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Home › Air Force › U.S. Capital Flag Program
U.S. Capital Flag Program
By Staff Writer on January 5, 2021
The U.S. Capital flag program began in 1937 when a Member of Congress requested a flag that had flown over the U.S. Capitol. The program has since been extended to the general public, specifically to military servicemen and women. Flying a flag for a new service member is very popular with recruit families. I was privileged to produce a video (below) for my friend's son, who recently became a U.S. Sailor.
Over the years the focus of the program gradually expanded to encompass the commemoration of national holidays and various special events, as well as to honor the work of groups such as schools and civic organizations. Requests for Capitol flags rapidly outgrew the supply; hence, the Architect of the Capitol (AOC) instituted a program of flying smaller flags that may be purchased through members' offices.
(Video produced by Andrew, CEO of MondayDelivery.com & USMC Veteran)
The AOC fulfills all flag requests from members of the United States Senate and the House of Representatives. Flags are flown daily year-round, weather permitting, excluding Thanksgiving Day, Christmas Day and New Year's Day.
Currently, the AOC fulfills on average more than 100,000 flag requests from Members of Congress annually, with the number of requests and the popularity of the Capitol Flag Program growing steadily each year. There are special flag poles where all flags are flown. After it is flown over the U.S. Capitol, each flag is issued a keepsake Certificate of Authenticity by the AOC.
AOC employees in the U.S. Capitol flag office. Image courtesy of AOC website.
Monday Delivery is not affiliated with AOC and does not submit requests on behalf of family members. The video above was produced independent of AOC. To learn more or apply please see program details.
Don't forget to send your recruit a postcard today!
‹ A Postcard Prayer
Recruit Family Bingo ›
Categories: Air Force, Army, Coast Guard, Navy, USMC | {
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Domify extension for aero.js library (WIP)
| {
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Color Run/Walk 5K Makes a Splash
FARGO – On July 20, over 130 people gathered for the 5th Annual Crosby Color Splash and raised over $10,000 to help the Crosby Country Club replace their old and broken irrigation system. The Color Splash was a collaboration between St Luke's Medical Center in Crosby and the Blue Cross Blue Shield of North Dakota Caring Foundation. Jody Nelson, CEO of St Luke's Medical Center, expressed how much of a difference the Rural Provider Health Grant from the Blue Cross Blue Shield of North Dakota Caring Foundation made in planning the color splash, sharing that she is "so grateful for the opportunity to collaborate with Blue Cross Blue Shield of North Dakota."
One highlight from the run was the community support from many angles. Several community businesses volunteered to throw color, getting creative with leaf blowers and lift boxes. The 50th Annual Threshing Bee also played into the Color Splash, helping bring more participants to the 5K. Caring Foundation manager Amber Blomberg expressed, "The planning committee did a wonderful job and tying it into the Threshing Bee was a great way to further bring in the community."
Nelson expressed how much the city of Crosby looks forward to the color splash every year, so keep an eye out for news of the 6th annual 5K next summer!
About the Blue Cross Blue Shield of North Dakota (BCBSND) Caring Foundation
Established in 1989, the BCBSND Caring Foundation (www.BCBSND.com/Caring-Foundation) is a private 501(c)(3) charitable organization. Through giving and investing, the Caring Foundation focuses on key health issues that help improve the health and well-being of North Dakotans and their communities. Caring Foundation staff work with grantees to support nonprofit sustainability, collaborate on shared outcomes and create lasting change.
Andrea Dinneen
Director, Brand & Marketing Communications
BLUE CROSS BLUE SHIELD OF NORTH DAKOTA, BISMARCK
701-282-1071 (work) | 320-894-4423 (mobile)
andrea.dinneen@bcbsnd.com| www.BCBSND.com | {
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Q: Why reads C# console application 'empty' strings from stdin if launched in a remote PS session I'm writing a serial port to TCP bridge in C#. It's a console application with 4 threads.
*
*main thread for reading commands from console (e.g. 'exit' to close the application)
*if a connection is established, it handles a client
*a TCP worker thread, which copies all received TCP data to COM3
*a UART worker thread, which copies all received UART data to TCP
This applications works fine if it's launched locally on my PC either in Visual Studio debug modus, in CMD.exe or even in PowerShell.
Now I want to grant one of my colleges the right to launch UARTServer.exe in a remote PowerShell session. PS is setup in the correct way and he can use PS like a ssh session (Enter-PSSession -ComputerName xxxx -Credentials yyyy).
But when he starts UARTServer.exe in his remote session, Console.ReadLine() returns with null.
Here is the C# snipped for the main thread, which reads from stdin to look for commands like 'exit'.
Console.WriteLine("Starting UARTServer...");
uartServerThread = new Thread(new ThreadStart(ServerWorker));
uartServerThread.Start();
Console.WriteLine("Enter 'exit' to stop this server.");
Console.Write("UARTServer> ");
while(true)
{
var input = Console.ReadLine();
if (input == "exit")
{
break;
}
else if (input == null)
{
Console.WriteLine("Null");
break;
}
else
{
Console.WriteLine("Unknown command");
Console.Write("UARTServer> ");
}
}
Console.WriteLine("Shutting down UARTServer...");
uartServerThread.Abort(true);
uartServerThread.Join();
Console.WriteLine("Exiting UARTServer");
This is the corresponding console output:
[localhost]: PS D:\Shares\SATA> .\UARTServer.exe
Starting UARTServer...
Enter 'exit' to stop this server.
UARTServer> Null
Shutting down UARTServer...
Exiting UARTServer
[localhost]: PS D:\Shares\SATA>
My questions:
*
*Why does it happen? Is Stdin not connected in a remote session?
*How can I fix this?
A: Many streams behave quite differently when they are on top of a network connection, such that .Read will return nothing if there is nothing available in the buffer. I suspect that when remoting on the command line then your Console is actually on top of a network stream. Note in this example they are checking for a null result from ReadLine and looping until they have something to actually process:
https://stackoverflow.com/a/1450496/84206
while((line = Console.ReadLine()) != null) {
// do something with line
}
I would suggest this approach so that you can loop until a line is received from the client. There are more refined approaches that avoid looping indefinitely and consuming CPU just for a loop that is waiting for data. But this demonstrates how you have to adjust for stream reads that sometimes return nothing.
| {
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Джек Сти́венс (; родился 27 января 1994) — английский футболист, защитник клуба «Саутгемптон».
В данный момент выступает на правах аренды за футбольный клуб Борнмут.
Клубная карьера
В 2005 году 11-летний Джек Стивенс начал выступать за молодёжную академию клуба «Плимут Аргайл». В 2009 году сыграл в Молочном кубке. В сентябре 2010 года начал тренироваться с основным составом клуба. 18 сентября 2010 года дебютировал в основном составе «Плимута» в матче против «Шеффилд Уэнсдей». Начал карьеру на позиции правого крайнего защитника, но также может сыграть и в центре обороны.
5 апреля 2011 года перешёл в «Саутгемптон» за 150 тысяч фунтов, подписав с клубом трёхлетний контракт. 7 января 2012 года Джек дебютировал за «святых» в матче Кубка Англии против «Ковентри Сити». Перед началом сезона 2012/13, в котором «Саутгемптон» вернулся в Премьер-лигу, Стивенс стал одним из четырёх игроков молодёжной академии клуба, переведённых в основной состав команды (наряду с Люком Шоу, Джеймсом Уорд-Проузом и Калумом Чеймберсом).
Дебют Стивенса в Премьер-лиге состоялся 2 января 2017 года в матче против «Эвертона», в котором он на 6-й минуте заменил получившего травму Седрика Суареша.
13 марта 2014 года Стивенс отправился в аренду в «Суиндон Таун» до 3 мая. 1 сентября того же года «Саутгемптон» вновь отправил Стивенса в аренду в Суиндон Таун до января 2015 года.
31 июля 2015 года Стивенс продлил свой контракт с «Саутгемптоном» до лета 2019 года, а затем отправился в сезонную аренду в клуб Чемпионшипа «Мидлсбро». 4 января 2016 года главный тренер «Саутгемптона» Рональд Куман отозвал Стивенса из аренды в связи с тем, что футболист получал недостаточно игровой практики. Куман заявил: «Он [почти] не играл там, поэтому зачем ему быть в аренде, если он не играет? Пусть лучше он вернётся».
1 февраля 2016 года Стивенс перешёл в клуб «Ковентри Сити» на правах аренды до окончания сезона. За время аренды в «Ковентри Сити» Стивенс сыграл 16 матчей.
Карьера в сборной
Джек Стивенс выступал за юношеские и молодёжные сборные Англии разных возрастов (до 18, 19, 20 и 21 года).
Статистика выступлений
Примечания
Ссылки
Профиль футболиста на официальном сайте «Саутгемптона»
Профиль футболиста на официальном сайте английской Премьер-лиги
Футболисты Англии
Игроки сборной Англии по футболу (до 21 года)
Игроки ФК «Плимут Аргайл»
Игроки ФК «Саутгемптон»
Игроки ФК «Суиндон Таун»
Игроки ФК «Мидлсбро»
Игроки ФК «Ковентри Сити» | {
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Q: Angular *ngif with Observable and data from async calls I'm using a few REST services to gather data and then change the value of an Observable in my component code. I'm trying to use an *ngIf to show/hide div tags based on the result. However, it isn't working, the *ngIf is not updating based on the change to the Observable.
Do I have to trigger change detection? What's the best way to achieve this in Angular 7? Here's one example I tried:
HTML:
<div *ngIf="isAnimal$">
<p>animal</p>
</div>
Component:
public isAnimal$: Observable<boolean> = of(false);
...
ngOnInit() {
checkAnimal();
}
private checkAnimal() {
this._dataService.getData()
.subscribe((result) => {
if (result === 'animal') {
this.isAnimal$ = of(true);
}
});
}
I also tried creating a simple service and that didn't work either. Like this: Angular *ngIf binding to function
A: I suggest the following:
HTML:
<div *ngIf="isAnimal">
<p>animal</p>
</div>
Component:
public isAnimal: boolean = false;
ngOnInit() {
checkAnimal();
}
private checkAnimal() {
this._dataService.getData().subscribe(
(result) => {
if (result === 'animal') {
this.isAnimal = true;
}
},
(error) => {
console.log(error);
}
);
}
I am not sure, but I think you could use the async pipe instead using your implementation:
<div *ngIf="isAnimal$ | async">
<p>animal</p>
</div>
The reason your solution is not working is because isAnimal$ is an observable that has to be resolved. The async pipe could do this, but I prefer the first solution to be honest. It does not seem natural to me to create another observable of(true) in your subscription.
| {
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{"url":"https:\/\/discuss.pytorch.org\/t\/custom-dataloader-structure\/29159","text":"I have a dataset which has images and corresponding texts.\n\nI want to cut part of a dataset to get input data in a shape of 4 sequence images with corresponding texts.\nlike data = [ [image1, image2, image3, image4] ,[text1, ...text4] ]\nSo, if I have 6 images, I can make 3 inputs data.\n\nmy dataset loader is like below\n\nclass dataset(Dataset):\ndef __init__(self, path, start, end):\nself.path = path\nself.data = self.getsubsequnce(start, end)\n\ndef _getMegabatch(self, start, end):\nfile = h5.File(self.path)[start:end]\nreturn file\n\ndef _getsubsequence(self,start,end):\nmegabatch = self._getMegabatch(start, end)\n\"\"\"\nreturns set of 4 length images\n\"\"\"\n\ndef __len__(self):\nreturn len(self.data)\n\ndef __getitem__(self, index):\ndata = self.data\nreturn data[index]\n\n\nI\u2019m wondering this is a common way of generating mini-batches from mega-batch data.\nCould you give me an advice??\n\nI\u2019m not sure I understand the implementation of your Dataset fully.\nAre you creating a new Dataset for each sequence, i.e. when are you passing new start and end values?\nIt looks like you are returning one sample from the sequence. As far as I\u2019ve understood your use case, I thought you would like to return 4 samples holding the images and text data?\n\nCould you post the shapes of your whole dataset and each sample?\n\n@ptrblck Okay I think above code is too simplified.\n\noverall, from my whole dataset, _getMegabatch will get Mega-batch dataset[start:end]\nthen from the megabatch I want to return minibatches.\n\nSpecifically, my whole dataset is a list [images, texts], images.shape = [20000, 224, 224, 3], texts.shape=[20000, 30]\n\nand _getMegabatch() slices whole data to mega-batch data [images, texts] with images.shape = [6, 224, 224, 3] texts.shape=[6,30] mega-batch size 6 is just an example.\n\nFor easy understanding let\u2019s represent this mega-batch data images (shape [6, 224, 224, 3] and texts as [a, b, c, d, e, f], [1,2,3,4,5, 6], each alphabet, number is image and text.\n\nFrom this mega-batch data I am gonna get all 4-length sub-sequence datas with _getsubsequence like self.data =[ [ [a, b, c, d], [b, c, d, e], [c, d, e, f] ], [ [1, 2, 3, 4], [2, 3, 4, 5], [3, 4, 5, 6] ] ]\n\nFinally, __getitem__ will return self.data[0][index], self.data[1][index] e.g. [a, b, c, d], [1, 2, 3, 4]\n\nI`m not sure this mega-batch \\in mini-batch dataloader structure is efficient or not.","date":"2022-05-19 02:21:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2880043089389801, \"perplexity\": 6371.582917330588}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662522741.25\/warc\/CC-MAIN-20220519010618-20220519040618-00063.warc.gz\"}"} | null | null |
module Nimbu #:nodoc
# Raised when Nimbu returns the HTTP status code 403
module Error
class Forbidden < ServiceError
http_status_code 403
def initialize(response)
super(response)
end
end
end # Error
end # Nimbu
| {
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Omnium Gatherum Announce North American Tour 2022 – News
The Finnish melodic death metal band will be hitting the road for their headline tour with support from Allegaeon and labelmates Black Crown Initiate.
Swallow The Sun Announces 2021 US Tour – News
Finnish death doom masters, Swallow The Sun, have announced their Moonflowers Tour 2021, which will include support slots from Abigail Williams and Wilderun.
Ignite releases Anti-Complicity Anthem EP + music video – News
Ignite release their new digital EP Anti-Complicity Anthem. The EP includes the title track as well as B-side, "Turn XXI," that is only available on this EP.
Swallow The Sun's 20 Years Of Gloom, Beauty and Despair – Live in Helsinki – News
Swallow The Sun announce their first ever live album, 20 Years of Gloom, Beauty And Despair – Live in Helsinki, set for release July 30th via Century Media.
Mayhem Release "Voces Ab Alta" Video – News
Mayhem release "Voces Ab Alta" and its video, a preview of music on the forthcoming Atavistic Black Disorder/Kommando EP, out July 9 on Century Media.
At The Gates Announces New Album, The Nightmare of Being – News
Swedish melodic death metal pioneers At The Gates announces their new full-length album, The Nightmare of Being. The album is set for release on July 2nd.
Samael: Vinyl Editions of Rebellion and Eternal – News
Transcending Records is pleased to issue limited edition vinyl pressings of Rebellion and Eternal from Swiss blackened experimental metallers, Samael.
EyeHateGod Debut "Built Beneath The Lies" – News
Eyehategod, whose new album, A History of Nomadic Behavior, arrives on March 12, share a third song from the forthcoming album, "Built Beneath the Lies."
Tribulation launches "Leviathans" video – News
On January 29th, Tribulation will release their new album, Where the Gloom Becomes Sound. For a first preview, see the video for the new single, "Leviathans."
Voivod – The End Of Dormancy – Music Stream
After jaw-dropping work by Dog Fashion Disco/Polkadot Cadaver, Jim Thirlwells' Foetus projects, and Tod A.'s Cop Shoot Cop/Firewater, this is rudimentary. | {
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Titta vi spökar () är en amerikansk skräck-/komedifilm från 1986 i regi av Steve Miner, med William Katt, George Wendt, Richard Moll och Kay Lenz i rollerna. Filmen fick tre uppföljare: Titta vi spökar 2 (1987), Titta vi spökar 3 (1989) och Titta vi spökar 4 (1992).
Handling
Filmen handlar om författaren Roger Cobb (William Katt) som efter sin son Jimmys försvinnande flyttar in i hans faster gamla hus, för att få vara ensam och skriva på sin nya bok, som ska handla om Rogers upplevelser i Vietnam. Efter ett tag börjar det hända underliga saker och Roger inser snart att det finns riktiga monster i huset.
Rollista
Externa länkar
Officiell webbsida
Filmer 1986
Amerikanska skräckfilmer
Amerikanska komedifilmer
Amerikanska spökfilmer
Engelskspråkiga filmer | {
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Club de Foot de Montreal
Joey Saputo
Kevin Gilmore
Union battle back from early deficit to grab draw in Montreal
By Matt Zampini May 14, 2016 7:00 pm
By Matt Zampini | May 14, 2016 7:00 pm
Photo by Jean-Yves Ahern/USA Today Sports
After playing a mid-week game on Wednesday, it could have been expected that the Philadelphia Union would come out a bit sluggish on Saturday afternoon away against the Montreal Impact, a match that featured the top-two teams in the Eastern Conference.
The Impact scored just three minutes in and dominated the first 10 minutes of the contest, however, the Union were able to equalize midway through the first half. The match finished with nothing between the two sides at 1-1.
The result is the fourth consecutive tie for the Impact and third tie in a row for the Union.
In the third minute, Didier Drogba scored his fourth goal of the season and third in as many games to give the Impact the early advantage.
Drogba found himself unmarked off a corner kick and C.J. Sapong's failed clearance of the initial corner fell to Drogba, who then fired a half volley into the top of the net.
Sapong would soon redeem himself, though, in the 24th minute. Impact defender Victor Cabrera was caught ball-watching inside the six-yard box, leaving Sapong wide open on the door step. Sebastien Le Toux played a cross back across the center of goal to Sapong, who calmly finished into the back of the net to tie the game.
Drogba would leave the game in the 70th minute with what seemed to be right-hip injury, although there hasn't any confirmation as to what hampered the striker.
The result was impressive for the Union, who got down early and handled a lot of pressure in the beginning of the game from the Impact. Defenseman Keegan Rosenberry played a big role for the Union, playing superb one-0n-one defense on the flanks.
The Union will play next next on Friday, May 20, against D.C. United. The Impact will return to the pitch next Saturday, May 21, when they visit Orlando City.
Union goalkeeper Andre Blake made a few key saves on Saturday night to keep the Impact at just one goal. Blake made an acrobatic save off a header and a free kick from Didier Drogba. Blake also did a great job of commanding the 18-yard box all game, grabbing any ball in the area. He finished with five saves.
Didier Drogba's goal in the third minute was a top-class finish from the 38-year-old Ivorian. Drogba also had many other chances on Saturday, forcing Union goalkeeper Andre Blake to make acrobatic saves.
Defender Laurent Ciman had a few mishaps in the back for the Impact on Saturday that led to chances for the Union. Luckily for Ciman, the Union weren't able to capitalize on them and the game ended in a draw.
MLS, Montreal Impact, Philadelphia Union, Major League Soccer, MLS- Montreal Impact, MLS- Philadelphia Union | {
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{"url":"https:\/\/chemistry.stackexchange.com\/questions\/19921\/ideal-gas-law-derivation-from-kinetic-theory","text":"# Ideal gas law derivation from kinetic theory\n\nConsider the derivation of the ideal gas law from kinetic theory presented here: http:\/\/en.wikipedia.org\/wiki\/Kinetic_theory\n\nI have some questions\n\n1. This derivation assumes the container is a cube, but the law holds for containers of any shape, right?\n\n2. This derivation assumes each particle has the same mass, but the law holds for a mixture of gasses of different masses, like air, right?\n\n3. The derivation assumes that the particle intercepts a give side with frequency $2L\/v_x$. Why can we make this assumption?\n\nThis derivation assumes the container is a cube, but the law holds for containers of any shape, right?\n\nThe resulting law holds for any shape of container. However, the details of the derivation depend on the geometry. A cube is easy, so we use it.\n\nThis derivation assumes each particle has the same mass, but the law holds for a mixture of gasses of different masses, like air, right?\n\nYes, it holds for a mixture of any ideal gases - the key is that only the kinetic energy of the gas molecules matters. Heavier molecules would be moving slower at the same temperature, but they would exert the same force (and therefore pressure) on the walls.\n\nThe derivation assumes that the particle intercepts a give side with frequency 2L\/vx. Why can we make this assumption?\n\nThis assumption works for only one particular case. If you notice the third sentence at the beginning of the pressure derivation (emphasis is mine):\n\nWhen a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed\n\nThey are looking at the special case of a collision at a 90 degree angle, which bounces off the opposite wall. In that case, it will take the particle t = L\/V time to reach the opposite wall, and t = L\/V time to bounce back - so the total is 2L\/V. This illustration may help explain it better:\n\nNow this might sound like an unreasonably strict condition, but the reason that it works is that the component velocity distribution for each particle is symmetrical (it's a normal distribution), which means that for every particle that impacts the wall at 95 degrees (for example) there is another that impacts 85 degrees. As a result, the average angle of impact is 90 degrees for each wall, and therefore the average collision time is given by looking at 90 degree impacts for each component of the velocity vector.\n\n\u2022 1.So how do we prove the law holds for any container? 2.If the law holds for a mixture of gasses of different masses (still acting as an ideal gas), how do we prove the law in this case? 3.I believe you are misunderstanding that quote from the article. It should read like: \"When a gas molecule collides with [a wall of the container that perpendicular to the x-axis]\". 4. How do you prove that the velocity component distribution for each particle is the normal distribution? \u2013\u00a0Joshua Benabou Nov 24 '14 at 0:20\n\u2022 Also can someone link me to a rigourous proof of the ideal gas law using kinetic theory? \u2013\u00a0Joshua Benabou Nov 24 '14 at 0:26\n\u2022 @JoshuaBenabou If you only want a proof that the ideal gas law holds for any container size it might be easier to simply show general applicability via thermodynamics. But if you want a general proof from kinetic theory I would imagine that this might not be easy and possibly quite lengthy. \u2013\u00a0Philipp Nov 24 '14 at 1:30\n\u2022 @JoshuaBenabou - 1. There are lots of ways, one way is to show that it holds empirically for many container shapes, then derive it from first principles for an arbitrary container (a cube in this case). I am not sure if there is way to rigorously show that it holds for all container geometries. 2. You would replace the kinetic energy terms with summations over the different masses. 3. Possibly. But, who is to say the x-axis isn't vertical? 4. You don't, that's empirically known. At any rate, it doesn't have to be normal, just symmetrical about a mean. \u2013\u00a0thomij Nov 24 '14 at 2:54\n\u2022 There is a nice general proof using statistical mechanics here: en.wikipedia.org\/wiki\/Ideal_gas_law. \u2013\u00a0Joshua Benabou Nov 24 '14 at 3:42","date":"2019-06-19 23:59:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7619315981864929, \"perplexity\": 347.9904682549076}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627999066.12\/warc\/CC-MAIN-20190619224436-20190620010436-00201.warc.gz\"}"} | null | null |
\subsubsection*{Notation}
To effectively present the derived expressions, we provide a guide to the notation that will be used in the following sections.
Let a transmission over a single link be an ``experiment." A binomial distribution with $n$ independent experiments, probability of success $1-p$, and number of success $m$ will be referred to as
\begin{equation}
B(n,m,p) = \binom{n}{m} (1-p)^{m}p^{n-m}.
\label{eq:B}
\end{equation}
Note that the probability $p$ is the probability of failure, not the probability of success.
The probability of at least one out of $n$ independent experiments failing will be denoted as
\begin{equation}
F(n,p) = 1 - (1 - p)^{n}.
\label{eq:F}
\end{equation}
A link with fading coefficient $h$ and bandwidth $W$ is considered ``good" (thus decodable) if the rate of transmission $R_i$ is less than or equal to the link's capacity, $C = W \log(1 + |h|^2\text{SNR})$. We assume that the nominal operating SNR is held consistent across the entire system. Consequently, for a rate $R$, the assumption of Rayleigh fading tells us that the probability of an unsuccessful transmission is defined as
\begin{equation}
p = P(R > C) = 1 - \exp{\left(-\frac{2^{R/W} -1}{\text{SNR}}\right)}.
\label{eq:pfail_singlelink}
\end{equation}
We assume that if $R$ exceeds capacity, the transmission will surely fail (with probability 1). If $R$ is less than capacity, the transmission will surely succeed and decode to the right codeword.
\subsubsection*{Set Notation}
We describe the various sets used in the analysis. Following general convention, the set itself will be represented in script font. The random variable representing the number of nodes in that set will be presented in uppercase letters. Finally, the instantiation of that random variable (the cardinality of the set), will be in lowercase letters. The sets being considered are:
\begin{itemize}
\item $\mathcal{A}$: the set of nodes successful in the downlink phase. Further divided into disjoint sets $\widetilde{\mathcal{A}}$ and $\widecheck{\mathcal{A}}$ such that $\mathcal{A} = \widetilde{\mathcal{A}} \bigcup \widecheck{\mathcal{A}}$.
\begin{itemize}
\item $\widetilde{\mathcal{A}}$: the set of nodes which succeed in downlink as well as uplink phases. This is further partitioned into $\widetilde{\mathcal{A}}_X$ (the set which connects to the controller in the XOR phase) and $\widetilde{\mathcal{A}}_U$ (the set which cannot connect to the controller in the XOR phase).
\item $\widecheck{\mathcal{A}}$: the set of nodes which do not succeed in uplink. This set is further partitioned into $\widecheck{\mathcal{A}}_X$ (which can connect to the controller in the XOR phase) and $\widecheck{\mathcal{A}}_D$ (which cannot connect to the controller in the XOR phase).
\end{itemize}
\item $\mathcal{B}$: the set of nodes that weren't successful in downlink phase but were successful in uplink phase. Further partitioned into disjoint sets $\widetilde{\mathcal{B}}$ (has link to the controller in the XOR phase) and $\widecheck{\mathcal{B}}$ (doesn't have link to controller in the XOR phase) such that $\mathcal{B} = \widetilde{\mathcal{B}} \bigcup \widecheck{\mathcal{B}}$.
\item $\mathcal{C}$: the set of nodes that succeed only in the XOR phase -- both uplink and downlink successes happen in this phase. They can only succeed through relays.
\end{itemize}
\subsection*{Analysis of XOR-CoW:}
Let the time allocated for the downlink phase be $T_D$, the uplink phase be $T_U$ and the XOR phase be $T_X$ such that $T_D + T_U + T_X = T$ where $T$ is the given cycle time. If we chose to do fixed scheduling then the transmission rates for downlink, uplink and XOR phases are fixed at $R_D = \frac{m \cdot n}{T_D}$, $R_U = \frac{m \cdot n}{T_U}$ and $R_X = \frac{m \cdot n}{T_X}$ respectively. If adaptive scheduling scheme is employed, then the transmission rates for downlink, uplink and XOR phases are given by $R_D = \frac{m \cdot n}{T_D}$, $R_U = \frac{(m + 1) \cdot n}{T_U}$ and $R_X = \frac{m \cdot (n-\widetilde{a})}{T_X}$ where $\widetilde{a}$ is the number of nodes that succeeded in both uplink and downlink phases. These $\tilde{\mathcal{A}}$ are called ``strong nodes'' and all the others need help. Without loss of generality we consider the flexible schedule scheme and proceed with the analysis. Depending on the time allocations for different phases and the number of strong nodes $\widetilde{a}$, we get the following theorem.
\begin{theorem}
Let the time allocated for downlink, uplink and XOR phases be $T_D$, $T_U$ and $T_X$ respectively, the number of non-controller nodes be $n$, and message size be $m$ bits.
The downlink and uplink transmission rates are given by $R_{D} = \frac{m \cdot n}{T_{D}}$ and $R_U = \frac{(m+1) \cdot n}{T_U}$ respectively. The corresponding probability of a single link failure, $p_{D}$ \& $p_{U}$, is given by Eq.~\eqref{eq:pfail_singlelink}. The XOR phase transmission rate is given by $R_X^{\widetilde{a}} = \frac{m\cdot (n-\widetilde{a})}{T_{X}}$ where $\widetilde{a}$ is the number of ``strong nodes'' in both downlink and uplink phases and the corresponding probability of a single failure $p_{X}$, is given by Eq.~\eqref{eq:pfail_singlelink}. The probability XOR-CoW{} failure is then
\begin{eqnarray*}
\begin{aligned}
P(\text{fail}) &= \sum_{a = 0}^{n} \Bigg[ \sum_{b = 0}^{n-a} P\left(\text{fail}_1\right) \mathds{1}\left(R_{D} \geq R_{U} > R_{X}\right) + \sum_{b = 0}^{n-a} \sum_{\widetilde{b} = 0}^{b} P\left(\text{fail}_2\right) \mathds{1}\left(R_{D} > R_{X} \geq R_{U}\right)\\
&\hspace{30pt} +\sum_{\widetilde{a} = 0}^{a} P\left(\text{fail}_3\right) \mathds{1}\left(R_{U} \geq R_{D} > R_{X}\right) + \sum_{\widetilde{a} = 0}^{a} \sum_{\widecheck{a}_X = 0}^{a - \widetilde{a}} P\left(\text{fail}_4\right) \mathds{1}\left(R_{U} > R_{X} \geq R_{D}\right)\\
&\hspace{30pt} +\sum_{\widetilde{a} = 0}^{a} \sum_{\widetilde{a}_X = 0}^{\widetilde{a}} P\left(\text{fail}_5\right) \mathds{1}\left(R_{X} \geq R_{U} > R_{D}\right) + \sum_{\widetilde{a}_X = 0}^{a} \sum_{b = 0}^{n - a} P\left(\text{fail}_6\right) \mathds{1}\left(R_{X} > R_{D} \geq R_{U}\right) \Bigg]
\label{eq:XOR}
\end{aligned}
\end{eqnarray*}
where,
$P\left(\text{fail}_1\right) = B(n, a, p_D) \times B(n - a, b, q_{UD}) \times F(n - a - b, p_U^a)$\\ is the probability of failure if the relationship between the rates is $R_{D} \geq R_{U} > R_{X}$,
$$P\left(\text{fail}_2\right) = B(n, a, p_D) \times B(n - a, b, q_{UD}) \times B(b, \widetilde{b}, r_{UX,UD}) \times F(n - a - \widetilde{b}, p_X^a)$$ is the probability of failure if the relationship between the rates is $R_{D} > R_{X} \geq R_{U}$,
$$P\left(\text{fail}_3\right) = B(n, a, p_D) \times B(a, \widetilde{a}, s_{UD}) \times F(n - a, p_U^a)$$ is the probability of failure if the relationship between the rates is $R_{U} \geq R_{D} > R_{X}$,
$$P\left(\text{fail}_4\right) = B(n, a, p_D) \times B(a, \widetilde{a}, s_{UD}) \times B(\widecheck{a}, \widecheck{a}_X, r_{DX,DU}) \times F(n - \widetilde{a} - \widecheck{a}_X, p_U^{\widetilde{a} + \widecheck{a}_X})$$ is the probability of failure if the relationship between the rates is $R_{U} > R_{X} \geq R_{D}$,
$$P\left(\text{fail}_5\right) = B(n, a, p_D) \times B(a, \widetilde{a}, s_{UD}) \times B(\widetilde{a}, \widetilde{a}_X, s_{XU}) \times (1 - P(\text{success}_5))$$
$$P(\text{success}_5) = (1 - p_X^{\widetilde{a}_X})^{\widecheck{a}} \times \left(\sum_{k = 1}^{\widetilde{a}_X} B(\widetilde{a}_X, k, p_U) \left( 1 - s_{XU}^k + s_{XU}^k(1 - p_X^{\widetilde{a}_U}) \right) \right)^{n - a}$$ are the probabilities of failure and success if the relationship between the rates is $R_{X} \geq R_{U} > R_{D}$,
$P\left(\text{fail}_6\right) = B(n, a, p_D) \times B(\widetilde{a}, \widetilde{a}_X, s_{XD}) \times B(n - a, b, q_{UD}) \times (1 - P(\text{success}_6))$
$$P(\text{success}_6) = (1 - p_X^a)^b \times \left(\sum_{k = 1}^{\widetilde{a}_X} B(\widetilde{a}_X, k, p_U) \left( 1 - s_{XU}^k + s_{XU}^k(1 - p_X^{\widetilde{a}_U}) \right) \right)^{n - a - b}$$
are the probabilities of failure and success if the relationship between the rates is $R_{X} > R_{D} \geq R_{U}$, where:
\begin{itemize}
\item $q_{UD} = P(C < R_U | C < R_D) = \frac{p_U}{p_D}$
\item $s_{UD} = P(C < R_U | C > R_D) = \frac{p_U - p_D}{1 - p_D}$
\item $s_{XU} = P(C < R_X | C > R_U) = \frac{p_X - p_U}{1 - p_U}$
\item $s_{XD} = P(C < R_X | C > R_D) = \frac{p_X - p_D}{1 - p_D}$
\item $r_{UX,UD} = P(R_U < C < R_X | R_U < C < R_D) = \frac{p_X - p_U}{p_D - p_U}$
\item $r_{DX,DU} = P(R_D < C < R_X | R_D < C < R_U) = \frac{p_X - p_D}{p_U - p_D}$
\end{itemize}
\end{theorem}
\begin{IEEEproof}
All potential relays get the schedules in the scheduling phase where the rate of transmission is the same as downlink rate as stated earlier in Sec.~\ref{sec:protocol}. This ensures that all potential relays (those that have the downlink information) know when to transmit. Additionally, all nodes that need help also know which packet is intended for them as their identity is built into the packet. We look at each case to understand the subtle effects that may arise.
\begin{figure}
\begin{center}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case1.jpeg}
\vspace{-10pt}
\caption{Case 1: $R_{D} \geq R_{U} > R_{X}$.}
\label{fig:case1}
\end{center}
\end{subfigure}
~\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case3.jpeg}
\vspace{-10pt}
\caption{Case 2: $R_{U} \geq R_{D} > R_{X}$.}
\label{fig:case3}
\end{center}
\end{subfigure}
\vspace{-30pt}
\caption{Different ways to succeed in XOR-CoW{} protocol. The links between the controller and nodes are annotated with the rates in which they are present. The links to $\mathcal{C}$ are only denoted for the rate at which the links are important.}
\vspace{-50pt}
\end{center}
\end{figure}
\input{case1}
\input{case3}
\begin{figure}
\begin{center}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case2.jpeg}
\vspace{-10pt}
\caption{Case 3: $R_{D} > R_{X} \geq R_{U}$.}
\label{fig:case2}
\end{center}
\end{subfigure}
~\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case4.jpeg}
\vspace{-10pt}
\caption{Case 4: $R_{U} > R_{X} \geq R_{D}$.}
\label{fig:case4}
\end{center}
\end{subfigure}
\vspace{-30pt}
\caption{Different ways to succeed in XOR-CoW{} protocol. The links between the controller and nodes are annotated with the rates in which they are present. The links to $\mathcal{C}$ are only denoted for the rate at which the links are important.}
\vspace{-50pt}
\end{center}
\end{figure}
\input{case2}
\input{case4}
\begin{figure}
\begin{center}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case5.jpeg}
\vspace{-10pt}
\caption{Case 5: $R_{X} \geq R_{U} > R_{D}$.}
\label{fig:case5}
\end{center}
\end{subfigure}
~\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[width = 0.75\textwidth]{case6.jpeg}
\vspace{-10pt}
\caption{Case 6: $R_{X} > R_{D} \geq R_{U}$.}
\label{fig:case6}
\end{center}
\end{subfigure}
\vspace{-30pt}
\caption{Different ways to succeed in XOR-CoW{} protocol. The links between the controller and nodes are annotated with the rates in which they are present. The links to $\mathcal{C}$ are only denoted for the rate at which the links are important.}
\vspace{-50pt}
\end{center}
\end{figure}
\input{case5}
\input{case6}
\end{IEEEproof}
\subsection{XOR-CoW{} Protocol Scheme for Arbitrary Topology}
The XOR-CoW{} scheme for a generic information topology can be summarized as follows. All nodes know the information topology -- the origin and destinations of the messages. Therefore, all nodes know which messages can be XORed. The schedule of messages $\mathcal{G}$ are determined and all nodes know the schedule. For the first phase, the schedule is simple: each message stream is allocated one slot. However, in the second phase (XOR phase), the schedule $\mathcal{G}_X$ is different: whenever bi-directional traffic exists in the information topology, allocate one slot for those two messages in $\mathcal{G}_X$, else allocate one slot for that single message in $\mathcal{G}_X$ (as shown in Fig.~\ref{fig:schedule}).
In the first phase, nodes take turn according to the schedule to transmit the messages. All nodes listen when they are not transmitting. In the XOR phase, all nodes that can transmit a message (or an XORed message) transmit according to the XOR phase schedule \emph{simultaneously} using a DSTC.
In the following section, we focus on the star topology as network coding yields maximum benefits when the traffic is bi-directional~\cite{fragouli2006network, li2004network}.
\section{Introduction}
\label{sec:intro}
\input{intro}
\vspace{-10pt}
\section{Background}
\label{sec:related}
\input{related}
\vspace{-10pt}
\section{Protocol Framework}
\label{sec:protocol}
\input{protocol}
\section{XOR-CoW{} for Generic Information Topology}
\label{sec:generic-xor}
\input{generic}
\section{XOR-CoW{} for Bi-directional Information Topology}
\label{sec:star-xor}
\input{star}
\vspace{-10pt}
\section{Analysis of XOR-CoW{}}
\label{sec:results}
\input{results}
\vspace{-10pt}
\section{Conclusions \& Future Work}
In this work, we designed a network coding based wireless communication protocol framework for high-performance control-like systems.
We have additionally shown that simple phase length allocations are sufficient and optimizations only provide marginal benefits.
In the future, we aim to address the impact of modeling assumptions such as spatial independence, quasi-static behavior, etc., on cooperative communication protocols.
Understanding the impact of imperfect sychronization as well as imperfect channel estimation would also be important in making these schemes practical.
\vspace{-10pt}
\subsection{Resource assumptions}
\label{subsubsec:assumptions}
We make a few assumptions about the network, channel characteristics and hardware to abstract away some of the details and to support the exposition. These assumptions hold for all schemes discussed in this paper.
The following assumptions are the same as the ones made in \cite{swamy2017real} for ``Occupy CoW'' protocol.
\begin{itemize}
\item We assume a local domain -- that while normally, the nodes are within range of each other, bad fading events can cause transmissions to fail.
Errors are caused only by bad fades.
\item All nodes know the information topology. They share a universal addressing scheme and order. Messages are of the same size and they contain their destination addresses.
\item Channels are assumed to be reciprocal. All nodes are half-duplex, but can switch instantly between transmit mode and receive mode.
\item Channels are assumed to be quasi-static and remain the same during a cycle.
\item Channel sounding to aid channel estimation is assumed to take a constant fraction of the cycle time $T$. All nodes are assumed to estimate channels that are being sounded. When multiple nodes simultaneously broadcast a message during the relaying phase, they would not need to spend time again sounding each channel and can do a short combined sounding as the intended receivers only need to identify which of the nodes that it can hear are transmitting.
\item Clocks on each of the nodes are perfectly synchronized in both time and frequency. One could achieve adequate synchronization with low overhead by adapting techniques such as \cite{5439770}. Thus we can schedule time slots for specific nodes without significant overhead.
\item The protocol relies on time/frequency synchronization to achieve simultaneous retransmission of messages by multiple relays. We assume that if $k$ relays simultaneously (with consciously introduced timing jitter\footnote{To transform spatial diversity into frequency-diversity \cite{5506252}.}) transmit the exact same information, then all receivers can realize signal diversity $k$.
\end{itemize}
\subsection{Overview of Occupy CoW Protocol Framework}
\label{subsec:occupy}
We briefly summarize the Occupy CoW protocol which would be the benchmark protocol for comparison purposes. For a detailed description, refer to \cite{swamy2017real}.
The XOR-CoW{} protocol shares the same network setup and aims to meet the same requirements as Occupy CoW.
In the Occupy CoW protocol, the source of different message streams transmit the message in a round-robin fashion.
After all messages have been transmitted once, each message is then re-transmitted \emph{simultaneously} by all the nodes which have the message (either the source or nodes that decoded the first transmission) using some appropriate distributed space-time code (DSTC) again in a round-robin fashion.
Consider how this would play out for a star information topology as shown in Fig.~\ref{fig:star}. There is a downlink and uplink phase (corresponding to the first time the messages are transmitted) of length $T_{D_1}$ and $T_{U_1}$ respectively. This is followed by a scheduling phase where all ``strong'' nodes get to know the state of each message (whether it has reached the intended destination or not). This is followed by the relaying phases -- first the downlink phases \upperRomannumeral{2} (length $T_{D_2}$) and \upperRomannumeral{3} (length $T_{D_3}$), where the controller and strong nodes alter the broadcast message to remove already-successful messages for the strong nodes and simultaneously broadcast the adapted packet. The unsuccessful nodes are listening.
At the end of this phase, the nodes who received their messages from the controller have also received the global ACK information. which allows these nodes to participate as relays in the uplink phases.
The uplink phases \upperRomannumeral{2} (length $T_{U_2}$) and \upperRomannumeral{3} (length $T_{U_3}$) are similar to their downlink counterparts. The protocol can either have two hops -- such that there are only two downlink and uplink phases or three hops -- where there are a total of three downlink phases and three uplink phases (as described above).
\subsection{Overview of XOR-CoW Protocol Framework}
Consider two nodes (say A and B) that have messages to each other i.e., node A has a message for B and node B has a message for A. If the direct channel exists (link AB), then A's message to B as well as B's message to A succeeds in reaching the destination.
If there is no direct channel, then A's message to B may succeed if there is at least one node (say C) that has connection to both A and B. If there exists such a node, then both A's message to B \emph{and} B's message to A succeeds via the same node (or set of nodes).
Essentially, when there is a bi-directional traffic, the paths of `success' in both direction are the same. When we have such bi-directional traffic patterns, then relay nodes can `XOR' the packets and broadcast the resulting packet \emph{simultaneously} using a DSTC as shown in Fig.~\ref{fig:netcode}. This is what we leverage in XOR-CoW{} -- opportunistically network code packets. Network coding also provides throughput benefits (and as a result reduction in latency or reduction in SNR needed) when the traffic patterns are multicast (messages need to reach multiple destinations). We consider this scenario in detail in Section.~\ref{sec:generic-xor}.
\subsection{Recent development in $5$G protocols}
\label{subsec:5g_disc}
The current vision of $5$G wireless standard not only focuses on increasing capacity and energy efficiency, but also on reducing latency. Tactile applications demanding latencies on the order of $1$ms may be enabled by using mmWave frequencies~\cite{what_will_5g_be,2020_beyond_4G}.
Recent works like \cite{levanen} concentrate on the proposed 5GETLA radio interface and show that latencies below $1$ ms for payloads of size $50$kb are achievable provided a bandwidth of 100MHz is available. Though the targeted latency is on the same order as required by industrial control, they do not consider reliability guarantees or retransmissions.
The feasibility, requirements, and design challenges of an OFDM based 5G radio interface that is suitable for mission-critical MTC (machine type communication) is discussed in~\cite{yilmaz2015analysis} where various modulation schemes as well as different MIMO configurations were considered. They concluded that for interference mitigation, multiple receive antennas were crucial. In similar spirit, the coverage and capacity aspects based on evaluations considering both noise-limited and interference-limited operations for MTC were considered in~\cite{brahmi2015deployment}.
Several works have studied the suitability of various signaling strategies for low-latency applications. Specifically alternatives for OFDM have been considered to relax synchronization requirements and reduce out-of-band (OOB) transmissions such as Filter Bank Multi-carrier~\cite{farhang2011ofdm}, Universal Filtered Multi-carrier~\cite{vakilian2013universal} and Generalized Frequency Division Multiplexing~\cite{michailow2014generalized}. In this paper, we do not consider explicit signaling strategies and push it for future work.
\subsection{Industrial Control}
\label{subsec:indus_control}
Communication in industrial control is supported by wired fieldbus systems like HART, PROFIBUS, WorldFIP, Foundation Fieldbus, and SERCOS \cite{richardzurawski} meet these requirements.
Several wireless extensions of these fieldbus systems such as \cite{1280615, 573264} (as well as WirelessHART \cite{WHART} and ISA100 \cite{ISA100}) which are based on wireless sensor network (WSN) techniques have been developed. They have worst-case latencies on the order of hundreds on milliseconds \cite{akerberg2011future} making them unsuitable for high-performance control applications.
WISA \cite{WISA} targeted wireless control by employing frequency hopping techniques but it achieves latency on the order of $10$ms with a reliability of $10^{-4}$ \cite{Willig07recentand}, which fails to meet the reliability achieved by wired fieldbuses.
\subsection{Cooperative communication and multi-user diversity}
\label{subsec:multi-user}
In our prior work \cite{swamy2017real}, we discussed some of the relevant references on cooperative communication and multi-user diversity in detail.
Low-latency applications that we target cannot use time diversity since the cycle time can be shorter than the coherence time. Additionally, TDMA-based schemes for industrial control considered in~\cite{Willig08, 6648015} do not scale well with network size.
Commonly used frequency diversity techniques in WSN-inspired technologies~\cite{ZandCDH12} like channel hopping and contention-based MACs aren't sufficient to obtain the required diversity as they cause unbounded delays.
As there are multiple nodes in the system, harvesting cooperative, multi-user diversity is a viable option.
Cooperation amongst distributed antennas can provide full diversity without physical arrays \cite{Laneman}.
Even with noisy inter-user channels, multi-user cooperation increases capacity and leads to achievable rates that are robust to channel variations \cite{UserDiversity}.
\subsection{Network Coding}
\begin{figure}
\centering
\includegraphics[width = \textwidth]{net_top.pdf}
\vspace{-30pt}
\caption{{Illustration of network coding along with simultaneous retransmissions where the C and S nodes have information to convey to each other through 3 relays 1 - 3. The bold lines are active links and the dotted lines are inactive links. The blue packets are the downlink packets, the orange packets are the uplink packets and the maroon packets are the XORed packets. The XOR scheme can communicate the same amount of information in a shorter time because the uplink and downlink demands are satisfied simultaneously.}}
\label{fig:netcode}
\vspace{-40pt}
\end{figure}
The seminal work of Ahlswede \emph{et al.,}~\cite{ahlswede2000network} showed that regarding information to be multicast as a ``fluid'' to be routed or replicated in general is not optimal and employing coding at nodes can lead to efficient use of bandwidth.
This idea was further studied in \cite{katti2008xors}, where a forwarding architecture for wireless mesh networks to improve throughput by introducing a coding layer in between the IP and MAC layers was proposed. They provide a practical implementation of network coding into the current network stack, addressing the common case of unicast traffic, and dynamic and potentially bursty flows.
Recent results in \cite{bagheri2011randomized} show that using randomized space-time block coding (RSTBC) in two-way relay networks improves throughput by exchanging data through a bi-directional relay network. Like most works using network coding, we aim to increase throughput which translates to lower latency. Fig.~\ref{fig:netcode} illustrates how we use network coding combined with simultaneous retransmissions in our work. Essentially, if there is a natural viability for XORing then, only those nodes with the necessary packets help by broadcasting the XORed packet.
The proposed wireless communication system combines cooperative communication and network coding techniques to achieve the desired QoS requirements by exploiting multi-user diversity and distributed space-time codes (such as those in \cite{Oggier, PVK, 5506252}, so that each receiver can harvest a large diversity gain) to achieve high-reliability and low latency.
The key idea here is that relays simultaneously broadcast coded packets (as long as they are coding the same set of packets).
\subsection{Behavioral assumptions for analysis}
\label{behav_assump}
Our analysis depends on the following behavioral assumptions in addition to the resource assumptions in Sec.~\ref{subsubsec:assumptions}. We assume a fixed nominal SNR on all links with independent Rayleigh fading on each link. We also assume channel reciprocity. Our model assumes a single-tap channel\footnote{Performance would improve if we reliably had more taps/diversity.} (hence flat-fading). Because the cycle-time is so short, we use the delay-limited-capacity framework \cite{737514, 293655}.
A link with channel coefficient $h$ and bandwidth $W$ is deemed good (thus no errors or erasures) if the rate of transmission $R$ is less than or equal to the link's capacity $C = W \log(1 + |h|^2\textit{SNR})$. Consequently, the probability of link failure is defined as $p_{\scriptscriptstyle link} = P(R > C) = 1 - \exp{\left(-\frac{2^{R/W} -1}{\textit{SNR}}\right)}$.
As in \cite{swamy2015cooperative}, if there are $k$ simultaneous transmissions, then each receiver harvests perfect sender diversity of $k$. For analysis, this is treated as $k$ independent tries that only fail in communicating the message if all the tries fail. We do not consider any dispersion-style finite-block-length effects on decoding. This can be justified in spirit by \cite{YangDKP14}. We assume that transmission related errors are always detected ~\cite{forney}.
\subsection{XOR-CoW{} probability of failure}
\label{subsec:xor_calc}
The complete analysis of the performance of the XOR-CoW{} protocol is described in the \ref{appendix}. In this section we mainly present the results and state two theorems which are useful in understanding the results.
\begin{theorem}
If an instance of fixed schedule two-hop Occupy CoW protocol (i.e., no rate adaptation in the relaying phases) with equal downlink and uplink phases ($T_{D_1} = T_{U_1} = T_{D_2} = T_{U_2} = T_M$) succeeds, then there is a common downlink and uplink success path for each node in the network.
\end{theorem}
\begin{IEEEproof}
If a node successfully decoded the downlink message in one hop, its uplink message also gets through successfully to the controller in one hop (due to channel reciprocity). If a node successfully decoded the downlink message in two hops via a relay $Z$, then the same relay helps uplink as well -- again due to channel reciprocity.
\end{IEEEproof}
\begin{theorem}
If an instance of fixed schedule two-hop Occupy CoW protocol with equal downlink and uplink phase 1 ($T_{D_1} = T_{U_1} = T_{D_2} = T_{U_2} = T_M$) and a given SNR succeeds, then the fixed scheduling version of XOR-CoW{} with downlink and uplink phase lengths both equal to $T_M$ and XOR phase length also equals to $T_M$ succeeds at the same SNR.
\end{theorem}
\begin{IEEEproof}
From Theorem 1 we know that the paths for downlink and uplink success when $T_{D_1} = T_{U_1} = T_{D_2} = T_{U_2} = T_M$ are the same -- i.e., either they directly succeed to the controller or they have the same relay helping in both downlink and uplink. These relays essentially have the capability the XOR the packets as they have both the packets as well as good links for transmission. Hence, as long as the rate in the XOR phase stays the same (this is ensured by $T_{D} = T_{U} = T_X = T_M$), the XOR-CoW{} protocol also succeeds at the same SNR.
\end{IEEEproof}
\emph{A corollary of Theorem 2 is that while two-hop Occupy CoW would require time $4\times T_M$ to succeed, XOR-CoW{} succeeds in time $3\times T_M$ -- i.e., a throughput improvement of $\frac{4}{3}$.}
\subsection{Results and comparison}
\label{subsec:comp}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{baseline_comparison_noRA.jpeg}
\end{center}
\vspace{-20pt}
\caption{{The performance of XOR-CoW{} for a star information topology compared with reference schemes for varying network size, and a $2$ms cycle time, aiming at $10^{-9}$ probability of failure for a $20$MHz channel. The numbers next to the frequency-hopping scheme show the frequency diversity needed and those next to the non-simultaneous retransmission scheme show the optimal number of relays per message stream.}}
\label{fig:hockeystick}
\vspace{-40pt}
\end{figure}
We explore the performance of XOR-CoW{} with parameters taken from a contemporary practical application, the industrial printer case described in \cite{Weiner}. The SERCOS \upperRomannumeral{3} protocol~\cite{SERCOS} supports the printer's cycle time of $2$ ms with system error probability of $10^{-8}$. We target the following system requirements for the application:
$30$ moving printing heads that move at speeds up to $3$ m/s over distances of up to $10$ m. Every cycle lasts $2$ms and in each cycle the controller transmits $20$ bytes of actuation data to each head and each of the $30$ sensors transmit $20$ bytes of sensory data to the controller. Assuming access to a single $20$MHz wireless channel, this $4.8$ Mbit/sec throughput corresponds to an overall spectral efficiency of approximately $0.25$ bits/sec/Hz. SERCOS supports a reliability of $10^{-8}$ and for our protocol we target a reliability of $10^{-9}$.
We define the cycle failure probability as the probability that any packet transmitted during the cycle did not reach at least one of its destinations.
Following \cite{swamy2015cooperative, swamy2017real} and the communication-theoretic convention, we use the minimum SNR required to achieve $10^{-9}$ reliability as our metric to compare XOR-CoW{} to other schemes. Fig.~\ref{fig:hockeystick} compares the performance of the following protocols a) XOR-CoW{}, b) Occupy-CoW (the cooperative-communication-based protocol not employing network coding), and c) Frequency hopping based protocols.
We see that optimized version of Occupy CoW (the best performance that can be obtained without using network coding) and XOR-CoW{} with a simple equal-time allocation to different phases perform comparably for $m = 160$ bits (the dot-dashed lines). The advantage of XOR-CoW{} is clear for high aggregate rates and large networks as shown by the solid in Fig.~\ref{fig:hockeystick}. We see that XOR-CoW{} beats the performance of Occupy CoW for $m= 480$ bits and network size $> 20$ while also being a simpler scheme.
The dotted purple curves represent a hypothetical (non-adaptive) frequency-hopping scheme that divides the bandwidth $W = 20$MHz into $k$ sub-channels that are assumed to be independently faded, for $m = 160$ bits and $m = 480$ bits. The curves are annotated with the optimal $k$. As $k$ (and thus frequency hops) increases, the available diversity increases, but the added message repetitions force each link's instantaneous data rate to be higher. For low $n$ the scheme prefers more frequency hops to exploit diversity benefits. The SNR cost of doing this is marginal because the throughput is low enough that we are still in the linear-regime of channel capacity. For networks with fewer than $7$ nodes, this says that using frequency-hopping is great --- as long as we can reliably count on about $20$ independently faded sub-channels to repeat across, which is not always practical.
\subsection{Optimization}
\subsubsection{Network Coding Optimization}
XOR-CoW{} scheme only allows for the opportunity to XOR two packets and not more. Are we making sub-optimal decisions by restricting to XORing only two packets? We are not and the reason is as follows.
In undirected network (wireless networks considered here can be modeled as undirected networks) the throughput improvement that network coding provides when compared to routing only schemes is upper bounded at $2$~\cite{li2006achieving}. We showed in Sec.~\ref{sec:results} that the throughput improvement for the best case i.e., the star-topology is actually $\frac{4}{3} < 2$.
Furthermore, we can model the generic information topology as a multicast session. It has been shown that asymptotically network coding provides no benefits when compared to a pure routing schemes~\cite{swamy2013asymptotically}. Additionally, even if we end up with a network realization which can provide significant network coding benefits (a rare event in itself), the coding points (which perform network coding operation) need to know the state of each packet and the network realization to compute the optimium code. The overhead of acquiring this network information state is significant (similar in spirit to why backpressure routing isn't implemented as-is in current networks).
\subsubsection{Phase Length Optimization}
\begin{figure}
\centering
\vspace{-10pt}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width = \textwidth]{comparison.pdf}
\caption{\small{SNR comparison of optimized flexile-schedule XOR-CoW{} and fixed-schedule XOR-CoW{} for $m=160$ bit and varying network size with $20$MHz bandwidth and a $2$ms cycle time, aiming at $10^{-9}$.}}
\label{fig:xor_opt}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width= 0.9\textwidth]{opt_phase.jpg}
\caption{\small{The phase allocation for optimized XOR-CoW{} with flexible scheduling for $m=160$ bit messages and varying network size with $20$MHz bandwidth and a $2$ms cycle time, aiming at $10^{-9}$ is shown.}}
\label{fig:xor_phase}
\end{subfigure}
\vspace{-20pt}
\caption{Optimization of XOR-CoW{} protocol}
\label{fig:opt}
\vspace{-40pt}
\end{figure}
We consider the XOR-CoW{} protocol and look at the optimal allocation of time which minimizes the SNR required to meet the performance specifications.
Although the phase length allocations are uneven (as seen in the figure~\ref{fig:xor_phase}), we find that the SNR saving that we achieve by having different lengths is minimal (as seen in the figure~\ref{fig:xor_opt}). The complexity of building a system which can operate at variable rates is extremely difficult and ultimately negates out the small SNR savings achieved by optimization.
The strength of the protocol lies in the fact that a simple scheme with equal time allocations with fixed schedule performs almost as good as the optimal scheme -- thus paving the way for a practical system.
\subsection{Downlink and Uplink Phases}
During these phases, all the nodes are listening whenever they are not transmitting.
The downlink phase is common in both the fixed and flexible scheduling schemes.
The cycle starts with a downlink phase in which the controller broadcasts a single packet consisting of all $m$-bit messages to all $n$ nodes at rate $R_{D} = \frac{m \cdot n}{T_{D}}$.
In Fig~\ref{fig:protocol} column 2, S1 and S2 successfully decode the entire downlink message. Their starred buffers are filled along with the downlink buffers corresponding to other nodes.\\
\emph{Fixed Schedule Scheme:} This is followed by the uplink phase, in which the individual nodes transmit their messages to the controller one by one according to a predetermined schedule at rate $R_{U} = \frac{m}{T_{U}/n} = \frac{m \cdot n}{T_{U}}$ by evenly dividing the time slots among all nodes.
In Fig~\ref{fig:protocol} column 3, the controller successfully decodes the uplink messages of S1 and S2 and the starred uplink buffers of the controller corresponding to these nodes are filled. Since all nodes are listening whenever they are not transmitting, S1 receives the uplink messages of S3 and S4 while S2 receives the uplink message of S4.
The nodes which have successfully received the downlink message as well as successfully transmitted their uplink message to the controller are referred to as \textbf{strong nodes}. In Fig.~\ref{fig:protocol}, S1 and S2 are the strong nodes.\\
\emph{Flexible Schedule Scheme:} In the uplink phase of the flexible scheduling scheme, the nodes also transmit a one bit ACK to the controller (indicating whether they've successfully received the downlink packet or not). Therefore, the individual nodes transmit their messages (including one bit for an ACK) to the controller one by one according to a predetermined schedule at rate $R_{U} = \frac{m+1}{T_{U}/n} = \frac{(m+1) \cdot n}{T_{U}}$ by evenly dividing the time slots among all nodes.
\subsection{Scheduling Phase}
This phase is \emph{crucial} when the flexible scheduling scheme is employed.
In this phase the controller transmits acknowledgments to the strong nodes (at the same rate as the downlink phase). This is just $2$ bits of information per node for downlink and uplink. The common-information about the system's state enables the strong nodes to share a common schedule for relaying messages for the remaining nodes. Note that the schedule only reaches the strong nodes but the nodes which need help do not know the schedule. How will they know which message is intended for them without the knowledge of the schedule? This can be addressed by building in identification of the destination node in the packet such that the nodes can figure out which packet was addressed to them while keeping the transmission rate the same. This approach has been discussed in detail in \cite{han1992new}. Thus, for the remainder of the paper we'll assume that the nodes know which packet was meant for them.
\subsection{XOR phase:}
Depending on the scheduling scheme, the time allocated for this phase can either be equally divided among all nodes -- corresponding to the rate of transmission is $R_{X} = \frac{m \cdot n}{T_X}$, or only those that need help -- corresponding to the rate of transmission is $R_{X} = \frac{m \cdot n_1}{T_X}$ where $n_1$ are the number of unsuccessful nodes.
In either case, the strong nodes XOR the downlink and uplink messages of each of the unsuccessful nodes they've heard.
During the slot of an unsuccessful node (say node $Y$), all the strong nodes that have successfully heard node $Y$ act as simultaneous broadcast relays and transmit the XORed packet using a DSTC.
In Fig.~\ref{fig:protocol}, S3 and S4 are the unsuccessful nodes. In the XOR slot allocated for S3 (Fig.~\ref{fig:protocol} column 3), S1 XORs the downlink and uplink packet of S3 (represented by the purple packet) and broadcasts it. Using the downlink packet of S3, the controller can now recover the uplink packet. Using its own uplink packet, S3 can now recover the downlink packet. The process for S4 is similar and the difference lies in the fact that S1 and S2 simultaneously transmit the XORed packet for S4.
\begin{comment}
\noindent\emph{Fixed scheduling scheme:}
In this scheme, the time allocated for the XOR scheme is evenly divided among \textbf{all} the nodes regardless of whether they succeeded and the rate of transmission is $R_{X} = \frac{m \cdot n}{T_X}$.
During the slot of a node (whether successful or not), the strong nodes simultaneously relay the XORed packet as described earlier. The redundant transmission of already successful messages in this phase can be minimized by employing a scheduling phase between the uplink and XOR phases such that all strong nodes know when not to transmit.
\noindent\emph{Flexible scheduling scheme:}
In this scheme, the time allocated for the XOR scheme is evenly divided among the unsuccessful nodes only (say $n_1$ of them) and the rate of transmission is $R_{X} = \frac{m \cdot n_1}{T_X}$. All nodes are assumed to be capable of instantly decoding variable-rate transmissions~\cite{VerduVariableLength} so they keep listening until they get the message intended for them.
\end{comment}
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* Set locale now
* @private
* @return {string}
*/
var _setLocale=function(locale){
_locale=locale;
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Q: Does the sequence $\{\sin(en)\}$ converge or diverge? Is it known if $\{\sin(en)\}$ converges or diverges?
Also, I have a more general question. For almost every rational $r$, I think we can say that $\{\sin(rn)\}$ diverges. Does that statement hold if we only assume $r$ is a real number?
A: The sequence $(\sin rn)_{n\in\mathbb N}$ converges if and only if $r$ is a multiple of $\pi$ (in which case it is the constant sequence $0,0,0,\ldots$).
If $r$ is any rational-but-not-integral multiple of $\pi$, then the sequence is periodic but not constant, and therefore doesn't converge.
If $r/\pi$ is irrational, then values in the sequence are dense in $(-1,1)$ -- in particular there are infinitely many elements in $(-1,-\frac12)$ and also infinitely many elements in $(\frac12,1)$, so it cannot converge in this case either.
A: Henning Makholm's answer is perfectly fine, but if you like another approach, it is straightforward to check that $\{\sin(en)\}$ cannot be a Cauchy sequence, since:
$$ \limsup_{n\in\mathbb{N}}\left| \sin((n+1)e)-\sin(ne) \right|= 2\sin\frac{e}{2}\cdot\limsup_{n\in\mathbb{N}}\left|\cos\left((2n+1)\frac{e}{2}\right)\right|\color{red}{>}1. $$
A: It's worth mentioning that if we consider arbitrary $s$ then the series
$$ \sum_{n=0}^{\infty} \sin(sn) $$
Can usually still be assigned a formal divergent sum in a way similar to how $1 - 1 + 1 - 1 ... = \frac{1}{2}$
Recall that $$\sin(n) = \frac{e^{in} - e^{-in}}{2i} $$
So the series
$$ \sum_{n=0}^{\infty} \sin(sn)x^n = \frac{1}{2i} \left(\sum_{n=0}^{\infty} e^{isn}x^n - \sum_{n=0}^{\infty} e^{-isn}x^n \right) = $$
$$ \frac{1}{2i} \left( \frac{1}{1-e^{is}x} - \frac{1}{1-e^{-is}x} \right) $$
Now evaluating at $x=1$ we find that
$$ \sum_{n=0}^{\infty} \sin(sn) = \frac{1}{2i} \left( \frac{1}{1-e^{is}} - \frac{1}{1-e^{-is}} \right) $$
For that particular problem then we have
$$ \sum_{n=0}^{\infty} \sin(en) = \frac{1}{2i} \left( \frac{1}{1-e^{ei}} - \frac{1}{1-e^{-ei}} \right) $$
A: It does not convergence for all $e$ and therfore diverges. We can easily see this if we pick $e = \frac{\pi}{2}$. For $n \in \mathbb{N}$ we now see that $\sin(\frac{\pi}{2} n) = 0$ if $n$ is even. While if $n$ is odd, this is either $-1$ or $1$. This is not convergent and thus divergent.
A: You can use
$$
\sum_{k=1}^n \sin(kx) = \tfrac{1}{2} \sin(nx) + \tfrac{1}{2} \sin(x) \frac{ 1 - \cos(nx) }{ 1 - \cos(x) }
$$
When you put in $x=e$, you get
$$
\sum_{k=1}^n \sin(ke) = \tfrac{1}{2} \sin(ne) + \tfrac{1}{2} \sin(e) \frac{ 1 - \cos(ne) }{ 1 - \cos(e) },
$$
so
$$
\lim_{n \rightarrow \infty} \sum_{k=1}^n \sin(ke)
= \lim_{n \rightarrow \infty} \tfrac{1}{2} \left\{ \sin(ne) + \sin(e)
\frac{ 1 - \cos(ne) }{ 1 - \cos(e) } \right\},
$$
which is undefined, i.e. it does not converge.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,060 |
\section{Introduction}\label{sec1}
Learning disabilities or learning disorders, a hypernym for a wide variety of learning problems hinders the skill acquisition activity of an individual. It is not a problem with intelligence. Nevertheless, it can have a negative impact on the self-esteem and confidence of children who are expected to acquire new information and skills day by day. Students with learning disabilities account for the major proportion of "special educational needs" category \cite{Knickenberg2020}. Since these disabilities affect the perception capability of a child, the difficulties may be either in reading, writing, doing math, or any other tasks. Figure \ref{fig1} shows different common learning disabilities found in children.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img1.jpg}
\caption{Types Of Learning Disabilities}\label{fig1}
\end{figure}
Dyslexia \cite{Lyon1995} is a kind of learning disability that includes difficulty in reading because of issues in recognizing the speech sounds as well as decoding them. Dyslexia is affected in the brain areas which is assigned for processing the language. Dyscalculia \cite{Keong2020} is a specific type of learning disorder that involves difficulty in understanding numbers and its related issues in learning mathematics. Dyspraxia \cite{Gibbs2007} a.k.a developmental coordination disorder affects the coordination skills of individuals which hinders them from accomplishing tasks such as playing sports, driving, etc that requires balance. Central auditory processing disorder (CAPD) \cite{Koravand2017} involves difficulty in hearing in children. This is because of the lack of coordination between the brain and ears. Children with CAPD have trouble in understanding when they hear sounds. Dysgraphia is primarily considered as a disorder in written expression. It can affect the spelling, grammar, organization, etc., in addition to the handwriting aspects \cite{Deuel1995}. Figure \ref{fig2} summarizes the basic categorization of dysgraphia based on the related symptoms . Although the exact prevalence depends on the definition of dysgraphia, between 10-30 \% of children face difficulty in handwriting \cite{Chung2020}.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img2.jpg}
\caption{Types Of Dysgraphia}\label{fig2}
\end{figure}
The typical diagnostic strategy for any learning disability follows a team-based assessment \cite{Chung2020}, which involves the support of an occupational therapist, speech therapist, special education teacher, psychologist etc. In addition, any other prevailing medical conditions such as poor vision, hearing problems, intellectual disability, lack of proper training should be ruled out with the help of a specialist. As far as dysgraphia is concerned, it is indeed important to consider the several contributing factors such as handwriting speed and legibility, inconsistency between spelling ability and verbal intelligence quotient, as well as the pencil grip and writing posture evaluation. However, there is no generalized medical testing strategy available for the diagnosis of dysgraphia. Concise Evaluation Scale for Children's Handwriting (BHK) for French \cite{LHamstra-BletzdeBieJ1987}, Detailed Assessment of Speed of Handwriting (DASH) in Latin \cite{Barnett2009} are some commonly used standards in the assessment of dysgraphia.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img3.jpg}
\caption{Handwriting samples of person with dysgraphia }\label{fig10}
\end{figure}
It is quite difficult to diagnose any of the learning disorders because of the multiple cues that should be considered in the assessment. Depending on the age and developmental stage, the warning signs and symptoms may vary too. More importantly, the predefined symptoms should persist for at least 6 months with parallel intervention measures being administered \cite{AmericanPsychiatricAssociation.2013}. Dysgraphia may appear in isolation or as a comorbidity with other learning disorders or even autism spectrum disorder, developmental coordination disorder (DCD) \cite{Lopez2018}, and Attention Deficit Hyperactivity Disorder. This
20
emphasizes the timely diagnosis of dysgraphia or handwriting disorders in particular. The early recognition and intervention lessen the task and efforts needed to correct the disorders.
The manual assessment techniques solely rely on the handwritten product for the final scoring and judgment. This paved the way to propose several automated techniques that can exploit the dynamic characteristics of handwriting as well. Digital tablets capable of capturing these multiple features of handwriting have yielded promising results in the related research. Meantime studies indicate that the writing sensation may be different for stylus-tablet setting in contrast to pencil-paper, which is the commonly used procedure during the skill acquisition phase \cite{Gargot2020}. Despite the complexity and the multifaceted approaches involved in the diagnosis of dysgraphia, several artificial intelligence-based and non artificial intelligence based automated technologies have been proposed in recent years\cite{rosenblum2018inter,vaivre2021phenotyping,chang2013handwriting}.
To the best of our knowledge, no other work in the literature has focused on the review of automated dysgraphia diagnosis systems until now. There are a few related review articles in the literature and their details are tabulated in table 1. Most of the review articles in the literature targeted all learning disabilities instead of one. And few of them have not addressed dysgraphia at all. Since the trend of automated systems for dysgraphia diagnosis is increasing in the last few years, there is a need for a review article on the same. It can help researchers and academic students to learn more about this topic.
This work presents a review of the existing automated dysgraphia diagnosis systems for children in the literature. The main focus of the work is to review artificial intelligence-based systems for dysgraphia diagnosis in children. This work discusses the data collection method, important handwriting features, machine learning algorithms employed in the literature for the diagnosis of dysgraphia. Apart from that, this article discusses some of the non-artificial intelligence-based automated systems and psychological methods also. Furthermore, this article discusses the drawbacks of existing systems and proposes a novel framework for dysgraphia diagnosis.
\begin{table}[]
\caption{Summary of related works}
\begin{tabular}{|l|p{8cm}|}
\hline
Reference & Article Summary \\ \hline
Vanitha and Kasthuri\cite{vanithadyslexia} & Review of machine learning algorithms used in Dyslexia prediction \\ \hline
Chakraborty et al.\cite{chakraborty2019survey} & It presents a survey paper on the topic machine learning algorithms for learning disability prediction. But the presentation was some what abstract level and no details about dysgraphia diagnosis systems \\ \hline
Vanjari et al.\cite{vanjari2019review} & A review on learning disabilities and technologies determining the severity of learning disabilities \\ \hline
Prabha and Bhargavi\cite{jothi2019prediction} & A similar work like \cite{vanithadyslexia} an breif review on prediction of dyslexia using machine learning. \\ \hline
Saxena and Saxena\cite{saxena2020machine} & Reviews and explains the role of machine learning in learning disabilities diagnosis but lacks the insight in to dysgraphia. \\ \hline
Jamhar et al. \cite{jamhar2019prediction} & A systematic review about machine learning methods for learning disorder prediction.But not addressed about dysgraphia. Further more the focus was just on the machine learning algortihms not about data collection or features. \\ \hline
\end{tabular}
\end{table}
\section{Research Methodology}\label{sec2}
This work followed a systematic approach for preparing the literature review. The systematic approach is followed to find out the specific issues in this research domain. In the systematic approach, we defined certain search terms and pre-selected a few digital databases to find the related research articles. After finding the related articles the most relevant works are sorted out and reviewed in this work.
A research question is a required element for conducting the review in a systematic approach. The generic research questions in this work are:
\begin{itemize}
\item What are the automated systems used for the prediction of dysgraphia in children?
\item How effectively the machine learning technology has employed in the literature for the diagnosis of dysgraphia in children?
\item How the handwritten data are collected for training machine learning models and what are the relevant handwritten features which can discriminate the abnormal and normal handwriting.
\end{itemize}
During the literature search, we utilized four popular digital libraries IEEE Digital Library, Web of Science, PubMed, and Springer Link to find the related articles. The search keywords are used for literature search are given below.
\begin{itemize}
\item Automated System Dysgraphia
\item Machine Learning Dysgraphia
\item Deep Learning Dysgraphia
\item Automated System Learning Disorder
\item Machine Learning for Learning Disorder
\end{itemize}
Among all the obtained search results by using the above-mentioned keywords separately in each database, we found that only about 50 papers are most relevant to the specific topic. Another interesting fact is that most of the papers in the literature are published after 2015.
\section{Dysgraphia Diagnosis Methods}\label{sec3}
In this section, we first discuss the psychological methods used to diagnose dysgraphia and the manual analysis of handwriting. We then review the notable contributions to automated analysis (machine learning methods) and non-ML based diagnosis systems of handwriting
for the detection of dysgraphia.
\subsection{Psychological Methods}\label{sec3-1}
Dysgraphia is a Greek word with the meaning, difficulty, or poor (Dys) writing (graph). Initially, the difficulty in handwriting or dysgraphia was described by Hamstra-Bletz and Bolte as difficulty in the construction of letters while writing which is closely related to the mechanics of writing \cite{hamstra1993longitudinal}. It has also been referred to as a specific learning disability \cite{brown1981learning,rosenblum2004handwriting}.
The conventional approaches for dysgraphia diagnosis consist of two sections: academic skills examination and cognitive skills analysis. The popular symptoms of dysgraphia include "messy handwriting, inconsistency in letter spacing and capitalization, pain or discomfort when writing, fine motor skill challenges, trouble with spelling, or trouble with composing written work". And further, it is also common that the students with dysgraphia can express themselves well while speaking but can't transfer them onto paper perfectly. The occupational therapist or trained psychologist evaluate different skills (falling under different category (figure \ref{fig0})) of the student to find out the existence of dysgraphia.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img0.jpg}
\caption{Psychological evaluation: Assessment categories }\label{fig0}
\end{figure}
The constructional ability assessment will look for the proficiency of the student to copy or reconstruct lines, shapes, or figures. One of the popular conventional methods utilized by trained psychologists for constructional ability assessment is "Beery Visual Motor Test of Integration – Sixth Edition (VMI-6)" \cite{beery2004beery}. In the VMI-6, the participants are asked to copy drawings on paper. The complexity of the drawings will be increased throughout each stage of the test and participants are not allowed to erase the drawings. The drawings given for the copying task include overlapped figures, angles, and three-dimensional images. But the visual motor skills purely depend on the age and it is expected that until age 16, the standards of the visual-motor skills will be different for every age until 16. Usually, many students with dysgraphia struggle with eye-hand coordination and planning. The visual-motor integration test conveys more information about the participant's capacity to understand a drawing and the motor ability to copy the information (motor response). Other popular constructional ability assessment tests are Bender Gestalt II \cite{mehrinejad2012investigation}, NEPSY-II \cite{korkman2014nepsy}.
Executive function skills enable the student to plan, concentrate, remember commands and organize multiple tasks. The student requires all the executive functions to intact for writing \cite{chung2015dysgraphia}. Rey Complex Figure Test \cite{meyers1995rey} and Behaviour Rating Inventory of Executive Function (BRIEF) \cite{roth2014assessment} are the popular examination test conducted for executive function assessment. In Rey Complex Figure Test, the students were asked to draw very complex figures. Based on their drawing output the capacity of executive functions is quantified. On the other hand, BRIEF provides a questionnaire to students' parent and teacher, and ask them to answer for that. The provided questionnaire is a rating form that consists of 86 items to assess eight clinical scales independently. The eight clinical scales include three behavioral regulation scales (Inhibit, shift, emotional control)and five metacognition scales (initiate, working memory, plan, monitor, organization of materials ).
Writing and spelling skills, phonological awareness, retrieval fluency are commonly found to be less in children with dysgraphia \cite{mccloskey2017developmental}. Wechsler Individual Achievement Test (WIAT-III)\cite{burns2010wechsler}, Woodcock Johnson-III Tests of Achievement (WJ-III) \cite{woodcock2007woodcock}, and Test of Written Language-4 (TOWL-4)\cite{hammill2009test} can assess the writing and spelling skills. Phonological awareness refers to the ability of the human to perceive and work with the audio, especially the sound in the spoken language. It includes understanding patterns like alliteration or rhymes, the ability to segment the sentences into words, understanding of phonemes and syllables, etc. Comprehensive Test of Phonological Processing\cite{wagner1999comprehensive}, NEPSYII phonological processing are the two popular phonological awareness assessment tests.
Working memory refers to the cognitive ability of the human memory system that has limited capacity and can keep information for short time. Generally, dysgraphia can occur with some degree of working memory problems\cite{crouch2007dysgraphia}. Test of Memory and Learning – 2 (TOMAL-2) \cite{reynolds2007test} and Wide Range Assessment of Memory and Learning-2 (WRAML-2)\cite{hartman2007wide} are two popular examinations for assessing the decline of working memory. Intelligence measure a.k.a intelligence quotient quantifies human intelligence which varies with the age. Wechsler Intelligence Scale for Children (WISC-IV)\cite{petermann2011wechsler} and Differential Ability Scales (DAS)\cite{elliott1990differential} are the popular tests for quantifying the intelligence of children.
The Concise Evaluation Scale for Children's Handwriting (BHK) is another popular manual handwriting analysis method conducted under the supervision of an occupational therapist or psychologist to quantify the speed and standard of the writing.
BHK test is usually conducted in the individual clinical setting and classical scholar setting. Initially, it was introduced for assessing the handwritten samples of 2nd and 3rd-grade students. Currently, BHK scales are used in the research for constructing the ground truth of the data used for training and evaluation of machine learning-based dysgraphia diagnosis systems.
\subsection{Machine Learning Based Methods}\label{sec3-2}
These section discusses machine learning based systems for diagnosis of dysgraphia from either the images of handwritten texts or features of writing dynamics. The general flow of machine learning based dysgraphia diagnosis system is shown in figure \ref{fig3} . As like in other machine learning applications, dysgraphia diagnosis systems also follow similar workflow. Data collection, preprocessing, feature extraction, feature selection, and training using machine learning classifier are important steps included in building the machine learning-based dysgraphia diagnosis system. Data is a pivot for any machine learning task. Cleaned and sufficient data is required for any machine learning algorithms to make the accurate prediction. In the case of the dysgraphia diagnosis system, two types of data collection methods are usually implemented. The first approach is offline-based data collection where the subject is asked to write or copy a few words or sentences on the paper or tablet and the resulting handwritten images are used for further analysis. On the other hand, the online-based data collection is focused on collecting the handwritting data during the run time which includes the trajectory of the pen /pencil, writing speed, pressure on pen tip, etc. Most of the existing dysgraphia diagnosis systems followed the online data collection strategy.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img4.jpg}
\caption{General work flow of machine learning based dysgraphia diagnosis system}\label{fig3}
\end{figure}
\subsubsection{Feature Extraction}\label{subsec2}
Extraction of features from the data is crucial for training the machine learning models. Multiple features are extracted from the collected handwritten data for further analysis in the existing systems. The important online handwritten features used for machine learning-based systems in the literature for dysgraphia detection are shown in figure \ref{fig5}.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img5.jpg}
\caption{Relevant handwriting features}\label{fig5}
\end{figure}
\begin{itemize}
\item Kinematic Features
\end{itemize}
The kinematics features mainly include the velocity, acceleration, and jerk of the writing. Velocity just quantifies the speed of writing and abnormal variability in writing speed is related to the underlying handwriting problems. Acceleration computes the variability in the velocity during writing. Jerk is the change in acceleration over time per stroke. Unusual changes in acceleration over time per stroke may be related to writing problems.
Various studies \cite{kushki2011changes,rosenblum2003air} in the literature have found that the kinematic aspects are affected in children with handwriting disabilities.Along with the average velocity of the whole writing process, few works considered the velocity or speed of writing each word or letter as a new derived feature for enhanced discrimination.
\begin{itemize}
\item Temporal Features
\end{itemize}
The temporal features mainly quantify the writing/drawing duration or time. Usually, the children with dysgraphia will take more time to write or draw compared to normal children. In this sense, the writing or drawing duration has significance in differentiating the normal and abnormal handwriting behavior. The total time required to complete the task, the total time spent on the paper, the ratio between the total time spent on the paper and total time of the task, etc are the popular temporal or time-related features utilized in the literature for dysgraphia detection.
\begin{itemize}
\item Spatial Features
\end{itemize}
The width, height, and length of the strokes as well as whole written data are the common spatial features extracted from handwritten data. The writing or drawing activities of children with dysgraphia always contain various inconsistencies. Irregular size of letters, irregular spacing between the words are the common inconsistencies found in the writing. The spatial feature can well discriminate the writing samples of children with dysgraphia.This features can be called as geometric features also. The offline handwritten based dysgraphia diagnosis systems mainly rely on these features to differentiate the normal and abnormal hand writings .
\begin{itemize}
\item Dynamic Features
\end{itemize}
The popular dynamic features are pressure, tilt, azimuth, etc. Among these, pressure features are quantified to find out the characteristics of pressure induced by the pen tip on the writing surface. Generally, the statistical measures such as mean, median, the standard deviation of pressure are quantified for feature construction. Speed of pressure change, speed of pressure change frequencies are the other derivable features from pressure values. The tilt feature is used to measure the inclination of the pen or pencil used for writing. The azimuth angle of pen or pencil with respect to the plane of written surface is also quantified for feature set construction.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img6.png}
\caption{Tilt and Azimuth angle }\label{fig6}
\end{figure}
\begin{itemize}
\item Other Features
\end{itemize}
Apart from the main four categories of features ( Kinematic , temporal , spatial ,dynamic), few works in the literature have considered some other features to improve the dysgraphia screening efficiency . It includes number of interruptions occurred during the writing , number of pen elevations , as well as number of times the child has made the mistake and erased it ( erase count ) etc. These features has also have relevance in screening dysgraphia , since children with dysgraphia are tends to make more mistakes than normal children while writing.
Most of the works in the literature has used combination of different feature categories to effectively identify the existence of dysgraphia in children using machine learning algorithms. Table \ref{tab:3} displays the different feature combinations used in the literature .
\begin{table}[h]
\caption{ Summary of feature combinations used in the literature}
\begin{tabular}{|l|lllll|}
\hline
\multirow{2}{*}{References} & \multicolumn{5}{c|}{Features} \\ \cline{2-6}
& \multicolumn{1}{l|}{Kinematic} & \multicolumn{1}{l|}{Temporal} & \multicolumn{1}{l|}{Spatial} & \multicolumn{1}{l|}{Dynamic} & Other \\ \hline
Mekyska et al.\cite{Mekyska2017} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}- & \multicolumn{1}{l|}- & \multicolumn{1}{l|}\checkmark &\checkmark\\ \hline
Asselborn et al. \cite{Asselborn2018}& \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
Gargot et al. \cite{Gargot2020} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
Drotar et al.\cite{Drotar2020} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \checkmark \\ \hline
Asselborn et al. \cite{Asselborn2020} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
Devillaine et al.\cite{Devillaine2021}& \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}-& \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \checkmark\\ \hline
Deschamps et al. \cite{Deschamps2021} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
Dankovicova et al.\cite{Dankovicova2019} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}- & \multicolumn{1}{l|}\checkmark & \checkmark \\ \hline
Rosenblum et al. \cite{rosenblum2016identifying} & \multicolumn{1}{l|}- & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
Mekyska et al. \cite{Mekyska2019} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \checkmark \\ \hline
Devi et al. \cite{devi2021early} & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}- & \multicolumn{1}{l|}\checkmark & - \\ \hline
Kedar \cite{kedar2021identifying}& \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & \multicolumn{1}{l|}\checkmark & - \\ \hline
\end{tabular}
\label{tab:3}
\end{table}
\subsection{Existing Machine Learning Based Systems}\label{subsec2}
Reference \cite{Mekyska2017} proposed a method for automated diagnosis of developmental dysgraphia and estimation of handwriting difficulty level using handwriting analysis. A digitized tablet is used to acquire the handwriting data from the students. A total of 54 students ( 27 normal and 27 dysgraphic ) are selected for experiments. The students are made to write seven semi Hebrew letters on A4-sized paper attached to the surface of WACOM Intuos II xy digitizing tablet (404 × 306 × 10 mm) using a wireless electronic pen with a pressure-sensitive tip (Model GP-110). The Computerized Penmanship Evaluation Tool (COMPET) is used for collecting the data. COMPET is a popular and standardized tool for online data acquisition and analysis. From the acquired handwriting data, three different kinds of features: ten Kinematic features, 34 non-linear dynamic features, and 7 other features are extracted for training artificial intelligence models for the classification of dysgraphic students. Random forest algorithm \cite{Biau2012} and Linear discriminant analysis algorithm \cite{Izenman2013} were used to train the machine learning models. The obtained results indicated that the Random forest classifier can classify the dysgraphic and normal subjects with 96\% sensitivity and specificity. Furthermore, the finding indicated that the altitude/tilt and pressure can discriminate the subjects well.
An automated dysgraphia diagnosis tool for primary school students using the consumer-level tablet was proposed in \cite{Asselborn2018}. In the proposed work, 298 students including 56 dysgraphic individuals were instructed to write on a sheet of paper affixed to a Wacom Intous tablet for 5 mins. Ductus software ( LPNC laboratory ) tool is used for data collection when the students are writing. From the handwriting task, 54 features that define various characteristics of the handwriting are extracted for further analysis. The handwriting features include Static features: purely geometric characteristics of a written text, Kinematic features: dynamics of handwriting path, Dynamics features: characteristics of the pressure recorded between the pen tip and the tablet surface, characteristics of the pen tilt. The extracted features are utilized to train a machine learning classifier, Random Forest, for intelligent decision-making. The proposed machine learning model has achieved excellent accuracy for diagnosing dysgraphia.
A study was conducted on 76 elementary school students in the Czech republic for a computerized assessment of graph motor difficulties or dysgraphia \cite{Mekyska2019}. The cohort of students was asked to draw 7 different figures such as the Archimedean spiral, connected loops , rainbow etc on an A4 paper, that was laid down and fixed to a digitizing tablet Wacom Intuos Pro L (PHT-80). Five different categories of features were extracted from the acquired handwriting data. The extracted features include spatial: width, height, and length of the whole product, as well as its strokes, stroke width, height, and length, temporal: duration of drawing, kinematic – velocity, acceleration, and jerk, dynamic: pressure, tilt, and azimuth, other – number of interruptions (pen elevations and relative number of interruptions). Mann-Whitney U test \cite{LaerdStatistics2015} was conducted to compute the very relevant features among extracted features. An ensemble machine learning algorithm XGBoost \cite{Chen2016} is trained with the extracted features for automated detection of dysgraphia in children.
A tablet-based smart application was developed for early prediction of the potential risk of handwriting alteration in children during the preliteracy stage \cite{Dui2020} . The main aim of the proposed approach was to develop a tool to anticipate dysgraphia screening in preschoolers. For the data collection, 104 preschoolers were asked to use an application named ' Play-Draw-Write' on the tablet device to draw basic shapes such as circles, squares, lines, etc. in different sizes using a stylus pen. Gesture smoothness, pressure, drawing kinematics features collected during writing and drawing task are collected for further analysis. The logistic regression \cite{Peng2002}]model is trained using these features for the classification task. Even though the proposed approach can classify the handwriting alteration it is not possible to confirm that real handwriting difficulties would arise in the following year schooling stage.
In \cite{Gargot2020} 280 school-going children ( average age about 9 years) are selected for conducting the BHK test on digital tablets. Twelve different features of handwriting tasks including the static, kinematic, and dynamic features are extracted for further analysis. An unsupervised method named K-means clustering is employed to distinguish the students with dysgraphia based on the severity levels ( mild or severe ). Linear regression models are used to predict handwritten quality scores.
A machine learning-based system for dysgraphia is proposed in \cite{Drotar2020}. The proposed approach focused on collecting a new dataset as well as analysis of the data to detect the existence of dysgraphia disability in school children. 120 school students have participated in their study for building the handwritten data set. Trained professionals from the center for children with special needs are employed for data collection from dysgraphic children. The participated children were asked to write specific letters, words, and sentence on the paper attached to the screen of the WACOM Intuos Pro Large tablet using the normal pen. The tablet can capture five different signals: pen movement in the x-direction, pen movement in the y-direction, the pressure of the pen on the tablet surface, and the azimuth and altitude of the pen during handwriting. 22 types of features including spatiotemporal and kinematic features were extracted from the collected data. Multiple machine learning algorithms were utilized in this work for classification and their performances are compared. The obtained results indicate that the AdaBoost algorithm \cite{Zizka2019} achieved the highest classification accuracy of 80\%. Among all the extracted features, pressure and pen lifts are the features with high discriminatory potential.
Images of handwritten text alone were used \cite{Richard2020} for detecting students with writing disabilities. The proposed work is based on the fact that in the handwritten task the dysgraphic children will face difficulties in the various process such as the formation of the letter, maintaining the size of letters, following the straight line, maintaining the consistent gap between letters as well as words. About 1400 handwritten images where each image contains four lines of handwritten text. From the images, different features including slant, pressure, amplitude, letter spacing, word spacing, slant regularity, size regularity, size regularity, horizontal regularity were manually extracted and four different machine learning classifiers are trained for dysgraphia prediction. Among four classifiers the Random forest classifier achieved the best accuracy of 96.
Reference \cite{Asselborn2020} proposed various scales to evaluate handwriting difficulties in a modernized way. In the proposed work, the participated 448 children were asked to write five sentences used in the BHK test using an application deployed in the IPad device. Dynamico software is used for collecting handwritten raw data. From the raw data, for each student 63 features falling under four categories: static, pressure, kinematic, tilt are extracted. Principle component analysis is employed for projecting the 63 features to three-dimensional feature space. An unsupervised machine learning algorithm K-Means clustering is employed for grouping the normal and dysgraphic subjects to different clusters.
The raw data obtained from graphomotor tests are analyzed using machine learning algorithms for the early detection of dysgraphia in children\cite{Devillaine2021}. Initially, 305 students were asked to participate in two different sets of experiments: the BHK test and the graph motor test. The BHK test is performed to label the students as dysgraphic or not for training the machine learning models. No features or data from the BHK test are used for machine learning-based analysis. After labeling, the students are asked to perform a graphomotor test, where each student has to write or draw specific predefined shapes or figures called stimuli on the paper attached to the screen of the tablet. Numerous features including time duration of the stimulus, duration of a stroke , duration of lift, velocity, jerk , Renyi Entropy of order 2, signal to noise ratio, etc were extracted from the raw data for further analysis. Among participating students, only 43 children were having dysgraphia, so the whole dataset ( raw data from graphomotor task) are divided into two: Active database and Z-score database. The active database contains the data of 43 children with dysgraphia and 43 other children without dysgraphia. The data of the remaining 219 non-dysgraphic children were added to the Z-score database. The data in the Z-score database is not used for training the machine learning model, instead, it is used for correcting the features in the active database. After correcting the features in the active database, relevant features are selected using Linear SVM and Extra Trees algorithm and utilized for training machine learning algorithms. Various machine learning algorithms including SVM \cite{Burges1998}, Random Forest, MLP\cite{Yegnanarayana1994}, Extra Trees, Gaussian Naïve Bayes\cite{Wu2008}, Ada Boost, etc are utilized in the proposed work for analyzing the features. Among all implemented machine learning algorithms the Random Forest classifier trained with features selected using the Linear SVM algorithm achieved the highest classification accuracy of 73.4
Reference \cite{Deschamps2021} aims at developing an automated pre-diagnosis tool for dysgraphia allowing a wide screening among children. Multiple tablets were utilized for data collection to ensure that the proposed system is not tablet/data collection tool dependent. To the best of our knowledge, this work has created the largest database for dysgraphia diagnosis by participating the 580 children for data collection. The children from 2nd to 5th grade are considered for the experiment. And among them, 122 children are having dysgraphia. The children were asked to perform BHK ( French version) on the tablet. 100 different features were extracted from the handwritten task. 80 \% of the feature samples are utilized to train the SVM machine learning classifier and the remaining 20\% has been used for testing. In the test set, the SVM classifier achieved a sensitivity of 91\% and specificity of 81\% for the detection of children with dysgraphia.
The performance of multiple machine learning algorithms for the detection of dysgraphia is analyzed in \cite{Et.al.2021} . Initially, a dataset was constructed by conducting various handwritten-based exercises in a group of 240 subjects where 142 have some sort of learning disabilities and 45 subjects have dysgraphia. These subjects were asked to complete a dot connecting exercise to analyze their motor skill. Furthermore, the samples of writing are analyzed for legibility and space knowledge. Boolean features are extracted for the following properties: sentence structure, word formation, visual-spatial response. Along with that, the images of writing are subjected to SSIM evaluation, and spellings are checked through a spell checker. Among all the extracted features the relevant features are selected using a feature selection technique known as Elastic Net\cite{Jenul2021}. The selected relevant features are used to train KNN \cite{Zhang2016}, Naïve Bayes, Decision tree\cite{Quinlan1986} , Random Forest, SVM classification model for prediction of dysgraphia. The Random forest classifier achieved the highest classification accuracy and it can differentiate the dysgraphic and non-dysgraphic subjects with an accuracy of 99\% in the experimented dataset.
A mobile-based system named 'Nana Shilpa' is proposed for screening and identifying the risk of dysgraphia as well as dyscalculia in primary school students \cite{Hewapathirana2021}. The proposed software solution is targeted at primary school students in Srilanka. For dysgraphia screening, the proposed method predicts the risk using two risk identification components letter level dysgraphia and numeric dysgraphia. To identify the risk of letter level dysgraphia the students were asked to write 4 letters and 4 words which are preselected by primary school teachers and the teacher's guide of Srilanka. A convolutional neural network model which consists of four two-dimensional convolutional layers are developed for initial screening of letter level dysgraphia. The letter and word data images collected from students are used to train the CNN model. Before training, the image data are subjected to preprocessing to segment the letters and words. The CNN model will predict the letter as well as words along with the confidence score. To identify the risk, the output of the CNN model including the letter level accuracy confidence of written letters, words, total correct count of letters and words, along with the total number of erase counts are given as features to train an SVM model. The trained SVM model will predict the risk of letter level dysgraphia. For numeric dysgraphia screening, the students were asked to write the number from 0 to 9 in the drawable canvas of the application. The correctness of the written number, erase count, time taken are considered as the features for the prediction of risk. The correctness or accuracy of the written number is determined using a CNN model trained in the MNIST dataset. These features are then used to train a machine learning model to predict the risk of numeric dysgraphia. The proposed methods can predict the risk of numeric and letter level dysgraphia with an accuracy of 9
A random forest-based machine learning method for dysgraphia identification from digitized handwriting is proposed in \cite{Dankovicova2019}. A dataset was constructed for handwritten analysis by making 78 subjects do various handwriting tasks in a Wacom Intous Pro Large graphic tablet. Among 78 subjects, 36 were having dysgraphia disability. Every handwriting was carefully analyzed by three specialists to create the ground truth labels. Various kinematic, temporal, and other features such as pen lifts, etc . are extracted for each written sample for further analysis. Three different machine learning classifiers Random Forest, SVC, and AdaBoost are trained using the extracted features for automated detection of dysgraphia. The obtained results indicate that Random forest displayed better classification performance.
A software system based on the SVM method and Android application is proposed for dysgraphia identification from handwriting data in elementary school students \cite{Sihwi2019}. An Android application employed with a handwriting recognition tool namely "WritePad" is used for data acquisition. The students were asked to write on the screen of the smartphone and the following data: time, pressure, spacing between the letters, size of letters, position of letters, consistency of boundaries, etc. are stored for further analysis. The ground-truth labels of classes include three classes of dysgraphia light, moderate, and severe as well as normal are constructed using an existing method proposed in \cite{Kurniawan2018}. SVM classifiers are implemented in two different ways: OVO and OVR approach for classification. The result, after using three different kernels in SVM such as Linear, Polynomial, and Radial Base Function kernel (RBF), shows that the RBF kernel produces better average accuracy and Cohen's kappa value compared to Linear and Polynomial kernels, where the average accuracy of each kernel is 78.56\% for Linear, 81.40\% for Polynomial, and 82.51\% for RBF.
Rosenblum et al. \cite{rosenblum2016identifying} proposed an online handwritten feature-based tool for the diagnosis of dysgraphia in 3rd-grade students. 99 students have participated in the study where 50 students are proficient in handwriting. The handwriting proficiency screening questionnaire (HPSQ) approach is utilized to classify the students into two groups ( with handwriting difficulties and without handwriting difficulties). The students are asked to do multiple tasks in a paper attached to a tablet to collect the online handwritten data. ComPET tool is utilized for data collection and analysis. Initially, the students were asked to write a six-word sentence from a well-known children's in the Hebrew language. Different forms of time features, static properties of written product, and pressure features were extracted during this task for each student. In the second task, the students were asked to write a different six-word sentence. The features obtained in the second task are highly correlated with the features obtained in the second task. Thus the features obtained in the second task are omitted from the dataset. In the third task, students were asked to write a single Hebrew letter seven to ten times continuously. From that total time as well as the total number of segments in the written and total number of completed letters are extracted as the features. The fourth task is associated with the doodling of connected loops. Time, as well as the geometry of the loops produced, are extracted from the data to build features. The extracted features are utilized to train an SVM linear model for classification. The classification results indicate that the proposed methodology classifies the students with dysgraphia and without dysgraphia with a specificity and sensitivity of 90\%. Furthermore, the proposed study found that time-related features and pressure-related features are having high relevance in discriminating the normal and abnormal handwritings.
Zvoncak et al. \cite{zvoncak2018effect}proposed a novel intra-writer normalization approach for reducing the error rate in computer-based development dysgraphia diagnosis. The data for the study was collected from 97 students by making them write a paragraph on a paper affixed to the screen of the digitizing tablet. The ground truth value is labeled using the HPSQ approach. Common spatial, kinematic, temporal, and dynamic features were extracted for each student during the handwriting task. Four intra-writer normalization approach, l1, l2, l infinity, and Z score has been proposed in this work. Instead of classification, a regression task is performed in this work to estimate the effectiveness of the normalization methods. A gradient boosted tree-based regression model is trained in this work for estimating the handwriting quality scores and it is validated against the ground truth value obtained during the HPSQ approach. Various experiments were conducted to analyze the effectiveness of different feature combinations and normalization methods. Observed results indicate that the l2 normalization can reduce the error rate by 5\%.
A deep learning-based mobile application named "Pubudu" \cite{kariyawasam2019pubudu}is proposed for screening and intervention multiple learning disorders like dyslexia, dysgraphia, and dyscalculia in school children. In dysgraphia diagnosis, the proposed application is targeted for screening numerical as well as letter dysgraphia. In the letter dysgraphia screening approach, a CNN model is trained with the 5000 handwritten image data ( 3 Sinhala letters) collected from non-dysgraphic children. After that, the students with dysgraphia and without dysgraphia were asked to write the same letters, and multiple features such as success probability of written letters, the total number of correct letters, total number of incorrect letters, number of attempts, total time take as well as erase count were extracted from those handwritten data. The trained CNN model is used for automated estimation of the success probability of the written letter, the total number of correct letters, and the total number of incorrect letters. These extracted features are used to train an SVM model for letter dysgraphia screening. A similar approach was employed for numerical dysgraphia screening, instead of the customized letter dataset, the MNIST dataset was used to train the CNN model for automated feature extraction purposes. The proposed approach obtained an accuracy of 88\% and 90\% for screening letter dysgraphia and numerical dysgraphia.
A deep learning-based method is proposed in \cite{yogarajah2020deep} for automated detection of dyslexia-dysgraphia from handwritten images. For dataset preparation, an initial screening was conducted among the student of 1st grade to 5th grade to find out the students with poor scholastic records. From that 54 students were identified as having dyslexia-dysgraphia with the help of a conventional approach. Instead of making these students write new words or draw something, handwritten samples from their Hindi notebooks are collected, and images of some predefined two-letter words and vowel signs are extracted to form the dataset. The dataset consists of a total of 267 handwritten images where 164 images are of class dyslexia-dysgraphia and 103 are of class normal. The normal images are collected from students of similar age without any learning disabilities. The images are converted to grayscale format and resized to a fixed height of 113 pixels. A convolutional neural network model with three 2D convolutional layers is trained with the random patches of the images for automated diagnosis. The developed CNN model achieved a classification accuracy of 86.14 \%.
Devi et al. \cite{devi2021early}proposed a learning disability diagnosis tool for children with age 7 to 8 by utilizing the decision tree algorithm. The proposed tool can diagnose the existence of different learning disabilities including dysgraphia, dyscalculia, dyslexia, etc. An E-learning framework named testing scale is constructed for data collection and examination. The children have to answer a set of questionnaires to validate the existence of a learning disability. The questions are extracted from the WOODCOCK JOHNSON test of cognitive abilities. For dysgraphia, the set of questions is related to visual auditory, drawing lines, figure tracing, matching shape encircling, etc. The response of the students are extracted and features are created to train the decision tree algorithm. Only 40 students have participated in the experiment for data collection and evaluation.
A digital handwriting analysis-based system for dysgraphia identification is proposed in \cite{kedar2021identifying}. Similar to other works, the proposed method makes use of a Wacom digital tablet affixed with white paper and an interactive stylus pen for data collection. The participating students were asked to perform five different tasks and the total time, tip pressure, X coordinate, and Y coordinate of the pen tip on the paper at each instant during the task are stored in the device. From the stored data nine different feature values such as the total number of on paper segments, in air segments, total time, total air time, total time spent on paper, mean pressure, the velocity of the pen, etc. were extracted for further analysis. Three different machine learning classifiers are trained using the extracted features. The first and second classifier is based on the decision tree and random forest algorithm respectively. The third classifier is an ensemble model based on a soft voting approach. The ensemble classifier consists of three separately trained individual classifiers: decision tree, random forest, and SVM. A total of 60 students have participated in the proposed work for dataset construction. In the constructed dataset the Random forest classier achieved a maximum accuracy of 92.59\%.
