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On September 25, 2003, the Associate Village Justice of this Court signed a search warrant directed to “any police officer of the County of Nassau.” A New York Criminal Lawyer said the search warrant provided: “Proof, by affidavit, having been this day made before me by Senior Building Inspector, Village of Westbury, Public Works, Village of Westbury and Department of Public Works, Village of Westbury that there is probable cause for the issuance of the search warrant, as set forth in the affidavit and Exhibits attached hereto and made a part hereof as if fully set forth herein; you are therefore, commanded to make a search with Senior Building Inspector and his agents, between 09/25/03 and 10/02/03 in the hours between 6:00 A.M. and 9:00 P.M. of the entire premises designated and described as 335 Princeton Street, Westbury, New York. “The seizure of the foregoing evidence shall be limited to the taking of still photographs and videotape pictures of the inside and outside of the premises. This warrant must be executed within 10 days of the date of signing and a return to court 10 days thereafter. “If you find the same or any part thereof you are hereby directed to return and deliver said evidence to the undersigned Justice of the Village Court without unnecessary delay.”
A Bronx Criminal Lawyer said that, the Senior Building Inspector submitted what may be described as an exhaustive affidavit in support of the application. This Court wonders why, in view of the thoroughness of this affidavit and the apparent pre-warrant investigation, that a search and search warrant were needed at all unless the Village is simply trying to test the legal waters in this case to determine whether they may have another tool at their disposal, namely search warrants, that they may use to enforce the Village’s zoning and building code laws. The application for and the execution of a search warrant may in themselves deter the proliferation of illegal housing. The execution of a search warrant is an extremely frightening event for those subject to it. The court questions the need for this warrant because there is no legal requirement that a warrant be obtained in order to take photos of the outside of the premises from a public thoroughfare in front of the home. However, this Court finds that the Village has acted in good faith attempting; for example, to obtain the homeowner’s consent for the search prior to seeking the warrant and no doubt believing that similar actions have been approved and utilized in other villages without challenge.
A Bronx Criminal Lawyer said that, the subject property, 335 Princeton Street, is a two story house within the Incorporated Village of Westbury, New York. As shown on the records of the Department of Buildings of the Village of Westbury it is located on a quiet residential block consisting of one (1) family homes neatly maintained on a tree lined block. | https://www.nycriminalattorneyblog.com/category/criminal-procedure/new-york-1/ |
Wonder
With all that is wrong in the world today I sometimes lose sight of the beauty and the majesty of creation. A friend and a reader of my stories noted a sense of pessimism in my Focus musings. I will admit that I am a glass-half-empty kind of guy. And though I don’t dwell on the past, I focus too much on the future when I should concentrate on the present. Fortunately, my wife is a glass-half-full gal who balances my masculine yang with her grounded and feminine yin.
I told my friend that it’s hard for a conservative like me to ignore the Washington swamp and our feckless leaders. Though I believe in the ultimate triumph of good over evil, it saddens me to think it may not happen in my lifetime or in America. My worldly concerns are the dissolution of morality, the disintegration of traditional institutions like the Democrat and Republican parties; even the NFL is falling apart. Babylon has existed in the Hollywood swamp for decades, so the recent exposé in tinsel town and its unraveling comes at no surprise to me. What astounds me is that so many can be manipulated by clueless movie stars and by the utterly corrupt media.
Too often too many walk with senses tuned to the latest fake news cycle instead of the greater reality around us, when wonder is just outside the door. I often encourage my grandson Oakley to walk with me in the woods. I tell him, “Come on Oaks, let’s go; we might see something.” And of course we usually do when we “look.”
I’m a stargazer and when the air is cold and clear wondrous things can be seen if you look up at the night sky instead of gazing down at your smart phone or at the idiot box. However, my eyes and telescope could not have seen the wondrous event recently recorded by scientists and the Hubble space telescope. Apparently, two neutron stars ended their cosmic dance and collided with each other 130,000,000 light years away in the distant past. The light from that cataclysmic explosion is just now reaching us. Astronomers and astrophysicists are studying this so-called “kilonova” and have measured gravity waves as predicted by Einstein.
Some might ask, what does it matter that scientists confirmed general relativity by measuring these gravity waves? What is the relevance of seeing farther and extending man’s perceptual horizons? Actually, I believe it’s far more important than reality TV or useless Congressional hearings which only afford a stage for Senate dinosaurs to grandstand, while the Rocket Man threatens the world and the mullahs in Iran build their bomb. The Jewish theologian, philosopher and rabbi, Abraham Joshua Heschel once said, “Never once in my life did I ask God for success or wisdom or power or fame. I asked for wonder, and he gave it to me.”
I saw the first Christmas decorations this week in a Bearden shop window. Folks, that’s just not right; we haven’t even had Halloween or Thanksgiving yet! Nonetheless, these decorations and the recent scientific revelations caused me to think of perhaps my favorite Christmas song. It’s called “I Wonder as I Wander (…out under the sky”) by John Jacob Niles. If you are unfamiliar with this song and it’s haunting melody and spiritual reflections, go to YouTube and tune your heart, mind and soul to wonder.
The universe is unimaginably vast and wondrous. We see distant stars and galaxies by the electromagnetic radiation they produce which reaches us across space and time as light. This radiation travels at the speed of 186,000 miles a second. Contrast that with the fastest transmission of a human nerve impulse which travels at 100 meters/sec or about one football field/second. The light from our own sun takes eight minutes to reach the earth, and the light from the next nearest star takes four years to reach our eyes and telescopes. It may be hard to get your mind around, but we actually see what stars looked like in the past when their light rays began their journey. By observing and measuring distant celestial objects, scientists look backwards toward the dawn of the universe estimated to be 13.8 billion years ago or 13.8 billion light years “distant.”
I learned last week that the Hubble space telescope is not only seeing farther, but seeing more. New estimates of the size of the universe boggle my mind. Our Milky Way galaxy has several hundred billion stars. (And we also know that many stars have planets circling them.) For some time we thought there were 100 billion other galaxies in the known universe. The latest estimate is that there are more than a trillion other galaxies! If you take an average of 100 billion stars per galaxy then there are 1022 stars in the universe, that’s a ten followed by twenty-two zeros. There are more stars than there are grains of sand on all the beaches of the world.
I once wrote a paper about the location of heaven. No, I’ve never visited there or heard voices revealing heaven’s coordinates. My ruminations were just a science based thought experiment. People are curious by nature, and for all of recorded time, people have wondered about death and if anything comes next. Actually, as I consider the vastness of space and the multidimensional wonders of creation’s space-time, I don’t worry about that “far country.” I just trust that it will be OK for me and others. Even before Jesus, Socrates thought so.
In the sixth century BC, Jerusalem and God’s Temple were destroyed by the Babylonians, and the Jews were carried off into slavery. Some thought the exile would quickly end. However, the ancient prophet Jeremiah told the conquered Hebrews that they should not focus on the future, but live in the present moment. Jeremiah told the exiles to plant gardens and vineyards, to raise families and trust in the Lord. They did and restoration eventually occurred.
This star gazer sees that lesson and Psalm 118:24 as relevant for today’s journey. | |
Students in Agricultural and Environmental Plant Sciences begin with core courses that provide a thorough introduction to seven concentrations.
The Cal Poly Horticulture and Crop Science Department epitomizes the University's "learn-by-doing" approach to studying and mastering a field of expertise.
We specialize in the three primary areas of Environmental Horticultural Science, Fruit and Crop Science, and Plant Protection Science. The HCS Department instills a sense of the "real world," including the nature of jobs, salaries and how to advance in your chosen field.
Read the 2015 report, "Want a Job? Major in Agriculture," a joint report of the USDA's National Institute of Food and Agriculture and Purdue University, which highlights the need for graduates in agriculture and the shortage of students majoring in fields related to plant science. Also, explore career opportunities at Seed Our Future, a joint effort of universities and the agricultural industry to support interest in and an appreciation for horticulture.
View a video by AEPS student, Grace Watson, in which she describes her Learn-by-Doing experiences in the Horticulture and Crop Science Department and how, as a part of that experience, she markets student-grown produce to local farmers' markets.
Learn more about the HCS Department at Cal Poly's Annual Open House on April 11-13, 2019. This annual event showcases the Cal Poly campus to new students, their families, and the community while offering a glimpse to life as a Mustang. | http://aeps.calpoly.edu/ |
DC IPL 2022 playoffs scenarios: The ongoing Indian Premier League (IPL 2022) season has reached its business end where the teams have to up their respective game in order to successfully qualify for the playoffs. There are a total of 14 matches remaining in the league stage of the tournament and none of the teams have booked a spot in the knockouts. The Delhi Capitals have reached the IPL playoffs all three times in the last three seasons and lets analyse, if the Rishabh Pant-led team can do similar wonders this year or not.
The Delhi Capitals lost their most recent game against Chennai Super Kings by a massive margin of 91 runs and that has dented their chances of qualifying for Indian Premier League 2022 playoffs. With a positive Net Run Rate of 0.150, the Rishabh Pant-led team is currently placed at the fifth spot on the points table. The Delhi Capitals have managed to win 5 out of their 11 matches this T20 season and now they desperately need to win all of their remaining three games against Rajasthan Royals, Punjab Kings and Mumbai Indians to have a chance of qualification.
|Requirement 1||DC will have to win all of their remaining matches against RR, PBKS and MI. They will also have to ensure that their Net Run Rate (0.150) remains positive and continues to increase.|
|Requirement 2||Delhi Capitals will hope that some other results go their way as they can finish with a maximum of 16 points.|
Related: Most sixes in IPL
|Teams||P||W||L||Pts||NRR|
|Lucknow Super Giants||11||8||3||16||+0.703|
|Gujarat Titans||11||8||3||16||+0.120|
|Rajasthan Royals||11||7||4||14||+0.326|
|Royal Challengers Bangalore||12||7||5||14||-0.115|
|Delhi Capitals||11||5||6||10||+0.150|
|Sunrisers Hyderabad||11||5||6||10||-0.031|
|Kolkata Knight Riders||12||5||7||10||-0.057|
|Punjab Kings||11||5||6||10||-0.231|
|Chennai Super Kings||11||4||7||8||+0.028|
|Mumbai Indians||11||2||9||4||-0.894|
Related: IPL Team owners
|Matches||RR||DC|
|Overall (25)||13||12|
|Last 5 IPL Games||2||3|
Related: IPL 2022 Prize Money
It has been an exciting head-to-head rivalry between the Rajasthan Royals and Delhi Capitals which will be reunited on 11 May, 2022 at the Dr DY Patil Sports Academy in Mumbai. During the previous game between these two teams earlier this season, the Rajasthan Royals defeated Delhi Capitals by 15 runs. | https://www.indiafantasy.com/cricket/cricket-news/ipl-playoffs-how-can-dc-qualify-for-ipl-2022-playoffs-check-complete-scenarios/ |
Escargots (French for snails) are a delicacy consisting of cooked edible land snails. They are often served as an hors d’oeuvre.
This simple and classic method for escargot was my first attempt at making my own escargot, it was such an easy-fix & done in less than 15 min.
The use of French brandy was the special inspired touch and should not be omitted. The garlicky buttery sauce in this dish is almost as delicious as the escargots themselves.
Canned snails work just as well as fresh expensive ones and turn this appetizer into one that can be whipped up anytime. | http://fast2eat.com/ingredient/black-pepper/ |
Use the phonic cards I provided the week starting 11th May to make the word ‘tug’. This is the link to the cards:
|https://www.st-lukesislington.co.uk/week-commencing-11th-may-2/|
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|
Once your child makes the connection between the word and this week’s book, challenge them to identify the word in every page of the book.
Activity 2
The next three activities will use the phonic cards for ‘sounding out’ and ‘blending’ words from the book like you did yesterday. Once you finish, help your child to identify the words in the story. The words today are:
|
|
can
|
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Tom (do not worry about capital letters, this is a phonic activity)
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get
Activity 3
Same as yesterday. Today’s words are:
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big
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and
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Nick
Activity 4
Same as yesterday. Today’s words are:
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Lin
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Sam
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Kit
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up
You need to do all this phonics activities to be able to complete activity 5 of English.
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Activity 5
Use the link below to access the sounds board. Your child will not know many of the sounds (they will be learn in reception), but they can choose familiar ones. | https://www.st-lukesislington.co.uk/week-commencing-1st-june-19/ |
|Location||44 Stillwater Rd.
|
Groton, Vermont
|Operated by||Vermont Department of Forests, Parks, and Recreation|
|Status||Memorial Day weekend - Labor Day weekend|
|Website||https://vtstateparks.com/stillwater.html|
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Stillwater State Park
|Location||126 Boulder Beach Rd., Groton, Vermont|
|Area||57 acres (23 ha)|
|Built||1938|
|Built by||CCC|
|Architectural style||CCC|
|MPS||Historic Park Landscapes in National and State Parks MPS|
|NRHP reference No.||02000278|
|Added to NRHP||March 29, 2002|
Stillwater State Park is a state park located on Lake Groton in Groton, Vermont. The park is located in Groton State Forest close to the Groton Nature Center, Boulder Beach State Park and Big Deer State Park. The park offers camping, picnicking, and access to water-related activities on Lake Groton. The park was developed in the 1930s by crews of the Civilian Conservation Corps (CCC). It is open to the public between Memorial Day weekend and Columbus Day weekend; fees are charged for day use and camping.
Features
The park is located in northern Groton, between Vermont Route 232 and Lake Groton off Boulder Beach Road. The park is 57 acres (23 ha) on the west side of the lake, which is ringed by vacation cottages and the facilities of the Groton Nature Center, Boulder Beach State Park and Big Deer State Park, also state parks. From the contact station a short way off Boulder Beach Road, the camp road divides to provide access to camping and recreational areas on either side of Stillwater Creek. To the left is a campground loop, originally built as a picnic area by the CCC. To the right is a second, larger campground loop, which includes day use facilities near the ranger's house at the northern end. The campground has a total of 62 tent/RV sites and 17 lean-tos. All rest rooms include hot showers. A sanitary dump station is available, but no hookups. There is a swimming beach, boat launch/dock facility, play area, shelter, and access to hiking trails.
History
The park facilities were developed in 1938 by the CCC, on state forest land the state began purchasing in 1919. The CCC was also responsible for building Vermont Route 232, which provides access into the forest. Crews built the park access roads, a picnic area on the north side of the creek, and a camp loop (about half the extant loop) on the south side. Surviving elements of their work include the ranger's house, the picnic shelter in the north loop, numerous stone fireplaces, and several leanto camping shelters. The north loop was converted into a campground in the late 1960s, when the southern loop was also enlarged. The park was listed on the National Register of Historic Places in 2002 for its well-preserved CCC architecture. | https://kids.kiddle.co/Stillwater_State_Park |
"I want to make friends all over the world! And I want many people to experience Japanese home cooking! I always think. Go to your friend 's house and eat pleasantly by making rice together! .... I'd like to have such a feeling. If you eat together, you get to know each other well, you can get along better. I am dreaming of making such encounters as people all over the world. I am very interested in international exchange since I was a child. I am hosting a host family of high school students from Switzerland, Australia and Spain, and now I am volunteering and enjoying studying Japanese for international students. Why do not you try new experiences with me who love to eat and cook? I am looking forward to it!"
Location
Aichi
Fumie's cooking classes
Aichi
Beautiful Japanese sweets "Nerikiri(wagashi)" and Matcha
¥5500
★ ★ ★ ★ ★
(3)
Experiences of making sweets (Wagashi) Cute Japanese sweets are not too sweet, and they match well with green tea or coffee. | https://airkitchen.me/users/?uid=24 |
New Delhi [India], August 24 (ANI): Afghan nationals held a protest on Monday in front of the United Nations High Commissioner for Refugees (UNHCR) office in New Delhi, seeking refugee cards and resettlement options in a third country. Scores of Afghan nationals gathered at the UNHCR office in Vasant Vihar and demanded refugee status/cards for resettlement. The Taliban declared the end of the two-decade war in Afghanistan last week, when the terror group entered Kabul, completing a weeks-long offensive across the country amid the departure of foreign troops. Since the Taliban took over the country, Afghan nationals are scrambling to get out of the war-ravaged country.
In response to the increasing number of Afghan nationals demanding refugee status, the UNHCR said refugees and asylum-seekers in India who have international protection needs to contact UNHCR or UNHCR's partners. "Less than 1 per cent of refugees are currently resettled globally, due to the limited number of places. For this reason, only the most vulnerable refugees are able to be prioritized for resettlement," UNHCR added. The UN agency further said that they are upscaling their support to Afghan refugees and prioritizing very vulnerable individuals. "We are upscaling our support to Afghan refugees and asylum- seekers, through registration and documentation for assistance, prioritizing very vulnerable individuals," said UNHCR. Many countries have resorted to evacuating their citizens and diplomatic personnel from Afghanistan due to the precarious security situation, and some have pledged to take in a limited number of Afghan asylum seekers. (ANI) | |
To dream of rain represents sadness, disappointment, difficulties, or depression. It may also reflect despair. Feelings about some area of your life being ruined or your happiness "rained on." Experiencing an unwanted change or that you are being swept along with by a problem. Grieving. A bad mood or feeling that something is going wrong. Feeling that bad times are ahead or that the future will not be very bright. Feeling that you are going miss someone or something in your life.
Rain water that builds up, or begins to rise reflects your feelings about problems becoming too much for you.
Example: A woman dreamed that it was raining so hard that the water began to leak into her home through the ceiling. In waking life her husband began to slowly lose his mind to a psychological condition. She felt hopeless about her husband's irreversible condition.
Example 2: A young man dreamed of walking down a street during the day and then turned a corner to find it began to rain. In waking life he was in the prime of his life and then suddenly fell ill to a disease that slowly destroyed his life.
Example 3: A woman dreamed of experiencing heavy rain fall. In waking life she was going through a divorce.
Example 4: A man dreamed of seeing rain. In waking life he had gotten turned down for a job and started to have a bad outlook on his future.
Example 5: A man dreamed of hard rain pouring down. In waking life he got a job in another city and was sad that he was going to miss all his friends once he moved away. | http://www.dreambible.com/search.php?q=Rain |
In the midst of the European debt crisis seven years ago, Irish filmmaker David Freyne had an idea for his first feature: a zombie movie.
The leap from eurozone austerity measures to flesh-eating creatures isn’t as big as one might think. The horror genre has long been a vehicle for social and political issues (just ask Jordan Peele, who has cited George Romero’s Night of the Living Dead as a main source of inspiration for his own racially driven thriller Get Out). With The Cured, Freyne is channeling the finger-pointing and populist politicians exploiting people’s fears during Europe’s economic downswing.
The Cured zeroes in on a town in Ireland that’s readjusting after a worldwide epidemic of a virus that turned people into the undead. A cure has restored a vast majority of the infected to normal, but their reintegration from quarantine is rife with vehement protests from the citizens who feel like the death and destruction the former zombies caused can’t be forgiven or forgotten. To make matters more complicated, there’s a small group of zombies who are resistant to the cure, which sets off a debate over whether to kill them, lest they spark another infection, or to continue to drain already strained resources in the pursuit of an antidote. Amid rising tensions, one of the cured, Conor (Tom Vaughan-Lawlor), becomes a leader for the disenfranchised group. But his extremist views clash with his protégé Senan (Sam Keeley) who just wants to quietly assimilate into the life he once had and keep his sister-in-law Abbie (Ellen Page) and his nephew safe.
“It just melded together, the idea of these cured who are being blamed for things beyond their control and being dehumanized,” Freyne says. “And how populist figures were manipulating people’s fear and anger and directing it at immigrants or asylum seekers and making them feel like they’re the problem, not banks.”
What Freyne so deftly accomplishes in The Cured is making sure his message never gets lost in clichéd scare tactics. The presence of zombies feels more atmospheric than something lurking in the shadows ready to pounce. A debate on categorizing his film as horror or drama could honestly go either way because, as Freyne mentions, it’s a character-driven film at its core.
“My hope with this film is that it’s very much so a horror-drama,” Freyne says. “[Horror is] such a great genre to tell a social story and that’s what it’s always done. What’s different now is people are beginning to really wake up to that and it’s becoming less and less niche, which is a really good thing.”
The Cured is in theaters and available on VOD now. | https://www.fastcompany.com/40537163/this-filmmaker-turned-the-euro-debt-crisis-into-a-zombie-movie |
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The fourth time is a charm for SpaceX, after successfully launching a rocket into orbit
Space Exploration Technologies, also known as SpaceX, successfully launched a commercial rocket into orbit carrying a dummy payload. On the company's fourth attempt, the Falcon I vehicle headed into space after launching from Omelek Island with a 364-pound dummy satellite.
SpaceX, backed by PayPal founder and Tesla Motors Chairman Elon Musk, wants to become the first company able to launch a privately developed rocket into Low Earth Orbit (LEO). Musk hopes the company is one day able to carry supplies -- and even astronauts or space tourists -- into space and to the International Space Station (ISS).
"
This really means a lot
," Musk said after the successful launch. "There's only a handful of countries on Earth that have done this. It's usually a country thing, not a company thing. We did it!"
Prior to the successful launch on Sunday, the latest attempt made it 135 miles above Earth's surface, but the rocket failed after the second stage was unable to separate from the first stage. This time around, the aluminum chamber designed to mimic a satellite will stay attached to the two-stage rocket as it begins to orbit Earth.
The Falcon 9 rocket could help NASA take cargo and astronauts into orbit in the future, assuming SpaceX can continue its successful launches. In addition, an injection of private capital makes it possible for SpaceX to keep attempting to prove the effectiveness of Falcon for one-tenth the total launch cost of commercial launches.
SpaceX plans to launch another Falcon 1 sometime in early 2009, with the Malaysian RazakSat satellite as its main cargo. If all goes according to plan, a Falcon 9 launch is expected sometime next summer.
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RE: Looks like...
By
Tsuwamono
on
9/30/08
,
Rating:
2
By
Tsuwamono
on
9/30/2008 12:00:31 PM
,
Rating:
2
I would think it would be easier actually... I always found it easier in a pool... that way nobody has to be held up.. Zero gravity means more energy for thrust instead of wasting energy holding one self up or your partner.
Parent
"Well, we didn't have anyone in line that got shot waiting for our system." -- Nintendo of America Vice President Perrin Kaplan
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Page 566, line 32, after "adults", insert "which includes the costs of administering and implementing the provisions of the Section 1115 waiver".
Page 566, line 35, after "providing coverage" insert ", which includes the costs of administering and implementing the provisions of the Section 1115 waiver,"
A. The Department of Medical Assistance Services (DMAS) is authorized to levy an assessment upon private acute care hospitals operating in Virginia in accordance with this item. Private acute care hospitals operating in Virginia shall pay a coverage assessment beginning on or after October 1, 2018. For the purposes of this coverage assessment, the definition of private acute care hospitals shall exclude public hospitals, freestanding psychiatric and rehabilitation hospitals, children's hospitals, long stay hospitals, long-term acute care hospitals and critical access hospitals.
B.1. The coverage assessment shall be used only to cover the non-federal share of the full cost for expanded Medicaid coverage for newly eligible individuals pursuant to 42 U.S.C. § 1396d(y)(1) of the Patient Protection and Affordable Care Act, including the administrative costs of collecting the coverage assessment, and implementing and operating the coverage for newly eligible adults.
2. The Department of Medical Assistance Services (DMAS) shall calculate each hospital's “coverage assessment" annually by multiplying the “coverage assessment percentage" times “net patient service revenue" as defined below.
3. The “coverage assessment percentage" shall be calculated as (i) 1.08 times the non-federal share of the “full cost of expanded Medicaid coverage" for newly eligible individuals under the Patient Protection and Affordable Care Act (42 U.S.C. § 1396d(y)(1)) divided by (ii) the total “net patient service revenue" for hospitals subject to the assessment. By May 1 of each year, DMAS shall report the estimated assessment payments by hospital and all assessment percentage calculations for the upcoming fiscal year to the Director, Department of Planning and Budget and Chairmen of the House Appropriations and Senate Finance Committees.
4. The “full cost of expanded Medicaid coverage" shall equal the amount estimated in the official Medicaid forecast due by November 1 of each year as required by paragraph A.1. of Item 307 of this Act. This Act estimates the non-federal share of the cost of coverage for FY 2019 as $80,823,953 and FY 2020 as $226,123,826.
5. Each hospital's “net patient service revenue" equals the amount reported in the most recent Virginia Health Information (VHI) “Hospital Detail Report" as of December 15 of each year. In the first year, net patient service revenue shall be prorated by the portion of the year subject to the tax.
6. Any estimated excess or shortfall of revenue from the previous year shall be deducted from or added to the “full cost of expanded Medicaid coverage" for the next year prior to the calculation of the “coverage assessment percentage."
7. DMAS shall be responsible for collecting the coverage assessment. Hospitals subject to the coverage assessment shall make quarterly payments to the department equal to 25 percent of the annual “coverage assessment" amount. In the first year, quarterly amounts for the remainder of the state fiscal year shall equal one-third of the coverage assessment. The payments are due not later than the first day of each quarter. In the first year, the first coverage assessment payment shall be due on or after October 1, 2018. Hospitals that fail to make the coverage assessment payments within 30 days of the due date shall incur a five percent penalty. Any unpaid coverage assessment or penalty will be considered a debt to the Commonwealth and DMAS is authorized to recover it as such.
8. DMAS shall submit a report due September 1 of each year to the Director, Department of Planning and Budget and Chairmen of the House Appropriations and Senate Finance Committees. The report shall include, for the most recently completed fiscal year, the revenue collected from the coverage assessment, expenditures for purposes authorized by this Item, and the year-end coverage assessment balance in the Health Care Coverage Assessment Fund.
9. All revenue from the coverage assessment including penalties shall be deposited into the Health Care Coverage Assessment Fund. Proceeds from the coverage assessment, including penalties, shall not be used for any other purpose than to cover the non-federal share of the full cost of enhanced Medicaid coverage for newly eligible individuals, pursuant to 42 U.S.S. § 1396d(y)(1) of the Patient Protection and Affordable Care Act, including the administrative costs of collecting the assessment, and implementing and operating the coverage for newly eligible adults.
10. Any provision of this Item is contingent upon approval by the Centers for Medicare and Medicaid Services if necessary.
B.1. The coverage assessment shall be used only to cover the non-federal share of the “full cost of expanded Medicaid coverage” for newly eligible individuals pursuant to 42 U.S.C. § 1396d(y)(1) of the Patient Protection and Affordable Care Act, including the administrative costs of collecting the coverage assessment and implementing and operating the coverage for newly eligible adults.
2.a. The “full cost of expanded Medicaid coverage” shall include: 1) any and all Medicaid expenditures related to individuals eligible for Medicaid pursuant to 42 U.S.C. § 1396d(y)(1) of the Patient Protection and Affordable Care Act, including any federal actions or repayments; and, 2) all administrative costs associated with providing coverage and collecting the coverage assessment.
b. The “full cost of expanded Medicaid coverage” shall be updated: 1) on November 1 of each year based on the official Medicaid forecast and latest administrative cost estimates developed by DMAS; 2) no more than 30 days after the enactment of this Act to reflect policy changes adopted by the latest session of the General Assembly; and 3) on March 1 of any year in which DMAS estimates that the most recent non-federal share of the “full cost of expanded Medicaid coverage” times 1.08 will be insufficient to pay all expenses in 2.a. for that year.
c. This Act estimates the non-federal share of the cost of Medicaid expansion to be $86,103,345 the first year and $293,192,716 the second year. However, these amounts shall not be construed as a limitation on collections or override the provisions of this item that allow for periodic updates of the full cost of coverage.
C. 1. The Department of Medical Assistance Services (DMAS) shall calculate each hospital's “coverage assessment amount" by multiplying the “coverage assessment percentage" times “net patient service revenue" as defined below.
2. The “coverage assessment percentage" shall be calculated as (i) 1.08 times the non-federal share of the “full cost of expanded Medicaid coverage" divided by (ii) the total “net patient service revenue" for hospitals subject to the assessment.
3. Each hospital's “net patient service revenue” equals the amount reported in the most recent Virginia Health Information (VHI) “Hospital Detail Report.” In FY 2019, net patient service revenue shall be prorated by the portion of the year subject to the tax. Hospitals shall certify that the net patient service revenue is hospital revenue and this amount shall be the assessment basis for the following fiscal year.
D.1. DMAS shall, at a minimum, update the “coverage assessment amount” to be effective on January 1, of each year. DMAS is further authorized to update the “coverage assessment amount” on a quarterly basis to ensure amounts are sufficient to cover the full cost of expanded Medicaid coverage based on the latest estimate. Hospitals shall be given no less than 30 days' notice prior to a change in its coverage assessment amount and be provided with associated calculations. Prior to any change to the coverage assessment amount, DMAS shall perform and incorporate a reconciliation of the Health Care Coverage Assessment Fund. Any estimated excess or shortfall of revenue since the previous reconciliation shall be deducted from or added to the “full cost of expanded Medicaid coverage" for the updated coverage assessment amount.
2. DMAS shall be responsible for collecting the coverage assessment amount. Hospitals subject to the coverage assessment shall make quarterly payments due no later than July 1, October 1, January 1 and April 1 of each state fiscal year. In FY 2019, quarterly amounts for the remainder of the state fiscal year shall equal one-third of the coverage assessment. In the first year, the first coverage assessment payment shall be due on or after October 1, 2018.
3. Hospitals that fail to make the coverage assessment payments within 30 days of the due date shall incur a five percent penalty that shall be deposited in the Virginia Health Care Fund. Any unpaid coverage assessment or penalty will be considered a debt to the Commonwealth and DMAS is authorized to recover it as such.
E. DMAS shall submit a report due September 1 of each year to the Director, Department of Planning and Budget and Chairmen of the House Appropriations and Senate Finance Committees. The report shall include, for the most recently completed fiscal year, the revenue collected from the coverage assessment, expenditures for purposes authorized by this Item, and the year-end coverage assessment balance in the Health Care Coverage Assessment Fund.
F. All revenue from the coverage assessment excluding penalties, shall be deposited into the Health Care Coverage Assessment Fund. Proceeds from the coverage assessment, excluding penalties, shall not be used for any other purpose than to cover the non-federal share of the full cost of expanded Medicaid coverage.
G. Any provision of this Item is contingent upon approval by the Centers for Medicare and Medicaid Services if necessary. | https://budget.lis.virginia.gov/amendment/2019/1/SB1100/Introduced/MR/3-5.15/1s/ |
8. Plan Route - Consider: Take off mins, enroute weather, obstacle clearances,
navigational aids
9. Enroute Consideration: - Obstructions clearance, Navigation aids, Freezing levels
10. Approaches - Consider: Weather mins, , is an alternate required (1-2-3 rule),
other ways to get
into field, check foot notes on approach plates
11. IFR – 1000ft 3 miles visibility
Filing and Picking Up Clearances
• File at least 30 minutes before you need it
• Pick up clearance 10 minutes before take-off
• Filed flight plans remain in system for 2 hours from ETD
• Void time allows you to depart IMC from an uncontrolled field
Filing an Alternate Airport:
• if 1 hour before and 1 hour after time of intended landing at primary airport the
weather is predicted to be below
2000’ ceiling and 3 SM visibility. (1-2-3 rule ,Filing an alternate airport is
required).
• Or if the primary airport does not have a published instrument approach. You
always have to file an alternate
airport regardless of the weather
• The exception to filing an alternate without an instrument approach is if a descent
can be made from the MEA
and land at the airport visually (VFR)
Alternate requirements
• When an alternate is required, check the approach plate to see if the airport can
legally be filed as an alternate.
Look for (A NA) It needs to have approved weather reporting.
• Alternate airport weather requirements at ETA : precision approach must have
600’ AGL ceiling and 2 miles
visibility. non-precision approach must have 800’ ceiling and 2 miles visibility if no
alternate weather minimums
are published.
Departure
Departure Procedures (DP’s) can be either
SIDs – Established for traffic flow, can be avoided by requesting "no sids’ on flight
plan. or...
DP’s – Established for obstacle clearance, must be followed.
ATC- Pilot Navigation – Formerly SIDs
ATC- Radar Vectors - Formerly SIDs
Obstacle Clearance (front or TRPPS) note terrible "T"s (
Standard take off min = 1 mile Vis for 2 eng or less ½ mile Vis for more than 1 eng.
Climb rate =Ground speed/60 xfeet/nm=climb rate
Approach Speeds are Based on Vso x 1.3
A= below 90 knots
B= 91 to 120 knots
C= 121-140 Knots
D= 141-165 knots
E= Greater than 165 knots
Flight review, Inspections & currency
• Biennial Flight Review 24 Calendar Months
• Medical Certificate 24 Calendar Months
• Transponder 24 Calendar Months
• Altimeter/Pitot/Static 24 Calendar Months (PAT 24)
• Annual 12 Calendar Months
• ELT 12 Calendar Months
• 100-Hour 100 hours-If for hire
• IFR Currency 6 Calendar Months
• VFR Currency 90 Days (3 TO AND LANDINGS TO CARRY PASSENGERS same
category and class)
• VOR Test 30 Days
1. The Transponder/Mode C and the Altimeter/Pitot/Static test are usually
performed
concurrently and recorded under a single logbook endorsement.
2. ELT
• Battery: Replace or recharge after more than 1 hour of continuous use or at ½ the
average
shelf-life
• To test: Tune to 121.5 during the first 5 min of every hour
3. VFR w/passengers
• 3 Take-offs and landings for day currency
• 3 Take-offs and landings to a full stop for night currency
4. IFR Currency
• All VFR requirements
• 6 approaches in the past 6 months with holding, intercepting and tracking courses.
Simulator or hood is acceptable.
• There is a 6-month grace period if approaches are not completed within the
previous 6
months; If not completed in next 6 months an instrument proficiency check by at
least a CFII
is required.
5. VOR Test Note date, place, bearing error, and signature
• Over airborne checkpoint: +/- 6
• Over ground checkpoint: +/- 4
• Against two VOR’s: 4
maximum difference.
• VOT test signal: +/- 4 with 180 TO indication or 360 from
Minimum equipment required for IFR flight. (GRABCARD)
Generator
Radios
Attitude indicator
Ball
Clock
Adjustable altimeter
Rate of turn indicator
Directional gyro
Partial-panel compass turns: UNOS
Undershoot
North
Overshoot
South
Compass dip: ANDS
Accelerate
North
Decelerate
South
Compass errors
compass deviation- caused by metals and electrical accessories in aircraft
variation- Difference between true north and magnetic north
Magnetic dip- magnetic north is below horizon due to curvature of the earth.
Flight Clearance: CRAFT
Clearance
Routing
Altitude
Frequency
Transponder
IFR mandatory reports: FAME Performance
Fixes: arriving or leaving
Altitude changes
Missed approach
Equipment: loss or problems
Performance: poor climb/descend, TAS change
RADIO FAILURE:
Much of the procedural elements of the IFR rating involve dealing with radio
communications failure. If we could completely count on radio contact, then such
things as initial clearances, procedure turns, and even expected arrival times would
be largely unnecessary, because we could simply expect ATC to give vectors to final
and keep aircraft separate in real time.
If you experience a radio failure, put the code 7600 on your transponder. If you
can still hear ATC (but not transmit), keep listening for instructions. (A standard
procedure for ATC is to ask if you can hear them and have you IDENT in reply.
Other questions can also be answered with an IDENT.) If radio reception is also a
problem, listen over nearby VOR and NDB channels, which ATC will also try.
The procedures for two-way radio communications are covered entirely by FAR
91.185, and, of course, here:
I. VFR: If communications failure happens in VMC, or if VFR conditions are
encountered after the failure and you can stay in VMC, you should continue the flight
under VFR and land as soon as practicable.
II. IFR: If the failure occurs in IFR conditions, then you should continue your
flight, and ATC will also assume that you are continuing, and clear airspace
accordingly. The three elements of the navigation are:
Route
Altitude
Leaving the clearance limit in order to shoot the approach
ROUTE
Think of "Avenue F": AVE F. This is the order of priority to your routing:
as Assigned
as Vectored
as Expected
as Filed.
1. Assigned: Fly the route assigned in the last ATC clearance received.
2. Vectored: If being radar vectored, fly directly to the fix, route, or airway
specified in the vectoring clearance.
3. Expected: In the absence of an assigned route, fly the route that ATC told you to
expect (in a further clearance).
4. Filed: In the absence of an assigned or expected routing, fly what you filed in
your flight plan.
ALTITUDE
Fly the highest of these three, for the segment of flight you're on:
Assigned Altitude
Expected Altitude
MEA
Assigned: The altitude assigned in your last clearance.
Expected: The altitude that ATC has told you to expect. ("Cherokee 2RJ expect
7,000 in ten minutes.")
MEA: The minimum enroute altitude for the segment you are on, as given on the
enroute chart.
In flying the highest of these three, your altitude may change repeatedly, because
the altitude assigned may be lower than the MEA for certain segments. In this case,
you should climb to the higher MEA, and then descend again when the MEA is lower
than your assigned or expected altitude.
LEAVING THE CLEARANCE LIMIT
Plan to leave the clearance limit or the IAF (if the limit was the airport itself) at the
time calculated from your flight plan. On the plan was an expected time enroute: add
that to your departure time off, and start your instrument approach procedure at
that time. If you arrive at the clearance limit before then, hold there until that
expected arrival time.
Preflight Planning Required by FARs §91.103:
• Weather reports and forecasts
• Known traffic delays as advised by ATC
• Runway lengths of intended use
• Alternatives if flight cannot be completed as planned
• Fuel requirements
• Takeoff and landing distance data in the approved aircraft flight manual
Definitions
Minimum Safe Altitude (MSA) Is the safe altitude within 25NM of the airport or
navaid and provides 1000’ obstacle clearance in both mountainous and non-
mountainous terrain. It is usually located within 30 miles of airport and is for
emergency use only.
Minimum Vectoring Altitude (MVA) Is the minimum altitude in which ATC can vector
an aircraft. This guarantees 1000’ obstacle clearance in non-mountainous, 2000’ in
mountainous, and 300’ within airspace.
Minimum Enroute Altitude (MEA) Is usually the lowest published altitude between
radio fixes that guarantees adequate navigation signal reception and obstruction
clearance (2000’ mountainous within 4NM, 1000’ elsewhere) Adequate communication
can be expected but not guaranteed. There may be gaps up to 65 miles as indicated
by "MEA GAP."
Minimum Obstruction Clearance Altitude (MOCA) guarantees obstacle clearance
(2000’ mountainous within 4NM, 1000’ elsewhere), but only guarantees navigation
signal coverage for 22 NM from the navigation facility. It is proceeded by a * on
NOS charts and a "T" on Jeppesen charts
Minimum Crossing Altitude is the lowest altitude at certain fixes at which an
aircraft must cross when proceeding in the direction of a higher minimum enroute
IFR altitude
Minimum Reception Altitude is the lowest altitude at which an intersection can be
determined.
Off-Route Obstruction Clearance Altitude (OROCA) gives 2000’ obstruction
clearance in mountainous areas
and 1000’ elsewhere within a latitude and longitude grid area.
Non-Precision Approach is a standard instrument approach procedure in which no
electronic glide slope is provided; for example NDB, VOR, TACAN, ASR, LDA, or
SDF
Precision Approach is an IAP in which an electronic glideslope is provided such as a
ILS, MLS, or PAR approaches.
Procedure Turn (PT) is a maneuver prescribed when it is necessary to reverse
direction in order to establish an aircraft on the intermediate approach segment or
on the final approach course. A procedure turn begins by overflying a facility or
fix. The maximum speed for a PT is 200 KIAS
Final Approach Fix (FAF) is at the glideslope intercept (lighting bolt) on a precision
approach. If ATC directs a glideslope intercept altitude which is lower than that
published, the actual point of glideslope intercept becomes the
FAF. The Maltese cross indicates the FAF on a non-precision approach.
Final Approach Point (FAP) applies only to non-precision with no designated FAF
such as on-airport VOR or NDB. It is the point at which an aircraft has completed
the procedure turn, is established inbound on the final approach course, and may
start the final descent. The FAP serves at the FAF and identifies the beginning of
the final approach segment.
Glideslope is a glide path that provides vertical guidance for an aircraft during
approach and landing. Applying the glideslope angle and the ground speed to the rate
of descent table gives a recommended vertical speed.
Height Above Touchdown (HAT) is the height above the highest point within the first
3000’ of the runway. It is
published in conjunction with straight-in approaches and appears next to the MDA or
DH of the approach plate.
Height Above Airport (HAA) is the height above the highest point on any of the
landing surfaces. It is published in conjunction with circling approaches and appears
next to the MDA of the approach plate.
Threshold Crossing Height (TCH) is the height above the threshold of the runway for
a given glideslope.
Touchdown Zone Elevation (TDZE) is the highest point within the first 3000’ of
runway
Field Elevation is the highest point on any of the landing surfaces. It is not the
highest point on the field, just the landing surface.
Minimum Descent Altitude (MDA) is the altitude on a non-precision approach in
which you must go missed or land visually and guarantees 300’ obstacle clearance.
Pilot can only go below MDA 30 degrees of the runway. Field Elevation + HAA=
MDA
Decision Height (DH) is the altitude on a precision approach while following a
glideslope in which you must go missed or land visually. HAT + TDZE =DH
• Cruise Clearance - Can fly between MEA and assigned altitude at Pilot’s
discretion but must
request lower once altitude attained.
• "CRUISE 6000"
• "You may climb and descend between your clearance altitude and MEA all you want
unless you report leaving an altitude. The key is to not report leaving an altitude!
• You are cleared to your destination airport and may shoot ANY of the instrument
approaches upon arrival without further clearance.
• Cannot get a cruise clearance on the ground.
• Review a sectional for terrain and obstacles to avoid CFIT.
• VFR on Top - Maintain visual separation but still IFR, and may want to get back
down. Must maintain cloud clearances (2000’ hor, 1000’ above, 500 below)
• Climb to VFR on Top - Cancel IFR once VFR on Top, your on your own, NOT
ALLOWED ABOVE FL 180.
• SVFR – Special VFR must be 1 mile clear of clouds, and can only be accepted at
night if pilot and aircraft are IFR. Allowed in Class B, C, D, E airspace. | http://flyaccelerated.com/general-info/ifr-ground-lessons.html |
According to the National Oceanic and Atmospheric Administration’s (NOAA) National Centers for Environmental Information, the 12 months between July 2018 and June 2019 saw record precipitation in the U.S., with an average of 37.86 inches, which is 7.9 inches above normal. With such brutal conditions to contend with, soil compaction has been a concern for many farmers this year.
While there are other causes of compaction, most of it comes from heavy machinery driving over fields when they are saturated. Studies have shown that compaction can reduce crop yields by 10-20%, depending upon a variety of factors, such as soil type, soil moisture, whether the machinery sports tracks or tires, the type of tire, tire inflation pressure and more.
No-Till Farmer editors asked several industry experts to share how producers can prevent, treat and survive compaction-related issues in 2019 and ensure the problems won’t linger into 2020.
Analysis First
No-till alone helps reduce compaction, simply by virtue of cutting out passes over the field. Where a conventionally-tilled field usually sees 7 or more passes over the course of a growing season, a no-till field can get by with as few as 2 or 3.
No-Till Takeaways
- To evaluate compaction, look at the plant roots. It they are growing horizontally, they most likely have hit a compaction layer.
- Tire pressures below 15 psi are unlikely to restrict root growth.
- To both prolong tire life and reduce the risk of causing compaction, tire inflation should be changed when switching between field work and road work.
Add in GPS and controlled traffic farming, and farmers can restrict compaction to limited areas. Nevertheless, prolonged wet conditions provide the perfect environment for damage.
“Compaction can happen anywhere if there are wet conditions, because it’s a function of the pore space between the organic matter, the water and the air in the soil,” says Brad Harris, manager of global agriculture field engineering at Firestone Ag. “And when we have water replacing the air molecules in the soil structure and then drive over them, that’s when we induce compaction into the soil.”
Tire ruts are an obvious sign of compaction but in fact compaction isn’t always visible. Jamie Patton, soil scientist with the University of Wisconsin, recommends a simple approach: “A shovel is a very valuable tool for evaluating soil compaction,” she says.
She suggests digging a hole that’s about 2 feet deep “because compaction can occur at the soil surface, but it also can compact at depth.” She looks at soil layers and structure, evaluating the soil aggregates to see if they are spherical or if they’ve been squished into blocks or plates.
In addition, Patton looks at the plant roots to see if they’re growing straight down or if they go sideways, which is a sign they’ve hit a compaction layer they can’t penetrate.
Another useful tool is a penetrometer, which measures resistance as it’s pushed down through the soil layers. Using a penetrometer can be tricky, though, because it will give different results depending upon soil moisture. In wet soils, the lubrication from the water will allow the tool to penetrate the soil with less pressure than would be required in dry soils.
Tires vs. Tracks
As equipment gets larger and heavier, whether farmers choose tires or tracks and how they use them plays a big role in the compaction story.
In 1987, Caterpillar released the first tractor with rubber tracks, providing a longer footprint, reduced soil compaction and better fuel consumption than was available at the time.
According to Jon Hanson, Midwest regional manager for the ag division of Soucy Track, “During harvest, for example, tracks let you get the crops out of the field with the least amount of damage, so you don’t have to do major rework in the field the following spring. And with these wet conditions, the windows of opportunity to get into the field seem to be getting shorter. Tracked units let farmers get into the field potentially sooner than they can with units on tires.”
But tracks have their downside, bringing higher maintenance costs, slower road speeds and difficulties with turning, according to Ryan Bales, field technician for Titan Intl.
Recent developments in tire technology have delivered new options for producers. The low-sidewall (LSW) tires that are now common on cars are touted as being equally beneficial in many ag applications, providing a larger footprint, better flotation and a smoother ride than their predecessors.
Extra-large LSW tires, like Titan Intl.’s Goodyear OptiTrac LSW 1400/30R46 ‘super singles,’ have been shown to have comparable performance to tracks in terms of traction, slippage and speed.
“Seven or eight years ago, your one option to solve compaction issues was a tracked machine. We’re offering another option now in a wheeled tractor that competes side-by-side with track,” says Bales.
Test results shared by Titan showed that a John Deere 9620R equipped with LSW 1400/30R46s consumed up to 15% less fuel and had an average of 16% less ground-bearing pressure than a John Deere 9RX track machine, at a potential machine cost savings of around $40,000.
The new CupWheel from Galileo, touted as running on lower air pressure, is also a promising new technology. “The large footprint reduces ground pressure and soil compaction,” says Avishay Novoplanski, co-founder of Galileo Wheel.
Under Pressure
For farmers using tires, the most impactful aspect to consider in terms of compaction is tire pressure.
“The most important thing with tires is to make sure you’re running the correct inflation pressure for the load you’re carrying,” says Harris. “You want enough pressure to ensure you’re not damaging the tires. Running ultra-low could cause problems. But if you’ve got too much inflation pressure in the tires, you start transferring more pressure into the ground, causing compaction.”
With standard radial tires, whatever pressure (measured as psi, or pounds per square inch) the tire is inflated to corresponds to the compaction that is being put into the soil. For example, if a tire is running at 20 psi, it is exerting 20-22 psi into the soil. If the actual load only requires 10 psi, operators are putting more pressure on the soil than is needed.
SUPER SINGLES. New extra-wide tires are being touted as a good alternative to tracks, and some studies show they can help reduce compaction by spreading pressure out over a larger footprint.
The good news is that pressures below 15 psi are unlikely to restrict root growth, according to Harris. “At 15-25 psi, you can start to see some crop damage if the soil or weather conditions get extremely stressful,” he says. “When you start getting above 25 psi, you need to start looking at larger tires, using dual tires or consider new tire technologies, like increased flexion (IF) and very high flexion (VF) tires.”
Specifically designed to help mitigate compaction problems, IF and VF tires can substantially reduce the amount of air pressure needed. Assuming a uniform load weight, IF tires require 20% less inflation pressure than standard tires; VF tires require 40% less inflation pressure than standard tires.
While determining the correct inflation pressure sounds fairly straight-forward, it gets a little more complicated because of the need to take into account the weight on each axle and the speed at which the machine will be operated. But few machines are operated under only one set of conditions.
Jim Enyart, manager of technical services at Ceat Specialty Tires explains, “We farm in the field at slow speeds and the recommended air pressures for ag tires are appropriate for that. But most farmers also drive their tractors down the road and they’re not going at slow speeds — they’re going as fast as they can. At that speed, they don’t have enough air pressure in the tire if the tires are inflated for in-field use.”
Of course, farmers can adjust tire pressure any time they’re moving equipment from the road to the field or vice versa. But research shows many farmers don’t do that. In fact, a survey done by Titan Intl. showed that nearly half of all growers report checking tire inflation pressures only twice per year.
Smart Tires
Now, wouldn’t it be great if farmers could just press a button and tires would inflate or deflate to the proper level? Well, the introduction of central tire inflation (CTI) has made that possible, and Bob Recker, for one, is a fan.
Recker, a no-tiller and owner of Cedar Valley Innovation in Waterloo, Iowa, says, “Running your tire pressure too high is like walking in soft soils wearing high heels. People know that and yet they want to go down the road safely without killing their tires. So they end up with high pressure because nobody wants to take the time to check 8 tires.” So he was excited for the opportunity to use a CTI system this spring.
“In two minutes I can check the air pressure and change them. That for me is pretty powerful because it’s going to give me longer life on my tires and I won’t have that compaction variable.”
Cover Crops
For farmers who are already dealing with compaction, new tires or tracks won’t do much other than prevent additional problems. Patton suggests that the location of the compaction, along with the producer’s management system, will dictate how they should go about addressing the problem.
COVER CROP CONDITIONING. A cover crop mix containing brassicas and grasses will help break up sub-surface soil compaction and re-aggregate damaged soils.
“If compaction is at the soil surface and you’re in an area that gets nice cold winters, some freeze/thaw action might help break that up,” she says. “If the compaction is lower in the profile, then planting cover crops with an array of root architectures may help.”
Patton recommends a combination of brassicas and grasses to break through sub-surface compaction and re-aggregate the surface soils. It does take time for cover crops to restore soil health, though, so a long-term mindset is helpful.
No-Till Deep Ripping
As a last resort, Patton says, no-till deep ripping may be necessary to deal with severe cases of compaction. | https://www.no-tillfarmer.com/articles/9202-how-to-combat-compaction-in-a-soggy-season |
A regular meeting for singers, rappers and musicians to come and register.
It's also a good time to meet other amateurs and train in the music business.
The studio is managed by RastaVin's, as there are many requests, the registration time is limited to 20 minutes per person and it is mandatory to register in advance on the registration form in the lobby of the Rex . | https://fiji.travlr.com/events/maoworkshoprec-rex20194 |
This game was invented by E. I. Csaszar in 1934. It differs from the perhaps more well known Baseline chess by its castling rules.
RulesThe game starts with only the pawns on their respective opening squares.
Then, players put turnwise one of their eight major pieces on an empty square at the first row at their side of the board. When all pieces are placed, the game starts. One plays using the usual rules of chess, but with the following castling rule.
A player can castle with a king and a rook, when the king has not moved, that rook has not moved, the king is not in check, and the king moves, while castling, not into check or over an attacked square, and all the squares the rook and king move over or to are empty. (These are more or less the usual conditions of castling.) When castling, the rook moves two squares in the direction of the king, and the king moves to the square that was passed over by the rook. So, for instance when a rook on c1 and a king on d1 castle, the rook moves to e1, and the king stays at d1. When a rook on c1 and a king on g1 castle, the rook moves to e1 and the king to d1. A rook on g1 cannot castle with a king on h1.
Written by: Hans Bodlaender.
WWW page created: January 22, 1997. Last modified: February 18, 2000. | http://server.chessvariants.com/diffsetup.dir/real.html |
One-hour workshops for pupils aged 11-16 to address the awkwardness around disability. The sessions aim to break down barriers that disabled people face and offer students the chance to ask questions they might not usually ask. Sessions are co-facilitated by a member of Scope staff and a role model volunteer (made up of disabled people from all walks of life).
Contact: [email protected]
EqualiTeachPrimary Secondary Parents
Workshops on all areas of equality, including racism, religious intolerance, sexism, homophobia, disability discrimination, global learning, critical thinking, immigration and human rights.
Telephone: 01480 470660
Show Racism the Red CardPrimary Secondary
Anti-racism workshops, and workshops focusing on hate crime and discrimination, as well as fitness sessions with ex-professional footballers that promote respect and teamwork in students.
Telephone: 01702 567166
|FREE|
Body and SoulPrimary Secondary
A locally based and national charity working with children, teenagers and families affected by HIV. Available to give talks and in-class sessions for secondary schools. Topics include HIV the basics, stigma, healthy relationships and empathy.
Telephone: 020 7923 6880
Hear2Change - Solace Women's AidFREESecondary
See Keeping safe and managing risk page for details. | https://www.islingtoncs.org/node/6989 |
Top 5 Private School Interview Questions
The private school interview is an important part of a student’s application; it’s a chance for students to show what they can contribute to the school and to learn more about the school. Below, you’ll find five of the most common questions asked at private school interviews and some important things to remember when answering these questions.
1. Why do you think our school would be a good fit for you?
This question is one of the most common questions asked during a private school interview. When you answer, tell the interviewer what you like about the school; be sure to mention specific aspects of their school that you like rather than general statements that could apply to all schools. If you’ve shadowed at the school or attended the open house, talk about specific things that you liked from your visit. You can also talk about certain classes and extracurricular activities you are interested in participating in at the school. If possible, talk about how you think the school can help you. Will the schools resources help you pursue certain academic interests? Will the small class sizes or engaging teachers help you become a better learner? Interviewers want to know specific things about their school that make it appealing to you.
2. What do you like to do in your free time?
This question is asked at almost every private school interview, so students should be prepared to talk about some hobbies they have outside of school. You don’t need to talk about everything that you do in your free time; pick a couple of hobbies that you are passionate about, and focus on those. These hobbies can include sports, drama, singing, writing, drawing, robotics, coding, creating Youtube videos, or anything else that your passionate about. You can also tell the interviewer how you will continue pursuing these hobbies at their school, or you can talk about a new interest that you want to pursue at their school.
3. What is your favorite subject, and what is your least favorite subject?
Be honest when answering this question. If you don’t like math and you really enjoy art, then say that! Don’t try to tell the interviewer what you think he/she wants to hear; the interviewer’s job is to craft a diverse class of students who have different interests. When talking about your favorite subject, be sure to explain why you enjoy that subject. For example, if you like art, you could say, “My favorite subject is art because it allows me to express myself in a creative way.” If you like math, you could say, “My favorite subject is math because I really enjoy coming up with new ways to solve problems.” When talking about your least favorite subject, be honest, but avoid being overly negative. For example, don’t talk about teachers that you don’t like, and don’t say things that make it seem like you have a bad attitude. Instead, try and turn your dislike of the subject into something positive. For example, if you don’t like reading you could say, “My least favorite subject is reading because I’m not the fastest reader. However, I’ve been meeting with my teacher to get tips on how to improve my speed and comprehension.” Answers like this show the interviewer that you are willing to work hard even if you are struggling or dislike a subject.
4. What are some of your strengths and weaknesses as a student?
This is a chance for you to show your academic strengths during your interview and to also honestly admit some things you could improve on. Everyone has areas in which they excel and areas that they can improve; the interviewer wants an honest answer about your strengths and weaknesses, so they can decide if their school is a good fit for you. When talking about your strengths, don’t be shy. You don’t need to brag, but if you’re a gifted writer and you’ve started your own blog, or if you’re extremely organized and manage your time well, tell the interviewer! When talking about your weaknesses, make sure you talk about how you want to work on your weaknesses, and try to talk about how you think their school can help you improve those areas of weakness.
5. Do you have any questions for me?
Most interviewers will end the interview with this question. It’s important to think of a few questions in advance, so you can have at least one question to ask the interviewer; asking questions shows the interviewer that you are genuinely interested in the school and that you’ve taken the time to do some research. Avoid generic questions such as, “What classes do you offer?” Instead, ask specific questions that show you’ve put in the time to research their school. Ask questions about things that you are genuinely interested in. For example, if you are interested in graphic design, you can say, “I read on your website that you have a graphic design club; I was wondering what types of graphic design students learn in this club?” If you are interested in community service, and you have a specific type of community service that interests you, such as working with kids, ask if the school has any opportunities in this area. Whatever the question, make sure it is specific and shows your interests. | https://www.elevateprep.com/post/top-5-private-school-interview-questions |
A healthy whole wheat bread made with fresh apples and walnuts.
Serves:
15 slices
Ingredients
1 Cup Greek Non Fat Plain Yogurt
2 Tbsp. Coconut Oil or Canola Oil
¾ cup NuNaturals Presweet Tagatose or ½ cup sugar
3 egg whites
1 tsp vanilla extract
1 Cup Whole Wheat Pastry Flour
¼ cup flax seed, ground
2 tbsp wheat germ
1 tsp baking powder
1 tsp baking soda
½ tsp salt
2 tsp cinnamon, ground
1 apple (1/2 cup) peeled diced small or shredded (if shredded don't squeeze out the juice)
¼ cup chopped walnuts (you can add up to ½ cup)
Instructions
Preheat oven to 350°
Grease 1 large loaf pan or 3 mini loaf pans with cooking spray; set a side.
In a medium bowl, mix yogurt, coconut oil, sugar, egg whites and vanilla extract; mix until until blended.
In a separate bowl add flour, flax seed, wheat germ, baking powder, baking soda, salt and cinnamon. Mix together.
Slowly incorporate the flour mixture into the yogurt mixture, be careful not to over mix.
Fold in apples and walnuts.
Spoon batter into a greased 9- x 5-inch loaf pan or 3 mini loafs. Use the back of the spoon to smooth batter out.
Bake for 50 minutes to 1 hour for 1 large loaf or 30 – 35 minutes for 3 mini loafs, or until a wooden pick inserted into center comes out clean. Cool in pan on a wire rack 10 minutes; remove from pan to wire rack.
Notes
Calories per slice with NuNaturals: 100.6, Fat: 4, Cholesterol: 0, Sodium: 209, Potassium: 50, Carbs: 18.4, Fiber: 2, Sugar: .06, Protein: 4.2
Calories per slice with Sugar: 1112, Fat: 4, Cholesterol: 0, Sodium: 209, Potassium: 50, Carbs: 15.5, Fiber: 2, Sugar: 7.3, Protein: 4.2
Nutrition Information
Serving size: | https://www.peanutbutterandpeppers.com/easyrecipe-print/10575-0/ |
Understanding spring solstice vs spring equinox (including spring equinox and vernal equinox) and 15 spring equinox lesson plans (vernal equinox lesson for kids), solstice and equinox lesson plans and STEM activities.
Here are some spring equinox facts you can discuss with the kids before getting to spring activities for kids.
Be sure to read through it because it covers common questions, especially on spring solstice versus spring equinox.
Let’s take a closer look at the spring equinox definition…
What is spring solstice?
Spring solstice is actually called spring equinox. (People mistakenly called it spring solstice sometimes.) It is when the sun crosses the equator going north. In the northern hemisphere, the sun’s rays are directly over the equator and making it have 12 hours of day and 12 hours of night (an equal amount). It marks the changing of the season to spring.
What are the solstices and equinoxes?
What is the difference between a solstice and an equinox? An equinox happens when the amount of day time is equal to the amount of night time. It happens twice per year — as a spring equinox and a fall equinox. A solstice happens when it is the longest day of the year (summer solstice) and the shortest day of the year (winter solstice). A solstice happens when the sun’s zenith is at the furthest point from the equator.
When is spring equinox this year?
In the Northern Hemisphere, Spring equinox occurs on Thursday, March 19, 2020.
What is the exact time of the spring equinox?
The spring equinox will occur at 11:59 p.m. Eastern Standard Time on March 19, 2020.
What is the beginning of spring called?
The first day of spring is called the spring equinox. It is also called the March equinox or the vernal equinox.
What is the spring equinox called?
In the Northern Hemisphere, the spring equinox is called the vernal equinox.
According to Merriam-Webster:
“Equinox comes from the Latin words aequi, which means equal, and nox, which means night. The vernal equinox is considered the first day of spring: finally, the day and night are of equal length.”
How do you celebrate the spring equinox?
Here are 13 fun ways to celebrate the spring (vernal) equinox with kids:
- Read a book and learn more about spring equinox.
- Talk a nature hike.
- Create a spring bird feeder.
- Plan a garden to plant later in the year.
- Visit a farm where baby animals were just born.
- Go for a bike ride.
- Make and hide kindness rocks during a spring equinox walk.
- Make some baby chick cupcakes.
- Pick or plant some flowers.
- Make a recipe with seasonal spring vegetables or fruits.
- Get organized with spring cleaning!
- Plant a container fairy garden.
- Go on a signs of spring scavenger hunt.
SEE OUR FULL LIST OF SPRING EQUINOX ACTIVITIES AND LESSON PLANS BELOW. KEEP SCROLLING!
It seems like every year I see parent and homeschooling groups trying the “egg stand up on the equinox” trick and then being amazed.
Let’s learn more about standing up an egg on equinox…
Does an egg stand up on the equinox?
There are claims that during the equinoxes you can stand an egg on end because of the pull of equal gravity toward both the North and South. However, the spring or fall equinox egg balance is actually a myth. You can, however, balance an egg any day of the year with some practice!
Try it!
You May Also Enjoy:
Full Moon Schedule and 22 Moon Activities for Kids
15 Spring Equinox (“Spring Solstice”) Activities, Crafts, and Lessons for Kids
- Watch a video about equinoxes and solstices
- Read a book with a multicultural perspective of spring equinox
- Watch the video Happy Equinox!
- Learn about the earth’s rotation in the roll of equinoxes
- Get solstice and equinox lesson plans (grades 6-8)
- Use the interactive map to learn why we have seasons
- Make your own solar system model to practice seasons
- Make a hand flower craft
- Talk about the egg balancing myth and then try balancing an egg!
- Start seedlings indoors to transplant outside or to a container garden later in the season
- Do a lesson plan on the seasons. | https://homeschoolsuperfreak.com/spring-solstice/ |
How to make fudgy brownies :
Keto brownies recipe (Fudgy):
Who doesn’t like brownies ?!, we all do, well most of us, now this is a really easy recipe to make fudy keto brownies of only 8 ingredients and the results are amazing, now put on that apron and get ready to enjoy the ride (before- during and after).
There are certainly all kinds of different ways to make brownies but this one just tops it all for me, from the taste to the mixture to the smell of Vanilla.
Ingrediens:
- 1/4 cup Coco Powder
- 2 tbsp Coconut flour
- 3 large eggs (room temperature)
- 12 tbsp Butter
- 1/2 cup erythritol
- 1/2 tsp vanilla
- pinch of salt
- 2 oz Unsweetened Bakers Chocolate
Preheat the oven to 400F°
Preparation Instructions :
Step 1 : Use a loaf pan 8*4 (if you want to use a normal brownies pan it’s going to take longer to make).
Step 2 : In a bowl add 1/4 of cocoa powder, add 2 tablespoon of coconut flour and a dash of salt, press it while mixing and set the bowl aside
Step 3 : in a seperate mixing bowl crack 3 eggs, and add half a cup of eryhrithol plus half a teaspoon of vanilla extract , you want to whip up the eggs untill they become 3 times the size. Set the bowl aside as well.
Step 4 : In another bowl add two ounces of bakers chocolate, microwave the chocolate and the butter together (feel free to use a double boiler instead) dont exceed 30 seconds so you don’t burn either. Add the eggs in 3 parts, mixing each time.
Step 5 : Switch from a whisk to a spatula to combine the mixture well.
Step 6 : For your final step: add in the dry ingredients in 3 parts, mixing with each time.
Once you get a pretty loose and thick better, this is exactly what you are looking for.
Step 7 : Grease the loaf pan with coconut oil spray (this is an important tip, do not forget to grease the pan)
Step 8 : Poor the batter in the pan and even it out and bake it at 350 F° from 55 to 60 Minutes
Step 9 : Let It cool for 15 to 20 minutes and cutt them to 10 mini-pieces.
Tools that you will need : | https://healthagenda.us/keto-brownies-recipes/ |
Your article on how to bring swords to a gunfight was very interesting, and I’m dealing with a (somewhat) similar idea except that it involves characters. The plot’s picking up speed. Things are getting serious. Your protagonist is in over their head. They don’t know what’s going on and suddenly they’re smack in the middle of a punching match / firefight / laser battle / duel / car chase / some combination of the above. Oh, noes! You want your character to be super awesome and have a super awesome throw down, or at least survive the encounter, but they’re just your average protagonist with no fighting expertise! What should you do? Basically, my question is: how do you bring an inexperienced protagonist to a firefight?
This can also be a problem with groups of protagonists, where maybe only one has real fighting expertise or training and the others are suddenly deadweight as soon as the fight starts. And if you’re writing, say, a thriller, your character probably wouldn’t realistically survive their first elbow brush with the baddies, much less learn how to fight over the relatively short time span that most novels take place in. Is the only solution to this problem to give characters fighting backgrounds? Is there a way to bring an inexperienced protagonist to the top in a fight without snapping believability? What’re your thoughts?
Thanks!
Bunny
Hey Bunny, nice to hear from you again!
I’ve seen the problem you’re describing in a lot of manuscripts. I’ve even started calling it the Frodo-Aragorn Problem: authors not only want their protagonist to be a sympathetic underdog, like Frodo, but also want them to be a super badass, like Aragorn, for all their cool battle sequences. Fortunately, there are a few solutions, though I wouldn’t call any of them quick fixes.
1. Make Your Character a Badass
If your story has a lot of fight scenes, it’s okay to revise the protagonist’s backstory so they’re good at fighting. This is usually easy to insert into their backstory, though the exact method will depend on your story’s setting. Perhaps the character has a military background, or perhaps they grew up fighting street battles with rival gangs. Even a background in sports like judo or boxing can go a long way in giving your character fighting skills.
The protagonist won’t have Frodo’s underdog sympathy, but that’s okay; there are other ways to make them sympathetic.
2. Avoid Fight Scenes
If your story is about a character who can’t fight, it’s okay not to have fight scenes. That way you don’t need to explain how a complete neophyte bests or escapes from a seasoned warrior. Some storytellers think they need action sequences to make their stories exciting, but they don’t. Action sequences are just one way to create conflict and tension. Your story can be about a small town’s political struggle to tear down an old building as long as you show how these stakes are compelling to the characters.
3. Focus on the Underdog’s Experience
Okay, if you’re set on fight scenes with a protagonist who isn’t good at fighting, you can still do that, but it’s tricky. When a fight scene starts, you have to focus on how the protagonist avoids danger, using what skills they have to make up for the skills they don’t. Returning to Lord of the Rings, when Frodo is in a fight, the story focuses on how he scrambles away from stronger enemies and hides in tiny crevices or uses his intelligence to outwit them. Your character might not have the skills necessary for a high-speed car chase, but they might know that they can take a shortcut through a local parking garage that never locks their gate.
There are limits to this method though. If the power between hero and villain is too skewed in the villain’s favor, it simply won’t be believable for the hero to escape. To address this, it can be helpful to include other characters who are better at fighting, the way LotR includes the rest of the fellowship. We don’t usually focus on Aragorn, especially early on; we just know he’s being a badass ranger while the hobbits are hiding. Tolkien makes this a little easier on himself by using an omniscient narration, but you can do it in limited as well. Just make sure the more fighty characters don’t overshadow the protagonist.
When using unconventional skills to get a protagonist out of trouble, it’s extra important to establish them ahead of time, since you’re asking the audience to accept that these skills will compensate for a serious power imbalance. If your hero is going to use their knowledge of the city to escape a car chase, the audience needs to know about that ahead of time.
4. Train the Underdog
If you’re going with option three, it can also be helpful to show the hero gaining combat skills as the story goes on. This works best in longer stories since it’ll seem painfully unrealistic if the hero suddenly acquires a black belt between scenes. Rather than a sudden change, you can show the character getting slowly better after each encounter as well as the occasional training sequence. That way your hero can start off not knowing which end of the sword to hold and end the story by defeating the villain in a duel.
You may have guessed by now, but this is what Lord of the Rings does. Frodo never reaches Aragorn’s level of badassery, but he and the other hobbits get noticeably better as the trilogy goes on, culminating in the Scouring of the Shire, where they lead the other hobbits in a battle to oust Saruman and his cronies. When done properly, this kind of slow transformation is incredibly satisfying.
Finally, a few articles that touch on the subjects mentioned above.
- Twelve Traits of a Lovable Hero
- How to Narrate a Riveting Fight Scene
- Five Archetypes That Can Steal a Hero’s Spotlight
- The Why & How of Foreshadowing
Hope that answers your question!
Do you have a question you’d like answered by Oren or Chris? Submit it here. Q&As are only made public if you give your permission and we decide to feature it. If you’d like more than an answer to a general question, you can hire us to look over your story. | https://mythcreants.com/blog/how-do-you-bring-an-inexperienced-hero-to-a-fight/?replytocom=390210 |
A tour of the Rías Baixas
A total of 62 town councils in 10 regions make up the Rías Baixas. We’re going to take a tour of some of the most interesting municipalities and locations:
Sanxenxo
Situated in the Salnés Valley, Sanxenxo is famous worldwide for the immense beauty of its landscape, its special microclimate and fantastic beaches which line its entire coast.
With 13 blue flags, year on year it is the Spanish municipality with the most recognition from the European Union.
It is worth visiting its impressive recreational port, Pazo de los Patiño manor house and Iglesia de San Xinés church.
Baiona
Its old quarter has been declared a Site of Historical Artistic Interest. The main attractions are the Museo de la Carabela La Pinta museum, the Fortaleza de Monterreal fortress (now a parador) and the large recreational port.
The town’s tranquillity and immense beauty are two of its main tourist attractions.
Combarro
A town of traditional granaries and ships, its traditional streets retain all the popular architecture of the 18th and 19th centuries, thanks to which it has been recognised as a historical-artistic site of interest.
Its small seaside houses with delicate stonework are a delight to behold for any visitor.
O Grove
This municipality consists of a small peninsula which geographically separates the Rias of Arousa and Pontevedra.
Its Shellfish Festival, considered of national interest to tourists, welcomes thousands of visitors from all over the world in the town’s port.
Cambados
This town is listed as a historical-artistic site of interest. It’s worth taking a walk around its three districts: the stately Fefiñáns, Cambados business and shopping district and the seaside area of Santo Tomé.
Cambados also has wonderful Albariño wineries where visitors are invited to taste this prized wine, which is only produced in this area. Its main monuments include the Pazo de Ulloa and Pazo de Fefiñáns manor houses and the As Vieiras monument by Manolo Paz.
Ons Island
The nearby Ons Island forms part of the Natural Park of the Atlantic Islands, next to the Cíes, Sálvora, and Cortegada islands.
Boats leave for Ons Island from Sanxenxo and Portonovo every 45 minutes from 10 in the morning. Thanks to its size, visitors can tour the entire island in just a day.
La Toja Island
A small and idyllic island situated in the Rías Baixas, across from O Grove and 30 km from Pontevedra.
It is famous for its magnificent spa, surrounded by stunning and idyllic scenery, with a practically virgin forest surrounding it. If you’re looking for somewhere quiet to get away from it all, relax and meditate, this is the place.
Pontevedra
As the capital of the province, its beautiful and well preserved old quarter has been declared a historical-artistic site of interest.
Its main monuments include the Real Basílica Santa María la Mayor basilica, the Pazo Provincial manor house, Santuario de las Apariciones sanctuary and the famous Museo Provincial museum.
The best beaches in Sanxenxo: Playa Canelas …
Canelas Beach
In the shape of a half shell, this beach is situated in a semi-urban area next to Portonovo and bounded by Seame point on its left side and Punta Cabicastro on its right side.
Caneliñas Beach
Situated in the urban area of Portonovo, bounded at its edges by Caneliñas viewpoint, a beautiful spot with panoramic views of the inland area of Pontevedra Ria and, by the Seame point, which separates it from Canelas beach.
Paxariñas Beach
It is situated in a rural area within the parish of Adina, between Cabicastro point and a small rocky projection known as Paxariñas point.
Its sand is white in colour and very fine.
Silgar Beach
Situated in the centre of Sanxenxo, between the Juan Carlos I recreational port and Punta Vicaño.
This is an urban style beach. There are very few sandy strips with these characteristics in the Rías Baixas.
Baltar Beach
Situated in the urban area of Portonovo it is in the shape of a half shell.
It is delimited by Vicaño point and the Muelle del Chasco, the recreational port of Portonovo.
Montalvo Beach
It is delimited by Montalvo point on its left side and Paxariñas point on its right side.
It is 1000 metres long with fine, white sand. It is very well-known because the swell of the sea is ideal for surfing or body boarding.
Areas Beach
Situated in the parishes of Bordóns and Dorrón it is situated in a semi-urban area. Bounded by the small Cala dos Mortos cove, with access when the sea level is low and Cabicastro point, it offers stunning views of Pontevedra Ria.
Bascuas Beach
It has the attraction of being the only nudist beach in the municipality of Sanxenxo. This is a small cove in a rectilinear shape, situated in a rural area, between the beaches of Montalvo and Pragueira, situated amongst cliffs.
A Lanzada Beach
It is situated in the municipalities of Sanxenxo and O Grove.
This is one of the most iconic and famous Galician beaches, as well as being one of the best as regards the quality of the sand and cleanliness of the water. It is also a popular amongst surfers, windsurfers and kitesurfers thanks to its perfect wave and wind conditions. | https://www.hotelduna.com/en/portonovo-sanxenxo/playa-canelas/ |
This recipe is a continuation on my obsession with pappardelle pasta. The sauce coats the ribbons of pasta perfectly. I can’t get over how good it is.
My family has always loved eating Italian sausages with sauteéd peppers and onions on a bun. I would say we had it at least once a week growing up. This recipe is kind of my take on our popular family recipe, but in pasta form.
I also added red wine..adds a great depth of flavor to this classic combo of flavors.
Serves 4-6
Prep time: 20 min
Cook Time: 30 min
-9 oz. pappardelle pasta (1 package)
-2 bell peppers, thinly sliced
-1 large yellow onion, thinly sliced
-3 cloves garlic, chopped
-1/2 tsp. dried basil
-1/2 tsp. dried thyme
-pinch of red pepper flake
-1 c. red wine (I used a red blend, but anything will work here)
-1 1/2 lbs. sweet Italian sausage
-2 c. white button mushrooms, sliced
-2 c. crimini mushrooms, sliced
-2 T. balsamic vinegar
-1/2 c. grated parmesan cheese
-1/4 c. fresh basil, chopped
-olive oil
-salt and pepper
Bring a large pot of water to boil for pasta and cook according to package directions or desired doneness. Begin by sautéing peppers, onions, and garlic in 2 T. olive oil over medium high heat. Season with salt and pepper, as well as dried basil, thyme, and red pepper flakes, and cook for 10 minutes or so until golden and caramelized. Add wine to pan and reduce, cooking for an additional 3-4 minutes. Set aside.
Using the same pan, heat another 2 T. olive oil over medium/high heat and sauté mushrooms for 5-6 minutes until caramelized and golden. Add balsamic vinegar, and cook for an additional 1-2 minutes until some of the vinegar has evaporated. Season with salt and pepper. Set aside.
Next, remove sausage from casings, and sauté in same pan for 6-7 minutes, until browned, breaking up with a spatula into small pieces. Set aside.
Place pasta back in the large pot, or a large serving bowl, drizzle with 2 T. olive oil, and add caramelized onions and peppers, mushrooms, sausage, parmesan cheese, and chopped fresh basil. Toss to combine, and garnish with additional chopped herbs and parmesan cheese, if desired. Serve and enjoy! | https://www.spicesinmydna.com/pappardelle-with-sweet-italian-sausage-peppers-onions-and-red-wine/ |
Since this social benefit was first applied, many have many doubts (especially when it comes to foreigners). But in this article we will solve them all. If you would like to know what exactly the minimum vital income in Spain is and how it works, what requirements you will have to meet in order to apply for it successfully (plus how to do it step by step); then keep on reading. In this article we answer all these questions and provide you with other useful information about this income.
What is the minimum vital income?
The minimum vital income is a benefit granted by the Government through the Spanish Social Security that aims to guarantee a minimum standard of living to those people who do not receive a level of income considered as basic (or who do not have huge wealth possessions) through a monthly monetary benefit.
In other words, the payment of a fixed amount of money at the end of each month.
The amount of this benefit, as well as all its conditions, are regulated in the Royal Decree-Law 20/2020, and they depend greatly on the particular case of the applicant.
Thus, depending on the situation of each family, the number of members living together, the socio-economic situation, and a set of extra factors, the amount received at the end of the month will vary (below we will see a table with the specific income brackets that are granted).
In addition, it will be very important that you look at all the conditions and requirements that must be met in order to apply for this benefit (which, as we have already mentioned, are quite extensive).
Without further ado, let’s analyze this social assistance in greater detail.
Who can apply for it?
Let’s look at the main groups that can apply for this benefit:
- First of all, we are talking about individuals who must be between 23 and 65 years old with full capacity to act, and who live alone. This would be the general case.
- In addition, women over the age of 18 who are victims of gender violence or human trafficking.
- In the case of receiving a residential, social, or health service benefit that of a permanent nature (financed through public funds), then you can’t apply for this monthly income.
- Those over 18 years old or below 18 but emancipated who have children or are in charge of minors in adoption.
- Individuals over 65 years old long as their cohabitation unit is formed only by persons over 65 years of age, or also including minors or persons who are judicially incapacitated.
But what happens if don’t live alone?
People who share a household may also apply for the minimum vital income; that is, when we find family units composed of more than one member, as long as:
- They are neither registered as a civil union with another person nor married (unless divorce proceedings have been initiated).
- Have not been living independently for the last 3 years, unless they have done so after initiating divorce proceedings or due to gender violence.
- Do not form part of another cohabitation unit.
- They have been registered with Social Security for at least 12 months (regardless of whether this period is continuous or not), and that during the last 3 years have lived at a different address from that of their parents or legal guardians.
Can more than one person at the same house or family unit apply?
No. That is not possible.
There can only be one person receiving the minimum vital income per household, even if there are several families or individuals in the same house.
In other words, only one member of your household can receive this benefit. Although, as we will see, the more people in the family unit, the higher the income received.
Main requirements
Now that we have seen who can apply for the minimum vital income, let’s understand what are the exact conditions or requirements you must meet in order to apply and receive this benefit.
Basically, there are four main conditions:
1. To have legal residence in Spain uninterruptedly for at least one year. Foreigners with a visa would not fall into this group, since the time in Spain with a visa is a time considered as of stay (and not of residency); therefore, it will be important to have a residence permit.
2. Secondly, and perhaps the most important point, if the average monthly income you receive is at least 10 euros less than the monthly amount that this minimum vital income will pay (and this depends on your family situation, as you will receive more or less income depending on that). This is the maximum amount of earnings you can receive if you want to get this benefit.
3. You must also be registered as a job seeker (as long as you are of legal age).
4. Finally, it is important to mention that no member of the family unit that is going to receive this income can be the owner of a company; nor have a patrimony superior to the established limit.
In order to effectively demonstrate that you are in a situation of economic vulnerability, you must comply with points 2. and 4. When it comes to the latter point, below you can see the maximum asset value (in euros) that you can have if you want to apply for this social benefit:
Minimum vital income simulator
So far we have seen all the situations in which you could apply for this monthly income.
However, it is very likely that you still are not quite sure about if you actually could apply or not, as there are many exceptions and different situations.
Thus, the government created a very useful tool that will allow you to know if you can benefit from this income or not: the simulator of the minimum living income.
You just have to enter the following link, answer the questions that appear there, and you will finally see if your particular case is applicable.
Click here to access the minimum vital income simulator.
Step by step application process
The process is relatively simple, and can be done completely online.
The first thing you will have to do is enter the Social Security Sede Electrónica, and go to “ciudadanos” > “familia”.
You can also access directly by clicking here.
Once inside, select the drop-down menu option “Ingreso Mínimo Vital”.
You will see that different options open up for you after scrolling down a bit; as you can apply with an electronic certificate, with cl@ve, or without a certificate:
Once you select the option according to your particular case, you will have to fill in the information that appears on the screen: name, date of birth, ID card, marital status, etc.
It is important that you answer all the questions correctly.
And upload your ID scanned, front and backside.
Finally, you must attach the accreditation of willingness to apply for the benefit (a document with your name and ID number detailing your intention to apply for this benefit); and verify that all the data you have entered is correct.
And that’s it!
Other important documents for your application
In addition to what we have seen so far, there are also a series of documents that will be essential to submit in order to prove that you meet the requirements to receive this benefit:
- Certificate of census registration (“empadronamiento”) from the town hall that corresponds to your address (in the case of not living alone, the rest of the members of the family unit must be registered in that same census).
- DNI, family book, or residence card (which must be valid at the time of the application).
- In the case of being registered as a civil partner with your partner, you must also submit your family book with the certificate from the civil registry that proves it.
- Document that verifies that you are actually a job seeker in Spain.
What is the exact amount to be received each month?
The exact amount to be received depends greatly on the number of individuals in each family unit or cohabitation unit (i.e., in the same household).
In the following table you can see the different amounts for 2021, divided into two groups: single-parent units and non-single-parent units:
Check the status of your application
Once you have submitted your application, it is very likely that you will want to know its status and find out if it has been approved or not.
And the truth is that you can do so easily and online.
Just enter the Social Security’s platform through on this link, click on the button “Aportar documentación / solicitar estado”, and enter the application code (which you will receive after sending all the documentation), and your ID card.
For any other questions, our lawyers are at your complete disposal through the following link: | https://balcellsgroup.com/minimum-vital-income/ |
This is a guest post by Benji Heywood, Director of Competitions for UK Ultimate
For more TD Tuesdays articles: http://leaguevine.com/blog/tags/td-tuesdays/
Photo courtesy of Windmill Windup
First things first - this isn't going to be a format-manual kind of article. That would be very long, and there's sure to be some particular constraint at your venue or with your number of teams that makes your case different... Instead, I'm just going to talk about some of the things that might help if you're inexperienced at writing schedules. First, and most important:
The Golden Rule of Scheduling:
You WILL get it wrong
I've written hundreds of schedules, for big events and small, and the most crucial advice I can give is to get someone else to check it. Really check it, not just glance at it and assume you've got it vaguely right. No matter how careful you are, there'll be an error in there. I still miss something every single time. It might be that a game is missing or played twice; it might be that the pools are seeded so as to have rematches in the Quarters; it might be that you moved some games around to avoid some other problem and now a team is playing two games at the same time.
It might be something tiny, like the order of games in a pool (usually best to play the most important games last) or the fact that you could rearrange it so that quarter-final opponents could watch each other. There are a million ways to make a complete mess of it, and also a million small improvements that could tidy up an already usable schedule. It's like writing an essay - there's always something else you could tweak, right up to the moment you hand it in.
An example from this year, which I particularly enjoyed: I put the women's matches on the far pitches at a big tournament, and got complaints that it was too far from the toilets - girls can't go in the woods so easily. There's always something...
Schedules are complicated, and you cannot keep the whole thing in your head; when you make a change, it's very hard to go through and check that you didn't cause another problem, because it's all so familiar already and checking is boring. Get someone to look at it with fresh eyes.
Games and game-breaks
Different countries accept different schedules. In the UK, we run schedules that mainland Europe wouldn't consider; USAU run schedules that we wouldn't touch with a 30-foot pole. The UPA format manual, for example, might have you playing in a pool of seven, then a bracket with as many as 3 games to finish - 9 in a weekend. The player base is perhaps used to that, and will bring huge squads that can cope with the demands of playing maybe 4 times in 5 slots; at UK tournaments, or indeed at fun tournaments with smaller squads, that sort of schedule is not going to be popular - we have an absolute horror of 3-in-a-row at official tournaments. In the UK we always try to play 3 games per day, 2 of them back-to-back (so that you only warm up twice); in much of Europe, playing even 2 in a row would be considered a shockingly bad schedule. So i guess a big thing to think about is your intended audience - squads of 20 or squads of 8? Athletes or drunks?
The type of schedule you run depends on the type of event - at fun tournaments it's crucial that everyone gets a similar number of games, whereas at a regional event it might be more important to qualify the correct 3 teams, and who cares if the guys who got knocked out on saturday just go home? Again, I can only speak for the UK, and say that any schedule in which the busiest team would play more than 2 more games than the least busy team would be no good to us. Whenever possible, we try to write so that there is no more than one game difference between the team who plays most and the team who plays least. You all pay the same entry fee, so you should get the same number of games (within the constraints of funny numbers of teams or additional qualification games).
Basic advice
You cannot write a fair schedule. All you can do is decide where to compromise. Anyone who's seen a full round-robin, like for example in English football, will know that teams rest players for certain games - so even the league is not completely fair. The order of matches matters. And it's clear that any schedule where you don't play every opponent is open to unfairness in seeding. There is no such thing as a fair schedule. Give up on that idea now. Constraints (such as a maximum number of games without exhausting players, or maximum number of fields or time-slots, or horrible odd numbers of teams) merely add to the unfairness that is already somewhere in there. But here's a couple of things that might help a little...
First off, unless you're running something complicated (like an event that qualifies a certain number of teams for another event - and let's face it, if you are part of a bigger championship there'll probably be scheduling help available anyway) then the first thing you look at is how many games you want people to play. Set a maximum and a minimum, and then choose your pool sizes, number of crossover rounds, and brackets to meet that number. It's a non-trivial task to fit a fair schedule to the right number of games, but it's always a far better idea than wasting your time inventing fabulously fair schedules with multiple crossovers and power-pools and then realizing you'll need until next Wednesday to play all the games.
Deal with teams in multiples of 8 wherever possible, and multiples of 4 at worst. If you've got funny numbers, 99% of the time you're better off pretending that you've got a multiple of 4 and putting byes in the schedule. We've tried a whole bunch of times to write schedules with clever bits where pools of 3 go into power-pools of 3 then a modified bracket etc... it almost never turns out well, and I don't think we've ever really used one of those schedules at an actual tournament. They lead to things like rested teams playing unrested teams, people having three games off in a row, fields lying empty, and all sorts of weird stuff. That may be fine for a qualifier where the most important thing is simply to make sure that the best x teams qualify, but it won't wash at an ordinary event. If you want to finish with neat brackets, start off simple, in 4s and 8s.
If you put in any form of crossovers, triple check what will happen in the next matches. Seeding the pools is non-trivial if you want to avoid rematches later on.
Remember that odd-numbered pools eat up pitches - for example, 2 pools of 4 (8 teams) can be played on 2 pitches in a day (6 time slots); 1 pool of 5 also requires 2 pitches all day (6 time slots again - 5 slots if you're prepared to make some people play 4 games in a row). It still frustrates me, but that's just the way it is. Thinking of nice 3, 5 or 7 team pools in your head is no use until you actually sit down and squeeze it onto your pitches - more often than you expect, it won't fit.
Anyway...
I could go on. I could write about 50,000 words on the intricacies of scheduling - for example, the UPA format manual is a fantastic document that tries to cope with any number of teams, and it's looong; and even then it doesn't come close to covering all the possible situations that might apply at an event (e.g. not enough pitches, constraints on back-to-back games, the team from far away can't start before midday) and doesn't touch the finesse parts of the schedule itself (as opposed to the format) like making sure that back-to-back games are not at opposite ends of the venue, making sure there's a decent lunch break for every team, making sure that the girls are near the toilets...
All I can say to you here is keep it simple, only accept entries in 4s where possible, and if it gets complicated, find an expert. Offer to pay the guy who wrote that great schedule at that other tournament you went to, even - you'll get no thanks for writing a good schedule, but the abuse you get for a bad one means it's worth doing what you can to get it right. A genuinely well-written schedule is of real value to every player who shows up, even if they only notice it when it's gone wrong - so if you need to pay someone to make sure it's done well, I'd say this isn't a place to be scared of spending a few dollars. A good schedule might not make people come back next year, but a bad one might well put them off.
Benji
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In which I take a look at some newly-found information about my 6x great-grandmother, Mary Howorth, who married Samuel Backler, Vicar of Ashwell (and Newnham), Herts. Not too much progress is made, but perhaps it will trigger some more information from some source, about who her father was!
BUT – I got carried sideways by re-acquainting myself with the Will of John Howorth, proved in 1692, naming both ‘my’ Samuel Backler and his wife Mary [Howorth] and citing as his Executor John Somerscales, who married Mary’s sister Elizabeth Howorth on the same day as Samuel and Mary’s wedding.
I had for some years been unable to trace the parents of the Howorth girls, but in a new search of PCC Wills, I came across that of their Mother, Margarett (or Margret) Howorth, widow, proved in April 1687. This Will named her children, coinciding exactly with the siblings mentioned just a few years later in John Howorth’s Will. Thus I am able to locate my 6x gt. grandmother’s siblings and mother, though not yet her father!
This is a period in which it is easy to be confused by dates. The Backler/Somerscales/Howorth marriages were in January 1686, which one might assume to be 11 months prior to the date of the girls’ mother’s death in November 1686 and her burial in Newnham, Herts. But we need to remember that the calendar year before the change from Julian to Gregorian styles in 1752, changed to the New Year at the beginning of April, not the beginning of January. Thus, January 1686 was about 6 weeks after their mother’s death in November. For more information about Mary, we can now turn to her mother’s Will.
Margarett Howorth: some questions and facts: the questions first – who was her husband? When had he died? Where was she born, what was her maiden name, and when and where were she and her yet-to-be-named husband married? When and where were her children born? Why were Samuel and Mary married in London? How and why had Elizabeth Howorth met John Somerscales, gent?
Some facts: She was buried in Newnham, Herts 28 November 1686, about 6 weeks before the marriage of her two daughters. She left a Will, naming her children, and showing that she was a very prosperous woman. The Will also shows that the more senior the child, the better pickings they had from the estate!
Her burial in Newnham was by her soon-to-be son-in-law, Samuel Backler. Could the Howorth family have had anything to do with his tenure there?
Margaret Howorth, Widow, of Hertford. Will dated 8 November 1683. She was buried in Newnham, Herts, 28 November 1686. Probate in London 13 April 1687 to Elizabeth Somerscale [Howorth] wife of John Somerscale NB: most commas and all bullet points are mineIt appears that three sons, Richard, Austin, and Nicholas were all under age 21 at the time Margarett wrote this Will. Her legacy first to Richard, then Austin [named as Augustus in John’s Will written 9 years later], then Nicholas was that the recipient of all the ffarme and messuage in Hardwick in the County of Cambridge bought from Ambrose Benning and now in occupation of Robert Challis, should pay £200 to each of the other two, on attainment of their 21 years of age. If none of them were to pay the £200 apiece to the other two, then the legacy should be divided equally among them, in a life share and proportionally to them and their heirs forever. If one were to die before coming of age, then his share to be divided equally among the other two. But if two were to die, then the share of the second deceasing to be shared equally among all the surviving sisters. If all three brothers die without coming of age, then the oldest son John shall have a share of the third brother deceasing equal to the shares of the sisters, and the other half divided equally among all the surviving sisters.
To my daughter Elizabeth Howorth [who would marry John Somerscales] I bequeath £320, my great tankard, my great plate, and half a dozen spoons, all my Childbed Lynnen with a spreading Mat and three Pillowbeers thereto belonging, the best suite of Diaper Table Lynnen being made up of two Cloths and a dozen and a halfe of Napkins, four paires of household sheets and one paire of fine Holland ones, my watch and my best ring.
To my fourth Sonne Nicholas I give my little silver Tankard, my little Salt, four silver Spoones, and a gold ring.
Further, it is my Will that if there shall happen to be any loss in my moneys above given to my daughters that such loss the three Oldest shall bear an equall share but there shall be noe deduction made from the youngest. And further it is my Will that if any of my daughters dye before Marriage their whole Portions shall be equally divided amongst the remaining Sisters. Or if anything herein given shall happen to fall short it shall be borne by all alike in equall shares except as before in Moneys given of which the youngest is to receive her fifty pound notwithstanding. All the rest of my Goodes if anything shall be found remaining more than is before bequeathed I give to my daughter Elizabeth Howorth whom I make the sole Executrix of this my last Will and Testament made upon the Eighth day of November in the Year of our Lord One Thousand Six Hundred Eighty and Three – the rent of Hardwick Estate until my sonnes Richard Austin and Nicholas shall come to age and the Interest of my Children’s Moneys to be received by my Executrix until their ages for their respective maintenances all just charges of my Executrix to be boarn out of my Estate. In witness whereof I have hereto sett my Hand and Seale the day and Yeare above written. Margrat [sic] Howorth/ Published with the addition of her Last Will in the presence of Ralph Battell Eliz Battell Hannah Sowter.
The Will of Margarett’s son John Howorth [PROB 11/414/55]: Written in February, 1692 and proved in March 1692, this is the Will of John Howorth, Gentleman, of St. Matthew, Friday Street, London. It confirms the relationships between the various siblings. And presumably it takes account of the fact that the mother’s Will had previously provided for the various siblings, including what appear to be unmarried daughters.
I give my brother in law John Somerscales ‘allthat my Capitall messuage or tenement lands and appurtenances thereunto belonging situate lying and being in South Walsham and elsewhere in the County of Norfolk and all other my estate Real and Personal. John Somerscales shall be my sole Executor.
Lastly I revoke all previous wills etc.
Signed 25 February 1692. Witnesses Peter Alder, James Wright and Cooke.
The Howorths: what happened to the other siblings? When did John Somerscales and his wife Elizabeth die? As far as I can see, they had only one surviving daughter, Elizabeth, born in 1696. | https://backlers.com/tag/john-somerscale/ |
This gives [families] the opportunity to see their children in awe.
”
-
Wish mom Shayna
When Jaxon was diagnosed with Leukemia at 3 years old, doctors told his family their life was going to “completely change.” Jaxon had to be taken out of daycare during treatment. He went from spending hours with friends to spending over a year and a half at home.
Shayna, Jaxon’s mom, said, “We had to take him away from his friends and stay at home.”
Jaxon and his dad would watch his favorite characters on Disney Channel, and Jaxon began to love Mickey Mouse.
“For a year and a half, Jaxon would ask me ‘Are we going to Disney?’” Shayna said. “When you first hear a diagnosis like cancer, you don’t see the future. It was hard for me to promise something I didn’t know was going to happen.
Jaxon and his family met with Make-A-Wish of Central & South Texas to choose a wish, and without hesitation, Jaxon wanted to meet his friend, who had stood by him throughout treatment, Mickey Mouse. But Jaxon didn’t just want to meet him, he wanted to play baseball with him.
“It’s really hard for me to describe that moment,” Shayna said. “It was so magical to be able to see the amazement in Jaxon’s eyes when he first saw Mickey. After a year of being in our house, without other kids, he was able to see his friend. He ran straight to him to give him the biggest hug.”
During his trip, Jaxon and his family were taken to the ESPN World of Sports arena. They saw “Welcome Molinar Family!” displayed on the large entrance screen. Jaxon was taken to the dugout with his brother, Jaedon. They heard Jaxon’s name announced over the loudspeaker, and Mickey ran out onto the field in a baseball uniform. Jaxon jumped up from the dugout and ran straight to him for another hug.
“It’s one thing to have a quick meet and greet with the characters,” Shayna said. “But its another thing to have that one-on-one time.”
Jaxon and Jaedon were able to play a game of baseball with Mickey. They ran the bases. They practiced throwing and catching. They were supposed to go to Hollywood Studios, but instead, they chose to stay and play for over an hour.
“The power of a wish means so much for the child, but it also means so much to the family,” Shayna said. “We were very fortunate to have family time together and time to focus on each other and have fun. It was a break from having to worry about everything that comes with cancer. The opportunity to have his wish granted is something that I believe really helped Jaxon’s spirit.”
Jaxon’s family looks forward to the future. They know this wish is something that will stay with Jaxon for the rest of his life.
“For families who are struggling with a diagnosis, this gives them a chance to get away from everything that has happened – this gives them the opportunity to see their children in awe.”
This story makes me feel
December 09, 2019
December 03, 2019
November 13, 2019
March 30, 2020
April 05, 2020
June 06, 2020
Help wish kids take flight.
Give your air miles now.
Stay in Touch!
Sign up to receive email from Make-A-Wish. | https://cstx.wish.org/wishes/wish-stories/i-wish-to-meet/jaxon |
We all know Porsche for its legendary sports cars and – recently – SUVs. But fans also like to joke about the propensity of this company to overcrowd its model series with as many variants as possible. Automotive industry aficionados with a knack for German sports cars will have ample reasons to remember this year’s edition of the iconic Los Angeles Auto Show. As for Porsche’s contribution, the c... (continue reading...) | |
Finding The Area Of An Object Using Photoshop
This is a way to estimate the area of any shape in Photoshop. I developed this technique because I didn’t have a better app or program to find the relative size of the sun (indicating its brightness) as it went through the solar eclipse in August 2017. I did this by filling the sun with little dots and then counting those dots and comparing the numbers for the sun at various stages of eclipse. In other uses, you would need to divide the number of dots in your shape of interest with the number of dots in a unit square (for example a square that was one inch on each side) to find the number of those units (square inches) that were in your shape. This technique will not give an answer with eight significant decimal places, but is good for quickly finding the area of complicated shapes. When I developed this technique, I had never used the pattern fill or count functions before.
Since I've surrendered to the 'cloud', I have trouble keeping track of which version of Photoshop I'm using, or for which versions this technique will work. Use at your own risk.
The Technique
- If you haven’t already done this, make a small pattern. For simplicity, I chose a small black dot, but if you really want to use the available small hollow red heart, that will work too.
- Select Custom Shape Tool (under the Rectangle Tool (shortcut U)).
- Select a shape. For the largest selection of shapes, you may have to click on the little gear in the upper right corner of the window and select "All".
- Make sure Pixels or Shape are selected instead of Path.
- Pick a color with some contrast to the background.
- Draw that shape on a blank layer.
- Use the Rectangular Marquee Tool to draw a rectangle closely around your shape.
- Click Edit ⇒ Define Pattern... Name your pattern and click "OK".
- Delete or erase your shape, layer, and selection rectangle.
- Select area to be measured.
- Fill the selected area with your pattern (you may want to do this on a blank layer above the desired shape in your photograph). Click Edit ⇒ Fill...
- Contents: Pattern
- Custom Pattern: select your simple pattern.
- Put a check in the Script: box
- Select Brick Fill
- Mode: Normal
- Opacity: 100%
- Click "OK".
- Pattern Scale: You may have to play with this. The smaller your pattern, the higher the count and the more precise your answer will be, but the Count Tool can only count to 4,094. Start around 0.5 for pattern scale. If this causes the Count Tool to max out in the next step, you can either increase the pattern scale, or divide your shape into segments.
- Offset between rows: I like 50%, but it doesn’t really matter.
- Leave all other settings at 0. Click "OK".
- Count objects.
- With the Magic Wand Tool, select your new fill objects to count (uncheck Anti-alias and Contiguous)
- Click Image ⇒ Analysis ⇒ Select Data Points ⇒ Custom
- In the Data Points area, under Selections, check "Count".
- Click "OK".
- Click Window ⇒ Measurement Log.
- In the Measurement Log window, click Record Measurements. Your answer will be in the new last line of the Measurement Log.
- Repeat Steps 2 through 4 to count fill objects of a reference area. This can be either a unit square or whatever else you want to compare your area to.
- Clean up your mess. Delete your dots and/or their layer, etc.
Well, that’s all there is to it. Congratulations. | http://www.beehappygraphics.com/find-area.html |
Q:
What is the relation between a policy which is the solution to a MDP and a policy like $\epsilon$-greedy?
In the context of reinforcement learning, a policy, $\pi$, is often defined as a function from the space of states, $\mathcal{S}$, to the space of actions, $\mathcal{A}$, that is, $\pi : \mathcal{S} \rightarrow \mathcal{A}$. This function is the "solution" to a problem, which is represented as a Markov decision process (MDP), so we often say that $\pi$ is a solution to the MDP. In general, we want to find the optimal policy $\pi^*$ for each MDP $\mathcal{M}$, that is, for each MDP $\mathcal{M}$, we want to find the policy which would make the agent behave optimality (that is, obtain the highest "cumulative future discounted reward", or, in short, the highest "return").
It is often the case that, in RL algorithms, e.g. Q-learning, people often mention "policies" like $\epsilon$-greedy, greedy, soft-max, etc., without ever mentioning that these policies are or not solutions to some MDP. It seems to me that these are two different types of policies: for example, the "greedy policy" always chooses the action with the highest expected return, no matter which state we are in; similarly, for the "$\epsilon$-greedy policy"; on the other hand, a policy which is a solution to a MDP is a map between states and actions.
What is then the relation between a policy which is the solution to a MDP and a policy like $\epsilon$-greedy? Is a policy like $\epsilon$-greedy a solution to any MDP? How can we formalise a policy like $\epsilon$-greedy in a similar way that I formalised a policy which is the solution to a MDP?
I understand that "$\epsilon$-greedy" can be called a policy, because, in fact, in algorithms like Q-learning, they are used to select actions (i.e. they allow the agent to behave), and this is the fundamental definition of a policy.
A:
for example, the "greedy policy" always chooses the action with the highest expected return, no matter which state we are in
The "no matter which state we are in" there is generally not true; in general, the expected return depends on the state we are in and the action we choose, not just the action.
In general, I wouldn't say that a policy is a mapping from states to actions, but a mapping from states to probability distributions over actions. That would only be equivalent to a mapping from states to actions for deterministic policies, not for stochastic policies.
Assuming that our agent has access to (estimates of) value functions $Q(s, a)$ for state-action pairs, the greedy and $\epsilon$-greedy policies can be described in precisely the same way.
Let $\pi_g (s, a)$ denote the probability assigned to an action $a$ in a state $s$ by the greedy policy. For simplicity, I'll assume there are no ties (otherwise it would in practice be best to randomize uniformly across the actions leading to the highest values). This probability is given by:
$$
\pi_g (s, a) =
\begin{cases}
1, & \text{if } a = \arg\max_{a'} Q(s, a') \\
0, & \text{otherwise}
\end{cases}
$$
Similarly, $\pi_{\epsilon} (s, a)$ could denote the probability assigned by an $\epsilon$-greedy strategy, with probabilities given by:
$$
\pi_{\epsilon} (s, a) =
\begin{cases}
(1 - \epsilon) + \frac{\epsilon}{\vert \mathcal{A}(s) \vert}, & \text{if } a = \arg\max_{a'} Q(s, a') \\
\frac{\epsilon}{\vert \mathcal{A}(s) \vert}, & \text{otherwise}
\end{cases}
$$
where $\vert \mathcal{A}(s) \vert$ denotes the size of the set of legal actions in state $s$.
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The design of the Paderewski Room Display was developed to preserve the Maestro’s possessions which were granted to Chicago’s Polish Museum of America, shortly after his death in 1941. The Museum’s director, Mieczyslaw Haiman, gladly received this property by Paderewski’s sister Antonina. Among the entrusted objects were Paderewski’s last practice Steinway piano, the pen that Paderewski used to sign the Versailles Treaty, jewelry, letters, documents, every day use objects and furnishings of his last living place, the suite at the Buckingham Hotel in New York city.
The Paderewski Association presented the display design development proposal in 2008 to the PMA. The ultimate goal of this project was to ensure proper safety and security of all the artifacts according to the highest levels of archival standards as well as to preserve the room’s original architectural artwork from 1937.
The design proposal included the following
recommendations:
- To establish a complete collection inventory along with a comprehensive conservation plan and exhibition options.
- The historic display cabinets be fitted with archival frames, mounting systems, safe LED and UV protective lighting distribution, and environmental monitoring elements without altering the beauty of original millwork.
- The room’s plaster crown moldings be repaired where necessary and finished to match the existing composition, texture and color.
- Areas of decorative ceiling murals and stencil work where plaster and paint were lost, would be conserved and in-painted.
- The implementation of new technologies such as projected digital imagery, sound and light, to be carefully designed and integrated to enhance the interpretation of the collection.
- The overall design proposal outlined the requirements of proper
monitoring for theft,
fire, humidity and temperature.
the generosity of professionals who performed their
work at reduced rates or donated their time free of
charge. We would like to express our gratitude
especially to Dr. Bronislaw Orawiec, M.D. for
a generous donation, and to all the Polish and
American artists and artisans who donated their
works during our three annual fund raising auctions
“Luxuries Befitting a Maestro” in 2006, 2007 and
2008, the volunteers who worked tirelessly
with auction logistics, and the PMA staff.
Image credit: Agnes Otworowski, computer rendition of the original historic display cabinet crossection with archival lighting and membranes implementation, based on CAD drawings
DESIGN CREDITS:
Dennis Novak, interior and display design
Margaret Wygoda, CAD drawings
Agnes Otworowski, computer renderings based on CAD drawings
Andrzej Dajnowski, preservation consulting
Stan Bernacki, project coordinator
Ron Steinberg, digital display consulting
Jerzy Chlopek, upholstery conservation consulting
The image indicates the placement of LED light source, laminated white opaque safety glass layer preventing the display area from direct light exposure, and openings for air flow. | http://www.paderewskiassociation.org/room.htm |
Albert Einstein was born at, March- 14- 1879 in Ulm, Germany. He was a Theoretical Physicist. His family was Jewish, and he was born very strangely. He was born with a big head and a small body and his arms were huge and flappy. He was educated with Catholic schools (mostly those were the only schools around him). He grew up with anger issues, his face would turn bright red and he would throw stuff and would “try” to beat up the ground. Whenever he would be angry at a teacher, he would throw a metal chair at a them (or worse, a computer). He was classified as AUTISTIC, and that’s one of the challenges that was hard for him to overcome.
Capurnicus, Isaac Newton, and James Clerk Maxwell inspired him into becoming who he was. One of his most favorite equations were E2 = MC2 and A2 + B2 = C2. He was very good in his Science and Math class but the other classes NOT SO MUCH, he was terrible in ELA because he couldn’t focus on reading, but he can read math and Science equations. Which was very confusing, but the fact is if someone gives me this equation:
E2 x CO2 + H2 x H20 – CO = Gas (basically)
But if I can solve that or read that why can’t I go to my ELA class and read as well like:
Planting plumber planted plenty plantains per plantation.
I should be easily able to read that, but the things that I just told you about were the things people were itching their head about. Not MOST people understood what his LEGACY was other than “A VERY SMART MAN”. But Albert didn’t stop there he went ABOVE & BEYOND.
He went on to learn more about science and even created his own theory but were not there yet. He married a Woman named, Mileva Mariç in 1903-1919, then married a woman named Elsa Einstein 1919-1936. He had 3 children, Eduard Einstein, Hans Albert Einstein, and Lieserl Einstein. Albert didn’t have a lot of workers that worked with him until when he learned about the ATOMIC NUCLEAR BOMB and started the “Emergency Committee of Atomic Scientists”, what Albert wanted to come out of his intelligence and theories was to help the world and recreate THE LAWS OF PHYSICS, which wasn’t easy the person Albert most worked with was his wife Elsa they were both smart and carried lots of intelligence in their brains. But the person he mostly was Similar too was Isaac Newton, not only were they both smart people but they both aced their math and science classes having a hard time especially with reading, but the only real clear difference was that Isaac Newton knew how to write. Yah, yah we all know that Albert Einstein knew how to write but he didn’t write as good as Isaac Newton did, if Albert Einstein was 10 years old and you asked him:
“HOW MANY SYLLABLES ARE NEUROLOGICAL DISORDER?
He would answer:
“8”
OR:
“HOW MANY SYLLABLES OR IN DEOXYBUNUCLIEC ACID?
“9”
But if you ask him:
“WHAT IS A COMPLEX SENTENCE?”
He would answer:
“I DON’T KNOW THAT ANSWER.”
You could probably say Albert Einstein was a “SPECIAL” scientist. Really all he wanted to do was prove his theories to the world and make the world know about the world they live in. And like he said, TWO THINGS ARE INFINITE: THE UNIVERSE AND HUMAN STUPIDITY; AND I’M NOT SURE ABOUT THE UNIVERSE THOUGH.
This man Inspires me because of his Intelligence and his hard working. The things that he has shown the world, the things that have never been revealed or thought of, and a habit that me and him have Is that whenever we are angry, we retaliate In a way that won’t be good but we both grew, we both lowered the aggressions and started to learn more of what Is right and what Is wrong. And I want to get more smarter and wiser as I grow and make the world a better place, I want to make theories just like he did and I want to prove things to people that nobody has ever noticed before, and I want to prove those bullies especially MY bullies, that they were wrong about me. The other thing Is, a lot of kids call the smart kids, quote on quote “nerds!”, and yet they aren’t making good grades, so my advice to smart people out there that have people that call them nerds, reply back, “maybe you should become one and maybe your grades will be better.” And remember BRAINS ALWAYS BEAT MUSCELS. | https://www.harvardinteractivemedia.org/essay/albert-einstein--theoretical-physicist-51251 |
Rice with Veggies Casserole!
By: Kecia’s Flavor Breakthrough!
Some days I really struggle to come up with something new that everyone will be happy with. Not because I am out of ideas, but because I just don’t have a lot of time or I want to make it ahead so we can reheat it later and not have a lot of work or just because I want to make everyone happy that night! Today was one of those days. I thought long and hard about this dish and I am still thinking it through so I decided I better sit down and write it out. Sometimes seeing it on the screen helps me really create the recipe. I don’t know how the rest of food bloggers and recipe writers do it, but I usually write up the recipe, print it out and follow it while I create it, then make adjustments a long the way. I guess that is the way I have always cooked, just wing it really. But now, writting it down it gives me more of a process and allows me to recreate the dishes we love again.
Serves 8
Prep Time: 30 minutes Cook Time: 45 minutes
Ingredients:
- 1 cup whole grain rice
- 2 cups chicken or vegetable broth
- 1 small zucchini, cut into small cubes
- 1 onion, julienned
- 2 cloves garlic, minced
- 1 bell pepper, diced
- 1 cup corn
- 1 cup black beans, drained and rinsed
- 1 tomato, diced
- 2 cups shredded pepper jack cheese, divided
- 1 tsp. Cilantro
- Salt and pepper to taste
Preheat oven to 350° F
In large sauce pan, place rice and broth. Cook according to directions on rice, undercooking by 5 minutes. Stir in the zucchini, onion, garlic, bell pepper, corn and black beans.
