Because physical exercise can be strenuous and subject to risk of serious injury, we urge you to obtain a physical examination from a doctor before using any exercise equipment or participating in any exercise activity. You agree that by participating in physical exercise or training activities, you do so entirely at your own risk. Any recommendation for changes in diet including the use of food supplements, weight reduction and/or body building enhancement products are entirely your responsibility and you should consult a physician prior to undergoing any dietary or food supplement changes. You agree that you are voluntarily participating in these activities and assume all risks of injury, illness, or death. We are also not responsible for any loss of your personal property.
By creating an account with Kai, I acknowledge that I understand its content and that this release cannot be modified orally. I also verify that I am 18 years of age, or that I am the parent or legal guardian of the individual whose name appears on the account and I am accepting this release on their behalf.


text:
Home » la-musica » Top 40 » Mental as Anything
Listen for free or download thousands of 80's Top 40 tracks for just 12 cents per song!
Mental As Anything, another Australian new wave rock band, formed at an art school in Sydney way back in 1976. This artistic background gave the band a leg up when it came to crafting their music videos and satirical, self-deprecating lyrics. Known as the "Mentals" by fans, Mental as Anything was a good-time band that certainly took themselves less than serious. The name of the band was a saying in late 70's Australian vernacular that essentially meant "having outlandish fun".
After making a name for themselves on the Syndey club scene, Mental as Anything landed a record deal and produced their debut EP, Mental as Anything Plays at Your Party in 1978. In 1979, the band released the album Get Wet, which featured the song "The Nips Are Getting Bigger", which became a hit in Australia. Mental As Anything had a number of hits throughout the 80's, most of which stayed on the Australian charts, but a few of which landed elsewhere. Some of their more popular songs included "Come Around", "If You Leave Me, Can I Come Too?", "Too Many Times", "Spirit Got Lost", "Working for the Man", "Your'e So Strong", "Live It Up", "Let's Go to Paradise", "He's Just No Good for You", "Rock and Roll Music", and "The World Seems Difficult".
Post 80's, Mental As Antyhing continue to play shows from time to time and release new material. The band was known to have been active as late as 2011, although not all original members were present. The Mental as Anything Discography is as follows:
1978: Mental as Anything Plays at Your Party
1979: Get Wet aka Mental As Anything
1980: Espresso Bongo
1981: Cats & Dogs aka If You Leave Me, Can I Come Too?
1983: Creatures of Leisure
1985: Fundamental
1986: Greatest Hits Volume 1
1987: Mouth to Mouth
1989: Cyclone Raymond
1993: Chemical Travel
1995: Liar Liar Pants on Fire
1995: Bicycle
1997: Minus Bonus
1998: Garàge
1999: Best of
2000: Beetroot Stains
2001: Borscht
2002: Road Case
2005: Plucked
2009: Tents Up
2009: iTunes Live from Sydney
2009: Essential As Anything – 30th Anniversary Edition
*New* Vinyl Record DESK CLOCKS Each Original Single Artists starting with 'M'
Mental As Anything Press Kit and Photo Fundamental
Mental as Anything Fundamental LP Flat Promo 12x12 Poster
MENTAL AS ANYTHING -Creatures Of Leisure- Vintage Promo Poster 1983 12 x 12
MENTAL AS ANYTHING--Creatures Of Leisure--Promo Poster
Mental as Anything Promotional 8x10 Press Photo and Bio Sheet 1985 Columbia
MENTAL AS ANYTHING Noiseworks MIDNIGHT OIL Wa Wa Nee 1987 PROMO POSTER AD
MENTAL AS ANYTHING - HAVE GONE BANANA - HALF PAGE MAGAZINE ADVERT CLIPPING 1986
Mental As Anything 1980 OLD MUSIC PHOTO
MENTAL AS ANYTHING Rare 1985 Oz Only Orig OOP Pop Sheet Music "Concrete & Clay"

redpajama_set_name: RedPajamaCommonCrawl


Shoes Calf Oh Cedar Vachetta Women's The Draw: Alex (Romain Duris) is a nice guy with a dirty job — breaking up unhappy relationships. His mission: Accompany Juliette (Vanessa Paradis) to Monaco and prevent her from marrying her English boyfriend.
8. Cedar Shoes Calf Oh Women's Vachetta Ne Quittez Pas!
Vachetta Oh Calf Women's Shoes Cedar The Draw: A documentary about French pastry chefs. Do we really need to spell out the appeal?
Cedar Oh Vachetta Calf Shoes Women's The Fact: France is the birthplace of the dessert. The word "dessert" is derived from the French word "desservir," meaning "to clear the table."


text:
Q: Поиск Min и Max в диагоналях матрицы Нужно вывести на экран(в консоле) матрицу А(размерность NxN)(числа введены случайным образом)найти первое max на главной диагонали, последнее min на побочной диагонали и поменять их местами
Я немогу найти первое max на главной диагонали, последнее min на побочной диагонали и поменять их местами первая часть кода есть:
program Project2;
{$APPTYPE CONSOLE}
uses SysUtils;
const n=6;
var a:array[1..n] of array[1..n] of integer;
  i,j:integer;
begin
Randomize;
for i:=1 to n do
  for j:=1 to n do a[i,j]:=Random(9)+1;
for i:=1 to n do begin
  for j:=1 to n do
    Write(a[i,j],' ');
    Writeln;
  end;
readln;
end.

A: var imax, nmax: integer;
    imin, nmin, jtmp: integer;
    buf: integer;
{ первое max на главной }
imax := 1;
for i:=1 to n do begin
  if (i=1) then
    nmax := a[i, i];
  if (a[i, i]>nmax) then begin
    imax := i;
    nmax := a[i, i];
    end;
  end;
// nmax = a[imax, imax]

{ первое min на побочной }
imin := 1;
for i:=1 to n do begin
  jtmp := n - i + 1;
  if (i=1) then
    nmin := a[1, jtmp];
  if (a[i, jtmp]<nmin) then begin
    imin := i;
    nmin := a[i, jtmp];
    end;
  end;
// nmin = a[imin, n - imin + 1]

{ меняем местами }
buf := a[imax, imax];
a[imax, imax] := a[imin, n - imin + 1];
a[imin, n - imin + 1] := buf;

Вывести хоть сможете?)
UPD: Поменять можно и проще, все равно же сохранили(тогда buf: integer из переменных можно убрать):
a[imax, imax] := nmin;
a[imin, n - imin + 1] := nmax;

redpajama_set_name: RedPajamaStackExchange


text:
Metals & Electronics›
Import value of parts of TV or radio reception apparatus in the UK 2011 to 2017
Published by N. Sönnichsen, Nov 12, 2019
This statistic displays the total import value of parts of TV or radio reception apparatus in the United Kingdom (UK), from 2011 to 2017 in thousand British pounds. The import value of these items increased over this period from approximately 277 million British pounds in 2011 to 436.8 million British pounds in 2017.
Import value of parts of TV or radio reception apparatus in the United Kingdom (UK) from 2011 to 2017 (in 1,000 GBP)
Import value in thousand GBP
According to the publisher the data is updated on a constant basis. No release date was given.
U.S. trade balance 2000-2018
Key Figures of Retail
Total value of U.S. trade in goods worldwide 2004-2018
Total value of U.S. trade in goods with China 2008-2018
California's imports and exports of goods 2017-2019
Statistics on "Trade in the U.S."
Value of the leading merchandise exporters worldwide 2018, by exporting nation
Share of the leading merchandise exporters worldwide, by exporting nation 2018
Leading global merchandise importers 2018, by importing nation
Share of the leading merchandise importers worldwide 2018, by importing nation
Total value of U.S. international trade 2000-2018
U.S. trade balance of goods and services 2000-2018
Growth rate of U.S. exports 1990-2024
Growth rate of U.S. imports 1990-2024
U.S. foreign trade - balance of trade goods and services by quarter 2010-2019
U.S. foreign trade - imports of trade goods and services by quarter 2010-2019
U.S. foreign trade - exports of trade goods and services by quarter 2010-2019
Total imports and exports of goods of the U.S. 2017-2019
Total value of U.S. trade balance 2015-2019, by month
Leading 15 trade partners with the United States 2018, by country
Leading 15 countries share of U.S. total trade in goods 2018
Total value of U.S. trade in goods with Canada 2004-2018
Total value of U.S. trade in goods with Mexico 2004-2018
Total value of U.S. trade in goods with Japan 2004-2018
Total value of U.S. trade in goods with Germany 2004-2018
Texas' imports and exports of goods 2017-2019
Illinois' imports and exports of goods 2017-2019
New York's imports and exports of goods 2017-2019
Michigan's imports and exports of goods 2017-2019
New Jersey's imports and exports of goods 2017-2019
Export value of parts of TV or radio reception apparatus in the UK 2011 to 2017
Import value of reflector antennas in the UK 2011 to 2017
Export value of reflector antennas in the UK 2011-2017
Import value of screenless television receivers in the UK 2011 to 2017
Import value of reception apparatus for color TV in the UK 2011-2017
Import value of motor vehicle radios in the United Kingdom (UK) 2011 to 2017
Export value of reception apparatus for color TV in the UK 2011 to 2017
Export value of screenless television receivers in the UK 2011 to 2017
Import value of electric water heaters in the United Kingdom (UK) 2011-2017
Import value of processors and controllers in the UK 2011-2017
Import value of burglar or fire alarms in the UK 2011-2017
Import value of cameras and video camera recorders in the UK 2011-2017
Import value of telephones for cellular networks in the UK 2011 to 2017
Import value of telecomm. reception and transmission machines in the UK 2011-2017
Import value of electronic integrated circuits as memories in the UK 2011-2017
Import value of radio and television transmission apparatus in the UK 2011 to 2017
Import value of headphones and headsets in the United Kingdom (UK) 2011-2017
Import value of ignition or other wiring sets for vehicles in the UK 2011-2017
Import value of LCD or LED indicator panels in the UK 2011-2017
Import value of electric smoothing irons in the United Kingdom (UK) 2011-2017
Flower industry in the Netherlands U.S. Agriculture
International Trade Centre. (July 12, 2018). Import value of parts of TV or radio reception apparatus in the United Kingdom (UK) from 2011 to 2017 (in 1,000 GBP) [Graph]. In Statista. Retrieved January 19, 2020, from https://www.statista.com/statistics/535288/uk-import-value-of-parts-of-tv-reception-apparatus/
International Trade Centre. "Import value of parts of TV or radio reception apparatus in the United Kingdom (UK) from 2011 to 2017 (in 1,000 GBP)." Chart. July 12, 2018. Statista. Accessed January 19, 2020. https://www.statista.com/statistics/535288/uk-import-value-of-parts-of-tv-reception-apparatus/
International Trade Centre. (2018). Import value of parts of TV or radio reception apparatus in the United Kingdom (UK) from 2011 to 2017 (in 1,000 GBP). Statista. Statista Inc.. Accessed: January 19, 2020. https://www.statista.com/statistics/535288/uk-import-value-of-parts-of-tv-reception-apparatus/
International Trade Centre. "Import Value of Parts of Tv or Radio Reception Apparatus in The United Kingdom (Uk) from 2011 to 2017 (in 1,000 Gbp)." Statista, Statista Inc., 12 Jul 2018, https://www.statista.com/statistics/535288/uk-import-value-of-parts-of-tv-reception-apparatus/
International Trade Centre, Import value of parts of TV or radio reception apparatus in the United Kingdom (UK) from 2011 to 2017 (in 1,000 GBP) Statista, https://www.statista.com/statistics/535288/uk-import-value-of-parts-of-tv-reception-apparatus/ (last visited January 19, 2020)


