text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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The systems we offer form the heart of our solutions. Our ingenuity in every solution comes from our deep understanding of these systems through years of rigorous research and implementation. Our diverse and rich portfolio of advanced engineering and digital technology systems helps us in delivering mission critical solutions every single time.
Having designed, deployed and supported systems built on various technologies, we have a track record of delivering world-class systems either as part of a turnkey solution or on a purely supply basis. | {
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} | 31 |
class UsersController < ApplicationController
skip_before_filter :require_user, only: [:new, :create, :confirm_email]
before_filter :require_no_user, only: [:new, :create]
def confirm_email
if user = User.find_using_email_token(params[:token], 5.days)
was_unactivated = !user.activated
user.confirm_email!
UserSession.create(user)
if was_unactivated
redirect_to root_url, success: 'Your account has been activated.'
else
redirect_to root_url, success: 'Your email address has been confirmed.'
end
else
render action: :bad_confirmation_email
end
end
def create
@user = User.new new_user_params
if @user.save
@user.maybe_deliver_email_confirmation! self
redirect_to root_url, success: "Thanks for joining. Please check your email for instructions on activating your account."
else
render action: :new
end
end
def edit
end
def new
@user = User.new
end
def update
@user = current_user
if @user.update_attributes existing_user_params
if @user.maybe_deliver_email_confirmation! self
notice =
'An email has been sent to your new address. Please follow the instructions in ' +
'that email to confirm your new address. Until then, we will continue to use ' +
'your old address.'
else
notice = 'Account settings saved.'
end
if params[:user][:password].present?
notice << ' Your password has been changed. Please log back in with your new password.'
redirect_to login_url, success: notice
else
redirect_to account_url, success: notice
end
else
render action: :edit
end
end
private
def existing_user_params
params.require(:user).permit([:new_email] + common_user_params)
end
def new_user_params
permitted = [:email] + common_user_params
permitted << [:login] if User.has_login_field?
params.require(:user).permit(permitted)
end
def common_user_params
[:password, :password_confirmation, :time_zone]
end
end | {
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} | 8,892 |
#Bali. Look at that huge plate, a big portion of really tasty & delicious Balinese Paela from @CasaLunaUbud served with lots of seafood. Well this plate sure makes my hand look small isn't it?
6 Cool Restaurants In BALI That Provide COOKING CLASSES! | {
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} | 2,310 |
Enjoy working on your ride? Considered a set of quality ramps? Then check out our 2018 best car ramps buyers guide.
When repairing or servicing your car or truck at home, most projects will require you to raise your vehicle a few inches to give you a better view and allow you access to those hard to reach areas. To maintain the vehicle, you will often have to access the underside.
This can be done just using car jacks but this can be awkward and unsafe. There is also no need to buy an expensive garage lift, as a car ramp can be trusted to do the job for a fraction of the cost.
You can easily make oil changes, flush or drain the radiator or complete other maintenance, saving you time and effort.
There are many types of car ramps available from low profile to heavy duty, depending on the size of the vehicle and what type of automotive repairs you want to make.
Headlamps looking a little dim? Our best headlight bulb guide is written just for you.
Therefore, it is paramount that you choose the product most suited to your car's weight or you may put additional strain on the ramps or damage your vehicle.
So, to help you navigate through this crowded market, please refer to our comprehensive car ramp buyers' guide below. The products are all available to purchase from Amazon.
1. Ground clearance: Not every ramp is suitable for every car. You must ensure that it can fit under the car without scratching it. Low profile vehicles, such as sports cars, race cars, and sedans need ramps that are longer and lower, so you gradually drive onto them, without scratching the car's undercarriage.
Similarly, high profile vehicles require high profile ramps, which are shorter.
2. Car weight capacity: Car ramps often have weight ratings, so if your automobile is light, then most ramps will fit it. However, if you put a heavy car on a lightweight ramp, then you may break it and place yourself in danger. The load rating for most products is between 6000 and 10000 pounds.
The weight capacity of car ramps is the maximum weight that they can safely support. Manufacturers show this on their products but the specification may vary as some car ramps state the single axle rating instead of the Gross Vehicle Weight (GVW). In this case, you will need to know the weight of each axle on your car before choosing suitable ramps.
Generally, the front axle is heavier than the rear because the engine is at the front of the car, leading to vehicle weight bias. But regardless of the rating used on the car ramps, you must never exceed the weight capacity because you will not only damage the device but you will put anyone working on the car at risk.
However, the Gross Vehicle Weight rating does not mean that the ramps will hold all that weight but that they are designed to lift and hold up one half of a vehicle that weighs that much. Therefore, a 12000 GVW rating means that the devices will hold up to 6000 pounds for the pair. Conversely, a single axle rating means exactly what it says. For example, a 3000 pound single axle rating means that the pair of car ramps will raise up to 3000 pounds safely.
3. Car ramp inclination angle: One you have worked out the weight load of the ramp, then you should also check the angle of lift. Low slung cars will need a low profile ramp with a small lift angle. Car ramps for higher profile vehicles, such as trucks and SUVs tend to have steeper inclines.
4. Quality of construction material: Whether the device is made of metal, foam, rubber, plastic or wood it will have various advantages. It is important to select the best material because it has a bearing on the durability, weight, portability and storage of the product.
5. Width of car ramps: Please ensure that the ramps are wide enough to accommodate your car tires and leave a few inches on either side. So before choosing the car ramps, you will need to measure the width of the tires. This will give you a margin for error and will reduce slippage when you are loading your car onto the ramps.
6. Surface of car ramps: The surface of your chosen device should provide enough traction for your vehicle to enable you to drive up the ramps whatever the weather and terrain. A smooth surface will lead to slippage, even if the ramps have a low incline. To combat this problem, you can opt for a non-slip ramps, which feature punched plates for extra traction.
So, if you want to use them in snowy or wet conditions, then you should choose this surface. The staggered, serrated holes on the surface of these ramps offer optimal traction for automobile, whatever the inclination angle.
7. Ease of use: The ramp should be simple to use. It should slide smoothly beneath the tires and be easy to remove. The best products are also easy to carry and maintain.
8. Price and warranty: Even if the car ramp is at the high end of the price range, you should still consider it, as the more expensive option will have more features. A decent warranty from a reputable company is also essential. But make sure you choose a car ramp that is built to last.
Remember, safety first when using car ramps! You must never use them on an unstable surface or an incline of more than around five degrees and use another support device. There is only a small margin of error, so the ramps need to be positioned correctly, or your vehicle may go off the edge.
Always put the brake on and use tire chocks to stop the car from rolling back off the ramp. Unless you are using non-collapsible car ramps, then it is advisable to place jack stands (a tire with a board across it) or another device beneath the vehicle to provide extra protection.
This will prevent injury to yourself or others, if the ramp were to collapse with someone under the car. When driving onto the amps, use a smooth throttle. It is not advisable to put your foot down, as you may drive over the edge of the ramps.
The ramps may even shoot back under the vehicle and damage it. It may help to use the parking brake alongside the throttle to keep the car from lurching forward.
So, now that you know what to consider before buying them and how to use them safely, please take a look at our review of the best car ramps for 2018.
This set of two low clearance vehicle ramps from the highly respected manufacturer Rhino Ramps incorporate a patented polymer internal core support, therefore, they are tough and built to last. They are designed to handle loads of up to six tons, which should be sufficient for most personal vehicles.
The ramps feature a sophisticated CoreTRAC tread pattern and non-skid base to reduce slippage, keeping them stable on most surfaces, except sealed concrete and epoxy coated garage floors. In this case, you can always invest in a couple of floor mats to put underneath them, to give more friction. The weight of the vehicle will prevent the ramps from slipping off the mats.
They are ideal for cars, SUVs, vans, and trucks but they are not suitable for high clearance vehicles. Their inclination angle of 17 degrees means they will elevate your vehicle 6.5 inches off the ground, so you can easily fit an oil drain pan underneath to change the oil.
So, you can quickly change the tires, the oil or do other vehicle maintenance. They feature a wide stance for maximum stability, have excellent weight distribution and proven strength.
The Rhino Ramps are of high quality and meet the Portable Automotive Lifting Device (PALD) standard and they are a best-seller due to their affordability. They are easy to store away as they nest together, saving space and despite their heavy build, they are very portable, so you can take them with you or keep them in your car.
However, they are not suitable for use in extremely cold temperatures, as they flex and crack and they are not UV protected. But if you store them in your garage or workshop, you will get many years out of them.
These sleek and simple twin ramps are designed for sports cars and are recommended by petrol heads, car clubs and racing organisations. They have a smooth angle of inclination, making them perfect for low profile vehicles, such as: BMWs, Corvettes, Jaguars, Mustangs, Porches, Cadillacs, minivans and pick-up trucks. If you want to reduce the approach angle even further Race Ramps also offer a range of Xtenders.
They have a strong and lightweight construction, giving them a solid core with no holes or hollows and ensuring durability, even if you overload them. The ramps are made using a high density expanded polystyrene foam and then sprayed with a hybrid polyurea coating to increase their strength and protect them from automotive chemicals.
They have a firm and safe grip, making them non-slip even on ice. Race Ramps are very compact and remain stable and are not prone to collapse like other plastic models. They can be used on most flat surfaces, including: grass, sand, dirt and are suitable for most tire widths, even truck tires.
Their innovative two-piece design allows you to remove the extended incline part of the ramp that you drive up.
Therefore, you can access the underside of your car from the side without the ramp getting in the way. They will lift your vehicle up to seven inches high and have no sharp edges, so will not damage your garage floor.
Race Ramps are easy to use even for first-timers and the integrated straps at the end of each ramp make them easy to lift, carry, position and store. They do not rust and will not be damaged if you leave them outside in bad weather.
These high quality solid steel heavy gauge ramps are much tougher than their plastic, foam and wooden rivals. They feature a powder coated black surface, which not only looks stylish but protects them from the elements.
This durable finish also prevents corrosion and wear and tear on the devices. Measuring in at 35 inches long, they also include side rails, so you can reverse your car onto them safely. The punched holes on the ramps and wide trail give excellent traction, when coupled with wheel chocks. Nicky Nice ramps are not designed for low profile cars, as they are quite high.
These lightweight but sturdy plastic ramps are suitable for small and medium-sized cars. Their solid and tough construction features a parabolic arch running through the centre. This unique addition enhances the strength and weight distribution of the base.
Cleverly designed to reduce slippage by including an open grid pattern on the ramps, they are perfect for snowy and wet conditions, as water, snow and dirt will just seep through. There is also little surface area on the ramps for particles to collect, so you can safely leave them outside.
This set has a smooth angle of inclination, so they should not be used on low profile vehicles, as the ridges on the outer edge may rub the car's underside.
The Scepter ramps will allow you to increase the ground clearance by up to six inches and are light enough to carry around. They measure 35 inches long and 9.8 wide, so do not take up much storage space.
These super affordable twin ramps are designed for leveling trailers and RVs. Finished in eye-catching canary yellow, they will certainly stand out if you use them at the roadside. They can be used to elevate your vehicle to three different heights, hence the name.
So, you can raise your trailer three, seven or eight inches, depending on the amount of clearance you need and put it on a more level footing. This will enable the sensors in your black and grey water holding tanks to get an accurate reading and they will allow other appliances, such as the refrigerator to operate more efficiently.
They feature a non-slip surface, ensuring your tires remain in place. To start the leveling process, check which side of the vehicle needs to be lifted using the sensors, then place the Tri-Leveler in front or behind the wheel.
Then slowly drive onto the ramps, until you reach the desired level. It may be useful to have a helper outside the trailer who can direct you. Then when you are finished, just slowly reverse down the ramps.
Constructed from lightweight resin, the Drive-on Tri-levelers are portable enough to carry around with you when not in use and compact enough to stow away in your vehicle. They are easy to operate, as they include built-in handles.
Their versatility means they are suitable to be used on most tires. The build quality and durability of this product is assured by the Manufacturer's One Year Limited Warranty. However, they are one-piece devices, so can only be used on one tire at a time and they are not the best products to make oil changes.
But don't let that put you off as these leveling blocks are very competitively priced and give great value for money if you just want to complete basic maintenance.
This may be a pricier option but it is well worth the extra cost. These are the best ramps for high profile vehicles and are easy to set up and dismantle without using tools.
They feature an ingenious four-piece detachable assembly system: first, you position the ramps, then drive onto them, then you set the locking system on each ramp to release the slopes, leaving these pieces to serve as safety tire chocks for other wheels.
This all-in-one kit is very convenient, as it is highly portable and frees up much needed storage space. Magnum are one of the leading manufacturers of car parts and accessories and this product is made right here in the USA from tough, environmentally friendly material. What more could you want!
As the name suggests, you are getting a good deal here. This two piece mini ramp set is designed to increase the ground clearance for low profile cars, allowing you to perform occasional repairs or display your vehicle.
They will raise your car 2.5 inches off the ground, which should be sufficient for you to access the underside. You can then fit car jacks under the vehicle and you can lift it up to the height you need. Discount Ramps are not suitable for oil changes but if you own a sports car or rally vehicle, then you can elevate the automobile during winter storage to keep it clean.
They are constructed from heavy duty, durable plastic with a honeycomb pattern, which has the combined advantage of deterring long-term moisture build-up and reducing the product's weight. The ramps are compact with each weighing less than five pounds, making them convenient to carry around with you and to store away.
Their integrated wheel grooves work as tire stops, which help to stabilize the vehicle and the non-slip surface provides firm support and reliability. They are suitable for domestic and professional use alike and will save you time and work.
Car ramps are essential tools for vehicle repairs and maintenance, as they are reliable and functional. They elevate the front or rear of your vehicle, giving you access to all those hard to reach spots.
You need to make sure you take your time and choose the right product, otherwise it may not hold your automobile and it may end up crushing whoever is beneath it.
DIY fan? Often work under your car? Then check out our car ramp and car dollies buying guides.
Car ramps are becoming increasingly popular as they have many uses and allow you to perform regular checks and repairs. They are stable and simple to use, even for beginners and are lightweight and easy to store.
Their price has come down in recent years and they are much safer than using car jacks alone because they offer a large surface area to place your tires and add traction to the ground and support the vehicle's wheels.
They will also save you money on professional mechanics, as you will be able to complete your own basic vehicle maintenance. If used responsibly, they will protect you and your vehicle from harm.
So, I hope by following our handy guide, you have been able to choose the right product for you! | {
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This template of High School Physics Sound Waves Worksheet Acdabc Bbcpc intended for Waves Worksheet Template can be saving in your computer by clicking resolution image in Download by size:. There is many more templates you can found in other pages. Don't forget to rate and comment after you save this template & wallpaper.
Related Posts of "High School Physics Sound Waves Worksheet Acdabc Bbcpc intended for Waves Worksheet Template" | {
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Фердинанд Хоринек (; , 1896, София — 17 января 1982, там же) — болгарский шахматист чешского происхождения.
Семья богатых промышленников Хоринеков переселилась в Болгарию вскоре после окончания Русско-Турецкой войны 1877—1878 гг. Отец шахматиста Антон Хоринек вместе с Димитром Беровым основал в 1906 г. ткацкую фабрику «Беров и Хоринек».
Фердинанд Хоринек получил высшее образование в Аахене. Окончил Рейнско-Вестфальский технический университет по специальности инженер-текстильщик.
В 1928 г. Хоринек стал председателем Болгарского шахматного союза. Вскоре эта организация была распущена. В 1931 г. Хоринек стал председателем вновь образованного Болгарского шахматного союза, предшественника Болгарской федерации шахмат.
В 1936 г. входил в состав сборной Болгарии на неофициальной шахматной олимпиаде в Мюнхене. Был заявлен вторым запасным, но не сыграл ни одной партии.
Книги
Ръководство по шахматната игра, изд. 1923 г. — первая болгарская шахматная книга.
Къси шахматни партии и задачи.
Примечания
Литература
Личностите в Българския шахмат, БФШ, 2008.
Ссылки
Родившиеся в 1896 году
Родившиеся в Софии
Умершие 17 января
Умершие в 1982 году
Умершие в Софии
Персоналии по алфавиту
Спортсмены по алфавиту
Шахматисты по алфавиту
Шахматисты Болгарии
Шахматисты XX века
Участники шахматных олимпиад
Шахматные функционеры
Выпускники Рейнско-Вестфальского технического университета Ахена
Шахматные арбитры Болгарии | {
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Robert Clifton Weaver (December 29, 1907 – July 17, 1997) was an American economist, academic, and political administrator who served as the first United States secretary of housing and urban development (HUD) from 1966 to 1968, when the department was newly established by President Lyndon B. Johnson. Weaver was the first African American to be appointed to a US cabinet-level position.
Prior to his appointment as cabinet officer, Weaver had served in the administration of President John F. Kennedy. In addition, he had served in New York State government, and in high-level positions in New York City. During the Franklin D. Roosevelt administration, he was one of 45 prominent African Americans appointed to positions and helped make up the Black Cabinet, an informal group of African-American public policy advisers. Weaver directed federal programs during the administration of the New Deal, at the same time completing his doctorate in economics in 1934 at Harvard University.
Background
Robert Clifton Weaver was born on December 29, 1907, into a middle-class family in Washington, D.C. His parents were Mortimer Grover Weaver, a postal worker, and Florence (Freeman) Weaver. They encouraged him in his academic studies. His maternal grandfather was Dr. Robert Tanner Freeman, the first African American to graduate from Harvard in dentistry.
The young Weaver attended the M Street High School, now known as the Dunbar High School. The high school for blacks at a time of racial segregation had a national reputation for academic excellence. Weaver went on to Harvard University, where he earned a Bachelor of Science and Master of Arts degree. He also earned a Doctor of Philosophy degree in Economics, completing his doctorate in 1934.
Career
Government
Washington
In 1934, Weaver was appointed as an aide to United States Secretary of the Interior Harold L. Ickes. In 1938, he became special assistant to the US Housing Authority. In 1942, he became administrative assistant to the National Defense Advisory Commission, the War Manpower Commission (1942), and director of Negro Manpower Service.
With a reputation for knowledge about housing issues, in 1934 the young Weaver was invited to join President Franklin D. Roosevelt's Black Cabinet. Roosevelt appointed a total of 45 prominent blacks to positions in executive agencies, and called on them as informal advisers on public policy issues related to African Americans, the Great Depression and the New Deal.
Weaver drafted the U.S Housing Program under Roosevelt, which was established in 1937. The program was intended to provide financial support to local housing departments, as a subsidy toward lowering the rent poor African Americans had to pay. The program decreased the average rent from $19.47 per month to $16.80 per month. Weaver claimed the scope of this program was insufficient, as there were still many African Americans who made less than the average income. They could not afford to pay for both food and housing. In addition, generally restricted to segregated housing, African Americans could not necessarily take advantage of other subsidized housing.
Chicago
In 1944, Weaver became director of the Commission on Race Relations in the Office of the Mayor of Chicago. In 1945, he became director of community services for the Chicago-based American Council on Race Relations through 1948.
New York
In 1949, Weaver become director of fellowship opportunities for the John Hay Whitney Foundation. In 1955, Weaver the first Black State Cabinet member in New York when he became New York State Rent Commissioner under Governor W. Averell Harriman. In 1960, he became vice chairman of the New York City Housing and Redevelopment Board.
Washington: HUD
In 1961, Weaver became administrator of the United States Housing and Home Financing Agency (HHFA).
After election, Kennedy tried to establish a new cabinet department to deal with urban issues. It was to be called the Department of Housing and Urban Development. Postwar suburban development, following the construction of highways, and economic restructuring had drawn population and jobs from the cities. The nation was faced with a stock of substandard, aged housing in many cities, and problems of unemployment.
In 1961, while trying to create HUD, Kennedy had done everything short of promising the new position to Weaver. He appointed him Administrator of the Housing and Home Finance Agency (HHFA), a group of agencies which Kennedy wanted to raise to cabinet status.
When Dr. Weaver joined the Kennedy Administration, whose Harvard connections extended to the occupant of the Oval Office, he held more Harvard degrees – three, including a doctorate in economics – than anyone else in the administration's upper ranks.
Some Republicans and southern Democrats opposed the legislation to create the new department. The following year, Kennedy unsuccessfully tried to use his reorganization authority to create the department. As a result, Congress passed legislation prohibiting presidents from using that authority to create a new cabinet department, although the previous Republican Dwight D. Eisenhower administration had created the cabinet-level U.S. Department of Health, Education, and Welfare under that authority.
He contributed the compilation housing bill in 1961. He took part in lobbying for the Senior Citizens Housing Act of 1962.
In 1965, Congress approved the department. At the time, Weaver was still Administrator of the HHFA. In public, President Lyndon B. Johnson reiterated Weaver's status as a potential nominee but would not promise him the position. In private, Johnson had strong reservations. He often held pro-and-con discussions with Roy Wilkins, Executive Director of the NAACP.
Johnson wanted a strong proponent for the new department. Johnson worried about Weaver's political sense. Johnson seriously considered other candidates, none of whom was black. He wanted a top administrator, but also someone who was exciting. Johnson was worried about how the new Secretary would interact with congressional representatives from the Solid South; they were overwhelmingly Democrat as most African Americans were still disenfranchised and excluded from the political system. This was expected to change as the federal government enforced civil rights and the provisions of the Voting Rights Act of 1965. As candidates, Johnson considered the politician Richard Daley, mayor of Chicago; and the philanthropist Laurence Rockefeller.
Ultimately, Johnson believed that Weaver was the best-qualified administrator. His assistant Bill Moyers had rated Weaver highly on potential effectiveness as the new secretary. Moyers noted Weaver's strong accomplishments and ability to create teams. Ten days after receiving the report, the president put forward the nomination, and Weaver was successfully confirmed by the United States Senate.
Weaver served as Secretary of United States Department of Housing and Urban Development from 1966 to 1968.
Weaver had expressed his concerns about African Americans' housing issue before 1930 in his article, "Negroes Need Housing", published by the magazine The Crisis of the NAACP after the Stock Market Crash. He noted there was a great difference between the income of most African Americans and the cost of living; African Americans did not have enough housing supply because of many social factors, including the long economic decline of rural areas in the South. He suggested a government housing program to enable all the African Americans the chance to buy or rent their house.
Academia
In 1945, Weaver began teaching at Columbia University.
In 1969, after serving under President Johnson, Weaver became president of Baruch College.
In 1970, Weaver became a distinguished professor of Urban Affairs at Hunter College in New York and taught there until 1978.
Personal life and death
In 1935, Robert C. Weaver married Ella V. Haith. They adopted a son, who died in 1962.
Weavers served on the boards of Metropolitan Life Insurance Company (1969–1978) and Bowery Savings Bank (1969–1980). He served in advisory capacities to the United States Controller General (1973–1997), the New City Conciliation and Appeals Board (1973–1984), Harvard University School of Design (1978–1983), the National Association for the Advancement of Colored People (NAACP) Legal Defense Fund and NAACP executive board committee (1978–1997).
Robert C. Weaver died at age 89 on July 17, 1997, in Manhattan, New York.
Honors
Weaver received more than 30 honorary university degrees, as well as the following:
1962: NAACP Spingarn Medal
1963: Russworm Award
1968:
Albert Einstein Commemorative Award
Merrick Moore Spaulding Award
1975: Public Service Award of the US General Accounting Office
1977: Frederick Douglass Award of the New York City Urban League
1978: Schomberg Collection Award
1985: Fellow, American Academy of Arts and Sciences
1987: Equal Opportunity Day Award of the National Urban League
Legacy
2000: Robert C. Weaver Federal Building HUD headquarters (which Weaver had dedicated in 1968)
2006: "Robert Clifton Weaver Way" NE in Washington, DC
Undated: "Robert Weaver Avenue" "Robert Weaver Circle" in Austin, Texas
Works
Weaver wrote a number of books regarding black issues and urban housing, including:
Negro Labor: A National Problem (1946)
The Negro Ghetto (1948)
The Urban Complex: Human Values in Urban Life (1964)
Dilemmas of Urban America (1965)
Herbert Aptheker reviewed The Negro Ghetto in the August 1948 issue of Masses and Mainstream (successor to the New Masses magazine).
See also
List of African-American United States Cabinet members
J. Raymond Jones
Harlem Clubhouse
References
Further reading
John C. Walker, The Harlem Fox: J. Raymond Jones at Tammany 1920–1970, New York: State University New York Press, 1989.
Primary sources
"Weaver, Robert Clifton". Infoplease
Speech by Robert Weaver given on April 8, 1969. Audio recording from The University of Alabama's Emphasis Symposium on Contemporary Issues
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Telephone Conversation between President Johnson and Martin Luther King Jr., 15 January 1965, 12:06pm, Citation # 6736, Recordings of Telephone Conversations, Lyndon B. Johnson Presidential Library
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\section{Introduction}
The integrable system introduced in \cite{Hit2} is now 35 years old but there are still unexplored features. A new issue arose in the recent paper \cite{HH} with Tam\'as Hausel and work which followed on from it \cite{H1}. It concerns fixed points $m$ of the $\mathbf{C}^*$-action on the moduli space ${\mathcal M}$ of Higgs bundles (these fixed point sets form a subject on which Oscar Garc\'ia-Prada has in particular made fundamental contributions e.g.\cite{GP1},\cite{GP2},\cite{GP3}). Such a point is given by a Higgs bundle $(E,\Phi)$ where $\Phi$ is necessarily nilpotent and so the functions $h=(h_1,\dots,h_n)$ defining the integrable system all vanish. Then one defines from the $h_i$ a commutative algebra associated to the fixed point $m$, which is called the {\it multiplicity algebra} \cite{H1}. When the fixed point is {\it very stable}, the algebra is finite-dimensional and the dimension gives the multiplicity of the component of the nilpotent cone containing $m$, hence the name. An interesting observation is that
for certain fixed points
at the upper end of the nilpotent cone the multiplicity algebras are isomorphic to the cohomology of homogeneous spaces and it is this which suggests further research into their structure in more generality.
This paper is about the algebras defined at the {\it lower} end of the nilpotent cone, namely the case of a very stable bundle, where $\Phi=0$. Here we learn about the algebra by studying examples, but some underlying structure remains in general: in all cases the $\mathbf{C}^*$-action defines a grading and yields a Poincar\'e duality ring -- a nondegenerate pairing on subspaces of complementary degree.
When $E$ has rank $2$ and fixed determinant the relations for the algebra are given by $3g-3$ homogeneous quadratic functions on a vector space of dimension $3g-3$, more invariantly given by
the quadratic map $\mathop{\rm tr}\nolimits \Phi^2: H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)\rightarrow H^0(C,K^2)$. Considering $H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)$ as the cotangent space at the point $[E]$ of the moduli space ${\mathcal N}$ of stable bundles, this is the definition of the integrable system.
The particular issue we address here is to exhibit algebras which, up to equivalence, contain continuous parameters unlike the integral structure of cohomology. We use the discriminant: a class $\alpha \in H^1(C,K^*)$ defines a homomorphism $\alpha: H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)\rightarrow H^1(C,\mathop{\rm End}\nolimits_0 E)$ which by Serre duality is equivalent to the quadratic map above. Then $\det \alpha=0$ describes a hypersurface of degree $3g-3$ in the projective space ${\rm P}(H^1(C,K^*))$ and the projective equivalence class of this is an invariant of the algebra, independent of any specific choice of generators and relations. In the terminology of classical algebraic geometry we have a family of quadrics and the discriminant is a familiar invariant.
In genus $g=2$ we have three quadratic functions on a 3-dimensional vector space which geometrically is a net of conics. In \cite{H} we showed that if the degree of $E$ is odd then there are just finitely many equivalence classes of algebras, the very stable ones having three generators $\xi_i$ with $\xi_i^2=0$. This is a case where the algebra is indeed isomorphic to the cohomology $H^*(({\mathbf C}{\rm P}^1)^3,\mathbf{C})$. In contrast we show in Section \ref{two} that when $E$ has even degree the discriminant is a cubic curve isomorphic to a plane section of a singular cubic surface, and the modulus of this elliptic curve varies as $E$ varies in the moduli space of stable bundles. To do this we make essential use of the description in \cite{vGP} of the integrable system using a classical relationship between certain curves of genus 2,3 and 5. The discriminant approach also gives a new viewpoint on the moduli space: the space of discriminants can be seen as the projection from a point on the Igusa quartic threefold, the quotient of the moduli space ${\rm P}^3$ of stable bundles (as described in the foundational paper \cite{Nar0}) by the group $H^1(C,\mathbf {Z}_2)$.
The geometry of these same three curves comes into play when we consider in Section \ref{three} a family of bundles of degree zero on a general nonhyperelliptic curve of genus $3$. The moduli space ${\mathcal N}$ for $\Lambda^2E\cong {\mathcal{O}}$ is well-known \cite{Nar1} to be a special singular quartic hypersurface in ${\rm P}^7$, initially studied by A.Coble. We consider a two-dimensional family by choosing a line bundle $U$ of order $2$ and looking at the fixed point set in ${\mathcal N}$ under the action $E\mapsto E\otimes U$. This is a quartic surface in ${\rm P}^3\subset {\rm P}^7$, in fact the Kummer surface of the Jacobian for a genus $2$ curve, part of the story of the previous section.
The results of Pauly \cite{P} allow us to write down the six relations for the algebra and to determine the discriminant variety more geometrically. We find that for our chosen family the degree 6 hypersurface is reducible to a singular quadric and a quartic. The degeneracy subspace of the quadric is a projective plane
and its intersection with the quartic hypersurface is a quartic curve. It is the modulus of this curve which varies as we vary $E$ in the family, but we use a genericity argument to show this rather than using the explicit formula. The geometry behind this introduces a quartic surface with 10 nodes associated to a pair of conics.
The original motivation for this study in \cite{HH} involves mirror symmetry for the Higgs bundle moduli space. The precise role of the multiplicity algebra in this context has yet to be determined.
\section{The quadratic map}
Let $C$ be a curve of genus $g>1$, $E$ a rank $2$ vector bundle over $C$ with $\Lambda^2 E$ fixed, and let $\mathop{\rm End}\nolimits_0E$ denote the bundle of trace zero endomorphisms of $E$. For $\Phi\in H^0(C, \mathop{\rm End}\nolimits_0 E\otimes K)$ we consider $\mathop{\rm tr}\nolimits \Phi^2\in H^0(C,K^2)$. When $E$ is stable, then the dimension of both these spaces is $3g-3$ and we consider the map
$$\mathop{\rm tr}\nolimits \Phi^2: H^0(C, \mathop{\rm End}\nolimits_0 E\otimes K)\rightarrow H^0(C,K^2)$$
which commutes with the scalar action of $\lambda\in \mathbf{C}^*$ on $\Phi$ and $\lambda^2$ on the right hand side. As $[E]$ varies over the moduli space
${\mathcal N}$ of stable bundles, linear functions on $H^0(C,K^2)$ generate the integrable system of \cite{Hit2}.
We can view the situation in different ways, one is to regard it as a family of quadrics in ${\rm P}^{3g-4}\cong {\rm P}(H^0(C, \mathop{\rm End}\nolimits_0 E\otimes K))$ parametrized by ${\rm P}^{3g-4}\cong {\rm P}(H^1(C,K^*))$, evaluating $\alpha\in H^1(C,K^*)=H^0(C,K^2)^*$ on $\mathop{\rm tr}\nolimits\Phi^2$. Thus in genus $2$ it is a net of conics using the classical term. The locus of singular quadrics is a hypersurface, the {\it discriminant}, in ${\rm P}(H^1(C,K^*))$.
A bundle $E$ is called {\it very stable} \cite{Laum} if there is no nonzero nilpotent Higgs field $\Phi\in H^0(C, \mathop{\rm End}\nolimits_0 E\otimes K)$. In rank 2 nilpotency is equivalent to $\mathop{\rm tr}\nolimits \Phi^2=0$, or a non-empty base locus for the family of quadrics. The very stable bundles are important, not only because they constitute the generic situation, but also because then the quadratic map is proper \cite{PPN}, a crucial feature in the more general setting of \cite{HH}.
One further point -- our question just involves the endomorphism bundle and not $E$ itself and so is insensitive to the action $E\mapsto E\otimes L$ where $L$ is a line bundle (with $L^2$ trivial if we want to fix $\Lambda^2E$). Equivalently, we are only concerned with the projective bundle ${\rm P}(E)$.
\section{Genus two curves} \label{two}
\subsection{Odd degree}
Let $C$ be a smooth curve of genus $2$, then Atiyah \cite{At} described the projective bundle associated to an indecomposable rank $2$ vector bundle $E$ of odd degree. The construction represents the projective bundle by a vector bundle which is an
extension ${\mathcal O}\rightarrow E\rightarrow L$ where $\mathop{\rm deg}\nolimits L=1$.
The extension class $[\alpha]$ lies in $H^1(C,L^*)$ and since $\dim H^1(C,L^*)=2$ it is uniquely determined up to a multiple by its annihilator, a section $s\in H^0(C,KL)$ with $[\alpha] s=0\in H^1(C,K)$. Then $s$ has divisor
$p+q+r$.
Now $E$ has 4 degree zero subbundles, and the four divisors are
$$ p+q+r,\quad p+\sigma(q)+\sigma(r),\quad q+\sigma(r)+\sigma(p),\quad r+\sigma(p)+\sigma(q)$$
where
$\sigma:C\rightarrow C$ is the hyperelliptic involution. Let $\pi:C\rightarrow {\rm P}^1$ denote the quotient map, then $(\pi(p),\pi(q),\pi(r))=(a_1,a_2,a_3)$ and out of the eight inverse images, the four above give the same projective bundle. Hence we have Atiyah's description of the moduli space as a branched double covering of the symmetric product $S^3{\rm P}^1$. This symmetric product may be considered as ${\rm P}(V)$ where $V$ is the 4-dimensional space of cubic polynomials $p(x)$ with roots $a_1,a_2,a_3$. If $x_1,\dots, x_6\in {\rm P}^1$ are the images of the fixed points of $\sigma$ (so that the curve has equation $y^2=(x-x_1)\dots(x-x_6)$) then the branch locus consists of the six planes $p(x_i)=0$.
In \cite{H} we calculated the three-dimensional space of sections of $\mathop{\rm End}\nolimits_0 E\otimes K$ in terms of a $C^{\infty}$ splitting of the extension: writing $E={\mathcal{O}}\oplus L$ with $\bar\partial$-operator $\bar\partial_E(f,g)=\bar\partial f +\alpha g$ where $\alpha\in \Omega^{0,1}(C, L^*)$ represents the extension class. The condition $[\alpha]s=0$ means we can write $\alpha=\bar\partial u/s$ where $u$ is supported in a neighbourhood of the zeros $p,q,r$ of $s$ and in fact can be chosen to vanish at $q,r$.
In concrete terms the space of Higgs fields has basis
$$(x-a_2)\begin{pmatrix} 1 & \frac{2}{s}(u-u_1\frac{(x-a_3)}{(a_1-a_3)})\cr
0 & -1\end{pmatrix}\frac{dx}{y},\quad
(x-a_3)\begin{pmatrix}1 & \frac{2}{s}(u-u_1\frac{(x-a_2)}{(a_1-a_2)})\cr
0 & -1\end{pmatrix}\frac{dx}{y}$$
$$\begin{pmatrix}-u & \frac{1}{s}(-u^2+u_1^2\frac{(x-a_2)(x-a_3)}{(a_1-a_2)(a_1-a_3)}\cr
s & u\end{pmatrix}\frac{dx}{y} $$
where $u_1=u(p)$.
\begin{remark}
Consider the map from ${\mathcal{O}}\oplus K^*$ to $E$ given by $(1,t)\mapsto (1-ut,st)$ where $t$ is a local holomorphic section of $K^*$. Since $\bar\partial (1-ut)+(\bar\partial u/s)st=0$ this is holomorphic and the quotient sheaf is ${\mathcal{O}}_{p+q+r}(L)$. So a Hecke transform on $E$ at the points $p,q,r$ gives the bundle ${\mathcal{O}}\oplus K^*$. But the formulas above show that each Higgs field $\Phi$ on $E$ preserves the trivial subbundle at $p,q,r$ and so they transform to the Higgs fields on ${\mathcal{O}}\oplus K^*$. It follows that $H^0(C,\mathop{\rm End}\nolimits_0E\otimes K)$, considered as a Lagrangian submanifold of the Higgs bundle moduli space ${\mathcal M}$, is a Hecke transform of the Hitchin section. Then the isomorphism of the multiplicity algebra with the cohomology of $({\mathbf C}{\rm P}^1)^3$ is a consequence of \cite{H1} Theorem 3.10, as well as the direct calculation in \cite{H}.
\end{remark}
The quadratic form on $H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)$ with values in $H^0(C,K^2)$ takes the form
$$f(a_1)\frac{(x-a_2)(x-a_3)}{(a_1-a_2)(a_1-a_3)}\xi_1^2+f(a_2)\frac{(x-a_3)(x-a_1)}{(a_2-a_3)(a_2-a_1)}\xi_2^2+f(a_3)\frac{(x-a_1)(x-a_2)}{(a_3-a_1)(a_3-a_2)}\xi_3^2$$
where $f(x)=(x-x_1)\dots (x-x_6)$ and we identify $H^0(C,K^2)$ with $\pi^*H^0({\rm P}^1,{\mathcal{O}}(2))$.
Our main focus in this article is the discriminant, and since the multiplicity algebra has relations $\xi_1^2=\xi_2^2=\xi_3^2=0$ this consists of three lines in ${\rm P}^2$. However, introducing the geometry of the curve itself we have three lines in ${\rm P}(H^1(C,K^*))\cong {\rm P}^2$, the dual space of $H^0(C,K^2)$. The linear system $K^2$ maps $C$ to a conic $C_0$ in ${\rm P}(H^1(C,K^*))$ -- evaluating $H^0(C,K^2)$ at a point $p\in C$ is the map. Considering the coefficients of $\xi_i^2$ in the above formula we see that the three lines form the sides of the triangle with vertices $a_1,a_2,a_3$ on the conic, isomorphic to ${\rm P}^1=\pi(C)$. Thus, apart from the double covering, the discriminant reproduces Atiyah's parametrization.
\subsection{Even degree} \label{even}
Now consider $E$ a stable bundle over $C$ with $\Lambda^2E$ trivial. The moduli space of (S-equivalence classes of) semi-stable bundles in this case is well-known to be ${\rm P}^3$ \cite{Nar0}, defined by considering the line bundles $L$ of degree $1$ such that $E\otimes L$ has a non-zero section: for a given $E$, $L$ varies in a $2\Theta$-divisor in $\mathop{\rm Pic}\nolimits^1(C)$. There are explicit formulas in \cite{Lor} for the three quadratic functions:
\begin{eqnarray*}
h_1&=& rst[\xi_0(u_0^2-1)+\xi_1(u_0u_1+u_2)+\xi_2(u_2u_0+u_1)]^2-\\
&& st[\xi_0(u_0u_1-u_2)+\xi_1(u^2_1+1)+\xi_2(u_1u_2+u_0)]^2+\\
&&4rs(\xi_0u_0+\xi_1u_1)^2-rt[\xi_0(u_0^2+1)+\xi_0(u_0u_1+u_2)+\xi_2(u_2u_0-u_1)]^2\\
h_2&=&t(u_0^2+u_1^2+u_2^2+1)[(\xi_0^2+\xi_1^2+\xi_2^2)+(\xi_0u_0+\xi_1u_1+\xi_2u_2)^2]+\\
&&st(u_0^2-u_1^2+u_2^2-1)[(\xi_0^2-\xi_1^2+\xi_2^2)-(\xi_0u_0+\xi_1u_1+\xi_2u_2)^2]+\\
&&4r(u_0u_2-u_1)[\xi_0\xi_2+(\xi_0u_0+\xi_1u_1+\xi_2u_2)\xi_1]+\\
&&4sr(u_2u_0+u_1)[\xi_2\xi_0-(\xi_0u_0+\xi_1u_1+\xi_2u_2)\xi_1]+\\
&&4s(u_1u_2+u_0)[\xi_1\xi_2-(\xi_0u_0+\xi_1u_1+\xi_2u_2)\xi_0]+\\
&&4rt(u_0u_1+u_2)[\xi_0\xi_1-(\xi_0u_0+\xi_1u_1+\xi_2u_2)\xi_2]\\
h_3&=&s[\xi_0(u_2u_0+u_1)+\xi_1(u_1u_2+u_0)+\xi_2(u_2^2-1)]^2-\\
&&[\xi_0(u_2u_0-u_1)+\xi_1(u_1u_2+u_0)+\xi_2(u_2^2+1)]^2-\\
&&t[\xi_0(u_0u_1+u_2)+\xi_2(u_1u_2-u_0)+\xi_1(u_2^2+1)]^2+4r(\xi_1u_1+\xi_2u_2)^2.
\end{eqnarray*}
Here $(u_0,u_1,u_2)$ are affine coordinates on ${\rm P}^3$, and $(\xi_0,\xi_1,\xi_2)$ the corresponding coordinates on the cotangent space. The genus $2$ curve $C$ is the double cover of ${\rm P}^2$ branched over the six points $0,1,\infty, r,s,t$. Then $E$ is determined by fixing $u_i$ and the relations in the algebra are given by the vanishing of the $h_i$.
It is difficult to draw conclusions about the structure of the multiplicity algebra from this mass of formulae so here we adopt a more geometric approach, drawing on the paper of van Geemen and Previato \cite{vGP}, which was the first investigation into explicit formulae for the integrable system. The genus 2/3/5 story below appears in various contexts (see \cite{Bruin}, \cite{Duc}, \cite{Mas},\cite{Verra}).
If $E\otimes L$ has a non-zero section, then by Riemann-Roch, Serre duality and the isomorphism $E^*\cong E\otimes \Lambda^2E^*\cong E$, we have $H^0(C,E\otimes KL^*)\ne 0$. It follows that each $2\Theta$-divisor is symmetric with respect to the involution $L\mapsto KL^*$ on $\mathop{\rm Pic}\nolimits^1(C)$. If the divisor is smooth, it is a curve $C_5$ of genus 5 and the involution acts freely with quotient $C_3$ of genus $3$ (we use the notation of \cite{vGP}). This is a plane section of a Kummer quartic surface, and if it is nonsingular then the bundle is certainly very stable, since, as in \cite{PP}, the complement -- the so-called ``wobbly" locus -- consists of either the strictly semistable bundles or the 16 hyperplanes which meet the Kummer surface in a double conic.
The double covering is defined by a line bundle $U$ on $C_3$ with $U^2$ trivial and we consider the 2-dimensional space $H^0(C_3,K_3U)$. The tangent space at $[E]\in {\mathcal N}$ is $H^1(C,\mathop{\rm End}\nolimits_0E)$ but in terms of the corresponding $2\Theta$-divisor $C_5$ it is identified with a subspace of sections of the normal bundle in $\mathop{\rm Pic}\nolimits^1(C)$. The normal bundle is the canonical bundle so there is a map from global sections of the tangent bundle of $\mathop{\rm Pic}\nolimits^1(C)$ to sections of the normal bundle i.e. $H^1(C,{\mathcal{O}})\rightarrow H^0(C_5,K_5)$. Deformations of $E$ are given by curves $C_5$ which are divisors of the $2\Theta$ line bundle and so are transverse to translations on $\mathop{\rm Pic}\nolimits^1(C)$, which is the image of $H^1(C,{\mathcal{O}})$. They are also symmetric with respect to the involution. It follows that $H^1(C,\mathop{\rm End}\nolimits_0 E)$ is isomorphic to the even sections of $K_5$ and $H^1(C,{\mathcal{O}})$ to the odd ones. In terms of the genus $3$ curve we have
\begin{equation} \label{isos}
H^1(C,\mathop{\rm End}\nolimits_0 E)\cong H^0(C_3,K_3),\qquad H^1(C,{\mathcal{O}})\cong H^0(C_3,UK_3)
\end{equation}
and taking duals $H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)\cong H^1(C_3,{\mathcal{O}})$.
If $s,t$ form a basis of $H^0(C_3,UK_3)$ then we have sections $q_1=s^2,q_2=st,q_3=t^2$ of $K_3^2$. If $C_3$ is nonhyperelliptic then every quadratic differential $q_i$ is uniquely a quadratic form $Q_i$ in elements of $H^0(C_3,K_3)$. Since $q_1q_3=s^2t^2=(st)^2=q_2^2$ the homogeneous quartic equation $Q_1Q_3-Q_2^2=0$ defines $C_3$ in its canonical embedding: $C_3\subset {\rm P}(H^1(C_3,{\mathcal{O}}))$.
The key result is:
\begin{prp} \cite{vGP} \label{quad} Under the isomorphism $H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K))\cong H^0(C_3,K)^*$ the net of conics is spanned by $Q_1,Q_2,Q_3$.
\end{prp}
\begin{remark}
\noindent 1. The three quadratic forms correspond to basis elements of $S^2H^0(C_3,UK_3)\cong S^2H^1(C,{\mathcal{O}})$ and since each quadratic differential in genus 2 is a quadratic in sections of $K$, this is $H^1(C,K^*)\cong H^0(C,K^2)^*$. This is the invariant form for the map but it is more convenient from our point of view to think of the multiplicity algebra as the 8-dimensional algebra generated by $1,x,y,z$ with relations $Q_i(x,y,z)=0$.
\noindent 2. The essential input for the authors of \cite{vGP} to prove the isomorphisms in (\ref{isos}) involves the section $s$ of $E\otimes L$ and $s'$ of $E\otimes KL^*$. Then $\Phi=s\otimes s'-\langle s, s'\rangle/2$ is a trace zero Higgs field.
\end{remark}
The discriminant in ${\rm P}^2={\rm P}(H^1(C,K^*))$ is the cubic curve $D$ defined by $\det (\sum_{i=1}^3y_iQ_i)=0$. The curve $C$, as before, is mapped to a conic $C_0\subset {\rm P}(H^1(C,K^*))$ by the linear system $K^2$ with six distinguished points $x_1,\dots, x_6$ the branch points of $\pi:C\rightarrow {\rm P}^1$. We shall use the following
\begin{prp}\label{W}
The six points $x_1,\dots, x_6$ lie on the discriminant.
\end{prp}
\begin{proof}
A fixed point $\tilde x_i$ of $\sigma$ is the divisor for the unique section $s$ of a square root $K^{1/2}$ of the canonical bundle. Hence $\dim H^0(C,K^{1/2})$ is odd. If $E$ is a rank 2 bundle of degree zero, and hence topologically trivial, then since $\mathop{\rm End}\nolimits_0 E$ is odd-dimensional and has an orthogonal structure given by $\mathop{\rm tr}\nolimits \phi^2$ it follows from the mod 2 index theorem \cite{AS} that $\dim H^0(C, \mathop{\rm End}\nolimits_0 E\otimes K^{1/2})$ is also odd (this is part of a more general story \cite{Ox}) and hence contains a non-zero section $\Psi$. So $s\Psi\in H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)$ vanishes at $\tilde x_i$ and hence so does $\mathop{\rm tr}\nolimits(s\Psi\Phi)$ for all $\Phi$. This means that evaluation of $H^0(C,K^2)$ at $\tilde x_i$, i.e. its image of $x_i$ in ${\rm P}(H^1(C,K^*))$ under the bicanonical map, gives a degenerate quadratic form, and hence lies on the discriminant. \end{proof}
\begin{remark} For higher genus $\dim H^0(C,\mathop{\rm End}\nolimits_0\otimes K)=3g-3> 3=\mathop{\rm rk}\nolimits (\mathop{\rm End}\nolimits_0\otimes K)$ and so for any point $x\in C$ there exists a Higgs field vanishing at $x$. Consequently the bicanonical image of $C$ always lies in the discriminant hypersurface.
\end{remark}
To describe the discriminant curve for a bundle $E$, from Proposition \ref{W} we need to understand the cubics through the six points $x_i$ on $C_0$.
Blow up ${\rm P}^2$ at these points, then the conic $C_0$ becomes a $-2$ curve. The linear system of cubics through the six points is then equivalent to $3H-E_1-\cdots-E_6$, where $E_i$ are the exceptional curves and $H$ the class of a line. This maps the surface to $S\subset {\rm P}^3$, a cubic surface with a node from the collapse of the $-2$ curve. Then any cubic curve through the $x_i$, $i=1,..,6$, is defined by a plane section of $S$. We shall show next that conversely a generic section arises from a stable bundle $E$.
\begin{prp} To each genus $2$ curve $C$ we associate a singular cubic surface $S$ as above. Then a generic plane section is the discriminant of the net of conics defined by a very stable rank 2 bundle $E$ on $C$.
\end{prp}
\begin{proof}
We start with the classical fact (see \cite{Bea2}) that the choice of a non-trivial line bundle of order $2$ on a nonsingular plane cubic curve $D$ provides an equation of $D$ as $\det (\sum_{i=1}^3 y_iQ_i)=0$ where $Q_1,Q_2,Q_3$ are symmetric $3\times 3$ matrices. Parametrize the conic $C_0\cong {\rm P}^1$ by $(y_1,y_2,y_3)=(1,2u,u^2)$ then the six points of intersection $b_i\in D\cap C_0$ are the roots of $\det (Q_1+2uQ_2+u^2Q_3)=0$.
Each $Q_i$ defines a conic $Q_i(x,y,z)=0$ in a projective plane ${\rm P}^2$ and $Q_1Q_3-Q_2^2=0$ a quartic curve $C_3$ which for generic $D$ will be smooth. Since ${\mathcal{O}}(1)$ on ${\rm P}^2$ is the canonical bundle $K_3$ on $C_3$ the equation says that $Q_1$ on $C_3$ is a section of $K^2_3$ with double zeros, that is $s^2$ for a section $s$ of $K_3U$ for some line bundle $U$ with $U^2$ trivial. But on $C_3$
$$(uQ_1+vQ_2)^2=u^2Q_1^2+2uvQ_1Q_2+v^2Q_2^2=Q_1(u^2Q_1+2uvQ_2+v^2Q_3)$$
so as $u,v$ vary the bundle $U$ is constant and we have $Q_1=s^2, Q_3=t^2, Q_2=st$ for a basis $s,t$ of sections of $K_3U$.
The line bundle $U$ defines an unramified covering, a curve $C_5$ of genus $5$ with an involution $\tau$. The Prym variety ${\rm P}(C_5,C_3)$ is a 2-dimensional abelian variety consisting of the divisor classes in $\mathop{\rm Pic}\nolimits^0(C_5)$ which are anti-invariant under $\tau$. It has two components and $x-\tau(x)$ embeds $C_5$ (if it is not hyperelliptic) in the non-trivial component. Then (see \cite{Verra} or the other references above) each component is isomorphic to the Jacobian of the genus 2 curve $y^2=\det(Q_1+2uQ_2+u^2Q_3)$ and $C_5$ is a symmetric $2\Theta$-divisor with $C_3$ a plane section of the Kummer surface. This defines a point in the moduli space which, thanks to Proposition \ref{quad}, has the given cubic $D$ as discriminant.
The isomorphism of the two abelian surfaces can be standardized up to elements of order two \cite{vGP}, for at the branch point $x_i$ the conic is a pair of lines each of which is a bitangent to $C_3$, that is sections $s_1, s_2$ of $K_3^{1/2}, K_3^{1/2}U$. Then pulled back to $C_5$ this gives a line bundle $K^{1/2}_5$ with $\dim H^0(C_5, K_5^{1/2})=2$. So $b_i\in \mathop{\rm Pic}\nolimits^1(C)$ corresponds to $K_5/2\in \mathop{\rm Pic}\nolimits^4(C_5)$ and $K_5/2+a-\tau(a)$ identifies with the Prym variety. Since we are concerned with $\mathop{\rm End}\nolimits_0 E\otimes K$, the choice of line bundle of order $2$ is irrelevant
\end{proof}
\begin{corollary} For a fixed genus $2$ curve, the isomorphism class of the multiplicity algebra for a very stable rank 2 bundle $E$ of even degree varies continuously as $E$ varies.
\end{corollary}
\begin{proof}
From \cite{Bea1}, apart from quadrics, smooth plane sections of a surface vary the modulus of the curve non-trivially. In our case each section is a cubic curve isomorphic to the discriminant of the net of conics associated to $E$.
\end{proof}
\begin{remark}
Passing from the discriminant to the bundle $\mathop{\rm End}\nolimits_0 E$ involved a degree 3 covering of the space ${\rm P}^3$ of plane sections of the singular cubic surface. This fits into a classical situation: two threefolds, dual to each other, the Igusa quartic $B$ in ${\rm P}^4$ and the Segre cubic in the dual projective space. The quartic has an interpretation as the GIT quotient of six points on a conic and a point $x$ on it represents by duality a hyperplane section of the cubic. The six points $x_i$ give us the genus 2 curve $C$ and the singular cubic surface $S$ is constructed, as in the proposition, by blowing up the points. Then the plane sections of $S$ correspond by duality to the lines through $x$. Projection from $x\in B$ is then a threefold covering of ${\rm P}^3$, the space of discriminants. But the Igusa quartic is also the quotient of ${\rm P}^3$ by the action of $H^1(C,\mathbf {Z}_2)$, or equivalently the moduli space of projective bundles ${\rm P}(E)$, so we have an analogue of Atiyah's description of the moduli space as a branched cover of ${\rm P}^3$, at least in the very stable situation.
\end{remark}
\section{Genus 3 curves}\label{three}
\subsection{The vector bundles}\label{vect}
Let $C$ be a nonhyperelliptic curve of genus $3$ -- a nonsingular plane quartic in the canonical embedding. The moduli space ${\mathcal N}$ of (semi)stable rank two bundles $E$ with $\Lambda^2E$ trivial is isomorphic to the Coble quartic hypersurface in ${\rm P}^7$ \cite{Nar1}. The decomposable bundles $E=L\oplus L^*$ describe a three dimensional Kummer variety, the quotient of the Jacobian $\mathop{\rm Jac}\nolimits(C)$ by $x\mapsto -x$, which is the singular locus. The Coble quartic is characterized by this property together with invariance by the action on $\mathbf{C}^8$ of a finite Heisenberg group which corresponds to $E\mapsto E\otimes U$ for $U\in H^1(C,\mathbf {Z}_2)$, a line bundle with $U^2$ trivial.
The discriminant here is a degree 6 hypersurface in ${\rm P}(H^0(C, K^*))={\rm P}^5$ which is quite complicated -- as noted above the bicanonical embedding of $C$ lies in it, and the rank of the quadratic form there is at most 3, so this is a curve of singularities. We shall consider instead the family of bundles which are defined by the fixed points in ${\mathcal N}$ of one element $U$. This is the intersection of a ${\rm P}^3\subset {\rm P}^7$ with the quartic hypersurface, hence a quartic surface, and we have a 2-dimensional family.
A fixed point in the moduli space means an isomorphism $\psi:E\rightarrow E\otimes U$. We can view this as a Higgs bundle twisted with $U$ rather than the usual $K$, which implies that we have a genus 5 spectral curve $\pi:\tilde C\rightarrow C$, the unramified covering corresponding to $U$, and $E$ is the direct image $\pi_*(L\otimes \pi^*U^{1/2})$ where $\sigma^*L\cong L^*$ i.e. $L$ lies in the Prym variety. The bundle $E$ is strictly semistable if $L^2$ is trivial.
A section $e$ of $E=\pi_*(L\otimes \pi^*U^{1/2})$ on an open set $V\subset C$ is by definition of the direct image a section $s\in H^0(\pi^{-1}(V),L\otimes \pi^*U^{1/2})$ and then $s\sigma^*s\in H^0(V,U)$ is a nondegenerate $U$-valued quadratic form $(e,e)$ on $E$. In the presence of this orthogonal structure, the trace zero endomorphisms $\mathop{\rm End}\nolimits_0E$ split into symmetric and skew symmetric components, so we have
\begin{equation}\label{sum}
\mathop{\rm End}\nolimits_0 E\cong U\oplus E'.
\end{equation}
The isomorphism $\psi:E\cong E\otimes U$ gives an involution on $\mathop{\rm End}\nolimits_0 E$ and the above summands are the eigenspaces. Moreover
the rank 2 bundle $E'$ has an orthogonal structure, this time with values in the trivial bundle. If we consider $\psi$ as a $U$-valued Higgs field on $E$, its eigenspaces in the adjoint representation are squares of the eigenspaces on $E$. This means that $E'=\pi_*L^2$.
The expression $\mathop{\rm End}\nolimits_0 E\cong U\oplus E'$ is an orthogonal decomposition with respect to the quadratic form $\mathop{\rm tr}\nolimits \phi^2$ so the map $$\mathop{\rm tr}\nolimits \Phi^2:H^0(C,\mathop{\rm End}\nolimits_0E\otimes K)\rightarrow H^0(C, K^2)$$ in this case takes $(u,e)\in H^0(C,UK) \oplus H^0(C,E'\otimes K)$ to a multiple of $u^2+(e,e)$.
We met the first term $H^0(C,UK)$ with $C=C_3$ in the previous section, giving quadratic forms $Q_1,Q_2,Q_3$. These will appear next in a different form.
%
\subsection{The net of quadrics}
In \cite{P}, Pauly associates to a rank 2 stable bundle $E$ with trivial determinant a bundle $F$ with $\Lambda^2F\cong K$ such that $\dim H^0(C,E\otimes F)=4$. It arises as follows: $E$ is defined as an extension
$L^*\rightarrow E \rightarrow L$
where $\mathop{\rm deg}\nolimits L=1$ and the extension class lies in $ H^1(C,L^{-2})$ which has dimension $4$. This class annihilates a 3-dimensional subspace $W\subset H^0(C,L^2K)$. Then $(F\otimes L)^*$ is defined as the kernel of the evaluation map ${\mathrm{ev}}: C\times W\rightarrow L^2K$.
In the long exact sequence
\begin{equation} \label{long}
0\rightarrow H^0(C,F\otimes L^*)\rightarrow H^0(C,F\otimes E)\rightarrow H^0(C, F\otimes L) \stackrel{\delta} \rightarrow H^1(C,F\otimes L^*)
\end{equation}
consider first
$H^0(C,F\otimes L^*)$. We have $(F\otimes L)^*\subset C\times W$ so since $F\cong F^*\otimes K$, $F\otimes L^*K^*\subset C\times W$. Tensoring with $K$ gives
$$H^0(C,F\otimes L^*)\rightarrow H^0(C,K)\otimes W\rightarrow H^0(C, K^2L^2)$$
and by Riemann-Roch $\dim H^0(C,F\otimes L^*)\ge 9-8=1$.
The connecting homomorphism $\delta$ in the sequence (\ref{long}) is zero on $W^*\subset H^0(C, F\otimes L)$ and, if the bundle is not a so-called exceptional one, $\dim H^0(C,F\otimes L^*)=1$ and this gives the four dimensions for $H^0(C,E\otimes F)$.
Importantly, it turns out that $F$ is independent of the choice of $L^*$, which is a maximal subbundle -- there are generically eight of these.
As shown in \cite{P}, if $E$ is stable then $F$ is stable unless $E$ is defined by a point on the Coble quartic which lies on a trisecant of the Kummer variety.
The isomorphism $\Lambda^2E\cong {\mathcal{O}}$ defines a skew form on $E$ and $\Lambda^2F\cong K$ gives a skew form on $F$ with values in $K$ so the tensor product has a $K$-valued {\it symmetric} form. Then the 4-dimensional space $V=H^0(C,E\otimes F)$ has a symmetric bilinear form $(\,,\,)$ with values in $H^0(C,K)$, geometrically a net of quadrics in ${\rm P}^3={\rm P}(V)$.
\subsection{The multiplicity algebra}\label{multialg}
Following on from this approach, in \cite{H} the author made use of
the natural map
$$\Lambda^2 (E\otimes F)\rightarrow S^2E\otimes \Lambda^2F\cong \mathop{\rm End}\nolimits_0E\otimes K$$
which induces one from $\Lambda^2 H^0(C,E\otimes F)$ to $H^0(C, \mathop{\rm End}\nolimits_0E\otimes K)$. Both spaces have dimension 6 and it was shown that this is an isomorphism.
Explicitly, if
we choose a basis $v_1,\dots, v_4$ of $V$ then the quadratic form $\mathop{\rm tr}\nolimits \Phi_1\Phi_2$ with values in $H^0(C,K^2)$ is given up to a scale by
\begin{align*}
(v_i\wedge v_j,v_i\wedge v_k)&=(v_i,v_i)(v_j,v_k)-(v_i,v_k)(v_j,v_i)\\
(v_1\wedge v_2, v_3\wedge v_4)&=(v_1,v_3)(v_2,v_4)-(v_1,v_4)(v_2,v_3)+\sqrt{\det(v_i,v_j)}.
\end{align*}
The square root means that the equation of the quartic is $\det(v_i,v_j)-Q^2$ for some quadratic $Q(x,y,z)$. Thus $\det(v_i,v_j)=0$ is a quartic curve meeting $C$ tangentially at 8 points, and Pauly describes in more detail the rational map from the moduli space of projective bundles ${\rm P}(E)$ to the space of tangential quartics.
In our case $E\cong E\otimes U$ and from the construction of the bundle $F$ it follows that $F\cong F\otimes U$ which means the composition of the isomorphisms defines an involution $\tau$ on $V=H^0(C,E\otimes F)$ which is orthogonal with respect to the $H^0(C,K)$-valued inner product. We also have the symmetric form on $E$ with values in $U$ which gives a skew form on $V$ with values in $H^0(C,UK)$ (this defines a homomorphism from $\Lambda^2(E\otimes F)$ to $UK$ which gives the decomposition (\ref{sum})).
\begin{remark} The analogy was drawn in \cite{H} with the differential geometry of four dimensions expressed in terms of the two spinor bundles $S_+,S_-$ with $E,F$ playing similar roles here. In pursuit of this analogy the situation we have here is parallel to K\"ahler geometry in two complex dimensions
with the involution being parallel to the complex structure and the skew form to the K\"ahler form.
\end{remark}
The involution $\tau$ on $V$ induces one on $\Lambda^2V\cong H^0(C,\mathop{\rm End}\nolimits_0V\otimes K)\cong H^0(C,UK)\oplus H^0(C, E'\otimes K)$. The first summand is a 2-dimensional $+1$ eigenspace, with a 4-dimensional $-1$ eigenspace as orthogonal complement.
It follows that $\tau$ splits $V=V^+\oplus V^-$ into two orthogonal 2-dimensional subspaces spanned by $v_1,v_2$ and $v_3,v_4$, the $\pm 1$ eigenspaces of $\tau$, and each has a quadratic form $A^+,A^-$ . Then $\det (v_i,v_j)=\det A^+\det A^-$ so the equation of the quartic is $\det A^+\det A^- - Q^2=0$ -- the familiar expression $Q_1Q_2=Q^2$ associated to $U$ as in the first section.
In Pauly's description of the moduli space of projective bundles on $C$ using tangential quartic curves, the case we are considering is where the tangential quartic is a pair of conics, defined by two sections of $KU$.
\subsection{Explicit forms} \label{explicit}
Take a basis $v_1,v_2,v_3,v_4$ as above and write $b_{ij}=(v_i,v_j)$, sections of $K$, and take a corresponding basis $v_{23},v_{31},v_{12},v_{41},v_{42},v_{43}$ where $v_{ij}=v_i\wedge v_j$ for the six-dimensional exterior product $\Lambda^2V\cong H^0(C,\mathop{\rm End}\nolimits_0 E\otimes K)$.
The involution acts as $1$ on $v_{12},v_{34}$ and $-1$ on the others. Then the matrix of the quadratic form $\mathop{\rm tr}\nolimits \Phi^2$ with values in $H^0(C,K^2)$
is
\begin{equation}\label{mat}
\begin{bmatrix} b_{22}b_{33} & -b_{12}b_{33} & 0 &Q -b_{12} b_{34} & -b_{22}b_{34} & 0\\
-b_{12}b_{33} & b_{11}b_{33} & 0 & b_{11}b_{34} & Q+b_{12}b_{34} & 0\\
0 & 0 & Q_1 & 0 & 0 &Q \cr
Q -b_{12} b_{34}&b_{11} b_{34}& 0 &b_{11} b_{44}& b_{12}b_{44} & 0 \\
-b_{22}b_{34} &Q+ b_{12}b_{34}& 0 & b_{12}b_{44} & b_{22} b_{44}& 0\\
0 & 0 & Q & 0 & 0 & Q_2
\end{bmatrix}
\end{equation}
where $Q_1=\det A^+=b_{11}b_{22}-b_{12}^2, Q_2=\det A^-=b_{33}b_{44}-b_{34}^2$. The discriminant is given by the vanishing of this determinant.
\begin{remark}To explain the roles of quadratic forms in this picture, recall that the discriminant lies in ${\rm P}(H^1(C,K^*))$ and by Serre duality $H^1(C,K^*)$ consists of linear functions on $H^0(C,K^2)$ and the latter are quadratic expressions in $x,y,z\in H^0(C,K)$. Then $x^2,y^2, z^2, xy \dots$ as sections of $H^0(C,K^2)$ are to be regarded as linear functions on its dual $H^1(C,K^*)$.
\end{remark}
It is clear from $(\ref{mat})$ that $Q_1Q_2-Q^2$ is a quadratic factor of the determinant and this gives a singular quadric whose degeneracy subspace is the ${\rm P}^2$ defined by the linear forms $Q_1=Q_2=Q=0$. The other component is a quartic hypersurface $X$. Then $X\cap {\rm P}^2$ is a quartic curve whose modulus is an invariant of the algebra.
We look at this slightly differently.
The intersection of the hypersurface $X$ with the ${\rm P}^3$ defined by $Q_1=Q_2=0$ is a surface given by the determinant of the $4\times 4$ matrix obtained from (\ref{mat}) by deleting rows and columns 3 and 6. Its intersection with $Q=0$ is the determinant of this matrix where we set $Q=0$ also. This is
\begin{equation}\label{mat1}
\begin{bmatrix} b_{22}b_{33} & -b_{12}b_{33} & -b_{12} b_{34} & -b_{22}b_{34} \\
-b_{12}b_{33} & b_{11}b_{33} & b_{11}b_{34} & b_{12}b_{34} \\
-b_{12} b_{34}&b_{11} b_{34} &b_{11} b_{44}& b_{12}b_{44} \\
-b_{22}b_{34} & b_{12}b_{34} & b_{12}b_{44} & b_{22} b_{44}\\
\end{bmatrix}.
\end{equation}
which now depends only on the two conics and not on the genus 3 curve $C$. It is the inner product on the 4-dimensional anti-invariant part of $\Lambda^2V$ induced from the inner product on $V$.
Its determinant defines a quartic surface $S\subset {\rm P}^3$ and the curve $X\cap {\rm P}^2$ we are seeking is its intersection with the plane $Q=0$.
\begin{remark} The determinant of this is to be understood as a polynomial in the quadratics in $x,y,z$ without imposing the relations $(xy)(yz)=(xz)(y^2)$ etc. Those relations define a Veronese surface in ${\rm P}^5$ and the determinant is then $(Q_1Q_2)^2$.
\end{remark}
To obtain a formula we take a generic pair $Q_1=x^2-y^2-z^2, Q_2=a^2x^2-b^2y^2-z^2$ or equivalently we take the quadratic form on $V\cong \mathbf{C}^4$ to be
\begin{equation}
\begin{bmatrix} A^+ & 0\\
0 & A^-
\end{bmatrix}=
\begin{bmatrix} x+y & z & 0 & 0\\
z & x-y & 0 & 0\\
0 & 0 & ax+by & z\\
0 & 0 & z & ax-by
\end{bmatrix} \label{4mat}
\end{equation}
From this point of view the six parameters in the choice of $Q(x,y,z)$ provide the $3g-3=6$ degrees of freedom for the genus 3 curve $C$: $(x^2-y^2-z^2) (a^2x^2-b^2y^2-z^2)=Q^2$. The extra parameters
$a,b$ yield the 2-dimensional family of vector bundles but, from the equation of the curve, they are interrelated.
The ${\rm P}^3$ is given by $Q_1=0=Q_2$ so we use the relations $x^2-y^2-z^2=0, a^2x^2-b^2y^2-z^2=0$ to give homogeneous coordinates $w_1=yz, w_2=zx, w_3=xy, z^2=(a^2-b^2)w_0$ and then the matrix (\ref{mat1}) is (with $c=(a-b)(1+ab), d=(a+b)(1-ab)$)
\begin{equation}\label{mat2}
M=\begin{bmatrix}
cw_0-(a-b)w_3 & -bw_1-aw_2 & (b^2-a^2)w_0 & w_1-w_2\\
-bw_1-aw_2 & dw_0 + (a+b)w_3 & w_1+w_2 & (a^2-b^2) w_0\\
(b^2-a^2) w_0& w_1+w_2 & cw_0+(a-b)w_3 & -bw_1+aw_2\\
w_1-w_2 & (a^2-b^2)w_0 & -bw_1+aw_2 & d w_0-(a+b) w_3
\end{bmatrix}
\end{equation}
Incorporating terms from the third and sixth rows and columns of (\ref{mat}) this yields the following explicit formulas for the algebra. It is
generated by $1, \xi_1,\xi_2,\xi_3,\eta_1,\eta_2,\eta_3$ subject to the following relations, where $\xi\mathbf{\cdot} \eta=\xi_1\eta_1+\xi_2\eta_2+\xi_3\eta_3$:
\newpage
$$(a-b)(1+ab)(\xi_1^2+\eta_1^2)+(a+b)(1-ab)(\xi_2^2+\eta_2^2) +2(a^2-b^2)(\xi_1\eta_1-\xi_2\eta_2)+ a_1 \xi\mathbf{\cdot} \eta$$
$(\xi_1\eta_2-\xi_2\eta_1)+a(\xi_1\xi_2-\eta_1\eta_2)+a_2 \xi\mathbf{\cdot} \eta,\qquad
(\xi_1\eta_2+\xi_2\eta_1)-b(\xi_1\xi_2+\eta_1\eta_2)+a_3 \xi\mathbf{\cdot} \eta$
$$(a+b)(\xi_2^2-\eta_2^2) -(a-b)(\xi_1^2-\eta_1^2)+ a_4 \xi\mathbf{\cdot} \eta,
\qquad\xi_3^2+a_5 \xi\mathbf{\cdot} \eta,\qquad
\eta_3^2+a_6 \xi\mathbf{\cdot} \eta.$$
Just as the formulas for genus 2 in Section \ref{even} were of little use in determining the isomorphism class of the algebra, in this case we consider a more degenerate version and observe that our family consists of a continuous deformation of it.
\subsection{A special case}\label{special}
The first algebra to be calculated using this method is in \cite{H} where the relations have the simple form
\begin{equation}\label{rel0}
\xi_i^2=a_i(\xi_1\eta_1+\xi_2\eta_2+\xi_3\eta_3),\qquad \eta_i^2=b_i(\xi_1\eta_1+\xi_2\eta_2+\xi_3\eta_3)
\end{equation}
This was based on taking $Q_1=xy, Q_2=z(x+y+z)$ two degenerate quadratic forms. The discriminant is then given by three singular quadrics and the algebra may be regarded as a deformation of the relations $\xi_i^2=\eta_i^2=0$ for the cohomology of $H^*(({\mathbf C}{\rm P}^1)^6,\mathbf{C})$. This is a continuous variation in the isomorphism class of the algebra, but the parameters describe a variation of the curve $C$ rather than the bundle $E$. In fact, if we take the approach of Section \ref{vect} then
the bundle $E'$ here corresponds to taking $L^2=\pi^*U_1$ where $U_1^2$ is trivial, hence for a given curve there are only finitely many bundles of this type.
In this case we have $\mathop{\rm End}\nolimits_0 E\cong U\oplus UU_1\oplus U_1$. We can check if $E$ is very stable by looking for base points of the family of quadrics, that is non-trivial solutions to (\ref{rel0}), but these only exist if
$\sqrt{a_1b_1}+\sqrt{a_2b_2}+\sqrt{a_3b_3}=1$. So for a generic curve $C$ a bundle of this type is very stable.
Our family of vector bundles is obtained by direct image from the Prym variety by deforming $L$ away from $U_1$ and so we can say already that a generic member will be very stable.
\begin{remark} In \cite{PP} the authors show that the ``wobbly" bundles are cut out by a hypersurface in ${\rm P}^7$ of degree $48$.
\end{remark}
For this special case the quartic curve $X\cap {\rm P}^2$ is a pair of conics. To show a variation in the modulus it is enough to show that as $a,b$ vary we get an irreducible curve. But by analyzing the surface $S$ we get rather more.
\section{The quartic surface $S$}
\subsection{Plane sections}
A variety such as $S$ ($\det M=0$ from (\ref{mat2})) which is defined by the determinant of a {\it symmetric} matrix of linear forms in $n$ variables has singularities when $n>3$. These hypersurfaces are classically known as symmetroids (see e.g. \cite{Dol}). For a quartic surface the generic symmetroid has 10 nodes.
To find the singular locus in our case first note that the plane section $w_0=0$ gives a quartic curve
$$(b^2-1)^2w_1^4+((a^2-1)w_2^2+(a^2-b^2)w_3^2)^2-2(b^2-1)w_1^2((a^2-1)w_2^2+(b^2-a^2)w_3^2)=0$$
which is reducible to a pair of conics:
$$(b^2-1)w_1^2=(a^2-1)w_2^2-(a^2-b^2)w_3^2\pm 2\sqrt{(1-a^2)(a^2-b^2)}w_2w_3.$$
These meet where $w_2=0$ or $w_3=0$.
If $w_0=w_3=0$ $\sqrt{b^2-1}w_1=\pm\sqrt{a^2-1}w_2$ is a singular point of this curve. The matrix is then
$$\begin{bmatrix} 0 & -bw_1-aw_2 & 0 & w_1-w_2\\
-bw_1-aw_2 & 0 & w_1+w_2 & 0\\
0 & w_1+w_2 & 0 & -bw_1+aw_2 \\
w_1-w_2 & 0 & -b w_1+aw_2 & 0
\end{bmatrix}$$
and two pairs of rows are linearly dependent so the matrix has rank $\le 2$. This is a singular point of $S$.
Calculating the Hessian of the quartic function $\det M$ at this point one sees that unless $a^2=b^2$ or $(a^2-1)(b^2-1)=1$ the singularity is a node, and similarly at the other point $x_0=x_2=0$. Moreover, at no other points on the curve $w_0=0$ is the rank of the matrix $\le 2$, so this means that $S$ has no codimension 1 singularities. Then a general plane section is a smooth quartic curve.
\begin{prp} Let ${\mathcal N}$ be the moduli space of rank 2 semi-stable bundles $E$ with trivial determinant on a non-hyperelliptic genus 3 curve $C$ and $U$ a line bundle with $U^2$ trivial. Then a generic bundle with $E\cong E\otimes U$ is very stable, and the isomorphism class of the multiplicity algebra varies non-trivially in this two-dimensional family, for a generic curve $C$.
\end{prp}
\begin{proof}
In Section \ref{explicit} we showed using \cite{P} that such a bundle $E$ is defined up to tensoring by a line bundle by a pair of tangential conics. A generic pair can be simultaneously diagonalized and hence defined by equations $Q_1(x,y,z)=x^2-y^2-z^2=0, Q_2(x,y,z)=a^2x^2-b^2y^2-z^2=0$. We calculated the discriminant of the family of quadrics to be a singular quadric and a quartic hypersurface in ${\rm P}^5$. The projective equivalence class of the curve of intersection of the degeneracy plane of the quadric with the quartic is an invariant of the isomorphism class of the multiplicity algebra.
We identified the quartic curve as the intersection of a quartic surface $S$ with a hyperplane in ${\rm P}^5$ defined by the conic $Q$ which gives $Q_1Q_2-Q^2=0$ as the equation of $C$. A generic section is a smooth quartic curve, but the special bundles
in Section \ref{special}, which are very stable for generic $C$, belong to the connected family of bundles $E$ considered. This family thus defines quartic curves which are both smooth and reducible and hence vary in modulus.
\end{proof}
\subsection{ Ten singularities}
Going back to the quartic surface $S$, by rescaling, the roles of $x,y,z$ interchange, and since $w_0$ corresponds to the $z^2$ term and $w_3$ to $xy$, we obtain nodes by the method above for the pairs $(y^2, zx)$ and $(x^2, yz)$. This yields six singular points.
To find the other four for the symmetroid, consider the intersection of the ${\rm P}^3$ given by $Q_1=0=Q_2$ with the Veronese surface. As an element of the dual space of quadratic polynomials in $x,y,z$, a point on the Veronese corresponds to evaluation at some point $(a,b,c)\in\mathbf{C}^3$. So the intersection with ${\rm P}^3$ is in this case the four points of intersection $a_1,a_2,a_3,a_4$ of the conics $x^2-y^2-z^2=0, a^2x^2-b^2y^2-z^2=0$. In homogeneous coordinates these are $(x,y,z)=(\pm\sqrt{1-b^2},\pm \sqrt{1-a^2},\pm \sqrt{a^2-b^2})$. The symmetric matrix on $V$ given by (\ref{4mat}) then has rank $2$ and correspondingly the inner product on the anti-invariant part of $\Lambda^2V$ (which is the matrix (\ref{mat2})) has rank $\le 2$. This is thus a singular point of the determinantal surface $S$.
The line in ${\rm P}^3$ joining evaluation at $a_1=(\sqrt{1-b^2}, \sqrt{1-a^2}, \sqrt{a^2-b^2})$ to evaluation at $a_2=(\sqrt{1-b^2}, \sqrt{1-a^2}, -\sqrt{a^2-b^2})$ meets the plane $z^2=0$ in the singularity evaluated in the previous section, so we see that the ten nodes are labelled by the four points of intersection of two conics and the six lines joining them.
\section*{Acknowledgments}
The author wishes to thank Tam\'as Hausel for raising this question and ICMAT for support, and in particular Oscar Garc\'ia-Prada for his continuing contributions to research into the geometry of vector bundles on curves.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,647 |
{"url":"http:\/\/www.justanswer.com\/computer\/2tk6e-seagate-freeagent-desk-external-drive-not-able.html","text":"\u2022 100% Satisfaction Guarantee\n\nsqr_root, Information Technology Manager\nCategory: Computer\nSatisfied Customers: 83\nExperience:\u00a0 20+ years experience.\n26750020\nsqr_root is online now\n\n# My Seagate FreeAgent Desk External Drive has not been able\n\n### Customer Question\n\nMy Seagate FreeAgent Desk External Drive has not been able to successfully back up. My Icon is Red and it says \"last backup failed\". This has been going on for about 2 weeks now.\nSubmitted: 5 years ago.\nCategory:\u00a0Computer\nExpert:\u00a0 sqr_root replied\u00a05 years ago.\n\nHi.\n\nThere are a number of reasons why the backup may have failed.\n\nFirst check and make sure that all of your cables are connected and not loose.\n\nThe next most likely cause is that one of the files that it was trying to backup was \"busy\" or \"locked, meaning that it was still in use when the backup ran.\n\nGo to the backup status icon and right click on it. You will see an option to \"View Backup Log\" . This log file will give you more information about why the backup failed.\n\nCustomer: replied\u00a05 years ago.\n\nThank you for replying Kelly....I have looked at the log and there is a huge list of things that were skipped and a few clusters of errors. Most of the errors are locked. Am I supposed to do something to resolve these errors and what about all those skipped. Basiclly, what do I do next if anything?\n\nExpert:\u00a0 sqr_root replied\u00a05 years ago.\n\nHi.\n\nCan you copy the contents of the error log and paste here for me to look at?\n\nCustomer: replied\u00a05 years ago.\nI can't figure out how to do that. I have copied it to the Desktop and clicked copy then when I come back here I can't get it to paste. I then put it into \"Documents\" and had the same results. Any suggestions?\nExpert:\u00a0 sqr_root replied\u00a05 years ago.\n\nIf it is on your desktop as a text file then doubleclick the text file and it should open up in notepad. You can then select the text, copy and paste it here.\n\nCustomer: replied\u00a05 years ago.\nSorry, I'm really having trouble. I tried it your way and it was too long, it would not send it. I have copied it to a file folder and have tried to copy and paste without success. I'm sorry this is taking so long, I will keep trying to find a way to get it to you unless you want to give up on this task.\nExpert:\u00a0 sqr_root replied\u00a05 years ago.\n\nNo need to apologize.\n\nAre you able to open the file in notepad successfully?\n\nIf so, if you can copy and paste as much of the end of the file as possible that may provide more clues.\n\nCustomer: replied\u00a05 years ago.\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Common\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Data\\Common\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Data\\Other\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Data\\RequestTypes\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Data\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Events\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Interface\\Animation\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Interface\\Interfaces\\Callback\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Interface\\Interfaces\\Full\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Interface\\Interfaces\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\Interface\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\PlatformSpecific\\Vista\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\AwsClasses\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\EN\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\FR\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\cams\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\current\\day\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\current\\night\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\forecast\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\mini\\day\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\mini\\night\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\more\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\other\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\backgrounds\\radar\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\loader\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\navigation\\left\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\navigation\\right\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\navigation\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\alerts\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\collapse\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\expand\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\forecasticons\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\tabs\\collapse (old)\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\tabs\\expand (new)\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\tabs\\live\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\other\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\small\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\update\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\images\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\JA\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\KO\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\ZH\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files\\Windows Sidebar\\Gadgets\\WeatherBug.Gadget\\ZH-sg\\vssver2.scc Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\master.mdf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\mastlog.ldf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\model.mdf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\modellog.ldf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\msdbdata.mdf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\msdblog.ldf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\tempdb.mdf Open or locked file skipped\nERROR >>> C:\\Program Files (x86)\\Microsoft SQL Server\\MSSQL.1\\MSSQL\\Data\\templog.ldf Open or locked file skipped\nSkipped C:\\Program Files (x86)\\Online Services\\MSN90\\Menu\\Thumbs.db Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files (x86)\\SMINST\\BOOTDISK\\boo.mgr Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files (x86)\\SMINST\\DVD Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Program Files (x86)\\SMINST\\HPCD.sys Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Application Data Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Desktop Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Documents Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Favorites Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Hewlett-Packard\\HP Advisor\\basefeeds\\hq\\93\\ec-base\\attach\\media\\bby\\Thumbs.db Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Hewlett-Packard\\HP Advisor\\basefeeds\\hq\\93\\ec-base\\attach\\media\\windows\\Thumbs.db Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\ProgramData\\Kaspersky Lab\\AVP8\\Data\\av71E2.tmp Open or locked file skipped\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_CValidator.H1D Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_AssetId.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_BestBet.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_LinkTerm.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_SubjectTerm.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MTOC_help.H1H Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MValidator.H1D Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MValidator.Lck Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Microsoft\\DRM\\Server Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\ProgramData\\Microsoft\\Search\\Data\\Applications\\Windows\\MSS.log Open or locked file skipped\nERROR >>> C:\\ProgramData\\Microsoft\\Search\\Data\\Applications\\Windows\\MSStmp.log Open or locked file skipped\nERROR >>> C:\\ProgramData\\Microsoft\\Search\\Data\\Applications\\Windows\\tmp.edb Open or locked file skipped\nERROR >>> C:\\ProgramData\\Microsoft\\Search\\Data\\Applications\\Windows\\Windows.edb Open or locked file skipped\nSkipped C:\\ProgramData\\Microsoft\\Windows\\DRM Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Start Menu Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\ProgramData\\Templates Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\All Users Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\Application Data Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\History Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\Microsoft\\Windows\\History Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\Microsoft\\Windows\\Temporary Internet Files Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\Temp Not Supported\/Folder skipped ( backup of Windows System Directory and temporary Internet Files is not supported)\nSkipped C:\\Users\\Default\\AppData\\Local\\Temporary Internet Files Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\AppData\\Roaming\\Microsoft\\Windows\\Cookies Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Application Data Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Cookies Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Documents\\My Music Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Documents\\My Pictures Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Documents\\My Videos Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Local Settings Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\My Documents Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\NetHood Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\NTUSER.DAT Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\NTUSER.DAT.LOG Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Default\\ntuser.dat.LOG1 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Default\\ntuser.dat.LOG2 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Default\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\PrintHood Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Recent Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\SendTo Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Start Menu Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default\\Templates Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Default User Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Application Data Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\History Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_CValidator.H1D Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_AssetId.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_BestBet.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_LinkTerm.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MKWD_SubjectTerm.H1W Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MTOC_help.H1H Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MValidator.H1D Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Assistance\\Client\\1.0\\en-US\\Help_MValidator.Lck Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Feeds Cache Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Internet Explorer\\DOMStore Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Internet Explorer\\Recovery\\Active\\RecoveryStore.{D8059234-DBA8-11DE-8956-0026187B10B4}.dat Open or locked file skipped\nERROR >>> C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Internet Explorer\\Recovery\\Active\\{4915C3E1-DBAA-11DE-8956-0026187B10B4}.dat Open or locked file skipped\nERROR >>> C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Internet Explorer\\Recovery\\Active\\{A39FD0C0-DBAB-11DE-8956-0026187B10B4}.dat Open or locked file skipped\nERROR >>> C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Internet Explorer\\Recovery\\Active\\{EC806901-DBA9-11DE-8956-0026187B10B4}.dat Open or locked file skipped\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\History Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\Temporary Internet Files Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat.LOG1 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat.LOG2 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{1a100a4d-bb64-11de-aeaf-0026187b10b4}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{1a100a4d-bb64-11de-aeaf-0026187b10b4}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{1a100a4d-bb64-11de-aeaf-0026187b10b4}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{792c9a30-9990-11de-9ca9-0026187b10b4}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{792c9a30-9990-11de-9ca9-0026187b10b4}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{792c9a30-9990-11de-9ca9-0026187b10b4}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{7fe39e4e-bb32-11de-ab6a-0026187b10b4}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{7fe39e4e-bb32-11de-ab6a-0026187b10b4}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows\\UsrClass.dat{7fe39e4e-bb32-11de-ab6a-0026187b10b4}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\Users\\Katherine\\AppData\\Local\\Microsoft\\Windows Defender\\FileTracker\\{28DFECD1-D7E0-4EAA-B390-373CC1B44BFC} Open or locked file skipped\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Temp Not Supported\/Folder skipped ( backup of Windows System Directory and temporary Internet Files is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\Temporary Internet Files Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Local\\VirtualStore\\Windows\\ftpcache Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\LocalLow\\Microsoft\\Internet Explorer\\DOMStore Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Internet Explorer\\UserData Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Protect\\CREDHIST Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Protect\\S-1-5-21-3939495418-199653993-2255716305-1000\\4a417d8a-88af-407d-bef0-db8689297f50 Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Protect\\S-1-5-21-3939495418-199653993-2255716305-1000\\Preferred Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Windows\\Cookies Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Windows\\IECompatCache\\Low Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Windows\\IETldCache Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\AppData\\Roaming\\Microsoft\\Windows\\PrivacIE Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Application Data Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Cookies Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Documents\\My Music Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Documents\\My Pictures Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Documents\\My Videos Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Local Settings Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\My Documents Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\NetHood Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat.LOG1 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat.LOG2 Not Supported\/FIle skipped ( backup of operating system files is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{25ea617f-bb62-11de-b07c-b37b94e49c91}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{25ea617f-bb62-11de-b07c-b37b94e49c91}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{25ea617f-bb62-11de-b07c-b37b94e49c91}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{7fe39e4a-bb32-11de-ab6a-0026187b10b4}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{7fe39e4a-bb32-11de-ab6a-0026187b10b4}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.dat{7fe39e4a-bb32-11de-ab6a-0026187b10b4}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TM.blf Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TMContainer00000000000000000001.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\NTUSER.DAT{c328fef1-6a85-11db-9fbd-cf3689cba3de}.TMContainer00000000000000000002.regtrans-ms Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\ntuser.ini Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\PrintHood Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Recent Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\SendTo Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Start Menu Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Katherine\\Templates Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nERROR >>> C:\\Users\\Public\\Documents\\Intuit\\QuickBooks\\Company Files\\Stuhr Products.QBW\nERROR >>> C:\\Users\\Public\\Documents\\Intuit\\QuickBooks\\Company Files\\Stuhr Products.QBW.TLG\nSkipped C:\\Users\\Public\\Documents\\My Music Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Documents\\My Pictures Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Documents\\My Videos Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArtSmall.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{208F236E-B511-4949-BDF9-3791602ED53A}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{208F236E-B511-4949-BDF9-3791602ED53A}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{2BEDE989-0477-48C8-8E85-D5FC97494EC0}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{2BEDE989-0477-48C8-8E85-D5FC97494EC0}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{81244B04-70BE-47F1-9A5E-2026093D598F}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{81244B04-70BE-47F1-9A5E-2026093D598F}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{CA6465E3-92B8-4969-B053-E091250B3E3E}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{CA6465E3-92B8-4969-B053-E091250B3E3E}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{D4213C57-0F32-4AED-82E0-A6560E1EA35F}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{D4213C57-0F32-4AED-82E0-A6560E1EA35F}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{DAAE5A7A-D07D-4C7C-AE7B-E926C737721B}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{DAAE5A7A-D07D-4C7C-AE7B-E926C737721B}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{F87D14E5-4DEB-4169-B9EA-D067EBCD4297}_Large.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\AlbumArt_{F87D14E5-4DEB-4169-B9EA-D067EBCD4297}_Small.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Music\\Sample Music\\Folder.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Recorded TV\\Sample Media\\Apollo 13.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\nSkipped C:\\Users\\Public\\Recorded TV\\Sample Media\\Vertigo.jpg Not Supported\/File skipped (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported)\n\nCopied 354 files, 49032808 bytes. Total files protected: 75108. Unable to backup 20 file(s)\n\nBackup completed Friday, November 27, 2009 10:00:55 PM\nExpert:\u00a0 sqr_root replied\u00a05 years ago.\n\nHi.\n\nGood news. - your backups are completing successfully.\n\nThere are two issues being seen in this log file.\n\n1. On lines where you see (backup of operating system files and folders ( those that have both system and hidden attributes) is not supported) this is nothing to worry about. You can adjust the backup settings of FreeAgent to have it stop trying to back up these files (and stop showing the red icon) if you'd like.\n\nHere is the link that explains how to do this:\n\nhttp:\/\/seagate.custkb.com\/seagate\/crm\/selfservice\/search.jsp?DocId=209671&NewLang=en&Hilite=red\n\nThere are some files that show as not being backed up because they were open or in use by some other application.\n\nThe easiest way around this is to make sure that you close all other applications before the backup runs.\n\nsqr_root, Information Technology Manager\nCategory: Computer\nSatisfied Customers: 83\nExperience:\u00a020+ years experience.\nCustomer: replied\u00a05 years ago.\nThank you, XXXXX XXXXX give this a try.\nCustomer: replied\u00a05 years ago.\nThank you so much..I now have a green icon and my backup was a success. You were very patient with me and I appreciate your helping me on this matter. Now you can go holiday shopping and not worry about me. Thank again!\n\nAsk-a-doc Web sites: If you've got a quick question, you can try to get an answer from sites that say they have various specialists on hand to give quick answers... Justanswer.com.\n...leave nothing to chance.\nTraffic on JustAnswer rose 14 percent...and had nearly 400,000 page views in 30 days...inquiries related to stress, high blood pressure, drinking and heart pain jumped 33 percent.\nTory Johnson, GMA Workplace Contributor, discusses work-from-home jobs, such as JustAnswer in which verified Experts answer people\u2019s questions.\nI will tell you that...the things you have to go through to be an Expert are quite rigorous.\n\n### What Customers are Saying:\n\n\u2022 My Expert answered my question promptly and he resolved the issue totally. This is a great service. I am so glad I found it I will definitely use the service again if needed. One Happy Customer New York\n< Last | Next >\n\u2022 My Expert answered my question promptly and he resolved the issue totally. This is a great service. I am so glad I found it I will definitely use the service again if needed. One Happy Customer New York\n\u2022 I am very happy with my very fast response. Eric is very knowledgeable in the subject area. Thank you! RP Austin, TX\n\u2022 Hi John, Thank you for your expertise and, more important, for your kindness because they make me, almost, look forward to my next computer problem. After the next problem comes, I'll be delighted to correspond again with you. I'm told that I excel at programing. But system administration has never been one of my talents. So it's great to have an expert to rely on when the computer decides to stump me. God bless, Bill Bill M. Schenectady, New York\n\u2022 The Expert answered my Mac question and was patient. He answered in a thorough and timely manner, keeping the response on a level that could understand. Thank you! Frank Canada\n\u2022 Wonderful service, prompt, efficient, and accurate. Couldn't have asked for more. I cannot thank you enough for your help. Mary C. Freshfield, Liverpool, UK\n\u2022 This expert is wonderful. They truly know what they are talking about, and they actually care about you. They really helped put my nerves at ease. Thank you so much!!!! Alex Los Angeles, CA\n\u2022 Thank you for all your help. It is nice to know that this service is here for people like myself, who need answers fast and are not sure who to consult. GP Hesperia, CA\n\n\u2022 ### Andy\n\n#### Satisfied Customers:\n\n5311\n11yr exp, Comp Engg, Internet expert, Web developer, SEO\n< Last | Next >\n\n### Andy\n\n#### Satisfied Customers:\n\n5311\n11yr exp, Comp Engg, Internet expert, Web developer, SEO\n\n### James\n\n#### Satisfied Customers:\n\n8376\n20 years of experience building, fixing and servicing PCs and operating systems.\n\n### Ryan H.\n\n#### Satisfied Customers:\n\n1741\nA+ Certified Technician - 10 Years experience working with all types of computer systems.\n\n### Jane Lefler\n\n#### Satisfied Customers:\n\n0\nComputer Programmer \/ Technician\/ Consultant 16+ years\n\n### RPI Solutions\n\n#### Satisfied Customers:\n\n3476\n5+ Years in IT, BS in Computer Science\n\n### B. Rath\n\n#### Satisfied Customers:\n\n8671\nCertified Computer\/Networking Support Specialist.\n\n### Frederick S.\n\n#### Satisfied Customers:\n\n7240\nComputer technician and founder of a home PC repair company.\n\n## Related Computer Questions\n\nChat Now With A Tech Support Specialist\nsqr_root\n80 Satisfied Customers\n20+ years experience.","date":"2014-12-19 08:02:39","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8619768619537354, \"perplexity\": 8594.422123078755}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-52\/segments\/1418802768309.18\/warc\/CC-MAIN-20141217075248-00158-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/193434\/why-is-an-integration-error-appearing-during-this-code","text":"# Why is an integration error appearing during this code?\n\nI'm really confused about this to be honest. If i evaluate approx[x,k] for some k, and then input the list into the Plot function, I get a plot. But if I try to use approx[x,k] directly, I get an Integration error, even though, outside of Plot, this error doesn't happen. I feel like I'm fundamentally missing something. NIntegrate fixes this, which I can use, but I'm worried that for more complex Inner Product evaluations that NIntegrate may not suffice. Any ideas on the error?\n\n**** DOES NOT WORK WITH N[] OR \/\/N ****\n\nClear[chebyinnerprod, approx]\nchebyinnerprod[k_] := chebyinnerprod[k]=\nIntegrate[(Sin[\\[Pi] x] ChebyshevT[k, x])\/Sqrt[1 - x^2], {x, -1, 1}]\/\nWhich[k == 0, \\[Pi], k > 0, \\[Pi]\/2]\n\napprox[x_, k_] := approx[x,k]=Sum[chebyinnerprod[t]*ChebyshevT[t, x], {t, 0, k}]\n\nPlot[approx[x, 2], {x, -1, 1}]\n\n\n**** WORKS ****\n\nClear[chebyinnerprod, approx]\n\nchebyinnerprod[k_] := chebyinnerprod[k]=\nNIntegrate[(Sin[\\[Pi] x] ChebyshevT[k, x])\/Sqrt[1 - x^2], {x, -1, 1}]\/\nWhich[k == 0, \\[Pi], k > 0, \\[Pi]\/2]\n\napprox[x_, k_] :=approx[x,k]= Sum[chebyinnerprod[t]*ChebyshevT[t, x], {t, 0, k}]\n\nPlot[approx[x, 2], {x, -1, 1}]\n\n\u2022 Try this: chebyinnerprod[k_Integer?NonNegative] := chebyinnerprod[k] = Block[{x}, Integrate[(Sin[\u03c0 x] ChebyshevT[k, x])\/Sqrt[1 - x^2], {x, -1, 1}]\/Which[k == 0, \u03c0, k > 0, \u03c0\/2]] \u2013\u00a0J. M. will be back soon Mar 17 '19 at 16:18\n\u2022 So you suspect localizing chebyinnerprod will fix the solution, if I understand you? I'm going to try it right now. \u2013\u00a0Shinaolord Mar 17 '19 at 16:19\n\u2022 That does fix it. I'm surprised that worked, to be honest. Thanks ! (: \u2013\u00a0Shinaolord Mar 17 '19 at 16:20\n\u2022 You might also be interested in this thread. \u2013\u00a0J. M. will be back soon Mar 17 '19 at 16:20\n\u2022 Mostly, your problem was in using x both in Integrate[] and Plot[]. \u2013\u00a0J. M. will be back soon Mar 17 '19 at 16:22","date":"2020-01-28 18:52:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.39605265855789185, \"perplexity\": 4175.780977812274}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251783000.84\/warc\/CC-MAIN-20200128184745-20200128214745-00072.warc.gz\"}"} | null | null |
Wrenne works, and works hard, for her art. Not only does she possess a startling, striking voice but this is an exciting performance obviously influenced by the theatrical productions of Bjork, Kate Bush and Sia (albeit on a much smaller scale).
There is a throughline to her show, it's like a concept album being performed live, dealing with ideas of self-discovery and acceptance. Admittedly some of the narrative gets lost in the dynamic staging as Wrenne shifts swiftly and elegantly around the deceptively simple stage using boxes, beams, projections and platforms to bring the story to vivid life.
The music itself moves from Ellie Goulding-style electro pop to deeper darker beats reminiscent of someone like Massive Attack.
Her vocals are incredibly sharp and pure over the backing track (which includes a collaboration with Idris Elba) capturing a real sense of passion and emotion. A couple of acoustic numbers also prove Wrenne is a fine guitarist. She ends meekly asking the audience if we have any questions before a final cover of a Scots pop / rock favourite (we won't spoil the surprise) reworked in her own unique style. A captivating performer with an extraordinary voice. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,751 |
Salvatore Livatino, University of Hertfordshire, U.K.
3D acquisition today: all's well that ends well?
Design and Implementation of a Low Cost Virtual Rugby Decision Making Interactive Alan Cummins, Cathy Craig.
Download here the Salento AVR 2016 scientific program. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,618 |
\section{Introduction}
This paper is devoted to studying the Dirichlet problem for elliptic systems in
the upper-half space with data in generalized H\"{o}lder spaces and generalized
Morrey-Campa\-nato spaces. As a byproduct of the PDE-based techniques developed here, we are able
to establish the equivalence of these function spaces. To be more specific requires introducing some notation.
Having fixed $n,M\in{\mathbb{N}}$ with $n\geq 2$ and $M\geq 1$, consider a homogeneous,
constant (complex) coefficient, $M\times M$ second-order system in ${\mathbb{R}}^n$
\begin{equation}\label{ars}
L:=\left(a_{jk}^{\alpha\beta}\partial_j\partial_k\right)_{1\leq\alpha,\beta\leq M}.
\end{equation}
Here and elsewhere, the summation convention over repeated indices is employed.
We make the standing assumption that $L$ is strongly elliptic, in the sense that
there exists $\kappa_0>0$ such that the following Legendre-Hadamard condition is satisfied:
\begin{equation}\label{LegHad}
\begin{array}{c}
\operatorname{Re}\left[a_{jk}^{\alpha\beta}\xi_j\xi_k\overline{\zeta_\alpha}\zeta_\beta\right]
\geq\kappa_0\left|\xi\right|^2\left|\zeta\right|^2\,\,\text{ for all}
\\[10pt]
\xi=\left(\xi_j\right)_{1\leq j\leq n}\in\mathbb{R}^n
\,\,\text{ and }\,\,\zeta=\left(\zeta_\alpha\right)_{1\leq\alpha\leq M}\in\mathbb{C}^M.
\end{array}
\end{equation}
Examples include scalar operators, such as the Laplacian $\Delta=\sum\limits_{j=1}^n\partial_j^2$
or, more generally, operators of the form ${\rm div}A\nabla$ with $A=(a_{rs})_{1\leq r,s\leq n}$ an
$n\times n$ matrix with complex entries satisfying the ellipticity condition
\begin{equation}\label{YUjhv-753}
\operatornamewithlimits{inf\vphantom{p}}_{\xi\in S^{n-1}}{\rm Re}\,\big[a_{rs}\xi_r\xi_s\bigr]>0,
\end{equation}
(where $S^{n-1}$ denotes the unit sphere in ${\mathbb{R}}^n$), as well as the complex
version of the Lam\'e system of elasticity in $\mathbb{R}^n$,
\begin{equation}\label{TYd-YG-76g}
L:=\mu\Delta+(\lambda+\mu)\nabla{\rm div}.
\end{equation}
Above, the constants $\lambda,\mu\in{\mathbb{C}}$ (typically called Lam\'e moduli),
are assumed to satisfy
\begin{equation}\label{Yfhv-8yg}
{\rm Re}\,\mu>0\,\,\mbox{ and }\,\,{\rm Re}\,(2\mu+\lambda)>0,
\end{equation}
a condition equivalent to the demand that the Lam\'e system \eqref{TYd-YG-76g}
satisfies the Legendre-Hadamard ellipticity condition \eqref{LegHad}.
While the Lam\'e system is symmetric, we stress that the results in this paper require no symmetry
for the systems involved.
With each system $L$ as in \eqref{ars}-\eqref{LegHad} one may associate a Poisson kernel, $P^L$,
which is a ${\mathbb{C}}^{M\times M}$-valued function defined in ${\mathbb{R}}^{n-1}$ described
in detail in Theorem~\ref{PoissonConvolution}. This Poisson kernel has played a pivotal role in
the treatment of the Dirichlet problem with data in $L^p$, $\mathrm{BMO}$, $\mathrm{VMO}$ and
H\"older spaces (see \cite{K-MMMM, BMO-MarMitMitMit16}). For now, we make the observation that the
Poisson kernel gives rise to a nice approximation to the identity in ${\mathbb{R}}^{n-1}$ by setting
$P^L_t(x')=t^{1-n}P^L(x'/t)$ for every $x'\in{\mathbb{R}}^{n-1}$ and $t>0$.
For every point $x\in\mathbb{R}^n$ write $x=(x',t)$, where $x'\in\mathbb{R}^{n-1}$ corresponds to
the first $n-1$ coordinates of $x$, and $t\in\mathbb{R}$ is the last coordinate of $x$. As is customary,
we shall let $\mathbb{R}^n_{+}:=\{x=(x',t)\in\mathbb{R}^n:\,x'\in\mathbb{R}^{n-1},\,t>0\}$ denote the
upper-half space in $\mathbb{R}^n$, and typically identify its boundary with $(n-1)$-dimensional Euclidean
space, via $\partial\mathbb{R}^n_{+}\ni(x',0)\equiv x'\in\mathbb{R}^{n-1}$. The cone with vertex at
$x'\in\mathbb{R}^{n-1}$ and aperture $\kappa>0$ is defined as
\begin{equation}\label{64rrf}
\Gamma_{\kappa}(x'):=\{y=(y',t)\in\mathbb{R}^n_{+}:\,|x'-y'|<\kappa t\}.
\end{equation}
When $\kappa=1$ we agree to drop the dependence on aperture and simply write $\Gamma(x')$.
Whenever meaningful, the nontangential pointwise trace of a vector-valued function $u$ defined
in $\mathbb{R}^n_{+}$ is given by
\begin{equation}\label{6tfF}
\left(\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}\right)(x'):=
\lim_{\substack{\mathbb{R}^n_+\ni y\to (x',0)\\ y\in\Gamma_\kappa (x')}}u(y),
\quad x'\in\mathbb{R}^{n-1}.
\end{equation}
The unrestricted pointwise trace of a vector-valued function $u$ defined in
$\mathbb{R}^n_+$ at each $x'\in\partial\mathbb{R}^n_+\equiv\mathbb{R}^{n-1}$ is taken to be
\begin{equation}\label{tfDSq56f}
\left(\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm lim}}\right)(x')
:=\lim_{\mathbb{R}^n_+\ni y\to(x',0)}u(y),\quad x'\in\mathbb{R}^{n-1},
\end{equation}
whenever such a limit exists exists.
\begin{definition}\label{6ttFSD}
Call a given mapping $\omega:(0,\infty)\to(0,\infty)$ a \texttt{growth function} if $\omega$ is
non-decreasing and $\omega(t)\to 0$ as $t\to 0^{+}$.
\end{definition}
\begin{definition}\label{7tFFD}
Let $E\subset\mathbb{R}^{n}$ be an arbitrary set (implicitly assumed to have cardinality at least $2$)
and let $\omega$ be a growth function. The \texttt{homogeneous $\omega$-H\"older space} on $E$ is defined as
\begin{equation}\label{7rsSS}
\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M):=\big\{u:E\to\mathbb{C}^M:\,
[u]_{\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)}<\infty\big\},
\end{equation}
where $[\cdot]_{\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)}$ stands for the seminorm
\begin{equation}\label{7tEEE}
[u]_{\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)}:=\sup_{\substack{x,y\in E\\ x\neq y}}\frac{|u(x)-u(y)|}{\omega(|x-y|)}.
\end{equation}
\end{definition}
Let us note that the fact that $\omega(t)\to 0$ as $t\to 0^+$ implies that if
$u\in\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)$ then $u$ is uniformly continuous.
The choice $\omega(t):=t^\alpha$ for each $t>0$, with $\alpha\in(0,1)$, yields the
classical scale of H\"older spaces.
Here and elsewhere in the paper, we agree to denote the $(n-1)$-dimensional Lebesgue measure of
given Lebesgue measurable set $E\subseteq{\mathbb{R}}^{n-1}$ by $|E|$. Also, by a cube $Q$ in $\mathbb{R}^{n-1}$
we shall understand a cube with sides parallel to the coordinate axes. Its side-length will be denoted by $\ell(Q)$,
and for each $\lambda>0$ we shall denote by $\lambda\,Q$ the cube concentric with $Q$ whose side-length is
$\lambda\,\ell(Q)$. For every function $h\in L^1_{\rm loc}(\mathbb{R}^{n-1},{\mathbb{C}}^M)$ we write
\begin{equation}\label{nota-aver}
h_Q:=\fint_{Q} h(x')\,dx':=\frac{1}{|Q|}\int_{Q}h(x')\,dx'\in{\mathbb{C}}^M,
\end{equation}
with the integration performed componentwise.
\begin{definition}\label{65RFF}
Given a growth function $\omega$ along with some integrability exponent $p\in[1,\infty)$,
the associated \texttt{generalized Morrey-Campanato space} in $\mathbb{R}^{n-1}$ is defined as
\begin{equation}\label{utfFFD}
\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)
:=\big\{f\in L^1_{\operatorname{loc}}(\mathbb{R}^{n-1},\mathbb{C}^M):\,
\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}<\infty\big\},
\end{equation}
where $\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}$ stands for the seminorm
\begin{equation}\label{987tF}
\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
:=\sup_{Q\subset\mathbb{R}^{n-1}}\frac{1}{\omega(\ell(Q))}\bigg(\fint_Q|f(x')-f_Q|^p\,dx'\bigg)^{1/p}.
\end{equation}
\end{definition}
The choice $\omega(t):=t^\alpha$ with $\alpha\in(0,1)$ corresponds to the classical Morrey-Campanato spaces,
while the special case $\omega(t):=1$ yields the usual space of functions of bounded mean oscillations ($\operatorname{BMO}$).
We also define, for every $u\in\mathscr{C}^1(\mathbb{R}^{n}_{+},\mathbb{C}^M)$ and $q\in(0,\infty)$,
\begin{equation}\label{7yGGG}
\norm{u}_{**}^{(\omega,q)}:=\sup_{Q\subset\mathbb{R}^{n-1}}
\frac{1}{\omega(\ell(Q))}\bigg(\fint_Q\bigg(\int_0^{\ell(Q)}|(\nabla u)(x',t)|^2\,t\,dt\bigg)^{q/2}\,dx'\bigg)^{1/q}.
\end{equation}
As far as this seminorm is concerned, there are two reasonable candidates for the end-point $q=\infty$
(see Proposition~\ref{prop:proper-sols} and Lemma~\ref{lem:appendix}). First, we may consider
\begin{equation}\label{ygFF}
\norm{u}_{**}^{(\omega,\exp)}:=\sup_{Q\subset\mathbb{R}^{n-1}}
\frac{1}{\omega(\ell(Q))}\bigg\| \Big( \int_0^{\ell(Q)}|(\nabla u)(\cdot,t)|^2\,t\,dt
\Big)^{1/2} \bigg\|_{\exp L,Q}
\end{equation}
where $\|\cdot\|_{\exp L,Q}$ is the version of the norm in the Orlicz space $\exp L$
localized and normalized relative to $Q$, i.e.,
\begin{equation}\label{eq:Lux-norm}
\|f\|_{\exp L,Q}:=\operatornamewithlimits{inf\vphantom{p}}\left\{t>0:\fint_{Q}\Big(e^{\tfrac{|f(x')|}{t}}-1\Big)\,dx'\le 1\right\}.
\end{equation}
Second, corresponding to the limiting case $q=\infty$ we may consider
\begin{equation}\label{yRRFF}
\norm{u}_{**}^{(\omega,\infty)}:=\sup_{(x',t)\in\mathbb{R}^{n}_{+}}\frac{t}{\omega(t)}|(\nabla u)(x',t)|.
\end{equation}
We are ready to describe our main result concerning the Dirichlet problems with
data in generalized H\"older and generalized Morrey-Campanato spaces for homogeneous
second-order strongly elliptic systems of differential operators with constant complex coefficients
(cf. \eqref{ars} and \eqref{LegHad}). In Section~\ref{section:well-general}
(cf. Theorems~\ref{mainthmBMO2b}-\ref{mainthmBMO2}), we weaken the condition \eqref{omega-cond:main}
and still prove well-posedness for the two Dirichlet problems. The main difference
is that in that case they are no longer equivalent as \eqref{maincorBMOeq} might fail
(see Example~\ref{example}).
\begin{theorem}\label{mainthmBMO}
Consider a strongly elliptic constant complex coefficient second-order $M\times M$ system $L$, as in
\eqref{ars}-\eqref{LegHad}. Also, fix $p\in[1,\infty)$ along with $q\in(0,\infty]$, and let $\omega$
be a growth function satisfying, for some finite constant $C_0\geq 1$,
\begin{equation}\label{omega-cond:main}
\int_0^{t}\omega(s)\frac{ds}{s}+t\,\int_t^{\infty}\frac{\omega(s)}{s}\,\frac{ds}{s}\leq C_0\,\omega(t)
\,\,\text{ for each }\,\,t\in(0,\infty).
\end{equation}
Then the following statements are true.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item\label{bvp-Hol-Dir:main}
The generalized H\"older Dirichlet problem for the system $L$ in $\mathbb{R}^{n}_{+}$, i.e.,
\begin{equation}\label{BVPb}
\left\lbrace
\begin{array}{l}
u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M),
\\[4pt]
Lu=0\,\,\text{ in }\,\,\mathbb{R}^n_{+},
\\[4pt]
\left[u\right]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}<\infty,
\\[4pt]
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm lim}}=f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)
\,\,\text{ on }\,\,\mathbb{R}^{n-1},
\end{array}
\right.
\end{equation}
is well-posed. More specifically, there is a unique solution which is given by
\begin{equation}\label{eqn-Dir-BMO:u}
u(x',t)=(P_t^L*f)(x'),\qquad\forall\,(x',t)\in{\mathbb{R}}^n_{+},
\end{equation}
where $P^L$ denotes the Poisson kernel for $L$ in $\mathbb{R}^{n}_+$ from Theorem~\ref{PoissonConvolution}.
In addition, $u$ belongs to the space $\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$,
satisfies $\restr{u}{\partial\mathbb{R}^n_{+}}=f$, and there exists a finite constant $C=C(n,L,\omega)\geq 1$
such that
\begin{equation}\label{mainthmBMOeq2}
C^{-1}[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq[u]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}
\leq C [f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
\item\label{bvp-MC-Dir:main}
The generalized Morrey-Campanato Dirichlet problem for $L$ in $\mathbb{R}^{n}_{+}$, formulated as
\begin{equation}\label{BVP}
\left\lbrace
\begin{array}{l}
u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M),
\\[4pt]
Lu=0\,\,\text{ in }\,\,\mathbb{R}^n_{+},
\\[4pt]
\norm{u}_{**}^{(\omega,q)}<\infty,
\\[4pt]
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}
=f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)\,\,\text{ a.e. on }\,\,\mathbb{R}^{n-1},
\end{array}
\right.
\end{equation}
is well-posed. More precisely, there is a unique solution \eqref{BVP} which is given by \eqref{eqn-Dir-BMO:u}.
In addition, $u$ belongs to $\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$, satisfies
$\restr{u}{\partial\mathbb{R}^n_{+}}=f$ a.e. on $\mathbb{R}^{n-1}$, and there exists a
finite constant $C=C(n,L,\omega,p,q)\geq 1$ such that
\begin{equation}\label{mainthmBMOeq1}
C^{-1}\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq\norm{u}_{**}^{(\omega,q)}
\leq C\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
Furthermore, all these properties remain true if $\norm{\cdot}_{**}^{(\omega,q)}$ is replaced everywhere
by $\norm{\cdot}_{**}^{(\omega,\exp)}$.
\item\label{equiv:main}
The following equality between vector spaces holds
\begin{equation}\label{maincorBMOeq}
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)
=\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)
\end{equation}
with equivalent norms, where the right-to-left inclusion is understood in the sense
that for each $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ there exists a unique
$\widetilde{f}\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ with the property that
$f=\widetilde{f}$ a.e. in $\mathbb{R}^{n-1}$.
As a result, the Dirichlet problems \eqref{BVPb} and \eqref{BVP} are equivalent. Specifically,
for any pair of boundary data which may be identified in the sense of \eqref{maincorBMOeq}
these problems have the same unique solution (given by \eqref{eqn-Dir-BMO:u}).
\end{list}
\end{theorem}
A few comments regarding the previous result. In Lemma~\ref{wlemma} we shall
prove that, for growth functions as in \eqref{omega-cond:main}, each
$u\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$ extends uniquely to
a function $u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$.
Hence, the ordinary restriction $\restr{u}{\partial\mathbb{R}^n_{+}}$ is well-defined in the
context of item {\rm (a)} of Theorem~\ref{mainthmBMO}.
In item {\rm (b)} the situation is slightly different. One can first show that $u$ extends to a
continuous function up to, and including, the boundary. Hence, the non-tangential pointwise
trace agrees with the restriction to the boundary everywhere. However, since functions in
$\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ are canonically identified whenever they
agree outside of a set of zero Lebesgue measure, the boundary condition in \eqref{BVP} is most naturally
formulated by asking that the non-tangential boundary trace agrees with the boundary datum almost everywhere.
The same type of issue arises when interpreting \eqref{maincorBMOeq}. Specifically, while the left-to-right inclusion
has a clear meaning, the converse inclusion should be interpreted as saying that each equivalence class
in $\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ (induced by the aforementioned identification)
has a unique representative from $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$. We would like to observe that \eqref{maincorBMOeq} extends the well-known result of N.G.~Meyers \cite{Mey} who considered the case $\omega(t)=t^\alpha$, $t>0$. Here we extend the class of growth functions for which \eqref{maincorBMOeq} holds and our alternative approach is based on PDE.
It is illustrative to provide examples of growth functions to which Theorem~\ref{mainthmBMO} applies.
In this vein, we first observe that \eqref{omega-cond:main} is closely related to the dilation indices
of Orlicz spaces studied in \cite{BenSha88, FioKrb97} in relation to interpolation in Orlicz spaces.
Concretely, given a growth function $\omega$ set
\begin{equation}\label{6gDDD.1}
h_{\omega}(t):=\sup_{s>0}\frac{\omega(st)}{\omega(s)},\qquad\forall\,t>0,
\end{equation}
and define the lower and upper dilation indices, respectively, as
\begin{equation}\label{6gDDD.2}
i_{\omega}:=\sup_{0<t<1}\frac{\log h_{\omega}(t)}{\log t}\,\,\text{ and }\,\,
I_{\omega}:=\operatornamewithlimits{inf\vphantom{p}}_{t>1}\frac{\log h_{\omega}(t)}{\log t}.
\end{equation}
One can see that if $0<i_\omega\leq I_\omega<1$ then \eqref{omega-cond:main} holds.
Indeed, it is not hard to check that there exists a constant $C\in(0,\infty)$ with the property that
$h_\omega(t)\leq C\,t^{i_\omega/2}$ for every $t\in(0,1]$, and $h_\omega(t)\leq C\,t^{(I_\omega+1)/2}$
for every $t\in[1,\infty)$. These, in turn, readily yield \eqref{omega-cond:main}.
Now, given $\alpha\in(0,1)$, if $\omega(t):=t^\alpha$ for each $t>0$ then $i_\omega=I_\omega=\alpha$ and, hence,
\eqref{omega-cond:main} holds. Note that in that case $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)
=\dot{\mathscr{C}}^{\alpha }(\mathbb{R}^n_{+},\mathbb{C}^M)$ is the standard homogeneous H\"older space of order
$\alpha$ and the particular version of Theorem~\ref{mainthmBMO} corresponding to this scenario
has been established in \cite{BMO-MarMitMitMit16}. This being said, there many examples of interest that are treated
here for the first time, such as $\omega(t) = t^\alpha\,(A+\log_{+}t)^\theta$ for $A:=\max \{ 1, -\theta/\alpha \}$ and each $t>0$, or
$\omega(t)= t^\alpha\,(A+\log_{+}(1/t))^\theta$ for $A:=\max \{ 1, \theta/\alpha \}$ and each $t>0$, with $0<\alpha<1$, $\theta\in\mathbb{R}$,
and $\log_{+}(t) := \max \{ 0 , \log t \}$.
In these situations $i_\omega=I_\omega=\alpha$ which guarantees that \eqref{omega-cond:main} holds.
Furthermore, if $\omega(t):=\max\{t^\alpha,t^\beta\}$, or $\omega(t):=\min\{t^\alpha,t^\beta\}$,
for each $t>0$, with $0<\alpha,\beta<1$, then in both cases we have $i_\omega=\min\{\alpha,\beta\}$
and $I_\omega=\max\{\alpha,\beta\}$, hence condition \eqref{omega-cond:main} is verified once again.
The following result, providing a characterization of the generalized H\"older and generalized Morrey-Campanato
spaces in terms of the boundary traces of solutions, is a byproduct of the proof of the above theorem.
\begin{corollary}\label{maincorBMO}
Let $L$ be a strongly elliptic, constant {\rm (}complex{\rm )} coefficient, second-order
$M\times M$ system in ${\mathbb{R}}^n$. Fix $p\in[1,\infty)$ along with $q\in(0,\infty)$,
and let $\omega$ be a growth function for which \eqref{omega-cond:main} holds. Then for every function
$u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)$ satisfying $Lu=0$ in $\mathbb{R}^{n}_{+}$
one has
\begin{equation}\label{eq:qfrafr}
\norm{u}_{**}^{(\omega,q)}\approx\norm{u}_{**}^{(\omega,\exp)}\approx\norm{u}_{**}^{(\omega,\infty)}
\approx[u]_{\dot{\mathscr{C}}^{w}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\end{equation}
where the implicit proportionality constants depend only on $L$, $n$, $q$, and the constant
$C_0$ in \eqref{omega-cond:main}. Moreover,
\begin{align}\label{HLMO}
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M) &=\big\{\restr{u}{\partial\mathbb{R}^n_{+}}:\,
u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M),\,\,Lu=0\text{ in }\mathbb{R}^{n}_{+},\,\,
\left[u\right]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}<\infty\big\}
\nonumber\\[7pt]
&=\big\{\restr{u}{\partial\mathbb{R}^n_{+}}:\,u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M),
\,\,Lu=0\text{ in }\mathbb{R}^{n}_{+},\,\,\norm{u}_{**}^{(\omega,q)}<\infty\big\}
\nonumber\\[7pt]
&=\big\{\restr{u}{\partial\mathbb{R}^n_{+}}:\,u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M),
\,\,Lu=0\text{ in }\mathbb{R}^{n}_{+},\,\,\norm{u}_{**}^{(\omega,\exp)}<\infty\big\}
\nonumber\\[7pt]
&=\big\{\restr{u}{\partial\mathbb{R}^n_{+}}:\,u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M),
\,\,Lu=0\text{ in }\mathbb{R}^{n}_{+},\,\,\norm{u}_{**}^{(\omega,\infty)}<\infty\big\}.