Now add the tomato, corn, beans, 1 1/2 cups cheese, cilantro and salt and pepper. Stir to combine.
Pour into a large casserole that has been sprayed with nonstick spray. Cover with foil and bake for 25 minutes.
Remove foil and sprinkle with remaining cheese and place back in oven for an additional 10 minutes or until cheese is bubbly.
Serve hot. This dish can be prepared ahead of time, reheat for 30 to 40 minute at 350°. | https://keciasflavorbreakthrough.com/rice-with-veggies-casserole/ |
Sonic Circus is known worldwide for its exclusive private stock of vintage recording consoles. With a deep knowledge of vintage and modern mixing desks, our technicians have a unique ability to restore classic audio recording consoles, and make them thrive in a modern recording world. We’re doing our best to bring back the warm organic analog sound. In addition to the classics, the Sonic Circus showroom also houses a selection of contemporary used consoles. Keep your eye on our ‘new arrivals’ page to see the latest additions to our comprehensive inventory. | http://shopping.na2.netsuite.com/s.nl/c.316965/sc.21/category.133628/.f |
Hello!
I have a bunch of apple poems to share with you. The clip art is from mycuteclipart.com. I’ve left lots of room on the templates for the children to draw their own illustrations.
Enjoy!
The poem above would be fun to act out with apples on craft sticks. You could have five children standing up with their apples then one sits down at the end of the poem. Repeat the poem singing “four little apples” etc. until there are zero apples on the tree. | https://www.gradeonederful.com/2013/09/apple-poem-freebies.html |
A side scrolling mecha shooting game, direct sequel to the previous game released on Megadrive and third in the “Assault Suit” series (the other title was released on Super Famicom).
The player uses a mech equipped with many different weapons, over 50 in total, a jet boost, a dash and shield. Before each missions a lot of parameters can be customized, speed, maneuverability and protection of shields, while some speciale upgrades can alter characteristics of the mech itsel.
During the game the camera can zoom in and out, letting the player to see more detail or a larger portion of the area, the latter is very helpful when facing huge mechs.
Assault Suit Leynos 2 was released only in Japan.
Additional information
|Weight||0.152 kg|
Reviews
There are no reviews yet. | https://japanvideogames.shop/product/sat-assault-suit-leynos-2/?v=437851873f06 |
Children form significant, lifelong memories of their interactions with the various adults who enter their lives, including their parents, grandparents, aunts and uncles, teachers and so on. Those memories, and the experiences from which they derive, shape the beliefs children hold with respect to themselves, others and the world in which they live. They also shape their behaviour.
The way adults treat any generation of children shapes the way those children will, in turn, treat the next generation when they are adults. It follows that if we are seeking to create a more gentle, humanistic world we adults need to pause and reflect on how we interact with the current generation of children.
Yesterday, I was returning to the Melbourne CBD on an over crowded tram after a day at the Australian Formula One Grand Prix. People were packed into the tram like sardines in a can. Shoulder-to-shoulder they stood in the aisles, swaying and brushing against each other with every jerk and bump. In this environment of uncomfortable levels of physical closeness to strangers eye-contact is minimal and conversation, when it exists, is brief and muted.
So it was that I could clearly hear in the carriage behind me a young girl of primary school age initiate a conversation with a complete stranger standing adjacent to her on the tram. The child had apparently noticed that this stranger had spoken with a heavy accent and had summoned the courage to inquire after its origin. The stranger, who I later observed to be an exotic-looking young woman, responded that her accent was Spanish. The child advised the young woman that she was learning Spanish. What followed over almost one hour was a child maintaining an animated and enthusiastic conversation about learning Spanish, to which the young woman responded with acceptance, warmth, patience and corresponding enthusiasm.
As a psychologist who has interacted with children over a long career I could not help but be impressed, and touched, by the manner in which the young woman engaged with the child. It left me sure that this child would remember fondly the day she interacted with a real-life Spanish-speaking adult, apart from her teacher. I thought immediately of what might be the legacy of this interaction for the child, and what had been the young woman’s own experiences of relating to adults when she was a child that had resulted in her warm, accepting and caring manner towards a previously unknown child. | https://colbypearce.net/2014/03/17/kindness-is-magic/?shared=email&msg=fail |
Photo: Fred Ferand – Skater: Jeff Hedges
The Vert Attack 8 halfpipe chaos which was once again sooo good is winding down. I’m sitting in the hotel lobby, saying bye to many of the vert skaters and new friends from the weekend after the final party last night. Three full days of vert ramp skating and watching the ramp being taken apart – and I am totally stoked by it all.
The level of skating was insane in all groups. Juniors, Girls, Masters (which was probably the largest group of skaters in a Masters division ever!) and of course the Pro/Ams group. Check out the first of many videos coming in the next few days.
All the best wishes to Jeff Hedges, get good quickly my man! | http://www.europeskate.com/vert-attack-8-wrapup/ |
SAVANNA, Ill. — In northern Illinois, a sunny spring day is the perfect opportunity for turtle hunting.
To spot the tiny reptiles, you'll have to be escorted through the old Savanna Army Depot. Past gated-off roads and through a field of old weapon bunkers, lies a 19-acre enclosure, built on a sand prairie.
There, a group of state threatened ornate box turtles are trying to make a comeback.
The "turtle resort" (of sorts) is run by their caretakers at the Upper Mississippi River National Wildlife and Fish Refuge. Researchers are trying to figure out how many reptiles roam the fenced-off region and study how well they're reproducing. All the while, a fence — which extends a foot below ground — helps keep predators out, and the little escape artists in.
At one time, ornate box turtles were found in half of Illinois' 102 counties. But today, researchers estimate the animals live in less than 10.
Where warm-blooded animals go into hibernate, cold-blooded beings go through brumation. Once winter hits, the turtles burrow into their brumation bunkers — typically a foot and a half below ground — to wait out the frigid weather.
Inside the enclosure, 15 turtles have radio transmitters on them, allowing researchers to track their habits and brumation sites.
To do so, researchers use an antenna in a 19-acre game of hot and cold. Biologists tune their radios to the frequency of the specific turtle they wish to find, then point the antenna and listen if the signal gets louder or softer in a specific direction.
Wildlife biologist Angela Dedrickson used this method to find a 21-year-old ornate box turtle named "Mama."
When we walked up, she had emerged from her brumation hole, crawled a few yards away, then burrowed back down up to her head, to escape the heat of the sun.
"They're state threatened and we don't want to lose populations, especially in an area such as this, which is their prime habitat," Dedrickson said.
Her job is to figure out how many turtles are in the enclosure, then figure out how well they're reproducing.
"Once we have answers to those questions, then the walls will be torn down and they'll be set free to go do what they need to do out in nature," Dedrickson said.
The reptiles were listed as state threatened back in 2009. One year later, the "turtle resort" was constructed. Between 2010 and 2016, biologists released 69 zoo-born turtles into the area.
Last year, the team of researchers were able to observe 32 turtles.
"But the estimate that I have, based on the information that I have, is around 80 turtles," Dedrickson said. "We need to have these populations healthy and productive because we don't want to lose these turtles."
Eventually, she hopes that won't be a worry anymore. But until then, turtles like Mama are content, relaxing in their turtle resort. | https://www.wqad.com/article/tech/science/environment/ornate-box-turtle-endangered-illinois-biologist-save/526-dd224f79-38bf-4f5e-8655-623e6ab10321 |
AboutMithril is located at the address 9 Pope Ct in Fairhope, Alabama 36532.
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For maps and directions to Mithril view the map to the right. For reviews of Mithril see below.
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Love the images! And, by the by, one might assume that those bands should have little to do with seasonal / radiative heating effects at that distance (if I'm off base, let me know). Aside from the fact that they're at BOTH poles!
Can you say Birkeland's terella figure 248b?
(Chapter VI. On Possible Electric Phenomena in Solar Systems and Nebulae; near the top of the page)
http://www.plasma-universe.com/index.ph ... nd_Nebulae
In layman's terms he said that an unmagnetized globe will have evenly distributed spots, as in Fig 248a. With even minimal magnetization, spots appear in band(s) parallel to the equator. The banding latitude varies with either the strength of magnetization or with the strength of current (discharge tension).We will now pass on to experiments that in my opinion have brought about the most important discoveries in the long chain of experimental analogies to terrestrial and cosmic phenomena that I have produced. In the experiments represented in figs. 248 a-e, there are some small white patches on the globe, which are due to a kind of discharge that, under ordinary circumstances, is disruptive, and which radiates from points on the cathode. If the globe has a smooth surface and is not magnetised, the disruptive discharges come rapidly one after another, and are distributed more or less uniformly all over the globe (see a). On the other hand, if the globe is magnetised, even very slightly, the patches from which the disruptive discharges issue, arrange themselves then in two zones parallel with the magnetic equator of the globe; and the more powerfully the globe is magnetised, the nearer do they come to the equator (see b, c, d). With a constant magnetisation, the zones of patches will be found near the equator if the discharge-tension is low, but far from the equator if the tension is high.
[Figure 248 (a-e)]
- If the magnetization level is variable, but not the current, then the stronger the magnetization the closer the band(s) come to the equator, the weaker the magnetization the closer the band(s) come to the poles.
- On the other hand, if the current is variable, but not the magnetization level, then the weaker the discharge current the closer the bands(s) come to the equator, the stronger the discharge current the closer the band(s) come to the poles.
A) the magnetic field is non-variable and the discharge current is high enough to move the banding toward the poles, or
B) the discharge current is stable (at whatever current it is receiving, through whatever mechanism) but the magnetic field is proportionately weak enough that the discharge currents move toward the poles.
Really it seems like there's just some kind of relationship between the magnetization level, strength of discharge current and the position of the bands. IE, if the units of current are over some threshold per unit of magnetism, then the bands appear at the poles. Or, if the current is under that threshold, the bands migrate toward the equator...
So, I guess sat sums up my potential implications for Neptune...
*Yanks his Yoghurt doll's string*
"May the Schwartz be with you!"
Cheers, | http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?p=6308 |
---
abstract: 'We prove the existence of the dynamics automorphism group for Hamiltonian QCD on an infinite lattice in $\R^3$, and this is done in a C\*-algebraic context. The existence of ground states is also obtained. Starting with the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph (cf. [@KR; @KR1]), we state its field algebra and a natural representation. We then generalize this representation to the infinite lattice, and construct a Hilbert space which has represented on it all the local algebras (i.e. kinematics algebras associated with finite connected sublattices) equipped with the correct graded commutation relations. On a suitably large C\*-algebra acting on this Hilbert space, and containing all the local algebras, we prove that there is a one parameter automorphism group, which is the pointwise norm limit of the local time evolutions along a sequence of finite sublattices, increasing to the full lattice. This is our global time evolution. We then take as our field algebra the C\*-algebra generated by all the orbits of the local algebras w.r.t. the global time evolution. Thus the time evolution creates the field algebra. The time evolution is strongly continuous on this choice of field algebra, though not on the original larger C\*-algebra. We define the gauge transformations, explain how to enforce the Gauss law constraint, show that the dynamics automorphism group descends to the algebra of physical observables and prove that gauge invariant ground states exist.'
author:
- |
[Hendrik Grundling]{}\
[Department of Mathematics,]{}\
[University of New South Wales,]{}\
[Sydney, NSW 2052, Australia.]{}\
[[email protected]]{}\
[FAX: +61-2-93857123]{}\
- |
[Gerd Rudolph ]{}\
[Institut für Theoretische Physik,]{}\
[Universität Leipzig,]{}\
[Postfach 100 920, D-4109 Leipzig.]{}\
[[email protected]]{}\
[FAX: +49-341-9732548]{}
title: '**Dynamics for QCD on an infinite lattice**'
---
Introduction
============
In a previous paper ([@GrRu]) we constructed in a C\*-algebraic context a suitable field algebra which can model the kinematics of Hamiltonian QCD on an infinite lattice in $\R^3$. It was based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph (cf. [@KR; @KR1]). We did not consider dynamics, and the construction and analysis of the dynamics for QCD on an infinite lattice is the main problem which we want to address in this paper. For reasons to be explained, we will not here directly use the C\*-algebra which we constructed before for dynamics construction, but will follow a different approach.
Whilst the algebra constructed in [@GrRu] contained all the information of the gauge structures required, and its representation space contained the physical representations, it suffered from the following defects.
- The true local algebras (i.e. the kinematics algebras for the model on finite sublattices) were not subalgebras of the constructed kinematics algebra. The kinematics algebra of [@GrRu] did contain isomorphic copies of the local algebras, whose multiplier algebras contained the local algebras, but these did not satisfy local graded-commutativity. That is, they did not graded-commute if they corresponded to disjoint parts of the lattice, unlike the true local algebras. This was due to a novel form of the infinite tensor product of nonunital algebras, where approximate identities replaced the identity in the infinite “tails" of the tensor products.
- As a consequence of this infinite tensor product, current methods of defining dynamics on lattice systems by suitable limits of the local dynamics did not apply, which made it very hard to construct dynamics for the full system.
Due to these problems, especially the latter one, we will here extend our focus to the multiplier algebra of our previous kinematics field algebra, and build an appropriate new kinematics field algebra in this setting. Our strategy will be to define a dynamics on a concrete C\*-algebra which is “maximally large” in the sense that it contains all the true local kinematics C\*-algebras, and it is contained in the multiplier algebra of our previous kinematics field algebra. We will then take our new kinematics field C\*-algebra to be the smallest subalgebra which contains all the true local kinematics C\*-algebras, and is preserved w.r.t. the dynamics. Thus, the dynamics itself, will create the kinematics algebra for the system. The methods we use for the proof come from the application and generalization of Lieb–Robinson bounds on lattice systems (cf. [@NaSi; @NaSi2]). Moreover, we will see that the dynamics is strongly continuous on our new kinematics algebra.
On the algebra we construct here, we are able to define both the dynamics and the gauge transformations of our model. We will prove that the dynamics automorphism group commutes with the action of the group of local gauge transformations, hence the dynamics automorphism group action descends to the algebra of physical observables. We will prove the existence of gauge invariant ground states, which therefore produces ground states on the algebra of physical observables.
The cost of using this new algebra, is that it contains infinitely many nonphysical representations, so one needs to restrict to the class of appropriate “regular” representations by hand. This should be compared with the use of the Weyl algebra for canonical systems, which contains many nonregular representations. It is a well-known practical necessity for the Weyl algebra to restrict representations by hand to regular representations when one analyzes physical systems. We are able to prove the existence of gauge invariant ground states which are regular.
Whereas each true local kinematics algebra (corresponding to a finite sublattice) has a unique ground state w.r.t. the local time evolution, for the infinite lattice limit, we do not presently have such a uniqueness property. The ground states are the weak \*-limit points of a sequence of “partial ground states”. This nonuniqueness needs further investigation. For the finite lattice, in a toy model the spectral problem for the lattice Hamiltonian has been solved exactly, yielding an explicit formula for the unique ground state, cf. [@HRS].
Our paper is organized as follows. In Sect. \[PM\] we state the model for the finite lattice taken from [@KR; @KR1], and give a very natural representation for it. We then generalize this representation for the infinite lattice in Sect. \[GFA\], and construct a Hilbert space $\cl H.$ which has represented on it all the local algebras ${\mathfrak A}_S$ (each associated with a finite sublattice $S$), with the correct (graded) commutation relations. We then define a conveniently large C\*-algebra $\al A._{\rm max}$ acting on this Hilbert space, and containing all the local algebras. In Sect. \[LDA\], we then define on $\al A._{\rm max}$ the “local automorphism groups" $\alpha_t^S$, i.e. those produced by the Hamiltonians of the finite sublattices $S$. Using a Lieb–Robinson bounds argument, we then prove in Sect. \[DMFA\] that for each $A\in \al A._{\rm max}$, that $\alpha_t^S(A)$ converges in norm as $S$ increases to the full lattice, to an element $\alpha_t(A)$, and this defines a one-parameter automorphism group $t\mapsto\alpha_t\in{\rm Aut}(\al A._{\rm max})$. This is the global automorphism group, and we use it to define in Sect. \[KARR\] our chosen minimal field algebra by $${\mathfrak A}_{\Lambda}:=C^*\Big( \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S ) \Big)
\subset\cl A._{\rm max} \subset\cl B.(\cl H.).$$ We also clarify the relation of ${\mathfrak A}_{\Lambda}$ with the kinematics algebra previously constructed in [@GrRu]. We prove the existence of regular ground states in Sect. \[PGDA\], and in Sect. \[GTGL\] we define gauge transformations and consider enforcement of the Gauss law constraint.
The Kinematics Field Algebra {#FieldAlgebra}
============================
We consider a model for QCD in the Hamiltonian framework on an infinite regular cubic lattice in $\Z^3.$ For basic notions concerning lattice gauge theories including fermions, we refer to [@Seiler] and references therein.
We first fix notation. For the lattice, define a triple $\Lambda:=(\Lambda^0, \Lambda^1, \Lambda^2)$ as follows:
- $\Lambda^0:=\{(n,m,r)\in\R^3\,\mid\,n,\,m,\,r\in\Z\}=\Z^3$ i.e. $\Lambda^0$ is the unit cubic lattice and its elements are called sites.
- Let $\wt\Lambda^1$ be the set of all directed edges (or links) between nearest neighbours, i.e. $$\wt\Lambda^1:= \{(x,y)\in\Lambda^0\times\Lambda^0\,\mid\,y=x\pm \b e._i\;\;\hbox{for some $i$}\}$$ where the $\b e._i\in\R^3$ are the standard unit basis vectors. Let $\Lambda^1\subset\wt\Lambda^1 $ denote a choice of orientation of $\wt\Lambda^1,$ i.e. for each $(x,y)\in\wt\Lambda^1$, $\Lambda^1$ contains either $(x,y)$ or $(y,x)$ but not both. Thus the pair $(\Lambda^0,\Lambda^1)$ is a directed graph, and we assume that it is connected.
- Let $\wt\Lambda^2$ be the set of all directed faces (or plaquettes) of the unit cubes comprising the lattice i.e. $$\wt\Lambda^2:= \{(\ell_1,\ell_2,\ell_3,\ell_4)\in\big(\wt\Lambda^1\big)^4\,\mid\,Q_2\ell_i=Q_1\ell_{i+1}\;
\hbox{for}\;i=1,2,3,\;\hbox{and}\; Q_2\ell_4=Q_1\ell_1\}$$ where $Q_i:\Lambda^0\times\Lambda^0\to\Lambda^0$ is the projection onto the $i^{\hbox{th}}$ component. Note that for a plaquette $p=(\ell_1,\ell_2,\ell_3,\ell_4)\in\wt\Lambda^2,$ it has an orientation given by the order of the edges, and the reverse ordering is $\overline{p}=(\overline\ell_4,\overline\ell_3,\overline\ell_2,\overline\ell_1)$ where $\overline\ell={(y,x)}$ if $\ell={(x,y)}\in \wt\Lambda^1$. In analogy to the last point, we let $\Lambda^2$ be a choice of orientation in $\wt\Lambda^2$.
- If we need to identify the elements of $\Lambda^i$ with subsets of $\R^3$, we will make the natural identifications, e.g. a link $\ell=(x,y)\in\Lambda^1$ is the undirected closed line segment from $x$ to $y$.
- We also need to consider subsets of the lattice, so given a connected subgraph $S\subset(\Lambda^0, \Lambda^1)$, we let $\Lambda_S^0$ be all the vertices in $S$, $\Lambda_S^1\subset\Lambda^1$ is the set of links which are edges in $S$, and $\Lambda_S^2\subset\Lambda^2$ is the set of those plaquettes whose sides are all in $S$.
We recall the lattice approximation on $\Lambda$ for a classical matter field with a classical gauge connection field acting on it. Fix a connected, compact Lie group $G$ (the gauge structure group), and let ${\big(\Cn,\,(\cdot,\cdot)_\Cn\big)}$ be a finite dimensional complex Hilbert space (the space of internal degrees of freedom of the matter field) on which $G$ acts smoothly as unitaries, so we take $G\subset U(\Cn)$. Then the classical matter fields are elements of $\prod\limits_{x\in\Lambda^0}\Cn$, on which the local gauge group $\prod\limits_{x\in \Lambda^0}G=G^{\Lambda^0}=\big\{\zeta:\Lambda^0\to G\big\}$ acts by pointwise multiplication. The classical gauge connections are maps $\Phi:\Lambda^1\to G$, i.e. elements of $\prod\limits_{\ell\in\Lambda^1}G$.
The full classical configuration space is thus $\big(\prod\limits_{x\in\Lambda^0}\Cn\big)\times
\big(\prod\limits_{\ell\in\Lambda^1}G\big)$, and the local gauge group $\prod\limits_{x\in \Lambda^0}G$ acts on it by $$\label{ClassGTr}
\big(\prod_{x\in\Lambda^0}v_x\big)\times\big(\prod_{\ell\in\Lambda^1}g_\ell\big)\mapsto
\big(\prod_{x\in\Lambda^0}\zeta(x)\cdot v_x\big)\times\big(\prod_{\ell\in\Lambda^1}\zeta(x_\ell)\,g_\ell\,\zeta(y_\ell)^{-1}\big)$$ where $ \ell=(x_\ell,y_\ell)$ and $\zeta \in\prod\limits_{x\in \Lambda^0}G$. Note that the orientation of links in $\Lambda^1$ was used in the action because it treats the $x_\ell$ and $y_\ell$ differently.
This is the basic classical kinematical model for which the quantum counterpart is given below for finite lattices.
The finite lattice model. {#PM}
-------------------------
In this subsection we want to state the model for finite lattice approximation of Hamiltonian QCD in $\R^3$ developed by Kijowski, Rudolph [@KR; @KR1]. A more expanded version of this section is in [@GrRu]. This model is based on the model constructed in the classical paper of Kogut [@K] (which elaborates the earlier one of Kogut and Susskind [@KS]).
Fix a finite connected subgraph $S$, and let $\Lambda_S:=(\Lambda^0_S, \Lambda^1_S, \Lambda^2_S)$. For ease of notation, we will omit the subscript $S$ in this section. Given such a finite lattice $\Lambda^0$, the model quantizes the classical model on $\Lambda^0$ above, by replacing for each lattice site $x\in\Lambda^0$, the classical matter configuration space $\Cn$ with the algebra for a fermionic particle on $\Cn$ (the quarks), and for each link $\ell\in\Lambda^1$ we replace the classical connection configuration space $G$ by an algebra which describes a bosonic particle on $G$ (the gluons).
Equip the space of classical matter fields $\prod\limits_{x\in\Lambda^0}\Cn=\{f:\Lambda^0\to\Cn\}$ with the natural pointwise inner product ${\langle}f, h {\rangle}=\sum\limits_{x\in\Lambda^0}\big({f(x)},\, h(x)\big)_{\Cn}$, and take for the quantized matter fields the CAR-algebra ${\mathfrak F}_{\Lambda} := {\rm CAR}\big(\prod\limits_{x\in\Lambda^0}\Cn)$. That is, for each classical matter field $f\in\prod\limits_{x\in\Lambda^0}\Cn$, we associate a fermionic field $a(f)\in{\mathfrak F}_{\Lambda}$, and these satisfy the usual CAR–relations: $$\{a(f), a(h)^*\} = {\langle}f, h {\rangle}{\mathbf{1}}\quad \hbox{ and } \quad \{a(f),a(h)\} = 0
\quad \mbox{ for } \quad f, h\in \prod\limits_{x\in\Lambda^0}\Cn$$ where $\{A,B\} := AB + BA$ and ${\mathfrak F}_{\Lambda}$ is generated by the set of all $a(f)$. As $\Lambda^0$ is finite, ${\mathfrak F}_{\Lambda}$ is a full matrix algebra, hence up to unitary equivalence it has only one irreducible representation.
In physics notation, the quark at $x$ is given by $a(\delta_x{\bf v}_i)=\psi_i(x)$ where ${\{{\bf v}_i\mid i=1,\ldots n\}}$ is an orthonormal basis for $\Cn$ and $\delta_x:\Lambda^0\to\R$ is the characteristic function of $\{x\}$. Further indices may be included if necessary, e.g. by putting $\Cn = \b W.\otimes\C^k$ where $\b W.$ has non–gauge degrees of freedom (such as the spinor part), and $\C^k$ has the gauge degrees of freedom.
To quantize the classical gauge connection fields $\prod\limits_{\ell\in\Lambda^1}G,$ we take for a single link $\ell$ a bosonic particle on $G$. This is given in a generalized Schrödinger representation on $L^2(G)$ by the set of operators ${\{U_g,\;T_f\mid g\in G,\;f\in L^\infty(G)\}}$ where: $$\label{GSRp}
(U_g\varphi)(h):=\varphi(g^{-1}h)\quad\hbox{and}\quad\big(T_f\varphi)(h):=f(h)\varphi(h)\quad\hbox{for}\quad
\varphi\in L^2(G),$$ $g,\,h\in G$ and $f\in L^\infty(G)$, and it is irreducible in the sense that the commutant of ${U_G\cup T_{L^\infty(G)}}$ consists of the scalars. Note that there is a natural ground state unit vector $\psi_0\in L^2(G)$ given by the constant function $\psi_0(h)=1$ for all $h\in G$ (assuming that the Haar measure of $G$ is normalized). Then $U_g\psi_0=\psi_0$, and $\psi_0$ is cyclic w.r.t. the \*-algebra generated by ${U_G\cup T_{L^\infty(G)}}$ (by irreducibility).
The generalized canonical commutation relations are obtained from the intertwining relation $U_gT_fU^*_g=T\s\lambda_g(f).$ where $$\label{leftact}
\lambda:G\to\aut C(G)\, , \quad
\lambda_g(f)(h) := f(g^{-1} h) \quad\hbox{for}\quad g,\,h\in G$$ is the usual left translation. In particular, given $X\in\mathfrak{g}$, define its associated momentum operator $$\begin{aligned}
P_X:C^\infty(G)\to C^\infty(G)\quad&\hbox{by}&\quad
P_X\varphi:=i{d\over dt}U(e^{tX})\varphi\Big|_{t=0}\,.\\[1mm]
\hbox{Then}\qquad
\big[P_X,\,T_f\big]\varphi
&=& iT\s X^R(f). \varphi\qquad\hbox{for}\quad
f,\,\varphi\in C^\infty(G),\end{aligned}$$ where $X^R\in\ot X.(G)$ is the associated right-invariant vector field. As $P_X=dU(X)$, it defines a representation of the Lie algebra $\mathfrak{g}$, and clearly $P_X\psi_0=0$.
To identify the quantum connection $\Phi(\ell)$ at link $\ell$ in this context, use the irreducible action of the structure group $G$ on $\C^k$ to define the function $\Phi_{ij}(\ell)\in C(G)$ by $$\label{Def-QuConn}
\Phi_{ij}(\ell)(g):=(e_i,ge_j),\quad g\in G,$$ where $\{ e_i\mid
i =1,\ldots,k\}$ is an orthonormal basis of $\C^k$. Then the matrix components of the quantum connection are taken to be the operators $T_{\Phi_{ij}(\ell)}$, which we will see transform correctly w.r.t. gauge transformations. As the $\Phi_{ij}(\ell)$ are matrix elements of elements of $G$, there are obvious relations between them which reflect the structure of $G$. The C\*-algebra generated by the operators $\{ T_{\Phi_{ij}(\ell)}\,\mid\, i,\,j=1,\ldots,k\}$ is $T\s C(G).$.
To define gauge momentum operators, we first assign to each link an element of $\mathfrak{g}$, i.e. we choose a map $\Psi:\Lambda^1\to \mathfrak{g}.$ Given such a $\Psi$, take for the associated quantum gauge momentum at $\ell$ the operator $P\s\Psi(\ell).:C^\infty(G)\to C^\infty(G)$. The generalized canonical commutation relations are $$\label{genCCR}
\big[P\s\Psi(\ell).,\,T\s\Phi_{ij}(\ell).\big]=i\sum_mT\s\Psi(\ell)_{im}\Phi_{mj}(\ell).\quad\hbox{on}\quad C^\infty(G),$$ where $\Psi(\ell)_{im}:=(e_i,\Psi(\ell)e_m).$
To obtain the G–electrical fields at $\ell$, choose a basis ${\{Y_r\mid r=1,\ldots,{\rm dim}( \mathfrak{g})\}\subset \mathfrak{g}}$, then substitute for $\Psi$ the constant map $\Psi(\ell)=Y_r$ and set $E_r(\ell):=P_{Y_r}$. In the case that $G=SU(3)$, these are the colour electrical fields, and one takes the basis $\{Y_r\}$ to be the traceless selfadjoint Gell–Mann matrices satisfying ${\rm Tr}(Y_rY_s)=\delta_{rs}$. We then define $$E_{ij}(\ell):=\sum_r(Y_r)_{ij}E_r(\ell)=\sum_r(Y_r)_{ij}P\s{Y_r}.$$ and for these we obtain from $(\ref{genCCR})$ the commutation formulae in [@KR; @KR1] for the colour electrical field. Of particular importance for the dynamics, is the operator $E_{ij}(\ell) E_{ji}(\ell)$ (summation convention). We have $$E_{ij}(\ell) E_{ji}(\ell)=\sum_{r,s}(Y_r)_{ij}P\s{Y_r}.(Y_s)_{ji}P\s{Y_s}.
=\sum_{r,s}(Y_rY_s)_{ii}P\s{Y_r}.P\s{Y_s}.=n\sum_rP\s{Y_r}.^2$$ i.e. it is the Laplacian for the left regular representation $U:G\to\cl U.(L^2(G))$ which therefore commutes with all $U_g$. Below in Equation (\[PTfell\]) we will see that a gauge transform just transforms the Laplacian to one w.r.t. a transformed basis of $\mathfrak{g}$, which leaves the Laplacian invariant.
The full collection of operators which comprises the set of dynamical variables of the model is as follows. The representation Hilbert space is $$\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)\qquad\hbox{ where}\qquad
{\pi_F:{\mathfrak F}_{\Lambda}}\to\cl B.(\cl H._F)$$ is any irreducible representation of ${\mathfrak F}_{\Lambda}$. As $\Lambda^1$ is finite, $\cl H.$ is well–defined. Then $\pi_F\otimes\un:{\mathfrak F}_{\Lambda}\to\cl B.(\cl H.)$ will be the action of ${\mathfrak F}_{\Lambda}$ on $\cl H.$. The quantum connection is given by the set of operators $$\{{\widehat}{T}_{\Phi_{ij}(\ell)}^{(\ell)}\,\mid\,\ell\in \Lambda^1,\;i,j=1,\ldots, k\}\quad
\hbox{where}\quad {\widehat}{T}_f^{(\ell)}:=\un\otimes\big(\un\otimes\cdots\otimes\un\otimes T^{(\ell)}_f\otimes\un\otimes\cdots\otimes\un\big)$$ and $T^{(\ell)}_f$ is the multiplication operator on the $\ell^{\rm th}$ factor, hence ${\widehat}{T}_{\Phi_{ij}(\ell)}^{(\ell)}$ acts as the identity on all the other factors of $\cl H.$. Likewise, for the gauge momenta we take $${\widehat}{P}^{(\ell)}_{\Psi(\ell)}:=\un\otimes\big(\un\otimes\cdots\otimes\un\otimes P^{(\ell)}_{\Psi(\ell)}\otimes\un\otimes\cdots\otimes\un\big),\quad
\ell\in \Lambda^1$$ where $P^{(\ell)}_X$ is the $P_X$ operator on the subspace $C^\infty(G)\subset L^2(G)$ of the $\ell^{\rm th}$ factor. Note that if we set $g=\exp(t\Psi(\ell))$ in ${\widehat}{U}_g^{(\ell)}:=\un\otimes\big(\un\otimes\cdots\otimes\un\otimes{U}_g^{(\ell)}\otimes\un\otimes\cdots\otimes\un\big)$ where $U^{(\ell)}_g$ is the $U_g$ operator on the $\ell^{\rm th}$ factor, then this is a unitary one parameter group w.r.t. $t$, with generator the gauge momentum operator ${\widehat}{P}^{(\ell)}_{\Psi(\ell)}$. Thus the quantum G–electrical field ${\widehat}{E}_r$ is a map from $\Lambda^1$ to operators on the dense domain $\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}C^\infty(G)$, given by ${\widehat}{E}_r(\ell):={\widehat}{P}^{(\ell)}_{Y_r}$.
Next, to define gauge transformations, recall from Equation (\[ClassGTr\]) the action of the local gauge group $\gauc \Lambda=\prod\limits_{x\in \Lambda^0}G=\{\zeta:\Lambda^0\to G\}$ on the classical configuration space. For the Fermion algebra we define an action $\alpha^1:\gauc \Lambda \to\aut{\mathfrak F}_{\Lambda}$ by $$\alpha_\zeta^1(a(f)):=a(\zeta\cdot f)\qquad \hbox{where}\qquad (\zeta\cdot f)(x):={\zeta(x)}f(x)\quad \hbox{for all}\quad x\in\Lambda^0,$$ and $f\in\prod\limits_{x\in\Lambda^0}\Cn$ since $f\mapsto \zeta\cdot f$ defines a unitary on $\prod\limits_{x\in\Lambda^0}\Cn$ where $\zeta\in\gauc \Lambda$. As ${\mathfrak F}_{\Lambda}$ has up to unitary equivalence only one irreducible representation, it follows that ${\pi_F:{\mathfrak F}_{\Lambda}}\to\cl B.(\cl H._F)$ is equivalent to the Fock representation, hence it is covariant w.r.t. $\alpha^1$, i.e. there is a (continuous) unitary representation $U^F:\gauc \Lambda\to \cl U.(\cl H._F)$ such that $$\pi_F(\alpha^1_\zeta(A))=U^F_\zeta\pi_F(A)U^F_{\zeta^{-1}}\quad\hbox{for}\quad A\in {\mathfrak F}_{\Lambda}.$$
On the other hand, if the classical configuration space $G$ corresponds to a link $\ell=(x_\ell,y_\ell)$, then the gauge transformation is $\zeta\cdot g=
\zeta(x_\ell)\,g\,\zeta(y_\ell)^{-1}$ for all $g\in G$. Using this, we define a unitary $W_\zeta:L^2(G)\to L^2(G)$ by $$(W_\zeta\varphi)(h):=\varphi(\zeta^{-1}\cdot h)=\varphi(\zeta(x_\ell)^{-1}\,h\,\zeta(y_\ell))$$ using the fact that $G$ is unimodular, where the inverse was introduced to ensure that $\zeta\to W_\zeta$ is a homomorphism. Note that $W_\zeta\psi_0=\psi_0$. So for the quantum observables ${U_G\cup T_{L^\infty(G)}}$, the gauge transformation becomes $$\label{GTfell}
T_f\mapsto W_\zeta T_f W_\zeta^{-1}=T_{W_\zeta f}\quad\hbox{and}\quad
U_g\mapsto W_\zeta U_g W_\zeta^{-1}=U_{\zeta(x_\ell)g\zeta(x_\ell)^{-1}}$$ for $f\in L^\infty(G)\subset L^2(G)$ and $g\in G$. Moreover each $W_\zeta$ preserves the space $C^\infty(G)$, hence Equation (\[GTfell\]) also implies that $$\label{PTfell}
W_\zeta P_X W_\zeta^{-1}=P\s{\zeta(x_\ell)X\zeta(x_\ell)^{-1}}.\quad\hbox{for}\quad
X\in\mathfrak{g}.$$
Thus for the full system we define on $\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)$ the unitaries $$\label{LocalW}
{\widehat}{W}_\zeta:=U^F_\zeta\otimes\big(\bigotimes_{\ell\in\Lambda^1}W^{(\ell)}_\zeta\big),\quad
\zeta\in\gauc \Lambda$$ where $W^{(\ell)}_\zeta$ is the $W_\zeta$ operator on the $\ell^{\rm th}$ factor. Then the gauge transformation produced by $\zeta$ on the system of operators is given by ${\rm Ad}({\widehat}{W}_\zeta)$.
In particular, recalling $W_\zeta T\s\Phi_{ij}(\ell). W_\zeta^{-1}=T\s{W_\zeta \Phi_{ij}(\ell)}.$ we see that $$\begin{aligned}
\big(W_\zeta \Phi_{ij}(\ell)\big)(g)&=&\Phi_{ij}(\ell)(\zeta(x_\ell)^{-1}\,g\,\zeta(y_\ell))
=\big(e_i,\zeta(x_\ell)^{-1}\,g\,\zeta(y_\ell)e_j\big)\nonumber\\[1mm]
\label{PhiTfs}
&=&\sum_{n,m}[\zeta(x_\ell)^{-1}]_{in}\,\Phi_{nm}(\ell)(g)\,[\zeta(y_\ell)]_{mj}\end{aligned}$$ where $[\zeta(x_\ell)]_{in}=(e_i,\zeta(x_\ell)e_n)$ are the usual matrix elements, so it is clear that the indices of the quantum connection $T_{\Phi_{ij}(\ell)}$ transform correctly for the gauge transformation $\zeta^{-1}$.
Finally, we construct the appropriate field C\*-algebra for this model. For the fermion part, we already have the C\*-algebra ${\mathfrak F}_{\Lambda} = {\rm CAR}\big(\prod\limits_{x\in\Lambda^0}\Cn)$. Fix a link $\ell$, hence a specific copy of $G$ in the configuration space. Above in (\[leftact\]) we had the distinguished action $\lambda:G\to\aut C(G)$ by $$\lambda_g(f)(h) := f(g^{-1} h) \, \, , \, \,
f \in C(G),\; g,h\in G.$$ The generalized Schrödinger representation $(T,U)$ above in (\[GSRp\]) is a covariant representation for the action $\lambda:G\to\aut C(G)$ so it is natural to take for our field algebra the crossed product C\*-algebra $C(G)\rtimes_\lambda G $ whose representations are exactly the covariant representations of the $C^*$-dynamical system defined by $\lambda$. The algebra $C(G)\rtimes_\lambda G $ is also called the generalised Weyl algebra, and it is well–known that $C(G)\rtimes_\lambda G\cong\cl K.\big(L^2(G)\big)$ cf. [@Rief] and Theorem II.10.4.3 in [@Bla1]. In fact $\pi_0\big(C(G)\rtimes_\lambda G\big)=\cl K.\big(L^2(G)\big)$ where $\pi_0:C(G)\rtimes_\lambda G\to\cl B.( L^2(G))$ is the generalized Schrödinger representation. Since the algebra of compacts $\cl K.\big(L^2(G)\big)$ has only one irreducible representation up to unitary equivalence, it follows that the generalized Schrödinger representation is the unique irreducible covariant representation of $\lambda$ (up to equivalence). Moreover, as $\psi_0$ is cyclic for $\cl K.\big(L^2(G)\big)$, the generalized Schrödinger representation is unitary equivalent to the GNS–representation of the vector state $\omega_0$ given by $\omega_0(A):={(\psi_0,\pi_0(A)\psi_0)}$ for $A\in
C(G)\rtimes_\lambda G$.
Note that the operators $U_g$ and $T_f$ in equation (\[GSRp\]) are not compact, so they are not in $\cl K.\big(L^2(G)\big)=\pi_0\big(C(G)\rtimes_\lambda G\big)$, but are in fact in its multiplier algebra. This is not a problem, as a state or representation on $C(G)\rtimes_\lambda G$ has a unique extension to its multiplier algebra, so it is fully determined on these elements. If one chose $C^*(U_G\cup T_{L^\infty(G)})$ as the field algebra instead of $C(G)\rtimes_\lambda G$, then this would contain many inappropriate representations, e.g. covariant representations for $\lambda:G\to\aut C(G)$ where the implementing unitaries are discontinuous w.r.t. $G$. Thus, our choice for the field algebra of a link remains as ${C(G)\rtimes_\lambda G}\cong\cl K.\big(L^2(G)\big)$. Clearly, as the momentum operators $P_X$ are unbounded, they cannot be in any C\*-algebra, but they are obtained from $U_G$ in the generalized Schrödinger representation.
We combine these C\*-algebras into the kinematic field algebra, which is $${\mathfrak A}_{\Lambda} :=
{\mathfrak F}_{\Lambda} \otimes \bigotimes_{\ell\in\Lambda^1}\big(C(G)\rtimes_\lambda G\big)$$ which is is well–defined as $\Lambda^1$ is finite, and the cross–norms are unique as all algebras in the entries are nuclear. (If $\Lambda^1$ is infinite, the tensor product ${\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\big(C(G)\rtimes_\lambda G\big)}$ is undefined, as $C(G)\rtimes_\lambda G$ is nonunital). Moreover, since $C(G)\rtimes_\lambda G\cong\cl K.\big(L^2(G)\big)$ and $\cl K.(\cl H._1)\otimes\cl K.(\cl H._2)\cong\cl K.(\cl H._1\otimes\cl H._2),$ it follows that $$\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\big(C(G)\rtimes_\lambda G\big)\cong\cl K.\big(
\mathop{\otimes}\limits_{\ell\in\Lambda^1}L^2(G)\big)\cong\cl K.(\cl L.)$$ as $\Lambda^1$ is finite, where $\cl L.$ is a generic infinite dimensional separable Hilbert space. So $${\mathfrak A}_{\Lambda} ={\mathfrak F}_{\Lambda} \otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\big(C(G)\rtimes_\lambda G\big)\cong
{\mathfrak F}_{\Lambda} \otimes\cl K.\big(\mathop{\otimes}\limits_{\ell\in\Lambda^1}L^2(G)\big)\cong\cl K.(\cl L.)$$ as ${\mathfrak F}_{\Lambda}$ is a full matrix algebra. This shows that for a finite lattice there will be only one irreducible representation, up to unitary equivalence. Also, ${\mathfrak A}_{\Lambda}$ is simple, so all representations are faithful.
The algebra ${\mathfrak A}_{\Lambda}$ is faithfully and irreducibly represented on $\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)$ by $\pi=\pi_F\otimes\big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\pi_{\ell}\big)$ where $\pi_\ell:C(G)\rtimes_\lambda G\to L^2(G)$ is the generalized Schrödinger representation for the $\ell^{\rm th}$ entry. Then $\pi\big({\mathfrak A}_{\Lambda}\big)$ contains in its multiplier algebra the operators ${\widehat}{T}^{(\ell)}_{\Phi_{ij}(\ell)},\;{\widehat}{U}^{(\ell)}_g$ for all $\ell\in \Lambda^1$.
To complete the picture, we also need to define the action of the local gauge group on ${\mathfrak A}_{\Lambda}$. Recall that in $\pi$ it is given by $\zeta\to{\rm Ad}({\widehat}{W}_\zeta)$, and this clearly preserves $\pi\big({\mathfrak A}_{\Lambda}\big)=\cl K.(\cl H.)$ and defines a strongly continuous action $\alpha$ of $\gauc \Lambda$ on $\pi\big({\mathfrak A}_{\Lambda}\big)$ (hence on ${\mathfrak A}_{\Lambda} $) as $\zeta\to{\widehat}{W}_\zeta$ is strong operator continuous. By construction ${(\pi,{\widehat}{W})}$ is a covariant representation for the C\*-dynamical system given by $\alpha:\gauc \Lambda \to\aut{\mathfrak A}_{\Lambda}$. As $\gauc \Lambda=\prod\limits_{x\in \Lambda^0}G$ is compact, we can construct the crossed product ${\mathfrak A}_{\Lambda}\rtimes_\alpha \gauc \Lambda$ which has as representation space all covariant representations of $\alpha:\gauc \Lambda \to\aut{\mathfrak A}_{\Lambda}$. As it is convenient to have an identity in the algebra, our full field algebra for the system will be taken to be: $$\al F._e:=({\mathfrak A}_{\Lambda}\oplus\C)\rtimes_\alpha (\gauc \Lambda)$$ where ${\mathfrak A}_{\Lambda}\oplus\C$ denotes ${\mathfrak A}_{\Lambda}$ with an identity adjoined. This has a unique faithful representation on ${\mathcal{H}}$ corresponding to the covariant representation ${(\pi,{\widehat}{W})}$.
We consider two types of gauge invariant observables for lattice QCD which appeared in the literature (cf. [@KS]).
- We start with gauge invariant variables of pure gauge type, and consider the well-known Wilson loops cf. [@Wilson]. To construct a Wilson loop, we choose an oriented loop $L=\{\ell_1,\ell_2,\ldots,
\ell_m\}\subset\Lambda^1$, $\ell_j=(x_j,y_j)$, such that $y_j=x_{j+1}$ for $j=1,\ldots,{m-1}$ and $y_m=x_1$. Let $G_k=G$ be the configuration space of $\ell_k$. Denoting the components of a gauge potential $\Phi$ by $\Phi_{ij}(\ell_k)\in C(G_k)$ as in equation , the matrix components of the quantum connection at $\ell_k$ are given by $T_{\Phi_{ij}(\ell_k)}$. To construct the gauge invariant observable associated with the loop, define (summing over repeated indices): $$\begin{aligned}
W(L)&:=&\Phi_{i_1i_2}(\ell_1)(g_1)\,\Phi_{i_2i_3}(\ell_3)(g_2)\cdots \Phi_{i_{m-1}i_1}(\ell_m)(g_m)\\[1mm]
&=&(e_{i_1},g_1g_2\cdots g_me_{i_1})= {\rm Tr}(g_1g_2\cdots g_m)\,.
\end{aligned}$$ (Note that to perform the product we need to fix identifications of $G_i$ with $G$.) This defines a gauge invariant element $W(L)\in C(G_1)\otimes\cdots\otimes C(G_m)={C(G_1\times\cdots G_m)}$. To see the gauge invariance, just note that $$\big(\zeta\cdot\Phi)(\ell)\big(\zeta\cdot\Phi)(\ell')=
\zeta(x_\ell)\,\Phi(\ell)\,\zeta(y_\ell)^{-1}\zeta(x_{\ell'})\,\Phi(\ell')\,\zeta(y_{\ell'})^{-1}
=\zeta(x_\ell)\,\Phi(\ell)\,\Phi(\ell')\,\zeta(y_{\ell'})^{-1}$$ if $y_\ell=x_{\ell'}$ (i.e. $\ell'$ follows $\ell$), and use the trace property for $W(L)$.
Wilson loops of particular importance are those where the paths are plaquettes, i.e. $L={(\ell_1,\ell_2,\ell_3,\ell_4)}\in\Lambda^2$ as such $W(L)$ occur in the lattice Hamiltonian. As remarked above, as $C(G_j)\subset M({C(G_j)\rtimes_\lambda G_j})$, it is not actually contained in $\cl L._{\ell_j}$. We embed $C(G_1)\otimes\cdots\otimes C(G_m)$ (hence $W(L)$) in $M({\mathfrak A}_{\Lambda})$ by letting it act as the identity in entries not corresponding to $\{\ell_1,\ell_2,\ell_3,\ell_4\}.$
- Another method of constructing gauge invariant observables, is by Fermi bilinears connected with a Wilson line (cf. [@KS]). Consider a path $C=\{\ell_1,\ell_2,\ldots,
\ell_m\}\subset\Lambda^1$, $\ell_j=(x_j,y_j)$, such that $y_j=x_{j+1}$ for $j=1,\ldots,m-1$. We take notation as above, so $G_k=G$ is the configuration space of $\ell_k,$ and $\Phi_{ij}(\ell_k)(g_k):=(e_i,g_ke_j)$, $g_k\in G_k$. To construct a gauge invariant observable associated with the path, consider (with summation convention): $$\begin{aligned}
Q(C)&:=& \psi^*_{i_1}(x_1)\,\Phi_{i_1i_2}(\ell_1)\,\Phi_{i_2i_3}(\ell_3)\cdots \Phi_{i_{m-1}i_m}(\ell_m)\, \psi_{ i_m}(y_m)\\[1mm]
&\in& {\mathfrak F}_{S}\otimes C(G_1)\otimes\cdots\otimes C(G_m)
\end{aligned}$$ where $S\subseteq\Lambda^0$ contains the path and we assume $\Cn=\C^k$ (otherwise $\Cn=\C^k\times{\bf W}$ and there are more indices). Then $ Q(C)$ is gauge invariant. As above, we embed ${\mathfrak F}_{S}\otimes C(G_1)\otimes\cdots\otimes C(G_m)$ in $M({\mathfrak A}_{\Lambda})$ in the natural way.
In the representation $\pi=\pi_F\otimes\big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\pi_{\ell}\big)$ it is also possible to build unbounded observables as gauge invariant operators. For example we can build gauge invariant combinations of the gluonic and the colour electric field generators, and in the finite lattice context, such operators were analyzed in [@KR1; @JKR]. As an example of such a gauge invariant operator, in the context of a finite lattice, we state the Hamiltonian, where we disregard terms by which $H$ has to be supplemented in order to avoid the fermion doubling problem. We first need to add spinor indices, hence take $\Cn = \b W.\otimes\C^k$ where $\b W.\cong\C^4$ will be the spinor part on which the $\gamma$-matrices act. If $\{w_1,\ldots,w_4\}$ is an orthonormal basis of $\b W.$ and $\{e_1,\ldots,e_k\}$ is an orthonormal basis of $\C^k$, then w.r.t. the orthonormal basis $\{w_j\otimes e_n\mid j=1,\ldots,4,\,
n =1,\ldots,k\}$ of $\Cn$, we obtain the indices $$a(w_j\otimes e_n\cdot\delta_x)=:\psi_{jn}(x)$$ for the quark field generators, where the subscript $j$ is the spinor index, and the $n$ is the gauge index. Then the Hamiltonian is $$\begin{aligned}
\label{Hamiltonian}
H & = & \tfrac{a}{2} \sum_{\ell \in \Lambda^1}
E_{ij}(\ell) E_{ji}(\ell)
+ \tfrac{1}{2 g^2 a}\sum_{p \in \Lambda^2}
( W (p) + W(p)^*) \nonumber \\
& + & i\tfrac{a}{2} \sum_{\ell \in \Lambda^1}
\bar\psi_{jn}(x_\ell) \big[\underline\gamma\cdot(y_\ell-x_\ell)\big]_{ji}
\Phi_{nm} (\ell)\psi_{im} (y_\ell) + h.c.
\nonumber \\
& + & ma^3 \sum_{x \in \Lambda^0} \bar \psi_{jn} (x) \psi_{jn}(x)\,,\end{aligned}$$ where $a$ is the assumed lattice spacing; $W (p)$ is the Wilson loop operator for the plaquette $p= (\ell_1, \ell_2 ,\ell_3, \ell_4)$; the vector $y_\ell-x_\ell$ for a link $\ell=(x_\ell,y_\ell)$ is the vector of length $a$ pointing from $x_\ell$ to $y_\ell$ and h.c. means the Hermitean conjugate. As usual for spinors, $ \bar \psi_{jn} (x)=\psi_{in}(x)^*(\gamma_0)_{ij}$, and we use the standard gamma-matrices. We have omitted the flavour indices. The summands occurring in are either Laplacians, Wilson loop operators or Fermi bilinears, hence they are all gauge invariant, hence are observables, some unbounded.
The above Hamiltonian suffers from the well-known fermion doubling problem (cf. [@FL01]);- the latter can be cured by passing e.g. to Wilson fermions [@Wilson2]. This modification does not affect the arguments below, hence we focus our analysis on the naive Hamiltonian given by (\[Hamiltonian\]). Below in Sect. \[LDA\] we will consider the dynamics produced by this Hamiltonian.
The Fermion algebra for an infinite lattice. {#FAlg}
--------------------------------------------
It is unproblematic to specify the Fermion field on an infinite lattice $\Lambda=(\Lambda^0, \Lambda^1, \Lambda^2)$:\
[**Assumption:**]{} [*Assume the quantum matter field algebra on $\Lambda$ is: $$\label{fermifieldalgebra}
{\mathfrak F}_{\Lambda} := \CAR \ell^2(\Lambda^0,\Cn). =C^*\big(\mathop{\bigcup}_{x\in\Lambda^0}{\mathfrak F}_x\big)$$ where ${\mathfrak F}_x:=\CAR V_x.$ and $V_x:=\{f\in\ell^2(\Lambda^0,\Cn)\,\mid\, f(y)=0\;\;\hbox{if}\;\; y\not=x\}\cong \Cn.$ We interpret ${\mathfrak F}_x\cong\CAR \Cn.$ as the field algebra for a fermion at $x.$ We denote the generating elements of $\CAR \ell^2(\Lambda^0,\Cn).$ by $a(f),$ $f\in\ell^2(\Lambda^0,\Cn),$ and these satisfy the usual CAR–relations: $$\label{eq:car}
\{a(f), a(g)^*\} = {\langle}f, g {\rangle}{\mathbf{1}}\quad \hbox{ and } \quad \{a(f),a(g)\} = 0
\quad \mbox{ for } \quad f, g \in \ell^2(\Lambda^0,\Cn)$$ where $\{A,B\} := AB + BA$.*]{}\
Note that the odd parts of ${\mathfrak F}_x$ and ${\mathfrak F}_y$ w.r.t. the fields $a(f)$ anticommute if $x\not=y.$ Moreover, as $\Lambda^0$ is infinite, ${\mathfrak F}_{\Lambda}$ has inequivalent irreducible representations.
We have the following inductive limit structure. Let $\cl S.$ be a directed set of finite connected subgraphs $S\subset(\Lambda^0, \Lambda^1)$, such that $\bigcup\limits_{S\in\cl S.}S=(\Lambda^0, \Lambda^1)$, where the partial ordering is set inclusion. Note that $S_1\subseteq S_2$ implies $\Lambda_{S_1}^i\subseteq\Lambda_{S_2}^i$ and $\bigcup\limits_{S\in\cl S.}\Lambda_S^i=\Lambda^i.$ Define ${\mathfrak F}_{S}:=C^*\big(\mathop{\cup}\limits_{x\in\Lambda_S^0}{\mathfrak F}_x\big)\subset{\mathfrak F}_{\Lambda}$ and then ${\mathfrak F}_{\Lambda}=\ilim{\mathfrak F}_{S}$ is an inductive limit w.r.t. the partial ordering in $\cl S..$
This defines the quantum matter fields on the lattice sites, and to obtain correspondence with the physics notation, we choose an appropriate orthonormal basis in $\Cn$ and proceed as before for a finite lattice.
A maximal C\*-algebra for dynamics construction. {#GFA}
------------------------------------------------
In this section we wish to define a concrete C\*-algebra which is “maximally large” in the sense that it contains all the true local C\*-algebras, as well as the (bounded) terms of the local Hamiltonians. In the next section we will then define the dynamics automorphisms on it, and finally we will then take our new field C\*-algebra to be the smallest subalgebra which contains all the true local C\*-algebras, and is preserved w.r.t. the dynamics. Thus, the dynamics itself will create the field algebra for the system. The gauge transformations will be added later.
Recall from above that for every link $\ell$ we have a generalised Weyl algebra ${C(G)\rtimes_\lambda G}
\cong\cl K.\big(L^2(G)\big)$, for which we have a generalized Schrödinger representation $\pi_0:C(G)\rtimes_\lambda G\to\cl B.\big(L^2(G)\big)$ such that $\pi_0\big(C(G)\rtimes_\lambda G\big)=\cl K.\big(L^2(G)\big)$. Explicitly, it is given by $\pi_0(\varphi\cdot f)=\pi_1(\varphi)\pi_2( f)$ for $\varphi\in L^1(G)$ and $f\in C(G)$ where $$(\pi_1(\varphi)\psi)(g):=\int\varphi(h)\psi(h^{-1}g)\,{\rm d}h \qquad\hbox{and}\qquad
(\pi_2(f)\psi)(g):=f(g)\psi(g)$$ for all $\psi\in L^2(G)$. Note that by irreducibility, the constant vector $\psi_0=1$ is cyclic and normalized (assuming normalized Haar measure on $G$).
We start by taking an infinite product of generalized Schrödinger representations, one for each $\ell\in\Lambda^1$, w.r.t. the reference sequence ${(\psi_0,\psi_0,\ldots)}$ where $\psi_0=1$ is the constant vector (cf.[@vN]). Thus the space $\al H._\infty$ is the completion of the pre–Hilbert space spanned by finite combinations of elementary tensors of the type $$\varphi_1\otimes\cdots\otimes \varphi_k\otimes\psi_0\otimes\psi_0\otimes\cdots,\quad\varphi_i\in \al H._i= L^2(G),\;
k\in\N$$ w.r.t. the pre–inner product given by $$\left(\varphi_1\otimes\cdots\otimes \varphi_k\otimes\psi_0\otimes\psi_0\otimes\cdots,\;
\varphi'_1\otimes\cdots\otimes \varphi'_k\otimes\psi_0\otimes\psi_0\otimes\cdots\right)_\infty
:=\prod_{i=1}^k(\varphi_i,\varphi'_i)\,,$$ and we assume the usual entry-wise tensorial linear operations for the elementary tensors. Denote the reference vector by $\psi_0^\infty:=\psi_0\otimes\psi_0\otimes\cdots$.
Fix a finite nonempty connected subgraph $S$ of $(\Lambda^0, \Lambda^1)$, and let $$\Lambda_S:=(\Lambda^0_S, \Lambda^1_S, \Lambda^2_S).$$ Then $\al L._S:={\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\big(C(G)\rtimes_\lambda G\big)}$ acts on $\al H._\infty$ as a product representation of $\pi_0$ where each factor $\al L._\ell:=C(G)\rtimes_\lambda G$ acts on the factor of $\al H._\infty$ corresponding to $\ell$. In fact if $[\cdot]$ denotes closed span, then $$[\al L._S\psi_0^\infty]=\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)\otimes
\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\psi_0\subset\al H._\infty.$$ Thus all $\al L._S$ are faithfully imbedded in $\al B.(\al H._\infty)$, and if $S$ and $S'$ are disjoint, $\al L._S$ and $\al L._{S'}$ commute.
Now consider the Fock representation $\pi_{\rm Fock}:{\mathfrak F}_{\Lambda}\to\al B.(\al H._{\rm Fock})$ of the CAR–algebra with vacuum vector $\Omega$. The Hilbert space on which we define our infinite lattice model is $$\al H.:=\al H._{\rm Fock}\otimes\al H._\infty.$$ Then by ${\mathfrak F}_{S}\subset{\mathfrak F}_{\Lambda}$, we also have a product representation of the local field algebras ${\mathfrak A}_S:={\mathfrak F}_{S}\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\cl L._\ell$ on $\al H.$. If $S$ and $S'$ are disjoint, then ${\mathfrak A}_S$ and ${\mathfrak A}_{S'}$ will graded–commute w.r.t. the Fermion grading. If we have containment i.e., $R\subset S$, then ${\mathfrak F}_{R}\subset{\mathfrak F}_{S}$, but we have $\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_R}\cl L._\ell\not\subset
\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\cl L._\ell$ because $\al K.(\al H._1)\otimes\un\not\subset\al K.(\al H._1\otimes\al H._2)$ if $\al H._2$ is infinite dimensional. However w.r.t. the natural operator product we have ${\mathfrak A}_R\cdot{\mathfrak A}_S={\mathfrak A}_S$, hence ${\mathfrak A}_R\subset M({\mathfrak A}_S)$, as the action of ${\mathfrak A}_R$ on ${\mathfrak A}_S$ is nondegenerate.
Note that the Fermion grading also produces a grading unitary $U_F$ on $\al H._{\rm Fock}$ as the second quantization of -1 acting on $\ell^2(\Lambda^0,\Cn)$. This grading coincides with the even-odd grading for $n\hbox{-particle}$ vectors in $\al H._{\rm Fock}$. Naturally $U_F$ extends to $\al H.:=\al H._{\rm Fock}\otimes\al H._\infty$ as $U_F\otimes\un$, which extends by conjugation the Fermion grading to all of $\cl B.(\al H.)$.
The local graded commutation properties above are crucial for our construction of local dynamics, and this property will be preserved in our definition of a maximal C\*-algebra on which we will construct the dynamics. For each finite connected subgraph $S$ of $(\Lambda^0, \Lambda^1)$, we define $$\al H._S:= [{\mathfrak F}_{S}\Omega]\otimes[\al L._S\psi_0^\infty],\qquad\cl B._S:= \cl B.(\al H._S).$$ To see how to imbed $ \cl B.(\al H._S)\subset \cl B.(\al H.)$, note that ${\mathfrak F}_{S}$ is finite dimensional, hence so is $[{\mathfrak F}_{S}\Omega]$ and as the restriction of $\pi_{\rm Fock}({\mathfrak F}_{S})$ to $[{\mathfrak F}_{S}\Omega]$ is just the Fock representation of ${\mathfrak F}_{S}$, we obtain from irreducibility that $\cl B.([{\mathfrak F}_{S}\Omega])= {\mathfrak F}_{S}$ on $[{\mathfrak F}_{S}\Omega]$. Thus $$\cl B._S=\cl B.(\al H._S)=\cl B.( [{\mathfrak F}_{S}\Omega]\otimes[\al L._S\psi_0^\infty])
=\cl B.([{\mathfrak F}_{S}\Omega])\otimes \cl B.([\al L._S\psi_0^\infty])
={\mathfrak F}_{S}\otimes\cl B.([\al L._S\psi_0^\infty])$$ where the third equality follows from [@KR83 Example 11.1.6]. Let ${\mathfrak F}_{S}\subset{\mathfrak F}_{\Lambda}$ act on $\al H._{\rm Fock}$ as part of the Fock representation of ${\mathfrak F}_{\Lambda}$. As $$[\al L._S\psi_0^\infty]=\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)\otimes
\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\psi_0\subset\al H._\infty,$$ we can extend $\cl B.([\al L._S\psi_0^\infty])$ to $\al H._\infty$ by letting its elements act as the identity on those factors of $\al H._\infty$ corresponding to $\ell\not\in\Lambda^1_S$, i.e. $$\label{BSisFB}
\cl B._S
=\pi_{\rm Fock}({\mathfrak F}_{S})\otimes\cl B.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)
\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\un.$$ Thus we have obtained the embedding $ \cl B.(\al H._S)\subset \cl B.(\al H.)$, and we have containments $ \cl B.(\al H._S)\subseteq \cl B.(\al H._T)$ if $S\subseteq T$. Note that the restriction of this embedded copy of $ \cl B.(\al H._S)$ to $\al H._S\subset\al H.$ gives a faithful representation, but it is nonzero outside $\al H._S$ as it contains the identity. Now we can define: $$\al A._{\rm max}:=\ilim \cl B._S = C^*\Big(\bigcup_{S\in\al S.} \cl B.(\al H._S) \Big)$$ where in the last inductive limit and union, $S$ ranges over the directed set $\al S.$ of all finite connected subgraphs of $(\Lambda^0, \Lambda^1)$. Here and below, we will assume that $\al S.$ does not contain the empty set, so we can ignore this trivial special case.
Note that if we use the grading unitary above to extend the Fermion grading to all of $\cl B.(\al H.)$, then if $S$ and $S'$ are disjoint, then $\cl B.(\al H._S)\supset{\mathfrak A}_S$ and $\cl B.(\al H._{S'})
\supset{\mathfrak A}_{S'}$ will graded–commute. Note that a similar inductive limit cannot be done for the local algebras ${\mathfrak A}_S$ as it is NOT true that $S\subset S'$ implies ${\mathfrak A}_S\subset{\mathfrak A}_{S'}$. An operator $A\in\cl B.(\al H.)$ will be said to have [*support in*]{} $S$ if $S$ is the smallest connected graph for which $A\in\cl B.(\al H._S)=\cl B._S$.
As $\al A._{\rm max}$ contains all the local algebras ${\mathfrak A}_S$, we aim to define the full dynamics on $\al A._{\rm max}$, and then generate the new field algebra in $\al A._{\rm max}$ from the orbits of the local ones. At the end of the next section we will show that $\al A._{\rm max}$ is contained in the multiplier algebra of the kinematics field algebra we constructed previously in [@GrRu].
Dynamics {#Dynamics}
========
Next, we want to define the dynamics on $\al A._{\rm max}:=\ilim \cl B._S $ corresponding to the heuristic Hamiltonian given by $$\begin{aligned}
\label{HamiltonianInft}
& + & ma^3 \sum_{x \in \Lambda^0} \bar \psi_{jn} (x) \psi_{jn}(x)\,,\end{aligned}$$ where the difference with (\[Hamiltonian\]) is that now the sums are over an infinite lattice, so are not yet properly defined. The unbounded operators in the first summand are defined on the appropriate factor of $\al H._\infty$, and all calculations will be done concretely, i.e. in terms of operators on $\al H.:=\al H._{\rm Fock}\otimes\al H._\infty$, and we will not explicitly indicate the representations $\pi_0$ and $\pi_{\rm Fock}$. Otherwise, notation is as before.
The summands occurring in are all gauge invariant (locally) and hence are observables, some unbounded. All summands also have Fermion degree zero, hence for these summands graded local commutativity becomes just ordinary local commutativity with elements of the algebra $\al A._{\rm max}:=\ilim \cl B._S $.
Below we will follow the familiar technique for defining the dynamics of lattice systems by first defining it for each $S\in\cl S.$, then proving that these “local automorphism groups" have a pointwise norm limit which defines a dynamics automorphism for the full algebra (cf. [@BR2]).
The local dynamics automorphism groups. {#LDA}
---------------------------------------
Now for a fixed $\ell$, in the representation $\pi_0$ on $L^2(G)$ the operator $E_{ij}(\ell) E_{ji}(\ell)$ (summation convention) is just the group Laplacian, hence it is defined on the domain $C^\infty(G)\subset L^2(G)$, and it is essentially selfadjoint as it produces a positive quadratic form. These are the only unbounded terms in $H$. Given $S\in\cl S.$ we define the local Hamiltonian $H_S$ by summing only over the restricted lattice $\Lambda_{S}$, thus: $$\begin{aligned}
H_S&=& H_S^{\rm loc}+H_S^{\rm int}\qquad\hbox{on}\qquad \cl H._{\rm Fock}\otimes\cl D._S
\qquad\hbox{where} \\[2mm]
H_S^{\rm loc} & := & \tfrac{a}{2} \sum_{\ell \in \Lambda^1_S}
E_{ij}(\ell) E_{ji}(\ell)+ma^3 \sum_{x \in \Lambda^0_S} \bar \psi_i (x) \psi_i(x)
\qquad\hbox{on}\qquad \cl H._{\rm Fock}\otimes\cl D._S \\[1mm]
H_S^{\rm int} & := & \tfrac{1}{2 g^2 a}\sum_{p \in \Lambda^2_S}
( W (p) + W(p)^*) +i\tfrac{a}{2} \sum_{\ell \in \Lambda^1_S}
\bar\psi_{jn}(x_\ell) \big[\underline\gamma\cdot(y_\ell-x_\ell)\big]_{ji}
\Phi_{nm} (\ell)\psi_{im} (y_\ell) + h.c.\end{aligned}$$ where $\cl D._S\subset\bigotimes\limits_{\ell\in\Lambda^1}L^2(G)$ is the span of those elementary tensors such that if $\ell\in\Lambda^1_S$, then in the $\ell\hbox{-th}$ entry it takes its value in $C^\infty(G)$ but otherwise it is unrestricted. One may write this as $\cl D._S=\bigotimes\limits_{\ell\in\Lambda^1_S}C^\infty(G)\otimes\bigotimes\limits_{\ell'\not\in\Lambda^1_S}L^2(G)$ (infinite tensor products are w.r.t. the reference vector $\psi_0^\infty$). Note that $H_S^{\rm loc}$ is an unbounded essentially selfadjoint (positive) operator which affects the individual sites and links independently (so produces free time evolution), and that the interaction term $H_S^{\rm int}$ is bounded. In fact the free time evolution is just a tensor product of the individual free time evolutions: $$U^{\rm loc}_S(t):=\exp(it\bar{H}_S^{\rm loc})=U^{\rm CAR}_S(t)\otimes
\bigotimes_{\ell\in\Lambda^1_S}U_\ell(t)\otimes\bigotimes\limits_{\ell'\not\in\Lambda^1_S}{\mathbf{1}}$$ where $U^{\rm CAR}_S(t)=\exp\big(itma^3 \sum\limits_{x \in \Lambda^0_S} \bar \psi_i (x) \psi_i(x)\big)\in{\mathfrak F}_{S}$ and $U_\ell(t):=\exp(it\overline{E_{ij}(\ell) E_{ji}(\ell)})$ (the overline notation $\bar{H}_S^{\rm loc}$ and $\overline{E_{ij}(\ell) E_{ji}(\ell)}$ indicates the closure of the essentially selfadjoint operators). The local free time evolutions $\alpha^{\rm loc}_S(t):={\rm Ad}(U^{\rm loc}_S(t))$ will preserve each $\cl B._{S'}, $ $S'\in\al S.$, hence preserves $\al A._{\rm max}=\ilim \cl B._S $ because it acts componentwise. However $\alpha^{\rm loc}$ is not strongly continuous i.e. pointwise norm continuous, as $H_S^{\rm loc}$ is unbounded and $\cl B._S=\cl B.(\al H._S)$ (cf. Prop. 5.10 in [@GrN14]).
As $U^{\rm CAR}_S(t)\in{\mathfrak F}_{S}\subset{\mathfrak F}_{\Lambda},$ we have that $\alpha^{\rm CAR}_S(t):={\rm Ad}(U^{\rm CAR}_S(t))$ clearly preserves both ${\mathfrak F}_{S}$ and ${\mathfrak F}_{\Lambda}.$ Moreover $t\mapsto\alpha^{\rm CAR}_S(t)$ is uniformly norm continuous as the generator $ma^3 \sum\limits_{x \in \Lambda^0_S} \bar \psi_i (x) \psi_i(x)$ is bounded. Observe that $\alpha^{\rm loc}_S(t)$ satisfies the compatibility condition that if $R\subset S$, then $\alpha^{\rm loc}_S(t)$ preserves the subalgebra $\al B._R\subset\al B._S$ and its restriction on it coincides with $\alpha^{\rm loc}_R(t)$, as is clear from the tensor product construction. We have that ${\rm Ad}(U^{\rm loc}_S(t))$ defines a one parameter group $t\mapsto\alpha^{\rm loc}_S(t)\in{\rm Aut}(\al A._{\rm max})$ which preserves $\al B._S$, and is the identity on any $\al B._R$ where $R\cap S=\emptyset$. Next, note that as the (bounded) interaction Hamiltonian $H_S^{\rm int}\in \al B._S,$ its commutator produces a derivation on $\al A._{\rm max}$, which preserves $\al B._S$ i.e. $ [H_S^{\rm int},\al B._S]\subseteq \al B._S$.
To be more precise, consider the individual terms in the finite sums comprising $ H_S^{\rm int}$. Recall that for $p={(\ell_1,\ell_2,\ell_3,\ell_4)}\in\Lambda^2$, we have $$\begin{aligned}
W(p)&\in& C(G_{\ell_1})\otimes\cdots\otimes C(G_{\ell_4})={C(G_{\ell_1}\times\cdots G_{\ell_4})}\subset M(
{\cal L}_{S}) \subseteq \al B._S \qquad
\hbox{if}\qquad p\subset S\\[1mm]
\hbox{and}\;
&& \bar\psi_{jn}(x_\ell) \big[\underline\gamma\cdot(y_\ell-x_\ell)\big]_{ji}
\Phi_{nm} (\ell)\psi_{im} (y_\ell)\in {\mathfrak F}_{S}\otimes C(G_\ell)\subset \al B._S
\end{aligned}$$ where $\ell=(x,y)\subset S$. Thus the commutator with $ H_S^{\rm int}$ defines a bounded derivation on $\al A._{\rm max}$, which preserves $\al B._S$. For each $S\in\cl S.$ we have a local time evolution $\alpha^S:\R\to{\rm Aut}\big(\al A._{\rm max}\big)$ which preserves $\pi(\al B._S)$ and acts trivially on $\al A._{\rm max}$ outside of it, given by $$\alpha^S_t:={\rm Ad}(U_S(t))\quad\hbox{and}\quad
U_S(t):=\exp(itH_S).$$ Due to the interaction terms, we do not expect that $R\subset S$ implies that $\alpha^S$ will coincide with $\alpha^R$ on $\al B._R\subset\al B._S$. By construction, $(\pi,U_S)$ is a covariant irreducible representation for $\alpha_S:\R\to{\rm Aut}(\al A._{\rm max})$. The generator $H_S= H_S^{\rm loc}+H_S^{\rm int}$ of its implementing unitary group $U_S$ has a positive unbounded part $H_S^{\rm loc}$ and a bounded interaction part $H_S^{\rm int}$, hence it is bounded from below. As $H_S$ is unbounded, $\alpha^S$ is not strongly continuous on $\al B._S$. However, on the local algebras ${\mathfrak A}_S\subset\al B._S$ we have:
\[localauto\] Let $R,\,S\in\al S.$ with $R\subseteq S$. For each $A\in{\mathfrak A}_S$ we have that $\; t\mapsto \alpha^R_t(A)\;$ is continuous, i.e. the restriction of $\alpha^R$ to ${\mathfrak A}_S$ is strongly continuous.
Recall that ${\mathfrak A}_S:={\mathfrak F}_{S}\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\cl L._\ell$ acts faithfully on $$\al H._S:= [{\mathfrak F}_{S}\Omega]\otimes[\al L._S\psi_0^\infty]\subset\cl H.\quad\hbox{where}\quad
[\al L._S\psi_0^\infty]=\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)\otimes
\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\psi_0\subset\al H._\infty.$$ As $S$ is finite, $[{\mathfrak F}_{S}\Omega]$ is finite dimensional, and so by $\al L._S
=\al K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)$ we conclude that ${\mathfrak A}_S\restriction\al H._S\subseteq\al K.(\al H._S)$. Now $\alpha^R_t={\rm Ad}(U_R(t))$ where $U_R(t)=\exp(itH_R)$, and $U_R(t)$ preserves $\al H._S$ and is the identity on any $\al H._{S'}$ where $S'$ is disjoint from $S$. Thus for $A\in{\mathfrak A}_S$, $\alpha^R_t(A)$ becomes just the conjugation of a compact operator by a strong operator continuous one parameter unitary group, and this is well known to be norm continuous in the parameter (the strict topology on $\al B.(\al H.)=M(\al K.(\al H.))$ is the $\sigma\hbox{--strong}$ \*–topology).
The analogous converse statement will be proven below in the proof of Theorem \[GlobDynCont\], i.e. that if $R\subset S$, then $\; t\mapsto \alpha^S_t(A)\;$ is continuous for $A\in{\mathfrak A}_R$. As ${\mathfrak A}_R\not\subset{\mathfrak A}_S$ and $\alpha^S_t$ need not preserve ${\mathfrak A}_R$, this is not obvious.
Existence of the global dynamics automorphism group. {#DMFA}
----------------------------------------------------
We want to apply arguments of Nachtergaele and Sims in [@NaSi] to establish the existence of the full infinite lattice dynamics on $\al A._{\rm max}$. Here we will supply all the necessary details to adapt their argument to our situation.
Return to the faithful representation $\pi=\pi_{\rm Fock}\otimes\pi_\infty$, on $\cl H.=\cl H._{\rm Fock}\otimes\cl H._\infty$ where $\cl H._\infty=\bigotimes\limits_{\ell\in\Lambda^1}L^2(G_\ell)$ is defined w.r.t. the reference vector $\psi_0\otimes\psi_0\otimes\cdots$. All calculations below will be done concretely in this representation, but to limit notation it will not usually be indicated. As before, we fix the directed set $\cl S.$ of finite connected subgraphs of $(\Lambda^0,\Lambda^1)$, where the partial ordering is set inclusion. Note that for each link $\ell\in\Lambda^1$, we can identify a graph in $\cl S.$ (also denoted by $\ell$) as the graph consisting of the endpoints $(x_\ell,y_\ell)$ for vertices, and the edge from $x_\ell$ to $y_\ell$. Likewise, we can identify any plaquette $p\in\Lambda^2$ with a graph in $\cl S.$. Then $$\al A._{\rm max}=\ilim \cl B._S = C^*\Big(\bigcup_{S\in\al S.} \cl B.(\al H._S) \Big)
\qquad\hbox{and}\qquad
{\mathfrak F}_{S}\otimes{\cal L}_{S}
\subset \cl B.(\al H._S).$$ Recall that for each $S\in\cl S.$ we have a local time evolution $\alpha^S:\R\to{\rm Aut}\big(\al A._{\rm max}\big)$ which preserves $\pi(\al B._S)$ and acts trivially on $\al A._{\rm max}$ outside of it, given by $\alpha^S_t={\rm Ad}(U_S(t))$. We will show below that for each $A\in \al B._R,$ $R\subset S$ that $\alpha^S_t(A)$ converges as ${S\nearrow\Z^3},$ where ${S\nearrow\Z^3}$ indicates that we take the limit over increasing sequences in $\cl S.$ such that the union of the graphs in the sequence is the entire connected graph ${(\Lambda^0,\,\Lambda^1)}$ for the lattice.
We now want to apply arguments in [@NaSi] to establish the existence of an infinite lattice dynamics on $ \al A._{\rm max}$. We first revisit notation. Fix $S\in\cl S.$, then $$\begin{aligned}
H_S&=& H_S^{\rm loc}+H_S^{\rm int}\qquad\hbox{on}\qquad \cl H._{\rm Fock}\otimes\cl D._S
\qquad\hbox{where} \\[2mm]
H_S^{\rm loc} & := & \sum_{\ell \in \Lambda^1_S}H_\ell+\sum_{x \in \Lambda^0_S}H_x\qquad\hbox{where}\\[2mm]
H_\ell&:=&\tfrac{a}{2}
E_{ij}(\ell) E_{ji}(\ell)\qquad\hbox{and}\qquad H_x:=ma^3 \bar \psi_i (x) \psi_i(x)\\[2mm]
H_S^{\rm int} & := & \sum_{p \in \Lambda^2_S}\widetilde{W}(p) + \sum_{\ell \in \Lambda^1_S}B(\ell)
\qquad\hbox{where}\quad \widetilde{W}(p):= \tfrac{1}{2 g^2 a}( W (p) + W(p)^*) \\[2mm]
\hbox{and}\qquad B(\ell)&:=&
i\tfrac{a}{2}
\bar\psi_{jn}(x_\ell) \big[\underline\gamma\cdot(y_\ell-x_\ell)\big]_{ji}
\Phi_{nm} (\ell)\psi_{im} (y_\ell) + h.c.\end{aligned}$$ Clearly $H_S^{\rm int}\in \al B._S$ and $\sum_{x \in \Lambda^0_S}H_x\in \al B._S.$ The only terms of $H_S$ not in the $ \al B._S$ are the unbounded $H_\ell$. Moreover the operator norm $\| \widetilde{W}(p)\|=:\|\widetilde{W}\|$ is independent of $p$ and $\|B(\ell)\|=:\|B\|$ is independent of $\ell.$ Below we will frequently need the following notation:- if $A\in\al B._S$ then $$A(t):=e^{itH_S^{\rm loc}}A\,e^{-itH_S^{\rm loc}}\in\al B._S\,.$$
\[GlobDynExist\] With notation as above, we have for all $A\in \al A._{\rm max}$ and $t\in\R$ that the norm limit $$\lim_{S\nearrow\Z^3}\alpha^S_t(A)=:\alpha_t(A)$$ exists, and defines an automorphism group $t\mapsto\alpha_t\in{\rm Aut}(\al A._{\rm max})$. Furthermore, for each $T>0$, the limit is uniform w.r.t. $t\in[-T,T]$.
[**Proof:**]{} Fix $T>0,$ a nonempty $R\subset S$ and let $A\in\al B._R$. First we want to show that the limit of $\alpha^S_t(A)$ as ${S\nearrow\Z^3}$ exists for $|t|<T$. Fix a strictly increasing sequence $\{S_n\}_{n\in\N}\subset\cl S.$ such that ${S_n\nearrow\Z^3}$ as $n\to\infty$. The limit we obtain below will be independent of the choice of sequence $\{S_n\}_{n\in\N}\subset\cl S.$ (given any two such sequences, each term of one sequence will be contained in some term of the other one, which allows one to form a new increasing sequence containing elements of both).
For the proof, we need to show that $\{\alpha^{S_n}_t(A)\}_{n\in\N}$ is Cauchy. Now for $R\subseteq S_n$: $$\alpha^{S_n}_t(A)=\tau^{S_n}_t\big( e^{itH_{S_n}^{\rm loc}} A e^{-itH_{S_n}^{\rm loc}} \big)
=\tau^{S_n}_t\big( e^{itH_R^{\rm loc}} A e^{-itH_R^{\rm loc}} \big)=\tau^{S_n}_t\big( A(t) \big)$$ where $\tau^{S}_t:={\rm Ad}(e^{itH_S}e^{-itH_S^{\rm loc}})$ so as $A(t)\in\al B._R$, it suffices to show that the sequence $\{\tau^{S_n}_t(A)\}_{n\in\N}$ is Cauchy for all $A\in\al B._R$.
Fix $S\in\cl S.$, consider the strong operator continuous map $U_S:\R\times\R\to\cl U.(\cl H.)$ given by $$U_S(t,s):=e^{itH_S^{\rm loc}}e^{i(s-t)H_S}e^{-isH_S^{\rm loc}}$$ and note that $U_S(t,t)=\un$, $U_S(t,s)^*=U_S(s,t)$ and $\tau^{S}_t(A)=U_S(0,t)AU_S(t,0)$. As $H_S$ and $H_S^{\rm loc}$ differ by a bounded operator, they have the same domain $\cl D.$ and this domain is preserved by both unitary groups $e^{itH_S}$ and $e^{itH_S^{\rm loc}}$, hence by $U_S(t,s)$ and so for $\psi\in\cl D.$ we have $$\frac{d}{dt}U_S(t,s)\psi=ie^{itH_S^{\rm loc}}( H_S^{\rm loc}- H_S )e^{i(s-t)H_S}e^{-isH_S^{\rm loc}}\psi
=-iH_S^{\rm int}(t)U_S(t,s)\psi$$ where $H_S^{\rm int}(t):=e^{itH_S^{\rm loc}}H_S^{\rm int}e^{-itH_S^{\rm loc}}$. Likewise, for $\psi\in\cl D.$ we have $$\frac{d}{ds}U_S(t,s)\psi=iU_S(t,s)H_S^{\rm int}(s)\psi.$$ Now by Lemma \[Lemma1\] in the Appendix, we conclude that these relations hold on all of $\cl H.$. Let $R\subset S_n\subset S_m$ (hence $n<m$) then by the fundamental theorem of calculus $$\tau^{S_m}_t(A)-\tau^{S_n}_t(A)=\int_0^t\frac{d}{ds}\Big(U_{S_m}(0,s)U_{S_n}(s,t)A\,U_{S_n}(t,s)U_{S_m}(s,0)\Big)\,ds$$ where the differential and integral is w.r.t. the strong operator topology. The integrand is for any $\psi\in\cl H.$: $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{d}{ds}\,U_{S_m}(0,s)U_{S_n}(s,t)A\,U_{S_n}(t,s)U_{S_m}(s,0)\psi\\[1mm]
&=&iU_{S_m}(0,s)\Big[(H_{S_m}^{\rm int}(s)-H_{S_n}^{\rm int}(s)), U_{S_n}(s,t)A\,U_{S_n}(t,s)\Big]U_{S_m}(s,0)\psi\\[1mm]
&=&iU_{S_m}(0,s)e^{isH_{S_n}^{\rm loc}}\Big[{N}(s), \alpha^{S_n}_{s-t}(A(t))\Big]e^{-isH_{S_n}^{\rm loc}}U_{S_m}(s,0)\psi
\qquad\hbox{where}\\[3mm]
{N}(s)&:=&e^{-isH_{S_n}^{\rm loc}}(H_{S_m}^{\rm int}(s)-H_{S_n}^{\rm int}(s))e^{isH_{S_n}^{\rm loc}}
=e^{isH_{S_m\backslash S_n}^{\rm loc}} H_{S_m}^{\rm int}e^{-isH_{S_m\backslash S_n}^{\rm loc}} -H_{S_n}^{\rm int}\\[1mm]
&=&{\rm Ad}\Big(e^{isH_{S_m\backslash S_n}^{\rm loc}} \Big)\Big(\sum_{p \in \Lambda^2_{S_m}
\backslash \Lambda^2_{S_n}}\widetilde{W}(p) + \sum_{\ell \in \Lambda^1_{S_m}\backslash \Lambda^1_{S_n}}B(\ell)\Big)\\[2mm]
&=&{\rm Ad}\Big(e^{isH_{S_m\backslash S_n}^{\rm loc}} \Big)\Big(\sum_{q \in \Lambda^i_{S_m}
\backslash \Lambda^i_{S_n}}\Psi(q)\Big)
\qquad\qquad\hbox{and}\\[2mm]
A(t)&=& e^{itH_R^{\rm loc}} A e^{-itH_R^{\rm loc}}=e^{itH_{S_n}^{\rm loc}} A e^{-itH_{S_n}^{\rm loc}}\end{aligned}$$ where the last sum is over $\Lambda^i_{S_m}\backslash \Lambda^i_{S_n}:=\Lambda^1_{S_m}\backslash \Lambda^1_{S_n}
\cup\Lambda^2_{S_m}\backslash \Lambda^2_{S_n}$ and $\Psi(p):=\widetilde{W}(p)$ for plaquette $p$, and $\Psi(\ell):=B(\ell)$ for a link $\ell$. Then $\|\Psi\|:=\max\{\|{\widetilde}{W}\|,\|B\|\}\geq\|\Psi(q)\|$ for all $q$.
The support of ${N}(s)$ is contained in $S_m\backslash(S_n)_0$ which is defined as the set of all the lattice points in $S_m$ (and links between them) obtained from either links in $\Lambda^1_{S_m}\backslash \Lambda^1_{S_n}$ or plaquettes in $\Lambda^2_{S_m}\backslash \Lambda^2_{S_n}$. Now ${N}(s)$ is a sum of terms with support in $q \in \Lambda^i_{S_m}
\backslash \Lambda^i_{S_n}$, and only those for which $q$ has a point in $S_n$ will have nonzero commutant with $\al B._{S_n} $. Thus for $B\in \al B._{S_n}$, we have $$\begin{aligned}
[N(s),B]&=& [{\widetilde}{N}(s),B]\qquad\hbox{where}\nonumber\\[1mm]
\label{primesum}
{\widetilde}{N}(s) & := &
{\rm Ad}\Big(e^{isH_{S_m\backslash S_n}^{\rm loc}} \Big)\Big(\mathop{\mathord{\sum}'}_{q \in \Delta_{S_m}(
S_n)}\Psi(q)\Big)\qquad\hbox{and}\\[1mm]
\label{bdry}
\Delta_{T}(R)&:=&\{Z\subset T\,\big|\,Z\cap R\not=\emptyset\not=Z\cap(T\backslash R)\},\end{aligned}$$ and the prime on the sum indicates that it is restricted by requiring $q$ to be a link or plaquette.
Thus, using Lemma \[Lemma2\] in the Appendix, and the fact that $\cl H.$ is separable, we get $$\begin{aligned}
\big\|\tau^{S_m}_t(A)-\tau^{S_n}_t(A)\| &\leq & \int_{t-}^{t^+}\Big\|\Big[{N}(s), \alpha^{S_n}_{s-t}(A(t))\Big]\Big\|\,ds\nonumber\\[2mm]
\label{ComIneq}
&=& \int_{t-}^{t^+}\Big\|\Big[{\widetilde}{N}(s), \alpha^{S_n}_{s-t}(A(t))\Big]\Big\|\,ds\end{aligned}$$ where $t^-={\rm min}\{0,t\}$ and $t^+={\rm max}\{0,t\}$, using the fact that $\alpha^{S_n}_{s-t}(A(t))\in \al B._{S_n}$. We will now estimate $\big\|\big[{\widetilde}{N}(s), \alpha^{S_n}_{s-t}(A(t))\big]\big\|$ (a method known as Lieb-Robinson bounds).
We first define the auxiliary function $$f(t):= \big[D, \alpha^{S_n}_{t}(\beta_{-t}(A))\big]\in \al B._{S_m},$$ where $\beta_t:={\rm Ad}(e^{it(H_{S_n}^{\rm loc}+H_R^{\rm int})})$, $A\in \al B._R$, $R\subset S_n$, and $D$ is any bounded operator with support in $S_m\backslash(S_n)_0$ (we have ${\widetilde}{N}(s)$ in mind). Then $\|f(0\|\leq 2\|A\|\|D\|\delta^{S_n}_R$ where $\delta^{S_n}_R=0$ if $R\cap S_m\backslash(S_n)_0=\emptyset$ (hence $\big[D, A\big]=0$) and one otherwise.
We claim that $\beta_t$ preserves $\al B._R$. To see this, note that both $\alpha^{\rm loc}_{S_n}(t):={\rm Ad}(e^{itH_{S_n}^{\rm loc}})$ and $\alpha^{\rm int}_{R}(t):={\rm Ad}(e^{itH_R^{\rm int}})$ preserve $\al B._R$, hence restrict to $\al K.(\al H._R)\subset \al B._R=\al B.(\al H._R)$. Then for these restrictions $\alpha^{\rm loc}_{S_n}$ is a $C_0$-group (i.e. strongly continuous), and $\alpha^{\rm int}_{R}$ is uniformly continuous (with bounded generator). Then the sum of the generators of these two groups is again a generator of a $C_0$-group on $\al K.(\al H._R)$ by Theorem 3.1.33 in [@BR1], and this $C_0$-group extends uniquely to an automorphic action of $\R$ on $M(\al K.(\al H._R))$. However the sum of the generators of the two groups on $\al K.(\al H._R)$ coincides with the generator of the W\*-action $\beta_t$, hence $\beta_t$ preserves $\al B._R$.
Now $\alpha^{S_n}_{t}\circ\beta_{-t}={\rm Ad}\big(V(t,-t)\big)$ where $V(s,t):=\exp{(isH_{S_n})}\exp{(it(H_{S_n}^{\rm loc}+H_R^{\rm int}))}$. As $V(s,t)$ is strong operator continuous in $s,\,t$ and both $H_{S_n}$ and $(H_{S_n}^{\rm loc}+H_R^{\rm int})$ have the same domain, which is preserved by these unitaries, it follows from strong differentiation on the domain and Lemma \[Lemma1\] in Appendix, that for all $\psi\in\cl H.$: $$\begin{aligned}
\frac{d}{dt}V(t,-t)\psi&=&ie^{itH_{S_n}}(H_{S_n}^{\rm int}-H_{R}^{\rm int})e^{-it(H_{S_n}^{\rm loc}+H_R^{\rm int})}\psi
=i(H_{S_n}^{\rm int}-H_{R}^{\rm int})(t)\,V(t,-t)\psi.
\\[2mm]
\hbox{Thus:}\qquad&&\\[2mm]
\frac{d}{dt}f(t)\psi&=&i\Big[D, \alpha^{S_n}_{t}(\big[(H_{S_n}^{\rm int}-H_{R}^{\rm int}),\,\beta_{-t}(A)\big])\Big]\psi\\[2mm]
&=&i\Big[D, \alpha^{S_n}_{t}\Big(\big[\sum_{q \in \Lambda^i_{S_n}
\backslash \Lambda^i_{R}}\Psi(q),
\,\beta_{-t}(A)\big]\Big)\Big]\psi\\[2mm]
&=&i\Big[D, \alpha^{S_n}_{t}\Big(\big[\mathop{\mathord{\sum}'}_{q\in \Delta_{S_n}(R)} \Psi(q),
\,\beta_{-t}(A)\big]\Big)\Big]\psi\end{aligned}$$ where we used $\beta_{-t}(A)\in \al B._R$ in the third step. Next, we define $$\begin{aligned}
{\widetilde}{H}_{R}^{\rm int}&:=&\mathop{\mathord{\sum}'}_{q\in \Delta_{S_n}(R)} \Psi(q),\quad\hbox{then}\\[2mm]
\frac{d}{dt}V(t,-t)\psi &=&i\Big[D, \alpha^{S_n}_{t}(\big[{\widetilde}{H}_{R}^{\rm int},
\,\beta_{-t}(A)\big])\Big]\psi\\[2mm]
&=&i\big[\alpha^{S_n}_{t}({\widetilde}{H}_{R}^{\rm int}),f(t)\big]\psi
-i\Big[ \alpha^{S_n}_{t}(\beta_{-t}(A)),\big[ D, \alpha^{S_n}_{t}({\widetilde}{H}_{R}^{\rm int}) \big]\Big]\psi.\end{aligned}$$ As the first term is a commutator with a bounded operator, it is a bounded linear map on $\al B._{S_m}$, hence by Lemma \[Lemma3\] in the appendix we have $$\begin{aligned}
\|f(t\|&\leq&\|f(0)\|+2\|A\|\int_{t^-}^{t^+}\left\| \big[ D, \alpha^{S_n}_{r}({\widetilde}{H}_{R}^{\rm int}) \big] \right\|\,dr,
\\[2mm]
\frac{\big\|\big[D, \alpha^{S_n}_{t}(\beta_{-t}(A))\big]\big\|}{2\|A\|} &\leq &\|D\|\delta^{S_n}_R +
\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)} \int_{t^-}^{t^+}\left\| \big[ D, \alpha^{S_n}_{r}(\Psi(q)) \big] \right\|\,dr.
$$ As $\beta_t:={\rm Ad}(e^{it(H_{S_n}^{\rm loc}+H_R^{\rm int})})$ preserves $\al B._R$, we can replace $A$ by $\beta_t(A)$ to get the estimate $$\label{CommToIterate}
\frac{\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|}{2\|A\|} \leq \|D\|\delta^{S_n}_R+
\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)} \int_{t^-}^{t^+}\left\| \big[ D, \alpha^{S_n}_{r}(\Psi(q)) \big] \right\|\,dr\,.
$$ If $\delta^{S_n}_R=0$, this inequality is potentially better than the naive inequality $$\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|\leq 2\|A\|\|D\|$$ which will be the case below when we let $S_n$ become large. The inequality $(\ref{CommToIterate})$ can now be iterated as $\Psi(q)\in \al B._{q}$. If we substitute $\Psi(q)$ for $A$ in $(\ref{CommToIterate})$ we get $$\frac{\big\|\big[D, \alpha^{S_n}_{t}(\Psi(q))\big]\big\|}{2\|\Psi\|} \leq
\|D\|\delta^{S_n}_{q} +
\mathop{\mathord{\sum}'}\limits_{q'\in\Delta_{S_n}(q)}
\int_{t^-}^{t^+}\left\| \big[ D, \alpha^{S_n}_{r}(\Psi(q')) \big] \right\|\,dr.$$ Substitution of this into the integrand of $(\ref{CommToIterate})$ and iterating produces: $$\begin{aligned}
\frac{\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|}{2\|A\|} &\leq &\|D\|\delta^{S_n}_R+
\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)}2\|D\|\|\Psi\|\delta^{S_n}_{q} \int_{t^-}^{t^+}1\,dr\\[1mm]
+\;2\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{\mathord{\sum}'}_{q'\in\Delta_{S_n}(q)}
\|\Psi\|\int_{t^-}^{t^+}\int_{r^-}^{r^+}\left\| \big[ D, \alpha^{S_n}_{r'}(\Psi(q')) \big] \right\|\,dr'\,dr\\[1mm]
&\leq&\|D\|\Big(\delta^{S_n}_R+2\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)}
\|\Psi\|\delta^{S_n}_{q} |t|\\[1mm]
&&\qquad\qquad+\;4\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)}
\mathop{\mathord{\sum}'}_{q'\in\Delta_{S_n}(q)}
\|\Psi\|^2\delta^{S_n}_{q'} |t|^2/2\Big)\\[1mm]
+\;4\mathop{\mathord{\sum}'}_{q\in\Delta_{S_n}(R)}\;\,
\mathop{\mathord{\sum}'}_{q'\in\Delta_{S_n}(q)}
&&\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{\mathord{\sum}'}_{q'' \in \Delta_{S_n}(q')}
\|\Psi\|^2\int_{t^-}^{t^+}\int_{r_1^-}^{r_1^+}\int_{r_2^-}^{r_2^+}
\left\| \big[ D, \alpha^{S_n}_{r_3}(\Psi(q'')) \big] \right\|\,dr_3\,dr_2\,dr_1\,.\end{aligned}$$ At the $N^{\rm th}$ iteration we have: $$\begin{aligned}
\frac{\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|}{2\|A\|} &\leq &
\|D\|\Big(\delta^{S_n}_R+\mathop{\mathord{\sum}'}_{k=1}^N\frac{(2\|\Psi\||t|)^k}{k!}a_k\Big)
+R_N\\[1mm]
\hbox{where}\qquad a_k := \mathop{\mathord{\sum}'}_{q_1\in\Delta_{S_n}(R)} &&\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{\mathord{\sum}'}_{q_2 \in\Delta_{S_n}(q_1)}\cdots
\mathop{\mathord{\sum}'}_{q_k \in\Delta_{S_n}(q_{k-1})}
\delta^{S_n}_{q_k} \\[1mm]
R_N:=2^N\|\Psi\|^N\!\!\!\!\mathop{\mathord{\sum}'}_{q_1\in\Delta_{S_n}(R)}
\mathop{\mathord{\sum}'}_{q_2 \in\Delta_{S_n}(q_1)}
\!\!\!\!\!\!&\cdots&\!\!\!\!\!\!\!\!
\mathop{\mathord{\sum}'}_{q_{N+1} \in\Delta_{S_n}(q_N) } \\[1mm]
\times &&
\!\!\!\!\!\!\!\!\int_{t^-}^{t^+}\int_{r_1^-}^{r_1^+}\!\!\!\!\cdots\!\!\int_{r^-_{N-1}}^{r_{N-1}^+}
\left\| \big[ D, \alpha^{S_n}_{r_{N+1}}(\Psi(q_{N+1})) \big] \right\|\,dr_{N+1}\!\!\cdots dr_1\,.\end{aligned}$$ To prove that the iteration converges, we need to estimate the remainder term. The integral is bounded by $ 2\|\Psi\|\,\|D\| |t|^{N+1}/(N+1)!$ so we concentrate on counting the number of terms in the sums.
Let $S_n$ be the lattice cube with corner vertices $(\pm n,\pm n,\pm n)$ and $R=S_d$ for $d$ fixed, then for $d<n$ we have that as a link $\ell \in \Delta_{S_n}(R)$ must have one point in $R$ and one point outside, the pairs of endpoints of these links are uniquely specified by the points in $S_n\backslash S_d$ with one nearest neighbour in $S_d$. For each of the six faces of $R=S_d$ there are $(2d+1)^2$ such points, hence $$|\Delta_{S_n}(R)\cap\Lambda^1|=6(2d+1)^2,$$ where we ignored the fact that links have two possible orientations, because $\Lambda^1$ only contains one orientation for each link. (We used the notation $|Z|$ for the cardinality of a set $Z$).
To estimate the number of plaquettes $p\in \Delta_{S_n}(R)$, note that as at least one side of a $p\in \Delta_{S_n}(R)$ must be a link $\ell \in \Delta_{S_n}(R)$, and given a link, there are 4 possible plaquettes it can belong to, we get $$|\Delta_{S_n}(R)\cap\Lambda^2|\leq 4\times|\Delta_{S_n}(R)\cap\Lambda^1|= 24(2d+1)^2.$$ We therefore have the estimate $$\label{sumest}
\mathop{\mathord{\sum}'}_{q \in \Delta_{S_n}(R)}1
\leq 30(2d+1)^2.$$ Next we want to estimate the size of $\Delta_{S_n}(q)\cap(\Lambda^1\cup\Lambda^2)$. If $q$ is a link, then there are 10 links in $\Delta_{S_n}(q)\cap\Lambda^1$ for $n$ large enough, and 20 plaquettes in $\Delta_{S_n}(q)\cap\Lambda^2$ (4 for which $q$ is a side, and 16 for which the plaquette has only one vertex in common with $q$). If $q$ is a plaquette, then there are 16 links in $\Delta_{S_n}(q)\cap\Lambda^1$ for $n$ large enough, and 32 plaquettes in $\Delta_{S_n}(q)\cap\Lambda^2$ (8 in the plane of $q$, and 24 perpendicular to the plane of $q$). Thus the number of terms in $\Delta_{S_n}(q)$ is less than or equal to the maximum of 30 and 48, i.e. 48. Thus we have $$\|R_N\|\leq 30(2d+1)^2 (48)^N
\|D\|( 2\|\Psi\| |t|)^{N+1}/(N+1)!$$ and it is clear for a fixed $t$ that this converges to $0$ as $N\to\infty$. We conclude that the iteration converges. Thus $$\label{iterarg}
\frac{\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|}{2\|A\|} \leq
\|D\|\Big(\delta^{S_n}_R+\sum_{k=1}^\infty\frac{(2\|\Psi\||t|)^k}{k!}a_k\Big).$$ Next, we want to estimate the coefficients $a_k$. Assuming as above that $S_n$ is the lattice cube with corner vertices $(\pm n,\pm n,\pm n)$ and $R=S_d$ for $d$ fixed, then using the estimates above, we have $$a_1=\mathop{\mathord{\sum}'}_{q \in \Delta_{S_n}(R)} \delta^{S_n}_{q}
\leq 30(2d+1)^2.$$ Recalling that $\delta^{S_n}_R=0$ if $R\cap S_m\backslash(S_n)_0=\emptyset$, if we keep $R$ fixed and let $n$ become large, then $a_1=0$. We can also use the estimates above for $$a_k = \mathop{\mathord{\sum}'}_{q_1 \in \Delta_{S_n}(R)}
\mathop{\mathord{\sum}'}_{q_2 \in \Delta_{S_n}(q_1)} \cdots
\mathop{\mathord{\sum}'}_{q_k \in \Delta_{S_n}(q_{k-1}) }
\delta^{S_n}_{q_k}
$$ Note that the sequence $$(q_1,q_2,\ldots,q_k)\quad\hbox{with}\quad q_i \in \Delta_{S_n}(q_{i-1})$$ specifies a continuous path where the steps are either links or plaquettes, starting from a $q_1\in \Delta_{S_n}(R)$ which has a point in $R$. As $\Delta_{S_n}(S_r)\cap(\Lambda^1\cup\Lambda^2)\subset S_{r+1}$ and $\delta^{S_n}_{S_r}=0$ if $r<n-2$ we conclude that $\delta^{S_n}_{q_k}=0$ whenever $k<n-d-2$. Thus $$a_k = \mathop{\mathord{\sum}'}_{q_1 \in \Delta_{S_n}(R)}
\mathop{\mathord{\sum}'}_{q_2 \in \Delta_{S_n}(q_1)} \cdots
\mathop{\mathord{\sum}'}_{q_k \in \Delta_{S_n}(q_{k-1}) }
\delta^{S_n}_{q_k}
\leq 30(2d+1)^2(48)^{k-1}\qquad\hbox{if}\quad k\geq n-d-2,$$ and $a_k=0$ otherwise. A substitution into (\[iterarg\]) produces: $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!
\frac{\big\|\big[D, \alpha^{S_n}_{t}(A)\big]\big\|}{2\|A\|} \leq
\|D\|\Big(\delta^{S_n}_{S_d}+\sum_{k=1}^\infty\frac{(2\|\Psi\||t|)^k}{k!}a_k\Big)\nonumber \\[1mm]
&\leq& \|D\|\Big(\delta^{S_n}_{S_d}+\sum_{k=n-d-2}^\infty\frac{(2\|\Psi\||t|)^k}{k!}
30(2d+1)^2(48)^{k-1}\Big) \nonumber \\[1mm]
\label{newLRs}
&=&\|D\|(2d+1)^2\sum_{k=n-d-2}^\infty\frac{5(96\|\Psi\||t|)^k}{8(k!)}
\qquad\hbox{if $n>d+4$.}
\end{aligned}$$ In order to estimate the integrand in (\[ComIneq\]), we make the substitutions into Eq. (\[newLRs\]) $$D\rightarrow {\widetilde}{N}(s)={\rm Ad}\Big(e^{isH_{S_m\backslash S_n}^{\rm loc}} \Big)\Big(\mathop{\mathord{\sum}'}_{q \in \Delta_{S_m}(
S_n)}\Psi(q)\Big),\qquad A\rightarrow A(t),\qquad t\rightarrow s-t,$$ where $s\in[t_-,t_+]$. Then we obtain $$\|{\widetilde}{N}(s)\|=\Big\|\mathop{\mathord{\sum}'}_{q \in \Delta_{S_m}(
S_n)}\Psi(q)\Big\|\leq\|\Psi\|\mathop{\mathord{\sum}'}_{q \in \Delta_{S_m}(
S_n)}1\leq \|\Psi\|30(2n+1)^2$$ by the estimates above. As it is obvious that $\|A(t)\|=\|A\|$ and $|s-t|\leq|t|$, we obtain from Eq. (\[newLRs\]) that if $n>d+4$ then $$\begin{aligned}
\big\|\big[{\widetilde}{N}(s), \alpha^{S_n}_{s-t}(A(t))\big]\big\| &\leq&
2\|{\widetilde}{N}(s)\|\|A\|(2d+1)^2\sum_{k=n-d-2}^\infty
\frac{5(96\|\Psi\||t|)^k}{8(k!)}\\[1mm]
&\leq& 2\|A\|\|\Psi\|30(2n+1)^2(2d+1)^2\sum_{k=n-d-2}^\infty
\frac{5(96\|\Psi\||t|)^k}{8(k!)}\,.\end{aligned}$$ Substitution of this into (\[ComIneq\])gives for $n>d+4$:\
$$\begin{aligned}
\label{NewCsumIneq}
&&\!\!\!\!\!\!\big\|\tau^{S_m}_t(A)-\tau^{S_n}_t(A)\|\nonumber\\[1mm]
&\leq&60\|A\|\|\Psi\|(2d+1)^2(2n+1)^2
\int_{t-}^{t^+}\sum_{k=n-d-2}^\infty\frac{5(96\|\Psi\||t|)^k}{8(k!)}\,ds\nonumber\\[1mm]
&=&\frac{75}{2}\|A\|\|\Psi\||t|(2d+1)^2(2n+1)^2\sum_{k=n-d-2}^\infty
\frac{(96\|\Psi\||t|)^{k}}{k!}\nonumber\\[1mm]
&\leq&\frac{75}{192}\|A\|(2d+1)^2(2n+1)^2
\frac{(96\|\Psi\||t|)^{n-d-1}}{(n-d-2)!}\exp\big(96\|\Psi\||t|\big)\,.\end{aligned}$$ It is clear that this converges to zero as $n\to\infty$ for any $t$. Furthermore, by first taking the limit $m\to\infty$, the estimate in (\[NewCsumIneq\]) also shows that the limit in $n$ is uniform for $t\in[-T,T]$ for a fixed $T$. This concludes the proof of Theorem \[GlobDynExist\].\
\[alphadiscont\] With notation as above, we have for some $A\in\al A._{\rm max}$ that $t\mapsto\alpha_t(A)$ is not norm continuous, i.e. $\alpha$ is not strongly continuous on $\al A._{\rm max}$.
Assume the contrary, i.e. that $\alpha:\R\to{\rm Aut}(\al A._{\rm max})$ is strongly continuous. Then it has a densely defined generator on $\al A._{\rm max}$, which we can perturb by a bounded generator $B$ to obtain a new strongly continuous action $\alpha^B:\R\to{\rm Aut}(\al A._{\rm max})$. Fix $S\in\cl S.$ as a large enough lattice cube and recall that $H_S= H_S^{\rm loc}+H_S^{\rm int}$ where $H_S^{\rm int}\in \al B._S\subset\al A._{\rm max}. $ Let the bounded perturbation $B$ then be $$B(A):=i[-H_S^{\rm int},A],\quad A \in \al A._{\rm max}.$$ Fix a strictly increasing sequence $\{S_n\}_{n\in\N}\subset\cl S.$ such that ${S_n\nearrow\Z^3}$ as $n\to\infty$, and $S\subseteq S_1$, and recall that $\alpha_t(A)=\lim\limits_{n\to \infty}\alpha^{S_n}_t(A)$. It suffices to prove that $\alpha_t^B(A)=\lim\limits_{n\to \infty}\alpha^{S_n,B}_t(A)$ where $\alpha^{S_n,B}_t:={\rm Ad}(\exp(it(H_{S_n}-H_S^{\rm int})))$. This is because on any subset $Z\subset S$ which cannot be reached by a link or plaquette with a point on the boundary of $S$, we have that $\alpha^{S_n,B}_t$ (hence $\alpha_t^B$) coincides with the free time evolution ${\rm Ad}(\exp(itH_Z^{\rm loc}))$. As the free time evolution preserves $\al B._Z\subset\al A._{\rm max}$ and is not strongly continuous on it, this contradicts with the strong continuity of $\alpha_t^B$. Recall the Dyson series for bounded perturbations (cf. [@BR1 Theorem 3.1.33]): $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\alpha_t^B(A)=\alpha_t(A)+\\
&&\!\!\!\!\sum_{n=1}^\infty (-i)^n\int_0^t dt_1\int_0^{t_1}dt_2\cdots\int_0^{t_{n-1}}dt_n\big[
\alpha_{t_n}(H_S^{\rm int}),\big[\alpha_{t_{n-1}}(H_S^{\rm int}),\ldots[\alpha_{t_1}(H_S^{\rm int}),\alpha_t(A)]\ldots\big]\big].\end{aligned}$$ Now each multiple commutator in the integrands $$\big[
\alpha_{t_n}(H_S^{\rm int}),\big[\alpha_{t_{n-1}}(H_S^{\rm int}),\ldots[\alpha_{t_1}(H_S^{\rm int}),\alpha_t(A)]\ldots\big]\big]$$ can be replaced by the norm limit $$\lim_{k\to \infty}\big[\alpha^{S_k}_{t_n}(H_S^{\rm int}),\ldots[\alpha^{S_k}_{t_1}(H_S^{\rm int}),\alpha^{S_k}_t(A)]\ldots\big]$$ and as these norm limits are uniform on compact intervals in the time parameters, the limits can be taken through the (weak operator convergent) integrals to produce: $$\begin{aligned}
\alpha_t^B(A)
&=&\lim_{k\to \infty}\Big\{\alpha^{S_k}_t(A)+\\
\sum_{n=1}^\infty\!\!\!\!\! &(-i)^n&\!\!\!\!\!\int_0^t dt_1\cdots\int_0^{t_{n-1}}dt_n
\big[\alpha^{S_k}_{t_n}(H_S^{\rm int}),\ldots[\alpha^{S_k}_{t_1}(H_S^{\rm int}),\alpha^{S_k}_t(A)]\ldots\big]\Big\}\\[2mm]
&=&\lim_{k\to \infty}\alpha^{S_k,B}_t(A)\end{aligned}$$ as required.