Top FOX and Sinclair Stations Push Renewals of New Syndicated Talk Show From Producer ITV Studios America Past 50% of U.S.
LOS ANGELES AND NEW YORK – Debmar-Mercury, teaming with producer ITV Studios America, has renewed The Jeremy Kyle Show for the 2012-13 season in more than 50% of the U.S., including the FOX stations in New York, Los Angeles, Chicago, Dallas, Washington, D.C., and Houston. Debmar-Mercury Co-Presidents Mort Marcus and Ira Bernstein made the announcement today.
The Jeremy Kyle Show features everyday people facing extraordinary challenges, ranging from dysfunctional relationships and addictions to family crises; a no-nonsense host who isn't afraid to tell them some hard truths; and a compassionate guest services program that works diligently to get them the help they need.
Kyle launched his eponymous program in the U.K. six years ago, where it quickly became the top daytime show and made him into an icon whose program wins viewer accolades, including receiving the TV Choice Award for Best Daytime Series in September 2011.
The U.S. version of The Jeremy Kyle Show is produced by ITV Studios America, distributed domestically in the U.S. by Debmar-Mercury and internationally by ITV Studios Global Entertainment.
Debmar-Mercury, a leading media company specializing in syndication, network, cable and ancillary markets, produces and distributes for daytime television The Wendy Williams Show and The Jeremy Kyle Show, and has U.S. distribution rights to Fremantle Media North America's Family Feud; distributes Tyler Perry sitcoms House of Payne, spinoff Meet the Browns and For Better or Worse (premiering on TBS in November), Lionsgate Television/Revolution Studios/Evolution Management's Anger Management (premiering on FX in summer 2012 and starring Charlie Sheen) and Revolution/CubeVision's Are We There Yet? (starring Ice Cube); and has U.S. TV distribution rights to South Park, Hell's Kitchen and a movie library featuring titles from Revolution Studios. Debmar-Mercury is a wholly owned subsidiary of Lionsgate (NYSE: LGF), the leading next-generation studio.
ITV Studios America, the US based production entity of ITV Studios in the UK, is one of the largest international producers for the US market and a major force in acquiring, developing and producing reality and scripted programming for US networks. ITV Studios America has a slate of formats in production with major networks and syndicators, including Prime Suspect, Hell's Kitchen, Kitchen Nightmares, First 48, After the First 48, Steven Seagal -Lawman, Four Weddings, America Now and The Jeremy Kyle Show. In addition, ITV Studios America's psychological documentary special I Love You To Death airs December 1 on A&E.
ITV Studios is a world leader in television production and distribution. It has international production centers in the UK, Germany, France, Spain, Scandinavia and Australia as well as America. It is a distributor of programs to over 155 territories worldwide.


text:
Despite constraints of COVID-19, sheriff's office recognizes, celebrates employees
Donna Thornton
The Gadsden Times
From the cozy family gatherings to the corporate Christmas fetes, COVID-19 is (or should be) necessitating adjustments to the way people celebrate this holiday season.
The Etowah County Sheriff's Office Christmas party last week was no exception, according to Sheriff Jonathon Horton.
For the Christmas/awards banquet, the sheriff's office had three separate meals, served in take-out plates so that social distance could be maintained to help keep everyone safe.
There was a traditional Christmas lunch and dinner for employees, as well as a breakfast meal offered. There were door prizes, and even a visit, Horton said, from the "most loved guest" — Santa Claus.
During a three-hour period, the jolly old elf visited the sheriff's office so deputies and other employees' children could space out visits and make their Christmas gift requests known.
The main purpose of the event was to show appreciation for the exceptional effort of all employees, Horton said in a press release. He gave special recognition to to several deserving individuals from both sides of the organization — law enforcement and corrections.
Capt. Narfleata Excford and Lt. Justin Plunkett each received the Sheriff's Award — awarded by Horton to those who have stood out the most for their exceptional service in law enforcement.
Deputy Shannon Thomas was honored as Deputy of the Year and Brandi Fuller was recognized as Investigator of the Year. Sgt. Steve Griffin was selected Law Enforcement Supervisor of the Year.
Deputy David Bankson was named School Resource Officer of the Year and Kelley McGinnis as Administrative Assistant of the Year.
The Top Gun Award — presented to the most efficient with a firearm during annual training — went to Capt. Josh Morgan.
Deputy Regina Benzel was named Corrections Deputy of the Year and Lt. Jessica Murray was honored as Correctional Supervisor of the Year.
Deputy Jeremy Smith was recognized as SOD Team Member of the Year. Deputy Stacey Vaughn was named Administrative Deputy of the Year and EMT Martha Lopez was honored as Medical Team Member of the Year.
Horton expressed his appreciation for those who donated gifts and time to prepare for the event, and the "outstanding employees of the Etowah County Sheriff's Office who make this a fantastic place to work and strive each day to make Etowah County a safe place for the citizens."
Contact Us Support Local Business Advertise Your Business Advertising Terms and Conditions Buy and Sell Help Center Subscriber Guide My Account My Print Account Give Feedback Licensing & Reprints
Subscribe Today Newsletters Facebook Twitter E-Edition Instagram
© 2021 www.gadsdentimes.com. All rights reserved.


It's not often you get really good Mail but last week, I did! I got a wee stack of DIY Pie goodies from the lovely Mahala LeMay of Edible/Wearable. I got a fabulous zine, full of lovely pie-themed illustrations, things to make and recipes to try; along with some DIY Pie postcards.
Edible/Wearable are Mahala Lemay and Hilary Grant– who both graduated from Duncan of Jordanstone College of Art and Design in 2007. The created DIY Pie for the Maker Faire festival in Newcastle in March this year. I've got a good feeling there may be some pie-shaped fun coming to Dundee in the very near future…fingers crossed!
In the mean time, i'll be giving the Puff-Pastry tips a go!


The Ranchland Trust of Kansas (RTK) invites professional and amateur photographers to submit photographs for its seventh annual photo contest to showcase the natural beauty of ranching and grazing lands across the state. The competition offers Kansans the chance to explore and capture the many activities, seasons and faces of Kansas' ranching tradition.
Participants should submit photos expressing the mission of RTK: "To preserve Kansas' ranching heritage and open spaces for future generations through the conservation of working landscapes." This may include landscape, livestock and people, with preference to grasslands, grazing land, ranchers, cattle and much more.
Participants may enter up to five photos. The entry deadline for the contest is August 15, 2018. Entries will be judged by a panel of professionals in the field of photography and by RTK representatives.
All prizes for the contest are sponsored by Wolfe's Camera of Topeka. Wolfe's will award 10 photo cards to all 10 category winners. Categories include: Grand Prize, Landscape, Livestock, People, Youth (ages 5-17), Kansas Livestock Association Member, Fan Favorite via Facebook voting and Honorable Mentions. The grand prize winner will receive their choice of an 11"x 14" stretched photo canvas or a 12" x 18" metal print of their winning photo.
For full contest rules and to enter, visit www.ranchlandtrustofkansas.org (News, Events, Photo Contest). For questions, please contact Samantha Weishaar at (785) 273-5115 or samantha@kla.org.


Television chef and food author Lesley Waters is helping with a major branding exercise to lift the profile of the Q Guild, the organisation that represents some of the best butchers in Britain, including Drake & Macefield. Lesley is devising a series of seasonal recipes for promotion in Q butcher shops throughout the country, showing consumers what can be achieved with British beef, lamb, pork and chicken.
The promotion coincides with a national branding exercise by the Guild in all of its 150 member shops to convince consumers that the best place to buy the best meat, which will be British and more than likely sourced locally, is at a Q butcher – the Q stands for quality. Q butchers shops like Drake & Macefield now have new external signage – a signal to consumers that they are shopping at one of the best independent meat retailers in the country. Q butchers are unique in that they voluntarily submit their businesses to a stringent and independent audit before attaining membership. They are then subjected to regular audits, spot checks and visits from mystery shoppers to ensure standards are adhered to.
Q membership is one of the meat trade's highest endorsements of excellence.
"We want to get that across to consumers. Many are fed up with the impersonal service, confusing labeling and variable quality frequently experienced in the multiples," said Drake & Macefield's partner Richard Teal. "Q Butchers support local retailers and local producers. There are less food miles, meat is fresher, there are quality products and quality service. We have our own individual brand, but being associated with the Q brand should send a strong message to local consumers.
Drake & Macefield won one of only seven diamond awards presented in the annual Smithfield Awards, run by the Q Guild.
We're absolutely thrilled. We normally do well in product evaluation events, but a diamond award is the pinnacle.
The Smithfield Awards are among the most prestigious in the meat industry and a true indication of excellence. More than 400 products from Q butchers across the UK were assessed by an independent panel of experts.
© 2019 Drake and Macefield. All Rights Reserved.


text:
"use strict";
const path = require("path");
const jsdom = require("../lib/jsdom");
const fs = require("fs");
const exceptionTable = require("../lib/jsdom/web-idl/dom-exception-table.json");
const request = require("request");

function toPathname(dirname, relativePath) {
  let pathname = path.resolve(dirname, relativePath).replace(/\\/g, "/");
  if (pathname[0] !== "/") {
    pathname = "/" + pathname;
  }
  return pathname;
}

function toFileUrl(dirname, relativePath) {
  return "file://" + toPathname(dirname, relativePath);
}

exports.toFileUrl = dirname => {
  return function (relativePath) {
    return toFileUrl(dirname, relativePath);
  };
};

exports.toPathname = dirname => {
  return function (relativePath) {
    return toPathname(dirname, relativePath);
  };
};

exports.load = dirname => {
  const fileCache = Object.create(null);

  return function (name, options) {
    options = options || {};

    const file = path.resolve(dirname, "files/" + name + ".html");

    if (!options.url) {
      options.url = toFileUrl(dirname, file);
    }

    const contents = fileCache[file] || fs.readFileSync(file, "utf8");
    const doc = jsdom.jsdom(null, options);
    const window = doc.defaultView;

    doc.parent = window;
    window.loadComplete = function () {};

    doc.write(contents);
    doc.close();

    fileCache[file] = contents;
    return doc;
  };
};

/**
 * @param {Document} document
 * @param {String} name
 * @return {Function} A function that tests if the given
 *         exception is a `DOMException` of the specified `code`
 *
 * @example t.throws(function () {
 *   // ...
 * }, domExceptionPredicate(doc, "NoModificationAllowedError"));
 */
exports.domExceptionPredicate = (document, name) => {
  return function (error) {
    return error instanceof document.defaultView.DOMException &&
           error.name === name &&
           error.code === exceptionTable[name].legacyCodeValue;
  };
};

exports.todo = (test, fn) => {
  fn({
    ok(value, message) {
      test.ok(!value, "Marked as TODO: " + message);
    }
    // add more as needed
  });
};

exports.injectIFrame = document => {
  return exports.injectIFrameWithScript(document);
};

exports.injectIFrameWithScript = (document, scriptStr) => {
  scriptStr = scriptStr || "";
  const iframe = document.createElement("iframe");
  document.body.appendChild(iframe);

  const scriptTag = iframe.contentWindow.document.createElement("script");
  scriptTag.textContent = scriptStr;
  iframe.contentWindow.document.body.appendChild(scriptTag);

  return iframe;
};

/**
 * Create a new Promise and call the given function with a resolver function which can be passed as a node
 * style callback.
 *
 * Node style callbacks expect their first argument to be an Error object or null. The returned promise will
 * be rejected if this first argument is set. Otherwise the promise will be resolved with the second argument of
 * the callback, or with an array of arguments if there are more arguments.
 *
 * @example nodeResolverPromise(nodeResolver => fs.readFile('foo.png', nodeResolver)).then(content => {});
 * @param {Function} fn
 * @returns {Promise}
 */
exports.nodeResolverPromise = fn => {
  return new Promise((resolve, reject) => {
    fn(function (error, result) {
      if (error) {
        reject(error);
      } else if (arguments.length > 2) {
        // pass all the arguments as an array,
        // skipping the error param
        const arrayResult = new Array(arguments.length - 1);
        for (let i = 1; i < arguments.length; ++i) {
          arrayResult[i - 1] = arguments[i];
        }
        resolve(arrayResult);
      } else {
        resolve(result);
      }
    });
  });
};

/**
 * Is this script currently running within a Web Worker context?
 * @returns {boolean}
 */
exports.inWebWorkerContext = () => {
  /* globals WorkerGlobalScope, self */
  return typeof WorkerGlobalScope !== "undefined" && self instanceof WorkerGlobalScope;
};