\end{align}
\end{corollary}
The plan of the paper is as follows. In Section~\ref{section:growth} we present some properties
of the growth functions and study some of the features of the generalized H\"{o}lder and
Morrey-Campanato spaces which are relevant to this work. Section~\ref{section:props-elliptic}
is reserved for collecting some known results for elliptic systems, and for giving the proof
of Proposition~\ref{prop:proper-sols}, where some a priori estimates for the null-solutions
of such systems are established. In turn, these estimates allow us to compare the seminorm
$\norm{\cdot}_{**}^{(\omega,q)}$ (corresponding to various values of $q$) with
$[\cdot]_{\dot{\mathscr{C}}^{w}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}$.
In Section~\ref{section:Existence} we prove the existence of solutions for the Dirichlet problems
with boundary data in $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and
$\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$. Section~\ref{section:Fatou}
contains a Fatou-type result for null-solutions of a strongly elliptic system $L$
belonging to the space $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$,
which will be a key ingredient when establishing uniqueness for the boundary value problems
formulated in Theorem~\ref{mainthmBMO}. Combining the main results of the previous two sections yields
two well-posedness results under different assumptions on the growth function: one for boundary data in
$\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and solutions in
$\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n}_{+},\mathbb{C}^M)$, and another one for
boundary data in $\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and solutions satisfying
$\norm{u}_{**}^{(\omega,q)}<\infty$ for some $0<q\le\infty$, or even in the case where $q$
is replaced by $\exp$. In concert, these two results cover all claims of Theorem~\ref{mainthmBMO}.
Finally, in Appendix~\ref{section:JN} we present a John-Nirenberg type inequality of real-variable
nature, generalizing some results in \cite{Hofmann-Mayboroda, HofMarMay} by allowing more flexibility
due to the involvement of growth functions. This is interesting and useful in its own right.
In addition, we are able to show exponential decay for the measure of the associated level sets
which, in turn, permits deriving estimates not only in arbitrary $L^q$ spaces but also in the space
$\exp L$. Our approach for deriving such results is different from \cite{Hofmann-Mayboroda, HofMarMay},
and uses some ideas which go back to a proof of the classical John-Nirenberg exponential integrability
for {\rm BMO} functions due to Calder\'{o}n. As a matter of fact, our abstract method yields easily
Calder\'on's classical result.
\section{Growth Functions, Generalized H\"{o}lder and Morrey-Campanato Spaces}\label{section:growth}
We begin by studying some basic properties of growth functions. As explained in the introduction, we ultimately
wish to work with growth functions satisfying conditions weaker than \eqref{omega-cond:main}.
Indeed, the two mains conditions that we will consider are
\begin{equation}\label{omega-cond:a}
\int_0^{1}\omega(s)\frac{ds}{s}<\infty,
\end{equation}
and
\begin{equation}\label{omega-cond:b}
t\,\int_t^{\infty}\frac{\omega(s)}{s}\,\frac{ds}{s}\leq C_\omega\,\omega(t),\qquad\forall\,t\in(0,\infty),
\end{equation}
for some finite constant $C_\omega\geq 1$. In what follows, $C_\omega$ will always denote the constant
in \eqref{omega-cond:b}. Clearly, if $\omega$ satisfies it satisfies \eqref{omega-cond:main} then both
\eqref{omega-cond:a} and \eqref{omega-cond:b} hold but the reverse implication is not true in general
(see Example~\ref{example} in this regard).
Later on, we will need the auxiliary function $W$ defined as
\begin{equation}\label{omega-cond:WDef}
W(t):=\int_0^{t}\omega(s)\frac{ds}{s}\,\,\text{ for each }\,\,t\in(0,\infty).
\end{equation}
Note that \eqref{omega-cond:a} gives that $W(t)<\infty$ for every $t>0$. Then \eqref{omega-cond:main}
holds if and only if \eqref{omega-cond:b} holds and there exists $C\in(0,\infty)$ such that
$W(t)\leq C\,\omega(t)$ for each $t\in(0,\infty)$.
The following lemma gathers some useful properties on growth functions satisfying
condition \eqref{omega-cond:b}.
\begin{lemma}\label{wlemma}
Given a growth function $\omega$ satisfying \eqref{omega-cond:b}, the following statements are true.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item\label{omega-t-incre}
Whenever $0<t_1\leq t_2<\infty$, one has
\begin{equation}\label{wnonincreasing}
\frac{\omega(t_2)}{t_2}\leq C_\omega\frac{\omega(t_1)}{t_1}.
\end{equation}
\item\label{item:wdoubling}
For every $t\in(0,\infty)$ one has
\begin{equation}\label{wdoubling}
\omega(2t)\leq 2C_\omega\,\omega(t).
\end{equation}
\item\label{wlimit}
One has $\lim_{t\to\infty}\omega(t)/t=0$.
\item\label{holderclosure}
For each set $E\subset\mathbb{R}^n$ one has
$\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)=\dot{\mathscr{C}}^\omega(\overline{E},\mathbb{C}^M)$,
with equivalent norms. More specifically, the restriction map
\begin{equation}\label{tgVVVa}
\dot{\mathscr{C}}^\omega(\overline{E},\mathbb{C}^M)\ni u\longmapsto u\big|_{E}
\in\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)
\end{equation}
is a linear isomorphism which is continuous in the precise sense that, under the canonically identification
of functions $u\in\dot{\mathscr{C}}^\omega(\overline{E},\mathbb{C}^M)$ with
$u\big|_{E}\in\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)$, one has
\begin{equation}\label{holderclosureeq}
[u]_{\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)}
\leq[u]_{\dot{\mathscr{C}}^\omega(\overline{E},\mathbb{C}^M)}
\leq 2C_\omega[u]_{\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)}
\end{equation}
for each $u\in\dot{\mathscr{C}}^\omega(E,\mathbb{C}^M)$.
\end{list}
\end{lemma}
\begin{proof}
We start observing that for every $t>0$
\begin{equation}\label{eq:fafr}
\frac{\omega(t)}{t}\leq\int_t^{\infty}\frac{\omega(s)}{s}\,\frac{ds}{s}
\leq C_\omega\,\frac{\omega(t)}{t}.
\end{equation}
The first inequality uses that $\omega$ is non-decreasing and the second is just \eqref{omega-cond:b}.
Then, given $t_1\leq t_2$, we may write
\begin{equation}\label{6gDDD.24}
\frac{\omega(t_2)}{t_2}\leq\int_{t_2}^{\infty}\frac{\omega(s)}{s}\,\frac{ds}{s}
\leq\int_{t_1}^{\infty}\frac{\omega(s)}{s}\,\frac{ds}{s}
\leq C_{\omega}\frac{\omega(t_1)}{t_1},
\end{equation}
proving \eqref{omega-t-incre}. The doubling property in \eqref{item:wdoubling} follows at once
from \eqref{omega-t-incre} by taking $t_2:=2t_1$ in \eqref{wnonincreasing}. Next, the claim in \eqref{wlimit}
is justified by passing to limit $t\to\infty$ in the first inequality in \eqref{eq:fafr}
and using Lebesgue's Dominated Convergence Theorem.
Turning our attention to \eqref{holderclosure}, fix an arbitrary $u\in\mathscr{C}^{\omega}(E,\mathbb{C}^M)$.
As noted earlier, this membership ensures that $u$ is uniformly continuous, hence $u$ extends uniquely
to a continuous function $v$ on $\overline{E}$. To show that $v$ belongs to
$\dot{\mathscr{C}}^\omega(\overline{E},\mathbb{C}^M)$ pick two arbitrary distinct points $y,z\in\overline{E}$
and choose two sequences $\{y_k\}_{k\in{\mathbb{N}}}$, $\{z_k\}_{k\in{\mathbb{N}}}$ of points in $E$
such that $y_k\to x$ and $z_k\to z$ as $k\to\infty$. By discarding finitely many terms, there is no
loss of generality in assuming that $|y_k-z_k|<2|y-z|$ for each $k\in{\mathbb{N}}$.
Relying on the fact that $\omega$ is non-decreasing and
\eqref{wdoubling}, we may then write
\begin{multline}
|v(y)-v(z)|=\lim_{k\to\infty}|u(y_k)-u(z_k)|
\leq[u]_{\dot{\mathscr{C}}^{\omega}(E,\mathbb{C}^M)}\limsup_{k\to\infty}\omega(|y_k-z_k|)
\\[4pt]
\leq[u]_{\dot{\mathscr{C}}^{\omega}(E,\mathbb{C}^M)}\,\omega(2|y-z|)
\leq 2C_\omega [u]_{\dot{\mathscr{C}}^{\omega}(E,\mathbb{C}^M)}\,\omega(|y-z|).
\end{multline}
From this, all claims in \eqref{holderclosure} follow, completing the proof of the lemma.
\end{proof}
In the following lemma we treat $W$ defined in \eqref{omega-cond:WDef} as a growth
function depending on the original $\omega$.
\begin{lemma}\label{wlemma3}
Let $\omega$ be a growth function satisfying \eqref{omega-cond:a} and \eqref{omega-cond:b},
and let $W(t)$ be defined as in \eqref{omega-cond:WDef}. Then $W:(0,\infty)\to(0,\infty)$ is a growth function
satisfying \eqref{omega-cond:b} with
\begin{equation}\label{wW.aaa}
C_W\leq 1+(C_\omega)^2.
\end{equation}
Moreover,
\begin{equation}\label{wW}
\omega(t)\leq C_\omega\,W(t)\,\,\text{ for each }\,\,t\in(0,\infty).
\end{equation}
\end{lemma}
\begin{proof}
By design, $W$ is a non-decreasing function and, thanks to Lebesgue's Dominated Convergence Theorem
and \eqref{omega-cond:a} we have $W(t)\to 0$ as $t\to 0^{+}$. Also, on account of \eqref{wnonincreasing},
for each $t\in(0,\infty)$ we may write
\begin{equation}\label{tfDDq.at}
\omega(t)=\int_0^t\frac{\omega(t)}{t}\,ds
\leq C_\omega\int_0^t\frac{\omega(s)}{s}\,ds=C_\omega W(t),
\end{equation}
proving \eqref{wW}. In turn, Fubini's Theorem, \eqref{omega-cond:b}, and \eqref{wW} permit us to estimate
\begin{align}\label{tfDDq}
t\int_t^\infty\frac{W(s)}{s}\frac{ds}{s}
&=t\int_t^\infty\left(\int_0^s\omega(\lambda)\frac{d\lambda}{\lambda}\right)\frac{ds}{s^2}
\nonumber\\[4pt]
&=t\int_0^t\left(\int_t^{\infty}\frac{ds}{s^2}\right)\omega(\lambda)\frac{d\lambda}{\lambda}
+t\int_t^\infty\left(\int_\lambda^{\infty} \frac{ds}{s^2}\right)\omega(\lambda)\frac{d\lambda}{\lambda}
\nonumber\\[4pt]
&=t\int_0^t\frac1{t}\omega(\lambda)\frac{d\lambda}{\lambda}
+t\int_t^{\infty}\frac{\omega(\lambda)}{\lambda}\frac{d\lambda}{\lambda}
\nonumber\\[4pt]
&\leq W(t)+C_\omega\,\omega(t)
\nonumber\\[4pt]
&\leq\left(1+(C_\omega)^2\right)W(t),
\end{align}
for each $t\in(0,\infty)$. This shows that $W$ satisfies \eqref{omega-cond:b}
with constant $C_W\leq 1+(C_\omega)^2$.
\end{proof}
Moving on, for each given function $f\in L^1_{\operatorname{loc}}(\mathbb{R}^{n-1},\mathbb{C}^M)$ define the $L^p$-based
mean oscillation of $f$ at a scale $r\in(0,\infty)$ as
\begin{equation}\label{tfDDq.rD}
\operatorname{osc}_p(f;r):=\sup_{\substack{Q\subset\mathbb{R}^{n-1}\\ \ell(Q)\leq r}}
\bigg(\fint_Q|f(x')-f_Q|^p\,dx'\bigg)^{1/p}.
\end{equation}
The following lemma gathers some results from \cite[Lemmas 2.1 and 2.2]{BMO-MarMitMitMit16}.
\begin{lemma}\label{M4lem}
Let $f\in L^1_{\operatorname{loc}}(\mathbb{R}^{n-1},\mathbb{C}^M)$.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item For every $p,q\in[1,\infty)$ there exists some finite $C=C(p,q,n)>1$ such that
\begin{equation}\label{estimatea}
C^{-1}\operatorname{osc}_p(f;r)\leq\operatorname{osc}_q (f;r)\leq C\operatorname{osc}_p (f;r),\quad\forall\,r> 0.
\end{equation}
\item For every $\varepsilon>0$,
\begin{equation}\label{estimateb}
\int_{1}^{\infty}\operatorname{osc}_1 (f;s)\frac{ds}{s^{1+\varepsilon}}<\infty
\,\Longrightarrow\,f\in L^1\left(\mathbb{R}^{n-1},\frac{dx'}{1+|x'|^{n-1+\varepsilon}}\right)^M.
\end{equation}
\end{list}
\end{lemma}
We augment Lemma~\ref{M4lem} with similar results involving generalized Morrey-Campanato spaces
and generalized H\"{o}lder spaces.
\begin{lemma}\label{wlemma2}
Let $\omega$ be a growth function and fix $p\in[1,\infty)$. Then the following properties are valid.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item\label{osc-f-Ewp}
If $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$, then
\begin{equation}\label{oscmorrey}
\operatorname{osc}_p(f;r)\leq\omega(r)\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\,\,\text{ for each }\,\,r\in(0,\infty).
\end{equation}
\item\label{inclusionitem}
If $\omega$ satisfies \eqref{omega-cond:b}, then for each
$f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ one has
\begin{equation}\label{easyinclusion}
\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq\sqrt{n-1}\,C_\omega [f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\end{equation}
and
\begin{equation}\label{inclusionE}
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)\subset
\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)\subset
L^1\left(\mathbb{R}^{n-1},\frac{dx'}{1+|x'|^{n}}\right)^M.
\end{equation}
\end{list}
\end{lemma}
\begin{proof}
Note that given any $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and $r>0$,
based on \eqref{tfDDq.rD}, the fact that $\omega$ is non-decreasing, and \eqref{987tF} we may write
\begin{align}\label{lemeq2}
\operatorname{osc}_p(f;r) &=\sup_{\substack{Q\subset\mathbb{R}^{n-1}\\ \ell(Q)\leq r}}
\omega(\ell(Q))\frac1{\omega(\ell(Q))}\bigg(\fint_Q|f(x')-f_Q|^p\,dx'\bigg)^{1/p}
\nonumber\\[6pt]
&\leq\omega(r)\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)},
\end{align}
proving \eqref{osc-f-Ewp}. Consider next the claims in \eqref{inclusionitem}.
Given any $f\in\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n-1},\mathbb{C}^M)$, a combination
of \eqref{987tF}, \eqref{7tEEE}, and \eqref{wnonincreasing} yields
\begin{align}\label{lemeq1}
\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
&\leq\sup_{Q\subset\mathbb{R}^{n-1}}\left(\fint_Q\fint_Q\bigg(\frac{|f(x')-f(y')|}
{\omega(\ell(Q))}\bigg)^p\,dx'\,dy'\right)^{1/p}
\nonumber\\[4pt]
&\leq\sup_{Q\subset\mathbb{R}^{n-1}}\frac{\omega(\sqrt{n-1}\ell(Q))}{\omega(\ell(Q))}\,
[f]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\nonumber\\[4pt]
&\leq\sqrt{n-1}C_\omega[f]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{align}
This establishes \eqref{easyinclusion}, hence also the first inclusion in \eqref{inclusionE}.
For the second inclusion in \eqref{inclusionE}, using Jensen's inequality, \eqref{oscmorrey},
and \eqref{omega-cond:b} we may write
\begin{align}\label{tfDRV}
\int_1^{\infty}\operatorname{osc}_1(f;s)\frac{ds}{s^2} &\leq
\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}\int_1^{\infty}\omega(s)\frac{ds}{s^2}
\nonumber\\[6pt]
&\leq C_\omega\,\omega(1)\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}<\infty.
\end{align}
The desired inclusion now follows from this and \eqref{estimateb} with $\varepsilon:=1$.
\end{proof}
\section{Properties of Elliptic Systems and Their Solutions}\label{section:props-elliptic}
The following result is a particular case of more general interior estimates
found in \cite[Theorem 11.9]{Mit13}.
\begin{theorem}\label{thmestimateder}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying \eqref{LegHad}.
Then for every $p\in(0,\infty)$, $\lambda\in(0,1)$, and $m\in\mathbb{N}\cup\{0\}$ there exists
a finite constant $C=C(L,p,m,\lambda,n)>0$ with the property that for every null-solution $u$
of $L$ in a ball $B(x,R)$, where $x\in\mathbb{R}^n$ and $R>0$, and every $r\in(0,R)$ one has
\begin{equation}\label{6gDDD.a4}
\sup_{z\in B(x,\lambda r)}|(\nabla^{m} u)(z)|\leq\frac{C}{r^m}\left(\fint_{B(x,r)}|u(x)|^p\,dx\right)^{1/p}.
\end{equation}
\end{theorem}
To proceed, introduce
\begin{align}\label{6gDDD.arf}
W^{1,2}_{\rm bdd}(\mathbb{R}^n_{+}) &:=\big\{u\in L^2_{\operatorname{loc}}(\mathbb{R}^n_{+}):\,
u,\partial_j u\in L^2\big(\mathbb{R}^n_{+}\cap B(0,r)\big)
\nonumber\\[0pt]
&\hskip 0.80in
\,\text{ for each }\,\,j\in\{1,\dots,n\}\,\,\text{ and }\,\,r\in(0,\infty)\big\},
\end{align}
and define the Sobolev trace $\operatorname{Tr}$, whenever meaningful, as
\begin{equation}\label{defi-trace}
(\operatorname{Tr}\,u)(x'):=\lim_{r\to 0^{+}}\fint_{B((x',0),r)\cap\mathbb{R}^n_{+}}u(y)\,dy,
\qquad\,x'\in\mathbb{R}^{n-1}.
\end{equation}
The following result is taken from \cite[Corollary 2.4]{MazMitSha10}.
\begin{proposition}\label{uniqprop}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying \eqref{LegHad},
and suppose $u\in W^{1,2}_{\rm bdd}(\mathbb{R}^n_{+})$ satisfies $Lu=0$ in $\mathbb{R}^n_{+}$ and
$\operatorname{Tr}\,u=0$ on $\mathbb{R}^{n-1}$. Then $u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M)$
and there exists a finite constant $C>0$, independent of $u$, such that for each
$x\in\overline{\mathbb{R}^n_{+}}$ and each $r>0$,
\begin{equation}\label{6gDDD.ar}
\sup_{\mathbb{R}^n_{+}\cap B(x,r)}|\nabla u|\leq\frac{C}{r}\sup_{\mathbb{R}^n_{+}\cap B(x,2r)}|u|.
\end{equation}
\end{proposition}
The following theorem is contained in \cite[Theorem 2.3 and Proposition 3.1]{BMO-MarMitMitMit16}.
\begin{theorem}\label{PoissonConvolution}
Suppose $L$ is a constant complex coefficient system as in \eqref{ars}, satisfying \eqref{LegHad}.
Then the following statements are true.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item There exists a matrix-valued function
$P^L=(P^L_{\alpha\beta})_{1\leq\alpha,\beta\leq M}:\mathbb{R}^{n-1}\to\mathbb{C}^{M\times M}$,
called the Poisson kernel for $L$ in $\mathbb{R}^{n}_{+}$, such that
$P^L\in{\mathscr{C}}^\infty(\mathbb{R}^{n-1})$, there exists some finite constant $C>0$ such that
\begin{equation}\label{PoissonDecay}
|P^L(x')|\leq\frac{C}{(1+|x'|^2)^{n/2}},\qquad\forall\,x'\in\mathbb{R}^{n-1},
\end{equation}
and
\begin{equation}\label{PoissonConvolutionId}
\int_{\mathbb{R}^{n-1}}P^L(x')\,dx'=I_{M\times M},
\end{equation}
where $I_{M\times M}$ stands for the $M\times M$ identity matrix.
Moreover, if for every $x'\in\mathbb{R}^{n-1}$ and $t>0$ one defines
\begin{equation}\label{7hfDD}
K^L(x',t):=P_t^L(x'):= t^{1-n}P^L(x'/t),
\end{equation}
then $K^L\in{\mathscr{C}}^\infty\big(\overline{{\mathbb{R}}^n_{+}}\setminus B(0,\varepsilon)\big)$
for every $\varepsilon>0$ and the function $K^L=\big(K^L_{\alpha\beta}\big)_{1\leq\alpha,\beta\leq M}$
satisfies
\begin{equation}\label{uahgab-UBVCX}
LK^L_{\cdot\beta}=0\,\,\text{ in }\,\,\mathbb{R}^{n}_{+}
\,\,\text{ for each }\,\,\beta\in\{1,\dots,M\},
\end{equation}
where $K^L_{\cdot\beta}:=\big(K^L_{\alpha\beta}\big)_{1\leq\alpha\leq M}$ is the $\beta$-th column in $K^L$.
\item\label{convo-poiss-sols}
For each function $f=(f_\beta)_{1\le\beta\le M}\in L^1\big(\mathbb{R}^{n-1},\frac{dx'}{1+|x'|^n}\big)^M$
define, with $P^L$ as above,
\begin{equation}\label{ConvDef}
u(x',t):=(P^L_t\ast f)(x'),\qquad\forall\,(x',t)\in\mathbb{R}^{n}_{+}.
\end{equation}
Then $u$ is meaningfully defined, via an absolutely convergent integral, and satisfies
\begin{equation}\label{exist:u2****}
u\in\mathscr{C}^\infty(\mathbb{R}^n_{+},{\mathbb{C}}^M),\quad\,\,
Lu=0\,\,\text{ in }\,\,\mathbb{R}^{n}_{+},\quad\,\,
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f\text{ a.e. on }\,\mathbb{R}^{n-1}.
\end{equation}
Furthermore, there exists a finite constant $C>0$ such that
\begin{equation}\label{ConvDer}
|(\nabla u)(x',t)|\leq\frac{C}{t}\int_1^{\infty}\operatorname{osc}_1(f;st)\frac{ds}{s^{2}},
\qquad\forall\,(x',t)\in\mathbb{R}^{n}_{+},
\end{equation}
and, for each cube $Q\subset\mathbb{R}^{n-1}$,
\begin{equation}\label{PoissonConvolutioneq}
\left(\int_0^{\ell(Q)}\fint_Q|(\nabla u)(x',t)|^2\,t\,dx'\,dt\right)^{1/2}
\leq C\int_1^{\infty}\operatorname{osc}_1(f; s\ell(Q))\frac{ds}{s^2}.
\end{equation}
\end{list}
\end{theorem}
Our next proposition contains a number of a priori estimates comparing
$\norm{u}_{**}^{(\omega,q)}$, corresponding to different values of $q$, for solutions
of $Lu=0$ in ${\mathbb{R}}^n_{+}$. To set the stage, we first state some simple estimates
which are true for any function $u\in\mathscr{C}^{1}(\mathbb{R}^n_{+},\mathbb{C}^M)$:
\begin{equation}\label{uqeq:1}
\norm{u}_{**}^{(\omega,p)}\leq\norm{u}_{**}^{(\omega,q)}
\leq C\norm{u}_{**}^{(\omega,\exp)},\qquad 0<p\le q<\infty,
\end{equation}
where $C=C(q)\geq 1$. Indeed, the first estimate follows at once from Jensen's inequality.
The second estimate is a consequence of the fact that $t^{\max \{1 , q \} }\leq C(e^{t}-1)$ (with $C>0$ depending on $\max \{ 1,q \}$) for each $t\in(0,\infty)$
and the definition of $\|\cdot\|_{\exp L,Q}$ (cf. \eqref{ygFF}).
\begin{proposition}\label{prop:proper-sols}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying the strong ellipticity
condition \eqref{LegHad}, and let $u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M)$ be such
that $Lu=0$ in $\mathbb{R}^n_+$. Then the following statements hold.
\begin{list}{\textup{(\theenumi)}}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm \itemsep=0.2cm \topsep=.2cm \renewcommand{\theenumi}{\alph{enumi}}}
\item\label{int-Whitney}
For every $q\in(0,\infty)$ there there exists a finite constant $C=C(L,n,q)\geq 1$ such that for
each $(x',t)\in\mathbb{R}^{n}_{+}$ one has
\begin{equation}\label{eq:inter-est}
t\,|(\nabla u)(x',t)|\leq C
\bigg(\fint_{|x'-y'|<\frac{t}{2}}\bigg(\int_{t/2}^{3t/2}|(\nabla u)(y',s)|^2\,s\,ds\bigg)^{q/2}\,dy'\bigg)^{1/q}.
\end{equation}
\item\label{con-le-carl}
There exists a finite constant $C=C(L,n)\geq 1$ such that for each cube $Q\subset\mathbb{R}^{n-1}$
and each $x'\in\mathbb{R}^{n-1}$ one has
\begin{align}\label{eq:conical-carl-q:1}
\bigg(\int_0^{\ell(Q)}|(\nabla u)(x',t)|^2\,t\,dt\bigg)^{1/2}
\leq C\,\bigg(\int_{0}^{2\ell(Q)}\int_{|x'-y'|<s}|(\nabla u)(y',s)\,s|^2\,dy'\frac{ds}{s^n}\bigg)^{1/2}.
\end{align}
Furthermore, whenever $2\leq q<\infty$ there exists a finite constant $C=C(L,n,q)\geq 1$ such that
for each cube $Q\subset\mathbb{R}^{n-1}$ and each $x'\in\mathbb{R}^{n-1}$ one has
\begin{multline}\label{eq:conical-carl-q:2}
\bigg(\fint_{Q}\bigg(\int_{0}^{\ell(Q)}\int_{|x'-y'|<s}
|(\nabla u)(y',s)\,s|^2\,dy'\frac{ds}{s^n}\bigg)^{q/2}\,dx'\bigg)^{1/q}
\\[4pt]
\leq C\,\bigg(\fint_{3Q}\bigg(\int_0^{3\ell(Q)}|(\nabla u)(x',t)|^2\,t\,dt\bigg)^{q/2}\,dx'\bigg)^{1/q}.
\end{multline}
\item\label{sup-Comega}
There exists a finite constant $C=C(L,n)\geq 1$ such that for each growth function $\omega$ one has
\begin{equation}\label{lemsmalleq}
\norm{u}_{**}^{(\omega,\infty)}\leq C[u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}.
\end{equation}
\item\label{sup-q-car}
For every $q\in(0,\infty)$ there exists a finite constant $C=C(L,n,q)\geq 1$ such that
for each growth function $\omega$ satisfying \eqref{omega-cond:b} one has
\begin{equation}\label{uqeq-infty}
\norm{u}_{**}^{(\omega,\infty)}\leq C\,C_\omega\norm{u}_{**}^{(\omega,q)}.
\end{equation}
\item\label{comp-car-p-q-2}
There exists a finite constant $C=C(L,n)\geq 1$ such that for each growth function
$\omega$ satisfying \eqref{omega-cond:b} one has
\begin{equation}\label{uqeq}
\norm{u}_{**}^{(\omega,\exp)}\leq C(C_\omega)^2\norm{u}_{**}^{(\omega,2)}.
\end{equation}
\item\label{CW-car-q}
Let $\omega$ be a growth function satisfying \eqref{omega-cond:a} as well as \eqref{omega-cond:b},
and define $W(t)$ as in \eqref{omega-cond:WDef}. Then
\begin{equation}\label{mainthmBMOeq4}
[u]_{\dot{\mathscr{C}}^{W}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\leq C_\omega (2+C_\omega)\norm{u}_{**}^{(\omega,\infty)},
\end{equation}
and, if the latter quantity is finite, $u\in\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$
in the sense of Lemma~\ref{wlemma}\eqref{holderclosure}.
\item\label{car-q-sup}
Let $\omega$ be a growth function satisfying
\begin{equation}\label{omega-cond:second}
\int_0^{t}\omega(s)\frac{ds}{s}\leq C_\omega'\,\omega(t),\qquad\forall\,t\in(0,\infty),
\end{equation}
for some finite constant $C_\omega'>1$. Then
\begin{equation}\label{uqeq-infty:converse}
\norm{u}_{**}^{(\omega,\exp)}\le (C_\omega')^{1/2}\norm{u}_{**}^{(\omega,\infty)}.
\end{equation}
\item\label{all-comp}
Let $\omega$ be a growth function satisfying \eqref{omega-cond:main}. Then for every $q\in(0,\infty)$
\begin{equation}\label{relation-all-norms:1}
\norm{u}_{**}^{(\omega,q)}\approx\norm{u}_{**}^{(\omega,\exp)}\approx\norm{u}_{**}^{(\omega,\infty)}
\approx [u]_{\dot{\mathscr{C}}^{w}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\end{equation}
where the implicit constants depend only on $L$, $n$, $q$, and the constant $C_0$
in \eqref{omega-cond:main}. In particular, if $\norm{u}_{**}^{(\omega,q)}<\infty$
for some $q\in(0,\infty]$, or $\norm{u}_{**}^{(\omega,\exp)}<\infty$, then
$u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$
in the sense of Lemma~\ref{wlemma}\eqref{holderclosure}.
\end{list}
\end{proposition}
\begin{proof}
We start by proving \eqref{int-Whitney}. Fix $(x',t)\in\mathbb{R}^{n}_{+}$ and let $Q_{x',t}$
be the cube in $\mathbb{R}^{n-1}$ centered at $x'$ with side-length $t$. Then from
Theorem~\ref{thmestimateder} (presently used with $m:=0$ and $p:=\min\{q,2\}$) and Jensen's inequality
we obtain
\begin{align}\label{step2}
|(\nabla u)(x',t)| &\leq C\left(\fint_{|(y',s)-(x',t)|<\frac{t}{2}}|(\nabla u)(y',s)|^p\,dy'\,ds\right)^{1/p}
\nonumber\\[4pt]
&\leq C\left(\fint_{|x'-y'|<\frac{t}{2}}\bigg(\fint_{(t/2,3t/2)}|(\nabla u)(y',s)|^2\,ds\bigg)^{p/2}\,dy'\right)^{1/p}
\nonumber\\[4pt]
&\leq C\bigg(\fint_{|x'-y'|<\frac{t}{2}}\bigg(\fint_{(t/2,3t/2)}|(\nabla u)(y',s)|^2\,ds\bigg)^{q/2}\,dy'\bigg)^{1/q}
\nonumber\\[4pt]
&=Ct^{-1}\bigg(\fint_{|x'-y'|<\frac{t}{2}}\bigg(\int_{t/2}^{3t/2}|(\nabla u)(y',s)|^2\,s\,ds\bigg)^{q/2} dy'\bigg)^{1/q},
\end{align}
proving \eqref{eq:inter-est}. Turning our attention to \eqref{con-le-carl}, fix
a cube $Q\subset\mathbb{R}^{n-1}$ along with a point $x'\in\mathbb{R}^{n-1}$.
First, integrating \eqref{eq:inter-est} written for $q:=2$ yields
\begin{align}\label{6gDDD.atrf}
\int_{0}^{\ell(Q)}|(\nabla u)(x',t)|^2\,t\,dt
&\leq C\int_{0}^{\ell(Q)}\frac1{t^{n+1}}\int_{t/2}^{3t/2}\int_{|x'-y'|<s}|(\nabla u)(y',s)|^2\,s\,dy'\,ds\,t\,dt
\nonumber\\[4pt]
&\leq C\int_{0}^{2\ell(Q)}\int_{|x'-y'|<s}|(\nabla u)(y',s)|^2\int_{2s/3}^{2s}\,t^{-n}\,dt\,dy'\,s\,ds
\nonumber\\[4pt]
&=C\int_{0}^{2\ell(Q)}\int_{|x'-y'|<s}|(\nabla u)(y',s)\,s|^2\,dy'\,\frac{ds}{s^n},
\end{align}
and this readily leads to the estimate in \eqref{eq:conical-carl-q:1}. To justify \eqref{eq:conical-carl-q:2},
observe that for each nonnegative function $h\in L^1_{\rm loc}({\mathbb{R}}^{n-1})$ we have
\begin{align}\label{eq:Con-vert}
&\fint_{Q}\bigg(\int_{0}^{\ell(Q)}\int_{|x'-y'|<s}|(\nabla u)(y',s)\,s|^2\,dy'\frac{ds}{s^n}\bigg)h(x')\,dx'
\nonumber\\[4pt]
&\qquad\leq 3^n\fint_{3Q}\int_{0}^{\ell(Q)}\bigg(\frac1{s^{n-1}}\int_{|y'-x'|<s}h(x')\,dx'\bigg)
|(\nabla u)(y',s)|^2\,s\,ds\,dy'
\nonumber\\[4pt]
&\qquad\leq C_n\fint_{3Q}\bigg(\int_{0}^{3\ell(Q)}|(\nabla u)(y',s)|^2\,s\,ds\bigg)(Mh)(x')\,dx',
\end{align}
where $M$ is the Hardy-Littlewood maximal operator in ${\mathbb{R}}^{n-1}$.
Note that if $q=2$ then \eqref{eq:Con-vert} gives at once \eqref{eq:conical-carl-q:2}
by taking $h=1$ in $Q$ and using that $Mh\leq 1$. On the other hand, if $q>2$, we
impose the normalization condition $\|h\|_{L^{(q/2)'}(Q,dx'/|Q|)}=1$ and
then rely on \eqref{eq:Con-vert} and H\"older's inequality to write
\begin{align}\label{eq:Con-vert-q>2}
&\fint_{Q}\bigg(\int_{0}^{\ell(Q)}\int_{|x'-y'|<s}|(\nabla u)(y',s)\,s|^2\,dy'\frac{ds}{s^n}\bigg)h(x')\,dx'
\nonumber\\[4pt]
&\qquad\leq C_n\bigg(\fint_{3Q}\bigg(\int_{0}^{3\ell(Q)}|(\nabla u)(y',s)|^2\,s\,ds\bigg)^{q/2}\,dx'\bigg)^{2/q}
\,\|Mh\|_{L^{(q/2)'}(Q,dx'/|Q|)}
\nonumber\\[4pt]
&\qquad\leq C\bigg(\fint_{3Q}\bigg(\int_{0}^{3\ell(Q)}|(\nabla u)(y',s)|^2\,s\,ds\bigg)^{q/2}\,dx'\bigg)^{2/q},
\end{align}
bearing in mind that $M$ is bounded in $L^{(q/2)'}(\mathbb{R}^{n-1})$, given that $q>2$.
Taking now the supremum over all such functions $h$ yields \eqref{eq:conical-carl-q:2}
on account of Riesz' duality theorem.
As regards \eqref{sup-Comega}, fix $(x',t)\in\mathbb{R}^n_{+}$ and use
Theorem~\ref{thmestimateder} together with the fact that $\omega$ is a non-decreasing function
to write
\begin{align}\label{BMOuniqeq1}
|(\nabla u)(x',t)| &=\big|\nabla(u(\cdot)-u(x',t))(x',t)\big|
\nonumber\\[4pt]
&\leq\frac{C}{t}\fint_{|(y',s)-(x',t)|<t/2}|u(y',s)-u(x',t)|\,dy'\,ds
\nonumber\\[4pt]
&\leq C[u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}\frac{\omega(t)}{t}.
\end{align}
In view of \eqref{yRRFF}, this readily establishes \eqref{lemsmalleq}.
The claim in \eqref{sup-q-car} is proved by combining \eqref{eq:inter-est} and \eqref{wdoubling}, which permit us
to estimate (recall that $Q_{x',t}$ denotes the cube in $\mathbb{R}^{n-1}$ centered at $x'$ with side-length $t$)
\begin{align}\label{step4dedawe}
\norm{u}_{**}^{(\omega,\infty)}
&\leq C\,\sup_{(x',t)\in\mathbb{R}^{n}_{+}}\frac{1}{\omega(t)}
\bigg(\fint_{(3/2) Q_{x',t}}\bigg(\int_{0}^{3t/2}|(\nabla u)(y',s)|^2\,s\,ds\bigg)^{q/2}\,dy'\bigg)^{1/q}
\nonumber\\[4pt]
&\leq C\,C_\omega\norm{u}_{**}^{(\omega,q)}.
\end{align}
Going further, consider the claim in \eqref{comp-car-p-q-2}. For starters, observe that the
convexity of the function $t\mapsto e^{t}-1$ readily implies that $2^{n-1}(e^{t}-1)\leq e^{2^{n-1}t}-1$
for every $t>0$ which, in view of \eqref{eq:Lux-norm}, allows us to write
\begin{equation}\label{eq:exp-Q-2Q}
\|f\|_{\exp L,Q}\leq 2^{n-1}\,\|f\|_{\exp L,2Q}
\end{equation}
for each cube $Q$ in ${\mathbb{R}}^{n-1}$ and each Lebesgue measurable function $f$ on $Q$.
Turning to the proof of \eqref{uqeq} in earnest, by homogeneity we may assume that
$\norm{u}^{(\omega,2)}_{**}=1$ to begin with. We are going to use Lemma~\ref{lem:appendix}.
As a prelude, define
\begin{equation}\label{6agRD4r7}
F(y',s):=|(\nabla u)(y',s)\,s|,\qquad\forall\,(y',s)\in\mathbb{R}^n_{+},
\end{equation}
and, for each cube $Q$ in ${\mathbb{R}}^{n-1}$ and each threshold $N\in(0,\infty)$, consider the set
\begin{equation}\label{6gDDD.a4r9jh}
E_{N,Q}:=\bigg\{x'\in Q:\,\frac{1}{\omega(\ell(Q))}\bigg(\int_0^{\ell(Q)}
\int_{|x'-y'|<\kappa s}|F(y',s)|^2\,dy'\frac{ds}{s^n}\bigg)^{1/2}>N\bigg\}.
\end{equation}
where $\kappa:=1+2\sqrt{n-1}$. Denoting $Q^{*}:=(2\kappa+1)Q=(3+4\sqrt{n-1})Q$, then
using Chebytcheff's inequality, and \eqref{wnonincreasing}, for each cube $Q$ in ${\mathbb{R}}^{n-1}$
and each $N>0$ we may write
\begin{align}\label{6gDDD.a433}
|E_{N,Q}| &\leq\frac{1}{N^2}\frac{1}{\omega(\ell(Q))^2}
\int_Q\int_0^{\ell(Q)}\int_{|x'-y'|<\kappa s}|F(y',s)|^2\,dy'\frac{ds}{s^n}\,dx'
\nonumber \\[4pt]
&\leq\frac{1}{N^2}\frac{1}{\omega(\ell(Q))^2}
\int_{Q^{*}}\int_0^{\ell(Q)}\Big(\int_{|y'-x'|<\kappa s}\,dx'\Big)\,|F(y',s)|^2\frac{ds}{s^n}\,dy'
\nonumber\\[4pt]
&\leq C\frac{1}{N^2}\frac{1}{\omega(\ell(Q))^2}\int_{Q^{*}}\int_0^{\ell(Q^{*})}
|(\nabla u)(y',s)\,s|^2\frac{ds}{s}\,dy'
\nonumber\\[4pt]
&\leq C\frac{|Q^{*}|}{N^2}\frac{\omega(\ell(Q^{*}))^2}{\omega(\ell(Q))^2}
\big(\norm{u}^{(\omega,2)}_{**}\big)^2
=C\frac{|Q^{*}|}{N^2}\Big[\frac{\omega(\ell(Q^{*}))}{\omega(\ell(Q))}\Big]^2
\nonumber\\[4pt]
&\leq C_0(C_{\omega})^2\frac{1}{N^2}|Q|,
\end{align}
for some finite constant $C_0>0$. Therefore, taking $N:=\sqrt{2C_0}C_{\omega}>0$, we conclude that
\begin{equation}\label{eq:lemmaapplies}
|E_{N,Q}|\leq\frac{1}{2}|Q|.
\end{equation}
This allows us to invoke Lemma~\ref{lem:appendix} with $\varphi:=\omega$, which together with
\eqref{eq:conical-carl-q:1}, \eqref{wdoubling}, and \eqref{eq:exp-Q-2Q}, gives
\begin{align}\label{iatfa}
\norm{u}^{(\omega,\exp)}_{**}
&\leq C\sup_{Q\subset\mathbb{R}^{n-1}}\frac{1}{\omega(\ell(Q))}
\bigg\|\bigg(\int_{0}^{\ell(2Q)}\int_{|\,\cdot\,-y'|<s}|F(y',s)|^2\,dy'\frac{ds}{s^n}\bigg)^{1/2}\bigg\|_{\exp L,Q}
\nonumber\\[4pt]
&\leq C(C_{\omega})^{2}.
\end{align}
This completes the proof of \eqref{comp-car-p-q-2}.
Turning our attention to \eqref{CW-car-q}, fix $x=(x',t)$ and $y=(y',s)$ in $\mathbb{R}^{n}_{+}$,
and abbreviate $r:=|x-y|$. Then,
\begin{align}\label{step4eq1}
\frac{|u(x)-u(y)|}{W(|x-y|)} &\leq\frac{1}{W(r)}|u(x',t)-u(x',t+r)|+\frac1{W(r)}|u(x',t+r)-u(y',s+r)|
\nonumber\\[4pt]
&\quad+\frac1{W(r)}|u(y',s+r)-u(y',s)|
\nonumber\\[4pt]
&=:I+II+III.
\end{align}
To bound $I$, we use the Fundamental Theorem of Calculus, \eqref{wnonincreasing}, and
\eqref{omega-cond:WDef} and obtain
\begin{multline}\label{step4eq2}
I=\frac1{W(r)}\left|\int_0^r(\partial_n u)(x',t+\xi)\,d\xi\right|
\leq\norm{u}_{**}^{(\omega,\infty)}\frac{1}{W(r)}\int_0^r\frac{\omega(t+\xi)}{t+\xi}\,d\xi
\\[4pt]
\leq C_\omega\norm{u}_{**}^{(\omega,\infty)}\frac{1}{W(r)}\int_0^r\frac{\omega(\xi)}{\xi}\,d\xi
=C_\omega\norm{u}_{**}^{(\omega,\infty)}.
\end{multline}
Note that $III$ is bounded analogously replacing $x'$ by $y'$ and $t$ by $s$. For $II$,
we use again the Fundamental Theorem of Calculus, together with \eqref{wnonincreasing} and \eqref{wW},
to write
\begin{align}\label{step4eq3}
II &=\frac{1}{W(r)}\left|\int_0^1\frac{d}{d\theta}[u(\theta(x',t+r)+(1-\theta)(y',s+r))]\,d\theta\right|
\nonumber\\[4pt]
&=\frac{1}{W(r)}\left|\int_0^1(x'-y',t-s)\cdot(\nabla u)\big(\theta(x',t+r)+(1-\theta)(y',s+r)\big)\,d\theta\right|
\nonumber\\[4pt]
&\leq\norm{u}_{**}^{(\omega,\infty)}\frac{r}{W(r)}\int_0^1
\frac{\omega\big((1-\theta)s+\theta t+r\big)}{(1-\theta)s+\theta t+r}\,d\theta
\nonumber\\[4pt]
&\leq C_\omega\norm{u}_{**}^{(\omega,\infty)}\frac{r}{W(r)}\int_0^1\frac{\omega(r)}{r}\,d\theta
\nonumber\\[4pt]
&\leq(C_\omega)^2\norm{u}_{**}^{(\omega,\infty)}.
\end{align}
As $x$ and $y$ were chosen arbitrarily, \eqref{step4eq1}, \eqref{step4eq2},
and \eqref{step4eq3} collectively justify \eqref{mainthmBMOeq4}.
To justify \eqref{car-q-sup}, observe that since $\omega$ is non-decreasing and satisfies
\eqref{omega-cond:second} we may write
\begin{align}\label{eq:76ergtr}
\bigg(\fint_Q\bigg(\int_0^{\ell(Q)}|(\nabla u)(x',t)|^2\,t\,dt\bigg)^{q/2}\,dx'\bigg)^{1/q}
&\leq\bigg(\int_0^{\ell(Q)}\omega(t)^2\frac{dt}{t}\bigg)^{1/2}\norm{u}_{**}^{(\omega,\infty)}
\nonumber\\[4pt]
&\leq(C_\omega')^{1/2}\omega(\ell(Q))\norm{u}_{**}^{(\omega,\infty)},
\end{align}
which readily leads to the desired inequality.
As regards \eqref{all-comp}, the idea is to combine \eqref{uqeq:1}, \eqref{car-q-sup},
and \eqref{sup-q-car} for the first three equivalences. In concert, \eqref{sup-Comega},
the fact that \eqref{omega-cond:main} gives $W\leq C_0\,\omega$, and \eqref{CW-car-q}
also give the last equivalence in \eqref{all-comp}. The proof of Proposition~\ref{prop:proper-sols}
is therefore complete.
\end{proof}
\section{Existence Results}\label{section:Existence}
In this section we develop the main tools used to establish the existence of solutions for the boundary
value problems formulated in the statement of Theorem~\ref{mainthmBMO}. We start with the
generalized H\"older Dirichlet problem.
\begin{proposition}\label{propfholder}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying the strong ellipticity
condition formulated in \eqref{LegHad}, and let $\omega$ be a growth function satisfying \eqref{omega-cond:b}.
Given $f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$, define
$u(x',t):=(P_t^L\ast f)(x')$ for every $(x',t)\in\mathbb{R}^{n}_{+}$.
Then $u$ is meaningfully defined via an absolutely convergent integral and satisfies
\begin{equation}\label{exist:u2****-Holder-omega}
u\in\mathscr{C}^\infty(\mathbb{R}^n_{+},{\mathbb{C}}^M),\quad\,\,
Lu=0\,\,\mbox{ in }\,\,\mathbb{R}^{n}_{+},\quad\,\,
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f\text{ a.e. on }\,\,\mathbb{R}^{n-1}.
\end{equation}
Moreover, there exists a finite constant $C=C(L,n)>0$ such that
\begin{equation}\label{fholdereq}
[u]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\leq C\,C_\omega(1+C_\omega)[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)},
\end{equation}
and $u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$
with $\restr{u}{\partial\mathbb{R}^n_{+}}=f$.
\end{proposition}
\begin{proof}
Let $f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and define
$u(x',t):=(P_t^L\ast f)(x')$ for every $(x',t)\in\mathbb{R}^{n}_{+}$. By \eqref{inclusionE} and
Theorem~\ref{PoissonConvolution}\eqref{convo-poiss-sols}, $u$ satisfies all
properties listed in \eqref{exist:u2****-Holder-omega}. To prove the estimate in
\eqref{fholdereq}, we first notice that for any $(x',t)\in\mathbb{R}_+^{n}$,
we can write
\begin{align}\label{iatuyhG}
(P_t^L\ast f)(x') &=\int_{\mathbb{R}^{n-1}} P_t^L(x'-y')f(y')\,dy'
=\int_{\mathbb{R}^{n-1}}t^{1-n}P^L\left(\frac{x'-y'}{t}\right)f(y')\,dy'
\nonumber\\[4pt]
&=\int_{\mathbb{R}^{n-1}}P^L(z')f(x'-tz')\,dz'.
\end{align}
Fix now $x=(x',t)$ and $y=(y',s)$ arbitrary in $\mathbb{R}^{n}_{+}$, and set $r:=|x-y|$.
By \eqref{PoissonDecay} and the fact that $\omega$ is non-decreasing we obtain
\begin{align}\label{fholdereq-1}
|u(x',t)-u(y',s)|&=|(P^L_t\ast f)(x')-(P^L_s\ast f)(y')|
\nonumber\\[4pt]
&\leq C\int_{\mathbb{R}^{n-1}}\frac1{(1+|z'|^2)^{n/2}}\, |f(x'-tz')-f(y'-sz')|\, dz'
\nonumber\\[4pt]
&\leq C[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\int_{\mathbb{R}^{n-1}}\frac1{(1+|z'|^2)^{n/2}}\,\omega((1+|z'|)r)\,dz'
\nonumber\\[4pt]
&\leq C [f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\int_0^{\infty}\frac{1}{(1+\lambda^2)^{n/2}}\,\omega\big((1+\lambda)r\big)\,
\lambda^{n-1}\frac{d\lambda}{\lambda}
\nonumber\\[4pt]
&\leq C[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\bigg(\int_0^1\omega(2 r)\,\lambda^{n-1}\frac{d\lambda}{\lambda}
+\int_1^{\infty}\frac{\omega(2\lambda r)}{\lambda}\frac{d\lambda}{\lambda}\bigg)
\nonumber\\[4pt]
&=C[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\bigg(\omega(2r)+2r\int_{2r}^{\infty}\frac{\omega(\lambda)}{\lambda}\frac{d\lambda}{\lambda}\bigg)
\nonumber\\[4pt]
&\leq C\,C_\omega(1+C_\omega)\omega(r)[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)},
\end{align}
where in the last inequality we have used \eqref{omega-cond:b} and \eqref{wdoubling}. Hence,
\eqref{fholdereq} holds. In particular, $u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$
by Lemma~\ref{wlemma}\eqref{holderclosure}. This and the fact that
$\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f$ a.e. in $\mathbb{R}^{n-1}$ with
$f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ then prove that, indeed,
$\restr{u}{\partial\mathbb{R}^n_{+}}=f$.
\end{proof}
The result below is the main tool in the proof of existence of solutions for the generalized Morrey-Campanato
Dirichlet problem.
\begin{proposition}\label{propstep1}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying the strong ellipticity
condition stated in \eqref{LegHad}, and let $\omega$ be a growth function satisfying \eqref{omega-cond:b}.
Given $1\leq p<\infty$, let $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and define
$u(x',t):=(P_t^L\ast f)(x')$ for every $(x',t)\in\mathbb{R}^{n}_{+}$. Then $u$ is meaningfully defined
via an absolutely convergent integral and satisfies
\begin{equation}\label{exist:u2****-Morrey-campanato}
u\in\mathscr{C}^\infty(\mathbb{R}^n_{+},{\mathbb{C}}^M),\quad\,\,
Lu=0\,\,\mbox{ in }\,\,\mathbb{R}^{n}_{+},\quad\,\,
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f\text{ a.e. on }\,\,\mathbb{R}^{n-1}.
\end{equation}
Moreover, for every $q\in(0,\infty]$ there exists a finite constant $C=C(L,n,p,q)>0$ such that
\begin{equation}\label{step1}
\norm{u}_{**}^{(\omega,q)}\leq C(C_\omega)^{4}\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
Furthermore, the same is true if $\norm{\cdot}_{**}^{(\omega,q)}$ is replaced
by $\norm{\cdot}_{**}^{(\omega,\exp)}$.
\end{proposition}
\begin{proof}
Given $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$, if $u(x',t):=(P_t^L\ast f)(x')$
for every $(x',t)\in\mathbb{R}^{n}_{+}$, from \eqref{inclusionE} and
Theorem~\ref{PoissonConvolution}\eqref{convo-poiss-sols} we see that
$u$ satisfies all properties listed in \eqref{exist:u2****-Morrey-campanato}.
Next, having fixed an arbitrary exponent $q\in(0,\infty)$, based on Proposition~\ref{prop:proper-sols}\eqref{sup-q-car},
\eqref{uqeq:1}, Proposition~\ref{prop:proper-sols}\eqref{comp-car-p-q-2}, \eqref{PoissonConvolutioneq},
\eqref{estimatea}, \eqref{oscmorrey} and \eqref{omega-cond:b} we may write
\begin{align}\label{pa6ff}
\norm{u}_{**}^{(\omega,\infty)} &\leq C\,C_\omega\norm{u}_{**}^{(\omega,q)}
\leq C\,C_\omega\norm{u}_{**}^{(\omega,\exp)}
\nonumber\\[4pt]
&\leq C(C_\omega)^3\norm{u}_{**}^{(\omega,2)}
\leq C(C_\omega)^3\sup_{t>0} \frac1{\omega(t)}\int_{1}^{\infty}\operatorname{osc}_1(f,st)\frac{ds}{s^2}
\nonumber\\[4pt]
&=C(C_\omega)^3\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\sup_{t>0}\frac{t}{\omega(t)}\int_t^{\infty}\omega(s)\frac{ds}{s^2}
\nonumber\\[4pt]
&\leq C(C_\omega)^{4}\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)},
\end{align}
which proves \eqref{step1} and the corresponding estimate for $\norm{u}_{**}^{(\omega,\exp)}$.
\end{proof}
\section{A Fatou-Type Result and Uniqueness of Solutions}\label{section:Fatou}
We shall now prove a Fatou-type result which is going to be the main ingredient in establishing
the uniqueness of solutions for the boundary value problems we are presently considering.
More precisely, the following result establishes that any solution in
$\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$
can be obtained as a convolution of its trace with the associated Poisson kernel.
\begin{proposition}\label{fatou}
Let $L$ be a constant complex coefficient system as in \eqref{ars} satisfying the strong ellipticity
condition stated in \eqref{LegHad}, and let $\omega$ be a growth function satisfying \eqref{omega-cond:b}.
If $u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)\cap
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$ is a function satisfying $Lu=0$ in $\mathbb{R}^n_{+}$,
then $\restr{u}{\partial\mathbb{R}^n_{+}}\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and
\begin{equation}\label{claimeq}
u(x',t)=\left(P_t^L\ast(\restr{u}{\partial\mathbb{R}^n_{+}})\right)(x'),\qquad\forall\,(x',t)\in\mathbb{R}^n_{+}.
\end{equation}
\end{proposition}
\begin{proof}
Let $u\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)\cap
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$ satisfy $Lu=0$ in $\mathbb{R}^n_{+}$.
By Lemma~\ref{wlemma}\eqref{holderclosure}, it follows that $u$ can be continuously extended to
a function (which we call again $u$) $u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$.
In particular, the trace $\restr{u}{\partial\mathbb{R}^n_{+}}$ is well-defined and belongs to the space
$\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$.
To proceed, fix an arbitrary $\varepsilon>0$ and define $u_\varepsilon=u(\cdot+\varepsilon e_n)$
in $\mathbb{R}^n_{+}$, where $e_n=(0,\dots,0,1)\in\mathbb{R}^n$. Then, by design,
$u_\varepsilon\in\mathscr{C}^{\infty}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)\cap
\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$, $Lu_{\varepsilon}=0$ in $\mathbb{R}^n_{+}$,
and $[u_\varepsilon]_{\dot{\mathscr{C}^{\omega}}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)}
\leq [u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}$. Moreover, using
Proposition~\ref{prop:proper-sols}\eqref{sup-Comega} and \eqref{wnonincreasing} we obtain
\begin{align}\label{ayttrf}
\sup_{(x',t)\in\mathbb{R}^n_{+}}|(\nabla u_\varepsilon)(x',t)|
&=\sup_{(x',t)\in\mathbb{R}^n_{+}}|(\nabla u)(x',t+\varepsilon)|
\nonumber\\[4pt]
&\leq C[u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\sup_{(x',t)\in\mathbb{R}^n_{+}} \frac{\omega(t+\varepsilon)}{t+\varepsilon}
\nonumber\\[4pt]
&\leq C\,C_\omega [u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\frac{\omega(\varepsilon)}{\varepsilon}.
\end{align}
This implies that $\nabla u_\varepsilon$ is bounded in $\mathbb{R}^n_{+}$, hence
$u_\varepsilon\in W^{1,2}_{\rm bdd}(\mathbb{R}^n_{+},\mathbb{C}^M)$.
Define next $f_\varepsilon(x'):=u(x',\varepsilon)\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$
and $w_\varepsilon(x',t):=(P_t^L\ast f_\varepsilon)(x')$ for each $(x',t)\in\mathbb{R}^n_{+}$.
Then, Proposition~\ref{propfholder} implies that that
$w_\varepsilon\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)\cap
\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$, $L w_\varepsilon=0$
in $\mathbb{R}^n_{+}$ and $\restr{w_\varepsilon}{\partial\mathbb{R}^n_{+}}= f_\varepsilon$.
Moreover, for every pair of points $x',y'\in\mathbb{R}^{n-1}$ we have, on the one hand,
\begin{equation}\label{ayfff}
|f_\varepsilon(x')-f_\varepsilon(y')|=|u(x',\varepsilon)-u(y',\varepsilon)|
\leq [u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}\,\omega(|x'-y'|),
\end{equation}
and, on the other hand, using the Mean Value Theorem and Proposition~\ref{prop:proper-sols}\eqref{sup-Comega},
\begin{align}\label{6ggff}
|f_\varepsilon (x')-f_\varepsilon(y')| &=|u(x',\varepsilon)-u(y',\varepsilon)|
\nonumber\\[4pt]
&\leq|x'-y'|\sup_{z'\in[x',y']}|(\nabla u)(z',\varepsilon)|
\nonumber\\[4pt]
&\leq C\,|x'-y'|\,[u]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\frac{\omega(\varepsilon)}{\varepsilon}.
\end{align}
Therefore, we conclude that $f_\varepsilon\in\dot{\mathscr{C}}^{\Psi}(\mathbb{R}^{n-1},\mathbb{C}^M)$,
with norm depending (unfavorably) on the parameter $\varepsilon$, where the growth function $\Psi$ is given by
\begin{equation}\label{psi}
\Psi(t):=\min\left\lbrace t,\frac{\omega(t)}{\omega(1)}\right\rbrace=\left\lbrace
\begin{array}{ll}
t &\text{ if }t\leq 1,
\\[4pt]
\omega(t)/\omega(1) &\text{ if }t>1.
\end{array}
\right.
\end{equation}
For every $R>1$ and $x=(x',t)$, let us now invoke \eqref{ConvDer},
\eqref{estimatea} and \eqref{oscmorrey}, with $\Psi$ in place of $\omega$, to write
\begin{align}\label{q34r34fr}
&\int_{B(0,R)\cap\mathbb{R}^n_{+}}|(\nabla w_\varepsilon)(x)|^2\,dx
\leq\int_{B(0,R)\cap\mathbb{R}^n_{+}}\left(\frac{C}{t}\int_1^{\infty}\operatorname{osc}_1(f_\varepsilon;st)
\frac{ds}{s^2}\right)^2\,dx
\nonumber\\[4pt]
&\qquad\leq C\norm{f_\varepsilon}_{\mathscr{E}^{\Psi,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\int_{B(0,R)\cap\mathbb{R}^n_{+}}\left(\int_1^{\infty}\frac{\Psi(st)}{st}\frac{ds}{s}\right)^2\,dx
\nonumber\\[4pt]
&\qquad\leq C\norm{f_\varepsilon}_{\mathscr{E}^{\Psi,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
R^{n-1}\int_0^R\left(\int_t^\infty\frac{\Psi(s)}{s}\frac{ds}{s}\right)^2\,dt,
\end{align}
and then use \eqref{omega-cond:b} to observe that
\begin{align}\label{ayredf5t}
\int_0^R\left(\int_t^\infty\frac{\Psi(s)}{s}\frac{ds}{s}\right)^2\,dt
&\leq\int_0^1\left(\int_t^1\frac{ds}{s}+\frac1{\omega(1)}\int_1^\infty\frac{\omega(s)}{s}\frac{ds}{s}\right)^2\,dt
\nonumber\\[4pt]
&\qquad\quad+\int_1^R\left(\frac1{\omega(1)}\int_1^\infty\frac{\omega(s)}{s}\frac{ds}{s}\right)^2\,dt
\nonumber\\[4pt]
&\leq\int_0^1\big(\log(1/t)+C_\omega\big)^2\,dt+(R-1)(C_\omega)^2<\infty,
\end{align}
Collectively, \eqref{q34r34fr} and \eqref{ayredf5t} show that
$w_\varepsilon\in W^{1,2}_{\rm bdd}(\mathbb{R}^n_{+},\mathbb{C}^M)$.
We now consider $v_\varepsilon:=u_\varepsilon-w_\varepsilon\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)
\cap\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)\cap
W^{1,2}_{\rm bdd}(\mathbb{R}^n_{+},\mathbb{C}^M)$, which satisfies $Lv_\varepsilon=0$ in $\mathbb{R}^n_{+}$ and
$\restr{v_\varepsilon}{\partial\mathbb{R}^n_{+}}=0$. Hence, $\operatorname{Tr}\,v_\varepsilon=0$ on
$\mathbb{R}^{n-1}$ (see \eqref{defi-trace}) and for each $x\in{\mathbb{R}}^n$ we have
\begin{align}\label{uatf}
|v_\varepsilon(x)| &\leq|v_\varepsilon(x)-v_\varepsilon(0)|+|v_\varepsilon(0)|
\nonumber\\[4pt]
&\leq\max\big\{[v_\varepsilon]_{\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}\,,\,
|v_\varepsilon(0)|\big\}(1+\omega(|x|)).
\end{align}
From this and Proposition~\ref{uniqprop} we then conclude that
\begin{equation}\label{uniqeq2}
\sup_{\mathbb{R}^n_{+}\cap B(0,r)}|\nabla v_\varepsilon|
\leq\frac{C}{r}\sup_{\mathbb{R}^n_{+}\cap B(0,2r)}|v_\varepsilon|\leq C_\varepsilon\frac{1+\omega(2r)}{r},
\end{equation}
and from Lemma~\ref{wlemma}\eqref{wlimit} we see that the right side of \eqref{uniqeq2} tends to
$0$ as $r\to\infty$. This forces $\nabla v_\varepsilon\equiv 0$, and since
$v_\varepsilon\in\mathscr{C}^{\infty}(\mathbb{R}^n_{+},\mathbb{C}^M)\cap
\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$
with $\restr{v_\varepsilon}{\partial\mathbb{R}^n_{+}}=0$ we ultimately conclude that
$v_\varepsilon\equiv 0$. Consequently,
\begin{equation}\label{claimepsilon}
u(x',t+\varepsilon)=(P_t^L\ast f_\varepsilon)(x'),\qquad\forall\,(x',t)\in\mathbb{R}^n_{+}.
\end{equation}
Since, as noted earlier,
$\restr{u}{\partial\mathbb{R}^n_{+}}\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$,
for every $x'\in\mathbb{R}^{n-1}$ and $\varepsilon>0$ we may now write
\begin{align}\label{ay6RR}
\big|u(x',t+\varepsilon)-\big(P_t^L\ast(\restr{u}{\partial\mathbb{R}^n_{+}})\big)(x')\big|
&=\big|\big(P_t^L * (f_\varepsilon-\restr{u}{\partial\mathbb{R}^n_{+}})\big)(x')\big|
\nonumber\\[4pt]
&\leq\|P_t^L\|_{L^1(\mathbb{R}^{n-1})}\sup_{y'\in\mathbb{R}^{n-1}}
\big|f_\varepsilon(y') -\restr{u}{\partial\mathbb{R}^n_{+}}(y')\big|
\nonumber\\[4pt]
&=\|P^L\|_{L^1(\mathbb{R}^{n-1})}\sup_{y'\in\mathbb{R}^{n-1}}
\big|u(y',\varepsilon) -\restr{u}{\partial\mathbb{R}^n_{+}}(y')\big|
\nonumber\\[4pt]
&\leq\|P^L\|_{L^1(\mathbb{R}^{n-1})}
[u]_{\dot{\mathscr{C}^{\omega}}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)}\,\omega(\varepsilon).
\end{align}
From \eqref{PoissonDecay} we know that $\|P^L\|_{L^1(\mathbb{R}^{n-1})}<\infty$.
Upon letting $\varepsilon\to 0^{+}$ and using that $\omega$ vanishes in the limit at the origin,
we see that \eqref{ay6RR} implies \eqref{claimeq}. This finishes the proof of Proposition~\ref{fatou}.
\end{proof}
\section{Well-Posedness Results}\label{section:well-general}
We are now ready to prove well-posedness results. We first consider the case in which
the boundary data belong to generalized H\"older spaces and we note that, in such a scenario,
the only requirement on the growth function is \eqref{omega-cond:b}.
\begin{theorem}\label{mainthmBMO2b}
Let $L$ be a constant complex coefficient $M\times M$ system as in \eqref{ars} satisfying the
strong ellipticity condition \eqref{LegHad}. Also, let $\omega$ be a growth function satisfying
\eqref{omega-cond:b}. Then the generalized H\"older Dirichlet problem for $L$ in
$\mathbb{R}^{n}_{+}$, formulated as
\begin{equation}\label{BVP2b}
\left\lbrace
\begin{array}{l}
u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M),
\\[4pt]
Lu=0\,\,\text{ in }\,\,\mathbb{R}^n_{+},
\\[4pt]
\left[u\right]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}<\infty,
\\[6pt]
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm lim}}=f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)
\,\,\text{ on }\,\,\mathbb{R}^{n-1},
\end{array}
\right.
\end{equation}
is well-posed. More specifically, there exists a unique solution which is given by
\begin{equation}\label{eqn-Dir-Holder:u}
u(x',t)=(P_t^L\ast f)(x'),\qquad\forall\,(x',t)\in{\mathbb{R}}^n_{+},
\end{equation}
where $P^L$ denotes the Poisson kernel for the system $L$ in $\mathbb{R}^{n}_+$ from Theorem~\ref{PoissonConvolution}.
In addition, $u$ extends to a function in $\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$ with
$\restr{u}{\partial\mathbb{R}^n_{+}}=f$, and there exists a finite constant $C=C(n,L,\omega)\geq 1$
such that
\begin{equation}\label{mainthmBMOeq12b}
C^{-1}[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq[u]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}
\leq C[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
\end{theorem}
\begin{proof}
Given $f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$, define $u$
as in \eqref{eqn-Dir-Holder:u}. Proposition~\ref{propfholder} then implies that
$u$ satisfies all conditions in \eqref{BVP2b}. Also, $u$ extends to a function
in $\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$ with
$\restr{u}{\partial\mathbb{R}^n_{+}}=f$, and the second inequality in
\eqref{mainthmBMOeq12b} holds. Moreover, \eqref{holderclosureeq} yields
\begin{equation}\label{mainthmBMOeq3}
[f]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
=[\restr{u}{\partial\mathbb{R}^n_{+}}]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq[u]_{\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)}
\leq 2C_\omega [u]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n}_{+},\mathbb{C}^M)},
\end{equation}
so that the first inequality in \eqref{mainthmBMOeq12b} follows.
It remains to prove that the solution is unique. However, this follows at once from
Proposition~\ref{fatou}. Indeed, the first three conditions in \eqref{BVP2b} imply \eqref{claimeq}
and since $\restr{u}{\partial\mathbb{R}^n_{+}}=f$ we conclude that necessarily
$u(x',t)=\left(P_t^L\ast f\right)(x')$ for every $(x',t)\in\mathbb{R}^n_{+}$.
\end{proof}
Here is the well-posedness for the generalized Morrey-Campanato Dirichlet problem.
In this case, the growth function is assumed to satisfy both \eqref{omega-cond:a}
and \eqref{omega-cond:b}.
\begin{theorem}\label{mainthmBMO2}
Let $L$ be a constant complex coefficient $M\times M$ system as in \eqref{ars} satisfying the
strong ellipticity condition \eqref{LegHad}. Fix $p\in[1,\infty)$ along with $q\in(0,\infty]$,
and let $\omega$ be a growth function satisfying \eqref{omega-cond:a} and \eqref{omega-cond:b}.
Then the generalized Morrey-Campanato Dirichlet problem for $L$ in $\mathbb{R}^{n}_{+}$, namely
\begin{equation}\label{BVP2}
\left\lbrace
\begin{array}{l}
u\in\mathscr{C}^{\infty}(\mathbb{R}^{n}_{+},\mathbb{C}^M),
\\[4pt]
Lu=0\,\,\text{ in }\,\,\mathbb{R}^n_{+},
\\[4pt]
\norm{u}_{**}^{(\omega,q)}<\infty,
\\[4pt]
\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)
\,\,\text{ a.e. on }\,\,\mathbb{R}^{n-1},
\end{array}
\right.
\end{equation}
is well-posed. More specifically, there exists a unique solution which is given by
\begin{equation}\label{eqn-Dir:M-C:u}
u(x',t)=(P_t^L\ast f)(x'),\qquad\forall\,(x',t)\in{\mathbb{R}}^n_{+},
\end{equation}
where $P^L$ denotes the Poisson kernel for the system $L$ in $\mathbb{R}^{n}_+$ from Theorem~\ref{PoissonConvolution}.
Moreover, with $W$ defined as in \eqref{omega-cond:WDef}, the solution $u$ extends to a function in
$\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)$ with
$\restr{u}{\partial\mathbb{R}^n_{+}}=f$ a.e. on $\mathbb{R}^{n-1}$, and there
exists a finite constant $C=C(n,L,\omega,p,q)\geq 1$ for which
\begin{equation}\label{mainthmBMOeq12}
C^{-1}\norm{f}_{\mathscr{E}^{W,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq\norm{u}_{**}^{(\omega,q)}\leq C\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
Furthermore, all results remain valid if $\norm{\cdot}_{**}^{(\omega,q)}$
is replaced everywhere by $\norm{\cdot}_{**}^{(\omega,\exp)}$.
\end{theorem}
\begin{proof}
Having fixed $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$, if $u$ is defined as in
\eqref{eqn-Dir:M-C:u} then Proposition~\ref{propstep1} implies the validity of all conditions
in \eqref{BVP2} and also of the second inequality in \eqref{mainthmBMOeq12}
(even replacing $q$ by $\exp$). In the case $q=\infty$ we invoke
Proposition~\ref{prop:proper-sols}\eqref{CW-car-q} to obtain that
$u\in\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)$
in the sense of Lemma~\ref{wlemma}\eqref{holderclosure}. Note that we also have
\begin{align}\label{8fafrafr}
[\restr{u}{\partial\mathbb{R}^n_{+}}]_{\dot{\mathscr{C}}^{W}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq[u]_{\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)}
\leq 2C_{W}[u]_{\dot{\mathscr{C}}^{W}(\mathbb{R}^{n}_{+},\mathbb{C}^M)}
\leq C(C_\omega)^4\norm{u}_{**}^{(\omega,\infty)}
\end{align}
thanks to \eqref{holderclosureeq} (for the growth function $W$), Lemma~\ref{wlemma3}, and \eqref{mainthmBMOeq4}.
Given that, on the one hand,
$\restr{u}{\partial\mathbb{R}^n_{+}}=\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}$
everywhere in $\mathbb{R}^{n-1}$ due to the fact that $u\in\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)$,
and that, on the other hand, $\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f $ a.e. in $\mathbb{R}^{n-1}$,
we conclude that $\restr{u}{\partial\mathbb{R}^n_{+}}=f$ a.e. in $\mathbb{R}^{n-1}$. In addition,
\eqref{easyinclusion} (applied to $W$), Lemma~\ref{wlemma3}, and \eqref{8fafrafr} permit us to estimate
\begin{multline}\label{mainthmBMOeq3b}
\norm{f}_{\mathscr{E}^{W,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
=\|\restr{u}{\partial\mathbb{R}^n_{+}}\|_{\mathscr{E}^{W,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq\sqrt{n-1}\,C_W[\restr{u}{\partial\mathbb{R}^n_{+}}]_{\dot{\mathscr{C}^{W}}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\\[4pt]
\leq C(C_\omega)^6\norm{u}_{**}^{(\omega,\infty)}\leq C(C_\omega)^7\norm{u}_{**}^{(\omega,q)}
\leq C(C_\omega)^7\norm{u}_{**}^{(\omega,\exp)},
\end{multline}
where $0<q<\infty$ and where we have also used Proposition~\ref{prop:proper-sols}\eqref{sup-q-car} and \eqref{uqeq:1}.
To prove that the solution is unique, we note that having $\norm{u}_{**}^{(\omega,q)}<\infty$
for a given $q\in(0,\infty]$, or even $\norm{u}_{**}^{(\omega,\exp)}<\infty$, implies that
$\norm{u}_{**}^{(\omega,\infty)}<\infty$ by Proposition~\ref{prop:proper-sols}\eqref{sup-q-car}
and \eqref{uqeq:1}. Having established this, Proposition~\ref{prop:proper-sols}\eqref{CW-car-q} applies
and yields that $u\in\dot{\mathscr{C}}^{W}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)$.
Consequently, $\restr{u}{\partial\mathbb{R}^n_{+}}=\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}$
everywhere in $\mathbb{R}^{n-1}$, and if we also take into account the boundary condition from \eqref{BVP2},
we conclude that $\restr{u}{\partial\mathbb{R}^n_{+}}=f$ a.e. on $\mathbb{R}^{n-1}$.
Moreover, since Lemma~\ref{wlemma3} ensures that $W$ is a growth function satisfying \eqref{omega-cond:b},
we may invoke Proposition~\ref{fatou} to write
\begin{equation}\label{7ttrR}
u(x',t)=\left(P_t^L\ast(\restr{u}{\partial\mathbb{R}^n_{+}})\right)(x')=\left(P_t^L\ast f\right)(x'),
\qquad\forall\,(x',t)\in\mathbb{R}^n_{+}.
\end{equation}
The proof of the theorem is therefore finished.
\end{proof}
\begin{remark}
Theorems~\ref{mainthmBMO2b} and \ref{mainthmBMO2} are closely related. To elaborate in this,
fix a growth function $\omega$ satisfying \eqref{omega-cond:a} and \eqref{omega-cond:b}.
From \eqref{easyinclusion} and Proposition~\ref{propstep1} it follows that, given any
$f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$, the unique solution of the
boundary value problem \eqref{BVP2b}, i.e., $u(x',t)=(P_t^L\ast f)(x')$ for $(x',t)\in\mathbb{R}^{n}_{+}$,
also solves \eqref{BVP2}, regarding now $f$ as a function in
$\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ (cf. \eqref{inclusionE})
with $p\in[1,\infty)$ and $q\in(0,\infty]$ arbitrary (and even with
$\norm{\cdot}_{**}^{(\omega,q)}$ replaced by $\norm{\cdot}_{**}^{(\omega,\exp)}$).
As such, $u$ satisfies \eqref{mainthmBMOeq12} whenever \eqref{omega-cond:a} holds.
This being said, the fact that $f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$
does not guarantee, in general, that the corresponding solution satisfies
$u\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$, even though we have
established above that the solution to the boundary value problem \eqref{BVP2} belongs
to $\dot{\mathscr{C}}^{W}(\mathbb{R}^n_{+},\mathbb{C}^M)$. Note that, as seen from
\eqref{7rsSS}-\eqref{7tEEE} and \eqref{wW}, the space $\dot{\mathscr{C}}^{W}(\mathbb{R}^n_{+},\mathbb{C}^M)$
contains $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)$.
This aspect is fully clarified with the help of Example~\ref{example} discussed further below, where
we construct some growth function $\omega$ satisfying \eqref{omega-cond:a}, \eqref{omega-cond:b}, and
for which the space $\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})$
is strictly bigger than $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C})$.
Its relevance for the issue at hand is as follows. Consider the boundary problem \eqref{BVP2}
formulated with $L$ being the Laplacian in ${\mathbb{R}}^n$ and with
$f\in\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})\setminus
\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C})$ as boundary datum. Its solution
$u$ then necessarily satisfies $u\notin\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C})$,
for otherwise Lemma~\ref{wlemma}\eqref{holderclosure} would imply
$u\in\dot{\mathscr{C}}^{\omega}(\overline{\mathbb{R}^n_{+}},\mathbb{C}^M)$ and since
$\restr{u}{\partial\mathbb{R}^n_{+}}=\restr{u}{\partial\mathbb{R}^n_{+}}^{{}^{\rm nt.lim}}=f$
a.e. on $\mathbb{R}^{n-1}$ and $f$ is continuous in $\mathbb{R}^{n-1}$ we would conclude that
$f$ coincides everywhere with
$\restr{u}{\partial\mathbb{R}^n_{+}}\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C})$, a contradiction.
\end{remark}
In spite of the previous remark, Theorem~\ref{mainthmBMO} states that the boundary problems
\eqref{BVP2b} and \eqref{BVP2} are actually equivalent under the stronger assumption \eqref{omega-cond:main}
on the growth function. Here is the proof of Theorem~\ref{mainthmBMO}.
\vskip 0.08in
\begin{proof}[Proof of Theorem~\ref{mainthmBMO}]
We start with the observation that \eqref{omega-cond:main} and Lemma~\ref{wlemma3}
yield $C_0^{-1}W(t)\leq\omega(t)\leq C_0 W(t)$ for each $t\in(0,\infty)$. Therefore,
\begin{equation}\label{spaceidentity1}
\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n}_{+},\mathbb{C}^M)
=\dot{\mathscr{C}^{W}}(\mathbb{R}^{n}_{+},\mathbb{C}^M),\quad
\dot{\mathscr{C}^{\omega}}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)
=\dot{\mathscr{C}^{W}}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M),
\end{equation}
and
\begin{equation}\label{spaceidentity2}
\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)
=\mathscr{E}^{W,p}(\mathbb{R}^{n-1},\mathbb{C}^M),
\end{equation}
as vector spaces, with equivalent norms.
Having made these identifications, we now proceed to observe that \eqref{bvp-Hol-Dir:main} follows
directly from Theorem~\ref{mainthmBMO2b}, while \eqref{bvp-MC-Dir:main} is implied by
Theorem~\ref{mainthmBMO2} with the help of \eqref{spaceidentity1} and \eqref{spaceidentity2}.
To deal with \eqref{equiv:main}, we first observe that the left-to-right inclusion follows from
Lemma~\ref{wlemma2}\eqref{inclusionitem}, whereas \eqref{easyinclusion} provides the accompanying
estimate for the norms. For the converse inclusion, fix
$f\in\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and set
$u(x',t):=(P_t^L\ast f)(x')$ for every $(x',t)\in\mathbb{R}^{n}_{+}$.
Theorem~\ref{mainthmBMO2} and \eqref{spaceidentity1} then imply that
$u\in\dot{\mathscr{C}^{\omega}}(\overline{\mathbb{R}^{n}_{+}},\mathbb{C}^M)$
with $\restr{u}{\partial\mathbb{R}^n_{+}}= f$ a.e. on $\mathbb{R}^{n-1}$.
Introduce $\widetilde{f}:=\restr{u}{\partial\mathbb{R}^n_{+}}$ and note that
$\widetilde{f}\in\dot{\mathscr{C}^{\omega}}(\mathbb{R}^{n-1},\mathbb{C}^M)$
with $\widetilde{f}=f$ a.e. on $\mathbb{R}^{n-1}$.
Then $u(x',t)=(P_t^L\ast\widetilde{f})(x')$ and, thanks to \eqref{mainthmBMOeq12b},
\eqref{relation-all-norms:1}, and \eqref{mainthmBMOeq12}, we have
\begin{equation}\label{y6tgg}
[\widetilde{f}\,]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)}
\leq C[u]_{\dot{\mathscr{C}}^{\omega}(\mathbb{R}^n_{+},\mathbb{C}^M)}
\leq C\norm{f}_{\mathscr{E}^{\omega,p}(\mathbb{R}^{n-1},\mathbb{C}^M)}.
\end{equation}
This completes the treatment of \eqref{equiv:main}, and finishes the proof of Theorem~\ref{mainthmBMO}.
\end{proof}
We are now in a position to give the proof of Corollary~\ref{maincorBMO}.
\vskip 0.08in
\begin{proof}[Proof of Corollary~\ref{maincorBMO}]
We start by observing that \eqref{eq:qfrafr} is a direct consequence of
Proposition~\ref{prop:proper-sols}\eqref{all-comp}. In particular, the last three equalities
in \eqref{HLMO} follow at once. Also, the fact that the second set in the first
line of \eqref{HLMO} is contained in $\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$
is a consequence of Lemma~\ref{wlemma}\eqref{holderclosure}.
Finally, given any $f\in\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C}^M)$, if $u$ is the solution
of \eqref{BVPb} corresponding to this choice of boundary datum, then $\restr{u}{\partial\mathbb{R}^n_{+}}=f$
and $u$ also satisfies the required conditions to be an element in the second set displayed in \eqref{HLMO}.
\end{proof}
The following example shows that conditions \eqref{omega-cond:a} and \eqref{omega-cond:b}
do not imply \eqref{maincorBMOeq}.
\begin{example}\label{example}
Fix two real numbers $\alpha,\beta\in(0,1)$ and consider the growth function
$\omega:(0,\infty)\to(0,\infty)$ defined for each $t>0$ as
\begin{equation}\label{6tgtfa}
\omega(t):=
\left\lbrace
\begin{array}{ll}
t^{\alpha}, &\text{ if }t\leq 1,
\\[4pt]
1+(\log t)^{\beta}, &\text{ if }t>1.
\end{array}
\right.
\end{equation}
Clearly, $\omega$ satisfies \eqref{omega-cond:a}, and we also claim that
$\omega$ satisfies \eqref{omega-cond:b}. Indeed, for $t\leq 1$,
\begin{equation}\label{65tr}
\int_t^{\infty}\frac{\omega(s)}{s}\frac{ds}{s}=\int_{t}^1 s^{\alpha-1}\frac{ds}{s}
+\int_1^{\infty}\frac{1+(\log s)^{\beta}}{s^2}\,ds\leq C(t^{\alpha-1}+1)\leq 2C\,t^{\alpha-1}.
\end{equation}
For $t\in[1,\infty)$, define
\begin{equation}\label{65fff}
F(t):=\frac{\displaystyle t\int_t^{\infty}\frac{\omega(s)}{s}\frac{ds}{s}}{\omega(t)}
=\frac{\displaystyle\int_t^{\infty}\frac{1+(\log s)^{\beta}}{s^2\,ds}}{\displaystyle\frac{1+(\log t)^{\beta}}{t}},
\end{equation}
which is a continuous function in $[1,\infty)$ and satisfies $F(1)<\infty$.
Moreover, using L'H\^{o}pital's rule,
\begin{equation}\label{6tfff}
\lim_{t\to\infty}F(t)=\lim_{t\to\infty}\frac{-(1+(\log t)^\beta)}{\beta(\log t)^{\beta-1}-(1+(\log t)^\beta)}=1.
\end{equation}
Hence, $F$ is bounded, which amounts to having $\omega$ satisfy \eqref{omega-cond:b}.
The function $W$, defined as in \eqref{omega-cond:WDef}, is currently given by
\begin{equation}\label{63dd}
W(t)=
\left\lbrace
\begin{array}{ll}
\dfrac1{\alpha} t^{\alpha}, &\text{ if }t\leq 1,
\\[10pt]
\dfrac{1}{\alpha}+\dfrac1{\beta+1}(\log t)^{\beta+1}+\log t, &\text{ if }t>1.
\end{array}
\right.
\end{equation}
Since \eqref{omega-cond:main} would imply $W(t)\leq C\omega(t)$ which is not the case for $t$ sufficiently large,
we conclude that the growth function $\omega$ satisfies \eqref{omega-cond:a} and \eqref{omega-cond:b}
but it does not satisfy \eqref{omega-cond:main}.
For this choice of $\omega$, we now proceed to check that
$\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})\neq\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C})$.
To this end, consider the function
\begin{equation}
f(x):=\log_{+}|x_1|,\qquad\forall\,x=(x_1,x_2,\dots,x_{n-1})\in\mathbb{R}^{n-1},
\end{equation}
where $\log_{+}t:=\max\{0,\log t\}$. With $e_1=:(1,0,\dots,0)\in\mathbb{R}^{n-1}$ we then have
\begin{equation}\label{6r4d5}
\sup_{x\neq y}\frac{|f(x)-f(y)|}{\omega(|x-y|)}
\geq\lim_{x_1\to\infty}\frac{|f(x_1e_1)-f(e_1)|}{\omega(|x_1e_1-e_1|)}
=\lim_{x_1\to\infty}\frac{\log x_1}{1+(\log (x_1-1))^{\beta}}=\infty,
\end{equation}
since $\beta<1$. This means that $f\notin\dot{\mathscr{C}}^{\omega}(\mathbb{R}^{n-1},\mathbb{C})$.
To prove that $f\in\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})$, consider
$\widetilde{Q}:=(a,b)\times Q \subset\mathbb{R}^{n-1}$, where $Q$ is an arbitrary cube in
$\mathbb{R}^{n-2}$ and $a,b\in\mathbb{R}$ are arbitrary numbers satisfying $a<b$. Then,
\begin{align}\label{eq:example1}
\norm{f}_{\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})}
&\leq\sup_{\widetilde{Q}\subset\mathbb{R}^{n-1}}\frac1{\omega(\ell(\widetilde{Q}))}
\fint_{\widetilde{Q}}\fint_{\widetilde{Q}}|f(x)-f(y)|\,dx\,dy
\nonumber\\[4pt]
&\leq\sup_{a<b}\frac{1}{\omega(b-a)}H(a,b),
\end{align}
where
\begin{equation}\label{eq:example1-hhh}
H(a,b):=\fint_a^b\fint_a^b\big|\log_{+}|x_1|-\log_{+}|y_1|\big|\,dx_1\,dy_1.
\end{equation}
We shall now prove that the right-hand side of \eqref{eq:example1} is finite considering several different cases.
\vskip 0.08in
\noindent{\bf Case~I: $\boldsymbol{ 1\leq a<b}$.} In this scenario, define
\begin{align}\label{eq:example2-ggg}
G(\lambda):=1+2\lambda-2\lambda(\lambda+1)\log\left(1+\frac1{\lambda}\right),\quad\forall\,\lambda>0.
\end{align}
Note that $G$ is continuous in $(0,\infty$), $G(0)=1$, and by
L'H\^{o}pital's rule, $\lim_{\lambda\to\infty}G(\lambda)=0$, hence $G$ is bounded.
Also,
\begin{align}\label{eq:example2}
H(a,b)=\frac{b^2-a^2-2ab\log (b/a)}{(b-a)^2}=G\big(a/(b-a)\big).
\end{align}
Consequently, whenever $b-a\geq 1$ we have
\begin{equation}\label{eq:example3}
H(a,b)=G\big(a/(b-a)\big)\leq C\leq C\big(1+(\log (b-a)\big)^{\beta})=C\omega(b-a).
\end{equation}
Again by L'H\^{o}pital's rule, $\lim_{\lambda\to\infty}\lambda^{\alpha} G(\lambda)=0$,
hence $\lambda^{\alpha}G(\lambda)\leq C$ for every $\lambda>0$. Therefore, whenever $0<b-a<1$ we may write
\begin{equation}\label{eq:example4}
H(a,b)=G\big(a/(b-a)\big)\leq C\left(\frac{b-a}{a}\right)^{\alpha}\leq C(b-a)^\alpha=C\omega(b-a).
\end{equation}
All these show that $H(a,b)\leq C\omega(b-a)$ in this case.
\vskip 0.08in
\noindent{\bf Case~II: $\boldsymbol{a<b\leq -1}$.} This case is analogous to the previous one by symmetry.
\vskip 0.08in
\noindent{\bf Case~III: $\boldsymbol{-1\leq a<b\leq 1}$.} This case is straightforward
since $H(a,b)=0$, given that $\log_{+}|x_1|=\log_{+}|y_1|=0$ whenever $a<x_1,y_1<b$.
\vskip 0.08in
\noindent{\bf Case~IV: $\boldsymbol{-1<a<1<b}$.} In this case we obtain
\begin{align}\label{eq:example5}
H(a,b) &=\frac{1}{(b-a)^2}\int_1^b\int_1^b|\log x_1-\log y_1|\,dx_1\,dy_1
\nonumber\\[4pt]
&\qquad+\frac1{(b-a)^2}\int_a^1\int_1^b\log x_1\,dx_1\,dy_1
+\frac{1}{(b-a)^2}\int_1^b\int_a^1\log y_1\,dx_1\,dy_1
\nonumber\\[4pt]
&\leq\frac{(b-1)^2}{(b-a)^2}H(1,b)+2\frac{(1-a)(b\log b-b+1)}{(b-a)^2}.
\end{align}
For the first term in the right-hand side of \eqref{eq:example5}, we use \eqref{eq:example3}
and \eqref{eq:example4} (written with $a:=1$) and obtain, keeping in mind that in this case $a<1$,
\begin{equation}\label{765FF}
\frac{(b-1)^2}{(b-a)^2}H(1,b)\leq C\omega(b-1)\left(\frac{b-1}{b-a}\right)^2\leq C\omega(b-a).
\end{equation}
To bound the second term in the right-hand side of \eqref{eq:example5}, we first use the
fact that $1-a<2$ and $\log t\leq t-1$ for every $t\geq 1$ to obtain
\begin{align}
\frac{(1-a)(b\log b-b+1)}{(b-a)^2} &\leq 2\frac{b(b-1)-b+1}{(b-a)^2}
\leq 2\left(\frac{b-1}{b-a}\right)^2\leq 2
\nonumber\\[4pt]
&\leq 2\big(1+(\log(b-a))^{\beta}\big)=2\omega(b-a),
\end{align}
whenever $b-a\geq 1$. To study the case when $b-a<1$, bring in the auxiliary function
\begin{equation}\label{ttFF}
\widetilde{G}(\lambda):=\frac{\lambda\log \lambda-\lambda+1}{(\lambda-1)^{1+\alpha}},\qquad\forall\lambda>1.
\end{equation}
By L'H\^{o}pital's rule, $\lim_{\lambda\to 1^{+}}\widetilde{G}(\lambda)=0$, hence $\widetilde{G}(\lambda)\leq C$
for each $\lambda\in(1,2]$. If $b-a<1$, we clearly have $1<b\leq 2$ which, in turn, permits us to estimate
\begin{align}\label{6ttffa}
\frac{(1-a)(b\log b-b+1)}{(b-a)^2}
&=\frac{(1-a)(b-1)^{1+\alpha}G(b)}{(b-a)^2}
\nonumber\\[4pt]
&\leq\frac{C(b-1)^{1+\alpha}}{b-a}\leq C(b-a)^{\alpha}=C\omega(b-a).
\end{align}
Consequently, we have obtained that $H(a,b)\leq C\omega(b-a)$ in this case as well.
\vskip 0.08in
\noindent{\bf Case~V: $\boldsymbol{a<-1<b<1}$.} This is analogue to Case~IV, again by symmetry.
\vskip 0.08in
\noindent{\bf Case~VI: $\boldsymbol{a<-1,\,\,b>1}$.} We break the interval $(a,b)$ into two
intervals $(a,0)$ and $(0,b)$ to obtain
\begin{equation}\label{6t5ff}
H(a,b)\leq\frac{1}{(b-a)^2}(I+II),
\end{equation}
where, using Case~IV and Case~V,
\begin{align}\label{6g5f5}
I &:=\int_a^0\int_a^0\big|\log_{+}|x_1|-\log_{+}|y_1|\big|\,dx_1\,dy_1
+\int_0^b\int_0^b|\log_{+}x_1-\log_{+}y_1|\, dx_1\,dy_1
\nonumber\\[4pt]
&=H(a,0)(0-a)^2+H(0,b)(b-0)^2\leq C|a|^2\omega(|a|)+C\,b^2\omega(b)
\nonumber\\[4pt]
&\leq 2C(b-a)^2\omega(b-a).
\end{align}
Similarly, by Case~IV,
\begin{align}\label{9g9h9h}
II &:=\int_a^0\int_0^b\big|\log_{+}|x_1|-\log_{+}|y_1|\big|\, dx_1\,dy_1
+\int_0^b\int_a^0\big|\log_{+}|x_1|-\log_{+}|y_1|\big|\,dx_1\,dy_1
\nonumber\\[4pt]
&\leq 2\big(\max\{|a|,b\}-0\big)^2H\big(\max\{|a|,b\},0\big)
\nonumber\\[4pt]
&\leq 2C\max\{|a|,b\}^2\,\omega(\max\{|a|,b\})
\nonumber\\[4pt]
&\leq 2C(b-a)^2\,\omega(b-a).
\end{align}
Thus, $H(a,b)\leq C\omega(b-a)$ in this case also.
Collectively, the results in Cases~I-VI prove that
$f\in\mathscr{E}^{\omega,1}(\mathbb{R}^{n-1},\mathbb{C})$.
\end{example}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,749 |
As more business continues to be done online, a top-of-the-line broadband business plan needs to be in place for your business. Whatever your bandwidth needs might be, Top Comms can help get you started with a plan that will save you money. We'll help you every step of the way, from getting you the right services to setting up the network in your office. And since we can also take care of all your business communications needs, you save even more by consolidating through a single provider.
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Broadband speeds can sometimes begin to lag for any user, even those with expensive connection plans in place. That's because as the number of users on a line increases, speeds can decrease as the system deals with the load. If your business relies heavily on constant high speed access, avoid headaches and service slowdowns with a leased lines package. Having a leased line means never having to deal with network slowdowns due to other users because there are no other users. You enjoy all the benefits of permanent uncontested service running through your own dedicated lines. Your speeds will never fluctuate and you'll never encounter those moments when crucial data transfers slow down because of other users. In many cases, bandwidth speeds can even be tailored and modified to help give you more during those busy periods when your traffic increases. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,798 |
Lelya is a monotypic genus of flowering plants belonging to the family Rubiaceae. It only contains one known species, Lelya osteocarpa Bremek.
Its native range is Tropical Africa. It is found in Angola, Malawi, Nigeria, Tanzania, Zambia and Zaïre.
The genus name of Lelya is in honour of Hugh Vandervaes Lely (1891–1947), an English botanist and forester in Nigeria. The Latin specific epithet of osteocarpa is derived from 2 words, osteo from the Greek word meaning bone and also carpa is derived from carpus meaning fruit.
Both genus and species were first described and published in Verh. Kon. Ned. Akad. Wetensch., Afd. Natuurk., Sect. 2, Vol.48 (Issue 2) on page 181 in 1952.
References
Rubiaceae
Rubiaceae genera
Plants described in 1952
Flora of South Tropical Africa
Flora of East Tropical Africa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,875 |
Daniel Loss (Winterthur, ) é um físico suíço. É atualmente professor de física téorica do estado sólido da Universidade de Basileia.
Condecorações
Prêmio Humboldt (2005)
Prêmio Marcel Benoist (2010)
Ligações externas
University of Basel biography & publications list
Professores da Universidade de Basileia
Físicos da Suíça
Físicos do século XX
Físicos do século XXI | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8 |
Table of Contents
Also by Roben Ryberg:
Title Page
Dedication
Acknowledgements
Foreword
The Science of Gluten-Free Cookies
Introduction
Chapter 1 - Successful Gluten-Free Baking
About Flour Blends
Gluten-Free Baking Simplified
The Cost of Flours
Making a Great Gluten-Free Cookie
Helpful Hints
Chapter 2 - Kitchen and Baking Tips
Cleanliness and Preventing Cross-Contamination
The Cookie Pantry
Equipment
Chapter 3 - Drop Cookies
Almond Flower Cookies
Applesauce-Raisin Cookies
Banana-Nut Cookies
Chocolate Cookies with White Chips
Chocolate Chip Cookies #1
Chocolate Chip Cookies #2
Coconut Macaroons
Cranberry Cookies
Ginger Spice Cookies
Hermits
Hot Chocolate Cookies
Lace Cookies, Oatmeal
Lemon-Poppy Seed Cookies
Molasses Cookies
Mrs. Fields- or Neiman Marcus-Style Cookies
Oatmeal-Raisin Cookies
Peanut Butter Blossom Cookies
Pecan Cookies
Pecan Lace Cookies
Sour Cream Breakfast Cookies
Chapter 4 - Bar Cookies
Almond Layered Cookie Bars
Apple Crumble Bars
Blondies #1
Blondies #2
Carrot Cake Bars
Cheesecake Bars
Cranberry-Orange-Almond Granola Bars
Decadent Brownies
Jam Bars
Lemon Bars
Peanut Butter-Chocolate Chip- Oatmeal Bars
Raspberry-Cream Cheese Brownies
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Shortbread
Toffee Bars
Trail Mix Bars
Turtle Bars
White Chocolate Bars
Chapter 5 - Rolled and Piped Cookies
Animal Crackers
Butter Cookies
Candy Cane Cookies
Chinese Marble Cookies
Chocolate Butter Cookies
Chocolate Graham Crackers
Chocolate Meringues
Chocolate Pinwheel Cookies
Chocolate Wafer Hearts
Cinnabon-Style Cookies
Crisp Almond Cookies
Gingerbread Men and Gingersnaps
Graham Crackers
Ice-Cream Sandwich Cookies
Lady Fingers
Lemon Meringues
Linzer Sandwich Cookies
Nut Meringue Wreaths
Rolled Sugar Cookies
Rolled Sugar Cookies, Dairy-Free
Stained-Glass Cookies
Chapter 6 - Great Fakes Cookies
Chips Ahoy!-Style Cookies
Chocolate Marshmallow "Scooter" Pies
Fig Newton-Style Cookies
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Lemon Chalet Cremes-Style Cookies
Girl Scout Samoas-Style Cookies
Girl Scout Tagalongs-Style Cookies
Girl Scout Thin Mints-Style Cookies
Girl Scout Trefoils-Style Cookies
Little Debbie Oatmeal Creme Pies-Style Cookies
Maple Leaf Cookies
Nutter Butter-Style Peanut Butter Sandwich Cookies
Oreos-Style Cookies
Pecan Sandies-Style Cookies
Pepperidge Farm Milano-Style Cookies
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
Chapter 7 - Sandwich, Shaped, and Filled Cookies
Almond Biscotti
Chocolate-Cherry Cookies
Chocolate Crinkles
Chocolate-Mint Sandwich Cookies
Chocolate Sandwich Cookies (Scooter Pies)
Filled Triangle Cookies
Lemon Tassies
Mocha Meltaways
Pecan Tassies
Pizzelle
Pizzelle, Almond
Pizzelle, Chocolate
Pumpkin Sandwich Cookies (Scooter Pies)
Red Velvet Sandwich Cookies (Scooter Pies)
Rugalach
Sand Balls
Snickerdoodles
Thumbprint Cookies
Chapter 8 - Egg-Free Cookies
Almond Joy-Style Cookies
Crackerdoodles
Egg-Free Applesauce Bars
Egg-Free Banana Bars with Browned Butter Icing
Egg-Free Brownies
Egg-Free Chocolate Chip Cookies
Egg-Free Chocolate Cookies
Egg-Free Gingerbread Men
Egg-Free Oatmeal Cookies
Egg-Free Peanut Butter Cookies
Egg-Free Pumpkin Bars
Egg-Free Rolled Sugar Cookies
Egg-Free Spice Cookies
Egg-Free Sugar Cookies
Fruitcake Nuggets
Orange-Cream Cheese Tassies
Pepper Jack Cookies
Chapter 9 - Cookies Made with Other Gluten-Free Flours
Blueberry-Cranberry Jumbles
Chocolate Biscotti
Chocolate Cookies, Dairy-Free
Chocolate Crinkles
Cloud Cookies
Coconut Chip Cookies
Fortune Cookies and Pirouettes
Iced Oatmeal Cookies
Love Letter Rolled Chocolate Sugar Cookies
Oatmeal Cookies
Pumpkin Cookies
Rolled Sugar Cookies, Dairy-Free
Rosettes
Spritz
Appendix - Gluten-Free Resources
Index
Copyright Page
**Also by Roben Ryberg:**
_You Won't Believe It's Gluten-Free!_
_The Gluten-Free Kitchen_
**Recipes by Roben Ryberg:**
_Eating for Autism_
_**To happy days,**_
_**counting blessings**_
_**old and new . . .**_
Coconut Macaroons, page 26
_**Acknowledgments**_
# **To those that guide my research:**
Thank you for your feedback on gluten-free foods; for wanting healthier, whole-grain flours; for demanding good taste; and for wanting foods that don't stale in just one day. Thank you for pushing me to do better simply.
# **To those that test:**
Thank you to the Boonsboro High School cross-country team for eating batch after batch of cookies; to friends who took cookies to bake sales to raise awareness of gluten sensitivity; to Crawford's diner for testing and sharing my cookies with customers; to other friends and family for fearlessly testing again and again.
# **To those that give:**
Thank you to Sara Boswell for sharing her expertise in food science; to Stacie Nitza, my fellow foodie; to Cassandra Gee for baking, editing, and playing cheerleader; to the gluten-free community, both online and in various states, for supporting my work; to the Girl Scouts of Central Maryland, Inc., for providing samples of their cookies so that I could test my "great fakes" side-by-side with their outstanding cookies.
_**Foreword**_
**By Dr. Stephen Wangen**
**Roben and I first met at a national** conference on gluten intolerance sponsored by GIG, the Gluten Intolerance Group of North America (www.gluten.net). Like everyone there, we shared a keen interest in everything gluten-free.
Roben has a passion for making gluten-free bread and baked goods. However, she isn't satisfied with "good" alternatives. She demands that they are as close to the real thing as possible. In a recent conversation where she was describing the painstaking process that she went through to develop one particular cookie, I was impressed with her persistence and her patience. She spent hours coming up with just the right mix of ingredients and techniques to perfectly replicate a traditional cookie. And that was just one of the more than one hundred cookies in this book!
We also share an interest in the significant health benefits of being gluten-free. My passion is in helping people to discover their gluten intolerance. Too many people remain undiagnosed. And many people think that gluten intolerance and celiac disease are essentially the same thing. But celiac disease only represents a fraction of those who are gluten intolerant.
Celiac disease is caused by an inability to digest gluten, a protein found in wheat, rye, (cross-contaminated) oats, and barley. This condition affects about 1 percent of the population, or around 3 million people in the United States. These people develop damage in the small intestine called villous atrophy. However, there are millions more who are suffering from gluten intolerance who do not get this type of damage in the small intestine because it only represents one possible result of a gluten intolerance. These people often get overlooked or misdiagnosed, as do many people who have celiac disease. As a medical community doctors are still only in the early stages of properly recognizing this problem.
It is worth noting that nonceliac gluten intolerance is not necessarily any less severe than that of celiac disease. And in many people it is more severe. In either case, avoiding gluten helps to resolve an incredibly large number of health problems. There are well over two hundred different conditions known to be associated with gluten intolerance, and the list grows weekly as more research is done. Many common complaints such as fatigue, headaches, arthritis, anemia, heartburn, diarrhea, constipation, gas, bloating, abdominal pain, eczema, osteoporosis, and even weight gain can be caused by a gluten intolerance. A more complete list can be found in my new book, _Healthier Without Wheat_ : _A New Understanding of Wheat Allergies, Celiac Disease, and Non-Celiac Gluten Intolerance_.
Of course, going gluten-free is easier said than done. This is where the ideal of living healthier without gluten meets the practicality of trying to do just that. It's a steep learning curve, and there are bound to be frustrations. There are also the emotional attachments to food that we develop over a lifetime. And that is never truer than it is for baking cookies, which is almost a national pastime.
For these reasons, we are incredibly fortunate to have Roben's book. Roben has dramatically shortened the learning curve for the rest of us, and given us hope that we can maintain some old traditions as long as we're willing to learn to do them a little differently. And she has done this for just about any cookie that one can imagine, or at least that I can imagine.
What's more, Roben has also incorporated the recent research by the Cereal Quality Lab at Texas A&M to bring us a healthier cookie. That may sound like a bit of an oxymoron, but every effort helps. Gluten-free foods are nutritionally often a shell of their wheat-based counterparts. Using brown rice flour and sorghum flour is a major improvement, and it doesn't detract from the taste of the cookie. These are still the real thing, not some health food substitute for a cookie.
I hope that you enjoy learning to bake these cookies, and I am thrilled that you'll be able to be gluten-free in the process. On behalf of all of us, thank you, Roben.
_Dr. Stephen Wangen_
_www.HealthierWithoutWheat.com_
_**The Science of Gluten-Free Cookies**_
**By Texas A &M University, Cereal Quality Lab, College Station, TX**
**Carbon, hydrogen, oxygen** —no, we are not in chemistry class. We are discussing the elemental ingredients in your cookies. A certain fuzzy blue monster once sang, "C is for Cookie and that's good enough for me," but when it comes to baking easy gluten-free and allergy-free cookies, one has to take a closer look at the science behind the combination of ingredients that we are using.
Your typical wheat-based cookie is made from what bakers call "soft flour." A soft flour means that it has a lower protein (gluten) content than a hard wheat flour that is used to make wheat breads. The gluten protein in the flour helps hold the dough and the final cookie or bread together. That's why your cookie is soft and easy to chew but strong enough to keep from crumbling. Gluten-free baking can often be frustrating, as you no longer have the majority of that protein structure. Gluten-free flours do not have the same type of proteins that hold your other contents together, so ingredients like xanthan gum or guar gum are added to gluten-free dough to help out instead. Fortunately, due to the low gluten content in cookie recipes, cookies are easier to mimic for gluten-free recipes than more complicated products.
Gluten-free flours are made from a wide variety of grains, pseudo-grains, seeds, nuts, tubers, and pretty much anything else you can think of that can be ground into a semi-powder form to be called a "flour." Some gluten-free flours are processed into what is sold as a starch, just a portion of the parent flour. These starches (sometimes referred to as flours by producers) from potato, tapioca, and corn are often criticized for being "unhealthy," so this poses the questions: What is starch? And why does it make my gluten-free baked goods tasty?
Starch is the primary constituent of all cookies, gluten-free or not. It is a naturally occurring carbohydrate found in fruits, vegetables, grains, and tubers that are made up of thousands upon thousands of glucose molecules linked together. The easiest way to visualize starch in its natural form is to grab a sheet of paper and draw a circle on it; this is your starch granule that is made up of the thousands of little glucose molecules. Any flour, when you look at it under a microscope, contains lots of these starch granules that make up the "bulk" of what you can see and feel of the flour.
Starch can take many different sizes, shapes, and forms depending on the source you are using, but they don't act the same. For instance, the starch in tapioca is not the same as the starch in rice. Some starch granules can be shaped like pentagons, some in nice round circles or ovals, and the sizes vary as well. In baking, when liquid is added to the batter or dough and then heated, the starch molecules soak up the water molecules, causing the chemical bonds within the starch to change. When the water-soaked starch reaches a certain temperature it melts (gelatinizes), which means that you can no longer see the original shape of the starch granule and it helps hold the crumb of the cookie or bread you are baking together, like glue. Starch, whether used in pure form or as a component of all flours, is a critical part of gluten-free baking as it makes up for the majority of the structure of your cookie or bread, since the traditional wheat proteins are missing.
The cookie recipes that Roben created for this book mainly use brown rice flour or sorghum flour. These are two easy to find, high fiber, nutritious, and economical flours. Her recipes will not fill your pantry with expensive flours often utilized in various gluten-free blends. We hope that you enjoy the recipes from this book as much as we do—and remember, you don't need seven different kinds of flour to make a good cookie!
_Sara Boswell, Research Assistant_
_Cassandra McDonough, M.S., Research Scientist_
_Dr. Lloyd Rooney, Regents Professor and Faculty Fellow_
_Texas A &M University, Cereal Quality Lab, College Station, TX_
_**Introduction**_
**As I write these words** , I think about you standing there, trying to figure out which gluten-free cookbook to buy and wondering if these cookies will taste good. You may even be wondering if purchasing ingredients will stress your budget, too. I wish you could be in my kitchen, having the scent of delicious, fresh-baked cookies hit you as you enter my home, and sampling three or four kinds while we chat. But you'll have to settle for my words, my reputation, and my photos. The picture of Girl Scout Thin Mints-Style Cookies on page 6 of the insert, one of my favorite great fakes, is representative that almost anything is deliciously possible in a gluten-free cookie.
Whether seeking a squishy scooter pie, a delicate butter cookie, or a crisp graham cracker, you will find it here. I thought about making a separate chapter for dairy-free cookies, but most of the cookies in this book are dairy-free. Dairy just doesn't always enhance a gluten-free cookie recipe. I have, however, made a separate chapter for egg-free cookies. In that chapter I opt to avoid egg substitutes and, as in the rest of this book, use ordinary ingredients to tempt your taste buds.
These amazing cookies are not only easy to make, they're as close as you can get to the real deal. I've been working in the gluten-free industry since the early 1990s, but I myself do not live with major dietary restrictions (although I limit dairy). I am a food science junkie and a person that simply loves food. I will test, test, and retest a gluten-free recipe until the nuances are achieved, often by eating the original alongside my version. I don't have to remember or wonder what something tastes like (although some foods are vivid in everyone's memory!). I simply taste the original cookie, dissect it, and do my best to duplicate the experience for you.
In the last thirty years, gluten-free baking has come a long way. (See Chapter 1 for a discussion of the evolution of gluten-free baking and how we ended up with complicated food.) Fortunately, by embracing food science and daring to think simpler, we now can have delicious, healthier treats, made with just _one_ bag of flour, a little xanthan gum, and everyday ingredients already in your kitchen.
Most of the cookies in this book are made with either brown rice flour or sorghum. Really—that's it! No complicated flour blends required. All I can say, is how cool is that? Brown rice flour is a healthy, affordable, whole-grain flour that provides a wonderful, neutral base on which to build cookie flavor! Sorghum flour, nearly whole-grain, has a lower glycemic index, good nutrition, and an understated fuller grain flavor that doesn't spar with other flavors. And both of these flours can produce incredible textures!
Setting aside food science, it just feels great knowing that you will have tasty cookies! But hopefully, this book gives you something more . . . fabulous cookies to take to a party, a marathon holiday baking session with loved ones, s'mores at a backyard barbecue, renewed fame for cookie-baking talents, or the most sought-after cookies on the tray. It is my goal that this book brings you the simple joy of cookies. And when you taste your first scrumptious cookie from this book, know that I'm smiling with you. Cookie, cookies, cookies. Life can be sweet.
1
_**Successful Gluten-Free Baking**_
**I first began experimenting with** gluten-free foods nearly two decades ago, when a friend of mine who was diagnosed with gluten intolerance innocently asked me if I could help make her something tasty to eat. At that time many gluten-free recipes were made with just rice flour. They were often gritty, dried out quickly, and didn't taste very good. Availability was sparse and the industry was just taking notice of the need for gluten-free foods.
Over the last fifteen years or so, thanks to advances in food science, gluten-free baking has evolved to produce really delicious desserts. These results were made possible by using blends of two, three, four . . . or even seven different alternative flours. Sometimes gelatin, ascorbic acid, and other ingredients were added to provide better results. And, to their credit, many recipes and gluten-free cookbooks use blends that create admirable products.
But using many different flours in a blend can quickly get complicated for you, the baker. After all, who wants to buy three different kinds of expensive flour to make one recipe, then several other flours to make another? Did the pendulum have to swing so far in such a difficult direction? Why choose complicated? So, I began to work toward improving single-flour gluten-free baking. And as it turns out, it can be done very successfully!
# _**About Flour Blends**_
Historically we've found that those early, simple gluten-free baking recipes provided less than ideal results. Complicated recipes that called for multiple grains and starches were more reliable. But why? Are complex blends really necessary?
First we must understand why blends work. It has everything to do with flour textures and baking properties. To oversimplify, if you think of a cup of traditional white (wheat) flour, you could say it is a perfect 5 on a scale of 1 to 10! Utilizing traditional white flour makes cakes and cookies that turn out just as we expect.
Traditional white flour is not too light, not too heavy. Accordingly, if you're baking cookies, this perfect 5 would beat out a 10—a light flour that rises greatly and which would be at the highest end of our scale—anytime. Likewise, a 2 or a 3 "heavy" flour (at the bottom of our scale) wouldn't be good, either. And, as early history has proven, a heavy flour, such as rice, substituted in a recipe cup for cup, would typically result in baked goods with a texture like a hockey puck or cardboard. It is funny only in hindsight!
Although using blends is generally not my philosophy, let's see how one is made. Remember that 5 is the known perfect on our scale. If you were to take one or more light alternative flours (such as tapioca starch, potato starch, or cornstarch) and combine them with several heavier flours (such as cornmeal, brown rice flour, or millet), you would end up with an averaged value pretty close to perfect. To take a very broad view, light flours = 10, heavy flours = 1. Add 10 plus 1 and then divide by 2, and you have just about achieved the perfect 5.
But it doesn't stop there. While we're building a better mousetrap, let's worry about empty flours (starches with limited dietary benefit—but easy on a tender gut), fiber levels (whole grains, flax, and others), protein levels (think soy and bean flours), and more attributes of other pseudograins! If you combine all these levels of flours, some light, some medium, some heavy, you can still stay near that perfect 5 and enhance nutrition at the same time! Wow, all of that in a flour blend! If flour represented our entire dietary intake (and if it were the only ingredient in a recipe), that would be ideal. We do, however, eat other nutritious foods. Have we ever expected so much from traditional breads, cakes, and cookies?
Should you choose to mix blends to use for baking, you will soon find your cupboards full of alternative flours. It may be difficult to evaluate which of them your body tolerates. It may be also difficult to know the true flavor of some of these flours, because they are part of a blend. Sometimes we can look at the source of the flour and get a feel for how it will taste. Bean flour does, actually, taste like beans. Coconut flour tastes like coconut with every bit of sweet juice removed (like tropical dirt, in my opinion). And some others taste worse (soy tastes like grass, montina like dried corn husks—actually good in a small quantity; quinoa even grassier; teff is earthy and grassy; etc.). I'm just sharing my honest opinions before you spend a lot of money on a flour you may not enjoy. Although one can make a cookie out of almost anything, that doesn't mean you want to eat it.
Gingerbread Men and Gingersnaps, page 81
And, let us not forget the irony of giving a person with a sensitive immune system a flour blend that contains several flours that are among the top eight allergens in the United States, for instance, tree nuts (e.g., almonds) or soybeans. And dare we further consider people with multiple food allergies, such as to dairy, eggs (both also among those top eight allergens), and corn? Does this make sense?
Alternatively, we have a simple solution. Use just one flour and use it in ratios that make sense. Don't just take a regular recipe and substitute the flour cup for cup. Instead, I've developed recipes that embrace and maximize that flour's unique characteristics and baking properties. Let's stop treating an apple like a banana!
# _**Gluten-Free Baking Simplified**_
I have listened closely to food concerns from the gluten-free community. In my first book, _The Gluten-Free Kitchen_ , great taste was the primary goal (and of course, remains so today). I utilized cornstarch and potato starch to make foods that closely resembled their traditional counterparts. It was my goal to make the reader eat safely, not feel deprived, and finally, eat other nutritious foods to round out a healthy diet.
The recipes in my second book, _You Won't Believe It's Gluten-Free!_ utilized one flour at a time to create almost any food desired. Cornstarch alone, potato starch alone, rice flour alone, and oat flour alone. It provided the reader with an opportunity to figure out what suited their taste buds and their gut. It also provided a huge array of everyday foods from appetizers to desserts.
My third major work, as contributing food writer for _Eating for Autism_ , pushed my gluten-free food making knowledge to the limits. Nutritious, dairy-free, soy-free, corn-free, refined sugar-free, etc. It forced me to rethink our gluten-free base flours once again. Ultimately a combination of brown rice flour and sorghum became a new proving ground for the special dietary guidelines suggested in this work. It was groundbreaking and delicious.
And, that brings us to today. I am delighted to tell you that the recipes in _The Ultimate Gluten-Free Cookie Book_ primarily use brown rice flour or sorghum flour. Most often they are used just one at a time. If you had asked me as little as two years ago if any heavy flour could be used to make a cookie that is light and fluffy, such as whoopie pies, I would have said, "Impossible." Today I know it is a reality!
Exploiting the unique characteristics of each of these flours makes "light and fluffy" possible. Also, "tender and crisp." The same goes for "soft and tender." These cookie qualities are all possible using nearly _any_ flour (or any blend, for that matter). The final chapter in this book uses other flours solo to make that point. Hopefully, those recipes will also enable you to use up that cupboard full of alternatives fondly called a tower of flours!
For good taste, great textures, good nutrition, and reasonable affordability, I am embracing the wonderful and hard-to-believe attributes of brown rice flour and sorghum flour.
# _**The Cost of Flours**_
From the CSA/USA Web site (www.csaceliacs.org), I gathered five well-known blend formulas. From the Bob's Red Mill Web site (www.bobsredmill.com), I gathered pricing information for types of flours sometimes used in gluten-free baking. The point in gathering this information is to show how purchase of multiple flours can quickly become quite an expense (not to mention utilize cabinet space!)
**TABLE 1.1** **THE COST OF FLOURS**
GENERAL BAKING MIX #1
**Carol Fenster**
(makes 2 cups)
**1 cup rice flour**
**½ to** **¾** **cup potato starch**
**¼** **cup tapioca starch**
_Initial cost for flours_ = $7.70; $0.62 per cup
GENERAL BAKING MIX #2
**Carol Fenster**
(makes 9 cups)
**3 cups garfava bean**
**2 cups potato starch**
**2 cups cornstarch**
**1 cup tapioca flour**
**1 cup sorghum flour**
_Initial cost for flours_ = $18.73; $0.91 per cup
ORIGINAL FORMULA
**Bette Hagman**
(makes 3 cups)
**2 cups rice flour**
**⅔** **cup potato starch**
**⅓** **cup tapioca starch**
_Initial cost for flours_ = $7.70; $0.66 per cup
FOUR FLOUR BEAN
**Bette Hagman**
(makes 3 cups)
**⅔** **cup garfava bean flour**
**⅓** **cup sorghum flour**
**1 cup cornstarch**
**1 cup tapioca starch**
_Initial cost for flours_ = $15.72; $0.93 per cup
FEATHERLIGHT
**Bette Hagman**
(makes 3 cups)
**1 cup rice flour**
**1 cup cornstarch**
**1 cup tapioca starch**
**1 tablespoon potato flour**
_Initial cost for flours_ = $12.64; $0.53 per cup.
Alternatively, a 1.5-pound bag of brown rice flour or sorghum flour costs about $3.00, or roughly $0.70 per cup.
And, finally, factor in the ratios of flours used. Blends are often substituted cup for cup; not so for my single-flour recipes. For example, a traditional chocolate cake may call for 1¾ cups of flour in addition to the cocoa. Two versions of chocolate cakes (in _You Won't Believe It's Gluten-Free!_ ) use just 3/4 cup of potato starch or 1 cup of rice flour. Food science tells us that our ratios do not need to mirror the rules of traditional baking. Gluten-free flours should have their own rules!
Obviously, we don't need to spend a lot on flours to bake delicious treats! Specifically, we only need one flour (or two if you like), a package of xanthan gum (used in very small quantity, so it lasts for a long time), and an adventurous spirit.
We don't need more blends. We don't need all of our nutritional needs met with unusual-tasting pseudograins. We don't need yet another flour discovery. We need common sense. Let us enjoy the whole-grain goodness of brown rice flour and sorghum flour. This book lets you set aside venturing into many flour directions (and avoid the, shall we say, interesting taste of many), and make some great cookies!
# _**Making a Great Gluten-Free Cookie**_
The cookie recipes in this book are, for the most part, quite easy to make. There is science behind the ratios, the choice of ingredients, and the order in which the ingredients are mixed.
To ensure success, first, measure the ingredients carefully. A scale provides more accurate measurements, but is not essential. Combine the sugar and fat and mix well. Scrape down the sides of the bowl.
Add the flour and mix well. This step is critical, even though your dough just looks like crumbles in the mixing bowl. Essentially, you are coating the individual flour particles with fat, thereby deterring the flour from stealing moistness from the finished cookie. This simple step greatly extends the moistness-life of any gluten-free cookie (or any other baked good for that matter!). This step also helps a cookie to retain its intended character. (For example, a crisp cookie can become soft in a day if this step is not followed!)
Add the remaining ingredients as directed. You don't have to worry too much about overbeating the batter as there is no gluten in our flours. It is the gluten that develops in traditional flour when you overbeat it, which can make a cookie tough. However, excess beating may cause increased viscosity of the xanthan gum, making the finished cookie hold shape a little stronger than desired. But the taste and texture should not change!
For rolled cookies, I strongly recommend a lightly oiled surface and rolling pin. We don't want to add any dryness (flour) to the exterior of our cookies! Gluten-free cookie dough may require a small amount of shaping with moist fingertips to have finished results that mirror their traditional counterparts.
Bake at the recommended temperature. Do not be tempted to speed along baking with a higher temperature. Browning and spread are both affected by temperature!
Finally, it is best to avoid substitutions in a recipe. If you can, try the recipe as written the first time and venture from there. The helpful hints that follow should help you as you become more adventurous in your baking.
# _**Helpful Hints**_
With each new cookbook I write, unwritten rules of gluten-free food theory become clear. I first disclosed a set of gluten-free food theories and helpful hints while speaking at a celiac conference in Dallas. It was my most popular handout, and requests for copies even followed me home. Following are my latest gluten-free cookie baking theories and helpful hints.