Kinematics algebras and regular representations. {#KARR}
------------------------------------------------
In this subsection we will define our new kinematics algebra, show its relation to the kinematics field algebra we previously constructed in [@GrRu], and consider the class of regular states and representations.
Having obtained the time evolution $\alpha:\R\to{\rm Aut}(\al A._{\rm max})$, we can now define our kinematics algebra as $${\mathfrak A}_{\Lambda}:=C^*\Big( \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S ) \Big)
\subset\cl A._{\rm max} \subset\cl B.(\cl H.).$$ which is the minimal C\*-algebra which contains all the local field algebras, and is preserved by the dynamics. Note that $\alpha$ does not preserve the local algebras. For the full field algebra, one should take a crossed product w.r.t. the actions of desirable transformations, such as gauge transformations (see below). First, we want to show that the dynamics group $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$ is in fact strongly continuous on ${\mathfrak A}_{\Lambda}$.
\[GlobDynCont\] With notation as above, we have for all $A\in {\mathfrak A}_{\Lambda}$ that $t\mapsto\alpha_t(A)$ is norm continuous, i.e. $\alpha$ is strongly continuous on ${\mathfrak A}_{\Lambda}$.
It suffices to prove continuity of $t\mapsto\alpha_t(A)$ for $A\in{\mathfrak A}_R$ and $R\in\al S.$ arbitrary. Fix an $A\in{\mathfrak A}_R$, then $$\|\alpha_t(A)-A\|\leq\|\alpha_t(A)-\alpha^S_t(A)\|+\|\alpha^S_t(A)-A\|$$ for any $S$. Fix a $\varepsilon>0$, then by Theorem \[GlobDynExist\] there is an $S\in\al S.$ such that ${\|\alpha_t(A)-\alpha^S_t(A)\|}<\varepsilon/2$ for all $t\in[-1,1]$, and it also holds for all larger $S$. Fix such an $S$ such that $S\supset R$, and assume that $t\mapsto\alpha_t^S(A)$ is norm continuous (this will be proven below). So $\lim\limits_{t\to 0}\alpha_t^S(A)=A$, hence there is a $\delta>0$ such that $\|\alpha^S_t(A)-A\|<\varepsilon/2$ for all $t\in[-\delta,\delta]$. Thus if $|t|<\min\{1,\delta\}$, we get that $\|\alpha_t(A)-A\|\leq\varepsilon$, i.e. $\lim\limits_{t\to 0}\alpha_t(A)=A$ as required.
It remains to prove that $t\mapsto\alpha_t^S(A)$ is norm continuous for $A\in{\mathfrak A}_R$, $R\subset S$ (note that $\alpha_t^S$ need not preserve ${\mathfrak A}_R$). Let $\tau^{S}_t:={\rm Ad}(e^{itH_S}e^{-itH_S^{\rm loc}})$, then $$\alpha^{S}_t(A)=\tau^{S}_t\big( e^{itH_{S}^{\rm loc}} A e^{-itH_{S}^{\rm loc}} \big)
=\tau^{S}_t\big( e^{itH_R^{\rm loc}} A e^{-itH_R^{\rm loc}} \big) $$ hence it suffices to show that the maps $t\mapsto\tau^{S}_t(B)$ and $t\mapsto{\rm Ad}\big(e^{itH_R^{\rm loc}}\big)(A)=:A(t)$ are both norm continuous for $A,\, B\in{\mathfrak A}_R$. For the map $t\mapsto A(t)$, recall that $$e^{itH_R^{\rm loc}}=U^{\rm CAR}_R(t)\otimes
\bigotimes_{\ell\in\Lambda^1_R}U_\ell(t)\otimes\bigotimes\limits_{\ell'\not\in\Lambda^1_R}{\mathbf{1}}$$ where $U^{\rm CAR}_R(t)=\exp\big(itma^3 \sum\limits_{x \in \Lambda^0_R} \bar \psi_i (x) \psi_i(x)\big)\in{\mathfrak F}_{R}$ and $U_\ell(t):=\exp(it\overline{E_{ij}(\ell) E_{ji}(\ell)})$. As the generator of $U^{\rm CAR}_R(t)$ is bounded, $t\mapsto U^{\rm CAR}_R(t)$ is norm continuous. As $${\mathfrak A}_R ={\mathfrak F}_R \otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_R}\al L._\ell
\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_R}\un$$ where $\al L._\ell:=C(G)\rtimes_\lambda G$ acts on the factor $\al H._\ell$ of $\al H._\infty$ as the algebra of compacts $\al K.(\al H._\ell)$ and $t\mapsto U_\ell(t)KU_\ell(-t)$ is norm continuous for a compact operator $K\in\al K.(\al H._\ell)$, it follows that the map $t\mapsto A(t)$ is norm continuous.
Finally, as $t\mapsto\alpha^{S}_t\in{\rm Aut}(\al B._S)$ is a bounded perturbation of the weak operator continuous one–parameter automorphism group $t\mapsto{\rm Ad}\big(e^{itH_S^{\rm loc}}\big)$ on $\al B._S\supset{\mathfrak A}_R$, we may express its cocycle $\tau^{S}_t={\rm Ad}(e^{itH_S}e^{-itH_S^{\rm loc}})$ as a Dyson series: (cf. [@BR1 Theorem 3.1.33]): $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\tau^{S}_t(B)=B+\\[1mm]
&&\!\!\!\!\sum_{n=1}^\infty i^n\int_0^t dt_1\int_0^{t_1}dt_2\cdots\int_0^{t_{n-1}}dt_n\big[
H_S^{\rm int}(t_n),\big[H_S^{\rm int}(t_{n-1}),\ldots[H_S^{\rm int}(t_1),B]\ldots\big]\big],\end{aligned}$$ where $H_S^{\rm int}(t):={\rm Ad}\big(e^{itH_S^{\rm loc}}\big)(H_S^{\rm int})$ and thus $\|H_S^{\rm int}(t)\|=\|H_S^{\rm int}\|<\infty$. Thus we obtain the estimate $$\big\|\tau^{S}_t(B)-\tau^{S}_{t'}(B)\big\|
\leq\sum_{n=1}^\infty \frac{\big|t^n-(t')^n \big|}{n!}
2^n\|H_S^{\rm int}\|^n\|B\|\,.$$ It is clear that this convergent series approaches zero when $t\to t'$, hence also $t\mapsto\tau^{S}_t(B)$ is continuous, from which it follows that $t\mapsto\alpha_t^S(A)$ is norm continuous for $A\in{\mathfrak A}_R$, as required.
Thus ${\mathfrak A}_{\Lambda}$ is a convenient field algebra, in fact we can construct for it the crossed product for the time evolution, which is not possible for $\alpha:\R\to{\rm Aut}(\al A._{\rm max})$.
Next we want to examine covariant representations for the automorphism group $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$, and consider the question of ground states. As the action $\alpha$ is a strongly continuous action of a locally compact group, it defines a C\*-dynamical system in the usual sense. Therefore we can construct its crossed product (cf. [@Ped]), and each representation of the crossed product produces a covariant representation for $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$. We therefore obtain a rich supply of covariant representations. However, for physics, the physically appropriate class of representations should be regular in the following sense:
A representation $\pi$ of ${\mathfrak A}_{\Lambda}$ is [**regular**]{} if its restriction to each local algebra ${\mathfrak A}_S$, $S\in \cl S.$, is nondegenerate (i.e. $\pi({\mathfrak A}_S)$ has no nonzero null spaces). A state is regular if its GNS representation is regular.
Note for a regular covariant representation $\pi$, that $\pi$ is also nondegenerate on all the time evolved local algebras $\alpha_t({\mathfrak A}_S )$.
The reasons why physical representations should be regular, are as follows. First, one requires that the local observables do not all have zero expectation values w.r.t. any (normalized) vector state in the representation, i.e. the local observables in the field algebra should be visible in any physically realizable state of the system. Second, observe that the local Hamiltonians $H_S$ in the defining representation have compact resolvents (see the next subsection below), hence ${(i\un-H_S)^{-1}}\in {\mathfrak A}_S$. Thus, if a representation $\pi$ of ${\mathfrak A}_{\Lambda}$ is degenerate on ${\mathfrak A}_S$, then ${\pi((i\un-H_S)^{-1})}$ has a nonzero kernel, hence it cannot be the resolvent of an operator (cf. Theorem 1 in [@Yos p 216]), thus the observable $H_S$ does not exist in this representation. Also observe, that this definition of regularity coincides with the one used in [@BuGr2; @GrN09].\
We now discuss the relation of the kinematics field algebra we constructed before in [@GrRu] with the kinematics field algebra ${\mathfrak A}_{\Lambda}$ we constructed here. (For the full field algebras we need to add identities and construct crossed products w.r.t. gauge transformations and time evolutions). The reader in a hurry may omit the rest of this subsection.
The kinematics field algebra constructed in [@GrRu] is ${\widehat}{\mathfrak A}_{\Lambda} :=
{\mathfrak F}_{\Lambda} \otimes {\cal L}[E]$ where ${\mathfrak F}_{\Lambda}$ is the Fermion algebra associated with $\Lambda$, and $ {\cal L}[E]$ is a new “infinite tensor product” of the algebra of compact operators. Concretely, ${\widehat}{\mathfrak A}_{\Lambda}$ is represented on $\al H.=\al H._{\rm Fock}\otimes\al H._\infty$ as a tensor product representation as follows. We let the Fermion algebra ${\mathfrak F}_{\Lambda}$ act in the Fock representation on $\al H._{\rm Fock}$. Next, to see how $ {\cal L}[E]$ acts on $\al H._\infty$, recall first that $\al H._\infty$ is the completion of the pre–Hilbert spanned by finite combinations of elementary tensors of the type $$\varphi_1\otimes\cdots\otimes \varphi_k\otimes\psi_0\otimes\psi_0\otimes\cdots,\quad\varphi_i\in \al H._i= L^2(G),\;
k\in\N\,.$$ Choose an increasing sequence of commuting finite dimensional projections $\{E_n\}_{n\in\N}\subset \cl K.\big(L^2(G)\big)
=\pi_0\big(C(G)\rtimes_\lambda G\big)$ which is an approximate identity for $\cl K.\big(L^2(G)\big)$ and with $\psi_0\in E_nL^2(G)$ for all $n$. (By [@GrRu] there exists a choice of $\{E_n\}_{n\in\N}$ which is invariant w.r.t. a certain action of $G\times G$, but we will not insist on this point here.) Then elementary tensors of the form $$A_1\otimes A_2\otimes\cdots\otimes A_k\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots,\qquad
A_i\in\cl B.(\al H._i),\; n_j\in\N$$ act entrywise on $\al H._\infty$ in a consistent way, so they define operators in $\cl B.(\al H._\infty)$. In particular $ {\cal L}[E]$ is the C\*-algebra generated by those elementary tensors where all $A_i\in\cl K.(\al H._i)$, and we now have represented ${\widehat}{\mathfrak A}_{\Lambda}$ on $\al H.$. (In [@GrRu] we allowed the approximate identity $\{E_n\}_{n\in\N}$ to be different for different entries, but this generality is not needed).
For a finite connected subgraph $S\in\al S.$ of $(\Lambda^0, \Lambda^1)$, we obviously have the subalgebra ${\widehat}{\mathfrak A}_S:={\mathfrak F}_{S}\otimes{\cal L}_{S}[E]$ where ${\cal L}_{S}[E]$ is the C\*-algebra generated in $\cl B.(\al H._\infty)$ by those elementary tensors of the type $$\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}K_\ell\Big)\otimes
\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}E_\ell,\qquad K_\ell\in\cl K.(\cl H._\ell)$$ where $E_\ell$ denotes an element of $\{E_n\}_{n\in\N}$ placed in the position of the tensor product corresponding to the link $\ell$. As $\cl K.(\cl H._1)\otimes\cl K.(\cl H._2)=\cl K.(\cl H._1\otimes\cl H._2)$ we have in fact that ${\cal L}_{S}[E]$ is the closure of the space spanned by $$\big\{K\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}E_\ell\;\Big|\;K\in \cl K.(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\cl H._\ell),\; E_\ell\in \{E_n\}_{n\in\N}\big\}.$$ Thus by compactness of the projections $E_n$, we have ${\widehat}{\mathfrak A}_S\subset {\widehat}{\mathfrak A}_T$ if $S\subset T$. This should be contrasted with ${\mathfrak A}_S\subset M({\mathfrak A}_T)$ for the local algebras. Now $${\widehat}{\mathfrak A}_{\Lambda} =\ilim{\widehat}{\mathfrak A}_S=\ilim\big({\mathfrak F}_{S}\otimes{\cal L}_{S}[E]\big).$$
Recall that for a finite connected subgraph $S\in\al S.$ of $(\Lambda^0, \Lambda^1)$, we defined $$\begin{aligned}
\al H._S&:=& [{\mathfrak F}_{S}\Omega]\otimes[\al L._S\psi_0^\infty],\qquad\cl B._S:= \cl B.(\al H._S)\subset \cl B.(\al H.),\\[1mm]
\al A._{\rm max}&:=&\ilim \cl B._S = C^*\Big(\bigcup_{S\in\al S.} \cl B.(\al H._S) \Big),\end{aligned}$$ where an element of $\cl B.(\al H._S)$ acts as the identity on the factors $\al H._\ell$ of $\al H._\infty$ corresponding to $\ell\not\in S$. By Equation (\[BSisFB\]), we realized $\cl B._S$ on $\cl H.$ by $$\cl B._S
=\pi_{\rm Fock}({\mathfrak F}_{S})\otimes\cl B.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)
\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\un.$$ The spanning elementary tensors for ${\widehat}{\mathfrak A}_{\Lambda}$ are all of the form $$A_F\otimes K_S\otimes K_Q\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots,\qquad A_F\in\pi_{\rm Fock}({\mathfrak F}_{R}),$$ for $R=\{1,2,\ldots,k\}\supset S$, $ Q=R\backslash S $ and $K_\lambda\in
\cl K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_\lambda}\al H._\ell\Big)$ for $\lambda\in\{S,Q\}$. Thus the product of an elementary tensor in $\cl B._S$ with such a tensor can only change the first two factors, and will again produce an elementary tensor of this kind. As the action of $\cl B.(\cl H.)$ on $\cl K.(\cl H.)$ is nondegenerate, this implies that $\cl B._S$ is in the multiplier algebra of ${\widehat}{\mathfrak A}_{\Lambda}$. Thus for the C\*-algebra they generate we also have $\al A._{\rm max}\subset M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$. As the kinematics field algebra ${\mathfrak A}_{\Lambda}$ we constructed here is contained in $\al A._{\rm max}$, we conclude that also ${\mathfrak A}_{\Lambda}\subset M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$. This implies that every representation of ${\widehat}{\mathfrak A}_{\Lambda}$ extends uniquely (on the same space) to ${\mathfrak A}_{\Lambda}$, but not conversely. Every representation of ${\mathfrak A}_{\Lambda}$ which is obtained from a nondegenerate representation of ${\widehat}{\mathfrak A}_{\Lambda}$ in this manner is regular, which is the content of the next lemma:-
\[LemRegRep\] Given, in the notation above that ${\mathfrak A}_{\Lambda}\subset M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$, let $\pi$ be a nondegenerate representation of ${\widehat}{\mathfrak A}_{\Lambda}$, and let ${\widetilde}\pi$ be the unique extension of $\pi$ on the same Hilbert space to $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$. Then ${\widetilde}\pi\restriction {\mathfrak A}_{\Lambda}$ is regular. Likewise, if ${\widetilde}\omega$ is the unique extension of a state $\omega$ on ${\widehat}{\mathfrak A}_{\Lambda}$ to a state on $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$, then ${\widetilde}\omega\restriction {\mathfrak A}_{\Lambda}$ is regular.
By Theorem A.2(vii) of [@GrN14], it suffices for the first part to show that ${\mathfrak A}_S\cdot{\widehat}{\mathfrak A}_{\Lambda}$ is strictly dense in ${\widehat}{\mathfrak A}_{\Lambda}$ for all $S\in{\cal S}$. Recall from above that on $\cl H.$, we have ${\mathfrak A}_S={\mathfrak F}_{S}\otimes\al K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)
\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\un$, and that ${\widehat}{\mathfrak A}_{\Lambda}$ is spanned by the elementary tensors $$A_F\otimes K_S\otimes K_Q\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots,\qquad A_F\in\pi_{\rm Fock}({\mathfrak F}_{R}),$$ for $R=\{1,2,\ldots,k\}\supset S$, $ Q=R\backslash S $ and $K_\lambda\in
\cl K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_\lambda}\al H._\ell\Big)$ for $\lambda\in\{S,Q\}$. Let $\{Y_i\}_{i\in\N}\subset\al K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)$ be an approximate identity for $\al K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)$, then $\un_F\otimes Y_i\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\un\in {\mathfrak A}_S$ where $\un_F$ is the identity of ${\mathfrak F}_{R}$. Then $$\begin{aligned}
\Big(\un_F\otimes Y_i\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\un\Big)\Big(
A_F\otimes K_S\otimes K_Q\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots\Big)\\[1mm]
\qquad\qquad =A_F\otimes Y_iK_S\otimes K_Q\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots\qquad\qquad\qquad\\[1mm]
\qquad\qquad\qquad \longrightarrow A_F\otimes K_S\otimes K_Q\otimes E_{n_{k+1}}\otimes E_{n_{k+2}}\otimes\cdots
\qquad\hbox{as}\; i\to\infty.\end{aligned}$$ Thus ${\mathfrak A}_S\cdot{\widehat}{\mathfrak A}_{\Lambda}$ is dense in ${\widehat}{\mathfrak A}_{\Lambda}$, and as the strict topology is weaker than the norm topology, this implies strict density. As $S$ is arbitrary, this proves that ${\widetilde}\pi\restriction {\mathfrak A}_{\Lambda}$ is regular.
If ${\widetilde}\omega$ is the unique extension of a state $\omega$ on ${\widehat}{\mathfrak A}_{\Lambda}$ to a state on $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$, then $\pi_{{\widetilde}\omega}$ is the unique extension of $\pi_\omega$ on the same Hilbert space to $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$. This is because ${\widetilde}\omega$ is strictly continuous on $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$, hence $\pi_{{\widetilde}\omega}:M\big({\widehat}{\mathfrak A}_{\Lambda}\big)\to\cl B.(\cl H._{{\widetilde}\omega})$ is strictly continuous w.r.t. the strong operator topology of $\cl B.(\cl H._{{\widetilde}\omega})$. By the strict density of ${\widehat}{\mathfrak A}_{\Lambda}$ in $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$ we get that $${[\pi_{{\widetilde}\omega}(M\big({\widehat}{\mathfrak A}_{\Lambda}\big))\Omega_{{\widetilde}\omega}]}
={[\pi_{{\widetilde}\omega}({\widehat}{\mathfrak A}_{\Lambda})\Omega_{{\widetilde}\omega}]}
={[\pi_{\omega}({\widehat}{\mathfrak A}_{\Lambda})\Omega_{\omega}]}
=\cl H._\omega.$$ Then $\pi_{{\widetilde}\omega}$ is regular by the first part, hence ${\widetilde}\omega\restriction {\mathfrak A}_{\Lambda}$ is regular.
By adapting the (lengthy) proof of Theorem 3.6 in [@GrN09], we can also prove the converse, i.e. that if a representation of ${\mathfrak A}_{\Lambda}$ is regular, then it is obtained from a nondegenerate representation of ${\widehat}{\mathfrak A}_{\Lambda}$ by the unique extension to $M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$ on the same space. However, this will not be needed here.
Ground states for the global dynamics. {#PGDA}
--------------------------------------
Next we want to examine covariant representations for the automorphism group ${\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})}$. and consider the question of ground states. For physics, only covariant representations where the generator of time translations is positive is acceptable, and even more, for these representations ground states are needed. As $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$ is a continuous action of an amenable group, it certainly has invariant states, but the difficult parts are to prove regularity and the spectrum condition for such an invariant state, establishing it as a regular ground state. To construct a ground state and establish its properties, we will follow a familiar method from [@BuGr2].
First, we need to consider the local Hamiltonians in greater detail. As the restriction to $\cl H._S$ of the embedded copy of $\cl B._S=\cl B.(\cl H._S)\subset\cl A._{\rm max}$ is faithful, we will do the analysis on $$\cl H._S=
[{\mathfrak F}_{S}\Omega]\otimes[\al L._S\psi_0^\infty]=
[{\mathfrak F}_{S}\Omega]\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}L^2(G)
\otimes\bigotimes\limits_{\ell'\not\in\Lambda^1_S}\psi_0\subset\cl H..$$ Separating the bounded and unbounded parts of $H_S$ on $\cl H._S$ we have: $$\begin{aligned}
H_S&=& H_S^{(0)}+H_S^{\rm bound}\qquad\hbox{on}\qquad
[{\mathfrak F}_{S}\Omega]\otimes{\widetilde}{\cl D.}_S\subset\cl H._S\subset
\cl H._{\rm Fock}\otimes\cl H._\infty\qquad\hbox{where:}\\[1mm]
H_S^{(0)} &:=& \tfrac{a}{2} \sum_{\ell \in \Lambda^1_S}
E_{ij}(\ell) E_{ji}(\ell),\qquad H_S^{\rm bound}\in\cl B.(\cl H._S)\qquad\hbox{and}\\[1mm]
{\widetilde}{\cl D.}_S&=&\bigotimes\limits_{\ell\in\Lambda^1_S}C^\infty(G)\otimes\bigotimes\limits_{\ell'\not\in\Lambda^1_S}\psi_0.\end{aligned}$$ Now $H_S^{(0)}=\un\otimes R_S\otimes\un$ where $R_S$ is the group Laplacian for the compact Lie group $G_S:=\prod\limits_{\ell\in\Lambda^1_S}G$ on $L^2(G_S)\cong\bigotimes\limits_{\ell\in\Lambda^1_S}L^2(G)$. Thus by the theory of elliptic operators on compact Riemannian manifolds, we conclude that for $R_S$ its eigenvalues are isolated, and its eigenspaces are finite dimensional, cf. Theorem III.5.8 in [@LM89] and [@BH79]. Thus it has compact resolvent, i.e. $(i\un-R_S)^{-1}\in\al K.(L^2(G_S))$. As $ [{\mathfrak F}_{S}\Omega]$ is finite dimensional, this is also true for $H_S^{(0)}$ on $ [{\mathfrak F}_{S}\Omega]\otimes{\widetilde}{\cl D.}_S\subset\cl H._S$, i.e. $\big(i\un-H_S^{(0)}\big)^{-1}\in\al K.(\cl H._S)$. Then $$(i\un-H_S)^{-1}=\big(i\un-H_S^{(0)}\big)^{-1}+ (i\un-H_S)^{-1} H_S^{\rm bound}\big(i\un-H_S^{(0)}\big)^{-1}\in\al K.(\cl H._S)$$ hence $H_S=H_S^{(0)}+H_S^{\rm bound}$ also has discrete spectrum with finite dimensional eigenspaces. As $ H_S^{(0)}$ is positive and unbounded, and $H_S^{\rm bound}$ is bounded, $H_S$ is bounded from below. Thus the lowest point in the spectrum of $H_S$ is an eigenvalue $\lambda_S^{\rm grnd}\in\R$ with finite dimensional eigenspace $\cl E._S\subset\cl H._S$. Fix a normalized eigenvector $\Omega_S\in\cl E._S\subset\cl H._S\subset\cl H.$. We conclude that the vector state $\omega_S(\cdot):={\big(\Omega_S,\,\cdot\,\Omega_S\big)}$ is a ground state for the local time evolution $\alpha^S:\R\to{\rm Aut}\big(\al A._{\rm max}\big)$ (and for its restriction to subalgebras such as ${\mathfrak A}_{\Lambda}$). In the original representation, $${\widetilde}{H}_S:=H_S-\un \lambda_S^{\rm grnd}$$ will be the positive Hamiltonian for $\alpha^S_t={\rm Ad}\big(\exp(it{\widetilde}{H}_S)\big)$ with smallest eigenvalue zero.
To construct a regular ground state for $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$ on ${\mathfrak A}_{\Lambda}$, we proceed as follows. Fix a strictly increasing sequence $\{S_n\}_{n\in\N}\subset\cl S.$ such that ${S_n\nearrow\Z^3}$ as $n\to\infty$. For each $n\in\N$ choose a state $\omega_n$ on $\cl B.(\cl H.)$ in the norm closed convex hull of vector states $$A\mapsto{\big(\Omega_{S_n},\, A\,\Omega_{S_n}\big)},\qquad A\in\cl B.(\cl H.), $$ where $\Omega_{S_n}\in\cl H._{S_n}$ ranges over the normalized vectors in the eigenspace $\cl E._{S_n}$ of $H_{S_n}$. This sequence $\{\omega_n\}_{n\in\N}$ need not converge, but by the Banach–Alaoglu theorem, the closed unit ball in $\cl B.(\cl H.)^*$ is compact in the weak \*–topology, hence the sequence $\{\omega_n\}_{n\in\N}$ has weak \*–limit points, and these limit points are states. From such weak \*–limit points we now want to show that we can obtain regular ground states on ${\mathfrak A}_{\Lambda}$. First we prove regularity.
\[RegSt\] In the context above, for the increasing sequence ${S_n\nearrow\Z^3}$ we fix $S_n$ to be the lattice cube with corner vertices $(\pm n,\pm n,\pm n)$, which produces the sequence $\{\omega_n\}_{n\in\N}$. Let $\omega_\infty$ be a weak \*–limit point of $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$. Then the restriction of $\omega_\infty$ to ${\mathfrak A}_{\Lambda}\subset\cl B.(\cl H.)$ is regular.
We have to prove that $\pi^o_\infty({\mathfrak A}_S)$ is nondegenerate on $\al H.^o_\infty$ for all $S\in\al S.$, where ${(\pi^o_\infty,\Omega^o_\infty,\al H.^o_\infty)}$ denotes the GNS–data of $\omega_\infty\restriction{\mathfrak A}_\Lambda$.
First consider $\omega_\infty$ on all of $\cl B.(\cl H.)$ with GNS–data ${({\widetilde}\pi_\infty,{\widetilde}\Omega_\infty,{\widetilde}{\al H.}_\infty)}$. The GNS–data set of $\omega_\infty$ restricted to any subalgebra $\al A.\subset\cl B.(\cl H.)$ is just given by the subspace ${[{\widetilde}\pi_\infty(\al A.){\widetilde}\Omega_\infty]}\subset{\widetilde}{\al H.}_\infty$ with the action of ${\widetilde}\pi_\infty(\al A.)$ on it, with cyclic vector ${\widetilde}\Omega_\infty$. In particular, $\al H.^o_\infty$ is identified (i.e. unitarily equivalent) to $$[{\widetilde}\pi_\infty({\mathfrak A}_\Lambda){\widetilde}\Omega_\infty]
=\Big[{\widetilde}\pi_\infty\Big(C^*\Big( \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S )\Big)\Big){\widetilde}\Omega_\infty \Big].
$$ We will prove below that $\|\omega_\infty\restriction{\mathfrak A}_{S_n}\|=1$ for all $n$. Assuming this, then on each $M({\mathfrak A}_{S_n})$, $\omega_\infty$ is uniquely determined by its restriction to ${\mathfrak A}_{S_n}$ (cf. Prop. 2.11.7 in [@Dix]). Let ${(\pi_\infty^{S_n},\Omega_\infty^{S_n},\al H._\infty^{S_n})}$ denote the GNS–data of $\omega_\infty\restriction M({\mathfrak A}_{S_n})$, then this means that $\pi_\infty^{S_n}$ is strictly continuous w.r.t. the strong operator topology of $\cl B.(\al H._\infty^{S_n})$. Then, using the strict density of ${\mathfrak A}_{S_n}$ in $M({\mathfrak A}_{S_n})$, we obtain as in the proof of Lemma \[LemRegRep\] that $$\left[\pi_\infty^{S_n}\big(M({\mathfrak A}_{S_n})\big)\Omega_\infty^{S_n}\right]
=\left[\pi_\infty^{S_n}\big({\mathfrak A}_{S_n}\big)\Omega_\infty^{S_n}\right]=\al H._\infty^{S_n}.$$ In fact, using strict density of $C^*\Big( \bigcup\limits_{S\subseteq S_n}\alpha_{\R}^{S_n}({\mathfrak A}_S ) \Big)
\subseteq M({\mathfrak A}_{S_n})$, we have $$\Big[\pi_\infty^{S_n}\big(C^*\Big( \bigcup_{S\subseteq S_n}\alpha_{\R}^{S_n}({\mathfrak A}_S )
\Big)\big)\Omega_\infty^{S_n}\Big]
=\left[\pi_\infty^{S_n}\big({\mathfrak A}_{S_n}\big)\Omega_\infty^{S_n}\right]=\al H._\infty^{S_n}.$$ Using the identification above of $\al H._\infty^{S_n}$ with a subspace of ${\widetilde}{\al H.}_\infty$, this means that $$\begin{aligned}
\label{pASnO}
\left[{\widetilde}\pi_\infty\big(M({\mathfrak A}_{S_n})\big){\widetilde}\Omega_\infty\right]
&=&
\Big[{\widetilde}\pi_\infty\big(C^*\Big( \bigcup_{S\subseteq S_n}\alpha_{\R}^{S_n}({\mathfrak A}_S )
\Big)\big){\widetilde}\Omega_\infty\Big] \nonumber \\[1mm]
\label{pASnO}
&=&\Big[{\widetilde}\pi_\infty\big({\mathfrak A}_{S_n}\big){\widetilde}\Omega_\infty\Big]=\al H._\infty^{S_n}
\subset{\widetilde}{\al H.}_\infty .\end{aligned}$$ Note that that if $n<m$ then $M({\mathfrak A}_{S_n})\subset M({\mathfrak A}_{S_m})$ hence equation (\[pASnO\]) implies that $$\Big[{\widetilde}\pi_\infty\big({\mathfrak A}_{S_n}\big){\widetilde}\Omega_\infty\Big]\subseteq
\Big[{\widetilde}\pi_\infty\big({\mathfrak A}_{S_m}\big){\widetilde}\Omega_\infty\Big].$$ Moreover, by Theorem \[GlobDynExist\] we have $\alpha_t(A)=\lim\limits_{S\nearrow\Z^3}\alpha^S_t(A)$ hence $\alpha_t(A)$ for $A\in {\mathfrak A}_\Lambda$ is in the norm closure of ${\bigcup\limits_{n\in\N} \bigcup\limits_{S\subseteq S_n}\alpha_{\R}^{S_n}({\mathfrak A}_S )}$ and so from (\[pASnO\]) we see $$\label{piALO}
\al H.^o_\infty=
[{\widetilde}\pi_\infty({\mathfrak A}_\Lambda){\widetilde}\Omega_\infty]
=\Big[{\widetilde}\pi_\infty\Big(C^*\Big( \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S )\Big)\Big){\widetilde}\Omega_\infty \Big]
=\Big[{\widetilde}\pi_\infty\Big(\bigcup_{n\in\N}{\mathfrak A}_{S_n} \Big){\widetilde}\Omega_\infty \Big].$$ Therefore, to prove for a fixed $S$ that $\pi^o_\infty({\mathfrak A}_S)$ is nondegenerate on $\al H.^o_\infty$, it suffices to prove that it is nondegenerate on each of the spaces ${\big[{\widetilde}\pi_\infty\big({\mathfrak A}_{S_n} \big){\widetilde}\Omega_\infty \big]}$ as they are increasing in $n$, and their union is dense in $\al H.^o_\infty$. Let $k\in \N$ be large enough so that $S\subset S_k$ then for all $n\geq k$ we have that ${\mathfrak A}_S\subset M({\mathfrak A}_{S_n})$ and as ${\mathfrak A}_S$ acts nondegenerately on ${\mathfrak A}_{S_n}$, we have for any approximate identity $\{e_\gamma\}_{\gamma\in\Gamma}\subset{\mathfrak A}_S$ that $\lim\limits_\gamma e_\gamma A=A$ for all $A\in {\mathfrak A}_{S_n}$. Thus $$\lim_\gamma\big({\widetilde}\pi_\infty(e_\gamma)-\un\big)\big[{\widetilde}\pi_\infty\big({\mathfrak A}_{S_n} \big){\widetilde}\Omega_\infty \big]
=0$$ for all $n\geq k$, hence on all of $\al H._\infty$. Thus $\pi^o_\infty({\mathfrak A}_S)$ is nondegenerate on $\al H.^o_\infty$ for all $S\in\al S.$, i.e. $\omega_\infty$ restricted to ${\mathfrak A}_{\Lambda}\subset\cl B.(\cl H.)$ is regular.
It remains to prove that $\|\omega_\infty\restriction{\mathfrak A}_{S_n}\|=1$ for all $n$. We will follow the proof of Lemma 7.3 in [@BuGr2]. Let $m>n\in\N$ and on $\cl H._{S_m}\subset\cl H.$ consider the operators ${\widetilde}H_n:={\widetilde}H_{S_n},\;{\widetilde}H_m:={\widetilde}H_{S_m}$ and ${\widetilde}H_{m\backslash n}:={\widetilde}H_{S_m\backslash S_n}$, and use analogous notation for $\lambda_m:=\lambda_{S_m}^{\rm grnd}$ etc. As $[{\widetilde}H_{m\backslash n},{\widetilde}H_n]=0$ these operators have a joint dense domain on which ${\widetilde}H_m$ is defined by $${\widetilde}H_m = {\widetilde}H_n + {\widetilde}H_{m\backslash n} + \mathop{\mathord{\sum}'}_{q\in \Delta_{S_m}(S_n)} \Psi(q) +
(\lambda_n+\lambda_{m\backslash n}-\lambda_m)\un$$ using notation from before in (\[primesum\]) and (\[bdry\]), as the additional terms are bounded. Let $\Omega$ be a normalized joint eigenvector for $ {\widetilde}H_n$ and ${\widetilde}H_{m\backslash n}$ for the eigenvalue $0$, then $$0\leq(\Omega,{\widetilde}H_m \Omega)=\Big(\Omega, \mathop{\mathord{\sum}'}_{q\in \Delta_{S_m}(S_n)} \Psi(q)\Omega\Big)
+\lambda_n+\lambda_{m\backslash n}-\lambda_m.$$ Now recalling the estimate (\[sumest\]), we have $$\mathop{\mathord{\sum}'}_{q \in \Delta_{S_m}(S_n)}\|\Psi\|
\leq 30(2n+1)^2\|\Psi\|.$$ Thus the previous inequality gives $$\lambda_n+\lambda_{m\backslash n}-\lambda_m\geq -30(2n+1)^2\|\Psi\|,$$ hence $${\widetilde}H_m + 60(2n+1)^2\|\Psi\|\un\geq {\widetilde}H_n.$$ Thus for all $\mu>0$ we have for the resolvents: $$\big({\widetilde}H_m + (\mu + 60(2n+1)^2\|\Psi\|)\un\big)^{-1}\leq ({\widetilde}H_n +\mu\un)^{-1}\leq 1/\mu\,.$$ From this we obtain $$\mu\big(\mu + 60(2n+1)^2\|\Psi\|\big)^{-1}
\leq \omega_m\big( \mu({\widetilde}H_n +\mu\un)^{-1} \big)
\leq 1.$$ As $\omega_\infty$ is a weak \*–limit point of $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$, there is a subsequence $\{\omega_{n_k}\}_{k\in\N}
\subset\{\omega_n\}_{n\in\N}$ which converges to $\omega_\infty$ in the weak \*–topology. We thus obtain $$\label{resolvineq}
\mu\big(\mu + 60(2n+1)^2\|\Psi\|\big)^{-1}\leq\lim_{k\to\infty}\omega_{n_k}(\mu({\widetilde}H_n +\mu\un)^{-1})
=\omega_\infty(\mu({\widetilde}H_n +\mu\un)^{-1})\leq 1.$$ Above we saw that on $\cl H._{S_n}$ we have $({\widetilde}H_n +\mu\un)^{-1}\in\cl K.(\cl H._{S_n})$, hence on $\cl H.$ we have $$({\widetilde}H_n +\mu\un)^{-1}\in {\mathfrak A}_{S_n}={\mathfrak F}_{S_n}\otimes\al K.\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_{S_n}}\al H._\ell\Big)
\otimes\mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_{S_n}}\un.$$ This proves by (\[resolvineq\]) that $\|\omega_\infty\restriction{\mathfrak A}_{S_n}\|=1$ for all $n$.
\[InvSt\] Assuming as above an increasing sequence ${S_n\nearrow\Z^3}$ of lattice cubes, with $\omega_\infty$ a weak \*–limit point of $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$, then $\omega_\infty\restriction\cl A._{\rm max}$ is invariant w.r.t. the automorphism group $\alpha:\R\to{\rm Aut}(\cl A._{\rm max})$.
Let $\{\omega_{n_k}\}_{k\in\N}$ be a subsequence weak \*–converging to $\omega_\infty$. Observe that for any $A\in\cl A._{\rm max}$ and $B\in\cl B.(\cl H.)$ that $$\begin{aligned}
\big|\omega_{n_k}(B\alpha^{S_{n_k}}_t(A)\! )&-&\!\omega_\infty(B\alpha_t(A))\big|\\[1mm]
&\leq& \big|\omega_{n_k}(B\alpha^{S_{n_k}}_t(A)-B\alpha_t(A))\big|
+\big|\omega_{n_k}(B\alpha_t(A))-\omega_\infty(B\alpha_t(A))\big|\\[1mm]
&\leq & \|B\|\big\|\alpha^{S_{n_k}}_t(A)-\alpha_t(A)\big\|
+\big|\omega_{n_k}(B\alpha_t(A))-\omega_\infty(B\alpha_t(A))\big|\,.\end{aligned}$$ As the last expression goes to $0$ for $k\to\infty$, we get that $$\label{lkioBS}
\lim\limits_{k\to\infty}\omega_{n_k}(B\alpha^{S_{n_k}}_t(A))=\omega_\infty(B\alpha_t(A)).$$ Next observe that as the vector states $A\mapsto{\big(\Omega_{S_n},\, A\,\Omega_{S_n}\big)}$ are invariant w.r.t. $\alpha^{S_n}$, so is any state in their norm closed convex hull, so $\omega_n$ is $\alpha^{S_n}\hbox{--invariant.}$ Thus $$\omega_\infty(\alpha_t(A))=\lim_{k\to\infty}\omega_{n_k}(\alpha^{S_{n_k}}_t(A))
=\lim_{k\to\infty}\omega_{n_k}(A)=\omega_\infty(A)\,,$$ hence $\omega_\infty$ is invariant for $\alpha$.
Consider the restriction of $\omega_\infty$ to ${\mathfrak A}_{\Lambda}\subset M\big({\widehat}{\mathfrak A}_{\Lambda}\big)$. Denote the GNS–data of $\omega\restriction {\mathfrak A}_{\Lambda}$ by ${(\pi^o_\infty,\Omega^o_\infty,\al H.^o_\infty,U^o_\infty)}$ where $$U^o_\infty(t)\pi^o_\infty(A)\Omega^o_\infty=\pi^o_\infty(\alpha_t(A))\Omega^o_\infty\quad\forall\,A\in{\mathfrak A}_{\Lambda}.$$ Then $(\pi^o_\infty, U^o_\infty)$ is a covariant pair for $\alpha:\R\to{\rm Aut}( {\mathfrak A}_{\Lambda})$, in particular $t\mapsto U^o_\infty(t)$ is a weak operator continuous unitary group. This follows directly from the strong continuity of $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$ obtained in Theorem \[GlobDynCont\].
Note that for $\omega_\infty\restriction\cl A._{\rm max}$, the analogous statement need not be true, because $\alpha:\R\to{\rm Aut}(\cl A._{\rm max})$ is not strongly continuous (cf. Lemma \[alphadiscont\]). Finally, to establish that $\omega_\infty$ is a ground state, we need to prove that the generator of $U^o_\infty(t)$ is nonnegative.
\[groundState\] Assuming as above an increasing sequence ${S_n\nearrow\Z^3}$ of lattice cubes, with $\omega_\infty$ a weak \*–limit point of $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$, then $\omega_\infty\restriction{\mathfrak A}_{\Lambda}$ is a regular ground state for $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$.
By the preceding lemmas, all that remains to be proven, is that the generator of $U^o_\infty$ is nonnegative. Let $h$ be a Schwartz function on $\R$, then we need to prove that $U^o_\infty(h):=\int h(t)U^o_\infty(t)\,dt =0$ when ${\rm supp}({\widehat}{h})\subset (-\infty,0)$. Fix a Schwartz function such that ${\rm supp}({\widehat}{h})\subset (-\infty,0)$. By (\[piALO\]) it suffices to prove for all $A\in {\mathfrak A}_{S_n}$ and $n\in\N$ that $$0= U^o_\infty(h) \pi^o_\infty(A)\Omega^o_\infty=\int h(t)U^o_\infty(t) \pi^o_\infty(A)\Omega^o_\infty\,dt
= \int h(t) \pi^o_\infty(\alpha_t(A))\Omega^o_\infty\,dt.$$ Let $B\in{\mathfrak A}_{\Lambda}$ arbitrary, and let $\{\omega_{n_k}\}_{k\in\N}$ be a subsequence weak \*–converging to $\omega_\infty$. Then by equation (\[lkioBS\]) we have $$\Big(\pi^o_\infty(B^*)\Omega^o_\infty,\, U^o_\infty(h) \pi^o_\infty(A)\Omega^o_\infty\Big) =
\int h(t) \omega_\infty(B\alpha_t(A))\,dt
=\lim_{k\to\infty}\int h(t)\,\omega_{n_k}(B\alpha^{S_{n_k}}_t(A))\,dt$$ using the dominated convergence theorem to take the limit through the integral. Observe that if $\omega_{n_k}$ is a vector state, i.e. $\omega_{n_k}(\cdot)={(\Omega_{S_{n_k}},\cdot\,\Omega_{S_{n_k}})}$ with $\Omega_{S_{n_k}}\in
\cl E._{S_{n_k}}$, then by ${\widetilde}{H}_{S_{n_k}}\Omega_{S_{n_k}}=0$ and $\alpha^{S_{n_k}}_t={\rm Ad}\big(e^{it{\widetilde}{H}_{S_{n_k}}}\big)$, we have: $$\begin{aligned}
\int h(t)\,\omega_{n_k}(B\alpha^{S_{n_k}}_t(A))\,dt
&=& \int h(t)\big(\Omega_{S_{n_k}},\,B e^{it{\widetilde}{H}_{S_{n_k}}} A\,\Omega_{S_{n_k}}\big)\,dt\\[1mm]
&=&\big(\Omega_{S_{n_k}},\,B\int h(t) e^{it{\widetilde}{H}_{S_{n_k}}}\,dt\,A\Omega_{S_{n_k}}\big)
=0\end{aligned}$$ where the strong operator integral $\int h(t) e^{it{\widetilde}{H}_{S_{n_k}}}\,dt=0$ because ${\widetilde}{H}_{S_{n_k}}\geq 0$ and ${\rm supp}({\widehat}{h})\subset (-\infty,0)$. It follows that this also holds if $\omega_{n_k}$ is any state in the norm closed convex hull of these vector states. Then $$\int h(t) \omega_\infty(B\alpha_t(A))\,dt
=\lim_{k\to\infty}\int h(t)\,\omega_{n_k}(B\alpha^{S_{n_k}}_t(A))\,dt=0\,.$$ As $B$ is arbitrary, it follows that $U^o_\infty(h) \pi^o_\infty(A)\Omega^o_\infty=0$ for all $A\in {\mathfrak A}_{S_n}$ and $n\in\N$ hence $U^o_\infty(h)=0$.
Note that there are several sources of nonuniqueness for ground states in this argument. Apart from the possibility of different weak \*–limit points of $\{\omega_n\}_{n\in\N}$, there are also different choices of $\omega_n$ as the lowest eigenspace $\cl E._{S_n}$ of $H_{S_n}$ over which $\Omega_{S_n}$ ranges may have dimension higher than one.
Gauge transformations and constraint enforcement. {#GTGL}
=================================================
To conclude this work, we need to define gauge transformations, enforce the Gauss law and identify the physically observable subalgebra. This analysis was essentially done in [@GrRu], but below we recall the details for completeness. After enforcement of constraints, we will consider how the time evolution automorphism group descends to the algebra of physical observables, and prove the existence of a ground state for it.
By construction ${\mathfrak A}_{\Lambda} \subset\cl B.(\cl H.)$, hence to define gauge transformations on ${\mathfrak A}_{\Lambda}$ it suffices to define a unitary representation of the gauge group on $\cl H.$ which implements the correct gauge transformations on the local subalgebras ${\mathfrak A}_S\subset{\mathfrak A}_{\Lambda}$.
The local gauge transformations on the lattice $\Lambda^0$ is the group of maps $\gamma:\Lambda^0\to G$ of finite support, i.e. $$\gauc\Lambda := G^{(\Lambda^0)}=\big\{\gamma:\Lambda^0\to G\,\mid\,\big|{\rm supp}(\gamma)\big|<\infty\big\},\qquad
{\rm supp}(\gamma):=\{x\in\Lambda^0\,\mid\,\gamma(x)\not=e\}.$$ This is an inductive limit indexed by the finite subsets $S\subset\cl S.$, of the subgroups ${\rm Gau}_S \Lambda:=\{\gamma:\Lambda^0\to G\,\mid\,{\rm supp}(\gamma)\subseteq S\}
\cong\prod\limits_{x\in S}G$, and we give it the inductive limit topology. It acts on each local field algebra ${\mathfrak A}_S={\mathfrak F}_{S}\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\cl L._\ell$ by a product action as above in Sect. \[PM\], implemented by a unitary (cf. (\[LocalW\])) $${\widehat}{W}_\zeta:=U^F_\zeta\otimes\big(\bigotimes_{\ell\in\Lambda^1_S}W^{(\ell)}_\zeta\big),\quad
\zeta\in\gauc \Lambda_S,$$ on $\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}L^2(G)$. Here $U^F_\zeta$ is the second quantization on the fermionic Fock space $\cl H._F$ of the transformation $f\to \zeta\cdot f$ where $(\zeta\cdot f)(x):={\zeta(x)}f(x)$ for all $x\in\Lambda_S^0$ and $f\in\Cn_S$, hence $U^F_\zeta$ implements the automorphism $\alpha^1:\gauc \Lambda_S \to\aut{\mathfrak F}_{\Lambda_S}$ by $\alpha_\zeta^1(a(f)):=a(\zeta\cdot f)$. The $W^{(\ell)}_\zeta$ are copies of the unitaries $W_\zeta:L^2(G)\to L^2(G)$ by $$(W_\zeta\varphi)(h):=\varphi(\zeta^{-1}\cdot h)=\varphi(\zeta(x_\ell)^{-1}\,h\,\zeta(y_\ell)).$$ For the full infinite lattice, these unitaries generalize naturally to $\al H.:=\al H._{\rm Fock}\otimes\al H._\infty$ by the same formulae, as each $\zeta\in\gauc \Lambda$ is of finite support, i.e. $${\widehat}{W}_\zeta:=U^F_\zeta\otimes\big(\bigotimes_{\ell\in\Lambda^1_{{\rm supp}'(\gamma)}}W^{(\ell)}_\zeta\big),\quad
\zeta\in\gauc \Lambda$$ where $U^F_\zeta$ is again the second quantization on the fermionic Fock space $\al H._{\rm Fock}$ of the map $f\to\zeta\cdot f$. Here ${\rm supp}'(\gamma)$ denotes the subgraph of $\Lambda$ consisting of all the links which have at least one point in ${\rm supp}(\gamma)$. Hence $\Lambda^1_{{\rm supp}'(\gamma)}$ consists of the links which have at least one point in ${\rm supp}(\gamma)$. In this notation, we assumed that ${\widehat}{W}_\zeta$ acts as the identity on those factors of $\cl H._\infty$ corresponding to $\ell\not\in{\rm supp}'(\gamma)$.
This produces a unitary representation ${\widehat}{W}:\gauc \Lambda\to\al U.(\al H.)$. Then the gauge transformation produced by $\zeta$ on the system of operators is given by ${\rm Ad}({\widehat}{W}_\zeta)$, and on the local algebras it produces the same gauge transformations as in Subsect. \[PM\]. It is clear also that these gauge transformations preserve $$\al A._{\rm max}=\ilim \cl B._S = C^*\Big(\bigcup_{S\in\al S.} \cl B.(\al H._S) \Big)$$ hence we can use ${\rm Ad}({\widehat}{W}_\zeta)$ to define gauge transformations on our maximal algebra.
Next we recall that the local Hamiltonians $H_S$ are constructed from gauge invariant terms, and hence ${\rm Ad}({\widehat}{W}_\zeta)(e^{itH_S})=e^{itH_S}$ and so for $\alpha_t^S:={\rm Ad}(e^{itH_S})$ we have ${\alpha_t^S\circ {\rm Ad}({\widehat}{W}_\zeta)}={\rm Ad}({\widehat}{W}_\zeta)\circ\alpha_t^S$. Thus for the global time evolutions on $ \al A._{\rm max}$ we get from Theorem \[GlobDynExist\] that for all $A\in \al A._{\rm max}$ and $\zeta\in\gauc \Lambda$ we have $$\begin{aligned}
{\widehat}{W}_\zeta\,\alpha_t(A)\,{\widehat}{W}_\zeta^*&=& {\widehat}{W}_\zeta\Big(\lim_{S\nearrow\Z^3}\alpha^S_t(A)\Big){\widehat}{W}_\zeta^*
=\lim_{S\nearrow\Z^3} {\widehat}{W}_\zeta\alpha^S_t(A) {\widehat}{W}_\zeta^*\\[1mm]
&=&\lim_{S\nearrow\Z^3}\alpha^S_t\big({\widehat}{W}_\zeta A{\widehat}{W}_\zeta^* \big)
=\alpha_t\big({\widehat}{W}_\zeta A{\widehat}{W}_\zeta^* \big)\end{aligned}$$ i.e. the global time evolution also commutes with the gauge transformations. This implies that the gauge transformations ${\rm Ad}({\widehat}{W}_\zeta)$ will preserve all orbits of the global time evolution, and hence ${\rm Ad}({\widehat}{W}_\zeta)$ preserves our kinematics algebra ${\mathfrak A}_{\Lambda}:=C^*\Big( \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S ) \Big)$. By restriction, the gauge transformations are therefore well-defined on ${\mathfrak A}_{\Lambda}$, and we will denote the action by $\beta:\gauc \Lambda\to{\rm Aut}({\mathfrak A}_{\Lambda})$. Note that using ${\mathfrak A}_{\Lambda}\subset M({\widehat}{\mathfrak A}_{\Lambda})$ this action is the canonical extension of the one on ${\widehat}{\mathfrak A}_{\Lambda}$ defined in [@GrRu].
Finally, we would like to enforce the Gauss law constraint and identify the physical subalgebra. In [@GrRu] we already did this for the local algebras ${\mathfrak A}_S$ in Theorem 4.12, and proved in Theorem 4.13 that the traditional constraint enforcement method - taking the gauge invariant part of the algebra, then factoring out the residual constraints - produced results coinciding with those of the T–procedure (cf. [@GrSrv]). As the time evolution automorphism group commutes with the gauge transformations, it will respect the constraint reduction, hence define a time evolution automorphism group on the algebra of physical observables.
The concrete constraint reduction in the defining representation of ${\mathfrak A}_{\Lambda} \subset\cl B.(\cl H.)$ starts with the representation ${\widehat}{W}:\gauc \Lambda\to\al U.(\al H.)$ of the gauge group which implements the gauge transformations. One defines the gauge invariant subspace $$\al H._G:=\{\psi\in\al H.\,\mid\,{\widehat}{W}_\zeta\psi=\psi\;\;\forall\,\zeta\in\gauc \Lambda\}$$ and observes that the cyclic vector $\Omega\otimes\psi_0^\infty$ is in $\al H._G$. Let $P_G$ be the projection onto $\al H._G$, then an $A\in\cl B.(\cl H.)$ commutes with $P_G$ iff both $A$ and $A^*$ preserve $\al H._G$. Thus our observables are in $$\{P_G\}'\cap {\mathfrak A}_{\Lambda} \subseteq P_G \cl B.(\cl H.)P_G + (\un-P_G) \cl B.(\cl H.)(\un-P_G).$$ The final step of constraining consists of restricting $\{P_G\}'\cap {\mathfrak A}_{\Lambda}$ to $\al H._G$, which means that we discard the second part of the decomposition above. On the local algebras $ {\mathfrak A}_S$ this will produce a copy of the algebra of compact operators on the gauge invariant part of $\al H._S$ by Theorem 4.13 in [@GrRu]. The algebra generated by the orbit of the time evolutions of these reduced local algebras will be a particularly important subalgebra of algebra of physical observables, as it is constructed purely from the original physical observables with no involvement of the gauge variables.\
[**Remark:**]{}\
The terminology we use here comes from the T–procedure (cf. [@GrSrv]), where the algebra consistent with the constraints is called the observable algebra (here it is $\{P_G\}'\cap {\mathfrak A}_{\Lambda}$), and the final algebra obtained from it by factoring out the ideal generated by the constraints is called the algebra of physical observables. This is different from the terminology in [@KR; @JKR] where the observable algebra is the final algebra obtained by factoring out the ideal generated by the constraints from the algebra of gauge invariant variables.\
To conclude this section, we will next prove that there are gauge invariant ground states. By Proposition 4.2 in [@GrRu] these produce Dirac states on the original algebra, hence states on the algebra of physical observables (cf. Theorem 4.5 in [@GrRu]). Such a state on the algebra of physical observables is a ground state w.r.t. the time evolution automorphism group on the algebra of physical observables which is descended from the one on ${\mathfrak A}_{\Lambda}$.
There is a gauge invariant regular ground state for $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$.
Recall from Theorem \[groundState\], that if we take an increasing sequence ${S_n\nearrow\Z^3}$ of lattice cubes, with $\omega_\infty$ a weak \*–limit point of $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$, then $\omega_\infty\restriction{\mathfrak A}_{\Lambda}$ is a regular ground state for $\alpha:\R\to{\rm Aut}({\mathfrak A}_{\Lambda})$. Here each $\omega_n$ is in the closed convex hull of vector states $$A\mapsto{\big(\Omega_{S_n},\, A\,\Omega_{S_n}\big)},\qquad A\in\cl B.(\cl H.)$$ and $\Omega_S\in\cl H._S$ ranges over the normalized eigenvectors of the lowest point in the spectrum of $H_S$. It is clear that if the sequence $\{\omega_n\}_{n\in\N}\subset\cl B.(\cl H.)^*$ consists of gauge invariant states, then so are its weak \*–limit points, hence it suffices to show for any $S_n\in\al S.$ that we can find $\omega_n$ chosen as above, which is invariant w.r.t. conjugation by the unitaries ${\widehat}{W}:\gauc \Lambda\to\al U.(\al H.)$.
As ${\widehat}{W}(\gauc \Lambda)$ commutes with each $H_S$, it leaves its eigenspaces invariant, and ${\widehat}{W}(\gauc \Lambda)$ restricted to $\al H._S$ is just ${\widehat}{W}({\rm Gau}_S \Lambda )$. This group is compact, hence by the Peter-Weyl theorem, each eigenspace of $H_S$ in $\al H._S$ decomposes into finite dimensional subspaces on which ${\widehat}{W}({\rm Gau}_S \Lambda )$ acts irreducibly. Choose for the lowest eigenspace $\cl E._S\subset\al H._S$ of $H_S$ such an irreducible component of ${\widehat}{W}({\rm Gau}_S \Lambda )$ contained in it, and denote the finite dimensional component space by $V_S\subseteq\cl E._S$. Denote the unit sphere of $V_S$ by $E_S$. By finite dimensionality of $V_S$, $E_S$ is compact. Consider the map $\eta:\al H.\to\al B.(\al H.)^*$ by $\eta(\psi)(A):={(\psi,A\psi)}$ for $\psi\in\al H.$, $A\in\al B.(\al H.)$. Then $\eta$ is continuous w.r.t. the Hilbert space topology and the w\*-topology of $\al B.(\al H.)^*$, hence $\eta$ takes compact sets to compact sets. In particular $\eta(E_S)\subset{\got S}(\al B.(\al H.))$ (denoting the state space of $\al B.(\al H.)$), is compact in the w\*-topology. Denote the norm closed convex hull of $\eta(E_S)$ by ${\got S}_S$, and observe that it is contained in the finite dimensional subspace of $\al B.(\al H.)^*$ spanned by the functionals $A\mapsto{(v_i,Av_j)}$, $A\in\al B.(\al H.)$, where $v_i$ and $v_j$ range over some orthonormal basis $\{v_1,\ldots,v_k\}$ of $V_S$. As all Hausdorff vector topologies of a finite dimensional vector space coincide (cf. [@Sch Theorem I.3.2]), we conclude from norm boundedness and closure, that ${\got S}_S$ is compact w.r.t. the w\*-topology.
As $E_S$ is invariant as a set w.r.t. the action of ${\widehat}{W}({\rm Gau}_S \Lambda )$, we conclude that $\eta(E_S)$ is invariant w.r.t. the action of ${\rm Gau}_S \Lambda$ on $\al B.(\al H.)^*$ by $$\varphi\mapsto\varphi_\zeta,\quad\hbox{where}\quad\varphi_\zeta(A):=\varphi(
{\widehat}{W}_\zeta A {\widehat}{W}_\zeta^*)$$ for $\zeta\in {\rm Gau}_S \Lambda$, $A\in \al B.(\al H.)$ and $\varphi\in \al B.(\al H.)^*$. As the action comes from automorphisms on $\al B.(\al H.)$, it extends to an affine action on the closed convex hull ${\got S}_S$, which is w\*-compact. This action is continuous in the w\*-topology, hence as ${\rm Gau}_S \Lambda$ is amenable (as it is compact), we obtain that there is an invariant point ${\widehat}\omega_S\in{\got S}_S$ (cf. [@Pie Theorem 5.4]). However, the action of ${\rm Gau}_S \Lambda$ on ${\got S}_S$ is just the restriction of the action of ${\rm Gau} \Lambda$ on ${\got S}(\al B.(\al H.))$ to ${\got S}_S$, hence ${\widehat}\omega_S\in{\got S}_S$ is gauge invariant, and can be restricted to ${\mathfrak A}_{\Lambda}$. This concludes the proof that there are gauge invariant ground states.
Conclusions.
============
In the preceding, we have proven the existence of the dynamical automorphism group for QCD on an infinite lattice, and obtained a suitable minimal field algebra on which it acts. This pair defines a C\*-dynamical system in the sense that it is strongly continuous. We proved the existence of regular ground states, and discussed how to enforce the Gauss law constraint.
Clearly much more remains to be done, e.g
- There is no uniqueness proven or analyzed for the ground states. One needs to determine the properties of the set of ground states.
- The form and structure of the physical observable algebra needs to be determined more explicitly.
- Existence of the dynamics is not enough, some useful approximation schemes are needed to connect with present analysis on finite lattices.
Appendix
========
\[Lemma1\] If $B,\; V:\R\to\al B.(\cl H.)$ are bounded strong operator continuous maps, and satisfy $$\label{ddtVA}
\frac{d}{dt}V(t)\psi=B(t)\psi\qquad\forall\,t\in\R$$ for all $\psi$ in some dense subspace $\al D.$ of $\al H.$, then the relation (\[ddtVA\]) holds for all $\psi\in\al H.$.
A version of this lemma is proven in Prop. 2.1 (iii) in [@NaSi2], but as that proof is indirect, we give a direct proof here. We will use the notation $\|V\|:=\sup\limits_{t\in\R}\|V(t)\|$ and $\|B\|:=\sup\limits_{t\in\R}\|B(t)\|$. By integration of (\[ddtVA\]) we obtain for all $\psi\in\al D.$ and $h\not=0$ that $$\begin{aligned}
\big(V(t+h)-V(t)\big)\psi &=& \int_t^{t+h}B(s)\psi\,ds \qquad\forall\,\psi\in\al D.,\qquad\hbox{and hence:}\\[1mm]
\Big\|\frac{1}{h}\big(V(t+h)-V(t)\big)\psi\Big\| &\leq& \frac{1}{|h|}\int_{t_-}^{t_+}\|B(s)\psi\|\,ds
\leq\|B\|\,\|\psi\|\end{aligned}$$ where $t_-:=\min(t, t+h)$ and $t_+:=\max(t, t+h)$. As $\frac{1}{h}\big(V(t+h)-V(t)\big)$ for fixed $t,\,h$ is bounded, and $\al D.$ is dense, this implies that $\|\frac{1}{h}\big(V(t+h)-V(t)\big)\|\leq\|B\|.$ Let $$D_h:=\frac{1}{h}\big(V(t+h)-V(t)\big)-B(t)\quad\hbox{hence}\quad \|D_h\|\leq 2\|B\|.$$ We want to prove that $\lim\limits_{h\to 0} D_h\varphi=0$ for all $\varphi\in\al H.$, i.e. that (\[ddtVA\]) holds on all of $\al H.$. Let $\varphi\in\al H.$ and choose a sequence $\psi_n\in\al D.$ such that $\psi_n\to\varphi$. Fix an $\varepsilon >0$. For any $n\in\N$: $$\|D_h\varphi\|\leq\|D_h(\varphi-\psi_n)\|+\|D_h\psi_n\|\leq 2\|B\|\,\|\varphi-\psi_n\|+\|D_h\psi_n\|.$$ Choose an $n$ such that $2\|B\|\,\|\varphi-\psi_n\|<\varepsilon/2$. By the limit in (\[ddtVA\]) there is a $\delta>0$ such that $|h|<\delta$ implies $\|D_h\psi_n\|<\varepsilon/2$, and hence $\|D_h\varphi\|<\varepsilon$. Thus $\lim\limits_{h\to 0} D_h\varphi=0$ as required.
\[Lemma2\] If $A:\R\to\al B.(\cl H.)$ is a measurable map w.r.t. the strong operator topology, and $\cl H.$ is separable, then $A:\R\to\al B.(\cl H.)$ is a measurable map w.r.t. the norm topology. Then for any bounded interval $I$ we have $$\Big\|\int_I A(t)\, dt\Big\|\leq \int_I\| A(t)\| dt.$$
(cf. Eq. (12) in [@NaSi2])\
By assumption, $t\mapsto\|A(t)\psi\|$ is measurable for each $\psi\in\cl H.$. Consider the supremum $$\|A(t)\|=\sup\{\|A(t)\psi\|\,\mid\,\psi\in\cl H.,\;\|\psi\|\leq 1\}.$$ As $\cl H.$ is separable, there is a countable dense set in the closed unit ball of $\cl H.$, which we can arrange into a sequence $(\psi_n)_{n\in\N}$ and hence obtain $$\|A(t)\|=\sup\{\|A(t)\psi_n\|\,\mid\,n\in\N\}.$$ However, the supremum of a sequence of measurable functions is measurable, hence $t\mapsto\|A(t)\|$ is measurable. For the Bochner integral of $t\mapsto A(t)\psi\in\cl H.$ we thus obtain $$\Big\|\Big(\int_I A(t)\, dt\Big)\psi\Big\|=\Big\|\int_I A(t)\psi\, dt\Big\|\leq
\int_I\| A(t)\psi\| dt\leq \int_I\| A(t)\| dt\cdot\|\psi\|.$$ By taking the supremum over $\psi$ in the closed unit ball of $\cl H.$ we obtain that\
$\Big\|\int_I A(t)\, dt\Big\|\leq \int_I\| A(t)\| dt.$
\[Lemma3\] Let $A,\; B:\R\to\al B.(\cl H.)$ be strong operator continuous maps, such that $A(t)^*=A(t)$ and $\|A(t)\|<M$ for all $t$ for a fixed $M$, and assume that $\cl H.$ is separable. Then for any $t_0\in\R$ and $f_0\in\al B.(\cl H.)$, there is a unique strong operator differentiable map $f:\R\to\al B.(\cl H.)$ such that $$\frac{d}{dt}f(t)\psi = i[f(t),A(t)]\psi+B(t)\psi\quad\forall\,\psi\in\cl H.,\quad
\hbox{and}\quad f(t_0)=f_0\in\al B.(\cl H.).$$ This solution $f$ of the IVP satisfies the estimate $$\|f(t)\|\leq \|f(t_0)\|+\int_{t_-}^{t_+}\|B(s)\|ds\qquad\forall\,t\in\R$$ where $t_-:=\min\{t_0,t\}$ and $t_+:=\max\{t_0,t\}$.
(cf. Lemma 2.2 in [@NaSi2])\
We first prove uniqueness of the solution. Given two solutions $f_1,\,f_2,$ then for all $\psi\in\cl H.$ we have $$(f_1(t)-f_2(t))\psi=\int_{t_0}^t\frac{d}{ds}\big(f_1(s)-f_2(s)\big)\psi \,ds
=\int_{t_0}^t i\big[f_1(s)-f_2(s),A(s)\big]\psi \,ds\,.$$ By Lemma \[Lemma2\] we thus obtain $$\|(f_1(t)-f_2(t))\|\leq \int_{t_-}^{t_+} \left\|\big[f_1(s)-f_2(s),A(s)\big]\right\| \,ds
\leq 2M \int_{t_-}^{t_+} \left\|f_1(s)-f_2(s)\right\| \,ds.$$ Gronwall’s Lemma then proves that $f_1(t)-f_2(t)=0$ and hence we have uniqueness.
Next, we prove existence of the solution $f$, and for this we recall the Dyson series for propagators, cf. Theorem X.69 in [@ReSi2], where the properties below are proven. The norm convergent Dyson series is $$U(t,t_0):=\un + \sum_{n=0}^\infty i^n\int_{t_0}^t\int_{t_0}^{t_1}\cdots\int_{t_0}^{t_{n-1}}A(t_1)\cdots A(t_n)\,dt_n\cdots dt_1$$ which is a unitary, and the integrals are defined w.r.t. the strong operator topology. Its basic properties are $U(t,t)=\un$, $U(t,s)^*=U(s,t)$, $U(r,s)U(s,t)=U(r,t)$ and $U(s,t)$ is strong operator differentiable in both entries. As $U(t,s)$ satisfies the equations $$i\frac{d}{dt}U(t,t_0)\psi=-A(t)U(t,t_0)\psi \qquad\hbox{and}\qquad i\frac{d}{dt}U(t,t_0)^*\psi = U(t,t_0)^*A(t)\psi$$ for all $\psi\in\cl H.$, it is easy to verify that $$f(t):=U(t,t_0)\Big(f_0+\int_{t_0}^tU(s,t_0)^* B(s)U(s,t_0)\,ds
\Big)U(t,t_0)^*$$ satisfies $$\frac{d}{dt}f(t)\psi = i[f(t),A(t)]\psi+B(t)\psi\quad\forall\,\psi\in\cl H.,\quad
\hbox{and}\quad f(t_0)=f_0\in\al B.(\cl H.).$$ (We used the fact that the product of strong operator differentiable maps is again strong operator differentiable). Then $$\|f(t)\|\leq\|f_0\|+\Big\| \int_{t_0}^tU(s,t_0)^* B(s)U(s,t_0)\,ds \Big\|$$ and application of Lemma \[Lemma2\] to the last integral produces the claimed estimate.
Acknowledgements. {#acknowledgements. .unnumbered}
=================
We wish to thank Professors B. Nachtergaele and R. Sims for clarifying their work in [@NaSi] to us in a number of emails and sending us their manuscript of [@NaSi2].
[99]{}
Blackadar, B.: Operator Algebras. Springer 2006
Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 1, Springer 1987 New York Inc.
Bratteli, O.; Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Equilibrium States, Models in Quantum Statistical Mechanics. Springer–Verlag, New York 1981
Bruening, J., Heintze, E.: Representations of compact Lie groups and elliptic operators. Invent. Math. [**50**]{}, 169–203 (1979)
Buchholz, D., Grundling, H.: The resolvent algebra: A new approach to canonical quantum systems. Journal of Functional Analysis [**254**]{}, 2725–2779 (2008)
Dixmier, J.: C\*-algebras. Amsterdam: North Holland Publishing Company 1977
Fang, Y-Z., Luo, X-Q. : arXiv: hep-lat/0108025v1 (2001)
Grundling, H., Neeb, K-H.: Full regularity for a C\*-algebra of the Canonical Commutation Relations, Rev. Math. Phys. [**21**]{} (2009), 587–613
Grundling, H., Rudolph, G.: QCD on an infinite lattice. Commun. Math. Phys. **318**, 717–766 (2013)
Grundling, H.: Quantum constraints. Rep. Math. Phys. [**57**]{}, 97-120 (2006)
Grundling, H., and K.-H. Neeb, [*Crossed products of $C^*$-algebras for singular actions*]{}, J. Funct. Anal. [**266**]{} (2014), 5199–5269
J. Huebschmann, G.Rudolph and M. Schmidt: A Gauge Model for Quantum Mechanics on a Stratified Space, Commun. Math. Phys. 286 (2009) 459-494
Jarvis, P. D., Kijowski, J. and Rudolph, G. : On the Structure of the Observable Algebra of QCD on the Lattice. J. Phys. A: Math. Gen. 38 (2005) 5359-5377
Kadison, R. V., and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras II, New York, Academic Press 1983
Kijowski, J., Rudolph, G.: On the Gauss law and global charge for quantum chromodynamics. J. Math. Phys. [**43**]{} (2002) 1796-1808
Kijowski, J., Rudolph, G.: Charge superselection sectors for QCD on the lattice, J. Math. Physics Vol. 46, 032303 (2005)
Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D [**11**]{}, 395–408 (1975)
Kogut, J.: Three Lectures on Lattice Gauge Theory. CLNS-347 (1976), Lecture Series Presented at the International Summer School, McGill University, June 21-26, 1976
Lawson, H.B., Michelson, M-L.: Spin Geometry. Princeton University Press, Princeton 1989
Nachtergaele, B., Sims, R.: Lieb-Robinson bounds in quantum many-body physics. Contemp. Math. 529, p141, Amer. Math. Soc., Providence, RI: (2010).\
This published argument contains some errors and omissions. A corrected version of the argument is in [@NaSi2].
Nachtergaele, B., Sims, R.: On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian. arXiv:1410.8174v1
G. K. Pedersen, $C^*$-Algebras and their Automorphism Groups. Academic Press 1989, London
J.–P. Pier, Amenable ocally compact groups. John Wiley & sons 1984, New York.
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol 2. Academic Press, San Diego, 1975.
Rieffel, M.A.: On the uniqueness of the Heisenberg commutation relations, Duke Mathematical Journal 39 (1972), 745–752
Schaefer, H., H.: Topological vector spaces. Macmillan company, 1966, New York
Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Phys., vol. 159, Springer (1982)\
Seiler, E.: “Constructive Quantum Field Theory: Fermions”, in Gauge Theories: Fundamental Interactions and Rigorous Results, eds. P. Dita, V. Georgescu, R. Purice
v. Neumann, J. On infinite direct products. Comp. Math. 6, 1–77. Collected Works, Vol. 3, Chapter 6. (ed. A.H. Taub), Pergamon Press, Oxford, New York, Paris 1961.
Wilson, K.G.: Confinement of quarks. Phys. Rev. D10, 2445 (1974)
Wilson, K.G.: Quarks and strings on a lattice, p69 in New phenomena in subnuclear physics. Part A. Proceedings of the International School of Subnuclear Physics, Erice 1975, A. Zichichi (ed.) Plenum Press 1977
Yosida, K.: Functional Analysis. Springer-Verlag, Berlin, Heidelberg, New York 1980.
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Even after obtaining what may be considered as ‘enough’ for personal fulfillment, many still feel a void inside them that yearns to be filled.
This is where the power of giving is then realized.
Anyone Can Give
Some people would be inclined to think that only those who are sufficiently endowed financially should give to charity. But one must always remember that giving starts form personal will or from the heart, a thing that every human being has no matter their financial or material standing. So long as whatever is given, whether a service or material gift, makes a positive impact and difference on the life of the recipient, then it qualifies to be a charitable gift. But it must be given unconditionally without any expectation of a reciprocal action from the receiver.
At the same time, you do not have to move far from where they are in order to find a worthy cause in which to exercise charity. Numerous statistics are produced each year that show various problems people face all over the world. These people are all around us. They are hardly a few meters away. For example, statistics show that one out of four people in the world are facing starvation. This means that starving people are just within our reach. What then prevents us from engaging in acts of charity?
Why Don’t Some People Give?
While a good number of people will give out selflessly despite their meagre capabilities, others who may even be more endowed do not give to charity. They claim that they first have to achieve substantially at a personal level before they can give to others. They seem to follow the idea developed in Maslow’s hierarchy of needs. In this, one rises through several levels of needs. After one need is fulfilled, then they move to the next level until they reach the peak in this pyramid of needs.
This kind of outlook would be too narrow to allow someone to give. This is because the complex nature of the human being forces us to seek for more, no matter what we may already have. As such, the urge to get more may never end. This is why some people never give to others. They do not realise that the path of self-fulfilment starts right at the bottom of the pyramid and that is where giving should start, not after you have fulfilled all of your possible personal needs.
The reason why some people do not give is because they act out of a consciousness of scarcity, they believe that they do not have it in them to give, that they have nothing of value to share with the rest of the world and that they have to be a certain type of person to give or that there is not enough to go around. All of this is far from the truth, anyone and everyone has something they can offer one another.
Giving Has Immense Personal Benefits
As stated earlier, giving does not necessarily mean giving out money or material items. People give not because they have but because they have that inherent urge to give. In fact, many well-known philanthropists did not start their charity work when they got rich. They probably didn’t know that they would become rich even when they started charitable work.
Even without any material gift to offer, sharing your ideas can have immense personal benefits. Ideas can transform others greatly. Many great things that have been achieved in the world emanated from simple ideas. The benefit of sharing your ideas is that the effect will somehow boomerang back to you. By listening to your own ideas repeatedly as you share with others, you may eventually be inclined to put them in practice yourself with wonderful results. You will be unlocking your own potential of self fulfilment. It may eventually seem like a miracle but that is how selfless giving works.
By giving out what you have without expecting anything in return, you start living a meaningful life. You get to realise your true calling in a life and world full of challenges. If you find meaning in the lives of those in need and do something about it, you will also find meaning in your own life. You find yourself in better health and peace and you achieve more happiness. That is the magic of selfless giving.
“You give but little when you give of your possessions. It is when you give of yourself that you truly give.” – Kahlil Gibran
Article By Joel Brown | Addicted2Success
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Mover inside Traders Limelight: Enerplus Corporation (ERF) closed Tuesday at $8.95 with -1.54%, Denison Mines Corp. (DNN)
Enerplus Corporation (ERF) stop its trading day at $8.95 with -1.54%. The firm exchanged a volume of 0.66 million shares at hands. In variance, the average volume was 0.68 million shares. At the time of calculation, Shares of the company recently traded 30.85% away 52-week low and noted price movement -35.47% away from the 52-week high level.
Many investors will opt to use several time periods when examining moving averages. Investors may also be paying close concentration to some simple moving average indicators on shares of Enerplus Corporation (ERF). The moving average uses the sum of all of the previous closing prices over a certain time period and divides the result by the number of prices used in the calculation. Recently, the stock has been noticed trading 2.64% away from the 20-day moving average. Going toward to the 50-day, we can see that shares are currently trading 4.41% off of that figure. Zooming out to the 200-day moving average, shares have been seen trading -12.74% away from that value.
The Stock currently has a consensus recommendation of 2.00. This rating uses a scale from 1 to 5. A recommendation of 1 or 2 would represent a consensus Buy. A rating of 4 or 5 would indicate a consensus Sell. A rating of 3 would signify a consensus Hold recommendation.
Investors who are keeping close eye on Enerplus Corporation (ERF) stock; watched recent volatility movements, they can see that shares have been recorded at 2.52% for the week, and 3.35% for the last month. Taking a look back at some historical performance numbers for Enerplus Corporation (ERF), we can see that the stock moved -0.44% for the last five trades. For the last month, company shares are 5.17%. For the last quarter, the stock has performed -2.61%. Over the past full-year, shares have performed -21.83%. If we look back year-to-date, the stock has performed 15.34%.
Shares of the company have shown an EPS growth of 45.10% in the last 5 years. Its sales stood at -0.30% a year on average in the period of last five years. The company maintains price to book ratio of 1.45. A P/B ratio of less than 1.0 can indicate that a stock is undervalued, while a ratio of greater than 1.0 may indicate that a stock is overvalued. The company was able to keep return on investment at 13.60% in the last twelve months.
Denison Mines Corp. (DNN) declined -2.43% and its total traded volume was 0.42 million shares contrast to the average volume of 0.52 million shares. The stock closed its day at $0.54. The closing price represents the final price that a stock is traded for on a trading day. It’s the most up-to-date valuation until trading begins again on the next day. However, most financial instruments are traded after hours which mean that the closing price of a stock might not match the after-hours price. Regardless, closing prices are a useful tool that investors use to quantify changes in stock prices over time. The closing prices are compared day-by-day to look for trends and can measure market sentiment for any security over the course of a trading day.
The company maintained ROI for the last twelve months at -12.00%. The stock has a beta value of 1.34. Beta can be useful to gauge stock price volatility in relation to the broader market. For the past 5 years, the stock’s EPS growth has been nearly 22.50%.
Denison Mines Corp. (DNN) observed trading 0.90% away from the 20-day moving average and 2.02% off from its 50-day simple moving average. Recently, the stock has been moved 1.43% from its 200-day simple moving average. The institutional ownership stake in the corporation is 7.30%. Its revenue has grown at an average annualized rate of about 8.40% during the past five years. Its RSI (Relative Strength Index) reached 50.20. The debt-to-equity ratio (D/E) was recorded at 0.00. However its weekly volatility is 4.48% and monthly volatility is 3.99%.
Let’s take a gaze at how the stock has been performing recently. Denison Mines Corp. (DNN) shares have moved -5.37% in the week and 2.15% in the month. Year to date is 16.66%, 9.35% over the last quarter, -13.18% for the past six months and 6.18% over the last 12 months. The average analysts gave this company a mean recommendation of 2.30. A rating of less than 2 means buy, “hold” within the 3 range, “sell” within the 4 range, and “strong sell” within the 5 range. | https://www.talktraders.com/2019/04/17/mover-inside-traders-limelight-enerplus-corporation-erf-closed-tuesday-at-8-95-with-1-54-denison-mines-corp-dnn/ |
There are many type of images, and we will look in detail about different types of images, and the color distribution in them.
The binary image
The binary image as it name states, contain only two pixel values.
0 and 1.
In our previous tutorial of bits per pixel, we have explained this in detail about the representation of pixel values to their respective colors.
Here 0 refers to black color and 1 refers to white color. It is also known as Monochrome.
Black and white image:
The resulting image that is formed hence consist of only black and white color and thus can also be called as Black and White image.
No gray level
One of the interesting this about this binary image that there is no gray level in it. Only two colors that are black and white are found in it.
Format
Binary images have a format of PBM ( Portable bit map )
2, 3, 4,5, 6 bit color format
The images with a color format of 2, 3, 4, 5 and 6 bit are not widely used today. They were used in old times for old TV displays, or monitor displays.
But each of these colors have more then two gray levels, and hence has gray color unlike the binary image.
In a 2 bit 4, in a 3 bit 8, in a 4 bit 16, in a 5 bit 32, in a 6 bit 64 different colors are present.
8 bit color format
8 bit color format is one of the most famous image format. It has 256 different shades of colors in it. It is commonly known as Grayscale image.
The range of the colors in 8 bit vary from 0-255. Where 0 stands for black, and 255 stands for white, and 127 stands for gray color.
This format was used initially by early models of the operating systems UNIX and the early color Macintoshes.
A grayscale image of Einstein is shown below:
Format
The format of these images are PGM ( Portable Gray Map ).
This format is not supported by default from windows. In order to see gray scale image, you need to have an image viewer or image processing toolbox such as Matlab.
Behind gray scale image:
As we have explained it several times in the previous tutorials, that an image is nothing but a two dimensional function, and can be represented by a two dimensional array or matrix. So in the case of the image of Einstein shown above, there would be two dimensional matrix in behind with values ranging between 0 and 255.
But thats not the case with the color images.
16 bit color format
It is a color image format. It has 65,536 different colors in it. It is also known as High color format.
It has been used by Microsoft in their systems that support more then 8 bit color format. Now in this 16 bit format and the next format we are going to discuss which is a 24 bit format are both color format.
The distribution of color in a color image is not as simple as it was in grayscale image.
A 16 bit format is actually divided into three further formats which are Red , Green and Blue. The famous (RGB) format.
It is pictorially represented in the image below.
Now the question arises, that how would you distribute 16 into three. If you do it like this,
5 bits for R, 5 bits for G, 5 bits for B
Then there is one bit remains in the end.
So the distribution of 16 bit has been done like this.
5 bits for R, 6 bits for G, 5 bits for B.
The additional bit that was left behind is added into the green bit. Because green is the color which is most soothing to eyes in all of these three colors.
Note this is distribution is not followed by all the systems. Some have introduced an alpha channel in the 16 bit.
Another distribution of 16 bit format is like this:
4 bits for R, 4 bits for G, 4 bits for B, 4 bits for alpha channel.
Or some distribute it like this
5 bits for R, 5 bits for G, 5 bits for B, 1 bits for alpha channel.
24 bit color format
24 bit color format also known as true color format. Like 16 bit color format, in a 24 bit color format, the 24 bits are again distributed in three different formats of Red, Green and Blue.
Since 24 is equally divided on 8, so it has been distributed equally between three different color channels.
Their distribution is like this.
8 bits for R, 8 bits for G, 8 bits for B.
Behind a 24 bit image.
Unlike a 8 bit gray scale image, which has one matrix behind it, a 24 bit image has three different matrices of R, G, B.
Format
It is the most common used format. Its format is PPM ( Portable pixMap) which is supported by Linux operating system. The famous windows has its own format for it which is BMP ( Bitmap ). | http://www.tutorialspoint.com/dip/types_of_images.htm |
CLAIM OF PRIORITY
FIELD OF INVENTION
BACKGROUND
SUMMARY
DETAILED DESCRIPTION
GRMG Infrastructure & Request Message
GRMG Application & Response Message
Alert Monitor and Error Tree Displays
The present application hereby claims the benefit of the filing date of a related Provisional Application filed on Oct. 24, 2003, and assigned Application Ser. No. 60/513,942.
The field of invention relates generally to Information Systems (IS); and, more specifically, to a graphical user interface (GUI) for displaying software component availability as determined by a messaging infrastructure.
The information systems (IS) of an enterprise are often responsible for performing the enterprise's database and business logic functions. Database functions involve the management or usage of the enterprise's records (such as accounting records, sales records, billing records, employee records, etc.). Business logic functions are underlying processes of the enterprise that have been reduced to automated execution (e.g., automatically calculating revenues, automatically scheduling services, etc.). Often, a business logic function depends upon the use of a database function (e.g., an automated billing system that invokes the customer order records of the enterprise). Moreover, database application software is often supplied with its own “business logic” software that enables business logic processes that invoke the core database function to be executed.
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In modern day enterprises, a complicated infrastructure of inter-networked computing systems and their corresponding software are typically orchestrated to perform, as a cooperative whole, the database and business logic tasks of the enterprise. An exemplary arrangement is depicted in . shows a network , which may be viewed as an enterprise's internal intranet or the Internet (or some combination thereof), to which an application server platform - and a Java based platform - are communicatively coupled. Through the immediately following discussion of each of these various functional elements - and some of their possible inter-relationships amongst each other, techniques employed by IS personnel in building the IS infrastructure of an enterprise should be better appreciated.
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An application server is often used to host a variety of applications (such as application ). Business logic application software and/or database application software are frequent types of application software that are hosted by an application server . Here, “hosting” generally means being responsible for interpreting and/or formatting messages received/sent to network so that the application is properly used by the enterprise. For example, in a basic case where application is a business logic application, the application server responds to a request from the network for application (i.e., a request from some entity that has expressed a need for application through network ) by properly invoking application in response to the request; and, forwards the result(s) of the application's execution to the requester.
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In other instances the application server may perform additional business logic/database functionality “on top of” basic functionality provided by application (e.g., so as to precisely respond to the request that was received from the network ). The additional business logic/database functionality may involve the invocation of other application software. In further instances the application server may physically assist in the downloading of executable application software to a requester. Many application servers are responsible for overseeing a plurality of different application software platforms. Moreover, one or more computing systems may be used to perform the application server function. These same one or more computing systems may also be used (depending on implementation preference) to execute one or more of the hosted applications themselves.
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Functional elements - depict a web server and its corresponding Java based “back-end” functionality -. The term “web server” is largely understood to mean being capable of presenting a web based interface (e.g., through the downloading of web pages scripted in HTML format) over a network . Accesses to specific web pages associated with the web based presentation are typically formatted in the HTTP protocol. Often, useful tasks that are dependent on business logic and/or database functions are made accessible through a web based presentation. suggests such an approach by way of the back end servlet engine , database (DB) and Enterprise Java Beans (EJB) applications, and J2EE server .
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A servlet is a body of software typically geared to perform a specific database and/or business logic function (or at least oversee the execution of a specific database and/or business logic function). A servlet engine is an entity capable of supporting the execution of a plurality of servlets and is the “target” for requests that invoke its constituent servlets. The architecture of suggests that one or more of the various servlets supported by the servlet engine depend upon separately packaged: 1) database software ; 2) business logic software implemented with Enterprise Java Beans ; and, 3) database and/or business logic software made accessible in a Java environment through a J2EE server .
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The servlet engine can also be used to generate web page matter that is forwarded to a user over the network by the web server . “Java Server Pages” (JSPs) are web pages having extended embedded software routines (which are often used for displaying dynamic content on a web page). The notion that the servlet engine is a JSP servlet engine indicates that the servlet engine of is capable of providing JSP type web pages.
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Enterprise Java Beans is a Java based application software development environment that produces software routines having a proficiency at being run in a distributed fashion (i.e., across multiple computing systems). Here, EJB and would be understood to correspond to a collection of programs (e.g., business logic programs) written with EJB. J2EE is a Java software platform for building applications that can be executed in a distributed fashion. EJB is a component of J2EE along with Java Server Pages (JSPs) and a variety of interfaces.
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J2EE servers are servers retrofitted with J2EE software and are largely used as “middleware” servers that allow legacy, non Java applications to be made accessible and useable in a Java based environment. For example, the J2EE server associated with EJB may be communicatively coupled to older non Java software that is still used to execute specific database and/or business logic routines. In this case, the J2EE server would be responsible for putting a “Java face” to the legacy software from the perspective of the servlet engine (e.g., by accepting Java commands and interpreting them into an format understandable to a legacy routine).
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Note that programs associated with EJB and database (DB) are configured to be accessible through a Java Native Interface (JNI) while programs associated with EJB are configured to be accessible through one or more of the native interfaces associated with J2EE. JNI is a programming interface that may be used for function calls such as the functions/programs implemented in database and EJB .
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The exemplary IS infrastructure of also shows an HTTP server communicatively coupled to a J2EE server . An HTTP server is a server that can respond to requests from a network authored in the HTTP protocol (which is the primary web page identification protocol—thus, HTTP server can also be viewed as a web server). The HTTP server is communicatively coupled to a J2EE server .
Many business logic processes require a number of different software components to be invoked in a specific sequence. For example, an automated billing process might first run a database application to check the customer order records of the enterprise and then run an automated scripting application to create a custom tailored invoice for each order. Many business logic processes invoke a significant number of different software components over the course of their execution.
An issue with enterprise information systems is the ability to continuously monitor the specific software components that are used by a particular business logic process. If a particular software component becomes unavailable (for whatever reason) so as to render a business logic process unworkable, the existence of the “problem” may not be known until after the next attempt to use the process after the component became unavailable. This represents an efficiency loss in cases where the “problem” could have been fixed (or at least routed around) during the time period that elapsed from the moment the component became unavailable to the next attempt to run the process.
An IS monitoring approach is described that is capable of monitoring the availability of various software components. A further capability is that the availability of the individual software components upon which a specific business logic process depends may each be individually and repeatedly checked, in a combined fashion that is referenced to the specific business logic process, so that the status of the business logic process itself (e.g., operable or non-operable) can be continuously determined on an on-going basis. Moreover, operability or non-operability can be established over a wide range of different business logic processes on a process by process basis.
In this manner, an IS administrator can keep abreast of the status of the IS infrastructure from a perspective that reflects an important purpose of the IS infrastructure: to execute business logic processes that depend upon lower level software components. In various embodiments, the results of the monitoring may be continuously updated and displayed in a display so that an IS administrator can visually ascertain the status of the enterprise's various business logic processes. The monitoring approach may also be capable of performing technical monitoring in which “foundational” operational features of the IS infrastructure (e.g., a JNI interface) are checked without reference to any particular business logic process.
An IS monitoring approach is described that is capable of monitoring the availability of various software components. A further capability is that the availability of the individual software components upon which a specific business logic process depends may each be individually and continuously checked, in a combined fashion that is referenced to the specific business logic process, so that the status of the business logic process itself (e.g., operable or non-operable) can be continuously determined on an on-going basis. Moreover, operability or non-operability can be established over a wide range of different business logic processes on a process by process basis.
In this manner, an IS administrator can keep abreast of the status of the IS infrastructure from a perspective that reflects an important purpose of the IS infrastructure: to execute business logic processes that depend upon lower level software components. In various embodiments, the results of the monitoring may be continuously updated and displayed in a display so that an IS administrator can visually ascertain the status of the enterprise's various business logic processes. The monitoring approach may also be capable of performing technical monitoring in which “foundational” operational features of the IS infrastructure (e.g., a JNI interface) are checked without reference to any particular business logic process.
Overview
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The monitoring approach of is an exemplary depiction that applies software monitoring techniques to the particular IS arrangement originally depicted in . According to the monitoring techniques depicted in , a Generic Request Message Generation (GRMG) infrastructure unit is responsible for repeatedly sending a GRMG request message (hereinafter, “request message”) to a GRMG application . The request message identifies the various software components of a higher level “scenario”. For example, in a typical implementation, the “scenario” might correspond to a business logic process that invokes a number of lower level software components (i.e., any one or more of: processes, programs, web pages); and, the request message for the scenario identifies each of these components.
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The GRMG application is a unit of software that is designed to receive the request message and “check into” the availability of each of the software components that are identified by the request message . The results of the inquiries into the software components are collected and placed into a GRMG response message (hereinafter “response message”). For example, a functional disposition (e.g., “OKAY” or “ERROR”) for each of the scenario's software components is included in the response message .
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In a typical situation the GRMG application is installed at a location where a plurality of business logic processes are overseen, managed and/or executed. In the exemplary depiction of , note that GRMG application is installed with the servlet engine . As discussed in the background, a servlet engine is an entity that is capable of supporting the execution of a plurality of servlets. As servlets are often used to perform business logic processes (which may or may not involve the preparation of web pages), the platform used to implement servlet engine is also an appropriate location for a GRMG application that is configured to test the availability of the software components that the servlets supported by servlet engine depend upon. For example, under the assumption that a number of servlets (whose execution is supported by servlet engine ) are designed to employ as sub-functions one or more database and/or EJB , software components, the GRMG application can best determine the availability of these same software components from the perspective of the servlet engine if the GRMG application is situated with the servlet engine itself.
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After the GRMG application forms the response message , it is sent to the GRMG infrastructure . In the particular embodiment of , the response message is sent by the servlet engine to the web server ; which, in turn, forwards the message into the network . The GRMG infrastructure , in response to its reception of the response message , provides the availability test results that were expressed within the response message to software that is responsible for generating images on a display . The results are then graphically depicted on the display (e.g., in an “alert monitor tree” ) so that an IS administrator can visually determine the status of the scenario (which, as discussed, may represent a business logic process).
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The display may also graphically depict (e.g., in an error tree ) which scenario monitoring schemes are working and which scenario monitoring schemes are not working. Here, a scenario monitoring scheme should be understood to include the entire monitoring process including: 1) request message generation by the GRMG infrastructure and transportation over the network ; 2) request message processing and software availability testing by the GRMG application ; 3) response message generation by the GRMG application and transportation over the network ; and, 4) response message processing by the GRMG infrastructure . Here, a scenario monitoring scheme may “fail” for reasons unrelated to the availability of its corresponding software components. For example, if network is “down” a scenario's request and response messages , can not be exchanged even if the scenario's corresponding software components do not have any availability problems with respect to the servlets that use them.
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Thus, having an alert monitor tree (which indicates which scenarios do not have software component availability problems and which scenarios have software component availability problems) and an error tree (which indicates which scenario monitoring schemes are “working” and which scenario monitoring schemes are “not working”) allows an IS administrator to distinguish between problems that cause software component unavailability and other problems (that are perhaps more fundamental to the workings of the IS infrastructure and the overall scenario monitoring scheme such as those involving network connections) that cause a scenario monitoring scheme to execute improperly.
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The GRMG infrastructure is a body of software that is responsible for the continuous sending of request messages on a scenario by scenario basis. As the above described example to which messages , were applied was written in the context of a single scenario, note that multiple scenarios may exist that each invoke repeated request and response message exchanges. For example, the GRMG infrastructure might be configured to implement a unique scenario for each unique business logic process that the servlet engine supports; and, orchestrate the sending and receiving of GRMG messages for each of these scenarios.
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The GRMG infrastructure may also be configured to communicate with multiple GRMG applications found in different locations across the enterprise. For example, another GRMG application might be included in application server (which is responsible for hosting application among possible others); and, another GRMG application might be included in HTTP server (which is responsible for providing web access for software components located on J2EE server ). Here, the GRMG infrastructure might be configured to not only orchestrate the sending/receiving of GRMG messages for each of the business logic processes supported by servlet engine but also orchestrate the sending/receiving of GRMG messages for each of the business logic processes supported by application server and HTTP server .
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The message exchange approaches discussed further below inherently support a range of deployment options with respect to “how many” GRMG applications are installed per scenario. At a first extreme, only one GRMG application exists at a location from which multiple scenarios are tested for component availability. For example, only the single GRMG application is called upon for all of the scenarios to be tested from servlet engine . As such request message , which is destined for GRMG application , could identify any of a plurality of different scenarios.
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Alternatively, different GRMG applications may be maintained for different scenarios at a same testing location. For example, at the other extreme, GRMG application may be implemented for only a particular scenario. Thus, request message , which is destined for GRMG application , would be capable of only identifying a single scenario. Continuing with such an extreme, for each scenario to be tested for component availability as a servlet from servlet engine , a separate GRMG application would be instantiated. Likewise, a different GRMG application may be maintained for each scenario testing scheme to be carried out from application server and HTTP server .
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Embodiments between the extremes discussed above are also inherently supported. Irrespective of how many GRM applications exist per scenario, GRM application may be implemented as a servlet (having its own unique URL) that is dedicated to execute the software component availability testing for its constituent scenario(s). Request message would therefore identify the URL of GRMG application so that it could be executed as a consequence.
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Also, as a monitoring option, “single” component scenarios are possible. Single component scenarios are useful for monitoring the availability of a software application as a whole. For example, the GRMG application of HTTP server might be configured to monitor the availability of an entire software application which is installed on J2EE server . In this case, because the J2EE server contains the application to be monitored, the HTTP server is an appropriate location from which to determine the availability of the application (e.g., because, during normal usage, the HTTP server is configured to “call” the application in response to a request that was received from the network ); and, therefore, the HTTP server is an appropriate location for GRMG application .
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As another side note, a complete GRMG application may be configured into a software package as part of its standard set of available services; or, a GRMG application may be custom tailored by IS personnel so as to service a custom arrangement of software components. For example, the business logic processes associated with servlet engine may have been “custom crafted” by IS personnel because they are unique to the enterprise that the IS infrastructure of serves. As a consequence, GRMG application may likewise be created by IS personnel so as to properly monitor these custom processes.
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By contrast, if the applications supported by application server are standard “off the shelf” applications that are supplied as part of a software vendor's standard product offering, GRMG application may likewise be part of the software vendor's product (that is, since the applications are supplied by the vendor, the vendor is also capable of developing a GRMG application to monitor them). In a further embodiment, which a standard product offering GRMG application may employ, a GRMG application (e.g., GRMG application ) is geared to call on specific “function modules” that perform specific monitoring functions. Here, a GRMG request message can be configured to call out (e.g., by name) a specific function module to be executed for its corresponding scenario. The GRMG request message may further identify the name of the identified function module being executed.
FIG. 2
209
The monitoring approach of may also be further used to support technical monitoring. Technical monitoring is the monitoring of foundational components of the IS infrastructure that support the execution of the business logic processes themselves (e.g., such as a Java Network Interface (JNI) through which certain software components are supposed to be accessible). Here, a request message would be sent by the GRMG infrastructure that describes a scenario which identifies one or more foundational components that are to be tested for availability.
FIG. 3
FIG. 2
FIG. 3
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outlines a high level methodology that is executed by the monitoring system observed in . According to the methodology of , the GRM infrastructure sends a request message for a scenario to a GRMG application. Upon receipt of the request message the GRMG application performs availability monitoring for the scenario, which may include individual monitoring of its constituent lower level components, and sends a response message back to the GRMG infrastructure . The response message provides the availability monitoring results.
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The GRMG infrastructure, upon receipt of the response message, forwards the results so that they can be displayed . The process then repeats for the scenario. The periodicity of the repetition of the message exchange may be targeted for a set interval (e.g., in minutes). In an embodiment, the process observed in is multi-dimensional in the sense that one such process is executed for each scenario to be monitored. For example, if 1,000 different scenarios are to be monitored, 1,000 instances of the methodology are effectively executed that may involve a plurality of GRMG applications distributed across various locations within the enterprise's IS infrastructure.
FIGS. 4
FIG. 4
FIG. 5
FIG. 5
FIGS. 2 and 5
5
5
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a
b
a
b
a
, and relate to GRMG infrastructure and request message embodiments. Specifically, provides an embodiment of an organization scheme that may be used for a customizing file; shows a process that may be executed by the GRMG infrastructure; and, shows an embodiment of an organization scheme that may be used for a request message. Referring briefly to , a customizing file is used as the base resource from which request messages are spawned. Specifically, the GRMG infrastructure reads information from the customizing file; and, based upon the information discovered in the customization file, the GRMG infrastructure generates a request message .
The process of reading the customizing file and generating a request message from the information that is read from the customizing file may be repeated across scenarios as well as for any particular scenario. Better said, if multiple monitoring scenarios are to be executed, multiple request messages will be executed for each scenario (i.e., a first scenario will result in the repetitive production of a first request message; a second scenario will result in the repetitive production of a second request message; etc.). Here, the customizing file may be partitioned into different regions where each region contains information for a specific scenario.
FIG. 4
401
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1
X
If so, the GRMG infrastructure is expected to refer to the appropriate region of the customizing file in order to generate a request message for a specific scenario (e.g., a first section of the customizing file is referred to in order to generate a request message for a first scenario; a second section of the customizing file is referred to in order to generate a request message for a second scenario; etc.). shows an embodiment of an organization scheme for a customizing file that includes information for each of a plurality of scenarios (so that a unique request message can be generated for each scenario). The organization scheme entails listing basic control information as well as the information for each the scenarios through . In an embodiment, the customizing file is in the format of a document that is capable of supporting the execution of software (e.g., an .XML document). As such, the information is embodied in the appropriate format for the document.
FIG. 4
401
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1
2
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1
According to the embodiment of , the basic control information is used to control the execution of the GRMG infrastructure itself and includes a “run” field ; a “runlog” field ; and, a “runerror” field . The run field specifies whether the GRMG infrastructure that would use the customizing file is running or not. In a further embodiment, the customizing file is the form of a document such as an XML document. Here, an “X” is marked at an appropriate location in the document to indicate whether or not the applicable GRMG infrastructure is running (e.g., X=running; no X=not running).
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FIG. 2
The runlog field specifies whether or not a log file is to be created at runtime for the GRMG infrastructure. The log file, once created, could include typical log entry information such as a log entry timestamp for each generated request message. Referring briefly back to , note that a database is drawn as being coupled to the GRMG infrastructure . Here, database could be used to implement the log file (e.g., as a table) that records the log entries. Similarly, the errorlog field specifies whether or not an error log file is to be created for the GRMG infrastructure. The error log file could include typical error log entry information such as a timestamp and description of each error that arose during execution of the GRMG infrastructure. Database could be used to implement the error log file (e.g., as a table) that records the error log entries.
FIG. 4
FIG. 4
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X
The exemplary customizing file embodiment of also indicates that messaging for a plurality of N scenarios are to be supported. As such a separate body of information is included for each scenario through ; where, for illustrative simplicity, only a breakdown of the information included for scenario is shown in detail. A similar collection of information should be included for the other scenarios. According to the embodiment of , the breakdown of information for a particular scenario includes: 1) the scenario name ; 2) the scenario version ; 3) the scenario instance ; 4) the scenario type ; 5) the starting URL for the scenario ; 6) the start module for the scenario ; 7) a description of the scenario ; 8) the language of the scenario description ; and, 9) a breakdown of information for each component through that is to be checked for availability for the particular scenario.
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Items through correspond to “control” items that apply to the scenario as whole while items through correspond to bodies of information that pertain to a specific component that is to be tested for availability. The scenario name field provides the name of the scenario. The scenario version field provides the version of the scenario.
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The scenario instance field provides the instance of the scenario. Here, for example, suppose that a particular business logic process is correlated with a particular scenario; yet, there are a number of different “ways” that components of the business logic process could be tested for availability (e.g., a first way includes a first group of components, a second way includes a second group of components (where some degree of overlap between the first and second groups may or may not exist), etc.). To handle this, multiple instances of the scenario that is identified with the business logic process may be created. The scenario instance field identifies “which instance” of the scenario the particular body of information it is included with corresponds to. In an embodiment, a different number is used to identify each instance of the scenario; and, therefore, the scenario instance field provides the number of the scenario instance it is included with the information of.
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FIG. 2
The scenario type field identifies how the appropriate GRMG application for the scenario instance is to be reached. Here, as there exist a number of different ways in which executable routines may be called upon, the manner that is identified in the scenario type field should be consistent with the manner in which the GRMG application that is to be executed for the scenario instance has been implemented. For example, if the appropriate GRMG application for the scenario instance is a Java servlet (e.g., to be executed by servlet engine of ) that is reachable with a URL address; then the scenario type field will indicate that a URL is to be specified in the request message. Alternatively, if the appropriate GRMG application for the scenario instance is reachable with an RFC destination, the scenario type field will indicate that HTTP should be used in the sending of the request message.
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The “start URL” field contains the specific address or destination identifier of the target GRMG application for the scenario. In an embodiment, the URL of the appropriate GRMG application is specified in the start URL field if the type field indicates a URL type; or, the RFC destination of the appropriate GRMG application is specified in the start URL field if the type field indicates an HTTP type. Here, the RFC destination may include an encrypted user name and password.
FIG. 2
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Recalling from the discussion of that a GRMG application (e.g., GRMG application ) may be geared to call on a specific “function modules” that perform specific monitoring functions. If the target GRMG application for the scenario is such a GRMG application, the start module field identifies the module to be invoked by the GRMG application. The description field contains a description of the scenario instance (e.g., a textual description of the scenario); and, the language field indicates what language the description that resides within the description field is written in.
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FIG. 4
FIG. 4
Items through correspond to bodies of information that pertain to each specific component that is to be tested for availability by the appropriate GRMG application during the test sequence of the scenario. An embodiment of information that may be associated with component is observed in . A similar collection of information should be included for the other components. According to the approach of , the body of information for a component includes: 1) a component name field ; 2) a component version field ; 3) a component description field ; 4) a component language field ; 5) and property fields - for the passing of a parameter for the component from the GRMG infrastructure (by way of the corresponding request message) to the appropriate GRMG application. The property fields include a property name field , a property type field ; and, a property value field .
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The component name field provides the technical name of the specific component to be monitored for availability. The component version field provides the specific version of the component to be monitored for availability. The component description field contains a description of the component (e.g., a textual description of the component); and, the component language field indicates what language the description that resides within the component description field is written in. In an embodiment, the description is used as the name of the component and contains the host name for the component and an instance number for the component.
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The property fields - are used to send a parameter pertaining to the component's availability testing from the GRMG infrastructure to the appropriate GRMG application (by way of a request message). The property name field identifies the name of the parameter, the property type field identifies the parameter's type, the property value field provides the value of the parameter.
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As an example of how these fields might be used, if the software component to which property fields - are dedicated corresponds to a web page, the URL of the web page would be provided to the GRMG application by embedding information contained in these fields - into the corresponding request message. For example, the name field may include the notation “URL” to indicate a URL is needed to fetch the web page component; and, the value field might provide the specific URL value of the web page (e.g., “http://localhost/index_test.html”).
As another example of how these fields might be used, if a login procedure is required to verify availability, a chain of property field groups could be used to provide the information needed to perform the login. For example, a first group of name and value fields would indicate that a client is to be used for the login (e.g., name=Client) as well as identify the specific client (e.g., value=“000”). A second group of name and value fields would indicate that a userid is to be used for the login (e.g., name=userid) as well as specify the actual userid to be used for the login (e.g., value=“KOJEVNIKOV”). A third group of name and value of fields would be used to indicate that a password is to be entered for the login (e.g., name=password) as well as specify the particular password to be used for the login (e.g., value=“tstpw”).
FIG. 5
FIG. 5
FIGS. 2 and 5
FIG. 4
FIG. 5
FIG. 5
a
b
a
a
a
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shows a process that may be executed by the GRMG infrastructure; and, shows an embodiment of an organization scheme that may be used for a request message. Again, referring briefly to , recall that a customizing file (such as a customizing file having the organization scheme depicted in ) is used as the base resource from which request messages are spawned. Specifically, the GRMG infrastructure reads information from the customizing file; and, based upon the information discovered in the customization file, the GRMG infrastructure generates a request message . In an embodiment, the customizing file is uploaded and its contents are written into a database table (such as table of database ). The GRMG infrastructure then reads from the database table in order to perform read of . Although the customizing file may be read multiple times (as suggested by ), alternatively, the customizing file may be read only once and its contents stored in database tables. The database tables are then referred to (rather than the customizing file itself) in order to generate a request message.
As already discussed, the process of reading the customizing file and generating a request message from the information that is read from the customizing file may be repeated across scenarios as well as for any particular scenario. Better said, if multiple monitoring scenarios are to be executed, multiple request messages will be executed for each scenario (i.e., a first scenario will result in the repetitive production of a first request message; a second scenario will result in the repetitive production of a second request message; etc.). Here, the customizing file may be partitioned into different regions where each region contains information for a specific scenario and the GRMG infrastructure is expected to refer to the appropriate region of the customizing file in order to generate a request message for a specific scenario (e.g., a first section of the customizing file is referred to in order to generate a request message for a first scenario; a second section of the customizing file is referred to in order to generate a request message for a second scenario; etc.).
FIG. 5
FIG. 5
FIG. 4
b
b
419
419
shows an embodiment of the contents and organization of the payload of a request message that may be sent for a particular scenario. The particular request message embodiment of can be viewed as the organization and content of the request message that is sent for the scenario that is described in section of the customizing file embodiment of . Note that it possesses a significant degree of overlap with respect to the organization and content of the section .
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th
Here, the request message could be crafted simply by copying the content of: 1) the scenario name field of the customizing file into the scenario name field of the request message; 2) the scenario version field of the customizing file into the scenario version field of the request message; 3) the scenario instance field of the customizing file into the scenario instance field of the request message; and, 5) attributes of the component specific information fields -of the customizing file into the component specific information fields -of the request message. For example, as applied to the “X” component, the content of the component name, component version, property name and property value fields of the customizing file , , , could be copied into the component name, component version, property name and property value fields of the request message , , , .
In an embodiment, the request message is a document such as an .XML document. Use of such documentation should make the copying of content from the customizing file to the request message a straightforward process.
FIG. 2
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Recall that the message exchange approaches discussed herein inherently support a range of deployment options with respect to “how many” GRMG applications are installed per scenario. At a first extreme, only one GRMG application exists at a location from which multiple scenarios are tested for component availability. For example, referring to , only the single GRMG application is called upon for all of the scenarios to be tested from servlet engine . As such request message , which is destined for GRMG application , could identify any of a plurality of different scenarios.
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Alternatively, different GRMG applications may be maintained for different scenarios at a same testing location. For example, at the other extreme, GRMG application may be implemented for only a particular scenario. Thus request message , which is destined for GRMG application , would be capable of only identifying a single scenario. Continuing with such an extreme approach, for each scenario to be tested for component availability as a servlet from servlet engine , a separate GRMG application would be instantiated. Likewise, a different GRMG application may be maintained for each scenario testing scheme to be carried out from application server and HTTP server .
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Embodiments between the extremes discussed above are also inherently supported. Irrespective of how many GRM applications exist per scenario, GRM application may be implemented as a servlet (having its own unique URL) that is dedicated to execute the software component availability testing for its constituent scenario(s). Request message would therefore identify the URL of GRMG application so that it could be executed as a consequence.
FIGS. 6
FIG. 6
FIG. 6
FIG. 6
FIG. 6
FIG. 2
a
c
a
b
c
a
6
601
through provide exemplary methodologies that may be designed into a particular GRMG application. shows an embodiment of a primary “end-to-end” GRMG application process. shows an embodiment of an availability test that may be applied to a specific software component. provides an exemplary “end-to-end” GRMG application flow that may be used within an object oriented environment such as a Java environment. According to the primary end-to-end process of , a GRMG application receives and processes a request message . As discussed above with respect to , the request message may contain the identity of multiple components that need to be tested for availability.
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FIG. 6
FIG. 7
a
As such, each component identified in the request message is monitored to see if it is available , . A recursive approach is depicted in . When a disposition has been reached on each of the components with respect to its availability (e.g., “OKAY” or “ERROR”), a response message is prepared and sent to the GRMG infrastructure that tabulates these results . An embodiment of the organization of a response message is provided and discussed in more detail below with respect to .