/**
 * Is this script currently running within a browser context?
 * Note: also returns true within a Web Worker context
 * @returns {boolean}
 */
exports.inBrowserContext = () => {
  /* globals window */
  return (typeof window === "object" && window === window.self) || exports.inWebWorkerContext();
};

/**
 * Resolves a path to a static fixture file to a file or http URL.
 * If running tests from node, a valid file url will be returned.
 * If running tests using karma, a http url to the file be returned (this file is served by karma)
 * @param {string} relativePath Relative path within the test directory. For example "jsdom/files/reddit.html"
 * @returns {string} URL
 */
exports.getTestFixtureUrl = relativePath => {
  /* globals location */
  if (exports.inBrowserContext()) {
    // location is a Location or WorkerLocation
    return location.origin + "/base/test" + (relativePath[0] === "/" ? "" : "/") + relativePath;
  }

  return toFileUrl(__dirname, relativePath);
};

/**
 * Reads a static fixture file as utf8.
 * If running tests from node, the file will be read from the file system
 * If running tests using karma, a http request will be performed to retrieve the file using karma's server.
 * @param {string} relativePath Relative path within the test directory. For example "jsdom/files/reddit.html"
 */
exports.readTestFixture = relativePath => {
  const useRequest = exports.inBrowserContext();

  return exports.nodeResolverPromise(nodeResolver => {
    if (useRequest) {
      request.get(exports.getTestFixtureUrl(relativePath), { timeout: 5000 }, nodeResolver);
    } else {
      fs.readFile(path.resolve(__dirname, relativePath), { encoding: "utf8" }, nodeResolver);
    }
  })
  // request passes (error, response, content) to the callback
  // we are only interested in the `content`
  .then(result => useRequest ? result[1] : result);
};

redpajama_set_name: RedPajamaGithub


Restraining Rigour of Prosecution on Administration Bonds, 115.
Declaring Punishment for Burning of Courthouses, 116.
For Stay of Executions after the Tenth Day of May, yearly, 267.
to settle, from the Payment of Duty on Negroes for any Domestic Slaves, 268.
Court to the Governor and Council, 270.
Confirming Lands devised for the Use of the Church, 473.
Regulating Prosecutions in Criminal Cases, 474.
For preventing the Destroying of Boundaries, etc., 475.
For the Advancement of Justice in Testamentary Affairs, 476.
Regulating Amerciaments in the Provincial and County Courts, etc., 478.
Prohibiting the Clandestine Running of Tobacco into Neighboring Colonies, 480.
Lands and obtaining Injunctions, 727.
To punish Blasphemers, Swearers, Drunkards and Sabbath-Breakers, 733.
Quieting the Possessions of the Indians on Nanticoke and Choptank Rivers, 738.


* The power is supplied by 3*AAA, it is suitable for compute, mp3,mp4 and so on. You can share the music by connecting the speaker and MP3, it is convenient to use and carry.
* The fabrics is filled with cotton in high quality that the main material is imported from Korea's woolen, it is suitable for all the people including the allergic ones.


Boggs v. Divested Atomic Corp.
Teresa Boggs and seven others (plaintiffs) lived within six miles of the Portsmouth Gaseous Diffusion Plant (Plant), which had been processing radioactive materials since the 1950s. The plaintiffs sued Divested Atomic Corporation and Martin Marietta Energy Systems (defendants), the former and current operators of the Plant, for personal injury and property damage caused by the Plant's unlawful emission of hazardous materials. The plaintiffs sought to proceed as a class action in which they would represent all current and former residents, property owners, and lessees who lived within six miles of the Plant and had been subjected to its harmful emissions. The defendants opposed class certification.


Samsung Galaxy J5 Prime comes with a 5-inch HD TFT touchscreen display with the resolution of 720 x 1280 pixel. The device is powered by a 1.4GHz Quad-core processor along with 2GB of RAM and 16/32 GB internal storage, expandable up to 256GB using microSD card. It houses 2400mAh Non-removable Battery and sports a 13-megapixel rear camera along with 5-megapixel selfie shutter. The Dual SIM device runs on the Android Marshmallow 6.0 Operating System. On the connectivity front, it offers 4G, 3G, Bluetooth, microUSB 2.0, Wi-Fi 802.11 b/g/n options.
Galaxy J5 Prime is blessed with the similar design language to the company's other flagship models, but its matte finish makes it a little different. The compact size of the device makes it handy and it's well arranged buttons makes it easy in use. Talking about the display, the smartphone comes with a 5-inch AMOLED display, which is capable of developing vivid and saturated colours.
The handset is fueled with a pretty nice combination of a Quad-core processor and 2GB RAM, capable of performing a minimum mid-level performance. The model is powered by a 1.4GHz processor and is blessed with 16GB of inbuilt storage which can be further expanded up to 128GB via microSD card.
Samsung Galaxy J5 Prime features a 13-megapixel camera with a wide aperture, which helps in absorbing good light while capturing the images. The camera of this device is capable of capturing good quality pictures even in low light. Camera interface of the smartphone packs seven different modes of shooting. The eye catching feature of the model is its double tapping camera mode on home screen, which has been proved to be very useful if someone wants to have a quick shot.
The battery part of the smartphone may be the greatest factor of its selling point. The handset is backed up by a 2600mAh battery, which can provide acceptable performance to the handset. The S Power Planning of Galaxy J5 Prime helps in saving battery for call and messages extending to the phone's life to its maximum level.
Samsung Galaxy J5 Prime is a pretty cool and nice looking budget-friendly smartphone and is blessed with attractive and ravishing features. If you plan to purchase it, don't think twice and just go for it.


Facts About Mobile Auto Detailing When it comes to mobile auto detailing, you should keep in mind that it's quite different with car washing. These services have their own advantages which is why it's best that you are aware of such things in the first place. In any case, mobile car washing will need a crew in order to efficiently wash about ten cars within an hour. When it comes to the mobile auto detailing, you only need to take care of a single car, but you will have to work alone and make sure that you do well with the job. Being aware of the crucial things about mobile auto detailing is something that you need to do. Also, before you decide to start this kind of job, it's necessary that you already have the right equipment. It's a fact that most car washing services that are available today will need expensive equipment, but the mobile auto detailing won't need such tools to begin with. Things like portable generator and vacuum would suffice for a standard mobile auto detailing work. Of course, the equipment that you will need depends on the kind of mobile detailing that you want to work with. Things like water pumps needs to be considered if you want to get the right one. You can always ask someone experienced in mobile detailing for tips that you can use if you want to make sure that you'll be able to get the right equipment in the first place. There are small pumps that you can use for the auto detailing and the best kind for standard detailing is the ones with low volume and low pressure. Still, you will need to make sure that you're considering some things first before you decide to buy this kind of equipment for the auto detailing. Knowing how the equipment work is also another thing that you have to keep in mind.
With the help of the online network, you should be able to look for the right equipment that you need. There should also be detailed information online on how each equipment works and if they are suitable for certain kinds of auto detailing work. You'll also have access to the function that helps you search for nearby retail stores and see if they are selling certain kinds of auto detailing equipment. There are also some lists online where you will be able to look for online shops that can provide you the equipment that you need for the auto detailing.
So if you want your clients to stay and get your service when they need it, you must have a reliable set of equipment for the auto detailing.


text:
Рок () — місто () в США, в окрузі Вуд штату Вісконсин. Населення — 855 осіб (2010).

Демографія
Згідно з переписом 2010 року, у місті мешкало 855 осіб у 318 домогосподарствах у складі 256 родин. Було 353 помешкання

Расовий склад населення:

До двох чи більше рас належало 0,7 %. Частка іспаномовних становила 2,5 % від усіх жителів.

За віковим діапазоном населення розподілялося таким чином: 23,4 % — особи молодші 18 років, 64,8 % — особи у віці 18—64 років, 11,8 % — особи у віці 65 років та старші. Медіана віку мешканця становила 43,1 року. На 100 осіб жіночої статі у місті припадало 104,5 чоловіків;  на 100 жінок у віці від 18 років та старших — 107,3 чоловіків також старших 18 років.

Середній дохід на одне домашнє господарство  становив  долари США (медіана — ), а середній дохід на одну сім'ю —  долари (медіана — ). Медіана доходів становила  доларів для чоловіків та  долари для жінок. За межею бідності перебувало 5,1 % осіб, у тому числі 4,3 % дітей у віці до 18 років та 11,3 % осіб у віці 65 років та старших.

Цивільне працевлаштоване населення становило 393 особи. Основні галузі зайнятості: освіта, охорона здоров'я та соціальна допомога — 36,1 %, виробництво — 18,6 %, роздрібна торгівля — 7,6 %, будівництво — 7,4 %.

Примітки

Джерела

Містечка Вісконсину
Населені пункти округу Вуд (Вісконсин)

redpajama_set_name: RedPajamaWikipedia


And while I spent most the week snapping images of the looks on the runway and behind the scenes, I also got a couple of chances to get in front of the camera thanks to Style Coalition, StyleItOnline and MyStyleGenie (which you'll be seeing soon!).
First was my second home during fashion week the Style Coalition Lounge at the Empire (first, being the Gem) , where I got to spend time with my favorites and hear designers discuss their upcoming collections, Carmen Marc Valvo being one.
We also were treated to on site manis, hair and makeup. How fun are these waves? And don't be fooled these bloggers, they're the real deal.
Of course we met Andre, and refrained from asking about Red Lobster, it was hard. really hard.
I also spied Lela Rose after the show, Look.At.That.Necklace!
And then there was my favorite look from The Rebecca Minkoff show (more to come)..
how fab is this?!!! you look amazing, BTW. i'm loving the fur and also those striped pants!!! what a fun time!
Cool. You're such a VIP!
Oh, my gosh! Looks like you had such a great time! Thanks for sharing!
girl, CUTE pictures of you and the scissors.


This page contains a list of Joanna Newsom's songs, Joanna Newsom's albums, and Joanna Newsom's videos. One of the most useful parts of this page is the list of artists that are similar to Joanna Newsom. Check them out, maybe you'll find your new favorite band!


Made of high quality non-woven material, so this is very durable and long-lasting. This is useful garment cover that can safe your coat/blazer or sherwani from dust, stains.
This cover has full length zip, shaped shoulders and top hole for hanger hook.
Ideal for Travel and long term storage of dresses, suits, pants and longer clothes, etc.
This cover has carry handles and can be folded in two for easy carrying or traveling.


text:
\section{Introduction}

In the field of uncertainty quantification it is common to study mathematical models with uncertain influences parameterized by countably infinite sequences of random variables. Consider, for instance, an abstract model $M\!:X\times U\to Y$ such that
\begin{equation}\label{eq:GuKa_model}
M(g({\boldsymbol{y}}),{\boldsymbol{y}})=0,
\end{equation}
where $X$ and $Y$ are separable Hilbert spaces and $U$ is a nonempty subset of the infinite-dimensional sequence space of parameters $\mathbb R^{\mathbb N}$. The solution $g({\boldsymbol{y}})\in X$ to~\eqref{eq:GuKa_model} for ${\boldsymbol{y}}\in U$, if it exists, may be computationally expensive to evaluate. To this end, it may be preferable to instead \emph{approximate} $g$ using a surrogate which is cheap to evaluate and hence enables, e.g., efficient sampling of the (approximated) solution.

Some possible surrogate models include, but are not limited to, Gaussian process regression~\cite{GPR1}, reduced basis approaches~\cite{ROM2,ROM1}, generalized polynomial chaos expansions \cite{GPC1,GPC2}, neural network approximations~\cite{NN4,NN2,NN3,NN1}, and kernel interpolation based on lattice point sets~\cite{kkkns22,Kernel1,Kernel2}. The results presented in this manuscript are particularly well-suited to the analysis of kernel methods used in conjunction with the so-called \emph{periodic model} discussed in~\cite{hhkks22,kkkns22,kks20}, and we will devote a section of this manuscript to explore the application of our dimension truncation results within this framework.