## **Fats**
1. Shortening makes for the crispest cookie. Butter makes for the richest and even a little softer cookie. Oil makes for a crisp and tender cookie. These are subtle differences, but very important depending upon the desired result.
2. Coating a flour with fat first dramatically extends the moisture-life of the baked item.
3. Use margarine with at least 80 percent fat if substituting for butter.
## **Sweeteners**
1. Sweeteners entirely change a cookie. Sugar makes for a dryer cookie. Honey makes for a soft cookie.
2. Sugar can be baked at a higher temperature than honey without burning. If using honey and you want a crisp cookie, you need a low temperature for a much longer time.
3. Honey tastes sweeter than corn syrup.
4. In some cookies, such as the animal crackers, you would think that you'd want sugar to produce that nice crisp texture. Not when you are baking with brown rice flour; using sugar throws you into a sugar cookie taste and texture. You must have honey, which adds the complication of how to get a cookie that's soft, but dry. Yes, I know they are crisp, but the starting dough is soft and the edges need to soften when baking. The honey also makes for tenderness.
## **Eggs**
1. Egg-free cookies benefit from additional moist ingredients such as pumpkin or applesauce.
2. Using honey or corn syrup makes eggs less necessary in a recipe.
## **Flours**
1. Tenderness is achieved with rice flour by using oil as the fat.
2. Rice flour carries the flavor of butter extraordinarily well, so less butter can be used.
3. Peanut butter must be viewed as both a fat and a flour. To oversimplify, consider 50 percent by volume as fat and 50 percent as solids (the latter, as you would a flour).
4. Lighten a recipe by using starch in place of some of the gritty flour. Or use extra egg whites, leavening, etc.
## **Binding**
1. Gluten-free dough wants to bind itself to fillings or toppings. While cookies will still be delicious assembled together, prebaking can help maintain the separateness of components. A gluten-free dough that has corn syrup as the sweetener is not as likely to bind itself to fillings. (See the Rugalach recipe, page 138.)
2. Cornstarch as a thickener is not as effective in the presence of an acid (such as lemon).
3. Xanthan gum does not exhibit strong binding properties when egg yolks are the only "liquid" in a recipe.
4. Xanthan gum exhibits stronger binding properties to a water-based liquid than to oil. (This may be the reason some of you notice a spongier texture in commercially available mixes, when replacing part of the fat with applesauce—acting as a water-based liquid.)
5. Xanthan gum exhibits stronger binding properties to corn syrup than to honey.
6. Use less xanthan gum if switching from dairy- to water-based liquids.
7. Xanthan gum has greater viscosity in the bowl than does guar gum. Guar gum has a greater binding effect in the oven. Guar gum can have a laxative effect.
8. Substitute 1¼ teaspoons of guar gum for 1 teaspoon of xanthan gum.
9. Not all xanthan gums are created equal! Surprisingly, some brands have less binding capabilities than others! Even the same brand can have an off batch on occasion.
10. Not all xanthan gum is grown on a corn base. Historically, some xanthan, grown on sugar cane base, has been available. Recently, it has been difficult to locate.
## **Rising**
1. Not all baking powders are created equal! Rumford provides more action in the bowl as compared to Clabber Girl. Clabber Girl provides more action in the oven. You may need to reduce baking powder by 25 percent if not using Rumford.
2. Baking soda is four times stronger than baking powder and needs an acid to activate. Acids include vinegar, yogurt, brown sugar, cocoa powder, and lemon juice.
## **Miscellaneous**
1. Use the same brand of ingredients as does the writer of the recipe, whenever possible, especially when first trying an author's work.
2. Generally speaking, don't worry about overbeating doughs and batters as there is no gluten to develop. Note, however, extensive beating can increase the viscosity of xanthan gum, making for a thicker dough or batter.
3. The use of sugar and/or baking soda in a recipe helps browning.
2
_**Kitchen and Baking Tips**_
**Even if you are an experienced baker,** I hope you will take a few minutes to peruse this chapter! While some things may be obvious, I hope to give you a feel for the methods and equipment used frequently in this book to make great gluten-free cookies.
# _**Cleanliness and Preventing Cross-Contamination**_
Without safeguards, lovingly prepared food can be dangerous to a person living with gluten-intolerance or celiac disease, as even the smallest traces of gluten must be avoided. To avoid cross-contamination, it is critical that your kitchen be a safe preparation and cooking area. The thought of wayward crumbs or flying flour dust is enough to make someone with celiac disease panic! Cleanliness and food safety are easy to implement, but you will find yourself seeking out gluten in unusual places.
In thoroughly cleaning a kitchen, I prefer to start up high and end low. (Don't forget to clean inside drawers and cabinets!) Soap and water are the basics you'll need.
If your kitchen is not already dedicated gluten-free, I strongly suggest taking the following steps before making gluten-free cookies:
1. Allow your kitchen to be free of traditional wheat baking for at least overnight. Any lingering flour dust must settle and be cleaned away.
2. Remove all the items from the entire work area, including the sink, and wash your dedicated work area well. This would be a good time to be sure you wear a fresh apron (if your old one was used for baking with wheat). Don't forget fresh towels and potholders, too!
3. Thoroughly wash the mixer and beaters.
4. Gather nonporous utensils, bowls, measuring spoons and cups, and baking sheets. Wood and scratched plastic may harbor tiny gluten-particles—don't forget the rolling pin! (Nonstick surfaces are controversial as they can become scratched and cross-contaminated; it is better to just avoid them.) Wash all of these items well. Kitchen drawers are notorious for hiding food crumbs—clean them out and wipe them down. Storage containers should also be washed well.
5. Gather the ingredients for your recipe. All the ingredient packages should be unopened and free of any wayward traditional baking dust. (Note that just a little wayward traditional flour on a measuring cup or spoon can easily cross-contaminate other ingredients. A separate, safe storage area should be made for _all_ ingredients used in gluten-free baking!) Also consider purchasing parchment paper. It is good to use on baking sheets (and cooling racks) to avoid potential cross-contamination from prior use with gluten-containing foods.
6. If in doubt about whether an ingredient is gluten-free, do not proceed! Read all ingredient labels carefully or call the manufacturer if you are unsure. Wheat should be clearly disclosed on any ingredient label in the United States.
7. Do not make any gluten-containing foods in the kitchen area until the cookies are packaged away. This is especially critical for flour dust! It travels despite the best of intentions.
# _**The Cookie Pantry**_
_Flours._ With the exception of cornstarch and oats, all flours utilized in this book have been tested using Bob's Red Mill brand. These and similar high-quality gluten-free flours are available in larger grocery stores, online, and in numerous health food stores. Oats and oat flour must be obtained from a safe source (see the Appendix, page 180).
Unfortunately, the growing and harvesting of most oats involve fields, mills, and/or transport trucks that are also used for wheat, rye, or barley. That cross-contamination makes most oats unsafe for the gluten-free diet.
_Baking Powder._ Rumford brand has been utilized in all recipes.
_Baking Soda._ Arm & Hammer brand has been utilized in testing all recipes in this book.
_Cocoa Powder._ Hershey's cocoa has been my brand of choice. I personally prefer it to the flavor of Nestlé's.
_Cream Cheese._ Philadelphia is my favorite brand.
_Decorations._ Be sure to read the labels on icings and sprinkles. While writing these recipes, I noticed for the first time that some icings sold in tubes contain wheat. Careful!
_Eggs._ Any brand, size large.
_Fats._ All the recipes have been tested using Crisco brand shortening, (any brand) canola oil, and (any brand) lightly salted butter. If you must substitute margarine for butter, it should contain 80 percent fat.
_Milk._ 2% milk is my preference, but any percentage should be fine.
_Salt._ Any brand. Iodized salt is sometimes avoided by people having the skin manifestation of celiac disease (dermatitis herpetiformis). If you have dermatitis herpetiformis, you may wish to discuss with your health-care provider whether to use iodized salt.
_Spices, Flavorings, Extracts._ When possible, I prefer to use extracts instead of artificial flavorings. In any case, McCormick is my preferred brand. This company has a reputation for clearly labeling its product ingredients.
_Sweeteners._ White, brown, and confectioners' sugars: generally speaking, any brand is fine. However, confectioners' sugar is usually processed with cornstarch. Although unlikely, it is worth double-checking to be sure that wheat doesn't appear on the ingredient listing. Honey and molasses: any brand is generally fine.
_Yogurt._ Low-fat plain yogurt was used in testing these recipes. I avoid nonfat yogurt as the resultant cookies seem a bit off to me.
# _**Equipment**_
Below is a list of the equipment I use in my kitchen. Most of my things are name brand but affordable. And as a side note, I highly recommend giving a baking-themed gift to anyone who has been diagnosed with dietary restrictions. A new rolling pin and a great sugar cookie recipe . . . a pizzelle iron and a bag of coffee . . . new spatulas and a bag of xanthan gum . . . all would touch the spirit.
_Baking Pans and Sheets._ I use aluminum or stainless steel baking pans and sheets. These are widely available in kitchen supply houses, craft stores, larger cooking departments, and kitchen specialty stores. I avoid nonstick surfaces, as they can become scratched and cross-contaminated. Nonstick cooking spray is ideal for lightly greasing the pans. Parchment paper works well with cookies and safeguards against cross-contamination from pans (especially nonstick) previously used for gluten-containing foods.
_Blender._ A blender is used in just a few recipes in this book (such as Fig Newton-Style Cookies, page 97). A handheld stick blender is ideal for this purpose. My Farberware model is now several years old and does a great job.
_Cooling Racks._ Any brand is fine. Be sure to cover the cooling racks with foil or parchment if they have been used previously for cooling gluten-containing foods. There are so may little nooks and crannies where cross-contamination could be an issue.
_Cutting Boards._ My collection of cutting boards is mostly plastic. Wooden boards are highly suspect for cross-contamination and should not be used (unless dedicated to only gluten-free use). Plastic boards with scratches may be suspect as well.
_Measuring Spoons and Cups._ Any name-brand measuring spoons and measuring cups are fine. I prefer metal measuring spoons and cups for durability and ease of cleaning. I use a Pyrex glass measuring cup. You may wish to avoid any plastic utensils (that have been previously used with gluten-containing foods) with scratches, due to possible cross-contamination.
_Microwave Oven._ Any brand. A microwave needs to be very, very clean! It is ideal for melting chocolates and making rice cereal bar cookies. Be sure to use microwave-safe bowls, cups, and pans; Pyrex is an ideal brand.
_Mixer._ I use a KitchenAid stand mixer. However, any strong hand mixer should work well with the recipes in this book. Also, nearly all of these recipes would do just fine mixed by hand.
_Mixing Bowls._ I use the large metal mixing bowl that came with my KitchenAid mixer. Metal and glass bowls are preferred for durability and ease of cleaning. You may wish to avoid any plastic bowls (used with gluten-containing foods) with scratches, due to possible cross-contamination.
_Rolling Pin._ I use a wooden "French" rolling pin. There are no mechanics to this rolling pin. It is simply a tapered piece of wood. It is one of my favorite kitchen tools. This rolling pin should be dedicated to gluten-free baking or covered with foil if previously used with gluten-containing doughs.
_Scale._ I use a Pelouze postage scale. Baking by weight is more accurate and faster. If using a scale, you will rarely need a measuring cup, as this scale (and now many others on the market) allow you to zero-out the weight of the bowl and weigh anew with each ingredient. The recipes in this book include weight measurements for dry ingredients.
_School Supplies._ A simple ruler and kitchen scissors come in handy again and again.
_Specialty Items._ Rosette irons, pizzelle makers, mini cookie cutters, etc. All of these items are available in larger cooking departments or kitchen specialty stores.
_Utensils._ Any brands and materials are fine as long as they are dedicated to gluten-free cooking only. Do not use wooden utensils, as these may be cross-contaminated. Spatulas can hide bits of gluten where the head meets the handle. Thorough cleaning or replacement is essential to avoid cross-contamination.
3
_**Drop Cookies**_
**Drop cookies are at the heart of any cookie platter.** They are quick to make and their flavors and textures vary tremendously. On occasion we all fall back on our longtime favorites, like Chocolate Chip or Oatmeal Raisin, and with good cause—they are delicious! However, I hope this chapter will expand your drop cookie repertoire. Perhaps you will be tempted by the soft and tender Almond Flower Cookies garnished with a splay of sliced almonds. Perhaps you'll make Coconut Macaroons. Or perhaps you will enjoy one of my new favorites, Lemon-Poppy Seed Cookies.
No matter which drop cookie you choose first, I trust you will be rewarded with oohs and aahs for your effort.
# _**Almond Flower Cookies**_
**brown rice flour and almond meal**
MAKES ABOUT 25 COOKIES
_This traditional version has sliced almonds arranged on top to resemble flowers._
_These cookies are soft and moist._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**2 eggs**
**½** **cup almond meal, 60 grams**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon almond extract**
TOPPING (OPTIONAL):
**¼ cup sliced almonds**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness. Arrange the almond slices on top into a flower or other nice pattern.
Bake the cookies for 8 to 10 minutes, until there is the slightest hint of color; the tops will be dry. Let cool on wire racks before serving.
# _**Applesauce-Raisin Cookies**_
**brown rice flour**
MAKES ABOUT 40 COOKIES
_It is such fun to have great ingredients in a tasty cookie. No guilt here!_
_Light in flavor and texture, they are sure to please._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**½** **cup applesauce**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**¼** **teaspoon vanilla extract (optional)**
**½** **teaspoon ground cinnamon**
**⅓** **cup roughly chopped raisins, 55 grams**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be soft.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Banana-Nut Cookies**_
**brown rice flour**
MAKES ABOUT 40 COOKIES
_Inspired by one of my family's favorite snacks ... banana bread._
_These cookies are an understated treat._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
1 **½** **cups brown rice flour, 185 grams**
**1 egg**
**1 (4-ounce) jar Beech-Nut baby food bananas (stage 2), 105 grams**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1** **¼** **teaspoons xanthan gum**
**½** **teaspoon vanilla extract**
**½** **cup chopped pecans, 60 grams**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the flour and beat well. Add the remaining ingredients and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Continue beating until the dough comes together; it will be soft.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness. The cookies will hold this shape during baking, so you can place the cookies close together on the pan.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Chocolate Cookies with White Chips**_
**brown rice flour**
MAKES ABOUT 35 COOKIES
_This cookie is the reverse of a chocolate chip cookie. It has the slightly drier_
_texture of a store-bought cookie. It's perfect with a cup of tea or glass of milk._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**¼** **cup unsweetened cocoa powder, 20 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**¾** **cup white chocolate chips, 130 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together. The dough will be very soft and a little sticky.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for about 9 minutes, until the tops are dry. Let cool on wire racks before serving.
# _**Chocolate Chip Cookies #1**_
**brown rice flour and cornstarch**
MAKES ABOUT 30 COOKIES
_A really tasty chocolate chip cookie!_
_Enjoy them hot from the oven . . . what a treat!_
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar (dark if possible), 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓** **cup cornstarch, 40 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 cup semisweet chocolate chips, 160 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except for the chocolate chips, and mix well. Stir in the chocolate chips.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness, for a better cookie shape. Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Chocolate Chip Cookies #2**_
**sorghum flour**
MAKES ABOUT 24 COOKIES
_Unlike most chocolate chip cookies, this recipe calls for regular sugar_
_instead of brown sugar. The whole-grain taste of the sorghum does a_
_fabulous job in making these cookies taste just right._
**⅓** **cup oil, 65 grams**
**2 tablespoons butter**
**½** **cup sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 cup semisweet chocolate chips, 160 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil, butter, and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together. The dough will be soft and seem oily.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for 8 to 10 minutes, until the edges are lightly browned. Let cool on wire racks before serving.
# _**Coconut Macaroons**_
**brown rice flour**
MAKES ABOUT 20 COOKIES
_This recipe is an adaptation of the Coconut Macaroons recipe on the back of the_
_Baker's Angel Flake Coconut bag. I don't want you to miss out on this tasty_
_cookie when making them is so easy! If you are substituting another brand of_
_coconut, please note that this coconut is sweetened._
**1 (7-ounce) package sweetened flaked coconut (2⅔ cups)**
**⅓** **cup sugar, 75 grams**
**2 tablespoons brown rice flour**
**Pinch of salt (scant ⅛ teaspoon)**
**Pinch of xanthan gum (scant** ⅛ **teaspoon)**
**2 egg whites**
**½** **teaspoon vanilla or almond extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine all the ingredients in the order given. Beat well. The mixture will look like loose, moistened coconut. Drop rounded teaspoonfuls of the dough onto the prepared pan.
Bake the cookies for 15 to 20 minutes, until the edges are golden brown. Let cool on wire racks before serving.
# _**Cranberry Cookies**_
**sorghum flour**
MAKES ABOUT 24 COOKIES
_These cookies are inspired by a Girl Scout favorite,_
_Thank U Berry Munch. The gluten-free version produces tender,_
_understated cookies with a hint of whole-grain flavor._
**⅓** **cup oil, 65 grams**
**2 tablespoons butter**
**½** **cup sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1** **½** **teaspoons vanilla extract**
**⅓** **cup finely chopped dried sweetened cranberries, 40 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil, butter, and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together. The dough will be soft and seem oily.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, flatten them to ¼-inch thickness.
Bake the cookies for 8 to 10 minutes, until the edges are lightly browned. Let cool on wire racks before serving.
# _**Ginger Spice Cookies**_
**sorghum flour**
MAKES ABOUT 24 COOKIES
_These cookies are tender and spicy, yet mellowed by a bit of vanilla_
_flavoring. Omit the vanilla for more "bite."_
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon ground ginger**
**1** **½** **teaspoons pumpkin pie spice**
**¼** **teaspoon vanilla extract**
TOPPING:
**2 tablespoons sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together. The dough will look like a traditional cookie dough.
Drop rounded teaspoonfuls of the dough onto the prepared pan. For the topping, place the 2 tablespoons of sugar in a bowl. Dip the bottom of a glass into water to moisten it, then dip it into the sugar. Use the sugared glass to press the dough balls to ⅛- to ¼-inch thickness.
Bake the cookies for 8 minutes, until they just begin to brown at the edges. These cookies will crisp a little during cooling. Let cool on wire racks before serving.
# _**Hermits**_
**sorghum flour**
MAKES ABOUT 24 COOKIES
_A classic combination of coffee, spices, raisins, and nuts._
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1** **½** **teaspoons pumpkin pie spice**
**1** **½** **teaspoons instant coffee dissolved in 1 teaspoon water**
**¼** **teaspoon vanilla extract**
**½** **cup raisins, 80 grams**
**½** **cup chopped pecans, 60 grams**
TOPPING (OPTIONAL):
**2 tablespoons confectioners' sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together. The dough will look like a traditional cookie dough.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moist fingertips, press them to ⅛- to ¼-inch thickness.
Bake the cookies for 8 minutes, until they just begin to brown at the edges. The cookies will crisp a little during cooling. Let cool on wire racks before serving.
Dust the tops of the cookies with confectioners' sugar if desired.
# _**Hot Chocolate Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_This cookie tastes like hot chocolate with tiny marshmallows melted on top._
_I hope you enjoy these as much as I enjoyed creating them. They are slightly_
_crisp on the outside and soft inside._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **¼** **cups brown rice flour, 155 grams**
**1 egg**
**½** **cup chocolate syrup (such as Hershey's)**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**⅓** **cup finely chopped mini marshmallows**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except the marshmallows, and beat well. Continue beating until the dough comes together; it will be soft and sticky. Gently mix in the marshmallows.
Drop rounded teaspoonfuls of the dough onto the prepared pan.
Bake the cookies for about 9 minutes, until the edges begin to brown. Let cool for a minute or so on the baking sheet to make the cookies easier to remove. (The cookies will flatten during cooling.) Let cool on wire racks before serving.
# _**Lace Cookies, Oatmeal**_
**oats**
MAKES ABOUT 20 COOKIES
_Suggested by my good friend Theresa, I've included these tasty cookies! This_
_dough will spread into a lacy, crisp cookie. If you work quickly, you can shape_
_them into tubes. If not, enjoy them flat. Although not traditional, I've included_
_a bit of vanilla to soften the flavor of this cookie._
**1** **¼** **cups rolled oats, 125 grams**
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1 egg**
**¼** **teaspoon salt**
**1 teaspoon baking powder**
**Pinch of xanthan gum**
**½** **teaspoon vanilla extract**
Preheat the oven to 350°F. Liberally grease a cookie sheet or a sheet of foil placed on the cookie sheet.
Place the oatmeal in a blender. Process until a third of the oatmeal is powdery; the rest may be of varying sizes.
Pour the oatmeal into a mixing bowl. Add the remaining ingredients and mix very well. A sticky-looking dough will form.
Place the dough by the tablespoonful onto the prepared pan. With moist fingertips, press the dough to ⅛-inch thickness.
Bake the cookies for 6 to 9 minutes, until they are lightly browned and appear cooked in the center. Keep the cookies on the pan for a minute or two to set. Gently ease the cookies from the cookie sheet with a spatula. The cookies will be pliable, yet have real substance. Let cool on wire racks before serving.
# _**Lemon-Poppy Seed Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_These cookies are inspired by lemon-poppy seed pound cake. A delicate cookie_
_full of poppy seeds and topped with an exceptional lemon glaze._
**⅓** **cup oil, 65 grams**
**½** **sugar, 100 grams**
**1** **¼** **cups brown rice flour, 155 grams**
**1 egg**
**¼** **baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**2 tablespoons frozen lemonade concentrate**
**1 teaspoon lemon extract**
**1 tablespoon poppy seeds**
TOPPING:
**½** **cup confectioners' sugar**
**2 tablespoons frozen lemonade concentrate**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together. The dough will be quite thick.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for 8 to 10 minutes, until they are lightly browned at the edges. Let cool on wire racks. Mix the confectioners' sugar with the 2 tablespoons of frozen lemonade concentrate. Drizzle the topping over the cookies.
# _**Molasses Cookies**_
**brown rice flour**
MAKES ABOUT 20 COOKIES
_Soft, moist, and lightly spiced. A hint of glaze adds to this favorite cookie,_
_should you like them extra sweet._
**⅓** **cup oil, 65 grams**
**¼** **cup sugar, 50 grams**
**1** **½** **cups brown rice flour, 185 grams**
**¼** **cup unsulfured molasses, 85 grams**
**1 egg**
**½** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**½** **teaspoon pumpkin pie spice**
**½** **cup raisins (optional), 80 grams**
TOPPING (OPTIONAL):
**1 cup confectioners' sugar**
**About 4 teaspoons milk**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough is well blended.
Drop rounded tablespoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks.
If desired, combine the topping ingredients to form a glaze and lightly ice the tops of the cookies.
# _**Mrs. Fields- or Neiman Marcus-Style Cookies**_
**sorghum flour and oats**
MAKES ABOUT 36 COOKIES
_Packed full of the good stuff you expect in these popular cookies. I chose to use_
_pecans in this recipe because it is my favorite nut for baking, but use your_
_favorite. Despite using healthy, whole grains, I cannot call these cookies_
_healthy. They are decadent. I urge portion control!_
**⅓** **cup oil, 65 grams**
**2 tablespoons butter**
**½** **cup brown sugar, 100 grams**
**¾** **cup sorghum flour, 100 grams**
**½** **cup rolled oats, pulsed in a blender to make smaller pieces of all sizes, 50 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**½** **cup semisweet chocolate chips, 80 grams**
**½** **cup chopped pecans, 50 grams**
**2 ounces milk chocolate (from a bar), grated (just over ½ cup)**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil, butter, and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except the chocolates and nuts, and mix well. Continue beating until the dough comes together. Mix in the chocolates and nuts. The dough will be soft and seem oily.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼- to ⅓-inch thickness.
Bake the cookies for 8 to 10 minutes, until the edges are lightly browned. Let cool on wire racks before serving.
# _**Oatmeal-Raisin Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_Rather than having you purchase individual spices for "spiced" cookies in this_
_book, I have opted for pumpkin pie spice, saving space and budget._
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**½** **cup rolled oats, pulsed in a blender to make smaller pieces, 50 grams**
**½** **cup raisins, 80 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**½** **teaspoon vanilla extract (optional)**
**1 teaspoon pumpkin pie spice**
**3 tablespoons water**
TOPPING (OPTIONAL):
**½** **cup confectioners' sugar**
**About 2 teaspoons milk**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks.
If desired, combine the topping ingredients to form a glaze. Drizzle it over the tops of the cookies.
# _**Peanut Butter Blossom Cookies**_
**brown rice flour and cornstarch**
MAKES ABOUT 26 COOKIES
_A classic peanut butter cookie with a milk chocolate kiss right in the middle. The_
_cookie has a light tenderness to it that blends perfectly with the understated milk_
_chocolate flavor._
**¼** **cup creamy peanut butter, 65 grams**
**2 tablespoons oil, 20 grams**
**½** **cup brown sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓** **cup cornstarch, 40 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**2 tablespoons water**
TOPPING:
**1 to 2 tablespoons sugar**
**26 milk chocolate kisses**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the peanut butter, oil, and sugar. Beat well. Add the brown rice flour and cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together.
Shape the dough into 1-inch balls. Roll them in sugar and place them on the prepared pan. Press the tops to flatten them slightly.
Bake the cookies for 10 to 12 minutes, until the edges begin to have a hint of browning. Remove the sheet from the oven. Immediately press a chocolate kiss into the center of each cookie. Let cool on wire racks before serving.
# _**Pecan Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_Whereas a Pecan Sandie is a dry, tender cookie, this is a soft, tender cookie._
**⅓** **cup shortening, 70 grams**
**½** **cup brown sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 egg, plus 1 egg yolk**
**½** **cup pecan meal, 40 grams**
**¼** **teaspoon baking soda**
**½** **teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a baking sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well; the dough will be quite thick.
Drop rounded teaspoonfuls of the dough onto the prepared pan.
Bake the cookies for 8 to 10 minutes, until there is the slightest hint of color; the tops of the cookies will be dry. Let cool on wire racks before serving.
# _**Pecan Lace Cookies**_
**pecan meal**
MAKES ABOUT 20 COOKIES
_A delicate version of lace cookies. They are delicious, bright, crisp on the edges,_
_and almost nougatlike in the middle. Their appearance is_
_more like a wafer than a lace cookie._
**¾** **cup finely chopped pecans, 85 grams**
**¼** **cup pecan meal (flour), 20 grams**
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**2 eggs**
**¼** **teaspoon salt**
**Pinch of xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Grease a cookie sheet very well.
Combine all the ingredients and mix very well. The dough will need a few minutes of beating to develop body; this is important. As you beat the mixture, it will first look like a thick table syrup, then, with continued beating, a sticky-looking dough will form.
Place the dough by the tablespoonful onto the prepared pan. With moistened fingertips, press the dough to a scant ⅛-inch thickness.
Bake the cookies for 6 to 9 minutes, until they are lightly browned and appear cooked in the center. Keep the cookies on the pan for a minute or two to "set." Gently ease them from the cookie sheet with a spatula. The cookies will be pliable, yet have real substance. Let cool on wire racks before serving.
**Note: If you have trouble finding pecan meal, you can grind pecans (with brief pulses) in a blender or coffee mill to a flourlike consistency.**
# _**Sour Cream Breakfast Cookies**_
**brown rice flour**
MAKES ABOUT 35 COOKIES
_Inspired by that great breakfast coffee cake, these cookies are a terrific excuse_
_for eating breakfast on the run! They are rich, moist, and flavorful._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**½** **cup sour cream**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**3 tablespoons brown sugar**
**½** **teaspoon ground cinnamon**
**2 tablespoons butter, 30 grams**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining cookie ingredients and beat well. Continue beating until the dough comes together; it will be soft. Set aside.
In a separate bowl, mix together the brown sugar, cinnamon, and butter to form a crumble. (If the butter is soft, the crumble may not crumble at all. This is okay.) Set aside.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness. Place a small amount of crumble mixture on top of each cookie.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
**Note: You may be tempted to just fold the topping crumble mixture into the dough mixture. Unfortunately, this creates areas where the topping will run onto the baking sheet. Adding the topping as directed will give you the desired result.**
4
_**Bar Cookies**_
**Are you ever in a hurry for something wonderful?** This chapter is full of delicious bar cookies. While I am a huge fan of any cookie dark with chocolate (such as my Decadent Brownies!), three of my favorites in this chapter are Carrot Cake Bars, Jam Bars, and White Chocolate Bars. And if you are in a super hurry, try either version of the Rice Cereal Bars. Sure, they are traditionally fast to make in the first place, but you can microwave the marshmallow mixture in just two minutes! Not even a messy pot to clean. I've topped the traditional version with just a little chocolate . . . which makes it so much better than you can imagine.
I used to be in a rut, making just one or two favorite bar cookies, but not anymore. I hope you find new favorites here, too.
# _**Almond Layered Cookie Bars**_
**brown rice flour and almond meal**
MAKES 15 COOKIES
_A tasty cookie bar full of good stuff! Let the bar cool fully to allow the structure_
_to set. Then enjoy the multitude of tempting layers!_
**½** **cup butter, 110 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 egg**
**½** **cup almond meal, 60 grams**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**1 (14-ounce) can sweetened condensed milk**
**1 cup sweetened flaked coconut, 120 grams**
**1 (4-ounce) package sliced almonds**
**1** **½** **cups chocolate chips, 240 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and continue to beat well. Add the remaining dough ingredients and mix well. The cookie dough will be heavy.
Press the dough evenly into the bottom of the prepared pan. Bake for about 15 minutes, until golden brown. Layer with the condensed milk, coconut, almonds, and chocolate chips, and bake for an additional 25 minutes. The toppings will take on a bit of color.
Let cool completely and cut into bars.
# _**Apple Crumble Bars**_
**brown rice flour**
MAKES 15 COOKIES
_This cookie bar is fashioned after those found in a little bakery in_
_Shepherdstown, West Virginia, a pretty college town near my home that is_
_peppered with quaint restaurants. Although not called for in the recipe, a few_
_tablespoons of finely chopped raisins would be a very nice addition to the_
_topping. These cookies are sweet!_
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **¼** **cups brown rice flour, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 tablespoon water**
**1 teaspoon vanilla extract**
**½** **teaspoon ground cinnamon**
TOPPING:
**1 apple (a sweet variety such as Gala or Golden Delicious), peeled and diced small (about 1 cup)**
**¼** **cup sugar**
**¼** **cup water**
**1** **½** **teaspoons cornstarch or potato starch**
**¼** **teaspoon ground cinnamon**
**⅓** **of dough reserved from above (135 grams)**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and beat well. Continue beating until the dough comes together; it will be quite thick.
Press two-thirds of the dough evenly into the prepared pan. Set aside.
In a microwave-safe bowl, combine all the topping ingredients, except the retained dough. Stir well. Microwave on HIGH for about 2 minutes, until the apples are soft and the sauce is clear and thick. Stir and check the topping after 1 minute. Spread the mixture over the top of the dough in the pan.
Top with bits of the remaining dough. (It should look crumbly.) Bake for 20 to 25 minutes, until lightly browned and a toothpick tests cleanly. Let cool before slicing and serving.
# _**Blondies #1**_
**brown rice flour**
MAKES 15 COOKIES
_This cookie is great even without the chocolate topping,_
_but I just couldn't resist._
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1** **¼** **cups brown rice flour, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**2 tablespoons water**
**1 teaspoon vanilla extract**
**½** **cup chopped pecans, 60 grams**
TOPPING:
**⅔** **cup mini chocolate chips, 60 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and beat well. Continue beating until the dough comes together; it will be quite thick.
Press the dough evenly into the prepared pan. Bake for about 20 minutes, until lightly browned and a toothpick tests cleanly.
While still hot, spread the chocolate chips over the top and cover with foil or a baking sheet. When the chocolate is melted, spread it across the tops of bars to coat evenly. Let cool completely and cut into bars.
# _**Blondies #2**_
**sorghum flour**
MAKES 18 COOKIES
_A dense, moist, brownielike bar with nuts, but not chocolate. I do include a_
_little melted chocolate to drizzle over the tops, but you may resist if you like._
_These cookies are delicious at room temperature, but not as much when warm._
**⅓** **cup oil, 65 grams**
**2 tablespoons butter**
**½** **cup brown sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**½** **cup chopped pecans or other nuts, 60 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**2 ounces semisweet or milk chocolate, melted**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil, butter, and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together; it will be a heavy batter.
Spread the dough evenly in the prepared pan. Bake for 20 minutes, until a toothpick tests cleanly and the edges begin to brown (and pull away from the sides of the pan).
Let cool. Cut into 18 bars. Drizzle melted chocolate over the top of the bars, if desired.
# _**Carrot Cake Bars**_
**brown rice flour**
MAKES 15 COOKIES
_This cookie bar has carrots, raisins, and nuts. I like these cookies without the_
_frosting, but have included a little cream cheese icing if you want to be spoiled!_
_By combining shortening with the cream cheese, the sticky sweetness of_
_traditional icing is avoided._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **¼** **cups brown rice flour, 155 grams**
**1 egg, plus 1 egg yolk**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1** **½** **teaspoons pumpkin pie spice**
**½** **cup packed grated carrots, 75 grams**
**½** **cup raisins, 80 grams**
**½** **cup chopped walnuts, 35 grams**
ICING:
**2 tablespoons shortening, 25 grams**
**½** **cup confectioners' sugar, 60 grams**
**3 ounces cream cheese, 85 grams**
**¼** **teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and beat well. Continue beating until the dough comes together; it will be quite thick.
Press the dough evenly into the prepared pan. Bake for 20 to 25 minutes, until lightly browned and a toothpick tests cleanly. Let cool.
To make the icing, mix the icing ingredients until well blended. Spread over the cookies. Cut into bars.
# _**Cheesecake Bars**_
**brown rice flour**
MAKES 15 COOKIES
_Enjoy these rich bars in small proportion! They have the richness of cheesecake_
_crossed with the tender crumb of a cookie._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**2 egg whites**
**1** **½** **cups brown rice flour, 185 grams**
**½** **teaspoon salt**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**1 teaspoon xanthan gum**
**½** **teaspoon vanilla extract**
TOPPING:
**1 (8-ounce) package cream cheese**
**⅓** **cup softened butter, 70 grams**
**1** **¼** **cups confectioners' sugar, 150 grams**
**1 egg**
**½** **teaspoon vanilla extract**
**⅓** **cup raspberry jam (or other favorite)**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and mix well until the dough comes together.
Press the dough evenly into the prepared pan. Bake for 5 minutes, then set aside. Reduce the heat to 325°F.
In a small bowl, combine the topping ingredients, except the jam. Beat until creamy. Pour the topping over the cookie dough. Stir the jam to make it easier to spread and scatter tablespoons of jam over the topping. Using a knife, swirl the jam through the topping.
Bake for about 45 minutes, until the top layer is lightly browned and set. Let cool completely and cut into bars.
# _**Cranberry-Orange-Almond Granola Bars**_
**oats and brown rice flour**
MAKES 12 COOKIES
_Inspired by my gluten-free friends, these granola bars are full of feel-good_
_ingredients. Substitute your favorite dried fruits and nuts for mine. If it sounds_
_good, it probably will be. Please note that these bars are like a soft, heavy_
_oatmeal cookie when first baked. Let them mellow for a day or two and_
_they are much better!_
**⅓** **cup melted butter, 70 grams**
**½** **cup honey, 160 grams**
**2 cups rolled oats, 160 grams**
**½** **cup brown rice flour, 60 grams**
**¼** **teaspoon baking soda**
**½** **teaspoon xanthan gum**
**½** **cup chopped sweetened, dried cranberries, 50 grams**
**½** **cup sliced almonds, 35 grams, toasted**
**1 tablespoon orange zest**
Preheat the oven to 325°F. Lightly grease an 8-inch square baking pan.
In a large bowl, combine the butter and honey. Mix well. Add all the remaining ingredients and mix well. Pour the batter into the prepared pan and press firmly. Bake for 20 to 25 minutes until lightly browned. Let cool and cut into bars.
**Notes: To toast sliced almonds, spray a small skillet with nonstick spray and heat to medium heat. Add the almonds and fry until lightly browned. Stir frequently and watch carefully to avoid burning the almonds. The almonds should be lightly browned and fragrant.**
**To make orange zest without a zester, just peel a bit of the outermost edge of an orange with a potato peeler. Then cut into very narrow strips, using a knife.**
# _**Decadent Brownies**_
**sorghum flour**
MAKES 16 BROWNIES
_A little cakey, a little gooey when warm, a little dense, and, oh so chocolaty._
_These are especially good warm from the oven. (I'm thinking brownie sundaes!)_
**⅓** **cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**¼** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**½** **cup mini semisweet chocolate chips, 90 grams**
TOPPING (OPTIONAL):
**¼** **cup mini semisweet chocolate chips, 45 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and cocoa and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together; it will be stiff and a little oily.
Press the dough evenly into the prepared pan. Bake for 15 minutes, until the cookie tests cleanly with a toothpick. If desired, immediately after removing the pan from the oven, sprinkle the mini chocolate chips over the brownies and cover with foil or a baking sheet. Once the chips melt, spread with a knife to form a very thin chocolate topping. Let cool completely and cut into bars.
# _**Jam Bars**_
**brown rice flour**
MAKES 15 COOKIES
_This cookie bar is so much easier to prepare than a Danish, while every bit as_
_tasty. I use raspberry jam, but substitute your favorite for mine!_
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**2 tablespoons water**
**1 teaspoon vanilla extract**
TOPPING:
**⅓** **of dough reserved from above (135 grams)**
**½** **cup seedless raspberry jam**
**2 tablespoons confectioners' sugar**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and beat well. Continue beating until the dough comes together; it will be quite thick.
Press two-thirds of the dough evenly into the prepared pan. Spread the jam over the dough and top with bits of the remaining dough. (I like to flatten the bits of dough between greased fingertips and lay them on top of the jam, much like for a cobbler.) Bake for 20 to 25 minutes, until lightly browned and a toothpick tests cleanly. Dust with confectioners' sugar. Let cool completely and cut into bars.
# _**Lemon Bars**_
**brown rice flour**
MAKES 12 COOKIES
_Lemon curd atop a lemon cookie base. Frozen lemonade concentrate provides_
_fresh lemon flavor. Serving these with a garnish of lemon peel is very pretty._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**2 tablespoons frozen lemonade concentrate**
**1 teaspoon lemon extract**
TOPPING:
**1 cup water**
**½** **cup frozen lemonade concentrate**
**1 egg**
**¼** **cup cornstarch or potato starch, 35 grams**
**1 tablespoon confectioners' sugar**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be quite thick.
Press the dough evenly into the prepared pan. Bake for about 20 minutes, until lightly browned and a toothpick tests cleanly. Let cool.
In a microwave-safe cup, combine the topping ingredients, except for the confectioners' sugar, and mix well. Microwave on HIGH 1 minute at a time, for about 3 minutes, until the mixture is thick, stirring after each minute to blend well. Spread the lemon topping over the cooled cookie layer. Let cool.
Dust the top of the cookies with confectioners' sugar. Let cool completely and cut into bars.
# _**Peanut Butter-Chocolate Chip- Oatmeal Bars**_
**brown rice flour**
MAKES 15 COOKIES
_This cookie was inspired by Annie at my publisher's office. We were looking for_
_a perfect cookie to round out the recipes in the book. We opted for her favorite._
_It may become your favorite, too! Chocolaty and crumbly!_
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**¼** **cup peanut butter, 65 grams**
**½** **cup rolled oats, 50 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1** **¼** **teaspoons xanthan gum**
**1 teaspoon vanilla**
**½** **cup semisweet chocolate chips**
Preheat the oven to 350°F. Lightly grease an 8-inch square pan.
In medium-size bowl, combine oil and sugar. Beat well to combine. Add flour and beat well. Scrape down sides of the mixing bowl at least once during mixing. Add remaining ingredients, except chips, and beat well. Continue beating until the dough comes together. Mix in the chips. The dough will be crumbly and shiny.
Press the dough into the pan. Bake for approximately 20 minutes, until lightly browned and a toothpick tests clean.
# _**Raspberry-Cream Cheese Brownies**_
**brown rice flour**
MAKES 12 BROWNIES
_I didn't think a true brownie could be made with a brown rice flour base. Time_
_and experimentation has once again proven that almost anything is possible with_
_almost any gluten-free flour. These brownies are a little cakey, a little gooey,_
_with a swirl of raspberry and cheesecake. Yum!_
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together. The dough will be very soft.
Spread the dough evenly in the prepared pan and set aside.
In a separate mixing bowl, beat the cream cheese, egg yolk, confectioners' sugar, and vanilla until well blended. Drop the mixture by the tablespoonful onto the brownie dough, then drop the raspberry jam atop the cream cheese mixture. Use the back of a spoon to swirl it into the brownie dough. Smooth out the top (it will look messy).
RASPBERRY-CREAM CHEESE SWIRL:
**4 ounces cream cheese**
**1 egg yolk**
**½** **cup confectioners' sugar, 60 grams**
**½** **teaspoon vanilla extract**
**⅓** **cup seedless raspberry jam**
TOPPING:
**¾** **cup semisweet chocolate chips, 120 grams**
Bake for about 25 minutes, until the top is dry and a toothpick tests cleanly. While hot, sprinkle the chocolate chips on top of the brownies and cover with foil or a pan. Once melted, spread the chocolate over the brownies. Let cool completely and cut into bars.
# _**Rice Cereal Bars**_
MAKES 16 COOKIES
_This is a very well-known favorite cookie bar. Mine is a not-too-big batch_
_and I've used the microwave to speed up making it! (Just one pan from start to_
_finish, too!) I've included an optional chocolate topping._
**½** **(10** ½ **-ounce) bag mini marshmallows, 150 grams (3** **¼** **cups)**
**2 tablespoons butter, cut into small chunks**
**4 cups rice cereal**
TOPPING (OPTIONAL):
**1 cup chocolate chips, 160 grams**
Lightly grease a microwave-safe 8- or 9-inch pan (Pyrex is great). Place the marshmallows and butter in the pan. Microwave on HIGH for 1½ to 2 minutes to melt the marshmallows, stopping after 1 minute to stir. When melted, stir well to fully incorporate the butter. Add the rice cereal and mix in until fully coated. With oiled fingertips, press the mixture flat into the pan.
If desired, melt the chocolate chips in a microwave-safe cup or bowl for about 1½ minutes. (Chocolate chips can retain their shape even though melted; it is important to stir them to see if they are fully melted.) Stir well and spread over the top of the rice cookies. Let cool completely and cut into bars.
# _**Rice Cereal Bars, Peanut Butter**_
MAKES 16 COOKIES
_I've included a little melted chocolate to drizzle over the top_
_to make these bars look pretty._
**½** **(10½-ounce) bag mini marshmallows, 150 grams (3¼ cups)**
**⅓** **cup peanut butter**
**½** **teaspoon vanilla extract**
**3** **½** **cups rice cereal**
**¼** **cup chocolate chips, 40 grams (optional)**
Lightly grease a microwave-safe 8- or 9-inch pan (Pyrex is great). Place the marshmallows, peanut butter, and vanilla in the pan. Microwave on HIGH for 1½ to 2 minutes to melt the marshmallows, stopping after 1 minute to stir. When melted, stir well to fully incorporate the peanut butter. Add the rice cereal and mix in until fully coated. With oiled fingertips, press the mixture flat into the pan.
If desired, melt the chocolate chips in a microwave-safe cup or bowl for 30 to 45 seconds and drizzle on top of the bars. Let cool completely and cut into bars.
# _**Shortbread**_
**brown rice flour**
MAKES 12 COOKIES
_Great traditional shortbread has so few ingredients: just butter, sugar, flour, and_
_maybe a little salt. It is necessary to use a few extra ingredients to duplicate that_
_incredible cookie, but it is so worth it. This would make an incredible base for a_
_fresh fruit tart, too!_
**⅓** **cup butter, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1** **½** **teaspoons xanthan gum**
**½** **teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a 9-inch round baking pan or springform pan.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add all the remaining ingredients and mix well. The dough will form lots of small clumps but will not quite come together to form a ball; this is okay. Scrape down the bowl and mix just a little longer to be sure all is mixed well.
Transfer the dough to the prepared pan and press to form a solid, even layer at the bottom of the pan. Prick the dough with a fork.
Bake the shortbread for about 15 minutes, until it has the slightest hint of color at the edge of the pan. Let cool. Cut into pretty wedges before serving.
# _**Toffee Bars**_
**brown rice flour**
MAKES 25 COOKIES
_Made with a shortbread cookie base, this cookie is topped with_
_toffee and melted chocolate. Delish._
**⅓** **cup butter, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1** **½** **teaspoons xanthan gum**
**½** **teaspoon vanilla extract**
TOPPING:
**½** **cup brown sugar**
**⅓** **cup butter**
**¾** **cup chopped pecans, 120 grams**
**1 cup chocolate chips, 160 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. The dough will form lots of small clumps but will not quite come together to form a ball; this is okay. Scrape down the bowl and mix just a little longer to be sure all is mixed well.
Transfer the dough to the prepared pan and press to form a solid, even layer at the bottom of the pan. Prick the dough with a fork.
Bake the shortbread for about 15 minutes, until it has the slightest hint of color at the edge of the pan. Set aside while preparing the topping.
To make the topping, combine the brown sugar and butter in a small saucepan. Bring to a boil and cook for 7 minutes, stirring constantly. (This mixture becomes exceedingly hot, so be careful that it does not come into contact with your skin! Also, please know that the sugar and butter will blend together, despite not seeming to at first.) Pour this toffee mixture over the cookie base and spread it evenly.
Immediately sprinkle the nuts and chocolate chips over the hot toffee layer. Cover the pan with foil or a baking sheet to allow the chips to melt, then spread them lightly with a knife. Let cool. Cut these into small pieces, as these bars are quite rich.
# _**Trail Mix Bars**_
MAKES 16 COOKIES
_This recipe is essentially a rice cereal bar that is chock full of good trail mix_
_ingredients. I think you will like it much more than the cereal bars_
_available in most grocery stores. This recipe uses the standard chocolate, nuts,_
_and raisins, but you can use other dried fruits, nuts, and even coconut_
_to make the recipe your own._
**½** **(10** ½ **-ounce) bag mini marshmallows, 150 grams (3¼ cups)**
**2 tablespoons butter, cut into small chunks**
**3** **½** **cups rice cereal**
**¾** **cup raisins, 120 grams**
**¾** **cup M &M's**
**¾** **cup salted peanuts, 85 grams**
Lightly grease a microwave-safe 8- or 9-inch pan (Pyrex is great). Place the marshmallows and butter in the pan. Microwave on HIGH for 1½ to 2 minutes to melt the marshmallows, stopping after 1 minute to stir. When melted, stir well to fully incorporate the butter. Add the rice cereal, raisins, and M&M's. Mix in until fully coated. Stir in the peanuts. With oiled fingertips, press the cookie mixture flat into the pan. Let cool completely and cut into bars.
# _**Turtle Bars**_
**brown rice flour**
MAKES 15 COOKIES
_A tender cookie layer topped with a chocolate cookie layer full of caramel,_
_chocolate, and nuts, all covered with chocolate._
BASE LAYER:
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**½** **teaspoon salt**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**1 teaspoon xanthan gum**
**½** **teaspoon vanilla extract**
SECOND LAYER:
**⅓** **of dough from base layer**
**1 tablespoon unsweetened cocoa powder**
**2 tablespoons oil**
**1 egg yolk**
**5 soft caramel candies, chopped, 40 grams**
**¼** **cup chopped chocolate chips, 40 grams**
**¼** **cup chopped peanuts (optional), 40 grams**
TOPPING:
**½** **cup chocolate chips, 80 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining base layer ingredients and mix well, until the dough comes together. Press two-thirds of the dough (about 260 grams) evenly into the prepared pan. Set aside.
For the second layer, combine the retained third of dough with the cocoa, oil, and egg yolk. Mix well. Mix in the caramel, chocolate pieces, and nuts (if desired).
Bake for about 25 minutes. After removing from the oven, sprinkle the bars with the chocolate chips and cover the pan with foil to melt the chips. Once soft, spread the chocolate over the bars. Let cool completely and cut into bars.
# _**White Chocolate Bars**_
**brown rice flour**
MAKES 20 COOKIES
_A tender cookie layer topped with a white chocolate fudgelike topping. For extra_
_flavor, add a layer of chopped raisins or macadamia nuts before applying the_
_white chocolate top layer. These are very rich cookie bars._
**⅓** **cup butter, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1** **½** **teaspoons xanthan gum**
**1 teaspoon vanilla extract**
OPTIONAL MIDDLE LAYER:
**½** **cup chopped raisins (optional), 80 grams**
**½** **cup chopped macadamia nuts (optional), 65 grams**
Preheat the oven to 350°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining base layer ingredients and mix well. The dough will form lots of small clumps but will not quite come together to form a ball; this is okay. Scrape down the bowl and mix just a little longer to be sure all is mixed well.
Transfer the dough to the prepared pan and press to form a solid, even layer at the bottom of the pan. Add the layer of chopped raisins and/or nuts, if desired.
TOPPING:
**1** **½** **tablespoons cherry juice**
**1 cup white chocolate chips**
**½** **cup sweetened condensed milk**
In a microwave-safe cup, combine the cherry juice, white chocolate chips, and condensed milk. Microwave on HIGH for 1½ to 2 minutes, stirring frequently, until the chocolate melts and is of spreading consistency. Pour the mixture over the cookie layers, tilting the pan to reach the edges.
Bake for about 18 minutes, until the edges begin to lightly brown. Let cool completely and cut into bars.
5
_**Rolled and Piped Cookies**_
**Beautiful cookies!** This chapter is full of beautiful cookies! The Butter Cookies and Chocolate Butter Cookies are easy to pipe spritz-style cookies—classics. And for smiles that only S'mores can bring, there are Graham Crackers, traditional and chocolate! One of the prettiest cookies of all is the Linzer Sandwich Cookie with jam peeking out of a window dusted with confectioners' sugar.
The most important thing to remember when working with these special cookies is patience, whether you are placing the cookie dough in the fridge for easier handling, or rolling and cutting out shapes.
I have found it easiest to roll out cookies on a lightly oiled surface, using a lightly oiled rolling pin. I use nonstick spray to coat the surfaces. And, although it is certainly not necessary, granite is my surface of choice. It stays naturally cool to the touch. An 18-inch square of granite tile or smooth ceramic tile can be acquired from your local hardware store at minimal cost.
# _**Animal Crackers**_
**brown rice flour and cornstarch**
MAKES ABOUT 150 COOKIES
_I think I've eaten one too many cookies! I was testing the "real" Barnum's_
_Animal Crackers and noticed an underlying corn taste. I hadn't noticed that in_
_all my years of eating these treats, so I checked the label. Yes, yellow corn flour_
_is one of the ingredients. We'll use a little cornstarch since we already have the_
_grit from brown rice flour. These cookies are a little more tender than the_
_original, but are delicious—not to mention fun to make. Finding tiny cookie_
_cutters might be the hardest thing about making these cookies!_
**⅓** **cup oil, 65 grams**
**½** **cup light brown sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓** **cup cornstarch, 40 grams**
**2 egg yolks**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1** **½** **tablespoons water**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be soft and oily to the touch.
Roll out the dough to ⅛-inch thickness. Use tiny 1-inch cookie cutters to cut the cookies. The dough is very easy to work with, but quite soft. Place the cookies on the prepared pan.
Bake the animal crackers for 10 to 12 minutes, until the tops are lightly browned. Avoid both overbaking and underbaking. Let cool on wire racks before serving.
# _**Butter Cookies**_
**brown rice flour**
MAKES ABOUT 60 COOKIES
_Very buttery, delicate, not-too-sweet cookies._
_I use a cookie press to make these pretty cookies._
**½** **cup butter, 110 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg, plus 1 egg yolk**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Very lightly grease a cookie sheet. (Overgreasing will prevent the cookie dough from adhering during pressing. Wipe off extra oil if necessary.)
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be soft and almost creamy.
Place the dough in a cookie press and "shoot" the cookies onto the prepared pan. Alternatively, the dough may be piped from a pastry bag.
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Candy Cane Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_These cookies are cut from strips of colored dough that are rolled flat into a_
_striped mat, making for a prettier cookie than twisting strips of dough together._
_As with the Chinese Marble Cookies, this recipe avoids using butter. By using_
_oil (shortening in the case of the Chinese Marble Cookies) the clean essence of_
_other flavors shine._
**⅓** **cup oil, 65 grams**
**½** **cup plus 2 tablespoons sugar, 125 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**3 tablespoons water**
**½** **teaspoon vanilla extract**
**½** **teaspoon peppermint extract**
**About** **¼** **teaspoon red paste food coloring**
TOPPING:
**1 tablespoon sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together.
Divide the dough in half. Set half of the dough (about 250 grams) aside. Mix a little red food coloring into the remaining half of the dough until the desired color is reached. Refrigerate both batches of dough for at least 30 minutes, for easier handling.
Roll each section of the dough flat, about ¼inch thick. Cut the dough into thin strips and then lay down alternating colored strips side by side to form a tight, striped dough mat. Roll it slightly to be sure that the dough strips seal to each other. Finally, cut the dough at a 45-degree angle to form thin strips with a slanted angle of stripes showing. Cut the strips into 5-inch lengths and place them on the prepared pan. Gently bend the top of each 5-inch length sideways to form the arch of the candy cane. Sprinkle the tops lightly with plain sugar.
Bake the cookies for about 10 minutes, until the tops just begin to take on a little color. Let cool on wire racks before serving.
# _**Chinese Marble Cookies**_
**brown rice flour**
MAKES ABOUT 18 COOKIES
_I was introduced to Chinese Marble Cookies at a beautiful dance pavilion in_
_Maryland, thanks to Steve, a great dance instructor. These cookies have_
_chocolate "marbling" throughout a very light-colored cookie base. These cookies_
_taste like a cross between a sugar cookie and shortbread, with dark chocolate_
_thrown in for good measure. The character of the cookie changes entirely if_
_butter is substituted, so stick with shortening. I love butter,_
_but not in these cookies._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**½** **teaspoon vanilla extract**
**1** **½** **teaspoons water**
**1 square (1 ounce) unsweetened chocolate**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except for the chocolate, and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball; this is okay.
Melt the chocolate in a microwave-safe bowl or cup in a microwave on HIGH for about 1 minute. Pour it over the dough and stir it in with a few swirls of a fork just until a marbled effect occurs (do not overmix).
Roll out the dough to ¼-inch thickness and cut it with a 2½-inch round cookie cutter (or other cookie cutter of your choice).
Bake the cookies for 9 to 10 minutes, until they have the slightest hint of color. The tops will be dry. Let cool on wire racks before serving.
# _**Chocolate Butter Cookies**_
**brown rice flour**
MAKES ABOUT 50 COOKIES
_It is almost unbelievable that such a cookie can be made using brown rice flour_
_as the base. Gluten-free just gets better and better! Choose your favorite_
_chocolate to drizzle over the top. This cookie flavor is best at room temperature;_
_it is more buttery when warm._
**½** **cup butter, 110 grams**
**½** **cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams**
**¼** **cup unsweetened cocoa powder, 20 grams**
**1 egg, plus 1 egg yolk**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 tablespoon water**
TOPPING (OPTIONAL):
**2 ounces chocolate**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be soft and almost creamy.
Place the dough in a pastry bag and pipe the cookies onto the prepared pan. Bake for about 12 minutes, until the edges begin to brown and the cookies lose their sheen. Let cool on wire racks.
For the topping, break up the chocolate into a microwave-safe cup or bowl. Melt, stirring often, in the microwave on HIGH for up to 1 minute. As the container may be very hot, use care as you dip the tines of a fork into the melted chocolate and drizzle it over the cookies.
# _**Chocolate Graham Crackers**_
**brown rice flour**
MAKES 10 TO 12 LARGE GRAHAM CRACKERS
_It's funny to use the word graham in the name of this cookie as graham flour is_
_wheat flour, but no matter, it is the very description you need to visualize this_
_cookie style. This wheat-free graham cracker is richly chocolaty and not too sweet._
**⅓** **cup shortening, 70 grams**
**1 cup brown rice flour, 125 grams**
**½** **cup brown sugar, 100 grams**
**⅓** **cup cocoa, 30 grams**
**1 egg white**
**Scant** **½** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**¼** **teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough is thick and heavy. Continue beating for an additional 30 seconds to 1 minute for additional dough stability.
Roll out the dough to ⅛-inch thickness. Using a pizza cutter, cut the dough into 5 by 2¼inch rectangles and place them on the prepared pan. Lightly score each cracker with a lengthwise and a crosswise indentation (to quarter each cracker), not cutting all the way through. Prick the tops of the crackers with a fork.
Bake for about 9 minutes. As the crackers crisp during cooling and it is hard to tell what browning is taking place under that cocoa color, the proof is in the cooled cracker. I suggest you bake one as a test to determine the exact, best baking time—not enough leaves a slightly soft (although delicious) cookie; too much leaves an overly crisp cracker. Let cool on wire racks before serving.
# _**Chocolate Meringues**_
MAKES 35 TO 40 COOKIES
_Most meringue recipes call for cream of tartar to stabilize the meringue_
_structure. In this recipe, we are able to omit it. These cookies are light and airy_
_on the outside and almost a little fudgy on the inside. A fat-free winner!_
**2 egg whites, 65 grams (at room temperature)**
**⅓** **cup sugar, 75 grams**
**½** **teaspoon vanilla extract**
**¼** **cup unsweetened cocoa powder, 20 grams**
Preheat the oven to 300°F. Lightly grease a cookie sheet.
In a medium-size bowl, beat the egg whites, sugar, and vanilla until the whites form stiff peaks (This takes several minutes, depending upon the strength of your mixer. It also takes longer for stiff peaks to form with the inclusion of the sugar than if the egg whites were beaten alone.) The batter will look like marshmallow cream. Sprinkle the cocoa on the batter and gently fold it in. The dough will then look like whipped cream.
Using a piping bag with a large star tip, pipe large stars of dough onto the prepared pan, or just drop rounded teaspoonfuls of dough onto the pan. Bake the cookies for 15 to 20 minutes, until the bottom edges just begin to brown (which is hard to tell on this chocolate version) and the tops appear dry. Let cool on wire racks before serving.
# _**Chocolate Pinwheel Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_A spiral of two tender cookies that complement each other well. What could be_
_better than both chocolate and vanilla in a not-too-sweet cookie?_
**½** **cup butter, 110 grams**
**½** **cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams, plus 2 tablespoons, 15 grams**
**1 egg, plus 1 egg yolk**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**½** **teaspoon baking powder**
**2 tablespoons unsweetened cocoa powder, 10 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the 1¼ cups of brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the egg, egg yolk, baking soda, salt, xanthan gum, and vanilla, and beat well. Continue beating until the dough comes together; it will be soft and almost creamy.
Remove half of the dough (220 grams) from the mixing bowl and set it aside in another bowl. To the dough remaining in the bowl, add the remaining 2 tablespoons of brown rice flour and the baking powder and mix well. Set aside. To the other half of the dough, add the cocoa and mix well. Set aside. Both batches of dough may be refrigerated for 30 minutes for easier handling.
Pat the lighter dough on a lightly greased surface (waxed paper is nice) to form a 12 by 6-inch rectangle. Top the lighter dough with the darker dough, creating one rectangle with two layers. Roll the dough into a long log, rolling from the longer side, to create a spiral effect. Refrigerate the log until firm, about 30 minutes.
Slice the dough into ⅓- to ½-inch slices and place them on the prepared pan.
Bake the pinwheels for 10 to 12 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Chocolate Wafer Hearts**_
**sorghum flour**
MAKES ABOUT 24 COOKIES
_Tender, very chocolaty cookies. Imagine a chocolate version of a rolled sugar_
_cookie_ ... _yep, that good. Drizzle with a little melted chocolate if desired._
_If you are avoiding rice flour, these make an ideal substitute for the base cookie_
_used in Oreos-Style Cookies (page 116)._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and cocoa and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together. The dough will be oily, yet seem almost too dry.
Roll out the dough to no more than ¼-inch thickness. Cut it with a 2¼-inch heart-shaped cookie cutter or other favorite cookie cutter.
Bake for 8 minutes, until the cookies lose their sheen. Let cool on wire racks before serving.
# _**Cinnabon-Style Cookies**_
**brown rice flour**
MAKES ABOUT 24 COOKIES
_Inspired by those famous buns at the mall, these cookies are buttery, tender,_
_and full of cinnamon flavor._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1** **½** **teaspoons water**
CINNAMON-SUGAR MIXTURE:
**2 tablespoons butter, very soft**
**1 teaspoon ground cinnamon**
**2 tablespoons brown sugar**
GLAZE:
**⅔** **cup confectioners' sugar**
**1 tablespoon water**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients. The cookie dough will form large clumps, but will not quite come together to form a ball; this is okay.
On a sheet of waxed paper, roll out the dough to a ¼-inch-thick, 8 by 11-inch rectangle. Spread the butter, then the cinnamon and sugar over the rolled dough. Roll up the dough into a snug, long tube, rolling from the longer side, and slice it into ¼-inch rounds. Place them on the prepared pan.
Bake the cookies for 11 to 13 minutes, until the tops are lightly browned. Let cool on wire racks.
Combine the glaze ingredients in a small cup and stir well. Drizzle the glaze over the tops of the cookies.
# _**Crisp Almond Cookies**_
**brown rice flour and almond meal**
MAKES ABOUT 25 COOKIES
_These crisp cookies have the wonderful flavor of almonds complemented by a_
_substantial partial dipping in dark chocolate. These cookies are delicious even_
_without the extra chocolate._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 egg, plus 1 egg yolk**
**½** **cup almond meal, 60 grams**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon almond extract**
TOPPING (OPTIONAL):
**4 to 6 ounces dark chocolate, melted**
**¼** **cup sliced almonds**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. The dough will be quite thick.
Roll the dough to ⅛- to ¼-inch thickness and cut it with a 2-inch round cookie cutter (or other cookie cutter of your choice).
Bake the cookies for 8 to 10 minutes, until they have the slightest hint of color; the tops will be dry. Let cool on wire racks.
In a microwave-safe dish, melt 2 ounces of chocolate at a time. Dip or brush half of each cookie with chocolate and place on waxed paper to firm up. Sprinkle with a few sliced almonds, if desired.
# _**Gingerbread Men and Gingersnaps**_
**brown rice flour and sorghum flour**
MAKES ABOUT 16 SMALL GINGERBREAD MEN OR 32 GINGERSNAPS
_As gingerbread men, these cookies are sturdy, crisp, not-too-sweet,_
_and quite tasty. I use a hint of vanilla to soften the bitterness of the_
_ground ginger. As gingersnaps, these cookies have a light sprinkling of sugar_
_and definitely have a snap to their texture._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**2 tablespoons unsulfured molasses**
**¾** **cup brown rice flour, 95 grams**
**¾** **cup sorghum flour, 100 grams**
**½** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**¾** **teaspoon ground ginger**
**½** **teaspoon vanilla extract (optional)**
**1 egg**
TOPPING FOR GINGERBREAD MEN:
**Icing and/or candies**
TOPPING FOR GINGERSNAPS:
**2 tablespoons sugar**
Preheat the oven to 375°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening, sugar, and molasses. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. The batter will form a soft dough; continue to beat it for about 30 seconds after it forms, to stabilize the dough.
For gingerbread men, roll out the dough on a lightly greased surface to ⅛-inch thickness. Cut with a cookie cutter and place the cookies on the prepared pan.
For gingersnaps, drop rounded teaspoonfuls of the dough on the prepared pan and press to just under ¼-inch thickness. Sprinkle lightly with sugar.
Bake the cookies for 7 to 8 minutes, until the edges are lightly browned.
The cookies will crisp during cooling. Let cool on wire racks. Decorate with icing and candies as desired.
# _**Graham Crackers**_
**brown rice flour**
MAKES 10 TO 12 LARGE GRAHAM CRACKERS
_These classic crackers are tender, crisp, and easy to make._
_S'mores are back on the bonfire menu! Top them with a little cinnamon sugar_
_for another tasty alternative._
**⅓** **cup shortening, 70 grams**
**1** **½** **cups brown rice flour, 185 grams**
**½** **cup brown sugar, 100 grams**
**1 egg white**
**½** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough is thick and heavy. It should not fall apart easily.
Roll out the dough to ⅛-inch thickness. Using a pizza cutter, cut it into large 5 by 2¼-inch rectangles and place them on the prepared pan. Lightly score each cracker with a lengthwise and a crosswise indentation (to quarter each cracker), not cutting all the way through. Prick the tops of the crackers with a fork.
Bake the graham crackers for 8 to 10 minutes, until the edges are lightly browned and the tops just begin to brown. The crackers will crisp during cooling. Underbaking leaves the cooled cracker slightly soft and overbaking makes it just a little too crisp. My test crackers were perfect at 9 minutes. Let cool on wire racks before serving.
**Note: Should you have difficulty in handling the dough, you can roll it out directly on the baking sheet, then cut the rectangles and remove the excess dough. I like to make other shapes as well: circles, hearts, whatever. Please note that the more you handle this dough, the easier it becomes to work with . . . with no ill effects on the cookie. (You can't say that about a traditional cookie!)**
# _**Ice-Cream Sandwich Cookies**_
**brown rice flour**
MAKES ABOUT 11 SANDWICH COOKIES
_Not quite cakelike, not quite cookielike. For simplicity, I make these cookies_
_round. Tasty alone, they're even better with a scoop of ice cream sandwiched_
_between two cookies just before serving._
**⅓** **cup oil, 65 grams**
**¾** **cup sugar, 150 grams**
**1 cup brown rice flour, 125 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**1 egg**
**¼** **cup plain yogurt, 60 grams**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Add the remaining ingredients and mix well. The dough will be very thick, but still thin enough for rolling out.
Drop tablespoonfuls of dough onto the prepared pan. Using moistened fingertips, press it to ⅛-inch thickness. Prick the tops of the cookies with a fork. Or, for a prettier presentation, press the tops with a fondant cutter.
Bake for 10 minutes, until the cookies take on a bit of color at the edges and the tops appear dry. Let cool. The cookies will be very soft when removing them from the baking sheet, but will firm during cooling. Let cool on wire racks before serving.
# _**Lady Fingers**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_Light and airy, I would call them even a little poofy. Sandwich two of these_
_together with the lemon curd created as part of the Lemon Tassies recipe on_
_page 130. Awesome._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 10 grams**
**1 cup brown rice flour, 125 grams**
**½** **cup plain yogurt, 120 grams**
**1 teaspoon baking soda**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 egg yolk**
**2 egg whites**
TOPPING:
**2 tablespoons confectioners' sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients, except the egg whites, and mix well. The dough will be like a thick cake batter. Set aside.
Separately, beat the egg whites until stiff peaks form. (This will take several minutes, depending upon the temperature of the egg whites and the strength of the mixer.) Gently fold them into the batter.
Transfer the batter into a zip-type plastic bag. Cut off a ½-inch angle at one corner. Pipe 3- to 3½-inch long fingers onto the prepared pan.
Bake for 8 to 10 minutes, until browned at edges and the sponge is firm. Let cool briefly on the pan for easier removal and then cool on wire racks before serving.
# _**Lemon Meringues**_
MAKES 35 TO 40 SMALL COOKIES
_Inspired by lemon meringue pie, I just couldn't resist trying this out! To make_
_these cookies extra special, make the lemon curd that is part of the Lemon_
_Tassies recipe on page 130, and sandwich a bit of it between two lemon_
_meringues. They taste like a lemon cloud._
**2 egg whites, 65 grams**
**⅓** **cup sugar, 75 grams**
**1 teaspoon lemon extract**
Preheat the oven to 300°F. Lightly grease a cookie sheet.
In a medium-size bowl, beat the egg whites and sugar until the whites form stiff peaks. (This will take several minutes, depending upon the temperature of the egg whites and the strength of the mixer. It takes longer for stiff peaks to form with the inclusion of the sugar than if egg whites were beaten alone.) The batter will look like marshmallow cream. Beat in the lemon extract. Do not underbeat, as the shape of cookie relies upon the ability of the meringue to hold its shape!
Using a piping bag with a large star tip, pipe a small star onto the prepared pan, or just drop rounded teaspoonfuls of the dough onto the pan. Bake the cookies for 15 to 20 minutes, until the bottom edges just begin to brown and the tops look dry. Let cool on wire racks before serving.
# _**Linzer Sandwich Cookies**_
**brown rice flour and almond meal**
MAKES ABOUT 17 SANDWICH COOKIES
_Your favorite jam sandwiched between two crisp nut cookies, dusted with a little_
_confectioners' sugar. Be sure to cut a small "window" in half of the cookies_
_so the jam peeks out. Very pretty! These cookies lose some crispness once filled,_
_so fill these shortly before serving if possible._
**⅓** **cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 egg**
**½** **cup almond meal, 60 grams**
**½** **teaspoon baking powder**
**½** **teaspoon salt**
**1** **½** **teaspoons xanthan gum**
**½** **teaspoon almond extract**
TOPPING:
**½** **cup seedless raspberry jam, or other favorite jam**
**¼** **cup confectioners' sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will form lots of small clumps, and, with continued beating, will come together.
Roll out the dough to ⅛-inch thickness and cut it into 2-inch circles. Using a smaller cookie cutter, cut a small window in the center of half of the cookies. Place the cookies on the prepared pan.
Bake the cookies for about 8 minutes, until the edges begin to lightly brown. Let cool on wire racks. Place a small amount of jam on the bottoms of the full-circle cookies. Sprinkle the cookie tops (the ones with windows) with confectioners' sugar. Gently sandwich in pairs to form sugar-topped, jam-filled cookies.
# _**Nut Meringue Wreaths**_
**almond meal**
MAKES 20 TO 30 COOKIES
_I've used almond meal for this recipe as it is readily available. Although these_
_meringues may be piped or spooned into any shape, wreaths make a pretty_
_presentation for the holiday season._
**2 egg whites, 65 grams**
**⅓** **cup sugar, 75 grams**
**¼** **teaspoon cream of tartar**
**½** **teaspoon vanilla extract**
**¼** **cup almond meal, 30 grams**
TOPPING (OPTIONAL):
**Almond slices or bits**
Preheat the oven to 300°F. Lightly grease a cookie sheet.
In a medium-size bowl, beat the egg whites, about half of the sugar, the cream of tartar, and vanilla, until the whites form stiff peaks. (This will take several minutes, depending upon the temperature of the egg whites and the strength of the mixer. It takes longer for stiff peaks to form with the inclusion of the sugar than if egg whites were beaten alone.) The batter will look like marshmallow cream. Sprinkle the almond meal and remaining sugar on the batter and gently fold them in. The dough will now look like whipped cream.
Using a piping bag with a circle tip (or a zip-type plastic bag with a corner cut off), pipe wreath-shaped cookies onto the prepared pan and garnish with almond slices or bits. Bake for 15 to 20 minutes, until the bottom edges just begin to brown and the tops look dry. Let cool on wire racks before serving.
# _**Rolled Sugar Cookies**_
**brown rice flour**
MAKES ABOUT 30 TWO-INCH COOKIES
_These are traditional buttery sugar cookies. You may use all butter, if desired_
_(instead of shortening). When planning to make this recipe, keep in mind that_
_the dough needs to be refrigerated for an hour and you must work very quickly_
_with small portions of dough to achieve sharp cutouts._
**¼** **cup butter, 55 grams**
**¼** **cup shortening, 50 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups brown rice flour, 185 grams**
**1 egg, plus 1 egg yolk**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**1** **¼** **teaspoons xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**Sprinkles or colored sugar**
**Icing (optional)**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the butter, shortening, and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together. At this stage the dough will be too soft to roll. Refrigerate for 1 hour.
Roll out the dough to between ⅛- and ¼-inch thickness. Cut it using cookie cutters of your choice and place the cut cookies on the prepared pan. Decorate with sprinkles or colored sugar as desired (or leave bare, if you are planning to ice them later).
Bake the cookies for about 10 minutes, until the edges begin to brown. Let cool on wire racks.
Ice and decorate, if desired.
# _**Rolled Sugar Cookies, Dairy-Free**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_Here is a dairy-free version of a rolled sugar cookie._
_They are simply delicious._
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg**
**1½ teaspoons baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 ½ teaspoons water**
TOPPING (OPTIONAL):
**Sprinkles or colored sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball. Press it together with your hands.
Roll out the dough to ⅛ to ¼-inch thickness and cut it with a 2-inch round cookie cutter (or other cookie cutter of your choice). Place the cookies on the prepared pan. Top with sprinkles or colored sugar, if desired.
Bake the cookies for 8 to 10 minutes, until they have the slightest hint of color and the tops are dry. Let cool on wire racks.
# _**Stained-Glass Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_Use your favorite clear candy (such as lollipops or individual hard candies) for_
_the "glass" in these pretty cookies. Use multiple colors or just one. And expect a_
_casualty or two when removing the cookies from the baking sheet._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg, plus 1 egg yolk**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
STAINED GLASS:
**About 8 crushed, clear candies**
TOPPING (OPTIONAL):
**Confectioners' sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough almost comes together. Press it together with your hands to form a ball. The dough will feel oily.
Roll out the dough flat, to less than ¼-inch thick. Cut the dough with a 2½-inch cookie cutter and then cut the center of each cookie with another cutter of the same shape, but smaller, to form the window, and remove the center scrap of dough. Place the large cutouts well apart on the prepared pan. The cookies will spread during baking. Fill the windows with a small amount of crushed candy. Reroll the scraps into additional cookies.
Bake for 9 to10 minutes, until the cookies begin to take on color. Let cool on the baking sheet to allow the window candy to harden.
Sprinkle the tops with confectioners' sugar, if desired.
**Note: To crush the candies, place them in a small plastic bag and hit them with a rolling pin or the dull end of a butter knife. Take care while eating these; the candy windowpanes can be sharp when bitten or broken.**
6
_**Great Fakes Cookies**_
**When I first started working on recipes** for this book, I wanted to be sure to meet our everyday cravings. Although everyone loves a homemade classic, sometimes a supermarket cookie or Girl Scout cookie is really what is desired! I re-created several of my favorite Girl Scout cookies during "cookie season," but time and time again, I was asked to add just one more.
The Girl Scouts of Central Maryland, Inc., came to my rescue! They provided me with sample packages of cookies so that I could compare my work side-by-side with their delicious varieties. I cannot help but smile that the Girl Scouts gave cookies to help those who will never be able to eat theirs. And, while of course the recipes included in this chapter are not identical to Girl Scout cookies, they are some awesome great fakes!
My local grocery store's cookie aisle (and Trader Joe's) held the rest of the inspiration for this chapter: Oreos, Chips Ahoy!, Nutter Butters, Fig Newtons, and Little Debbie Oatmeal Creme Pies, to name a few. Diverse and delicious in their own way, I have also re-created these beloved favorites.
In making each cookie, I deconstructed the original. Many cookies are not what you think. Sometimes it is just the filling that has the flavor. Other times, you think you are eating a sweet-base cookie only to find it is rather plain and not sweet at all. And oh, the nuances of cream filling! These substitutes are perhaps the greatest proof that almost anything is possible in a gluten-free cookie.
# _**Chips Ahoy!-Style Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_Wandering down the cookie aisle at the grocery store, I thought a lot about_
_which cookies to duplicate. These original, crunchy ones were the first in my_
_cart. I've made these slice-and-bake for the no-fuss kitchen._
**⅓ cup shortening, 70 grams**
**½ cup light brown sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg**
**½ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**½ teaspoon vanilla extract**
**1 tablespoon water**
**½ cup semisweet chocolate**
**morsels, chopped, 80 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and mix well. Add the remaining ingredients, except for the chocolate chips, and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball; this is okay. Add the chocolate chips and mix well.
On a sheet of waxed paper or plastic wrap, shape the dough into a log about 2¼ inches thick and 7 inches long. Refrigerate for 2 hours or more. Freeze for later use, if desired.
Slice the dough to a scant ¼-inch thick. (These are thin cookies, if we're being true to the original!) Bake for 12 to 15 minutes, until the cookies are lightly browned; the tops will be dry. Let cool on wire racks before serving.
# _**Chocolate Marshmallow "Scooter" Pies**_
**brown rice flour**
MAKES ABOUT 15 SANDWICH COOKIES
_Inspired by the Little Debbie Chocolate Marshmallow Pies, these cookies feature_
_two tender cookies with a marshmallow sandwiched in the middle, covered in_
_chocolate. I daresay that these are an upscale version of the original. They are_
_smaller, measuring 2 inches in diameter as compared to the original's 3-inch_
_diameter. Milk chocolate chips are closer to the original, but semisweet chocolate_
_chips make for a more interesting glaze._
**⅓ cup shortening, 70 grams**
**⅓ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg**
**½ teaspoon salt**
**¼ teaspoon baking soda**
**2 teaspoons baking powder**
**1 teaspoon xanthan gum**
**½ teaspoon vanilla extract**
FILLING :
**15 regular-size marshmallows**
GLAZE:
**1 cup semisweet chocolate chips, 160 grams**
**2 tablespoons shortening**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well until the dough comes together. The dough will be soft, but manageable.
Roll out the dough to ⅛-inch thickness and cut it with a 2-inch circle cookie cutter. Place the slices on the prepared pan.
Bake the cookies for 9 to 10 minutes, until the edges are lightly browned. Let cool on wire racks.
For the filling, use your rolling pin to roll out each marshmallow into a circle about 2 inches in diameter.
For the glaze, combine the chocolate chips and shortening in microwave-safe cup or bowl. Microwave it on HIGH for about 2 minutes to melt, stopping to stir it periodically.
To assemble the cookies, place the marshmallow disk between two cookies and cover completely with chocolate and place on waxed paper to firm.
**Note: You can refrigerate the dough for 30 minutes prior to rolling for easier handling, if desired. I just take my time and ease the cut dough from the rolling surface with a spatula.**
# _**Fig Newton-Style Cookies**_
**brown rice flour**
MAKES 32 COOKIES
_I wanted to find an easy way to make this cookie yet stay true to the flavor of_
_the famous ones found in your local grocery store. These are a little sweeter in_
_filling and the crust is a bit more pastrylike. A very good cookie._
**⅓ cup shortening, 70 grams**
**1¾ cups brown rice flour, 220 grams**
**¼ cup corn syrup, 80 grams**
**1 egg**
**¼ cup sugar, 50 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon vanilla extract**
**1½ teaspoons xanthan gum**
FILLING :
**8 ounces dried figs, chopped finely**
**½ cup apple jelly, 135 grams**
**½ cup water**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
To make the dough, combine the shortening and brown rice flour in a medium-size bowl. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. The cookie dough will be quite heavy in texture. Once the dough comes together, continue beating for an additional minute or so to make the dough easier to handle. At this point, the dough will be very soft and should be refrigerated for at least 30 minutes.
For the filling, combine the figs, apple jelly, and water in a small saucepan over low heat. Puree it with a stick blender and continue cooking until a thick paste remains, about 1 cup of filling. Set it aside to cool.
Roll a quarter of the dough into a 5 by 8-inch rectangle. Spread ¼ cup of filling on the center third of the dough. Fold the two other thirds over the top to fully wrap the filling with dough. Place each cookie seam side down on the prepared pan. Do the same with the remaining dough.
Bake for 15 to 20 minutes until golden brown. Let cool completely on wire racks before slicing into 1-inch cookies.
# _**Girl Scout Do-Si-Dos-Style Cookies**_
**sorghum flour**
MAKES ABOUT 24 SANDWICH COOKIES
_It is strange how one peanut butter cookie can have such nuances when com-_
_pared to others. In these, the base cookies are soft in peanut butter flavor and_
_almost crackerlike in texture. The cream filling is very flavorful and spread quite_
_thinly. The combination is delicious—it is no wonder they are so popular._
**¼ cup peanut butter, 65 grams**
**2 tablespoons oil, 20 grams**
**½ cup sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**½ teaspoon vanilla extract**
**1 tablespoon water**
FILLING :
**¼ cup confectioners' sugar, 30 grams**
**¼ cup peanut butter, 65 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the peanut butter, oil, and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. Continue beating until the dough comes together; it will be oily, yet seem a little dry.
Roll scant teaspoonfuls of dough into balls about ½-inch in diameter and drop them onto the prepared pan. Press to no more than ¼-inch thickness, using the bottom of a glass or butter press. Bake for 8 minutes, until the cookies just begin to brown. Let cool on wire racks.
For the filling, combine the confectioners' sugar and peanut butter. Beat until fully combined. The cream filling will be stiff and almost dry. Place a marble-size bit of creamy filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Girl Scout Lemon Chalet Cremes-Style Cookies**_
**brown rice flour**
MAKES ABOUT 25 SANDWICH COOKIES
_Amazingly, this little lemon sandwich cookie does not start with two_
_lemon-flavored cookies. They taste slightly of lemon and a little cinnamon._
_But the filling is very lemony._
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1½ cup brown rice flour, 185 grams**
**2 egg whites**
**½ teaspoon salt**
**¼ teaspoon baking soda**
**1½ teaspoons xanthan gum**
**½ teaspoon lemon extract**
**½ teaspoon ground cinnamon**
FILLING :
**⅓ cup shortening, 70 grams**
**1 teaspoon lemon extract**
**¾ cup confectioners' sugar, 90 grams**
**Several drops yellow food coloring (optional)**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well until the dough comes together; it will be very soft, yet manageable. The dough may be refrigerated for 30 minutes, for easier rolling.
Roll out the dough to a ⅛-inch thickness and cut it with a 1½-inch cookie cutter. Place the cutouts on the prepared pan and press them with a butter press, or make a pretty pattern with the tines of a fork.
Bake the cookies for about 12 minutes, until lightly browned. Let cool on wire racks.
For the filling, combine all ingredients and mix until creamy. Place a marble-size bit of creamy filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Girl Scout Samoas-Style Cookies**_
**brown rice flour**
MAKES ABOUT 45 COOKIES
_Duplicating the essence of this cookie was not the easiest task. But knowing that_
_Samoas are one of the top-selling Girl Scout cookies made me plunge forward_
_anyway. The components are a hard, sweet, little wreath-shaped cookie,_
_heavily vanilla flavored, with an indentation down the middle. It's covered with_
_a sticky caramel sauce, finely cut toasted coconut, and a drizzle of very sweet_
_chocolate that's striped over the front and totally covers the base. For those of_
_you that are still with me . . . let's make all of the components and then com-_
_bine to make a very tasty, even more tender version of this fabulous cookie._
**⅓ cup shortening, 70 grams**
**1½ cups brown rice flour, 185 grams**
**½ cup sugar, 100 grams**
**1 egg**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**2 teaspoons vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be soft like play dough.
Roll out the dough to ¼-inch thickness. Use a 2-inch cutter for the outside of the circle and use a 1½-inch round cookie cutter to cut away the inside of the cookie, forming a wreath shape. Place the cookies on the prepared pan. Make a narrow indentation across the center of the wreath. This trench will eventually be filled with caramel sauce. (I used two coins—nickels—held together to achieve the right thickness. A board game checker may also work. Please wash this item well before using—or wrap in foil—and overlook the nonkitchen aspect of the tool.) Then prick the center of the cookie trench to help it retain its shape. Bake the cookies for 7 to 8 minutes, until the edges just begin to brown. The cookies should be crisp upon cooling. Let cool on wire racks before proceeding. Keep the oven heated to 350°F for toasting the coconut.
CARAMEL FILLING:
**2 cups heavy whipping cream (unwhipped)**
**2 teaspoons vanilla extract**
**¾ teaspoon xanthan gum**
**2 cups sugar, 400 grams**
TOASTED COCONUT:
**2 cups sweetened flaked coconut (200 grams)**
CHOCOLATE COATING:
**12 ounces broken milk chocolate bars, 340 grams**
**¼ cup confectioners' sugar**
To make the caramel sauce, mix the cream, vanilla, and xanthan gum in a small cup. Set aside. Place the sugar in a metal saucepan and heat over high heat. Cook it until it is melted and golden brown in color. _Do not_ touch the hot sugar or caramel with wooden or plastic utensils or your fingers. It will be dangerously hot. With _great caution_ , immediately pour all of the cream mixture into the saucepan and stir well using a metal utensil. Bring the mixture to a boil. Remove it from the heat and continue stirring until any large pieces of cooked sugar are dissolved. Let cool. The mixture will continue to thicken during cooling.
To make the toasted coconut, place the coconut on a cutting board and chop it into very small pieces. (I like to "walk" the knife through the coconut. This only takes a minute or so to complete.) Spread it onto an ungreased baking sheet and place in a 350°F oven. Stir it after 3 minutes, to prevent burning. The coconut will be lightly browned when toasted for 6 to 7 minutes. Set it aside to cool.
For the drizzled and dipped chocolate coating, I suggest making two batches of coating (using half of the ingredients at a time) so that it will not harden before you have used it all. Place half the chocolate into a microwave-safe bowl and heat on HIGH for 1 to 2 minutes, until melted. Stir in half the sugar, until creamy.
To assemble the cookies, dip or spread a cookie base with melted chocolate to fully cover the base. Place it on waxed paper. Then overfill the cookie trench with caramel sauce (place the sauce in zip-type plastic bag and snip the corner) and spread it over the sides of the cookie with a knife. Sprinkle liberally with toasted coconut and press it into the underlying caramel. Allow to cool thoroughly. Peel the cookies off the waxed paper and transfer them to a clean sheet of waxed paper, shaking off the excess coconut.
Finally, prepare the remaining chocolate topping, and place it into a zip-type plastic bag and snip the corner. Pipe thin chocolate stripes across the tops of the cookies. Allow the cookies to cool again.
**Note: If you lose your spunk after making twenty or thirty of these, please know that the extra dough can be used to make good sugar cookies. Just roll and bake (about 8 minutes).**
# _**Girl Scout Tagalongs-Style Cookies**_
**brown rice flour**
MAKES ABOUT 40 COOKIES
_In the original Tagalongs, the underlying cookie is rather thin, a little hard, high_
_in vanilla flavor, and not too sweet. The peanut butter is plain and the chocolate_
_coating is sweet. It combines for a delicious cookie that you may have enjoyed_
_prior to dietary restriction._
**⅓ cup shortening, 70 grams**
**1½ cups brown rice flour, 185 grams**
**¼ cup sugar, 50 grams**
**1 egg**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**2 teaspoons vanilla extract**
CHOCOLATE COATING:
**12 ounces broken milk chocolate bars, 340 grams**
**2 tablespoons confectioners' sugar**
FILLING :
**½ cup peanut butter**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be soft like play dough.
Roll out the dough to ⅛-inch thickness. Use a 1½-inch round cookie cutter to cut the cookies and place them on the prepared pan. Depress the center of each cookie a bit with the back of a measuring spoon; this will eventually help hold the peanut butter. Then prick the center of the cookie to help the bottom of the cookie stay flat. Bake for 8 to 9 minutes, until the edges begin to brown. The cookies should be crisp upon cooling. Let cool well before proceeding.
For the chocolate coating, I suggest making two batches of coating (using half of the ingredients at a time) so that it does not harden before you have used it all. Place half the chocolate into a microwave-safe bowl and cook on HIGH for 1 to 2 minutes, until melted. Stir in half the sugar. Stir until creamy.
Place about ½ teaspoon of peanut butter on top of each cookie and spread, leaving a margin of ¼ to ⅓ inch from the edge of the cookie. Dip the cookies into the chocolate coating (remove the excess, especially from the bottom) and place on waxed paper to cool. The coating should be as thin as possible to mimic the original.
Shortbread, page 57
Blondies #2, page 45
Carrot Cake Bars, page 46
Chocolate Chip Cookies #2, page 25
Almond Flower Cookies, page 20
Gingerbread Men, page 81
Chinese Marble Cookies, page 70
Love Letter Rolled Chocolate Sugar Cookies, page 172
Graham Crackers, page 82
Linzer Sandwich Cookies, page 87
Girl Scout Thin Mints-Style Cookies, page 107
Rolled Sugar Cookies, page 89
Pumpkin Sandwich Cookies (Scooter Pies), page136
Pizzelle, page 133
Fortune Cookies and Pirouettes, page 169
Rosettes, page 176
# _**Girl Scout Thin Mints-Style Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_In the original Thin Mints cookies, the underlying chocolate cookie is quite thin,_
_not too sweet, and rather compact. It is the thin milk chocolate covering_
_that is very minty._
**⅓ cup oil, 65 grams**
**¼ cup sugar, 50 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg**
**⅛ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
CHOCOLATE COATING:
**12 ounces broken milk chocolate bars, 340 grams**
**2 tablespoons confectioners' sugar**
**½** **teaspoon mint extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be soft and oily to the touch.
Roll out the dough to ⅛-inch thickness. Use a 1½-inch round cookie cutter to cut the cookies and place them on the prepared pan. Bake the cookies for 8 to 9 minutes, until the tops are dry. Let cool well before coating; the cookies should be crisp upon cooling.
For the chocolate coating, I suggest making two batches of coating (using half of the ingredients at a time) so that it does not harden before you have used it all. Place half the chocolate into a microwave-safe bowl and cook on HIGH for 1 to 2 minutes, until melted. Stir in half the sugar and half the mint extract. Stir until creamy. Dip the cookies into the chocolate and place on waxed paper to cool. The coating should be as thin as possible to mimic the original.
**Note: Milk chocolate bars melt "thinner" than do milk chocolate chips. Milk chocolate chips may be used, but will make for a thicker chocolate coating.**
# _**Girl Scout Trefoils-Style Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_I realized something very interesting about the original cookies. They don't taste_
_like a traditional shortbread. They are a little sweeter, have more vanilla flavor,_
_and are a little smoother in texture. Besides being obviously delicious, I noted_
_that they are like a cross between an animal cracker and traditional shortbread_
_in flavor. I hope they bring back fond memories for you! I used an old-fashioned_
_butter press for the design on the top. Please note that a little bit of butter makes_
_a big flavor difference!_
**¼ cup oil, 50 grams**
**2 tablespoons butter**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**2 egg yolks**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 ½ teaspoons vanilla extract**
**3 tablespoons water**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil, butter, and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together.
For each cookie, shape a slightly rounded teaspoonful of dough into a ball and place it on the prepared cookie sheet. Using the bottom of a glass or a butter press that has been sprayed lightly with nonstick spray, press the ball into a circle about 1¾-inches wide and less than ¼-inch thick.
Bake the cookies for about 15 minutes, until the tops are lightly browned. Underbaking makes for a sugar cookie taste (which is also very good). Correct baking provides the drier cookie that is desired. Let cool on wire racks before serving.
# _**Little Debbie Oatmeal Creme Pies-Style Cookies**_
**oats**
MAKES ABOUT 7 SANDWICH COOKIES
_Unfortunately for my hips, it took quite a few cookies for me to figure this_
_cookie out! Even duplicating the cream filling was elusive. Like the original, the_
_oatmeal cookie is extremely soft and the filling is super sweet._
**½ tablespoon raisins, 5 grams**
**1¼ cups rolled oats, 125 grams**
**⅓ cup oil, 65 grams**
**¾ cup brown sugar, 150 grams**
**1 egg**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 tablespoon water**
FILLING :
**⅓ cup shortening, 70 grams**
**¾ cup confectioners' sugar, 90 grams**
**½ cup marshmallow cream, 50 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
Finely mince the raisins; using a little bit of oats mixed with the raisins will make this much easier. Place the raisins and oats in a blender. Process until most of the oats are powdery, yet a good amount of small pieces remain.
Pour the oat mixture in a mixing bowl. Add the remaining batter ingredients and mix well. A sticky-looking dough will form. Scrape down the sides of the mixing bowl at least once during mixing.
Place tablespoonfuls of dough on the prepared pan and, using moistened fingertips, press down to ¼-inch thickness.
Bake the cookies for 8 to 10 minutes, until the edges are lightly browned. Keep the cookies on the pan for a minute to "set" before transferring to a cooling rack. The cookies will be very pliable.
For the filling, combine the shortening and confectioners' sugar and mix until creamy. Add the marshmallow cream and mix well. Spread the bottom of one cookie with the filling and press it together with the bottom of another cookie. Repeat to make the other sandwiches.
# _**Maple Leaf Cookies**_
**brown rice flour**
MAKES ABOUT 15 SANDWICH COOKIES
_Inspired by the Maple Leaf Cookies at Trader Joe's, these cookies have a creamy_
_center that is so full of maple flavor that it seems as if the entire cookie has_
_maple syrup in it. Like the original, it is an understated, plain, hard vanilla_
_cookie on the outside. For a slightly healthier cookie, omit the filling and use_
_maple flavoring in the base cookie. Very enjoyable!_
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**2 egg whites**
**½ teaspoon salt**
**¼ teaspoon baking soda**
**1½ teaspoons xanthan gum**
**½ teaspoon vanilla extract**
FILLING :
**⅓ cup shortening, 70 grams**
**1 teaspoon maple flavoring**
**¾ cup confectioners' sugar, 90 grams**
**Note: Roll these cookies as thinly as possible, as they do rise just a little in the oven. Also, you can refrigerate the dough for 30 minutes for easier handling, if desired. I just take my time and ease the cut dough from the rolling surface with a spatula.**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well until the dough comes together. The dough will be very soft, yet manageable.
Roll out the dough to ⅛-inch thickness and cut it with maple leaf-shaped cookie cutters (symmetrical for sandwiching!) or other preferred cookie cutter. Place them on the prepared pan.
Bake the cookies for about 13 minutes, until lightly browned. Let cool on wire racks.
For the filling, combine all ingredients and mix until creamy. Place a marble-size bit of creamy filling between two cookies (bottom sides facing) and press them together to form a sandwich. Repeat to make the other sandwiches.
# _**Nutter Butter-Style Peanut Butter Sandwich Cookies**_
**brown rice flour and cornstarch**
MAKES ABOUT 15 SANDWICH COOKIES
_A bent 2½-inch circle cookie cutter does a pretty good imitation of the peanut_
_shape of these cookies. I prepared these cookies on the first day of school, which_
_left me all alone in my kitchen. I had no one to share the excitement of a nearly_
_perfect cookie. Dancing alone in my house was good, but these cookies are_
_better than that!_
**¼ cup creamy peanut butter, 65 grams**
**2 tablespoons oil, 20 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup cornstarch, 40 grams**
**2 egg yolks**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**2½ tablespoons water**
FILLING :
**6 tablespoons peanut butter, 100 grams**
**½ cup confectioners' sugar, 60 grams**
**¼ teaspoon vanilla extract**
**A few drops of water (only if needed)**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the peanut butter, oil, and sugar. Beat well. Add the brown rice flour and cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well until the dough almost comes together. The dough will hold together when pressed in the hand and will be oily to the touch.
Roll out the dough to ⅛-inch thickness. Use the back of a fork to crisscross wavy indentations (to mimic peanut texture). Use a 2-inch round or a cookie cutter bent into a peanut shape to cut the cookies. The dough is very easy to work with, but quite soft. Place the cutouts on the prepared pan.
Bake the cookies for 10 to 12 minutes, until the edges just begin to brown. Let cool on wire racks.
To prepare the filling, combine the peanut butter, confectioners' sugar, and vanilla. Mix until creamy. Add a few drops of water, if necessary, to the mixture. Spread the bottom of one cookie with a little filling and press it together with the bottom of another cookie. Repeat to make the other sandwiches.
# _**Oreos-Style Cookies**_
**brown rice flour**
MAKES ABOUT 25 SANDWICH COOKIES
_An original Oreo is very sweet and very chocolaty; these are, too. Grab a glass_
_of milk for dunking. I used a decorative fondant punch set from Wilton to make_
_pretty designs on the tops of the cookies, but a simple design can be drawn with_
_a toothpick or the tines of a fork as well._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 tablespoon water**
FILLING :
**⅓ cup shortening, 70 grams**
**½ teaspoon vanilla extract**
**¾ cup confectioners' sugar, 90 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be soft and oily to the touch.
Roll out the dough to ⅛-inch thickness. Use a 1½-inch round cookie cutter to cut the cookies. Place them on the prepared pan and press the tops with a decorative stamp if you have one.
Bake the cookies for 10 minutes, until the tops are dry. The cookies should be crisp upon cooling. Let cool on wire racks.
For the filling, combine all ingredients and mix until creamy. Place a marble-size bit of creamy filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Pecan Sandies-Style Cookies**_
**brown rice flour and pecan meal**
MAKES ABOUT 25 COOKIES
_Although I prefer traditional shortbread with butter, we are duplicating these_
_famous cookie-aisle cookies, so I stay true to the flavor by using shortening_
_instead. If you loved the Keebler cookie, you will love these, too. (But if you_
_want to use butter, substitute ⅓ cup butter for the shortening, decrease the_
_water to 2 teaspoons, and increase the xanthan gum to 1⅓ teaspoons.)_
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1¼ cups brown rice flour, 220 grams**
**¼ cup pecan meal, 20 grams (or substitute ¼ cup brown rice flour)**
**1 egg**
**2 teaspoons baking powder**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**½ teaspoon vanilla extract**
**1 tablespoon water**
**½ cup chopped pecans**
When ready to bake, preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the brown rice flour and mix well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except for the pecan pieces, and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball; this is okay. Add the chopped pecans and mix well.
On a sheet of waxed paper or plastic wrap, shape the dough into a log about 2 inches thick and 7½-inches long. Refrigerate for 2 hours or more. You can also freeze for later use, if desired.
Slice the dough into ¼-inch slices and place on the prepared pan. Bake the cookies for about 15 minutes, until they are lightly browned around the edges and the tops are dry. Let cool on wire racks before serving.
# _**Pepperidge Farm Milano-Style Cookies**_
**brown rice flour**
MAKES ABOUT 18 SANDWICH COOKIES
_Light and crisp, sandwiched with a little dark chocolate. This cookie is not quite_
_as delicate as the original because it is made with whole grain. For a closer_
_replica, try the starch-based version on page 178._
**⅓ cup oil, 65 grams**
**⅓ cup sugar, 75 grams**
**1 cup brown rice flour, 125 grams**
**1 egg, plus 1 egg white**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**½ teaspoon vanilla extract**
FILLING :
**4 ounces broken dark chocolate bar**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be a thick, cakelike batter.
Place the batter into a zip-type plastic bag. Cut off one corner at a ⅓-inch angle. Pipe 2- to 2½-inch-long fingers well apart onto the prepared pan. The cookies will spread during baking.
Bake for 10 to 12 minutes, until the edges of the cookies are very golden. (Underbake and the cookies will not be crisp throughout—but still tasty!) Let cool on wire racks.
Place the chocolate into a microwave-safe bowl and heat on HIGH for 1 to 2 minutes, until melted. Spread the bottom of one cookie with the melted chocolate and press it together with the bottom of another cookie. Repeat to make the other sandwiches.
# _**Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies**_
**brown rice flour**
MAKES ABOUT 22 COOKIES
_A big, sweet, crispy, tender cookie, with white chocolate chunks and_
_macadamia nuts (our version has a little extra). Note the lower baking tempera-_
_ture, which is necessary for the center to cook before the edges burn._
**¼ cup butter, 55 grams**
**¼ cup oil, 50 grams**
**½ cup plus 2 tablespoons sugar, 125 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg, plus 1 egg yolk**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**⅓ cup roughly chopped**
**macadamia nuts, 50 grams**
**⅓ cup roughly chopped white**
**chocolate chips, 60 grams**
Preheat the oven to 325°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the butter, oil, and sugar. Beat well. Add the flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be soft and almost creamy. Set aside.
Drop rounded tablespoonfuls of the dough onto the prepared pan. Press to ¼-inch thickness. Bake the cookies for 10 to 12 minutes, until the edges begin to brown. Let cool on wire racks before serving.
7
_**Sandwich, Shaped, and Filled Cookies**_
**This chapter is an eclectic mix** of unique cookies. From deep-fried Rosettes and nicely pressed Pizzelle to picture-perfect Thumbprint Cookies, you will be cookie-ready for nearly any occasion.
Among my favorites are the Chocolate-Cherry Cookies and the classic Snickerdoodles. The Almond Biscotti is a great coffeetime treat and all of the sandwich cookies are soft and tender with a tasty creamy filling.
# _**Almond Biscotti**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_This recipe is fashioned after the mini biscotti at Trader Joe's. Almond and_
_butter team up for a delicious cookie._
**⅓ cup butter, 70 grams**
**⅓ cup sugar, 75 grams**
**1½ cups brown rice flour, 185 grams**
**2 eggs**
**¼ teaspoon baking soda**
**2 teaspoons baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1½ teaspoons almond extract**
**½ cup sliced almonds**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and mix well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be soft and heavy.
Shape about half the dough into a flattened log about ½-inch thick and place on the prepared pan. Do the same with the other half of the dough.
Bake the logs for 20 minutes. Let cool completely on wire racks. Cut each log into slices about ½-inch thick and place these back on the prepared pan. Bake the biscotti for an additional 10 to 15 minutes until quite dry. Let cool again on wire racks before serving
# _**Chocolate-Cherry Cookies**_
**brown rice flour**
MAKES ABOUT 27 COOKIES
_This recipe is my gluten-free re-creation of the Chocolate Cherry Cookies_
_contained in the 100 Best Cookies by Better Homes and Gardens. They taste_
_like a rich chocolate cookie crossed with a cherry cordial. Delicious._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**2 eggs**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPINGS:
**1 (10-ounce) jar maraschino cherries**
**1 cup chocolate chips**
**½ cup sweetened condensed milk**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together; it will be very soft.
Drop rounded teaspoonfuls of the dough onto the prepared pan. With moistened fingertips, press the dough to ⅓-inch thickness and make a depression in the middle, using your fingertip.
Place a cherry (stem removed) in the hollowed-out middle of each unbaked cookie. Save the cherry juice!
In a microwave-safe cup, combine 1½ tablespoons of the cherry juice, the chocolate chips, and the condensed milk. Microwave on HIGH for 1½ to 2 minutes, stirring frequently, until the chocolate melts and is of spreading consistency. (The topping may be thinned with additional cherry juice if necessary.) Drop a rounded teaspoonful of the chocolate mixture over each cherry to cover. (This will seem like a large amount.) It is important to fully cover the cherry.
Bake the cookies for about 10 minutes, until the edges are dry. Let cool on wire racks before serving.
# _**Chocolate Crinkles**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_A very chocolaty cookie, covered with confectioners' sugar, then baked._
_This cookie is a favorite of my older son._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**2 eggs**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**¼ cup confectioners' sugar, 30 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together; it will be very soft.
Drop rounded teaspoonfuls of the dough into the confectioners' sugar (or shape the dough into small balls and drop them in) and gently cover them with confectioners' sugar. Place them on the prepared pan and press the dough balls down to ¼-inch thickness. (This step is very important.)
Bake the cookies for 10 minutes, until dry. Let cool on wire racks before serving.
**Note: A heavy hand in applying the confectioners' sugar makes for a more visible crinkle to the cookie.**
# _**Chocolate-Mint Sandwich Cookies**_
**sorghum flour**
MAKES ABOUT 24 SANDWICH COOKIES
_Much like the popular Oreo, these cookies have that predictable creamy filling,_
_but the cookie is just a bit crisper and minty!_
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon mint extract**
FILLING :
**⅓ cup shortening, 70 grams**
**½ teaspoon vanilla extract**
**¾ cup confectioners' sugar, 90 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together. The dough will be oily and dry at the same time.
Roll out the dough to ⅛-inch thickness. Use a 1½-inch round cookie cutter to cut the cookies. Place them on the prepared pan and press the tops with a decorative stamp if you have one.
Bake the cookies for 8 minutes, until the tops are dry and lose their sheen. Let cool on wire racks.
For the filling, combine all the ingredients and mix until creamy. Place a marble-size bit of creamy filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Chocolate Sandwich Cookies (Scooter Pies)**_
**brown rice flour**
MAKES ABOUT 11 SANDWICH COOKIES
_Essentially a soft, cakelike cookie with a cream cheese filling._
**⅓ cup oil, 65 grams**
**¾ cup sugar, 150 grams**
**1 cup brown rice flour, 125 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg**
**½ cup plain yogurt, 120 grams**
**1 teaspoon baking soda**
**½ teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
FILLING :
**⅓ cup shortening, 70 grams**
**¾ cup confectioners' sugar, 90 grams**
**4 ounces cream cheese, 110 grams**
**½ teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well; the dough will be like a thick cake batter.
Drop rounded tablespoonfuls of the dough onto the prepared pan. The cookies will spread during baking.
Bake for 9 to 10 minutes, until the cookies take on a bit of color at the edges and the tops appear dry. Let cool on wire racks.
For the filling, blend all the ingredients until light and fluffy. Place a tablespoon of the filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Filled Triangle Cookies**_
**brown rice flour**
MAKES BETWEEN 25 AND 30 COOKIES
_This recipe uses the Rolled Sugar Cookie recipe, but opts to use all butter. The_
_extra buttery flavor is nice to compliment the flavor of the jam. I chose_
_strawberry, but any jam may be substituted._
**½ cup butter, 110 grams**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg plus 1 egg yolk**
**¼ teaspoon baking soda**
**½ teaspoon salt**
**1¼ teaspoons xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**½ cup jam**
**2 tablespoons confectioners' sugar**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the butter and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and beat well. Continue beating until the dough comes together. The dough will be too soft to roll. Refrigerate for 1 hour.
Roll out the dough to between ⅛- and ¼-inch thickness. Working with just one cookie at a time, use a cookie cutter to cut a 3-inch circle. Place the cutout on the prepared pan and place about 1 teaspoon of jam in the center of the cookie. Imagine a triangle whose points extend to the circumference of the dough. Fold the curved parts outside the triangle upward and partially over the jam. Bring the corners of these folds together to form the three corners and pinch the edges together so the jam does not leak out. (This is much easier than it sounds.) Continue forming cookies with the remaining dough and jam until the sheet is filled.
Bake the cookies for 10 to 12 minutes, until the edges begin to brown. Let cool on wire racks. Dust with confectioners' sugar.
# _**Lemon Tassies**_
**brown rice flour**
MAKES BETWEEN 20 AND 24 TASSIES
_Like the Pecan Tassies, this cookie is made with a very slight modification to my_
_piecrust (rice-based) recipe in_ You Won't Believe It's Gluten-Free! _These_
_cookies are great to make if you've made meringues and want to use the leftover_
_egg yolks. They are also great to make if you just love lemon, like I do._
**4 ounces cream cheese**
**2 tablespoons butter**
**¾ cup brown rice flour, 95 grams**
**pinch of salt**
**1 teaspoon sugar**
**½ teaspoon xanthan gum**
**¼ teaspoon baking soda**
FILLING:
**3 egg yolks**
**½ cup sugar, 100 grams**
**⅓ cup frozen lemonade concentrate, 95 grams**
**3 tablespoons butter, 40 grams**
TOPPING (OPTIONAL):
**Confectioners' sugar**
Preheat the oven to 400°F. Lightly grease a 24-count mini-muffin pan.
For the tart shells, combine all the batter ingredients in a medium-size bowl and mix well until the dough comes together.
Drop rounded teaspoonfuls of the dough into the cups of the prepared pan and press the center with the back of a spoon to evenly form tart shells. Prick the bottom of the shells with a fork to discourage air pockets. Bake for about 10 minutes, until golden brown. Remove the shells from the oven and cool on wire racks.
For the filling, combine all the filling ingredients in a small saucepan and mix well. (The butter will melt during cooking.) Over medium heat and stirring constantly, bring the mixture to a boil and continue cooking for about a minute, then remove it from the heat to let cool. The mixture will thicken upon standing. Spoon the filling into the tart shells. Dust with confectioners' sugar, if desired, before serving.
# _**Mocha Meltaways**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_Another melt-in-your-mouth cookie hidden beneath a light coating of_
_confectioners' sugar. Cocoa and coffee combine to create this delicious flavor._
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams**
**1 egg**
**¼ cup unsweetened cocoa powder, 20 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1½ teaspoons instant coffee dissolved in 1 teaspoon hot water**
TOPPING:
**⅔ cup confectioners' sugar**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine shortening and sugar. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the brown rice flour and beat well. Add the remaining ingredients and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball. Press it together with your hands.
Roll the dough into 1-inch balls, about 1 rounded teaspoonful. (Alternatively, you can roll out the dough to ⅓-inch thickness and cut it with 1-inch cookie cutters. This makes for a very pretty presentation.) Place the cookies on the prepared pan.
Bake the cookies for 12 to 15 minutes, until lightly browned. Immediately roll them in confectioners' sugar. Let cool on wire racks before serving.
# _**Pecan Tassies**_
**brown rice flour**
MAKES BETWEEN 20 AND 24 TASSIES
_This cookie is made with a very slight modification to my piecrust (rice-based)_
_recipe in_ You Won't Believe It's Gluten-Free! _The filling is rich,_
_but not too sweet._
**4 ounces cream cheese**
**2 tablespoons butter**
**¾ cup brown rice flour, 95 grams**
**Pinch of salt**
**1 teaspoon sugar**
**½ teaspoon xanthan gum**
**¼ teaspoon baking soda**
FILLING:
**1 egg, plus 1 egg yolk**
**½ cup brown sugar, 100 grams**
**¼ teaspoon salt**
**¼ cup light corn syrup, 80 grams**
**½ teaspoon vanilla extract**
**¼ teaspoon white vinegar**
**¾ cup chopped pecans, 65 grams**
Preheat the oven to 400°F. Lightly grease a 24-count mini-muffin pan.
For the tart shells, combine all the batter ingredients in a medium-size bowl and mix well until the dough comes together.
Drop rounded teaspoonfuls of the dough into the cups of the prepared pan and press the center of each dough ball with the back of a spoon to evenly form tart shells. Place them in the hot oven for 5 minutes. Remove the pan from the oven and cool on wire racks.
Reduce the oven temperature to 350°F.
For the filling, combine all the ingredients and mix very well. Spoon the filling into the center of the shells.
Bake the cookies for about 10 minutes, until the filling is golden brown. Let cool on wire racks before serving.
**Note: Corn syrup adds a gooeyness to this pie filling and that makes it the sweetener of choice to pair with the brown sugar. Also, you can fill the shells before baking them, but the filling will not be as separate.**
# _**Pizzelle**_
**brown rice flour**
MAKES ABOUT 13 FOUR-INCH COOKIES
_A pizzelle is a wonderful, crisp, extremely thin cookie made with the use of a_
_pizzelle maker. Essentially, the machine makes two flat, circular, wafflelike_
_cookies and is very easy to use. These cookies can be wrapped (warm) around a_
_dowel to make cannoli shells, draped over a cup to form cookie bowls, used to_
_make ice cream sandwiches, or simply enjoyed as is._
**⅓ cup sugar, 75 grams**
**2 eggs**
**¼ cup oil, 50 grams**
**1 cup brown rice flour, 125 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**Pinch of salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Turn on the pizzelle machine.
In a medium-size bowl, combine the sugar and eggs. Beat until light and thick, then set aside. In another bowl, combine the oil and brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, including the egg sugar mixture, and beat well. The dough will thicken as you mix.
Place about 1 tablespoon of dough onto each pizzelle spot. Cook until very lightly browned (this is a 3½ setting on my Cuisinart model). Let cool on wire racks before serving.
# _**Pizzelle, Almond**_
**brown rice flour and almond meal**
MAKES ABOUT 14 FOUR-INCH COOKIES
_These pizzelle are perhaps the thinnest, lightest, and crispest of the ones_
_presented here. I couldn't stop eating them! With the use of vanilla, the almond_
_flavor is subtle. For a stronger flavor, opt for almond extract._
**⅓ cup sugar, 75 grams**
**2 eggs**
**¼ cup oil, 50 grams**
**¾ cup brown rice flour, 95 grams**
**¼ cup almond meal, 30 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**Pinch of salt**
**½ teaspoon xanthan gum**
**½ teaspoon vanilla or almond extract**
Turn on the pizzelle machine.
In a medium-size bowl, combine the sugar and eggs. Beat until light and thick, then set aside. In another bowl, combine the oil and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, including the egg mixture, and beat well. The dough will thicken as you mix; it should be thick.
Place about 1 tablespoon of dough onto each pizzelle spot. Cook until very lightly browned (this is a 3½ setting on my Cuisinart model). Let cool on wire racks before serving.
# _**Pizzelle, Chocolate**_
**brown rice flour**
MAKES ABOUT 14 FOUR-INCH COOKIES
_If you enjoy pizzelle, please do not limit yourself to the traditional version._
_I am a firm believer that chocolate makes almost everything better._
_This version is not very sweet. Increase the sugar by a tablespoon or so if_
_you'd like a sweeter cookie._
**⅓ cup sugar, 75 grams**
**2 eggs**
**¼ cup oil, 50 grams**
**¾ cup brown rice flour, 95 grams**
**¼ cup unsweetened cocoa powder, 20 grams**
**¼ teaspoon baking soda**
**½ teaspoon baking powder**
**Pinch of salt**
**¼ teaspoon xanthan gum**
**½ teaspoon vanilla extract**
Turn on the pizzelle machine.
In a medium-size bowl, combine the sugar and eggs. Beat until light and thick, then set aside. In another bowl, combine the oil and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, including the egg mixture, and beat well. The dough will thicken as you mix.
Place about 1 tablespoon of dough onto each pizzelle spot. Cook until very lightly browned (this is a 3½ setting on my Cuisinart model). Let cool on wire racks before serving.
# _**Pumpkin Sandwich Cookies (Scooter Pies)**_
**sorghum flour**
MAKES ABOUT 11 SANDWICH COOKIES
_Are you a fan of the pumpkin cake roll that has a cream cheese filling? This soft_
_pumpkin cookie is a cross between a whoopie pie and a pumpkin cake roll._
**⅓ cup oil, 65 grams**
**½ cup brown sugar, 100 grams**
**1 cup sorghum flour, 135 grams**
**1 egg**
**½ cup canned pumpkin**
**1 teaspoon baking soda**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon pumpkin pie spice**
**½ teaspoon vanilla extract**
FILLING :
**⅓ cup shortening, 70 grams**
**¾ cup confectioners' sugar, 90 grams**
**4 ounces cream cheese, 110 grams**
**½ teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the sorghum flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together; it will be soft and airy.
Drop rounded tablespoonfuls of the dough onto the prepared pan. Press to ¼-inch thickness with moist fingertips.
Bake for 10 minutes, until the cookies just begin to brown at the edges. Let cool on wire racks.
For the filling, blend all the ingredients until light and fluffy. Press a tablespoon of filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Red Velvet Sandwich Cookies (Scooter Pies)**_
**brown rice flour**
MAKES ABOUT 10 SANDWICH COOKIES
_This is a popular sandwich cookie in my area: a soft red velvet cookie with a_
_cream cheese filling. The flavor is hard to define . . . although it does contain_
_cocoa, it's not chocolate._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 tablespoon unsweetened cocoa powder**
**1 egg**
**½ cup plain yogurt, 120 grams**
**1 teaspoon baking soda**
**½ teaspoon salt**
**¾ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1 teaspoon liquid red food coloring**
FILLING :
**⅓ cup shortening, 70 grams**
**¾ cup confectioners' sugar, 90 grams**
**4 ounces cream cheese, 110 grams**
**½ teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. The dough will be a thick, cakelike batter.
Drop rounded tablespoonfuls of the dough well apart onto the prepared pan. The cookie will spread during baking.
Bake for 9 to 10 minutes, until the cookies take on a bit of color at the edges and the tops appear dry. Let cool on wire racks.
For the filling, blend all the ingredients until light and fluffy. Press a tablespoon of filling between two cookies (bottom sides facing) and press together to form a sandwich. Repeat to make the other sandwiches.
# _**Rugalach**_
**brown rice flour**
MAKES 24 COOKIES
_Creating a gluten-free dough that does not bind with a filling is difficult. Often,_
_prebaking a shell or crust is the answer to this problem. But I had to figure out_
_something different for rugalach. Finally, an answer bounced into my head—to_
_use corn syrup as part of the sweetener. A brown sugar-nut-raisin filling is_
_used here, but your favorite jam would make a great alternative!_
**⅓ cup shortening, 70 grams**
1 **¾** **cups brown rice flour, 220 grams**
**¼ cup corn syrup, 80 grams**
**1 egg**
**¼ cup sugar, 50 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon vanilla extract**
**1½ teaspoons xanthan gum**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
For the dough, in a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining batter ingredients and mix well. The cookie dough will be quite heavy in texture. Once the dough comes together, continue beating it for an additional minute or so to make the dough easier to handle. It will be very soft and should be refrigerated for at least 30 minutes.
For the filling, combine all the ingredients in a small bowl and mix well. Set aside.
FILLING :
**½** **cup chopped pecans, 60 grams**
**¼ cup raisins, chopped finely, 40 grams**
**2 tablespoons brown sugar**
Divide the dough in half. Roll out each half into a 9-inch circle. Spread each circle with the filling, then cut it into twelve wedges (like pie slices). Roll each wedge into a crescent shape, rolling from the broad perimeter to the point, and place it on the prepared pan.
Bake the rugalach for 15 to 20 minutes, until golden brown. Let cool on wire racks before serving.
# _**Sand Balls**_
**brown rice flour and almond meal**
MAKES ABOUT 30 COOKIES
_This recipe is inspired by a melt-in-your-mouth holiday nut ball cookie. I've_
_used almond meal to achieve the melt-in-your-mouth texture. Although the_
_components are simple, avoid substitution. Using butter for shortening removes_
_the melting texture of the cookie. Other nut meals do not provide adequate_
_dough structure. And, finally, egg is an important part of this recipe, despite not_
_seeming correct._
**⅓ cup shortening, 70 grams**
**½ cup sugar, 100 grams**
**1 cup brown rice flour, 125 grams**
**1 egg**
**½ cup almond meal, 60 grams**
**¼ teaspoon baking soda**
**2 teaspoons baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**⅔ cup confectioners' sugar**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the brown rice flour and beat well. Add the remaining ingredients and mix well. The cookie dough will form large clumps, but will not quite come together to form a ball. Press it together with your hands.
Roll the dough into 1-inch balls, about 1 rounded teaspoonful each. Place on the prepared pan.
Bake for 12 to 15 minutes, until lightly browned. Immediately roll in confectioners' sugar. Let cool on wire racks before serving.
# _**Snickerdoodles**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_A soft, tender, buttery sugar cookie rolled in cinnamon sugar._
_As a side note, this is among the prettiest doughs in this book._
_Soft and very traditional in appearance._
**¼ cup butter, 110 grams**
**¼ cup oil, 50 grams**
**½ cup sugar, 100 grams**
**1½ cups brown rice flour, 185 grams**
**1 egg plus 1 egg yolk**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**¾ teaspoon ground cinnamon**
**3 tablespoons sugar**
Preheat the oven to 350°F. Very lightly grease a cookie sheet.
In a medium-size bowl, combine the butter, oil, and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. Continue beating until the dough comes together. The dough will be soft and almost creamy. Set aside.
For the topping, combine the cinnamon and sugar. Mix well.
Place a rounded teaspoonful of dough in the cinnamon-sugar mixture (or gently shape into balls and place in the mixture). Gently roll the dough in the sugar mixture to cover it. Place it on the prepared pan. Press it down to ¼-inch thickness.
Bake for about 10 minutes, until the edges begin to brown. Let cool on wire racks before serving.
# _**Thumbprint Cookies**_
**brown rice flour**
MAKES ABOUT 25 COOKIES
_A soft tender cookie covered with nuts and enhanced by a bit of jam._
_A very nice cookie._
**⅓ cup oil, 65 grams**
**½ cup sugar, 100 grams**
**1¼ cups brown rice flour, 155 grams**
**2 eggs**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½ teaspoon salt**
**1¼ teaspoons xanthan gum**
**1 teaspoon vanilla extract**
TOPPING:
**¾ cup very finely chopped pecans, 90 grams**
**½ cup seedless raspberry jam**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the brown rice flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and beat well. The batter will be very thick and sticky.
Drop rounded teaspoonfuls of the dough into the nuts. Gently coat the dough balls and place them on the prepared pan. Press each down to ⅓-inch thickness and then press the center with your thumb to make room for the jam.
Bake for about 10 minutes, until the cookies begin to take on color. If the cookie centers are not very indented, gently them press again with your thumb. Let cool on wire racks. Place about ½ teaspoon of jam into the center of each cookie and sprinkle with nuts, if desired.
