Just a few months are left for IBPS RRBs Mains It is time to pace up your preparation of Banking Awareness for IBPS RRBs PO and Clerk Mains. These Banking questions will also help you in preparing for other upcoming banking recruitment examination.
Sol. Public Sector Banks (PSBs) are banks where a majority stake (i.e. more than 50%) is held by a government. The shares of these banks are listed on stock exchanges.
Q2. Which of the following works is/are done by Credit Information Companies?
Sol. Credit Information Companies collects and maintains records of an individual's payments pertaining to loans and credit cards. These records are submitted to Credit Information Companies by banks and other lenders, on a monthly basis. This information is then used to create Credit Information Reports (CIR) and credit scores which are provided to lenders in order to help evaluate and approve loan applications.
Q3. By which rate is the domestic current rate of currency converted into foreign currency?
Sol. An exchange rate- Between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country's currency in terms of another currency.
(b) Gold (c) SDRs (d) Reserve Tranche Position in the IMF.
Q5. Fiscal policy is concerned with which of the following?
Sol. Fiscal policy is the policy relating to government revenues from taxes and expenditure on various projects. Monetary Policy, on the other hand, is mainly concerned with the flow of money in the economy.
Q6. In which of the following types of banking, there is a direct execution of the transaction between a bank and its consumers?
Sol. Retail banking also known as Consumer Banking is the provision of services by a bank to individual consumers, rather than to companies, corporations or other banks. Services offered include savings and transactional accounts, mortgages, personal loans, debit cards, and credit cards.
Sol. Loans/ advances granted to individuals against the security of shares, debentures and PSU bonds should not exceed Rs.10 lakh and Rs.20 lakh, if the securities are held in physical form and dematerialized form respectively.
Sol. Monetary Policy is the process by which monetary authority of a country, generally a central bank controls the supply of money in the economy by exercising its control over interest rates in order to maintain price stability and achieve high economic growth. In India, the central monetary authority is the Reserve Bank of India (RBI). is so designed as to maintain the price stability in the economy.
Sol. ECGC Limited (Formerly Export Credit Guarantee Corporation of India Ltd) is a company wholly owned by the Government of India based in Mumbai, Maharashtra. It provides export credit insurance support to Indian exporters and is controlled by the Ministry of Commerce.
Q10. Who amongst the following is the regulator in the financial system of the country?
Sol. The financial system in India is regulated by independent regulators in the field of banking, insurance, capital market, commodities market, and pension funds. Example of Financial Regulators: RBI, IRDAI, SEBI, PFRDA.
Q11. Which of the following is the negotiable instrument?
Sol. A demand draft is a negotiable instrument similar to a bill of exchange. A bank issues a demand draft to a client (drawer), directing another bank (drawee) or one of its own branches to pay a certain sum to the specified party (payee).
Q12. Bank Holidays are covered by which of the following?
Sol. Bank Holidays are declared by Central/State Governments/Union Territory under the Negotiable Instruments (NI) Act, 1881.
Sol. A Non-performing asset (NPA) refers to a classification for loans on the books of financial institutions that are in default or are in arrears on scheduled payments of principal or interest. In most cases, debt is classified as nonperforming when loan payments have not been made for a period of 90 days.
Sol. Alterations, if any, on cheques are required to be authenticated by the drawer's signature against each such alteration.
Q15. What is the purpose of KYC in banking?
Sol. Know your customer (KYC) is the process of a business verifying the identity of its clients. Know your customer policies are becoming much more important globally to prevent identity theft, financial fraud, money laundering and terrorist financing.


text:
clap back
Microsoft says Slack 'suffered' from 'absence of video-conferencing' in response to anticompetitive allegations
Microsoft says people embraced Teams during the current pandemic while Slack "suffered."
Sean Endicott
Microsoft responded to Slack's allegations of anticompetitive practices.
Slack accused Microsoft of abusing its market dominance to eliminate competition for Microsoft Teams.
A Microsoft spokesperson said that Slack "suffered" due to its lack of video conferencing.
Microsoft responded to Slack's accusation of anticompetitive practices. Earlier today, Slack filed a complaint to the EU which claimed Microsoft abused its market dominance to eliminate competition for Microsoft Teams. Now, a Microsoft spokesperson responded to the accusations.
Here's the complete statement from the Microsoft spokesperson with emphasis added:
We created Teams to combine the ability to collaborate with the ability to connect via video, because that's what people want. With COVID-19, the market has embraced Teams in record numbers while Slack suffered from its absence of video-conferencing. We're committed to offering customers not only the best of new innovation, but a wide variety of choice in how they purchase and use the product.
We look forward to providing additional information to the European Commission and answering any questions they may have.
The statement is more direct than many responses shared from Microsoft spokespersons. We'll continue to update our coverage of these accusations from Slack and Microsoft's response as more details come out.
Much work to be done
The Anacrusis is an awkward adventure that needs more time in the oven
The Anacrusis is an Xbox console exclusive title joining Xbox Game Pass in Game Preview. This interesting four-player co-op game has an appealing premise and a fair amount of promise, but currently fails to deliver an enjoyable experience in several ways.
A BIG step forward
Everything you need to know about AMD Radeon RX 6000 GPUs
AMD has unveiled its new lineup of Radeon RX 6000 Series graphics cards, with three new pieces of hardware to look forward to. Here's what you need to know.

redpajama_set_name: RedPajamaCommonCrawl


text:
China's Final Module Docks With Space Station
BEIJING (AP) — China's third and final module docked with its permanent space station Tuesday to further a more than decade-long effort to maintain a constant crewed presence in orbit, as its competition with the U.S. grows increasingly fierce.
The Mengtian module arrived at the Tiangong station early Tuesday morning, state broadcaster CCTV said, citing the China Manned Space agency.
Mengtian was blasted into space on Monday afternoon from the Wenchang Satellite Launch Center on the southern island province of Hainan. It was expected to take about 13 hours to complete the flight and docking mission.
A large crowd of amateur photographers, space enthusiasts and others watched the lift-off from an adjoining beach.
Many waved Chinese flags and wore T-shirts emblazoned with the characters for China, reflecting the deep national pride invested in the space program and the technological progress it represents.
"The space program is a symbol of a major country and a boost to the modernization of China's national defense," said Ni Lexiong, a professor at Shanghai University of Political Science and Law, underscoring the program's close military links.
"It is also a boost to the confidence of the Chinese people, igniting patriotism and positive energy," Ni said.
Mengtian, or "Celestial Dream," joins Wentian as the second laboratory module for the station, collectively known as Tiangong, or "Celestial Palace." Both are connected to the Tianhe core module where the crew lives and works.
Like its predecessors, Mengtian was launched aboard a Long March-5B carrier rocket, a member of China's most powerful family of launch vehicles.
Tiangong is currently populated by a crew of two male and one female astronauts, according to the China Manned Space Agency.
Chen Dong, Cai Xuzhe and Liu Yang arrived in early June for a six-month stay on board, during which they will complete the station's assembly, conduct space walks and carry out additional experiments.
Following Mengtian's arrival, an additional uncrewed Tianzhou cargo craft is due to dock with the station next month, with another crewed mission scheduled for December, at which time crews may overlap as Tiangong has sufficient room to accommodate six astronauts.
Mengtian weighs in at about 23 tons, is 17.9 meters (58.7 feet) long and has a diameter of 4.2 meters (13.8 feet). It will provide space for science experiments in zero gravity, an airlock for exposure to the vacuum of space, and a small robotic arm to support extravehicular payloads.
China's increasing space capabilities also featured in the latest Pentagon defense strategy released Thursday.
"In addition to expanding its conventional forces, the PLA is rapidly advancing and integrating its space, counterspace, cyber, electronic, and informational warfare capabilities to support its holistic approach to joint warfare," the strategy said.


ROC NATION Sports is pleased to announce that Two-Time World Champion and top-rated pound-for-pound fighter Andre Ward (28-0, 15 KOs) will return to the ring on Saturday, Nov. 21 in the co- feature to the Miguel Cotto vs. Canelo Alvarez middleweight world championship showdown that will be produced and distributed live by HBO Pay-Per-View. Ward has signed an exclusive multi-fight agreement with HBO that will commence with Ward's Nov. 21 fight at The Mandalay Bay Events Center in Las Vegas. Having cleaned out the super middleweight division, Ward, 31, will be making his debut in the 175-pound light heavyweight division against an opponent to be announced.
Known for his strong character and integrity outside the ring and his warrior's instinct inside it, Ward's skill and talent were apparent early in his outstanding amateur career. He racked up every title in the books, culminating with a gold medal in the light heavyweight division at the 2004 Olympic Games in Athens. As the only male American boxer to claim Olympic gold since 1996, Ward joined the likes of Muhammad Ali, Sugar Ray Leonard and Oscar De La Hoya. He turned professional on December 18, 2004, scoring a second round technical knockout victory over Chris Molina at Staples Center in a fight that was televised live on HBO. He has gone on to rack up 27 more victories since then, building an ever-growing legion of fans in the process. After becoming the Ring Magazine and WBA Super Middleweight World Champion, rising to the number two spot on the pound-for-pound list and winning the 2011 Fighter of the Year Award (ESPN, Sports Illustrated, Ring Magazine and the Boxing Writers Association of America), it was announced that Ward signed an exclusive promotional agreement with Roc Nation Sports in January 2015, opening a new chapter in his storied boxing career. The Bay Area product returned to the ring on June 20, 2015 at the Oracle Arena in front of his hometown fans in Oakland, California and scored a ninth-round knockout over Paul Smith, continuing his unbeaten streak which dates back to when he was a 13 year old amateur. Ward recently stepped in front of the camera for a completely different role, a part in the upcoming New Line/Metro-Goldwyn-Mayer/Warner Bros. feature "Creed," hitting theaters on November 25, a few days following fight week. Fellow Bay Area natives, director Ryan Coogler and the film's star Michael B. Jordan, reached out to Ward so that he could provide his expertise for the film. In addition to working in front of the camera with Ward, Jordan spent time at Ward's home gym in preparation for his role as Apollo Creed's son.


text:
Q: python pypdf2 pdfreader pdfwriter close file after write I am trying to open the file using pypdf2 pdfreader and pdfwriter and finally close the file after processing it. As I want to move that file to another folder.
Following is my code
import sys
import os
import shutil

from PyPDF2 import PdfReader, PdfWriter

pdf_path = "12-1805-0001842092-00.pdf"
reader = PdfReader(pdf_path)
writer = PdfWriter()
for i in range(0, len(reader.pages) - 1, 2):
    lhs = reader.pages[i]
    rhs = reader.pages[i + 1]
    lhs.mergeTranslatedPage(rhs, float(lhs.mediabox.right), 0, True)
    writer.add_page(lhs)
    sys.stdout.flush()

with open(pdf_path, "wb") as fp:
    fl = writer.write(fp)

fp.close()

if os.path.exists("1805") == False:
    os.mkdir("1805")
shutil.move(pdf_path, "prodcode\\" + pdf_path)

I am getting following error
Traceback (most recent call last):
  File "C:\Program Files\Python38\lib\tkinter\__init__.py", line 1892, in __call__
    return self.func(*args)
  File "D:/PycharmProjects/pyGUI/main.py", line 30, in maxim
    maximus()
  File "D:\PycharmProjects\pyGUI\maximus.py", line 601, in maximus
    genfpgpdf()
  File "D:\PycharmProjects\pyGUI\maximus.py", line 475, in genfpgpdf
    os.remove(pdf_path)
PermissionError: [WinError 32] The process cannot access the file because it is being used by another process: '12-1805-0001842092-00.pdf'

redpajama_set_name: RedPajamaStackExchange


I was born into a family who not only taught the Bible, but set a strong Christian example in their lives. So as a child on our dry land farm in West Texas, through God's grace and mercy, His Holy Spirit began to draw me to Himself, and I accepted Him as my Savior and Lord. That was many years ago now, and I can say with the song writer, "the longer I serve Him, the sweeter He grows."
Serving the Lord, has also meant serving on the staff of FBC Clyde for many years. The members of this church have certainly become my "church family." This church family has had its ups and downs over the years. There have been many mountaintop experiences in our church's life, but also times when we have been heart broken by tragedy. However, through it all, God has been faithful to give us strength when we were weak, courage when we were afraid, and comfort when our hearts were broken. This is a wonderful church family.
My hope and prayer for this church is that it will continue to be a loving church family that proclaim God's grace, mercy, and love within the church walls, to the community of Clyde, and throughout the world around us until Christ returns.