A natural first step for the numerical treatment of~\eqref{eq:GuKa_model} is the approximation by a dimensionally-truncated model $M_s\!:X\times U_s\to Y$ such that
\begin{equation*}
M_s(g_s({\boldsymbol{y}}_{\leq s}),{\boldsymbol{y}}_{\leq s})=0,
\end{equation*}
where $\varnothing\neq U_s\subseteq \mathbb R^s$ and $g_s({\boldsymbol{y}}_{\leq s})\in X$ for all ${\boldsymbol{y}}_{\leq s}\in U_s$.  Consider the problem of finding a surrogate solution $g_{s,n}:=A_n(g_s)$ using an algorithm $A_n$ which uses $n$ point evaluations of the $s$-dimensional function $g_s$, where the surrogate belongs to $X$ such that
$$
\|g_s-g_{s,n}\|_{L_{{\boldsymbol{\mu}}}^2(U;X)} \xrightarrow{n\to\infty}0
$$
with some known convergence rate and ${{\boldsymbol{\mu}}}$ indicating a probability measure on $U$. The total error of the approximation obtained in this fashion can be estimated using the triangle inequality
$$
\|g-g_{s,n}\|_{L_{{\boldsymbol{\mu}}}^2(U;X)}\leq \|g-g_s\|_{L_{{\boldsymbol{\mu}}}^2(U;X)}+\|g_s-g_{s,n}\|_{L_{{\boldsymbol{\mu}}}^2(U;X)}.
$$
In this manuscript we focus on the first term---the \emph{dimension truncation error}---which is independent of the chosen approximation scheme $A_n$.

Dimension truncation error rates are typically studied for problems involving partial differential equations (PDEs) with random inputs. For integration problems a dimension truncation rate is derived in~\cite{kss12} for the source problem with an affine parameterization of the diffusion coefficient. This rate was then improved by \cite{gantner} in the generalized context of affine parametric operator equations. Dimension truncation has also been studied for coupled PDE systems arising in optimal control problems under uncertainty~\cite{guth21}, in the context of the periodic model of uncertainty quantification for both numerical integration~\cite{kks20} and kernel interpolation~\cite{kkkns22}, as well as for Bayesian inverse problems governed by PDEs~\cite{gantner2,hks21}. The results in these papers have been proved using Neumann series, which is known to work well in the affine parametric setting, but may lead to suboptimal results if the problem depends nonlinearly on the parameters.

In the non-affine setting, using Taylor series makes it possible to derive dimension truncation error rates by exploiting the parametric regularity of the problem, whereas the Neumann series approach relies fundamentally on the parametric structure of the model. The Taylor series approach was first applied in \cite{GGKSS2019}, and motivated the authors in \cite{guth22} and \cite{gk22} to derive dimension truncation error rates for sufficiently smooth, Banach space valued integrands, and with parameters following a generalized $\beta$-Gaussian distribution. An overview of the various dimension truncation error bounds studied in the literature is given in Table~\ref{table:glossary}.

\begin{table}[!t]
\begin{center}
\begin{tabular}{l|l|l}
 &Integration&Function approximation\\
\hline Affine parametric&\cite{gantner,kss12}  &\cite{kkkns22}\\
operator equation setting&rate $\mathcal O(s^{-\frac{2}{p}+1})$&rate $\mathcal O(s^{-\frac{1}{p}+\frac12})$\\
\hline Non-affine parametric&\cite{GGKSS2019,gk22}&this paper\\
operator equation setting&rate $\mathcal O(s^{-\frac{2}{p}+1})$&rate $\mathcal O(s^{-\frac{1}{p}+\frac12})$
\end{tabular}
\end{center}
\caption{An overview of various dimension truncation results.}\label{table:glossary}
\end{table}

Our manuscript is structured as follows. Subsection~\ref{sec:not} introduces the multi-index notation used throughout the paper. The problem setting is introduced in Section~\ref{sec:prob}, including the central assumptions for the ensuing dimension truncation analysis. Section~\ref{sec:dimtrunc} contains
the $L^2$ dimension truncation theorem for Hilbert space valued functions, and in Section~\ref{sec:invariance} we discuss the invariance of the dimension truncation rate under certain transformations of the variables. Numerical experiments assessing the sharpness of our theoretical results are presented in Section~\ref{sec:numex}. The paper ends with some conclusions in Section~\ref{sec:concl}.

\subsection{Notations and preliminaries}\label{sec:not}
Throughout this manuscript, boldfaced symbols are used to denote multi-indices while the subscript notation $m_j$ is used to refer to the $j$-th component of multi-index ${\boldsymbol{m}}$.
Let
$$
\mathcal F:=\{{\boldsymbol{m}}\in \mathbb N_0^{\mathbb N}:|{\boldsymbol{m}}|<\infty\}
$$
denote the set of finitely supported multi-indices, where the order of multi-index ${\boldsymbol{m}}$ is defined as
$$
|{\boldsymbol{m}}|:=\sum_{j\geq 1}m_j.
$$
Moreover, we denote
$$
|{\boldsymbol{m}}|_\infty := \max_{j\geq 1}m_j,
$$
and, for any sequence ${\boldsymbol{x}} := (x_j)_{j =1}^{\infty}$ of real numbers and ${\boldsymbol{m}} \in \mathcal{F}$, we define
$$
{\boldsymbol{x}}^{{\boldsymbol{m}}} := \prod_{j\geq 1} x_j^{m_j},
$$
where we use the convention $0^0 := 1$.

\section{Problem setting}\label{sec:prob}

Let $X$ be a real separable Hilbert space, $U:=[-\tfrac12,\tfrac12]^{\mathbb N}$ a set of parameters, and suppose that $g({\boldsymbol{y}})\in X$ is a parameterized family of functions with smooth dependence on ${\boldsymbol{y}}\in U$. We define $g_s({\boldsymbol{y}}) := g({\boldsymbol{y}}_{\leq s},\boldsymbol{0}) := g(y_1,\ldots,y_s,0,0,\ldots)$ and assume that ${\boldsymbol{\mu}}(\mathrm d{\boldsymbol{y}}) := \bigotimes_{j\geq1}\mu(\mathrm dy_j)$ is a countable product probability measure, i.e., ${\boldsymbol{\mu}}(U) = 1$.
We suppose that
\begin{enumerate}
\item \label{assumpA1}For ${\boldsymbol{\mu}}$-a.e.~${\boldsymbol{y}}\in U$, there holds
$$
\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X\xrightarrow{s\to\infty}0.
$$
\item \label{assumpA2}Let $(\Theta_k)_{k\geq 0}$ and ${\boldsymbol{b}}:=(b_j)_{j\ge 1}$ be sequences of nonnegative numbers such that ${\boldsymbol{b}}\in \ell^p(\mathbb N)$ for some $p\in (0,1)$ and $b_1\geq b_2\geq\cdots$. Suppose that $g$ is continuously differentiable up to order $k+1$, with
$$
\|\partial^{{\boldsymbol{\nu}}}g({\boldsymbol{y}})\|_X\leq \Theta_{|{\boldsymbol{\nu}}|}{\boldsymbol{b}}^{{\boldsymbol{\nu}}}
$$
for all ${\boldsymbol{y}} \in U$ and for all ${\boldsymbol{\nu}}\in\mathcal F_k := \{{\boldsymbol{\nu}} \in {\mathbb{N}}_0^{{\mathbb{N}}}:|{\boldsymbol{\nu}}|\leq k+1\}$, where $k:=\lceil \frac{1}{1-p}\rceil$.
\item \label{assumpA3} There holds $\int_{-1/2}^{1/2}y_j\,\mu({\rm d}y_j)=0$ and there exists a constant $C_{\mu}\geq 0$ such that $\int_{-1/2}^{1/2}|y_j|^k\,\mu({\rm d}y_j)\leq C_\mu$ for all $k\geq 2$.
\end{enumerate}

If Assumption~\ref{assumpA2} holds, then we infer that ${\boldsymbol{y}} \mapsto G(g({\boldsymbol{y}}))$ for all $G\in X'$ is continuous as a composition of continuous mappings. Hence ${\boldsymbol{y}} \mapsto G(g({\boldsymbol{y}}))$ is measurable for all $G \in X'$, i.e., ${\boldsymbol{y}} \mapsto g({\boldsymbol{y}})$ is weakly measurable. Since $X$ is assumed to be a separable Hilbert space, by Pettis' theorem (cf., e.g., \cite[Chapter 4]{Yosida}) we obtain that ${\boldsymbol{y}} \mapsto g({\boldsymbol{y}})$ is strongly measurable. The upper bound in Assumption~\ref{assumpA2} is ${\boldsymbol{\mu}}$-integrable. Thus we conclude from Bochner's theorem (cf., e.g., \cite[Chapter 5]{Yosida}) and Assumption~\ref{assumpA2} that $g$ is ${\boldsymbol{\mu}}$-integrable over $U$.

Further, ${\boldsymbol{\mu}}$-a.e.~equality defines an equivalence relation among strongly ${\boldsymbol{\mu}}$-measurable functions.
By $L^2_{{\boldsymbol{\mu}}}(U;X)$ we denote the Hilbert space of equivalence classes of strongly ${\boldsymbol{\mu}}$-measurable functions $f:U\to X$ with norm
\begin{align*}
    \|f\|_{L^2_{{\boldsymbol{\mu}}}(U;X)} := \bigg(\int_U \|f({\boldsymbol{y}})\|_X^2\,{\boldsymbol{\mu}}(\mathrm d{\boldsymbol{y}})\bigg)^{\frac12} <\infty.
\end{align*}
Moreover, under the Assumptions~\ref{assumpA1} and~\ref{assumpA2} it can be shown that $g,g_s \in L^2_{{\boldsymbol{\mu}}}(U;X)$ and $$
\lim_{s\to \infty} \|g({\boldsymbol{y}}) - g({\boldsymbol{y}}_{\leq s},\boldsymbol{0})\|_{L_{{\boldsymbol{\mu}}}^2(U;X)} = \lim_{s\to \infty} \bigg(\int_{U} \|g({\boldsymbol{y}}) - g({\boldsymbol{y}}_{\leq s},\boldsymbol{0})\|_X^2\,{\boldsymbol{\mu}}(\mathrm d{\boldsymbol{y}})\bigg)^{\frac12} = 0,
$$
by applying Lebesgue's dominated convergence theorem (see, e.g., \cite[Theorem 1]{infiniteintegration} and \cite[Section 26]{Halmos}) to
$$F^s({\boldsymbol{y}}) := \|g({\boldsymbol{y}}) - g({\boldsymbol{y}}_{\leq s},\boldsymbol{0})\|_X^2,$$
which converges ${\boldsymbol{\mu}}$-a.e.~to zero by Assumption~\ref{assumpA1}, and can be bounded by $(2\Theta_{0})^2$ by Assumption~\ref{assumpA2}. We use the superscript to avoid confusion with the notation used to denote dimensionally-truncated functions elsewhere in the document.