8
_**Egg-Free Cookies**_
**This chapter covers the basics** when it comes to making egg-free versions of your favorite cookies. (In addition to the recipes in this chapter, there are also a number of naturally egg-free recipes elsewhere in this book, including the Trail Mix Bars and Rice Cereal Bars.) For simplicity, I opted to avoid egg substitutes and the sometimes recommended flaxseed alternative. To retain the moistness and texture of these cookies, I simply used honey or corn syrup as the primary sweetener. Applesauce, bananas, and pumpkin are also naturals in the egg-free baking pantry.
As a side note, honey is sweeter than corn syrup in baking. And corn syrup gives a bit of hardness to the exterior of a cookie. Corn syrup also provides a neutral flavor base for certain cookies.
Egg-free baking doesn't have to be very different from traditional baking. It only takes ordinary ingredients to benefit this special dietary need.
# _**Almond Joy-Style Cookies**_
**cornstarch or potato starch**
MAKES ABOUT 20 COOKIES
_This cookie is inspired by the Almond Joy candy bar. They require few_
_ingredients and will satisfy any sweet tooth!_
**1 (7-ounce) bag sweetened flaked coconut (2⅔ cups)**
**2 tablespoons cornstarch or potato starch**
**½ cup sweetened condensed milk**
**About 20 almonds**
CHOCOLATE COATING:
**12 ounces broken milk chocolate bars, 340 grams**
**½ teaspoon vanilla extract**
Preheat oven to 325°F. Lightly grease a mini-muffin pan.
In a medium-size bowl, combine the coconut, cornstarch, and condensed milk. Beat very well.
Drop rounded teaspoonfuls of the dough into each cup of the muffin pan. Top each cookie with an almond.
Bake the cookies for 13 to 15 minutes, until the tops just begin to brown. Allow them to cool briefly before removing them from the pan. (To do so, loosen them with the tip of a knife if necessary.) Set aside.
For the chocolate coating, I suggest making two batches of coating so that it does not harden before you have used it all. Place half the chocolate bars in a microwave-safe bowl and cook on HIGH for 1 to 2 minutes, until melted. Stir in half the of vanilla. Stir until creamy.
Dip the cookies into the chocolate and place them on waxed paper to cool.
# _**Crackerdoodles**_
**brown rice flour**
MAKES ABOUT 60 ONE-INCH SQUARE COOKIES
_These cookies are a cross between a cracker and a snickerdoodle. When you_
_need just a little something sweet, these cookies are perfect. My friends Erin and_
_Michael Good are responsible for the creative title for these treats._
**½ cup butter**
**1½ cups brown rice flour, 185 grams**
**⅓** **cup sugar, 75 grams**
**½** **teaspoon baking soda**
**½** **teaspoon salt**
**1½ teaspoons xanthan gum**
**1 teaspoon vanilla extract**
**½** **cup plain low-fat yogurt, 120 grams**
TOPPING:
**1 tablespoon sugar**
**½** **teaspoon ground cinnamon**
Preheat the oven to 375°F. Lightly grease a baking sheet.
In a medium-size bowl, combine all the batter ingredients, except the yogurt. Beat until fine crumbs form. Add the yogurt and beat until the dough comes together.
Press the dough as thinly and evenly as possible onto a baking sheet. Ideally, the dough will just about cover a 12 by 17-inch sheet pan, at ⅛-inch thickness or less. Use a sharp knife or pizza wheel to deeply score a grid pattern across the dough to create 1-inch squares. Use a fork to pierce holes throughout the tops of the cookies.
Combine the sugar and cinnamon. Sprinkle the dough with the cinnamon-sugar mixture.
Bake the cookies for 15 to 20 minutes, until the tops are golden brown. The cookies should be crisp. Break the cookies along the scored lines. Let cool on wire racks before serving.
**Note: An empty salt shaker is ideal for mixing and sprinkling the cinnamon-sugar mixture.**
# _**Egg-Free Applesauce Bars**_
**brown rice flour**
MAKES 15 COOKIES
_I designed these lower-fat, not-too-sweet bars to be reminiscent of bread_
_pudding, but with a dense, cakelike texture. I hope you enjoy them as much as I_
_enjoyed creating them! I've also included an optional cream cheese icing, should_
_you like to counteract any low-fat benefits._
**¼ cup oil, 50 grams**
**1½ cups brown rice flour, 185 grams**
**⅓** **cup honey, 105 grams**
**¾** **cup applesauce**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon ground nutmeg**
**¾** **teaspoon xanthan gum**
**½** **cup chopped raisins, 80 grams**
ICING:
**2 tablespoons shortening, 25 grams**
**½** **cup confectioners' sugar, 60 grams**
**3 ounces cream cheese, 85 grams**
Preheat the oven to 325°F. Lightly grease an 8-inch square baking pan.
In a medium-size bowl, combine the oil and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining cookie ingredients and mix well. The dough will be the consistency of thick batter.
Spread the dough evenly in the prepared pan. Bake for about 25 minutes until a toothpick tests cleanly. Cool.
For the icing, mix the ingredients together until well blended and spread it over the cookies. Let cool completely and cut into bars.
# _**Egg-Free Banana Bars with Browned Butter Icing**_
**brown rice flour**
MAKES 15 COOKIES
_The browned butter icing adds a nice contrast to this not-too-sweet cookie bar._
**¼ cup oil, 50 grams**
**1½ cups brown rice flour, 185 grams**
**⅓** **cup honey, 105 grams**
**¾** **cup smashed bananas (2 to 3), or 6-ounce jar baby food bananas**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla**
**¾** **teaspoon xanthan gum**
**½** **cup chopped nuts (optional)**
FOR ICING:
**¼ cup butter**
**1½ tablespoons milk**
**1½ cups powdered sugar**
Preheat the oven to 325°F. Lightly grease an 8-inch square pan.
In medium-size bowl, combine the oil and brown rice flour. Beat well to fully coat flour. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be the consistency of thick batter.
Spread in the prepared pan. Bake for approximately 25 minutes until a toothpick tests dry. Cool.
For icing, cook the butter over medium heat in a small pan, until lightly browned. The butter will become fragrant and almost nutty. Place the melted butter and the milk in the mixing bowl and gradually add the powdered sugar, while mixing, to form a smooth icing. Spread over the cookies.
# _**Egg-Free Brownies**_
**brown rice flour**
MAKES ABOUT 15 BROWNIES
_These are moist, cakelike brownies, identical to egg-containing brownies . . ._
_no. Very tasty . . . yes!_
**⅓** **cup shortening, 70 grams**
**1½ cups brown rice flour, 185 grams**
**½** **cup honey, 160 grams**
**¼** **cup sugar, 50 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**⅓** **cup water, 65 grams**
**½ teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon xanthan gum**
Preheat the oven to 325°F. Lightly grease an 8-inch square pan.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The brownie dough will be quite heavy in texture, but will come together with continued beating.
Spread the batter evenly in the prepared pan. Bake for about 25 minutes, until a toothpick tests cleanly. (The toothpick is unlikely to hold any batter, but will seem wet, when tested, if the brownies are not done.) When fully baked, the brownies will also begin to pull away from the sides of the pan.
Let cool completely and cut into bars.
# _**Egg-Free Chocolate Chip Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_For greater "brown sugar" flavor, you may wish to use a tablespoon of molasses_
_in place of 1 tablespoon of honey, use a darker color honey,_
_or simply enjoy as they are._
**⅓ cup shortening, 70 grams**
**1¾ cups brown rice flour, 220 grams**
**½** **cup honey, 160 grams**
**½ teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**2 teaspoons vanilla extract**
**1 teaspoon pumpkin pie spice**
**1¼ teaspoons xanthan gum**
**½** **cup chocolate chips, 80 grams**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be quite heavy in texture.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Using your fingertips, press them to ¼-inch thickness. Bake the cookies for 7 to 9 minutes, until the bottom edges begin to brown and the tops take on a little color.
# _**Egg-Free Chocolate Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_These cookies are probably my favorite egg-free cookie in this book, although the_
_chocolate chip cookies are a close second. This cookie has a very chewy brownie_
_texture. For a more traditional texture, use honey instead of corn syrup and_
_omit_ **¼** _cup of the sugar._
**⅓** **cup shortening, 70 grams**
**1½ cups brown rice flour, 185 grams**
**½** **cup corn syrup, 160 grams**
**½ cup sugar, 100 grams**
**⅓** **cup unsweetened cocoa powder, 30 grams**
**½** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**¾** **teaspoon xanthan gum**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be quite heavy in texture.
Drop rounded teaspoonfuls of the dough well apart onto the prepared pan. The cookies will spread during baking. Bake for 7 to 9 minutes until the bottom edges begin to brown and the tops take on a little color. Allow the cookies to cool briefly on the pan for easier removal.
# _**Egg-Free Gingerbread Men**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_This is a mild gingerbread cookie. For a richer flavor, substitute a few_
_tablespoons of unsulfured molasses for a few tablespoons of the honey._
_Refrigeration of this dough is critical for rolling. For faster cookies, simply drop_
_teaspoonsful of dough on the cookie sheet and bake._
**⅓** **cup shortening, 70 grams**
**1¾ cups brown rice flour, 220 grams**
**½** **cup honey, 160 grams**
**¼ teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½ teaspoon vanilla extract**
**1¼ teaspoons ground ginger**
**2 teaspoons xanthan gum**
TOPPING:
**Icing, decorative candies, and/or raisins**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Once the dough comes together, continue beating for an additional minute or so to make the dough easier to handle. The dough will be very soft and ideally should be refrigerated for 30 minutes if you plan to roll it out and cut it, rather than make drop cookies. (The most adept at rolling cookies will prove me wrong on this point.)
Roll out the dough to a scant ¼-inch thickness and cut it with gingerbread men cookie cutters. Place decorative candies or raisins on the cookies if desired. Bake for 7 to 9 minutes, until the edges just begin to brown. Let cool on wire racks. The cookies will stiffen while cooling. Decorate with icing, if desired.
# _**Egg-Free Oatmeal Cookies**_
**oats and brown rice flour**
MAKES ABOUT 30 COOKIES
_These cookies are not-too-sweet, soft, and slightly chewy. They are the perfect_
_balance between light and heavy, being right in the middle._
_I can't imagine not enjoying these._
**¾** **cup rolled oats, 70 grams**
**⅓** **cup shortening, 70 grams**
**1** **½** **cups brown rice flour, 185 grams**
**½** **cup honey, 160 grams**
**½** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**2 teaspoons vanilla extract**
**1 teaspoon xanthan gum**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
Place the oats in a blender and pulse until they are partially broken down. Pieces of all sizes should remain. Set aside. In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, including the oats, and mix well. The cookie dough will be very heavy to the beaters, but soft to the touch.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Using your fingertips, press them to ¼-inch thickness. The cookies will not spread much during baking. Bake for 8 to 10 minutes, until the bottom edges begin to brown. Let cool on wire racks before serving.
# _**Egg-Free Peanut Butter Cookies**_
**brown rice flour**
MAKES ABOUT 35 COOKIES
_You can add_ ½ _cup of chocolate chips to these if you've got a chocolate_
_and peanut butter craving. Or keep them plain for an especially good_
_traditional cookie! The slight undertone of the honey works nicely with_
_the overall peanut butter flavor._
**⅓** **cup shortening, 70 grams**
**½** **cup peanut butter, 140 grams**
**1** **½** **cups brown rice flour, 185 grams**
**½** **cup honey, 160 grams**
**½** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon xanthan gum**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening, peanut butter, and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be tacky and soft.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Using the tines of a fork dipped in brown rice flour (or sprayed lightly with nonstick spray), press them to ¼-inch thickness. Bake for 8 to 10 minutes, until the bottom edges begin to brown and the tops take on a little color. Let cool on wire racks before serving.
# _**Egg-Free Pumpkin Bars**_
**brown rice flour**
MAKES ABOUT 15 COOKIES
_I developed these bars just before the Thanksgiving holiday. I love pumpkin pie_
_and put that flavor in these quick cookie bars. These bars have a slightly dense,_
_cakelike texture and are bright in flavor. Although a little icing would be pretty,_
_a liberal sprinkling of confectioners' sugar is more than sufficient. If you want a_
_slightly chewier texture, increase the xanthan gum to 1 teaspoon._
**⅓** **cup shortening, 70 grams**
**1** **½** **cups brown rice flour, 185 grams**
**½** **cup honey, 160 grams**
**¼** **cup brown sugar, 50 grams**
**½** **cup pumpkin puree, 100 grams**
**½** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon pumpkin pie spice**
**¾** **teaspoon xanthan gum**
TOPPING:
**½** **cup confectioners' sugar**
Preheat the oven to 325°F. Lightly grease an 8-inch square pan.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be extremely soft.
Spread the dough evenly in the prepared pan. Bake for about 30 minutes, until a toothpick tests cleanly. (If the bars are not done, the toothpick is unlikely to hold any batter, but will seem wet.) Let cool.
Sprinkle the bars liberally with confectioners' sugar before slicing. Let cool, then cut into bars.
**Note: These cookies will crumble if cut when still hot. Allow to cool for prettier bars!**
# _**Egg-Free Rolled Sugar Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_As with the plain sugar cookies, corn syrup is used here for the neutral flavor_
_base. The corn syrup gives almost a bit of chew to the cookie. And although_
_they are sweet, they are not sugary sweet._
**⅓** **cup shortening, 70 grams**
**1** **¾** **cups brown rice flour, 220 grams**
**½** **cup corn syrup, 160 grams**
**¼** **cup sugar, 50 grams**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1** **½** **teaspoons xanthan gum**
TOPPING:
**Icing and/or sprinkles**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be quite heavy in texture. Once the dough comes together, continue beating for an additional minute or so to make it easier to handle. The dough will be very soft and may be refrigerated for easier handling.
Roll out the dough to a scant ¼-inch thickness and cut it with your favorite cookie cutters. Top with sprinkles, if desired. Bake the cookies for 7 to 9 minutes, until the edges just begin to brown. Let cool on wire racks. Decorate with icing, if desired.
# _**Egg-Free Spice Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_This lightly flavored spice cookie is soft and pleasant._
_I prefer it with the raisins, but I do chop them for what I think is better flavor._
_The cookies have a soft yet tight crumb._
**⅓** **cup shortening, 70 grams**
**1** **¾** **cups brown rice flour, 220 grams**
**½** **cup honey, 160 grams**
**½** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon pumpkin pie spice**
**1** **¼** **teaspoons xanthan gum**
**⅓** **cup roughly chopped raisins (optional), 55 grams**
Preheat the oven to 325°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be quite heavy in texture.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Use your fingertips to press them to ¼-inch thickness. Bake for 7 to 9 minutes, until the bottom edges begin to brown and the tops take on a little color. Let cool on wire racks before serving.
# _**Egg-Free Sugar Cookies**_
**brown rice flour**
MAKES ABOUT 30 COOKIES
_These cookies are a little light and crispy. Corn syrup is not as sweet as honey, but_
_provides the neutral base needed for a plain sugar cookie. You may use a very,_
_very light honey instead, but you will need to omit the sugar if you go that route._
**⅓** **cup shortening, 70 grams**
**1** **¾** **cups brown rice flour, 220 grams**
**½** **cup corn syrup, 160 grams**
**¼** **cup sugar, 50 grams**
**½** **teaspoon baking soda**
**1 tablespoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1 teaspoon xanthan gum**
TOPPING:
**2 tablespoons sugar or sprinkles**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The cookie dough will be quite heavy in texture.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Press to ¼-inch thickness with your fingertips. Sprinkle the tops with sugar or sprinkles, as desired. Bake the cookies for 7 to 9 minutes, until the bottom edges just begin to brown. Let cool on wire racks before serving.
# _**Fruitcake Nuggets**_
**cornstarch or potato starch**
MAKES ABOUT 20 COOKIES
_These cookies would be great to make at the same time as the Almond Joy-Style_
_Cookies (page 144) as each use just_ ½ _cup of sweetened condensed milk. These_
_will be enjoyed by most guests at your holiday party, not just the fruitcake lover!_
**1 (7-ounce) bag sweetened flaked coconut (2** **⅔** **cups)**
**2 tablespoons cornstarch or potato starch**
**½** **cup sweetened condensed milk**
**1 cup candied fruit**
**1 cup chopped walnuts**
**1 teaspoon vanilla extract**
Preheat the oven to 325°F. Lightly grease a mini-muffin pan.
In a medium-size bowl, combine the coconut and starch. Stir well. Add the remaining ingredients and mix well. Drop rounded teaspoonfuls of the dough into the cups of the prepared pan. With moist fingertips, press down the cookie dough a little so that the cookies stick together.
Bake for 13 to 15 minutes, until the tops just begin to brown. Allow the cookies to cool briefly before removing them from the pan. You can loosen them with the tip of a knife if necessary.
# _**Orange-Cream Cheese Tassies**_
**brown rice flour**
MAKES ABOUT 24 TASSIES
_Soft, orange cheesecakey filling surrounded by a crisp sugar cookie cup._
**⅓** **cup shortening, 70 grams**
**1** **¾** **cups brown rice flour, 220 grams**
**½** **cup corn syrup, 160 grams**
**¼** **cup sugar, 50 grams**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon vanilla extract**
**1** **½** **teaspoons xanthan gum**
FILLING :
**1 (8-ounce) package cream cheese**
**½** **cup confectioners' sugar, 60 grams**
**⅓** **cup orange marmalade, 120 grams**
Preheat the oven to 350°F. Lightly grease a 24-count mini-muffin pan.
For the tart shells, in a medium-size bowl, combine the shortening and brown rice flour. Beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining dough ingredients and mix well. The cookie dough will be quite heavy in texture. Once the dough comes together, continue beating it for an additional minute or so to make the dough easier to handle; it will be very soft.
Drop rounded teaspoonfuls of the dough into the cups of the prepared pan and press the center with the back of a spoon to evenly form tart shells. Set aside.
For the filling, combine all the ingredients and mix very well. Place rounded teaspoonfuls of the filling into the center of the shells.
Bake the cookies for about 9 minutes, until the shell is golden and the filling is set. Allow the cookies to cool in the pan for about 5 minutes, for easier removal. Let cool on wire racks before serving.
# _**Pepper Jack Cookies**_
**brown rice flour**
MAKES ABOUT 55 COOKIES
_To the palate, these cookies begin as a delicate, slightly sweet cookie, followed by a bit of growing, mellow heat. The salt adds to the play on the tongue. These are great served with a glass of wine or as an unexpected treat on a cookie tray._
**4 ounces pepper Jack cheese, finely grated**
**¼** **cup butter, chopped in small chunks**
**⅔** **cup brown rice flour, 85 grams**
**¼** **cup sugar, 50 grams**
**½** **teaspoon xanthan gum**
**½** **teaspoon baking powder**
**½** **teaspoon salt**
TOPPING:
**Salt**
Preheat the oven to 375°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine all the ingredients. Beat (or mix with your fingertips) until the dough forms a soft ball.
Roll out the dough to ¼-inch thickness. Cut it into cookies with a 1-inch cookie cutter and place them on the prepared pan. Bake for 8 to 10 minutes, until the bottoms of the cookies are tinged with color. Let cool on wire racks before serving.
**Note: Be careful as you bake these cookies, as they are very easy to burn on the bottom. The browning will barely be visible from a side view.**
9
_**Cookies Made with Other Gluten-Free Flours**_
**In baking gluten-free, we are often tempted** to purchase many different flours, either because they are new and exciting (as was the case when coconut flour first came out), for nutritional content (such as bean flours, or certain pseudograins), or simply because they are called for in a blend. It is hard to get to know the nature of a flour when it is just one of many. The old theory that you must use a blend to bake good foods hasn't helped, either.
While the other recipes in this book call for either brown rice flour or sorghum flour, the cookies in this chapter use a variety of flours, just one at a time, to let their character show through. We may even make a dent in that tower of flours you've accumulated over time. You can make a cookie out of almost any gluten-free flour. But that doesn't mean you want to eat them. And, for that reason, a number of stronger-tasting flours, such as soy and teff, have simply not been utilized in this chapter.
Please be sure to try the Chocolate Crinkles made with garbanzo bean flour. They are one of my favorites. Oatmeal, potato starch, cornstarch, and even tapioca starch do a cookie proud in this chapter!
And, finally, at the risk of being extremely blunt, I would describe the flavor of coconut flour on its own as tasting like tropical dirt. Used with other tropical flavors, it is much more palatable. It is also not very responsive to leavening. However, it does make a very pretty, flat cookie. It would be ideal to use in making cookie ornaments: expensive, but ideal.
If you have the time, I strongly encourage you to bake with just one flour at a time. It surely helps in figuring out what you like in a flour, what suits your tummy, and ultimately which flours are worth the price.
# _**Blueberry-Cranberry Jumbles**_
**potato starch**
MAKES ABOUT 25 COOKIES
_A bright and healthy change from chocolate chip, the fruit flavor in this cookie_
_shines. If you are a fan of blueberry pancakes, try the recipe with just the dried_
_blueberries. Tasty! The cookie base is very tender, quite sweet, and a good_
_contrast to the chewy bits of fruit._
**⅓** **cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1 cup potato starch, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**⅓** **cup coarsely chopped dried blueberries, 60 grams**
**⅓** **cup coarsely chopped dried, sweetened cranberries, 40 grams**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the potato starch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients, except the blueberries and cranberries, and mix well. The dough will be like a thick cake batter. Stir in the berries.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Bake the cookies for 8 to 9 minutes, until the bottom edges are browned. Let cool on wire racks before serving.
# _**Chocolate Biscotti**_
**oat flour**
MAKES ABOUT 30 COOKIES
_A simple, traditional biscotti, adorned with just a little flavored icing. Nuts,_
_chocolate chips, or other tasty bits would make a nice addition._
**⅓ cup shortening, 70 grams**
**¼ cup sugar, 100 grams**
**1 cup oat flour, 120 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**Pinch of cayenne (optional)**
TOPPING:
**⅔** **cup confectioners' sugar**
**1 tablespoon of your favorite gluten-free liqueur (or water)**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the remaining ingredients and mix well. Scrape down the sides of the mixing bowl at least once during mixing. The dough will be soft and heavy.
Divide the dough in half. Shape each portion into a flattened log about ½-inch tall and place on the prepared pan. (This is a good opportunity to mix in any nuts, chocolate chips, or other tasty goodies, if desired.) Do the same with the other half of the dough.
Bake the logs for 20 minutes. Let cool completely. Cut each log into slices about ½-inch thick and place these back on the prepared pan. Bake the biscotti for an additional 10 to 15 minutes, until quite dry. Let cool again.
Combine the confectioners' sugar and liqueur in a small cup and stir to dissolve. Spread the mixture over the tops of the cookies.
# _**Chocolate Cookies, Dairy-Free**_
**cornmeal**
MAKES ABOUT 30 COOKIES
_Corn teams up with chocolate for a crisp, tender cookie that is almost chewy,_
_with a hint of grit. This is a chocolate cookie that will please most anyone. Add_
_the optional drizzle of chocolate to make them prettier if you like._
**⅓** **cup shortening, 65 grams**
**¾** **cup sugar, 150 grams**
1 **¼** **cups cornmeal, 150 grams (120 grams per cup)**
**⅓ cup cocoa, 30 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**2 teaspoons baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING (OPTIONAL):
**2 ounces chocolate**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the cornmeal and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be gooey and pasty.
Drop rounded teaspoonfuls of the dough onto the prepared baking pan. Using wet fingertips, press to ¼-inch thickness.
Bake for 8 to 10 minutes, until the edges look dry and the tops are slightly crackled. Do not overbake. Allow the cookies to cool on wire racks. Place the chocolate in a microwave-proof plastic cup and cook on HIGH for 1 minute, then drizzle the tops of the cookies with the melted chocolate, if desired.
# _**Chocolate Crinkles**_
**garbanzo bean flour**
MAKES ABOUT 25 COOKIES
_This cookie has a rich chocolate flavor and is a little chewy. You will be_
_hard-pressed to identify the bean flour. All in all, a very good cookie._
**⅓ cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1¼ cups garbanzo bean flour, 135 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½ teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**2 tablespoons water**
TOPPING:
**¼** **cup confectioners' sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the garbanzo bean flour and cocoa and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be heavy and sticky.
Place the confectioners' sugar in small bowl. Drop rounded teaspoonfuls of the dough into the confectioners' sugar and roll to coat well. Place them onto the prepared pan and use your fingertips to press them to about ¼-inch thickness.
Bake for 8 to 9 minutes, until the tops are dry. The cookies spread little, if any, during baking. Let cool on wire racks before serving.
# _**Cloud Cookies**_
**cornstarch**
MAKES ABOUT 25 COOKIES
_Pillow soft and not too sweet. These cookies are topped with finely chopped_
_chocolate or nuts. They are both plain and sophisticated at the same time._
**⅓ cup oil, 65 grams**
**⅓ cup sugar, 75 grams**
**1¼ cups cornstarch, 155 grams**
**1 egg yolk**
**2 egg whites**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½ teaspoon xanthan gum**
**½ teaspoon vanilla extract**
TOPPING:
**½** **cup finely chopped chocolate chips**
**and/or**
**½** **cup finely chopped nuts**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be as thin as cake batter.
Drop rounded teaspoonfuls of the dough well apart onto the prepared pan. Sprinkle the tops with chocolate bits and/or nuts. Bake for 8 to 9 minutes, until the bottom edges are browned. The cookies will spread during baking. Let cool on wire racks before serving.
**Note: I like to melt a small chocolate bar and spread it on the top of the cookies and sprinkle them with nuts.**
# _**Coconut Chip Cookies**_
**coconut flour**
MAKES ABOUT 30 COOKIES
_The chopped coconut and chocolate chips make this cookie come together._
_Without those embellishments, these cookies would have a "muddy" coconut_
_undertone that doesn't shine as well. (But alone, that base cookie would make a_
_good substitute for the base cookies in the great fake Samoas, page 102)._
**⅓ cup oil, 65 grams** i
**½** **cup sugar, 100 grams**
**1 cup coconut flour, 140 grams**
**2 eggs**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**¼** **cup water**
**¾ cup coarsely chopped chocolate chips**
**½** **cup coarsely chopped sweetened flaked coconut**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine all the ingredients and mix well. Scrape down the sides of the mixing bowl at least once during mixing. The dough will come together and be very thick and soft.
Roll out to ¼-inch thickness and cut into desired shapes. Bake for 8 to 9 minutes, until lightly browned at the edges. Let cool on wire racks before serving.
# _**Fortune Cookies and Pirouettes**_
**tapioca starch**
MAKES ABOUT 15 COOKIES
_In making these, the goal is to have a pliable disk of cookie that can be quickly_
_shaped before it cools into a crisp cookie. Please know that these cookies will be_
_an effort in patience and diligence to get them baked for just the right amount of_
_time—too long and the cookie will be burnt; too short and the cookie will have_
_an unwanted softness to the bite. Baking several single test cookies should give_
_you the exact timing for your oven. Remember to include great fortunes!_
**2 egg whites**
**½** **cup sugar, 100 grams**
**1 cup tapioca starch, 90 grams**
**1 tablespoon shortening**
**3 tablespoons milk**
**¼** **teaspoon xanthan gum**
**Pinch of baking soda**
Preheat the oven to 325°F. Lightly grease a cookie sheet. Prepare slips of paper with fortunes, if desired.
Mix all the ingredients in a medium-size bowl, until well blended. (A whisk is useful for this.) Be sure no lumps remain.
For each cookie, drop scant tablespoonfuls of the batter well apart onto the prepared pan. The batter will be thin. If the batter thickens upon standing, spread it out into a 3-inch circle. Place up to four rounds of batter on the prepared pan.
Bake for 8 to 10 minutes, until the edges of the cookies are golden brown. The bottom of the cookies will also be browned.
For fortune cookies: Remove a cookie from the pan with a spatula and place a fortune in the center. Fold the cookie in half and then bring the two ends together (while pushing the center fold area with your finger) to shape the cookie. Allow to cool.
For pirouette cookies: Remove a cookie from the pan with a spatula and quickly roll the cookie around a pencil or similar thin item. Let cool on wire racks before serving.
**Note: Tapioca starch measures quite differently from the amounts listed on the package. If at all possible, please measure your flour by weight.**
# _**Iced Oatmeal Cookies**_
**oats and oat flour**
MAKES ABOUT 30 COOKIES
_This cookie is fashioned after the Iced Oatmeal Cookies from Trader Joe's. They_
_are hard, crispy, and full of both vanilla and cinnamon flavors. Please_
_remember to use only safe oats in your gluten-free baking!_
**1 cup rolled oats, 85 grams**
**⅓ cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1 cup oat flour, 120 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**½** **teaspoon ground cinnamon**
TOPPING:
**⅔** **cup confectioners' sugar**
**1 tablespoon water**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
Place the oats in a blender and pulse until most pieces are about a quarter of the original size. In a medium-size bowl, combine the shortening and sugar. Beat well. Add the cut oats and the remaining ingredients and mix well. Scrape down the sides of the mixing bowl at least once during mixing. The dough will be soft and heavy.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Press them with your fingertips (or the base of a glass) to about ⅛-inch thickness.
Bake the cookies for 10 to 11 minutes, until lightly browned. Let cool on wire racks.
Combine the confectioners' sugar and water in a small cup. Stir to dissolve. Spread the mixture over the tops of the cookies.
# _**Love Letter Rolled Chocolate Sugar Cookies**_
**tapioca starch**
MAKES ABOUT 14 COOKIES
_Cut into mini envelopes, these cookies are a pretty way to enjoy a crisp, tender,_
_chocolaty cookie. Decorate with a little icing to bring the envelopes to life._
**⅓ cup shortening, 70 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups tapioca starch, 135 grams**
**⅓ cup unsweetened cocoa powder, 30 grams**
**1 egg plus 1 egg yolk**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In medium-size bowl, combine the shortening and sugar. Beat well. Add the tapioca starch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will just come together and be quite heavy.
Roll out the dough to ⅛-inch thickness and cut into 2 by 4-inch rectangles. Bake for 8 to 9 minutes, until the edges take on a little color and the tops are dry. Let cool on wire racks. Decorate with icing to mimic a letter or envelope.
# _**Oatmeal Cookies**_
**cornstarch and oats**
MAKES ABOUT 30 COOKIES
_These cookies are soft with simple vanilla flavoring. Feel free to spice them up_
_with a little cinnamon and/or nutmeg! Or, add my kids' favorite: chocolate chips._
**⅓ cup oil, 65 grams**
**½** **cup brown sugar, 100 grams**
**1** **¼** **cups cornstarch, 155 grams**
**⅔** **cup rolled oats**
**2 eggs**
**½** **teaspoon baking soda**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**½** **teaspoon vanilla extract**
**1 cup chocolate chips (optional)**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the cornstarch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will seem like a thick, sticky batter.
Drop rounded teaspoonfuls of the dough well apart onto the prepared pan. With moistened fingertips, press them to ¼-inch thickness. Bake for 8 to 9 minutes, until the bottom edges are browned. The cookies will spread during baking. Let cool on wire racks before serving.
# _**Pumpkin Cookies**_
**garbanzo bean flour**
MAKES ABOUT 25 COOKIES
_This cookie is a moist, light, and nicely spiced cookie. But this is not raw_
_dough that you will want to nibble on. Not even a little bit—it's beany! Bean_
_flour is a taste that's not for everyone. However, baked and cooled, this is a_
_pretty good cookie._
**⅓ cup oil, 65 grams**
**½** **cup sugar, 100 grams**
**1** **¼** **cups garbanzo bean flour, 135 grams**
**1 egg**
**½** **cup pumpkin puree**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**½** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
**1** **½** **teaspoons pumpkin pie spice**
**½** **cup coarsely chopped raisins**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the oil and sugar. Beat well. Add the garbanzo bean flour and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be soft and not too heavy.
Drop rounded teaspoonfuls of the dough onto the prepared pan. Press them with your fingertips to a scant ¼-inch thickness.
Bake the cookies for 8 to 9 minutes, until the edges are lightly browned. Let cool on wire racks before serving.
# _**Rolled Sugar Cookies, Dairy-Free**_
**cornmeal**
MAKES ABOUT 30 COOKIES
_This gritty flour makes a very tasty cookie. Not surprisingly, the cookie tastes_
_pleasantly of corn. Sprinkles are nice, but the addition of a light icing_
_substantially diminishes the corn flavor._
**⅓ cup shortening, 65 grams**
**½** **cup sugar, 100 grams**
**1** **½** **cups cornmeal, 180 grams (120 grams per cup)**
**1 egg**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon salt**
**1 teaspoon xanthan gum**
**1 teaspoon vanilla extract**
TOPPING (OPTIONAL):
**Sprinkles or colored sugar**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the cornmeal and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. Continue beating until the dough comes together.
Roll out the dough to ⅛-inch thickness (for crispier cookies) or to ¼-inch thickness (for a bit softer) and cut it with a 2-inch round cookie cutter (or other cookie cutter of your choice).
Place the cookies on the prepared cookie sheet and top with sprinkles or colored sugar, if desired. Bake for 8 to 10 minutes, until they have the slightest hint of color.
Let cool on wire racks.
# _**Rosettes**_
**potato starch and cornstarch**
MAKES ABOUT 100 COOKIES
_There are times when some flours are just better suited to do a job. This is_
_especially true when a light batter is fried. Enjoy these pretty, light cookies with_
_just a bit of confectioners' sugar sprinkled on top. Rosettes are the "cookie"_
_version of a funnel cake. In making these, I used an antique rosette mold_
_from my mother-in-law, Lucile._
**½** **cup potato starch, 80 grams**
**½** **cup cornstarch, 65 grams**
**1 tablespoon sugar**
**¼** **teaspoon baking soda**
**1 teaspoon baking powder**
**½** **teaspoon xanthan gum**
**½** **teaspoon vanilla extract**
**Pinch of salt**
**1 egg**
**⅔** **cup milk**
FOR FRYING:
**2 cups canola oil**
TOPPING:
**½** **cup confectioners' sugar**
Prior to frying, heat the oil to 370°F.
In a medium bowl, combine the potato starch, cornstarch, sugar, baking soda, baking powder, xanthan gum, vanilla, and salt. Add the egg. Slowly add the milk, mixing well to remove all lumps. Transfer the batter to a shallow bowl. The batter will thicken a lot over the course of making the rosettes; this is fine.
Dip the rosette iron into hot oil to heat for about 30 seconds. (Lift to drain off excess oil.) Then, dip the rosette iron into the batter to coat it most of the way up its sides. (It is very likely that you will need to dip, lift up, and dip again to get sufficient coating.) Do not cover the top of the iron or the cookie will not slide off. Return the iron to the hot oil and fry the rosette for 25 to 30 seconds, until golden brown. Lift it from the oil and, using a fork, gently ease the rosette from the iron and place it on paper towels to drain. Continue until all the batter is used.
Dust the tops with confectioners' sugar.
# _**Spritz**_
**potato starch**
MAKES ABOUT 36 COOKIES
_Pretty, delicate, and delicious._
_Dip them in dark chocolate for a special treat!_
**⅓ cup shortening, 70 grams**
**⅓ cup sugar, 75 grams**
**1 cup potato starch, 155 grams**
**1 egg**
**¼** **teaspoon baking soda**
**½** **teaspoon salt**
**¾** **teaspoon xanthan gum**
**1 teaspoon vanilla extract**
Preheat the oven to 350°F. Lightly grease a cookie sheet.
In a medium-size bowl, combine the shortening and sugar. Beat well. Add the potato starch and beat well. Scrape down the sides of the mixing bowl at least once during mixing. Add the remaining ingredients and mix well. The dough will be soft.
Using a pastry bag, pipe the cookies well apart onto the prepared pan. As the cookies spread a lot during baking, be sure to leave room between the cookies. Bake for 8 to 9 minutes, until the bottom edges are browned. Let cool on wire racks before serving.
Fig Newton-Style Cookies, page 97
_**Appendix**_
_**Gluten-Free Resources**_
# **National Gluten-Free Support Groups**
**American Celiac Society**
www.americanceliacsociety.org
PO Box 23455
New Orleans, LA 70183
504-737-3293
**Celiac Disease Foundation**
www.celiac.org
13251 Ventura Boulevard, #1
Studio City, CA 91604
818-990-2354
**Celiac Sprue Association/USA Inc.**
www.csaceliacs.org
PO Box 31700
Omaha, NE 68131
877-CSA-4CSA
**The Gluten Intolerance Group of North America**
www.gluten.net
31214 124th Ave SE
Seattle, WA 98092
253-833-6655
# **Local Celiac Support Groups**
**Celiac.com**
www.celiac.com
Scroll down the home page to locate index,
then click on support groups.
# **Gluten-Free Mail Order Suppliers**
We are fortunate that there are now numerous gluten-free food suppliers in the United States. Even better, most of what you need can be found at your local grocery store or health food store. A Web search of "gluten-free foods" will give you hundreds of options for high-quality gluten-free food suppliers, but you really only need a few. Here are several of the best:
Hot Chocolate Cookies, page 30
**Amazon.com**
www.amazon.com
A surprising home to many gluten-free foods and baking supplies. You'll save if ordering in quantity, but be sure you like the item before you order in bulk. Many gluten-free books can be ordered quite reasonably there as well.
**Breads from Anna by Gluten Evolution, LLC**
www.glutenevolution.com
Iowa City, Iowa
319-354-3886
877-354-3886
This small company has mixes for some very good breads. In my opinion, they made the best overall bread at the last conference I attended. And, that is why they are listed here.
**Celiac.com**
www.celiac.com
Home to the "celiac mall," which includes numerous suppliers of gluten-free foods, books, and so on.
**The Gluten-Free Pantry**
www.glutenfree.com
PO Box 840
Glastonbury, CT 06033
860-633-3826
Gluten-free baking supplies and mixes.
**Ener-G Foods**
www.energyfoods.com
PO Box 84487
5960 1st Avenue South
Seattle, WA 98124
800-331-5222
# **Manufacturers of Safe Oats**
**Bob's Red Mill**
www.bobsredmill.com
5209 S.E. International Way
Milwaukie, OR 97222
800-553-2258
Gluten-free oats, baking supplies, etc.
**Gifts of Nature, Inc.**
www.giftsofnature.net
810 7th St. E, #17
Polson, MT 59860
888-275-0003
**Cream Hill Estates**
www.creamhillestates.com
9633 rue Clement
LaSalle, Quebec
Canada H8R 4B4
514-363-2066
1-866-727-3628
**Gluten Free Oats**
www.glutenfreeoats.com
578 Lane 9
Powell, WY 82435
307-754-2058
# **My Favorite Gluten-Free Books**
_Celiac Disease: A Hidden Epidemic_ by Dr. Peter Green. Dr. Green takes the reader through the sometimes complicated and intimidating world of gluten-free living. The serious medical content of this book is softened by Dr. Green's straightforward, down-to-earth writing style. The questions and struggles of real patients peppered throughout the work put a human face on the disease.
_The Gluten-Free Kitchen_ by Roben Ryberg. I am naturally biased toward my first book. Several of my very favorite recipes reside there, including angel and chiffon cakes, breakfast gravies, raised doughnuts, and a number of main courses.
_You Won't Believe It's Gluten-Free!: 500 Delicious, Foolproof Recipes for Healthy Living_ by Roben Ryberg. I am most proud of this work because it meets so many dietary needs by using just one flour at a time. From appetizers through desserts, just one gluten-free flour can be amazing!
# **My Favorite Gluten-Free Magazine**
_**Living Without**_
www.livingwithout.com
PO Box 2126
Northbrook, IL 60065
# **Additional Resources for the Gluten-Free Community**
In addition to the national and local support groups, www.celiac.com is a wonderful resource for medical studies, recipes, diagnosis steps, and so on.
My favorite online discussion board is www.forums.delphi.com/celiac/start. Most important, they have adopted a "zero-tolerance" policy for inclusion of any gluten in the diet (i.e., simply picking croutons off a salad is not safe!). It is a great place to talk with other individuals who live the celiac diet every day. There is no fee for basic membership. You will sometimes find me there.
Another very good on-line discussion board is www.glutenfreeforum.com.
For vacation getaways without worry, visit www.bobandruths.com.
Maple Leaf Cookies, page 113
**Note: Thousands of helpful organizations, companies and Web sites are available to the gluten-free community. Mountains of information are readily available. After making your home safe, the next step should be joining a support group—whether national, local, or online—and learning more. And, if you're not the support-group type, learn more by visiting the Web sites included in this appendix.**
_**Index**_
## _**A**_
Almond meal
Almond Flower Cookies
Almond Layered Cookie Bars
Crisp Almond Cookies
Linzer Sandwich Cookies
Nut Meringue Wreaths
Pizzelle, Almond
Sand Balls
Almonds
Almond Biscotti
Almond Flower Cookies
Almond Joy-Style Cookies
Almond Layered Cookie Bars
Cranberry-Orange-Almond Granola Bars
Crisp Almond Cookies
Nut Meringue Wreaths
Animal Crackers
Apple Crumble Bars
Applesauce Bars, Egg-Free
Applesauce-Raisin Cookies
## _**B**_
Banana Bars with Browned Butter Icing, Egg-Free
Banana-Nut Cookies
Bar cookies
Almond Layered Cookie Bars
Apple Crumble Bars
Blondies #1
Blondies #2
Carrot Cake Bars
Cheesecake Bars
Cranberry-Orange-Almond Granola Bars
Decadent Brownies
Egg-Free Applesauce Bars
Egg-Free Banana Bars with Browned Butter Icing
Egg-Free Brownies
Egg-Free Pumpkin Bars
Jam Bars
Lemon Bars
Peanut Butter-Chocolate Chip- Oatmeal Bars
Raspberry-Cream Cheese Brownies
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Shortbread
Toffee Bars
Trail Mix Bars
Turtle Bars
White Chocolate Bars
Binders for gluten-free cookies
Biscotti, Almond
Biscotti, Chocolate
Blondies #1
Blondies #2
Blueberry-Cranberry Jumbles
Breakfast Cookies, Sour Cream
Brown rice flour
Almond Biscotti
Almond Flower Cookies
Almond Layered Cookie Bars
Animal Crackers
Apple Crumble Bars
Applesauce-Raisin Cookies
Blondies #1
Butter Cookies
Candy Cane Cookies
Carrot Cake Bars
Cheesecake Bars
Chinese Marble Cookies
Chips Ahoy!-Style Cookies
Chocolate Butter Cookies
Chocolate Chip Cookies #1
Chocolate Cookies with White Chips
Chocolate Crinkles
Chocolate Graham Crackers
Chocolate Marshmallow "Scooter" Pies
Chocolate Pinwheel Cookies
Chocolate Sandwich Cookies (Scooter Pies)
Chocolate-Cherry Cookies
Cinnabon-Style Cookies
Crackerdoodles
Cranberry-Orange-Almond Granola Bars
Crisp Almond Cookies
Egg-Free Applesauce Bars
Egg-Free Banana Bars with Browned Butter Icing
Egg-Free Brownies
Egg-Free Chocolate Chip Cookies
Egg-Free Chocolate Cookies
Egg-Free Gingerbread Men
Egg-Free Oatmeal Cookies
Egg-Free Peanut Butter Cookies
Egg-Free Pumpkin Bars
Egg-Free Rolled Sugar Cookies
Egg-Free Spice Cookies
Egg-Free Sugar Cookies
Fig Newton-Style Cookies
Filled Triangle Cookies
Gingerbread Men and Gingersnaps
Girl Scout Lemon Chalet Cremes- Style
Girl Scout Samoas-Style Cookies
Girl Scout Tagalongs-Style Cookies
Girl Scout Thin Mints-Style Cookies
Girl Scout Trefoils-Style Cookies
Graham Crackers
Hot Chocolate Cookies
Ice-Cream Sandwich Cookies
Jam Bars
Lady Fingers
Lemon Bars
Lemon Tassies
Lemon-Poppy Seed Cookies
Linzer Sandwich Cookies
Maple Leaf Cookies
Mocha Meltaways
Molasses Cookies
Nutter Butter-Style Peanut Butter Sandwich Cookies
Oatmeal-Raisin Cookies
Orange-Cream Cheese Tassies
Oreos-Style Cookies
Peanut Butter Blossom Cookies
Peanut Butter-Chocolate Chip- Oatmeal Bars
Pecan Cookies
Pecan Sandies-Style Cookies
Pecan Tassies
Pepper Jack Cookies
Pepperidge Farm Milano-Style Cookies
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
Pizzelle
Pizzelle, Almond
Raspberry-Cream Cheese Brownies
Red Velvet Sandwich Cookies (Scooter Pies)
Rolled Sugar Cookies
Rolled Sugar Cookies, Dairy-Free
Rugalach
Sand Balls
Shortbread
Snickerdoodles
Sour Cream Breakfast Cookies
Stained-Glass Cookies
Thumbprint Cookies
Toffee Bars
Turtle Bars
White Chocolate Bars
Brownies, Decadent
Brownies, Egg-Free
Brownies, Raspberry-Cream Cheese
Butter Cookies
## _**C**_
Candy Cane Cookies
Caramel
Girl Scout Samoas-Style Cookies
Turtle Bars
Carrot Cake Bars
Cheese
Carrot Cake Bars
Cheesecake Bars
Chocolate Sandwich Cookies (Scooter Pies)
Egg-Free Applesauce Bars
Lemon Tassies
Orange-Cream Cheese Tassies
Pecan Tassies
Pepper Jack Cookies
Pumpkin Sandwich Cookies (Scooter Pies)
Raspberry-Cream Cheese Brownies
Red Velvet Sandwich Cookies (Scooter Pies)
Cheesecake Bars
Cherry-Chocolate Cookies
Chinese Marble Cookies
Chips Ahoy!-Style Cookies
Chocolate. _See also_ White chocolate
Almond Joy-Style Cookies
Almond Layered Cookie Bars
Blondies #1
Blondies #2
Chinese Marble Cookies
Chips Ahoy!-Style Cookies
Chocolate Biscotti
Chocolate Butter Cookies
Chocolate Chip Cookies #1
Chocolate Chip Cookies #2
Chocolate Cookies, Dairy-Free
Chocolate Cookies with White Chips
Chocolate Crinkles
Chocolate Graham Crackers
Chocolate Marshmallow "Scooter" Pies
Chocolate Meringues
Chocolate Pinwheel Cookies
Chocolate Sandwich Cookies (Scooter Pies)
Chocolate Wafer Hearts
Chocolate-Cherry Cookies
Chocolate-Mint Sandwich Cookies
Cloud Cookies
Coconut Chip Cookies
Crisp Almond Cookies
Decadent Brownies
Egg-Free Brownies
Egg-Free Chocolate Chip Cookies
Egg-Free Chocolate Cookies
Egg-Free Peanut Butter Cookies (variation)
Girl Scout Samoas-Style Cookies
Girl Scout Tagalongs-Style Cookies
Girl Scout Thin Mints-Style Cookies
Hot Chocolate Cookies
Ice-Cream Sandwich Cookies
Love Letter Rolled Chocolate Sugar Cookies
Mocha Meltaways
Mrs. Fields-or Neiman Marcus-Style Cookies
Oatmeal Cookies
Oreos-Style Cookies
Peanut Butter Blossom Cookies
Peanut Butter-Chocolate Chip- Oatmeal Bars
Pepperidge Farm Milano-Style Cookies
Pizzelle, Chocolate
Raspberry-Cream Cheese Brownies
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Toffee Bars
Trail Mix Bars
Turtle Bars
Cinnabon-Style Cookies
Cloud Cookies
Coconut
Almond Joy-Style Cookies
Almond Layered Cookie Bars
Coconut Chip Cookies
Coconut Macaroons
Fruitcake Nuggets
Girl Scout Samoas-Style Cookies
Coconut flour, for Coconut Chip Cookies
Coffee
Hermits
Mocha Meltaways
Cornmeal
Chocolate Cookies, Dairy-Free
Rolled Sugar Cookies, Dairy-Free
Cornstarch
Animal Crackers
Chocolate Chip Cookies #1
Cloud Cookies
Lemon Bars
Nutter Butter-Style Peanut Butter Sandwich Cookies
Oatmeal Cookies
Peanut Butter Blossom Cookies
Rosettes
Crackerdoodles
Cranberry Cookies
Cranberry-Blueberry Jumbles
Cranberry-Orange-Almond Granola Bars
Cream cheese
Carrot Cake Bars
Cheesecake Bars
Chocolate Sandwich Cookies (Scooter Pies)
Egg-Free Applesauce Bars
Lemon Tassies
Orange-Cream Cheese Tassies
Pecan Tassies
Pumpkin Sandwich Cookies (Scooter Pies)
Raspberry-Cream Cheese Brownies
Red Velvet Sandwich Cookies (Scooter Pies)
Crisp Almond Cookies
## _**D**_
Dairy-free
Almond Flower Cookies
Animal Crackers
Apple Crumble Bars
Applesauce-Raisin Cookies
Banana-Nut Cookies
Blueberry-Cranberry Jumbles
Candy Cane Cookies
Chocolate Biscotti
Chocolate Crinkles
Chocolate Graham Crackers
Chocolate Meringues
Chocolate-Mint Sandwich Cookies
Chocolate Wafer Hearts
Coconut Macaroons
Egg-Free Brownies
Egg-Free Chocolate Cookies
Egg-Free Oatmeal Cookies
Egg-Free Peanut Butter Cookies
Egg-Free Pumpkin Bars
Egg-Free Spice Cookies
Egg-Free Sugar Cookies
Fig Newton-Style Cookies
Ginger Spice Cookies
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Lemon Chalet Cremes- Style Cookies
Graham Crackers
Hermits
Hot Chocolate Cookies
Iced Oatmeal Cookies
Jam Bars
Lace Cookies, Oatmeal
Lemon Bars
Lemon Meringues
Lemon-Poppy Seed Cookies
Linzer Sandwich Cookies
Little Debbie Oatmeal Creme Pies- Style Cookies
Love Letter Rolled Chocolate Sugar Cookies
Maple Leaf Cookies
Mocha Meltaways
Nut Meringue Wreaths
Nutter Butter-Style Peanut Butter Sandwich Cookies
Oreos-Style Cookies
Pecan Cookies
Pecan Lace Cookies
Pecan Sandies-Style Cookies
Pizzelle
Pizzelle, Almond
Pizzelle, Chocolate
Pumpkin Cookies
Rolled Sugar Cookies, Dairy-Free
Rugalach
Sand Balls
Spritz
Stained-Glass Cookies
Thumbprint Cookies
Decadent Brownies
Do-Si-Dos-Style Cookies, Girl Scout
Drop cookies
Almond Flower Cookies
Applesauce-Raisin Cookies
Banana-Nut Cookies
Blueberry-Cranberry Jumbles
Chocolate Chip Cookies #1
Chocolate Chip Cookies #2
Chocolate Cookies with White Chips
Chocolate Sandwich Cookies (Scooter Pies)
Cloud Cookies
Coconut Macaroons
Cranberry Cookies
Egg-Free Chocolate Chip Cookies
Egg-Free Chocolate Cookies
Egg-Free Oatmeal Cookies
Egg-Free Spice Cookies
Egg-Free Sugar Cookies
Ginger Spice Cookies
Gingersnaps
Hermits
Hot Chocolate Cookies
Iced Oatmeal Cookies
Lace Cookies, Oatmeal
Lemon Meringues
Lemon-Poppy Seed Cookies
Little Debbie Oatmeal Creme Pies- Style Cookies
Molasses Cookies
Mrs. Fields-or Neiman Marcus-Style Cookies
Oatmeal Cookies
Oatmeal-Raisin Cookies
Pecan Cookies
Pecan Lace Cookies
Pumpkin Cookies
Pumpkin Sandwich Cookies (Scooter Pies)
Red Velvet Sandwich Cookies (Scooter Pies)
Sour Cream Breakfast Cookies
## _**E**_
Egg-free cookies
Almond Joy-Style Cookies
Crackerdoodles
Cranberry-Orange-Almond Granola Bars
Egg-Free Applesauce Bars
Egg-Free Banana Bars with Browned Butter Icing
Egg-Free Brownies
Egg-Free Chocolate Chip Cookies
Egg-Free Chocolate Cookies
Egg-Free Gingerbread Men
Egg-Free Oatmeal Cookies
Egg-Free Peanut Butter Cookies
Egg-Free Pumpkin Bars
Egg-Free Rolled Sugar Cookies
Egg-Free Spice Cookies
Egg-Free Sugar Cookies
Fruitcake Nuggets
Orange-Cream Cheese Tassies
Pepper Jack Cookies
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Trail Mix Bars
## _**F**_
Fats for gluten-free cookies
Fig Newton-Style Cookies
Filled cookies. _See_ Sandwich and filled cookies
Filled Triangle Cookies
Flours, about
blends
cost
helpful hints
Fortune Cookies and Pirouettes
Fruitcake Nuggets
## _**G**_
Garbanzo bean flour
Chocolate Crinkles
Pumpkin Cookies
Ginger Spice Cookies
Gingerbread Men, Egg-Free
Gingerbread Men and Gingersnaps
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Lemon Chalet Cremes-Style
Girl Scout Samoas-Style Cookies
Girl Scout Tagalongs-Style Cookies
Girl Scout Thin Mints-Style Cookies
Girl Scout Trefoils-Style Cookies
Gluten-free cookies
cleanliness and food safety
equipment for
helpful hints
mixing and baking
pantry items for
Graham Crackers
Graham Crackers, Chocolate
Granola Bars, Cranberry-Orange-Almond
## _**H**_
Hermits
Hot Chocolate Cookies
## _**I**_
Ice-Cream Sandwich Cookies
Iced Oatmeal Cookies
## _**J**_
Jam
Cheesecake Bars
Filled Triangle Cookies
Jam Bars
Linzer Sandwich Cookies
Orange-Cream Cheese Tassies
Raspberry-Cream Cheese Brownies
Thumbprint Cookies
Jumbles, Blueberry-Cranberry
## _**L**_
Lace Cookies, Oatmeal
Lace Cookies, Pecan
Lady Fingers
Leaveners for gluten-free cookies
Lemon
Girl Scout Lemon Chalet Cremes- Style
Lemon Bars
Lemon Meringues
Lemon Tassies
Lemon-Poppy Seed Cookies
Linzer Sandwich Cookies
Little Debbie Oatmeal Creme Pies-Style Cookies
Love Letter Rolled Chocolate Sugar Cookies
## _**M**_
Macadamia nuts
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
White Chocolate Bars
Macaroons, Coconut
Maple Leaf Cookies
Marshmallow cream, for Little Debbie Oatmeal Creme Pies-Style Cookies
Marshmallows
Chocolate Marshmallow "Scooter" Pies
Hot Chocolate Cookies
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Trail Mix Bars
Meltaways, Mocha
Meringue Wreaths, Nut
Meringues, Chocolate
Meringues, Lemon
Milano-Style Cookies, Pepperidge Farm
Mint
Candy Cane Cookies
Chocolate-Mint Sandwich Cookies
Girl Scout Thin Mints-Style Cookies
M&M's, for Trail Mix Bars
Mocha Meltaways
Molasses Cookies
Mrs. Fields-or Neiman Marcus-Style Cookies
## _**N**_
Nut meals
Almond Flower Cookies
Almond Layered Cookie Bars
Crisp Almond Cookies
Linzer Sandwich Cookies
Nut Meringue Wreaths
Pecan Cookies
Pecan Lace Cookies
Pecan Sandies-Style Cookies
Pizzelle, Almond
Sand Balls
Nuts. _See also_ Coconut
Almond Biscotti
Almond Flower Cookies
Almond Joy-Style Cookies
Almond Layered Cookie Bars
Banana-Nut Cookies
Blondies #1
Blondies #2
Carrot Cake Bars
Cloud Cookies
Cranberry-Orange-Almond Granola Bars
Crisp Almond Cookies
Egg-Free Banana Bars with Browned Butter Icing
Fruitcake Nuggets
Hermits
Mrs. Fields-or Neiman Marcus-Style Cookies
Nut Meringue Wreaths
Pecan Lace Cookies
Pecan Sandies-Style Cookies
Pecan Tassies
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
Rugalach
Thumbprint Cookies
Toffee Bars
Trail Mix Bars
Turtle Bars
White Chocolate Bars
Nutter Butter-Style Peanut Butter Sandwich Cookies
## _**O**_
Oat flour
Chocolate Biscotti
Iced Oatmeal Cookies
Oats
Cranberry-Orange-Almond Granola Bars
Egg-Free Oatmeal Cookies
Iced Oatmeal Cookies
Lace Cookies, Oatmeal
Little Debbie Oatmeal Creme Pies- Style Cookies
Mrs. Fields-or Neiman Marcus-Style Cookies
Oatmeal Cookies
Oatmeal-Raisin Cookies
Orange-Cranberry-Almond Granola Bars
Orange-Cream Cheese Tassies
Oreos-Style Cookies
## _**P**_
Peanut butter
Egg-Free Peanut Butter Cookies
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Tagalongs-Style Cookies
Nutter Butter-Style Peanut Butter Sandwich Cookies
Peanut Butter Blossom Cookies
Peanut Butter-Chocolate Chip- Oatmeal Bars
Rice Cereal Bars, Peanut Butter
Peanuts
Trail Mix Bars
Turtle Bars
Pecan meal
Pecan Cookies
Pecan Lace Cookies
Pecan Sandies-Style Cookies
Pecans
Banana-Nut Cookies
Blondies #1
Blondies #2
Hermits
Mrs. Fields-or Neiman Marcus-Style Cookies
Pecan Lace Cookies
Pecan Sandies-Style Cookies
Pecan Tassies
Rugalach
Thumbprint Cookies
Toffee Bars
Pepper Jack Cookies
Pepperidge Farm Milano-Style Cookies
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
Pinwheel Cookies, Chocolate
Piped cookies
Butter Cookies
Chocolate Butter Cookies
Chocolate Meringues
Lady Fingers
Lemon Meringues
Nut Meringue Wreaths
Pepperidge Farm Milano-Style Cookies
Spritz
Pirouettes and Fortune Cookies
Pizzelle
Pizzelle, Almond
Pizzelle, Chocolate
Poppy Seed-Lemon Cookies
Potato starch
Blueberry-Cranberry Jumbles
Lemon Bars
Rosettes
Spritz
Pressed cookies. _See_ Piped cookies; Shaped and molded cookies
Pumpkin Bars, Egg-Free
Pumpkin Cookies
Pumpkin Sandwich Cookies (Scooter Pies)
## _**R**_
Raisins
Applesauce-Raisin Cookies
Carrot Cake Bars
Egg-Free Applesauce Bars
Egg-Free Spice Cookies
Hermits
Little Debbie Oatmeal Creme Pies- Style Cookies
Molasses Cookies
Oatmeal-Raisin Cookies
Pumpkin Cookies
Rugalach
Trail Mix Bars
White Chocolate Bars
Raspberry-Cream Cheese Brownies
Red Velvet Sandwich Cookies (Scooter Pies)
Rice cereal
Rice Cereal Bars
Rice Cereal Bars, Peanut Butter
Trail Mix Bars
Rolled cookies
Animal Crackers
Candy Cane Cookies
Chinese Marble Cookies
Chocolate Graham Crackers
Chocolate Marshmallow "Scooter" Pies
Chocolate Pinwheel Cookies
Chocolate Wafer Hearts
Chocolate-Mint Sandwich Cookies
Cinnabon-Style Cookies
Coconut Chip Cookies
Crisp Almond Cookies
Egg-Free Gingerbread Men
Egg-Free Rolled Sugar Cookies
Filled Triangle Cookies
Gingerbread Men and Gingersnaps
Girl Scout Lemon Chalet Cremes- Style
Girl Scout Samoas-Style Cookies
Girl Scout Tagalongs-Style Cookies
Graham Crackers
Ice-Cream Sandwich Cookies
Linzer Sandwich Cookies
Love Letter Rolled Chocolate Sugar Cookies
Maple Leaf Cookies
Mocha Meltaways
Nutter Butter-Style Peanut Butter Sandwich Cookies
Oreos-Style Cookies
Pepper Jack Cookies
Rolled Sugar Cookies
Rolled Sugar Cookies, Dairy-Free
Rugalach
Stained-Glass Cookies
Rosettes
Rugalach
## _**S**_
Samoas-Style Cookies, Girl Scout
Sand Balls
Sandwich and filled cookies
Chocolate Marshmallow "Scooter" Pies
Chocolate Sandwich Cookies (Scooter Pies)
Chocolate-Cherry Cookies
Chocolate-Mint Sandwich Cookies
Fig Newton-Style Cookies
Filled Triangle Cookies
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Lemon Chalet Cremes- Style
Jam Bars
Lemon Tassies
Linzer Sandwich Cookies
Little Debbie Oatmeal Creme Pies- Style Cookies
Maple Leaf Cookies
Nutter Butter-Style Peanut Butter Sandwich Cookies
Orange-Cream Cheese Tassies
Oreos-Style Cookies
Pecan Tassies
Pepperidge Farm Milano-Style Cookies
Pumpkin Sandwich Cookies (Scooter Pies)
Red Velvet Sandwich Cookies (Scooter Pies)
Rugalach
Thumbprint Cookies
Scooter Pies
Chocolate Marshmallow "Scooter" Pies
Chocolate Sandwich Cookies (Scooter Pies)
Pumpkin Sandwich Cookies (Scooter Pies)
Red Velvet Sandwich Cookies (Scooter Pies)
Shaped and molded cookies. _See also_ Piped cookies; Rolled cookies
Almond Biscotti
Almond Joy-Style Cookies
Chocolate Biscotti
Chocolate Crinkles
Crackerdoodles
Egg-Free Peanut Butter Cookies
Fortune Cookies and Pirouettes
Girl Scout Do-Si-Dos-Style Cookies
Girl Scout Trefoils-Style Cookies
Lemon Tassies
Mocha Meltaways
Orange-Cream Cheese Tassies
Peanut Butter Blossom Cookies
Pecan Tassies
Pizzelle
Pizzelle, Almond
Pizzelle, Chocolate
Rosettes
Sand Balls
Snickerdoodles
Thumbprint Cookies
Shortbread
Girl Scout Trefoils-Style Cookies
Shortbread
Toffee Bars
Slice-and-bake cookies
Chips Ahoy!-Style Cookies
Chocolate Pinwheel Cookies
Cinnabon-Style Cookies
Pecan Sandies-Style Cookies
Snickerdoodles
Sorghum flour
Blondies #2
Chocolate Chip Cookies #2
Chocolate Wafer Hearts
Chocolate-Mint Sandwich Cookies
Cranberry Cookies
Decadent Brownies
Ginger Spice Cookies
Girl Scout Do-Si-Dos-Style Cookies
Hermits
Mrs. Fields-or Neiman Marcus-Style Cookies
Pumpkin Sandwich Cookies (Scooter Pies)
Sour Cream Breakfast Cookies
Spice Cookies, Egg-Free
Spice Cookies, Ginger
Spritz
Stained-Glass Cookies
Sugar cookies
Egg-Free Rolled Sugar Cookies
Egg-Free Sugar Cookies
Girl Scout Samoas-Style Cookies (note)
Love Letter Rolled Chocolate Sugar Cookies
Rolled Sugar Cookies
Rolled Sugar Cookies, Dairy-Free
Sweetened condensed milk
Almond Joy-Style Cookies
Almond Layered Cookie Bars
Chocolate-Cherry Cookies
Fruitcake Nuggets
White Chocolate Bars
Sweeteners for gluten-free cookies
## _**T**_
Tagalongs-Style Cookies, Girl Scout
Tapioca starch
Fortune Cookies and Pirouettes
Love Letter Rolled Chocolate Sugar Cookies
Tassies, Lemon
Tassies, Orange-Cream Cheese
Tassies, Pecan
Thin Mints-Style Cookies, Girl Scout
Thumbprint Cookies
Toffee Bars
Trail Mix Bars
Trefoils-Style Cookies, Girl Scout
Turtle Bars
## _**W**_
Walnuts
Carrot Cake Bars
Fruitcake Nuggets
White chocolate
Chocolate Cookies with White Chips
Pepperidge Farm White Chocolate Macadamia Crispy-Style Cookies
White Chocolate Bars
www.fda.gov/Food/FoodSafety/FoodAllergens
The cost of xanthan gum is not included in the analysis of these blends. That would add an additional cost of $11.96 per 8-ounce package. A package of xanthan gum lasts a very long time! Nor do any of these blends contain some of the more recently embraced alternative flours, which are typically quite costly and would substantially increase the cost.
Often referred to as whole grain by many, sorghum is technically not whole grain, but carries many nutritious properties.
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Da Capo Press was aware of a trademark claim, the designations have been
Copyright © 2010 by Roben Ryberg
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. For information, address Da Capo Press, 11 Cambridge Center, Cambridge, MA 02142.
Library of Congress Cataloging-in-
Publication Data
Ryberg, Roben.
The ultimate gluten-free cookie book :
125 favorite recipes / Roben Ryberg.—1st
Da Capo Press ed.
p. cm.
Includes index.
eISBN : 978-0-738-21442-9
paper) 1. Gluten-free diet—Recipes. I. Title.
RM237.86.R932 2010
641.5'638—dc22
2010029918
Published by Da Capo Press
A Member of the Perseus Books Group
www.dacapopress.com
Note: The information in this book is true and complete to the best of our knowledge. This book is intended only as an informative guide for those wishing to know more about health issues. In no way is this book intended to replace, countermand, or conflict with the advice given to you by your own physician. The ultimate decision concerning care should be made between you and your doctor. We strongly recommend you follow his or her advice. Information in this book is general and is offered with no guarantees on the part of the authors or Da Capo Press. The authors and publisher disclaim all liability in connection with the use of this book. The names and identifying details of people associated with events described in this book have been changed. Any similarity to actual persons is coincidental.
Da Capo Press books are available at special discounts for bulk purchases in the U.S. by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA, 19103, or call (800) 810-4145, ext. 5000, or e-mail special .markets@perseusbooks.com.
| {
"redpajama_set_name": "RedPajamaBook"
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{"url":"http:\/\/gmatclub.com\/forum\/disabled-do-not-just-want-sympathy-158140.html","text":"Disabled do not just want sympathy : GMAT Critical Reasoning (CR)\nCheck GMAT Club Decision Tracker for the Latest School Decision Releases http:\/\/gmatclub.com\/AppTrack\n\n It is currently 22 Jan 2017, 22:03\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# Disabled do not just want sympathy\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nManager\nJoined: 15 Apr 2013\nPosts: 86\nLocation: India\nConcentration: Finance, General Management\nSchools: ISB '15\nWE: Account Management (Other)\nFollowers: 1\n\nKudos [?]: 93 [0], given: 61\n\nDisabled do not just want sympathy\u00a0[#permalink]\n\n### Show Tags\n\n18 Aug 2013, 03:44\n2\nThis post was\nBOOKMARKED\n00:00\n\nDifficulty:\n\n75% (hard)\n\nQuestion Stats:\n\n37% (02:06) correct 63% (01:06) wrong based on 142 sessions\n\n### HideShow timer Statistics\n\nDisabled do not just want sympathy in state 'Y'.\nThey want to be treated as equals. In state 'Y'\nthere is a certain company which has recently\nstarted hiring disabled people. Company 'Y' has\nstarted a sensitivity training program for its\nemployees because\nA. The employees are happy when training programs are conducted.\nB. The employees are schooled in the old way of treating disabled people.\nC. Company Y has an image of treating all employees equally.\nD. Companies which have such sensitivity training programs have a congenial work atmosphere.\nE. Employees change for the better when they undergo a sensitivity training program.\n[Reveal] Spoiler: OA\nIf you have any questions\nNew!\nPrinceton Review Representative\nJoined: 17 Jun 2013\nPosts: 163\nFollowers: 140\n\nKudos [?]: 303 [1] , given: 0\n\nRe: Disabled do not just want sympathy\u00a0[#permalink]\n\n### Show Tags\n\n18 Aug 2013, 05:57\n1\nKUDOS\nExpert's post\nWhile this argument would probably not appear on the GMAT as written because there are many reasons why the company could have started training, not all of them logical, Assuming that you are looking for the answer that matches the premise - that disabled people want to be treated as equals, you would need an answer that shows the training somehow helps disabled people be treated like equals\npavan2185 wrote:\nDisabled do not just want sympathy in state 'Y'.\nThey want to be treated as equals. In state 'Y'\nthere is a certain company which has recently\nstarted hiring disabled people. Company 'Y' has\nstarted a sensitivity training program for its\nemployees because\nA. The employees are happy when training programs are conducted.the happiness of all the employees is not given as a problem for them\nB. The employees are schooled in the old way of treating disabled people.assuming the old school way of treating them is to not treat them as equals then the training program would be used to help over come this. Again, this answer choice requires you to assume more than a GMAT answer would\nC. Company Y has an image of treating all employees equally.this appears correct but if everyone were already treating them equally, the company would not need a training course\nD. Companies which have such sensitivity training programs have a congenial work atmosphere.the question is looking to treat disabled people equally, not for a congenial work atmosphere so this is out of scope\nE. Employees change for the better when they undergo a sensitivity training program.This does not specifically address the treatment of disabled people\n\n_________________\n\nSpecial offer! Save $250 on GMAT Ultimate Classroom, GMAT Small Group Instruction, or GMAT Liveonline when you use the promo code GCVERBAL250. Or, save$150 on GMAT Self-Prep when you use the code GCVERBAL150. Enroll at www.princetonreview.com\n\nGMAT Club Legend\nJoined: 01 Oct 2013\nPosts: 10542\nFollowers: 919\n\nKudos [?]: 204 [0], given: 0\n\nRe: Disabled do not just want sympathy\u00a0[#permalink]\n\n### Show Tags\n\n28 Mar 2016, 17:20\nHello from the GMAT Club VerbalBot!\n\nThanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).\n\nWant to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.\nRe: Disabled do not just want sympathy \u00a0 [#permalink] 28 Mar 2016, 17:20\nSimilar topics Replies Last post\nSimilar\nTopics:\n1 A person is more likely to become disabled as that person ages. Among 4 24 Dec 2016, 22:18\nJust wondering, when doing CR questions does anyone write 3 30 Jan 2011, 11:25\n1 What do you think? Anyone who wants to argue option E 6 13 Aug 2010, 09:24\nCR - Help wanted 1 09 Jul 2008, 05:45\nIts little tricky. Just wanted to share with you all. The 12 28 Aug 2007, 07:06\nDisplay posts from previous: Sort by","date":"2017-01-23 06:03:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18230891227722168, \"perplexity\": 6074.772748575155}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560282110.46\/warc\/CC-MAIN-20170116095122-00317-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
New Delhi: Traces of PETN (Penta Erithrotol Tetra Nitrate) particles found at a Gurukul/temple blast site in Mewat has created concern among security agencies. It is believed that the Gurukul located at Bhadas village in Haryana's Mewat district was planning for a major blast operation but because of the sudden explosion the major operation was abandoned. Central agencies ordered a second search after finding PETN particles. The second forensic report is awaited. Sources claimed that the Gurukul was under the observation of Intelligence Bureau (IB) prior to the Ayodhya verdict.
The blast took place in the evening of 22 September this year shattering the walls of the mandir. Initially the local police and intelligence tried to hush up the matter but the news slowly trickled to Chandigarh and Delhi and the IB had to take notice. Bhadas village is located on the Delhi-Alwar-Gurgaon-Sohna Road under Nagina police station at a distance of around 17 kms from Nuh. The mandir is managed by a pujari called Anand Mishra, popularly called "Baba" in the area. He is believed to be a Bangali. Mishra is absconding ever since. Sources informed the MG that around 50-60 youths from other parts of Haryana and other states live in this mandir which is believed to be a Hindutva activity centre in the area. The gurukul is said to be affiliated to a Jhajjar-based organisation which runs a similar gurukul in Faridabad. It is believed that the explosives were kept in the gurukul for use in anticipation of Ayodhya verdict if it went against Hindus.
A related incident exposes the preparations made to vitiate the communal situation in Muslim dominated-Mewat in recent months. A number of attacks on idols in mandirs have taken place in the area but no arrests were made. The most sinister of these incidents took place in Malab village in the night of 12 October when one Ramesh son of Sampat Singh, who runs a sweets shop, instigated a kumhar called Parjapat Sitaram son of Sohan Lal to smash an idol in a local temple in the dead of night. Obviously the plan was to blame Muslims and subject them to police atrocities. The sound in the dead of night caused by Parjapat Sitaram's smashing of the idol caused local Hindu banias to rush to the place and catch Parjapat Sitaram red-handed. They thrashed him and called the police. Jai Chand, chowki incharge of Akeda, rushed to the place and arrested the culprit. The police filed a case implicating Parjapat Sitaram and Ramesh (sweet shop owner) in crimes under articles 120B and 295 of the criminal code. They took the culprit to Nuh court which remanded him to judicial custody in Bhondsi jail while Ramesh is absconding. | {
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{"url":"https:\/\/www.quizover.com\/course\/section\/exercises-trigonometry-by-openstax","text":"# 0.13 Trigonometry\n\n Page 3 \/ 3\n\nComment: Triangulation is a process that can be applied to solve problems in a number of areas of engineering including surveying, construction management, radar, sonar, lidar, etc.\n\n## Refraction\n\nRefraction is a physical phenomenon that occurs when light passes from one transparent medium (such as air) through another (for example, glass.) It is known that light travels at different speeds through different transparent media. The index of refraction of a medium is a measure of how much the speed of light is reduced as it passes through the medium. In the case of glass, the index of refraction is approximately 1.5. This means that light travels as a speed of $\\frac{1}{1\\text{.}5}=\\frac{2}{3}$ times the speed of light in a vacuum.\n\nTwo common properties of transparent materials can be attributed to the index of refraction. One is that light rays change direction as they pass from one medium through another. Secondly, light is partially reflected when it passes from one medium to another medium with a different index of refraction. We will focus on the first of these properties in this reading.\n\nIn optics, which is a field of physics, you will learn about Snell's law, which is also known as Descartes' law after the scientist, Rene Descartes . Snell\u2019s law takes the form of an equation that states the relationship between the angle of incidence and the angle of refraction for light passing from one medium to another. Stated mathematically, Snell\u2019s law is\n\n$\\frac{\\text{sin}\\left({\\theta }_{\\text{incidence}}\\right)}{\\text{sin}\\left({\\theta }_{\\text{refraction}}\\right)}=\\frac{{c}_{\\text{incidence}}}{{c}_{\\text{refraction}}}$\n\nIt follows that\n\n$\\frac{\\text{sin}\\left({\\theta }_{\\text{incidence}}\\right)}{\\text{sin}\\left({\\theta }_{\\text{refraction}}\\right)}=\\frac{{I}_{2}}{{I}_{1}}$\n\nwhere ${I}_{1}\\text{and}{I}_{2}$ are the Index of Refraction of medium 1 and medium 2 respectively.\n\nConsider a situation where light rays pass are shined from air through a tank of water. This situation is illustrated below.\n\nThe Index of refraction for air is 1.0003 and that of water is 1.3000. Let us assume that the angle that light enters the water is 21 0 40\u2019, what is the angle of refraction, w ?\n\nFrom Snell\u2019s law, we know\n\n$\\frac{{I}_{W}}{{I}_{A}}=\\frac{\\text{sin}\\left(a\\right)}{\\text{sin}\\left(w\\right)}$\n$\\text{sin}\\left(w\\right)=\\frac{{I}_{A}}{{I}_{W}}\\text{sin}\\left(a\\right)$\n\n$\\text{sin}\\left(w\\right)=\\frac{{I}_{A}}{{I}_{W}}\\text{sin}\\left(a\\right)$\n\nSubstituting in the numerical values for I A , I W and a yield\n\n$\\text{sin}\\left(w\\right)=0\\text{.}\\text{2841}$\n\nWe now make use of the inverse sine function\n\n$w={\\text{sin}}^{-1}\\left(0\\text{.}\\text{2841}\\right)$\n\nThis leads to the result\n\n$w={\\text{16}}^{0}{\\text{30}}^{\\text{'}}$\n\nWe conclude that the refracted ray will travel through the water at an angle of refraction of 16 0 30\u2019 .\n\n## Exercises\n\n1. A 50 ft ladder leans against the top of a building which is 30 ft tall. Determine the angle the ladder makes with the horizontal. Also determine the distance from the base of the ladder to the building.\n2. A straight trail leads from the Alpine Hotel at elevation 8,000 feet to a scenic overlook at elevation 11,100 feet. The length of the trail is 14,100 feet. What is the inclination angle \u03b2 in degrees? What is the value of \u03b2 in radians?\n3. A ray of light moves from a media whose index of refraction is 1.200 to another whose index of refraction is 1.450. The angle of incidence of the ray as it intersects the interface of the two media is 15 0 . Sketch the geometry of the situation and determine the value of the angle of refraction.\n4. One-link planar robots can be used to place pick up and place parts on work table. A one-link planar robot consists of an arm that is attached to a work table at one end. The other end is left free to rotate about the work space. If l = 5 cm, sketch the position of the robot and determine the ( x, y ) coordinates of point p ( x , y ) for the following values for \u03b8: (50\u02da, 2\u03c0\/3 rad, -20\u02da, and -5 \u03c0\/4 rad).\n\n#### Questions & Answers\n\nhow to know photocatalytic properties of tio2 nanoparticles...what to do now\nit is a goid question and i want to know the answer as well\nMaciej\nDo somebody tell me a best nano engineering book for beginners?\nwhat is fullerene does it is used to make bukky balls\nare you nano engineer ?\ns.\nwhat is the Synthesis, properties,and applications of carbon nano chemistry\nMostly, they use nano carbon for electronics and for materials to be strengthened.\nVirgil\nis Bucky paper clear?\nCYNTHIA\nso some one know about replacing silicon atom with phosphorous in semiconductors device?\nYeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.\nHarper\nDo you know which machine is used to that process?\ns.\nhow to fabricate graphene ink ?\nfor screen printed electrodes ?\nSUYASH\nWhat is lattice structure?\nof graphene you mean?\nEbrahim\nor in general\nEbrahim\nin general\ns.\nGraphene has a hexagonal structure\ntahir\nOn having this app for quite a bit time, Haven't realised there's a chat room in it.\nCied\nwhat is biological synthesis of nanoparticles\nwhat's the easiest and fastest way to the synthesize AgNP?\nChina\nCied\ntypes of nano material\nI start with an easy one. carbon nanotubes woven into a long filament like a string\nPorter\nmany many of nanotubes\nPorter\nwhat is the k.e before it land\nYasmin\nwhat is the function of carbon nanotubes?\nCesar\nI'm interested in nanotube\nUday\nwhat is nanomaterials\u200b and their applications of sensors.\nwhat is nano technology\nwhat is system testing?\npreparation of nanomaterial\nYes, Nanotechnology has a very fast field of applications and their is always something new to do with it...\nwhat is system testing\nwhat is the application of nanotechnology?\nStotaw\nIn this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google\nAzam\nanybody can imagine what will be happen after 100 years from now in nano tech world\nPrasenjit\nafter 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments\nAzam\nname doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world\nPrasenjit\nhow hard could it be to apply nanotechnology against viral infections such HIV or Ebola?\nDamian\nsilver nanoparticles could handle the job?\nDamian\nnot now but maybe in future only AgNP maybe any other nanomaterials\nAzam\nHello\nUday\nI'm interested in Nanotube\nUday\nthis technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15\nPrasenjit\ncan nanotechnology change the direction of the face of the world\nhow did you get the value of 2000N.What calculations are needed to arrive at it\nPrivacy Information Security Software Version 1.1a\nGood\nBerger describes sociologists as concerned with\nCan someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution","date":"2018-10-18 11:26:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 10, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5735704898834229, \"perplexity\": 1079.7574987375756}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583511806.8\/warc\/CC-MAIN-20181018105742-20181018131242-00510.warc.gz\"}"} | null | null |
Congresul PNL a avut loc la data de 25 Septembrie 2021. Alegerile din PNL sunt primele alegeri care au loc în această toamnă în partidele care formează guvernul Cîțu. Totodată, la data de 2 octombrie a fost aleasă și conducerea USR PLUS, fostul partener al PNL de la guvernare, care a părăsit coaliția de guvernare în 3 Septembrie și au părăsit guvernul în 7 Septembrie.
Context
Conform statutului PNL, alegerile pentru alegerea conducerii partidului, au loc o dată la 4 ani.
La ultimele alegeri din PNL, Ludovic Orban a fost ales președinte din primul tur de scrutin.
La finalul anului 2020, Florin Vasile Cîțu a fost numit prim-ministru al Romaniei cu susținerea PNL, în urma negocierilor cu USR PLUS și UDMR.
Alegeri pe filiale si numărul de delegați ai fiecărei filiale
Alba - 146 delegați
Arad - 128 delegați
Arges - 103 delegați
Bacău - 105 delegați
Bihor - 191 delegați
Bistrița Năsăud - 98 delegați
Botoșani - 89 delegați
Brașov - 128 delegați
Brăila - 67 delegați
Buzău - 67 delegați
Caraș Severin - 113 delegați
Călărași - 97 delegați
Cluj - 180 delegați
Constanța - 144 delegați
Covasna -22 delegați
Dîmbovița 111 delegați
Dolj 132 delegați
Galați - 101 delegați
Giurgiu 145 delegați
Gorj - 91 delegați
Harghita 19 delegați
Hunedoara - 83 delegați
Ialomița 68 delegați
Iași 147 delegați
Ilfov 138 delegați
Maramureș - 108 delegați
Mehedinți 113 delegați
Mureș - 79 delegați
Neamț - 105 delegați
Olt - 94 delegați
Prahova 170 delegați
Satu Mare 75 delegați
Sălaj - 84 delegați
Sibiu 154 delegați
Suceava 142 delegați
Teleorman 141 delegați
Timiș 133 delegați
Tulcea 82 delegați
Vaslui 90 delegați
Vâlcea 106 delegați
Vrancea 121 delegași
București – Sector 1 - 71 delegați
București – Sector 2 - 68 delegați
București – Sector 3 -73 delegați
București – Sector 4 - 61 delegați
București – Sector 5 -57 delegați
București – Sector 6 - 74 delegați
Diaspora - 78 delegați
Sistem electoral
7 C. Alegerile în Congresul PNL Art. 74
1. CON este legal constituită dacă sunt prezenți mai mult de două treimi din numărul delegaților. 2. În CON se aleg prin votul secret al delegaților prezenți: a) una dintre moțiunile prezentate și Președintele PNL; b) presedintele CA; c) președintele CNRC.
Art. 75
Moțiunea câștigătoare devine Programul Politic de acțiune al PNL.
Partidul Național Liberal
Page24
Art. 76
1. Pentru alegerea moțiunii se organizează dezbateri în plenul CON și eventual pe secțiuni. 2. Moțiunile sunt prezentate, în perioade de timp egale, de către liderul moțiunii și reprezentanții săi desemnați
conform Art. 73 alin (2). Perioada de timp este stabilită de către COCON. Ordinea de prezentare se stabilește de
COCON prin tragere la sorți în prezența a câte unui reprezentant din partea fiecărei moțiuni acceptate.
Dezbaterile sunt conduse de președintele COCON care nu poate face parte din grupul de promovare a niciunei
moțiuni. 3. În cursul dezbaterilor, liderul și reprezentanții săi vor răspunde și la întrebările delegaților.
Art. 77
1. Pentru alegerile din CON se întocmesc trei buletine de vot pe care se înscriu: a) titlurile moțiunilor care se dezbat și numele liderilor acestora; b) candidații pentru funcția de președinte al CA; c) candidații pentru funcția de președinte al CNRC. 2. Votul delegaților este secret. Ei vor opta: a) pentru sau împotriva moțiunii, când există o singură moțiune, și unul dintre candidații pentru funcția de
președinte al CA, respectiv unul dintre candidații pentru funcția de președinte al CNRC; b) pentru una dintre moțiunile prezentate, când există mai multe moțiuni, unul dintre candidații pentru funcția
de președinte al CA, respectiv unul dintre candidații pentru funcția de președinte al CNRC. 3. Dacă niciuna dintre moțiuni nu obține majoritatea voturilor valabil exprimate, se realizează un al doilea tur de
scrutin, între primele două moțiuni, în ordinea voturilor. 4. Sunt aleși președinți ai CA și CNRC candidații cu cel mai mare număr de voturi. 5. Liderul moțiunii care obține majoritatea voturilor valabil exprimate, în turul 1 sau în turul al doilea, este declarat
Președinte al PNL. Cei 35 de membri ai grupului de promovare devin membri ai CN. 6. De la moțiunile care nu au obținut majoritatea voturilor, dar au obținut cel puțin 10% din voturile valabil
exprimate la primul tur de scrutin, liderul și un numar proporțional de susținători dintre membrii grupului de
promovare, dar nu mai puțin de 5, desemnați de către liderul moțiunii, vor deveni membri ai CN.
Candidații pentru conducerea PNL și moțiunile lor
Referințe
PNL
Partidul Național Liberal | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 970 |
\section{Introduction}
Modern representation theory is a harmonious confluence of algebra, geometry and combinatorics. There are now many examples of algebraic problems which exhibit complementary geometric and combinatorial solutions. In this article we introduce a natural morphism between certain algebraic varieties which are ubiquitous in representation theory, and we describe this morphism combinatorially, thus continuing the trialogue between these fields.
The theory of Lusztig--Spaltenstein (LS) induction is a geometric process for constructing nilpotent orbits in the Lie algebra of a reductive group from the nilpotent orbits in Levi subalgebras. Nilpotent orbits are one of the most pervasive geometric objects appearing in representation theory, taking a starring role in:
\begin{enumerate}
\item Springer's construction of Weyl group representations \cite{CG}
\item associated varieties of primitive ideals of enveloping algebras \cite[\textsection 9]{JaNO}.
\item conical symplectic singularities \cite{Bea}.
\item modular representation theory of Lie algebras \cite{BMR}.
\end{enumerate}
It is therefore unsurprising that LS induction has played a important role in these theories, serving as a geometric analogue of parabolic induction of representations.
Let $G$ be a reductive algebraic group over an algebraically closed field $\k$ of any characteristic and assume the standard hypotheses \cite[\textsection 2.9]{JaNO}. Perhaps the most famous projective variety appearing in representation theory is the flag variety $G/B$ of a reductive algebraic group $G$. This is the base space of the Springer resolution $T^* G/B$, and can be identified with the set of Borel subalgebras in $\mathfrak{g}$. The fibres of the resolution $T^* G/B \to \mathcal{N}(\mathfrak{g})$ are the the {\it Springer fibres}: for $e\in \mathcal{N}(\mathfrak{g})$ the fibre is described as
$$\mathcal{B}_e := \{gB \in G/B \mid e\in \operatorname{Ad}(g) \b\}$$
They carry a wealth of representation theoretic information, placing them at the very core of geometric representation theory. We begin this paper by showing that LS induction of orbits induces a morphism of relative dimension zero between the corresponding Springer fibres (Proposition~\ref{P:LSwelldefined}). Let us state this more precisely. If $\mathfrak{g}_0 \subseteq \mathfrak{p} \subseteq \mathfrak{g}$ is a Levi factor of a parabolic and $\O_0 \subseteq \mathcal{N}(\mathfrak{g}_0)$ is an orbit then for $e_0 \in \O_0$ and $e \in (e_0 + \mathfrak{n}) \cap \operatorname{Ind}_{\mathfrak{g}_0}^\mathfrak{g}(\O_0)$ we define a morphism
\begin{eqnarray}
\label{e:LSintro}
\LS : \mathcal{B}_{e_0}^0 \longrightarrow \mathcal{B}_e
\end{eqnarray}
where $\mathcal{B}^0 $ is the flag variety for $G_0$. Since this definition is elementary this construction is probably well-known to experts, however we cannot find a reference in the literature.
When $G = \GL_N$ the theory of Springer fibres is especially nice. Spaltenstein showed that there is a map $\Phi : \mathcal{B}_e \to \Std(\lambda)$ where $\lambda \vdash N$ and $\Std(\lambda)$ denotes the set of standard tableaux of shape $\lambda$. For $\sigma \in \Std(\lambda)$ the closure $C_\sigma:= \overline{\Phi^{-1}(\sigma)}$ is an irreducible component of $\mathcal{B}_e$. Furthermore the combinatorics of the tableaux control aspects of the geometry of the sets $\Phi^{-1}(\sigma)$ and their closures \cite{Spa, FM}. We abuse notation viewing $\Phi$ as a bijection $\Comp \mathcal{B}_e \to \Std(\lambda)$.
If $\mathfrak{g}_0$ is a Levi subalgebra then $\mathfrak{g} \cong \mathfrak{gl}_{\lambda_1} \times \cdots \times \mathfrak{gl}_{\lambda_n}$ for some $\lambda \vDash N$, and the nilpotent orbits in $\mathfrak{g}_0$ are classified by a tuple of partitions $(\mu_1,...,\mu_n)$ with $\mu_i \vdash \lambda_i$. If $\O_0 \subseteq \mathcal{N}(\mathfrak{g}_0)$ is classified by $(\mu_1,...,\mu_n)$ then Spaltenstein's construction gives a bijection $\Phi_0 : \Comp \mathcal{B}_{e_0}^0 \to \prod_{i=1}^n \Std(\mu_i)$.
The purpose of this paper is to describe the combinatorics lying behind the Lusztig--Spaltenstein morphism \eqref{e:LSintro}. In Section~\ref{ss:combinatorics} we introduce a new map $\stk : \prod_{i=1}^n \Std(\mu_i) \to \Std(\lambda)$, combinatorially defined, which we call {\it the stacking map}. This is our main result (Cf. Theorem~\ref{T:stackwithproof}).
\begin{Stkthm}
The combinatorics of components of Springer fibres under $\LS$ is determined by the stacking map, so the following diagram commutes
\begin{center}
\begin{tikzpicture}[node distance=2.8cm, auto]
\node (A) {$\Comp \mathcal{B}^0_{e_0}$};
\node (B) [below of= A] {$\prod_{i=1}^n \Std(\mu_i)$};
\node (C) [right of= A] {$\Comp \mathcal{B}_e$};
\node (D) [right of= B]{$\Std(\lambda).$};
\draw[->] (A) to node [swap]{$\Phi_0$} (B);
\draw[->] (A) to node {$\LS$} (C);
\draw[->] (B) to node [swap]{$\stk$} (D);
\draw[->] (C) to node {$\Phi$} (D);
\end{tikzpicture}
\end{center}
\end{Stkthm}
Springer fibres bear a very close relationship to orbital varieties (see \cite[\textsection~9]{JaNO}, for example). In \cite{FMe} Fresse and Melnikov explain that Lusztig--Spaltenstein induction leads to a morphism of orbital varieties, similar to our construction. It is natural to expect that our induction of Springer fibres is compatible with their construction.
\subsection*{Acknowledgements} Many of the ideas which we present in this paper were initiated at the workshop ``Springer fibres and Geometric Representation Theory'', organised by the authors, and held at the University of Greenwich in 2019. The authors would like to thank the Heilbronn Institute for Mathematical Research for their generous support which made that event possible, and the participants for sharing their insights. The first author's research is supported by UKRI grant MR/S032657/1. The second author benefited from the LMS Scheme 4, grant number 42037.
\section{Combinatorics }
\subsection{Partitions, compositions and Young tableaux}
\label{ss:combinatorics}
Throughout the paper $N > 0$. Let $\lambda = (\lambda_1,...,\lambda_n)$ be a tuple of positive integers. If $\sum_i \lambda_i = N$ then we write $\lambda \vDash N$ and call $\lambda$ a {\it composition of $N$}, and if $\lambda$ is a non-increasing ordered composition then we write $\lambda \vdash N$ and say that $\lambda$ is a {\it partition of $N$}. We write $\ell(\lambda) = n$ for the length of $\lambda$. Denote the set of partitions of $N$ by $\mathcal{P}_{N}$ \\
When $\lambda \vdash N$, a {\it Young diagram of shape $\lambda$} is an array of boxes where the $i$-th row consists of $\lambda_{i}$ boxes (the top row is the first and the bottom row is the $n$-th). A {\it standard tableau} of shape $\lambda$ is a filling of a Young diagram of shape $\lambda$ with the alphabet $\{1, 2, \ldots, N\}$ in such a way that the numbers are increasing along rows and down columns. The set of standard tableaux of shape $\lambda$ is denoted $\Std(\lambda)$. For $T^{\lambda} \in \Std(\lambda)$ we write $t_{i,j}^{\lambda}$ for the $(i,j)$ entry of $T^{\lambda}$, for $1 \leq i \leq \ell(\lambda)$ and $1 \leq j \leq \lambda_{i}$.
\subsection{Ordering on tableaux} \label{subsection:ordering} For this section, we follow the notational convention of Spaltenstein \cite{Spa}. Fix $\lambda \vdash n$ and let $\sigma$ denote a standard Young tableau of shape $\lambda$. For each $i \in \{1,\ldots, n\}$, let $\sigma_i$ denote the column of $\sigma$ in which the entry $i$ occurs. Observe that $\sigma$ is completely determined by the sequence $\sigma_1, \ldots, \sigma_n$. \\
For $\sigma, \tau \in \Std(\lambda)$ declare that $\sigma < \tau$ if for some $1 \leq i \leq n$, we have $\sigma_i < \tau_i$ and for each $i \leq j \leq n$, $\sigma_j=\tau_j$. This defines a total ordering on $\Std(\lambda)$.
\begin{example}
For $\lambda=(3,2)$, the following represents the total ordering:
$$
\young(123,45)< \young(124,35)< \young(134,25)< \young(125,34) < \young(135,24)
$$
\end{example}
\subsection{Stacking tableaux}
Let $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_n) \vDash N$. Now for each $i =1,...,n$ we pick a partition $\mu_i = (\mu_{i,1},...,\mu_{i, m_i})$ of $\lambda_i$ and write $m := \max m_i$. Define a new partition
\begin{eqnarray}
\label{e:musigmadefn}
\mu^\Sigma = (\mu^\Sigma_1, \mu^\Sigma_2, ..., \mu^\Sigma_{m}),\\
\mu^\Sigma_j := \sum_{i=1}^m \mu_{i, j}.
\end{eqnarray}
where we adopt the convention $\mu_{i,j} = 0$ for $j > m_i$.\\
Now we define a map
\begin{eqnarray}
\stk : \prod_{i=1}^n \Std(\mu_i) \hookrightarrow \Std(\mu^\Sigma)
\end{eqnarray}
which we call {\it the stacking map}. This is the key new combinatorial construction of this paper. \\
Now for $\lambda \vDash N$ and $\mu_{i} \vdash \lambda_i$ as above, let $(T^{\mu_{1}}, T^{\mu_{2}}, \ldots, T^{\mu_{n}})$ be a tuple of tableaux in $\prod_{i=1}^n \Std(\mu_i)$. We describe the image of $(T^{\mu_{1}}, T^{\mu_{2}}, \ldots, T^{\mu_{m}})$ under the stacking map by determining its $(i,j)$-th entry, as follows: for a fixed $1\le i \le m$ and $1\le j\le \mu^\Sigma_{i}$ let $k$ be the maximal index such that $\mu_{1,i} + \mu_{2,i} + \ldots + \mu_{k-1,i} < j$ (once again $\mu_{i,j} = 0$ for $j > m_i$). Put
\begin{equation} \label{eq:stack}
\displaystyle \tilde{j}:= j -(\mu_{1,i} + \mu_{2,i} + \ldots + \mu_{k-1,i} ).
\end{equation}
Then define
\begin{equation}
t_{i,j}^{\Sigma}:=t_{i,\tilde{j}}^{\mu_{k}} + \sum_{l=1}^{k-1} \lambda_{l}
\end{equation}
to be the $(i,j)$-th entry of the tableau $T^{\Sigma}$, and we set
$$
\stk(T^{\mu_{1}}, T^{\mu_{2}}, \ldots, T^{\mu_{m}} ) := T^\Sigma.
$$
Since each $T^{\mu_{i}}$ is standard, it follows that $T^{\Sigma}$ is standard of shape $\mu^{\Sigma}$.
\begin{example}
Let $N=15$ and consider the partition $\lambda=(6,5,4)$. Define three further partitions of the parts of $\lambda$ as follows: $\mu_{1}=(3,3), \mu_{2}=(2,2,1)$ and $\mu_{3}=(1^4)$; and now consider three standard Young tableaux of shape $\mu_{i}$ for $i=1,2,3$.
$$
\left( \young(134,256), \young(13,25,4), \young(1,2,3,4)\right)
$$
Then the new partition $\mu^{\Sigma}$ is $(6,6,2,1)$ and the image under the stacking map will be
$$
T^{\Sigma}=\young(1347912,25681113,1014,15)
$$
We explicitly confirm the $(3,2)$ entry of $T^{\Sigma}$. Set $i=3$ and $j=2$. Then the maximal $k$ such that $\mu_{1,3}+\mu_{2,3} + \ldots + \mu_{k-1,3} < 2$ is $k=3$ (where we note that $\mu_{1,3}=0)$. Hence $\tilde{j}=2-(\mu_{1,3}+\mu_{2,3})=2-(0+1)=1$ and so
$$
t_{3,2}^{\Sigma}=t_{3,1}^{\mu_{3}} + \sum_{l=1}^{2} \lambda_l = 3+(6+5)=14.
$$
Let us also calculate the $(3,1)$ entry of $T^{\Sigma}$. Set $i=3$ and $j=1$. Then the maximal $k$ such that $\mu_{1,3}+\mu_{2,3} + \ldots + \mu_{k-1,3} < 1$ is $k=2$, since $\mu_{1,3}=0$ and $\mu_{2,3}=1$. Hence $\tilde{j}=1 - (\mu_{1,3})=1-0=1$ and so:
$$
t_{3,1}^{\Sigma}=t_{3,1}^{\mu_{2}}+\sum_{l=1}^{1}\lambda_{l}=4+6=10
$$
\end{example}
\begin{proposition}
The tableau $T^{\Sigma}=\stk(T^{\mu_{1}}, T^{\mu_{2}}, \ldots, T^{\mu_{m}})$ is standard.
\end{proposition}
\begin{proof}
We first show that $T^{\Sigma}$ is increasing along columns. Fix $i,j$ and $j'$ in the appropriate range and suppose that $j < j'$. Let $k$ be maximal such that $\mu_{1,i}+ \ldots + \mu_{k-1,i} < j$ and let $k'$ be maximal such that $\mu_{1,i}+ \ldots + \mu_{k'-1,i} < j'$. Since $j < j'$, it follows that $k \leq k'$. Suppose that $k=k'$. Put $\displaystyle \tilde{j}=j-\sum_{l=1}^{k-1} \lambda_{l}$ and $\tilde{j'}=j-\sum_{l=1}^{k'-1} \lambda_{l}$. It is clear that $\tilde{j} < \tilde{j'}$ and since $T^{\mu_{k}}$ is standard, we have $t_{i,\tilde{j}}^{\mu_{k}} < t_{i,\tilde{j'}}^{\mu_{k}}$ and so $t_{i,j}^{\Sigma} < t_{i,j'}^{\Sigma}$. If $k<k'$, then since $1\leq t_{i,\tilde{j}}^{\mu_{k}} \leq \lambda_{k}$ we have:
$$
t_{i,j}^{\Sigma} = t_{i,\tilde{j}}^{\mu_{k}} + \sum_{l=1}^{k-1}\lambda_{l} < \sum_{l=1}^{k'-1}\lambda_{l} < t_{i,\tilde{j'}}^{\mu_{k'}} + \sum_{l=1}^{k-1}\lambda_{l} = t_{i,j'}^{\Sigma}
$$
Hence $T^{\Sigma}$ is increasing along columns. \vspace{5pt}
We now show that $T^{\Sigma}$ is increasing down rows. Now fix $i, i'$ and $j$ in the appropriate range and suppose that $i < i'$. Then $k \leq k'$. If $k=k'$, then since $T^{\mu_{k}}$ is a standard tableau, it follows that $t_{i,j}^{\mu_{k}} < t_{i',j}^{\mu_{k}}$. If $k < k'$ then again since $1 \leq t_{i,j}^{\mu_{k}} \leq \lambda_{k}$, we have:
$$
t_{i,j}^{\Sigma} = t_{i,\tilde{j}}^{\mu_{k}} + \sum_{l=1}^{k-1}\lambda_{l} < \sum_{l=1}^{k'-1}\lambda_{l} < t_{i',\tilde{j}}^{\mu_{k'}} + \sum_{l=1}^{k-1}\lambda_{l} = t_{i',j}^{\Sigma}
$$
and so $T^{\Sigma}$ is increasing down rows. Hence $T^{\Sigma}$ is a standard tableau.
\end{proof}
\section{Lie algebras and Springer fibres}
\subsection{Lie algebras of reductive algebraic groups}
Fix a $p \ge 0$ either zero or prime. Pick once and for all an algebraically closed field $\k$ of characteristic $p$. All vector spaces, algebras and algebraic varieties will be defined over $\k$. If $X$ is any variety over $\k$ then we write $\Comp(X)$ for the set of irreducible components. Let $G$ be a reductive algebraic group over $\k$ and assume the standard hypotheses (\cite[§2.9]{JaNO}) so that, in particular, $p$ is a good prime for the root system of $G$.\\
Write $\mathfrak{g} = \Lie(G)$. When $p > 0$ we write $x \mapsto x^{[p]}$ for the natural $G$-equivariant restricted structure on $\mathfrak{g}$. The definition of a nilpotent element of $\mathfrak{g}$ depends on whether or not $p = 0$. For $p = 0$ a {\it nilpotent element} $e\in \mathfrak{g}$ is one which acts nilpotently on every finite dimensional representation. For $p > 0$ a {\it nilpotent element} is one satisfying $e^{[p]^i} = 0$ for $i \gg 0$. For example when $G = \GL_N$ nilpotent elements are just the matrices which act nilpotently on the natural representation $\k^N$ of $\mathfrak{g}$ (this description is independent of $p$). Write $\mathcal{N}(\mathfrak{g}) \subseteq \mathfrak{g}$ for the {\it nilpotent cone}, defined to be the closed algebraic subvariety consisting of nilpotent elements.\\
When $x\in G$ we write $G^x$ and $\mathfrak{g}^x$ for the stabliser and centraliser respectively. By \cite[§2.9]{JaNO} we have
\begin{eqnarray}
\label{e:centraliservsLiealg}
\Lie(G^x) = \mathfrak{g}^x
\end{eqnarray}
for all $x\in \mathfrak{g}$.\\
As usual $\mathcal{B}$ denotes the projective algebraic variety consisting of all Borel subalgebras of $\mathfrak{g}$, the flag variety of $\mathfrak{g}$. If we pick a Borel subgroup $B = T \ltimes N$, maximal torus $T$ and unipotent radical $N$, and Lie algebra $\b = \t \oplus \mathfrak{n}$ then we may identify $G/B \overset{\sim}{\longrightarrow} \mathcal{B}$ via the morphism of varieties $gB \mapsto g\cdot\b$ (see \cite[§10]{JaNO} for example). If $e \in \mathfrak{g}$ is nilpotent then the {\it Springer fibre of $e$} is the closed subvariety
\begin{eqnarray}
\mathcal{B}_e := \{ \b \in \mathcal{B} \mid e\in \b\}
\end{eqnarray}
Identifying with $G/B$ this is equal to $\{gB \in G/B \mid e\in \Lie(g\cdot \b)\}$.
This is equal to the fibre over $e$ of Springer's resolution $\widetilde \mathcal{N}(\mathfrak{g}) \twoheadrightarrow \mathcal{N}(\mathfrak{g})$ of the nilpotent cone. The dimensions of Springer fibres are conveniently described as follows.
\begin{thm} \cite[Theorem~10.11]{JaNO}
\label{L:Springerdimension}
$\mathcal{B}_e$ is of pure dimension, and every irreducible component has dimension
$$\dim G/B - \frac{1}{2} \dim (G\cdot e) = \frac{1}{2}\mathrm{codim}_{\mathcal{N}(\mathfrak{g})} (G\cdot e).$$
\end{thm}
The goal of this paper is to study the combinatorics of $\Comp(\mathcal{B}_e)$ for $G = \GL_n$ using the theory of induced nilpotent orbits.
\subsection{Lusztig--Spaltenstein induction for Springer fibres}
The theory of (Lusztig--Spaltenstein) induced unipotent classes was first introduced in \cite{LS} over $\mathbb{C}$. In this section we recap properties of induced nilpotent orbits for the Lie algebra of a reductive group, under the standard hypothesis, following \cite{PS}. We go on to explain how induction of orbits gives rise to a closed morphism of relative dimension zero between Springer fibres (Proposition~\ref{prop:LSmap}); this result was presumably well-known to experts. \\
A {\it Levi subalgebra of $\mathfrak{g}$} is the Lie algebra of a Levi factor of a parabolic subgroup of $G$. Let $\mathfrak{g}_0 \subseteq \mathfrak{g}$ be a Levi subalgebra and let $\mathfrak{p}$ be a parabolic subalgebra admitting $\mathfrak{g}_0$ as a Levi factor. Write $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{n}$ for a Levi decomposition of $\mathfrak{p}$, where $\mathfrak{n}$ is the nilradical of $\mathfrak{p}$. \\
If $\O_0$ is any nilpotent orbit in $\mathfrak{g}_0$ then it is easily seen that $\operatorname{Ad}(G)(\O_0 + \mathfrak{n})$ contains a unique dense $G$-orbit which we denote $\operatorname{Ind}_{\mathfrak{g}_0}^\mathfrak{g}(\O_0)$, and call the {\it induced orbit from $(\mathfrak{g}_0, \O_0)$}. As the notation suggests it only depends on the $G$-orbit of the pair $(\mathfrak{g}_0, \O_0)$ and not on the choice of parabolic $\mathfrak{p}$ containing $\mathfrak{g}_0$ as a Levi factor. Furthermore induction satisfies the following two important properties:
\begin{lem}
\label{L:inductionproperties}
\begin{enumerate}
\setlength{\itemsep}{4pt}
\item (Transitivity) If $\mathfrak{g}_0 \subseteq \mathfrak{g}_1 \subseteq \mathfrak{g}$ are Levi subalgebras and $\O_0 \subseteq \mathfrak{g}_0$ is a nilpotent orbit then $\operatorname{Ind}_{\mathfrak{g}}^\mathfrak{g}(\O_0) = \operatorname{Ind}_{\mathfrak{g}_0}^{\mathfrak{g}_1} \operatorname{Ind}_{\mathfrak{g}_1}^\mathfrak{g}(\O_0).$
\item (Preservation of codimension) With $\O_0\subseteq \mathfrak{g}_0$, \ $e_0 \in \mathfrak{g}_0$ and $e\in \operatorname{Ind}_{\mathfrak{g}_0}^\mathfrak{g}(\O_0)$ we have $\dim \mathfrak{g}_0^{e_0} = \dim \mathfrak{g}^e$.
\item $\O_\mathfrak{p}$ is a single $P$-orbit.
\end{enumerate}
\end{lem}
\begin{proof}
The first two parts were first observed under the standard hypotheses in \cite[§2.5]{PS}. The third part was proven by Lusztig--Spaltenstein \cite[Theorem~1.3(c)]{LS} in the setting of complex algebraic groups, and the same proof works in our setting, applying \cite[Theorem~2.6(iv)]{PrKR}.
\end{proof}
Now let $\mathfrak{p}\subseteq \mathfrak{g}$ be a parabolic subalgebra and $\mathfrak{g}_0$ a Levi factor. Pick a nilpotent orbit $\O_0\subseteq \mathfrak{g}_0$ and write $\O_\mathfrak{p} = \operatorname{Ind}_{\mathfrak{g}_0}^\mathfrak{g}(\O_0) \cap \mathfrak{p}$. Pick an element $e \in \O_\mathfrak{p}$ and write $e = e_0 + e_1$ for the decomposition of $e$ across $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{n}$. In this paper we study the following map
\begin{eqnarray}
\label{e:LSdefn}
\begin{array}{rcl}
\LS & : & \mathcal{B}^0 \longrightarrow \mathcal{B},\\
& & \b_0 \longmapsto \b_0 \oplus \mathfrak{n}.
\end{array}
\end{eqnarray}
We call this the {\it Lusztig--Spaltenstein morphism of Springer fibres}. Some basic properties are listed here.
\begin{proposition}
\label{P:LSwelldefined}
\begin{enumerate}
\setlength{\itemsep}{4pt}
\item $\LS$ is well-defined $G_0$-equivariant morphism of algebraic varieties.
\item $\LS$ restricts to a map $\mathcal{B}_{e_0}^0 \to \mathcal{B}_e$ which is closed and of relative dimension zero.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $B \subseteq P$ be any Borel subgroup and pick a torus $T \subseteq B$. Let $\Phi \subseteq X^*(T)$ be the corresponding root system and $\Delta \subseteq \Phi$ the set of simple roots corresponding to $B$. Thanks to the classification of parabolic subgroups and their Levi factors (see \cite[Theorems~30.1\&~30.2]{Hum}, for example) we can choose $T$ so that $P$ is a standard parabolic with respect to $\Delta$. This means that after choosing a Levi factor $G_0$ containing $T$, and writing $\Phi_0 \subseteq \Phi$ for the root system of $G_0$, we have that $\Delta_0 := \Phi_0 \cap \Delta$ is a set of simple roots in $\Phi_0$.
We have implicitly chosen positive roots $\Phi_0^+, \Phi^+$. Write $\Phi^+_1 = \Phi^+ \setminus \Phi^+_0$. Pick a set $\{x_\alpha \mid \alpha \in \mathfrak{g}\} \subseteq \Phi$ of root vectors. Since $P$ is standard with respect to $\Delta$ we have $\mathfrak{p} = \underline{\b} + \mathfrak{g}_0$ and so $\mathfrak{n} = \sum_{\alpha \in \Phi^+_1} \k x_\alpha$.
Since $G_0$ preserves $\mathfrak{n}$ it follows from the definition of $\LS$ that for any Borel $\b_0 \subseteq \mathfrak{g}_0$ we have $\LS(g\cdot \b_0) = g\cdot \LS(\b_0)$ for all $g\in G_0$. So $\LS$ is a $G_0$-equivariant map sending vector spaces of $\mathfrak{g}_0$ to subspaces of $\mathfrak{g}$.
Now let $\b_0 \subseteq \mathfrak{g}_0$ be the unique Borel subalgebra containing $\t$ and $\{x_\alpha \mid \alpha \in \Phi_0^+\}$ for $\alpha \in \Phi_0^+$. By construction $\LS(\b_0) = \b$. Since $\mathcal{B}^0 = G_0 \cdot \b_0$ the $G_0$-equivariance implies that $\LS(\mathcal{B}^0)\subseteq \mathcal{B}$. So $\LS$ is well-defined.
We now check that $\LS$ is a morphism. Let $B_0$ be the Borel subgroup of $G_0$ with Lie algebra $\b_0$ chosen in the previous paragraph, and $B_0^- \subseteq G_0$ be the Borel subgroup of $G_0$ opposite to $B_0$. Write $U_0^-$ for the unipotent radical. Recall that the big cell $\Omega_0 \subseteq \mathcal{B}_0$ is the image of $U_0^-$ under the map $G_0 \to \mathcal{B}^0$ given by $g\mapsto g\cdot \b_0$ (see \cite[28.5]{Hum} for example), and that the map $U_0^- \to \mathcal{B}^0$ is injective. Since $\LS$ is $G_0$-equivariant and $\mathcal{B}^0$ has an affine cover by $G_0$-translates of $\Omega_0$ it will suffice to show that $\LS|_{\Omega_0}$ is a morphism. Writing $U^-$ for the unipotent radical of the opposite Borel to $B$ we have a similar description of the big cell $\Omega\subseteq \mathcal{B}$. It follows from $B_0 \subseteq B$ that $\LS(\Omega_0) \subseteq \Omega$ and so it suffices to show that the pullback $\LS^* : \k[\Omega] \to \k[\Omega_0]$ is a homomorphism.
For $\alpha\in \Phi$ let $u_\alpha : \k \to G$ be the corresponding 1-parameter subgroup. For a fixed total order on $\Phi$ we have isomorphisms $\mathbb{A}^{\Phi^+} \overset{\sim}{\longrightarrow} U^-$ and $\mathbb{A}^{\Phi^+_0} \overset{\sim}{\longrightarrow} U^-_0$ given by $(t_{\alpha})_{\alpha \in \Phi^+} \mapsto \prod_{\alpha \in \Phi^+} u_{-\alpha}(t_{\alpha})$, and similar for $U^-_0$. Now after identifying through the isomorphism $\Omega \cong U^- \cong \mathbb{A}^{\Phi^+}$ and $\Omega_0 \cong U_0^- \cong \mathbb{A}^{\Phi_0^+}$ we see that $\LS^*$ is just the projection homomorphism $\k[\mathbb{A}^{\Phi^+}] \twoheadrightarrow \k[\mathbb{A}^{\Phi_0^+}]$ corresponding to the inclusion $\Phi_0^+ \subseteq \Phi_0$. Thus $\LS$ is a morphism.
The relative dimension of $\mathcal{B}_{e_0}^0 \to \mathcal{B}_e$ is zero thanks to \eqref{e:centraliservsLiealg}, Lemma~\ref{L:Springerdimension} and Lemma~\ref{L:inductionproperties}(2). Since $\LS$ is a morphism of projective (hence complete) varieties, it follows from \cite[Proposition~6.1]{Hum} that it is closed. This completes the proof.
\end{proof}
\begin{cor}
\label{cor:inducedmaponcomponents}
The map on Springer fibres induces a map on the sets of components: for every $C \in \Comp(\mathcal{B}^0_{e_0})$ we have $\LS(C) \in \Comp(\mathcal{B}_e)$.
\end{cor}
\begin{rem}
One of the most surprising features of Lusztig--Spaltenstein induction of nilpotent orbits is that it depends on the conjugacy class of $\mathfrak{g}_0$, not on the conjugacy class of $\mathfrak{p}$. It would be interesting to know whether a similar independence statement can be formulated for \eqref{e:LSdefn}.
\end{rem}
\section{General linear lie algebras}
For the rest of the paper keep $N > 0$ fixed and choose an algebraically closed field of any characteristic $p \ge 0$. Let $G = \GL_N(\k)$ and $\mathfrak{g} = \Lie(G) = \mathfrak{gl}_N(\k)$. Let $\kappa$ denote the trace form associated to the natural representation $V = \k^N$. We note that the standard hypotheses are satisfied for $G$.
\subsection{Nilpotent Orbits, Levi subalgebras and induction}
\label{ss:nilpotentorbitslevisandinduction}
The nilpotent elements of $\mathfrak{g}$ are those which act nilpotently on the natural representation, and we denote the set of such elements $\mathcal{N}(\mathfrak{g})$. If $e\in \mathcal{N}(\mathfrak{g})$ then we can decompose $V$ non-uniquely into indecomposable $\k[e]$-modules $V = \bigoplus_{i=1}^n V_i$ which we refer to as (a choice of) {\it Jordan block spaces for $e$}. A {\it Jordan basis for $e$} is a basis $\{v_{i,j} \mid i=1,...,n, \ j = 1,\ldots, \dim V_i\}$ such that $V_i$ has basis $\{v_{i,j} \mid j=1,\ldots, \dim(V_i)\}$
$$ev_{i,j} = \left\{\begin{array}{cc} v_{i, j-1} & \text{ if } j > 1 \\ 0 & \text{ if } j = 1\end{array} \right.\vspace{6pt}$$
The $G$-orbits on $\mathcal{N}(\mathfrak{g})$ are classified by partitions: for each $\lambda = (\lambda_1,...,\lambda_n) \vdash N$ we let $\O_\lambda$ denote the $G$-orbit consisting of elements with Jordan block spaces of dimension $\lambda_1, \lambda_2, ...,\lambda_n$.\\
The Levi subalgebras of $\mathfrak{g}$ are also classified by partitions of $N$: for $\lambda \vdash N$ choose any vector space decomposition $V = \bigoplus_{i=1}^n V_i$ where $\dim V_i = \lambda_i$, let $\mathfrak{g}_\lambda \cong \bigoplus \mathfrak{gl}_{\lambda_i}$ be the subalgebra of $\mathfrak{g}$ which preserves each $V_i$. This defines a bijection from partitions of $N$ to conjugacy classes of Levi subalgebras (see \cite[§7.2]{CM} for example).\\
Now we are in a position to describe Lusztig--Spaltenstein induction of nilpotent orbits. Let $\lambda \vdash N$ and let $\mathfrak{g}_\lambda$ be a choice of Levi subalgebra, as described above. Suppose that $G_\lambda \subseteq G$ is a Levi subgroup with $\mathfrak{g}_\lambda = \Lie(G_\lambda)$. The basic result for describing induced nilpotent $G$-orbits is due to Kraft and Ozeki--Wakimoto, independently.
\begin{lem} \cite[7.2.3]{CM}
\label{L:inducedfromzero}
If $\O_0$ is the zero orbit in $\mathfrak{g}_\lambda$ then the partition associated to $\operatorname{Ind}_{\mathfrak{g}_\lambda}^\mathfrak{g}(\mathcal{O}_0)$ is the transpose $\lambda^\top$.
\end{lem}
We need to upgrade this result to describe induction from non-zero orbits. For each $i=1,...,n$ choose a partition $\mu_i = (\mu_{i,1},...,\mu_{i,m_i}) \vdash \lambda_i$. By the above remarks there is a nilpotent $G_0$-orbit $\O_\mu \subseteq \mathfrak{g}_0$ such that the projection to the factor $\mathfrak{gl}_{\lambda_i}$ is the $\GL_{\lambda_i}$-orbit classified by partition $\mu_i$.\\
Recall the definition of $\mu^\Sigma$ from \eqref{e:musigmadefn}.
\begin{cor}
\label{cor:partitionsigma}
The partition of $\operatorname{Ind}_{\mathfrak{g}_\lambda}^\mathfrak{g}(\mathcal{O}_\mu)$ is $\mu^\Sigma$.
\end{cor}
\begin{proof}
Writing $\mathfrak{g}_\lambda = \bigoplus_{i=1}^n \mathfrak{gl}_{\lambda_i}$ we pick a Levi subalgebra $\mathfrak{g}_{\mu_i^\top}$ of $\mathfrak{gl}_{\lambda_i}$ which has partition of type $\mu_i^\top$. Write $\mathfrak{g}_{\mu^\top} = \bigoplus_{i=1}^n \mathfrak{g}_{\mu_i^\top} \subseteq \bigoplus_{i=1}^n \mathfrak{gl}_{\lambda_i}$ and note that $\mathfrak{g}_{\mu^\top}$ is a Levi subalgebra of $\mathfrak{g}$. Write $\O_0$ for the zero orbit in $\mathfrak{g}_{\mu^\top}$. Then according to Lemma~\ref{L:inducedfromzero} $\O_\mu = \operatorname{Ind}_{\mathfrak{g}_{\mu^\top}}^{\mathfrak{g}_\lambda}(\O_0)$. Now by the transitivity of induction (Lemma~\ref{L:inductionproperties}) we see that $\operatorname{Ind}^\mathfrak{g}_{\mathfrak{g}_\lambda}(\O_\mu) = \operatorname{Ind}_{\mathfrak{g}_{\mu^\top}}^\mathfrak{g}(\O_0)$.
Now observe that if we concatenate $\mu_1^\top, \mu_2^\top,...,\mu_n^\top$ and reorder to make a partition $\nu \vdash N$ then $\nu^\top = \mu^\Sigma$. The proof concludes by applying Lemma~\ref{L:inducedfromzero} once more, to show that the partition of $\operatorname{Ind}_{\mathfrak{g}_{\mu^\top}}^\mathfrak{g}(\O_0)$ is $\nu^\top$.
\end{proof}
\subsection{A representative for the induced orbit}
\label{ss:representative}
In this Section we indicate a nice choice of representative for a nilpotent orbit $\O_0 \subseteq \mathfrak{g}_0$ in a Levi subalgebra, and for the induced orbit $\O := \operatorname{Ind}_{\mathfrak{g}_0}^\mathfrak{g}(\O_0)$. This will be useful for illustrating some of our arguments below.