Established in the heart of the Italian Alps, 1000 meters above sea level, Loacker is now one of the most popular brands in Italy. What makes Loacker Quadratini so famous? The secret is in the ingredients. Loacker only select high quality ingredients: finest Italian hazelnuts, freshly roasted by Loacker, real "Bourbon" vanilla pods, luxurious aromatic cocoa, the finest quality milk from the Alps and fresh mountain water. Result? Irresistibly delicious crispy wafers! Enjoy this dark chocolate made by the famous Loacker!
Facts 8.8 oz. Product of Austria.
Ingredients Unbleached wheat flour, coconut oil, glucose syrup, sugar, cocoa processed with alkali, dextrose, bittersweet chocolate (sugar, chocolate liquor), soy flour, whey powder (milk), skimmed milk powder, barley malt extract, leavening agents (sodium hydrogen carbonate, disodium diphosphate), hazelnuts, emulsifier soy lecithin, salt, natural vanilla pods, spices. Contains hazelnuts, milk, soy and wheat (gluten). May contain almonds. No colourings and no preservatives.


Tune in tomorrow for an exclusive story! Insider reports indicate that infamous speeder, and all around nutcase, MaximaSpeed may be acquiring a new "set of wheels". What are these wheels and will anything be attached to them? Find out all these details and more tomorrow after noon EST. Only here on Faux News.


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Trump hosts Hanukkah party at White House
December 8, 2017 admin Politics 0
The United States President Donald Trump hosted a Hanukkah celebration at the White House where he talked about his decision to recognise Jerusalem as the capital of Israel. He also announced that he will be moving the U.S. embassy from Tel Aviv to Jerusalem.
New THQ Nordic horror game 'Fade to Silence' hits Steam ahead of The Game Awards
White House on Jerusalem move: 'A very courageous and bold action'


Two faculty members from the Texas A&M University Health Science Center received 2018 Regents Professor Awards: Thomas J. McDonald, PhD, professor in the School of Public Health, and Jian Q. Feng, PhD, professor at the College of Dentistry.
Established in 1996, this award from the Texas A&M University System Board of Regents recognizes faculty members who have provided stellar contributions to their university or agency and to the people of Texas. Recipients have the title added to their current designation or rank. To date, 239 Texas A&M system faculty members have been recognized with the Regents Professor Award.
McDonald earned his doctorate in oceanography from Texas A&M. He studies environmental chemistry, petroleum geochemistry and organic chemistry in relation to public health. He has served as head of the Environmental and Occupational Health Department and as interdisciplinary faculty of toxicology. McDonald is currently leading the Research Translation Core of the Texas A&M Superfund Research Center composed of scientists from across campus conducting four environmental research projects funded by a $10 million grant from the National Institute of Environmental Health Sciences.
Feng earned his doctorate in physiology from the University of Connecticut. He teaches physiology to dental and graduate students and researches craniofacial, tooth and bone developmental biology. His work has challenged the dogma of bone formation and improved understanding of the transformation of cartilage into bone.
The selection process for the awards begins with a call for nominations from the chancellor. Final nominations are put forth to the chief executive officer of each respective entity. They are then subject to a system-level review consisting of academic vice chancellors and past recipients of the awards. Finally, nominations are forwarded to the chancellor and the board for final approval.


Azteca Mexican Restaurant is a top merchant due to its average rating of 4.5 stars or higher based on a minimum of 400 ratings.
Promotional value expires 120 days after purchase. Amount paid never expires. Not valid for Happy Hour, or other specials and discounts. Must have two or more people to use $20 voucher; must have four or more people to use $40 voucher. Limit 1 per person, may buy 1 additional as gift(s). May be repurchased every 180 days. Limit 1 per visit. Limit 1 per table. Valid only for option purchased. Merchant is solely responsible to purchasers for the care and quality of the advertised goods and services.
It's not the bright orange walls, the colorful tiles, or the murals of rural Mexico. More than anything, it's the food that makes Azteca Mexican Restaurant such a faithful representation of Mexico and its traditions. It all starts with the first bite of chips and salsa—or maybe cheese dip, if you're so inclined. From there, you'll likely have a hard time choosing from a menu that features steak tacos, fajitas, and quesadillas stuffed to the brim with delicious ingredients. Whatever you decide on, pair it with a strawberry margarita for a true taste of Mexico.


If you still need to convince yourself to buy the upcoming Halo 4 game then you'd probably want to watch the trailer after the break. The Halo 4 UNSC Weapons Trailer will give you a quick rundown of some of the weaponry you can find in the game which you can use to frag other players. I'm sure most of you out there have already decided to buy the game so if you're trying to convince yourself not to buy Halo 4 then it might be a good time to click on another post around here.


text:
Q: Paython script not finding files I have following script which is supposed to list all picture etc from my disk drives, but it failes to find any file. Any Idea what I may be doing wrong. I am doing in PyCharm environment:
    files = []
    for root, dirs, files in os.walk("."):
        for file in files:
            if file.endswith((".jpg", ".jpeg", ".mp3", ".mp4")):
                files.append(os.path.join(root, file))

    # Sort the files by their current folder name
    files.sort(key=lambda x: os.path.dirname(x))

    # Display a list of the files
    for file in files:
        print(file)

A: Your for-loop will never end as you add the found file name into files which you are looping through. Use another variable to store the list.
This works fine:
import os

results = []
for root, dirs, files in os.walk("."):
    print(f"Got: {len(results)} files, current dir: {root}")
    for file in files:
        if file.endswith((".jpg", ".jpeg", ".mp3", ".mp4")):
            results.append(os.path.join(root, file))

# Sort the files by their current folder name
results.sort(key=lambda x: os.path.dirname(x))

# Display a list of the files
for file in results:
    print(file)


Lazy morning at the cottage, and I'm just planning out a fantastic photo shoot I'm doing on Tuesday afternoon with a great group of girls and and spectacular location here in Southampton.
Well the last few days I spent time wondering the streets of Southampton photographing spasms of colour. Tonight after the strong winds blew all our kayaks and boards all over, and my son and I had to collect them I got inspired by the colours "in my own backyard:)"
Sunset cruise through the bay with his pup ... Easton somehow comes alive at dusk and always wants to go out and kayak, and there is no leaving the dog behind !!
There's nothing like an evening kayak when the waves are rolling and you can watch a storm in the background. Sunlight peaking out of the clouds to the south, and rain directly west. The marvels of mother nature at work !!
Going for a walk along the coast this morning, always such a beautiful walk surrounded by Nature ... Blue sky, crystal clear water, green trees and grass .... sun shining.
This morning I couldn't help but be captivated by the bright red tug on the river. It must be freshly painted, and surrounded by such beautiful, earthy colours, it just POPPED.
Nothing like a POP of colour to inspire a photo shoot .. gonna take my camera out this afternoon and look for more wonderful images !!
Since I had to be in the city for a week .. at least I got to enjoy some fun with my son and his team. Roller Hockey is a very civilized and relaxed version of hockey ... Warm rinks, shorts and t-shirts, and lots of fun!! No crazy parents screaming to pass the puck, no crazy coaches telling the kids to get onside .. just 4 on 4 high speed hockey.
We had the privilege to play on the big ice at Hershey Centre where the pro's usually play ... 3 wins so far .. and lots of opportunity to take some fun and awesome 'soRoller' pictures !!!
After spending the last 5 days back in the city for work I now look back on this photo and remember why I enjoy the cottage so much during the summer months. Cool breezes, great sunsets, great friends, lots of water and always something to photograph !!
Nothing seeing your BFF after a week away. Nice casual walk and hang in Port Credit for an afternoon before going off to camp again.
soLifestyle image at it's finest !!
We are very excited to be launching our new services guide of all offering, samples of work and pricing for the studio.


Home » DIY & Hacks » DIY » DIY Sidewalk Chalk – only three ingredients needed!
DIY sidewalk chalk it is cheap to make and you can make any shape you want! You only need three ingredients and a mold and you are on your way to non-stop summer fun!
Sure one could just go buy chalk at a dollar store. But this way, one can get the kids to help make it and give them a project to do as well! My grand babies would love making their own sidewalk chalk! Besides a great way to spend time with the kids without boring them or parking them in front of a TV or some kind of game system!!
Hey Gabby – I haven't had that problem. I actually left it in the mold for a week once and it didn't eat through.
OK awesome I have new Lego brick silicone molds I just didn't want to ruin them. Thank you and I love your posts!
Will food coloring work or does it need to be the paint?
Has anyone used this as sidewalk chalk paint? I know you'd need to use as you go before it sets up.
Could I possibly get a recipe with ingredients and directions please?
This looks like so much fun to make and give! My grandbabies love to write and draw with chalk on the sidewalk. I shared this craft on my Facebook craft page with a link back to this page.
How much chalk does this recipe end up making?


Larry C. Liston, 73, of Ashland died Saturday, August 16, 2008 at Hospice House. On March 19,1935, he was born to the late Eugene and Mary Ellen (Green) Liston and has been a resident of Ashland County all his life. He was employed at Abbott Labs as a journeyman machinist and retiring in 1998. He served in the Army as a sharpshooter and was a member of VFW Post 1067, Eagles 2178 and Am Vets Post 96.
Larry is survived by three daughters and a son-in-law Peggy Dillon of Garland, Texas, Robin and Billy Bauchman of Ashland and Tuesday Bodiford of Ashland, two sons and a daughter-in-law Brian and Nichole Liston of Marion, Ohio and Douglas Liston of Dallas, Texas, 14 grandchildren and 11 great-grandchildren. He is also survived by sisters Sharon Frary, Peggy (Ben) Kelly, Vikki (Gary) Richard and Deborah (Mike) Fite and brothers Ricky (Bear) Liston, James E."Gene" Rufener and Denny Rufener and several cousins, nieces and nephews. He was preceded by his parents, an infant son Larry and an infant sister Sandy Sue Liston.
Services will be held at Heyl Funeral Home, 227 Broad Street, Ashland on Wednesday, August 20, 2008 at 11:00 am with Larry's son Brian and Pastor Doug Ford officiating followed by military services by The Ashland Veterans Honor Guard. Burial will be at a later date. Friends may visit with the family Tuesday, August 19, 2008 from 6 to 8 pm at the funeral home. The family suggests contributions be made to Hospice of North Central Ohio, 1050 Dauch Drive, Ashland, Ohio.


