakfast, lunch or elaborate dinners. This way you get can eat in the comfort of your accommodations without having to worry about preparing food. You may hire chefs for just a single dinner or for your entire stay in Cabo.
Grocery stocking services are also available through caterers or concierge companies if you are staying in a vacation rental in Los Cabos. Most companies will have a list for you to choose what you want ahead of time. When you arrive in Cabo, your fridge and pantry will be filled with all of your favorite food and drinks!
Whether you are planning an elaborate destination wedding or a simple business lunch meeting, there are a variety of catering options in Los Cabos. Most catering companies are also able to provide event staffing and bartenders.
If you need to setup corporate events at a preferred hotel or inn, this service will guarantee perfection. All you do is make a phone call, and all the preparations are taken care of for you. When you arrive, a personal chauffeur will pick you up and take you directly to your hotel. Of course, after the meeting, allow the chauffeur to give you a personal tour of the local area.
Catering private parties is easier than ever. In fact, you will host a dynamic party that will be in the memories of everyone for a lifetime. When you dream of the ultimate wedding reception, a concierge will provide the best in conveniences. Be a guest at your own party where a private chef and wait staff has been hired to deliver your specific instructions.


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IECA Staff
Heather Buerger
Executive Vice President, North America
Heather Buerger currently serves as Executive Vice President of North America. Heather is a former energy executive having spent nine years as a Contracts and Credit Analyst at Superior Natural Gas Corporation where she supported the treasury and accounting departments before transitioning to a role in credit and contacts. Prior to her work in the energy sector, Heather spent nine years working in the travel industry as a corporate trainer and quality assurance auditor for the Hertz Corporation.
Heather attended Eastern New Mexico University where she received a B.S. in Mass Communication with an emphasis in broadcast journalism and production. Heather has been an active member of the IECA since 2012 and served as the co-chair of the Marketing and Communications Committee where she was a consistent contributor to the Journal and Insights publications and was the recipient of the 2014 IECA Outstanding Member Award.
When not at work, Heather is actively involved with a number of community events and charities including her local alumni chapter of Chi Omega, a national sorority, and the Waller County Demolition Derby which raises money for Laurel's Army Foundation, an organization that helps support childhood cancer research and wellness programs for children with cancer. Heather resides in Houston with her husband Nicholas and daughter Lauren and one of their favorite family activities is practicing Taekwondo.
Heather wears many hats within the IECA and can field questions pertaining to sponsorship, marketing opportunities, education, networking opportunities and general IECA questions.
1120 Route 73, Suite 200
Email: hbuerger@ieca.net
JOHN TREASURE
Executive Vice President, Europe
John Treasure formally commenced as Executive Vice President for Europe in April 2021. John was Managing Director, International Trade and Transaction Banking at Credit Agricole Corporate and Investment Bank London where he was employed for 20 years.
He has been an IECA member since 1994 and served on the Board of Directors from (2014 - 2017). John was also Deputy Chair of The European Committee for many years until 2017. In 2014 he established the IECA Education fund (together with Maria Taylor) which provides (independent means-tested) financial support for deserving students to attend St. Catherine's College Oxford.
He was recently appointed a 'Freeman of the City of London' in recognition of his services to International Trade following a 40-year career, but is now prohibited from driving sheep over London Bridge! He is a lover of all sport (as are his son Stuart, and daughter Megan) and still plays squash (after 50 years on court), but would describe it as ' a bit like watching an old black and white film, only in slow motion. He lives by the old City code of 'My word is my Bond ' and the even older 'Carpe Diem....'
Email: evpeurope@ieca.net
Tatiana Veres
ASSOCIATE MEETING MANAGER
Tatiana Veres is the Associate Meeting Manager for the International Energy Credit Association and joined Association Headquarters, Inc. in April 2017, starting out in membership and meetings roles. As such, she has a full range of knowledge for all kinds of IECA questions. As the current main meeting manager for IECA, Tatiana oversees the high-level meetings operations for the Spring and Annual Conferences. She holds a B.A. in French Language and Literature from Dickinson College and currently resides in Mullica Hill, NJ.
Mt. Laurel, NJ 08054
Ph: 856-793-0910 Fax: 856-439-0525
Email: tveres@ieca.net
DAVID GRIMM
MEMBERSHIP & MEETING COORDINATOR
David Grimm is the Meeting Coordinator for the International Energy Credit Association. David provides internal meetings and membership support to the team. He is the front line of membership and meetings, so you can go to him with any of your IECA questions, from membership inquiries to registration questions. David graduated with a B.A. in English/Digital Studies from Rutgers University and currently resides in the Philadelphia area.
Email: dgrimm@ieca.net
I enjoy the opportunity to collaborate with like-minded people of the industry. Together we work to give back to the industry through educational advancement opportunities and networking. There are great opportunities to learn from some of the most distinguished members of the Energy Credit industry.
Michael Barberio
Exelon Corp

redpajama_set_name: RedPajamaCommonCrawl


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Singapore's Wildlife Parks are Teeming with Adorable Newborn Animals
There's hope for the world's endangered species yet.
By Angelica Gutierrez | Jan 29, 2018
IMAGE Wildlife Reserves Singapore
There's absolutely no shame in partaking of a little eye-bleach from time to time, and here we hit the motherlode. Just in time to cure the Monday blues, Singapore Wildlife Reserves released photos of their newborns from 2017.
Last year Jurong Bird Park, Night Safari, River Safari, and Singapore Zoo reported 540 animal births, and over a fourth of them are endangered species. These animals aren't just cute—they're symbols of hope.
Khansa the Bornean Orangutan Baby
Khansa was born in Singapore Zoo last April. Her birth is good news for Bornean orangutans, since their IUCN conservation status was changed from endangered to critically endangered in 2016. Fortunately, Singapore Zoo has a very good track record when it comes to breeding orangutans—Khansa is the 46th orangutan to be born in the park.
Maru the King Penguin Chick
Last October, Jurong Bird Park welcomed the birth of Maru, whose name means "round" in Japanese (Sanrio character Badtz-Maru comes to mind). He was raised by a dedicated team at Jurong's Breeding and Research Center before being introduced to his family at Christmas. Given that he's the park's first successfully bred king penguin chick in nearly 10 years, it's not surprising that they doted on him with some Christmas presents too.
Squish the Greater Flamingo
You might have seen viral videos of Squish waddling around Jurong Bird Park in blue booties. They were used to protect his foot pads while he learned to walk—as any kid would know, walking barefoot on concrete on a hot day is no joke.
Jaguar Cub
This little furball was born to first-time mom Aswa at the River Safari on November 16. He's the first jaguar to be born in the park since 2009, and will stay in the nursery for about two more months before he's named and revealed to the public.
Abina the Pygmy Hippo
This baby hippo was also born in November at Singapore Zoo. Her name, Abina, means "born on Tuesday," and she is the 24th pygmy hippo to be successfully bred at the zoo.
Asian Small-Clawed Otters
These four adorable otters are among 14 that were bred at Singapore Zoo and the Night Safari.
Oban the White Rhino Calf
Oban is the first male white rhino to born at Singapore Zoo in five years. He's also his mother Donsa's 11th baby, and his name means "king" in Yoruba.
light reads
Angelica Gutierrez
Angelica is currently Editorial Assistant for Esquiremag.ph.
View Other Articles From Angelica
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Cape Kidnappers is a headland at the southeastern extremity of Hawke's Bay on the east coast of New Zealand's North Island and sits at the end of an 8 kilometres peninsula which protrudes into the Pacific Ocean. Set atop 6,000 rolling acres of stunning pasture-land in Hawke's Bay, New Zealand, The Farm at Cape Kidnappers is the perfect setting for romantic getaways whether it is celebrating your wedding, vow renewals, engagements or anniversaries since everything can be tailored for the special day to be unique and magical.
Continue reading Contemplate Visiting The Farm at Cape Kidnappers in New Zealand at Covet Edition.