Fix a composition $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_n)$ of $N$. For each $i =1,...,n$ we pick a partition $\mu_i = (\mu_{i,1},...,\mu_{i, m_i})$ of $\lambda_i$. Now from this data we define a set
\begin{eqnarray}
\mathbb{I} = \{(i,j,k) \mid i=1,...,n, \ j=1,...,m_i, \ k=1,...,\mu_{i,j}\}
\end{eqnarray}
and we let $V$ be the $N$-dimensional complex vector space with basis $\{v_{i,j,k} \mid (i,j,k) \in \mathbb{I}\}$. We identify $V$ with the natural representation of the general linear Lie algebra $\mathfrak{g} := \mathfrak{gl}_N$, and so $\mathfrak{g}$ admits a basis
\begin{eqnarray}
& &\{e_{i_1,j_1,k_1; i_2,j_2,k_2} \mid (i_1,j_1,k_1), (i_2,j_2,k_2) \in \mathbb{I}\},\\
& & e_{i_1,j_1,k_1; i_2,j_2,k_2}v_{i_3,j_3,k_3} = \delta_{i_2, i_3} \delta_{j_2, j_3} \delta_{k_2, k_3} v_{i_1, j_1, k_1}.
\end{eqnarray}
Now we define the Levi subalgebra $\mathfrak{g}_0 \subseteq \mathfrak{g}$ to be the subalgebra spanned by elements $\{e_{i, j_1, k_1; i, j_2, k_2} \mid i=1,...,n, \ \ (i, j_1, k_1), (i, j_2, k_2) \in \mathbb{I}\}$. This algebra is isomorphic to $\mathfrak{gl}_{\lambda_1}\oplus \cdots \oplus \mathfrak{gl}_{\lambda_m}$.
There is a corresponding decomposition of $V$: for $i=1,...,n$ fixed we let $V_i$ be the subspace spanned by $v_{i,j,k}$, allowing $j,k$ to vary. Then we have $V = \bigoplus_i V_i$, and $V_i$ identifies with the natural representation of $\mathfrak{gl}_{\lambda_i}$
There is a parabolic subalgebra $\mathfrak{p} \subseteq \mathfrak{g}$ admitting $\mathfrak{g}_0$ as a Levi factor, which is spanned by elements $e_{i_1, j_1, k_1; i_2, j_2, k_2}$ where $i_2 \ge i_1$. The nilradical $\mathfrak{n} \subseteq \mathfrak{p}$ consists of elements with $i_2 > i_1$.
Now we define some nilpotent elements. We let
\begin{eqnarray}
\label{e:e0defn}
e_0 := \sum_{i=1}^m \sum_{j=1}^{m_i} \sum_{k=1}^{\mu_{i,j} - 1} e_{i,j,k; i,j,k+1} \in \mathfrak{g}_0.
\end{eqnarray}
This is a nilpotent element of $\mathfrak{g}_0$. When restricted to the subspace $V_i$ the associated nilpotent operator has partition $\mu_i$. Now we define an element $e_1 \in \mathfrak{n}$. Let $d_j$ be the number of indexes $1\le i \le n$ such that $m_i \ge j$ and let $\{i_1^j,...,i_{d_j}^j\} \subseteq \{1,...,n\}$ be the indexes satisfying $m_{i_k} \ge j$ for $k = 1,...,d_j$. Now we let
\begin{equation}
\label{e:e1defn}
e_1 := \sum_{j > 0} \sum_{k=1}^{d_j-1} e_{i_{k}^j, j, \mu_{i_{k}^j, j}; i_{k+1}^j, j, 1}
\end{equation}
and define
\begin{eqnarray}
e := e_0 + e_1.
\end{eqnarray}
The elements $e_0, e_1$ are easily understood pictorially, identifying the elements of the Jordan basis with the boxes in a tuple of Young diagrams. We illustrate this with an example.
\begin{example}
\label{ex:representativediagram}
Let $\lambda=(7, 5, 12)$ be a composition of $24$ and let $$(\mu_1, \mu_2, \mu_3)=((4,2,1), (3,2), (3^2,2^2,1^2)).$$ In the following diagram, the elements of the Jordan basis $v_{i,j,l}$ are identified with the boxes of the Young diagrams in the obvious manner. The action of $e_0$ is illustrated in black, and $e_1$ in blue, as follows:
\begin{center}
\begin{tikzpicture}[scale=1]
\node at (0,0) {$\yng(4,2,1)$};
\node at (3,0.25) {$\yng(3,2)$};
\node at (6,-0.7) {$\yng(3,3,2,2,1,1)$};
\draw[->] (0.7,0.8) to[bend right]node[above]{\tiny $e_0$} (0.3,0.8);
\draw[->] (0.2,0.8) to[bend right] (-0.2,0.8);
\draw[->] (-0.3,0.8) to[bend right] (-0.7,0.8);
\draw[->] (-0.8,0.8) to[bend right] (-1.2,0.8) node[ above]{\tiny $0$};
\draw[->] (3.5,0.8) to[bend right]node[above]{\tiny $e_0$} (3.1,0.8);
\draw[->] (3,0.8) to[bend right] (2.6,0.8);
\draw[->] (2.5,0.8) to[bend right] (2.1,0.8)node[ above]{\tiny $0$};
\draw[->] (6.5,0.8) to[bend right]node[above]{\tiny $e_0$} (6.1,0.8);
\draw[->] (6,0.8) to[bend right] (5.4,0.8);
\draw[->] (5.3,0.8) to[bend right] (4.9,0.8) node[ above]{\tiny $0$};
\draw[blue, ->] (2,0.5) to node[below] {\tiny $e_{1}$} (1.3,0.5);
\draw[blue, ->] (2,0) to (0.3,0);
\draw[blue, ->] (5,0.5) to node[below] {\tiny $e_{1}$} (4.3,0.5);
\draw[blue, ->] (5,0) to (3.3,0);
\draw[blue, ->] (5,-0.45) to (-0.3,-0.45);
\draw[blue, ->] (5,-.9) to (4.5,-.9)node[left]{\tiny $0$};
\draw[blue, ->] (5,-1.4) to (4.5,-1.4)node[left]{\tiny $0$};
\draw[blue, ->] (5,-1.9) to (4.5,-1.9) node[left]{\tiny $0$};
\end{tikzpicture}\\
{\sf Diagram 1: The action of $e_0$ and $e_1$ on the natural representation.}
\end{center}
Hence we may think of $e_0$ and $e_1$ as acting on the boxes where:
\begin{itemize}
\item $e_0$ moves a box in a given Young diagram to the box immediately to its left, unless it is in the first column of a Young diagram, in which case it sends it to $0$; and
\item $e_1$ sends a box in a given Young diagram to $0$ unless it is in the first column and there exists a box in the Young diagram immediately to to the left,
in which case it sends it to the box to its immediate left.
\end{itemize}
At this point, it is also worth illustrating how $e:=e_0 + e_1$ acts on the $N$-dimensional space $V$, via the `stacked' Young diagram. We picture the action as follows:
\begin{center}
\begin{tikzpicture}
\draw[->,black] (-2,1.5) to[bend right]node[above]{\tiny $e$} (-2.4,1.5);
\foreach \i [evaluate={\j=\i-0.4; }] in {2,1.5,1,...,-1.5}
\draw[->,black] (\i,1.5) to[bend right] (\j,1.5);
\end{tikzpicture}\vspace{2pt}\\
\hspace{15pt}{\tiny
\ydiagram[*(darkgray)]
{4,2,1}
*[*(gray)]{7,4}
* [*(white)]{10,7,3,2,1,1}}\\
{\sf Diagram 2: The action of $e$ on the natural representation.}
\end{center}
Hence we may see that for each $ 1 \leq j \leq 10 =\sum_{k=1}^{3}\mu_{k,1}$, we may picture $\ker(e^j)$ as the span of the boxes in the first $j$ columns of the stacked Young diagram. The colours of the boxes will be useful for illustrating certain arguments later on.
\end{example}
\begin{lem}
$e\in \O_\mathfrak{p}$.
\end{lem}
\begin{proof}
For $j=1,...,\max m_i$ fixed the basis vectors $v_{i,j,l}$ (allowing $i,l$ to vary) span a single Jordan block for $e$ of size $\sum_{i=1}^n \mu_{i,j} = \mu^\Sigma_j$. It follows that the $G$-orbit of $e$ has partition $\mu^\Sigma$ and the Lemma follows, thanks to Corollary~\ref{cor:partitionsigma}
\end{proof}
\subsection{Flags and Spaltenstein's description of the components of Springer fibres}
Let $e$ be a nilpotent element in $\mathfrak{gl}_{N}$ of Jordan type $\lambda$ and let $\mathcal{B}_e$ be the corresponding Springer fibre over $e$. We recall that the points of $\mathcal{B}$ are described in terms of full flags of $V = \k^N$: a flag $F_\bullet = (0\subsetneq F_1\subsetneq \cdots \subsetneq F_{N-1} \subsetneq \k^N)$ corresponds to the Borel subalgebra consisting of elements of $\mathfrak{g}$ preserving each $F_i$. Thus we may explicitly describe the Springer fibre in terms of flags as follows
$$
\mathcal{B}_e = \{ 0 \subset F_1 \subset \cdots \subset F_{N-1} \subset \mathbb{C}^N \, | \, \dim(F_i)=i, \ e(F_i) \subseteq F_{i-1} \} .
$$
For a fixed flag $F_{\bullet} \in \mathcal{B}_{e}$, by considering the Jordan type of $e$ restricted to the subspaces $F_i$, we obtain a natural map from the fibre to sequences of partitions:
\begin{eqnarray}
\label{e:Spalmapdefn}
\begin{array}{rcl}
\Phi & : & \mathcal{B}_e \longrightarrow \mathcal{P}_1 \times \mathcal{P}_2 \times \ldots \times \mathcal{P}_{N} \vspace{4pt} \\
& & F_{\bullet} \longmapsto (\Type(e_{|F_1}), \ldots, \Type(e_{|F_{n}}))
\end{array}
\end{eqnarray}
We identify partitions of $N$ with their corresponding Young diagrams.
\begin{lemma} \label{lemma:spaltenstein}
Let $e$ be a nilpotent of Jordan type $\lambda$ and let $H$ be an $e$-stable hyperplane of $\mathbb{C}^n$. Then $\Type(e|_{H})$ is obtained by removing the last box from the $j$-column of $\lambda$ where $j$ is maximal such that $H \supseteq \ker(e^{j-1})$.
\end{lemma}
\begin{proof}
We may choose a Jordan basis for the nilpotent $e$ on $V$ such that $H$ is spanned by all but one of those Jordan basis vectors. The result follows.
\end{proof}
By a {\it nested sequence of partitions} we mean a sequence of partitions, or Young diagrams, for each $1 \leq k \leq n$ such that each Young diagram is obtained from the previous by adding a box to an available row, where available means that the diagram obtained by adding the box is indeed a Young diagram. \\
Lemma \ref{lemma:spaltenstein} shows that the image of $\Phi$ is the set of nested sequences of partitions which are in bijection with $\Std(\lambda)$. For instance, the standard tableau
\begin{eqnarray}
\label{e:atab}
\young(134,25)
\end{eqnarray}
corresponds to the nested sequence:
$$
\left(\varnothing, \yng(1), \yng(1,1), \yng(2,1), \yng(3,1), \yng(3,2) \right)
$$
Hence we may identify the image of $\Phi$ with a subset of $\Std(\lambda)$ and a simple induction argument shows that $\Phi$ is a surjection onto $\Std(\lambda)$ and we have the following due to Spaltenstein:
\begin{theorem}\cite{Spa}
Let $e$ be nilpotent of Jordan type $\lambda$. Then:
\begin{enumerate}
\setlength{\itemsep}{4pt}
\item there is a bijection between $\Comp(\mathcal{B}_{e})$ and $\Std(\lambda)$ induced by $\Phi$ where for any $\sigma \in \Std(\lambda)$, $\overline{\Phi^{-1}(\sigma)}$ is an irreducible component.
\item (Cf. Theorem~\ref{L:Springerdimension}). $\mathcal{B}_{e}$ is of pure dimension and for all $\sigma \in \Std(\lambda)$, $$\dim(\overline{\Phi^{-1}(\sigma)})=\frac{1}{2}\sum_{i \geq 1} \lambda_i^\top(\lambda^\top_i-1).$$
\end{enumerate}
\end{theorem}
For $\sigma \in \Std(\lambda)$, put $X_{\sigma}:=\Phi^{-1}(\sigma)$. Using the ordering on standard tableaux given in Section~\ref{subsection:ordering} we have the following important property of closures.
\begin{lem}
\label{lem:Xsigmaintersectsclosure}
We have $X_\sigma \cap \overline{X}_\tau = \emptyset$ for $\tau > \sigma$.
\end{lem}
\begin{proof}
By \cite[Proposition, (a)]{Spa} we know that $X_\sigma$ is locally closed and $\bigcup_{\tau > \sigma} X_\sigma$ is closed in $\mathcal{B}_e$. Therefore $\overline X_\tau \subseteq \bigcup_{\tau' \ge \tau} X_{\tau'}$ and the lemma follows from the fact that the fibres of Spaltenstein's map are disjoint.
\end{proof}
As we observed in Section~\ref{ss:nilpotentorbitslevisandinduction} the $G_0$-orbit $\O_0$ is determined by partitions $\mu_1,...,\mu_n$ such that $\mu_i \vdash \lambda_i$. Note that Spaltenstein's map gives a map $\Phi_0 : \mathcal{B}_{e_0}^0 \to \prod_{i=1}^n \Std(\mu_i)$, and the fibre over a tuple $(\sigma^{(1)},...,\sigma^{(n)})$ of standard tableaux is denoted $X_{\sigma^{(1)},...,\sigma^{(n)}}$.
The next Lemma, combined with Lemma~\ref{L:inductionproperties}(3), shows that in order to understand $\LS$ for any element $e\in \O_\mathfrak{p}$ it suffices to understand the morphism for a single element. Thus our representative chosen above may be seen as a typical element for our purposes. It follows directly from the definition of $\LS$ and $\Phi$, see \eqref{e:LSdefn} and \eqref{e:Spalmapdefn}.
\begin{proposition}
\label{L:reductionlemma}
Suppose that $\mathfrak{p} = \Lie(P)$, $g \in P$ and let $g = g_0 u \in G_0U$ where $U = \mathrm{Rad}(P)$. Then we have a commutative diagrams
\begin{center}
\begin{tikzpicture}[node distance=2cm, auto]
\node (A) {$\mathcal{B}^0_{e_0}$};
\node (B) [below of= A] {$\mathcal{B}^0_{g_0\cdot e_0}$};
\node (C) [right of= A] {$\mathcal{B}_e$};
\node (D) [right of= B]{$\mathcal{B}_{g\cdot e}$};
\draw[->] (A) to node [swap]{$g_0$} (B);
\draw[->] (A) to node {$\LS$} (C);
\draw[->] (B) to node [swap]{$\LS$} (D);
\draw[->] (C) to node {$g$} (D);
\node (E) [right of= C] {$\mathcal{B}_{e}$};
\node (F) [below of= E] {$ $};
\node (G) [right of= F] {$\Std(\lambda)$};
\node (H) [right of= E] {$ $};
\node (I) [right of= H] {$\mathcal{B}_{g\cdot e}$};
\draw[->] (E) to node [swap]{$\Phi$} (G);
\draw[->] (E) to node {$g$} (I);
\draw[->] (I) to node {$\Phi$} (G);
\node (J) [right of= I] {$\mathcal{B}_{e_0}$};
\node (K) [below of= J] {$ $};
\node (L) [right of= K] {$\prod_{i=1}^n \Std(\mu_i)$};
\node (M) [right of= J] {$ $};
\node (N) [right of= M] {$\mathcal{B}_{g_0\cdot e_0}$};
\draw[->] (J) to node [swap]{$\Phi_0$} (L);
\draw[->] (J) to node {$g_0$} (N);
\draw[->] (N) to node {$\Phi_0$} (L);
\end{tikzpicture}
\end{center}
\end{proposition}
\subsection{The Lusztig--Spaltenstein map on fibres}
\label{ss:LSonfibres}
Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$ and pick a Levi factor $\mathfrak{g}_0 \subseteq \mathfrak{p}$. Let $\lambda = (\lambda_1,...,\lambda_n)$ denote the ranks of the general linear factors of $\mathfrak{g}_0$. Observe that the decomposition $\mathfrak{g}_0 \cong \bigoplus_{j=1}^n \mathfrak{gl}_{\lambda_j}$ gives a decomposition $\mathbb{C}^N = \bigoplus_{j=1}^N V_j$ where any $V_j$ is stable under the action of every $\mathfrak{gl}_{\lambda_k}$ and the restriction of $\mathfrak{gl}_{\lambda_j}$ to endomorphisms of $V_j$ is an isomorphism. We order the spaces $V_j$ in such a way that $\mathfrak{p}$ maps $V_j$ to $\bigoplus_{k=1}^{j} V_j$, and we note that $\lambda$ is a composition of $N$, not a partition in general.\\
We pick a nilpotent orbit $\O_0 \subseteq \mathfrak{g}_0$ and choose an element $e\in \operatorname{Ind}^\mathfrak{g}_{\mathfrak{g}_0}(\O_0) \cap \mathfrak{p}$. Let $e = e_0 + e_1$ be the decomposition of $e$ across the direct sum $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathrm{Rad}(\mathfrak{p})$. Now we can consider the map $$\LS : \mathcal{B}_{e_0}^0 \longrightarrow \mathcal{B}_e.$$
By our choices above we note that setting $W_j = \bigoplus_{k=1}^j V_k$ determines a partial flag of $\k^N$, and $\mathfrak{p}$ is described by $\mathfrak{p} = \{x\in \mathfrak{g} \mid xW_j \subseteq W_j \text{ for all } j\}$. In the following we regard the elements of $\mathcal{B}$ as full flags of $\mathbb{C}^N$, and similarly we regard an element of $\mathcal{B}^0$ as a tuple $(F^{(1)},...,F^{(n)})$ where each $F^{(j)}$ is a flag of $V_j$. In order to describe $\LS$ in this language we must rephrase the map in terms of flags.
\begin{lem} We have
\label{L:LSonflags}
$$
\LS(F_{\bullet}^{(1)}, \ldots, F_{\bullet}^{(n)}): ={\bar{F}}_{\bullet} =(0 \subset \bar{F}_{1} \subset \bar{F}_{2} \subset \cdots \subset \bar{F}_{N-1} \subset \k^{N})
$$
where
$$
\bar{F}_{i} = \begin{cases} F_{i}^{(1)} & \text{for} \; 1 \leq i \leq \lambda_{1} \\ W_{j-1} + F_{k}^{(j)} & \text{for}\; i > \lambda_{1} \; \text{where $j$ is maximal such that} \; \displaystyle \sum_{l=1}^{j-1} \lambda_{l} < i\; \text{and} \; k=i - \sum_{l=1}^{j-1} \lambda_{l}. \end{cases}
$$
\end{lem}
As we observed in Section~\ref{ss:nilpotentorbitslevisandinduction} the $G_0$-orbit $\O_0$ is determined by partitions $\mu_1,...,\mu_n$ such that $\mu_i \vdash \lambda_i$. Note that Spaltenstein's map gives a map $\Phi_0 : \mathcal{B}_{e_0}^0 \to \prod_{i=1}^n \Std(\mu_i)$, and the fibre over a tuple $(\sigma^{(1)},...,\sigma^{(n)})$ of standard tableaux is denoted $X_{\sigma^{(1)},...,\sigma^{(n)}}$. The next lemma follows directly from the definition of $\LS$ and $\Phi$, see \eqref{e:LSdefn} and \eqref{e:Spalmapdefn}.
Our Main Theorem will follow fairly quickly from the next Proposition.
\begin{proposition} \label{prop:LSmap}
For each $i=1,...,n$ we choose a standard tableau $\sigma^{(i)}$ for $\mu_i$. Then
$$\LS(X_{\sigma^{(1)},...,\sigma^{(n)}}) \subseteq \bigcup_{\tau \ge \stk(\sigma^{(1)}, \ldots, \sigma^{(n)})} X_\tau.$$
\end{proposition}
\begin{proof}
Let $(F^{(1)},\cdots,F^{(n)}) \in \mathcal{B}^0_{e_0}$, and suppose that $(F^{(1)},\cdots,F^{(n)})\in \Phi_0^{-1}(\sigma^{(1)},...,\sigma^{(n)})$. Let $\LS(F^{(1)},...,F^{(n)}) = \bar{F}_\bullet$, which is described explicitly in Lemma~\ref{L:LSonflags}. Suppose that $\bar{F}_\bullet \in \Phi^{-1}(\tau)$ for some $\tau \in \Std(\lambda)$. Put $\sigma^{\Sigma}= \stk(\sigma^{(1)},...,\sigma^{(n)})$. In order to prove the proposition we must show that $\tau \ge \sigma^\Sigma$. If $\tau = \sigma$ then we are done, so we may assume that $\tau \ne \sigma$.\\
Recall that $\tau_i$ is the column of $\tau$ where $i$ occurs. To establish the Proposition, we need to show that there exists an index $i \in \{1,\ldots, N\}$ such that $\tau_{i} > \sigma_{i}^{\Sigma}$ and $\tau_{k} = \sigma_{k}^{\Sigma}$ for $k > i$. Let $i$ be the maximal index such that $\tau_i \ne \sigma^\Sigma_i$. Now define $V' := \bar{F}_i$ and we consider $\mathfrak{g}' = \mathfrak{gl}(V')$, $e' := e|_{V'} \in \mathfrak{g}'$, $\mathfrak{g}_0' := \mathfrak{g}_0|_{V'}$, $G_0' := G_0|_{V'}$ and $\mathfrak{p}' := \mathfrak{p}|_{V'}$. Also write $\bar{F}'_\bullet = (0\subseteq \bar{F}_1 \subseteq \cdots \subseteq \bar{F}_{i-1} \subseteq V')$ which is a full flag of $V'$. Write $\mathcal{B}'$ for the flag variety of $\mathfrak{g}'$ and $\mathcal{B}'_{e'}$ for the Springer fibre, and $\Phi'$ for Spaltenstein's map. Note that $\bar{F}'_\bullet \in \mathcal{B}'_{e'}$ and that $\Phi'(\bar{F}_\bullet')$ consists of the boxes of $\tau$ with labels $1,...,i$.\\
Now let $j \in \{0,...,n-1\}$ be the largest index such that $\sum_{k=1}^j \lambda_k < i$, and set $d := i-\sum_{k=1}^{j-1}$. By Lemma~\ref{L:LSonflags} we know that $V_1,...,V_{j-1} \subseteq V'$ and $\tilde V_j := \bar{F}_i \cap V_j$ satisfies $V_1 \oplus \cdots V_{j-1} \oplus \tilde V_{j}$. Therefore $\mathfrak{p}'$ is the parabolic subalgebra of $\mathfrak{g}'$ stabilising the partial flag $W'_\bullet$ of $V'$ given by $W'_k = W_k$ for $k=1,..,j-1$ and $W'_j = V'$, whilst $\mathfrak{g}_0'$ is a Levi factor of $\mathfrak{p}'$. It follows that the projection $e_0'$ of $e'$ across $\mathfrak{p}' = \mathfrak{g}_0' \oplus \nil(\mathfrak{p}')$ is equal to the restriction $e_0|_{V'}$. The $G_0'$-orbit of $e_0'$ is determined by partitions $\mu_1',...,\mu_j'$ which are described as follows: $\mu_k = \mu_k'$ for $k=1,...,j-1$ and $\mu_j'$ is the shape of the tableau $\sigma^{(j)}{}'$ obtained from $\sigma^{(j)}$ by considering only the boxes labelled $1,2,...,d$. \\
We claim that $e'$ lies in the nilpotent orbit in $\mathfrak{g}'$ induced from $(\mathfrak{g}_0', G_0'\cdot e_0')$. To see this write $\sigma^{\Sigma}{}'$ for the tableau formed from $\sigma^\Sigma$ by considering only the boxes with entries $1,...,i$. Similarly define $\tau'$. Using Corollary~\ref{cor:partitionsigma} and the remarks of the previous paragraph we see that the shape of $\sigma^{\Sigma}{}'$ is the Young diagram of the induced orbit $\operatorname{Ind}_{\mathfrak{g}_0'}^{\mathfrak{g}'}(G_0' \cdot e_0')$. On the other hand the shape of $\tau'$ is the Young diagram of the orbit of $e'$. Since we assumed that $\tau_j = \sigma^\Sigma_j$ for $j=i+1,...,N$ it follows that the shape of $\tau'$ is the shape of $\sigma^{\Sigma}{}'$, and this confirms the claim.\\
Let $F^{(1)}{}'_\bullet,...,F^{(j)}{}'_\bullet$ be the collection of flags of $V_1,...,V_{j-1}, \tilde V_j$ given by $F^{(k)}{}'_\bullet := F^{(k)}_\bullet$ for $k=1,..,j-1$ and $F^{(j)}{}'_\bullet := (0 \subseteq F^{(j)}_1\subseteq \cdots \subseteq F^{(j)}_{d-1} \subseteq \tilde V_j)$. By Lemma~\ref{L:LSonflags} the Lusztig--Spaltenstein map on fibres sends $(F^{(1)}{}'_\bullet,...,F^{(j)}{}'_\bullet)\in\mathcal{B}^0_{e_0'}{}'$ to $\bar{F}'_\bullet \in \mathcal{B}'_{e'}$. The standard tableau $\sigma^{(1)}{}',...,\sigma^{(j)}{}'$ corresponding to $F^{(1)}{}'_\bullet,...,F^{(j)}{}'_\bullet$ are determined from $\sigma^{(1)},...,\sigma^{(j)}$ in the obvious manner. The tableau $\sigma^{\Sigma}{}' := \stk(\sigma^{(1)}{}',...,\sigma^{(j)}{}')$ is the numbered diagram obtained from $\sigma^\Sigma$ by considering only the boxes labelled $1,...,i$. Since we have assumed $\sigma_i^\Sigma \ne \tau_i$, it follows that $\sigma_i^{\Sigma}{}' \ne \tau_i'$. To prove the Proposition we must show that $\sigma_i^\Sigma{}' > \tau'_i$.\\
Since we are in precisely the same situation as the start of the proof, we might as well simplify our notation by assuming that $i = N$, and working with $\mathfrak{g}, \mathfrak{p}, e, \mathfrak{g}_0, e_0$ rather than $\mathfrak{g}', \mathfrak{p}', e', \mathfrak{g}_0', e_0'$. We shall assume $\sigma_N^\Sigma \ne \tau_N$ to prove $\sigma_N^\Sigma > \tau_N$. \\
Suppose that $\lambda_n$ is in the $(i,j)$-th position of $\sigma^{(n)}$ (so we have now fixed $i$ and $j$). Then by the formula in \eqref{eq:stack}, $N$ occupies the $(i,j')$-th position in $\sigma^{\Sigma}$, where $ j' = \sum_{k=1}^{n-1} \mu_{k,i} + j$; that is, $\sigma_{N}^{\Sigma}=j'$. Hence we need to show that $\bar{F}_{N-1} \supseteq \ker(e^{j'-1})$ to deduce that $\tau_{N} \geq j' =\sigma_{N}^{\Sigma}$. From the outset we assumed that $\tau_N \ne \sigma^\Sigma_N$ and so this will imply that $\tau > \sigma^\Sigma$ and complete the proof.\\
By Lemma~\ref{L:LSonflags}, we have $\bar{F}_{N-1} = W_{n-1} \oplus F^{(n)}_{\lambda_{n}-1}$ and we have that $j$ is the maximal index such that $F^{(n)}_{\lambda_n-1} \supseteq \ker(e_{0|V_{n}}^{j-1})$. At this point, we use Proposition \ref{L:reductionlemma} to make an explicit choice for $e_0$ and $e_1$ that comes with a Jordan basis as in Section~\ref{ss:representative}. Thus we picture the $N$-dimensional vector space as the span of all the boxes in the tuple of Young diagrams corresponding to the partitions $(\mu_1, \ldots, \mu_{n})$, and hence we may identify $W_{n-1}$ as the span of the boxes in the first $n-1$ Young diagrams and $F_{\lambda_{n}-1}^{(N)}$ as the span of all the boxes in the last Young diagram except for the box in the $(i,j)$-th position (or the box at the bottom of column $j$ in last Young diagram). This pictorial interpretation is explained clearly in Example~\ref{ex:representativediagram}. \\
Since may we diagrammatically represent $\ker(e^{j'-1})$ as the span of the boxes in the first $j'-1$ columns of the stacked Young diagram (see Diagram 2 in Section~\ref{ss:representative}), it follows that $\ker(e^{j'-1}) \subseteq W_{n-1} \oplus \ker(e_{0|V_{n}}^{j-1})$. Since $\ker(e_{0|V_{n}}^{j-1}) \subseteq F^{(n)}_{\lambda_n-1}$ by construction, it follows that
$$
\ker(e^{j'-1}) \subseteq W_{n-1} \oplus F^{(n)}_{\lambda_{n}-1}= \bar{F}_{N-1}.
$$
Therefore $\tau_{N} \geq j' =\sigma_{N}^{\Sigma}$ and so $\tau > \sigma^{\Sigma}$, and the proof is complete.
\end{proof}
The following example illustrates that $\LS(X_{\sigma_1,...,\sigma_n})$ is not contained in $X_{\stk(\sigma_1,....,\sigma_n)}$ in general, and so the previous proposition is best possible.
\begin{example}
Let $\k^5$ have basis $\{v_1, v_2, v_3, v_4, v_5\}$ and let $V_1 = \Span\{v_1, v_2, v_3\}$ and $V_2 = \Span\{ v_4, v_5\}$. Let $e_{i,j}$ be the standard basis for $\mathfrak{g} = \mathfrak{gl}_5$ with respect to this basis of $\k^5$ and let $\mathfrak{g}_0$ be the Levi factor of $\mathfrak{g}$ preserving the decomposition $\k^5 = V_1\oplus V_2$. Let $\mathfrak{p}$ be the parabolic preserving the flag $0 \subseteq V_1 \subseteq \k^5$. We define $e_0 := e_{1,2}$ and $e_1 = e_{3,4} + e_{2,5}$. One can check that these data satisfy the set up of Proposition~\ref{prop:LSmap}.
Then if $F^{(1)}$ is any full flag of $V_1$ preserved by $e_0$, satisfying $e_0|_{F^{(1)}_{2}} = 0$, and $$F^{(2)} = (0 \subseteq \k v_5 \subseteq V_2)$$ then $(F^{(1)}, F^{(2)}) \in \mathcal{B}^0_{e_0}$. On the other hand, $\LS(F^{(1)}, F^{(2)})$ lies in $X_\tau$ where
$$\tau = \young(135,24)$$
However the stacked tableau $\sigma$ corresponding to $F^{(1)}, F^{(2)}$ is the one appearing in \eqref{e:atab}. In the notation of Section~\ref{subsection:ordering} we have $\tau_5 = 3 > 2 = \sigma_5$ and so $\tau \gneq \sigma$.
\end{example}
\subsection{The stacking theorem}
Retain the notation and choices made at the start of Section~\ref{ss:LSonfibres}. Let $\sigma^{(1)}, \ldots, \sigma^{(n)}$ be a choice of standard tableaux of shape $\mu_1,...,\mu_n$ respectively.
By Spaltenstein's theorem, the closure $\overline{X}_{\sigma^{(1)}, \ldots, \sigma^{(n)}}$ is an irreducible component of $\mathcal{B}^0_{e_0}$ which we denote $C_{\sigma^{(1)}, \ldots, \sigma^{(n)}}$. Similarly for $\sigma$ a standard tableau of shape $\mu^\Sigma$ we write $C_\sigma := \overline{X}_\sigma \subseteq \mathcal{B}_e$, which is an irreducible component of $\mathcal{B}_e$.
The following is our main result.
\begin{thm}
\label{T:stackwithproof}
$\LS(C_{\sigma^{(1)}, \ldots, \sigma^{(n)}}) = C_{\stk(\sigma^{(1)}, \ldots, \sigma^{(n)})}$.
\end{thm}
\begin{proof}
The first step is to show that there is an element $(F^{(1)}_\bullet,...,F^{(n)}_\bullet) \in \mathcal{B}^0_{e_0}$ such that $\LS(F^{(1)}_\bullet,...,F^{(n)}_\bullet) \in X_{\stk(\sigma^{(1)},...,\sigma^{(n)})}$. Notice that using the argument in the first paragraph of the proof of Proposition~\ref{prop:LSmap} it suffices to find such a flag for a particular choice of $e\in \O_\mathfrak{p}$, and it follow for all such elements. Pick a basis $\{v_{j,l,m} \mid j, l, m\}$ for $\k^N$ such that $\{v_{j,l,m} \mid l, m\}$ spans the spaces $V_j$ described in Section~\ref{ss:LSonfibres} and then choose an element $e = e_0 + e_1 \in \O_\mathfrak{p}$ using formulas \eqref{e:e0defn} and \eqref{e:e1defn}. For each $k=1,...,n$ we define a full flag $F^{(k)}$ of $V_j$ by letting $F^{(k)}_j$ be the span of those vectors $v_{k,l,m}$ such that the $(l,m)$-entry of $\sigma^{(k)}$ is less than or equal to $j$.\vspace{6pt}
\noindent {\bf Claim.} \ $(F^{(1)}_\bullet, \ldots, F^{(n)}_\bullet) \in \mathcal{B}^0_{e_0}$ and $\LS(F^{(1)}_\bullet, \ldots, F^{(n)}_\bullet) \in X_{\stk(\sigma^{(1)},...,\sigma^{(n)})}$.\vspace{6pt}
The first part of the claim is obvious from the construction. To prove the second part we describe the Jordan blocks of $e|_{\bar{F}_j}$ where $\bar{F}_{\bullet} := \LS(F^{(1)}_\bullet, ..., F^{(n)}_\bullet)$. For $j = 1,...,N$ we let $k \ge 0$ be such that $\sum_{i=1}^{k-1} \lambda_i < j \le \sum_{i=1}^{k} \lambda_i$. Write
$$\mathcal{X}_j := \{(i,l,m)\mid i < k \text{ or } i = k \text{ and the } (l,m)\text{-entry of } \sigma^{(k)} \text{ is less than or equal to }\tilde j := j - \sum_{r=1}^{k-1} \lambda_r\}.$$
Then $\bar{F}_j$ is spanned by $\{v_{i,l,m} \mid (i,l,m) \in \mathcal{X}_j\}$. Now if we fix $1\le i \le \max \ell(\mu_s)$ then the span of $\{v_{i,l,m} \mid (i,l,m) \in \mathcal{X}_j\}$ is a single Jordan block for $e$. We invite the reader to check this claim using Diagram 2 in Example~\ref{ex:representativediagram}. Combining this description of the Jordan blocks of $e|_{\bar{F}_k}$ with the relationship between standard tableaux and sequences of nested partitions given in Section~\ref{ss:LSonfibres} it follows that $\Phi(\bar{F}_\bullet) = \stk(\sigma^{(1)},...,\sigma^{(n)})$. This proves the claim.
By Corollary~\ref{cor:inducedmaponcomponents} and Proposition~\ref{prop:LSmap} we know that $\LS(\overline{X}_\sigma^{(1)},...,\sigma^{(n)})$ is equal to $\overline{X}_{\tau}$ for some $\tau \ge \stk(\sigma^{(1)},...,\sigma^{(n)})$. Combining the above Claim with Lemma~\ref{lem:Xsigmaintersectsclosure} we deduce that $\tau = \stk(\sigma^{(1)},...,\sigma^{(n)})$, which completes the proof.
\end{proof}
\begin{rem}
A direct consequence of the stacking theorem is that $\stk$ is an associative operation on tableaux, which can also be checked directly by a combinatorial argument.
\end{rem}
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,148 |
'use strict';
let expect = require('chai').expect,
books = require('../lib/books');
let booksCount = 0;
beforeEach(function() {
booksCount = parseInt(books.count());
});
describe('Books module', () => {
//test books.get function
it('returns requested book', function() {
var result = books.get('Dragonflight');
expect(result[0]).to.deep.equal({title: 'Dragonflight',
author: 'Anne McCaffrey',
publisher: 'Del Rey Books/Ballantine Books',
year: '1978',
isbn: '034527749X',
volumes: 1});
});
it('fails with invalid book', () => {
var result = books.get('JXJ4XJD6SJ23VEF75ZZ');
expect(result.length > 0).to.equal(false);
});
//test books.add function
it('adds new book successfully', function() {
var result = books.add({
title: 'A Great New Book',
author: 'Great Author'
});
expect(parseInt(books.count())).to.equal(booksCount+1);
});
it('fails to add if book exists', () => {
var result = books.add({
title: 'Dragonflight',
author: 'Anne McCaffrey'
});
expect(parseInt(books.count())).to.equal(booksCount);
});
//test books.delete function
it('deletes requested book', function() {
var result = books.delete('Dragonflight');
expect(parseInt(books.count())).to.equal(booksCount-1);
});
it('fails to delete with invalid book', () => {
var result = books.delete('JXJXJDSJVEFZZ');
expect(parseInt(books.count())).to.equal(booksCount);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,183 |
\section{INTRODUCTION}
The parent compound for high transition-temperature superconductors,
La$_2$CuO$_4$, is an antiferromagnetic insulator. Magnetic exchange
interaction $J$ between the nearest neighbor $S$=1/2 spins of Cu$^{2+}$ ions
in the CuO$_2$ plane is several orders of magnitude stronger than the
interplane exchange interaction, making quantum spin fluctuations an
essential ingredient for magnetic properties in the quasi-two-dimensional
(2D) Heisenberg system\cite{la2dv2,2dheis,2dheiqc}.
The N\'{e}el temperature $T_N$ of La$_2$CuO$_4$ is suppressed rapidly
to zero by $x_c=2$$-$3\% hole dopants such as Sr, Ba or Li\cite{ht_pd,nagano,Li214phs},
while it is suppressed with isovalent Zn
substitution at a much higher concentration close to the
site dilution percolating threshold of $\sim$30\%\cite{ZnSr214}.
The strong effect of holes has been shown to be related to induced
magnetic vortices, which are topological defects in
2D systems\cite{h_haas,h_ctb}. The paramagnetic phase exposed by hole doping
at $T \ll J/k_B$ is dominated by zero-point quantum spin
fluctuations and is referred to as a quantum spin liquid\cite{2dheis}.
Detailed predictions for spin dynamics have been made for the quantum
spin-liquid\cite{2dheis,2dheiqc}.
However, in a wide doping range of La$_{2}$Cu$_{1-x}$Li$_x$O$_4$
below $\sim$10~K, a spin-glass transition has
been reported in muon spin rotation ($\mu$SR)\cite{Li214phs},
nuclear quadrupole resonance (NQR)\cite{Li214NQR} and
magnetization\cite{Li214phs2} studies.
A similar magnetic phase diagram has also been reported for
La$_{2-x}$Sr$_x$CuO$_4$ and
Y$_{1-x}$Ca$_x$Ba$_2$Cu$_3$O$_6$\cite{sg_drh,bjbquasi,fchou,muchn,gqcp,sg_fcc}.
In conventional spin glasses, magnetic interactions are more or
less isotropic in space, and the entire spin system is believed
to be frozen in the spin-glass phase\cite{byoung}.
Such was also the conclusion of a comprehensive magnetization study
on La$_{2-x}$Sr$_x$CuO$_4$\cite{sg_fcc}. Although magnetization
can only account for a tiny fraction of spins, theoretical
pictures were proposed for spin-freezing in the whole
sample\cite{sg_fcc,sg_rjg}.
If the spin-glass phase in hole-doped cuprates behaved as in
conventional spin-glasses, the ground state would be a spin-glass, instead of
the N\'{e}el order for doping smaller than $x_c$, or a quantum spin liquid
for doping larger than $x_c$. Thus, as pointed out
by Hasselmann et~al.\cite{sg_cn}, the quantum critical point of
the antiferromagnetic phase at $x_c\approx 2$$-$3\% would be preempted.
In widely circulating ``generic'' phase-diagram for laminar cuprates,
the ``reentrant'' spin-glass transition below the N\'{e}el temperature
is generally ignored. Also generally ignored is the spin-glass transition below the superconducting transition. The spin glass
phase exists side by side with the N\'{e}el order at lower doping and the superconducting order at higher doping in this neat picture.
This ``generic'' picture does not conform to experimental results,
and serves to support the theory that the spin-freezing is an extrinsic dirt effect. However, there are other theories which consider spin-freezing
intrinsic to the doped cuprates\cite{slf,sg_sahc}.
Physical quantities in the doping regime, including spin excitation
spectra, have also been calculated from microscopic
model\cite{march1,march2}.
Recently, 2D spin fluctuations in La$_{2}$Cu$_{1-x}$Li$_x$O$_4$
($0.04\leq x\leq 0.1$) were observed to remain liquid-like below
the spin-glass transition temperature\cite{bao02c,bao04a}, $T_g\sim 9$~K,
which can be reliably detected using the $\mu$SR technique\cite{Li214phs}.
The characteristic energy of 2D spin fluctuations
saturates at a finite value below $\sim$50~K\cite{bao02c,bao04a}, as expected for a
quantum spin liquid\cite{2dheis}, instead of becoming zero
at $T_g$ as for spin-glass materials\cite{byoung,FeAl_msm}.
To reconcile these apparently contradicting experimental results,
we have conducted a thorough magnetic neutron scattering investigation
of La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$ to search for spin-glass
behavior. We found that in addition to the
liquid-like 2D dynamic spin correlations, the rest of spins which participate
in almost 3D and quasi-3D correlations become frozen in the
spin-glass transition. This partial spin freezing in the laminar
cuprate is distinctly different from total spin freezing
in conventional 3D spin-glass materials.
The observed phase separation into spin glass and spin liquid components
of {\em different dimensionality} sheds light on a long-standing confusion
surrounding the magnetic ground state in hole-doped cuprates.
The remaining of the paper is organized as the follows.
Section II covers experimental details concerning the sample and neutron scattering instrumentation. Section III covers small angle neutron
scattering, which is the ideal tool to detect ferromagnetic spin
clusters proposed in some theories for the spin-freezing state.
Section IV covers cold neutron triple-axis measurements. The
excellent energy resolution is important for this study. Finally,
in Section V, we discuss and summarize our results.
\section{EXPERIMENTAL DETAILS}
The same single crystal sample of La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$
used in the previous higher energy study\cite{bao02c} was investigated in this work.
$T_g$$\approx$8~K was determined in $\mu$SR study\cite{Li214phs}
and is consistent with magnetization work\cite{Li214phs2}.
The lattice parameters of the orthorhombic $Cmca$ unit cell are
$a$=5.332$\AA$, $b$=13.12$\AA$ and $c$=5.402$\AA$ at 15~K.
Wave-vector
transfers {\bf q} near (000) and (100) in both the ($h0l$) and ($hk0$)
reciprocal planes were investigated at NIST using the 30 meter high
resolution small angle neutron scattering (SANS) instrument at NG7,
and cold neutron triple-axis spectrometer SPINS.
We set the array detector of NG7-SANS to
1 and 9 m, corresponding to a $q$ range from 0.012 to 0.39~$\AA^{-1}$
and from 0.0033 to 0.050~$\AA^{-1}$, respectively.
At SPINS, the (002) reflection of pyrolytic
graphite was used for both the monochromator and analyzer. Horizontal
Soller slits of 80$^{\prime}$ were placed before and after the sample.
A cold BeO or Be filter was put before the analyzer to eliminate
higher order neutron in the fixed $E_f$=3.7 or 5 meV configuration,
respectively.
Sample temperature was controlled by a pumped $^4$He cryostat which could
reach down to 1.5~K.
\section{Small angle neutron scattering}
Hole induced ferromagnetic exchange has been theoretically proposed
in the CuO$_2$ plane\cite{sg_rjg,sg_cn}. It is regarded as competing with
the original antiferromagnetic exchange, thus, leading to the spin-glass
transition.
Although long-range ferromagnetic order has never been observed,
there is the possibility of short-range ferromagnetic spin clusters
which freeze in the spin-glass state in this class of spin-glass
models\cite{sg_rjg,sg_lin}. SANS has been demonstrated as a powerful tool to probe such clusters\cite{FeAl_msm}.
Two reciprocal zones of La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$
were studied, with incident
beam parallel to the (001) or (010) direction. Therefore,
any spin orientation in the sample can be detected in our experiment. The
experiments were carried out at 3, 10, 15, 30 and 80~K. A
collection time of 1 or 2 hours per temperature provides good
statistics.
No temperature dependence in the scattering patterns
could be detected. The inset to Fig.~\ref{fig1} shows SANS patterns at
3 K and 30 K with incident beam parallel to the (001) direction.
Intensity at 3 and 30~K in the rectangular box on the
SANS pattern is shown in the main frame.
The difference intensity (circles) fluctuates around zero,
and its standard deviation sets a upper limit of
1.5$\times$10$^{-7}$ bn or
1.4$\times$10$^{-3} \mu_B$ per Cu for ferromagnetic moments
in the clusters.
\begin{figure}[t]
\vskip -1ex
\centerline{
\psfig{file=fig1.ps,width=\columnwidth,angle=90,clip=}}
\vskip -3ex
\caption{ \label{fig1} (color)
Measured SANS cross-section in the 10 pixels
wide rectangle shown in the inset at 3~K (squares) and 30~K
(diamonds), and the difference of the intensity between 3 K and 30 K
(circles) as a function of wave vector transfer $q$. Lines
are guide to the eye. Inset: the SANS pattern at 30 and 3~K,
with the intensity color scale at the left.
}
\end{figure}
This result provides serious constrain on the class of theoretical
models for the spin-glass transition in doped cuprates\cite{sg_rjg}
which lead to formation of ferromagnetic clusters. Instead of
this ``large spin fixed point'', models leading to other fixed points
such as ``Griffiths fixed point'' as discussed by Lin et al.\cite{sg_lin} may be considered.
\section{triple-axis neutron scattering}
While no appreciable ferromagnetic signal was detected for
La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$, as in other Li-doped
La$_2$CuO$_4$\cite{bao99a,bao04a}, antiferromagnetic
scattering was readily observed along the rods
perpendicular to the CuO$_2$ plane and intercepting the
plane at the commensurate ($\pi,\pi$)-type Bragg points
of the square lattice. This means that antiferromagnetic
correlations in the CuO$_2$ plane are chessboard-like,
which is similar to electron-doped La$_2$CuO$_4$\cite{edoped1,edoped2},
but different from the more complex, incommensurate ones
in La$_{2-x}$Sr$_x$CuO$_4$ at similar hole doping\cite{Waki_Sr3}.
Scans through such a rod in the CuO$_2$ plane at various temperatures
with the SPINS spectrometer set at $E$=0 are shown in Fig.~\ref{fig2}(a).
\begin{figure}[t]
\vskip -1ex
\centerline{
\psfig{file=fig2.ps,width=\columnwidth,angle=90,clip=}}
\vskip -2ex
\caption{ \label{fig2}
Representative magnetic quasielastic and elastic scattering along a) an
in-plane direction and b) the interlayer direction between 1.5 and 180~K.
Open squares in b) were measured at (1.06,$k$,0)
and represent background.
The solid lines are resolution convoluted
$S^{3D}({\bf q},E)+S^{q3D}({\bf q},E)$ in Eq.~(\ref{eq1})-(\ref{eq2}).
}
\end{figure}
Inelastic scans have been reported previously in a related but different study which focuses on scaling in different quantum regimes\cite{bao02c}. There is little change in the peak width
in these scans, consistent with previous results of
temperature independent in-plane correlation length
for La$_2$Cu$_{0.95}$Li$_{0.05}$O$_4$\cite{la2dv2}
and La$_{2-x}$Sr$_x$CuO$_4$ (0.02$\leq x\leq$0.04)\cite{la2bkb}
below 300~K. Modeling the width of the rod in
Fig.~\ref{fig2}(a) with Lorentzian
\begin{equation}
\mathcal{L}^{\xi}(q)=\frac{\xi}{\pi[1+(q\xi)^2]},
\label{eq0}
\end{equation}
the {\em lower limits} from deconvolution is
$\xi_{\Box}\ge 274 \AA$, where the
$\Box$ indicates the correlation length as in-plane.
These large antiferromagnetic clusters in the CuO$_2$ plane correlate in three different
ways in the interlayer direction, giving rise to
almost 3D, quasi-3D and 2D magnetic correlations.
Let us now examine the three components.
Scans along the rod in the interlayer
direction, with the SPINS spectrometer set at $E$=0,
were measured at various temperatures
from 1.5 to 180~K. A few of them, at 1.5, 49 and 180~K, respectively,
are shown in Fig.~\ref{fig2}(b).
Magnetic intensity is composed of both broad and sharp
peaks at magnetic Bragg points (100) and
(120) of the parent compound.
The (110) peak is temperature-independent thus nonmagnetic.
Fitting the broad
peaks to Eq.~(\ref{eq0}), we obtained an interlayer correlation
length $\xi^{q3D}= 6.2(4)$~$\AA$. Again, no temperature dependence
can be detected for $\xi^{q3D}$ below 49~K. Above 49~K, signal is
too weak to have a reliable determination of $\xi^{q3D}$.
Thus, the quasi-3D spin correlations are typically three planes thick.
For the sharp peak at (100) or (120), only the {\em lower limit}
for the correlation length can be reliably estimated:
$\xi^{3D}\ge 168\AA$, since the width is close to instrumental resolution.
Therefore, the number of correlated antiferromagnetic planes is more
than 50, resembling a 3D antiferromagnetic order.
Both the broad and sharp peaks in Fig.~\ref{fig2}(b)
are energy-resolution-limited
with the half-width-at-half-maximum $=$~0.07 meV.
The energy scan in Fig.~\ref{fig3} is an example and more
can be found in Fig.~2 in reference [\cite{bao02c}].
However, these peaks should not be regarded automatically
as from {\em static} magnetic order. Static magnetic signal
was observed only below $T_g$=8~K at the spin glass transition
in $\mu$SR study\cite{Li214phs}, which has a much better energy
resolution. Thus, the quasi-3D and almost 3D
correlations are slowly dynamic for $T> 8$~K, with their spectra
faster than 1 MHz\cite{Li214phs,bjbquasi}, the zero-field $\mu$SR static cutoff frequency, but slower than 17 GHz=0.07 meV/$h$, the frequency
resolution at spectrometer SPINS.
The 2D antiferromagnetic correlations have been investigated
in detail\cite{bao02c}. The dynamic magnetic structure factor,
\begin{equation}
S^{2D}({\bf q},E)=\sum_{\bm{\tau}}
\mathcal{L}^{\xi_{\Box}}(\bm{\kappa}_{\Box})
\frac{\chi''(E)}{\pi \left(1-e^{-\hbar\omega/k_BT}\right)},
\label{eq4}
\end{equation}
where ${\bm{\tau}}$ is a magnetic Bragg wave-vector
and $\bm{\kappa}\equiv {\bf q}-{\bm{\tau}}$, has been determined from measurements in the energy range, $E \le 4.2$~meV, between 1.5 and
150~K.
Eq.~(\ref{eq4}) is independent of $k$, befitting to a 2D magnetic
correlation, see the flat $k$ scan at 1.2 meV in Fig.~\ref{fig3}.
The almost 3D and quasi-3D spin correlations described in previous
paragraphs can be written as
\begin{equation}
S^{3D}({\bf q},E)=I^{3D} \sum_{\bm{\tau}}
\mathcal{L}^{\xi_{\Box}}(\bm{\kappa}_{\Box})
\mathcal{L}^{\xi^{3D}}(k-\tau_k) \mathcal{L}^{1/\epsilon}(E) \label{eq1}
\end{equation}
and
\begin{equation}
S^{q3D}({\bf q},E)=I^{q3D} \sum_{\bm{\tau}}
\mathcal{L}^{\xi_{\Box}}(\bm{\kappa}_{\Box})
\mathcal{L}^{\xi^{q3D}}(k-\tau_k) \mathcal{L}^{1/\epsilon}(E), \label{eq2}
\end{equation}
respectively, where $\epsilon < 0.07$ meV, the spectrometer energy resolution.
Note that we use conventional Lorentzian function, Eq.~(\ref{eq0}),
to model sharp peaks which we could not experimentally resolve,
in addition to $\mathcal{L}^{\xi^{q3D}}$ in Eq.~(\ref{eq2}) which we could
resolve. We are fully aware that the true
peak profile can be different for these unresolved peaks.
The use of Eq.~(\ref{eq0})
is for the purpose of calculating resolution convolution of
Eq.~(\ref{eq4})-(\ref{eq2}), which is used in the following paragraphs to
obtain correct normalization of $I^{2D}$, $I^{q3D}$ and $I^{3D}$.
The choice of the function will
not affect the result as long as the function describes a sharp
peak significantly narrower than instrument resolution.
With negligible ferromagnetic correlations, the total
dynamic structure factor is a summation of Eq.~(\ref{eq4})-(\ref{eq2}),
\begin{equation}
S({\bf q},E)=S^{2D}({\bf q},E)+S^{q3D}({\bf q},E)+S^{3D}({\bf q},E). \label{eq5}
\end{equation}
Of the four variables of $S({\bf q},E)$, {\bf q}$_{\Box}$ are fixed
at the ($\pi,\pi$)-type Bragg points by the sharply peaked
$\mathcal{L}^{\xi_{\Box}}(\bm{\kappa}_{\Box})$\cite{note}.
To comprehend the composition of $S({\bf q},E)$, it is
sufficient to plot $S({\bf q},E)$ as a function of $E$ and the
interlayer wavenumber $k$. Such a plot of measured $S({\bf q},E)$
at 1.5~K is shown with a logarithmic intensity scale in Fig.~\ref{fig3}.
The temperature and {\bf q} independent incoherent scattering
and other background at $E=0$ has been subtracted, which can be
determined, e.g., by the 180~K scan in Fig.~\ref{fig2}(b).
\begin{figure}[t]
\vskip -5ex
\centerline{
\psfig{file=fig3.ps,width=1.1\columnwidth,angle=90,clip=}}
\vskip -4ex
\caption{ \label{fig3} (color)
Measured $S({\bf q},E)$ as a function of $E$ and interlayer $k$ at 1.5~K
with a logarithmic intensity scale,
showing three color-coded magnetic components in Eq.~(5).
$S^{3D}({\bf q},E)$ (red) and $S^{q3D}({\bf q},E)$ (blue)
are energy-resolution limited
at $E=0$, representing very slow spin dynamics which is associated with
the spin-glass freezing. They are modulated along the interlayer $k$
direction. The spin-liquid component, $S^{2D}({\bf q},E)$ (green),
has a finite energy scale of about 1~meV
below 50~K, and 0.18$k_B T$ above 50~K\cite{bao02c}.
It is flat along the $k$ direction.
A few representative scans at 1.5~K are shown with yellow symbols. The black
surface indicates background of $\sim$1.3 counts/min.
}
\end{figure}
The sharp peak fitted by the red curve is from $S^{3D}({\bf q},E)$,
the narrow blue ridge at $E$=0 from $S^{q3D}({\bf q},E)$, and
the green surface from $S^{2D}({\bf q},E)$. The red peak at (100)
is about one order of magnitude stronger than the peak intensity
of the blue surface, and three orders of magnitude stronger
than the peak intensity of the green surface. Thus, $S^{3D}({\bf q},E)$
is the easiest component to be observed in a neutron scattering experiment,
and is often mistakenly attributed to a {\em static} magnetic order.
The spectral weights
$\int d{\bf q} dE\, S^{3D}({\bf q},E)$$\equiv$$I^{3D}$ and
$\int d{\bf q} dE\, S^{q3D}({\bf q},E)$$\equiv$$I^{q3D}$ can be obtained
by fitting resolution-convoluted Eq.~(\ref{eq1})-(\ref{eq2})
to scans such as those shown in Fig.~\ref{fig2}(b).
They are shown as a function of
temperature in Fig.~\ref{fig4}, with $I^{3D}$ magnified by a
factor of 5 for clarity. For the 2D component, the spectral weight is
\begin{equation}
I^{2D}\equiv \int dE\, \frac{2\chi''(E)}{\pi \left(1-e^{-\hbar\omega/k_BT}\right)},
\label{eq6}
\end{equation}
where the integration limits are $\pm \infty$.
Green squares in Fig.~\ref{fig4} represent the lower bound of
$I^{2D}$ with the energy integration limited in
$|E| \le 10$~meV, using the analytical expression
of $\chi''(E)$ in reference [\cite{bao02c}] to extrapolate to $E$=10 meV,
where spin fluctuations were observed in La$_2$Cu$_{0.9}$Li$_{0.1}$O$_4$
using a thermal neutron spectrometer\cite{bao99a}.
$I^{3D}$ and $I^{q3D}$ appear simultaneously below $\sim$150~K.
Their concave shape in Fig.~\ref{fig4} differ drastically
from the usual convex-shape of a squared order parameter, orange circles,
which was observed in $\mu$SR study below $T_g$=8~K\cite{Li214phs}.
\begin{figure}[t]
\centerline{
\psfig{file=fig4.ps,width=\columnwidth,angle=90,clip=}}
\vskip -2ex
\caption{ \label{fig4} (color)
Temperature dependence of spectral weights
$I^{2D}$ (green), $I^{q3D}$ (blue) and $I^{3D}$ (red) in the same unit for three experimentally separable antiferromagnetic components
in Eq.~(\ref{eq5}). $I^{3D}+I^{q3D}$ is the total spectral weight of
the spin-glass component.
$I^{2D}$ is the spectral weight of the spin-liquid component
within $|E|< 10$ meV, thus the lower limit of its total spectral weight. The orange circles represent squared {\em static} order parameter
of the spin-glass transition, which was measured by
$\mu$SR\cite{Li214phs} and equals to $I^{3D}+I^{q3D}$
at T=0.}
\end{figure}
They are typical neutron scattering signal from
slow {\em dynamic} spin correlations in spin-glasses\cite{FeAl_msm,sg_mh},
which fluctuate in the frequency window between 1 MHz
and 17 GHz for $T> 8$~K, and below 1 MHz for $T< 8$~K.
Previously, energy-resolution-limited
neutron scattering from
La$_{1.94}$Sr$_{0.06}$CuO$_4$ was observed to
have a similar temperature dependence as $I^{3D}$ in
Fig.~\ref{fig4} and was attributed to
spin freezing\cite{bjbquasi}.
The kink of $I^{3D}$ at 20~K reflects an increased $T_g$ from 8~K
to 20~K when probing frequency is increased from 1 MHz
to 17 GHz\cite{bjbquasi,byoung}. At 0 Hz, $T_g\approx 6$~K
from DC magnetization measurements\cite{Li214phs2}. The increase
of $T_g$ with measurement frequency is a hallmark of glassy
systems\cite{byoung}.
\section{Discussions and summary}
The fact that $I^{3D}$ decreases below $T_g$(17GHz)$\approx$20~K
while $I^{q3D}$ continues to increase indicates that
the ``Edwards-Anderson order parameter''\cite{byoung,FeAl_msm,sg_mh}
distributes only
along lines such as the (1$k$0). In conventional spin-glasses,
the ``Edwards-Anderson order parameter'' is more
isotropically distributed in the {\bf q}-space\cite{byoung,sg_mh,FeAl_msm}.
Thus, the spin-glass state in La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$
is characterized mainly by interlayer disorder which upsets phase correlation
between large antiferromagnetic clusters in different CuO$_2$ planes.
This picture offers a possible alternative to the conventional
competing antiferromagnetic/ferromagnetic interaction model for
spin freezing in doped cuprates. In addition, it suggests that
the weak interlayer exchange interaction likely plays an important
role in the finite temperature spin-glass transition in the quasi-2D
Heisenberg magnetic systems.
Another important difference from conventional spin-glasses in which
all spins are believe to freeze at low temperature is that only a fraction of spins
freeze in La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$.
Other spins in 2D correlations
remain fluctuating down to 1.5~K. This is consistent with
numerical evidence that quantum fluctuations prevent spin-glass
transition for 2D $S$=1/2 Heisenberg system\cite{bhatt}.
The spin-glass component in our sample has to acquire interlayer
correlations to achieve a higher dimension in order to be realized.
It appears that the lower critical dimension for a $S$=1/2 Heisenberg
quantum spin glass is between 2 and 3.
A further difference from conventional spin-glasses, for
which one can measure the narrowing of magnetic
spectrum toward $E$=0\cite{FeAl_msm}, is that
when $S^{3D}({\bf q},E)$ and $S^{q3D}({\bf q},E)$ in
La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$ become detectable at about 150~K,
they are already energy-resolution-limited, with spins fluctuating
much slower than 17 GHz.
This property of $S^{3D}({\bf q},E)$ and $S^{q3D}({\bf q},E)$ resembles
the classic central peak phenomenon in the soft phonon
transition\cite{SrTiO,SrTiOr}.
The disparate dynamics of the central peak and phonon are explained by
Halperin and Varma\cite{hcav} using a phase separation model:
defect cells contribute to the slow relaxing central peak
while coherent lattice motions (phonons) to the resolved
inelastic channel.
This mechanism has been applied with success to a wide class of
disordered relaxor ferroelectrics\cite{Courtens82,Burns83}.
For La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$,
we envision that disorder accompanying doping
prevents the long-range order of the antiferromagnetic phase mainly by
upsetting interlayer magnetic phase coherence,
see Fig.~\ref{fig3} for the {\bf q}-distribution of
frozen spins. This upsetting is not uniform in the Griffiths
fashion\cite{qsg_hy} with weak and strong coupling parts in the
sample. In our laminar material, however, the weak and strong coupling
parts have different dimensionality: 2D and nearly 3D, respectively.
The 2D part is a spin liquid and represents essentially the whole
system at high temperature, see Fig.~\ref{fig4}.
Part of sample with stronger interplane coupling tends to order
three dimensionally below $\sim$150~K, producing
$S^{3D}({\bf q},E)$ and $S^{q3D}({\bf q},E)$.
The condensation of the 2D spin liquid at $\sim$ 150~K
into the quasi-3D dynamic clusters
of diminishing energy scale, instead of a true long-range order,
may reflect the divergent fluctuations which destabilize
static order at finite temperature
for 2D random $XY$ or Heisenberg systems\cite{bhatt,Hertz79,2d_gls}.
The nearly 3D spin-glass instead of a 3D antiferromagnet finally orders
at a much reduced $T_g\approx 20$~K, when $I^{q3D}+I^{3D}$
approaches the 2D spectral weight (Fig.~\ref{fig4}).
The coexistence of spin liquid and spin glass
components at low temperatures may be a general
consequence of no ``mobility edge''
separating finite and infinite range correlations for a 2D
random system\cite{Hertz79}.
Recently, Monte-Carlo simulations of a doped 2D classical
antiferromagnet suggest that there are two populations of
spins: one with fast and the other with slow dynamics\cite{sg_mpk}.
This is consistent with our experimental results and the
Griffiths picture for random magnetic systems. A
phenomenological Halperin and Varma model may be built
for spin dynamics in doped cuprates based on these microscopic
insights.
In summary, spins in La$_2$Cu$_{0.94}$Li$_{0.06}$O$_4$
develop {\it dynamic} antiferromagnetic order in the CuO$_2$ plane
with very long $\xi_{\Box}$ below 180~K. The characteristic
energy of the 2D spin fluctuations is 0.18$k_B T$ for $T>50$~K and
1~meV for $T<50$~K\cite{bao02c}. Below $\sim$150~K,
interlayer phase coherence appears between some of these planar
antiferromagnetic clusters with an energy scale smaller than 70~$\mu$\,eV.
While the 2D antiferromagnetic correlations in an individual plane remain
liquid down to 1.5~K, coherent multiplane
antiferromagnetic correlations become frozen below $T_g$.
The phase separation into 2D spin-liquid and spin-glass
of higher dimension with an unusual {\bf q}-structure for
the ``Edwards-Anderson order parameter'' is most likely
related to quasi-2D nature of magnetic exchange in
the cuprates and is distinctly different from conventional spin-glasses.
A theory of spin-glass in doped cuprates should include
interlayer coupling. Theory explaining both the partial spin freezing and
the observed crossover\cite{bao02c,bao04a} of quantum
spin fluctuations are called for. The heterogeneous magnetic
correlations, instead of a uniform magnetic phase,
suggests the possibility that superconductivity and the almost
3D antiferromagnetic order may reside in
different phases in La$_{2-x}$Sr$_x$CuO$_4$
and Y$_{1-x}$Ca$_x$Ba$_2$Cu$_3$O$_{6+y}$.
Similar, detailed {\bf q}, $E$ and $T$ dependent cold neutron
spectroscopic study on these cuprates are desirable.
We thank R.H.\ Heffner, P.C.\ Hammel, S.M.\ Shapiro, C. Broholm,
L.\ Yu, Z.Y. Weng, A.C. Castro Neto, O. Sushkov, J. Ye,
F.C. Zhang, X. G. Wen, T. Senthil, P. C. Dai and C.M. Varma for useful discussions.
SPINS and NG7-SANS at NIST are supported partially by NSF. Work at LANL is
supported by U.S. DOE.
| {
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,954 |
Нащо́кинский домик — кукольный домик, выполненный по заказу Павла Нащокина (1801—1854), точная копия дома Нащокиных в Москве, в Воротниковском переулке (ныне № 12), который семейство снимало у губернской секретарши Аграфены Ивановны Ивановой. Это последний московский адрес Александра Пушкина — он гостил у Нащокина с 3 по 20 мая 1836 года.
«Мой маленький домик» — как называл его сам владелец — заключён в футляр 2,5 × 2 м с раздвижными зеркальными стеклами. Все вещи в домике являются действующими моделями реальных вещей и были изготовлены на тех же фабриках, что и настоящие. Так, миниатюрную мебель выполнили в мастерской братьев Гамбс, фарфоровый сервиз — на заводе А. Г. Попова, рояль, на котором играли с помощью вязальных спиц, принадлежит фирме «Фишер».
А. С. Пушкин был свидетелем создания домика. Он неоднократно упоминал о нём в письмах к жене Наталье Николаевне. Так, 8 декабря 1831 года он писал: «Дом его (помнишь?) отделывается; что за подсвечники, что за сервиз! он заказал фортепьяно, на котором играть можно будет пауку, и судно, на котором испразнится разве шпанская муха». Через год Пушкин сообщал супруге: «С Нащокиным вижусь всякий день. У него в домике был пир: подали на стол мышонка в сметане под хреном в виде поросенка. Жаль, не было гостей. По своей духовной домик этот отказывает он тебе»; 4 мая 1836 года он писал: «Домик Нащокина доведен до совершенства — недостает только живых человечиков. Как бы Маша им радовалась».
Однако, домик не был передан семье Пушкиных. Вскоре после смерти поэта Нащокин был вынужден заложить его, после чего реликвия несколько раз меняла своих владельцев. Позднее домик обнаружил и восстановил художник и коллекционер С. А. Галяшкин.
В 1919 году домик был реквизирован, после чего попал в Государственный музейный фонд, находившийся в здании Английского клуба на Тверской улице. В начале 1921 года в двух залах бывшего клуба был организован музей «Старая Москва», год спустя ставший филиалом Исторического музея. В октябре 1922 года, когда здание на Тверской было передано под выставку «Красная Москва», домик, в составе других фондов музея, переехал в Юсуповский дворец в Большом Харитоньевском переулке. В 1926 году, после ликвидации музея Старой Москвы, он вошёл в коллекцию Исторического музея.
В 1937 году в рамках «Пушкинского юбилея», приуроченного к 100-летию со дня гибели поэта, в Историческом музее проходила Всесоюзная пушкинская выставка, материалы которой стали основой для образования Музея А. С. Пушкина. Домик был перевезён в Ленинград, и после эвакуации в период войны, вновь предстал в экспозиции музея, разместившейся в 17 залах Государственного Эрмитажа. С 1967 года домик экспонировался в одном из 27 залов экспозиции «А. С. Пушкин. Личность, жизнь и творчество» в церковном флигеле Большого Екатерининского дворца в Пушкине.
Позднее, в рамках подготовки к 200-летию со дня рождения Пушкина, домик переехал в экспозицию Всероссийского музея А. С. Пушкина, на Мойке, 12.
В XXI веке домик дважды экспонировался в Москве: в 2001 году в галерее «Дом Нащокина» в доме, который был его прообразом, и в 2017 году в Государственном музее А. С. Пушкина на Пречистенке, на выставке «Нащокинский домик — путешествие в Москву. К 60-летию со дня основания Государственного музея А. С. Пушкина».
См. также
Кукольный домик Петронеллы Ортман
Примечания
Ссылки
Нащокинский домик — путешествие в Москву. К 60-летию со дня основания Государственного музея А. С. Пушкина
Нащокина
Произведения 1830-х годов
Произведения декоративно-прикладного искусства XIX века
Произведения декоративно-прикладного искусства Российской империи
Экспонаты Всероссийского музея А. С. Пушкина
Интерьеры России
Интерьеры XIX века
Нащокины
Предметы, связанные с Александром Пушкиным | {
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} | 7,831 |
Q: select2 apply entered text as tag before user choose it I use select2 v4.0.7 with possibility to search by text (tags: true)
$query.select2({
minimumInputLength: 2,
ajax: {
url: '/'+ LOCALE +'/search-suggest',
dataType: 'json',
delay: 450,
cache: true,
data: function (params) {
return {q: params.term}
}
},
language: LOCALE,
tags: true,
data: selectData,
allowClear: true,
placeholder: ''
});
Catch some problem when I paste text with Ctrl+v. In this case select2 requests ajax url and apply WHOLE entered text. When I push Enter nothing happens, cause tag text equal with entered text and select2:select doesn't fire.
How to stop auto applying the first result when input is empty and tags option is true?
A: *
*remove tags: true, cause in apply text as result without firing any events.
*add escaped user query as first result in ajax controller/url
$queryToDisplay =
htmlspecialchars(
strip_tags(
stripslashes(
$queryOriginal
)
),
ENT_QUOTES
)
;
$this->data['results'][] = ['id' => $queryToDisplay, 'text' => $queryToDisplay];
| {
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Dynasty TV Show
Shoulder pads, popularized by Joan Collins and Linda Evans from the soap opera Dynasty, remained popular throughout the 1980s and even the first three years of the 1990s. The reason behind the sudden popularity of shoulder pads for women in the 1980s may be that women in the workplace were no longer unusual, and wanted to "power dress" to show that they were the equals of men at the office. Many women's outfits had Velcro on the inside of the shoulder where various sized shoulder pads could be attached.
The Dynasty television show, watched by over 250 million viewers around the world in the 1980s, influenced fashion in mainstream America and perhaps most of the Western world. The show influenced women to wear glitzy jewelry as a way of flaunting wealth. Synthetic fabrics went out of style in the 1980s. Wool, cotton, and silk returned to popularity for their perceived quality.
Men's business attire saw a return of pinstripes for the first time since the 1970s. The new pinstripes were narrower and subtler than in 1930s and 1940s suits but were similar to the 1970s styles. Three-piece suits gradually went out of fashion in the early '80s and lapels on suits became very narrow (similar to 1950s styles). While vests in the 1970s had commonly been worn high with six or five buttons, those made in the early 1980s often had only four buttons and were made to be worn low. Neckties also became narrower in the 1980s and skinny versions, some made of leather, briefly were stylish among men interested in New Wave music. Button-down collars made a return, for both business and casual wear.
Meanwhile women's fashion and business shoes revisited the pointed toes and spiked heels that were popular in the 1950s and early 1960s with . Some stores stocked canvas or satin covered fashion shoes in white and dyed them to the customer's preferred color. While the most popular shoes amongst young women were bright colored high heels, a trend started to emerge which saw 'Jellies'—colorful, transparent plastic flats—become popular. The top fashion models of the 1980s were Brooke Shields, Christie Brinkley, Joan Severance, , Kim Alexis, Carol Alt, Renée Simonsen, Kelly Emberg, Tatjana Patitz, Elle McPherson, and Paulina Porizkova.
Images courtesy of http://www.fashion-era.com/power_dressing.htm and http://thegrumpyowl.com/2009/06/25/jacket/
Source: "1980s in Fashion." Wikipedia, the Free Encyclopedia. Wikimedia Foundation, Inc. 14 Apr. 2011. Web. 15 Apr. 2011. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Białe Jeziorki ist ein Dorf in der polnischen Woiwodschaft Ermland-Masuren. Es liegt im Kreis Gołdap (Goldap) und gehört zur Landgemeinde Dubeninki (Dubeningken, 1938–1945 Dubeningen).
Białe Jeziorki liegt 17 Kilometer südöstlich der Kreisstadt Gołdap südlich der Rominter Heide (). Das Dorf ist zu erreichen über den Abzweig Cisówek an der Nebenstraße von Dubeninki über Rakówek nach Przerośl. Eine Bahnanbindung besteht nicht.
Vor 1938 lag Białe Jeziorki auf polnischem Gebiet unmittelbar hinter der Grenze zum Deutschen Reich. Einen deutschen Namen kennt der Ort daher nicht. Zwischen 1975 und 1998 war er in die Woiwodschaft Suwałki, seither ist er in die Woiwodschaft Ermland-Masuren eingegliedert. Białe Jeziorki ist heute eine Ortschaft mit Schulzenamt (Sołectwo) im Verbund der Gmina Dubeninki im Powiat Gołdapski.
Die katholischen Einwohner Białe Jeziorki sind in die Pfarrei Przerośl eingegliedert, die zum Dekanat Filipów im Bistum Ełk (Lyck) der Katholischen Kirche in Polen gehört. Der evangelische Teil der Bevölkerung ist in die Kirchengemeinde Suwałki (Suwalken) eingepfarrt und damit der Diözese Masuren der Evangelisch-Augsburgischen Kirche in Polen zugehörig.
Ort der Woiwodschaft Ermland-Masuren
Gmina Dubeninki | {
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{"url":"http:\/\/blog.davidchudzicki.com\/2013\/07\/mine-depth-make-your-axes-meaningful.html","text":"## Tuesday, July 2, 2013\n\n### Mine Depth (Make your axes meaningful!)\n\nAt the Empire Mine State Historic Park, you can walk 30 feet into the old gold mine, meet a quirky and knowledgeable blacksmith named David, see an awesome scale model of the mine (above--very secret during the years of operation!), and see this chart visualizing the depth of the mine over time:\n\nI usually like axes to be as meaningful as possible, so I redrew the chart (code below) representing the year with the horizontal axis:\nThe original version probably has its own advantages as well, but some observations and questions comes a little more quickly from my version:\n\n\u2022 From 1886 onward, the mine seems to get deeper as a roughly constant rate (averaged over time).\n\u2022 Are they digging it deeper smoothly over time, or does the depth increase in spurts?\n\u2022 Why do we have so much more data in the period from 1886 to 1892 than the rest?\n\nBefore drawing the chart this way, I figured that depth would increase faster in more recent years (with better technology), which isn't true at all. But note that this is just the maximum depth -- maybe they can dig much more easily with better technology, but that's spread across more tunnels, so the depth doesn't increase any faster? Or maybe each bar on the original chart represents a short, concentrated digging effort (meaning they can increase the depth much faster in recent years, but also go for much longer without increasing it at all)?\n\nThe code:\n\nlibrary(\"ggplot2\")\n\nmineDepths.string ='\"Year\",\"Depth\"\n1854,102\n1865,201\n1874,1250\n1886,1600\n1888,1700\n1890,1900\n1892,2100\n1901,3000\n1914,4600\n1924,6200\"\n1956,11000'\n\ndepths$Depth = -depths$Depth","date":"2017-11-21 21:07:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.47058653831481934, \"perplexity\": 1744.1338570261676}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806426.72\/warc\/CC-MAIN-20171121204652-20171121224652-00000.warc.gz\"}"} | null | null |
Prisonology spoke with Patricia Griffin, PhD, Bureau Of Prisons RDAP Coordinator - Retired, on the RDAP program and how to improve your chances of getting into it.
The Residential Drug Abuse Program (RDAP) is a 500-hour program over nine months for inmates incarcerated in select institutions. The program is intensive therapy addressing past alcohol and drug abuse issues for those enrolled. Completion of the program can result in up to a one year reduction off of an inmates prison term and six months of halfway house. It is the only program, if completed, in the BOP that offers a reduction of time off of a sentence.
You can qualify for up to 12 months off of your sentence by participating in RDAP but overcrowding and the duration of your sentence may result in less time being taken off of your prison term.
Even if a federal judge recommends you to the RDAP program, and if it is in your PSR, you still need to apply. Soon after you arrive at a federal institution, you need to contact the Drug Abuse Program Coordinator in the Psychology Department to apply for RDAP.
Once you have applied, your documentation will be reviewed by staff to see if you qualify for the RDAP. The Presentence Report (PSR) is especially relied up during the screening process.
If there is no documentation of substance abuse or dependence in the PSR in the year before your arrest, you will be denied admission in the RDAP.
If there is no documentation in your PSR about substance abuse, documentation from physicians and counselors prior to prison may be submitted. However the review process for this is stringent and, frequently, not successful.
It is up to the specialists at the prison as to whether or not to admit an inmate into the RDAP program.
Not every federal prison has an RDAP. Be sure check the list of facilities that do have RDAP.
Every inmate who applies for the RDAP is going to be interviewed by a Drug Treatment Specialist for screening into the program.
Once in the RDAP program, inmates are held to a higher level of accountability, so rules that may not have been enforced in other living Units will be strictly enforced during the program.
Stay focused and dedicated to the program. Staff does not appreciate RDAP participants that are just going through the motions. If you are engaged, the program will be easier for you.
Hang out with people in your cohort as much as possible. They know the rules and want to make it to the finish line as much as you do. Inmates who are not in the program may not share that same view.
There are a number of conditions that may exclude a person from participating in the program, but two primary ones are having a sentence of 24 months or less, or not being a U.S. Citizen.
So here I am a few days into this new camp for RDAP. Let's just say that - so far at least - it's not particularly fun.
My first notice that this wasn't an ordinary prison camp came when I stepped away from my bunk for a moment to ask a neighbor a question. Before I could cross the aisle, several guys were in my face informing me that I had just broken an important rule. As I soon learned, you cannot leave your area before 4 p.m. unless all your items are in stowed away in your locker; no shirt or book or brush can be anywhere to be seen. Then I made my bed, only to be informed that it was completely wrong: the pillow can't touch the headboard and the blanket has to be folded down exactly 1 inch. Later, I walked to the bathroom to wash my face. Before the water touched my skin someone was informing me that I was using the wrong sink. In the evening, I stepped outside only to be yelled at for stopping on a painted yellow square. The worst crime? Stepping off the path onto the white gravel that's replaced the grass in this drought-prone area.
I'd heard stories about the place before I came in - about all the rules, about the particular culture. RDAP is a favorite topic of conversation at the 'ordinary camp': rumors abound about how they brainwash you, about the endless rules, about a punishment called a "pull-up" in which you stand up before the entire population and admit your transgression, however seemingly minor. But I wondered to myself how hard it could be. I consider myself a courteous person and tend (despite my crime) to follow rules. I also understood the basic concept: to create a self-contained, law-abiding community out of a disparate group of lawbreakers and addicts.
The first problem for me was that very few of the rules were written down. We were apparently just expected to know them. The second was that I was not used to getting etiquette and behavior lessons from fellow inmates. So, I'm ashamed to say, I got a little defensive. Especially when someone criticized how I blew my nose, how I brushed my teeth and how I flushed the toilet. I started thanking my fellow participants for their comments through gritted teeth until, at 4 a.m. the next morning as I prepared quietly for work amidst a sea of snoring inmates, I was informed by a fellow early riser that I had not washed my hands properly after sneezing. I grunted and turned away without a 'thank you'. Later I was told that my behavior was not pro-social.
But I'm doing my best - the first few days are considered a "grace period" before the actual punishment begins. I'm using up, it seems, my allotment of free passes until the pull-ups start to fly. But it's not as bad as hazing week at the fraternity in college: no green underwear, beer bongs or screaming in my ears ... or my first week in prison at the other camp, which was nearly infinitely worse. In some ways it's even fun and instructive: a useful lesson in humility and how to follow the rules. I just wish there weren't so many of them."
"I would need a new notebook with 100 pages to answer this question. Inmates are constantly breaking rules every day. Remember, I am in RDAP and if we break the rules or get in trouble, we could jeopardize our Good Time [time off for good behavior and time off for participating the RDAP program]. The RDAP program is very strict in my Unit. So it is like two different societies and prisons here at [prison camp]; RDAP and non-RDAP. The prisoners not in RDAP steal from the kitchen, break all the common small rules like buying and selling food, shoes, clothes and radios from each other. They also eat and drink in places they should not then of course we have the much more serious rule violations. That includes using cell phones, smoking cigarettes, marijuana and alcohol use. We do not have a fence so inmates sneak off to the nearest road to get packages that are dropped off. It is risky but some do it. So there is plenty of contraband items around, I just stay as far away from it as I can.
The ones who usually bring in packages are the inmates who have been in prison for the longest period of time. If an inmate gets caught with contraband, or is caught leaving the compound to pick up a package, they get shipped to another higher security facility. I've seen that happen several times within my first year here. This is the worst thing in the world so I am just astonished that guys do this stuff. Honestly, it is just easier being in the RDAP program and obeying the rules and knowing that everyone around you is obeying them as well. We all just want to go home." | {
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Q: Can I drive a Crystal Report in a standalone application with .NET objects? I'm trying to learn Crystal Reports (VS 2005, VB) and per this question I was trying to drive the report with my own data objects instead of through a DB connection.
I found this tutorial and it looked promising as an answer to my question, but after diving in it only seems to apply to web development.
Am I out of luck in trying to power a Crystal Report with .NET objects in a standalone application?
A: Based on this step, I think the report might have to be a strongly typed report. Where did it indicate it was only for the web?
A: Please see my answer for your previous question.
How to use Crystal Reports without a tightly-linked DB connection?
| {
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Shchyolkovskaya () is a Moscow Metro station on the Arbatsko-Pokrovskaya Line. It is an Eastern terminus of the line. It opened in 1963.
Name
Its name owes to the location near the Shchyolkovo highway.
Building
It was built at below the ground to the standardized column tri-span design, which was commonly used from the 1960s till 1990s. The pillars are faced with dark green marble. The walls were originally tiled by yellow and black ceramic tiles, but a modern metalloplastic cladding was applied in 2002, giving the station a cleaner look. The architects were Ivan Taranov and Nadezhda Bykova.
Traffic
The station is highly loaded due to nearby Moscow Central Bus Terminal.
References
Moscow Metro stations
Railway stations in Russia opened in 1963
Arbatsko-Pokrovskaya Line
Railway stations located underground in Russia | {
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} | 6,369 |
Q: How to iterate the backend api map response in angular or typescript I have an api which will return the map response like .
{thomas: 3, test70: 2, tim: 2, elin: 2, sumeet12: 1}
I want to iterate in angular but unfortunately i got error while trying this.
Error Got ::
This expression is not callable. Type 'Number' has no call
signatures.ts(2349)
this.testObject.forEach((value: number, key: string) => {
console.log(key, value);
});
Iterating over Typescript Map
But i am able to iterate in the html using below code..
In html
<div *ngFor="let item of testObject | keyvalue: originalOrder; let i=index">
<span *ngIf="i<3">
Key: <b>{{item.key}}</b> and Value: <b>{{item.value}}</b>
</span>
</div>
In .ts file
testObject: { [key: string]: number };
showresult: any;
getQuizResult(id){
this.service.getResult(id).subscribe((stndrdResp: Response) => {
this.showresult = stndrdResp.obj; // this will return a object
this.testObject = this.showresult; // assigning the key value response to testObject
});
}
Basically i want to iterate the backend api response inside the getQuizResult so that i can perform some action. Appreciate your valuable suggestion.
A: I think this is what you are looking for: https://stackoverflow.com/a/24440475/12193298
A comment; in one of your code above you have key: string in the second parameter, but in the forEach callback the second one is the index, which always is a number. So that should be looking like this:
this.testObject.forEach((value: number, index: number) => {
console.log(index, value);
});
A: Because testObject is a plain object,not a Map instance.You need to create a map object or array.
const res = {thomas: 3, test70: 2, tim: 2, elin: 2, sumeet12: 1};
var arr = Object.entries(res);//[[ "thomas", 3 ]...]
var map = new Map(arr);//use arr or create a map
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,104 |
The ocean views and surrounding scenery provide a peaceful ambiance to your vacation home in Talamone. This city is walkable and welcoming, recognized for its fishing, boating and restaurants. If you need more space to accommodate a family vacation or friends trip, rental homes often come with separate dining areas and spacious living rooms.
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Can I rent Houses in Talamone?
Can I find a holiday accommodation with pool in Talamone?
Yes, you can select your preferred holiday accommodation with pool among our 117 holiday homes with pool available in Talamone. Please use our search bar to access the selection of holiday rentals available. | {
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/* --------------------------------------------------------------------------
* Name: image-observer.h
* Purpose: Informs clients when image objects change
* ----------------------------------------------------------------------- */
#ifndef APPENGINE_IMAGE_OBSERVER_H
#define APPENGINE_IMAGE_OBSERVER_H
#include "appengine/graphics/image.h"
typedef int imageobserver_change;
enum
{
imageobserver_CHANGE_ABOUT_TO_DESTROY,
imageobserver_CHANGE_PREVIEW,
imageobserver_CHANGE_ABOUT_TO_MODIFY,
imageobserver_CHANGE_MODIFIED,
imageobserver_CHANGE_SAVED, /**< When image safely saved to disc */
imageobserver_CHANGE_HIDDEN,
imageobserver_CHANGE_REVEALED,
imageobserver_CHANGE_GAINED_FOCUS,
imageobserver_CHANGE_LOST_FOCUS,
};
typedef union imageobserver_data
{
struct
{
image_modified_flags flags;
}
modified;
}
imageobserver_data;
typedef void (imageobserver_callback)(image_t *image,
imageobserver_change change,
imageobserver_data *data,
void *opaque);
int imageobserver_register(image_t *image,
imageobserver_callback *callback,
void *opaque);
int imageobserver_deregister(image_t *image,
imageobserver_callback *callback,
void *opaque);
/* 'Greedy' functions are called for changes on all images. */
int imageobserver_register_greedy(imageobserver_callback *callback,
void *opaque);
int imageobserver_deregister_greedy(imageobserver_callback *callback,
void *opaque);
int imageobserver_event(image_t *image,
imageobserver_change change,
imageobserver_data *data);
#endif /* APPENGINE_IMAGE_OBSERVER_H */
| {
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Home / Africa Weekly / Business
Shenzhen Energy powers new path in Africa
By Zhang Yue (China Daily Africa)
A new electrical plant being built with Chinese development fund is helping meet Ghana's needs
Through the joint effort by Shenzhen Energy Group and the China-Africa Development Fund in 2007, a major power plant is providing electricity for the city of Tema in the Republic of Ghana.
Located 29 kilometers from the capital, Accra, the plant is run by Sunon-Asogli Power (Ghana) Ltd, and cofounded by Shenzhen Energy and the fund.
Power plant run by Sunon-Asogli Power (Ghana) Ltd and cofounded by Shenzhen Energy and the China-Africa Development Fund provides electricity for Tema, Ghana. Photos provided to China Daily
Staff members at the power plant run by Sunon-Asogli Power (Ghana) Ltd in Tema in the Republic of Ghana.
The power plant is the first built overseas by Shenzhen Energy and covers an area of 200,000 square meters. Today, it is an important source of power for Ghana.
The fund and Shenzhen Energy co-established Sunon-Asogli Power as part of China's "reaching-out" strategy of investing and expanding overseas.
Shenzhen Energy had been interested in investing in Africa since late 2006, when Ghana tribal leader Togbe Afede XIV invited Shenzhen Energy to look into the West African nation's power market.
The total capacity of the plant will be 560 mW from gas-steam power generators, built in two phases. The first phase of construction of two 100 mW gas-steam generators was completed and put into operation in 2010, mitigating Ghana's urgent power needs.
The first phase of the project generates about 15 percent of the country's total electricity and benefits more than 2 million residents.
Shenzhen Energy has recently expanded its energy market investments in Ghana. Feasibility studies for the second phase of the power plant is in progress and construction is to start in the spring. The company is also planning a coal-fired power plant and a wind farm in Ghana.
Currently, the company has 91 Chinese employees and 51 local employees.
The successful experience in Ghana has encouraged the company to expand their business to other countries in Africa. In recent years, it has participated in building power plants in Kenya and Zambia with the help of the China-Africa Business Council and the Shenzhen Trade Promotion Association.
Still, the company faces difficulties in Africa, says Li Xiaohai, head of Sunon-Asogli Power (Ghana) Ltd.
"Developing a business abroad is comparatively more risky because we have less experience and it is less predictable. We need to be prepared for all kinds of circumstances. The culture and political environment in African countries are very different," he says. "It is a big sacrifice for our project managers to work in a foreign country as they have great workloads and sometimes even face danger. Therefore, we sincerely need more support and understanding from the local government."
He says recruiting talent is key to expanding the company's progress in foreign markets.
"We are paying lots of attention to developing our teams in management, marketing and technology," he says. "Only when we make good use of our talent in different areas can we successfully blend into the international market."
He says the company needs more support from the local government in terms of financing and fair trading.
zhangyue@chinadaily.com.cn
(China Daily Africa Weekly 01/16/2015 page20)
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1st Shanghai-Kathmandu route | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 245 |
Nomindra arenaria är en spindelart som beskrevs av Norman I. Platnick och Baehr 2006. Nomindra arenaria ingår i släktet Nomindra och familjen Prodidomidae.
Artens utbredningsområde är Northern Territory, Australien. Inga underarter finns listade i Catalogue of Life.
Källor
Spindlar
arenaria | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,420 |
Les études de cinéma et d'audiovisuel englobent tant les métiers techniques de l'image et du son que les métiers artistiques. Ces formations ont souvent et de plus en plus de points communs, avec le développement de l'informatique, au cinéma, la télévision, les jeux vidéo, le multimédia et le Web.
En Europe
La plupart des écoles dispensent tant des enseignements pour l'image que pour le son ou encore d'autres métiers du cinéma et de l'audiovisuel.
Allemagne : Deutsche Film und Fernsehakademie Berlin (DFFB)
Belgique : Il existe plusieurs écoles publiques et privées délivrant des diplômes d'ingénierie du son : INSAS, IAD, INRACI, SAE.
Danemark : École nationale du film de Copenhague
Grande-Bretagne : London Film School - National Film and Television School (NFTS) - Royal College of Arts
Pologne : École nationale de cinéma de Łódź
Russie : Institut d'État fédéral de cinématographie
Suisse : en Suisse, la formation d'ingénieur du son se fait en école privée : Suisse Romande : CFMS (Lausanne) ; SAE (Genève), Suisse Allemande : TTS (Aarau) ; SAE (Zürich)
France
Le nombre de formations au cinéma et à l'audiovisuel est en augmentation et en diversification constantes. De nouveaux métiers apparaissent comme ceux du monde du jeu vidéo, ceux liés à l'évolution de l'environnement informatique et des réseaux, etc. Les débouchés professionnels ne semblent pas toujours augmenter dans les mêmes proportions. Environ deux-tiers des métiers techniques de l'audiovisuel se pratiquent en France sous le statut particulier d'un CDD (notamment CDD d'usage) qui est celui des intermittents du spectacle.
En Afrique
Maroc
Au Maroc, la formation audiovisuelle se fait à l'ISMAC, AAMA et à l'ISMC .
Bénin
Au Bénin, une formation se fait à l'Institut Supérieur des Métiers de l'Audiovisuel (ISMA). Elle se termine par un BTS en Métiers du son, de l'image et de la production puis une licence en journalisme et en réalisation TV et cinéma.
En Amérique
Québec
Au Québec, la formation d'ingénieur du son se fait au niveau collégial. Le Cégep de Drummondville, le Collège d'Alma, L'institut d'enregistrement du Canada à Montréal, Musitechnic, Institut Trebas ainsi que le Campus Notre-Dame-De-Foy disposent d'une AEC (Attestation d'Études Collégiales) à temps plein, échelonnée sur 1 an.
Au Québec, pour pouvoir porter le titre d'ingénieur, l'on se doit d'être diplômé d'une université en ingénierie et de faire partie de l'ordre des ingénieurs du Québec.
Notes et références
Annexes
Articles connexes
Métiers du cinéma
Liste d'écoles de cinéma d'animation
Observatoire européen de l'audiovisuel
Enseignement du cinéma | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,910 |
(function($) {
$.fn.bootstrapValidator.i18n.different = $.extend($.fn.bootstrapValidator.i18n.different || {}, {
'default': 'Please enter a different value'
});
$.fn.bootstrapValidator.validators.different = {
html5Attributes: {
message: 'message',
field: 'field'
},
/**
* Return true if the input value is different with given field's value
*
* @param {BootstrapValidator} validator The validator plugin instance
* @param {jQuery} $field Field element
* @param {Object} options Consists of the following key:
* - field: The name of field that will be used to compare with current one
* - message: The invalid message
* @returns {Boolean}
*/
validate: function(validator, $field, options) {
var value = $field.val();
if (value === '') {
return true;
}
var compareWith = validator.getFieldElements(options.field);
if (compareWith === null || compareWith.length === 0) {
return true;
}
if (value !== compareWith.val()) {
validator.updateStatus(options.field, validator.STATUS_VALID, 'different');
return true;
} else {
return false;
}
}
};
}(window.jQuery));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,183 |
'use strict';
const chai = require('chai'),
expect = chai.expect,
Support = require('../support'),
DataTypes = require('../../../lib/data-types');
describe(Support.getTestDialectTeaser('Model'), () => {
beforeEach(function() {
this.User = this.sequelize.define('User', {
username: DataTypes.STRING,
age: DataTypes.INTEGER
});
this.Project = this.sequelize.define('Project', {
name: DataTypes.STRING
});
this.User.hasMany(this.Project);
this.Project.belongsTo(this.User);
return this.sequelize.sync({force: true});
});
describe('count', () => {
beforeEach(function() {
return this.User.bulkCreate([
{username: 'boo'},
{username: 'boo2'}
]).then(() => {
return this.User.findOne();
}).then(user => {
return user.createProject({
name: 'project1'
});
});
});
it('should count rows', function() {
return expect(this.User.count()).to.eventually.equal(2);
});
it('should support include', function() {
return expect(this.User.count({
include: [{
model: this.Project,
where: {
name: 'project1'
}
}]
})).to.eventually.equal(1);
});
it('should return attributes', function() {
return this.User.create({
username: 'valak',
createdAt: (new Date()).setFullYear(2015)
})
.then(() =>
this.User.count({
attributes: ['createdAt'],
group: ['createdAt']
})
)
.then(users => {
expect(users.length).to.be.eql(2);
// have attributes
expect(users[0].createdAt).to.exist;
expect(users[1].createdAt).to.exist;
});
});
it('should not return NaN', function() {
return this.sequelize.sync({ force: true })
.then(() =>
this.User.bulkCreate([
{ username: 'valak', age: 10},
{ username: 'conjuring', age: 20},
{ username: 'scary', age: 10}
])
)
.then(() =>
this.User.count({
where: { age: 10 },
group: ['age'],
order: ['age']
})
)
.then(result => {
expect(parseInt(result[0].count)).to.be.eql(2);
return this.User.count({
where: { username: 'fire' }
});
})
.then(count => {
expect(count).to.be.eql(0);
return this.User.count({
where: { username: 'fire' },
group: 'age'
});
})
.then(count => {
expect(count).to.be.eql([]);
});
});
it('should be able to specify column for COUNT()', function() {
return this.sequelize.sync({ force: true })
.then(() =>
this.User.bulkCreate([
{ username: 'ember', age: 10},
{ username: 'angular', age: 20},
{ username: 'mithril', age: 10}
])
)
.then(() =>
this.User.count({
col: 'username'
})
)
.then(count => {
expect(parseInt(count)).to.be.eql(3);
return this.User.count({
col: 'age',
distinct: true
});
})
.then(count => {
expect(parseInt(count)).to.be.eql(2);
});
});
it('should be able to use where clause on included models', function() {
const queryObject = {
col: 'username',
include: [this.Project],
where: {
'$Projects.name$': 'project1'
}
};
return this.User.count(queryObject).then(count => {
expect(parseInt(count)).to.be.eql(1);
queryObject.where['$Projects.name$'] = 'project2';
return this.User.count(queryObject);
}).then(count => {
expect(parseInt(count)).to.be.eql(0);
});
});
it('should be able to specify column for COUNT() with includes', function() {
return this.sequelize.sync({ force: true }).then(() =>
this.User.bulkCreate([
{ username: 'ember', age: 10},
{ username: 'angular', age: 20},
{ username: 'mithril', age: 10}
])
).then(() =>
this.User.count({
col: 'username',
distinct: true,
include: [this.Project]
})
).then(count => {
expect(parseInt(count)).to.be.eql(3);
return this.User.count({
col: 'age',
distinct: true,
include: [this.Project]
});
}).then(count => expect(parseInt(count)).to.be.eql(2));
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,758 |
Magnesium is een scheikundig element met symbool Mg en atoomnummer 12. Het is een zilverwit aardalkalimetaal.
Geschiedenis
Magnesium is genoemd naar het district Magnesia in Thessalië in Griekenland. Magnesium was in de vorm van magnesiumoxide al heel lang bekend, maar pas in 1755 werd door de Schotse wetenschapper Joseph Black onderkend dat men bij magnesiumoxide (het werd toen nog niet zo genoemd) met een nieuwe stof te maken had. Tot die tijd werden magnesiumoxide en calciumoxide met elkaar verward en beide ongebluste kalk genoemd.
Toen in 1803 in Moravië een aanzienlijke afzetting van natuurlijk magnesiumcarbonaat werd ontdekt, werd dit door C.F. Ludwig aanvankelijk talcum carbonatum genoemd. De term magnesiet werd voor het eerst gebruikt in 1808 door L.G. Karsten. In datzelfde jaar werd door Humphry Davy ontdekt dat magnesiumoxide het oxide was van een metaal dat tot dan toe onbekend was. Naar verluidt zou hij ook de eerste persoon zijn geweest die een kleine hoeveelheid metallisch magnesium (als een amalgaan met kwik) wist te winnen door elektrolytische reductie van magnesiumoxide. De ontdekking van magnesium in zijn zuivere vorm zou echter op het conto komen van de Franse chemicus Antoine Bussy, die er in 1828 of 1831 in slaagde om niet alleen een grotere, maar ook een zuiverder hoeveelheid magnesium te prepareren.
Om magnesium te verkrijgen, liet Bussy magnesiumchloride (MgCl2) reageren met metallisch kalium. In 1833 probeerde Michael Faraday eveneens magnesium te produceren en wel door gedehydrateerd vloeibaar MgCl2 te elektrolyseren tot vloeibaar magnesium en chloorgas. In 1852 bouwde de Duitser Robert Bunsen een kleine cel voor de elektrolyse van MgCl2. Deze vinding leidde in Duitsland in 1886 tot de eerste commerciële productie van magnesium en tot uitgebreid onderzoek naar de toepassing van het materiaal. Tijdens de Tweede Wereldoorlog resulteerde dit in een relatief veelvuldig gebruik van magnesium in de oorlogsindustrie. Gebaseerd op Duitse patenten waarin dolomiet onder vacuüm gereduceerd was met een ijzer-siliciumverbinding, zette de Canadees Lloyd Montgomery Pidgeon in 1940 de eerste industrie op rond metallothermische magnesiumextractie. Dit zogeheten Pidgeonproces zou 60 jaar later nog steeds op grote schaal gebruikt worden.
Productie en recycling
Magnesium kan geproduceerd worden middels elektrolyse van MgCl2 of door metallothermische reductie van MgO. Anno 1988 was de totale productie van magnesium afkomstig uit de omzetting van dolomiet (42%), metaalzouten (36%), zeewater (18%) en magnesiet (4%). Het productieproces voor de vorming van één kilo magnesium kost relatief veel energie, vergeleken met de conventionele materialen in de lichtgewicht-materiaal-industrie. Gekeken naar het volume is het echter juist energiezuiniger dan aluminium of zink en kan het zelfs de concurrentie met polymeren aan.
Bij het recyclen van magnesium zijn er meerdere problemen die omzeild moeten worden:
Ongewenste insluitsels in het materiaal: niet-metallische deeltjes, intermetallische verbindingen en delen van oxidehuid.
reactiviteit van magnesium: de smelt moet goed afgeschermd worden om de vorming van een oxidehuid te voorkomen. Hiervoor worden smeltzouten gebruikt.
Legering/naamgeving
De naam van een magnesiumlegering bestaat uit vier delen.
twee letters, die slaan op de twee legeringselementen die het meest in het materiaal aanwezig zijn.
een (afgerond) getal, dat aangeeft hoeveel procent van de legering uit de twee belangrijkste legeringselementen bestaat.
een letter uit het alfabet, die aan de legering wordt toegekend zodra hij voortaan als standaardlegering beschouwd wordt. De letters worden in oplopende volgorde uitgereikt, behalve de letter I en O.
een letter, gevolgd door een nummer, dat iets zegt over de warmtebehandeling of versteviging die het materiaal heeft ondergaan.
Een gangbare magnesiumlegering is bijvoorbeeld de gietlegering AZ91A, die onder meer wordt toegepast in de auto-industrie. Het bevat 90 (88,2-91,2) %(m/m) Mg (magnesium), 9 (8,3-9,7) % Al (aluminium) en minimaal 0,13 % Mn (mangaan). Verder maximaal 0,5 % Si (silicium), maximaal 0,1 % Cu (koper), maximaal 0,03 % Ni (nikkel).
Kristalstructuur
Zuiver magnesium heeft een hexagonale kristalstructuur, zoals is weergegeven in de afbeelding hiernaast. Door deze ingewikkelde structuur (ten opzichte van die van bijvoorbeeld aluminium of staal, die respectievelijk de kvg- en de krg-structuur bezitten) heeft het slechts beperkte mogelijkheden tot vervorming. In de jaren 30 en 40 ontdekte men dat magnesium bij kamertemperatuur tweelingen kan vormen aan de {}-vlakken in de -richting, en dat er afschuiving optreedt aan het (0001)-vlak in de -richting.
Toepassingen
Omdat het met een fel wit licht brandt, wordt magnesium dikwijls in vuurwerk gebruikt als reductor en werd het in het verleden veel in flitslichten voor de fotografie toegepast.
Het metaal speelt een belangrijke rol in de organische synthese. Het reageert met een organohalogeniden tot een organomagnesiumhalogenide: een klasse verbindingen die bekendstaan als de Grignard-reagentia.
Op het gebied van gewicht, sterkte en verwerkingseigenschappen voldoet magnesium aan de eisen om toe te passen in (lichtgewicht) constructies. Een voorbeeld van een constructieve toepassing van magnesium is de stadsfiets van de Engelse ontwerper Richard Thorpe. Een keramische deklaag beschermt het magnesium daar tegen corrosie. Een ander vlak waar magnesium als relevant gezien wordt, is de auto-industrie. Lichtgewicht voertuigen van magnesium zouden het brandstofverbruik sterk kunnen terugdringen. Experimenten met magnesium in de auto-industrie worden al gedaan sinds de ontwikkeling van de "Auto van de Toekomst" van 1952, de Buick LeSabre, maar vooralsnog blijft het toepassen van magnesium grotendeels voorbehouden aan onderdelen in voertuigen die op kleine schaal geproduceerd worden. De technieken voor het vervormen van magnesium en de bescherming van het materiaal die nodig is tegen corrosie, maken het metaal vooralsnog namelijk erg kostbaar.
Verder is het aantal bestaande legeringen nog niet zo uitgebreid en wordt er nog actief onderzoek verricht naar de verschillende vormingstechnieken van magnesium. Doordat de kennis over magnesium nog niet zo groot is, is men in de industrie nog geneigd tot het uitwijken naar de conventionele materialen zoals plastic en aluminium. Onderzoek wordt onder meer gedaan naar de extrusie van magnesium tot profielen in constructies en naar het spuitgieten van magnesium in thixotrope toestand (thixomoldingproces). Magnesium is thixotropisch (stroperig) bij 100 graden onder het smeltpunt.
Opmerkelijke eigenschappen
Het metaal is nog een derde lichter dan aluminium en het is onontbeerlijk voor lichtgewicht legeringen met dit metaal. Deze legeringen worden veel gebruikt in de luchtvaartindustrie.
Magnesiumchemie wordt gekenmerkt door de elektronenconfiguratie [Ne]3s2: Het is een aardalkalimetaal. Het atoom raakt bij voorkeur de twee buitenste elektronen kwijt om het tweewaardige ion Mg2+ te vormen. Het metaal is een sterke reductor, hoewel minder sterk dan de alkalimetalen. In tegenstelling tot het beryllium ion Be2+ is Mg2+ niet bijzonder klein en het heeft dan ook niet de bijzonder giftige eigenschappen van zijn buurelement.
Magnesiumverbindingen spelen een beduidende biologische rol. Zo bevat chlorofyl, het pigment dat in planten zorgdraagt voor de fotosynthese, een magnesiumatoom. Het hydroxide Mg(OH)2 is een matige base en wordt wel als middel tegen maagzuur gebruikt. Het magnesiumsulfaat MgSO4.7H2O (bitterzout) werd als laxeermiddel en wordt bij toxicose (zwangerschapsvergiftiging) gebruikt. Bitterzout wordt ook gebruikt om te voorkomen dat naaldgewassen bruine naalden krijgen. Kieseriet is een magnesiumhoudende kunstmeststof en wordt gebruikt in de tuinbouw.
Normaal bevat een mol magnesiumsulfaat 7 mol kristalwater. Anhydrisch magnesiumsulfaat (dus zonder kristalwater) is sterk hygroscopisch en wordt in de organische chemie als droogmiddel gebruikt.
Verschijning
Magnesium komt niet in zijn vrije vorm in de natuur voor, daar is het metaal veel te onedel voor. Het is het achtste element naar voorkomen op aarde. Er zijn wijdverspreide afzettingen van dolomiet en magnesiet waarin het in grote hoeveelheden voorkomt. Het wordt meestal gewonnen uit pekel die uit ondergrondse zoutlagen gehaald wordt. Het metaal kan daaruit door elektrolyse van het chloride MgCl2 gewonnen worden.
Het vormt een dun oxidehuidje aan de lucht, maar het metaal kan gemakkelijk tot ontbranden gebracht worden en brandt dan verwoed met een helle witte vlam. Een magnesiumbrand kan niet met water geblust worden doordat het metaal ook met water reageren kan via:
Mg + 2H2O -> Mg2+ + 2OH- + H2
Het gevormde waterstof is ook brandbaar. Magnesiumbranden kunnen alleen met zand worden geblust.
Isotopen
Magnesium komt in de natuur als drie stabiele isotopen voor. 24Mg maakt daarvan ongeveer 80% uit. De overige 20 procent bestaat uit 25Mg en 26Mg. 26Mg is een vervalproduct van aluminium-26 dat een halveringstijd van 717 duizend jaar heeft. 26Mg wordt regelmatig in grote hoeveelheden aangetroffen in sommige meteorieten. Uit de verhouding tussen aluminium-26 en magnesium-26 kan dus bepaald worden hoe oud een meteoriet is.
Toxicologie en veiligheid
</font>
Magnesium en zijn legeringen oxideren snel. Zowel fijn verdeeld als in gesmolten toestand kan het zeer heftig reageren met water en is het materiaal uiterst brandbaar. Bij verbranding ontstaat een opmerkelijk fel wit licht dat schade aan de ogen kan toebrengen. Om tijdens de verwerking van vloeibaar magnesium te voorkomen dat er een reactie met zuurstof optreedt, wordt er een beschermend gas gebruikt om het materiaal mee af te dekken. Fijn verdeeld magnesium wordt gedroogd en samengedrukt opgeslagen om zelfontbranding of een explosie te voorkomen. Deze eigenschappen van magnesium zorgen er overigens wel voor dat het geschikt is voor gebruik in de pyrotechniek.
Klinische betekenis
Magnesium is een mineraal dat aanwezig is in iedere cel van het lichaam. Het is onmisbaar voor de energieproductie, de werking van spieren en zenuwen en voor het behoud van de stevigheid van botten. Magnesium speelt een belangrijke rol bij de werking van enzymen in het lichaam en is betrokken bij de aanmaak van hormonen. Ongeveer de helft van de magnesiumvoorraad in het lichaam bevindt zich (in combinatie met calcium en fosfaat) in het bot.
Voeding is de bron van magnesium. Het is aanwezig in vele voedingsmiddelen, met name in noten, granen, groene groenten zoals postelein, spinazie en erwten. De hoeveelheid magnesium in bloed, cellen en bot wordt door het lichaam constant gehouden. De regulatie gebeurt door aanpassing van opname (via de darmen) en uitscheiding (met de urine, via de nieren).
De dokter kan een bepaling van magnesium aanvragen als de patiënt aanhoudend een verlaagd calcium of kalium heeft, voor patiënten met symptomen die passen bij een verlaagd magnesium, onderzoek naar malabsorptie, ondervoeding, diarree of alcoholmisbruik, bij gebruik van sommige geneesmiddelen die de uitscheiding van magnesium door de nieren bevorderen of ter controle van de nierfunctie.