Comparative analysis of few relevant ML based dysgraphia screening systems are provided in table \ref{tab:4}
\begin{landscape}[]
\begin{longtable}{|p{0.6cm}|p{1.7cm}|p{1.7cm}|p{1.7cm}|p{1.7cm}|p{1.7cm}|p{1.7cm}|p{1.7cm}|p{4cm}|}
\caption{Comparative analysis of machine learning based dysgraphia diagnosis systems}\label{tab:4}\\
\hline
Ref. & Age group & Participants & Tools & Task & Features & ML Algorithm & Performance & Remarks \\ \hline
\cite{Mekyska2017} & 3rd grade & 54 subjects ( 27 are having dysgraphia) & WACOM Intuos II xy digitizing tablet, Model GP-110 wireless electronic pen & Write a sequence of seven semi-HET (letters) & Kinematic measures, 34 nonlinear dynamic, other 7 Features & Linear discriminant analysis, Random forest & Sensitivity : 96\% & (+)Altitude/tilt and pressure features discriminate well \\ \hline
\cite{Asselborn2018} & Primary school & 298 subjects (56 are having dysgraphia) & Wacom graphic tablet & Copying a text for 5 mins & Static, Kinematic, Pressure, Tilt & Random forest & Sensitivity : 96.5 \% & - \\ \hline
\cite{Isa2019} & 7 - 12 years & - & Scanner & Samples of handwritten text & OCR,MSER & Artificial neural network & Accuracy : 71 \% & (-) Only image of handwritten text , which is suitable for spatial dysgraphia screening alone. \\ \hline
\cite{Mekyska2019} & 3rd and 4th grade & 76 subjects ( 15 are having dysgraphia) & Wacom Intuos Pro L (PHT-80) tablet & Seven drawing activities & Spatial, temporal, kinematic, dynamic, other – pen elevations and relative number of interruptions & XG-Boost & Specificity : 90\% & (+) Kinematic features are found to be more relevant for classification , specifically the kinematic features in vertical projection. \\ \hline
\cite{Zvoncak2019} & 3rd and 4th grade & 65 subjects ( 33 are having dysgraphia) & Digitizing tablet Wacom Intuos Pro L (PHT-80) digitizer with Wacom Inking pen. & Copy a short paragraph (63 words, 371 characters including spaces) & Kinematic, temporal, spatial, and dynamic & Support vector machine and Random forest & Sensitivity : 88\% & (+) The features from vertical movement and pressure are having highest discrimination power. It indicates that subject with dysgraphia are having difficulties in maintaining constant force on the pen tip and their vertical movement are less fluent \\ \hline
\cite{dui2020tablet} & Pre-schoolers & 104 subjects ( 28 are having dysgraphia) & An application in Unity 2018.3.2f1, for an iPad 6, with Apple Pencil & Copy square, copy sequence of symbols, Questionare & Gesture smoothness, pressure(mean value), Drawing kinematics & Logistic regression & AUC : 0.82 & (-) Since done in preschool stage, it is not possible to confirm that real handwriting difficulties would arise in the following year school stage \\ \hline
\cite{Gargot2020} & Around 9 years & 280 subjects (62 are having dysgraphia) & Sheet of paper affixed to a Wacom Intuos 4 graphic tablet & - & Static, kinematic, pressure, and tilt & Multiple linear regression models and K-means clustering & - & (+) Pressure on the pen tip and tilt features were found to be significant \\ \hline
\cite{Drotar2020} & 8–15 years & 120 subjects & WACOM Intuos Pro Large tablet & Writing letters, words, sentence & Spatiotemporal and kinematic features & AdaBoost & Accuracy : 80\% & (+)Pressure of pen tip on the surface and pen lifts were among the features with high discriminatory potential . The dataset includes subjects aged 8–15 years, which is basically a broad range and it has affected the overall accuracy. \\ \hline
\hline
\cite{rosenblum2016identifying} & 8–9 years & 90 subjects (49 are non-proficient ) subjects & WACOM Intuos II x-y digitizing tablet & Writing letters, words, sentence & Spatiotemporal , dynamic, kinematic and other features & Support vector machines & Accuracy : 90\% & (+)Pressure of pen tip on the surface and time were among the features with high discriminatory potential . \\ \hline
\cite{Sihwi2019} & 3rd to 6th grade & 32 subjects &Smartphone & Drawing and writing & Spatial, temporal , dynamic and other features & Support Vector machines & Accuracy : 82.51\% & (-) Less subjects
(+) Instead of binary classification , the proposed system can classify the data to different levels of dysgraphia such as mild , moderate , severe etc. \\ \hline
\cite{Dankovicova2019}& 10-13 years & 72 subjects (36 are having dysgraphia ) & WACOM Intuos Pro large & Writing letters, words, sentence & Spatial,temporal , dynamic, kinematic and other features & Support Vector machines & Sensitivity : 75.5\% & (-) Less subjects
(-) Not an acceptable sensitivity \\ \hline
\cite{devi2021early} & 7-8 years & 40 subjects & A web based framework & drawing lines, tracing figures , encircling matching shapes & Not explicitly mentioned & Decision tree & Not mentioned & (-) Number of data samples are very less. (-) Children with other learning disorders such as dyscalculia , dyslexia are also included in the experiment\\ \hline
\cite{kedar2021identifying} & 7-12 years & 60 subjects & WACOM digital tablet & Writing and Drawing . & Spatiotemporal, dynamic and kinematic features & Random forest & Accuracy : Sensitivity 92.85\% & (-) Number of data samples ( only 60 ) are less compared to other works. \\ \hline
\cite{Asselborn2020} & 5 - 12 years & 448 subjects (58 are having dysgraphia) & iPad tablet & Write the five first sentences used in the BHK test & Pressure, tilt and azimuth angle of the pen & PCA and unsupervised methods & & (+) Kinematic and pressure features are found to be more important than static or tilt features. (+)Age treated as a continuous variable since it is evident that handwriting evolves at a more rapid pace. (+)Handwriting quality measured on a numeric scale, allowing a new categorization of handwriting difficulties \\ \hline
\cite{Deschamps2021} & 2nd to 5th grade & 580 subjects (128 are having dysgraphia) & Four different graphic tablets & Perform french version of BHK test on tablet. & 100 features including spatial ,kinematic , dynamic etc. & SVM & Sensitivity :91\% & (+) Different types of tablets are used for data collection to ensure that the data is not device dependent . \\ \hline
\end{longtable}
\end{landscape}
\subsection{Automated Diagnosis Systems Without ML}
Non machine learning based automatede digital screening systems are also proposed in the literature for diagnosing the dysgraphia. This section briefly discus about popular Non-ML based digital dysgraphia screening systems. A tablet-based app named Play Draw Write has been proposed in \cite{dui2020tablet} for screening the handwriting skills of children in the preliteracy stage. The proposed tablet app will quantify three handwriting laws isochrony, homothety, and speed-accuracy trade-off for assessing the existence of dysgraphia markers. "The isochrony principle states that bigger gesture execution is accompanied by an increase in average movement speed to keep the movement time approximately constant \cite{viviani1982trajectory}. The homothety principle predicts that the fraction of time devoted to each letter of a word is kept constant and is independent of the total word duration \cite{viviani1982trajectory,lashley1951problem}." These principles seem to be altered in the subjects with dysgraphia along with dyslexia. The app is developed for Samsung Galaxy Tab A and children can use S-Pen with a rubber tip write or draw on the screen of the tablet App. Two tasks are available in the app, where the first one is a copy game and the second is a tunnel game. In the copy game, the students are asked to copy a few symbols and words to the canvas provided in the app using the stylus. The copying game is targeted for testing the isochrony and homothety. The tunnel game is targeted for evaluating the speed-accuracy trade-off.
Giordano and Maiorana \cite{giordano2014addressing}have developed a web-based software system for multiple handwritten gesture recognition which can be extended for online dysgraphia screening. The proposed software system is designed in a client-server manner and it can be used in any type of smartphone or tablet and computers. A modified version of the dynamic wrapping algorithm is employed in the server to recognize multiple hand gestures. The proposed software system offers multiple functions to execute different writing and drawing tasks. For each task, the system will store the time taken, degree of similarity with the reference line, amount of points outside the reference line in terms of percentage, average horizontal distance, etc. From the stored data, the administrator of the software tool can derive other parameters such as time taken for each stroke or their average time, length of the path, velocity, total air time, trajectory analysis, etc. These features can be utilized for the development of machine learning classifiers for dysgraphia diagnosis.
Raza et al.\cite{raza2017interactive} proposed an interactive mobile application for dysgraphia screening in children from age 5 to 12. The mobile application offers different activities to assess the handwriting quality of the student, assess the phonological dysgraphia, assess the surface dysgraphia, assess the copying ability. The student can use any compatible stylus pen to write on the screen of the device. The software utilizes a handwritten recognition method to recognize the spelling and after recognition, it is compared with the ground truth value to provide a score based on the correctness. A total of twenty words are given for students to write where seven words are used to estimate the existence of phonological dysgraphia, eight words for surface dysgraphia, and the remaining for copying ability.
TestGraphia \cite{Dimauro2020}is a software system proposed for the early diagnosis of dysgraphia. The proposed system is based on the conventional BHK method. They extended the conventional BHK method to the software device for automated diagnosis. In the BHK test, the students are asked to copy a few given texts to a paper in 5 minutes. From the handwritten document, scores are given to the 13 features including writing size, skewed writing, sharp angles, etc., and scoring these features is the crucial step in the BHK test. The TestGraphia will automatically compute the scores for 9 features ( "non-aligned left margin", "skewed writing" , "insufficient space between two words" , "sharp angles" , " broken links between two letters" , "irregular size of letters", " inconsistent height between letters with extension and letters without extension", "ambiguous letters"). Scores of the remaining features can be added by the doctors or occupational therapists manually using the dashboard of the software. For the automated scoring, initially, the images of the writing are subjected to segmentation ( individual line segmentation) using an image processing algorithm based on horizontal and vertical histograms. Multiple simple image processing algorithms are employed on the segmented lines to compute the feature score. The proposed system was tested with 109 students ( 2nd – 5th grade). The obtained result indicates that TestGraphia can diagnose the existence of dysgraphia with a sensitivity value of 0.83.
\subsection{Commercial Systems for dysgraphia diagnosis}
In this section, we briefly discuss the available software systems in the marketplace for dysgraphia diagnosis. There are very few commercially available systems for dysgraphia screening. 'Lexcercise' is a company founded in 2008 for supporting students with learning disabilities. Their web application provides an easy and cost-free way for preliminary assessment of dysgraphia in children \cite{Le}. The whole procedure for assessment consists of three sections. The parents or caretakers of the student are asked to fill out the answers to the provided questions about the student. In the first section, the questions are related to letter and number writing (such as reverse writing, messy writing, mixing up of lowercase and uppercase). The questions in the second section and third sections are targeted for assessing the writing convention and writing proficiency of the student respectively. They will provide the service of the dysgraphia therapist also. Upon completion of the assessment, the parents or caretaker will be asked to submit the actual writing samples of the student to a dysgraphia therapist, if the assessment displays any risk. \cite{Addit} also, provide a web-based tool for the free assessment of dysgraphia. It is similar to \cite{Le}, just a screener test to decide whether the child is having the symptoms of dysgraphia. The assessment consists of 16 basic questions where most of which are considered to be the prevalent symptoms of dysgraphia. All the questions are prepared based on the criteria from the learning disabilities association of America. Unlike \cite{Le}, here they will not provide the therapist, but they will give suggestions for the parents to check with an occupational therapist if the test is positive.
Dyscreen \cite{Dy} is a smartphone application developed by the Australian company Dystech in 2020 for self-screening of dysgraphia and dyslexia using the power of artificial intelligence. Dystech claims that they are the pioneer in this field to introduce the first AI-based smartphone application for dysgraphia screening. Currently, the application is available for both iOS and Android. The dysgraphia screening functionality of the application is very simple. Just take the picture of the student's handwritten text and upload it via the application. The intelligent machine learning model deployed in the cloud will classify the handwritten sample into positive or negative classes and provide feedback to the user. Since the processing of the images is done in the cloud, internet connectivity is required for dysgraphia screening. Dystech claims that the machine learning model can predict the existence of dysgraphia with an accuracy of 95.6\%. The dysgraphia screening in Dyscreen is free of cost. The same application provides other services such as dyslexia screening, but it is not free of cost.
A team of professionals and engineers from Hongkong university of science and technology has developed an artificial intelligence-based smartphone application named " AI Dysgraphia Pre-screening' for dysgraphia screening \cite{AIdy}. The application is free and available on both Android as well as iOS. The application provides a 15-minute drawing exercise for the student. Upon completing the drawing exercise the essential dynamic and static features are extracted from the handwritten data and processed for data analysis using trained ML algorithms. Internet connectivity is required to access the app services. Compared to Dyscreen this app considered the dynamics of writings also for screening dysgraphia.
To the best of our knowledge, these are the available software tools for dysgraphia screening in the market. All are free to access, however have many limitations. The Dyscreen just considers the images only, so the dynamics of the handwriting are not considered. On the other hand, in \cite{AIdy} the data are collected via smartphone screen, and it is very different from a normal paper, and friction while writing will be very different. The marketplace lacks a sophisticated smart device that can intelligently detect dysgraphia.
\section{Discussion and Future recommendations}
The popularity and the built-in capabilities of digitizing tablets to acquire information such as the position of the pen tip, on surface/ in-air pen position, pen tip pressure, the azimuth angle of pen with respect to the tablet surface, the tilt of pen, timestamp \cite{Mekyska2017} made it a suitable tool for the handwriting analysis. Stroke dimensions, Velocity, Acceleration, Jerk, Pressure, Tilt, Temporal, Azimuth angle, Number of pen elevations are the features that have more or less equal prominence in the dysgraphia discrimination systems \cite{Mekyska2017,Asselborn2018,Mekyska2019,Zvoncak2019,Dui2020,Drotar2020,Asselborn2020,Gargot2020} . The most relevant features that are spotted out in a multitude of works are Kinematic and Pen tip pressure.
Compared to the approaches merely based on handwriting images,\cite{wu2019automated,hen2019characteristics} the tablet-based techniques could explore more characteristics of handwriting, which turned out to be significant for the detection of dysgraphia. The data acquisition in the latter approach involved writing with a normal pen or an electronic pen on paper overlaid on the tablet. However, the lower friction surface of tablet computers modifies graphomotor execution, which in turn contradicts the purpose \cite{Guilbert2019}. Moreover, the pressure sensitivity of these tablets may vary depending on the model \cite{Prunty2020}. Even though the grip force between the hands and writing instrument has a substantial correlation to the fine motor performance \cite{Lin2017} and the improper handling of the pen is prevalent in motor dysgraphia \cite{Biotteau2019}, it is not considered as an attributing factor for dysgraphia prediction. The existing methodologies are meant to provide an overall diagnostic assessment as being dysgraphic/ non-dysgraphic except a few with a grading scale for handwriting difficulties . In addition, a single task (copying/ writing/ drawing) is not sufficient for the judgment .
In our knowledge there are number of gaps in the automated dysgraphia detection system domain. Some of the recommendations for future works are :
\begin{itemize}
\item Since the pressure sensitivity of the tablets varies by models and tablets have lower friction surfaces, future dysgraphia diagnosis systems can consider normal paper (writing on paper) based systems rather than writing on tablets.
\item The future systems can explore the importance of improper handling of pen/pencil and grip force between hand and writing instruments as relevant features for discriminating normal and abnormal handwriting.
\item Instead of developing binary systems ( dysgraphia or not) for dysgraphia diagnosis, future work should consider developing systems for recognizing different types of dysgraphia or grading the levels of dysgraphia. Because the symptoms of dysgraphia vary with different types and levels of disability.
\end{itemize}
\section{New frameworks for dysgraphia daignosis}
To address some of the limitations in the existing frameworks, we propose new frameworks for dysgraphia diagnosis. Overview of the framework is shown in figure \ref{fig7-1}. The framework consists of 4 steps. In the first step, children are classified into normal and dysgraphic children using psychological methods. Then, an automatic system is developed that takes into account data collection and processing. In this step, depending on the type of approach used, other processing steps can be added. For example, for handwriting on paper, preprocessing should be added, and a fusion method for features and classifiers can be used. Furthermore, instead of
feature extraction and classifier, deep learning methods can also be used. In the collection step, the content that the participants are asked to write or draw is very important. Depending on what equipment is available, a combination of this equipment can also be considered. In feature extraction, off-line and on-line features are considered based on the approach of data collection.
\begin{landscape}
\begin{figure}[h]%
\centering
\includegraphics[width=1.6\textwidth]{img7.png}
\caption{Overview of proposed framework}\label{fig7-1}
\end{figure}
\end{landscape}
It should be noted that step 1 is considered to obtain training data. In step 3, you can start adding test data to evaluate the system based on the trained model and decide to which classes the new case (test data) belongs. In this step, we can assign different weights to each classifier depending on the type of data collection. For example, since handwriting on paper can lead to low accuracy, we consider a lower weight. In the last step, which is very important, an algorithm is proposed for tracking the dysgraphia cases
and their treatment process.
This step can be carried out in the periods proposed by the psychologist.
After dysgraphia cases are detected, treatment should be performed. Impact of the treatment method can be explored during the tracking step. It is possible to add new treatments and study their effectiveness.
It should be noted that some tasks and steps can be added to the framework such as detecting type of dysgraphia. The new framework consist of two novel methods (figure \ref{fig7} ) that can be used along with existing methods for automatic screening of spatial dysgraphia and motor dysgraphia, respectively.For the former one, the dynamic characteristics of handwriting are captured with the help of two types of sensors- (1) Force-sensitive resistors (FSR) attached to the pencil for capturing pencil grip patterns (2) sEMG sensor attached to the hand for capturing muscle activity. The acquired data is preprocessed followed by feature extraction and binary classification with the help of prebuilt machine learning models.
Handwriting is a prominent fine motor skill acquired during the early developmental stage. Several types of research have been carried out to understand the detailed aspects and characteristics of handwriting. Earlier in 1961, Herrick and Otto have signified the barrel pressure variability as an important distinguishing aspect of individual handwriting \cite{herrick1961pressure}. The grip force variability and its effect on handwriting legibility are studied in \cite{falk2010grip}. The demonstrated results indicated that grip force variability during the entire writing task was lower for non-proficient writers. Reference \cite{lin2017comprehension} investigated the pen grip kinetics of school-age children to validate the hypothesis of correlation between force control when handling a pen and fine motor performance. The significance of the middle finger compared to the thumb and index finger for the force control is also demonstrated. The writing speed being unaffected by the pencil grip force and muscular activity holds promise for the application of these cues in the remedial program for disabled persons.Surface Electromyography (sEMG) has proved out to study muscle coordination \cite{hug2018surface}, which can be exploited to understand the poor muscle tone in motor dysgraphia. sEMG signals generated during handwriting have been already applied for character recognition \cite{linderman2009recognition,beltran2020multi}and Parkinson's disease diagnosis \cite{loconsole2019model}. It has been used in Arabic handwriting character recognition as well \cite{lansari2003novel}.
For the latter one, a mobile phone camera is utilized to capture the image of handwritten piece. Relevant features are extracted from the image and further classification is carried out with the aid of machine learning models. The illegibility aspect of handwriting due to the defect in understanding of space results in spatial dysgraphia. Suitable line/ word segmentation techniques \cite{alma2006recognition} can be utilized to observe and compare the line as well as the word spacing. Further, document analysis algorithms \cite{Mekyska2017} serves to identify the dysgraphic characteristics such as skewed writing, irregular writing size, etc.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{Img8.jpg}
\caption{Work flow of novel methods}\label{fig7}
\end{figure}
\section{Conclusion}
Learning disabilities are often unrecognized, which makes it often misinterpreted as a lack of intelligence. This comes from the fact that the screening process is quite complex. In this work, we presented survey of existing tools and method for the preliminary screening of motor and spatial dysgraphia, which is characterized by impaired handwriting skills. We discussed many AI based and Non AI based automated systems for dysgraphia screening. We provided a comparative analysis of these systems and highlighted the strength and weakness . Later we proposed a novel frame work for automated dysgraphia screening by combining the existing methods along with new methods .
\section*{Declarations}
\begin{itemize}
\item Funding
This publication was supported by Qatar University Graduate Assistant
Grant. The contents of this publication are solely the responsibility of the
authors and do not necessarily represent the official views of Qatar University.
\item Conflict of interest/Competing interests
The authors declare that they have no competing interests.
\item Ethics approval
Not applicable
\item Consent to participate
Not applicable
\item Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
\item Authors' contributions
JK have made major contribution in designing the work, reviewing the
literature and writing the manuscript. The concept of the work was contributed by SA and JK . SA was the principal investigator of the project.
SA as well as YA revised and edited the manuscript.SK provided guidance
and as well helped in drafting the manuscript. All authors read and approved
the final version of manuscript.
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 29 |
\section*{Introduction}
We will consider $W^{2,p}$-viscosity solutions $u:\Omega\subset\mathds{R}^n\longrightarrow\mathds{R}$ for
\begin{equation}\label{pde}
\begin{cases}
F(D^2u(x))=g\Big(\vert u\geq u(x)\vert\Big) & \text{in $\Omega$}\\
u=\psi & \text{on $\partial\Omega$}
\end{cases}
\end{equation}
where $\Omega\subset\mathds{R}^n$ is an open, bounded and connected set, with $C^{1,1}$ boundary. The operator $F:\mathcal{S}\longrightarrow\mathds{R}$, is a convex, uniformly elliptic operator with ellipticity constants $0<\lambda\leq\Lambda$, where $\mathcal{S}:=\{$real n$\times$n symmetric matrices$\}$. For simplicity we will assume $F(0)=0$. We will also require that $F$ satisfies the following structure condition
\begin{equation}\label{structurecondition}
\mathcal{M}^-(M-N)\leq F(M)-F(N)\leq\mathcal{M}^+(M-N),
\end{equation}
for all $M,N\in\mathcal{S}$. Here, $\mathcal{M}^-$ and $\mathcal{M}^+$ are the extremal Pucci operators
$$\mathcal{M}^-(M)=\lambda\sum_{e_i>0}e_i+\Lambda\sum_{e_i<0}e_i,$$
and
$$\mathcal{M}^+(M)=\Lambda\sum_{e_i>0}e_i+\lambda\sum_{e_i<0}e_i,$$
where $e_i=e_i(M)$ are the eigenvalues of $M$. In the right hand side of \eqref{pde}, $\vert\cdot\vert$ denotes the n-dimensional Lebesgue measure, and $g:[0;\vert\Omega\vert]\longrightarrow\mathds{R}$ is a continuous function. We will adopt the notation
$$u\geq u(x):=\{y\in \Omega:u(y)\geq u(x)\}$$
for the superlevel sets of $u$. Finally, we consider a boundary value $\psi\in W^{2,p}(\Omega)$ for $p>n$.
\bigskip
The motivation to study this problem is to generalize Grad equations in plasma physics, and its approximations, to nonlinear operators. These equations were introduced by Harold Grad in \cite{grad}, and appear in the literature as Queer Differential Equations(QDE), or Grad Equations. They arise in modelling plasma, which is confined under magnetic forces in a toroidal container. Grad noticed that a simplified version of plasma equations was possible using $u^*$, the increasing rearrangement of $u$:
$$u^{*}(t):=\inf\{s:\vert u<s\vert\geq t\}.$$
Here is where we start building a connection with our approximation problem (\ref{pde}). Notice that heuristically, $u^*$ is the inverse of the measure of the sublevel sets of $u$. In \cite{grad}, he demonstrated that there are profile functions $\mu$ and $\nu$ which are prescribed by the dynamics of the plasma; consequently his equation reads
$$\Delta u=-\mu'(u)(u^*{'})^\gamma-\gamma\mu(u)(u^{*}{'})^{\gamma-2}u^{*}{''}-\frac{1}{2}(\nu^2(u)){'}(u^{*}{'})^2-\nu^2(u)u^{*}{''}$$
for some power $\gamma$. For clarity we avoided the arguments: $u$ and its derivatives are evaluated at some point $x$ while the rearrangements and its derivatives are evaluated at $t:=\vert u<u(x)\vert$. Many authors attacked the problem trying to approximate these equations. The first one was introduced by Roger Temam in \cite{temam}, and then improved by Mossino and Temam in \cite{mossinotemam}. They studied properties of directional derivatives of the rearrangement function, and proved existence results for
$$\Delta u(x)=g(\vert u<u(x)\vert,u(x))+f(x).$$
Years later, Laurence and Stredulinsky, in \cite{lsnewapproach} and \cite{lsbootstrap}, studied a model equation, closer to Grad's formulation. They considered the particular case when $\gamma=2$, $\mu\equiv 1/2$ and $\nu\equiv 0$ obtaining
$$\Delta u(x)=-u^{*}{''}(\vert u<u(x)\vert).$$
Even this simplified case presents many difficulties. The authors introduced a very interesting approach to the problem: they described an approximation with solutions to a $N-$free boundary problem. In order to apply this process they assumed extra regularity for the level sets of a solution, which is mentioned later in Section 3.
\bigskip
The idea behind this paper is the following: all of these previous papers addressed the problem with a variational method for the Laplacian; instead, we will use a viscosity approach for a general family of fully nonlinear operators. A similar equation to the one of Mossino and Temam is studied, and \textbf{even for the case with the Laplacian we improve the regularity results}.
\bigskip
The paper is organized as follows: In the first section we cite some preliminary definitions. Mainly, we state the basics of $W^{2,p}$-viscosity solutions. The classic viscosity solutions' theory does not apply to this particular problem because of our right hand side in (\ref{pde}). Disregarding the regularity of $g$ and $u$, we notice that having $\vert u=c\vert>0$ for some constant $c$ makes the right hand side of our equation discontinuous. Therefore, we adopt this $W^{2,p}$-viscosity notion defined in \cite{ccks} by Caffarelli, Crandall, Kocan and Swiech which allows merely measurable ``ingredients''. In their paper they proved existence and interior $W^{2,p}$-estimates for solutions to an equation with a fixed right hand side $f(x)$. Strongly based on their results, Winter in \cite{winter} extended this regularity up to the boundary proving global $W^{2,p}$-estimates for viscosity solutions and an existence result for $W^{2,p}$-strong solutions. For clarity in the presentation, the results from the literature that will be used through the paper will be addressed at the Appendix A.
In section 2 we state and prove the main theorem of existence and global regularity. The idea of the proof is to
\begin{itemize}
\item freeze $u$ in the right hand side
\item solve the resulting equation using \cite{ccks} theory
\item build a sequence of right hand sides and solutions
\item use a fixed point argument and a convergence theorem to find a solution
\end{itemize}
In section 3 we prove more regularity under additional hypothesis. As long as $\vert\nabla u\vert$ is uniformly bounded below, or equivalently, if we have a uniform interior ball condition for the level sets of $u$, then we have $C^{0,\alpha}$ regularity for the right hand side. This estimate turns into $C^{2,\alpha}$ regularity for the solution $u$. We cannot ensure regularity for the level sets, but if we start with a regular enough domain, say $\partial\Omega$ with a uniform interior ball condition, then we gain $C^{2,\alpha}$ regularity for $u$ in a neighborhood of the boundary.
\section{Preliminary Definitions}
First we are going to present the definitions of viscosity solutions for fully nonlinear equations with measurable ingredients, described in the paper of Caffarelli-Crandall-Kocan-Swiech \cite{ccks}. In this setting we work with the problem
\begin{equation}\label{pdegeneral}
\begin{cases}
F(D^2u(x))=f(x) & \text{in $\Omega$}\\
u=\psi & \text{on $\partial\Omega$}
\end{cases}
\end{equation}
where our right hand side is a fixed measurable function $f$.
\begin{defn}
Let be $F$ a uniformly elliptic operator, $f\in L^{p}(\Omega)$ for $p>n/2$. Let $u:\Omega\longrightarrow\mathds{R}$ be a continuous function, we say it is a \textbf{$W^{2,p}$-viscosity subsolution} of \eqref{pdegeneral} in $\Omega$, if $u\leq \psi$ on $\partial\Omega$ and the following holds: for all $\varphi\in W^{2,p}(\Omega)$ such that $u-\varphi$ has a local maximum at $x_0\in \Omega$ then
$$ess\limsup_{x\rightarrow x_0}F(D^2\varphi(x))-f(x)\geq 0.$$
We define supersolutions in the same way; $u$ is a \textbf{$W^{2,p}$-viscosity supersolution} of \eqref{pdegeneral} in $\Omega$, if $u\geq \psi$ on $\partial\Omega$ and the following holds: for all $\varphi\in W^{2,p}(\Omega)$ such that $u-\varphi$ has a local minimum at $x_0\in \Omega$ then
$$ess\liminf_{x\rightarrow x_0}F(D^2\varphi(x))-f(x)\leq 0.$$
\end{defn}
\begin{rmk}
We can also use this alternative definition for $W^{2,p}$-viscosity subsolutions. For all $\varphi\in W_{loc}^{2,p}(\Omega)$, for all $\varepsilon>0$, and $O\subset\Omega$ open such that
$$F(D^2\varphi(x))-f(x)\leq -\varepsilon,$$
a.e. in $O$, then $u-\varphi$ cannot have a local maximum in $O$.
\end{rmk}
Because we will use Winter's results, we also add the definition of $W^{2,p}$-strong subsolutions.
\begin{defn}
In the same setting as before, $u$ is a \textbf{$W^{2,p}$-strong subsolution} of \eqref{pdegeneral} in $\Omega$, if $u\leq \psi$ on $\partial\Omega$ and
$$F(D^2u(x))\geq f(x)$$
a.e. in $\Omega$.
\end{defn}
\section{Main result}
In this section we state and prove existence and a first global regularity result.
\begin{thm}\label{main}
Our problem (\ref{pde})
\begin{equation*}
\begin{cases}
F(D^2u(x))=g\Big(\vert u\geq u(x)\vert\Big) & \text{in $\Omega$}\\
u=\psi & \text{on $\partial\Omega$}
\end{cases}
\end{equation*}
with the setting given in the introduction, has a $W^{2,p}$-viscosity solution $u$. Furthermore, $u\in W^{2,p}(\Omega)$ and we have the following estimate
$$\Vert u\Vert_{W^{2,p}(\Omega)}\leq C\Big[\Vert u\Vert_{L^{\infty}(\Omega)}+\Vert \psi\Vert_{W^{2,p}(\Omega)}+\Vert g(\vert u\geq u(x)\vert)\Vert_{L^{p}(\Omega)}\Big].$$
\end{thm}
\begin{cor}
Using Sobolev embedding theorem we get that a solution is in $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha<1$, provided that $\psi\in W^{2,p}$ for every $p>n$.
\end{cor}
\bigskip
The structure of the proof for Theorem \ref{main} is somehow simple; we set an approximating problem \eqref{pdeepsilon}, we prove the existence of a solution for it and then we take the limit to obtain the solution to \eqref{pde}. Before presenting this approximating problem in Lemma \ref{approxlemma}, we give a quick explanation on the reasoning behind it. Recall that the results from the appendix will be used next: existence and uniqueness, fixed point, and convergence.
If in \eqref{pde} we freeze a function $v\in Lip(\Omega)$ for the right hand side, i.e. $f_v(x):=g\Big(\vert \{y\in \Omega:v(y)\geq v(x)\}\vert\Big)$, we get
\begin{equation}\label{pdefixedrighthandside}
\begin{cases}
F(D^2u(x))=f_v(x) & \text{in $\Omega$}\\
u=\psi & \text{on $\partial\Omega$}.
\end{cases}
\end{equation}
Then the hypothesis of Theorem \ref{winter} are satisfied and there exists a unique $W^{2,p}$-viscosity solution $u$ to \eqref{pdefixedrighthandside}. The next step would be to apply the fixed point Theorem \ref{fixed} for the application $T(v)=u$. The problem is that we cannot ensure continuity for $T$ because of the right hand side of \eqref{pdefixedrighthandside}. Not even if we require more regularity for $v$(not even $C^\infty$ works). We will overcome this inconvenient solving an auxiliary problem with a smoothened right hand side which allows us to perform the fixed point argument. Given $v\in Lip(\Omega)$, $\varepsilon>0$, consider
\begin{equation}\label{pdefixedrighthandsideepsilon}
\begin{cases}
F(D^2u(x))=f_v^\varepsilon(x):=g\Big(\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert v\geq v(x)-h\vert dh\Big) & \text{in $\Omega$}\\
u=\psi & \text{on $\partial\Omega$}.
\end{cases}
\end{equation}
Because $f_v^\varepsilon\in L^p(\Omega)$, using Theorem \ref{winter} we have existence and uniqueness of a $W^{2,p}$-viscosity solution $u\in W^{2,p}(\Omega)$ to \eqref{pdefixedrighthandsideepsilon} with the estimate
\begin{equation*}
\Vert u\Vert_{W^{2,p}(\Omega)}\leq C\Big[\Vert u\Vert_{L^{\infty}(\Omega)}+\Vert \psi\Vert_{W^{2,p}(\Omega)}+\Vert f_v^\varepsilon\Vert_{L^{p}(\Omega)}\Big].
\end{equation*}
Now we can state our approximation lemma.
\begin{lemma}\label{approxlemma}
Given $\varepsilon>0$, there exists a $W^{2,p}$-viscosity solution $u_\varepsilon$ to
\begin{equation}\label{pdeepsilon}
\begin{cases}
F(D^2u(x))=f_u^\varepsilon(x), & \text{in $\Omega$}.\\
u=\psi, & \text{on $\partial\Omega$}.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
The existence is proved, as we remarked, using the fixed point Theorem \ref{fixed}. We define $T:Lip(\Omega)\longrightarrow Lip(\Omega)$ as the application defined by \eqref{pdefixedrighthandsideepsilon} and the existence and uniqueness theorem, i.e., $T(v)=u$. In order to prove the hypothesis required for $T$, we will make use of the convergence Theorem \ref{convergence}.
\textit{Continuity of T:} If we consider $v_k\overset{Lip}{\longrightarrow}v$, then, does $u_k:=T(v_k)\overset{Lip}{\longrightarrow}T(v)$?
We know that $u_k\in W^{2,p}(\Omega)$ and
\begin{equation*}
\Vert u_k\Vert_{W^{2,p}(\Omega)}\leq C\Big[\Vert u_k\Vert_{L^{\infty}(\Omega)}+\Vert \psi\Vert_{W^{2,p}(\Omega)}+\Vert f_v^\varepsilon\Vert_{L^{p}(\Omega)}\Big]\leq \widetilde{C}
\end{equation*}
with $\widetilde{C}$ independent on $k$. This is achieved using Alexandroff-Bakelman-Pucci (ABP) estimates
\begin{equation*}
\sup_{\Omega}u_k\leq \sup_{\partial\Omega}u_k+C\Vert f_v^\varepsilon\Vert_{L^{p}(\Omega)}
\end{equation*}
and the equivalent for the $\inf_{\Omega}u_k$. This ABP version for measurable ingredients is stated in Caffarelli-Crandall-Kocan-Swiech(Proposition 3.3 in \cite{ccks}). We also have the estimate
\begin{equation*}
\Vert f_v^\varepsilon\Vert_{L^{p}(\Omega)}\leq \vert\Omega\vert^{1/p}g\big(\vert\Omega\vert\big)
\end{equation*}
which makes $\widetilde{C}$ even independent on $\varepsilon$.
Now consider $u_{k_j}$ any subsequece of $u_k$. Using Rellich-Kondrachov theorem we can find a subsequence $u_{k_{j_i}}$ of $u_{k_j}$(for simplicity we will use the notation $u_i:=u_{k_{j_i}}$) converging to some $u_\varepsilon$ in the Lipschitz norm, i.e., $u_i\overset{Lip}{\longrightarrow}u_\varepsilon$. If we can prove that $u_\varepsilon$ is the unique $W^{2,p}$-viscosity solution to \eqref{pdefixedrighthandsideepsilon}($T(v)=u_\varepsilon$), then we have the convergence $u_k=T(v_k)\overset{Lip}{\longrightarrow}u_\varepsilon=T(v)$. Therefore, we obtain the continuity for $T$.
Every $u_i\in C^0(\Omega)$ is the unique $W^{2,p}$-viscosity solution to \eqref{pdefixedrighthandsideepsilon} with $v_i$ in the right hand side. We have $\Omega_i=\Omega$, and $F_i=F$ fixed for every $i$. The convergence $u_i\overset{Lip}{\longrightarrow}u_\varepsilon$ implies the locally uniformly convergence. So we only need to check the convergence
$$\Vert f^\varepsilon_v(x)-f^\varepsilon_{v_i}(x)]\Vert_{L^{p}(B_r(x_0))}\longrightarrow 0$$
in order to satisfy all the hypothesis in the convergence Theorem \ref{convergence}. We know that $v_i\overset{Lip}{\longrightarrow}v$, then $\delta_i:=\Vert v_i-v\Vert_{L^{\infty}}\longrightarrow 0$. Thus let $x$ and $h$ fixed,
$$\vert v_i\geq v_i(x)-h\vert\leq\vert v+\delta_i\geq v(x)-\delta_i-h\vert\searrow\vert v\geq v(x)-h\vert$$
as $i\longrightarrow\infty$ and also
$$\vert v_i\geq v_i(x)-h\vert\geq\vert v-\delta_i\geq v(x)+\delta_i-h\vert\nearrow\vert v>v(x)-h\vert.$$
We can show that $\vert v>v(x)-h\vert=\vert v\geq v(x)-h\vert$ for a.e. $h$ in $[0;\varepsilon]$. This happens if and only if $\vert v=v(x)-h\vert=\vert v^{-1}(v(x)-h)\vert=0$ for a.e. $h\in[0;\varepsilon]$. A corollary of Rademacher theorem, says that if $v$ is a Lipschitz function then for a.e. $y\in v^{-1}(\alpha)$, $\nabla v(y)=0$. Therefore $\vert v^{-1}(v(x)-h)\vert=\vert v^{-1}(v(x)-h)\cap\{\nabla v(y)=0\}\vert$. Using a corollary of the Coarea formula we get also that $\mathcal{H}^{n-1}\Big( v^{-1}(v(x)-h)\cap\{\nabla v(y)=0\}\Big)=0$. Here $\mathcal{H}^{n-1}$ stands for the $(n-1)$-dimensional Hausdorff measure. Then for every $x\in\Omega$ we get the convergence
$$\vert v_i\geq v_i(x)-h\vert\longrightarrow\vert v\geq v(x)-h\vert$$
for a.e. $h$. Applying the dominated convergence theorem first, and the continuity of $g$ we get
\begin{equation}\label{gconvergence}
g\Big(\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert v_i\geq v_i(x)-h\vert dh\Big)\longrightarrow g\Big(\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert v\geq v(x)-h\vert dh\Big)
\end{equation}
as $i\longrightarrow\infty$. This last result is the pointwise convergence of $f^\varepsilon_{v_i}$ to $f^\varepsilon_{v}$. Again, applying the dominated convergence theorem we get the $L^p$ convergence needed. So all the hypothesis are satisfied to apply the theorem and therefore $T$ is continuous.
\textit{Compactness of T:} Let $v_k$ a bounded sequence in $Lip(\Omega)$ then $u_k:=T(v_k)\in W^{2,p}(\Omega)$ is bounded as before. After Rellich-Kondrachov there exists a convergent subsequence.
\textit{Boundedness of the eigenvectors:} We have to prove that the set
$$\Gamma:=\{v\in Lip(\Omega): \exists\gamma\in[0;1] \text{ such that }v=\gamma T(v)\}$$
is bounded. Suppose by contradiction that it is not. First we note that $0\in\Gamma$ with
$\gamma=0$, and for every $0\neq v\in\Gamma$ the $\gamma$ associated with $v$ is not zero. Suppose then that there exist a sequence of nonzero elements $v_k\in\Gamma$, and a respective sequence $\gamma_k$ such that $v_k=\gamma_kT(v_k)$ and $\Vert v_k\Vert_{Lip(\Omega)}\longrightarrow\infty$. Because $v_k\in Lip(\Omega)$, then $v_k/\gamma_k\in W^{2,p}(\Omega)$ and
$$\Vert v_k\Vert_{Lip(\Omega)}\leq\Vert \dfrac{v_k}{\gamma_k}\Vert_{Lip(\Omega)}\leq C\Vert \dfrac{v_k}{\gamma_k}\Vert_{W^{2,p}(\Omega)}\leq \widetilde{C}$$
which is a contradiction. Therefore $\Gamma$ is bounded.
The hypothesis of Schaefer's theorem are satisfied, so there exists a Lipschitz fixed point $u_\varepsilon$ for $T$, i.e., $u_\varepsilon=T(u_\varepsilon)$. Moreover, by Theorem \ref{winter}, $u_\varepsilon$ is a $W^{2,p}$-viscosity solution to \eqref{pdeepsilon}, which is in $W^{2,p}(\Omega)$.
\end{proof}
The purpose of finding such a $u_\varepsilon$, was to approximate a solution for \eqref{pde}. Then the following question is if we can take the limit $\varepsilon\longrightarrow\infty$.
\begin{proof}[Proof of Theorem \ref{main}]
For every $\varepsilon>0$ we have a solution $u_\varepsilon\in W^{2,p}(\Omega)$ with uniformly bounded $W^{2,p}$ norm(with respect to $\varepsilon$). Then there exists a subsequence(that we will also call $u_\varepsilon$) and a Lipschitz function $u$ such that $u_\varepsilon\overset{Lip}{\longrightarrow} u$. So $u_\varepsilon\longrightarrow u$ locally uniformly and we will be able to apply again the convergence Theorem \ref{convergence}. In this case we have on the right hand sides, the $L^p$ functions
$$f_u(x):=g\big(\vert u\geq u(x)\vert\big)$$
and
$$f_{u_\varepsilon}^\varepsilon(x):=g\Big(\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u_\varepsilon\geq u_\varepsilon(x)-h\vert dh\Big).$$
We are left to prove the convergence
$$\Vert f_u-f^\varepsilon_{u_\varepsilon}(x)\Vert_{L^{p}(B_r(x_0))}\longrightarrow 0.$$
By triangle inequality
$$\Vert f_u-f^\varepsilon_{u_\varepsilon}(x)\Vert_{L^{p}(\Omega)}\leq\Vert f_u-f^\varepsilon_{u}(x)\Vert_{L^{p}(\Omega)}+\Vert f_u^\varepsilon-f^\varepsilon_{u_\varepsilon}(x)\Vert_{L^{p}(\Omega)}.$$
We have that the second term goes to zero as in previous calculations \eqref{gconvergence}. We will use a similar argument for bounding the first term.
\begin{align*}
\vert u\geq u(x)\vert&\leq \vert u\geq u(x)-h\vert\\
&=\vert u\geq u(x)\vert+\vert u(x)>u\geq u(x)-h\vert\\
&\leq\vert u\geq u(x)\vert+\vert u(x)>u\geq u(x)-\varepsilon\vert.
\end{align*}
Then
\begin{align*}
\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u\geq u(x)-h\vert dh&\geq\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u\geq u(x)\vert dh\\
&=\vert u\geq u(x)\vert
\end{align*}
and
\begin{align*}
\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u\geq u(x)-h\vert dh&\leq
\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u\geq u(x)\vert+\vert u(x)>u\geq u(x)-\varepsilon\vert dh\\
&=\vert u\geq u(x)\vert+\vert u(x)>u\geq u(x)-\varepsilon\vert.
\end{align*}
Therefore
\begin{equation*}
\dfrac{1}{\varepsilon}\int_0^\varepsilon\vert u\geq u(x)-h\vert dh\searrow \vert u\geq u(x)\vert
\end{equation*}
as $\varepsilon\longrightarrow\infty$. Accordingly we obtained pointwise convergence for $f^\varepsilon_u$ to $f_u$, which after the dominated convergence theorem implies the convergence on the $L^p$ norm.
Hypothesis of Theorem \ref{convergence} are satisfied and we finally obtain our main result: $u$ a $W^{2,p}$-viscosity solution to \eqref{pde}, in $W^{2,p}(\Omega)$ and with the corresponding estimates.
\end{proof}
The last remark of this section is that we obtain an explicit formula for the $0$ Dirichlet problem in a ball.
\begin{ex}\label{ball}
When $\Omega=B_r(x_0)$, $F=\Delta$, $g(t)=-t$ and $\psi=0$ we have the solution:
\begin{equation*}
\tilde{u}(x)=\frac{\omega_n}{2n(n+2)}\Big[r^{n+2}-\vert x-x_0\vert^{n+2}\Big]
\end{equation*}
where $\omega_n$ is the measure of the n-dimensional unit ball. In a similar way we can prove that $\frac{1}{\Lambda}\tilde{u}$ is a solution when $F=\mathcal{M}^-$(respectively $\frac{1}{\lambda}\tilde{u}$ for $F=\mathcal{M}^+$). We will use this example in the next section to build subsolutions that can be used as barriers to prove gradient bounds.
\end{ex}
\section{Further Regularity}
In order to gain more regularity for our solution $u$ we probably need to get some regularity for the right hand side $f_u$. So far, in the case when $u$ has flat regions, $f_u$ is not even continuous. In principle, this discontinuity does not depend on the regularity of $u$ but on its flat regions. We can prove that under the negativity of $g$, $u$ is not allowed to have these flat regions with positive measure.
\begin{rmk}
Let be $u$ a solution for (\ref{pde}) with right hand side $g<0$ in $(0;\vert\Omega\vert]$, then
$$\vert u=a\vert = 0$$
for every constant $a\in\mathbb{R}$.
\end{rmk}
\begin{proof}
Suppose that there exists an $a\in\mathds{R}$ such that $A:=\{u=a\}$ satisfies that $\vert A\vert>0$. Then, by a classic result from Stampacchia we obtain that
\begin{equation*}
\nabla u(x)=0 \quad\text{for a.e. $x\in A$}.
\end{equation*}
Now we can define $A':=A\cap\{\nabla u=0\}$, and apply Stampacchia's result again
\begin{equation}\label{laplacianocero}
D^2u(x)=0 \quad\text{for a.e. $x\in A'$}.
\end{equation}
We are left with the set $A'':=A'\cap\{D^2u=0\}$ with the same measure $\vert A''\vert=\vert A'\vert=\vert A\vert>0$. By the definition of $u$ being a $W^{2,p}$-strong supersolution of \eqref{pde}, we have that for a.e $x$ in $\Omega$
$$F(D^2 u(x))-g\Big(\vert u\geq u(x)\vert\Big)\leq 0.$$
Moreover, for $x_0\in A''$, the argument inside $g$ is strictly positive; $\vert u\geq u(x_0)\vert\geq\vert A''\vert>0$. So in the particular case when $g<0$ in $(0;\vert\Omega\vert]$ we get the contradiction
$$F(D^2 u(x_0))-g\Big(\vert u\geq u(x_0)\vert\Big)> 0.$$
\end{proof}
Then, for this specific case we obtain continuity for $f_u$. But we need at least $C^{0,\alpha}$ regularity on $f_u$ in order to apply Schauder type estimates to obtain $u\in C^{2,\alpha}$. We will have this regularity in two particular cases listed in the next two theorems. The first one is an adaptation(simplification) to Laurence and Stredulinsky's theorem and requires an additional lower bound for the gradient.
\begin{thm}[Theorem 3.1 in \cite{lsbootstrap}]\label{gradientbound}
Let $u\in W^{2,p}_0(\Omega)$ with a uniform lower bound $\vert\nabla u\vert>c_0>0$ in the set $\Omega_{t_0}:=\{y\in \Omega:u(y)< t_0\}$, where $t_0<\Vert u \Vert_{L^\infty}$ and $c_0=c_0(t_0)$. Then $f_u\in C^1(\Omega_{t_0})$.
\end{thm}
In other words, the theorem asserts that if we have an uniform lower bound for the gradient(away from the maximum of $u$), then we get: regularity for the level sets of $u$ and we discard a possible ``flatness'' which ruins the smoothness of $f_u$. The proof presented in \cite{lsbootstrap} includes an approximation argument by $C^\infty_0$ functions and coarea formula.
This last theorem translates into regularity for our problem. We state this in the following corollary.
\begin{cor}
If we have a solution $u$ to our problem (\ref{pde}), with $0$ boundary condition and a gradient lower bound as in Theorem \ref{gradientbound}, then $f_u\in C^1(\Omega_{t_0})$, and therefore $u\in C^{2,\alpha}(\Omega_{t_0})$.
\end{cor}
\begin{proof}
In order to get $C^{2,\alpha}$ estimates we just need to apply the classical theory of viscosity solutions for fully nonlinear equations as in Chapter 8 from \cite{cc}. Recall that at this point we have a right hand side in $C^1(\Omega_{t_0})$ which allows us to use classical viscosity solutions instead of $W^{2,p}$-viscosity solutions.
\end{proof}
The second theorem states that, under certain conditions, a barrier argument implies lower bounds as in Theorem \ref{gradientbound}.
\begin{thm}
If $\Omega$ has a uniform inner ball condition (i.e., for any point $y$ in $\partial\Omega$, there exists a ball $B_\varepsilon\subset\Omega$ with $\varepsilon>\varepsilon_0>0$ and $y\in\partial B_\varepsilon$), then $\vert\nabla u\vert>c_0>0$ in a neighborhood of $\partial\Omega$, where $c_0=c_0(\Vert u\Vert_{C^{1,\alpha}(\overline{\Omega})})$. We consider the case where $g(t)=-t$ and $u=0$ on $\partial\Omega$.
\end{thm}
\begin{proof}
If we pick any point $y\in\partial\Omega$ we can touch it with a ball $B_{\varepsilon_0}\subset\Omega$. As in Example \ref{ball} we can build a an explicit solution $\tilde{u}$ in $B_{\varepsilon_0}$ for $F=\mathcal{M}^-$. Now we apply comparison between $u$ and $\tilde{u}$ in order to get gradient estimates. Without loss of generality we can take $\varepsilon_0$ small enough, such that
$$\vert u\geq u(x)\vert\geq \frac{1}{2}\vert\Omega\vert\geq\vert B_{\varepsilon_0}\vert$$
for every $x\in B_{\varepsilon_0}$. This is possible because of the continuity of $u$ and of the right hand side, i.e.,
$$\vert u\geq t\vert \xrightarrow[t\longrightarrow 0]{}\vert\Omega\vert.$$
If this is the case then
\begin{align*}
\mathcal{M}^-(D^2 \tilde{u}(x))&=-\vert \{\tilde{u}\geq \tilde{u}(x)\}\cap B_{\varepsilon_0}\vert\\
&\geq -\frac{1}{2}\vert\Omega\vert\\
&\geq -\vert u\geq {u}(x)\vert\\
&=F(D^2u(x))\\
&\geq \mathcal{M}^-(D^2u(x))
\end{align*}
in $ B_{\varepsilon_0}$. In addition, we have that $0=\tilde{u}\leq u$ at $\partial B_{\varepsilon_0}$. So comparison applies and forces $\tilde{u}\leq u$ in $ B_{\varepsilon_0}$. Therefore we also have a lower bound for the gradient at the boundary, with the estimate
$$\vert\nabla u\vert\geq \tilde{u}_{-\nu}=\frac{\omega_n}{2n\Lambda}{\varepsilon_0}^{n+1}=c_0>0$$
where $-\nu$ is the inner normal to $\partial\Omega$. Finally we can extend a lower bound(say $c_0/2$) to a neighborhood of the boundary of $\Omega$ which will depend on the $C^{1,\alpha}(\overline{\Omega})$ norm of $u$.
\end{proof}
\begin{rmk}
We can repeat this argument as long we have uniform inner ball conditions for the level sets $\{u=t\}$, and so, $C^{2,\alpha}$ regularity for the solution in that annulus.
\end{rmk}
\begin{rmk}
We expect this condition to be satisfied for convex domains, where we deduce the solutions will have convex level sets. On the other hand, for nonconvex domains, in particular for dumbbell shaped domains, we expect to have a singular critical point where the superlevel sets separate into two components.
\end{rmk}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,382 |
(-) Remove <label class='research-domain' title='Applied Life Sciences and Non-Medical Biotechnology'>LS9 (10)</label> filter LS9 (10)
Project acronym 3D-OA-HISTO
Project Development of 3D Histopathological Grading of Osteoarthritis
Researcher (PI) Simo Jaakko Saarakkala
Host Institution (HI) OULUN YLIOPISTO
Summary "Background: Osteoarthritis (OA) is a common musculoskeletal disease occurring worldwide. Despite extensive research, etiology of OA is still poorly understood. Histopathological grading (HPG) of 2D tissue sections is the gold standard reference method for determination of OA stage. However, traditional 2D-HPG is destructive and based only on subjective visual evaluation. These limitations induce bias to clinical in vitro OA diagnostics and basic research that both rely strongly on HPG. Objectives: 1) To establish and validate the very first 3D-HPG of OA based on cutting-edge nano/micro-CT (Computed Tomography) technologies in vitro; 2) To use the established method to clarify the beginning phases of OA; and 3) To validate 3D-HPG of OA for in vivo use. Methods: Several hundreds of human osteochondral samples from patients undergoing total knee arthroplasty will be collected. The samples will be imaged in vitro with nano/micro-CT and clinical high-end extremity CT devices using specific contrast-agents to quantify tissue constituents and structure in 3D in large volume. From this information, a novel 3D-HPG is developed with statistical classification algorithms. Finally, the developed novel 3D-HPG of OA will be applied clinically in vivo. Significance: This is the very first study to establish 3D-HPG of OA pathology in vitro and in vivo. Furthermore, the developed technique hugely improves the understanding of the beginning phases of OA. Ultimately, the study will contribute for improving OA patients' quality of life by slowing the disease progression, and for providing powerful tools to develop new OA therapies."
"Background: Osteoarthritis (OA) is a common musculoskeletal disease occurring worldwide. Despite extensive research, etiology of OA is still poorly understood. Histopathological grading (HPG) of 2D tissue sections is the gold standard reference method for determination of OA stage. However, traditional 2D-HPG is destructive and based only on subjective visual evaluation. These limitations induce bias to clinical in vitro OA diagnostics and basic research that both rely strongly on HPG. Objectives: 1) To establish and validate the very first 3D-HPG of OA based on cutting-edge nano/micro-CT (Computed Tomography) technologies in vitro; 2) To use the established method to clarify the beginning phases of OA; and 3) To validate 3D-HPG of OA for in vivo use. Methods: Several hundreds of human osteochondral samples from patients undergoing total knee arthroplasty will be collected. The samples will be imaged in vitro with nano/micro-CT and clinical high-end extremity CT devices using specific contrast-agents to quantify tissue constituents and structure in 3D in large volume. From this information, a novel 3D-HPG is developed with statistical classification algorithms. Finally, the developed novel 3D-HPG of OA will be applied clinically in vivo. Significance: This is the very first study to establish 3D-HPG of OA pathology in vitro and in vivo. Furthermore, the developed technique hugely improves the understanding of the beginning phases of OA. Ultimately, the study will contribute for improving OA patients' quality of life by slowing the disease progression, and for providing powerful tools to develop new OA therapies."
Project acronym ANTILEAK
Project Development of antagonists of vascular leakage
Researcher (PI) Pipsa SAHARINEN
Summary Dysregulation of capillary permeability is a severe problem in critically ill patients, but the mechanisms involved are poorly understood. Further, there are no targeted therapies to stabilize leaky vessels in various common, potentially fatal diseases, such as systemic inflammation and sepsis, which affect millions of people annually. Although a multitude of signals that stimulate opening of endothelial cell-cell junctions leading to permeability have been characterized using cellular and in vivo models, approaches to reverse the harmful process of capillary leakage in disease conditions are yet to be identified. I propose to explore a novel autocrine endothelial permeability regulatory system as a potentially universal mechanism that antagonizes vascular stabilizing ques and sustains vascular leakage in inflammation. My group has identified inflammation-induced mechanisms that switch vascular stabilizing factors into molecules that destabilize vascular barriers, and identified tools to prevent the barrier disruption. Building on these discoveries, my group will use mouse genetics, structural biology and innovative, systematic antibody development coupled with gene editing and gene silencing technology, in order to elucidate mechanisms of vascular barrier breakdown and repair in systemic inflammation. The expected outcomes include insights into endothelial cell signaling and permeability regulation, and preclinical proof-of-concept antibodies to control endothelial activation and vascular leakage in systemic inflammation and sepsis models. Ultimately, the new knowledge and preclinical tools developed in this project may facilitate future development of targeted approaches against vascular leakage.
Dysregulation of capillary permeability is a severe problem in critically ill patients, but the mechanisms involved are poorly understood. Further, there are no targeted therapies to stabilize leaky vessels in various common, potentially fatal diseases, such as systemic inflammation and sepsis, which affect millions of people annually. Although a multitude of signals that stimulate opening of endothelial cell-cell junctions leading to permeability have been characterized using cellular and in vivo models, approaches to reverse the harmful process of capillary leakage in disease conditions are yet to be identified. I propose to explore a novel autocrine endothelial permeability regulatory system as a potentially universal mechanism that antagonizes vascular stabilizing ques and sustains vascular leakage in inflammation. My group has identified inflammation-induced mechanisms that switch vascular stabilizing factors into molecules that destabilize vascular barriers, and identified tools to prevent the barrier disruption. Building on these discoveries, my group will use mouse genetics, structural biology and innovative, systematic antibody development coupled with gene editing and gene silencing technology, in order to elucidate mechanisms of vascular barrier breakdown and repair in systemic inflammation. The expected outcomes include insights into endothelial cell signaling and permeability regulation, and preclinical proof-of-concept antibodies to control endothelial activation and vascular leakage in systemic inflammation and sepsis models. Ultimately, the new knowledge and preclinical tools developed in this project may facilitate future development of targeted approaches against vascular leakage.
Project acronym aQUARiUM
Project QUAntum nanophotonics in Rolled-Up Metamaterials
Researcher (PI) Humeyra CAGLAYAN
Host Institution (HI) TAMPEREEN KORKEAKOULUSAATIO SR
Summary Novel sophisticated technologies that exploit the laws of quantum physics form a cornerstone for the future well-being, economic growth and security of Europe. Here photonic devices have gained a prominent position because the absorption, emission, propagation or storage of a photon is a process that can be harnessed at a fundamental level and render more practical ways to use light for such applications. However, the interaction of light with single quantum systems under ambient conditions is typically very weak and difficult to control. Furthermore, there are quantum phenomena occurring in matter at nanometer length scales that are currently not well understood. These deficiencies have a direct and severe impact on creating a bridge between quantum physics and photonic device technologies. aQUARiUM, precisely address the issue of controlling and enhancing the interaction between few photons and rolled-up nanostructures with ability to be deployed in practical applications. With aQUARiUM, we will take epsilon (permittivity)-near-zero (ENZ) metamaterials into quantum nanophotonics. To this end, we will integrate quantum emitters with rolled-up waveguides, that act as ENZ metamaterial, to expand and redefine the range of light-matter interactions. We will explore the electromagnetic design freedom enabled by the extended modes of ENZ medium, which "stretches" the effective wavelength inside the structure. Specifically, aQUARiUM is built around the following two objectives: (i) Enhancing light-matter interactions with single emitters (Enhance) independent of emitter position. (ii) Enabling collective excitations in dense emitter ensembles (Collect) coherently connect emitters on nanophotonic devices to obtain coherent emission. aQUARiUM aims to create novel light-sources and long-term entanglement generation and beyond. The envisioned outcome of aQUARiUM is a wholly new photonic platform applicable across a diverse range of areas.
Novel sophisticated technologies that exploit the laws of quantum physics form a cornerstone for the future well-being, economic growth and security of Europe. Here photonic devices have gained a prominent position because the absorption, emission, propagation or storage of a photon is a process that can be harnessed at a fundamental level and render more practical ways to use light for such applications. However, the interaction of light with single quantum systems under ambient conditions is typically very weak and difficult to control. Furthermore, there are quantum phenomena occurring in matter at nanometer length scales that are currently not well understood. These deficiencies have a direct and severe impact on creating a bridge between quantum physics and photonic device technologies. aQUARiUM, precisely address the issue of controlling and enhancing the interaction between few photons and rolled-up nanostructures with ability to be deployed in practical applications. With aQUARiUM, we will take epsilon (permittivity)-near-zero (ENZ) metamaterials into quantum nanophotonics. To this end, we will integrate quantum emitters with rolled-up waveguides, that act as ENZ metamaterial, to expand and redefine the range of light-matter interactions. We will explore the electromagnetic design freedom enabled by the extended modes of ENZ medium, which "stretches" the effective wavelength inside the structure. Specifically, aQUARiUM is built around the following two objectives: (i) Enhancing light-matter interactions with single emitters (Enhance) independent of emitter position. (ii) Enabling collective excitations in dense emitter ensembles (Collect) coherently connect emitters on nanophotonic devices to obtain coherent emission. aQUARiUM aims to create novel light-sources and long-term entanglement generation and beyond. The envisioned outcome of aQUARiUM is a wholly new photonic platform applicable across a diverse range of areas.
Project acronym BHIVE
Project Bio-derived HIgh Value polymers through novel Enzyme function
Researcher (PI) Emma Rusi Master
Summary Recent advances in systems-level study of cells and organisms have revealed the enormous potential to live more sustainably through better use of biological processes. Plants sustainably synthesize the most abundant and diverse materials on Earth. By applying recent advances in life science technology, we can better harness renewable plant resources and bioconversion processes, to develop environmentally and politically sustainable human enterprise and lifestyles. At the same time, the global market for high-value biochemicals and bioplastics from forest and agricultural sources is rapidly increasing, which presents new opportunities for forest and agricultural sectors. The overall aim of BHIVE is to illuminate uncharted regions of genome and metagenome sequences to discover entirely new protein families that can be used to sustainably synthesize novel, high-value biomaterials from renewable plant resources. The approach will include three parallel research thrusts: 1) strategic analysis of transcriptome and metagenome sequences to identify proteins with entirely unknown function relevant to biomass (lignocellulose) transformation, 2) mapping of uncharted regions within phylogenetic trees of poorly characterized enzyme families with recognized potential to modify the chemistry and biophysical properties of plant polysaccharides, and 3) the design and development of novel enzyme screens to directly address the increasing limitations of existing assays to uncover entirely new protein functions. BHIVE will be unique in its undivided focus on characterizing lignocellulose-active proteins encoded by the 30-40% of un-annotated sequence, or genomic "dark matter", typical of nearly all genome sequences. In this way, BHIVE tackles a key constraint to fully realizing the societal and environmental benefits of the genomics era.
Recent advances in systems-level study of cells and organisms have revealed the enormous potential to live more sustainably through better use of biological processes. Plants sustainably synthesize the most abundant and diverse materials on Earth. By applying recent advances in life science technology, we can better harness renewable plant resources and bioconversion processes, to develop environmentally and politically sustainable human enterprise and lifestyles. At the same time, the global market for high-value biochemicals and bioplastics from forest and agricultural sources is rapidly increasing, which presents new opportunities for forest and agricultural sectors. The overall aim of BHIVE is to illuminate uncharted regions of genome and metagenome sequences to discover entirely new protein families that can be used to sustainably synthesize novel, high-value biomaterials from renewable plant resources. The approach will include three parallel research thrusts: 1) strategic analysis of transcriptome and metagenome sequences to identify proteins with entirely unknown function relevant to biomass (lignocellulose) transformation, 2) mapping of uncharted regions within phylogenetic trees of poorly characterized enzyme families with recognized potential to modify the chemistry and biophysical properties of plant polysaccharides, and 3) the design and development of novel enzyme screens to directly address the increasing limitations of existing assays to uncover entirely new protein functions. BHIVE will be unique in its undivided focus on characterizing lignocellulose-active proteins encoded by the 30-40% of un-annotated sequence, or genomic "dark matter", typical of nearly all genome sequences. In this way, BHIVE tackles a key constraint to fully realizing the societal and environmental benefits of the genomics era.
Project acronym Brain Health Toolbox
Project The Brain Health Toolbox: Facilitating personalized decision-making for effective dementia prevention
Researcher (PI) Alina Gabriela SOLOMON
Host Institution (HI) ITA-SUOMEN YLIOPISTO
Summary Preventing dementia and Alzheimer disease (AD) is a global priority. Previous single-intervention failures stress the critical need for a new multimodal preventive approach in these complex multifactorial conditions. The Brain Health Toolbox is designed to create a seamless continuum from accurate dementia prediction to effective prevention by i) developing the missing disease models and prediction tools for multimodal prevention; ii) testing them in actual multimodal prevention trials; and iii) bridging the gap between non-pharmacological and pharmacological approaches by designing a combined multimodal prevention trial based on a new European adaptive trial platform. Disease models and prediction tools will be multi-dimensional, i.e. a broad range of risk factors and biomarker types, including novel markers. An innovative machine learning method will be used for pattern identification and risk profiling to highlight most important contributors to an individual's overall risk level. This is crucial for early identification of individuals with high dementia risk and/or high likelihood of specific brain pathologies, quantifying an individual's prevention potential, and longitudinal risk and disease monitoring, also beyond trial duration. Three Toolbox test scenarios are considered: use for selecting target populations, assessing heterogeneity of intervention effects, and use as trial outcome. The project is based on a unique set-up aligning several new multimodal lifestyle trials aiming to adapt and test non-pharmacological interventions to different geographic, economic and cultural settings, with two reference libraries (observational - large datasets; and interventional - four recently completed pioneering multimodal lifestyle prevention trials). The Brain Health Toolbox covers the entire continuum from general populations to patients with preclinical/prodromal disease stages, and will provide tools for personalized decision-making for dementia prevention.
Preventing dementia and Alzheimer disease (AD) is a global priority. Previous single-intervention failures stress the critical need for a new multimodal preventive approach in these complex multifactorial conditions. The Brain Health Toolbox is designed to create a seamless continuum from accurate dementia prediction to effective prevention by i) developing the missing disease models and prediction tools for multimodal prevention; ii) testing them in actual multimodal prevention trials; and iii) bridging the gap between non-pharmacological and pharmacological approaches by designing a combined multimodal prevention trial based on a new European adaptive trial platform. Disease models and prediction tools will be multi-dimensional, i.e. a broad range of risk factors and biomarker types, including novel markers. An innovative machine learning method will be used for pattern identification and risk profiling to highlight most important contributors to an individual's overall risk level. This is crucial for early identification of individuals with high dementia risk and/or high likelihood of specific brain pathologies, quantifying an individual's prevention potential, and longitudinal risk and disease monitoring, also beyond trial duration. Three Toolbox test scenarios are considered: use for selecting target populations, assessing heterogeneity of intervention effects, and use as trial outcome. The project is based on a unique set-up aligning several new multimodal lifestyle trials aiming to adapt and test non-pharmacological interventions to different geographic, economic and cultural settings, with two reference libraries (observational - large datasets; and interventional - four recently completed pioneering multimodal lifestyle prevention trials). The Brain Health Toolbox covers the entire continuum from general populations to patients with preclinical/prodromal disease stages, and will provide tools for personalized decision-making for dementia prevention.
Project acronym BRAIN2BRAIN
Project Towards two-person neuroscience
Researcher (PI) Riitta Kyllikki Hari
Summary Humans interact with other people throughout their lives. This project aims to demonstrate that the complex social shaping of the human brain can be adequately tackled only by taking a leap from the conven-tional single-person neuroscience to two-person neuroscience. We will (1) develop a conceptual framework and experimental setups for two-person neuroscience, (2) apply time-sensitive methods for studies of two interacting persons, monitoring both brain and autonomic nervous activity to also cover the brain body connection, (3) use gaze as an index of subject s attention to simplify signal analysis in natural environments, and (4) apply insights from two-person neuroscience into disorders of social interaction. Brain activity will be recorded with millisecond-accurate whole-scalp (306-channel) magnetoencepha-lography (MEG), associated with EEG, and with the millimeter-accurate 3-tesla functional magnetic reso-nance imaging (fMRI). Heart rate, respiration, galvanic skin response, and pupil diameter inform about body function. A new psychophysiological interaction setting will be built, comprising a two-person eye-tracking system. Novel analysis methods will be developed to follow the interaction and possible synchronization of the two persons signals. This uncoventional approach crosses borders of neuroscience, social psychology, psychophysiology, psychiatry, medical imaging, and signal analysis, with intriguing connections to old philosophical questions, such as intersubjectivity and emphatic attunement. The results could open an unprecedented window into human human, instead of just brain brain, interactions, helping to understand also social disorders, such as autism and schizophrenia. Further applications include master apprentice and patient therapist relationships. Advancing from studies of single persons towards two-person neuroscience shows promise of a break-through in understanding the dynamic social shaping of human brain and mind.
Humans interact with other people throughout their lives. This project aims to demonstrate that the complex social shaping of the human brain can be adequately tackled only by taking a leap from the conven-tional single-person neuroscience to two-person neuroscience. We will (1) develop a conceptual framework and experimental setups for two-person neuroscience, (2) apply time-sensitive methods for studies of two interacting persons, monitoring both brain and autonomic nervous activity to also cover the brain body connection, (3) use gaze as an index of subject s attention to simplify signal analysis in natural environments, and (4) apply insights from two-person neuroscience into disorders of social interaction. Brain activity will be recorded with millisecond-accurate whole-scalp (306-channel) magnetoencepha-lography (MEG), associated with EEG, and with the millimeter-accurate 3-tesla functional magnetic reso-nance imaging (fMRI). Heart rate, respiration, galvanic skin response, and pupil diameter inform about body function. A new psychophysiological interaction setting will be built, comprising a two-person eye-tracking system. Novel analysis methods will be developed to follow the interaction and possible synchronization of the two persons signals. This uncoventional approach crosses borders of neuroscience, social psychology, psychophysiology, psychiatry, medical imaging, and signal analysis, with intriguing connections to old philosophical questions, such as intersubjectivity and emphatic attunement. The results could open an unprecedented window into human human, instead of just brain brain, interactions, helping to understand also social disorders, such as autism and schizophrenia. Further applications include master apprentice and patient therapist relationships. Advancing from studies of single persons towards two-person neuroscience shows promise of a break-through in understanding the dynamic social shaping of human brain and mind.
Project acronym BRAINCANNABINOIDS
Project Understanding the molecular blueprint and functional complexity of the endocannabinoid metabolome in the brain
Researcher (PI) István Katona
Host Institution (HI) INSTITUTE OF EXPERIMENTAL MEDICINE - HUNGARIAN ACADEMY OF SCIENCES
Summary We and others have recently delineated the molecular architecture of a new feedback pathway in brain synapses, which operates as a synaptic circuit breaker. This pathway is supposed to use a group of lipid messengers as retrograde synaptic signals, the so-called endocannabinoids. Although heterogeneous in their chemical structures, these molecules along with the psychoactive compound in cannabis are thought to target the same effector in the brain, the CB1 receptor. However, the molecular catalog of these bioactive lipids and their metabolic enzymes has been expanding rapidly by recent advances in lipidomics and proteomics raising the possibility that these lipids may also serve novel, yet unidentified physiological functions. Thus, the overall aim of our research program is to define the molecular and anatomical organization of these endocannabinoid-mediated pathways and to determine their functional significance. In the present proposal, we will focus on understanding how these novel pathways regulate synaptic and extrasynaptic signaling in hippocampal neurons. Using combination of lipidomic, genetic and high-resolution anatomical approaches, we will identify distinct chemical species of endocannabinoids and will show how their metabolic enzymes are segregated into different subcellular compartments in cell type- and synapse-specific manner. Subsequently, we will use genetically encoded gain-of-function, loss-of-function and reporter constructs in imaging experiments and electrophysiological recordings to gain insights into the diverse tasks that these new pathways serve in synaptic transmission and extrasynaptic signal processing. Our proposed experiments will reveal fundamental principles of intercellular and intracellular endocannabinoid signaling in the brain.
We and others have recently delineated the molecular architecture of a new feedback pathway in brain synapses, which operates as a synaptic circuit breaker. This pathway is supposed to use a group of lipid messengers as retrograde synaptic signals, the so-called endocannabinoids. Although heterogeneous in their chemical structures, these molecules along with the psychoactive compound in cannabis are thought to target the same effector in the brain, the CB1 receptor. However, the molecular catalog of these bioactive lipids and their metabolic enzymes has been expanding rapidly by recent advances in lipidomics and proteomics raising the possibility that these lipids may also serve novel, yet unidentified physiological functions. Thus, the overall aim of our research program is to define the molecular and anatomical organization of these endocannabinoid-mediated pathways and to determine their functional significance. In the present proposal, we will focus on understanding how these novel pathways regulate synaptic and extrasynaptic signaling in hippocampal neurons. Using combination of lipidomic, genetic and high-resolution anatomical approaches, we will identify distinct chemical species of endocannabinoids and will show how their metabolic enzymes are segregated into different subcellular compartments in cell type- and synapse-specific manner. Subsequently, we will use genetically encoded gain-of-function, loss-of-function and reporter constructs in imaging experiments and electrophysiological recordings to gain insights into the diverse tasks that these new pathways serve in synaptic transmission and extrasynaptic signal processing. Our proposed experiments will reveal fundamental principles of intercellular and intracellular endocannabinoid signaling in the brain.
Project acronym BrainDrain
Project Translational implications of the discovery of brain-draining lymphatics
Researcher (PI) Kari ALITALO
Summary In 2010, 800 billion Euros was spent on brain diseases in Europe and the cost is expected to increase due to the aging population. – Here I propose to exploit our new discovery for research to alleviate this disease burden. In work selected by Nature Medicine among the top 10 "Notable Advances" and by Science as one of the 10 "Breakthroughs of the year" 2015, we discovered a meningeal lymphatic vascular system that serves brain homeostasis. We want to reassess current concepts about cerebrovascular dynamics, fluid drainage and cellular trafficking in physiological conditions, in Alzheimer's disease mouse models and in human postmortem tissues. First, we will study the development and properties of meningeal lymphatics and how they are sustained during aging. We then want to analyse the clearance of macromolecules and protein aggregates in Alzheimer's disease in mice that lack the newly discovered meningeal lymphatic drainage system. We will study if growth factor-mediated expansion of lymphatic vessels alleviates the parenchymal accumulation of neurotoxic amyloid beta and pathogenesis of Alzheimer's disease and brain damage after traumatic brain injury. We will further analyse the role of lymphangiogenic growth factors and lymphatic vessels in brain solute clearance, immune cell trafficking and in a mouse model of multiple sclerosis. The meningeal lymphatics could be involved in a number of neurodegenerative and neuroinflammatory diseases of considerable human and socioeconomic burden. Several of our previous concepts have already been translated to clinical development and we aim to develop proof-of-principle therapeutic concepts in this project. I feel that we are just now in a unique position to advance frontline European translational biomedical research in this suddenly emerging field, which has received great attention worldwide.
In 2010, 800 billion Euros was spent on brain diseases in Europe and the cost is expected to increase due to the aging population. – Here I propose to exploit our new discovery for research to alleviate this disease burden. In work selected by Nature Medicine among the top 10 "Notable Advances" and by Science as one of the 10 "Breakthroughs of the year" 2015, we discovered a meningeal lymphatic vascular system that serves brain homeostasis. We want to reassess current concepts about cerebrovascular dynamics, fluid drainage and cellular trafficking in physiological conditions, in Alzheimer's disease mouse models and in human postmortem tissues. First, we will study the development and properties of meningeal lymphatics and how they are sustained during aging. We then want to analyse the clearance of macromolecules and protein aggregates in Alzheimer's disease in mice that lack the newly discovered meningeal lymphatic drainage system. We will study if growth factor-mediated expansion of lymphatic vessels alleviates the parenchymal accumulation of neurotoxic amyloid beta and pathogenesis of Alzheimer's disease and brain damage after traumatic brain injury. We will further analyse the role of lymphangiogenic growth factors and lymphatic vessels in brain solute clearance, immune cell trafficking and in a mouse model of multiple sclerosis. The meningeal lymphatics could be involved in a number of neurodegenerative and neuroinflammatory diseases of considerable human and socioeconomic burden. Several of our previous concepts have already been translated to clinical development and we aim to develop proof-of-principle therapeutic concepts in this project. I feel that we are just now in a unique position to advance frontline European translational biomedical research in this suddenly emerging field, which has received great attention worldwide.
Project acronym CholAminCo
Project Synergy and antagonism of cholinergic and dopaminergic systems in associative learning
Researcher (PI) Balazs Gyoergy HANGYA
Summary Neuromodulators such as acetylcholine and dopamine are able to rapidly reprogram neuronal information processing and dynamically change brain states. Degeneration or dysfunction of cholinergic and dopaminergic neurons can lead to neuropsychiatric conditions like schizophrenia and addiction or cognitive diseases such as Alzheimer's. Neuromodulatory systems control overlapping cognitive processes and often have similar modes of action; therefore it is important to reveal cooperation and competition between different systems to understand their unique contributions to cognitive functions like learning, memory and attention. This is only possible by direct comparison, which necessitates monitoring multiple neuromodulatory systems under identical experimental conditions. Moreover, simultaneous recording of different neuromodulatory cell types goes beyond phenomenological description of similarities and differences by revealing the underlying correlation structure at the level of action potential timing. However, such data allowing direct comparison of neuromodulatory actions are still sparse. As a first step to bridge this gap, I propose to elucidate the unique versus complementary roles of two "classical" neuromodulatory systems, the cholinergic and dopaminergic projection system implicated in various cognitive functions including associative learning and plasticity. First, we will record optogenetically identified cholinergic and dopaminergic neurons simultaneously using chronic extracellular recording in mice undergoing classical and operant conditioning. Second, we will determine the postsynaptic impact of cholinergic and dopaminergic neurons by manipulating them both separately and simultaneously while recording consequential changes in cortical neuronal activity and learning behaviour. These experiments will reveal how major neuromodulatory systems interact to mediate similar or different aspects of the same cognitive functions.
Neuromodulators such as acetylcholine and dopamine are able to rapidly reprogram neuronal information processing and dynamically change brain states. Degeneration or dysfunction of cholinergic and dopaminergic neurons can lead to neuropsychiatric conditions like schizophrenia and addiction or cognitive diseases such as Alzheimer's. Neuromodulatory systems control overlapping cognitive processes and often have similar modes of action; therefore it is important to reveal cooperation and competition between different systems to understand their unique contributions to cognitive functions like learning, memory and attention. This is only possible by direct comparison, which necessitates monitoring multiple neuromodulatory systems under identical experimental conditions. Moreover, simultaneous recording of different neuromodulatory cell types goes beyond phenomenological description of similarities and differences by revealing the underlying correlation structure at the level of action potential timing. However, such data allowing direct comparison of neuromodulatory actions are still sparse. As a first step to bridge this gap, I propose to elucidate the unique versus complementary roles of two "classical" neuromodulatory systems, the cholinergic and dopaminergic projection system implicated in various cognitive functions including associative learning and plasticity. First, we will record optogenetically identified cholinergic and dopaminergic neurons simultaneously using chronic extracellular recording in mice undergoing classical and operant conditioning. Second, we will determine the postsynaptic impact of cholinergic and dopaminergic neurons by manipulating them both separately and simultaneously while recording consequential changes in cortical neuronal activity and learning behaviour. These experiments will reveal how major neuromodulatory systems interact to mediate similar or different aspects of the same cognitive functions. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,219 |
\section{Introduction}
Consider a government funding a research grant for a vaccine for a new
disease. Several firms have prior experience of developing vaccines for
closely related diseases and can apply for the grant. Being mature firms in
the same sector, firms are familiar (owing to prior experience) with the
quality of research processes of the other firms. However, candidate
vaccines for the new disease (which serve as evidence of the quality of a
firm's research) are products of the firm's internal research processes, and
are, therefore, private information. Firms want the grant to be awarded to
them irrespective of the true quality of their research processes while the
government wants to maximize the probability of obtaining a functional
vaccine.
Alternately, consider a personal injury trial where a judge faces the
problem of determining the level of compensation that a defendant should pay
a plaintiff. Suppose that the judge wants to set a level of compensation
commensurate to the damage that has been caused, but does not know what that
damage is. Each agent knows the extent of the damage, and irrespective of
the level of damage, the plaintiff wants more compensation and the defendant
less. In such a setting, evidence is likely to be the result of
investigation and therefore uncertain. However, both the defendant and the
plaintiff may have some notion of the possible evidence which may arise.
The common threads that underlie the two situations described above are the
following - there is a (decision relevant) state commonly known to the
agents but not to the designer, agents have state independent incentives
which are not aligned with those of the designer, and while there is
evidence in the setting, it is uncertain, so that agents do not know
precisely what evidence other agents may possess. This characterizes a wide
variety of practical situations and thus it is important to study how to
obtain credible information for decision making in these settings.
In this paper, we study a setup inspired by \cite{BL2012}, but different in
one important aspect - the evidence is uncertain. This means that articles
of evidence are drawn (independently) from a distribution, and the actual
realization of evidence is private information for the agents. The setup is
Bayesian in the evidence dimension, i.e. while agents do not know what
evidence other agents have, the distribution from which this evidence is
drawn is common knowledge. This is more realistic than complete information
along all dimensions, evidence often being the product of research, which is
fundamentally uncertain in nature. It is however, conceivable, that even
though the exact outcome of the search for evidence may be unknown, prior
experience may allow agents to roughly predict what evidence may arise in a
certain situation.
Evidence variation is used in two distinct ways in the mechanisms we
construct - first, in presenting evidence, an agent directly \emph{refutes}
all the states where she may not possess such evidence with positive
probability; second, if an agent cannot present the evidence for a state
claim (perhaps because some evidence is not available with enough
probability), the distribution induced on her evidence report varies from
the expected distribution under truth-telling - a difference which we will
show can be used to flag incorrect reporting using a system of \emph{bets}.
Our result relies on a novel classification of lies (incorrect state claims)
based on their refutability with evidence, and for some lies which cannot be
directly refuted with evidence, the latter technique involving bets is used.
When constructing mechanisms, agents may attempt to deceive the designer by
either misreporting the state, or withholding some part of their evidence
endowment, since their incentives are not necessarily aligned with those of
the designer. Such strategies, which deviate from truth-telling, are called
\emph{deceptions}. Sometimes, deceptions might exist which perfectly mimic
another state in both the state and evidence dimensions. In such cases, we
will show that the designer cannot separate these states by using a
mechanism, and this yields our implementing condition - No Perfect
Deceptions (NPD). More precisely, NPD can be summarized as follows: if
states $s$ and $s^{\prime }$ are such that $s^{\prime }$ is a lie in state
s $ which cannot be directly refuted by evidence, and at state $s$, it is
possible for all agents to induce the same distribution on their evidence as
they would under truth-telling in state $s^{\prime }$, then the social
choice function must induce the same outcome at $s$ and $s^{\prime }$.
To establish the sufficiency of NPD for mixed-strategy implementation, we
present a finite mechanism which fully implements any social choice function
(SCF) which satisfies NPD in mixed-strategy Bayesian Nash equilibria. The
mechanism works with two or more agents and uses off-equilibrium transfers.
A striking feature of this mechanism is that while most Bayesian
implementation results (see for example \cite{J1991} and \cite{SV2010})
require three or more agents and involve "...questionable features..." such
as integer or modulo games\footnote
See \cite{J1992} for a description of the issues which arise from the use of
these questionable features.}, we obtain exact implementation with only two
agents and without the use of such "...questionable features...". We note
however, that unlike in \cite{J1991}, where each agent knows only her own
type, the domain of the social choice function is common knowledge in our
setup, and agents only contend with uncertainty along the evidence dimension.
A strictly weaker condition, which we call No Pure-Perfect Deceptions (NPPD)
is necessary and sufficient for pure-strategy implementation. It requires
that the strategies the agents use to mimic the true distribution at another
state be pure, and is therefore strictly weaker than NPD. This result spells
out a clear distinction in this setup between pure and mixed-strategy
implementation, with the set of mixed-strategy implementable SCFs being
strictly smaller than the set of pure-strategy implementable SCFs.\footnote
This is consistent with Bayesian implementation with preference variation,
where Mixed Bayesian Monotonicity is strictly stronger than Bayesian
Monotonicity. It is, however, different from implementation with complete
information, where Maskin monotonicity characterizes implementation with
preference variation, and measurability characterizes implementation with
evidence irrespective of pure or mixed strategies.} This condition is still
stronger than a stochastic version of the measurability condition in \cit
{BL2012} (which we term \emph{stochastic measurability})\emph{\ }however,
elucidating an important point - when uncertain evidence is considered,
conditions in the flavour of measurability (which only require that the
evidence distribution vary between states with different desired outcomes)
no longer suffice, and stronger conditions are required even when limiting
attention to pure strategies.
At a methodological level, we introduce two significant innovations. First,
in line with our treatment of deterministic evidence in \cite{BCS2020}, we
use a novel classification of lies (incorrect state claims) in terms of
their refutability with evidence and eliminate these lies successively in a
mechanism. This eschews the canonical approach of first obtaining consistent
reports using an integer game and then using the implementing condition to
argue that consistent reports may only obtain on the truth. The versatility
of this approach is evidenced by its utility in both deterministic and
uncertain evidence settings. To motivate the second innovation, we first
draw attention to the fact that our setting allows for the existence of lies
which cannot be eliminated with positive probability by evidence possessed
by any agent.\footnote
This is possible even though the evidence distribution varies between two
states. In particular, the support of the evidence distribution could be the
same, while the probabilities of particular collections of evidence vary
amongst the states.} In these cases, the necessary condition (NPD) yields
that some agents presenting these lies will not be able to present every
article of evidence with the same probability as they would be expected to
under truthful reporting if the lie were indeed the truth. In this case, it
becomes possible for agents to profitably signal to the designer that a lie
is being presented by placing monetary bets on the evidence realizations of
other agents. This is a form of arbitrage over probability distributions and
is critical to the treatment of mixed strategies alongside Bayesian
uncertainty. The technique is somewhat close in spirit to the approach in
\cite{CML1988}, where it finds use in efficient partial implementation from
agents about correlated valuations. This paper marks, to our knowledge, the
first use of such an approach in the field of full implementation of general
SCFs.\footnote
Note that \cite{CML1988} focuses on efficient allocation with cheap talk
messages while we focus on implementation of SCFs.}
We then turn to a more general setup where agents' private information is
represented by a general type, and evidence is contained within this type.
Further, arbitrary correlations are allowed in these types, so that evidence
draws may now be correlated. In this context, we present a mechanism which
implements a given SCF in rationalizable strategies but with arbitrarily
small transfers alongside. This requires that at least one of the agents be
able to separate between states (type profiles) at which different outcomes
are prescribed in terms of their beliefs at some finite order, a condition
we term \emph{higher-order measurability. }While this condition bears a
resemblance to the measurability condition in \cite{AM92B}, we spell out the
difference here. In an environment without preference variation, only
constant SCFs satisfy the measurability condition in \cite{AM92B}. However,
in such a scenario, with sufficient evidence variation, we can build belief
hierarchies which allow for implementing non-constant SCFs. In a setting
with independent evidence endowments, such as the one we study for Bayesian
Nash implementation, higher order measurability is identical to stochastic
measurability, and therefore weaker than both NPD and NPPD. We show that
higher-order measurability is necessary and sufficient for implementation in
rationalizable strategies with small transfers, alongside the usual
incentive compatibility condition. This significantly expands the scope of
obtaining implementation with evidence, albeit at the cost of small
transfers alongside the desired outcome and a significantly more complicated
mechanism.
The rest of the paper is organized as follows - In Section \ref{Example}, we
describe a illustrating example more formally to provide context for the
analysis in later sections. Section \ref{Model} defines the setup formally
and introduces some properties of the evidence structure. Section \re
{BayesianImplementation} deals with the Bayes-Nash implementation result.
Section \ref{PBNI} discusses pure-strategy Bayes-Nash implementation.
Section \ref{IST} details rationalizable implementation in the setup.
Finally, Section \ref{RL} reviews and compares our results with the
literature.
\section{\label{Example}Illustrating Example - Research Grant}
For the aforementioned problem of research grants for a vaccine, suppose
there are two firms, $A$ and $B$, which apply for the grant. Firm $A$ is a
startup whose research process quality may either be low ($L$), medium ($M$)
or high ($H$). A low quality research process is characterized either by
having a low quality candidate vaccine (e.g. with major side effects). A
firm with a medium quality research process has a probability $0.4$ of
having a high quality candidate (one with only minor side effects). A firm
with a high quality research process produces a high quality candidate with
a probability $0.6$. Firm $B$ is an incumbent with (commonly known) medium
research quality ($M$). The government is not aware of Firm $A$'s research
process quality, so that the states of the world are given by $L,M,$ or $H$.
In state $L$, the government wants to allot the grant to firm $B$, in state
H$ to firm $A$, and in state $M$ to firm $B$. As in the introduction, we
argue that the research process quality of firm A is common knowledge
amongst the firms, but unknown to the government. With the social choice
function so defined, we turn our attention to the following formal
representation of the evidence structure.
\begin{equation*}
\begin{tabular}{|l|l|}
\hline
State & Evidence (Firm A) \\ \hline
$H$ & $0.6\times \{\{M,H\},\{L,M,H\}\}+0.4\times \{\{L,M,H\}\}$ \\ \hline
$M$ & $0.4\times \{\{M,H\},\{L,M,H\}\}+0.6\times \{\{L,M,H\}\}$ \\ \hline
$L$ & $\{\{L,M,H\}\}$ \\ \hline
\end{tabular
\end{equation*}
Note that the articles of evidence in the above example have been associated
with subsets of the state space. In general, we could have abstract articles
of evidence, such as $e_{1}$ and $e_{2}~$instead of $\{L,M,H\}$, and
\{M,H\} $ respectively. These "names" are obtained by associating an article
of evidence with the states in which they appear with positive probability.
This form of nomenclature is used in \cite{BL2012}, albeit in a
deterministic setting. We use this nomenclature to simplify the exposition
(especially within examples) in several places in the paper. The articles of
evidence above are interpreted thus - $\{L,M,H\}$ is interpreted as
possessing a low quality candidate, and $\{M,H\}$ as possessing a high
quality candidate. Thus, firm A, in presenting the article $\{M,H\}$ proves
that its research quality is not $L$. The above table essentially captures
the intuition presented in the opening of this section - as we move upwards
in research quality, we see an increasing probability of having a high
quality candidate. Notice that firm A of type $H$ maybe unable to separate
itself from a firm A type $M$, since either the type $M$ may have the high
quality candidate or the type $H$ may not possess the high quality
candidate. This yields that an unraveling result of the flavour of \cite{M81}
or \cite{G81} may have no bite in this setup. This is a property possessed
by the setup in \cite{BL2012}, and shared by the more general setup we study
here.
From the above intuition of evidence, it is clear that no agent can
eliminate state $M$ at state $H$. This provides a major challenge in
achieving implementation. To see this, suppose the mechanism were direct,
requiring only a state and an evidentiary claim. Further, consider the
following strategies for firm A in state $H$ possessed with
\{\{M,H\},\{LMH\}\}$: Report $\{\{M,H\},\{LMH\}\}$ with probability $\frac{
}{3}$ and report $\{\{LMH\}\}$ with probability $\frac{1}{3}$; In the state
dimension, always claim that the state is $M$. With this strategy, the
agents would induce the same distribution on their reports as if the true
state were $M$, and they were reporting truthfully. This form of strategy is
later dubbed a \emph{perfect} deception. If perfect deceptions exist, it is
not possible to separate states in any mechanism (direct or indirect) which
relies on Bayesian Nash Implementation. This result is formally proved in
Section \ref{NPD} and followed by an intuitive description. Note that there
is also a perfect deception for type $M$ to type $L$, but $L$ is refutable
at $M$ with positive probability, so that types $M$ and $L$ are separable by
a mechanism.
With the above evidence structure then, it is not possible for the
government to allot the grant to firm $A$ in state $H$\ and firm $B$ in
state $M$. Consider though the following mild perturbation of the evidence
structure: An additional article of evidence $\{\{H\}\}$ is available to
firm $A$ with a small probability when its research quality is $H$. For
instance, the distribution in state $H$ may be $0.1\times
\{\{H\}\}+0.5\times \{\{M,H\},\{LMH\}\}+0.4\times \{\{L,M,H\}\}$. In this
case, the government can actually implement different outcomes in states $M$
and $H\ $- while in state $M$, $H$ is nonrefutable, the article $\{\{H\}\}$
is not available, so that a perfect deception is impossible; in state $H$,
while it is possible to perfectly match the evidence distribution for state
M$, $M$ is actually refutable with positive probability.
\section{\label{Model}Model and Preliminaries}
\subsection{Setup}
There is a set of agents $\mathcal{I=}\left\{ 1,...,I\right\} ,$ $I\geq 2$,
a set of outcomes $A$ and a finite set of states $S.$ Agents have bounded
utility over the outcomes given by $\bar{u}_{i}:A\times S\rightarrow \mathbb
R}$. Their preferences are quasilinear, so that $u_{i}(a,\bar{\tau}_{i},s)
\bar{u}_{i}(a,s)+\bar{\tau}_{i}$, where $\bar{\tau}_{i}$ is the transfer of
money to agent $i$. The bound of $\bar{u}_{i}$ can be normalized so that it
is strictly less than $1$, that is, an agent can be persuaded to accept any
outcome if the the alternative were any other outcome with a penalty of $1$
dollar.
Each agent $i$ is endowed with a collection (mathematically, a set) of
articles of evidence $E_{i}$ which can vary from state to state. A profile
of collections (one for each agent) is denoted by $E$. The set of all
possible collections of evidence for agent $i$ is denoted by $\mathbb{E}_{i}$
and we denote by $\mathbb{E}$ ($=\Pi _{i\in \mathcal{I}}\mathbb{E}_{i}$),
the set of all possible profiles of evidence collections. As in the
illustrating example, each collection of evidence can be associated with a
set of subsets of the state space - namely those subsets in which the
article occurs with positive probability.
We also assume that it is possible for an agent to submit the entire
collection of evidence she is endowed with. This condition, which is
commonly called \emph{normality}, is present in \cite{BL2012} as well, and
is, generally speaking, a common feature of single stage mechanisms in such
settings. In case of uncertain evidence though, there emerges a subtle
difference. In \cite{BL2012}, normality is formulated by requiring that a
single "most informative" evidence message be available in any state to any
agent.\footnote
In context of the example for instance, the article $\{M,H\}$ is more
informative than the article $\{L,M,H\}$. In general, the most informative
article of evidence is given by the intersection of all the available
articles of evidence, when they are named with subsets of the state space
(as in the example).} In this setting however, it is essential that agents
be allowed to submit an entire collection of evidence, rather than just the
most informative message. Intuitively, this is because several different
collections of evidence may yield the same \textquotedblleft most
informative\textquotedblright\ evidence message. However, it may be possible
to distinguish between these collections when each member is observed by the
designer. We discuss this point formally in Appendix \ref{Conn}.
Evidence at state $s$ is distributed according to the commonly known prior
p:S\rightarrow \Pi _{i\in \mathcal{I}}\Delta (\mathbb{E}_{i})$. Denote the
marginals of $p(s)$ on $\mathbb{E}_{i}$ and $\Pi _{j\neq i}\mathbb{E}_{j}$
by $p_{i}(s)$ and $p_{-i}(s)$, respectively. Deterministic evidence is a
special case of this (the above distribution is degenerate). Note that $p$
is defined as a product distribution, eschewing the possibility of the
evidence draws being correlated. This assumption is maintained in most of
the paper, but relaxed in Section \ref{IST}.
Let $P_{i}$ denote the set of all possible evidence distributions for agent
i$, i.e. $P_{i}=\{p_{i}(s):s\in S\}$. We denote by $P$, the set $\Pi _{i\in
\mathcal{I}}P_{i}$.
\subsection{Mechanisms and Implementation}
Any mechanism $\mathcal{M}$ is represented by the tuple $(M,g,\tau )$ where
M$ is the message space, $g:M\rightarrow A$ is the outcome function, and
\tau =\left( \tau _{i}\right) _{i\in \mathcal{I}}$ is the profile of
transfer rules with $\tau _{i}:M\rightarrow
\mathbb{R}
$. At state $s$, and with a prior $p$, a mechanism $\mathcal{M}$ induces a
Bayesian game $G(\mathcal{M},u,s,p)=\left\langle \mathcal{I
,s,p,\left\langle M_{i},u_{i},S\times \mathbb{E}_{i}\right\rangle _{i\in
\mathcal{I}}\right\rangle $ with the following properties:
\begin{itemize}
\item The type space of a typical agent $i$ is : $S\times \mathbb{E}_{i}$.
\item Every type of every agent forms beliefs on $\mathbb{E}_{-i}$ according
to $p_{-i}$.
\item The action space of a type $(s,E_{i})$ is given by $M_{i}(E_{i})$ (as
some messages may be contingent on evidence availability). We only consider
finite mechanisms , so that $M_{i}(E_{i})$ is finite for any $E_{i}$.
\item A typical (mixed) strategy for an agent $i$ at the state $s$ is a
function: $\sigma _{i}:S\times \mathbb{E}_{i}\rightarrow \Delta (M_{i})$
such that $\sigma _{i}(s,E_{i})\in M_{i}(E_{i})$.
\end{itemize}
The common prior type space is specified by $p$. Given a game $G(\mathcal{M
,u,s,p)$, say $\sigma _{i}(s,\cdot )$ is a best response to $\sigma
_{-i}(s,.)$ if
\begin{gather*}
\sigma _{i}(s,E_{i})\left( m_{i}\right) >0\Rightarrow \\
m_{i}\in \arg \max_{m_{i}^{\prime }\in M_{i}}\sum_{E_{-i}\in \mathbb{E
_{-i}}p_{-i}(s)(E_{-i})\sum_{m_{-i}\in M_{-i}}\sigma
_{-i}(s,E_{-i})(m_{-i})[u_{i}(g(m_{i}^{\prime },m_{-i}),\tau
_{i}(m_{i}^{\prime },m_{-i}),s)].
\end{gather*
Next, we define the notion of a Bayesian Nash equilibrium in this particular
setting.
\begin{definition}
A Bayesian Nash equilibrium (henceforth BNE) of the game $G(\mathcal{M
,u,s,p)$ is defined as a profile of strategies $\sigma $ such that $\forall
i,s$, $\sigma _{i}(s,\cdot )$ is a best response to $\sigma _{-i}(s,.)$.
\end{definition}
A pure-strategy BNE is a BNE $\sigma $ such that for any $i$ and $s$,
\sigma _{i}\left( s,E_{i}\right) \left( m_{i}\right) =1$ for some $m_{i}$.
We then define the main notion of implementation we work with in this paper.
\begin{definition}
An SCF $f$ is implementable in mixed-strategy BNE by a finite mechanism if
there exists a finite mechanism $\mathcal{M}=(M,g,\tau )$ such that for any
profile of bounded utility functions $\bar{u}$, at any state $s$, we have
g(m)=f(s)$ and $\tau (m)=0$ for every message $m\in M$ in the support of
\sigma (s)$ of any BNE $\sigma $ of the Bayesian game $G(\mathcal{M},u,s,p)$.
\end{definition}
That is, for any BNE of the Bayesian game induced by the mechanism, the
outcome is correct, and there are no transfers. We stress here that
implementation should obtain regardless of the realized profile of evidence
collections. Since we wish to implement by depending solely on evidence,
rather than depending on preference variation, the notion of implementation
above requires that implementation obtain regardless of the profile of
bounded utility functions.
\subsection{Lies and their classification}
At the heart of the Bayesian implementation result is a classification of
lies (states not identical to the truth), which we illustrate below.
\begin{definition}
A collection of evidence $E_{i}$ refutes a state $s$ (denoted
E_{i}\downarrow s$) if agent $i$ does not have $E_{i}$ or a superset of
E_{i}$ with positive probability at $s$. Otherwise, $E_{i}\not\downarrow s$.
\end{definition}
That is, $E_{i}\downarrow s$ if $p_{i}(s)(E_{i}^{\prime })=0$ $\forall
E_{i}^{\prime }\supseteq E_{i}$. In a mechanism, agents will always have the
opportunity to withhold some articles from a collection. Therefore, even if
a collection does not occur at $s$ with positive probability, being
presented with this collection does not directly imply that $s$ is false.
Indeed the evidence presented may be part of a larger collection which can
occur. Therefore, only when a collection \emph{or any superset} does not
occur at $s$, can we claim that it refutes $s$.
In the context of the leading example, in state $M$ or $H$, the article
\{M,H\}$ refutes the state $L$. A difference from the deterministic evidence
setting is that it is not available with probability 1. For instance, if
some firm claims that another firm's research process is of low quality
(when it is of medium quality), the latter firm cannot necessarily prove
this claim to be false by presenting the article $\{M,H\}$ because it is not
available with probability 1. This is different from a setting of
deterministic evidence, where the necessary condition (measurability)
entails that the realization of the evidence vary between states, so that
the article $\{M,H\}$ would have to be available with probability 1
\footnote
In a setting of complete information (e.g. \cite{BL2012}) , an SCF $f$
satisfies measurability if whenever $f(s)\neq f(s^{\prime })$, there is an
agent $i$ whose evidence endowment varies between $s$ and $s^{\prime }$.}
Also, if firm $A$, when possessed with the article $\{H\}$ claims a medium
quality process, then it is the only agent who can prove this claim to be
false.\footnote
Note that in context of this example, it would not be to the advantage of
this firm to make such a claim, but since we do not rely on preference
variation, in the general case, such issues may easily arise.} This
scenario, referred to as a self-refutable lie, also exists under
deterministic evidence, but under deterministic evidence, the firm would be
able to refute a lie about itself with probability 1, rather than being able
to do so only with some probability (as in the above example).
The following conditions arise as a result of the definition of refutation
used in our setup:
\begin{enumerate}
\item (se1) For any agent $i$,$\ p_{i}(s)(E_{i}\mathcal{)}=0$ if
E_{i}\downarrow s$. That is, at any state, the probability that collections
of evidence which refute the truth are available is zero. In short, proof is
true.
\item (se2) If $E_{i}\not\downarrow s^{\prime }$ then $p_{i}(s^{\prime
})(E_{i}^{\prime })>0$ for some $E_{i}^{\prime }\supseteq E_{i}$. That is,
if a collection of evidence $E_{i}$ does not refute $s^{\prime }$, then it
(or a superset) is available in state $s^{\prime }$ as well.
\end{enumerate}
In Appendix \ref{Conn}, we study the connections between this setup and the
deterministic evidence setup from \cite{BL2012}.
\begin{definition}
A lie $s^{\prime }\in S$ is said to be\ refutable by $i$ at $s$ if $\exists
E_{i}\in \mathbb{E}_{i}$ s.t. $p_{i}(s)(E_{i})>0$ and $E_{i}\downarrow
s^{\prime }$.