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You are here: HomenextPlacesnextANCIENT WICKLOWnextCounty Wicklow in early Christian Times
County Wicklow in early Christian Times
By Michel Wolgemut, Wilhelm Pleydenwurff (Text: Hartmann Schedel) - Self-scanned, Public Domain, https://commons.wikimedia.org/w/index.php?curid=4421408
Wicklow's Holy Men & Women
Our perception of the early holy men and women who were later to become the early saints and the churches they established needs to be revised for the purposes of this article.
Many of them were local and natives, others arrived from Britain and Wales. These early holy men and women craved isolation and seclusion well away from the temptations of men and women. They knew well that the devil stalked them and they found their favourite places and locations in the woods and mountains of what would later become Co. Wicklow.
They were a determined hard breed of people who lived off the land, fishing and hunting with their dogs. They wore animal skins and used their isolation for fasting and prayer. They could not hide forever and word would soon spread that a holy man was in the area. People would then flock to them for blessings and to become followers. At first they lived in caves and constructed hides and for St. Kevin, the hollow of a tree suited his needs for a time. They were more Grizzly Adams than Padre Pio types in their quest for survival and the hermit lifestyle. Later they built cells or churches when their isolation ended. Many would then leave their followers to these churches and then go in search of isolation once more. Mochorog was one such man.
The story of Mochorog
Son of a British King and a Saxon Queen, he came to Ireland as a monk in search of St. Kevin. He established churches at Enniskerry and Delgany. St. Kevin to whom he delivered the last rites helped him construct a cell to the east of Glendalough. Mochorog is also associated with Kilmacurrough, (Cill-mo Cred). Cred was recorded as being the wife of Finnach, both parents of St. Baithin of Inisbaoithin. Cred was a daughter of Ronan, 11th Christian King of Leinster, who founded a church at modern day Kilmacurragh. The area being a holy location and a place of isolation is probably what drew Mochorog there.
Small early churches
The churches established by the early saints were not large buildings and usually only housed the priest, a few of the older amongst the locals and perhaps a disabled local or two. The congregation who gathered at these small churches stayed outside. One such church Teach na Cille or house of the church gives the origin to the place name of Tinakilly. The early churches were small and varied in size, such as 42ft x 20ft or 50ft x 40ft.
Glendalough and other churches
Mochorog's brother, Canoc, was also one of the early saints. Mochorog is credited with St. Berach and St. Kevin with founding the monastic city of Glendalough and turning it into a centre of learning. Other early churches include Kilmartin, the saint was said to be a holy man from Wales. St. Credan was said to have established a church at the location named after him, Macreddan. He was a disciple of St. Petroc and was known to St. Degan who had established his church at Ennorilly. St. Cathach's Church was founded at Kilcarra. Tegan, later St. Tegan, was instructed by St. Patrick to found a church at the location now named after him, Kiltegan. St. Ernene founded his churches at Portmarnock, ( the 'marnock' segment in the place name Portmarnock is called after him – 'mo Ernoc,' one of the many variations of his name.) He is also associated with Kiladreenan in Newtown.
He spent time in isolation at Newrath and Glenealy. An ancient church also stood at Kilfey at the entrance to the Devil's Glen. St. Mhantain established his church at Wicklow Town, which was second in importance to that of St. Molibbo or Molibba's church which is situated in the middle of the graveyard on Church Hill in Wicklow Town. Kilnamanagh in Glenealy was also an important site, where monks were said to have erected a wooden church in the woods. Kilnamanagh in Glenealy was also claimed as the site where St. Kevin was educated rather than Kilnamanagh in Tallaght. Cellbritton in the Glenealy was said to be the church of a Briton or Welsh monk. He is claimed to be Petroc, the famous Abbot of Padstow in Cornwall. He may even be the founder of Kilnamanagh in Glenealy. The area is also associated with St. Enan also known as St. Menoc or Momenoc and St. Molibbo.
From Wales to Wicklow
Many Welsh monks made the straight crossing from Wales to the Wicklow coastline. St. Molig built a church at Kilmanin, close to Powerscourt. St. Finnian founded a religious settlement at Aghowle. At Ballykine is an ancient church and cemetery possibly founded by St. Caemhan, brother to St. Kevin.
Recorded in folklore
Much of what we know of these early saints and the locations of their churches were recorded hundreds of years after their deaths. Much of the folklore surrounding these early men and women dates from after their deaths or were recorded by pilgrims of the time and carried across Europe. These are but a few of the large number of the early churches established in the county with female church sites, with dedications to local saints, to the east of Bray and the Penal Church at Wicklow town.
By Stan J. O'Reilly
By Stan J. O'Reilly
communitysites-custom-layout (266)
ANCIENT WICKLOW
A Route Through the Past
Aghowle Church
Burial Traditions
Ceremonial Monuments
Defended Settlements
Early Christian Wicklow - Sites to Visit
Evidence for the Earliest Christians
Fortifying Their Land
Fulachta Fiadh
Glendalough Monastic Site: Guide to sources
Medieval Wicklow - Sites to Visit
NEW! Guides to sources for the archaeology of County Wicklow.
Norman Churches
The Beginnings of Farming - The Neolithic
The Beginnings of Metal-Working - The Bronze Age
The Dawn of History
Wicklow - The Heart of The Ancient East
Wicklow In Prehistory
Wicklow In The Grip Of An Ice Age
Wicklow in the Later Middle Ages


"A smoothie a day keeps tube delays at bay"… is perhaps wishful thinking. However, there is something to be said about a fantastically refreshing homemade smoothie. This summer, plant-based brand Alpro is helping us create delicious smoothies of our own using the tasty smoothie inspiration tool. So not only do you save money on the *ahem* pricey *ahem* smoothies from various hipster cafes we aren't going to mention (#LondonProblems), but you also get to switch up your flavours every day. Below we've picked out a different fruit and veg-packed smoothie recipe to solve more of your #LondonProblems.
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London problem 3: Let's face it, London's weather this summer has so far been one huge disappointment. You desperately need a bit of sunshine in your life and, no matter how much you check the forecast, your dreams of lounging around on London Fields this weekend probably won't happen.
Smoothie Solution: The Sun Seeker. Capture the sunshine in a glass with a bright, cheerful cup of yellow goodness. Made using Alpro Coconut Original, passion fruit, yellow pepper and sunflower seeds, you can have a taste of the tropics without splurging on an expensive flight overseas – turns out smoothie ingredients prove far more affordable. And if the sun does make an appearance, then what better than a refreshing zesty blend to enjoy it with? Check out the full recipe here.
Smoothie Solution: Green With Envy. What you need alongside your swanky new workout kit is the perfect gym-friendly accessory: a bright green, veg-packed smoothie. Whizz up kiwi, cucumber, spinach, basil and Alpro's Simply Plain plant-based alternative to yogurt to take with you to the gym. Just walk around the mat area a little, flaunting your new outfit and your tasty blend of greens and looking like a complete god/goddess of health. Go on, you fitness guru – try this recipe here.
Got any other #LondonProblems? Need a smoothie to cure your rent-day woes? For a more personalised smoothie solution, check out Alpro's Smoothie inspiration tool. You'll be mixing up your own recipes in no time. Send us pictures of your favourite problem-solving #AlproSmoothie by tagging us (@secret.london) and Alpro (@Alpro) on Instagram!


I wanted to introduce you to easybib.com just in case this is a resource that might enhance your studies in this online course.
EasyBib offers citation support for your bibliography your works cited pages as you can see on easybib.com there are features for creating citations in MLA APA.
for our course we will be using Easybib APA. MLA is free Chicago is free the others are free Easybib APA does come with a charge now I don't know if you want to use Easybib APA or not but you might benefit from the two-day trial to see if it's something that you might like to add to your toolbar but I just want to show it to you just in case it's something that might help you.
so I have an account with the Easybib APA feature so that I can go through and create projects in APA.
process by researching the sources that are needed for whatever it is that I'm writing.
as you can see when I find and locate that information whether it's online or whether it's from a resource in the library or from a database I would take that entry and place it in this white box and copy and paste it and then I would click this green button cite this and the APA citation will appear so let's go ahead and just practice starting a new project.
I need to give the project a name I can fill in the subject if I choose and I can go ahead and put my thesis statement in this box so that I'm reminded as I go through the entire process what my project is about.
as you can see we're going to do an APA bibliography.
everything I research in site will be added to this project and that way all of the information will be easy to access when I log into EasyBib.
I simply go to this project that I'm working on and continue adding resources to field and creating the citation I'm going to practice one with you.
I just typed into Google American Psychiatric Nurses Association and here is my research fine.
let's go ahead and just click on this first one just as practice and just to get us to a citation for practice I'm goanna click on free continuing.
this takes me to some articles that are written in their free continuing education program I'm just going to click on one of them.
so I've just clicked on the first one here's the information I copied the URL I pasted the URL into my EasyBib project and I have APH checked.
EasyBib has searched through the database and found the information and I'm going to click continue.
now it's goanna bring up fields for me to check in case I need to add information these aren't error messages they are boxes for me to fill out if I have that information if I don't have that information then I move on but for APA I should locate that information. I located her name the author's name the electronically published date I filled in all the fields that I can I'm ready to create the citation.
there it is my pretty citation all in Easybib APA format and as you can see that has been added to my bibliography after I finish doing all my research I will click the ones that I want to print and click more and create the citation page.


This mini tee shirt dress is made from soft, breathable cotton, and is just waiting to become your favorite casual dress. It is presented in your choice of black or blue and features a large, contemporary Day of the Dead logo over the front of the garment. It is mini length, with the hemline ending at the upper portion of the thigh. This dress features a simple crew neckline and has long sleeves. It is washable in warm water and may be tumbled dry on a low temperature setting. It is available for purchase in sizes Small, Medium and Large.


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Q: Are $M,N \subset G$ necessarily subgroups for their elements to commute, given that $x^{-1} M x = M$ and $x^{-1} N x = N$? I am trying to solve the following problem.

Let $G$ be a group. If $M, N \subset G$ are such that $x^{-1} M x = M$
  and $x^{-1} N x = N$ for all $x \in G$ and $M \cap N = \{1\}$, prove
  that $m n = n m$ for all $m \in M, n \in N$.

I have already proven it for the specific case in which $M, N$ are subgroups of $G$ (see proof below). However, I can't prove it without this hypothesis. I think $M, N$ must be subgroups for the result to hold, but I can't find a counterexample to show this as well.
Any help with a proof or counterexample is much appreciated.

Current proof outline:

$$ (m \cdot n)\cdot (n  \cdot m)^{-1} = (m  \cdot n)\cdot (m^{-1}
 \cdot  n^{-1})  =: k$$
  $ k = (m \cdot  n \cdot  m^{-1})  \cdot n^{-1}
 \in N$, because $(m \cdot  n \cdot  m^{-1}) \in N $ and $n^{-1} \in
 N$ (and $N$ is subgroup).
$ k = m \cdot  (n \cdot  m^{-1}  \cdot n^{-1}) \in M$, because $m \in
 M $ and $(n \cdot  m^{-1}  \cdot n^{-1}) \in M$ (and $M$ is subgroup).
So $k \in M \cap N = \{1\}$, which implies $k = 1$ and
  $$ (m \cdot n)\cdot (n  \cdot m)^{-1} = 1 $$
  $$ (m \cdot n)\cdot (n  \cdot m)^{-1}\cdot (n  \cdot m) = 1 \cdot (n  \cdot m) $$
  $$ m \cdot n = n  \cdot m $$

A: The assertion doesn't hold when $M$ or $N$ is not a subgroup of $G$.
Here is a counterexample.
Let $M=\{e,(12),(13),(23)\}$ and $N=\{e, (123),(132)\}$ in $G=S_3$.
Then $x^{-1}Mx=M$ and $x^{-1}Nx=N$ for all $x\in G$
and $M\cap N=\{e\}$.
$M$ is not a subgroup of $G$ and $(123)(12)\neq (12)(123)$.