\section{Dimension truncation error}\label{sec:dimtrunc}
We will require the following parametric regularity bound for the main dimension truncation result.
\begin{lemma}\label{lemma:regularity}
Under Assumption~\ref{assumpA2}, there holds
$$
|\partial^{{\boldsymbol{\nu}}}\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2|\leq \bigg(\max_{0\leq \ell\leq |{\boldsymbol{\nu}}|}\frac{2\Theta_\ell}{\ell!}\bigg)^2 (|{\boldsymbol{\nu}}|+1)!{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\quad\text{for all}~{\boldsymbol{\nu}}\in\mathcal F_k~\text{and}~{\boldsymbol{y}}\in U.
$$
\end{lemma}
\begin{proof}Let ${\boldsymbol{\nu}}\in\mathcal F_k$. We apply the Leibniz product rule with respect to the inner product of the Hilbert space $X$ to obtain
\begin{align*}
\partial^{{\boldsymbol{\nu}}}\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2&=\partial^{{\boldsymbol{\nu}}}\langle g({\boldsymbol{y}})-g_s({\boldsymbol{y}}),g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\rangle_X\\
&=\sum_{{\boldsymbol{m}}\leq{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\langle \partial^{{\boldsymbol{m}}}(g({\boldsymbol{y}})-g_s({\boldsymbol{y}})),\partial^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}(g({\boldsymbol{y}})-g_s({\boldsymbol{y}}))\rangle_X.
\end{align*}
Using the Cauchy--Schwarz inequality together with Assumption~\ref{assumpA2} yields
\begin{align*}
|\partial^{{\boldsymbol{\nu}}}\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2|&\leq \sum_{{\boldsymbol{m}}\leq{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\|\partial^{{\boldsymbol{m}}}(g({\boldsymbol{y}})-g_s({\boldsymbol{y}}))\|_X\|\partial^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}(g({\boldsymbol{y}})-g_s({\boldsymbol{y}}))\|_X\\
&\leq 4 \sum_{{\boldsymbol{m}}\leq{\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\Theta_{|{\boldsymbol{m}}|}{\boldsymbol{b}}^{{\boldsymbol{m}}}\Theta_{|{\boldsymbol{\nu}}|-|{\boldsymbol{m}}|}{\boldsymbol{b}}^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}\\
&=4{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\sum_{\ell=0}^{|{\boldsymbol{\nu}}|}\Theta_{\ell}\Theta_{|{\boldsymbol{\nu}}|-\ell}\sum_{\substack{|{\boldsymbol{m}}|=\ell\\ {\boldsymbol{m}}\leq {\boldsymbol{\nu}}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\\
&=4{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\sum_{\ell=0}^{|{\boldsymbol{\nu}}|}\Theta_{\ell}\Theta_{|{\boldsymbol{\nu}}|-\ell}\frac{|{\boldsymbol{\nu}}|!}{\ell!(|{\boldsymbol{\nu}}|-\ell)!}\\
&\leq 4\bigg(\max_{0\leq \ell\leq |{\boldsymbol{\nu}}|}\frac{\Theta_\ell}{\ell!}\bigg)^2 (|{\boldsymbol{\nu}}|+1)!{\boldsymbol{b}}^{{\boldsymbol{\nu}}},
\end{align*}
where we used the Vandermonde convolution $\sum_{\substack{|{\boldsymbol{m}}|=\ell\\ {\boldsymbol{m}}\leq{\boldsymbol{\nu}}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}=\binom{|{\boldsymbol{\nu}}|}{\ell}=\frac{|{\boldsymbol{\nu}}|!}{\ell!(|{\boldsymbol{\nu}}|-\ell)!}$.
\end{proof}
The main result of this document is stated below.
\begin{theorem}\label{thm:1}
Let $g({\boldsymbol{y}})\in X$, ${\boldsymbol{y}}\in U$, satisfy Assumptions~\ref{assumpA1}--\ref{assumpA3}. Then
$$
\|g-g_s\|_{L_{\boldsymbol\mu}^2(U;X)}=\mathcal O(s^{-\frac{1}{p}+\frac12}),
$$
where the implied coefficient is independent of $s$.
\end{theorem}
\begin{proof}
Let $s\geq 1$ and define
$$
F^{{s}}({\boldsymbol{y}}):=\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2\quad\text{for}~{\boldsymbol{y}}\in U.
$$
In the special case of the uniform distribution ${\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})={\rm d}{\boldsymbol{y}}$, we can apply~\cite[Theorem~4.2]{gk22} to obtain
$$
\|g-g_s\|_{L^2(U;X)}^2=\bigg|\int_U(F^s({\boldsymbol{y}})-F^s({\boldsymbol{y}}_{\leq s},\mathbf 0))\,{\rm d}{\boldsymbol{y}}\bigg|=\mathcal O(s^{-\frac{2}{p}+1}),
$$
from which the claim follows. For completeness, we present the proof below for the probability measure ${\boldsymbol{\mu}}$ and because parts of the argument will also be useful to establish the invariance of the dimension truncation rate in Section~4.

Developing the Taylor expansion of $F^s$ about $({\boldsymbol{y}}_{\leq s},\mathbf 0)$ and observing that $F^s({\boldsymbol{y}}_{\leq s},\mathbf 0)=0$, we obtain
\begin{align}
\begin{split}
F^s({\boldsymbol{y}})&=\sum_{\ell=1}^k \sum_{\substack{|{\boldsymbol{\nu}}|=\ell\\ \nu_j=0~\forall j\leq s}}\frac{{\boldsymbol{y}}^{{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},\mathbf 0)\\
&\quad + \sum_{\substack{|{\boldsymbol{\nu}}|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{k+1}{{\boldsymbol{\nu}}!}{\boldsymbol{y}}^{{\boldsymbol{\nu}}}\int_0^1 (1-t)^k \partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},t{\boldsymbol{y}}_{>s})\,{\rm d}t,
\end{split}\label{eq:taylorexp}
\end{align}
where ${\boldsymbol{y}}_{>s}:=(y_j)_{j>s}$. Integrating both sides over ${\boldsymbol{y}}\in U$ yields
\begin{align*}
\int_U F^s({\boldsymbol{y}})\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})&=\sum_{\ell=1}^k \sum_{\substack{|{\boldsymbol{\nu}}|=\ell\\ \nu_j=0~\forall j\leq s}}\frac{1}{{\boldsymbol{\nu}}!}\int_U {\boldsymbol{y}}^{{\boldsymbol{\nu}}}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},\mathbf 0)\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})\\
&\quad +\sum_{\substack{|{\boldsymbol{\nu}}|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{k+1}{{\boldsymbol{\nu}}!}\int_U \int_0^1 (1-t)^k {\boldsymbol{y}}^{{\boldsymbol{\nu}}}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},t{\boldsymbol{y}}_{>s})\,{\rm d}t\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}}).
\end{align*}
If ${\boldsymbol{\nu}}\in\mathcal F_k$ is such that $\nu_j=1$ for any $j>s$, then Fubini's theorem together with Assumption~\ref{assumpA3} imply for the summands appearing in the first term that
$$
\int_U {\boldsymbol{y}}^{{\boldsymbol{\nu}}}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},\mathbf 0)\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})=\underset{=0}{\underbrace{\bigg(\prod_{j>s}\int_{-\frac12}^{\frac12} y_j^{\nu_j}\,\mu({\rm d}y_j)\bigg)}}\int_{[-\frac12,\frac12]^s}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{y}}_{\leq s},\mathbf 0)\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}}_{>s}).
$$
Therefore all multi-indices with any component equal to 1 can be removed from the first sum (especially, we can omit all multi-indices with $|{\boldsymbol{\nu}}|=1$). Further, applying the regularity bound proved in Lemma~\ref{lemma:regularity} and writing open the definition of $F^s$ yields
\begin{align}
\begin{split}
\int_U \|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})&\leq C_\mu^k\bigg(\max_{0\leq\ell\leq k}\frac{2\Theta_\ell}{\ell!}\bigg)^2(k+1)!\sum_{\ell=2}^k \sum_{\substack{|{\boldsymbol{\nu}}|=\ell\\ \nu_j=0~\forall j\leq s\\ \nu_j\neq 1~\forall j>s}}{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\\
&\quad +C_\mu^{k+1}\bigg(\max_{0\leq\ell\leq k+1}\frac{2\Theta_\ell}{\ell!}\bigg)^2(k+2)!\sum_{\substack{|{\boldsymbol{\nu}}|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{1}{{\boldsymbol{\nu}}!}{\boldsymbol{b}}^{{\boldsymbol{\nu}}},
\end{split}\label{eq:splitsum}
\end{align}
where we used $\int_0^1 (1-t)^k\,{\rm d}t=\frac{1}{k+1}$ and Assumption~\ref{assumpA3}. The second term in~\eqref{eq:splitsum} can be estimated from above using the multinomial theorem in conjunction with Stechkin's lemma:
$$
\sum_{\substack{|\boldsymbol \nu|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{1}{{\boldsymbol{\nu}}!}{{\boldsymbol{b}}}^{\boldsymbol \nu}\leq \sum_{\substack{|\boldsymbol \nu|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{|\boldsymbol \nu|!}{\boldsymbol \nu!}{{\boldsymbol{b}}}^{\boldsymbol \nu}=\bigg(\sum_{j>s}b_j\bigg)^{k+1}\leq s^{(k+1)(-\frac{1}{p}+1)}\bigg(\sum_{j\geq 1}{b}_j^p\bigg)^{\frac{k+1}{p}}.
$$
On the other hand, the first term in~\eqref{eq:splitsum} can be estimated similarly to~\cite{gantner}:
\begin{align*}
    &\sum_{\substack{2\leq |\boldsymbol \nu|\leq k\\ \nu_j=0~\forall j\leq s\\ \nu_j\neq 1~\forall j> s}}{{\boldsymbol{b}}}^{\boldsymbol \nu} \leq \sum_{\substack{0\neq |\boldsymbol{\nu}|_{\infty}\leq k\\ \nu_j=0~\forall j\leq s\\ \nu_j\neq 1~\forall j> s}}{{\boldsymbol{b}}}^{\boldsymbol \nu}
    = -1 + \prod_{j>s} \bigg(1+ \sum_{\ell = 2}^{k} {b}_j^\ell\bigg)
    = -1 + \prod_{j>s} \bigg(1+ {b}_j^2 \sum_{\ell = 0}^{k-2} {b}_j^\ell\bigg) \\
    &\leq -1 + \prod_{j>s} \bigg(1+ {b}_j^2 \underbrace{\sum_{\ell = 0}^{k-2} {b}_1^\ell}_{=: \beta_k}\bigg)
    \leq  -1 + \exp{\Big(\beta_k \sum_{j> s} {b}_j^2\Big)} = \sum_{\ell \geq 1} \frac{1}{\ell!}\Big(\beta_k \sum_{j> s} {b}_j^2\Big)^{\ell}.
\end{align*}
Using $\sum_{j>s} b_j^2 \leq s^{-\frac{2}{p}+1} (\sum_{j\geq 1}{b}_j^p)^{\frac{2}{p}}$, which follows from  Stechkin's lemma, we further estimate
\begin{align*}
    \sum_{\ell \geq 1} \frac{1}{\ell!}\Big(\beta_k \sum_{j> s} {b}_j^2\Big)^{\ell} \leq s^{-\frac{2}{p}+1} \sum_{\ell \geq 1} \frac{1}{\ell!} (\beta_k \|{{\boldsymbol{b}}}\|_p^{2})^\ell = s^{-\frac{2}{p}+1} (-1 + \exp(\beta_k \|{{\boldsymbol{b}}}\|_p^2)
\end{align*}
since $s^{-\frac{2}{p}+1} \geq (s^{-\frac{2}{p}+1})^{\ell}$ for all $\ell \geq 1$.