\end{definition}
That is, at $s$, $s^{\prime }$ is refutable by $i$ if $i$ may have a
collection of evidence $E_{i}$ such that $E_{i}\downarrow s^{\prime }$.
Clearly from (se1), $E_{i}$ is not available at $s^{\prime }$ (if it were
available, then $E_{i}$ cannot refute $s^{\prime }$). Then, the support of
the evidence distribution for agent $i$ contains a collection under the
truth which is missing under a refutable lie.
\begin{definition}
A lie $s^{\prime }\in S$ is said to be refutable at $s$ if there is an agent
$i$ so that $s^{\prime }$ is refutable by $i$ at $s$. Otherwise, it is
nonrefutable.
\end{definition}
Returning to the leading example, $H$ is nonrefutable at $M$. This brings up
another difference from the deterministic evidence setting, which is that it
is not possible for a firm to prove that it's research process is of high
quality rather than being of medium quality. This is because there is no
additional article of evidence available at state $H$ which is not available
at state $M$. This scenario, called a nonrefutable lie (e.g. $H$ is
nonrefutable at $M$), also occurs under deterministic evidence, but under
deterministic evidence, it is necessary that some agent has additional
evidence under a non-refutable lie than under the truth with probability 1.
Indeed the treatment of non-refutable lies is one of the main differences
between the deterministic and uncertain evidence setups.
We note that if $s^{\prime }$ is a nonrefutable lie at $s$, then no agent
may possess (with positive probability) any articles of evidence in state $s$
which they may not possess at state $s^{\prime }$, since these articles
would refute $s^{\prime }$. Therefore, if $s^{\prime }$ is nonrefutable at
s $, there are two possibilities with respect to the support of the evidence
distribution at $s^{\prime }$. Either it strictly includes the support under
$s$, so that $s$ is refutable (with positive probability) at $s^{\prime }$,
or the supports are identical, so that only the probabilities with which
various collections of evidence are available has changed. The latter
scenario poses a major challenge in the implementation result we pursue.
\section{\label{BayesianImplementation}Mixed-Strategy Bayesian Nash
Implementation}
\subsection{\label{NPD}A Necessary Condition: No Perfect Deceptions}
Given that under deterministic evidence, the implementing condition (called
measurability) entails only that at least some agent is endowed with a
different set of evidence, it would be natural to expect that a stochastic
version of this condition, which entails requiring $p(s)\neq p(s^{\prime })$
whenever $f(s)\neq f(s^{\prime })$ would suffice as the implementing
condition for Bayesian implementation. We term this condition \emph
stochastic measurability}. It turns out however, that not every
non-refutable lie can be eliminated by a mechanism, necessitating a stronger
implementing condition. We establish the condition and prove its necessity
after some preliminaries below.
First, we define a deception for an agent of type $(s,E_{i})$.
\begin{definition}
A deception for an agent $i$ is a function $\alpha _{i}:S\times \mathbb{E
_{i}\rightarrow \Delta (S\times \mathbb{E}_{i}\mathbb{)}$ such that if
(s^{\prime },E_{i}^{\prime })\in \supp\alpha _{i}(s,E_{i})$, then
E_{i}^{\prime }\subseteq E_{i}$.
\end{definition}
We interpret $\alpha _{i}$ as being a function from the agent's true type to
a distribution over types she can mimic (which requires that true type has a
weakly larger collection of evidence than the mimiced type). A profile of
deceptions is denoted by $\alpha =(\alpha _{i})_{i\in \mathcal{I}}$. This
notion is similar to the one with the same name in \cite{J1991}.
Note that a deception can also be interpreted as a strategy for an agent in
a direct mechanism (a mechanism where agents only report their types, i.e.
M_{i}=S\times \mathbb{E}_{i}$ for each agent).
\begin{definition}
A deception $\alpha _{i}$ is perfect for state $s^{\prime }$ at state $s$
for agent $i$ if,
\begin{equation*}
\forall E_{i}^{\prime }\text{,}\sum_{E_{i}\in \mathbb{E}_{i}}p_{i}(s)(E_{i}
\alpha _{i}(s,E_{i})(s^{\prime },E_{i}^{\prime })=p_{i}(s^{\prime
})(E_{i}^{\prime })\text{.}
\end{equation*}
\end{definition}
Likewise, a profile of deceptions $\alpha =(\alpha _{i})_{i\in \mathcal{I}}$
is perfect for state $s^{\prime }$ at state $s$ if $\alpha _{i}$ is perfect
for state $s^{\prime }$ at state $s$ for each agent $i$.
That is, a deception is perfect for state $s^{\prime }$ at state $s$ if
every agent induces the same distribution over her types as she would if the
true state were $s^{\prime }$ and if she were reporting truthfully.
We denote by $s\rightarrow _{i}s^{\prime }$ the notion that agent $i$ has a
perfect deception from $s$ to $s^{\prime }$. Therefore, if there is a
perfect deception from $s$ to $s^{\prime }$, then, $s\rightarrow
_{i}s^{\prime }$ $\forall i\in \mathcal{I}$. Further, $s\not\rightarrow
_{i}s^{\prime }$ denotes the notion that $i$ does not have a perfect
deception from $s$ to $s^{\prime }$ and $s\not\rightarrow s^{\prime }$
denotes that there is no perfect deception from $s$ to $s^{\prime }$.
As mentioned earlier, it is not possible to eliminate non-refutable lies
such that there is a perfect deception for the non-refutable lie at the true
state. We now define the necessary condition for implementation.
\begin{definition}
(\textbf{Condition NPD - No Perfect Deceptions}) We say that a social choice
function $f$ satisfies \textbf{NPD }whenever for two states $s$ and
s^{\prime }$, if at $s$, state $s^{\prime }$ is nonrefutable, and\ there
exists a perfect deception for $s^{\prime }$, then $f(s)=f(s^{\prime })$.
\end{definition}
That is, to have different desirable outcomes between $s$ and $s^{\prime }$,
either more evidence must be available at $s$ than is available at
s^{\prime }$, or it must be impossible to have a perfect deception for state
$s^{\prime }$ at state $s$. We note that one way for a perfect deception to
be impossible is if there are articles of evidence at $s^{\prime }$ which
are not available at $s$. We can therefore also interpret the above
condition as follows: we are able to eliminate all kinds of lies other than
nonrefutable lies so that agents can mimic the lie both in the state and the
evidence dimensions perfectly.
We also wish to stress the fact that the existence of a perfect deception
from $s$ to $s^{\prime }$ does not by itself preclude our ability to
separate the states by a mechanism. Indeed it must be the case that
s^{\prime }$ is nonrefutable at $s$. To see this, consider a scenario where
the evidence structure is identical between $s$ and $s^{\prime }\emph{except}
$ that each collection at $s$ is appended with an additional article of
evidence $e_{i}$ for each agent $i$. Clearly, a perfect deception involves
each agent claiming the state is $s^{\prime }$ and withholding the article
e_{i}$. However, it is clear that these states can be separated by
incentivizing refutation of a state claim by presenting any $e_{i}$.
For an example of a social choice function and an evidence structure under
which the evidence distribution varies between two states, we refer the
reader to the leading example in Section \ref{Example}. We reproduce the
evidence structure below:
\begin{equation*}
\begin{tabular}{|l|l|}
\hline
State & Evidence (Firm A) \\ \hline
$H$ & $0.6\times \{\{M,H\},\{L,M,H\}\}+0.4\times \{\{L,M,H\}\}$ \\ \hline
$M$ & $0.4\times \{\{M,H\},\{L,M,H\}\}+0.6\times \{\{L,M,H\}\}$ \\ \hline
$L$ & $\{\{L,M,H\}\}$ \\ \hline
\end{tabular
\end{equation*}
It is clear that the distribution of evidence does change between every pair
of states, so that every SCF is stochastically measurable. However, it can
be seen that it is possible to mimic the distribution at $M$ when the true
state is $H$, as the article $\{M,H\}$ is available with extra probability
at $H$. For instance, the deception $\alpha
_{A}(H,\{\{M,H\},\{L,M,H\}\})\rightarrow \frac{1}{3}(M,\{\{L,M,H\}\})+\frac{
}{3}(M,\{\{M,H\},\{L,M,H\}\})$ suffices to do so. However, it is not
possible to mimic the distribution at $H$ when the state is $M$, as the
article $\{M,H\}$ isn't available often enough. This establishes an
important property of the NPD condition, that is, it is not bidirectional.
Therefore, the structure induced by the social choice function on the state
space is not partitional. We stress here that this is a stark point of
differentiation between deterministic and uncertain evidence, since the
necessary condition under deterministic evidence - measurability, induces a
partitional structure on the state space.
We present the main result of the paper below.
\begin{theorem}
\label{MAIN} A social choice function is implementable in BNE if and only if
it satisfies NPD.
\end{theorem}
We present the proof of necessity below, while the implementing mechanism
which constitutes the sufficiency proof is presented in Section \re
{MECHANISM}.
\begin{proof}
\textbf{(Necessity of NPD)} Suppose a mechanism $\mathcal{M}=(M,g,\tau )$
implements $f$ and consider a pair of states $s$ and $s^{\prime }$ so that
s^{\prime }$ is nonrefutable at $s$ and there is a perfect deception from $s$
to $s^{\prime }$.
First, we claim that
\begin{eqnarray}
&&\text{For any collection }E_{i}\text{ in the support of the evidence
distribution at }s\text{,}E_{i}\text{, or a } \TCItag{*} \label{*} \\
&&\text{superset thereof must be in the support of the evidence distribution
at s}^{\prime }\text{.} \notag
\end{eqnarray
This follows from the nonrefutability of $s^{\prime }$ at $s$.
Now, consider any equilibrium $\sigma $ of the mechanism and the following
strategy for agent $i$ at state $s$ when endowed with the collection $E_{i}
. First, the agent plays according to the deception $\alpha _{i}(s,E_{i})$.
This yields another type realization, which we denote by $(s^{\prime
},E_{i}^{\prime })$. Now, the agent plays $\sigma _{i}(s^{\prime
},E_{i}^{\prime })$. Since there is no additional collection of evidence at
s$, this object is well defined. With some abuse of notation, we denote the
profile of such strategies by $\sigma \circ \alpha $.
We claim that $\sigma \circ \alpha $ forms an equilibrium at $s$ with
outcome $f(s^{\prime })$. To see this, consider an arbitrary agent $i$.
First, given the strategies of other agents $j\neq i$, the belief induced
over the actions of other agents $j$ is the same under $\sigma \circ \alpha $
at $s$ as that under $\sigma $ at the state $s^{\prime }$. Given these
beliefs, at state $s^{\prime }$, $\sigma $ induces the outcome $f(s^{\prime
})$ with no transfers. Each type of agent $i$ can obtain this outcome in
state $s$ by playing according to $\sigma _{i}\circ \alpha _{i}$. If any
type $E_{i}$ of agent $i$ at $s$ has a profitable deviation, then there is
another type of agent $i$ at $s^{\prime }$ which has weakly more evidence
than $E_{i}$ (from the claim (\ref{*}) above) and can also deviate
profitably. This contradicts the optimality of $\sigma _{i}$ against $\sigma
_{-i}$ at $s^{\prime }$. Therefore, playing according to $\sigma _{i}\circ
\alpha _{i}$ continues to be a best response for the type. Thus, if there is
a perfect deception $\alpha $ from $s$ to $s^{\prime }$, then any
implementable social choice function must have $f(s)=f(s^{\prime })$.
\end{proof}
Informally, under $\alpha $, agents pretend to be other types and play the
equilibrium strategy under $\sigma $ of the mimiced type. The reason a type
is best responding even though she may be playing the equilibrium strategy
of another type is that the outcome is the same and the preferences of
agents do not change from state to state. That is, even though the concerned
type may have extra evidence, and therefore be able to play other messages,
there was another type at $s^{\prime }$ who also had that extra evidence and
that type's best response still led to the outcome $f(s^{\prime })$ which
this agent is also getting from her strategy under $\sigma (\alpha (s))$, so
that she continues to best respond. Therefore, since an equilibrium outcome
at $s$ is $f(s^{\prime })$, then $f$ is implementable only if
f(s)=f(s^{\prime })$.
We note here that whereas the discussion above has centered around
deceptions where an agent pretends to mimic herself at another state, this
applies to mimicing a profile of distributions as well. More precisely,
suppose that in the true state $s^{\ast }$, which is characterized by the
profile of distributions $(p_{i}^{\ast })_{i\in \mathcal{I}}$, agents
pretend to mimic themselves at another state $s^{\prime }$, which is
characterized by a profile of distributions $(p_{i}^{\prime })_{i\in
\mathcal{I}}$. If they can perfectly mimic this other state (in terms of
playing their equilibrium strategies for state $s^{\prime }$), then by
Theorem \ref{MAIN}, $f$ is implementable only if $f(s^{\ast })=f(s^{\prime
}) $.
This yields an interesting insight, which is that since it is always
possible to mimic any cheap talk messages in the message space,
non-refutable lies $s^{\prime }$ for which agents do not have perfect
deceptions from the truth must require presentation of evidence which is
impossible in the true state. Then, if an agent $i$ does not have a perfect
deception at state $s$ for $s^{\prime }$, then she cannot mimic the
distribution $p_{i}^{^{\prime }}$, as otherwise it would be possible for her
to play the equilibrium strategies for the state $s^{\prime }$. This
highlights the role of evidence in this setup, which is twofold. Either
agents can refute distributional claims using evidence, or agents are unable
to mimic incorrect states perfectly owing to their inability to mimic the
distribution of evidence.
\subsection{\label{CHALLENGE}A Challenge Scheme}
The above discussion established that when agents report certain
nonrefutable lies, they are unable to mimic the distribution of evidence for
these lies. This brings up the question of how we can exploit this fact in
the construction of mechanisms to facilitate the elimination of nonrefutable
lies. To elucidate the underlying idea, we first consider an analogous
situation.
Consider a setting with two agents. The first agent has a deck of cards,
from which she draws a card and presents it. The second agent knows that the
firsts' deck of cards is missing one black card. How does she convey this
information credibly to the designer if it is not possible to check the
entire deck? If she simply says there is a card missing, this claim may not
be credible. Rather, the second agent can place a bet of the following form
- if you draw a black card, take a dollar from me. If you draw a red card,
give me 99 cents. This bet clearly loses the second agent money if there is
no card missing. If she believes there is a black card missing though, then
this wins her money, as $\frac{26}{51}\times 0.99+\frac{25}{51}\times (-1)>0
. The fact that the second agent is willing to take this bet credibly
signals to the designer that there is something wrong with the deck.
Note that the deck of cards is analogous to agent 1's mixed-strategy.
Therefore, it is impossible to check the deck of cards and the second agent
must find this alternate solution to inform the designer that the first
agent is not playing the truth-telling strategy. Individual cards are
analogous to collections of evidence, and the inability to induce the
correct distribution of evidence is analogous to not having a black card.
For the implementation problem then, agents can blow the whistle on an
incorrect (nonrefutable) state report by identifying which agent does not
have a perfect deception from the truth to the nonrefutable lie, and then
betting sums of money on that agent's plausible collections of evidence so
that - (i) If the nonrefutable lie were to be the truth, then the bet loses
money, and (ii) Since this is not the case, the bet actually wins money. As
we will discuss later, this can be thought of as analogous to a monotonicity
like condition, yielding a \textquotedblleft reversal\textquotedblright\
between true and false states. Returning to the illustrating example, if
firm A tried to claim $H$ when the true state was $M$, it would, at best be
able to induce probability $0.4$ on the article $\{M,H\}$ as that is the
maximum probability with which it is available in state $M$. Then, consider
the following bet: $1$ dollar on $\{\{L,M,H\}\}$ and $1$ dollar against
\{\{L,M,H\},\{M,H\}\}$. In state $H$ under truthful reporting, it yields an
expected loss of 20 cents, while in state $M$, it yields a minimum expected
profit of 20 cents. We will show below that this is possible whenever there
is no perfect deception. Before that, we will provide a simple way for an
agent to place such a bet.
Asking agents to place bets on (say) agent $i$'s evidence would mean that
agents would have to provide the designer a vector of numbers (one for each
collection in $\mathbb{E}_{i}$). This yields a complex message space. In
fact, the designer can make the appropriate bets for agents. All that she
needs to know is what state (say $s^{\prime }$) is being claimed, and what
is the true state. With this information, the designer can deduce which
agent has no perfect deception for the claimed state and place an
appropriate bet against the evidence of that agent on behalf of the agent
who wishes to raise the challenge. We will denote these bets by
b_{i}:S\times S\rightarrow \mathbb{R}^{|\mathbb{E}_{i}|}$.
\begin{remark}
The above can also be viewed as the building block of a monotonicity like
condition. That is, for a pair of states $s$ and $s^{\prime }$ such that
s\not\rightarrow _{i}s^{\prime }$, if all agents claim that the state is
s^{\prime }$, then there is an agent $j$ for whom $(f(s^{\prime })+b_{\hat
\imath}}(s^{\prime },s)(E_{\hat{\imath}}))$ is strictly preferred to
f(s^{\prime })$ in state $s$ while $f(s^{\prime })$\ is weakly preferred to
(f(s^{\prime })+b_{\hat{\imath}}(s^{\prime },s)(E_{\hat{\imath}}))$ in state
$s^{\prime }$. The allocation $f(s^{\prime })+b_{\hat{\imath}}(s^{\prime
},s)(E_{\hat{\imath}})$ therefore, forms a test allocation which yields a
reversal for an agent $j$ with $f(s^{\prime })$ between the states $s$ and
s^{\prime }$.
\end{remark}
We formalize this notion below. First, denote by $\mathcal{A}_{i}(s,E_{i})$
the set of all deceptions $\alpha _{i}(s,E_{i})$ for agent $i$ when endowed
with evidence $E_{i}$ at state $s$. Let $\alpha _{i}(s)$ be defined as
\alpha _{i}(s)=(\alpha _{i}(s,E_{i}))_{E_{i}^{\prime }\in \mathbb{E}_{i}}$,
and analogously, let $\mathcal{A}_{i}(s)$ be the set of all possible
profiles $\alpha _{i}(s)$. Further, denote by $p_{\alpha
_{i},s}(E_{i}^{\prime })$ the probability induced over evidence collection
E_{i}^{\prime }$ induced by agent $i$ under $\alpha _{i}(s)$. That is,
\begin{equation*}
p_{\alpha _{i},s}(E_{i}^{\prime })=\sum_{E_{i}\in \mathbb{E
_{i}}p_{i}(s)(E_{i})\alpha _{i}(s,E_{i})(s^{\prime },E_{i}^{\prime })\text{.}
\end{equation*}
Defining probability distributions over evidence, i.e. $(p_{\alpha
_{i},s}(E_{i}^{\prime }))_{E_{i}^{\prime }\in \mathbb{E}_{i}}$by $p_{\alpha
_{i},s}$, we then denote by $P_{i}(s)$ the set $\{p_{\alpha _{i},s}:\alpha
_{i}\in \mathcal{A}_{i}\}$, which is the set of all possible such
probability distributions which agent $i$ can induce by playing any
deception in state $s$. Among these distributions is also $p_{\alpha
_{i}^{\ast },s}$, the distribution over agent $i$'s evidence at $s$, which
corresponds to the \textquotedblleft no deception\textquotedblright\
scenario. At state $s^{\prime }$ let $p_{\alpha _{i}^{\ast },s^{\prime }}$
denote the induced distribution over evidence by the \textquotedblleft no
deception\textquotedblright\ scenario.
With these preliminaries established, we now present the challenge scheme in
the following lemma.
\begin{lemma}
\label{SEP_HYP}For every agent $i$, there is a finite set
B_{i}=\{b_{i}(s,s^{\prime }):s\not\rightarrow _{i}s^{\prime }\}$ such that
for any pair of states $s$ and $s^{\prime }$ for which $s\not\rightarrow
_{i}s^{\prime }$, we have $b_{i}(s,s^{\prime })\cdot p_{\alpha _{i}^{\ast
},s^{\prime }}<0$ and $b_{i}(s,s^{\prime })\cdot p>0$ $\forall p\in P_{i}(s)
.
\end{lemma}
\begin{proof}
Since $s\not\rightarrow _{i}s^{\prime }$, $p_{\alpha _{i}^{\ast },s^{\prime
}}\notin P_{i}(s)$. Further, $P_{i}(s)$ is closed and convex. Then, from the
separating hyperplane theorem, there is a hyperplane $b_{i}$ such that
b_{i}\cdot p_{\alpha _{i}^{\ast },s^{\prime }}<0$ and $b_{i}\cdot p>0$
\forall p\in P_{i}(s)$.
For a pair of states $s$ and $s^{\prime }$ one such $b_{i}$ is sufficient.
Iterating over all possible pairs of states which satisfy the premise of no
perfect deception yields the set $B_{i}$. Therefore, each $B_{i}$ is finite.
\end{proof}
\begin{remark}
The above lemma is interpreted as follows: there is a vector of real numbers
(one for each possible collection of evidence for $i$) so that under
truthful revelation in state $s^{\prime }$, the expectation of this vector
is negative whereas for any possible strategy of agent $i$ in state $s$, the
expectation is positive. Accordingly, if an agent $j$ were to receive/give
sums of money based on $b_{i}$ according to agent $i$'s evidence
presentation in the direct mechanism, then in state $s^{\prime }$ under
truthful revelation by agent $i$, agent $j$ loses money while under any
strategy for agent $i$ in state $s$, agent $j$ gains money. This yields a
monotonicity like interpretation - if $s^{\prime }$ is a lie, the agent $j$
can "blow the whistle" and indeed chooses to do so since it is profitable.
\end{remark}
We remark here that the set $B_{i}$ is a set of possible bets, one for each
pair of states such that there is no prefect deception for one to another.
For the set to be usable in a mechanism, we need a way for an agent to
utilize members of this set to challenge unanimous but incorrect state
claims. This operationalization is achieved as follows. First, define a
function $\hat{\imath}:S\times S\rightarrow \mathcal{I}$ such that if
s\not\rightarrow s^{\prime }$, $s\not\rightarrow _{\hat{\imath}(s,s^{\prime
})}s^{\prime }$. That is, $\hat{\imath}$ identifies the agent who does not
have a perfect deception from state $s$ to $s^{\prime }$. Then, $b_{\hat
\imath}}:S\times S\rightarrow B_{\hat{\imath}}$ is defined such that $b_
\hat{\imath}}(s,s^{\prime })\cdot p_{\sigma _{\hat{\imath}}^{\ast
},s^{\prime }}<0$ and $b_{\hat{\imath}}(s,s^{\prime })\cdot p>0$ $\forall
p\in P_{\hat{\imath}}(s)$. The existence of such functions follows
immediately from Lemma \ref{SEP_HYP}.
That is, when an agent $j$ wants to bet against a report of $s^{\prime }$ in
state $s$, then merely telling the designer that the true state is $s$
suffices. This is because the designer can use the function $\hat{\imath}$
as defined above to infer which agent has no perfect deception from $s$ to
s^{\prime }$ and can appropriately place a bet against $\hat{\imath}$'s
evidence on behalf of $j$ using the function $b_{\hat{\imath}}$ so that the
bet loses money for $j$ if $s^{\prime }$ were indeed true and $\hat{\imath}$
were reporting truthfully, and wins money for $j$ if $s$ were true
irrespective of what strategy $\hat{\imath}$ employs.
\subsection{\label{SKETCH}A Sketch of the Proof}
In Section \ref{NPD}, we have established that NPD\ is necessary for
implementation. In what follows, we will construct an implementing mechanism
to establish that it is also sufficient. To fix ideas, we begin with a
sketch of the proof first.
The mechanism asks each agent for the following reports: A distribution
report about herself, a distribution report for the agent to her right, an
evidence report, and a state report (used for placing bets). The proof
utilizes the following steps. The first step is an intermediate result that
is used frequently, and thus forms one of the main building blocks of the
proof. In essence, the result states that if each type of a particular agent
(say $i$) submits all the evidence it is endowed with, then every agent ($i$
herself, and all others) present truthful reports about $i$'s evidence
distribution. This result is obtained as follows. First, we use a proper
scoring rule ($\tau ^{2}$ in the proof) to incentivize other agents to
predict the evidence report by agent $i$. Since agent $i$ presents maximal
evidence in all her types, the optimal predictor for other agents is the
true distribution for agent $i$. This truthful report is used (by means of a
crosschecking penalty) to discipline agent $i$'s report about herself. This
is denoted by $\tau ^{3}$ in the proof. To reiterate, agent $i$'s maximal
evidence presentation incentivizes other agents to make truthful predictions
about her, whereupon a crosscheck incentivizes agent $i$ to make truthful
reports about her own distribution. A direct corollary of this result is
that if all agents present maximal evidence, then the entire profile of
reports is truthful.
A key novelty of this argument is in the solution to the central tension
when using scoring rules - on the one hand, the scoring rule must be active
on all realizations of evidence to elicit the true distribution (otherwise
it elicits the distribution conditional on being active, which may not yield
truthful elicitation), but on the other hand, we require zero transfers on
equilibrium. This is solved by breaking the scoring rule transfer into two
parts - a direct part, which incentivizes predictions, and an offset part,
which is used to cancel the direct part in equilibrium. When an agent makes
a prediction, she only controls the direct part, so that from her point of
view, the scoring rule is always active. This incentivizes her to make
truthful predictions about others. However, her truthful report is used to
discipline other agent's reports about themselves, and this disciplined
report is used to offset the scoring rule, so that it does not result in a
transfer on equilibrium.
Armed with this intermediate result, the second step argues that the report
profile must be pure (recall that agents are allowed to mix) and must be
consistent with some state $s$. This step depends on an incentive for
evidence presentation which is triggered if agents disagree with each other.
This is denoted by $\tau ^{1}$ in the proof. Broadly, if agents mix, then
agents expect disagreement in the distribution reports, and if the message
profile is inconsistent (due to such disagreements), then each agent is
incentivized to present their maximal evidence. Then, from the above
intermediate result, we can argue that the message profile cannot be
inconsistent, since agents must make truthful reports if all evidence
presentation is maximal and truthful reports must, by definition, be
consistent with the true state.
The third step establishes that there cannot be a consensus on a refutable
lie. We institute a penalty ($\tau ^{4}$ in the proof) which penalizes
agents if this commonly agreed upon state $s$ is refuted by another agent.
There is also a small reward for refuting this consensus (this is implicit
within $\tau ^{1}$, which is also activated upon refutation). This penalty (
\tau ^{4}$) is chosen to be large enough to deter agents from agreeing on a
refutable lie.
The fourth and final step establishes that there cannot be a consensus on a
nonrefutable lie which has a different outcome than the truth. If this were
to be the case, then from Condition NPD, there is an agent (say $i$) who
does not have a perfect deception from the true state to $s$. Then, the
mechanism provides a way for agents to make a bet, as discussed in Section
\ref{CHALLENGE}. Further $\tau ^{1}$ is activated when there is an active
bet as well. This leads to the submission of maximal evidence by all agents,
which by the first step means that the profile of reports must have been
true, so that there could not have been a consensus on a nonrefutable lie in
the first place.
Thus, we are left only the possibility of a joint pure report of the true
state (or a state which is outcome equivalent to the truth), so that
implementation obtains. In Section \ref{IMPL}, we also show that there are
no transfers in equilibrium.
\subsection{\label{MECHANISM}Mechanism}
\subsubsection{Message Space}
Suppose the agents are seated around a circular table, so that for each
agent $i$, there is an agent to her right (this is agent $i+1$). For agent
I $, the agent considered as \textquotedblleft being to the right
of\textquotedblright\ her is agent $1$. With this, the message space is
defined as follows,
\begin{equation*}
M_{i}=P_{i}\times P_{i+1}\times \mathbb{E}_{i}\times S\text{,}
\end{equation*}
and a typical message ($m_{i}$) is given by
m_{i}=(p_{i,i},p_{i,i+1},E_{i},s_{i})$.
That is, each agent makes a claim about her own evidence distribution (
p_{i,i}$) , a claim about the evidence distribution of the agent to her
right ($p_{i,i+1}$), submits a collection of evidence and makes a state
claim for (possibly) betting against other's evidence reports.
\subsubsection{Outcome}
A message profile $m$ is said to be \emph{consistent} with state $s$ if each
distributional claim matches the true distribution in state $s$. That is,
m=(p_{i,i},p_{i,i+1},E_{i},s_{i})_{i\in \mathcal{I}}$ is consistent with $s$
if $\forall i,j$, $p_{i,j}=p_{j}(s)=p_{j,j}$. Otherwise, it is \emph
inconsistent}.
The outcome is given by $f(s)$ if $m$ is consistent with a state $s$, and an
arbitrary outcome $a\in A$ otherwise.
\subsubsection{Transfers}
The mechanism uses five transfers. There are scaling parameters with most of
the transfers, and we will define the appropriate scaling in Section \re
{SCALING}.
The first transfer provides an agent the incentive to submit evidence when
there is inconsistency, or an active bet, or if a consistent profile has
been refuted by an agent. That is,
\begin{equation*}
\tau _{i}^{1}\left( m\right) =\left\{
\begin{tabular}{ll}
$\varepsilon \times |E_{i}|$, & if $m$ involves inconsistency, or an active
bet, \\
& or if $m$ is consistent with $s$ and $E_{i}\downarrow s$; \\
$0$ & otherwise
\end{tabular
\right.
\end{equation*}
where $\varepsilon >0$ is a small positive number.
The second transfer is a modified proper scoring rule as defined below.
First, define $Q_{j}(p_{i,j},E_{j})=[2p_{i,j}(E_{j})-p_{i,j}\cdot p_{i,j}]$.
That is, $Q_{j}(p_{i,j},E_{j})$ gives the value of the scoring rule if $j$
presents the collection $E_{j}$ and agent $i$'s report about $j$ is $p_{i,j}
. With this, define
\begin{equation*}
\tau _{i,i+1}^{2}(m)=\left\{ \underline{\tau }\times \lbrack
Q_{i+1}(p_{i,i+1},E_{i+1})-Q_{i+1}(p_{i+1,i+1},E_{i+1})]\text{,}\right.
\end{equation*}
Where $\underline{\tau }>0$. This transfer provides agent $i$ the incentive
to truthfully report agent $i+1$'s distribution (due to the first term) when
agent $i+1$ presents all her evidence. In equilibrium, the reports are
consistent and therefore the transfer is off.
\begin{equation*}
\tau _{i}^{3}(m)=\left\{
\begin{tabular}{ll}
$-\underline{\tau }$, & if $p_{i,i}\neq p_{i-1,i}$ \\
$0$ & otherwise
\end{tabular
\right.
\end{equation*}
Every agent's report about herself is cross-checked with the report about
her by the agent to her left. If the reports are not in agreement, then the
agent is fined $\underline{\tau }$.
The fourth transfer is related to lies which other agents can refute.
\begin{equation*}
\tau _{i}^{4}\left( m\right) =\left\{
\begin{tabular}{ll}
$-\bar{\tau}$, & if $p_{i,i+1}=p_{i+1}(s)$ $\forall i$, and $E_{j}\downarrow
s$ for some $j\neq i$ \\
$0$ & otherwise
\end{tabular
\right.
\end{equation*}
That is, an agent $i$ is fined $\overline{\tau }$ if another agent has
refuted state $s$ obtained from the vector of reports by agents about the
agents to their right. Note that this transfer does not apply to\ an agent's
report about herself. Further, this transfer is inactive when the profile
(p_{i,i+1})_{i\in \mathcal{I}}$ is not consistent.
$\tau ^{5}$ is as defined below.
\begin{equation*}
\tau _{i}^{5}\left( m\right) =\left\{
\begin{tabular}{ll}
$\varepsilon \times b_{j}(s^{\prime },s_{i})(E_{j})$, where $j=\hat{\imath
(s^{\prime },s_{i})$ & if $m$ is consistent with $s^{\prime }$ \\
$0$ & otherwise
\end{tabular
\right.
\end{equation*}
where $\varepsilon >0$ is a small positive number as defined earlier. Note
that $\tau ^{1}$ and $\tau ^{5}$ are scaled using the same number
\varepsilon $.
That is, if agent $i$ bets against a unanimous report consistent with state
s^{\prime }$ with a bet claiming state $s_{i}$, the designer evaluates the
challenge in accordance with the set $B_{j}$, where $j=\hat{\imath
(s^{\prime },s_{i})$ and the function $b_{j}$ defined in Section \re
{CHALLENGE} and pays out the revenue from the bet to agent $i$.
\subsubsection{\label{SCALING}Scaling}
In this section, we will define some parameters used for appropriately
scaling the transfers, and then establish that there are values for
\overline{\tau }$, $\varepsilon $, and $\underline{\tau }$ so that the
appropriate scaling is possible.
First, we provide an intuitive overview of the scaling based on the proof
sketch in Section \ref{SKETCH}. Recall that we are working in a normalized
setup in which the utility of the outcome has been normalized to a number
less than 1. Since an arbitrarily small $\varepsilon $ provides sufficient
incentive for agents to submit evidence and make bets, we can choose
\varepsilon $ small enough so that a penalty of 1 dollar is enough to
dominate not only any change in outcome, but also any gains or losses from
\tau ^{1}$ or $\tau ^{5}$. The scoring rule $\tau ^{2}$ is at the medium
scale ($\underline{\tau }$) and dominates any effect from a change of
outcome.\footnote
This is, strictly speaking, larger than it needs to be, since $\tau ^{2}$
needs only to dominate the transfers at $\varepsilon $ scale (as in setting
her report about the agent to her right, an agent may cause (or break)
consistency). However, we scale it at the level of $\underline{\tau }$ for
notational parsimony.} The crosscheck $\tau ^{3}$ is also at this scale
since similar considerations apply to the agent's report about herself.
Since we wish to prioritize the elimination of lies which others can refute
above other incentive considerations, $\tau ^{4}$ dominates all other
transfers, so that it belongs to the highest transfer scale ($\overline{\tau
}$). We define the transfer scales formally below.
Since $\tau ^{2}$ is a proper scoring rule, if an agent $i+1$ presents all
her evidence, agent $i$ maximizes her revenues from $\tau ^{2}$ by
truthfully reporting agent $i+1$'s evidence distribution. We require that
the minimum expected loss for an agent $i$ from lying about agent $i+1$'s
distribution dominates any effect from a change in the outcome. That is,
\begin{equation}
\min_{s\in S}\min_{p_{i,j}\neq p_{j}(s)}\sum_{E_{j}\in \mathbb{E
_{j}}p_{j}(s)(E_{j})\underline{\tau
[Q_{j}(p_{j}(s),E_{j})-Q_{j}(p_{i,j},E_{j})]>1 \label{B}
\end{equation}
Turning to the crosscheck, it must dominate any incentive from the outcome.
Since the minimum loss from lying about agent $i+1$'s distribution scales
linearly in $\underline{\tau }$, we can choose $\underline{\tau }>1$ such
that inequality \ref{B} is satisfied.
Further, define as $\bar{\tau}^{2}$ the maximum possible change in utility
from $\tau ^{2}$. That is, i.e.
\begin{equation*}
\bar{\tau}^{2}=\max_{i\in \mathcal{I}}\max_{E_{i+1}\in \mathbb{E
_{i+1}}\max_{p_{i,i+1},p_{i,i+1}^{\prime }\in P_{i+1}}\underline{\tau
[Q_{i+1}(p_{i,i+1},E_{i+1})-Q_{i+1}(p_{i,i+1}^{\prime },E_{i+1})]
\end{equation*
Now we define the fourth scaling inequality as follows:
\begin{equation}
\overline{\tau }\geq \frac{1+\bar{\tau}^{2}}{\min_{s^{\prime }\in
RL_{i}(s)}\{p_{i}(s)(E_{i}):E_{i}\downarrow s^{\prime }\}},\forall s\in
S,\forall i\in \mathcal{I}\text{.} \label{D}
\end{equation}
That is, $\tau ^{4}$ provides sufficient incentive to avoid being caught
lying even for the smallest probability with which refutable lies may be
refuted by the evidence of other agents.
It is then clear that the parameters $\varepsilon $, $\underline{\tau }$ and
$\overline{\tau }$ can be chosen to satisfy the above inequalities
simultaneously.
\subsection{Proof of Implementation}
In what follows, we assume that the true state is $s^{\ast }$ and that it
induces the profile of evidence distributions $(p_{i}^{\ast })_{i\in
\mathcal{I}}$.
\subsubsection{Preliminary Results}
\begin{lemma}
\label{main}If agent $i$ presents all her evidence in $E_{i}$, then each
agent $j\in \mathcal{I}$ presents the truth in each message $p_{j,i}$.
\end{lemma}
\begin{proof}
If agent $i$ presents all her evidence in $E_{i}$, then for agent $i-1$,
setting $p_{i-1,i}=p_{i}^{\ast }$ maximizes her payoff from $\tau ^{2}$
(since truthful reporting is optimal under a scoring rule). However, this
could also cause losses from $\tau ^{1}$ (by creating consistency) and $\tau
^{5}$ (by activating a bet via creating consistency). However, since
\underline{\tau }>1$ (by choice), $p_{i-1,i}=p_{i}^{\ast }$ is the best
response for agent $i-1$ in equilibrium. Then, since $\underline{\tau }>1$,
p_{i,i}=p_{i}^{\ast }$ is the best response for agent $i$.
\end{proof}
\subsubsection{\label{CONS}Consistency}
\begin{claim}
\label{CONSISTENCY}In any equilibrium, there is a state $s$ such that the
message profile is consistent with $s$.
\end{claim}
\begin{proof}
If all agents present all their evidence, then Lemma \ref{main} yields that
all agents present true distributional claims in their reports and thus the
profile of reports must be consistent with the truth, so that $s=s^{\ast }$.
Now, suppose that there is inconsistency but not all agents present all
their evidence. We claim that this is only possible if precisely one agent
i $ randomizes in her reports $p_{i,i}$ or $p_{i,i+1}$. To see this, notice
that if two or more agents randomize in their distribution reports, then
each type of each agent expects inconsistency with positive probability and
presents all their evidence. So suppose that only agent $i$ mixes. Then in
each message, each agent $j\neq i$ expects inconsistency with positive
probability and presents all her evidence ($\varepsilon >0$). Then, from
Lemma \ref{main}, $p_{k,j}$ is the truth for all $k\in \mathcal{I}$ and
j\neq i$. Therefore, $i$ can only mix in $p_{i,i}$. However, by hypothesis,
each agent $j\neq i$ presents a pure report in $p_{j,i}$. Therefore, in any
message $m_{i}$ where $p_{i,i}\neq p_{j,i}$, she incurs a loss from $\tau
^{3}$. Note that there are no losses from $\tau ^{4}$ since agent $i$ is
only changing $p_{i,i}$ and $\tau ^{4}$ does not apply to an agent's report
about herself. Consider a deviation to match $p_{j,i}$. This avoids the loss
from $\tau ^{4}$, but may cause losses from $\tau ^{2}$ (by creating
consistency) and $\tau ^{5}$ (by activating a bet via creating consistency).
Since $\underline{\tau }>1$, this is a profitable deviation. Therefore, we
have a contradiction.
\end{proof}
\subsubsection{\label{norefutable}Eliminating Refutable Lies}
\begin{claim}
\label{NOORL}There is no equilibrium where the distribution reports are
consistent with a refutable lie.
\end{claim}
\begin{proof}
Suppose instead that $m$ is consistent with a refutable lie $s$. If it is
refutable by two or more agents, then $\tau ^{1}$ provides the incentive to
refute the lie, and so all agents expect the lie to be refuted with positive
probability so that they present maximal evidence. In turn, Lemma \ref{main}
implies the profile must be consistent with the truth, a contradiction.
So suppose $s$ can only be refuted by some agent $i$. Then, all agents other
than $i$ expect agent $i$ to refute $s$ with positive probability and
therefore present maximal evidence. Thus, from Lemma \ref{main},
p_{.,j}=p_{j}^{\ast }$ $\forall j\neq i$. Then, $p_{i-1,i}\neq p_{i}^{\ast }$
(otherwise the profile would be consistent with the truth). Therefore, in
presenting $p_{i-1,i}$, agent $i-1$ knows that this claim will be refuted
with probability at least $\min_{E_{i}\in \mathbb{E}_{i}}\{p_{i}(s^{\ast
})(E_{i}):E_{i}\downarrow s\}$. However, if she deviates to
p_{i-1,i}=p_{i}^{\ast }$, the profile has $p_{i,i+1}=p_{i+1}(s^{\ast })$
\forall i$ and therefore cannot be refuted. So, the agent gains $\overline
\tau }$ from $\tau ^{4}$. Also, the agent is deviating away from
consistency, so that $\tau ^{1}$ yields no losses. Since $\overline{\tau }$
dominates all the other incentives (Inequality (\ref{D})), deviating to
p_{i}^{\ast }$ instead of presenting $p_{i-1,i}$ is a profitable deviation
for agent $i$.
\end{proof}
\subsubsection{\label{NOBET}No Bets in Equilibrium}
\begin{claim}
\label{NONRL}In any equilibrium, the distribution reports are not consistent
with a non-refutable lie.
\end{claim}
\begin{proof}
From Claim \ref{CONSISTENCY}, the agents must all present pure distribution
reports which are consistent with a state $s$. From Claim \ref{NOORL}, $s$
is not refutable by any agent, since this would mean that an agent is
presenting a lie which another agent can refute. If $f$ prescribes a
different outcome at $s$ than at the truth, then there must be an agent $i$
such that $s^{\ast }\not\rightarrow _{i}s$, that is, agent $i$ does not have
a perfect deception from the truth to $s$. Then, any agent $j\neq i$ has the
incentive to bet with probability 1, since $\varepsilon >0$, whereupon all
evidence presentation is maximal (due to $\tau ^{1}$) and Lemma \ref{main}
presents a contradiction.
\end{proof}
\subsubsection{\label{IMPL}Implementation}
\begin{claim}
In any equilibrium, the outcome is $f(s^{\ast })$ and there are no transfers.
\end{claim}
\begin{proof}
From Claims \ref{CONSISTENCY}-\ref{NONRL}, the report profile is consistent
with the true state. Therefore, the outcome is $f(s^{\ast })$. Clearly, such
a consensus is not refutable, since any collection available under the truth
cannot (by definition) refute it.
Now, we claim that there are no bets in equilibrium. To see this, notice
that if an agent $i$ bets in equilibrium, then it must be that agents have
not presented all their evidence in $E$, since from Lemma \ref{SEP_HYP}, a
bet yields a loss if placed against a truthful reporting strategy. If some
agent $i$ finds it profitable to place a bet in any of her messages, then
she must find it profitable to bet in all of her messages. Then, each agent
finds it optimal to present all their evidence in $E$, a contradiction.
Since the message profile is consistent, and features no bets in
equilibrium, $\tau ^{1}$, $\tau ^{3}$ and $\tau ^{5}$ are inactive as well.
Finally, notice that the absence of any inconsistency yields that each
agent's claims about herself must be the same as those of the agent to her
left. That is, $p_{i,i+1}=p_{i+1,i+1}$ $\forall i$. Then, $\tau ^{2}$ is
inactive as well. Thus, there are no transfers.
\end{proof}
\section{\label{PBNI}Pure-Strategy Bayesian Nash Implementation}
In this section, we take a brief digression to discuss pure-strategy
implementation. While NPD was shown to be necessary for mixed-strategy
implementation, a weaker version, called No Pure-Perfect Deceptions (NPPD)
suffices for pure-strategy implementation. It is essentially identical to
NPD except for the additional requirement that any deception $\alpha $ be
degenerate, since playing a non-degenerate deception requires mixing by the
agents. A pure-perfect deception is defined below.
\begin{definition}
A deception $\alpha =(\alpha _{i})_{i\in \mathcal{I}}$ is pure-perfect for
state $s^{\prime }$ at state $s$ if it is a perfect deception for $s$ at
s^{\prime }$ and $\alpha $ is degenerate.
\end{definition}
We now define the No Pure-perfect deceptions condition.
\begin{definition}
(\textbf{Condition NPPD - No Pure-Perfect Deceptions}) We say that a social
choice function $f$ satisfies \textbf{NPPD }whenever for two states $s$ and
s^{\prime }$, if at $s$, state $s^{\prime }$ is nonrefutable, and\ there
exists a pure-perfect deception for $s^{\prime }$, then $f(s)=f(s^{\prime })
.
\end{definition}
We now state the pure-strategy implementation result.
\begin{theorem}
\label{PS_SM}A social choice function is implementable in pure-strategy BNE
if and only if it satisfies NPPD.
\end{theorem}
The proof of this result is derived as a simplification of our
mixed-strategy implementation result, and is relegated to Appendix \re
{PS_SM_NECC&SUFF}.
No Pure-perfect deceptions is shown to be strictly weaker than NPD and
strictly stronger than Stochastic Measurability in Appendix \re
{NOPUREDECEPTION}. Therefore, in our setup, mixed-strategy implementation is
a strictly harder problem than pure-strategy implementation. We note that
this distinguishes this setup from the complete-information setup in \cit
{BL2012} where mixed-strategy Nash Implementation does not require a
stronger condition on the social choice function. However, it is consistent
with the rest of the Bayesian implementation literature though, in that
mixed-strategy Nash implementation in \cite{SV2010} requires Mixed Bayesian
Monotonicity, which is stronger than Bayesian Monotonicity which is required
for pure-strategy implementation (\cite{J1991}).
\section{\label{IST}Implementation with a General Type Space}
In this section, we study a more general model than the one considered
above. We allow agents to have a type $t_{i}\in T_{i}$ (where $T_{i}$ is
finite) and allow agents' types to be correlated. A type profile $t$
determines agents' preferences over the outcomes in a set $A$, so that $\bar
u}_{i}(a,t)\,$\ is agent $i$'s bounded utility when the outcome is $a$ and
the type profile is given by $t$. Further, $u_{i}(a,\bar{\tau}_{i},t)=$
\bar{u}_{i}(a,t)\,+\bar{\tau}_{i}$ is the quasilinear extension of $\bar{u
_{i}$. The type profile $t$ also determines agents' evidence, which is
represented by $\hat{E}_{i}(t_{i})\in \mathbb{E}_{i}$. States of the world
are represented by type profiles $t=(t_{i})_{i\in \mathcal{I}}$. Each agent
has a belief $q_{i}:T_{i}\rightarrow \Delta (T_{-i})$ where $T_{-i}=\Pi
_{j\neq i}T_{j}$. Such a model is represented by $\mathcal{T=}(T_{i},\hat{E
_{i},q_{i})_{_{i\in \mathcal{I}}}$. A social choice function maps a state to
the desirable outcome in that state, so that $f:T\rightarrow A$, where
T=\Pi _{i}T_{i}$. Once again, we are interested in implementing $f$ without
relying on preference variation, so that we consider constant preferences as
a possibility.
In this section, we will adapt the solution concept of Interim Correlated
Rationalizability from \cite{DFM2007} to our setting. Consider a mechanism
\mathcal{M}=(M,g,\tau )$, which induces a static Bayesian game denoted by
G\left( \mathcal{M},\mathcal{T},u\right) $. Define the set of messages
feasible for an agent $i$ of type $t_{i}$ by $M_{i,t_{i}}$. Denote by
R_{i}^{G}\left( t_{i}\right) $ use the set of rationalizable messages for
agent $i$ of type $t_{i}$. Then, we have $R_{i,0}^{G}\left( t_{i}\right)
=M_{i,t_{i}}$;
\begin{equation*}
R_{i,k+1}^{G}\left( t_{i}\right) =\left\{
\begin{array}{l|l}
m_{i}\in M_{i,t_{i}} &
\begin{array}{l}
\text{ there exists }\pi _{i}\in \Delta \left( T_{-i}\times M_{-i}\right)
\text{ such that } \\
(1)\pi _{i}\left( t_{-i},m_{-i}\right) >0\Rightarrow m_{-i}\in \Pi _{j\neq
i}R_{j,k}^{G}\left( t_{j}\right) \\
(2)m_{i}\in \underset{m_{i}^{\prime }\in M_{i,t_{i}}}{\arg \max
\sum_{E_{-i},m_{-i}}u_{i}(g\left( m_{i}^{\prime },m_{-i}\right) ,\tau
_{i}\left( m_{i}^{\prime },m_{-i}\right) ,t) \\
(3)\sum_{m_{-i}}\pi _{i}\left( t_{-i},m_{-i}\right) =q_{i}(t_{i})(t_{-i}
\end{array
\end{array
\right\} \text{;}
\end{equation*
and $R_{i}^{G}\left( t_{i}\right) =\cap _{k=1}^{\infty }R_{i,k}^{G}\left(
t_{i}\right) $. Then, denote by $R^{G}\left( t\right) =\Pi _{i\in
I}R_{i}^{G}\left( t_{i}\right) $ as the profile of all rationalizable
messages.
With this solution concept, we formalize the implementation notion below.
\begin{definition}
A social choice function $f$ is rationalizably implementable with
arbitrarily small transfers if for any $\varepsilon >0$, there is a
mechanism $\mathcal{M}=(M,g,\tau )$ such that for any profile of bounded
utility functions $\bar{u}=\left( \bar{u}_{i}\right) _{i\in \mathcal{I}}$,
any type profile $t$, and any profile of rationalizable messages $m\in
R^{G}\left( t\right) $ of the game $G\left( \mathcal{M},\mathcal{T},u\right)
$, we have $g(m)=f(t)$. Further, $|\tau _{i}(m)|\leq \varepsilon $ for every
message profile $m$.
\end{definition}
That is, given an arbitrarily small (strictly) positive number $\varepsilon
, it should be possible to obtain the socially optimal outcome at any
profile of rationalizable messages, and limit the transfers for any message
profile (rationalizable or otherwise) to being no higher than $\varepsilon
. This represents a relaxation of the prior requirement that the transfers
should be zero in equilibrium, but yields a relaxation of the solution
concept to rationalizability, allows correlation among types, and (we will
show) yields a weaker implementing condition than either NPD or NPPD.
We now describe the agent's higher order beliefs. Since we wish to implement
without relying on preference variation, so that constant preferences are a
possibility, we restrict agents to form beliefs only on evidence to obtain a
necessary condition which is strong enough to allow implementation when
preferences are constant. Note that the implementing mechanism we will
present later will allow for preference variation, but will not rely on it
for implementation. To this end, define $Z_{i}^{0}=\mathbb{E}_{-i}$, and for
$k\geq 1$, define $Z_{i}^{k}=Z_{i}^{k-1}\times \Delta (Z_{-i}^{k-1})$. For
each $i$, let $T_{i}^{\ast }$ denote the set of $i$'s collectively coherent
hierarchies (\cite{MZ85}). For each $t_{i}\in T_{i}$, let $\hat{\pi
_{i}^{0}(t_{i})=\hat{E}_{i}(t_{i})$; construct mappings $\hat{\pi
_{i}^{k}:T_{i}\rightarrow \Delta (Z_{i}^{k-1})$ recursively for all $i\in
\mathcal{I}$ and $k\geq 1$, such that $\hat{\pi}_{i}^{1}$ is the
push-forward of $q_{i}(t_{i})$ given by the map from $T_{-i}$ to $Z_{-i}^{0}
, such that
\begin{equation*}
t_{-i}\longmapsto \hat{\pi}_{-i}^{0}(t_{-i})\text{,}
\end{equation*}
and $\hat{\pi}_{i}^{k}(t_{i})$ is the push-forward of $q_{i}(t_{i})$ given
by the map from $T_{-i}$ to $Z_{-i}^{k-1}$ such that
\begin{equation*}
t_{-i}\longmapsto (\hat{\pi}_{-i}^{0}(t_{-i}),\hat{\pi}_{-i}^{1}(t_{-i}),...
\hat{\pi}_{-i}^{k-1}(t_{-i}))\text{,}
\end{equation*}
where for any $k=0,1,2,..$, $\hat{\pi}_{-i}^{k}(t_{-i})=\Pi _{j\neq i}\hat
\pi}_{j}^{k}(t_{j})$.
The mappings thus constructed, i.e. $\hat{\pi}_{i}^{\ast }(t_{i})=(\hat{\pi
_{i}^{0}(t_{i}),\hat{\pi}_{i}^{1}(t_{i}),...)$ assign to each type a
hierarchy of beliefs. We now propose a new implementing condition which we
term higher-order measurability.
\begin{definition}
A social choice function $f$ satisfies higher-order measurability if for any
pair of states $t$ and $t^{\prime }$ such that $f(t)\neq f(t^{\prime })$,
there is $k\in
\mathbb{N}
$ such that there is an agent $i$ whose $k^{th}$ order belief $\hat{\pi
_{i}^{k}(t_{i})\neq \hat{\pi}_{i}^{k}(t_{i}^{\prime })$.
\end{definition}
The necessity of higher-order measurability is demonstrated in Appendix \re
{VirtualImplementation}. Intuitively, higher-order measurability requires
that the social choice function $f$ be measurable on a partition of $T$
where each cell of the partition is such that if $t$ and $t^{\prime }$
belong to the same cell, then each agent has the same belief hierarchy in
both $t$ and $t^{\prime }$.
We now consider the setup we work with for Bayesian Nash implementation.
Suppose an SCF $f$ satisfies stochastic measurability, so that if $f(s)\neq
f(s^{\prime })$, then there is an agent $i$ such that $p_{i}(s)\neq
p_{i}(s^{\prime })$. Then, the first order belief of any agent $j\neq i$ is
different between $s$ and $s^{\prime }$, so that $f$ must satisfy
higher-order measurability. Conversely, if there are two states $s$ and
s^{\prime }$ so that for each agent $i$, $p_{i}(s)=p_{i}(s^{\prime })$, and
evidence distributions are independent, then $f(s)$ must be identical to
f(s^{\prime })$ if $f$ is to satisfy higher-order measurability. Therefore,
in such a setup, higher-order measurability is identical to stochastic
measurability, and therefore weaker than either NPD or NPPD.
Towards establishing that higher-order measurability is also sufficient for
implementation with the above implementation notion, we will define the
relevant version of incentive compatibility in this setting. To do so,
consider the direct mechanism with respect to $\mathcal{T=}(T_{i},\hat{E
_{i},q_{i})_{_{i\in \mathcal{I}}}$, where each agent $i$ makes a report
t_{i}\in T_{i}$ and the outcome chosen is $f((t_{i})_{i\in \mathcal{I}}$. No
transfers are induced.
\begin{definition}
A social choice function $f$ satisfies evidence incentive compatibility if
for each agent $i$, and each type $t_{i}\in T_{i}$,
\begin{equation*}
t_{i}\in \arg \max_{t_{i}^{\prime }\in \hat{T}_{i}(t_{i})}\sum_{t_{-i}\in
T_{-i}}q_{i}(t_{i})(t_{-i})\bar{u}_{i}(f(t_{i}^{\prime
},t_{-i}),(t_{i},t_{-i}))
\end{equation*}
\noindent where $\hat{T}_{i}(t_{i})=\{t_{i}^{\prime }:\hat{E
_{i}(t_{i}^{\prime })\subseteq \hat{E}_{i}(t_{i})\}$.
\end{definition}
That is, if agent $i$ is endowed with type $t_{i}$, and anticipates every
other agent to report their type truthfully, then it is among her best
responses to also report truthfully. This is consistent with \cite{DS2008},
who show that if agents can send every combination of their available
messages, then the above condition is necessary and sufficient for
truthtelling to be a Bayesian Nash equilibrium. Theorems \ref{MAIN} and \re
{PS_SM}, however, do not require this condition since the use of large
transfers causes it to be automatically satisfied. Now, we present the main
result of this section.
\begin{theorem}
\label{rat_imp}A social choice function is rationalizably implementable with
arbitrarily small transfers iff it satisfies higher-order measurability and
evidence incentive compatibility.
\end{theorem}
The sufficiency of higher-order measurability is demonstrated through the
construction of an implementing mechanism which is presented in Appendix \re
{VirtualImplementation}. Here, we describe the intuition behind the
construction, which follows ideas from \cite{AM92B}. The first step is to
use a small incentive to extract the evidence from each agent. Irrespective
of the other reports they make, it will be a strictly dominant strategy to
present all their evidence. In the second step, we ask each agent to predict
the evidence presentation of the other agents and score their predictions
using a quadratic scoring rule. The input to the scoring rule is the agent's
own type, since common knowledge of the type allows the designer to compute
the posterior distribution $q_{i}(t_{i})$ and score the prediction
appropriately. Since each agent expects every other agent will have
presented all their evidence, she presents her true belief about the
evidence presentations of the other agent in order to maximize the returns
from this scoring rule. This elicits the other agent's true first order
belief. In the third step, we extract the agents true second order belief by
asking them to predict the first order beliefs extracted in the previous
step and scoring their predictions using a quadratic scoring rule.
Higher-order measurability assures us that if we repeat this process enough
times, we will have extracted the true types of the agents. This mechanism
performs the role of the \emph{dictatorial choice function} in \cite{AM92B}.
Each transfer here can be made small enough so that the overall transfer can
be bounded below any preselected $\varepsilon $. In what follows, we refer
to this extracted profile of types as $t^{0}$ and to agent $i$'s component
of $t^{0}$ as $t_{i}^{0}$.
After achieving extraction, we proceed towards the goal of implementing the
social choice function. The steps taken are essentially identical to those
in \cite{AM92B}. Agents are asked for $J$ additional reports of their type,
denoted $(t_{i}^{j})_{j=1,2,..,J}$ in what follows. Each profile of reports
determines the outcome with a probability of $\frac{1}{J}$, but the
following penalties are applied. To the first agent $i$ who deviates in
their report $t_{i}^{j}$ from $t_{i}^{0}$, a penalty is applied which is
large enough to dominate the incentive from affecting the outcome with a
probability of $\frac{1}{J}$ but not so large as to incentivize lying in
t^{0}$. That such a balance can be achieved can be deduced from the fact
that the designer can choose $J$ to be as large as required to reduce the
incentive to affect the outcome. Then, no agent wants to be the first to
deviate from the truthful report in $t^{0}$ and implementation obtains while
the penalties remain switched off.
This mechanism however inherits some of the well known critiques of the
Abreu-Matsushima type of mechanisms, most notably that it requires a high
depth of reasoning, involves a fairly complex message space, and is more
vulnerable to renegotiation (than say the mechanism proposed in the proof of
Theorem \ref{MAIN}) owing to the use of small transfers.
\section{\label{RL}Related Literature}
In this section, we discuss our paper's results alongside some other related
papers, to provide further insight into our contributions.
An early reference in the field of mechanism design with evidence is \cit
{GL86} who study the principal-agent problem when the agent cannot
manipulate the truth arbitrarily. This corresponds to a notion of evidence -
an agent, by presenting messages which are only available to her in a
certain set of states, can prove that the true state is within that set.
Early contributions in the field of game theory and evidence include \cit
{M81}, \cite{G81} and \cite{D85}. More recent papers involving mechanism
design include \cite{GR2004}, \cite{GR2006}, \cite{BW2007}, \cite{DS2008},
\cite{HKP2017}, and \cite{BDL2019}. \cite{BDL2021} study a setup involving
stochastic evidence. While we assume that agents are endowed with evidence
to begin with, \cite{BDL2021} study the process of evidence acquisition and
the value of commitment. In addition, they focus on partial implementation
whereas our focus is on full implementation.
\cite{BL2012}, \cite{KT2012}, and \cite{BCS2020} study full implementation
with hard evidence as well. While \cite{BL2012} and \cite{BCS2020} explictly
study hard evidence, \cite{KT2012} model hard evidence within their more
general costly evidence framework when articles of evidence are constrained
to be either costless or infinitely costly. Specifically, in \cite{BCS2020}
we provide a direct mechanism to fully implement SCFs in mixed strategies
with two or more agents. The chief addition to this line of inquiry in \cit
{BL2012} and \cite{BCS2020} that the present paper contributes is
uncertainty in the dimension of evidence. We are able to provide a finite
mechanism that fully implements in mixed-strategy equilibria, but need an
indirect mechanism to achieve this goal.\footnote
In a classical setting with preference variation and no evidence, \cite{MAM}
also provides a finite (indirect) mechanism for mixed-strategy
implementation. It focuses on a complete-information setting however.}
Restricting attention to direct mechanisms, \cite{P2019} establishes
necessary and sufficient conditions for full Bayesian implementation with
uncertain evidence; however, he also points out that the restriction to
direct mechanisms entails loss of implementability.
Turning away from evidence, \cite{J1991} is a seminal paper in the theory of
Bayesian implementation, focusing on exploiting preference variation under
incomplete information to implement SCFs. While focusing on pure strategies,
it suggests that looking into mixed strategies is important. \cite{SV2010}
indeed extend the analysis in \cite{J1991} to mixed strategies. The
mechanisms in \cite{J1991} use modulo games and those in \cite{SV2010} use
integer games. In our setup, preference variation does not necessarily exist
and yet our implementation notion is robust to preference variation. We
present in this paper, to our knowledge, the first mechanism in the
literature that achieves implementation in a finite mechanism in a Bayesian
setting which explicitly accounts for mixed-strategy equilibrium.
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{"url":"https:\/\/math.meta.stackexchange.com\/questions\/17421\/how-to-ask-a-popular-question","text":"# How to ask a popular question\n\nProblem. Today I saw a question that gathered more than $500$ views less than in $10$ hours. It gives some privilege to the asking person since different people can look at your question from different angles. Usually my questions gather at most $100$ views. So what is the clue? Can somebody explain?\n\nMy attempt. Probably all of high-viewed questions are from the number theory field and they are usually easy for understanding.\n\nThanks a lot at least for reading!\n\n\u2022 Ask questions that everyone can understand, give it a \"catchy\" (not sure what the right word is) title, and don't use mathjax in the title so that it can be featured in the \"hot network questions\" list. Compare \"what is 1+1?\" with \"how to compute the Hochschild homology of $S(x,y,z)\/(y^2=xz)$\". \u2013\u00a0Najib Idrissi Nov 23 '14 at 14:39\n\u2022 Those views come from Hot Network List and have little relation to \"different angles\": it's programmers with passing interest in math stopping by to read and upvote. \u2013\u00a0user147263 Nov 23 '14 at 16:22","date":"2020-10-26 22:16:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5047656893730164, \"perplexity\": 1426.59733714244}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107892062.70\/warc\/CC-MAIN-20201026204531-20201026234531-00151.warc.gz\"}"} | null | null |
Set of 12 storage caddies. Includes 3 of each: Blue, Red, Green and Yellow. 2 Compartments with holes for drainage. Non-toxic. 12.5 in. L x 11 in. W x 6.8 in. H. Organize those arts and crafts supplies with a 4-pack of durable copolymer plastic caddies. Features 2 compartments with holes for drainage and a built-in carry handle. | {
"redpajama_set_name": "RedPajamaC4"
} | 668 |
Q: Android Studio Failure [INSTALL_PARSE_FAILED_MANIFEST_MALFORMED] on Run I experience this error when trying to test the app on my phone on Run. I dunno why this error pop out but I have try to do some fix based based on some solution from stack but it is not working.
my pkg: /data/local/tmp/com.example.wolex_000.freesim
MY Manifest code is:
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="com.chat.freshim"
android:versionCode="1"
android:versionName="0.1" >
<uses-permission android:name="android.permission.READ_CONTACTS" />
<uses-permission android:name="android.permission.INTERNET"/>
<uses-permission android:name="android.permission.ACCESS_NETWORK_STATE" />
<uses-permission android:name="android.permission.GET_ACCOUNTS" />
<uses-permission android:name="android.permission.WAKE_LOCK" />
<!-- FOR QB -->
<uses-permission android:name="android.permission.CAMERA"/>
<uses-permission android:name="android.permission.FLASHLIGHT"/>
<uses-feature android:name="android.hardware.camera"/>
<uses-feature android:name="android.hardware.camera.autofocus"/>
<uses-feature android:name="android.hardware.camera.flash"/>
<uses-permission android:name="android.permission.RECORD_AUDIO"/>
<uses-permission android:name="android.permission.MODIFY_AUDIO_SETTINGS"/>
<!-- For GCM -->
<uses-permission android:name="com.google.android.c2dm.permission.RECEIVE" />
<!-- For GCM
<permission android:name=".permission.C2D_MESSAGE"
android:protectionLevel="signature" /> -->
<uses-permission android:name=".permission.C2D_MESSAGE"
android:protectionLevel="signature" />
<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.ACCESS_COARSE_LOCATION" />
<uses-permission android:name="android.permission.ACCESS_FINE_LOCATION" />
<application
android:allowBackup="true"
android:icon="@drawable/icon_fresh"
android:label="fssl"
android:theme="@style/MyTheme">
<activity
android:name="com.chat.freshim.LoginActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"
android:label="fssl"
android:theme="@style/NoActionBar">
<intent-filter>
<action android:name="android.intent.action.MAIN" />
<category android:name="android.intent.category.LAUNCHER" />
</intent-filter>
</activity>
<activity
android:name="com.chat.freshim.FriendsListActivity"
android:screenOrientation="portrait"
android:launchMode="singleTask"
android:theme="@style/NoActionBar"/>
<activity
android:name="com.chat.freshim.FriendsAllListActivity"
android:screenOrientation="portrait"
android:launchMode="singleTask"/>
<activity
android:name="com.chat.freshim.FriendsDetailActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.RegisterActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.UserDetailActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.StatusActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.SettingsActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.EditProfileActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.EditPasswordActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.GroupCreateActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.GroupListActivity"
android:screenOrientation="portrait"
android:launchMode="singleTask"/>
<activity
android:name="com.chat.freshim.GroupMessageDetailActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.GroupChooseFriendListActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.GroupDetailActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.GroupEditMemberListActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.UploadFileActivity"
android:screenOrientation="portrait"/>
<activity
android:name="com.chat.freshim.CountryCodesActivity"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.InviteFriends"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<activity
android:name="com.chat.freshim.AddContactFriend"
android:screenOrientation="portrait"
android:windowSoftInputMode="stateHidden"/>
<!-- QB ACTIVITY AND SERVICES -->
<activity
android:name="com.chat.freshim.qb.ActivityVideoChat"
android:label="fssl"
android:screenOrientation="portrait"
android:theme="@style/NoActionBar"/>
<activity
android:name="com.chat.freshim.qb.ActivityCallUser"
android:label="fssl"
android:screenOrientation="portrait"/>
<service android:name="com.quickblox.module.videochat.core.service.QBVideoChatService"/>
<!-- For GCM -->
<receiver
android:name="com.chat.freshim.GcmBroadcastReceiver"
android:permission="com.google.android.c2dm.permission.SEND" >
<intent-filter>
<action android:name="com.google.android.c2dm.intent.RECEIVE" />
<category android:name="com.chat.imapp" />
</intent-filter>
</receiver>
<service android:name="com.chat.freshim.services.GcmIntentService" />
<service android:name="com.chat.freshim.services.LoadImageService" />
<service android:name="com.chat.freshim.services.LoadFileService" />
<service android:name="com.chat.freshim.services.ResponseRequestService" />
<service android:name="com.chat.freshim.services.FriendRequestService" />
<service android:name="com.chat.freshim.services.DownLoadFileService" />
<service android:name="com.chat.freshim.services.DownloadImageService" />
<service android:name="com.chat.freshim.services.LastSeenGetService" />
<service android:name="com.chat.freshim.services.LastSeenSetService" />
<service android:name="com.chat.freshim.services.QbRegisterService" />
<!--service android:name="com.quickblox.module.chat.videochat.VideoChatService"/-->
<!-- For GCM -->
<meta-data
android:name="com.google.android.gms.version"
/>
</application>
I have try some help but have not get the solution to it.
A: Your <meta-data> tag is missing the required android:value entry. It should read:
<meta-data android:name="com.google.android.gms.version"
android:value="@integer/google_play_services_version" />
A: Try removing this line :
<meta-data
android:name="com.google.android.gms.version"
/>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,864 |
Browse: Home » Cowbell Maths Competition 2012: Final Round
Cowbell Maths Competition 2012: Final Round
Promasidor Nigeria Limited, makers of Cowbell Milk, has held the second and final stage exams of its Cowbell National Secondary School Mathematics Competition (NASSMAC) in four zones in the country.
A total of 74 pupils, who scored the highest marks in the senior and junior categories in the first stage exams (state champions) held in March in 200 centres in the country (including the Federal Capital Territory, FCT) participated in the final exam in the Lagos, Enugu, Port Harcourt and Abuja zones.
Fourteen pupils from the Southwest, with two from each state (one junior and one senior champion) took part in the competition in the Lagos zone. The states are Lagos, Ogun, Ondo, Ekiti, Kwara, Oyo and Osun.
The pupils representing the Southeast sat for the exam in Enugu. Those representing the Southsouth sat for the exam in Port Harcourt while those representing the Northern sat for the exam in Abuja.
Mr Kachi Onubogu, Marketing Director of the company, said the best pupils in each category (junior and senior) from each state and the FCT participated in the final exam, adding that a total of 74 pupils participated in it.
He also said the final exam would produce the national winner as well as other best 10 pupils from each category.
On the zoning of the exam centres, Mr Orunsolu Ismail, a top official of the state's Ministry of Education, who represented the Commissioner of Education, said the company has decentralised its exam centres as against the former practice of converging all the pupils on Lagos.
"The usual practice before was that all the 74 pupils would converge on Lagos. But now, there are about four zones – Lagos, Port Harcourt, Enugu and Abuja", he said.
The best pupil in the senior category from Iganmode Grammar School, representing Ogun State, Akinjide Ayokunle, described the final exam as easy, adding that the competition is an eye opener.
He advised the government and other companies to emulate Promasidor's initiative.
Similarly, Mr Anthony Okogi, a Mathematics teacher with Baptist Model High School, Ilorin, who accompanied his pupil, Akerekan Olayinka, winner of the senior category in the state, said the competition has assisted pupils and teachers to know their abilities, as it prepares them to compete with others in the country.
The Cowbell National Secondary School Mathematics Competition (NASSMAC) is a platform for identifying outstanding pupils and encouraging excellence among secondary school pupils in the country.
It started in 1998 with the aim of helping pupils to demystify mathematics and has since then, become a reference point in educational corporate social responsibility.
Source:74 attend Cowbell maths competition
June 28, 2012 | Editor | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/healpy.readthedocs.io\/en\/latest\/generated\/healpy.projector.SphericalProj.html","text":"# healpy.projector.SphericalProj\u00b6\n\nclass healpy.projector.SphericalProj(rot=None, coord=None, flipconv=None, **kwds)\n\nThis class defines functions for spherical projection.\n\nThis class contains class method for spherical projection computation. It should not be instantiated. It should be inherited from and methods should be overloaded for desired projection.\n\nAttributes\narrayinfo\n\nDictionary with information on the projection array\n\nMethods\n\n ang2xy(self,\u00a0theta[,\u00a0phi,\u00a0lonlat,\u00a0direct]) From angular direction to position in the projection plane (%s). get_center(self[,\u00a0lonlat]) Get the center of the projection. get_extent(self) Get the extension of the projection plane. get_fov(self) Get the field of view in degree of the plane of projection ij2xy(self[,\u00a0i,\u00a0j]) From image array indices to position in projection plane (%s). projmap(self,\u00a0map,\u00a0vec2pix_func[,\u00a0rot,\u00a0coord]) Create an array containing the projection of the map. set_flip(self,\u00a0flipconv) flipconv is either \u2018astro\u2019 or \u2018geo\u2019. vec2xy(self,\u00a0vx[,\u00a0vy,\u00a0vz,\u00a0direct]) From unit vector direction to position in the projection plane (%s). xy2ang(self,\u00a0x[,\u00a0y,\u00a0lonlat,\u00a0direct]) From position in the projection plane to angular direction (%s). xy2ij(self,\u00a0x[,\u00a0y]) From position in the projection plane to image array index (%s). xy2vec(self,\u00a0x[,\u00a0y,\u00a0direct]) From position in the projection plane to unit vector direction (%s).\n get_proj_plane_info mkcoord set_proj_plane_info","date":"2019-11-14 10:59:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31976431608200073, \"perplexity\": 1303.4713078395941}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496668416.11\/warc\/CC-MAIN-20191114104329-20191114132329-00072.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.theinfolist.com\/php\/SummaryGet.php?FindGo=Archimedean_property","text":"TheInfoList\n\nIn abstract algebra and\nanalysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...\n, the Archimedean property, named after the ancient Greek mathematician\nArchimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...\n\nof\nSyracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana *Syracuse, Kansas *Syracuse, Missou ...\n, is a property held by some\nalgebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...\ns, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers x and y, there is an integer n so that nx > y. It also means that the set of\nnatural numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...\nis not bounded above. Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes\u2019 ''\nOn the Sphere and Cylinder ''On the Sphere and Cylinder'' ( el, \u03a0\u03b5\u03c1\u1f76 \u03c3\u03c6\u03b1\u03af\u03c1\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03c5\u03bb\u03af\u03bd\u03b4\u03c1\u03bf\u03c5) is a work that was published by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an ex ...\n''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as\nDavid Hilbert David Hilbert (; ; 23 January 1862 \u2013 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...\n's axioms for geometry, and the theories of ordered groups,\nordered fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...\ns, and local fields. An algebraic structure in which any two non-zero elements are ''comparable'', in the sense that neither of them is\ninfinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...\nwith respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with slightly different formulations. For example, in the context of\nordered fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...\ns, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.\n\n# History and origin of the name of the Archimedean property\n\nThe concept was named by Otto Stolz (in the 1880s) after the ancient Greece, ancient Greek geometer and physicist\nArchimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...\n\nof\nSyracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana *Syracuse, Kansas *Syracuse, Missou ...\n. The Archimedean property appears in Book V of Euclid's Elements, Euclid's ''Elements'' as Definition 4: Because Archimedes credited it to Eudoxus of Cnidus it is also known as the \"Theorem of Eudoxus\" or the ''Eudoxus axiom''. Archimedes's use of infinitesimals, Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.\n\n# Definition for linearly ordered groups\n\nLet and be Linearly_ordered_group#Definitions, positive elements of a linearly ordered group ''G''. Then is infinitesimal with respect to (or equivalently, is infinite with respect to ) if, for every natural number , the multiple is less than , that is, the following inequality holds: ::: $\\underbrace_ < y. \\,$ This definition can be extended to the entire group by taking absolute values. The group is Archimedean if there is no pair such that is infinitesimal with respect to . Additionally, if is an\nalgebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...\nwith a unit (1) \u2014 for example, a ring (mathematics), ring \u2014 a similar definition applies to . If is infinitesimal with respect to 1, then is an infinitesimal element. Likewise, if is infinite with respect to 1, then is an infinite element. The algebraic structure is Archimedean if it has no infinite elements and no infinitesimal elements.\n\n## Ordered fields\n\nOrdered fields have some additional properties: * The rational numbers are Embedding, embedded in any ordered field. That is, any ordered field has Characteristic (algebra), characteristic zero. * If is infinitesimal, then is infinite, and vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. * If is infinitesimal and r is a rational number, then is also infinitesimal. As a result, given a general element , the three numbers , , and are either all infinitesimal or all non-infinitesimal. In this setting, an ordered field is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: : \"Let be any element of . Then there exists a natural number such that .\" Alternatively one can use the following characterization: ::: $\\forall\\, \\varepsilon \\in K\\big\\left(\\varepsilon > 0 \\implies \\exists\\ n \\in N : 1\/n < \\varepsilon\\big\\right).$\n\n# Definition for normed fields\n\nThe qualifier \"Archimedean\" is also formulated in the theory of Valuation ring, rank one valued fields and normed spaces over rank one valued fields as follows. Let be a field endowed with an absolute value function, i.e., a function which associates the real number 0 with the field element 0 and associates a positive real number $, x,$ with each non-zero and satisfies $, xy, =, x, , y,$ and $, x+y, \\le , x, +, y,$. Then, is said to be Archimedean if for any non-zero there exists a natural number such that ::: $, \\underbrace_, > 1. \\,$ Similarly, a normed space is Archimedean if a sum of terms, each equal to a non-zero vector , has norm greater than one for sufficiently large . A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality, ::: $, x+y, \\le \\max\\left(, x, ,, y, \\right)$, respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean. The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74\u201384.\n\n# Examples and non-examples\n\n## Archimedean property of the real numbers\n\nThe field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function $, x, =1,$ when , the more usual $, x, = \\sqrt$, and the -adic absolute value functions. By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete. The completion with respect to the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.Neal Koblitz, \"p-adic Numbers, p-adic Analysis, and Zeta-Functions\", Springer-Verlag,1977. On the other hand, the completions with respect to the other non-trivial absolute values give the fields of -adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the ultrametric property, then the -adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields). In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows. Denote by the set consisting of all positive infinitesimals. This set is bounded above by 1. Now proof by contradiction, assume for a contradiction that is nonempty. Then it has a least upper bound , which is also positive, so . Since is an upper bound of and is strictly larger than , is not a positive infinitesimal. That is, there is some natural number for which . On the other hand, is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between and , and if then is not infinitesimal. But , so is not infinitesimal, and this is a contradiction. This means that is empty after all: there are no positive, infinitesimal real numbers. The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.\n\n## Non-Archimedean ordered field\n\nFor an example of an\nordered fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...\nthat is not Archimedean, take the field of rational functions with real coefficients. (A rational function is any function that can be expressed as one polynomial divided by another polynomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now if and only if ''f''\u00a0\u2212\u00a0''g'' > 0, so we only have to say which rational functions are considered positive. Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function 1\/''x'' is positive but less than the rational function 1. In fact, if is any natural number, then ''n''(1\/''x'') = ''n''\/''x'' is positive but still less than 1, no matter how big is. Therefore, 1\/''x'' is an infinitesimal in this field. This example generalizes to other coefficients. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say , produces an example with a different order type.\n\n## Non-Archimedean valued fields\n\nThe field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990.\n\n## Equivalent definitions of Archimedean ordered field\n\nEvery linearly ordered field contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of , which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in . The following are equivalent characterizations of Archimedean fields in terms of these substructures. 1. The natural numbers are cofinal (mathematics), cofinal in . That is, every element of is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound. 2. Zero is the infimum in of the set . (If contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.) 3. The set of elements of between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of infinitesimals is closed. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between. Consequently, any non-Archimedean ordered field is both incomplete and disconnected. 4. For any in the set of integers greater than has a least element. (If were a negative infinite quantity every integer would be greater than it.) 5. Every nonempty open interval of contains a rational. (If is a positive infinitesimal, the open interval contains infinitely many infinitesimals but not a single rational.) 6. The rationals are Dense set, dense in with respect to both sup and inf. (That is, every element of is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.","date":"2021-12-09 01:22:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 9, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8917022347450256, \"perplexity\": 474.431244486007}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363641.20\/warc\/CC-MAIN-20211209000407-20211209030407-00624.warc.gz\"}"} | null | null |
Q: Need to rewrite unless a rule is matched I need to rewrite to another host, unless the url starts with /store or /utils
I have tried looking at the apache docs, but I can't work out how to do this
A: Ok assuming you want to redirect to a host otherhost.example.com this should do the trick under Apache.
RewriteCond %{REQUEST_URI} !^/(store|utils)/.*
RewriteRule (.*) http://otherhost.example.com$1 [R=301,L]
The conditional part says "if the request URI doesn't begin with /store/ or /utils/ apply the Rewrite. The rewrite rule will then generate a 301 redirect with the captured URI appended to the request.
A: As ickymettle succintly responded, the ! flag signifies it should consider the input as matched if the conditions of the rule are NOT met. So, breaking down the syntax:
! = 'Not' prefix
^ = start of string
/ = The first forward slash tells the rule to match only if the input string
begins with a /
| = the pipe symbol denotes 'or' inside the bracketed group section
(as in 'store or utils')
The second forward slash denotes that the string ends with a forward slash
. = any single character
* = zero or more of the preceding character (so .* = 0 or more of any character, effectively passing on the rest of the URI
The AddedBytes.com mod_rewrite cheatsheet is dead handy, I have it taped to the wall next to where I sit (as I dive into htaccesses almost daily and desperately try to teach myself mod_rewrite voodoo!) There's also a [great detailed article on PerishablePress with loads of tips, tricks etc. The one nice thing about mod_rewrite is that it isn't deprecated with new releases constantly, so old articles on the topic remain perfectly valid :-)
Might as well go for broke and also mention [the mod_rewrite / .htaccess cheatsheet on TheJackol, which has some useful quick code snippets and a few bits of info.
Other URLs:
*
*perishablepress.com/press/2006/01/10/stupid-htaccess-tricks/
*thejackol.com/htaccess-cheatsheet/
Sorry I couldn't hyperlink them, I don't have 10 Rep points yet so the system blocked me from hyperlinking all three :-(
A: RewriteCond URL (?!.*/utils).*
RewriteCond URL (?!.*/store).*
RewriteRule (.*) http\://{your new host here}/$3 [I,R]
Note: This is the syntax for ISAPI_Rewrite, which should be compatible with mod-rewrite.
| {
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December 4, 2019 / 6:06 AM / a month ago
Jared Kushner, Trump's son-in-law, takes bigger role in China trade talks
David Lawder, Jeff Mason
WASHINGTON (Reuters) - President Donald Trump's son-in-law Jared Kushner has added another role to his long list of White House duties - U.S.-China trade negotiator - as Washington and Beijing try to reach an initial agreement to avoid new U.S. tariffs on Dec. 15.
FILE PHOTO: White House senor adviser Jared Kushner looks on as U.S. President Donald Trump speaks in the Oval Office of the White House in Washington, U.S., July 26, 2019. REUTERS/Leah Millis
People familiar with the talks said Kushner, who helped bring the U.S.-Mexico-Canada trade agreement (USMCA) to fruition, has increased his direct involvement in the negotiations with China over the past two weeks.
While the talks have made some progress, these people said the two sides have not yet agreed on the extent to which the United States will remove existing tariffs on Chinese goods and on specific commitments by China to increase purchases of U.S. agriculture products.
A White House official confirmed Kushner's involvement, but declined to provide specific details on the influence he has had on the negotiations. Speaking on condition of anonymity, the official said Kushner has recently met with Cui Tiankai, the Chinese ambassador to the United States.
The two have met multiple times since Trump took office, establishing a kind of back-channel relationship, trade experts say.
U.S. Trade Representative Robert Lighthizer and Treasury Secretary Steven Mnuchin have been leading negotiations with Chinese Vice Premier Liu He for the past two years over a range of U.S. complaints about China's trade and subsidy practices, including the forced transfer of American technology to Chinese firms.
"Jared has been engaged in the process from the beginning in full coordination and in support of Ambassador Lighthizer's and Secretary Mnuchin's efforts," the White House official said.
Kushner played a pivotal role in the later stages of U.S. trade negotiations with Canada and Mexico in 2018 to replace the North American Free Trade Agreement, helping to resolve final differences. Lighthizer said the USMCA deal "would not have happened if it wasn't for Jared."
Former Mexican foreign Minister Luis Videgaray, with whom Kushner met frequently, said Kushner patched up the negotiations more than once after they fell apart.
Kushner, who is married to Trump's daughter Ivanka, has taken on many challenges during the past three years, including trying to develop a Middle East peace plan, working on changes to U.S. immigration policies and advising Trump on dealing with opioid addiction and problems with Department of Veterans Affairs.
But sealing a deal with the China could prove daunting. U.S. Commerce Secretary Wilbur Ross, speaking to Reuters on Tuesday, rejected any deadlines for a deal and launched a fresh attack on Chinese telecom equipment giant Huawei, accusing it of telling suppliers to move operations overseas to skirt U.S. sanctions.
Reporting by David Lawder; Editing by Raju Gopalakrishnan | {
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\section{Введение}
Шашки Фейнмана --- одна из простейший моделей движения электрона. Она была придумана Р. Фейнманом и опубликована в 1965 году в ~\cite{feyn} (см. современные обзоры в~\cite{Konno-20, article}). А. Амбайнис, Э. Бах, А. Наяк, Э. Вишванатх и Дж. Ватроуз установили блестящие результаты об этой модели в ~\cite{Amb}. Хотя они изучали одномерное квантовое блуждание и блуждание Адамара, их модель была эквивалентна шашкам Фейнмана.
В 1972 году Дж. Нарликар нашел непрерывный предел модели. Этот предел совпадает с известным выражением, которое получается при решении уравнения Дирака на прямой. Точные определения всех объектов, участвующих в формулировке этой и последующих теорем, будут даны в разделе~\ref{opredel}.
\begin{theorem}[{\cite[теорема 8]{article}}]
\label{problim}
Пусть $m, \varepsilon>0 ,|x| < t$, причем $x/2\varepsilon, t/2\varepsilon\in \mathbb{Z}$. Тогда на $2$-мерной квадратной решетке с шагом $\varepsilon$, деленная на $4\varepsilon^2$ вероятность обнаружить в точке $(x,t)$ электрон массой $m$, испущенный из точки $(0,0)$, стремится при $\varepsilon~\to~0$~к $$\frac{m^2}{4}\left( J_0\left(m \sqrt{t^2-x^2}\right)^2 + \frac{t+x}{t-x}J_1\left(m \sqrt{t^2-x^2}\right)^2 \right).$$
\end{theorem}
Здесь $J_0(z) := \sum\limits_{j=0}^{\infty}(-1)^j\frac{(z/2)^{2j}}{(j!)^2}$ и $J_1(z) := \sum\limits_{j=0}^{\infty}(-1)^j\frac{(z/2)^{2j+1}}{(j!)(j+1)!}$ --- \emph{функции Бесселя} первого рода порядков 0 и 1 соответственно. Слагаемые, содержащие $J_1$ и $J_0$, отвечают вероятности обнаружить электрон с исходной и измененной \emph{хиральностью} соответственно \cite[\S4]{article}.
Элементарное математическое доказательство этой теоремы приведено в~\cite[приложение B]{article}. Для случая достаточно гладкого начального условия аналогичный результат доказан в~\cite{MasMay-AkiSuz-2019}.
В начале 2000-х Н. Конно и Г. Гриммет–С. Джэнсон–П. Скудо нашли предел модели при больших временах, а именно, предельное распределение положения электрона.
\begin{theorem}[{\cite[теорема 1]{Grimmet-Janson-Scudo-04}}]
\label{wl-Grimmet}
Пусть $m,\varepsilon,t>0, v\in \mathbb{R}$, причем $t/\varepsilon\in \mathbb{Z}$. Тогда на $2$-мерной квадратной решетке с шагом $\varepsilon$ вероятность обнаружить в момент $t$ электрон массой $m$, испущенный из точки $0$ в момент времени $0$, левее точки $vt$ стремится при $t\to \infty$~к
$$
F(v):=
\begin{cases}
0 ,& \text{при } v < -\frac{1}{\sqrt{1+m^2\varepsilon^2}};\\
\frac{1}{\pi}\arccos{\frac{1-(1+m^2\varepsilon^2)v}{\sqrt{1+m^2\varepsilon^2}(1-v)}}, &\text{при } |v| \le \frac{1}{\sqrt{1+m^2\varepsilon^2}};\\
1 , &\text{при } v > \frac{1}{\sqrt{1+m^2\varepsilon^2}}.
\end{cases}
$$
\end{theorem}
Короткое доказательство этой теоремы можно найти в \cite[\S 12.1]{article}.
Этот результат имеет многочисленные вариации и обобщения \cite[\S 3.2]{article}, \cite{Kuyanov-Slizkov}. Сформулируем одну из таких вариаций, принадлежащую И. Богданов (ср. \cite[теорема 2]{Novikov}).
\begin{theorem}[{\cite[теорема 2]{IB-20}}]
\label{chiral-rev}
Пусть $0 \le m\varepsilon \le 1$. Тогда вероятность обнаружить электрон, испущенный из точки 0 в момент времени 0, с измененной хиральностью в момент времени $t$ стремится при $t \to \infty$ и целом $t/\varepsilon$ к $\frac{m\varepsilon}{2\sqrt{1+m^2\varepsilon^2}}$.
\end{theorem}
Существует множество модификаций этой модели. Например, в 1990х годах в работах~\cite{shul} и~\cite{Ord} в модель было добавлено электромагнитное поле. Модификация из~\cite{shul}, рассматриваемая в этой статье, эквивалентна неоднородному квантовому блужданию, которому посвящен обзор~\cite{Konno-20}. См. также \cite{Dmitriev-22, Skopenkov-Ustinov-22}.
В данной работе теоремы~\ref{problim}, \ref{wl-Grimmet}, \ref{chiral-rev} перенесены на случай простейшего нетривиального электромагнитного поля (см. рис.~\ref{Limit},~\ref{fig:problim} и~\ref{fig: chir-rev-f} и теоремы \ref{fproblim}, \ref{wl-intr}, \ref{chir-rev-f} соответственно).
\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=0.65\paperwidth]{bessel12.png}
\end{tabular}
\end{center}
\caption{\footnotesize{{Деленная на $4\varepsilon^2$ вероятность обнаружения электрона массой $1$ в точке $(x,t)$, испущенного из точки $(0,0)$ и движущегося в однородном электромагнитном поле, при $\varepsilon=0.1$ (чёрточки) и при $\varepsilon \to 0$ (кривая). Здесь $t=10$ и $20x \in \mathbb{Z}$. См. определение~1.}}}
\label{Limit}
\end{figure}
\begin{theorem}
\label{fproblim}
Пусть $m, \varepsilon>0, |x| < t,$ причем $x/4\varepsilon, t/4\varepsilon \in \mathbb{Z}$. Тогда на $2$-мерной решетке с шагом $\varepsilon$, деленная на $4\varepsilon^2$ вероятность обнаружить в точке $(x,t)$ электрон массой $m$, испущенный из точки $(0,0)$ и движущийся в однородном электромагнитном поле поле, стремится при $\varepsilon \to 0$ к $$\frac{m^2}{4}\left( J_0\left(m \sqrt{\frac{t^2-x^2}{2}}\right)^2 + 2\frac{t+x}{t-x}J_1\left(m \sqrt{\frac{t^2-x^2}{2}}\right)^2 \right).$$
\end{theorem}
В ходе доказательства этой теоремы мы впервые получим <<явное>> выражение для волновой функции электрона, движущегося в однородном электромагнитном поле, для дискретной модели, то есть её точное решение (см. предложение~\ref{formula}). В отличие от исходной модели без электромагнитного поля, это выражение не получается вывести из простых комбинаторных соображений. А получив <<явное>> выражение, мы применяем метод доказательства теоремы 1 выше \cite[приложение B]{article}.
Движение в таком электромагнитном поле является простейшим примером неоднородного квантового блуждания, в котором неоднородность имеет период 2 по времени и пространству. Периодические квантовые блуждания изучались в~\cite[\S 3.1]{Cantero-etal-12}, для них ранее не было известно никаких асимптотических результатов.
\begin{figure}[httb]
\centering
\includegraphics[width=.8\linewidth]{weak-limit1.png}
\caption{\footnotesize{График вероятности $F_t(v)$ обнаружить электрон массой 1, испущенный из точки $0$ и движущийся в однородном электромагнитном поле, левее точки $vt$ в момент времени $0$ при $|v|\le 1/2$ и $t=25$ (точки). Предел этой вероятности при $t\to\infty$ (кривая). См. определение 1.}}
\label{fig:problim}
\end{figure}
\begin{theorem}
\label{wl-intr}
Пусть $m,\varepsilon,t>0, v\in \mathbb{R}$, причем $t/\varepsilon \in \mathbb{Z}$. Тогда на $2$-мерной квадратной решетке с шагом $\varepsilon$ вероятность обнаружить в момент $t$ электрон массой $m$, испущенный из точки $0$ в момент времени $0$ и движущийся в однородном электромагнитном поле, левее точки $vt$ стремится при $t\to \infty$ к
$$
F(v):=
\begin{cases}
0 ,& \text{при } v < -\frac{1}{1+m^2\varepsilon^2};\\
\frac{1}{\pi}\arccos{\frac{1-(1+m^2\varepsilon^2)^2v}{(1+m^2\varepsilon^2)(1-v)}}, &\text{при } |v| \le \frac{1}{1+m^2\varepsilon^2};\\
1 , &\text{при } v > \frac{1}{1+m^2\varepsilon^2}.
\end{cases}
$$
\end{theorem}
Таким образом, результат для модели с полем (теорема~\ref{wl-intr}) связан с аналогичным для модели без поля (теоремой~\ref{wl-Grimmet}) \emph{перенормировкой} массы:
$$
(1+m^2\varepsilon^2)^2 = 1 +m_0^2\varepsilon^2,
$$
где $m$ --- масса в модели с полем, а $m_0$ --- в модели без поля. Устремляя $\varepsilon$ к $0$, получаем следующее соотношение:
$$
1+2m^2\varepsilon^2 + o(\varepsilon^2)= (m_0\varepsilon)^2.
$$
Таким образом, $m\sim m_0/\sqrt{2}$ при $\varepsilon \to 0$. Именно такому соотношению удовлетворяют аргументы функций Бесселя в теоремах \ref{problim} и \ref{fproblim}.
\begin{figure}[!hbt]
\centering
\includegraphics[width=.75\linewidth]{Chir_rev.png}
\caption{\footnotesize{(из ~\cite{article}) График вероятности смены хиральности электрона массой 1, испущенного из начала координат и движущегося в однородном электромагнитном поле. См. определение 1.}}
\label{fig: chir-rev-f}
\end{figure}
\begin{theorem}
\label{chir-rev-f}
Пусть $m,\varepsilon >0$. Тогда вероятность обнаружить электрон, испущенный из точки 0 в момент времени 0 и движущийся в однородном электромагнитном поле, с измененной хиральностью в момент времени $t$ стремится при $t \to \infty$ и фиксированной четности числа~$t/\varepsilon$~к
$$
\begin{cases}
\frac{m\varepsilon}{(1+m^2\varepsilon^2)\sqrt{m^2\varepsilon^2 + 2}}, & \text{ при } \frac{t}{\varepsilon} \equiv_2 1;\\
\frac{m\varepsilon}{\sqrt{m^2\varepsilon^2 + 2}}, & \text{ при } \frac{t}{\varepsilon} \equiv_2 0.
\end{cases}
$$
\end{theorem}
Этот результат демонстрирует \emph{прецессию спина электрона в электромагнитном поле}: вероятность смены хиральности стремится к периодической функции (а не к постоянной, как в отсутствии поля; см. теорему \ref{chiral-rev} и \cite[пример 5]{article}). Тем самым решена проблема 11 из~\cite{article} и доказана гипотеза 1 из~\cite{IB-20} (в этих работах рассматривалось поле, калибровочно эквивалентное нашему полю $u_{\varepsilon}$, а потому дающее те же вероятности).
В \S 3 мы сформулируем теоремы \ref{fproblim}, \ref{wl-intr}, \ref{chir-rev-f} в более сильной форме, а в последующих разделах докажем результаты из \S 3.
\section{Определения}
\label{opredel}
Сначала дадим неформальное описание модели ''шашки Фейнмана'', потом --- её модификации с электромагнитным полем, и наконец, приведём точное определение. Большая часть этого вводного раздела заимствована из обзора~\cite{article}.
\par
Зафиксируем $m \ge 0$, называемое \emph{массой} электрона. Рассмотрим бесконечную шахматную доску со стороной клетки $\varepsilon$.
Шашка ходит на соседнюю по диагонали клетку, влево-вверх или вправо-вверх. Каждому пути $s$ шашки сопоставим вектор ${a}(s)$ на плоскости следующим образом (рис.~\ref{Checker-paths}). В начале движения этот вектор направлен вверх и имеет длину 1. Пока шашка движется прямо, вектор не меняется, а после каждого поворота шашки он поворачивается на $90^{\circ}$ по часовой стрелке и домножается на $m\varepsilon$. В конце движения вектор сжимается в $(1+m^2\varepsilon^2)^{\frac{t/\varepsilon - 1}{2}}$ раз, где $t/\varepsilon$ - общее число ходов шашки. Полученный в итоге вектор и есть $a(s)$.
Обозначим $a(x,t,m,\varepsilon) :=\sum_{s}a(s)$, где суммирование ведётся по всем путям шашки из клетки $(0,0)$ в клетку $(x,t)$ , начинающихся с хода вправо-вверх. Квадрат длины вектора $a(x,t,m,\varepsilon)$ называется \textit{вероятностью обнаружения в клетке} $(x,t)$ \textit{электрона}, \textit{испущенного из клетки} $(0,0)$, а сам вектор $a(x,t,m,\varepsilon)$ называется \textit{стрелкой} или \textit{волновой функцией}.
Например, на рисунке~\ref{Checker-paths} имеем: $a(s_0)=(1/2,0), a(s)=(0,-1/2)$, а $a(1,3,1,1) = (1/2,-1/2)$.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{l}
\end{tabular}
\begin{tabular}{lccc}
\includegraphics[width=1.4cm]{movie8a.png} &
\includegraphics[width=1.5cm]{movie5.png} &
\includegraphics[width=1.5cm]{movie6.png} &
\includegraphics[width=1.5cm]{movie7.png} \\
\includegraphics[width=1.5cm]{movie0a.png} &
\includegraphics[width=1.5cm]{movie1.png} &
\includegraphics[width=1.5cm]{movie2.png} &
\includegraphics[width=1.5cm]{movie3.png}
\end{tabular}
\begin{tabular}{r}
\includegraphics[width=3.5cm]{8a1.png}
\end{tabular}
\end{center}
\vspace{-0.6cm}
\caption{\footnotesize{(из ~\cite{article}) {Пути шашки (слева). Векторы, сопоставленные путям (справа).}}}
\label{Checker-paths}
\vspace{-0.3cm}
\end{figure}
\par
В этой модели внешнее электромагнитное поле не добавляется искусственно, а возникает естественным образом.
У нас вектор $a(s)$ не поворачивался, пока шашка двигалась прямо. Напрашивается изменение модели: поворачивать вектор равномерно во время движения. Это не изменит модель по сути: поскольку все пути шашки из начального положения в конечное имеют одинаковую длину, то и векторы поворачиваются на один и тот же угол, что не влияет на вероятности. Более интересная модификация получается, когда текущий угол поворота зависит от положения шашки.
Именно это и делает электромагнитное поле. В дальнейшем, для простоты, угол поворота будет принимать только два значения $0^\circ$ и $180^\circ$, что означает домножение на $\pm 1$.
Таким образом, электромагнитное поле понимается как фиксированная расстановка $u$ чисел $ \pm 1 $ в вершинах квадратов-клеток (см. рис.~\ref{C}). Модифицируем определение вектора ${a}(s)$, меняя его направление на противоположное каждый раз, когда путь проходит через вершину с электромагнитным полем $-1$. Обозначим получившийся вектор как ${a}(s,u)$. Определим ${a}(x,t,m,\varepsilon,u)$ и $P(x,t,m,\varepsilon,u)$ аналогично ${a}(x,t,m,\varepsilon)$ и $P(x,t,m,\varepsilon)$, заменяя в их определении ${a}(s)$ на ${a}(s,u)$. Например, если $u$ тождественно равно $+1$, то $P(x,t,m,\varepsilon,u)=P(x,t,m,\varepsilon)$. На рисунке~\ref{C} изображено другое поле и нарисованы стрелки $a(s,u)=(0,1/2)$, $a(s_0,u)=(1/2,0)$.
Резюмируем эту конструкцию следующим точным определением.
\textbf{Определение 1.}
Зафиксируем действительные $\varepsilon > 0$ и $m\ge 0$ --- \textit{шаг решетки} и \textit{массу электрона} соответственно.
Рассмотрим решетку $\varepsilon \mathbb{Z}^2 = \{ (x,t): x/\varepsilon, t/\varepsilon \in \mathbb{Z} \}$.
\textit{Путь шашки} --- конечная последовательность таких точек решетки $\varepsilon \mathbb{Z}^2$, что вектор из каждой точки (кроме последней) к следующей равен либо $(\varepsilon,\varepsilon)$, либо $(-\varepsilon,\varepsilon)$. \textit{Поворот} --- это такая точка пути (не первая и не последняя), что вектор, соединяющий эту точку с предыдущей, ортогонален вектору, соединяющему её со следующей. Обозначим через $\mathrm{turns}(s)$ количество поворотов пути $s$.
\emph{Вспомогательным ребром} назовём отрезок, соединяющий две ближайших точки $(x_1,t_1)$ и $(x_2,t_2)$ решетки $\varepsilon \mathbb{Z}^2$ такие, что $(x_1 + t_1)/\varepsilon$ и $(x_2 + t_2)/\varepsilon$ четны. Пусть $u$ --- некоторое отображение из множества всех вспомогательных ребер в $\{+1,-1\}$. Обозначим через
$$
a(x,t,m,\varepsilon,u):=(1+m^2\varepsilon^2)^{(1-t/\varepsilon)/2}\,i\,\sum_s (-im\varepsilon)^{\mathrm{turns}(s)}u(s_0s_1)u(s_1s_2)\dots u(s_{t-1}s_t)
$$
сумму по всем путям $s=(s_0,s_1,\dots,s_{t/\varepsilon})$ шашки, у которых $s_0=(0,0)$, $s_1=(\varepsilon,\varepsilon)$, $s_{t/\varepsilon}=(x,t)$. Здесь и далее пустая сумма по определению считается равной нулю.
Обозначим
\begin{align*}
P(x,t,m,\varepsilon,u) &:= |a(x,t,m,\varepsilon,u)|^2, \\
a_1(x,t,m,\varepsilon,u)&:=\mathrm{Re}\, a(x,t,m,\varepsilon,u), \\
a_2(x,t,m,\varepsilon,u)&:=\mathrm{Im}\,a(x,t,m,\varepsilon,u).\\
\end{align*}
Величина $P(x,t,m,\varepsilon,u)$ называется \textit{вероятностью обнаружения на решетке с шагом $\varepsilon$ электрона массой $m$ в точке $(x,t)$ (или, по-другому, в точке $x$ в момент времени $t$), испущенного из $(0,0)$ и движущегося в поле $u$}. Величина $\sum\limits_{x \in \varepsilon\mathbb{Z}}a_1(x,t,m,\varepsilon,u)^2$ называется \emph{вероятностью смены направления движения (или хиральности) в момент времени $t$}. (Смысл такой терминологии поясняется в \cite[\S4]{article}.)