A: Let $G$ be the dihedral group of order $8$, with $b$ an element of order $2$ and $a$ an element of order $4$. For $x,y\in G$ let $x^y=y^{-1}xy$. By direct calculation we have:
$$\begin{array}{rcc}
x:&e&b&a&a^2&a^3&ab&a^2b&a^3b&\\ \hline
a^x:&a&a^3&a&a&a&a^3&a^3&a^3\\
b^x:&b&b&a^2b&b&a^2b&a^2b&b&a^2b
\end{array}$$
Let $M=\{1,a,a^3\}$ and $N=\{1,b,a^2b\}$; then $M$ and $N$ satisfy the stated conditions, but $ba=a^3b\ne ab$.

redpajama_set_name: RedPajamaStackExchange


On Monday I posted the final, edited version of Everyday Rails Testing with RSpec, available now on Leanpub for $12 US.
First, let me again thank everyone who purchased the early access, beta version of the book. I appreciate the support and all the feedback. As I shared on the Leanpub mailing list yesterday, releasing the book so early in its development cycle perhaps frustrated a couple of readers, but it resulted in a very different (and in my opinion, better) book than I'd originally set out to write. I'm planning to write a more complete retrospective in my personal blog later this week, for anyone interested in the lean publishing approach.
Next, let me remind you what's in the book as it now stands: 115 pages (as the PDF is formatted), 12 chapters (and a short appendix), and a code sample that iterates from chapter to chapter. The content is more or less a retelling of how I learned to test (specifically, with RSpec). It's not for people who've never worked with Rails. It's not for people who have developed good testing habits and patterns for themselves already. It is for people who are still trying to build those habits and patterns. I'm not saying my way is the way, but hopefully I've shared enough in the book to help you get comfortable with testing and improve your coverage.
If you're still on the fence, check out the preview available through Leanpub, or refer to the testing series on this blog that started the whole thing.
Once Rails 4.0 begins hitting the release candidate phase, I'm going to revisit the content of the book and the libraries shared within it, with the hope that an updated version of the book can be available within a month of Rails 4's final release. This version will be a free update to anyone who's purchased the book. I'm doing that because I know how quickly software books can become obsolete, and that Rails 4 is going to hit sooner rather than later.
I know that the book isn't available right now in countries Paypal doesn't support. I know also that there are people who just don't like Paypal and don't want to use it. This is a limitation placed by Leanpub, not me. My intention is to make the book available elsewhere as soon as possible, but I want to talk to the folks at Leanpub to determine the best way to go about it. More news as I get it.


Maat Saxophone Quartet is a 100% Portuguese ensemble, founded and based in Amsterdam, Netherlands.
Prize-winners of the most prestigious classical music competition in Portugal "Prémio Jovens Músicos", the ensemble combines four musicians from the same country and was born from the great desire and passion of making music together.
With an ongoing activity between Portugal and The Netherlands, this young quartet already performed at Willem Twee Concertzaal in – s'Hertogenbosch, Cenakel in Tilburg, "Grande Auditório" of Coimbra Music Conservatory, Casa da Música and "Grande Auditório" of Fundação Calouste Gulbenkian in Portugal.
In 2018 they are one of the finalists at the "Willem Twee Chamber Music Competition", at the Willem Twee Concertzaal in –s'Hertogenbosch, won the 1st prize in "National Chamber Music Competition" of Vila Verde, Portugal, and are finalists of the Grachtenfestival Competition.
Students from the saxophone class of Arno Bornkamp, at Conservatorium van Amsterdam, the quartet has guidance from Arno Bornkamp and Willem van Merwijk, having also worked regularly with Ties Mellema, Lars Niederstrasser, Nuno Pinto, Ricardo Silva and António José Pereira.
The quartet works regularly with new and young composers, promoting the development of new saxophone quartet repertoire, as well as the development of the new pieces for this formation by young composers, such as Nuno Lobo, Adam Lukawski and Ramin Amin Tafreshi.
In 2018 they are selected for the "RaboNEXTStage" program, in a partnership with the Rabo Bank and TivoliVredenburg Utrecht, having the possibility to perform in 2019 at TivoliVredenburg and work with the renowed string quartets Dudok Kwartet and Quatuor Danel.
Maat Saxophone Quartet has the desire to bring abroad and develop the great music found in Portugal, as well as bring with them the great music found in the Netherlands, making this artistic and cultural connection through the music.


text:
BEP Delays Sale of Cleveland $2 Single Note
by CoinNews.net on May 29, 2008 · 0 comments
The Bureau of Engraving and Printing (BEP) had announced Monday's release of the 2008 Cleveland $2 Single Notes for collectors. However, a follow-up message sent to customers on Tuesday, May 27, indicates there will be an unspecified delay in their sale.
The BEP message said in part,
"Due to system maintenance the Bureau of Engraving and Printing will postpone the release of the 2008 $2 Single Cleveland Note. Please check our website"
After two days, there is no indication of a new date for release, and the original sale date for the note was changed to: "to be determined".
The always sellout $2 bills offer collectors crisp, uncirculated notes with early serial numbers and from each of the Federal Reserve Banks throughout the year.
The Cleveland notes are the sixth $2 collector set for release in 2008. The Atlanta $2 single notes were first off printing presses in January followed by the New York $2 notes in February, the San Francisco $2 notes in March, the Chicago $2 notes in April and the Dallas $2 notes in May.
The BEP's proactive announcement is greatly appreciated and a vast improvement compared to earlier in the month when their online system was taken down during the initial day of the Dallas $2 offering. Notices went out only after the fact, resulting in many frustrated collectors.
2010 $2 Single Note Collection: Final Installment on Sale
2008 Cleveland $2 Single Notes are SOLD OUT!
Philadelphia Note Joins Cleveland $2 Single Delay
Gold Dips 0.2%; Silver Marks 20-Week High
Gold Ends Near Two-Week High on Monday, July 15
Gold, Silver and Platinum Log Weekly Gains
Gold Falls 0.4% As Core Inflation Firms; Palladium Slides From Record
Gold Climbs 0.9%; Palladium Scores New Record
Coin Web Sites
Melt Values for Silver Coins


I graduated from California Institute of the Arts college in Valencia, California in 1990 with a certificate of fine arts degree. Right out of college, I was picked up by Project X in Studio City to do character & prop designs for "Peter Pan and the Pirates" animated series. Soon after, I was doing storyboards for the show mentioned.
A year after, I started working for Colossal Pictures in San Francisco to do character designs & layouts for "Back to the Future" animated series for two seasons. I continued to work at Colossal for another four years animating on various projects which mostly consist of commercial ads that include Nike, Coke, & Pepsi, just to name a few.
In 1996, I was promoted to an animation director working for Curious Pictures, West Coast in San Francisco until the end of 1999. There, I animation directed a couple of shorts for the Cartoon Network, which includes, "Droopy- Thanks a Latte", and directed commercial ads. In between those projects, I was creating a couple of shorts for Curious' company reel, which I called "Catburglar". (Check it out in the "Animation" and "Frames" section of the menu to the left.) Unfortunately, it was never completed.
I then, went on to Wildbrain Inc, also in San Francisco, to animate on various commercial jobs including directing the animation for "Sony: Pass It On - 'Adventure' " spot in the year 2000. In the same year, I went to PDI / Dream Works in Palo Alto to do the 2D animation for a flash animated short called "Ron's Dream".
From there, I did storyboards on "Phantom Investigators" TV series at Wholesome Products, San Francisco in 2001 for Sony Pictures Entertainment, Culver City. Then, I took a month off to create my website portfolio, www.stewardlee.com.
In 2002, I storyboarded on "Jackie Chan Adventures" TV series also for Sony Pictures Entertainment. Towards the end of the year, I started doing storyboards on "James Bond 007: Everything Or Nothing" game for Electronic Arts, Redwood City.
2003- I finally got around to creating my first self-published comic, "It Flows Like... Wataah! #01". Also in that year I storyboarded on a game called, "Rise To Honor", a Jet Li game for Sony Computer Entertainment America, Foster City.
2004- Sony Computer Entertainment America called me back to storyboard on their 2nd upcoming Untitled Jet Li game, and I continued that until the end of 2005. During this time, I also was working on "The Batman" TV series for Warner Bros. Animation.
2006, I joined the team up at Lucasfilm Animation, storyboarding on their Star Wars: The Clone Wars TV series.
2007, I have been promoted to Episodic Director at Lucasfilm Animation, and directing episodes on Star Wars: The Clone Wars TV series. Check out my video profile on starwars.com.
2013, I'm back at Lucasfilm Animation as Episodic Director on Star Wars: Rebels .
Examples of my professional work are in the "Animation", "Frames", and "Storyboards" section of the menu to the left.


text:
Discovering the Fourth Dimension
Four doors, four seats — is it a Porsche? It is a Porsche, and what a Porsche it is! The Stuttgart sports-car maker's fourth model series is an exploration of the fourth dimension. With the Panamera, Porsche has demonstrated that even in a four-seater, luxury and sportiness are not mutually exclusive. The Panamera is truly in a class of its own.
Light on the Horizon
The newcomers: the Cayman and the Boxster promise even more pleasure and lower consumption thanks to their six-cylinder boxer engines with direct fuel injection and double-clutch transmission.
Make Way for Clay
A new era begins at the Porsche Tennis Grand Prix: from April 25 to May 3, 2009, Stuttgart will host the world's first indoor clay court tournament.
Direct from the Original Owner
Old-time Porsche tractors have attained cult status. That passion was on full display in Germany's Tauber Valley, where 76 of the classic tractors showed their stuff to the public.
A Porsche Pictorial: The Christophorus Poster
Due to popular demand, Christophorus is now offering its readers a special poster featuring all Porsche sports cars from 1948 to 2008 — including the new 911s.
The Ultimate Tour de France
The fortunate members of the Porsche Travel Club don't just get to drive beautiful cars down sublime stretches of road. Their tour of Southern France had plenty to offer to all the senses.
News: Record Values
News: Premiere for the New Generation 911 in Paris
News: Six Titles for Porsche
News: Two Cayennes on a Special Mission
News: Top Grades from Customers