FIG. 6
FIG. 5
FIG. 6
b
b
b
601
shows an embodiment of an availability test that may be used to test the availability of a web page component. Here, for example, a business logic process may require a web page to be fetched at some point during its execution. In this case, the web page is deemed a component of the scenario for whom a request message was received ; and, as described above (e.g., with respect to ), the request message would identify the web page component (e.g., by name and version). In this case, the methodology of could be used by the GRMG application to determine whether or not the web page is available.
FIG. 6
b
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According to the methodology of the URL address of the web page component is fetched and an attempt is made to connect to the URL address , . If a web page is returned as a consequence of the connection attempt the component web page is deemed available and a positive disposition is registered (e.g., “OKAY”). If a web page is not returned as a consequence of the connection attempt the component web page is deemed unavailable and a negative disposition is registered (e.g., “ERROR”).
FIG. 2
Recall from the discussion of that the GRMG application may be custom made by IS personnel or simply purchased from a software vendor. In those instances where the IS personnel are expected to custom craft their own GRMG application, a software vendor may nevertheless provide a suite of tools used to support the IS personnel in their custom GRMG application development efforts. For example, in one embodiment that applies to an object oriented environment (such as Java), a predefined set of classes are provided for GRMG application development. Such classes may include: 1) a class for a request message; 2) a class for a response message; 3) a class for a scenario; 4) a class for a component; 5) a class for a component parameter; and 6) a class for a component message.
FIG. 6
c
In a potentially related embodiment, a servlet is used to implement the GRMG application itself. Thus, in a Java environment, a Java servlet may be used to implement the GRMG application. Here, continuing with the idea from above that a suite of tools may be provided to help create a custom GRMG application, illustrates a detailed embodiment of a flow for code that could be formed with supplied Java classes for building a custom GRMG application implemented as a Java servlet.
FIG. 6
c
620
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According to the code flow of , an “HTTPServlet Request” is used to invoke the GRMG application. In response to the invocation the GRMG application, with the above mentioned request message class, creates a “request” object from the input stream of the servlet (where the input stream of the servlet corresponds to the content of the received request message). As the request object would identify the scenario, a “scenario” object is further created from the request object (e.g., with the above mentioned scenario class).
FIG. 6
b
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Then custom code used to perform availability testing (such as the process described in ) is executed . Here, the results of the availability testing would be associated with the “scenario” object. After the availability testing is complete a “response” object is created (e.g., with the above mentioned response class) using the scenario object for information that will be used to makeup the content of the request message. A document is then prepared and sent as the servlet output stream which corresponds to the sending of the response message . In a further embodiment, the document is an .XML document.
FIG. 7
FIG. 7
703
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1
X
X
1
2
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6
shows an embodiment of an organization scheme for the layout of the payload of a response message. According to the layout embodiment of , the response message includes: 1) a scenario name ; 2) a scenario version ; 3) a scenario instance ; and, 4) information regarding each component in the scenario that was tested -. The information for a component includes (depicted only for component for purposes of illustrative simplicity): a) the component's name ; b) the component's version ; c) the component's host ; d) the component's instance ; and, e) message information that provides information regarding the testing results for the component . The messaging information includes: i) an alert message ; ii) a severity parameter ; and, iii) area, number, parameters and text fields for transporting a specific message -. A discussion of each of these is provided immediately below. In an embodiment, the response message is embodied as a document such as an .XML document.
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FIG. 5
b
The scenario name, scenario version and scenario instances , , “repeat” the information provided in the scenario name, version and instance parameters , , originally provided in the request message (e.g., as depicted in ). Here, as a response message effectively replies to a request message, the scenario name, version and instance act as a signature for a specific group of one or more component tests. With the signature, the GRMG Infrastructure is able to keep track of a plurality of requests for different groups and a plurality of received responses for different groups. From the perspective of the GRMG application, the preparation of a response message may involve copying the scenario name, version and instance fields from the request message that is being responded to. In embodiments where the request and response messages are in a document format (such as .XML) this should be a straightforward procedure.
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FIG. 5
b
With respect to the component specific information , the component name and component version parameters , identify the particular component for which availability results are being presented. Again, in an embodiment, these values are copied directly from the corresponding values , (e.g., as depicted in ) found in the request message to which the response message is responding. The component host parameter identifies the name of the particular host or server that the software component is running on. The component instance parameter is a reference number that identifies the instance of the component on the host/server on which the component is running.
721
In an embodiment, both of these parameters are determined by the GRMG application and are not provided in the request. Component instances may derive from multiple or redundant software components on a single host or server in the IS infrastructure. For example, if three “copies” of the same software component exist on a host; then, it is possible that any of these three copies may be used to support the business logic flow/application that the scenario corresponds to. As such, three separate instances of the component exist; and, field would identify which one was tested for availability.
FIG. 7
722
722
722
722
1
1
2
The component specific information depicted in the embodiment of also includes a number of “message” parameters used for explaining the result(s) of the component's availability test. Here, the alert parameter indicates whether the test result is “OKAY” or is deemed to be an “ERROR” (e.g., by providing “OKAY” or “ERROR” text in the alert parameter field ). The severity parameter indicates how serious an error is. In an embodiment, the severity parameter is a number within a specified range (e.g., a number between a range of 0 and 255) where the severity of the error scales in a specific direction over the range. For example, the higher the number the more serious the error.
722
722
722
722
722
3
5
3
4
5
The area, number and parameters fields -are used to provide numerically encoded messages. Here a specific encoding scheme is envisioned where the numeric code is broken down into three sections (area, number and parameters). The area field numerically defines a class of messages. The number field numerically defines a specific message within the class specified by the area value. The parameters field are for parameters that are to be included with a numerically encoded message that contemplates the inclusion of parameters with the message itself. If numerically encoded messages are employed, it is assumed that the GRMG Infrastructure is configured with functionality sufficient for decoding the messages.
722
722
722
6
3
6
The text field is used to provide textual messages rather than numerically encoded messages. In an embodiment, the GRMG application has the option of sending both a numerically encoded message and a textual message; or, sending only a numerically encoded message or textual message. As described in more detail below with respect to the display, message information may be displayed (e.g., so that IS personnel can graphically read the message sent from the GRMG application). According to an embodiment, if the number provided in the area field is not recognized, the display will present whatever is presented in the text field .
215
214
215
216
FIG. 2
FIG. 2
Recall that alert monitor and error tree displays were first discussed with respect to . Recall from the discussion of that availability results received by the GRMG infrastructure through the response messages may be graphically depicted in an alert monitor tree that is presented on a display (e.g., as part of a graphical user interface (GUI) so that an IS administrator can visually determine the status of a scenario.
216
214
211
209
201
210
212
210
201
209
The display may also graphically depict in an error tree (again as part of a GUI) which scenario monitoring schemes are not working properly. A scenario monitoring scheme should be understood to include the entire monitoring process including: 1) request message generation by the GRMG infrastructure and transportation over the network ; 2) request message processing and software availability testing by the GRMG application ; 3) response message generation by the GRMG application and transportation over the network ; and, 4) response message processing by the GRMG infrastructure .
201
211
212
A scenario monitoring scheme may “fail” for reasons unrelated to the availability of its corresponding software components. For example, if network is “down” a scenario's request and response messages , can not be exchanged even if the scenario's corresponding software components do not have any availability problems with respect to the servlets that use them.
215
214
Thus, having an alert monitor tree (which indicates which scenarios have software component availability problems) and an error tree (which indicates which scenario monitoring schemes are “not working”) allows an IS administrator to distinguish between problems that cause software component unavailability and other problems that cause a scenario monitoring scheme to execute improperly.
Alert Monitor Tree
FIG. 8
FIG. 9
FIG. 8
FIGS. 4
801
403
503
703
5
7
409
801
b
shows an embodiment of an alert monitor tree and shows an embodiment of an error tree display. According to the embodiment depicted in the alert monitor tree includes a primary node which identifies the name of the scenario. Here, the name of the scenario may be “passed up” from the GRMG infrastructure with ease because of its dealings with an identical parameter in the customizing file , request messaging and response messaging (as previously depicted in , and , respectively). In an alternative embodiment, information from the “scenario description” is passed up from the GRMG infrastructure and presented at primary node .
802
801
802
802
414
416
802
1
X
FIG. 4
FIG. 4
As discussed, a scenario may be configured to correspond to a business logic process or other software application; and, therefore, the name given to the scenario may be identical to that given to the business logic process or other software application. The secondary nodes that are connected in the alert monitoring tree to the primary node correspond to the individual components of the scenario -. In an embodiment, the names for the components that are displayed next to the component nodes in the alert monitoring tree originate from the component description field of the GRMG infrastructure's customizing file (e.g., as previously depicted in ). In an alternative embodiment, information describing a component (e.g., as provided from location of ) is provided by the GRMG infrastructure and displayed at the secondary nodes .
802
803
720
720
803
X
th
FIG. 8
FIG. 8
FIG. 7
A subtree is capable of being displayed from each component node. For illustrative convenience, only the node associated with the Xcomponent is observed in . In an embodiment to which corresponds, the primary node of the subtree specifies the host or server that the component is running on. Recall from the discussion of that the response message may provide this information (i.e., component host ). Thus, in a further embodiment, the host or server that is identified by the component host value in the received response message for the scenario is identified as a node in the component's subtree.
803
803
5
7
FIGS. 4
b
In an embodiment, the technical name of the component is added as a prefix to the host name observed in the subtree at node . For example, if the technical name of the component is “SWP_Comp” and the host name is “Host3753”; then, the text next to node might would read “SWP_Comp Host3753”. The technical name for the component may be provided from the component name value 412, 512, 712 values found in the customizing file, request message and response message for the scenario (e.g., as respectively depicted in , and ).
FIG. 7
FIG. 7
721
Recall from the discussion of the response message with respect to that the GRMG application may also include in the response message the instance of the component . In an embodiment a unique node exists in the component's subtree for each unique instance of the component that runs on a particular host (recalling from the request message discussion concerning that component instances may derive from multiple or redundant software components on a single host within the IS infrastructure).
th
803
804
804
804
FIG. 8
Thus, for example, if three separate instances existed for the Xcomponent (and response messages that identified them where received by the GRMG infrastructure); then, two additional nodes would exist in the subtree beneath node other than node . In an embodiment, the technical name of the component and the name of the host is added as a prefix to the numeric identifier associated with the instance. For example, continuing with the example above where the component name is “Comp_SWP” and the host name is “Host3753”, if the reference number for the instance is “1” (which may be done for a single instance as depicted in ); then, the text next to node would read “SWP_Comp Host3753 1”. In an alternate embodiment, the status of the availability testing service for the instance is identified as either being “off” or “on” at tree node (e.g., “Run Status: Broadcast Messaging Server on”).
805
806
805
The availability and heartbeat nodes , present the actual monitoring results for the component. In an embodiment the percentage availability of the component (e.g., the number of OKAY responses normalized by the number of attempted availability tests for the component) is displayed next to the availability node . In an embodiment, the percentage availability corresponds to the percentage availability that has been demonstrated over a set period of elapsed time (e.g., the number of OKAY responses normalized by the number of attempted availability tests for the component that have occurred within the past fifteen minutes).
806
722
722
722
806
722
FIG. 7
FIG. 7
3
5
6
Actual messages pertaining to the component that were contained in received response messages can be displayed next to the heartbeat node . Recalling the discussion of the response messages that was provided above, and referred to , recall that items for conveying messages were specifically provided for. Here, for example, the actual textual message (e.g., “component is alive”) to which a numeric encoding was provided by way of the area, number (and perhaps number) fields -in a received response message for the component may be displayed next to the heartbeat node in the display. Likewise, a textual message received in a response message for a component (e.g., as contained in field of the response message embodiment of ) can also be displayed. In an alternative embodiment, a status indication is provided rather than message text.
221
723
723
FIG. 2
FIG. 7
FIG. 7
1
1
In a further embodiment, received messages are logged by being stored into a database (e.g., database of ). Here, a pushbutton may be provided for in a toolbar in the GUI to which the display pertains. When the heartbeat icon is marked and the pushbutton clicked on, the log of received messages for the component are displayed. Colors may also be used to convey which components are available and which components are unavailable. For example, in one embodiment, messages associated with a component deemed “OKAY” (e.g., as indicated in the alert field of the response message embodiment of ) are presented in green; and, messages associated with a component deemed “OKAY” (e.g., as indicated in the alert field of the response message embodiment of ) are presented in red.
722
2
FIG. 7
Also, the ordering of the display alert monitor tree may be affected by the severity of an error as expressed in a response message. For example, according to one approach, alerts given a higher degree of severity (e.g., as expressed in severity field in the response message embodiment of ) are moved upward in the alert monitor tree.
Error Tree
FIG. 9
FIG. 9
FIG. 9
901
901
shows an embodiment of an error monitor tree which indicates whether or not a scenario is working. According to the error tree embodiment of , a primary node is used to indicate that the GRMG system as a whole is being self monitored. That is, consistent with the preceding discussion that was provided immediately above, the alert monitor and error trees may be part of a larger integrated IS monitoring system that is responsible for displaying other “alerts” or problems with trees presented in a GUI display. Here, the error tree of is drawn as being a subtree of a larger tree structure (i.e., primary node has a tree structure to its left).
901
902
904
902
904
The primary node therefore is responsible for indicating to the viewer that its corresponding tree (e.g., with nodes through ) is for recognizing errors in the GRMG monitoring scheme as a whole. As such the “application name” associated with the primary node should be given a name that identifies the GRMG monitoring system as a whole (e.g., “GRMG Self Monitoring). The remaining subtree components are used simply to display error messages relating to specific scenarios. Thus, if a scenario is deemed non operational, text next to the scenario error node identifies a scenario that is experiencing an error and the text next to the heartbeat node displays a specific error message.
902
903
904
In an alternate embodiment that allows working and non working scenarios to be represented, the status as to whether or not a scenario is currently being tested is displayed at node (e.g., “Run Status: GRMG messaging service on), the availability node indicates the percentage of scenario testing schemes that have executed successfully for the scenario; and, the heartbeat node displays a specific error message.
903
In an embodiment, the underlying software for the alert monitoring tree and the error monitor tree overlap in structure so that a heartbeat node is displayed; however, no real use is made of the heartbeat node because only “unavailable” scenarios having error alert messages for represented for display.
Possible cause(s): a) the URL specified in the scenario customizing file points to a non-existent host or port (check that the URL is valid); b) the HTTP server specified in the URL is not running . . . start the HTTP server if it is not running . . . if the GRMG Application runs on the same server as the monitored components, then this error message also means that the tested components are no available.
1. Error Messages: “HTTP POST Failure: HTTP Communication Error”; “HTP POST Failure: Connect Failed”; or, “HTTP POST Failure: Timeout Occurred”.
Possible cause: the GRMG application is responding without reporting on the state of any of the components requested to be monitored . . . check that the components specified in the GRMG customizing file match those checked by the GRMG application.
2. Error Message: “Scenario Failure: No Response For Any Component in Request”
The following is pair of possible error messages that may be displayed for a non working scenario and their corresponding causes.
Embodiments of the invention may include various steps as set forth above. The steps may be embodied in machine-executable instructions which cause a general-purpose or special-purpose processor to perform certain steps. Alternatively, these steps may be performed by specific hardware components that contain hardwired logic for performing the steps, or by any combination of programmed computer components and custom hardware components.
Elements of the present invention may also be provided as a machine-readable medium for storing the machine-executable instructions. The machine-readable medium may include, but is not limited to, hard disk drives, flash memory, optical disks, CD-ROMs, DVD ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, propagation media or other type of machine-readable media suitable for storing electronic instructions. For example, the present invention may be downloaded as a computer program which may be transferred from a remote computer (e.g., a server) to a requesting computer (e.g., a client) by way of data signals embodied in a carrier wave or other propagation medium via a communication link (e.g., a modem or network connection).
Throughout the foregoing description, for the purposes of explanation, numerous specific details were set forth in order to provide a thorough understanding of the invention. It will be apparent, however, to one skilled in the art that the invention may be practiced without some of these specific details.
Accordingly, the scope and spirit of the invention should be judged in terms of the claims which follow.
FIGURES
The present invention is illustrated by way of example and not limitation in the figures of the accompanying drawings, in which like references indicate similar elements and in which:
FIG. 1
shows components of an exemplary enterprise information system;
FIG. 2
shows an Information Systems (IS) monitoring approach capable of monitoring the availability of different software applications;
FIG. 3
FIG. 3
shows a methodology that can be executed by the IS monitoring approach of ;
FIG. 4
FIG. 2
213
shows an embodiment of a customizing file such as customizing file of ;
FIG. 5
FIG. 2
a
209
shows a methodology that can be executed by the GRMG infrastructure of ;
FIG. 5
FIG. 2
b
211
shows an embodiment of a GRMG request message such as the GRMG request message of ;
FIG. 6
FIG. 2
a
210
shows a methodology that can be executed by the Generic Request and Message Generation (GRMG) application of ;
FIG. 6
FIG. 4
b
a;
shows a methodology for the monitoring of a component in scenario as initially depicted in
FIG. 6
c
shows an embodiment of an object oriented GRMG application flow;
FIG. 7
FIG. 2
212
shows an embodiment of a GRMG response message such as the GRMG response message of ;
FIG. 8
FIG. 2
216
shows an embodiment of an alert monitor tree structure that indicates scenario component availability that may be displayed on a display such as display of ;
FIG. 9
FIG. 2
9
216
shows s an embodiment of an error tree structure that indicates scenario availability that may be displayed on a display such as display of . | |
Rein, Martin and Höhler, Gebhard and Schütte, Andreas and Bergmann, Andreas and Löser, Thomas (2006) Ground-based simulation of complex maneuvers of a delta-wing aircraft. In: AIAA paper, AIAA Paper (2006-3149), 1--7. 24th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, 2006-06-05 - 2006-06-08, San Francisco.
Full text not available from this repository.
Official URL: http://pdf.aiaa.org/preview/CDReadyMATGT06_1185/PV2006_3149.pdf
Abstract
In the process of aircraft development numerical approaches are gaining more and more importance. Not only steady flight conditions need to be modelled but also dynamic derivatives and last but not least realistic flight maneuvers. In particular in the case of delta-wing aircrafts a small change in the flight conditions can strongly influence the vortex dominated flow field about the wings and thus result in large changes of the aerodynamic loads. Numerical tools that are developed for predicting such behaviours need to be validated by experimental data. In order to obtain a data base for validation ground-based simulations of complex maneuvers of a model of the X-31 aircraft have been performed in the low-speed wind tunnel NWB of the German-Dutch Wind Tunnels DNW. In the wind tunnel tests a newly installed novel test rig with six degree of freedoms (DOF) was used for the first time for moving the model. Furthermore, the model was equipped with eight remotely controlled moving flaps. In this manner realistic flight maneuvers could be reproduced in a ground-based facility. Both, the specific technical equipment of the model and the novel six DOF test rig will be reviewed. Thereafter, experimental results obtained will be discussed and compared with numerical results. The X-31 is a single-engine, single-place cockpit, delta-wing aircraft. For control the aircraft had a small, forward-mounted canard; single vertical tail with conventional rudder, wing leading flaps and trailing-edge flaps (elevons). A fully equipped wind tunnel model of the X-31, the so-called X-31 remote-control model, was developed and built to a scale of about 1/7.25 at the German Aerospace Center DLR (cf. Fig. 1). The model is made from steel and carbon fiber reinforced plastic. Its control surfaces can be moved via a remote control system. The main part of the X-31 model is a wing-fuselage section including eight servo motors for changing the angles of canard, leading-edge inner and outer flaps, trailing-edge flaps and rudder. Dynamic surface pressures are measured by miniature piezo-resistive pressure sensors located at 60% and 70% chord length on the upper surface of the delta wing and on the leading edge flaps. Forces and moments are obtained by a 6-component strain gauge also included in the main part of the model. Data are transferred back and force between the model and the external data acquisition system by a 64-channel telemetric system. The transfer rate of the telemetric system is about 3 kHz. The novel configuration of the six DOF dynamic test rig used in the present tests for simulating real flight maneuvers was developed at DNW for its low-speed wind tunnel NWB located in Braunschweig and is called a “Model Positioning Mechanism” (MPM) hereafter. The MPM is based on the concept of a Stewart platform. This platform is linked to the wind tunnel fixed base by six constant-length struts that are connected to six carriages which can move along two parallel guiding rails so that the position and orientation of the platform is adjusted (cf. Fig. 1). The six carriages run independently of each other on the guiding rails thus allowing a displacement within all six degrees of freedom. Because each guiding rail is shared by three carriages, the design is simplified and has fewer components than conventional systems. The six linear motors used for moving the carriages allow accelerations up to 2.5 g. The workspace spans 1100 mm in the flow direction, 300 mm in the lateral direction and 500 mm in the heave direction. The range of pitching or rolling motions can be enlarged by an additional actuator on the MPM and a corresponding joint between the ventral sting and the internal balance. The accuracy of the system, for example, in pivoting angles is better than 0.005°. At the top of the sting the first eigenfrequency is above 20 Hz. The MPM allows for a payload of up to 5000 N. The complex three-dimensional motion of the model is controlled by an optical position tracking system. All measurements were performed in the open test section of the 2.85 X 3.20 m2 low speed atmospheric wind-tunnel NWB of DNW. The X-31 remote-control model was connected to the MPM by a belly sting (cf. Fig. 1). Already during the commissioning phase of the MPM complex maneuvers were successfully simulated. An example based on a real flight maneuver corresponding with steady-heading sideslip test points is shown in Fig. 2. On the left side, the variation in time of the angles of pitch, yaw and roll that were performed by the MPM and the corresponding motions of the flaps that were realized by the remotely controlled servo motors in the model, are shown. As an example, the coefficients of the lateral force and the rolling moment resulting from this maneuver are shown on the right side of Fig. 2. In the same manner many more scientific data have been gathered that are to be used for validating a numerical simulation framework that is under development at DLR for calculating a freely flying maneuvering combat aircraft. In the numerical approach a maneuver is realized by a time-accurate coupling of aerodynamics, structural mechanics and flight mechanics. Results of first numerical simulations will be compared with experimental data obtained under unsteady flow conditions. Thus, a ground-based simulation method has been successfully developed and tested that provides the possibility of simulating complex maneuvers in a subsonic facility. | https://elib.dlr.de/21774/ |
What is IRR and how do you calculate it?
The ability to evaluate the profitability of your investments can play an important role in your company’s long-term planning. Many businesses and investors utilise IRR to measure return on potential investments, allowing you to compare and rank different projects based on their projected yield. Find out everything you need to know about the IRR formula – including how to calculate IRR – with our simple guide. First off, what is IRR?
IRR meaning
IRR stands for internal rate of return. It measures your rate of return on a project or investment while excluding external factors. It can be used to estimate the profitability of investments, similar to accounting rate of return (ARR). Generally, a high IRR is preferable to a low IRR, as it signals that a potential project or investment is likely to add value to your business. If you’re using IRR to rank prospective projects, the investment with the highest IRR is probably the one that should be undertaken first (assuming the cost of investment for each project is equal).
How to calculate IRR
Understanding how to calculate IRR can be a challenge, as the IRR formula is a little more complex than many other financial metrics. Here’s the IRR formula you can use in your calculations:
0 = NPV = t∑t=1 Ct/(1+IRR)t − C0
Where:
Ct = Net cash inflow during period t
C0 = Initial investment cost
IRR = Internal rate of return
t = Number of time periods
That may look a little complex, so let’s break it down. As you can see, the IRR formula equates the net present value (NPV) of future cash flows to zero. In other words, if you calculate the NPV from a potential project and use IRR as the discount rate, subtracting out the original investment, the NPV of the project would equate to zero.
While you can learn how to calculate IRR by hand, it’s important to note that it’s a complex method based on trial and error. This is because you’re trying to work out the rate that makes the NPV equal zero. You may be better served by using Microsoft Excel or other types of business software to complete your calculations.
How to use IRR
The IRR method is often used by businesses to determine which project or investment is worth funding. For example, if you’re trying to work out whether to purchase a new piece of equipment or invest in a new product line, IRR can help you understand the option that’s most likely to yield a healthy rate of return. Although the actual rate of return is likely to differ significantly from the estimated IRR, projects which have a much higher IRR than competing options are likely to offer better value.
There are several different scenarios where the IRR method is particularly useful. If you’re comparing the profitability of expanding existing operations with establishing new operations, your company could use the IRR calculation formula to decide which is the more profitable option. Furthermore, IRR can be helpful for companies considering a stock buyback program; if the company’s stock has a lower IRR than other potential projects, a stock buyback may not be the best idea.
Limitations of IRR
Although IRR can be an excellent tool for estimating the profitability of future projects or investments, it can be a little misleading if you use it on its own.
Projects with a low IRR may have high NPV, indicating that although the rate of return may be slower than other projects, the investment itself is likely to yield significant value for your business. By the same token, IRR may not be the best tool for evaluating projects of different lengths.
Furthermore, IRR assumes that the positive cash flows associated with an investment will be reinvested at the project’s rate of return. This may not be the case, and as a result, the IRR method may not be the most accurate reflection of a project’s cost and profitability.
Overall, the IRR calculation formula is a valuable metric, but it’s important not to place too much weight on it when making your final decision. There’s another formula called the modified internal rate of return (MIRR) that corrects these issues and may be worth investigating if you’re considering using the IRR formula to evaluate projects.
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FreeTimeCalculator.CalculateFreeTime(TimeInterval, Resource) Method
Finds all time intervals that are not in use for the specified resource.
Namespace: DevExpress.XtraScheduler.Tools
Assembly: DevExpress.XtraScheduler.v18.2.Core.dll
Declaration
public TimeIntervalCollection CalculateFreeTime( TimeInterval interval, Resource resource )
Public Function CalculateFreeTime( interval As TimeInterval, resource As Resource ) As TimeIntervalCollection
Parameters
|Name||Type||Description|
|interval||TimeInterval||
|
A TimeInterval object, representing the time period in which the search is performed.
|resource||Resource||
|
A Resource object, which specifies a particular resource to narrow a search.
Returns
|Type||Description|
|TimeIntervalCollection||
|
A TimeIntervalCollection collection, containing all spare time intervals.
Remarks
The CalculateFreeTime method performs a search within the specified interval for the specified resource only.
The method returns a collection of intervals or the empty collection, if free intervals are not found. | https://docs.devexpress.com/CoreLibraries/DevExpress.XtraScheduler.Tools.FreeTimeCalculator.CalculateFreeTime(DevExpress.XtraScheduler.TimeInterval-DevExpress.XtraScheduler.Resource)?v=18.2 |
Last Wednesday just before 6am, I woke up to a strange light coming through the slit between my curtains. Later, I’d determine the light’s three sources: a firetruck, a firefighter’s high-output flashlight, and the flames shooting up from the townhouse in front of mine, 20 steps away. Heading downstairs, I smelled smoke, and, more subtly, cedar. Just seven hours ago, I’d finished building a cedar chest. The garage was still covered in cedar. Cedar shavings on the miter saw, cedar dust around the orbital sander, cedar off-cuts in a cardboard box I’d labeled “Cedar only!”. The cedar chest had been the last thing I’d worked on before going to bed, and now, even as I exited my apartment with all my valuables in a backpack, it was still on my mind. I looked back at the chest one more time and imagined the cedar lid catching sparks.
That was a week ago. To make sense of how both the cedar chest and the fire fit into the van build’s story, I’m going back even further. Weeks ago, I wrote that one of the things I planned to do next was connect the solar battery to the solar panels. That didn’t happen. It’s impractical to install a 120-pound battery and its associated wiring without first building the battery’s permanent housing. I plan to store the solar electronics in a compartment underneath the bed. So first I should build the bed. But before I build the bed, I should install the floor that goes under the bed. And before I install the floor, I should put up the walls and ceiling, so as not to scuff up the new floor while fiddling with the ceiling. Sometimes the hardest part of a build is determining the order of operations, but I think I’ve got it now: Ceiling’s next.
Even before I started looking for a van, I knew exactly what I’d use for the ceiling: cedar planks, with which I have only positive associations—saunas, rich-people closets, the Schitt’s Creek cedar chest episode. (I also strongly considered beetle kill pine but couldn’t find it in planks that were thin enough. I’ll use beetle kill wood for my furniture instead.) Various forum and blog posts recommended buying cheap cedar at Menards, a discount home improvement store that has a location in southern Wyoming, about an hour from me.
I don’t want to like Wyoming (the state voted 67% for Trump; still doesn’t have laws against hate crimes; does not accept refugees), but, the truth is, I’ll take any excuse to go there. I spent most of my childhood in a busy part of Pennsylvania where, no matter which direction you drove or for how long, you were basically always within walking distance of a Wawa. All I ever wanted was to get out of there, and now I understand why: the only time I feel 100% sane is when I’m able to make total sense of my surroundings. Ideally, my immediate environment should contain no more than one point of interest, and that point should be unmoving or (worst case scenario) moving slowly. Dullness—that’s something Wyoming delivers. Even in the capital city, Cheyenne, the only bright block of color is the red Wrangler building. Often, the only sound is the cargo train. There’s always a guy waiting to cross the train tracks, but never more than the one guy. The only coffee worth drinking is at Paramount Cafe, which used to be a theater with a sloping tiled floor. The best art I’ve seen in Cheyenne is that floor, with its decades of boot grime caught between its chipped tiles. (I’ve never been to Cheyenne’s art galleries, because every time I’ve visited, they’ve been closed.) I am not the only coffee shop patron who stares at that floor; I’ve caught others doing it too, including one greyhound who promptly fell asleep. On one of the highways leading out of Cheyenne, there’s a tree growing out of a crack in a rock. The tree is fenced off and marked “Tree Rock.” People stop there.
So two weekends ago, I happily visited Cheyenne and bought nine boxes of cedar planks at Menards.
Let me tell you about those planks. They are red and white. They smell as spicey as the promise of homemade dinner. And they sparkle! The thin layer of sap coating their faces actually glitters like fresh snow. You know the calm magic that Lucy felt at the doorstep to Narnia? Before the lion, before the witch? When she walked through the wardrobe and suddenly found herself “standing in the middle of a wood at night-time with snow under her feet and snowflakes falling through the air?” That’s basically what I felt as I unboxed the planks in my garage the evening after I got back from Menards. I meant to just take a quick peek at the wood before making dinner but ended up running my hand along each of the 108 planks. Suddenly, it was midnight.
I couldn’t wait to work the wood. I decided that building a quick cedar chest never hurt anyone. To be clear, this chest wouldn’t be for the van (it would be too big; I’d have to put it in my living room), but my van could spare a few planks. Anyway, I needed the woodworking practice. When was the last time I’d used a miter saw, high school woodshop? It would be best to get all my mistakes out of the way on a project that mattered even less than the van. I can rationalize anything when I want to.
I built the cedar chest with the worst of the planks—the curvy, cracking, or brownish ones. My design was intentionally simple, because I like my furniture how I like my literature—underwrought. Add some trim and decorative handles, and the storage chest begins to look like a box for laying in and nailing shut (at least to me). I used a miter saw to cut the wood, a pocket hole jig to screw the frame together, and wire brads to nail the cedar planks to the frame. Then I sanded the whole thing.
Here’s what I learned:
- Start with the right materials. These planks are gorgeous, but they’re too thin and brittle for a chest. To give the chest integrity, I added an internal frame. That’s why this chest looks more like a crate.
- Always plan ahead. I did not work off a project plan because I just felt like winging it. This created two problems. Problem one, I neglected to account for the thickness of the planks in several key places. For example, I made the width of the frame equal to the width of exactly five planks, forgetting to consider that the end plank would have to extend a bit past the edge of the frame, so as to form a nice corner with the plank meeting it from the perpendicular side. The photo below might make it easier to visualize the problem. To solve this issue, I ended up spacing the planks a bit further apart than I originally intended. (This actually turned out to be a good thing, giving the planks room to expand in higher humidity, but I didn’t realize this benefit at the time.) Problem two, the chest is oddly proportioned. Doesn’t it look a bit too tall? I chose this height for practical purposes—it’s what you get when you cut a cedar plank in half. I should have remembered that aesthetical considerations are practical, too; if I’m bothered by how something looks, I’m less likely to use it. To solve both of these problems, I could have modeled the chest in a program like Sketchup before starting my build.
- Simpler is harder. Now I get why furniture looks the way it does. There’s a reason carpenters started edging furniture with trim—to give them more room for error. Since I wanted to avoid trim, I had to make sure that every one of the chest’s 68 planks lined up precisely with all of its neighbors. What a pain, especially when you consider that the planks I used for this project were the worst of the bunch, many of them curving in on themselves. In a vain attempt to create order, I spent hours sanding the chest, repositioning planks, sanding again. When nailing the planks to the frame, I fussed with the positioning of the brads, trying to make them equidistant—a last-ditch effort to create the illusion of discipline.
- Pocket hole jigs are great, but not for me. A pocket hole jig helps make strong joints by guiding screws into wood at just the right angle. Many van builders swear by the jig because it makes woodworking fast and easy. But the jig has its limitations. It’s useable only on wood .5″-1.5″ thick and at least 1.25″ (preferably 1.5″) wide. I also don’t like how basic the resulting joints look. Because I have the privilege of time, I’ve decided to learn how to make pretty finger and twisted dovetail joints using the CNC at my local maker space before moving on to building my van’s furniture.
So last Tuesday night I finished building the chest, reflected on what I’d learned (i.e., that I wanted to learn to work wood better), and went to sleep. Seven hours later, I woke up to that fire. The fire ended tragically, though not for me. The burn was contained to the townhouse where it started and its two immediate neighbors. It took eight hours for firefighters to completely extinguish it. One person and two cats died.
This tragedy has nothing to do with me except proximity, though that seems to be enough to keep me thinking about it a week later. Strangely, the police still haven’t released the victim’s identity. For days, they cordoned off our neighborhood with police tape, though I could still wander within the tape’s boundaries, since my front door was within it. While sorting my mail and languorously tying my shoe, I eavesdropped on the police investigators who talked loudly to my neighbors and each other until they noticed me and lowered their voices.
In the absence of information, I’m spinning stories from the few things I overheard. I have no hard evidence to support the hypothesis that the victim of the fire was the man who used to work in his garage long into the night, whose project car is the only car still parked on the property (maybe it just doesn’t run). Six other people lived in that townhouse, too. When the police asked neighbors what the man was like, the neighbors said “loud” and “a piece of work.” The man and I never said anything interesting to each other, though we sometimes ran into each other at night, when we were two of the only people still up. One night not long ago, I woke up on my living floor, having accidentally fallen asleep there, and realized I’d forgotten to water the tomato plants. Stumbling out onto the porch with a watering can, I was not surprised to find the man’s silhouette against his lit garage. 2am, and there he still was, banging two hunks of metal together. He never annoyed me much, largely because his banging drowned out my power tools, making my noise more palatable to the other neighbors. Also, I felt I understood why he was out there. Most likely, he hadn’t meant to stay out so late. There are days you intend to make dinner, and instead spend the night arranging cedar planks, desperately seeking the right configuration, until you find it: two planks that, placed side by side, form a yin-yang. Momentarily, everything feels alright, as perfectly settled as a Wyoming Sunday afternoon. Finally, you can go back upstairs and eat a banana.
I don’t know what you do after witnessing a tragedy that’s not yours. I guess one thing people do is acknowledge their random good luck, and the victim’s shit hand. When I looked back at the chest as I exited the apartment last Wednesday morning, I tried to feel devastated about the thought of losing something I’d put so much effort into; it seemed to be what the moment required. Really, I felt okay—I already knew I’d build a better chest one day, had already placed “The Art of Japanese Joinery” on hold at the library, was already positioned a safe distance from the fire, already looked forward to the nights I’d spend alone in the garage. | http://bunyovan.com/index.php/2018/11/01/about-a-cedar-chest-and-a-house-fire-wait-what-about-the-van/ |
How do I create a peer review assignment?
When creating an assignment, you can require students to complete a peer review of another student's work. Learn more about peer review assignments.
For peer reviews, you can manually assign peer reviews or choose to have Canvas automatically assign peer reviews for you. You can also choose to allow students to see other students' names in peer reviews or make them anonymous. When anonymous peer reviews are enabled, instructors and TAs can still view the names of student reviewers in SpeedGrader and in the student submission page. However, if anonymous grading is enabled in SpeedGrader, the names of both students will be hidden in SpeedGrader but not in the student submission page.
To complete the peer review, students are required to leave at least one comment. If you include a rubric, they are only required to complete the rubric.
To learn how assignment and peer review due dates appear in a student's To Do list, view the Peer Review Tips PDF.
Peer reviews cannot be used with External Tool assignments.
Students can see peer review comments when an assignment is muted. However, students cannot see instructor comments until after the assignment is unmuted.
Enter a name and description for your assignment, as well as any other assignment details .
Note: The Rich Content Editor includes a word count display below the bottom right corner of the text box.
In the Submission Type drop-down menu , select your preferred submission type .
Note: The External Tool submission type does not support peer review assignments.
Select the checkboxes for the types of online entries allowed in the assignment.
Peer reviews can be used with group assignments. If you want to create a group assignment, click the This is a Group Assignment checkbox.
Click the Require Peer Reviews checkbox . By default, peer reviews are assigned manually .
If you want to assign peer reviews automatically, select the Automatically Assign radio button .
Note: Peer reviews must be manually assigned for On Paper and No Submission assignment types.
If you automatically assign peer reviews, the menu displays additional options. In the Reviews Per User field , enter the number of reviews each student will be required to complete.
In the Assign Reviews field , use the calendar icon to select a date or manually enter the date for student peer reviews to be assigned. The Assign date must come on or after the assignment due date. If left blank, Canvas will use the assignment due date.
In group assignments, you also have the option to allow intra-group peer reviews.
Peer reviews require a student to review an individual submission by another student. However, group assignment submissions are made by one group member on behalf of the entire group, and all group members have the same submission.
By default, the Allow intra-group peer reviews checkbox is not selected, which means Canvas will filter out members of the same group when automatically assigning the reviews. However, selecting the checkbox allows assignments to be truly random and disregard student group associations.
To allow Canvas to automatically assign a peer review to a student from within the student's own group, select the Allow intra-group peer review checkbox.
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Table of Contents > Assignments > How do I create a peer review assignment?
Re: Peer Response on Canvas?
Re: Is it possible to set up a rubric that students can fill out to evaluate each other's speeches? | https://community.canvaslms.com/docs/DOC-10094-415254249 |
3. The immune system is a complex network of cells known as immune cells that include Lymphocytes.
4. Lymphatic system, network of vessels and other tissues, including the tonsils, spleen, and thymus, that maintains fluid balance and fights infection
5. Extracellular fluid in the Lymphatic system is known as Lymph
6. Lymph contains disease-fighting cells called Lymphocytes, which are supplied by the Lymphatic system.
7. The Lymphatic system consists of Lymph vessels, ducts, nodes, and other tissues
8. The Lymphs blood test is part of a complete blood count [CBC]
9. Lymphocytes are one of the white blood …
10. Low Lymph percent at blood test result high white blood cell count Thyroid ultrasound shows prominent Lymph nodes in cervical chain Tests Results for: Lymph% 45.4
11. What does that mean? understanding bloodwork results Thyroid blood test What is the best treatment for swollen Lymph nodes in …
12. Lymph is the fluid that flows throughout the Lymphatic system; it is made up of different components as it goes through different parts of the body
13. Regardless of the content of the Lymph at any given time, it must flow freely to ensure that waste products do not build
14. Lymphocytopenia is a common diagnosis from a complete blood count test
15. Lymph is a protein-rich fluid that moves throughout your body in Lymph vessels
16. It scoops up things like bacteria, viruses, and waste, and carries them to your Lymph nodes
17. The Lymph system is a network of organs, Lymph nodes, Lymph ducts, and Lymph vessels that make and move Lymph from tissues to the bloodstream
18. The Lymph system is a major part of the body's immune system.
19. Lymph: Lymph, also called Lymphatic fluid, is a collection of the extra fluid that drains from cells and tissues (that is not reabsorbed into the capillaries) plus other substances
20. The Lymphatic system, or Lymphoid system, is an organ system in vertebrates that is part of the circulatory system and the immune system.It is made up of a large network of Lymph, Lymphatic vessels, Lymph nodes, Lymphatic or Lymphoid organs, and Lymphoid tissues
21. The vessels carry a clear fluid called Lymph (the Latin word Lympha refers to the deity of fresh water, "Lympha") towards the heart.
22. When Lymph nodes become infected, it's usually because an infection started somewhere else in your body
23. Rarely, Lymph nodes can enlarge due to cancer
24. You have about 600 Lymph nodes in your body, but normal Lymph nodes may only be felt below your jaw, under your arms, and in your groin area
25. A normal Lymph node is small and firm.
26. What is the Lymph System? Your Lymphatic, or Lymph system is a critical part of your immune system that is essential for protecting you from inflammation and illness
27. The main role of your Lymph system is to keep your fluid levels in balance while protecting you from infections, bacteria, cancers, and other potential threats.
28. What Are Normal Lymph Levels? Normal Lymphocyte levels can vary based on age, race, gender and state of health
29. Generally speaking, however, normal Lymph ranges fall between 1,000 and 4,800 Lymphocytes in 1 microliter
30. This varies with children, and normal levels for children are between 3,000 and 9,500 Lymphocytes in 1 µL of blood.
31. Non-Hodgkin Lymphoma can cause Lymph nodes to become enlarged
32. Enlarged Lymph nodes close to the surface of the body (such as on the sides of the neck, in the groin or underarm areas, or above the collar bone), may be seen or felt as lumps under the skin
33. Lymph in the Lymph vessels eventually reaches a Lymph node-- there are about 100 nodes scattered throughout the body
34. Lymph nodes filter the Lymph and also contain large numbers of white blood cells (a big part of the immune system ), which remove foreign cells and debris from the Lymph.
35. The Lymph then flows through the Lymph vessels into the Lymph glands, which filter out any bacteria and damaged cells
36. From the Lymph glands, the Lymph moves into larger Lymphatic vessels that join up
37. These eventually reach a very large Lymph vessel at the base of the neck called the thoracic duct.
38. A Lymph node biopsy is a procedure used to remove a sample of tissue to be tested
39. Healthcare providers may remove Lymph cells through a needle or remove one or more Lymph nodes during surgery
40. How is Lymphadenopathy treated? Your symptoms may go away without treatment.
41. Lymph, pale fluid that bathes the tissues of an organism, maintaining fluid balance, and removes bacteria from tissues; it enters the blood system by way of Lymphatic channels and ducts
42. Prominent among the constituents of Lymph are Lymphocytes and macrophages, the primary cells of the immune
43. Swollen Lymph nodes are much more likely to be caused by infections or a disease that affects your immune system
44. The Lymph gets filtered at the Lymph nodes
45. The spleen, tonsils, adenoids and the thymus all forms a part of the Lymphatic system
46. The spleen is considered as the largest Lymphatic organ in the system, which is located under the ribcage, above the stomach, and exactly in …
47. Lymphatic: [adjective] of, relating to, or produced by Lymph, Lymphoid tissue, or Lymphocytes
48. Lymph is the fluid in your Lymphatic system
49. Lymph definition, a clear yellowish, slightly alkaline, coagulable fluid, containing white blood cells in a liquid resembling blood plasma, that is derived from the tissues of the body and conveyed to the bloodstream by the Lymphatic vessels
50. Lymphoma is a group of blood malignancies that develop from Lymphocytes (a type of white blood cell)
51. Signs and symptoms may include enlarged Lymph nodes, fever, drenching sweats, unintended weight loss, itching, and constantly feeling tired
52. The enlarged Lymph nodes are usually painless.
53. In Moderna’s study, 11.6 percent of patients reported swollen Lymph …
54. Lymph nodes are part of the immune system
55. During a physical exam, your doctor may find that certain Lymph nodes are swollen
56. In Lymphocytopenia, the Lymph nodes may hold on to too many Lymphocytes instead of releasing them into the bloodstream
57. To test a Lymph node, you may need to have it removed.
58. The Lymph glands, also known as Lymph nodes, are mostly on the side of the neck
59. The Lymph glands are the body’s sophisticated sewer system
60. And so your Lymph coming from the liver has a much higher concentration of protein than your Lymph coming from elsewhere
61. You can have 10 times as much protein in the Lymph coming from the liver compared to other places.
62. If you've ever had a surgery on your Lymph nodes, your doctor may have suggested Lymphatic drainage massage
63. Lymph is another type of circulatory fluid of the animal body, it flows through the Lymphatic system, which consists of Lymph nodes, Lymph vessels
64. I.e., the fluid that lies in the interstitial spaces of all body tissues, which is collected through Lymph capillaries.
65. Swollen Lymph nodes in the neck can appear as small as a pea or as large as a cherry
66. Regular massage sessions can help reduce swollen Lymph nodes and swelling
67. This will stimulate the Lymph nodes to improve Lymphatic function.
68. 6 hours ago · A Lymph node collection kit can help surgeons attain compete resection and improve long-term survival after curative-intent lung cancer surgery, according to a study published in the Journal of
69. Lymph nodes contain Lymphocytes, which is a type of WBC
70. At the time of infection, in response to the antigen, Lymphocytes make antibodies which target the pathogens and destroy them
71. The fluid transported by Lymphatic vessels is called Lymph
72. Lymph is a clear fluid that comes from blood plasma that exits blood vessels at capillary beds
73. Lymph vessels collect and filter this fluid before directing it …
74. Lymph is normally a clear colourless fluid, but Lymph draining the intestine during absorption is often milky in appearance because of its high lipid content
75. The Lymphatic capillaries merge to form thicker-walled vessels which resemble venules and medium-sized veins.
76. Lymph drains away from your breasts through the Lymphatic system, which is made up of Lymphatic channels and Lymph nodes
77. The vein is a one-lane highway taking away the used blood, and the Lymphatic channels are the other one-lane highway
78. Properties of Lymph: Lymph should be regarded as modified tissue fluid
79. Lymph is the clear watery-appearing fluid found in Lymphatic vessels and is formed by the passage of substances from blood capillaries into tissue spaces
80. Lymph is the fluid that circulates throughout the Lymphatic system
81. It is formed when the interstitial fluid is collected through Lymph capillaries
82. It is then transported through larger Lymphatic vessels to Lymph nodes, where it is cleaned by Lymphocytes before emptying ultimately into the right or left subclavian vein, where it mixes back
83. Lymph nodes play an important role in cancer staging, which determines the extent of cancer in the body
84. One of the most commonly used systems for staging cancer is the TNM system, which is based on the extent of the tumor (T), the extent of spread to the Lymph nodes (N), …
Lymph is a clear fluid containing white blood cells (WBCs) and dead and diseased tissue for disposal. The primary function of lymph nodes is to harbor the body’s disease-fighting cells and to filter lymph before it reenters circulation.
Lymph Composition. Lymph contains a variety of substances, including proteins, salts, glucose, fats, water, and white blood cells. Unlike your blood, lymph does not normally contain any red blood cells.
The main difference between lymph node and spleen is that a lymph node is the small mass of tissues located along the lymphatic vessels whereas spleen is an organ found in vertebrates, which is structurally similar to a large lymph node. Furthermore, lymph nodes filter pathogens in the lymph while the spleen filters the pathogens in the blood.
Whereas lymph is a colorless liquid , found mostly in the inter-cellular spaces of a tissue. Blood has RBC's, WBC's, platelets and a fluid called plasma. Whereas lymph has WBC's and watery fluid. They both have immune and also circulatory functions in them. | https://useenglishwords.com/lymph/ |
The term of employment is limited to months.
Location: United States of America : Osceola, AR
Function: Engineering
Career Level: Experienced professionals (2-5 years)
Legal Entity: Röhm America LLC
Business Line: Operations Management Methacrylates
WHAT WE OFFER
Roehm is a market leader in methacrylate chemicals (found in polymer plastics), we are working on exciting topics and projects. At Roehm, our strength comes from our employees. Their ideas, creativity, and passion are crucial for the further development of our business. Are you ready to become a part of that?
By becoming part of our team, we offer opportunities and perspectives in an international environment – both for career changers and experienced professionals. Waiting for you is an exciting and creative role within a team, on-the-job training, development opportunities as well as attractive compensation and benefits!
RESPONSIBILITIES
Our Project Engineer is to provide engineering expertise and support for the Manufacturing facility; coordinate improvements of process safety and environmental protection, through daily review of operations, continuous improvement in product quality and reduction of off-grade; and facilitate installation and modification of equipment, documentation of the process, and training of the workforce.
Responsibilities at a high level will include but not limited to:
- Work closely with maintenance and project engineering to coordinate the installation of engineering projects and assist in timely resolution of maintenance problems.
- Maintain safety assurance through a continuous safety training and awareness program, maintenance of safety procedures, and safety contact with workers.
- Provide technical support to operators in manufacturing departments as needed, to maintain maximum efficiency of personnel and equipment in a safe manner.
- Develop, coordinate, and run reports on long-term and short-term product and process development schedules as part of an ongoing program to improve quality, cost, and productivity such that budgeted goals and planned objectives are met.
- Coordinate investigations, HAZOPS, and report on process upsets and product problems utilizing guidance from all operating personnel and support groups as necessary to improve systems and procedures thereby preventing reoccurrence of the same or similar events.
- Act as team facilitator in the manufacturing departments to cultivate and encourage the team and individual participation in meeting plant quality, safety, and productivity goals.
- Develop and implement capital projects to enhance safety, improve quality and throughput, reduce manufacturing costs, and improve methods of operation.
- Review operations daily with manufacturing, resolve problems as a team and provide off-shift support when necessary.
- Possess a solid understanding of the maintenance and purchasing functions in the SAP process.
- Complete other duties as assigned
REQUIREMENTS
- Possess a Bachelor's in Engineering or possess a degree within a related field
- Possess 5 years or more of hands on experience within a manufacturing environment
- High level of interpersonal and writing skills with the ability to work successfully in a team environment
- Self-motived individual, adept at balancing multiple tasks or projects including managing competing priorities and successfully meeting deadlines
- Adept at using standard Microsoft Office programs and tools, especially Excel, AutoCAD, and SAP
YOUR APPLICATION
To ensure the fastest process of your application and to protect the environment, please apply online via our careers portal at https://www.roehm.com/en/career.
Please address your application to Sissly Harris. If you have any questions regarding the application process, please contact us.
VACANCY REFERENCE NUMBER 117667
Please note that Röhm will not accept any unsolicited application documents sent by staffing firms. Röhm works in conjunction with preferred service providers and will not pay any fee to staffing firms in the absence of an appropriate framework agreement. Should Röhm receive a candidate profile from a staffing firm with which it has no framework agreement, and should this candidate subsequently be considered in the recruitment process or offered employment, no claims from the staffing firm will be entertained in this regard. | https://jobs.roehm.com/job/Osceola%2C-AR-Project-Engineer-AR-72370/617882501/ |
Ultra Music Festival, considered by many EDM fans as one of the best electronic music festivals in the world, might not occur this year. This is due to concerns regarding the coronavirus pandemic that is currently affecting the world. A meeting between Ultra representatives and elected Miami officials on Wednesday morning discussed the possibility of postponing the festival. Miami-Dade County Mayor Carlos A. Giménez addressed yesterday on Twitter that the county will not cancel any major events such as Ultra Music Festival. According to the Miami Herald, the festival may postpone for a full year if all parties agreed to postpone it.
Reasons behind the possible cancellation
The weekend of Mar. 20-Mar. 22, 2020 is when the festival will kick off. Ultra Music Festival is returning to Bayfront Park this year after Virginia Key became the host of last year’s festival. For the past couple months, the novel coronavirus outbreak, also known as COVID-19, has influenced people and organizations to be very cautious when maintaining their hygiene and health as the virus continues to spread. Large gatherings of people at events such as sports games and music festivals tend to increase probability of individuals getting contagious diseases. Aside from Ultra Music Festival, there are rumors that EDC Vegas could possibly get postponed as well.
What to anticipate next for Ultra Music Festival
Before the news of the cancellation broke out, the fans couldn’t wait to see the headliners at the festival. Among the headliners at this year’s UMF are Zedd, Martin Garrix, David Guetta, Afrojack, Gryffin, and Armin Van Buuren. Due to the news surrounding the potential cancellation, Martin Garrix announced that he has pulled out of the festival. Likewise, David Guetta followed him as well. Shortly after the meeting, Mayor Francis Suarez and Commissioner Joe Carollo told reporters that there will be an official announcement on Friday morning. The City of Miami and Ultra will make their joint announcement at 9 am EST on Friday morning. | https://oneedm.com/edm-news/possible-cancellation-of-ultra-music-festival-due-to-coronavirus/ |
INCORPORATION BY REFERENCE TO ANY PRIORITY APPLICATIONS
This application is a continuation of U.S. application Ser. No. 16/450,893, filed on Jun. 24, 2019, which is a continuation-in-part of U.S. application Ser. No. 15/898,210, now U.S. patent Ser. No. 10/327,526, filed on Feb. 15, 2018, which claims the benefit of U.S. provisional U.S. Application No. 62/459,950, filed on Feb. 16, 2017, all of which are incorporated herein in their entirety by reference.
Any and all applications for which a foreign or domestic priority claim is identified in the Application Data Sheet as filed with the present application are hereby incorporated by reference under 37 CFR 1.57.
BACKGROUND OF THE INVENTION
Field of the Invention
Background of the Invention
The present invention relates in general to carrying devices with at least one built-in embedded security unit for monitoring the contents therein. More particularly, the invention relates to carrying devices with a security system for real time and security monitoring of belongings therein.
Everyone wonders what happens to their luggage when their bags are checked at the airport. Due to security regulations, the Transportation Security Administration (“TSA”) must be able to open and inspect your luggage, which most times is not done in your presence and items can be misplaced or stolen. It is also possible that baggage handlers can go through your luggage too and steal your valuables or mistreat your suitcase by throwing it across the room. Even if you had a lock on your carry bag a thief can easily break the locks.
Another big problem when traveling is missing luggage. It is an inconvenient event to have arrived at a new travel destination and your suitcase to be lost in transit.
U.S. Publication No. 2015/0136552 to Mercado discloses a luggage tracking and surveillance system device having a replaceable zipper pull and interior retaining fastening member. The replaceable zipper pull has a touch sensor, power system, GPS and system for communicating with the retaining fastener. The fastening member contains a camera, GPS, touch sensor, power system and system for communicating with the zipper pull.
U.S. Pat. No. 8,964,037 to Petricoin Jr. discloses a battery-powered camera mounted within the interior of a suitcase. The security system includes a switch center, camera, controller for storing data, motion detector, touch sensor, motion sensor and a weight sensor at various places inside the surface of the suitcase.
U.S. Patent Publication No. 2015/0337565 to DiBella et al. is a security device which can be added to the luggage.
It is known in the prior art that there are luggage pieces with cameras, but most of these cameras are removable and don't allow the user to view their luggage and contents at all times and keep constant watch over them.
The invention solves these problems by providing carrying devices with a built-in security system. The system includes a security unit having all components housed in an integral unit embedded in the carrying device. The components include a camera with a light sensor, a speed sensor, a distance sensor, a GPS tracker, a microcontroller, a transponder and a rechargeable power supply to provide real time images to a user of the status of their carrying device and the contents therein.
The purpose of invention device is to protect people's belongings from theft and keep their belongings safe. Users of the carrying devices, which include carry bags such as luggage, handbags, backpacks, briefcases, golf bags and the like, with the built-in security system, can view the cameras at any time from any personal digital assistant or computer to view footage to see who has opened their luggage or other carrying device and taken their belongings.
A purpose of the invention is that it will be a deterrent to theft and make a thief think twice before opening your bag.
Another purpose of the invention is for baggage handlers to handle your bags with more caution knowing they are being viewed.
Another purpose of the invention is to track your bag and/or contents to monitor if any items are stolen.
Yet another purpose of the invention is to provide the user with a monitoring system and alarm system to alert the user when their carry bag is open and to permit them to see the individual who opened the carry bag.
Another purpose is to provide a musical instrument case with a built-in security unit both inside and outside the case to monitor the location and status of the instrument stored inside.
Another purpose of the invention is to provide a backpack with a built-in security unit both inside and outside the backpack to monitor the location and status of the backpack and contents therein.
Yet another purpose of the invention is to provide a backpack with a built-in security unit both inside and outside the backpack, where the outer security unit aids in bully prevention since it provides the user with images from behind when the backpack is on their back.
Another purpose of the invention is to provide a briefcase with a built-in security unit both inside and outside the briefcase to monitor the location and status of the briefcase and contents therein.
Another purpose of the invention is to provide a golf bag with a built-in security unit both inside and outside the briefcase to monitor the location and status of the golf bag and contents therein.
SUMMARY OF THE INVENTION
In the present invention, these purposes, as well as others which will be apparent, are achieved generally by a carrying device comprised of at least one built in non-removable embedded security unit for monitoring the contents of the carrying device.
Each security unit includes a camera, rechargeable power supply, light, speed and distance sensors, a microcontroller and a speaker/receiver. At least one security unit is built-in and embedded in the outer surface of the carry bag and at least one security unit is built-in and embedded in the inner surface of the carry bag. This allows the user to have access to views both inside and outside the carry bag.
The built-in security unit has a transponder for sending and accepting digital data to a central processor and storage of digital data outside of the unit.
The invention device uses WiFi and/or Bluetooth connections to communicate with a remote control and/or a personal digital assistant (PDA) such as a mobile phone. The remote control allows the user to turn sensors, cameras, alarms and messages on or off. The PDA device can also be used to turn sensors, cameras, alarms and messages on or off.
The user can use a PDA to access a web-based platform which permits the user to track the carrying device and monitor the contents therein. The system detects, by light sensors, when the carrying device is opened, and by distance sensors, when the device is moved away an undesired distance from the user and is programed to send an alarm to the owner of the carrying device.
Other objects, features and advantages of the present invention will be apparent when the detailed description of the preferred embodiments of the invention are considered with reference to the drawings, which should be construed in an illustrative and not limiting sense.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
FIG. 1A
FIG. 1B
illustrates the embedded camera according to the invention; is an illustration of the carrying device in a closed position and the outside view of the embedded security unit according to the invention; is an illustration of the carrying device in an open position and the inside view of the embedded security unit according to the invention;
FIG. 2A
FIG. 2B
is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a musical instrument case in an open position; and is this embodiment in a closed position;
FIG. 3
is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a backpack;
FIG. 4A
FIG. 4B
is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a briefcase in an open position; and is this embodiment in a closed position;
FIG. 5
is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a golf bag;
FIG. 6
is a schematic illustration of the security monitoring system according to the invention;
FIG. 7
FIG. 7A
FIG. 7B
21
22
is an illustration of the eject feature of the security unit according to the invention; illustrates the eject button on the back side of the security unit and illustrates the motherboard and
FIG. 8
is an illustration of the embodiment of the invention where the carrying device is a package delivery bag.
DETAILED DESCRIPTION OF THE INVENTION
1
—camera;
2
—light sensor on the camera;
3
—shatter proof/bullet proof glass covering over the camera lens;
4
—speed sensor;
5
—rechargeable power supply;
6
—USB charging port;
7
—transponder;
8
—GPS tracker;
9
—distance sensor;
10
—Security unit;
11
—Secure digital memory card;
12
—Speaker/receiver;
13
—Hard plastic outer surface of the security unit;
14
—solar panel;
15
—micro-controller;
16
—camera lens;
17
—remote control;
18
—personal digital assistant (PDA);
19
—central processor (remote);
20
—outer surface of the carry bag′
21
—eject button to open the unit to remove the power supply;
22
—Motherboard, PCB printed circuit board;
23
—hard plastic reusable delivery bag;
25
—internal cavity of the carry bag; and
30
—inner surface of the carry bag.
The invention describes a carrying device with a built-in security unit. The security unit enables the user to monitor the status, location and contents of their carrying device. The components of the invention device are defined as indicated with the referenced numerals as follows and shown in the accompanying figures:
FIG. 1
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As illustrated in , the security unit has the following components a camera with a light sensor ; a speed sensor ; a distance sensor ; a GPS tracker ; a transponder for sending and accepting digital data; and a rechargeable power supply in electrical communication with the microcontroller , camera , the speed sensor , the distance sensor , the GPS tracker and the transponder .
1
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1
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The main component of the security unit of the invention is the camera with a light sensor . Generally, when the carry bag is in an open position the light sensor senses the change in light and is triggered to turn on the camera in the security unit . When the carry bag is closed it's dark inside and the light sensor is triggered to switch the camera off. The embodiment with the light sensor is used in both situations where the security unit is embedded on the inner surface and outer surface of the carry device.
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The camera itself is non-removable from the security unit embedded in the carry bag. The camera is preferably the size of a dime or less, similar in size to the cameras used in mobile devices. The power supply is essentially the same size. The camera may be hidden or in plain view. Typically, the camera in the outer units are in view to discourage potential thieves while the inner units are hidden or less visible. The lens of the camera is covered by a shatter proof cover . The clear covering can be bullet proof glass, shatter proof glass, plexiglass, clear hard plastic or any material that will not break or shatter upon impact. This material protects the camera and camera lens to ensure that the camera does not get damaged if the carry bag is thrown, dropped or bounced around. The lens is also waterproof and anti-fogging. The lens can also be an infrared eye lens and a night vision lens.
The camera itself has a panoramic 360 degree view and is similar to the cameras that are built in at the rear of a motor vehicle used to view the surroundings when backing up. Due to the wide view of the camera itself there is no need for the user to have to control the angle or reposition the camera. The only control over the camera is to turn it on or off for capturing photos or streaming video. Any number of spy/wireless cameras currently on the market can be used in the invention as long as they are built in and non-removable. The make and size of the camera depend on the carrying device and can vary. Cameras that can be used in the security unit include, but are not limited to, the Lentenda Mini Remote SPY Camera for Iphone Android Ipad Pc MiniWifi Ip Wireless Spy Surveillance Camera Remote Cam. Other cameras or any image recording devices are intended to be part of the invention.
FIG. 1
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As seen in all components of the security unit are housed within an integral unit having a shatter and bullet proof glass covering over the camera lens and a hard plastic outer surface . The hard plastic case built is built around the security unit for protection and is embedded in the carry bag. Preferably the hard plastic case is slim, less than ½ inch and the entire security unit approximately 4 inches by 3 inches.
The sensors used in the invention include light sensors, speed sensors, distance sensors and in the suitcase embodiment an altitude sensor. While these are the preferred sensors used in the invention, other sensors or devices achieving the same function can be used. In general, sensors are a type of transducer, however, the main difference is that a transducer is a device that can convert energy from one form to another, whereas a sensor is a device that can detect a physical quantity and convert the data into an electrical signal.
In general, a light sensor is an electronic device used to detect light. Several types of light sensors are known and can be used in the invention. A photocell or photo resistor is a small sensor which changes its resistance when light shines on it. A CCD (charged coupled device) transports electrically charged signals and is usually found in digital cameras and night-vision devices. Photomultipliers detect light and multiply it.
Motion detector sensors are known to be used in advanced security systems and include passive infrared (PIR) motion detectors. These sensors “see” the infrared energy emitted by an intruder's body heat. When an intruder walks into the field of view of the detector, the sensor detects a sharp increase in infrared energy. These sensors can be incorporated into the security unit.
In the embodiment where the carry bag is luggage or a suitcase and the security unit further includes an altitude sensor. Altitude is typically determined based on the measurement of atmospheric pressure. The greater the altitude, the lower the pressure. A barometer and gyroscope are used for air pressure measurements. The altitude sensor is preferably preprogramed so that once the carry bag reaches an altitude height of 5,000 feet the camera is turned off. The camera can then be programmed to turn on and off at certain altitudes. When stored in flight, no one will be opening bags above 10,000 feet and if the camera is off the power supply battery life will be saved.
4
The speed sensor is preprogramed so that once the carry bag hits speeds of 150 miles per hour the camera is turned off. The security unit can include a accelerometer to determine speed.
9
8
The distance sensor is preprogramed to track the carry bag a desired distance and is used in combination with the GPS tracker for monitoring the location of the carry bag. When the chosen distance is achieved an alert is sent to the user via text message or as a ping on a personal digital assistant device.
The distance sensor can be an ultrasonic sensor which measures the distance to an object by using sound waves. It measures distance by sending out a sound wave at a specific frequency and listening for that sound wave to bounce back. By recording the elapsed time between the sound wave being generated and the sound wave bouncing back, it is possible to calculate the distance between the sonar sensor and the object. The Ultrasonic Sensor sends out a high-frequency sound pulse and then times how long it takes for the echo of the sound to reflect back. The sensor has 2 openings on its front. One opening transmits ultrasonic waves, (like a tiny speaker), the other receives them, (like a tiny microphone).
Laser sensors are used where small objects or precise positions are to be detected. They are designed as through-beam sensors, retro-reflective sensors or diffuse reflection sensors. Laser light consists of light waves of the same wave length with a fixed phase ratio (coherence).
The invention also includes Global Positioning System (GPS) Sensors. GPS sensors are receivers with antennas that use a satellite-based navigation system with a network of multiple satellites in orbit around the earth to provide position, velocity, and timing information. The orbits are arranged so that at any time, anywhere on Earth, there are at least four satellites “visible” in the sky. A GPS receiver locates four or more of these satellites, figure out the distance to each, and use this information to deduce its own location. Generally, GPS itself does not require an internet connection but it does if used in smartphones because it needs access to geographic map data.
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The transponder sends and accepts digital data collected from the security unit. The transponder sends digital data to a remote central processor wherein a user can access the central processor via a personal digital assistance device to view the digital data transmitted. A transponder is a wireless communications, monitoring, or control device that picks up and automatically responds to an incoming signal. The term is a contraction of the words transmitter and responder. Transponders can be either passive or active.
5
The rechargeable power supply can be selected from a variety of sources including rechargeable batteries, such as lithium or NiCd, or solar powered.
21
21
FIG. 7
The eject button is used to open and remove the battery of the unit if needed, to comply with TSA requirements and standards. While the entire unit is built into the carry case this feature enables the user to pull out components such as the battery if need be. This feature is shown in , the security unit is embedded in the outer surface of the carry bag. The eject button is on the back panel of the security unit on the inside surface of the carry bag. Thus, if the battery is removed the potential thief has no idea if the camera is operational or not.
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The eject button can be on the part of the unit where the battery is housed such that when the button is pressed only the battery opens for removal and/or replacement. Alternatively, when the button is pressed the entire security unit pops open or slides out so that the battery can be removed. In both embodiments the unit itself is built in to the carry bag and the unit itself cannot be removed. In both embodiments the eject button is on the inside surface of the carry bag so a potential thief would have no idea that the battery is not present.
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The power supply is in electrical communication with the camera , the speed sensor , the distance sensor , the GPS tracker and the transponder either by a wired connection or a wireless connection.
6
The security unit further consists of a charging port for the rechargeable power supply.
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In an alternate embodiment a solar panel is located on the outer surface of the carry bag and is in electrical communication with the rechargeable power supply .
7
The invention includes a remote control for the security unit which is in communication with the transponder . Communication is by Wi-Fi or Bluetooth depending on the distance and whether the device used has been paired with the security unit.
FIG. 1A
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The carrying device according to the invention includes a carry bag having an open and closed position. shows the carry bag as a suitcase/luggage in a closed position having an outer surface . The security unit of the invention is built in and embedded on this outer surface . The security unit is flush on the outer surface and is preferably visible to the naked eye to act as a deterrent to potential thieves. The security unit may be anywhere on the outer surface put is preferably located at the top of the carrying device near the handle.
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In an alternate embodiment, a solar panel is on the outer surface for recharging the power supply .
FIG. 1B
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shows the carry bag in an open position. An internal cavity is shown having an inner surface . The security unit of the invention is built in and embedded on this inner surface . Up to five security units can be built-in and embedded in the inner surface of the luggage or suitcase.
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In all embodiments of the invention at least one security unit is built-in and embedded in the outer surface of the carry bag and at least one security unit is built-in and embedded in the inner surface of the carry bag.
A microprocessor generally does not have RAM, ROM and IO pins. It usually uses its pins as a bus to interface to peripherals such as RAM, ROM, Serial ports, Digital and Analog IO. It is expandable at the board level due to this. A microcontroller is ‘all in one’, the processor, ram, IO all on the one chip, as such you cannot (say) increase the amount of RAM available or the number of IO ports. The controlling bus is internal and not available to the board designer. In general, this means that a microprocessor is generally capable of being built into bigger general purpose applications than a microcontroller. The microcontroller is usually used for more dedicated applications such as used in the present invention. However, in some applications a microprocessor can be used.
The microcontroller of the invention has the following properties: typically 8 to 32 bit; runs at speeds less than 200 MHz; uses very little power; provides current to operate an LED; is a useful interface with sensors and motors; constrained for RAM and flash storage.
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The invention also includes a printed circuit board (PCB) or motherboard to house the microcontroller .
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The security unit of the carrying device further includes a secure digital (SD) memory card . The secure digital memory card preferably has a capacity between 4 GB and 128 32 GB. The SD card is generally the size of a dime and can be preprogrammed to take a series of photos when the camera is activated. The user will have options to choose the number of photos taken or continuous photos or video until the movement stops and the camera is shut off.
12
A speaker/receiver is present on the security unit. This speaker can be used to sound an alarm to prevent a potential thief from stealing contents or the carry bag itself. The owner of the carry bag can also send prerecorded sounds to the carry bag such as “Drop it now” or “I can see you” to again prevent the potential thief from stealing the carry bag or the contents therein. When the carry bag is on a plane this feature would be disabled for safety reasons.
12
The speaker/receiver can also be used to communicate in real-time with a potential thief. The owner of the carry bag can play screeching music or speak directly to the intruder to deter a potential thief.
FIG. 1A
In alternate embodiments, multiple security units can be incorporated particularly wherever the device can be opened. For example, as seen in , if the luggage piece has dual openings, one camera can be built on the side and one camera on the front. At least two cameras are preferably since the luggage can be opened in these two places. In the luggage embodiment it is possible to have up to five cameras can be built in to the interior surface of the carry bag for maximum viewing. Positions of the security unit would be one in the front, one in the back, one on each side and one on the inside middle.
FIG. 2A
FIG. 2B
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is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a musical instrument case in an open position. The security unit is positioned in the middle of the case. The camera of the unit is visible. is this embodiment in a closed position illustrating a security unit on the outer surface with the camera visible to the naked eye. The owner of the case can program the system in such a way that if the case is opened screeching music can stream from the speaker loud enough to be heard over 20 feet away.
FIG. 3
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is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a backpack. The backpack itself can have a solid back and made of bullet proof material. As shown multiple security units are built in all over the backpack on the backside and on the sides. The camera and recharging USB ports for the power supply are shown. Not illustrated but included in this embodiment is a security unit on the interior surface.
Where the carry bag is a backpack the security unit also can be used in bully prevention. A school aged teenage can monitor their digital device to watch the cameras in the security unit to see if someone is approaching from behind. It's as if they have eyes in the back of their head.
FIG. 4A
FIG. 4B
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is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a briefcase in an open position. The security unit is positioned in the middle of the case. The camera of the unit is visible. is this embodiment in a closed position illustrating a security unit on the outer surface with the camera visible to the naked eye. The camera and recharging USB ports for the power supply are shown.
In embodiments where the carry bag is a handbag or purse, the security units would be built in on the outer and inners surfaces as described in the other embodiments.
FIG. 5
FIG. 5
10
is an illustration of the embodiment of the invention where the carrying device with a built-in security system is a golf bag. One or multiple security units can be built into the golf bag. As shown in there is a security unit at the top of the golf bag, one on the front pouch, one on the side pouch and one on the bottom of the bag. Each unit provides the user with a different perspective view. It enables the owner to not only monitor the golf bag and contents when they step away from the bag to use the restroom or even on the golf course itself.
Once the suitcase is opened the light triggers the light sensor on the camera to activate the camera to start taking photos/video. Using the remote control device the user can automatically turn on the cameras, either all of them or just a select number. The cameras could also be already activated by the respective sensors. The cameras in the security unit can be set manually or be set to be activated by the light sensor.
The camera on the outside of the carry bag can be on all the time depending on the user's needs.
Amazon, Fedex, and UPS have a huge problem with stolen delivered merchandise. When they deliver their packages, people steal them from doorsteps, front lawns, etc. The invention security device can be embedded into a secure delivery package. The user will leave it on their lawn or front door and when packages are delivered the delivery person would secure the package inside. Every time the bag would be returned a monetary credit to the user's account would be made, similar to a CRV on bottles.
FIG. 8
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illustrates the package delivery bag embodiment in more detail. The package delivery bag is made of a carry bag; a security unit () embedded in the carry bag comprising components consisting of: a camera () with a light sensor (); a microcontroller (); a transponder () for sending and accepting digital data; and a rechargeable power supply () in electrical communication with said microcontroller (), camera () and said transponder (); wherein said components are housed entirely within a single integral unit having a shatter proof clear covering () over the camera lens () and a hard plastic outer surface () over the rest of said unit;
23
The package delivery bag is sturdy enough to stand on its own and house a security unit according to the invention. The delivery bag is preferably a thin, sturdy, hard plastic bag.
The unit would be connected through the carrier (i.e. Amazon or Fedex) where the user would get a notification the carrier is delivering a package to the owner. The user would than accept, on a delivery platform, the delivery notification, and would then receive a corresponding number that would activate the carry case holding your package. Each carry case will have a tracking number with a camera. So when the package arrive the camera is already activated. When the user comes home, or to wherever the package is delivered, they pick up the package on my doorstep and send a message on the delivery platform “I've received my package.” Optionally, the delivery carrier can provide incentives and say for example you will receive $2 off upon return. The owner can also hold the package delivery bag until they order something else and until the next package is delivered. The delivery driver drops off the new package, scans the empty/waiting carry case on the doorstep and a monetary credit is issued. The process is similar to the old milk delivery system where the milkman delivers and picks up the empty bottles, the delivery person would pick up the empty package delivery bags. By having a security unit embedded therein the user can monitor the package delivery up to the point they pick it up for themselves.
17
The invention includes a remote control that can turn on and off the security unit components as desired. For example, where the carry bag is luggage or a suitcase, the user can set the security unit to automatically turn off when the plane accelerates or is at a certain altitude or when the GPS tracker shows it's stored under the plane.
There is a choice to not turn on those sensors. This is important for carry-on luggage since many items are stolen from overhead compartments, especially when the owner can't keep the carry on near them.
FIG. 6
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The security system of the invention is illustrated in the schematic of . The remote control can be paired by Bluetooth to the security unit of the carrying device or if too far away can be connected via Wi Fi. The remote control can be used to turn the cameras, sensors and alarms on and off as desired by the user.
Wi-Fi, originally meant “wireless fidelity,” is primarily about connecting one or many devices to the Internet or creating a local wireless network that can link multiple devices. It depends on a central base station (or multiple stations) that sends out a network signal strong enough and wide enough to cover, say, an office or home, a coffee shop or even an airport. It sends out what might be thought of as invisible Internet “rays” around the globe that can be tapped into by any laptops, smartphones or tablets within their range to get online. Bluetooth is much shorter-range, usually around 10 to 30 feet. It rarely involves getting onto the Internet and doesn't depend on any central device like a router. It is almost always used to connect two devices together in some useful way.
Like mobile phones, a W-Fi network makes use of radio waves to transmit information across a network. The computer or PDA includes a wireless adapter that translates data sent into a radio signal. This same signal will be transmitted, via an antenna, to a decoder known as the router.
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As mentioned earlier digital output from the security unit is sent to a remote central processor for storage of the images. These images are accessed by the owner by a web-based platform. The invention cameras are in communication with the web platform wherein the user can monitor the handling and opening of the carrying device in real time or the stored images. The invention has both Bluetooth and WI-Fi capability to feed activity from the camera for viewing at all times via a web-based platform.
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A PDA or any device that can access the internet can be used to access the camera feeds from the security unit , using the web-based platform. There is a user login for viewing the feed from the cameras.
10
Once a user has logged into the platform a number of functions can be performed by the user. The platform enables the user to view each camera feed from the security units in the carry bag. It also enables the user to be able to turn the security unit components on or off, similar to the remote control functions. For example, the user can selectively turn on or off a camera or sensor as desired in any of the security units of the carry bag.
12
The platform also enables the user to send out an alarm or message to the speaker/receiver on the carry device itself. The user can send an alarm or a pre-recorded voice message or command.
This platform allows users of the carrying devices according to the invention to be able to view their bag at all times if they choose.
Users are able to review and record footage up to a certain amount of time. Unless instructed otherwise, the cameras in the security units are constantly recording and are able to hold up to a certain amount of footage, typically up to 4 days. After 4 days, it automatically purges the saved footage unless the footage has been backed up to another device. However, the user can go back and view footage and then record whatever footage they choose and save it.
The user also gets an alert on their digital device if their carrying device is being opened. The user can also set an alarm to go off if the carry device is opened if so desired. In all embodiments of the invention allows the user to track their carrying device.
The following examples indicate how the invention device can be used:
Example 1
A person is driving to Atlantic City and leaves their luggage in the car. The car is locked but the person knows someone can break into the car, but they don't feel like carrying a 49 lb. luggage inside to the bathroom. They activate the invention security unit to start the cameras recording and initiate the GPS tracking. They also activate the distance sensor to “PING ME IF MY LUGGAGE MOVES MORE THAN 10 FEET.” The user can choose 10 feet, 100 feet, 1,000 feet as desired.
12
If the thief steals the luggage, but doesn't open it, the user still gets the alert/ping that my luggage is moving when it's supposed to be locked in my trunk. Not only do they know it's being stolen, but it is video recording where the thief is going. And then when the thief opens the luggage, he's being video recorded. This digital data is all being sent to a central processor for access by the user in real time if desired. A voice alert or an alarm can be sent to the carrying device broadcasting through the speaker “Why are you opening my luggage?” or “This is being reported to the police”.
While the invention has been described with examples of preferred embodiments the invention includes other variations. Such as the security unit of the invention can be built in and incorporating inside and outside of a motor vehicle, boat, or even a pet carrying case.
The foregoing description of various and preferred embodiments of the present invention has been provided for purposes of illustration only, and it is understood that numerous modifications, variations and alterations may be made without departing from the scope and spirit of the invention as set forth in the following claims. | |
The Bills have finally hit the field with pads on in 2020.
After a delay to their offseason schedule due to the ongoing coronavirus pandemic, Bills players finally have started to get on the field and hit each other. With preseason games still canceled, each day of practice held in Orchard Park is that much more important before the team’s season opener against the Jets on Sept. 13.
Tuesday saw the second day of such physical practices. With that, here are three main takeaways from Day 2 of Bills training camp padded practices:
Micah Hyde picks off Josh Allen
The big storyline from Day 1 of training camp was the way that it ended. Quarterback Josh Allen chucked a ball deep down the field, found wide receiver John Brown for a long score with Tre’Davious White in coverage. Flash forward and the highlighted effort via various outlets on Tuesday was from the defense, and specifically, safety Micah Hyde.
During team sessions, Hyde reportedly intercepted Allen during 7-on-7 work. Syracuse.com indicates that the pass was intended for Stefon Diggs. Hyde correctly read and jumped the route, perhaps like a pick-six scenario.
Even with the strong talent the Bills have added on offense as of late, the team’s bread and butter is still going to be their defense in 2020… and leading that defensive effort? The secondary, just as it has since Sean McDermott took over in 2017. The group probably took things very personal following that prior mentioned Brown score, too. WGR-550 radio reports that practice on Tuesday ended on a defensive stop in the end zone. The group celebrated, just like the offense did a day earlier at the end of practice when Brown scored. | https://billswire.usatoday.com/2020/08/18/takeaways-day-2-buffalo-bills-training-camp-micah-hyde-josh-allen-isaiah-hodgins-stephen-hauschka/ |
Discipline is more thorough and consistent when it is built on a set of core beliefs, which are turned into principles, that take into consideration the social needs of students and the contents of the students’ Quality Worlds.
All participants must believe in and use these principles when making classroom decisions. Core beliefs are based on helping students develop their internalized sense of control, rather than trying to control students with rewards and punishments. The emphasis is on a an ethical core, not a rules-based system. “Consistency comes, not from trying to force everybody to do the same thing at the same time - but by living by a set of core beliefs.” Jim Fay and David Funk. Teaching with Love and Logic.
Creating A Code of Conduct Based on Core Beliefs
Step One:
- Individually or in subgroups of the larger group brainstorm a list of five to seven beliefs related to ways of behaving.
Use stems such as: I believe students should ... , Behavior should ... , Treat .... ,
- Next, each person or subgroup in turn shares their beliefs a collated list is displayed. Length is unimportant. If a teacher holds a belief, it should go on the list.
Step Two:
- Alone or in small groups teachers rank order the collated list from most important (1) to least important (the highest number being the number of beliefs you have).
- Take each ranked order and find the totals for each belief.
- Decide how to arrive at a manageable number. For example, six beliefs. Then review the list and decide which to keep and which to discard. Might select the six lowest scores and discarded the rest. Or may want to consolidate, edit and vote again.
Step Three:
- Have each person or group think about their purpose for disciplining students.
- Have each examine each the six beliefs to identify which belief fits with his or her purposes for discipline
Step Four:
- Regroup to come to consensus on which four of the six beliefs are most important.
- Review them for ethical considerations such as: appropriate for and protective of students, instructive, protecivet and respectful of self, others, and property.
- These four beliefs become the FINAL FOUR - OUR UNCHANGING CORE around which all activity, all change, and all behavior takes place.
Core Beliefs:
- Belief 1
- Belief 2
- Belief 3
- Belief 4
Step Five:
List six to ten of the most common misbehaviors that are dealt with on a daily basis.
Step Six:
- For each of the common misbehaviors describe how each is presently handled (present interventions used).
- Explain how each present intervention is or is not compatible with the four core beliefs.
Step Seven:
Identify current interventions, which are compatible and effective in helping students choose and use mastery oriented behaviors. These should be kept and others which are not compatible or not effective, should be eliminated or changed.
Identify different mastery oriented behaviours, which are consistent with the four core beliefs and students can use successfully.
Step Eight:
Select a behavior and describe its performance at an acceptable outcome level.
Describe what student behaviors Look Like, Sound Like, and Feel Like.
Step Nine:
Using the acceptable outcome levels consider how to perform acceptable behaviors and reduce their use of unacceptable behaviors. Describe in detail a procedure to successfully select, innitiate, and perform the behaviors.
Step Ten:
Describe three types of consequences for behaviors related to each code if it is not followed.
- Natural Consequences
- Given Consequences
- Logical Consequences
Step Eleven:
Write step by step detailed learning sequences to use to help students learn appropriate behaviors for the codes.
Outcomes of this Process:
The main outcome of this process is to use intervention strategies that focus on developing an internalized sense of responsibility in the students with whom we work. To use interventions that are instructive so as to develop students' ability to choose and use mastery oriented behaviors with a sense of responsibility and a feeling of shared control. | https://www.homeofbob.com/cman/tchrTls/makingCodeOfConductTogether.html |
Korean photographer JeeYoung Lee was born in 1983, and earned both her undergraduate and graduate degrees at Seoul’s Hongik University. Since 2007, Lee has been shooting whimsical images that represent either her experiences, dreams and memories, or represent traditional Korean folk tales and legends.
Seeing Lee’s work for the first time, most viewers will presume her colorful, fantasy world images are the product of a large amount of digital manipulation. Yet each of her photos is created through the meticulous construction of elaborate sets by the artist herself, rather than use of Photoshop. In the middle of each image you can always find the artist herself, as Lee’s work is a type of unconventional self-portraiture.
In “Resurrection” Lee appears inside a lotus portraying rebirth. The image references Shim Cheongin, a Korean fable about a girl who throws herself into the sea and comes back to life inside a blooming lotus. Lee created this dreamlike image by painting paper lotus and flooding the room with fog and carbonic ice.
What boggles the mind is that Lee creates all the scenes in her images by hand – in a tiny studio that measures a mere 3.6 x 4.1 x 2.4 meters. Starting with an idea born in her imagination, Lee will labor for weeks, sometimes months, constructing a surreal set for the sake of taking a single photograph. For each of her photographs the artist fills every square inch of space with hand-made props, set pieces, and backdrops
When the set is complete, Lee inserts herself in the scene and then takes multiple test shots. After carefully examining the test shots and making any adjustments she deems necessary, Lee takes the final shot with a 4×5 large format film camera. Lee then disassembles the set once the final photograph is produced.
To create “Treasure Hunt”, Lee devoted three months to crafting the wire grassland, which carpets her studio to evoke a child-like wonderland. She spent nearly eight hours a day weaving bits of craft wire to a mesh screen to complete the grass flooring.
Lee’s avoidance of the use of Photoshop is based on her belief that the building and breaking-down of the set is an integral part of her artwork. She only uses Photoshop when she has suspended objects from the ceiling of her studio, in which case she uses the program to erase the fishing lines used for suspension.
“My Chemical Romance” with its maze of pipes and yellow & black danger tape, Lee depicts the anxiety and disappointments felt by herself or those around her, and how they can lead to conflict and clashes of personality.
Lee is unique in that in addition to the role of photographer, she also assumes the roles of set designer, sculptor, installation artist, and performer. The results are magical, as can be seen in this small selection of a few of her work.
“Panic Room” shows the artist hiding herself inside a cupboard to protect and shelter herself from the confusion outside – symbolized by the dizzying atmosphere Lee created by bending the perspective in her studio. (For
Recipient of multiple artistic awards, JeeYoung Lee is recognized as one of the most promising up-and-coming artists in Korea. Her work has also received extensive coverage outside her home country by global news outlets such as Huffington Post, NBC news, CNN international, France 3 National news, China Daily, etc. as well as on various art/photo websites. | https://kanakukui.com/2016/07/01/korean-photographer-jeeyoung-lee/ |
The Joy of Ballpark Food: From Hot Dogs to Haute Cuisine
To my father who took me to my first San Francisco Giants baseball game in 1965, and instilled in me a continuing love of baseball.
To my wife Debbie who finally became a baseball fan during the 2010 San Francisco Giants pennant race.
To my daughter Aviva and daughter-in-law Marya who have little interest in baseball, are vegetarian-inclined, but still were able to enjoy a Fenway Frank when I took them to a Red Sox game.
Acknowledgments
Celebrate The Joy Of Ballpark Food While Helping Those In Need Of Food
Introduction
The Hot Dog Comes To America
The Hot Dog Comes To Baseball
Harry M. Stevens
Hot Dogs vs Sausages
Take Me Out To The Ball Game
Cracker Jack
Peanuts
Old-Time Food Offerings And Prices
The Start Of The New Food Era
Nachos And Garlic Fries
Healthy And Vegetarian Options
Kosher Food And Knishes
Official Cheese Doodle Of The New York Mets
The Big Six And The Business Of Concessions
Colorful Vendors
Culinary Tour Of The Major League Ballparks
Conclusion
Sources
ACKNOWLEDGMENTS
I would like to thank all my family and friends who encouraged me in this project.
Special acknowledgments:
to Tim Wiles and the staff of the Research Library at the National Baseball Hall of Fame for providing assistance and access to their collections.
to Fran Galt who used his extensive knowledge of both baseball and English grammar to edit the final text.
to N.E. for her assistance with research and helping me to get my thoughts down on paper.
to Ellen Gilmore for editing my initial drafts.
to Elizabeth, Kay, Gary, and the staff at the San Jose Giants concession stand who make being a ballpark vendor an incredibly fun experience.
to Mother Nature for allowing us to visit all 30 stadiums within one season without a rainout.
to Andy Nichols for providing the book layout and design.
to my wife Debbie for accompanying me to the stadiums, being my official food photographer, taste-testing foods that I didn’t want to eat, and encouraging me when I needed it.
CELEBRATE THE JOY OF BALLPARK FOOD WHILE HELPING THOSE IN NEED OF FOOD
The goal of this book is to celebrate the joy of ballpark food while helping those in need of food. All of the royalties from the sales of this book are being donated directly to the Second Harvest Food Bank of Santa Clara and San Mateo Counties of California.
Second Harvest Food Bank of Santa Clara and San Mateo Counties is the trusted leader dedicated to ending local hunger. Since its inception in 1974, Second Harvest has become one of the largest food banks in the nation, providing food to more than 250,000 people each month. The Food Bank mobilizes individuals, companies, and community partners to connect people to the nutritious food they need. More than half of the food distributed is fresh produce. Second Harvest also plays a leading role in promoting federal nutrition programs and educating families on how to make healthier food choices.
Dreamland, Coney Island, N.Y., 1904. Library of Congress, LC-USZ62-115624.
Literary scholars have debated for decades whether Shakespeare actually wrote the plays attributed to him. Similarly, there are opposing theories, each with its own advocates, of how the hot dog as we know it came to be.
The familiar American hot dog is a type of German sausage served in a roll and handheld for eating. It is widely believed that many butchers in America commonly sold the sausage (known as a dachshund sausage) during the nineteenth century. What is in dispute, however, is who came up with the idea of putting the dachshund sausage in a roll or bun and calling it a hot dog.
Was it...
… in the 1860s when a German immigrant whose name remains unknown sold them from a pushcart in New York City’s Bowery?
… in 1871 when Charles Feltman opened up a stand on Coney Island? (An employee of Feltman’s, Nathan Handwerker, eventually went on to found Nathan’s Famous Hot Dogs.)
… in 1893 when they were introduced at the Chicago Columbian Exposition by Austrian immigrants Emil Reichel and Sam Ladany?
… in 1893 when Chris Von de Ahe, St. Louis bar owner and owner of the St. Louis Browns baseball team, sold them at his ballpark?
… in 1901 when Harry M. Stevens sold them at the Polo Grounds, home of the New York Giants, on a cold day when ice cream was not selling well? (More about this in the next chapter.)
… in 1904 when a Bavarian sausage seller named Anton Feuchtwanger sold them at the St. Louis Louisiana Purchase Exposition? (As the story goes, he loaned white gloves to his customers to hold his hot sausages. Most of the gloves were not returned. He reportedly asked his brother-in-law, a baker, for help. The baker improvised long, soft rolls that fit the meat, thus inventing the hot dog bun.)
There are also reports of references to “hot dogs” appearing in college publications in the 1880s and 1890s.
As with the Shakespeare authorship dispute, the definitive answer to this mystery will probably never be known.
A popular story states that at a New York Giants baseball game on a cold April day in 1901, pioneer vendor Harry M. Stevens was unable to sell ice cream. Instead he ordered his staff to purchase dachshund sausages from all the surrounding butcher shops. He then stuffed the sausages into bread rolls and shouted “Get your red hots!” Tad Dorgan, the sports cartoonist for the New York Evening Journal, was reputed to have been unable to spell dachshund, so he wrote hot dogs instead.
It is a wonderful story except for the fact that it is not true. Professor Gerald Cohen of the Missouri University of Science and Technology published a 293-page book entitled Origin of the Term Hot Dog. In the book Professor Cohen points out that, despite the widespread acceptance of this story, no copy of the cartoon has ever been found. Moreover, it appears that Dorgan was not even working for the New York Evening Journal in 1901.
Additionally, in an interview conducted by Fred Lieb of The Sporting News in 1926, Harry M. Stevens stated, “I have been given credit for introducing the hot dog in America. Well, I don’t deserve it. It was my son Frank who first got the idea of selling hot dogs and wanted to try it on one of the early six-day bicycle race crowds at Madison Square Garden....At the time, we had been selling mostly sandwiches, and I told Frank that the bike fans preferred ham and cheese. He insisted that we try it out for a few days, and at last I consented. His insistence had all America eating hot dogs.”
The cartoon myth probably would have died out if not for an article by Quentin Reynolds in Colliers Magazine in 1935. Harry M. Stevens had died in 1934 and Reynolds’ article was a tribute to him. It is believed that Stevens’ sons, out of loving memory to their father, did not object to Reynolds portraying the story as fact.
The first hot dog at a baseball game could have been at an 1893 St. Louis Browns (a member of the short-lived American Association baseball league) game, or could have been at a New York Giants game during the 1900s, or somewhere else.
What is clear, however, is that no one did more to popularize hot dogs and connect them to baseball than Harry M. Stevens.
“Hot Dogs” for Fans Waiting for Gates to Open at Ebbets Field, Oct. 6, 1920.
Library of Congress, LC-USZ62-58784.
Although we can’t give Harry M. Stevens credit for inventing the hot dog, no one did more to connect hot dogs with baseball and to establish the connection between baseball and food.
In 1941, on the 50th anniversary of the founding of the Harry M. Stevens Catering Company, a celebratory dinner was held in New York. Harry M. Stevens’ sons (Harry had passed away a few years earlier) received the following Western Union telegram:
“I have always known that there was more to your business than just selling hot dogs because a part of your fine personalities seemed to creep in to the flavor of the dogs. If my health permitted I would be with you tonight, but when I get going again, you will find me in the stadium rooting for McCarthy and the boys, with a season pass from Barrow in one hand and a Stevens hot dog in the other, and pride in my heart for all of you.”
Lou Gehrig. National Baseball Hall of Fame Library, Cooperstown, NY.
The telegram was signed by Lou Gehrig.
The author of a newspaper column in the El Paso Herald in 1916 wanted to use a metaphor to indicate his subject’s wealth. The metaphor he chose was “He has more money than Harry Stevens has hot dogs.”
Harry Mosley Stevens was born in Litchurch (a village in Derbyshire), England, in either 1855 or 1856 (sources disagree). As a young man he immigrated to America and settled in Columbus, Ohio, where he obtained employment as a book salesman. Although Stevens is best known for his food empire, that was not his start in the baseball world.
Stevens attended baseball games of the Columbus Buckeyes in the Ohio State League. He was frustrated by the lack of information in the scorecards. There was very little discussion about the home team players and no mention at all of the visiting team players. Through his work as a book salesman, he had become acquainted with Ralph Lazarus, one of the owners of the Columbus Buckeyes. Lazarus sold Stevens the rights to produce and sell a new scorecard for the Buckeye games.
Stevens bought the scorecard rights for $500. By the next afternoon, he had sold $700 worth of advertising space in the scorecard. So before he even sold his first scorecard, he had already made a nice profit.
Legend has it that Harry came up with the slogan “You can’t tell the players without a scorecard!” when he was peddling his scorecards in Columbus.
To call attention to the new scorecards, Stevens dressed in a red coat and straw hat. As he went through the grandstands with his scorecards, he recited lines from William Shakespeare and poet Lord Byron. Fans were able to have printed programs and lineup cards for the first time.
The scorecard was a success in Columbus, so Harry branched out first to Toledo and Milwaukee, and then to Pittsburgh, Cleveland, Washington, Boston, and Philadelphia.
Harry M. Stevens was always looking for new ways to expand his business. According to the September 2012 issue of Columbus Monthly, in 1887 Stevens opened the first food concession stand at a ballpark. This was at the Columbus minor league ballpark.
Stevens’ big break came in 1894 when he obtained not only the scorecard rights for the New York Giants at the Polo Grounds but also the rights to food concessions in the ballpark. This would be the beginning of the Harry M. Stevens catering empire.
Before Harry M. Stevens came on the scene, what was sold at baseball games was mostly sandwiches, ice cream, and lemonade. By the turn of the twentieth century, these items had been replaced by hot dogs, peanuts, and soda pop. Stevens also was the concessionaire for horse races, bicycle races, and other New York area sporting events. In time Stevens branched out to providing food concessions at other major league baseball stadiums.
Harry M. Stevens died in 1934 in New York City. His obituary in the New York Times stated, “The thousands of uniformed men who proffer their wares, principally wieners, peanuts and pop, at all sorts of athletic contests in all parts of the country, owe their vocation to Mr. Stevens. He was the first of them and the most successful.”
After his death the company, Harry M. Stevens, Inc., was taken over by his three sons and later by his grandsons and a fourth generation. In 1994, after over a hundred years in business, the company was bought by Aramark Sports and Entertainment Services.
Aramark is one of six companies (along with Delaware North Companies, Levy Restaurants, Centerplate, Legends, and Ovations Food Services) that now control the food concessions at all 30 major league baseball stadiums. In addition, they provide food services at many other stadiums and arenas.
New Polo Grounds, August 1911. Harry M. Stevens (left), John Foster, builder (center), John T. Brush, owner of New York Giants (right). National Baseball Hall of Fame Library, Cooperstown, NY.
What is a hot dog? And what is the difference between a hot dog and a sausage? These may seem like simple questions, but after spending an entire afternoon of Internet searching, I can assure you that they are not.
Since I began thinking about writing this book, I asked everyone I know (and some people I didn’t know) what they thought the difference was between a hot dog and a sausage. Many answers I received reminded me of the famous quote from Supreme Court Justice Potter Stewart that “hard-core pornography” was hard to define but “I know it when I see it.”
I asked this perplexing question to a crusty vendor with a hot dog and sausage cart on Yawkey Way outside of Fenway Park. His succinct response was to point to the hot dog and say “This is a hot dog” and point to the sausage and say “This is a sausage.”
After searching through many sources, the best definition I have found is from the website www.differencebetween.net :
1. Sausage is an encompassing term for any processed meat with fat, spices, and preservatives that is encased into an animal’s intestines or commercial wrapping. Many types of sausages are made and available in many markets; one of them is the popular American hot dog.
2. The hot dog is not an original sausage but merely an American adoption of German sausages, frankfurters, and wieners.
3. The texture of a hot dog is smooth and paste-like while sausages have a more composite mixture of miniscule bits of meat.
4. A hot dog is usually a food for leisure time while a sausage can be eaten for the same purpose and can also be used for main dishes.
According to food historian Bruce Kraig, author of Hot Dog: A Global History, “The hot dog species of sausage might be defined as an ‘emulsified,’ or very finely chopped or ground meat product. As a further subspecies, the hot dog is a precooked sausage. In its truly defined state, the hot dog is meant to be eaten out of the hand encased in a bun. In this sense, the hot dog crosses food categories and becomes one of America’s singular foods, a sandwich.”
Oftentimes children’s book authors present concepts in a more cogent manner. Adrienne Sylver, the author of Hot Diggity Dog: The History of the Hot Dog, states “Hot dogs are a kind of sausage....The meats or veggies are chopped into small parts and blended with bread crumbs, flour, and seasonings. The gooey hot dog batter is pumped into a thin plastic tube to hold it together. Then it’s cooked or smoked. Finally, the hot dog takes a bath in cool water and the plastic is peeled away.”
And finally, for a clear and easy to understand definition, what better source to turn to than the United States Code of Federal Regulations:
9 CFR 319.180 - Frankfurter, frank, furter, hotdog, weiner, vienna, bologna, garlic bologna, knockwurst, and similar products.
§ 319.180
Frankfurter, frank, furter, hotdog, weiner, vienna, bologna, garlic bologna, knockwurst, and similar products.
(a) Frankfurter, frank, furter, hot-dog, wiener, vienna, bologna, garlic bologna, knockwurst and similar cooked sausages are comminuted, semisolid sausages prepared from one or more kinds of raw skeletal muscle meat or raw skeletal muscle meat and raw or cooked poultry meat, and seasoned and cured, using one or more of the curing agents in accordance with a regulation permitting that use in this subchapter or in 9 CFR Chapter III, Subchapter E, or in 21 CFR Chapter I, Subchapter A or Subchapter B. They may or may not be smoked. The finished products shall not contain more than 30 percent fat. Water or ice, or both, may be used to facilitate chopping or mixing or to dissolve the curing ingredients but the sausage shall contain no more than 40 percent of a combination of fat and added water. These sausage products may contain only phosphates approved under part 318 of this chapter. Such products may contain raw or cooked poultry meat and/or Mechanically Separated (Kind of Poultry) without skin and without kidneys and sex glands used in accordance with § 381.174, not in excess of 15 percent of the total ingredients, excluding water, in the sausage, and Mechanically Separated (Species) used in accordance with § 319.6. Such poultry meat ingredients shall be designated in the ingredient statement on the label of such sausage in accordance with the provisions of § 381.118 of this chapter.
(b) Frankfurter, frank, furter, hot-dog, wiener, vienna, bologna, garlic bologna, knockwurst and similar cooked sausages that are labeled with the phrase “with byproducts” or “with variety meats” in the product name are comminuted, semisolid sausages consisting of not less than 15 percent of one or more kinds of raw skeletal muscle meat with raw meat byproducts, or not less than 15 percent of one or more kinds of raw skeletal muscle meat with raw meat byproducts and raw or cooked poultry products; and seasoned and cured, using one or more of the curing ingredients in accordance with a regulation permitting that use in this subchapter or in 9 CFR Chapter III, Subchapter E, or in 21 CFR Chapter I, Subchapter A or Subchapter B. They may or may not be smoked. Partially defatted pork fatty tissue or partially defatted beef fatty tissue, or a combination of both, may be used in an amount not exceeding 15 percent of the meat and meat byproducts or meat, meat byproducts, and poultry products ingredients. The finished products shall not contain more than 30 percent fat. Water or ice, or both, may be used to facilitate chopping or mixing to dissolve the curing and seasoning ingredients, the sausage shall contain no more than 40 percent of a combination of fat and added water. These sausage products may contain only phosphates approved under part 318 of this chapter. These sausage products may contain poultry products and/or Mechanically Separated (Kind of Poultry) used in accordance with § 381.174, individually or in combination, not in excess of 15 percent of the total ingredients, excluding water, in the sausage, and may contain Mechanically Separated (Species) used in accordance with § 319.6. Such poultry products shall not contain kidneys or sex glands. The amount of poultry skin present in the sausage must not exceed the natural proportion of skin present on the whole carcass of the kind of poultry used in the sausage, as specified in § 381.117(d) of this chapter. The poultry products used in the sausage shall be designated in the ingredient statement on the label of such sausage in accordance with the provisions of § 381.118 of this chapter. Meat byproducts used in the sausage shall be designated individually in the ingredient statement on the label for such sausage in accordance with § 317.2 of this chapter.
(c) A cooked sausage as defined in paragraph (a) of this section shall be labeled by its generic name, e.g., frankfurter, frank, furter, hotdog, wiener, vienna, bologna, garlic bologna, or knockwurst. When such sausage products are prepared with meat from a single species of cattle, sheep, swine, or goats they shall be labeled with the term designating the particular species in conjunction with the generic name, e.g., “Beef Frankfurter,” and when such sausage products are prepared in part with Mechanically Separated (Species) in accordance with § 319.6, they shall be labeled in accordance with § 317.2(j)(13) of this subchapter.
(d) A cooked sausage as defined in paragraph (b) of this section shall be labeled by its generic name, e.g., frankfurter, frank, furter, hotdog, wiener, vienna, bologna, garlic bologna, or knockwurst, in conjunction with the phrase “with byproducts” or “with variety meats” with such supplemental phrase shown in a prominent manner directly contiguous to the generic name and in the same color on an identical background.
(e) Binders and extenders as provided in § 319.140 of this part may be used in cooked sausage that otherwise comply with paragraph (a) or (b) of this section. When any such substance is added to these products, the substance shall be declared in the ingredients statement by its common or usual name in order of predominance.
(f) Cooked sausages shall not be labeled with terms such as “All Meat” or “All (Species),” or otherwise to indicate they do not contain nonmeat ingredients or are prepared only from meat.
(g) For the purposes of this section: Poultry meat means deboned chicken meat or turkey meat, or both, without skin or added fat; poultry products mean chicken or turkey, or chicken meat or turkey meat as defined in § 381.118 of this chapter, or poultry byproducts as defined in § 381.1 of this chapter; and meat byproducts (or variety meats), mean pork stomachs or snouts; beef, veal, lamb, or goat tripe; beef, veal, lamb, goat, or pork hearts, tongues, fat, lips, weasands, and spleens; and partially defatted pork fatty tissue, or partially defatted beef fatty tissue.
[38 FR 14742, June 5, 1973]
“Take Me Out to the Ball Game” is the third most frequently played song in America (after “Happy Birthday” and “The Star-Spangled Banner”) according to Tim Wiles, formerly of the National Baseball Hall of Fame and co-author of Baseball’s Greatest Hit: The Story of Take Me Out to the Ball Game.
And what is the first thing ball game fans want to do when they get to the ballpark? “Buy me some peanuts and Cracker Jack.”
This American classic song was written by Jack Norworth in 1908 with music composed by Albert Von Tilzer. Norworth had written many popular songs including “Shine on Harvest Moon.” Von Tilzer also founded a publishing company and was the first to publish compositions by both Irving Berlin and George Gershwin. Although both Norworth and Von Tilzer have strong credentials in the field of music, neither of them knew anything about baseball at the time the song was written.
Norworth was riding on the New York subway when he got the idea for the song lyrics from a sign advertising a baseball game at the Polo Grounds. Von Tilzer matched the lyrics to a tune he had previously composed. Neither of them had ever attended a baseball game. They were spot on about the peanuts and Cracker Jack. Harry M. Stevens had been selling peanuts at baseball games for years. And Cracker Jack was first sold at a ballpark in 1896. It would be 32 years before Norworth attended his first baseball game. Unfortunately, we have no record of what he had to eat.
Norworth was married for a time to singer-actress Nora Bayes. Together they were a well-known celebrity couple of the day. Nora was the first to sing “Take Me Out to the Ball Game.” Others followed, and the song became popular in vaudeville shows and at intermissions in movie theaters -- but not yet at baseball games.
The first time “Take Me Out to the Ball Game” was sung at a baseball game was in 1934 at a Los Angeles high school game. The first time it was played at a major league game was the performance of the St. Louis Cardinals band led by third baseman Pepper Martin before game four of the 1934 World Series.
It is unclear when “Take Me Out to the Ball Game” was first sung during the seventh inning stretch, as is now the tradition. Although it was sung at some stadiums prior to the 1970s, legendary broadcaster Harry Caray popularized the song by singing it first to the crowds at the Chicago White Sox games during the 1970s, and later at the Chicago Cubs games.
The lyrics that we sing during the seventh inning stretch are actually the chorus of the song. The verses of the song tell the story of a girl who prefers to be taken to a ball game rather than a show.
Jack Norworth at the Piano. National Baseball Hall of Fame, Cooperstown, NY.
Changes were made to the lyrics in 1927 but the chorus that we all know was not changed. The most famous line of the song remains “Buy me some peanuts and Cracker Jack.”
1908 version:
Katie Casey was baseball mad.
Had the fever and had it bad;
Just to root for the home town crew,
Ev’ry sou Katie blew.
On a Saturday, her young beau
Called to see if she’d like to go,
To see a show but Miss Kate said,
“No, I’ll tell you what you can do.”
Chorus
“Take me out to the ball game,
Take me out with the crowd.
Buy me some peanuts and Cracker Jack,
I don’t care if I never get back,
Let me root, root, root for the home team,
If they don’t win it’s a shame.
For it’s one, two, three strikes, you’re out,
At the old ball game.”
Katie Casey saw all the games,
Knew the players by their first names;
Told the umpire he was wrong,
All along good and strong.
When the score was just two to two,
Katie Casey knew what to do,
Just to cheer up the boys she knew,
She made the gang sing this song:
Chorus
“Take me out to the ball game,
Take me out with the crowd.
Buy me some peanuts and Cracker Jack,
I don’t care if I never get back,
Let me root, root, root for the home team,
If they don’t win it’s a shame.
For it’s one, two, three strikes, you’re out,
At the old ball game.”
1927 version:
Nelly Kelly loved baseball games,
Knew the players, knew all their names,
You could see her there ev’ry day,
Shout “Hurray,” when they’d play.
Her boy friend by the name of Joe
Said, “To Coney Isle, dear, let’s go,”
Then Nelly started to fret and pout,
And to him I heard her shout.
Chorus
“Take me out to the ball game,
Take me out with the crowd.
Buy me some peanuts and Cracker Jack,
I don’t care if I never get back,
Let me root, root, root for the home team,
If they don’t win it’s a shame.
For it’s one, two, three strikes, you’re out,
At the old ball game.”
Nelly Kelly was sure some fan,
She would root just like any man,
Told the umpire he was wrong,
All along, good and strong.
When the score was just two to two,
Nelly Kelly knew what to do,
Just to cheer up the boys she knew,
She made the gang sing this song.
Chorus
“Take me out to the ball game,
Take me out with the crowd.
Buy me some peanuts and Cracker Jack,
I don’t care if I never get back,
Let me root, root, root for the home team,
If they don’t win it’s a shame.
For it’s one, two, three strikes, you’re out,
At the old ball game.”
Take Me Out to the Ball Game, 1908.
Library of Congress Baseball Sheet Music, 200033481.
Along with hot dogs and peanuts, Cracker Jack® popcorn is one of the early and traditional baseball foods. Although its famous connection with baseball through the song “Take Me Out to the Ball Game” did not occur until 1908, an 1896 scorecard for a game played in Atlantic City, New Jersey, between the Atlantic City Baseball Club and the Cuban Giants contained a Cracker Jack advertisement.
Frederick Rueckheim immigrated to Chicago from Germany in 1871. Frederick and his brother Louis sold popcorn from a cart in the streets of Chicago. Later they added a caramel coating and peanuts to create the popcorn candy which eventually was marketed as Cracker Jack.
It has been variously reported that the Rueckheims distributed their new product at the 1893 Chicago Columbian Exposition. There is no record of the brothers having a stand at the Expo, but it is possible that they hawked their product on foot throughout the fair.
Cracker Jack Popcorn.
Courtesy of Frito-Lay North America Inc.
In the nineteenth century “crackerjack” was a slang expression that meant “something very pleasing or excellent.” The story goes that a customer, upon tasting the pop corn concoction, exclaimed “That’s crackerjack!” and the Rueckheims took that as the trade name. The Cracker Jack brand name was registered in 1896.
In 1912 toy surprises were first put into every Cracker Jack box. In 1914 and 1915 a baseball card was placed in each Cracker Jack box. Customers were unhappy because there was less space for the caramel corn, and the baseball card distribution was soon discontinued. However, a 1915 Ty Cobb card from a Cracker Jack box sold in 2005 for $94,709.
Each box of Cracker Jack has a picture of Sailor Jack and his dog Bingo. Sailor Jack was modeled after Robert Rueckheim, an eight-year-old nephew or grandson of Frederick (sources vary). Tragically, Robert died of pneumonia shortly after his image first appeared.
Cracker Jack remained a family business until it was sold to Borden Inc. in 1964. In 1997 ownership of the brand was transferred to Frito-Lay North America, Inc.