Для полуцелых $x/\varepsilon,t/\varepsilon$ обозначим через $u(x,t)$ значение поля $u$ на вспомогательном ребре с серединой $(x,t)$.
Поле $u_{\varepsilon}$, заданное формулой $u_{\varepsilon}(x+\varepsilon/2,t+\varepsilon/2)=-1$, если $(t-x)/4\varepsilon \in \mathbb{Z}$ , и $u_{\varepsilon}(x+\varepsilon/2,t+\varepsilon/2)=+1$ иначе, называется \textit{однородным электромагнитным полем} (рис.~\ref{C}).
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=.95\linewidth]{emf.png}
\end{tabular}
\end{center}
\vspace{-0.6cm}
\caption{\footnotesize{{Однородное электромагнитное поле (слева). Векторы, сопоставленные путям с рис.~\ref{Checker-paths} в этом поле (справа).}}}
\label{C}
\vspace{-0.3cm}
\end{figure}
\begin{table}[!h]
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$4\varepsilon$&$\frac{-m \varepsilon}{(1+m^2\varepsilon^2)^{3/2}}$&&$\frac{(m \varepsilon+ m^3 \varepsilon^3) + m^2 \varepsilon^2 i}{(1+m^2\varepsilon^2)^{3/2}}$&&$\frac{-m \varepsilon }{(1+m^2\varepsilon^2)^{3/2}}$&&$\frac{1}{(1+m^2\varepsilon^2)^{3/2}} i$\\
\hline
$3\varepsilon$&&$\frac{-m \varepsilon}{1+m^2\varepsilon^2}$&&$\frac{m \varepsilon + m^2 \varepsilon^2 i}{1+m^2\varepsilon^2}$&&$\frac{-1}{(1+m^2\varepsilon^2)} i$&\\
\hline
$2\varepsilon$&&&$\frac{-m\varepsilon}{\sqrt{1+m^2\varepsilon^2}}$&&$\frac{1}{\sqrt{1+m^2\varepsilon^2}} i$&&\\
\hline
$\varepsilon$&&&&$-i$&&&\\
\hline
\diagbox[dir=SW,height=21pt]{$t$}{$x$}&$-2\varepsilon$&$-\varepsilon$&$0$&$\varepsilon$&$2\varepsilon$&$3\varepsilon$&$4\varepsilon$ \\
\hline
\end{tabular}
\caption{\footnotesizeЗначения $a(x,t,m,\varepsilon, u_{\varepsilon})$ в однородном поле при малых $x$ и $t$.}
\label{table-amu}
\end{table}
\par
\begin{remark}
\label{Ueq}
Для любых целых $x,t$ выполнено $a(\varepsilon x, \varepsilon t, m,\varepsilon,u_{\varepsilon}) = a(x,t,m\varepsilon,1,u_{1}).$
\end{remark}
\begin{remark}[{Ср. с \cite[замечание 5]{article}}]
У нас поле $u$ --- это фиксированное классическое внешнее поле, электрон не воздействует на него.
Это определение аналогично исторически первому построению калибровочной теории, принадлежащему Вейлю--Фоку--Лондону, и даёт связь шашек Фейнмана с решеточной $\mathbb{Z}/2\mathbb{Z}$ калибровочной теорией Вегнера--Вильсона.
Например, поле $u_{\varepsilon}$ на рис.~\ref{C} имеет вид $u_{\varepsilon}(s_1s_2)= \exp\left(-i\int\limits_{s_1,s_2}\left(A_0 \mathrm{d} t + A_1 \mathrm{d} x\right) \right)$ для каждого вспомогательного ребра $s_1s_2$, где $(A_0, A_1) := (\pi/ \varepsilon)(\frac{t-x}{2} + 1, \frac{t-x}{2} + 1)$ --- вектор-потенциал постоянного однородного электромагнитного поля. Это поле становится все сильнее при убывании $\varepsilon$, поэтому предел в теореме \ref{fproblim} не удается интерпретировать как волновую функцию электрона в каком-то непрерывном внешнем поле. Ср. \cite[проблема 12]{article}.
\end{remark}
В таблице~\ref{table-amu} показаны значения волновой функции электрона в модели с однородным электромагнитным полем при малых $x,t$.
\section{Формулировки результатов}
\label{results}
Сформулируем наш основной результат (теорему~\ref{fproblim} выше) в более сильной форме.
\begin{theorem}
\label{wavelimit}
Пусть $u_\varepsilon$ --- однородное электромагнитное поле. Тогда для любых действительных $m > 0$ и $|x|<t$ выполнено:
\begin{align*}
&\lim\limits_{\varepsilon \searrow 0}\frac{1}{2\varepsilon} a_1\left(4\varepsilon\left\lfloor{\frac{ x}{4\varepsilon}}\right\rfloor,4\varepsilon\left\lfloor{\frac{ t}{4\varepsilon}}\right\rfloor,m,\varepsilon,u_{\varepsilon}\right) =\frac{m}{2} J_0\left(m \sqrt{\frac{t^2-x^2}{2}}\right),\\
&\lim\limits_{\varepsilon \searrow 0}\frac{1}{2\varepsilon} a_2\left(4\varepsilon\left\lfloor{\frac{ x}{4\varepsilon}}\right\rfloor,4\varepsilon\left\lfloor{\frac{ t}{4\varepsilon}}\right\rfloor,m,\varepsilon,u_{\varepsilon}\right) =-\frac{m}{\sqrt{2}} \sqrt{\frac{t+x}{t-x}}J_1\left(m \sqrt{\frac{t^2-x^2}{2}}\right).
\end{align*}
\end{theorem}
На рисунке~\ref{Limit} чёрточками изображен график нормализованной вероятности \newline $\frac{1}{4\varepsilon^2}P\left(4\varepsilon\left\lfloor{\frac{ x}{4\varepsilon}}\right\rfloor,4\varepsilon\left\lfloor{\frac{ t}{4\varepsilon}}\right\rfloor,m,\varepsilon,u_{\varepsilon}\right)$ при $t = 10, m = 1, \varepsilon = 0.1$ и произвольном $|x| < t$, кратном $0.2$, в то время как кривая --- график выражения с функциями Бесселя из теоремы~\ref{fproblim}, т.е. поточечного предела нормализованной вероятности при $\varepsilon \to 0$.
Для доказательства теоремы ~\ref{wavelimit} нам потребуется следующее предложение.
Обозначим через $\delta_2(b)$ остаток, который даёт целое число $b$ при делении на 2. Также будем считать ${a\choose j}=0$ при целых $a<0, j \ge 0$.
\begin{predl}[{Ср. \cite[предложение 11]{article} }]
\label{formula}
Для любого действительного $m \ge 0$ и целых $\xi,\eta \ge 0$ выполнено
\begin{align*}
&a_1(\xi - \eta + 1,\xi + \eta + 1,m,1,u_{1})=\\
&=(-1)^{\xi+1}\frac{m(1+m^2)^{\delta_2(\xi(\eta+1))} }{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}{\lfloor\frac{\xi}{2}\rfloor\choose j}{\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j ;\\
&a_2(\xi - \eta + 1,\xi + \eta+1,m,1,u_{1})=\\
&=\frac{(-1)^{\xi+1}}{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}\left( {\lfloor\frac{\eta}{2}\rfloor\choose j}(1 +m^2)^{\delta_2(\xi\eta)}-\right.\\ &-\left.{\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1+m^2)^{\delta_2(\xi(\eta+1))} \right){\lfloor\frac{\xi}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j.
\end{align*}
\end{predl}
\begin{remark} В частности, $a_2(\xi - \eta + 1,\xi + \eta+1,m,1,u_{1}) =0$ при $\eta - 1\equiv_2 \xi \equiv_2 0$.
\end{remark}
Формулы из предложения~\ref{formula} можно переписать с помощью гипергеометрических функций.
Для целых $a,b,c$, где $b \le 0$ и $c>0$, многочлен
$${}_2F_1\left(a,b;c;z\right) := 1 + \sum_{k=1}^{\infty}\prod_{l=0}^{k-1}\frac{(a+l)(b+l)}{(1+l)(c+l)}z^k$$
называется \emph{гипергеометрической функцией} (или \emph{многочленом Якоби}).
Заметим, что для целого $b \le 0$ эта сумма на самом деле содержит лишь конечное число ненулевых слагаемых.
\begin{predl}[{Ср. \cite[замечание 3]{article}}]
\label{hyper-form}
Обозначим $z = 1 - (1+m^2)^2$. Для любого действительного $m \ge 0$ и целых $\xi, \eta \ge 0$ выполнено
\begin{align*}
&a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)=\\
&=(-1)^{ \xi+1}m(1+m^2)^{-\frac{\xi+\eta}{2}+\delta_2((1+\eta)\xi)}\cdot{}_2F_1\left( - \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor;1;z\right);\\
&a_2(\xi-\eta+1,\xi+\eta+1,m,1,u_1)=\\
&=\begin{dcases}
\frac{-\xi}{2}(1+m^2)^{-\frac{\xi+\eta}{2}}z\cdot{}_2F_1\left( - \frac{\eta}{2} + 1 ,-\frac{\xi}{2}+ 1;2;z\right), &\text{при } \xi \equiv_2 \eta \equiv_2 0,\\
-\frac{\xi z\cdot{}_2F_1\left( - \frac{\eta}{2} + 1 ,-\frac{\xi-1}{2} + 1;2;z\right) +2m^2\cdot{}_2F_1\left( - \frac{\eta}{2}+1,-\frac{\xi-1}{2};1;z\right)}{2(1+m^2)^{\frac{\xi+\eta}{2}}}, &\text{при } \xi - 1 \equiv_2 \eta \equiv_2 0,\\
0 , &\text{при } \xi \equiv_2 \eta - 1 \equiv_2 0,\\
(1+m^2)^{-\frac{\xi+\eta}{2}}m^2\cdot{}_2F_1\left( - \frac{\eta-1}{2},-\frac{\xi-1}{2};1;z\right), &\text{при } \xi \equiv_2\eta \equiv_2 1.
\end{dcases}
\end{align*}
\end{predl}
Мы приведем два доказательства предложения~\ref{formula}. Одно из них (см.~\S~\ref{first-proof}) элементарное, но при этом не объясняет, откуда берется формула, второе же (см.~\S~\ref{second-proof}) использует метод производящих функций (ср. ~\cite[ приложение A]{article}) и аппарат гипергеометрических функций.
Сформулируем теоремы~\ref{wl-intr} и \ref{chir-rev-f} в более сильной форме.
\begin{theorem}[{ср.~\cite[теорема 1]{article}}]
\label{large-time-lim} Для любых действительных $m, \varepsilon> 0$ выполнены утверждения:
\begin{enumerate}
\item[(A)] Для любого действительного $v$ верно, что $$\lim_{\substack{t \to \infty \\ t \in \varepsilon\mathbb{Z}}}\sum_{\substack{x\le v t\\ x \in \varepsilon\mathbb{Z}}}P(x,t,m,\varepsilon,u_{\varepsilon})= F(v):=
\begin{cases}
0 ,& \text{при } v < -\frac{1}{1+m^2\varepsilon^2};\\
\frac{1}{\pi}\arccos{\frac{1-(1+m^2\varepsilon^2)^2v}{(1+m^2\varepsilon^2)(1-v)}}, &\text{при } |v| \le \frac{1}{1+m^2\varepsilon^2};\\
1 , &\text{при } v > \frac{1}{1+m^2\varepsilon^2}.
\end{cases} $$
\item[(B)] Для любого действительного $v$ имеет место следующая сходимость по распределению при $t\to\infty$, где $t \in \varepsilon\mathbb{Z}$:
$$
t P(\left\lceil \frac{vt}{\varepsilon}\right\rceil\varepsilon, t,m,\varepsilon,u_{\varepsilon}) \overset{d}{\to} F^{\prime}(v) =
\begin{cases}
\frac{\sqrt{(1+m^2\varepsilon^2)^2 - 1}}{\pi(1-v)\sqrt{1-(1+m^2\varepsilon^2)^2v^2}}, & \text{при } |v| \le \frac{1}{1+m^2\varepsilon^2};\\
0 , & \text{при } |v| > \frac{1}{1+m^2\varepsilon^2}.
\end{cases}
$$
\item[(C)] Для любого целого $r \ge 0$ верно, что
\\
$\lim\limits_{\substack{t \to \infty \\ t \in \varepsilon\mathbb{Z}}}\sum\limits_{x \in \varepsilon\mathbb{Z}}\frac{x^r}{t^r}P(x,t,m,\varepsilon,u_{\varepsilon})=\int\limits_{-1}^{1}v^r F^{\prime}(v) dv.$
\end{enumerate}
\end{theorem}
\begin{theorem}[{ср. \cite[теорема 2]{IB-20}}]
\label{prob-rev-field}
При любых $m,\varepsilon, t > 0$, где $t\in \varepsilon\mathbb{Z}$, выполнена следующая асимптотика:
\begin{align*}
\sum_{\substack{x \in \varepsilon\mathbb{Z}}}a_1^2(x,t,m,\varepsilon,u_{\varepsilon}) =
\begin{cases}
\frac{m\varepsilon}{(1+m^2\varepsilon^2)\sqrt{2 + m^2\varepsilon^2}} + O_{m,\varepsilon}\left(t^{-1/3}\right) ,& \text{при } \frac{t}{\varepsilon} \equiv_{2} 1;\\
\frac{m\varepsilon}{\sqrt{2 + m^2\varepsilon^2}} + O_{m,\varepsilon}\left(t^{-1/3}\right),& \text{при } \frac{t}{\varepsilon} \equiv_{2} 0.
\end{cases}
\end{align*}
\end{theorem}
Здесь и далее запись $f(x,t,m,\varepsilon) = g(x,t,m,\varepsilon) + O_{m,\varepsilon}(h(x,t))$ означает, что существует константа $C(m,\varepsilon)$ (зависящая от $m,\varepsilon$, но не зависящая от $x,t$) такая, что для всех $x,t$, удовлетворяющих условию теоремы, выполнено $|f(x,t,m,\varepsilon) - g(x,t,m,\varepsilon)| \le C(m,\varepsilon)h(x,t)$.
При $m=\varepsilon=1$ этот результат подтверждает гипотезу, выдвинутую И. Богдановым в \cite[гипотеза 1]{IB-20}.
Наш метод доказательства теоремы \ref{prob-rev-field} отличается от предложенных в \cite[\S 12.7]{article} и \cite[\S 3]{IB-20}. Для доказательства теорем~\ref{large-time-lim} и~\ref{prob-rev-field} мы используем метод, предложеный в~\cite[~\S 2]{Grimmet-Janson-Scudo-04}, а также следующий новый результат.
\begin{predl}[{ср.~\cite[предложение 12]{article}}]
\label{IntForm}
Положим $$\omega_p:= \frac{1}{2\varepsilon}\arcsin{\frac{\sin{2p\varepsilon}}{1+m^2\varepsilon^2}}.$$ Тогда
для любого $m > 0$ и $(x,t)\in \varepsilon\mathbb{Z}^2$, где $t>0$, выполнено
\begin{align*}
&a_1(x,t,m,\varepsilon,u_{\varepsilon}) =\\
&=\footnotesize{\begin{cases}
(-1)^{\frac{x-\varepsilon}{2\varepsilon} + \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{m\varepsilon^2}{\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/ \varepsilon}^{\pi/ \varepsilon}e^{ipx}\sin{\omega_p(t-\varepsilon)}\cdot\frac{\sin{p\varepsilon}}{\cos{2\omega_p\varepsilon}} dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 1,\\
(-1)^{\frac{x}{2\varepsilon} + \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\cdot\frac{i\sin{\omega_p(t-2\varepsilon) - \cos{\omega_pt} }}{\cos{2\omega_p\varepsilon}} dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 2,\\
(-1)^{\frac{x-\varepsilon}{2\varepsilon} + \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{im\varepsilon^2}{\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\cos{\omega_p(t-\varepsilon)}\cdot\frac{\sin{p\varepsilon}}{\cos{2\omega_p\varepsilon}} dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 3,\\
(-1)^{\frac{x+2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\frac{\cos{\omega}_p(t-2\varepsilon) - i\sin{\omega}_pt}{\cos{2\omega_p\varepsilon}} dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 0.
\end{cases}}\\
&a_2(x,t,m,\varepsilon,u_{\varepsilon}) = \\
&=\begin{cases}
(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{\varepsilon}{2\pi}\int\limits_{-\pi/ \varepsilon}^{\pi/ \varepsilon}e^{ip(x-\varepsilon)}\left(i\sin{\omega_p(t-\varepsilon)}\cdot\frac{m^2\varepsilon^2 + \cos{2p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} - \cos{\omega_p(t-\varepsilon)}\right) dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 1,\\
(-1)^{\frac{x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\cdot\frac{ie^{2ip\varepsilon}\sin{\omega_p(t-2\varepsilon)} - \cos{\omega_pt} }{\cos{2\omega_p\varepsilon}} dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 2,\\
(-1)^{\frac{x+\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\left(\cos{\omega_p(t-\varepsilon)}\cdot\frac{m^2\varepsilon^2 +\cos{2p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} - i\sin{\omega_p(t-\varepsilon)}\right) dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 3,\\
(-1)^{\frac{x}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\cdot
\frac{e^{2ip\varepsilon}\cos{\omega_p(t-2\varepsilon)} - i\sin{\omega_pt}}{\cos{2\omega_p\varepsilon}}
dp, &\text{при } \frac{t}{\varepsilon} \equiv_4 0.
\end{cases}
\end{align*}
\end{predl}
Здесь мы считаем для определенности, что $(-1)^{n/2}:=i^n$ для нечетного $n$ (хотя это выражение встречается только в виде комбинации $(-1)^{n/2}\cdot 0 =0$).
Данные интегралы представляют волну, испущенную точечным источником, как суперпозицию волн с длиной волны $2\pi/p$ и частотой $\omega_p$.
\section{Первое доказательство предложения~\ref{formula}}
\label{first-proof}
Для доказательства предложения~\ref{formula} воспользуемся известным результатом. Он доказывается совершенно аналогично~\cite[предложение 5]{article}.
\begin{lemma}[Уравнение Дирака в электромагнитном поле на решетке] (ср.~\cite[предложения 5 и 14]{article})
\label{dirak}
Для любых натуральных $\xi,\eta$ выполнено
$$
a_1(\xi - \eta + \varepsilon,\xi + \eta + \varepsilon,m,\varepsilon,u_{\varepsilon}) =
$$
$$
=\frac{1}{\sqrt{1+m^2\varepsilon^2}} u_{\varepsilon}\left(\xi - \eta +\frac{3\varepsilon}{2}, \xi + \eta +\frac{\varepsilon}{2}\right)(a_1(\xi - \eta +2\varepsilon, \xi +\eta , m,\varepsilon,u_{\varepsilon}) + m\varepsilon a_2(\xi -\eta +2\varepsilon, \xi + \eta,m,\varepsilon,u_{\varepsilon})),$$\\
$$
a_2(\xi - \eta + \varepsilon,\xi + \eta + \varepsilon,m,\varepsilon,u_{\varepsilon}) =
$$
$$
=\frac{1}{\sqrt{1+m^2\varepsilon^2}} u_{\varepsilon}\left(\xi - \eta +\frac{\varepsilon}{2}, \xi + \eta +\frac{\varepsilon}{2}\right)(-m\varepsilon a_1(\xi - \eta ,\xi + \eta,m,\varepsilon,u_{\varepsilon}) + a_2(\xi - \eta,\xi + \eta,m,\varepsilon,u_{\varepsilon})).
$$
\end{lemma}
\vspace{.3cm}
\textit{Доказательство предложения~\ref{formula}.}
Проведем доказательство индукцией по~$\xi~+~\eta$.\\
\textit{База} индукции $\xi=\eta=0$ очевидна.
\textit{Переход.} Имеем: \par
$$a_1(\xi - \eta + 1, \xi + \eta + 1, m,1 ,u_1) = \frac{a_1(\xi - \eta + 2, \xi + \eta, m,1, u_1) + ma_2(\xi - \eta + 2, \xi + \eta, m,1, u_1)}{\sqrt{1 + m^2}}=$$
$$
=\frac{(-1)^{\xi+1}}{(1+m^2)^{\frac{\xi + \eta}{2}}}\left(m(1+m^2)^{\delta_2(\xi\eta) }\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}{\lfloor\frac{\xi}{2}\rfloor\choose j}{\lfloor\frac{\eta-2}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j + \right.
$$
$$
\hspace{-0.3cm}+\left.m\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}\left( {\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1+m^2)^{\delta_2(\xi(\eta-1))} - {\lfloor\frac{\eta-2}{2}\rfloor\choose j}(1+m^2)^{\delta_2(\xi\eta) }\right){\lfloor\frac{\xi}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j \right)=
$$
$$
=\frac{(-1)^{\xi+1}m}{(1+m^2)^{\frac{\xi+\eta}{2}}}(1+m^2)^{\delta_2((\eta+1)\xi)}\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}{\lfloor\frac{\xi}{2}\rfloor\choose j}{\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j.
$$
Здесь первое равенство выполнено в силу уравнения Дирака (лемма~\ref{dirak}) и определения поля $u_{
\varepsilon}$, второе --- в силу предположения индукции, а третье --- приведение подобных при $(1 - (1+m^2)^2)^j$.
Проверим индукционный переход для $a_2(\xi - \eta+1, \xi + \eta+1,m,1,u_1)$. Для этого рассмотрим 2 случая.
\textit{Случай 1:} $\eta \equiv_2 1$ или $\xi \equiv_2 1$. Тогда:
$$a_2(\xi - \eta + 1, \xi + \eta + 1, m,1 ,u_1) = \frac{(-1)^{\eta+1}(-ma_1(\xi - \eta, \xi + \eta, m,1, u_1) + a_2(\xi - \eta, \xi + \eta, m,1, u_1))}{\sqrt{1 + m^2}}=$$
$$
=\frac{(-1)^{\xi+\eta+1}}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\left(-m^2\sum_{j=0}^{\lfloor \frac{\xi-1}{2}\rfloor}{\lfloor\frac{\xi-1}{2}\rfloor\choose j}{\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j + \right.
$$
$$
+\left.\sum_{j=0}^{\lfloor\frac{\xi-1}{2}\rfloor}\left( {\lfloor\frac{\eta}{2}\rfloor\choose j}(1+m^2)^{\delta_2((\xi-1)\eta)} - {\lfloor\frac{\eta-1}{2}\rfloor\choose j}\right){\lfloor\frac{\xi-1}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j \right)=
$$
$$
\hspace{-0.3cm}=\frac{(-1)^{\xi+\eta+1}}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\lfloor\frac{\xi-1}{2}\rfloor}\left( {\lfloor\frac{\eta}{2}\rfloor\choose j}(1+m^2)^{\delta_2((\xi-1)\eta)} - (1+m^2){\lfloor\frac{\eta-1}{2}\rfloor\choose j}\right){\lfloor\frac{\xi-1}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j =
$$
$$
\hspace{-0.3cm}=\frac{(-1)^{\xi+1}}{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\lfloor\frac{\xi}{2}\rfloor}\left( {\lfloor\frac{\eta}{2}\rfloor\choose j}(1+m^2)^{\delta_2(\xi\eta)} - {\lfloor\frac{\eta-1}{2}\rfloor\choose j}(1+m^2)^{\delta_2(\xi(\eta+1))}\right){\lfloor\frac{\xi}{2}\rfloor\choose j}(1 - (1+m^2)^2)^j.
$$
Здесь первое равенство, как и в предыдущем случае, следует из уравнения Дирака (лемма~\ref{dirak}) и определения поля $u_{\varepsilon}$, второе верно в силу предположения индукции, третье --- приведение подобных при $(1 - (1+m^2)^2)^j$, а для проверки четвертого нужно рассмотреть 3 подслучая.
\textit{Подслучай 1:} $\xi \equiv_2 \eta \equiv_2 1$. Тогда левая и правая части четвертого равенства равны почленно:
$$
\hspace{-0.3cm}\frac{-1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-1}{2}}\left( {\frac{\eta-1}{2}\choose j} - (1+m^2){\frac{\eta-1}{2}\choose j}\right){\frac{\xi-1}{2}\choose j}(1 - (1+m^2)^2)^j =
$$
$$
\hspace{-0.3cm}=\frac{1}{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-1}{2}}\left( {\frac{\eta-1}{2}\choose j}(1+m^2) - {\frac{\eta-1}{2}\choose j}\right){\frac{\xi-1}{2}\choose j}(1 - (1+m^2)^2)^j.
$$
\textit{Подслучай 2:} $\xi - 1 \equiv_2 \eta \equiv_2 1$. Тогда
$$
\hspace{-0.3cm}\frac{1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-2}{2}}\left( {\frac{\eta-1}{2}\choose j}(1+m^2) - (1+m^2){\frac{\eta-1}{2}\choose j}\right){\frac{\xi-2}{2}\choose j}(1 - (1+m^2)^2)^j =
$$
$$
\hspace{-0.3cm}=0=\frac{-1}{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi}{2}}\left( {\frac{\eta-1}{2}\choose j} - {\frac{\eta-1}{2}\choose j}\right){\frac{\xi}{2}\choose j}(1 - (1+m^2)^2)^j.
$$
\textit{Подслучай 3:} $\xi \equiv_2 \eta-1 \equiv_2 1$. Тогда
$$
\hspace{-0.3cm}\frac{1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-1}{2}}\left( {\frac{\eta}{2}\choose j} - (1+m^2){\frac{\eta-2}{2}\choose j}\right){\frac{\xi-1}{2}\choose j}(1 - (1+m^2)^2)^j =
$$
$$
\hspace{-0.3cm}=\frac{1}{(1+m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-1}{2}}\left( {\frac{\eta}{2}\choose j} - {\frac{\eta-2}{2}\choose j}(1+m^2)\right){\frac{\xi-1}{2}\choose j}(1 - (1+m^2)^2)^j.
$$
Тем самым случай 1 разобран.
\textit{Случай 2:} $\eta \equiv_2 \xi \equiv_2 0$. Тогда:
$$a_2(\xi - \eta + 1, \xi + \eta + 1, m, 1, u_1) = \frac{(-1)^{\eta+1}(-ma_1(\xi - \eta, \xi + \eta, m,1, u) + a_2(\xi - \eta, \xi + \eta, m,1, u))}{\sqrt{1 + m^2}}=$$
$$
=\frac{(-1)^{\xi+\eta+1}}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\left(-m^2(1+m^2)\sum_{j=0}^{ \frac{\xi-2}{2}}{\frac{\xi-2}{2}\choose j}{\frac{\eta-2}{2}\choose j}(1 - (1+m^2)^2)^j + \right.
$$
$$
+\left.\sum_{j=0}^{\frac{\xi-2}{2}}\left( {\frac{\eta}{2}\choose j} - (1+m^2){\frac{\eta-2}{2}\choose j}\right){\frac{\xi-2}{2}\choose j}(1 - (1+m^2)^2)^j \right)=
$$
$$
=\frac{-1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi-2}{2}}\left( {\frac{\eta}{2}\choose j} - (1+m^2)^2{\frac{\eta-2}{2}\choose j}\right){\frac{\xi-2}{2}\choose j}(1 - (1+m^2)^2)^j =
$$
$$
=\frac{-1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\left(\sum_{j=0}^{\frac{\xi-2}{2}}\left( {\frac{\eta}{2}\choose j} - {\frac{\eta-2}{2}\choose j}\right){\frac{\xi-2}{2}\choose j}(1 - (1+m^2)^2)^j \right. +
$$
$$
+\left. \sum_{j=1}^{\frac{\xi}{2}}{\frac{\eta-2}{2}\choose j-1}{\frac{\xi-2}{2}\choose j-1}(1 - (1+m^2)^2)^j\right) =
$$
$$
=\frac{-1}{(1 + m^2)^{\frac{\xi+\eta}{2}}} \sum_{j=1}^{\frac{\xi}{2}} {\frac{\eta-2}{2}\choose j-1}\left({\frac{\xi-2}{2}\choose j}+{\frac{\xi-2}{2}\choose j-1}\right)(1 - (1+m^2)^2)^j =
$$
$$
=\frac{-1}{(1 + m^2)^{\frac{\xi+\eta}{2}}}\sum_{j=0}^{\frac{\xi}{2}} \left({\frac{\eta}{2}\choose j}-{\frac{\eta-2}{2}\choose j}\right){\frac{\xi}{2}\choose j}(1 - (1+m^2)^2)^j.
$$
Здесь первое равенство следует из уравнения Дирака (лемма~\ref{dirak}) и определения поля $u_{\varepsilon}$, второе верно в силу предположения индукции, третье --- приведение подобных членов при $(1 - (1+m^2)^2)^j$, четвертое получается, если добавить и вычесть $\sum_{j=0}^{\frac{\xi-2}{2}}{\frac{\eta-2}{2}\choose j}{\frac{\xi-2}{2}\choose j}(1 - (1+m^2)^2)^j$ и перегруппировать слагаемые, пятое --- применение правила Паскаля и приведение подобных при $(1- (1+m^2)^2)^j$, шестое --- два раза примененное правило Паскаля.
Тем самым последний случай, а с ним и предложение, доказаны.
$\square$
\section{Второе доказательство предложения~\ref{formula}}
\label{second-proof}
В этом разделе нам понадобятся следующие обозначения:
\begin{align*}
\hat{A}_1(p,q) := \sum_{\xi,\eta \ge 0}(1+m^2)^{\frac{\xi+\eta}{2}}a_1(\xi-\eta + 1, \xi + \eta + 1,m,1,u_1)p^{\xi}q^{\eta},\\
\hat{A}_2(p,q) := \sum_{\xi,\eta \ge 0} (1+m^2)^{\frac{\xi + \eta}{2}}a_2(\xi-\eta + 1, \xi +\eta + 1,m,1,u_1)p^{\xi}q^{\eta} .
\end{align*}
\begin{lemma}
\label{generSum}
Имеют место равенства формальных степенных рядов:
$$ \hat{A}_1(p,q) = -\frac{m q \left(1 - p + q - pq\left(m^2+1\right)\right)}{1-\left(q^2 + p^2 -\left(m^2+1\right)^2p^2q^2\right) },$$
$$
\hat{A}_2(p,q) = \frac{(1-q) \left(1 - p + q - pq\left(m^2+1\right)\right)}{1-\left(q^2 + p^2 -\left(m^2+1\right)^2p^2q^2\right) }.
$$
\end{lemma}
\begin{proof}
Для $g=0$ или $1$ и $i=1$ или $2$ обозначим
$$B^{g}_i(p,q):= \sum\limits_{\xi, \eta \ge 0: \eta \equiv_{2} g}(1+m^2)^{\frac{\xi + \eta}{2}}a_i(\xi - \eta + 1, \xi + \eta +1 ,m,u_1)p^{\xi}q^{\eta}, $$
Тогда из уравнения Дирака (лемма~\ref{dirak}) получаем
$$
\begin{pmatrix}
-1& 0& q& mq\\
pm&-1-p&0&0\\
q& mq&-1&0\\
0&0&-pm& -1+p\\
\end{pmatrix}
\begin{pmatrix}
B^0_1(p,q)\\
B^0_2(p,q)\\
B^1_1(p,q)\\
B^1_2(p,q)\\
\end{pmatrix}
=-
\begin{pmatrix}
B^0_1(p,0)\\
B^0_2(0,q)\\
B^1_1(p,0)\\
B^1_2(0,q)\\
\end{pmatrix}=
\begin{pmatrix}
0\\
1\\
0\\
0\\
\end{pmatrix}.
$$
Решая данную систему и выражая $ \hat{A}_i(p,q)$ через $B_i^{j}$, где $j=0,1$, получаем требуемое.
\end{proof}
\begin{lemma}Для любых целых $\xi,\eta \ge 0$ имеем
\label{firstclosedf}
$$
(1+m^2)^{\frac{\xi+\eta}{2}}a_1(\xi - \eta +1, \xi + \eta +1,m,1,u_1) =
$$
$$
=m(1 + m^2)^{\xi + \eta-2 + \delta_2((1+\xi)\eta)}(-1)^{\lfloor\frac{3\xi}{2}\rfloor + \lfloor\frac{\eta+1}{2}\rfloor}\sum\limits_{l=\max(\lfloor\frac{\xi}{2}\rfloor,\lfloor\frac{\eta-1}{2}\rfloor)}^{\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor}{l\choose \lfloor{\frac{\xi}{2}}\rfloor}{\lfloor{\frac{\xi}{2}}\rfloor\choose \lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor - l}(1 + m^2)^{-2l}(-1)^{l}.
$$
\end{lemma}
\begin{proof}
По лемме~\ref{generSum} имеем:
$$
\hat{A}_1(p,q) = -mq\left(1 - p + q - pq\left(m^2+1\right)\right)\sum\limits_{l=0}^{\infty}\left(p^2 + q^2 - p^2q^2(1+m^2)^2 \right)^l.
$$
Теперь посчитаем коэффициент при $p^{\xi}q^{\eta}$. B зависимости от выбора $1, p, q, (1+m^2)pq$ из множителя $(1 - p + q -(1+m^2)pq)$ мы должны посчитать в оставшейся сумме коэффициент при $p^{\xi}q^{\eta-1},p^{\xi-1}q^{\eta-1},p^{\xi}q^{\eta-2},p^{\xi-1}q^{\eta-2}$ соответственно. Заметим, что в оставшейся сумме присутствуют только мономы c чётными степенями. Тогда есть 4 случая:
\begin{align*}
&\frac{(1+m^2)^{\frac{\xi+\eta}{2}}}{m}a_1(\xi - \eta + 1, \xi + \eta + 1, m,1,u_1) =\\
&=\begin{cases}
-\sum\limits_{l=\max(\frac{\xi}{2}, \frac{\eta-2}{2})}^{\frac{\xi+\eta-2}{2}}{l\choose\frac{\xi}{2}}{\frac{\xi}{2}\choose\frac{\xi + \eta -2}{2}-l}\left(-(1+m^2)^2\right)^{\frac{\xi + \eta -2}{2}-l}, &\text{при } \xi \equiv_2 \eta \equiv_2 0, \\
(1+m^2)\sum\limits_{l=\max (\frac{\xi-1}{2}, \frac{\eta-2}{2})}^{\frac{\xi+\eta -3}{2}}{l \choose\frac{\xi-1}{2}}{\frac{\xi-1}{2}\choose\frac{\xi + \eta -3}{2} - l}\left(-(1+m^2)^2\right)^{\frac{\xi+\eta-3}{2}-l}, &\text{при } \xi-1\equiv_2\eta\equiv_2 0, \\
-\sum\limits_{l=\max(\frac{\xi}{2}, \frac{\eta-1}{2})}^{\frac{\xi+\eta-1}{2}}{l \choose \frac{\xi}{2}}{\frac{\xi}{2}\choose\frac{\xi+\eta-1}{2}-l}\left(-(1+m^2)^2\right)^{\frac{\xi+\eta-1}{2}-l}, &\text{при } \xi-1 \equiv_2 \eta\equiv_2 1, \\
\sum\limits_{l=\max(\frac{\xi-1}{2}, \frac{\eta-1}{2})}^{\frac{\xi+\eta-2}{2}}{l\choose\frac{\xi-1}{2}}{\frac{\xi-1}{2}\choose\frac{\xi+\eta-2}{2}-l}\left(-(1+m^2)^2\right)^{\frac{\xi+\eta-2}{2}-l}, &\text{при }\xi\equiv_2\eta\equiv_2 1. \\
\end{cases}
\end{align*}
\end{proof}
Теперь для доказательства предложения~\ref{formula} мы будем использовать гипергеометрические функции, а также некоторые известные тождества, с ними связанные.
Функция ${}_2F_1\left(a,b;c;z\right)$ при целых $c<b\le 0$ определяется аналогично случаю $c > 0$ (см.~\S\ref{results}), только суммирование распространяется только до $k=|b|$.
\begin{lemma}
\label{hyper}
Для любого действительного $m \ge 0$ и целых $\xi,\eta \ge 0$ выполнено
$$a_1(\xi - \eta +1, \xi+\eta+1,m,1,u_1) =$$
$$=(-1)^{\xi+1}\frac{m(1+m^2)^{\delta_2((1+\eta)\xi)}}{(1+m^2)^{\frac{\xi+\eta}{2}}}{\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor \choose \lfloor \frac{\xi}{2}\rfloor} \cdot$$
$$\cdot{}_2F_1\left(- \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor; -\left\lfloor\frac{\xi}{2}\right\rfloor - \left\lfloor\frac{\eta-1}{2}\right\rfloor ;(1+m^2)^2\right).$$
\end{lemma}
\begin{proof}
Преобразуем выражение из леммы~\ref{firstclosedf}.
Заметим, что $\xi + \eta - 2 + \delta_2((1+\xi)\eta) = 2(\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor) + \delta_2((1+\eta)\xi)$ и проведем замену индекса суммирования ($l$ заменим на $\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor - l = j$). Получим
$$
(1+m^2)^{\frac{\xi+\eta}{2}-\delta_2((1+\eta)\xi)}a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)=
$$
$$(-1)^{\xi+1}m\sum_{j=0}^{\min(\lfloor\frac{\eta-1}{2}\rfloor, \lfloor\frac{\xi}{2}\rfloor)}\frac{(\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor - j)!}{j!(\lfloor\frac{\eta-1}{2}\rfloor -j)!(\lfloor\frac{\xi}{2}\rfloor-j)!}(1 + m^2)^{2j}(-1)^{-j}.$$
Приведём этот ряд к гипергеометрическому виду. Отношение $(j+1)$-го слагаемого в полученной сумме к $j$-му равно
$$- \frac{( \lfloor\frac{\eta-1}{2}\rfloor -j)(\lfloor\frac{\xi}{2}\rfloor-j)}{(\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor-j)(j+1)}(1+m^2)^2.
$$
Но для гипергеометрической функции отношение $(j+1)$-го коэффициента к $j$-му по определению равно $\frac{(j+a)(j+b)}{(j+c)(j+1)}$. В нашем случае $a = - \lfloor\frac{\eta-1}{2}\rfloor$, $b = -\lfloor\frac{\xi}{2}\rfloor$, $c = -\lfloor\frac{\xi}{2}\rfloor - \lfloor\frac{\eta-1}{2}\rfloor$.
Таким образом,
$$(1+m^2)^{\frac{\xi+\eta}{2}-\delta_2((1+\eta)\xi)}a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)=$$
$$ =(-1)^{\xi+1}m{\left\lfloor\frac{\xi}{2}\right\rfloor + \left\lfloor\frac{\eta-1}{2}\right\rfloor \choose \left\lfloor\frac{\xi}{2}\right\rfloor}\cdot{}_2F_1\left(- \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor;-\left\lfloor\frac{\xi}{2}\right\rfloor - \left\lfloor\frac{\eta-1}{2}\right\rfloor;(1+m^2)^2\right).$$
\end{proof}
\textit{Доказательство предложений~\ref{formula} и \ref{hyper-form}.}
Случаи $\eta=0,1,2$ проверяются прямой подстановкой $\eta$ в условие леммы~\ref{firstclosedf}, поэтому считаем $\eta \ge 3$.
Применим известное тождество \newline ${}_2F_1\left( a,-n, c;z\right) =\left(\prod\limits_{l=0}^{n-1}\frac{l - c+ a}{l + c}\right){}_2F_1\left( a,-n, 1 - n + a - c; 1- z\right) $ (см.~\cite[(5.105)]{math}):
$$
{}_2F_1\left(- \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor;-\left\lfloor\frac{\xi}{2}\right\rfloor - \left\lfloor\frac{\eta-1}{2}\right\rfloor;(1+m^2)^2\right)=
$$
$$
=\left(\prod\limits_{l=0}^{\lfloor\frac{\xi}{2}\rfloor-1}\frac{l-\lfloor\frac{\xi}{2}\rfloor}{l-\lfloor\frac{\xi}{2}\rfloor - \lfloor\frac{\eta-1}{2}\rfloor }\right)\cdot{}_2F_1\left(- \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor;1;1-(1+m^2)^2\right).
$$
Так как $$\prod\limits_{l=0}^{\lfloor\frac{\xi}{2}\rfloor}\frac{l-\lfloor\frac{\xi}{2}\rfloor}{l-\lfloor\frac{\xi}{2}\rfloor - \lfloor\frac{\eta}{2}\rfloor} = \frac{1}{{\lfloor\frac{\xi}{2}\rfloor + \lfloor\frac{\eta-1}{2}\rfloor\choose \lfloor \frac{\xi}{2}\rfloor}},$$
то по лемме~\ref{hyper}
$$
a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)=
$$
$$
=\frac{(-1)^{ \xi+1}m}{(1+m^2)^{\frac{\xi+\eta}{2}-\delta_2((1+\eta)\xi)}}\cdot{}_2F_1\left( - \left\lfloor\frac{\eta-1}{2}\right\rfloor ,-\left\lfloor\frac{\xi}{2}\right\rfloor;1;1-(1+m^2)^2\right).
$$
Это доказывает формулу для $a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)$ из предложения~\ref{hyper-form}. Теперь, представляя эту гипергеометрическую функцию как ряд с конечным числом ненулевых членов, получаем формулу из предложения~\ref{formula} при $\eta \ge 3$.
Формула для $a_2(\xi-\eta+1,\xi+\eta+1,m,1,u_1)$ получается из доказанной формулы для $a_1(\xi-\eta+1,\xi+\eta+1,m,1,u_1)$ с помощью уравнения Дирака (леммы~\ref{dirak}).
$\square$
\section{Доказательство теорем ~\ref{fproblim} и~\ref{wavelimit}}
В этом разделе мы используем следующие обозначения: $A:=\left\lfloor{\frac{x}{4\varepsilon}}\right\rfloor + \left\lfloor{\frac{ t}{4\varepsilon}}\right\rfloor,$ $B:= \left\lfloor{\frac{t}{4\varepsilon}}\right\rfloor - \left\lfloor{\frac{ x}{4\varepsilon}}\right\rfloor$.
\smallskip \textit{Доказательство теоремы~\ref{wavelimit}, считая леммы~\ref{norm} --~\ref{binom} доказанными.}
Первое равенство в теореме следует из цепочки формул:
\footnotesize{\begin{align*}
&\frac{1}{2\varepsilon}a_1\left(4\varepsilon\left\lfloor\frac{x}{4\varepsilon}\right\rfloor, 4\varepsilon\left\lfloor\frac{t}{4\varepsilon}\right\rfloor, m, \varepsilon, u_{\varepsilon}\right) = \frac{1}{2\varepsilon}a_1\left(2(A-B), 2(A+B), m\varepsilon, 1, u_1\right) = \\
&=\frac{1}{2\varepsilon}m\varepsilon\left(1+(m\varepsilon)^2\right)^{3/2-2\left\lfloor\frac{ t}{4\varepsilon}\right\rfloor} \sum_{j=0}^{A-1}(-1)^j{A - 1\choose j}{B - 1\choose j}\left(2 + (m\varepsilon)^2\right)^j(m\varepsilon)^{2j} \sim \\
&\sim\frac{m}{2}\sum_{j=0}^{\infty}(-1)^j{A - 1\choose j}{B - 1\choose j}\left(2 + (m\varepsilon)^2\right)^j(m\varepsilon)^{2j} =\\
&\hspace{-0.3cm}=\frac{m}{2}\sum_{\substack{j=0\\ 2|j} }^{\infty}{A - 1\choose j}{B - 1\choose j}\left(2 + (m\varepsilon)^2\right)^j(m\varepsilon)^{2j} -\\
&-\frac{m}{2}\sum_{\substack{j=0 \\ 2 \nmid j}}^{\infty}{A - 1\choose j}{B - 1\choose j}\left(2 + (m\varepsilon)^2\right)^j(m\varepsilon)^{2j} \to \\
&\to \frac{m}{2}\left(\sum_{\substack{j=0\\ 2 |j} }^{\infty}\frac{(x+t)^j(t-x)^j}{8^j(j!)^2}m^{2j} - \sum_{\substack{j=0\\ 2 \nmid j}}^{\infty}\frac{(x+t)^j(t-x)^j}{8^j(j!)^2}m^{2j}\right) = \\
&=\frac{m}{2}J_0\left(m \sqrt{\frac{t^2-x^2}{2}}\right).
\end{align*}}
Здесь первое равенство выполнено в силу замечания~\ref{Ueq}, второе --- в силу предложения~\ref{formula} для $\xi=2A-1$ и $\eta = 2B$, эквивалентность следует из леммы~\ref{norm}. Cледующее равенство выполнено, так как все три суммы --- конечные, а предельный переход следует из лемм~\ref{lim20}--\ref{binom} (аналогов \cite[леммы 20 и 21]{article}) и \cite[лемма 22]{article}. Леммы~\ref{norm}--\ref{binom} будут доказаны ниже.
Для $a_2(x,t,m,\varepsilon,u_{\varepsilon})$ теорема доказывается аналогично.
$\square$
\begin{lemma}
\label{norm}
Для любых $m,t > 0$ выполнено следующее равенство: $$\lim_{\varepsilon \to 0}(1+(m\varepsilon)^2)^{3/2-2\left\lfloor\frac{t}{4\varepsilon}\right\rfloor} = 1.$$
\end{lemma}
\begin{proof} Лемма следует из следующей цепочки формул:
$$
1 \le \left(1+(m\varepsilon)^2\right)^{2\left\lfloor\frac{t}{4\varepsilon}\right\rfloor - 3/2} \le (1+(m\varepsilon)^2)^{\frac{ t}{2\varepsilon}} = \left(1+(m\varepsilon)^2\right)^{\frac{t\varepsilon}{2\varepsilon^2}} \sim \left(\exp{\frac{m^2t}{2}}\right)^{\varepsilon} \underset{\varepsilon \to 0}{\to} 1.$$
Применяя лемму о двух миллиционерах, получаем требуемое.
\end{proof}
\begin{lemma}
\label{lim20}
Для любого натурального $r$ выполнено
$$\lim\limits_{\varepsilon \to 0}{A-1\choose r}{B-1 \choose r}(2 + (m\varepsilon)^2)^{r}m^{2r+1}\varepsilon^{2r} = \frac{(x+t)^{r}(x-t)^{r}m^{2r+1}}{2^{3r}(r!)^2}.$$
\end{lemma}
\begin{proof}
Имеем
\begin{align*}
&{A-1 \choose r}{B-1 \choose r}(2 + (m\varepsilon)^2)^{r}m^{2r+1}\varepsilon^{2r} = \\ &=\frac{(A-1)\dots(A-r)\cdot(B-1)\dots(B-r)}{(r!)^2}\cdot(2 + (m\varepsilon)^2)^{r}m^{2r+1}\varepsilon^{2r} \to \left(\frac{x+t}{4}\right)^r\left(\frac{t-x}{4}\right)^r\frac{2^rm^{2r+1}}{(r!)^2}
\end{align*}
при $\varepsilon \to 0$, так как для любого $1 \le j \le r$
\begin{align*}
\lim\limits_{\varepsilon \to 0}(A-j)\varepsilon = \lim\limits_{\varepsilon \to 0}\left(\left\lfloor\frac{x}{4\varepsilon}\right\rfloor + \left\lfloor\frac{t}{4\varepsilon}\right\rfloor - j\right)\varepsilon = \lim\limits_{\varepsilon \to 0}\left(\frac{x}{4\varepsilon} + \frac{t}{4\varepsilon}\right)\varepsilon = \frac{x+t}{4}.
\end{align*}
Аналогично, $\lim\limits_{\varepsilon \to 0}(B-j)\varepsilon = \frac{t-x}{4}.$
\end{proof}
\begin{lemma}
\label{binom}
Для любых $|x|<t$ при достаточно малых $\varepsilon > 0$ для любого целого $j \ge 0$ верно неравенство
$$
{A - 1\choose j}{B - 1 \choose j}(2 + (m\varepsilon)^2)^j(m\varepsilon)^{2j} \le \frac{(x+t)^j(t-x)^jm^{2j}}{2^{3j}(j!)^2}.
$$
\end{lemma}
\begin{proof} Лемма следует из цепочки неравенств:
$$
{A - 1\choose j}{B - 1 \choose j}(2 + (m\varepsilon)^2)^j(m\varepsilon)^{2j} \le \frac{\left(x+t-4\varepsilon\right)^j\left(t-x-4\varepsilon\right)^j}{(4\varepsilon)^{2j}(j!)^2}\left(2 + (m\varepsilon)^2\right)^j(m\varepsilon)^{2j} =
$$
$$
=\frac{\left(x+t-4\varepsilon\right)^j\left(t-x-4\varepsilon\right)^j}{(4)^{2j}(j!)^2}\left(2 + (m\varepsilon)^2\right)^jm^{2j} \le \frac{(x+t)^j(t-x)^jm^{2j}}{2^{3j}(j!)^2},
$$
где последнее неравенство выполнено при $\varepsilon \le \min\{\frac{8}{m^2(x+t)}, \frac{x+t}{4}, \frac{t-x}{4}\},$
так как при таком $\varepsilon$ выполнена следующая цепочка неравенств:
$$
\left(1 - \frac{4\varepsilon}{x+t}\right) \left(1 + \frac{(m\varepsilon)^2}{2}\right) \le 1-\frac{4\varepsilon}{x+t} + \frac{(m\varepsilon)^2}{2}\le 1.
$$
\end{proof}
\begin{proof}
[Доказательство теоремы~\ref{fproblim}] Это сразу следует из теоремы~\ref{wavelimit}.
\end{proof}
\section{Доказательство предложения~\ref{IntForm}}
\label{IntFormProof}
\quad\textit{Случай 1:} $x/\varepsilon$ и $t/\varepsilon$ разной четности.
Заметим, что
$$\omega_{p+\pi/\varepsilon} = \frac{1}{2\varepsilon}\arcsin \frac{\sin 2(p\varepsilon+\pi)}{1+m^2\varepsilon^2}= \frac{1}{2\varepsilon}\arcsin \frac{\sin 2p\varepsilon}{1+m^2\varepsilon^2} = \omega_{p}. $$
Значит, подинтегральное выражение для $a_k(x,t,m,\varepsilon,u_{\varepsilon})$, где $k=1,2$, меняет знак при замене $p \to p+\pi/\varepsilon$ (оно домножается на $(-1)^{x/\varepsilon + 1}$ при $t/\varepsilon$ нечетном и на $(-1)^{x/\varepsilon}$ при $t/\varepsilon$ четном). Таким образом, весь интеграл равняется $0$, и случай 1 разобран.
\textit{Случай 2:} $x/\varepsilon$ и $t/\varepsilon$ одной четности.
Проведем доказательство индукцией по $t/\varepsilon$.
\textit{База.} Пусть $t=\varepsilon, x\in \varepsilon\mathbb{Z}$.
Так как $\sin{\omega_p0}=0$, то подынтегральное выражение для $a_1(x,t,m,\varepsilon,u_{\varepsilon})$ равняется $0$. Тем самым база индукции для $a_1(x,t,m,\varepsilon,u_{\varepsilon})$ проверена.
Для $a_2(x,t,m,\varepsilon,u_{\varepsilon})$ имеем:
\begin{align*}
&-\frac{\varepsilon(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+\varepsilon}{4\varepsilon}\right\rfloor}}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}dp = \\
&=\begin{dcases}
-1, &\text{при } x=\varepsilon,\\
-\frac{\varepsilon(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+\varepsilon}{4\varepsilon}\right\rfloor}\sin{\frac{\pi}{\varepsilon}(x-\varepsilon)}}{(x-\varepsilon)\pi}, &\text{при } x\ne\varepsilon;\\
\end{dcases} = \begin{dcases}
-1, &\text{при } x=\varepsilon,\\
0, &\text{при } x\ne\varepsilon.\\
\end{dcases} = a_2(x,t,m,\varepsilon,u_{\varepsilon}).
\end{align*}
Здесь предпоследнее равенство выполнено, так как $x\in \varepsilon\mathbb{Z}$.
\textit{Переход.} Рассмотрим 4 подслучая.
\textit{Подслучай 1:} $t/\varepsilon \equiv_4 2$.
Тогда для $a_1(x,t,m,\varepsilon, u_{\varepsilon})$ имеем следующую цепочку равенств:
\footnotesize{\begin{align*}
&a_1(x,t,m,\varepsilon, u_{\varepsilon}) = \frac{a_1(x+\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon}) + m\varepsilon a_2(x+\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon})}{\sqrt{1+m^2\varepsilon^2}} =\\
&=\frac{(-1)^{\frac{x}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}}{\sqrt{1+m^2\varepsilon^2}}\left(\frac{m\varepsilon^2}{2\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}2 e^{ip(x+\varepsilon)}\sin\omega_p(t-2\varepsilon)\cdot\frac{\sin p\varepsilon}{\cos2\omega_p\varepsilon}dp + \right.\\
&\left.+\frac{m\varepsilon^2}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\left(i\sin\omega_p(t-2\varepsilon)\cdot\frac{m^2\varepsilon^2+\cos 2p\varepsilon}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon}-\cos\omega_p(t-2\varepsilon)\right)dp \right) =\\
&=\frac{(-1)^{\frac{x}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\left(\sin\omega_p(t-2\varepsilon)\cdot\frac{2e^{ip\varepsilon}\sin p\varepsilon + i(m^2\varepsilon^2+\cos 2p\varepsilon)}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} - \cos\omega_p(t-2\varepsilon)\right)dp=\\
&=\frac{(-1)^{\frac{x}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\left(i\sin\omega_p(t-2\varepsilon)\cdot\frac{1 - i\sin2\omega_p\varepsilon }{\cos2\omega_p\varepsilon} - \cos\omega_p(t-2\varepsilon)\right)dp =\\
& = (-1)^{\frac{x}{2\varepsilon} + \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\cdot\frac{i\sin{\omega_p(t-2\varepsilon) - \cos{\omega_pt} }}{\cos{2\omega_p\varepsilon}} dp.
\end{align*}}
Здесь первое равенство следует из уравнения Дирака (лемма~\ref{dirak}), так как $u_{\varepsilon}(x+\varepsilon/2, t-\varepsilon/2)=1$ при целом $\frac{x+t}{2\varepsilon}$. Второе верно в силу предположения индукции. Третье~---~использование линейности интеграла и приведение подобных при $\sin\omega_p(t-2\varepsilon)$ и $\cos\omega_p(t-2\varepsilon)$ соответственно. Четвертое верно в силу следующей цепочки равенств:
\begin{align*}
&\frac{2e^{ip\varepsilon} \sin p\varepsilon + i(m^2\varepsilon^2 + \cos 2p\varepsilon)}{1+m^2\varepsilon^2} = \frac{i(1 - e^{2ip\varepsilon} + m^2\varepsilon^2 + \cos 2p\varepsilon)}{1+m^2\varepsilon^2} = \\
& =i\frac{1+m^2\varepsilon^2 - i\sin 2p\varepsilon}{1+m^2\varepsilon^2} = i(1 - i\sin 2\omega_p\varepsilon).
\end{align*}
Пятое следует из формулы для косинуса суммы.
Для $a_2(x,t,m,\varepsilon, u_{\varepsilon})$ имеем следующую цепочку равенств:
\footnotesize{\begin{align*}
&a_2(x,t,m,\varepsilon, u_{\varepsilon}) = (-1)^{\frac{t-x-2\varepsilon}{2\varepsilon}}\frac{a_2(x-\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon}) - m\varepsilon a_1(x-\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon})}{\sqrt{1+m^2\varepsilon^2}} =\\
&=\frac{(-1)^{\frac{x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t - 2\varepsilon}{4\varepsilon}\right\rfloor + \frac{t-x-2\varepsilon}{2\varepsilon} }}{\sqrt{1+m^2\varepsilon^2}}
\left(\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\left(i\sin\omega_p(t-2\varepsilon)\cdot\frac{m^2\varepsilon^2+\cos 2p\varepsilon}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon}-\cos\omega_p(t-2\varepsilon)\right)dp \right. -\\
&-\left.m\varepsilon\frac{m\varepsilon^2}{2\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}2 e^{ip(x-\varepsilon)}\sin\omega_p(t-2\varepsilon)\cdot\frac{\sin p\varepsilon}{\cos2\omega_p\varepsilon}dp \right)=
\end{align*}
\begin{align*}
&=\frac{(-1)^{\frac{x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\left(\sin\omega_p(t-2\varepsilon)\cdot\frac{i(m^2\varepsilon^2+\cos 2p\varepsilon) - 2m^2\varepsilon^2e^{ip\varepsilon}\sin p\varepsilon}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} - \right. \\
&- \left.\cos\omega_p(t-2\varepsilon)\right)dp= \\
&=\frac{(-1)^{\frac{x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\left(i\sin\omega_p(t-2\varepsilon)\cdot\frac{\cos 2p\varepsilon + m^2\varepsilon^2e^{2ip\varepsilon}}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} - \cos\omega_p(t-2\varepsilon)\right)dp = \\
&=(-1)^{\frac{x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-2\varepsilon)}\cdot\frac{ie^{2ip\varepsilon}\sin{\omega_p(t-2\varepsilon)} - \cos{\omega_pt} }{\cos{2\omega_p\varepsilon}} dp.
\end{align*}}
Здесь первое равенство следует из уравнения Дирака (лемма~\ref{dirak}), так как $u_{\varepsilon}(x-\varepsilon/2, t-\varepsilon/2)=(-1)^{\frac{t-x-2\varepsilon}{2\varepsilon}}$. Второе верно в силу предположения индукции. Третье --- использование линейности интеграла и приведение подобных при $\sin\omega_p(t-2\varepsilon)$ и $\cos\omega_p(t-2\varepsilon)$ соответственно. Знак в третьем равенстве получается из того, что $t/\varepsilon\equiv_{4}2$ и следующей цепочки равенств:
\begin{align*}
\left\lfloor\frac{x+t - 2\varepsilon}{4\varepsilon}\right\rfloor + \frac{t-x-2\varepsilon}{2\varepsilon} \equiv_2
\begin{dcases}
\frac{x+t-2\varepsilon}{4\varepsilon}, &\text{при }\frac{x}{\varepsilon}\equiv_{4} 0,\\
\frac{x+t}{4\varepsilon}, &\text{при }\frac{x}{\varepsilon}\equiv_{4} 2;
\end{dcases} = \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor.
\end{align*}
Четвертое получается выражением $\sin p\varepsilon$ через $e^{\pm ip\varepsilon}$. Пятое получается из следующей цепочки равенств:
\begin{align*}
&i\sin\omega_p(t-2\varepsilon)\cdot\frac{\cos 2p\varepsilon + m^2\varepsilon^2e^{2ip\varepsilon}}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} - \cos\omega_p(t-2\varepsilon) =\\
&= i\sin\omega_p(t-2\varepsilon)\cdot\frac{e^{2ip\varepsilon} -i\sin 2\omega_p\varepsilon}{\cos2\omega_p\varepsilon} - \cos\omega_p(t-2\varepsilon) =\frac{ie^{2ip\varepsilon}\sin\omega_p(t-2\varepsilon) -\cos\omega_pt}{\cos2\omega_p\varepsilon}.
\end{align*}
Тем самым подслучай 1 доказан.
\textit{Подслучай 2:} $t/\varepsilon \equiv_4 3$. Тогда для $a_1(x,t,m,\varepsilon, u_{\varepsilon})$ имеем следующую цепочку равенств:
\footnotesize{\begin{align*}
&a_1(x,t,m,\varepsilon, u_{\varepsilon}) = \frac{a_1(x+\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon}) + m\varepsilon a_2(x+\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon})}{\sqrt{1+m^2\varepsilon^2}} =\\
&=\frac{1}{\sqrt{1+m^2\varepsilon^2}}\left((-1)^{\frac{x+\varepsilon}{2\varepsilon} + \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor}\frac{m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x+\varepsilon)}\cdot\frac{i\sin{\omega_p(t-3\varepsilon) - \cos{\omega_p(t-\varepsilon)} }}{\cos{2\omega_p\varepsilon}} dp + \right.\\
&\left.+\frac{(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}m\varepsilon^2}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\cdot\frac{ie^{2ip\varepsilon}\sin{\omega_p(t-3\varepsilon)} - \cos{\omega_p(t-\varepsilon)} }{\cos{2\omega_p\varepsilon}} dp\right) =\\
&=\frac{(-1)^{\frac{x+\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}m\varepsilon^2}{2\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\cdot\frac{(1 - e^{2ip\varepsilon})\cos \omega_p(t-\varepsilon)}{\cos 2\omega_p\varepsilon}=\\
&=(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}\frac{im\varepsilon^2}{\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ipx}\cos{\omega_p(t-\varepsilon)}\cdot\frac{\sin{p\varepsilon}}{\cos{2\omega_p\varepsilon}}dp.
\end{align*}}
Здесь первое равенство верно в силу уравнения Дирака (лемма~\ref{dirak}), так как $u_{\varepsilon}(x+\varepsilon/2, t-\varepsilon/2)=1$ при целом $\frac{t+x}{2\varepsilon}$. Второе --- в силу предположения индукции. Третье --- использование линейности интеграла и приведение подобных при $\sin\omega_p(t-3\varepsilon)$ и $\cos\omega_p(t-\varepsilon)$ соответственно. Четвертое следует из выражения $\sin p\varepsilon$ через $e^{\pm ip\varepsilon}$.
Для $a_2(x,t,m,\varepsilon, u_{\varepsilon})$ имеем следующую цепочку равенств:
\footnotesize{\begin{align*}
&a_2(x,t,m,\varepsilon, u_{\varepsilon}) = (-1)^{\frac{t-x-2\varepsilon}{2\varepsilon}}\frac{a_2(x-\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon}) - m\varepsilon a_1(x-\varepsilon,t-\varepsilon,m,\varepsilon, u_{\varepsilon}) }{\sqrt{1+m^2\varepsilon^2}} =\\ &=\frac{(-1)^{\frac{t-x-2\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t - 2\varepsilon}{4\varepsilon}\right\rfloor}}{\sqrt{1+m^2\varepsilon^2}}\left(\frac{(-1)^{\frac{x-3\varepsilon}{2\varepsilon}}\varepsilon}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-3\varepsilon)}\cdot\frac{ie^{2ip\varepsilon}\sin{\omega_p(t-3\varepsilon)} - \cos{\omega_p(t-\varepsilon)} }{\cos{2\omega_p\varepsilon}} dp - \right.\\
&\left.- \frac{(-1)^{\frac{x-\varepsilon}{2\varepsilon}}m^2\varepsilon^3}{2\pi\sqrt{1+m^2\varepsilon^2}}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\cdot\frac{i\sin{\omega_p(t-3\varepsilon) - \cos{\omega_p(t-\varepsilon)} }}{\cos{2\omega_p\varepsilon}} dp\right) =\\
&=\frac{\varepsilon(-1)^{\frac{x-\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}}{2\pi(1+m^2\varepsilon^2)}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\frac{i\sin{\omega_p(t-3\varepsilon)}(1+m^2\varepsilon^2) - (m^2\varepsilon^2 + e^{-2ip\varepsilon})\cos{\omega_p(t-\varepsilon)}}{\cos{2\omega_p\varepsilon}} dp =\\
&=\frac{(-1)^{\frac{x+\varepsilon}{2\varepsilon}+\left\lfloor\frac{x+t }{4\varepsilon}\right\rfloor}\varepsilon }{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}e^{ip(x-\varepsilon)}\left(\cos{\omega_p(t-\varepsilon)}\cdot\frac{m^2\varepsilon^2 +\cos{2p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} - i\sin{\omega_p(t-\varepsilon)}\right) dp.
\end{align*}}
Здесь первое равенство верно в силу уравнения Дирака (лемма~\ref{dirak}), так как $u_{\varepsilon}(x-\varepsilon/2, t-\varepsilon/2)=(-1)^{\frac{t-x-2\varepsilon}{2\varepsilon}}$ при целом $\frac{t-x}{2\varepsilon}$. Второе --- в силу предположения индукции. Третье --- использование линейности интеграла и приведение подобных при $\sin\omega_p(t-3\varepsilon)$ и $\cos\omega_p(t-\varepsilon)$ соответственно. Знак в третьем равенстве получается из того, что $t/\varepsilon \equiv_{4}3$ и следующей цепочки равенств:
\begin{align*}
\left\lfloor\frac{x+t - 2\varepsilon}{4\varepsilon}\right\rfloor + \frac{t-x}{2\varepsilon} \equiv_2
\begin{dcases}
\frac{x+t}{4\varepsilon}, &\text{при }\frac{x}{\varepsilon}\equiv_{4} 1,\\
\frac{x+t-2\varepsilon}{4\varepsilon}, &\text{при }\frac{x}{\varepsilon}\equiv_{4} 3;
\end{dcases} = \left\lfloor\frac{x+t}{4\varepsilon}\right\rfloor.
\end{align*}
Четвертое следует из следующей цепочки равенств:
\begin{align*}
& \frac{i\sin{\omega_p(t-3\varepsilon)}(1+m^2\varepsilon^2) - (m^2\varepsilon^2 + e^{-2ip\varepsilon})\cos{\omega_p(t-\varepsilon)}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} = \\
&=i\sin{\omega_p(t-\varepsilon)} - \frac{(i\sin2p\varepsilon + m^2\varepsilon^2 + e^{-2ip\varepsilon})\cos{\omega_p(t-\varepsilon)}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} = \\
&=i\sin{\omega_p(t-\varepsilon)} - \frac{( m^2\varepsilon^2 + \cos{2p\varepsilon})\cos{\omega_p(t-\varepsilon)}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}},
\end{align*}
где мы воспользовались формулой синуса разности, определением $\omega_p$ и формулой Эйлера для $e^{-2ip\varepsilon}$.
Таким образом, подслучай 2 разобран.
\emph{Подслучаи 3 и 4}: $\frac{t}{\varepsilon}\equiv_4 0 $ и $\frac{t}{\varepsilon} \equiv_4 1$ доказываются аналогично подслучаям 1 и 2 с заменой $\cos \omega_p(t-k\varepsilon)$ на $\sin\omega_p(t-k\varepsilon)$ и наоборот, где $k=2$ или $3$. $\square$
\begin{remark}
Придумать формулы из предложения \ref{IntForm} можно, решая уравнение Дирака методом Фурье.
\end{remark}
\section{Доказательство теорем~\ref{wl-intr}~и~\ref{large-time-lim}}
Введем следующие обозначения для части подынтегральных выражений в предложении~\ref{IntForm}:
\begin{align}
\begin{split}
\label{Fourier_form}
\hat a_1(p,t)& :=
\begin{dcases}
2e^{ip\varepsilon}m\varepsilon\sin{\omega_p(t-\varepsilon)}\cdot \frac{\sin{p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 1 ;\\
e^{ip\varepsilon}m\varepsilon\cdot\frac{i\sin{\omega_p(t-2\varepsilon) - \cos{\omega_pt} }}{\sqrt{1+m^2\varepsilon^2}\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 2;\\
2ie^{ip\varepsilon}m\varepsilon\cos{\omega_p(t-\varepsilon)}\cdot \frac{\sin{p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 3;\\
e^{ip\varepsilon}m\varepsilon\cdot\frac{\cos{\omega}_p(t-2\varepsilon) - i\sin{\omega}_pt}{\sqrt{1+m^2\varepsilon^2}\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 0;
\end{dcases}\\
\hat{a}_2(p,t) &:=
\begin{dcases}
i\sin{\omega_p(t-\varepsilon)}\cdot\frac{m^2\varepsilon^2 + \cos{2p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} - \cos{\omega_p(t-\varepsilon)}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 1;\\
e^{-ip\varepsilon}\cdot\frac{ie^{2ip\varepsilon}\sin{\omega_p(t-2\varepsilon)} - \cos{\omega_pt} }{\sqrt{1+m^2\varepsilon^2}\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 2;\\
\cos{\omega_p(t-\varepsilon)}\cdot\frac{m^2\varepsilon^2 + \cos{2p\varepsilon}}{(1+m^2\varepsilon^2)\cos{2\omega_p\varepsilon}} - i\sin{\omega_p(t-\varepsilon)}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 3;\\
e^{-ip\varepsilon}\cdot
\frac{e^{2ip\varepsilon}\cos{\omega_p(t-2\varepsilon)} - i\sin{\omega_pt}}{\sqrt{1+m^2\varepsilon^2}\cos{2\omega_p\varepsilon}}, &\text{ при } \frac{t}{\varepsilon} \equiv_{4} 0;
\end{dcases}
\end{split}
\end{align}
Тогда $a_1(x,t,m,\varepsilon,u_{\varepsilon})$ и $a_2(x,t,m,\varepsilon,u_{\varepsilon})$ с точностью до знака суть коэффициенты ряда Фурье для функций $\hat a_1(p,t)$ и $\hat a_2(p,t)$ соответственно.
\begin{proof}[Доказательство теоремы~\ref{large-time-lim}]
В силу ~\cite[лемма 2]{article} достаточно доказать пункт (C). Он следует из цепочки равенств, которую мы прокомментируем ниже:
\begin{align*}
&\sum\limits_{x \in \varepsilon\mathbb{Z}}\frac{x^r}{t^r}P(x,t,m,\varepsilon,u_{\varepsilon})= \sum\limits_{x \in\varepsilon\mathbb{Z}}\begin{pmatrix}
a_1(x,t,m,\varepsilon,u_{\varepsilon})\\
a_2(x,t,m,\varepsilon,u_{\varepsilon})
\end{pmatrix}^{*}\frac{x^r}{t^r}\begin{pmatrix}
a_1(x,t,m,\varepsilon,u_{\varepsilon})\\
a_2(x,t,m,\varepsilon,u_{\varepsilon}) \\
\end{pmatrix} = \\
&=
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\begin{pmatrix}
\hat{a}_1(p,t)\\
\hat{a}_2(p,t)
\end{pmatrix}^{*}\frac{i^r}{t^r}\frac{\partial^r}{\partial p^r}\begin{pmatrix}
\hat{a}_1(p,t)\\
\hat{a}_2(p,t) \\
\end{pmatrix}d p = \\
=&\begin{dcases}
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\left( \omega_p^{\prime}\right)^{r}
\left|\begin{pmatrix}
\hat{a}_1(p,t)\\
\hat{a}_2(p,t)
\end{pmatrix}\right|^{2} dp + O_{m,\varepsilon,r}\left(\frac{1}{t}\right), & \text{ при четном } r, \\
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\left( \omega_p^{\prime}\right)^{r-1}
\begin{pmatrix}
\hat{a}_1(p,t)\\
\hat{a}_2(p,t)
\end{pmatrix}^{*}\frac{i\partial}{\partial p}\begin{pmatrix}
\hat{a}_1(p,t)\\
\hat{a}_2(p,t)
\end{pmatrix} dp + O_{m,\varepsilon,r}\left(\frac{1}{t}\right),& \text{ при нечетном } r,
\end{dcases} =\\
\end{align*}
\begin{align*}
&=\begin{dcases}
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\left( \omega_p^{\prime} \right)^{r}dp +O_{m,\varepsilon,r}\left(\frac{1}{t}\right),& \text{ при четном } r,\\
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\left( \omega_p^{\prime} \right)^{r} \frac{m^2\varepsilon^2 + \cos 2p\varepsilon}{(1+m^2\varepsilon^2)\cos 2\omega_p\varepsilon} dp+O_{m,\varepsilon,r}\left(\frac{1}{t}\right),& \text{ при нечетном } r,
\end{dcases} =\\
&=\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon}\left( \omega_p^{\prime} \right)^{r+\delta_2(r)}dp +O_r\left(\frac{1}{t}\right) = \frac{2\varepsilon}{\pi}\int\limits_{0}^{\pi/2\varepsilon}\left( \omega_p^{\prime} \right)^{r+\delta_2(r)}dp +O_{m,\varepsilon,r}\left(\frac{1}{t}\right)=
\end{align*}
\begin{align*}
&=\int\limits_{-\frac{1}{1+m^2\varepsilon^2}}^{\frac{1}{1+m^2\varepsilon^2}}
v^{r+\delta_2(r)}\frac{\sqrt{(1+m^2\varepsilon^2)^2-1}}{\pi(1-v^2)\sqrt{1 - (1+m^2\varepsilon^2)^2v^2}} dv + O_{m,\varepsilon,r}\left(\frac{1}{t}\right) =\\ &=\int\limits_{-\frac{1}{1+m^2\varepsilon^2}}^{\frac{1}{1+m^2\varepsilon^2}}
(v^{r+\delta_2(r)}+v^{r+\delta_2(r+1)})\frac{\sqrt{(1+m^2\varepsilon^2)^2-1}}{\pi(1-v^2)\sqrt{1 - (1+m^2\varepsilon^2)^2v^2}}dv +O_{m,\varepsilon,r}\left(\frac{1}{t}\right) = \\
&=\int\limits_{-1}^{1}
v^{r}F^{\prime}(v) dv +O_{m,\varepsilon,r}\left(\frac{1}{t}\right).
\end{align*}
Здесь первое равенство верно из определения $P(x,t,m,\varepsilon,u_{\varepsilon})$. Второе следует из предложения~\ref{IntForm}, свойства производных ряда Фурье и теоремы Парсеваля. Третье равенство следует из следующей асимптотики для производной:
\begin{align*}
\frac{\partial^r}{t^r\partial p^r} \hat{a}_k(p,t)
= \begin{cases}
(-1)^{r/2}\left( \omega_p^{\prime} \right)^r \hat{a}_k(p,t) +O_{m,\varepsilon,r}\left(\frac{1}{t}\right),& \text{при четном } r,\\
(-1)^{(r-1)/2}\left( \omega_p^{\prime} \right)^{r-1} \frac{\partial}{\partial p}\hat{a}_k(p,t) +O_{m,\varepsilon,r}\left(\frac{1}{t}\right), & \text{при нечетном } r.
\end{cases}
\end{align*}
Действительно, дифференцируем $r$ раз выражение~(\ref{Fourier_form}) с помощью правила Лейбница. Для того, чтобы получить главный член, мы должны каждый раз дифференцировать множитель, содержащий $t$, то есть либо $\sin\omega_p(t-k\varepsilon)$, либо $\cos\omega_p(t-k\varepsilon)$, где $k=0,1$ или $2$. Остальные слагаемые дадут $O_{m,\varepsilon,r}\left(\frac{1}{t}\right)$, так как это гладкие периодические функции.
Четвертое равенство проверяется прямым вычислением в \cite[раздел 1]{computations}. Пятое и шестое равенства следуют из того, что $\omega_p^{\prime} =\frac{\cos 2p\varepsilon}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} $ и $\omega_{p+\pi /2\varepsilon}^{\prime} = -\omega_p^{\prime} $. Седьмое равенство --- замена переменной $v = \omega_p^{\prime}=\frac{\cos 2p\varepsilon}{\sqrt{(1+m^2\varepsilon^2)^2 - \sin^2 2p\varepsilon}}$ (см. \cite[разделы 0 и 5]{computations}). Такая замена переменной возможна, так как $\omega_p^{\prime}$ убывает на $(0, \pi/2\varepsilon)$ (см. \cite[раздел 0]{computations}). Восьмое равенство верно, так как следующая функция нечетна и поэтому не даёт вклада в интеграл:
$$
v^{r+\delta_2(r+1)}\frac{\sqrt{(1+m^2\varepsilon^2)^2-1}}{\pi(1-v^2)\sqrt{1 - (1+m^2\varepsilon^2)^2v^2}}.
$$
Тем самым теорема~\ref{large-time-lim} доказана.
\end{proof}
\begin{proof}
[Доказательство теоремы~\ref{wl-intr}] Это сразу следует из теоремы~\ref{large-time-lim}.
\end{proof}
\section{Доказательство теорем~\ref{chir-rev-f} и \ref{prob-rev-field}}
\begin{proof}[Доказательства теоремы \ref{prob-rev-field}, считая лемму \ref{cos-approx1} доказанной] Рассмотрим 4 случая:
\textit{Случай 1:} $\frac{t}{\varepsilon} \equiv_{4} 1$.
Имеем следующую цепочку равенств:
\begin{align*}
&\sum\limits_{x \in \varepsilon\mathbb{Z}}a_1^2(x,t,m,\varepsilon,u_{\varepsilon})= \sum\limits_{x \in\varepsilon\mathbb{Z}}\left|a_1(x,t,m,\varepsilon, u_{\varepsilon})\right|^2 = \\
&=
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \left|\hat{a}_1(p,t)\right|^2 d p = \frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{4m^2\varepsilon^2 \sin^2 \omega_p(t-\varepsilon) \sin^2 p\varepsilon}{(1+m^2\varepsilon^2)^2 \cos^2 2\omega_p\varepsilon} d p= \\
&=
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{2m^2\varepsilon^2 \sin^2 \omega_p(t-\varepsilon)}{(1+m^2\varepsilon^2)^2 \cos^2 2\omega_p\varepsilon} d p = \frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{m^2\varepsilon^2}{(1+m^2\varepsilon^2)^2 \cos^2 2\omega_p\varepsilon} d p -\\ &-\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{m^2\varepsilon^2\cos 2(t-\varepsilon)\omega_p d p}{(1+m^2\varepsilon^2)^2 \cos^2 2\omega_p\varepsilon} = \frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{m^2\varepsilon^2d p}{(1+m^2\varepsilon^2)^2 \cos^2 2\omega_p\varepsilon} + O_{m,\varepsilon}\left(t^{-1/3}\right) =\\
&=\frac{2\varepsilon}{\pi}\int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{m^2\varepsilon^2d p}{(1+m^2\varepsilon^2)^2 - \sin^2 2p\varepsilon} + O_{m,\varepsilon}\left(t^{-1/3}\right)
= \frac{m\varepsilon}{(1+m^2\varepsilon^2)\sqrt{2+m^2\varepsilon^2}} + O_{m,\varepsilon}\left(t^{-1/3}\right).
\end{align*}
Здесь первое равенство выполнено в силу того, что $a_1(x,t,m,\varepsilon, u_{\varepsilon}) \in \mathbb{R}$. Второе равенство следует из предложения~\ref{IntForm}, обозначения \eqref{Fourier_form} и теоремы Парсеваля. Третье --- из определения $\hat{a}_1(p,t)$. Четвертое следует из формулы понижения степени и того, что $\omega_{p +\pi/2\varepsilon} = \frac{1}{2\varepsilon}\arcsin \frac{\sin{(2p\varepsilon + \pi)}}{1+m^2\varepsilon^2} = -\omega_p$. Пятое --- из формулы понижения степени и линейности интеграла. Шестое следует из леммы \ref{cos-approx1}, которую мы докажем ниже. Седьмое следует из того, что $\omega_p=-\omega_{p+\pi/2\varepsilon}$ и определения величины $\omega_p$. Восьмое --- прямое вычисление, проведенное в \cite[раздел 4]{computations}.
\textit{Случай 2:} $\frac{t}{\varepsilon} \equiv_{4} 2$.
Имеем следующую цепочку равенств:
\begin{align*}
&\sum\limits_{x \in \varepsilon\mathbb{Z}}a_1^2(x,t,m,\varepsilon,u_{\varepsilon})= \sum\limits_{x \in\varepsilon\mathbb{Z}}\left|a_1(x,t,m,\varepsilon, u_{\varepsilon})\right|^2 = \\
&=
\frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \left|\hat{a}_1(p,t) \right|^2d p = \frac{\varepsilon}{2\pi}\int\limits_{-\pi/\varepsilon}^{\pi/\varepsilon} \frac{m^2\varepsilon^2 (\sin^2 \omega_p(t-2\varepsilon) +\cos^2 \omega_pt)}{(1+m^2\varepsilon^2) \cos^2 2\omega_p\varepsilon} d p= \\
&=
\frac{2\varepsilon}{\pi}\left(\int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{m^2\varepsilon^2 d p }{(1+m^2\varepsilon^2) \cos^2 2\omega_p\varepsilon} - \int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{m^2\varepsilon^2 \cos2 \omega_p(t-2\varepsilon) d p}{2(1+m^2\varepsilon^2) \cos^2 2\omega_p\varepsilon} + \right. \\
&+ \left. \int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{m^2\varepsilon^2 \cos 2\omega_pt d p}{2(1+m^2\varepsilon^2) \cos^2 2\omega_p\varepsilon} \right) = \frac{2\varepsilon}{\pi}\int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{(1+m^2\varepsilon^2) m^2\varepsilon^2 d p}{(1+m^2\varepsilon^2)^2 - \sin^2 2p\varepsilon} + O_{m,\varepsilon}\left(t^{-1/3}\right) =
\end{align*}
\begin{align*}
&=\frac{m\varepsilon}{\sqrt{2+m^2\varepsilon^2}} + O_{m,\varepsilon}\left(t^{-1/3}\right).
\end{align*}
Здесь первые два равенства аналогичны случаю 1. Третье следует из обозначения \eqref{Fourier_form}.
Четвертое следует из формулы понижения степени, линейности интеграла и того, что $\omega_{p +\pi/2\varepsilon} = \frac{1}{2\varepsilon}\arcsin \frac{\sin{(2p\varepsilon + \pi)}}{1+m^2\varepsilon^2} = -\omega_p$. Пятое следует из леммы \ref{cos-approx1}, которую мы докажем позже, и определения величины $\omega_p$. Шестое аналогично седьмому равенству случая 1.
\emph{Случаи $\frac{t}{\varepsilon}\equiv_{4} 3$ и $\frac{t}{\varepsilon}\equiv_{4} 0$} аналогичны случаям 1 и 2 соответственно с заменой $\sin \omega_p(t-k\varepsilon)$ на $\cos \omega_p(t-k\varepsilon)$, где $k = 1$ и $2$, и наоборот. $\square$
\end{proof}
Для завершения доказательства осталось доказать лемму~\ref{cos-approx1} --- грубую оценку в духе метода стационарной фазы \cite[лемма 3]{article}.
\begin{lemma}
\label{cos-approx1} Путь $m,\varepsilon,t >0$, тогда
\begin{align*}
\int\limits_{-\pi/4\varepsilon}^{\pi/4\varepsilon} \frac{\cos2 \omega_pt}{\cos^2 2\omega_p\varepsilon} d p = O_{m,\varepsilon}\left(t^{-1/3}\right).
\end{align*}
\begin{proof}
Будем считать, что $t > (\frac{4\varepsilon}{\pi})^3$, иначе лемма очевидна, так как подынтегральное выражение ограничено сверху:
\begin{align*}
\left|\frac{\cos2 \omega_pt}{\cos^2 2\omega_p\varepsilon}\right| \le \left|\frac{1}{1 - \frac{\sin^2 2p\varepsilon}{(1+m^2\varepsilon^2)^2}}\right| \le \frac{(1+m^2\varepsilon^2)^2}{(1+m^2\varepsilon^2)^2 - 1}.
\end{align*}
Разобьем интеграл на три части $I_1, I_2, I_3$:
\begin{align*}
I_1 := \int\limits_{-\pi/4\varepsilon}^{-\pi/4\varepsilon + t^{-1/3}} \frac{\cos2 \omega_pt}{\cos^2 2\omega_p\varepsilon} d p,~
I_2 := \int\limits_{-\pi/4\varepsilon+t^{-1/3}}^{\pi/4\varepsilon-t^{-1/3}} \frac{\cos2 \omega_pt}{\cos^2 2\omega_p\varepsilon} d p,~
I_3 := \int\limits_{\pi/4\varepsilon-t^{-1/3}}^{\pi/4\varepsilon} \frac{\cos2 \omega_pt}{\cos^2 2\omega_p\varepsilon} d p.
\end{align*}
Заметим, что $I_1 + I_3 = O_{m,\varepsilon}(t^{-1/3})$, так как подынтегральное выражение ограничено сверху величиной $\frac{(1+m^2\varepsilon^2)^2}{(1+m^2\varepsilon^2)^2 - 1}$.
Найдем асимптотику для $I_2$. Для этого домножим и разделим подинтегральное выражение на $\omega_p^{\prime}$ и возьмем этот итеграл по частям. Это действие законно, так как $\omega_p^{\prime} = \frac{\cos 2p\varepsilon}{(1+m^2\varepsilon^2)\cos2\omega_p\varepsilon} $ не обращается в 0 на $[-\pi/4\varepsilon + t^{-1/3}, \pi/4\varepsilon - t^{-1/3}]$ при $t > \left(\frac{4\varepsilon}{\pi}\right)^3$.
Имеем:
\begin{align}
\label{integr}
&I_2 = \left.\frac{\sin2\omega_pt}{2t\omega_p^{\prime}\cos^2 2\omega_p\varepsilon }\right|^{\pi/4\varepsilon-t^{-1/3}}_{-\pi/4\varepsilon+t^{-1/3}} - \int\limits^{\pi/4\varepsilon-t^{-1/3}}_{-\pi/4\varepsilon+t^{-1/3}} \frac{\sin2\omega_pt}{2t}\cdot\left(\frac{1}{\omega_p^{\prime}\cos^2 2\omega_p\varepsilon}\right)^{\prime} dp
\end{align}
Сначала оценим первое слагаемое в правой части равенства \eqref{integr}:
\begin{align*}
& \left|\left.\frac{\sin2\omega_pt}{2t\omega_p^{\prime}\cos^2 2\omega_p\varepsilon }\right|^{\pi/4\varepsilon-t^{-1/3}}_{-\pi/4\varepsilon+t^{-1/3}}\right| = \left|\left.\frac{(1+m^2\varepsilon^2)^2 \sin2\omega_{p}t}{t\cos2p\varepsilon\sqrt{(1+m^2\varepsilon^2)^2 - \sin^2 2p\varepsilon}}\right|_{p=-\pi/4\varepsilon+t^{-1/3}}\right| \le\\
& \le \left|\frac{(1+m^2\varepsilon^2)^2}{2t\sin(2\varepsilon t^{-1/3})\sqrt{(1+m^2\varepsilon^2)^2 - 1}}\right| \le \left|\frac{(1+m^2\varepsilon^2)^2}{2\varepsilon t^{2/3}\sqrt{(1+m^2\varepsilon^2)^2 - 1}}\right| = O_{m,\varepsilon}\left(t^{-2/3}\right).
\end{align*}
Здесь первое равенство следует из определения величины $\omega_p^{\prime}$. Второе следует из ограниченности синуса. Третье следует из того, что $|\sin{2\varepsilon t^{-1/3}}| \ge |\varepsilon t^{-1/3}|$ при $t > \left(\frac{4\varepsilon}{\pi}\right)^3$.
Теперь оценим второе слагаемое в правой части равенства \eqref{integr}.
Заметим, что
\begin{align*}
&\left(\frac{1}{\omega_p^{\prime}\cos^2 2\omega_p\varepsilon}\right)^{\prime} = \left(\frac{1+m^2\varepsilon^2}{\cos 2p\varepsilon\cos 2\omega_p\varepsilon}\right)^{\prime}= 2\varepsilon(1+m^2\varepsilon^2)\cdot\frac{\sin2p\varepsilon}{\cos^2 2p\varepsilon\cos 2\omega_p\varepsilon} + 2\varepsilon \frac{\sin2\omega_p\varepsilon}{\cos^3 2\omega_p\varepsilon}.
\end{align*}
Так как $|\cos 2\omega_p\varepsilon| \ge \frac{(1+m^2\varepsilon^2)^2}{(1+m^2\varepsilon^2)^2-1}$ и $\frac{1}{\cos^2{2p\varepsilon}} \le \frac{1}{t\sin^2(2\varepsilon t^{-1/3})} \le \frac{t}{\varepsilon^2t^{1/3}}$ на $[-\pi/4\varepsilon + t^{-1/3}, \pi/4\varepsilon - t^{-1/3}]$ при $t \ge \left(\frac{4\varepsilon}{\pi}\right)^3$, то $I_2 = O_{m,\varepsilon}\left(t^{-1/3}\right)$.
Теперь, складывая $I_1, I_2,I_3$, получаем требуемое.
\end{proof}
\end{lemma}
\begin{proof}[Доказательство теоремы~\ref{chir-rev-f}]
Это сразу следует из теоремы~\ref{prob-rev-field}.
\end{proof}
\section{Обобщения}
В заключение предложим метод вычисления производящих функций последовательностей $a_1(x,t,m,1,u)$ и $a_2(x,t,m,1,u)$ для произвольного двоякопериодического поля $u$.
Рассмотрим решетку $\mathbb{Z}^2$ с шагом 1. Для $2h,2v \in \mathbb{Z}$ через $u(h,v)$ обозначим значение функции $u$ на вспомогательном ребре с серединой в $(h - v + 1, h + v + 1)$. Отождествим $u\colon \left(\mathbb{Z}\cup (\mathbb{Z}+1/2)\right)\times (\mathbb{Z}\cup (\mathbb{Z}+1/2)) \to \{\pm1\}$ с отображением $u$ из множества всех ребер в $\{\pm1\}$.
\begin{hyp}
\label{generSumPeriodic} Пусть даны
действительное $m > 0$ и двоякопериодическая функция $u\colon \left(\mathbb{Z}\cup (\mathbb{Z}+1/2)\right)\times (\mathbb{Z}\cup (\mathbb{Z}+1/2)) \to \{\pm1\}$ с целыми периодами $s_h,s_v\ge 2$ по первому и второму аргументу соответственно. Тогда вектор
$$\Bar{B} := (B^{0,0}_1, B^{0,0}_2, B^{0,1}_1, B^{0,1}_2 \dots B^{0, s_v-1}_1, B^{0,s_v-1}_2, B^{1,0}_1, \dots B^{s_h-1, s_v-1}_2),$$ где
$$B^{f,g}_i(p,q):= \sum\limits_{h,v \ge 0, h \equiv_{s_h} f, v \equiv_{s_v} g }a_i(h-v + 1, h+v+1 ,m,1,u)p^{h}q^{v}, $$
является решением системы $\Bar{B} - Q\Bar{B} = \Bar{a} $, где
$$
Q =
\begin{pmatrix}
Q^0&0&\dots &\dots &0&R^{s_h-1}\\
R^0&Q^1 & \dots& \dots&\dots &0\\
\dots& R^1& \dots&\dots&\dots &\dots\\
\dots& \dots& \dots&\dots&\dots &\dots\\
0&\dots&R^{i-1} &Q^i&\dots &0\\
\dots& \dots& \dots&\dots&\dots &\dots\\
0& 0 &\dots&\dots &R^{s_h-2}& Q^{s_h-1}
\end{pmatrix},
$$
$$
Q^i =
\begin{pmatrix}
0&0&\dots&\dots& W^{i,s_v-1}\\
W^{i,0}&0&\dots&\dots&0\\
0&W^{i,1}&\dots&\dots&0\\
\dots& \dots& \dots&\dots&\dots\\
0&0&\dots&W^{i,s_v-2}&0\\
\end{pmatrix},
\qquad
R^i =
\begin{pmatrix}
G^{i,0}&0&\dots&0\\
0&G^{i,1}&\dots&0\\
\dots& \dots& \dots&\dots\\
0&0&\dots&G^{i,s_v-1}\\
\end{pmatrix},
$$
$$
W^{i,j} = q(1+m^2)^{-1/2}
\begin{pmatrix}
u(i,j+1/2)& mu(i,j+1/2)\\
0&0\\
\end{pmatrix},
$$
$$
G^{i,j} = qp(1+m^2)^{-1/2}
\begin{pmatrix}
0&0\\
-mu(i+1/2,j)& u(i+1/2,j)\\
\end{pmatrix},
$$
$$
\Bar{a}:= (a^{0,0}_1, a^{0,0}_2, a^{0,1}_1, a^{0,1}_2 \dots a^{0, s_v-1}_1, a^{0,s_v-1}_2, a^{1,0}_1, \dots a^{s_h-1, s_v-1}_2),
$$
$$
a^{f,g}_i:=
\begin{cases}
q^ga_2(-g+1, g+1, m,1 ,u) &\text{если }f=0\text{ и }i=2,\\
p^gq^ga_1(f+1, f+1, m,1 ,u) &\text{если }g=0\text{ и }i=1,\\
0&\text{иначе.}
\end{cases}
$$
\end{hyp}
\begin{op}[{ср. с \cite[теорема 1]{Novikov}}]
Обращается ли $P(x,t,m,\varepsilon, u_{\varepsilon})$ в ноль при каких-нибудь $m,\varepsilon>0$, $|x|<t$ с четным $(x+t)/\varepsilon$?
\end{op}
\section{Благодарности}
Исследование выполнено за счет гранта Российского научного фонда № 22-41-05001, https://rscf.ru/project/22-41-05001/.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,500 |
A Portal is Born
If you build it, will they come? The marketers at Global Entertainment Holding/Equities, Inc. hope so. As competition heats up in the online gambling biz, it's becoming increasingly difficult to round up enough bettors to pay the bills. A popular answer is to build online communities and/or useful sites to attract potential online bettors and hopefully funnel them into the pay sites.
Global Entertainment's stab at this technique is a new site, www.TheSportsDaily.com, where sports fans and bettors can get free, up-to-date sports information, along with links to online news and online betting sites, including VIPsports.com, the online sportsbook operated by Global subsidiary Interactive Gaming and Wagering.
To beef up the new site, Global has partnered with Predict It!, an innovative site that offers users an opportunity to share picks and earn cash, and DBC Sports, which provides much of the statistical content. The site features a news-feed, lines, odds, transactions, injury and match-up reports, game schedules and scoreboards.
"We created TheSportsDaily.com to offer online gaming enthusiasts a complete source for all the current information required to make educated wagers," said Steven Abboud, president and CEO of Global Entertainment Holdings. "There is no other site that gathers all of the relevant sports information on the Web and makes it available to users in one place. It's a one-stop resource for free wagering intelligence." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 668 |
Documenting Natural Phenomena
By Michael Kauffmann from the CNPS Blog
Kids need nature, and we as parents, educators, and caring adults, need to provide access to it for them. It's a simple statement, but one that has become harder and harder to achieve in the world of standardized tests, electronics, and organized sports.
To help kids get out and enjoy nature more often, CNPS worked with nature educator John (Jack) Muir Laws a few years ago to publish his nature journaling curriculum. The book guides kids through a combination of art, writing, and science-based activities.
For years, Jack has been developing his curriculum to engage students of all ages in sharpening their observational powers through sketching in the field. He has found that this combination of visual and kinesthetic learning reaches even students who had given up on their artistic abilities long ago.
More recently, the Language Arts component completed the experience. Jack began to work with Emily Breunig, an English and writing instructor, to incorporate exercises such as writing haikus, creating narrative stories, and formulating hypotheses to complement the outdoor observational activities.
This interdisciplinary combination of art, science, writing, and observation exemplifies the California Native Plant Society's goals in creating educational programs: to engage students of all ages in the incredible natural world of California, to inspire them to keen observations of the wild places in their own backyards, and to foster a desire to protect these unique habitats.
Opening the World Through Nature Journaling has been available for seven years and in that time has guided children and adults throughout California in connecting with their natural surroundings. In classrooms, parks, and vacant lots around the state, more and more people are taking up their sketchbooks and letting nature journaling guide their appreciation of the outdoors. As Beverly Black, a fourth grade teacher in the San Francisco East Bay Area, reports, "Since I've been sharing nature observation through journaling with my students, I've noticed that they are much more in tune with their surroundings. They're seeing more, asking more questions, and I think appreciating the natural world more. Nature journaling has been rewarding to me personally, and I hope that I can encourage my students to make it a lifelong passion as well."
May these activities bring the same passion to you and the children you love! Bring it along on your summer adventures.
—Download this FREE curriculum
Chapter Blog RSS
Source: https://grownatives.cnps.org/2017/06/30/documenting-natural-phenomena/
Tagged: journaling
Educational, Repost
Annual Wildflower Tests Challenge Conventional Wisdom
By Lee Gordon, CNPS-San Diego Propagation Committee, from CNPS Blog
Conventional wisdom says that the best time to sow annual wildflower seeds is in the fall, just before the rains, and that seeds should be covered with a thin layer of soil to protect them from predation. This conventional wisdom may be wrong. Tests in Scripps Ranch and Poway (San Diego County) suggest that it is better to sow wildflower seeds months in advance of the fall rain, and that covering seeds may actually prevent them from germinating.
Tagged: wildflowers, seeds, propagation
Repost, Educational
Citizen Science with Mary Ellen Hannibal
By Michael Kauffmann from California Native Plant Society Blog
As human animals, we're drawn to the natural world. The impulse to observe, touch, and understand begins at birth. It's no wonder then that, throughout human history, laypeople—philosophers, gardeners, and vagabonds alike—have contributed to the most meaningful scientific knowledge we have. As Joseph Campbell said, myth is nature speaking, and the goal of human life is to align with nature.
Source: https://grownatives.cnps.org/2017/04/28/citizen-science-with-mary-ellen-hannibal/#more-2238
Tagged: citizen science
Educational, Gardening
General Statement on Ants
By Greg Rubin, CNPS-San Diego Garden Native Committee member
Many people have been experiencing problems with many native species, such as Ceanothus, manzanita, mallow-like plants, and mounding perennials. One of the primary causes, surprisingly, appears to be invasion by Argentine ants! The increasingly hot, monsoonal weather of recent years greatly promotes them. What these ants are doing is placing insects like scale and aphids all over the ROOTS, which literally suck the life out of the plant from below, often undetected to those without the experience to pick up on the subtle clues...
Argentine ants appear to be responsible for other horticultural threats. They plant innumerable types of weeds, including Veldt grass, spotted spurge, petty spurge, purslane, scarlet pimpernel, chickweed, brass buttons, and dandelion, as well as natives like Miner's lettuce and Purple three awn grass (they're not very picky). Often massive infestations of weeds can be associated with ant activity; they give themselves away by the weeds they plant.
An additional concern, of grave consequence, is that these same ants may be spreading pathogens like Phytophthora. This is especially significant as there are virtually no treatments available for these water molds (Sudden Oak Death is just one example of this devastating group of pathogens).
Tagged: Argentine ants
Watering Strategies from South Bay Botanic Garden
By Susan Krzywicki from grownatives.cnps.org
After so much debate about how to water native plant gardens, you'd think it had all been said. Let me add some tips and techniques from Eddie Munguia, who is the Horticultural Lab Technician at the South Bay Botanic Garden, located on the campus of Southwestern College in Chula Vista. Eddie installed a native garden over four years ago and one of the key objectives of the botanic garden is to do just this sort of closely observed research and analysis.
Source: https://grownatives.cnps.org/2016/12/22/watering-strategies-from-south-bay-botanic-garden/
Tagged: watering, gardening
Styles of the New California Garden
Peyton Ellas, Quercus Landscape Design
It used to be that a California native garden meant only a wild-looking, informal garden, or that you could add some California native plants among your existing non-native (exotic) plants in standard planting beds. California landscaping has gone through a phase where a dry creek had to be part of a native-plant garden, and I still add dry-creeks and similar water-theme features in some of my landscape designs, but it's no longer mandatory. We've seen wildflower meadows and native-grass-as-turf-substitute styles come and go.
The new California garden seems to be developing along the following basic styles. See if any of these fit with your yard or goals.
Source: https://grownatives.cnps.org/2015/06/19/styles-of-the-new-california-garden/
Tagged: landscaping
Native Moonshine
By Susan Kryzwicki for the California Native Plant Society Blog
Last year, I gave you my tried and true shortbread recipe. This year, I experimented with a new fun project: Tecate Cypress-infused bourbon.
Source: https://grownatives.cnps.org/2014/12/17/native-moonshine/
Tagged: recipes, tecate cypress, Hesperocyparis forbesii
California Phenology Project species profile for Honey Mesquite (Prosopis glandulosa)
From California Phenology Project
Source: https://www.usanpn.org/cpp/sites/www.usanpn.org.cpp/files/pdfs/PRGL-Aug12.pdf
Tagged: honey mesquite, prosopis glandulosa
Plant of the Month (February) : California polypody fern – Polypodium californicum
Article reposted from Mother Nature's Backyard
My, how it's rained recently; almost 9 inches in January alone. That's as much as we sometimes get in an entire year! The ground is really moist, and some of our perennials are looking the best they have in years. Coming into its own right now is the one native fern we currently grow – the California polypody, Polypodium californicum. The scientific name is pronounced pol-ee-PODE-ee-um ka-li-FOR-ni-kum.
Source: http://mother-natures-backyard.blogspot.com/2017/02/plant-of-month-february-california.html
Tagged: Polypodium californicum, native ferns
Creating a native wildflower meadow
Article reposted from California Native Plant Society Blog
Few things evoke magical memories like spring wildflowers. Whether it is a desire to recapture a serendipitous discovery of a color-laden flower field from our past, or simply re-living that scene from the "Wizard of Oz", nothing stirs our passion for nature like a beautiful field of flowers. California was once celebrated for its annual floral shows; unfortunately, these delightful events are becoming a thing of the past. The great Kate Sessions lamented that wildflowers were disappearing from San Diego's foothills by the early 1900s. Even her attempts to include wildflower displays at Balboa Park repeatedly failed. Why?
The answer is that European settlement in California altered our delicate ecology so profoundly that it was lost at all levels. Nothing is quite so fragile as a wildflower meadow. These annuals serve as pioneers that help re-establish ecology should a disturbance wipe out climax shrubs and trees. Being so low in lignin, they disappear after dying, returning all of their nutrients back to the ecology. Additionally, they fill holes not occupied by shrubby plants, and persist in places inhospitable to anything with deep roots, such as in the shallow soils of true native grasslands. What the Europeans brought were non-native weeds: competitive plants unhindered by native bio-controls while putting all of their life energy into reproducing themselves. These non-native seed banks now reach 10-100,000 dormant seeds per cubic foot! The wildflowers never stood a chance.
Seed was spread to create overlapping drifts of color, which is essential to maintain drama in such a large planting of wildflowers. Photo by Greg Rubin.