It's been 3 years since the Google Nexus Player went on sale, but it's still the only first-party Android TV box sold by Google (even if it was actually manufactured by Asus). While there are plenty of newer, more powerful Android TV devices on the market today, Google has continued to release software updates for the Nexus Player, bringing new features and security patches.
The latest update was ostensibly just a November security patch. But Android Police noticed it includes a bonus feature: Google Assistant support.
Google recently added Assistant support to its Android TV software and you can already use the company's voice assistant on the NVIDIA Shield or some Sony TVs with Android TV software.
Now you can also use the included voice remote control or the Android TV Remote app on your phone to speak to the Nexus Player. Among other things, that means you can search for videos or launch apps by voice, ask for weather or news updates, or generally use the Nexus Player like a slightly less convenient Google Home smart speaker (less convenient because your TV needs to be on and you need to speak into a handheld microphone).
Google doesn't actually sell the Nexus Player anymore, and if you're in the market for a new Android TV box you're probably better off spending $69 on a Mi Box or $179 on an NVIDIA Shield anyway. But it's nice to know that there's one less reason for folks who already have a Nexus Player to upgrade.
Likely won't use it. But unfortunately, it came with a side effect. There is now a 3 – 5 second delay between a button press on the remote and any action taking place. It used to be nearly instantaneous.


text:
The Power of Power Naps
The last couple of weeks have been an adventure of a small kind, but good preparation for big one later this year. A short trip around the UK covering about one tenth of the total distance to Mongolia came with a few salutary lessons. In a former life your correspondent sold medical equipment in England for a living. Long before carbon footprinting was even a concept he did his best to cover as much of the country each week as possible. Being a young buck with a Marks & Spencer suit, a Samsonite briefcase and little experience, the old-timers in the company gave him some good advice. "Always fill your car up the night before so you're not faffing around in the morning when you run into a traffic delay," they said. "Double check your appointments for the week and reschedule ahead of time. Make sure that you get enough sleep and pull over if you're tired." This last piece of advice saved my life on more than one occasion. Thrashing a Vauxhall Cavalier up and down the motorways became an almost pleasing routine back in 1984. In an era before cell phones a busy day was bookended by time for reflection as the miles whizzed by on my way home.
Today things have changed a bit. Speed cameras have taken the place of jam sandwiches (police cars with their distinctive horizontal stripe) as the primary means of controlling the 'ton-up boys.' Coffee is now drinkable at the rest stops and the web available 24x7, wherever you are. Yet, there's nothing quite like that 20 minute power nap in the middle of the afternoon. It's so much better than driving off the road at 70 miles an hour.
The Adventurists Film Festival
A triffic time had by all at the Adventurists Film Festival today. By extending a personal trip to London by a couple of days, the chance to take in the hilarity and real art of expedition film making was unmissable. All the films were riveting in their own way and the judges had clearly done their homework. One particular standout in the Public category (i.e. not from an Adventurists adventure) was, Across Europe in a Paper Boat, directed by Julius Markevicius from Lithuania. Using 6mm plywood, Julius and a couple of friends built a small boat in their apartment and glued it together using epoxy and fiberglass. Surprisingly seaworthy, they powered her with a four horse outboard and proceeded to navigate her, almost non-stop, from Belarus to Rotterdam via the lakes, rivers, and canals of Europe. Julius' commentary was dead-pan hilarious. You can see the all the films at The Adventurists Film Festival web site.
Ed Stafford, the bloke that walked the entire length of the Amazon in two and half years, provided us insight into what it takes to meet a modern challenge including determination and a good partner. His partner, Cho, joined him at a particularly dangerous point as Ed made his way through indigenous Peru, then stayed with him until they reached the Atlantic ocean.
In addition, there were a couple of educational sessions centered around making your own documentary. Perhaps the two most useful pieces of advice were 1) ensure that you get good sound and 2) communicate with your audience. From a technical standpoint, winners used everything from $100 Flip video cameras to sophisticated digital SLR's. It appears the success of a movie like this lies as much in the editing as anything else.
With so many Adventurists on hand, there was expert advice available. Turns out that not only is the Nissan Micra the most popular car for starting the rally, it's also the one that crosses the finish line more than any other - largely due to their popularity in Eastern Asia and the availability of spare parts. The roads in Kazakhstan are vastly variable, the ones in Russia generally predictable. We shouldn't miss the Altai region or attempt the northern route across Mongolia if we want a prayer of doing this in less than eight weeks. Topped off with Hendrick's Gin and a set of three official car stickers it was a splendid day. http://filmfestival.theadventurists.com/


text:
Klix.ba: "Hadžović o članku Foreign Affairsa: Tekst je površan, EU i SAD jasno podržavaju cjelovitost BiH"
NewsPublic Events
"European Security and Defence Policy and Bosnia and Herzegovina" Mostar, 22nd January 2008
Centre for Security Studies, with support from the Government of the Republic of Hungary organised a third seminar on the topic of: "European Security and Defence Policy (ESDP) and Bosnia and Herzegovina", on 22nd of January, in Hotel Ero, Mostar.
The overall objective of the seminar was to provide an overview of the EU Common Foreign and Security Policy, with a particular focus on the aspects of security and defence. Specific aim was to disseminate thorough information concerning the structures which influence the creation of the EU policies. At the same time we recognised a need for Bosnia and Herzegovina to be fully aware of EU external policy developments and its impact on the process of European integration. The ESDP elements play an important role in Bosnia and Herzegovina, bearing in mind their active role in BiH, through the work of the Office of the High Representative, EUFOR and the EUPM.
Introductory remarks at the seminar were provided by Mr. Miomir Brajic from the Ministry of Foreign Affairs of Bosnia and Herzegovina, Mr. Safet Softic Speaker of the House of Representatives of the Parliament of the Federation of Bosnia and Herzegovina and H.E. Imre Varga, Ambassador of the Republic of Hungary in Bosnia and Herzegovina. The topics of the seminar were further discussed by the representatives from the Royal Institute for International Relations (Brussels) and the German Institute for International and Security Affairs (Berlin).
The seminar was attended by the representatives of the legislative and executive institutions, the international community and NGOs, scholars, students and media in Bosnia and Herzegovina.
DOWNLOAD: LIST OF PARTICIPANTS, AGENDA,
Treći sastanak Radne grupe za razvoj pravosudnog priručnika o posebnim istražnim radnjama / Third meeting of the Working Group on the Development of Judicial Benchbook on Special Investigation Measures
Drugi sastanak radne grupe za razvoj pravosudnog priručnika o posebnim istražnim radnjama / Second meeting of the Working Group for the development of the Judicial Benchbook on Special Investigative Measures


text:
<?php

namespace Oryzone\Bundle\OauthBundle\Authorization\Error;

use OAuth\Common\Http\Exception\TokenResponseException;

/**
 * Class TokenExceptionError
 * @package Oryzone\Bundle\OauthBundle\Authorization\Error
 */
class TokenExceptionError extends AbstractError
{
    /**
     * @var TokenResponseException $exception
     */
    protected $exception;

    /**
     * Constructor
     *
     * @param string                 $provider
     * @param string                 $targetPath
     * @param TokenResponseException $exception
     */
    public function __construct($provider, $targetPath, TokenResponseException $exception)
    {
        $this->exception = $exception;
        $description = 'Cannot retrieve or parse the auth token';
        parent::__construct($provider, $description, $targetPath);
    }

    /**
     * Get the related exception
     *
     * @return TokenResponseException
     */
    public function getException()
    {
        return $this->exception;
    }

    /**
     * {@inheritDoc}
     */
    public function getType()
    {
        return ErrorInterface::TYPE_TOKEN_EXCEPTION;
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}

redpajama_set_name: RedPajamaGithub


Hello, Poppytalk readers! I'm back with another Dispatch from Dublin - this time, a visit to the Dublin Flea Market and the Jam Art Factory.
On the last Sunday of every month, you can find the Dublin Flea Market Newmarket Square in Dublin. Filled with vintage clothes, bric-a-brac, and even a seamstress who's on hand to make alterations, Dublin Flea Market is held in an indoor and outdoor venue - perfect for the unpredictable Irish weather. Each month, the organizers commission a local artist to create the Flea Market's poster - and each of the previous month's posters can be bought at the poster stall. My favorites are the Christmas market posters.
The Dublin Flea Market also has great vegetarian lunch options - I'd recommend the falafel wrap that's packed with veggies. And you might even be in for a surprise treat - musicians often serenade flea market visitors with impromptu sets.
If you don't find what you're looking for at the Flea Market, head around the corner to the Jam Art Factory, a great little store with a collection of work from up and coming Irish artists. The Jam Art Factory is just down the hill from Christ Church Cathedral, and on Saturday afternoons, they invite up and coming Irish bands to play in the store.
One last tip? When you're through shopping, stop into the Bull and Castle pub just up the hill for one of their home-brewed beers - it's the perfect way to top off a day of shopping!
Thanks, ladies! It's a special place, for sure!
I was in Dublin for Christmas 2009 and have my Dublin Flea poster framed on the wall to remind me of that cold icy day (yes, i slipped on the cobblestones!). We had wonderful spicy warm chai and chocolate vegan brownies from the little cafe while I tried to work out how to bring home all the amazing vintage crockery to Australia! Can't wait to go back there again!
I have the Christmas '09 poster in my office still waiting to be framed... its so lovely and a great reminder of wonderful day we had there while on our visit home to my boyfriends family. I can't wait to make it back to the Flea Market one day!


text:
\section{Introduction}

Sobolev norms and semi-norms play a central role in the numerical analysis
of discretization methods for partial differential equations. For instance,
standard finite element error analysis is essentially a combination of
the Bramble-Hilbert lemma and transformation properties of Sobolev (semi-) norms.
These properties are also central to the area of preconditioners for (and
based on) variational methods. More precisely, arguments based on finite dimensions
of local spaces are inherently connected with scaling arguments to keep dimensions
bounded. Norms are usually not scalable. That is, the corresponding equivalence numbers
behave differently with respect to a scaling parameter like the diameter $D_{{\cal O}}$
of the domain ${\cal O}$ when the domain under consideration is transformed by an affine map
that maintains shape regularity (i.e., the ratio of $D_{{\cal O}}$ and the ``inner diameter''
of ${\cal O}$ is bounded). This can be usually fixed only when
essential boundary conditions are present. An example is using the $H^1$-semi-norm
as norm in $H^1_0$. More generally, semi-norms have better scaling properties: usually
they can be defined so that equivalence numbers are of the same order with
respect to $D_{{\cal O}}$ under shape-regular affine transformations of the domain.

Whereas properties of Sobolev (semi-) norms under smooth transformations or
simple scalings are straightforward as long as their orders are integer, things
are getting more complicated for fractional-order Sobolev norms. Such norms appear,
e.g., in a natural way when considering boundary integral equations of the
first kind \cite{McLean_00_SES,HsiaoW_08_BIE}
or when studying the regularity of elliptic problems in non-convex
polygonal domains \cite{Grisvard_85_EPN}.
There are different
ways to define fractional-order Sobolev norms and they all have advantages and
disadvantages (standard references are \cite{LionsMagenes,Adams}).
Different norm variants are known to be equivalent. But dependence
of the equivalence constants on the order and the domain are more involved.

In this paper we analyze the equivalence of different variants of fractional-order
semi-norms of positive orders bounded by one. The use of semi-norms is essential
to guarantee scaling properties and we don't know of any publication that analyzes
their equivalence.