Altogether, the above discussion yields the bound
$$
\|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_{L^2_{{\boldsymbol{\mu}}}(U;X)}^2=\int_U \|g({\boldsymbol{y}})-g_s({\boldsymbol{y}})\|_X^2\,{\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}})=\mathcal O(s^{-\frac{2}{p}+1}+s^{(k+1)(-\frac{1}{p}+1)}),
$$
where the implied coefficient is independent of $s$. Since we assumed that $k=\lceil \frac{1}{1-p}\rceil$, the assertion follows by taking the square root on both sides.
\end{proof}

\section{Invariance of the dimension truncation rate under transformations of variables}\label{sec:invariance}
An interesting consequence of the Taylor series argument used in Theorem~\ref{thm:1} is that the \emph{dimension truncation rate remains invariant under certain transformations of the variables}. This has been previously observed in the context of dimension truncation for integration problems under the periodic model~\cite{hhkks22}. To make this notion precise, let us consider a mapping ${\boldsymbol{\xi}}\!:U\to U$, ${\boldsymbol{\xi}}({\boldsymbol{y}}):=(\xi(y_1),\xi(y_2),\ldots)$, which satisfies the following conditions:
\begin{enumerate}
    \item[4.] There hold $\xi(0)=0$ and $\int_{-1/2}^{1/2} \xi(y)\,{\rm d}y=0$.
    \item[5.] There exists $C_\xi\geq 0$ such that $\int_{-1/2}^{1/2} |\xi(y)|^k\,{\rm d}y\leq C_\xi$ for all $k\geq 2$.
\end{enumerate}
Then we obtain the following as a consequence of Theorem~\ref{thm:1}.
\begin{corollary}\label{cor:1} Let $g({\boldsymbol{y}})\in X$, ${\boldsymbol{y}}\in U$, satisfy Assumptions~\ref{assumpA1}--\ref{assumpA3} and let ${\boldsymbol{\xi}}\!:U\to U$ satisfy Assumptions~4--5. Define the ${\boldsymbol{\xi}}$-transformed function $g_{{\boldsymbol{\xi}}}$ by
$$
g_{{\boldsymbol{\xi}}}({\boldsymbol{y}}):=g({\boldsymbol{\xi}}({\boldsymbol{y}})),\quad {\boldsymbol{y}}\in U,
$$
and its dimension truncation by $g_{{\boldsymbol{\xi}},s}({\boldsymbol{y}}):=g_{\boldsymbol{\xi}}({\boldsymbol{y}}_{\leq s},\mathbf 0)$ for ${\boldsymbol{y}}\in U$. Then
$$
\|g_{{\boldsymbol{\xi}}}-g_{{\boldsymbol{\xi}},s}\|_{L_{\boldsymbol\mu}^2(U;X)}=\mathcal O(s^{-\frac{1}{p}+\frac12}),
$$
where the implied coefficient is independent of $s$.
\end{corollary}
\begin{proof}We introduce $F_{\boldsymbol\xi}^s({\boldsymbol{y}}):=\|g_{\boldsymbol\xi}({\boldsymbol{y}})-g_{\boldsymbol\xi,s}({\boldsymbol{y}})\|_X^2$ for ${\boldsymbol{y}}\in U$. By carrying out the change of variable ${\boldsymbol{y}}\leftarrow\boldsymbol\xi({\boldsymbol{y}})$ in~\eqref{eq:taylorexp}, we obtain
\begin{align*}
F_{{\boldsymbol{\xi}}}^s({\boldsymbol{y}})&=\sum_{\ell=1}^k \sum_{\substack{|{\boldsymbol{\nu}}|=\ell\\ \nu_j=0~\forall j\leq s}}\frac{{\boldsymbol{\xi}}({\boldsymbol{y}})^{{\boldsymbol{\nu}}}}{{\boldsymbol{\nu}}!}\partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{\xi}}({\boldsymbol{y}}_{\leq s},\mathbf 0))\\
&\quad + \sum_{\substack{|{\boldsymbol{\nu}}|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{k+1}{{\boldsymbol{\nu}}!}{\boldsymbol{\xi}}({\boldsymbol{y}})^{{\boldsymbol{\nu}}}\int_0^1(1-t)^k \partial^{{\boldsymbol{\nu}}}F^s({\boldsymbol{\xi}}({\boldsymbol{y}}_{\leq s},t{\boldsymbol{y}}_{>s}))\,{\rm d}t.
\end{align*}
Integrating the above formula on both sides over ${\boldsymbol{y}}\in U$ and utilizing Lemma~\ref{lemma:regularity} as well as Assumption~5, we obtain---in complete analogy with the proof of Theorem~\ref{thm:1}---that
\begin{align*}
\int_U \|g_{{\boldsymbol{\xi}}}({\boldsymbol{y}})-g_{{\boldsymbol{\xi}},s}({\boldsymbol{y}})\|_X^2\,{\rm d}{\boldsymbol{y}}&\leq C_\xi^k\bigg(\max_{0\leq\ell\leq k}\frac{2\Theta_\ell}{\ell!}\bigg)^2(k+1)!\sum_{\ell=2}^k \sum_{\substack{|{\boldsymbol{\nu}}|=\ell\\ \nu_j=0~\forall j\leq s\\ \nu_j\neq 1~\forall j>s}}{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\\
&\quad + C_\xi^{k+1}\bigg(\max_{0\leq\ell\leq k+1}\frac{2\Theta_\ell}{\ell!}\bigg)^2(k+2)!\sum_{\substack{|{\boldsymbol{\nu}}|=k+1\\ \nu_j=0~\forall j\leq s}}\frac{1}{{\boldsymbol{\nu}}!}{\boldsymbol{b}}^{{\boldsymbol{\nu}}}.
\end{align*}
The desired result follows by exactly the same argument as in the proof of Theorem~\ref{thm:1}.
\end{proof}

As an application, with $U:=[-\tfrac12,\tfrac12]^{\mathbb N}$, let ${\boldsymbol{\xi}}\!:U\to U$ satisfy the Assumptions~4 and~5, let $D\subset \mathbb R^d$, $d\in\{1,2,3\}$, be a bounded Lipschitz domain, and let $f\!:D\to \mathbb R$ be a fixed source term. Consider the parametric PDE problem
\begin{align}
\begin{cases}
-\nabla\cdot(a_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})\nabla u_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}}))=f({\boldsymbol{x}}),&{\boldsymbol{x}}\in D,~{\boldsymbol{y}}\in U,\\
u_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})=0,&{\boldsymbol{x}}\in\partial D,~{\boldsymbol{y}}\in U,
\end{cases}\label{eq:xipde}
\end{align}
endowed with the ${\boldsymbol{\xi}}$-transformed diffusion coefficient
$$
a_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}}):=a_0({\boldsymbol{x}})+\sum_{i=1}^\infty \xi(y_i)\psi_i({\boldsymbol{x}}),\quad {\boldsymbol{x}}\in D,~{\boldsymbol{y}}\in U,
$$
which is assumed to satisfy the following:
\begin{enumerate}
    \item[6.] There exist $a_{\min},a_{\max}>0$ such that $0<a_{\min}\leq a_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})\leq a_{\max}<\infty$ for all ${\boldsymbol{x}}\in D$ and ${\boldsymbol{y}}\in U$.
    \item[7.] $a_0\in L^\infty(D)$ and $\psi_i\in L^\infty(D)$ for all $i\in\mathbb N$.
    \item[8.] $\sum_{i=1}^\infty \|\psi_i\|_{L^\infty(D)}^p<\infty$ for some $p\in(0,1)$.
\end{enumerate}

In this case, the transformation ${\boldsymbol{\xi}}({\boldsymbol{y}}):=(\frac{1}{\sqrt 6}\sin(2\pi y_j))_{j\geq 1}$ corresponds to the so-called \emph{periodic model} studied in~\cite{hhkks22,kkkns22,kks20}. Let $X:=H_0^1(D)$ be equipped with the norm $\|v\|_{X}:=\int_D \|\nabla v({\boldsymbol{x}})\|_{\mathbb R^d}^2\,{\rm d}{\boldsymbol{x}}$. In this special case, it is known that the weak solution $u(\cdot,{\boldsymbol{y}})\in X$ to~\eqref{eq:xipde} for ${\boldsymbol{y}}\in U$ satisfies the parametric regularity bound
$$
\|\partial^{{\boldsymbol{\nu}}}u_{{\boldsymbol{\xi}}}(\cdot,{\boldsymbol{y}})\|_{X}\leq (2\pi)^{|{\boldsymbol{\nu}}|}\frac{\|f\|_{X'}}{a_{\min}}\sum_{{\boldsymbol{m}}\leq {\boldsymbol{\nu}}}|{\boldsymbol{m}}|!{\boldsymbol{b}}^{{\boldsymbol{m}}}\prod_{j\geq 1}S(\nu_j,m_j)
$$
for all ${\boldsymbol{\nu}}\in \mathcal F$ and ${\boldsymbol{y}}\in U$, where the source term $f\in X'$, $S(\cdot,\cdot)$ denotes the Stirling number of the second kind, and ${\boldsymbol{b}}:=(b_j)_{j\geq 1}$ is defined by setting $b_j:=\frac{\|\psi_j\|_{L^\infty(D)}}{\sqrt 6 a_{\min}}$ for all $j\geq 1$. Let ${\boldsymbol{\mu}}({\rm d}{\boldsymbol{\mu}})={\rm d}{\boldsymbol{y}}$. Then Corollary~\ref{cor:1} can be used to deduce that
$$
\|u_{{\boldsymbol{\xi}}}-u_{{\boldsymbol{\xi}},s}\|_{L^2_{{\boldsymbol{\mu}}}(U;X)}=\mathcal O(s^{-\frac{1}{p}+\frac12}),
$$
where the constant is independent of the dimension $s$. In fact, if $X_h$ is a conforming finite element subspace of $X$, $u_{{\boldsymbol{\xi}},h}(\cdot,{\boldsymbol{y}})\in X_h$ denotes the finite element discretization of $u_{{\boldsymbol{\xi}}}(\cdot,{\boldsymbol{y}})\in X$ for all ${\boldsymbol{y}}\in U$, and $u_{{\boldsymbol{\xi}},h,s}(\cdot,{\boldsymbol{y}})\in X_h$ denotes the dimension truncation of $u_{{\boldsymbol{\xi}},h}(\cdot,{\boldsymbol{y}})$ for all ${\boldsymbol{y}}\in U$, then we have
$$
\|u_{{\boldsymbol{\xi}},h}-u_{{\boldsymbol{\xi}},h,s}\|_{L^2_{{\boldsymbol{\mu}}}(U;X)}=O(s^{-\frac{1}{p}+\frac12}),
$$
independently of $s$.

Finally, we present an example illustrating how our results can be applied to nonlinear quantities of interest.

\emph{Example.} Let $X:=H_0^1(D)$ as above. Consider the nonlinear quantity of interest
\begin{align}
G_{\mathrm{nl}}(v):=\|v\|_{X}^2:=\int_D \|\nabla v({\boldsymbol{x}})\|_{\mathbb R^d}^2\,{\rm d}{\boldsymbol{x}},\quad v\in X.\label{eq:nonlinearqoi}
\end{align}
If $u(\cdot,{\boldsymbol{y}})\in X$ is the solution to~\eqref{eq:xipde} with $U=[-\tfrac12,\tfrac12]^{\mathbb N}$, ${\boldsymbol{\mu}}({\rm d}{\boldsymbol{y}}):={\rm d}{\boldsymbol{y}}$, and ${\boldsymbol{\xi}}({\boldsymbol{y}}):={\boldsymbol{y}}$, then it is known to satisfy Assumptions~\ref{assumpA1}--\ref{assumpA3} with the regularity bound $$\|\partial^{{\boldsymbol{\nu}}}u(\cdot,{\boldsymbol{y}})\|_{X}\leq C|{\boldsymbol{\nu}}|!{\boldsymbol{b}}^{{\boldsymbol{\nu}}},$$
where the constant $C>0$ only depends on $\|f\|_{X'}$ and $a_{\min}$. By the Leibniz product rule, there holds
\begin{align*}
\partial^{{\boldsymbol{\nu}}}G_{\mathrm{nl}}
(u(\cdot,{\boldsymbol{y}}))&=\int_D\sum_{{\boldsymbol{m}}\leq {\boldsymbol{\nu}}} \binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}} \nabla\partial^{{\boldsymbol{m}}}u({\boldsymbol{x}},{\boldsymbol{y}})\cdot \nabla \partial^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}u({\boldsymbol{x}},{\boldsymbol{y}})\,{\rm d}{\boldsymbol{x}}\\
&\leq \sum_{{\boldsymbol{m}}\leq {\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\|\partial^{{\boldsymbol{m}}}u(\cdot,{\boldsymbol{y}})\|_{X}\|\partial^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}u(\cdot,{\boldsymbol{y}})\|_{X}\\
&\leq C^2\sum_{{\boldsymbol{m}}\leq {\boldsymbol{\nu}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}|{\boldsymbol{m}}|!{\boldsymbol{b}}^{{\boldsymbol{m}}}|{\boldsymbol{\nu}}-{\boldsymbol{m}}|!{\boldsymbol{b}}^{{\boldsymbol{\nu}}-{\boldsymbol{m}}}\\
&=C^2{\boldsymbol{b}}^{{\boldsymbol{\nu}}}\sum_{\ell=0}^{|{\boldsymbol{\nu}}|}\ell!(|{\boldsymbol{\nu}}|-\ell)!\sum_{\substack{{\boldsymbol{m}}\leq {\boldsymbol{\nu}}\\|{\boldsymbol{m}}|=\ell}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}\\
&=C^2{\boldsymbol{b}}^{{\boldsymbol{\nu}}}(|{\boldsymbol{\nu}}|+1)!,
\end{align*}
where we used the Vandermonde convolution $\sum_{\substack{|{\boldsymbol{m}}|=\ell\\ {\boldsymbol{m}}\leq {\boldsymbol{\nu}}}}\binom{{\boldsymbol{\nu}}}{{\boldsymbol{m}}}=\binom{|{\boldsymbol{\nu}}|}{\ell}=\frac{|{\boldsymbol{\nu}}|!}{\ell!(|{\boldsymbol{\nu}}|-\ell)!}$.