Fortunately, knowledge is the best tool, and we have ways to turn back the clock. Eliminating weeds is foremost. It is usually not sufficient to clear a space and drop seed, as Ms. Sessions learned. Instead, the seed bank must be addressed, either through repeated watering and killing of emerged weeds, or the use of chemicals called pre-emergents that kill seed in the soil when watered in. This must be done months in advance of planting. Solarization with clear plastic can also be used, but the effect is usually temporary. After treatment, seeds can be spread and either gently raked in or covered lightly in decomposed granite (this avoids disturbance and deters birds). Wildflower seeds can be purchased at most garden centers and online; however, be sure that the word "NATIVE" is somewhere in the title, and that the species are native to your locale, or you will end up with a mix of weedy introduced flowers, the worst being Alyssum. (ED note: You can check the local appropriateness of your chosen seed mix ingredients by entering your address in CNPS's Calscape app, available at calscape.org or on CNPS's homepage.) Common native mixes include California Poppies, Lupines, Goldfields, Desert Bluebells, Gilia, Baby Blue Eyes, Tidy Tips, and Farewell-to-Spring. You can add Owls Clover, Five Spot, and Thistle Sage if available. Keep the plot lightly moist until germination, continue watering twice a week if rainfall is lacking, and CONTROL WEEDS! The outcome will be thrilling!
CNPS Member Greg Rubin is the founder and owner of California's Own Landscape Design, Inc. (www.calown.com) and a popular speaker. A specialist in the use of native plants in the landscape, he is responsible for over 700 native landscapes in San Diego County and co-authored (with Lucy Warren) two books on native landscaping: The California Native Landscape: the Homeowner's Design Guide to Restoring its Beauty and Balance and The Drought Defying California Garden: 230 Native Plants for a Lush, Low-water Landscape, both on Timber Press.
Source: https://grownatives.cnps.org/2016/09/26/creating-a-native-wildflower-meadow/
Tagged: wildflowers, gardening, seeds, planting | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,475 |
Olympic Peninsula Women's Fellowship
Creating unity among women across the Peninsula
Wow! What a weekend it was!
Inspirational Weekends
Inspirational Weekend 2018
Team members represent 5 different decades from 5 of the area churches:
Calvary Chapel Sequim, Dungeness Community Church, The Gateway Church of Sequim, King's Way Foursquare Church, and Sequim Community Church
TESTIMONIALS FROM SOME OF OUR TEAM MEMBERS
"I honestly had no intention of getting involved but . . . Jesus changed my heart. I love 'vision casting' with women and listening to Jesus together." ~ Nancy S
"God touched my heart four years ago to begin praying for unity among the churches." ~ Branette
"I look forward to joining the other women the Lord has called to be part of this amazing, God glorifying mission of unification of the saints on the peninsula." ~ Eileen Damian, new team member
To be a catalyst for unifying and encouraging Christian women on the Olympic Peninsula through Biblically-based annual events.
Order OPWF Logo Products
We do not stock inventory of items with our logo; however, you may purchase anything from the Lands' End Business store and have the OPWF logo applied (for a fee of $5.95-8.95, depending on size). From time to time, Lands' End runs a special with free logo application. To shop at their store, click here.
OPWF is a non-denominational evangelical organization. Our leadership team consists of women from many different Bible-believing churches in the Olympic Peninsula to ensure that no one church owns or directs the organization.
We are a registered 501(c)(3) organization. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,960 |
package org.apache.guacamole.auth.duo;
import com.google.inject.Inject;
import java.util.Collections;
import javax.servlet.http.HttpServletRequest;
import org.apache.guacamole.GuacamoleException;
import org.apache.guacamole.auth.duo.api.DuoService;
import org.apache.guacamole.auth.duo.conf.ConfigurationService;
import org.apache.guacamole.auth.duo.form.DuoSignedResponseField;
import org.apache.guacamole.form.Field;
import org.apache.guacamole.language.TranslatableGuacamoleClientException;
import org.apache.guacamole.language.TranslatableGuacamoleInsufficientCredentialsException;
import org.apache.guacamole.net.auth.AuthenticatedUser;
import org.apache.guacamole.net.auth.Credentials;
import org.apache.guacamole.net.auth.credentials.CredentialsInfo;
/**
* Service for verifying the identity of a user against Duo.
*/
public class UserVerificationService {
/**
* Service for retrieving Duo configuration information.
*/
@Inject
private ConfigurationService confService;
/**
* Service for verifying users against Duo.
*/
@Inject
private DuoService duoService;
/**
* Verifies the identity of the given user via the Duo multi-factor
* authentication service. If a signed response from Duo has not already
* been provided, a signed response from Duo is requested in the
* form of additional expected credentials. Any provided signed response
* is cryptographically verified. If no signed response is present, or the
* signed response is invalid, an exception is thrown.
*
* @param authenticatedUser
* The user whose identity should be verified against Duo.
*
* @throws GuacamoleException
* If required Duo-specific configuration options are missing or
* malformed, or if the user's identity cannot be verified.
*/
public void verifyAuthenticatedUser(AuthenticatedUser authenticatedUser)
throws GuacamoleException {
// Pull the original HTTP request used to authenticate
Credentials credentials = authenticatedUser.getCredentials();
HttpServletRequest request = credentials.getRequest();
// Ignore anonymous users
if (authenticatedUser.getIdentifier().equals(AuthenticatedUser.ANONYMOUS_IDENTIFIER))
return;
// Retrieve signed Duo response from request
String signedResponse = request.getParameter(DuoSignedResponseField.PARAMETER_NAME);
// If no signed response, request one
if (signedResponse == null) {
// Create field which requests a signed response from Duo that
// verifies the identity of the given user via the configured
// Duo API endpoint
Field signedResponseField = new DuoSignedResponseField(
confService.getAPIHostname(),
duoService.createSignedRequest(authenticatedUser));
// Create an overall description of the additional credentials
// required to verify identity
CredentialsInfo expectedCredentials = new CredentialsInfo(
Collections.singletonList(signedResponseField));
// Request additional credentials
throw new TranslatableGuacamoleInsufficientCredentialsException(
"Verification using Duo is required before authentication "
+ "can continue.", "LOGIN.INFO_DUO_AUTH_REQUIRED",
expectedCredentials);
}
// If signed response does not verify this user's identity, abort auth
if (!duoService.isValidSignedResponse(authenticatedUser, signedResponse))
throw new TranslatableGuacamoleClientException("Provided Duo "
+ "validation code is incorrect.",
"LOGIN.INFO_DUO_VALIDATION_CODE_INCORRECT");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,072 |
{"url":"https:\/\/www.tutorialspoint.com\/difference-between-structure-and-union-in-c-program","text":"# Difference between Structure and Union in C Program\n\nCServer Side ProgrammingProgramming\n\nIn C we have container for both i.e. for same type data and multiple type data. For storage of data of same type C provides concept of Array which stores data variables of same type while for storing data of different type C has concept of structure and union that can store data variable of different type as well.\n\nSince both Structure and Union can hold different type of data in them but now on the basis of internal implementation we can find several differences in both of these containers.\n\nFollowing are the important differences between Structure and Union.\n\nSr. No.KeyStructureUnion\n1DefinitionStructure is the container defined in C to store data variables of different type and also supports for the user defined variables storage.On other hand Union is also similar kind of container in C which can also holds the different type of variables along with the user defined variables.\n2Internal implementationStructure in C is internally implemented as that there is separate memory location is allotted to each input memberWhile in case Union memory is allocated only to one member having largest size among all other input variables and the same location is being get shared among all of these.\n3SyntaxSyntax of declare a Structure in C is as follow :\nstruct struct_name{\ntype element1;\ntype element2;\n.\n.\n} variable1, variable2, ...;\nOn other syntax of declare a Union in C is as follow:\nunion u_name{\ntype element1;\ntype element2;\n.\n.\n} variable1, variable2, ...;\n4SizeAs mentioned in definition Structure do not have shared location for its members so size of Structure is equal or greater than the sum of size of all the data members.On other hand Union does not have separate location for each of its member so its size or equal to the size of largest member among all data members.\n5Value storageAs mentioned above in case of Structure there is specific memory location for each input data member and hence it can store multiple values of the different members.While in case of Union there is only one shared memory allocation for all input data members so it stores a single value at a time for all members.\n6InitializationIn Structure multiple members can be can be initializing at same time.On other hand in case of Union only the first member can get initialize at a time.\nPublished on 10-Jan-2020 10:17:01","date":"2021-06-23 21:41:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.194528266787529, \"perplexity\": 1242.1171167331047}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488540235.72\/warc\/CC-MAIN-20210623195636-20210623225636-00411.warc.gz\"}"} | null | null |
Manchester United vs Southampton Premier League Match | My Hack Stuff
Manchester United vs Southampton Premier League Match
Manchester United vs Southampton Premier League Match Preview or Key Statistics 📊
Manchester United are unbeaten in their last nine Premier League meetings with Southampton (W4 D5), though are looking to complete their first league double over them since their title winning campaign of 2012-13.
Having lost 13 of their first 14 Premier League away games against Man Utd (D1), Southampton have lost just two of their last seven visits to Old Trafford (W2 D3).
Manchester United have come from behind to win 10 different Premier League games against Southampton, including a 3-2 victory in the reverse fixture at St Mary's this season. No side has come from behind to beat another more often in the competition's history (Man Utd also 10 vs Newcastle).
Of all teams to have played at least 10 Premier League games in a specific month, no-one has a higher win rate than Manchester United do in February, with the Red Devils winning 68 of their 100 February games in the competition.
Manchester United have lost four of their 10 Premier League home games so far this season (W4 D2) – it's already the third highest number of defeats the Red Devils have suffered in a single Premier League campaign, after 2001-02 (6) and 2013-14 (7).
Southampton have failed to score in each of their last two Premier League away games, having only failed to find the net in two of their previous 25 on the road. Saints last went three without scoring in October 2018 under Mark Hughes.
Manchester United have drawn four Premier League matches goalless this season – in only three previous seasons have they had more: five in 2015/16 and six in both 2004/05 and 2016/17.
Southampton have lost their last three Premier League games, more than they had in their previous 15 (W8 D5 L2). They last lost four in a row in the competition in April 2018.
Southampton midfielder James Ward-Prowse's direct free-kick goal against Manchester United in the November meeting between the sides was the eighth direct free-kick scored in this Premier League fixture, the joint-most of any fixture in Premier League history along with Liverpool versus Man Utd.
Manchester United striker Edinson Cavani came off the bench to score twice and assist once against Southampton in their 3-2 win in November. The last Red Devils player to score 2+ goals in consecutive Premier League appearances against an opponent was Cristiano Ronaldo versus West Ham in October 2008.
Featured Football Manchester United Premier League Sports Trending
My Hack Stuff: Manchester United vs Southampton Premier League Match
Manchester United are unbeaten in their last nine Premier League meetings with Southampton (W4 D5), though are looking to complete their first league
https://www.myhackstuff.com/2021/02/manchester-united-vs-southampton.html | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,148 |
zkLedger: privacy-preserving auditing for distributed ledgers
May 31, 2018 ~ Adrian Colyer
zkLedger: privacy-preserving auditing for distributed ledgers Narula et al., NSDI'18
Somewhat similarly to Solidus that we looked at late last year, zkLedger (presumably this stands for zero-knowledge Ledger) provides transaction privacy for participants in a permissioned blockchain setting. zkLedger also has an extra trick up its sleeve: it provides rich and fully privacy-preserving auditing capabilities. Thus a number of financial institutions can collectively use a blockchain-based settlement ledger, and an auditor can measure properties such as financial leverage, asset illiquidity, counter-party risk exposures, and market concentration, either for the system as a whole, or for individual participants. It provides a cryptographically verified level of transparency that's a step beyond anything we have today.
The goals of zkLedger are to hide the amounts, participants, and links between transactions while maintaining a verifiable transaction ledger, and for the Auditor to receive reliable answers to its queries. Specifically, zkLedger lets banks issue hidden transfer transactions which are still publicly verifiable by all other participants; every participant can confirm a transaction conserves assets and assets are only transferred with the spending bank's authority.
A zkLedger system comprises n banks and an auditor that verifies certain operational aspects of transactions performance by the basks. A depositor or set of depositors can also issue and withdraw assets from the system. Issuance and withdrawal of assets are global public events.
The main action takes place when banks exchange assets by creating transfer transactions. A transfer moves v shares of some asset t to a given recipient bank (or banks). Agreements to transfer are arranged outside of the system, and settled on zkLedger. All transactions are submitted to a globally-ordered append-only ledger, which could be a blockchain.
Cryptographic building blocks
To protect their privacy, banks do not broadcast payment details in the clear. Instead, banks post commitments to the ledger, using Pedersen commitments. Pedersen commitments are perfectly hiding and computationally binding, they are also additively homomorphic, a fact which zkLedger makes extensive use of. (By additively homomorphic we mean that given commitments to values v1 and v2, there is an operation we can perform on those commitments to produce a commitment to the value v1 + v2. )
Every bank has a Schnorr signature keypair and distributes their public key to all other system participants.
Assertions about payment details are made based on non-interactive zero-knowledge proofs (NIZKs). In an NIZK scheme a prover can convince a verifier of some property about private data the prover holds, without revealing the private data itself. The binary string proof can be appended to the ledger and verified by any party of the system without interaction between the prover and the verifier.
In theory, NIZK proof systems exist for all properties in NP whereas the practical feasibility of NIZKs is highly dependent on the complexity of the property at hand… The design of zkLedger is carefully structured so that all NIZK proofs have particularly efficient constructions.
The zkLedger
At a high level zkLedger looks like this:
Banks maintain their own private state, and for efficiency a commitment cache which holds a rolling product of commitments by row and by asset so that it can quickly produce proofs and answer questions from auditors. The ledger itself has own entry (row) per transaction, and every row contains one column for each participating bank. (Banks can be added or removed by appending special signed transactions to the ledger).
Suppose bank A wants to transfer 100 shares of an asset to bank B. The transaction row conceptually contains a -100 entry in A's column, 100 in B's column, and zero in every other column. The values are not posted in the clear though, instead the column entries are Pedersen commitments for the respective amounts. Since there is no way for an outsider to tell the difference between a commitment to zero and any other value, both the transaction amounts and participants are protected.
Keeping values and participants private is a good start, but we also need to maintain overall integrity via the following invariants:
Transfer transactions cannot create or destroy assets
The spending bank must give consent to the transfer, and must own enough of the particular asset to execute the transaction
The first invariant is upheld via a proof of balance, and the second invariant is upheld using a proof of assets.
For proof of balance it suffices to show that the values in a given row sum to zero. If the the prover chooses the random inputs r to the commitments such that all of the r values also sum to zero, then a verifier can confirm that the the committed values all sum to zero by showing that the product of the commitments is 1.
A common approach to showing proof-of-assets is to use Unspent Transaction Objects (UTXOs). In a system that doesn't use zk-SNARKs though, this leaks the transaction graph. zk-SNARKs require a trusted third party for setup, which zkLedger wants to avoid: "the consequences of incorrect or compromised setup are potentially disastrous…"
In zkLedger, a bank proves it has assets by creating a commitment to the sum of the value for the asset in its column, including this transaction. If the sum is greater than or equal to 0, then the bank has the assets to transfer. Note that this is true since the bank's column represents all the assets it has received or spent, and the Pedersen commitments can be homomorphically added in columns as well as in rows.
In addition, in its own entry (where the value is negative), a bank includes proof of the knowledge of its secret key as a proof of authorisation. Thus we have a disjunctive proof – either the committed value for an entry is greater than or equal to zero, or the creator of the transaction knows the secret key for the entry.
There's one more issue we still need to consider: commitments rely on modulus. If we're using modulus N, we need to make sure that committed values are within 0..N-1. Range proofs are used to show that values are within range, and zkLedger supports asset value amounts up to a trillion. Now the only thing I can really tell you about range proofs is that they're the most expensive part of generating the transaction and if we're not careful we need two of them: one for the commitment value and one for the sum of assets in the column. With a level of indirection zkLedger manages to get this back down to just one range proof per transaction.
The auditor can ask a query of a Bank, such as "How many Euros did you hold at time t?," and the bank responds with an answer and a proof that the answer is consistent with the transactions on the ledger. The auditor can multiply commitments in the bank's column for Euros, and verify the proof and answer with the total. Given the table construction, the auditor knows that they are seeing every asset transfer – there is no way for a bank to 'hide' assets on the ledger.
Given that every bank is affected by every transaction (because each row contains a commitment for every bank, even if to the value zero), each bank needs to be able to total and prove all of the commitments in its column. To do this, the bank needs to know the random input used for each of those commitments, otherwise it won't be able to open up commitments to the auditor. To meet this requirement, the spending bank is required to include a publicly verifiable Token in every entry, which is based on a combination of the bank's public key and the random input. The token construction enables the bank to show that the asset total is correct without actually needing to know the random input (details in §4.2 of the paper). Alongside the token, we also need a proof of consistency that the same random input was used both in construction of the token and in forming the value commitment.
Through the use of sums, means, ratios, variance, co-variance, and standard deviation, an auditor in zkLedger can determine the following, among other measurements: leverage ratios (how much of an asset a bank has on its books compared to other holdings); concentration (using a measure called the Herfindahl-Hirschman Index – HHI – to measure how competitive an industry is); real-timet price indexes.
Sums are supported via the additive structure of Pedersen commitments. For everything else there is map/reduce. Take as an example an auditor that wants to calculate the mean transaction size for a given bank and asset. A commitment to the total value is obtained by summing the column, but we don't know what the denominator should be, because we don't know which entries are actually commitments to zero. Map/reduce solves this: in the map step the bank produces new commitments per row indicating whether or not the bank was involved in the transaction (1 if the bank is involved, zero otherwise). In the reduce step these commitments are summed and the result is sent to the auditor along with the corresponding proofs. More complex queries may require multiple map and reduce computations (see the example in §5 of the paper for computing variance).
For a transfer transaction, each entry in the row contains:
A Pedersen commitment to the value being transferred
An audit token so that audit requests can be answered without knowing the random input to the commitment
A proof -of-balance
A proof-of-assets
A proof-of-consistency between tokens and commitments
Banks can also add additional metadata – either encrypted or in plaintext.
A Go prototype of zkLedger shows that it is possible to create the needed proofs in milliseconds.
However, the cost of verifying transactions increases quadratically with the number of banks, and all transactions must be strictly serialised. Banks can verify transaction in parallel, so the time to process transactions increases linearly.
(Enlarge)
With 10 banks, we're already down to around 2 transactions per second. "We are optimistic that a faster range proof implementation will directly improve performance." Realistically though, it looks like with the current state-of-the-art we're limited to fairly low volume markets with limited numbers of participants.
Using the commitment cache (online auditor below), auditing time is roughly constant. Without it (offline auditor) audit time is linear in the number of transactions in the ledger.
zkLedger is the first distributed ledger system to provide strong transaction privacy, public verifiability, and complete, provably correct auditing. zkLedger supports a rich set of auditing queries which are useful to measure the financial health of a market.
Posted in Uncategorized Blockchain
< Previous Towards a design philosophy for interoperable blockchain systems
Next > ForkBase: an efficient storage engine for blockchain and forkable applications | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,535 |
Product Material: Terminal made of high quality copper, good conductivity, high quality ROHS-PVC jacket made of durable anti-aging.
Product Features: Good quality, no crack after cracking open, -40 degrees heat to 150 degrees normal use.
Copper terminal HUP discount merchandise, a limited number of while supplies last; Large price is better. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,235 |
***
### 目 录
##### 测试环境
1. 服务端测试环境
2. 客户端测试环境
##### 测试结果
1. 多进程多角色同时战斗场景(最多486个角色)
2. 多进程多角色同时行走场景(最多800个角色)
3. 多进程多角色同时战斗/行走场景(最多558个角色)
***
### 测试环境
##### 1.1 服务端测试环境
<table class="table table-bordered table-striped table-condensed">
<th width="15%">服务</th>
<th width="30%">机器</th>
<th width="55%">硬件配置</th>
<tr>
<td>GameServer<br>
&<br>
WebServer
</td>
<td>pomelo3.server.163.org</td>
<td>
云主机<br>
1CPU 8核心<br>
CPU型号 GenuineIntel QEMU Virtual CPU version 1.1.2@2.0GHz<br>
16G 内存<br>
1网卡<br>
linux/64位 OS<br>
</td>
</tr>
</table>
###### 说明:
* 服务端采用Node.js开发, 磁盘IO主要为log写入, 所有game-server服务器均配置toobusy(MAXLAG为默认的70ms), 出现瓶颈的部分在CPU.
##### 1.2 客户端测试环境
<table class="table table-bordered table-striped table-condensed">
<th width="15%">服务</th>
<th width="30%">机器</th>
<th width="55%">硬件配置</th>
<tr>
<td>Clients</td>
<td>pomelo16~25.server.163.org</td>
<td>
云主机<br>
1CPU 1核心<br>
CPU型号 GenuineIntel Westmere E56xx/L56xx/X56xx (Nehalem-C)@2.6GHz<br>
1G 内存<br>
1网卡<br>
linux/64位 OS<br>
</td>
</tr>
</table>
###### 说明:
* 客户端是全部走内网的。
***
### 测试结果
##### 1. 多进程多角色同时战斗场景(最多486个角色同时处于area3场景)
* 9台云主机上各启动1个客户端进程, 每个客户端进程上运行1个代理, 每个代理负责55个角色, 这样会有495个角色登录游戏. 9个客户端代理并发, 每2秒发起一次角色登录操作, 最终会有486个角色同时处于area3场景, 登录成功的角色每隔2~5秒发起一次攻击(攻击对象为附近的其他角色或者怪物)或者捡道具操作. 攻击的成功率为98.29%. 如下图:
* 每个角色的出生点随机分布于15个点上, 如下图:
* 486个角色都登录成功并开始发起攻击时, 服务器和pomelo16客户端的top:
* 测试完毕时服务器和pomelo16客户端的top:
* 测试过程中服务器状态:
***
##### 2. 多进程多角色同时移动场景(最多800个角色同时处于area3场景)
* 9台云主机上各启动1个客户端进程, 每个客户端进程上运行1个代理, 每个代理负责90个角色, 这样会有810个角色登录游戏. 9个客户端代理并发, 每2秒发起一次角色登录操作, 最终会有800个角色同时处于area3场景, 登录成功的角色每隔2~5秒发起一次移动操作. 移动的成功率为99.95%. 如下图:
* 这800个角色随机出生于地图上的15个点. 每一个角色的移动操作都是有规律的, 8次移动为一轮动作循环, 每次的步长为随机值(1~walkSpeed), 8次移动后该角色大致可以回到起点. 这么做的目的是尽量将角色限定在其出生点附近, 以使得所有角色尽量均匀分布于整张地图. 如下图:
* 800个角色都登录成功并开始移动时, 服务器和pomelo16客户端的top:
* 测试完毕时服务器和pomelo16客户端的top:
* 测试过程中服务器状态:
***
##### 3. 多进程多角色同时战斗/行走场景(最多558个角色)
* 9台云主机上各启动1个客户端进程, 每个客户端进程上运行1个代理, 每个代理负责63个角色, 这样会有567个角色登录游戏. 9个客户端代理并发, 每2秒发起一次角色登录操作, 最终会有558个角色同时处于area3场景, 登录成功的角色每隔2~5秒发起一次攻击(50%)或者移动(50%)操作. 攻击的成功率为99.50%. 移动的成功率为99.57%. 如下图:
* 这558个角色随机出生于地图上的15个点. 角色攻击时的对象为附近的其他角色或者怪物. 角色的移动操作都是有规律的, 8次移动为一轮动作循环, 每次的步长为随机值(1~walkSpeed), 8次移动后该角色大致可以回到起点. 这么做的目的是尽量将角色限定在其出生点附近, 以使得所有角色尽量均匀分布于整张地图. 如下图:
* 558个角色都登录成功并开始攻击/移动时, 服务器和pomelo16客户端的top:
* 测试完毕时服务器和pomelo16客户端的top:
* 测试过程中服务器状态:
***
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YouDao:
# Pomelo-performance test (take LordOfPomelo as test object)
***
### Contents
##### Test environment
1. The server test environment
2. The client test environment
##### Test results
1. Multiple roles in multiprocess fight at the same time (up to 486 characters)
2. Multiple roles in multiprocess walk at the same time (up to 800 characters)
3. Multiple roles in multiprocess fight/walk at the same time (up to 558 characters)
***
### Test environment
##### 1.1 server test environment
<table class="table table-bordered table-striped table-condensed">
<th width="15%">Service</th>
<th width="30%">Machine</th>
<th width="55%">Hardware configuration</th>
<tr>
<td>GameServer<br>
&<br>
WebServer
</td>
<td>pomelo3.server.163.org</td>
<td>
cloud host<br>
1 CPU, 8 cores<br>
CPU Type: GenuineIntel QEMU Virtual CPU version 1.1.2@2.0GHz<br>
16G memory<br>
1 network card<br>
linux/64bit OS<br>
</td>
</tr>
</table>
###### Description:
* Server side is developed using Node.js. Disk I/O is mainly used for the log. All game-server are configured toobusy (MAXLAG is the default 70ms). The bottlenecks is the CPU's computing power.
##### 1.2 client test environment
<table class="table table-bordered table-striped table-condensed">
<th width="15%">Service</th>
<th width="30%">Machine</th>
<th width="55%">Hardware configuration</th>
<tr>
<td>Clients</td>
<td>pomelo16~25.server.163.org</td>
<td>
cloud host<br>
1 CPU, 1 core<br>
CPU Type: GenuineIntel Westmere E56xx/L56xx/X56xx (Nehalem-C)@2.6GHz<br>
1G memory<br>
1 network card<br>
linux/64bit OS<br>
</td>
</tr>
</table>
###### Description:
* The clients are all within intranet.
***
### Test results
##### 1. Multiple roles in multiprocess fight at the same time (up to 486 characters in area3 scene at the same time)
* We launch a client process on each of 9 cloud hosts. A proxy is running on each client process. Each agent is responsible for 55 characters. So there will be 495 characters login game. 9 client proxies act concurrently. Each proxy launches a role login operation every 2 seconds. Eventually, there will be 486 roles in area3 scene at the same time. The role which logined successfully launches an attack (the target is other role or monster nearby) or pick up item every 2~5 seconds. The success rate of attack is 98.29%. As shown below:
* Each character spawn randomly distributed in 15 points, as shown below:
* After logined successfully, 486 characters began to attack targets. Server and pomelo16's top is:
* When the test is completed, server and pomelo16's top is:
* In the process of the testing, server's status is:
***
##### 2. Multiple roles in multiprocess walk at the same time (up to 800 characters in area3 scene at the same time)
* We launch a client process on each of 9 cloud hosts. A proxy is running on each client process. Each agent is responsible for 90 characters. So there will be 810 characters login game. 9 client proxies act concurrently. Each proxy launches a role login operation every 2 seconds. Eventually, there will be 800 roles in area3 scene at the same time. The role which logined successfully launches an move every 2~5 seconds. The success rate of move is 99.95%. As shown below:
* 800 characters spawn randomly distributed in 15 points on the map. Every character's movement operating is regular. 8 times movements are an action cycle. Each step is a random value(1~walkSpeed). After 8 times movements, the character can be roughly back to the beginning point. The purpose of doing so is to try to define a character in the vicinity of its birth. So that all characters were distributed evenly over the whole map as far as possible. As shown below:
* After logined successfully, 800 characters began to move targets. Server and pomelo16's top is:
* When the test is completed, server and pomelo16's top is:
* In the process of the testing, server's status is:
***
##### 3. Multiple roles in multiprocess fight/walk at the same time (up to 558 characters in area3 scene at the same time)
* We launch a client process on each of 9 cloud hosts. A proxy is running on each client process. Each agent is responsible for 63 characters. So there will be 567 characters login game. 9 client proxies act concurrently. Each proxy launches a role login operation every 2 seconds. Eventually, there will be 558 roles in area3 scene at the same time. The role which logined successfully launches an attack(50%) or move(50%) every 2~5 seconds. The success rate of attack is 99.50%. The success rate of move is 99.57%. As shown below:
* 558 characters spawn randomly distributed in 15 points on the map. The target which is attacked is other role or monster nearby. Every character's movement operating is regular. 8 times movements are an action cycle. Each step is a random value(1~walkSpeed). After 8 times movements, the character can be roughly back to the beginning point. The purpose of doing so is to try to define a character in the vicinity of its birth. So that all characters were distributed evenly over the whole map as far as possible. As shown below:
* After logined successfully, 558 characters began to attack or move. Server and pomelo16's top is:
* When the test is completed, server and pomelo16's top is:
* In the process of the testing, server's status is:
***
| {
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Медуница лекарственная () — вид невысоких многолетних травянистых растений из рода Медуница семейства . У растений этого вида наблюдается нечастое среди цветковых растений явление изменения окраски венчика в процессе цветения: розовые в начале, к концу цветения они становятся синими.
Медуница лекарственная с древних времён использовалась в качестве лекарственного растения.
Культивируется как декоративное садовое растение. Медонос.
Распространение
Медуница лекарственная растёт в лесах Центральной и Восточной Европы, на лугах и в болотистых местах Британских островов. На территории России встречается в Калининградской области.
Биологическое описание
Растения этого вида — многолетние травянистые корневищные растения высотой до .
Стебель прямостоячий, покрыт волосками.
Листья длиной до , со светлыми пятнами. Пятна представляют собой сильно разрыхлённую ткань с большим количеством устьиц.
Цветок с двойным околоцветником. Венчик актиноморфный, пятидольный, в начале цветения розово-красный, постепенно становится синим. Растение начинает цвести в марте.
Тычинок пять. Столбик цельный, голый, с цельным головчатым рыльцем. Завязь верхняя. Опыление происходит с помощью насекомых; нектар в цветке защищён от всех неэффективных опылителей длинной трубкой венчика.
В качестве приспособления для перекрёстного опыления для всех видов медуницы характерна диморфная гетеростилия: у разных растений может быть различная длина столбиков и тычиночных нитей (у одних растений столбики короче тычинок, у других — тычинки короче столбиков). Как самоопыление, так и переопыление между растениями одного морфологического типа почти не даёт семян, эффективно переопыление только между особями с разным типом цветков. Опыляются преимущественно мухами-жужжалами.
Плод — орешек с мясистыми придатками (ариллусами), привлекающими муравьёв.
Химический состав
В медунице лекарственной содержатся сапонины и танины.
Применение
Значительный и один из самых ранних медоносов (причём красные цветки выделяют нектара намного больше синих).
Применение в кулинарии
Листья медуницы лекарственной добавляют в салаты и супы, они придают блюдам привкус вермута.
Медицинское применение
Этот вид медуницы с древних времён использовался в качестве лекарственного растения для лечения лёгочных заболеваний. Такое использование было связано и с так называемой , долгое время применявшейся в фитотерапии (согласно этой доктрине внешний вид растения определяет его полезные свойства, а поскольку листья медуницы лекарственной, покрытые светлыми пятнами, в некоторой степени похожи на человеческие лёгкие, растение использовали для лечения именно лёгочных заболеваний), и с реальными медицинскими свойствами растения (медуница лекарственная содержит сапонины и танины, которые действуют как отхаркивающие и смягчающие средства при инфекциях слизистых оболочек). Сейчас в фитотерапии медуница лекарственная не используется, поскольку есть другие, существенно более эффективные растения с аналогичным действием.
Применение в садоводстве
Медуницу лекарственную выращивают в садах ради её декоративных пятнистых листьев и раннего цветения. Выведено немало декоративных сортов. Один из самых известных сортов — : растение с крупными пятнистыми листьями и белыми цветками, не меняющими окраску в процессе цветения.
Агротехника
Медуницу лучше выращивать в полутени, в прохладном месте (на сухих солнечных местах растение нередко поражается мучнистой росой). Желательно, чтобы почва была влажной и богатой гумусом. Размножение — семенами, черенками или делением.
Примечания
Литература
Ссылки
Pulmonaria officinalis на сайте USDA NRCS
Бурачниковые
Декоративные садовые растения
Лекарственные растения
Травы
Флора Евразии
Многолетние травы
Растения, впервые описанные в книге Species Plantarum | {
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{"url":"https:\/\/simple.wikipedia.org\/wiki\/Hexadecagon","text":"TypeRegular polygon\nEdges and vertices16\nSchl\u00e4fli symbol{16}, t{8}, tt{4}\nCoxeter diagram\nSymmetry groupDihedral (D16), order 2\u00d716\nInternal angle (degrees)157.5\u00b0\nDual polygonSelf\nPropertiesConvex, cyclic, equilateral, isogonal, isotoxal\n\nAn hexadecagon or hekkaidecagon is a shape with 16 sides and 16 corners.\n\nA regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. Its Schl\u00e4fli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.\n\n### Area\n\nThe area of a regular hexadecagon with edge length t is\n\n{\\displaystyle {\\begin{aligned}A=4t^{2}\\cot {\\frac {\\pi }{16}}=&4t^{2}\\left(1+{\\sqrt {2}}+{\\sqrt {4+2{\\sqrt {2}}}}\\right)\\\\=&4t^{2}({\\sqrt {2}}+1)({\\sqrt {4-2{\\sqrt {2}}}}+1).\\end{aligned}}}\n\nBecause the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius R by truncating Vi\u00e8te's formula:\n\n${\\displaystyle A=R^{2}\\cdot {\\frac {2}{1}}\\cdot {\\frac {2}{\\sqrt {2}}}\\cdot {\\frac {2}{\\sqrt {2+{\\sqrt {2}}}}}=4R^{2}{\\sqrt {2-{\\sqrt {2}}}}.}$\n\nSince the area of the circumcircle is ${\\displaystyle \\pi R^{2},}$ the regular hexadecagon fills approximately 97.45% of its circumcircle.\n\n## Dissection\n\nCoxeter states that every parallel-sided 2m-gon can be divided into m(m-1)\/2 rhombs. For the regular hexadecagon, m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petrie polygon projection of a 8-cube, with 28 of 1792 faces. [1] The list A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.\n\n{8}#{ } { \u200b83 }#{ } { \u200b85 }#{ }\nA regular skew hexadecagon is seen as zig-zagging edges of a octagonal antiprism, a octagrammic antiprism, and a octagrammic crossed-antiprism.\n\nA skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A skew zig-zag hexadecagon has vertices alternating between two parallel planes.\n\nA regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, [2+,16] symmetry, order 32. The octagrammic antiprism, s{2,16\/3} and octagrammic crossed-antiprism, s{2,16\/5} also have regular skew octagons.\n\n## In art\n\nThe hexadecagonal tower from Raphael's The Marriage of the Virgin\n\nIn the early 16th century, Raphael was the first to construct a perspective image of a regular hexadecagon: the tower in his painting The Marriage of the Virgin has 16 sides, elaborating on an eight-sided tower in a previous painting by Pietro Perugino.[2]\n\nA hexadecagrammic pattern from the Alhambra\n\nHexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra.[3]","date":"2020-03-29 19:37:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 3, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6539386510848999, \"perplexity\": 5816.821840057245}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370495413.19\/warc\/CC-MAIN-20200329171027-20200329201027-00496.warc.gz\"}"} | null | null |
CK Beckett founded in 1984 by 2 brothers and a lot of commitment started off in the general waste management trade. Several years later gained a highly regarded reputation for the outstanding level of customer service and punctuation. From this they moved into the power distribution equipment ? Removal & disposal focusing on distribution and power transformers. This is a very bespoke market, not many people in the UK can do what we do. Removing transformers safely from 2ton up to 700ton units. Over the course of the last 20+ years we have expanded the recycling centre by over 15 staff, 5 Wagons and additional land. With transformers comes transformer oil. 4 years a go we opened our very own oil recycling centre CK Beckett Oil Processing. Due to the increased demand and rigid environmental legislation, CK Beckett now hold a PPC license, which is a legal requirement when 10T or more of waste oil is stored on site. CK Beckett have invested in a brand new Oil Recycling and storage facility. This purpose built oil recycling plant is now fully operational and is currently reprocessing in excess of 3,000,000 litres per annum of used transformer oil.
Carter James and Co, approved Auctioneer on the worlds leading commercial online Auctioneering Platforms. With a truly worldwide reach, get maximum value for your commercial assets, quickly! From a single asset to a full office/factory clearance, working with our partners we have the resources to act quickly and effectively throughout the UK 24/7. Contact us today for free, no obligation advice.
We are an established and respected Auction House specialising in Early Oak Furniture sales that include a rich variety of carvings, tapestries and paintings. We also hold Fine Furniture sales of mahogany and walnut furniture with bronzes, jewellery, silver & clocks. With only five or six sales per year we are able to concentrate on high quality sales that attract buyers from all over the world.We are situated in South Yorkshire and are able to collect across the UK and Europe and arrange delivery worldwide.We are a small firm with a dedicated team of enthusiastic, reliable and courteous people offering a unique service. | {
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Q: Don't match if character is present in javascript I'm trying to test if an input field contains a valid "slug" (a string that can only contain dashes and lowercase letters, used in URLs)
My issue is that the user can write a valid slug, but then they can write invalid characters after the valid ones, and it will still match.
My regex looks like this: /[a-z\-]+/
This should match: 'my-slug-is-valid'
But this should not match: 'my-SLUG-is not valid'
Yet when I test the regex against that last string, true is returned.
var re = /[a-z\-]+/,
str = 'my-SLUG-is not valid';
if (re.test(str) && str !== '') {
console.log('Valid');
}
Is it possible to return false if there are characters present that are not part of the char-class?
A: It looks like you are trying to match the entire string. In that case, add anchors to the regex:
re = /^[a-z\-]+$/
A: var re = /^[a-z-]+$/;
You need the ^ and $ to anchor the beginning and end of the string.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Wordpress & AJAX - Loading Internal Pages with AJAX prevents Javascript from Running on Internal Pages I am very sorry for the long title, but its an adequate representation of the problem.
I am attempting to use AJAX on my WordPress website to load inner pages. I have this functionality working, however, when loading inner pages, any Javascript/jQuery that is meant to load and be run on these inner pages is not running/loading.
There are two steps in order to see and then recreate the problem:
1st - Take a look at this page: http://www.trekradio.net/dev/schedule - this page contains a schedule - the days are tabs, also when you click on a schedule item, it opens a jQuery popup. Javascript is also used on the right sidebar, in the "support us" widget, the dropdown is JS based, and also, the twitter feed loads using the Twitter API script. As you can see, all scripting is present and working fine.
2nd - http://www.trekradio.net/dev/ - goto the main page, then click "schedule" on the main menu - it loads up the schedule page using AJAX, however, all scripts on the page are now missing or have not been executed.
I've tried a number of things to solve this - on the schedule page, several of the scripts are embedded into the page itself, I tried moving these into the header, hoping that, because the header is not reloaded, the scripts might run - they did not.
Also, I have attempted the solution mentioned here: Executing <script> inside <div> retrieved by AJAX - it talks about using a dynamic script pattern, however, I either lack the ability to get it working correctly, or, it does not work properly in the context of what I am trying to accomplish.
I also tried using the solutions suggested here: jquery html() strips out script tags however, again, I could not get it to work.
Here is the contents of my ajax.js which is powering the ajax functionality:
jQuery(document).ready(function($) {
var $mainContent = $("#ajax-content-container"),
siteUrl = "http://" + top.location.host.toString(),
url = '';
$(document).delegate("a[href^='"+siteUrl+"']:not([href*=/wp-admin/]):not([href*=/wp-login.php]):not([href$=/feed/])", "click", function() {
location.hash = this.pathname;
return false;
});
$("#searchform").submit(function(e) {
location.hash = '?s=' + $("#s").val();
e.preventDefault();
});
$(window).bind('hashchange', function(){
url = window.location.hash.substring(1);
if (!url) {
return;
}
url = url + " #main-content-container";
$mainContent.animate({opacity: "0.1"}).html('<p>Please wait...</>').load(url, function() {
$mainContent.animate({opacity: "1"});
});
});
$(window).trigger('hashchange');
});
From what I have found by Google-ing - many many folks have this type of problem with their AJAX websites, struggling to get scripts running on loaded pages, so I am hoping that folks here can help come up with a solution which, not only I can apply to my website, but others can use on theirs too.
Would appreciate if folks would include code with their answers or at least explain clearly how I can modify the code to accomplish what is needed. I am still learning jQuery/AJAX, so perhaps I am being stupid and overlooking a relativly simple solution, but well documented answers would be appreciated by, I suspect, the entire community as well as myself.
A: It's because you provided a selector expression as a part of your URL:
http://api.jquery.com/load/
Per the documentation:
However, in the following case, script blocks in the document being
loaded into #b are stripped out and not executed:
$('#b').load('article.html #target')
Your URL has the selector:
url = url + " #main-content-container";
A: I think I have have solved this for myself - I'm not sure if its the most elegant solution, however it seems to work.
Basically, I've taken all of the scripts that would load on my inner pages and put them into a file called "inner-scripts.js" and modified ajax.js to this:
jQuery(document).ready(function($) {
var $mainContent = $("#ajax-content-container"),
siteUrl = "http://" + top.location.host.toString(),
url = '';
$(document).delegate("a[href^='"+siteUrl+"']:not([href*=/wp-admin/]):not([href*=/wp-login.php]):not([href$=/feed/])", "click", function() {
location.hash = this.pathname;
return false;
});
$("#searchform").submit(function(e) {
location.hash = '?s=' + $("#s").val();
e.preventDefault();
});
$(window).bind('hashchange', function(){
url = window.location.hash.substring(1);
if (!url) {
return;
}
url = url + " #main-content-container";
$mainContent.animate({opacity: "1.0"}).html('<div id="loading">Loading the next page for you. Please stand by!</div>').load(url, function() {
$mainContent.animate({opacity: "1"});
$.getScript('http://www.trekradio.net/dev/wp-content/themes/tr2012/js/inner-scripts.js', function() {});
$.getScript('http://ui.jquery.com/latest/ui/effects.core.js', function() {});
$.getScript('http://twitter.com/javascripts/blogger.js', function() {});
$.getScript('https://api.twitter.com/1/statuses/user_timeline.json?screen_name=trekradio&include_rts=1&count=3&callback=twitterCallback2', function() {});
});
});
$(window).trigger('hashchange');
});
A: I know you've answered the question yourself, but here's another way of doing it.
The conventional way to make ajax requests in wordpress is to use the admin-ajax interface as documented here: http://codex.wordpress.org/AJAX_in_Plugins.
This allows you to attach a function to an ajax request, and the function executes more or less in the context of WordPress. So you could write something roughly like this in your functions.php:
add_action('wp_ajax_trekradio_get_ajax_page', 'trekradio_get_ajax_page');
add_action('wp_ajax_nopriv_trekradio_get_ajax_page', 'trekradio_get_ajax_page');
function trekradio_get_ajax_page() {
$page_id = $_POST['page_id']; // send this in the ajax request
if ( the page is a certain page that needs scripts ) {
print_relevant_script_tags();
}
print_page_content( $page_id );
die(); // you have to do this at the end
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,147 |
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| {
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\section{Introduction} \label{s:s1}
\IEEEPARstart{B}{ayesian} Networks (BNs) are a popular class of graphical models whose structure is represented by a DAG $\mathcal{G}$. BNs have been used in many applications such as economics,
finance, biology, etc \cite{Swanson1997, neil2005, needham2007, hyvarinen2010, dallakyan2020}.
In recent years the following two approaches have been evolved to learn the structure of the underlying DAG from data: Independence-based (also called constraint-based) methods \cite{spirtes1991, pearl2009} and score-based methods \cite{heckerman1995, Chickering2002, teyssier2005, loh2014}. Here, structure learning refers to recovering DAG from observational data.
Independence-based methods, such as the inductive causation (IC) \cite{pearl2009} and PC (Peter-Clark) \cite{spirtes1991} algorithm, utilize conditional independence tests to detect the existence of edges between each pair of variables. The method assumes that the distribution is Markovian and faithful with respect to the underlying DAG, where $\mathcal{P}$ is faithful to the DAG $\mathcal{G}$ if all conditional independencies in $\mathcal{P}$ are entailed in $\mathcal{G}$ and Markovian if the factorization property (\ref{eq:markov}) is satisfied.
In contrast, score-based methods measure the goodness of fit of different graphs over data by optimizing a score function with respect to the unknown (weighted) adjacency matrix $B$ with a combinatorial constraint that the graph is DAG. Then a search procedure is used to find the best graph. Commonly used search procedures include hill-climbing \cite{heckerman1995, tsamardinos2006}, forward-backward search \cite{Chickering2002}, dynamic, and integer programming \cite{silander2006, koivisto2006, jaakkola10, studeny2014,hemmecke2012}. Recently, \cite{zheng2018, zheng2020} proposed a fully continuous optimization for structure learning by introducing a novel characterization of acyclicity constraint.
Generally, the DAG search space is intractable for a large number of nodes $p$ and the task of finding a DAG is NP-hard \cite{Chickering2002}. Consequently, approximate methods have been proposed with additional assumptions such as bounded maximum indegree of the node \cite{cooper1992} or tree-like structures \cite{chow1968}. Alternatively, the ordering space (or the space of topological ordering) has been exploited for score-based methods \cite{teyssier2005, vandegeer2013,aragam2015, ye2019} where the topological ordering is considered as a parameter \cite{teyssier2005}.
The order-based search has two main advantages: the ordering space ($2^{O(p\log p)}$) is significantly smaller than the DAG search space ($2^{O(p^2)}$), and the existence of ordering guarantees satisfaction of the acyclicity constraint.
The recent Annealing on Regularized Cholesky Score (ARCS) algorithm in \cite{ye2019} is based on representing an ordering by the corresponding permutation matrix $P$, and then given the order, encoding the weighted adjacency matrix $B$ into the Cholesky factor $L$ of the inverse covariance matrix. ARCS optimizes a regularized likelihood score function to recover sparse DAG structure and utilizes simulated annealing (SA) to search over the permutation matrix space. In SA, using a pre-specified constant $m$ and a temperature schedule $\{T^{(i)}, i = 0, \dots, N\}$, in the $i$th iteration a new permutation matrix $P^{*}$ is proposed by flipping a fixed-length $m$ random interval in the current permutation $\hat P$, and checking whether to stay at the current $\hat P$ or move to the proposed $P^{*}$ with some probability.
Motivated by the ARCS two-step framework, we propose an order-based method for learning Gaussian DAGs by optimizing a non-convex regularized likelihood score function with the following distinct features and advantages:
First, we use a relaxation technique instead of the expensive search for a permutation matrix $P$ in the non-convex space of permutation matrices. More precisely, we project $P$ onto the Birkhoff polytope (the convex space of doubly stochastic matrices) and then find the ``closest'' permutation matrix to the optimal doubly stochastic matrix (See Figure~\ref{f:geom}).
Second, given $P$, we resort to the cyclic coordinatewise algorithm to recover the DAG structure entailed in the Cholesky factor $L$. We show that the optimization reduces to $p$ decoupled penalized regressions where each iteration of the cyclic coordinatewise algorithm has a closed form solution.
Third, we show consistency of our Cholesky factor estimator for the non-convex score function when the true permutation matrix is known. To the best of our knowledge, consistency results for the sparse Cholesky factor estimator were established only for convex problems \cite{yu2017, khare2016}.
The paper is organized as follows: Section 2 introduces background on Gaussian BNs and structural equation models (SEMs). In Section 3, we derive and discuss the form of the score function. In Section 4, we introduce our Relaxed Regularized Cholesky Factor (RRCF) framework. The analyses of the simulated and real macro-economic datasets are contained in Section 5. For the real data analysis, we apply RRCF to solve the price puzzle, a classic problem in the economics literature. Section 6 provides statistical consistency of our estimator, and we conclude with a discussion in Section 7.
\section{Bayesian Networks} \label{s:bn}
We start by introducing the following graphical concepts. If the graph $\mathcal{G}$ contains a directed edge from the node $k \rightarrow j$, then $k$ is a parent of its child $j$. We write $\Pi^{\mathcal{G}}_{j}$ for the set of all parents of a node $j$. If there exist a directed path $k \rightarrow \dots \rightarrow j$, then $k$ is an ancestor of its descendant $j$.
A \textit{Bayesian Network} is a directed acyclic graph $\mathcal{G}$ whose nodes represent random variables $X_1, \dots, X_p$. Then $\mathcal{G}$ encodes a set of conditional independencies and conditional probability distributions for each variable. The DAG $\mathcal{G} = (V,E)$ is characterized by the node set $V = \{1, \dots, p\}$ and the edge set $E = \{(i,j): i \in \Pi^{\mathcal{G}}_j\} \subset V \times V$. It is well-known that for a BN, the joint distribution factorizes as:
\begin{equation} \label{eq:markov}
P(X_1,\dots,X_p) = \prod_{j=1}^{p}P(X_j| \Pi^{\mathcal{G}}_{j})
\end{equation}
\subsection{Gaussian BN and Structural Equation Models}
It is known that a Gaussian BN can be equivalently represented by the linear SEM \cite{pearl2009}:
\begin{equation} \label{eq:sem}
X_j = \sum_{k \in \Pi^{\mathcal{G}}_j} \beta_{jk} X_k + \varepsilon_j, \; j = 1, \dots, p,
\end{equation}
where $\varepsilon_j \sim N(0, \omega^2_j)$ are mutually independent and independent of $\{X_k: k \in \Pi^{\mathcal{G}}_j \}$. Denoting $B = (\beta_{jk})$ with zeros along the diagonal, the vector representation of (\ref{eq:sem}) is
\begin{equation} \label{eq:vsem}
X = BX + \varepsilon,
\end{equation}
where $\varepsilon: = (\varepsilon_1, \dots, \varepsilon_p)^t$ and $X: = (X_1, \dots, X_p)^t$.
Thus, one can characterize the linear SEM $X \sim (B, \Omega)$ by the weighted adjacency matrix $B$ and the noise variance matrix $\Omega = \mbox{diag}(\omega^2_1, \dots, \omega^2_p)$. From (\ref{eq:vsem}), the inverse covariance matrix of $X \sim N_p(0, \Sigma)$ is $\Sigma^{-1} = (I-B)^t \Omega^{-1} (I-B)$,
and the edge set of the underlying DAG is equal to the support of the weighted adjacency matrix $B$; i.e., $E = \{(k,j): \beta_{jk} \neq 0\}$, which defines the structure of DAG $\mathcal{G}$. Consequently, $B$ should satisfy the acyclicity constraint so that $\mathcal{G}$ is indeed a DAG.
It is known that a DAG admits a topological ordering $\pi$, to which one may associate a $p \times p$ permutation matrix $P_{\pi}$ such that $P_{\pi}x = (x_{\pi(1)}, \dots, x_{\pi(p})$, for $x \in R^p$. The existence of a topological order leads to the permutation-similarity of $B$ to a strictly lower triangular matrix $B_{\pi} = P_{\pi}BP^t_{\pi}$ by permuting rows and columns of $B$, respectively \cite{bollen1989} (see Figure~\ref{fig:exdag} for the illustrative example). Therefore, the stringent acyclicity constraint on $B$ transforms into the constraint that $B_{\pi}$ is a strictly lower triangular matrix, then the linear SEM can be rewritten as
\begin{equation} \label{eq:pvsem}
P_{\pi}X = B_{\pi} P_{\pi}X + P_{\pi}\varepsilon,
\end{equation}
using the fact that $P^t_{\pi}P_{\pi} = I$. From (\ref{eq:pvsem}), the inverse covariance matrix can be expressed as
\begin{equation} \label{eq:pcov}
\Sigma^{-1}_{\pi} = (I - B_{\pi})^t \Omega^{-1}_{\pi} (I - B_{\pi}),
\end{equation}
where $\Omega_{\pi} = P_{\pi}\Omega P^t_{\pi}$.
Using~(\ref{eq:pvsem}) and (\ref{eq:pcov}) and defining $L_{\pi} = \Omega_{\pi}^{-1/2}(I - B_{\pi})$, the relationship between the Cholesky factor $L_{\pi}$ of the inverse covariance matrix $\Sigma^{-1}_{\pi}= L^t_{\pi}L_{\pi}$ and the matrix $B_{\pi}$ is
\begin{equation} \label{eq:lb}
\begin{split}
(L_{\pi})_{ij} &= -(B_{\pi})_{ij} / \sqrt{\omega_{j}},\;\mbox{and}\;\\
(L_{\pi})_{ij} &= 0 \iff (B_{\pi})_{ij} = 0\; \mbox{for every}\; i \geq j
\end{split}
\end{equation}
Hence, $L_{\pi}$ preserves the DAG structure of $B_{\pi}$; i.e., non-zero elements in $L_{\pi}$ correspond to directed edges in DAG $\mathcal{G}$.
\begin{figure}[t]
\centering
\includegraphics[width= 9cm, height = 5cm]{bnexample}
\caption{Illustration of DAG $\mathcal{G}$, corresponding coefficient matrix $B$, permutation matrix $P$, and permuted strictly lower triangular matrix $B_{\pi}$.}
\label{fig:exdag}
\end{figure}
\section{The Score Function} \label{s:rrcf}
In this section, given data from the Gaussian BN (or SEM), we derive the form of the score function used to recover the underlying DAG structure. A natural choice for such function is the log-likelihood function, which will be used for the estimation of the permutation and Cholesky factor matrices.
We assume that each row of data matrix $\mathbf{X} = (X_1, \dots, X_p) \in R^{n \times p}$ is an i.i.d observation from (\ref{eq:sem}). Using reformulation (\ref{eq:pvsem}),
\begin{equation} \label{eq:pvsemmat}
\mathbf{X}P_{\pi}^t = \mathbf{X} P_{\pi}^t B_{\pi}^t + \mathbf{E}P_{\pi}^t,
\end{equation}
where each row of $\mathbf{E}$ is an i.i.d $N_p(0, \Omega)$ vector. Thus, each row of $\mathbf{X}P_{\pi}^t$ is, again, an i.i.d from $N_p(0, \Sigma_{\pi})$, and the negative log-likelihood for (\ref{eq:pvsemmat}) is:
\begin{equation} \label{eq:loglik}
\begin{split}
\ell (B_{\pi}, \Omega_{\pi}, P_{\pi} | \mathbf{X}) &= \frac{1}{2} \mbox{tr} \Big (P_{\pi} \mathbf{X}^t\mathbf{X}P_{\pi}^t(I - B_{\pi})^t \Omega_{\pi}^{-1}(I - B_{\pi}) \Big ) \\
& + \frac{n}{2} \log |\Omega_{\pi}|,
\end{split}
\end{equation}
using the facts that $\Sigma_{\pi} = \mbox{cov}(P_{\pi}X) = P_{\pi}\Sigma P^t_{\pi} = (I - B_{\pi})^{-1} \Omega_{\pi} (I - B_{\pi})^{-t}$, and $B_{\pi}$ is a strictly lower triangular matrix.
From now on, whenever there is no confusion, we drop the subscript $\pi$ from $P_{\pi}, B_{\pi}, \Omega_{\pi}$ and $\Sigma_{\pi}$.
After reparametrizing (\ref{eq:loglik}) in terms of $L$ and $P$, it reduces to
\begin{equation} \label{eq:parloglik}
\ell (L, P | \mathbf{X}) = \frac{1}{2} \mbox{tr} \Big (P SP^tL^t L \Big ) - \sum_{j = 1}^p\log L_{jj},
\end{equation}
where $S = \mathbf{X}^t \mathbf{X} / n$ is the sample covariance and $|\Omega|^{-1/2}= |L| = \prod_{j =1}^p L_{jj}$. Unfortunately, as stated in the next proposition, $\ell(L,P| \mathbf{X})$ is permutation invariant and maximum likelihood does not favor any particular ordering. Consequently, all maximum likelihood DAGs corresponding to a different permutation produce the same value of the Gaussian log-likelihood function.
\begin{prop} \label{p:prop1}
The log-likelihood function $\ell(L,P| \mathbf{X})$, defined in (\ref{eq:loglik}), is permutation invariant, i.e., $\ell(L, I| \mathbf{X}) = \ell(L, P| \mathbf{X})$, where $I$ is the $p \times p$ identity matrix.
\end{prop}
We note that \cite[Proposition 1]{ye2019} provide a similar result, but a closer look at their proof reveals that it is valid only for the (lower dimensional) case $n > p$. Our result is more general and its proof can be utilized to verify the permutation invariance for other score functions. The key idea in our proof is showing Schur-convexity \cite[Chapter 3]{marshall11} of the score function.
We follow \cite{ye2019} to break the permutation invariancy in (\ref{eq:loglik}) and regularize the negative log-likelihood function to favor sparse DAGs, hence learning a better model \cite[Chapter 18.1]{koller2009}. We consider the following penalized score function:
\begin{equation} \label{eq:frrcf}
\begin{split}
\min_{L \in \mathcal{L}_p ,P \in \mathcal{P}_p}Q(L,P) &= \min_{L,P} \Big \{\frac{1}{2} \mbox{tr} \Big (P SP^tL^t L \Big ) \\ &- \sum_{j = 1}^p\log L_{jj}\\ & + \sum_{1 \leq j \leq i \leq p} \rho(|L_{ij}|; \lambda) \Big \},
\end{split}
\end{equation}
where $\mathcal{P}_p,\, \mathcal{L}_p$ are the sets of all $p \times p$ permutation and lower triangular matrices with positive diagonal entries and the penalty function $\rho(\cdot,\lambda): \mathcal{R} \rightarrow \mathcal{R}$ satisfies conditions listed in \cite{loh2015} and reiterated in Supplementary for convenience. These conditions are required for establishing theoretical properties of our estimators in Section~\ref{s:statprop}.
\section{A minimization algorithm}
In this section, we introduce our two-step algorithm to minimize the score function (\ref{eq:frrcf}), named Relaxed Regularized Cholesky Factor (RRCF). First, we propose a relaxation to solve the optimization problem in line 5 through a gradient projection algorithm (see Algorithm~\ref{a:ppm}). Then estimate a Cholesky factor in line 6 utilizing a cyclic coordinatewise algorithm (see Algorithm \ref{a:cca}). We show that in the first step, a convex relaxation can be achieved when the number of observations exceeds the number of variables.
\begin{algorithm}[ht!]
\caption{RRCF algorithm}\label{a:RRCF}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$\lambda,k_{max}$} \gets \textit{Tuning Parameter, iteration}$
\State $\textit{${L}^{(0)}, P^{(0)}$} \gets \textit{Initial matrices }$
\State\hskip-\ALG@thistlm \emph{while $k < k_{max}$}:
\State $\quad \quad \hat P^{(k)} = \argmin_{P \in \mathcal{P}_p} Q_{RRCF}(L^{(k - 1)}, P) $ \label{a:SSC:1}
\State $\quad \quad \hat L^{(k)} = \argmin_{L \in \mathcal{L}_p} Q_{RRCF}(L, P^{(k)}) $ \label{s:hi}
\State $ \quad \quad k = k+1$
\State\hskip-\ALG@thistlm \emph{Output}:$\;(\hat L, \hat P)$
\end{algorithmic}
\end{algorithm}
\subsection{Optimization over the permutation space}
A paramount issue in finding an optimal permutation matrix is that the size of the search space is $p!$.
\cite{ye2019} mitigate the problem by using a simulated annealing technique to search over the permutation space. Our approach is significantly different and relies on enlarging the non-convex set of permutation matrices to the convex set of doubly stochastic matrices (Birkhoff polytope) and finding the ``closest'' permutation matrix to the optimal doubly stochastic matrix. In view of the recent advances in Seriation \cite{fogel2013} and Graph Matching problems \cite{zaslavskiy2009,wolstenholme2016}, our approach amounts to a relaxation of the hard combinatorial problem.
\subsubsection{A Convex Relaxation}
The impetus of the work in this section is the framework in \cite[Section 3.2]{fogel2013}.
The optimization in line 5 of Algorithm~\ref{a:RRCF} can be written as:
\begin{equation} \label{p:perm}
\begin{aligned}
\min_{P} \quad & \frac{1}{2}\mbox{tr}(LPSP^tL^t) \\
\mbox{s.t.} \quad & P \in \mathcal{P}_p ,
\end{aligned}
\end{equation}
where we eliminate terms that are constant with respect to $P$. We denote the Birkhoff polytope by $\mathcal{D}_p$ (the space of doubly stochastic matrices), where $\mathcal{D}_p = \{A \in R^{p \times p}: A \geq 0, A \mathbf{1} = \mathbf{1}, A^t \mathbf{1} = \mathbf{1}\}$, it has $p!$ vertices and dimension of $(p - 1)^2$. It is informative to note that every permutation matrix is a doubly stochastic matrix, and a matrix is a permutation if and only if it is both doubly stochastic and orthogonal; i.e., $\mathcal{P}_p = \mathcal{D}_p \cap \mathcal{O}_p$, where $\mathcal{O}_p$ is the set of $p \times p$ orthogonal matrices. Moreover, from Birkhoff's Theorem, every doubly stochastic matrix can be written as a convex combination of permutation matrices and the set of doubly stochastic matrices is the convex hull of the set of permutation matrices \cite[Theorem 8.7.2]{Horn2012}, where permutation matrices are vertices (extreme points) of the polytope. More on Birkhoff polytopes and its properties can be found in \cite{brualdy1974}.
Since the sample covariance matrix $S \succcurlyeq 0 $ is positive semi-definite, we can introduce a convex relaxation to the combinatorial problem~(\ref{p:perm}) by replacing $\mathcal{P}_p$ with its convex hull $\mathcal{D}_p$:
\begin{equation} \label{p:stoc}
\begin{aligned}
\min_{P} \quad & \frac{1}{2}\mbox{tr}(LPSP^tL^t) \\
\mbox{s.t.} \quad & P \in \mathcal{D}_p ,
\end{aligned}
\end{equation}
However, as shown in Corollary~\ref{c:optsol}, the solution of (\ref{p:stoc}) is not an acceptable candidate. In the next lemma, we list well-known properties of doubly stochastic matrices that are used to establish the framework for the convex relaxation. Since we are not aware of a source to cite, a proof is given in the Supplementary for completeness. We denote by $J \in \mathcal{D}_p$ the $p \times p$ matrix all of whose entries are $1$.
\begin{lemma} \label{l:dsprop}
For any $p \times p$ doubly stochastic matrix $P \in \mathcal{D}_p$,
\item \[1 \leq \|P\|_F \leq \sqrt{p}\]
The left and right equalities hold if and only if $P = J/p$ and $P$ is a permutation matrix, respectively.
\end{lemma}
From Lemma~\ref{l:dsprop}, the following corollary easily follows.
\begin{corollary} \label{c:optsol}
The optimal solution of (\ref{p:stoc}) is $ \hat P = J/p$.
\end{corollary}
Thus, the solution of (\ref{p:stoc}) is the center of the Birkhoff polytope \cite[page 20]{ziegler1995} and far from vertices where permutation matrices are located. To force it to move closer to the vertices, we utilize Lemma~\ref{l:dsprop} to motivate and add a proper penalty to the objective function (See Figure~\ref{f:geom} for the geometric depiction.)
\begin{equation} \label{p:stocort}
\begin{aligned}
\min_{P} \quad & \frac{1}{2}\mbox{tr}(LPSP^tL^t) - \frac{1}{2}\mu\|P\|^2_F\\
\mbox{s.t.} \quad & P \geq 0, P \mathbf{1} = \mathbf{1}, P^t \mathbf{1} = \mathbf{1}.
\end{aligned}
\end{equation}
Note that for larger $\mu > 0$, $\|P\|^2_F$ is pushed toward its upper bound $p$ (Lemma~\ref{l:dsprop}), so that the larger $\mu$, the closer the solution of (\ref{p:stocort}) is to a permutation matrix.
\begin{figure}[t]
\centering
\resizebox{4cm}{3cm}{%
\begin{tikzpicture}
\newdimen\R
\R=2.7cm
\node[inner sep = 0.5pt, circle, draw] {J/p};
\draw (0:\R) \foreach \x in {60,120,...,360} { -- (\x:\R) };
\foreach \x/\l/\p in
{ 60/{(2,3,1)}/above,
120/{(2,1,3)}/above,
180/{(1,2,3)}/left,
240/{(1,3,2)}/below,
300/{(3,1,2)}/below,
360/{(3,2,1)}/right
}
\node[inner sep=1pt,circle,draw,fill,label={\p:\l}] at (\x:\R) {};
\end{tikzpicture}%
}
\caption{A geometric depiction of relaxation (\ref{p:stocort}) for $\mathcal{D}_3$ Birkhoff polytope. Here, vertices represent permutations, and matrix $J/p$ indicates the center of the polytope.}
\label{f:geom}
\end{figure}
Similar to \cite[Proposition 3.5]{fogel2013}, the next lemma shows that the convexity of the objective function (\ref{p:stocort}) depends on the inextricably intertwined values of $\mu$ and the smallest eigenvalue of $S$ and $L^tL$. As a result, the convexity is untenable when $n<<p$. We mitigate this problem by introducing an additional transformation to maintain convexity when $n \approx p$ (Lemma~\ref{l:l1}(b)). The following notation is used in the lemma: we write $\lambda_1 < \lambda_2 < \dots < \lambda_m$ as an ordered, distinct eigenvalues of the $p \times p$ matrix. The proof is provided in the Supplementary for completeness.
\begin{lemma} \label{l:l1}
\begin{enumerate}
\item[a.] If $\mu \leq \lambda_1(S)\lambda_1(L^tL)$, the optimization problem (\ref{p:stocort}) is convex in $P$.
\item[b.] If $\mu \leq \lambda_2(S)\lambda_1(L^tL)$ and $T = \mathbf{I} - \frac{1}{p} \mathbf{1}\mathbf{1}^t$ is the projection matrix into the orthogonal complement of $\mathbf{1}$, then the optimization problem
\begin{equation} \label{p:trstoc}
\begin{aligned}
\min_{P} \quad & \frac{1}{2}\mbox{tr}(LPSP^tL^t) - \frac{1}{2}\mu\|TP\|^2_F\\
\mbox{s.t.} \quad & P \geq 0, P \mathbf{1} = \mathbf{1}, P^t \mathbf{1} = \mathbf{1},
\end{aligned}
\end{equation}
is equivalent to problem (\ref{p:stocort}) and is convex in $P$.
\item[c.] If $\mu > \lambda_m(S)\lambda_m(L^tL)$, the optimization problem (\ref{p:stocort}) is concave in $P$ and the solution is a permutation matrix.
\end{enumerate}
\end{lemma}
From Lemma~\ref{l:l1}(a) and (b), for $n <<p$, $\lambda_2(S)$ is zero, and there is no $\mu >0$ that validates convexity of (\ref{p:trstoc}).
The question we investigate next is whether, under the convexity assumption of Lemma~\ref{l:l1}(a) or (b), there is a value of $\mu$ that asymptotically achieves ``closeness'' to the permutation matrix in terms of Frobenius norm.
We provide the answer only for (\ref{p:stocort}), but the similar result holds for (\ref{p:trstoc}) by analogy.
Recall that we tacitly assume the condition $n > p$ to maintain convexity.
\begin{lemma} \label{l:lbound}
Under convexity condition in Lemma~\ref{l:l1}(a), for $\mu > 0$
\begin{equation} \label{e:lbound}
\|\hat P - P\|_F \not \rightarrow 0,\, \mbox{as}\; n \rightarrow \infty,
\end{equation}
where $P \in \mathcal{P}_p$ and $\hat P$ is the solution of (\ref{p:stocort}).
\end{lemma}
The proof can be found in the Supplementary. Lemma~\ref{l:lbound} suggests that under a convexity condition, the solution of (\ref{p:stocort}) does not get ``close'' to the permutation matrix, even when $n \rightarrow \infty$.
The result may encourage the use of higher values of $\mu$, resulting in a non-convex objective function. However, this approach is not recommended. Our empirical results suggest that for comparably large $\mu$, the RRCF algorithm becomes independent from the data and highly dependent on the initial choice of $P$.
Consequently, it gets stuck at one of the extreme points of the Birkhoff polytope.
The choice of $\mu$ for this setting is an open question and left for further investigation. Here, when $n <p$, we propose to treat $\mu$ as a tuning parameter and use information criteria or cross-validation for the selection.
\subsubsection{Gradient Projection Algorithm}
We provide details for solving (\ref{p:trstoc}), but the procedure similarly applies to (\ref{p:stocort}). Optimization (\ref{p:trstoc}) is a quadratic program (QP), and rich literature exists on solving this class of problems.
In this section, we rely on the Gradient Projection \cite{bertsekas2015} method and show the convergence of the algorithm. Algorithm~\ref{a:agp} outlines general steps, where $[\cdot]^+$ denotes projection on the space of doubly stochastic matrices $\mathcal{D}_p$.
\begin{algorithm}[ht!]
\caption{Gradient Projection}\label{a:agp}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$k_{max},\mu, \eta$} \gets \textit{the number of iterations and positive scalars}$
\State $\textit{$L, P^{(0)}$} \gets \textit{Cholesky and Initial Permutation matrix }$
\State\hskip-\ALG@thistlm \emph{while $\|P^{(k+1)}- P^{(k)}\|>\epsilon$ }:
\State $\quad \hat P^{(k+1)} = [P^{(k)} - \eta \nabla Q_{RRCF}(P^{(k)}, L) ]^+\;\;$ \textit{via Algorithm~\ref{a:dsm}}
\State $\quad P^{(k+1)} = P^{(k)} + \alpha^k (\hat P^{(k+1)} - P^{(k)})$
\State $ \quad k = k+1$
\State\hskip-\ALG@thistlm \emph{Output}:$\; \textit{Doubly Stochastic Matrix}\; P $
\end{algorithmic}
\end{algorithm}
Line 5 of the algorithm requires projection onto the Birkhoff polytope, which can be efficiently implemented by the block coordinate ascent, where each iteration has a closed form solution. The details on the block coordinate ascent algorithm are given in the next section. The convergence of the algorithm to a global minimum easily follows from the \cite[Proposition 6.1.2]{bertsekas2015}.
\subsubsection{Projection onto the Birkhoff Polytope}
Here, we give details on solving line 5 of Algorithm~\ref{a:agp}. For a given matrix $P_0$, its projection onto $\mathcal{D}_p$ is defined by
\begin{equation} \label{p:proj}
\begin{aligned}
\min_{P} \quad & \frac{1}{2}\|P - P_o\|^2_F\\
\mbox{s.t.} \quad & P \geq 0, P \mathbf{1} = \mathbf{1}, P^t \mathbf{1} = \mathbf{1}.
\end{aligned}
\end{equation}
The Lagrangian of (\ref{p:proj}) is \cite{bertsekas2015}
\[
\begin{split}
\mathcal{L}(P,u,v,U) &= \frac{1}{2}\|P - P_0\|^2_F + u^t (P \mathbf{1} - \mathbf{1} )\\
&+ v^t(P^t \mathbf{1} - \mathbf{1}) - tr(U^t P) ,
\end{split}\]
and the dual objective function is defined as:
\begin{equation} \label{e:dual}
\mathcal{L}_{*}(u,v,U) = \inf_{P}\mathcal{L}(P,u,v,U)
\end{equation}
Consequently, the dual problem of (\ref{p:proj}) is (see Supplementary
for details)
\begin{equation} \label{p:dual}
\begin{aligned}
\max_{u,v,U} \quad & -\frac{1}{2}\|u\mathbf{1}^t + \mathbf{1}v^t - U\|^2_F - tr(U^tP_0)\\
& +u^t(P_0\mathbf{1} - \mathbf{1}) + v^t(P_0^t\mathbf{1} - \mathbf{1}) \\
\mbox{s.t.} \quad & U \geq 0 ,
\end{aligned}
\end{equation}
Following \cite[Section 4.2]{fogel2013}, we use the block coordinate ascent algorithm to optimize the dual problem (\ref{p:dual}). We show that each block update has a closed form solution.
Details of the algorithm and the derivation of closed form solutions are relegated to the Supplementary.
In the next section, we propose a framework to find the ``closest'' permutation matrix to the doubly stochastic matrix solution (\ref{p:stocort}) or (\ref{p:trstoc}).
\subsubsection{Sampling Permutations from the Space of Doubly Stochastic Matrices}
Since the solution of a convex relaxation (\ref{p:stocort}) is not a permutation matrix, we need to project it to the ``closest'' matrix $P \in \mathcal{P}_p$.
Let $\tilde P$ be the doubly stochastic matrix solution of (\ref{p:stocort}), then a common method to project this matrix onto the space of permutation matrices is through the following optimization \cite[Section 2.1]{zaslavskiy2009}:
\begin{equation} \label{e:hung}
\argmin_{P \in \mathcal{P}_p} \|\tilde P - P\|^2_F = \argmax _{P \in \mathcal{P}_p} tr \{\tilde P^t P\},
\end{equation}
which is a linear assignment problem and usually solved by the Hungarian algorithm \cite[Section 4.2.1]{burkard2012} and takes $O(p^3)$ operations.
Unfortunately, (\ref{e:hung}) suffers a serious drawback as it only delivers one candidate solution to (\ref{p:stocort}), and if it is not ``close'' to the true permutation matrix $P$, it is unclear how to continue \cite[Section 3]{wolstenholme2016}. A viable alternative is a permutation sampling procedure initially proposed for the orthogonal matrices in \cite{barvinok2005}. The idea is to ``round'' an orthogonal matrix $Q$ to a permutation matrix $P$ by considering its action on a random vector sampled from a Gaussian distribution. Consider a sample $x \in R^p$ from a Gaussian distribution and an ordering vector $r(x)$ such that $r(x)_i = k$ where $x_i$ is the $k$th smallest value of $x$. For example, if $x = [4.7,-2.1, 2.5 ]^t \Rightarrow r(x) =[3,1,2]^t $. Barvonik argues, if the permutation matrix $P$ satisfies
\begin{equation} \label{e:samp}
P(r(x)) = r(Qx)
\end{equation}
then it is ``close'' in Frobenious norm to $Q$ with respect to $x$, as they both act on $x$ in a similar way \cite[Theorem 1.6]{barvinok2005}. In other words, $P$ matches the $k$th smallest coordinate of $x$ with the $k$th smallest coordinate of $Qx$, and $P$ represents a ``rounding'' of $Q$. This provides a framework to project an orthogonal matrix to a distribution of permutation matrices.
A close examination of proof of \cite[Theorem 1.6]{barvinok2005} reveals that it is not restricted to orthogonal matrices and can be successfully extended to doubly stochastic matrices \cite[Section 4A]{wolstenholme2016}.
We use (\ref{e:samp}), selecting a doubly stochastic matrix $\tilde P$ instead of $Q$, to generate $N$ permutation matrices each ``close'' to the doubly stochastic matrix $\tilde P$. Then a common way to select the ``best'' permutation matrix from the $N$ sampled matrices is to pick a matrix that provides the lowest cost to (\ref{p:stoc}) \cite[Section 3.2.4]{fogel2013}.
Finally, Algorithm~\ref{a:ppm} combines necessary steps to estimate a permutation matrix $P$ in line 5: estimation of the doubly stochastic matrix (\ref{p:trstoc}), and its approximation to the ``closest'' permutation matrix via (\ref{e:samp}).
\begin{algorithm}[ht!]
\caption{Optimization over permutation matrices}
\label{a:ppm}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$N_{max}$} \gets \textit{max. number of sampling}$
\State $\textit{Find $\tilde P$ via Algorithm~\ref{a:agp}} $
\State\hskip-\ALG@thistlm \emph{if $\tilde P \not \in \mathcal{P}_p$}:
\State\hskip-\ALG@thistlm $\quad$ \emph{while $ j < N_{max}$}:
\State $\quad \quad \textit{Sample:}\; x^{(j)} \sim N(0, \mathbf{I}_p)\}$
\State $\quad \quad \textit{Solve for $P^{(j)}$ using (\ref{e:samp}):}
\State\hskip-\ALG@thistlm \emph{\textit{From $\{P^{(j)}\}_{j = 1}^{N_{max}}$ choose} $P$ \text{that minimizes (\ref{p:stoc}) }}.
\State\hskip-\ALG@thistlm \emph{Output}:$\; \textit{Permutation Matrix}\; P $
\end{algorithmic}
\end{algorithm}
\subsection{Cholesky Factor Estimation}
This section focuses on Cholesky factor $L$ estimation from line 6 of Algorithm~\ref{a:RRCF}. That is we fix a permutation matrix $P$ and update the Cholesky factor $L$ using a \textbf{non-convex} objective function (\ref{eq:frrcf}). It is informative to recall that a Cholesky factor $L$ entails the DAG structure, and by learning $L$, accordingly, we learn the DAG structure in $B$.
Given an ordering, \cite{shojaie2010} and convex sparse Cholesky selection (CSCS) algorithm proposed in \cite{khare2016} estimate sparse Cholesky factor $L$ using a lasso-based penalty and convex objective function.
Here, for the fix permutation matrix $P$, we propose a cyclic coordinatewise algorithm to learn the Cholesky factor $L$ from the non-convex objective function (\ref{eq:frrcf}).
We show that the objective function can be decoupled into $p$ parallel penalized regression problems. The latter can be compared with the decomposable property of the score function in the BN literature, since non-zero values in each $i$th row of $L$ correspond to the parents of the $i$th node in DAG. Recall that a score function $f(\mathcal{G}, X)$ is decomposable if it can be written as $f(\mathcal{G}, X) = \sum_{i=1}^p f(X_i| \Pi^{\mathcal{G}}_{i})$ \cite[Definition 18.2]{koller2009}.
Denoting $S^P = PSP^t$,\; $S^P_i$ the $i \times i$ sub-matrix of $S^p$, $L_{i \cdot}$ the $i$th row of $L$, and $\beta^i$ non-zero values of the $L_{i \cdot}$, it follows from (\ref{eq:frrcf})
\begin{equation}
\begin{aligned}
Q_{RRCF}(L)& =
tr(LS^pL^t) - 2 \sum_{i=1}^{p}\log L_{ii}\\
&+ \sum_{1\leq j < i \leq p} \rho(|L_{ij}|, \lambda) = \sum_{i = 1}^{p} (\beta^i)^t S^P_i \beta^i\\
&- 2 \sum_{i =1}^p \log (\beta^i_i)
+ \sum_{i =2}^{p} \sum_{j =1}^{i-1} \rho(|\beta^i_j|, \lambda)\\
&= \sum_{i =1}^p Q_{RRCF,i}(\beta^i),
\end{aligned}
\end{equation}
where in arguments of $Q_{RRCF}(\cdot)$ we omit the dependence from $P$, and
\begin{equation} \label{eq:rrcfi}
\begin{split}
Q_{RRCF,i}(\beta^i) &= (\beta^i)^t S^P_i \beta^i - 2 \log \beta^i_i \\ & + \sum_{j = 1}^{i -1} \rho(|\beta^i_j|, \lambda)
\end{split}
\end{equation}
for $2 \leq i \leq p$, and
\begin{equation}\label{eq:rrcf1}
Q_{RRCF,1}(L_{11}) = L^2_{11}S^P_{11} - 2 \log L_{11}
\end{equation}
In (\ref{eq:rrcfi}), we focus on the class of penalties called the minimax concave penalty (MCP) \cite{zhang2010}. MCP exploits convexity of the penalized loss near the sparse regions and concavity outside. It includes $\ell_1$ and $\ell_0$ as extreme cases and with two parameters $(\gamma,\lambda)$ takes the form
\begin{equation}\label{eq:mcp}
\rho(\theta, \lambda, \gamma) = \begin{cases} \lambda |\theta| - \frac{\theta^2}{2\gamma} & |\theta| < \gamma \lambda\\
\frac{1}{2} \gamma \lambda^2 & |\theta| \geq \gamma \lambda,
\end{cases}
\end{equation}
where $\lambda \geq 0$ and $\gamma >1$.
Next, we derive steps to minimize the score function $Q_{RRCF}(L)$ with respect to non-zero values of $L$ for the fixed $P$. We assume that diagonal entries of the sample covariance matrix $S$ are strictly positive. Since $\{\beta^i\}^p_{i=1}$ disjointly partition the parameters in $L$, then optimizing $Q_{RRCF}(L)$ can be implemented by separately optimizing $Q_{RRCF}(\beta^i)$ for $1 \leq i \leq p$.
We define a generic function $h: R^{k-1} \times R_+ \rightarrow R$ of the form
\begin{equation} \label{e:gen}
h_{k,A,\lambda,\gamma}(x) = - 2 \log x_k + x^tAx + \sum_{i =1}^{k -1} \rho(|x_i|, \lambda, \gamma),
\end{equation}
where $\lambda>0,\, \gamma > 1$ and $A$ is a positive semi-definite matrix with positive diagonal entries. It is instructive note that $Q_{RRCF,i}(\beta^i) = h_{i,S_i,\lambda,\gamma}(\beta^i)$ for every $1 \leq i \leq p$, and it suffices to develop an algorithm which minimizes a function of the form $h_{i,A,\lambda,\gamma}$. For every $1 \leq j \leq k$, we define
\[x^*_j = \inf_{x_j}h_{k,A,\lambda,\gamma}(x).\]
Next lemma shows that $\{x^*_j\}_{j=1}^{k}$ can be computed in the closed form. The proof is given in Supplementary.
\begin{lemma} \label{l:xstar}
The optimal solution $\{x^*_j\}_{j =1}^{k}$ can be computed in the closed form.
\begin{equation} \label{eq:xk}
x^*_k = \frac{-\sum_{l \neq k}A_{lk}x_l + \sqrt{(\sum_{l \neq k}A_{lk}x_l)^2 +4A_{kk}}}{2A_{kk}}
\end{equation}
and for $1 \leq j \leq k-1$,
\begin{equation}\label{eq:xj}
x^*_j = \frac{S_{\lambda}(-2\sum_{l \neq k}A_{lk}x_l)}{2A_{jj} - 1/\gamma}
\end{equation}
\end{lemma}
Here, $S_{\lambda}$ is the soft-thresholding operator given by $S_{\lambda}(x) = sign(x)(|x| - \lambda)_{+}$. From Lemma~\ref{l:xstar}, Algorithm~\ref{a:cca} provides a cyclic coordinatewise minimization algorithm for $h_{k,A,\lambda,\gamma}$.
We use it to minimize $Q_{RRCF}(\beta^i)$ for $1 \leq i \leq p$, and combine outputs to obtain the estimated Cholesky factor $L$ in Algorithm~\ref{a:ecf}
\begin{algorithm}[ht]
\caption{Cyclic coordinatewise algorithm}\label{a:cca}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$k_{max},A, \lambda, \gamma, \epsilon$}$
\State $x^{(0)} \gets \textit{Initial estimate} $
\State $\textit{Set}\; x^{\textit{current}} = x^{(0)};\;\mbox{Converged}=\mbox{FALSE} $
\State\hskip-\ALG@thistlm \emph{while $\mbox{Converged} ==\mbox{FALSE}$ or $ k < k_{max}$ }:
\State $\quad x^{old} \gets x^{current}$
\State $\quad \textit{For}\; $j = 1,2,\dots, k - 1$ $
\State\hskip-\ALG@thistlm \emph{$\quad \quad x^{current}_j = x^*_j\;\;\textit{via (\ref{eq:xj})}$}
\State $\quad x^{current}_k = x^*_k\;\;\textit{via (\ref{eq:xk})}$
\State $\quad \emph{if}\; \|x^{current} - x^{old}\| < \epsilon $
\State $\quad \quad \mbox{Converged = TRUE}$
\State $\quad \emph{else} \quad k = k+1$
\State\hskip-\ALG@thistlm \emph{Output}:$\; x$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[ht]
\caption{Cholesky Factor Estimation}\label{a:ecf}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$k_{max}, \mathbf{X}, \lambda, \gamma, \epsilon$}$
\State $L^{(0)} \gets \textit{Initial Cholesky factor} $
\State $ \textit{For}\; $i = 1,2,\dots, p$ $
\State\hskip-\ALG@thistlm $\quad \beta^{i} = \argmin_{\beta_i} Q_{RRCF,i}(\beta^i)\;\;\textit{via Algorithm~\ref{a:cca}}$
\State $\textit{Construct $L \in \mathcal{L}_p$ by setting its non-zero values as $\beta^i$}$
\State\hskip-\ALG@thistlm \emph{Output}:$\; \textit{Lower diagonal matrix $L$}$
\end{algorithmic}
\end{algorithm}
\subsubsection{Convergence of the Cyclic Coordintewise Algorithm}
As discussed, the score function $Q_{RRCF}(L)$ is non-convex with respect to $L$, and the convergence of iterates in Algorithm~\ref{a:ecf} can be guaranteed only to a local minimum. Next lemma shows that for the fixed permutation matrix $P$, the objective function $Q_{RRCF}(L)$ is lower bounded, a local minimum lies in the space of lower triangular matrices $\mathcal{L}_p$ with positive diagonal entries, and for certain values of $\gamma$, the generic function $h(\cdot)$ is strictly convex.
\begin{lemma} \label{l:convanal}
\begin{itemize}
\item[a.] If $A_{ii} > 0$, for $1 \leq i \leq p$
\[h_{k,A,\lambda,\gamma}(x) \geq 2x_k - 2.\]
\item[b.] For $\gamma > \max\{1/2A_{ii},1\}$, $h_{k,A,\lambda,\gamma}(x)$ is a strictly convex function of $x_i$ for $1 \leq i \leq k - 1$.
\item[c.] For every $n$ and $p$
\[
\begin{aligned}
\inf_{L \in \mathcal{L}_p}Q_{RRCF}(L) &= \sum_{i =1}^p \inf_{\beta^i} Q_{RRCF,i}(\beta^i)\\
&\geq -2p > - \infty
\end{aligned}
\]
and any local minimum of $Q_{RRCF}$ over the open set $\mathcal{L}_p$ lies in $\mathcal{L}_p$.
\end{itemize}
\end{lemma}
From this lemma, we can establish the convergence of the cyclic coordintewise algorithm.
\begin{theorem}\label{t:conv}
Under assumptions of Lemma~\ref{l:convanal}, Algorithm~\ref{a:ecf} converges to a local minimum of $Q_{RRCF}(L)$.
\end{theorem}
\section{Simulation and Data Analysis}
In this section, we study the empirical performance of our estimator on simulated and macro-economic datasets. The simulation results indicate that for a fixed p, if the number of edges in the DAG increases, i.e., the DAG is denser, the performance of RRCF tends to improve. The macro-economic data analysis provides the application of RRCF to solve the price puzzle \cite{Sims1992}, a well known problem in economics.
\subsection{Simulation Study}
We compare the performance of our algorithm with the three recent BN learning algorithms: \textbf{ARCS}:\cite{ye2019}, \textbf{CCDr}:\cite{aragam2015}, and
\textbf{NOTEARS} \cite{zheng2018}.
The performance is measured, both in terms of the structure learning, and how well the weighted adjacency matrix $\hat B$ estimates $B$.
The weighted adjacency matrix $B$ is constructed following \cite[Section 4.1]{kalisch2008} framework. We adapt parameterization $L = \Omega^{-1/2}(I - B)$ to generate data, where $\Omega = I_p$. The dimension of the data varies $p \in \{100, 200\}$ and the expected sparsity levels are $s \in \{p, 2p\}$. The latter corresponds to the expected number of edges in the DAG. In all simulations, the sample size is $n = 150$ and each sample follows $p$-dimensional normal distribution $N(0, (L^tL)^{-1})$.
Each of the simulation settings $(p,s)$ is repeated over 20 datasets. The tuning parameters $\eta, \mu, \lambda$ for the RRCF algorithm are selected using the extended BIC criterion \cite{foygel2010} over the specified grid (See Supplementary for details). We note that, for the case $p \approx n$, one computational disadvantage of RRCF, compare to CCDr or NOTEARS, is a need to tune three tuning parameters which can be computationally costly for high dimensional datasets.
\subsubsection{Structure Learning and Estimation Accuracy}
We compare the four algorithms using the following four metrics: True Positive Rate (TPR), False Positive Rate (FPR), Structural Hamming Distance (SHD), and scaled Frobenius norm, which estimates how far the weighted adjacency matrix $\hat B$ is from $B$; i.e., $\frac{1}{p}\|\hat B - B\|_F$.
\begin{table}[ht!]
\centering
\caption{Average of three metrics over 20 replication for four $(p,m)$ settings. For TPR, a larger value indicates better performance; for FPR and FRB. NORM, a smaller value indicates better performance.}
\label{t:res1}
\setlength\extrarowheight{-3pt}
\begin{tabular}{c|c|c|c|c}
\hline
\rule{0pt}{12pt}
$(p,s)$ &Method & TPR & FPR & FRB. NORM\\
\hline
\rule{0pt}{12pt}
\multirow{4}{*}{ (100,100)}&ARCS & 0.601 & 0.001 & 7.052\\
&CCDr & \textbf{0.621} & \textbf{0.001} & 9.930\\
&RRCF & 0.603 & 0.001 & \textbf{6.868}\\
&NOTEARS & 0.612 & 0.001 & 9.930\\
\hline
\rule{0pt}{12pt}
\multirow{4}{*}{(100,200)}&ARCS & 0.637 & \textbf{0.003} & \textbf{10.569}\\
&CCDr & 0.636 & 0.005 & 14.305\\
&RRCF & \textbf{0.649} & 0.009 & 10.599\\
&NOTEARS & 0.645 & 0.004 & 11.349\\
\hline
\rule{0pt}{12pt}
\multirow{4}{*}{(200,200)}
&ARCS & 0.611 & \textbf{0.001} & 12.883\\
&CCDr & 0.651 & 0.007 & 19.845\\
&RRCF & 0.623 & 0.003 & 12.509\\
&NOTEARS & \textbf{0.657} & \textbf{0.001} & \textbf{12.233}\\
\hline
\rule{0pt}{12pt}
\multirow{4}{*}{(200,400)}
&ARCS & 0.635 & 0.001 & 13.043\\
&CCDr & 0.658 & \textbf{0.001} & 17.679\\
&RRCF & 0.643 & 0.001 & \textbf{12.752}\\
&NOTEARS & \textbf{0.655} & 0.002 & 14.617\\
\hline
\end{tabular}%
\end{table}
Table~\ref{t:res1} and Figure~\ref{fig:shd} report the simulation results. The best average score for each metric and $(p,s)$ setting is highlighted in bold. Results suggest that RRCF performance improves when $s$ is higher for fixed $p$. In particular, for the $(100,100)$ case, CCDr provides the best results for the TPR and FPR metrics, followed by NOTEARS and RRCF. The situation changes for the $(100, 200)$ case, where RRCF provides the best TPR average score, and ARCS provides the best FPR average score. NOTEARS perform the best when the dimension increases from the 100 to 200. RRCF provides the best scaled Frobenius norm result for $(100,100)$ and $(200, 200)$ settings.
\begin{figure}[ht!]
\centering
\includegraphics[width= 6.5cm, height = 5.8cm]{shdresnew}
\caption{Structural Hamming Distance boxplot for four $(p,s)$ settings. A smaller SHD value indicates better performance.}
\label{fig:shd}
\end{figure}
From Figure~\ref{fig:shd}, overall, the performance of RRCF is compatible with the considered algorithms.
\subsection{Macro-Economic Application}
We illustrate the application of the RRCF algorithm to the macro-economic dataset. In particular, we utilize our methodology to estimate the contemporaneous causal influences in the structural vector autoregression (SVAR) model. Then utilize impulse response functions to analyze the dynamics in SVAR models \cite[Section 2.3.2]{Lutkepohl2007} and discover the future effects of a shock on variables.
For a $K \times 1$ vector $Y_t = (y_{1t}, \dots, y_{Kt})^t$ the SVAR with $p_0$ lags is defind as
\begin{equation}
Y_t = B_0 Y_t +\sum_{i=1}^{p_0}B_iY_{t-i} + \varepsilon_t,
\end{equation}
where $E(\varepsilon_t \varepsilon^t_t) = \Omega$ is a diagonal matrix and $B_i$'s are $K \times K$ matrices. Notice that the relations among the contemporaneous components of $Y_t$ are embedded in the matrix $B_0$ and such causal structure can be represented by a DAG, whose vertices are the elements of the vector $Y_t$. That is there is a directed edge pointing from $y_{it}$ to $y_{jt}$ if and only if $(B_0)_{ij} \neq 0$. The knowledge of the ordering among contemporaneous error terms is used for the estimation of the impulse response functions (for details, see \cite[Section 2]{hyvarinen2010}). \cite{bessler1998, Demiralp2003} exploit PC algorithm for Gaussian data and \cite{hyvarinen2010, moneta2013, dallakyan2020} propose methods for non-Gaussian data to learn the contemporaneous ordering. \cite{chu2008, malinsky2018} propose methods for settings with unmeasured confounding.
Algorithm~1 in \cite{dallakyan2020} summarizes steps on the use of DAGs for the SVAR estimation. We iterate it in Algorithm~\ref{alg:dagsvar} by incorporating the RRCF step in line 7 to recover the ordering of error terms.
\begin{algorithm}[!ht]
\caption{SVAR procedure with RRCF} \label{alg:dagsvar}
\begin{algorithmic}[1]
\Procedure{}{}
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$y_1,\dots,y_t$} \gets \text{$K$ dimensional }\textit{stationary series}$
\State\hskip-\ALG@thistlm \emph{top}:
\State \textbf{Estimate} \text{the VAR model} \textit{$y_t = A_1y_{t-1}+ \dots + A_p y_{t-p} + u_t$,
\State \textbf{Estimate} \textit{the residuals $\hat u_t = y_t - \hat A_1y_{t-1}- \dots - \hat A_p y_{t-p}$}
\State \textbf{Perform} \textit{ RRCF algorithm on residuals to recover ordering among residuals} $\hat u_{1t}, \dots, \hat u_{Kt}$}
\State\hskip-\ALG@thistlm \emph{Output}:
\State $B_0$
\EndProcedure
\end{algorithmic}
\end{algorithm}
We use RRCF incorporated Algorithm~\ref{alg:dagsvar} to solve the price puzzle. The price puzzle in a structural autoregression (SVAR) system is known as an inability to explain the positive relationship between an innovation(shock) in the federal funds rate (FFR) and inflation \cite{Bernanke1992,Sims1992,Balke1994,Hoover2014}. It is a puzzle since an increase in the federal funds is expected to be followed by a decrease in the price level rather than an increase (See Figure~\ref{fig:FFRirfPC}).
\cite{dallakyan2020} showed that utilization of the recent DAG techniques to recover the ordering of error terms in VAR mitigates the price puzzle problem. Here, we show that using the sparse VAR approach and RRCF algorithm to recover the DAG structure of error terms leads to the complete disappearance of the price puzzle (See Figure~\ref{fig:FFRirf}).
To analyze the price puzzle, we use a relatively rich dataset from \cite{Hoover2014}. Data consist of 12 monthly series for the United States that run from 1959:02 to 2007:06. Data sources and details are provided in \cite{Hoover2014}.
In the dataset, monetary policy is represented both by the Federal funds rate (FFR) and two reserve components: (the logarithms of) borrowed reserves (BORRES) and nonborrowed reserves (NBORRES). Financial markets are represented by two monetary aggregates (the logarithms of) M1 and (the non-M1 components of) M, as well as by three interest rates: the own-rate of interest on M2 (M2OWN), the 3-month Treasury bill rate (R3M), and the 10-year Treasury bond rate (R10Y ). Prices are represented by (the logarithms of) the consumer price index (CPI) and an index of sensitive commodity prices (COMPRICE). Finally, the real economy is represented by the (logarithm of) industrial production (INDPRO) and the output gap (GAP). Our sample period runs from January 1990 until 2009. The sample period is chosen such that to avoid a policy break \cite{Hoover2014}.
To introduce sparsity in the VAR estimation, we impose a lasso penalty on the VAR coefficient matrix \cite{Song2011,Nicholson2016}. For the sparse VAR estimation, we use the \texttt{BigVAR} package in \texttt{R} \cite{bigvar}, with the number of lags equal to 4. Then, using contemporaneous time restrictions obtained from Algorithm~\ref{alg:dagsvar}, estimate impulse response functions. For comparison, we include the impulse response function obtained from the procedure proposed in \cite{bessler1998}(BA for short).
Figure~\ref{fig:svarirf} plots the responses of Consumer Price Index (CPI) to Federal Fund Rate (FFR) obtained from the BA and RRCF algorithms, respectively. From Figure~\ref{fig:FFRirfPC}, the prize puzzle is apparent when the response is estimated using the BA's procedure. However, it disappears when the response is estimated using the RRCF algorithm (see Figure~\ref{fig:FFRirf}). The latter result is consistent with the macro-economic literature.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.25\textwidth}
\centering
\includegraphics[width=.8\linewidth, height = 3cm]{pcirfplot}
\caption{BA}
\label{fig:FFRirfPC}
\end{subfigure}%
\begin{subfigure}{.25\textwidth}
\centering
\includegraphics[width=.8\linewidth, height = 3cm]{irfplotgg}
\caption{RRCF}
\label{fig:FFRirf}
\end{subfigure}
\caption{The response of the Consumer Price Index to the Federal Fund Rate shock.}
\label{fig:svarirf}
\end{figure}
\section{Statistical Properties} \label{s:statprop}
In this section, we study the consistency of the RRCF estimator, assuming that the true permutation matrix $P$ is known; i.e., data have known order. Under this assumption, the dependence of $Q_{RRCF}$ on $P$ is omitted and focus is only on the consistency of a Cholesky factor estimator in (\ref{eq:frrcf}).
\cite{khare2016, yu2017} provide consistency of the sparse Cholesky factor estimator for the convex objective function. However, our objective function is \textit{non-convex} and it may possess multiple local optima that are not global. Therefore, the standard statistical techniques are not applicable for establishing consistency
We establish upper bounds on the Frobenius norm between \textbf{any local optimum} of the empirical estimator and the unique minimizer of the population. Even though the non-convex function may possess multiple local optima, our theoretical results guarantee that, from a statistical perspective, all local optima are fundamentally as good as a global optimum. The theoretical analysis relies on the following assumptions:
\begin{itemize}
\item{A1} \textit{Marginal sub-Gaussian assumption:} The sample matrix $X \in \mathcal{R}^{n \times p}$ has $n$ independent rows with each row drawn from the distribution of a zero-mean random vector $X = (X_1, \dots, X_p)^t$ with covariance $\Sigma$ and sub-Gaussian marginals; i.e.,
\[E[\mbox{exp}(tX_j/ \sqrt{\Sigma_{jj}})] \leq \mbox{exp}(Ct^2) \]
for all $j=1,\dots,p, \, t \leq 0$ and for some constant $C > 0$.
\item{A2} \textit{Sparsity Assumption:} The true Cholesky factor $L \in \mathcal{R}^{p \times p}$ is the lower triangular matrix with positive diagonal elements and support $\mathcal{S}(L) =\{(i,j), i \neq j| L_{ij} \neq 0\} $. We denote by $s = |S|$ cardinality of the set $S$.
\item{A3} \textit{Bounded eigenvalues:} There exist a constant $\kappa$ such that
\[0 < \kappa^{-1} \leq \lambda_{min}(L) \leq \lambda_{max}(L) \leq \kappa \]
\end{itemize}
Before providing our main result, we recall that a matrix $\hat L \in \mathcal{L}_p$ is a stationary point for $Q_{RRCF}$ if it satisfies \cite{bertsekas2015}
\begin{equation} \label{eq:stat}
\langle\nabla \mathcal{L}_n(\hat L) + \nabla \rho(\hat L, \lambda), L - \hat L \rangle \geq 0,\; \mbox{for}\, L \in \mathcal{L}_p,
\end{equation}
where $\mathcal{L}_n(L) = \mbox{tr}(SLL^t) - 2\log|L|$ and $\nabla \rho(\cdot,\cdot)$ is the subgradient.
\begin{theorem} \label{t:stprop}
Under Assumptions A1-A3, with tuning parameter $\lambda$ of scale $\sqrt{\frac{\log p}{n}}$, and $\frac{3}{4 \gamma} < (\kappa + 1)^{-2}$, the scaling $(s+p)\log p = o(n)$ is sufficient for any stationary point $\hat L$ of the non-convex program $Q_{RRCF}$ to satisfy the following estimation bounds:
\[
\begin{aligned}
\|\hat L - L\|_F &= \mathcal{O}_p \Big (\sqrt{\frac{(s + p)\log p}{n}} \Big ) \\
\|\hat \Sigma^{-1} - \Sigma^{-1}\|_F &= \mathcal{O}_p \Big (\sqrt{\frac{(s + p) \log p}{n}} \Big ) \\
\end{aligned}
\]
\end{theorem}
The proof is provided in Supplementary.
\section{Conclusion}
The present paper proposes two-step algorithm to learn a DAG from the regularized score function when data are generated from a Gaussian, linear SEM. The first step introduces a permutation matrix as a new parameter to represent variable ordering.
For its estimation, we utilize a relaxation technique in which we relax the non-convex space of permutation matrices by the convex space of doubly stochastic matrices. Then choose the ``closest'' permutation matrix to the optimal doubly stochastic matrix.
In the second step, given the variable ordering, the algorithm estimates a Cholesky factor, which entails the DAG structure. For each step, we provide necessary conditions that guarantee convergence of the proposed algorithm. The numerical results study the advantage and potential applications of the algorithm in recovering DAG structure.
As future work, we left the proof of the statistical consistency of the RRCF estimator when the permutation matrix $P$ is unknown, selection of the penalty parameter $\mu$ when $n <<p$, and the possible extension of the proposed method to learn DAGs from the non-linear SEMs.
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\section{Proofs} \label{a:proof}
\subsection*{Proof of Proposition~\ref{p:prop1}}
Before delving into details, we define two important concepts that will be used in the proof: majorization and Schur-convexity. For details, see \cite{marshall11}.
\begin{definition}
For $x,y \in \mathcal{R}^p$, $x$ is said to be majorized by $y$
\[ x \prec y \; \mbox{if}\, \begin{cases} \sum_{i =1}^k x_{[i]} \leq \sum_{i=1}^{k} y_{[i]}, & k=1,\dots p-1 \\
\sum_{i=1}^p x_{[i]} = \sum_{i = 1}^p y_{[i]},
\end{cases}\]
where $x_{[1]} \geq \dots \geq x_{[n]}$ denotes the components of $x$ in decreasing order.
\end{definition}
\begin{definition}
A real-valued function $f$ defined on a set $\mathcal{A} \subset \mathcal{R}^p$ is said to be Schur-convex on $\mathcal{A}$ is
\[x \prec y \; \mbox{on} \; \mathcal{A} \Rightarrow f(x) \leq f(y) \]
\end{definition}
Now, since from \cite[page 20]{marshall11} a Schur-convex function is permutation invarient, we only need to show that $\ell(L,I| \mathbf{X})$ is Schur-convex. That is we show if $\mathbf{x} = \mbox{vec}(\mbox{X}) \prec \mathbf{y} = \mbox{vec}(\mbox{Y})$ then $\ell(L,I| \mathbf{X}) \leq \ell(L,I| \mathbf{Y})$, where for the $n \times p$ matrix $X$, the $np \times 1$ vector $ \mbox{vec}(X)$ is its standard vectorization formed by stacking up its column vectors.