During the 1980s the New York Yankees hosted an Old-timers Game which was sponsored by the Cracker Jack brand. As part of the festivities, Cracker Jack was cooked on site for players and officials. According to game promoter Marty Appel, “The scent of hot Cracker Jack was almost indescribably wonderful. I’ll always associate Cracker Jack with baseball, and the smell of hot Cracker Jack with the fun of those old-timers games.”
In 2004 the New York Yankees decided to stop selling Cracker Jack and chose instead to offer another brand of caramel corn called Crunch ‘n Munch. The decision left fans stunned and upset. Several months later, the Yankees corrected their error and brought back Cracker Jack. The Yankees’ chief operating officer, Lonn Trost, gave the reason for the return of Cracker Jack: “The fans have spoken.” Cracker Jack is currently sold at almost all of the major league stadiums.
1915 Cracker Jack Popcorn Ty Cobb Baseball Card.
Courtesy of Love of the Game Auctions.
Much credit for the popularity of Cracker Jack at baseball games can be given to “Take Me Out to the Ball Game.” During the seventh inning of every baseball game across the country, fans are reminded of Cracker Jack. Although the Cracker Jack brand owes much of its success to the song, there is no evidence that songwriter Jack Norworth was ever compensated by the Rueckheims. Norworth was simply looking for a rhyme. If “back” did not rhyme with “jack,” who knows what kind of snack baseball fans would be eating today.
Just as with the hot dog, Harry M. Stevens has a prominent role in the association of peanuts with baseball games. In 1895 the Cavagnaros peanut company wished to place an advertisement in one of Harry M. Stevens’ scorecards but did not have money to pay for the ad. Instead Stevens offered to take payment in the form of peanuts, which he then sold at the stadium. According to Amusement Business magazine, this was the origin of the phrase “working for peanuts.”
Hampton Farms Major League Baseball Peanuts Bags
Courtesy of Hampton Farms.
Peanuts became an instant success at baseball stadiums, and by the turn of the century they were standard ballpark fare along with hot dogs, popcorn, and Cracker Jack. Because of the increasing popularity of ballpark peanuts, Stevens later purchased land in Virginia and had peanuts grown for him there. He brought them to all his ballparks in truckloads.
About the same time that Stevens began to sell peanuts, three brothers, Marvin, Charles, and Louis Jacobs, were selling peanuts at Coney Island in New York. The brothers then moved to Buffalo, New York, and began selling peanuts at the local ballpark. In the early twentieth century they expanded to other ballparks that were not being served by Harry M. Stevens. In 1915 the brothers founded a company called Delaware North. Delaware North Companies is now one of the six major companies that provide food concessions to the 30 major league baseball stadiums.
According to the Washington Star, Homer Rose (grandson of Harry M. Stevens), discussing peanuts, said “They’ve never been popular at race tracks because people need to keep their hands free for betting. In baseball, the tension builds slowly. Eating peanuts is part of the nervous habit -- it gives you something to do with your hands.”
During the 1934 season, pitching stars Dizzy and Daffy Dean won 49 games between them for the world-champion St. Louis Cardinals. That same season their older brother Elmer was “pitching” bags of peanuts to customers in the stands at the Houston Buffaloes minor league park.
The off-Broadway musical Diamonds, directed by Broadway legend Harold Prince, made its debut in 1984. The show featured two characters: a child who wants to be a star slugger, and a peanut vendor in the stands. Although the show was not a commercial hit, this is yet another example of the importance of peanuts in baseball culture.
After retiring, many major league players have second careers relating to baseball. Hall-of-Fame pitchers Jim “Catfish” Hunter and Gaylord Perry chose a different career. They both became peanut farmers.
Hampton Farms is one of the leading retailers of peanuts. In grocery stores throughout the country, they are now selling “Major League Baseball” peanuts personalized with the logo of nearby major league teams. According to the Hampton Farms website, “Summer just wouldn’t be the same without baseball and roasted in-shell peanuts. This classic combination has been delighting fans for generations, so it seemed only natural to combine America’s favorite peanuts with America’s favorite summer game.”
However, eating peanuts is not an option for many ballpark fans. The Asthma and Allergy Foundation of America reports that approximately 2% of the U.S. population is allergic to peanuts. On June 29, 2013, the Chicago White Sox hosted an Allergy Awareness Day featuring nut-free seating sections. Brooks Boyer, White Sox Senior Vice President of Sales and Marketing, said: “Hopefully some of our fans and families who typically could not attend a White Sox game due to potential allergic reactions will be able to enjoy an afternoon of White Sox baseball at U.S. Cellular Field.” Likewise, the Baltimore Orioles have a “peanut allergy suite” available at reduced prices for those who need it. Similar considerations are catching on at other major and minor league parks.
Jim “Catfish” Hunter. National Baseball Hall of Fame Library, Cooperstown, NY.
I have been happily married for over 30 years and my wife and I seldom disagree about anything. A major disagreement we have, however, occurs when we go to a baseball game together and I purchase a bag of peanuts. I believe it is part of the great baseball tradition to throw your peanut shells on the floor. She believes this is rude and insists that I put my peanut shells in a bag and carry them out to the garbage can. Judging from the accumulation of peanut shells on stadium floors after the crowds file out, it seems that most people agree with me.
Peanuts were popular at ballparks before the song “Take Me Out to the Ball Game” became a standard, but having them mentioned at every seventh inning stretch certainly doesn’t hurt sales.
During the course of my research at the National Baseball Hall of Fame Library, I was pleased to come across pictures of old-time scorecards and programs that included food listings and prices, as well as advertisements and other information. I have included the following nine examples spanning the years from 1937 through 1974. Poring over these pictures is like looking through a window into our national past.
It is interesting to compare the prices for similar items that you can find at today’s ballparks. In the 1930s a hot dog cost 10 cents; by 1974 it was 50 cents. Peanuts increased from 10 cents in 1937 to 30 cents in 1974.
In the 1938 program, directions to the “retiring rooms” for women say that there is a “matron always in attendance” -- a practice that has disappeared today.
Some surprising food offerings include filet of sole for 40 cents in 1952 from the Cincinnati Reds and a peanut butter sandwich for 10 cents in 1954 from the Orioles.
PHILADELPHIA ATHLETICS, 1937
Philadelphia Athletics Program. National Baseball Hall of Fame Library, Cooperstown, NY.
BOSTON BRAVES, 1938
Boston Braves Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
CHICAGO CUBS, 1947
Chicago Cubs Program. National Baseball Hall of Fame Library, Cooperstown, NY.
PITTSBURGH PIRATES, 1948
Pittsburgh Pirates Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
CLEVELAND INDIANS, 1951
Cleveland Indians Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
CINCINNATI REDS, 1952
Cincinnati Reds Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
BALTIMORE ORIOLES, 1954
Baltimore Orioles Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
PHILADELPHIA PHILLIES, 1967
Philadelphia Phillies Scorecard. National Baseball Hall of Fame Library, Cooperstown, NY.
MINNESOTA TWINS, 1974
Minnesota Twins Program. National Baseball Hall of Fame Library, Cooperstown, NY.
The traditional ballpark food consisting of hot dogs, peanuts, popcorn, and Cracker Jack did not change much from the late nineteenth century through the 1970s. During the 1980s new foods gradually began to be introduced. However, there was a proliferation of food offerings during the 1990s fueled by the opening of twelve new major league ballparks.
Chicago-Style Pizza at Wrigley Field, Chicago, IL.
In the 1980s ballparks gradually began to expand their food offerings to include regional specialties. In 1982 food editor and author Elaine Corn wrote an extensive feature in the Louisville Courier -Journal entitled “Fans Fare: Expanding the Classics of Baseball.” “Now, instead of simply the usual, fans can choose from a wide selection of regional delicacies from Boston to Hawaii. Fans have been stuffing themselves with Bratwurst at Bloomington, Minnesota, kosher products at New York’s Shea Stadium, fried chicken at Pittsburgh, and manapua (a meat-filled or bean-paste-filled bun) in Honolulu.”
For the start of the 1988 season, the New York Yankees dramatically increased their food-choice concessions. They spent 2.5 million dollars during the winter to renovate the food stands and revitalize the menus. Their new menu items included pizza, roast beef and pastrami sandwiches, egg rolls, shrimp, chicken fingers, hamburgers, and French fries. According to Craig S. Trimble, Vice President for Volume Services (now Centerplate), “…We have no intention of challenging the supremacy of the hot dog.…We’re just trying to add a little variety.”
Grilled Wild Alaskan Salmon Sandwich at Safeco Field, Seattle, WA.
In a 1988 New York Times article entitled “New Yankee Call: Getcha Fresh Shrimp,” it was reported that fan reaction was mixed. Yankee fan Robert Brown, while biting into a hot dog, asked, “Why would I come to Yankee Stadium and have a pastrami sandwich? If I wanted a pastrami sandwich, I’d go to a deli.” Season ticket holder Martin Boeck said, “This roast beef is tender and delicious.…usually when you go to a stadium you get garbage.…it seems like you’re in the World’s Fair.” As it turned out, there are far more Martin Boecks than Robert Browns at the ballparks.
Between 1990 and 2000, twelve new major league ballparks were built. Rather than having to redesign the food stands as the Yankees had done in 1988, the new ballparks offered the opportunity to have optimally designed spaces for concession stands.
Tropicana Field in Tampa Bay offers Cuban sandwiches. Coors Field in Colorado has Rocky Mountain oysters. Wrigley Field in Chicago features Chicago-style pizza. Minute Maid Park in Houston has the famous Texas barbeque brisket. Safeco Field in Seattle has a grilled wild Alaskan salmon sandwich.
Oriole Park at Camden Yards in Baltimore and AT&T Park in San Francisco are among several ballparks that feature concessions stands owned by famous retired players from those teams. Former Oriole first baseman and 1970 American League Most Valuable Player Boog Powell owns Boog’s Corner BBQ. Former San Francisco Giant and Hall of Famer Orlando Cepeda owns Orlando’s Caribbean BBQ, featuring its Cha Cha bowl (jerk chicken, beans and rice).
By the end of the twentieth century, all ballparks were featuring a wide variety of food options, well beyond the wildest dreams of Harry M. Stevens.
In 1900 the most common snack foods served at ballparks were peanuts and Cracker Jack. By the end of the twentieth century, they had been supplanted by nachos and garlic fries.
Pulled Pork Nachos at Progressive Field, Cleveland, OH.
Nachos were invented in 1943 by head waiter Ignacio “Nacho” Anaya in Piedras Negras, Coahuila, Mexico. One day after his restaurant (Victory Club) had closed and the chef had gone home, a group of women arrived. They were wives of U.S. soldiers stationed at Fort Duncan in nearby Eagle Pass, Texas. Wanting to serve them quickly, Anaya used what he had on hand – tortillas, cheese, and sliced jalapeños. After cutting the tortillas into small triangles, he covered them with shredded cheddar cheese and heated them so that the cheese melted and then topped them with the sliced jalapeños.
When asked what the dish was called, he replied “Nacho’s Especiales,” i.e., a specialty of Ignacio “Nacho” Anaya.
The popularity of the dish spread quickly throughout Texas and the Southwest. In 1959 nachos were introduced at the El Cholo Mexican restaurant in Los Angeles.
Nachos could be found only at Mexican restaurants until 1973, when they were brought to the old Arlington Stadium, home of the Texas Rangers. According to Carey Risinger, Director of Food, Beverage and Retail at Arlington Stadium, “The standard concessions stand in the 70s was simple. We sold the basics: hot dogs, peanuts, and popcorn.” Risinger wanted to add something new so he decided to open a nachos stand at the ballpark. To make the nachos, Cheez Whiz was melted in a hot fudge warmer and the mixture was ladled over circular corn chips. To help with consistency, jalapeño juice was added to the Cheez Whiz mix. The rest is history, and nachos are now sold at almost every sporting event throughout the country.
Garlic fries were created by Dan Gordon, one of the founders of Gordon Biersch Brewery Restaurants, while he was studying beer making at the Technical University in Munich, Germany. At the request of a professor, he spent a day doing research in a garlic field. At the end of the day he was served a meal featuring ten different dishes made with garlic. He was so inspired by the experience that he created the garlic and deep-fried potato combination as a late-night snack during his final exams.
Garlic fries were first sold at the Gordon Biersch Brewery Restaurant in Palo Alto, California, in 1988. Gordon referred to them as “the perfect carb partner for our beers.”
Gordon Biersch Garlic Fries at AT&T Park, San Francisco, CA.
Garlic fries made their way into a ballpark in 1994 when Gordon Biersch opened a concession stand at Candlestick Park, former home of the San Francisco Giants. Garlic fries have now become a staple at almost all ballparks. At AT&T Park in San Francisco you don’t have to worry about offending your seatmates with garlic breath, since each order of garlic fries comes with two breath mints.
If Jack Norworth were writing his classic “Take Me Out to the Ballgame” song today, he might have written “…Buy me some nachos and garlic fries, I don’t care if we have cloudy skies…”
According to the Centers for Disease Control and Prevention, just over two-thirds of adults and children in the United States today are overweight or obese. This has led to a call by many organizations for healthier eating habits and has become a major emphasis for First Lady Michelle Obama. The United States Department of Agriculture created the website www.choosemyplate.gov to promote healthy eating habits. A 2008 study published by Vegetarian Times reports that over 30 million Americans are either vegetarians or follow a vegetarian-inclined diet. Ballpark concessionaires have acknowledged this trend by increasing the variety of healthy and vegetarian options available at their food stands.
One of the pioneers in convincing the major ballpark concessionaires to offer a veggie dog alternative to the hot dog was actress Johanna McCloy (Ensign Calloway in Star Trek: The Next Generation). Veggie Happy (the organization started by McCloy and originally known as Soy Happy) states that its members are “advocates for vegetarian/vegan options on mainstream concession menus and…known for opening the door to vegetarian hot dogs and frankfurters at MLB [major league baseball] stadiums.”
In 2005 RFK Stadium, then home of the Washington Nationals, was the first major league venue to offer a veggie dog. Rob Sunday, Aramark Resident District Manager, noted “Aramark is pleased to announce vegetarian hot dogs to our baseball fans at RFK. Aramark understands that the palate of today’s fan is more diverse than ever and takes great pride in offering options that reflect these expanding tastes. Through consulting with the advocacy service, Soy Happy, Aramark believes that offering veggie dogs at RFK is an idea whose time has come.”
From this beginning, there has been an expansion of healthy and vegetarian foods introduced at ballparks around the country. The following are a sampling of what one can find at major league ballparks today.
Grilled Vegetable Panini at Angel Stadium (Los Angeles Angels of Anaheim) is made with zucchini, carrots, mushrooms, tomatoes, and pesto sauce
Vegetarian Sushi at Great American Ballpark (Cincinnati Reds)
South Beach Fruit Salad at Marlins Park (Miami Marlins) contains fresh local fruits including grapefruit, watermelon, pineapple, mango, and yellow pear tomato
Portobello Sandwich at Fenway Park (Boston Red Sox) has a large portobello mushroom topped with arugula, tomato jam, and fried onions
Roasted Tomato Hummus with Pita Chips (a Mediterranean dish) at Petco Park (San Diego Padres)
Wild Pacific Salmon at PNC Park (Pittsburgh Pirates), roasted on a cedar plank and seasoned with sea salt and cracked black pepper
South Beach Fruit Salad at Marlins Park, Miami, FL.
Roasted Vegetable Spinach Wrap at Globe Life Park in Arlington (Texas Rangers)
Veggie Kabobs at Target Field (Minnesota Twins) include summer squash and bell peppers, seasoned with pesto
Mushroom Quesadilla at Turner Field (Atlanta Braves)
Edamame at AT&T Park (San Francisco Giants)
Portobello Mushroom Sandwich at Fenway Park, Boston, MA.
The Jewish Virtual Library estimated the American Jewish population in 2012 to be 6.7 million. Sources estimate around 20% of that population observes kosher dietary laws. The majority of Jews who keep kosher live in the Northeast, Southern Florida, major Midwestern cities, or the Los Angeles area.
To be certified kosher, all ingredients in every product, and the process of preparing the product, must be certified for kosher compliance. Several major league ballparks now offer kosher food.
Kosher hot dogs, Italian sausages, pretzels, peanuts, and beer can be found at Oriole Park at Camden Yards. The Miami Marlins park features a concession stand called “Kosher Korner” with hamburgers, cheeseburgers (with soy cheese), corned beef sandwiches and potato knishes. Kosher stands can also be found at Citi Field and Yankee Stadium in New York, Dodger Stadium in Los Angeles, and other venues.
At Fenway Park in Boston, fans have access to a kosher vending machine dispensing hot dogs and knishes. The vending machine is provided by Hot Nosh Boston, a company founded by Wayne Feder, an avid sports fan and orthodox Jew. Feder often had trouble finding kosher food at sporting events, so he decided to invest in a machine that provides quick access to hot, fresh kosher food.
Hot Nosh Boston, as well as other companies including Kosher Sports Inc. and Keep It Kosher LLC, subcontract with the major concession companies to provide their kosher products.
I grew up in California and visited my Aunt Sylvie and Uncle Abe in New York City almost every summer. One of my fondest memories is eating knishes sold at that time from street corner carts. I also enjoyed knishes when Uncle Abe took me to see the New York Mets play at Shea Stadium.
A knish is a Jewish snack food made popular in North America by Eastern European immigrants. It consists of a filling covered with dough that is baked, grilled, or deep fried. Although knishes can have a variety of fillings, the most common is potato. I believe a knish is a perfect baseball snack food. It is a non-messy finger food that allows for concentrating on the game while eating. When prepared like my mother used to make, the potato filling would literally melt in your mouth.
Kosher food is tasty and healthy. Lenny Kohn (of Kohn’s Kosher Deli) runs a kosher stand at Busch Stadium in St. Louis and estimates that 95% of his customers are non-kosher. They simply prefer the kosher food for its taste and quality.
Health Magazine produced a feature on its website entitled “Healthy Eats at 30 Major League Baseball Parks.” At only 180 calories the magazine’s healthy choice for Citi Field in New York was the potato knish.
Potato Knish at Yankee Stadium, New York, NY.
On June 15, 2005, Cheez Doodles, manufactured by Wise Foods Inc., became the official cheese doodle of the New York Mets. “Our partnership with Wise snacks is a home run,” said Mets Senior Executive Vice President Jeff Wilpon. “Nothing signals the arrival of summer more than watching Mets baseball and enjoying delicious Wise snacks.” To this day, Cheez Doodles remains the official cheese doodle of the New York Mets.
For those of you not familiar with cheese doodles, they are a snack food made with corn meal that has been puffed, baked, and coated with cheddar cheese. Morrie Yohai invented the cheese doodle in the 1940s. After serving in the U.S. Marines in World War II, he returned home to the Bronx, New York, and took over his family’s snack food business. The business was later purchased by Wise Foods. Today over 15 million pounds of Cheez Doodles are manufactured each year.
Bag of Cheez Doodles. Courtesy of Wise Foods Inc.
According to Hallmark, March 5 is National Cheese Doodle Day. This corresponds to the approximate start of spring exhibition games. It is hard to determine which of these events is more highly anticipated by baseball fans.
Now that you know more about cheese doodles than you ever wanted to know, are you wondering what the function of an “official” cheese doodle is versus an “unofficial” cheese doodle? Do you want to understand how other major league teams cope without an official cheese doodle? As with the mystery surrounding the invention of the hot dog, we may never know the answer.
In certain American consumer markets, a limited number of companies dominate a large share of that particular market. The majority of cellular phone service is provided by Verizon, AT&T, Sprint, or T-Mobile. Most of the breakfast cereals sold in grocery stores are products manufactured by Kellogg, General Mills, Post, or Quaker. A similar situation exists with major league baseball concessionaires.
Concessions services at all 30 major league parks are provided by six companies: Aramark, Delaware North Companies, Levy Restaurants, Centerplate, Legends, and Ovations Food Services.
Aramark
Atlanta Braves (Turner Field)
Boston Red Sox (Fenway Park)
Colorado Rockies (Coors Field)
Houston Astros (Minute Maid Park)
Kansas City Royals (Kauffman Stadium)
New York Mets (Citi Field)
Philadelphia Phillies (Citizens Bank Park)
Pittsburgh Pirates (PNC Park) -- [except luxury suites and club restaurants]
Toronto Blue Jays (Rogers Centre)
Delaware North Companies
Baltimore Orioles (Oriole Park at Camden Yards)
Cincinnati Reds (Great American Ball Park)
Cleveland Indians (Progressive Field)
Delaware North Companies continued
Detroit Tigers (Comerica Park)
Milwaukee Brewers (Miller Park)
Minnesota Twins (Target Field)
San Diego Padres (Petco Park)
St. Louis Cardinals (Busch Stadium)
Texas Rangers (Globe Life Park in Arlington)
Levy Restaurants
Arizona Diamondbacks (Chase Field)
Chicago Cubs (Wrigley Field)
Chicago White Sox (U.S. Cellular Field)
Los Angeles Dodgers (Dodger Stadium)
Miami Marlins (Marlins Park)
Pittsburgh Pirates (PNC Park) -- [luxury suites and club restaurants only]
Washington Nationals (Nationals Park)
Centerplate
San Francisco Giants (AT&T Park)
Seattle Mariners (Safeco Field)
Tampa Bay Rays (Tropicana Field)
Legends
Los Angeles Angels of Anaheim (Angel Stadium of Anaheim)
New York Yankees (Yankee Stadium)
Ovations Food Services
Oakland Athletics (O.co Coliseum)
Additionally, these six companies have the concessions contracts for many minor league baseball teams as well as at other sport venues. Some minor league teams handle their concessions in-house, without contracting with an outside company.
Aramark Corporation provides food service and facilities management to businesses, educational institutions, prisons, and health care institutions, as well as sports facilities. It is headquartered in Philadelphia, Pennsylvania, and had annual 2012 revenues of 13.5 billion dollars. Aramark was founded in 1936 by brothers Davre and Henry Davidson, who started with vending services for employees in Southern California’s aviation industry. By purchasing Harry M. Stevens, Inc. in 1994, the company increased their ballpark presence.
Delaware North (also known by one of its division names, Sportservice) is a food service and hospitality company that operates hotels as well as concessions at airports, gaming and entertainment venues, and sports venues. It is headquartered in Buffalo, New York, and had annual 2012 revenues of 2.6 billion dollars. Delaware North was founded in 1915 by brothers Charles, Marvin, and Louis Jacobs, who started selling peanuts at Coney Island in New York, and then expanded to the Buffalo Bisons’ ballpark.
Levy Restaurants specializes in providing vending and food services to entertainment and sports venues. It is based in Chicago, Illinois, and is a subsidiary of Compass Group, a British multinational contract food service and support services company. Levy Restaurants was founded by Larry Levy who began with D.B. Kaplan’s Delicatessen in Chicago in 1978. In 1982 his company obtained the concessions contract at Comiskey Park, then home of the Chicago White Sox. In 1985 it got the contract for the Chicago Cubs at Wrigley Field. In 2006 Levy Restaurants was purchased by Compass Group. In 2012 annual revenues for Compass Group were 27.4 billion dollars.
Although Centerplate serves only three major league baseball parks, it is the largest food service provider to the teams of the National Football League as well as to soccer teams in the United Kingdom. Centerplate was formerly known as Volume Services America and was a division of the Flagstar Companies. In 1995 Flagstar sold Volume Services to the Blackstone Group. The name was changed to Centerplate in 2004. In 2009 it became an independent privately-owned company headquartered in Spartanburg, South Carolina. The company traces its roots back to 1929 when Nathaniel Leverone founded the Automated Canteen Company of America, selling candy bars, nuts and chewing gum from vending machines.
Legends, the smallest of the six major league ballpark concessionaires, is jointly owned by the New York Yankees, the Dallas Cowboys football team, and Checketts Partners Investment Fund. The company acquired the contract for a second major league baseball park when the Los Angeles Angels signed with Legends to replace Aramark at the end of the 2013 season. Headquartered in New York City, Legends has been selected to operate the observation deck opening in 2015 at the top of One World Trade Center.
Ovations Food Services provides food and beverage service to arenas, stadiums, amphitheaters, fairgrounds, and convention centers throughout the country. Although it has served AAA teams including the Fresno Grizzlies at Chukchansi Park and the Sacramento River Cats at Raley Field, its January 2014 contract with the Oakland A’s is its first with a major league team. Ovations Food Services is a subsidiary of Comcast Spectacor, a global leader in the sports management industry. Comcast Spectacor in turn is a subsidiary of Comcast Corporation, the nation’s largest video and high-speed Internet provider.
A recent trend in ballpark food service is to have well-known local restaurants or brands represented in the concession options. Ghirardelli Chocolate Company provides sundaes at AT&T Park in San Francisco. Seattle’s Ivar’s Seafood Restaurant offers its famous clam chowder at Safeco Field. Nathan’s Famous hot dogs are sold at Yankee Stadium. In most cases the local restaurant or brand contracts directly with the ballpark’s concessionaire.
According to SportsBusiness Journal, annual revenue from on-site game day concessions in 2012 was 10.7 billion dollars. This included football, basketball, and hockey as well as baseball venues. Aramark revenues accounted for 31% of the market, Levy Restaurants had 24%, Delaware North had 19%, and Centerplate had 16%. (Figures are not available for Legends and Ovations Food Services.)
During the era of traditional ballpark food (approximately 1900 to 1980), Harry M. Stevens was a giant in the business. However, his lifetime revenues represent only a small fraction of the big business of ballpark concessions today.
Fans at baseball games are accustomed to seeing food vendors making their way throughout the stands, climbing up and down the steep stairs as they call out their wares and pass the purchased products down the row of seats. Some vendors are particularly good at attracting attention by their dress or actions.
The tradition of colorful vendors hawking their goods goes back all the way to Harry M. Stevens himself. In his red coat and straw hat, Stevens was a noticeable figure in the ballpark stands, quoting Shakespeare and Byron as he offered his scorecards and food for sale.
The best-known vendor of modern times is Roger Owens of Dodger Stadium, known as “The Peanut Man.” Roger has worked at Dodger Stadium from its opening in 1962 until the present day. He became known for his trick tossing of peanut bags to customers up to 30 rows away. His antics gained him much renown including an appearance on The Tonight Show Starring Johnny Carson in 1976. Although many ballplayers have had biographies written about them, Roger Owens may be the only peanut vendor to have achieved this honor. The Perfect Pitch: The Biography of Roger Owens, The Famous Peanut Man at Dodger Stadium, by Daniel S. Green, was published in 2004.
At the age of 24 in 1938, Dan Ferrone was picked from among hundreds of young men waiting for a vending job with the Chicago Cubs at Wrigley Field. Although he had other jobs as well, he remained at Wrigley Field until 1995. He was a familiar face to generations of fans as he peddled first programs, then beer and peanuts. Sadly, during his 58-year career, he never was able to watch the Cubs win a World Series.
Hot dog vendor Charley Marcuse worked for the Detroit Tigers for the last decade. He was known as the “Singing Hot Dog Man” for singing the words “hot dog” in operatic falsetto. He was also known for his adamant dislike of ketchup, wanting all of his customers to put mustard on their hot dogs. This condiment controversy led to a dispute with management and to Charley leaving the Tigers in 2013.
Fans in the outfield bleachers at Coors Field in Denver are served by beer vendor Brent Doeden, also known as “Captain Earthman.” He hands out cards with both his cell phone number and his planetary location number. No one quite knows what it means, but he has many loyal customers.
In the 1970s at the Texas Rangers Arlington Stadium, Ray Jones was known as “The Birdman of Pennants.” Jones used a bird call to attract attention to the banners he was selling. He also made a yelping sound like a dog and performed “The Eyes of Texas” in a shrill whistle.
Many other vendors find subtle ways to attract customers. At a recent Arizona Diamondbacks game we attended, the man with a tray of drink cups bellowed out “Lemonade, lemonade, just like your grandma made.” Both my wife and I bought his lemonade.
“The Peanut Man,” Roger Owens. Courtesy of Roger Owens.
To paraphrase Laozi and Mao Zedong, a journey of 30 ballparks begins with a single hot dog. During the 2014 baseball season my wife and I traveled to each of the major league stadiums to investigate the variety of food offerings from hot dogs and sausages to the specialities of the new food era.
In the early days, when baseball was becoming the American national pastime, a fan who wanted a hot dog at the ballpark got just that -- a hot dog. There was no choice; a hot dog was a hot dog. Times have changed. The Great American Hot Dog Book: Recipes and Side Dishes From Across America, by Becky Mercuri, includes hot dog recipes from every state in the union. Many ballparks have signature hot dogs as well as a variety of other hot dogs from which to choose.
Still not convinced that you want a hot dog? You have many other choices. Some stadiums have gone all out to showcase unique, gourmet-style food. Many parks emphasize regional food as well as having offerings from well-known local restaurants. There are also several ballparks where retired ballplayers are shaping new careers as signature food purveyors.
Some stadiums have luxury levels open only to fans who have purchased the most expensive seats. I have elected not to include concession stands located on these luxury levels. I have also chosen not to include sit-down restaurants located inside a ballpark. Instead the focus is on food available to general ticketed fans throughout the ballpark.
My culinary tour is not intended to be comprehensive nor a ratings guide, but rather to celebrate the variety of foods now available at major league ballparks. The era of hot dogs, peanuts and Cracker Jack alone is now just a memory, as you will soon discover.
ARIZONA DIAMONDBACKS
Chase Field opened in 1998 and has been the only home of the Diamondbacks since they were added to the National League as an expansion team. The field was one of the first major league stadiums built with a retractable roof.
Featured Hot Dog/Sausage
The Venom Dog is named after the venomous nature of the western diamondback rattlesnake. The Venom Dog is a specially-crafted habanero sausage with black beans, pico de gallo, and sour cream. Habanero is a very hot chili pepper that originated many centuries ago in the Amazon region of South America.
Venom Dog
More Hot Dogs and Sausages
The Sonoran Dog gives fans a taste of old Mexico. Mesquite-smoked bacon is grilled and wrapped around a hot dog, which is then topped with pico de gallo and ranch-style beans. Mayonnaise is drizzled over the top. This special hot dog gets its name from the Mexican state of Sonora where it originated.
The D-Bat corn dog, at 18 inches long, is the Diamondbacks contribution to the supersize trend. The hot dog is filled with cheddar cheese, bacon, and jalapeños, coated in cornmeal batter and deep fried, then served on a bed of French fries.
Macayo’s Mexican Kitchen
The first Macayo’s restaurant opened in 1946 in the Phoenix area. Still owned and operated by the Macayo family, there are now 14 Macayo locations in Arizona, including one at Chase Field. A wide variety of burritos, tacos and fajitas are offered. The Macayo Bowl contains your choice of chicken or beef, rice, beans, cheese, sour cream, guacamole, and pico de gallo. Here you can also get spinach con queso (spinach cheese dip).
Fatburger
Although founded in Los Angeles, the first Fatburger at a ballpark opened in 2008 at Chase Field. The iconic hamburger chain was started in 1946 by Lovie Yancey. Today Fatburgers can be found in over 30 countries around the world. Magic Johnson, Queen Latifah, Kanye West, and Montel Williams have all, at one point, been involved in the ownership or operation of Fatburger franchises. The D-backs Double cheeseburger features two one-third pound patties and two slices of cheese with lettuce, tomatoes, and onions on a toasted bun. In addition to the beef burgers Fatburger also offers a turkey burger.
Fatburger D-backs Double Cheeseburger
Rey Gloria
Rey Gloria’s Tamale Stand is run by former Diamondback security guard Rey Cota. In 2006 Diamondback president Derrick Hall tasted one of Rey’s tamales, and he liked it so much that he offered his employee a stand at Chase Field. Cota’s red chili and green corn tamales are based on a recipe from his mother, Gloria.
Bisbee Tamale
Vegans can find a tamale to their liking known as the Bisbee Tamale provided by the Tucson Tamale Company. It is stuffed with soy chorizo, potatoes, and pinto beans, and served with guacamole and salsa roja (made from red chilies and tomatillos).
Bisbee Tamale
Antipasto and Greek Salads
At the Streets of New York pizza stand, two specialty salads are offered. The antipasto salad has capicola ham, Genoa salami, pepperoni, tomatoes, cucumbers, bell peppers, black and green olives, pepperocini (a mild yellowish-green pepper), and includes a homemade Italian dressing. The Greek salad contains romaine lettuce, tomatoes, cucumbers, bell peppers, red onions, Kalamata olives, feta cheese, and comes with a traditional Greek dressing.
Caramel Apples and New York Cheesecake
Whoever first said “An apple a day keeps the doctor away” probably wasn’t thinking about caramel apples. These colorful treats on sticks can be ordered with caramel alone, with additional nuts, or with either chocolate or rainbow sprinkles.
For many people, New York brings up images of the Statue of Liberty or the Empire State Building. But for cheesecake lovers the first thing that comes to mind is New York-style cheesecake. New York cheesecake is rich with a dense, smooth, and creamy consistency. The Streets of New York stand at Chase Field serves its slices with cherry sauce.
New York Cheesecake
ATLANTA BRAVES
Turner Field was originally built as Centennial Olympic Stadium in 1996 as part of the Summer Olympic Games in Atlanta. The field is named for Ted Turner, founder of CNN cable news network and former owner of the Atlanta Braves. The Braves are planning to build a new stadium in suburban Cobb County. Completion is expected in 2016.
Featured Hot Dog/Sausage
The Taste of the Majors stand at Turner Field features two Southern-themed hot dogs. One is the Dixie Dog which is twelve inches long, weighs one-half pound, and (reflecting its Southern heritage) is flash fried. Then it’s topped with pulled pork, cole slaw, pickles and barbeque sauce.
More Hot Dogs and Sausages
The Taste of the Majors also offers the Georgia Dog with cole slaw, relish, and Vidalia onions. Vidalia onions, the official state vegetable of Georgia since 1990, were developed accidentally in Georgia during the Great Depression. When farmers planted onions in Georgia’s sandy soil, what grew was a strange onion that was sweeter than ordinary onions. The sweet onion was promoted by the Piggly Wiggly grocery store chain, which was headquartered in the town of Vidalia.
Dixie Dog
One of the hot dog stands at Turner Field is called Nat’s Grand Slam Franks Trolley Car 207, and is shaped like a trolley car. The Nat’s is for National Deli, the provider of Turner Field hot dogs. After extensive research, I could find no significance to the number 207, but the red trolley car design is eye-catching.
Waffle House
Waffle House restaurants, founded in 1955, are located throughout the South. On the www.oneforthetable.com website, Ann Nichols refers to the Waffle House as the “unofficial flower of the Southern interstate exit.” Since 1955 the over 1,500 Waffle House restaurants have served 880 million waffles and 1.3 billion cups of coffee. The Waffle House at Turner Field serves its classic waffle, chocolate chip waffle, and peanut butter waffle. The popular Double Hash Browns come with onions, cheese, ham, and peppers.
Chocolate Chip Waffle
Holeman & Finch Public House
Atlanta’s Holeman & Finch Public House (called H&F) is known for its handcrafted double patty cheeseburgers served each night at 10 PM. Only 24 burgers are prepared each night, and are reserved ahead of time by discerning customers. Now burger fans have another opportunity to purchase these burgers at the H&F Stand at Turner Field, where larger numbers of the famous cheeseburger are served up, along with homemade French fries.
Kevin Rathbun
Restaurateur and chef Kevin Rathbun owns three award-winning restaurants in Atlanta. His restaurants have been featured in The New York Times and USA Today as well as on the The Today Show and Good Morning America. Mr. Rathbun has worked with well-known chefs including Emeril Lagasse. Kevin Rathbun Steak at Turner Field serves two items: a steak sandwich and a Big Kev. The Big Kev is a double steak sandwich for those who want a pound of meat. A viewing window beside the Kevin Rathbun Steak stand allows customers to watch a butcher cutting the steaks.
Mayfield Dairy Farms
In 1910, T. B. Mayfield, Jr., purchased 45 Jersey cows and began delivering milk to customers in his hometown of Athens, Tennessee (about 150 miles north of Atlanta.) Today Scottie and Rob Mayfield are the fourth generation of the Mayfield family to run Mayfield Dairy Farms. Its well-known ice cream is served at Turner Field in 13 flavors including chocolate chip cookie dough, and butter pecan.
Frozen Pints
Frozen Pints produces craft beer ice cream. According to their creators, “Someone spilled a beer near the ice cream maker, and in a moment of slightly inebriated inspiration, we found our calling.” Four flavors are offered at the Turner Field stand: peach lambic (light, tart, sweet peach, champagne-like); vanilla bock (creamy vanilla, hints of banana and clove); brown ale chip (hints of roasted hazelnut, dark chocolate chip); malted milk chocolate stout (chocolate malted milk, hints of coffee.) The alcohol by volume (ABV) is between 1% and 2% and fans must be at least 21 years of age in order to purchase the ice cream.
Yicketty Yamwich
Chipper Jones and the Yicketty Yamwich
The term “yicketty” means to hit a home run. It was first introduced by Atlanta Braves third baseman Chipper Jones via a July 25, 2012, tweet. Chipper, who retired in 2012, played his entire 20-year career with the Atlanta Braves and holds the team record for career on-base percentage. In honor of Chipper Jones is the Yicketty Yamwich. The Yamwich contains boneless short ribs, Brie cheese, apple butter spread, baby spinach, and cheddar cheese, a combination not likely found elsewhere in a sandwich.
Cinnamon-Glazed Pecans
According to the United States Department of Agriculture, Georgia is the leading producer of pecans with over 40% of the nation’s crop. Turner Field offers fresh, warm cinnamon-glazed pecans. Fans can watch the pecans swirling in the mixer while they are being glazed.
Cinnamon-Glazed Pecans
BALTIMORE ORIOLES
Oriole Park at Camden Yards opened in 1992. It was built at the beginning of the “retro” major league ballpark trend that occurred during the 1990s and early 2000s in which the ballparks were built for baseball only and each park has a unique character. It is believed that Babe Ruth’s father once owned a saloon on a plot of land that is now center field of Oriole Park.
Birdland Hot Dog
Featured Hot Dog/Sausage
Baltimore Magazine refers to Stuggy’s as a “Baltimore institution.” Stuggy’s restaurants are located in the Fell’s Point and Federal Hill neighborhoods of Baltimore. In a mission to create the most delicious and nutritious hot dog for their hometown of Baltimore, a father and son traveled to many countries researching hot dogs. They came up with an all-beef, gluten-free, kosher-style hot dog that has become an area favorite. At the Stuggy’s stand at Oriole Park, fans can try some unique creations. The Birdland Hot Dog has smoked brisket, pepperoni hash, tomato jam, and frizzled onions on top of the hot dog.
More Hot Dogs and Sausages
Two other hot dogs of note are available at the Stuggy’s stand. The Crab Mac ‘N Cheese Hot Dog, as its name suggests, is covered with crab and macaroni and cheese. The Early Bird Hot Dog comes with a fried egg, cheddar cheese and crispy bacon.
The Sausage Haus serves Natty Boh bratwurst. Natty Boh is short for National Bohemian (a local beer that was first brewed in Baltimore in 1885) which is added to the German-style handmade bratwurst.
Polock Johnny’s has been serving Polish sausages in Baltimore since the early part of the twentieth century. Since 1921 its motto has been “Polock Johnny is my name; Polish sausage is my game.” “Polock” is an alternate spelling of “polack” which the American Heritage Dictionary refers to as “offensive.” Polock Johnny’s Polish sausage comes with “the works”: green peppers, onions, cucumbers, celery, and relish.
The traditional hot dogs at Oriole Park are made by Esskay. Esskay was founded in 1858 by German immigrant William Schluderberg. Esskay has been a leader in marketing innovation, being the first company to feature the Muppets puppets in its advertisements. Esskay remained an independent company for 127 years, until it was sold in 1985 to Smithfield Foods.
Gino’s Hamburgers
Gino’s Hamburgers was a fast food restaurant chain founded in Baltimore by Baltimore Colts’ defensive end Gino Marchetti and running back Alan Ameche. The chain had grown to 359 restaurants when it was purchased by the Marriott Corporation in 1982. Marriott discontinued the brand and converted the locations to Roy Rogers Restaurants. A new version known as Gino’s Burgers and Chicken was opened in 2010 by Tom Romano, who was chief operating officer of Gino’s Hamburgers in 1982 when the chain was sold. At Oriole Park, Gino’s serves hamburgers and chicken tenders. The Camden Giant Burger has a crab cake on top of the hamburger. The Bang Bang Chipotle Burger comes with fried onions and jalapeños.
Soft-Shell Crab Sandwich and Maryland Crab Soup
The Old Bay Seafood stand features products made with Old Bay seasoning. For over 70 years, the Old Bay seasoning’s blend of 18 spices and herbs has brought the flavor of the Chesapeake Bay area to the rest of the country. At Oriole Park, along with traditional crab cakes, fans can sample a soft-shell crab sandwich and Maryland crab soup. For marine biologists, a soft-shell crab is a crab which has recently molted its exoskeleton; for the rest of us, this means almost the entire crab can be eaten rather than having to shell the crab first in order to reach the meat. The soft-shell crab sandwich is served with a choice of cocktail or tartar sauce.
Soft-Shell Crab Sandwich
Maryland crab soup is made from blue crabs found along the mouth of Chesapeake Bay in Virginia and North Carolina. It was first made by Native Americans who lived along these shores. They would combine the crab meat with vegetables and steam them together in large pots.
Flying Dog
In 1983 George Stranahan climbed the dangerous K2 peak in Pakistan, the second highest mountain in the world. Later while out having a drink and celebrating his success, he happened to notice a large oil painting of a dog that appeared to be flying. At this point, you are probably asking what this has to do with ballpark food in Baltimore. The answer is: in 1990 Stranahan founded a brewpub in Aspen, Colorado, and named it the Flying Dog. Later he opened a full-fledged brewery in Denver. (We’re still not to the Baltimore ballpark, but we’re getting there.) In 1994 the brewery moved to Frederick, Maryland. Finally, back at Oriole Park, the Flying Dog Brewery stand offers Chesapeake Waffle Fries. Its fries are topped with a crab dip made with Old Bay seasoning.
Boog Powell’s BBQ
Boog Powell was a first baseman with the Baltimore Orioles from 1961-74. He won the American League Most Valuable Player Award in 1970. He later appeared in television commercials for Miller Lite Beer. Boog is now the owner of Boog’s BBQ serving beef, pork and turkey sandwiches. The beef sandwich is Maryland pit beef (charcoal-grilled top roast, sliced thinly), a regional specialty.
Boog’s Pit Beef Sandwich
Jack Daniels
The Jack Daniels stand, in addition to selling whiskey drinks, offers pulled pork sandwiches, hot dogs, and bacon on a stick. For fans who can’t make up their minds, the Triple Crown Sandwich is available featuring pulled pork, a grilled hot dog and bacon on a stick combined into a pork lover’s delight.
Tako Korean BBQ
According to the 2010 Census, Baltimore has the third largest Korean-American population in the United States. Tako Korean BBQ at Oriole Park serves Kogi beef and Kogi chicken takos. The shell in which the tako is served is similar to that of the Mexican soft taco, reflecting the fusion of Korean and Mexican food found in the taco food trucks of the Los Angeles area. The takos come with a choice of sweet Asian slaw or kimchi. Kimchi is a traditional fermented Korean side dish usually made from cabbage. Also available are steamed Asian buns filled with either barbequed pork or edamame, and Pad Thai Cold Noodle Salad.
Kogi Chicken Takos
Reuben Sandwich
The Baseline Chop House stand serves a traditional Reuben sandwich. A Reuben sandwich contains corned beef, Swiss cheese, Russian dressing and sauerkraut, all grilled and served on rye bread. Two stories exist as to the origin of the Reuben name. One holds that Reuben Kulakofsky was the inventor as part of a group effort by members of his weekly poker game held in the Blackstone Hotel in Omaha, Nebraska, during the 1920s. Another account says that the sandwich was created in 1914 by Arnold Reuben, a New York City delicatessen owner. As is the case with the origin of the hot dog, we may never know for sure.
BOSTON RED SOX
Fenway Park, built in 1912, is the oldest stadium in the major leagues. Although its age contributes to the old-time charm, it limits the amount of space for food concessions. As a result, nearly all the food booths are on the concourse level or on Yawkey Way just outside the park. During games, Yawkey Way is considered part of the stadium and can be accessed only with a game ticket.
Featured Hot Dog/Sausage
The Fenway Frank (along with the Dodger Dog) is the best known of the “classic” ballpark hot dogs. It is made by Kayem Foods in nearby Chelsea. Kayem Foods was founded in a small storefront in 1909 by Polish immigrants Kazimierz and Helena Monkiewicz. The hot dog is boiled and grilled and served on a traditional New England-style bun (crustless on the side with a split top). Fenway Franks are sold throughout the ballpark. The Monster Dog is an extra-large version of the Fenway Frank.
Fenway Frank
More Hot Dogs and Sausages
Located on Yawkey Way is The Best Sausage Company. It’s hard to miss this stand when its employee, wearing a oversized sausage hat, is actively waving and hawking at potential customers. The sausage varieties are Italian (sweet or hot), Cajun, or chicken. It also serves steak tips and pepperoni pizza. I noticed that The Best Sausage Company did not serve sausage pizza, and I inquired about this. I was told that they grill the sausage but they are not going to bother putting it on pizza. (Maybe one has to be a native New Englander to understand this).
Legal Sea Foods
The Fenway Fish Shack features sea food from Legal Sea Foods, a Cambridge, Massachusetts, restaurant and fish market run by the Berkowitz family. The “Legal” part of the name comes not from any law but from the connection of the original family grocery store with Legal Stamps (an early trading stamp incentive for cash-paying customers). At the Fenway Fish Shack fans can get fish and chips, fish sandwiches, fried clams, and clam chowder. With the slogan “If it isn’t fresh, it isn’t Legal,” you can count on having fresh fish every time.
Nicky’s Peanut Wagon
Nicky’s Peanut Wagon can be found at the entry to Yawkey Way. Nicky’s grandfather George began the family peanut business in 1912. In addition to fresh roasted peanuts, Nicky sells pistachios, cashews, salted almonds, and honey-roasted peanuts.
Pork Jerky
In addition to the normal packaged snack items such as Cracker Jack and candy bars, stands throughout Fenway sell Krave pork jerky. According to the Krave website, 2014 Boston Marathon winner Meb Keflezighi trained on Krave jerky products.
Whoopie Pie
Besides ice cream, Scoop Scoop Scoop sells the New England dessert favorite, Whoopie Pie, the “official state treat” of Maine. Said to have originated with the Pennsylvania Amish, the Whoopie Pie has two rounded mounds of cake (usually chocolate) with a cream filling between them.
Whoopie Pie
El Tiante
The El Tiante stand is named for former Boston Red Sox pitcher Luis Tiant, known as El Tiante. Tiant, a native of Cuba, pitched the opening game of the 1975 World Series after having a stunning year for the Red Sox. The signature item offered at the El Tiante stand is the Cuban sandwich: layers of ham, pork, cheese, pickles and mustard on a grilled bun.
Cuban Sandwich
World Fare
Sandwich lovers have several choices at Fenway’s World Fare. Choices include corned beef on marble rye; hot Reuben with sauerkraut, Russian dressing, and Swiss cheese; hot pastrami with spicy mustard; and hot Italian beef au jus served with fresh mozzarella and banana peppers.
Lobster Roll
The Home Plate Grill serves the traditional New England lobster roll, with the lobster meat soaked in butter and placed in a steamed roll with the opening on the top. According to The Encyclopedia of American Food and Drink by John F. Mariani, the lobster roll was created at Perry’s restaurant in Milford, Connecticut, in the 1920s.
Lobster Roll
Visitors Veggies
Located on the visitors’ side of the concourse is Visitors Veggies. Veggie burgers, veggie dogs, and hummus and chips can be found there. Is it located on the visitors’ side because true members of Red Sox Nation don’t like their veggies?
Big Concourse Sandwiches
Big Concourse Sandwiches features two interesting specialties. The Breakfast Burger is a hamburger patty topped with a fried egg, fresh mozzarella and spicy chipotle sauce. The Portobello Sandwich has a large portobello mushroom topped with arugula, tomato jam, and fried onions.
Kosher Vending Machine
Although there is not a kosher stand at Fenway Park, there is a kosher vending machine which dispenses hot cheese pizza slices, mozzarella sticks, pizza pockets, onion rings, and potato knishes. It is the only kosher vending machine located at a major league ballpark.
Fenway Fry Bar
The Fenway Fry Bar sells French fries with a variety of interesting toppings: baked beans, poutine gravy, chili con carne, and nacho cheese. It also sell Fenway Spiral Fries which are spiral-shaped potato chips served on a wooden skewer.
CHICAGO CUBS
Wrigley Field opened in 1914 and is celebrating its centennial in 2014. The stadium is known for its ivy-covered outfield wall. Wrigley Field and Boston’s Fenway Park are the only stadiums remaining from the early days of the American and National Leagues. However, unlike Fenway Park, Wrigley Field is still waiting to see its first World Series championship home team.
TV Dinner Dog
Featured Hot Dog/Sausage
The Decade Dogs stand celebrates the history of Wrigley Field by offering hot dogs from various decades. The 1950s dog is the TV Dinner Dog. The TV dinner was first introduced by Swanson in 1954. It fulfilled two post-war trends: interest in time-saving activities and fascination with television. Swanson’s sold over ten million TV dinners in its first year. The TV Dinner Dog is topped with mashed potatoes, corn, fried onions, and gravy.
More Hot Dogs and Sausages
Other Decade Dogs include the 1910s Reuben Dog topped with sliced corned beef, sauerkraut, Thousand Island dressing, and Swiss cheese. The 1920s Chicago Dog comes with tomato wedges, pickle spears, diced onions, neon-green relish, sport peppers, mustard and celery salt on a poppy seed bun. The 1960s Buffalo Wing Dog is topped with diced chicken, buffalo sauce, and blue cheese slaw. Finally, the 1970s Pulled Pork Dog is covered with pulled pork, cole slaw, and barbeque sauce.
Vienna Beef provides the beef hot dogs and sausages at Wrigley Field. Vienna Beef hot dogs were first introduced at the 1893 Chicago Columbian Exposition by Austrian immigrants Emil Reichel and Sam Ladany. They opened their first store in 1894 on Chicago’s Near West Side and began selling their products to other restaurants and markets in 1900. It remains, 121 years later, Chicago’s best-selling and best known hot dogs.
The Maxwell Street Polish consists of a fried Polish sausage topped with grilled onions, green peppers, and yellow mustard. It traces its origins back to Chicago’s Maxwell Street Market, where the sandwich was created by Jimmy Stefanovic at his hot dog stand in 1939. Chicago Sun-Times food writer Sandy Thorne Clark called the Maxwell Street Polish “a classic food synonymous with Chicago.” Fans at Wrigley Field can order this classic at the Big Dawgs stand.
Chicagoans and visitors are not limited to Vienna Beef hot dogs and sausages. High Plains Bison is “the official lean meat of the Chicago Cubs.” Its stand at Wrigley Field offers bison hot dogs, bison bratwurst, and bison Italian sausage. Bison is naturally lean and lower in saturated fat than beef, chicken, pork, or salmon. High Plains bison graze on grasses, sagebrush, and other native vegetation, and the meat contains no additives or fillers.
Giordano’s Stuffed Pizza
Chicago is known for its deep-dish pizza and its stuffed pizza. Deep-dish pizza has a very high crust creating a pizza that resembles a pie more than flatbread. Unlike other pizza, the extra ingredients (e.g. sausage or pepperoni) are layered on the crust with the sauce and cheese on top. A stuffed pizza is similar to a deep-dish pizza but with an additional layer of dough on top, covered with more tomato sauce. Giordano’s may have been the originator of stuffed pizza, though there is a competing claim from Nancy’s Pizza of Chicago. At Wrigley Field, Giordano’s sells stuffed sausage and cheese pizzas. Giordano’s has been named Chicago’s best pizza by NBC, The New York Times, Chicago Tribune, and Home & Garden Magazine.
Uncle Dougie’s Barbeque Sauce
In 1989 Chicago resident Doug Tomek invented a marinade for preparing tasty chicken wings without frying. Friends and family members loved the marinade and asked Doug to make his wings at all of their events. This encouraged Doug to start a business, and Uncle Dougie’s was born. Today Uncle Dougie’s produces many different marinades and sauces. Pulled pork sandwiches with Uncle Dougie’s barbeque sauce are available throughout Wrigley Field.
Prairie City Bakery
Many ballparks have an official hot dog, but Wrigley Field also has an official cookie. Prairie City Bakery’s Giant Chocolate Chip Cookie is indeed “the official cookie of the Chicago Cubs.” Prairie City Bakery was founded in nearby Vernon Hills in 1994 by Bill Skeenes and Bob Rosean. Both Bill and Bob previously worked for Sara Lee Bakery. Today Prairie City Bakery products are found in over 20,000 locations nationwide. In addition to the chocolate chip cookie, fans at Wrigley Field can also buy a Big n’ Fudgy Brownie.
Big n’ Fudgy Brownie
Asian Pork Burger
A specialty item available at the Decades Diner is the Asian Pork Burger, a meat patty made with ground pork instead of beef or turkey. The burger is topped with Asian slaw (cole slaw with a sweet vinaigrette dressing) and served on a toasted Hawaiian bun. Hawaiian buns or bread have a fluffy texture and a sweet flavor. The bread is similar to Portuguese sweet bread and is believed to have been brought to Hawaii by Portuguese immigrants.
Asian Pork Burger
“Nuts On Clark”
The “Nuts On Clark” corporate offices and main retail store are located two blocks north of Wrigley Field on Clark Street. Naturally, it has a cart at Wrigley Field. But if you guessed that it sells nuts, you would be wrong. Instead it sells gourmet popcorn. Fans can get original popcorn, kettle corn, cheese corn, caramel corn, or the Chicago mix (a combination of cheese and caramel corn).
Sloppy Jane
Sloppy Jane
A Sloppy Joe is a sandwich consisting of ground beef, onions, tomato sauce, Worcestershire sauce and other seasonings served on a hamburger bun. According to The American Century Cookbook: The Most Popular Recipes of the 20th Century by Jean Anderson, research suggests that the Sloppy Joe began in a Sioux City, Iowa, café as a loose-meat sandwich created in 1930 by a cook named Joe. The Decades Diner offers a vegetarian version of the Sloppy Joe called a Sloppy Jane. Instead of beef, the Sloppy Jane uses tempeh, a soy product originally from Indonesia. It is made by a natural culturing and fermentation process that binds soy beans into a cake form.
Italian Beef Sandwich
The Italian Beef Sandwich is believed to have originated in Chicago in the 1930s. The beef is sliced thin and wet-roasted in beef broth with garlic, oregano, and other spices. The meat is served dripping wet and is therefore placed on a chewy bread, as a soft bread would disintegrate. The cooking process was historically a way to make less expensive and tougher cuts of beef more tender and tasty. The Hey Hey Hits Grill offers fans its version of the Italian Beef Sandwich.
Pretzel Baguettes
The Blue W stand serves two sandwiches on pretzel baguettes. Using pretzel dough in place of bread has become trendy in the fast food arena. According to the QSR (Quick Service Restaurants) Magazine website, “Quick-service restaurants across the country are capitalizing on a new pretzel-bread trend, taking one of the most popular summer snacks and working it into innovative new menu options.” The Blue W offers a turkey pretzel baguette and a veggie pretzel baguette with cucumbers, lettuce, tomato and hummus.
CHICAGO WHITE SOX
U. S. Cellular Field opened in 1991 on Chicago’s South Side. It holds the distinction of having the highest top row stadium seating in the major league ballparks. The park is built next to the site of the old Comiskey Park which served the White Sox from 1910 through 1990.
Featured Hot Dog/Sausage
As is the case for the Chicago Cubs at Wrigley Field, Vienna Beef provides the hot dogs at U.S. Cellular Field. The Comiskey Dog stand (named for Charles Comiskey, owner of the Chicago White Sox from 1901 to 1931) sells a traditional Chicago-style hot dog. The Vienna Beef frank is topped with mild yellow mustard, neon relish, chopped onions, tomato wedges, dill pickle spears, sport peppers, and celery salt, and served on a poppy seed bun. Neon relish is brightly colored and sweeter than ordinary pickle relish. Although the origins of neon relish are somewhat unclear, Superdawg Drive-In of Norwood Park, Illinois, claims to have been serving it since 1949 and is believed to be the first to introduce it to the Chicago-style hot dog.
Chicago-Style Hot Dog
More Hot Dogs and Sausages
Sausages, as well as pork hot dogs, are provided by The Bobak Sausage Company of Chicago. Bobak’s refers to its company as “Chicago’s sausageologists.” (I just checked my Scrabble app but unfortunately “sausageologist” is not acceptable.) Frank Bobak immigrated to Chicago in the 1960s from Zakopane, Poland, where he was a shepherd. He began making sausages in his basement and smoked them in his garage. This arrangement worked well until one day the Chicago Fire Department came to put out a fire. This led Bobak to open a commercial location. At U.S. Cellular Field, Bobak’s provides Polish sausage, Italian sausage, spicy jalapeño cheddar sausage, adobo mango chicken sausage, and bratwurst.
Beggars Pizza
Beggars Pizza has served the Chicago area since 1976. Today there are more than twenty locations throughout the area. At U.S. Cellular field fans can get either cheese, sausage, or pepperoni slices -- a much smaller selection than at its restaurants. But after all, beggars can’t be choosers.
Hot Asian Buns
A bao is a ball-shaped steamed bun usually containing meat or vegetables. It originated in China as a way to help feed large groups of people. Wow Bao, with six locations in the Chicago area, advertises that it serves “hot Asian buns.” The stand at U.S. Cellular Field offers chicken, barbeque pork, and veggie buns.
Hooters
Across from the “hot Asian buns” sign is the Hooters stand. (Feel free to let your mind wander.) Hooters restaurants are known for its scantily-clad female waitresses. Many individuals and organizations believe this practice is sexist and demeaning. According to the Hooters website, “Hooters girls are the very essence of Hooters. Trained to excel in customer service, they provide the energy, charisma, and engaging conversation that keep guests coming back. Much more than just a pretty face, Hooters girls have game.” Although Hooters was founded in Florida, U.S. Cellular Field is the only major league stadium that offers its food. Fans have their choice of chicken wings or tenders with either barbeque, honey, or Thai sauces.
Minnie Minoso, The Cuban Comet
Minnie Minoso, born in Havana, Cuba, had a long major league career including many years with the Chicago White Sox. He is one of only two players in major league history to play in five different decades (1940s-1980s). Thirteen years after retiring in 1964, Minoso made a three-day comeback at the age of 50 with the Chicago White Sox. Four years later he made a second comeback, this one for two games, at the age of 54. Due to his great speed, he was nicknamed “The Cuban Comet.” The Cuban Comet stand at U.S. Cellular Field sells a pressed hot Cuban sandwich made from sliced ham, shredded pork, Swiss cheese, mustard, sliced pickles and mojo sauce. Mojo sauce consists of olive oil, salt, water, garlic, paprika, coriander, and local pepper varieties.
Pork Chop Sandwich
The South Side Hitmen and the Pork Chop Sandwich
The 1977 Chicago White Sox team hit a total of 192 home runs, an American League record which stood until 1996. Due to their great power, the team was nicknamed “The South Side Hitmen.” The South Side Hitmen Grille at U.S. Cellular Field offers hamburgers, turkey burgers, veggie burgers, chicken sandwiches, and a pork chop sandwich. One might wonder how it is possible to eat a pork chop sandwich, since a pork chop has a bone in it. The answer is that it is not really a pork chop but a pork fillet. However, pork chop sandwich seems to sound much better at a stand named South Side Hitmen.
Chicken Flautas
Barbacoa Style
The Tex-Mex stand offers burritos with the choice of seasoned beef, chicken carnitas, or pork barbacoa. Barbacoa is a form of cooking that originated in the Caribbean. In contemporary Mexico it refers to meats slow-cooked over an open fire. Different meats are cooked barbacoa style, but pork is the most popular in the Yucatan Peninsula area of Mexico.
Mac & Cheese Bites
Tamale & Flauta Hut
The Tamale & Flauta Hut offers chicken and pork tamales and chicken and potato flautas. A flauta (also known as a taquito) is a Mexican dish consisting of a fried rolled-up tortilla with a filling. It is shaped like a flute (‘flauta’ is Spanish for flute). The Tamale & Flauta Hut serves its offerings with tomato, lettuce, cheese, sour cream and salsa tropiquena.
Strawberry Churro
Mac & Cheese Bites
The Triple Play Café puts a new twist on macaroni and cheese. It offers Mac & Cheese Bites, triangular-shaped cakes of macaroni and cheese covered with batter and deep fried. Also available are Irish nachos (French fries covered with sour cream, cheese, chives, and bacon.) Although several American ballparks sells Irish nachos, don’t plan on buying some the next time you’re in Dublin. Dubliners have never heard of them.
Strawberry Churro
A churro is a fried dough pastry shaped into a grooved stick. It is sometimes referred to as a “Spanish doughnut.” In Spain churros are eaten for breakfast, dipped in hot chocolate or coffee. Chicago Style Churros at U.S. Cellular Field customizes its churros by inserting a filling of chocolate cream, vanilla cream, or strawberry jelly.
CINCINNATI REDS
Great American Ball Park is located in downtown Cincinnati near the banks of the Ohio River. It is named for the Great American Insurance Group which is headquartered in Cincinnati. The stadium address is 100 Joe Nuxhall Way, named for the long-time Cincinnati broadcaster who died in 2007. Also, at the age of fifteen, he was the youngest person to ever play in the Majors.
Cheese Coney Dog
Featured Hot Dog/Sausage
At Great American Ball Park, Skyline Chili sells the Cincinnati classic Cheese Coney, a Coney Island-style hot dog topped with chili, cheddar cheese, onions and mustard. The original Skyline Chili was opened in Cincinnati in 1949 by Greek immigrant Nicholas Lambrinides. That restaurant location had a great view of the Cincinnati skyline; hence the name of the business. Lambrinides originated “Cincinnati-style chili,” a regional chili characterized by the use of seasonings such as cinnamon, cloves, allspice, or chocolate. The chili is commonly served over spaghetti or hot dogs (the choice at Great American Ball Park.)
More Hot Dogs and Sausages
Kahn’s is the official hot dog of the Cincinnati Reds. Elias Kahn founded a neighborhood retail meat market in Cincinnati in 1883. The American Beauty rose was chosen as Kahn’s trademark and remains so to this day.
Queen City Sausage is the official sausage of the Cincinnati Reds. (Cincinnati is nicknamed the Queen City of the West.) Since the 1800s, Cincinnati’s West End has been known as “Porkopolis,” with over forty shops producing meat products for the rest of the nation. At the age of twelve, West End native Elmer Hensler worked in the meat business. In 1965, Hensler joined forces with spice man Alois Stadler and master sausage-maker George Nagel to create Queen City Sausage. Today Porkopolis’ forty shops have been replaced by Queen City Sausage.
Marge Schott was the controversial owner of the Cincinnati Reds from 1984 to 1999. During her tenure she insisted that a basic hot dog should cost only one dollar. That tradition continues to this day. At the High 5 Grill located in a corner on the top deck, the $1 Value Dog can still be purchased. However, one should note that it does not take more than two or three bites to consume a Value Dog.
Frisch’s Big Boy
Frisch’s Big Boy serves hamburgers, chicken fingers, and French fries. Its hamburger comes with tartar sauce and fans can even buy an additional side order of tartar sauce. Apparently folks in Cincinnati really like tartar sauce. Outside the stand is a statue of “Frisch’s Big Boy,” a lad wearing red-checked overalls. I grew up in California and commented to my wife that the statue looked just like the iconic Bob’s Big Boy statue with which I was familiar. As it turns out, it is the same fellow. After the bankruptcy of Bob’s Big Boy and many legal maneuvers, Frisch’s Restaurants was granted the use of the Big Boy name in Ohio, Kentucky, Indiana, and Tennessee.
Montgomery Inn
In 1951 Ted and Matula Gregory opened the Montgomery Inn in nearby Montgomery, Ohio, and turned it into a barbeque restaurant. Over the next sixty years it earned a reputation as one of the best barbeque restaurants in the area. On President Obama’s 52nd birthday, Speaker of the House John Boehner ordered ribs for Obama by mail from the Montgomery Inn – perhaps the greatest sign of bipartisanship in the last ten years. Although the Montgomery Inn does not have a stand at Great American Ball Park, several other stands sell its pulled pork sandwich.
Waffle Bowl Sundae
In 1938 Carl Lindner, Sr., opened a small dairy in Norwood, Ohio. At that time, nearly all milk was home delivered. Carl, however, had a new concept: he would process milk and other dairy products and sell them at his own dairy store. It is believed that he also was one of the first to sell milk in gallon bottles instead of quarts. His original store was known as the United Dairy Farmers. Today its dairy products, including ice cream, can be found in many grocery stores. At Great American Ball Park, United Dairy Farmers sells ice cream either in a waffle cone or a waffle bowl. The ice cream is topped with chocolate or caramel sauce, peanuts, whipped cream, sprinkles, and cherries.
Waffle Bowl Sundae
Goetta
At Great American Ball Park, a local specialty known as a Goetta Burger is available. Goetta is a breakfast sausage of German-American origin, popular in the Cincinnati area. Goetta is ground beef combined with steel-cut oats. The Goetta Burger is served on a pretzel bun with onions and peppers.
Goetta Burger
Crunch ‘n Munch
In the Cracker Jack chapter of this book, it was noted that in 2004 the New York Yankees replaced Cracker Jack popcorn with Crunch ‘n Munch, a similar concoction. A fan outcry led to the Yankees returning to Cracker Jack. Great American Ball Park is now selling Crunch ‘n Munch instead of Cracker Jack. So far I have not heard of any fan uprising.
Smoked Turkey Leg
Mr. Red’s Smokehouse
Barbeque is served at Mr. Red’s Smokehouse. Daily offerings include a smoked prime rib sandwich, a smoked turkey leg, a smoked pulled pork sandwich, and smoked chicken wings. There was also a specialty item for each of the Reds 26 homestands during the 2014 season. On the day we visited, the homestand special was a smoked salmon burger made from ground Atlantic salmon blended with chipotle aioli. Also available is Mr. Red’s version of nachos known as The Chipper. The Chipper uses homemade potato chips instead of the usual tortilla chips. The chips are topped with pulled pork, cheese sauce, barbeque sauce, onions, red and green bell peppers, and jalapeños.
The Chipper
CLEVELAND INDIANS
Progressive Field, which opened in 1994, is in downtown Cleveland. From 1994 to 2008 it was known as Jacobs Field after team owners Richard and David Jacobs. It was ranked as major league baseball’s best ballpark in a 2008 Sports Illustrated fan poll.
Featured Hot Dog/Sausage
The C Dawg is a one-half pound all-beef hot dog, butterflied and grilled, and topped with chili, diced onions, and shredded cheddar cheese. It is advertised as “the hot dog you eat with a knife and fork.” The spelling of “dawg” probably comes from the bleacher section at the nearby Cleveland Browns football stadium, which their fans call “The Dawg Pound.”
C Dawg
More Hot Dogs and Sausages
Also available at the C Dawg stand is a Super Italian Sausage or a Super Bratwurst. Fans can choose any four toppings from a list of either hot or cold items. Hot toppings available are chili, bacon bits, pulled pork, peppers and onions, sauerkraut, and baked beans. Cold toppings include shredded cheddar cheese, cole slaw, diced onions, Fritos, and barbeque sauce.
Hometown Sausages sells an Italian sausage, a spicy Cajun sausage, and a broccoli and cheddar sausage where ground-up broccoli is mixed with the meat.
In Cleveland, the great debate is not about which hot dog to have, but rather which mustard to choose. The two rival brands are Stadium Mustard and Bertman Original Ballpark Mustard. Both mustards have a distinctive spicy, tangy taste and are a special color of brown. Stadium Mustard is the official mustard of Progressive Field.
Kosher hot dogs and sausages are available at a kosher stand, but the day we visited, though open for business, a posted sign read “This stand is not kosher-certified today.” This reminds me of the joke: What do you call a fly without any wings? Answer: a walk.
Veggie Burger
The Burgers and Fries stand sells cheeseburgers, hamburgers, grilled chicken sandwiches, chicken tenders, French fries, and veggie burgers. Although veggie burgers are popular today, the first commercially sold veggie burger appeared just 32 years ago. Gregory Sams, owner of a vegetarian restaurant in London, experimented with a variety of ingredients to come up with the “VegeBurger” patty that was sold dry and needed to be rehydrated before being cooked. The VegeBurgers were sold under the brand name of Harmony Foods. Over time, in the United States, “vege” became “veggie.” The veggie burgers at Progressive Field are made from black beans and corn.
Veggie Burger
Sweet and Salty Chipper
The Chipper
The Chipper displays its offerings as “Our spin on the loaded nacho using house made kettle chips fried fresh daily.” The variations on the Chipper include three meat-centric items (the CleOH, the BBQ and the Mexican) and one dessert offering (the Sweet and Salty.) The CleOH has chunks of corned beef and a cheese sauce, covered with pico de gallo, jalapeños, scallions, and sour cream. The BBQ replaces the corned beef with chunks of pulled pork and adds barbeque sauce. The Mexican uses seasoned ground beef in place of the corned beef. The Chipper also sells The Sweet and Salty. Chocolate and caramel sauce are drizzled on top of the kettle chips, with a side of whipped cream for dipping. According to 2013 article on the Good Housekeeping website, sweet and savory is one of the “five hottest trends in snack foods.”
Spuds
Spuds and Suds sells fresh-cut fries and garlic fries. Customers can watch the process from whole fresh potatoes being sliced and then fried. Perhaps you haven’t, but I have always wondered why potatoes are called “spuds.” Research has uncovered a story as interesting as Harry M. Stevens and the origin of the hot dog in America. A commonly cited legend claims there was a nineteenth-century activist group called The Society for the Prevention of an Unwholesome Diet (SPUD), which was formed to keep potatoes out of Britain. Well-known linguist Mario Andrew Pei wrote a bestseller in 1949 entitled The Story of Language, in which he passed on the SPUD story. However, according to linguist David Wilton, there is no evidence that the practice of pronouncing acronyms began before the twentieth century. Another theory states that a spade used to dig up large rooted plants (such as potatoes) was known in the mid-nineteenth century as a “spud.” Like the origin of the hot dog, no one knows for sure.
Fried Cookie Dough
Fried Delights sells funnel cake, corn dogs, and fried chocolate chip cookie dough. Raw cookie dough is deep fried, similar to the concept of deep-fried Twinkies.
Pierre’s Ice Cream
Pierre’s Ice Cream Shop opened in 1932 in Cleveland at the corner of East 82nd Street and Euclid Avenue. Alexander Pierre Basset manufactured the ice cream in the back of his shop. The company has expanded throughout the years, but each time it built a new facility, it has been located within three miles of the original shop. In 2011 Pierre’s opened a 35,000-square-foot factory located one mile west of the original shop. At Progressive Field, Pierre’s sells soft serve as well as hard dip ice cream. Flavors include strawberry chip, mint chocolate chip, cookie dough, and moose tracks (fudge ripple ice cream filled with crushed peanut butter cups).
Fried Chocolate Chip Cookie Dough
COLORADO ROCKIES
Coors Field opened in 1995 in downtown Denver. One of the first names considered for the new stadium was “Jurassic Park” because of the seven-foot-long triceratops skull found on the site during construction. This led to the selection of a dinosaur as the Rockies mascot, “Dinger.” Denver’s reputation as “the mile-high city” is noted by the twentieth row of seats in the upper deck being a different color, marking the one-mile elevation.
Elk Bratwurst
Featured Hot Dog/Sausage
At most ballparks, fan choices for hot dogs and sausages are either beef, pork, or vegetarian. At Coors Field buffalo dogs and elk bratwurst are available. The elk bratwurst (known as “Elk Brat”) comes with grilled onions and peppers. According to the North American Elk Breeders Association, elk meat is high in protein and low in fat, cholesterol and calories compared to beef.
More Hot Dogs and Sausages
As the name of the stand implies, Sausage on a Stick serves (you guessed it!) sausages on a stick similar to the way corn dogs are served. The varieties are Cheddarwurst, Spicy Polish, and Sweet Italian. The sausages are provided by the Gold Star Sausage Company of Denver, a family-owned and operated business since 1936 and the largest purveyor of sausages in the Rocky Mountain region.
At Xtreme Dogs fans can choose from six varieties of gourmet hot dogs. The Denver Dog comes with “stinkin’” green chili, shredded cheddar cheese, and jalapeños. The Diablo Dog has red chili, diced red onions, shredded pepper jack cheese and jalapeños. The New York Dog has sauerkraut, spicy brown mustard, diced onions, and sliced peppers. The Santa Fe Dog has sour cream, red chili, shredded cheddar cheese, and jalapeños. The Chicago Dog has relish, diced onions, sport peppers, wedge tomatoes, cucumbers, and celery salt. Finally, the Bacon Blue Dog has blue cheese, diced onions, and bacon.
Chocolate Bacon
Famous Dave’s and Chocolate Bacon
Barbeque lovers can enjoy a variety of food items at the Famous Dave’s stand at Coors Field. Dave Anderson, the founder of this nationally known pit barbeque restaurant chain, did not do well academically in high school. But he had a dream to “create the best BBQ America ever tasted.” Author and motivational speaker Zig Zigler dedicated his Success for Dummies book to Famous Dave, and featured Dave’s life story. In addition to St. Louis-style ribs, Texas beef brisket, and Georgia chopped pork, Famous Dave’s also offers its meat in dessert format: chocolate covered bacon. Three pieces of bacon dipped in melted milk chocolate are served chilled.
#17 Helton Burger Shack
The #17 Helton Burger Shack is named for former Colorado Rockies infielder Todd Helton, who not only wore the number 17 on his uniform but played his entire 17-year major league career with the Rockies. The Helton Burger is served with white American cheese, Thousand Island dressing, pickles and onions. Also available are fresh-cut fries and jumbo onion rings.
Rocky Mountain Oysters
With Coors Field being located in the Rocky Mountains, it is no surprise that Rocky Mountain Oysters are sold at the Blake Street Grill. Rocky Mountain oysters are not really oysters, but bull calf testicles coated in flour, pepper and salt, and then deep-fat fried. They are served with cocktail sauce. Rocky Mountain Oysters are common in cattle ranching areas of the American West, Canada, and Argentina.
Rocky Mountian Oysters
Wazee Market
The Wazee Market features three meaty sandwich creations. The Pretzel Pastrami Sandwich is made with pastrami, Swiss cheese and sauerkraut on a pretzel roll. The Smoked Brisket Bacon Melt is grilled and includes brisket, bacon, and Swiss cheese. The Rocky Mountain Ribeye Steak Sandwich is topped with cheddar cheese and fried onion rings. Meat and potato lovers can also order loaded tots to accompany their sandwiches. The tots are topped with bacon, green onions, sour cream, and cheese.
Loaded Tater Tots
Infield Greens
Infield Greens advertises “fresh salads built your way.” Fans create their own salads in four steps: (1) Choose your green: spinach, romaine, iceberg, or spring mix; (2) Add a protein: chicken, shrimp, or tofu; (3) Choose up to five toppings from the more than a dozen available; (4) Top it with one of six dressings. Some of the ingredients may come from the sustainable herb and vegetable garden initiated near Gate A at Coors Field in 2013, where produce is grown on site.
Berrie Kabob
Berrie Kabobs
Fruit and chocolate lovers will be satisfied with the dessert at The Original Berrie Kabobs. Chunks of bananas and whole strawberries are skewered and then drizzled with white and dark chocolate. The stand also serves frozen cheesecake, dipped in milk chocolate, on a stick.
DETROIT TIGERS
Comerica Park opened in 2000 in the Grand Circus Park neighborhood of Detroit. The entrance to the stadium is adorned with multiple bigger-than-life-size statues of tigers in various fierce poses. The carnival atmosphere continues inside the stadium, where a carousel and Ferris wheel offer rides during the baseball games.
Featured Hot Dog/Sausage
The hot dog stand inside the Big Cat Food Court offers the Late Night Dog. The hot dog is topped with a fried egg, bacon bits, and shredded cheese (the perfect snack when the Tigers night game goes into the 18th inning).
Late Night Dog
More Hot Dogs and Sausages
Five other specialty dogs are available within the Big Cat Food Court. The Frank N’ Beans Dog is topped with baked beans, shredded cheese, and bacon bits. The Coney Dog is topped with Coney sauce and onions. Detroit Coney sauce is an all-meat no-bean chili with onions and yellow mustard. The recipe does not come from Coney Island, New York, but rather from the American Coney Island restaurant in Detroit. The restaurant was founded in 1907 by Greek immigrant (by way of New York’s Ellis Island) Gust Keros.
The Slaw Dog is a Coney Dog that adds cole slaw to the topping. The Chicago Dog is topped with relish, pickles, peppers, tomatoes, onions, and celery salt. Finally, the Poutine Dog is topped with French fries, gravy, and cheese curds. Poutine is a popular Canadian dish, and for those geographically challenged readers, Detroit is only two miles from Windsor, Ontario, Canada. Windsor is actually south of Detroit, but this is probably more geography than you wanted to know.
Each of the specialty dogs are advertised as coming with “a natural casing.” Natural casings are edible, derived from the intestinal tract of farmed animals (yum). Natural casings breathe and allow cooking flavors to infuse the meat.
Throughout the park, carts sell Winter’s Italian sausages and kosher hot dogs. Winter Sausage Manufacturing Company, located in nearby Eastpointe, was founded in 1951 by Eugene Winter, a master sausage maker from Germany. The company is managed today by Eugene’s daughter, Rose Mary Wuerz.
Hot dogs at Comerica Park are provided by Ball Park Franks. In 1959 Hygrade Food Products became the exclusive supplier of hot dogs at the Detroit Tigers stadium. The company sponsored a contest to come up with a name for the hot dogs, and Mary Ann Kurk won the contest with the name “Ball Park Franks.” Her prize was a leather living room chair and $25 in cash. Since then, Hygrade Food Products has been sold to Hanson Industries, who then sold it to Sara Lee Corporation, who transferred the product to its Hillshire Farms brand, who then sold it to Tyson Foods.
Salsa
At Comerica Park, nachos are all about the salsa. Jack and Annette Aronson started making fresh all-natural salsa at their small restaurant in nearby Ferndale. The owner of an upscale grocery chain tried some and was so impressed that he asked them to package it for his stores. Fifteen years later, their Garden Fresh Gourmet salsa is the country’s number one selling refrigerated salsa. Food manufacturers no longer need to hire high-priced consultants to write a mission statement, as they can just copy the Aronson’s. “Gather the world’s best ingredients, craft them into delicious, high-quality, proprietary products and offer them at a reasonable price.” At this point it is probably anticlimactic, but in addition to Garden Fresh Gourmet salsa, the nachos grande at Comerica Park are topped with all the usual nacho ingredients.
“Pizza! Pizza!”
When ballpark pizza is sold by a large national chain, I have not included it in this book. An exception is made for Comerica Park, because Little Caesar’s pizza originated in Garden City, Michigan, near Detroit. Mike and Marian Ilitch invested their $10,000 life savings to open the first Little Caesar’s in 1959. Today Little Caesar’s is the largest carry-out pizza chain in the world, and Mike Ilitch, once a minor-league player for the Tigers, owns the Detroit Tigers as well as the Detroit Red Wings of the National Hockey League. At Comerica Park, fans can get cheese, pepperoni, the three-meat treat (sausage, bacon, pepperoni), or classic veggie (onion, green pepper, mushroom).
Hudsonville Creamery
Hudsonville Creamery started making ice cream in 1926 in Holland, Michigan, on the shore of Lake Michigan. The initial flavors in 1926 were vanilla, chocolate, strawberry, butter pecan, orange pineapple, and tootie fruitie. The Creamery is still family owned. Flavors available at Comerica Park include Tiger Traxx and Grand Traverse Bay. Tiger Traxx is cherry vanilla ice cream loaded with chocolate covered pretzels shaped as baseballs and finished with a thick fudge swirl. Traverse City, located in the Traverse Bay region of Michigan, claims to be the “cherry capital of the world.” Grand Traverse Bay ice cream is amaretto flavored with cherry pieces and a thick fudge swirl.
Bumpy Cake
Sanders Chocolates was opened in Detroit by Fred Sanders in 1875. Although none exists today, at one time there were 57 stores throughout the Great Lakes region. In 2002 the brand name was sold to Morley Candy Makers. In 1913 Sanders Chocolates had begun making what it called “Devil’s Food Buttercream Cake.” The chocolate layer cake was topped with thick rows of piped buttercream frosting, then covered with a fudge frosting. Although Sanders called it a “Devil’s Food Buttercream Cake,” families around Detroit simply knew it as “a bumpy cake.” Although the bumpy cake sold at Comerica Park is not made by Sanders, the creation is still enjoyed today.
Bumpy Cake
Infamous Bacon Burger
The 313 Burger Company (named for Detroit’s area code of 313) offers fans The Infamous Bacon Burger. Unlike other bacon burgers, which are simply hamburgers topped with bacon, the Infamous Bacon Burger’s patty is made with 50% ground beef and 50% ground bacon. The burger is then topped with more bacon as well as onion rings and barbeque sauce.
The Infamous Bacon Burger
313 Taco Company
The 313 Taco Company cart advertises that the company has been selling “authentic street tacos since 2014.” (Hopefully this claim will be a bit more impressive a few years down the road.) The street tacos are served in corn tortillas and, in addition to the normal toppings, contain Cotija cheese. Cotija is a cow’s milk cheese that comes from Cotija, Michoacan, Mexico. The cheese is salty with a taste similar to Greek feta cheese, a sheep or goat’s milk product.
Elephant Ears
Side Kicks sells corn dogs, chicken fingers, and elephant ears. Chicken fingers are not the fingers of chickens, and elephant ears are not the ears of elephants. In fact, elephant ears don’t even come from elephants. Elephant ears, a common carnival food, are made from dough fried in the shape of an elephant ear and covered with powdered sugar.
Hicory-Smoked Soy Riblet
Hickory-Smoked Soy Riblet
Although the Brush Fire Grill serves beef brisket, pulled pork, cheeseburgers and grilled chicken sandwiches, it also has a very large vegetarian menu. It offers a veggie dog, a veggie Italian sausage, and two types of veggie burgers (black bean or garden burger). Additionally, it has a hickory-smoked riblet sandwich made from soy which resembles the McDonald’s McRib.
Bavarian Pretzel Sticks
Also available at the Brush Fire Grill are grilled Bavarian pretzel sticks. Bavarian pretzels are crunchy on the outside and soft on the inside. Legend has it that in 1839 Anton Nepomuk Pfannenbrenner, a baker for the Royal Coffee House located in Munich, was preparing sweet pretzels for his guests. Instead of brushing them with sugar water, he accidentally brushed them with a baking soda-lye solution used to clean and disinfect the bakery counter tops. The pretzels came out of the oven with a brown crust, soft center, and delicious taste. The Brush Fire Grill serves its warm Bavarian Pretzel Sticks with a cheese dipping sauce.
Bavarian Pretzel Sticks
HOUSTON ASTROS
Minute Maid Park, nicknamed The Juice Box, opened in 2000 in downtown Houston. The ballpark was originally known as Enron Field, until Enron declared bankruptcy in one of America’s largest business scandals. The park has a retractable roof to protect fans and players from Houston’s extremely humid weather.
Featured Hot Dog/Sausage
The Extreme Dog stand sells five regional hot dogs including the Texas Dog. The Texas Dog is wrapped in bacon, topped with jalapeño relish, and served on Texas toast. Texas toast is usually made from white bread sliced twice as thick as usual sliced bread. It is believed that Texas toast was first served at Kirby’s Pig Stand restaurant in Dallas in the 1920s. It has remained a popular side dish throughout Texas.
Texas Dog
More Hot Dogs and Sausages
The other regional dogs sold at the Extreme Dog stand are the Cincinnati Cheese Coney (chili, cheddar cheese and diced onions), the Georgia Dog (creamy coleslaw, barbeque sauce and diced onions), the Coney Island Dog (chili, chopped onions and spicy mustard), and the Ken Hoffman New York City Dog (grilled sauerkraut and spicy mustard). Ken Hoffman is a food columnist for the Houston Chronicle. According to the Houston Chronicle website, “Ken has written more than 800 fast food reviews. His cholesterol is higher than yours.”
The Astros Sizzling Grill serves sausages. The Grill follows the Extreme Dog in offering its sausages in the same five regional versions. Each version is available either as a mild sausage or as a jalapeño hot sausage.
Fans who want pork hot dogs or sausages are out of luck at Minute Maid Park. All hot dogs and sausages are beef and are provided by Nolan Ryan All-Natural Beef. During Nolan Ryan’s 27-year major league career, he pitched for the Houston Astros and the Texas Rangers as well as the New York Mets and the California Angels. He holds the major league record with seven career no-hitters and was inducted into the National Baseball Hall of Fame in 1999. Ryan is the only major league player to have his number retired by three teams (Astros, Rangers, Angels). On his website, Nolan Ryan states “While you may know me as a baseball player, the cattle business has always been my first passion. I started raising cattle when I was very young. Even then I was just as dedicated to producing the best beef possible as I am today!” Many stands throughout the park sell Nolan Ryan’s regular hot dog and jumbo foot-long hot dog, as well as his brisket sausage.
Blue Bell Creameries
Blue Bell Creameries, originally known as the Brenham Creamery Company, was founded in 1907 in Brenham, Texas (about 75 miles northwest of Houston.) The company began making ice cream in the 1920s and soon changed its name to Blue Bell Creameries, after the native Texas blue bell wildflower. Blue Bell ice cream is sold in supermarkets throughout the South. At Minute Maid Park fans can enjoy Blue Bell ice cream in the following flavors: homestyle vanilla, Dutch chocolate, cookies and cream, or birthday cake (vanilla ice cream with pieces of chocolate cake, a chocolate icing swirl and multi-colored sprinkles.)
Texas Smoke
The state of Texas is famous for its brisket. The Texas Smoke stand at Minute Maid Park offers both sliced and chopped barbeque brisket sandwiches. A cooked brisket has two parts known as “the point” and “the flat.” The sliced meat comes from the flat or meaty side of the brisket; the chopped meat comes from the point or fatter side.
New York Strip Steak Sandwich
New York Strip Steak Sandwich
The Texas Legends Grill features hamburgers made with—you guessed it—Nolan Ryan’s beef. In addition to a basic hamburger, the Legends Grill offers cheeseburgers, bacon cheeseburgers, and a mushroom Swiss burger. The New York Strip Steak Sandwich comes with white cheese, horseradish, and fried onions. A New York strip steak is cut from the short loin of the beef and consists of a muscle that does little work and therefore is particularly tender.
Chicken Fajitas
Little Bigs
The Little Bigs stand serves sliders advertised as “The best things in life are in 3.” Here is one of the few places at Minute Maid Park where pork is available. The sliders, which are either hamburgers or pulled pork, are served on soft rolls. The burgers are topped with cheese and onions; the pulled pork sliders are topped with cole slaw.
Tex-Mex
The El Real Grill serves Tex-Mex food, a unique blend of traditional American and Mexican-American cuisines. The El Real Grill offers beef and chicken fajitas. Fans can watch the fresh tortillas being patted out and run through the machine as they wait for their order.
Green Fork
The Green Fork serves four fresh salads. The Taco Salad has iceberg lettuce, chick peas, roasted corn, black beans, red onion, spicy French dressing, and taco meat served in a tortilla shell. The Astros Signature Caesar has romaine lettuce, Caesar dressing, croutons, and Parmesan cheese. The Texas Cobb has iceberg lettuce, blue cheese, tomatoes, cucumbers, eggs, olives, bacon, avocados, and blue cheese dressing. The Heart Healthy Salad has spinach, bell peppers, cucumbers, carrots, cilantro, grape tomatoes, avocados, red onions, and balsamic dressing. All salads can be topped with grilled chicken or taco meat, if desired.
Chicken Caesar Salad
Street Eats
“Houston’s hot and local” Street Eats is based on the food truck concept. The Texas chuckwagon is considered a precursor to the American food truck. At Street Eats fans can get lobster rolls, pulled pork tacos, smoked pork sandwiches, or the Texas Hold’em Sandwich (named for the popular poker game.) The Texas Hold’em has barbequed chicken, cheddar cheese, tomato, jalapeños, and cole slaw piled on Texas toast.
KANSAS CITY ROYALS
Kauffman Stadium, originally called Royals Stadium, opened in 1973. A major stadium renovation took place from 2007 to 2009. The stadium is named for Ewing Kauffman, the original owner of the Royals. Kauffman Stadium is part of the Harry S Truman Sports Complex along with Arrowhead Stadium, home of the Kansas City Chiefs football team.
Featured Hot Dog/Sausage
The Dugout Dog House offers seven specialty dogs. Since Kansas City is well known for its barbeque, the All-Star BBQ Dog is our featured hot dog. It comes with pulled pork, cole slaw, pickles, and barbeque sauce.
More Hot Dogs and Sausages
At the Dugout Dog House, the Kansas City Dog has Swiss cheese, grilled sauerkraut and Boulevard Pale Ale Mustard. Boulevard is the best-selling craft beer in the Midwest. The Blazing Buffalo Dog has pulled chicken tossed in spicy buffalo sauce and topped with cole slaw. The Royal Bacon Blue Dog has blue cheese crumbles, chopped bacon, and red onion.
All-Star BBQ Dog
Three of the hot dogs at Dugout Dog House are named for other areas of the country. The Texas Dog has chili, cheddar cheese, diced onions, and Frito bits. The New York City Dog has sauerkraut and spicy mustard and is served on a poppy seed bun. The Chicago Dog has mustard, onions, sport peppers, tomatoes, pickles, celery salt and relish.
Crown Classics sells a foot-long Sheboygan bratwurst. Bratwurst is believed to have originated in Nuremberg, Germany, and is very popular in Wisconsin. The city of Sheboygan is known as the “Bratwurst capital of the world” and celebrates Sheboygan Bratwurst Days each August.
Burnt Ends
Kansas City Barbeque
When most people think of Kansas City, barbeque comes to mind. In the early 1920s Henry Perry, known as “The Father of Kansas City Barbeque,” moved to a barn on Highland Street and started barbecuing in an outdoor pit. Perry served slabs of barbequed meat wrapped in newspaper. Perry’s disciples include Arthur Bryant, George Gates, Otis Boyd, John Harris, and Sherman Thompson. They all learned Perry’s technique and then went on to create their own unique blends of Kansas City barbeque. Kansas City-style barbeque is slow-smoked over wood, usually hickory.
Kauffman Stadium has its own barbeque pit located behind right field. Using oak and hickory wood from the Ozarks, Kauffman’s smoker produces up to 400 pounds of smoked meat on busy days. The barbeque is served at Sweet Baby Ray’s. Sweet Baby Ray’s is a Chicago-based company that sells its barbeque sauces and meats in grocery stores throughout the United States.
Sweet Baby Ray’s serves barbeque sandwiches of ham, brisket, turkey or pulled pork, in addition to chopped burnt ends or ribs. Burnt ends are a traditional part of Kansas City barbeque. The entire brisket is cooked whole, then the point end is removed and cooked further. The longer cooking time gives rise to the name “burnt ends.” A proper burnt end should display a small amount of charred meat on at least one side. The burnt ends are served either in a sandwich or in a basket with a skewer.
Cheesy Corn
In addition to the usual cole slaw, baked beans, and potato salad sides, Sweet Baby Ray’s has a Kansas City specialty of cheesy corn. Cheesy corn is kernels of sweet corn served in a cheddar cheese sauce with ham and bacon bits. (Unlike cheese grits, the dish is known as “cheesy corn” and not “cheese corn.” In the 2012 presidential campaign Mitt Romney, in trying to show his Southern side, said how much he liked cheesy grits. However, in the South, cheesy grits are bad grits and cheese grits are grits with cheese.)
Cheesy Corn
Belfonte Dairy
The Belfonte Ice Cream Shop is an air-conditioned indoor ice cream stand. On warm days fans seem to go there not only for ice cream, but to stay cool. Sal Belfonte, a door-to-door milkman for many years, started Belfonte Dairy as a family business. Today Belfonte ice cream is found in over 500 supermarkets and restaurants in Kansas City and surrounding areas. At Kauffman Stadium, Belfonte offers twelve “scoopilishish” flavors (including strawberry cheesecake and cookies and cream) and eight toppings.
Ice Cream with Sprinkles
Brisket-acho
Cheesy Corn Brisket-acho is Kauffman Stadium’s version of nachos. Brisket, baked beans, cheesy corn, and cole slaw are piled high over a bed of chips and then topped with plenty of barbeque sauce. Unlike with traditional nachos, you will need a fork to eat a Cheesy Corn Brisket-acho.
Royal Bacon Blue Fries
Royal Bacon Blue Fries
The Fry Works sells four version of what it calls “Extreme Fries”: BBQ Pulled Pork Fries, Chili Cheese Fries, Buffalo Ranch Fries, and Royal Bacon Blue Fries. The Royal Bacon Blue Fries contain bacon, ranch dressing, blue cheese and green onions.
Nutty Bavarian
The Nutty Bavarian offers almonds, cashews, pecans, and peanuts coated with sugar and cinnamon. Fans can watch the nuts swirling around in the coating machine while the sugar and cinnamon are added.
LOS ANGELES ANGELS OF ANAHEIM
Angel Stadium of Anaheim opened in 1966, located within several miles of Disneyland. In front of the stadium is the landmark “Big A” sign and electronic marquee. The halo located near the top of the 230-foot tall, 210-ton sign is illuminated following games the Angels win, which gives rise to the fan expression, “Light up the Halo!”
Featured Hot Dog/Sausage
For many years the signature hot dog at Angel Stadium was the Halo Dog (an all-beef hot dog wrapped with bacon and topped with charro beans, shredded Monterey Jack cheese, and pico de gallo salsa). In 2014 it has been replaced by the Farmer John Jumbo Hot Dog.
Farmer John hot dogs have a history in Southern California dating back to 1931. The company claims that its product is “seasoned with every culture in the world,” as represented by the population of Southern California. Farmer John hot dogs have been sold at sporting and other entertainment venues across the state, including the famous Dodger Dog at Dodger Stadium. At Angel Stadium the Jumbo Hot Dog comes “undressed” and fans can add their own condiments. It is sold at Farmer John Grilling Stations along with bratwurst, Italian sausages, and hot links.
Farmer John Hot Dog
Oggi’s Pizza
Oggi’s Pizza and Brewing Company serves slices of pepperoni and cheese pizza at Angel Stadium. “Oggi” is Italian for “today,” chosen to represent the fresh quality of the ingredients chosen by George and John Hadjis when they opened their first restaurant in Del Mar, California. The brothers, who had formerly worked in the technology industry, expanded their sports-themed operation into a microbrewery in 1995. A few years later it was named the Champion Small Brewing Company at the World Beer Cup.
Chronic Tacos
The first Chronic Tacos location opened in Newport Beach in 2002. Here one can get Mexican street tacos (served on a soft tortilla rather than in a fried shell) with a choice of marinated grilled chicken or steak, slow-cooked pork, or all-veggie. Chronic Tacos is well known for its pork nachos. The chips are topped with pork, melted cheese sauce, shredded jack and cheddar cheese, rice, beans, onions, cilantro, and jalapeños.
Chicken Tacos
BBQ Brisket Sandwich
Smoke Ring BBQ
The Smoke Ring BBQ features a variety of meats cooked over an open flame. Along with a smoked barbeque brisket sandwich, fans can order smoked half chicken, smoked kielbasa link, or St. Louis pork ribs. Side dishes include cole slaw, macaroni and cheese, and cornbread.
Grilled Cheese Sandwich with Tomato Soup
The Big Cheese
The Big Cheese stand offers four varieties of grilled cheese sandwiches. In addition to the traditional, fans can order thick-cut bacon, short rib, or tomato and arugula. The cheese is a mixture of cheddar and Monterey Jack. Grilled cheese sandwiches and tomato soup are often considered classic comfort food. In fact, there is a Midwest restaurant chain named Tom+Chee that specializes in this combo. Here at Angel Stadium The Big Cheese sells tomato soup to accompany any of its grilled cheese sandwiches.
Carne Asada Waffle Fries
Spuds is the place for those who like their French fries out of the ordinary. The Carne Asada Waffle Fries come with beef covered with sour cream, guacamole, jalapeños, and pico de gallo. Chili Cheese Fries are topped with short rib chili, and house-made-beer cheese. And for those who prefer sweet instead of savory, there are the Sweet Potato Fries with cinnamon and sugar.
Carne Asada Waffle Fries
LOS ANGELES DODGERS
Dodger Stadium, which opened in 1962, is the third oldest of the major league stadiums, following Fenway Park (1912) and Wrigley Field (1914). With seating for 56,000 fans, Dodger Stadium can accommodate more attendees then any other major league stadium.
Unlike most other stadiums today, Dodger Stadium does not have stands from local area merchants. At one time my favorite deli (and in my opinion, the best Jewish deli west of New York), Canter’s Deli, had a stand at Dodger Stadium. It closed in 2010. Fortunately, deli fans can visit Canter’s Deli on nearby Fairfax Avenue in Los Angeles before or after any Dodgers game since it is open 24 hours a day. Additionally, fans at Dodger Stadium are allowed access only to the level where their seats are located and cannot patronize food stands on other levels.
Featured Hot Dog/Sausage
Harry M. Stevens would be pleased to know that nearly every concession stand at Dodger Stadium offers hot dogs. The well-known Dodger Dog, a pork wiener, is available either steamed or grilled. In addition to the traditional Dodger Dog, fans can choose an all-beef Super Dodger Dog, or a Brooklyn Dodger Dog, which is all beef with a thicker casing.
Along with the Fenway Frank, the Dodger Dog is the best-known ballpark hot dog. The Dodger Dog is ten inches long (though sometimes referred to as a foot-long) and sticks out at both ends of the bun. Thomas Arthur, food concessions manager at Dodger Stadium from 1962 to 1991, created the concept of the Dodger Dog. He coined the name “Dodger Dog” after being criticized for calling it a “foot-long dog” when it was actually only ten inches long. Dodger Dogs were originally made by the Morrell Meat Company, but are now made by Farmer John, which was purchased by Hormel in 2004.
Dodger Dog
More Hot Dogs and Sausages
For fans who want a non-traditional hot dog, the Extreme Loaded Dogs stand has five varieties from which to choose. The Doyer Dog Jr., named for the Spanish pronunciation of “Dodgers,” comes with chili, nacho cheese, jalapeños, and pico de gallo. LA’s Extreme Bacon-Wrapped Dog is a one-third pound all-beef hot dog wrapped with three slices of applewood-smoked bacon, smothered with grilled peppers and topped with onions. During each Dodger homestand, the Extreme Loaded Dogs stand offers a special dog. When we were there, the specialty hot dog was topped with French fries and cole slaw. Extreme Loaded Dogs features not only one but two varieties of hot dogs topped with Frito corn chips. The Frito Pie Dog has chili, cheese, and Fritos. The Big Kid Dog comes topped with gooey mac and cheese and Fritos.
Pasta Platter
Tommy Lasorda’s Trattoria
New to Dodger Stadium in 2014 is the Tommy Lasorda Trattoria, an informal Italian eatery. Tommy Lasorda was best known as a Hall-of-Fame manager of the Dodgers from 1976 to 1996. He was also known for his love of Italian food. At the stand named for Tommy Lasorda, Italian specialties include the Italian Meatball Marinara Sandwich, the Chicken Parmesan Sandwich, and the Lasorda’s Pasta Platter.
The Italian Meatball Marinara Sandwich is made with hand-formed all-beef meatballs with Italian seasonings. The Chicken Parmesan Sandwich is a breaded chicken breast with provolone, grated Parmesan, and zesty marinara sauce, served on a toasted Italian roll. The Pasta Platter has penne pasta with zesty marinara sauce, Italian beef meatballs and Parmesan cheese. Also available at the Trattoria are cheese and pepperoni pizza and garlic fries smothered with a fresh garlic marinade.
Pistachio Gelato
Gelato, described at Tommy Lasorda’s Trattoria as “Italy’s original ice cream,” is sold in unique flavors. The day we were there, flavors offered were cappuccino crunch, white chocolate raspberry swirl, pistachio, and panna cotta (an Italian dessert made with cream, milk, sugar and gelatin).
Elote
Elote
The Think Blue Bar-B-Que (named for the blue color of the Dodger uniform) provides barbeque beef sandwiches and Louisiana hot sausages. Also available is a Mexican-style corn on the cob known as elote. The ear of corn is roasted and seasoned with lemon, mayonnaise, Parmesan, and powdered barbeque seasonings.
Chef Merito
The L.A. Taqueria stand features Chef Merito seasonings, “the Official Seasoning of the Los Angeles Dodgers” according to the Chef Merito website. The Taqueria serves street-style tacos with a choice of carne asada or fish, as well as chicken taquitos (a small rolled-up tortilla filled with chicken and deep fried).
Elysian Park Grill
Elysian Park is the second largest park in the city of Los Angeles and encompasses the Dodger Stadium area. It was perhaps named after Elysian Fields in Hoboken, New Jersey, the site of the first organized baseball game in 1845. The Elysian Park Grill at Dodger Stadium offers two hamburger specialties. The Dodger “Blue Heaven” Burger is a one-third pound premium burger with crumbled blue cheese, caramelized onions, and sliced tomatoes topped with a pasilla chili and chipotle aioli on a bun. The Elysian Park Cheeseburger comes with American cheese, caramelized onions, dill pickles and special dressing.
Cool-A-Coo Ice Cream Sandwich
Long a favorite with Dodger fans, the Cool-A-Coo ice cream sandwich is once again available at Dodger Stadium after an absence of more than ten years. Made in Southern California, the Cool-A-Coo features vanilla ice cream between two oatmeal cookies, dipped in a chocolate coating.
MIAMI MARLINS
Marlins Park opened in 2012 on the site of the former Miami Orange Bowl. It is located in the Little Havana section of Miami, about two miles from city center. The stadium, with its retractable roof, was rated by the U.S. Green Building Council as the greenest major league ballpark.
Pizza Dog
Featured Hot Dog/Sausage
Each Marlins game features a “hot dog specialty of the game.” On the day we visited, the specialty dog was topped with pizza sauce, tomatoes, mozzarella cheese, mushrooms, and chopped sausage.
More Hot Dogs and Sausages
Hot dogs and sausages play a lesser role in the food offerings at Marlins Park than at many major league stadiums. The hot dogs are made by Kayem Foods which makes the famous Fenway Franks in Boston. Here Kayem provides a traditional hot dog, a chili cheese dog, and an Italian sausage.
Ceviche
Ceviche
Cuban and Latin American specialties are found in the section of Marlins Park called Taste of Miami. Don Camaron Seafood Grill & Market, a Miami restaurant, has a stand here. It offers a snapper sandwich, fried shrimp basket, half-dozen oysters, ceviche, conch fritters, and malanga chips.
Ceviche is a seafood dish popular in the coastal regions of Central and South America. It is made from fresh raw fish cured in citrus juices. The ceviche at Don Camaron is made with sea bass and includes sweet potatoes, Peruvian corn kernels, cilantro, onions and lime juice. Conchs are marine gastropod mollusks. The meat of conchs is eaten raw in salads or (as at Marlins Park) cooked as fritters, breaded and fried. Malanga is a root vegetable from the tropics of South America. It has been grown commercially in South Florida since 1963 to meet the demands of Latin Americans living in the region. Malanga chips take the place of potato chips at Don Camaron.
Papo Llega y Pon
The original Papo Llega y Pon was opened by retired Cuban boxer Miguel Alfonso in Miami. In Cuba, a “llega y pon” is a place where country folk would come to town and set up food for sale, often in a ramshackle hut. It has long been a favorite place for Miamians and has been favorably reviewed by food critics. Miguel’s daughter Julia now runs Papo Llega y Pon. She was convinced by several Marlins employees to open a stand at Marlins Park. The featured item at Marlins Park is the roasted pork sandwich made from pork shoulder layered with extra crispy skin, salt, onions, mojo marinade, and hot sauce.
Tequenos and Empanadas
Also in the Taste of Miami section is Panna Café Express. Panna Café Express serves authentic Latin American cuisine at several locations throughout the Miami area. At Marlins Park it offers tequeños (a spear of bread dough with a white cheese filling) and empanadas (a turnover-shaped pastry with a filling of chicken or beef). The empanadas are served with aji sauce, a spicy mixture containing tomatoes, cilantro, aji peppers, and onions.
Chicken Empanadas
Goya Latin Café
Founded in 1936, Goya Foods is America’s largest Hispanic-American owned food company. It sells over 2,000 food products from the Caribbean, Mexico, Spain, and parts of Central and South America. The Goya Latin Café at Marlins Park offers a pressed Cuban sandwich. Instead of the Cuban sandwich being on a roll, it is on pressed bread similar to a panini. The Café also has a veggie burger described as “a home-made black bean, maduros and yellow rice patty with shredded cabbage, sliced tomato, and chipotle aioli.” Maduros are sweet, ripe plantains.
Black Bean Burger
Latin American Grill
The Latin American Grill serves Cuban sandwiches (layers of ham, pork, cheese, pickles and mustard on a grilled roll). It also has a Midnight Sandwich, which is a Cuban sandwich on a sweet roll. The sandwiches come with a side of plantain chips. Plantain is a banana-like fruit that grows in the tropics.
Kosher Korner
The Kosher Korner serves pastrami sandwiches, hot dogs, hamburgers, and cheeseburgers with soy cheese (kosher dietary laws do not allow for dairy to be mixed with meat.) Also available are fried potato knishes from Gabila’s of New York, the same manufacturer of knishes sold at the Mets Citi Field.
Meatball Marinara Fries
Burger 305 offers an Italian twist on French fries with its Meatball Marinara Fries. Fried potatoes are topped with meatballs and marinara sauce.
SoBe Fruit Salad
SoBe is the nickname for the South Beach neighborhood in the city of Miami Beach. The SoBe fruit salad contains fresh local fruits (grapefruit, watermelon, pineapple, mango) plus cucumbers and yellow pear tomatoes. The salad is accompanied by a sweet pineapple-based dressing.
MILWAUKEE BREWERS
Miller Park, which opened in 2001, features major league baseball’s only fan-shaped retractable roof, which allows for opening or closing in less than ten minutes. Large panes of glass allow for natural grass to grow inside the stadium.
Featured Hot Dog/Sausage
South Paw Dogs offers fans The Beast, a grilled foot-long bratwurst stuffed with a hot dog and wrapped in bacon. It is topped with beer-braised onions and sauerkraut, stadium sauce and mustard, and served on a pretzel bun. The hot dog is very narrow, which allows for it to be inserted inside the bratwurst.
The Beast
More Hot Dogs and Sausages
Fans not quite up to The Beast can get either the Milwaukee Dog or the Crab Mac n’ Cheese Polish. The Milwaukee Dog is topped with bacon strips and bits, Bernie’s Barrelman Ale cheese sauce, and fried cheese curds. Bernie’s Barrelman Ale is the name for a new beer that celebrates the partnership between the Milwaukee Brewers and the Leinenkugel Brewing Company. The limited-release beer is available exclusively at Miller Park.
The Crab Mac n’ Cheese Polish is a Polish sausage topped with crab meat, macaroni and cheese, and sriracha mayonnaise and served on a pretzel bun.
Klement’s Sausage Company is the official provider at Miller Park for both the sausages and hot dogs. In addition to Polish sausages and bratwurst, fans can purchase a chorizo sausage or a cheddar bratwurst. Hot dogs come in both jumbo and junior sizes. The brothers John, George and Ron Klement opened a small sausage company in Milwaukee’s South Side in 1956. Their goal was to bring “the taste of the Old Country” to the people of Milwaukee. Klement’s has grown to become Milwaukee’s largest producer of sausage products. In 2005 the brothers were inducted into the Wisconsin Meat Industry Hall of Fame. Although it’s not quite the same as making it at Cooperstown, note that the first member inducted into the Wisconsin Meat Industry Hall of Fame was Oscar Mayer (the Babe Ruth of the Wisconsin meat industry.)
Sargento Cheese Company
Milwaukee is known for beer, sausages, and cheese. This book does not cover beer; we already talked about sausages; now it’s time for the cheese. The BBQ and Burgers stand features a cheddar burger made with cheese from the Sargento Cheese Company. The Sargento Cheese Company began as the Plymouth Cheese Counter, a small delicatessen located in Plymouth, about 50 miles north of Milwaukee. Its initial product line included mozzarella, provolone, and romano cheeses. The company name comes from a combination of the founding partners’ last names, Leonard Gentine and Joseph Sartori. In 1965 Gentine bought out his partner. Nearly fifty years later, the company is run by Leonard’s grandson, Louie Gentine.
Palermo’s Pizza
Palermo’s Pizza stand offers cheese and pepperoni slices. Palermo’s story is a twist on the “local pizza parlor goes national” story. Italian immigrants Gaspare and Zina Fallucca opened a pizzeria in 1969. Several years later Palermo’s was named Milwaukee’s best pizza by the Milwaukee Journal. In addition to pizza, Palermo’s also sold pizza bread based on a family recipe from Zina’s relatives. A local grocer urged the Falluccas to offer frozen pizza which could be baked at home. In 1989 Palermo’s revolutionized the frozen pizza industry by offering the first frozen pizza with a crust that rises while it bakes. Today Palermo’s frozen products can be found in grocery stores throughout the Midwest.
Cedar Crest Ice Cream
Miller Park offers its fans Cedar Crest ice cream. Cedar Crest, located 90 miles north in Manitowoc, makes its ice cream in small batches. Its only business is ice cream and it considers itself a “craft” ice cream business (similar to a microbrewery, but for ice cream.) The company’s best known flavor is Elephant Tracks chocolate ice cream with shards of peanut butter cups inserted throughout. In addition to Elephant Tracks, fans can get butter pecan, chocolate chip cookie dough, cookies and cream, and many other flavors.