The rest of the paper is organized as follows. In Section~\ref{sec_S} we
collect all definitions and technical results.
In Section~\ref{sec_S1} we recall two definitions of norms and
define three different semi-norms: one of the Sobolev-Slobodeckij type, one by
interpolation, and one of a quotient space type. Section~\ref{sec_S3} is devoted
to basic equivalence estimates. In particular, we present Poincar\'e-Friedrichs'
inequalities for the Sobolev-Slobodeckij and interpolation semi-norms
(Propositions~\ref{prop_PF1} and \ref{prop_PF2}).
A direct proof in the case of the Sobolev-Slobodeckij semi-norm
is cited from Faermann \cite{Faermann_02_LAS}. An indirect proof for
the interpolation semi-norm is standard, and is given for completeness.
Affine transformation properties of norms
(also given for completeness) and semi-norms are analyzed in Section~\ref{sec_scaling}.
Eventually, in Section~\ref{sec_main} we combine the intermediate results
to show the equivalence of all the semi-norms under consideration, with
explicit equivalence numbers depending on the domain via its transformation from
a reference domain (Theorems~\ref{thm1}--\ref{thm3}).
In Theorem~\ref{thm_main} we resume the results in a form that is appropriate
for affine maps that maintain shape-regularity of the domain.
In particular, it shows (i) the uniform (with respect to $D_{{\cal O}}$)
equivalence of the Sobolev-Slobodeckij and the interpolation semi-norms,
(ii) that the Sobolev-Slobodeckij and quotient space semi-norms are uniformly
equivalent as long as the diameter of the domain is bounded from
above, and (iii) that the interpolation and quotient space semi-norms
can be uniformly bounded mutually in one direction depending on whether the diameter
of the domain is bounded from above or from below.
Finally, in Corollary~\ref{cor_scale} we collect the scaling properties
of all the norms and semi-norms studied in this paper that have this property.

\section{Sobolev norms} \label{sec_S}

In this section we recall definitions of several Sobolev (semi-) norms and collect
technical results that are needed to prove our main results in Section~\ref{sec_main},
or which are interesting in its own.

Throughout the paper, ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ denotes a generic
bounded connected Lipschitz domain.
We consider the usual $L^2({\cal O})$- and
$H^1({\cal O})$-norms with notations $\|\cdot\|_{0,{\cal O}}$ and
$\|\cdot\|_{1,{\cal O}}$, respectively and the $H^1({\cal O})$-semi-norm
$|\cdot|_{1,{\cal O}}$. Here and in the following, in all types of norms,
the underlying domain of definition ${\cal O}$ will be occasionally dropped from the
notation when not being ambiguous.

\subsection{Fractional-order norms and semi-norms} \label{sec_S1}

There are several ways to define Sobolev norms. We use the ones defined by
a double integral (Sobolev-Slobodeckij) and by interpolation. For the latter we
use the so-called real K-method, cf.~\cite{BerghL_76_IS}.
For $0<s<1$, the interpolation norm in the fractional-order Sobolev space $H^s({\cal O})$
is defined by
\[
   \|v\|_{[L^2({\cal O}),H^1({\cal O})]_s}
   :=
   \|v\|_{L^2({\cal O}),H^1({\cal O}),s}
   :=
   \left( \int_0^\infty
          t^{-2s} \inf_{v=v_0+v_1}
          \Bigl(\|v_0\|_{0,{\cal O}}^2 + t^2\, \|v_1\|_{1,{\cal O}}^2\Bigr)
          \frac{dt}{t}
   \right)^{1/2}.
\]
Here and in the following, the notation
$\inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0,{\cal O}}^2 + t^2\, \|v_1\|_{1,{\cal O}}^2\Bigr)$
implies that the infimum is taken over $v_0\in L^2({\cal O})$ and $v_1\in H^1({\cal O})$,
or corresponding spaces as indicated by the respective norms.

We also define the interpolation space
\[
   \tilde H^s({\cal O}) = \left[ L^2({\cal O}), H^1_0({\cal O}) \right]_s
\]
with corresponding notation for the norm.
The notation $\tilde H^s$ is used by Grisvard and is common in the
boundary element literature, whereas the notation
$H_{00}^s=\tilde H^s$
is used by Lions and Magenes and is common in the finite element literature.

The Sobolev-Slobodeckij variant of these norms is defined (for $0<s<1$) by
\begin{equation} \label{SS-norm}
  \|v\|_{H^s({\cal O})}
  :=
  \|v\|_{s,{\cal O}}
  :=
  \left(\|v\|_{L^2({\cal O})}^2
  +
  \int_{{\cal O}} \int_{{\cal O}} \frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}}\,dx\,dy
  \right)^{1/2},
\end{equation}
\[
  \|v\|_{\tilde H^s({\cal O})}
  :=
  \|v\|_{\sim,s,{\cal O}}
  :=
  \left(
  \|v\|_{H^s({\cal O})}^2
  +
  \|\frac{v(x)}{\mathop{\rm dist}\nolimits(x,\partial {\cal O})^s}\|_{L^2({\cal O})}^2
  \right)^{1/2}
  \qquad\text{(preliminary version)}.
\]
The corresponding semi-norms are
\[
   |v|_{[L^2({\cal O}),H^1({\cal O})]_s}
   :=
   |v|_{L^2({\cal O}),H^1({\cal O}),s}
   :=
   \left( \int_0^\infty t^{-2s}
             \inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0,{\cal O}}^2 + t^2\, |v_1|_{1,{\cal O}}^2\Bigr)
          \frac{dt}{t}
   \right)^{1/2}
\]
and
\[
  |v|_{H^s({\cal O})}
  :=
  |v|_{s,{\cal O}}
  :=
  \left(
  \int_{{\cal O}} \int_{{\cal O}} \frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}}\,dx\,dy
  \right)^{1/2}.
\]
Additionally, it is useful to define the semi-norm of quotient space type
\[
  |v|_{s,{\cal O},\inf} := \|v\|_{H^s({\cal O})/\mbox{\footnotesize\rm I\kern-.18em R}}
                      = \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{s,{\cal O}}.
\]

\subsection{Equivalence of semi-norms on a fixed domain} \label{sec_S3}

The aim of this section is to study equivalences of the semi-norms previously defined,
on a fixed domain. Together with mapping properties (provided in Section~\ref{sec_scaling})
these estimates are needed to prove our main results in Section~\ref{sec_main}.
Proofs are based on a standard norm equivalence and specific Poincar\'e-Friedrichs'
inequalities, which are also recalled here.

It is well known that for Lipschitz domains different definitions of Sobolev norms are
equivalent. However, equivalence constants depend usually on the order and the
domain under consideration. In particular, for a bounded
Lipschitz domain ${\cal O}$, the norms $\|\cdot\|_{s,{\cal O}}$ and
$\|\cdot\|_{L^2({\cal O}),H^1({\cal O}),s}$ are equivalent for $0<s<1$,
cf.~\cite{LionsMagenes,Grisvard_85_EPN,McLean_00_SES}.
Such equivalences are shown by corresponding equivalences
on $\mbox{\rm I\kern-.18em R}^n$ and the use of appropriate extension operators,
cf.~\cite{BrennerS_94_MTF}, see also \cite{DevoreS_93_BSD} for non-Lipschitz
domains. In particular, the norms previously defined are uniformly equivalent
for $s$ in a closed subset of $(0,1)$, see~\cite{Heuer_01_ApS}.

Here, for the norms, we don't elaborate on the dependence of the equivalence
constants on $s$ and ${\cal O}$.
We rather give them specific names to be used in estimates to follow.

\begin{proposition}[equivalence of norms] \label{prop_norms}
For a bounded Lipschitz domain ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ and for given
$s\in(0,1)$ there exist constants $k(s,{\cal O}), K(s,{\cal O})>0$ such that
\[
   k(s,{\cal O})\,\|v\|_{L^2({\cal O}),H^1({\cal O}),s}
   \le \|v\|_{s,{\cal O}}
   \le K(s,{\cal O})\, \|v\|_{L^2({\cal O}),H^1({\cal O}),s}
   \qquad\forall v\in H^s({\cal O}).
\]
\end{proposition}

For a proof see, e.g., \cite{McLean_00_SES}.

It is well known that, on bounded Lipschitz domains, lower-order norms can be bounded
by higher-order semi-norms plus finite rank terms. Such estimates are referred to
as Poincar\'e-Friedrichs' inequalities. For integer-order norms
there are direct proofs with explicit constants (depending on orders and
domains) \cite[Th\'eor\`eme 1.3]{Necas_67_MDT} and attention has received
finding best constants and deriving improved weighted estimates, see, e.g.,
\cite{PayneW_60_OPI,Verfuerth_99_NPA} and \cite{DrelichmanD_08_IPI}, respectively.
We need such a Poincar\'e-Friedrichs' inequality for fractional-order
norms on bounded domains (for unbounded domains, see \cite{MouhotRS_11_FPI}),
and refer to \cite[Lemma 3.4]{Faermann_02_LAS} for a proof.
This proof is given for two dimensions but immediately extends to the general case.

\begin{proposition}[Poincar\'e-Friedrichs inequality, Sobolev-Slobodeckij semi-norm]
\label{prop_PF1}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded domain, and $s\in(0,1)$. Then there holds
\[
   \|v\|_{0,{\cal O}} \le C_{\rm PF,SS}(s,{\cal O}) \Bigl(|v|_{s,{\cal O}} + |\int_{\cal O} v|\Bigr)
   \qquad\forall v\in H^s({\cal O})
\]
with
\[
   C_{\rm PF,SS}(s,{\cal O}) = |{\cal O}|^{-1/2} \max\{1,2^{-1/2} D_{{\cal O}}^{n/2+s}\}.
\]
Here, $|{\cal O}|$ denotes the area of ${\cal O}$ and, as mentioned in the introduction,
$D_{{\cal O}}$ is its diameter.
\end{proposition}

\begin{lemma} \label{la1}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded, connected Lipschitz domain.
Then there holds
\[
   |v|_{s,{\cal O}}^2
   \le |v|_{s,{\cal O},\inf}^2
   =
   |v|_{s,{\cal O}}^2 + \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0,{\cal O}}^2
   \le
   (1+C_{\rm PF,SS}^2) |v|_{s,{\cal O}}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $C_{\rm PF,SS}=C_{\rm PF,SS}(s,{\cal O})$ is the number from Proposition~\ref{prop_PF1}.
\end{lemma}

\begin{proof}
By definition of $|\cdot|_{s,{\cal O}}$ there holds for any $c\in\mbox{\rm I\kern-.18em R}$
and any $v\in H^s({\cal O})$ (we now drop ${\cal O}$ from the notation)
\[
   |v|_{s} = |v+c|_{s}.
\]
Therefore
\[
   |v|_{s} \le \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{s} = |v|_{s,\inf}
\]
which is the first assertion.
By the initial argument and the definition of the Sobolev-Slobodeckij norm
one also finds that
\[
   |v|_{s,\inf}^2
   =
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{s}^2
   = \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0}^2 + |v|_{s}^2.
\]
This is the second assertion.