We take a different approach to safety training programs. Whether we are delivering standard or customized training courses, we integrate scenario-based learning using our state-of-the-art simulator to ensure our clients will achieve intended learning outcomes and be prepared for the real world. Irwin's demands the highest standard of health and safety training, and exceeds legislative requirements to educate and empower workers to be aware and safe on the jobsite. We have even designed a 'Train-the-Trainer' course for our Irwin's specific courses! Our courses can be delivered online, on-site, or at one of our networked training centres. We all have a reason to get home safe… What's yours?
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text:
Senior Move Manager
Factors to Consider When Selecting a Senior Move Manager in Lincolnshire
If you are considering moving to a smaller house, you should hire a senior move manager to help with the emotional aspects of downsizing. There are however some factors that you should review when looking for a reliable senior move manager in Lincolnshire. This is because in the last five to ten years, there has been a rise in the number of senior move managers available in the industry. It is therefore important to assess the reliability of various senior move managers in Lincolnshire before you make a choice. This will enable you to pick the most positively reviewed senior move manager in the market. It is also vital to keep the experience of the chosen senior move manager in mind when making your selection. People who hire the first senior move manager they come across in the market end up losing a lot of money. If possible, you should consider with an individual that has worked with the chosen senior move manager before. In this article, we will address the factors to consider when looking for the most reliable senior move manager lincolnshire.
First, review the reliability of the chosen senior move manager. You should use the services of the most reputable senior move manager in Lincolnshire. You can learn about the senior move manager's reliability by reading their customer reviews. Another way to ensure that you select the most reputable senior move manager in the industry is consulting with their clients. This will give you a chance to interact with the senior move manager's clients. The reviews should act as a guide for selecting the most reliable senior move manager in the industry. It is vital to do away with any senior move manager with a lot of bad ratings. Make sure to also hire the services of the senior move manager with the most positive reviews in Lincolnshire.
Secondly, review the experience of the chosen senior move manager. It is vital to pick a senior move manager that has been in this sector for a long time. Contacting the senior move manager's references will enable you to learn about their experience. It is advisable to use a senior move manager with more than seven years of experience. Make sure to also check the licensing of the chosen senior move manager. You should select a senior move manager with an active license from all the relevant boards. Going through the site of the chosen senior move manager will enable you to learn about their certification. It is also advisable to check the senior move manager's license to ensure that it is up-to-date.
Finally, review the pricing of the chosen senior move manager. You should compare the cost of using different senior move managers in Lincolnshire during the decision-making process. This will help you find the senior move manager with the most reasonable rates in Lincolnshire. Going through the senior move manager's official site will also help you to learn about the cost of their services. The cost of using the chosen senior move manager should only be reviewed when one has assessed the quality of their services.
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text:
Why a Self-Directed IRA?
iLearn
Guides and eBooks
Home » Blog » IRA Investing » The New Definition of "Accredited Investor"
The New Definition of "Accredited Investor"
Roberta Hudy
IRA Investing
For nearly 40 years- spanning six different US Presidents, nine different SEC Chairs and five bull markets- the definition of "accredited investor" has not experienced any substantial modification.
That all changed on August 26, 2020, when the Securities and Exchange Commission (SEC) adopted final amendments to the "accredited investor" definition which fundamentally broadens the qualification standards and increases access to investments in the private capital markets.
These amendments are game-changing and, for the first time, include both new, non-wealth based, categories for investors that include financial professionals as well as several new entity categories.
The new changes signal the latest step in a general direction that the U.S. has been heading towards facilitating better access to capital for startups and wider access to alternative investments through the easing of restrictions on individual investors.
Who Can Now Qualify as an Accredited Investor?
The rules governing accredited investors are designed to protect individual investors. They've operated on the idea that the private markets are riskier and less transparent than public markets and that only higher net-worth investors should thus be able to participate. But the regulations have also drawn criticism for being too exclusionary and keeping investment opportunities among the wealthiest people.
Until now, accredited investors were defined solely by monetary metric: The thresholds stood at a net worth of at least $1 million excluding the value of primary residence, or income at least $200,000 each year for the last two years (or $300,000 combined income if married). The new rules expand the definition to those with knowledge of the financial markets, such as licensed brokers or employees of financial institutions.
The new amendments to the accredited investor definition also increases the list of entities that can qualify as accredited investors.
Let's take a look at the immediate additions to the list of Accredited Investors:
All holders in good standing of the Series 7, Series 65, and Series 82 licenses (for broker-dealers and investment advisers) without regard to their net worth or annual income;
"Knowledgeable Employees" of a private investment fund without regard to their net assets or annual income (but this only applies to investments in the investment fund that employs them);
Limited liability companies with $5 million in assets that was not formed for the specific purpose of investing in the securities offered (even if some of its owners are not accredited);
"Family Offices" with at least $5 million in assets under management and their "Family Clients" (as each term is defined under the Investment Advisers Act);
Registered (either state or SEC) investment advisers, exempt reporting advisers, and rural business investment companies (RBICs);
Any entity, including Indian tribes, governmental bodies, funds, and entities organized under the laws of foreign countries, that own "investments," as defined in Rule 2a51-1(b) under the Investment Company Act, in excess of $5 million and that was not formed for the specific purpose of investing in the securities offered; and
"Spousal Equivalents" may pool their finances for the purpose of qualifying as Accredited Investors.
Why is this Significant?
For some time the SEC has been kicking around ideas to, in its own words, "simplify, harmonize, and improve the exempt offering framework, thereby expanding investment opportunities while maintaining appropriate investor protections and promoting capital formation." In particular, since June of 2019, the SEC has released a series of proposals/reports discussing potential amendments to the existing "accredited investor" definition and opening the same to public comment.
Their efforts culminated in late August with the adoption and release of its final rule which significantly amends Rule 501. In the final release, SEC Chairman Jay Clayton is quoted as saying:
"Today's amendments are the product of years of effort by the Commission and its staff to consider and analyze approaches to revising the accredited investor definition, …For the first time, individuals will be permitted to participate in our private capital markets not only based on their income or net worth, but also based on established, clear measures of financial sophistication. I am also pleased that we have expanded and updated the list of entities, including tribal governments and other organizations, that may qualify to participate in certain private offerings."
While the above-discussed bump in the pool of individual investors is significant, the potential effect of the modifications to the entity categories has the potential to be transformational. Putting it into context, there was roughly $2.7 trillion in investment funds raised via private securities transactions in 2019. The pool of individual investors that can now access these private offerings is deeper than ever before.
These amendments have been a long time coming and represent the culmination of years of heated debate. While there have been various bills and other legislation put forth over the years to accomplish one or more of the final rule amendments, none have succeeded until now. We are truly on the precipice of witnessing a mass opening of the private securities market.
Ultimately, this will materially improve the private capital markets as well as allow more investors to participate in them.
The more inclusive definition may have the effect of expanding the opportunities for investors to participate in private funds and those investment opportunities that have historically been relegated to investors meeting a strict requirement of liquid wealth. As a result, many sophisticated investors seeking to make "fund" investments could only do so through registered securities offerings (mutual funds).
The wealth test for accreditation presumed that those who had wealth were cerebral enough to fend for themselves in investing in private offerings. However, those who were sophisticated and well-educated on the subject of private offerings, but lacked the net worth, were not.
The investment universe is increasingly opening up to investors of varying wealth, and with it, alternatives and private offerings are being democratized and made accessible to investors of varying financial backgrounds.
About Johnathan Leiter of Excelsior Capital
Jonathan's primary responsibilities include business development and investor relations. Prior to joining Excelsior, he spent 14 years in New York City working as an investment advisor for Citigroup and HSBC Securities.
Jonathan then transitioned into private equity, opening the New York office for SharesPost, a Silicon Valley
based broker-dealer providing accredited investors with access to the venture capital asset class.
Having covered multiple industries, he enters real estate private equity with a proven track record of aligning decision makers and facilitating investment into private offerings. He has been a contributor to several civic and philanthropic organizations in his community.
Learn more about Excelsior Capital here: https://www.excelsiorgp.com/management-team/
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text:
\section{Introduction}

Double-layer two dimensional electron gas (2DEG) systems,
where electrons are confined to nearby
parallel planes,  are expected to exhibit many novel phenomena
due to interlayer electron-electron
interaction.  For example, in strong magnetic fields
Coulomb effects are
expected to produce new incompressible ground states
that exhibit the fractional quantum Hall effect\cite{ahm1} and
to cause the collapse of certain integer quantum Hall
effect gaps\cite{ahm2}.  In zero magnetic field, it has
been suggested that Wigner crystallization in double layer
systems is favored by interlayer Coulomb interactions\cite{neilson}
and that in the case of electron-hole systems, excitonic
superfluity could result\cite{spfl} from interlayer interactions.
In recent experiments by
Gramila et al. on electron-electron double layer
systems\cite{gramila1,gramila2} and in
similar experiment by Sivan et al. on electron-hole systems\cite{e-h},
the strength of interlayer interactions
was studied directly by measuring the frictional drag of
one two-dimensional electron gas layer on another.
In these experiments a current flowing in one layer tends to
induce a current in nearby layers.
If no current is allowed to flow in the nearby layer
an electric field develops whose influence cancels the
frictional force between the layers.  The transresistance, defined as
the ratio of the induced voltage in the second layer to
the applied current in the first layer, directly measures
the rate at which momentum is transferred from the current
carrying 2DEG to its neighbor.