It follows from Theorem~\ref{thm:1} that
$$
\|G_{\mathrm{nl}}(u)-G_{\mathrm{nl}}(u_s)\|_{L_{{\boldsymbol{\mu}}}^2(U;\mathbb R)}=O(s^{-\frac{1}{p}+\frac12}),
$$
independently of $s$. Moreover, if ${\boldsymbol{\xi}}({\boldsymbol{y}}):=(\frac{1}{\sqrt 6}\sin(2\pi y_j))_{j\geq 1}$, then it follows from Corollary~\ref{cor:1} that
$$
\|G_{\mathrm{nl}}(u_{{\boldsymbol{\xi}}})-G_{\mathrm{nl}}(u_{{\boldsymbol{\xi}},s})\|_{L_{{\boldsymbol{\mu}}}^2(U;\mathbb R)}=O(s^{-\frac{1}{p}+\frac12}),
$$
independently of $s$.

\section{Numerical experiments}\label{sec:numex}
Let $D=(0,1)^2$ be a spatial domain, $U=[-\tfrac12,\tfrac12]^{\mathbb N}$, and $f({\boldsymbol{x}}):=x_1$ a fixed source term. Let ${\boldsymbol{\xi}}\!:U\to U$, ${\boldsymbol{\xi}}({\boldsymbol{y}})=(\frac{1}{\sqrt 6}\sin(2\pi y_j))_{j\geq 1}$. We consider the PDE problem
\begin{align}
\begin{cases}
-\nabla\cdot(a_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})\nabla u_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}}))=f({\boldsymbol{x}}),&{\boldsymbol{x}}\in D,~{\boldsymbol{y}}\in U,\\
u_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})=0,&{\boldsymbol{x}}\in\partial D,~{\boldsymbol{y}}\in U,
\end{cases}\label{eq:pdenumex}
\end{align}
equipped with the diffusion coefficient
\begin{align*}
a_{{\boldsymbol{\xi}}}({\boldsymbol{x}},{\boldsymbol{y}})=\frac32+\sum_{j\geq 1}\xi(y_j)j^{-\vartheta}\sin(j\pi x_1)\sin(j\pi x_2),\quad {\boldsymbol{x}}\in D,~{\boldsymbol{y}}\in U,~\vartheta>1
\end{align*}
The PDE~\eqref{eq:pdenumex} is spatially discretized using a first-order conforming finite element method with mesh size $h=2^{-5}$.

We consider the dimension truncation error for the full PDE solution using the formula
\begin{align*}
\|u_{{\boldsymbol{\xi}}}-u_{{\boldsymbol{\xi}},s}\|_{L^2(U;L^2(D))}\approx\bigg(\int_{[-\frac12,\frac12]^{s'}}\|u_{{\boldsymbol{\xi}},s'}(\cdot,{\boldsymbol{y}})-u_{{\boldsymbol{\xi}},s}(\cdot,{\boldsymbol{y}})\|_{L^2(D)}^2\,{\rm d}{\boldsymbol{y}}\bigg)^{\frac12}
\end{align*}
and we also consider the nonlinear quantity of interest~\eqref{eq:nonlinearqoi}, estimating the dimension truncation error using the formula
\begin{align*}
\|G_{\rm nl}(u_{\boldsymbol{\xi}})-G_{\rm nl}(u_{{\boldsymbol{\xi}},s})\|_{L^2(U)}\approx \bigg(\int_{[-\frac12,\frac12]^{s'}}|G_{\rm nl}(u_{{\boldsymbol{\xi}},s'}(\cdot,{\boldsymbol{y}}))-G_{\rm nl}(u_{{\boldsymbol{\xi}},s}(\cdot,{\boldsymbol{y}}))|^2\,{\rm d}{\boldsymbol{y}}\bigg)^{\frac12}
\end{align*}
In both cases, we choose $s'\gg s$ and the high-dimensional integrals are approximated using a randomly shifted rank-1 lattice rule with $2^{20}$ cubature nodes and a single random shift. As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule~\cite[lattice-39101-1024-1048576.3600]{KuoLattice} and use the same random shift for each value of $\vartheta$. As the reference solution, we use the PDE solution corresponding to $s'=2^{11}$.

The numerical results for dimensions $s\in\{2^k:k=1,\ldots,9\}$ and decay rates $\vartheta\in\{1.5,2.0,3.0\}$ corresponding to the full PDE solution and the nonlinear quantity of interest are displayed in Figures~\ref{fig:1} and~\ref{fig:2}, respectively. The theoretical convergence rates in each case are $-1.0$, $-1.5$, and $-2.5$, respectively, and they are displayed alongside the numerical results.

The convergence graphs corresponding to the full PDE solution in Figure~\ref{fig:1} display an aliasing behavior between $10\leq s\leq 100$, which may be explained by the contributions of the finite element discretization error as well as the use of an ``off-the-shelf'' lattice rule (in contrast to a ``tailored'' lattice rule). This behavior appear to be exacerbated in the convergence graphs corresponding to the nonlinear quantity of interest in Figure~\ref{fig:2}. Nonetheless, in all cases the theoretically anticipated convergence rates are easily observed in practice. We remark that the convergence graphs corresponding to the affine and uniform model with $\boldsymbol\xi({\boldsymbol{y}}):=(y_j)_{j\geq 1}$ are extremely similar to the results corresponding to the periodic model, and have thus been omitted.

\begin{figure}[H]
    \centering
    \includegraphics[width=.8\textwidth]{fullpde_periodic.eps}
    \caption{The dimension truncation errors of the full PDE solution corresponding to a periodically parameterized input random field with decay parameters $\vartheta\in\{1.5,2.0,3.0\}$. The expected dimension truncation error rates are $-1.0$, $-1.5$, and $-2.5$, respectively.}
    \label{fig:1}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=.8\textwidth]{nonlinearqoi_periodic.eps}
    \caption{The dimension truncation errors of the nonlinear quantity of interest corresponding to a periodically parameterized input random field with decay parameters $\vartheta\in\{1.5,2.0,3.0\}$. The expected dimension truncation error rates are $-1.0$, $-1.5$, and $-2.5$, respectively.}
    \label{fig:2}
\end{figure}

\section{Conclusions}\label{sec:concl}
Unlike many studies which have considered the dimension truncation error rate within the context of high-dimensional numerical integration, we considered the $L^2$ dimension truncation error rate for parametric Hilbert space valued functions. Our theory covers both affine parametric as well as non-affine parametric problems with sufficiently smooth dependence on a sequence of bounded, parametric variables. The main dimension truncation results presented in this work can be applied to nonlinear quantities of interest of parametric model problems, provided that they satisfy the conditions of our framework. In addition, the Hilbert space can be chosen to be a finite element subspace, indicating that our dimension truncation results are also valid for conforming finite element approximations of parametric PDEs.

The $L^2$ dimension truncation error rates considered in this work arise, e.g., in the study of high-dimensional function approximation of parametric PDEs. An example of such an approximation scheme is the kernel method over lattice point sets considered in~\cite{kkkns22}. The kernel method was analyzed in the context of the so-called \emph{periodic model}, in which a countable number of independent random variables enter the input random field of the PDE as periodic functions. Our second main result shows that the $L^2$ dimension truncation error rate remains invariant under certain transformations of the parametric variables: especially, the $L^2$ dimension truncation rate considered in this work holds for periodically parametrized model problems such as those studied in~\cite{hhkks22,kkkns22,kks20}.

redpajama_set_name: RedPajamaArXiv


Koyorad is proud to introduce a high-performance all-aluminum radiator for the Naturally Aspirated 90-96 300ZX with 3.0L engine with manual transmissions. This new application features Koyorad's 36mm high density Hyper core. This core was selected to increase the thickness of the stock radiator to boost the 300ZX's heat capacity and endurance which is especially helpful during track conditions. In addition, the all-aluminum construction replaces the weak plastic tanks found on the stock unit. Koyorad has engineered this radiator with precision, directly replacing the stock OEM radiator, maintaining provisions for the factory fans and accepting OEM hoses for a quick and simple installation. Koyorad's advanced engineering and design methods, impeccable welds, experienced furnace brazing, and triple pressure testing for quality has earned their excellent reputation in performance cooling worldwide.


Singing sensation Susan Boyle is back with her eagerly anticipated new album 'Someone to watch over me'.
Susan shot to fame following her stunning audition on Britain's Got Talent in 2009 and has been unstoppable ever since selling a whopping 14 million albums worldwide!
'Someone to Watch Over Me', Susan's third album, features classics such as 'Unchained Melody' and 'Both Sides Now' as well as some new tracks. Susan's song choices for the album were inspired by the life stories and experiences of fans from around the world who have written letters to her.
Susan said, "There are certain songs that I liked that resonated with the letters that people had written and sent to me. I knew that's what my record needed to be about. They wrote about grief, love, happy and sad times, it was all deeply moving and the songs mirror the emotion and life experience."