Stormin’ Gorman Thomas
Gorman Thomas was a designated hitter/outfielder for the Milwaukee Brewers from 1973 to 1983, plus his last major league year of 1986. During the period from 1978 to 1983 he hit more home runs than any other player in the American League. He also struck out frequently and had a low batting average. He was known as Stormin’ Gorman and is considered one of the all-time fan favorites in Milwaukee. Gorman’s Corner stand at Miller Park, which sells traditional ballpark fare, honors Thomas. The stand also sells jumbo kosher pickles (and I mean jumbo) which are displayed in a large jar on the counter.
Meatloaf Link
The Smokehouse serves carved sandwiches of beef brisket, turkey, and Italian roast pork. It advertises “All of our meat is smoked in-house for 12 hours.” During our visit there, it offered a homestand special: meatloaf sandwich. The sandwich is made from a grilled meatloaf link with bacon chipotle catsup glaze on a hoagie bun. Since Milwaukee is the sausage capital of America, the Smokehouse decided to put the meatloaf into a sausage link.
Meatloaf Link Sandwich
Walking Tacos
Texas League Tacos offers a twist on the walking taco concept. Instead of using a tortilla, the contents of the taco (beef, chicken or pork, pico de gallo, scallions and sour cream) are served inside a bag of Cool Ranch Doritos. Just a hunch, but I think traditionalists may not be pleased by this offering.
Spaghetti-in-a-Meatball
Spaghetti-in-a-Meatball
The Double Clutch stand is designed to look like a food truck, saluting the food truck movement. A sign reads “Sweet eats from the street to your seat.” Many places offer spaghetti and meatballs, but the Double Clutch serves Spaghetti-in-a-Meatball. A softball-sized ground beef, pork and veal meatball is stuffed with spaghetti and mozzarella cheese and topped with marinara sauce and provolone.
Fried Cheese Curd Canoe
Fried Cheese Curd Canoe
The Home Fire Grill serves a Fried Cheese Curd Canoe. (I believe Wisconsin state law requires all businesses to sell fried cheese curds in some form.) Cheese curds are the solid parts of soured milk, sometimes referred to as “squeaky cheese.” The curds squeak against the teeth when bitten into, due to air trapped inside the porous material. They have a somewhat salty, mild flavor. The fried cheese curds at the Home Fire Grill are served in a cardboard canoe-shaped box. My extensive research was not able to find a reason for the canoe shape.
Turtle Stick
Turtle Stick
The feature offering at Heavenly Roasted Nuts is the Turtle Stick. A traditional “turtle” candy (pecans in chocolate-covered caramel, shaped somewhat like a turtle) is served on a salted pretzel stick. The original turtle candy was made in 1918 by Johnson’s Candy Company which later became DeMet’s Candy. The turtle name was coined by an unknown candy dipper who first thought that the candy looked like a turtle.
MINNESOTA TWINS
Target Field, located in the Warehouse District near downtown Minneapolis, opened in 2010. It was designed to be a neutral park, intended to favor neither hitters nor pitchers. The park is an open-air field, creating an often chilly environment for April games.
Featured Hot Dog/Sausage
Although the scope of this book does not include beverages, we will make an exception when the beverage is advertised as “A Meal in a Cup.” The Bigger Better Bloody Mary stand offers a Bloody Mary with a bratwurst on a skewer stuck into the drink. Bloody Mary is a cocktail containing vodka, tomato juice, and spices such as Worcestershire or Tabasco sauce. The cocktail ingredients give a unique flavor to the bratwurst. Both New York’s 21 Club and Harry’s New York Bar in Paris, France, claim to have invented the Bloody Mary.
Bloody Mary Bratwurst
More Hot Dogs and Sausages
Wasyl and Anna Kramarczuk immigrated from Ukraine in the late 1940s. Wasyl was a skilled sausage maker and Anna was an experienced baker. Together they opened Kramarczuk’s Restaurant and Delicatessen in Minneapolis in 1954. Every product it sells is made by hand from scratch. At Target Field the stand sells bratwurst and Polish sausage.
Halsey’s Sausage Haus is named in honor of former Twins broadcaster Halsey Hall. Hall originated the “Holy Cow!” as a home-run call long before it was used by Harry Caray. Halsey’s Sausage Haus serves Italian and Polish sausages, and cheddar and plain bratwurst. The sausages are provided by the Sheboygan Sausage Company.
The hot dogs at Target Field are provided by Schweigert Meats of Minnesota. This company has a quirky website. The timeline on its website shows the first store opening in 1937. It then lists two events between 1937 and 2014. The first event is “R&D team proposes new line of specialty sausages with trendy ingredients.” The second event, “Ray [Schweigert] fires R&D team.” Nothing like sticking to the basics.
Murray’s Restaurant
Murray’s Restaurant, located several blocks from Target Field, was opened in 1946 by Art and Marie Murray. Art met Marie when she was his waitress at the Schroeder Hotel in Milwaukee. Murray’s signature dish is the Silver Butter Knife Steak, purported to be so tender that it can be cut with a silver butter knife. The restaurant is still owned and operated by the Murray family. The Murray’s steak sandwich is available at the Mill City Grill stand at Target Field. The sandwich is made with choice sirloin and provolone cheese, served on a ciabatta roll. Mill City Grill also serves Murray’s famous garlic toast.
Andrew Zimmern of the Travel Channel
Andrew Zimmern is creator and host of the Travel Channel series Bizarre Foods with Andrew Zimmern. He has twice won the prestigious James Beard Foundation Award. For five years he was executive chef at the Café Un Deux Trois in Minneapolis. Andrew Zimmern’s Canteen at Target Field advertises itself as “a food adventure.” It offers two distinctive entrees: a Minnesota crispy belly bacon sandwich with vinegar slaw and jalapeño jelly; and a smoked meat sandwich with vinegar slaw and maple syrup hot sauce. The slice of bacon is the thickest slice of bacon I have ever seen in my life. The smoked meat is a very thinly cut brisket. Andrew Zimmern’s Canteen menu literally goes “through thick and thin.”
Turkey to Go and Ken Davis Barbeque Sauces
Turkey to Go “proudly serves turkey raised by Minnesota turkey farmers.” Turkey To Go, which is owned by the Minnesota Turkey Growers Association, offers its turkey sandwich at many Minnesota festivals. At Target Field the pulled turkey sandwich is served with Ken Davis barbeque sauces. Jazz musician Ken Davis opened a barbeque restaurant in Minneapolis in the late 1960s. He spent several years trying to duplicate the barbeque sauce his grandmother had made. Once he mastered the recipe, he decided to close the restaurant and sell barbeque sauces instead. Ken died in 1991, but his barbeque sauce company is still in business.
Minnesota Crispy Belly Bacon Sandwich
Señor Smokes
District Del Sol is a Hispanic neighborhood of St. Paul. The first residents were migrant agricultural workers who came in the early twentieth century. The community remained small until the 1980s when many immigrants from Mexico and Central America settled in the neighborhood. El Burrito Mercado, located in District Del Sol, provides burritos and other Mexican dishes to the Señor Smokes stand at Target Field. The Señor Smokes stand offers burritos and a Walk-A-Taco (an ice-cream-cone shaped tortilla filled with traditional taco ingredients). Señor Smoke was the nickname of former Twins pitcher Juan Berenguer.
Loon Café Chili
In addition to sausages, Halsey’s Sausage Haus sells chili from the Loon Café in Minneapolis. This classic Texas style chili contains cubed sirloin steak, onions, Tex-Mex spices and chili pepper. The Loon Café has been featured in magazines such as Esquire, Rolling Stone, and Bon Appétit. It is also noted as one of America’s best singles bars and, according to Playboy magazine, the clientele includes “office workers, docs, lawyers, art folks, visiting celebs, ladies openly on the move.”
Walleye and Chips
Walleye and Chips
The State Fair Classics stand features items from three local food purveyors. Mac’s Fish & Chips was opened in 1991 in St. Paul by former Minnesota North Star hockey player Tom McCarthy. At Target Field fans can get Mac’s Walleye and Chips. Walleye is a freshwater fish native to Canada and the northern United States.
Pork Chop on a Stick
J.D. Hoyt’s Supper Club is located several blocks from Target Field and caters to the “before and after game” crowd. During the game, fans can order Hoyt’s Pork Chop on a Stick. I could try to describe it further, but the name says it all.
Minneapple Pie
Minneapple Pie
The famous 1974 Chevrolet ad talked about “baseball, hot dogs, apple pie and Chevrolet.” George Atsidakos has taken his Minneapple Pie to Target Field. George’s parents, Andy and Libby, ran several restaurants in the Minneapolis area. George modified his father’s apple pie recipe and created a deep-fried apple pie in the shape of a mini turnover. His creation became known as the Minneapple Pie, sold at fairs and festivals. Fans at Target Field can get the Minneapple Pie at the State Fair Classics stand.
Izzy’s Ice Cream
Attorney Lara Hammel and her husband Jeff Sommers, a middle-school teacher, decided they wanted to open an ice cream shop. After a great deal of research, they opened Izzy’s in St. Paul in 2000. In May of 2005 Readers’ Digest named Izzy’s the best ice cream shop in America. Flavors at Target Field include salted caramel, peppermint bonbon, cookies and cream, and church elder berry.
Tony Oliva
Tony Oliva played his entire 15-year career with the Minnesota Twins. He was the American League Rookie of the Year in 1964 and won several batting titles during his career. Tony grew up in Pinar del Rio, Cuba. A scout for the Minnesota Twins saw him play in Cuba and invited him to play for the Twins organization. He did not want to leave home, but his father encouraged him to “become rich and famous in America.” At Target Field Tony is remembered at the Tony O’s stand where Cuban sandwiches are sold. The Target Field version comes with ham, pulled pork, Swiss cheese, Dijon mustard and pickles.
Frank Viola
Frank Viola was a pitcher for the Minnesota Twins from 1982 to 1989. He was the World Series MVP in 1987 and won the American League Cy Young Award in 1988. Viola was nicknamed “Sweet Music” by a Minnesota sports writer who said when Viola pitched, there was sweet music in the dome (referring to the Metrodome, former home of the Twins.) The Italian food stand at Target Field is named Frankie V’s in honor of Viola. It serves house made meatballs, calzones, and pizza from Papa John’s.
Minnesota Wild Rice
Wild rice is not related to the more familiar Asian rice. The plant grows widely in shallow water in small lakes and streams in Minnesota. The grain is a steady diet for ducks and other aquatic wildlife. Several Native American cultures including the Ojibwa, consider wild rice to be a sacred plant. Target Field offers Minnesota Wild Rice Soup made with chunks of chicken and assorted vegetables. While visiting the stadium in the summer, I was able to imagine having a warming bowl on a cold April day.
Minnesota Wild Rice Soup
Rib Tip Basket
Located in the courtyard just inside the main gate of Target Field is the Butcher and the Boar stand. The flavors from its smoker greets fans as they enter the stadium. The Butcher sells a Smoke and Fire Rib Tip Basket. Rib tips are chewy strips cut from the lower ends of spareribs.
NEW YORK METS
Citi Field opened in 2009 next to the site of Shea Stadium, former home of the Mets and location of the 1964-65 World’s Fair. Visitors entering Citi Field through the front entrance pass through the Jackie Robinson Rotunda, where the Brooklyn Dodgers legendary player is honored.
Featured Hot Dog/Sausage
The Hot Pastrami and Rye stand announces “We make our own hot pastrami, hand-carved and piled high on rye bread, served with deli mustard and a kosher dill pickle.” For those who love New York pastrami but still believe you should have a hot dog at a ball game, there is a perfect solution: Hot Pastrami and Rye serves a Pastrami Dog. A Nathan’s hot dog is covered with chopped pastrami and served in a hot dog bun.
Pastrami Dog
More Hot Dogs and Sausages
The Kosher Grill serves all-beef hot dogs as well as all-beef sweet sausages with grilled onions and peppers. Fried potato knishes from Gabila’s (known as the original Coney Island square knish) are available at the Kosher Grill.
Pat LaFrieda
In 1922 Italian immigrant Anthony LaFrieda opened a butcher shop in Brooklyn. Anthony taught the business to his five sons. In 1950 they expanded to a shop in New York City’s meatpacking district on West 14th Street. The shop was on the second floor of a building with no elevator and the LaFrieda brothers had to carry 200-pound hind saddles of beef on their backs, up a flight of stairs. Today the business, now known as Pat LaFrieda Meat Purveyors, is run by the next generation of LaFriedas. Citi Field features three stands offering Pat LaFrieda meat dishes.
Meatball Sliders
Pat LaFrieda’s Original Filet Mignon Steak Sandwich stand advertises, “The original LaFrieda family recipe showcases hand cut 100% Black Angus seared filet mignon, topped with Vermont Monterey Jack cheese and sweet caramelized onions served on a custom made toasted French baguette.” Pat LaFrieda’s Signature Meatball Sliders stand announces “A trio of Grandpa LaFrieda’s all-beef meatball sliders in traditional homemade tomato sauce topped with whipped ricotta cheese served on a locally-baked toasted roll.” Also available at the Pat LaFrieda’s Burgers stand are classic hamburgers and cheeseburgers.
Chef Danny Meyer
New York City chef and restaurant owner Danny Meyer has opened four stands at Citi Field. Shake Shack and Blue Smoke are outposts of his restaurants; El Verano Taqueria and Box Frites are stands created for Citi Field (and also Nationals Park in Washington, DC).
Shake Shack
The Shake Shack has become one of the hottest hamburger spots in New York City. Shake Shack started as a hot dog cart in Madison Square Park. The cart was a huge success, with fans lined up daily for three summers. Since then the Shake Shack has expanded to restaurants across the Northeast U.S. and around the world. The Shake Shack stand at Citi Field offers their classic ShackBurger. Its burgers are 100% all-natural Angus beef with “no hormones and no antibiotics ever.” It also serves a vegetarian ‘ShroomBurger, a crisp fried portobello mushroom filled with melted Muenster and cheddar cheeses and topped with their special Shack Sauce. End your Shake Shack meal with a frozen custard, a dense and creamy ice cream spun fresh daily.
Blue Smoke
Danny Meyer envisioned opening a restaurant in New York that would bring “…the soulful cuisine of America’s South and Midwest.” This vision became Blue Smoke. The flagship Blue Smoke is combined with the club Jazz Standard on East 27th Street in New York City. At Citi Field fans can enjoy a North Carolina applewood-smoked pulled pork sandwich, a grilled or fried chicken sandwich with buttermilk ranch sauce, a hickory-smoked beef brisket sandwich with spicy Kansas City sauce, or smoked chipotle chicken wings. Sides include thick-cut French fries dusted with Blue Smoke “magic dust” seasoning, slow-cooked pit beans with pulled pork, vinegar-based slaw, and cornbread served with chipotle butter.
Meat the Mets Pizza
El Verano Taqueria
El Verano Taqueria offers tacos, quesadillas, burritos, and nachos. Each of these four menu items can be made with a choice of steak (char-grilled sirloin), chicken pipian (grilled chicken breast with green mole), carnitas (slow-cooked pork shoulder), or vegetarian (roasted portobello mushrooms, zucchini, corn, and poblano peppers).
Two Boots Pizza
Two Boots Pizza serves Louisiana-style Italian pizza. The name comes from the shapes of both Louisiana and Italy, which look like boots. Although Two Boots serves the usual cheese and pepperoni slices, it also serves slices not found elsewhere. The Happy Recap slice comes with homemade andouille meatballs, ricotta, mozzarella and piquant sauce on a round Sicilian crust. The Meat the Mets slice contains Creole chicken, pepperoni, sweet Italian sausage, jalapeños, ricotta and mozzarella. The V for Vegan slice comes with artichokes, shiitake mushrooms, red onions, pesto, and Daiya non-dairy cheese.
Box Frites
Box Frites (French for “chips”) specializes in gourmet French fries. The thick-cut Idaho fries or sweet potato fries come with a choice of Buffalo blue cheese, smokey bacon, pesto, chipotle barbeque, or rosemary ranch dipping sauces. Also available at Box Frites are garlic fries and cheddar bacon fries, topped with crumbled Applewood-smoked bacon, cheddar sauce, and fresh scallions.
Antipasto Salad
Mama’s of Corona
Mama’s of Corona sells hero sandwiches and salads. Its signature hero, known as Mama’s Italian Special, has pepper ham, salami, homemade mozzarella, mushrooms, and peppers. The antipasto salad features salami, pepperoni, green and black olives, carrots, red cabbage, marinated mushrooms, artichoke hearts, and pepper cheese on top of iceberg lettuce. Mama’s is also a place for dessert with its traditional cannoli, an Italian pastry dessert meaning “little tube.” The tube is stuffed with a sweet creamy filling containing ricotta, then served either plain or chocolate covered.
Cannoli
Keith Hernandez and His Grill
Keith’s Grill is named for Keith Hernandez, former MVP first baseman for the New York Mets. Keith won the Gold Glove award for eleven years in a row. Hernandez also guest starred in several classic episodes of the Seinfeld television show. The featured item at Keith’s Grill is the Gold Glove Burger, a six-ounce hamburger patty on a toasted sesame bun with cheddar cheese, lettuce, tomato, dill pickle, raw onion, mayonnaise, with ketchup on the top bun and mustard on the bottom bun. Also offered is the Mets Burger, “created by Keith and his culinary team,” with cheddar and Jack cheese, bacon, guacamole, chipotle aioli and jalapeños.
NEW YORK YANKEES
The new Yankee Stadium opened in 2009 next to the site of the iconic original Yankee Stadium built in 1923. The design of the new stadium follows the form of the old stadium. If Babe Ruth were around in 2009, he probably would have said “Some ball yard!” when he saw the new Yankee Stadium for the first time, as he said in 1923 when the original Yankee Stadium opened. Considering his prolific appetite, “The Babe” would approve of the food offerings at the new stadium.
Featured Hot Dog/Sausage
Nathan Handwerker started selling his hot dogs at Coney Island in 1916. They have been the best-known hot dogs in New York for nearly a century. Nathan’s also sponsors the world-famous Fourth of July hot dog eating contest. Defending champion Joey Chestnut holds the record (set in 2013) for eating 69 Nathan’s Famous hot dogs and buns in ten minutes. It is only fitting that Yankee Stadium has chosen Nathan’s Famous as its hot dog provider. Varieties include the original, extra long, or natural casing hot dogs. The Bronx Bomber version (taking its name from the nickname of the Yankee team) includes chili, cheese, onion, sauerkraut, and relish. To enhance its hot dogs, Nathan’s offers crinkle-cut fries and dipping sauces of chipotle aioli, bacon mayonnaise, or roasted garlic herb aioli.
Nathan’s Hot Dog
More Hot Dogs and Sausages
For those who prefer an all-beef hot dog, Hebrew National has a stand at Yankee Stadium, where the hot dogs are topped with sauerkraut.
From the vast array of sausages manufactured by Premio Foods (a decades-old family business), two varieties are available at Yankee Stadium: Sweet/Mild Italian Sausage and Hot Italian Sausage. Both are served with onions and peppers.
Lobel’s of New York
The Lobel family developed a meat business in Austria in the 1840s. When a grandson of the founder came to the U.S. in 1911, he brought the family butcher shop business to New York City, where it is now owned and operated by the fourth and fifth generations of the Lobel family. At Yankee Stadium, the Lobel stand offers exactly one item: a USDA prime beef steak sandwich. Judging by the long line, the one item offered is worth the wait.
Parm
Parm’s restaurant (Mulberry Street in Manhattan) brings its classic meatball Parmesan sandwich to Yankee Stadium. Additionally, Parm’s has a fresh mozzarella sandwich made with eggplant, cheese, tomato, fresh basil, and house spicy dressing. Both sandwiches are served hot.
Fresh Mozzarella Sandwich with Eggplant
Bazzini Nuts
Anthony Bazzini came to America from Italy in the 1800s. His first job was in a nut factory. After years of hard work, he bought the factory which has now operated under his name for more than 100 years. Bazzini peanuts and pistachios are sold throughout the stadium.
Masahiro Tanaka and the Tanaka Roll
The ONE Sushi serves a variety of rolls including California, veggie, spicy tuna, salmon, and shrimp tempura. Their signature roll is the Tanaka Roll, named for New York Yankees pitcher Masahiro Tanaka. Before coming to the Yankees in 2014, Tanaka pitched in Japan for the Tohoku Rakuten Golden Eagles and had a 24-0 record in 2013. The Tanaka Roll has shrimp tempura topped with spicy tuna, lobster meat salad, and wasabi mayonnaise.
Tanaka Roll
Chicken and Waffles
The first stand encountered when entering through the main gate at Yankee Stadium is Chicken and Waffles. Fried chicken patties are served between two waffles with maple syrup as a condiment. They come as an order of three sliders. The same stand offers a vanilla ice cream sandwich with waffles taking the place of traditional cookies.
Brother Jimmy’s BBQ
With a slogan of “Put Some South in Yo’ Mouth,” Brother Jimmy’s BBQ brings legendary North Carolina slow-smoking barbeque to Yankee Stadium. Fans can choose either a Carolina pulled-pork sandwich or a pulled barbeque chicken sandwich.
Malibu Roof Top Deck
The Malibu Roof Top Deck can be found on the top level at Yankee Stadium. This area is a popular hangout before the game for fans who want island-inspired grilled chicken jerk wings served with corn on the cob, or coconut-rum-glazed barbeque smoked ribs with cilantro and slaw. Also available is a bacon cheddar stuffed burger on a potato roll.
Pretzel Twist
New York soft pretzels are sold throughout Yankee Stadium. In addition to the usual pretzel shape, they are available as a pretzel twist which resembles a loaf of bread.
Pretzel Twist
Strictly Kosher
Strictly Kosher Inc. provides kosher food at Yankee Stadium. The stand offers a glatt kosher hot dog. For meat to be kosher, it must come from a kosher animal and be slaughtered in a kosher way. For meat to be glatt kosher, it must also come from an animal with adhesion-free or smooth lungs. Kosher laws require certification by a rabbi. This stand posts a certificate from Rabbi Vaad Harabonim of Queens. In addition to the hot dog, Strictly Kosher Inc. serves chicken nuggets, roast beef and chicken deli wraps, and potato knishes.
OAKLAND ATHLETICS
O.co Coliseum opened in 1966 as the home of the Oakland Raiders football team. It became home to the Athletics in 1968 when the team moved to Oakland from Kansas City. Today it is the only stadium that is home to both a Major League Baseball team and a National Football League team.
Featured Hot Dog/Sausage
The Oakland Athletics Atomic Hot Sausage has the reputation of being the hottest American-style sausage. It’s the fresh onions and extra hot spices added to the beef and pork sausage that give it this reputation. Hickory smoked in a natural pork casing, the sausage is grilled and then topped with peppers and onions. The Atomic Hot Sausage is sold by Saag’s. Saag’s has produced authentic German sausages using old-world recipes since 1933 when George Saag opened a small butcher shop in downtown Oakland.
Atomic Hot Sausage
More Hot Dogs and Sausages
Saag’s other ballpark sausages are the Sweet Italian, the Polish Sausage, the Hot Link, and the Bratwurst.
Also available at the New Belgium Brewing Company stand are three specialty hot dogs. The All Star Dog has macaroni and nacho cheese with jalapeños. The Diablo Dog has nacho cheese, bacon, and fried onion strings. The Bay Bridge Dog has chili and cheese.
There are several A’s Grill carts, located throughout the stadium, serving jumbo dogs and Italian and Polish sausages with grilled onions and peppers.
Queen Margherita and Her Pizza
For pizza lovers, the White Elephant Brick Oven offers a margherita pizza with buffalo mozzarella cheese. A margherita pizza, with its red tomatoes, green basil, and white mozzarella, replicates the colors of the Italian flag. A popular legend holds that the margherita pizza was created to honor Queen Margherita when she was visiting the Royal Palace of Capodimonte in Naples in 1889. However, according to the BBC Food website, Zachary Nowak, Assistant Director of Food Studies at the Umbra Institute in Perugia, Italy, raises doubts on the authenticity of the legend.
Margherita Pizza
Seared Salmon Cake
Sold from a cart and prepared to order (takes about ten minutes) is the Seared Salmon Cake with frizzled onions and lemon dill aioli on a salt-and-pepper bun. A salt-and-pepper bun contains both sea salt and cracked pepper.
Cheeseburger Poppers
Ball Park Poppers
Ball Park Poppers are bite-size balls of deep-fried dough. They come in three varieties: jalapeño, cheeseburger, and corn dog. The poppers are topped with parsley and Parmesan cheese. Ball Park Poppers also sells full-size corn dogs. At most ballparks, corn dogs are prepared off site and frozen, then reheated at the park. At O.co Coliseum the hot dogs are dipped in batter and fried on site, as the customer watches.
Sweet Potato Pie
Sweet Potato Pie
Sweet potato pie is a traditional side dish in the Southern United States, a soul food staple that probably came out of the African traditions of black slaves. It is often served during the American holiday season, especially at Thanksgiving. O.co’s Ribs & Things stand serves it with whipped cream.
Gourmet Popcorn
Gourmet popcorn is available in seven flavors: caramel corn, cheddar cheese, spicy cheese, white cheddar, chocolate drizzle, birthday cake, and apple cinnamon. The popcorn cart also offers what it refers to as “Chicago style” popcorn, which is a mixture of cheddar cheese and caramel.
PHILADELPHIA PHILLIES
Citizens Bank Park opened in 2004 and is part of the Philadelphia Sports Complex. The homes of the other three major Philadelphia sports teams are located within the Complex. The Philadelphia Eagles of the National Football League play at Lincoln Financial Field; the 76ers of the National Basketball Association and the Flyers of the National Hockey League play at the Wells Fargo Center.
Summer Dog
Featured Hot Dog/Sausage
The Summer Dog comes with cucumber slices, peppers, onion, relish, and ancho chili sauce (a type of mole sauce made with mildly spicy ancho chilies.) It is available at The Philly Frank and Stein concessions stand. With the play on “Frankenstein” in the name of the stand, it’s clear that not only hot dogs but beer is available here.
More Hot Dogs and Sausages
The Old Bay Signature Sausage is a sweet Italian sausage covered with grilled peppers and onions and topped with Old Bay Seasoning. This blend of 18 herbs and spices is a standard in the Chesapeake Bay region, found on most restaurant and home tables along with salt and pepper. Old Bay’s name was inspired by a steamship line that once traveled between Maryland and Virginia.
As one would expect in Philadelphia, there is a Cheesesteak Dog at the ballpark. The hot dog is topped with chopped steak, onions, and melted cheese.
The Schmitter
The Schmitter
There is a common misbelief that The Schmitter is named for Mike Schmidt, Hall-of-Famer third baseman who played for the Phillies from 1973 to 1989. In reality, it was invented in the 1920s at McNally’s Quick Lunch in the Chestnut Hill area of Philadelphia. The Schmitter was named after Schmidt’s beer. It is described as “Philadelphia’s Big League Sandwich.” Sliced beef is topped with cheese, fried onions, tomatoes, grilled salami, and Schmitter sauce and then served on a flash-broiled Conshohocken kaiser roll. The Schmitter sauce, which has the look of Thousand Island dressing, is a “secret” recipe. The Conshohocken Italian bakery opened in the Philadelphia suburb of Conshohocken in 1973.
Philadelphia Cheesesteak
The food that people most associate with Philadelphia is the cheesesteak. Two of Philadelphia’s best known cheesesteak providers have stands at Citizens Bank Park.
Tony Luke’s describes its cheesesteaks as “the real taste of South Philly.” It uses thinly sliced Black Angus beef carefully chosen from humanely raised cattle. Customers can add the cheese of their choice (American, Cheez Whiz, or provolone). Another choice at Tony Luke’s is its roast pork sandwich. Garlicky pork, sliced thin, is served au jus on a roll.
Campo’s advertises itself as “the best place for authentic Philadelphia food.” Its cheesesteak sandwich choices are The Heater or The Works. The Heater is a spicy cheesesteak made with jalapeño cheddar cheese and buffalo hot sauce. The Works adds sweet bell peppers, mushrooms, fried onions and provolone cheese. The cheesesteaks can be accompanied by local favorite Herr’s potato chips.
Chickie’s and Pete’s Crabfries
The most crowded stand on the day we visited Citizens Bank Park was Chickie’s and Pete’s Crabfries. When Chickie’s and Pete’s opened in Philadelphia in 1977 as a crab house and sports bar, it sold crab mainly in the summer. Pete wanted a way to use the leftover crab seasoning during the winter, so he experimented with putting it on French fries. After two winters of trying different combinations and asking his customers for their opinions, Pete settled on what has become his signature crinkle-cut Crabfries.
Crabfries
Federal Donuts
Federal Donuts is a Philadelphia institution that serves fried chicken with donuts. Every order of chicken includes Japanese cucumber pickles and a honey donut. The chicken can be original style or cooked with either buttermilk-ranch seasoning or a chili-garlic glaze. For those who want their donuts chicken-less, vanilla spice and cinnamon brown sugar varieties are available.
Pennsylvania Dutch Funnel Cake
The Pennsylvania Dutch Funnel Cake stand serves its deep-fried cakes topped with powdered sugar and either strawberries or apples. The funnel cake mix comes from the Funnel Cake Factory which traces its origin back to Lorraine Wilson. Mrs. Wilson was a supper-club singer who founded a family-run funnel cake business in 1974. Using her grandmother’s recipe, she began selling at local fairs in Pennsylvania.
Funnel Cake with Strawberries
Water Ice
The Philadelphia Water Ice Factory stand sells three flavors of water ice (sometimes known as Italian ice). Water ice is a frozen dessert similar to ice cream but made without dairy products or eggs. Flavorings come from fruit concentrates, juices or purees. Flavors available at Citizens Bank Park are cherry, lemon, and mango.
Greg “The Bull” Luzinski’s BBQ
Left fielder Greg “The Bull” Luzinski, an All-Star player with the Phillies, opened Bull’s BBQ at Citizens Bank Park. He tries to attend every home game to meet with fans and share his love of barbeque. The Bull offers pulled pork sandwiches, pit beef sandwiches, ribs, turkey legs, hot dogs, and half-roast-chicken plates. Baked beans and cole slaw are served as side dishes. All the meats are cooked with The Bull BBQ Sauce, which is also sold by the bottle at the stand and in many area grocery stores.
BBQ Chicken Platter
Hoagies
In addition to cheesesteak, Philadelphia is known for hoagies. Called by different names throughout the country, a hoagie is a sandwich built on a long bread roll and filled with a variety of meats, cheeses, vegetables and seasonings. Citizens Bank Park serves several varieties of hoagies: turkey, ham and cheese, roast beef, or vegetarian.
PITTSBURGH PIRATES
PNC Park opened in 2001. It is located along the Allegheny River on the North Shore of Pittsburgh, with a panoramic view of the Pittsburgh skyline. The architects modeled the stadium after historic Forbes Field, home of the Pittsburgh Pirates from 1909 to 1970.
Featured Hot Dog/Sausage
Polish Hill is a residential neighborhood in Pittsburgh which is also home to the Immaculate Heart of Mary church, the oldest and largest in Pittsburgh. The neighborhood was settled in the nineteenth century by Polish immigrants who came to work in the steel industry. The Polish Hill Dog is topped with mini potato pierogies and homemade onion straws, with cole slaw spread on the bun.
More Hot Dogs and Sausages
The Federal Street Grill offers a Polish kielbasa and an Italian sausage, both topped with grilled peppers and onions. Sausages at PNC Park are provided by Silver Star Meats of nearby McKees Rocks. Silver Star Meats, founded in 1964, provides products from Eastern European traditions dating back to the late 1800s.
Polish Hill Dog
Buns for hot dogs at PNC Park are made by Cellone’s Italian Bread Company of Pittsburgh. The Cellone family came to Pittsburgh in 1911 from Torino, Italy. They started baking breads in their home and delivering them door-to-door throughout the neighborhood, using a horse-drawn wagon. Cellone’s is said to be the first bakery in the United States to produce the egg bun, which has become an American bakery standard.
Primanti Brothers
In the early 1930s, Joe Primanti set up a cart in Pittsburgh, selling sandwiches to hungry truckers who were coming and going at all times of the night. Encouraged by his success, he opened a storefront on 18th Street, along with his brothers Dick and Stanley and their nephew John DePriter. They became the Primanti Brothers, operating a chain of restaurants throughout the Pittsburgh area. Primanti Brothers is best known for serving sandwiches with French-fried potatoes and cole slaw incorporated into the sandwich filling. At PNC Park, two of its specialty sandwiches are offered, both topped with French fries, cole slaw and tomatoes on white French bread. The choice is either roast beef and cheese or capicola and cheese. Capicola is a traditional Italian cold cut made from the dry-cured muscle running from the neck to the fourth or fifth rib of a pork shoulder. (It probably tastes better than it sounds.) Primanti Brothers also offers a traditional cheesesteak sandwich.
Primanti Brothers Roast Beef Sandwich
Chicken Gyros
Papa Dukes Gyros
George and Dorothea Papas, along with their adult children Perry and Frances, opened their first restaurant in 1982. It was known as Papa Duke’s Paris Grill. A second location, Papa Duke’s Bar and Grill, opened in 2008. At PNC Park Papa Duke’s has a gyros stand. Gyros is a Greek dish with meat roasted on a vertical spit. The meat, thinly sliced off the spit, is served in a pita bread pocket with tomato, onion, lettuce and tzatziki sauce. Tzatziki is made from strained yogurt mixed with cucumbers, garlic, salt, olive oil and lemon juice. Papa Duke’s gyros come with a choice of beef or chicken.
Pierogi
A pierogi (with many different spellings) is a filled dumpling of unleavened dough, first broiled and then baked or fried. Pierogies are traditionally stuffed with cheese, potato filling, ground meat, or fruit. They are usually semicircular in shape, but can be made in other shapes. They are of Eastern European origin. At PNC Park, fans can order Mrs. T’s cheese pierogies, a four-cheese medley of aged cheddar, Parmesan, romano, and Swiss cheese. Mrs. T (Mary Twardzik) and her son Ted started making pierogies in 1952 in Shenandoah, Pennsylvania. Today Mrs. T’s sells over one-half billion pierogies a year in grocery stores and at restaurants throughout the country.
Seaweed Salad
Seaweed Salad
Nakama Japanese Steakhouse and Sushi Bar has been voted the best sushi in Pittsburgh for eight consecutive years by Pittsburgh Magazine. At PNC Park, Nakama Express offers sushi, hibachi meats, udon noodles, egg rolls, fried rice, and seaweed salad. (I believe Harry M. Stevens never hawked seaweed salad himself.) Seaweed is a good source of fiber and has many other nutritional values including being one of the best sources of iodine. The seaweed salad at Nakama Express is made with sesame seeds and a vinaigrette dressing.
Pirates Buried Treasure
Cold Cow Ice Cream is a family-owned business based in Pittsburgh, specializing in using locally-sourced ingredients. One special flavor offered at PNC Park is Pirates Buried Treasure, fudge ripple ice cream blended with peanut butter cups.
Rita’s Italian Ice
In 1984 Pennsylvania former firefighter Bob Tumolo started selling Italian ice from a small porch window in Bensalem, Pennsylvania, with the goal of earning a little extra income. He named the business after his wife, Rita. He more than achieved his goal, and Rita’s Italian Ice now has over 500 locations. At PNC Park this dairy-free frozen dessert comes in flavors such as mango, mint chocolate chip, cookies and cream, strawberry colada, and Swedish Fish (soft chewy fish-shaped candies).
Willie Stargell and Pops Plaza
Willie Stargell, nicknamed Pops, played his entire 21-year major league career in Pittsburgh as an outfielder and first baseman. He led his team to World Series championships in 1971 and 1979 and was inducted into the Baseball Hall of Fame in 1988. Four concessions stands are part of Pops Plaza, honoring Willie Stargell. Joining Nakama Express and Chickie’s and Pete’s at Pops Plaza are Chicken on the Hill and the Familee BBQ.
Tatchos
In addition to chicken tenders and French fries, Chicken on the Hill offers its unique creation, Tatchos. Tatchos are tater tots covered with sour cream, chili, cheese, and jalapeños.
Familee BBQ
“We Are Family,” a song by Sister Sledge, was the theme song for the 1979 World Series champion Pittsburgh Pirates. This might be the inspiration for the Familee BBQ stand at Pops Plaza. The stand serves all-beef jumbo hot dogs, one-half-pound burgers, pulled pork nachos, and a pulled pork pierogi stacker (a pulled pork sandwich with pierogies on top.)
Manny Sanguillen
Manny Sanguillen played for the Pittsburgh Pirates as a catcher during 13 of his 14 years in the major leagues. His .296 lifetime batting average is tenth highest for catchers in major league history. Legendary Pittsburgh Pirate Roberto Clemente was killed in 1972 in an airline disaster while taking relief supplies to victims of an earthquake in Nicaragua. Clemente had invited his best friend, Manny Sanguillen, to join him on the flight. When Sanguillen was ready to go to the airport he could not find his car keys, thus missing the flight and saving his life.
Manny’s BBQ stand at PNC Park is known for its grilled Angus burger, a chargrilled one-half-pound Angus burger topped with American cheese and Manny’s signature barbeque sauce. Angus beef comes originally from Angus County in Scotland. Black Angus is now the most common breed of beef cattle in the United States. Manny’s BBQ also sells a pulled pork sandwich.
Shrooms Burger
BR-GR (Burger)
The BR-GR stand says, “Our beef is a handcrafted blend of sirloin, chuck, rib eye, and strip ground fresh daily.” It offers four specialty hamburgers. The Abso-Bac’n-Lutely Burger adds bacon to three cheeses (American, provolone, pepper jack). The Shrooms Burger has forest mushrooms, caramelized onions, provolone, and whole grain mustard aioli. The Fire in the Hole Burger has guacamole, jalapeños, pepper jack cheese, chipotle aioli, and sriracha sauce. The California Lovin’ Burger is actually a turkey burger. It is topped with provolone, oven-roasted tomatoes, pesto mayo, alfalfa sprouts, and guacamole.
Quaker Steak and Lube
At the Quaker Steak and Lube stand boneless wings and onion rings are offered. (I am not sure why you would name a food stand as a takeoff on motor oil. I am further puzzled by the fact that the stand sells chicken rather than steak.) The sauces for the wings are Arizona Ranch, Louisiana Lickers, BBQ Hot and BBQ Medium. Its onion rings are known as O-Rings, a pun on o-rings, a common automobile component.
Healthy Options
The Just4U stand offers “healthy options.” Several salads are available including the tomato and mozzarella salad with oven-dried grape tomatoes tossed in olive oil and basil, served with fresh mozzarella and balsamic dressing. The Caprese Toaster is a sandwich with sliced tomatoes and mozzarella served on gluten-free bread with fresh basil and a balsamic glaze.
SAN DIEGO PADRES
Petco Park opened in 2004 in downtown San Diego, part of the trend of building ballparks in the middle of urban centers. There was no need for a retractable roof as the Padres average one rainout for every seven years.
“K” Basa
Featured Hot Dog/Sausage
Randy Jones, a former Padres pitcher and Cy Young Award winner, moved from baseball to a career in catering and restaurants. He opened the All-American Sports Grill in downtown San Diego. The Randy Jones Grill at Petco Park features hot dogs, sausages, and sliders. The “K” Basa is a one-half pound kielbasa sausage. “K” is the symbol for strikeout on a baseball scorecard.
More Hot Dogs and Sausages
Also available at the Randy Jones Grill are The Slugger Dog, a one-half pound hot dog, and the Hi Heat Link, a one-third pound hot link sausage.
Friar Franks
The traditional hot dogs at Petco Park are called Friar Franks and are provided by the hot dog chain Wienerschnitzel. Wienerschnitzel (sometimes known as Der Wienerschnitzel) has hundreds of locations throughout California and the Southwest. The Wienerschnitzel website has a timeline of the history of the hot dog. Interestingly, it incorrectly credits Harry M. Stevens and cartoonist Tad Dorgan with coining the phrase “hot dog.” (See “The Hot Dog Comes to Baseball” chapter.)
El Toro Tri-Tip Sandwich
El Toro Tri-Tip Sandwich
In 2013, USA Today asked fans of all 30 major league teams to vote on their favorite ballpark food in the Stadium Food King Challenge. The winner was the El Toro Tri-Tip Sandwich at Petco Park. The sandwich is provided by Phil’s BBQ, a local San Diego favorite. Phil’s BBQ stand also serves baby back ribs and pulled pork sandwiches.
Seafood Burritos
For the many fans of Mexican cuisine in the San Diego area, Petco Park has a variety of options. Both Lucha Libre and Miguel’s Cocina sell versions of meat and seafood burritos. The Lucha Libre’s Surfin’ California Burrito was featured on the Travel Channel television show Man v. Food. This burrito is stuffed with grilled steak, shrimp, French fries, avocado, pico de gallo, cheese, and super-secret chipotle sauce. Miquel’s Cocina describes their Surf ‘n Turf Burrito as “an epic 1 lb. burrito.”
Hodad’s
Local San Diego burger place Hodad’s has a cart at Petco Field. The cart is decorated with license plates from many states. The menu is simple: hamburgers, cheeseburgers, and fries. www.cnn.com listed Hodad’s as one of its “Five Tasty Burger Joints Worth Visiting.”
Anthony’s Fish Grotto
When World War II ended in 1945, Anthony and Tod Ghio and their friend Roy Weber returned home to help “Mama” Ghio open a seafood restaurant on the San Diego waterfront. Using old-world recipes from her days in Italy, “Mama” Ghio’s menu has been served for over sixty years. At Petco Park, Anthony’s Fish Grotto continues the tradition with fish and chips, shrimp and chips, shrimp avocado salad, and clam chowder.
Bumble Bee Tuna
Bumble Bee Seafoods moved its corporate headquarters in 2014 to an unoccupied historic building located adjacent to Petco Park. Bumble Bee is one of the leading sellers of canned tuna. The stand at Petco Park offers a grilled tuna melt, a tuna Nicoise salad (a French salad popular in the seaside town of Nice), and a tuna salad sandwich with vine-ripened tomatoes, golden raisins, and arugula served on a ciabatta roll.
Southpaw Sliders
In addition to hot dogs and sausages, the Randy Jones Grill at Petco Park offers two varieties of Southpaw Sliders. Randy was a left-handed pitcher (referred to as a southpaw). The garlic marinated steak sliders are topped with blue cheese spread. The chipotle chicken sliders have chimichuri spread (a green sauce of Argentinian origins using minced garlic, parsley, olive oil, oregano and wine vinegar). Both sliders come with sides of summer squash and kettle chips.
Seaside Market
The Seaside Market serves a burgundy pepper tri-tip sandwich on a brioche roll. It comes with a choice of two sides from a list of gooey mac ‘n cheese, crispy potato wedges, pineapple cole slaw, or country applewood-smoked bacon potato salad. Also available are nachos topped with sliced tri-tip. The Seaside Market also serves unique salads including Spicy Tofu Noodle Salad with sweet peppers, cilantro, orzo, sun-dried tomatoes and feta cheese or California Kale and Avocado Salad with oregano vinaigrette.
California Kale and Avocado Salad
Rimel’s Rotisserie
Rimel’s Rotisserie provides fish tacos made with grilled chunks of fresh mahi-mahi and served with black beans and rice. As its name implies, rotisserie chicken is available fresh off the spit. In the California fusion food tradition, Rimel’s also offers Chinese-style dishes such as pot stickers and “wok’d” bowls. The “wok’d” bowls include vegetables, rice, and the choice of chicken or mahi-mahi.
Mahi-Mahi Taco
Chocolat Bistro Creperie Cremerie
The Chocolat Bistro Creperie Cremerie serves sweet and savory homemade crepes. The sweet crepes may be chocolate or Nutella and come with a choice of bananas or strawberries. The savory crepes are ham and cheese or a four-cheese combination.
The Baked Bear
The Baked Bear sells custom ice cream cookie sandwiches. Each customer chooses a cookie for the top, a cookie for the bottom, and a flavor of ice cream. The varieties of cookies and ice cream flavors change daily. We chose a red velvet top cookie, a snickerdoodle bottom, and mint chip ice cream.
Ice Cream Cookie Sandwich
SAN FRANCISCO GIANTS
AT&T Park opened in 2000, located in the “South of Market” neighborhood of San Francisco, overlooking San Francisco Bay. McCovey Cove (named for former Giants first baseman Willie McCovey) is the unofficial name of the section of San Francisco Bay beyond the right field wall of AT&T Park. During games small boats anchor in the Cove hoping to retrieve home run balls known as “splash hits.”
Featured Hot Dog/Sausage
Tres Agave (literally, three agave plants) is the San Francisco restaurant that supplies AT&T Park with the Tres Agave Dog. This hot dog would be at home on a street cart in Tijuana, Mexico. The wiener has a bacon wrap, spicy chipotle mayonnaise, sweet grilled onions, and cucumber pico de gallo.
Tres Agave Dog
More Hot Dogs and Sausages
The Chicago Dog stand (serving Hebrew National products) offers three regional hot dogs. The Chicago Dog is topped with mustard, relish, onion, tomato, pickle, sport peppers and celery salt. The Coney Island Dog comes with chili, cheddar cheese and onions. The San Francisco Dog has Swiss cheese, sauerkraut, onion, pickle spears, and Thousand Island sauce.
The Doggie Diner at AT&T Park is named after the iconic fast food restaurant chain that flourished in the San Francisco Bay Area from 1948 to 1986. The AT&T Doggie Diner features hot dogs from the Eisenberg Sausage Company, founded in 1929.
At the Say Hey! sausage stand (baseball great Willie Mays was known as “The Say Hey Kid”), fans have their choice of Italian sausage, bratwurst (regular or cooked with beer), Louisiana hot link, chicken apple sausage, pineapple sausage, Polish kielbasa, pepper jack hickory-smoked sausage, as well as an all-beef colossal dog.
McCovey’s Restaurant serves a traditional stadium dog, a Polish dog, and a hot link.
Edsel Ford Fong
Edsel Ford Fong was a well-known waiter at the Sam Wo Restaurant in San Francisco’s Chinatown. He was often called the world’s rudest, worst, and most insulting waiter. Legendary San Francisco Chronicle columnist Herb Caen included Fong as #58 in his guide of things to do in San Francisco: “See the world’s rudest waiter.” The Edsel Ford Fong stand at AT&T Park serves standard Chinese fare including crispy orange chicken, beef with broccoli, and vegetable fried rice. It also serves edamame (young soybeans in the pod).
Edamame
Ghirardelli Chocolate
One of the best-known San Francisco treats is Ghirardelli chocolate. The Ghirardelli Chocolate Company began in San Francisco in 1852 and per their website is “one of the few original and continuously operating businesses in California.” Domingo Ghirardelli, its founder, was the son and apprentice of an Italian chocolatier. In 1849 he moved to San Francisco first to join the gold prospectors, and later to sell tired miners chocolate candies. Part of Ghirardelli’s great success for so long is due to its complete control of the manufacturing process (from cocoa bean selection to finished product) and innovative use of advertising. At AT&T Park, fans can order either the classic hot fudge sundae or ice cream served in a waffle cone.
Ghiradelli Sundae
Orlando’s Caribbean BBQ
Orlando’s Caribbean BBQ is owned by former Giant slugger Orlando Cepeda. Its signature item is the Cha Cha Bowl which consists of jerk chicken, white rice, black beans, and pineapple salsa. It also serves a Cuban sandwich (pork loin, ham, Swiss cheese and pickles on a panini roll) as well as a wide variety of nachos. The sweet potato fries are topped with a cinnamon chipotle sprinkle.
North Beach
North Beach is an area of San Francisco known as “Little Italy” and the childhood home of Joe DiMaggio. North Beach – A Taste of San Francisco at AT&T Park offers three classics found in North Beach restaurants. The catchiest name goes to the Stinking Rose 40 Clove Garlic Chicken Sandwich. It also serves a meatball sandwich and a ravioli bowl.
Crazy Crab’z
The Crazy Crab’z serves a traditional San Francisco crab sandwich consisting of fresh Dungeness crab salad on grilled sourdough bread with tomatoes. Salad lovers can get a crab and shrimp salad or a Crab Louie salad.
Crab Sandwich
Murph’s Irish Pub
Murph’s Irish Pub serves a classic grilled Reuben sandwich (corned beef, Swiss cheese, and sauerkraut on rye). It also sells Irish Nachos -- French fries covered with chili, cheddar cheese, and jalapeños.
Pier 44 Chowder House
The Pier 44 Chowder House offers the famous San Francisco clam chowder in a sourdough bread bowl. Seafood lovers can also choose traditional fish and chips, shrimp and chips, or fried calamari.
Lamb
At most ballparks, lamb lovers would be out of luck. At AT&T Park, however, there is not one but two locations serving lamb. The Anchor Grill and the California Cookout each serves its own variety of lamb sausage.
‘Outta Here Cheesesteak
‘Outta Here Cheesesteak is named for the familiar baseball announcer’s call for a home run. In addition to the original Philly Cheesesteak, it serves a San Francisco variety with grilled chicken, mushrooms, onions, Cheez Whiz, and the choice of sweet or hot peppers. The veggie cheesesteak features Cheez Whiz, shredded zucchini, grilled tomatoes, mushrooms, onions and peppers. All of the cheesesteaks are served on traditional Amoroso rolls. Amoroso’s Baking Company is a five-generation family-owned Philadelphia company producing hearth-baked rolls and bread.
Fresh Roasted Peanuts
The fresh roasted peanut stand has its own peanut roasting machine where you can watch the peanuts whirling around as they roast. The roasting machine was manufactured by C. Cretors & Co. of Chicago, which entered the concessions machine business in 1885.
Biscotti and Madeleines
For those who want an out-of-the-ordinary sweet treat, the @Café, an internet café within the ballpark, serves biscotti and madeleines.
SEATTLE MARINERS
Safeco Field is a retractable-roof stadium located in the SoDo district of Seattle. SoDo is the “South of Downtown” area of Seattle, near the historic Pioneer Square. Next door is the site of CenturyLink Field, home of the Seattle Seahawks football team.
Featured Hot Dog/Sausage
Ethan Stowell is a celebrated chef and owner of restaurants in the Seattle area. Stowell was named by Food & Wine Magazine as one of the “Best New Chefs in America” in 2008. Since 2010 Ethan has been working with the Seattle Mariners on the food offerings at Safeco Field. Ethan Stowell’s Hamburg + Frites, as the name implies, sells hamburgers and French fries. It also offers, however, a signature hot dog known as the Pen Seattle Dog. The stand is located in The Pen, an area in centerfield behind the Mariners’ bullpen. The Pen Seattle Dog comes with cream cheese, banana peppers, and onions. I’ve often seen cream cheese on bagels and occasionally turkey sandwiches, but this is the first time I’ve seen it on a hot dog.
Pen Seattle Dog
More Hot Dogs and Sausages
The traditional hot dog at Safeco Field comes in three sizes with names as confusing as Starbucks’ tall, grande, and venti. (Starbucks was founded in Seattle.) The “tall” hot dog is called the SoDo Dog. The “grande” version is called the Mariners Dog. The “venti” is called the Foot-long Dog.
At the Safeco Field Sausage Company stand are cheddar bratwurst, sweet Italian sausages, and Polish sausages. They are all served with grilled onions, banana peppers, and Riesling sauerkraut. According to the Total Wine & More website, “Washington State produces high-quality Riesling.”
Veggie Dogs and Burgers
The Natural stand sells veggie burgers and veggie dogs. The veggie dog is made by local Seattle company Field Roast. Seattle chef David Lee learned of the Asian tradition of using grains as the foundations of vegetarian “meat.” By adding European flavors, he developed a new version of grain meat used at Field Roast.
Swingin’ Wings
Swingin’ Wings is another stand created by Chef Ethan Stowell. Chicken wings are available in classic, honey serrano, or barbeque varieties. For those who like fried foods other than potatoes, fried cheese curds and fried pickles are interesting alternatives.
Ivar’s
Ivar’s is a Seattle seafood icon. Ivar Haglund, known in Seattle in the 1930s as a folksinger, opened the first Seattle aquarium on the Puget Sound waterfront in 1938, exhibiting sea life that he had collected from the Sound. Seeing that aquarium visitors were hungry, he soon added Ivar’s Fish Bar where he sold clam chowder and fish and chips. Today Ivar’s has expanded to include full-service restaurants and fish bars in many locations.
At Safeco Field, in addition to clam chowder, salads, and fish and chips, Ivar’s offers two signature sandwiches. Advertised as “Seattle’s Fish Sandwich,” the Ivar Dog is fried and freshly breaded Pacific cod, cole slaw and tartar sauce on a roll. The other specialty is the Grilled Wild Alaska Salmon Sandwich. Ivar’s uses salmon from the Yukon River.
Edgar’s Cantina
Growing up in Mexico City, Roberto Santibañez learned to make tamales and salsa from his grandmother. Migrating to the U.S. in 1997, he brought his restaurant ideas first to Texas, then New York City. His stand at Safeco field, Edgar’s Cantina, makes torta sandwiches. A Mexcian torta sandwich is served on an oblong crusty sandwich roll. Edgar’s tortas are toasted and filled with either carne asada, pork carnitas, chicken Milanese or salchicha (sausage). The stand is named for former Seattle Mariner great Edgar Martinez.
Chicken Caesar Pizzetti
Chicken Caesar Pizzetti
Modern Apizza has long been a local tradition in New Haven, Connecticut. Now, 2,932 miles from New Haven, owner and chef Bill Pustari has opened a concession stand at Safeco Field called Apizza. Pizza varieties are cheese, pepperoni, and wild mushroom. Appiza also serves pizzetti, a single-serving Neapolitan-style thin crust pizza, topped with Caesar salad or chicken Caesar salad.
Thai Ginger
Local Seattle restaurant Thai Ginger has opened the Thai Ginger International Wok at Safeco Field. Offerings include garlic beef, chicken curry, cashew chicken, and pad Thai vegetables.
ShiskaBerry’s
ShiskaBerry’s has been selling chocolate-dipped fruit on a stick at sporting events and concerts since 1998 in the western United States. At Safeco Field fans can get the Double Play, fresh strawberries dipped in white and milk chocolate. The Berry I’bananez is a skewer of banana chunks and strawberries dipped in a choice of white, milk or dark chocolate. This treat is named after former Seattle Mariner Raúl Ibañez.
Vegan Steamed Buns
Bao Choi serves vegan steamed buns. The buns are similar to the style often served with Peking duck. Two choices of fillings are offered. The first contains black vinegar-glazed portobello mushrooms, green chili and cucumber salad, basil, and siracha mayonnaise. The second has gochujang-glazed eggplant, cilantro cabbage slaw, and kimchee mayonnaise. (Gochujang is a savory fermented Korean condiment made from red chili, rice and soybeans.)
Vegan Steamed Buns
Banana Peppers
Uncle Charlie’s Cheesesteak serves the usual cheese-steak except for the type of pepper. Mild, tangy banana peppers are used instead of the usual bell peppers. I counted four different stands at Safeco Field that use banana peppers but cannot find a Seattle banana pepper connection.
Fried Twinkie
Fair Territory sells favorites traditionally sold at county and state fairs. Ballpark fans can choose from funnel cake, churros, corn dogs, or the infamous fried Twinkies. Fortunately for those who think a Twinkie is too healthy for you, one can now add the deep-fried element.
Fried Twinkie
ST. LOUIS CARDINALS
Busch Stadium opened in 2006 in downtown St. Louis. Outside the stadium is St. Louis Ballpark Village, a newly developed area containing retail shops, restaurants, and residential units. The highest attendance for a sporting event at Busch Stadium was not a baseball game, but rather a soccer match between Chelsea Football Club and Manchester City Football Club played in 2013.
Bacon-Wrapped Dog
Featured Hot Dog/Sausage
Hunter Hot Dogs are “the official hot dog of the St. Louis Cardinals.” The Hunter bacon-wrapped dog is a jumbo dog topped with baked beans, French-fried onion strings, sauerkraut, pico de gallo, and barbeque sauce.
More Hot Dogs and Sausages
For fans who prefer beef hot dogs, several carts throughout the stadium sell Nathan’s All-Beef Jumbo Dogs. For fans who want the hot dog experience without red meat, chicken bratwurst is available.
Meat Knish
Pretzels and bratwurst are popular ballpark foods. The Triple Play stand decided to go for a double play with the Bratzel, a bratwurst wrapped inside a pretzel.
Kohn’s Kosher Knishes and Knockwurst
Kohn’s offers kosher food at Busch Stadium. Kohn’s is a kosher restaurant founded in St. Louis by Simon and Bobbie Kohn in 1963. At Busch Stadium pastrami and corned beef sandwiches are served with potato salad. Kosher hot dogs and knockwurst, along with meat and potato knishes, are also sold. The knishes are made locally and are baked instead of fried.
MoonPies and Cherry Licorice
The Plaza Grill offers two regional sweets. MoonPies have been made at the Chattanooga Bakery since 1917. MoonPies consist of two round graham cracker cookies with marshmallow filling between them, then dipped in chocolate. Cherry licorice at the Plaza Grill comes from the Switzer Candy Company. Frederick Switzer was born in St. Louis in 1865. His father died when Frederick was young, and the boy needed to raise money to help support his family. He would walk the riverfront area of St. Louis, peddling candies from a cart. Eventually he started the Switzer Candy Company, which remains a St. Louis business to this day.
Smoked Brisket Sandwich
Smoked Beef Brisket Sandwich
Smoked beef brisket with Cardinal Nation Chips (homemade kettle-cooked barbequed potato chips) is sold from a cart where the brisket is hand-sliced as you wait. Cardinal Nation refers to the fans of the St. Louis Cardinals throughout portions of the Midwest and South. “Our house-smoked beef brisket is served on a fresh Kaiser roll with red onions, dill pickles, and bourbon BBQ sauce.”
Island Grill
The Island Grill offers three unique specialties. Mahi-Mahi Tacos come with lettuce, pico de gallo and chipotle mayo. Crab Cake Sliders are served with pickles and roasted red pepper remoulade. The Spicy Shrimp Hoagie sandwich comes with lettuce, pico de gallo and lemon-caper aioli.
Spicy Shrimp Hoagie
Dizzy’s Diner
Dizzy’s Diner is named for former St. Louis ballplayer Dizzy Dean. Dizzy Dean was the last National League pitcher to win 30 games in one season, and was elected to the Baseball Hall of Fame in 1953. Available at Dizzy’s Diner are hot dogs, bratwurst, corn dogs, Polish sausage, hamburgers, cheeseburgers, chicken sandwiches and chicken tenders.
Asian Stir Fry
Made-to-order Chinese food is available at the Asian Stir Fry stand. Fans choose either fried rice or noodles and either chicken, beef or shrimp. The selection is freshly cooked on site in a wok with green and red peppers, cabbage, mushrooms, onions, peapods and carrots.
Double Play Tap and Grill
At the Double Play Tap and Grill the nachos are so large that they are called Four Hands Nacho Platter (maybe because you need four hands to carry the plate). The chips are topped with Monterey Jack and shredded queso fresco cheeses, pico de gallo, scallions, jalapeños, sour cream, and a choice of beef, chicken, or pork. The Double Play also serves flatbread pizzas. The barbequed pork pizza has red onions, banana peppers, Gouda and mozzarella cheeses. The carnita chicken pizza has olives, salsa, jalapeños, Jack and cheddar cheeses. The spinach artichoke pizza comes with Alfredo sauce, Parmesan and mozzarella cheeses.
Spinach Artichoke Flatbread Pizza
Free Ice Cream
There may be no such thing as a free lunch but on Sundays at Busch Stadium there is free vanilla ice cream. Prairie Farms offers all fans an ice cream cup as part of their ice cream “Sunday” promotion.
TAMPA BAY RAYS
Tropicana Field is the only domed stadium in the major leagues. It has been the host of the Tampa Bay Rays since the expansion team’s inaugural season in 1998. The stadium is located in St. Petersburg, one of the major cities of the Tampa Bay area.
Featured Hot Dog/Sausage
As is the case at Marlins Park, hot dogs and sausages play a minor role in the food offerings at Tropicana Field. In addition to the traditional hot dog and sausage, fans can get an Italian Sausage with onions and peppers served on pizza bread instead of a traditional bun.
More Hot Dogs and Sausages
Hot dogs are provided by Kayem Foods, makers of the famous Fenway Frank. The Kayem hot dog topped with chili and cheese is known as The Heater. The Slaw Dog is topped with cole slaw, nacho cheese, salsa, and jalapeños, and served on a poppyseed bun.
Italian Sausage on Pizza Bread
Bloomin’ Onion
Outback Steakhouse, the Australian-themed steakhouse restaurant, has a stand at Tropicana Field. It serves a steak sandwich with mushrooms and grilled chicken on the barbie. The grilled chicken is prepared in strips and served with a barbeque sauce. Also available is Outback’s signature item, the Bloomin’ Onion. According to its website, “Our special onion is hand-carved by a dedicated bloomologist, cooked until golden and ready to dip into our spicy signature bloom sauce.” When served, the Bloomin’ Onion looks like a flower with its petals spread.
Bloomin’ Onion
Beignets
Carni Classics sells hot beignets (squares of fried dough sprinkled with powdered sugar.) The most famous beignets are sold at the Café du Monde in New Orleans. The original Café du Monde coffee stand opened in 1862 in the New Orleans French Market.
Everglades BBQ Company
The Everglades BBQ Company offers pulled smoked pork sandwiches and pulled smoked pork nachos. The server at the stand literally pulls the pork by hand to make each sandwich. Provided at the nearby condiment stand are eight varieties of Everglades BBQ sauces.
Cuban Burger
Tropicana Field offers several Cuban-style items. At the Burger Up! stand, the Cuban Burger consists of two all-beef patties topped with ham, Genoa salami, Cuban pork, Swiss cheese, pickle planks, and a secret mustard sauce. The Grand Slam Grill serves Cuban sandwiches (layers of ham, pork, cheese, pickles and mustard on a grilled roll.)
Cuban Burger
Gourmet Grilled Cheese Sandwich
The Gourmet Grilled Cheese Sandwich stand serves roast beef sandwiches. (Just kidding, but I’m not sure what makes these grilled cheese sandwiches “gourmet.”) This stand serves grilled cheese sandwiches either plain, with bacon, or with tomato.
Italian Beef
The Italian Beef Sandwich stand offers “slow oven-roasted, perfectly Italian seasoned beef, sliced thin and served with mild giardiniera.” Giardiniera, which means “under vinegar,” is an Italian-American relish of pickled vegetables in vinegar or oil.
Turkey on Marble Rye
The Carvery serves hand-carved roasted turkey sandwiches on marble rye bread. Contrary to popular belief, the dark swirl in marble rye is not pumpernickel but is made darker with the addition of cocoa powder. (This is a fun fact that you can share at your next office party.)
Boiled Peanuts
In addition to the ordinary roasted peanuts, Tropicana Field offers boiled peanuts, a Southern specialty. After boiling in salt water, the peanuts (still in their shells) take on a strong salty taste and become soft, somewhat resembling a cooked pea or a bean. Since the eighteenth century, boiling peanuts has been a folk cultural practice in the southern United States. Peanuts are sometimes called “goober peas.”
Boiled Peanuts
Black and White Cookie
Black and White Cookie
Ray’s Café offers many dessert items. Among them are raspberry coffee cream cheese cake, cinnamon crumble cake, Rice Krispie treats, and the black and white cookie. A black and white cookie is a soft, sponge-cake-like shortbread which is iced on one half with vanilla and on the other half with chocolate. The cookies are often sold in Jewish delis in New York. The black and white cookie was featured in a well-known 1994 episode of the Seinfeld television show entitled “The Dinner Party.”
TEXAS RANGERS
Globe Life Park in Arlington is located between the cities of Dallas and Fort Worth. The park opened in 1994 and has had multiple names before becoming Globe Life Park just before the 2014 season. The Dallas Cowboys football team also plays in Arlington at nearby AT&T Stadium. Globe Life Park is not roofed and is open to the Texas heat.
Featured Hot Dog/Sausage
The Texas Taco Dog combines the two food classics of hot dogs and tacos. The hot dog is placed inside a taco shell and topped with ground taco meat and traditional taco toppings. The entire taco is then put inside a traditional hot dog bun. It is sold at the Casa de Fuego stand.
More Hot Dogs and Sausages
The Home Plate Butcher Block serves what is called a Sausage Sundae. The sausage is topped with mashed potatoes, macaroni and cheese, barbequed brisket and a red pickled pepper (which looks like the cherry on top of a sundae).
Texas Taco Dog
The Texas Big Dog stand offers three regionally themed foot-long hot dogs. The foot-long Texas Dog comes with chili, cheddar cheese, and grilled onions. The foot-long Big Apple Dog has spicy brown mustard, grilled onions and sauerkraut. The foot-long Chicago Dog comes with pickles, relish, tomato, sport peppers, celery salt and mustard.
The American Dog stand serves a Frito Dog topped with chili, cheese, and Fritos corn chips. The Bacon-Wrapped Dog is wrapped in crispy bacon and smothered in grilled onions. The Bringing the Heat Dog is topped with hot relish, cheddar cheese and jalapeños.
As at the Houston ballpark, all of the beef hot dogs and beef sausages at Globe Life Park are provided by Nolan Ryan All-Natural Beef. In addition to the beef, pork sausages are sold at Globe Life Park. The pork sausages are provided by Earl Campbell Meat Products of Waelder, Texas (south of Austin.) Earl Campbell won the Heisman trophy (college football’s highest honor) in 1977 while playing for the University of Texas Longhorns. He then had a long career with the Houston Oilers and was inducted into the Pro-Football Hall of Fame in 1991.
Cholula Hot Wings
Cholula Hot Sauce is a brand of chili-based sauce manufactured in Chapala in the state of Jalisco, Mexico. The recipe, which blends pequin peppers (almost ten times hotter than jalapeños) and red peppers, is a well-guarded secret that is over 100 years old. At Globe Life Park, the Cholula Hot Wing stand offers wing baskets with the signature Cholula Hot Sauce. Wings are also available in barbeque or garlic flavors.
Cholula Hot Wings
Blue Bell Creameries
Blue Bell Creameries is a Texas-based company that also has a stand at Houston’s Minute Maid Park. In addition to ice cream cups and cones, the Blue Bell Ice Cream stand at Globe Life Park sells ice cream floats. An ice cream float consists of flavored syrup mixed with ice cream in either a soft drink or carbonated water. The ice cream float was invented by Robert McCay Green in Philadelphia in 1874 during a local celebration. According to a 1910 article in Soda Fountain Magazine, Green wanted to create a new treat to attract customers away from another vendor who had a bigger and fancier soda fountain.
Texas Sized 24
It is often said that everything is bigger in Texas. The Texas Sized 24 stand features supersized items that are 24 inches long. Three of them are named for Texas Rangers players.
The Boomstick is a two-foot-long hot dog topped with a layer of chili covered in caramelized onions, a layer of melted jalapeño cheese and jalapeño peppers, served on a potato bun. The hot dog is named for the bat of former Texas Ranger Nelson Cruz. (He moved to the Baltimore Orioles in 2014.) Cruz is known for his long home runs. In a commercial that he did in 2010 for the MLB 2K10 baseball video game, he called his bat “The Boomstick.”
Ka-Boom Kabob
The Choomongous is a two-foot-long Korean barbeque sandwich with chopped beef, spicy slaw, and sriracha mayo on a sweet bun. Sriracha is a chili pepper hot sauce originating in Thailand. The Choomongous is named for Shin-Soo Choo, a Ranger outfielder and native of South Korea. Dallas Morning News sports reporter Marcus Murphree decided to eat this sandwich all by himself. He managed to down it in twelve minutes. Murphree said “In baseball speak, this felt like stepping up to the plate and knowing that nothing a pitcher could throw would go by me.”
The Ka-Boom Kabob is a two-foot-long shish kabob served on a bed of rice. Pieces of grilled chicken are skewered with cherry tomatoes, peppers, onions and pineapple chunks.
Although not 24 inches long, the Beltre Buster Burger is a one-pound hamburger with bacon, Jack cheese, and grilled onions. The burger is named for third baseman Adrian Beltre who is known for his signature swing of dropping to one knee while belting out a home run.
For fans who want to try a Boomstick or a Choomongous but don’t have companions with them to share the supersize serving, mini versions are available at the Right Field Grill.
Totally Rossome Nachos
Totally Rossome Nachos are available at several stands throughout Globe Life Park. The nachos come in a pink souvenir batting helmet and are topped with a choice of brisket or chicken, lettuce, pico de gallo, queso blanco, and sour cream. They are named for pitcher Robbie Ross. Although Ross was sent down to the minors in June of the 2014 season, his nachos remain for sale in the majors.
Smokehouse Mac
Smokehouse 557 serves smoked brisket and turkey sandwiches along with macaroni and cheese. However, the macaroni and cheese is not just a side dish. Both the brisket or the turkey are served on top of the mac and cheese and known as a “Smokehouse Mac.”
Bacon on a Stick
The Home Plate Butcher Block offers Baked Bacon on a Stick, in three variations. Fans can get the bacon covered with a maple glaze, with Old Bay Seasoning or with whipped cream and chocolate syrup.
Smokehouse Mac
Frito Pie
Barbeque is provided at the Sweet Baby Ray’s stand. Sweet Baby Ray’s is a Chicago-based company that sells its barbeque sauces and meats in grocery stores. Barbequed chicken and beef sandwiches, smoked turkey legs, and Frito chili pie (Fritos covered with chili, cheese, and onion) are on the menu there. According to a Texas Monthly Magazine article by Michael Hilton, “Frito pies are still a mainstay at football games.” Apparently they go well at baseball games as well.
The Chipper
The Chipper stand makes fresh kettle chips. There are four ways to order your Chippers, all of them available with a choice of either chicken or brisket. The Bases Loaded Chipper is topped with nacho cheese, bacon, sour cream and chives. The Southwest Chipper is topped with queso blanco, pico de gallo, and sour cream. The Ballpark BBQ Chipper is topped with barbeque sauce, onions and jalapeños. And lastly, with a Canadian touch, the Texas Poutine Nachos are topped with creamy gravy, bacon, jalapeños, chives and shredded cheddar cheese.
Big Game Pretzel
Big Game Pretzel
Not only do the sandwiches and the kabobs come big in Texas, but fans here can also get a giant pretzel. The Big Game Pretzel is one foot in diameter and comes with nacho cheese, honey mustard, and marinara dipping sauces…a fitting finale to a supersized eating experience.
TORONTO BLUE JAYS
Rogers Centre, originally known as SkyDome, opened in 1989 in downtown Toronto next to the CN Tower. The stadium was the first to have a fully retractable motorized roof. A 348-room hotel is attached. Rogers Centre is also home to the Toronto Argonauts of the Canadian Football League.
Featured Hot Dog/Sausage
The Garrison Creek Foot-Long Hot Dog, an all-beef wiener accompanied by grilled onions and peppers, is officially described by the Blue Jays as their signature hot dog. The hot dog is named for Garrison Creek, a stream that once flowed into Toronto Harbor. It has been largely covered over and filled in, though geographical traces still remain.
Garrison Creek Foot-Long Hot Dog
More Hot Dogs and Sausages
At the Garrison Creek Flat Grill fans can also get a Farmers Sausage. Farmers Sausage is a form of smoked raw pork sausage of Mennonite origin. It is very popular with the Mennonite settlers of southern Ontario and differs from the sausages available in Mennonite communities in the United States.
Shopsy’s Deli
Shopsy’s Deli, known as the “King of Sandwiches” since the early 1920s, was opened in Toronto in 1921 by Harry Shopsowitz and his wife Jenny. The couple ran their restaurant until Harry died in 1945, and the business was passed on to their sons, Sam and Izzy. Sam and Izzy decided to expand the business in 1947 by opening a meat processing plant. The restaurant, although no longer owned by the Shopsowitz family, remains a Toronto institution. The meat processing business was sold in 1992 to Maple Leaf Foods.
Montreal Smoked Meat Sandwich
Shopsy’s at Rogers Centre offers a Montreal Smoked Meat Sandwich. Montreal-style smoked meat is made by salting and curing beef brisket with spices for at least a week, and then smoking and steaming the meat. Warm Montreal-smoked meat is always sliced by hand since a meat slicer would cause the tender meat to disintegrate. The sandwich is traditionally served on rye bread with mustard. Also available are pastrami sandwiches, corned beef sandwiches, and the Bill Cosby Triple Decker Sandwich (pastrami and corned beef with three pieces of bread.) Actor and comedian Bill Cosby is a big fan of Shopsy’s.
Sunflower Seeds
Sunflower seeds have become popular with baseball players. Traditionally ballplayers chewed tobacco but as the harmful effects of tobacco became known, many players switched to sunflower seeds. By the 1980s, the seeds had surpassed tobacco in on-field popularity. This popularity spread to the fans in the stands, and soon sunflower seeds were competing with peanuts. Rogers Centre sells Spitz Sunflower Seeds. In 1982 Alberta family farmers Tom and Emmy Droog decided to switch from planting wheat to sunflowers, and established Spitz Sunflower Seeds. Their small business kept growing until it was sold to Frito-Lay in 2008.
#12 Roberto Alomar
The 12 Kitchen is named in honor of former Toronto Blue Jay infielder Roberto Alomar, who wore jersey number 12. Alomar was the first Toronto Blue Jay to be inducted into the National Baseball Hall of Fame. The 12 Kitchen serves the Hall of Famer Chicken Wrap. A spinach tortilla is filled with spiced fire-roasted chicken, crushed cilantro mojo, romaine hearts, chorizo crackle, Asiago cheese and Caesar dressing.
Hall of Famer Chicken Wrap
#12 Nachos
The 12 Nachos use homestyle kettle potato chips instead of tortilla chips. The chips are topped with warm cheddar cheese, sliced Canadian peameal bacon, charred corn and pineapple salsa, diced red tomatoes, shredded pickles, and jalapeño-infused island sauce. Peameal bacon is a type of bacon that originated in Toronto. The name reflects the historic practice of rolling the cured meat in dried and ground yellow peas. Today it is rolled in yellow cornmeal instead.
Poutine
Poutine
Poutine is a Canadian dish originating in Quebec. French fries are topped with a light brown gravy and cheese curds. The dish is now popular in all parts of Canada as well as some places in the northern United States. The Hog Town Grill serves traditional poutine at Rogers Centre.
The Quaker Steak and Lube stand serves a version of poutine known as Wedge & Wing Poutine. Potato wedges and chunks of breaded chicken are combined with blue cheese dressing and nacho cheese. The stand also serves fish tacos made with tilapia.
Adobo Chicken Trio
Muddy York
An early and disparaging nickname for Toronto was “Muddy York.” At the time there were no storm drains and the streets were unpaved. During rainfall, water would accumulate on the dirt roads, transforming them into impassable muddy avenues. Two of the stands at Rogers Centre use the Muddy York name.
The Muddy York Market serves various specialities including a Little Italy Meatball Hoagie, a Big Smoke Jerk Chicken Sandwich, a Distillery Beef Brisket Sandwich, and the Adobo Chicken Trio. The Adobo Chicken Trio consists of three barbequed chicken drumsticks cooked “adobo” style. The raw chicken is immersed in a sauce composed of paprika, oregano, salt, garlic, and vinegar to preserve and enhance its flavor. This practice is native to Iberia, Spain.
The Muddy York Cantina offers a five-step “Create Your Own Cantina Box.” Step l: Choose your base (tortilla chips, burrito, or romaine lettuce). Step 2: Choose your starter (warm cheddar cheese, three-bean chili, black beans, fresh pico de gallo or cantina rice). Step 3: Choose your topping (jerk chicken, smoked chopped pork, barbeque beef brisket; vegetarians can choose a second item from Step 2). Step 4: Choose up to four favorite market ingredients (salsa verde, red cabbage slaw, roasted corn salsa, pico de gallo, green onions, sun-dried tomatoes, jalapeños, black olives, guacamole, chili lime sour cream, or sautéed peppers and onions). Step 5: “Enjoy your cantina box!”
Craft Chop House Burger
Fresh Burger offers a Craft Chop House Burger and invites fans to “make it your own.” There is a wide variety of sauces and cold and hot toppings, advertised as over 6,500 possible combinations. (In the interest of space, I won’t list all 6,500 choices.)
Cavendish Farms
Potatoes can be found at various stands at Rogers Centre. They come in the form of poutine, regular French fries, curly fries, and fries with various toppings. Many ballparks have an official hot dog, but Rogers Centre also has an official potato provider. Cavendish Farms, located on Prince Edward Island, provides all the potatoes throughout the stadium. In 2001 Cavendish Farms became the first frozen potato processing company to switch all their products to non-hydrogenated oil. It sells over one billion pounds of potatoes annually.
WASHINGTON NATIONALS
The Nationals Park is located along the Anacostia River in the Navy Yard neighborhood of Washington, DC. The stadium opened in 2008 as the new home of the Nationals, who had played at RFK Stadium after their move from Montreal to Washington in 2005.
Featured Hot Dog/Sausage
Ben’s Chili Bowl serves its well-known Original Half-Smoke. Ben and Virginia Ali opened Ben’s Chili Bowl in 1958. They renovated the Minnehaha Theater, which first housed silent movies in 1910. Ten days before his inauguration in 2009, President-Elect Barack Obama visited and ate lunch there.
Half-Smoke
A half-smoke is similar to a hot dog but spicier and made with more coarsely-ground meat. The hot dog, made from half pork and half beef combined with herbs, onions, and chili sauce, is smoked slowly. At Nationals Park, the Original Half-Smoke is served “all the way” accompanied by chili, cheese, onions, and mustard.
More Hot Dogs and Sausages
The Grand Slam Grill serves the DMV. I spent the evening wondering why a hot dog would be named for the Department of Motor Vehicles. My research discovered that DMV refers to the DC metro area and stands for DC, Maryland, and Virginia. The DMV is a foot-long half-smoke (from DC) smothered with Maryland crab dip and topped with Virginia ham.
Senator Sausages serves bratwurst, Italian sausages, and andouille sausages. Andouille sausage, of French origin, is most often associated with Acadians and Cajun cooking of Louisiana.
All-beef Hebrew National hot dogs are available at Nationals Park. Hebrew National Kosher Sausage Factory was founded by a Russian immigrant, Theodore Krainin, in 1905 in the Lower East Side of Manhattan. In the 1940s, the company created products especially for grocery stores and the suburban market. Its products appeal to all consumers and not just to those who keep kosher.
Shawafel
The Shawafel stand at Nationals Park serves many of the same Lebanese specialties that local customers will find at their H Street restaurant location of the same name. Its name comes from the combination of two popular Middle Eastern foods, shawarma and falafel. Shawarma is meat slow-cooked on a vertical spit for many hours. Shavings are cut off the block of meat. Falafel is a deep-fried ball or patty made from ground spiced chickpeas and/or fava beans. Shawarma and falafel are commonly served in a pita bread pocket or wrap.
Cauliflower Sandwich Wrap
At Nationals Park, the Shawafel stand serves chicken or lamb and beef shawarma. The falafel is made from chick peas. Both are served in pita pockets with tahini sauce made from ground sesame seeds. Also served is a Cauliflower Sandwich Wrap made with fried cauliflower, parsley, lettuce, tomatoes, pickles and tahini sauce. Side dishes include Baba Ghanoush, Lebanese Fries and Hummus. Baba Ghanoush is a puree made from eggplant and sesame paste. The eggplant is peeled and mashed, then mixed with sesame paste, garlic, salt, pepper, cumin, lemon juice and olive oil. Lebanese Fries are French fries topped with zaatar (also referred to as za’atar), a spice mixture made with dried herbs including marjoram, thyme, oregano, sumac, cumin, and sesame seeds. Hummus is a puree of chickpeas, garlic, tahini and lemon juice.
Italian Hero
G by Mike Isabella
Chef Mike Isabella opened G Sandwich Shop next to his Northern Greek restaurant, Kapnos, along the 14th Street corridor in Washington, DC. At Nationals Park, G by Mike Isabella serves four of his gourmet sandwiches. The Chicken Parm sandwich has chicken thigh ragu, provolone, and Thai basil. The Italian Hero has capicola (an Italian cold cut made from the pork shoulder or neck), soppressata (Italian dry salami), prosciutto, mozzarella, oil and vinegar, mayonnaise, and pickled vegetables. The Roasted Cauliflower sandwich has almond romesco (a Catalan condiment made with almonds, roasted tomatoes, olive oil and vinegar), pickled vegetables, and paprika. The Drewno sandwich is named for Scott Drewno, the executive chef at The Source restaurant in the Newseum in Washington, DC. The Drewno has kielbasa sausage, roast beef, and sauerkraut.
Dolci Gelati
Dolci Gelati sells gelato throughout the Washington, DC area. The company was founded in 2006 by pastry chef Gianluigi Dellaccio, who uses ingredients from local dairy farms and fruit orchards. Its chocolate is imported from a small sustainable farm in Ecuador. Flavors available at Nationals Park are chocolate, cookies and cream, stracciatella (chocolate chip), baccio (chocolate hazelnut), peanut butter, strawberry, lemon, and mango.
Teddy Roosevelt Cupcake
Fluffy Thoughts Bakery was founded by Lara Stuckey, combining her two passions of art and baking. Her stand at Nationals Park sells brownies, cookies, and cupcakes. What would be more appropriate in Washington, DC, than a presidential cupcake? The Teddy Roosevelt Cupcake has a salted caramel ganache filling and vanilla frosting. Teddy’s glasses are made of hard icing, and he has a chocolate moustache.
Teddy Roosevelt Cupcake
Union Square Hospitality Group
New York restaurateur Danny Meyer’s Union Square Hospitality Group has four stands at New York’s Citi Field. These four – Shake Shack, Box Frites, Blue Smoke Barbeque, and El Verano Taqueria – are also found at Nationals Park.
Chesapeake Crab Cake Specialties
Chesapeake Crab Cake Specialties serves jumbo lump crab cakes with mixed greens, tomatoes, and Old Bay remoulade. (Remoulade, invented in France, is similar to tartar sauce. It is mayonnaise based and can be flavored with curry, horseradish, paprika, anchovies, or other items.) Fans can also get a crab grilled cheese sandwich on sourdough bread with Brie. Homemade tortilla chips are served with crab queso and corn cilantro salsa.
Change Up Chicken
The Change Up Chicken stand features chicken in a waffle cone. Pieces of chicken are breaded with waffle crumbs, then served in an ice cream cone.
Field of Greens
The Field of Greens serves a wide variety of vegetarian options, greater than what is found in most ballparks. Its menu includes vegan crab cakes, grilled portobello sandwich, grilled veggie wrap, veggie cheese steak, veggie hot dog, house made veggie burger, and a variety of meatless salads.
Taste of the Majors
The Taste of the Majors offers foods from other parts of the country including the Miami Cuban Sandwich, the Arizona Quesadilla, and the New York Pastrami Sandwich.
Jammin’ Island BBQ
The Jammin’ Island BBQ serves Caribbean food. Jerk chicken and jerk ribs are accompanied by rice and beans, plantains, and yucca fries. Yucca is a desert plant grown in arid sections of North America.
Intentional Wok
The Intentional Wok features Chicken Pad Thai Noodles and Beef Drunken Noodles. Asian vegetables mixed with soy sauce, eggs, and diced peanuts complement the Chicken Pad Thai Noodles. The Beef Drunken Noodles has beef mixed with Asian vegetables and a Thai basil sauce.
Crab Pretzel
Pretzels come in many varieties at Nationals Park. The Pretzel Loaf is a pretzel stuffed with steak and cheese. The Half-Smoke Pretzel Log is -- you guessed it -- a half-smoke hot dog inside a pretzel. Finally, the Crab Pretzel is an oblong twisted pretzel topped with large amounts of a crab meat mixture.
Crab Pretzel
If you are reading this now you have just survived a culinary tour of the 30 MLB ballparks without gaining a pound—something my wife and I can not truthfully say.
The new food era has brought such a wonderful gustatory experience at the ballparks with chef-prepared masterpieces, vegetarian and kosher delights, as well as amped up riffs on the hot dog and sausage.
While it is unlikely that food choices and prices will revert to the simpler fare of the early days of baseball during the Harry M. Stevens era, we can still celebrate and remember the beginning of the food-baseball connection—a firmly entrenched connection that is here to stay.
SOURCES
Notes:
All websites were accessed between January and October of 2014.
Background information was also obtained from restaurants and food purveyors.
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“Welcome to Veggie Happy.” Veggie Happy. www.veggiehappy.com
“What Are Rocky Mountain Oysters?” WiseGEEK: Clear Answers for Common Questions. www.wisegeek.org
“What Exactly is Pastrami?” How Stuff Works. www.howstuffworks.com
“What is Shawarma?” WiseGEEK: Clear Answers for Common Questions. www.wisegeek.org
“What is Tempeh?” Tempeh. www.tempeh.info
“What is the Difference Between a Hot Dog, Wiener, Frank, and Sausage?” WiseGEEK: Clear Answers for Common Questions. www.wisegeek.org
“What is the Difference Between Hot Dog and Sausage?” Food Journey Singapore. www.foodjourneysg.blogspot.com
“Willie Mays.” National Baseball Hall of Fame. www.baseballhall.org
“Willie Stargell.” National Baseball Hall of Fame. www.baseballhall.org
“Wise Foods Named Official Potato Chip and Cheez Doodle Sponsor for the New York Mets.” MLB. www.mlb.com
Woo, Elaine. “Thomas G. Arthur, 84; Made Dodger Dogs a Staple of L.A. Stadium Experience.” Los Angeles Times, 27 Jun. 2006.
Wood, Bob. Dodger Dogs to Fenway Franks: And All the Wieners In Between. New York: McGraw-Hill, 1988.
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Zimmer, Erin. “What’s a Half-Smoke?” Serious Eats. www.seriouseats.com | |
The Advanced Placement Music Theory students will develop the ability to recognize, understand, and describe the basic materials and processes of music that are heard or presented in a score. The student will develop aural, sight-singing, written, compositional, and analytical exercises. The AP Music Theory student will solve compositional problems and become proficient in part-writing. Students will receive ear training and skills for aural identification and dictation notation. The curriculum for this course has been designed in cooperation with the AP Testing Service and will prepare students to take the AP Music Theory exam in May.
BEGINNING PIANO
Instructors: Jeri Hockensmith, Carrie Mascaro
Beginning Piano is a one-semester class, offered in the fall, which focuses upon rudimentary piano playing skills. The class is intended for students who always wanted to learn how to play the piano, but never had the time or resources. Students will learn the basics including: hand position, posture, note reading, and how to use common music software applications to visualize performance accuracy. Popular folk songs and traditional melodies are utilized to teach basic concepts. No previous musical experience is necessary.
MUSIC TECH.
Instructor: Gregg Winters
Music technology is a one-semester class, offered in the spring, for students in grades 9-12. Over the course of the semester, students utilize music software and digital instruments to create, compose, and respond to various musical elements and fundamentals. Songs and musical arrangements are studied to identify how each piece utilizes the basic elements of music (tempo, rhythm, pitch, melody, harmony, form, and style). Students use music applications to create, edit, manipulate and arrange musical compositions in a style similar to the ones studied in class. Students demonstrate their understanding of these musical components by generating original compositions and/or arrangements in a variety of musical styles (jazz, rock, Latin, reggae, techno, pop, etc.).
POPULAR MUSIC AND DIVERSITY IN AMERICAN SOCIETY (OPTION FOR UCONN ECE - UCONN MUSI1003)
Instructor: Philip Giampietro
Note: Popular Music and Diversity in American Society may count toward fulfilling the Visual and Performing Arts, Open Humanities, OR Open STEAM distribution requirement.
This course examines American popular music within its historical and social context, primarily throughout the 20th century. It will encourage students to think critically and creatively about popular music in relation to topics of diversity. We will study significant styles of American popular music, with a focus on select songs that exemplify their respective genres, and explore several recurring themes throughout the course. The course is intended to enhance students’ enjoyment and understanding of the music they already know, as well as to introduce less familiar styles and genres. Students will develop critical listening skills and become more informed consumers of popular music. | https://www.staplesmusic.org/classroom-music |
Winters can be very variable. Mild some years, bitterly cold in others … but they often switch from cold to warm to cold again, sometimes more than once.
You’d think the “warm” part—a midwinter thaw—would be good news and I suppose it could be if you hate cold weather or feel like barbecuing on your balcony. But a thaw can often cause as much damage to your garden plants as a cold snap, sometimes even more.
The Best-Case Situation
In a climate where winters are going to be cold anyway, the ideal situation for plants is when temperatures drop gradually in the fall and then remain relatively cold throughout the winter, then warm up slowly in spring. This allows the plants to harden off (acclimatize to the cold) gradually and to remain dormant throughout the winter, remaining more or less oblivious to cold until spring comes around. And cold hardy plants—plants like trees, shrubs and perennials, the ones you probably grow in your garden if you live in zone 7 and less—do best when they have a long, fairly cold winter. That’s simply their thing!
When Things Go Wrong
But the situation is rarely ideal. After a long, mild autumn, winter can arrive suddenly, before the plants have time to harden off. Or temperatures can go up and down like a pendulum all winter: warm, cold, warm, cold, etc.
And that’s what a midwinter thaw is, the mythical “January thaw” (although it sometimes occurs in February!). An exceptionally warm period during an otherwise cold winter. By definition, temperatures have to have been below freezing for awhile, then rise above it, for it to be considered a thaw.
Seesawing temperatures are never good for plants, even in summer, but the effect is much worse in winter, especially if the thaw lasts long enough for plants to start losing the cold acclimation they so carefully acquired during the fall. The longer the thaw lasts and the warmer it becomes, the greater the potential harm to the plants.
What Happens to Gardens During a Thaw?
Here are a few of the things that can happen in your garden during a thaw.
- The snow becomes wet and heavy. When snow falls, it’s often light and fluffy or powdery, full of insulating air spaces, and it thus offers plants excellent cold protection, because the stagnant air trapped among the snow particles is a poor conductor of cold. During a thaw, however, the air spaces fill with water, either from snow melt or from the rain that often accompanies a thaw. This greatly diminishes the insulating quality of snow and its capacity to protect plants.
- The snow melts. Hopefully it won’t all melt, because snow really is beneficial for hardy plants, but it likely will melt here and there, leaving the plants in those patches even more exposed to future harsh conditions.
- The snow turns icy. When a snow charged with water (and lacking air spaces) is exposed to cold again, it turns icy, further reducing its insulating capacity. It looks like snow, but it’s essentially closer to ice: you can often walk on it without it collapsing. This causes a secondary problem. Even when plants are fully dormant, there is still a bit of respiration and ventilation taking place. If the snow turns to ice, though, any respiration and gas transfer pretty much end and this can weaken or kill the plants … something you’ll only discover come spring.
- Melting snow can lead to flooding, turning your garden into a pond. Very few land plants will tolerate spending the next few months under water and root rot can easily set in.
- Meltwater can cover the soil and then freeze solid with the return of cold weather. Solid ice contains even less air than icy snow and won’t let the plants breathe at all. Also, it’s a fairly poor insulator and any extreme cold to come will hit the plants full force. This is the worst possible situation and it often causes plant death through the double whammy of asphyxiation and cold damage.
- Frost cracks can appear on trees. This often occurs bark when a thaw is followed by extreme cold, especially when temperatures swing regularly between freezing and thawing. It mostly harms relatively young trees or trees with thin bark. Sometimes the split can be 3 feet (90 cm) or more in length and it never heals very readily.
- Branches caught in the snow are released. This is the only real benefit I can see of a winter thaw. If any branches were bent over or trapped in snow or ice during an earlier storm, the snow/ice may melt back enough so that they can start to straighten up. The branches too, once frozen and fragile, regain their flexibility. You can help by staking the branch so it isn’t caught a second time during an upcoming ice or snow storm.
- Freezing rain (which sometimes accompanies thawing) can cover branches with ice and cause them to bend or break. Don’t try to straighten ice-covered branches: you’ll only damage them. As in the previous point, if the thaw continues and the ice melts, allowing the branches to regain their flexibility, you can help straighten them out. If not, just let the branches bend all winter if necessary. Trying to break off the ice that covers branches will do much more serious damage than leaving the branch bent over.
- The plant’s cold acclimation decreases, especially if the thaw persists. Even a hardy plant, capable of surviving -40˚ F (-40˚ C) temperatures when properly hardened off, can be seriously damaged at 0˚ F (-17˚ C) if it starts to lose its cold acclimation. This will mostly be noticeable after a fairly long thaw, a week or so in duration.
- Plants actually wake up and start growing, even as early as January. Hey, why not? If the thaw lasts several weeks, many plants will figure it’s spring and start to sprout. Not all plants will be duped, though. Many hardy plants have a built-in calendar that keeps them dormant until they’ve undergone at least three months of cold, but others have no such restriction. Some early bloomers, like hellebores and bulbs such as snowdrops and narcissus, won’t be harmed if the cold returns even after they’ve started to grow, but other plants (most other perennials and shrubs) can be seriously damaged or even killed.
What Can You Do to Help?
There is relatively little you can do to help your plants during a thaw, but you can protect any especially fragile plants when the thaw has exposed them by covering them with snow or mulch, even a rose cone or plastic-lined geotextile. Of course, this is rather uncomfortable work to do in mushy, melting snow and nippy weather. You could have saved yourself a lot of work by mulching and protecting such plants in the fall.
If your Christmas tree is still around, chop it up and used its branches to cover fragile plants. If not, perhaps you have conifers from which you could harvest a few branches.
Try to walk as little as possible on thawing soils, because your weight will compact them terribly and can also damage plants and their roots, so stay on paths if you can. If you have to go work in the garden in the winter (ideally, you wouldn’t have to), wear snowshoes: they’ll spread out your weight considerably and help prevent damage.
Let Nature Cull the Weak Ones
If there are plants that are seriously suffering from a winter thaw, I’d personally tend to let just let them go rather than rushing out to protect them. I’m not one for preserving plants artificially. If any plants don’t appreciate my conditions, I’ll just replace them come spring with plants that do. So, I see a winter thaw as a hardiness test and highly appreciate the plants come through it in perfect shape. I guess that’s the laidback gardener in me coming out!
However, whatever your gardening persuasion, if you’re being hit with a thaw, I wish you the best of luck! | https://laidbackgardener.blog/2018/01/12/beware-mid-winter-thaws-can-damage-your-plants/ |
Banks to jerk up lending rate for corporates in Q1’19
By Babajide Komolafe
Banks will increase lending rates for corporate organisations in the first quarter (Q1’19) of next year even as the industry recorded increased loan default by households in the fourth quarter of this year (Q4’18).
The Central Bank of Nigeria (CBN) disclosed this yesterday in its Credit Conditions survey report for Q4’18.
Among other things the report showed increased supply of loans by banks to households and corporates in Q4’18, driven by improved economic outlook and quest for increased market share.
The report however showed that while banks approved more loans to households during the quarter, more households defaulted on their loans, leading to a rise in default rates for secured and unsecured loans to households during the quarter.
The report also revealed that while banks kept lending rates constant for loans to households and corporates in Q4’18, they will however increase lending rates for loans to corporates in the next quarter, Q1’19.
Secured Credit to Households
The report stated: “In the current quarter relative to the previous quarter, lenders reported an increase in the availability of secured credit to households. Improving economic outlook and higher appetite for risk were major factors behind the increase. Availability of secured credit was expected to increase in the next quarter as well, with improving economic outlook and market share objectives as the likely contributory factors
“Lenders maintained the same credit scoring criteria in Q4 2018, but the proportion of loan applications approved in the quarter decreased. Lenders expect to loosen the credit scoring criteria in the next quarter, yet still expect an increase in the proportion of approved households’ loan applications in Q1 2019.
“Households demand for lending for house purchase decreased in Q4 2018 but was expected to increase in the next quarter. For the current quarter, households demand for prime lending and buy to let lending decreased, while demand for other lending increased. Demand for prime lending and demand for other lending were expected to increase, while demand for buy to let was expected to increase, in the next quarter.
“Households demand for consumer loans rose in the current quarter and is expected to rise in the next quarter. Demand for mortgage/remortgaging from households fell in Q4 2018 but is expected to rise in Q1 2019.
“Secured loan performance, as measured by default rates, worsened in Q4’18 but is expected to improve in Q1’19. However, bank lenders reported lower losses given default by households in both the current next quarters”.
Unsecured credit to Households
The report further revealed that, “Availability of unsecured credit provided to households rose in the current quarter and is expected to rise in the next quarter. Lenders reported market share objectives and higher appetite for risk as the major factors that contributed to the increase in Q4 2018.
“Despite lenders’ resolve to leave the credit scoring criteria for total unsecured loan applications unchanged in the review quarter, the proportion of approved total loan applications for households increased. Lenders expect to tighten the credit scoring criteria in the next quarter, but anticipate an increase in the total loans applications to be approved in Q1 2019.
“Lenders reported that spreads on credit card lending widened in Q4 2018 but were expected to remain unchanged in the next quarter. Spreads on unsecured approved overdrafts/personal loans applications widened in the current quarter and was expected to widen in the next quarter. Spreads on overall unsecured lending widened in the current quarter, and was expected to widen in the next quarter
“Lenders experienced higher default rates on credit card and on overdrafts/personal lending to households in the current quarter. They however, expect improvement in default rates in the next quarter for all loan types. Losses given default on total unsecured loans to households improved in Q4’18 and were expected to improve in the next quarter.”
Credit to Corporates
According to the report, “The overall availability of credit to the corporate sector increased in Q4 2018 and was expected to increase in Q1 2019. This was driven by favourable economic conditions, changing sector-specific risks, improved liquidity conditions, market share objectives and changing appetite for risk. “Lenders reported that the prevailing commercial property prices negatively influenced credit availability of the commercial real estate sector in the current quarter. However, lenders expect the prevailing commercial property prices to positively influence secured lending to PNFCs in the current quarter.
“Spreads between bank rates and Monetary Policy Rates (MPR) on approved new loan applications widened for all business sizes except for loans to other non financial corporations (OFCs) in Q4 2018, but were expected to widen for all business sizes in Q1 2019.
“The proportion of loan applications approved for all business sizes increased in the current quarter, and are expected to further increase in Q1 2019.
“Lenders required stronger loan covenants from all firm sized businesses in the current quarter. However, they reported that they would require stronger loan covenants for all firm sized businesses except for small business, which they plan to leave unchanged, in the next quarter.
“For the current quarter, fees/commissions on approved new loan applications fell for medium public non-financial corporations (PNFCs) and OFCs, and rose for small and large businesses; while for Q1 2019 lenders expect fees/commissions on approved new loan applications to fall for medium PNFCs, remain unchanged for small and large PNFCs, and rise for OFCs.
“Small and medium PMFCs benefitted from an increase in maximum credit lines on approved new loan applications in Q4 2018, while Large PNFCs and OFCs did not. Similarly, Small PNFCs and OFCs are expected to benefit from an increase in maximum credit lines on approved new loan applications in Q1 2019, while medium and large PNFCs are not.
“More collateral requirements were demanded from all firm sizes on approved new loan application in Q4 2018. Similarly, lenders will demand for more collateral from all firm sizes in the next quarter.” | |
This application is a continuation of U.S. application Ser. No. 16/589,693, filed Oct. 1, 2019 and claims the benefit of U.S. Provisional Appln. No. 62/739,893, filed Oct. 2, 2018, U.S. Provisional Appln. No. 62/753,224, filed Oct. 31, 2018, and U.S. Provisional Appln. No. 62/826,074, filed Mar. 29, 2019, the contents of each of which are incorporated herein by reference.
FIELD OF THE INVENTION
This invention relates to breadboard prototyping. More particularly, the invention relates to an automated breadboard wiring assembly.
BACKGROUND OF THE INVENTION
FIGS. 1 and 2
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Referring to , a breadboard is a solderless device for temporary prototyping with electronics and testing circuit designs. The breadboard includes a physical board with a plurality of holes defined therein. The breadboard has strips of metal (not shown) underneath the board that connect the holes on the top of the board within a given node ,
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Most electronic components , , (e.g. resistors, capacitors, integrated circuits, etc.) in electronic circuits can be interconnected by inserting their leads or terminals into the holes and then making connections through wires where appropriate. As illustrated in , in some circuits the number of wires necessary to complete the circuit may be significant. Placing of the wires is a time-consuming and error-prone process. Even one small mistake in placement of one of the wire ends can lead to hours of debugging to fix the circuit.
SUMMARY OF THE INVENTION
In at least one embodiment, the present invention provides an automated breadboard wiring assembly. The assembly includes a breadboard with holes therein defining at least two nodes. The assembly includes a breadboard with holes therein defining at least two nodes and at least a primary wiring board. The primary wiring board has a wiring matrix composed of a plurality of interconnected wiring segments, each wiring segment having a switch therealong. A plurality of contacts are interconnected with the wiring matrix with a switch positioned between each contact and the wiring matrix. Each contact is configured to engage a respective one of the breadboard nodes. An input device is configured to indicate desired wires between nodes and the locations of the desired wires define wiring information. A microprocessor configured to receive wiring information from the input device and open selective ones of the switches such that an electrical path along selective ones of the contacts and the wire segments is defined to correspond to each desired wire set forth in the wiring information.
In at least one embodiment, the present invention provides an automated breadboard wiring assembly. The assembly includes a breadboard with holes therein defining at least two nodes and at least a primary wiring board. The primary wiring board has a wiring matrix composed of a plurality of interconnected wiring segments, each wiring segment having a switch therealong. A plurality of contacts are interconnected with the wiring matrix with a switch positioned between each contact and the wiring matrix. Each contact is configured to engage a respective one of the breadboard nodes. An input device has a screen configured to display a virtual breadboard corresponding to the breadboard. The input device is configured to position virtual circuit components and virtual wires on the virtual breadboard with the locations of the virtual wires defining wiring information. A microprocessor is configured to receive the wiring information from the input device and open selective ones of the switches such that an electrical path along selective ones of the contacts and the wire segments is defined to correspond to each virtual wire set forth in the wiring information.
BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are incorporated herein and constitute part of this specification, illustrate the presently preferred embodiments of the invention, and, together with the general description given above and the detailed description given below, serve to explain the features of the invention. In the drawings:
FIG. 1
is a top plan view of an illustrative prior art breadboard.
FIG. 2
FIG. 1
is a perspective view of the breadboard of with an illustrative circuit wired thereon.
FIG. 3
is a schematic diagram of an automated breadboard wiring assembly in accordance with an embodiment of the disclosure.
FIG. 4
FIG. 3
is a perspective view of a portion of a primary wiring board of the automated breadboard wiring assembly of .
FIG. 5
FIG. 3
is a perspective view of a portion of a secondary wiring board of the automated breadboard wiring assembly of .
FIG. 6
is a perspective view illustrating the secondary wiring board interconnected with the primary wiring board.
FIG. 7
is a schematic diagram of an illustrative electrically controlled switch in accordance with an embodiment of the disclosure.
FIGS. 8A and 8B
are each a schematic diagram illustrating interconnection between nodes of the primary wiring board with nodes of a secondary wiring board along a single row of the breadboard.
FIG. 9
is a schematic diagram illustrating interconnection between nodes of the secondary wiring board.
FIG. 10
is a schematic diagram illustrating interconnection between nodes of the primary wiring board with nodes of a secondary wiring board and also amongst nodes of the secondary wiring board.
FIGS. 11A and 11B
are each a schematic diagram illustrating interconnection between nodes of the primary wiring board with nodes of first and second secondary wiring boards along a single row of the breadboard.
FIG. 12
is a schematic diagram illustrating interconnection between nodes of the second secondary wiring board.
FIG. 13
is a schematic diagram illustrating interconnection between nodes of the primary wiring board with nodes of the first and second secondary wiring board and also amongst nodes of the second secondary wiring board.
FIGS. 14-19
are schematic diagrams illustrating an LED indicator assembly for visually indicating interconnected nodes.
FIG. 20
is a schematic diagram illustrating another manner of visually indicting interconnected nodes utilizing an LED indicator assembly in accordance with an embodiment of the invention.
FIG. 21
is a schematic diagram illustrating yet another manner of visually indicting interconnected nodes utilizing an LED indicator assembly in accordance with another embodiment of the invention.
FIGS. 22 and 23
are screen shots illustrating virtual wiring of a circuit on a breadboard in accordance with an embodiment of the disclosure.
FIGS. 24 and 25
are screen shots illustrating virtual wiring of a circuit on a breadboard in accordance with another embodiment of the disclosure.
FIG. 26
is a plan view of a standalone breadboard assembly in accordance with an embodiment of the invention.
FIG. 27
FIG. 26
is a schematic view of an illustrative stylus and contact pin assembly of the breadboard assembly of .
FIGS. 28-35
FIG. 26
are plan views of the breadboard assembly of illustrating various wiring sequences.
DETAILED DESCRIPTION OF THE INVENTION
In the drawings, like numerals indicate like elements throughout. Certain terminology is used herein for convenience only and is not to be taken as a limitation on the present invention. The following describes preferred embodiments of the present invention. However, it should be understood, based on this disclosure, that the invention is not limited by the preferred embodiments described herein.
FIG. 3
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Referring to , an automated breadboard wiring assembly in accordance with an embodiment of the disclosure is shown. The assembly generally includes an input device , at least one wiring board , and a microcontroller . The microcontroller may be, for example, a Raspberry Pi processor which is connected to the breadboard via a GPIO expander or the like. It is recognized that the disclosure is not limited to this specific processor and other processing devices may be utilized. The input device and microcontroller are illustrated with a Wi-Fi interconnection, however, other communication interconnections may be utilized including wired and non-wired interconnections.
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The input device includes a screen for displaying a virtual breadboard . As will be described hereinafter, the input device is utilized to position circuit components and wiring on the virtual breadboard . The input device may be any type of processing device that includes a screen and allows for input of the circuit components and the wiring, for example, a personal computer, a laptop, a tablet, a smart phone or the like. The microcontroller receives wiring information from the input device and controls switches (as described hereinafter) on the wiring boards , to automatically electronically interconnect circuit components positioned on the physical breadboard . As such, the automated breadboard wiring assembly reduces the time necessary for wiring of the breadboard and also reduces the likelihood of wiring error and the associated debugging time.
FIG. 4
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Referring to , a primary wiring board in accordance with an embodiment of the disclosure will be described. The primary wiring board includes a board which supports an electronically interconnected wiring matrix . The board and wiring matrix may be, for example, in the form of a PCB, however, other structures may be utilized. The wiring matrix is composed of a plurality of interconnected wire segments . Each of the wire segments has a switch thereon to control flow of electricity through the wire segment . Each of the switches is configured to be controlled by the microcontroller and has a default closed condition.
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A plurality of contacts , are supported by the board . Each of the contacts is configured to electrically connect with a respective node of the breadboard and each of the contacts is configured to electrically connect with a respective node of the breadboard. Each of the contacts , are electrically connected to a switch which in turn is electrically connected to the wiring matrix . Again, each of the switches is configured to be controlled by the microcontroller and has a default closed condition. As such, when the primary wiring board is interconnected with the breadboard , each of the contacts , engages a respective node , , however, none of the nodes , is electrically interconnected with the wiring matrix . Upon instruction to initiate automatic wiring, the microcontroller opens selected switches proximate the necessary contacts , to interconnect the associated nodes , with the wiring matrix . The microcontroller also opens the switches along the necessary wiring segments to interconnect the nodes , , and thereby the circuit components positioned therein, based on the wiring information received from the input device . As an example, with reference to , to electronically wire contact with contact , the microcontroller would open switches , , , ,
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For some circuits, the primary wiring board may be sufficient to achieve the desired circuit wiring, however, for complex circuits, additional wiring options may be required. To facilitate such, secondary wiring boards may be connected with the primary wiring board . In the illustrative example in , two secondary wiring boards are connected to the primary wiring board , however, more or fewer secondary wiring boards may be utilized. Referring to , each secondary wiring board has a structure similar to the primary wiring board , except that the contacts for nodes are not included. As such, the secondary wiring board includes a board which supports another electronically interconnected wiring matrix ′. Again, the matrix ′ is made up of a plurality of wire segments with each segment having a control switch therealong. A plurality of contacts ′ are supported by the board and are configured to engage the wiring matrix of the wiring board positioned thereabove, as shown in . A switch is provided adjacent each contact ′ and controls electrical connection between the contact ′ and the wiring matrix ′. Again, each of the switches has a default closed condition. As such, the secondary wiring matrix ′ is not electrically interconnected with the primary wiring matrix until at least one of the switches associated with one of the contacts ′ on the secondary wiring board is opened by the microcontroller . Any desired number of secondary wiring boards may be stacked below and interconnected with the primary wiring board to achieve a wiring matrix of a desired complexity.
FIG. 7
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Referring to , an illustrative switch for use along the wire segments and proximate each of the contacts , will be described. The illustrated switch is a bidirectional, voltage invariant switch. The switch includes a nmos gate in parallel with a pmos gate. Each of the gates includes a pair of oppositely facing transistors, with each transistor of the pair acting as a switch in one direction while the other transistor acts as a switch in the opposite direction. The back-to-back transistors (both nmos in one of the gates and both pmos in the other gate) allows for bidirectional switching.
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Use of both a nmos gate and a pmos gate in parallel allows the switch to be voltage invariant. More specifically, a nmos transistor only conducts when gate voltage (Vg) minus source voltage (Vs) is greater than a voltage threshold (e.g. approximately 1.5 to 3.0 volts). In the open state Vg is a static 5 volts and Vs must be less than 3.5 volts to achieve the voltage threshold. Conversely, the pmos transistor conducts when Vg−Vs is less than −1.5. In the open state Vg is 0 volts and the pmos transistor will conduct when Vs is greater than 1.5 volts. With this configuration, any voltage presented at the switch input will appear undistorted at the switch output. It is noted that the switch input and switch output can be interchanged without any effect on the behavior of the switch. By controlling gate voltages, the microcontroller can open or close a switch. It is noted that the disclosure is not limited to this specific switch design and other bidirectional switches/relays may be utilized.
FIGS. 8-10
FIGS. 8-10
FIG. 8A
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Referring to , an illustrative wire routing utilizing a primary wiring board, referred to as the “breadboard layer”, and a secondary wiring board, referred to as the “virtual column layer”. In , the breadboard nodes are shown as black circles and the routing vertices are shown as gray circles. shows a horizontal slice through row 1 of the breadboard layer, connecting 1 row of breadboard nodes to 1 row of routing vertices while FIG. B shows a horizontal slice through row 2 of the breadboard layer, again wherein 1 row of breadboard nodes is connected to 1 row of routing vertices. Such an interconnection is formed for each row of the breadboard.