Drag between spatially separated electron systems
due to Coulomb interactions between carriers was first considered by
Pogrebinskii\cite{sov} and Price\cite{price}.
The experiments of Gramila et al.\cite{gramila1,gramila2} and of
Sivan et al.\cite{e-h} have stimulated recent theoretical
attention, especially to the case of drag between two-dimensional
electron layers at low
temperatures\cite{gramila1,gramila2,solomon,jauho,mahan1,rojo,takis}.
The interlayer Coulomb drag is caused by fluctuations in
the density of electrons in each layer since 2D layers with
charge uniformly distributed will not exert any frictional
forces upon each other.
In this paper we examine for the
first time the possibility of enhanced frictional drag
between disordered layers due to the diffusive nature of
long-wavelength long-time electron density fluctuations.
Disorder is known\cite{aa,plee} to enhance interaction effects
and to lead to violations of Fermi liquid theory for
individual two-dimensional electron gas layers.
In the diffusive regime,
where wavelengths are longer than the mean free path of
the electrons and times are longer than the electron scattering time,
the electron density-density response function possesses a
diffusion pole\cite{forster} in momentum-frequency space.
In perturbation theory the diffusion pole arises from
dressing the electron-electron vertex\cite{plee}
with corrections arising from impurity potential scattering.
At shorter distances or shorter times the disorder vertex
corrections are not important.
We find that disorder enhances the interlayer drag
at low temperatures, changing the temperature dependence
of the drag from $T^2$ to $-T^2 \ln T$.
In very high-mobility samples (on which existing
experiments have been performed) or in samples with small
layer separations the corrections due to disorder become
important only at extremely low temperatures.
For samples with lower mobility or more widely separated
layers the influence on the interlayer scattering rate
from disorder scattering should be easily measurable.

Most\cite{gramila1,solomon,jauho} previous work on interlayer
friction has been based on Boltzmann transport theory which
cannot capture disorder enhanced interaction effects.
(An exception is the work of Vasilopoulos and co-workers\cite{takis}.)
In Section II we present a derivation of the expression
for the frictional transresistance based on the memory
function formalism\cite{forster} which is sufficiently
general to treat the case of disordered 2DEG layers.
Although phonon-mediated\cite{gramila2,takis,myphmed}
interlayer interactions are not discussed explicitly
in this paper the expression we derive in Section II
are sufficiently general that other coupling mechanisms
can be incorporated as an effective-interlayer interaction
potential.  Similarly particle-particle interaction
vertex corrections, which are
probably quantitatively important especially
for electron-hole double layer systems\cite{e-h},
can also be incorporated into the results we derive in
Section II as a contribution to the effective
electron-electron interaction\cite{we}.  Readers interested only
in the application to disordered double-layer
systems should proceed to Section III where we
discuss how disorder within the layers influences the interlayer
friction.  The interlayer scattering rate of typical
electron-electron double layer samples is evaluated numerically
in Section IV.  We show that the temperatures
below which disorder becomes important decreases very
rapidly with increasing mobility.  A
brief summary of our findings concludes the paper in
Section V.

\section{Memory Function Formalism Derivation}

The memory function formalism provide a very convenient method
for deriving a flexible expression for the transresistance of double
layer 2DEG systems.
The Kubo current-current correlation function formula
for the conductivity is
first converted into a force-force
correlation function expression for the resistivity
by making use of Mori's\cite{mori} projection operator.
This force-force correlation function is then evaluated at
lowest order in the screened interlayer interaction to
obtain an approximate expression for the transresistance.
The advantage of using the force-force correlation function
rather than the current-current correlation function
is that it yields a reasonable approximation even when
evaluated at lowest order\cite{mf1,mf2}.  At this level
the results are physically equivalent to
momentum-balance\cite{takis} approximations or to
relaxation-time approximations in a Boltzmann-transport
approach\cite{mahan2}.  The derivation for the situation
of present interest is sketched in the following paragraphs.

For notational simplicity we restrict ourself to the case of
zero magnetic field so that the
current is in the same direction as the
applied electric field and the conductance,
therefore, forms a 2$\times$2
matrix with respect to the layer indices.
(The final expression for the transresistance
is equally valid\cite{magfield}
in the presence of a magnetic field.)
We need to consider only the long wavelength limit of the conductance.
To use the memory function formalism it is convenient to
write the Kubo formula in the form
\begin{equation}
\sigma_{ij}(\omega) = {\beta\over\nu} \int_{0}^{\infty}dt
e^{i\omega t} ( {\hat J}_i(t), {\hat J}_j )
\label{ku}
\end{equation}
where $\beta=1/k_B T$, $\nu$ is the
cross section area of the 2DEG layers, and
the indices $i$ and $j$ are layer labels.
$\sigma_{ij}(\omega)$ gives the current density induced in
layer $i$ due to an electric field in layer $j$.
$\hat J$ is the zero wavevector Fourier component of
the current density operator.
The inner product appearing in Eq.~(\ref{ku}) is defined by
\begin{eqnarray}
C_{AB}(t) &\equiv & ({\hat A(t)},{\hat B}) \nonumber \\
&\equiv & \beta^{-1}\int_{0}^{\beta}d\lambda\langle {\hat A}^\dagger(t),
{\hat B}(i\hbar\lambda)\rangle
\label{ip}
\end{eqnarray}
In Eq.~(\ref{ip}) the angle brackets denote thermal averages.
The following relationship can be used to change Eq.~(\ref{ku}) into
the more familiar form of the Kubo formula\cite{mf2}:
\begin{equation}
i\beta\partial_tC_{AB}(t) = {1\over\hbar}\langle[{\hat A}(t),{\hat
B}]\rangle.
\label{dtc}
\end{equation}

The projection operator method is now
used to obtain an expression for the matrix inverse
in layer indices of $C_{J_iJ_j}$.
We define a superoperator ${\cal P}$ which `projects' an operator
$\hat O$ onto the current density, and its complement ${\cal Q}$, by
\begin{eqnarray}
{\cal P}\hat O
&\equiv & \sum_k{ {\hat J}_k ({\hat J}_k,\hat O) \over
 ({\hat J}_k,{\hat J}_k ) }
\\
{\cal P} \hat O & \equiv & \hat O-{\cal Q} \hat O
\label{po}
\end{eqnarray}
It is useful to define a matrix $\chi_{ij}$:
\begin{equation}
\chi_{ij} = {\beta\over\nu} C_{J_iJ_j}(0) = ({\hat J}_i,{\hat J}_j)
= {n_ie^2\over m}\delta_{ij}.
\label{ch}
\end{equation}
Here $n_i$ is the areal density of 2D electrons in the ${i}$-th layer.
Following the usual development of the memory function
formalism\cite{forster}, we obtain an equation for the matrix
inverse of the Fourier transform of $C_{J_iJ_j}(t)$:
\begin{equation}
[C_{J_iJ_j}(z)]^{-1} = {\beta\over\nu}\chi^{-1}[-iz{\bf 1}+M(z)]
\label{em}
\end{equation}
where
\begin{equation}
M_{ij}(z) = {\beta\over\nu} (\dot{J}_i,{i\over z-{\cal Q}{\cal L}}
\dot{J}_j)\chi_{jj}^{-1}.
\label{si}
\end{equation}
A dot over an operators denotes its time derivative at $t=0$
and the Liouville superoperator is defined by ${\cal L}
{\hat O}   = [\hat H,\hat O]$, where $\hat H$ is the Hamiltonian.
In obtaining Eq.~(\ref{em}),
the time reversal invariance condition ${\cal P}\dot{J}=0$
has been applied. Combining Eq.~(\ref{ku}), Eq.~(\ref{em})
and Eq.~(\ref{si}) we obtain an expression for the resistivity
matrix:
\begin{eqnarray}
\rho_{ij}(z) &=& \chi_{ii}^{-1} M _{ij}(z)\nonumber \\
&=& \chi_{ii}^{-1}\chi_{jj}^{-1}{\beta\over\nu}
\int_{0}^{\infty}dt e^{izt}(\dot{J}_i,e^{-i{\cal Q}Lt}\dot{J}_j)
\label{ro}.
\end{eqnarray}
$\rho_{ij}$ relates the electric field in layer $i$ to the
current density in layer $j$.
With the relation $\dot{J}_i = -e/m F_i$, we obtain the force-force
correlation function expression for the trans-resistance,
\begin{equation}
\rho_{LR} = {\beta\over n_Ln_Re^2\nu}\int_{0}^{\infty}dte^{izt}
(\hat F_L,e^{-i{\cal Q}{\cal L}t}\hat F_R).
\label{rolr}
\end{equation}

It is easy to demonstrate that $\rho_{LR}$ is identically zero
in the absence of interlayer coupling, since the forces in
left and right layers are uncorrelated.
The leading contribution to the force operator
from interlayer interactions
can be expressed in terms of the
interlayer interaction potential $U_e(q)$ and the electron
density $\varrho(q)$
\begin{equation}
{\bf F}_{R(L)} = \pm {i\over\nu}\sum_{\vec q}{\vec  q}\varrho_L(\vec q)
\varrho_R(-\vec q) U_e(q).
\label{fe}
\end{equation}
To leading order in interlayer interactions
$ e^{-i{\cal Q}{\cal L}t}\hat F_i$ in Eq.~(\ref{rolr}) can be
replaced by $e^{-i{\cal L}t}\hat F_i$.
This replacement leads to the desired
force-force correlation function expression for the trans-resistance
\begin{equation}
\rho_{LR}(z) = {\beta\over n_Ln_Re^2\nu}
\int_{0}^{\infty}dt e^{izt} (F_L(t),F_R)_0.
\label{ffr}
\end{equation}
The subscript on the inner product in Eq.~(\ref{ffr})
indicates that it should be evaluated in the
absence of interlayer interactions.  Substituting the
explicit expression for the interlayer forces gives
\begin{equation}
\rho_{LR}(z) = {\beta\over2n_Ln_Re^2\nu^3}\sum_{\vec q}q^2
\int_{0}^{\infty}e^{izt}dt|U_e(q)|^2({\hat A(t)},
{\hat A}^{\dagger}(0))_0
\label{r1}
\end{equation}
with
\begin{equation}
{\hat A(t)} = \varrho_L(-\vec q,t)\varrho_R(\vec q,t).
\label{as}
\end{equation}