Afhankelijk van de voeding kan een magnesiumtekort ontstaan. Dit is bijvoorbeeld het geval in veel voedselproducten in de Westerse maatschappij. Door het voortdurend gebruik van eenzijdige kunstmest kan de grond zo arm worden aan magnesium, dat er een tekort in onze voeding kan ontstaan. De symptomen van een tekort zijn algehele lusteloosheid of vermoeidheid. Bij een langdurig tekort aan magnesium treden klachten op als irritatie van de zenuwen in de spieren, hartritmestoornissen en maagkrampen. Een lage concentratie magnesium (hypomagnesiëmie) kan het gevolg zijn van:
onvoldoende inname van magnesium via de voeding, vooral bij ouderen, mensen met ondervoeding en bij alcoholmisbruik
onvoldoende opname van magnesium via de darmen bijvoorbeeld als gevolg van de ziekte van Crohn (ontstekingen in het slijmvlies van de darmen)
te hoge uitscheiding van magnesium via de nieren
te hoge of te lage hoeveelheden glucose (suiker) in het bloed (ongecontroleerde diabetes)
verminderde activiteit van de bijschildklier (hypoparathyreoïdie)
langdurig gebruik van plaspillen (diuretica)
langdurige diarree
na een chirurgische ingreep
bij ernstige brandwonden
bij zwangerschapsvergiftiging
Bij een teveel aan magnesium ontstaat lichte diarree. Een verhoogde concentratie magnesium is zelden aan de voeding te wijten. Meestal is een verhoogd magnesium het resultaat van problemen bij uitscheiding of van kunstmatige toediening. Een verhoogd magnesium kan worden gevonden bij:
nierfalen
overactieve bijschildklier (hyperparathyreoïdie)
slecht werkende schildklier
uitdroging
verzuring (te lage pH) van het bloed bij diabetes (diabetische acidose)
ziekte van Addison
gebruik van magnesium bevattende laxeermiddelen
Externe links
Elementenlijst
Mineraal en sporenelement | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,264 |
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<language type="sat" draft="contributed">santali</language>
<language type="sc" draft="contributed">sard</language>
<language type="scn" draft="contributed">sicilian</language>
<language type="sco" draft="contributed">scot</language>
<language type="sd" draft="contributed">sindhi</language>
<language type="se" draft="contributed">sami dal nord</language>
<language type="sel" draft="contributed">selkup</language>
<language type="sem" draft="contributed">lingua semitica</language>
<language type="sg" draft="contributed">sango</language>
<language type="sga" draft="contributed">irlandais vegl</language>
<language type="sgn" draft="contributed">lingua da segns</language>
<language type="sh" draft="contributed">serbo-croat</language>
<language type="shn" draft="contributed">shan</language>
<language type="si" draft="contributed">singalais</language>
<language type="sid" draft="contributed">sidamo</language>
<language type="sio" draft="contributed">linguas sioux</language>
<language type="sit" draft="contributed">linguas sino-tibetanas</language>
<language type="sk" draft="contributed">slovac</language>
<language type="sl" draft="contributed">sloven</language>
<language type="sla" draft="contributed">lingua slava</language>
<language type="sm" draft="contributed">samoan</language>
<language type="sma" draft="contributed">sami dal sid</language>
<language type="smi" draft="contributed">lingua sami</language>
<language type="smj" draft="contributed">sami lule</language>
<language type="smn" draft="contributed">sami inari</language>
<language type="sms" draft="contributed">sami skolt</language>
<language type="sn" draft="contributed">shona</language>
<language type="snk" draft="contributed">soninke</language>
<language type="so" draft="contributed">somali</language>
<language type="sog" draft="contributed">sogdian</language>
<language type="son" draft="contributed">songhai</language>
<language type="sq" draft="contributed">albanais</language>
<language type="sr" draft="contributed">serb</language>
<language type="srn" draft="contributed">sranan tongo</language>
<language type="srr" draft="contributed">serer</language>
<language type="ss" draft="contributed">swazi</language>
<language type="ssa" draft="contributed">lingua nilo-sahariana</language>
<language type="st" draft="contributed">sotho dal sid</language>
<language type="su" draft="contributed">sundanais</language>
<language type="suk" draft="contributed">sukuma</language>
<language type="sus" draft="contributed">susu</language>
<language type="sux" draft="contributed">sumeric</language>
<language type="sv" draft="contributed">svedais</language>
<language type="sw" draft="contributed">suahili</language>
<language type="syc" draft="contributed">siric classic</language>
<language type="syr" draft="contributed">siric</language>
<language type="ta" draft="contributed">tamil</language>
<language type="tai" draft="contributed">lingua tai</language>
<language type="te" draft="contributed">telugu</language>
<language type="tem" draft="contributed">temne</language>
<language type="ter" draft="contributed">tereno</language>
<language type="tet" draft="contributed">tetum</language>
<language type="tg" draft="contributed">tadjik</language>
<language type="th" draft="contributed">tailandais</language>
<language type="ti" draft="contributed">tigrinya</language>
<language type="tig" draft="contributed">tigre</language>
<language type="tiv" draft="contributed">tiv</language>
<language type="tk" draft="contributed">turkmen</language>
<language type="tkl" draft="contributed">tokelau</language>
<language type="tl" draft="contributed">tagalog</language>
<language type="tlh" draft="contributed">klingonic</language>
<language type="tli" draft="contributed">tlingit</language>
<language type="tmh" draft="contributed">tamasheq</language>
<language type="tn" draft="contributed">tswana</language>
<language type="to" draft="contributed">tonga</language>
<language type="tog" draft="contributed">lingua tsonga</language>
<language type="tpi" draft="contributed">tok pisin</language>
<language type="tr" draft="contributed">tirc</language>
<language type="ts" draft="contributed">tsonga</language>
<language type="tsi" draft="contributed">tsimshian</language>
<language type="tt" draft="contributed">tatar</language>
<language type="tum" draft="contributed">tumbuka</language>
<language type="tup" draft="contributed">linguas tupi</language>
<language type="tut" draft="contributed">linguas altaicas</language>
<language type="tvl" draft="contributed">tuvalu</language>
<language type="tw" draft="contributed">twi</language>
<language type="ty" draft="contributed">tahitian</language>
<language type="tyv" draft="contributed">tuvinian</language>
<language type="udm" draft="contributed">udmurt</language>
<language type="ug" draft="contributed">uiguric</language>
<language type="uga" draft="contributed">ugaritic</language>
<language type="uk" draft="contributed">ucranais</language>
<language type="umb" draft="contributed">mbundu</language>
<language type="und" draft="contributed">linguas betg determinadas</language>
<language type="ur" draft="contributed">urdu</language>
<language type="uz" draft="contributed">usbec</language>
<language type="vai" draft="contributed">vai</language>
<language type="ve" draft="contributed">venda</language>
<language type="vi" draft="contributed">vietnamais</language>
<language type="vo" draft="contributed">volapuk</language>
<language type="vot" draft="contributed">votic</language>
<language type="wa" draft="contributed">vallon</language>
<language type="wak" draft="contributed">linguas wakash</language>
<language type="wal" draft="contributed">walamo</language>
<language type="war" draft="contributed">waray</language>
<language type="was" draft="contributed">washo</language>
<language type="wen" draft="contributed">sorb</language>
<language type="wo" draft="contributed">wolof</language>
<language type="xal" draft="contributed">kalmuk</language>
<language type="xh" draft="contributed">xhosa</language>
<language type="yao" draft="contributed">yao</language>
<language type="yap" draft="contributed">yapais</language>
<language type="yi" draft="contributed">jiddic</language>
<language type="yo" draft="contributed">yoruba</language>
<language type="ypk" draft="contributed">linguas yupik</language>
<language type="za" draft="contributed">zhuang</language>
<language type="zap" draft="contributed">zapotec</language>
<language type="zbl" draft="contributed">simbols da Bliss</language>
<language type="zen" draft="contributed">zenaga</language>
<language type="zh" draft="contributed">chinais</language>
<language type="zh_Hans" draft="contributed">chinais simplifitgà</language>
<language type="zh_Hant" draft="contributed">chinais tradiziunal</language>
<language type="znd" draft="contributed">zande</language>
<language type="zu" draft="contributed">zulu</language>
<language type="zun" draft="contributed">zuni</language>
<language type="zxx" draft="contributed">nagins cuntegns linguistics</language>
<language type="zza" draft="contributed">zaza</language>
</languages>
<scripts>
<script type="Arab" draft="contributed">arab</script>
<script type="Armi" draft="contributed">arameic imperial</script>
<script type="Armn" draft="contributed">armen</script>
<script type="Avst" draft="contributed">avestic</script>
<script type="Bali" draft="contributed">balinais</script>
<script type="Batk" draft="contributed">batak</script>
<script type="Beng" draft="contributed">bengal</script>
<script type="Blis" draft="contributed">simbols da Bliss</script>
<script type="Bopo" draft="contributed">bopomofo</script>
<script type="Brah" draft="contributed">brahmi</script>
<script type="Brai" draft="contributed">scrittira da Braille</script>
<script type="Bugi" draft="contributed">buginais</script>
<script type="Buhd" draft="contributed">buhid</script>
<script type="Cakm" draft="contributed">chakma</script>
<script type="Cans" draft="contributed">simbols autoctons canadais unifitgads</script>
<script type="Cari" draft="contributed">carian</script>
<script type="Cham" draft="contributed">cham</script>
<script type="Cher" draft="contributed">cherokee</script>
<script type="Cirt" draft="contributed">cirth</script>
<script type="Copt" draft="contributed">coptic</script>
<script type="Cprt" draft="contributed">cipriot</script>
<script type="Cyrl" draft="contributed">cirillic</script>
<script type="Cyrs" draft="contributed">slav da baselgia vegl</script>
<script type="Deva" draft="contributed">devanagari</script>
<script type="Dsrt" draft="contributed">deseret</script>
<script type="Egyd" draft="contributed">egipzian demotic</script>
<script type="Egyh" draft="contributed">egipzian ieratic</script>
<script type="Egyp" draft="contributed">ieroglifas egipzianas</script>
<script type="Ethi" draft="contributed">etiopic</script>
<script type="Geok" draft="contributed">kutsuri</script>
<script type="Geor" draft="contributed">georgian</script>
<script type="Glag" draft="contributed">glagolitic</script>
<script type="Goth" draft="contributed">gotic</script>
<script type="Grek" draft="contributed">grec</script>
<script type="Gujr" draft="contributed">gujarati</script>
<script type="Guru" draft="contributed">gurmukhi</script>
<script type="Hang" draft="contributed">hangul</script>
<script type="Hani" draft="contributed">han</script>
<script type="Hano" draft="contributed">hanunoo</script>
<script type="Hans" draft="contributed">scrittira chinaisa simplifitgada</script>
<script type="Hant" draft="contributed">scrittira chinaisa tradiziunala</script>
<script type="Hebr" draft="contributed">ebraic</script>
<script type="Hira" draft="contributed">hiragana</script>
<script type="Hmng" draft="contributed">pahawn hmong</script>
<script type="Hrkt" draft="contributed">katanaka u hiragana</script>
<script type="Hung" draft="contributed">ungarais vegl</script>
<script type="Inds" draft="contributed">indus</script>
<script type="Ital" draft="contributed">italic vegl</script>
<script type="Java" draft="contributed">javanais</script>
<script type="Jpan" draft="contributed">giapunais</script>
<script type="Kali" draft="contributed">kayah li</script>
<script type="Kana" draft="contributed">katakana</script>
<script type="Khar" draft="contributed">kharoshthi</script>
<script type="Khmr" draft="contributed">khmer/cambodschan</script>
<script type="Knda" draft="contributed">kannada</script>
<script type="Kore" draft="contributed">corean</script>
<script type="Kthi" draft="contributed">kaithi</script>
<script type="Lana" draft="contributed">lanna</script>
<script type="Laoo" draft="contributed">laot</script>
<script type="Latf" draft="contributed">latin (scrittira gotica)</script>
<script type="Latg" draft="contributed">latin (scrittira gaelica)</script>
<script type="Latn" draft="contributed">latin</script>
<script type="Lepc" draft="contributed">lepcha</script>
<script type="Limb" draft="contributed">limbu</script>
<script type="Lina" draft="contributed">linear A</script>
<script type="Linb" draft="contributed">linear B</script>
<script type="Lyci" draft="contributed">lichic</script>
<script type="Lydi" draft="contributed">lidic</script>
<script type="Mand" draft="contributed">mandaic</script>
<script type="Mani" draft="contributed">manicheic</script>
<script type="Maya" draft="contributed">ieroglifas maya</script>
<script type="Mero" draft="contributed">meroitic</script>
<script type="Mlym" draft="contributed">malaisian</script>
<script type="Mong" draft="contributed">mongolic</script>
<script type="Moon" draft="contributed">moon</script>
<script type="Mtei" draft="contributed">meetei mayek</script>
<script type="Mymr" draft="contributed">burmais</script>
<script type="Nkoo" draft="contributed">n'ko</script>
<script type="Ogam" draft="contributed">ogham</script>
<script type="Olck" draft="contributed">ol chiki</script>
<script type="Orkh" draft="contributed">orkhon</script>
<script type="Orya" draft="contributed">oriya</script>
<script type="Osma" draft="contributed">osman</script>
<script type="Perm" draft="contributed">permic vegl</script>
<script type="Phag" draft="contributed">phags-pa</script>
<script type="Phli" draft="contributed">pahlavi dad inscripziuns</script>
<script type="Phlp" draft="contributed">pahlavi da psalms</script>
<script type="Phlv" draft="contributed">pahlavi da cudeschs</script>
<script type="Phnx" draft="contributed">fenizian</script>
<script type="Plrd" draft="contributed">fonetica da Pollard</script>
<script type="Prti" draft="contributed">partic dad inscripziuns</script>
<script type="Qaai" draft="contributed">ertà</script>
<script type="Rjng" draft="contributed">rejang</script>
<script type="Roro" draft="contributed">rongorongo</script>
<script type="Runr" draft="contributed">runic</script>
<script type="Samr" draft="contributed">samaritan</script>
<script type="Sara" draft="contributed">sarati</script>
<script type="Saur" draft="contributed">saurashtra</script>
<script type="Sgnw" draft="contributed">lingua da segns</script>
<script type="Shaw" draft="contributed">shavian</script>
<script type="Sinh" draft="contributed">singalais</script>
<script type="Sund" draft="contributed">sundanais</script>
<script type="Sylo" draft="contributed">syloti nagri</script>
<script type="Syrc" draft="contributed">siric</script>
<script type="Syre" draft="contributed">siric estrangelo</script>
<script type="Syrj" draft="contributed">siric dal vest</script>
<script type="Syrn" draft="contributed">siric da l'ost</script>
<script type="Tagb" draft="contributed">tagbanwa</script>
<script type="Tale" draft="contributed">tai le</script>
<script type="Talu" draft="contributed">tai lue</script>
<script type="Taml" draft="contributed">tamil</script>
<script type="Tavt" draft="contributed">tai viet</script>
<script type="Telu" draft="contributed">telugu</script>
<script type="Teng" draft="contributed">tengwar</script>
<script type="Tfng" draft="contributed">tifinagh</script>
<script type="Tglg" draft="contributed">tagalog</script>
<script type="Thaa" draft="contributed">thaana</script>
<script type="Thai" draft="contributed">tailandais</script>
<script type="Tibt" draft="contributed">tibetan</script>
<script type="Ugar" draft="contributed">ugaritic</script>
<script type="Vaii" draft="contributed">vaii</script>
<script type="Visp" draft="contributed">alfabet visibel</script>
<script type="Xpeo" draft="contributed">persian vegl</script>
<script type="Xsux" draft="contributed">scrittira a cugn sumeric-accadica</script>
<script type="Yiii" draft="contributed">yi</script>
<script type="Zmth" draft="contributed">notaziun matematica</script>
<script type="Zsym" draft="contributed">simbols</script>
<script type="Zxxx" draft="contributed">linguas na scrittas</script>
<script type="Zyyy" draft="contributed">betg determinà</script>
<script type="Zzzz" draft="contributed">scrittira nunenconuschenta u nunvalaivla</script>
</scripts>
<territories>
<territory type="001" draft="contributed">mund</territory>
<territory type="002" draft="contributed">Africa</territory>
<territory type="003" draft="contributed">America dal Nord</territory>
<territory type="005" draft="contributed">America dal Sid</territory>
<territory type="009" draft="contributed">Oceania</territory>
<territory type="011" draft="contributed">Africa dal Vest</territory>
<territory type="013" draft="contributed">America Centrala</territory>
<territory type="014" draft="contributed">Africa da l'Ost</territory>
<territory type="015" draft="contributed">Africa dal Nord</territory>
<territory type="017" draft="contributed">Africa Centrala</territory>
<territory type="018" draft="contributed">Africa Meridiunala</territory>
<territory type="019" draft="contributed">America dal Nord, America Centrala ed America dal Sid</territory>
<territory type="029" draft="contributed">Caribica</territory>
<territory type="030" draft="contributed">Asia da l'Ost</territory>
<territory type="034" draft="contributed">Asia dal Sid</territory>
<territory type="035" draft="contributed">Asia dal Sidost</territory>
<territory type="039" draft="contributed">Europa dal Sid</territory>
<territory type="053" draft="contributed">Australia e Nova Zelanda</territory>
<territory type="054" draft="contributed">Melanesia</territory>
<territory type="057" draft="contributed">Regiun Micronesica</territory>
<territory type="061" draft="contributed">Polinesia</territory>
<territory type="062" draft="contributed">Asia Centrala dal Sid</territory>
<territory type="142" draft="contributed">Asia</territory>
<territory type="143" draft="contributed">Asia Centrala</territory>
<territory type="145" draft="contributed">Asia dal Vest</territory>
<territory type="150" draft="contributed">Europa</territory>
<territory type="151" draft="contributed">Europa Orientala</territory>
<territory type="154" draft="contributed">Europa dal Nord</territory>
<territory type="155" draft="contributed">Europa dal Vest</territory>
<territory type="172" draft="contributed">Communitad dals Stadis Independents</territory>
<territory type="200" draft="contributed">Tschecoslovachia</territory>
<territory type="419" draft="contributed">America Latina e Caribica</territory>
<territory type="830" draft="contributed">Inslas dal Chanal da la Mongia</territory>
<territory type="AD" draft="contributed">Andorra</territory>
<territory type="AE" draft="contributed">Emirats Arabs Unids</territory>
<territory type="AF" draft="contributed">Afghanistan</territory>
<territory type="AG" draft="contributed">Antigua e Barbuda</territory>
<territory type="AI" draft="contributed">Anguilla</territory>
<territory type="AL" draft="contributed">Albania</territory>
<territory type="AM" draft="contributed">Armenia</territory>
<territory type="AN" draft="contributed">Antillas Ollandaisas</territory>
<territory type="AO" draft="contributed">Angola</territory>
<territory type="AQ" draft="contributed">Antarctica</territory>
<territory type="AR" draft="contributed">Argentinia</territory>
<territory type="AS" draft="contributed">Samoa Americana</territory>
<territory type="AT" draft="contributed">Austria</territory>
<territory type="AU" draft="contributed">Australia</territory>
<territory type="AW" draft="contributed">Aruba</territory>
<territory type="AX" draft="contributed">Inslas Aland</territory>
<territory type="AZ" draft="contributed">Aserbaidschan</territory>
<territory type="BA" draft="contributed">Bosnia ed Erzegovina</territory>
<territory type="BB" draft="contributed">Barbados</territory>
<territory type="BD" draft="contributed">Bangladesch</territory>
<territory type="BE" draft="contributed">Belgia</territory>
<territory type="BF" draft="contributed">Burkina Faso</territory>
<territory type="BG" draft="contributed">Bulgaria</territory>
<territory type="BH" draft="contributed">Bahrain</territory>
<territory type="BI" draft="contributed">Burundi</territory>
<territory type="BJ" draft="contributed">Benin</territory>
<territory type="BL" draft="contributed">Son Barthélemy</territory>
<territory type="BM" draft="contributed">Bermudas</territory>
<territory type="BN" draft="contributed">Brunei</territory>
<territory type="BO" draft="contributed">Bolivia</territory>
<territory type="BQ" draft="contributed">Territori Antarctic Britannic</territory>
<territory type="BR" draft="contributed">Brasila</territory>
<territory type="BS" draft="contributed">Bahamas</territory>
<territory type="BT" draft="contributed">Bhutan</territory>
<territory type="BV" draft="contributed">Insla Bouvet</territory>
<territory type="BW" draft="contributed">Botswana</territory>
<territory type="BY" draft="contributed">Bielorussia</territory>
<territory type="BZ" draft="contributed">Belize</territory>
<territory type="CA" draft="contributed">Canada</territory>
<territory type="CC" draft="contributed">Inslas Cocos</territory>
<territory type="CD" draft="contributed">Republica Democratica dal Congo</territory>
<territory type="CF" draft="contributed">Republica Centralafricana</territory>
<territory type="CG" draft="contributed">Congo</territory>
<territory type="CH" draft="contributed">Svizra</territory>
<territory type="CI" draft="contributed">Costa d'Ivur</territory>
<territory type="CK" draft="contributed">Inslas Cook</territory>
<territory type="CL" draft="contributed">Chile</territory>
<territory type="CM" draft="contributed">Camerun</territory>
<territory type="CN" draft="contributed">China</territory>
<territory type="CO" draft="contributed">Columbia</territory>
<territory type="CR" draft="contributed">Costa Rica</territory>
<territory type="CS" draft="contributed">Serbia e Montenegro</territory>
<territory type="CT" draft="contributed">Inslas Canton ed Enderbury</territory>
<territory type="CU" draft="contributed">Cuba</territory>
<territory type="CV" draft="contributed">Cap Verd</territory>
<territory type="CX" draft="contributed">Insla da Christmas</territory>
<territory type="CY" draft="contributed">Cipra</territory>
<territory type="CZ" draft="contributed">Republica Tscheca</territory>
<territory type="DD" draft="contributed">Germania da l'Ost</territory>
<territory type="DE" draft="contributed">Germania</territory>
<territory type="DJ" draft="contributed">Dschibuti</territory>
<territory type="DK" draft="contributed">Danemarc</territory>
<territory type="DM" draft="contributed">Dominica</territory>
<territory type="DO" draft="contributed">Republica Dominicana</territory>
<territory type="DZ" draft="contributed">Algeria</territory>
<territory type="EC" draft="contributed">Ecuador</territory>
<territory type="EE" draft="contributed">Estonia</territory>
<territory type="EG" draft="contributed">Egipta</territory>
<territory type="EH" draft="contributed">Sahara Occidentala</territory>
<territory type="ER" draft="contributed">Eritrea</territory>
<territory type="ES" draft="contributed">Spagna</territory>
<territory type="ET" draft="contributed">Etiopia</territory>
<territory type="EU" draft="contributed">Uniun europeica</territory>
<territory type="FI" draft="contributed">Finlanda</territory>
<territory type="FJ" draft="contributed">Fidschi</territory>
<territory type="FK" draft="contributed">Inslas dal Falkland</territory>
<territory type="FM" draft="contributed">Micronesia</territory>
<territory type="FO" draft="contributed">Inslas Feroe</territory>
<territory type="FQ" draft="contributed">Territoris Meridiunals ed Antarctics Franzos</territory>
<territory type="FR" draft="contributed">Frantscha</territory>
<territory type="FX" draft="contributed">Frantscha Metropolitana</territory>
<territory type="GA" draft="contributed">Gabun</territory>
<territory type="GB" draft="contributed">Reginavel Unì</territory>
<territory type="GD" draft="contributed">Grenada</territory>
<territory type="GE" draft="contributed">Georgia</territory>
<territory type="GF" draft="contributed">Guyana Franzosa</territory>
<territory type="GG" draft="contributed">Guernsey</territory>
<territory type="GH" draft="contributed">Ghana</territory>
<territory type="GI" draft="contributed">Gibraltar</territory>
<territory type="GL" draft="contributed">Grönlanda</territory>
<territory type="GM" draft="contributed">Gambia</territory>
<territory type="GN" draft="contributed">Guinea</territory>
<territory type="GP" draft="contributed">Guadeloupe</territory>
<territory type="GQ" draft="contributed">Guinea Equatoriala</territory>
<territory type="GR" draft="contributed">Grezia</territory>
<territory type="GS" draft="contributed">Georgia dal Sid e las Inslas Sandwich dal Sid</territory>
<territory type="GT" draft="contributed">Guatemala</territory>
<territory type="GU" draft="contributed">Guam</territory>
<territory type="GW" draft="contributed">Guinea-Bissau</territory>
<territory type="GY" draft="contributed">Guyana</territory>
<territory type="HK" draft="contributed">Regiun d'administraziun speziala da Hongkong, China</territory>
<territory type="HM" draft="contributed">Inslas da Heard e da McDonlad</territory>
<territory type="HN" draft="contributed">Honduras</territory>
<territory type="HR" draft="contributed">Croazia</territory>
<territory type="HT" draft="contributed">Haiti</territory>
<territory type="HU" draft="contributed">Ungaria</territory>
<territory type="ID" draft="contributed">Indonesia</territory>
<territory type="IE" draft="contributed">Irlanda</territory>
<territory type="IL" draft="contributed">Israel</territory>
<territory type="IM" draft="contributed">Insla da Man</territory>
<territory type="IN" draft="contributed">India</territory>
<territory type="IO" draft="contributed">Territori Britannic en l'Ocean Indic</territory>
<territory type="IQ" draft="contributed">Irac</territory>
<territory type="IR" draft="contributed">Iran</territory>
<territory type="IS" draft="contributed">Islanda</territory>
<territory type="IT" draft="contributed">Italia</territory>
<territory type="JE" draft="contributed">Jersey</territory>
<territory type="JM" draft="contributed">Giamaica</territory>
<territory type="JO" draft="contributed">Jordania</territory>
<territory type="JP" draft="contributed">Giapun</territory>
<territory type="JT" draft="contributed">Atoll Johnston</territory>
<territory type="KE" draft="contributed">Kenia</territory>
<territory type="KG" draft="contributed">Kirghisistan</territory>
<territory type="KH" draft="contributed">Cambodscha</territory>
<territory type="KI" draft="contributed">Kiribati</territory>
<territory type="KM" draft="contributed">Comoras</territory>
<territory type="KN" draft="contributed">Saint Kitts e Nevis</territory>
<territory type="KP" draft="contributed">Corea dal Nord</territory>
<territory type="KR" draft="contributed">Corea dal Sid</territory>
<territory type="KW" draft="contributed">Kuwait</territory>
<territory type="KY" draft="contributed">Inslas Cayman</territory>
<territory type="KZ" draft="contributed">Kasachstan</territory>
<territory type="LA" draft="contributed">Laos</territory>
<territory type="LB" draft="contributed">Libanon</territory>
<territory type="LC" draft="contributed">Saint Lucia</territory>
<territory type="LI" draft="contributed">Liechtenstein</territory>
<territory type="LK" draft="contributed">Sri Lanka</territory>
<territory type="LR" draft="contributed">Liberia</territory>
<territory type="LS" draft="contributed">Lesotho</territory>
<territory type="LT" draft="contributed">Lituania</territory>
<territory type="LU" draft="contributed">Luxemburg</territory>
<territory type="LV" draft="contributed">Lettonia</territory>
<territory type="LY" draft="contributed">Libia</territory>
<territory type="MA" draft="contributed">Maroc</territory>
<territory type="MC" draft="contributed">Monaco</territory>
<territory type="MD" draft="contributed">Moldavia</territory>
<territory type="ME" draft="contributed">Montenegro</territory>
<territory type="MF" draft="contributed">Saint Martin</territory>
<territory type="MG" draft="contributed">Madagascar</territory>
<territory type="MH" draft="contributed">Inslas da Marshall</territory>
<territory type="MI" draft="contributed">Inslas da Midway</territory>
<territory type="MK" draft="contributed">Macedonia</territory>
<territory type="ML" draft="contributed">Mali</territory>
<territory type="MM" draft="contributed">Myanmar</territory>
<territory type="MN" draft="contributed">Mongolia</territory>
<territory type="MO" draft="contributed">Regiun d'administraziun speziala Macao, China</territory>
<territory type="MP" draft="contributed">Inslas Mariannas dal Nord</territory>
<territory type="MQ" draft="contributed">Martinique</territory>
<territory type="MR" draft="contributed">Mauretania</territory>
<territory type="MS" draft="contributed">Montserrat</territory>
<territory type="MT" draft="contributed">Malta</territory>
<territory type="MU" draft="contributed">Mauritius</territory>
<territory type="MV" draft="contributed">Maldivas</territory>
<territory type="MW" draft="contributed">Malawi</territory>
<territory type="MX" draft="contributed">Mexico</territory>
<territory type="MY" draft="contributed">Malaisia</territory>
<territory type="MZ" draft="contributed">Mosambic</territory>
<territory type="NA" draft="contributed">Namibia</territory>
<territory type="NC" draft="contributed">Nova Caledonia</territory>
<territory type="NE" draft="contributed">Niger</territory>
<territory type="NF" draft="contributed">Insla Norfolk</territory>
<territory type="NG" draft="contributed">Nigeria</territory>
<territory type="NI" draft="contributed">Nicaragua</territory>
<territory type="NL" draft="contributed">Pajais Bass</territory>
<territory type="NO" draft="contributed">Norvegia</territory>
<territory type="NP" draft="contributed">Nepal</territory>
<territory type="NQ" draft="contributed">Terra da la Regina Maud</territory>
<territory type="NR" draft="contributed">Nauru</territory>
<territory type="NT" draft="contributed">Zona neutrala</territory>
<territory type="NU" draft="contributed">Niue</territory>
<territory type="NZ" draft="contributed">Nova Zelanda</territory>
<territory type="OM" draft="contributed">Oman</territory>
<territory type="PA" draft="contributed">Panama</territory>
<territory type="PC" draft="contributed">Territori fiduziar da las Inslas dal Pacific</territory>
<territory type="PE" draft="contributed">Peru</territory>
<territory type="PF" draft="contributed">Polinesia Franzosa</territory>
<territory type="PG" draft="contributed">Papua Nova Guinea</territory>
<territory type="PH" draft="contributed">Filippinas</territory>
<territory type="PK" draft="contributed">Pakistan</territory>
<territory type="PL" draft="contributed">Pologna</territory>
<territory type="PM" draft="contributed">Saint Pierre e Miquelon</territory>
<territory type="PN" draft="contributed">Pitcairn</territory>
<territory type="PR" draft="contributed">Puerto Rico</territory>
<territory type="PS" draft="contributed">Territori Palestinais</territory>
<territory type="PT" draft="contributed">Portugal</territory>
<territory type="PU" draft="contributed">Diversas inslas dals Stadis Unids da l'America en il Pacific</territory>
<territory type="PW" draft="contributed">Palau</territory>
<territory type="PY" draft="contributed">Paraguai</territory>
<territory type="PZ" draft="contributed">Zona dal Canal da Panama</territory>
<territory type="QA" draft="contributed">Katar</territory>
<territory type="QO" draft="contributed">Oceania Periferica</territory>
<territory type="RE" draft="contributed">Réunion</territory>
<territory type="RO" draft="contributed">Rumenia</territory>
<territory type="RS" draft="contributed">Serbia</territory>
<territory type="RU" draft="contributed">Russia</territory>
<territory type="RW" draft="contributed">Ruanda</territory>
<territory type="SA" draft="contributed">Arabia Saudita</territory>
<territory type="SB" draft="contributed">Salomonas</territory>
<territory type="SC" draft="contributed">Seychellas</territory>
<territory type="SD" draft="contributed">Sudan</territory>
<territory type="SE" draft="contributed">Svezia</territory>
<territory type="SG" draft="contributed">Singapur</territory>
<territory type="SH" draft="contributed">Sontg'Elena</territory>
<territory type="SI" draft="contributed">Slovenia</territory>
<territory type="SJ" draft="contributed">Svalbard e Jan Mayen</territory>
<territory type="SK" draft="contributed">Slovachia</territory>
<territory type="SL" draft="contributed">Sierra Leone</territory>
<territory type="SM" draft="contributed">San Marino</territory>
<territory type="SN" draft="contributed">Senegal</territory>
<territory type="SO" draft="contributed">Somalia</territory>
<territory type="SR" draft="contributed">Surinam</territory>
<territory type="ST" draft="contributed">São Tomé e Principe</territory>
<territory type="SU" draft="contributed">Uniun Sovietica</territory>
<territory type="SV" draft="contributed">El Salvador</territory>
<territory type="SY" draft="contributed">Siria</territory>
<territory type="SZ" draft="contributed">Swaziland</territory>
<territory type="TC" draft="contributed">Inslas Turks e Caicos</territory>
<territory type="TD" draft="contributed">Tschad</territory>
<territory type="TF" draft="contributed">Territoris Franzos Meridiunals</territory>
<territory type="TG" draft="contributed">Togo</territory>
<territory type="TH" draft="contributed">Tailanda</territory>
<territory type="TJ" draft="contributed">Tadschikistan</territory>
<territory type="TK" draft="contributed">Tokelau</territory>
<territory type="TL" draft="contributed">Timor da l'Ost</territory>
<territory type="TM" draft="contributed">Turkmenistan</territory>
<territory type="TN" draft="contributed">Tunesia</territory>
<territory type="TO" draft="contributed">Tonga</territory>
<territory type="TR" draft="contributed">Tirchia</territory>
<territory type="TT" draft="contributed">Trinidad e Tobago</territory>
<territory type="TV" draft="contributed">Tuvalu</territory>
<territory type="TW" draft="contributed">Taiwan</territory>
<territory type="TZ" draft="contributed">Tansania</territory>
<territory type="UA" draft="contributed">Ucraina</territory>
<territory type="UG" draft="contributed">Uganda</territory>
<territory type="UM" draft="contributed">Inslas pitschnas perifericas dals Stadis Unids da l'America</territory>
<territory type="US" draft="contributed">Stadis Unids da l'America</territory>
<territory type="UY" draft="contributed">Uruguay</territory>
<territory type="UZ" draft="contributed">Usbekistan</territory>
<territory type="VA" draft="contributed">Citad dal Vatican</territory>
<territory type="VC" draft="contributed">Saint Vincent e las Grenadinas</territory>
<territory type="VD" draft="contributed">Vietnam dal Nord</territory>
<territory type="VE" draft="contributed">Venezuela</territory>
<territory type="VG" draft="contributed">Inslas Verginas Britannicas</territory>
<territory type="VI" draft="contributed">Inslas Verginas Americanas</territory>
<territory type="VN" draft="contributed">Vietnam</territory>
<territory type="VU" draft="contributed">Vanuatu</territory>
<territory type="WF" draft="contributed">Wallis e Futuna</territory>
<territory type="WK" draft="contributed">Insla Wake</territory>
<territory type="WS" draft="contributed">Samoa</territory>
<territory type="YD" draft="contributed">Republica Democratica Populara da Jemen</territory>
<territory type="YE" draft="contributed">Jemen</territory>
<territory type="YT" draft="contributed">Mayotte</territory>
<territory type="ZA" draft="contributed">Africa dal Sid</territory>
<territory type="ZM" draft="contributed">Sambia</territory>
<territory type="ZW" draft="contributed">Simbabwe</territory>
<territory type="ZZ" draft="contributed">Regiun betg encouschenta u nunvalaivla</territory>
</territories>
<variants>
<variant type="1901" draft="contributed">ortografia tudestga tradiziunala</variant>
<variant type="1994" draft="contributed">ortografia standardisada da Resia</variant>
<variant type="1996" draft="contributed">nova ortografia tudestga</variant>
<variant type="1606NICT" draft="contributed">franzos mesaun tardiv (fin 1606)</variant>
<variant type="1694ACAD" draft="contributed">franzos modern tempriv (a partir da 1694)</variant>
<variant type="AREVELA" draft="contributed">armen oriental</variant>
<variant type="AREVMDA" draft="contributed">armen occidental</variant>
<variant type="BAKU1926" draft="contributed">alfabet tirc unifitgà</variant>
<variant type="BISKE" draft="contributed">dialect da San Giorgio</variant>
<variant type="BOONT" draft="contributed">dialect boontling</variant>
<variant type="FONIPA" draft="contributed">alfabet fonetic internaziunal (IPA)</variant>
<variant type="FONUPA" draft="contributed">alfabet fonetic da l'Ural (UPA)</variant>
<variant type="GAULISH" draft="contributed">gallic</variant>
<variant type="GUOYU" draft="contributed">mandarin/chinais da standard</variant>
<variant type="HAKKA" draft="contributed">hakka</variant>
<variant type="LIPAW" draft="contributed">dialect lipovaz da Resia</variant>
<variant type="LOJBAN" draft="contributed">lojban</variant>
<variant type="MONOTON" draft="contributed">monotonic</variant>
<variant type="NEDIS" draft="contributed">dialect da Natisone</variant>
<variant type="NJIVA" draft="contributed">dialect da Gniva</variant>
<variant type="OSOJS" draft="contributed">dialect da Oscacco</variant>
<variant type="POLYTON" draft="contributed">politonic</variant>
<variant type="POSIX" draft="contributed">computer</variant>
<variant type="REVISED" draft="contributed">ortografia revedida</variant>
<variant type="ROZAJ" draft="contributed">dialect da Resia</variant>
<variant type="SAAHO" draft="contributed">Saho</variant>
<variant type="SCOTLAND" draft="contributed">englais da standard scot</variant>
<variant type="SCOUSE" draft="contributed">dialect scouse</variant>
<variant type="SOLBA" draft="contributed">dialect da Stolvizza</variant>
<variant type="TARASK" draft="contributed">ortografia taraskievica</variant>
<variant type="VALENCIA" draft="contributed">valencian</variant>
<variant type="XIANG" draft="contributed">xiang/hunanais</variant>
</variants>
<keys>
<key type="calendar" draft="contributed">chalender</key>
<key type="collation" draft="contributed">zavrada</key>
<key type="currency" draft="contributed">munaida</key>
</keys>
<types>
<type type="big5han" key="collation" draft="contributed">chinaisa tradiziunala - Big5</type>
<type type="buddhist" key="calendar" draft="contributed">chalender budistic</type>
<type type="chinese" key="calendar" draft="contributed">chalender chinais</type>
<type type="direct" key="collation" draft="contributed">reglas directas</type>
<type type="gb2312han" key="collation" draft="contributed">chinaisa simplifitgada - GB2312</type>
<type type="gregorian" key="calendar" draft="contributed">chalender gregorian</type>
<type type="hebrew" key="calendar" draft="contributed">chalender ebraic</type>
<type type="indian" key="calendar" draft="contributed">chalender naziunal indic</type>
<type type="islamic" key="calendar" draft="contributed">chalender islamic</type>
<type type="islamic-civil" key="calendar" draft="contributed">chalender islamic civil</type>
<type type="japanese" key="calendar" draft="contributed">chalender giapunais</type>
<type type="phonebook" key="collation" draft="contributed">cudesch da telefon</type>
<type type="pinyin" key="collation" draft="contributed">Pinyin</type>
<type type="roc" key="calendar" draft="contributed">chalendar da la republica chinaisa</type>
<type type="stroke" key="collation" draft="contributed">urden dals stritgs</type>
<type type="traditional" key="collation" draft="contributed">reglas tradiziunalas</type>
</types>
<measurementSystemNames>
<measurementSystemName type="metric" draft="contributed">unitads metrics</measurementSystemName>
<measurementSystemName type="US" draft="contributed">unitads americans</measurementSystemName>
</measurementSystemNames>
<codePatterns>
<codePattern type="language" draft="contributed">Lingua: {0}</codePattern>
<codePattern type="script" draft="contributed">Scrittira: {0}</codePattern>
<codePattern type="territory" draft="contributed">Regiun: {0}</codePattern>
</codePatterns>
</localeDisplayNames>
<characters>
<exemplarCharacters draft="contributed">[a à b c d e é è f g h i ì j k l m n o ò p q r s t u ù v w x y z]</exemplarCharacters>
<exemplarCharacters type="auxiliary" draft="contributed">[á à ă â å ä ā æ ç é è ĕ ê ë ē í ì ĭ î ï ī ñ ó ò ŏ ô ö ø ō œ ú ù ŭ û ü ū ÿ]</exemplarCharacters>
<exemplarCharacters type="currencySymbol" draft="contributed">[a b c d e f g h i j k l m n o p q r s t u v w x y z]</exemplarCharacters>
<exemplarCharacters type="index" draft="contributed">[A À B C D E É È F G H I Ì J K L M N O Ò P Q R S T U Ù V W X Y Z]</exemplarCharacters>
</characters>
<delimiters>
<quotationStart draft="contributed">«</quotationStart>
<quotationEnd draft="contributed">»</quotationEnd>
<alternateQuotationStart draft="contributed">‹</alternateQuotationStart>
<alternateQuotationEnd draft="contributed">›</alternateQuotationEnd>
</delimiters>
<dates>
<dateRangePattern draft="contributed">{0}−{1}</dateRangePattern>
<calendars>
<calendar type="gregorian">
<months>
<monthContext type="format">
<monthWidth type="abbreviated">
<month type="1" draft="contributed">schan.</month>
<month type="2" draft="contributed">favr.</month>
<month type="3" draft="contributed">mars</month>
<month type="4" draft="contributed">avr.</month>
<month type="5" draft="contributed">matg</month>
<month type="6" draft="contributed">zercl.</month>
<month type="7" draft="contributed">fan.</month>
<month type="8" draft="contributed">avust</month>
<month type="9" draft="contributed">sett.</month>
<month type="10" draft="contributed">oct.</month>
<month type="11" draft="contributed">nov.</month>
<month type="12" draft="contributed">dec.</month>
</monthWidth>
<monthWidth type="wide">
<month type="1" draft="contributed">schaner</month>
<month type="2" draft="contributed">favrer</month>
<month type="3" draft="contributed">mars</month>
<month type="4" draft="contributed">avrigl</month>
<month type="5" draft="contributed">matg</month>
<month type="6" draft="contributed">zercladur</month>
<month type="7" draft="contributed">fanadur</month>
<month type="8" draft="contributed">avust</month>
<month type="9" draft="contributed">settember</month>
<month type="10" draft="contributed">october</month>
<month type="11" draft="contributed">november</month>
<month type="12" draft="contributed">december</month>
</monthWidth>
</monthContext>
<monthContext type="stand-alone">
<monthWidth type="narrow">
<month type="1" draft="contributed">S</month>
<month type="2" draft="contributed">F</month>
<month type="3" draft="contributed">M</month>
<month type="4" draft="contributed">A</month>
<month type="5" draft="contributed">M</month>
<month type="6" draft="contributed">Z</month>
<month type="7" draft="contributed">F</month>
<month type="8" draft="contributed">A</month>
<month type="9" draft="contributed">S</month>
<month type="10" draft="contributed">O</month>
<month type="11" draft="contributed">N</month>
<month type="12" draft="contributed">D</month>
</monthWidth>
</monthContext>
</months>
<days>
<dayContext type="format">
<dayWidth type="abbreviated">
<day type="sun" draft="contributed">du</day>
<day type="mon" draft="contributed">gli</day>
<day type="tue" draft="contributed">ma</day>
<day type="wed" draft="contributed">me</day>
<day type="thu" draft="contributed">gie</day>
<day type="fri" draft="contributed">ve</day>
<day type="sat" draft="contributed">so</day>
</dayWidth>
<dayWidth type="wide">
<day type="sun" draft="contributed">dumengia</day>
<day type="mon" draft="contributed">glindesdi</day>
<day type="tue" draft="contributed">mardi</day>
<day type="wed" draft="contributed">mesemna</day>
<day type="thu" draft="contributed">gievgia</day>
<day type="fri" draft="contributed">venderdi</day>
<day type="sat" draft="contributed">sonda</day>
</dayWidth>
</dayContext>
<dayContext type="stand-alone">
<dayWidth type="narrow">
<day type="sun" draft="contributed">D</day>
<day type="mon" draft="contributed">G</day>
<day type="tue" draft="contributed">M</day>
<day type="wed" draft="contributed">M</day>
<day type="thu" draft="contributed">G</day>
<day type="fri" draft="contributed">V</day>
<day type="sat" draft="contributed">S</day>
</dayWidth>
</dayContext>
</days>
<quarters>
<quarterContext type="format">
<quarterWidth type="wide">
<quarter type="1" draft="contributed">1. quartal</quarter>
<quarter type="2" draft="contributed">2. quartal</quarter>
<quarter type="3" draft="contributed">3. quartal</quarter>
<quarter type="4" draft="contributed">4. quartal</quarter>
</quarterWidth>
</quarterContext>
</quarters>
<dayPeriods>
<dayPeriodContext type="format">
<dayPeriodWidth type="wide">
<dayPeriod type="am">am</dayPeriod>
<dayPeriod type="pm">sm</dayPeriod>
</dayPeriodWidth>
</dayPeriodContext>
</dayPeriods>
<eras>
<eraNames>
<era type="0" draft="contributed">avant Cristus</era>
<era type="1" draft="contributed">suenter Cristus</era>
</eraNames>
<eraAbbr>
<era type="0" draft="contributed">av. Cr.</era>
<era type="1" draft="contributed">s. Cr.</era>
</eraAbbr>
</eras>
<dateFormats>
<dateFormatLength type="full">
<dateFormat>
<pattern draft="contributed">EEEE, d. MMMM y</pattern>
</dateFormat>
</dateFormatLength>
<dateFormatLength type="long">
<dateFormat>
<pattern draft="contributed">d. MMMM y</pattern>
</dateFormat>
</dateFormatLength>
<dateFormatLength type="medium">
<dateFormat>
<pattern draft="contributed">dd.MM.yyyy</pattern>
</dateFormat>
</dateFormatLength>
<dateFormatLength type="short">
<dateFormat>
<pattern draft="contributed">dd.MM.yy</pattern>
</dateFormat>
</dateFormatLength>
</dateFormats>
<timeFormats>
<timeFormatLength type="full">
<timeFormat>
<pattern draft="contributed">HH:mm:ss zzzz</pattern>
</timeFormat>
</timeFormatLength>
<timeFormatLength type="long">
<timeFormat>
<pattern draft="contributed">HH:mm:ss z</pattern>
</timeFormat>
</timeFormatLength>
<timeFormatLength type="medium">
<timeFormat>
<pattern draft="contributed">HH:mm:ss</pattern>
</timeFormat>
</timeFormatLength>
<timeFormatLength type="short">
<timeFormat>
<pattern draft="contributed">HH:mm</pattern>
</timeFormat>
</timeFormatLength>
</timeFormats>
<dateTimeFormats>
<availableFormats>
<dateFormatItem id="d" draft="contributed">d</dateFormatItem>
<dateFormatItem id="Ed" draft="contributed">E d.</dateFormatItem>
<dateFormatItem id="EEEd" draft="contributed">d. EEE</dateFormatItem>
<dateFormatItem id="H" draft="contributed">H</dateFormatItem>
<dateFormatItem id="HHmm" draft="contributed">HH:mm</dateFormatItem>
<dateFormatItem id="HHmmss" draft="contributed">HH:mm:ss</dateFormatItem>
<dateFormatItem id="Hm" draft="contributed">H:mm</dateFormatItem>
<dateFormatItem id="M" draft="contributed">L</dateFormatItem>
<dateFormatItem id="Md" draft="contributed">d.M.</dateFormatItem>
<dateFormatItem id="MEd" draft="contributed">E, d.M.</dateFormatItem>
<dateFormatItem id="MMd" draft="contributed">d.MM.</dateFormatItem>
<dateFormatItem id="MMdd" draft="contributed">dd.MM.</dateFormatItem>
<dateFormatItem id="MMM" draft="contributed">LLL</dateFormatItem>
<dateFormatItem id="MMMd" draft="contributed">d. MMM</dateFormatItem>
<dateFormatItem id="MMMEd" draft="contributed">E d. MMM</dateFormatItem>
<dateFormatItem id="MMMMd" draft="contributed">d. MMMM</dateFormatItem>
<dateFormatItem id="MMMMdd" draft="contributed">dd. MMMM</dateFormatItem>
<dateFormatItem id="MMMMEd" draft="contributed">E d. MMMM</dateFormatItem>
<dateFormatItem id="mmss" draft="contributed">mm:ss</dateFormatItem>
<dateFormatItem id="ms" draft="contributed">mm:ss</dateFormatItem>
<dateFormatItem id="y" draft="contributed">y</dateFormatItem>
<dateFormatItem id="yM" draft="contributed">yyyy-M</dateFormatItem>
<dateFormatItem id="yMEd" draft="contributed">EEE, yyyy-M-d</dateFormatItem>
<dateFormatItem id="yMMM" draft="contributed">MMM y</dateFormatItem>
<dateFormatItem id="yMMMEd" draft="contributed">EEE, d. MMM y</dateFormatItem>
<dateFormatItem id="yMMMM" draft="contributed">MMMM y</dateFormatItem>
<dateFormatItem id="yQ" draft="contributed">Q yyyy</dateFormatItem>
<dateFormatItem id="yQQQ" draft="contributed">QQQ y</dateFormatItem>
<dateFormatItem id="yyMM" draft="contributed">MM.yy</dateFormatItem>
<dateFormatItem id="yyMMdd" draft="contributed">dd.MM.yy</dateFormatItem>
<dateFormatItem id="yyMMM" draft="contributed">MMM yy</dateFormatItem>
<dateFormatItem id="yyQ" draft="contributed">Q yy</dateFormatItem>
<dateFormatItem id="yyQQQQ" draft="contributed">QQQQ yy</dateFormatItem>
<dateFormatItem id="yyyy" draft="contributed">y</dateFormatItem>
<dateFormatItem id="yyyyMMMM" draft="contributed">MMMM y</dateFormatItem>
</availableFormats>
<intervalFormats>
<intervalFormatFallback draft="contributed">{0} - {1}</intervalFormatFallback>
<intervalFormatItem id="d">
<greatestDifference id="d" draft="contributed">d.-d.</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="h">
<greatestDifference id="a" draft="contributed">HH-HH</greatestDifference>
<greatestDifference id="h" draft="contributed">HH-HH</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="hm">
<greatestDifference id="a" draft="contributed">HH:mm-HH:mm</greatestDifference>
<greatestDifference id="h" draft="contributed">HH:mm-HH:mm</greatestDifference>
<greatestDifference id="m" draft="contributed">HH:mm-HH:mm</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="hmv">
<greatestDifference id="a" draft="contributed">HH:mm-HH:mm v</greatestDifference>
<greatestDifference id="h" draft="contributed">HH:mm-HH:mm v</greatestDifference>
<greatestDifference id="m" draft="contributed">HH:mm-HH:mm v</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="hv">
<greatestDifference id="a" draft="contributed">HH-HH v</greatestDifference>
<greatestDifference id="h" draft="contributed">HH-HH v</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="M">
<greatestDifference id="M" draft="contributed">M.-M.</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="Md">
<greatestDifference id="d" draft="contributed">dd.MM. - dd.MM.</greatestDifference>
<greatestDifference id="M" draft="contributed">dd.MM. - dd.MM.</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="MEd">
<greatestDifference id="d" draft="contributed">E, dd.MM. - E, dd.MM.</greatestDifference>
<greatestDifference id="M" draft="contributed">E, dd.MM. - E, dd.MM.</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="MMM">
<greatestDifference id="M" draft="contributed">MMM-MMM</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="MMMd">
<greatestDifference id="d" draft="contributed">d.-d. MMM</greatestDifference>
<greatestDifference id="M" draft="contributed">d. MMM - d. MMM</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="MMMEd">
<greatestDifference id="d" draft="contributed">E, d. - E, d. MMM</greatestDifference>
<greatestDifference id="M" draft="contributed">E, d. MMM - E, d. MMM</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="MMMM">
<greatestDifference id="M" draft="contributed">LLLL-LLLL</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="y">
<greatestDifference id="y" draft="contributed">y-y</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yM">
<greatestDifference id="M" draft="contributed">MM.yy - MM.yy</greatestDifference>
<greatestDifference id="y" draft="contributed">MM.yy - MM.yy</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMd">
<greatestDifference id="d" draft="contributed">dd.MM.yy - dd.MM.yy</greatestDifference>
<greatestDifference id="M" draft="contributed">dd.MM.yy - dd.MM.yy</greatestDifference>
<greatestDifference id="y" draft="contributed">dd.MM.yy - dd.MM.yy</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMEd">
<greatestDifference id="d" draft="contributed">E, dd.MM.yy - E,
dd.MM.yy</greatestDifference>
<greatestDifference id="M" draft="contributed">E, dd.MM.yy - E,
dd.MM.yy</greatestDifference>
<greatestDifference id="y" draft="contributed">E, dd.MM.yy - E,
dd.MM.yy</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMMM">
<greatestDifference id="M" draft="contributed">MMM-MMM y</greatestDifference>
<greatestDifference id="y" draft="contributed">MMM y - MMM y</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMMMd">
<greatestDifference id="d" draft="contributed">d.-d. MMM y</greatestDifference>
<greatestDifference id="M" draft="contributed">d. MMM - d. MMM y</greatestDifference>
<greatestDifference id="y" draft="contributed">d. MMM y - d. MMM y</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMMMEd">
<greatestDifference id="d" draft="contributed">E, d. - E, d. MMM y</greatestDifference>
<greatestDifference id="M" draft="contributed">E, d. MMM - E, d. MMM y</greatestDifference>
<greatestDifference id="y" draft="contributed">E, d. MMM y - E, d. MMM
y</greatestDifference>
</intervalFormatItem>
<intervalFormatItem id="yMMMM">
<greatestDifference id="M" draft="contributed">MM – MM.yyyy</greatestDifference>
<greatestDifference id="y" draft="contributed">MM.yyyy – MM.yyyy</greatestDifference>
</intervalFormatItem>
</intervalFormats>
</dateTimeFormats>
<fields>
<field type="era">
<displayName draft="contributed">epoca</displayName>
</field>
<field type="year">
<displayName draft="contributed">onn</displayName>
</field>
<field type="month">
<displayName draft="contributed">mais</displayName>
</field>
<field type="week">
<displayName draft="contributed">emna</displayName>
</field>
<field type="day">
<displayName draft="contributed">Tag</displayName>
<relative type="-3" draft="contributed">squarsas</relative>
<relative type="-2" draft="contributed">stersas</relative>
<relative type="-1" draft="contributed">ier</relative>
<relative type="0" draft="contributed">oz</relative>
<relative type="1" draft="contributed">damaun</relative>
<relative type="2" draft="contributed">puschmaun</relative>
<relative type="3" draft="contributed">squartmaun</relative>
</field>
<field type="weekday">
<displayName draft="contributed">di da l'emna</displayName>
</field>
<field type="dayperiod">
<displayName draft="contributed">mesadad dal di</displayName>
</field>
<field type="hour">
<displayName draft="contributed">ura</displayName>
</field>
<field type="minute">
<displayName draft="contributed">minuta</displayName>
</field>
<field type="second">
<displayName draft="contributed">secunda</displayName>
</field>
<field type="zone">
<displayName draft="contributed">zona d'urari</displayName>
</field>
</fields>
</calendar>
</calendars>
<timeZoneNames>
<hourFormat draft="contributed">+HH:mm;-HH:mm</hourFormat>
<gmtFormat draft="contributed">GMT{0}</gmtFormat>
<regionFormat draft="contributed">{0}</regionFormat>
<fallbackFormat draft="contributed">{1} ({0})</fallbackFormat>
<zone type="Etc/Unknown">
<exemplarCity draft="contributed">zona betg enconuschenta</exemplarCity>
</zone>
<zone type="America/Anguilla">
<exemplarCity draft="contributed">The Valley</exemplarCity>
</zone>
<zone type="Europe/Tirane">
<exemplarCity draft="contributed">Tirana</exemplarCity>
</zone>
<zone type="Asia/Yerevan">
<exemplarCity draft="contributed">Jerevan</exemplarCity>
</zone>
<zone type="America/Curacao">
<exemplarCity draft="contributed">Curaçao</exemplarCity>
</zone>
<zone type="Antarctica/South_Pole">
<exemplarCity draft="contributed">Pol dal Sid</exemplarCity>
</zone>
<zone type="Antarctica/Syowa">
<exemplarCity draft="contributed">Showa</exemplarCity>
</zone>
<zone type="Antarctica/DumontDUrville">
<exemplarCity draft="contributed">Dumont d'Urville</exemplarCity>
</zone>
<zone type="Antarctica/McMurdo">
<exemplarCity draft="contributed">Mac Murdo</exemplarCity>
</zone>
<zone type="America/Argentina/San_Juan">
<exemplarCity draft="contributed">San Juan, Argentinia</exemplarCity>
</zone>
<zone type="America/Argentina/Ushuaia">
<exemplarCity draft="contributed">Ushuaïa</exemplarCity>
</zone>
<zone type="America/Jujuy">
<exemplarCity draft="contributed">San Salvador de Jujuy</exemplarCity>
</zone>
<zone type="America/Argentina/Tucuman">
<exemplarCity draft="contributed">Tucumán</exemplarCity>
</zone>
<zone type="America/Cordoba">
<exemplarCity draft="contributed">Córdoba</exemplarCity>
</zone>
<zone type="America/Aruba">
<exemplarCity draft="contributed">Oranjestad</exemplarCity>
</zone>
<zone type="Europe/Brussels">
<exemplarCity draft="contributed">Bruxelles</exemplarCity>
</zone>
<zone type="Atlantic/Bermuda">
<exemplarCity draft="contributed">Bermudas</exemplarCity>
</zone>
<zone type="Asia/Brunei">
<exemplarCity draft="contributed">Bandar Seri Begawan</exemplarCity>
</zone>
<zone type="America/Eirunepe">
<exemplarCity draft="contributed">Eirunepé</exemplarCity>
</zone>
<zone type="America/Cuiaba">
<exemplarCity draft="contributed">Cuiabá</exemplarCity>
</zone>
<zone type="America/Belem">
<exemplarCity draft="contributed">Belém</exemplarCity>
</zone>
<zone type="America/Araguaina">
<exemplarCity draft="contributed">Araguaína</exemplarCity>
</zone>
<zone type="America/Sao_Paulo">
<exemplarCity draft="contributed">São Paulo</exemplarCity>
</zone>
<zone type="America/Noronha">
<exemplarCity draft="contributed">Fernando de Noronha</exemplarCity>
</zone>
<zone type="America/Belize">
<exemplarCity draft="contributed">Belmopan</exemplarCity>
</zone>
<zone type="America/St_Johns">
<exemplarCity draft="contributed">Saint John's</exemplarCity>
</zone>
<zone type="Indian/Cocos">
<exemplarCity draft="contributed">West Island</exemplarCity>
</zone>
<zone type="Europe/Zurich">
<exemplarCity draft="contributed">Turitg</exemplarCity>
</zone>
<zone type="Pacific/Easter">
<exemplarCity draft="contributed">Insla da Pasca</exemplarCity>
</zone>
<zone type="Asia/Kashgar">
<exemplarCity draft="contributed">Kaschgar</exemplarCity>
</zone>
<zone type="Asia/Urumqi">
<exemplarCity draft="contributed">Ürümqi</exemplarCity>
</zone>
<zone type="America/Costa_Rica">
<exemplarCity draft="contributed">San José</exemplarCity>
</zone>
<zone type="Atlantic/Cape_Verde">
<exemplarCity draft="contributed">Cap Verd</exemplarCity>
</zone>
<zone type="Indian/Christmas">
<exemplarCity draft="contributed">Flying Fish Cove</exemplarCity>
</zone>
<zone type="Africa/Djibouti">
<exemplarCity draft="contributed">Dschibuti</exemplarCity>
</zone>
<zone type="Europe/Copenhagen">
<exemplarCity draft="contributed">Copenhaghen</exemplarCity>
</zone>
<zone type="Africa/Algiers">
<exemplarCity draft="contributed">Algier</exemplarCity>
</zone>
<zone type="Pacific/Galapagos">
<exemplarCity draft="contributed">Galápagos</exemplarCity>
</zone>
<zone type="Africa/El_Aaiun">
<exemplarCity draft="contributed">El Ayun</exemplarCity>
</zone>
<zone type="Africa/Asmera">
<exemplarCity draft="contributed">Asmara</exemplarCity>
</zone>
<zone type="Atlantic/Canary">
<exemplarCity draft="contributed">Inslas Canarias</exemplarCity>
</zone>
<zone type="Africa/Addis_Ababa">
<exemplarCity draft="contributed">Addis Abeba</exemplarCity>
</zone>
<zone type="Pacific/Fiji">
<exemplarCity draft="contributed">Fidschi</exemplarCity>
</zone>
<zone type="Atlantic/Stanley">
<exemplarCity draft="contributed">Port Stanley</exemplarCity>
</zone>
<zone type="Pacific/Truk">
<exemplarCity draft="contributed">Weno</exemplarCity>
</zone>
<zone type="Pacific/Ponape">
<exemplarCity draft="contributed">Palikir</exemplarCity>
</zone>
<zone type="Pacific/Kosrae">
<exemplarCity draft="contributed">Tofol</exemplarCity>
</zone>
<zone type="Atlantic/Faeroe">
<exemplarCity draft="contributed">Inslas Feroe</exemplarCity>
</zone>
<zone type="Europe/London">
<exemplarCity draft="contributed">Londra</exemplarCity>
</zone>
<zone type="Europe/Guernsey">
<exemplarCity draft="contributed">Saint Peter Port</exemplarCity>
</zone>
<zone type="America/Godthab">
<exemplarCity draft="contributed">Godthåb</exemplarCity>
</zone>
<zone type="America/Guadeloupe">
<exemplarCity draft="contributed">Basse-Terre</exemplarCity>
</zone>
<zone type="Europe/Athens">
<exemplarCity draft="contributed">Athen</exemplarCity>
</zone>
<zone type="Atlantic/South_Georgia">
<exemplarCity draft="contributed">Georgia dal Sid</exemplarCity>
</zone>
<zone type="Pacific/Guam">
<exemplarCity draft="contributed">Hagåtña</exemplarCity>
</zone>
<zone type="Asia/Hong_Kong">
<exemplarCity draft="contributed">Hongkong</exemplarCity>
</zone>
<zone type="Asia/Makassar">
<exemplarCity draft="contributed">Macassar</exemplarCity>
</zone>
<zone type="Europe/Isle_of_Man">
<exemplarCity draft="contributed">Douglas</exemplarCity>
</zone>
<zone type="Indian/Chagos">
<exemplarCity draft="contributed">Diego Garcia</exemplarCity>
</zone>
<zone type="Asia/Baghdad">
<exemplarCity draft="contributed">Bagdad</exemplarCity>
</zone>
<zone type="Asia/Tehran">
<exemplarCity draft="contributed">Teheran</exemplarCity>
</zone>
<zone type="Europe/Rome">
<exemplarCity draft="contributed">Roma</exemplarCity>
</zone>
<zone type="Europe/Jersey">
<exemplarCity draft="contributed">Saint Helier</exemplarCity>
</zone>
<zone type="America/Jamaica">
<exemplarCity draft="contributed">Giamaica</exemplarCity>
</zone>
<zone type="Asia/Tokyo">
<exemplarCity draft="contributed">Tokio</exemplarCity>
</zone>
<zone type="Asia/Bishkek">
<exemplarCity draft="contributed">Bischkek</exemplarCity>
</zone>
<zone type="Pacific/Tarawa">
<exemplarCity draft="contributed">South Tarawa</exemplarCity>
</zone>
<zone type="Indian/Comoro">
<exemplarCity draft="contributed">Comoras</exemplarCity>
</zone>
<zone type="America/St_Kitts">
<exemplarCity draft="contributed">Saint Kitts</exemplarCity>
</zone>
<zone type="Asia/Seoul">
<exemplarCity draft="contributed">Seul</exemplarCity>
</zone>
<zone type="America/Cayman">
<exemplarCity draft="contributed">Inslas Cayman</exemplarCity>
</zone>
<zone type="Asia/Aqtobe">
<exemplarCity draft="contributed">Aqtöbe</exemplarCity>
</zone>
<zone type="America/St_Lucia">
<exemplarCity draft="contributed">Santa Lucia</exemplarCity>
</zone>
<zone type="Europe/Luxembourg">
<exemplarCity draft="contributed">Luxemburg</exemplarCity>
</zone>
<zone type="Asia/Choibalsan">
<exemplarCity draft="contributed">Tschoibalsan</exemplarCity>
</zone>
<zone type="Asia/Macau">
<exemplarCity draft="contributed">Macao</exemplarCity>
</zone>
<zone type="America/Montserrat">
<exemplarCity draft="contributed">Brades</exemplarCity>
</zone>
<zone type="Indian/Maldives">
<exemplarCity draft="contributed">Maldivas</exemplarCity>
</zone>
<zone type="America/Mazatlan">
<exemplarCity draft="contributed">Mazatlán</exemplarCity>
</zone>
<zone type="America/Mexico_City">
<exemplarCity draft="contributed">Citad da Mexico</exemplarCity>
</zone>
<zone type="America/Merida">
<exemplarCity draft="contributed">Mérida</exemplarCity>
</zone>
<zone type="America/Cancun">
<exemplarCity draft="contributed">Cancún</exemplarCity>
</zone>
<zone type="Pacific/Noumea">
<exemplarCity draft="contributed">Nouméa</exemplarCity>
</zone>
<zone type="Africa/Niamey">
<exemplarCity draft="contributed">Niger</exemplarCity>
</zone>
<zone type="Pacific/Norfolk">
<exemplarCity draft="contributed">Kingston</exemplarCity>
</zone>
<zone type="Pacific/Nauru">
<exemplarCity draft="contributed">Yaren</exemplarCity>
</zone>
<zone type="Pacific/Niue">
<exemplarCity draft="contributed">Alofi</exemplarCity>
</zone>
<zone type="Asia/Muscat">
<exemplarCity draft="contributed">Mascate</exemplarCity>
</zone>
<zone type="Pacific/Gambier">
<exemplarCity draft="contributed">Rikitea</exemplarCity>
</zone>
<zone type="Asia/Karachi">
<exemplarCity draft="contributed">Karatschi</exemplarCity>
</zone>
<zone type="Europe/Warsaw">
<exemplarCity draft="contributed">Varsovia</exemplarCity>
</zone>
<zone type="America/Miquelon">
<exemplarCity draft="contributed">Saint Pierre</exemplarCity>
</zone>
<zone type="Atlantic/Azores">
<exemplarCity draft="contributed">Azoras</exemplarCity>
</zone>
<zone type="Europe/Lisbon">
<exemplarCity draft="contributed">Lissabon</exemplarCity>
</zone>
<zone type="Pacific/Palau">
<exemplarCity draft="contributed">Melekok</exemplarCity>
</zone>
<zone type="America/Asuncion">
<exemplarCity draft="contributed">Asunción</exemplarCity>
</zone>
<zone type="Indian/Reunion">
<exemplarCity draft="contributed">Réunion</exemplarCity>
</zone>
<zone type="Europe/Bucharest">
<exemplarCity draft="contributed">Bucarest</exemplarCity>
</zone>
<zone type="Europe/Moscow">
<exemplarCity draft="contributed">Moscau</exemplarCity>
</zone>
<zone type="Asia/Yekaterinburg">
<exemplarCity draft="contributed">Jekaterinburg</exemplarCity>
</zone>
<zone type="Asia/Novosibirsk">
<exemplarCity draft="contributed">Nowosibirsk</exemplarCity>
</zone>
<zone type="Asia/Krasnoyarsk">
<exemplarCity draft="contributed">Krasnojarsk</exemplarCity>
</zone>
<zone type="Asia/Yakutsk">
<exemplarCity draft="contributed">Jakutsk</exemplarCity>
</zone>
<zone type="Asia/Sakhalin">
<exemplarCity draft="contributed">Sachalin</exemplarCity>
</zone>
<zone type="Asia/Kamchatka">
<exemplarCity draft="contributed">Kamtschatka</exemplarCity>
</zone>
<zone type="Asia/Riyadh">
<exemplarCity draft="contributed">Riyad</exemplarCity>
</zone>
<zone type="Pacific/Guadalcanal">
<exemplarCity draft="contributed">Honiara</exemplarCity>
</zone>
<zone type="Africa/Khartoum">
<exemplarCity draft="contributed">Khartum</exemplarCity>
</zone>
<zone type="Asia/Singapore">
<exemplarCity draft="contributed">Singapur</exemplarCity>
</zone>
<zone type="Atlantic/St_Helena">
<exemplarCity draft="contributed">Sontg'elena</exemplarCity>
</zone>
<zone type="Africa/Mogadishu">
<exemplarCity draft="contributed">Mogadischu</exemplarCity>
</zone>
<zone type="Africa/Sao_Tome">
<exemplarCity draft="contributed">São Tomé</exemplarCity>
</zone>
<zone type="America/El_Salvador">
<exemplarCity draft="contributed">Salvador</exemplarCity>
</zone>
<zone type="America/Grand_Turk">
<exemplarCity draft="contributed">Cockburn Town</exemplarCity>
</zone>
<zone type="Indian/Kerguelen">
<exemplarCity draft="contributed">Port-aux-Français</exemplarCity>
</zone>
<zone type="Africa/Lome">
<exemplarCity draft="contributed">Lomé</exemplarCity>
</zone>
<zone type="Asia/Dushanbe">
<exemplarCity draft="contributed">Duschanbe</exemplarCity>
</zone>
<zone type="Asia/Ashgabat">
<exemplarCity draft="contributed">Aşgabat</exemplarCity>
</zone>
<zone type="Pacific/Tongatapu">
<exemplarCity draft="contributed">Nukuʻalofa</exemplarCity>
</zone>
<zone type="America/Port_of_Spain">
<exemplarCity draft="contributed">Port-of-Spain</exemplarCity>
</zone>
<zone type="Africa/Dar_es_Salaam">
<exemplarCity draft="contributed">Daressalam</exemplarCity>
</zone>
<zone type="Europe/Uzhgorod">
<exemplarCity draft="contributed">Uschgorod</exemplarCity>
</zone>
<zone type="Europe/Zaporozhye">
<exemplarCity draft="contributed">Saporischja</exemplarCity>
</zone>
<zone type="America/Anchorage">
<exemplarCity draft="contributed">Alasca</exemplarCity>
</zone>
<zone type="America/North_Dakota/New_Salem">
<exemplarCity draft="contributed">North Dakota (New Salem)</exemplarCity>
</zone>
<zone type="America/North_Dakota/Center">
<exemplarCity draft="contributed">North Dakota (Central)</exemplarCity>
</zone>
<zone type="America/Indiana/Vincennes">
<exemplarCity draft="contributed">Vincennes</exemplarCity>
</zone>
<zone type="America/Indiana/Petersburg">
<exemplarCity draft="contributed">Petersburg</exemplarCity>
</zone>
<zone type="America/Indiana/Tell_City">
<exemplarCity draft="contributed">Tell City</exemplarCity>
</zone>
<zone type="America/Indiana/Knox">
<exemplarCity draft="contributed">Knox</exemplarCity>
</zone>
<zone type="America/Indiana/Winamac">
<exemplarCity draft="contributed">Winamac</exemplarCity>
</zone>
<zone type="America/Indiana/Marengo">
<exemplarCity draft="contributed">Marengo</exemplarCity>
</zone>
<zone type="America/Indiana/Vevay">
<exemplarCity draft="contributed">Vevay</exemplarCity>
</zone>
<zone type="America/Kentucky/Monticello">
<exemplarCity draft="contributed">Monticello</exemplarCity>
</zone>
<zone type="Asia/Samarkand">
<exemplarCity draft="contributed">Samarcanda</exemplarCity>
</zone>
<zone type="Asia/Tashkent">
<exemplarCity draft="contributed">Taschkent</exemplarCity>
</zone>
<zone type="Europe/Vatican">
<exemplarCity draft="contributed">Citad dal Vatican</exemplarCity>
</zone>
<zone type="America/St_Vincent">
<exemplarCity draft="contributed">Saint Vincent</exemplarCity>
</zone>
<zone type="America/Tortola">
<exemplarCity draft="contributed">Road Town</exemplarCity>
</zone>
<zone type="America/St_Thomas">
<exemplarCity draft="contributed">Saint Thomas</exemplarCity>
</zone>
<zone type="Asia/Saigon">
<exemplarCity draft="contributed">Ho Chi Minh</exemplarCity>
</zone>
<zone type="Pacific/Wallis">
<exemplarCity draft="contributed">Matāʻutu</exemplarCity>
</zone>
</timeZoneNames>
</dates>
<numbers>
<symbols>
<decimal draft="contributed">.</decimal>
<group draft="contributed">'</group>
<list draft="contributed">;</list>
<percentSign draft="contributed">%</percentSign>
<nativeZeroDigit draft="contributed">0</nativeZeroDigit>
<patternDigit draft="contributed">#</patternDigit>
<plusSign draft="contributed">+</plusSign>
<minusSign draft="contributed">−</minusSign>
<exponential draft="contributed">E</exponential>
<perMille draft="contributed">‰</perMille>
<infinity draft="contributed">∞</infinity>
<nan draft="contributed">NaN</nan>
</symbols>
<decimalFormats>
<decimalFormatLength>
<decimalFormat>
<pattern draft="contributed">#,##0.###</pattern>
</decimalFormat>
</decimalFormatLength>
</decimalFormats>
<scientificFormats>
<scientificFormatLength>
<scientificFormat>
<pattern draft="contributed">#E0</pattern>
</scientificFormat>
</scientificFormatLength>
</scientificFormats>
<percentFormats>
<percentFormatLength>
<percentFormat>
<pattern draft="contributed">#,##0 %</pattern>
</percentFormat>
</percentFormatLength>
</percentFormats>
<currencyFormats>
<currencyFormatLength>
<currencyFormat>
<pattern draft="contributed">#,##0.00 ¤</pattern>
</currencyFormat>
</currencyFormatLength>
<unitPattern count="one" draft="contributed">{0} {1}</unitPattern>
<unitPattern count="other" draft="contributed">{0} {1}</unitPattern>
</currencyFormats>
<currencies>
<currency type="ADP">
<displayName draft="contributed">peseta andorrana</displayName>
</currency>
<currency type="AED">
<displayName draft="contributed">dirham dals Emirats Arabs Unids</displayName>
</currency>
<currency type="AFA">
<displayName draft="contributed">afghani (1927–2002)</displayName>
</currency>
<currency type="AFN">
<displayName draft="contributed">afghani</displayName>
</currency>
<currency type="ALK">
<displayName draft="contributed">lek albanais (1947–1961)</displayName>
</currency>
<currency type="ALL">
<displayName draft="contributed">lek</displayName>
</currency>
<currency type="AMD">
<displayName draft="contributed">dram armen</displayName>
</currency>
<currency type="ANG">
<displayName draft="contributed">flurin da las Antillas Olandaisas</displayName>
</currency>
<currency type="AOA">
<displayName draft="contributed">kwanza angolan</displayName>
</currency>
<currency type="AOK">
<displayName draft="contributed">kwanza angolan (1977-1990)</displayName>
</currency>
<currency type="AON">
<displayName draft="contributed">nov kwanza angolan</displayName>
</currency>
<currency type="AOR">
<displayName draft="contributed">kwanza angolan reajustado</displayName>
</currency>
<currency type="ARA">
<displayName draft="contributed">austral argentin</displayName>
</currency>
<currency type="ARL">
<displayName draft="contributed">peso argentin ley</displayName>
</currency>
<currency type="ARM">
<displayName draft="contributed">peso argentin moneda nacional</displayName>
</currency>
<currency type="ARP">
<displayName draft="contributed">peso argentin (1983-1985)</displayName>
</currency>
<currency type="ARS">
<displayName draft="contributed">peso argentin</displayName>
</currency>
<currency type="ATS">
<displayName draft="contributed">schilling austriac</displayName>
</currency>
<currency type="AUD">
<displayName draft="contributed">dollar australian</displayName>
</currency>
<currency type="AWG">
<displayName draft="contributed">flurin da l'Aruba</displayName>
</currency>
<currency type="AZM">
<displayName draft="contributed">manat aserbaidschanic (1993-2006)</displayName>
</currency>
<currency type="AZN">
<displayName draft="contributed">manat aserbaidschanic</displayName>
</currency>
<currency type="BAD">
<displayName draft="contributed">dinar da la Bosnia ed Erzegovina</displayName>
</currency>
<currency type="BAM">
<displayName draft="contributed">marc convertibel bosniac</displayName>
</currency>
<currency type="BAN">
<displayName draft="contributed">nov dinar da la Bosnia ed Erzegovina</displayName>
</currency>
<currency type="BBD">
<displayName draft="contributed">dollar da Barbados</displayName>
</currency>
<currency type="BDT">
<displayName draft="contributed">taka bangladais</displayName>
</currency>
<currency type="BEC">
<displayName draft="contributed">franc beltg (convertibel)</displayName>
</currency>
<currency type="BEF">
<displayName draft="contributed">franc beltg</displayName>
</currency>
<currency type="BEL">
<displayName draft="contributed">franc beltg (finanzial)</displayName>
</currency>
<currency type="BGL">
<displayName draft="contributed">lev bulgar</displayName>
</currency>
<currency type="BGM">
<displayName draft="contributed">lev bulgar socialistic</displayName>
</currency>
<currency type="BGN">
<displayName draft="contributed">nov lev bulgar</displayName>
</currency>
<currency type="BGO">
<displayName draft="contributed">vegl lev bulgar</displayName>
</currency>
<currency type="BHD">
<displayName draft="contributed">dinar dal Bahrain</displayName>
</currency>
<currency type="BIF">
<displayName draft="contributed">franc dal Burundi</displayName>
</currency>
<currency type="BMD">
<displayName draft="contributed">dollar da las Bermudas</displayName>
</currency>
<currency type="BND">
<displayName draft="contributed">dollar dal Brunei</displayName>
</currency>
<currency type="BOB">
<displayName draft="contributed">boliviano</displayName>
</currency>
<currency type="BOL">
<displayName draft="contributed">vegl boliviano</displayName>
</currency>
<currency type="BOP">
<displayName draft="contributed">peso bolivian</displayName>
</currency>
<currency type="BOV">
<displayName draft="contributed">mvdol bolivian</displayName>
</currency>
<currency type="BRB">
<displayName draft="contributed">cruzeiro novo brasilian (1967-1986)</displayName>
</currency>
<currency type="BRC">
<displayName draft="contributed">cruzado brasilian</displayName>
</currency>
<currency type="BRE">
<displayName draft="contributed">cruzeiro brasilian (1990-1993)</displayName>
</currency>
<currency type="BRL">
<displayName draft="contributed">real brasilian</displayName>
</currency>
<currency type="BRN">
<displayName draft="contributed">cruzado novo brasilian</displayName>
</currency>
<currency type="BRR">
<displayName draft="contributed">cruzeiro brasilian</displayName>
</currency>
<currency type="BRZ">
<displayName draft="contributed">vegl cruzeiro brasilian</displayName>
</currency>
<currency type="BSD">
<displayName draft="contributed">dollar da las Bahamas</displayName>
</currency>
<currency type="BTN">
<displayName draft="contributed">ngultrum butanais</displayName>
</currency>
<currency type="BUK">
<displayName draft="contributed">Kyat burmais</displayName>
</currency>
<currency type="BWP">
<displayName draft="contributed">pula da la Botswana</displayName>
</currency>
<currency type="BYB">
<displayName draft="contributed">nov rubel bieloruss (1994-1999)</displayName>
</currency>
<currency type="BYR">
<displayName draft="contributed">rubel bieloruss</displayName>
</currency>
<currency type="BZD">
<displayName draft="contributed">dollar dal Belize</displayName>
</currency>
<currency type="CAD">
<displayName draft="contributed">dollar canadais</displayName>
</currency>
<currency type="CDF">
<displayName draft="contributed">franc congolais</displayName>
</currency>
<currency type="CHE">
<displayName draft="contributed">euro WIR</displayName>
</currency>
<currency type="CHF">
<displayName draft="contributed">franc svizzer</displayName>
</currency>
<currency type="CHW">
<displayName draft="contributed">franc WIR</displayName>
</currency>
<currency type="CLE">
<displayName draft="contributed">escudo chilen</displayName>
</currency>
<currency type="CLF">
<displayName draft="contributed">unidades de fomento chilenas</displayName>
</currency>
<currency type="CLP">
<displayName draft="contributed">peso chilen</displayName>
</currency>
<currency type="CNY">
<displayName draft="contributed">yuan renminbi chinais</displayName>
</currency>
<currency type="COP">
<displayName draft="contributed">peso columbian</displayName>
</currency>
<currency type="COU">
<displayName draft="contributed">unidad de valor real</displayName>
</currency>
<currency type="CRC">
<displayName draft="contributed">colon da la Costa Rica</displayName>
</currency>
<currency type="CSD">
<displayName draft="contributed">vegl dinar serb</displayName>
</currency>
<currency type="CSK">
<displayName draft="contributed">cruna tschecoslovaca</displayName>
</currency>
<currency type="CUP">
<displayName draft="contributed">peso cuban</displayName>
</currency>
<currency type="CVE">
<displayName draft="contributed">escudo dal Cap Verd</displayName>
</currency>
<currency type="CYP">
<displayName draft="contributed">glivra cipriota</displayName>
</currency>
<currency type="CZK">
<displayName draft="contributed">cruna tscheca</displayName>
</currency>
<currency type="DDM">
<displayName draft="contributed">marc da la Germania da l'Ost</displayName>
</currency>
<currency type="DEM">
<displayName draft="contributed">marc tudestg</displayName>
</currency>
<currency type="DJF">
<displayName draft="contributed">franc dal Dschibuti</displayName>
</currency>
<currency type="DKK">
<displayName draft="contributed">cruna danaisa</displayName>
</currency>
<currency type="DOP">
<displayName draft="contributed">peso dominican</displayName>
</currency>
<currency type="DZD">
<displayName draft="contributed">dinar algerian</displayName>
</currency>
<currency type="ECS">
<displayName draft="contributed">sucre equadorian</displayName>
</currency>
<currency type="ECV">
<displayName draft="contributed">unitad da scuntrada da l'Ecuador</displayName>
</currency>
<currency type="EEK">
<displayName draft="contributed">cruna estona</displayName>
</currency>
<currency type="EGP">
<displayName draft="contributed">glivra egipziana</displayName>
</currency>
<currency type="ERN">
<displayName draft="contributed">nakfa eritreic</displayName>
</currency>
<currency type="ESA">
<displayName draft="contributed">peseta spagnola (conto A)</displayName>
</currency>
<currency type="ESB">
<displayName draft="contributed">peseta spagnola (conto convertibel)</displayName>
</currency>
<currency type="ESP">
<displayName draft="contributed">peseta spagnola</displayName>
</currency>
<currency type="ETB">
<displayName draft="contributed">birr etiopic</displayName>
</currency>
<currency type="EUR">
<displayName draft="contributed">euro</displayName>
</currency>
<currency type="FIM">
<displayName draft="contributed">marc finlandais</displayName>
</currency>
<currency type="FJD">
<displayName draft="contributed">dollar dal Fidschi</displayName>
</currency>
<currency type="FKP">
<displayName draft="contributed">glivra dal Falkland</displayName>
</currency>
<currency type="FRF">
<displayName draft="contributed">franc franzos</displayName>
</currency>
<currency type="GBP">
<displayName draft="contributed">glivra sterlina</displayName>
</currency>
<currency type="GEK">
<displayName draft="contributed">kupon larit georgian</displayName>
</currency>
<currency type="GEL">
<displayName draft="contributed">lari georgian</displayName>
</currency>
<currency type="GHC">
<displayName draft="contributed">cedi ghanais (1979-2007)</displayName>
</currency>
<currency type="GHS">
<displayName draft="contributed">cedi ghanais</displayName>
</currency>
<currency type="GIP">
<displayName draft="contributed">glivra da Gibraltar</displayName>
</currency>
<currency type="GMD">
<displayName draft="contributed">dalasi gambic</displayName>
</currency>
<currency type="GNF">
<displayName draft="contributed">franc da la Guinea</displayName>
</currency>
<currency type="GNS">
<displayName draft="contributed">syli da la Guinea</displayName>
</currency>
<currency type="GQE">
<displayName draft="contributed">ekwele da la Guinea Equatoriala</displayName>
</currency>
<currency type="GRD">
<displayName draft="contributed">drachma greca</displayName>
</currency>
<currency type="GTQ">
<displayName draft="contributed">quetzal da la Guatemala</displayName>
</currency>
<currency type="GWE">
<displayName draft="contributed">escudo da la Guinea Portugaisa</displayName>
</currency>
<currency type="GWP">
<displayName draft="contributed">peso da la Guinea-Bissau</displayName>
</currency>
<currency type="GYD">
<displayName draft="contributed">dollar da la Guyana</displayName>
</currency>
<currency type="HKD">
<displayName draft="contributed">dollar da Hongkong</displayName>
</currency>
<currency type="HNL">
<displayName draft="contributed">lempira hondurian</displayName>
</currency>
<currency type="HRD">
<displayName draft="contributed">dinar croat</displayName>
</currency>
<currency type="HRK">
<displayName draft="contributed">kuna croata</displayName>
</currency>
<currency type="HTG">
<displayName draft="contributed">gourde haitian</displayName>
</currency>
<currency type="HUF">
<displayName draft="contributed">forint ungarais</displayName>
</currency>
<currency type="IDR">
<displayName draft="contributed">rupia indonaisa</displayName>
</currency>
<currency type="IEP">
<displayName draft="contributed">glivra indonaisa</displayName>
</currency>
<currency type="ILP">
<displayName draft="contributed">glivra israeliana</displayName>
</currency>
<currency type="ILR">
<displayName draft="contributed">vegl sheqel israelian</displayName>
</currency>
<currency type="ILS">
<displayName draft="contributed">sheqel</displayName>
</currency>
<currency type="INR">
<displayName draft="contributed">rupia indica</displayName>
</currency>
<currency type="IQD">
<displayName draft="contributed">dinar iracais</displayName>
</currency>
<currency type="IRR">
<displayName draft="contributed">rial iranais</displayName>
</currency>
<currency type="ISJ">
<displayName draft="contributed">veglia cruna islandaisa</displayName>
</currency>
<currency type="ISK">
<displayName draft="contributed">cruna islandaisa</displayName>
</currency>
<currency type="ITL">
<displayName draft="contributed">lira taliana</displayName>
</currency>
<currency type="JMD">
<displayName draft="contributed">dollar giamaican</displayName>
</currency>
<currency type="JOD">
<displayName draft="contributed">dinar jordanic</displayName>
</currency>
<currency type="JPY">
<displayName draft="contributed">yen giapunais</displayName>
</currency>
<currency type="KES">
<displayName draft="contributed">schilling kenian</displayName>
</currency>
<currency type="KGS">
<displayName draft="contributed">som kirghis</displayName>
</currency>
<currency type="KHR">
<displayName draft="contributed">riel cambodschan</displayName>
</currency>
<currency type="KMF">
<displayName draft="contributed">franc comorian</displayName>
</currency>
<currency type="KPW">
<displayName draft="contributed">won da la Corea dal Nord</displayName>
</currency>
<currency type="KRH">
<displayName draft="contributed">hwan da la Corea dal Sid</displayName>
</currency>
<currency type="KRO">
<displayName draft="contributed">vegl won da la Corea dal Sid</displayName>
</currency>
<currency type="KRW">
<displayName draft="contributed">won da la Corea dal Sid</displayName>
</currency>
<currency type="KWD">
<displayName draft="contributed">dinar dal Kuwait</displayName>
</currency>
<currency type="KYD">
<displayName draft="contributed">dollar da las Inslas Cayman</displayName>
</currency>
<currency type="KZT">
<displayName draft="contributed">tenge casac</displayName>
</currency>
<currency type="LAK">
<displayName draft="contributed">kip laot</displayName>
</currency>
<currency type="LBP">
<displayName draft="contributed">glivra libanaisa</displayName>
</currency>
<currency type="LKR">
<displayName draft="contributed">rupia da la Sri Lanka</displayName>
</currency>
<currency type="LRD">
<displayName draft="contributed">dollar liberian</displayName>
</currency>
<currency type="LSL">
<displayName draft="contributed">loti dal Lesotho</displayName>
</currency>
<currency type="LTL">
<displayName draft="contributed">litas lituan</displayName>
</currency>
<currency type="LTT">
<displayName draft="contributed">talonas lituan</displayName>
</currency>
<currency type="LUC">
<displayName draft="contributed">franc convertibel luxemburgais</displayName>
</currency>
<currency type="LUF">
<displayName draft="contributed">franc luxemburgais</displayName>
</currency>
<currency type="LUL">
<displayName draft="contributed">franc finanzial luxemburgais</displayName>
</currency>
<currency type="LVL">
<displayName draft="contributed">lats letton</displayName>
</currency>
<currency type="LVR">
<displayName draft="contributed">rubel letton</displayName>
</currency>
<currency type="LYD">
<displayName draft="contributed">dinar libic</displayName>
</currency>
<currency type="MAD">
<displayName draft="contributed">dirham marocan</displayName>
</currency>
<currency type="MAF">
<displayName draft="contributed">franc marocan</displayName>
</currency>
<currency type="MCF">
<displayName draft="contributed">franc monegas</displayName>
</currency>
<currency type="MDC">
<displayName draft="contributed">cupon moldav</displayName>
</currency>
<currency type="MDL">
<displayName draft="contributed">leu moldav</displayName>
</currency>
<currency type="MGA">
<displayName draft="contributed">ariary madagasc</displayName>
</currency>
<currency type="MGF">
<displayName draft="contributed">franc madagasc</displayName>
</currency>
<currency type="MKD">
<displayName draft="contributed">dinar da la Macedonia</displayName>
</currency>
<currency type="MKN">
<displayName draft="contributed">vegl dinar macedon</displayName>
</currency>
<currency type="MLF">
<displayName draft="contributed">franc dal Mali</displayName>
</currency>
<currency type="MMK">
<displayName draft="contributed">Kyat dal Myanmar</displayName>
</currency>
<currency type="MNT">
<displayName draft="contributed">tugrik mongolic</displayName>
</currency>
<currency type="MOP">
<displayName draft="contributed">pataca dal Macao</displayName>
</currency>
<currency type="MRO">
<displayName draft="contributed">ouguiya da la Mauretania</displayName>
</currency>
<currency type="MTL">
<displayName draft="contributed">lira maltaisa</displayName>
</currency>
<currency type="MTP">
<displayName draft="contributed">glivra maltaisa</displayName>
</currency>
<currency type="MUR">
<displayName draft="contributed">rupia dal Mauritius</displayName>
</currency>
<currency type="MVP">
<displayName draft="contributed">rupia da las Maledivas</displayName>
</currency>
<currency type="MVR">
<displayName draft="contributed">rufiyaa da las Maledivas</displayName>
</currency>
<currency type="MWK">
<displayName draft="contributed">kwacha dal Malawi</displayName>
</currency>
<currency type="MXN">
<displayName draft="contributed">peso mexican</displayName>
</currency>
<currency type="MXP">
<displayName draft="contributed">peso d'argient mexican (1861-1992)</displayName>
</currency>
<currency type="MXV">
<displayName draft="contributed">unidad de inversion mexicana (UDI)</displayName>
</currency>
<currency type="MYR">
<displayName draft="contributed">ringgit da la Malaisia</displayName>
</currency>
<currency type="MZE">
<displayName draft="contributed">escudo dal mozambican</displayName>
</currency>
<currency type="MZM">
<displayName draft="contributed">vegl metical mozambican</displayName>
</currency>
<currency type="MZN">
<displayName draft="contributed">metical dal mozambican</displayName>
</currency>
<currency type="NAD">
<displayName draft="contributed">dollar namibian</displayName>
</currency>
<currency type="NGN">
<displayName draft="contributed">naira nigeriana</displayName>
</currency>
<currency type="NIC">
<displayName draft="contributed">cordoba nicaraguan</displayName>
</currency>
<currency type="NIO">
<displayName draft="contributed">cordoba oro nicaraguan</displayName>
</currency>
<currency type="NLG">
<displayName draft="contributed">flurin ollandais</displayName>
</currency>
<currency type="NOK">
<displayName draft="contributed">cruna norvegiaisa</displayName>
</currency>
<currency type="NPR">
<displayName draft="contributed">rupia nepalaisa</displayName>
</currency>
<currency type="NZD">
<displayName draft="contributed">dollar da la Nova Zelanda</displayName>
</currency>
<currency type="OMR">
<displayName draft="contributed">rial da l'Oman</displayName>
</currency>
<currency type="PAB">
<displayName draft="contributed">balboa dal Panama</displayName>
</currency>
<currency type="PEI">
<displayName draft="contributed">inti peruan</displayName>
</currency>
<currency type="PEN">
<displayName draft="contributed">nov sol peruan</displayName>
</currency>
<currency type="PES">
<displayName draft="contributed">sol peruan</displayName>
</currency>
<currency type="PGK">
<displayName draft="contributed">kina da la Papua Nova Guinea</displayName>
</currency>
<currency type="PHP">
<displayName draft="contributed">peso filippin</displayName>
</currency>
<currency type="PKR">
<displayName draft="contributed">rupia pakistana</displayName>
</currency>
<currency type="PLN">
<displayName draft="contributed">zloty polac</displayName>
</currency>
<currency type="PLZ">
<displayName draft="contributed">zloty polac (1950-1995)</displayName>
</currency>
<currency type="PTE">
<displayName draft="contributed">escudo portugais</displayName>
</currency>
<currency type="PYG">
<displayName draft="contributed">guarani paraguaian</displayName>
</currency>
<currency type="QAR">
<displayName draft="contributed">riyal da Katar</displayName>
</currency>
<currency type="RHD">
<displayName draft="contributed">dollar rodesian</displayName>
</currency>
<currency type="ROL">
<displayName draft="contributed">vegl leu rumen</displayName>
</currency>
<currency type="RON">
<displayName draft="contributed">leu rumen</displayName>
</currency>
<currency type="RSD">
<displayName draft="contributed">dinar serb</displayName>
</currency>
<currency type="RUB">
<displayName draft="contributed">rubel russ (nov)</displayName>
</currency>
<currency type="RUR">
<displayName draft="contributed">rubel russ (vegl)</displayName>
</currency>
<currency type="RWF">
<displayName draft="contributed">franc ruandais</displayName>
</currency>
<currency type="SAR">
<displayName draft="contributed">riyal saudit</displayName>
</currency>
<currency type="SBD">
<displayName draft="contributed">dollar da las Salomonas</displayName>
</currency>
<currency type="SCR">
<displayName draft="contributed">rupia da las Seychellas</displayName>
</currency>
<currency type="SDD">
<displayName draft="contributed">dinar sudanais</displayName>
</currency>
<currency type="SDG">
<displayName draft="contributed">glivra sudanaisa</displayName>
</currency>
<currency type="SDP">
<displayName draft="contributed">glivra sudanaisa (1956–2007)</displayName>
</currency>
<currency type="SEK">
<displayName draft="contributed">cruna svedaisa</displayName>
</currency>
<currency type="SGD">
<displayName draft="contributed">dollar dal Singapur</displayName>
</currency>
<currency type="SHP">
<displayName draft="contributed">glivra da Sontg'Elena</displayName>
</currency>
<currency type="SIT">
<displayName draft="contributed">tolar sloven</displayName>
</currency>
<currency type="SKK">
<displayName draft="contributed">cruna slovaca</displayName>
</currency>
<currency type="SLL">
<displayName draft="contributed">leone da la Sierra Leone</displayName>
</currency>
<currency type="SOS">
<displayName draft="contributed">schilling somalian</displayName>
</currency>
<currency type="SRD">
<displayName draft="contributed">dollar surinam</displayName>
</currency>
<currency type="SRG">
<displayName draft="contributed">flurin surinam</displayName>
</currency>
<currency type="STD">
<displayName draft="contributed">dobra da São Tomé e Principe</displayName>
</currency>
<currency type="SUR">
<displayName draft="contributed">rubel sovietic</displayName>
</currency>
<currency type="SVC">
<displayName draft="contributed">colon da l'El Salvador</displayName>
</currency>
<currency type="SYP">
<displayName draft="contributed">glivra siriana</displayName>
</currency>
<currency type="SZL">
<displayName draft="contributed">lilangeni dal Swaziland</displayName>
</currency>
<currency type="THB">
<displayName draft="contributed">baht tailandais</displayName>
</currency>
<currency type="TJR">
<displayName draft="contributed">rubel dal Tadschikistan</displayName>
</currency>
<currency type="TJS">
<displayName draft="contributed">somoni dal Tadschikistan</displayName>
</currency>
<currency type="TMM">
<displayName draft="contributed">manat turkmen</displayName>
</currency>
<currency type="TND">
<displayName draft="contributed">dinar tunesian</displayName>
</currency>
<currency type="TOP">
<displayName draft="contributed">pa'anga da Tonga</displayName>
</currency>
<currency type="TPE">
<displayName draft="contributed">escudo da Timor</displayName>
</currency>
<currency type="TRL">
<displayName draft="contributed">lira tirca</displayName>
</currency>
<currency type="TRY">
<displayName draft="contributed">nova lira tirca</displayName>
</currency>
<currency type="TTD">
<displayName draft="contributed">dollar da Trinidad e Tobago</displayName>
</currency>
<currency type="TWD">
<displayName draft="contributed">nov dollar taiwanais</displayName>
</currency>
<currency type="TZS">
<displayName draft="contributed">schilling tansanian</displayName>
</currency>
<currency type="UAH">
<displayName draft="contributed">hryvnia ucranais</displayName>
</currency>
<currency type="UAK">
<displayName draft="contributed">karbovanetz ucranais</displayName>
</currency>
<currency type="UGS">
<displayName draft="contributed">schilling ucranais</displayName>
</currency>
<currency type="UGX">
<displayName draft="contributed">schilling ugandais</displayName>
</currency>
<currency type="USD">
<displayName draft="contributed">dollar dals Stadis Unids da l'America</displayName>
</currency>
<currency type="USN">
<displayName draft="contributed">dollar dals Stadis Unids da l'America (proxim di)</displayName>
</currency>
<currency type="USS">
<displayName draft="contributed">dollar dals Stadis Unids da l'America (medem di)</displayName>
</currency>
<currency type="UYI">
<displayName draft="contributed">peso da l'Uruguay (unidades indexadas)</displayName>
</currency>
<currency type="UYP">
<displayName draft="contributed">nov peso da l'Uruguay (1975-1993)</displayName>
</currency>
<currency type="UYU">
<displayName draft="contributed">peso da l'Uruguay</displayName>
</currency>
<currency type="UZS">
<displayName draft="contributed">sum usbec</displayName>
</currency>
<currency type="VEB">
<displayName draft="contributed">bolivar venezuelan</displayName>
</currency>
<currency type="VEF">
<displayName draft="contributed">bolivar fuerte venezuelan</displayName>
</currency>
<currency type="VND">
<displayName draft="contributed">dong vietnamais</displayName>
</currency>
<currency type="VNN">
<displayName draft="contributed">vegl dong vietnamais</displayName>
</currency>
<currency type="VUV">
<displayName draft="contributed">vatu dal Vanuatu</displayName>
</currency>
<currency type="WST">
<displayName draft="contributed">tala da la Samoa</displayName>
</currency>
<currency type="XAF">
<displayName draft="contributed">franc CFA BEAC</displayName>
</currency>
<currency type="XAG">
<displayName draft="contributed">argient</displayName>
</currency>
<currency type="XAU">
<displayName draft="contributed">aur</displayName>
</currency>
<currency type="XBA">
<displayName draft="contributed">unitad europeica cumponida</displayName>
</currency>
<currency type="XBC">
<displayName draft="contributed">unitad dal quint europeica (XBC)</displayName>
</currency>
<currency type="XBD">
<displayName draft="contributed">unitad dal quint europeica (XBD)</displayName>
</currency>
<currency type="XCD">
<displayName draft="contributed">dollar da la Caribica Orientala</displayName>
</currency>
<currency type="XDR">
<displayName draft="contributed">dretgs da prelevaziun spezials</displayName>
</currency>
<currency type="XFO">
<displayName draft="contributed">franc d'aur franzos</displayName>
</currency>
<currency type="XFU">
<displayName draft="contributed">franc UIC franzos</displayName>
</currency>
<currency type="XOF">
<displayName draft="contributed">franc CFA BCEAO</displayName>
</currency>
<currency type="XPD">
<displayName draft="contributed">palladi</displayName>
</currency>
<currency type="XPF">
<displayName draft="contributed">franc CFP</displayName>
</currency>
<currency type="XPT">
<displayName draft="contributed">platin</displayName>
</currency>
<currency type="XRE">
<displayName draft="contributed">fonds RINET</displayName>
</currency>
<currency type="XTS">
<displayName draft="contributed">code per verifitgar la valuta</displayName>
</currency>
<currency type="XXX">
<displayName draft="contributed">valuta nunenconuschenta</displayName>
</currency>
<currency type="YDD">
<displayName draft="contributed">dinar dal Jemen</displayName>
</currency>
<currency type="YER">
<displayName draft="contributed">rial dal Jemen</displayName>
</currency>
<currency type="YUD">
<displayName draft="contributed">dinar jugoslav (1966-1990)</displayName>
</currency>
<currency type="YUM">
<displayName draft="contributed">nov dinar jugoslav</displayName>
</currency>
<currency type="YUN">
<displayName draft="contributed">dinar jugoslav convertibel</displayName>
</currency>
<currency type="YUR">
<displayName draft="contributed">dinar jugoslav refurmà</displayName>
</currency>
<currency type="ZAL">
<displayName draft="contributed">rand sidafrican (finanzial)</displayName>
</currency>
<currency type="ZAR">
<displayName draft="contributed">rand sidafrican</displayName>
</currency>
<currency type="ZMK">
<displayName draft="contributed">kwacha da la sambia</displayName>
</currency>
<currency type="ZRN">
<displayName draft="contributed">nov zaire dal Zaire</displayName>
</currency>
<currency type="ZRZ">
<displayName draft="contributed">zaire dal Zaire</displayName>
</currency>
<currency type="ZWD">
<displayName draft="contributed">dollar dal Simbabwe</displayName>
</currency>
</currencies>
</numbers>
<units>
<unit type="day">
<unitPattern count="one" draft="contributed">{0} di</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} di</unitPattern>
<unitPattern count="other" draft="contributed">{0} dis</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} dis</unitPattern>
</unit>
<unit type="hour">
<unitPattern count="one" draft="contributed">{0} ura</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} ura</unitPattern>
<unitPattern count="other" draft="contributed">{0} uras</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} uras</unitPattern>
</unit>
<unit type="minute">
<unitPattern count="one" draft="contributed">{0} minuta</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} min.</unitPattern>
<unitPattern count="other" draft="contributed">{0} minutas</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} mins.</unitPattern>
</unit>
<unit type="month">
<unitPattern count="one" draft="contributed">{0} mais</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} mais</unitPattern>
<unitPattern count="other" draft="contributed">{0} mais</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} mais</unitPattern>
</unit>
<unit type="second">
<unitPattern count="one" draft="contributed">{0} secunda</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} sec.</unitPattern>
<unitPattern count="other" draft="contributed">{0} secundas</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} secs.</unitPattern>
</unit>
<unit type="week">
<unitPattern count="one" draft="contributed">{0} emna</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} emna</unitPattern>
<unitPattern count="other" draft="contributed">{0} emnas</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} emnas</unitPattern>
</unit>
<unit type="year">
<unitPattern count="one" draft="contributed">{0} onn</unitPattern>
<unitPattern count="one" alt="short" draft="contributed">{0} onn</unitPattern>
<unitPattern count="other" draft="contributed">{0} onns</unitPattern>
<unitPattern count="other" alt="short" draft="contributed">{0} onns</unitPattern>
</unit>
</units>
<posix>
<messages>
<yesstr draft="contributed">gea:g</yesstr>
<nostr draft="contributed">na:n</nostr>
</messages>
</posix>
</ldml>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,471 |
Q: WSUS not working on clients - deployed via GPO I've deployed WSUS on a server called INet
This has downloaded the required updates that I selected.
I've configured automatic updates in the GPO -
Set the following:
*
*Allow Automatic Updates immediate installation
*Allow non-administrators to receive update notifications
*Specify intranet Microsoft update service location:
(I set this to http://INet)
I also set the time to check for updates to 4pm (for testing)
I linked the GPO in the relevant OU, then signed on to a test computer (windows xp pro)
Logged in, checked the Update tab in My Computer, and it was grayed out, with 4pm set as the update time (so i know the GPO had worked)
4pm came and went, but no updates....
Checked on INet server in the WSUS console, still showed 0 computers...
What have I missed?
**
Edit / Update
**
I've now run "wuauclt /detectnow" and it has put some errors in the error log
Failed to find updates with error code 80244019
I also notice the url it is searching for is:
http://INet/ClientWebService/client.asmx
When i try this URL in IE from the client, I get a 404.
on looking on IIS on INet - i see the folder, but it is emtpy..
A: Ok,
I figured it out...
WSUS was installed on port 8530 for some reason... this may be the default port?
By changing my url in the GPO to http://INet:8530 it then worked.
I ran the wuauclt /detectnow command, and checked the error log.
There were a bunch of errors about Windows Installer 3.1 - then a balloon popped up, saying updates were ready etc..
I presume one of these updates will be the required Windows Installer 3.1
A: The %SystemRoot%\WindowsUpdate.log file is your friend. Assuming the "Automated Updates" service is started, you'll see the diagnostic output from the process there.
That's all I can really say w/o knowing more detail.
A: The time in the Automatic Updates configuration is the time when downloaded updates are installed, the check will stay on its usual schedule (see Automatic Updates detection frequency, default every 22 hours). Running 'wuauclt /detectnow' will trigger an immediate detection.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 461 |
using System;
using System.Collections.Generic;
using System.Text;
using System.Drawing.Imaging;
namespace System.Drawing {
public abstract class Image : MarshalByRefObject, IDisposable {
public static Image FromFile(string filename) {
if (string.IsNullOrEmpty(filename)) {
throw new ArgumentNullException("filename");
}
int width, height;
PixelFormat pixelFormat;
IntPtr native = LibIGraph.BitmapFromFile(filename, out width, out height, out pixelFormat);
return new Bitmap(native, width, height, pixelFormat);
}
internal IntPtr native = IntPtr.Zero;
protected int width, height;
protected PixelFormat pixelFormat;
~Image() {
this.Dispose();
}
public void Dispose() {
if (this.native != IntPtr.Zero) {
LibIGraph.DisposeImage(this.native);
GC.SuppressFinalize(this);
this.native = IntPtr.Zero;
}
}
public int Width {
get {
return this.width;
}
}
public int Height {
get {
return this.height;
}
}
public PixelFormat PixelFormat {
get {
return this.pixelFormat;
}
}
public Size Size {
get {
return new Size(this.width, this.height);
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,475 |
This is an article of notable issues relating to the terrestrial environment of Earth in 2020. They relate to environmental events such as natural disasters, environmental sciences such as ecology and geoscience with a known relevance to contemporary influence of humanity on Earth, environmental law, conservation, environmentalism with major worldwide impact and environmental issues.
Events
MT New Diamond
Environmental disasters
Environmental sciences
Geosciences, biotechnology, anthropology and geoengineering
Environmental policy
From 1 January 2020, ships will only be permitted to use fuel oil with a very low sulfur content. The International Maritime Organization estimates that the new limit of 0.5% sulfur content, down from 3.5%, will cut sulfur dioxide emissions from ships by about 8.5 million tonnes.
The Pacific nation of Palau bans sun cream that is harmful to corals and sea life in January 2020.
COVID-19 pandemic
In February - March 2020 campaigners say that governments should act with the same urgency on climate as on the coronavirus. The health crisis is reducing carbon emissions more than any policy.
Global air traffic decreased by 4.3% in February 2020 with cancellations of tens of thousands of flights to affected areas.
Chinese measures against coronavirus in Feb 2020 reduced coal consumption at power plants 36%, oil refining capacity 34% and satellite-based NO2 levels 37%.
US seafood imports and exports for fresh products dropped during the pandemic while frozen products were less affected. Demand for seafood at restaurants also dropped but there were increases in seafood delivery and takeout.
For further information see the tag [COVID-19] above.
Predicted and scheduled events
International goals
A list of − mostly self-imposed and legally voluntary or unenforceable − goals due by 2020 as decided by multinational corporate associations and international governance entities and their status:
See also
2020s in environmental history
2020 in climate change
List of large volcanic eruptions in the 21st century
Energy development
Timeline of solar cells#2020
2020 in space
Lists of extinct animals#Recent extinction
:Category:Species described in 2020
:Category:Protected areas established in 2020
Human impact on the environment
List of environmental issues
Outline of environmental studies
References
Environmental | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,533 |
Q: How can I do squence labeling and entities relationships labeling at the same time Is there an NLP annotation tool can do both of them?
Btw, I can't install Brat, the download page is 404 page.
I have doccane and tagtog but it seems that they can only do one kind of labeling.
A: I can recommend INCEpTION (https://inception-project.github.io/) for the kind of parallel annotation you're looking for.
It's got many features and takes some setting up, but it can handle more complex tasks and I've found it to be more reliable than Doccano for example. Its developers are from the Computer Science Department at TU Darmstadt and it seems well-suited for NLP research.
A: LightTag can, and it's free to use for moderate sized projects.
I happen to have made LightTag and this feature in particular. It's set up so that you can do entities and then drag and drop them onto each other to create a relationship tree. You can express constituency grammars or dependency grammars.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,108 |
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta http-equiv="X-UA-Compatible" content="ie=edge">
<title>Flip card</title>
<script src="https://npmcdn.com/react@15.3.0/dist/react.js"></script>
<script src="https://npmcdn.com/react-dom@15.3.0/dist/react-dom.js"></script>
<script src="https://npmcdn.com/babel-core@5.8.38/browser.min.js"></script>
<link href="https://fonts.googleapis.com/css?family=Galada" rel="stylesheet">
<link rel="stylesheet" type="text/css" href="style.css">
</head>
<body>
<div id="root"></div>
<script type="text/babel" src="main.js"></script>
</body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,647 |
@implementation TGFastBtn
-(void)setupUI{
self.titleLabel.textAlignment = NSTextAlignmentCenter;
}
- (instancetype)initWithFrame:(CGRect)frame{
if (self = [super initWithFrame:frame]) {
[self setupUI];
}
return self;
}
- (void)awakeFromNib{
[super awakeFromNib];
[self setupUI];
}
- (void)layoutSubviews{
[super layoutSubviews];
self.imageView.y = 0;
self.imageView.centerX = self.width * 0.5;
//self.imageView.width = 50;
//self.imageView.height = self.imageView.width;
self.titleLabel.y = self.height - self.titleLabel.height ;
[self.titleLabel sizeToFit];
self.titleLabel.centerX = self.width * 0.5;
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,263 |
Q: How do I find objects with a property inside another object in JavaScript I have a object with all my users, like so:var users = {user1:{}, user2:{}}, And every user has a isPlaying property. How do I get all users that have isPlaying false?
A: You should use Object.keys, Array.prototype.filter and Array.prototype.map:
// This will turn users object properties into a string array
// of user names
var userNames = Object.keys(users);
// #1 You need to filter which users aren't playing. So, you
// filter accessing users object by user name and you check that
// user.isPlaying is false
//
// #2 Using Array.prototype.map, you turn user names into user objects
// by projecting each user name into the user object!
var usersNotPlaying = userNames.filter(function(userName) {
return !users[userName].isPlaying;
}).map(function(userName) {
return users[userName];
});
If it would be done using ECMA-Script 6, you could do using arrow functions:
// Compact and nicer!
var usersNotPlaying = Object.keys(users)
.filter(userName => users[userName].isPlaying)
.map(userName => users[userName]);
Using Array.prototype.reduce
As @RobG has pointed out, you can also use Array.prototype.reduce.
While I don't want to overlap his new and own answer, I believe that reduce approach is more practical if it returns an array of user objects not playing.
Basically, if you return an object instead of an array, the issue is that another caller (i.e. a function which calls the one doing the so-called reduce) may need to call reduce again to perform a new operation, while an array is already prepared to fluently call other Array.prototype functions like map, filter, forEach...
The code would look this way:
// #1 We turn user properties into an array of property names
// #2 Then we call "reduce" on the user property name array. Reduce
// takes a callback that will be called for every array item and it receives
// the array reference given as second parameter of "reduce" after
// the callback.
// #3 If the user is not playing, we add the user object to the resulting array
// #4 Finally, "reduce" returns the array that was passed as second argument
// and contains user objects not playing ;)
var usersNotPlaying = Object.keys(users).reduce(function (result, userName) {
if (!users[userName].isPlaying)
result.push(users[userName]);
return result;
}, []); // <-- [] is the new array which will accumulate each user not playing
Clearly using Array.prototype.reduce concentrates both map and filter in a single loop and, in large array, reducing should outperform "filter+map" approach, because looping a large array twice once to filter users not playing and looping again to map them into objects again can be heavy...
Summary: I would still use filter+map over reduce when we talk about few items because sometimes readability/productivity is more important than optimization, and in our case, it seems like filter+map approach requires less explanations (self-documented code!) than reduce.
Anyway, readability/productivity is subjective to who does the actual coding...
A: Iterate through your users object:
var list = [];
for (var key in users) {
if (users[key].isPlaying === false) {
list.push(key);
}
}
This will give you a list of all users who have an isPlaying property that is false.
If you would like all of the user objects where isPlaying is false, you can add the objects themselves instead:
var list = [];
for (var key in users) {
if (users[key].isPlaying === false) {
list.push(users[key]);
}
}
A: This can also be achieved using Array.prototype.reduce, which is a great all round tool. It starts with getting an array of the names:
var userNames = Object.keys(users);
To return an array just the user names where isPlaying is false, you can do:
var usersNotPlaying = userNames.reduce(function(names, name) {
if (!users[name].isPlaying) {
names.push(name);
}
return names}, []);
To return an object of user objects with their names as keys is similar:
var usersNotPlaying = userNames.reduce(function(names, name) {
if (!users[name].isPlaying) {
names[name] = users[name];
}
return names}, {});
You could also use forEach in a similar way, however since it returns undefined the object or array collecting the members must be initialised in an outer scope first:
var usersNotPlaying = {};
userNames.forEach(function(name) {
if (!users[name].isPlaying) {
usersNotPlaying[name] = users[name];
}
});
You can also use for..in:
var usersNotPlaying = {};
for (var user in users) {
if (users.hasOwnProperty(user) && !users[user].isPlaying) {
usersNotPlaying[user] = users[user];
}
}
All of the above can return an array of names, array of user objects or object of user objects. Choose whatever suits. ;-)
A: Please try the JS code below: set all the isPlaying to false.
var json_object={};//{"user1":{"isPlaying":"false"},"user2":{"isPlaying":"ture"}};
json_object['user1']={"isPlaying":"false"};
json_object['user2']={"isPlaying":"ture"};
console.log(json_object);
for(var key in json_object){
if(json_object[key].isPlaying === "false"){/*do what you want*/}
}
console.log(json_object);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,518 |
Q: angular.js ng-repeat + infinite scroll causing too many watchers I have this code:
<section infinite-scroll='load()'>
<div class="product" ng-repeat="product in products | limitTo:total">
..
</div>
</section>
In controller:
$scope.products = []; // array of all products
$scope.load = function() {
$scope.total += 5;
};
When an user scrolls to the bottom controller increases limitTo variable so he would see 5 more products. If there are 40 products site has 4700 watchers. If he keeps scrolling there might be even 12k watchers.. Any ideas how to fix that? Each product has various inputs for adding to cart action.
I was taking a look for virtual repeat(e.g. this), but it's using a container with fixed height so it would be ugly and completely wrong to scroll products like that.
A: Angular will place a watch on every binding expression that you have. You'll have to reduce the number of binding expressions in your code to see a decrease in the number of watches.
However you can try to alleviate the problem by using track by to help angular figure out what shouldn't be re-evaluated.
<section infinite-scroll='load()'>
<div class="product" ng-repeat="product in products | limitTo:total track by $index">
..
</div>
</section>
In order to remove unneeded watches consider rendering only what is visible to the user and replacing everything else with empty elements if possible.
A: You don't say whether the list is supposed to be editable or not. That said, I don't think a monster list (thousands of elements) editable is feasible anyway.
In a project I'm working on we have the same use case as the one you present. So far, with about 5000 elements in the list, it seems to be working great: We have gone to great lengths to use one-time bindings and create efficient directories which frees up their watchers as quickly as possible.
For editing we pass on the element which is clicked on and open that in a separate context.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,077 |
Q: Struggle with typing redux-form component I am trying to export my component A by decorating it with redux-form in order to have access on my form-state, which is mainly filled by another component.
When trying to export my component, I get this typing-error:
TS2322 Type A is not assignable to type B
(Property 'onAbortHandler' does not exist on type 'IntrinsicAttributes & IntrinsicClassAttributes<FormInstance<{}.......)
I also tried to connect my Component and then to decorate my connected Component with reduxForm. Thats also not working.
This is my code
interface OwnProps {
onAbortHandler: () => void;
onSubmitHandler: () => void;
}
class MyComponent extends React.Component
<OwnProps & Partial<InjectedFormProps>> {
render() {
return (
<div>
<Button
className={'btn}
onClick={this.props.onAbortHandler}
>
<FormattedMessage id={'xx'}/>
</Button>
<Button
className={'btn'}
type="submit"
onClick={this.props.onSubmitHandler}
>
<FormattedMessage id={'xx'}/>
</Button>
</div>
);
}
}
export default reduxForm({
form: 'myFormName',
})(MyComponent);
Where is my mistake and why does this seem to be not straight forward?
A: This should work
interface OwnProps {
onAbortHandler: () => void;
onSubmitHandler: () => void;
}
class MyComponent extends React.Component
<OwnProps & InjectedFormProps<{}, OwnProps>> {
render() {
return (
<div>
<Button
className={'btn}
onClick={this.props.onAbortHandler}
>
<FormattedMessage id={'xx'}/>
</Button>
<Button
className={'btn'}
type="submit"
onClick={this.props.onSubmitHandler}
>
<FormattedMessage id={'xx'}/>
</Button>
</div>
);
}
}
export default reduxForm<{}, OwnProps>({
form: 'myFormName',
}
InjectedProps contains handleSubmit method that should know about the props you will pass to form. Check library types here for more info.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,845 |
{"url":"https:\/\/projecteuclid.org\/euclid.bj\/1458132990","text":"Bernoulli\n\n\u2022 Bernoulli\n\u2022 Volume 22, Number 3 (2016), 1520-1534.\n\nPerformance of empirical risk minimization in linear aggregation\n\nAbstract\n\nWe study conditions under which, given a dictionary $F=\\{f_{1},\\ldots,f_{M}\\}$ and an i.i.d. sample $(X_{i},Y_{i})_{i=1}^{N}$, the empirical minimizer in $\\operatorname{span}(F)$ relative to the squared loss, satisfies that with high probability\n\n$R(\\tilde{f}^{\\mathrm{ERM}})\\leq\\inf_{f\\in\\operatorname{span}(F)}R(f)+r_{N}(M),$ where $R(\\cdot)$ is the squared risk and $r_{N}(M)$ is of the order of $M\/N$.\n\nAmong other results, we prove that a uniform small-ball estimate for functions in $\\operatorname{span}(F)$ is enough to achieve that goal when the noise is independent of the design.\n\nArticle information\n\nSource\nBernoulli, Volume 22, Number 3 (2016), 1520-1534.\n\nDates\nRevised: February 2015\nFirst available in Project Euclid: 16 March 2016\n\nPermanent link to this document\nhttps:\/\/projecteuclid.org\/euclid.bj\/1458132990\n\nDigital Object Identifier\ndoi:10.3150\/15-BEJ701\n\nMathematical Reviews number (MathSciNet)\nMR3474824\n\nZentralblatt MATH identifier\n1346.60075\n\nCitation\n\nLecu\u00e9, Guillaume; Mendelson, Shahar. Performance of empirical risk minimization in linear aggregation. Bernoulli 22 (2016), no. 3, 1520--1534. doi:10.3150\/15-BEJ701. https:\/\/projecteuclid.org\/euclid.bj\/1458132990\n\nReferences\n\n\u2022 [1] Anthony, M. and Bartlett, P.L. (1999). Neural Network Learning: Theoretical Foundations. Cambridge: Cambridge Univ. Press.\n\u2022 [2] Audibert, J.-Y. (2007). Progressive mixture rules are deviation suboptimal. In Proceedings of the Twenty-First Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 36, 2007. Advances in Neural Information Processing Systems (NIPS) 20 41\u201348. Vancouver: MIT Press.\n\u2022 [3] Audibert, J.-Y. and Catoni, O. (2011). Robust linear least squares regression. Ann. Statist. 39 2766\u20132794.\n\u2022 [4] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford: Oxford Univ. Press.\n\u2022 [5] Breiman, L., Friedman, J.H., Olshen, R.A. and Stone, C.J. (1984). Classification and Regression Trees. Wadsworth Statistics\/Probability Series. Belmont, CA: Wadsworth Advanced Books and Software.\n\u2022 [6] Bunea, F., Tsybakov, A.B. and Wegkamp, M.H. (2007). Aggregation for Gaussian regression. Ann. Statist. 35 1674\u20131697.\n\u2022 [7] Catoni, O. (2004). Statistical Learning Theory and Stochastic Optimization. Lecture Notes in Math. 1851. Berlin: Springer. Lecture notes from the 31st Summer School on Probability Theory held in Saint-Flour, July 8\u201325, 2001.\n\u2022 [8] Chafa\u00ef, D., Gu\u00e9don, O., Lecu\u00e9, G. and Pajor, A. (2012). Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry. Panoramas et Synth\u00e8ses [Panoramas and Syntheses] 37. Paris: Soci\u00e9t\u00e9 Math\u00e9matique de France.\n\u2022 [9] Dalalyan, A.S. and Tsybakov, A.B. (2008). Aggregation by exponential weighting, sharp pac-Bayesian bounds and sparsity. Mach. Learn. 72 39\u201361.\n\u2022 [10] Devroye, L., Gy\u00f6rfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Applications of Mathematics (New York) 31. New York: Springer.\n\u2022 [11] de la Pe\u00f1a, V.H. and Gin\u00e9, E. (1999). Decoupling: From Dependence to Independence. Probability and Its Applications (New York). Randomly Stopped Processes. $U$-Statistics and Processes. Martingales and Beyond. New York: Springer.\n\u2022 [12] Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer Series in Statistics. New York: Springer.\n\u2022 [13] Juditsky, A. and Nemirovski, A. (2000). Functional aggregation for nonparametric regression. Ann. Statist. 28 681\u2013712.\n\u2022 [14] Juditsky, A., Rigollet, P. and Tsybakov, A.B. (2008). Learning by mirror averaging. Ann. Statist. 36 2183\u20132206.\n\u2022 [15] Lecu\u00e9, G. (2013). Empirical risk minimization is optimal for the convex aggregation problem. Bernoulli 19 2153\u20132166.\n\u2022 [16] Lecu\u00e9, G. and Mendelson, S. (2009). Aggregation via empirical risk minimization. Probab. Theory Related Fields 145 591\u2013613.\n\u2022 [17] Lecu\u00e9, G. and Mendelson, S. (2010). Sharper lower bounds on the performance of the empirical risk minimization algorithm. Bernoulli 16 605\u2013613.\n\u2022 [18] Lecu\u00e9, G. and Mendelson, S. (2013). Learning subGaussian classes: Upper and minimax bounds. Technical report, CNRS, Ecole polytechnique and Technion.\n\u2022 [19] Lecu\u00e9, G. and Mendelson, S. (2013). Minimax rate of convergence and the performance of ERM in phase recovery. Technical report. Electron. J. Probab. To appear.\n\u2022 [20] Lecu\u00e9, G. and Mendelson, S. (2014). Sparse recovery under weak moment assumptions. Technical report. J. Eur. Math. Soc. To appear.\n\u2022 [21] Lecu\u00e9, G. and Rigollet, P. (2014). Optimal learning with $Q$-aggregation. Ann. Statist. 42 211\u2013224.\n\u2022 [22] Mendelson, S. (2013). Learning without concentration. Technical report. J. ACM. Available at arXiv:1401.0304. To appear.\n\u2022 [23] Mendelson, S. (2014). Learning without concentration for general loss functions. Technical report, Technion, Israel and ANU, Australia. Available at arXiv:1410.3192.\n\u2022 [24] Mendelson, S. (2014). A remark on the diameter of random sections of convex bodies. In Geometric Aspects of Functional Analysis (GAFA Seminar Notes). Lecture Notes in Math. 2116 395\u2013404.\n\u2022 [25] Mendelson, S. and Koltchinskii, V. (2013). Bounding the smallest singular value of a random matrix without concentration. Technical report, Technion and Georgia Tech. Available at arXiv:1312.3580.\n\u2022 [26] Nemirovski, A. (2000). Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1738. Berlin: Springer. Lectures from the 28th Summer School on Probability Theory held in Saint-Flour, August 17\u2013September 3, 1998, edited by Pierre Bernard.\n\u2022 [27] Rigollet, Ph. and Tsybakov, A.B. (2007). Linear and convex aggregation of density estimators. Math. Methods Statist. 16 260\u2013280.\n\u2022 [28] Schapire, R.E. and Freund, Y. (2012). Boosting: Foundations and Algorithms. Adaptive Computation and Machine Learning. Cambridge, MA: MIT Press.\n\u2022 [29] Steinwart, I. and Christmann, A. (2008). Support Vector Machines. Information Science and Statistics. New York: Springer.\n\u2022 [30] Tsybakov, A.B. (2003). Optimal rate of aggregation. In Computational Learning Theory and Kernel Machines (COLT-2003). Lecture Notes in Artificial Intelligence 2777 303\u2013313. Heidelberg: Springer.\n\u2022 [31] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer. Revised and extended from the 2004 French original, translated by Vladimir Zaiats.\n\u2022 [32] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.\n\u2022 [33] Yang, Y. (2000). Mixing strategies for density estimation. Ann. Statist. 28 75\u201387.\n\u2022 [34] Yang, Y. (2001). Adaptive regression by mixing. J. Amer. Statist. Assoc. 96 574\u2013588.\n\u2022 [35] Yang, Y. (2004). Aggregating regression procedures to improve performance. 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Q: exception while using nativeJPQL delete Query I tried to delete using nativeQuery but I have this exсeption:
org.springframework.dao.InvalidDataAccessApiUsageException: Executing an update/delete query; nested exception is javax.persistence.TransactionRequiredException: Executing an update/delete query
at org.springframework.orm.jpa.EntityManagerFactoryUtils.convertJpaAccessExceptionIfPossible(EntityManagerFactoryUtils.java:403)
...................
Caused by: javax.persistence.TransactionRequiredException: Executing an update/delete query
at org.hibernate.internal.AbstractSharedSessionContract.checkTransactionNeededForUpdateOperation(AbstractSharedSessionContract.java:413)
at org.hibernate.query.internal.AbstractProducedQuery.executeUpdate(AbstractProducedQuery.java:1668)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:62)
...............
My Employe Entity:
@Entity
public class Employe implements Serializable{
private static final long serialVersionUID = 1L;
@Id
@GeneratedValue(strategy = GenerationType.IDENTITY)
private int id;
private String prenom;
private String nom;
private String email;
private int isActif;
@Enumerated(EnumType.STRING)
private Role role;
@OneToOne(mappedBy ="employe", cascade = CascadeType.REMOVE)
Contrat contrat;
@ManyToMany( mappedBy="employes", fetch= FetchType.EAGER , cascade = {CascadeType.PERSIST })
private List<Departement> departements;
@ManyToMany(cascade = {CascadeType.PERSIST, CascadeType.MERGE ,CascadeType.REMOVE})
List<Mission> missions;
.........
My Department Entity:
@Entity
public class Departement implements Serializable{
private static final long serialVersionUID = 1L;
@Id
@GeneratedValue(strategy = GenerationType.IDENTITY)
private int id;
private String name;
@ManyToOne( cascade = {CascadeType.PERSIST, CascadeType.MERGE} )
private Entreprise entreprise;
@OneToMany( mappedBy="departement", cascade = {CascadeType.PERSIST, CascadeType.MERGE})
private List<Mission> missions;
@ManyToMany(fetch = FetchType.EAGER, cascade = {CascadeType.PERSIST, CascadeType.MERGE})
private List<Employe> employes;
My Service :
@Autowired
IEmployeRepository empRep;
......
@Override
public void deleteEmployerById(int id) {
Employe e = empRep.findById(id).orElse(null);
// remove all associations for this author
empRep.deleteEmployerDepartementsAssociations(id);
empRep.delete(e);
}
My EmployeRepository:
@Repository
public interface IEmployeRepository extends CrudRepository<Employe, Integer>{
@Modifying
@Query(value = "DELETE FROM departement_employes WHERE employes_id = :id ", nativeQuery = true)
void deleteEmployerDepartementsAssociations(@Param("id") int id );
}
Only the delete Query do not work , I am trying to delete all the departments that are associated with a certain employe.
The association for the employe and the departments is many to many bidirectional association.
I am trying to delete the associations before deleting the employe
A: Try to correct your query in this way:
@Modifying
@Query(value = "DELETE FROM departement_employes WHERE employes_id = :id ", nativeQuery = true)
void deleteEmployerDepartementsAssociations(@Param("id") int id );
You have to wrap any database-modifying operation in a transaction. Look for example to this article.
So, as one of possible approaches you can annotate your service method by @Transactional annotation:
@Override
@Transactional
public void deleteEmployerById(int id) {
// ...
}
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Замок Кунетицка-Гора (, ) — средневековый замок в Чехии, основанный в XIV веке и сыгравший заметную роль во время гуситских войн. Замок расположен на одноимённой возвышенности недалеко от Пардубице.
История замка
Археологические исследования показали, что замок на Кунетицка-Горе существовал уже во 2-й половине XIV века. Вполне возможно, учитывая его стратегическое положение, что замок первоначально принадлежал королю Чехии и был возведён по его указанию.
В 1420 году замок перешёл под контроль гуситских войск и стал одной из их важнейших стратегических опорных точек. В замке разместился гуситский гетман Дивиш Боржек из Милетина, при котором в 1423 году замок был перестроен в горную гуситскую крепость, способную принять на зимовку большое количество войск. Замок стал военно-административным центром окрестных владений Дивиша Боржека, полученных им в результате завоеваний (в состав которых, в частности, вошли земли разорённого Опатовицкого монастыря).
На сословном сейме 1436 года в Праге король Сигизмунд Люксембургский официально подтвердил право собственности Дивиша Боржека на фактически принадлежавшие ему замок Кунетицка-Гора, городок Богуданеч и более 50-ти деревень бывшего Опатовицкого монастыря. После смерти Дивиша Боржека в 1437 году замок Кунетицка-Гора унаследовал его сын Собеслав Калека, верный сторонник Йиржи из Подебрад, которому он в 1464 году и продал замок.
Король Йиржи из Подебрад передал замок Кунетицка-Гора своим сыновьям, один из которых, Йиндржих I из Подебрад, в 1491 году продал замок вместе с кунетицким панством Вилему II из Пернштейна. Во владении Пернштейнов замок в 1491—1548 годах претерпел масштабную реконструкцию в ренессансном стиле с элементами поздней готики. В 1509 году во внутреннем дворе замка была возведена цилиндрическая башня. В 1560 году Ярослав из Пернштейна продал кунетицкое панство вместе с замком королевской казне.
Во время Тридцатилетней войны замок приобрёл стратегическое значение и 17 ноября 1645 года был захвачен шведскими войсками генерала Торстенссона. Замок был разграблен, сожжён и сильно разрушен. Разрушение замка постепенно продолжалось вплоть до начала XX века, прежде всего, благодаря тому, что замок буквально разбирали по камням (особенно с юго-западной стороны), которые использовали на строительстве других объектов. Это привело к тому, что в 1884 году часть замка обрушилась с западной и юго-западной стороны. Несмотря на то, что в XIX веке всё же предпринимались различные усилия по спасению и охране впечатляющих руин замка, это не привело к остановке его продолжающегося разрушения. Пожар 1896 года уничтожил все хозяйственные постройки у подножия замка.