From (\ref{eq:parloglik}),
\[\ell(L,I| \mathbf{X}) = \mathbf{x}^t (L^tL \otimes I) \mathbf{x} + C,\]
where $C$ contains terms constant with respect to $X$. Thus, to proof the Schur-convexity, we need to show that $\mathbf{x}^t (L^tL \otimes I) \mathbf{x} \leq \mathbf{y}^t (L^tL \otimes I) \mathbf{y}$, which easily follows from the \cite[Chapter 4, Proposition B.9]{marshall11}.
\subsection*{Proof of Lemma~\ref{l:dsprop}}
For the proof of only if side; i.e., $P$ is a permutation matrix or $P = J/p$, the equality holds trivially from the definition of the Frobenius norm \cite[Chapter 5.6]{Horn2012}.
For the if part, it is known that the maximum spectral radius and the spectral norm $\|P\|_2$ of the doubly stochastic matrix $P \in \mathcal{D}_p$ are equal 1 \cite[Chapter 8.7]{Horn2012}. From the matrix norm inequality \cite[Corollary 5.4.5]{Horn2012}
\begin{equation} \label{eq:frbds}
1 = \|P\|^2_2 \leq \|P\|^2_F =\sum_{j = 1}^{p}\sigma^2_j(P) \leq p \|P\|^2_2 \leq p,
\end{equation}
where $\sigma_j(\cdot)$ is the $j$th singular value. Thus, $\|P\|_F = \|P\|_2 = 1$ if and only if it is a matrix of rank one; i.e., $P = J/p$ from the \cite[Theorem 4]{marcus1962}.
On the other hand, it is easy to see that $\|P\|^2_F = p$ equality holds if and only if $\sigma_j(p) = 1,\, j= 1, \dots, p$, that is $P \in \mathcal{P}_p$, and any doubly stochastic matrix with Frobenius norm $\sqrt{p}$ is a permutation matrix.
\subsection*{Proof of Corollary~\ref{c:optsol}}
We write for $P \in \mathcal{D}_p$
\[
\begin{aligned}
\mbox{tr}(LPSP^tL^T) & = \mbox{vec}(P)^t(L^tL \otimes S) \mbox{vec}(P)\\
&\geq \lambda_1(L^tL \otimes S)\|P\|^2_F,
\end{aligned}
\]
where $\lambda_1(A)$ is the smallest eigenvalue of the symmetric matrix $A$ , $\mbox{vec}(\cdot)$ is the usual matrix vectorization operator and the minimum value achieved when $P = J/p$ from the Lemma~\ref{l:dsprop}.
\subsection*{Proof of Lemma~\ref{l:l1}}
\paragraph*{(a):} The Hessian of the objective function can be found by noting that
\[
\begin{aligned}
\frac{1}{2} \mbox{tr}(LPSP^tL^t) - \frac{1}{2}\mu\|P\|^2_F = &\frac{1}{2}\mbox{vec}(P)^t (L^tL \otimes S) \mbox{vec}(P)\\
&- \frac{1}{2} \mu \mbox{vec}(P)^t \mbox{vec}(P).
\end{aligned}
\]
Thus the Hessian is
\[L^tL \otimes S - \mu \mathbf{I},\]
and the result easily follows from the definition of convexity \cite{Boyd2004}.
\paragraph*{(b):} We first show that the transformation (\ref{p:trstoc}) is equivalent to (\ref{p:stocort}). Following \cite{fogel2013}, we write
\[\|TP\|^2_F = \mbox{tr}(P^tT^tTP) = \mbox{tr}(P^tP - \mathbf{1}\mathbf{1}^t/p) = \|P\|^2_F - 1, \]
where we use idempotent property of the projection matrix $T$. Thus, (\ref{p:trstoc}) has the same objective function as (\ref{p:stocort}) up to a constant.
To show convexity we look on the Hessian of the objective function. Note that
\[
\begin{aligned}
\frac{1}{2}\mbox{tr}(LPSP^tL^t) - \frac{1}{2}\mu\|TP\|^2_F &= \frac{1}{2}\mbox{vec}(P)^t (L^tL \otimes S) \mbox{vec}(P) \\
&- \frac{1}{2} \mu \mbox{vec}(P)^t(\mathbf{I} \otimes T) \mbox{vec}(P),
\end{aligned}
\]
where $vec(\cdot)$ is the usual matrix vectorization operator. Thus the Hessian is
\begin{equation} \label{e:hess}
L^tL \otimes S - \mu \mathbf{I} \otimes T
\end{equation}
and under $\mu \leq \lambda_2(S)\lambda_1(L^tL)$ convexity holds.
\paragraph*{(b):}
Similarly, from (\ref{e:hess}) if $\mu > \lambda_m(S)\lambda_m(L^tL)$ then the objective function of (\ref{p:trstoc}) is concave. Thus, from the \cite[Corrollary 8.7.4]{Horn2012} the minimum of concave function over the set of doubly stochastic matrices is attained at a permutation matrix and the proof follows.
\subsection*{Proof of Lemma~\ref{l:lbound}}
We start by assuming contradiction. The proof exploits strategy introduced in \cite{rothman2008}. Let for $\tilde P \in \mathcal{D}_p$ belonging to a Birkhoff polytope
\begin{equation} \label{e:lbound1}
\begin{aligned}
Q(\tilde P) &= \mbox{tr}(L \tilde P S \tilde P^t L^t) - \frac{1}{2} \mu \|\tilde P\|^2_F \\
& - \frac{1}{2} tr(LPSP^tL^t) - \frac{1}{2} \mu \|P\|^2_F\\
&=\mbox{tr}(L(\tilde P - P)S(\tilde P - P)^t L^t)\\
&- \frac{1}{2}\mu(\|\tilde P\|^2_F - \|P\|^2_F),
\end{aligned}
\end{equation}
where $P \in \mathcal{P}_p$ is a true permutation matrix, $\|P\|^2_F = p$ from Lemma~\ref{l:dsprop}.
Our estimate $\hat P$ minimizes $Q(\tilde P)$, or equivalently $\hat \Delta = \hat P - P$ minimizes $G(\Delta) = Q(P + \Delta)$, where
$\Delta = \tilde P - P$. Under convexity condition in Lemma~\ref{l:l1}(a), $G(\Delta)$ is a convex function and $G(\hat \Delta) \leq G(0) = 0$.
Next we introduce the set
\[\Theta_n = \{\Delta: \|\Delta\|_F = r_n\},\]
where $r_n \rightarrow 0$.
Now if we show that $\inf \{G(\Delta): \Delta \in \Theta_n\} >0$ then $\hat \Delta \in \Theta_n$ and $\|\hat \Delta\|_F \leq r_n$.
We write
\begin{equation} \label{e:lbound2}
G(\Delta) = I + II,
\end{equation}
where $I = \mbox{tr}(L \Delta S \Delta^t L^t)$ and $II = \frac{\mu}{2} (\|P\|^2_F - \|P + \Delta\|^2_F)$.
For the part $I$,
\begin{equation} \label{e:lbound5}
\begin{aligned}
I = \mbox{tr}(L^tL \Delta S \Delta^t) &= \mbox{vec}(\Delta)^t (L^tL \otimes S) \mbox{vec}(\Delta)\\
&\geq \lambda_1(L^tL \otimes S) \| \Delta \|^2_F
\end{aligned}
\end{equation}
To find lower bound for $II$, we denote by $C = \{(i,j):P_{ij} = 1\}$ non-zero entries of the permutation matrix $P$. We also note that the cardinality $|C| = p$. Thus,
\[
\begin{aligned}
\|P + \Delta\|^2_F &= \sum_{(i,j) \in C}|1 + \Delta_{ij}|^2 + \sum_{(m,n) \not \in C} |\Delta_{mn}|^2\\
&= p + 2 \sum_{(i,j) \in C}|\Delta_{ij}| + \|\Delta\|^2_F
\end{aligned}
\]
and
\begin{equation} \label{e:lbound7}
\begin{aligned}
\|P + \Delta\|^2_F - \|P\|^2_F &= 2 \sum_{(i,j) \in C}|\Delta_{ij}| + \|\Delta\|^2_F \\
& \leq 2p \max_{i,j}|\Delta_{ij}| + \|\Delta\|^2_F\\
&\leq 2p + \|\Delta\|^2_F ,
\end{aligned}
\end{equation}
where we used the norm inequality $\|A\|_{max} \leq \|A\|_F$ and $\max_{i,j}|\Delta_{ij}| \leq 1$.
It follows from (\ref{e:lbound5}) and \ref{e:lbound7} the lower bound for (\ref{e:lbound2}) is
\begin{equation} \label{e:lbound8}
\begin{aligned}
G(\Delta) & \geq \lambda_1(L^tL \otimes S) \|\Delta\|^2_F - 2p \mu - \mu \|\Delta\|^2_F \\
& = \|\Delta\|^2_F ( \lambda_1(L^tL \otimes S) - \mu - \frac{2p}{r^2_n}\mu)
\end{aligned}
\end{equation}
Thus, $G(\Delta) > 0$ condition holds if and only if
\[ \lambda_1(L^tL \otimes S) - \mu - \frac{2p}{r^2_n}\mu > 0, \]
from which follows that
\[\mu < \frac{ \lambda_1(L^tL \otimes S) }{1 + \frac{2p}{r^2_n}}\]
and from $r_n \rightarrow 0$, it follows $\mu \rightarrow 0$, which contradicts the initial statement.
\subsection*{Proof of Lemma~\ref{l:xstar}}
We start by writing for $1 \leq j \leq k -1$
\begin{equation} \label{eq:objfncj}
h_{k,A,\lambda,\gamma} = x_j^2A_{jj} + 2x_j (\sum_{l \neq j} A_{lj}x_l) + \rho(|x_j|, \lambda,\gamma) + C_j,
\end{equation}
where $C_j$ includes terms independent of $x_j$. Taking derivative with respect to $x_j$ and noting that the subdifferential
\begin{equation}\label{eq:mcp}
\rho^{'}(|x_j|, \lambda, \gamma) = \begin{cases} \lambda s - \frac{x_j}{\gamma} & |x_j| < \gamma \lambda \\
0 & |x_j| \geq \gamma \lambda
\end{cases}
\end{equation}
Here, the subgradient $s = \mbox{sign}(x_j)$ if $x_j \neq 0$ and take values in $[-1,1]$ otherwise. Thus it follows
\[x^*_j = \frac{S_\lambda(-2\sum_{l \neq j} A_{lj}x_l)}{2A_{jj} - 1/\gamma}\]
Similarly,
\[h_{k,A,\lambda,\gamma} = x_k^2A_{kk} + 2x_k (\sum_{l \neq k} A_{lj}x_l) -2\log x_k+ C_k,\]
where $C_k$ includes terms independent of $x_k$ and after taking derivatives with respect to $x_k$
\[
\begin{aligned}
&\frac{-2}{x_k} + 2x_kA_{kk} + 2 \sum_{l \neq k}A_{lk}x_l = 0 \iff \\
& x^2_k A_{kk} + \sum_{l \neq k}A_{lk}x_l x_k -1 =0,
\end{aligned}
\]
and (\ref{eq:xk}) follows after retaining the positive root of the above quadratic equation.
\subsection*{Proof of Lemma~\ref{l:convanal}}
\paragraph*{(a):}
From (\ref{e:gen})
\[h_{k,A,\lambda,\gamma}(x) \geq 2x_k - 2,\]
where we used that $A$ is positive semidefinite and $x^tAx \geq 0,\; \rho(|x_i|, \lambda, \gamma) \geq 0$ and for $y > 0$, $\log y \leq 1 - y$.
\paragraph*{(b):}
We denote by $D_{u}h$ and $D^2_{u}h$ the derivative and the second derivative of $h$ in the direction of $u$. Thus, the proof follows from (\ref{eq:objfncj}) and (\ref{eq:mcp}) by writing
\[\min \{D^2_{\beta^{-}_{i}}h_{k,A,\lambda,\gamma}(\beta),D^2_{\beta^{+}_{i}}h_{k,A,\lambda,\gamma}(\beta)\} \geq 2A_{ii} - \frac{1}{\gamma} \]
\paragraph*{(c):}
Note that
\[\inf_{L \in \mathcal{L}_p}Q_{RRCF}(L) \geq -2p > - \infty\]
directly follows from the part (b) of the proof. Moreover, from (\ref{eq:rrcf1}) if $|\beta^i_j| \rightarrow \infty$ or $\beta^i_i = 0$, then $Q_{RRCF} \rightarrow \infty$. Therefore, any local minimum of $Q_{RRCF}$ over the open set $\mathcal{L}_p$ lies in $\mathcal{L}_p$.
\subsection*{Proof of Theorem~\ref{t:conv}}
For the proof of Theorem~\ref{t:conv}, we exploit \cite[Theorem 5.1]{tseng2001}, where the author established sufficient conditions for the convergence of cyclic coordinate descent algorithms to coordinate-wise minima. The strict convexity of (\ref{eq:rrcfi}) with respect to each coordinate direction and lower boundedness established in Lemma~\ref{l:convanal} satisfy the required conditions in Theorem 5.1. Thus, convergence to a coordinate-wise minimum point is guaranteed. Moreover, since all directional derivatives exist, every coordinate-wise minimum is also a local minimum.
\subsection*{Conditions for the penalty function} \label{pr:condF}
The penalty function $\rho(\cdot, \lambda)$ satisfies the following conditions:
\begin{itemize}
\item The function $\rho(\cdot, \lambda)$ satisfies $\rho(0,\lambda) = 0$ and is symmetric around zero.
\item On the non negative real line, the function $\rho(\cdot, \lambda)$ is nondecreasing.
\item For $t > 0$, the function $t \rightarrow \rho(\cdot, \lambda) / t$ is nonincreasing in $t$.
\item The function $\rho(\cdot, \lambda)$ is differentiable for all $t \neq 0$ and subdifferentiable at $t = 0$, with $\mbox{lim}_{t \rightarrow 0^+} \rho'(t, \lambda) = \lambda C$.
\item There exists $\mu > 0$ such that $\rho_{\mu}(t, \lambda) = \rho(t, \lambda) + \frac{\mu}{2}t^2$ is convex.
\end{itemize}
\subsection*{Proof of Theorem~\ref{t:stprop}}
We start by showing that $\mathcal{L}_n$ satisfies RSC conditions. Recall that the differentiable function $\mathcal{L}_n: \mathcal{R}^{p \times p} \rightarrow \mathcal{R}$ satisfies RSC condition if:
\[
\langle \nabla \mathcal{L}_n(L + \Delta) - \nabla \mathcal{L}_n(L), \Delta \rangle \geq
\]
\begin{numcases}{\geq}
\alpha_1 \| \Delta \|^2_F - \tau_1 \frac{\log p}{n} \| \Delta \|^2_1, & $\forall \| \Delta \|_F \leq 1$ \label{eq:rsc1}\\
\alpha_2 \| \Delta \|_F - \tau_2 \sqrt{\frac{\log p}{n}} \| \Delta \|_2, & $\forall \| \Delta \|_F \geq 1$ \label{eq:rsc2}
\end{numcases}
where the $\alpha_j$'s are strictly positive constants and the $\tau_j$'s are nonnegative constants. From \cite[Lemma 4]{loh2015}, under conditions of Theorem~\ref{t:stprop}, if (\ref{eq:rsc1}) holds then (\ref{eq:rsc2}) holds. Thus, we concentrate only on showing that (\ref{eq:rsc1}) holds for $\| \Delta \|_F \leq 1$.
Recall that
\begin{equation} \label{e:ln}
\mathcal{L}_n(L) = \mbox{tr}(SL^tL) - 2\log|L|
\end{equation}
\begin{lemma} \label{l:rsc}
The cost function (\ref{e:ln}) satisfies RSC condition (\ref{t:stprop}) with $\alpha_1 = (\kappa + 1)^{-2}$ and $\tau_1 = 0$; i.e.,
\begin{equation} \label{eq:rscp}
\langle \nabla \mathcal{L}_n(L + \Delta) - \nabla \mathcal{L}_n(L), \Delta \rangle \geq (\kappa + 1)^{-2} \| \Delta \|^2_F, \: \forall \| \Delta \|_F \leq 1
\end{equation}
\end{lemma}
The proof is provided in Appendix~\ref{a:rsc}.
From the penalty conditions listed above, $\rho_{\mu}(L,\lambda) = \rho(L, \lambda) + \frac{\mu}{2} \|L\|^2_F$ is convex, where in case of MCP $\mu = 1/ \gamma$. Thus,
\[
\begin{aligned}
\rho_{\mu}(L,\lambda) - \rho_{\mu}(\hat L,\lambda) &\geq \langle \nabla \rho_{\mu}(\hat L,\lambda), L - \hat L \rangle \\
&=\langle \nabla \rho(\hat L,\lambda) + \mu \hat L, L - \hat L \rangle,
\end{aligned}
\]
which implies that
\begin{equation} \label{eq:rho}
\langle \nabla \rho( \hat L,\lambda), L - \hat L \rangle \leq \rho( L,\lambda) - \rho( \hat L,\lambda) + \frac{\mu}{2} \|\hat L - L\|^2_F
\end{equation}
From stationarity condition (\ref{eq:stat})
\[\langle \nabla \mathcal{L}_n (\hat L), L - \hat L \rangle \geq - \langle \nabla \rho(\hat L, \lambda), L - \hat L \rangle\]
and combining above result with (\ref{eq:rscp})
\[ \begin{aligned}
(1 + \kappa)^{-2}\|\Delta\|^2_F &\leq \langle \mathcal{L}_n(\hat L), \Delta \rangle - \langle \nabla \mathcal{L}_n(L), \Delta \rangle \\
& \leq \langle \nabla \rho(\hat L, \lambda), L - \hat L \rangle - \langle \nabla \mathcal{L}_n(L), \Delta \rangle \\
& \leq \rho(L, \lambda) - \rho(\hat L, \lambda) + \frac{\mu}{2} \|\hat L - L\|^2_F \\
&- \langle \nabla \mathcal{L}_n(L), \Delta \rangle
\end{aligned}
\]
After rearrangement and H{\''o}lder inequality
\[
\begin{aligned}
\Big((1 + \kappa)^{-2}) - \frac{\mu}{2} \Big) \|\Delta\|^2_F & \leq \rho(L, \lambda) - \rho(\hat L, \lambda)\\ & + \|\nabla \mathcal{L}_n(L)\|_{\infty} \|\Delta\|_1
\end{aligned}
\]
From \cite[Lemma 4]{loh2015}
\[\lambda \|\Delta\|_1 \leq \rho(\Delta,\lambda) + \frac{\mu}{2}\|\Delta\|^2_F\]
and from \cite[Lemma 15]{yu2017} under the assumed scaling of $\lambda$
\[\|\nabla \mathcal{L}_n(L)\|_{\infty} \leq \frac{\lambda}{2}\]
with probability going to 1. Combining above two results and using subadditive property; i.e., $\rho(\Delta,\lambda) \leq \rho(L, \lambda) + \rho(\hat L, \lambda)$:
\[
\begin{aligned}
\Big( (1 + \kappa)^{-2} - \frac{\mu}{2} \Big ) \|\Delta\|^2_F &\leq \rho(L, \lambda) - \rho (\hat L, \lambda) + \frac{\lambda}{2} \|\hat \Delta\|_1\\
& \leq \rho(L, \lambda) - \rho (\hat L, \lambda) \\
&+ \frac{\rho (\Delta, \lambda)}{2} + \frac{\mu}{4} \|\Delta\|^2_F \\
& \leq \rho(L, \lambda) - \rho(\hat L, \lambda)\\
&+ \frac{\rho (L, \lambda) + \rho(\hat L, \lambda)}{2} + \frac{\mu}{4} \|\Delta\|^2_F
\end{aligned}
\]
After rearranging and using $3/4\mu \leq (1+ \kappa)^{-2} $
\begin{equation} \label{eq:cnd1}
0 \leq \Big( (1 + \kappa)^{-2} - \frac{3}{4}\mu \Big) \|\Delta\|^2_F \leq 3 \rho(L, \lambda) - \rho(\hat L, \lambda)
\end{equation}
From (\ref{eq:cnd1}) and \cite[Lemma 5]{loh2015} follows
\[ \rho(L, \lambda) - \rho(L, \lambda) \leq 2\lambda \|\Delta_S\| - \lambda \|\Delta_{S^c}\| \Rightarrow \|\Delta_{S^c}\|_1 \leq 3 \|\Delta_S\|_1\]
Thus,
\[
\begin{aligned}
\Big(2 (1 + \kappa)^{-2} - \frac{3}{2}\mu \Big) \|\Delta\|^2_F & \leq \lambda \|\Delta_S\|_2 - \lambda \|\Delta_{S^c}\|_1 \\
& \leq \lambda \|\Delta_S\|_1 \leq \lambda \sqrt{p + s} \|\Delta\|_F,
\end{aligned}
\]
from which we conclude that
\begin{equation} \label{eq:res1}
\|\Delta\|_F \leq \frac{6\lambda \sqrt{p + s}}{4(1 + \kappa)^{-2} - 3 \mu},
\end{equation}
and the result follows from the chosen scaling of $\lambda$.
For the precision matrix bound, from page 45 of \cite{yu2017} we note that
\[ \hat L^t \hat L - L^tL = (\hat L - L)^t (\hat L - L) + (\hat L - L)^TL + L^t (\hat L - L)\]
and
\[\|L^t (\hat L - L)\|_F \leq |||L|||_2 \|\hat L - L\|_F\]
From submultiplicativity property of matrix norm
\[\|(\hat L - L)^t (\hat L - L)\|_F \leq \|(\hat L - L)\|^2_F \]
Therefore,
\begin{equation} \label{eq:res2}
\|\hat L^t \hat L - L^tL\|_F \leq (\|\hat L - L\|_F + 2 |||L|||_2)\|\hat L - L\|_F
\end{equation}
\subsection*{Proof of Lemma~\ref{l:rsc}} \label{a:rsc}
The following facts will be useful in the proof.
\paragraph*{Fact 1}
\begin{enumerate}
\item $(K_{pp})^{-1} = K_{pp}$
\item $\lambda_{max}(K_{pp}) = 1$
\item $tr(ABCD) = vec(D^t)(C^t \otimes A)vec(B)$
\item $\lambda_{max}(A \otimes B) = \lambda_{max}(A)\lambda_{max}(B)$
\end{enumerate}
where $K_{pp}$ is the commutation matrix such that $vec(L) = K_{pp} vec(L^t)$. The proof of the facts can be found in \cite[Section 4]{magnus1986}.
To show the RSC condition, we rely on the directional derivatives (for example see \cite[Section 6.3]{tao2016}). In particular, if we denote by $D_{\Delta}\mathcal{L}_n(L)$ the directional derivative with respect to the direction $\Delta$, then from \cite[Lemma 6.3.5]{tao2016} :
\begin{equation} \label{e:dder1}
\langle \nabla \mathcal{L}_n(L), \Delta \rangle = D_{\Delta}\mathcal{L}_n(L) = 2\mbox{tr}[(SL^t - L^{-1})\Delta]
\end{equation}
Similarly
\begin{equation} \label{e:dder2}
\begin{aligned}
\langle \nabla \mathcal{L}_n(L + \Delta), \Delta \rangle &= D_{\Delta}\mathcal{L}_n(L + \Delta) \\
&= 2\mbox{tr}[(S(L+\Delta)^t - (L+ \Delta)^{-1})\Delta]
\end{aligned}
\end{equation}
From Woodbury identity \cite{Horn2012}
\[(L + \Delta)^{-1} = L^{-1} - L^{-1}\Delta(L + \Delta)^{-1}\]
Plugging back into (\ref{e:dder2}) and after some algebra
\begin{equation} \label{e:dder2.1}
\begin{aligned}
\langle \nabla \mathcal{L}_n(L + \Delta), \Delta \rangle &= 2\mbox{tr}[(S(L+\Delta)^t - (L+ \Delta)^{-1})\Delta] \\
&+ 2\mbox{tr}[S \Delta^t\Delta + L^{-1}\Delta(L + \Delta)^{-1}\Delta]
\end{aligned}
\end{equation}
Thus, from (\ref{e:dder1}) and (\ref{e:dder2.1})
\begin{equation} \label{eq:dder3}
\begin{aligned}
\langle \nabla \mathcal{L}_n(L + \Delta) &- \nabla \mathcal{L}_n(L), \Delta \rangle =\\ &= 2\mbox{tr}[\Delta^tS\Delta + L^{-1}\Delta(L+\Delta)^{-1}\Delta] \\
& \geq \mbox{vec}(\Delta)^t K_{pp}((L + \Delta)^{-t}\otimes L^{-1}) \mbox{vec}(\Delta)\\
& = \mbox{vec}(\Delta)^t [((L + \Delta)^{t}\otimes L)K^{-1}_{pp}]^{-1} \mbox{vec}(\Delta)\\
& \geq \lambda_{min}([((L + \Delta)^{t}\otimes L)K^{-1}_{pp}]^{-1})\|\Delta\|^2_F,
\end{aligned}
\end{equation}
where for the first inequality we used the fact that $S$ is positive semi-definite and the second equality follows from the Fact 1.
Now, since
\begin{equation} \label{eq:preres}
\begin{aligned}
\lambda_{min}([((L + \Delta)^{t}&\otimes L)K^{-1}_{pp}]^{-1}) =\\
&= \lambda^{-1}_{max}[((L + \Delta)^{t}\otimes L)K^{-1}_{pp}] \\
&\geq \lambda^{-1}_{max}(K^{-1}_{pp}) \lambda^{-1}_{max}(L) \lambda^{-1}_{max}(L + \Delta)\\
&\geq (\kappa + 1)^{-2} ,
\end{aligned}
\end{equation}
where the first inequality follows from the submultiplicativity property of the norm and lower-triangularity of the $L$ and $\Delta$. The second inequality follows from the triangular property, the fact that $\|\Delta\|_2 \leq \|\Delta\|_F \leq 1$ and, properties of the $K_{pp}$ stated in Fact 1. After plugging (\ref{eq:preres}) into (\ref{eq:dder3}), the result follows.
\section{Algorithms and related derivations} \label{a:alg}
\subsection*{Algorithm and derivation of the closed form solution for itarates in (\ref{p:dual})} \label{a:dual}
For each iteration $k$, we use the following notation $ f(P^{(k)}) = \frac{1}{2} \|P^{(k)} - P_0\|$, and $f_*(u^{(k)},v^{(k)},U^{(k)}) = -\frac{1}{2}\|u^{(k)}\mathbf{1}^t + \mathbf{1}(v^{(k)})^t - U^{(k)}\|^2_F - tr((U^{(k)})^tP_0) $.
\begin{algorithm}[H]
\caption{Projection on doubly stochastic matrices}\label{a:dsm}
\begin{algorithmic}[1]
\State\hskip-\ALG@thistlm \emph{input}:
\State $\textit{$k_{max}, \epsilon$} \gets \textit{max. number of iteration and stopping criteria}$
\State $\textit{$U^{(0)} \in R^{p \times p}_{+}, u^{(0)},v^{(0)} \in R^{p}$} \gets \textit{Initial dual variables}$
\State $\textit{converged = FALSE}$
\State\hskip-\ALG@thistlm \emph{while converged == False $\;\textit{and}\; k < k_{max}$}:
\State $\quad U^{(k)} \gets \max \{0, u^{(k-1)}\mathbf{1}^t + \mathbf{1}(v^{(k-1)})^t - P_0\}$
\State $\quad u^{(k)} \gets \frac{1}{p}(P_0 \mathbf{1} - ((v^{(k-1)})^t \mathbf{1}+ 1)\mathbf{1} + U^{(k)} \mathbf{1}) $
\State $\quad v^{(k)} \gets \frac{1}{p}(P^t_0 \mathbf{1} - ((u^{(k)})^t\mathbf{1} + 1)\mathbf{1} + (U^{(k)})^t \mathbf{1})$
\State $\quad \textit{Update primal variable:\;} P^{(k)} = P_0 - u^{(k)}\mathbf{1}^t - \mathbf{1}(v^{(k)})^t + U^{(k)}$
\State $\quad \textit{If}\; |f(P^{(k)}) - f_*(u^{(k)},v^{(k)},U^{(k)})| < \epsilon$
\State $\quad \quad \textit{converged = TRUE}$
\State $\quad \textit{else}$
\State $ \quad \quad k = k+1$
\State\hskip-\ALG@thistlm \emph{Output}:$\; \textit{Doubly Stochastic Matrix}\; P $
\end{algorithmic}
\end{algorithm}
The \textbf{convergence of Algorithm~\ref{a:dsm} is guaranteed} since the objective function is differentiable and strictly concave in each block component when all other block components are held fixed \cite[Proposition 6.5.2]{bertsekas2015}.
\subsection*{Dual function derivation}
The Lagrangian of (\ref{p:proj}) is \cite{bertsekas2015}
\[
\begin{split}
\mathcal{L}(P,u,v,U) &= \frac{1}{2}\|P - P_0\|^2_F + u^t (P \mathbf{1} - \mathbf{1} )\\
&+ v^t(P^t \mathbf{1} - \mathbf{1}) - tr(U^t P)
\end{split}\]
The dual objective function is defined as
\begin{equation} \label{e:dual}
\mathcal{L}_{*}(u,v,U) = \inf_{P}\mathcal{L}(P,u,v,U)
\end{equation}
Thus,
\[ P = P_0 - u \mathbf{1}^t - \mathbf{1}v^t + U \]
Plugging this back into (\ref{e:dual})
\[ \begin{aligned}
\mathcal{L}_{*}(u,v,U) &= \frac{1}{2}\|u\mathbf{1}^{t} + \mathbf{1}v^{t} - U\|^{2}_F + u^{t} (P_0 \mathbf{1} -\mathbf{1})\\
&+ v^t(P_0^t \mathbf{1} - \mathbf{1}) - tr(U^{t}P_0) \\
& -\mbox{tr}(u^{t}u\mathbf{1}^{t}\onevec + u^{t}\onevec v^{t} \onevec + u^{t}U\onevec + v^{t}\onevec u^{t}\onevec \\
&+ v^{t}v\onevec^{t}\onevec + v^{t}u^{t}\onevec \\
& - U^{t}u\onevec^t - U^{t}u\onevec^t - U^t \onevec v^t - U^{t}U)
\end{aligned}
\]
Then the (\ref{p:dual}) follows by noting that the expression in the trace function is equal to $\|u\mathbf{1}^t - \mathbf{1}v^t + U\|^2_F$. Now taking derivative with respect to $u,v, U$, the corresponding expressions for iterations in the Algorithm~\ref{a:dsm} follows.
\section{Tuning parameter selection} \label{a:ebic}
We use extended BIC (eBIC) criterion \cite{foygel2010} for the tuning parameters $\theta = (\lambda, \gamma)$ selection in Algorithm~\ref{a:RRCF}. Ideally, we want to tune the parameters for each update of $\hat L^{(k)}$ in line 6, however, when the convergence takes more iteration this approach is computationally costly. In practice, the tuning parameters are selected before starting the iterations \cite{ye2019}, hence a particular scoring function is fixed throughout the algorithm.
The eBIC criterion takes the form
\[\mbox{BIC}_{\gamma_{BIC}}(\mathcal{S}(L)) = -2 \mathcal{L}_n(\hat L) + s\log n + 4 s\gamma_{bic} \log p,\]
where $\mathcal{S}(L)$ is the support of matrix $L$, $s = |\mathcal{S}(L)|$, $\mathcal{L}_n(\hat L)$ is the maximized log-likelihood function and $\gamma_{BIC} \in [0,1]$. A larger value of $\gamma_{BIC}$ results to a stronger penalization of $L$, and the case $\gamma_{BIC} = 0$ corresponds to the classical BIC. In general, the value of $\gamma_{eBIC}$ is unknown, but relying on simulation results, authors suggest $\gamma_{eBIC} = 0.5$ as a candidate value.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,262 |
Q: Asp Net Core, Custom Model Binding for nested Models not working In Summary: I Want to replace IFormFile IFormFileCollection with my own classes not attached to Asp .Net because my view models are in different project with poco classes. My custom classes are ICommonFile, ICommonFileCollection, IFormFile (not Asp .net core class) and IFormFileCollection.
i will share it here:
ICommonFile.cs
/// <summary>
/// File with common Parameters including bytes
/// </summary>
public interface ICommonFile
{
/// <summary>
/// Stream File
/// </summary>
Stream File { get; }
/// <summary>
/// Name of the file
/// </summary>
string Name { get; }
/// <summary>
/// Gets the file name with extension.
/// </summary>
string FileName { get; }
/// <summary>
/// Gets the file length in bytes.
/// </summary>
long Length { get; }
/// <summary>
/// Copies the contents of the uploaded file to the <paramref name="target"/> stream.
/// </summary>
/// <param name="target">The stream to copy the file contents to.</param>
void CopyTo(Stream target);
/// <summary>
/// Asynchronously copies the contents of the uploaded file to the <paramref name="target"/> stream.
/// </summary>
/// <param name="target">The stream to copy the file contents to.</param>
/// <param name="cancellationToken">Enables cooperative cancellation between threads</param>
Task CopyToAsync(Stream target, CancellationToken cancellationToken = default(CancellationToken));
}
ICommonFileCollection.cs
/// <inheritdoc />
/// <summary>
/// Represents the collection of files.
/// </summary>
public interface ICommonFileCollection : IReadOnlyList<ICommonFile>
{
/// <summary>
/// File Indexer by name
/// </summary>
/// <param name="name">File name index</param>
/// <returns>File with related file name index</returns>
ICommonFile this[string name] { get; }
/// <summary>
/// Gets file by name
/// </summary>
/// <param name="name">file name</param>
/// <returns>File with related file name index</returns>
ICommonFile GetFile(string name);
/// <summary>
/// Gets Files by name
/// </summary>
/// <param name="name"></param>>
/// <returns>Files with related file name index</returns>
IReadOnlyList<ICommonFile> GetFiles(string name);
}
IFormFile.cs
/// <inheritdoc />
/// <summary>
/// File transferred by HttpProtocol, this is an independent
/// Asp.net core interface
/// </summary>
public interface IFormFile : ICommonFile
{
/// <summary>
/// Gets the raw Content-Type header of the uploaded file.
/// </summary>
string ContentType { get; }
/// <summary>
/// Gets the raw Content-Disposition header of the uploaded file.
/// </summary>
string ContentDisposition { get; }
}
IFormFileCollection.cs
/// <summary>
/// File Collection transferred by HttpProtocol, this is an independent
/// Asp.net core implementation
/// </summary>
public interface IFormFileCollection
{
//Use it when you need to implement new features to Form File collection over HttpProtocol
}
I finally created my model binders successfully, i will share it too:
FormFileModelBinderProvider.cs
/// <inheritdoc />
/// <summary>
/// Model Binder Provider, it inspects
/// any model when the request is triggered
/// </summary>
public class FormFileModelBinderProvider : IModelBinderProvider
{
/// <inheritdoc />
/// <summary>
/// Inspects a Model for any CommonFile class or Collection with
/// same class if exist the FormFileModelBinder initiates
/// </summary>
/// <param name="context">Model provider context</param>
/// <returns>a new Instance o FormFileModelBinder if type is found otherwise null</returns>
public IModelBinder GetBinder(ModelBinderProviderContext context)
{
if (context == null) throw new ArgumentNullException(nameof(context));
if (!context.Metadata.IsComplexType) return null;
var isSingleCommonFile = IsSingleCommonFile(context.Metadata.ModelType);
var isCommonFileCollection = IsCommonFileCollection(context.Metadata.ModelType);
if (!isSingleCommonFile && !isCommonFileCollection) return null;
return new FormFileModelBinder();
}
/// <summary>
/// Checks if object type is a CommonFile Collection
/// </summary>
/// <param name="modelType">Context Meta data ModelType</param>
/// <returns>If modelType is a collection of CommonFile returns true otherwise false</returns>
private static bool IsCommonFileCollection(Type modelType)
{
if (typeof(ICommonFileCollection).IsAssignableFrom(modelType))
{
return true;
}
var hasCommonFileArguments = modelType.GetGenericArguments()
.AsParallel().Any(t => typeof(ICommonFile).IsAssignableFrom(t));
if (typeof(IEnumerable).IsAssignableFrom(modelType) && hasCommonFileArguments)
{
return true;
}
if (typeof(IAsyncEnumerable<object>).IsAssignableFrom(modelType) && hasCommonFileArguments)
{
return true;
}
return false;
}
/// <summary>
/// Checks if object type is CommonFile or an implementation of ICommonFile
/// </summary>
/// <param name="modelType"></param>
/// <returns></returns>
private static bool IsSingleCommonFile(Type modelType)
{
if (modelType == typeof(ICommonFile) || modelType.GetInterfaces().Contains(typeof(ICommonFile)))
{
return true;
}
return false;
}
}
FormFileModelBinder.cs
/// <inheritdoc />
/// <summary>
/// Form File Model binder
/// Parses the Form file object type to a commonFile
/// </summary>
public class FormFileModelBinder : IModelBinder
{
/// <summary>
/// Expression to map IFormFile object type to CommonFile
/// </summary>
private readonly Func<Microsoft.AspNetCore.Http.IFormFile, ICommonFile> _expression;
/// <summary>
/// FormFile Model binder constructor
/// </summary>
public FormFileModelBinder()
{
_expression = x => new CommonFile(x.OpenReadStream(), x.Length, x.Name, x.FileName);
}
/// <inheritdoc />
/// <summary>
/// It Binds IFormFile to Common file, getting the file
/// from the binding context
/// </summary>
/// <param name="bindingContext">Http Context</param>
/// <returns>Completed Task</returns>
// TODO: Bind this context to ICommonFile or ICommonFileCollection object
public Task BindModelAsync(ModelBindingContext bindingContext)
{
dynamic model;
if (bindingContext == null) throw new ArgumentNullException(nameof(bindingContext));
var formFiles = bindingContext.ActionContext.HttpContext.Request.Form.Files;
if (!formFiles.Any()) return Task.CompletedTask;
if (formFiles.Count > 1)
{
model = formFiles.AsParallel().Select(_expression);
}
else
{
model = new FormFileCollection();
model.AddRange(filteredFiles.AsParallel().Select(_expression));
}
bindingContext.Result = ModelBindingResult.Success(model);
return Task.CompletedTask;
}
}
Actually Everything is working good except when i have Nested Models. I share an example of my models I'm using and I'll do some comments with working scenarios and which don't
Test.cs
public class Test
{
//It's Working
public ICommonFileCollection Files { get; set; }
//It's Working
public ICommonFileCollection Files2 { get; set; }
//This is a nested model
public TestExtra TestExtra { get; set; }
}
TestExtra.cs
public class TestExtra
{
//It's not working
public ICommonFileCollection Files { get; set; }
}
Actually when i make a request to my API I've got the following (Screenshot):
I'm sharing a screenshot of my postman request too for clarifying my request is good.
If there is any subjection to make this work with nested model it would be great.
A: Asp Net Core Model Binder won't bind model with one property only, if you have one property in a class, that property will be null but when you add two or more will bind it. my mistake i had one one property in a nested class. The entire code is correct.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,145 |
\section{Introduction}
Recent analyses of meteorites confirm the presence
of Fe isotopes in the early solar nebula (Tachibana \& Huss 2003).
This fact suggests
that the Sun was formed in a high-mass star forming region that
has expericenced at least one supernova explosions
(Hester et al. 2004).
It is important to understand
the formation process of low-mass stars in environments under the influence
of massive stars.
Bright-rimmed clouds (BRCs) are considered to be such sites located in
the peripheries of H\emissiontype{II} regions excited by massive stars.
Sugitani, Tamura, Ogura (1995) obtained near-infrared (NIR) images of
44 bright-rimmed clouds which harbor IRAS point sources in the SFO catalog
(Sugitani, Fukui, Ogura 1991). They
found that small NIR clusters are situated from the IRAS source toward
the tips of some BRCs but mostly behind their bright rims. This asymmetric
distribution of the clusters strongly suggests star formation propagation from
the side of the H\emissiontype{II} region toward the IRAS source due
to UV from massive
stars, i.e., small-scale sequential star formation. This idea was further
supported by a grism spectroscopic survey of H$\alpha$ emission stars
of BRCs (Ogura, Sugitani, Pickles 2002).
However, the limiting magnitude of Sugitani et al. (1995) was not
deep enough to
detect sub-solar mass young stellar objects (YSOs).
In fact, the detected sources are considered to be mainly YSOs of
$\sim$1-2 \MO
based on their $K$-band magnitude.
It is necessary to make deeper NIR observations in order to fully reveal
the cluster members associated with the BRCs and to examine the details of
star formation there.
BRC 14 (IC 1848A) is considered to be one prominent
example of such sequential star formation.
The bright rim of BRC 14 is an ionization front at the interface
between the hot ionized gas of the H\emissiontype{II} region IC 1848 (S 199)
and the cold dense material of the molecular cloud, where
a bright infrared source AFGL 4029 (IRAS 02575+6017) is embedded.
BRC 14 is a part of active high-mass star
forming region IC 1848, whose age is estimated to be $\leqq$
1 Myrs (Harris 1976; Feinstein et al. 1986; Deharveng et al. 1997).
Its photometric distance is 2.2 $\pm$ 0.2 kpc
(Becker \& Fenkart 1971; Moffat 1972; Deharveng et al. 1997).
Here, we present the results of the deep NIR observations of BRC 14.
Based on the photometry of NIR sources including those of sub-solar masses,
we will show clear evidence for the sequential star formation of this
cloud located
in the periphery of IC 1848.
\section{Observations \& Data Reduction}
The observations were carried out in 2001 August 26 with the
Simultaneous-3 color InfraRed Imager for Unbiased Survey (SIRIUS)
mounted on the 2.2 m telescope of University of Hawaii.
SIRIUS takes $JHKs$ images simultaneously
using three 1024 $\times$ 1024
arrays with a pixel scale of $\timeform{0.288''}$.
Details of the camera are described in Nagashima et al. (1999) and
Nagayama et al. (2003).
One quadrant of the $J$-band array was not functional at the time of this
observation.
We imaged a $\timeform{4.9'}\times\timeform{4.9'}$ area
centered on $\alpha=\timeform{03h01m31s}, \delta=\timeform{+60D28'52''}$
(J2000), and the same size field with $\timeform{-100''}$ offset in RA,
in order to compensate for failure of the $J$-band array.
We obtained 30 dithered images per each area with an exposure time
of 30 s for each frame.
A part (9.8 arcmin$^{2}$) has twice as much total integration
time as else.
Sky frames, offset $(\alpha, \delta) =
(\timeform{-120''}, \timeform{+500''})$
from BRC 14, were taken just before and after the object frames.
We observed the standard star P9107 (Persson et al. 1998)
for photometric calibration.
The typical seeing size was measured to be $\timeform{1.0''}$ at $Ks$.
Dome flat frames were obtained at the beginning of the night.
To reduce the data, we used the Imaging Reduction and Analysis Facility
(IRAF) software.
First, linearity was corrected for all frames.
After subtraction of the median-combined sky frame,
each image was divided by the normalized flat frame.
Then we combined them into one frame for each band.
To identify NIR sources, we used the DAOFIND task in IRAF
with a 3 $\sigma$ detection threshold above the background.
The APPHOT package was used for photometry with a $\timeform{3.5''}$
diameter aperture (6 pixel radius).
The annular sky area was chosen with an inner radius of $\timeform{2.9''}$ and
an outer radius of $\timeform{4.3''}$.
All magnitudes are transformed into the CIT $JHK$ system.
The 10 $\sigma$ limiting magnitudes for the photometry
are 19.2, 18.5, and 17.7
at the $J$, $H$, and $K$ bands, respectively.
Bright sources with $J,H\lesssim12$, or $K\lesssim11$ are saturated.
The coordinates of the objects
are determined by three stars listed in the USNO-B1.0 catalog with
the accuracy of about $\timeform{0.5''}$.
\section{Results}
The $JHKs$ composite image of BRC 14 is shown in figure \ref{image}.
The bright rim of BRC 14 is clearly seen in the NIR image.
The position of the NIR rim is coincident with the rim in the optical
wavelengths.
Nebular emission components of AFGL 4029,
IRS 1 (west) and IRS 2 (east), are much brighter in the NIR wavelengths
than in the optical wavelengths.
Many invisible sources are also detected in the NIR wavelengths.
We have detected 607 sources within 10 $\%$ photometric accuracies
in all three bands.
We select YSO candidates, based on their NIR excess
originated from a circumstellar envelope and/or disk.
An upper-left panel of Figure \ref{col-col} is the ($J-H$, $H-K$)
color-color diagram for the entire survey region.
We define three regions in the color-color diagram (Itoh et al. 1996).
The ``P'' region is the region between the reddening lines extending from
the loci of main-sequence stars and giants.
The objects plotted in this region are interpreted as main-sequence stars,
giants, supergiants, Class III sources, or Class II sources with
small NIR excess.
The ``D'' region, where Class II sources are mainly plotted,
is sandwiched between the ``P'' region and the reddening line projected
from the point of $(J-H, H-K)=(1.1, 1.0)$.
This point corresponds to the reddest intrinsic color of
classical T Tauri stars (CTTSs, Meyer et al. 1997).
Redward of the ``D'' region is the ``E'' region, in which Class I sources
are plotted.
We classify YSO candidates with NIR-excess (hereafter NIR-YSO candidates)
as the objects plotted in the ``D'' or ``E'' region
only if its 1 $\sigma$ photometric error bars lie entirely within those regions.
With this criterion, 74 sources are identified as NIR-YSO candidates
(table E1).
The objects are identified between the previous NIR observations
and our observations,
if differences of the position and magnitudes of the object
are less than $\timeform{0.5"}$ and 0.5 mag.
No NIR-YSO candidate corresponds to the NIR sources in Deharveng et al. (1997),
because most of them are plotted
in the ``P'' (field star) region and some bright sources are saturated
in our observation.
Ogura et al. (2002) found 44 emission-line stars in our field of view.
Among them, only 10 objects are identified as NIR-YSO candidates by our NIR survey.
One reason for this low frequency
is that weak-line T Tauri stars have the similar NIR color to dwarfs.
26 emission-line stars have weak H$\alpha$ emission line
(W(H$\alpha$)$<$ 10 \AA).
The other reason for low frequency is that even for CTTSs
half of them cannot be distinguished from field stars only with NIR photometry
(Meyer et al. 1997).
\section{Discussion}
The H$\alpha$ emission stars are found as an aggregate in the vicinity of
the bright rim with offset toward the exciting stars of S 199
(Ogura et al. 2002).
The alignment of the objects from west to east,
i.e. the exciting stars, the H$\alpha$ stars,
the bright rim, and AFGL 4029, implies that star formation proceeds from
the H\emissiontype{II} region to the AFGL 4029.
In order to conform the propagation of star formation,
we investigate characteristics of the YSO candidates in three areas. Each area has
about 1 arc-minute square as shown in figure \ref{image}.
The area A centered on ($\alpha$, $\delta$)=($\timeform{03h 01m 33s}$,
$\timeform{+60D 29' 19''}$) contains the luminous source AFGL 4029.
The area B is in the bright rim and west of the area A.
The area C is west of the rim.
Figure \ref{col-col} represents color-color diagrams of the
objects in the three areas.
We discuss three indicators; extinctions of all sources,
fractions of the NIR-YSO candidates,
and NIR-excesses of
the NIR-YSO candidates (table \ref{parameters}).
Average of the extinctions of the sources in the area
represents the column density of the molecular cloud.
An extinction of each Class II source is derived from the distance
between the intrinsic color of CTTSs and the observed color
on the color-color diagram (Meyer et al. 1997).
On the other hand,
it is difficult to estimate the extinctions of the field-stars.
Since we do not know spectral types of field-stars, we can not
distinguish between a late-type dwarf with small extinction and an
early-type dwarf with large extinction.
A star count model of the Galaxy (e.g. Jones et al. 1981) predicts
that the majority of the field-stars detected by such a deep NIR survey
are late-type dwarfs.
We assume that the intrinsic color of the field-stars is
the line connecting the intrinsic colors of M4V and K7V.
The sources we detected are categorized into NIR-YSOs, YSO without NIR excess,
and field stars.
We estimate the number of the field stars in each area.
First we count the 2MASS sources down to $K < 14$ mag in
the region $(l,b)=(\timeform{138D.3}, \timeform{-1D.56})$,
the opposite side of the Galactic plane.
Then, we calculate the number of the field stars using a star count model of
the Galaxy (Jones et al. 1987).
The number of the field star is overestimated, because
the model is valid only for high galactic latitute.
The number is well reproduced (difference $<$ 5 \%),
if we take an additional extinction of $A_{\rm K}=0.2$ mag.
By extrapolating this model and taking average extinction
in each area, we estimate the number of field star to be
15, 18, and 24 in the areas A, B, and C, respectively.
The numbers of YSOs are then estimated to be 50, 24, and 32 in the
areas A, B, and C, respectively.
Fraction of NIR-YSO candidates is a ratio of the number of NIR-YSO candidates
to the estimated number of YSOs (see table 1).
Haisch et al. (2001) analyzed infrared ($JHKL$)
colors for several young clusters
then obtained the relation that the fraction of NIR-YSOs
to all YSOs decreases with increasing the age of the cluster.
NIR-excess index of each YSO candidate is also derived from the color-color diagram.
We define the index as the distance from the color of the object to
the boundary between the ``P'' and ``D'' regions (the PD boundary),
which is then normalized by the distance between the PD boundary and
the boundary between the ``D'' and ``E'' regions (the DE boundary).
An object with the NIR-excess of 0 is plotted on the PD boundary,
while that with the NIR-excess of 1 is plotted on the DE boundary.
The NIR-excess is a function of many parameters, such as
accretion rate, an inner radius of a disk,
and an inclination angle (Hillenbrand et al. 1998).
Assuming random distribution in inclination,
we consider that accretion rate and an inner radius of a disk
mainly influence on the NIR-excess.
As discussed in Oasa et al. (2006), younger YSOs have larger NIR-excesses.
If a cluster is as young as the associated YSOs which have optically-thick
disks and are heavily embedded in the parent molecular cloud,
all indicators above are expected to be large.
In contrast, if a cluster is as old as the YSOs which do not
have optically-thick circumstellar disks and are no longer heavily
embedded, all indicators should be small.
In BRC 14, all indicators decrease in order of the areas A, B, and C.
Many sources in the area A have large extinctions,
and the area A contains many NIR-YSO candidates.
In the area B, many sources also have large extinctions,
but the fraction of the NIR-YSO candidates as well as the
NIR-excesses of the YSO candidates are
smaller than those in the area A.
In the area C, the fraction of the NIR-YSO candidates is small,
and the sources with large extinctions are less abundant compared to
the areas A and B.
Although these indicators have large uncertainties,
all indicators are large for the area A and small for the area C,
implying that the area A is young and the area C is relatively old.
We emphasize that all indicators of the area A are significantly larger
than those of the areas B and C.
The extinctions of all sources and the NIR-excesses of
the YSO candidates are also shown as functions of
the right ascension of the objects (figure \ref{RA}).
We notice that the extinctions of the sources and
the NIR excesses of the YSO candidates increase toward the east.
The extinctions change significantly at the rim.
The figures again indicate that the YSOs with circumstellar structures
are associated with the molecular cloud.
We claim that the sequential star formation previously proposed by the
optical study (Ogura et al. 2002) also occurs inside of the rim
where NIR light can penetrate.
In all areas, low-mass objects are formed.
To make a census of low-mass YSOs in BRC 14,
we construct an extinction-corrected
NIR luminosity function of the Class II candidates
plotted in the ``D'' region in the color-color diagram (figure \ref{lf}).
Note that heavily embedded YSO candidates cannnot be detected
even if their intrinsic luminosity is brighter
than the limiting magnitude.
To remove this bias of extinction, we discuss $J$-band luminosity function
only for the Class II sources with $A_{\rm V} \leqq 8$ mag.
As $A_{\rm J} = 0.288 A_{\rm V}$ (Bessell \& Brett 1988), $A_{\rm V} = 8$ mag
corresponds to $A_{\rm J} = 2.3$ mag.
The apparent $J$-band limiting magnitude is then 16.9 mag, when
we consider the extinction bias.
With the distance to BRC 14 of 2.2 kpc,
the limiting magnitude in the $J$-band luminosity
function is about 5.2 mag in absolute
magnitude.
By the 1 Myr isochrone of the evolutionary track of low-mass
objects (Baraffe et al. 1998),
the limiting magnitude corresponds to 0.12\MO.
Figure \ref{lf} indicates that
luminous YSO candidates are located only in the area A.
They might be younger or massive sources.
On the other hand, low-luminosity YSO candidates are found in all of the
three areas.
An aggregate of NIR sources has been discovered by
Sugitani et al. (1995).
Its asymmetric distribution leads them to an idea of
sequential star formation in the BRC 14 region,
though only qualitative discussion was presented.
Ogura et al. (2002) found many H$\alpha$ emission stars in the
BRC 14 region. Most of them are located outside of the rim,
i.e. in the H II region between the exciting star and the rim. They
proposed small scall sequential star formation from the
exciting star to the rim.
However, only a small number of the emission-line stars were detected inside
the rim due to heavy extinction of the cloud.
We detect the aggregate of NIR sources inside the rim.
By applying the YSO indicators as above,
we firstly find quantitative evidence that
low-mass stars,
including sub-solar mass ones, in BRC 14
sequentially form from the outside of the rim to the inside of the molecular
cloud.
\bigskip
\bigskip
\bigskip
We are grateful to S. Sato, T. Nagata, A. J. Pickles, Y. Nakajima,
H. Nakaya, T. Nagayama, C. Nagashima, for the observations.
This work is supported by "The 21st Century COE program: The Origin and
Evolution of Planetary Systems" of the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT). Y.I. is supported by a
Grant-in-Aid for Scientific Research No. 16740256.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,578 |
package org.araqne.logstorage.repair;
import java.io.BufferedReader;
import java.io.BufferedWriter;
import java.io.Closeable;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileOutputStream;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.OutputStreamWriter;
import java.util.concurrent.ConcurrentHashMap;
import java.util.concurrent.ConcurrentMap;
import java.util.concurrent.locks.Condition;
import java.util.concurrent.locks.Lock;
import java.util.concurrent.locks.ReentrantLock;
import org.apache.felix.ipojo.annotations.Component;
import org.apache.felix.ipojo.annotations.Provides;
import org.apache.felix.ipojo.annotations.Validate;
import org.araqne.logstorage.LogFileRepairer;
import org.araqne.logstorage.LogFileRepairerRegistry;
import org.araqne.logstorage.LogFileRepairerService;
import org.osgi.framework.BundleContext;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
@Component(name = "logstorage-log-file-repairer-registry")
@Provides
public class LogFileRepairerRegistryImpl implements LogFileRepairerRegistry {
private final static Logger logger = LoggerFactory.getLogger(LogFileRepairerRegistryImpl.class);
private BundleContext bc;
private ConcurrentHashMap<String, WaitEvent> availableRepairers = new ConcurrentHashMap<String, WaitEvent>();
private ConcurrentMap<String, LogFileRepairerService> repairers = new ConcurrentHashMap<String, LogFileRepairerService>();
public LogFileRepairerRegistryImpl(BundleContext bc) {
this.bc = bc;
String opt = System.getProperty("araqne.logstorage.repair");
if (opt != null && (opt.equalsIgnoreCase("disabled") || !Boolean.parseBoolean(opt)))
bc.registerService(IntegrityChecker.class.getName(), new DummyIntegrityChecker(), null);
}
@Override
public void register(LogFileRepairerService repairer) {
String type = repairer.getType();
logger.info("araqne logstorage: loaded file repairer [{}]", type);
repairers.put(type, repairer);
WaitEvent ev = new WaitEvent(type, true);
WaitEvent old = availableRepairers.putIfAbsent(type, ev);
if (old != null) {
old.setReady();
} else {
rewriteRepairerListFile();
}
}
@Override
public void unregister(LogFileRepairerService repairer) {
String type = repairer.getType();
WaitEvent ev = availableRepairers.get(type);
if (ev == null)
throw new UnsupportedOperationException("not supported repairer: " + type);
ev.ready = false;
repairers.remove(type);
logger.info("araqne logstorage: unloaded file repairer [{}]", type);
}
@Override
public void uninstall(String type) {
WaitEvent ev = availableRepairers.remove(type);
if (ev == null)
throw new IllegalStateException("not installed repairer: " + type);
rewriteRepairerListFile();
}
@Override
public String[] getRepairerTypes() {
return availableRepairers.keySet().toArray(new String[0]);
}
@Override
public String[] getInstalledTypes() {
return availableRepairers.keySet().toArray(new String[0]);
}
@Override
public LogFileRepairer newRepairer(String type) {
WaitEvent ev = availableRepairers.get(type);
if (ev == null)
throw new UnsupportedOperationException("not supported repairer: " + type);
ev.await();
LogFileRepairerService logFileRepairerService = repairers.get(type);
return logFileRepairerService.newRepairer();
}
@Validate
public void start() throws IOException {
File f = getRepairerListFile();
if (!f.exists())
return;
// load file engine list
FileInputStream is = null;
BufferedReader br = null;
try {
is = new FileInputStream(f);
br = new BufferedReader(new InputStreamReader(is, "utf-8"));
while (true) {
String line = br.readLine();
if (line == null)
break;
String type = line.trim();
availableRepairers.put(type, new WaitEvent(type));
}
} finally {
ensureClose(br);
ensureClose(is);
}
}
private void ensureClose(Closeable c) {
if (c != null) {
try {
c.close();
} catch (IOException e) {
}
}
}
private File getRepairerListFile() {
return bc.getDataFile("repairer.lst");
}
private void rewriteRepairerListFile() {
// update engine list file
FileOutputStream fos = null;
BufferedWriter bw = null;
try {
File f = getRepairerListFile();
f.delete();
fos = new FileOutputStream(f);
bw = new BufferedWriter(new OutputStreamWriter(fos, "utf-8"));
for (String name : availableRepairers.keySet())
bw.write(name + "\n");
} catch (IOException e) {
logger.error("araqne logstorage: cannot update engine list", e);
} finally {
if (bw != null) {
try {
bw.close();
} catch (IOException e) {
}
}
if (fos != null) {
try {
fos.close();
} catch (IOException e) {
}
}
}
}
// XXX : copy from LogFileServiceRegistryImpl
private static class WaitEvent {
private String type;
private Lock lock = new ReentrantLock();
private Condition cond = lock.newCondition();
private volatile boolean ready;
public WaitEvent(String type) {
this(type, false);
}
public WaitEvent(String type, boolean ready) {
this.type = type;
this.ready = ready;
}
public void await() {
if (ready)
return;
lock.lock();
try {
while (!ready) {
try {
logger.info("araqne logstorage: waiting file engine [{}]", type);
cond.await();
} catch (InterruptedException e) {
}
}
} finally {
lock.unlock();
}
logger.info("araqne logstorage: file engine [{}] ready!", type);
}
public void setReady() {
ready = true;
lock.lock();
try {
cond.signalAll();
} finally {
lock.unlock();
}
}
}
private class DummyIntegrityChecker implements IntegrityChecker {
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,264 |
\section{Introduction}
\label{sec:intro}
Ever since Pieter Zeeman (1897) discovered the splitting of spectral lines
in a magnetic field, astrophysicists have tried to take advantage of the
diagnostic capabilities of this effect. Thanks to the Zeeman effect, G.E. Hale
(1908) was able to demonstrate the presence of strong magnetic fields in
sunspots and over the years countless Zeeman observations (of ever increasing
precision) of lines originating in all parts of the solar atmosphere have
provided deep insights into the physics of the outer layers of the sun. The
discovery by Friedrich Paschen and Ernst Back (1921) that in strong fields
some transitions change from the anomalous Zeeman effect to the normal Zeeman
effect, has not resulted in any revision of the traditional interpretation of
the observations, because even in the strongest solar magnetic fields (about
0.4\,T) only very few lines are noticeably affected. Not until almost 50 years
later was the question of the Paschen-Back (PB) effect in the solar spectrum first
addressed by Engvold et al. (1970). Similarly, after the discovery by
H.W. Babcock, starting 1947, of very strong magnetic fields in some upper-main
sequence chemically peculiar stars (Ap stars), no serious thought was given to
the Paschen-Back effect. Although it is clear that in a 3.4\,T field like
that of HD\,215441 (Babcock 1960), many spectral lines would exhibit the
transition from the Zeeman to the PB regime (which we shall call the incomplete
or partial PB effect), the first papers on the Paschen-Back effect in a stellar
context appeared only much later (Kemic 1975; Stift 1977).
Recently, interest in the Paschen-Back effect has revived, mainly in the
solar context and for molecular lines. As to atomic transitions, let us
note that the He\,{\sc i}\,10830\,{\AA} triplet has received special attention
(Socas Navarro et al. 2004), as has for example the Mn\,{\sc i}\,15262.7\,{\AA}
line (Asensio Ramos et al. 2007). In the stellar context, the work of Mathys
(1990) constitutes a particularly detailed modelling attempt and discussion of
the incomplete PB effect in the Fe\,{\sc ii} lines at $\lambda$\,6147.7
and $\lambda$\,6149.2. The PB effect on hyperfine structure and its
observational consequences have been investigated by Landolfi et al. (2001).
It should be kept in mind that these results have generally been obtained in the
Milne-Eddington approximation, and that heavy blending as often found in Ap stars
could not be taken into account.
Our interest in the Paschen-Back effect has been stimulated by the difficulties
we encountered when trying to match high resolution spectral observations of
strongly magnetic Ap stars with synthetic spectra (see Fig.\,\ref{fig:shift}).
The Zeeman doublet of Fe\,{\sc ii} at 6149\,{\AA} in particular not only
displays non-symmetric relative intensities of the components -- already noted
by Mathys (1990) -- but it also proves impossible to model the velocity
shift (relative to the reference RV determined from magnetic null lines of iron)
within the Zeeman regime. From Fig.\,\ref{fig:shift} it emerges that even at the
very moderate field strength of 0.36\,T this shift is already observable.
Assuming that the Paschen-Back effect is responsible for both asymmetries and
shifts, the question naturally arises whether the measurement of the
magnetic field modulus $H_s$ (the absolute field strength integrated over the
visible hemisphere), which is based on the interpretation of the observed
splitting of the $\lambda\,6149$ line in terms of simple Zeeman splitting, may
be systematically affected by the PB effect. This has not been studied so far,
and we are also not aware of any discussion in the literature of the observed
shift of the $\lambda\,6149$ line.
In the past, neither computers nor codes were powerful enough to embark on
any ambitious program of modelling stellar Stokes profiles in the incomplete
PB regime. Now that multi-processor and/or multi-core architectures
have become affordable, COSSAM (Stift 2000) -- a state-of-the-art object-oriented
and fully parallel Stokes code -- has been modified in such a way as to allow
the calculation for realistic stellar atmospheres of a multiplet in the
incomplete PB regime, with full blending from the remaining spectral
lines which are assumed to display classical anomalous Zeeman patterns.
These tools make it possible to systematically explore how the observed Stokes
profiles are affected by the partial PB effect, to have a close look
at the diagnostic content of Paschen-Back lines, and to model in detail selected
spectral intervals in Ap stars with very strong fields. First important results
are presented in this paper.
\begin{figure}
\includegraphics[width=84mm]{HD66318_l6149.eps}
\includegraphics[width=84mm]{HD188041_l6149.eps}
\caption{
RV-corrected Stokes\,$I$ spectra in the vicinity of the Fe\,{\sc ii} lines at 6147
and 6149\,{\AA} of the stars HD\,66318 (top) and HD\,188041 (bottom). The magnetic
field modulus is 1.5\,T for the former star, 0.36\,T for the latter. The dashed
red lines give the zero field positions of the lines of Cr, Fe and Nd. The full
red lines are plotted for the only purpose of illustrating the line shift in
$\lambda\,6149$; they result from a straightforward spectral synthesis, assuming
normal Zeeman splitting. The RV has been determined from the magnetic null lines
($g_{\rm eff} = 0.0$) of Fe\,{\sc i} at $\lambda\,5434.52$ and $\lambda\,5576.09$.}
\label{fig:shift}
\end{figure}
\section{The incomplete Paschen-Back effect}
\label{sec:paschen}
Named after the two German physicists Friedrich Paschen and Ernst Back, this
effect generalises to magnetic fields of arbitrary strengths the better known
Zeeman effect. The effect was discovered in the laboratory in various
multiplets (Paschen \& Back 1921), including the
$4s \, 4d \, ^3D \rightarrow 4s \, 4p \, ^3P$ multiplet of Zn\,{\sc i} that was later
studied in major detail by van Geel (1928). The Paschen-Back effect has been
successfully interpreted within the framework of quantum mechanics, and
nowadays this interpretation can be found in classical textbooks of atomic
and/or molecular spectroscopy (see e.g. Condon \& Shortley 1935).
In order to recall the basic physical facts, let us consider a term of an atom,
characterised by the quantum numbers $L$ and $S$. The term is composed of
$2S+1$ (or $2L+1$ if $L < S$) $J$-levels whose energy separation is due to
the spin-orbit interaction. When a weak magnetic field is present, each
$J$-level splits into $2J+1$ magnetic sublevels that can be identified by
the further quantum number $M$. In as far as the magnetic splitting is much lower
than the fine-structure separation between different $J$-levels, such splitting
turns out to be proportional to the magnetic field, and the atom is said to be
in the Zeeman effect regime. However, when the magnetic field starts to be
comparable, or even larger than the fine-structure separation between
$J$-levels, the linearity property is lost and the atom enters the
regime of the (incomplete) Paschen-Back effect. Here the situation is
more complicated because the quantum number $J$ is no longer a good quantum
number and the atomic levels can thus be characterised only by the magnetic
quantum number $M$, not by $J$.
The energy values of the levels (the eigenvalues) and the corresponding
eigenvectors can be obtained by diagonalisation of a set of matrices.
The details of this procedure and the relevant equations are
described in full detail in Sect. 3.4 of the monograph by Landi Degl'Innocenti
\& Landolfi (2004). In the same book one can also find the expressions for the
splitting and the strengths of the various components ($\sigma_{\rm red}, \,
\pi, \; {\rm and} \; \sigma_{\rm blue}$) that arise in the transition between
a lower and an upper term, both in the Paschen-Back regime. The remarkable fact
is the appearance of some extra-components that can be referred to as
"satellite" or "ghost" components. These components are strictly forbidden
in the absence of a magnetic field, and have negligible strengths in the
Zeeman effect regime, because of the selection rule
$\Delta J = \pm \, 1, \, 0$, $0 \rightarrow \!\!\!\!\!\!\! / \;\;\; 0$.
Referring for instance to the above-mentioned multiplet of Zn\,{\sc i}, the
multiplet contains 6 lines in the Zeeman regime ($J_{\rm low} = 0 \rightarrow
J_{\rm up} = 1, \,
1 \rightarrow 1, \, 1 \rightarrow 2, \, 2 \rightarrow 1, \ 2 \rightarrow 2,
\, 2\rightarrow 3$), but the number of lines increases to 9 in the Paschen-Back
regime. At the same time, the number of magnetic components increases
from 54 (18 $\sigma_{\rm red}$, 18 $\pi$, and 18 $\sigma_{\rm blue}$) in the
Zeeman regime to 71 (23 $\sigma_{\rm red}$, 25 $\pi$, and 23
$\sigma_{\rm blue}$) in the Paschen-Back regime.
The fact that strong
magnetic fields are capable of bringing out lines which violate the ordinary
selection rule on $J$ was indeed the most important hint which led to the
original discovery of the Paschen-Back effect. Examples of this behaviour
and further properties concerning the strengths and the splittings of the
components are described in the following sections of the present paper.
\begin{figure}
\includegraphics[width=84mm]{fe_split.eps}
\caption{
Magnetic splittings of the lowermost 2 levels of the ${^4}D$ term
of Fe\,{\sc ii} multiplet 74 as a function of field strength.}
\label{fig:6147_49_split}
\end{figure}
\section{Energy level splittings and relative component intensities}
\label{sec:energy}
The theory of the incomplete Paschen-Back effect reveals that both
the splittings of the energy levels in a multiplet and the relative
intensities of the subcomponents can change in a non-linear way with
the magnetic field strength. Whether or not this happens at field
strengths typical for Ap stars (0.1 - 3.5\,T) depends on the detailed
fine structure splittings of the terms involved. It is useful to
recall that a field strength of 1\,T corresponds to a splitting of
$0.47\,g\,M\,{\rm cm}^{-1}$, where $g$ denotes the Land{\'e}-factor
and $M$ the magnetic quantum number. Taking as an example the
${^4}D$ term of Fe\,{\sc ii} multiplet 74 (which gives rise to
the well-known lines at $\lambda\,6147$ and at $\lambda\,6149$),
the magnetic splitting of the $J = 3/2$ level can exceed the
distance to the neighbouring $J = 1/2$ level for fields of about
4.7\,T. From detailed calculations it emerges that deviations from
simple Zeeman splitting occur much earlier: very close scrutiny of
the respective positions of the blue and red doublet components
relative to the zero field wavelength reveals a difference of about
20\,m{\AA} (the total splitting is 330\,m{\AA}) at only 0.7\,T (see
Fig.\,\ref{fig:6147_49_split}).
The famous Li\,{\sc i}\,$\lambda\,6707$ doublet is another interesting
case that has attracted some attention over the past decades
(let us mention Engvold et al. (1970), Mathys (1991), and Leone (2007)).
Its fine structure splitting amounts to a mere 0.34\,cm$^{-1}$
and can be exceeded by magnetic splitting already at field strengths
of 0.5\,T. Again, detailed calculations show that deviations from
simple Zeeman splitting become clearly visible somewhat earlier than
0.2\,T (Fig.\,\ref{fig:li_split}). The surprise however comes
when one looks at the relative intensities of the 4 sub-components of
$\lambda$\,6707.912: already at a bare 0.05\,T, the two $\pi$-components
differ by 20\,\%, and the respective $\sigma$-components by 10\,\%! At
the still very moderate field strength of 0.1\,T, these values rise to
44\,\% and 20\,\%.
\begin{figure}
\includegraphics[width=84mm]{li_split.eps}
\caption{
Wavelength splittings and relative intensities of the subcomponents of
the Li\,{\sc i} $\lambda\,6707$ doublet as a function of magnetic
field strength. For clarity, the respective $\pi$ components are
plotted in black and red, the $\sigma_{\rm blue}$ components in magenta
and blue, and the $\sigma_{\rm red}$ components in green and brown.}
\label{fig:li_split}
\end{figure}
How many spectral lines will then be subject to the partial
Paschen-Back effect in strongly magnetic Ap stars? This is no easy
question to answer but a quick search in the NIST energy level
tables already provides a substantial number of candidate terms for
some elements: there are for example 18 terms of Cr\,{\sc i} with
energies up to 52000\,cm$^{-1}$ where at least 1 pair of neighbouring
levels are separated by less than 5\,cm$^{-1}$. It is mostly possible
to identify the lines originating from these terms in the Kurucz
(1993b) line-list and the surprising result is that a staggering 18\,\%
or 2320 lines out of some 13000 may be affected by the partial Paschen-Back
effect in Ap stars with strong magnetic fields. The lowest ${^5}G$
term alone where the splittings between adjacent levels vary between
0.25\,cm$^{-1}$ and 2.77\,cm$^{-1}$ gives rise to 846 lines. Are there
important lines among these? Have they been used for example for
magnetic Doppler imaging or for abundance analyses? For the moment we
do not know; we need not be overly alarmist but the possibility cannot
be ruled out and it is not quite clear what the consequences would be.
\begin{figure}
\includegraphics[width=84mm]{fe_solar_I.eps}
\caption{
Stokes\,$I$ profiles for the Fe\,{\sc ii} lines at $\lambda\,6238.392$ and
$\lambda\,6239.953$ belonging to multiplet 74, both in the Zeeman and in the
partial Paschen-Back regime. Calculations have been carried out for a 12000\,K,
$\log g = 4.0$ Kurucz (1993a) atmosphere, the magnetic field strength is
2\,T, and the respective angles between field direction and line-of-sight are
0, 30, 45, 60, 75, and 90\degr (from top to bottom). The corresponding Zeeman
and Paschen-Back patterns of the two lines are shown in the conventional form
of vertical bars, with the $\pi$-components above and the $\sigma$-components
below the axis.}
\label{fig:6238_I}
\end{figure}
\section{Partial Paschen-Back effect and Stokes profile modelling}
\label{sec:model}
The tool for our modelling investigations into the partial
Paschen-Back effect is CossamPaschen, a new version of the
COSSAM code developed by Stift (2000). COSSAM is a state-of-the-art
line synthesis code (Wade et al. 2001) that allows the calculation
of full Stokes spectra either in the solar case (one specific point
on the solar surface) or the stellar case (spatial integration of
the profile over the visible hemisphere of the star). The code is
characterised by LTE in a plane-parallel atmosphere, continuous
opacities taken from Atlas9 or Atlas12 (Kurucz 1993a, 2005), full
component by component opacity sampling (CoCoS), and by the choice
between the Zeeman Feautrier (Alecian \& Stift 2004) and the DELO
(Rees et al. 1989) formal polarised solvers.
In the Zeeman regime where subcomponent splittings can be assumed
to scale linearly with magnetic field strength, one just has to
store the respective anomalous Zeeman patterns for each line in
the atomic data list. Splittings for any point on the stellar
surface are determined by simple multiplication with a
field-dependent factor. Relative intensities are independent of
field strength which implies that both in the solar and in the
stellar case, storage requirements are restricted to just one set
of Zeeman splittings and relative intensities of the Zeeman
subcomponents.
Not so in the partial Pachen-Back regime. We have seen that
splittings and relative subcomponent intensities go non-linearly
with field strength. For each point on the stellar surface (and
there have to be several hundreds to a few thousands of points
depending on field geometry and on rotation) one has to determine
the exact splittings and relative intensities for the given local
field strength by the usual diagonalisation of the total Hamiltonian
as outlined before. One then has the choice of synthesising
separately all the local profiles before combining them to an
integrated profile or to store the local Paschen-Back patterns in
some appropriate structure and proceed first with spatial integration
at each wavelength point. Memory requirements will be substantial,
especially in the latter case.
Since few if any of the lines of astrophysical interest which exhibit
the transitions to the Paschen-Back effect are entirely unblended,
it is important to account for blending and to include the full list
of lines appropriate to the stellar atmosphere. The final version of
CossamPaschen thus allows the modelling of the full Stokes profiles
of the transitions in one multiplet under the partial Paschen-Back
regime, blended with all other transitions found in the interval in
question and treated in the Zeeman approximation.
\begin{figure}
\includegraphics[width=84mm]{fe_solar_V.eps}
\caption{
The same as in Fig.\,\ref{fig:6238_I}, but for Stokes\,$V$. The
case of 90\degr is omitted.}
\label{fig:6238_V}
\end{figure}
\section{Stokes profiles: Zeeman vs. Paschen-Back}
Differences between Stokes profiles calculated in the simple
Zeeman regime and profiles calculated in the partial Paschen-Back
regime can range from subtle to spectacular. Zeeman components
are shifted, get stronger or fade, symmetries are destroyed,
blends start to look quite different. The lines $\lambda$\,6238.39
and $\lambda$\,6239.95 of the Fe\,{\sc ii} multiplet 74
beautifully illustrate these changes for a field strength of 2\,T.
The former line, a ${^4}D_{3/2} - {^4}P_{3/2}$ transition, and
the latter, a ${^4}D_{1/2} - {^4}P_{3/2}$ transition, have together
6 $\pi$ components and 5 $\sigma_{\rm blue}$ and $\sigma_{\rm red}$
components.
The most spectacular changes occur for a longitudinal field. The component
near 6240.4\,{\AA} disappears while at the same time the component near
6239.6\,{\AA} increases in strength (see Fig.\,\ref{fig:6238_I}). The
outer $\sigma_{\rm blue}$ component becomes stronger too, and the blend
with the $\sigma_{\rm red}$ components of $\lambda$\,6238.39 completely
changes shape. In Stokes $V$ we also note the conspicuous weakening of
the $\sigma_{\rm red}$ components of $\lambda$\,6239.95; a positive $V$
signal emerges to the right of the blend (Fig.\,\ref{fig:6238_V}).
Changes remain clearly visible in a transversal field, although
they are no longer as important as in the longitudinal case. Note that
in the Paschen-Back regime, some components hardly change intensity with
angle.
The reason for the disappearance of the component near 6240.25\,{\AA} is
easily explained: $\pi$- and $\sigma_{\rm red}$-components are almost exactly
at the same position, but the intensity of the $\sigma_{\rm red}$-component
is now by more than a factor of 30 lower than in the Zeeman regime. The
intensity of the red $\pi$-component grows by about a factor of 1.6. The inner
$\sigma_{\rm blue}$-component of $\lambda$\,6239.95 displays an intensity
increase by a factor of 2.5, the outer one by a factor of 1.8, whereas the
blue $\pi$-component weakens.
\begin{figure}
\includegraphics[width=84mm]{cog_solar_00.eps}
\caption{
Wavelengths of the respective centres-of-gravity in Stokes $I$ of the
synthesised Fe\,{\sc ii} lines at ${\lambda}\,6149$ (bottom) and at
${\lambda}\,6147$ (middle) as a function of magnetic field strength
and of abundance. The curves (black, red, green, blue, brown, magenta)
correspond to decreasing abundances in steps of 0.5\,dex. Wavelengths
of the centre-of-gravity of the 2 lines taken together are shown in
the top panel. The field is taken to be longitudinal.}
\label{fig:cog_solar_00}
\end{figure}
\section{Selected results}
There has not been in the past any systematic investigation of the
incomplete Paschen-Back effect in the Stokes spectra of strongly
magnetic Ap stars. Only the pioneering work by Mathys (1990) has
provided valuable insight and raised interesting questions. We have
tried to extend these results and we want to illustrate here some
salient points of our findings, especially concerning the
Fe\,{\sc ii} doublet at 6149\,{\AA} and its use in the determination
of the magnetic field modulus.
\subsection{Line shifts}
\label{sec:shifts}
The findings presented in Sec.\,\ref{sec:energy} warrant detailed
modelling of selected lines in order to determine the wavelength
shifts that could show up in the incomplete Paschen-Back regime.
Indeed, the notable deviations from linear Zeeman splitting in a
number of well-studied lines, together with the rapidly changing
relative intensities of the subcomponents should necessarily not
only lead to asymmetries in the line profiles but also to line
shifts of various amounts and different signs. Such shifts are
found e.g. for the Fe\,{\sc ii} line at 6149\,{\AA} in HD\,126515
(see also Figs.\,7 and 8 of Mathys (1990) and cannot be
modelled in a pure Zeeman regime.
For instructive purposes we have therefore chosen the Fe\,{\sc ii}
${\lambda}\,6147$ and ${\lambda}\,6149$ lines and calculated high
resolution synthetic profiles for various field strengths up to 3\,T
and for a number of abundances in steps of 0.5\,dex. Considering
the longitudinal and the transverse case, we obtained a grid of 228
models which cover enough of parameter space to allow a meaningful
discussion.
Based on our polarised spectral line synthesis, we note for a
longitudinal field of 2\,T and in the weak-line limit a marginal
blue-shift (-0.027\,{\AA}) of the centre-of-gravity (COG) of the
${\lambda}\,6149$ doublet and a clear red-shift (+0.11\,{\AA})
of ${\lambda}\,6147$ (see Fig.\,\ref{fig:cog_solar_00}). The COG of
the 2 lines combined remains almost unaffected (-0.01\,{\AA}). The
COGs are determined from the Stokes $I$ profiles. Interestingly, for
very strong lines and 1\,T (larger field strengths lead to blending),
the situation is partially inversed with the ${\lambda}\,6149$ doublet
slightly (+0.015\,{\AA}) red-shifted and ${\lambda}\,6147$ essentially
unchanged. The COG of the 2 combined lines however is now blue-shifted
by a remarkable -0.18\,{\AA}).
\begin{figure}
\includegraphics[width=84mm]{cog_solar_90.eps}
\caption{
The same as in Fig.\,\ref{fig:cog_solar_00} but for a transverse field.}
\label{fig:cog_solar_90}
\end{figure}
In the case of a transverse 2\,T field, the weak-line limit leads to a
noticeable (-0.07\,{\AA}) blue-shift of ${\lambda}\,6149$ and a red-shift
of exactly the same size (+0.07\,{\AA}) of ${\lambda}\,6147$ (see
Fig.\,\ref{fig:cog_solar_90}). Again, the COG of the 2 lines combined
remains almost unaffected (-0.01\,{\AA}). Very strong lines and a 1\,T field
yield a marginally (+0.01\,{\AA}) red-shifted ${\lambda}\,6149$ doublet and
${\lambda}\,6147$ unchanged. The blue-shift of the COG of the 2 lines combined
is reduced to -0.10\,{\AA}) but surprisingly we now also encounter a red-shifts
of up to 0.025\,{\AA} for fields of less than 0.3\,T.
Decidedly, the behaviour of the various lines of multiplet 74 of
Fe\,{\sc ii} in the incomplete Paschen-Back regime cannot easily be
predicted. Line shifts, intensities and asymmetries vary with magnetic
field strength, field inclination and total line opacity in the
multiplet. The same holds true for the numerous other lines in the
solar spectrum and in stellar spectra which show the incomplete
Paschen-Back effect.
\begin{figure}
\includegraphics[width=84mm]{ghost.eps}
\caption{
Splittings and relative intensities of the subcomponents of the
``ghost line'' at $\lambda$\,6408.9 (belonging to multiplet 74 of
Fe \,{\sc ii}). The respective $\sigma_{\rm blue}$, $\pi$ and
$\sigma_{\rm red}$ components are plotted in different colours.
}
\label{fig:Fig_6408}
\end{figure}
\subsection{Ghost components}
\label{sec:ghosts}
As pointed out in Condon \& Shortley (1935) and in more detail in
Landi Degl'Innocenti \& Landolfi (2004), lines appear in the
incomplete Paschen-Back regime which are normally forbidden under the
selection rules for electric dipole transitions. This has to be
ascribed to the fact that $J$ is no longer a good quantum number and
there occurs J-mixing of the various levels belonging to a particular
term. These lines are sometimes referred to as ``forbidden lines'',
but we prefer here to call them ``ghost components'' or ``ghost lines''
in order to avoid confusion with the well known forbidden lines in
astrophysics which arise from metastable levels.
Take multiplet 74 of Fe\,{\sc ii} as an example. Without a
magnetic field we have 8 transitions at $\lambda\,6456.383,$
$6416.919,$ $6407.251,$ $6247.557,$ $6239.953,$ $6238.392,$
$6147.741,$ and $6149.258$, but J-mixing leads to an additional
4 transitions at $\lambda\,6408.897,$ $6284.959,$ $6192.961,$
and $6156.642$. At a field strength of 4\,T, the total strength
of the $\lambda 6408.897$ ``ghost line'' reaches 15\% of the
strength of the nearby allowed $\lambda 6407.251$ line. Generally,
the ``ghost lines'' are extremely weak in fields of the order of
0.1\,T but their strength can become comparable to that of
many allowed components once field strengths of 10\,T are
exceeded.
This led Mathys (1990) to speculate that some ghost lines
originating in strong multiplets of abundant elements could
possibly become observable in selected Ap stars whose atmospheres
are permeated by fields in the range of $3 - 5$\,T in some parts.
We are rather sceptical as for the observability for reasons
obvious from Fig.\,\ref{fig:Fig_6408}: ``ghost lines'' can show
up only in fairly strong fields so that the observable profiles
will invariably be severely diluted unless the star is not only
endowed with a reasonably extended strong-field region but also
exhibits only weak field gradients.
\subsection{The Fe\,{\sc ii} doublet at 6149\,{\AA} and the
magnetic field modulus}
\label{seq:doublet}
Originally, the main rationale for our investigation was to establish
whether or not measurements of the magnetic field modulus $H_{s}$
from the splitting of the Fe\,{\sc ii} $\lambda\,6149$ Zeeman
doublet could be affected by the incomplete Paschen-Back regime and
possibly lead to incorrect results when interpreted in terms of
standard Zeeman splitting.
\begin{figure}
\includegraphics[width=84mm]{fe_mean.eps}
\caption{
A comparison between the mean profiles from 9 different Oblique Rotator
models as detailed in the text, calculated in the partial Paschen-Back
regime (red) and in the Zeeman regime (blue). The wavelength shift
PB minus Zeeman is -0.048\,{\AA}. The dashed profile (black) represents
the PB-profile shifted by this amount. allowing a better comparison of
the respective {\em shapes} of the profiles.}
\label{fig:fe_mean}
\end{figure}
To this end we have calculated a grid of profiles for 9 Oblique Rotator
models with various inclinations and excentricities, all with a field
modulus of $H_{s} = 1.45$\,T, but with dipole strengths ranging from
0.86 to 1.40\,T (in a centred dipole model, the polar field strengths
would be twice these values). Means were taken over the wavelengths of
the 9 respective centres-of-gravity, blue and red component
separately, both for the incomplete Paschen-Back and the Zeeman
regime. PB-profiles are found to be {\em always} red-shifted by about
0.05\,{\AA} relative to the Zeeman-profiles, but there is no change in
the splittings (see Table\,\ref{tab:COG_positions}); the value of
$H_{s} = 1.442$\,T in the former regime is nearly the same as
$H_{s} = 1.457$\,T in the latter. PB-profiles can become dramatically
asymmetric: a deep and relatively narrow blue component stands in
sharp contrast to a shallower red component characterised by a strong
and extended redward wing (Fig.\,\ref{fig:fe_mean}).
These models (and all other models calculated on a random basis) have
not revealed any undesirable behaviour of the doublet as far as the
determination of $H_s$ is concerned, although, as discussed above,
sizeable wavelength shifts are encountered. With all the necessary
caution, we deem it highly probable that these findings hold true for
all magnetic geometries.
Let us note that the scatter about the mean COG wavelengths in
Table\,\ref{tab:COG_positions} is extremely small; PB-profiles exhibit
somewhat larger differences between each other than the
Zeeman-profiles. This suggests that at least some of the lines
formed in the incomplete Paschen-Back regime can provide increased
diagnostic content for magnetic field mapping. Inclusion and correct
treatment of these lines is expected to increase the reliability of
magnetic maps.
\subsection{Doppler mapping}
\label{seq:doppler}
A few words of caution are appropriate for the aficionados of (magnetic) Doppler
imaging. In fact, if one looks at the list of 12 Cr\,{\sc i} lines used by
Kochukhov et al. (2004) for the mapping of HR\,3831, the lines at 6881.65,
6882.48, and 6883.00\,{\AA}, all belonging to multiplet 222, are definitely in
the partial Paschen-Back regime, given the
narrow wavelength spread of these 3 lines. The quite uncertain oblique
rotator model with a centred dipole proposed by these authors -- detailed
analyses based on full Stokes profiles usually find non-negligible quadrupole
contributions -- predicts a polar field strength of 0.25\,T. We calculated local
intensity profiles for this field and various angles between field vector and
line of sight. The respective Zeeman and PB intensity profiles differ by up to
20\% as can be seen in Fig.\,\ref{fig:diff}.
\begin{figure}
\includegraphics[width=84mm]{cr_diff.eps}
\caption{Intensity profiles of the lines of multiplet 222 of Cr\,{\sc i} in the
spectral interval $6881 - 6884$\,{\AA} for a 7500\,K, $\log g = 4.0$ Kurucz
(1993a) model. The magnetic field strength is 0.25\,T, the respective
angles between the field direction and the line of sight are 0, 30, 60,
and 90\degr (red, green, blue, brown). Note the perfect symmetry of the
components in the Zeeman regime (bottom) and its deformation by the partial
Paschen-Back effect (middle). The relative difference between the two profiles
reaches about 20\% (top). Since the theoretical line strengths and positions do
not perfectly match the Kurucz line data, very small differences occur even for
zero field strength (dashed black line in the top panel). For clarity we have
indicated both the wavelength and the velocity scale.}
\label{fig:diff}
\end{figure}
\begin{table}
\caption{Mean wavelength positions in the Zeeman and in the partial
Paschen-Back regime of the centres-of-gravity of the blue and red components
respectively of the $\lambda$\,6149 Zeeman doublet for different oblique
rotator geometries but a constant field modulus of $H_{s} = 1.45\,T$.
Wavelengths are given in {\AA}, the scatter $\sigma$ in m{\AA}.}
\label{tab:COG_positions}
\begin{tabular}{l|r|r|r|r|r|r}
& \multicolumn{1}{c}{${\lambda\,}^{COG}_{blue}$} & \multicolumn{1}{|c}{$\sigma$}
& \multicolumn{1}{c}{${\lambda\,}^{COG}_{red}$} & \multicolumn{1}{|c}{$\sigma$}
& \multicolumn{1}{c}{${\Delta\lambda\,}^{COG}$} & \multicolumn{1}{|c}{$\sigma$} \\
\hline
Zeeman & 6148.916 & 0.4 & 6149.602 & 0.3 & 0.686 & 0.5\\
P.-B. & 6148.968 & 0.5 & 6149.647 & 0.5 & 0.679 & 0.5
\end{tabular}\\
\end{table}
Translated to velocity space, the use of Zeeman profiles instead of the correct
PB profiles will thus result in spurious over- and under-abundances at relative
velocities ranging from $\pm 3$\,km\,s$^{-1}$ to $\pm 6$\,km\,s$^{-1}$ in these
3 lines. Note that the differences between PB and Zeeman profiles hardly depend
on the angle in the case of $\lambda\,6881.65$, whereas a strong dependency is
found for $\lambda\,6883.00$. In the latter line, differences are largest for a
longitudinal field but just the opposite is true for $\lambda\,6882.48$. Looking
at the sign of the differences, one finds a global blue shift of the
$\lambda\,6883.00$ PB profile and a global red shift of the other 2 lines.
Given this complex behaviour, it is difficult if not impossible to predict to
what degree the Cr abundance maps derived by Kochukhov et al. (2004) may have
been affected by the simplifying assumption of normal Zeeman splitting for
these 3 lines. Close scrutiny of their Fig.\,6 reveals that whereas
$\lambda\,6881.65$ and $\lambda\,6882.48$ are reasonably well reproduced by
the chromium Doppler map shown in their Fig.\,8, there are significant deviations
from the observed profiles in the vicinity of $\lambda\,6883.00$ near phases
0.30 and 0.78 (when the magnetic equator dominates the visible hemisphere and
the poles are near the limb). Given the special behaviour of $\lambda\,6883.00$,
can we attribute these differences to pure chance or is there some
responsibility of the PB effect? The question remains open and further detailed
investigations are necessary to assess the true importance of this potential
problem.
\section{Conclusions}
\label{seq:conclusion}
High resolution spectral observations of magnetic Ap stars make it clear that
the Paschen-Back effect cannot be neglected in the interpretation and modelling
of a number of spectral lines, whether it be the Li\,{\sc i} $\lambda\,6707$
resonance line or the Fe\,{\sc ii}\,$\lambda\,6149$ Zeeman doublet. Because of
its simple structure, the latter is particularly useful for the determination of
the magnetic field modulus $H_s$. We have calculated in detail the splittings of
the spectroscopic terms involved in these transitions and the relative
intensities of the subcomponents and found, somewhat to our surprise, that some
lines can enter the incomplete Paschen-Back regime already at field strengths
of a bare 0.05\,T as is the case for the lithium resonance lines. Looking
further to other chemical elements, we have discovered that in extreme cases
(Cr\,{\sc i} is an example), up to an estimated 18\% of the lines may exhibit
the transition to the partial Paschen-Back regime in Ap stars with strong
magnetic fields of the order of 2\,T.
The development of CossamPaschen, a modified version of COSSAM -- the polarised
line synthesis code established by Stift (2000) -- has made it possible
over the last few years to explore the effect of the partial Paschen-Back
regime on Stokes spectra. CossamPaschen allows the detailed and realistic
modelling of a PB multiplet, blended with lines which are treated in the Zeeman
approximation. First ever detailed Stokes profiles in realistic stellar
atmospheres have been presented at the CP\#AP Workshop in Vienna (Stift 2007)
and provide valuable insight into the sometimes spectacular changes of selected
spectral lines. These calculations show that the centres-of-gravity of the
Fe\,{\sc ii}\,$\lambda\,6149$ line and of its sister line at $\lambda\,6147$ are
shifted in magnetic fields, either towards the red or towards the blue,
depending on line strength, magnetic field strength, and field direction. A
comparison between the observations plotted in Fig.\,\ref{fig:shift} and the
theoretical results displayed in Figs.\,\ref{fig:6238_I} and \,\ref{fig:6238_V}
shows gratifying agreement.
Another interesting finding is the confirmation of the existence of normally
forbidden ``ghost components'' in strong magnetic fields. The intensity of these
components is a very strong function of magnetic field strength, and in Ap stars
some of them may actually significantly contribute to the {\em local} spectra in
places with very strong magnetic fields (of the order of 4\,T). However, the
strong dependency of intensity and position on field strength will probably lead
to the {\em global} signature of the ``ghost components'' begin washed out.
An important result of our investigation is the confirmation that $H_s$
measurements obtained from the splitting of the Fe\,{\sc ii}\,$\lambda\,6149$
line and interpreted in terms of classical Zeeman splitting do not need to
be corrected for the PB effect. For all practical purposes, even in the
strongest fields encountered in Ap stars, the PB splitting is not different from
the Zeeman splitting. As for maps derived by means of (magnetic) Doppler
mapping, we cannot provide such a reassuring confirmation and there is a real
danger that the inclusion of lines subject to the partial PB regime may lead to
spurious results. What can be a serious nuisance may also provide improved
diagnostic capabilities and so we are looking forward to improved imaging codes
incorporating the full treatment of the partial Paschen-Back effect.
\section*{Acknowledgements}
MJS acknowledges support by the {\sf\em Austrian Science Fund (FWF)},
project P16003-N05 ``Radiation driven diffusion in magnetic stellar
atmospheres''.
\bsp
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,438 |
\section{Introduction}
\label{intro}
\quad Applications of machine learning are prevalent throughout the modern world. While their tasks vary greatly in purpose and scale, all machine learning models must be trained before they can be deployed for practical use. In some cases, the training process is extremely time consuming, even on the most powerful classical computers. This is particularly true for models with NP-hard or NP-complete training problems such as $k$-means clustering \cite{Aloise2009np}, neural networks \cite{Blum1988np}, decision tree learning \cite{Hyafil1976np}, etc.
Quantum computers offer an alternative platform for efficiently solving computationally challenging problems. For instance, the D-Wave 2000Q adiabatic quantum computer approximately solves the NP-complete quadratic unconstrained binary optimization (QUBO) problem efficiently. The D-Wave quantum computer has already been used for a number of machine learning tasks including training a support vector machine \cite{Willsch2020svm}, training a restricted Boltzmann machine \cite{dixit2020rbm} \cite{date2019classical}, linear regression \cite{date2020adiabatic} and matrix factorization for feature learning \cite{O_Malley2018featurelearning}. While modern quantum computers are too small and error-prone to effectively solve large problems, their scale and fidelity are expected to improve dramatically in time \cite{Preskill2018nisq}.
In this paper, we use the D-Wave 2000Q adiabatic quantum computer to perform a special case of $k$-means clustering. $k$-means clustering is a popular machine learning model that partitions a set of $N$ data points into $k$ clusters such that each cluster is made up of similar points. Similarity is measured by the statistical variance within each cluster. We focus on balanced $k$-means clustering, which requires that each cluster contains approximately the same number of points. Balanced clustering models are used in a variety of domains including network design \cite{gupta2003networks}, marketing \cite{Ghosh2005marketing}, and document clustering \cite{Banerjee2003documentclustering}.
Classically, it is computationally challenging to find the exact solution to the balanced $k$-means training problem. Thus, existing algorithms converge after finding a locally optimal solution. In the worst case, this can still require large computational resources, especially as problem size scales. Due to these challenges, we explore the prospect of training the balanced $k$-means model on an adiabatic quantum computer. First, we outline a QUBO formulation of the balanced $k$-means clustering training problem. We then theoretically analyze our formulation, comparing our quantum approach to current classical algorithms. Next, we empirically analyze the clustering performance and scalability of our quantum approach on synthetic classification data sets. Finally, we analyze the clustering performance of our approach on portions of the Iris benchmark data set.
\section{Related Work}
\quad The $k$-means clustering model is one of the most widely used unsupervised machine learning techniques. Classically, the model is usually trained through an iterative approach known as Lloyd's algorithm. Hartigan and Wong show that the time complexity of this approach is $\mathcal{O}(Nkdi)$ where $N$ is the number of data points, $k$ is the number of clusters, $d$ is the dimension of the data set, and $i$ is the number of iterations before the algorithm converges \cite{Hartigan1979KMeans}. Arthur and Vassilvitskii prove that for random cluster initialization, $i = 2^{\Omega(\sqrt{n})}$ with high probability \cite{Arthur2006worstcasei}. Therefore, Lloyd's algorithm has superpolynomial time complexity.
Many different implementations and variations of Lloyd's algorithm have been proposed to avoid long training times or poor clustering performance. Na et al. propose an efficient implementation that reduces the number of required distance calculations without compromising clustering quality \cite{Na2010efficient}. Celebi et al. compare the impact of several different centroid initialization methods on clustering performance and run time \cite{Celebi2012initialization}. Kapoor and Singhal observe a reduction in run time and superior clustering results when sorting input data before training the $k$-means model \cite{Kapoor2017sorted}. The Scikit-learn implementation of Lloyd's algorithm bounds the number of iterations by a constant, effectively reducing the time complexity to $\mathcal{O}(Nkd)$ \cite{Pedregosa2011scikit}. We have used this implementation as a point of comparison to our quantum approach.
Constrained $k$-means models, such as balanced $k$-means clustering, are common in applications where additional knowledge regarding the training data or the form of a plausible solution is known. Sometimes constrained $k$-means models are also used in instances where the generic $k$-means algorithm is likely to converge to a suboptimal solution \cite{bradley2000constrained}. Bradley et al. propose an algorithm that enforces a minimum bound on cluster size \cite{bradley2000constrained}. This approach reduces to balanced clustering when the minimum cluster size is $\operatorname{floor}(N / k)$. Ganganath et al. present a constrained $k$-means clustering algorithm in which the size of each cluster is specified prior to training the model \cite{Ganganath2014samesizekclustering}. Malinen et al. propose an efficient balanced $k$-means clustering algorithm that runs in $\mathcal{O}(N^3)$ time \cite{Malinen2014balancedcomplexity}. This algorithm will be used as a point of comparison to our quantum approach.
Quantum approaches to training clustering models have been proposed as well. Khan et al. implement a quantum algorithm similar to Lloyd's algorithm on the IBMQX2 universal quantum computer \cite{khan2019kmeans}. Ushijima-Mwesigwa et al. demonstrate partitioning a graph into $k$ parts concurrently using quantum annealing on the D-Wave 2X machine \cite{UshijimaMwesigwa2017partitioning}. Neukart et al. propose a quantum-classical hybrid approach to clustering \cite{neukart2018quantumassisted}. Wereszczynski et al. demonstrate the performance of a novel quantum clustering algorithm on small data sets using the D-Wave 2000Q \cite{weresz2018clustering}. Bauckhage et al. propose a QUBO formulations to binary clustering ($k = 2$) \cite{bauckhage2018k2clustering} and $k$-medoids clustering \cite{bauckhage2019qubo}. Kumar et al. present a QUBO formulation for $k$-clustering that approximates the $k$-means model \cite{Kumar2018combinatorialclustering}. We have previously formulated three machine learning problems as QUBO problems \cite{date2020qubo}.
While many quantum clustering algorithms have been proposed, none target the exact solution to the $k$-means or balanced $k$-means clustering model. Instead, they are heuristic approaches that approximate the $k$-means optimization problem. We propose a QUBO formulation that is identical to the balanced $k$-means training problem. We also tested our approach on both synthetic and benchmark data.
\section{QUBO Formulation}
\quad Adiabatic quantum computers are able to find the global minimum of the quadratic unconstrained binary optimization (QUBO) problem, which can be stated as follows:
\begin{align}
\min_{z \in \mathbb{B}^M} z^T A z \label{eq:qubo}
\end{align}
where
$\mathbb{B} = \{0, 1\}$ is the set of binary numbers,
$z \in \mathbb{B}^M$ is the binary decision vector,
and $A \in \mathbb{R}^{M \times M}$ is the real-valued $M \times M$ QUBO matrix. Our goal is to convert the balanced $k$-means training problem into this form.
The $k$-means clustering model, aims to partition a data set $X = \{ x_1, x_2, ..., x_N \}$ into $k$ clusters $\Phi = \{\phi_1, \phi_2, ..., \phi_k\}$. The centroid of cluster $\phi_i$ is denoted as $\mu_i$. Formally, training the $k$-means clustering model is expressed as:
\begin{align}
\min_{\Phi} \sum_{i = 1}^k \sum_{x \in \phi_i} ||x - \mu_i||^2 \label{eq:kclusters1}
\end{align}
Utilizing the law of total variance, the training problem can be rewritten as:
\begin{align}
\min_{\Phi} \sum_{i = 1}^k \frac{1}{2|\phi_i|} \sum_{x, y \in \phi_i} || x - y ||^2 \label{eq:kclusters2}
\end{align}
In the case that each cluster is of equal size (i.e. balanced), $|\phi_i|$ is constant, and Problem \ref{eq:kclusters2} reduces to:
\begin{align}
\min_{\Phi} \sum_{i = 1}^k \sum_{x, y \in \phi_i} || x - y ||^2
\label{eq:kmeanssamesize}
\end{align}
To formulate Problem \ref{eq:kmeanssamesize} as a QUBO problem, it will be useful to define a matrix $D \in \mathbb{R}^{N \times N}$ where each element is given by $d_{ij} = ||x_i - x_j||^2$.