The correlation function for decoupled layers appearing
in Eq.~(\ref{r1}) is related to the isolated layer density
fluctuations.  Using a representation of
exact eigenstates that in the $z=0$ limit it is easy to show that
\begin{equation}
\rho_{LR} = { \pi \beta \over 2 n_L n_R e^2 }
\int {d^2 \vec q \over (2 \pi)^2 } q^2 |U_e(q)|^2
\int_{-\infty}^{\infty} d \omega
S_L(\vec q, \omega) S_R( - \vec q, - \omega)
\label{ahm1}
\end{equation}
where $S_i(\vec q,\omega) $ is the dynamic structure factor
for layer $i$:
\begin{equation}
S_i(\vec q, \omega) \equiv { 1 \over \nu }
\sum_{n,m} \exp (- \beta E_n) |\langle n | \rho_i(\vec q) | m \rangle |^2
\delta (\omega - (E_m-E_n)/\hbar )
\label{ahm2}
\end{equation}
It is usually more convenient to express the resistance in
terms of individual layer response functions rather than the dynamic
structure factor.
We relate $S_i(q,\omega)$
to the retarded density-density response function for layer
$i$, $\chi_{i}(\vec q, \omega) $ by
applying the fluctuation-dissipation theorem\cite{forster}
\begin{equation}
S_i(q,\omega) = {\hbar\over1-e^{-\hbar\omega\beta}}
{\rm Im} \chi_i(\vec q,\omega).
\label{fdearly}
\end{equation}
This gives us the final form of our expression for the drag
resistivity, which is summarized
diagrammatically in Fig.~(\ref{fa}).
\begin{equation}
\rho_{LR} = {\hbar^2\beta\over\pi n_Ln_Re^2}{1\over\nu}
\sum_{\bf q}q^2|U_e(q)|^2\int_{0}^{\infty}d\omega
{{\rm Im}\chi_{R}(q,\omega)
{\rm Im}\chi_{L}(q,\omega)\over
e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}.
\label{r3}
\end{equation}

\section{Disorder and Screening}

The electron density-density response
function $\chi(q,\omega)$ in Eq.~(\ref{r3})
can be obtained by applying many-body perturbation
theory methods to a 2DEG whose Hamiltonian
contains disorder and (or) interaction terms.
For a non-interacting disorder-free 2DEG the
response function can be evaluated analytically\cite{2degrf}.
Disorder leads to an enhancement in ${\rm Im}\chi(q,\omega)$
at low frequencies and long wavelengths and, as we discuss
below, can enhance the interlayer friction.  The enhancement
reflects the increased spatial correlation of states with
nearby energies in disordered systems.   The effect of
disorder on the friction can be described without
making a specific model of disorder by invoking the
Einstein relation between the conductivity and the diffusive
density-density response at long wavelengths and
low frequencies\cite{arbitrary}.
We introduce a phenomenological intralayer electron
(transport) scattering time $\tau$.
$\tau$ is related to the mobility by $\mu=e\tau/m$ and
at low temperatures is related to the mean-free-path
by $l  = \tau\hbar k_f/m$.
For $q l > 1$ or $  \omega \tau > 1$ we assume that disorder is
unimportant and approximate the density-density response
function by the non-interacting electron result\cite{2degrf}.
At zero temperature and zero disorder
\begin{eqnarray}
&&\chi_i(q,\omega) \equiv
\chi_{i}^{B}(q,\omega)\nonumber \\
&=& {dn\over d\mu}{m\over q^2}\{{q^2\over m} -
C_+|(k_fq/m)^2-(\omega+\varepsilon_q)^2/\hbar^4|^{1/2}
-C_-|(k_fq/m)^2-(\omega-\varepsilon_q)^2/\hbar^4|^{1/2}\}
\label{xb}
\end{eqnarray}
\noindent where $\varepsilon_q=\hbar q^2/2m$,
 $\mu$ is the chemical potential and the $C_{\pm}$ are:
 \begin{eqnarray}
 C_{\pm} &=& {\rm sign}(\varepsilon_q\pm\omega)\ \ \ \ \ \ {\rm if} \ \ \
 (k_fq/m)^2-(\omega\pm\varepsilon)^2/\hbar^2<0,
 \nonumber\\
 C_{\pm} &=&\pm i\ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm if}\ \ \
 (k_fq/m)^2-(\omega\pm\varepsilon_q)^2/\hbar^2.
 >0\nonumber
 \end{eqnarray}
However for $q l < 1$ and $\omega \tau < 1$ disorder becomes
important.  In this diffusive regime the electron
density-density response function is completely characterized
by the diffusion constant\cite{plee}:
\begin{eqnarray}
\chi_i(q,\omega) &\equiv &
\chi_{i}^{D}(q,\omega) \nonumber \\
&=& {dn\over d\mu}{Dq^2\over Dq^2-i\omega}
\label{xd}
\end{eqnarray}
\vskip 1ex
\centerline{ $\ \ {q<{1/l}}$\ \ \ \ \  \ \ \ \
$\omega<{1 / \tau}$}
\vskip 1ex
\noindent where $D=l^2/ 2 \tau$ is the diffusion
constant.

The derivation of the expression for the transresistivity in
the previous section is
valid up to second order in the interlayer interaction.
For the system of physical interest the interlayer interaction
is Coulombic and it is essential to include screening in
order to get qualitatively correct results.  In this
paper we adopt the usual expediency of employing
the second-order expression with the interlayer interaction
replaced by a screened interlayer interaction and argue that
this includes the most important higher-order effects.
In the disorder free limit our expression for the transresistivity
then becomes identical to those derived using other approaches
in earlier work\cite{gramila1,solomon,jauho,mahan1,takis}.
The random-phase-approximation (RPA) screened interlayer
interaction is
\begin{equation}
U_e(q,\omega) = {V_e(q)\over[1+V_a(q)\chi_L
(q,\omega)][1+V_a(q)\chi_R(q,\omega)]-V_e^2(q)\chi_L(q,\omega)
\chi_R(q,\omega)}
\label{rpa}
\end{equation}
where the bare intra- and inter-layer electron-electron interaction
potentials are $V_a(q)=2\pi e^2/q$ and $V_e(q)=V_a(q)e^{-qd}$
where $d$ is the separation between the layers.
In the above expression either the ballistic or the diffusive
form for $\chi_{L(R)}$ should be used as appropriate.  Note
that the interlayer interaction is cut-off by the factor
$e^{-qd}$ for $q > 1/d$.   The layer separation dependence
of the friction in both diffusive and ballistic limits
results from this cutoff.  Physically the cutoff reflects the
fact that charge fluctuations in one layer with a wavelength shorter
than the layer separation get averaged out when viewed from
the other layer.

The expression for the trans-resistance of Eq.~(\ref{r3})
can be split into contributions from the ballistic and diffusive
regimes. With $\rho_{LR}^{-1} \equiv n_Re^2\tau_{LR}/m$, we have
\begin{equation}
\tau_{RL}^{-1} = \tau_B^{-1} + \tau_{\Delta}^{-1}
\label{rlbd}
\end{equation}
where
\begin{equation}
\tau_B^{-1} = {\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{\infty}
dqq^3\int_{0}^{\infty}d\omega |U_e|^2
{{\rm Im}[\chi_L^B(q,\omega)] {\rm Im}[\chi_R^B(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tbe}
\end{equation}
and
\begin{equation}
\tau_{\Delta}^{-1} =\tau_D^{-1} -
{\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{1\over l}
dqq^3\int_{0}^{1\over\tau}d\omega |U_e|^2
{{\rm Im}[\chi_L^B(q,\omega)] {\rm Im}[\chi_R^B(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tdel}
\end{equation}
with
\begin{equation}
\tau_D^{-1} =
{\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{1\over l}
dqq^3\int_{0}^{1\over\tau}d\omega |U_e|^2
{{\rm Im}[\chi_L^D(q,\omega)] {\rm Im}[\chi_R^D(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tde}
\end{equation}
$\tau_B^{-1}$ is the result for
a disorder-free 2DEGs and
$\tau_{\Delta}^{-1}$ is the correction due to the
the enhanced fluctuations at long wavelengths and low frequencies
in disordered systems.
In the next section we discus the evaluation of these expressions.

\section{Numerical Results and Discussion}

The dependence of the interlayer scattering rate on temperature
and on layer separation depends on whether
the interlayer scattering is dominated by $\tau_B^{-1}$ or
$\tau_D^{-1}$.
In Fig.~(\ref{fb}) and Fig.~(\ref{fc})
we show numerical results for $\tau_B$ and
$\tau_{\Delta}$ as a functions of temperature calculated
for two different values of layer separation for a high
mobility two-dimensional electron gas sample.
The data in these figures are obtained from numerical evaluation of
Eq.~(\ref{tbe}) and Eq.~(\ref{tde}) with the input
parameters taken from the experiment of ref.~\cite{gramila1}.
In Fig.~(\ref{fb}) we see that $\tau_B^{-1}  \sim
T^2/d^4$ at low temperatures, as pointed out in
Ref.\cite{gramila1}. (The $d^{-4}$ dependence can be
recognized by noticing that the scattering rate decreases by
a factor of approximately four when the layer separation
increases by a factor of $\sqrt{2}$.) From Fig.~(\ref{fc}) one can see
that $\tau_{\Delta}^{-1}$ falls off more slowly
with both temperature and layer separation
as $T\rightarrow 0$.   The inset to Fig.~(\ref{fc})
establishes that for $T\ll T_{\tau}$,
$\tau_D^{-1} \sim - T^2{\rm ln}T/d^2$.  The dimensionless
temperature scale for Fig.~(\ref{fb}) is the Fermi
temperature ($T_F \equiv E_F / k_B$) which is about $60$K for
this sample, while
the dimensionless temperature scale for Fig.~(\ref{fc}) is
the disorder temperature
($ T_{\tau} \equiv \hbar / k_B \tau $) which is
$\sim 56$mK for this sample.   Comparing Fig.~(\ref{fb})
and Fig.~(\ref{fc}) we see that for this high-mobility
sample the disorder correction is smaller than
one part in $10^5$ at temperatures above $\sim 10mK$.