FIG. 9
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For the purpose of this explanation, the dimensions of the breadboard nodes are 15×2, but different dimensions can be selected. Furthermore, note that the rows of routing vertices match 1:1 with the rows of breadboard nodes. But, the number of columns of routing vertices is independent of the number of breadboard node columns. In this illustrative case, there are 15 rows of routing vertices, but 3 columns. is a top-down view of the virtual routing layer. Adjacent routing vertices within the same column are connected by switches . As shown in , the routing vertices allow the rows to be interconnected.
Wire routing between breadboard pins happens in the following way. The user selects a starting point (RA, CA) and an ending point (RB, CB). These coordinates correspond to two breadboard nodes. Without loss of generality, assume that RA<RB for the following explanation.
A virtual column X is selected that meets the following condition: routing vertices [(RA, X), (RA+1, X), . . . , (RB, X)] are unassigned. Once a virtual column has been identified, the corresponding switches/edges as identified by either a breadboard node and a virtual vertex, or two virtual vertices, are turned on. In the illustrated embodiment, the following switches would be turned on:
(RA, CA, breadboard node)<->(RA, X, virtual vertex)
(RA, X, virtual vertex)<->(RA+1, X, virtual vertex)
. . .
(RB−1, X, virtual vertex)<->(RB, X, virtual vertex)
(RB, X, virtual vertex)<->(RB, CB, breadboard node)
In addition to turning on all of these switches, each of the routing vertices that are endpoints of these switches are marked as assigned to this wire so that two wires aren't crossed. If it is desired to remove a wire, such is accomplished by turning off these switches and marking the associated routing vertices as unassigned.
To facilitate routing a breadboard pin to power/ground, each breadboard node is connected to a unique switch which is connected to power and a unique switch which is connected to ground. It is contemplated that each breadboard node may have two unique power switches, one connected to 5V power and one connected to 3.3V power. Therefore, a pin can be routed to power by turning on the switch that connects it to power (either 5V or 3.3V). Similarly, a pin can be routed to ground by turning on the switch that connects it to ground.
FIGS. 11-13
FIGS. 11-13
Referring to , an illustrative wire routing utilizing a primary wiring board, referred to as the “breadboard layer”, and a first secondary wiring board, referred to as the “bridge layer” and a second secondary wiring board, referred to as the “virtual column layer”. In , the breadboard nodes are shown as black circles, the bridge vertices are shown as gray circles and the routing vertices are shown as white circles.
The breadboard layer connects directly to each breadboard node and can create electrical connections between breadboard nodes and vertices in the bridge layer. The bridge layer sits between the breadboard layer and the virtual column layer and can create electrical connections between the breadboard layer and the virtual column layer. Note that the breadboard layer and the bridge layer only create electrical connections within the context of a single breadboard row. Finally, the virtual column layer sits below the bridge layer. Similar to the previous embodiment, the virtual column layer allows electrical connections to span across multiple breadboard rows.
FIG. 11A
FIG. 11B
shows a horizontal slice through row 1 of the breadboard layer, connecting 1 row of breadboard nodes to 1 row of bridge layer nodes which in turn connects to one row of routing vertices. shows a horizontal slice through row 2 of the breadboard layer, again wherein 1 row of breadboard nodes is connected to 1 row of bridge layer nodes which in turn connects to one row of routing vertices. For the purpose of this explanation, the dimensions of the breadboard nodes are 15×2, but different dimensions can be selected. The above structure is replicated 15 times, once for each of the 15 breadboard rows.
FIG. 12
FIG. 13
Importantly, the number of bridge vertices for a single row (3 in the illustrated embodiment) is independent of the number of breadboard columns (2 in the illustrated embodiment). Furthermore, each bridge vertex is connected to its own set of virtual column vertices. Each bridge vertex is assigned the same number of virtual column vertices, although there are no limitations to this number. In the illustrated embodiment, there are 2 virtual columns for each bridge vertex such that there are 6 virtual columns as illustrated in . As a reminder, the virtual column layer sits below the bridge layer, and the virtual column layer spans across multiple breadboard rows as shown in .
Wire routing happens in the following way. The user selects a starting point (RA, CA) and an ending point (RB, CB). These coordinates correspond to two breadboard nodes. Without loss of generality, assume that RA<RB for the following explanation.
A virtual column CX is selected that meets the following conditions: virtual column vertices [(RA, CX), (RA+1, CX), (RB, CX)] within the virtual column layer are unassigned. Furthermore, bridge vertices (RA, C{floor((X+1)/2)}) and (RB, C{floor((X+1)/2)}) are available.
Once a virtual column X has been identified, the corresponding switches/edges, as identified by a breadboard node and a bridge vertex, a bridge vertex and a virtual column vertex, or two virtual column vertices, are turned on. So, in the example of a connection between (RA, CA) and (RB, CB), the following switches would be turned on:
(RA, CA, breadboard node)<->(RA, C{floor((X+1)/2)}), branch vertex)
(RA, C{floor((X+1)/2)}, branch vertex)<->(RA, CX, virtual column vertex)
(RA, CX, virtual column vertex)<->(RA+1, CX, virtual column vertex) . . . .
(RB−1, CX, virtual column vertex)<->(RB, CX, virtual column vertex)
(RB, CX, virtual column vertex)<->(RB, C{floor((X+1)/2)}, bridge vertex)
(RB, C{floor((X+1)/2)}, bridge vertex)<->(RB, CB, breadboard node)
FIGS. 12 and 13
Consider the following concrete example wherein no wires have been placed yet, so all bridge and virtual column vertices are available. Referring to , a wire from R1C1 to R3C2 can be routed by turning on the following edges:
(R1, C1—breadboard node)<->(R1, C1—branch vertex)
(R1, C1—branch vertex)<->(R1, C1—virtual column vertex)
(R1, C1—virtual column vertex)<->(R2, C1—virtual column vertex)
(R2, C1—virtual column vertex)<->(R3, C1—virtual column vertex)
(R3, C1—virtual column vertex)<->(R3, C1—bridge vertex)
(R3, C1—bridge vertex)<->(R3, C2—breadboard node)
In addition to turning on all of these switches, each of the routing vertices that are endpoints of these switches are marked as assigned to this wire so that two wires aren't crossed. If it is desired to remove a wire, such is accomplished by turning off these switches and marking the associated routing vertices as unassigned.
To facilitate routing a breadboard pin to power/ground, each breadboard node is connected to a unique switch which is connected to power and a unique switch which is connected to ground. It is contemplated that each breadboard node may have two unique power switches, one connected to 5V power and one connected to 3.3V power. Therefore, a pin can be routed to power by turning on the switch that connects it to power (either 5V or 3.3V). Similarly, a pin can be routed to ground by turning on the switch that connects it to ground.
FIGS. 14-19
80
Referring to , alternative assemblies and methods for identifying and displaying an available wiring path will be described. In each of these embodiments, the wiring path is shown on a 2D LED matrix and a breadth-first search (BFS) algorithm is utilized to identify open paths. In each of the figures, a black dot represents an unlit LED, and a white circle represents a breadboard pin. In the illustrated embodiment, the dimensions of the breadboard pins are 3 rows×2 columns, however, more or fewer of each may be utilized.
With this assumption, the LEDs are organized in the following way. First, if we think about this in terms of columns of LEDs. Notice that to the left of the first column of LEDs that is interspersed with breadboard pins, there are 4 columns just of LEDs. Then a column where the LEDs alternate with breadboard pins. Between two columns of LEDs interspersed with breadboard pins, there are 5 columns of just LEDs. Finally, after the last column of LEDs interspersed with breadboard pins, there are another 4 columns just of LEDs.
Now, if we consider this in terms of rows of LEDs, a row of just LEDs alternates with a row of LEDs interspersed with breadboard pins. Therefore, the number of columns of LEDs in a design with 3×2 breadboard pins, there are 4+1+5+1+4=15 columns of LEDs. The number of rows of LEDs in this design is 3*2+1=7.
When tracing wires, a few rules are imposed upon our BFS search so that the wires can be displayed in a visually appealing manner and the number of turns taken by a wire is minimized:
FIG. 15
1—Wires connect breadboard pins (2 or more). In order to indicate that a wire is connected to a breadboard pin, the LED within the same row as that breadboard pin and directly adjacent (to the left and/or right) is lit-up. So, as an example, if a wire is connected to R1C1, then one of the two (or both) LEDs may be lit-up, as shown as a star in , to indicate this connection.
FIG. 16
2—A wire path consists of a contiguous path of LEDs. That is, an LED is bordered in the top, bottom, left or right by either two LEDs assigned to the same wire, or a breadboard pin and an LED. In , the lit-up LEDs (stars) represent a wire which connects R1C1 and R2C2.
FIG. 17
3—LEDs positioned vertically between breadboard pins can only be assigned to indicate a wire that is moving horizontally. These LEDs are found in the columns with the breadboard pins as indicated by the arrows in . These LEDs allow pins in the same row to be connected without passing through a breadboard pin. For example, in connecting R1C1 and R1C3, the path, as indicated by the stars, needs to pass through one of such LEDs so as to avoid passing through a breadboard pin, namely, through R1C2.
FIG. 18
4—When tracing multiple wires, it is inevitable that wires will cross over each other. To reconcile this in a visually appealing manner, if two wires overlap, they will overlap by at most a single LED. For example, shows a scenario in which a first wire (indicated by stars) is connecting R1C1 to R2C2 and a second wire (indicated by pluses) is connecting R2C1 to R1C2. The circled LED, which is circled only for ease of description, indicates where the wires overlap. In the illustrated embodiment, at the point where the two wires overlap, the LED assumes the color of the last wire to be drawn.
FIG. 19
FIG. 19
It is recognized that a wire can connect 2 or more breadboard pins. Referring to , a wire is drawn first between R1C1 and R1C2, and then a wire is drawn between R1C1 and R2C2. Electrically, these three pins are all connected together and part of the same net—it doesn't matter how they are wired together. illustrates one valid way of displaying such a wiring configuration.
FIG. 20
FIG. 20
80
If there are too many wires, such that displaying all of the wires is too complicated/messy, it may be beneficial to utilize a modified display as illustrated in . In this embodiment, the same LED matrix is utilized, however, only certain ones of the LEDs will be lit, for example, only the 3 LEDs next to each pin as indicated by the boxes in . Each wire will be assigned a unique color or sequence of colors. If pin is connected via that wire, the assigned color or sequence will be lit up within the three boxes associated with that pin. In the illustrated example, a wire with the assigned sequence of off (black circle), red (star), blue (plus) connects the pins R1C1, R1C2 and R2C2 and a wire with the sequence of blue (plus), off (black circle), red (star) connects the pins R3C2 and R3C3. Various colors (created using RGB LEDs) and sequences may be utilized to represent each wire.
FIG. 21
80
80
The embodiment illustrated in is similar to the previous embodiment except that the LED matrix ′ excludes the LEDs that are not utilized. That is, only the 3 (or however many chosen) LEDs next to each pin are provided. In other aspects, the LED matrix ′ functions in the manner described with respect to the previous embodiment.
FIGS. 22 and 23
FIG. 22
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Referring to , a method of inputting wiring information into the input device will be described. The component input screen is shown in . A virtual breadboard which corresponds to the physical breadboard is displayed on the screen . A menu of circuit components is shown on the screen and the user may drag and drop the components on the virtual breadboard in the same manner in which the components are positioned on the physical breadboard . It is understood that input device is configured to allow the user to adjust sizes, rotate or otherwise manipulate the components . Additionally, while a drag and drop interface is illustrated, other interfaces, for example coordinate input, may be utilized.
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FIG. 23
Once all of the components have been placed on the virtual breadboard such that it matches the physical breadboard , the user clicks at to advance to the wiring screen as shown in . The user clicks a wire start arrow or the like on the starting hole of a desired wire and then clicks a wire finish arrow ′ on the finish hole of the wire. The system is configured to draw a virtual wire extending between the holes . The user can quickly and easily see that the wires are in the correct, desired positions. If any additional components are required, the user can click as indicated at to return to the component input screen. Once all of the wires are placed, the user can then click as indicated at to finish and initiate the auto wiring of the physical breadboard . It is further contemplated that the user may have the option to test/simulate the virtual circuit prior to initiating the auto wiring. The input device may be configured to test the virtual circuit and provide visual feedback to the user of any in continuities in the wiring and allow the user to inspect the simulation and observe waveforms and other electrical properties of nodes throughout the circuit.
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Upon initiation of the auto wiring process, the microprocessor receives the wiring information from the input device . The microprocessor then opens each switch necessary to route electrical interconnection along the contacts , and wire segments corresponding to the wires in the wiring information. Once the auto wiring process is complete, the components on the physical breadboard will be electrically interconnected in the manner set forth on the virtual breadboard. Any changes in wiring can be made easily through the input device .
FIGS. 24 and 25
FIG. 24
FIG. 25
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Referring to , an alternative method of inputting wiring information into the input device ′ will be described. In this embodiment, the user inputs a circuit diagram into the input device ′ with the components identified in a manner readable by the input device processor. For example, the circuit diagram may be created utilizing a schematic design tool which is compatible with the input device ′ such that the input device ′ recognizes each of the components and their interconnection. As illustrated in , the input device ′ analyzes the circuit to identify each component and the interconnections between the components. Once the circuit is analyzed, the input device ′ is configured to lay out the virtual components and virtual wiring on the virtual breadboard as illustrated in . Once the layout on the virtual breadboard is complete, the user can place the components on the physical breadboard in the same positions as shown on the virtual breadboard and then send the wiring information to the microprocessor to complete auto wiring.
FIGS. 26-35
30
30
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Referring to , a standalone breadboard assembly ′ in accordance with an embodiment of the invention will be described. The standalone breadboard assembly ′ incorporates a system of LEDs and button/contact interfaces which act as the input device ′ such that the device can operate standalone and allow users to build complete circuits. It is noted that the standalone breadboard assembly ′ may also interface with a separate input device, e.g. a laptop or the like, to operate in a manner as described above.
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The breadboard assembly ′ includes a housing which holds one or more breadboards . Within the housing, one or more wiring boards and a processor (not shown), for example, a Raspberry Pi processor, are interconnected with the breadboards to facilitate automated wiring as described above. The input device ′, as will be described, is integral with the housing and allows the user to directly indicate which nodes are to be interconnected.
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In the present embodiment, the input device ′ includes a plurality of breadboard LEDs , a plurality of breadboard contacts , a plurality of function LEDs , a plurality of function contacts and a stylus . A breadboard LED and breadboard contact pair is associated with each breadboard pin . Similarly, a plurality of function LED and function contact pairs are provided to carry out desired functions as will be described hereinafter. The illustrated assembly includes contacts for the following functions: Clear/Delete; GND; VCC 3.3; VCC 5.5; and Undo. Each of the LEDs is preferably an RGB LED such that it may be lit with various colors.
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FIG. 25
The user must use the stylus to press the contacts , . Referring to , the stylus plugs into an input jack in the housing and has a metal tip which is electrically grounded. In the illustrated embodiment, the contacts , are metal pins which are connected to pull-up resistors. Each contact , is connected to a GPIO input pin, which monitors the voltage of the contact , . When un-pressed, a contact , is pulled up to 5 volts. If the user presses the stylus to a contact , , an electrical connection is made between the contact , and ground, and the contact's voltage becomes 0V. The changes in state (5V/un-pressed and 0V/pressed) of the contacts , are monitored in software, which interprets these state changes to provide high-level functionality to the user. The software looks for changes between 0V and 5V, but also measures the time between state changes of a contact , , providing a lot of flexibility for creating different functionality (for example timed press operations like holding down the stylus on the reset function pin for 3 seconds to reset the board state).
Together, the breadboard contacts and LEDs, and the function contacts and LEDs offer the user the following functionality.
A user can place a wire connecting two breadboard pins. This can be accomplished by touching the stylus to the breadboard contact adjacent to one of the two pins, and then pressing the breadboard contact adjacent to the second pin. Once the user presses the second contact, a wire is routed in hardware in a manner described above, and the two breadboard LEDs next to these two breadboard pins are illuminated with the same RGB color, indicating that they are electrically connected.
FIGS. 28 and 29
FIG. 28
FIG. 29
48
1
1
48
Referring to , placing a wire from breadboard pin (row=1, col=1) to breadboard pin (20,4) will be illustrated. First, the stylus is pressed to breadboard contact (1,1). In , breadboard LED (,) is illuminated yellow (indicated by a star). In the illustrated embodiment, yellow is used as an intermediate color to indicate that the user has selected a breadboard pin to perform a wiring operation. Turning to , the stylus is next pressed to breadboard contact (20,4). Breadboard LEDs (1,1) and (20,4) are assigned their final color, for example, blue (indicated by an X).
By placing multiple wires, a user can create an electrical short/connection between 2 or more breadboard pins. Breadboard pins that are electrically shorted/connected are referred to herein as a net. Breadboard pins are considered electrically connected if they are reachable (in a graph algorithm sense) by traversing virtual wires.
If a user connects a breadboard pin with no connected wires to a net (including 2 or more bread board pins), the assembly places a wire to create an electrical connection between this pin and the net, and the LED adjacent to this new breadboard pin is illuminated with the same color as the net, indicating that it is now electrically connected with all breadboard pins in the net.
FIGS. 30 and 31
FIG. 30
FIG. 31
48
For example, illustrated the process of placing a wire from (5,5) to (20,4). As a first step, the user utilizes the stylus to press the contact at pin (5,5) and the associated LED turns yellow (star), as shown in . The user than utilizes the stylus to press the contact at pin (20,4) as shown in . Breadboard pin (5,5) is added to the original net and its associated LED turns to the color of the net, in this case blue (X).
FIGS. 32 and 33
FIG. 32
A user can also join two nets. By first selecting any one pin from net A, and then selecting any one pin from net B, a wire is placed between net A and B, such that nets A and B are electrically connected. Furthermore, arbitrarily, the color of the breadboard LEDs in net B are updated to the color of the breadboard LEDs in net A. Such a connection is illustrated with reference to . In , the user creates a wire between (5,3) and (28,3) by pressing the breadboard contacts in that order. The associated LEDs shine the same color to represent the new net, for example, a greenish-yellow (indicated by a gear shape). Note that there are currently two nets, net A is blue (X) and net B is a greenish-yellow (gear shape).
FIG. 33
Turning to , the user next places a wire between (5,3) and (5,5). The two nets are joined into a single net and each of the associated LEDs shines the same color, namely, blue (X).
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FIGS. 34 and 35
A user can also place a wire between a breadboard pin and power (3.3V or 5V) or ground (0V). They can do so by selecting a breadboard pin contact , and then pressing either the GND, VCC 3.3, or VCC 5 function contact . They can also perform this operation by pressing the function contact first, and then the breadboard contact . illustrate a wire placed between GND and breadboard pin (35,1), wherein each associated LED is lit the same color, for example, green (indicated by a minus).
As an additional complexity, an entire net can be routed to GND, VCC 3.3, or VCC 5. If the user were to have routed one breadboard pin from a net (e.g. pin (1,1) to GND), then that net would have been electrically connected to ground, and the color of all LEDs would have been updated to green (minus) to indicate that all of the wires in that net are grounded. In one embodiment, VCC 3.3 is represented by orange, and VCC 5 is represented by red.
47
A user can remove all electrical connections to a breadboard pin. If they press the Delete/Clear function contact , and then the breadboard contact (or these operations in reverse order), any wires between that pin and GND, VCC 3.3, and VCC5 will be removed. Furthermore, if that breadboard pin is a member of a net, all wires connecting this pin to that net are removed such that the pin is not connected to the net, but the remainder of the net remains connected. Note that when removing a breadboard pin from a net, it is possible that such removal will break electrical connections within the net, such that the net is not fully connected (in a graph sense). In order to rectify this, additional wires are routed automatically to guarantee that the breadboard pins remaining in the net remain electrically connected. The LED color of the removed pin is reset to RGB (0, 0, 0).
A user can clear the state of the entire board by holding the stylus to the Delete/Clear function contact for at least 3 seconds. Once a timer has elapsed, all LEDs will be reset to (R=0, G=0, B=0) and all electrical connections will be removed.
The user can undo an immediately previous operation by pressing the stylus to the undo function contact.
These and other advantages of the present invention will be apparent to those skilled in the art from the foregoing specification. Accordingly, it will be recognized by those skilled in the art that changes or modifications may be made to the above-described embodiments without departing from the broad inventive concepts of the invention. It should therefore be understood that this invention is not limited to the particular embodiments described herein but is intended to include all changes and modifications that are within the scope and spirit of the invention as defined in the claims. | |
This content was reproduced from the employer’s website on February 20, 2022. Please visit their website below for the most up-to-date information about this position.
What’s so interesting about this internship?
Customer Experience is at the heart of Grindr’s mission. The User Insights Intern will work under the supervision of Grindr’s Customer Support Program Manager to review customer feedback from Grindr’s global uberbase and document trends and key results. This role is a perfect opportunity for someone with an open and analytical mind, who is passionate about serving the LGBTQ+ community.
What you’ll do:
- Monitor inbound feature requests submitted to Uservoice, our community feedback forum
- Organize existing feature requests and user feedback into common themes
- Identify follow up opportunities for users with since-fulfilled requests
- Document trends in feedback and requests for posterity
- Research and document user sentiment across social media platforms such as Twitter and Reddit
- Reading zendesk tickets
- Propose edits to our help center and email response language based on your findings
- Shadow Grindr’s Customer Experience (CX) team in meetings where appropriate
- Present your findings to CX and other stakeholders at the end of the internship findings
- Part-time or full–time opportunity
- To run June-August, 2022
What we’ll love about you:
- Ability to comprehend large amounts of reading and exercise critical thinking
- Strong written and verbal communication skills
- Familiarity with Grindr
- Committed to protecting user data and privacy
- You are a champion of the LGBTQ+ community and its allies
What you’ll love about us:
- Global impact. Grindr is the world leading LGTBQ+ social networking service. Your role will impact the lives of millions of LGTBQ+ people around the world.
- High Growth. Like the company itself, this role offers significant room for growth and development
- Remote First: We have offices in LA, NYC, SF, Chicago, and Taipei, though the company is set-up not just as remote-friendly, but remote-first. More than 30% of our employees work outside the cities where we have offices.
What we hope you’ll take away from this:
- Experience in tech for non-tech backgrounds
- A deeper understanding of our community
- Experience with support specific tooling
To apply for this job please visit boards.greenhouse.io. | https://www.tspa.org/job/user-insights-intern/ |
Emotion strengthens the subjective experience of recollection. However, these vivid and confidently remembered emotional memories may not necessarily be more accurate. We investigated whether the subjective sense of recollection for negative stimuli is coupled with enhanced memory accuracy for contextual details using the remember/know paradigm. Our results indicate a double-dissociation between the subjective feeling of remembering, and the objective memory accuracy for details of negative and neutral scenes. "Remember" judgments were boosted for negative relative to neutral scenes. In contrast, memory for contextual details and associative binding was worse for negative compared to neutral scenes given a "remember" response. These findings show that the enhanced subjective recollective experience for negative stimuli does not reliably indicate greater objective recollection, at least of the details tested, and thus may be driven by a different mechanism than the subjective recollective experience for neutral stimuli. | https://nyuscholars.nyu.edu/en/publications/emotion-enhances-the-subjective-feeling-of-remembering-despite-lo |
The Geek Culture Forums!: Whiffling goose!
» The Geek Culture Forums! » Techno-Talking » Science! » Whiffling goose!
For some reason I find this so cool. A greylag goose was captured by wildlife photographer Brian McFarlane in a manoeuvre known as whiffling, where the bird flips upside down in order to rapidly lose altitude.
__________________ Hi All __ Since we have a full horizon view, minus a few degrees. We get to watch a lot of birds. A couple of years ago I pointed out two sparrows tormenting a hawk, eventually driving it away. Now the Mrs. thinks that sparrows are mean birds, I keep telling her that they are just trying to protect their own.
Your wife is right, MoMan. Sparrows are mean. They have to be, since they aren't a natural part of the food chain. They're not native to the United States. Our park naturalists have told me many stories about how sparrows torment the native songbirds.
Sparrows... ...aren't a natural part of the food chain.
__________________ The Famous Druid __ Speaking about Cats. The real comedy was last year our cat (Muffin) was actually stalking a Sand Hill Crane. I am unsure what she would do if she came face to face with a bird of that size, I think it would have been fun to watch.
When I was a lad I watched the woods Owls get barn rats (after we put the dog in the barn the rats would run) there is no sound until the rat screams. My Uncle called putting the dog in the barn feeding the Owls, this was back before TV and we would be setting on the porch of the farm house, Uncle Chuck would say "time to feed the Owls, put the dog in the barn", we would have about ten minutes until the owls stopped swooping.
Watching an owl swoop or take off is good watching, they sort of lean forward and topple towards the ground about half way to the Earth they start opening their wings and glide to the target.
When I was a kid, Clan McDruid had a pair of cats with attitude, they used to lie in wait for dogs passing our house, and ambush them. The sight of 2 cats chasing a german shepherd half way up the hill was hilarious.
We had a cat that chased our dog, but this cat takes the cake for treeing a bear twice, even though he'd been declawed.
My aunt's cat catches lizards. My cat stalks and pounces the light from a laser pointer. And gets mighty confused when, after he's "caught" the red dot, it appears right on top of his paw! | http://www.geekculture.com/cgi-bin/ultimatebb/ultimatebb.cgi?ubb=get_topic;f=28;t=000278 |
There are many different Postal Communication & Commercial entities that use the name of Europe/Europa in their names or institutions.
As a continent Europe trade organsiations also create partnerships with other 'continental' bodies.
EUROPA / CEPT
The Europa postage stamp (also known as Europa - CEPT until 1992) is an annual joint issue of stamps with a common design or theme by postal administrations of member countries of the European Communities (1956-1959), the European Conference of Postal and Telecommunications Administrations (CEPT) from 1960 to 1992, and the PostEurop Association since 1993. Europe is the central theme.
EUROPA stamps underlines cooperation in the posts domain, taking into account promotion of philately. They also build awareness of the common roots, culture and history of Europe and its common goals. As such, EUROPA stamp issues are among the most collected and most popular stamps in the world.
Since the first issue in 1956, EUROPA stamps have been a tangible symbol of Europe’s desire for closer integration and cooperation.
Subcategories
This category has only the following subcategory.
E
- ► Europe (40)
Pages in category "Europe"
The following 40 pages are in this category, out of 40 total. | https://www.stampsoftheworld.co.uk/wiki/Category:Europe |
In the present era, general knowledge is crucial to your children’s development and growth. There are a number of choices accessible nowadays. The key is to learn and participate in various areas. It is particularly important for them to learn outside the school environment. Moving beyond education lets children succeed in a number of fields. The success of a child depends on what he or she learns from the surrounding environment.
The knowledge your kid assembles and the learning that he or she has will build his or her intellectual processes. This directly influences how he or she performs not only in class but also in professional and personal life at a later date. As parents, we just want the best for our children.
In this blog, you will learn about various easy GK questions along with some riddles for kids.
Learning courses for your kids! Get free trial here
General Knowledge Questions for Kids
General awareness that lets your child learn in a holistic way is also a very broad subject. It will explore a wide range of subjects. In basic language, general knowledge refers to information on a wide variety of subjects. Most parents also face obstacles such as:
- What do we cover on the basis of the age of the child?
- Depth of knowledge for each theme picked
- Methods to simplify knowledge and educate children in an interactive way
- Techniques to improve comprehension so that the kid holds as much knowledge as possible
A kid is like a sponge able to consume all the knowledge presented. As a mother or father, you need to make sure that your child absorbs the right things and moves in the correct direction.
GK Questions for Kids in English
It’s very difficult to pick just a few GK questions for children, because it’s a big field and, in actuality, limitless. Several of the GK questions will seem very simple for higher-level students, but it’s always a good idea to amend what you understand before adding more information to your base of knowledge.
Learn some basic GK questions for kids with some riddles for kids below.
Easy GK Questions: 4 to 7 Years
Children who belong to this age category are intrigued about everything in their environment. So interacting with concerns about General Knowledge would be a great support to them. In the section below, we have included some easy GK questions for children to explore and broaden their level of general knowledge.
Also Read: Worried about Lockdown’s Adverse Effect on Your Kid? Here’s RealSchool at Your Rescue
GK Questions for Kids with Answers: 4 to 7 Years
#1. How many days do we have in a week?
Answer: Seven
#2. How many days are there in a normal year?
Answer: 365 (not a leap year)
#3. How many colours are there in a rainbow?
Answer: 7
#4. Which animal is known as the ‘Ship of the Desert?’
Answer: Camel
#5. How many letters are there in the English alphabet?
Answer: 26
#6. How many consonants are there in the English alphabet?
Answer: 21
#7. How many sides are there in a triangle?
Answer: Three
#8. Which month of the year has the least number of days?
Answer: February
#9. Which are the vowels in the English alphabet series?
Answer: A, E, I, O, U
#10. Which animal is called King of Jungle?
Answer: Lion
#11. How many primary colours are there?
Answer: Three (red, yellow, blue)
#12. How many days are there in the month of February in a leap year?
Answer: 29 days
Learning courses for your kids! Get free trial here
Quiz Questions for Kids: 8 to 10 Years
Once children reach this age bracket, they begin to learn new concepts, notice the changes surrounding them, and try to find solutions to their queries by learning more about GK questions and much more. Other than their teaching in the classroom, they also gain general knowledge from books and online. Here are a few important questions for children between the ages of 8-10 years.
Also Read: Overseeing Kids during Isolation: Here’s Your Way out of the Juggling Situation
GK Questions for Kids with Answers: 8 to 10 Years
#1. In which direction does the sunrise?
Answer: East
#2. Which is the world’s largest flower?
Answer: Rafflesia
#3. How many zeros are there in one hundred thousand?
Answer: Five
#4. How many hours are there in two days?
Answer: 48 hours (24+24)
#5. How many months of the year have 31 days?
Answer: 7 (January, March, May, July, August, October and December)
#6. How many weeks are there in one year?
Answer: 52
#7. Which are the colours in a rainbow?
Answer: Violet, Indigo, Blue, Green, Yellow, Orange, Red
#8. How many bones does an adult human have?
Answer: 206
#9. Who was the first man to walk on the moon?
Answer: Neil Armstrong
#10. How many millimetres are there in 1cm?
Answer: 10
#11. Which is the nearest star to planet earth?
Answer: Sun
#12. Which is the longest river on the earth?
Answer: Nile
#13. Which is the principal source of energy for the earth?
Answer: Sun
Conclusion
Parents put all their hard work into doing what is best for their children. It’s crucial to know what general knowledge implies to your children and how they profit from it. All the awareness and life skills that your child has can help develop courage and skills.
Learn more about children growth factors with The Real School.
Also Read: What is Project-Based Learning? Does PBL Really Work for Kids? | https://therealschool.in/blog/gk-questions-for-kids-with-answers-check-basic-children/ |
GILDE Gallery are three-dimensional art pieces created from metal, which are intricately produced by hand. The large-scale pieces leave a strong impression. They are snapshots of places all around the world and art impressions. GILDE Gallery artworks impress with delicate individual parts, from which the theme gradually becomes a large-scale relief. The design, shape and colour combine to create a fascinating, unique piece. Cool metal transforms into great vibrancy, which evokes pure emotion.
GILDE Gallery is unique handicraft with modern interpretation and intricate staging. GILDE Gallery 3D pictures and figures are crafted by hand. The individual parts are carefully cut to size, hammered and polished. After this they are welded with further fragments, to produce shapes and relief designs according to the sketches of the GILDE designers. Finally, the pictures and objects are painted numerous times, so that new shading results each time. Each piece is an inimitable one-off from the GILDE Gallery collection. | https://www.gildehandwerk.com/en/gilde-collection/gilde-gallery/ |
the restoration of five mosques in the Makkah region,
according to the Saudi Press Agency on Sunday.
The mosques’ architecture has been impacted by the changing
climate throughout the centuries, and this initiative intends
to safeguard and rehabilitate them so that they can last for many more years.
The second phase of the Makkah development project begins with
the Al-Baiah Mosque, which was constructed by the Abbasid Caliph
Abu Jafar Al-Mansour is located close to Jamarat Al-Aqaba in Mina.
The mosque will maintain its current size and capacity of 68 worshippers
after the refurbishment is complete, with a total space of 457.56 square meters.
Two of the five mosques are located in the Jeddah governorate:
the Abu Inbeh Mosque in Harat Al-Sham and the Al-Khadr
Mosque in the Al-Balad area, both on Al-Dhahab Street.
In Makkah Five Mosques To Be Restored
The Abu Inbeh Mosque dates back over a thousand years
and occupies a space of 339.98 square meters.
The proposed renovations will increase the size to 335.31 square meters,
with enough room for 357 worshippers.
Approximately 66 kilometers from the Grand Mosque in
Makkah is the Al-Khadr Mosque, which dates back to
the early Islamic era (around the seventh century).
Once completed, the renovations will bring the square footage
up to 355.09 m2, allowing for seating for 355 worshippers.
According to tradition, the Prophet Muhammad visited the Al-Fath Mosque
in the governorate of Al-Jamoum the year he captured Makkah.
The current space can only accommodate 218 people for worship,
but with the planned expansion to 553.50 square meters,
that number should rise to 333.
More than 300 years old, the Al-Jubail Mosque will have a remodeled area of
310 square meters as part of the project. It will still be able to hold 45 worshipers.
The second stage of the development project, which spans the entire Kingdom,
will entail the construction of 30 mosques.
The project’s goals are to raise awareness of Saudi Arabia’s cultural heritage,
rehabilitate old mosques for religious purposes, protect the architectural integrity of historic mosques,
and elevate the cultural and religious significance of these buildings. | https://azofficial.org/makkah-five-mosques-restored-part-development-project/ |
What is resource allocation problem?
The resource allocation problem seeks to find an optimal allocation of a fixed amount of resources to activities so as to minimize the cost incurred by the allocation. The amount of resources to be allocated to each activity is treated as a continuous or integer variable, depending on the cases.
What is resource allocation explain?
Resource allocation is the process of assigning and managing assets in a manner that supports an organization’s strategic goals. Resource allocation includes managing tangible assets such as hardware to make the best use of softer assets such as human capital.
What is resource allocation problem in project management?
Resource allocation is the process of assigning and scheduling available resources in the most effective and economical way possible. The task, therefore, lies with the project manager to determine the proper timing and allocation of those resources within the project schedule.
What is an allocation sheet?
This is a read-only sheet that totals resource data by role. Allocations are shown for each role. At the company level, allocations are shown for each role against one or more projects. The combination of role and project is always unique.
What is resource allocation in data structure?
To assign the available resources in an economic way is known as resource allocation. The planning of the activities and the resource required by these activities while taking into consideration both resources availability and project time is termed as resource allocation in project management.
What are the types of resources in resource allocation?
Generally, there are five types of resources:
- Labor. Human resources are an integral part of most projects.
- Equipment. Tools and equipment that are used to produce the product, but don’t become part of it, must be identified and allocated to each task.
- Materials.
- Facilities.
- Miscellaneous.
What is resource allocation matrix?
The Resource Allocation Matrix frames the time perspective and proactivity of efforts. The Matrix can be used both descriptively, i.e. to illustrate how IT resources are allocated within the company, and prescriptively, i.e. to actively guide the allocation of resources between different concurrent projects.
How is resource allocation done?
Resource allocation is the process of assigning and scheduling resources to project tasks….There are 6 steps to performing a proper resource allocation:
- Divide the Project into Tasks.
- Assign the Resources.
- Determine resource attributes.
- Resource Leveling.
- Re-allocate as necessary.
- Track resource utilization.
How do you show resource allocation?
To view resource allocation in Project using the “Resource Usage” view, click the “Task” tab in the Ribbon. Then click the “Gantt Chart” drop-down button in the “View” group and then select the “Resource Usage” command. This view shows resources, work contours, and resource allocation issues.
What are the methods of resource allocation?
Top 6 Means of Resource Allocation
- BCG-Based Budgeting: The BCG matrix can also be used for resources allocation.
- Strategic Budgeting: Budgeting is a common technique used as a planning coordination, and control device in management.
- Zero-Based Budgeting:
- PLC-Based Budgeting:
- Capital Budgeting:
- Parta System:
What is importance of resource allocation?
Resource allocation is the process of determining the best way to use available assets or resources in the completion of a given project. Companies attempt to allocate resources in a manner that helps to minimize costs while maximizing profits, typically by using strategic planning methods to structure the operation,…
What is an over allocated resource?
Over-allocation generally refers to situations where resources are allocated at excessive levels. In the context of IT, the resources often refer to hardware or software capabilities such as processing power, memory, data management, bandwidth or other specifications.
What is the definition of resource allocation?
In economics, resource allocation is the assignment of available resources to various uses. In the context of an entire economy, resources can be allocated by various means, such as markets or central planning.
What is efficient allocation of resources?
Efficient Allocation of Resources. An efficient allocation of resources is: That combination of inputs, outputs and distribution of inputs, outputs such that any change in the economy can make someone better off (as measured by indifference curve map) only by making someone worse off ( pareto efficiency ). | https://penelopethemovie.com/what-is-resource-allocation-problem/ |
Job Category: Manufacturing | FMCGOur client, a leading chemical and food ingredient distributor, is looking for a dedicated and dynamic Supply Chain & Operation Manager. The successful candidate will lead the Supply Chain and Operations Team to Ensure that in-bound logistics are optimized to ensure consistent availability of raw materials at minimum cost and that inventory planning is optimized to ensure optimal working capital deployment at all times in-line with customer demand forecasts. Experience in the food/beverage processing industry or specialty materials distribution is STRONGLY PREFERRED as well as having worked for a Multinational company.
Key Responsibilities:
- Oversee the customer service function
- Oversee the in-bound logistics/procurement function
- Oversee the warehousing function
- Ensure the procurement of appropriate storage and logistics services and optimize inventory
- Ensure maximum efficiency, consistency of product, quality of packaging and elimination of leakage
- Support the review and implementation of appropriate and cost-effective ERP/WMS software
- Support the expansion of the business through identification of appropriate premises/facilities
- Monitor the overall supply pipeline for the business
- Create and foster an environment of continuous improvement, teamwork, accountability and innovation
- To actively solicit and contribute ideas for safer, more cost effective approaches or new techniques
- Responsible for meeting goals within time and budget constraints and for success of activities
Qualification:
- Minimum of a bachelor’s degree in Supply chain & logistics or related field
- Minimum of 7 years’ relevant experience in Logistics, Supply Chain/Operations management
- At least 3 years’ experience managing individuals across different functional areas
- Relevant experience in East African companies and Intra-East Africa logistics and PVOC operations
- Strong Analytical and Strategic skills
- Exceptional problem solving and negotiation skills
- Strong planning skills and a self starter
- Outstanding communication (both verbal and written) and relationship management skills
- Willingness and ability to travel
Education: Bachelor’s Degree
Job Type: Permanent
Location: Nairobi City Kenya
Career Level: Senior Management
Deadline: April 28, 2019
Applications
Only short listed candidates will be contacted.
Please do not apply if you do not meet the requirements of the job. | https://keweb.co/summit-experienced-supply-chain-operation-manager-job-2019/ |
Jino Restaurant is seeking F/T Cooks.
Position: Cook
Terms of employment: Full-time, Permanent
Wage: $14.50 - $17.00/hour (depending on experience)
Hours: 30 - 40 hours/week
Benefits:
5.77% vacation pay
Job Duties:
Prepare, season and cook dishes
Plan menus, determine size of food portions, estimate food requirements and costs
Monitor stock of supplies, ingredients, and place orders when there are shortages
Clean and sanitize the kitchen
Maintain inventory and records of food, supplies and equipment
Train kitchen staff and supervise their activities
May hire staff and train them in preparation and cooking and handling of food
Manage kitchen operations
Educational Requirements:
Completion of Secondary school
Experience Requirements:
2~3 years of commercial cooking experience are required.
Language Requirements:
English
Number of Positions: | https://newcomersjob.ca/index.php/jobseekers/search-jobs/jobs?layout=listjobs&location_id=617 |
Ballot measures There were wins for reproductive freedom in five states:
California – Proposition 1: “Amends California Constitution to expressly include an individual’s fundamental right to reproductive freedom, which includes the fundamental right to choose to have an abortion and the fundamental right to choose or refuse contraceptives. This amendment does not narrow or limit the existing rights to privacy and equal protection under the California Constitution.”
The Yeses won
Kentucky – Amendment 2: Would add to the Kentucky Constitution: “To protect human life, nothing in this Constitution shall be construed to secure or protect a right to abortion or require the funding of abortion.”
The Nos won
Michigan – Proposal 3: “This proposed constitutional amendment would:
- Establish new individual right to reproductive freedom, including right to make and carry out all decisions about pregnancy, such as prenatal care, childbirth, postpartum care, contraception, sterilization, abortion, miscarriage management, and infertility;
- Allow state to regulate abortion after fetal viability, but not prohibit if medically needed to protect a patient’s life or physical or mental health;
- Forbid state discrimination in enforcement of this right; prohibit prosecution of an individual, or a person helping a pregnant individual, for exercising rights established by this amendment;
- Invalidate state laws conflicting with this amendment.”
The Yeses won
Montana – Legislative Referendum 131: “An act adopting the born-alive infant protection act; providing that infants born alive, including infants born alive after an abortion, are legal persons; requiring health care providers to take necessary actions to preserve the life of a born-alive infant; providing a penalty; providing that the proposed act be submitted to the qualified electors of Montana; and providing an effective date.”
The Nos won
Vermont – Proposal 5: Would add to the Vermont Constitution: “That an individual’s right to personal reproductive autonomy is central to the liberty and dignity to determine one’s own life course and shall not be denied or infringed unless justified by a compelling State interest achieved by the least restrictive means.”
The Yeses won
Governor seats that flipped from Republican to Democrat
Maryland: Wes Moore (Democrat) defeated Dan Cox
Moore’s stance on repro rights: “Wes recognizes that all Marylanders deserve the autonomy to make their own decisions about their reproductive healthcare. Wes and Aruna will fight to make access to reproductive care more affordable and accessible in every corner of our state…” (source)
Massachusetts: Maura Healey (Democrat) defeated Geoffrey Diehl (Republican)
Healey’s stance on repro rights: “With our reproductive rights under attack like never before, Maura will ensure patients and providers are protected here in Massachusetts…” (source)
Senate seat that flipped from Republican to Democrat
Pennsylvania: John Fetterman (Democrat) defeated Mehmet Oz (Republican)
Fetterman’s stance on repro rights: “A woman’s right to make her own health care decisions is sacred and non-negotiable. Period.” (source)
Some House seats that flipped from Republican to Democrat
Michigan’s 3rd District: Hillary Scholten (Democrat) defeated John Gibbs (Republican)
Scholten’s stance on repro rights: “Protecting reproductive health care choices is fundamentally a matter of privacy and freedom from government control. It is a kitchen table issue, a worker’s rights issue, a child welfare issue, and a healthcare worker protection issue.” (source)
New Mexico’s 2nd District: Gabriel Vasquez (Democrat) defeated Yvette Herrell (Republican)
Vasquez’s stance on repro rights: “Access to healthcare is a right and politicians in Congress and in state Legislatures shouldn’t stand between women and their healthcare. Gabe will oppose any attempt by Republicans in Congress to ban access to abortion, birth control, or to prevent women from having the right to choose.” (source)
North Carolina’s 13th District: Wiley Nickel (Democrat) defeated Bo Hines (Republican)
Nickel’s stance on repro rights: “I believe that politicians have no business getting in the middle of healthcare decisions, and that those decisions belong between a woman and her doctor. That’s why I helped lead the fight in North Carolina to stop Republican attacks on the right to choose, and why I’ll fight hard to protect that right in Congress.” (source)
Ohio’s 1st District: Greg Landsman (Democrat) defeated Steve Chabot (Republican)
Landsman’s stance on repro rights: “Greg will oppose any efforts to undermine the privacy between a woman and her doctor, and he supports the codification of the rights afforded by Roe v. Wade on the federal level.” (source)
Other important victories: Governor seat remains blue in these key states: | https://www.democratsabroad.org/wc_pro_choice_victories_nov2022_elections |
Avinash thanks God after bee attack
GOVERNMENT Senator Avinash Singh is in good spirits by his own admission and thanking God for sparing his life after he was stung by approximately 200 bees while working in his Chaguanas garden on Saturday.
“My condition remains the same but the doctors advise another 24 hours monitoring and treatment as my breathing has not gone back to normalcy as yet,” he wrote in a text message yesterday.
Singh, 28, the parliamentary secretary in the Ministry of Agriculture, Land and Fisheries, recounted the attack in a vivid Facebook post on Sunday, saying the stings had left him virtually unresponsive.
He said around 1pm on Sunday he was helping his father in the garden when he accidentally drove over a beehive with a tractor.
“Upon realising the danger I was in, I immediately left the tractor and ran but was attacked by close to 200 bees, I ran towards a river for safety and cried for help while I felt each pierce of stings entering my face, head and body,” he wrote.
“I collapsed short of the river as my body became unresponsive and being further attacked on the ground, determined to go in the river, I got up and leaped in the river where 90 per cent of my body was submerged in water, mud and grass, only my face was exposed and the more I cried for help the more the bees stung me.”
He heard a voice telling him people were calling an ambulance and going for help, because they could not come to him as they had also been stung by the bees circling above Singh.
By that time, his body was “totally unresponsive, speech impaired, body in shock and in danger of drowning.” His father and nearby farmers bodily removed him from the river, put him in the tray of his van and drove towards a pick-up point for the ambulance.
Luckily, said Singh, they stopped by a village doctor who, upon learning of the emergency, immediately left his patients and gave him an injection for allergic reactions. The ambulance then took him to the Chaguanas Health Centre, where medical staff stabilised him.
“I only realized how many stings I got as the medical team began tweezing out the small needle-like stings which (were) attached to sacs of venom from the bees. Stabilising became challenging as there were so many stings and I began to vomit continuously what appeared to be a yellow liquid with a strong chemical smell.”
He was subsequently taken to Mt Hope Hospital, given additional treatment and warded.
He said he was feeling much better and “thanking God for life” but was still receiving oxygen support to help him breathe.
Singh thanked everyone who had helped him.
TT Beekeepers Association public relations officer Vearna Gloster warned farmers to take precautionary measures while working in bushy areas.
“Whenever any farmer is ploughing their garden and there is a lot of bush and you wouldn’t observe a beehive there, it is precautionary that farmers who are using tractors to make sure their tractors are enclosed, because this can happen to anyone,” she said adding the windows should remain closed at all times.
“Once you interfere with a beehive in a rough manner, this will occur because you upset the nest, they will become defensive. They will move to protect their queen and their colony...they only become defensive when they become upset or when someone affects their nest.
She said the reaction would be the same whether they were Africanised bees or regular honey bees.
Asked what someone should do if they were attacked, she said a good defence was to get out of the area as fast as possible or jump into a large body of water, as Singh had done.
“Try to get out of the area as fast as possible, and after they have calmed down you can go back with smoke or call the Abatement Unit of the Ministry of Agriculture,” she said.
She said such attacks were not regular occurrences and noted that bees were important assets in the pollination of crops. | https://newsday.co.tt/2018/03/09/avinash-thanks-god-after-bee-attack/ |
We accept bookings only from minimum 3 nights. In case of booking of the complete apartment-house for 6-8 guests, we can accept shorter period bookings too. At this facility we accept children over 3 years.
High season
Easter, Whit Monday, 20th of August, Christmas, New Year’s Eve. For the high season periods we charge special rates, our general rates are not valid during the high season.
Booking
Via our contact form under the contact menu. Until our written booking confirmation we consider every requests only as inquiries.
Payment
Payment upon arrival. Cash only. (EUR or HUF) The nearest ATM (OTP Bank): 9737 Bük, Kossuth u. 1.
Cancellation policy
If cancelled up to 7 days before date of arrival, no fee will be charged. If cancelled later or in case of no-show, 100% of the first two nights will be charged. | http://www.marinabuk.hu/en/booking/ |
STANDARD TERMS & CONDITIONS
- Any additional flying in the area on client request, over-and-above quoted times, will be charged at the applicable hourly rate, and to be settled in full before leaving from Grand Central Airport after the flight.
- A 100% deposit is payable to secure the booking at least 48 hours prior to departure.
- Prices based on specific helicopter availability for the flight.
- We retain the right to charge a cancellation fee (50% of the quoted price) should the charter be cancelled at short notice (24 hours before the intended flight) or a no-show situation arises.
- A written order is required to cancel a charter.
- Departure delays due to bad weather or air traffic are unfortunately out of our control.
- All flights are subject to availability of aircraft and the relevant flight clearances being obtained.
- Due to the risk incurred if an aircraft is overloaded, luggage is limited to one bag per passenger not exceeding 10kg's and a maximum combined weight limit for passengers, that is helicopter type specific , will be applicable. | http://www.helicoptercharters.co.za/terms-conditions/ |
The official currency is the Kyat (MMK) but US dollars are accepted everywhere. Current notes are: ks 1000, 5000, 500, 200, 100, 50, 20, 10 and 5 bank notes.
At the time of writing the exchange rate is US$ 1 = 988 Kyat and 1 EURO = 1327 Kyat (December 2013, please note that this can change significantly on a daily basis).
It is now possible to find money changer from official private banks in every airport and in city centre. They usually offer better rates than the black market and they accept US Dollars as well as Euros (maximum amount: 2000USD). | http://asiaholidays.info/en/prepare-your-travel/myanmar/currency |
Red Nose Day Mad Hatter’s Tea Party
In aid of Red Nose Day on 24th March, Ajuda hosted a Mad Hatter’s Tea Party to raise money for the very important charity. The money that is raised on Red Nose Day funds more than 2,000 projects throughout the UK, addressing issues such as homelessness, mental health, dementia and vulnerable young people. The money also helps 11.7 million people across Africa, tackling issues such as immunisation, malaria, education, maternal health and much more. Since launching in 1985, Comic Relief has raised over £1 billion to help drive positive change using the power of entertainment.
After weeks of deliberating in the office about what we could do to raise money for this great cause, Managing Director Dawn came up with the idea of the Mad Hatter’s Tea Party (we’re all mad here!).
Considering the fact that we are only a small office, we are incredibly proud that between us, we fundraised £50. Every member of staff got involved by dressing up and bringing snacks to work and a fantastic afternoon was had by all. The total amount of money raised by Comic Relief this year is a staggering £73,026,234 so far.
Here are some pictures of the day: | https://www.ajuda.org.uk/red-nose-day-mad-hatters-tea-party/ |
It is that time of the year again! School is starting back up for children across the country. Sending a typically developing child off to school can be challenging enough, but sending a child with Rett syndrome can be downright scary. I know firsthand because my daughter, Jilly, returned to school this week. She is now 18 years old, so although neither the diagnosis of Rett syndrome nor starting a new school year is new, we still have the myriad of emotions that come along with this wonderful yet nerve wracking experience.
I have had the chance to get to know many Rett families over the years, and when I talk with them about how to start the school year off right, we almost always come back to one thing: communication. No matter what age your child is, it’s important to let teachers, staff, and fellow students know what to expect when spending time with them and, most importantly, how to treat them with dignity and respect despite their differences. I suggest writing everything in a well organized letter that, although jam-packed with information, is simple and easy to read.
Here are several pieces of information that I suggest you include in a letter to anyone who will be working with your child:
- Greetings and wishes for a great school year ahead full of creativity, motivation, and excellent care
- A description of your child, for example using words such as happy, bright-eyed, smart, and eager to learn
- An overview of what Rett syndrome is
- A list of your child’s favorite activities
- Techniques to soothe your child if she’s not happy
- Descriptions of your child’s family
- A vision for your child’s school year, for example: People working with your child will give her time to respond and engage, will be respectful by talking TO her and not about her, and will “presume competence” when working with her.
- Concerns that we have as parents, such as wanting peer acceptance and respect, making sure your child always has a clean face with no drool or food on her, and making sure she has no soiled clothing and no mention of diapers
- Daily feedback that we’d like shared with us, including a summary of therapies, academics and projects, peer interaction, time in the stander, and bathroom info.
A friend and fellow Rett mom, Jesse, whose daughter, Ruby, is 12 and is entering sixth grade, says she always includes a ton of pictures of Ruby with her family to help normalize and humanize what must seem to be an overwhelming packet of school paperwork on her little warrior. Jesse creates a “Ruby Manual” each year that shows that her daughter is like every other typical child her age, and that she hears and understands everything but cannot always respond. She emphasizes that the golden rule applies here as it would for anyone else: Please treat her as you would want to be treated if you were her.
Another friend and fellow Rett mom, Megan, whose daughter, Emma, is 5 and is entering kindergarten, writes a letter that goes to her daughter’s classmates. However old your child is, it can be just as important (if not more so) to explain things to their peers as it is to communicate with their teachers. Here are some ideas for communicating with classmates:
- Write from a first-person perspective so your child’s personality can shine through as they introduce themselves to peers.
- Explain how Rett syndrome affects ability, e.g. “I have Rett syndrome and it affects my ability to walk and talk.”
- Note that your child can still communicate and describe how, e.g. “When I get excited I can make happy, spontaneous noises, and if I’m sad or confused I might close my eyes and shake my head back and forth.”
- List some things that your child enjoys, such as being read to or listening to music.
- Explain that just like it’s important for the classmates themselves to spend time with friends, it’s also important for your child, and that she is hoping to make some great friends in school this year.
- Include your contact information so that classmates can share it with their parents if anyone has any questions.
I think one thing all Rett parents and caregivers want everyone to know is that our loved ones are worth the effort to get to know. They are full of personality, love, curiosity, and an eagerness to connect with friends and learn in the school environment. | https://reverserett.org/news/articles/communication-tips-for-the-new-school-year/ |
What is a project?
Create a new project
Update a project
Delete a project
Connect a dataset
Export data
Import Data
Dataset
What is a dataset?
Create a new dataset
Update a dataset
Connect with AWS S3
Delete a dataset
Manage images
Connect to a project
Use the review mode
Label configuration
What is a label configuration?
Configuration entry
Entry types
Creating an entry
Editing an entry
Duplicating an entry
Label mode
What is the label mode?
Entry-list
Value-list
Task control
Toolbar
Workspace
AI-assisted labeling
Video labeling
Tasks
What is a task?
Process a task
Manage Tasks
AI-Assistant
AI-assisted pre-labeling
Object Classes
API Token
API
Manage API Token
Account-Management
Account Settings
Organisation-Management
Python API
Getting Started
Projects
Labeled data
Datasets
Images
Label configuration
Uploading COCO
Changelog
Powered By
GitBook
What is a project?
Let me introduce you: a project
Description:
A project holds a label configuration and is connected to one or multiple datasets that need to be labeled with the given label configuration.
In order to label images, a label configuration has to be defined and the images must be assigned to the project. This assignment is done within the project. There's a 1:1 relation between a project and its label configuration. However, many images can be assigned to the project through the connected
datasets
.
The combination of the label configuration and one image is managed as a task. These
tasks
are managed within the project and can be managed by the administrator.
Related documents:
Create a new project
Update a project
Delete a project
Connect a dataset
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Getting Started
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Description:
Related documents: | https://docs.datagym.ai/documentation/project/what-is-a-project |
Implementing Best Practices: Water Safety Plans7 December 2018
According to the World Health Organisation (WHO) diarrhoeal diseases which are largely derived from poor water sanitation account for 2.4 million deaths per year (WHO 1999). Implementing effective water safety plans and conducting thorough risk assessments is key to ensuring protection against waterborne diseases.
What Is A Water Safety Plan (WSP)?
A Water Safety Plan (WSP) is a comprehensive risk assessment and management plan that encompasses all steps in water supply, from source to consumption, in order to ensure the safety of potable water.
The Water Safety Plans (WSP) application was first formally adopted in 2000 during the revision of WHO Guidelines for Drinking Water Quality. The risk management practices used in the WSP was based on many aspects of HACCP (Hazard Analysis and Critical Control Points) used in food hygiene and manufacturing.
WSP and risk assessments should comprise of:
- A System Assessment - to determine if the water quality throughout the system meets health-based targets
- Control Measures - identifying the risks and implementing effective risk assessments and operational controls to control the risks and monitor system performance
- Effective Management - implementing plans that describe the required actions that need to be taken for both normal operations and incidents along with required upgrades, testing, and monitoring. Part of the management is to cohesively communicate the plan to relevant stakeholders.
- Monitoring - a system of independent surveillance that verifies the above bullet points are operating correctly
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According to WHO Water Safety Plans, the WSP objective is to ensure that drinking water is safe. This can be achieved through good water supply practice which encompasses:
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When considering hazard identification when conducting risk assessments to form the WSP it is important to consider these factors:
- Variation In Weather
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Hazardous events can lead to contamination and can risk the health of humans and the environment. Knowing where hazards can occur and implementing effective risk control measures are critical steps in protecting teams using the WSP.
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Want to know more about Water Safety Plans, Certificates of Compliance and Audits? Discover more here: Lucion Marine: Water Safety Plans. | https://lucionservices.com/insights/implementing-best-practices |
Feral pigeons are derived from domestic pigeons that have returned to the wild. The domestic pigeon was originally bred from the wild Rock dove, which naturally inhabits sea-cliffs and mountains. Feral pigeons find the ledges of buildings to be a substitute for sea cliffs, and have become adapted to urban life and are abundant in towns and cities throughout much of the world.
All pigeons are one species (columba livia). Pigeons breed when the food supply is good, which in cities can be any time of the year. Laying of eggs can take place up to six times per year. Pigeons mate for life, and are often found in pairs during the breeding season, but usually the pigeons are gregarious preferring to exist in flocks of from 50 to 500 birds.
Nests are rudimentary, similar to other ground or cliff nesting birds. Abandoned buildings are favorite nesting areas. Mass nesting is common as pigeons are a community flocking bird, often dozens of birds will share a building. Loose tiles and broken windows give pigeons access; they are good at spotting new access points, for example following property damage caused by strong winds. Pigeons are particularly fond of roof spaces. These often contain water tanks. Any water tank or cistern on a roof must therefore be secured and sealed off to keep the pigeons out of them. On undamaged property, the gutters, window air conditioners and empty air conditioner containers, chimney pots and external ledges are used as nesting sites. Many building owners try to limit roosting by using bird control spikes and netting to cover ledges and potential nesting places on buildings. This has little effect on the size of the pigeon population, but it can reduce the accumulation of droppings on and around a particular building location.
Feral pigeons can be seen eating grass seeds and berries in parks and gardens in the spring, but there are plentiful sources throughout the year from scavenging (e.g., dropped fast-food cartons) and they will also take insects and spiders. Further food is also usually available from the disposing of stale bread in parks by restaurants and supermarkets, from tourists buying and distributing birdseed, etc. Pigeons tend to congregate in large, often thick flocks when going for discarded food, and many have been observed flying skilfully around trees, buildings, telephone poles and cables, and even moving traffic just to reach it.
Many city squares are famous for their large pigeon populations, for example, the Piazza San Marco in Venice, and Trafalgar Square in London. For many years, the pigeons in Trafalgar Square were considered a tourist attraction, with street vendors selling packets of seeds for visitors to feed the pigeons. The feeding of the Trafalgar Square pigeons was controversially banned in 2003 by London mayor Ken Livingstone. However, activist groups flouted the ban, feeding the pigeons from an area south of Nelson’s Column in which the ban does not apply. They eventually agreed to feed the pigeons only once a day, at 7:30 a.m.
Feral pigeons are often considered a pest or even vermin, owing to concerns that they spread disease (however, it is rare that a pigeon will transmit a disease to humans due to their immune system). While pest exterminators use poison, hawks and nets have also been employed at ground level to control urban pigeon populations, though this generally achieves only a limited, temporary effect. Long-term reduction of feral pigeon populations can be achieved by restricting food supply, which in turn involves legislation and litter (garbage) control. Some cities have deliberately established favorable nesting places for pigeons – nesting places that can easily be reached by city workers who regularly remove eggs, thereby limiting their reproductive success. In addition, pigeon populations may be reduced by bird control systems that successfully reduce nesting sites.
Peregrine Falcons which are also originally cliff dwellers have also adapted to the big cities, living on the window ledges of skyscrapers and often feeding exclusively on pigeons. Some cities actively encourage this through falcon breeding programs. Larger birds of prey occasionally take advantage of this population as well. In New York City, the abundance of pigeons (and other small animals) has created such a conducive environment for predators that the Red-Tailed Hawk has begun to return in very small numbers, the most famous of which is Pale Male, who lives near Central Park (bird watchers have followed him since he was hatched in 1990).
The use of poisons has been proven to be fairly ineffective, however, as pigeons can breed very quickly — up to six times a year — and their numbers are determined by how much food is available; that is, they breed more often when more food is provided to them. An additional problem with poisoning is that it also kills pigeon predators. Due to this, in cities with Peregrine Falcon programs it is typically illegal to poison pigeons. A more effective tactic to reduce the number of feral pigeons is deprivation. Cities around the world have discovered that not feeding their local birds results in a steady population decrease in only a few years. Pigeons, however, will still pick at garbage bags containing discarded food or at leftovers carelessly dropped on the ground. Feeding of pigeons is banned in parts of Venice, Italy.
In 1998, in response to conservation groups and the public interest, the National Wildlife Research Center, a USDA laboratory, started work on nicarbazin, a promising compound for avian contraception. Avian contraception has the support of a range of animal welfare groups including the Humane Society of the United States, the ASPCA, and PETA. USDA continues to develop wildlife contraceptives for deer, birds, and small mammals. The new field of wildlife contraceptives is developing rapidly and promises humane management of animal populations. | https://thedailyomnivore.net/2012/02/18/street-pigeon/ |
---
abstract: 'In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $ \mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2$.'
address: 'Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal'
author:
- Ophélie Rouby
bibliography:
- 'biblio.bib'
nocite: '[@*]'
title: 'Berezin-Toeplitz quantization and complex Weyl quantization of the torus $ \mathbb{T}^2$.'
---
[^1]
Introduction {#introduction .unnumbered}
============
The object of this paper is to construct a new semi-classical quantization of the torus $ \mathbb{T}^2$ by adapting Sjöstrand’s complex Weyl quantization of $ \mathbb{R}^2$ and to give the correspondence between this quantization and the well-known Berezin-Toeplitz quantization of $ \mathbb{T}^2$. When the phase space is $ \mathbb{R}^{2n}$, the pseudo-differential Weyl quantization allows us to relate a classical system to a quantum one through the symbol map; thus pseudo-differential operators have become an important tool in quantum mechanics. On the mathematical side, these operators have been introduced in the mid-sixties by André Unterberger and Juliane Bokobza [@MR0176360] and in parallel by Joseph Kohn and Louis Nirenberg [@MR0176362] and have been investigated by Lars Hörmander [@MR0180740; @MR0233064; @MR0383152]. They allow to study physical systems in positions and momenta. On the other hand, Berezin-Toeplitz operators have been introduced by Feliks Berezin [@MR0411452] and investigated by Louis Boutet de Monvel and Victor Guillemin [@MR620794] as a generalization of Toeplitz matrices. The study of these operators has been motivated by the fact that pseudo-differential operators take into account only phases spaces that can be written as cotangent spaces, whereas in mechanics, there are physical observables like spin that naturally lives on other types of phases spaces, like compact Kähler manifolds, which can be quantized in the Berezin-Toeplitz way. In fact, it was realized recently that the Berezin-Toeplitz quantization applies to even more general symplectic manifolds, and thus has become a tool of choice for applications of symplectic geometry and topology, see [@2016arXiv160905395C].\
\
In this paper, we give a relation between the Berezin-Toeplitz quantization of the torus, studied for instance by David Borthwick and Alejandro Uribe in [@MR2014161] and the complex Weyl quantization of the torus, which we introduce as a variation of Sjöstrand’s quantization of $ \mathbb{R}^2$. The complex Weyl quantization of $ \mathbb{R}^2$ has been investigated by Johannes Sj[ö]{}strand in [@SJ], then by Anders Melin and Johannes Sj[ö]{}strand in [@MR1957486; @MR2003421], also by Michael Hitrik and Johannes Sj[ö]{}strand in [@MR2036816] and in their mini-courses [@HJ] and by Michael Hitrik, Johannes Sj[ö]{}strand and San Vũ Ngc in [@MR2288739]. This quantization of the real plane $ \mathbb{R}^2$ allows to study pseudo-differential operators with complex symbols, and therefore is particularly useful for problems involving non self-adjoint operators or quantum resonances. It is defined by a contour integral over an $IR$-manifold ($I$-Lagrangian and $R$-symplectic) which plays the role of the phase space. Here, we define an analogue of this notion in the torus case.\
If we consider the complex plane as a phase space, there exists a correspondence between the complex Weyl and the Berezin-Toeplitz quantizations (this correspondence uses a variant of Bargmann’s transform and can be found, for instance, in the book [@MR2952218 chapter 13] of Maciej Zworski); using this result, we are able to obtain Bohr-Sommerfeld quantization conditions for non-selfadjoint perturbations of self-adjoint Berezin-Toeplitz operators of the complex plane $ \mathbb{C}$ by first proving the result in the case of pseudo-differential operators (see [@lapin]). Therefore, we expect that this new complex quantization of $ \mathbb{T}^2$, together with its relationship to the Berezin-Toeplitz quantization, will be crucial in obtaining precise eigenvalue asymptotics of non-selfadjoint Berezin-Toeplitz operators on the torus.\
*Structure of the paper:*
- in Section \[section\_resultat\], we state our result;
- in Section \[section\_demo\], we give the proof of our result which is divided into three parts, the first one consists in recalling the Berezin-Toeplitz quantization of the torus, the second one in introducing the complex Weyl quantization of the torus and the last one in relating these two quantizations.
$ $
**Acknowledgements.** The author would like to thank both San Vũ Ngc and Laurent Charles for their support and guidance. Funding was provided by the Université de Rennes 1 and the Centre Henri Lebesgue.
Result {#section_resultat}
======
Context {#subsection_context}
-------
In this section, we recall the definition of the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2$ (see for example [@MR3349834]) and we give a definition of the complex Weyl quantization of a symbol on the torus. Let $ 0 < \hbar \leq 1$ be the semi-classical parameter. By convention the Weyl quantization involves the semi-classical parameter $ \hbar$, contrary to the Berezin-Toeplitz quantization which involves the inverse of this parameter, denoted by $k$. In the whole paper, we will use these two parameters.\
`Notation:` let $k$ be an integer greater than $1$. Let $u$ and $v$ be complex numbers of modulus $1$.
- If $z \in \mathbb{C}$, we denote by $z = (p, q) \in \mathbb{R}^2$ or $z = p + iq$ via the identification of $ \mathbb{C}$ with $ \mathbb{R}^2$.
- $ \mathbb{T}^2$ denotes the torus $ (\mathbb{R}/ 2 \pi \mathbb{Z}) \times ( \mathbb{R}/ \mathbb{Z})$.
- $ \mathcal{G}_k$ is the space of measurable functions $g$ such that: $$\int_0^{2 \pi} \! \! \! \int_0^1 \left| g(p, q) \right|^2 e^{-k q^2} dp dq < + \infty,$$ which are invariant under the action of the Heisenberg group (for more details, see Subsection \[subsection\_quantif\_toeplitz\]), *i.e.* for all $(p, q) \in \mathbb{R}^2$, we have: $$g(p+2 \pi, q) = u^k g(p, q)\quad \text{and} \quad g(p, q+1) = v^k e^{-i(p+iq)k+k/2} g(p, q).$$
- $ \mathcal{H}_k$ is the space of holomorphic functions in $ \mathcal{G}_k$, *i.e.*: $$\mathcal{H}_k = \left\lbrace g \in {\mathrm{Hol}}( \mathbb{C}); \quad g(p+2 \pi, q) = u^k g(p, q),\quad g(p, q+1) = v^k e^{-i(p+iq)k+k/2} g(p, q) \right\rbrace.$$
- $\Pi_k$ is the orthogonal projection of the space $ \mathcal{G}_k$ (equipped with the weighted $L^2$-scalar product on $[0, 2 \pi] \times [0, 1]$) on the space $\mathcal{H}_k$.
$ $
- The spaces $ \mathcal{G}_k$ and $ \mathcal{H}_k$ depend on the complex numbers $u$ and $v$.
- In [@MR2014161], they consider the torus $ \mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$ and they choose an other quantization which leads to an other space of holomorphic functions, also called $ \mathcal{H}_k$, defined as follows: $$\mathcal{H}_k = \left\lbrace g \in {\mathrm{Hol}}( \mathbb{C}); \quad \forall (m, n) \in \mathbb{Z}^2, g(z+m+in) = (-1)^{kmn} e^{k \pi ( z(m-in) + (1/2)(m^2 + n^2))} g(z) \right\rbrace.$$
\[defi\_asymp\_expan\] Let $f_k \in \mathcal{C}^{ \infty}( \mathbb{R}^2)$. We say that $f_k$ admits an asymptotic expansion in powers of $1/k$ for the $ \mathcal{C}^{ \infty}$-topology of the following form: $$f_k(x,y) \sim \sum_{l \geq 0} k^{-l} f_l(x,y) ,$$ if:
1. $ \forall l \in \mathbb{N}$, $f_l \in \mathcal{C}^{ \infty}( \mathbb{R}^2)$;
2. $ \forall L \in \mathbb{N}^*$, $ \forall (x, y) \in \mathbb{R}^2$, $ \exists C > 0$ such that: $$\left| f_k(x,y) - \sum_{l=0}^{L-1} k^{-l} f_l(x,y) \right| \leq C k^{-L} \quad \text{for large enough $k$}.$$
We denote by $ \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ the space of such functions.
\[defi\_quantif\_BT\_tore\] Let $f_k \in \mathcal{C}^{ \infty}_k(\mathbb{R}^2)$ be a function such that, for $(x,y) \in \mathbb{R}^2$, we have: $$f_k(x+ 2 \pi, y) = f_k(x, y) = f_k(x, y+1 ) .$$ Define the Berezin-Toeplitz quantization of the function $f_k$ by the sequence of operators $ T_{f_k} := (T_k)_{k \geq 1}$ where, for $k \geq 1$, the operator $T_k$ is given by: $$T_k = \Pi_k M_{f_k} \Pi_k : \mathcal{H}_k \longrightarrow \mathcal{H}_k,$$ where $M_{f_k} : \mathcal{G}_k \longrightarrow \mathcal{G}_k$ is the multiplication operator by the function $f_k$.\
We call $f_k$ the symbol of the Berezin-Toeplitz operator $T_{f_k}$.
Now, we define the complex Weyl quantization of a symbol on the torus. We will explain in details in Subsection \[subsection\_complex\_weyl\_quantization\] why we consider such a notion. First, we introduce some notations.\
`Notation:` let $ \Phi_1$ be the strictly subharmonic quadratic form defined by the following formula for $z \in \mathbb{C}$: $$\Phi_1(z) = \dfrac{1}{2} \Im(z)^2 .$$
- $ \Lambda_{ \Phi_1}$ denotes the following set: $$\Lambda_{ \Phi_1} = \left\lbrace \left(z, \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z}(z) \right); z \in \mathbb{C} \right\rbrace = \lbrace (z, - \Im(z)); z \in \mathbb{C} \rbrace \simeq \mathbb{C}.$$
- $L(dz)$ denotes the Lebesgue measure on $ \mathbb{C}$, *i.e.* $L(dz)= \dfrac{i}{2} dz \wedge d \overline{z}$.
- $L^2_{\hbar}( \mathbb{C}, \Phi_1) := L^2( \mathbb{C}, e^{-2 \Phi_1(z)/ \hbar} L( d z))$ is the set of measurable functions $f$ such that: $$\int_{ \mathbb{C}} |f(z)|^2 e^{-2 \Phi_1(z)/ \hbar} L(dz) < + \infty.$$
- $H_{ \hbar}( \mathbb{C}, \Phi_1) := {\mathrm{Hol}}( \mathbb{C}) \cap L^2_{\hbar}( \mathbb{C}, \Phi_1)$ is the set of holomorphic functions in the space $L^2_{ \hbar}( \mathbb{C}, \Phi_1)$.
- $ \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ denotes the set of smooth functions on $ \Lambda_{ \Phi_1}$ admitting an asymptotic expansion in powers of $ \hbar$ for the $ \mathcal{C}^{ \infty}$-topology in the sense of Definition \[defi\_asymp\_expan\] (by replacing $1/k$ by $ \hbar$ and $ \mathbb{R}^2$ by $ \Lambda_{ \Phi_1}$).
There are several definitions of the Bargmann transform. Here we chose the weight function $ \Phi_1 (z) = \dfrac{1}{2} \Im(z)^2$ instead of $ |z|^2$ because it is well-adapted to the analysis of the torus.
\[defi\_quantif\_complex\_tore\] Let $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ be a function such that, for $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_{ \hbar}(z + 2 \pi, w) = b_{ \hbar}(z, w) = b_{ \hbar}(z+i, w-1).$$ Define the complex Weyl quantization of the function $b_{ \hbar}$, denoted by $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$, by the following formula, for $u \in H_{ \hbar}( \mathbb{C}, \Phi_1)$: $${\mathrm{Op}}_{ \Phi_1}^w(b_{ \hbar}) u(z) = \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} b_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right) u(w) dw d \zeta,$$ where the contour integral is the following: $$\Gamma(z) = \left\lbrace (w, \zeta) \in \mathbb{C}^2; \zeta = \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z} \left( \dfrac{z+w}{2} \right) = - \Im \left( \dfrac{z+w}{2} \right) \right\rbrace.$$ We call $b_{ \hbar}$ the symbol of the pseudo-differential operator $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$.
We will show that for $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ satisfying the hypotheses of Definition \[defi\_quantif\_complex\_tore\], the complex Weyl quantization defines an operator $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$ which sends the space of holomorphic functions $ \mathcal{H}_k$ on itself (see Proposition \[prop\_action\_quantif\_weyl\_complexe\_sur\_SS’(C)\]). Therefore, the Berezin-Toeplitz and the complex Weyl quantizations give rise to operators acting on the space of holomorphic functions $ \mathcal{H}_k$.
Main result
-----------
\[theoA\] Let $ f_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for $(x, y) \in \mathbb{R}^2$, we have: $$f_k(x + 2 \pi, y ) = f_k(x, y) = f_k(x, y+1) .$$ Let $T_{f_k} = ( T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of symbol $ f_k$. Then, for $k \geq 1$, we have: $$T_k = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) + \mathcal{O}(k^{- \infty}) \quad \text{on $ \mathcal{H}_k$},$$ where $b_{\hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{\Phi_1})$ is given by the following formula, for $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$: $$b_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) (f_k(z)).$$ This formula means that $b_{ \hbar}$ is the solution at time $1$ of the following ordinary differential equation: $$\begin{cases}
\partial_t b_{ \hbar}(t, z) = \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \left( b_{ \hbar}(t, z) \right), \\
b_{ \hbar}(0,z) = f_k(z).
\end{cases}$$ Besides, $b_{ \hbar}$ satisfies the following periodicity conditions, for $(z,w) \in \Lambda_{ \Phi_1}$: $$b_{ \hbar}(z+ 2 \pi, w) = b_{ \hbar}(z, w) = b_{ \hbar}(z+i, w-1).$$
This result is analogous to Proposition \[prop\_Toeplitz=pseudo\_H(Phi1)\] (see for example [@MR2952218 Chapter 13]) which relates the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane. The important difference here is that the phase space is the torus.
As a corollary of this result, we can establish a connection between the Berezin-Toeplitz and the *classical* Weyl quantizations of the torus (see Corollary \[coro\]).
Proof {#section_demo}
=====
The structure of the proof is organized as follows:
- in Subsection \[subsection\_quantif\_toeplitz\], we recall the Berezin-Toeplitz quantization of the torus;
- in Subsection \[subsection\_complex\_weyl\_quantization\], we introduce the complex Weyl quantization of the torus;
- in Subsection \[subsection\_links\_between\_quantizations\], we relate the Berezin-Toeplitz quantization of the torus to the complex Weyl quantization of the torus.
Berezin-Toeplitz quantization of the torus $ \mathbb{T}^2$ {#subsection_quantif_toeplitz}
----------------------------------------------------------
In this paragraph, we recall the geometric quantization of the torus (see for example the article [@MR3349834] of Laurent Charles and Julien Marché).\
Consider the real plane $ \mathbb{R}^2$ endowed with the euclidean metric, its canonical complex structure and with the symplectic form $ \omega = dp \wedge dq$. Let $L_{ \mathbb{R}^2} = \mathbb{R}^2 \times \mathbb{C}$ be the trivial complex line bundle endowed with the constant metric and the connection $ \nabla = d + \dfrac{1}{i} \alpha$ where $ \alpha$ is the $1$-form given by: $$\alpha = \dfrac{1}{2} \left( p dq - q dp \right).$$ The holomorphic sections of $L_{ \mathbb{R}^2}$ are the sections $f$ satisfying the following condition: $$\nabla_{ \overline{z}} f = \dfrac{\partial}{\partial \overline{z}} f + \dfrac{1}{4} z f = 0 .$$
We are interested in the holomorphic sections of the torus $ \mathbb{T}^2 = ( \mathbb{R}/ 2 \pi \mathbb{Z}) \times ( \mathbb{R}/ \mathbb{Z})$. Let $x = 2 \pi \dfrac{\partial}{\partial p}$, *i.e.* if we denote by $t_{x}$ the translation of vector $x$, it is defined by the following formula: $$\begin{aligned}
t_{x} : \mathbb{R}^2 & \longrightarrow \mathbb{R}^2 \\
(p,q) & \longrightarrow (p+2 \pi, q).\end{aligned}$$ And let $y = \dfrac{\partial}{\partial q}$ which corresponds to the translation $t_y$ given by: $$\begin{aligned}
t_y: \mathbb{R}^2 & \longrightarrow \mathbb{R}^2 \\
(p, q) & \longmapsto (p, q+1).\end{aligned}$$ Note that the $ \omega$ volume of the fundamental domain of the lattice is $ 2 \pi$.\
Let $k \geq 1$, the Heisenberg group at level $k$ is $ \mathbb{R}^2 \times U(1)$ with the product: $$(x, u) \cdot (y, v) = \left( y+x, u v e^{(ik/2) \omega(x, y)} \right),$$ for $(x, u), (y, v) \in \mathbb{R}^2 \times U(1)$ (where $U(1)$ denotes the set of complex numbers of modulus one). This formula defines an action of the Heisenberg group on the bundle $L_{ \mathbb{T}^2}^{ \otimes k}$ endowed with the product measure. We identify the space of square integrable sections of $L_{ \mathbb{T}^2}^{ \otimes k}$ which are invariant under the action of the Heisenberg group with the space $ \mathcal{G}_k$ (defined in Subsection \[subsection\_context\]). In fact, if $ \psi$ denotes such a section, we associate to it a function $g \in \mathcal{G}_k$ using the following application: $$\begin{aligned}
L^2( \mathbb{T}^2, L^{ \otimes k}_{ \mathbb{T}^2}) & \longrightarrow \mathcal{G}_k \\
\psi & \longmapsto g( \tilde{x}),\end{aligned}$$ where $ \tilde{x} \in \mathbb{R}^2$ and $ \tilde{x} = x_0 + (n_1,n_2)$ with $ x_0 \in [0, 2 \pi] \times [0,1]$, $(n_1, n_2) \in \mathbb{Z}^2$ and where: $$( \tilde{x}, g( \tilde{x})) = ((n_1, n_2),1) \cdot (x_0, \psi(x_0)) .$$ Similarly, we identify the space of holomorphic sections of $L_{ \mathbb{T}^2}^{ \otimes k}$ with the following Hilbert space: $$\mathcal{H}_k = \left\lbrace g \in {\mathrm{Hol}}( \mathbb{C}); \quad g(p+2 \pi, q) = u^k g(p, q), \quad g(p, q+1) = v^k e^{-i(p+iq)k+k/2} g(p, q) \right\rbrace,$$ endowed with the $L^2$-weighted scalar product on $[0, 2 \pi] \times [0,1]$. The complex numbers $u$ and $v$ are called Floquet indices. The Hilbert space $ \mathcal{H}_k$ admits an orthogonal basis, given for $l \in \lbrace 0, \ldots, k-1 \rbrace$, by the functions $e_l$ which are defined, for $z \in \mathbb{C}$, as follows: $$\label{eq_base_el}
e_l(z) = u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left( v^{-k} e^{-l-kj/2} u^{ik/(2 \pi)} \right)^j e^{i(l+jk)z} .$$
Complex Weyl quantization of the torus $ \mathbb{T}^2$ {#subsection_complex_weyl_quantization}
------------------------------------------------------
In this paragraph, we introduce the notion of complex Weyl quantization of the torus which, to our knowledge, is new. To do so, we follow these three steps:
1. we recall the definition of the classical Weyl quantization of the torus;
2. we recall the definition of the semi-classical Bargmann transform and we look at some of its properties;
3. we introduce the complex Weyl quantization as the range of the classical Weyl quantization by the Bargmann transform.
### Classical Weyl quantization of the torus
The classical Weyl quantization of a symbol on the torus has been studied, for example, by Monique Combescure and Didier Robert in the book [@MR2952171 Chapter 6]. We need to introduce the following notation.\
`Notation:`
- $ \mathcal{S}( \mathbb{R})$ denotes the Schwartz space, *i.e.*: $$\mathcal{S}( \mathbb{R}) = \left\lbrace \phi \in \mathcal{C}^{ \infty}( \mathbb{R}); \|\phi \|_{\alpha, \beta} := \sup_{ x \in \mathbb{R}} | x^{ \alpha} \partial^{ \beta}_x \phi(x) | < + \infty, \forall \alpha, \beta \in \mathbb{N} \right\rbrace;$$
- for $ \phi \in \mathcal{S}( \mathbb{R})$, $ \mathcal{F}_{ \hbar} \phi$ denotes the semi-classical Fourier transform of the function $ \phi$ and it is defined by the following equality: $$\mathcal{F}_{ \hbar} \phi(\xi) = \int_{ \mathbb{R}} e^{-(i/ \hbar) x \xi} \phi(x) dx$$ this transform is an isomorphism of the Schwartz space and its inverse is given by: $$\mathcal{F}_{ \hbar}^{-1} \phi(x) = \dfrac{1}{2 \pi \hbar} \int_{ \mathbb{R}} e^{(i/ \hbar) x \xi} \phi( \xi) d \xi ;$$
- $ \mathcal{S}'( \mathbb{R})$ denotes the space of tempered distributions, it is the dual of the Schwartz space $ \mathcal{S}( \mathbb{R})$, *i.e.* it is the space of continuous linear functionals on $ \mathcal{S}( \mathbb{R})$;
- $ \langle \cdot, \cdot \rangle_{ \mathcal{S}', \mathcal{S}}$ denotes the duality bracket between $ \mathcal{S}'( \mathbb{R})$ and $ \mathcal{S}( \mathbb{R})$;
- for $ \psi \in \mathcal{S}'( \mathbb{R})$, $ \mathcal{F}_{ \hbar} \psi$ denotes the semi-classical Fourier transform of a tempered distribution and it is defined by the following equality, for $ \phi \in \mathcal{S}( \mathbb{R})$: $$\langle \mathcal{F}_{ \hbar} \psi, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = \langle \psi, \mathcal{F}_{ \hbar} \phi \rangle_{ \mathcal{S}', \mathcal{S}} ;$$
- for $a \in \mathbb{R}$, we denote by $\tau_a$ the translation of vector $a$ defined as follows: $$\begin{aligned}
\tau_a : \mathbb{R} & \longrightarrow \mathbb{R} \\
x & \longmapsto x+a,\end{aligned}$$ recall that the translation of a tempered distribution $ \psi \in \mathcal{S}'( \mathbb{R})$ is defined as follows, for $ \phi \in \mathcal{S}( \mathbb{R})$: $$\langle \tau_a \psi, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = \langle \psi, \tau_{-a} \phi \rangle_{ \mathcal{= \mathbb{R} / \mathbb{Z}^2S}', \mathcal{S}}$$ the distribution $ \psi$ is called $a$-periodic if $\tau_a \psi = \psi$, in this case, $ \psi$ can be written as a convergent Fourier series in $ \mathcal{D}'( \mathbb{R})$ (see for example the book of Jean-Michel Bony [@Bony]): $$\psi = \sum_{l \in \mathbb{Z}} \psi_l e^{ilt 2 \pi/a} ,$$ where the sequence $( \psi_l)_{l \in \mathbb{Z}}$ is such that, there exists an integer $N \geq 0$ such that: $$| \psi_l| \leq C (1 + |l|)^N \quad \forall l \in \mathbb{Z}.$$
$ $\
Recall now the definition of the subspace of tempered distributions that corresponds to the natural space on which pseudo-differential operators of the torus act (see [@MR2952171 Chapter 6]). For $k \geq 1$ and for $u, v \in U(1)$, we consider the following space: $$\mathcal{L}_k = \left\lbrace \psi \in \mathcal{S}'( \mathbb{R}); \quad \tau_{2 \pi} \psi = u^k \psi, \quad \tau_1 \mathcal{F}_{ \hbar}( \psi) = v^{-k} \mathcal{F}_{ \hbar}( \psi) \right\rbrace .$$
$ $
- The definition of the space $ \mathcal{L}_k$ involves two complex numbers $u$ and $v$. We will see that they correspond to the Floquet indices seen in the definition of the space $ \mathcal{H}_k$.
- In [@MR2679813], they consider the torus $ \mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ and they choose $u = v= 1$.
This space admits a basis (see for example [@MR2952171 Chapter 6]), given for $l \in \lbrace 0, \ldots, k-1 \rbrace$, by the distributions $ \epsilon_l$ which are defined as follows: $$\label{eq_base_epsilon_l}
\epsilon_l = u^{kt/(2 \pi)} \sum_{j \in \mathbb{Z}} \left( v^{-k} \right)^j e^{i(l+jk)t} .$$ We consider the structure of Hilbert space such that the family $( \epsilon_l)_{l \in \mathbb{Z}}$ is an orthonormal basis of the space $ \mathcal{L}_k$. Recall now two different definitions of the Weyl quantization of a symbol on the torus $ \mathbb{T}^2$. In the whole paper $S( \mathbb{R}^2)$ denotes the following class of symbols on $ \mathbb{R}^2$: $$S( \mathbb{R}^2) = \left\lbrace a \in \mathcal{C}^{ \infty}( \mathbb{R}^2); \forall \alpha \in \mathbb{N}^2, \text{there exists a constant $C_{ \alpha} > 0$ such that:} \, \left| \partial^{ \alpha} a \right| \leq C_{ \alpha} \right\rbrace.$$
\[rema\_symbole\_perio\_dans\_S(R2)\] Let $a_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{ \hbar}(x + 2 \pi, y) = a_{ \hbar}(x, y) = a_{ \hbar}(x, y+1).$$ Then the function $a_{ \hbar}$ belongs to the class of symbols $S( \mathbb{R}^2)$.
\[defi\_quantif\_weyl\_tore\_integrale\] Let $a_{ \hbar } \in \mathcal{C}^{ \infty}_{ \hbar}(\mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{ \hbar}(x+2 \pi, y) = a_{ \hbar}(x, y) = a_{ \hbar}(x, y+1) .$$ Define the Weyl quantization of the symbol $a_{ \hbar}$, denoted by $ {\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x)$, by the following integral formula, for $u \in \mathcal{S}( \mathbb{R})$: $${\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x) u(x) = \dfrac{1}{2 \pi \hbar}
\int_{ \mathbb{R}} \int_{ \mathbb{R}} e^{i(x-y) \xi/ \hbar} a _{ \hbar}\left( \dfrac{x+y}{2}, \xi \right) u(y) dy d \xi .$$ We call $a_{ \hbar}$ the symbol of the pseudo-differential operator $ {\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x)$.
Recall that if $a_{ \hbar} \in S( \mathbb{R}^2)$, then (see for example the book of Maciej Zworski [@MR2952218 Chapter 3]):
1. $ {\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x): \mathcal{S}( \mathbb{R}) \longrightarrow \mathcal{S}( \mathbb{R})$;
2. $ {\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x): \mathcal{S}'( \mathbb{R}) \longrightarrow \mathcal{S}'( \mathbb{R})$;
are continuous linear transformations and the action of $ {\mathrm{Op}}^w(a_{ \hbar})$ on $ \mathcal{S}'( \mathbb{R})$ is defined, for $\psi \in \mathcal{S}'( \mathbb{R})$ and $\phi \in \mathcal{S}( \mathbb{R})$, by: $$\label{equa_action_operateur_pseudo_sur_S'(R)}
\langle {\mathrm{Op}}^w(a_{ \hbar}) \psi, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = \langle \psi, {\mathrm{Op}}^w( \tilde{a}_{ \hbar}) \phi \rangle_{ \mathcal{S}', \mathcal{S}},$$ where, for $(x, y) \in \mathbb{R}^2$, $ \tilde{a}_{ \hbar}(x, y) := a_{ \hbar}(x, - y) \in S(\mathbb{R}^2)$. This property allows to easily prove the following proposition (see [@MR2679813]).
Let $a_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{ \hbar }(x + 2 \pi,y) = a_{ \hbar}(x, y) = a_{ \hbar}(x, y+1) .$$ Then, if $ \hbar = \dfrac{1}{k}$ for $k \geq 1$, we have: $ {\mathrm{Op}}^w(a_{ \hbar})(x, \hbar D_x): \mathcal{L}_k \longrightarrow \mathcal{L}_k$.
Since we consider a symbol $a_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \mathbb{R}^2)$ which is periodic, we can rewrite it as a Fourier series, for all $(x, y) \in \mathbb{R}^2$: $$\label{equation_ecriture_somme_symbole_tore}
a_{\hbar}(x, y) = \sum_{(m, n) \in \mathbb{Z}^2} a_{m, n}^{\hbar} e^{ixn} e^{-i 2 \pi y m}$$ where $\left(a_{m, n}^{\hbar}\right)_{(m, n) \in \mathbb{Z}^2}$ is a sequence of complex coefficients depending on the semi-classical parameter $\hbar$. Recall an other definition of the Weyl quantization of a symbol on the torus, linked to Equation , found in the book of Monique Cobescure and Didier Robert [@MR2952171 Chapter 6]. By convention, this definition uses the parameter $k$, which is the inverse of the semi-classical parameter $ \hbar$. Throughout this text, we will make the abuse of notation of using $a_k$ and $a_{ \hbar}$ for the same object where $\hbar = 1/k$.
\[defi\_quantif\_weyl\_tore\_somme\] Let $a_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_k(x+ 2 \pi, y) = a_k(x, y) = a_k(x, y+1).$$ Define the Weyl quantization of the symbol $a_k$, denoted by $ {\mathrm{Op}}^w_k(a_k)$, by the following formula: $${\mathrm{Op}}^w_k(a_k) = \sum_{(m, n) \in \mathbb{Z}^2} a_{m, n}^k \hat{T} \left( \dfrac{2 \pi m}{k}, \dfrac{n}{k} \right),$$ where the sequence $\left(a_{m,n}^k \right)_{(m,n) \in \mathbb{Z}^2}$ is defined by Equation and where $ \hat{T}(p, q)$ is the Weyl-Heisenberg translation operator by a vector $(p, q) \in \mathbb{R}^2$ defined, for $ \phi \in \mathcal{S}( \mathbb{R})$, by: $$\hat{T}(p, q) \phi(x) = e^{-iqpk/2} e^{ixqk} \phi(x-p).$$
\[prop\_qauntif\_weyl\_Lk\_avec\_somme\] Let $a_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_k(x + 2 \pi, y) = a_k(x, y) = a_k(x, y+1).$$ Then, we have: $ {\mathrm{Op}}^w_k(a_k) : \mathcal{L}_k \longrightarrow \mathcal{L}_k$.
Definition \[defi\_quantif\_weyl\_tore\_integrale\] and Definition \[defi\_quantif\_weyl\_tore\_somme\] coincides in the sense that, if $a_{ \hbar} = a_k \in \mathcal{C}^{ \infty}_{ \hbar}( \mathbb{R}^2)$ is a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{\hbar}(x + 2 \pi, y) = a_{\hbar}(x, y) = a_{\hbar}(x, y+1).$$ Then, $ {\mathrm{Op}}^w(a_{ \hbar}) = {\mathrm{Op}}^w_k(a_k)$ on the space $ \mathcal{L}_k$.
### Bargmann transform
In this paragraph, we recall the definition of the semi-classical Bargmann transform and we study some of its properties. The principal difference with the transform introduced by Valentine Bargmann in the article [@MR0201959] is the weight function that we choose. The semi-classical Bargmann transform has been studied by Anders Melin, Michael Hitrik and Johannes Sjöstrand in [@MR1957486; @MR2003421; @MR2036816] and by the last two authors in the mini-course [@HJ]. Here, we investigate the action of the semi-classical Bargmann transform on the Schwartz space, on the tempered distributions space and on the space $ \mathcal{L}_k$.\
First, we recall the definition of the Bargmann transform and its first properties (see for example the book of Maciej Zworski [@MR2952218 Chapter 13]).
\[defi\_Bargmann\_transform\] Let $ \phi_1$ be the holomorphic quadratic function defined, for $(z, x) \in \mathbb{C} \times \mathbb{C}$, by: $$\phi_1(z,x) = \dfrac{i}{2} (z-x)^2 .$$ The Bargmann transform associated with the function $ \phi_1$ is the operator, denoted by $T_{ \phi_1}$, defined on $ \mathcal{S}( \mathbb{R})$ by: $$T_{ \phi_1} u(z) = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} e^{(i/\hbar) \phi_1(z,x)} u(x) dx = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} e^{-(1/2 \hbar)(z - x)^2} u(x) dx,$$ where: $$\label{formule_constante_c_phi}
c_{ \phi_1} = \dfrac{1}{2^{1/2} \pi^{3/4}} \dfrac{| \det \partial_x \partial_z \phi_1|}{(\det \Im \partial^2_x \phi_1 )^{1/4}} = \dfrac{1}{2^{1/2} \pi^{3/4}}.$$ Define the canonical transformation associated with $T_{ \phi_1}$ by: $$\begin{aligned}
\kappa_{ \phi_1}: \mathbb{C} \times \mathbb{C} & \longrightarrow \mathbb{C} \times \mathbb{C}, \\
(x, - \partial_x \phi_1(z, x)) =: (x, \xi) & \longmapsto (z, \partial_z \phi_1(z, x)) = (x- i\xi, \xi).\end{aligned}$$
We have the following properties on the Bargmann transform (see for example [@MR2952218 Chapter 13]).
\[prop\_ecriture\_integrale\_Pi\_Phi1\] $ $
1. $ T_{ \phi_1}$ extends to a unitary transformation: $L^2( \mathbb{R}) \longrightarrow H_{ \hbar}( \mathbb{C}, \Phi_1)$.
2. If $ T_{ \phi_1}^* :L^2_{ \hbar}( \mathbb{C}, \Phi_1) \longrightarrow L^2( \mathbb{R})$ denotes the adjoint of $T_{ \phi_1}: L^2( \mathbb{R}) \longrightarrow L^2_{ \hbar}( \mathbb{C}, \Phi_1)$, then it is given by the following formula, for $v \in L^2_{ \hbar}( \mathbb{C}, \Phi_1)$: $$T_{ \phi_1}^* v(x) = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{C}} e^{-(1/2 \hbar) ( \overline{z} - x)^2} e^{-2 \Phi_1(z)/ \hbar} v(z) L( dz).$$
3. Let $ \psi_1$ be the unique holomorphic quadratic form on $ \mathbb{C} \times \mathbb{C}$ such that, for all $z \in \mathbb{C}$, we have: $$\psi_1(z, \overline{z}) = \Phi_1(z).$$ Then the orthogonal projection $ \Pi_{ \Phi_1, \hbar} : L^2_{ \hbar}( \mathbb{C}, \Phi_1) \longrightarrow H_{ \hbar}( \mathbb{C}, \Phi_1)$ is given by the following formula: $$\Pi_{ \Phi_1, \hbar} u(z) = \dfrac{2 \det \partial^2_{z, w} \psi_1}{\pi \hbar} \int_{ \mathbb{C}} e^{2( \psi_1(z, \overline{w})- \Phi_1(w))/ \hbar} u(w) dw d \overline{w}.$$ Moreover, $ \Pi_{ \Phi_1, \hbar} = T_{ \phi_1} T_{ \phi_1}^*$.
The following proposition gives a connection between the Weyl quantization of $ \mathbb{R}^2$ and the complex Weyl quantization of $ \mathbb{R}^2$ (see for example the mini-course [@HJ]).
\[prop\_transfor\_Bargmann\_et\_qauntif\_weyl\] Let $a_{ \hbar} \in S( \mathbb{R}^2)$ be a function admitting an asymptotic expansion in powers of $ \hbar$. Let $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) := T_{ \phi_1} {\mathrm{Op}}^w(a_{ \hbar}) T_{ \phi_1}^*$. Then:
1. $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) : H_{ \hbar}( \mathbb{C}, \Phi_1) \longrightarrow H_{ \hbar}( \mathbb{C}, \Phi_1)$ is uniformly bounded with respect to $\hbar$;
2. $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$ is given by the following contour integral: $${\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u(z) = \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{\Gamma(z)} e^{(i/\hbar) (z-w)\zeta} b_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right) u(w) dw d\zeta,$$ where $ \Gamma(z) = \left\lbrace (w, \zeta) \in \mathbb{C}^2; \zeta = \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z} \left( \dfrac{z+w}{2} \right) = - \Im \left( \dfrac{z+w}{2} \right) \right\rbrace$, where the symbol $b_{ \hbar}$ is given by $ b_{ \hbar} = a_{ \hbar} \circ \kappa_{ \phi_1}^{-1}$ and where the canonical transformation $ \kappa_{ \phi_1}$ is defined by: $$\begin{aligned}
\kappa_{ \phi_1}: \mathbb{R}^2 & \longrightarrow \Lambda_{ \Phi_1} = \lbrace (z, - \Im(z)); z \in \mathbb{C} \rbrace \\
(x, \xi) & \longmapsto (x-i \xi, \xi).\end{aligned}$$
We study now the action of the Bargmann transform on the Schwartz space in the spirit of the article of Valentine Bargmann [@MR0201959], except that in our case, we introduce a semi-classical parameter and a different weight function. Therefore, for the sake of completeness, we recall the theory. To do so, we introduce some new notations.\
`Notation:`
- for $ j \in \mathbb{N}$: $$\mathcal{S}^j( \mathbb{R}) := \left\lbrace \phi \in \mathcal{C}^j( \mathbb{R}); \| \phi \|_j := \max_{m \leq j} \left( \sup_{x \in \mathbb{R}} | (1+x^2)^{(j-m)/2} \partial^m_x \phi(x) | \right) < + \infty \right\rbrace ;$$ thus, the Schwartz space can be rewritten as follows: $$\mathcal{S}( \mathbb{R}) = \bigcap_{j=0}^{ \infty} \mathcal{S}^j( \mathbb{R}) = \left\lbrace \phi \in \mathcal{C}^{ \infty}( \mathbb{R}); \forall j \in \mathbb{N}, \| \phi \|_j < + \infty \right\rbrace ;$$
- for $j \in \mathbb{N}$: $$\mathfrak{S}^j( \mathbb{C}) := \left\lbrace \psi \in {\mathrm{Hol}}( \mathbb{C}); | \psi|_j := \sup_{z \in \mathbb{C}} \left( \left(1+ |z|^2 \right)^{j/2} e^{- \Phi_1(z)/ \hbar} | \psi(z)| \right) < + \infty \right\rbrace;$$
- we finally define: $$\mathfrak{S}( \mathbb{C}) := \bigcap_{j=0}^{ \infty} \mathfrak{S}^j( \mathbb{C}) = \left\lbrace \psi \in {\mathrm{Hol}}( \mathbb{C}); \forall j \in \mathbb{N}, | \psi |_j < + \infty \right\rbrace.$$
\[prop\_Tphi\_envoie\_S(R)\_sur\_SS(C)\] $ $
1. Let $j \in \mathbb{N}$, let $ \phi \in \mathcal{S}^j( \mathbb{R})$, then, for all $z \in \mathbb{C}$, we have the following estimate: $$\label{equation_estimation_Tphi1_sur_Sj(R)}
| T_{ \phi_1} \phi(z) | \leq a_j^{ \hbar} \left( 1 + |z|^2 \right)^{-j/2} e^{ \Phi_1(z)/ \hbar} \| \phi \|_j,$$ where $a_j^{ \hbar}$ is a constant depending on $j$ and on the semi-classical parameter $ \hbar$. As a result: $T_{ \phi_1} \mathcal{S}^j( \mathbb{R}) \subset \mathfrak{S}^j( \mathbb{C})$.
2. $T_{ \phi_1} \mathcal{S}( \mathbb{R}) \subset \mathfrak{S}( \mathbb{C})$.
We give a sketch of the proof ; for more details, see the article of Valentine Bargmann [@MR0201959] where the argument can be adapted to the new weight.\
**Step 1:** we prove using simple integral estimates that for $j=0$ and for $ \phi \in \mathcal{S}^0( \mathbb{R})$, there exists a constant $a_0^{ \hbar}$ such that, for all $z \in \mathbb{C}$, we have: $$| T_{ \phi_1} \phi(z) | \leq a_0^{ \hbar} e^{ \Phi_1(z)/ \hbar} \| \phi \|_0 .$$ **Step 2:** we prove that for $j \geq 1$ and for $ \phi \in \mathcal{S}^j( \mathbb{R})$, there exists a constant $a_j^{ \hbar}$ such that, for all $z \in \mathbb{C}$, we have: $$| T_{ \phi_1} \phi(z) | \leq a_j^{ \hbar} \left( 1 + |z|^2 \right)^{-j/2} e^{ \Phi_1(z)/ \hbar} \| \phi \|_j.$$ This step can be divided into 7 steps.
- **Step 2.1:** it follows from the definition of $\| \phi \|_j$ that:
1. $ | \partial^m_x \phi(x) | \leq \| \phi \|_j$ for $m \leq j$;
2. $ | \phi(x)| \leq \| \phi \|_j (1 + x^2)^{-j/2}$.
- **Step 2.2:** for $z \in \mathbb{C}$ and for $ \tau \in \mathbb{R}$, let: $F( \tau) = (T_{ \phi_1} \phi) ( \tau z)$, thus: $F(1) = (T_{ \phi_1} \phi)(z)$ and we can use the function $F$ to decompose $(T_{ \phi_1} \phi) (z)$ into two functions: $$F(1) = p_j(z) + r_j(z) ,$$ where: $$\left\lbrace
\begin{split}
p_j(z) & = \sum_{l=0}^{j-1} \dfrac{F^{(l)}(0)}{l!}, \\
r_j(z) & = \int_0^1 \dfrac{(1- \tau)^{j-1}}{(j-1)!} F^{(j)}( \tau) d \tau.
\end{split}
\right.$$ Then, we deduce the following estimates: $$\left\lbrace
\begin{split}
\left|p_j(z) \right| & \leq \sum_{l=0}^{j-1} |z|^l \eta_l(0), \\
\left| r_j(z) \right| & \leq \int_0^1 \dfrac{(1- \tau)^{j-1}}{(j-1)!} |z|^j \eta_j( \tau z) d \tau,
\end{split}
\right.$$ where $ \eta_l(z)$ is a bound on $ \partial^l (T_{ \phi_1} \phi) (z)$.\
- **Step 2.3:** we prove that the function $ \eta_l$ satisfies the following equality for $z \in \mathbb{C}$: $$\eta_l(z) = \beta e^{ \Phi_1(z)/ \hbar} \quad \text{for $l \leq j$},$$ where $ \beta = (\pi \hbar)^{-1/4} \| \phi \|_j$ using Lebesgue’s theorem, Step 2.1 (a) and integral estimates.\
- **Step 2.4:** we deduce from Step 2.2 and Step 2.3 that, for $z \in \mathbb{C}$, we have the following estimates: $$\left\lbrace
\begin{split}
|p_j(z)| & \leq \beta \sum_{l=0}^{j-1} |z|^l, \\
|r_j(z)| & \leq \beta |z|^j (2 \hbar)^j e^{1/2 \hbar} (1+ \Im(z)^2)^{-j} e^{\Phi_1(z)/ \hbar}.
\end{split}
\right.$$
- **Step 2.5:** we prove using Step 2.4 that, for $z \in \mathbb{C}$, we have: $$| T_{ \phi_1} \phi (z) | \leq \rho'_{ \hbar} \| \phi \|_j (1+|z|^2)^{j/2} (1+\Im(z)^2)^{-j} e^{ \Phi_1(z)/ \hbar} \quad \text{where $\rho'_{ \hbar}$ is a constant.}$$
- **Step 2.6:** we prove using Step 2.1 (b) that, for $z \in \mathbb{C}$, we have: $$| T_{ \phi_1} \phi(z) | \leq \rho_{ \hbar}'' \| \phi \|_j (1+\Re(z)^2)^{-j/2} e^{\Phi_1(z)/ \hbar} \quad \text{where $\rho_{ \hbar}''$ is an other constant.}$$
- **Step 2.7:** we compare the estimates of Step 2.5 and Step 2.6 and we deduce that, for $z \in \mathbb{C}$, we have: $$| T_{ \phi_1} \phi(z) | \leq \rho_{ \hbar} \| \phi \|_j (1+|z|^2)^{-j/2} e^{\Phi_1(z)/ \hbar} \quad \text{where $ \rho_{ \hbar} = \max (2^j \rho_{ \hbar}', 2^{j/2} \rho_{ \hbar}'')$.}$$
**Step 3:** the fact that $T_{ \phi_1} \mathcal{S}^j( \mathbb{R}) \subset \mathfrak{S}^j( \mathbb{C})$ is a corollary of Equation and $T_{ \phi_1} \mathcal{S}( \mathbb{R}) \subset \mathfrak{S}( \mathbb{C})$ can be deduced from the first assertion of the proposition and the definitions of the spaces $ \mathcal{S}( \mathbb{R})$ and $ \mathfrak{S}(\mathbb{C})$.
\[rema\_S(C)\_inclus\_dans\_H(C,Phi1)\] Since $ \mathcal{S}( \mathbb{R}) \subset L^2( \mathbb{R})$, then according to Propositions \[prop\_ecriture\_integrale\_Pi\_Phi1\] and \[prop\_Tphi\_envoie\_S(R)\_sur\_SS(C)\], we have $ \mathfrak{S}( \mathbb{C}) \subset H_{ \hbar}( \mathbb{C}, \Phi_1)$.
Conversely, we have the following proposition.
\[prop\_T\_(phi1)\*\_envoie\_S(C)\_sur\_S(R)\] $ $
1. Let $ \mu = 1+j+ \tau$ with $j \in \mathbb{N}$ and $ \tau \in \mathbb{N}^*$, then, for all $ \psi \in \mathfrak{S}^{ \mu}( \mathbb{C})$, we have the following estimate: $$\label{equation_estimation_Tphi1*(SSmu(C))}
\| T_{ \phi_1}^* \psi \|_j \leq a^{ \hbar}_{j, \tau} | \psi |_{ \mu},$$ where $a^{ \hbar}_{j, \tau}$ is a constant depending on the semi-classical parameter $ \hbar$ and on the integers $j$ and $ \tau$. As a result: $ T_{ \phi_1}^* \mathfrak{S}^{ \mu}( \mathbb{C}) \subset \mathcal{S}^j( \mathbb{R})$.
2. $T_{ \phi_1}^* \mathfrak{S}( \mathbb{C}) \subset \mathcal{S}( \mathbb{R})$.
We give a sketch of the proof, for more details see [@MR0201959].\
**Step 1:** we give an estimate on $ \partial^m T_{ \phi_1}^* \psi$ for $m \leq j$ by following these steps.
- **Step 1.1:** we prove using Lebesgue’s theorem that, for $x \in \mathbb{R}$, we have: $$\left| \partial^m_x (T_{ \phi_1}^* \psi(x)) \right| \leq |c_{ \phi_1}| \hbar^{-3/4} \int_{ \mathbb{C}} B_m(z, x) L(dz),$$ where for $(z, x) \in \mathbb{C} \times \mathbb{R}$: $$B_m(z, x) = \left| \partial^m_x \left( e^{-(1/2 \hbar)( \overline{z}-x)^2} \right) e^{-2 \Phi_1(z)/ \hbar} \psi(z) \right|.$$
- **Step 1.2:** we prove that, for $(z, x) \in \mathbb{C} \times \mathbb{R}$ and for $m \leq j$, we have: $$B_m(z, x) \leq \delta_{\hbar}^j 2^m \left( 1 + \dfrac{1}{2 \hbar}(\Re(z)-x)^2 \right)^{m/2} e^{-(1 /2 \hbar)(\Re(z)-x)^2} (1+|z|^2)^{(m- \mu)/2} | \psi |_{ \mu},$$ where $ \delta_{\hbar}^j$ is a constant depending on $j$ and $ \hbar$. To do so, we use estimates on Hermite polynomials for the term $ \left| \partial^m_x \left( e^{-(1/2 \hbar)( \overline{z}-x)^2} \right) \right| $ and the following estimate, for $z \in \mathbb{C}$: $$| \psi(z) | \leq (1+|z|^2)^{- \mu/2} e^{ \Phi_1(z)/ \hbar} | \psi |_{ \mu} ,$$ resulting from the fact that $ \psi \in \mathfrak{S}^{ \mu}( \mathbb{C})$.
- **Step 1.3:** according to Step 1.1 and Step 1.2, for $x \in \mathbb{R}$, we have: $$\begin{aligned}
| \partial_x^m (T_{\phi_1}^* \psi(x))|
& \leq a_{j, \tau}^{ \hbar} (1+x^2)^{(m-j)/2} | \psi |_{ \mu}, \end{aligned}$$ where $a_{j, \tau}^{ \hbar}$ is a constant depending on $j$, $ \tau$ and $ \hbar$.