We also define a binary matrix $\hat{W} \in \mathbb{B}^{N \times k}$ such that $\hat{w}_{ij} = 1$ if and only if point $x_i$ belongs to cluster $\phi_j$. This use of binary variables is identical to the ``one-hot encoding" quantum clustering method proposed by Kumar et al. \cite{Kumar2018combinatorialclustering}. Since we are assuming clusters of the same size, each column in $\hat{W}$ should have approximately $N / k$ entries equal to 1. Additionally, since each data point belongs to exactly one cluster, each row in $\hat{W}$ must contain exactly one entry equal to 1. Using this notation, the inner sum in Problem \ref{eq:kmeanssamesize} can be rewritten:
\begin{align}
\sum_{x, y \in \phi_j} || x - y ||^2 = \hat{w}{'}_j^T D \hat{w}'_j
\label{eq:sumrelation}
\end{align}
where $\hat{w}'_j$ is the $j^\text{th}$ column in $\hat{W}$. From this relation, we can cast Problem \ref{eq:kmeanssamesize} into a constrained binary optimization problem.
First, we vertically stack the $Nk$ binary variables in $\hat{W}$ as follows:
\begin{align}
\hat{w} =
[\hat{w}_{11} \ldots \hat{w}_{N1} \ \hat{w}_{12} \ldots \hat{w}_{N2} \ldots \hat{w}_{1k} \ldots \hat{w}_{Nk}]^T \label{eq:kmeansz}
\end{align}
Provided the constraints on $\hat{w}$ are upheld, Problem \ref{eq:kmeanssamesize} is equivalent to:
\begin{align}
\min_{\hat{w}} \hat{w}^T (I_k \otimes D) \hat{w}
\label{eq:kmeansconstrained}
\end{align}
where $I_k$ is the $k$-dimensional identity matrix.
We can remove the constraints on $\hat{w}$ by including penalty terms that are minimized when all conditions are satisfied. First, we account for the constraint that each cluster must contain approximately $N/k$ points. For a given column $\hat{w}'_j$ in $\hat{W}$, this can be enforced by including a penalty of the form:
\begin{align}
\alpha (\hat{w}{'}_j^T \hat{w}'_j - N/k)^2
\label{eq:krow1}
\end{align}
where $\alpha$ is a constant factor intended to make the penalty large enough that the constraint is always upheld. Dropping the constant term $\alpha(N/k)^2$, this penalty is equivalent to $\hat{w}{'}_j^T \alpha F \hat{w}'_j$ where $F$ is defined as:
\begin{align}
F = 1_N - \frac{2N}{k} I_N
\end{align}
In the expression above, $1_N$ refers to an $N \times N$ matrix where each element is equal to 1. Using this formulation, the sum of all column constraint penalties is:
\begin{align}
\hat{w}^T (I_k \otimes \alpha F ) \hat{w} \label{eq:kmeanscolumns}
\end{align}
Next, we account for the constraint that each point belongs to exactly $1$ cluster. For a given row $\hat{w}_i$, this can be enforced by including a penalty of the form:
\begin{align}
\beta (\hat{w}_i^T \hat{w}_i - 1)^2
\end{align}
where $\beta$ is a constant with the same purpose as $\alpha$ in Equation \ref{eq:krow1}. Dropping the constant term, this penalty is equivalent to $\hat{w}_i^T \beta G \hat{w}_i$ where $G$ is defined as:
\begin{align}
G = 1_k - 2 I_k
\end{align}
To find the sum of all row constraint penalties, we first convert the binary vector $\hat{w}$ into the form $\hat{v}$ shown below:
\begin{align}
\hat{v} = [w_{11} \ldots w_{1k} \ w_{21} \ldots w_{2k} \ldots w_{N1} \ldots w_{Nk}]^T
\end{align}
This can be accomplished through a linear transformation $Q \hat{w}$ where each element in $Q \in \mathbb{B}^{Nk \times Nk}$ is defined as:
\begin{align}
q_{ij} =
\begin{cases}
1 & j = N \operatorname{mod}(i - 1, k) + \lfloor \frac{i - 1}{k} \rfloor + 1 \\
0 & \text{else} \\
\end{cases}
\end{align}
After the transformation, the sum of all row constraint penalties is given by $\hat{v}^T (I_N \otimes \beta G) \hat{v}$. This sum can be equivalently expressed as:
\begin{align}
\hat{w}^T Q^T (I_N \otimes \beta G) Q \hat{w} \label{eq:kmeansrow}
\end{align}
Combining the column and row penalties with the constrained binary optimization problem from Equation \ref{eq:kmeansconstrained}, Problem \ref{eq:kmeanssamesize} can be rewritten as:
\begin{align}
\min_{\hat{w}} \hat{w}^T (I_k \otimes (D + \alpha F) + Q^T (I_N \otimes \beta G) Q) \hat{w} \label{eq:finalkmeans}
\end{align}
This is identical to Equation \ref{eq:qubo} with $z = \hat{w}$ and $A = (I_k \otimes (D + \alpha F) + Q^T (I_N \otimes \beta G) Q)$. Thus, we have converted the balanced $k$-means training problem (Equation \ref{eq:kmeanssamesize}) into a QUBO problem which can be solved on adiabatic quantum computers. Provided $N$ is divisible by $k$, and $\alpha$ and $\beta$ are large enough to ensure all constraints are upheld, Problem \ref{eq:finalkmeans} and Problem \ref{eq:kmeanssamesize} share the same global solution.
\subsection{Implementation Details}
\quad In order to achieve good performance on quantum hardware, $\alpha$ and $\beta$ must be chosen such that the penalty for violating a constraint is large, but not so large as to overshadow the importance of minimizing within cluster variance. In practice we achieved the best performance when defining $\alpha$ and $\beta$ as follows:
\begin{align}
\alpha = \frac{\operatorname{max}(D)}{2(N / k) - 1} \\
\beta = \operatorname{max}(D)
\end{align}
where $\operatorname{max}(D)$ is the maximum element in $D$.
By choosing these values, we scale $F$ and $G$ such that the maximum value in each scaled matrix is equal to the maximum value in $D$. Assuming the training data set has well defined clusters, the maximum element of $D$ is much larger than the average squared distance between two points in a given cluster. Therefore, these multipliers assure that the penalty for violating a constraint is almost always larger than the penalty for a poor clustering assignment. By multiplying $F$ by a smaller factor than $G$, we also guarantee that row constraints are more strictly enforced than column constraints. This is desirable since we would like to permit small violations of the equal-size cluster constraint when $N$ is not divisible by $k$.
In practice the quantum annealing process is not perfect, and instances occur in which a point is assigned to multiple clusters or not assigned to any cluster at all. If quantum annealing assigns a point to multiple clusters, we consider the point to belong to the cluster with the smaller index. If quantum annealing does not assign a point to any cluster, we consider the point to belong to the first cluster.
\section{Results and Analysis}
\begin{figure*}[t!]
\centering
\includegraphics[scale = 0.35]{Figures/quantum_demo.png}
\caption{Example of a synthetic data set containing $N = 15$ points partitioned into 3 classes (left). The quantum algorithm correctly partitioned all but one point (right).}
\label{fig:demo}
\end{figure*}
\subsection{Theoretical Analysis}
\label{sub:theory}
\quad The generic $k$-means clustering problem stated in Equation \ref{eq:kclusters1} and the balanced $k$-means clustering problem stated in Equation \ref{eq:kmeanssamesize} both contain $\mathcal{O}(Nd)$ data and $\mathcal{O}(N)$ variables (where each variable indicates the cluster assignment of a given data point). In our QUBO formulation of balanced $k$-means clustering, we introduce $k$ binary variables for each variable in the original problem. Thus, the total number of variables in Equation \ref{eq:finalkmeans} is $\mathcal{O}(Nk)$. This translates to a quadratic qubit footprint of $\mathcal{O}(N^2 k^2)$ using an efficient embedding algorithm such as \cite{date2019efficiently}.
It has been shown to require $\mathcal{O}(N^{kd + 1})$ time to exactly solve the generic $k$-means clustering problem (Problem \ref{eq:kclusters1}) \cite{Inaba1994kworstcase}. Alternatively, a locally optimal solution can be found in $\mathcal{O}(Nkdi)$ time using Lloyd's algorithm. The Scikit-learn approach to $k$-means is able to effectively reduce the time complexity to $\mathcal{O}(Nkd)$ by bounding the number of iterations by a constant and performing Lloyd's algorithm multiple times from different centroid initializations. While this approach cannot guarantee a locally optimal solution, it achieves high quality clustering performance in practice.
The time complexity required to exactly solve the balanced $k$-means clustering problem has not been thoroughly analyzed. However, a locally optimal solution to Problem \ref{eq:kmeanssamesize} can be found in $\mathcal{O}(N^3)$ time using the classical approach proposed by Malinen et al. \cite{Malinen2014balancedcomplexity}. To compare this to our quantum approach, we first determine the time complexity for converting Equation \ref{eq:kmeanssamesize} into a QUBO problem. To do so, we rewrite Equation \ref{eq:finalkmeans} as follows:
\begin{align}
& \min_{W} \sum_{l = 1}^k \sum_{j = 1}^N \sum_{i = 1}^N \sum_{m = 1}^d w_{il} (x_{im} - x_{jm})^2 w_{jl}
\nonumber \\ &
+ \alpha \sum_{l = 1}^k \sum_{j = 1}^N \sum_{i = 1}^N w_{il} f_{ij} w_{jl}
+ \beta \sum_{l = 1}^N \sum_{j = 1}^k \sum_{i = 1}^k w_{li} g_{ij} w_{lj}
\label{eq:kmeanscomplexity}
\end{align}
From Equation \ref{eq:kmeanscomplexity}, the worst case time complexity is $\mathcal{O}(N^2 k d)$, which is dominated by the first term. For practical purposes, solving the QUBO problem through quantum annealing can be done in constant time. Therefore, the total time complexity of the quantum algorithm is $\mathcal{O}(N^2 k d)$. Provided $kd < N$, this time complexity is better than the time complexity of the best classical balanced $k$-means clustering algorithm $(\mathcal{O}(N^3))$. However, it is worse than the Scikit-learn implementation of generic $k$-means clustering $(\mathcal{O}(Nkd))$.
\subsection{Empirical Analysis}
\label{sub:empirical}
\subsubsection{Methodology and Performance Metrics}
Our quantum approach was tested on the D-Wave 2000Q adiabatic quantum computer. We compare the performance of our approach to the Scikit-learn implementation of classical $k$-means as well as our own implementation of the classical balanced k-means algorithm with the best time complexity \cite{Malinen2014balancedcomplexity}. Note that the Scikit-learn implementation of $k$-means searches for a solution to Problem \ref{eq:kclusters1}, while the classical balanced $k$-means algorithm and our quantum approach search for a solution to Problem \ref{eq:kmeanssamesize}. The Scikit-learn algorithm is still a valid point of comparison since the solution to both problems should be very similar for all data sets used in our experiments.
We use two performance metrics to compare the three algorithms: (i) adjusted rand index and (ii) total computing time. In the quantum approach, total computing time is composed of the time required to convert the problem into a QUBO problem, the time required to embed the QUBO problem on the hardware, the time for the quantum computer to solve the QUBO problem (annealing time), and the time required to extract the clustering information from the binary solution (postprocessing time).
\subsubsection{Data Generation} \label{sub:synthetic}
We tested our algorithm on synthetic classification data sets created using the \textit{make\_classification} function in the Scikit-learn datasets package. Each data set contains $N$ points, $k$ classes, $1$ cluster per class, and $d$ features. This function generates a data set where each cluster is centered at one of the vertices of a $d$-dimensional hypercube with side length $2.0$. The points are then generated from a normal distribution (standard deviation of $1.0$) about their cluster center. For all experiments, each class was made up of exactly $N/k$ points.
\subsubsection{Hardware Configuration}
Preprocessing and postprocessing for our quantum approach and entire classical approach were run on a machine with 2.7 GHz Dual-Core Intel i5 processor and 8 GB 1,867 MHz DDR3 memory. The quantum approach also used the D-Wave 2000Q quantum computer, which had 2,048 qubits and about 5,600 inter-qubit connections. For all experiments, each quantum annealing operation is performed 100 times, and only the ground state is used.
\begin{table*}[t!]
\centering
\caption{Number of binary variables and average number of qubits used in the quantum approach.}
\label{tab:numqubits}
\begin{tabular}{c|ccccccccc}
\hline\noalign{\smallskip}
($N$, $k$) & (16, 2) & (24, 2) & (32, 2) & (12, 3) & (15, 3) & (21, 3) & (8, 4) & (12, 4) & (16, 4) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\shortstack{Variables} & 32 & 48 & 64 & 36 & 45 & 63 & 32 & 48 & 64 \\
\shortstack{Qubits} & 185 & 429 & 794 & 244 & 381 & 743 & 209 & 456 & 806 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table*}
\subsubsection{Adjusted Rand Index}
The adjusted rand index (ARI) is a metric used to compare the similarity of two partitions of a data set. This metric ranges from -1 to 1, with larger values indicating that the two partitions are similar. We use the adjusted rand index to compare the ground truth labels of a classification data set to the partitioning produced by a clustering algorithm. A value of 1 indicates that the algorithm perfectly partitioned the data, and values close to 0 are reflective of random clustering.
If $\Phi = \{ \phi_1, \phi_2, ..., \phi_k \}$ is the partitioning produced by a given clustering algorithm, and $Y = \{ Y_1, Y_2, ... Y_k \}$ is the partitioning produced by the target function, the overlap of $\Phi$ and $Y$ is given in the contingency table $[n_{ij}]$ where $n_{ij} = |\phi_i \cap Y_j|$. We denote the sum over all entries in the $i^\text{th}$ row of the table as $a_i = \sum_{m=1}^k n_{im}$ and the sum over all entries in the $j^\text{th}$ column of the table as $b_j = \sum_{m=1}^k n_{mj}$. Using this notation, the adjusted rand index is defined below:
\begin{align}
ARI = \frac{\sum_{ij} {n_{ij} \choose 2} - \left[ \sum_i {a_i \choose 2} \sum_j {b_j \choose 2} \right] / {N \choose 2}}
{\frac{1}{2} \left[ \sum_i {a_i \choose 2} + \sum_j {b_j \choose 2}\right] - \left[ \sum_i {a_i \choose 2} \sum_j {b_j \choose 2} \right] / {N \choose 2}}
\end{align}
\subsubsection{Clustering Synthetic Data Sets}
\label{sec:synth}
We compare the clustering quality produced by classical $k$-means, classical balanced $k$-means, and quantum balanced k-means on a number of small synthetic data sets. For each problem type (defined by the number of points and number of clusters), all three algorithms were run on $50$ synthetic classification data sets. The average adjusted rand index of each clustering algorithm is reported in Figure \ref{fig:randsynth}.
\begin{figure}[t!]
\centering
\includegraphics[scale = 0.54]{Figures/rand_synth2.png}
\caption{Adjusted rand index of clustering solutions produced by classical k-means (orange bar), classical balanced k-means (green bar), and quantum balanced k-means (blue bar).}
\label{fig:randsynth}
\end{figure}
For most experiments, the classical balanced k-means algorithm had the best performance. This is not surprising since the Scikit-learn implementation of classical k-means does not always produce clusters of equal size, and the quantum approach is running on imperfect hardware. The relative drop in performance between the classical and quantum approach is particularly apparent when $k = 2$. We suspect that the classical algorithms perform better for small values of $k$ because the number of ways to partition a data set increases dramatically as $k$ increases. When there are less possible ways to cluster a data set, a local solution to the training problem is more likely to be the correct partitioning of the data set.
These experiments also show that the performance of our quantum algorithm degrades as problem size increases. We believe this is a reflection of the hardware that solved the QUBO problem rather than a flaw in our approach. The strong performance of the quantum algorithm on problems of size (8, 4), (12, 3), and (12, 4) give hope that as the fidelity and scale of quantum computers improves, our quantum approach may outperform its classical alternatives.
\begin{table*}[t!]
\centering
\caption{Time required to perform classical $k$-means, classical balanced $k$-means, and our QUBO formulation on datasets of increasing size. We also report approximate embedding time for the quantum approach.}
\label{tab:scalabilityN}
\begin{tabular}{c|cccc}
\hline\noalign{\smallskip}
\shortstack{Number \\ of points} & \shortstack{Classical \\ $k$-means} & \shortstack{Classical balanced \\
$k$-means} & \shortstack{QUBO \\ formulation} & \shortstack{Embedding \\ (estimated)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
64 & 0.0218 $\pm$ 0.0017 & 0.0028 $\pm$ 0.0008 & 0.0008 $\pm$ 0.0003 & 0.1252 \\
128 & 0.0256 $\pm$ 0.0022 & 0.0073 $\pm$ 0.0025 & 0.0070 $\pm$ 0.0008 & 0.4973 \\
256 & 0.0334 $\pm$ 0.0035 & 0.0315 $\pm$ 0.0143 & 0.0192 $\pm$ 0.0018 & 1.9833 \\
512 & 0.0414 $\pm$ 0.0060 & 0.1637 $\pm$ 0.0607 & 0.1154 $\pm$ 0.0024 & 7.9224 \\
1024 & 0.0521 $\pm$ 0.0085 & 1.5577 $\pm$ 1.0501 & 0.4624 $\pm$ 0.0095 & 31.6696 \\
2048 & 0.0684 $\pm$ 0.0134 & 10.8928 $\pm$ 5.5405 & 1.8409 $\pm$ 0.0201 & 126.6392 \\
4096 & 0.1006 $\pm$ 0.0231 & 95.4876 $\pm$ 58.0103 & 7.6902 $\pm$ 0.0581 & 506.4798 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table*}
\begin{table*}[t!]
\centering
\caption{Time required to perform classical $k$-means, classical balanced $k$-means, and our QUBO formulation for datasets with an increasing number of clusters. We also report approximate embedding time for the quantum approach.}
\label{tab:scalabilityk}
\begin{tabular}{c|cccc}
\hline\noalign{\smallskip}
\shortstack{Number \\ of clusters} & \shortstack{Classical \\ $k$-means} & \shortstack{Classical balanced \\
$k$-means} & \shortstack{QUBO \\ formulation} & \shortstack{Embedding \\ (estimated)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
2 & 0.02707 $\pm$ 0.0042 & 0.0276 $\pm$ 0.0101 & 0.0080 $\pm$ 0.0011 & 0.4973 \\
4 & 0.0427 $\pm$ 0.0058 & 0.0417 $\pm$ 0.0164 & 0.0198 $\pm$ 0.0021 & 1.9833 \\
8 & 0.0584 $\pm$ 0.0052 & 0.0399 $\pm$ 0.0113 & 0.1273 $\pm$ 0.0053 & 7.9224 \\
16 & 0.0873 $\pm$ 0.0089 & 0.0390 $\pm$ 0.0094 & 0.5129 $\pm$ 0.0271 & 31.6696 \\
32 & 0.1349 $\pm$ 0.0120 & 0.0271 $\pm$ 0.0052 & 1.9598 $\pm$ 0.0308 & 126.6392 \\
64 & 0.2341 $\pm$ 0.0090 & 0.0201 $\pm$ 0.0023 & 7.6511 $\pm$ 0.0695 & 506.4798 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table*}
\subsubsection{Scalability with Number of Data Points (N)}
We also perform a scalability study to determine how the run time of our quantum approach varies as the number of data points increases. Due to the qubit limitations of modern adiabatic quantum computers, problems that require more than 64 binary variables ($N k > 64$) are impossible on the D-Wave 2000Q. However, we can approximate the run time of our algorithm on larger problems by measuring the time required to formulate the QUBO problem and the time required to postprocess a plausible solution. We estimate the time required to embed the problem ($t_e$) as well as annealing time ($t_a$).
\begin{figure}[t!]
\centering
\includegraphics[scale = 0.6]{Figures/embedding.png}
\caption{Time required to embed small problems on the D-Wave using the embedding algorithm proposed by Date et. al. \cite{date2019efficiently}. Embedding time scales quadratically with the number of binary variables.}
\label{fig:embedding}
\end{figure}
The runtime of the efficient embedding algorithm proposed in \cite{date2019efficiently} scales quadratically with the number of binary variables in the QUBO problem.
Extrapolating upon the performance of this embedding algorithm on small problems, we approximate embedding time (in seconds) using the following equation:
\begin{align}
t_e &= 1.887 \times 10^{-6} (Nk)^2 + 4.632 \times 10^{-6} (Nk) \nonumber \\
& \qquad + 4.022 \times 10^{-4}
\end{align}
\begin{figure}[t!]
\centering
\includegraphics[scale = 0.6]{Figures/scalabilityN.png}
\caption{Total computing time of classical $k$-means (orange bar), classical balanced $k$-means (green bar), and quantum balanced $k$-means (blue bar) as the number of points ($N$) in the training data set varies. Embedding and annealing times are approximate.}
\label{fig:scalabilityN}
\end{figure}
As mentioned before, annealing is performed in constant time. Therefore, we assume that the annealing time for larger problems ($t_a$) is equal to the average annealing time for the small clustering experiments discussed in Section \ref{sec:synth}.
\begin{align}
t_a = 0.03481 \pm 0.00008
\end{align}
\begin{figure}[t!]
\centering
\includegraphics[scale = 0.6]{Figures/scalabilityk.png}
\caption{Total computing time of classical $k$-means (orange bar), classical balanced $k$-means (green bar), and quantum balanced $k$-means (blue bar) as the number of clusters varies ($k$). Embedding and annealing times are approximate.}
\label{fig:scalabilityk}
\end{figure}
We performed classical $k$-means, classical balanced $k$-means, and our QUBO formulation on data sets of increasing size. For a given problem type (defined by the number of points), all three approaches were run on 50 synthetic classification data sets. Each data set contained $k = 4$ classes, and each data point had $d = 2$ features. The average run time of each clustering approach is reported in Figure \ref{fig:scalabilityN} and Table \ref{tab:scalabilityN}.
\begin{table*}[t!]
\centering
\caption{Time required to perform classical $k$-means, classical balanced $k$-means, and our QUBO formulation on data sets with an increasing number of features. We also report approximate embedding time for the quantum approach.}
\label{tab:scalabilityd}
\begin{tabular}{c|cccc}
\hline\noalign{\smallskip}
\shortstack{Number \\ of features} & \shortstack{Classical \\ $k$-means} & \shortstack{Classical balanced \\
$k$-means} & \shortstack{QUBO \\ formulation} & \shortstack{Embedding \\ (estimated)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
2 & 0.0508 $\pm$ 0.0089 & 1.5068 $\pm$ 0.6899 & 0.4742 $\pm$ 0.0185 & 31.6696 \\
4 & 0.0681 $\pm$ 0.0137 & 1.7589 $\pm$ 0.6546 & 0.4771 $\pm$ 0.0189 & 31.6696 \\
8 & 0.0803 $\pm$ 0.0105 & 2.2591 $\pm$ 1.0435 & 0.4737 $\pm$ 0.0101 & 31.6696 \\
16 & 0.4190 $\pm$ 0.1065 & 2.0672 $\pm$ 0.6473 & 0.4760 $\pm$ 0.0102 & 31.6696 \\
32 & 0.5411 $\pm$ 0.1171 & 2.1599 $\pm$ 0.6157 & 0.4895 $\pm$ 0.0179 & 31.6696 \\
64 & 0.6598 $\pm$ 0.1048 & 1.8983 $\pm$ 0.4888 & 0.5033 $\pm$ 0.0178 & 31.6696 \\
128 & 1.0369 $\pm$ 0.1577 & 1.6768 $\pm$ 0.4551 & 0.5283 $\pm$ 0.0221 & 31.6696 \\
256 & 1.2474 $\pm$ 0.1726 & 1.4060 $\pm$ 0.2184 & 0.5759 $\pm$ 0.0207 & 31.6696 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table*}
In each case, the quantum approach performed slower than both classical algorithms. However, the quantum run time was dominated by the embedding time. Embedding is extremely difficult on modern quantum computers due to limited qubit connectivity. As hardware improves, we expect embedding to be a considerably faster process. Therefore, on a future quantum computer, the quantum algorithm may outperform the classical balanced k-means algorithm for $N \geq 1024$, depending on how well the embedding process is optimized. Of the three approaches, our results indicate that the Scikit-learn implementation of classical $k$-means scales the best. This is expected since the time complexity of the Scikit-learn implementation of classical $k$-means ($\mathcal{O}(Nkd)$) is better than classical balanced $k$-means ($\mathcal{O}(N^3)$) or quantum balanced $k$-means ($\mathcal{O}(N^2kd)$).
\subsubsection{Scalability with Number of Clusters (k)}
\label{par:numclusters}
Following the same procedure, we analyze the scalability of each algorithm as the number of clusters $k$ is increased. For each problem type, all three clustering algorithms were run on 50 synthetic data sets. Each data set consisted of $N = 256$ points, and all points had $d = 8$ features. The average run time of each clustering approach is reported in Figure \ref{fig:scalabilityk} and Table \ref{tab:scalabilityk}.
In all cases, the quantum approach had a longer run time than both classical algorithms. Additionally, the quantum run time scaled worse as the number of clusters increased. This is expected since the third term in the QUBO formulation (Equation \ref{eq:kmeanscomplexity}) has time complexity $\mathcal{O}(Nk^2)$. Alternatively, classical $k$-means scales linearly with the number of clusters ($\mathcal{O}(Nkd)$), and balanced $k$-means clustering scales independently of the number of clusters ($\mathcal{O}(N^3)$). It is somewhat surprising that the average run time of the balanced $k$-means clustering approach decreases for $k > 16$. However, we suspect this is due to the smaller cluster sizes when $k$ is large.
\subsubsection{Scalability with Number of Features (d)}
Finally, we analyze the scalability of each algorithm with respect to the dimension of the training data set. For each problem type, all three clustering approaches were run on 50 synthetic classification data sets. Each data set consisted of $N = 1024$ points separated into $k = 4$ clusters. The average run time of each clustering approach is reported in Figure \ref{fig:scalabilityd} and Table \ref{tab:scalabilityd}.
\begin{figure}[h]
\centering
\includegraphics[scale = 0.6]{Figures/scalabilityd.png}
\caption{We report the average total computing time of classical $k$-means (orange bar), classical balanced $k$-means (green bar), and quantum balanced $k$-means (blue bar) as the number of features ($d$) varies. Embedding and annealing times are approximate.}
\label{fig:scalabilityd}
\end{figure}
As before, the quantum algorithm had the longest run time in all cases. However, on future hardware the quantum approach could perform better than classical $k$-means for $d \geq 128$ and better than classical balanced $k$-means for $d \leq 256$, depending on how well the embedding process is optimized. In Figure \ref{fig:scalabilityd}, it appears that the quantum approach scales better than classical $k$-means as $d$ increases. This is not surprising since the QUBO formulation only requires one computation related to the dimension of the data set (calculation of the distance matrix), while classical $k$-means requires distance calculations with each iteration. On the other hand, the quantum approach scales worse than classical balanced $k$-means. This is expected since the time complexity of classical balanced $k$-means is independent of $d$. It is somewhat surprising that the average run time of classical balanced $k$-means begins to decrease for $d > 32$, but we suspect this is due to cluster centers being farther apart on average.
\subsection{Clustering a Benchmark Data Set}
As a final proof of concept, we clustered portions of the Iris benchmark data set using our quantum clustering approach. This data set contains 150 points (each with 4 features) divided into 3 equal-size classes. Unfortunately, due to qubit limitations on modern hardware, it is impossible to perform quantum balanced k-means clustering on the entire data set. Therefore, we generate smaller data sets by picking $N / k$ points at random from $2 \leq k \leq 3$ of the data set's classes. For a given problem type (denoted by the number of points and number of clusters), all three clustering algorithms were run on 50 subsets of the Iris data set. Note that when $k=2$, all points were chosen from the first and second classes, which are linearly separable.
\begin{figure}[t!]
\centering
\includegraphics[scale = 0.6]{Figures/rand_iris.png}
\caption{Average adjusted rand index of classical $k$-means (orange bar), classical balanced $k$-means (green bar), and quantum balanced $k$-means (blue bar) on portions of the Iris data set.}
\label{fig:randiris}
\end{figure}
For $k = 2$, the classical algorithms performed better than the quantum approach. This becomes particularly apparent as the number of binary variables ($Nk$) increases. For $k = 3$, the quantum algorithm has similar performance to classical balanced $k$-means and outperforms the Scikit-learn implementation of classical $k$-means for small data sets. Again, the performance of the quantum algorithm degrades as problem size increases. These results mirror the performance seen on the synthetic data sets discussed in Section \ref{sec:synth}.
\section{Conclusion}
\quad As new applications of machine learning models continue to emerge, it is of great interest to improve upon existing training algorithms. Adiabatic quantum computers are a promising alternative platform for solving NP-hard or NP-complete training problems efficiently. In this paper, we propose a quantum approach to training the balanced $k$-means clustering model. We analyze our approach theoretically, showing that it targets the global solution of the training problem better than its classical alternatives. We also show that our approach scales favorably on large data sets when compared to current classical balanced $k$-means algorithms. We test our approach using the D-Wave 2000Q adiabatic quantum computer and compare it to the Scikit-learn implementation of classical $k$-means as well as our own implementation of the classical balanced $k$-means algorithm with the best time complexity. We demonstrated that our quantum approach partitions data with similar accuracy to the classical approaches, even when running on imperfect hardware. As quantum hardware continues to improve in both fidelity and scale, we expect our approach to become a viable alternative to existing classical balanced clustering algorithms.
In the future, we hope to generalize our QUBO formulation to satisfy the generic $k$-means clustering training problem (Problem \ref{eq:kclusters1}). We also look to use elements of our approach to formulate quantum algorithms to similar clustering models, such as $k$-medoids clustering or fuzzy $C$-means clustering. Finally, we plan to investigate quantum approaches to clustering larger datasets within the qubit constraints of modern hardware.
\section*{Conflict of interest}
\quad The authors declare that they have no conflict of interest.
\bibliographystyle{IEEEtran}
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<para>
<classname>Zend_Mime</classname> is a support class for handling multipart
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<classname>Zend_Mime</classname> defines a set of constants commonly used with
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<para>
<constant>Zend_Mime::TYPE_OCTETSTREAM</constant>: 'application/octet-stream'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::TYPE_TEXT</constant>: 'text/plain'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::TYPE_HTML</constant>: 'text/html'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::ENCODING_7BIT</constant>: '7bit'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::ENCODING_8BIT</constant>: '8bit'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::ENCODING_QUOTEDPRINTABLE</constant>: 'quoted-printable'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::ENCODING_BASE64</constant>: 'base64'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::DISPOSITION_ATTACHMENT</constant>: 'attachment'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::DISPOSITION_INLINE</constant>: 'inline'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::MULTIPART_ALTERNATIVE</constant>:
'multipart/alternative'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::MULTIPART_MIXED</constant>: 'multipart/mixed'
</para>
</listitem>
<listitem>
<para>
<constant>Zend_Mime::MULTIPART_RELATED</constant>: 'multipart/related'
</para>
</listitem>
</itemizedlist>
</para>
</section>
<section xml:id="zend.mime.mime.instantiation"><info><title>Instantiating Zend_Mime</title></info>
<para>
When Instantiating a <classname>Zend_Mime</classname> Object, a <acronym>MIME</acronym>
boundary is stored that is used for all subsequent non-static method calls on that
object. If the constructor is called with a string parameter, this value
is used as a <acronym>MIME</acronym> boundary. If not, a random <acronym>MIME</acronym>
boundary is generated during construction time.
</para>
<para>
A <classname>Zend_Mime</classname> object has the following Methods:
<itemizedlist>
<listitem>
<para>
<methodname>boundary()</methodname>: Returns the <acronym>MIME</acronym>
boundary string.
</para>
</listitem>
<listitem>
<para>
<methodname>boundaryLine()</methodname>: Returns the complete
<acronym>MIME</acronym> boundary line.
</para>
</listitem>
<listitem>
<para>
<methodname>mimeEnd()</methodname>: Returns the complete
<acronym>MIME</acronym> end boundary line.
</para>
</listitem>
</itemizedlist>
</para>
</section>
</section>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,418 |
\section{Introduction}
\setcounter{equation}{0}
Supersymmetric field theories in curved space have attracted many interests in recent years. For such theories supersymmetric localization technique \cite{Nekrasov:2002qd,Pestun:2007rz} allows the non-perturbative exact computation of some interesting physical quantities such as the partition function and BPS Wilson loops, which
can be used to test the duality conjectures like the AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}. In this paper we focus on supersymmetric field theories with a $U(1)_R$ symmetry on
curved manifolds in 3+1 dimensions.
According to \cite{Festuccia:2011ws,Dumitrescu:2012ha, Klare:2012gn} (see also \cite{Dumitrescu:2016ltq} for a recent review), one can formulate a ${\cal N}=1$ theory with a $U(1)_R$ symmetry in 4D curved space via coupling to the new-minimal supergravity \cite{Sohnius:1981tp,Sohnius:1982fw}: One first couples the ${\cal R}$-multiplet to the new-minimal supergravity multiplet and then take a \emph{rigid} limit sending the Newton's constant to zero, so that the supergravity is decoupled while the fields in the supergravity multiplet are sent to fixed backgrounds. In constructing supersymmetric field theories in curved space, background fields are typically chosen to be bosonic, and consistency with supersymmetry requires the supersymmetry variation of the gravitino in the gravity multiplet to vanish, which leads to a \emph{generalized Killing spinor} (GKS) equation.
For each solution of the GKS equation there exists a conserved supercharge. In the case of 4D ${\cal N}=1$ theory with a $U(1)_R$ symmetry, the GKS equation can have a solution if and only if the background manifold admits an integrable complex structure and a compatible Hermitian metric.
Important exact results were obtained for the 4D ${\cal N}=1$ supersymmetric field theory with an $R$-symmetry in \cite{Closset:2013vra,Closset:2014uda,Assel:2014paa,Cassani:2013dba, Assel:2015nca,Cassani:2014zwa}, using localization technique. Here we list some of them:
\begin{itemize}
\item When there are two supercharges with opposite $U(1)_R$ charge, supersymmetric renormalization scheme is free of ambiguities and the partition function is invariant under the deformation of the Hermitian metric for a fixed complex structure.
\item When the background manifold is $S_{r_3}^3\times S_\beta^1$, the supersymmetric Casimir energy becomes
\begin{equation}\label{eqn:e_susy}
E_{\rm susy}\equiv -\lim_{\beta\rightarrow\infty}\frac{\mbox{d}}{\mbox{d}\beta}\log Z_{\rm susy}=\frac{4}{27r_3}(a+3c),
\end{equation}
where $a$ and $c$ are two trace anomaly coefficients in four dimensions.
\end{itemize}
We recall that the field theory computations in \cite{Closset:2013vra,Closset:2014uda,Assel:2014paa,Cassani:2013dba, Assel:2015nca,Cassani:2014zwa}
were carried out by using the supersymmetric Ward identities and the supersymmetry algebra which are
derived from the \emph{classical} new-minimal supergravity.
In \cite{Gates:1981yc} (see also section 7.10 of \cite{Gates:1983nr}) it was argued via the superspace formalism that the new-minimal supergravity
can be in general quantum-mechanically inconsistent, due to the appearance of the ``superscale'' anomalies (consisting of the conformal anomaly,
the $U(1)_R$ chiral anomaly and the $\gamma$-trace of the supercurrent) that are inconsistent with the local supersymmetry Ward identities.
Then one could also question the quantum consistency of the new-minimal supergravity formulation of the rigid
${\cal N}=1$ field theories with an $R$-symmetry.
However, \cite{Gates:1981yc} considered only the one-point function of the supercurrent superfield and
one could expect that there is no inconsistency for the backgrounds on which the anomalies in the Ward identities (for current one-point functions) are \emph{numerically} vanishing.
In particular, the backgrounds considered in \cite{Assel:2014paa, Cassani:2013dba ,Assel:2015nca,Cassani:2014zwa,Genolini:2016sxe,Genolini:2016ecx} are anomaly-free in this sense.
Nevertheless, the anomalies might appear in the higher-point correlation functions as the \emph{contact terms}, some of which may be physically meaningful.
This motivates us to study the higher-point correlation functions to investigate the quantum consistency.
In this paper we analyze the two-point and higher-point correlation functions
of the current operators in the ${\cal R}$-multiplet.
Assuming that the supercurrent Ward identity is non-anomalous and that the vacuum is supersymmetric, we show that the unbroken supersymmetry is inconsistent at the quantum level unless the coefficient of the $U(1)_R$ anomaly vanishes.
Note that the anomaly coefficients depend only on the field content of the theory.
Since the Ward identities and the rigid supersymmetry algebra are a direct consequence of the new-minimal supergravity,
this implies that the 4D ${\cal N}=1$ field theory with an $R$-symmetry can be consistently formulated in terms of the new-minimal supergravity only for some special systems
with the field content that leads to vanishing coefficient of $U(1)_R$ anomaly. One example of this is
a theory that consists of a free chiral multiplet with $R$-charge 1, since in this case the Weyl fermion in the chiral multiplet is uncharged under the $U(1)_R$ symmetry and gives no contribution to the $U(1)_R$ chiral anomaly.
The rest of this note is organized as follows. In section \ref{sec:nm-sugra} we briefly review the symmetries of the new-minimal supergravity, the definition of the generalized Killing spinor (GKS) and the construction of the Killing vector from the GKSs. We then derive the Ward identities of ${\cal N}=1$ supersymmetric field theories with an $R$-symmetry in section \ref{sec:ward}, the results of which are used to reproduce the rigid supersymmetry algebra in section \ref{SUSY}. In section \ref{sec:quantum-consistency} we show that in order for ${\cal N}=1$ field theories with an $R$-symmetry to be consistently formulated via the new-minimal supergravity the (pure) $U(1)_R$ chiral anomaly coefficient should vanish.
Finally, we end with concluding remarks in section \ref{sec:discussion}.
Appendix \ref{app:conventions} contains our conventions, while appendix \ref{app:free-chiral} explicitly derives
the transformation laws of the current operators in ${\cal R}$-multiplet
in a ${\cal N}=1$ supersymmetric theory with a free chiral multiplet on $\mathbb R\times S^3$.
\section{New minimal supergravity and Killing spinor}
\label{sec:nm-sugra}
\setcounter{equation}{0}
In this section, we briefly review the symmetries of the new-minimal supergravity and define a GKS and a Killing vector, as preliminaries for deriving the Ward identities of the ${\cal N}=1$ field theory with an $R$-symmetry in the next section.
We begin with the construction of the new-minimal supergravity\cite{Sohnius:1981tp,Sohnius:1982fw}. It is formulated by first minimal-coupling the ${\cal R}$-multiplet
(consisting of energy-momentum tensor ${\cal T}^{\mu\nu}$, supercurrent ${\cal S}_\alpha^\mu$, $U(1)_R$ current ${\cal J}^\mu$ and closed two-form ${\cal F}_{\mu\nu}$)
to the supergravity multiplet (containing metric $g_{\mu\nu}$, gravitino $\psi_{\alpha\mu}$, $U(1)_R$ gauge field $A_\mu$ and two-form gauge field $B_{\mu\nu}$)
to obtain the linear Lagrangian, which is then completed to the non-linear form.
The linear couplings take the form
\bbepsilon
-\frac12 {\cal T}^{\mu\nu}g_{\mu\nu}+\bar\psi_\mu {\cal S}^\mu+{\cal J}^\mu (A_\mu-\frac32 V_\mu)+\frac14\varepsilon^{\mu\nu\rho\lambda} {\cal F}_{\mu\nu}B_{\rho\lambda},
\end{equation}
where $V_\mu=\frac14\varepsilon_\mu{}^{\nu\rho\lambda}\pa_\nu B_{\rho\lambda}$. By definition, the vector field $V^\mu$ is conserved, i.e. $\nabla_\mu V^\mu=0$.
In general backgrounds the operators in the ${\cal R}$-multiplet are defined in terms of the functional derivatives of the action $S$ with respect to corresponding fields in the supergravity multiplet, namely
\begin{subequations}\label{eqn:operators}
\begin{align}
&{\cal T}^{\mu\nu}(x)=-\frac{2}{\bm e}\frac{\delta S}{\delta g_{\mu\nu}(x)},\\
&{\cal S}^{\mu}(x)=\frac{1}{\bm e}\frac{\delta S}{\delta \bar\psi_{\mu}(x)},\\
& {\cal J}^{\mu}(x)=\frac{1}{\bm e}\frac{\delta S}{\delta A_\mu(x)},\\
& {\cal F}_{\mu\nu}(x)=\frac{1}{\bm e}\varepsilon_{\mu\nu\rho\lambda}\frac{\delta S}{\delta B_{\rho\lambda}(x)},
\end{align}
\end{subequations}
where ${\bm e}\equiv|\det e^a_\mu|$ with $e^a_\mu$ being the vierbein.
At the classical level the new-minimal supergravity possesses the local supersymmetry as well as the $U(1)_R$ chiral symmetry and the diffeomorphism invariance. The corresponding transformation laws for the fields in the supergravity multiplet are given as follows (for conventions see Appendix \ref{app:conventions}):
\begin{itemize}
\item Local supersymmetry transformation
\begin{subequations} \label{eqn:nm-susy}
\begin{align}
& \delta_\epsilon e^a_\mu=-\frac12\bar\psi_\mu\gamma^a\epsilon,\\
& \delta_\epsilon\psi_\mu={\cal D}_\mu\epsilon+\frac{i}{2}\gamma_\mu(V^\rho\gamma_\rho\gamma_*\epsilon),\\
& \delta_\epsilon B_{\mu\nu}=\frac12(\bar\psi_\mu\gamma_\nu-\bar\psi_\nu\gamma_\mu)\epsilon,\\
&\delta_\epsilon A_\mu=\frac{i}{4}({\cal D}_\lambda\bar\psi_\sigma+\frac{i}{2}\bar\psi_\sigma\gamma_*\gamma_\kappa V^\kappa\gamma_\lambda)\gamma_*\gamma^{\lambda\sigma}\gamma_\mu\epsilon.
\end{align}
\end{subequations}
\item $U(1)_R$ chiral gauge transformation
\begin{subequations} \label{eqn:nm-gauge}
\begin{align}
& \delta_\Lambda A_\mu=\pa_\mu \Lambda,\\
& \delta_\Lambda\psi_\mu=-i\gamma_*\psi_\nu\Lambda.
\end{align}
\end{subequations}
\item Diffeomorphism
\begin{subequations}\label{eqn:nm-diffeo}
\begin{align}
& \delta_\xi g_{\mu\nu}={\cal L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu,\\
&\delta_\xi A_\mu={\cal L}_\xi A_\mu=-F_{\mu\nu}\xi^\nu+\nabla_\mu(\xi^\nu A_\nu),\\
&\delta_\xi B_{\rho\lambda}={\cal L}_\xi B_{\rho\lambda}=\xi^\kappa\nabla_\kappa B_{\rho\lambda}+\nabla_\rho\xi^\kappa B_{\kappa\lambda}+\nabla_\lambda\xi^\kappa B_{\rho\kappa},\\
& \delta_\xi \psi_\mu=\xi^\nu\pa_\nu\psi_\mu+(\pa_\mu\xi^\nu)\psi_\nu.
\end{align}
\end{subequations}
\end{itemize}
Here $F_{\mu\nu} \equiv \nabla_\mu A_\nu-\nabla_\nu A_\mu$.
We have omitted the higher-order terms in the gravitino, since they are irrelevant to our analysis. One can confirm the above transformation laws by checking invariance of the pure new-minimal supergravity action, given by
\bbepsilon \label{eqn:nmsugra-action}
S_{\rm nm}=\frac{1}{2}\int\mbox{d}^4x\;{\bm e}\(R+6V_\mu V^\mu-8A_\mu V^\mu-\bar\psi_\mu\gamma^{\mu\nu\rho}{\cal D}_\nu\psi_\rho+\text{(4 fermion terms)}\),
\end{equation}
where $R$ is a Ricci scalar.
As mentioned in the Introduction, a field theory with rigid supersymmetry is defined on the supersymmetric backgrounds, i.e. the ones that admit at least one solution of the GKS equation. For ${\cal N}=1$ field theories with an $R$-symmetry the GKS equation becomes
\begin{equation}\label{eqn:GKS}
\delta_\epsilon\psi_\mu={\cal D}_\mu\epsilon+\frac{i}{2}\gamma_\mu(V^\rho\gamma_\rho\gamma_*\epsilon)=0.
\end{equation}
Now let us denote a solution of \eqref{eqn:GKS} as $\zeta$. As we will see in section \ref{sec:ward}, there exists a conserved supercharge corresponding to $\zeta$.
For the discussion of the supersymmetry algebra given in section \ref{sec:ward}, we need to define a real vector\footnote{Notice that $\bar\epsilon\gamma_\mu\epsilon=0$, which follows from the property of the Majorana conjugation.} (i.e. $K^*=K$)
\begin{equation} \label{K}
K= K^\mu \pa_\mu \quad \text{with} \quad K^\mu= \bar\eta\gamma^\mu\zeta .
\end{equation}
Here $\eta\equiv i\gamma_*\zeta$ is also a GKS. By using the Fierz identity and the integrability condition for the GKS $\zeta$, one can derive following relations \cite{Cassani:2012xx}:
\begin{subequations}\label{eqn:killing-vector-relation}
\begin{align}
& K^\mu\gamma_\mu\zeta=0,\quad K^\mu K_\mu=0,\\
&{\cal L}_K\zeta=K^\mu\nabla_\mu\zeta+\frac14\nabla_\mu K_\nu\gamma^{\mu\nu}\zeta=-K^\mu A_\mu(i\gamma_*\zeta),\label{eqn:lie-derivative-zeta}\\
& \nabla_\mu K_\nu=-\varepsilon_{\mu\nu\rho\sigma}V^\rho K^\sigma,\quad K^\mu\nabla_\mu V_\nu=0,\\
& F_{\mu\nu}K^\mu=0.
\end{align}
\end{subequations}
It therefore follows that the background fields are invariant with respect to the null vector $K$ up to a gauge transformation for the $U(1)_R$ gauge field $A$, namely
\begin{subequations}
\label{eqn:killing-eqns}
\begin{align}
& {\cal L}_K g_{\mu\nu}=\nabla_\mu K_\nu+\nabla_\nu K_\mu=0,\\
& {\cal L}_K A_\mu=-F_{\mu\nu}K^\nu+\nabla_\mu(K^\nu A_\nu)=\nabla_\mu(K^\nu A_\nu),\\
&{\cal L}_K V_\mu=-V_{\mu\nu}K^\nu+\nabla_\mu(V^\nu K_\nu)=0,
\end{align}
\end{subequations}
where $V_{\mu\nu} \equiv\nabla_\mu V_\nu-\nabla_\nu V_\mu$.
Note that the Killing condition for $V_\mu$ is not equivalent to that for $B_{\mu\nu}$. In fact, we do not need the Killing condition for the background field $B_{\mu\nu}$, as we will see in the next section.
\section{Ward identities and correlation functions
}
\label{sec:ward}
\setcounter{equation}{0}
In this section, we derive the Ward identities of the ${\cal N}=1$ field theory with an $R$-symmetry in 4D curved space and then comment on some properties of correlation functions, which will be basis of the discussions of the next sections.
\subsection{Ward identities}
The Ward identities corresponding to the symmetries discussed in the previous section can be obtained by using the local renormalization group formalism \cite{Osborn:1991gm} (see also \cite{Papadimitriou:2016yit} for a recent review). To this end one first defines the generating functional of connected correlation functions
\begin{equation}
W[g_{\mu\nu},\psi_\mu, A_\mu,B_{\mu\nu}]= -i \log Z[g_{\mu\nu},\psi_\mu, A_\mu,B_{\mu\nu}],
\end{equation}
where $Z[g_{\mu\nu},\psi_\mu, A_\mu,B_{\mu\nu}]$ is the partition function in the presence of the non-dynamical background sources, i.e.
$Z=\int[{\cal D}\Phi]\exp{i S[\Phi;g_{\mu\nu},\psi_\mu, A_\mu,B_{\mu\nu}]}$ ($\Phi$ represents generic matter fields), and the usual expectation values are defined as $\langle... \rangle\equiv Z^{-1}\int[{\cal D}\Phi]...\exp{i S}$.
The gravitino background $\psi_\mu$ is set to zero at the end of the computations, since we consider the bosonic backgrounds. Recall that the variation of the generating functional $W$ is given by
\bbepsilon
\delta W=\int\mbox{d}^4x\; {\bm e}\[-\frac12\braket{{\cal T}^{\mu\nu}}\delta g_{\mu\nu}+ \delta \bar \psi_\mu \braket{{\cal S}^\mu} +\braket{{\cal J}^\mu}\delta G_\mu+\frac14\varepsilon^{\mu\nu\rho\lambda}\braket{{\cal F}_{\mu\nu}}\delta B_{\rho\lambda}\],\label{eqn:var-W}
\end{equation}
where $G_\mu\equiv A_\mu-\frac32 V_\mu$. \eqref{eqn:var-W} gives the definition of the one-point functions of the operators in the presence of arbitrary sources.
Namely, \eqref{eqn:var-W} implies that
\begin{subequations}\label{eqn:one-point-operators}
\begin{align}
& \braket{{\cal T}^{\mu\nu}(x)}_{g_{\mu\nu}, \psi_\mu, A_\mu,B_{\mu\nu}}=-\frac{2}{\bm e}\frac{\delta W}{\delta g_{\mu\nu}(x)},\\
& \braket{{\cal S}^{\mu}(x)}_{g_{\mu\nu}, \psi_\mu, A_\mu,B_{\mu\nu}}=\frac{1}{\bm e}\frac{\delta W}{\delta \bar\psi_{\mu}(x)},\\
& \braket{{\cal J}^{\mu}(x)}_{g_{\mu\nu}, \psi_\mu, A_\mu,B_{\mu\nu}}=\frac{1}{\bm e}\frac{\delta W}{\delta A_\mu(x)},\\
& \braket{{\cal F}_{\mu\nu}(x)}_{g_{\mu\nu}, \psi_\mu, A_\mu,B_{\mu\nu}}=\frac{1}{\bm e}\varepsilon_{\mu\nu\rho\lambda}\frac{\delta W}{\delta B_{\rho\lambda}(x)}.
\end{align}
\end{subequations}
The Ward identities corresponding to the symmetries \eqref{eqn:nm-susy}, \eqref{eqn:nm-gauge} and \eqref{eqn:nm-diffeo} are obtained
by requiring $\delta W=0$ up to potential quantum anomalies for the variations \eqref{eqn:nm-susy}-\eqref{eqn:nm-diffeo}, and the results are as follows:
\begin{align*} 0&=-{\cal D}_\mu\braket{\bar{\cal S}^\mu}+\frac{i}{2}\braket{\bar{\cal S}^\mu}\gamma_\mu(V^\rho\gamma_\rho\gamma_*)+\frac12\bar\psi_\mu\gamma_\nu\braket{{\cal T}^{\mu\nu}}+\frac38\Big[(\bar\psi_\mu\gamma_\nu+\bar\psi_\nu\gamma_\mu) V^\nu\braket{{\cal J}^\mu}\\
&\hskip1em-\bar\psi_\rho\gamma^\rho V_\mu\braket{{\cal J}^\mu}-\bar\psi_\lambda\gamma_\rho\varepsilon^{\mu\nu\rho\lambda}\nabla_\nu\braket{{\cal J}_\mu}
-i \varepsilon^{\mu\rho\sigma\nu} \braket{{\cal J}_\sigma} V_\rho\bar\psi_\mu\gamma_*\gamma_\nu\Big]\\
&\hskip1em
+\frac{i}{4}{\cal D}_\lambda\bar\psi_\sigma (-i \varepsilon^{\lambda\sigma}{}_{\mu\nu}\gamma^\nu+\delta^\sigma_\mu\gamma_*\gamma^\lambda-\delta_\mu^\lambda\gamma_*\gamma^\sigma)\braket{{\cal J}^\mu}+\frac14\bar\psi_\mu\gamma_\nu\varepsilon^{\mu\nu\rho\lambda}\braket{{\cal F}_{\rho\lambda}},\numberthis\label{eqn:cls-ward-susy}\\
{\cal A}_{\rm diffeo}&=\nabla^\mu \braket{{\cal T}_{\mu\nu}}-\braket{{\cal J}^\mu} G_{\mu\nu}-\nabla_\mu \braket{{\cal J}^\mu} G_\nu+\frac14\varepsilon^{\mu\kappa\rho\lambda}\braket{{\cal F}_{\mu\kappa}}(\nabla_\nu B_{\rho\lambda}+\nabla_\rho B_{\lambda\nu}+\nabla_\lambda B_{\nu\rho}),\numberthis\label{eqn:cls-ward-diffeo}\\
{\cal A}_{\rm chiral}&=\nabla_\mu\braket{{\cal J}^\mu}. \numberthis\label{eqn:qt-ward-gauge}
\end{align*}
Here
\begin{equation}
G_{\mu\nu}\equiv \nabla_\mu G_\nu-\nabla_\nu G_\mu,\quad {\cal D}_\nu\bar{\cal S}^\mu\equiv \nabla_\nu\bar{\cal S}^\mu-G_\nu(i\gamma_*\bar{\cal S}^\mu)\,
\end{equation}
and the gravitino-dependent terms are omitted in \eqref{eqn:cls-ward-diffeo} and \eqref{eqn:qt-ward-gauge}.
In \eqref{eqn:cls-ward-susy} it is assumed that there is no supersymmetry anomaly.
The diffeomorphism anomaly ${\cal A}_{\rm diffeo}$ and the $U(1)_R$ chiral anomaly ${\cal A}_{\rm chiral}$ need further explanation. The chiral anomaly is usually accompanied by the mixed gravitational anomaly, which breaks the classical diffeomorphism invariance. The anomalies are given by
\begin{align}
& {\cal A}_{\rm chiral}=\frac14\varepsilon^{\kappa\sigma\alpha\beta}\[c_A G_{\kappa\sigma}G_{\alpha\beta}+(1-\alpha)c_m R^\nu{}_{\lambda\kappa\sigma}R^\lambda{}_{\nu\alpha\beta} \],
\label{ca}\\
& {\cal A}_{\rm diffeo}=-\alpha c_m g^{\mu\nu}\frac{1}{\sqrt{-g}}\pa_\lambda\[\sqrt{-g}\frac12\varepsilon^{\kappa\sigma\alpha\beta}G_{\kappa\sigma}\pa_\alpha\Gamma^\lambda_{\mu\beta} \],
\end{align}
see e.g. \cite{Jensen:2012kj} for a recent review. Here the coefficients $c_A$ and $c_m$ are determined according to the field content of the theory and are related to the central charges. The scheme parameter $\alpha$ is the coefficient of the diffeomorphism and gauge non-invariant contact counterterm that determines where
the mixed anomaly appears: If $\alpha=0$, then the mixed anomaly appears only in the $U(1)_R$ Ward identity, while if $\alpha=1$ it appears only in the diffeomorphism Ward identity. In this note we choose a scheme $\alpha=0$ such that the mixed anomaly does not appear in the diffeomorphism Ward identity.
We emphasize that
even if we mainly consider bosonic backgrounds in quantum field theory, we must keep the gravitino background field in \eqref{eqn:cls-ward-susy} in order to compute two-point functions of the supercurrent operator. In principle, the Ward identities \eqref{eqn:cls-ward-diffeo} and \eqref{eqn:qt-ward-gauge} also contain the gravitino-dependent terms, which we ignore since they are irrelevant unless
we differentiate \eqref{eqn:cls-ward-diffeo} and \eqref{eqn:qt-ward-gauge} with respect to the gravitino.
\subsection{Higher-point correlation functions}
Taking further derivatives of \eqref{eqn:one-point-operators} with respect to the sources gives higher-point correlation functions, for which we use double bra-ket notation $\langle \langle... \rangle \rangle$, i.e.
\bbepsilon \label{correlator}
\langle \langle {\cal O}_{j_1}(x_1){\cal O}_{j_2}(x_2)\cdots {\cal O}_{j_n}(x_n)\rangle \rangle \equiv \frac{\Delta }{\Delta {\cal B}_{j_n}(x_n)}...\frac{\Delta \, \langle {\cal O}_{j_1}(x_1) \rangle}{\Delta {\cal B}_{j_2}(x_2)}
= \frac{\Delta }{\Delta {\cal B}_{j_n}(x_n)}...\frac{\Delta }{\Delta {\cal B}_{j_2}(x_2)} \frac{\Delta \,(i W) }{\Delta {\cal B}_{j_1}(x_1)}.
\end{equation}
Here ${\cal O}_j$ stands for any operator in the $\mathcal{R}$-multiplet, i.e. ${\cal O}_j = \left\{ {\cal T}^{\mu\nu},\;{\cal S}_\alpha^\mu,\;{\cal J}^\mu,\; {\cal F}^{\mu\nu} \right\}$, and $\frac{\Delta }{\Delta {\cal B}_j}$ is a suitably defined functional derivative
with respect to the background sources ${\cal B}_j= \left\{g_{\mu\nu},\bar\psi_{\mu}, A_\mu, B_{\rho\lambda} \right\}$ as
\bbepsilon
\frac{\Delta }{\Delta {\cal B}_j(x)} = \left\{-\frac{2}{i\bm e}\frac{\delta }{\delta g_{\mu\nu}(x)},\,\, \frac{1}{i\bm e}\frac{\delta }{\delta \bar\psi_{\mu}(x)}, \,\, \frac{1}{i\bm e}\frac{\delta }{\delta A_\mu(x)}. \,\,
\frac{\varepsilon_{\mu\nu\rho\lambda}}{i\bm e}\frac{\delta }{\delta B_{\rho\lambda}(x)}
\right\},
\end{equation}
so that ${\cal O}_j(x) = \frac{ \Delta (i S)}{\Delta {\cal B}_j(x)}$ according to \eqref{eqn:operators}.
For instance,
\begin{align}
& \langle \langle {\cal T}^{\mu\nu}(x) {\cal J}^{\rho}(y) \rangle \rangle \equiv \frac{1}{i\bm e}\frac{\delta \braket{{\cal T}^{\mu\nu}(x) }}{\delta A_\rho(y)}
= \frac{1}{i\bm e}\frac{\delta }{\delta A_\rho(y)} \left(- \frac{2}{\bm e}\frac{\delta W}{\delta g_{\mu\nu}(x)}\right) , \\
& \langle \langle {\cal S}^{\mu}(x) {\cal J}^{\nu}(y) \rangle \rangle \equiv \frac{1}{i\bm e}\frac{\delta \braket{{\cal S}^{\mu}(x) }}{\delta A_\nu(y)}
= \frac{1}{i\bm e}\frac{\delta }{\delta A_\nu(y)} \left( \frac{1}{\bm e}\frac{\delta W}{\delta \bar\psi_{\mu}(x)} \right), \\
& \langle \langle {\cal S}^{\mu}(x) {\cal J}^{\nu}(y) {\cal J}^{\rho}(z) \rangle \rangle \equiv \frac{1}{i\bm e}\frac{\delta \langle \langle {\cal S}^{\mu}(x) {\cal J}^{\nu}(y) \rangle \rangle}{\delta A_\rho(z)}
= \frac{1}{i\bm e}\frac{\delta }{\delta A_\rho(z)} \left( \frac{1}{i\bm e}\frac{\delta }{\delta A_\nu(y)} \left( \frac{1}{\bm e}\frac{\delta W}{\delta \bar\psi_{\mu}(x)} \right) \right) .
\end{align}
Notice that we use different notation $\langle \langle... \rangle \rangle$ than the usual one $\langle ... \rangle_C$ (the subscript $C$ stands for ``connected''), though $\langle \langle... \rangle \rangle$ is clearly a \emph{connected} correlation function as $W$ is the generating functional for connected correlation functions. The reason of using $\langle \langle ... \rangle \rangle$ rather than $\langle ... \rangle_C$ to denote two- and higher-point functions\footnote{For one-point functions we use the usual bra-ket notation $\langle \cdot \rangle$ since in this case both quantities are identical, i.e. $\langle \langle {\cal O}(x) \rangle \rangle = \langle {\cal O}(x) \rangle$} is that the quantity $\langle \langle... \rangle \rangle$ can differ from $\langle... \rangle_C$ by contact terms when the operators depend on the sources due to the non-linear dependence of the action on the sources. For instance,
using the definition of $\langle \langle... \rangle \rangle$ given above it can be easily seen that
\begin{eqnarray}
&& \langle \langle {\cal J}^{\mu}(x) {\cal J}^{\nu}(y) \rangle \rangle = \frac{1}{i\bm e}\frac{\delta \langle {\cal J}^{\mu}(x) \rangle }{\delta A_\nu(y)}
= \left \langle \left(\frac{1}{\bm e} \frac{\delta S}{\delta A_\nu(y)} + \frac{1}{i\bm e} \frac{\delta}{ \delta A_\nu(y)} \right) {\cal J}^{\mu}(x) \right \rangle_C \nonumber \\
&& = \left \langle {\cal J}^{\mu}(x) {\cal J}^{\nu}(y) \right \rangle_C -i \delta^4(x,y) \left \langle\frac{\partial {\cal J}^\mu(x)}{\partial A_\nu(x)} \right \rangle,
\end{eqnarray}
where $\delta^4(x,y)= \delta^4(x-y)/\bm e$ is the invariant Dirac delta function. In obtaining the last term in the second line above it is used that the action (and hence operator ${\cal J}$) does not depend on the derivatives of the background sources which are non-dynamical.
The contact term above corresponds to the second functional derivative of action $S$ with respect to $A$, i.e. $\left \langle \frac{\delta}{\bm e \delta A_\nu(y)}\frac{\delta S}{\bm e \delta A_\mu(x)} \right \rangle$, which is non-vanishing
when the action $S$ depends non-linearly on the gauge field $A$. For higher-point functions there can be more contact terms, and in general it is not obvious whether or not the contact terms play any role and can be ignored in the calculations.
In order to investigate the potential consequences of the contact terms, let us consider $(n+1)$-point functions obtained
by taking functional derivatives of an one-point function of an operator ${\cal O}(x)$
with respect to the background sources (see \eqref{correlator}), i.e.
\begin{align}
\langle \langle {\cal O}(x){\cal O}_{j_1}(x_1)\cdots {\cal O}_{j_n}(x_n)\rangle \rangle & = \frac{\Delta }{\Delta {\cal B}_{j_n}(x_n)}...\frac{\Delta }{\Delta {\cal B}_{j_2}(x_2)} \frac{\Delta }{\Delta {\cal B}_{j_1}(x_1)} \langle {\cal O}(x) \rangle \nonumber \\
& = \left \langle \left({\cal O}_{j_n}(x_n) + \frac{\Delta }{\Delta {\cal B}_{j_n}(x_n)} \right)...\left({\cal O}_{j_1}(x_1) + \frac{\Delta }{\Delta {\cal B}_{j_1}(x_1)} \right) {\cal O}(x) \right \rangle_C \nonumber \\
& = \left \langle {\cal O}_{j_n}(x_n)\cdots {\cal O}_{j_1}(x_1) {\cal O}(x) \right \rangle_C + \textrm{contact terms} , \label{O-n}
\end{align}
where the contact terms contain connected correlation functions with less than $n+1$ operators and are proportional to $\delta$-functions that arise whenever $ \frac{\Delta }{\Delta {\cal B}}$ acts
on operators. The contact terms can be split into two parts according to whether it contains $\delta^{4}(x,x_k)$ or not, namely
\bbepsilon \label{split-contact}
\textrm{contact terms} =\quad \textrm{terms with } \delta^{4}(x,x_k) \quad + \quad \langle {\cal O}(x)... \rangle_C \, ,
\end{equation}
where the first part on the RHS of \eqref{split-contact} consists of terms that contain $\delta^{4}(x,x_k)$ (and their products), which results from $ \frac{\Delta }{\Delta {\cal B}_{j_k}}$ acting on ${\cal O}(x)$, while the rest is collected into the second term, which does not contain the derivatives of ${\cal O}(x)$ and hence can be written in the form of $\langle {\cal O}(x)... \rangle_C$
The second part in \eqref{split-contact} contains $\delta^4(x_k,x_l)$ (and their products) with $k\neq l$ and is absent when $n=1$ (i.e. for two-point functions). Splitting in the form \eqref{split-contact} will be useful in the following discussions.
Now we suppose that
${\cal O}(x)$ corresponds to a conserved current $X^\mu$, i.e. ${\cal O}(x)=X^\mu(x)$ with $\nabla_\mu \langle X^\mu(x) \rangle=0$. When acting
$\nabla_\mu = \bm e^{-1} \frac{\partial}{\partial x^\mu} \bm e$ on \eqref{split-contact} and taking integration $\int\mbox{d}^4x\;{\bm e}$, the first part in \eqref{split-contact} does not contribute since it leads to the integration of the total derivative of $\delta^4(x-x_j)$ that vanishes. In the case of two-point functions, the second part in \eqref{split-contact} does not exist. Therefore, for the two-point functions all contact terms drop out through the operations mentioned above, so that we have
\bbepsilon \label{2pt-X-Q}
\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle \langle X^\mu(x) {\cal O}_{j_1}(y)\rangle \rangle= \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(y)\rangle_C
= \langle [Q_X, {\cal O}_{j_1}(y)]\rangle ,
\end{equation}
where $Q_X$ is the corresponding conserved charge defined as $Q_X \equiv \int_{\cal C}\mbox{d}\sigma_\mu\;X^\mu$ with ${\cal C}$
being the Cauchy surface.\footnote{One comment is in order on the second equality of \eqref{2pt-X-Q}. The usual relation is $\langle [Q_X, {\cal O}_j(y)]\rangle= \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_j(y)\rangle$ without subscript $C$. However, since $\nabla_\mu \langle X^\mu(x) \rangle=0$, the disconnected connected part vanishes, so that $\langle [Q_X, {\cal O}_j(y)]\rangle= \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_j(y)\rangle_C$.}
\eqref{2pt-X-Q} will be often used in section \ref{SUSY}. For three-point and higher-point functions the second part of the contact terms in \eqref{split-contact} can contribute and needs a careful treatment.
Let us consider the case when $Q_X$ annihilates the vacuum state $\ket{\Omega}$, i.e. $Q_X \ket{\Omega}=0$. In this case it follows that
\bbepsilon \label{npt-X-Q}
Q_X \ket{\Omega}=0
\,\,\implies
\,\, \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle \langle X^\mu(x) {\cal O}_{j_1}(x_1)\cdots {\cal O}_{j_n}(x_n)\rangle \rangle =0,
\end{equation}
This is analogous to the usual formula
$\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) ...\rangle =0$ when $Q_X \ket{\Omega}=0$.
\eqref{npt-X-Q} can be shown as follows. First, note that $Q_X \ket{ \Omega}=0$ leads to $\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x)...\rangle_C=0$, which follows
from $\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) ...\rangle =0$ and the definition of the connected correlation functions.\footnote{As an illustration of the statement that $\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x)...\rangle_C=0$
if $\nabla_\mu \langle X^\mu(x) ...\rangle=0$,
let us consider connected three-point function
$ \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(x_1) {\cal O}_{j_2}(x_2)\rangle_C$. From the definition of the connected correlation function we have
$ \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(x_1) {\cal O}_{j_2}(x_2)\rangle_C= \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(x_1) {\cal O}_{j_2}(x_2)\rangle -
\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) \rangle\langle {\cal O}_{j_1}(x_1) {\cal O}_{j_2}(x_2)\rangle-
\langle {\cal O}_{j_1}(x_1) \rangle \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_2}(x_2)\rangle-
\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(x_1)\rangle \langle {\cal O}_{j_2}(x_1) \rangle
+ 2 \int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) \rangle\langle {\cal O}_{j_1}(x_1) \rangle\langle {\cal O}_{j_2}(x_2)\rangle
$, each term of which contains
$\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x)...\rangle=0$,
so
we have
$\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle X^\mu(x) {\cal O}_{j_1}(x_1) {\cal O}_{j_2}(x_2)\rangle_C=0$. This fact is easily generalized to the arbitrary higher-point functions.
}
Then, the first term (non-contact part) in \eqref{O-n} (with ${\cal O}(x)=X^\mu(x)$) and the second part of the contact terms
in \eqref{split-contact} do not contribute to $\int\mbox{d}^4x\;{\bm e} \nabla_\mu \langle \langle X^\mu(x) {\cal O}_{j_1}(x_1)\cdots {\cal O}_{j_n}(x_n)\rangle \rangle$. Since
the first part of the contact terms in \eqref{split-contact} does not contribute either as mentioned above, we end up with \eqref{npt-X-Q}, which will be employed in section \ref{sec:quantum-consistency}.
\section{Rigid supersymmetry algebra} \label{SUSY}
Now we recover the rigid supersymmetry algebra on the curved backgrounds by deriving the transformation laws of the supercurrent and $U(1)_R$ current with respect to the rigid supersymmetry.\footnote{We suspect that these transformation rules should be known, but we found only their flat-space version in the literature, see e.g. \cite{Sohnius:1981tp,Closset:2013vra}.}
In order to set up the general strategy, we first deal with the diffeomorphism Ward identity.
We multiply \eqref{eqn:cls-ward-diffeo} by an arbitrary vector field $\xi^\nu(x)$ and take a functional derivative $\frac{\delta }{i{\bm e}\, \delta A_\rho(y)}$ to obtain
\bala
0&=i\xi^\nu\Big[\nabla^\mu\braket{\braket{{\cal T}_{\mu\nu}(x){\cal J}^\rho(y)}}-G_{\mu\nu}\braket{\braket{{\cal J}^\mu(x){\cal J}^\rho(y)}}-G_\nu\nabla_\mu\braket{\braket{{\cal J}^\mu(x){\cal J}^\rho(y)}}\\
&\hskip1em+\frac14\varepsilon^{\mu\kappa\rho\lambda}(\nabla_\nu B_{\rho\lambda}+\nabla_\rho B_{\lambda\nu}+\nabla_\lambda B_{\nu\rho})\braket{\braket{{\cal F}_{\mu\kappa}(x){\cal J}^\rho(y)}}\Big]\\
&\hskip1em-\delta^4(x,y)[\nabla_\nu(\xi^\nu\braket{{\cal J}^\rho})-\braket{{\cal J}^\nu}\nabla_\nu \xi^\rho]
+e^{-1}\pa_\nu[\delta^4(x-y)(\xi^\nu\braket{{\cal J}^\rho}-\xi^\rho\braket{{\cal J}^\nu})].\numberthis\label{eqn:stress-R-two-point}
\eala
The integration over $x$-space (i.e. $\int\mbox{d}^4x\;{\bm e}$) of the above gives
\begin{align}
&\nabla_\nu(\xi^\nu\braket{{\cal J}^\rho})-\braket{{\cal J}^\nu}\nabla_\nu \xi^\rho=\int\mbox{d}^4x\;{\bm e}\;i\xi^\nu\Big[\nabla^\mu\braket{\braket{{\cal T}_{\mu\nu}(x){\cal J}^\rho(y)}}-
G_{\mu\nu}\braket{\braket{{\cal J}^\mu(x){\cal J}^\rho(y)}}-\nonumber \\
&\hskip3em-G_\nu\nabla_\mu\braket{\braket{{\cal J}^\mu(x){\cal J}^\rho(y)}}+\frac14\varepsilon^{\mu\kappa\rho\lambda}(\nabla_\nu B_{\rho\lambda}+\nabla_\rho B_{\lambda\nu}+
\nabla_\lambda B_{\nu\rho})\braket{\braket{{\cal F}_{\mu\kappa}(x){\cal J}^\rho(y)}}\Big],\label{eqn:diffeo-var-R}
\end{align}
where the left-hand side
actually corresponds to the variation of the operator ${\cal J}^\rho$ under the diffeomorphism associated with the vector $\xi^\mu$, see e.g. section 5.2.3 in \cite{DiFrancesco:1997nk}. It then follows from \eqref{eqn:diffeo-var-R} that the $U(1)_R$ current ${\cal J}^\mu$ transforms as a vector density under the diffeomorphism, which
has to do with the fact that
the quantity conjugate to the vector source is not a vector but a vector density operator (see e.g. \cite{Papadimitriou:2010as}).
Multiplying \eqref{eqn:cls-ward-diffeo} by $K^\nu$ defined in \eqref{K} and assuming that $K^\mu A_\mu$ is made constant\footnote{When $K^\mu A_\mu$ is not constant, it seems that there are several inconsistencies in constructing the supersymmetry algebra. For instance, one can not define a conserved charge associated with the Killing vector $K$, and the Lie derivative of $\zeta$ with respect to $K$ does not satisfy the generalized Killing condition.} by a suitable $U(1)_R$ gauge transformation,
we obtain\footnote{A careful derivation is necessary for the term containing ${\cal F}_{\rho\lambda}$. First of all, we have $\nabla_\mu(\varepsilon^{\mu\nu\rho\lambda}K_\nu{\cal F}_{\rho\lambda})=4V^\mu K^\nu{\cal F}_{\mu\nu}=\varepsilon^{\mu\rho\sigma\lambda}K^\nu(\nabla_\rho B_{\sigma\lambda}){\cal F}_{\mu\nu}$. Combining this with $\varepsilon^{[\mu\rho\sigma\lambda}K^{\nu]}(\nabla_\rho B_{\sigma\lambda}){\cal F}_{\mu\nu}=0$, which implies $\varepsilon^{\mu\rho\sigma\lambda}K^\nu(\nabla_\rho B_{\sigma\lambda}){\cal F}_{\mu\nu}=\frac12{\cal F}_{\mu\nu}\varepsilon^{\sigma\lambda\mu\nu}K^\rho(\nabla_\rho B_{\sigma\lambda}+\nabla_\lambda B_{\rho\sigma}+\nabla_\sigma B_{\lambda\rho})$, gives
$\nabla_\mu(\varepsilon^{\mu\nu\rho\lambda}K_\nu{\cal F}_{\rho\lambda})=\frac12{\cal F}_{\mu\nu}\varepsilon^{\sigma\lambda\mu\nu}K^\rho(\nabla_\rho B_{\sigma\lambda}+\nabla_\lambda B_{\rho\sigma}+\nabla_\sigma B_{\lambda\rho})$. Notice that we do not need the Killing condition for $B_{\mu\nu}$.}
\begin{equation} \label{cc_K}
\nabla_\mu \braket{{\cal C}_K^\mu(x)}=0,
\end{equation}
where
\begin{equation} \label{cc_K_def}
{\cal C}^\mu_K(x)\equiv K_\nu\Big[{\cal T}^{\mu\nu}-{\cal J}^\mu(A^\nu-\frac32 V^\nu)+\frac12\varepsilon^{\mu\nu\rho\lambda}{\cal F}_{\rho\lambda} \Big](x).
\end{equation}
This allows us to define a conserved charge
\begin{equation}
Q_K\equiv \int_{\cal C}\mbox{d}\sigma_\mu\;{\cal C}^\mu_K,
\end{equation}
where ${\cal C}$ is any Cauchy surface. Now using \eqref{2pt-X-Q} and replacing $\xi$ by $K$ in \eqref{eqn:diffeo-var-R}, we obtain
\begin{equation}\label{eqn:diffeo-K-J}
i\langle [Q_K,{\cal J}^\rho]\rangle=\int\mbox{d}^4x\;{\bm e}\;i\nabla_\mu\langle \langle{\cal C}^\mu_K(x){\cal J}^\rho(y)\rangle \rangle=\nabla_\nu(K^\nu \langle{\cal J}^\rho\rangle)-\langle{\cal J}^\nu\rangle \nabla_\nu K^\rho.
\end{equation}
Now we use the above strategy to recover the rigid supersymmetry algebra for ${\cal N}=1$ field theories with an $R$-symmetry.
Multiplying \eqref{eqn:cls-ward-susy} by the GKS $\zeta(x)$ and setting the gravitino to zero gives
\begin{equation}
\nabla_\mu\left( \langle \bar{\cal S}^\mu \zeta(x) \rangle \right) =0 ,
\end{equation}
which allows us to define a conserved supercharge associated with the GKS $\zeta$ as
\begin{equation}
{\cal Q}_\zeta\equiv\int_{\cal C} \mbox{d}\sigma_\mu\;\bar{\cal S}^\mu\zeta .
\end{equation}
Independence of ${\cal Q}_\zeta$ on the choice for the Cauchy surface, i.e. conservation of ${\cal Q}_\zeta$ is an immediate consequence of the Ward identity \eqref{eqn:cls-ward-susy} on the bosonic background. Note that we can also define a conserved supercharge ${\cal Q}_\eta$ associated with the GKS $\eta=i\gamma_*\zeta$.
Now we multiply \eqref{eqn:cls-ward-susy} by the GKS $\zeta$ and taking the functional derivative $\frac{1}{i\bm e} \frac{\delta}{\delta\bar\psi_\mu(y)}$
to obtain
\begin{align*}
i\nabla_\nu\langle \langle(\bar{\cal S}^\nu\zeta)(x){\cal S}^\mu(y)\rangle \rangle &=\delta^4(x,y)\Big[\frac14\gamma_\nu\zeta\Big(2\braket{{\cal T}^{\mu\nu}}+3\braket{{\cal J}^\mu} V^\nu
+\braket{{\cal J}^\nu} V^\mu-g^{\mu\nu}\braket{{\cal J}^\rho} V_\rho+\varepsilon^{\mu\nu\rho\lambda}\braket{{\cal F}_{\rho\lambda}}\\
&\hskip1em -\frac12 \varepsilon^{\mu\nu\lambda\sigma}\nabla_\lambda\braket{{\cal J}_\sigma}\Big)+\frac{i}{4}\gamma_*\gamma_\nu\zeta(g^{\mu\nu}\nabla_\rho\braket{{\cal J}^\rho}-\nabla^\nu\braket{{\cal J}^\mu}+\varepsilon^{\mu\nu\sigma\kappa}\braket{{\cal J}_\sigma} V_\kappa)\Big]\\
&\hskip1em+\frac{i}{4} \bm e^{-1}\pa_\lambda\big[\delta^4(x-y)(-i \varepsilon^{\lambda\mu}{}_{\sigma\nu}\gamma^\nu+\delta^\mu_\sigma\gamma_*\gamma^\lambda-\delta_\sigma^\lambda\gamma_*\gamma^\mu)\zeta\braket{{\cal J}^\sigma}\big],\numberthis\label{eqn:supercurrent-two-point}
\end{align*}
where we
have set the gravitino background to zero at the end.
Integrating \eqref{eqn:supercurrent-two-point} over $x$-space and using \eqref{2pt-X-Q} give
\bala
& i\langle [{\cal Q}_\zeta,{\cal S}^\mu]\rangle=\int\mbox{d}^4x\;{\bm e}\;i\nabla_\nu\langle \langle (\bar{\cal S}^\nu\zeta)(x){\cal S}^\mu(y)\rangle \rangle\\
&=\frac14\gamma_\nu\zeta\left(2\braket{{\cal T}^{\mu\nu}}+3\braket{{\cal J}^\mu }V^\nu+\braket{{\cal J}^\nu} V^\mu-g^{\mu\nu}\braket{{\cal J}^\rho} V_\rho+\varepsilon^{\mu\nu\rho\lambda}\braket{{\cal F}_{\rho\lambda}}
-\frac12 \varepsilon^{\mu\nu\lambda\sigma}\nabla_\lambda\braket{{\cal J}_\sigma}\right)+\\
&
\quad +\frac{i}{4}\gamma_*\gamma_\nu\zeta\big(g^{\mu\nu}\nabla_\rho\braket{{\cal J}^\rho}-\nabla^\nu\braket{{\cal J}^\mu}+\varepsilon^{\mu\nu\sigma\kappa}\braket{{\cal J}_\sigma} V_\kappa\big). \numberthis\label{eqn:supercurrent-transformation}
\eala
Note that using \eqref{eqn:supercurrent-transformation} we can rewrite \eqref{eqn:supercurrent-two-point} in a simple form as
\begin{equation}
\nabla_\nu\braket{\braket{(\bar{\cal S}^\nu\zeta)(x){\cal S}^\mu(y)}}=
\delta^4(x,y)\braket{[{\cal Q}_\zeta,{\cal S}^\mu]}+\frac{1}{4}\bm e^{-1}\pa_\lambda\big[\delta^4(x-y)(-i \varepsilon^{\lambda\mu}{}_{\sigma\nu}\gamma^\nu+\delta^\mu_\sigma\gamma_*\gamma^\lambda-\delta_\sigma^\lambda\gamma_*\gamma^\mu)\zeta\braket{{\cal J}^\sigma}\big].\label{eqn:susy-s-s}
\end{equation}
Multiplying \eqref{eqn:supercurrent-transformation} by $\bar\zeta$ and $\bar\eta$ gives respectively
(omitting the bra-ket notation $\braket{\cdot}$)
\begin{align}
& i[{\cal Q}_\zeta,\bar\zeta{\cal S}^\mu]=\frac14\nabla_\rho(K^\mu{\cal J}^\rho-K^\rho{\cal J}^\mu),\label{eqn:Qzeta-zetaS}\\
& i[{\cal Q}_\zeta,\bar\eta{\cal S}^\mu]=\frac12K_\nu[{\cal T}^{\mu\nu}-G^\nu{\cal J}^\mu+\frac12\varepsilon^{\mu\nu\rho\lambda}{\cal F}_{\rho\lambda}]-\frac18\nabla_\lambda(\varepsilon^{\mu\nu\lambda\sigma}K_\nu{\cal J}_\sigma)+\frac12(K^\nu A_\nu){\cal J}^\mu.\label{eqn:Qzeta-etaS}
\end{align}
It then follows that
\begin{align}
& \nabla_\mu[{\cal Q}_\zeta,\bar\zeta{\cal S}^\mu]=0,\quad [{\cal Q}_\zeta,{\cal Q}_\zeta]=0,\\
& i[{\cal Q}_\zeta,{\cal Q}_\eta]=\frac12Q_K+\frac12(K^\nu A_\nu)Q_R,
\end{align}
where $Q_R$ is the $U(1)_R$ charge.
Now we multiply \eqref{eqn:cls-ward-susy} by the GKS $\zeta(x)$, differentiate it with respect to $A_\mu(y)$ and $B_{\sigma\lambda}(y)$, respectively (cf. \eqref{eqn:one-point-operators}), and set the gravitino to zero. Then, we get
\bbepsilon
\nabla_\nu\langle \langle(\bar{\cal S}^\nu\zeta){\cal J}^\mu\rangle \rangle= \braket{\bar{\cal S}^\mu}\gamma_*\zeta\delta^4(x,y),\quad \nabla_\rho\langle \langle(\bar{\cal S}^\rho\zeta){\cal F}_{\mu\nu}\rangle \rangle=
\frac{1}{2}\Big[\nabla_\mu(\braket{\bar{\cal S}^\rho}\gamma_\rho\gamma_\nu\gamma_*\zeta)-(\mu\leftrightarrow\nu)\Big]\delta^4(x,y),
\end{equation}
which lead to
\begin{align}
&[{\cal Q}_\zeta,{\cal J}^\mu]=\bar{\cal S}^\mu\gamma_*\zeta,\quad i[{\cal Q}_\zeta,Q_R]={\cal Q}_\eta,\label{eqn:susy-var-j-main}\\
&[{\cal Q}_\zeta,{\cal F}_{\mu\nu}]=\frac{1}{2}\Big[\nabla_\mu(\bar{\cal S}^\rho\gamma_\rho\gamma_\nu\gamma_*\zeta)-(\mu\leftrightarrow\nu)\Big].\label{eqn:susy-var-cf-main}
\end{align}
These transformation laws of the currents ${\cal S}^\mu$, ${\cal J}^\mu$ and ${\cal F}_{\mu\nu}$ under the rigid supersymmetry are explicitly checked in appendix \ref{app:free-chiral} for a free chiral theory on $\mathbb R\times S^3$.
Transformation law for the supercharge ${\cal Q}_\zeta$ under the diffeomorphism associated with the Killing vector $K$ can be obtained by differentiating the diffeomorphism Ward identity with respect to the gravitino source. For this, the diffeomorphism Ward identity should be extended to involve the gravitino-dependent terms. We do not present the details of its calculation here, but give the final result as
\begin{equation}
i[Q_K,{\cal Q}_\zeta]=-(K^\mu A_\mu){\cal Q}_\eta \quad \text{or}\quad [Q_K+(K^\mu A_\mu)Q_R,{\cal Q}_\zeta]=0,
\end{equation}
which is consistent with \eqref{eqn:lie-derivative-zeta}.
In summary, the
supersymmetry algebra is
\begin{subequations}\label{eqn:susyalgebra}
\begin{align}
& i[{\cal Q}_\zeta,{\cal Q}_\eta]=\frac12Q_K+\frac12(K^\nu A_\nu)Q_R,\label{eqn:susyalgebra-zeta-eta}\\
& [Q_K+(K^\mu A_\mu)Q_R,{\cal Q}_\zeta]=0,\quad i[{\cal Q}_\zeta,Q_R]={\cal Q}_\eta,\\
&[Q_K+(K^\mu A_\mu)Q_R,{\cal Q}_\zeta]=0,
\end{align}
\end{subequations}
see e.g. \cite{Dumitrescu:2012ha}.
We end this section by addressing the quantum consistency of the ${\cal N}=1$ rigid supersymmetry algebra \eqref{eqn:susyalgebra}. First, note that by differentiating \eqref{eqn:qt-ward-gauge} with respect to the source field $A_\nu(y)$, we get
\begin{equation}
i\nabla_\mu\braket{\braket{{\cal J}^\mu(x){\cal J}^\nu(y)}}=c_A\varepsilon^{\mu\nu\rho\sigma}\nabla_\mu\delta^4(x,y)G_{\rho\sigma},
\end{equation}
where the right-hand side is a total derivative. It then follows that
\begin{equation}
\int\mbox{d}^4x\;{\bm e}\;\nabla_\mu\braket{\braket{{\cal J}^\mu(x){\cal J}^\nu(y)}}=0,
\end{equation}
which implies that $[Q_R,{\cal J}^\nu]=0$. Using this, we find from \eqref{eqn:susyalgebra-zeta-eta} and \eqref{eqn:diffeo-K-J} that
\begin{equation}
2[[{\cal Q}_\zeta,{\cal Q}_\eta],{\cal J}^\mu]=-i[Q_K,{\cal J}^\mu]=-\nabla_\nu(K^\nu{\cal J}^\mu)+{\cal J}^\nu\nabla_\nu K^\mu.\label{eqn:zeta-eta-J-1}
\end{equation}
On the other hand, \eqref{eqn:susy-var-j-main} and \eqref{eqn:Qzeta-zetaS} imply that
\begin{equation}
2[{\cal Q}_\zeta,[{\cal Q}_\eta,{\cal J}^\mu]]=2i[{\cal Q}_\zeta,\bar\zeta{\cal S}^\mu]=\frac12\nabla_\nu(K^\mu{\cal J}^\nu-K^\nu{\cal J}^\mu),
\end{equation}
and therefore
\begin{equation}
2[[{\cal Q}_\zeta,{\cal Q}_\eta],{\cal J}^\mu]=2[{\cal Q}_\zeta,[{\cal Q}_\eta,{\cal J}^\mu]]-2[{\cal Q}_\eta,[{\cal Q}_\zeta,{\cal J}^\mu]]=\nabla_\nu(K^\mu{\cal J}^\nu-K^\nu{\cal J}^\mu).\label{eqn:zeta-eta-J-2}
\end{equation}
Since $K$ is a nowhere vanishing vector \cite{Dumitrescu:2012ha}, \eqref{eqn:zeta-eta-J-1} can be consistent with \eqref{eqn:zeta-eta-J-2} only when
\begin{equation}
\nabla_\mu\braket{{\cal J}^\mu}=0, \label{eqn:new-constraint-J}
\end{equation}
or equivalently (see \eqref{eqn:qt-ward-gauge} and \eqref{ca})
\begin{equation}
\varepsilon^{\mu\nu\rho\sigma}G_{\mu\nu}G_{\rho\sigma}=0, \quad \varepsilon^{\kappa\sigma\alpha\beta}R^\nu{}_{\lambda\kappa\sigma}R^\lambda{}_{\nu\alpha\beta}=0, \label{eqn:new-constraint}
\end{equation}
if we assume $c_A\neq 0$ and $c_m\neq0$. These are
additional constraints imposed on the background sources since the GKS condition \eqref{eqn:GKS} does not automatically imply \eqref{eqn:new-constraint} \cite{Cassani:2013dba}. Does the condition \eqref{eqn:new-constraint} suffice
for consistent construction of ${\cal N}=1$ field theory with an $R$-symmetry when $c_A\neq 0$ and $c_m\neq0$? As we will see in the next section the answer is no,
due to a problem that manifests itself in the higher-point correlation functions.
\section{Quantum consistency
}
\label{sec:quantum-consistency}
\setcounter{equation}{0}
In this section we show that the $U(1)_R$ coefficient $c_A$
should vanish in order for the new-minimal supergravity formulation of the ${\cal N}=1$ field theory with an $R$-symmetry to be quantum-mechanically consistent.
Condition \eqref{eqn:new-constraint} is related to the ${\cal N}=1$ rigid supersymmetry algebra \eqref{eqn:susyalgebra}, which relies on the assumption that $K^\mu A_\mu$ is constant.
Now we would like to
pursue our investigation
without using this assumption.
Instead, following \cite{Assel:2015nca} we suppose that the vacuum state $\ket{\Omega}$
is supersymmetric, i.e.
\begin{equation}
{\cal Q}_\zeta\ket{\Omega}={\cal Q}_\eta\ket{\Omega}=0, \label{Q_z_h}
\end{equation}
which implies $\langle\delta_\zeta(...) \rangle=\langle\delta_\eta(...) \rangle=0$ with $\delta_\zeta(...)\equiv [{\cal Q}_\zeta , \, ...]$ and $\delta_\eta(...)\equiv [{\cal Q}_\eta , \,...]$.
Notice that when $ K^\mu A_\mu $ is not constant, the right-hand side of \eqref{cc_K} becomes non-zero, i.e.
$ \nabla_\mu {\cal C}^\mu=-{\cal J}^\mu \nabla_\mu (A_\nu K^\nu)$,
and \eqref{eqn:Qzeta-etaS} therefore leads to
\begin{equation}
0=i\nabla_\mu\braket{[{\cal Q}_\zeta,\bar\eta{\cal S}^\mu]}=\frac12 K^\nu A_\nu\nabla_\mu\braket{{\cal J}^\mu}= \frac12 K^\nu A_\nu
{\cal A}_{\rm chiral} \label{eqn:one-point-inconsistency}
\end{equation}
on the supersymmetric vacuum. This implies that
\begin{equation}
K^\nu A_\nu=0 \quad \text{or}\quad \nabla_\mu\braket{{\cal J}^\mu}={\cal A}_{\rm chiral}=0.
\end{equation}
Yet this does not seem to cause a serious problem, as we saw the similar constraints
in the previous section, see \eqref{eqn:new-constraint-J}-\eqref{eqn:new-constraint}.
However, it turns out that the real problem shows up in the higher-point functions.
To see this, we multiply \eqref{eqn:cls-ward-susy} by any spinor $\epsilon(x)$ and differentiate with respect to $\bar \psi_\mu(x_1)$ and then $A_\nu(x_2)$. We then set the gravitino to zero and obtain
\bala
& i{\cal D}_\rho\langle \langle \bar{\cal S}^\rho\epsilon(x){\cal S}^\mu(x_1){\cal J}^\nu(x_2)\rangle \rangle +\frac{1}{2}\langle \langle\bar{\cal S}^\lambda\gamma_\lambda V^\rho\gamma_{\rho}\gamma_*\epsilon(x){\cal S}^\mu(x_1){\cal J}^\nu(x_2)\rangle \rangle=\\
&=\frac12\delta^4(x_1,x)\gamma_\lambda\epsilon\Big \langle \Big \langle \Big[{\cal T}^{\mu\lambda}+\frac34(V^\lambda{\cal J}^\mu+V^\mu{\cal J}^\lambda-g^{\mu\lambda}V_\rho {\cal J}^\rho-\varepsilon^{\mu\lambda\rho\kappa}\nabla_\rho{\cal J}_\kappa)+\frac12\varepsilon^{\mu\lambda\rho\kappa}{\cal F}_{\rho\kappa}\Big](x_1){\cal J}^\nu(x_2)
\Big \rangle \Big \rangle\\
&\hskip1em+\frac{3i}{8}\delta^4(x_1,x)\gamma_*\gamma_\lambda\epsilon\;\varepsilon^{\mu\lambda\rho\sigma}V_\sigma \langle \langle {\cal J}_\rho(x_1) {\cal J}^\nu(x_2)\rangle \rangle \\
&\hskip1em-\frac{i}{4} \delta^4(x_1,x)(-i \varepsilon^{\lambda\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\lambda-\delta_\sigma^\lambda\gamma_*\gamma^\mu){\cal D}_\lambda\big(\epsilon\langle \langle {\cal J}^\sigma(x_1){\cal J}^\nu(x_2) \rangle \rangle \big)\\
&\hskip1em +\frac{i}{4} \bm e^{-1}\pa_\lambda\big[\delta^4(x_1-x)(-i \varepsilon^{\lambda\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\lambda-\delta_\sigma^\lambda\gamma_*\gamma^\mu)\epsilon \langle \langle {\cal J}^\sigma(x_1){\cal J}^\nu(x_2)\rangle \rangle \big]\\
&\hskip1em+\frac14\delta^4(x_1,x)\delta^4(x_1,x_2)(-i \varepsilon^{\nu\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\nu-\delta_\sigma^\nu\gamma_*\gamma^\mu)\gamma_*\epsilon\;\braket{{\cal J}^\sigma}
+i\delta^4(x_2,x)\langle \langle\bar{\cal S}^\nu\gamma_*\epsilon(x)\;{\cal S}^\mu(x_1)\rangle \rangle.\numberthis\label{eqn:3point-general}
\eala
As mentioned before, it is important to keep all contact terms in the above computation.
Now we let $\epsilon(x)$ be the GKS $\zeta$ and multiply \eqref{eqn:3point-general} by $ \bar \eta= i\bar\zeta(x_1)\gamma_*$. We then obtain
\bala
&i\nabla_\rho\braket{\braket{\bar{\cal S}^\rho\zeta(x)\;\bar\eta{\cal S}^\mu(x_1)\;{\cal J}^\nu(x_2)}}= \\
&=\frac12\delta^4(x_1,x) \Big\langle \Big \langle \Big[K_\kappa({\cal T}^{\mu\kappa}-G^\kappa{\cal J}^\mu+\frac12\varepsilon^{\mu\kappa\rho\lambda}{\cal F}_{\rho\lambda})-\frac14\nabla_\lambda(\varepsilon^{\mu\kappa\lambda\sigma}K_\kappa{\cal J}_\sigma)+(K^\rho A_\rho){\cal J}^\mu\Big](x_1){\cal J}^\nu(x_2)
\Big\rangle \Big \rangle \\
&\quad +i\delta^4(x_2,x)\braket{\braket{\bar{\cal S}^\nu\gamma_*\zeta(x_2)\;\bar\eta{\cal S}^\mu(x_1)}}+\frac{i}{4}\delta^4(x_1,x)\delta^4(x_1,x_2)(K^\nu\braket{{\cal J}^\mu}-K^\mu\braket{{\cal J}^\nu}).\numberthis
\eala
Let us integrate this over $x$-space, using (see \eqref{npt-X-Q})
\begin{equation}
0=\Braket{\delta_\zeta\(\bar\eta{\cal S}^\mu(x_1)\;{\cal J}^\nu(x_2)\)} \,\,\implies \,\,0=\int\mbox{d}^4x\;{\bm e}\;\nabla_\rho\langle \langle \bar{\cal S}^\rho\zeta(x)\;\bar\eta{\cal S}^\mu(x_1)\;{\cal J}^\nu(x_2)\rangle \rangle,
\end{equation}
and take a covariant divergence with respect to $x_1$ to obtain
\begin{align}
0&=\nabla_\mu \int\mbox{d}^4x\;{\bm e}\;i\nabla_\rho\langle \langle\bar{\cal S}^\rho\zeta(x)\;\bar\eta{\cal S}^\mu(x_1)\;{\cal J}^\nu(x_2)\rangle \rangle
\nonumber\\
&=\frac12(K^\lambda A_\lambda )\nabla_\mu\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\nu(x_2)}}-\frac{i}{2}\delta^4(x_1,x_2)K^\nu\nabla_\rho\braket{{\cal J}^\rho(x_2)}\nonumber\\
&=\frac{K^\lambda}{2} \frac{\delta}{i\bm e\, \delta A_\nu(x_2)}\Big( A_\lambda\nabla_\mu\braket{{\cal J}^\mu} \Big). \label{two-point-consistency}
\end{align}
For the second equality we used the relation \eqref{eqn:susy-s-s} (where the first term on the RHS vanishes due to the assumption of the supersymmetric vacuum) and
\begin{align}
0&=i\nabla_\mu\Big[K_\nu\Big(\braket{\braket{{\cal T}^{\mu\nu}(x_1){\cal J}^\rho(x_2)}}+\frac32\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\rho(x_2)}} V^\nu
+\frac12\varepsilon^{\mu\nu\rho\lambda}\braket{\braket{{\cal F}_{\rho\lambda}(x_1){\cal J}^\rho(x_2)}}\Big)\Big]\nonumber\\
&\hskip3em -iK^\nu A_\nu\nabla_\mu\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\rho(x_2)}}-\delta^4(x_1,x_2)\Big[\nabla_\nu(K^\nu\braket{{\cal J}^\rho})-\braket{{\cal J}^\nu}\nabla_\nu K^\rho\Big]
\nonumber\\
&\hskip3em +\bm e^{-1}\pa_\nu\Big[\delta^4(x_1-x_2)(K^\nu\braket{{\cal J}^\rho}-K^\rho\braket{{\cal J}^\nu})\Big],\label{eqn:diffeo-T-J}
\end{align}
which is obtained from \eqref{eqn:stress-R-two-point} by replacing $\xi^\nu$ by the Killing vector $K^\nu$ and using the Killing equations \eqref{eqn:killing-eqns}. Note that in the above computation there occurs a complete cancellation between the contact terms. Using \eqref{eqn:qt-ward-gauge}, \eqref{two-point-consistency} implies
\begin{equation}
K^\lambda \frac{\delta}{i\bm e\, \delta A_\nu(x_2)}\Big( A_\lambda{\cal A}_{\rm chiral}(x_1) \Big)=0 , \label{eqn:two-point-consistency}
\end{equation}
which is another constraint in addition to \eqref{eqn:one-point-inconsistency}.
The analysis up to now is insufficient to say about the quantum inconsistency with the known $U(1)_R$ anomaly,
because there may exist very restrictive backgrounds on which the constraints \eqref{eqn:one-point-inconsistency} and \eqref{eqn:two-point-consistency} are satisfied.
Therefore we need to go further to higher-point functions.