The origin of the temperature and layer dependence
seen in Fig.~(\ref{fb}) and Fig.~(\ref{fc}) can be
understood by looking at the limit of large layer separations where
the interlayer scattering rates can be evaluated analytically.
The evaluation of
$\tau_{LR}^{-1}$ at large layer separations and low
temperatures for the
disorder-free limit,
where the relation $\tau_B\propto T^2/d^4$ holds,
has been carried out previously
by several authors\cite{gramila1,solomon,jauho}.
(The results quoted in Ref.~(\cite{gramila1}) and
Ref.~(\cite{solomon}) are in error by a factor of two.)  We
rederive those results here to allow a comparison with the
disordered case.  For $T  \ll T_F$ and $ d \gg k_F^{-1}$
only the low frequency and long wavelength limit of
${\rm Im}\chi$ contributes importantly to Eq.~(\ref{r3}).
($k_F$ is the Fermi wavelength.) From Eq.~(\ref{xb}) it
follows that in this limit
$ {\rm Re} \chi (q,\omega) = dn/d\mu$, and ${\rm Im} \chi (q,\omega) =
dn/d\mu (2 \hbar  \omega /E_F) (k_F/q)$.  For $T \ll T_F$,  we can
ignore the contribution of ${\rm Im}\chi$ to screening the
interlayer interaction and it follows from Eq.~(\ref{rpa}) that
we can replace the interlayer interaction by
\begin{equation}
U_e(q) = { \pi e^2 q \over k_{TF}^2 \sinh{qd} }
\label{uesq}
\end{equation}
where $k_{TF} \equiv 2 \pi e^2  dn/d\mu$ is the single
layer Thomas-Fermi screening wavevector.  Note that for
$q \to 0$ the effective screening wavevector is $2 k_{TF}^2 d$
, which is proportional to the layer separation.  (For GaAs
$k_{TF} = 0.2 {\rm nm}^{-1}$ independent of electron density.)
With these approximations the integral over frequency and
wavevector are known and we obtain
\begin{equation}
\tau_B^{-1} = {- \pi\zeta(3)(k_BT)^2\over16\hbar\varepsilon_f
(k_{TF}d)^2(k_Fd)^2}.
\label{equ:tb2}
\end{equation}
In this result two powers of $d^{-1}$ may be associated with
the enhanced screening of the interlayer interaction at large
separations and two powers of $d^{-1}$ with the combination
of phase space considerations which cause the integrand to
to vary as $q^1$ for small $q$.
For layer separations larger than the mean free path the
electron response is diffusive over the entire range of
wavevectors contributing importantly to Eq.~(\ref{r3}).
For low temperatures it follows from Eq.~(\ref{xd}) that we
may use ${\rm Re} \chi (q,\omega) = dn / d \mu$ and
${\rm Im} \chi (q,\omega) = dn / d\mu [ \omega D q^2 /
( \omega^2 + (D q^2)^2) ]$.  Again we may ignore the
contribution to screening from ${\rm Im}\chi$ so that
the relevant limit of the screened interlayer interaction
is unchanged.  At small $\omega$, ${\rm Im}\chi \propto q^{-2}$
compared to the $q^{-1}$ of the ballistic case.  The
integrand of the wavevector integral thus goes as $q^{-1}$
at small wavevector.  This logarithmically divergent wavevector
integral is cutoff at $q \sim {\omega/D}^{1/2}$.
The remaining frequency integral is elementary and we obtain
\begin{equation}
\tau_D^{-1} = {- \pi(k_BT)^2{\rm ln}(T/T_{\tau})
\over 12 \hbar\varepsilon_f(q_{TF}d)^2(k_Fl)^2}.
\label{equ:td2}
\end{equation}
The change in the layer separation from $d^{-4}$ in the
ballistic case to $d^{-2}$ in the diffusive case can be traced
directly to the change in the wavevector dependence of
${\rm Im}\chi$ from $q^{-1}$ to $q^{-2}$.

For $d \ll l$ Eq.~(\ref{equ:td2}) correctly gives the
contribution to the drag from $q \ll l^{-1}$  and
Eq.~(\ref{equ:tb2}) gives the contribution from
$l^{-1} < q < d^{-1}$.  Because of the different
temperature dependence it is still true that the
contribution from the diffusive regime will dominate at
sufficiently low temperatures.  Comparing Eq.~(\ref{equ:td2})
and Eq.~(\ref{equ:tb2}) we can estimate the crossover temperature:
\begin{equation}
T_c \sim T_{\tau} \exp [ - 3 (l/d)^2 / 4 \zeta (3)]
\label{tcross}
\end{equation}
For high-mobility samples the diffusive enhancement of the
drag will be observable only at extremely low temperatures.
For example, in GaAs $T_{\tau}$ is about $0.2$K and $l \sim 10 \mu$
for a sample with a mobility of $\sim 10^6cm^{2}/sV$ and
a typical density.  For a layer separation of $\sim 500 A^{\circ}$
this implies that $T_c \sim 10^{-100} K$.
For the samples in the experiment of ref.~\cite{gramila1},
which are of extremely high mobility and small layer separations,
the value of $T_{\tau}$ is $56mK$ and the $d/l$ is about $10^{-3}$.
$ \tau_{\Delta}^{-1}$ is smaller than $\tau_B^{-1}$
by a factor of $10^{6\sim7}$ at $T\sim T_{\tau}$.
The correction term, $\tau_{\Delta}^{-1}$, will be
difficult to observe for accessible temperatures.
In samples with lower mobility and/or thicker
barriers between the layers, $\tau_{\Delta}^{-1}$ and $\tau_B^{-1}$
have comparable amplitudes for $T\sim T_{\tau}$ and the contribution
from the diffusive regime dominates at lower temperatures.
In Fig.~(\ref{fd}) we plot the relative contribution to
the drag from the diffusive regime vs. mobility for a layer
separation of $50 {\rm nm}$ and $n =1.5 \times 10^{11} {\rm cm}^{-2}$.
These results show that the effect of disorder
will become easily observable at typical low temperatures
for samples with mobilities below $\sim 10^{5} {\rm cm}^2 {\rm V}^{-1}
{\rm s}^{-1}$.

It is possible to fabricate double-layer systems in which
one layer is much more disordered than the other.
In particular one may have $l\gg d$ for one layer
and $l\le d$ for the other layer.
Following the same steps leading to Eq.~(\ref{equ:td2}),
it is possible to derive an expression for the low-temperature
transresistance in such a system
by using one diffusive response
function and one free electron response function:
\begin{equation}
\tau_{LR}^{-1} = {\pi^3(k_BT)^2\over72\hbar\varepsilon_f
(k_{TF}d)^2(k_Fd)(k_Fl)}.
\label{equ:tbd}
\end{equation}
The dependences on temperature and layer
separation, $\tau_{LR}^{-1} \sim
T^2/d^3$, are easily understood by comparing it to Eq.~(\ref{equ:tb2})
and Eq.~(\ref{equ:td2})
and noticing that the integrand in this case approaches a
constant at small wavevector transfers.
In Fig.~(\ref{fd}), results are shown for
the relative correction due to
disorder enhancement for the case where one layer
consists of free electrons while the other layer has
a finite mobility.
The relative correction is essentially independent of temperatures
at low temperatures and it is weaker than in
the case where both layers are disordered.

\section{summary}

Using the memory-function method,
we have derived an expression
for the transresistance of double layer systems
which is sufficiently general to treat the case of
disordered layers.  The expression has been evaluated
as a function of temperature, layer separation (d)
and in-plane mobility $\mu$.
Both the case where only one layer is disordered and
the case where both layers are disordered have been considered.
We find that the drag varies as $d^{-4}$, $d^{-3}$ and $d^{-2}$
at low temperatures for clean, single-layer disorder, and
double-layer disorder cases respectively.   In the case of
double-layer disorder the transresistance varies as
$ - T^2 \ln (T)$ at low temperatures, otherwise the
transresistance is proportional to $ T^2$.   The low-temperature
drag is proportional to $\mu^{-2}$ and $\mu^{-1}$ for
double-layer disorder and single-layer disorder respectively.
Except for the case of extremely low temperatures the
crossover from clean to disordered regimes occurs
when the mean-free-path within a layer becomes smaller
than the layer separation.
In very high mobility two-dimensional electron gas systems,
where the transresistance has been studied
experimentally up to the present,
the effect of disorder on the drag is negligible at available
temperatures.  We predict that the enhancement due to disorder
and the associated crossovers in the temperature-dependence
and layer-separation dependence will be observable at
low temperatures in moderate and low mobility samples.