**Step 2:** according to Step 1, for $x \in \mathbb{R}$ and $ \mu = 1+j + \tau$, we have: $$\begin{aligned}
(1+x^2)^{(j-m)/2} | \partial_x^m(T_{\phi_1}^* \psi(x))| \leq a_{j, \tau}^{ \hbar} | \psi |_{ \mu} .\end{aligned}$$ Thus: $$\sup_{x \in \mathbb{R}} \left( (1+x^2)^{(j-m)/2} | \partial_x^m(T_{\phi_1}^* \psi(x))| \right) \leq a_{j, \tau}^{ \hbar} | \psi |_{ \mu}.$$ Consequently, we have: $$\| T_{\phi_1}^* \psi \|_j = \max_{m \leq j} \left( \sup_{x \in \mathbb{R}} \left( (1+x^2)^{(j-m)/2} | \partial_x^m(T_{\phi_1}^* \psi(x))| \right) \right) \leq a_{j, \tau}^{ \hbar} | \psi |_{ \mu}.$$ **Step 3:** the fact that $T_{ \phi_1}^* \mathfrak{S}^{ \mu}( \mathbb{C}) \subset \mathcal{S}^j( \mathbb{R})$ is a corollary of Equation and $T_{ \phi_1}^* \mathfrak{S}( \mathbb{C}) \subset \mathcal{S}( \mathbb{R})$ can be deduced from the first assertion of the proposition and the definitions of the spaces $ \mathcal{S}( \mathbb{R})$ and $ \mathfrak{S}(\mathbb{C})$.
We are interested now in the action of the Bargmann transform on the tempered distributions space. Here again we can adapt the techniques of [@MR0201959].\
`Notation:`
- $ \mathfrak{S}'( \mathbb{C})$ denotes the dual of the space $ \mathfrak{S}( \mathbb{C})$ (equipped with the topology of the semi-norms $|\cdot |_j$) *i.e.* the space of continuous linear functionals on $ \mathfrak{S}( \mathbb{C})$;
- $ \langle \cdot, \cdot \rangle_{ \mathfrak{S}', \mathfrak{S}}$ denotes the duality bracket between $ \mathfrak{S}'( \mathbb{C})$ and $ \mathfrak{S}( \mathbb{C})$;
- for $f, g \in {\mathrm{Hol}}( \mathbb{C})$, we denote by $ \langle g, f \rangle$ the following product: $$\langle g, f \rangle = \int_{ \mathbb{C}} \overline{g(z)} f(z) e^{- 2 \Phi_1(z)/ \hbar} L(dz),$$ when this integral converges.
\[rem\_crochet\_defini\_sur\_SSj(C)\] $ $
- The bracket defined above coincides with the scalar product $ \langle g, f \rangle_{L^2_{ \hbar}( \mathbb{C}, \Phi_1)}$ when $g, f \in H_{ \hbar}( \mathbb{C}, \Phi_1)$.
- If $g \in \mathfrak{S}^{ \rho}( \mathbb{C})$ and $f \in \mathfrak{S}^{ \sigma}( \mathbb{C})$ with $ \rho + \sigma > 2$, then the bracket $ \langle g, f \rangle$ is well-defined.
We use this bracket to describe the elements of the space $ \mathfrak{S}'( \mathbb{C})$ similarly to the article of Valentine Bargmann [@MR0201959].
\[prop\_ecriture\_elements\_de\_SS’(C)\] Every continuous linear functional $L$ on $ \mathfrak{S}( \mathbb{C})$ can be written, for all $f \in \mathfrak{S}( \mathbb{C})$, as follows: $$L(f) = \langle g, f \rangle = \int_{ \mathbb{C}} \overline{g(z)} f(z) e^{-2 \Phi_1(z)/ \hbar} L(dz),$$ where $g$ is a function in $ \mathfrak{S}^{-l}( \mathbb{C})$ for $l \in \mathbb{N}$ and is uniquely defined, for all $a \in \mathbb{C}$, by $g(a) = \overline{L(e_a)}$ (where for all $z \in \mathbb{C}$, $e_a(z) = e^{-( \overline{a}-z)^2/(4 \hbar)}$).\
Conversely, every functional of the form $L(f) = \langle g, f \rangle$ with $g \in \mathfrak{S}^{-l}( \mathbb{C})$ defines a continuous linear functional on $ \mathfrak{S}( \mathbb{C})$.
We give a sketch of the proof, for more details see [@MR0201959].\
**Step 1:** for all $L \in \mathfrak{S}'( \mathbb{C})$, there exists $C>0$ and $l \in \mathbb{N}$ such that: $|L(f)| \leq C |f|_l$, for all $f \in \mathfrak{S}( \mathbb{C})$.\
**Step 2:** for $a \in \mathbb{C}$, let $g$ be the function defined by $g(a) = \overline{L(e_a)}$. We prove using Step 1 that $e_a \in \mathfrak{S}( \mathbb{C})$ and $g \in \mathfrak{S}^{-l} ( \mathbb{C})$ ($e_a$ is a reproducing kernel for the space $H_{ \hbar}( \mathbb{C}, \Phi_1)$ and for $ \mathfrak{S}( \mathbb{C})$).\
**Step 3:** let $L_1$ be the continuous linear functional defined, for $f \in \mathfrak{S}( \mathbb{C})$, by $L_1(f) = \langle g, f \rangle$. We show that, for all $a \in \mathbb{C}$, we have $ L_1(e_a) = L(e_a)$ then we deduce that $L=L_1$ using the density of the set of finite linear combinations of elements of $ \mathcal{B} = \lbrace e_a, a \in \mathbb{C} \rbrace$ in $ \mathfrak{S}( \mathbb{C})$.
As in the article of Valentine Bargmann [@MR0201959], we prove that the Bargmann transform $T_{ \phi_1}$ and its adjoint $T_{ \phi_1}^*$ act on the spaces $ \mathcal{S}'( \mathbb{R})$ and $ \mathfrak{S}'( \mathbb{C})$ respectively.
\[prop\_Tphi\_envoie\_S’(R)\_sur\_SS’(C)\]$ $
1. $T_{ \phi_1}$ extends to an operator: $\mathcal{S}'( \mathbb{R}) \longrightarrow \mathfrak{S}'( \mathbb{C})$, which satisfies for $v \in \mathcal{S}'( \mathbb{R})$ and $f \in \mathfrak{S}( \mathbb{C})$: $$\langle T_{ \phi_1} v, f \rangle_{ \mathfrak{S}', \mathfrak{S}} = \langle v, T_{ \phi_1}^* f \rangle_{ \mathcal{S}', \mathcal{S}} .$$
2. $T_{ \phi_1}^*$ extends to an operator: $\mathfrak{S}'( \mathbb{C}) \longrightarrow \mathcal{S}'( \mathbb{R})$, which satisfies for $L \in \mathfrak{S}'( \mathbb{C})$ and $ \phi \in \mathcal{S}( \mathbb{R})$: $$\langle T_{ \phi_1}^* L, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = \langle L, T_{ \phi_1} \phi \rangle_{ \mathfrak{S}', \mathfrak{S}}.$$
We give a sketch of the proof, for more details see [@MR0201959].\
Let $v \in \mathcal{S}'( \mathbb{R})$, let $L(f)$ be the functional defined by: $$L(f) = \langle v, \phi \rangle_{ \mathcal{S}', \mathcal{S}},$$ where $f = T_{ \phi_1} \phi \in \mathfrak{S}( \mathbb{C})$, then $L(f)$ is a continuous linear functional on $ \mathfrak{S}( \mathbb{C})$. Conversely if $L \in \mathfrak{S}'( \mathbb{C})$ and if we define $v$ by: $$v( \phi) = \langle v, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = L(f) \quad \text{where $f =T_{ \phi_1} \phi$},$$ then $v$ is a continuous linear functional on $ \mathcal{S}( \mathbb{R})$. Then, according to Proposition \[prop\_ecriture\_elements\_de\_SS’(C)\], for $l \in \mathbb{N}$, there exists $g \in \mathfrak{S}^{-l}( \mathbb{C})$ such that: $$L(f) = \langle g, f \rangle .$$ Thus, for all $v \in \mathcal{S}'( \mathbb{R})$ and for all $ \phi \in \mathcal{S}( \mathbb{R})$, we have: $$\langle v, \phi \rangle_{ \mathcal{S}', \mathcal{S}} = \langle g, T_{ \phi_1} \phi \rangle .$$ This equality gives a bijection between the spaces $ \mathcal{S}'( \mathbb{R})$ and $ \mathfrak{S}'( \mathbb{C})$ with: $$g := T_{ \phi_1} v \quad \text{and} \quad v =: T_{ \phi_1}^* g .$$
\[prop\_reecriture\_action\_Tphi1\_sur\_S’(R)\] For $\psi \in \mathcal{S}'( \mathbb{R})$, we can rewrite $T_{ \phi_1} \psi$ as follows (see for example [@HJ] or [@MR2952218]): $$T_{ \phi_1} \psi(z) = \left\langle \psi, c_{ \phi_1} \hbar^{-3/4} e^{-(1/ 2 \hbar)(z-.)^2} \right\rangle_{ \mathcal{S}', \mathcal{S}} .$$
We are now looking at the range by the Bargmann transform of the space $ \mathcal{L}_k$ and we prove that this range is the space $ \mathcal{H}_k$, where we recall that: $$\begin{aligned}
\mathcal{L}_k & = \left\lbrace \psi \in \mathcal{S}'( \mathbb{R}); \quad \tau_{2 \pi} \psi = u^k \psi, \quad \tau_1 \mathcal{F}_{ \hbar}( \psi) = v^{-k} \mathcal{F}_{ \hbar}( \psi) \right\rbrace, \\
\mathcal{H}_k & = \left\lbrace g \in {\mathrm{Hol}}( \mathbb{C}); \quad g(p+2 \pi, q) = u^k g(p, q), \quad g(p, q+1) = v^k e^{-i(p+iq)k+k/2} g(p, q) \right\rbrace.\end{aligned}$$ To our knowledge, this result is new in the literature and it constitutes a fundamental step in our proof of Theorem \[theoA\].
\[prop\_formule\_C\] Let $k \geq 1$. Then, we have:
1. $T_{ \phi_1}: \mathcal{L}_k \longrightarrow \mathcal{H}_k $;
2. $T_{ \phi_1}^*: \mathcal{H}_k \longrightarrow \mathcal{L}_k $.
According to Proposition \[prop\_Tphi\_envoie\_S’(R)\_sur\_SS’(C)\], $T_{ \phi_1}: \mathcal{S}'( \mathbb{R}) \longrightarrow \mathfrak{S}'( \mathbb{C})$. Since $ \mathcal{L}_k \subset \mathcal{S}'( \mathbb{R})$, then $T_{ \phi_1}$ is well-defined on this space. Let’s prove that the Bargmann transform $T_{ \phi_1}$ sends the basis $( \epsilon_l)_{l \in \mathbb{Z}/ k \mathbb{Z}}$ of $ \mathcal{L}_k$ (see Equation ) on the basis $( e_l)_{l \in \mathbb{Z}/ k \mathbb{Z}}$ of $ \mathcal{H}_k$ (see Equation ). Let $c$ be the real number such that $u = e^{ic}$. Let $l \in \lbrace 0, \ldots, k-1 \rbrace$, using Remark \[prop\_reecriture\_action\_Tphi1\_sur\_S’(R)\], we have: $$\begin{aligned}
T_{ \phi_1} \epsilon_l(z) & = \left\langle \epsilon_l, c_{ \phi_1} \hbar^{-3/4} e^{-(1/ 2 \hbar)(z-.)^2} \right\rangle_{ \mathcal{S}', \mathcal{S}}, \\
& = \left\langle u^{k./(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k}\right)^j e^{i(l+jk).}, c_{ \phi_1} \hbar^{-3/4} e^{-(1/ 2 \hbar)(z-.)^2} \right\rangle_{ \mathcal{S}', \mathcal{S}}, \\
& = c_{ \phi_1} k^{3/4} \sum_{j \in \mathbb{Z}} \left(v^{-k} \right)^j \left\langle u^{k./(2 \pi)} e^{i(l+jk).}, e^{-(k/ 2)(z-.)^2} \right\rangle_{ \mathcal{S}', \mathcal{S}} \quad \text{since $k = \dfrac{1}{\hbar}$}, \\
& = c_{ \phi_1} k^{3/4} u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} \right)^j e^{i(l+jk)z} \left\langle u^{k./(2 \pi)} e^{i(l+jk).} , e^{-(k/2) (.)^2} \right\rangle_{ \mathcal{S}', \mathcal{S}}, \\
& = c_{ \phi_1} k^{3/4} u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left( v^{-k} \right)^j e^{i(l+jk)z} \sqrt{\dfrac{2 \pi}{k}} \exp \left(- \dfrac{1}{2k} \left( \dfrac{ck}{2 \pi}+l+jk \right)^2\right).\end{aligned}$$ By a simple computation, we obtain: $$\dfrac{1}{2 k}\left( \dfrac{ck}{2 \pi}+l+jk \right)^2 = \dfrac{1}{2k} \left( \dfrac{ck}{2 \pi} +l \right)^2 + \dfrac{j^2k}{2} + jl + \dfrac{jck}{2 \pi} .$$ Therefore, we have: $$\begin{aligned}
T_{ \phi_1} \epsilon_l(z) & = c_{ \phi_1} k^{3/4} u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} \right)^j e^{i(l+jk)z} \sqrt{\dfrac{2 \pi}{k}} \exp \left(- \dfrac{ 1}{2k} \left( \dfrac{ck}{2 \pi}+l+jk \right)^2 \right), \\
& = c_k^l u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} e^{-jk/2-l} u^{ik/(2 \pi)} \right)^j e^{i(l+jk)z}, \\
& = c_k^l e_l(z),\end{aligned}$$ where $c_k^l$ is given by the following equality: $$c_k^l = c_{ \phi_1} \sqrt{2\pi} k^{1/4} \exp \left( - \dfrac{1}{2k} \left( \dfrac{ck}{2 \pi} +l \right)^2 \right) .$$ Conversely, we compute $T_{ \phi_1}^* e_l$. First, since $ \mathcal{H}_k \subset \mathfrak{S}^0( \mathbb{C})$ and since the function $z \longmapsto e^{-(1/ 2 \hbar)(z-x)^2}$ belongs to the space $\mathfrak{S}( \mathbb{C})$, then for $v \in \mathcal{H}_k$, we have (according to Remark \[rem\_crochet\_defini\_sur\_SSj(C)\]): $$\left\langle c_{ \phi_1} \hbar^{-3/4} e^{-(1/ 2 \hbar)(.-x)^2}, v \right\rangle = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{C}} e^{-1/(2 \hbar)( \overline{z} - x)^2} e^{-2 \Phi_1(z)/ \hbar} e_l(z) L(dz) < + \infty .$$ Let $l \in \lbrace 0, 1, \ldots, k-1 \rbrace$, then we have: $$\begin{aligned}
& T_{ \phi_1}^* e_l(x) \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{C}} e^{-1/(2 \hbar)( \overline{z} - x)^2} e^{-2 \Phi_1(z)/ \hbar} e_l(z) L(dz), \\
& = c_{ \phi_1} k^{3/4} \int_{ \mathbb{C}} e^{-(k/2)( \overline{z} -x)^2} e^{-k ( \Im z)^2} e_l(z) L(dz) \quad \text{because $ \Phi_1(z) = \dfrac{1}{2} ( \Im z)^2$ and $k = \dfrac{1}{\hbar}$ }, \\
& =c_{ \phi_1} k^{3/4} \int_{ \mathbb{C}} e^{-(k/2)( \overline{z} -x)^2} e^{-k ( \Im z)^2} u^{kz/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} e^{-l-jk/2} u^{ik/(2 \pi)} \right)^j e^{i(l+jk)z} L(dz), \\
& = c_{ \phi_1} k^{3/4} \sum_{j \in \mathbb{Z}} \left(v^{-k} e^{-l-jk/2} u^{ik/(2 \pi)} \right)^j \int_{ \mathbb{C}} e^{-(k/2)(\overline{z})^2} e^{-k ( \Im (z+x))^2} u^{k(z+x)/(2 \pi)} e^{i(l+jk)(z+x)} L(dz), \\
& = c_{ \phi_1} k^{3/4} u^{kx/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} e^{-l-jk/2} u^{ik/(2 \pi)} \right)^j e^{i(l+jk)x} \int_{ \mathbb{C}} e^{-(k/2)(\overline{z})^2} e^{-k ( \Im z)^2} e^{iz(l+jk+ck/(2 \pi))} L(dz).\end{aligned}$$ We have to compute the following integral (after the change of variables $z=p+iq$): $$\begin{aligned}
& \int_{ \mathbb{R}} \int_{ \mathbb{R}} e^{-(k/2)(p-iq)^2} e^{-kq^2} e^{i(p+iq)(l+jk+ck/(2 \pi))} dp dq \\
& = \int_{ \mathbb{R}} \int_{ \mathbb{R}} e^{-k p^2/2} e^{-kq^2/2} e^{ikpq} e^{ip(l+jk+ck/(2 \pi))} e^{-q(l+jk+ck/(2 \pi))} dp dq.\end{aligned}$$ By a simple computation, we obtain: $$\begin{aligned}
\int_{ \mathbb{R}} e^{-k p^2/2} e^{ikpq} e^{ip(l+jk+ck/(2 \pi))} dp = \sqrt{\dfrac{2\pi}{k}} \exp \left( - \dfrac{1}{2k} \left( l+jk+ \dfrac{ck}{2 \pi} \right)^2 - \dfrac{kq^2}{2} - q \left( l+jk+ \dfrac{ck}{2 \pi} \right) \right).\end{aligned}$$ Thus, by an other simple computation, we obtain: $$\begin{aligned}
\int_{ \mathbb{R}} \int_{ \mathbb{R}} e^{-kp^2/2} e^{-kq^2/2} e^{ikpq} e^{ip(l+jk+ck/(2 \pi))} e^{-q(l+jk+ck/(2 \pi))} dp dq = \sqrt{2} \dfrac{\pi}{k} e^{(l+jk+ck/(2 \pi))^2/(2k)}.\end{aligned}$$ Consequently, we obtain: $$\begin{aligned}
& T_{ \phi_1}^* e_l(x) \\
& = c_{ \phi_1} k^{3/4} \sqrt{2} \dfrac{\pi}{k} u^{kx/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} e^{-l-jk/2} u^{ik/(2 \pi)} \right)^j e^{i(l+jk)x} e^{(l+jk+ck/(2 \pi))^2/(2k)}, \\
& = c_{ \phi_1} k^{-1/4} \sqrt{2} \pi e^{(l+ck/(2 \pi))^2/(2k)} u^{kx/(2 \pi)} \sum_{j \in \mathbb{Z}} \left(v^{-k} \right)^j e^{i(l+jk)x}, \\
& = \tilde{c}_k^l \epsilon_l(x),\end{aligned}$$ where $\tilde{c}_k^l = c_{ \phi_1} k^{-1/4} \sqrt{2} \pi e^{(l+ck/(2 \pi))^2/(2k)}$.
Since the Bargmann transform is a unitary transformation between the spaces $L^2( \mathbb{R})$ and $H_{ \hbar}( \mathbb{C}, \Phi_1)$, we study this feature between the spaces $ \mathcal{L}_k$ and $ \mathcal{H}_k$.
\[prop\_TphiTphi\*=id\_sur\_Hk\_et\_Tphi\*Tphi=id\_sur\_Lk\] $ $
1. $T_{ \phi_1}^* T_{ \phi_1} = {\mathrm{id}}$ on $\mathcal{L}_k$.
2. $T_{ \phi_1} T_{ \phi_1}^*= {\mathrm{id}}$ on $ \mathcal{H}_k$.
Let $c$ be the real number such that $u = e^{ic}$. According to the proof of Proposition \[prop\_formule\_C\], we have, for $l \in \lbrace 0, \ldots, k-1 \rbrace$: $$T_{ \phi_1} \epsilon_l = c_k^l e_l \quad \text{with $c_k^l = c_{ \phi_1} \sqrt{2\pi} k^{1/4} e^{-( ck/(2 \pi) +l)^2/(2k)}$}.$$ Let $ \mathcal{C} = {\mathrm{diag}}(c_k^0, \ldots, c_k^{k-1})$ be the matrix of the operator $T_{\phi_1}$ in the basis $(e_l)_{l \in \mathbb{Z}/ k \mathbb{Z}}$. According to the proof of Proposition \[prop\_formule\_C\], we also have, for $l \in \lbrace 0, \ldots, k-1 \rbrace$: $$T_{\phi_1}^* e_l = \tilde{c}_k^l \epsilon_l \quad \text{with $\tilde{c}_k^l = c_{ \phi_1} k^{-1/4} \sqrt{2} \pi e^{(l+ck/(2 \pi))^2/(2k)}$}.$$ Let $ \mathcal{C}^* = {\mathrm{diag}}( \tilde{c}_k^0, \ldots, \tilde{c}_k^{k-1}) $ be the matrix of the operator $T_{ \phi_1}^*$ in the basis $(\epsilon_l)_{l \in \mathbb{Z}/ k \mathbb{Z}}$. We want to prove that: $ \mathcal{C} \mathcal{C}^* = \mathcal{C}^* \mathcal{C} = I_k$. Let $k \geq 1$ and let $l \in \lbrace 0, 1, \ldots, k-1 \rbrace$, we have: $$\begin{aligned}
c_k^l \tilde{c}_k^l & = c_{ \phi_1} \sqrt{2\pi} k^{1/4} e^{-( ck/(2 \pi) +l)^2/(2k)} c_{ \phi_1} k^{-1/4} \sqrt{2} \pi e^{(l+ck/(2 \pi))^2/(2k)}, \\
& = c_{ \phi_1}^2 2 \pi^{3/2}, \\
& = \left( \dfrac{1}{2^{1/2} \pi^{3/4}} \right)^2 2 \pi^{3/2} \quad \text{according to Definition \ref{defi_Bargmann_transform}}, \\
& = 1, \\
& = \tilde{c}_k^l c_k^l.\end{aligned}$$ Therefore, we have: $ \mathcal{C} \mathcal{C}^* = \mathcal{C}^* \mathcal{C} = I_k $.
### Complex Weyl quantization of the torus
In this paragraph, we define the complex Weyl quantization of a symbol on the torus. As in the classical Weyl quantization case, we have two definitions for the complex Weyl quantization. With an Egorov theorem analogous to Proposition \[prop\_transfor\_Bargmann\_et\_qauntif\_weyl\] in the $ \mathbb{R}^2$-case, we exhibit the notion of complex Weyl quantization of the torus. First, we introduce a new class of symbols. Recall that $ \Lambda_{ \Phi_1}$ denotes the following space: $$\Lambda_{ \Phi_1} = \left\lbrace \left( z, \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z}(z) \right); z \in \mathbb{C} \right\rbrace = \left\lbrace (z, - \Im(z)); z \in \mathbb{C} \right\rbrace.$$ And that the canonical transformation $ \kappa_{ \phi_1}$ is defined as follows: $$\begin{aligned}
\kappa_{ \phi_1}: \mathbb{R}^2 & \longrightarrow \Lambda_{ \Phi_1} \\
(x, y) & \longmapsto (z, w) := (x-iy, y).\end{aligned}$$ Notice that, if $a_{k} \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ is a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{k}(x + 2 \pi, y) = a_{k}(x, y) = a_{k}(x, y+1) ;$$ and if $b_{k}$ is the function defined by the following relation, for $(z, w) \in \Lambda_{ \Phi_1}$: $$b_{k} (z, w) := a_{k} \circ \kappa_{ \phi_1}^{-1}(z, w).$$ Then $b_{k} \in \mathcal{C}^{ \infty}_k( \Lambda_{ \Phi_1})$ is a function such that, for $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_{k}(z+2 \pi, w) = b_{k}(z, w) = b_{k}(z+i, w-1).$$ Besides, thanks to the identification of $ \Lambda_{ \Phi_1}$ with $ \mathbb{C}$, we can rewrite the symbol $b_{k}$ as a convergent series, for $z \in \mathbb{C} \simeq \Lambda_{ \Phi_1}$: $$\label{equation_ecriture_somme_symbole_sur_LambdaPhi1}
b_{k}(z) = \sum_{(m, n) \in \mathbb{Z}^2} b_{m,n}^{ k} e^{in \Re(z)} e^{2 i \pi m \Im(z)},$$ where $\left(b_{m,n}^{k} \right)_{(m,n) \in \mathbb{Z}^2}$ is defined by the following formula: $$b_{m,n}^{k} = a_{m,n}^{k} \quad \text{where for $(x, y) \in \mathbb{R}^2$,} \quad a_{k}(x, y) = \sum_{(m,n) \in \mathbb{Z}^2} a_{m,n}^{k} e^{inx} e^{-2 i \pi my} .$$
Since $ \Lambda_{ \Phi_1} \simeq \mathbb{C}$, then the class of symbols $S( \Lambda_{ \Phi_1})$ can be identified with $S( \mathbb{C}) \simeq S( \mathbb{R}^2)$.
We can now deduce the following Egorov theorem.
\[prop\_quantif\_complexe\_somme\_definie\_sur\_S(C)\] $ $\
Let $a_{k} \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for all $(x, y) \in \mathbb{R}^2$, we have: $$a_{k}(x+2 \pi, y) = a_{k}(x, y) = a_{k}(x, y+1).$$ Then, we have: $$T_{ \phi_1} {\mathrm{Op}}^w_k(a_{k}) = {\mathrm{Op}}^w_{ \Phi_1, k}(a_{k} \circ \kappa_{ \phi_1}^{-1}) T_{ \phi_1} \quad \text{on $ \mathcal{S}( \mathbb{R})$},$$ where $ {\mathrm{Op}}^w_{ \Phi_1, k}$ is defined by the following formula, for $u \in \mathfrak{S}( \mathbb{C})$: $${\mathrm{Op}}_{\Phi_1, k}^w(a_{k} \circ \kappa_{ \phi_1}^{-1}) u(z) = \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{-n^2/2k} e^{inz} u\left( z- \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right) ,$$ where $\left(a_{m,n}^k \right)_{(m, n) \in \mathbb{Z}^2}$ is the sequence of coefficients defined in Equation .
According to Definition \[defi\_quantif\_weyl\_tore\_somme\], $ {\mathrm{Op}}^w_k(a_k) : \mathcal{S}( \mathbb{R}) \longrightarrow \mathcal{S}( \mathbb{R})$. Let $\phi \in \mathcal{S}( \mathbb{R})$, then we have: $$\begin{aligned}
& T_{ \phi_1} ( {\mathrm{Op}}^w_k(a_k) \phi)(z) \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} e^{-(1/2 \hbar)(z-x)^2} ({\mathrm{Op}}^w_k (a_k) \phi)(x) dx, \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} e^{-(1/2 \hbar)(z-x)^2} \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{ixn} \phi \left( x - \dfrac{2 \pi m}{k} \right) dx, \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{izn} e^{-(1/2 \hbar)(z-x)^2} e^{-i(z-x)n} \phi \left( x - \dfrac{2 \pi m}{k} \right) dx, \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{izn} e^{-(1/2 \hbar)(z-x+in \hbar)^2} e^{-n^2 \hbar/2} \phi \left( x - \dfrac{2 \pi m}{k} \right) dx, \\
& = c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{izn} e^{-(1/2 \hbar)(z-x+in \hbar-2 \pi m \hbar)^2} e^{-n^2 \hbar/2} \phi \left( x \right) dx, \\
& =\sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{-n^2 \hbar/2} e^{izn} c_{ \phi_1} \hbar^{-3/4} \int_{ \mathbb{R}} e^{-(1/2 \hbar)(z-x+in \hbar-2 \pi m \hbar)^2} \phi \left( x \right) dx, \\
& = \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{-n^2/2k} e^{izn} (T_{ \phi_1} \phi) \left( z - \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right) \quad \text{because $ \hbar = \dfrac{1}{k}$}.\end{aligned}$$ Therefore, for $u \in \mathfrak{S}( \mathbb{C})$, we define $ {\mathrm{Op}}_{ \Phi_1,k}^w(a_k \circ \kappa_{ \phi_1}^{-1})$ as follows: $${\mathrm{Op}}_{ \Phi_1, k}^w(a_k \circ \kappa_{ \phi_1}^{-1}) u(z) = \sum_{(m, n) \in \mathbb{Z}^2} a_{m,n}^k e^{-i \pi mn/k} e^{-n^2/2k} e^{izn} u \left( z - \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right).$$
The previous proposition leads us to define the notion of complex Weyl quantization of a symbol on the torus as follows.
\[defi\_weyl\_complex\_tore\_1\] Let $b_k \in \mathcal{C}^{ \infty}_k( \Lambda_{ \Phi_1})$ be a function such that, for all $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_k(z+2 \pi, w) = b_k(z, w) = b_k(z+i, w-1).$$ Define the complex Weyl quantization of the symbol $b_k$, denoted by $ {\mathrm{Op}}^w_{ \Phi_1,k}(b_k)$, by the following formula, for $u \in \mathfrak{S}( \mathbb{C})$: $${\mathrm{Op}}_{\Phi_1, k}^w(b_k) u(z) = \sum_{(m, n) \in \mathbb{Z}^2} b_{m,n}^k e^{-i \pi mn/k} e^{-n^2/2k} e^{inz} u\left( z- \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right) ,$$ where the sequence $\left(b_{m,n}^k \right)_{(m,n) \in \mathbb{Z}^2}$ is given by Equation .
Let’s prove a basic property on this notion of quantization. The operator $ {\mathrm{Op}}^w_{\Phi_1,k}(b_k)$ defined above acts on the space $ \mathfrak{S}( \mathbb{C})$. We are now going to show that it also acts on the space $ \mathcal{H}_k$ (as expected since the Bargmann transform sends the space $ \mathcal{L}_k$ on the space $ \mathcal{H}_k$).
\[prop\_Op\_Weyl\_k\_agit\_sur\_Hk\] Let $b_k \in \mathcal{C}^{ \infty}_k( \Lambda_{ \Phi_1})$ be a function such that, for all $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_k(z+ 2 \pi, w) = b_k(z, w) = b_k(z+i, w-1) .$$ Then $ {\mathrm{Op}}^w_{\Phi_1, k}(b_k)$ can be extended into an operator which sends the space $\mathcal{H}_k$ on itself.
According to Proposition \[prop\_quantif\_complexe\_somme\_definie\_sur\_S(C)\], $ {\mathrm{Op}}^w_{\Phi_1, k}(b_k) : \mathfrak{S}( \mathbb{C}) \longrightarrow \mathfrak{S}( \mathbb{C})$. Let $u, v \in \mathfrak{S}( \mathbb{C})$, then we have: $$\begin{aligned}
& \langle {\mathrm{Op}}^w_{\Phi_1, k}(b_k) u, v \rangle_{ \mathfrak{S}, \mathfrak{S}} \\
& = \int_{ \mathbb{C}} \overline{{\mathrm{Op}}^w_{\Phi_1, k}(b_k) u(z)} v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{m,n}^k} e^{i \pi mn/k} e^{-n^2/2k} e^{-in\overline{z}} \overline{u\left( z- \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right) } v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{m,n}^k} e^{-i \pi mn/k} e^{-n^2/2k} e^{-in\overline{z}} \overline{u\left( z + \dfrac{in}{k} \right) } v \left( z + \dfrac{2 \pi m}{k} \right) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \overline{u\left( z \right) } \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{m,n}^k} e^{-i \pi mn/k} e^{-n^2/2k} e^{-inz} v \left( z + \dfrac{2 \pi m}{k} - \dfrac{in}{k} \right) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \overline{u\left( z \right) } \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{-m,n}^k} e^{i \pi mn/k} e^{-n^2/2k} e^{-inz} v \left( z - \dfrac{2 \pi m}{k} - \dfrac{in}{k} \right) e^{-2 \Phi_1(z)/ \hbar} L(dz) \quad \text{via $m \longmapsto -m$}, \\
& = \int_{ \mathbb{C}} \overline{u\left( z \right) } \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{-m,-n}^k} e^{-i \pi mn/k} e^{-n^2/2k} e^{inz} v \left( z - \dfrac{2 \pi m}{k} + \dfrac{in}{k} \right) e^{-2 \Phi_1(z)/ \hbar} L(dz) \quad \text{via $n \longmapsto -n$}, \\
& = \int_{ \mathbb{C}} \overline{u\left( z \right) } {\mathrm{Op}}^w_{ \Phi_1,k}( \overline{b}_k) v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \langle u, {\mathrm{Op}}^w_{ \Phi_1,k}( \overline{b}_k) v \rangle_{ \mathfrak{S}, \mathfrak{S}},\end{aligned}$$ where $ \overline{b}_k \in \mathcal{C}^{ \infty}_k( \Lambda_{ \Phi_1})$ is defined, for $(z, w) \in \Lambda_{ \Phi_1}$, by: $$\begin{aligned}
\overline{b}_k(z, w) & = \sum_{(m, n)\in \mathbb{Z}^2} \overline{b_{-m, -n}^k} e^{in(z+iw)} e^{-2i \pi m w}, \\
& = \sum_{(m, n)\in \mathbb{Z}^2} \overline{b_{m, n}^k} e^{-in(z+iw)} e^{2i \pi m w} \quad \text{via $(m,n) \longmapsto (-m,-n)$}, \\
& = \sum_{(m, n)\in \mathbb{Z}^2} \overline{b_{m, n}^k} e^{-in\Re(z)} e^{-2i \pi m \Im(z)} \quad \text{because $(z, w) \in \Lambda_{ \Phi_1}$, thus $w = - \Im(z)$},\\
& = \overline{b_k(z, w)}.\end{aligned}$$ Since $v \in \mathfrak{S}( \mathbb{C})$ and $ \overline{b}_k \in \mathcal{C}^{ \infty}_k( \Lambda_{ \Phi_1})$, then $ {\mathrm{Op}}^w_{ \Phi_1,k}( \overline{b}_k) v \in \mathfrak{S}( \mathbb{C})$ and the complex Weyl quantization $ {\mathrm{Op}}^w_{ \Phi_1, k}(b_k)$ is well-defined on $ \mathfrak{S}'( \mathbb{C})$ by the following formula, for $u \in \mathfrak{S}'( \mathbb{C})$ and for $v \in \mathfrak{S}( \mathbb{C})$: $$\langle {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) u, v \rangle_{ \mathfrak{S}', \mathfrak{S}} : = \langle u, {\mathrm{Op}}^w_{ \Phi_1, k}( \overline{b}_k) v \rangle_{ \mathfrak{S}', \mathfrak{S}}.$$ Afterwards, since $ \mathcal{H}_k \subset \mathfrak{S}'( \mathbb{C})$, then for $g \in \mathcal{H}_k$, $ {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) g \in {\mathrm{Hol}}( \mathbb{C})$. Moreover, for $g \in \mathcal{H}_k$, using simple computations we prove that: $$\left\lbrace
\begin{split}
& {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) g(z+ 2 \pi) = u^k {\mathrm{Op}}^w_{ \Phi_1, k}(b_k) g(z), \\
& {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) g(z+i) = v^k e^{-ikz+k/2} {\mathrm{Op}}^w_{ \Phi_1, k}(b_k) g(z).
\end{split}
\right.$$
Let’s give a second definition of the complex Weyl quantization of the torus. This notion is analogous to the already existing one in the $ \mathbb{R}^2$-case (see for example [@HJ] or [@MR2952218]). We believe that it is the first time that such a contour integral is used in a context of the quantization of a compact phase space.
\[defi\_weyl\_complex\_tore\_2\] Let $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ be a function such that, for all $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_{\hbar}(z+2 \pi, w) = b_{\hbar}(z, w) = b_{\hbar}(z+i, w-1).$$ Define the complex Weyl quantization of the symbol $b_{ \hbar}$, denoted by $ {\mathrm{Op}}^w_{ \Phi_1}(b_{\hbar})$, by the following formula, for $u \in \mathfrak{S}( \mathbb{C})$: $${\mathrm{Op}}_{ \Phi_1}^w(b_{ \hbar}) u(z) = \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} b_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right) u(w) dw d \zeta,$$ where the contour integral is the following: $$\Gamma(z) = \left\lbrace (w, \zeta) \in \mathbb{C}^2; \zeta = \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z} \left( \dfrac{z+w}{2} \right) = - \Im \left( \dfrac{z+w}{2} \right) \right\rbrace.$$
This second definition expresses the fact that the complex Weyl quantization of $ \mathbb{R}^2$ (seen in Proposition \[prop\_transfor\_Bargmann\_et\_qauntif\_weyl\]) can be extended to a symbol defined on the torus. Similarly to Proposition \[prop\_Op\_Weyl\_k\_agit\_sur\_Hk\], we have the following property.
\[prop\_action\_quantif\_weyl\_complexe\_sur\_SS’(C)\] Let $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ be a function such that, for all $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_{ \hbar}(z + 2 \pi, w) = b_{ \hbar}(z, w) = b_{ \hbar}(z+i, w-1).$$ Then, $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$ can be extended into an operator which sends $\mathcal{H}_k$ on itself.
Let $a_{ \hbar}:= b_{ \hbar} \circ \kappa_{ \phi_1}$, then $a_{ \hbar} \in S( \mathbb{R}^2)$ and $ {\mathrm{Op}}^w(a_{ \hbar}): \mathcal{S}( \mathbb{R}) \longrightarrow \mathcal{S}( \mathbb{R})$. According to Proposition \[prop\_Tphi\_envoie\_S(R)\_sur\_SS(C)\], $ T_{ \phi_1} :\mathcal{S}( \mathbb{R}) \longrightarrow \mathfrak{S}( \mathbb{C})$ and according to Proposition \[prop\_transfor\_Bargmann\_et\_qauntif\_weyl\], we have: $${\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) = T_{ \phi_1} {\mathrm{Op}}^w(a_{ \hbar}) T_{ \phi_1}^* : \mathfrak{S}( \mathbb{C}) \longrightarrow \mathfrak{S}( \mathbb{C}) .$$ Afterwards, let $u, v \in \mathfrak{S}( \mathbb{C})$, then we have: $$\begin{aligned}
& \langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u, v \rangle_{ \mathfrak{S}, \mathfrak{S}} \\
& = \int_{ \mathbb{C}} \overline{({\mathrm{Op}}^w_{ \Phi_1} (b_{ \hbar}) u)(z)} v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{ \Gamma(z)} e^{(-i/ \hbar) \overline{(z-w) \zeta}} \overline{ b_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right)} \overline{ u(w)} dw d \zeta v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz), \\
& = \int_{ \mathbb{C}} \dfrac{C}{2 \pi \hbar} \int_{ \mathbb{C}} e^{(-i/ \hbar) \overline{(z-w)(- \Im(z+w/2))}} \overline{ b_{ \hbar} \left( \dfrac{z+w}{2}, - \Im \left( \dfrac{z+w}{2} \right) \right)} \overline{ u(w)} L(dw) v(z) e^{-2 \Phi_1(z)/ \hbar} L(dz),\end{aligned}$$ where we used the definition of the contour integral $ \Gamma(z)$ and where $C>0$ is a constant. Then, for $z \in \mathbb{C} \simeq \Lambda_{ \Phi_1}$, we have: $$b_{ \hbar}(z, - \Im(z)) = \sum_{(m, n) \in \mathbb{Z}^2} b_{m,n}^{ \hbar} e^{in \Re(z)} e^{2i \pi m \Im(z)}.$$ Thus, for $z \in \Lambda_{ \Phi_1}$, we obtain: $$\overline{b_{ \hbar}(z, - \Im(z))} = \sum_{(m, n) \in \mathbb{Z}^2} \overline{b_{m,n}^{ \hbar}} e^{-in \Re(z)} e^{-2i \pi m \Im(z)} = \overline{b}_{ \hbar}(z, - \Im(z)) .$$ Therefore, we can rewrite the integral as follows, for $u, v \in \mathfrak{S}( \mathbb{C})$: $$\begin{aligned}
& \langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u, v \rangle_{ \mathfrak{S}, \mathfrak{S}} \\
& = \dfrac{C}{2 \pi \hbar} \int_{ \mathbb{C}} \int_{ \mathbb{C}} e^{(i/ \hbar) ( \overline{z}- \overline{w})\Im(z+w/2)} \overline{b}_{ \hbar} \left( \dfrac{z+w}{2}, - \Im \left( \dfrac{z+w}{2} \right) \right) \overline{ u(w)} v(z) e^{-2 \Phi_1(z)/ \hbar} L(dw) L(dz).\end{aligned}$$ With a short computation, we prove the following equality: $$\begin{aligned}
\dfrac{i}{\hbar} \left( \overline{z} - \overline{w} \right) \Im \left( \dfrac{z+w}{2} \right) - \dfrac{2}{\hbar} \Phi_1(z)
& = \dfrac{i}{h}(z-w) \Im \left( \dfrac{z+w}{2} \right) - \dfrac{2}{\hbar} \Phi_1(w).\end{aligned}$$ Consequently, for all $u, v \in \mathfrak{S}( \mathbb{C})$, we have: $$\begin{aligned}
& \langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u, v \rangle_{ \mathfrak{S}, \mathfrak{S}} \\
& = \int_{ \mathbb{C}} \overline{u(w)} \dfrac{C}{2 \pi \hbar} \int_{ \mathbb{C}} e^{(i/ \hbar) (w-z)(-\Im(z+w/2))} \overline{b}_{ \hbar} \left( \dfrac{z+w}{2}, - \Im \left( \dfrac{z+w}{2} \right) \right) v(z) L(dz) e^{-2 \Phi_1(w)/ \hbar} L(dw) , \\
& = \int_{ \mathbb{C}} \overline{u(w)} \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{ \Gamma(w)} e^{(i/ \hbar) (w-z)\zeta} \overline{b}_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right) v(z) L(dz) e^{-2 \Phi_1(w)/ \hbar} L(dw) , \\
& = \int_{ \mathbb{C}} \overline{u(w)} ({\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar})v)(w) e^{-2 \Phi_1(w)/ \hbar} L(dw), \\
& = \langle u, {\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar}) v \rangle_{ \mathfrak{S}, \mathfrak{S}},\end{aligned}$$ where $ \Gamma(w) = \left\lbrace (z, \zeta) \in \mathbb{C}^2; \zeta = - \Im \left( \dfrac{z+w}{2} \right) \right\rbrace$. Since, for $v \in \mathfrak{S}( \mathbb{C})$, $ {\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar}) v \in \mathfrak{S}( \mathbb{C})$, then the operator $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$ is well-defined on $\mathfrak{S}'( \mathbb{C})$ by the following equality, for $u \in \mathfrak{S}'( \mathbb{C})$ and $v \in \mathfrak{S}( \mathbb{C})$: $$\langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u, v \rangle_{ \mathfrak{S}', \mathfrak{S}} = \langle u, {\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar}) v \rangle_{ \mathfrak{S}', \mathfrak{S}}.$$ As a result, the operator $ {\mathrm{Op}}^w_{ \Phi_1}(b_{\hbar})$ is well-defined on $ \mathcal{H}_k$ because it is a subspace of $ \mathfrak{S}'( \mathbb{C})$. Afterwards, since for $ g \in \mathcal{H}_k$, $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) g \in \mathfrak{S}'( \mathbb{C})$, then we have $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) g \in {\mathrm{Hol}}( \mathbb{C})$. Besides using simple computations, for $g \in \mathcal{H}_k$, we prove that: $$\left\lbrace
\begin{split}
& {\mathrm{Op}}^w_{ \Phi_1}(b_{\hbar}) g(z+ 2 \pi) = u^k {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) g(z), \\
& {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) g(z+i) = v^k e^{-ikz+k/2} {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) g(z).
\end{split}
\right.$$
To conclude this paragraph, we link Definition \[defi\_weyl\_complex\_tore\_1\] and Definition \[defi\_weyl\_complex\_tore\_2\].
Let $b_{ \hbar} =b_k \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ be a function such that, for all $(z, w) \in \Lambda_{ \Phi_1}$, we have: $$b_{ \hbar}(z+ 2 \pi, w) = b_{ \hbar}(z, w) = b_{ \hbar}(z+i, w-1).$$ Then: $${\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) = {\mathrm{Op}}^w_{\Phi_1,k}(b_k) \quad \text{on $ \mathcal{H}_k$.}$$
According to Equation , for $(z, w) \in \Lambda_{ \Phi_1}$, we can rewrite $b_{ \hbar}$ as follows: $$b_{ \hbar}(z, w) = \sum_{(m, n) \in \mathbb{Z}^2} b_{m,n}^{ \hbar} e^{in(z+iw)} e^{-2 i \pi m w}.$$ Consequently, we obtain, for $u \in \mathfrak{S}( \mathbb{C})$: $$\begin{aligned}
& {\mathrm{Op}}_{ \Phi_1}^w(b_{ \hbar}) u(z) \\
& =\dfrac{1}{2 \pi \hbar} \int \! \! \!\int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} \sum_{(m,n) \in \mathbb{Z}^2} b_{m,n}^{ \hbar} e^{in((z+w)/2+i \zeta)} e^{-2i \pi m \zeta} u(w) dw d \zeta, \\
& = \dfrac{1}{2 \pi \hbar} \int \! \! \!\int_{ \Gamma(z)} \sum_{(m,n) \in \mathbb{Z}^2} e^{(i/ \hbar)(z-w) \zeta} b_{m,n}^{ \hbar} e^{in((z+w)/2 +i \zeta)} e^{-i\pi m n/k} u \left( w - \dfrac{2 \pi m}{k} \right) dw d \zeta, \\
& =\dfrac{1}{2 \pi \hbar} \int \! \! \!\int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} \sum_{(m,n) \in \mathbb{Z}^2} b_{m,n}^{ \hbar} e^{-i\pi m n/k} e^{-n^2/2k} u \left( w + \dfrac{in}{k} - \dfrac{2 \pi m}{k} \right) dw d \zeta, \\
& = \dfrac{1}{2 \pi \hbar} \int \! \! \!\int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} ({\mathrm{Op}}^w_{\Phi_1,k} (b_k) u)(w) dw d \zeta, \\
& = {\mathrm{Op}}^w_{\Phi_1,k} (b_k) u(z),\end{aligned}$$ where we used the change of variables: $ \Gamma(z) \ni (w, \zeta) \longmapsto \left( w+ \dfrac{in}{k}, \zeta - \dfrac{n}{2k} \right) \in \Gamma(z)$. Therefore, for $u \in \mathfrak{S}'( \mathbb{C})$ and $v \in \mathfrak{S}( \mathbb{C})$, we have: $$\begin{aligned}
\langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) u, v \rangle_{ \mathfrak{S}', \mathfrak{S}} & = \langle u, {\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar}) v \rangle_{ \mathfrak{S}', \mathfrak{S}}, \\
& = \langle u, {\mathrm{Op}}^w_{ \Phi_1,k}( \overline{b}_k) v \rangle_{ \mathfrak{S}', \mathfrak{S}}, \\
& = \langle {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) u, v \rangle_{ \mathfrak{S}', \mathfrak{S}}.\end{aligned}$$ In others words, we have: $${\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) = {\mathrm{Op}}^w_{ \Phi_1,k}(b_k) \quad \text{on $ \mathfrak{S}'( \mathbb{C})$}.$$ Finally, since $ \mathcal{H}_k \subset \mathfrak{S}'( \mathbb{C})$, then by restriction and according to Propositions \[prop\_Op\_Weyl\_k\_agit\_sur\_Hk\] and \[prop\_action\_quantif\_weyl\_complexe\_sur\_SS’(C)\], we obtain the result.
Connections between the quantizations of the torus $ \mathbb{T}^2$ {#subsection_links_between_quantizations}
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In this paragraph, we relate the different notions of quantization of the torus. To do so, we follow these steps:
1. we recall the connection between a Berezin-Toeplitz operator and a complex pseudo-differential operator of the complex plane;
2. within the Berezin-Toeplitz setting, we relate the quantization of the torus to the quantization of the complex plane;
3. we establish a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus.
### Berezin-Toeplitz and complex Weyl quantizations of the complex plane
First, we recall the definition of the Berezin-Toeplitz quantization of a symbol on the complex plane and then the definition of the complex Weyl quantization of a symbol on $ \Lambda_{ \Phi_1}$ (see for example [@MR2952218]).
\[defi\_quantif\_BT\_du\_plan\_complexe\] Let $f_k \in S( \mathbb{C})$ be a function admitting an asymptotic expansion in powers of $1/k$. Define the Berezin-Toeplitz quantization of $f_k$ by the sequence of operators $T_{f_k} := (T_k)_{ k \geq 1}$, where for $k \geq 1$, $T_k$ is defined by: $$T_k = \Pi_{ \Phi_1,k} M_{f_k} \Pi_{ \Phi_1,k} ,$$ where $M_{f_k}: L^2_k( \mathbb{C}, \Phi_1) \longrightarrow L^2_k( \mathbb{C}, \Phi_1)$ is the multiplication operator by the function $f_k$ and where we recall that $ \Pi_{ \Phi_1, k}$ is the orthogonal projection of the space $L^2_k( \mathbb{C}, \Phi_1)$ on $H_k( \mathbb{C}, \Phi_1)$ defined in Proposition \[prop\_ecriture\_integrale\_Pi\_Phi1\].\
We call $f_k$ the symbol of the Berezin-Toeplitz operator $T_{f_k}$.
Let $b_{ \hbar} \in S( \Lambda_{ \Phi_1})$ be a function admitting an asymptotic expansion in powers of $ \hbar$. Define the complex Weyl quantization of $b_{ \hbar}$, denoted by $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$, by the following formula, for $u \in H_{ \hbar}( \mathbb{C}, \Phi_1)$: $${\mathrm{Op}}_{ \Phi_1}^w(b_{ \hbar}) u(z) = \dfrac{1}{2 \pi \hbar} \int \! \! \! \int_{ \Gamma(z)} e^{(i/ \hbar)(z-w) \zeta} b_{ \hbar} \left( \dfrac{z+w}{2}, \zeta \right) u(w) dw d \zeta,$$ where the contour integral is the following: $$\Gamma(z) = \left\lbrace (w, \zeta) \in \mathbb{C}^2; \zeta = \dfrac{2}{i} \dfrac{\partial \Phi_1}{\partial z} \left( \dfrac{z+w}{2} \right) = - \Im \left( \dfrac{z+w}{2} \right) \right\rbrace.$$
Recall now the result relating these two quantizations (see for example [@MR2952218 Chapter 13]).
\[prop\_Toeplitz=pseudo\_H(Phi1)\] $ $
1. Let $f_k \in S( \mathbb{C})$ be a function admitting an asymptotic expansion in powers of $1/k$. Let $T_{f_k} = (T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of symbol $f_k$. Then, for $k \geq 1$, we have: $$T_k = {\mathrm{Op}}^w_{ \Phi_1}( b_{ \hbar}) \quad \text{on $H_k( \mathbb{C}, \Phi_1)$},$$ where $b_{ \hbar} \in S( \Lambda_{ \Phi_1})$ is a function admitting an asymptotic expansion in powers of $ \hbar$ given by the following formula, for all $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$: $$b_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) ( f_k(z)).$$
2. Let $b_{ \hbar} \in S( \Lambda_{ \Phi_1})$ be a function admitting an asymptotic expansion in powers of $ \hbar$. Then, there exists $f_k \in S( \mathbb{C})$ a function admitting an asymptotic expansion in powers of $1/k$ such that for $k \geq 1$: $${\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) = T_k + \mathcal{O}(k^{- \infty}) \quad \text{on $H_{ \hbar}( \mathbb{C}, \Phi_1)$} ,$$ where $(T_k)_{k \geq 1} = T_{f_k}$ is the Berezin-Toeplitz operator of symbol $f_k$ and where, for all $N \in \mathbb{N}$ and for $z \in \mathbb{C}$, $f_k$ is given by: $$f_k(z) = \sum_{j=0}^N \dfrac{\hbar^j}{j!} \left( D_z D_{ \overline{z}} \right)^j (b_{ \hbar}(z)) + \mathcal{O}(\hbar^{N+1}) .$$
### Berezin-Toeplitz quantization of the torus and Berezin-Toeplitz quantization of the complex plane
In this paragraph, we study a Berezin-Toeplitz operator of the complex plane whose symbol is $2 \pi$-periodic with respect to its first variable and $1$-periodic with respect to its second variable. Previously, we looked at the action of a Berezin-Toeplitz operator of the complex plane on the spaces $ \mathfrak{S}( \mathbb{C})$ and $ \mathfrak{S}'( \mathbb{C})$.
\[prop\_toeplitz\_sur\_C\_defini\_sur\_S’\] Let $f_k \in S( \mathbb{C})$ be a function admitting an asymptotic expansion in powers of $1/k$. Let $T_{f_k} = (T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of symbol $f_k$. Then, for $k \geq 1$, we have:
1. $T_{k}$ can be defined as an operator which sends $\mathfrak{S}( \mathbb{C})$ on itself by: $$T_k v = \Pi_{ \Phi_1,k}(f_k v) \quad \text{for $v \in \mathfrak{S}( \mathbb{C})$,}$$ where $ \Pi_{\Phi_1,k}$ is seen as an operator which sends $ \mathfrak{S}( \mathbb{C})$ on itself.
2. $T_{k}$ can be extended into an operator which sends $\mathfrak{S}'( \mathbb{C})$ on itself by: $$\langle T_k u, v \rangle_{ \mathfrak{S}', \mathfrak{S}} = \langle u, \tilde{T}_k v \rangle_{ \mathfrak{S}', \mathfrak{S}} \quad \text{for $u \in \mathfrak{S}'( \mathbb{C})$ and for $v \in \mathfrak{S}( \mathbb{C})$,}$$ where $ (\tilde{T}_k)_{k \geq 1} =: T_{ \overline{f}_k}$ is the Berezin-Toeplitz operator of symbol $ \overline{f}_k$.
Since $T_{f_k} = (T_k)_{k \geq 1}$ is a Berezin-Toeplitz operator of the complex plane, then according to Proposition \[prop\_Toeplitz=pseudo\_H(Phi1)\], there exists $b_{ \hbar } \in S( \Lambda_{ \Phi_1})$ such that, for $k \geq 1$: $$T_k = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) \quad \text{on $H_k( \mathbb{C}, \Phi_1)$}.$$ Besides, according to Proposition \[prop\_transfor\_Bargmann\_et\_qauntif\_weyl\], we know that: $$T_{ \phi_1}^* {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) T_{ \phi_1} = {\mathrm{Op}}^w( b_{ \hbar} \circ \kappa_{ \phi_1}) : L^2( \mathbb{R}) \longrightarrow L^2( \mathbb{R}).$$ Since $b_{ \hbar} \circ \kappa_{ \phi_1} \in S( \mathbb{R}^2)$, then we have: $${\mathrm{Op}}^w(b_{ \hbar} \circ \kappa_{ \phi_1}) : \mathcal{S}( \mathbb{R}) \longrightarrow \mathcal{S}( \mathbb{R}) \quad \text{and} \quad {\mathrm{Op}}^w(b_{ \hbar} \circ \kappa_{ \phi_1}) : \mathcal{S}'( \mathbb{R}) \longrightarrow \mathcal{S}'( \mathbb{R}) .$$ Moreover, according to Propositions \[prop\_Tphi\_envoie\_S(R)\_sur\_SS(C)\], \[prop\_T\_(phi1)\*\_envoie\_S(C)\_sur\_S(R)\] and \[prop\_Tphi\_envoie\_S’(R)\_sur\_SS’(C)\], the Bargmann transform and its adjoint satisfy: $$\left\lbrace
\begin{split}
& T_{ \phi_1}: \mathcal{S}( \mathbb{R}) \longrightarrow \mathfrak{S}( \mathbb{C}) \quad \text{and} \quad T_{ \phi_1}^*: \mathfrak{S}( \mathbb{C}) \longrightarrow \mathcal{S}( \mathbb{R}), \\
& T_{ \phi_1}: \mathcal{S}'( \mathbb{R}) \longrightarrow \mathfrak{S}'( \mathbb{C}) \quad \text{and} \quad T_{ \phi_1}^*: \mathfrak{S}'( \mathbb{C}) \longrightarrow \mathcal{S}'( \mathbb{R}).
\end{split}
\right.$$ As a result, for $k \geq 1$, we obtain: $$T_k: \mathfrak{S}( \mathbb{C}) \longrightarrow \mathfrak{S}( \mathbb{C}) \quad \text{and} \quad T_k: \mathfrak{S}'( \mathbb{C}) \longrightarrow \mathfrak{S}'( \mathbb{C}) .$$ Then, by definition $ \Pi_{ \Phi_1,k} = T_{ \phi_1} T_{ \phi_1}^*$ and according to Proposition \[prop\_Tphi\_envoie\_S(R)\_sur\_SS(C)\], the operator $ \Pi_{ \Phi_1, k}$ can be extended into an operator which sends $ \mathfrak{S}( \mathbb{C})$ on itself. Since $ \mathfrak{S}( \mathbb{C}) \subset H_{ \hbar}( \mathbb{C}, \Phi_1)$ (see Remark \[rema\_S(C)\_inclus\_dans\_H(C,Phi1)\]), then for $v \in \mathfrak{S}( \mathbb{C})$ and for $f_k \in S( \mathbb{C})$, we have: $$\Pi_{ \Phi_1,k} v =v \quad \text{and} \quad \Pi_{ \Phi_1, k}(f_k v) \in \mathfrak{S}( \mathbb{C}).$$ Therefore, the Berezin-Toeplitz operator $T_{f_k} = (T_k)_{ k \geq 1}$ is defined as follows, for $k \geq 1$ and for $v \in \mathfrak{S}( \mathbb{C})$: $$T_k v = \Pi_{ \Phi_1,k} M_{f_k} \Pi_{ \Phi_1,k} v = \Pi_{ \Phi_1,k}(f_k v),$$ where $ \Pi_{ \Phi_1,k}$ is an operator which sends $ \mathfrak{S}( \mathbb{C})$ on itself. Finally, for $v = T_{\phi_1} \psi \in \mathfrak{S}'( \mathbb{C})$ and for $ u = T_{ \phi_1} \phi \in \mathfrak{S}( \mathbb{C})$, we have: $$\begin{aligned}
\langle T_k v, u \rangle_{ \mathfrak{S}', \mathfrak{S}} & = \langle {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) v, u \rangle_{ \mathfrak{S}', \mathfrak{S}} \quad \text{according to Proposition \ref{prop_Toeplitz=pseudo_H(Phi1)}}, \\
& = \langle v, {\mathrm{Op}}^w_{ \Phi_1}( \overline{b}_{ \hbar}) u \rangle_{ \mathfrak{S}', \mathfrak{S}} \quad \text{according to Proposition \ref{prop_action_quantif_weyl_complexe_sur_SS'(C)}}, \\
& =: \langle v, \tilde{T}_k u \rangle_{ \mathfrak{S}', \mathfrak{S}}.\end{aligned}$$ Then, according to Proposition \[prop\_Toeplitz=pseudo\_H(Phi1)\], for $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$, we have: $$b_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) ( f_k(z)).$$ Consequently, for $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$, we obtain: $$\overline{b}_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) (\overline{f_k(z)}).$$ In others words, the sequence of operators $( \tilde{T}_k)_{k \geq 1}$ is a Berezin-Toeplitz operator of symbol $ \overline{f}_k$.
Let $f_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for $ (x, y) \in \mathbb{R}^2$, we have: $$f_k(x+2 \pi, y) = f_k(x, y) = f_k(x, y+1).$$ Let $T_{f_k} = (T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of the complex plane of symbol $f_k$. Then, for $k \geq 1$, the operator $T_{k}$ is well-defined on $ \mathcal{H}_k$ according to Proposition \[prop\_toeplitz\_sur\_C\_defini\_sur\_S’\] since $ \mathcal{H}_k \subset \mathfrak{S}'( \mathbb{C})$.
The following proposition gives a connection between the orthogonal projection $ \Pi_{ \Phi_1,k}$ which appears in the definition of a Berezin-Toeplitz operator of the complex plane (see Definition \[defi\_quantif\_BT\_du\_plan\_complexe\]) and the orthogonal projection $ \Pi_k$ which appears in the definition of a Berezin-Toeplitz operator of the torus (see Definition \[defi\_quantif\_BT\_tore\]). This proposition is fundamental for understanding the relation between these two quantizations.
\[prop\_Pi\_Phi\_bien\_defini\_sur\_Hk\] $ $\
Let $ \Pi_{ \Phi_1, k}$ be the orthogonal projection of $L^2_k( \mathbb{C}, \Phi_1)$ on $ H_k( \mathbb{C}, \Phi_1)$. Then:
1. $ \Pi_{ \Phi_1, k}$ can be extended into an operator which sends $\mathcal{G}_k$ on $\mathcal{H}_k$ (defined in Subsection \[subsection\_context\]);
2. $ \Pi_{ \Phi_1, k}= {\mathrm{id}}$ on $ \mathcal{H}_k$.
Consequently, $ \Pi_{ \Phi_1, k}$ coincides with $ \Pi_k$ on $ \mathcal{G}_k$.
First, let’s prove that $ \Pi_{ \Phi_1,k}$ is well-defined on $ \mathcal{G}_k$. The main difficulty to prove this result comes from the fact that $ \mathcal{G}_k$ is not included in $ \mathfrak{S}'( \mathbb{C})$. Recall the formula defining $ \Pi_{ \Phi_1, k}$ for $g \in L^2_{k}( \mathbb{C}, \Phi_1)$ (see Proposition \[prop\_ecriture\_integrale\_Pi\_Phi1\]): $$\Pi_{ \Phi_1, k} g(z) = \int_{ \mathbb{C}} e^{-(1/ 4 \hbar)(z- \overline{w})^2} g(w) e^{-2 \Phi_1(w)/ \hbar} L(dw) .$$ Let $g \in \mathcal{G}_k$, by a simple computation, we notice that, for $(m, n) \in \mathbb{Z}^2$ and for $z \in \mathbb{C}$, we have: $$\left\lbrace
\begin{split}
& g(z+ 2 \pi m) = \left( u^k \right)^m g(z), \\
& g(z+in) = \left( v^k e^{-izk+kn/2} \right)^n g(z).
\end{split}
\right.$$ Then, for $g \in \mathcal{G}_k$, we can write an estimate of the integral defining $ \Pi_{ \Phi_1,k}$ as follows: $$\begin{aligned}
& \int_{ \mathbb{C}} \left| e^{-(1/ 4 \hbar)(z- \overline{w})^2} g(w) e^{-2 \Phi_1(w)/ \hbar} \right| L(dw) , \\
& = \sum_{m \in \mathbb{Z}} \int_{[0, 2 \pi] + i \mathbb{R}} \left| e^{-(1/ 4 \hbar)(z- (\overline{w + 2 \pi m}))^2} g(w + 2 \pi m) e^{-2 \Phi_1(w + 2 \pi m)/ \hbar} \right| L(dw) , \\
& = \sum_{m \in \mathbb{Z}} \int_{[0, 2 \pi] + i \mathbb{R}} \left| e^{-(1/ 4 \hbar)(z- \overline{w} - 2 \pi m)^2} \left( u^k \right)^m g(w) e^{-2 \Phi_1(w)/ \hbar} \right| L(dw), \\
& = \sum_{(m, n) \in \mathbb{Z}^2} \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- (\overline{w +in}) - 2 \pi m)^2} \left( u^k \right)^m g(w+in) e^{-2 \Phi_1(w+in)/ \hbar} \right| L(dw), \\
& = \sum_{(m, n) \in \mathbb{Z}^2} \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} \left( u^k \right)^m \left( v^k e^{-iwk+kn/2} \right)^n g(w) \right. \\
& \left. e^{-2 \Phi_1(w)/ \hbar} e^{-n^2/ \hbar} e^{-2n \Im(w) / \hbar} \right| L(dw), \\
& = \sum_{(m, n) \in \mathbb{Z}^2} \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} \left( u^k \right)^m \left( v^k \right)^n e^{-kn^2/2} e^{-ink \overline{w}} g(w) e^{-2 \Phi_1(w)/ \hbar} \right| L(dw).\end{aligned}$$ For all $z \in \mathbb{C}$, we have: $$\begin{aligned}
& \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} \left( u^k \right)^m \left( v^k \right)^n e^{-kn^2/2} e^{-ink \overline{w}} g(w) e^{-2 \Phi_1(w)/ \hbar} \right| L(dw) \\
& = \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} e^{-kn^2/2} e^{-ink \overline{w}} g(w) e^{-2 \Phi_1(w)/ \hbar} \right| L(dw) \\
& \leq \| g \|_{ \mathcal{G}_k}^2 \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} e^{-kn^2/2} e^{-ink \overline{w}} \right|^2 e^{-2 \Phi_1(w)/ \hbar} L(dw) \\
& \text{using Cauchy-Schwartz in $L^2( [0, 2 \pi]+ i [0,1], e^{-2 \Phi_1(z)/ \hbar} L(dz))$}, \\
& = \| g \|_{ \mathcal{G}_k}^2 \int_{[0, 2 \pi] + i [0, 1]} \left| e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} e^{-ink \overline{w}} \right|^2 e^{-kn^2} e^{-2 \Phi_1(w)/ \hbar} L(dw), \\
& \leq C \| g \|_{ \mathcal{G}_k}^2 e^{- k \Re (z^2)/2} e^{-2 k \pi^2 m^2} e^{-k n^2/2} \max \left( e^{2 \pi m k \Re(z)}, e^{2 \pi (m+1) k \Re(z)}\right) \max \left( e^{n k \Im(z)}, e^{(n+1) k \Im(z)}\right), \end{aligned}$$ where $C$ is a constant independent of $k$. We recognize the general term of a convergent series in $m$ and $n$, thus according to Fubini’s theorem, $ \Pi_{\Phi_1, k}$ is well-defined on $ \mathcal{G}_k$ by the following formula, for $g \in \mathcal{G}_k$: $$\begin{aligned}
& \Pi_{ \Phi_1, k} g(z) = \int_{ \mathbb{C}} e^{-(1/ 4 \hbar)(z- \overline{w})^2} g(w) e^{-2 \Phi_1(w)/ \hbar} L(dw), \\
& = \sum_{(m, n) \in \mathbb{Z}^2} \int_{[0, 2 \pi] + i [0, 1]} e^{-(1/ 4 \hbar)(z- \overline{w} +in - 2 \pi m)^2} \left( u^k \right)^m \left( v^k \right)^n e^{-kn^2/2} e^{-ink \overline{w}} g(w) e^{-2 \Phi_1(w)/ \hbar} L(dw).\end{aligned}$$ Now, since the range of $ \Pi_{\Phi_1,k}$ consists of holomorphic functions, then for $g \in \mathcal{G}_k$, $ \Pi_{ \Phi_1,k} g \in {\mathrm{Hol}}( \mathbb{C})$. Then, via the change of variables $w \longmapsto w + 2 \pi$, we prove with simple integral equalities that, for $g \in \mathcal{G}_k$, we have $\Pi_{ \Phi_1, k} g (z+ 2 \pi) = u^k \Pi_{ \Phi_1, k} g(z)$ and using the change of variables $w \longmapsto w +i$, we also obtain that, for $g \in \mathcal{G}_k$, $\Pi_{ \Phi_1, k} g(z+i) =v^k e^{-ikz+k/2} \Pi_{ \Phi_1, k} g(z)$. Finally, we recall that $ \Pi_{ \Phi_1, k} = {\mathrm{id}}$ on $ \mathcal{H}_k$ comes from Proposition \[prop\_TphiTphi\*=id\_sur\_Hk\_et\_Tphi\*Tphi=id\_sur\_Lk\].
We can now define the action of a Berezin-Toeplitz operator of the complex plane on the space $ \mathcal{H}_k$.
\[prop\_action\_toeplitz\_sur\_C\_sur\_Hk\] Let $f_k \in \mathcal{C}^{ \infty}_k( \mathbb{C})$ be a function such that, for $ z \in \mathbb{C}$, we have: $$f_k(z +2 \pi) = f_k(z) = f_k(z+i).$$ Let $T_{f_k} = (T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of the complex plane of symbol $f_k$. Then, for $k \geq 1$ and for $v \in \mathcal{H}_k$, we have: $$T_k v = \Pi_{ \Phi_1, k} (f_k v ),$$ where $ \Pi_{ \Phi_1, k}$ is seen as the operator which sends $ \mathcal{G}_k$ on $ \mathcal{H}_k$ (see Proposition \[prop\_Pi\_Phi\_bien\_defini\_sur\_Hk\]).
According to Proposition \[prop\_toeplitz\_sur\_C\_defini\_sur\_S’\], for $v \in \mathcal{H}_k$, for $ u \in \mathfrak{S}( \mathbb{C})$ and for $k \geq 1$, we have: $$\begin{aligned}
\langle T_k v, u \rangle_{ \mathfrak{S}', \mathfrak{S}} & = \langle v, \tilde{T}_k u \rangle_{ \mathfrak{S}', \mathfrak{S}}, \\
& = \langle g, \tilde{T}_k u \rangle \quad \text{with $g \in \mathfrak{S}^{-l}( \mathbb{C})$ for $l \in \mathbb{N}$ according to Proposition
\ref{prop_ecriture_elements_de_SS'(C)}}, \\
& = \langle g, \Pi_{ \Phi_1, k} M_{\overline{f}_k} \Pi_{ \Phi_1, k} u \rangle \quad \text{by definition of $ \tilde{T}_k$ on $ \mathfrak{S}( \mathbb{C})$}, \\
& = \langle \Pi_{ \Phi_1, k} M_{\overline{ \overline{f}}_k} \Pi_{ \Phi_1, k} g, u \rangle \quad \text{since $ \Pi_{ \Phi_1,k}^* = \Pi_{ \Phi_1,k}$}.\end{aligned}$$ Thus, for $v \in \mathcal{H}_k$ and for $k \geq 1$, we obtain, according to Proposition \[prop\_Pi\_Phi\_bien\_defini\_sur\_Hk\]: $$T_k v = \Pi_{ \Phi_1, k} M_{f_k} \Pi_{ \Phi_1, k} v = \Pi_{ \Phi_1, k} (f_k v ).$$
We deduce from Proposition \[prop\_action\_toeplitz\_sur\_C\_sur\_Hk\] and Proposition \[prop\_Pi\_Phi\_bien\_defini\_sur\_Hk\], a result which relates a Berezin-Toeplitz operator of the complex plane and a Berezin-Toeplitz operator of the torus. To our knowledge, this fact is new in the literature and it is also fundamental to prove Theorem \[theoA\].
\[prop\_toeplitz\_sur\_C\_egal\_toeplitz\_sur\_Hk\] Let $f_k \in \mathcal{C}^{ \infty}_k( \mathbb{C})$ be a function such that, for $ z \in \mathbb{C}$, we have: $$f_k(z +2 \pi) = f_k(z) = f_k(z+i).$$ Let $T_{f_k}^{ \mathbb{C}} = ( T_k^{ \mathbb{C}})_{k \geq 1}$ be the Berezin-Toeplitz operator of the complex plane of symbol $ f_k$ and let $T_{f_k}^{ \mathbb{T}^2} = (T_k^{ \mathbb{T}^2})_{k \geq 1}$ be the Berezin-Toeplitz operator of the torus of symbol $f_k$. Then, for $k \geq 1$, we have: $$T_k^{ \mathbb{C}} = T_k^{ \mathbb{T}^2} + \mathcal{O}(k^{- \infty}) \quad \text{on $ \mathcal{H}_k$}.$$ Consequently, a Berezin-Toeplitz operator of the complex plane whose symbol is periodic coincides with a Berezin-Toeplitz operator of the torus.
### Berezin-Toeplitz quantization and complex Weyl quantization of the torus
Finally, we are able to establish and to prove the following proposition which corresponds to Theorem \[theoA\].
Let $ f_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for $(x, y) \in \mathbb{R}^2$, we have: $$f_k(x + 2 \pi, y ) = f_k(x, y) = f_k(x, y+1) .$$ Let $T_{f_k} = ( T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of the torus of symbol $ f_k$. Then, for $k \geq 1$, we have: $$T_k = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) + \mathcal{O}(k^{- \infty}) \quad \text{on $ \mathcal{H}_k$},$$ where $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ is defined by the following formula, for $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$: $$\label{eq_formule_fk_en_fonction_ah}
b_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) (f_k(z)).$$ Besides, $b_{ \hbar}$ satisfies the following periodicity conditions, for $(z,w) \in \Lambda_{ \Phi_1} $: $$b_{ \hbar}(z+ 2 \pi,w) = b_{ \hbar}(z,w) = b_{ \hbar}(z+i,w-1).$$
Since in particular $f_k \in S( \mathbb{C})$, if we denote by $T_{f_k}^{ \mathbb{C}} = (T_k^{ \mathbb{C}})_{k \geq 1}$ the Berezin-Toeplitz operator of the complex plane of symbol $f_k$, then according to Proposition \[prop\_Toeplitz=pseudo\_H(Phi1)\], there exists $b_{ \hbar} \in S( \Lambda_{ \Phi_1})$ such that, for $k \geq 1$, we have: $$T_k^{ \mathbb{C}} = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) \quad \text{on $H_{ \hbar}( \mathbb{C}, \Phi_1)$},$$ where $b_{ \hbar}$ is given by the following formula, for $z \in \Lambda_{ \Phi_1} \simeq \mathbb{C}$: $$b_{ \hbar}(z) = \exp \left( \dfrac{1}{k} \partial_z \partial_{ \overline{z}} \right) (f_k(z)).$$ Since $ \mathfrak{S}( \mathbb{C})$ is included into $H_{ \hbar}( \mathbb{C}, \Phi_1)$ (see Remark \[rema\_S(C)\_inclus\_dans\_H(C,Phi1)\]) then, by restriction, we obtain: $$T_k^{ \mathbb{C}} = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) \quad \text{on $\mathfrak{S}( \mathbb{C})$}.$$ By duality, we have: $$T_k^{ \mathbb{C}} = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) \quad \text{on $\mathfrak{S}'( \mathbb{C})$}.$$ Since $ \mathcal{H}_k \subset \mathfrak{S}'( \mathbb{C})$, we obtain: $$T_k^{ \mathbb{C}} = {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar}) \quad \text{on $\mathcal{H}_k$}.$$ Notice that the periodicity conditions on $f_k$ and Equation give the periodicity conditions on $b_{ \hbar}$, consequently, $ {\mathrm{Op}}^w_{ \Phi_1}(b_{ \hbar})$ is well-defined on $ \mathcal{H}_k$.\
Finally, according to Proposition \[prop\_toeplitz\_sur\_C\_egal\_toeplitz\_sur\_Hk\], if we denote by $ T_{f_k}^{ \mathbb{T}^2} = (T_k^{ \mathbb{T}^2})_{k \geq 1}$ the Berezin-Toeplitz operator of the torus of symbol $f_k$, we have, for $k \geq 1$: $$T_k^{ \mathbb{C}} = T_k^{ \mathbb{T}^2} + \mathcal{O}(k^{- \infty}) \quad \text{on $ \mathcal{H}_k$}.$$ This concludes the proof.
Thanks to Theorem \[theoA\] and Proposition \[prop\_Toeplitz=pseudo\_H(Phi1)\], we deduce the following corollary.
\[coro\] Let $ f_k \in \mathcal{C}^{ \infty}_k( \mathbb{R}^2)$ be a function such that, for $(x, y) \in \mathbb{R}^2$, we have: $$f_k(x + 2 \pi, y ) = f_k(x, y) = f_k(x, y+1) .$$ Let $T_{f_k} = ( T_k)_{k \geq 1}$ be the Berezin-Toeplitz operator of the torus of symbol $ f_k$. Then, for $k \geq 1$, we have: $$T_{ \phi_1}^* T_k T_{ \phi_1} = {\mathrm{Op}}^w(a_{ \hbar}) + \mathcal{O}(\hbar^{\infty}) \quad \text{on $ L^2( \mathbb{R})$},$$ where $a_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}(\mathbb{R}^2)$ is defined by the following formula: $$a_{ \hbar} = b_{ \hbar} \circ \kappa_{ \phi_1},$$ where $b_{ \hbar} \in \mathcal{C}^{ \infty}_{ \hbar}( \Lambda_{ \Phi_1})$ is defined by Equation . Besides, $a_{ \hbar}$ satisfies the following periodicity conditions, for $(x, y) \in \mathbb{R}^2$: $$a_{ \hbar}(x+ 2 \pi, y) = a_{ \hbar}(x, y) = a_{ \hbar}(x, y+1).$$
[^1]:
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Approaches to solving the rigid receptor problem by identifying a minimal set of flexible residues during ligand docking.
Using fixed receptor sites derived from high-resolution crystal structures in structure-based drug design does not properly account for ligand-induced enzyme conformational change and imparts a bias into the discovery and design of novel ligands. We sought to facilitate the design of improved drug leads by defining residues most likely to change conformation, and then defining a minimal manifold of possible conformations of a target site for drug design based on a small number of identified flexible residues. The crystal structure of thymidylate synthase from an important pathogenic target Pneumocystis carinii (PcTS) bound to its substrate and the inhibitor, BW1843U89, is reported here and reveals a new conformation with respect to the structure of PcTS bound to substrate and the more conventional antifolate inhibitor, CB3717. We developed an algorithm for determining which residues provide 'soft spots' in the protein, regions where conformational adaptation suggests possible modifications for a drug lead that may yield higher affinity. Remodeling the active site of thymidylate synthase with new conformations for only three residues that were identified with this algorithm yields scores for ligands that are compatible with experimental kinetic data. Based on the examination of many protein/ligand complexes, we develop an algorithm (SOFTSPOTS) for identifying regions of a protein target that are more likely to accommodate plastically to regions of a drug molecule. Using these indicators we develop a second algorithm (PLASTIC) that provides a minimal manifold of possible conformations of a protein target for drug design, reducing the bias in structure-based drug design imparted by structures of enzymes co-crystallized with inhibitors.
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Influenced by nostalgic meals and cooking for loved ones, Megan Davies has written this book for the eco-minded home cook. She includes invaluable tips on how to make ingredients stretch; from potato peel chips to pickled cucumber and beets. Megan also features ways to turn leftovers into a new meal, such as a Roasted Fennel, Chive, and Dill Pasta Bake or Frittata, both from a leftover Raw Fennel, Chive, and Dill Salad. Multi-tasking recipes include brunch and late-night dishes such as Bircher Pancakes and Sweet Potato Baked Eggs. Suppers for Sharing that can be scaled up to feed a crowd or down for a more intimate occasion range from Roasted Squash with Almonds and Tarragon to the best Roast Chicken recipe with Pan Pastry Croutons (plus, of course, ways to use up any uneaten chicken!) From On the Side accompaniments and stunning Sweet Things such as Pot Luck Tarte Tatin this collection of ingenious recipes will give you all the inspiration you need to help run a more sustainable home kitchen, reduce your carbon footprint, and make the sort of small changes at home that can make a big difference to our world.
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ISBN - 13:9781788791991
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Around the world the highest mountains have been climbed, walked and measured and lists produced for various purposes. Often such lists provide a challenge for the more adventurous to climb all of the mountains on any particular list. In the UK the highest mountains are those over 3000 feet (914 metres). England has 4 such mountains (all in the Lake District), Wales has 15 (all in Snowdonia) and Scotland 284 (in the Highlands north of Glasgow and Edinburgh and on the islands of Mull and Skye). It should be obvious from these numbers that England and Wales can each be completed in a weekend. Scotland on the other hand presents an extended challenge, usually taking years to complete.
Dave Butcher and his wife Jan have always walked the hills and this increased considerably when they moved to the north of England when his job with Ilford Photo was relocated to Cheshire, 15 miles southwest of Manchester. The big peaks in the Lake District and Snowdonia became just a couple of hours away. The text and table below is a summary of Dave Butchers mountain hiking diary / log for the Munro’s of Scotland.
As we became more familiar with these areas we looked further afield and the Scottish Highlands beckoned. Before our first real hill walking foray we had been to Scotland once together and one other time for me (on the Pennine Way, a walk from Edale in the English Peak District to Kirk Yetholm in the Scottish borders) so we had an idea as to how beautiful the mountains were.
We chose Torridon for our first trip mainly because the local bookshop sold the OS 1:25000 map of the area and it sounded spectacular. It was amazing and still remains one our favourite areas in Scotland. It is near the west coast, with Loch Torridon being a sea loch, and has lots of streams, with mountains soaring up above the glens one behind another, with the slopes covered in white quartz which makes them look snow covered all the time, even in summer. Our first Munro was Beinn Alligin (the Jewel of Torridon as it is known) on 10th June 1984. We loved it and after that spent time every year in the Highlands walking the mountains.
It was not long before we heard about the Munro’s, I bought a copy of Munro’s Tables, the bible for all Munroists – the definitive list of every Munro (the list is maintained by the Scottish Mountaineering Council, SMC) and this steered us towards the 3000 foot high mountains and planning quite long days, backpacks and even a few ski mountaineering trips.
We quickly became Munroists, keenly seeking out the next challenge from new mountains and new areas. It took us almost 12 years to complete the ascent of all 284 unaided (yes we even walked up underneath the Aonach Mor ski lifts carrying our skis so that we could climb Aonach Mor and then ski down). Our last Munro was Ben More on the Isle of Mull which we did in gales, rain and low cloud on 1st June 1996 and became Compleatists 1569 (Jan) and 1570 (Dave). Click here to see us celebrating on Ben More in the rain!
Friday night would often find us doing a full day’s work and then driving up to the Highlands so that we were well placed for a weekend of walking the Munro’s. We quickly found a great bed and breakfast (Breadalbane House) in the village of Killin that was easy to reach in 5 hours from the Ilford factory in Cheshire and this became our staging post for trips further north and our base for the Southern Highlands. Danni Grant, the landlady, looked after us like we were part of the family.
We learnt the skills needed to venture onto the hills in winter (on a Glenmore Lodge course in the Cairngorms and with Alan Adshead, a friend from Ilford who sadly died in a climbing accident) with ice axes and crampons and I became experienced in the use of maps and compass for navigation, even in atrocious weather conditions. In total we climbed about 20% (one in five) in winter conditions (reasonable snow cover) and the rest spread throughout the months of the year.
A summary from our Munro log is shown below to give you an idea of the commitment needed to complete all of the Munro’s.
It would take too long to add notes from all the trips we made to Scotland but the image gallery has many dozens of images from, and of, the Munro’s and each has additional information as a caption. To find them on subsequent visits go to the UK Landscapes Gallery and scroll down to Scotland or type Munro into the search box. The thumbnail images just have a title but if you click on the thumbnail to see the enlarged image there is a detailed caption for each of them describing the location, etc.
I hope you enjoy browsing through them and if you are interested in an image of a specific Munro not displayed in the Landscape Gallery then please ask. I have an extensive library of images taken on our Munro trips but just a few are displayed here.
Equivalent to climbing Everest 18 times! | https://www.davebutcher.co.uk/scotland-munros-mountain-hiking-diary/ |
One of universally loved stories that college professors tell undergraduate students in intriguingly entitled courses like “Introduction to Language and Culture” is “The Great Snow Debate”.
As the story goes, one year a set of intrepid yet naïve scientists plowed their way through the frozen tundra to investigate and preserve the language of the indigenous peoples. During their interviews, the researchers noted that the Inuit have a surprisingly large number of names for snow…light snow, wet snow, heavy snow, deep snow… You name it, the Inuit have a moniker for it. Excited by their discovery, some of the scientists decided to concentrate their efforts on snow names. And the more they asked, the more names they uncovered. Soon, more researchers arrived to capture what seemed to be an inexhaustible onomastic store of names. The publications multiplied, careers were made, and contests were waged. No sooner had one researcher reported having identified dozens of snow names than another one came claiming to have recovered more.
At the end of it, the race for names finally collapsed under the weight of the scholarly attention placed upon the sometimes flimsy methodological scaffolding. The fact that interviewees were paid by the name meant that native speakers invented some of the names they reported. Another problem was the simple fact that the researchers grossly underestimated the number of names which many non-indigenous people have for snow. Consider, for example, the breadth and depth of names which meteorologists in Minnesota or skiers in Boulder have to describe snow.
This does not mean to say that all was lost. Despite the lasting chagrin which surrounds the Great Snow Debate, one of the benefits was the development and spread of one of the leading theories of modern linguistics: The Linguistic Theory of Relativity. One of the most important tenet of this theory is that people’s interaction with their physical environment will have a significant affect upon the names they develop to label their perceptual experience. | https://www.americannamesociety.org/the-great-snow-debate/ |
CUMMING -- For the second year, a Forsyth County elementary school student raised money for animals in need.
Earlier this month, 9-year-old Madison Thieke, who is getting ready to enter fourth grade at Cumming Elementary School, donated more than $2,000 she raised at a second annual yard sale to the Dawsonville Humane Society.
“My goal was $1,500, and I got $2,275,” she said.
Thieke raised the money through selling items and donations from those who came to the yard sale and presented a homemade ceremonial check for the donation.
She worked to raise the money with her grandmother, who lives in Dawson County and explained to her the importance of no-kill shelters, like the one to which she donated.
“When I moved down here, she was 5 and I introduced her to the animal shelter in Dawsonville and explained to her that it was a no kill shelter and no matter what happened they wouldn’t put the dogs to sleep,” said her grandmother, Suzzie Thieke. “I explained to her that through people donating money and doing things that keeps that going.”
Though she might be young, Thieke has earned a lot of experience helping the shelter.
At age 6, she collected and recycled cans in her neighborhood to raise money and the following year held a bake sale, along with her grandmother, at a family member’s yard sale, earning more than $200 for the shelter.
The next year she wanted to try something new.
“Last year, she wanted to do bigger,” said her grandmother. “She and I decided to do a yard sale.”
“Last year, I raised $1,225,” Thieke added.
Leftovers from the yard sale were donated to the society’s resell shop.
Thieke’s fundraising doesn’t stop with the yard sale.
“I used my birthday money and my Christmas money and walked dogs and I bathed dogs,” she said. “I made $376.”
“She has a heart of gold, and she loves animals,” Suzzie Thieke said, adding that Madison’s younger brother was starting to help out, as well. “They go in and love on the cats and feed the cats … and give them attention they don’t get. We just started walking dogs a little bit.”
Thieke said she doesn’t plan to stop doing things for the animals. | https://www.forsythnews.com/local/lifestyles/cumming-4th-grader-helps-local-animal-shelter/ |
Q:
Exponential Integrals: What is the derivative of the exponential integral $E_n(x)$?
The exponential integral is defined as: $$E_n=x^{n-1} \int_x^\infty{\frac{e^{-t}}{t^n}}dt,$$ and its derivative with respect to $x$ should be: $$\frac{dE_n}{dx}=-E_{n-1}.$$
However, when I tried to prove this relation I got stuck at: $$\frac{dE_n}{dx}=(n-1)x^{n-2}\int_x^\infty{\frac{e^{-t}}{t^n}}dt-\frac{1}{xe^x},$$ and I don't know how this should equal $-E_{n-1}.$
A:
Note that: $$\int_{x}^{\infty}\frac{e^{-t}}{t^{n}}dt = \frac{e^{-t}t^{-n+1}}{-n+1}\bigg{|}_{x}^{\infty}+\frac{1}{n-1}\int_{x}^{\infty}\frac{e^{-t}}{t^{n-1}}dt.$$ Thus,
$$(n-1)x^{n-2}\int_{x}^{\infty}\frac{e^{-t}}{t^{n}}dt = \frac{1}{xe^{x}}-x^{n-2}\int_{x}^{\infty}\frac{e^{-t}}{t^{n-1}}dt = \frac{1}{xe^{x}}-E_{n-1}.$$
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Dollar-a-Dish on NBC San Diego!
Thank you to NBC San Diego's Catherine Garcia for getting our Food 4 Kids Backpack Program fundraiser on the 5 p.m. newscast yesterday. She did a great job with it, as did Diane Stopford and the folks at Alchemy, who are one of our Dollar-a-Dish participants (And kudos to our PR wizard, Peyton Robertson of Bay Bird, Inc.).
If you missed the segment when it aired (I can tell you my mother didn't. I got the "turn on the TV now" call!), you can watch it on the NBC San Diego website. | http://www.sandiegofoodstuff.com/2010/08/dollar-dish-on-nbc-san-diego.html |
The 100-Foot Journey
Why “The 100-Foot Journey”?
I have been a watercolour artist for many years, with intensive training in watercolours and drawing, and working mainly in the studio. About one year ago I decided to add oil as a medium, since it is perfect for painting plein air painting (meaning outside vs. the studio). Ross Whitlock inspired me to do a painting a day, well “inspired” is not the right word: “This is your homework and I want to see 60 paintings by Christmas!”. At that time it was the beginning of November. Three colours were allowed (yellow, red, blue, plus white) and to begin with only the palette knife. So, off I went, trying to paint everything that came in front of my canvas. Step by step I became more confident with colour mixing, selection of subjects, looseness, composition, etc.
“The 100-Foot Journey” shows my journey in daily paintings and how my skills developed. Daily painting became my passion, whether I paint watercolours or oils doesn’t matter, as long as I can paint something. With my busy family life I find it difficult to concentrate on big painting projects and have a better sense of accomplishment with the small, but gorgeous paintings. And guess what, by now I have accomplished way more than 60 paintings, although I must admit sometimes life gets in the way and I cannot paint every day, but almost. | https://www.nicolerussellart.com/2018/05/06/the-100-foot-journey/ |
According to the latest World Bank projections, global growth is expected to slow by 2.7 percentage points between 2021 and 2024—a slowdown more than double what was recorded between 1976 and 1979. To avoid a prolonged period of stagflation, political leaders around the world should The world focus their efforts on five main goals.
Just over two years later global recession (a) Due to the COVID-19 pandemic, the worst since World War II, the global economy is once again threatened, this time by the dual phenomenon of high inflation and low growth. , with potentially destabilizing consequences for low- and middle-income countries. Unless the supply is increased significantly.
In a context marked by war in Ukraine, accelerating inflation and rising interest rates, global economic growth is expected to decline in 2022. The latest forecasts from the World Bank, Posted today, reflects a significant deterioration in the outlook: a sharp slowdown in global growth is expected this year, falling from 5.7% in 2021 to 2.9%. This number also represents a decrease of about a third compared to January 2022 forecastWhich forecast growth of 4.1% this year. This downward adjustment is mainly due to the sharp rise in energy and food prices, along with turbulence (a) Supply and trade resulting from the war in Ukraine and the necessary normalization of interest rates just beginning.
The COVID-19 pandemic has already brought about big hit Income growth and poverty reduction in developing economies. Repercussions of the war in Ukraine Highlighting Challenges (a) For many of these countries, which should achieve 3.4% overall growth in 2022. This is barely half the rate recorded in 2021 and well below the average observed between 2011 and 2019. Similarly, projections of Growth on average – income countries in 2022 were significantly reduced, losing 1.3 percentage points from the January forecast.
Real per capita income in 2023 will remain below its pre-COVID-19 level in about 40% of developing economies. For many countries, the Recession It will be difficult to avoid. with restrictionsnatural gas supplywhich particularly affects fertilizer production and power grids in the poorest countries, announcing strong increases in world production will be necessary to restore non-inflationary growth.
inflation (a) It has now reached multi-decade highs in many countries and the expansion in supply is likely to be slow, and the pace of price growth is likely to remain higher for longer than is currently expected.. Global growth is expected to slow by 2.7 percentage points between 2021 and 2024 – a slowdown more than double what was recorded between 1976 and 1979. It is likely to remain sluggish over the next decade due to lower investment in most parts of the world. knowing that
Moreover, the external public debt (a) Developing economies are now at record levels. Most of this debt is owned by private creditors, and most of it comes at variable interest rates that can rise sharply. With the tightening of global financing conditions and the devaluation of currencies, Over-indebtednesswhich was previously confined to low-income countries, is spreading to middle-income countries.
budget support (a) that were put in place in 2020 to combat the pandemic will have been removed, although debt levels will remain high. With the end of accommodative policies, it will be important to reduce inequality and seek higher incomes for all by using fiscal and monetary policy tools that strengthen supply chains, small businesses, and the process of allocation of capital., is another major detrimental factor for the developing world. In addition, in the next two years, the majority of
But the current conditions are also present Big differences from the 1970s (to me). The dollar, which was very weak in the seventies, is strong (To today. While they had quadrilateral (a) in 1973-1974 and Double (a) in 1979-80, and oil price (a) It is today only two-thirds of its 1980 level (adjusted for inflation). The balance sheets of major financial institutions are also generally strong, while they were fragile in the 1970s.
And economies in general are more resilient than they were in the 1970s, with less structural rigors in wages and labor markets, while political leaders are in a better position today to fend off stagflation. Monetary policy frameworks are more credible: central banks in advanced economies, such as those in many developing countries, have price stability state clearly defined. This factor, combined with the fact that existing technology and capital can lead to an exponential increase in supply, has helped solidify long-term inflation expectations.
To reduce the risks of stagflation, world leaders will need to take targeted action.
- First, they will have to limit the damage to those affected by the war in Ukraine. This will require coordinating responses to the crisis, including providing emergency food, medical and financial assistance in war-torn areas, burden-sharing housing, support, and even resettlement of refugees and internally displaced persons.
- Second, political leaders will have to fight to rise (a) Oil and food prices. It is necessary to increase the supply of basic food and energy products. Markets anticipate that simple announcements of future supply will help lower prices and inflation expectations. All countries should strengthen their social safety nets and avoid import and export restrictions that drive up prices.
- Third, there is an urgent need to intensify debt relief efforts. Debt-related vulnerabilities in low-income countries are already very high before the epidemic. while the Over-indebtedness The risks to the global economy are spreading to middle-income countries, and will only increase in the absence of rapid, comprehensive and widespread relief.
- Fourth, officials should enhance health preparedness and step up efforts to contain COVID-19. L ‘Expand vaccination campaigns (a) Low-income countries should be the world’s top priority.
- Fifth, the transition to low-carbon energy sources must be accelerated. to relieve Reliance on fossil fuels (a) There will be a need for more investment in electricity networks, cleaner energy sources and improved energy efficiency. National leaders must create climate-smart regulatory frameworks, adjust incentive structures and strengthen land-use rules.
There is good reason to believe that when the war in Ukraine ends, efforts to rebuild Ukraine’s economy and revitalize global growth – including at the level of the World Bank Group – will be intensified. Meanwhile, political leaders must mitigate other threats to global development: rising food and energy prices, persistent inflationary pressures, increasingly dangerous debt burdens, growing inequality and instability, and the many risks arising from climate change.
this is amphitheater Based on the June 2022 edition of the World Bank Group’s Global Economic Prospects. | https://www.thevalleypost.com/show-at-the-heart-of-the-solution-against-stagflation-david-malpass/ |
In the face of ‘geostrategic competition’ between world powers, the Pacific region is at a critical juncture in its history. Dame Meg Taylor, Secretary General of the Pacific Islands Forum Secretariat, says there ‘has never been a more important, nor opportune, time’ to act as one Blue Pacific continent.
The Pacific Islands Secretariat’s Dame Meg Taylor.
We have witnessed a recasting of geostrategic competition and cooperation under the rubric of the ‘Indo-Pacific’, with the Indian and Pacific Oceans increasingly being seen by a number of our traditional partners as one single strategic space.
The Pacific Islands have rarely featured in the discussions except from a perspective of vulnerability to China’s influence and therefore as a part of the Indo-Pacific that needs to be ‘secured’ by, and for, external partners.
Exercising stronger strategic autonomy as one Blue Pacific continent requires being clear on who we are as the Pacific.
Only once we clearly claim our collective geography, identity and resources will we be able to effectively secure the place and agency of the Pacific in the fast-changing global context.
By framing Our Islands as one ocean continent we can enhance the strategic autonomy of our region by identifying and leveraging the value that the Blue Pacific continent holds—not just for us, but for those seeking access to it.
This value is derived from various sources: our control of almost 10 per cent of UN voting rights; the world’s largest ocean resources including US$6 billion in fisheries; ‘unvalued’ ecosystem services and biodiversity; our people and our rich cultural heritage; our geostrategic positioning; and a range of geo-economic opportunities.
A concrete task is securing our maritime boundaries. The Pacific Island Countries and Territories manage 20 per cent of the world’s ocean in their Exclusive Economic Zones.
Of the 47 shared boundaries in the Pacific, 35 treaties have been concluded.
The settlement of maritime boundaries provides certainty to the ownership of our ocean space, as Pacific people taking control of our domain.
It is critical to managing our ocean resources, biodiversity, ecosystems and data—as well as for fighting the impacts of climate change.
The continued rise of China, and the unconventional politics of the Trump administration in the US, are creating great shifts and uncertainty.
A range of actors portray China’s growing presence in our region as an either/or predicament for the Pacific; that is, insisting that the Pacific must choose sides.
Such divisive politics are inconsistent with our region.
We need to talk more openly about what the rise of China means for our region, and indeed how we can engage with all partners in a manner that advances our development, our security, prosperity and harmony.
Key parts of the discussion might include the pros and cons of engaging as a region with China’s Belt and Road Initiative—or as I prefer to rename it for our purposes because of the reality of our place, the Belt and Seaways Initiative; closer dialogue with China (and other dialogue partners) over the removal of harmful fisheries subsidies; or how partners can work together to ensure sustainable, resilient infrastructure for the Pacific.
Closely linked to discussions on China is the issue of debt sustainability.
There have been suggestions that the leaders of Pacific Island Countries don’t understand the risks of taking on loans from China.
The single experience of Sri Lanka is used to warn us of the threat of Chinese loans, serving as a signal to the region.
However, many of the large infrastructure projects in the region are with development partners (including China) and the large financial institutions of our own region.
We must seek to understand the underlying reasons that enable such issues to emerge in the first place.
Despite all the aid received in the Pacific we still remain economically vulnerable and dependent on the good will of others.
We need to ensure that the assistance we are receiving builds on the capacities of our nations to enhance their socio-economic self-reliance and resilience.
That means investing in capacity and infrastructure.
These building blocks of growth are expensive and require long-term commitments in supporting complementary institutional and policy development.
It is one of the reasons why Chinese assistance is attractive to the Pacific. In response to China’s growing influence, we see competing infrastructure initiatives emerging from Japan, the US and Australia.
While this could be good news for the Pacific, we must tackle the problem at its cause, rather than getting drawn into the political jousting of others.
Dame Meg Taylor is the Secretary-General to the Pacific Islands Forum Secretariat. This is an excerpt from her recent speech to the 2018 State of the Pacific Conference at the Australian National University. | https://www.businessadvantagepng.com/opinion-the-pacific-islands-should-speak-as-one/ |
About Our Program
Students applying for admission to the Master of Arts in History should have some background in history, though not necessarily a BA in the subject.
Our program prepares students for a wide variety of professions, including teaching, government service, museum management, and historic preservation, as well as further degree work in history, law, librarianship and business. The department encourages applications from individuals of any age interested in resuming their education.
History students will
- Identify people, events, and processes significant to their courses of study
- examine similarities and differences across chronologies, geographies, and themes
- explain how past peoples understood their worlds and how those understandings shaped the ways they acted
- analyze the range of social, cultural, political, and economic possibilities available to people in particular contexts
- analyze why change occurs
2.1 Inquiry and Analysis – History students will
- develop a creative, focused, and manageable question for historical research
- synthesize evidence representing a variety of perspectives
- explain the challenges of constructing historical narratives using incomplete and contradictory evidence
- formulate a thesis and conclusion substantiated by primary and secondary source analysis
- critique alternative conclusions
2.2 Critical Thinking – History students will
- identify and analyze the central issues, arguments, and points of view in primary and secondary sources
- evaluate authors’ arguments and assess their evidence and conclusions
- critique their own and others’ assumptions and the contexts in which they develop those assumptions
- use the concept of historiography, in order to compare and contrast a variety of scholarly texts
- analyze the ways the histories historians write are products of particular historical contexts
2.3 Written Communication – History students will
- establish the context, audience, and purpose of their written assignments
- master the conventions of historical writing, including: clear paper organization (thesis, evidence, conclusion); logical paragraph organization; clear, direct, and engaging language; proper citation methods, using Chicago style
- compose papers employing narrative, descriptive, and analytical writing to convey their historical knowledge and analytical skills
2.4 Information literacy – History students will
- determine the types of sources that are relevant to a research question
- locate and evaluate appropriate materials for historical research, using book catalogs (Skyline, Prospector, WorldCat), article databases (particularly America: History and Life, Historical Abstracts, and JSTOR), and interlibrary loan
- demonstrate understanding of the ethical and legal issues surrounding the use of published and unpublished materials, including what constitutes plagiarism and how to cite sources
3.1 Intercultural knowledge and competence – History students will
- evaluate how their cultural biases inform their understandings of history
- evaluate the ways that historians of different cultural perspectives produce different histories
- interpret historical evidence with consideration to historical actors’ various cultural perspectives
3.2 Ethical reasoning and action – History students will
- analyze the ethical issues embedded in historical events and processes
- evaluate different ethical choices present in historical decision-making
- evaluate the ethical assumptions of the texts they read
History students will
- demonstrate connections between different courses and readings
- synthesize academic experiences with their experiences outside the classroom
- seek out applications of their historical knowledge and skills beyond the classroom
Plans of Study
Guidelines for all graduate programs can be found in the Graduate Student Handbook
The Master of Arts in History requires 36 semester hours (12 courses + Comprehensive Exam).
Public History with a Thesis Public History with a Project
Concentrations and Research Focuses
Select your major and minor fields from two of the following three groups. You may choose a major field in any of the following three groups. Then, your minor field must concentrate on one of the other two groups. Example: Major in Latin American History, with a minor in Cultural History.
- East Asia
- Latin America
- Mexico
- South America
- Middle East
- Europe
- Germany
- France
- Britain
- The Mediterranean
- United States
- Colonial
- Early Republic
- Nineteenth Century U.S.
- The West
- Twentieth Century US Foreign Policy
- Colorado
- Global (See also thematic fields)
- Atlantic World
- Pacific World
Note: Majors in Public History must follow the Plan of Study for Public History. | https://clas.ucdenver.edu/history/graduate |
Simulation: Cutting the Corner on Machine Learning
The Author: Shilpa Mesineni is currently a senior research engineer at the American Bureau of Shipping (ABS), leading efforts in System Simulation and Digitalization Projects in Digital Engineering team. Shilpa holds a MSc in Electrical Engineering from the University of New Orleans (UNO) at New Orleans, Lousiana and B.Tech in Electronics and Communications Engineering from the Jawaharlal Nehru Technological University at Hyderabad, India
Shilpa has an Electrical Automation and Control systems
As the offshore oil and gas industry becomes more competitive, it actively pursues increased efficiency through innovative approaches while streamlining production, reducing costs, and improving safety. Many companies are looking at digitization to insulate themselves from market shocks, remain profitable at lower oil prices, and generate competitive advantage during recovery. The path forward lies in leveraging machine learning-based technologies that are maturing quickly and are being adopted across the value chain. The use of Machine Learning (ML) models is particularly promising for the resolution of problems involving processes that are not completely understood or where it is not feasible to run mechanistic models at desired resolutions in space and time. With these growing technologies and solutions to complex science and engineering problems require novel methodologies that can integrate physics-based modeling approaches with state-of-the-art ML techniques. This paper provides an overview of the use of physics-based simulation models to test, correct, and retest ML algorithms under a range of scenarios and at a scale not practicable with physical testing.
Machine Learning (ML) ML is the use and development of computer systems that use algorithms and statistical models to analyze and draw interferences from patterns in data to learn and adapt automatically through experience. It is seen as a subset of artificial intelligence. ML algorithms build a mathematical model based on sample data, known as “training data”, in order to make predictions or decisions without being explicitly programmed to do so. ML models search through a sample space of possible mathematical models, utilize methods to discern and adapt between such model choices with data to arrive at final model that best describes the data. This decision is based often times on pre-defined criterion to guide the search process.
Choosing the right ML model ML models are beginning to play an important role in advancing discoveries in complex engineering applications that are traditionally dominated by mechanistic models. Selecting the right ML algorithm to accommodate the complexities of a real system that are not completely understood can be challenging. No single ML algorithm fits all scenarios. There are several factors that can affect the selection of a ML algorithm, including the complexity of the problem, type of data (structured, un-structured, texts, time series, images, etc.), and latency requirement for decision making (real-time, or offline analysis). A high volume of real-world data and testing is usually required but the data may not always be available. However, the absence of high-quality labelled data may result in inaccurate ML algorithms. In such scenarios, solving complex problems requires novel methodologies that can integrate prior knowledge from physics-based models with ML techniques.
Hybrid physics ML Model The hybrid modeling approach is a way of combining physics-based models with ML where both can operate simultaneously and can be decoupled in some way to improve the performance separately. See fig1.
Improving predictions beyond physics First principal physics-based models are extensively used in wide range of engineering applications. However, physics-based models require strict boundaries and assumptions to create an idealized approximation of reality. These approximations can be hindered by incomplete knowledge of certain processes or unaccounted physical phenomena which can introduce additional bias. Often, the input parameters may have to be estimated through observed data.
When provided enough data, ML neural network models have shown to outperform physics-based models where complexity prohibits the explicit programming of the system’s exact physical nature. ML models can find structure and patterns in complex problems where physical processes are not fully understood. In resolving and improving the performance of complex engineering systems, physics-based models can be combined with state-of-the-art ML models to leverage their complementary strengths. Such integrated physics-ML models are expected to better capture the dynamics of complex systems and advance the understanding of underlying physical processes.
Hybrid ML model approaches There are many ways to combine physics-based models with ML models, to train the ML model or to solve complex engineering applications. The three most common hybrid approaches (combining ML with SIM models) are discussed below:
ML after the simulation run The results from the physics-based simulation model can be used as data input for the ML model in a system of approaches. This can be used in advanced ML algorithm training, such as fully autonomous operations. The simulation results incorporate known knowledge and constraints from physics domain (e.g. parameter correlations, decision metrics, importance and weighting on parameters, etc.) into the ML algorithm to constrain the learning space. You train the models not only with real system (Physics Models) but let the ML “drive” through complex solutions.
ML prior to simulation ML models can be used to provide parameterization inputs to physics-based models. In a system of systems, ML may be applied on certain systems, but other connected systems may be represented with a physics-based model. Complex physics-based models often use an approach for parameterization to account for missing physics. In parameterization, specific complex dynamical processes are replaced by simple physics approximations whose associated parameter values are estimated from data. The failure to correctly identify parameter values can make the model less robust, and errors that result can also feed into other components of the entire physics-based model and deteriorate the modeling of important physical process. The ML model can be used to learn new parameterizations directly from observations and/or model simulation. A major benefit that can come from using ML based parameterization is the reduction of computation time compared to traditional physics-based simulations.
ML assisted simulation This architecture explores the best combination of using ML and data emanating from the physics-based simulations. In this approach ML can be integrated into physics-based simulation by connecting the output of the Simulation model into particular nodes of the ML or by making the ML model learn aspects of the physics-based system and apply that learning directly. This approach effectively provides data-driven decisions to the entire system of systems.
Data Driven ML control for DP Dynamic Positioning (DP) can be one example where the operation of vessel systems can benefit from ML algorithms.
Ships that are involved in safety-critical operations related to drilling, cargo-transfer, subsea crane operations and pipe-laying typically have an extended actuator setup to allow for redundancy in case of system errors. During such operations, the vessel is required to control its position and heading. DP of ships is a control mode that seeks to maintain a specific position (station keeping) or perform low-speed maneuvers.
A data driven control ML approach can be applied to resolve the problem of DP in an over-actuated ship subject to environmental forces. This control approach improved the overall ships performance criteria leaving the human decision to the top of the hierarchical DP control structure. A hybrid modelling theme as discussed in this paper, that combines the ML data model with the prior knowledge physics-based model can help find solutions and improve the performance.
SIM tool to train and test ML Physics based SIM models can help, to train and test the ML algorithms by:
1.Simulating near real world data 2.Testing different ML algorithms 3.Testing the quality of the data sources
This approach provides full control over the data provided to the ML, both parameters contained within the data and the volume and frequency of the data. The SIM tool neural network module as shown in fig 5 & fig 6, can help to train and test the ML algorithm. If output of the ML is not acceptable changes can be made to the algorithm and the ML module can use these changes to upgrade and segment the rules in the ML model. This process continues until a satisfactory output is obtained. The Neural Network Builder in SIM tool creates neural networks from simulation models or external data sets. Neural networks provide a functional and compact representation of input-output relationships which mimics the complex behavior of the underlying model. Creation of a neural network is executed in 3 main steps as shown in fig 6:
1.Import of simulation results from a physics model 2.Use results from sim model for training of multi-layer networks 3.Validation of trained networks with fidelity metrics and plots
Once trained, the network can be exported as a submodel or as an ONNX file. The Marine and Offshore industries are adopting state of the art ML concepts to improve vessel efficiency and performance. This paper describes the combination of ML and physics-based simulation as a hybrid approach fostering intelligent analysis of applications that can benefit from a combination of data-driven and knowledge-based approaches. Choosing the right ML algorithm to accommodate the complexities of a real system is challenging. A high volume of real-world data and testing is usually required in choosing the ML, but the data may not always be available. In this paper ABS describes how SIM tools can be used in solving the problem of real-world data hunt by connecting prior knowledge physics-based models with an ML model. We explain how these sim tools can be used to train the ML, test the ML, and make corrections and retest the ML algorithm under a range of scenarios at a scale not practicable with physical testing. A hybrid model workflow connecting physics-based model to train the multi-layer neural network (one of the most known ML algorithms) is shown as an example. ABS as a Class Society believes simulation will play an important role as the industry adopts ML.
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Video streaming has finally become the killer application the research community has been hoping it would become for the past 10-15 years. Today, the statistics around video streaming services are mind-boggling; for instance, YouTube is currently streaming 6 billion hours of video each month. In this article, we discuss some of the technological and non-technological factors that have resulted in the phenomenal uptake of streaming video. We discuss how the shift from traditional viewing platforms such as broadcast/cable television is impacting viewing habits, and how these changes can potentially lead to substantial innovation in content creation, delivery, and the interaction between the content’s consumers and the interfaces delivering it to their devices.
Data61; NICT; Video streaming; Peer-to-Peer; Content Distribution Networks; Personalization
https://doi.org/10.1109/MIC.2014.16
English
nicta:7624
Mahanti, Anirban. The Evolving Streaming Media Landscape. IEEE Internet Computing. 2014-01-15; 18(1):4-6. <a href="https://doi.org/10.1109/MIC.2014.16" target="_blank">https://doi.org/10.1109/MIC.2014.16</a>
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