The last relation and the Poincar\'e-Friedrichs' inequality (Proposition~\ref{prop_PF1})
lead to
\[
   |v|_{s,\inf}^2
   \le
   C_{\rm PF,SS}^2 \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \Bigl( |v|_s + |\int_{\cal O} (v+c)|\Bigr)^2 + |v|_{s}^2
   =
   (1+C_{\rm PF,SS}^2) |v|_{s}^2.
\]
This finishes the proof.
\end{proof}

\begin{lemma} \label{la2}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded Lipschitz domain. There holds
\[
   k^2
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   |v|_{s,{\cal O},\inf}^2
   \le
   3K^2 |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   +
   \frac {K^2}{s(1-s)}
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0,{\cal O}}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $k=k(s,{\cal O})$ and $K=K(s,{\cal O})$ are the numbers from Proposition~\ref{prop_norms}.
\end{lemma}

\begin{proof}
Let $v\in H^s({\cal O})$, and let $c_0, c_1$ denote generic constants.
For any $t>0$ there holds
\begin{align*}
   \inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)
   &=
   \inf_{v=v_0+c_0+v_1+c_1} \Bigl(\|v_0+c_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)\\
   &=
   \inf_{c_1, v-c_1=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr),
\end{align*}
that is
\begin{align*}
   \inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)
   &=
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \inf_{v+c=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)\\
   &\le
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \inf_{v+c=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 \|v_1\|_{1}^2\Bigr).
\end{align*}
(Recall that our convention for the notation
$\inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)$
implies that the infimum is taken with respect to $v_0\in L^2({\cal O})$ and $v_1\in H^1({\cal O})$.)
We conclude that
\begin{align*}
   |v|_{L^2,H^1,s}^2
   &=
   \int_0^\infty t^{-2s}
       \inf_{v=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2\, |v_1|_{1}^2\Bigr)
   \frac{dt}{t}
   \\
   &\le
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}}
   \int_0^\infty t^{-2s}
       \inf_{v+c=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2\, \|v_1\|_{1}^2\Bigr)
   \frac{dt}{t}
   =
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{L^2,H^1,s}^2.
\end{align*}
By Proposition~\ref{prop_norms}
\[
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{L^2,H^1,s}^2
   \le
   k^{-2} \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{s}^2
   =
   k^{-2} |v|_{s,\inf}^2,
\]
so that the first assertion follows.

By definition and using Proposition~\ref{prop_norms} there holds
\begin{align} \label{1}
   |v|_{s,\inf}^2
   &=
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{s}^2
   \le
   K^2 \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{L^2,H^1,s}^2
   \nonumber\\
   &=
   K^2
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}}
   \int_0^\infty t^{-2s}
      \inf_{v+c=v_0+v_1} \Bigl(\|v_0\|_{0}^2 + t^2 \|v_1\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)
   \frac{dt}{t}.
\end{align}
We bound the integrand separately for $t<1$ and $t\ge 1$.

For $t<1$ we use the representation $v+c=v_0+v_1$ to bound
\begin{align*}
   \|v_0\|_{0}^2 + t^2 \|v_1\|_{0}^2 + t^2 |v_1|_{1}^2
   &\le
   \|v_0\|_{0}^2 + 2 t^2\bigl( \|v+c\|_{0}^2 + \|v_0\|_{0}^2\bigr)
                     + t^2 |v_1|_{1}^2
   \\
   &\le
   3 \|v_0\|_{0}^2 + 2 t^2 \|v+c\|_{0}^2 + t^2 |v_1|_{1}^2.
\end{align*}
If $t\ge 1$ then we select $v_0:=v+c$ to conclude that
\[
   \inf_{v+c=v_0+v_1}
   \Bigl(\|v_0\|_{0}^2 + t^2 \|v_1\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)
   \le
   \|v+c\|_{0}^2.
\]
Together this yields
\begin{align} \label{pf_la2_1}
   &\int_0^\infty t^{-2s}
       \inf_{v+c=v_0+v_1}
       \Bigl(\|v_0\|_{0}^2 + t^2 \|v_1\|_{0}^2 + t^2 |v_1|_{1}^2\Bigr)
    \frac{dt}{t}
   \nonumber\\
   \le
   &\int_0^1 t^{-2s}
       \inf_{v+c=v_0+v_1}
       \Bigl( 3 \|v_0\|_{0}^2 + 2 t^2 \|v+c\|_{0}^2 + t^2 |v_1|_{1}^2 \Bigr)
    \frac{dt}{t}
   +
   \int_1^\infty t^{-2s} \|v+c\|_{0}^2 \frac{dt}{t}
   \nonumber\\
   =
   &\int_0^1
   t^{-2s} \inf_{v+c=v_0+v_1}
   \Bigl( 3 \|v_0\|_{0}^2 + t^2 |v_1|_{1}^2 \Bigr) \frac{dt}{t}
   +
   \|v+c\|_{0}^2
   \Bigl( \int_0^1 2 t^{1-2s} \,dt + \int_1^\infty t^{-1-2s} \,dt \Bigr)
   \nonumber\\
   &\le
   3 |v|_{L^2,H^1,s}^2 + \frac 1{s(1-s)} \|v+c\|_{0}^2.
\end{align}
Therefore, recalling \eqref{1}, we obtain
\[
   |v|_{s,\inf}^2
   \le
   3K^2 |v|_{L^2,H^1,s}^2 + \frac {K^2}{s(1-s)} \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0}^2,
\]
which is the second assertion.
\end{proof}

From the proof of the previous lemma one can conclude that the semi-norm
$|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ is indeed the principal part of a norm in
$H^s({\cal O})$. This will be useful to deduce a Poincar\'e-Friedrichs inequality
with this semi-norm. First let us specify what we mean by the semi-norm
being principal part of a norm.

\begin{corollary} \label{cor}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded Lipschitz domain. There holds
\[
   \|v\|_{s,{\cal O}}^2
   \le
   \frac {K^2}{s(1-s)}
   \|v\|_{0,{\cal O}}^2
   +
   3K^2 |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $K=K(s,{\cal O})$ is the number from Proposition~\ref{prop_norms}.
\end{corollary}

\begin{proof}
This is a combination of the second bound from Proposition~\ref{prop_norms}
and \eqref{pf_la2_1} with $c=0$.
\end{proof}

We are now ready to prove a second Poincar\'e-Friedrichs inequality.

\begin{proposition}[Poincar\'e-Friedrichs inequality, interpolation semi-norm]
\label{prop_PF2}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded connected Lipschitz domain, and $s\in(0,1)$. Then there
exists a constant $C_{\rm PF,I}>0$, depending on ${\cal O}$ and $s$, such that
\[
   \|v\|_{0,{\cal O}} \le C_{\rm PF,I}(s,{\cal O}) \Bigl(|v|_{L^2({\cal O}), H^1({\cal O}), s} + |\int_{\cal O} v|\Bigr)
   \qquad\forall v\in H^s({\cal O}).
\]
\end{proposition}

\begin{proof}
Assume that the inequality is not true. Then there is a sequence
$(v_j)\subset H^s({\cal O})$ such that
\[
   \|v_j\|_{0,{\cal O}}=1,\quad |v_j|_{L^2({\cal O}), H^1({\cal O}), s} + |\int_{\cal O} v_j|\to 0\ (j\to\infty).
\]
Therefore, by  Corollary~\ref{cor}, $(v_j)$ is bounded in $H^s({\cal O})$
with respect to the Sobolev-Slobodeckij norm.
Then, by Rellich's theorem (see \cite[Theorem 3.27]{McLean_00_SES})
there is a convergent subsequence (again denoted by $(v_j)$) in $L^2({\cal O})$.
Since $|v_j|_{L^2({\cal O}), H^1({\cal O}), s}\to 0$ this sequence is Cauchy and with limit $v$ in $H^s({\cal O})$.
It holds $|v|_{L^2({\cal O}), H^1({\cal O}), s}=0$ so that $v$ is constant.
Furthermore, since $\int_{\cal O} v=0$ and ${\cal O}$ is connected we conclude that $v=0$,
a contradiction to $\|v_j\|_{0,{\cal O}}=1$.
\end{proof}

With the help of Proposition~\ref{prop_PF2} we can now turn the estimate by
Lemma~\ref{la2} into a semi-norm equivalence.

\begin{lemma} \label{la3}
Let ${\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a connected bounded Lipschitz domain. There holds
\[
   k^2
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   |v|_{s,{\cal O},\inf}^2
   \le
   K^2
   \Bigl( 3 + \frac {C_{\rm PF,I}^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $k=k(s,{\cal O})$, $K=K(s,{\cal O})$ are the numbers from Proposition~\ref{prop_norms},
and $C_{\rm PF,I}=C_{\rm PF,I}(s,{\cal O})$ is the number from Proposition~\ref{prop_PF2}.
\end{lemma}

\begin{proof}
The lower bound is the one from Lemma~\ref{la2}. The upper bound is a combination
of the upper bound from the same lemma and the Poincar\'e-Friedrichs' inequality from
Proposition~\ref{prop_PF2}. To this end note that the infimum
$\inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}}\|v+c\|_{0,{\cal O}}$ is achieved by the same constant $c$ that
eliminates the integral in the bound of the Poincar\'e-Friedrichs' inequality for $v+c$.
\end{proof}

Meanwhile we have accumulated quite some parameters in the semi-norm estimates
that depend on the order $s$ and the domain ${\cal O}$ under consideration.
Our goal is to show equivalence of semi-norms which is uniform for a family of
affinely transformed domains. We therefore study transformation properties of
semi-norms in the following section. In this way, parameters from this section enter
final results only via their values on a reference domain.

\subsection{Transformation properties of norms and semi-norms} \label{sec_scaling}

Obviously, both norms in $H^s({\cal O})$ defined previously,
$\|\cdot\|_{L^2({\cal O}),H^1({\cal O}),s}$ and $\|\cdot\|_{s,{\cal O}}$, are not scalable.
This could be achieved by weighting the $L^2({\cal O})$-contributions according
to the diameter of ${\cal O}$, for instance, cf. \cite{ErvinH_06_ABS}.
Of course, in this way one does not obtain uniformly equivalent norms
(of un-weighted and weighted variants) under transformation of the domain.

This is different for the norm in $\tilde H^s({\cal O})$. It can be easily
fixed (to be scalable) by using that the semi-norm $|\cdot|_{1,{\cal O}}$ is
a norm in $H^1_0({\cal O})$, and re-defining
\[
   \|v\|_{[L^2({\cal O}),H^1_0({\cal O})]_s}
   :=
   \|v\|_{L^2({\cal O}),H^1_0({\cal O}),s}
   :=
   \left( \int_0^\infty
          t^{-2s} \inf_{v=v_0+v_1, v_1\in H^1_0({\cal O})}
                  \Bigl(\|v_0\|_{0,{\cal O}}^2 + t^2\, |v_1|_{1,{\cal O}}^2\Bigr)
          \frac{dt}{t}
   \right)^{1/2}
\]
in the case of interpolation. In the case of the Sobolev-Slobodeckij norm
one can ensure scalability by re-defining
\[
  \|v\|_{\tilde H^s({\cal O})}
  :=
  \|v\|_{\sim,s,{\cal O}}
  :=
  \left(
  |v|_{H^s({\cal O})}^2
  +
  \|\frac{v(x)}{\mathop{\rm dist}\nolimits(x,\partial {\cal O})^s}\|_{L^2({\cal O})}^2
  \right)^{1/2}
\]
since the last term guarantees positivity. In the following we will make
use of these re-defined norms.