text:
Home » Travel » Bars and Libraries
Bars and Libraries
Travel, Victoria, Australia
Melbourne has a fun bar scene, many of them being, "secret." In one case, you pull a refrigerator door inside of a sub shop and it opens to a hidden bar. The library was a beautiful piece of art too.
I was in Melbourne and planned to go to Adelaide for another house/cat sit in a week. Once I finished the sit, I wasn't sure how I'd spend the next five months in Australia. Each morning, I took some time searching options but also wanted to make the most of my time in Melbourne while I was there. I decided to take the tram to the downtown State Library.
The building was a work of art! It was free to enter, and I wandered around. I ended up in the central circular part. It was a giant octagon with nothing directly above it so that I could see the balconies of several stories above it. On the ground floor, there were rows of tables that originated from the center of the circle. The tables were like something from a movie – lamps lit the table above people studying. Bookshelves lined the outside of the ring.
There was a small museum that was on the floors above. It followed the octagon around, almost like a wide hallway. When you're up there, you can see down below to the people reading away. The museum talked about the history of literature, including comic books. There was even a timeline of Ned Kelly's life.
One sign at the museum read, "The 1950s was a golden era for Australian pulp fiction. Import restrictions on U.S. books and magazines in the 1940s and 1950s created an opportunity for local publishers to meet the growing demand for American style commercial novels." It goes on to talk about Australian publishers and how much they produced during that time. Then it continues, "With the arrival of television, and the lifting of import restrictions in 1959, the demand for locally produced pulp fiction declined. The next generation of gumshoes – characters such as Cliff Hardy and Phryne Fisher – plied their trade in distinctly local settings, a sign of Australia's growing cultural confidence."
After the library, I walked around the downtown area. Melbourne is an excellent city for walking. There are shops, restaurants, trams, and it's safe and clean. Some buildings are modern, while others have the old Victorian design.
For dinner, I walked to the night market that goes on twice a week. It was huge, with hundreds of people. There was a vast canopy protecting all of the booths from any rain that might come. I waited in line for a food booth called The Black Sheep. The food was worth that wait! I ordered the lamb that came with pita bread, fries, and a salad. It was so juicy and flavorful that I still about it to this day.
I walked around the incredibly packed market and realized it was mostly food vendors. There was the occasional booth with handmade goods. In the center, crowded tables were filled with people gorging on their food. There was also a section for some entertainment. At one point, an older man had a circular crowd around him while he did tricks on a super tall tricycle. Later, there was a silent disco party with people wearing headphones dancing on a small dance floor.
I ate some dessert, which consisted of ice cream inside of a donut. Later I got some melted cheese on bread. The man scraped it from a cheese block that was hanging over a fire directly onto my food. It was delicious, but I couldn't finish it because I was so full. After walking around, I decided to ditch the crowd. On the way out, I noticed a water trough with water spouts for people to fill up water bottles or cups. I wish more places had free tap water available like that.
The next evening, I signed up for a hidden bar tour. It was with the same company that gave the walking tour I was on a few days prior. This time, our tour guide was the owner, Jon. He was from Minnesota (USA), appeared to be in his early 30s, and was tall. In 2009, he traveled for a year and ended up in England and Wales long-term. For work, he gave tours and really enjoyed it. His girlfriend was from Australia, so he had recently moved there. Jon started the tour company there two years prior, and they provide walking tours and bar tours.
There were 13 people in the group: four girls from Columbia, two couples from Melbourne, one couple from Sydney, and one couple from Perth. One of the pair from Melbourne was on the tour in hopes of finding wedding venue options, as well as to explore a little bit of their city. The other Melbourne couple was looking for opportunities for their son's 18th birthday. The couple from Perth lived in the countryside and were just on vacation.
The first bar that we went to was called Whiskey Den. It was tiny, and I sat at the bar, drinking my whiskey with a lager. Jon gave us some information about whiskey and the bar scene in Melbourne, and then gave us time to get to know each other.
I sat next to Tawny and Johnny from Sydney. They were on vacation and were really friendly. Tawny was 43 years old and came to Australia from Vietnam when she was three years old. She was tall with long, black hair and worked in finance. Johnny was 41 years old and came to Australia from Hong Kong when he was an older child, and he worked in IT. They met on Bumble and had been dating for ten months. They were so cute, always holding hands and looking smitten with each other. It gave me hope in online dating.
To get to the next bar, we walked through the graffiti-filled alleyways. The following bar was called the White Hart Bar. It was still daylight outside, and we sat outside in the covered patio. There was a food truck with a painted picture of President Trump with him saying, "Best Tacos north of the wall. Wow. Tremendous."
I talked with different people in the group while enjoying a beer. The couple who was looking for wedding venues said they must have live music and a bottle of whiskey on each table at their wedding reception. They expected 110-120 guests. Surprised by prices, they commented, "It's going to cost $20,000 to host dinner for our friends?"
The next bar was located underground, called Beneath Driver Lane. I ordered the Grand Finale, which had nitrogen in it, and was fun to watch the bartender make a show of it. The final bar was called Loch and Key. I had been there before (after the walking tour), so I ordered some fries for people to try because they're delicious. We sat outside on the balcony and chatted while enjoying our drinks.
The tour was over, but I was hungry for dinner, and so were Johnny and Tawnie. We walked to a ramen place, which turned out to be closed down, so we Ubered to one farther away. They helped me order the ramen because they are foodies who have quality pallets. The ramen hit the spot! The flavors were just what I needed.
The three of us talked about life in our countries. They have been to the U.S. (Texas, New York, and the West Coast). They said they had excellent experiences there and people were friendly. The conversation turned into health insurance woes and differences in our systems. It was a fun time and felt like I had known them for a long time. Even though I travel solo, I am thrilled when I get opportunities to hang out with new friends.
Thanks for reading! Hit the Like button or leave a comment below!
PrevPreviousWho Names a Swimming Pool After a Prime Minister Who Drowned?
NextSolo Travel, But Often Not AloneNext


text:
a box office force awakens Dec. 20, 2015
7 Key Numbers Behind The Force Awakens' Record-Setting Opening
By Kevin Lincoln
You've got to Han it to him.
After months of speculation and secrecy, it's finally official: Star Wars: The Force Awakens is the highest-opening domestic movie of all time. Exactly where that opening will land is still a little bit unclear — Disney is saying $238 million, while Deadline is suggesting it could be as high as $246 million — but regardless, J.J. Abrams's seventh entry in the Franchise That Lucas Built has obliterated Jurassic World's young record of $209 million for the most money made in the first weekend of a movie's release.
Plenty of noise will be made over the magnitude of this performance, so, before the hangover sets in and the internet falls to squabbling, I wanted to zero in on the seven most interesting numbers from The Force Awakens' record-setting weekend, most of which are drawn from Deadline's comprehensive coverage.
What to Make of Star Wars' Insane Ticket Sales
$84,617,303: The previous highest opening weekend for a movie released in December, courtesy of The Hobbit: An Unexpected Journey. Prior to this year, conventional wisdom dictated that the months to launch a $100 million–plus opener were between April and July, with November as an exception for YA adaptations. December was for long runs and good returns, a strategy that did help Avatar become the best-grossing film ever made. (For comparison's sake: Avatar opened to $77 million on December 18, 2009.) But The Force Awakens is the last stage in a process that's been happening for a decade now: You can release a blockbuster at any time if it's a known property. And since audiences these days rarely see unknown properties, you can release a blockbuster at any time. September and October are now the only months to have never produced a $60 million–plus opening weekend, but expect that to fall soon enough.
6: The number of movies that have grossed more than The Force Awakens has this year. Have I mentioned that it's been out for three days? I have, right?
47 percent: How much of The Force Awakens' revenue came from 3-D. Deadline writes, "When Avatar opened there were 3,100 RealD screens in the U.S./Canada; now there are 14,000 with the majority of them playing Force Awakens." Considering that Jurassic World made 65 percent of its money on its still-record $524 million international opening weekend from 3-D showings, including an insane 95 percent of revenue in China, 3-D has solidified its role as a major component of blockbuster filmmaking.
$77,000. The average per-screen gross from Imax locations, adding up to $30.1 million — $9.2 million more than Jurassic World's previous opening-weekend domestic record. It's the same story as 3-D: You don't release a tentpole anymore without a significant Imax presence.
54 percent: Portion of the total $517 million gross that came from international markets, with $279 million rolling in across 30,000 screens. That $517 million figure is just short of Jurassic World's record $524 million, but The Force Awakens becomes only the second movie ever to haul in more than $500 million worldwide in a single weekend. But this isn't quite a one-to-one comparison: Unlike Jurassic World, which brought in a healthy $99 million from China on its opening weekend, The Force Awakens has yet to open in that country. Take away that $99 million, and The Force Awakens beats Jurassic World by more than $60 million internationally. As The Wall Street Journal pointed out, though, it isn't quite that simple: Star Wars lacks much of a pull in China, and its earning power in the country is still sort of a mystery, even with Disney's all-out marketing blitz ahead of its Chinese-market release on January 9. Fortunately, Stephen Colbert is doing his part.
33 percent and 66 percent: The proportion of ticket buyers who were 18 to 24 years old (33 percent) and male (66 percent), according to the exit-polling service Postrak, cited by The Hollywood Reporter. The first number is noteworthy because it proves that The Force Awakens didn't just pull in every adult man yearning for the nostalgia of his youth: Kids who were just 5 years old when The Phantom Menace came out are just as stoked for Star Wars as their older counterparts. (Overall, 47 percent of the attendance came from moviegoers younger than 25.)
Meanwhile, that means there are a lot of women out there who have yet to buy a ticket. It would be insane to expect women to come out for The Force Awakens in the same numbers that men do, but the hugely positive response — highlighted by an A Cinemascore and 95 percent rating on Rotten Tomatoes — might help convince otherwise-unpersuaded women (and men) to patronize the biggest movie in American history.
3.1: The multiplier by which Jurassic World grew from its opening weekend through the end of its domestic run. Box Office Mojo is forecasting a 3–3.5 multiplier for The Force Awakens, which would result in a total of $714–833 million. But an even higher total isn't out of the question: We're in uncharted waters here. Avatar's domestic record is $749 million, and its foreign record is $2 billion. Those are the numbers Disney wants to hit. Is anything less a disappointment? Well, disappointment's relative. Money is not.
star wars countdown
a box office force awakens
Breaking Down The Force Awakens' Record Opening


The concept of left-brain and right-brain thinking originated in the late 1960s from the research of Roger W Sperry. Sperry developed what is known as the Split Brain Theory. His Split Brain Theory divides the brain into two hemispheres; right and left. His research found that each hemisphere is specialized for specific behaviors and controls different types of thinking.
Based on his research, Sperry found that the left side of the brain tends to control many aspects of language and logic, while the right side tends to control spatial information and visual comprehension.
Since Sperry's original research, many other studies have shown that the brain is not nearly as divided and separate as Sperry thought. Research has since shown that the two sides of the brain collaborate to perform a variety of tasks and that the two sides communicate constantly. Research has also shown that a person's abilities are strongest when both halves of the brain work together.
Although the research has shown that there is not the extreme differential as first thought, there is still value in understanding that people have certain preferences. These preferences are often based on a person's brain dominance. People are typically more comfortable with a specific way of thinking. One person may want to hear directions, while another may want to have them written or drawn on a piece of paper. So, even though the Split Brain Theory has slowly been debunked over the years, it is still true that people have preferences. Therefore, there is still value in understanding the underlying theory.
According to the theory of left-brain or right-brain dominance, each side of the brain controls different types of thinking. It is thought that people prefer one type of thinking over the other. For example, a person who is "left-brained" is thought to be more logical, analytical, and objective. A person who is "right-brained" is said to be more intuitive, creative, emotional, thoughtful, and subjective.
Below is a list of the attributes associated with each hemisphere of the brain.


Rose Cottage, Auchendennan in Arden, Alexandria, Dunbartonshire sleeps 4 people. Situated in the heart of Loch Lomond National Park, these stunning holiday properties (refs SRRG, SXXO [Chestnut Cottage], SXXP [Willow Cottage], SXXQ [Primrose Cottage], SXXR [Ivy Cottage], SVVE [Burnbeag], SVVD [Courtyard Studio]) have been lovingly restored and furnished to provide wonderful retreats. Auchendennan Farm Cottage (SLLX) is a single-storey holiday cottage; within the house there is a large family sauna which can be used as an ideal way to unwind. Outside on the sun decking, there is a sunken hot tub which is large enough for all the family and powerful enough to give tired muscles a proper massage. Rose Cottage (SRRG) is a single-storey conversion adjoining the owner's farmhouse. Combining modern and traditional furnishings, the property boasts a superb hot tub and external sun decked area for relaxing and unwinding.
For golfing enthusiasts, the famous courses of Loch Lomond are just a short distance from the property, with an abundance of courses for all standards within easy reach. Cameron House, the 5 star golf/spa resort is just minutes from the property where one can enjoy fine dining, day spa packages and championship golf. Lomond Shores, again a short distance from the property, offers designer shopping, loch cruises and an aquarium. The cities of Glasgow and Edinburgh are easily accessible by car boasting award winning restaurants, theatres, galleries and concert venues, designer shopping as well as many tourist attractions including the breathtaking Edinburgh Castle. Shops 2 miles, pub ¼ mile.
Living room with kitchen area. Double bedroom with 6ft bed and en-suite bathroom with over-bath shower and toilet. Twin bedroom. Shower room with toilet.
For a better idea of where Auchendennan Farmhouse - Rose Cottage is, you can see Auchendennan Farmhouse - Rose Cottage on the map below.
Auchendennan Farmhouse - Rose Cottage is in Dumbartonshire; should you be on the lookout for a special offer, you can find all the special offers in Dumbartonshire. It's always worth checking to see what is on offer at the moment.
Do you want to holiday near Auchendennan Farmhouse - Rose Cottage (SRRG), you can see other cottages within 25 miles of Auchendennan Farmhouse - Rose Cottage; have a look to find your perfect holiday location.