Ситуация начала меняться, когда в 1917 году замок был арендован, а в 1919 году выкуплен Пардубицким музейным обществом. Попечение над руинами замка взяло на себя учреждённое в 1920 году кунетицкое товарищество. В 1923—1928 годах был проведён ремонт замка по проекту архитекторов Д. Юрковича и Й. Пацлова. Целью ремонта было, однако, не восстановление исторического памятника, а использование замка в коммерческих целях: были восстановлены некоторые стилизованные комнаты замкового дворца, где разместили экспозиции, и построен ресторан.
После Второй Мировой войны в замке был проведен капитальный ремонт и проведены работы по укреплению скалы в целях предотвращения обрушения. В 1953 году замок был национализирован правительством Чехословакии. В 70-х годах по причине неудовлетворительного состояния замок был закрыт для посещения. Замок открыли после реставрации только в 1993 году, однако восстановительные работы продолжаются и по сей день. В настоящее время замок является крупным туристическим центром в районе Пардубице, на его территории регулярно проводятся театральные, музыкальные и исторические мероприятия.
Литература
Ссылки
История замка на официальном сайте
Кунетицка-Гора
Национальные памятники культуры Чехии | {
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Flavio Pace (* 29. července 1977, Monza) je italský římskokatolický kněz a podsekretář Dikasteria pro východní církve.
Život
Narodil se 29. července 1977 v Monze.
Po studiu na gymnáziu ve svém rodném městě vstoupil roku 1996 do Arcibiskupského semináře v Miláně. Dne 29. září 2001 byl kardinálem Carlem Maria Martinim vysvěcen na jáhna a 8. června 2002 na kněze. Po vysvěcení působil nějakou dobu jako farní vikář v Abbiategrasso. Staral se o mládež, vyučoval náboženství a byl kaplanem v místním hospici. Poté začal studovat na Papežském institutu arabistických studií a islamologie, kde získal certifikát z islamologie, následně spolupracoval na některých iniciativách ve vztazích arcidiecéze Miláno s islámem.
Roku 2011 začal působit na Kongregaci pro východní církve. Mimo jiné byl kaplanem Malých sester Boží prozřetelnosti.
Dne 3. února 2020 jej papež František jmenoval podsekretářem Kongregace pro východní církve.
Externí odkazy
GCatholic
Catholic hierarchy
Press Vatican
Italští římskokatoličtí duchovní
Prefekti a sekretáři Dikasteria pro východní církve
Narození v roce 1977
Narození 29. července
Narození v Monze
Žijící lidé
Muži | {
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Winter is here – and it brings along some puzzle fun! Just in time for the coziest time of the year, winter is granting us a new never-ending logic hit. Winter Mosaics will mesmerize you with a heart-warming winterly imagery, relaxing tunes and most of all innumerous mosaic puzzles! Make yourself comfortable and dive into a winterly world of puzzles.
Experience hours of pure entertainment on cold winter days.
• Hard Drive: 58,9 Mb.
Часы чистого развлечения в холодные зимние дни. | {
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Junri Namigata and Erika Sema were the defending champions, having won the event in 2013, however both players chose to defend their titles with different partners. Namigata partnered with Akiko Yonemura but lost in the quarterfinals whilst Sema partnered with Miki Miyamura but lost in the first round.
Shuko Aoyama and Eri Hozumi won the title, defeating Naomi Broady and Eleni Daniilidou in the final, 6–3, 6–4.
Seeds
Draw
References
Draw
Fukuoka International Women's Cup - Doubles
Fukuoka International Women's Cup | {
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John Robinson (27 March 1823, Skipton – 9 July 1902, Westwood Hall, Leek, Staffordshire) was a British locomotive engineer.
He was apprenticed in 1839 to Sharp, Roberts and Co, and in 1843 he became a partner.
He became vice-chairman and co-managing director, with the chairman, C. P. Stewart.
Upon Stewart's death in 1882, Robinson became Chairman, which position he retained until his retirement in 1890, the year after the company moved to Glasgow.
He was a Member of the Institution of Mechanical Engineers from 9 April 1872, a Member of Council from 1866, a Vice-President from 1870. He was elected President in 1878, and in 1879.
He was a Justice of the Peace for Staffordshire, and was High Sheriff of Staffordshire for 1882.
Family
He married Helen Lees, (c 1827 - 1908) in 1848; they had children:
Emily Lees Robinson (22 May 1849 - 1900)
Rev. Arthur Edward Robinson (1851 - )
John Frederick Robinson (24 May 1853 -)
Helen Susan Robinson (1855 - )
Herbert M. Robinson (1859 - )
Edith Marianne Robinson (1860 - 1876)
References
1823 births
1902 deaths
English railway mechanical engineers
High Sheriffs of Staffordshire
People from Skipton
Engineers from Yorkshire | {
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The study focused on an Appraisal of Promotional Strategies of Barbing Salons in Enugu Metropolis with particular note on Top Class Barbing Salon with Enugu.
· To evaluate the promotional strategies adopted by Top Class Barbing Salon in order to determine these profitability.
· To determine the impact of public relations adopted by Top Class Barbing Salon in customers patronage.
· To determine if advertising strategies adopted by Top Class Barbing Salon in Enugu metropolis create customer's awareness of their services. Based on these four hypotheses was formulated each focusing on the impact of a specific promotion.
Data were sourced from two principle sources primary and secondary data. Extensive literature review on textbooks, Journals, and other related materials were carried out. The population includes management and relevant staff and customers of Top Class Barbing Salon within Enugu metropolis.
Data collected were presented analysed and interpreted using table, frequencies and percentages while the hypotheses were tested using chi-square.
Based on the analysis. The following findings were made. That barbing salon including the case organization are yet to fully appreciate the importance of designing an optimal combination of promotional mix to enhance it's performance.
That public relation impacts positively on customers patronage and billboards were poorly designed and not strategically located.
– Apart from the use of billboards advertising in radio and newspapers should be used. The advertising message should be based on quality of service offered, availability and moderate charges.
– Other promotional activities like sales promotion, public relations, publicity and personal selling should be adequately blended for improved performance. The researcher opined that if the recommendations were judiciously implemented the case organization should be able to serve their customer better with profit.
2.7 Evaluation of Promotional Strategies.
3.8 Method of Questionnaire Administration.
Ebue (1990 : 1) stated that if you produce the best product, package it brilliantly, price it right distribute it well and position it to best meet the needs of customer you must have wanted all the marketing skills if nobody knows you have done the things, and you customer and prospects can only know through promotional activities.
Adirika E, Ebue B. and Nnolim P. (2996 : 35) see promotion as the component used by the organization to inform, educate and persuade the market regarding the company's offerings, advertising, personal selling, sale promotion, publicity and public relations are the major variables of promotion.
Promotion is a vital ingredient of survival and development, without adequate promotion products may not sell when they sell their continuity is in doubt. The art and science of marketing promotion, which comprises advertising, personal selling, sales promotion, public relations and direct marketing is often associated with glamour and flamboyance. Infact most of the budget of some companies is spent on promotions because of the need to survive in the competitive marketing environment.
Onyeke K. J. (2003 : 88) confirm that the promotional tools serve as supreme vehicles in competition and provide the only way a market richer can, hope to penetrate an established market. He went further to state that for a company to excel above others in the competitive market such a company must value the import of promotion.
It may have been an unfelt want, there may have been no want at until business actions created it by advertising sales promotion, by investing something new.
Modern marketing companies are increasingly recognizing the value of an effective communication and promotion programme for their entire public. This includes Barbing Salon service companies.
Olakunori (2000 : 214) stated that the success of a company in a society of imperfect competition to a larger extent depends on the effectiveness and efficiency of marketing promotional activities.
The Emerson Theory is generally now being regarded as being misleading in today" world of noise and competition. A company must blow its trumpet he emphasized. Ebue noted that modern marketing does not stop at developing a good product, pricing it attractively and making it really available to target customers. The company must communicate to its audience, tell good stories, disseminate information about the products, existence, features terms and benefit to the target market.
Top Class Barbing Salon is a leading barbing salon in Enugu Metropolis. The salon started operation in 1989 with Okeigwe in Abia State.
The company does not involve so much in promotional activities; the only visible promotion is mounting of signboard which is a mere recognition of the location of the business premises. But with the proliferation of barbing salons in Enugu metropolis there is the need to appraise the promotional strategies used by the company in facing competition in the industry, which is becoming keener everyday. Many actouner and producers of goods and services are now aware that promotion does not only inform and persuade, but can strive towards profit making through increased sales.
It is in the light of his importance attached to promotion that the researcher seeks to make an Appraisal of promotional strategies of barbing salon in Enugu metropolis using Top Class Barbing Salon as a case study.
Inadequate sales are often given by entrepreneur, as major causes of their failure. A careful review of their circumstances often reveal perfectly ignorance of the need for promotional skill or deliberate neglect of the necessity for co-ordinate promotional strategies.
Many a time barbing salons are very sure/optimistic about sales (patronage) they concern the wrong notion that their goods will sell themselves, forgetting that even the best product, services still need to be stimulated in order to move out of the stores. Infact, the creation of effective promotional strategies is an essential move towards creating a market.
The above scenario applies equally to Top Class Barbing Salon despite the numerous advantages inherent is effective promotion, and the fact that the adage which says, if you don't say I am nobody will say you are still very true or you are here, the availability of promotional facilities, the need to used promotion, to higher competition in this industry, the use of co-ordinated/effective promo tools by Top Class Barbing Salon in its operation is barely evident and is negatively imposture on its operation.
In view of the above the researcher tried to appraise the promotional strategies of hair dressing salons in Enugu metropolis with special interest on Top Class Barbing Salon Enugu.
To determine the extent to which barbing salon owners in Enugu metropolis are aware of promo tools.
To determine the impact of direct marketing on customer patronage at Top Class Barbing Salon in Enugu metropolis.
To appraise the promotional strategies adopted by Top Class Barbing Salon in Enugu Metropolis in increase profitability.
The following hypothesis were formulated and tested.
Ho: Advertising strategies adopted by Top Class Barbing Salon in Enugu metropolis do not create consumers awareness of barbing salon services.
Hi: Advertising strategies adopted by Top Class Barbing Salon in Enugu metropolis create consumers awareness of barbing salon services.
Ho2: Public relation strategies adopted by Top Class Barbing Salon in Enugu metropolis have not helped to increase customer patronage.
Ho3 Personal selling strategies adopted by Top Class Barbing Salon in Enugu metropolis have no positive impact on your patronage.
Ho4 Promotional strategies adopted by Top Class Barbing Salon in Enugu metropolis have not increase their profitability.
Ho4 Promotional strategies adopted by Top Class Barbing Salon in Enugu metropolis have increase their profitability.
The study will be of great benefit to the operations of any firm at barbing salon. The study will show some of the cost effective and efficient promotion game plans that would be adopted to increase the performance of their service operation.
The study will equally help Top Class Barbing Salon to better their operations in Enugu metropolis. Consumers who patronize Top Class Barbing Salon will be better informed and educated not only on product sold but on their services. The study will act as a source of document to readers, who might find the study useful. Above all the study will benefit the researcher as it will provide the researcher with an in-depth knowledge and under standing in the area which could stir up further study.
The study which involves an appraisal of promotional strategies of barbing hair care salons took an in-depth look at promotional strategies applied by barbing salon. Again, the study is limited to Top Class Barbing Salon and its customers within Enugu metropolis.
1) Promotional strategy – It is controlled integrated programme of communication method and materials designed present a company and it products to prospective customer.
2) Advertising – Any term of non-personal communicating or presentation or goods ideas and services conducted through the mass media that is paid for by an identified sponsor.
3) Sales promotion – This consists of all activities aimed of promoting immediate sales. it is designed to achieve fast sales or consumer response.
4) Public relation – The deliberate planed and sustained effort to establish and maintain mutual understanding between an organization and its public.
5) Direct marketing _ If is an interaction marketing system that use one or more advertising media to effect a measurable response.
6) Personal selling- Fall to fall interaction with one or more prospective purchases for the purpose of making presentation answering questions and producing orders.
7) Source – Any activity benefit or satisfaction that is offered for sales it does not result in the ownership or anything.
8) Marketing- The management process responsible for identifying, anticipating and satisfying customers requirement profitably. | {
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A couple of weeks ago, I dusted off my favorite old cameras and headed up US 29 to Old Dominion Speedway, in Manassas, Virginia, for Bug Out 69. The "69" in this Bug Out's name simply indicates that it was the 69th semi-annual gathering of the clan. That's right, this venerable event is over 30 years old!
Despite what you're seeing in this photo, the day started with a heavy overcast. But the Bus in the Trees does capture the "rat rod" vibe that many owners embraced. "Rat rod" might be too strong a term, but, as you'll see, a bunch of the cars and van had a fine patina.
Samuel Cox, '56 Oval Window Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
Nothing could have been further from the rat rod aesthetic than Samuel Cox's gorgeous '56 Beetle. It looked like it had just been driven off a dealer's showroom floor. Better, actually. I wasn't the only person who liked it. The judges awarded Cox first place in Bug Sedan Modified ('62 and older) and Best of Show (air-cooled).
Bill Berry, '61 Beetle Volksrod. Bug Out 69, Manassas, Virginia, 29 May 2011.
Here's another car that impressed the judges. Bill Berry's Volksrod took second place in the Radical category. And, yes, that hot rod is street legal. Berry drove it to the show.
Hans Alvarez, '73 Super Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
As far as I can tell, nobody has a bad time at a Bug Out.
PFC Daniel Harrison, '62 Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
Daniel Harrison is the crew chief of a US Army Black Hawk helicopter. He was on leave for the weekend, showing off his beautiful black Beetle and visiting his family.
Carol Hoover, '70 Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
Carol Hoover calls her call "Carol's Lite/N Bug."
Adrian, Kari, and Gabi Giorgi, '69 Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
Adrian has worked on old VWs for over 20 years. He's always trying to come up with something new. So he's given this Bug an Afrika Korps feel.
Hoods for sale. Bug Out 69, Manassas, Virginia, 29 May 2011.
The swap meet and the vendors are a big part of any Bug Out. It takes a lot of work to keep elderly Vans and Beetles running.
Parts jumble. Bug Out 69, Manassas, Virginia, 29 May 2011.
That's a tiny American flag in the kid's hand. It was Memorial Day weekend, and this vendor was handing them out for free. He wanted to remind people what the holiday is all about. Bug Out 69, Manassas, Virginia, 29 May 2011.
Richard Butcher and Steph Novak, '74 VW Thing. Bug Out 69, Manassas, Virginia, 29 May 2011.
Like many of the owners at Bug Out (probably most), Richard Butcher works on his car himself.
Kate Vincent, '67 Double Cab Transporter. Bug Out 69, Manassas, Virginia, 29 May 2011.
Kate Vincent's Transporter won the third place trophy in the Truck Type 2 class. She's done a lot of the work on the truck, especially the wiring and interior. It doesn't stay in the garage -- it's her daily driver!
Jeremy Vincent, '67 Single Cab Pick Up Truck. Bug Out 69, Manassas, Virginia, 29 May 2011.
You guessed right. Jeremy is Kate's husband. His truck is also a daily driver. It's the work vehicle for his landscaping business. Like Kate, he works on the truck himself.
Andrew Marquesen, '65 Transporter. Bug Out 69, Manassas, Virginia, 29 May 2011.
Hard to believe, maybe, but Andrew Marquesen's Transporter is another daily driver -- it's his work truck. He's had it for about three years and does all the work on it himself. Marquesen comes from a VW family (like so many of the owners that I met). His father has owned several vans.
Bill Wright, '64 Type 2 Pick Up Truck. Bug Out 69, Manassas, Virginia, 29 May 2011.
The sun finally came out -- good for the photos, not so great for the body. It got hot quick.
In any case, Bill Wright's pick up was a gem, taking second place in the Truck Type 2 class. It's no trailer queen. He drove it up to Manassas from south of Richmond.
Drag racing Beetles in the staging lanes. Bug Out 69, Manassas, Virginia, 29 May 2011.
Here's something that always surprises outsiders -- drag racing Beetles.
A Beetle in staging. Bug Out 69, Manassas, Virginia, 29 May 2011.
But, why not? After all, we're at a race track (the first purpose-built drag strip east the the Mississppi, by the way).
A burnout, before a trip down the track. Bug Out 69, Manassas, Virginia, 29 May 2011.
Bruce Ridgeway, '72 Beetle. Bug Out 69, Manassas, Virginia, 29 May 2011.
Take Bruce Ridgeway's Beetle, for instance. It won the Super Pro class, covering the 1/8th mile track in less than seven seconds, at a top speed of nearly 100 miles per hour. Doesn't sound like much? That quarter-million-dollar Ferrari in your driveway would struggle to keep up. In fact, it probably couldn't.
By the way, if you're wondering how I got photos like these with my funky old cameras... I didn't. There are some things that Rolleiflex TLRs and Mamiya 6s just don't like to do.
Fender rainbow. Bug Out 69, Manassas, Virginia, 29 May 2011.
But back to the Rollei and the Mamiya.
Dana Kerrigan, '02 New Beetle Turbo S. Bug Out 69, Manassas, Virginia, 29 May 2011.
Dana Kerrigan was one of the nicest people I met at Bug Out, which is really saying something, since everyone was in a great mood. She had a fine day, to say the least, winning both the New Beetle Sedan and Convertible class and Best Display.
Miss Bug Out, finalist. Bug Out 69, Manassas, Virginia, 29 May 2011.
What's a car show without pretty young women? Well, there's actually no such thing.
Aurora Fegley, Miss Bug Out 69. Bug Out 69, Manassas, Virginia, 29 May 2011.
Aurora Fegley, Miss Bug Out 69, during the slalom competition. Bug Out 69, Manassas, Virginia, 29 May 2011.
And you thought that Aurora was just a pretty face. The girl can drive.
A Beetle heads down the drag strip. Bug Out 69, Manassas, Virginia, 29 May 2011.
Whenever there's a Bug Out 69, there's sure to be a 70. So mark your calendars for September 4th. | {
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Joseph Aníbal Carvallo Torres (Lota, Región del Biobío, Chile, 1 de mayo de 1988) delantero proveniente del Club Deportes Linares, equipo que jugaba en la Tercera División A donde el jugador fue comprado por Colo-Colo, pasó a las inferiores de Colo-Colo, luego pasó a la Primera B en calidad de préstamo a Deportes Temuco.
Luego pasó a Deportes Puerto Montt para fichar al año siguiente en Rangers de Talca de la Primera División de Chile.
En el mes julio del año 2009 es contratado por el club Municipal Iquique, donde permanece hasta septiembre de 2009, regresando en 2010 a Deportes Puerto Montt.
A fines del 2010 ficha por la Universidad de Concepción por expresa petición del director técnico Jaime Vera, quien lo entrenó en Deportes Puerto Montt.
El 18 de agosto de 2016 es confirmado como nuevo refuerzo de Deportes Melipilla para el campeonato de Segunda División Profesional 2016/17.
Selección nacional
En 2012 fue internacional con la Selección de fútbol de Chile, debutando en la derrota por 2-0 ante Paraguay el 15 de febrero de 2012.
Partidos internacionales
<center>
{|class="wikitable collapsible collapsed" style="width:100%"
|-
! colspan="45" | Partidos internacionales |- bgcolor=#DDDDDD style="background:beige"
! N.º
! Fecha!! Estadio!! Local!! Resultado!! Visitante!! Goles
!Asistencias
!DT!! Competición
|- style="text-align: center;"
| 1 || 15 de febrero de 2012 || Estadio Feliciano Cáceres, Luque, Paraguay || align=center| || bgcolor=Salmon| 2-0 || || || || Claudio Borghi || Amistoso
|- style="text-align: center;"
|Total || || || Presencias || 1 || Goles || 0
| ||
|-
|}
Clubes
Palmarés
Campeonatos nacionales
Referencias
Nacidos en Lota
Futbolistas de Chile
Futbolistas de la selección de fútbol de Chile en los años 2010
Futbolistas del Club de Deportes Linares
Futbolistas del Club de Deportes Temuco en los años 2000
Futbolistas del Club de Deportes Puerto Montt en los años 2000
Futbolistas del Club Social de Deportes Rangers en los años 2000
Futbolistas del Club de Deportes Iquique en los años 2000
Futbolistas del Club de Deportes Puerto Montt en los años 2010
Futbolistas del Club Deportivo Universidad de Concepción en los años 2010
Futbolistas del Club de Deportes Melipilla en los años 2010
Futbolistas del Club de Deportes La Serena en los años 2010
Futbolistas del Club Deportivo Ñublense en los años 2010
Futbolistas del Club Deportivo Ñublense en los años 2020 | {
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concentrate on the development of TV Android
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{"url":"https:\/\/multithreaded.stitchfix.com\/blog\/2017\/06\/28\/patterns-of-soa-denormalized-cache\/","text":"Patterns of Service-oriented Architecture: Denormalized Cache\n\nJune 28, 2017 - New York, NY\n\nNext up in our \u201cPatterns of Service-oriented Architecture\u201d series we\u2019ll talk about dealing with highly normalized data that spans many tables and services, or otherwise has a large object graph that reaches beyond just a simple database, by caching a denormalized version of it.\n\nIntent\n\nBuild custom objects in a cache from querying several different tables so that the data can be provided quickly via a single fetch from the cached data store.\n\nMotivation\n\nAn operational database that stores canonical data should be stored in a normalized schema. Such a schema is designed to prevent duplication and bad data, but it results in many interconnected tables that can be difficult to query. Although most database libraries ease the pain of querying such data, it can become quite expensive to do so.\n\nFor example, suppose you have users, shipping addresses, and orders (that have addresses):\n\nSuppose you need to create an \u201caccount\u201d view that shows the user\u2019s name, email, shipping address, and most recent order (along with its shipping address). That would require several queries to return that data. If you need to display this information for many users, it would require even more queries. If the data lives in a remote service (or multiple such services), the process is even more involved. At some point, this process could be too slow, resulting in timeouts.\n\nIf instead, you did the queries offline and built up a denormalized object (in this example, called Account), you could query that from a cache much more quickly:\n\nA further motivating example and discussion is on \u201cElasticSearch and Denormalization in Rails\u201d\n\nApplicability\n\nUse this when you are experiencing performance problems building up a complex object from a normalized, relational store. This can also be used if data is required from more than one source (e.g. a database and a remote service).\n\nStructure\n\nThe bulk of the logic lives in a Denormalizer, which knows what data sources to use in building up the complex object.\n\nThe denormalized object should ideally be a well-defined type, like a struct or simple object, as opposed to a hash or dictionary. This results in a much clearer definition of what should be in the object, and makes changes to the denormalization process explicit through managed code changes.\n\nAs with all caching, there\u2019s no easy to way to keep the cached object up to date. Two common strategies (which can be combined) are rebuilding when the underlying data changes, and rebuilding on a schedule.\n\nRebuilding on a schedule is simple, as you can walk a list of root objects and ask the Denormalizer to rebuild its cache for each object. The problem with this approach is when the data being indexed is so large that it takes too long to rebuild the entire thing.\n\nA better approach is to rebuild specific objects when their underlying data changes. This requires the ability to know when that data changed, which is not always easy. Often, your database can tell you when data changes, but if you are using a messaging system, you can adopt a convention to send messages when shared data is changing.\n\nIn any case, the selection of underlying technology is important. You might be tempted to use a database view (or a materialized view, which caches the results of a view in an intermediate table). Those work well if your data is all inside the same data store and requires no calculation or transformation. If, however, your data comes from an external service, or is the result of some sort of transformation, you will need some code to produce the denormalized object.\n\nAnti-Patterns and Gotchas\n\n\u2022 Cached data is naturally out of sync with reality and can lag. You will need to take this into account when designing your features. It could be sufficient to store the date when the data was last updated and surface that to a user, but you may need to do more sophisticated things like partially rebuild the cache if a user action would change the data they are looking at. Caching drives product decisions.\n\u2022 If you rebuild your cache based on events, be aware of the ordering of events. You may get events out of chronological order (depending on the type of event system and the way you process events).\n\u2022 Be cognizant of the durability of your cache. If rebuilding the cache is expensive, be sure to choose a technology (and configure it) for fault-tolerance and quick recovery. For example, if your cache is simply in the memory of your webserver, you could lose that at any moment, taking out your application as it struggles to rebuild the cache. Often, your relational database can hold your cache, since it\u2019s likely the most durable data store you have. Other common options are Elasticsearch, Redis, and memcache.","date":"2019-04-25 07:44:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2668200731277466, \"perplexity\": 1171.6761308766675}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578711882.85\/warc\/CC-MAIN-20190425074144-20190425100144-00476.warc.gz\"}"} | null | null |
Q: How do I set up a source code control system for myself? I program on my desktop in my office, but also sometimes at home in a different room on my laptop, and even away from home. What I need is a system that automatically or on-demand syncs my work from one to the other, at need.
I do not have a home network setup, and although I guess I could do it, that would be a question for another board, perhaps. I've thought about some kind of system that would keep the source code in the cloud, but I don't know enough about this to get started. I need a kind of free or cheap way to do this.
I work in .NET (Windows Phone 7, in fact).
A: You can use DVCS like Git or Mercurial that can create local repository, then install Dropbox and put your project folder (including the repository folder of course) into the dropbox folder.
Dropbox will handle the synchronization and it can handle offline situation as long as you are modifying only in one place between synchronization.
BTW Dropbox will not expose your files to public by default, but you can still expose them if you want.
EDIT
Concerning repository integrity in case Dropbox misses synchronizing a file or two, you can create a master repository outside Dropbox folder in your main PC, and push to it. So if the local repo inside Dropbox folder gets corrupted, just restore from the main PC. But I haven't experienced repository corruption.
A: Here are some quick instructions on how to set up a distributed VCS. The benefit of using git or mercurial is that you don't need to set up a server to get it all working as the repository is just a file system. You have the local repository in your computer next to your code (in git there is one .git folder and in mercurial there is one .hg folder in the project path).
Using git
Step 1: Download and install git. For windows you may want to use TortoiseGit. Some setup notes are here.
Step 2: Follow the git community book to initialize local repository for your project and commit to the repository.
Step 3: To put things on the "cloud" that is github you can follow this tutorial.
Using Mercurial
Step 1: Download and install mercurial. For windows you may want to use TortoiseHg.
Step 2: Follow the Quick Start guide to create repository for your project and commit to the repository.
Step 3 To put things on the "cloud" that is bitbucket you can follow this tutorial.
If you're working on .NET you might want to use Mercurial because of (sort of) better support in Windows at the writing moment.
A: The easiest way is to use one of the online systems. Checkout GitHub or BitBucket. For more information on Git or Mercurial, check out Git Reference and Hg Init, respectively.
A: Almost any DVCS would help.
The most popular are Git, mercurial, and a few others; but I really like Fossil. It's a single (small!) executable, easily portable, self-contained, multi-platform, and includes a wiki, web-based GUI, ticket system, documentation handler, etc.
A: As already @peter-rowell and others have rightfully said, The easiest way is to use any one of the DVCS like Git and the corresponding online systems like Github or Unfuddle. I personally use Git and Unfuddle.
Since, you've mentioned that getting Internet connection at home also isn't a problem for you, there is this new online IDE which runs in the browser and lives on the cloud. I didn't know anything was available like this before, but this looks very interesting and cool. Although, I haven't used it much, but its cool.
A: The option I would suggest would be Kiln. It is made by the same exact people that make FogBugz and Stack Exchange. So it should be pretty good, also it is free for up to 3 users I believe. The scale is just like FogBugz and the two can be intergrated so you can keep track of bugs and features as well.
Just like Joel had said in one of his blog posts bug tracking and source control are always a plus, even to a lone developer.
A: If privacy of your code is not as issue then Google Code might a viable solution. It takes care of the hosting part and it is free and fairly easy to setup. It supports Subversion, Mercurial and Git, which all have Windows clients. It also integrates with Google ID's so you can easily add contributors to the project. I host my hobby projects there using Mercurial. I have the TortoiseHg client installed on both my home and work computer, so I could work on my projects during breaks.
A: It depends on what your knowledge about setting up a server is and what time you're willing to invest.
Personally I've rented a small virtual server from a hoster and installed an Apache Webserver an integrated a Subversion repository (which I'm now converting to git). The setup really isn't that much and once it's up and running, you no longer have to worry about it. This setup has the advantage that you're not only be able to setup a version control repository but also other kind of stuff that you want to access from whereever you are (I've installed a WIKI and an issue tracker for example).
If that's too much overhead for you you can always use the already suggested online systems like GitHub.
A: The easiest is probably to go with git provider, github has the possibility to pay for a private account.
But don't forget about svn, it is simple and easy to use.
You could install a svn server somewhere that you can reach it, probably at work. (And project like Visual svn server makes this really easy on windows.)
As a client you could use either a Visual Studio plugin or tortoisesvn.
*
*http://www.visualsvn.com/server/
*http://tortoisesvn.net/
The only thing you need to remember is that you need to commit and update when you switch computers.
A: I use Beanstalk. It has worked great for me as a personal SVN server. They also have Git hosting but I haven't tried it. They have a free account to start with, and then you can graduate to a paid account if necessary.
A: Dropbox
All the other solutions are over-blow for your needs:
Just install Dropbox on both computers.
A full blown source control is nice but in your situation its not needed IMHO.
This is the most easiest simplest path to take -> you just end up with auto-synced folders on as many machines as you want.
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{% extends "house/house.html" %}
{% block content2 %}
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Biografia
Nacque a Napoli, dove frequentò il liceo classico "Vittorio Emanuele". Conseguita la maturità classica nel 1918, si iscrisse alla facoltà di giurisprudenza, ma verrà chiamato lo stesso anno alle armi, prestando quindi servizio in artiglieria. .
L'anno successivo abbandonò gli studi di giurisprudenza e si dedicò ai suoi interessi artistici e letterari, pubblicando nel 1923 il racconto filosofico La città degli uomini d'oggi edito dall'editrice Quattrini di Firenze.
Amico di Piero Gobetti, con cui collaborò alle riviste La Rivoluzione Liberale e Il Baretti, si trasferì nel 1927 a Torino dove visse tra penose ristrettezze lavorando come uomo di fatica alla FIAT.
Dopo molti sacrifici, fondò una propria casa editrice. Qui conobbe Lionello Venturi e sostenne un gruppo di artisti che saranno noti come i "Sei di Torino".
Nel 1929 Persico si trasferì a Milano, dove collaborò alla rivista Belvedere e intorno al 1930 fondò la Galleria del Milione: nel 1931 diresse con Giuseppe Pagano la rivista Casabella. Fu chiamato ad insegnare all'ISIA di Monza, una scuola di arte applicata.
Dal 1934 volse il suo interesse verso l'architettura, aderì al Movimento Razionalista, realizzò arredi di interni e allestimenti per esposizioni.
Persico venne trovato morto nella sua casa nel gennaio 1936; la morte, avvenuta fra il 10 e l'11 gennaio, presenta alcuni lati oscuri espressi nel dopoguerra, fra gli altri, da Oreste Del Buono in due articoli apparsi su Tuttolibri nel 1993, e da Andrea Camilleri in un libro inchiesta del 2012. Quest'ultimo ipotizzò che la morte di Edoardo Persico potesse essere stata causata da sicari fascisti.
Scritti
La Città degli uomini d'oggi, Firenze: Casa Ed. Italiana A. Quattrini, 1923.
Edoardo Persico (a cura di), Arte romana: la scultura romana e quattro affreschi della villa dei misteri, Milano: Domus, 1935.
Profezia dell'architettura, con una nota di Alfonso Gatto, Milano: Muggiani, 1945; a cura di Francesco D'Episcopo, Roma: Ripostes, 1990; con nota di Attilio Pracchi, Bologna: Ogni uomo e tutti gli uomini, 2010, ISBN 978-88-96691-05-2; Milano: Skira, 2012, ISBN 978-88-572-1395-8.
Edoardo Persico : Scritti critici e polemici (a cura di Alfonso Gatto), Rosa e Ballo Editori, Milano, 1947.
Giulia Veronesi (a cura di), Edoardo Persico: Tutte le opere : (1923-1935), 2 voll. Vol. I: "Politica, letteratura, pittura, scultura, teatro, fotografia, grafica, varie", Vol. II: "Architettura", Milano: Comunità, 1964.
Elena Pontiggia (a cura di), Destino e modernità. Scritti d'arte (1929-1935), Milano: Medusa, 2001, ISBN 978-88-88130-27-9.
Marcello Del Campo (a cura di), Edoardo Persico: scritti di architettura, Torino: Testo & immagine, 2004, ISBN 88-8382-114-9.
Note
Bibliografia
Francesco Tentori, Edoardo Persico. Grafico e architetto, Napoli: Clean, 2006, ISBN 88-8497-019-9.
Elena Pontiggia (a cura di), Edoardo Persico e gli artisti 1929-1936: il percorso di un critico dall'impressionismo al primitivismo, Milano: Electa, 1998.
Altri progetti
Collegamenti esterni
Antifascisti italiani
Architetti razionalisti
Direttori di periodici italiani
Sepolti nel cimitero maggiore di Milano | {
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\section{Introduction}
Let $\mathcal{A}_{g,n}$ be the moduli space of principally polarised abelian varieties with level $n \ge 3$ structure over $\overline{\mathbb{F}}_p$, and let $p$ be a prime number coprime to $n$. Recall the following classical result of Chai:
\begin{Thm}[Chai \cite{ChaiOrdinarySiegel}] \label{Thm:Chai}
Let $x \in \mathcal{A}_{g,n}(\overline{\mathbb{F}}_p)$ be a point corresponding to an ordinary principally polarised abelian variety. Then the prime-to-$p$ Hecke orbit of $x$ is Zariski dense in $\mathcal{A}_{g,n,\overline{\mathbb{F}}_p}$.
\end{Thm}
Our main result is a generalisation of this result to Shimura varieties of abelian type, for which we will need to introduce some notation.
\subsection{Main results}
Let $(G,X)$ be an Shimura datum of abelian type with reflex field $E$ and let $p>2$ be a prime that such that $G_{\mathbb{Q}_p}$ is quasi-split and unramified. Let $U_p \subset $ $\operatorname{Sh}_{G}$ be a hyperspecial compact open subgroup and let $U^p \subset G(\mathbb{A}_f^p)$ be a sufficiently small compact open subgroup. Let $\operatorname{Sh}_G/\overline{\mathbb{F}}_p$ be the special fiber of the canonical integral model of the Shimura variety of level $U^pU_p$ at a prime $v$ above $p$ of $E$. We will consider prime-to-$p$ Hecke orbits of points in the (dense open) $\mu$-ordinary locus $\operatorname{Sh}_{G,\mu\text{-ord}} \subset \operatorname{Sh}_G$.
Let $E_v$ be the $v$-adic completion of $E$, which is a finite extension of $\mathbb{Q}_p$. If $(G,X)$ is of Hodge type then there is a closed immersion $\operatorname{Sh}_G \to \mathcal{A}_{G,n}$ for some $n$ (see \cite{xu2020normalization}). If $E_v=\mathbb{Q}_p$, then the $\mu$-ordinary locus is equal to the intersection of $\operatorname{Sh}_G$ with the ordinary locus of $\mathcal{A}_{g,n}$.
\begin{mainThm} \label{Thm:Ordinary}
Suppose that $E_v=\mathbb{Q}_p$, then the prime-to-$p$ Hecke orbit of a point $x \in \operatorname{Sh}_{G,\mu\text{-ord}}(\overline{\mathbb{F}}_p)$ is Zariski dense in $\operatorname{Sh}_G$.
\end{mainThm}
Our result generalises results of Maulik-Shankar-Tang \cite{MaulikShankarTang} who deal with $\operatorname{GSpin}$ Shimura varieties associated to quadratic space over $\mathbb{Q}$ and $GU(1,n-1)$ Shimura varieties associated to imaginary quadratic fields $E$ with $p$ split in $E$; their methods are completely disjoint from ours. There is also work of Shankar \cite{ShankarEtranges} for Shimura varieties of type C, using a group-theoretic version of Chai's strategy of using hypersymmetric points and reducing to the case of Hilbert modular varieties. Shankar crucially proves that the Hodge map $\operatorname{Sh}_G \to \mathcal{A}_{g,n}$ is a closed immersion over the ordinary locus via canonical liftings, whereas we just cite Xu's work \cite{xu2020normalization}. Last we mention work of Zhou \cite{ZhouMotivic}, who proves our theorem for quaternionic Shimura varieties.
\subsection{A sketch of the proof}
A standard reduction argument shows that it suffices to prove the theorem when $(G,X)$ is of Hodge type and $G^{\text{ad}}$ is $\mathbb{Q}$-simple.
Let $y \in \operatorname{Sh}_{G, \text{ord}}(\overline{\mathbb{F}}_p)$ be an ordinary point, then it follows from Serre-Tate theory that the formal completion $\mathcal{A}_{g,n}^{/y}$ of $\mathcal{A}_{g,n}$ at $y$ is a formal torus and it follows from the main result of \cite{ShankarZhou} that
\begin{align}
\operatorname{Sh}_{G, U^p}^{/y} \subset \mathcal{A}_{g,n}^{/y}
\end{align}
is a formal subtorus. We will give an alternative proof of this fact using the results of \cite{KimLeaves}; this will also give us a precise description of the formal subtorus in question.
Now let $Z$ be the Zariski closure of the prime-to-$p$ Hecke orbit of $y$ and let $V$ be the smooth locus of $Z$, which is again prime-to-$p$ Hecke stable, and let $x \in V(\overline{\mathbb{F}}_p)$. Work of Chai on the deformation theory of ordinary $p$-divisible groups \cite{ChaiOrdinary} tells us that the dimension of the smallest formal subtorus of $\operatorname{Sh}_G^{/x}$ containing $V^{/x}$ is encoded in the unipotent part of the $p$-adic monodromy of the isocrystal $\mathcal{E}$ associated to the universal abelian variety $A/V$.
In order to control the monodromy of this isocrystal we first prove that $\ell$-adic monodromy of the universal abelian variety $A$ over $V$ is `maximal'. We do this by proving a general monodromy theorem for smooth Hecke-stable subvarieties of Shimura varieties of Hodge type, generalising work of Chai \cite{ChaiElladicmonodromy} in the Siegel case and others \cite{KasprowitzMonodromy}, \cite{LXiao} in the PEL case. Using results of D'Addezio \cite{d2020monodromy, AdezzioII} we get precise control over the $p$-adic monodromy of the isocrystal associated to $A[p^{\infty}]$. It follows that the smallest formal subtorus of $\operatorname{Sh}_G^{/x}$ containing $V^{/x}$ is equal to $\operatorname{Sh}_G^{/x}$.
We conclude by proving that the formal completion $V^{/x}$ is a formal subtorus of $\operatorname{Sh}_{G, U^p}^{/x}$. The rigidity theorem for $p$-divisible formal groups of Chai \cite{ChaiRigidity} reduces this to a computation in representation theory. More precisely, we know that the stabiliser $I_x(\mathbb{Z}_{(p)})$ of $x$ in $G(\mathbb{A}_f^p)$ acts on $V^{/x}$ and if we can prove that this action is `strongly non-trivial' then the rigidity theorem of Chai proves that $V^{/x}$ must be a formal subgroup of $\operatorname{Sh}_{G, U^p}^{/x}$.
\subsection{Acknowledgements}
It is clear that our approach owes a substantial intellectual debt to the work of Chai and Oort, and in fact our main results were conceived after reading a remark in \cite{ChaiConjecture}.
\section{An \texorpdfstring{$\ell$}{l}-adic monodromy theorem for Shimura varieties of Hodge type}
Let $(G,X)$ be a Shimura datum of Hodge type with reflex field $E$ such that $G^{\text{ad}}$ is $\mathbb{Q}$-simple and let $p>2$ be a prime such that $G_{\mathbb{Q}_p}$ is quasi-split and split over an unramified extension. Let $U_p \subset G(\mathbb{Q}_p)$ be a parahoric subgroup and let $U^p \subset G(\mathbb{A}_f^p)$ be a sufficiently small compact open subgroup. Let $\mathbf{Sh}_{G,U}/E$ be the Shimura variety of level $U=U^p U_p$ and for a prime $v | p$ of $E$ let $\mathscr{S}_{G,U}/\mathcal{O}_{E,v}$ be the canonical integral model of $\mathbf{Sh}_{G,U}$ constructed in \cite{KisinModels}.
Let $\operatorname{Sh}_{G, U^p}$ be the basechange to $\overline{\mathbb{F}}_p$ of this integral canonical model for some choice of map $\mathcal{O}_{E,v} \to \overline{\mathbb{F}}_p$. Let $\Sigma$ be a finite set of places of $\mathbb{Q}$ containing $p,\infty$ and the primes $\ell$ where $G_{\mathbb{Q}_{\ell}}$ is not totally isotropic. Let $G^{\text{sc}} \to G^{\text{der}}$ be the simply-connected cover of the derived group of $G$; we will almost always identify groups like $G^{\text{sc}}(\mathbb{A}_f^p)$ and $G^{\text{sc}}(\mathbb{Q}_{\ell})$ with their images in $G(\mathbb{A}_f^p)$ and $G(\mathbb{Q}_{\ell})$.
\subsection{Prime-to-\texorpdfstring{$p$}{p} monodromy}
There is a pro-\'etale $U^p$-torsor
\begin{align}
\pi:\varprojlim_{K^p \subset G(\mathbb{A}_f^p)} \operatorname{Sh}_{G,K^p} \to \operatorname{Sh}_{G, U^p}
\end{align}
over $\operatorname{Sh}_{G, U^p}$ such that the action of $U^p \subset G(\mathbb{A}_f^p)$ extends to an action of $G(\mathbb{A}_f^p)$. Let $Z \subset \operatorname{Sh}_{G, U^p}$ be a locally closed subvariety and let $\tilde{Z}$ be the inverse image of $Z$ under $\pi$. We say that $Z$ is stable under the prime-to-$p$-Hecke operators, or that $Z$ is $G(\mathbb{A}_f^p)$-stable, if $\tilde{Z}$ is $G(\mathbb{A}_f^p)$-stable. Similarly we will consider $Z$ that are stable under $G^{\text{sc}}(\mathbb{A}_f^p)$ or $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$. These are much more likely to be connected since $G^{\text{sc}}(\mathbb{A}_f^p)$ acts trivially on $\pi_0(\operatorname{Sh}_{G, U^p})$ while $G(\mathbb{A}_f^p)$ acts transitively (by Lemma 2.25 of \cite{KisinModels}).
\begin{Lem} \label{Lem:DevSmooth}
Let $Z \subset \operatorname{Sh}_{G, U^p}$ be a locally closed subvariety that is stable under the action of $G(\mathbb{A}_f^p)$ (or any other variant), then the smooth locus $U \subset Z$ is also stable under this action.
\end{Lem}
\begin{proof}
For $g \in G(\mathbb{A}_f^p)$ and $K^p \subset G(\mathbb{A}_f^p)$ there is a finite \'etale correspondence
\begin{equation}
\begin{tikzcd}
& \operatorname{Sh}_{G, K^p \cap g K^p g^{-1}} \arrow{dl}{p_1} \arrow{dr}{p_2} \\
\operatorname{Sh}_{G,K^p} & & \operatorname{Sh}_{G, g K^p g^{-1}} \arrow{r}{g} & \operatorname{Sh}_{G,K^p}.
\end{tikzcd}
\end{equation}
and the Hecke operator attached to $g$ is $g \circ p_2 \circ p_1^{-1}$. If this Hecke operators preserves $Z$ then it is clear that it takes smooth points of $Z$ to smooth points of $Z$ because all the maps are finite \'etale.
\end{proof}
It follows from the theory of connected Shimura varieties that the natural pro-\'etale $U^p$-torsor over $\operatorname{Sh}_{G, U^p}$ has a reduction to a $U^p_{\text{sc}}:=U^p \cap G^{\text{sc}}(\mathbb{A}_f^p)$-torsor over each connected component of $\operatorname{Sh}_{G, U^p}$ and moreover that there is a reduction which is stable under the action of $G^{\text{sc}}(\mathbb{A}_f^p)$. This fact will be used implicitly to state and prove our monodromy theorem.
\begin{Thm} \label{Thm:Monodromy}
Let $Z \subset \operatorname{Sh}_{G, U^p}$ be a $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable smooth locally closed subvariety and let $Z^{\circ}$ be a connected component of $Z$ that is not contained in the basic locus. Then the image of the $\ell$-adic monodromy
\begin{align}
\rho_{\ell, Z^ {\circ}}: \pi_1^{\text{\'et}}(Z^{\circ}) \to U_{\ell}^{\text{sc}}
\end{align}
is compact open for all $\ell$ and surjective for $\ell \not \in \Sigma$. Moreover, if $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$ acts transitively on the connected components of $Z$ then $Z$ is connected.
\end{Thm}
\begin{Cor}
Let $Z \subset \operatorname{Sh}_{G, U^p}$ be a $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable reduced locally closed subvariety that is not contained in the basic locus and suppose that $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$ acts transitively on the irreducible components of $Z$. Then $Z$ is irreducible.
\end{Cor}
\begin{proof}
Apply Theorem \ref{Thm:Monodromy} to the dense open smooth locus $U \subset Z$, which is $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable by Lemma \ref{Lem:DevSmooth}.
\end{proof}
This theorem is a generalisation of the main results of \cite{ChaiElladicmonodromy} in the Siegel case, the main results of \cite{KasprowitzMonodromy} in the PEL case (see also Section 4 of \cite{LXiao}). Our proofs are also generalisations of the proofs in the works cited, although we have do extra work at several points and so we give full details.
\begin{proof}
We start by proving that the $\ell$-adic monodromy cannot be finite.
\begin{Lem}[See Lemma 3.4 and Corollary 3.5 \cite{ChaiElladicmonodromy}] \label{Lem:Infinite}
Let $\ell \not=p$ be a prime of $\mathbb{Q}$ and let $Z^{\circ}$ be a connected component of $Z$ that is not contained in the basic locus. Then the image of $\rho_{\ell, Z^{\circ}}$ is infinite.
\end{Lem}
\begin{proof}
Choose a Hodge embedding for $(G,X)$ and let $\operatorname{Sh}_{G, U^p} \xhookrightarrow{} \mathcal{A}_{g,n}$ be the corresponding closed immersion (see \cite{xu2020normalization}) and $\pi:A \to \operatorname{Sh}_{G, U^p}$ the corresponding family of abelian varieties. The set of connected components $Z^{\circ}$ of $Z$ such that the image of $\rho_{\ell, Z^{\circ}}$ is finite is stable under $G^{\text{sc}}(\mathbb{A}_f^p)$. Indeed, if $g \in G^{\text{sc}}(\mathbb{A}_f^p)$ maps a connected component $Z_1$ to a connected component $Z_2$ then $Z_1$ and $Z_2$ have a common finite \'etale cover over which the two pullbacks of the universal abelian variety are isogenous. So we may replace $Z$ by the union of connected components which have finite $\ell$-adic monodromy, we will assume that $Z$ is nonempty for the sake of contradiction.
Consider the closure $\overline{Z}$ of $Z$ in (some choice of) toroidal compactification of $\operatorname{Sh}_{G, U^p}$, defined as in \cite{MP} by normalisation in a toroidal compactification of $\mathcal{A}_{g,n}$. We are going to show that $\overline{Z}$ is contained in the open Shimura variety $\operatorname{Sh}_{G, U^p}$, and for this it is enough to show that the image of $\overline{Z}$ doesn't intersect the boundary of the toroidal compactification of $\mathcal{A}_{g,n}$. In other words, we need to show that the universal family of semi-abelian schemes over $Z$ is actually an abelian scheme.
Let $x$ be a point in $\overline{Z}$ and let $y \in Z$, then by Corollary 1.9 of \cite{PoonenBertini} we can find an irreducible curve $C$ in $\overline{Z}$ containing both $x$ and $y$. Let $C' \subset C$ be the smooth locus of $C$, then we know that the $\ell$-adic monodromy of $A_{C'}$ is finite, and therefore it follows from Theorem 2.1 of \cite{OortCurvesIsotrivial} that there is an abelian variety $B/\overline{\mathbb{F}}_p$ and an \'etale cover $C'' \to C'$ such that $A_{C''}$ is isogenous to $B_{C''}$. Now take the normalisation $D$ of $C''$ in $C$ and let $D'$ be the (absolute) normalisation of $D$, which contains $C''$ as an open subscheme. It follows from a result of Faltings-Chai that the universal semi-abelian scheme over $D$ is isogenous to $B_{D}$. Therefore it follows that $x$ does not lie on the boundary, and hence $\overline{Z}$ doesn't intersect the boundary.
For a fixed $y \in \overline{Z}$ and any $x \in \overline{Z}$ as before we let $D$ be a smooth projective curve as above (constructed using \cite{PoonenBertini}). Then the $F$-isocrystal $\mathcal{E}_D$ over $D$ corresponding to $A_D[p^{\infty}]$ is constant on $D$. We know that $\mathcal{E}$ underlies an $F$ isocrystal with $G$-structure on $\operatorname{Sh}_{G, U^p}$, in other words there is an exact faithful tensor functor
\begin{align} \label{eq:Tannakian}
\operatorname{Rep}_{\mathbb{Q}_p}(G) \to \operatorname{F-Isoc}_{\operatorname{Sh}_{G, U^p}}
\end{align}
such that the representation $V$ corresponding to $G_{\mathbb{Q}_p} \to \operatorname{GSp}_V \to \operatorname{GL}_V$ is sent to the $F$-isocrystal $\mathcal{E}$ associated to the universal abelian variety over $\operatorname{Sh}_{G, U^p}$. It follows from Corollary 3.1.13 of \cite{KMPS} that the image factors of \eqref{eq:Tannakian} factors through the Tannakian subcategory of $\operatorname{F-Isoc}_{\operatorname{Sh}_{G, U^p}}$ generated by $\mathcal{E}_V$. It follows that the $F$-isocrystal with $G$ structure over $D$ is constant and thus that the corresponding map $|D| \to B(G)$ from \cite{RapoportRichartz} is constant. Since $x$ was arbitrary it follows that map $|\overline{Z}| \to B(G)$ is constant and hence that $\overline{Z}$ is contained in a single Newton stratum of $\operatorname{Sh}_{G, U^p}$; by assumption this Newton stratum is not the basic Newton stratum.
Next we show that $\overline{Z}$ intersects the basic locus of $\operatorname{Sh}_{G, U^p}$ to obtain a contradiction. For this we need to use the Ekedahl-Oort stratification on $\operatorname{Sh}_{G, U^p}$, see Theorem B of \cite{ShenZhang} for the basic properties that we will use below.
There is an Ekedahl-Oort stratum $\operatorname{Sh}_{G, U^p}^{w}$ of minimal dimension such that $\operatorname{Sh}_{G, U^p}^{w}$ intersects $\overline{Z}$ and by the minimality and the closure relations it follows that this intersection is closed in $\overline{Z}$, hence proper. Because $\operatorname{Sh}_{G, U^p}\{w\}$ is quasi-affine and $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable, if follows that this intersection is also quasi-affine and $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable. Therefore, the intersection is finite (quasi-affine+proper) and $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-stable, and because all finite $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$-orbits are contained in the basic locus\footnote{This is proved as in Lemma 3.3.2 and Proposition 3.1.5 of \cite{ZhouMotivic}, using Corollary 1.4.13 and Proposition 2.1.3 of \cite{KisinPoints} in the proofs.} we find that $\overline{Z}$ intersects the basic locus.
\end{proof}
Now let $\tilde{Z}_{\ell} \to Z$ be the $U_{\ell}^{\text{sc}}$ torsor with its $G^{\text{sc}}(\mathbb{Q}_{\ell})$-action and let $\tilde{Z^{\circ}_{\ell}} \to Z^{\circ}$ be its restriction to $Z^{\circ}$. Choose a connected component $z_{\ell}$ of $\tilde{Z^{\circ}_{\ell}}$, then the action maps give us homeomorphisms of (profinite) topological spaces (see the proof of Lemma 2.8 of \cite{ChaiElladicmonodromy})
\begin{align}
\pi_0 (\tilde{Z^{\circ}_{\ell}}) &\simeq U_{\ell}^{\text{sc}} / M \\
\pi_0 (\tilde{Z}_{\ell}) \supset G^{\text{sc}}(\mathbb{Q}_{\ell}) \cdot z_{\ell} &\simeq G^{\text{sc}}(\mathbb{Q}_{\ell})/P_{\ell},
\end{align}
where $P_{\ell} \subset G^{\text{sc}}(\mathbb{Q}_{\ell})$ is the stabiliser our chosen component (a closed topological subgroup). Now let $\mathbb{M}$ be the neutral component of the Zariski closure of $M$ in $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$. Note that $\mathbb{M}$ is a connected reductive group by the semi-simplicity of $\ell$-adic monodromy and so the neutral component $\mathbb{N}$ of the normaliser of $\mathbb{M}$ in $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$ is also connected reductive.
\begin{Claim}[See the proof of Proposition 4.1 of \cite{ChaiElladicmonodromy}]
The group $\mathbb{N}$ contains $P_{\ell}$.
\end{Claim}
\begin{proof}
It suffices to show that the Lie algebra of $\mathbb{M}$ is stable under conjugation by elements of $P_{\ell}$. The $p$-adic Lie algebra of the $p$-adic Lie group $M$ is equal to a $\mathbb{Z}_p$-lattice in the $p$-adic Lie algebra of its Zariski closure $\mathbb{M}$. Therefore it is enough to prove that the (rational) Lie algebra of $M$ is stable under conjugation by elements of $P_{\ell}$.
Let $G^{\text{sc}}(\mathbb{Z}_{\ell})$ be a compact open subgroup of $G^{\text{sc}}(\mathbb{Q}_{\ell})$ containing $M$, then we know that $G^{\text{sc}}(\mathbb{Z}_{\ell}) \cap P_{\ell}=M$. Moreover for every $\gamma \in P_{\ell}$ we can find an open subgroup $U \subset G^{\text{sc}}(\mathbb{Z}_{\ell})$ such that $\gamma U \gamma^{-1} \subset G^{\text{sc}}(\mathbb{Z}_{\ell})$. For example, we can just take the intersection of $G^{\text{sc}}(\mathbb{Z}_{\ell})$ with $\gamma G^{\text{sc}}(\mathbb{Z}_{\ell}) \gamma^{-1}$. It follows that $M \cap U$ is an open subgroup of $M$ that is stable under conjugation by $\gamma$. Since $M \cap U$ has the same rational Lie algebra as $M$ and therefore as $\mathbb{M}$, the Lie algebra of $\mathbb{M}$ is stable under conjugation by $\gamma$. \end{proof}
From our claim we deduce that $G/\mathbb{N}(\mathbb{Q}_{\ell})$ is compact, because it is a quotient of $G^{\text{sc}}(\mathbb{Q}_{\ell})/P_{\ell}$ which is compact. Since $\mathbb{N}$ is connected it follows from Propositions 8.4 and 9.3 of \cite{BorelTits} that it contains a parabolic subgroup of $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$ and because it is reductive it follows that $\mathbb{N}=G^{\text{sc}}_{\mathbb{Q}_{\ell}}$. Therefore we find that $\mathbb{M}$ is a normal subgroup of $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$ and hence it is equal to a simple factor of $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$.
Because $G^{\text{ad}}$ is $\mathbb{Q}$-simple, it is isomorphic to the restriction of scalars $\operatorname{Res}_{F/\mathbb{Q}} H$, where $H$ is an absolutely simple group over a number field $F$ (it follows from the classification of abelian type Shimura data that $F$ is totally real). This means that for a prime $\ell$ the simple factors of $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$ are isomorphic to $H^{\text{sc}}_{v}$ where $v$ ranges over places $F$ over $\ell$.
Thus if we choose a prime $\ell \not=p$ that remains inert in $F$, then the Zariski closure of the image of $\rho_{\ell, Z^{\circ}}$ is equal to $G^{\text{sc}}_{\mathbb{Q}_{\ell}}$. Since this Zariski closure does not depend on $\ell$, for example by Theorem 1.2.1 of \cite{d2020monodromy} (c.f. \cite{PalMonodromy}), we deduce that the Zariski closure of the image of $\rho_{\ell, Z^{\circ}}$ is equal to $G^{\text{sc}}$ for all $\ell$ and $Z^{\circ}$.
It follows that $\pi_0(\tilde{Z^{\circ}_{\ell}})$ and hence $\pi_0(\tilde{Z}_{\ell})$ is finite and therefore for $\ell \not \in \Sigma$ the action of $G^{\text{sc}}(\mathbb{Q}_{\ell})$ on $\pi_0(\tilde{Z}_{\ell})$ is trivial, because the group
$G^{\text{sc}}(\mathbb{Q}_{\ell})$ has no nontrivial finite quotients by Theorem 7.1 and Theorem 7.5 of \cite{PR}. It follows that we get an equality $P_{\ell}=G^{\text{sc}}(\mathbb{Q}_{\ell})$ for $\ell \not \in \Sigma$. This implies that the image of $\rho_{\ell, Z^{\circ}}$ is equal to $U_{\ell}^{\text{sc}}$ for all $\ell \not \in \Sigma$ because it is equal to $P_{\ell} \cap U_{\ell}^{\text{sc}}$.
To show that $Z$ is connected, we consider the $U^{\Sigma, \text{sc}}:=U^{\Sigma} \cap G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$ torsors
\begin{align}
\tilde{Z} &\to Z \\
\tilde{Z^{\circ}} &\to Z^{\circ},
\end{align}
which come, after a choice of basepoint, with homeomorphisms
\begin{align}
\pi_0(\tilde{Z}) &\simeq G^{\text{sc}}(\mathbb{A}_f^{\Sigma}) / P \\
\pi_0(\tilde{Z^{\circ}}) &\simeq U^{\Sigma, \text{sc}}/M,
\end{align}
where $M$ is the image of the prime-to-$\Sigma$ adic monodromy representation
\begin{align}
\rho: \pi_1^{\text{\'et}}(Z^{\circ}) \to U^{\Sigma, \text{sc}}.
\end{align}
Recall that we can write $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$, in the category of topological groups and in the category of topological spaces, as a colimit of the open subgroups
\begin{align}
G^{\text{sc}}(\mathbb{A}_f^{\Sigma}) &\simeq \varinjlim_{S} G^{\text{sc}}(\mathbb{A}(S)) \\
G^{\text{sc}}(\mathbb{A}(S))&:=\prod_{v\in S} G^{\text{sc}}(\mathbb{Q}_v) \times \prod_{v \not \in (S \cup \Sigma \cup \infty)} G^{\text{sc}}(\mathbb{Z}_{v}).
\end{align}
where $S$ runs over finite sets of places of $\mathbb{Q}$ disjoint from $\Sigma$ and $\infty$. Since the quotient map
\begin{align}
G^{\text{sc}}(\mathbb{A}_f^{\Sigma}) \to G^{\text{sc}}(\mathbb{A}_f^{\Sigma})/P=\pi_0(\tilde{Z})
\end{align}
is open the images of $G^{\text{sc}}(\mathbb{A}(S))$ are open and so we can write $\pi_0(\tilde{Z})$ as an increasing union of open subsets indexed by $S$. By compactness of $X$, it follows that we can choose $S$ such that $G^{\text{sc}}(\mathbb{A}(S))$ acts transitively on $\pi_0(\tilde{Z})$. We can take $S$ as big as we want, in particular we can assume that $G^{\text{sc}}(\mathbb{Z}_{v})=U_v^ {\text{sc}}$ for all $v \not \in (S \cup \Sigma \cup \infty)$. For such a choice of $S$, we find that
\begin{align}
\prod_{v\in S} G^{\text{sc}}(\mathbb{Q}_v) \label{eq:ProductGroups}
\end{align}
acts transitively on the connected components of
\begin{align}
\tilde{Z}_{S}:=\tilde{Z} / \prod_{v \not \in (S \cup \Sigma \cup \infty)} U_v^ {\text{sc}}.
\end{align}
Note that $\tilde{Z}_S \to Z$ is a torsor for
\begin{align}
U_S^{\text{sc}}:=\prod_{v \in S} U_v^{\text{sc}},
\end{align}
with corresponding monodromy representations
\begin{align}
\rho_S: \pi_1^{\text{\'et}}(Z^{\circ}) \to U_S^{\text{sc}}
\end{align}
for each connected component $Z^{\circ}$ of $Z$. If we can show that the image of $\rho_S$ is compact open, then it follows that $\pi_0(\tilde{Z}_S)$ is finite and therefore it is trivial because it has a transitive action by \eqref{eq:ProductGroups}. We know that for each $\ell \in S$ the image of $\rho_{\ell}$ contains a compact open pro-$\ell$ subgroup $V_{\ell}$ of $U_{\ell}^{\text{sc}}$ and it clearly suffices to show that the image of $\rho_S$ contains
\begin{align}
V_{S}:=\prod_{\ell \in S} V_{\ell}.
\end{align}
But we know that the pro-$\ell$ Sylow subgroup of $\pi_1^{\text{\'et}}(Z^{\circ})$ surjects onto $V_{\ell}$ for each $\ell \in S$, and therefore their product surjects onto $V_{S}$. We conclude that $\pi_0(\tilde{Z}_S)$ is trivial and hence $Z$ is connected.
\end{proof}
\subsection{\texorpdfstring{$p$}{p}-adic monodromy}
In this subsection we record a consequence of Theorem \ref{Thm:Monodromy} in combination with the main reuslts of \cite{d2020monodromy, AdezzioII}. For this we need to remark that our choice of Hodge embedding gives us an overconvergent $F$-isocrystal $\mathcal{E}^{\dagger}$ with underlying $F$-isocrystal $\mathcal{E}$. Recall (see \cite{Kottwitz1}) that for $b \in B(G)$ there is a fractional cocharacter $\nu_b$ with associated parabolic $P_{\nu_b}$ and Levi $M_b$.
\begin{Cor} \label{Cor:padicMonodromy}
Let $Z \subset \operatorname{Sh}_{G, U^p}$ be a $G(\mathbb{A}_f^p)$-stable smooth locally closed smooth and let $Z^{\circ}$ be a connected component of $Z$ that is not contained in the basic locus. Then the $p$-adic monodromy of $\mathcal{E}^{\dagger}$ is isomorphic to $G^{\text{der}}$. If $Z$ is contained in a central leaf $C$ inside a non-basic Newton stratum $\operatorname{Sh}_{G,b,U^p}$, then the $p$-adic monodromy of $\mathcal{E}$ is the parabolic subgroup $P(\nu_b)$ of $G^{\text{der}}$ with Levi $M_{b}$.
\end{Cor}
\begin{proof}
It follows from Theorem 1.0.1 that the $\ell$-adic monodromy of the abelian variety over $Z$ is equal to $G^{\text{der}}$, and so it follows from Theorem 1.2.1 of \cite{d2020monodromy} that the geometric monodromy group of the overconvergent isocrystal $\mathcal{E}^{\dag}$ is also equal to $G^{\text{der}}$. Because we are contained in a single central leaf our $p$-divisible group has a slope filtration by Proposition 2.5 of \cite{ChaiSustained} and hence the $F$-isocrystal $\mathcal{E}$ has a slope filtration. Theorem 1.1.1 of \cite{AdezzioII} then shows that the geometric monodromy group of the underlying isocrystal $\mathcal{E}$ is equal to the parabolic subgroup $P(\lambda_b) \cap G^{\text{der}} \subset G^{\text{der}}$, where $\nu_b$ is the fractional cocharacter of $G$ coming from the slope filtration on $\mathcal{E}$.
\end{proof}
\begin{Rem} \label{Rem:DimensionUnipotent}
It follows that the dimension of the unipotent radical of the $p$-adic monodromy group is equal to $\langle 2 \rho, \nu_b \rangle$, which is equal to the dimension of $C$ by Proposition 2.6 of \cite{Hamacher}.
\end{Rem}
\section{The ordinary locus and ordinary Hecke orbits}
\renewcommand{\operatorname{Sh}_{G, U^p}}{\operatorname{Sh}_G}
As before we assume that $G_{\mathbb{Q}_p}$ is quasi-split and that $U_p$ is a hyperspecial compact open subgroup of $G$ and we will drop the level $U^p$ from the notation. Let $\operatorname{Sh}_G \to \mathcal{A}_{g,n}$ be the closed immersion (see \cite{xu2020normalization}) corresponding to a choice of Hodge embedding for $(G,X)$ and define ordinary locus of $\operatorname{Sh}_{G, U^p}$ by $\operatorname{Sh}_{G, \text{ord}}:=\operatorname{Sh}_G \cap \mathcal{A}_{g,n}^{\text{ord}}$.
We will need to put extra assumptions on the place $v|p$ to make sure that the ordinary locus is nonempty for some (equivalently any) choice of Hodge embedding. The following result is a generalisation of the main result of \cite{BultelOrdinary}, see also Section 3.3 of \cite{KisinFarb}.
\begin{Lem} \label{Lem:Ord}
The ordinary locus $\operatorname{Sh}_{G, \text{ord}}:=\operatorname{Sh}_G \cap \mathcal{A}_{g,n}^{\text{ord}}$ is nonempty if and only if $E_v = \mathbb{Q}_p$.
\end{Lem}
\begin{proof}
There is $G(E)$-conjugacy class of cocharacters $\{\mu\}$ of $G$ coming from the Shimura datum $X$ and our choice of $v | p$ defines a $G(E_v)=G(\mathbb{Q}_p)$-conjugacy class of cocharacters of $G_{\mathbb{Q}_p}$. Because $G_{\mathbb{Q}_p}$ is quasi-split and unramified it follows that there is a representative $\mu$ of $\{\mu\}$ defined over $\mathbb{Q}_p$.
The $\mu$-ordinary locus is the largest Newton stratum in $\operatorname{Sh}_{G, U^p}$ and corresponds to the $\sigma$-conjugacy class of $\mu(p)$ inside of $B(G)$ and the image image of the $\mu$-ordinary locus under the Hodge embedding lies inside the Newton stratum corresponding to the $\sigma$-conjugacy class of $\mu(p)$ inside of $B(\operatorname{GSp}_{2g, \mathbb{Q}_p})$. The ordinary locus corresponds to the $\sigma$-conjugacy class of $\nu(p)$, where $\nu$ is a representative over $\mathbb{Q}_p$ of the $G(\mathbb{Q}_p)$ conjugacy class of cocharacters $\{\nu\}$ coming from the Shimura datum on $\operatorname{GSp_{2g}}$.
But the conjugacy classes of cocharacters $\{\nu\}$ and $\{\mu\}$ of $\operatorname{GSp}_{2g}$ are conjugate over $\overline{\mathbb{Q}}_p$ and therefore over $\mathbb{Q}_p$ because $\operatorname{GSp}_{2g}$ is split. Indeed, we can conjugate them both into the same maximal torus, and then use the fact that the relative Weyl group and the absolute Weyl group agree. It follows that the $\mu$-ordinary locus coincides with the ordinary locus.
Conversely Theorem 0.4 of \cite{KisinPoints} tells us every Newton stratum contains the reduction of a special point $(T,h) \subset (G,X)$ and Lemma 2.2 of \cite{BultelOrdinary} tells us that this reduction cannot be ordinary if $E_v \not= \mathbb{Q}_p$, so we are done.
\end{proof}
\subsection{The formal neighborhood of an ordinary point}
Let $x \in \operatorname{Sh}_{G, U^p}(\overline{\mathbb{F}}_p)$ be an ordinary point and consider the closed immersion of formal neighbourhoods
\begin{align}
\operatorname{Sh}_{G, U^p}^{/x} \subset \mathcal{A}_{g,N}^{/x}.
\end{align}
We know that the latter is canonically isomorphic to $\tfrac{g(g+1)}{2}$-copies of $\hat{\mathbb{G}}_m$. It is slightly easier to think of $\operatorname{Sh}_{G, U^p}^{/x}$ as a closed formal subscheme of the $g^2$-dimensional Serre-Tate torus of which $\mathcal{A}_{g,N}^{/x}$ is the fixed points under the involution coming from the polarisation. This $g^2$-dimensional Serre-Tate torus can be thought of as the internal-hom $p$-divisible group between $(\mathbb{Q}_p/\mathbb{Z}_p)^{\oplus g}$ and $(\mu_{p}^{\infty})^{\oplus g}$. Something that generalises slightly better is to think of it as the connected part of the internal hom $p$-divisible group $\mathcal{H}$ of
\begin{align}
(\mathbb{Q}_p/\mathbb{Z}_p)^{\oplus g} \oplus (\mu_{p}^{\infty})^{\oplus g}.
\end{align}
We can think of the rational Dieudonn\'e-module of this latter formal group as the non-positive slope (with the convention that slopes of Dieudonn\'e-modules of $p$-divisible groups are the additive inverses of the slopes of their $p$-divisible groups, as in Section 4.1 of \cite{CaraianiScholze}) part of the following $F$-isocrystal: Take the Lie algebra $\mathfrak{gl}_{2g}$ equipped with the Frobenius $\operatorname{Ad}\nu(p) \sigma$, where $\nu$ is the Hodge cocharacter of $\operatorname{GSp}_{2g} \subset \operatorname{GL}_{2g}$ as above. %
There is a subgroup $\mathcal{H}^G \subset \mathcal{H}$ of this internal endomorphism $p$-divisible group given by the tensor-preserving endomorphisms, constructed in Section 3.1 of \cite{KimLeaves}. By Proposition 3.1.4 of loc. cit. we can identify the rational Dieudonn\'e module of $\mathcal{H}^G$ with the non-positive slope part of the following $F$-isocrystal
\begin{align}
(\mathfrak{g}, \operatorname{Ad} \mu(p) \sigma).
\end{align}
The slope filtration of this $F$-isocrystal can be described as follows, the slope $t$ part is given by
\begin{align} \label{Eq:SlopeFiltrationIsocrystal}
\bigoplus_{\substack{\alpha_0 \in \phi_0^{+} \\ \langle \alpha_0, \mu \rangle=t}} \mathfrak{u}_{\alpha_0},
\end{align}
where $\alpha_0$ runs over the relative roots of $G$ with respect to a maximal split torus contained in a Borel $B$ with respect to which $\mu$ is dominant; because $\mu$ is minuscule all the slopes are contained in $\{-1,0,1\}$.
We see that the slope $\le 0$ part of the isocrystal is precisely given by the Lie algebra of the standard parabolic subgroup $P_{\mu}$ containing $B$, that the slope $<0$ part is given by the unipotent radical of $P_{\mu}$ and that the slope equal to zero is given by the Lie algebra of the standard Levi $M_{\mu}$. It follows that the dimension of the connected part of $\mathcal{H}^{\mathcal{G}}$ is equal to $\langle 2 \rho, \mu \rangle$ which is precisely the dimension of $\operatorname{Sh}_{G, U^p}$. Thus it makes sense to claim that the formal subgroup $\mathcal{H}^{G, \circ} \subset \mathcal{H}^{\circ}$ is equal to $\operatorname{Sh}_{G, U^p}^{/x}$.
To prove this we use the results of \cite{KimLeaves}: It is proven there that $\operatorname{Sh}_{G, U^p}^{/x} \subset \hat{\mathbb{G}}_m^{\oplus g^2}$ is stable under the action of
\begin{align} \label{Eq:Qisog}
\operatorname{Qisog}^{\circ}_G \left((\tfrac{\mathbb{Q}_p}{\mathbb{Z}_p})^{\oplus g} \oplus \mu_{p^{\infty}}^{\oplus g}) \right) \subset \operatorname{Qisog}^{\circ} \left((\tfrac{\mathbb{Q}_p}{\mathbb{Z}_p})^{\oplus g} \oplus \mu_{p^{\infty}}^{\oplus g}) \right)
\end{align}
on $\hat{\mathbb{G}}_m^{\oplus g^2}$. Here $\operatorname{Qisog}$ means the group of self quasi-isogenies of a $p$-divisible group, which acts on the universal deformation space of that $p$-divisible group as explained in Section 4.3 of \cite{KimLeaves}. Because we are in the two slope case the inclusion \eqref{Eq:Qisog} can be identified with
\begin{align}
\tilde{\mathcal{H}} \subset \tilde{\mathcal{H}^{\circ}},
\end{align}
where the tilde means taking the universal cover of a $p$-divisible group as in \cite{ScholzeWeinstein}. The action of $\tilde{\mathcal{H}}^{\circ}$ on $\hat{\mathbb{G}}_m^{\oplus g^2}$ factors through the quotient
\begin{align}
\tilde{\mathcal{H}^{\circ}} \to \mathcal{H}^{\circ}
\end{align}
and is then just given by the multiplication map. Therefore the action of the universal cover of $\mathcal{H}^{G, \circ}$ on $\operatorname{Sh}_{G, U^p}^{/x}$ factors through $\mathcal{H}^{G, \circ}$ and for dimension reasons it follows that this identifies $\operatorname{Sh}_{G, U^p}^{/x}$ with $\mathcal{H}^{G, \circ}$.
\subsection{The formal neighborhood of a prime-to-\texorpdfstring{$p$}{p} Hecke orbit}
Let $x \in \operatorname{Sh}_{G, U^p}(\overline{\mathbb{F}}_p)$ be an ordinary point and let $Z$ be the Zariski closure in $\operatorname{Sh}_{G, \text{ord}}$ of the $G(\mathbb{A}_f^p)$-orbit of $x$, this is a reduced locally closed subvariety of $\operatorname{Sh}_{G, U^p}$ that is stable under $G(\mathbb{A}_f^p)$. Let $z \in Z(\overline{\mathbb{F}}_p)$ smooth point of $Z$.
\begin{Prop} \label{Prop:SubTorus}
Let $Z_{/z}$ be the formal completion of $Z$ at $z$, then $Z^{/z}$ is a formal subtorus of $\operatorname{Sh}_{G, U^p}^{/z}$.
\end{Prop}
\begin{proof}
Let $I_z(\mathbb{Q})$ be the group of self quasi-isogenies of $z$ respecting the tensors and let $\operatorname{Aut}(A_z[p^{\infty}])$ be the group of automorphisms of the $p$-divisible group associated to $z$ preserving the tensors, it is a compact open subgroup of $J_b(\mathbb{Q}_p)$ which doesn't depend on $z$. Let $I_z(\mathbb{Z}_{(p)})$ be the intersection of $I_z(\mathbb{Q})$ with $\operatorname{Aut}(A_z[p^{\infty}])$ inside $J_b(\mathbb{Q}_p)$; this is the stabiliser of $z$ under the $G(\mathbb{A}_f^p)$-action. Indeed, by Theorem 0.3 of \cite{KisinPoints} the isogeny class $\mathscr{I}_z$ has the following shape (Rapoport-Zink uniformisation):
\begin{align}
I_z(\mathbb{Q}) \backslash G(\mathbb{A}_f^p) \times X_{\mu}(b_{\text{ord}}) / U^p \simeq \mathscr{I}_z,
\end{align}
where
\begin{align}
X_{\mu}(b_{\text{ord}}) = \frac{J_b(\mathbb{Q}_p)}{\operatorname{Aut}(A_z[p^{\infty}])}.
\end{align}
The stabiliser of the point $z$ under the prime-to-$p$ Hecke action is then precisely $I_z(\mathbb{Z}_{(p)})$.
We know that $\operatorname{Aut}(A_z[p^{\infty}])$ acts on the deformation space $\operatorname{Sh}_{G, U^p}^{/z}$ and we see that $Z^{/z}$ is stable under the action of $I_z(\mathbb{Z}_{(p)})$ and hence of its closure in $\operatorname{Aut}(A_z[p^{\infty}])$. We know that the algebraic group $I_{\mathbb{Q}_p} \subset J_b$ has the same rank of $J_b$ by Corollary 2.1.7 of \cite{KisinPoints}. It follows that the closure of $I_z(\mathbb{Z}_{(p)})$ contains an open subgroup of a maximal torus $T$ of $J_b(\mathbb{Q}_p)$. In order to show that $Z^{/z}$ is a formal subtorus we want to apply the formal rigidity results of \cite{ChaiRigidity}, and for this we need to understand the action of $T$ on the Dieudonn\'e module of the formal group $\mathcal{H}^{G}$.
Using the discussion from the previous section, we can identify $T$ and its action on the Dieudonn\'e-module of the formal group $\operatorname{Sh}_{G, U^p}^{/z}$ with a maximal torus $T$ of $M_{\mu}$ acting on the Lie algebra of the unipotent radical of $P_{\mu}$ by conjugation. This action doesn't have any trivial subquotients, and so it follows from Theorem 4.3 of \cite{ChaiRigidity} that $Z^{/z}$ is a formal subgroup of $\operatorname{Sh}_{G, U^p}^{/z}$.
\end{proof}
\section{Chai's canonical coordinates and monodromy}
\subsection{Canonical coordinates}
In this section we let $Z/\overline{\mathbb{F}}_p$ be a reduced scheme together with a map $f:Z \to \mathcal{A}_{g,n}^{\text{ord}}$. For a point $x \in Z(\overline{\mathbb{F}}_p)$ we will see that the maximal formal subtorus of $\mathcal{A}_{g,n}^{/f(x)}$ containing the image of $Z^{/x}$ under $f$ is related to the $p$-adic monodromy of the isocrystal $\mathcal{E}$ associated to the $p$-divisible group $X=A[p^{\infty}]$ over $Z$. This section is more or less a summary of Sections 2,3 and 4 of \cite{ChaiOrdinary}.
There is a short exact sequence of $p$-divisible groups
\begin{align}
0 \to \hat{X} \to X \to X^{\text{\'et}} \to 0.
\end{align}
We will write $T_n=\hat{X}[p^n]$ and $E_n=X^{\text{\'et}}[p^n]$, which also sit in an extension of fppf sheaves
\begin{align}
0 \to T_n \to X[p^n] \to E_n \to 0.
\end{align}
Let $X_{\ast}(T_n)$ and $X^{\ast}(T_n)$ be the finite \'etale group schemes of cocharacters respectively characters of $T_n$ and similarly we let $T_p(E_n)$ be finite \'etale group scheme of maps $\mathbb{Z}/p^n \mathbb{Z} \to E_n$ with dual $T_p^{\vee}(E_n)$.
Let $Z_{\text{\'et}}$ be the small \'etale site of $Z$, let $Z_{\operatorname{\text{fppf}}}$ denote the small fppf site of $Z$ and let $\pi:Z_{\operatorname{\text{fppf}}} \to Z_{\text{\'et}}$ be the natural map of sites. We now define
\begin{align}
\nu_{p^n}=R^1 \pi_{\ast} \mu_{p^n}.
\end{align}
Using the Kummer sequence
\begin{align}
1 \to \mu_{p^n} \to \mathbb{G}_m \to \mathbb{G}_m \to 1
\end{align}
on the fppf site of $Z$ and the associated long exact sequence on $Z_{\text{\'et}}$ we find that this is the sheaf associated to the presheaf
\begin{align}
R \mapsto R^{\times}/(R^{\times})^{p^n}.
\end{align}
Chai proves using Kummer theory and the Leray spectral sequence for $\pi$ that (corollary 2.4.1 of \cite{ChaiOrdinary})
\begin{align}
\operatorname{Ext}^1_{\mathbb{Z}/p^n\mathbb{Z},\text{fppf}}(E_{n},T_{n})) &= \operatorname{Hom}_{Z_{\text{\'et}}}(T_p(E_n) \otimes_{\mathbb{Z}/p^n \mathbb{Z}} X^{\ast}(T_n), \nu_{p^n}).
\end{align}
Taking the inverse limit over $n$ we obtain an element in
\begin{align}
q(X):\operatorname{Hom}_{Z_{\text{\'et}}}(T_p(X^{\text{et}}) \otimes_{\mathbb{Z}_p} X^{\ast}(\hat{X}), \nu_{\infty}).
\end{align}
We let $N(G)^{\vee}$ be the kernel of this morphism, Chai proves (Proposition 4.2.1 of \cite{ChaiOrdinary} that it is a smooth sheaf of $\mathbb{Z}_p$ modules on $Z_{\text{\'et}}$ and that the quotient
\begin{align}
\frac{T_p(X^{\text{et}}) \otimes_{\mathbb{Z}_p} X^{\ast}(\hat{X})}{N(G)^{\vee}}
\end{align}
is torsion free. Moreover he proves that
\begin{align}
N(G) \subset T_p(X^{\text{et}})^{\vee} \otimes_{\mathbb{Z}_p} X_{\ast}(\hat{X})
\end{align}
is a smooth sheaf of free $\mathbb{Z}_p$-modules on $Z_{\text{\'et}}$. In our case we can use the polarisation on the universal abelian variety to identify $X_{\ast}(\hat{X}) \simeq T_p(X^{\text{\'et},t})$.
If $Z=\operatorname{Spec} R$ with R a complete Noetherian normal domain with maximal ideal $\mathfrak{m}$, then $\nu_{\infty}$ is the constant sheaf associated to $1+\mathfrak{m}$ and $q(X)$ recovers the usual Serre-Tate coordinates. In this case $N(G)^{\vee}$ is the kernel of the map
\begin{align}
q(X):T_p(X^{\text{et}})^{\vee} \otimes_{\mathbb{Z}_p} X_{\ast}(\hat{X}) \to 1+\mathfrak{m}
\end{align}
and its dual defines a subtorus
\begin{align}
N(G) \otimes_{\mathbb{Z}_p} \hat{\mathbb{G}}_m \subset T_p(X^{\text{et}})^{\vee} \otimes_{\mathbb{Z}_p} T_p(X^{\text{\'et},t})^{\vee} \otimes \hat{\mathbb{G}}_m
\end{align}
through which the map from $\operatorname{Spf} R$ factors. In fact, Chai proves that this is the smallest such subtorus (Remark 3.1.4 of \cite{ChaiOrdinary}).
\subsection{Monodromy and conclusion}
Chai proves in Section 4 (Theorem 4.4) of \cite{ChaiOrdinary} that the dimension of the unipotent radical of the $p$-adic monodromy group of $\mathcal{E}$ over $Z$ is equal to the rank of $N(G)$.
\begin{proof}[Proof of Theorem \ref{Thm:Ordinary}]
We first prove the theorem when $(G,X)$ is of Hodge type and $G^{\text{ad}}$ is $\mathbb{Q}$-simple.
Let $x \in \operatorname{Sh}_{G, U^p}(\overline{\mathbb{F}}_p)$ be an ordinary point and let $Z$ be the Zariski closure (inside $\operatorname{Sh}_{G, \text{ord}}$) of its $G(\mathbb{A}_f^p)$ Hecke orbit. Let $U \subset Z$ be the smooth locus, which is dense because $Z$ is reduced and which is $G(\mathbb{A}_f^p)$-stable by Lemma \ref{Lem:DevSmooth}. Let $X/U$ be the $p$-divisible group of the universal abelian variety and let $\mathcal{E}$ be the associated $F$-isocrystal over $U$. Then we know that the dimension of the unipotent radical of the monodromy group of $\mathcal{E}$ is equal to the dimension of the $\operatorname{Sh}_{G, U^p}$ by Corollary \ref{Cor:padicMonodromy} and Remark \ref{Rem:DimensionUnipotent}. This implies that the rank of $N(G)$ over $U$ is also equal to the dimension of $\operatorname{Sh}_{G, U^p}$.
Therefore for a point $z \in U(\overline{\mathbb{F}}_p)$, the smallest formal subtorus of $C^{/z}$ containing $Z^{/z}$ is equal to $\operatorname{Sh}_{G, U^p}^{/z}$. Proposition \ref{Prop:SubTorus} tells us that $Z^{/z}$ is a formal subtorus and we conclude that $Z^{/z}=\operatorname{Sh}_{G, U^p}^{/z}$. Since this is true for a dense set of points, it follows that $Z$ is a union of irreducible (hence connected) components of $\operatorname{Sh}_{G, \text{ord}}$. Since $G(\mathbb{A}_f^p)$ acts transitively on $\pi_0(\operatorname{Sh}_{G, U^p})$ by Lemma 2.2.5 of \cite{KisinModels} it follows that $Z=\operatorname{Sh}_{G, \text{ord}}$. We conclude that the prime-to-$p$ Hecke orbit of $x$ is dense in $\operatorname{Sh}_{G, U^p}$ since $\operatorname{Sh}_{G, \text{ord}}$ is dense in $\operatorname{Sh}_{G, U^p}$.
Now let $(G,X)$ be as in the assumptions of Theorem \ref{Thm:Ordinary}. Then the reflex field $E(G,X)$ is a finite extension of the reflex field $E(G^{\text{ad}},X^{\text{ad}})$ of the adjoint Shimura datum $(G^{\text{ad}},X^{\text{ad}})$ and our place $v$ of $E(G,X)$ induces a unique place $v$ of $(G^{\text{ad}},X^{\text{ad}})$. Lemma 4.2.1 tells us that it suffices to prove the theorem for $(G^{\text{ad}},X^{\text{ad}},v)$ instead.
Writing $G^{\text{ad}}$ as a product of simple groups over $\mathbb{Q}$ we get
\begin{align}
(G^{\text{ad}},X^{\text{ad}})=(G_1,X_1) \times \cdots \times (G_n,X_n).
\end{align}
The reflex fields $E(G_i,X_i)$ are naturally contained in $E(G^{\text{ad}},X^{\text{ad}})$ and thus our place $v$ induces height one places $v_i$ of $E(G_i,X_i)$. We similarly get a product decomposition of Shimura varieties and prime-to-$p$ Hecke operators and therefore the theorem for $(G^{\text{ad}},X^{\text{ad}},v)$ follows from the theorem for the $(G_i,X_i,v_i)$ for all $i$. By Proposition 2.3.10 of \cite{Deligne} we can Shimura data $(H_i,Y_i)$ of Hodge type together with a central isogenies
\begin{align}
(H_i,Y_i) \to (G_i,X_i).
\end{align}
for all $i$. Moreover we can choose $(H_i,Y_i)$ such that all the places of $E(G_i,X_i)$ above $p$ split in $E(H_i,Y_i)$. We have already shown above that the theorem is true for $(H_i,Y_i,v_i)$, and we can conclude by applying Lemma \ref{Lem:CentralIsogenyt}.
\end{proof}
\begin{Lem} \label{Lem:CentralIsogenyt}
Let $(G,X) \to (H,Y)$ be a central isogeny of abelian type Shimura varieties and let $v,v' |p$ be places of $E(G,X)$ respectively $E(H,Y)$ above $p$. Then Theorem \ref{Thm:Ordinary} holds for $(G,X,v)$ if and only if it holds for $(H,Y,v')$.
\end{Lem}
\begin{proof}
The universal property of canonical integral models gives us a $G^{\text{sc}}(\mathbb{A}_f^p) \simeq H^{\text{sc}}(\mathbb{A}_f^p)$-equivariant map
\begin{align}
\operatorname{Sh}_G \to \operatorname{Sh}_{H}
\end{align}
which induces an isomorphism between a connected component of the source and a connected component of the target (see Section 3.4 of \cite{KisinModels}), and which preserves the $\mu$-ordinary locus (see Section 2.3 of \cite{ShenZhang}). Therefore if the Zariski closure of the $G^{\text{sc}}(\mathbb{A}_f^p)$-orbit of a point in the $\mu$-ordinary locus of $\operatorname{Sh}_{G}$ is equal to a connected component of $\operatorname{Sh}_G$, then the same is true for the $H^{\text{sc}}(\mathbb{A}_f^p)$-orbit of its image. The result follows because the prime-to-$p$ Hecke-operators acts transitively on connected components.
\end{proof}
\DeclareRobustCommand{\VAN}[3]{#3}
\bibliographystyle{amsalpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,118 |
John Webster was an English footballer who played in the Football League for Gainsborough Trinity, Rotherham Town and The Wednesday.
References
Date of death unknown
English footballers
Association football forwards
English Football League players
Attercliffe F.C. players
Sheffield Wednesday F.C. players
Rotherham Town F.C. (1878) players
Gainsborough Trinity F.C. players
Year of birth missing | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,623 |
La bandera de Baden estaba representada por una combinación de amarillo y rojo, los colores heráldicos del anterior Estado alemán de Baden.
Generalidades
Una bandera bicolor roja-amarilla fue introducida como bandera del Gran Ducado de Baden (1806-1918) en 1871. Esta fue remplazada por una bandera de tres bandas de amarillo-rojo-amarillo en 1891. Tras la abolición de la monarquía al fin de la I Guerra Mundial, fue establecida la República de Baden, que continuó utilizando esta bandera de tres bandas. Después de la llegada al poder del partido nazi en Alemania en 1933, los Estado alemanes individuales y sus símbolos fueron finalmente suprimidos. Después de la II Guerra Mundial, la mitad sur de Baden se convirtió en parte de la Zona de Ocupación Francesa en donde estableció el Estado de Baden del Sur. Baden del Sur utilizó la bandera de tres bandas amarillo-rojo-amarillo como su bandera hasta el desmantelamiento del Estado en 1952, cuando se convirtió en parte del moderno Estado federado de Baden-Wurtemberg.
La bandera todavía es común de verla en la región de Baden en la actualidad, ya que es utilizada por muchos ciudadanos a título privado.
Referencias
Baden
Baden
Historia de Baden-Wurtemberg | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,879 |
israelandstuffcom | May 28, 2013
Moscow to arm Assad with S-300 missiles to prevent 'foreign' (Israeli) intervention
Top Russian official says Russia will commence with sale of S-300 anti-aircraft missiles to Assad's army.
Israeli Defense Minister Ya'alon says that if advanced weapon systems reach Syria, Israel will respond.
By The Associated Press, Gili Cohen and Reuters
A top Russian diplomat confirmed that Moscow will provide Syria with state-of-the art air defense missiles to prevent foreign intervention in the country
A Russian S-300 in Moscow. – Photo: AP
Deputy Foreign Minister Sergei Ryabkov wouldn't say whether Russia has shipped any of the long-range S-300 air defense missile systems, but added that Moscow isn't going to abandon the deal despite strong Western and Israeli criticism.
Ryabkov said the deal helps restrain some "hot heads" considering a military intervention in Syria.
Russia has been the key ally of Syrian President Bashar Assad's regime, protecting it from the United Nations sanctions and providing it with weapons despite the civil war there that has claimed over 70,000 lives.
Ryabkov's statement comes a day after European Union's decision to lift an arms embargo to Syrian opposition.
After a marathon negotiating session in Brussels, EU governments failed to bridge their differences and let a ban on arming the opposition expire, with France and Britain scoring a victory at the expense of EU unity.
Britain and France have made a commitment not to deliver arms to the Syrian opposition "at this stage," an EU declaration said. But EU officials said the commitment effectively expires on August 1.
France said on Tuesday it reserved the right to send arms immediately to Syrian rebels fighting a two-year-old insurgency but had no plans to do so, despite an agreement by European countries to put off potential deliveries until August 1.
French foreign ministry spokesman Philippe Lalliot told reporters Paris hoped there would be a breakthrough in finding a political solution over the next two months, but that the EU decision was a political declaration that had no legal basis.
When asked if that meant France could deliver weapons before August 1 if it deemed it necessary, Lalliot said: "Yes."
Britain said on Tuesday it did not have to wait until an August 1 meeting of European Union foreign ministers before taking a decision to arm Syria's rebels, but stressed it had not yet taken any decision.
"I must correct one thing of concern. I know there has been some discussion of some sort of August deadline. That is not the case," Foreign Secretary William Hague told BBC radio, adding that Britain was "not excluded" from arming the rebels before August, and that it would not act alone if it chose to do so.
Defense Cheifs Ya'alon & Hagel after discussing Syria – Photo Yaron Brener
Defense Minister Moshe Ya'alon on Tuesday implied that Israel wOULD retaliate in Syria should the weapons systems reach the war-torn country.
Ya'alon said that Russia's intent to supply Assad's army with the advanced anti-aircraft systems is "a threat, as far as we're concerned," but asserted that the weapons have yet to be shipped out.
"I can't say there's been an acceleration [in weapons delivery]," he told reporters. "The shipments haven't set out yet and I hope they won't. If they do arrive in Syria, God forbid, we'll know what to do."
The defense minister's statement appears to contradict remarks made by IAF chief Maj. Gen. Amir Eshel, who said last week that Assad's regime has invested millions in purchasing anti-aircraft missiles, and that the S-300 shipment "is on its way."
Russia's foreign minister said earlier this month that Moscow had no new plans to sell the S-300 to Syria but left open the possibility of delivering such systems under an existing contract.
Israel is concerned that the weapons meant for Syria's arsenal could fall into the hands of Hezbollah, which is fighting alongside Assad against the rebels in Syria.
Last month Israel reportedly launched air strikes in Syria, targeting medium-range missiles that had arrived from Iran and were destined for Hezbollah.
View original HAARETZ publication at: http://www.haaretz.com/news/diplomacy-defense/diplomat-russia-to-arm-syria-regime-with-anti-aircraft-missiles-to-prevent-foreign-intervention-1.526425
Category: Israel News | Tags: Bashar Assad, EU, Russia, S-300 Missiles, Syria | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,600 |
gotta be careful of those ghosts, they can be deadly!
Sion went to recuperate after his suicide attempt. He never expected to be visited by the ghost of a dead villager, nor to fall in love with the young man who is his descendant. Mattie is home for the summer, and when HE sees the ghosts too, a long held curse comes to mind, and the two men wonder who will be the next victim of the ghosts.
Firstly, an apology. My keyboard won't type the correct spelling of Sion, so this review will have the wrong spelling throughout. It probably does, but I just don't know how to do it.
I loved this book, I really did. You don't get all of Sion's history in one go, or why he jumped off that bridges all those months ago. You don't get what Mattie had to deal with at university and how he felt about that. It creeps up on you, kinda like those ghosts, and it makes you shiver just typing about them!
You don't get the full story of Matthew, and Joshua, and what happened all those years ago, to cause two men to lose their lives off that cliff. Not all in one go, anyway, and I loved being made to wait for the full history to come to light. I loved that I did not see which ghost was the most dangerous, til it was revealed in the book, and I loved that maybe, just maybe, there won't be any more deaths.
One thing stopped it getting 5 stars. Just one. Single Person Point Of View! OH how I wanted, no, NEEDED to hear from Mattie! I really did. It would have finished this book with those 5 full and slightly seaweed smelling stars but It won't get them, because only Sion has a say. And I know I say it all the time, but thats the whole point of a review, its MY opinion, and MY opinion is, I NEEDED to hear what Mattie was thinking, when he met Sion, when he first en-counted the ghosts, when Sion told him why he was there, lots of keys points along the way I NEEDED Mattie.
Although the bay where the book is set does not exist, the book does visit real places along the Yorkshire coast. Places I've really been to, and spent time in, and I loved that. And I walked those steps up to the Abbey in Whitby, and it does take a hell of a lot longer to go up than it does down! I loved that I could really relate, and see these places in my mind, and through the eyes of Sion, just because I have been there.
First I have read of this author, and I would like to read more. | {
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} | 9,973 |
Q: How to apply a mapping function to CouchDB replication? I have a database on a powerful but distant CouchDB server (imagine a typical "cloud server").
I want to replicate this database to a local but less powerful computer (imagine a typical "mobile device" connected over a "slow network").
In the process, I'd like a mapping function (view?) to be applied to each source document on the powerful server prior to being transmitted over the network to the local machine. The intent is to reduce the size of the document by removing data elements which will not be required by processes that will be run on the local machine.
I have read the documentation for /db/_changes and noticed you can specify a view, but I'm completely unsure how to configure this as part of the replication source.
My use-case is different from the more common filtered replication that keeps coming up on Google searches, because that's focused on not replicating certain documents, while I want to replicate all documents in a given database, but not all data elements in each document.
A: You're on the right track. You need to create a view on the CouchDB server and then specify to use the view for the replication.
Inside that view, use a map function. For example, if you only want to retrieve the _id and the properties key1 and key2, you can create a design doc via the Fauxton UI with a map function like this:
map: function (doc) {
emit(doc._id, { key1: doc.key1, key2: doc.key2 });
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,660 |
Home > From the President > Speeches > Drawing a Larger Circle
Drawing a Larger Circle
Peter Salovey, President of Yale University
Baccalaureate Address, Yale College Class of 2018
Graduates of the Class of 2018, family members, and friends. It is a pleasure to be here with you today, a day filled with joy for the present and hope for the future.
There is a wonderful Yale tradition that I would like to honor right now:
May I ask all of the families and friends here today to rise and recognize the outstanding—and graduating—members of the Class of 2018?
And now, may I ask the Class of 2018 to consider all those who have supported your arrival at this milestone, and to please rise and recognize them?
These are the months and years when people tend to make a lot of plans. Some are practical: you schedule flights and rent apartments and consider where you will live, work, or study after graduation. Others are more aspirational: you imagine your future life and what you wish to accomplish in the years ahead.
I want to begin by sharing a passage Pauli Murray wrote in 1945 about her aspirations. At the time, she was a young lawyer and civil rights activist.
"I intend to destroy segregation by positive and embracing methods," Murray wrote. "When my brothers try to draw a circle to exclude me, I shall draw a larger circle to include them. Where they speak out for the privileges of a puny group, I shall shout for the rights of all mankind."[1]
So today I ask you: How large will you draw your circle?
Will you draw a circle that is large, inclusive, and vibrant? Or will it be small, "puny," and privileged?
The work of inclusion is difficult, but the rewards are great.
Let me suggest ways you might follow the example of Pauli Murray—and many other Yale graduates—when you leave this campus.
First, make sure your circles are truly large.
In today's world, where you can have 700 followers on Twitter and a thousand friends on Facebook, it may seem easy to have a large circle. But if you're bombarded with the same stories, memes, and opinions from all your so-called friends, then your world may in fact be quite narrow. A conversation with six friends in real life actually may lead to a greater variety of ideas and perspectives.
In my years at Yale, I have been privileged to know some of the most brilliant minds in the world. I have learned that the greatest scholars draw large circles. They read widely and are interested in ideas well beyond the scope of their own research and beliefs.
Robert Dahl, who was a Sterling Professor of Political Science, taught at Yale for forty years. One of the most respected political scientists of his generation, Professor Dahl was an authority on democracy and democratic institutions. And he was a beloved teacher and mentor.
After his death in 2014 at the age of 98, tributes from his former students poured in. One of his graduate students, Jeffrey Isaac, recalled how he vehemently disagreed with some of Dahl's arguments, even though he loved taking his classes. For his dissertation, Isaac proposed writing a critique of Dahl's theories. Much to his surprise, the most enthusiastic and supportive faculty member in the department was Dahl himself! He agreed to supervise the dissertation.
Isaac wrote, "Bob Dahl spent countless hours in his office talking with me about my principal theoretical antagonist—him! We would discuss this guy 'Dahl' in the third person, considering the limits of his arguments, speculating about how he might respond to my arguments."[2]
Professor Dahl embraced his critics, listened to them, and conversed with them, a model of open and engaged scholarship and teaching—the best we can aspire to at Yale.
The lesson extends beyond our campus. Our greatest challenges as a society—climate change, poverty, insecurity, and violence—demand innovative and creative solutions. Yet political polarization is making it more difficult than ever to solve these problems. We must be able to talk with our opponents even though we disagree with them.
We might start by emulating Professor Dahl—and so many other wise and generous thinkers who have drawn large circles and so added to the sum of human understanding.
My second piece of advice—and here I am taking some liberties with the metaphor—is to draw as many circles as you can.
One circle will be your work. Make sure you enjoy it, but make sure you have other circles as well.
We know one of the keys to happiness is developing a passion—even an expertise—outside of work. Sharing that passion with others gives us great joy, and it connects us to other circles of friends and associates who might be very different from the ones we would meet otherwise.
As many of you are aware, I am quite passionate about music from the Appalachian Mountain region. My love of traditional country and bluegrass music has allowed me to visit places such as southwest Virginia and eastern Kentucky, to chair the board of the International Bluegrass Music Museum, and to play bass—for thirty years now—with the Professors of Bluegrass. It enables me to share stories and songs with perfect strangers at summertime bluegrass festivals. Most significantly, though, it has led to circles of friendship beyond the towns in which I grew up, beyond the universities I attended, and beyond my profession of psychology.
I am proud, of course, to be a psychologist, and my discipline does in fact provide some empirical evidence to support my personal experience.
Patricia Linville is a social psychologist who studies how people think of themselves and how these self-perceptions influence well-being. She is now at Duke, but she was my teacher here at Yale when she completed several studies of what she terms "self-complexity."
Greater "self-complexity," according to Linville, means a person has a variety of aspects to his or her self. In other words, he or she draws many circles. For example, a woman who thinks of herself as a student, a marathon runner, a theater-goer, a reader of the New Yorker magazine, and—let's say—a bass player in a bluegrass band would demonstrate greater self-complexity than someone who thinks of himself only as a lawyer.
Professor Linville, in her research, found that greater self-complexity acts as a "buffer" against negative experiences. For example, if you define yourself almost entirely in terms of your job, getting passed over for a promotion might be devastating for your sense of self-worth. Linville calls this "putting all your eggs in one cognitive basket." People such as our marathon-running bass player, on the other hand, bounce back more quickly after a setback. Linville even found that college students with greater self-complexity were less likely to get sick or experience depression or stress.[3]
Third and finally, let me suggest one important way we can expand our circles—by reaching out and engaging with others.
Here I would like to turn again to Pauli Murray and one of her more surprising relationships.
Murray's papers contain thousands of letters—a reflection of a full life, animated by many interests, commitments, and relationships. A life of many circles.
During her time at Yale Law School, Murray received a letter from William S. Beinecke, a member of the Yale College Class of 1936. Now the name will sound familiar to everyone here. The Beinecke Rare Book & Manuscript Library is named for William's father and two uncles, and many other programs and places at Yale have benefited from the family's remarkable philanthropy.
Bill Beinecke passed away just last month; he was nearly 104 years old. In 1963 when he wrote Murray, he was chairman of the Sperry and Hutchison Company, a venerable American company founded by his grandfather. (Your parents and grandparents probably remember S&H Green Stamps.) Beinecke was a leader in corporate America and a wealthy and powerful man.
He had met Murray at an event at Yale, and not long after that meeting, he wrote her a letter. He enclosed a clipping from Time magazine about race relations in the United States and asked what she thought.
Murray responded. A few weeks later he sent her another article and asked her opinion again, this time about school integration. She wrote back. At one point, Murray wrote Beinecke a four-page, single-spaced, typed letter on what she called the "imponderables on the issue of race." Their correspondence continued for several weeks, with interesting and frank letters on both sides.
Beinecke and Murray—both exemplars of the Yale tradition—were able to sustain a conversation despite differences in gender, family background, race, class, and more. We don't know whether or not they entirely agreed with one another, but we can imagine they learned a lot from the exchange. All because two individuals decided to reach beyond their normal circles.[4]
Beinecke's decision to write Murray did not take place in a vacuum. In the 1950s, he attended a discussion at Yale Law School on the topic of American race relations. Not long after, he decided to look into Sperry and Hutchinson's hiring practices. He learned that the employment agency vetting applicants for his company was screening out African Americans, removing them from the pool before their applications ever reached Sperry & Hutchinson. Beinecke ended the practice.
He also supported scholarships for underprivileged high school students and established a fellowship for students of color at Yale Law School. It was in the course of this work that he met Murray and initiated their correspondence, hoping to bridge the gulf that separated his experience from hers.[5]
Bill Beinecke's life was made up of many different circles. He led efforts to improve New York's Central Park, he supported environmental causes, he was dedicated to the game of golf, and he remained an ardent champion of Yale and its students, among other interests.
And what about Pauli Murray, who as a young person promised to "draw a larger circle" in her life? One month after writing her last letter to Bill Beinecke, she participated in the historic March on Washington, which she helped organize. While finishing her doctor of jurisprudence degree here at Yale, she drafted an influential legal memo, helping to ensure that "sex" was included in the Civil Rights Act of 1964. Murray's other circles included writing poetry and teaching. At the age of 67, she became the first African-American woman ordained as an Episcopal priest, continuing her lifelong commitment to reconciliation and understanding.
Enlarging our circles is far from easy. It requires courage, surely, but also imagination and curiosity about our fellow human beings. It rejects fear and suspicion. It demands that we listen to one another. It measures the limits of our humanity.
Both Pauli Murray and Bill Beinecke drew such large circles—and so many circles—that their lives intersected. I urge you to do the same. Draw many circles; make them large in all kinds of ways. You will find life richer, fuller, and more meaningful, and you will bring to the world the empathy and understanding we so desperately need.
Members of the Class of 2018 (please rise):
As you go out on to a "world [that is] all before [you] … hand in hand with wandering steps and slow,"[6] bring to that world all that your Yale education has given you: the ability to engage critically even while listening respectfully, to respond creatively to challenges and obstacles; to embrace your responsibilities while finding happiness, and to draw ever wider the circle of belonging and understanding in this world.
We are delighted to salute your accomplishments, and we are proud of your achievements. Remember to give thanks for all that has brought you to this day. And go forth from this place with grateful hearts, paying back the gifts you have received here by using your minds, voices, and hands to strengthen your new communities and your world.
Congratulations, Class of 2018!
[1] Murray, P. (1945). An American Credo. Common Ground 5(2), 24.
[2] Isaac, J. (2014, February 11). Robert Dahl as mentor. Washington Post. https://www.washingtonpost.com/news/monkey-cage/wp/2014/02/11/robert-dahl-as-mentor/?utm_term=.e19c870c221c
[3] Linville, P.W. (1985). Self-complexity and affective extremity: Don't put all your eggs in one cognitive basket. Social Cognition, 3, 94-120.
[4] Pauli Murray Papers, Schlesinger Library, Harvard University.
[5] Beinecke, W. S., & Kabaservice, G. M. (2000). Through Mem'ry's Haze: A Personal Memoir. New York: Prospect Hill Press, 417-421.
[6] This quotation is from John Milton's Paradise Lost, a favorite passage of mine to read on Commencement weekend at Yale: "Some natural tears they dropped, but wiped them soon; The world was all before them, where to choose Their place of rest, and Providence their guide; They, hand in hand, with wandering steps and slow, Through Eden took their solitary way." | {
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Given it's rich history and cultural heritage, it's not surprising there a number of museums across Macau. They cover a wide variety of subjects, from local history to tea culture to motor racing. This list covers the top museums in Macau you may want to consider on your visit.
Dr. Sun Yat-Sen Memorial House
The Dr. Sun Yat-Sen Memorial House (澳門國父紀念館) is a distinctive looking building and museum where former family members of Sun Yat-Sen ('the father of modern China') used to live. Originally known as 'Mansion of the Sun', the house was built in 1912 for his first wife, Lu Muzhen.
Macao Museum
Macao Museum (Museu de Macau, 澳門博物館) details the history of Macau and is built on the site where the former Meteorological Service was located.
Macau Grand Prix (and Museum)
The Macau Grand Prix (Museu do Grande Prémio, 大賽車博物館) is held on the second or third weekend of November each year, and is the only street racing circuit where both cars and motorcycle races race on the same city track.
Macau Tea Culture House
The Macau Tea Culture House (Casa Cultural de Cha de Macau, 澳門茶文化館) is a museum and cultural centre dedicated to the role tea has played in the cultural development of Macau.
The Maritime Museum (Museu Marítimo, 海事博物館) was opened in 1987, with spacious new exhibition hall added in 1990. The 3 level museum is dedicated to the maritime activities of Macau, Portugal and China.
Museum of Taipa and Coloane History
The Museum of Taipa and Coloane History (Museu da História da Taipa e Coloane, 路氹歷史館) is located in Taipa village, housed in what used to be the Island Municipal Council building (and old public administration center). The original building dates from the 1920s, built in the Portuguese style.
St. Dominic's Church (Igreja de São Domingos, 玫瑰堂) was established in 1587 by 3 Spanish Dominican priests who arrived in Macau from Acapulco, Mexico, this church has been the scene of several violent incidents.
Tap Seac Gallery
Tap Seac Gallery is an art gallery hosted in a lovely European style building which lies on Tap Seac Square. Continue reading
← Top Gardens in Macau
Top Historical Attractions In Macau → | {
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La fiaca es una película argentina de comedia dramática de 1969 escrita y dirigida por Fernando Ayala. Está basada en la obra teatral homónima de Ricardo Talesnik, y es protagonizada por Norman Briski, Norma Aleandro, Jorge Rivera López y Lydia Lamaison. Fue estrenada el 6 de marzo de 1969.
Sinopsis
Néstor Vignale, un empleado de una oficina citadina, una mañana decide faltar a su trabajo en un intento de rebelión. Vignale explica a su esposa Martha que sufre de «fiaca» (pereza), negándose a seguir su rutina laboral mantenida hasta entonces y dedicando todo su tiempo a la simple diversión y entretenimiento. Vignale sufre los reproches de su madre (quien vive con el matrimonio), recordándole su rol de "empleado ejemplar", e inclusive sus colegas y jefes acuden a su casa a instarle a retomar sus actividades, hasta que las realidades de su vida cotidiana le hacen meditar sobre su decisión.
Reparto
Participaron del filme los siguientes intérpretes:
Premio
Juan Carlos Gené fue galardonado en 1970 con el Premio Cóndor de Plata al Mejor actor de reparto por su trabajo en tres películas, una de las cuales era La fiaca.
Comentarios
Confirmado escribió:
La Prensa opinó:
Primera Plana dijo:
Referencias
Enlaces externos
La fiaca en Cine Nacional
Películas basadas en obras de teatro | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,875 |
{"url":"http:\/\/akteam.top\/wordpress\/index.php\/2018\/05\/27\/lct\/","text":"# \u8bc1\u660e\u4f38\u5c55\u6811\u4e0e\u52a8\u6001\u6811\u7684\u65f6\u95f4\u590d\u6742\u5ea6\n\n#### Splay\u65f6\u95f4\u590d\u6742\u5ea6\u8bc1\u660e\n\n###### \u8bc1\u660e.\n\n$2\\lg(x)-\\lg(y)-\\lg(z)$\n$=\\lg((y+z+1)^2)-\\lg(y)-\\lg(z)$\n$=\\lg(\\frac{(y+z+1)^2}{yz})$\n$=\\lg(\\frac{2yz}{yz}+\\frac{y^2+z^2+1+2y+2z}{yz})$\n$\\geq \\lg(2+\\frac{y}{z}+\\frac{z}{y})$\n$\\geq \\lg(2+2)$\n$=2$\n\n###### \u8bc1\u660e.\n\n$r'(x)+r'(y)+r'(z)-r(x)-r(y)-r(z)$\n$=r'(y)+r'(z)-r(x)-r(y)$\n$\\leq r'(x)+r'(z)-2r(x)$ \u7531\u4e8e$size'(x)=size(x)+size'(z)+1$,\u6839\u636e\u5b9a\u74060,\u52a0\u4e0a2r'(x)-r'(z)-r(x)-2\u3002\n$\\leq 3(r'(x)+r(x))-2$\n\n$r'(x)+r'(y)+r'(z)-r(x)-r(y)-r(z)$\n$=r'(y)+r'(z)-r(x)-r(y)$\n$\\leq r'(y)+r'(z)-2r(x)$,\u7531\u4e8esize'(x)=size'(y)+size'(z)+1,\u6839\u636e\u5b9a\u74060,\u52a0\u4e0a2r'(x)-r'(y)-r'(z)-2\n$\\leq 2(r'(x)-r(x))-2$\n\n### 1 thought on \u201c\u8bc1\u660e\u4f38\u5c55\u6811\u4e0e\u52a8\u6001\u6811\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u201d\n\n1. Thanks for all your valuable hard work on this site. Gloria enjoys conducting investigations and it\u2019s obvious why. My spouse and i know all about the powerful tactic you render practical guides via the blog and as well as boost contribution from people on this topic and my child is without question discovering a lot. Take advantage of the remaining portion of the new year. You have been carrying out a stunning job.","date":"2018-08-18 22:19:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18622063100337982, \"perplexity\": 2092.9943577669665}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221213794.40\/warc\/CC-MAIN-20180818213032-20180818233032-00227.warc.gz\"}"} | null | null |
Lizotte and Associates Real Estate Inc. is a real estate firm based in Edmonton, founded in 1979 by Ross Lizotte. At that time, the business grew to approximately 50 agents and office staff working on residential, commercial and property management accounts. The current focus of the company is now strictly commercial real estate sales and leasing. With a long history of working with building owners and landlords, Lizotte and Associates has developed an extensive client list and a working knowledge of many properties. Timely and detailed servicing of all our listings has given us a solid reputation in Edmonton's commercial real estate market. Call us today for more information. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,333 |
Q: Facebook fan gate not working So I create a Facebook app for a page tab with "fan gate", but the problem is that the signed_request is not working, so I cannot receive users page like status. I always get the Not in iframe! message, but the page opens in iframe...
This is my code of the main page:
<?php
function parsePageSignedRequest() {
if (isset($_REQUEST['signed_request'])) {
$encoded_sig = null; $payload = null;
list($encoded_sig, $payload) = explode('.', $_REQUEST['signed_request'], 2);
$sig = base64_decode(strtr($encoded_sig, '-_', '+/'));
$data = json_decode(base64_decode(strtr($payload, '-_', '+/'), true));
return $data;
}
return false;
}
if($signed_request = parsePageSignedRequest()) {
if($signed_request->page->liked) {
include 'fan.php';
} else {
include 'no_fan.php';
}
} else {
echo 'Not in iframe!';
}
?>
A: The signed_request is available only for the first page loaded in the page tab.
If, in your iframe, you perform a redirection before calling your script, the signed_request will disappear from $_REQUEST.
The solution is to put the signed_request (at least the infos you need) in the session everytime it's available, and then fetch what you want in this session instead of the signed_request directly.
And maybe you could try to use the Facebook PHP SDK (https://developers.facebook.com/docs/reference/php/4.0.0). It's quite simple to use, and there is a getSignedRequest() method that will avoid you to make mistakes while manipulating it.
A: You need to check for the signed_request on only the initial page load of the app - facebook post's data only to the first page it loads into it's iframe.
I usually work around this using sessions:
if (isset($_REQUEST['signed_request'])) {
$encoded_sig = null;
$payload = null;
list($encoded_sig, $payload) = explode('.', $_REQUEST['signed_request'], 2);
$sig = base64_decode(strtr($encoded_sig, '-_', '+/'));
$data = json_decode(base64_decode(strtr($payload, '-_', '+/'), true));
if($data->page->liked) {
$_SESSION['liked'] = true;
}
} else {
$_SESSION['liked'] = false;
}
}
Bear in mind with this approach though that because Facebook require SSL you will have problems with some versions of Safari blocking all 3rd party cookies - so your app in the iframe wont have permission to create the session cookie needed. This will mean soon as your users navigate off the first page the signed_request from facebook and the session both vanish...
To get around this I tend to go old school & append the session_id to each URL in the app - so you need to append ?PHPSESSID=<?php echo session_id(); ?> to each URL or form action etc...
Additionally in your scripts or config you need to tell PHP to use cookie-less sessions.
I use:
ini_set('session.use_cookies', 0);
ini_set('session.use_only_cookies', 0);
ini_set('session.use_trans_sid', 1);
Normally appending the session ID would be a bit ugly - but as you're in an iframe noone really sees the URL anyway.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,390 |
ODNI enlists telecoms to find metadata storage solution
By Frank Konkel
| February 6, 2014
A request for information seeks suggestions on how to keep telephone metadata accessible to spy agencies without the government storing it.
Operational Policy
The Office of the Director of National Intelligence wants to gather industry input regarding how the government can continue to collect bulk telephone records metadata without actually storing the information.
ODNI issued a request for information on Feb. 5 that seeks suggestions on how to keep the telephone metadata accessible to the government under Section 215 of the Patriot Act without the use of government facilities to hold or maintain it.
In the RFI, ODNI states that responses will be "reviewed and may help to shape the framework for the future telephony metadata program to include the potential for non-government maintenance of that data," referencing several limitations on signals intelligence President Barack Obama issued Jan. 17.
More from FCW
White House review group issues intelligence reform report
Obama trims intelligence gathering
The RFI is a direct response to one of the president's directives to develop "options for a new approach that can match the capabilities and fill the gaps that the Section 215 program was designed to address without the government holding this metadata."
U.S. companies are asked to submit a two-page paper detailing commercially available capabilities that "could provide viable alternative approaches to the current Section 215 program without the government holding the metadata, while maintaining the current capabilities of that system and the existing protections for U.S. persons."
Possible approaches include:
Near-real-time access to data from the original source.
Correlation of data with varying provider data formats.
Simultaneous or near-simultaneous real-time access to data across multiple provider-stored datasets.
Secure storage of and access to U.S. telephone metadata records for a sufficient period of time.
Compliance with rigorous security and auditability standards to ensure that no queries take place without appropriate authorization and no data is provided to the government unless in response to an authorized query while maintaining 99.9 percent availability.
NEXT STORY: Big data today will be the norm tomorrow | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,198 |
Q: How do I return a scalar variable from a select statement I am altering a table by adding a new column, this column must have a default value taken from a single entry in another table
ALTER TABLE Bonus_Profile
ADD Orgunit varchar(50) NOT NULL
DEFAULT (select top 1 OrgUnit from OrgUnits where ReportTo is null)
The above statement gives Subqueries are not allowed in this context. Only scalar expressions are allowed. as and error
What should be used to get a scalar variable of OrgUnit
A: Create a parameterless scalar function first
CREATE FUNCTION ReturnOrgUnit ()
RETURNS VARCHAR(50)
BEGIN
Declare @orgUnit varchar(50)
Select TOP 1 @orgUnit= OrgUnit from OrgUnits where ReportTo is null
Return @orgUnit
END
Then you can use above udf in alter statement
ALTER TABLE Bonus_Profile
ADD Orgunit varchar(50) NOT NULL
DEFAULT (ReturnOrgUnit())
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,434 |
package org.hibnet.webpipes.processor.uglify2;
import java.util.Arrays;
import org.hibnet.jsourcemap.SourceMap;
import org.hibnet.webpipes.Webpipe;
import org.hibnet.webpipes.WebpipeOutput;
import org.hibnet.webpipes.WebpipeUtils;
import org.hibnet.webpipes.js.JsProcessor;
import org.hibnet.webpipes.processor.ProcessingWebpipe;
import org.hibnet.webpipes.processor.ProcessingWebpipeFactory;
import org.hibnet.webpipes.resource.WebJarHelper;
/**
* Compress js using uglifyJs utility.
*/
public class UglifyJs2Processor extends JsProcessor {
private static String[] libs;
static {
String uglifyjsPath = WebJarHelper.getWebJarAssetLocator().getFullPath("uglifyjs");
String uglifyjsDir = "/" + uglifyjsPath.substring(0, uglifyjsPath.length() - 13);
// @formatter:off
libs = new String[] {
uglifyjsDir + "/lib/utils.js",
uglifyjsDir + "/lib/ast.js",
uglifyjsDir + "/lib/parse.js",
uglifyjsDir + "/lib/transform.js",
uglifyjsDir + "/lib/scope.js",
uglifyjsDir + "/lib/output.js",
uglifyjsDir + "/lib/compress.js",
uglifyjsDir + "/lib/sourcemap.js",
uglifyjsDir + "/lib/mozilla-ast.js",
uglifyjsDir + "/lib/propmangle.js"
};
// @formatter:on
}
@Override
protected void initEngine() throws Exception {
addSourceMap();
eval("MOZ_SourceMap = sourceMap;");
for (String lib : libs) {
evalFromClasspath(lib);
}
evalFromClasspath("/org/hibnet/webpipes/processor/uglify2/webpipes_runner.js");
}
private WebpipeOutput process(Webpipe webpipe, boolean uglify) throws Exception {
SourceMap inSourceMap = webpipe.getOutput().getSourceMap();
String content = webpipe.getOutput().getContent();
WebpipeOutput output = callRunner(uglify, webpipe.getPath(), content, WebpipeUtils.toJsonString(inSourceMap));
if (inSourceMap == null) {
output.getSourceMap().sourcesContent = Arrays.asList(content);
}
return output;
}
private final class UglifyJs2Webpipe extends ProcessingWebpipe {
private boolean uglify;
private UglifyJs2Webpipe(String path, Webpipe webpipe, boolean uglify) {
super(WebpipeUtils.idOf(UglifyJs2Processor.class, webpipe, uglify), path, "uglify2", webpipe);
this.uglify = uglify;
}
@Override
protected WebpipeOutput fetchOutput() throws Exception {
return process(getChildWebpipe(), uglify);
}
}
public Webpipe createProcessingWebpipe(String path, Webpipe source, boolean uglify) {
return new UglifyJs2Webpipe(path, source, uglify);
}
public ProcessingWebpipeFactory createFactory(final boolean uglify) {
return new ProcessingWebpipeFactory() {
@Override
public Webpipe createProcessingWebpipe(String path, Webpipe source) {
return new UglifyJs2Webpipe(path, source, uglify);
}
};
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 103 |
#include <iostream>
#include "kernel.hpp"
#include "map.hpp"
#include "schedule.hpp"
#include "optdef.hpp"
Schedule::Schedule( Map &map ) : map_ref( map )
{
//TODO, see if we want to keep this
handlers.addHandler( raft::quit, Schedule::quitHandler );
}
Schedule::~Schedule()
{
/** nothing to do at the moment **/
}
void
Schedule::init()
{
for( raft::kernel *kern : map_ref.all_kernels )
{
(this)->scheduleKernel( kern );
}
}
raft::kstatus
Schedule::quitHandler( FIFO &fifo,
raft::kernel *kernel,
const raft::signal signal,
void *data )
{
/**
* NOTE: This should be the only action needed
* currently, however that may change in the futre
* with more features and systems added.