For a domain $\hat{\cal O}\in\mbox{\rm I\kern-.18em R}^n$ we denote by ${\cal O}=F(\hat{\cal O})$ the affinely transformed domain
\begin{equation} \label{trafo}
   {\cal O}:=\{F\hat x;\, \hat x\in\hat{\cal O}\}\quad\text{with}\quad F\hat x=x_0+B\hat x,\
   x_0\in\mbox{\rm I\kern-.18em R}^n,\ B\in\mbox{\rm I\kern-.18em R}^{n\times n}.
\end{equation}
Here, $B$ is assumed to be invertible.
Correspondingly, for a given real function $v$ defined on ${\cal O}$,
\[
   \hat v:\left\{\begin{array}{lll}
            \hat{\cal O} & \to & \mbox{\rm I\kern-.18em R}\\
            \hat x & \mapsto & v(F\hat x)
       \end{array}\right.
\]
is the function transformed onto $\hat{\cal O}$.

\begin{lemma}[transformation properties of norms] \label{la_scale}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded Lipschitz domain and let ${\cal O}$ be the affinely
transformed domain defined by \eqref{trafo}. Then there hold the transformation properties
\begin{equation} \label{scale_int}
   |\det B|\,\|B\|^{-2s}\; \|\hat v\|_{L^2(\hat {\cal O}),H^1_0(\hat {\cal O}),s}^2
   \le
   \|v\|_{L^2({\cal O}),H^1_0({\cal O}),s}^2
   \le
   |\det B|\,\|B^{-1}\|^{2s}\; \|\hat v\|_{L^2(\hat {\cal O}),H^1_0(\hat {\cal O}),s}^2,
\end{equation}
\begin{equation} \label{scale_tilde}
\begin{split}
   |\det B|\,\|B\|^{-2s}\min\{|\det B|\,\|B\|^{-n},1\}\; \|\hat v\|_{\sim,s,\hat{\cal O}}^2
   \le
   \|v\|_{\sim,s,{\cal O}}^2
\qquad\qquad\qquad\qquad\qquad
\\
\qquad\qquad\qquad\qquad\qquad
   \le
   |\det B|\,\|B^{-1}\|^{2s}\max\{|\det B|\,\|B^{-1}\|^n,1\}\; \|\hat v\|_{\sim,s,\hat{\cal O}}^2
\end{split}
\end{equation}
for any $\hat v\in \tilde H^s(\hat{\cal O})$ and $s\in (0,1)$.
\end{lemma}

\begin{proof}
For the interpolation norm and $\hat{\cal O}$, ${\cal O}$ being a cubes, this property (with an unspecified
equivalence constant) has been shown in \cite{Heuer_01_ApS}. It is simply the scaling
properties of the $L^2$ and $H^1_0$-norms together with the exactness of the K-method
of interpolation (employed here).
The proof generalizes to affine mappings in a straightforward way as follows.
In Euclidean norm one has $\|\nabla v(x)\| \le \|B^{-1}\|\,\|\nabla \hat v(\hat x)\|$ so
that the following relations are immediate,
\[
   \|v\|_{L^2({\cal O})}^2 = |\det B|\,\|\hat v\|_{L^2(\hat{\cal O})}^2,\quad
   |v|_{H^1({\cal O})}^2 \le |\det B|\, \|B^{-1}\|^2 |\hat v|_{H^1(\hat{\cal O})}^2.
\]
Then, with transformation $r=\|B^{-1}\|\,t$, we deduce that
\begin{align*}
   \|v\|_{L^2({\cal O}),H^1_0({\cal O}),s}^2
   &=
   \int_0^\infty t^{-2s} \inf_{v=v_0+v_1, v_1\in H^1_0({\cal O})}
          \Bigl(\|v_0\|_{0,{\cal O}}^2 + t^2\, |v_1|_{1,{\cal O}}^2\Bigr)
          \frac{dt}{t}
   \\
   &\le
   |\det B|
   \int_0^\infty t^{-2s} \inf_{\hat v=\hat v_0+\hat v_1, \hat v_1\in H^1_0(\hat {\cal O})}
          \Bigl(\|\hat v_0\|_{0,\hat {\cal O}}^2 + t^2\, \|B^{-1}\|^2|\hat v_1|_{1,\hat {\cal O}}^2\Bigr)
          \frac{dt}{t}
   \\
   &=
   |\det B|
   \int_0^\infty (\|B^{-1}\|^{-1}\,r)^{-2s} \inf_{\hat v=\hat v_0+\hat v_1, \hat v_1\in H^1_0(\hat {\cal O})}
          \Bigl(\|\hat v_0\|_{0,\hat {\cal O}}^2 + r^2\, |\hat v_1|_{1,\hat {\cal O}}^2\Bigr)
          \frac{dr}{r}
   \\
   &=
   |\det B|\,\|B^{-1}\|^{2s} \|\hat v\|_{L^2(\hat {\cal O}),H^1_0(\hat {\cal O}),s}^2.
\end{align*}
This proves the upper bound in \eqref{scale_int}. The lower bound is verified by using
the inverse transformation $F^{-1}$ with matrix $B^{-1}$.

The transformation property of the second norm is obtained similarly, see also
\cite[page 461]{DupontS_80_PAF} for the term of the double integral.
\begin{align*}
\lefteqn{
   \|v\|_{\sim,s,{\cal O}}^2
   =
   \int_{{\cal O}} \int_{{\cal O}} \frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}}\,dx\,dy
   +
   \int_{{\cal O}} \Bigl(\frac{v(x)}{\mathop{\rm dist}\nolimits(x,\partial {\cal O})^s}\Bigr)^2 \,dx
}
   \\
   &\le
   |\det B|^2
   \int_{\hat{\cal O}} \int_{\hat{\cal O}}
   \frac{|\hat v(\hat x)-\hat v(\hat y)|^2}{\|B^{-1}\|^{-n-2s}|\hat x-\hat y|^{n+2s}}
   \,d\hat x\,d\hat y
   +
   |\det B|
   \int_{\hat{\cal O}} \Bigl(\frac{\hat v(\hat x)}
                             {\|B^{-1}\|^{-s}\mathop{\rm dist}\nolimits(\hat x,\partial \hat{\cal O})^s}
                  \Bigr)^2 \,d\hat x
   \\
   &\le
   |\det B|\,\|B^{-1}\|^{2s}\max\{|\det B|\,\|B^{-1}\|^n,1\} \|\hat v\|_{\sim,s,\hat{\cal O}}^2.
\end{align*}
This is the upper bound in \eqref{scale_tilde}. Analogously one finds that
\[
   \|\hat v\|_{\sim,s,\hat{\cal O}}^2
   \le
   |\det B^{-1}|\,\|B\|^{2s}\max\{|\det B^{-1}|\,\|B\|^n,1\} \|v\|_{\sim,s,{\cal O}}^2.
\]
This proves the lower bound in \eqref{scale_tilde}.
\end{proof}

\begin{lemma}[transformation properties of semi-norms] \label{la_scale_semi}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded Lipschitz domain and let ${\cal O}$ be the affinely
transformed domain defined by \eqref{trafo}. Then there hold the transformation properties
\begin{equation} \label{scale_semi_int}
   |\det B|\,\|B\|^{-2s}\; |\hat v|_{L^2(\hat {\cal O}),H^1(\hat {\cal O}),s}^2
   \le
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   |\det B|\,\|B^{-1}\|^{2s}\; |\hat v|_{L^2(\hat {\cal O}),H^1(\hat {\cal O}),s}^2,
\end{equation}
\begin{equation} \label{scale_semi_tilde}
   |\det B|^2\,\|B\|^{-n-2s}\; |\hat v|_{s,\hat{\cal O}}^2
   \le
   |v|_{s,{\cal O}}^2
   \le
   |\det B|^2\,\|B^{-1}\|^{n+2s}\; |\hat v|_{s,\hat{\cal O}}^2
\end{equation}
for any $\hat v\in H^s(\hat{\cal O})$ and $s\in (0,1)$.
\end{lemma}

\begin{proof}
The proof is basically identical to the one of Lemma~\ref{la_scale}.
\end{proof}

The third semi-norm, $|\cdot|_{s,{\cal O},\inf}$, behaves under affine transformations
as follows.

\begin{lemma} \label{la_scale_semi3}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded Lipschitz domain and let ${\cal O}$ be the affinely
transformed domain defined by \eqref{trafo}. Then there hold the transformation properties
\[
\begin{split}
   |\det B|^2\,\|B\|^{-n-2s}\; |\hat v|_{s,\hat{\cal O}}^2
   +
   |\det B|\, \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|\hat v+c\|_{0,\hat{\cal O}}^2
   \le
   |v|_{s,{\cal O},\inf}^2
\qquad\qquad\qquad\qquad\qquad
\\
\qquad\qquad\qquad\qquad\qquad
   \le
   |\det B|^2\,\|B^{-1}\|^{n+2s}\; |\hat v|_{s,\hat{\cal O}}^2
   +
   |\det B|\, \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|\hat v+c\|_{0,\hat{\cal O}}^2
\end{split}
\]
for any $\hat v\in H^s(\hat{\cal O})$ and $s\in (0,1)$.
\end{lemma}

\begin{proof}
This result is immediate from the representation of the semi-norm
given in Lemma~\ref{la1} and the transformation properties of the
$|\cdot|_{s}$-semi-norm by Lemma~\ref{la_scale_semi} and of the $L^2$-norm.
\end{proof}

\section{Main results} \label{sec_main}

We are now ready to state and prove our main results on certain equivalences
of fractional-order Sobolev semi-norms.
We use the notation \eqref{trafo} from Section~\ref{sec_scaling} for
affine transformations. In particular, we assume that the domain ${\cal O}$ under
consideration is the affine image of a bounded Lipschitz domain $\hat{\cal O}$.
The following results specify how equivalence constants depend on the affine map.
At the end of this section we conclude the equivalence of some semi-norms which
is uniform for a family of so-called shape regular domains (Theorem~\ref{thm_main})
and some scaling properties (Corollary~\ref{cor_scale}).
These results are of importance for the approximation
theory of piecewise polynomial spaces in fractional-order Sobolev spaces.

The first theorem shows the equivalence of
the semi-norms $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ and $|\cdot|_{s,{\cal O}}$.

\begin{theorem} \label{thm1}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded, connected Lipschitz domain and let ${\cal O}$
be the affinely transformed domain defined by \eqref{trafo}.
Then there hold the following relations.
\begin{enumerate}
\item[(i)]
\[
   |v|_{s,{\cal O}}^2
   \le
   |\det B|\,\|B^{-1}\|^{n+2s} \|B\|^{2s}
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$
with $K(s,\hat{\cal O})$ from Proposition~\ref{prop_norms} and
     $C_{\rm PF,I}(s,\hat{\cal O})$ from Proposition~\ref{prop_PF2}.
\item[(ii)]
\[
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   |\det B|^{-1} \|B\|^{n+2s} \|B^{-1}\|^{2s}
   k(s,\hat{\cal O})^{-2} \Bigl( 1 + C_{\rm PF,SS}(s,\hat{\cal O})^2 \Bigr)
   |v|_{s,{\cal O}}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$
with $k(s,\hat{\cal O})$ from Proposition~\ref{prop_norms} and
     $C_{\rm PF,SS}(s,\hat{\cal O})$ from Proposition~\ref{prop_PF1}.
\end{enumerate}
\end{theorem}

\begin{proof}
On a fixed domain $\hat{\cal O}$ we obtain, by combining Lemmas~\ref{la1} and \ref{la3},
the equivalence of semi-norms:
\begin{equation} \label{pf_thm1_1}
   |\hat v|_{s,\hat{\cal O}}^2
   \le
   |\hat v|_{s,\hat{\cal O},\inf}^2
   \le
   K(s,\hat{\cal O})^2
   \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |\hat v|_{L^2(\hat{\cal O}),H^1(\hat{\cal O}),s}^2
\end{equation}
and
\begin{equation} \label{pf_thm1_2}
   |\hat v|_{L^2(\hat{\cal O}),H^1(\hat{\cal O}),s}^2
   \le
   k(s,\hat{\cal O})^{-2}
   |\hat v|_{s,\hat{\cal O},\inf}^2
   \le
   k(s,\hat{\cal O})^{-2}
   \Bigl(1+C_{\rm PF,SS}(s,\hat{\cal O})^2\Bigr) |\hat v|_{s,\hat{\cal O}}^2.
\end{equation}
The first assertion of the theorem then follows by combining \eqref{pf_thm1_1} with the
transformation properties of the semi-norms by Lemma~\ref{la_scale_semi}:
\begin{align*}
   |v|_{s,{\cal O}}^2
   &\le
   |\det B|^2 \|B^{-1}\|^{n+2s} |\hat v|_{s,\hat{\cal O}}^2
   \\
   &\le
   |\det B|^2 \|B^{-1}\|^{n+2s}
   K(s,\hat{\cal O})^2
   \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |\hat v|_{L^2(\hat{\cal O}),H^1(\hat{\cal O}),s}^2
   \\
   &\le
   |\det B|\, \|B^{-1}\|^{n+2s} \|B\|^{2s}
   K(s,\hat{\cal O})^2
   \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2.
\end{align*}
The second assertion of the theorem is proved by a combination of \eqref{pf_thm1_2}
with the transformation properties by Lemma~\ref{la_scale_semi}.
\end{proof}