\acknowledgments

The authors are grateful for stimulating
interactions with T.J. Gramila and J.P. Eisenstein and
with Jun Hu and Anthony Chan.  Informative
discussions with B.I. Altshuler during the initial stages of
this work and helpful communications with A.P. Jauho,
A.G. Rojo and G.D. Mahan are also acknowledged.
This work was supported by the National Science Foundation under
Grant No. DMR-9113911.

redpajama_set_name: RedPajamaArXiv


text:
Q: Why is My Minimax Not Expanding and Making Moves Correctly? I am implementing minimax in Python 2.7.11 in a basic game of Pacman. Pacman is the maximizing agent, and one or more ghosts (depending on the test layout) is/are the minimizing agent(s).
I must implement minimax so that there can be potentially more than one minimizing agent, and so that it can create a tree of n plies (depth). Ply 1, for example, would be each ghost taking a turn minimizing the terminal state utilities of their possible moves, as well as pacman taking his turn maximizing what the ghosts have already minimized. Graphically, ply 1 looks like this:

If we had the following arbitrary utilities assigned to the green terminal states (left to right):
-10, 5, 8, 4, -4, 20, -7, 17
Pacman should return -4 and then move in that direction, creating an entirely new minimax tree based on that decision.
First, a list of variables and functions needed for my implementation to make sense:
# Stores everything about the current state of the game
gameState

# A globally defined depth that varies depending on the test cases.
#     It could be as little as 1 or arbitrarily large
self.depth

# A locally defined depth that keeps track of how many plies deep I've gone in the tree
self.myDepth

# A function that assigns a numeric value as a utility for the current state
#     How this is calculated is moot
self.evaluationFunction(gameState)

# Returns a list of legal actions for an agent
#     agentIndex = 0 means Pacman, ghosts are >= 1
gameState.getLegalActions(agentIndex)

# Returns the successor game state after an agent takes an action
gameState.generateSuccessor(agentIndex, action)

# Returns the total number of agents in the game
gameState.getNumAgents()

# Returns whether or not the game state is a winning (terminal) state
gameState.isWin()

# Returns whether or not the game state is a losing (terminal) state
gameState.isLose()

This is my implementation:
"""
getAction takes a gameState and returns the optimal move for pacman,
assuming that the ghosts are optimal at minimizing his possibilities
"""
def getAction(self, gameState):
    self.myDepth = 0

    def miniMax(gameState):
        if gameState.isWin() or gameState.isLose() or self.myDepth == self.depth:
            return self.evaluationFunction(gameState)

        numAgents = gameState.getNumAgents()
        for i in range(0, numAgents, 1):
            legalMoves = gameState.getLegalActions(i)
            successors = [gameState.generateSuccessor(j, legalMoves[j]) for j, move
                                                           in enumerate(legalMoves)]
            for successor in successors:
                if i == 0:
                    return maxValue(successor, i)
                else:
                    return minValue(successor, i)

    def minValue(gameState, agentIndex):
        minUtility = float('inf')
        legalMoves = gameState.getLegalActions(agentIndex)
        succesors = [gameState.generateSuccessor(i, legalMoves[i]) for i, move
                                                      in enumerate(legalMoves)]
        for successor in successors:
            minUtility = min(minUtility, miniMax(successor))

        return minUtility

    def maxValue(gameState, agentIndex)
        self.myDepth += 1
        maxUtility = float('-inf')
        legalMoves = gameState.getLegalActions(agentIndex)
        successors = [gameState.generateSuccessor(i, legalMoves[i]) for i, move
                                                       in enumerate(legalMoves)]
        for successor in successors:
            maxUtility = max(maxUtility, miniMax(successor))

        return maxUtility

    return miniMax(gameState)

Does anyone have any ideas why my code is doing this? I'm hoping there are a few Minimax/Artificial Intelligence experts out there that can identify my issues.
Thanks in advance.
UPDATE: by instantiating my self.myDepth value as 0 instead of 1, I have irradicated the exception throwing issue. However, the overall incorrectness of my implementation still remains.

A: I have finally figured out a solution to my issue. The main problem was the fact that I was not referencing the depth correctly to keep track of the plies. Instead of incrementing depth within the maxValue method, it should instead be passed as a parameter to each function, and only incremented when being passed into maxValue. There were several other logical errors such as not correctly referencing numAgents, and also the fact that my miniMax method was not returning an action. Here is my solution which turned out to work:
def getAction(self, gameState):

    self.numAgents = gameState.getNumAgents()
    self.myDepth = 0
    self.action = Direction.STOP # Imported from a class that defines 5 directions

    def miniMax(gameState, index, depth, action):
        maxU = float('-inf')
        legalMoves = gameState.getLegalActions(index)
        for move in legalMoves:
            tempU = maxU
            successor = gameState.generateSuccessor(index, move)
            maxU = minValue(successor, index + 1, depth)
            if maxU > tempU:
                action = move
        return action

    def maxValue(gameState, index, depth):
        if gameState.isWin() or gameState.isLose() or depth == self.depth:
            return self.evaluationFunction(gameState)

        index %= (self.numAgents - 1)
        maxU = float('-inf')
        legalMoves = gameState.getLegalActions(index)
        for move in legalMoves:
            successor = gameState.generateSuccessor(index, move)
            maxU = max(maxU, minValue(successor, index + 1, depth)
        return maxU

    def minValue(gameState, index, depth):
        if gameState.isWin() or gameState.isLose() or depth == self.depth:
            return self.evaluationFunction(gameState)

        minU = float('inf')
        legalMoves = gameState.getLegalActions(index)
        if index + 1 == self.numAgents:
            for move in legalMoves:
                successor = gameState.generateSuccessor(index, move)
                # Where depth is increased
                minU = min(minU, maxValue(successor, index, depth + 1)
        else:
            for move in legalMoves:
                successor = gameState.generateSuccessor(index, move)
                minU = min(minU, minValue(successor, index + 1, depth)
        return minU

    return miniMax(gameState, self.index, self.myDepth, self.action)

And presto! Our final working multi-agent minimax implementation.


text:
Indian Crypto Exchange WazirX Says Userbase Has Grown Over 10 Times in 2021
By Hoddle November 12, 2021 November 13th, 2021 News
Binance-owned Indian crypto exchange WazirX said its userbase has grown by over 10 times to 10 million, this year.
WazirX recorded trading volume of over $38 billion YTD, equating to an average month-on-month growth of 44%, the exchange announced on Friday.
The company reported growth of 2,648% in user signups from Tier II and Tier III cities – in other words, large cities in the country except the six biggest: Mumbai, Delhi, Kolkata, Chennai, Bengaluru, Hyderabad.
The strong user growth was also backed by tremendous adoption by women, from the Tier II and Tier III cities, the company said.
The exchange's performance this year demonstrates the growing popularity of crypto in India despite lingering uncertainty about whether it would be banned by the government.
At the start of this year, India's government was set to outright ban crypto, but has since pivoted to a position of regulating it as an asset class.
WazirX, alongside other crypto players, has been invited to meet with the government next week to share its views on the industry.
Read more: Crypto Adoption Numbers in India Could Be 'Exaggerated,' Says RBI Governor
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- Button 'Send resources' in the World Wonder screen should no longer overlap other UI elements.
- Several bugfixes were also applied to improve the game's performance.
I have played this game for 3 1/2 years. Yes, you have to spend real money to grow faster. However, in the current world I'm in, I have first hand witnessed corruption, bias, and theft. After sending requested info multiple times, grepolis has shafted me out of $50. In addition, the moderators and administrators show favoritism and give intel to the alliance that spends the most real money. Basically, you buy the win. I have had colonies that were started to be founded destroyed by enemy alliances that were over 7 hours away. There is no way the info of my movements could have been given to them except from those that can see everything in the game. Aka mods and admins. It can be a fun game, but not when the cards are stacked against you.
Update: nothing has changed. Unless you want to spend hundreds of real dollars to win the game, don't get it.
Another update. If you get "banned" in the game for foul language or whatever they feel like, your cities can still get attacked and conquered but there's nothing you can do about it.
Like chess, Stratego, and Risk on Steroids.
It's a fun game. Takes a little while to get used to how to play, but the more you play the better you'll get. Yes you could get wiped off the map and have to start over but keep trying. Really to win you need the be in a good alliance of nice and dedicated team players. Some players spend money to help themselves get stronger but you don't have to. Being in a good alliance is the key to having fun. Some players will spend money and some won't but that's okay most alliances have a mix of spenders and nonspenders. Even if you decide to spend game gold bought with real money, it is still cheaper than being a golfer or joining a bowling league. There are all ages that play this game, but I'm surprised how many players are over 50.
Serious issues with moderators is causing worlds to die. I've played this game for years and the world size is a fraction of what it used to be when the game was good.
In US69-Gonnos a single mod, Medic, used her power to gain revenge. This mod was a member of our alliance, but listed for internalization because she didn't update forum posts and lost cities without asking for help. After her departure she personally targeted players in our alliance by banning g them from he game. She didn't respond to support tickets, used her access as a mod to give information to other teams, and refused to take action on reports of players openly cheating by using bots. Her actions alone caused about two dozen veteran players to quit the game.
If you want to play a game that allows cheaters and handles support unprofessionally then this is the game for you. Until these issues are resolved, I can't recommend this game to anyone. It is a shame because I used to enjoy playing it.


There was a big crowd on Quai Louis II for the first day of the Monaco Optimist Team Race, organised with the support of FxPro, Eventica, Fon and clothing supplier SLAM.
This original match-race regatta has attracted 64 under 14s for the seventh year in a row, with a veritable forest of Optimists, the cute single-handed one-design dinghy, invading the Yacht Club de Monaco and surrounds.
Swedes, Finns, Danes, Croats, Italians, Monegasques… in total 16 top-level teams divided into two competed in the 21 races held today. The team from the Royal Danish Yacht Club lead the provisional overall ranking, as do the Germans from the Bodensee Yacht Club Ueberlingen in their group. The Monegasques led by Andrea Tortella, Logan Viciana, Luca Frederiksen and Mathieu Choquet are currently lying 6th.
It all begins again for the Optimists tomorrow at 10.00am, with the Melges 20, J/70 and M32 fleets joining the fray at 12 noon for the start of the Monaco Sportsboat Winter Series regatta!


sterling silver jewellery york fashion jewellery Multi-Tone Costume Jewellery: Rose Gold Tone Beaded Toggle Bracelet Sterling silver jewellery range of Fashion and Danon jewellery.
This bracelet has a easy-to-use toggle fastening which allows the bracelet to have an adjustable size. This beaded style has silver, gold and rose gold elements so you'll be able to pair this bracelet with nearly anything!


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redpajama_set_name: RedPajamaGithub