*/
fifo.invalidate();
return( raft::stop );
}
void
Schedule::invalidateOutputPorts( raft::kernel *kernel )
{
auto &output_ports( kernel->output );
for( auto &port : output_ports )
{
port.invalidate();
}
return;
}
raft::kstatus
Schedule::checkSystemSignal( raft::kernel * const kernel,
void *data,
SystemSignalHandler &handlers )
{
auto &input_ports( kernel->input );
raft::kstatus ret_signal( raft::proceed );
for( auto &port : input_ports )
{
if( port.size() == 0 )
{
continue;
}
const auto curr_signal( port.signal_peek() );
if( __builtin_expect(
( curr_signal > 0 && curr_signal < raft::MAX_SYSTEM_SIGNAL ),
0 ))
{
port.signal_pop();
/**
* TODO, right now there is special behavior for term signal only,
* what should we do with others? Need to decide that.
*/
if( handlers.callHandler( curr_signal,
port,
kernel,
data ) == raft::stop )
{
ret_signal = raft::stop;
}
}
}
return( ret_signal );
}
bool
Schedule::scheduleKernel( raft::kernel *kernel )
{
/** does nothing **/
return( false );
}
bool
Schedule::kernelHasInputData( raft::kernel *kernel )
{
auto &port_list( kernel->input );
if( ! port_list.hasPorts() )
{
/** only output ports, keep calling till exits **/
return( true );
}
for( auto &port : port_list )
{
if( port.size() > 0 )
{
return( true );
}
}
return( false );
}
bool
Schedule::kernelHasNoInputPorts( raft::kernel *kernel )
{
auto &port_list( kernel->input );
/** assume data check is already complete **/
for( auto &port : port_list )
{
if( ! port.is_invalid() )
{
return( false );
}
}
return( true );
}
bool
Schedule::kernelRun( raft::kernel * const kernel,
volatile bool &finished,
jmp_buf *gotostate,
jmp_buf *kernel_state )
{
if( kernelHasInputData( kernel ) )
{
const auto sig_status = kernel->run();
if( sig_status == raft::stop )
{
invalidateOutputPorts( kernel );
finished = true;
}
}
/**
* must recheck data items again after port valid check, there could
* have been a push between these two conditional statements.
*/
if( kernelHasNoInputPorts( kernel ) && ! kernelHasInputData( kernel ) )
{
invalidateOutputPorts( kernel );
finished = true;
}
return( true );
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,226 |
Q: Getting the value of an input textbox and adding it to a textarea I am a beginner at jQuery, and am wondering if this is possible to do. I have a form that has a name input textbox. I want to grab what the user types into that textbox and add it to the first sentence in my textarea. So that text area would say Dear (whatever the user typed). Any push in the right direction would be appreciated.
<form>
<label>First Name:</label>
<input type="text" class="fName" />
<br /><br />
<label>Last Name:</label>
<input type="text" class="lName">
<br /><br />
<textarea>Dear +firstname+</textarea>
</form>
My form is fairly simple. I haven't tried anything yet, but you are so helping me go in the right direction. Thanks.
A: Try this:
$('input').keyup(function() {
$('textarea').text($(this).val());
});
A: You can try something like this:
<script>
$(function() {
$( "#name" ).change(function() {
$("#textAreaId").append($( "#name" ).val());
});
});
</script>
Hope it helps to you!
A: Assuming the input has an ID of name and the textarea has an ID of area then:
$("#name").on("input", function(){
var name = $(this).val();
var msg = "Your message here";
var content = "Dear "+name+", \n" + msg;
$("#area").text(content);
});
I would strongly advise against using change1 or keyup2 as input3 is a much more comprehensive check of whether the user has changed anything.
Here is a JSFiddle example
Footnotes:
*
*change only responds on blur (i.e. the user presses tab or clicks outside the input).
*keyup only works on key presses, so it doesn't register when the user has pasted using their mouse, for instance.
*input registers any change to an input.
Update
As per my discussion with @GeorgeMauer, here is the recommended approach:
$("input.full-name").on("input", function(){
var full_name = $(this).val();
var form = $(this).closest("form");
var message = form.find(":input.message");
var content = message.val();
var lines = content.split("\n");
lines[0] = "Dear " + full_name + ",";
var new_message = lines.join("\n");
message.val(new_message);
});
Please see the updated JSFiddle example.
This has the added benefit of re-using the form without adding any extra JavaScript. And it also works a lot better (e.g. you can edit both inputs freely).
A: Edit: For your situation, it doesn't look like you are setting up your form correctly. Forms are meant for submitting a group of information, then putting this information somewhere else - where it needs to go. Hence, I moved your textarea outside of the form.
I would recommend setting up your HTML like so:
<form id="target" action>
<label>First Name:</label>
<input type="text" id="fName" />
<br /><br />
<label>Last Name:</label>
<input type="text" id="lName" /><br><br>
<input type="submit" value="Submit"/>
<br /><br />
</form>
<textarea id="message"></textarea>
And here is the jQuery I used:
$('form#target').submit(function(e) {
e.preventDefault();
var message, firstName, lastName;
message = "Dear ";
firstName = $('input#fName').val();
lastName = $('input#lName').val();
message += firstName + ' ';
message += lastName + ',';
$('textarea#message').text(message);
});
Note: I changed your class properties to id because id's are meant for unique tags. If you had multiple items that you wanted to grab all at once with jQuery, classes would be the way to go.
Preview Here.
Hopefully this helps!
A: So this is a deceptively difficult question because you are trying to update PART OF the text inside a textarea. There is no structure inside of a textarea, it is just "text". So what what you're asking is similar to having a bucket of ping pong balls, writing a name on one of them, mixing up the bucket, then telling someone that they have to continuously replace the named ball with a new one.
While there are ways of doing this, it raises a bunch of questions that I doubt you want to answer - what if they delete the first line? What if they change their name directly in the textarea? What if they write their name multiple times in the text area? What if their browser uses a language that writes top-down?
Unless you are prepared to answer these queries (and if you are you should probably start a more specific question) I recommend changing your approach. For example, I doubt you want them to have the ability to change the first line. So instead you can structure your form as follows:
<form>
<label>First Name: <input class="first-name" /></label>
<label>Last Name: <input class="last-name" /></label>
<aside class="salutations">Dear <span class="first-name-salutation"></span></aside>
<textarea class="message"></textarea>
</form>
Notice I moved your inputs into the labels (this is the technically correct way of doing it without using ids), removed the <br> tags (you should use css for this) and moved the salutation into non user-editable text. Then your javascript can be simply
$(function(){
$('input.first-name').on('input', function(){
$('.first-name-salutation').text($(this).text());
});
})
And then prepend "Dear "+firstName line to the message on the server.
As an aside I'll point out that it's not 2009 anymore and there's little reason to use jquery directly for things like this when there are excellent frameworks that don't force you to do so many things manually like Knockout and Angular.
A: Please are you conversant with AJAX? If yes thatis the best approach to use.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,310 |
Lars Arvid Nilsen (* 15. April 1965 in Notodden) ist ein ehemaliger norwegischer Kugelstoßer.
Bei den Europameisterschaften 1986 und bei den Leichtathletik-Hallenweltmeisterschaften 1987 wurde er jeweils Fünfter. 1987 wurde er wegen der Verwendung von Probenecid und anabolen Steroiden für zwei Jahre gesperrt.
Einem sechsten Platz bei den Europameisterschaften 1990 in Split folgte 1991 ein fünfter bei den Hallenweltmeisterschaften in Sevilla. Im Sommer belegte er bei den Weltmeisterschaften in Tokio den dritten Platz, rückte aber dann auf den Silberrang vor, nachdem der Zweitplatzierte, sein Landsmann Georg Andersen, wegen Dopings disqualifiziert wurde.
Im Jahr darauf wurde bei einer Dopingkontrolle Nilsen erneut auf anabole Steroide positiv getestet. Er wurde daraufhin als Wiederholungstäter lebenslang gesperrt.
1989 und 1991 wurde er Norwegischer Meister.
Persönliche Bestleistungen
Kugelstoßen: 21,22 m, 6. Juni 1986, Indianapolis
Halle: 20,39 m, 28. Februar 1987, Lubbock
Weblinks
Fußnoten
Kugelstoßer (Norwegen)
Dopingfall in der Leichtathletik
Doping in Norwegen
Norweger
Geboren 1965
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,080 |
Since their first play in 2013, The Big House Theatre Company has worked with care-leavers and young people at risk, helping them to express themselves in entirely new ways.
This week, Phoenix Rising opens. A reimagining of their first play with a new cast, this iteration is being staged in memory of an original cast member who died earlier this year. Kieran Taylor, also known as Dwayne Kieran Nero, was a student at Wac Arts and died after collapsing in his car in Camden Street. Tutors at Wac Arts launched a memorial fund in his name this August.
The lead role is being played by Aston McCauley, who Maggie describes as a "breathtakingly talented young man".
A recent development is a programme called The Big House Means Business, a branch of the company where the young people are taken into the corporate and social care worlds. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,640 |
The sites may link to and/or contain advertisements about non-SPE owned or controlled websites or other Internet or mobile resources. You acknowledge and understand that SPE does not endorse or sponsor such other third party websites or other Internet resources and SPE EXPRESSLY AND SPECIFICALLY DISCLAIMS ANY RESPONSIBILITY AND LIABILITY FOR ANY CONTENT, SOFTWARE, FUNCTIONALITY, SERVICES OR ADVERTISED PRODUCTS OR SERVICES FOUND ON OR RELATED TO ANY SUCH THIRD PARTY WEB SITE OR OTHER INTERNET OR MOBILE RESOURCES.
YOU EXPRESSLY AGREE THAT YOUR USE OF THE SITE IS AT YOUR SOLE RISK. THE SITE (AND ANY PORTION OF THE SITE) IS PROVIDED "AS IS" "WITH ALL FAULTS" AND "AS AVAILABLE." TO THE FULLEST EXTENT PERMITTED BY LAW, SPE AND ITS AFFILIATES EXPRESSLY DISCLAIM ANY AND ALL WARRANTIES OF ANY KIND, WHETHER EXPRESS OR IMPLIED (INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR USE OR PURPOSE AND NON-INFRINGEMENT). Neither SPE nor any of its affiliates make any warranties or representations about the accuracy or completeness of content available on or through the site or the content of any websites, personal social media or other Internet or mobile resources linked to the site and assume no liability or responsibility for any: (i) errors, mistakes, or inaccuracies of content; (ii) personal injury or property damage, of any nature whatsoever, resulting from your access to or use of the site (or any parts thereof); (iii) any unauthorized access to or use of SPE's secure servers and/or any and all personal information stored therein; (iv) any interruption or cessation of transmission to or from the site; (v) any bugs, viruses, Trojan horses, or the like which may be transmitted to or through the site (or any parts thereof) by any third party; and/or (vi) for any loss or damage of any kind incurred as a result of the use of any User Submissions and/or other site content posted, shared, forwarded, emailed, transmitted, or otherwise made available on or by means of the site and/or otherwise through your or any other site users' exercise of any rights under any of the licenses granted by SPE herein. SPE reserves the right, in its sole and exclusive discretion, to change, modify, add, remove or disable access to any portion of the site (including, without limitation any of the site Services). | {
"redpajama_set_name": "RedPajamaC4"
} | 4,977 |
NodeJs wrapper for ideviceinstaller.
For use on OSX.
## Getting Started
First, make sure you have ideviceinstaller installed, if you don't you can get it with brew, simply run in you Terminal:
```
brew install ideviceinstaller
```
Install package with npm:
```
npm install ideviceinstaller-js --save
```
## Usage
import the module:
```
var ideviceinstaller = require('ideviceinstaller-js')
```
Note: Providing the device uuid is optional.
* ideviceinstaller.installApp
```javascript
ideviceinstaller.installApp('path/to/appFile.app/ipa','my-device-udid')
```
* ideviceinstaller.uninstallApp
```javascript
ideviceinstaller.uninstallApp('myapp.bundle.id','my-device-udid')
```
* ideviceinstaller.reinstallApp
Will try to uninstall first then install
```javascript
ideviceinstaller.reinstallApp('path/to/appFile.app/ipa','myapp.bundle.id','my-device-udid')
```
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,360 |
Nestle and Chinese university join forces to improve coffee quality and yield
by Weida Li Dec 18, 2018 16:12 MARKETS INDUSTRY FOOD
Over 100 tasters gathered in Yunnan in March to try and find the province's best coffee from a range of 123 coffee beans grown in places such as Pu'er, Lincang, Baoshan and Xishuangbanna. China News Service
LATEST IN LIFE & CULTURE
by GBTIMES TAMPERE Sponsored
by GBTIMES BEIJING SHANGHAI Sponsored
in the 2012 anti-corruption campaign, corrupt high-level officials were known as 'tigers' and those at grassroots were called 'flies'. Read more
Transnational food and drink company Nestle has teamed up with Yunnan Agriculture University to improve coffee plantations in the southwest Chinese city of Pu'er, Yunnan Province.
Nestle Coffee Centre (NCC) and the university signed a memorandum of understanding on Monday, in a move designed to reinforce the local coffee industry's competitiveness and sustainable development.
"Cooperation between the two sides will combine international and indigenous technology, enhance the recognition and competitiveness of Pu'er coffee on the international market and benefit more coffee growers," said Lu Han, director of Pu'er's Tea and Coffee Industry Bureau.
Starbucks partners with Alibaba for coffee delivery in China
Aug 02, 2018 14:14 alibaba non-alcoholic restaurants beijing Shanghai
Chinese city becomes world's major coffee distribution hub
May 05, 2017 09:25 coffee in China e-commerce trade non-alcoholic chongqing
Is China falling in love with coffee?
Jun 19, 2015 17:06 coffee in China non-alcoholic finance markets
According to Wang Hai, general manager of the coffee centre, within the next six years the two sides will transplant new coffee tree varieties cultivated at the facility to different areas in Yunnan in order to find out the area's most suitable varieties.
Moreover, the first stage of cooperation will involve the two sides conducting field trials in three counties across Pu'er in 2019.
The NCC, which covers an area of 30,000 square metres, was opened in 2016 at the Pu'er Industrial Park. Rumoured to be the biggest of its kind in China, the coffee centre is equipped with research, processing and training facilities.
Yunnan accounts for 95 percent of China's coffee harvest, with around half coming from the mist-shrouded landscape around Pu'er. The region's temperate climate is believed to be perfect for growing arabica beans.
Currently, China is the 13th biggest coffee producer in the world — rising from zero output three decades ago to 140,000 tonnes annually in 2016.
Weida Li
YUNNAN PROVINCE (109)
Taking Chinese viewers on a cultural journey across Italy
Almost six million people were introduced to Italian towns and villages in a video series that was promoted on social media networks throughout China in cooperation with GBTIMES.
by GBTIMES TRAVEL TOURISM
Helga Zepp-LaRouche discusses China's Belt and Road initiative | Video
Helga Zepp-LaRouche, founder and president of the Schiller Institute, talks about the Second Belt and Road Forum, the initiative's growing criticism and its role in helping Europe....
by GBTIMES BEIJING TRADE
China-EU ties: reinforcing cooperation vector at government and business levels
The Europe-China Economic and Trade Forum is building relationships within the EU by gathering experts to share advice with local entrepreneurs looking to expand into China.
by Jaycee Lui INVESTMENT MARKETS
China and France mark 55th anniversary of bilateral ties
Chinese President Xi Jinping and French President Emmanuel Macron on Sunday exchanged congratulations on the 55th anniversary of the establishment of diplomatic relations.
by Weida Li BEIJING TRADE
China tests new generation of maglev train
China on Saturday tested a new generation of maglev train with a top speed of 160 kilometres per hour in Changsha, the capital of central China's Hunan Province.
by Weida Li INDUSTRY HUNAN PROVINCE | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,601 |
<?xml version="1.0" encoding="ISO-8859-1"?>
<?xml-stylesheet href="latest_ob.xsl" type="text/xsl"?>
<current_observation version="1.0"
xmlns:xsd="http://www.w3.org/2001/XMLSchema"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://www.weather.gov/view/current_observation.xsd">
<credit>NOAA's National Weather Service</credit>
<credit_URL>http://weather.gov/</credit_URL>
<image>
<url>http://weather.gov/images/xml_logo.gif</url>
<title>NOAA's National Weather Service</title>
<link>http://weather.gov</link>
</image>
<suggested_pickup>15 minutes after the hour</suggested_pickup>
<suggested_pickup_period>60</suggested_pickup_period>
<location>Logan, Logan-Cache Airport, UT</location>
<station_id>KLGU</station_id>
<latitude>41.78278</latitude>
<longitude>-111.85389</longitude>
<observation_time>Last Updated on Jan 4 2016, 7:51 pm MST</observation_time>
<observation_time_rfc822>Mon, 04 Jan 2016 19:51:00 -0700</observation_time_rfc822>
<weather>Overcast</weather>
<temperature_string>24.0 F (-4.4 C)</temperature_string>
<temp_f>24.0</temp_f>
<temp_c>-4.4</temp_c>
<relative_humidity>77</relative_humidity>
<wind_string>Calm</wind_string>
<wind_dir>North</wind_dir>
<wind_degrees>0</wind_degrees>
<wind_mph>0.0</wind_mph>
<wind_kt>0</wind_kt>
<pressure_string>1017.4 mb</pressure_string>
<pressure_mb>1017.4</pressure_mb>
<pressure_in>29.91</pressure_in>
<dewpoint_string>18.0 F (-7.8 C)</dewpoint_string>
<dewpoint_f>18.0</dewpoint_f>
<dewpoint_c>-7.8</dewpoint_c>
<visibility_mi>7.00</visibility_mi>
<icon_url_base>http://forecast.weather.gov/images/wtf/small/</icon_url_base>
<two_day_history_url>http://www.weather.gov/data/obhistory/KLGU.html</two_day_history_url>
<icon_url_name>novc.png</icon_url_name>
<ob_url>http://www.weather.gov/data/METAR/KLGU.1.txt</ob_url>
<disclaimer_url>http://weather.gov/disclaimer.html</disclaimer_url>
<copyright_url>http://weather.gov/disclaimer.html</copyright_url>
<privacy_policy_url>http://weather.gov/notice.html</privacy_policy_url>
</current_observation>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,470 |
from auto_gen import DBVistrail as _DBVistrail
from auto_gen import DBAdd, DBChange, DBDelete, DBAbstractionRef, DBModule
from id_scope import IdScope
class DBVistrail(_DBVistrail):
def __init__(self, *args, **kwargs):
_DBVistrail.__init__(self, *args, **kwargs)
self.idScope = IdScope(remap={DBAdd.vtType: 'operation',
DBChange.vtType: 'operation',
DBDelete.vtType: 'operation',
DBAbstractionRef.vtType: DBModule.vtType})
self.idScope.setBeginId('action', 1)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,465 |
Q: How can I power a USB cord from a 12v 2a power supply? I'm new to the forums and have a problem I cannot find an answer to.
I am making a lamp with LED strips inside powered by a 12v 2a power supply.
I want to also power a USB cord with this power supply so I can charge my phone at night.
Is there an easy way of going about this so the same power supply can both power the lights and charge my phone?
A: A buck converter is cheap and easy way. There are lots of them on eBay, baggood, amazon, AliExpress any of those sites.
Search for DC to DC converter 12 volt to 5 volt.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,348 |
Garret Pendergrass Pottery is a fully equipped, fun, family centered, pottery studio in the heart of Fort Worth. Kids and adults of all ages and skill levels are welcome to learn and fall in love with the art of ceramics in my laid back studio. You will quickly see your confidence and skills increase while the clay molds under your hands into unique works of art. If art is your passion and you have an interest in learning the ins and outs of ceramics, feel free to see what classes we have to offer here! We have multiple settings for you to choose from. From group classes to summer camps, I'm sure we have something for you! Done this before? Simply looking to brush up on your skills? We now host Open Studio time on Saturday's. | {
"redpajama_set_name": "RedPajamaC4"
} | 563 |
Inside the World Premiere of Matthew Weiner's New Show <em>The Romanoffs </em>
Inside the World Premiere of Matthew Weiner's New Show The Romanoffs
The Mad Men showrunner's new anthology series centers around the descendants of the Romanov family.
By Caroline Hallemann
David M. BenettGetty Images
This evening, Amazon Prime premiered its newest anthology series, The Romanoffs with an event in London. The show's cast, including Ines Melab, Marthe Keller, and Hugh Skinner, and guests like Jon Hamm and the stars of The Marvelous Mrs. Maisel all walked the carpet in support of the show, which centers around the modern-day descendants of the Romanov family. Read on for a few photos from tonight's party, then tune in to The Romanoffs on Amazon Prime October 12.
Ines Melab
Tim P. WhitbyGetty Images
Adele Anderson
While the Mad Men star isn't appearing in The Romanoffs, he was still there to celebrate his old boss, Matthew Weiner.
Jing Lusi
Showrunner Matthew Weiner
Eamonn M. McCormackGetty Images
Hugh Skinner
The Cast of the Marvelous Mrs. Maisel.
Michael Zegen, Rachel Brosnahan, Tony Shalhoub and Marin Hinkle all walked the red carpet in support of their new Amazon Prime colleagues.
More: Everything We Know About The Marvelous Mrs. Maisel's Second Season
Caroline Hallemann Digital Director As the digital director for Town & Country, Caroline Hallemann covers culture, entertainment, and a range of other subjects
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'Poker Face' Inverts Traditional Murder Mysteries | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,720 |
require 'spec_helper'
RSpec.describe Tinybucket::Error::ServiceError do
context 'initialize' do
let(:env) do
{
response_headers: [],
method: 'POST',
url: 'https://api.example.org/path/to',
status: 500,
body: 'Internal Server Error'
}
end
subject { Tinybucket::Error::ServiceError.new(env) }
it { expect(subject).to be_an_instance_of(Tinybucket::Error::ServiceError) }
it { expect(subject.http_method).to eq(env[:method]) }
it { expect(subject.request_url).to eq(env[:url]) }
it { expect(subject.status_code).to eq(env[:status]) }
it { expect(subject.response_body).to eq(env[:body]) }
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,479 |
{"url":"https:\/\/wikimili.com\/en\/Econometrics","text":"# Econometrics\n\nLast updated\n\nEconometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. [1] More precisely, it is \"the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference\". [2] An introductory economics textbook describes econometrics as allowing economists \"to sift through mountains of data to extract simple relationships\". [3] The first known use of the term \"econometrics\" (in cognate form) was by Polish economist Pawe\u0142 Ciompa in 1910. [4] Jan Tinbergen is considered by many to be one of the founding fathers of econometrics. [5] [6] [7] Ragnar Frisch is credited with coining the term in the sense in which it is used today. [8]\n\nIn linguistics, cognates are words that have a common etymological origin. Cognates are often inherited from a shared parent language, but they may also involve borrowings from some other language. For example, the English words dish and desk and the German word Tisch (\"table\") are cognates because they all come from Latin discus, which relates to their flat surfaces. Cognates may have evolved similar, different or even opposite meanings, but in most cases there are some similar sounds or letters in the words, in some cases appearing to be dissimilar. Some words sound similar, but do not come from the same root; these are called false cognates, while some are truly cognate but differ in meaning; these are called false friends.\n\nJan Tinbergen was an important Dutch economist. He was awarded the first Nobel Memorial Prize in Economic Sciences in 1969, which he shared with Ragnar Frisch for having developed and applied dynamic models for the analysis of economic processes. He is widely considered to be one of the most influential economists of the 20th century and one of the founding fathers of econometrics. It has been argued that the development of the first macroeconometric models, the solution of the identification problem, and the understanding of dynamic models are his three most important legacies to econometrics. Tinbergen was a founding trustee of Economists for Peace and Security. In 1945, he founded the Bureau for Economic Policy Analysis (CPB) and was the agency's first director.\n\n## Contents\n\nA basic tool for econometrics is the multiple linear regression model. [9] Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. [10] [11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting.\n\nThe theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches. Within a given approach, statistical theory gives ways of comparing statistical procedures; it can find a best possible procedure within a given context for given statistical problems, or can provide guidance on the choice between alternative procedures.\n\nMathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.\n\nIn statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule, the quantity of interest and its result are distinguished.\n\n## Basic models: linear regression\n\nA basic tool for econometrics is the multiple linear regression model. [9] In modern econometrics, other statistical tools are frequently used, but linear regression is still the most frequently used starting point for an analysis. [9] Estimating a linear regression on two variables can be visualised as fitting a line through data points representing paired values of the independent and dependent variables.\n\nFor example, consider Okun's law, which relates GDP growth to the unemployment rate. This relationship is represented in a linear regression where the change in unemployment rate (${\\displaystyle \\Delta \\ {\\text{Unemployment}}}$) is a function of an intercept (${\\displaystyle \\beta _{0}}$), a given value of GDP growth multiplied by a slope coefficient ${\\displaystyle \\beta _{1}}$ and an error term, ${\\displaystyle \\varepsilon }$:\n\nIn economics, Okun's law is an empirically observed relationship between unemployment and losses in a country's production. The \"gap version\" states that for every 1% increase in the unemployment rate, a country's GDP will be roughly an additional 2% lower than its potential GDP. The \"difference version\" describes the relationship between quarterly changes in unemployment and quarterly changes in real GDP. The stability and usefulness of the law has been disputed.\n\n${\\displaystyle \\Delta \\ {\\text{Unemployment}}=\\beta _{0}+\\beta _{1}{\\text{Growth}}+\\varepsilon .}$\n\nThe unknown parameters ${\\displaystyle \\beta _{0}}$ and ${\\displaystyle \\beta _{1}}$ can be estimated. Here ${\\displaystyle \\beta _{1}}$ is estimated to be \u22121.77 and ${\\displaystyle \\beta _{0}}$ is estimated to be 0.83. This means that if GDP growth increased by one percentage point, the unemployment rate would be predicted to drop by 1.77 points. The model could then be tested for statistical significance as to whether an increase in growth is associated with a decrease in the unemployment, as hypothesized. If the estimate of ${\\displaystyle \\beta _{1}}$ were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related. The variance in a prediction of the dependent variable (unemployment) as a function of the independent variable (GDP growth) is given in polynomial least squares.\n\nIn statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis. More precisely, a study's defined significance level, denoted \u03b1, is the probability of the study rejecting the null hypothesis, given that the null hypothesis were assumed to be true; and the p-value of a result, p, is the probability of obtaining a result at least as extreme, given that the null hypothesis were true. The result is statistically significant, by the standards of the study, when . The significance level for a study is chosen before data collection, and typically set to 5% or much lower, depending on the field of study.\n\nIn mathematical statistics, polynomial least squares comprises a broad range of statistical methods for estimating an underlying polynomial that describes observations. These methods include polynomial regression, curve fitting, linear regression, least squares, ordinary least squares, simple linear regression, linear least squares, approximation theory and method of moments. Polynomial least squares has applications in radar trackers, estimation theory, signal processing, statistics, and econometrics.\n\n## Theory\n\nEconometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. [10] [11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. An estimator is unbiased if its expected value is the true value of the parameter; it is consistent if it converges to the true value as the sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size. Ordinary least squares (OLS) is often used for estimation since it provides the BLUE or \"best linear unbiased estimator\" (where \"best\" means most efficient, unbiased estimator) given the Gauss-Markov assumptions. When these assumptions are violated or other statistical properties are desired, other estimation techniques such as maximum likelihood estimation, generalized method of moments, or generalized least squares are used. Estimators that incorporate prior beliefs are advocated by those who favour Bayesian statistics over traditional, classical or \"frequentist\" approaches.\n\nIn statistics, the bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, \"bias\" is an objective property of an estimator. Unlike the ordinary English use of the term \"bias\", it is not pejorative even though it's not a desired property.\n\nIn the comparison of various statistical procedures, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. This article primarily deals with efficiency of estimators.\n\nIn statistics, a consistent estimator or asymptotically consistent estimator is an estimator\u2014a rule for computing estimates of a parameter \u03b80\u2014having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to \u03b80. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to \u03b80 converges to one.\n\n## Methods\n\nApplied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting. [12]\n\nEconometrics may use standard statistical models to study economic questions, but most often they are with observational data, rather than in controlled experiments. [13] In this, the design of observational studies in econometrics is similar to the design of studies in other observational disciplines, such as astronomy, epidemiology, sociology and political science. Analysis of data from an observational study is guided by the study protocol, although exploratory data analysis may be useful for generating new hypotheses. [14] Economics often analyses systems of equations and inequalities, such as supply and demand hypothesized to be in equilibrium. Consequently, the field of econometrics has developed methods for identification and estimation of simultaneous-equation models. These methods are analogous to methods used in other areas of science, such as the field of system identification in systems analysis and control theory. Such methods may allow researchers to estimate models and investigate their empirical consequences, without directly manipulating the system.\n\nOne of the fundamental statistical methods used by econometricians is regression analysis. [15] Regression methods are important in econometrics because economists typically cannot use controlled experiments. Econometricians often seek illuminating natural experiments in the absence of evidence from controlled experiments. Observational data may be subject to omitted-variable bias and a list of other problems that must be addressed using causal analysis of simultaneous-equation models. [16]\n\nIn addition to natural experiments, quasi-experimental methods have been used increasingly commonly by econometricians since the 1980s, in order to credibly identify causal effects. [17]\n\n## Example\n\nA simple example of a relationship in econometrics from the field of labour economics is:\n\n${\\displaystyle \\ln({\\text{wage}})=\\beta _{0}+\\beta _{1}({\\text{years of education}})+\\varepsilon .}$\n\nThis example assumes that the natural logarithm of a person's wage is a linear function of the number of years of education that person has acquired. The parameter ${\\displaystyle \\beta _{1}}$ measures the increase in the natural log of the wage attributable to one more year of education. The term ${\\displaystyle \\varepsilon }$ is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, ${\\displaystyle \\beta _{0}{\\mbox{ and }}\\beta _{1}}$ under specific assumptions about the random variable ${\\displaystyle \\varepsilon }$. For example, if ${\\displaystyle \\varepsilon }$ is uncorrelated with years of education, then the equation can be estimated with ordinary least squares.\n\nIf the researcher could randomly assign people to different levels of education, the data set thus generated would allow estimation of the effect of changes in years of education on wages. In reality, those experiments cannot be conducted. Instead, the econometrician observes the years of education of and the wages paid to people who differ along many dimensions. Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education. For example, people born in certain places may have higher wages and higher levels of education. Unless the econometrician controls for place of birth in the above equation, the effect of birthplace on wages may be falsely attributed to the effect of education on wages.\n\nThe most obvious way to control for birthplace is to include a measure of the effect of birthplace in the equation above. Exclusion of birthplace, together with the assumption that ${\\displaystyle \\epsilon }$ is uncorrelated with education produces a misspecified model. Another technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render ${\\displaystyle \\beta _{1}}$ identifiable. [18] An overview of econometric methods used to study this problem were provided by Card (1999). [19]\n\n## Journals\n\nThe main journals that publish work in econometrics are Econometrica , the Journal of Econometrics , the Review of Economics and Statistics , Econometric Theory , the Journal of Applied Econometrics , Econometric Reviews , the Econometrics Journal , [20] Applied Econometrics and International Development , and the Journal of Business & Economic Statistics .\n\n## Limitations and criticisms\n\nLike other forms of statistical analysis, badly specified econometric models may show a spurious relationship where two variables are correlated but causally unrelated. In a study of the use of econometrics in major economics journals, McCloskey concluded that some economists report p-values (following the Fisherian tradition of tests of significance of point null-hypotheses) and neglect concerns of type II errors; some economists fail to report estimates of the size of effects (apart from statistical significance) and to discuss their economic importance. She also argues that some economists also fail to use economic reasoning for model selection, especially for deciding which variables to include in a regression. [21] [22]\n\nIn some cases, economic variables cannot be experimentally manipulated as treatments randomly assigned to subjects. [23] In such cases, economists rely on observational studies, often using data sets with many strongly associated covariates, resulting in enormous numbers of models with similar explanatory ability but different covariates and regression estimates. Regarding the plurality of models compatible with observational data-sets, Edward Leamer urged that \"professionals ... properly withhold belief until an inference can be shown to be adequately insensitive to the choice of assumptions\". [23]\n\n## Notes\n\n1. M. Hashem Pesaran (1987). \"Econometrics,\" The New Palgrave: A Dictionary of Economics , v. 2, p. 8 [pp. 8\u201322]. Reprinted in J. Eatwell et al., eds. (1990). Econometrics: The New Palgrave, p. 1 [pp. 1\u201334]. Abstract Archived 18 May 2012 at the Wayback Machine (2008 revision by J. Geweke, J. Horowitz, and H. P. Pesaran).\n2. P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone (1954). \"Report of the Evaluative Committee for Econometrica,\" Econometrica 22(2), p. 142. [p p. 141-146], as described and cited in Pesaran (1987) above.\n3. Paul A. Samuelson and William D. Nordhaus, 2004. Economics . 18th ed., McGraw-Hill, p. 5.\n4. \"Archived copy\". Archived from the original on 2 May 2014. Retrieved 1 May 2014.CS1 maint: archived copy as title (link)\n5. \"1969 - Jan Tinbergen: Nobelprijs economie - Elsevierweekblad.nl\". elsevierweekblad.nl. 12 October 2015. Archived from the original on 1 May 2018. Retrieved 1 May 2018.\n6. Magnus, Jan & Mary S. Morgan (1987) The ET Interview: Professor J. Tinbergen in: 'Econometric Theory 3, 1987, 117\u2013142.\n7. Willlekens, Frans (2008) International Migration in Europe: Data, Models and Estimates. New Jersey. John Wiley & Sons: 117.\n8. \u2022 H. P. Pesaran (1990), \"Econometrics,\" Econometrics: The New Palgrave, p. 2, citing Ragnar Frisch (1936), \"A Note on the Term 'Econometrics',\" Econometrica, 4(1), p. 95.\n\u2022 Aris Spanos (2008), \"statistics and economics,\" The New Palgrave Dictionary of Economics , 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine\n9. Greene, William (2012). \"Chapter 1: Econometrics\". Econometric Analysis (7th ed.). Pearson Education. pp.\u00a047\u201348. ISBN \u00a0 9780273753568. Ultimately, all of these will require a common set of tools, including, for example, the multiple regression model, the use of moment conditions for estimation, instrumental variables (IV) and maximum likelihood estimation. With that in mind, the organization of this book is as follows: The first half of the text develops fundamental results that are common to all the applications. The concept of multiple regression and the linear regression model in particular constitutes the underlying platform of most modeling, even if the linear model itself is not ultimately used as the empirical specification.\n10. Greene, William (2012). Econometric Analysis (7th ed.). Pearson Education. pp.\u00a034, 41\u201342. ISBN \u00a0 9780273753568.\n11. Wooldridge, Jeffrey (2012). \"Chapter 1: The Nature of Econometrics and Economic Data\". Introductory Econometrics: A Modern Approach (5th ed.). South-Western Cengage Learning. p.\u00a02. ISBN \u00a0 9781111531041.\n12. Clive Granger (2008). \"forecasting,\" The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine\n13. Wooldridge, Jeffrey (2013). Introductory Econometrics, A modern approach. South-Western, Cengage learning. ISBN \u00a0 978-1-111-53104-1.\n14. Herman O. Wold (1969). \"Econometrics as Pioneering in Nonexperimental Model Building,\" Econometrica, 37(3), pp. 369-381.\n15. For an overview of a linear implementation of this framework, see linear regression.\n16. Edward E. Leamer (2008). \"specification problems in econometrics,\" The New Palgrave Dictionary of Economics. Abstract. Archived 23 September 2015 at the Wayback Machine\n17. Angrist, Joshua D; Pischke, J\u00f6rn-Steffen (May 2010). \"The Credibility Revolution in Empirical Economics: How Better Research Design is Taking the Con out of Econometrics\". Journal of Economic Perspectives. 24 (2): 3\u201330. doi:10.1257\/jep.24.2.3. ISSN \u00a0 0895-3309.\n18. Pearl, Judea (2000). Causality: Model, Reasoning, and Inference. Cambridge University Press. ISBN \u00a0 978-0521773621.\n19. Card, David (1999). \"The Causal Effect of Education on Earning\". In Ashenfelter, O.; Card, D. (eds.). Handbook of Labor Economics. Amsterdam: Elsevier. pp.\u00a01801\u20131863. ISBN \u00a0 978-0444822895.\n20. \"The Econometrics Journal \u2013 Wiley Online Library\". Wiley.com. Retrieved 8 October 2013.\n21. McCloskey (May 1985). \"The Loss Function has been mislaid: the Rhetoric of Significance Tests\". American Economic Review. 75 (2).\n22. Stephen T. Ziliak and Deirdre N. McCloskey (2004). \"Size Matters: The Standard Error of Regressions in the American Economic Review,\" Journal of Socio-economics, 33(5), pp. 527-46 Archived 25 June 2010 at the Wayback Machine (press +).\n23. Leamer, Edward (March 1983). \"Let's Take the Con out of Econometrics\". American Economic Review. 73 (1): 31\u201343. 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\section{Introduction}
A del Pezzo surface is a smooth, proper algebraic
surface~$S$
over a field
$K$
with very ample anti-canonical sheaf
$\calK^{-1}$.
Over~an algebraically
closed field, every del Pezzo surface of degree
$d \leq 7$
is isomorphic to
$\Pb^2$,
blown up in
$(9-d)$
points in general position~\cite[The\-o\-rem~24.4.iii)]{Ma}.
According to the adjunction formula, a smooth complete intersection of two quadrics in
$\Pb^4$
is del Pezzo. The converse is true, as~well. For every del Pezzo surface of degree four, its anticanonical image is the complete intersection of two quadrics in
$\Pb^4$~\cite[Theorem~8.6.2]{Do}.\smallskip
Although del Pezzo surfaces over number fields are generally expected to have many rational points, they do not always fulfill weak~approximation. Even~the Hasse principle may~fail. The~first example of a degree four del Pezzo surface violating the Hasse principle has been devised by B.~Birch and Sir Peter Swinnerton-Dyer \mbox{\cite[Theorem~3]{BSD}}. It~is given
in~$\Pb^4_\bbQ$
by the equations
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - 5T_3^2 \, ,\\
(T_0+T_1)(T_0+2T_1) & = & T_2^2 - 5T_4^2 \, .
\end{eqnarray*}
Meanwhile,~more counterexamples to the Hasse principle have been constructed, see, e.g.,~\cite[Examples~15 and~16]{BBFL}. Only~quite recently, N.\,D.\,Q.~Nguyen \mbox{\cite[Theorem~1.1]{Ng}} proved that the degree four del Pezzo surface, given~by
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - (64k^2+40k+5)T_3^2 \, ,\\
(T_0+(8k+1)T_1)(T_0+(8k+2)T_1) & = & T_2^2 - (64k^2+40k+5)T_4^2
\end{eqnarray*}
is a counterexample to the Hasse principle if
$k$
is an integer such that
$64k^2+40k+5$
is a prime~number. In~particular, under the assumption of Schinzel's hypothesis, this family contains infinitely many members that violate the Hasse principle.\medskip
In this paper, we prove that del Pezzo surfaces of degree four that are counterexamples to the Hasse principle are Zariski dense in the moduli~scheme. In~particular, we establish, for the first time unconditionally, that their number up to isomorphism is~infinite.
Although~certainly the case of the base
field~$\bbQ$
is of particular interest, we work over an arbitrary number
field~$K$.
Before we can state our main results, we need to recall some notation and facts about the coarse moduli scheme of degree four del Pezzo surfaces.\smallskip
For this we consider a del Pezzo surface
$X$
of degree four given as the zero set of two quinary quadrics
$$Q_1(T_0, \ldots, T_4) = Q_2(T_0, \ldots, T_4) = 0 \, .$$
The pencil $\smash{(uQ_1+vQ_2)_{(u:v) \in \Pb^1}}$
of quadrics defined by the forms
$Q_1$
and~$Q_2$
contains exactly five degenerate~elements. The~corresponding five values
$t_1, \ldots, t_5 \in \Pb^1(\overline{K})$
of~$t := (u:v)$
are uniquely determined by the surface
$X$,
up to permutation and the natural operation of
$\smash{\Aut(\Pb^1) \cong \PGL_2(\overline{K})}$.
Let~$\calU \subset (\Pb^1)^5$
be the Zariski open subset given by the condition that no two of the five components~coincide. Then~there is an~isomorphism
$$j\colon \calU\!/(S_5 \times \PGL_2) \stackrel{\cong}{\longrightarrow} \calM$$
to the coarse moduli
scheme~$\calM$
of degree four del Pezzo surfaces~\cite[Section~5]{HKT}
The~quotient of
$\calU$
modulo
$S_5$
alone is the space of all binary quintics without multiple roots, up to multiplication by~constants. This~is part of the stable locus in the sense of Geometric Invariant Theory, which is formed by all quintics without roots of multiplicity
$\geq\! 3$~\cite[Proposition~4.1]{MFK}.\smallskip
Furthermore,~classical~invariant theory teaches that, for binary quintics, there are three fundamental invariants
$I_4$,
$I_8$,
and~$I_{12}$
of degrees
$4$,
$8$,
and
$12$,
respectively, that define an open embedding
$$\iota\colon \calU\!/(S_5 \times \PGL_2) \hookrightarrow \Pb(1,2,3)$$
into a weighted projective~plane. This~result is originally due to Ch.~Hermite~\cite[Section~VI]{He}, cf.\ \cite[Paragraphs 224--228]{Sa}. A~more recent treatment from a computational point of view is due to A.~Abdesselam~\cite{Ab}.\smallskip
Altogether, this yields an open embedding
$I\colon \calM \hookrightarrow \Pb(1,2,3)$.
More~generally, every family
$\pi\colon \calS \rightarrow B$
of degree four del Pezzo surfaces over a base scheme
$B$
induces a morphism
$$I_\pi=I \colon B \to \Pb(1,2,3) \, ,$$
which we call the {\em invariant map\/} associated
with~$\pi$.
\begin{rem}
There cannot be a fine moduli scheme for degree four del Pezzo surfaces, as every such
surface~$X$
has at least 16 automorphisms~\cite[Theorem~8.6.8]{Do}. (The statement of Theorem 8.6.8 in \cite{Do} contains a misprint, but it is clear from the proof that the described quotient group may be isomorphic to one of the listed groups or may be trivial).
\end{rem}
We can now state our first main result in the following form.
\begin{theo}\label{theo1}
Let\/~$K$
be any number field,
$U_\reg \subset \Gr(2,15)_K$
the open subset of the Gra\ss mann scheme that parametrizes degree four del Pezzo surfaces, and\/
$\calHC_K \subset U_\reg(K)$
be the set of all degree four del Pezzo surfaces
over\/~$K$
that are counterexamples to the Hasse~principle.\smallskip
\noindent
Then~the image of\/
$\calHC_K$
under the invariant map
$$\smash{I\colon U_\reg \longrightarrow \Pb(1,2,3)_K}$$
is Zariski~dense.
\end{theo}
In Theorem \ref{theo1} we identify the space
$S^2((K^5)^*)$
of all quinary quadratic forms with coefficients
in~$K$
with~$K^{15}$.
This~is clearly a non-canonical~isomorphism.
To~give an intersection of two quadrics
in~$\Pb^4_K$
is then equivalent to giving a
$K$-rational
plane through the origin
of~$K^{15}$,
i.e.\ a
$K$-rational
point on the Gra\ss mann scheme
$\Gr(2,15)_K$.
The~open subset
$U_\reg \subset \Gr(2,15)_K$
that parametrizes non-singular surfaces is isomorphic to the Hilbert scheme~\cite{Gr1} of del Pezzo surfaces of degree four
in~$\Pb^4_K$.
We~will not go into the details of this as they are not necessary for our~purposes.
\begin{rem}
An analogous result for cubic surfaces has recently been established by A.-S.~Elsenhans together with the first author~\cite{EJ}. Our~approach is partly inspired by the methods applied in the cubic surface~case. The concrete construction of del Pezzo surfaces of degree four that violate the Hasse principle is motivated by the work \cite{Ng} of N.\,D.\,Q.\ Nguyen.
\end{rem}
In~fact, more is true than stated in Theorem~\ref{theo1}. Our second main result seems to be a strengthening of the first one, but is, in fact, more or less~equivalent. Our~strategy will be to prove Theorem~\ref{theo1} first and then to deduce Theorem~\ref{theo2} from~it.
\begin{theo}\label{theo2}
Let\/~$K$
be any number field,
$U_\reg \subset \Gr(2,15)_K$
the open subset of the Gra\ss mann scheme that parametrizes degree four del Pezzo surfaces, and\/
$\calHC_K \subset U_\reg(K)$
be the set of all degree four del Pezzo surfaces
over\/~$K$
that are counterexamples to the Hasse~principle.\smallskip
\noindent
Then\/~$\calHC_K$
is Zariski dense in\/
$\Gr(2,15)_K$.
\end{theo}
\begin{rem}[Particular
$K3$~surfaces
that fail the Hasse principle]
In his article~\cite{Ng}, N.\,D.\,Q.\ Nguyen also provides families of
$K3$~surfaces
of degree eight that violate the Hasse~principle.
These~$K3$~surfaces
allow a morphism
$p\colon Y \rightarrow X$
that is generically 2:1 down to a degree four del Pezzo
surface~$X$
that fails the Hasse~principle.
Since~$X(K) = \emptyset$,
the existence of the morphism alone ensures that
$Y(K) = \emptyset$.
Nguyen's construction easily generalizes to our setting. One~has to intersect the cone
$CX \subset \Pb^5$
over the del Pezzo surface with a quadric that avoids the~cusp. The~intersection $Y$
is then a degree eight
$K3$~surface,
provided it is smooth, which it is generically according to Bertini's theorem.
Thus,~$Y$
is a counterexample to the Hasse principle provided it has an adelic~point.
For~$Y$,
the failure of the Hasse principle may be explained by the Brauer-Manin obstruction (cf.~Section~\ref{sec_Brauer} for details).
If~$\alpha \in \Br(X)$
explains the failure
for~$X$
then
$p^*\alpha$
does so
for~$Y$.
However,~the
$K3$~surfaces
obtained in this way do clearly not dominate the moduli space of degree eight
$K3$~surfaces.
Indeed,~the pull-back homomorphism
$p^*\colon \Pic(X_{\overline{K}}) \to \Pic(Y_{\overline{K}})$
doubles the intersection numbers and is, in particular,~injective. This~means that
$Y$
has geometric Picard rank at least six, while a general degree eight
$K3$~surface
is of geometric Picard rank~one.
\end{rem}
\section{A family of degree four del Pezzo surfaces}
We~consider the surface
$S := S^{(D;A,B)}$
over a
field~$K$,
given by the equations
\begin{eqnarray}
T_0T_1 & = & T_2^2 - DT_3^2 \, , \label{eq_eins} \\
(T_0+AT_1)(T_0+BT_1) & = & T_2^2 - DT_4^2 \label{eq_zwei}
\end{eqnarray}
for
$A,B,D \in K$.
We~will typically assume that
$D$
is not a square
in~$K$
and that
$S$
is non-singular.
If~$S$
is non-singular, then
$S$
is a del Pezzo surface of degree~four.
\begin{prop}
\label{nonsing}
Let\/~$K$
be a field of characteristic\/
$\neq \!2$
and\/
$A,B,D \in K$.
\begin{abc}
\item
Then~the surface\/
$S^{(D;A,B)}$
is non-singular if and only if\/
$ABD\neq 0$,
$A\neq B$,
and\/
$A^2-2AB+B^2-2A-2B+1 \neq 0$.
\item
If\/~$D \neq 0$
then
\/
$S^{(D;A,B)}$
has not more than finitely many singular~points.
\end{abc}\smallskip
\noindent
{\bf Proof.}
{\em
a)
If
$D=0$
then
$S$
is a cone over a cone over four points
in~$\Pb^2$.
In~this case,
$S$
is singular, whether some of the four points coincide or~not.
Let~us suppose that
$D \neq 0$
from now~on.
A~point
$(t_0 : \ldots :t_4) \in S(K)$
is singular if and only if the Jacobian matrix
$$
\left(
\begin{array}{ccccc}
t_1 & t_0 & -2t_2 & 2Dt_3 & 0 \\
2t_0+(A+B)t_1 & (A+B)t_0+2ABt_1 & -2t_2 & 0 & 2Dt_4
\end{array}
\right)
$$
is not of full~rank. In~particular, we immediately see that
$(0:1:0:0:0) \in S(K)$
is a singular point in the case that
$A=0$
or~$B=0$.
Thus,~we may assume that
$A \neq 0$
and
$B \neq 0$.
Furthermore,~if
$(t_0 : \ldots :t_4) \in S(K)$
is singular then
$t_0^2 = ABt_1^2$
and
$t_2t_3 = t_2t_4 = t_3t_4 = 0$.
There~is clearly no point
on~$S$
that fulfills
$t_1 = 0$
together with these equations. Hence,~we may normalize the coordinates to
$t_1 = 1$,
i.e.~to
$t_0 = \pm\sqrt{AB}$,
and distinguish three cases.\smallskip
\noindent
{\em First case.}
$t_2=t_3=0$.
\noindent
Then
$t_0t_1=0$
by relation (\ref{eq_eins}), which is a~contradiction.\smallskip
\noindent
{\em Second case.}
$t_2=t_4=0$.
\noindent
Then
$t_0+At_1=0$ or $t_0+Bt_1=0$,
i.e.\
$\pm\sqrt{AB}+A=0$ or $\pm\sqrt{AB}+B=0$,
by the second equation. This immediately yields
$A=B$.\smallskip
\noindent
On~the other hand,
if~$A=B$
then
$\smash{((-A):1:0:\pm\sqrt{A/D}:0) \in S(\overline{K})}$
are singular~points.\smallskip
\noindent
{\em Third case.}
$t_3=t_4=0$.
\noindent
Then the relations (\ref{eq_eins}) and (\ref{eq_zwei}) together show that
$(t_0+At_1)(t_0+Bt_1) = t_0t_1$,
i.e.\
$$(\pm\sqrt{AB}+A)(\pm\sqrt{AB}+B) = \pm\sqrt{AB} \, .$$
This~equality clearly implies
$2AB \pm\sqrt{AB}(A+B-1) = 0$,
hence
$(A+B-1)^2 = 4AB$
and
$A^2-2AB+B^2-2A-2B+1 = 0$.
On the other hand, if
$A^2-2AB+B^2-2A-2B+1 = 0$
then
$\frac{1-A-B}2$
is a square root
of~$AB$
and a direct calculation shows that
$$\textstyle \big( \frac{1-A-B}2:1:\pm\sqrt{\frac{1-A-B}2}:0:0 \big)$$
are two singular points
on~$S^{(D;A,B)}$.\smallskip
\noindent
b)
Every singular point satisfies the relation
$t_0/t_1 = \pm\sqrt{AB}$.
Furthermore,~at least two of the coordinates
$t_2$,
$t_3$,
and~$t_4$
must~vanish. Together these~conditions define six lines in
$\Pb^4$,
which collapse to three in the case that
$AB = 0$.
If~there were infinitely many singular points then at least one of these lines would be entirely contained
in~$S$.
But~this is not the case, as, on each of the six lines, one equation of the~form
$$F(T_1) = T_2^2, \quad F(T_1) = -DT_3^2, \quad {\rm or} \quad F(T_1) = -DT_4^2$$
remains from the equations
of~$S$.
}
\eop
\end{prop}
\begin{rem}
Assume that
$D \in K$
is a non-square and that
$S^{(D;A,B)}$
is non-singular. Then~there is neither a
\mbox{$K$-rational}
point
$(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) \in S(K)$
such that
$t_0 = t_1 = 0$,
nor one such that
$t_0 + At_1 = t_0 + Bt_1 = 0$.
Indeed,~the first condition implies
$(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) = (0 \!:\! 0 \!:\! 0 \!:\! 0 \!:\! 1)$,
while, in view of
$A \neq B$,
the second one implies
$(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) = (0 \!:\! 0 \!:\! 0 \!:\! 1 \!:\! 0)$.
Both~points do not lie
on~$S$.
\end{rem}
\section{A class in the Grothendieck-Brauer group}
\label{sec_Brauer}
It~is a discovery of Yu.\,I.~Manin~\cite[\S47]{Ma} that a non-trivial element
$\alpha \in \Br(S)$
of the Gro\-then\-dieck-Brauer group \cite{Gr2}, \cite[Chapter~IV]{Mi} of a
variety~$S$
may cause a failure of the Hasse principle. Today,~this phenomenon is called the Brauer-Manin obstruction. Its~mechanism works as~follows.
Let
$K$
be a number field,
$\frakl \subset \calO_K$
a prime ideal, and
$K_\frakl$
be the corresponding completion. The~Grothendieck-Brauer group is a contravariant functor from the category of~schemes to the category of abelian groups. In~particular, for an arbitrary
scheme~$S$
and a
\mbox{$K_\frakl$-rational}
point
$x\colon \Spec K_\frakl \to S$,
there is a restriction~homomorphism
$x^*\colon \Br(S) \to \Br(\Spec K_\frakl) \cong \bbQ/\bbZ$.
For~a Brauer class
$\alpha \in \Br(X)$,
we call
$$\ev_{\alpha,\frakl}\colon S(K_\frakl) \longrightarrow \bbQ/\bbZ \, , \quad x \mapsto x^*(\alpha)$$
the local evaluation map, associated
to~$\alpha$.
Analogously, for
$\sigma\colon K \hookrightarrow \bbR$
a real prime, there is the local evaluation map
$\ev_{\alpha,\sigma}\colon S(K_\sigma) \to \frac12\bbZ/\bbZ$.
\begin{prop}[The Brauer-Manin obstruction to the Hasse principle]
\label{BM}
Let\/~$S$
be a projective variety over a number
field\/~$K$
and\/~$\alpha \in \Br(S)$
be a Brauer class.\smallskip
\noindent
For~every prime ideal\/
$\frakl \subset \calO_K$,
suppose that\/
$S(K_\frakl) \neq \emptyset$
and that the local evaluation map\/
$\ev_{\alpha,\frakl}$
is constant. Analogously, assume that, for every real prime\/
$\sigma\colon K \hookrightarrow \bbR$,
one has\/
$S(K_\sigma) \neq \emptyset$
and that the local evaluation map\/
$\ev_{\alpha,\sigma}$
is constant.
Denote the values of\/
$\ev_{\alpha,\frakl}$
and\/
$\ev_{\alpha,\sigma}$
by\/
$e_\frakl$
and\/
$e_\sigma$,
respectively.
If,~in this~situation,
$$\sum_{\frakl\subset\calO_K} \!\!e_\frakl + \!\!\!\sum_{\sigma\!\colon\!\! K \hookrightarrow \bbR} \!\!\!\!\!e_\sigma \neq \,0 \in \bbQ/\bbZ$$
then\/
$S$
is a counterexample to the Hasse~principle.\medskip
\noindent
{\bf Proof.}
{\em
The assumptions imply, in particular, that
$S$
is not the empty scheme. Consequently, there are
\mbox{$K_\tau$-rational}
points
on~$S$
for every complex prime\/
$\tau\colon K \hookrightarrow \bbC$.
The~Hasse principle would assert that
$S(K) \neq \emptyset$.
On~the other hand, by global class field theory~\cite[Section~10, Theorem~B]{Ta} one has a short exact sequence
$$0\rightarrow \Br(K)\rightarrow \bigoplus_{\nu} \Br(K_\nu) \rightarrow \bbQ/\bbZ\rightarrow 0 \, ,$$
where the direct sum is taken over all places
$\nu$
of the number field
$K$.
Assume that there is a point
$x\colon \Spec K \to S$.
Then
$x^*(\alpha) \in \Br(\Spec K)$
is a Brauer class that naturally maps to an element of
$\bigoplus_\frakl \Br(K_\frakl) \oplus \bigoplus_\sigma \!\Br(K_\sigma) \cong \bigoplus_\frakl \!\bbQ/\bbZ \oplus \bigoplus_\sigma \!\frac12\bbZ/\bbZ$
of a non-zero~sum, which is a contradiction to the exactness of the above sequence.
}
\eop
\end{prop}
\begin{prop}
\label{Brauerklasse}
Let\/~$K$
be any field of characteristic\/
$\neq \!2$
and\/
$A,B,D \in K \!\setminus\! \{0\}$
be arbitrary~elements. Set\/
$S := S^{(D;A,B)}$.
Suppose that\/
$D$
is a non-square and that\/
$S$
is non-singular. Put,~finally,
$\smash{L := K(\sqrt{D})}$.
\begin{abc}
\item
Then the quaternion algebra (see\/ \cite[Section 15.1]{Pi} for the notation)
$$\textstyle \calA := \big( L(S), \tau, \frac{T_0+AT_1}{T_0} \big)$$
over the function field\/
$K(S)$
extends to an Azumaya algebra over the whole
of\/~$S$.
Here,~by
$\tau \in \Gal(L(S)/K(S))$,
we denote the nontrivial~element.
\item
Assume that\/
$K$
is a number field and denote by\/
$\alpha \in \Br(S)$
the Brauer class, defined by the extension
of\/~$\calA$.
Let\/
$\frakl$
be any prime
of\/~$K$.
\begin{iii}
\item
Let\/~$(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) \in S(K_\frakl)$
be a point and assume that at least one of the quotients\/
$(t_0+At_1)/t_0$,
$(t_0+At_1)/t_1$,
$(t_0+Bt_1)/t_0$,
and\/
$(t_0+Bt_1)/t_1$
is properly defined and non-zero. Denote~that
by\/~$q$.
Then\/
$$\ev_{\alpha,\frakl}(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) =
\left\{
\begin{array}{cl}
0 & \text{~if\/~} (q,D)_\frakl = 1 \, , \\
\frac12 & \text{~if\/~} (q,D)_\frakl = -1 \, ,
\end{array}
\right.
$$
for\/
$(q,D)_\frakl$
the Hilbert~symbol.
\item
If\/~$\frakl$
is split in\/
$L$
then the local evaluation map\/
$\ev_{\alpha,\frakl}$
is constantly~zero.
\end{iii}
\end{abc}\smallskip
\noindent
{\bf Proof.}
{\em
a)
First of all,
$\calA$
is, by construction, a cyclic algebra of degree two. In~particular,
$\calA$
is simple~\cite[Section~15.1, Corollary~d]{Pi}.
Furthermore,~$\calA$
is obviously a central
\mbox{$K(S)$-algebra}.
To~prove the extendability assertion, it suffices to show that
$\calA$
extends as an Azumaya algebra over each valuation ring that corresponds to a prime divisor
on~$S$.
Indeed,~this is the classical Theorem of Auslander-Goldman for non-singular surfaces \cite[Proposition~7.4]{AG}, cf.~\cite[Chapter~IV, Theorem~2.16]{Mi}.
For~this, we observe that the principal divisor
$\div((T_0+AT_1)/T_0) \in \Div(S)$
is the norm of a divisor
on~$S_L$.
In~fact, it is the norm of the difference of two prime divisors, the conic, given by
$\smash{T_0+AT_1 = T_2 - \sqrt{D}T_4 = 0}$,
and the conic, given by
$\smash{T_0 = T_2 - \sqrt{D}T_3 = 0}$.
In~particular,
$\calA$
defines the zero element in
$H^2(\langle\sigma\rangle, \Div(S_L))$.
Under~such circumstances, the extendability
of~$\calA$
over the valuation ring corresponding to an arbitrary prime divisor
on~$S$
is worked out in
\cite[Paragraph 42.2]{Ma}.\smallskip
\noindent
b.i)
The~quotients
$$\textstyle \frac{T_0+AT_1}{T_0} \!/\! \frac{T_0+AT_1}{T_1} = \frac{T_2^2-DT_3^2}{T_0^2} \, , \;\;
\frac{T_0+BT_1}{T_0} \!/\! \frac{T_0+BT_1}{T_1} = \frac{T_2^2-DT_3^2}{T_0^2} \, ,
\;\, {\rm and} \;\;
\frac{T_0+AT_1}{T_0} \!/\! \frac{T_0+BT_1}{T_0} = \frac{T_2^2-DT_4^2}{(T_0+BT_1)^2}$$
are norms of rational~functions. Thus,~each of them defines the zero class in
$H^2(\langle\sigma\rangle, K(S_L)^*) \subseteq \Br K(S)$,
and hence
in~$\Br S$.
In~particular, the four expressions
$(T_0+AT_1)/T_0$,
$(T_0+AT_1)/T_1$,
$(T_0+BT_1)/T_0$,
and
$(T_0+BT_1)/T_1$
define the same Brauer~class.
The~general description of the evaluation map, given in \cite[Paragraph~45.2]{Ma} shows that
$\ev_{\alpha,\frakl}(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4)$
is equal to
$0$
or
$\frac12$
depending on whether
$q$
is in the image of the norm map
$N_{L_\frakL/K_\frakl}\colon L_\frakL^* \to K_\frakl^*$,
or not, for
$\frakL$
a prime of
$L$
lying
above~$\frakl$.
This~is exactly what is tested by the Hilbert
symbol~$(q,D)_\frakl$.\smallskip
\noindent
ii)
If~$\frakl$
is split in
$L$
then the norm map
$\smash{N_{K(S_{L_\frakL})/K(S_{K_\frakl})}\colon K(S_{L_\frakL})^* \to K(S_{K_\frakl})^*}$
is surjective. In~particular,
$\smash{\frac{T_0+AT_1}{T_0}} \in K(S_{K_\frakl})^*$
is the norm of a rational function
on~$S_{L_\frakL}$.
Therefore, it defines the zero class in
$H^2(\langle\sigma\rangle, K(S_{L_\frakL})^*) \subseteq \Br K(S_{K_\frakl})$,
and thus
in~$\Br S_{K_\frakl}$.
Finally,~we observe that every
\mbox{$K_\frakl$-rational}
point
$x\colon \Spec K_\frakl \to S$
factors via
$S_{K_\frakl}$.
}
\eop
\end{prop}
Geometrically,~on a rank four quadric
in~$\Pb^4$,
there are two pencils of~planes. In~our situation, these are conjugate to each other under the operation
of~$\smash{\Gal(K(\sqrt{D})/K)}$.
The~equation
$T_0=0$
cuts two conjugate planes out of the quadric~(\ref{eq_eins}) and the same is true
for~$T_1 = 0$.
The~equations
$T_0+AT_1=0$
and~$T_0+BT_1=0$
each cut two conjugate planes out of~(\ref{eq_zwei}).
\begin{rem}
A.~V\'arilly-Alvarado and B.~Viray \cite[Theorem~5.3]{VAV} prove for a certain class of degree four del Pezzo surfaces that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to weak~approximation. Their~result is conditional under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves and based on ideas of O.~Wittenberg \cite[Th\'eor\`eme~1.1]{Wi}.
The~class considered in~\cite{VAV} includes our family~(\ref{eq_eins},\,\ref{eq_zwei}).
\end{rem}
One might formulate our strategy to prove
$S^{(D;A,B)}(K) = \emptyset$
for
$K$
a number field and particular choices of
$A$,
$B$,
and~$D$
in a more elementary way as~follows.
Suppose~that there is a point
$(t_0 \!:\! t_1 \!:\! t_2 \!:\! t_3 \!:\! t_4) \in S(K)$.
Then~$(t_0, t_1) \neq (0,0)$.
Among
$(t_0+At_1)/t_0$,
$(t_0+At_1)/t_1$,
$(t_0+Bt_1)/t_0$,
and
$(t_0+Bt_1)/t_1$,
consider an
expression~$q$
that is properly defined and non-zero.
Then~show that, for every prime
$\frakl$
of~$K$
including the Archimedean ones, but with the exception of exactly an odd number, the Hilbert symbol
$(q,D)_\frakl$
is equal
to~$1$.
Finally,~observe that such a behaviour contradicts the Hilbert reciprocity law~\cite[Chapter~VI, Theorem~8.1]{Ne}.
In~other words, the element
$q \in K_\frakl$
belongs to the image of the norm map
$N \colon L_\frakL \to K_\frakl$,
for~$\smash{L := K(\sqrt{D})}$
and
$\frakL$
a prime
of~$L$
lying
above~$\frakl$,
for all but an odd number of~primes. And~this is incompatible with \cite[Chapter~VI, Corollary~5.7]{Ne} or~\cite[Theorem~5.1 together with 6.3]{Ta}.
\section{Unramified primes}
\begin{lem}
\label{P2fuenfPunkte}
Let\/~$K$
be any field of characteristic\/
$\neq \!2$
and\/
$A,B,D \in K$
be elements such that\/
$D \neq 0$.
Then~the minimal resolution of singularities\/
$\widetilde{S}$
of\/~$S := S^{(D;A,B)}$
is geometrically isomorphic to\/
$\Pb^2$,
blown up in five~points.\medskip
\noindent
{\bf Proof.}
{\em
{\em First step.}
$S_{\overline{K}}$
is a rational~surface.
\noindent
For this, we observe that the quadric hypersurface defined by equation~(\ref{eq_eins}) has
$(0 \!:\! 0 \!:\! 0 \!:\! 0 \!:\! 1)$
as its only singular point, while the hypersurface defined by~(\ref{eq_zwei}) is regular at that~point. According~to \cite[Book~IV, Paragraph~XIII.11, Theorem~3]{HP}, this implies that
$S_{\overline{K}}$
is a rational~surface.\smallskip
\noindent
{\em Second step.}
$S_{\overline{K}}$
has only isolated singularities, each of which is a rational double~point.
\noindent
Proposition \ref{nonsing}.b) implies that
$S_{\overline{K}}$
has only isolated~singularities. Moreover,~we observe that
$S_{\overline{K}}$
contains only finitely many~lines. Indeed,~there are only finitely many lines through each singular point, since
$S_{\overline{K}}$
is not a~cone. On~the other hand, a line contained in
$\smash{S_{\overline{K},\reg}}$
has self-intersection number
$(-1)$
according to the adjunction formula, and is therefore~rigid.
Now~consider the projection
$S'$
of
$\smash{S_{\overline{K}}}$
from a non-singular
$\smash{\overline{K}}$-rational
point~$p$
that does not lie on any of the~lines. By~construction,
$S'$
is a cubic surface
over~$\overline{K}$.
The~projection map
$\pr\colon S_{\overline{K}} \!\setminus\! \{p\} \to S'$
blows up the point
$p$
and is an isomorphism everywhere~else.
Indeed,
$\pr$
separates points since
$\pr(p_1) = \pr(p_2)$
implies that
$p$,
$p_1$,
and~$p_2$
are collinear, which enforces that the line through these three points must be contained in the quadrics defined by (\ref{eq_eins}) and (\ref{eq_zwei}), a
contradiction. The~same argument shows that
$\pr$
separates tangent~directions.
In~other words, the blowup
$\Bl_p(S_{\overline{K}})$
is a cubic~surface. Clearly,~it has as many singular points
as~$S_{\overline{K}}$.
In~particular,
$\Bl_p(S_{\overline{K}})$
is normal~\cite[Theorem~23.8]{Mt}. Moreover,~as
$S_{\overline{K}}$
is rational,
$\Bl_p(S_{\overline{K}})$
is a rational~surface.
It~is well known that there are two kinds of normal cubic surfaces.
Either~$\Bl_p(S_{\overline{K}})$
is the cone over an elliptic curve or it belongs to one of the 21 cases having only double points being
$ADE$,
as listed in \cite[Table 9.1]{Do}. The~former case is impossible as this is not a rational~surface.
Further,~$ADE$-singularities
are rational~\cite[page~135]{Ar}.\smallskip
\noindent
{\em Third step.}
Conclusion.
\noindent
Let~now
$\smash{\pi\colon \widetilde{S} \to S}$
be the resolution~map. The~adjunction formula shows that
$\smash{K_{\widetilde{S}} = \pi^*i^*(-H) + E}$,
where $H$ is a hyperplane section and
$E$
a divisor
on~$\smash{\widetilde{S}}$
supported on the exceptional fibers
of~$\pi$.
But,~as the singularities of
$S$
are rational double points, one necessarily has
$E = 0$~\cite[Proposition 8.1.10]{Do}.
This~yields that
$C \!\cdot\! K_{\widetilde{S}} \leq 0$
for every curve
$C \subset \widetilde{S}$.
Moreover,~$\smash{K_{\widetilde{S}}^2 = [i^*(-H)]^2 = 4}$.
In~other words,
$\smash{\widetilde{S}}$
is a generalized del Pezzo surface~\cite{CT} of degree~five. By~an observation of Demazure \cite[Proposition~0.4]{CT}, this implies that
$\smash{\widetilde{S}}$
is geometrically isomorphic to
$\smash{\Pb^2}$,
blown up in five~points.
}
\eop
\end{lem}
\begin{coro}
\label{ratpoint}
Let\/~$\bbF_{\!\ell}$
be a finite field of characteristic\/
$\neq \!2$
and\/
$A,B,D \in \bbF_{\!\ell}$
such that\/
$D \neq 0$.
Then\/~$S := S^{(D;A,B)}$
has a regular\/
$\bbF_{\!\ell}$-rational
point.\medskip
\noindent
{\bf Proof.}
{\em
By~Lemma~\ref{P2fuenfPunkte}, the minimal resolution of singularities
$\smash{\widetilde{S}}$
of~$S$
is geometrically isomorphic to
$\Pb^2$,
blown up in five~points. In~such a situation, the Weil conjectures have been established by A.~Weil himself \cite[page~557]{We}, cf.~\cite[Theorem~27.1]{Ma}.
At~least one of the eigenvalues of Frobenius
on~$\smash{\Pic(\widetilde{S}_{\overline\bbF_{\!\ell}})}$
is equal
to~$(+1)$.
Say,~the number of eigenvalues
$(+1)$
is exactly
$n \geq 1$.
The~remaining
$(6-n)$
eigenvalues are of real part
$\geq\!\! (-1)$.
Hence,~$\smash{\#\widetilde{S}(\bbF_{\!\ell}) \geq \ell^2 + (2n-6)\ell + 1}$.
Among~these, at most
$(n-1)(\ell+1)$
points may have originated from blowing up the singular points
of~$S_{\frakl}$.
Indeed,~each time an
\mbox{$\bbF_{\!\ell}$-rational}
point is blown up, a
\mbox{$(+1)$-eigenspace}
is added to the Picard~group.
Therefore,
$$\#S_\reg(\bbF_{\!\ell}) \geq \ell^2 + (2n-6)\ell + 1 - (n-1)(\ell+1) = \ell^2 - 5\ell + 2 + n(\ell-1) \geq \ell^2 - 4\ell + 1 \, .$$
For~$\ell \geq 5$,
this is positive.
Thus,~it only remains to consider the case that
$\ell = 3$.
Then~$S$
is the closed subvariety
of~$\Pb^4_{\!\bbF_{\!3}}$,
given~by
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - DT_3^2 \, ,\\
(T_0+aT_1)(T_0+bT_1) & = & T_2^2 - DT_4^2
\end{eqnarray*}
for
$D = \pm1$
and
certain~$a,b \in \bbF_{\!3}$.
Independently~of the values of
$a$
and~$b$,
$S$
has the regular
$\bbF_{\!3}$-rational
point
$(1\!:\!0\!:\!1\!:\!1\!:\!0)$
in the case that
$D=1$
and~$(1\!:\!0\!:\!0\!:\!0\!:\!1)$
in the case that
$D=-1$.
}
\eop
\end{coro}
\begin{prop}[Unramified primes]
\label{inert}
Let\/~$K$
be a number field,
$A,B,D \in \calO_K$,
and\/
$\frakl \subset \calO_K$
be a prime ideal that is unramified under the field extension\/
$K(\sqrt{D})/K$.
Consider~the
surface\/
$S := S^{(D;A,B)}$.
\begin{abc}
\item
If\/~$\#\calO_K/\frakl$
is not a power
of\/~$2$
then\/
$S(K_{\frakl}) \neq \emptyset$.
\item
Assume~that\/
$A \not\equiv B \pmod \frakl$,
that\/
$S$
is non-singular, and that\/
$S(K_\frakl) \neq \emptyset$.
Let\/~$\alpha \in \Br(S)$
be the Brauer class, described in Proposition~\ref{Brauerklasse}.a). Then~the local evaluation map\/
$\ev_{\alpha,\frakl} \colon S(K_\frakl) \to \bbQ/\bbZ$
is constantly~zero.
\end{abc}\smallskip
\noindent
{\bf Proof.}
{\em
We
put~$\ell := \#\calO_K/\frakl$.
Furthermore,~we normalize
$D$
to be a unit
in~$\calO_{K_\frakl}$.
This is possible because
$\frakl$
is~unramified.\smallskip
\noindent
a)
It~suffices to verify the existence of a regular
$\bbF_{\!\ell}$-rational
point on the reduction
$S_\frakl$
of~$S$.
For~this, we observe that
$(D \bmod \frakl\calO_{K_\frakl}) \neq 0$,
which shows that Corollary~\ref{ratpoint} applies.\smallskip
\noindent
b)
If~$\frakl$
is split then the assertion directly is Proposition~\ref{Brauerklasse}.b.ii).
Otherwise,~let
$(t_0\!:\!t_1\!:\!t_2\!:\!t_3\!:\!t_4) \in S(K_\frakl)$
be an arbitrary~point. Normalize the coordinates such that
$t_0, \ldots, t_4 \in \calO_{K_\frakl}$
and at least one is a~unit.
We~first observe that one of
$t_0$
and
$t_1$
must be a~unit. Indeed,~otherwise one has
$\frakl|t_0,t_1$.
According~to equation~(\ref{eq_eins}), this implies that
$\smash{\frakl|N_{K_\frakl(\sqrt{D})/K_\frakl}(t_2 + t_3\sqrt{D})}$.
Such a divisibility is possible only when
$\frakl|t_2,t_3$,
since
$\smash{K_\frakl(\sqrt{D})/K_\frakl}$
is an unramified, proper extension and
$\smash{\sqrt{D} \in K_\frakl(\sqrt{D})}$
is a~unit. But~then
$t_4$
is a unit, in contradiction to equation~(\ref{eq_zwei}).
Second,~we claim that
$t_0+At_1$
or
$t_0+Bt_1$
is a~unit.
Indeed,~since
$A \not\equiv B \pmod \frakl$,
the assumption
$\frakl|t_0+At_1, t_0+Bt_1$
implies
$\frakl|t_0,t_1$.
We~have thus shown that one of the four expressions
$(t_0+At_1)/t_0$,
$(t_0+At_1)/t_1$,
$(t_0+Bt_1)/t_0$,
and
$(t_0+Bt_1)/t_1$
is a~unit.
Write~$q$
for that~quotient. As~the local extension
$\smash{K_\frakl(\sqrt{D})/K_\frakl}$
is unramified of degree two, we see that
$(q,D)_\frakl = 1$.
Proposition~\ref{Brauerklasse}.b.i) implies the~assertion.
}
\eop
\end{prop}
If~$\frakl$
is a split prime then an even stronger statement is~true.
\begin{lem}[Split primes]
\label{split}
Let\/~$K$
be a number field,
$A,B,D \in \calO_K$,
and\/
$\frakl \subset \calO_K$
a prime ideal that is split under\/
$\smash{K(\sqrt{D})/K}$.
Consider~the
surface\/~$S := S^{(D;A,B)}$.
\begin{abc}
\item
Then\/
$S(K_{\frakl}) \neq \emptyset$.
\item
Furthermore,~if\/
$S$
is non-singular and\/
$\alpha \in \Br(S)$
is the Brauer class, described in Proposition \ref{Brauerklasse}.a), then the local evaluation map\/
$\ev_{\alpha,\frakl} \colon S(K_\frakl) \to \bbQ/\bbZ$
is constantly~zero.
\end{abc}\smallskip
\noindent
{\bf Proof.}
{\em
a)
The~assumption that
$\frakl$
is split under the field extension
$\smash{K(\sqrt{D})/K}$
is equivalent to
$\smash{\sqrt{D} \in K_\frakl}$.
Therefore, the point
$\smash{(1\!:\!0\!:\!1\!:\!\frac1{\sqrt{D}}\!:\!0)}$
is defined over
$K_\frakl$.
In~particular,
$S(K_\frakl)\neq \emptyset$.\smallskip
\noindent
b)
This is the assertion of Proposition~\ref{Brauerklasse}.b.ii).
}
\eop
\end{lem}
\begin{rem}
If
$\frakl$
is inert,
$0 \not\equiv A \equiv B \pmod \frakl$,
and
$(A/D \bmod \frakl) \in \calO_K/\frakl$
is a non-square then the assertion of~Proposition~\ref{inert}.b) is true,~too.
Indeed,~$t_0$
or~$t_1$
must be a unit by the same argument as~before. On~the other hand, the assumption
$\frakl|t_0+At_1, t_0+Bt_1$
does not lead to an immediate contradiction, but only to
$\frakl|t_2,t_4$
and
$t_0/t_1 \equiv -A \pmod \frakl$.
In~particular, both
$t_0$
and~$t_1$
must be~units. But~then equation~(\ref{eq_eins}) implies the congruence
$-At_1^2 \equiv -Dt_3^2 \pmod \frakl$.
\end{rem}
\begin{rem}[Inert primes--the case of residue characteristic~$2$]
\label{char2}
\leavevmode\\
We note that a statement analogous to Proposition~\ref{inert}.a) is true for any inert
prime~$\frakl$
under some more restrictive conditions on the coefficients
$A$
and
$B$.
For this suppose~that
$A$,
$B$,
$D \in \calO_K$
and that
$\frakl \subset \calO_K$
is a prime ideal that is inert under
$\smash{K(\sqrt{D})/K}$.
Let~$e$
be a positive integer such that
$x \equiv 1 \pmod {\frakl^e}$
is enough to imply that
$x \in K_\frakl$
is a~square. Assume~that
$\nu_\frakl(B-1) = f \geq 1$
and that
$\nu_\frakl(A)$
is an odd number such that
$\nu_\frakl(A) \geq 2f+e$.
Then~$S(K_\frakl) \neq \emptyset$.
Indeed, let us show that there exists a point
$(t_0\!:\!t_1\!:\!t_2\!:\!t_3\!:\!t_4) \in S(K_\frakl)$
such that
$t_3=t_4$
and~$t_1 \neq 0$.
This~leads to the equation
$(T_0+AT_1)(T_0+BT_1) = T_0T_1$,
or
$$T_0^2 + (A+B-1)T_0T_1 + ABT_1^2 = 0 \, .$$
The~discriminant of this binary quadric is
$(A+B-1)^2 - 4AB = (B\!-\!1)^2 + A(A-2B\!-\!2)$,
which is a square
in~$K_\frakl$
by virtue of our~assumptions. Thus,~there are two solutions
in~$K_\frakl$
for
$T_0/T_1$
and their product is
$AB$,
which is of odd~valuation. We~may therefore choose a solution
$t_0/t_1$
such that
$\nu_\frakl(t_0/t_1)$
is~even. This~is enough to imply that
$(t_0+At_1)(t_0+Bt_1) = t_0t_1$
is a norm
from~$\smash{K_\frakl(\sqrt{D})}$.
\end{rem}
\begin{rem}[Archimedean primes]
\label{Archimedean}
\begin{iii}
\item
Let~$\sigma\colon K \hookrightarrow \bbR$
be a real~prime. Then,~for
$A$,
$B \in K$
ar\-bitrary and
$D \in K$
non-zero, one has
$S_\sigma(\bbR) \neq \emptyset$.
Indeed,~we can put
$t_1 := 1$
and choose
$t_0 \in \bbR$
such that
$t_0$,
$t_0 + \sigma(A)$,
and
$t_0 + \sigma(B)$
are~positive. Then
$C := t_0 > 0$
and
$C' := (t_0 + \sigma(A))(t_0 + \sigma(B)) > 0$
and we have to show that the system of equations
\begin{eqnarray*}
T_2^2 - \sigma(D) T_3^2 & = & C \\
T_2^2 - \sigma(D) T_4^2 & = & C'
\end{eqnarray*}
is solvable in~$\bbR$.
For this one may choose
$t_2$
such that
$t_2^2 \geq \max(C,C')$
if
$\sigma(D) > 0$
and such that
$t_2^2 \leq \min(C,C')$,
otherwise. In~both cases it is clear that there exist real numbers
$t_3$
and
$t_4$
such that the resulting point is contained
in~$S_\sigma(\bbR)$.
Moreover~if
$\sigma(D) > 0$
then the local evaluation map
$\ev_{\alpha,\sigma}\colon S(K_\sigma) \to \frac12\bbZ/\bbZ$
is constantly~zero. Indeed,~then one has
$(q,D)_\sigma = 1$
for every
$q \in K_\sigma \cong \bbR$,
different from~zero.
\item
For~$\tau\colon K \hookrightarrow \bbC$
a complex prime and
$A$,
$B$,
and
$D \in K$
arbitrary, we clearly have that
$S(K_\tau) \neq \emptyset$.
Furthermore,~$(q,D)_\tau = 1$
for every non-zero
$q \in K_\tau \cong \bbC$.
\end{iii}
\end{rem}
\section{Ramification--Reduction to the union of four planes}
The goal of this section is to study the evaluation of the Brauer class at ramified primes
$\frakl$.
Under~certain congruence conditions on the parameters
$A$
and~$B$
we deduce that the evaluation map is constant on the
$K_\frakl$-rational
points on
$S$,
and we determine its value depending on
$A$
and~$B$.
\begin{prop}[Ramified primes in residue characteristic $\neq \!2$]
\label{ramified}
\leavevmode\\
Let\/~$K$
be a number field,
$A,B,D \in \calO_K$,
and\/
$\frakl \subset \calO_K$
a prime ideal such that\/
$\#\calO_K/\frakl$
is not a power
of\/~$2$
and that is ramified under the field extension\/
$\smash{K(\sqrt{D})/K}$.
Suppose~that\/
\mbox{$\overline{A} := (A \bmod \frakl)$}
$\in \calO_K/\frakl$
is a square, different from\/
$0$
and\/
$(-1)$,
that\/
$\smash{\overline{A}^2 + \overline{A} + 1 \neq 0}$,
and~that
$$\textstyle B \equiv -\frac{A}{A+1} \pmod \frakl \, .$$
Consider~the
surface\/~$S := S^{(D;A,B)}$.
\begin{abc}
\item
Then\/
$S(K_{\frakl}) \neq \emptyset$.
\item
Assume~that\/
$S$
is non-singular and let\/~$\alpha \in \Br(S)$
be the Brauer class, described in Proposition \ref{Brauerklasse}.a).
\begin{iii}
\item
If\/
$\overline{A}+1 \in \calO_K/\frakl$
is a square then the local evaluation map\/
$\ev_{\alpha,\frakl}\colon S(K_\frakl) \to \bbQ/\bbZ$
is constantly~zero.
\item
If\/
$\overline{A}+1 \in \calO_K\!/\frakl$
is a non-square then the local evaluation map\/
$\ev_{\alpha,\frakl}\colon S(K_\frakl) \to \bbQ/\bbZ$
is constant~of
value~$\frac12$.
\end{iii}
\end{abc}\smallskip
\noindent
{\bf Proof.}
{\em
First~of all, we note that
$\nu_\frakl(D)$
is~odd.
Indeed,~assume the contrary. We~may then normalize
$D$
to be a unit and write
$K_\frakl^\n$
for the unramified quadratic extension
of~$K_\frakl$.
Then~$(D \bmod \frakl\calO_{K_\frakl^\n})$
is a square and, since
$\calO_{K_\frakl^\n}/\frakl\calO_{K_\frakl^\n}$
is a field of characteristic different
from~$2$,
Hensel's Lemma ensures that
$D$
is a square in
$K_\frakl^\n$.
I.e.,~$\smash{K_\frakl(\sqrt{D}) \subseteq K_\frakl^\n}$,
a~contradiction.
Let~us normalize
$D$
such that
$\nu_\frakl(D) = 1$.
Then~the reduction
$S_\frakl$
of~$S$
is given by the equations
\begin{eqnarray}
T_0T_1
& = & T_2^2 \, , \label{eq_drei} \\
\textstyle
\smash{(T_0+\overline{A}T_1)(T_0-\frac{\overline{A}}{\overline{A}+1}T_1)}
& = & T_2^2 \, , \label{eq_vier}
\end{eqnarray}
which geometrically define a cone over a cone over four points
in~$\Pb^2$.\smallskip
\noindent
a)
We~write
$\ell := \#\calO_K/\frakl$.
It~suffices to verify the existence of a regular
$\bbF_{\!\ell}$-rational
point
on~$S_\frakl$.
For~this, it is clearly enough to show that one of the four points
in~$\Pb^2$,
defined by the equations (\ref{eq_drei}) and~(\ref{eq_vier}), is simple and defined
over~$\bbF_{\!\ell}$.
Equating the two terms on the left hand side, one finds the equation
$$\textstyle T_0^2 + \frac{\overline{A}^2-\overline{A}-1}{\overline{A}+1}T_0T_1 - \frac{\overline{A}^2}{\overline{A}+1}T_1^2 = 0 \, ,$$
which obviously has the two solutions
$T_0/T_1 = 1$
and
$\smash{T_0/T_1 = - \frac{\overline{A}^2}{\overline{A}+1}}$.
By~virtue of our assumptions, both are
\mbox{$\bbF_{\!\ell}$-rational}
points
in~$\Pb^1$,
different from
$0$
and
$\infty$.
They~are different from each other, since
$\smash{\overline{A}^2 + \overline{A} + 1 \neq 0}$.
Consequently,~the four points defined by the equations (\ref{eq_drei}) and~(\ref{eq_vier}) are all~simple. The~two points corresponding to
$(t_0\!:\!t_1) = 1$
are defined
over~$\bbF_{\!\ell}$.
The~two others are defined
over~$\bbF_{\!\ell}$
if and only if
$(-\overline{A}-1) \in \bbF_{\!\ell}$
is a~square.\smallskip
\noindent
b)
Let
$(t_0\!:\!t_1\!:\!t_2\!:\!t_3\!:\!t_4) \in S(K_\frakl)$
be any point. We~normalize the coordinates such that
$t_0,\ldots,t_4 \in \calO_{K_l}$
and at least one of them is a~unit.
Then
$\frakl$
cannot divide both
$t_0$
and
$t_1$.
Indeed,~this would imply
$\frakl^2 | t_2^2 - Dt_3^2$
and
$\frakl^2 | t_2^2 - Dt_4^2$
and,
as~$\nu_\frakl(D) = 1$,
this is possible only for
$\frakl | t_2, t_3, t_4$.
Therefore,~$((t_0 \!+\! At_1)/t_1 \bmod \frakl) = \overline{A} \!+\! (t_0/t_1 \bmod \frakl)$
is either equal to
$\smash{(\overline{A} \!+\! 1)}$
or to
$\smash{\overline{A} \!-\! \frac{\overline{A}^2}{\overline{A}+1} = \frac{\overline{A}}{\overline{A}+1}}$.
Both~terms are squares
in~$\bbF_{\!\ell}$
under the assumptions of b.i), while, under the assumptions of~b.ii), both are non-squares.
As~a unit
in~$\calO_{K_\frakl}$
is a norm from the ramified
extension~$\smash{K_\frakl(\sqrt{D})}$
if and only if its residue
modulo~$\frakl$
is a square, for
$q := (t_0+At_1)/t_1$,
we find that
$(q,D)_\frakl = 1$
in case~i) and
$(q,D)_\frakl = -1$
in case~ii). Proposition~\ref{Brauerklasse}.b.ii) implies the~assertion.
}
\eop
\end{prop}
\section{The main result}
We are now in the position to formulate sufficient conditions on
$A,B,D$,
under which the corresponding surface
$S^{(D;A,B)}$
violates the Hasse principle.
\begin{theo}
\label{Data_Hasse_Gegen}
Let\/~$D \in K$
be non-zero and\/
$(D) = (\frakq_1^{k_1}\cdot\ldots\cdot\frakq_l^{k_l})^2\, \frakp_1\cdot\ldots\cdot\frakp_k$
its decomposition into prime~ideals. Suppose~that the primes\/
$\frakp_1, \ldots, \frakp_k$
are~distinct.
\begin{abc}
\item
Assume~that
\begin{iii}
\item
$k \geq 1$,
\item
the quadratic extension\/
$\smash{K(\sqrt{D})/K}$
is unramified at all primes
of\/~$K$
lying over the rational
prime\/~$2$,
\item
for~every real prime\/
$\sigma\colon K \hookrightarrow \bbR$,
one has\/
$\sigma(D) > 0$.
\end{iii}
\item
For~every
prime\/~$\frakl$
of\/~$K$
that lies over the rational
prime\/~$2$
and is inert under\/
$\smash{K(\sqrt{D})/K}$,
assume~that
$$\nu_\frakl(B-1) = f_\frakl \geq 1\,, \quad{\it that\/~} \nu_\frakl(A) {\it ~is~odd \,, ~} \quad {\it and~that\/~} \nu_\frakl(A) \geq 2f_\frakl+e_\frakl\,,$$
for\/~$e_\frakl$
a positive integer such that\/
$x \equiv 1 \pmod {\frakl^{e_\frakl}}$
is enough to ensure that\/
$x \in K_\frakl$
is a~square.
\item
For~every\/
$i = 1,\ldots,k$,
assume~that
\begin{iii}
\item
$(A \bmod \frakp_i) \in \calO_K/\frakp_i$
is a square, different from\/
$0$,
$(-1)$,
and the primitive third roots of~unity.
If\/~$\#\calO_K/\frakp_i$
is a power
of\/~$3$
then assume\/
$(A \bmod \frakp_i) \neq 1$,~too.
\item
$B \equiv -\frac{A}{A+1} \pmod {\frakp_i}$.
\item
$1+(A \bmod \frakp_i) \in \calO_K/\frakp_i$
is a non-square for\/
$i=1,\ldots,b$,
for an odd
integer\/~$b$,
and a square
for\/
$i=b+1,\ldots,k$.
\end{iii}
\item
Finally,~assume that\/
$(A-B)$
is a product of only split~primes.
\end{abc}\smallskip
\noindent
Then\/
$S^{(D;A,B)}(\bbA_K) \neq \emptyset$.
However,~if\/
$S^{(D;A,B)}$
is non-singular then\/
$S^{(D;A,B)}(K) = \emptyset$.
\end{theo}
\begin{rem}
Without any change, one may assume that the
$\frakq_1, \ldots, \frakq_l$
are distinct,~too. Note,~however, that we do not suppose
$\{\frakp_1, \ldots, \frakp_k\}$
and
$\{\frakq_1, \ldots, \frakq_l\}$
to be disjoint.
\end{rem}
\noindent
{\bf Proof of Theorem~\ref{Data_Hasse_Gegen}.}
By~a.i),
$D$
is not a square
in~$K$,
hence
$K(\sqrt{D})/K$
is a proper quadratic field~extension. It~is clearly ramified at
$\frakp_1, \ldots, \frakp_k$.
According~to a.ii), these are the only ramified~primes.
In~view of assumption~b),
$S(\bbA_\bbQ) \neq \emptyset$
follows from Proposition~\ref{inert}.a) and Proposition~\ref{ramified}.a), together with Lemma \ref{split}.a), Remark \ref{char2}, and Remark~\ref{Archimedean}.
On~the other hand, let
$\alpha \in \Br(S)$
be the Brauer class, described in Proposition~\ref{Brauerklasse}.a). Then,~in view of assumptions~d), c) and~a.iii), Proposition~\ref{inert}.b) and Proposition~\ref{ramified}.b), together with Lemma \ref{split}.b) and Remark~\ref{Archimedean}, show that the local evaluation map
$\ev_{\alpha,\frakl}$
is constant of
value~$\frac12$
for
$\frakl = \frakp_1,\ldots,\frakp_b$
and constantly zero for all others. Proposition~\ref{BM} proves that
$S$
is a counterexample to the Hasse~principle.
\eop
\begin{ex}
Let~$S$
be the surface
in~$\Pb^4_\bbQ$,
given~by
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - 17T_3^2 \, ,\\
(T_0+9T_1)(T_0+11T_1) & = & T_2^2 - 17T_4^2 \, .
\end{eqnarray*}
Then
$S(\bbA_\bbQ) \neq \emptyset$
but~$S(\bbQ) = \emptyset$.\medskip
\noindent
{\bf Proof.}
We~have
$K=\bbQ$
and
$D=17$.
Furthermore,
$A = 9$
and
$B = 11$
such that Proposition~\ref{nonsing} ensures that
$S = S^{(D;A,B)}$
is non-singular.
The~extension
$\smash{L := \bbQ(\sqrt{17})/\bbQ}$
is real-quadratic,
i.e.~$D > 0$,
and ramified only
at~$17$.
Under~$\smash{\bbQ(\sqrt{17})/\bbQ}$,
the prime
$2$
is split, which completes the verification of~a) and shows that b) is fulfilled~trivially.
For~c), note that
$17 \not\equiv 1 \pmod 3$,
such that there are no nontrivial third roots of unity
in~$\bbF_{\!17}$.
Furthermore,
$9 \neq 0,(-1)$
is a square
modulo~$17$,
but~$10$
is not,
and~$\smash{11 \equiv -\frac9{10} \pmod {17}}$.
Finally, for~d), note that
$(A-B) = (-2) = (2)$
is a prime that is split
in~$\smash{\bbQ(\sqrt{17})}$.
\eop
\end{ex}
\begin{rem}
The assumption
on~$S$
to be non-singular may be removed from Theorem~\ref{Data_Hasse_Gegen}. Indeed, the elementary argument described at the very end of section~\ref{sec_Brauer} works in the singular case,~too.
\end{rem}
The goal of the next lemma is to construct discriminants
$D \in K$,
for which we will later be able to construct counterexamples to the Hasse principle, via the previous theorem.
\begin{lem}
\label{D_exists}
Let\/~$K$
be an arbitrary number field and\/
$\frakp, \frakr_1, \ldots, \frakr_n$
be distinct prime ideals such that\/
$\calO_K/\frakp$
and\/~$\calO_K/\frakr_i$
are of characteristics different
from\/~$2$.
Then~there exists some\/
$D \in K$
such~that
\begin{iii}
\item
the prime\/
$\frakp$
is ramified
in\/~$\smash{K(\sqrt{D})}$,
\item
all primes lying over the rational
prime\/~$2$
are split
in\/~$\smash{K(\sqrt{D})}$.
\item
For~every real prime\/
$\sigma\colon K \hookrightarrow \bbR$,
one has\/
$\sigma(D) > 0$.
\item
The~primes\/
$\frakr_i$
are unramified
in\/~$\smash{K(\sqrt{D})}$.
\end{iii}
\noindent
In~particular, assumptions~a) and~b) of Theorem~\ref{Data_Hasse_Gegen} are~fulfilled.\medskip
\noindent
{\bf Proof.}
{\em
Let
$\frakl_1,\ldots,\frakl_m$
be the primes
of~$K$
that lie over the rational
prime~$2$.
We~impose the congruence conditions
$D \equiv 1 \pmod {\frakl_1^{e_1}}, \ldots, D \equiv 1 \pmod {\frakl_m^{e_m}}$,
for~$e_1, \ldots, e_m$
large enough that this implies that
$D$
is a square
in~$K_{\frakl_1}, \ldots, K_{\frakl_m}$.
Furthermore,~the assumptions imply that
$\frakp, \frakr_1, \ldots, \frakr_n$
are different from
$\frakl_1,\ldots,\frakl_m$.
We~impose, in addition, the conditions
$D \in \frakp \!\setminus\! \frakp^2$
and
$D \not\in \frakr_1, \ldots, \frakr_n$.
According to the Chinese remainder theorem, these conditions have a simultaneous
solution~$D'$.
Put
$D := D' + k\cdot\#\calO_K/\frakl_1^{e_1}\ldots\frakl_m^{e_m}\frakp^2\frakr_1\ldots\frakr_n$,
for~$k$
an integer that is sufficiently large to ensure
$\sigma(D) > 0$
for every real prime
$\sigma\colon K \hookrightarrow \bbR$.
Then~assertion~iii) is~true.
Furthermore,~the congruences
$D \equiv 1 \pmod {\frakl_i^{e_i}}$
imply~ii), while
$D \in \frakp \!\setminus\! \frakp^2$
yields assertion~i) and
$D \not\in \frakr_1, \ldots, \frakr_n$
ensures that~iv) is~true.
}
\eop
\end{lem}
Before we come to the next main theorem of this section, we need to formulate two technical lemmata.
\begin{lem}
\label{Chebotarev}
Let\/~$K$
be a number field,
$I \subset \calO_K$
an ideal,
and\/~$x \in \calO_K$
an element relatively prime
to\/~$I$.\smallskip
\noindent
Then~there exists an infinite sequence of pairwise non-associated elements\/
$y_i \in \calO_K$
such that, for
each\/~$i \in \bbN$,
one has that\/
$(y_i)$
is a prime ideal and\/
$y_i \equiv x \pmod I$.\medskip
\noindent
{\bf Proof.}
{\em
It~is well known that there exist infinitely many prime ideals\/
$\frakr_i \subset \calO_K$
with the property~below.\smallskip
There~exist some\/
$u_i, v_i \in\calO_K$,
$u_i \equiv v_i \equiv 1 \pmod I$
such~that
$$\frakr_i \!\cdot\! (u_i) = (x) \!\cdot\! (v_i) \, .\medskip$$
\noindent
Indeed,~the invertible ideals
in~$K$
modulo the principal ideals generated by elements from the residue class
$(1 \bmod I)$
form an abelian group that is canonically isomorphic to the ray class group
$Cl_K^I \cong C_K/C_K^I$
of~$K$
\cite[Chapter~VI, Proposition~1.9]{Ne}. Thus,~the claim follows from the Chebotarev density theorem applied to the ray class field
$K^I\!/\!K$,
which has the Galois group
$\Gal(K^I\!/\!K) \cong Cl_K^I$.
Take~one of these prime~ideals.
Then~$\frakr_i \!\cdot\! (u_i) = (x) \!\cdot\! (v_i) = (xv_i)$.
As~$\frakr_i \subset \calO_K$,
this shows that
$xv_i$
is divisible
by~$u_i$.
Put~$y_i := xv_i/u_i$.
Then~$(y_i) = \frakr_i$.
Further,
\mbox{$y_i \equiv x \pmod I$}.
}
\eop
\end{lem}
\begin{lem}
\label{sq_nonsq_ex}
Let\/~$\bbF_{\!q}$
be a finite field of characteristic\/
$\neq \!2$
having\/
$> \!25$~elements.
Then~there exist elements\/
$a_{00}$,
$a_{01}$,
$a_{10}$,
and
$a_{11} \in \bbF_{\!q}$,
different from\/
$0$,
$(-1)$,
$(-2)$
and such that\/
$a_{ij}^2 + a_{ij} + 1 \neq 0$,
that fulfill the conditions~below.
\begin{iii}
\item
$a_{00}$,
$(a_{00}+1)$,
and\/
$(a_{00}+2)$
are squares
in\/~$\bbF_{\!q}$.
\item
$a_{01}$
and\/
$(a_{01}+1)$
are squares
in\/~$\bbF_{\!q}$,
but\/
$(a_{01}+2)$
is~not.
\item
$a_{10}$
and\/
$(a_{10}+2)$
are squares
in\/~$\bbF_{\!q}$,
but\/
$(a_{10}+1)$
is~not.
\item
$a_{11}$
is a square
in\/~$\bbF_{\!q}$,
but\/
$(a_{11}+1)$
and\/
$(a_{11}+2)$
are~not.
\end{iii}\smallskip
\noindent
{\bf Proof.}
{\em
Let~$\smash{C_1 \in \bbF_{\!q}^*}$
be a square in the cases i) and ii), and a non-square,~otherwise. Similarly,~let
$\smash{C_2 \in \bbF_{\!q}^*}$
be a square in the cases i) and iii), and a non-square,~otherwise. The~problem then translates into finding an
\mbox{$\bbF_{\!q}$-rational}
point on the
curve~$E$,
given
in~$\Pb^3$~by
\begin{eqnarray*}
U_1^2 + \phantom{1}U_0^2 & = & C_1U_2^2 \,, \\
U_1^2 + 2U_0^2 & = & C_2U_3^2 \,,
\end{eqnarray*}
such that
$U_i \neq 0$
for
$i = 0,\ldots,3$
and
$(U_1/U_0)^4 + (U_1/U_0)^2 + 1 \neq 0$.
Note~that the conditions
$U_2 \neq 0$
and
$U_3 \neq 0$
imply that
$\smash{\big(\frac{U_1}{U_0}\big)^2 \neq -1,-2}$.
Since~the characteristic of the base field is different from two, a direct calculation shows that
$E$~is
non-singular, i.e.~a smooth curve of
genus~$1$.
The~extra conditions define an open subscheme
$\smash{\widetilde{E} \subset E}$
that excludes not more than 32~points. Thus,~Hasse's bound yields
$\smash{\#\widetilde{E}(\bbF_{\!q}) \geq q - 2\sqrt{\mathstrut q} - 31}$.
This~is positive
for~$q > 44$.
An~experiment shows that
$\bbF_{\!3}$,
$\bbF_{\!5}$,
$\bbF_{\!7}$,
$\bbF_{\!9}$,
$\bbF_{\!13}$,
$\bbF_{\!17}$,
and
$\bbF_{\!25}$
are the only fields in
characteristic~$\neq \!2$,
for which the assertion is~false.
}
\eop
\end{lem}
The following theorem provides us with Hasse counterexamples in the family $S^{(D;A,B)}$
for suitable
discriminants~$D$.
For us, the important feature is that one may choose the parameters
$A$
and~$B$
to lie in (almost) arbitrary congruence classes modulo some prime ideal
$\frakl \subset \calO_K$,
unramified
in
$K(\sqrt{D})$,
provided only that
$A \not\equiv B \pmod \frakl$.
\begin{theo}
\label{Hasse_congr_ex}
Let\/~$K$
be an arbitrary number field and\/
$D \in K$
a non-zero~element. Write\/
$(D) = (\frakq_1^{k_1}\cdot\ldots\cdot\frakq_l^{k_l})^2 \frakp_1\cdot\ldots\cdot\frakp_k$
for its decomposition into prime ideals, the\/
$\frakp_i$
being~distinct. Assume~that
\begin{iii}
\item
$k \geq 1$,
\item
all primes lying over the rational
prime\/~$2$
are split
in\/~$\smash{K(\sqrt{D})}$,
\item
for~every real prime\/
$\sigma\colon K \hookrightarrow \bbR$,
one has\/
$\sigma(D) > 0$,
\item
all primes with residue field\/
$\bbF_{\!3}$
are unramified
in\/~$\smash{K(\sqrt{D})}$.
\end{iii}
\noindent
Suppose~further that among the primes\/
$\frakp$
of\/~$K$
that are ramified
in\/~$\smash{K(\sqrt{D})}$,
there is one such that\/
$\#\calO_K/\frakp > 25$.\smallskip
\noindent
Then,~for every prime\/
$\frakl \subset \calO_K$,
unramified
in\/~$\smash{K(\sqrt{D})}$,
and all\/
$a,b \in \calO_K/\frakl$
such that\/
$a \neq b$,
there exist
$A,B \in \calO_K$
such that\/
$(A \bmod \frakl) = a$,
$(B \bmod \frakl) = b$, and $S^{(D;A,B)}(\bbA_K) \neq \emptyset$,
but\/
$S^{(D;A,B)}(K) = \emptyset$.\medskip
\noindent
{\bf Proof.}
{\em
{\em First step.}
Construction of
$A$
and~$B$.
\noindent
Let~$M \in \{1,\ldots,k\}$
be such that
$\#\calO_K/\frakp_M > 25$.
Besides
\begin{equation}
\label{cong1}
(A \bmod \frakl) = a \qquad {\rm and} \qquad (B \bmod \frakl) = b \, ,
\end{equation}
we will impose further congruence~conditions on
$A$
and~$B$.
For~each
$i \neq M$,
we choose a square
$a_i \in \calO_K/\frakp_i$
such that
$a_i \neq 0, (-1), (-2)$
and
$a_i^2 + a_i + 1 \neq 0$.
This~is possible since
$\calO_K/\frakp_i$
is of
char\-ac\-ter\-is\-tic~$\neq \!2$
and~$\#\calO_K/\frakp_i > 3$.
We~require
\begin{equation}
\label{cong2}
(A \bmod \frakp_i) = a_i \qquad {\rm and} \qquad (B \bmod \frakp_i) = -\frac{a_i}{a_i+1} \, .
\end{equation}
Finally,~we choose a square
$a_M \in \calO_K/\frakp_M$
such that
$a_M \neq 0, (-1), (-2)$
and
$a_M^2 + a_M + 1 \neq 0$,
satisfying the additional conditions~below.
\begin{iii}
\item[$\bullet$ ]
If,~among the elements
$a_1+1, \ldots, a_{M-1}+1, a_{M+1}+1, \ldots, a_k+1$,
there are an odd number of non-squares then
$a_M+1$
is a~square.
Otherwise,~$a_M+1$
is a non-square.
\item[$\bullet$ ]
If,~among the elements
$a_1+2, \ldots, a_{M-1}+2, a_{M+1}+2, \ldots, a_k+2$,
there are an odd number of non-squares then
$a_M+2$
is a~square.
Otherwise,~$a_M+2$
is a non-square.
\end{iii}
Lemma~\ref{sq_nonsq_ex} guarantees that such an element
$a_M \in \calO_K/\frakp_M$~exists.
We~impose the final congruence~condition
\begin{equation}
\label{cong3}
(A \bmod \frakp_M) = a_M \qquad {\rm and} \qquad (B \bmod \frakp_M) = -\frac{a_M}{a_M+1} \, .
\end{equation}\smallskip
According~to the Chinese remainder theorem, one may choose an algebraic integer
$B \in \calO_K$
such that the conditions on the right hand sides of (\ref{cong1}), (\ref{cong2}), and (\ref{cong3}) are~fulfilled. Then,~by Lemma~\ref{Chebotarev}, there exist infinitely many non-associated elements
$y_i \in \calO_K$
such that
$(y_i)$
is a prime ideal and
$(y_i+B,B)$
a simultaneous solution of the system of congruences~(\ref{cong1},\,\ref{cong2},\,\ref{cong3}).
We~choose some
$i \in \bbN$
such that
$\frakr := (y_i)$
is of residue characteristic different
from~$2$,
that
$\frakr \neq \frakp_1, \ldots, \frakp_k, \frakq_1, \ldots, \frakq_l$,
and such that
$A^2-2AB+B^2-2A-2B+1 \neq 0$
for
$A := y_i+B$.
Note~that
$\frakr \neq \frakp_1, \ldots, \frakp_k, \frakq_1, \ldots, \frakq_l$
is equivalent to
$\frakr \not\ni D$.\smallskip
\noindent
{\em Second step.}
The surface
$S := S^{(D;A,B)}$
is a counterexample to the Hasse~principle.
\noindent
To~show this, let us use Theorem~\ref{Data_Hasse_Gegen}.
Our~assumptions
on~$D$
imply that assumptions~a) and~b) of Theorem~\ref{Data_Hasse_Gegen} are~fulfilled. Assumption~c) is satisfied, too, by consequence of the construction of the\
elements~$a_i$.
Observe,~in particular, that among the elements
$a_1+1, \ldots, a_k+1$,
there are an odd number of~non-squares.
Furthermore,~$S$
is non-singular.
It~therefore remains to check assumption~d). The~only
prime~$\frakp \subset \calO_K$,
for which
$A \equiv B \pmod \frakp$,
is
$\frakp = \frakr \;\,(=\!(A\!-\!B))$.
We~have to show that
$\frakr$
is split
under~$\smash{K(\sqrt{D})/K}$.\smallskip
For~this, we observe that,
for~$i = 1,\ldots,k$,
$$\textstyle A-B \equiv A + \frac{A}{A+1} = A\frac{A+2}{A+1} \pmod {\frakp_i} \, .$$
As~$A$
is a square
modulo~$\frakp_i$,
this shows
$$\prod_{i=1}^k (A-B,D)_{\frakp_i} = \prod_{i=1}^k (A+2,D)_{\frakp_i} \Big/ \prod_{i=1}^k (A+1,D)_{\frakp_i} \, .$$
Here,~by our construction, both
$1 + (A \bmod \frakp_i)$
and
$2 + (A \bmod \frakp_i)$
are non-squares, an odd number of~times. Consequently,
$$\prod_{i=1}^k (A-B,D)_{\frakp_i} = 1 \, .$$\smallskip
On~the other hand,
$D$~is
a square
in~$K_{\frakl_i}$
for~$l_i$
the primes of residue
characteristic~$2$
and for every real~prime, by assumption iii).
Thus,~$(A-B,D)_\frakl = 1$
unless
$\frakl$
divides either
$(A-B)$
or~$D$.
I.e.~for
$\frakl \neq \frakr, \frakp_1, \ldots, \frakp_k, \frakq_1, \ldots, \frakq_l$.
Moreover,~$(A-B,D)_\frakq = 1$
for~$\frakq \in \{\frakq_1, \ldots, \frakq_l\} \!\setminus\! \{\frakp_1, \ldots, \frakp_k\}$
since both arguments of the Hilbert symbol are of even
\mbox{$\frakq$-adic}
valuation.
The~Hilbert reciprocity law~\cite[Chapter~VI, Theorem~8.1]{Ne} therefore reveals the fact that
$$(A-B,D)_\frakr \cdot \prod_{i=1}^k (A-B,D)_{\frakp_i} = 1 \, .$$
Altogether, this implies
$(A-B,D)_\frakr = 1$.
Consequently,~the prime ideal
$\frakr$
splits
in~$\smash{K(\sqrt{D})}$.
}
\eop
\end{theo}
\begin{subl}
\label{01infty}
The~rational map
$\kappa\colon \Ab^{\!2}/S_2 \ratarrow \calU\!/(S_5 \times \PGL_2)$,
given on points~by
$$\overline{(a_1,a_2)} \mapsto \overline{(a_1,a_2,0,-1,\infty)} \, ,$$
is~dominant.\medskip
\noindent
{\bf Proof.}
{\em
It~suffices to prove that the rational map
$\widetilde\kappa\colon \Ab^{\!2} \ratarrow \calU\!/(S_5 \times \PGL_2)$,
given by
$\smash{(a_1,a_2) \mapsto \overline{(a_1,a_2,0,-1,\infty)}}$
is~dominant. For~this, recall that dominance may be tested after base extension to the algebraic~closure. Moreover,~it is well known that three distinct points
on~$\smash{\Pb^1_{\overline{K}}}$
may be sent to
$0$,~$(-1)$,
and~$\infty$
under the operation
of~$\smash{\PGL_2(\overline{K})}$.
}
\eop
\end{subl}
\begin{lem}
\label{inv_dom}
Let\/~$K$
be any field of
characteristic\/~$\neq \!2$
and\/
$0 \neq D \in K\!$.
Let\/
$\pi\colon \calS \to U$
be the family of degree four del Pezzo surfaces over an open subscheme\/
$U \subset \Ab^{\!2}_K$,
given by
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - DT_3^2 \, ,\\
(T_0+a_1T_1)(T_0+a_2T_1) & = & T_2^2 - DT_4^2 \, .
\end{eqnarray*}
I.e.,~the fiber of\/
$\pi$
over\/
$(a_1,a_2)$
is exactly the surface\/
$S^{(D;a_1,a_2)}$.
Then~the invariant map
$$I_\pi\colon U \longrightarrow\Pb(1,2,3) $$
associated
with\/~$\pi$
is~dominant.\medskip
\noindent
{\bf Proof.}
{\em
As~dominance may be tested after base extension to the algebraic closure, let us assume that the base
field~$K$
is algebraically~closed. Write
\begin{eqnarray*}
Q_1(a_1, a_2; T_0,\ldots,T_4) & := & T_0T_1 - (T_2^2 - DT_3^2) \hspace{3.5cm} {\rm and} \\
Q_2(a_1, a_2; T_0,\ldots,T_4) & := & (T_0+a_1T_1)(T_0+a_2T_1) - (T_2^2 - DT_4^2), \,
\end{eqnarray*}
and consider the pencil
$\smash{(uQ_1+vQ_2)_{(u:v) \in \Pb^1}}$
of quadrics, parametrized by
$(a_1, a_2) \in \Ab^{\!2}(K)$.
We~see that, independently of the values of the parameters, degenerate quadrics occur for
$(u:v) = 0$,
$\infty$,
and~$(-1)$.
The~two other degenerate quadrics appear for
$(u:v)$
the zeroes of the determinant
$$
\left|
\begin{array}{cc}
1 & (a_1+a_2+t)/2 \\
(a_1+a_2+t)/2 & a_1a_2
\end{array}
\right|
= \textstyle -\frac14 [t^2 + 2(a_1+a_2)t + (a_1-a_2)^2] \, .
$$
Thus,~$I_\pi$
is the composition of the rational map
$\rho\colon \Ab^{\!2} \supset U \ratarrow \Ab^{\!2}/S_2$,
sending
$(a_1,a_2)$
to the pair of roots
of~$t^2 + 2(a_1+a_2)t + (a_1-a_2)^2$,
followed by the rational map
$\kappa\colon \Ab^{\!2}/S_2 \ratarrow \calU\!/(S_5 \times \PGL_2)$,
studied in Sublemma~\ref{01infty}, and the open embedding
$\iota\colon \calU\!/(S_5 \times \PGL_2) \hookrightarrow \Pb(1,2,3)$,
defined by the fundamental~invariants. It~remains to prove that
$\rho\colon U \ratarrow \Ab^{\!2}/S_2$
is~dominant.
For~this, as coordinates on
$\Ab^{\!2}/S_2$
one may choose the sum and the product of the coordinates
on~$\Ab^{\!2}$.
Indeed,~these generate the field of
\mbox{$S_2$-invariant}
functions
on~$\Ab^{\!2}$.
Thus,~we actually claim that the map
$\Ab^{\!2} \to \Ab^{\!2}$, given by
$(a_1,a_2) \mapsto (-2(a_1+a_2), (a_1-a_2)^2)$
is dominant, which is~obvious.
}
\eop
\end{lem}
We~are now, finally, in the position to prove that the set of counterexamples to the Hasse principle is Zariski dense in the moduli scheme of del Pezzo surfaces of degree four. For~this, we will consider the family
$S^{(D;A,B)}$
for some fixed
discriminant~$D$
and use Theorem~\ref{Hasse_congr_ex}. It~will turn out to provide us with enough counterexamples to rule out the possibility that all pairs
$(A,B)$
leading to counterexamples might be contained in some finite union of~curves. Moreover, Lemma~\ref{inv_dom} shows that, for some fixed
$D \neq 0$,
the family
$S^{(D;A,B)}$
already dominates the moduli scheme of all degree four del Pezzo~surfaces.
\begin{theo}
Let\/~$K$
be any number field,
$U_\reg \subset \Gr(2,15)_K$
the open subset of the Gra\ss mann scheme that parametrizes degree four del Pezzo surfaces, and\/
$\calHC_K \subset U_\reg(K)$
be the set of all degree four del Pezzo surfaces
over\/~$K$
that are counterexamples to the Hasse~principle.\smallskip
\noindent
Then~the image of\/
$\calHC_K$
under the invariant map
$$\smash{I\colon U_\reg \longrightarrow \Pb(1,2,3)_K}$$
is Zariski~dense.\medskip
\noindent
{\bf Proof.}
{\em
According~to Lemma~\ref{D_exists}, there exists an algebraic integer
$D \in \calO_K$
fulfilling the assumptions of Theorem~\ref{Hasse_congr_ex}. Assume~that the image
of~$I$
would not be Zariski~dense. By~Lemma~\ref{inv_dom}, this implies that there exists a (possibly reducible) curve
$C \subset \Ab^{\!2}$
of certain
degree~$d$
such that, for all surfaces of the~form
\begin{eqnarray*}
T_0T_1 & = & T_2^2 - DT_3^2 \, , \\
(T_0+AT_1)(T_0+BT_1) & = & T_2^2 - DT_4^2
\end{eqnarray*}
that violate the Hasse principle, one
has~$(A,B) \in C(K)$.
On~the other hand, let
$\frakl \subset \calO_K$
be an unramified prime and put
$\ell := \#\calO_K/\frakl$.
Then,~by Theorem~\ref{Hasse_congr_ex}, we know counterexamples to the Hasse principle having
$\ell(\ell-1)$
distinct reductions
modulo~$\frakl$.
But~an affine plane curve of
degree~$d$
has
$\leq \!\ell d$
points
over
$\bbF_{\!\ell}$~\cite[the lemma in Chapter~1, Paragraph 5.2]{BS}.
For~a prime
ideal~$\frakl$
such that
$\ell \geq d+2$,
this is~contradictory.
}
\eop
\end{theo}
\section{Zariski density in the Hilbert scheme}
This~section is devoted to Zariski density of the counterexamples to the Hasse principle in the Hilbert~scheme. Our~result is, in fact, an application of the Zariski density in the moduli scheme established~above.
\begin{theo}
Let\/~$K$
be any number field,
$U_\reg \subset \Gr(2,15)_K$
the open subset of the Gra\ss mann scheme that parametrizes degree four del Pezzo surfaces, and\/
$\calHC_K \subset U_\reg(K)$
be the set of all degree four del Pezzo surfaces
over\/~$K$
that are counterexamples to the Hasse~principle.\smallskip
\noindent
Then\/~$\calHC_K$
is Zariski dense in\/
$\Gr(2,15)_K$.\medskip
\noindent
{\bf Proof.}
{\em
Let~us fix an algebraic closure
$\overline{K}$
and an embedding
of~$K$
into~$\overline{K}$.
Assume that, contrary to the assertion,
$\calHC_K \subset U_\reg \subset \Gr(2,15)_K$
would not be Zariski~dense. It~is well-known that the Gra\ss mann scheme
$\Gr(2,15)_K$
on the right hand side is irreducible and projective of dimension
$(15-2) \!\cdot\! 2 = 26$.
The~subset
$\calHC_K$
must therefore be contained in a closed subscheme
$H \subset \Gr(2,15)_K$
of dimension at
most~$25$.
By Theorem~\ref{theo1}, the invariant map
$H \to \Pb(1,2,3)$
is dominant. Its generic fiber may thus be, possibly reducible, of dimension at
most~$23$.
In~particular, outside of a finite union of curves
$C \subset \Pb(1,2,3)$,
the special fibers are of dimension
$\leq \!23$,
as~well.
Now,~let us choose a
\mbox{$K$-rational}
point
$s \in [\Pb(1,2,3) \!\setminus\! C](K)$
that is the image of a degree four del Pezzo
surface~$S \in\calHC_K$
under the invariant~map. The~geometric fiber
$I^{-1}(s)_{\overline{K}}$
over~$s$
of the full invariant map
$\smash{I\colon U_\reg \to \Pb(1,2,3)}$
parametrizes all reembeddings of
$S$
into~$\smash{\Pb^4_{\overline{K}}}$
and is therefore a torsor under
$\PGL_5(\overline{K})/\Aut(S_{\overline{K}})$.
In particular, it is of
dimension~$24$.
This implies that
$I^{-1}(s) \not\subseteq H$.
But~the orbit
of~$s$
under
$\PGL_5(K)$
para\-metrizes counterexamples to the Hasse principle, and is therefore contained
in~$H$.
As~$\PGL_5(K)$
is Zariski dense in
$\PGL_5(\overline{K})$,
this is a~contradiction.
}
\eop
\end{theo}
\frenchspacing
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