The next two theorems study the other pairs of semi-norms for equivalence
in combination with affine maps, $(|\cdot|_{s,{\cal O}}, |\cdot|_{s,{\cal O},\inf})$ and
$(|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}, |\cdot|_{s,{\cal O},\inf})$.

\begin{theorem} \label{thm2}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded, connected Lipschitz domain and let ${\cal O}$ be the affinely
transformed domain defined by \eqref{trafo}. Then there hold the following relations.
\begin{enumerate}
\item[(i)]
\[
   |v|_{s,{\cal O}}
   \le
   |v|_{s,{\cal O},\inf}
   \qquad\forall v\in H^s({\cal O}),\ \forall s\in (0,1),
\]
\item[(ii)]
\[
   |v|_{s,{\cal O},\inf}^2
   \le
   \Bigl( 1 + |\det B|^{-1} \|B\|^{n+2s} C_{\rm PF,SS}(s,\hat{\cal O})^2 \Bigr)
   |v|_{s,{\cal O}}^2
   \qquad\forall v\in H^s({\cal O}),\ \forall s\in (0,1)
\]
with $C_{\rm PF,SS}(s,\hat{\cal O})$ being the number from Proposition~\ref{prop_PF1}.
\end{enumerate}
\end{theorem}

\begin{proof}
Assertion (i) is a repetition of the first estimate in Lemma~\ref{la1}.

To show the second assertion we use Proposition~\ref{prop_PF1} and
Lemma~\ref{la_scale_semi} to deduce that
\begin{align*}
   \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0,{\cal O}}^2
   &=
   |\det B|\, \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|\hat v+c\|_{0,\hat {\cal O}}^2
   \le
   |\det B|\, C_{\rm PF,SS}(s,\hat{\cal O})^2 |\hat v|_{s,\hat{\cal O}}^2
   \\
   &\le
   |\det B|^{-1} \|B\|^{n+2s} C_{\rm PF,SS}(s,\hat{\cal O})^2 |v|_{s,{\cal O}}^2.
\end{align*}
The assertion then follows by the definition of the semi-norm $|\cdot|_{s,{\cal O},\inf}$.
\end{proof}

\begin{theorem} \label{thm3}
Let $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ be a bounded, connected Lipschitz domain and let ${\cal O}$
be the affinely transformed domain defined by \eqref{trafo}.
Then there hold the following relations.
\begin{enumerate}
\item[(i)]
\[
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   \|B^{-1}\|^{2s} \max\{|\det B|^{-1} \|B\|^{n+2s}, 1\}\;
   k(s,\hat{\cal O})^{-2}
   |v|_{s,{\cal O},\inf}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$
with $k(s,\hat{\cal O})$ from Proposition~\ref{prop_norms},
\item[(ii)]
\[
   |v|_{s,{\cal O},\inf}^2
   \le
   \|B\|^{2s} \max\{|\det B|\, \|B^{-1}\|^{n+2s}, 1\}\;
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$
with $K(s,\hat{\cal O})$ from Proposition~\ref{prop_norms} and
     $C_{\rm PF,I}(s,\hat{\cal O})$ from Proposition~\ref{prop_PF2}.
\end{enumerate}
\end{theorem}

\begin{proof}
By Lemmas~\ref{la_scale_semi}, \ref{la3}, and \ref{la_scale_semi3} we obtain
\begin{align*}
\lefteqn{
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   \le
   |\det B|\,\|B^{-1}\|^{2s}
   |\hat v|_{L^2(\hat{\cal O}),H^1(\hat{\cal O}),s}^2
   \le
   |\det B|\,\|B^{-1}\|^{2s}
   k(s,\hat{\cal O})^{-2} |\hat v|_{s,\hat{\cal O},\inf}^2
}
   \\
   &\le
   |\det B|\,\|B^{-1}\|^{2s}
   k(s,\hat{\cal O})^{-2}
   \Bigl(|\det B|^{-2}\|B\|^{n+2s} |v|_{s,{\cal O}}^2 +
         |\det B|^{-1} \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|v+c\|_{0,{\cal O}}^2\Bigr)
   \\
   &\le
   \|B^{-1}\|^{2s} \max\{|\det B|^{-1} \|B\|^{n+2s}, 1\}\;
   k(s,\hat{\cal O})^{-2} |v|_{s,{\cal O},\inf}^2.
\end{align*}
This is the first assertion. The second one follows analogously by the same lemmas:
\begin{align*}
   |v|_{s,{\cal O},\inf}^2
   &\le
   |\det B|^2 \|B^{-1}\|^{n+2s} |\hat v|_{s,\hat{\cal O}}^2
   +
   |\det B|\, \inf_{c\in\mbox{\footnotesize\rm I\kern-.18em R}} \|\hat v+c\|_{0,\hat{\cal O}}^2
   \\
   &\le
   |\det B|\, \max\{|\det B|\, \|B^{-1}\|^{n+2s}, 1\}\;
   |\hat v|_{s,\hat{\cal O},\inf}^2
   \\
   &\le
   |\det B|\, \max\{|\det B|\, \|B^{-1}\|^{n+2s}, 1\}\;
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |\hat v|_{L^2(\hat{\cal O}),H^1(\hat{\cal O}),s}^2
   \\
   &\le
   \max\{|\det B|\, \|B^{-1}\|^{n+2s}, 1\}\; \|B\|^{2s}
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2.
\end{align*}
\end{proof}

We end this section with establishing uniform equivalence of the semi-norms
$|\cdot|_{s,{\cal O}}$ and $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ for shape-regular domains.
Three of the four remaining bounds for other combinations of semi-norms are
uniform under further restrictions on the diameter of the domain.

Let us introduce some notation. We consider a bounded, connected Lipschitz domain
$\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$ and maps of $\hat{\cal O}$ onto domains ${\cal O}$ where
the ratio $\rho_{{\cal O}}:=D_{{\cal O}}/d_{{\cal O}}$ is controlled.
Here, $D_{{\cal O}}$ denotes the diameter of ${\cal O}$ and $d_{{\cal O}}$ is the supremum of
the diameters of all balls contained in ${\cal O}$. In the case of finite elements
(or convex polygons) boundedness of $\rho$ is referred to as {\em shape regularity}
of ${\cal O}$. Also, when defining $d_{{\cal O}}$ with balls with respect to which ${\cal O}$ is
star-shaped, then $\rho_{{\cal O}}$ is referred to as {\em chunkiness parameter}.

Using the notation \eqref{trafo} there holds
\begin{equation} \label{aff1}
   \|B\|             \le \frac{D_{{\cal O}}}{d_{\hat{\cal O}}}
                     =   \frac{D_{{\cal O}}}{D_{\hat{\cal O}}}\, \rho_{\hat{\cal O}},\quad
   \|B^{-1}\|        \le \frac{D_{\hat{\cal O}}}{d_{{\cal O}}}
                     =   \frac{D_{\hat{\cal O}}}{D_{{\cal O}}}\, \rho_{{\cal O}},\quad
   \|B\|\,\|B^{-1}\| \le \rho_{{\cal O}}\,\rho_{\hat{\cal O}},
\end{equation}
cf., e.g., \cite{Braess_97_FET}. Furthermore, we conclude that
\begin{equation} \label{aff2}
   |\det B| = \frac {|{\cal O}|}{|\hat{\cal O}|} \le \frac {D_{{\cal O}}^n}{d_{\hat{\cal O}}^n},\quad
   |\det B|^{-1} \le \frac {D_{\hat{\cal O}}^n}{d_{{\cal O}}^n}
                 = \rho_{{\cal O}}^n \frac {D_{\hat{\cal O}}^n}{D_{{\cal O}}^n}.
\end{equation}
With this notation, the results of Theorems~\ref{thm1}--\ref{thm3} imply the following.

\begin{theorem} \label{thm_main}
Let ${\cal O}$ be the affine map of a bounded connected Lipschitz domain $\hat{\cal O}\subset\mbox{\rm I\kern-.18em R}^n$,
cf. \eqref{trafo}.

\begin{enumerate}
\item[(i)]
The semi-norms $|\cdot|_{s,{\cal O}}$ and $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ are uniformly equivalent
for a family of shape-regular domains ${\cal O}$:
\begin{align*}
   |v|_{s,{\cal O}}^2
   &\le
   \rho_{{\cal O}}^{n+2s} \rho_{\hat{\cal O}}^{n+2s}
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2,
   \\
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   &\le
   \rho_{{\cal O}}^{n+2s} \rho_{\hat{\cal O}}^{n+2s}
   k(s,\hat{\cal O})^{-2} \Bigl( 1 + C_{\rm PF,SS}(s,\hat{\cal O})^2 \Bigr)
   |v|_{s,{\cal O}}^2
\end{align*}
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $k(s,\hat{\cal O})$, $K(s,\hat{\cal O})$ are the numbers from Proposition~\ref{prop_norms} and
$C_{\rm PF,SS}(s,\hat{\cal O})$, $C_{\rm PF,I}(s,\hat{\cal O})$ are as in Propositions~\ref{prop_PF1}, \ref{prop_PF2},
respectively.

\item[(ii)]
The semi-norms $|\cdot|_{s,{\cal O}}$ and $|\cdot|_{s,{\cal O},\inf}$ are uniformly equivalent
for a family of uniformly bounded, shape-regular domains ${\cal O}$:
\begin{align*}
   |v|_{s,{\cal O}}
   &\le
   |v|_{s,{\cal O},\inf},
   \\
   |v|_{s,{\cal O},\inf}^2
   &\le
   \Bigl( 1 + \frac {D_{{\cal O}}^{2s}}{D_{\hat{\cal O}}^{2s}} \rho_{{\cal O}}^n \rho_{\hat{\cal O}}^{n+2s}
              C_{\rm PF,SS}(s,\hat{\cal O})^2 \Bigr)
   |v|_{s,{\cal O}}^2
\end{align*}
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
Here, $C_{\rm PF,SS}(s,\hat{\cal O})$ is the number from Proposition~\ref{prop_PF1}.

\item[(iii)]
\begin{enumerate}
\item[a)]
For a family of shape-regular domains ${\cal O}$ whose diameters are bounded from below by a positive
constant, the semi-norm $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ is uniformly bounded by
$|\cdot|_{s,{\cal O},\inf}$:
\begin{align*}
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
   &\le
   \max\{\rho_{{\cal O}}^n \rho_{\hat{\cal O}}^{n-2s}, D_{{\cal O}}^{-2s} D_{\hat{\cal O}}^{2s}\}\;
   \rho_{{\cal O}}^{2s}
   k(s,\hat{\cal O})^{-2}
   |v|_{s,{\cal O},\inf}^2
\end{align*}
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
\item[b)]
For a family of uniformly bounded, shape-regular domains ${\cal O}$,
the semi-norm $|\cdot|_{s,{\cal O},\inf}$ is uniformly bounded by $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$:
\[
\begin{split}
   |v|_{s,{\cal O},\inf}^2
   \le
   \max\{\rho_{{\cal O}}^{n+2s} \rho_{\hat{\cal O}}^n, D_{{\cal O}}^{2s} D_{\hat{\cal O}}^{-2s}\}\;
   \rho_{\hat{\cal O}}^{2s}
\hspace{8em}
   \\
   K(s,\hat{\cal O})^2 \Bigl( 3 + \frac {C_{\rm PF,I}(s,\hat{\cal O})^2}{s(1-s)} \Bigr)
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\end{split}
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
\end{enumerate}
Here, $k(s,\hat{\cal O})$, $K(s,\hat{\cal O})$ are the parameters from Proposition~\ref{prop_norms},
and $C_{\rm PF,I}(s,\hat{\cal O})$ is the number from Proposition~\ref{prop_PF2}.
\end{enumerate}
\end{theorem}

\begin{proof}
The assertions (i)--(iii) are a combination of Theorems~\ref{thm1}--\ref{thm3},
respectively, with the bounds provided by \eqref{aff1}, \eqref{aff2}.
\end{proof}

The uniform equivalence of the semi-norms $|\cdot|_{s,{\cal O}}$ and
$|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ for shape-regular domains is based on what one calls
their {\em scaling property}. It means that both semi-norms for functions on a domain
${\cal O}$ are uniformly equivalent to the respective semi-norm of the affinely
transformed functions onto a fixed domain $\hat{\cal O}$, when one of the semi-norms is
multiplied by an appropriate number (it is a power of the diameter of ${\cal O}$).
This property applies also to the norms $\|\cdot\|_{L^2({\cal O}),H^1_0({\cal O}),s}$
and $\|\cdot\|_{\sim,s,{\cal O}}$, cf.~Lemma~\ref{la_scale}. Scaling properties are relevant
for the error analysis of piecewise polynomial approximations. We formulate the
result as a corollary to Lemmas~\ref{la_scale} and \ref{la_scale_semi}.

\begin{corollary} \label{cor_scale}
The norms $\|\cdot\|_{L^2({\cal O}),H^1_0({\cal O}),s}$, $\|\cdot\|_{\sim,s,{\cal O}}$
and semi-norms $|\cdot|_{s,{\cal O}}$, $|\cdot|_{L^2({\cal O}),H^1({\cal O}),s}$ are scalable
of order $D_{{\cal O}}^{n-2s}$:
\[
\begin{split}
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{-n} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{-2s}
   \|\hat v\|_{L^2(\hat {\cal O}),H^1_0(\hat {\cal O}),s}^2
   \le
   \|v\|_{L^2({\cal O}),H^1_0({\cal O}),s}^2
\hspace{6em}
   \\
   \le
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{2s} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{n}
   \|\hat v\|_{L^2(\hat {\cal O}),H^1_0(\hat {\cal O}),s}^2,
   \\[1em]
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{-n} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{-2s}
   \min\{\rho_{{\cal O}}^{-n} \rho_{\hat{\cal O}}^{-n},1\}\;
   \|\hat v\|_{\sim,s,\hat{\cal O}}^2
   \le
   \|v\|_{\sim,s,{\cal O}}^2
\hspace{6em}
   \\
   \le
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{2s} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^n
   \max\{\rho_{{\cal O}}^n \rho_{\hat{\cal O}}^n,1\}\;
   \|\hat v\|_{\sim,s,\hat{\cal O}}^2
\end{split}
\]
for any $v\in \tilde H^s({\cal O})$ and $s\in (0,1)$, and
\[
\begin{split}
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{-n} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{-2s}
   |\hat v|_{L^2(\hat {\cal O}),H^1(\hat {\cal O}),s}^2
   \le
   |v|_{L^2({\cal O}),H^1({\cal O}),s}^2
\hspace{9em}
   \\
   \le
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{2s} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{n}
   |\hat v|_{L^2(\hat {\cal O}),H^1(\hat {\cal O}),s}^2,
   \\[1em]
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{-2n} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{-n-2s}
   |\hat v|_{s,\hat{\cal O}}^2
   \le
   |v|_{s,{\cal O}}^2
\hspace{9em}
   \\
   \le
   D_{{\cal O}}^{n-2s} \rho_{{\cal O}}^{n+2s} D_{\hat{\cal O}}^{2s-n} \rho_{\hat{\cal O}}^{2n}
   |\hat v|_{s,\hat{\cal O}}^2
\end{split}
\]
for any $v\in H^s({\cal O})$ and $s\in (0,1)$.
\end{corollary}

\begin{proof}
The bounds are a combination of Lemmas~\ref{la_scale} and \ref{la_scale_semi}
with \eqref{aff1}, \eqref{aff2}.
\end{proof}

\begin{remark}
The estimate by Theorem~\ref{thm_main} (iii) a) breaks down
when $D_{{\cal O}}\to 0$. In fact, for a family of scaled domains ${\cal O}_h$ with $D_{{\cal O}_h}=h$
and a non-constant function $v$ scaled to a family $\{v_h\}$ of functions on $\{{\cal O}_h\}$,
$|v_h|_{L^2({\cal O}_h),H^1({\cal O}_h),s}^2 \simeq h^{n-2s}$ by Corollary~\ref{cor_scale}
whereas $|v_h|_{s,{\cal O}_h,\inf}^2\ge \inf_{c\in\mbox{\rm I\kern-.18em R}}\|v_h-c\|_{0,{\cal O}_h}^2\simeq h^n$.
Therefore, the dependence on $D_{{\cal O}}$ like $D_{{\cal O}}^{-2s}$ of the upper bound
in Theorem~\ref{thm_main} (iii) a) is optimal.
\end{remark}

\noindent
{\bf Acknowledgment.} We thank an anonymous referee for constructive remarks.

redpajama_set_name: RedPajamaArXiv


Broadband is now a necessity for many of us, in the same way that phones and televisions were once considered a luxury but are now an indispensable part of our everyday lives. We use broadband for entertainment, to keep in touch with our loved ones, to network and for many, it's also how they earn a living too.
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Here at Hoppy, we're all about people saving as much money as possible on their everyday costs. That's why we allow you to simply and effectively compare broadband providers – along with the likes of energy providers, tradespeople and much more – so that it only takes a few clicks to see what the right deal is for you. We know that a lot of people end up losing money simply because things aren't made clear to them and this oversight with line rental is exactly the type of thing Hoppy is here to help with. All we do is make the choice clear and simple.
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text:
module.exports = {
      name : 'Paper chaser',
      hypothesis : '5g is the best weight for planes',
      kind : 'ad_hoc',
      dependentVar: {
        name : 'Flight length',
        measures : [{
          name : 'Distance',
          kind : 'numeric',
          scale : null,
          list : null,
          samples : [{
            value: 3.2
          },
          {
            value: 2.5
          }],
          request : {
            freq : null,
            question : 'how far did it fly?'
          }
        }]
      },
      independentVars : [{
        name: 'Weight',
        actionStart : null,
        actionWarning : null, //should likely be some datetime thing....
        consecutiveActions : 1,
        options : ['3.4','6.0'],
        remind : {
          freq : null, // whatever datetime thing we decide on
          reminder: 'Put this amount of weight in your plane'
        }
      }]
    };


text:
USAG Bavaria wins Secretary of the Army Environmental Award
Community Contributor	March 13th, 2017
By Christine Fox and Corina Habermann, USAG Bavaria Environmental Division
GRAFENWOEHR, Germany — The U.S. Army recently announced U.S. Army Garrison Bavaria as winner of Environmental Quality in the overseas category for their annual Secretary of the Army Environmental Awards.
USAG Bavaria will now compete with the Navy, Air Force and Marines as the Army's representative in the Secretary of Defense Environmental Awards program.
For the past few years, USAG Bavaria has seen an increase in visiting rotational units, with approximately 180,000 troops from more than 35 countries coming each year to one of the four military communities to train. This places high demands on training lands and military infrastructure.
In a combined team effort with the Utilities Branch of the Directorate of Public Works, 7th Army Training Command, host nation authorities and other organizations, the Environmental Division's staff has successfully handled the garrison's increase in rotational forces, as well as continued to provide quality support to permanent units while complying with environmental laws.
It is these efforts that earned them the Environmental Quality category award, which recognizes the excellence of an environmental program as a whole.
Cooperation and collaboration with all stakeholders, a lot of hard work from a dedicated and highly motivated team of professionals, and a well-balanced and mission-oriented environmental program is what Manfred Rieck, chief, Environmental Division, attributes to their success.
What makes this program most effective is the full garrison leadership support it receives and everyone's continual focus on the military mission, he said.
The program's committed relationship with tenant and rotational units is one of many reasons it was recognized with the award.
Caecilia Kastl-Meier, staff member, Environmental Division, shows Vilseck High School students a mummified Zwergfledermaus, or common pipistrelle, bat during a tour of the Grafenwoehr Training Area. Photo by Christine Fox.
Environmental staff assist military trainers in evaluating training scenarios to avoid negative environmental impacts, like soil and groundwater contamination due to hazardous materials spills. Providing quality environmental training, offering technical assistance, and establishing a hazardous waste disposal system make environmental protection easier for Soldiers. The program has installed numerous strategically-placed refueling sites, as well as hazardous waste storage and disposal facilities, to make safety and compliance more convenient.
In addition to offering training and guidance to military personnel, the garrison — in cooperation with 7th ATC and other DPW divisions — effectively manages military lands to support a realistic training environment in compliance with laws, while also allowing more than 800 legally-protected flora and fauna species to flourish within the garrison's boundaries.
"In recent years, the garrison has experienced the re-emergence of highly threatened species that have been absent since the 19th century," said Rieck. "The white-tailed eagle, European lynx, fish otter, wild cat and greater horseshoe bat are just a few of the species that are now thriving. Currently, we are excited about European wolf sightings in the Grafenwoehr Training Area. It was first spotted in September 2016 by a wildlife camera and has been breaking news in German media."
Another boost for wildlife are the numerous habitat restoration or enhancement projects that have been completed throughout USAG Bavaria.
In the Hohenfels Training Area, a European Union-funded project is currently ongoing to stabilize roosting habitats for the greater horseshoe bat, the only reproducing population in Germany. The environmental program's coordination with the Federal Forest Service; the county and district nature protection authorities; and a local nature protection non-governmental organization has ensured the success of this project.
"The garrison's environmental program is thriving because of its trustful partnerships with host nation authorities. We have earned and maintained their trust by sharing our expertise in managing military training lands while conserving nature and protecting endangered species," said Rieck. "In turn, our host nation has supported the garrison by providing permits to conduct necessary military operation activities and, by monitoring soil, improve surface water and groundwater quality."
The garrison's environmental outreach program has built partnerships not only with local authorities and organizations but also international groups.
Visiting military forces from foreign countries like Lithuania, Latvia, Estonia, Israel and France, and students from universities in Prague and Australia have come to USAG Bavaria to learn about environmental projects and how the program is managed. In exchange, Manfred Rieck and his staff were invited to give presentations at workshops and conferences in France and Estonia.
This sharing of information and support helps other organizations all over the world to flourish in their protection of the environment, without compromising military operations.
Another highlight of the environmental program is its focus on modernizing energy infrastructure and improving energy efficiency within the garrison.
Grafenwoehr Training Area covers diverse wetland and forest habitats and is home to more than 3,000 plant and animal species. Photo by Stefan Haertl.
In Hohenfels, the recent introduction of a heat and power cogeneration system at Camp Nainhof has reduced annual greenhouse gas emissions by 1,300 tons, saving approximately $900,000 in overall energy costs each year. This is just one of many upgrades that have decreased the garrison's energy footprint.
The USAG Bavaria environmental program will continue to rise above challenges, Rieck said, upholding the motto: "Protect the environment, support the mission, sustain the future."