To this end, we differentiate \eqref{eqn:3point-general} once more with respect to the gauge field source
$A_\lambda(x_3)$ to obtain
\bala
&i{\cal D}_\rho\langle \langle\bar{\cal S}^\rho\epsilon(x){\cal S}^\mu(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)\rangle \rangle +\frac{1}{2}\langle \langle\bar{\cal S}^\kappa\gamma_\kappa V^\rho\gamma_\rho\gamma_*\epsilon(x){\cal S}^\mu(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)\rangle \rangle=\\
&=\frac12\delta^4(x,x_1)\gamma_\rho\epsilon\Big\langle \Big \langle\Big[{\cal T}^{\mu\rho}+\frac34(V^\rho{\cal J}^\mu+V^\mu{\cal J}^\rho-g^{\mu\rho}V_\kappa {\cal J}^\kappa-\varepsilon^{\mu\rho\sigma\kappa}\nabla_\sigma{\cal J}_\kappa)+ \nonumber \\
& \hspace{7.5cm}+\frac12\varepsilon^{\mu\rho\sigma\kappa}{\cal F}_{\sigma\kappa}\Big](x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)
\Big\rangle \Big \rangle\\
&\hskip1em+i\delta^4(x,x_2)\langle \langle\bar{\cal S}^\nu\gamma_*\epsilon(x)\;{\cal S}^\mu(x_1){\cal J}^\lambda(x_3)\rangle \rangle+i\delta^4(x,x_3)\langle \langle\bar{\cal S}^\lambda\gamma_*\epsilon(x)\;{\cal S}^\mu(x_1){\cal J}^\nu(x_2)\rangle \rangle \\
&\hskip1em +\frac{i}{4}\bm e^{-1}\pa_\rho\big[\delta^4(x-x_1)(-i \varepsilon^{\rho\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\rho-\delta_\sigma^\rho\gamma_*\gamma^\mu)\epsilon\langle \langle{\cal J}^\sigma(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)\rangle\big]\\
&\hskip1em-\delta^4(x,x_1)\frac{i}{4}(-i \varepsilon^{\rho\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\rho-\delta_\sigma^\rho\gamma_*\gamma^\mu){\cal D}_\rho\big(\epsilon\langle \langle{\cal J}^\sigma(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)\rangle \rangle\big)\\
&\hskip1em+\frac{3i}{8}\delta^4(x,x_1)\gamma_*\gamma_\rho\epsilon\;\varepsilon^{\mu\rho\kappa\sigma}V_\sigma \langle \langle {\cal J}_\kappa (x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)\rangle \rangle \\
&\hskip1em-\frac{i}{4}\delta^4(x,x_1)\delta^4(x,x_2)(-i \varepsilon^{\nu\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\nu-\delta_\sigma^\nu\gamma_*\gamma^\mu)\gamma_*\epsilon\;\langle \langle{\cal J}^\sigma(x){\cal J}^\lambda(x_3)\rangle \rangle\\
&\hskip1em-\frac{i}{4}\delta^4(x,x_1)\delta^4(x,x_3)(-i \varepsilon^{\lambda\mu}{}_{\sigma\kappa}\gamma^\kappa+\delta^\mu_\sigma\gamma_*\gamma^\lambda-\delta_\sigma^\lambda\gamma_*\gamma^\mu)\gamma_*\epsilon\;\langle \langle{\cal J}^\sigma(x){\cal J}^\nu(x_2)\rangle \rangle .\numberthis\label{eqn:4point-general}
\eala
By following essentially the same steps as done to reach
\eqref{eqn:two-point-consistency} from \eqref{eqn:3point-general}
for the 3-point function $\langle \langle \bar{\cal S}^\rho\epsilon(x){\cal S}^\mu(x_1){\cal J}^\nu(x_2)\rangle \rangle$, one can obtain
\bala
0&=\nabla_\mu \int\mbox{d}^4x\;{\bm e}\;i\nabla_\rho \langle \langle\bar{\cal S}^\rho\zeta(x)\;\bar\eta{\cal S}^\mu(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3) \rangle \rangle
\\
&=\frac12(K^\rho A_\rho)\nabla_\mu\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\nu(x_2){\cal J}^\lambda(x_3)}}-\frac{i}{2}\delta^4(x_1,x_2)K^\nu\nabla_\mu\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\lambda(x_3)}}\\
& \quad
-\frac{i}{2}\delta^4(x_1,x_3)K^\lambda\nabla_\mu\braket{\braket{{\cal J}^\mu(x_1){\cal J}^\nu(x_2)}}\\
&=\frac{K^\rho}{2}\frac{\delta}{i\bm e \,\delta A_\lambda(x_3)} \frac{\delta}{i\bm e\, \delta A_\nu(x_2)}\Big( A_\rho \nabla_\mu\braket{{\cal J}^\mu(x_1)} \Big)\\
&=\frac{K^\rho}{2}\frac{\delta}{i\bm e\, \delta A_\lambda(x_3)} \frac{\delta}{i\bm e \, \delta A_\nu(x_2)}\Big(A_\rho {\cal A}_{\rm chiral}(x_1) \Big).\numberthis \label{eqn:2-deriv}
\eala
The above procedures can be straightforwardly
extended to the higher-point functions obtained by differentiating \eqref{eqn:3point-general} successively with respect to the gauge fields,
giving rise to
constraints (putting together \eqref{eqn:one-point-inconsistency}, \eqref{eqn:two-point-consistency} and \eqref{eqn:2-deriv} here)
\begin{align}
0 & = K^\rho A_\rho {\cal A}_{\rm chiral}(x), \label{eqn:0-deriv}\\
0 & = K^\rho \frac{\delta}{\bm e \, \delta A_\nu(x_1)}\Big( A_\rho {\cal A}_{\rm chiral}(x) \Big), \label{eqn:1-deriv} \\
0 & = K^\rho\frac{\delta}{\bm e \, \delta A_\lambda(x_2)} \frac{\delta}{\bm e \, \delta A_\nu(x_1)}\Big( A_\rho {\cal A}_{\rm chiral}(x) \Big), \label{2-deriv} \\
0 &= K^\rho \frac{\delta}{\bm e \, \delta A_\alpha(x_3)} \frac{\delta}{\bm e\, \delta A_\lambda(x_2)} \frac{\delta}{\bm e \, \delta A_\nu(x_1)}\Big( A_\rho {\cal A}_{\rm chiral}(x) \Big),
\label{eqn:higher-order-constraints} \\
0 &= K^\rho \frac{\delta}{\bm e \, \delta A_\beta(x_4)} \frac{\delta}{\bm e \, \delta A_\alpha(x_3)} \frac{\delta}{\bm e\, \delta A_\lambda(x_2)} \frac{\delta}{\bm e \, \delta A_\nu(x_1)}\Big(A_\rho {\cal A}_{\rm chiral}(x) \Big)
\label{eqn:4-deriv}
\end{align}
and so on. These constraints are consequences of the Ward identities \eqref{eqn:cls-ward-susy}-\eqref{eqn:qt-ward-gauge} and relation \eqref{npt-X-Q}
with the supersymmetric vacuum condition \eqref{Q_z_h}. Constraint \eqref{eqn:4-deriv} and the subsequent ones with the higher functional derivatives are trivially satisfied,
since ${\cal A}_{\rm chiral}$ is quadratic in the gauge field, while
\eqref{eqn:0-deriv}-\eqref{eqn:higher-order-constraints} are nontrivial constraints.
In particular,
constraint \eqref{eqn:higher-order-constraints} with \eqref{ca} gives
\begin{align}
0&= K^\rho \frac{\delta}{\bm e\, \delta A_\alpha(x_3)}\frac{\delta}{\bm e \, \delta A_\lambda(x_2)}\frac{\delta}{\bm e \, \delta A_\nu(x_1)}\Big( A_\rho {\cal A}_{\rm chiral}(x) \Big)\nonumber\\
&=2c_A\Big[K^\alpha\delta^4(x,x_3)\varepsilon^{\kappa\lambda\sigma\nu}\pa_\kappa\delta^4(x,x_2)\pa_\sigma\delta^4(x,x_1)+\text{(permutations)}\Big],
\end{align}
which can be satisfied if and only if $c_A=0$.
Thus, the anomaly coefficient $c_A$ should vanish. This makes constraints \eqref{eqn:0-deriv}-\eqref{2-deriv} satisfied automatically.
\section{Discussions
}
\label{sec:discussion}
\setcounter{equation}{0}
In this note we have studied the quantum consistency
of the new-minimal supergravity formulation of the ${\cal N}=1$ supersymmetric theories with an $R$-symmetry in 3+1 dimensional curved space. By investigating the rigid supersymmetry algebra and the correlation functions obtained via differentiation
of the Ward identities (with respect to the background gravitino and $R$-gauge fields),we have shown that
the pure $U(1)_R$ chiral anomaly coefficient $c_A$ should vanish to be consistent with the supersymmetry.
Our result indicates that the supersymmetry is broken at the quantum level unless $c_A = 0$.
We emphasize that the anomaly coefficient
$c_A$ depends only on the field content of the theory.
There exist some special cases where $c_A=0$. For instance, in the ${\cal N}=1$ superconformal theories, the anomaly coefficient $c_A$ becomes
\begin{equation}
c_A=5a-3c,
\end{equation}
where the central charges $a$ and $c$ (for free theory) is given by \cite{Christensen:1978gi, Anselmi:1997ys}
\begin{equation}
a=\frac{1}{48}(9N_V+N_\chi),\quad c=\frac{1}{24}(3N_V+N_\chi).
\end{equation}
Here $N_V$ and $N_\chi$ are the number of gauge and chiral multiplets, respectively. Therefore, $c_A$ becomes vanishing
when $27 N_V=N_\chi$, for example when $N_V=1$ and $N_\chi=27$.
Another simple example for a theory with $c_A=0$ is the system that consists of a free chiral multiplet with $R$-charge 1 (see e.g. appendix \ref{app:free-chiral}). In this case, the Weyl fermion in the chiral multiplet is actually uncharged under the $U(1)_R$ symmetry and thus does not contribute to the $U(1)_R$ chiral anomaly.
In this work we have focused on the quantum consistency with respect to the pure $U(1)_R$ anomaly. We expect a similar result for the mixed $U(1)_R$ anomaly coefficient $c_m$. Namely, we anticipate that higher-point correlation functions involving both of the supercurrent and the stress-energy tensor can be consistent only when $c_m=0$. But
it needs more involved computations to show this, which
we leave for the future work.
We have assumed that
the supersymmetry Ward identity is non-anomalous, see \eqref{eqn:cls-ward-susy}, and our results bring forward a question about the supersymmetry anomaly, which is also related to the holography.
In \cite{Papadimitriou:2017kzw,An:2017ihs} the holographic renormalization \cite{Henningson:1998gx,Balasubramanian:1999re,deBoer:1999tgo,Kraus:1999di,deHaro:2000vlm,Bianchi:2001de,Bianchi:2001kw,Martelli:2002sp,Skenderis:2002wp,Papadimitriou:2004ap} was carried out for both of the bosonic and fermionic sector of the 5D ${\cal N}=2$ gauged supergravity, a virtual candidate for holographic dual of 4D ${\cal N}=1$ superconformal field theory (that has an $R$-symmetry).
By doing so, it was derived that the supersymmetric Ward identities for 4D ${\cal N}=1$ superconformal field theories (SCFTs) contain anomaly-terms, which lead to the anomalous variation law of supercurrent operators under the rigid supersymmetry transformation.\footnote{Recently, a similar result was obtained in \cite{An:2018xxx}, where it was shown in the context of AdS$_3$/CFT$_2$ that in 2D ${\cal N}=(1,1)$ superconformal field theories the supercurrent operator transforms anomalously under the rigid supersymmetry transformation.}
If one assumes that the supersymmetry Ward identity \eqref{eqn:cls-ward-susy} receives anomaly corrections,
those anomaly terms would introduce additional (contact) terms in the correlations functions considered in section \ref{SUSY} and \ref{sec:quantum-consistency}, which in turn may lead to complete cancellations on the right-hand sides of \eqref{eqn:0-deriv}-\eqref{eqn:4-deriv} restoring the consistency of the theory with $c_A\neq 0$.\footnote{One relevant question is whether there could exist local counterterms added to the action in such a way that the consistency of the theory is maintained without requiring $c_A=0$. In order for the local counterterms to change the argument of this paper and restore the consistency of the theory with $c_A\neq 0$, they should modify the Ward identities \eqref{eqn:cls-ward-susy}-\eqref{eqn:qt-ward-gauge} by introducing appropriate additional local terms to the Ward identities. However, the local counterterms that modify the Ward identities necessarily break the corresponding symmetries. Therefore, as far as one does not want to break supersymmetry and diffeomorphism invariance explicitly, we expect that local counterterms can not play a role in retrieving the consistency of the theory with $c_A\neq 0$.}
It would be interesting to explore if the anomaly corrections obtained in \cite{An:2017ihs,Papadimitriou:2017kzw} by holographic renormalization could do the job (see \cite{Katsianis:2019hhg, Papadimitriou:2019gel} for recent field-theoretical studies on the supersymmetry anomalies).
We hope to pursue this question in the future work.
\section*{Acknowledgments}
We thank Gwang Il Kim and Ui Ri Mun for useful discussions and comments. This research was supported in part by NSTC Project No. 130-01.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,485 |
package com.jummyshapefile.utils;
import java.nio.ByteBuffer;
/**
* Utility class to work with bytes.
*/
public final class ByteUtils {
/**
* Returns an int representing the bytes.
*
* @param bytes
* the byte array
* @param length
* the length of the chunk
* @param offset
* the offset of the chunk
* @return an int representing the bytes
*/
public static final int readInt(final byte[] bytes, final int length,
final int offset) {
int output = -1;
if (bytes == null) {
return output;
}
final byte[] intBytes = new byte[length];
for (int i = 0; i < intBytes.length; i++) {
intBytes[i] = bytes[offset + i];
}
output = ByteUtils.toInt(intBytes, 0);
return output;
}
/**
* Returns an int representing the bytes (least significant first).
*
* @param bytes
* the byte array
* @param length
* the length of the chunk
* @param offset
* the offset of the chunk
* @return an int representing the bytes (least significant first)
*/
public static final int readIntLeastSignificantFirst(final byte[] bytes,
final int length, final int offset) {
int output = -1;
if (bytes == null) {
return output;
}
final byte[] intBytes = new byte[length];
for (int i = 0; i < intBytes.length; i++) {
intBytes[i] = bytes[offset + length - i - 1];
}
output = ByteUtils.toInt(intBytes, 0);
return output;
}
/**
* Returns an String representing the bytes.
*
* @param bytes
* the byte array
* @param length
* the length of the chunk
* @param offset
* the offset of the chunk
* @return an String representing the bytes
*/
public static final String readString(final byte[] bytes, final int length,
final int offset) {
String output = null;
if (bytes == null) {
return output;
}
final byte[] rowCountBytes = new byte[length];
for (int i = 0; i < rowCountBytes.length; i++) {
rowCountBytes[i] = bytes[offset + i];
}
output = new String(rowCountBytes);
return output;
}
/**
* Returns a double representing the bytes.
*
* @param bytes
* the byte array
* @param length
* the length of the chunk
* @param offset
* the offset of the chunk
* @return a double representing the bytes
*/
public static final double readDouble(final byte[] bytes, final int length,
final int offset) {
double output = -1d;
if (bytes == null) {
return output;
}
final byte[] doubleBytes = new byte[length];
for (int i = 0; i < doubleBytes.length; i++) {
doubleBytes[i] = bytes[offset + i];
}
output = ByteBuffer.wrap(doubleBytes).getDouble();
return output;
}
/**
* Returns a double representing the bytes (least significant first).
*
* @param bytes
* the byte array
* @param length
* the length of the chunk
* @param offset
* the offset of the chunk
* @return a double representing the bytes (least significant first)
*/
public static final double readDoubleLeastSignificantFirst(
final byte[] bytes, final int length, final int offset) {
double output = -1d;
if (bytes == null) {
return output;
}
final byte[] doubleBytes = new byte[length];
for (int i = 0; i < doubleBytes.length; i++) {
doubleBytes[i] = bytes[offset + length - i - 1];
}
output = ByteBuffer.wrap(doubleBytes).getDouble();
return output;
}
/**
* Returns the integer representation of an array of bytes from a given
* offset.
*
* @param bytes
* an array of bytes
* @param offset
* the offset of the chunk
* @return the integer representation bytes
*/
private static final int toInt(final byte[] bytes, final int offset) {
int output = 0;
if (bytes == null) {
return output;
}
for (int i = 0; i < 4 && i + offset < bytes.length; i++) {
output <<= 8;
output |= bytes[i] & 0xFF;
}
return output;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,997 |
Q: Scrape product details from samsclub.com I'm scraping data from SamsClub.com using PHP
$res = file_get_contents('http://www.samsclub.com/sams/bath-towel-apple-gr-100-cotton/prod10450797.ip');
I have create function using PHP explode to fetch data.
function getData($content,$start,$end){
$str = explode($start,$content);
$str = explode($end,$str[1]);
return $str[0];
}
All required data fetched successfully but only one thing is remaining. That is variations of product mean other colors as u can see in snapshot, there are different colors available.
When we select an other color the item # & Model # of product also changed as shown in following snapshot
I just want to also fetch information like "item # & model #" of other colors too.
Waiting for your great response guys.
A: to do this you will need to use a library (PHP Simple HTML DOM Parser). Just upload simple_html_dom.php somewhere were you are able to include it (in my code, it's in the same folder).
<?php
$url = 'http://www.samsclub.com/sams/bath-towel-apple-gr-100-cotton/prod10450797.ip';
include('simple_html_dom.php');
$html = file_get_html($url);
$colour = array(); $item = array(); $model = array();
$script = $html->find('div[id=variance] script', 0)->innertext;
$script = preg_replace('/\s+/', ' ', $script);
$scripts = explode (";", $script);
$script = $scripts[2];
$id = $scripts[4];
$type = $scripts[5];
$script = str_replace("skuJson.skuVariantJson = $.parseJSON('", "", $script);
$script = str_replace("')", "", $script);
$colours = json_decode($script);
preg_match("/'([a-z0-9]*)'/", $type, $types); $type = $types[1];
preg_match("/'([a-z0-9]*)'/", $id, $ids); $id = $ids[1];
$script = $html->find('script', -1)->innertext;
$scripts = explode (";", $script);
$time = $scripts[0];
preg_match('/"([0-9]*)"/', $time, $times); $time = $times[1];
foreach ($colours as $key => $value) {
$url = 'http://www.samsclub.com/sams/shop/product/ajax/ajaxSkuVariant.jsp?skuId='. $value .'&productId='. $id .'&productType='. $type .'&_='. $time;
$html = file_get_html($url);
preg_match('/"legacyItemNumber":"([0-9]*)"/', $html, $match); $item[] = $match[1];
preg_match('/"model":"([a-z-]*)"/i', $html, $match); $model[] = $match[1];
$colour[] = substr($key, 0, -1);
}
//Print results
echo "<pre>"; print_r($colour); echo "</pre>";
echo "<pre>"; print_r($item); echo "</pre>";
echo "<pre>"; print_r($model); echo "</pre>";
?>
The only thing you need to change is the $url variable at the beginning. Why all this code, you may ask ... because the data you are looking for isn't on the same page and it's called via ajax every time you click on a colour, so basically we are making a lot of requests (one for each color). This is the output:
Array
(
[0] => White
[1] => Burgundy
[2] => Apple Green
[3] => Lilac
[4] => Chocolate
[5] => Sage
[6] => Grey
[7] => PckBlue
[8] => Linen
[9] => null
[10] => Plum
[11] => Clay
[12] => Light Blue
)
Array
(
[0] => 252368
[1] => 252505
[2] => 252414
[3] => 433076
[4] => 252389
[5] => 117268
[6] => 252438
[7] => 613317
[8] => 252382
[9] => 433083
[10] => 252541
[11] => 117175
[12] => 252400
)
Array
(
[0] => SAMW-B
[1] => SAMB-B
[2] => SAMA-B
[3] => SAMLC-B
[4] => SAMCH-B
[5] => SAMSS-B
[6] => SAMGR-B
[7] => SAMPB-B
[8] => SAMLI-B
[9] => SAMDR-B
[10] => SAMP-B
[11] => SAMTC-B
[12] => SAMLB-B
)
A: I would recommend scraping using .NET and Browser Class. This way you can make the bot click on each color and then grab the required value.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,752 |
in: Work, Pages with broken file links, Speculative Fiction Creator Index,
Trope Makers
File:JRRTolkien.jpg
He's a smiling Oxford don with a pipe. How can you not like and trust him?
"How, given little over half a century of work, did one man become the creative equivalent of a people?"
—The Guardian, concerning The Silmarillion
John Ronald Reuel Tolkien (1892-1973). English linguist (born in Bloemfontein, South Africa), university professor (Leeds and Oxford), Anglo-Saxon historian, CBE, and writer. The man who brought High Fantasy (and, it could be argued, literary Speculative Fiction as a whole) to the modern public. His most famous complete works are his tales of "Middle-earth": The Lord of the Rings and its prefatory novel, The Hobbit. A later work, The Silmarillion, was published in 1977, shortly after he died. In 2007, a fourth book about Middle-earth was edited from many manuscripts to form a consistent narrative, and published as The Children of Hurin.
Tolkien was a polyglot who spoke well over a dozen languages and had some comprehension of up to forty. He even made up a few of his own. Let's just say there aren't many authors who kept interfering with the foreign translations of their books to (correctly, see for instance the article on translator Åke Ohlmarks) point out how the translators aren't translating things properly into their native languages...
Tolkien's Legendarium
The collective term for all the stories about the world of Middle-earth (not actually the world's name, but the name of a super-continent in a world generally referred to as "Arda" by the peoples therein; it is used here for convenience's sake, being the name far better known to the general reader).
The earliest drafts of the great stories of the legendarium were written around the time of World War I, and continued to grow from there on. Tolkien worked on the legendarium for most of his life, continually exploring it further, developing and changing it again and again.
The first book published, The Hobbit, actually wasn't intended as part of the legendarium, only to borrow some material. When Tolkien began writing the Hobbit-sequel that was to become The Lord of the Rings, he moved the story of both books into the Middle-earth setting. This fact is responsible for the seeming inconsistencies in tone and canon between The Hobbit and the other Middle-earth works; this is often mistaken for the world and story having matured up by those who do not know it existed before. He also made some minor changes in a later edition of The Hobbit to match better with The Lord of the Rings, while also providing an in-universe justification for the original discrepancies in the latter.
The published books are:
The Hobbit (1937)
The Lord of the Rings (originally published in three volumes, 1954--55)
The Adventures of Tom Bombadil (collection of "in-universe" poetry, 1962)
The Silmarillion (1977)
Unfinished Talesof Numenor and Middleearth (1980)
The History of Middle Earth (12 volumes, 1983--96)
The Children of Hurin (2007)
The History of The Hobbit (2007)
Only the first three were published during his lifetime; the rest were published posthumously by his son Christopher. Of these, The Silmarillion and The Children of Húrin consist of a single narrative edited together from Tolkien's texts, while the rest are collections of Tolkien's material (with commentaries and notes by his son), ranging from complete narratives to early and new drafts, to essays.
Additionally, there are several titles used for collective bodies of stories (which are used in the fictional world, and also real-world terms to include all relevant material independently of published books): the Ainulindalë and Valaquenta (the creation of the world), the (Quenta) Silmarillion (the First Age), and the Akallabêth (History and Downfall of Númenor in the Second Age).
His other works include several shorter tales (including several written for his children) and his academic writings; among these works are.
Mr. Bliss
Smith of Wootton Major
Farmer Giles of Ham
The Father Christmas Letters
Beowulf: The Monsters and the Critics: This lecture redefined the importance of Beowulf as a poem, rather than "a relic of the past".
On Fairy Stories: This lecture essentially inaugurated the academic study of Fairy Tales; we thus have it to thank for stories as varied as Shrek, Princess Tutu, Nights at the Circus, and Revolutionary Girl Utena.
The Legend of Sigurd and Gudrún
Translations of medieval literature that he did in his spare time, including the best-known modern versions of Sir Gawain and the Green Knight, Pearl and Sir Orfeo.
The collected Letters of J.R.R. Tolkien are a valuable source of information, both regarding Middle-earth and his personal life and views. Tolkien also aided in compiling the Oxford English Dictionary, and worked on the Jerusalem Bible, a respected Roman Catholic translation.
His greatest fiction was based off his linguistic research and invention. His work on this subject filled well over a dozen volumes.
Think you'd like to have a legacy like this guy's? Start here!
All There In The Appendices And Posthumously Published Work
The Appendices made up nearly half of The Return of the King.
Always Chaotic Evil: Tolkien himself was troubled by the Unfortunate Implications.
Applicability: The Trope Namer
Artifact of Doom
Attention Deficit Creator Disorder: Not so much because there were too many projects, but because Tolkien was a perfectionist and had a day job as a university professor. Christopher Tolkien is still publishing the works never released in his father's lifetime.
Author Appeal: In case you missed it, Tolkien likes linguistics, trees, music, and dark-haired grey-eyed women.
Author Avatar: Word of God points to Beren and Faramir.
Author Phobia: Tolkien was bitten by a poisonous spider in his youth in South Africa and narrowly escaped death. Many of his works feature giant, malevolent arachnids, including the spiders of Mirkwood, Shelob, and Ungoliant.
Authority Equals Asskicking
Badass Bookworm
Fought in the Somme during World War I until trench fever made him unfit for further combat duty. He started writing about what would become The Silmarillion while recuperating.
He didn't actually fight in World War II, unlike some of his children. But he did accept a tentative offer from British military intelligence to advise their cryptography department — though it turned out they never needed his services. Does this mean Tolkien almost got a shot at cracking Enigma?
Tolkien was a Lieutenant in the Signal Corps. While as such he didn't participate in the actual bang-bang fighting, he was deployed at the front line and lived in the trenches. He later stated we all were Orcs and modelled the Orc mode of fighting after the human wave attacks in WWI.
He wrote to one of his sons serving in the military that he rather wished he was out there fighting as well, given the degree to which Hitler ruined a lot of otherwise valid ideas (like the old Germanic theory of courage) through his association with them.
Badass Santa: Father Christmas
Bittersweet Ending
Black Speech: The Lord of the Rings is the Trope Namer.
Blessed with Suck
Bring News Back: As described in the extended account of the disaster of Gladden Fields.
But Now I Must Go
Cash Cow Franchise: With all of the books about Middle-earth out, along with several movies (with a live-action, two-part Hobbit coming in 2012/13), several games, and tons of merchandise based on films and books, quite a bit of money has been made on Tolkien's world.
Changing of the Guard: Between The Hobbit and The Lord of the Rings
Cold-Blooded Torture: Both Sauron and Morgoth are fond of it.
Color-Coded Wizardry
Con Lang
Constructed World
Cool Old Guy
Crapsack World: Middle-Earth, especially during the Second Age.
Creation Myth: The Silmarillion begins with one, called Ainulindalë, or "The Music (literally "singing") of the Ainur". The Ainur are basically the equivalent of the angels in Christianity.
Cryptic Background Reference
Cue the Sun
Dark Is Not Evil / Light Is Not Good: Well, it usually is; there are exceptions.
Dear Negative Reader: In his introduction to the second edition of The Lord of the Rings:
"Some who have read the book, or at any rate have reviewed it, have found it boring, absurd, or contemptible, and I have no cause to complain, since I have similar opinions of their works, or of the kinds of writing that they evidently prefer."
Destructive Saviour: The people of Beleriand are relentlessly harried and killed by the evil Morgoth. The desperate remnant calls upon the Valar — extremely powerful gods or angels. The Valar come in force, launch the "War of Wrath" [1] and utterly defeat Morgoth — but in the process, nearly all of Beleriand is flooded and sinks under the sea, only a few mountain tops surviving as small islands.
And what would become the Elven kingdom of Lindon, which was originally the eastern edge of Beleriand (the Blue Mountains being the border of Beleriand).
The setting also has a Ragnarok equivalent in which the evil of Morgoth will be entirely purged from Middle Earth. Fortunately or not it will also be The End of the World as We Know It as all of Ea will be remade.
Dogged Nice Guy
Doorstopper
Dug Too Deep
Easter Egg: All of Tolkien's works about Middle-earth, as well as the many volumes of unpublished works edited by his son, have inscriptions (usually on the title page) that can be transliterated from his fictional alphabets.
Eldritch Abomination: Ungoliant in The Silmarillion.
Also, "Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things".
The Everyman: Hobbits, who also double as the Audience Surrogate in a world of mighty wizards and brave warriors.
Evil Cannot Comprehend Good: Invoked.
Evil Is Deathly Cold
Evil Tower of Ominousness
Evil Weapon
Exploring the Evil Lair: In The Lord of the Rings and The Hobbit.
The Fair Folk: Tolkien's treatment of the High Elves was a reaction to the way elves were dealt with in contemporary fiction — either as this or as childish fairies. In Tolkien only ignorant Men like Boromir regard Elves as The Fair Folk. However, Tolkien's conception then caught on among later fantasy writers and in the end people like Terry Pratchett reacted in turn back towards The Fair Folk.
Fairy Tale: Tolkien's lecture "On Fairy-Stories" was a groundbreaking work in the academic study of fairy tales.
Fantasy Counterpart Culture: With the exception of the Shire itself, which was modeled on the idealized 19th-century English countryside, the cultures of Middle-Earth are roughly equivalent to those of Dark Age Europe based on political situations and cultural aspects.
The political situation of Gondor and Arnor may remind one of Byzantium and Rome, who faced threats from the East (Huns, Ottomans, etc.) at various times in their history. Strangely, when Tolkien was asked about this comparison, he said that he regarded Gondor as being closer to Ancient Egypt.
Gondor was a direct descendant of Númenor, whose culture sounds Punic. The fact they were bilingual (speaking both a Semitic-like Adûnaic language and Elvish Sindarin), were a seafaring people and worshipped an evil god named originally Melkor ("He who arises in might") match Ancient Carthage: speaker of both Punic and Greek, seafaring, worshipped an evil god who demanded human sacrifices and was named Melkart ("Mighty one").
The Rohirrim have aspects of Anglo-Saxon culture, and have been compared to Vikings that rode horses rather than ships. Their Eotheod ancestors are based on the then-perception of Goths as a people of Germanic horse-warriors. The Rohirrim military is still this while their language has developed into the later Germanic language of Anglo-Saxon. The fact they had been a people of warrior-peasants whose entire culture ran around the horse and who lived on plains (as opposed to the hilly landscape of the British Isles) also makes them comparable 16th-19th century Russian Cossacks.
The Southrons are a vague, nonspecific representation of African peoples, as in the medieval writings Tolkien emulated, which always spoke of these in exotic terms. Similarly, the Easterlings are a vague representation of nomadic peoples from the East (ie Huns, Tartars, Mongols). However, the Easterlings of Khand are called Variags, a term used for Viking mercenaries in Constantinople.
The Dwarvish language is inspired by Semitic languages and their displacement throughout Middle-earth draws comparisons with the Jewish diaspora, but the Dwarvish culture resembles more that of Early Middle Ages Germanic peoples: metalworkers, builders, axe-armed.
Please note that the languages he based his invented languages on do not necessarily determine the cultural equivalence of the people who use them. Sindarin was based on Welsh, and Quenya on Finnish, but Grey Elves aren't Welsh, and High Elves aren't Finns.
Fantasy Pantheon
Fantasy World Map
Fire-Forged Friends
Five Races
Franchise Zombie: As explained above, The Lord of the Rings came into existence only because the publishers wanted a sequel for the immensely popular Hobbit while Tolkien was more interested in working on his legendarium. Tolkien avoided the negative effects of this trope by incorporating both LOTR and The Hobbit into his mythos that was part of the (then unpublished) Silmarillion.
Gentleman and a Scholar: He was every inch one of these.
Giant Spiders: Ungoliant in The Silmarillion, Mirkwood's spiders in The Hobbit, and Shelob in The Lord of the Rings.
Gondor Calls for Aid
Gollum Made Me Do It
Gotterdammerung
Green Aesop: He loved nature and stated that the internal combustion engine was mankind's most evil invention. The destruction of nature by industry is a common theme in his work.
Happily Married: To Edith, whom he considered his "Luthien."
Hidden Elf Village
Hobbits: The Trope Namer, if not the Trope Maker.
Honor Before Reason
Howl of Sorrow: In Farmer Giles of Ham. When Giles rides off to slay a dragon, his dog Garm howled all night because he thought Giles would be killed.
Humans Are Warriors: All humans except the men of Bree are warriors. Hobbits also avoid warfare, but even these get a few good blows in from time to time.
The Edain, the three main human tribes allied to the Elves fit this exactly. They got to Elven country by hacking their way through Morgoth's servants. They were allowed land in exchange for fighting for the Elves, which they did with great vigor.
I Am X, Son of Y: The standard form of Warrior Prince self-address in the Tolkien universe.
I Gave My Word
I Have Many Names
Inter-people Romance: Aegnor/Andreth (Star-Crossed Lovers... Oh, how!), Finduilas/Túrin (one-sided, Type 5 with Gwindor->F->T), Lúthien/Beren (married and mortal), Idril/Tuor (married and immortal), Arwen/Aragorn (married and mortal), Mithrellas/Imrazôr (married until she pulled a Missing Mom), Melian/Thingol (angel and immortal elf).
But Thingol got himself killed eventually, and Melian wandered off to, most likely, Fade Into Nothingness.
Inhumanly Beautiful Race: Elves in Tolkien's works are almost invariably described as being good looking. The three best looking females in Middle-earth are all Elves. The Valar also count, although they cheat, since their bodies are artificial and custom-made, so their beauty is limited only by ego and imagination.
Lady of War
Left Justified Fantasy Map
Literary Agent Hypothesis: In-universe, the Middle-earth books are translations of the writings of Bilbo (who wrote The Hobbit and translated The Silmarillion out of Elvish), Frodo and Sam (who wrote The Lord of the Rings), and the Anglo-Saxon sailor Ælfwine (who stumbled upon Tol Eressëa in the Middle Ages and learned of the Elder Days from the Elves).[1]
The Lost Woods
Manly Tears
"Real" men don't cry: Oh the HELL they don't.
To cite just one example: Aragorn is so overwhelmed with grief at the death of Boromir, weeping bitterly over the latter's body, that when Gimli and Legolas come upon the scene, they think at first that Aragorn himself has been perhaps mortally wounded.
Also, Gandalf encourages Sam, Pippin, and Merry to weep when Frodo is going away forever.
Mithril: the original
Most Wonderful Sound / Hell Is That Noise: Part of Tolkien's aim in devising the Elvish languages and the Black Speech. He deliberately tried to make one sound beautiful and the other sound ugly.
Mysterious Backer: Eru and the Valar in all of his works.
Only One Name, Patronymics
Theme Naming, various kinds: Theme Family Naming (including Theme Twin Naming and Alphabetical Theme Naming), naming conventions along a dynasty (be they birth or ruling names), or general ones (e.g. the hobbit tradition of naming girls after flowers or gemstones).
Near Villain Victory: Tolkien coined the word "eucatastrophe" to describe this trope.
No One Gets Left Behind
No Man of Woman Born: The Lord of the Rings is co-Trope Namer with Shakespeare.
Our Elves Are Better: Very much not, even if many people mistakenly think so. Yes, Elves are in many ways more powerful, "magical" and skilled than humans (they had better be, as they had long enough to practice), but they are just as capable as any human to be stupid, chauvinist, and screw up monumentally – possibly more able than humans in fact, as greater power can have bigger results. In addition, humans are created to be Immune to Fate, with the ultimate destiny of ourselves and the world left undetermined.
Doubters are referred, for a start, to the story of Feanor, the greatest creative genius in the history of the Elves, whose stubbornness and selfishness led to the millenia-long exile of almost his entire branch of the High-elven people, the Noldor, from the Blessed Lands, to civil war in those same Blessed Lands between two of the three tribes of the High-elves, to the destruction of the Elven kingdoms of Beleriand and of Beleriand itself, and to the deaths of himself and almost all his sons.
They do, however, appear to be this in The Lord of the Rings, since it isn't concentrating on Elvish history, and so most of their bigger mistakes are found elsewhere. Now, since it's the most popular and well-known of Tolkien's works, this means it's easy for people to get the wrong impression. This being said, Galadriel is shown to be just flawed as susceptible to the One Ring's power as anyone, although she is one of the handful who resist its power.
This is partly justified by the fact that elven leaders in the Hobbit and Lord of the Rings: Elrond (Rivendell), Thranduil (Mirkwood) and Galadriel and Celeborn (Lothlorien) have all lived to see the decline of their race in Middle-Earth, and are trying to do what is best for their people while offering what assistance they can to the other races. Also, considering that Galadriel and Celeborn were alive during the war against Morgoth that destroyed Beleriand (caused by Feanor, as noted above), and that Elrond saw what the folly of his people could do (it was Fëanor's grandson whom Sauron taught how to make the rings of power), they had all probably wisened up to not repeat the mistakes of their ancestors.
Oxbridge: Quite possibly the most widely-read writer to come out of Oxford's walls.
Palantir Ploy: The Palantir devices in The Lord of the Rings are the trope-namers.
People of Hair Color
Prophecy Twist/No Man of Woman Born
Proud Warrior Race Guy
Recurring Dreams: Tolkien had his 'Atlantis dream', featuring a huge wave coming over the land. He stated that it had a part in inspiring the Downfall of Númenor.
Rightful King Returns
Ring of Power
Sacred Hospitality
Samus Is a Girl
Second Hand Storytelling
Serious Business
Shadow Archetype
Short Cuts Make Long Delays
Shown Their Work
Sssssnaketalk
Speak Friend and Enter: The Lord of the Rings is the Trope Namer.
Supporting Leader: Former Trope Namer (The Aragorn) and possibly Trope Maker.
Sweet Polly Oliver
In response to a letter from a potential German translator/publisher of The Hobbit, who wished to know whether Tolkien was of "Arisch" extraction (which infuriated him considerably):
"I regret that I am not clear as to what you intend by arisch. I am not of Aryan extraction: that is Indo-Iranian; as far as I am aware none of my ancestors spoke Hindustani, Persian, Gypsy, or any related dialects. But if I am to understand that you are enquiring whether I am of Jewish origin, I can only reply that I regret that I appear to have no ancestors of that gifted people."
Tolkien sent his publisher two versions of this letter — this one, and a stronger version — and let the publisher choose which letter to send. Since the "mild" version is the one which remained in Tolkien's file, it may be presumed that the "wild" version was used.
Tolkien also sent a Strongly Worded Letter to AA Milne, complaining about the Adaptation Decay from The Wind in the Willows in Toad of Toad Hall and saying that his children were appalled. Somewhat Hilarious in Hindsight as Tolkien fans are famously equally stringent about Adaptation Decay in Tolkien's own works.
Tender Tears: Tolkien is rare amongst Western artists for creating consistently sensitive and soft-hearted men who do not see crying as shameful or dishonorable.
Thunderbolt Iron
Time Abyss
The Time of Myths
Throwing Down the Gauntlet
Translation Convention: All of our real-world languages do not exist in Middle-earth, and so the common Translation Convention applies. When not convention-translated, names and speech make use of either Tolkien's constructed languages, or of a real-world language used as stand-in for a fictional one. The latter ones are not chosen randomly, but to represent the relation between the respective "proper" languages, or a certain image. Languages regularly replaced by stand-in languages in the text are: "Westron" a.k.a. the "Common Speech" is always rendered as English (as it is the Third-Age-novel's POV-character's language), the Rohirric language by Anglo-Saxon a.k.a. Old English (to appear vaguely familiar to the hobbits' Westron-English), and the language used by the Dwarves and the Men of Dale by Old Norse. Information on the "translation" and what these languages "really" look like, can be found in various appendices and additional texts.
Tsundere: Towards CS Lewis and, even more severely, Dorothy L. Sayers.
Turn the Other Cheek
Turtle Island: Fastitocalon
Universe Chronology
Vitriolic Best Buds: Tolkien was this with C. S. Lewis, to the point that he frequently criticized his fellow author's books, such as The Chronicles of Narnia. But, that's vitriolic best buds for you...
Walk Into Mordor: The Movie of The Lord of the Rings is the Trope Namer.
War Is Hell: Constant theme since Tolkien is a World War One vet.
Warrior Prince: By the bucketload.
What Could Have Been
Tolkien once proposed coauthoring a scholarly book on linguistics with his academic colleague and friend C. S. Lewis. Lewis started the manuscript, but unfortunately they never got around to finishing it. [2]
For that matter, it's hard to keep from feeling wistful when reading the many fragments of unfinished stories and poems collected by his son Christopher in The History of Middle Earth. There's even a whole volume of them entitled Unfinished Tales.
When Trees Attack: The Ents were created because Tolkien had seen a production of Macbeth as a child and was disappointed when it turned out that the forest itself didn't actually attack.
"Well Done, Son" Guy: Faramir could be the poster boy for this trope.
World Shapes
World Tree
You Shall Not Pass: Gandalf's speech in the movie The Fellowship of the Ring, while a slightly paraphrased variation of the speech in the book, is the Trope Namer.
↑ Ælfwine was written out of the published Silmarillion by Christopher Tolkien, but since he appears in JRRT's writings after LotR, he apparently never abandoned the idea.
Speculative Fiction Creator Index | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,793 |
\section{Introduction}
Periodic orbits in the circular Restricted Three-Body Problem (RTBP), such as Lissajous and Halo type trajectories, are of high interest for space mission design applications. In particular, many missions, such as the cislunar space gateway concept, have been proposed that utilize these orbits. The proposed orbit for the gateway is a Near-Rectilinear Halo Orbit (NRHO), which is a non-Keplerian trajectory with the favorable properties of a continuous line of sight coverage for communications with Earth and fuel-efficient access to the lunar surface~\cite{whitley2016options}. Although operating in the RTBP region while utilizing non-Keplerian trajectories has its benefits, it remains challenging to develop, analyze, and perform guidance, navigation, and control for these missions due to the nonlinearities of the RTBP region. Therefore, new approaches for the analytical analysis of these missions are needed.
Analytical approaches exist for analyzing the RTBP and developing solutions for their trajectories. In particular, Richardson~\cite{richardson1980analytic} developed a Lindstedt–Poincaré procedure for computing the Halo orbits through matching the $x$-$y$ periodic frequency with that of the $z$ direction motion. Lindstedt–Poincaré methods are powerful perturbation approaches, but they require extensive algebraic computation and can be difficult to derive for higher-order solutions. In addition to Lindstedt–Poincaré method, Lie-Deprit methods have been successfully applied to the central gravity and RTBP problems~\cite{elipe1987rotation}, with extensive applications, from reduction center manifold~\cite{jorba1999dynamics} to long term propagation~\cite{lara2008computing}. Particularly, the evaluation of periodic orbits around the libration points is quite efficient when approached in complex variables~\cite{lara2018higher}.
A general framework for the analysis, estimation, and control of nonlinear systems remains an engineering grand challenge~\cite{junkins2004nonlinear}. Dealing with nonlinear dynamics and measurement models has been an active area of research~\cite{terejanu2008uncertainty,horwood2011gaussian,madankan2013polynomial}. A promising direction for the analysis of nonlinear systems is to seek a transformation that embeds the nonlinear dynamics in a global linear representation~\cite{hurtadousing}. The Koopman Operator (KO)~\cite{koopman1931hamiltonian} provides such a framework and has been applied to many high dimensional and nonlinear problems in engineering~\cite{mezic2013analysis,
mezic2004comparison,mezic2020koopman,budivsic2012applied,mezic2005spectral,williams2015data}, as well as to the zonal harmonics problem in astrodynamics~\cite{zonal_koopman}. This paper focuses instead on the application of the KO theory to obtain analytical orbit methodologies for the circular Restricted Three-Body Problem.
The goal of the proposed KO-based method is reformulating nonlinear problems that arise in astrodynamics into a linear framework that can be solved with available linear techniques. Motivated by Koopman~\cite{koopman1931hamiltonian} and Von Neumann~\cite{neumann1932operatorenmethode}, a new ``Heisenberg picture" was introduced in classical dynamics. In the work by Koopman, it was observed that a KO for a Hamiltonian system is unitary in an $L_2$ Hilbert space~\cite{koopman1931hamiltonian}. Later Von Neumann~\cite{neumann1932operatorenmethode} was able to make a connection between the spectrum of the KO and that of ergodicity of classical dynamical systems. The key result was demonstrating that there exists an infinite-dimensional linear operator, given by $\mathcal{K}$, that evolves all observation functions $g({\bf x})$ of the state, ${\bf x}$, for any nonlinear system. The evolution of these observables and the KO is defined by the chain rule, which is a linear operator, giving KO its linear properties.
The linearity of the KO is very appealing, but this benefit is contrasted with the fact that it is infinite-dimensional. However, this issue can be overcome by capturing the evolution on a finite subspace spanned by a finite set of basis functions instead of capturing the evolution of all measurement functions in a Hilbert space~\cite{brunton2016koopman}. In effect, this is a truncation of the KO to a finite subspace. Additionally, a Koopman invariant subspace is spanned by a set of eigenfunctions of the Koopman operator. A Koopman eigenfunction, $\phi_i({\bf x})$, corresponding to eigenvalue $\lambda_i$ is invariant under the Koopman operator (but for a normalized constant). As such, the evolution of the Koopman eigenfunctions can be expressed as $\frac{d}{dt}\phi_i({\bf x})=\lambda_i\phi_i({\bf x})$. The technique also provides an accurate spectral analysis of partial differential equations (PDEs)~\cite{nakao2020spectral,kutz2016koopman}.
The main contribution of the paper is the development of an analytical solution of the RTBP through the KO. Consequently, the dynamics are linearized and described through well-selected eigenfunctions, which describe the evolution of the state of the system over time. A detailed study of the eigenvalues of the KO matrix gives information regarding the stable and unstable part of the system, with a particular focus on its frequencies. As such, the modes are evaluated and spectral analysis is conducted. The robustness and accuracy of the newly developed technique is assessed through a convergence analysis for different KO orders. Moreover, the paper presents a new particle filtering technique where the propagation of samples is achieved through the Koopman solution and a new model predictive controller that optimizes the feedback control input using the KO to predict, in time, the state of the system.
The organization of this paper is as follows. First, the KO theory and the Galerkin method used in this paper are summarized. Next, the dynamics near the $\mathcal L_1$, $\mathcal L_2$, and $\mathcal L_3$ points are described along with the complex normal form Hamiltonian for the RTBP model. Then, simulation results are provided for both two-dimensional and three-dimensional periodic orbits, proposing the KO solution for Lyapounov and Halo orbits. The accuracy of the solution is assessed through a convergence analysis. The spectral behavior of the system is studied in terms of its modes and frequencies. A comparison between the Galerkin methodology and the Extended Dynamics Model Decomposition is also performed. Lastly, a station-keeping application uses new developed KO-based filter and controller to keep a spacecraft orbiting around the desired Halo orbit.
\section{Koopman Operator Theory}
A classical definition of nonlinear dynamical systems is given by the initial value problem, which can be represented by a set of coupled autonomous ordinary differential equations in the form:
\begin{equation}\label{problem}
\left\{ \begin{tabular}{l}
$\displaystyle\frac{d}{dt}{\bf x}(t) = {\bf f}({\bf x})$ \\
${\bf x}(t_0) = {\bf x_0}$
\end{tabular} \right.
\end{equation}
where ${\bf x}\in\mathbb{R}^d$ is the state which depends on the time evolution $t$, ${\bf f}: \mathbb{R}^d\rightarrow \mathbb{R}^d$ is the nonlinear dynamics model, $d$ is the number of dimensions in which the problem is defined, and ${\bf x_0}$ is the initial condition of the system at time $t_0$. The KO $(\mathcal{K})$ is an infinite-dimensional linear operator that evolves all observable functions $\mathcal{G}({\bf x})$ of the state, allowing to define any problem of classical mechanics in operator form.
Let $\mathcal{F}$ be a vector space of observable functions, where $\mathcal{G}({\bf x})\in \mathcal{F}$. Since the KO is an infinite-dimensional linear operator, this space of functions $\mathcal{F}$, in which the observables are defined on, is also infinite-dimensional. Therefore, if $g\subseteq\mathcal{G}({\bf x})$ is a given observable in this space, the evolution of $g$ in the dynamical system is represented by:
\begin{equation}
\mathcal{K}\left(g({\bf x})\right) = \frac{d}{dt}g({\bf x}) = \left( \nabla_{{\bf x}} g({\bf x})\right)\frac{d}{dt}{\bf x}(t) = \left( \nabla_{{\bf x}} g({\bf x})\right){\bf f}({\bf x}),
\end{equation}
where $\nabla_{{\bf x}} g = (\partial g/\partial x_1,\partial g/\partial x_2,\dots,\partial g/\partial x_d)$. That way, the evolution of any observable subjected to the dynamical system is provided by the Koopman Operator:
\begin{equation}
\mathcal{K}\left(\cdot\right) = \left( \nabla_{{\bf x}} \cdot\right){\bf f}({\bf x}).
\end{equation}
Note that the evolution of the observables is provided by the application of the chain rule for the time derivative of $g({\bf x})$. Consequently, the defined operator is linear, in that:
\begin{equation}
\mathcal{K}\left(\beta_1g_1({\bf x})+\beta_2g_2({\bf x})\right)=\beta_1\mathcal{K}\left(g_1({\bf x})\right)+\beta_2\mathcal{K}\left(g_2({\bf x})\right),
\end{equation}
for any pair of observables $g_1\subseteq\mathcal{G}({\bf x})$ and $g_2\subseteq\mathcal{G}({\bf x})$ and any arbitrary constants $\beta_1$ and $\beta_2$. This property was outlined in Koopman's paper~\cite{koopman1931hamiltonian}. The linearity of the Koopman Operator is very appealing, but this benefit is contrasted with the fact that it is infinite-dimensional. However, this issue can be overcome by capturing the evolution of the system on a finite subspace spanned by a finite set of basis functions instead of all measurement functions in a Hilbert space~\cite{brunton2016koopman}. In effect, this is a truncation of the Koopman operator to a finite subspace $\mathcal{F}_D$ of dimension $m$, where $\mathcal{F}_D \in \mathcal{F}$. This subspace $\mathcal{F}_D$ can be spanned by any set of eigenfunctions $\phi_i\in\mathcal{F}_D$, with $i\in\{1,2,\dots,m\}$, defined as:
\begin{equation}\label{koopman}
\mathcal{K}\left(\phi_i({\bf x})\right) =\frac{d}{dt}\phi_i({\bf x})=\lambda_i \phi_i({\bf x}),
\end{equation}
where $\lambda_i$ are the eigenvalues associated with the eigenfunctions $\phi_i$, and $m$ is the number of eigenfunctions chosen to represent the space. Therefore, the Koopman eigenfunctions can be used to form a transformation of variables that linearizes the system. Particularly, let ${\Phi}({\bf x}) = \left(\phi_1({\bf x}), \dots, \phi_m({\bf x}) \right)^T$ be the set of eigenfunctions of the KO in $\mathcal{F}_D$. Then, using the relation in Eq.~\eqref{koopman}, it is possible to write the evolution of ${\bf \Phi}$ as:
\begin{equation}\label{time_eigenf}
\mathcal{K}\left(\bf \Phi\right) = \frac{d}{dt}{\bf \Phi}=\Lambda {\bf \Phi},
\end{equation}
where $\Lambda=\text{diag}([\lambda_1, \dots,\lambda_m])$ is the diagonal matrix containing the eigenvalues of the system in $\mathcal{F}_D$. This transformation is called the Koopman Canonical Transform~\cite{surana2016linear}. The solution of Eq.~\eqref{time_eigenf} is:
\begin{equation}\label{eigen_time}
{\bf \Phi}(t) = \exp(\Lambda t){\bf \Phi}(t_0),
\end{equation}
where ${\Phi}(t_0)$ is the value of the eigenfunctions at the initial time $t_0$. This result will be used later to solve the complete system once the eigenfunctions of the operator are obtained.
In general, we are interested in the identity observable, that is, ${\bf g}({\bf x})={\bf x}$. Therefore, it is required to be able to represent these observables in terms of the KO eigenfunctions. This is achieved using the Koopman modes, i. e., the projection of the full-state observable onto the KO eigenfunctions. If this projection can be found, the evolution of the state is represented by means of the evolution of the KO eigenfunctions, and thus, an approximate solution to the system can be provided. However, it is important to note that the challenge resides in computing these eigenfunctions and eigenvalues of the system.
\subsection{Computing the Koopman eigenfunctions via Galerkin Method}
This section discusses the use of the Galerkin method for computing the eigenfunctions of the KO. First, the KO is used to define a Partial Differential Equation (PDE) for the time evolution of a scalar function $u({\bf x},t)$ (note that in general $u({\bf x},t)$ is a function of time, $t$, and ${\bf x}$). The Galerkin method is then used to convert the time evolution PDE to a matrix form using a series expansion for $u({\bf x},t)$ over a predefined basis set. This matrix form can be used to solve for the eigenfunctions and eigenvalues of the KO.
The Koopman Operator defines a first-order PDE for the time evolution of a scale function $u({\bf x},t)$
\begin{equation}\label{koopman_PDE1}
\frac{d u({\bf x},t)}{dt}
={ f}_1\left({\bf x}\right)\frac{\partial }{\partial x_1} u({\bf x},t) +\cdots+ { f}_d\left({\bf x}\right)\frac{\partial }{\partial x_d}u({\bf x},t),
\end{equation}
Additionally, the eigenfunctions of the Koopman Operator give rise to a set of linear first-order PDEs for the eigenfunctions in the form:
\begin{equation}\label{koopman2}
\mathcal{K}(\phi_i) = \left(\nabla_x \phi_i({\bf x})\right) {\bf f}({\bf x}) =\lambda_i \phi_i({\bf x}),
\end{equation}
or in a more expanded notation:
\begin{equation}\label{koopman_PDE}
\frac{d \phi_i({\bf x})}{dt}={ f}_1\left({\bf x}\right)\frac{\partial }{\partial x_1} \phi_i({\bf x}) +\cdots+ { f}_d\left({\bf x}\right)\frac{\partial }{\partial x_d}\phi_i({\bf x}) =\lambda_i \phi_i({\bf x}),
\end{equation}
where ${\bf f}({\bf x}) = (f_1({\bf x}), f_2({\bf x}), \dots, f_d({\bf x}))^T$. The above PDE is an advection equation for the ``concentration" function $\phi_i({\bf x})$ with velocity ${\bf f}({\bf x})$. This equation is a linear first-order PDE and in general has no closed-form solution. However, it is possible to approximate the solution of this linear PDE using the Galerkin method.
This work makes use of normalized Legendre polynomials as proposed by Arnas and Linares~\cite{zonal_koopman}, due to the advantages they provide in the computation of the Koopman matrix. Legendre polynomials are a set of orthogonal polynomials defined in a Hilbert space that generate a complete basis. The idea of this methodology is to represent any function of the space by using this set of basis functions. This is done by the use of inner products and the correct normalization of the Legendre polynomials. Let $f$ and $g$ be two arbitrary functions from the Hilbert space considered. Then, the inner product between these two functions is defined as:
\begin{equation}
\langle f, g \rangle =\int_{\Omega} f({\bf x})g({\bf x}) w({\bf x})d{\bf x},
\end{equation}
where $w({\bf x})$ is a positive weighting function defined on the space domain $\Omega$. For the case of Legendre polynomials, the weighting function is a constant $w({\bf x}) = 1$, and the domain for each variable ranges between $[-1,1]$. In addition, the normalized Legendre polynomials are defined such that:
\begin{equation}
\langle L_i, L_j \rangle =\int_{\Omega} L_i({\bf x})L_j({\bf x}) w({\bf x})d{\bf x} = \delta_{ij},
\end{equation}
where $L_i$ and $L_j$ with $\{i,j\}\in\{1,\dots,m\}$ are two given normalized Legendre polynomials from the set of basis functions selected, and $\delta_{ij}$ is Kronecker's delta.
By using this set of orthogonal multivariate polynomials, the function $u({\bf x},t)$ and the KO eigenfunctions, $\phi_i({\bf x})$, can be represented as a series expansion in terms of this set of basis functions:
\begin{subequations}\label{series}
\begin{align}
u({\bf x},t)=\sum_{\ell=1}^{\infty}c_{\ell}(t)L_{\ell}({\bf x})\approx \sum_{\ell=1}^{m}c_{\ell}L_{\ell}({\bf x})={\bf c}^T(t){\bf L}({\bf x}) = {\bf L}^T({\bf x}){\bf c}(t),\\
\phi_i({\bf x},t)=\sum_{\ell=1}^{\infty}p_{i\ell}(t)L_{\ell}({\bf x})\approx \sum_{\ell=1}^{m}p_{i\ell}L_{\ell}({\bf x})={{\bf p}_i}^T(t){\bf L}({\bf x}) = {\bf L}^T({\bf x}){\bf p}_i(t),
\end{align}
\end{subequations}
where $c_{\ell}(t)$ describes the time evolution of the function $u({\bf x},t)$ over the basis $L_{\ell}({\bf x})$ and $p_{i\ell}(t)$ are the coefficients associated with the eigenfunction $\phi_i$ and the basis $L_{\ell}$. Moreover, ${\bf c}(t)$, ${\bf p}_{i}(t)$, and ${\bf L}$ are three column vectors containing the set of coefficients $c_{\ell}(t)$, $p_{i\ell}(t)$ and the whole set of basis functions, respectively. Note that although the series is infinite, a truncation was performed using $m$ different basis functions, and thus, this represents an approximation of the eigenfunctions.
The PDE provided in Eq.~\eqref{koopman_PDE} for the computation of the Koopman eigenfunctions can be approximated using the Galerkin method and the series expansion from Eq.~\eqref{series}. The general concept of Galerkin methods is to project the operator into the subspace $\mathcal{F}_D$ using a weighted residual technique such that the residual of Eq.~\eqref{koopman_PDE} is orthogonal to the span of $\mathcal{F}_D$.
We start by approximating the solutions of Eq.~\eqref{koopman_PDE1} by
letting $u({\bf x},t) = {\bf c}^T(t){\bf L}({\bf x})$ be the truncated series defined by Eq.~\eqref{series}. We define the residual error of the Koopman PDE in q.~\eqref{koopman_PDE1} as:
\begin{equation}\label{residual0}
e({\bf x},t) = \frac{d u ({\bf x},t)}{dt}-\mathcal{K}\left(u({\bf x},t)\right)
\end{equation}
where $\frac{d u ({\bf x},t)}{dt}$ is approximated by $\dot{u}({\bf x},t) \approx \dot{\bf c}^T(t){\bf L}$, with $\dot{\bf c}(t)$ a
vector of size $m$, that is, we want the derivative to be expressed in the same set of basis functions as the eigenfunctions.
Moreover, the solution sought should be orthogonal to $\mathcal{F}_D$, being the orthogonality condition defined as:
\begin{equation}\label{residual}
\langle L_j({\bf x}), e({\bf x},t) \rangle =0, \quad \forall \ j \in \{1,2,\dots,m\}.
\end{equation}
The Koopman operator applied to $u({\bf x},t)$ is given by:
\begin{equation}\label{residual2}
\mathcal{K}\left(u({\bf x},t)\right) = \left(\nabla_{\bf x} {\bf L}^T{\bf c}(t)\right)^T {\bf f} = \left({\bf f}^T\nabla_{\bf x}{\bf L}^T\right) {\bf c}(t)= {\bf f}^T {\bf L_ x^T}{\bf c}(t),
\end{equation}
where the term ${\bf L_ x^T}=\nabla_{\bf x}{\bf L}^T$ is defined as:
\begin{equation}\label{derivatives}
{\bf L_ x^T}=\left(\begin{array}{ccc}
\displaystyle\frac{\partial }{\partial x_1}L_1({\bf x}) & \cdots & \displaystyle\frac{\partial }{\partial x_d}L_1({\bf x}) \\
\vdots & \ddots & \vdots\\
\displaystyle\frac{\partial }{\partial x_1}L_m({\bf x}) & \cdots& \displaystyle\frac{\partial }{\partial x_d}L_m({\bf x})\end{array}\right).
\end{equation}
It is important to note that all of the terms in the equation above can be calculated in closed-form since both ${\bf L}$ and ${\bf L_ x^T}$ are only comprised by polynomials. Therefore, it is possible to apply Eq.~\eqref{residual2} into Eq.~\eqref{residual0} to obtain:
\begin{eqnarray}
\langle L_j({\bf x}), e({\bf x},t) \rangle & = & \langle L_j({\bf x}), {\bf L}^T({\bf x})\dot{\bf c}(t) \rangle - \langle L_j({\bf x}),{\bf f}^T {\bf L_ x^T}{\bf c}(t) \rangle = \nonumber \\
& = & \langle L_j({\bf x}), {\bf L}^T({\bf x}) \rangle \dot{\bf c} - \langle L_j({\bf x}),{\bf f}^T {\bf L_ x^T} \rangle {\bf c}(t) = 0,
\end{eqnarray}
Above we define a set of $m$ equations for each $L_j({\bf x})$, we now gather these equations into a matrix vector representation to obtain:
\begin{equation} \label{time_matrix_form}
\frac{d {\bf c}(t)}{dt} =G^{-1}K{\bf c}(t),
\end{equation}
where:
\begin{equation}
G_{ij}= \langle L_i({\bf x}), L_j({\bf x})\rangle,
\end{equation}
However, due to the orthogonality of the basis functions we have that $G=I_{m\times m}$. Therefore, the $K$ matrix is the matrix representation of the KO over the basis $L_i({\bf x})$, a $m \times m$ sized matrix whose components are:
\begin{equation} \label{kmatrix}
K_{ij} = \langle L_i({\bf x}), {\bf f}^T \nabla_{\bf x}L_j({\bf x})\rangle.
\end{equation}
Equation \eqref{time_matrix_form} determines the time evolution of $u({\bf x},t)$ by solving for ${\bf c}(t)$. It is a set of coupled ordinary differential equations that can be solved using the matrix exponential, ${\bf c}(t)=\exp \left(Kt\right){\bf c}(t_0)$ where ${\bf c}(t_0)$ is the initial coefficient vector.
Note that by using Eq.~\eqref{time_matrix_form} we can obtain the solution for any state of the system. In order to do that, we start by defining the initial set of coefficient vectors as $C_0=[{\bf c}_0^1,\cdots,{\bf c}_0^m]$, where $C_0\in\mathbb{R}^{m \times d}$, ${\bf c}_0^i\in\mathbb{R}^{m \times 1}$, and ${\bf c}_0^i={\bf c}_i(t_0)$ is the initial coefficient vector for the $i^\text{th}$ coordinate, $x_i$, and $x_i(t_0)=L^T({\bf x}){\bf c}_0^i$. Then, the initial condition can be expanded in $L_i$ as ${\bf x}^T(t_0)=L^T({\bf x})C_0$ and the solution for each of these coefficient vectors is given by $C(t)=\exp(Kt)C_0$. These results can be merged to obtain the approximate evolution of the state:
\begin{equation} \label{decomposition}
{\bf x}(t)=C_0^T\exp(K^Tt)L({\bf x}_0),
\end{equation}
where the transpose of $C_0$ and $K$ is needed since ${\bf x}(t)$ is a column vector. This solution for the state will be expanded on in the next section to use the KO eigenfunctions. Note that the coefficient $C_0$ can be computed analytically using the Legendre polynomials basis functions. Additionally, $K$ is also computed analytically, and $L({\bf x}_0)$ are the collection of normalized Legendre basis functions up to a given maximum order. Therefore, the solution in Eq.~\eqref{decomposition} is an approximate analytical solution to Eq.~\eqref{problem} given the initial condition ${\bf x}_0$ at time $t_0$.
If we repeat this process for Eq.~\eqref{koopman_PDE} and consider all the basis functions with $j\in\{1,\dots,m\}$, we can show that the eigenvectors of the matrix $K$ are the coefficient vectors of the KO eigenfunctions with the same corresponding eigenvalue. This relationship is given by
\begin{equation} \label{decomposition2}
P K = \Lambda P,
\end{equation}
where $P$ is the matrix containing all the eigenvectors of the matrix $K$, that is, these eigenvectors are the coefficients $p_{ij}$ of the KO eigenfunctions. Therefore, the KO eigenfunctions are then given by
\begin{equation} \label{decomposition3}
\Phi({\bf x}) = L^T({\bf x})P
\end{equation}
Additionally, $\Lambda$ is a diagonal matrix of eigenvalues $\lambda_i$ of the matrix $K$, where the eigenvalue $\lambda_i$ relates to the $i$th eigenfunction. Note that Eq.~\eqref{decomposition} is in fact defining an eigendecomposition of matrix $K$, where $P$ is the matrix containing the left eigenvectors of $K$, and $\Lambda$ the associated eigenvalues. Therefore, it is possible to first obtain $K$ in closed-form solution from Eq.~\eqref{kmatrix}, and then perform the eigendecomposition to obtain $P$, and thus, the approximated eigenfunctions $\phi_i$ of the system through Eq.~\eqref{series}.
\subsection{Koopman Operator Theory: Linear System}
Let us consider the Koopman eigenvalues and eigenfunctions of a Linear Time Invariant (LTI) system. LTI systems are a particular case of the initial value problem where it is possible to compute the Koopman eigenfunctions analytically; thus, these systems are referred to as solvable. Consider the following LTI system:
\begin{equation}\label{linear}
\dot{\bf x}=\Xi{\bf x},
\end{equation}
with ${\bf x}\in\mathbb{R}^d$ and $\Xi\in\mathbb{R}^{d\times d}$. The eigendecomposition of this linear system yields complete information about the underlying dynamics provided that $\Xi$ has a complete set of eigenvectors, left and right eigenvectors ${\bf w}_i$ and ${\bf v}_i$ respectively, with corresponding eigenvalues $\zeta_i$. Given that the dynamics are linear, it is expected that the Koopman approach should contain the eigendecomposition of the matrix $\Xi$.
To show this, the KO for the linear system is
$
\frac{d}{dt}\phi({\bf x})=\mathcal{K}\left(\phi({\bf x})\right)=(\Xi{\bf x})^T\left[ \nabla\phi({\bf x})\right]
$. Assuming that $\Xi$ has a complete set of eigenvectors, there are $d$ left eigenvectors, ${\bf w}_i$, that satisfy ${\bf w}^T_i\Xi = \mu_i{\bf w}^T_i$. It is easy to verify that $\phi_i({\bf x})={\bf w}_i^T{\bf x}$ is an eigenfunction of the KO for the linear system in Eq.~\eqref{linear} by computing the KO
\begin{equation}
\mathcal{K}\left(\phi_i({\bf x})\right)={\bf x}^T \Xi ^T\nabla\left({\bf w}^T_i{\bf x}\right)={\bf x}^T \Xi ^T{\bf w}=\zeta_i\phi_i({\bf x}).
\end{equation}
Therefore, each one of the eigenvectors of the matrix $\Xi$ can be used to construct a first order polynomial in the state variable, $\phi_i({\bf x})={\bf w}_i^T{\bf x}$, which is a KO eigenfunction for this linear system. Then, thanks to the algebraic property of the KO, an infinite number of eigenfunctions for this linear system can be expressed as a polynomial function of the state (from the product of the first order polynomials $\phi_i({\bf x})={\bf w}_i^T{\bf x})$.
Thus, the function
\begin{equation}
\phi_\ell({\bf x})=\prod_{i=1}^{d}\left({\bf w}^T_i{\bf x}\right)^{n_{i,\ell}}
\end{equation}
is an eigenfunction of the KO with associated eigenvalue, $\lambda_{\ell}=\sum_{i}^{d}n_{i,\ell}\zeta_i$.
\subsection{Koopman modes}
The Koopman modes are the representation of the observables into the set of eigenfunctions ${\bf \Phi}$ of the system. Let ${\bf g}({\bf x})$ be the set of observables in which we are interested. These can be any function of the original variables ${\bf x}$, including the states themselves, called the identity observables. These observables can be expressed as a series expansion in the set of eigenfunctions obtained before:
\begin{equation}
g_i({\bf x}) \approx \sum_{\ell=1}^m b_{i\ell}\phi_{\ell}({\bf x}),
\end{equation}
where $b_{i\ell}$ is a set of constant coefficients that can be computed by projecting the observable into the set of basis functions:
\begin{equation}
b_{i\ell} = \left\langle g_i, \phi_{\ell} \right\rangle.
\end{equation}
This means that ${\bf g}({\bf x})$ can also be represented in matrix notation as:
\begin{equation}\label{observable}
{\bf g}({\bf x}) = B {\bf \Phi}({\bf x}),
\end{equation}
where $B$ is a matrix of size $q \times m$ containing the coefficients $b_{i\ell}$. The parameter $q$ represents the number of observables considered, for instance, in the particular case of the identity observables, $q = d$, that is, the number of dimensions of the original system of equations. Therefore, it is possible to obtain the evolution of any observable ${\bf g}({\bf x})$ by means of the evolution of the eigenfunctions of the system. Particularly, from Eq.~\eqref{eigen_time}:
\begin{equation}
{\bf g}({\bf x}(t)) = B {\bf \Phi}({\bf x}(t)) = B \exp(\Lambda t){\bf \Phi}({\bf x}(t_0)),
\end{equation}
or expressed in terms of the basis functions used:
\begin{equation} \label{solution}
{\bf g}({\bf x}(t)) = B {\bf \Phi}({\bf x}(t)) = B \exp(\Lambda t) P {\bf L}({\bf x}(t_0)),
\end{equation}
where ${\bf x}(t_0)$ are the values of the states in the initial condition.
Note that the observables can also be represented by the set of basis functions ${\bf L}$:
\begin{equation} \label{obs}
{\bf g}({\bf x}) = A {\bf L}({\bf x}) \rightarrow g_i({\bf x}) \approx \sum_{\ell=1}^m a_{i\ell}L_{\ell}({\bf x}), \quad \text{where} \quad a_{i\ell} = \left\langle g_i, L_{\ell} \right\rangle,
\end{equation}
where $A_{ij}=a_{ij}$, whose solution under the system of differential equations is:
\begin{equation}\label{KO_solution}
{\bf g}({\bf x}(t)) = A {\bf L}({\bf x}(t)) = A P^{-1} \exp(\Lambda t) P {\bf L}({\bf x}(t_0)).
\end{equation}
Additionally, it is important to note that the result from Eq.~\eqref{solution} can also be used to obtain the state transition matrix of the system. In that regard, it is important to note that if the identity observable is used, that is, ${\bf g}({\bf x}) = {\bf x}$ then we have that $C_0^T=A$, and thus:
\begin{equation} \label{state_trans}
\displaystyle\frac{\partial {\bf x}(t_f)}{\partial {\bf x}(t_0)} = B\exp(\Lambda t_f)P\frac{\partial {\bf L}({\bf x}(t_0))}{\partial {\bf x}(t_0)}.
\end{equation}
Higher-order state transition matrix terms can also be computed from Eq.~\eqref{solution}. In this work, we make use of Eq.~\eqref{state_trans} to approximate periodic solutions to the RTBP using a simple differential corrector method.
\section{Dynamics for the Motion near $\mathcal L_1$ or $\mathcal L_2$ in the Circular Restricted Three Body Problem}
The circular restricted three body problem (RTBP) equations can be expressed using the following form:
\begin{subequations}
\begin{align}
\ddot{x}&=\frac{\partial \Omega}{\partial x}+2\dot{y},\\
\ddot{y}&=\frac{\partial\Omega }{\partial y}-2\dot{x},\\
\ddot{z}&=\frac{\partial \Omega }{\partial z},
\end{align}
\end{subequations}
where $\Omega=1/2\left(x^2+y^2\right)+(1-\mu)/r_1+\mu/r_2$, $\mu$ is the mass of the secondary divided by the mass of the primary object, and $r_1$ and $r_2$ are the distances to the primary and secondary masses, respectively. The distance from the libration points, $\mathcal L_j$ to the primary body, $\gamma$, is found by solving the Euler quintic equation:
\begin{subequations} \label{eqgamma}
\begin{align}
\gamma_j^5&\mp \left(3-\mu\right)\gamma_j^4+\left(3-2\mu\right)\gamma_j^3-\mu\gamma_j^2\pm2\mu\gamma_j-\mu=0, &\quad & \text{if} \quad j=1,2;\\
\gamma_j^5&+ \left(2-\mu\right)\gamma_j^4+\left(1-2\mu\right)\gamma_j^3-\left(1-\mu\right)\gamma_j^2-2\left(1-\mu\right)\gamma_j-\mu-1=0, &\quad & \text{if} \quad j=3.
\end{align}
\end{subequations}
The circular Restricted Three-Body Problem (RTBP) model that used for this work follows the work from Richardson~\cite{richardson1980analytic}. In particular, the equations of motion for a satellite moving near the $\mathcal L_1$ or $\mathcal L_2$ points can be obtained by translating the origin to the location of $\mathcal L_1$ or $\mathcal L_2$. This coordinate transformation is given by the following expressions:
\begin{subequations} \label{coord}
\begin{align}
\bar{x}&=\frac{x-1+\mu\pm\gamma}{\gamma},\\
\bar{y}&=\frac{y}{\gamma},\\
\bar{z}&=\frac{z}{\gamma},
\end{align}
\end{subequations}
where $\mu$ is the mass ratio of the primary to the secondary object, and the $\pm$ symbol relates to the dynamic of the system in the following manner: the upper sign is a dynamic about $\mathcal L_1$, while the lower sign is for $\mathcal L_2$.
Additionally, the variables $x$, $y$, and $z$ are the position coordinates for the RTBP, with the origin at the primary body, while $\bar{x}$, $\bar{y}$, and $\bar{z}$ are the position variables in the shifted and scaled coordinates. In $\bar{x}$, $\bar{y}$, and $\bar{z}$, the new origin is either the $\mathcal L_1$ or the $\mathcal L_2$ point, provided that these variables are normalized such that the distance between $\mathcal L_1$ or $\mathcal L_2$ and the primary is 1. This scaled coordinate system was introduced by Richardson~\cite{richardson1980analytic} and was shown to improve the numerical properties of the RTBP model. For convenience of notation, $x$, $y$, $z$ will refer to $\bar{x}$, $\bar{y}$, and $\bar{z}$ variables henceforth.
The full nonlinear RTBP equations can be expanded in terms of polynomials as Richardson~\cite{richardson1980analytic} showed. Additionally, Ref.~\cite{richardson1980analytic} showed that using Legendre polynomials, $P_n$, for expanding the RTBP equations can lead to computational advantages. The main nonlinear terms in the RTBP model involve $\frac{1-\mu}{r_1}+\frac{\mu}{r_2}$, where $r_1$ and $r_2$ are the distances to the primary and secondary objects, can be approximated using the following relation:
\begin{equation}
\frac{1}{\sqrt{(x-A)^2+(y-B)^2+(z-C)^2}}=\frac{1}{D}\sum_{n=0}^{\infty}\left(\frac{\rho}{D}\right)^nP_n\left(\frac{Ax+By+Cz}{D\rho}\right),
\end{equation}
where $D^2=A^2+B^2+C^2$ and $\rho^2=x^2+y^2+z^2$. Therefore, following the development in Ref.~\cite{richardson1980analytic}, the nonlinear RTBP equations can be expressed with respect to the $\mathcal L_1$ or $\mathcal L_2$ points as:
\begin{subequations}\label{RTBP}
\begin{align}
\ddot{x}-2\dot{y}-(1+2c_2)x&=\frac{\partial}{\partial x}\sum_{n\geqslant 3} c_n \rho^n P_n\left(\frac{x}{\rho}\right),\\
\ddot{y}+2\dot{x}+(c_2-1)y&=\frac{\partial}{\partial y}\sum_{n\geqslant 3} c_n \rho^n P_n\left(\frac{x}{\rho}\right),\\
\ddot{z}+c_2z&=\frac{\partial}{\partial z}\sum_{n\geqslant 3} c_n \rho^n P_n\left(\frac{x}{\rho}\right),
\end{align}
\end{subequations}
Note that the left-hand side contains only linear terms. The coefficients $c_n$ are given by:
\begin{equation}\label{c_equation}
c_n=\frac{1}{\gamma^3}\left((\pm)^n\mu+(-1)^n\frac{(1-\mu)\gamma^{n+1}}{(1\mp\gamma)^{n+1}}\right),
\end{equation}
where the upper sign is for $\mathcal L_1$ and the lower one for $\mathcal L_2$. This work considers the Sun-Earth system. In the case of $\mathcal L_1$, the values of the problem constants are $\gamma=0.0099703255$ and
$\mu=3.0034106426\times 10^{-6}$, which can then be used with Eq.~\eqref{c_equation} to compute $c_2 = 4.060821911$, $c_3 = 3.019929488$, and $c_4 = 3.030412038$.
This model can be calculated using an elegant recursive formula~\cite{koon2000dynamical}. In particular, Legendre polynomials can be calculated recursively using the well-known three-term recursive formula, which can be in turn used to define a recursion for the RTBP model. Consider the following function:
\begin{equation}
T_n(x,y,z) =\rho^n P_n\left(\frac{x}{\rho}\right),
\end{equation}
where $T_n$ is a homogeneous polynomial of degree $n$ in variables $x$, $y$, and $z$. Using the Legendre polynomial three-term recursive formula, it can be shown that:
\begin{equation}
T_n =\frac{2n-1}{n}xT_{n-1}-\frac{n-1}{n}\left(x^2+y^2+z^2\right)T_{n-2},
\end{equation}
where the recursion starts with $T_0 = 1$ and $T_1 = x$.
Legendre polynomials had already been applied to approximate the inverse potential for the N-body problem~\cite{aarseth1967collisionless}, with particular attention to the circular RTBP, leading to multiple Legendre polynomial constructions (listed in~\cite{laskar2010explicit}) . Richardson analyzed the presented polynomial approach and best suited the representation for the RTBP, truncating Eq.~\eqref{RTBP} at order $n=3$. This work uses this formula to define the polynomial model of the RTBP, and it is henceforth referred to as the Richardson's formulation. The Richardson's model used for the comparisons in the numerical applications presented in this paper involves truncating Eq.~\eqref{RTBP} at order $n=10$.
\subsection{Hamiltonian in Normal Form}
Jorba and Masdemont~\cite{jorba1999dynamics} provided a summary of the normal form computation for the RTBP model and this section follows the notation and process given in that work. The equations of motion of the CRTBP following this model are:
\begin{equation}\label{H1}
H=\frac{1}{2}\left(p_x^2+p_y^2+p_z^2\right)+yp_x-xp_y-\sum_{n\geq2} c_n \rho^n P_n\left(\frac{x}{\rho}\right),
\end{equation}
where the variables are defined as pseudo momenta, $p_x=\dot{x}-y$, $p_y=\dot{y}+x$, and $p_z=\dot{z}$. Then, the normal form for this system is found by using a symplectic linear change of variables that utilizes the eigenvectors of the linearized matrix to simply Eq.~\eqref{H1}. The linearized system is given by
\begin{equation}
\dot{\bf v}=M{\bf v},
\end{equation}
where ${\bf v}=[{x}, {y}, {z}, p_x, p_y, p_z]^T$ and $M$ is determined by evaluating the sum in Eq.~\eqref{RTBP} for only the $n=2$ terms, leading to:
\begin{equation} \label{M}
M=\begin{bmatrix}
0 & 1 & 0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1\\
2c_2 & 0 & 0 & 0 & 1 & 0\\
0 & -c_2 & 0 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -c_2\\
\end{bmatrix}.
\end{equation}
Note that the $z$ dimension is decoupled from $x$ and $y$. Therefore, the characteristic polynomial for the $x$-$y$ motion is given by:
\begin{equation}
p(\lambda)=\lambda^4+\left(2-c_2\right)\lambda^2+\left(1+c_2-2c_2^2\right)=0,
\end{equation}
where the solutions for the characteristic polynomial in the $x$-$y$ direction are:
\begin{equation}
\omega_1^2=\frac{c_2-2-\sqrt{9c_2^2-8c_2}}{2}, \quad \lambda_1^2=\frac{c_2-2+\sqrt{9c_2^2-8c_2}}{2}.
\end{equation}
Under the assumption that $\mu>0$, the $x$-$y$ motion characteristic equation has four roots, two real and two imaginary given by $(\pm \lambda_1, \pm \omega_1)$. The $z$ direction motion is characterized by two imaginary eigenvalues $\pm \omega_2$ given by $\omega_2^2=c_2$. The linear behavior near the collinear libration point is that of a saddle×center×center. The eigenvectors for the $x$-$y$ coupled linearized system are given by ${\bf e}_{\lambda}=[2\lambda,\lambda^2-2c_2-1,0,\lambda^2+2c_2+1,\lambda^3\left(1-2c_2\right),0]^T$. The $z$ eigenvector is given by ${\bf e}_{z}=[0,0,1,0,0,0]^T$ and ${\bf e}_{\dot{z}}=[0,0,0,0,0,1]^T$. By using the eigenvector, we can form a symplectic transformation $C$ that normalizes the $x$-$y$ motion such that $C^TJC=J$, where $J$ is given by
\begin{equation}
J=\begin{bmatrix}
0_{3\times3} & I_{3\times3}\\
-I_{3\times3} & 0_{3\times3}
\end{bmatrix}.
\end{equation}
Additionally, using the normalization discussed in Jorba and Masdemont~\cite{jorba1999dynamics}, the symplectic transformation can be expressed as:
\begin{equation} \label{C}
C=\begin{bmatrix}
\frac{2\lambda_1}{s_1} & 0 & 0 & -\frac{2\lambda_1}{s_1} & \frac{2\omega_1}{s_2} & 0\\
\frac{\lambda_1^2-2c_2-1}{s_1} & \frac{-\omega_1^2-2c_2-1}{s_2} & 0 & \frac{\lambda_1^2-2c_2-1}{s_1} & 0 & 0\\
0 & 0 & \frac{1}{\sqrt{\omega_2}} & 0 & 0 & 0\\
\frac{\lambda_1^2+2c_2+1}{s_1} & \frac{-\omega_1^2+2c_2+1}{s_2} & 0 & \frac{\lambda_1^2+2c_2+1}{s_1} & 0 & 0\\
\frac{\lambda_1^3+\left(1-2c_2\right)\lambda_1}{s_1} & 0 & 0 & \frac{-\lambda_1^3-\left(1-2c_2\right)\lambda_1}{s_1} & \frac{-\omega_1^3+\left(1-2c_2\right)\omega_1}{s_2} & 0\\
0 & 0 & 0 & 0 & 0 & \sqrt{\omega_2}\\
\end{bmatrix}
\end{equation}
where $s_1=\sqrt{2\lambda_1\left(\left(4+3c_2\right)\lambda_1^2+4+5c_2-6c_2^2\right)}$ and $s_2=\sqrt{\omega_1\left(\left(4+3c_2\right)\omega_1^2-4-5c_2-6c_2^2\right)}$. Applying a transformation such that ${\bf v}'=C{\bf v}$ is the new set of variables, we can rewrite the Hamiltonian in the following form:
\begin{equation}\label{H2}
H_2=\lambda_1xp_x+\frac{\omega_1}{2}\left(y^2+p_y^2\right)+\frac{\omega_2}{2}\left(z^2+p_z^2\right)-\sum_{n\geq2} c_n \rho^n P_n\left(\frac{x}{\rho}\right),
\end{equation}
where $x$ and $\rho$ can be computed in terms of the new variable using the transformation $C$.
Finally, the complex normal transformation can be achieved with the following transformation:
\begin{subequations}\label{CRTBP}
\begin{align}
x=q_1, \quad y=\frac{q_2+\sqrt{-1}p_2}{\sqrt{2}}, \quad z=\frac{q_3+\sqrt{-1}p_3}{\sqrt{2}}\\
p_x=p_1, \quad p_y=\frac{\sqrt{-1}q_2+p_2}{\sqrt{2}}, \quad p_z=\frac{\sqrt{-1}q_3+p_3}{\sqrt{2}}
\end{align}
\end{subequations
This final transformation can be written in matrix form and we will denote the complexification transformation as $C_2$. Therefore the final variable in the complex normal form is given by ${\bf v}''=C_2C{\bf v}$. The complex normal form Hamiltonian is then given by:
\begin{equation}\label{H3}
H_3=\lambda_1 q_1 p_1 +\sqrt{-1}\omega_1 q_2 p_2 +\sqrt{-1} \omega_2 q_3p_3-\sum_{n\geq2} c_n \rho^n P_n\left(\frac{q_1}{\rho}\right),
\end{equation}
where $q_1$ and $\rho$ can be computed in term of the new variable using the transformation $C_2C$. Equation \eqref{H3} can then be used with Hamilton's equation, $\dot{\bf v}''=J\nabla H_3$, to compute the equations of motion in the ${\bf v}''$ variables. The Hamiltonian $H_3$ is used to compute the KO of the RTBP since it decouples the variables in the linear leading terms, and it also decouples the effects of the stable, unstable, and periodic manifolds. Additionally, we introduce an additional scaling for the variables, where each variable is multiplied by a small constant, $\alpha$. The scaling by $\alpha$ further reduces the magnitude of the coefficients $c_n$ for the higher-order terms in the series shown in Eq.~\eqref{H3}, increasing the accuracy of the Koopman approximation, and minimizing numerical problems. A value of $\alpha = 0.01$ has been chosen for the results presented in the following sections.
\subsection{Complex Computation of Koopman Operator}
The state vector in the original reference frame, i.e. ${\bf x}(t)$, has been translated and scaled to be consistent with Richardson's Hamiltonian formulation, $H$, which depends of a different state vector ${\bf v}(t)$. Afterwards, two separate transformations rewrite the Hamiltonian into its normal form, $H_3$, where the state of the transformed system is ${\bf v}(t)'' =[{q_1}, {q_2}, {q_3}, p_1, p_2, p_3]^T = C_2 C{\bf v}(t) $, which can take complex outcomes. The final Hamiltonian formulation has complex components and therefore, the Koopman operator will as well. We compute the Koopman operator by dividing the resulting dynamical system into complex and real parts given by
\begin{equation}
{\bf f}({\bf v''})={\bf f}_\text{Re}({\bf v''})+\sqrt{-1}{\bf f}_\text{Img}({\bf v''}),
\end{equation}
where ${\bf f}_\text{Re}({\bf v''})$ and ${\bf f}_\text{Img}({\bf v''})$ are the real and imaginary parts of the dynamic function, respectively. The dynamics ${\bf f}({\bf v''})$ are evaluated by applying Hamilton's equation to the complex Hamiltonian $H_3({\bf v''})$, Eq.~\eqref{H3}. Then the Koopman operator complex computation can be performed with Legendre polynomials using the follow expression:
\begin{equation} \label{kmatrix2}
K_{ij} = \langle L_i({\bf v''}), {\bf f}_\text{Re}^T \nabla_{\bf v''}L_j({\bf v''})\rangle+\sqrt{-1}\langle L_i({\bf v''}), {\bf f}_\text{Img}^T \nabla_{\bf v''}L_j({\bf v''})\rangle.
\end{equation}
The polynomial approximation of the dynamics leads to a clean separation between real and imaginary monomials in the equations of motion. As such, each entry of the Koopman matrix is evaluated by adding the integrals (through the inner products) of each single monomial of $\bf f (\bf v'')$ multiplied by the Legendre polynomials. There is no need to separate the Legendre polynomials into their real and imaginary parts since it is assumed that the state variable $\bf v''$ is complex. Consequently, the basis functions cover all possible outcomes in the state space, and provide a complex output given a complex input. The division of the dynamics into ${\bf f}_\text{Re}({\bf v''})$ and ${\bf f}_\text{Img}({\bf v''})$ simplifies the evaluation of the Koopman matrix by working individually on each single monomial of the polynomial representation of the dynamics and then summing all the contributions together.
The influence of a complex representation of the ODEs affects, also, the evaluation of the observable matrix $A$ from Eq.~\eqref{obs}. Each entry of $A$ is calculated as the inner product between the observable function $\bf g(\bf v'')$ and the basis functions $\bf L(\bf v'')$. In order to obtain the time evolution of the state of the system, we are required to select the identity observable. However, due to the multiple transformations applied to the Hamiltonian, the identity observable must be transformed to be consistent with the new variable set ${\bf v}''= C_2 C {\bf v}$, which identifies the ``normal form" representation of the state. Therefore, the observable matrix A is evaluated by separating real and complex contributions as for the Koopman matrix.
\begin{equation}
a_{ij} = \left\langle g_{i,\text{Re}}({C_2 C\bf v}), L_{j} ({\bf v''})\right\rangle + \sqrt{-1}\langle g_{i,\text{Img}}({C_2 C\bf v}), L_j({\bf v''})\rangle.
\end{equation}
Lastly, given the initial condition of the state ${\bf x}(t_0)$ in the original reference frame, the state vector is translated and scaled according to Eq.~\eqref{coord}, giving ${\bf v}_0$. Afterwards, the initial condition undergoes the inverse of transformation $C_2 C$ before being evaluated by the basis functions during the calculation of the Koopman solution. Thus, the Legendre polynomials complex evaluation is applied only to get the final solution of the system, and not during the integration in the inner product. In this way, Eq.~\eqref{KO_solution} outputs directly the state into its not-transformed representation $({\bf v})$, as it was calculated directly from the Hamiltonian defined in Eq.~\eqref{H1}, working with observable ${\bf g}({\bf v})$ over ${\bf g}({\bf v''})$. This property can be appreciated by analyzing the matching of the initial condition with the Koopman solution. Given the identity observable, at time $t_0$ the solution is:
\begin{align}
{\bf g}({\bf v}_0) &= A P^{-1} \exp(\Lambda t_0) P {\bf L}((C_2 C)^{-1} {\bf v}_0) \\
&= A P^{-1} P {\bf L}((C_2 C)^{-1} {\bf v}_0) \\
&= A {\bf L}((C_2 C)^{-1} {\bf v}_0);
\end{align}
where, for the $i$th observable, calling with $A_i$ the $i$th row of matrix $A$:
\begin{align}
g_i({\bf v}_0) &= A_i {\bf L}((C_2 C)^{-1} {\bf v}_0)\\
&= \sum_{j=1}^{m} \bigg( \left\langle g_{i,\text{Re}}({C_2 C\bf v}), L_{j} ({\bf v''})\right\rangle + \sqrt{-1}\langle g_{i,\text{Img}}({C_2 C\bf v}), L_j({\bf v''})\rangle \bigg) L_j((C_2 C)^{-1} {\bf v}_0);
\end{align}
where, once again, $m$ is the number of basis functions. The transformations in the observable, $C_2 C$, and in the basis functions, $(C_2 C)^{-1}$, simplify themselves, giving, as solution, the state of the system as if it were evaluated from the Hamiltonian $H$. The evaluation of the Legendre polynomials ${\bf L}((C_2 C)^{-1} {\bf v}_0)$ is a simple variable substitution, where the transformed state given as input is complex and it leads to a complex output of the polynomial evaluation.
After applying KO to the Hamiltonian in normal form and obtaining the final solution as if from $H$ (thanks to some mathematical manipulations), the state of the system is scaled and translated back to the original reference frame by inverting the coordinate transformation Eq.~\eqref{coord}, getting ${\bf x}(t)$ from ${\bf v}(t)$. As the final step, the velocity is derived directly from the pseudo momenta: $\dot{x} = p_x + y$, $\dot{y} = p_y - x$, and $\dot{z} = p_z$.
\section{Lyapunov Orbit Approximation Around $\mathcal L_1$}
The Koopman approximation of the dynamics is here applied for periodic orbits (POs) around the libration points $\mathcal L_1$ and $\mathcal L_2$. In the following paragraphs, the Koopman solution is shown to be an accurate propagator for Lyapunov and Halo orbits. The Koopman Operator theory is analyzed in terms of eigenvalues and eigenfunctions: the frequencies and modes of the system are observed, and a comparison among different expansion orders is presented. Lastly, the Koopman theory is incorporated in a station-keeping application, where the Koopman solution is the contribution for a newly-developed estimator and controller.
The KO solutions proposed in these sections have been achieved using a 10th order Richardson model to represent the dynamics of the RTBP. The same polynomial model, integrated numerically (with a Runge Kutta 4/5 or 7/8 propagator), would diverge. The Richardson solution is not able to achieve an accurate approximation of either the Lyapunov or Halo orbits. The accuracy of Richardson's polynomial solution decreases as the time propagation continues and Richardson's approximation of the periodic orbits diverges before achieving a full revolution.
The KO is first applied to solve the Lyapunov orbits around the first Lagrangian point, $\mathcal L_1$. These families of periodic orbits are planar and lie in the $(x,y)$ plane. Figure~\ref{fig:L1} shows the Koopman approximation, represented by a set of continuous curves, of the solution of the RTBP dynamics. As can be seen, the continuous lines overlap the truth solution, which is represented by dashed lines and was evaluated numerically by a Runge-Kutta integration scheme of the original dynamics (without any approximation).
\begin{figure}[h!]
\centering
\includegraphics[width=0.93\textwidth]{LyapL1.png}
\caption{Koopman solution (Order 5) of the $\mathcal L_1$ Lyapunov orbits (left) and relative position (top) and velocity (bottom) error magnitudes. }
\label{fig:L1}
\end{figure}
The right side of Figure~\ref{fig:L1} reports the errors for the position and velocity,top and bottom of the figure respectively. The errors are evaluated as norms:
\begin{align}
\epsilon_p = \sqrt{(x_{KO} - x_T)^2 + (y_{KO} - y_T)^2 + (z_{KO} - z_T)^2}; \label{eq:pos}\\
\epsilon_v = \sqrt{(\dot x_{KO} - x_T)^2 + (y_{KO} - y_T)^2 + (z_{KO} - z_T)^2};
\end{align}
where the subscript $KO$ refers to the Koopman approximation, while the subscript $T$ is the Runge-Kutta solution.
By analyzing the POs from Figure~\ref{fig:L1}, it is noted that the accuracy of the solution decreases as the initial condition moves away from the equilibrium point. The KO approximates the orbits as a linear combination of the eigenfunctions of the system. Therefore, in order to increase accuracy for initial conditions that are farther from the equilibrium point, the order of the KO should be increased as well. The figure shows the solution of the KO using basis functions up to order 5, which corresponds to a total of 462 different eigenfunctions. However, the authors have observed~\cite{koopmanzonal} that the KO has similar performance characteristics as perturbation theory methods, where a robust solution is achieved by considering small perturbations of the original system and, therefore, high accuracy levels for far initial conditions are harder to achieve. As the figure shows, the position and velocity errors of the KO solutions increase the farther the initial condition of the PO is from the $\mathcal L_1$ equilibrium point. The gray lines in the error plots, represented with a gradient getting darker the closer the initial condition is to the $\mathcal L_1$ point, show a similar behavior along with the whole orbit revolution, and they settle on different levels of accuracy. Therefore, for small Lyapunov orbits, the KO solution has shown to be an extremely accurate approximation.
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth]{eigs.png}
\caption{First 24 eigenfunctions of the 3rd order KO for the RTBP and Lyapunov initial conditions. }
\label{fig:eig}
\end{figure}
The eigenfunction analysis and the study of the minimum points (minima) are important when seeking stability and periodic orbits in a chaotic application, such as the RTBP. The KO orbits are able to maintain their periodicity due to the eigendecomposition of the dynamics, where the system is diagonalized and each eigenfunction is studied in terms of magnitude and shape. Therefore, if desired, unstable contributions from eigenfunctions connected to unstable eigenvalues can be separated. Figure~\ref{fig:eig} reports the first 24 eigenfunctions from the 3rd order KO solution. The six-dimensional functions are represented with just two dimensions in the $(x,\dot y)$ plane since this information sufficiently describes each Lyapunov orbit. The eigenfunctions are centered at the $\mathcal L_1$ point, meaning that position (0,0,0) corresponds to $\mathcal L_1$ instead of the primary celestial body. Examining a gray-scale gradient, the figure shows the locations of minima in bright white. These curves represent equilibrium conditions, or more precisely, their approximation through a 3rd order KO. Therefore, when we include the initial conditions from the orbits from Figure \ref{fig:L1} in the $(x,\dot y)$ plane as white dots, their locations settle close to the minimum lines. Thus, the Koopman approximation is correctly approximating the dynamics of the system, in terms of its eigenfunctions. However, the accuracy of the approximation decreases as the initial condition becomes larger: the farthest white points are not as well approximated as the white points near the origin. Moreover, the location of the minima is not unique. Many eigenfunctions show well-marked bright lines in the bottom part of the $(x,\dot y)$ plane, for low velocities. These initial conditions are still connected to periodic orbits. The resulting pathway leaves the Lagrangian $\mathcal L_1$ point and starts orbiting around the main celestial body with a periodic behaviour~\cite{restrepo2018database}. Therefore, even if the eigenfunctions become less accurate the farther they are from the equilibrium point, their minima are still able to connect to different kinds of POs. Furthermore, when looking for periodic orbits, the minima location gives an extremely efficient first guess for the implementation of a differential corrector that solves for the actual correct periodic initial condition of the KO.
Note also that Figure~\ref{fig:eig} reports only POs connected to negative $x$ and positive $\dot y$. However, thanks to the symmetric properties of the RTBP, the domain of the eigenfunctions can be widened to support retrograde orbits, and all the remaining POs from axial and central symmetries, as studied and shown by Russel in Ref.~\cite{restrepo2018database}.
The accuracy of the solutions obtained can be improved by increasing the order of the basis functions at the expense of higher computational costs. The computational burden of the KO solution is a function of the selected order and the dimension of the dynamics of the system, with complexity growing exponentially due to the combinatorial nature of the proposed solution. However, once the Koopman matrix has been calculated and diagonalized, the dynamics of the problem are fully described by the linear combinations of the eigenfunctions. Therefore, the propagation of any state can be performed with ease, making the Koopman theory suitable for spacecraft on-board applications such as navigation (filtering) and attitude control. Most of the computational work can be performed offline, creating a detailed characterization of the dynamics. Any software would just need to have access to the eigenfunctions to evaluate the propagated solution given each starting state, without performing any numerical integration.
\section{Halo Orbits Approximation Around $\mathcal L_1$ and $\mathcal L_2$}
The Koopman Operator has also been applied to approximate a second family of POs: the Halo orbits at $\mathcal L_1$ and $\mathcal L_2$. Figure~\ref{fig:halo} reports a set of 30 different orbits calculated through a KO using basis functions of order 5. The figure includes the projection of the orbits in the three reference planes and an isometric three-dimensional view of the trajectories. The black point in the picture represents the secondary celestial body, which is the Earth in the selected Sun-Earth system. As the figure shows, the Koopman solution achieves a robust representation of the orbits. The accuracy of this solution will be later assessed. The periodicity is given by an accurate selection of the initial conditions, which have been found through the implementation of a differential corrector in such a way that the proposed solution describes the same orbits for multiple revolutions.
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{halosolution.png}
\caption{$\mathcal L_1$ Halo orbit family solved by a 5th Order Koopman Operator. }
\label{fig:halo}
\end{figure}
A similar pattern is reported in Figure~\ref{fig:halo2}, showing a family of Halo orbits at $\mathcal L_2$ solved using the 5th order KO. Once again, the black point in the figure represents the secondary body in the RTBP system. The KO is able to accurately represent orbits around both libration points. Since the value of $\gamma$, from the system of Equations~\eqref{eqgamma}, is different when describing the dynamics around $\mathcal L_2$, Richardson's coefficients are different as well, and thus, the computed Koopman matrix is changed for the cases of motion about the $\mathcal L_1$ and $\mathcal L_2$ points.
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{L2halos.png}
\caption{$\mathcal L_2$ Halo orbit family solved by a 5th Order Koopman Operator. }
\label{fig:halo2}
\end{figure}
Moreover, convergence analysis of the Koopman operator has been applied by analyzing how the approximation error reduces as the order of the basis functions used increases. Therefore, after selecting a Halo orbit with an initial condition given by:
\begin{equation}
\bf{ x_0} =
\begin{bmatrix}
0.988882322146701 & 0 & 0.000809201887342 & 0 & 0.008904188320067 & 0
\end{bmatrix}^T, \label{horbit}
\end{equation}
different orders of the solution have been calculated. Figure~\ref{fig:halo_order} reports the position error, evaluated as in Eq.~\eqref{eq:pos}, for a complete orbit revolution at different orders of basis functions. The continuous line, representing order 3, behaves two orders of magnitude worse than the dotted curve, which is related to the 6th order KO solution. Order 4, dashed line, and order 5, dash-point line, are in between the performance of the other two curves, indicating that the KO accuracy of the solution improves constantly as the number of basis functions becomes larger and thus, we can better represent the non-linearities of the dynamical system. Moreover, while the 3rd order approximation of the Halo solution becomes less accurate as of the propagation proceeds, the 6th order solution maintains the same level of accuracy during the whole orbit, meaning that the second revolution is able to start with accuracy levels comparable to the first one.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{convergenceVSorder.png}
\caption{$\mathcal L_1$ Halo orbit accuracy analysis through different KO orders. }
\label{fig:halo_order}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=.6\textwidth]{barerror.png}
\caption{Average position error for different KO orders }
\label{fig:bar}
\end{figure}
Figure~\ref{fig:bar} shows the average position error from the KO approximation studied in Figure~\ref{fig:halo_order}. The histogram highlights how the 6th order solution is 22 times more accurate than the 3rd order approximation for the Halo orbit. In the RTBP, the 6th order Koopman Operator approximates the dynamics by using 11 times more basis eigenfunctions than the 3rd order. Therefore, non-linearities are better represented and the performance of the Koopman solution improves.
It is important to note that the benefits of a more precise propagation are paid in terms of computational effort. However, obtaining the KO is a one-time computational effort and thus, once the system has been represented through the eigenfunctions and the observables have been selected, any solution is rapidly evaluated as a simple numerical evaluation of the eigenfunctions at a given time. Therefore, once the basis functions are available and the system has been processed, there is no computational time delay in applying a 3rd or a 6th order solution.
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth]{eigenvalues.png}
\caption{$\mathcal L_1$ Halo orbit accuracy analysis through different KO orders. }
\label{fig:rombo}
\end{figure}
In addition to the study of the performance of the KO solution, a spectral study can also be done. Figure~\ref{fig:rombo} compares the eigenvalues, computed using the Galerkin method, from the 3rd order and 6th order Koopman matrices. In this particular application, the eigenvalues of the 3rd order solution are a subset of the 6th order approximation. Thus, the 6th order solution incorporates previous orders' information and enhances it with additional nonlinear functions. As such, the location of the high-order eigenvalues is at the edge of the diamond-shaped figure. By looking at the intensity of every single marker, it can be noted how eigenvalues closer to the origin and the imaginary axis are repeated multiple times. Indeed, it is possible to know in advance the position of the unperturbed eigenvalues, for any given order, by the simple composition of the eigenvalues that have already been calculated for lower orders. Therefore, position and repetitiveness are explained in terms of statics and combinatorics (the theory of combinations), where all the new eigenvalues are found as a linear combination of the original ones. However, the eigenvalue composition holds valid only for the linear motion or the unperturbed cases (see Ref.~\cite{koopmanzonal}). Moreover, the linear approximation near the $\mathcal L_1$ is known analytically, as well as its eigenvalues, which have been portrayed in Figure~\ref{fig:rombo} as circles. The linear part of the KO solution matches the linear contribution of the true dynamics. The six linear eigenvalues are perfectly represented and the high-order approximations evolve from the latter to best represent the nonlinear terms of the system of ODEs.
\begin{figure}[h!]
\centering
\includegraphics[width=.85\textwidth]{modes6.png}
\caption{$\mathcal L_1$ Halo orbit mode magnitudes for the 6th Order Koopman Operator. }
\label{fig:modes}
\end{figure}
The frequencies and spectral behavior of the system provided by the KO can also be used to assess the convergence of the methodology. This can be done by the study of the Koopman modes, that is, the projection of the observables in the operator. Figure~\ref{fig:modes} presents the intensity of the Koopman modes of the system. Particularly, it shows how the eigenvalues with the highest module are the least influential in the solution. However, the small contributions from the modes connected to large eigenvalues are crucial to achieving an accurate approximation of the dynamics, and they make the difference in accuracy when comparing the order 3 solution with the one of order 6. Therefore, many terms other than the linear ones are non-zero, with influence that decreases as the order of the eigenfunctions increases. This also implies that the Koopman modes decay with larger frequency values, leading to a possible truncation of the expansion for eigenfunctions with, particularly high frequencies.
A similar convergence analysis can be done on the Halo POs represented in Fig.~\ref{fig:halo}. Particularly, for each PO, the position error has been evaluated by comparing the Koopman solution with a classical Runge-Kutta numerical integration. Figure~\ref{fig:cool} shows, with different line styles, the accuracy of the approximation on each orbit for different orders of the KO. As expected, as the Halo orbits become larger, the position error increases consequently. Therefore, for any given order, the position error convergence analysis describes a set of curves. In that regard, order 3, represented by a continuous line, is the least accurate solution, while order 6 is the most accurate. However, it is worth noticing how, for each orbit, the gain in accuracy obtained by increasing the KO order behaves differently depending on the initial condition. The KO order 6 solution is two orders of magnitude more accurate than the 3rd order for orbits close to $\mathcal L_1$. However, as the initial condition of the Halo orbit moves away from the equilibrium point, the gain in accuracy is reduced, settling around a single order of magnitude. Indeed, for very large initial conditions, increasing the order of the basis functions to over a given value stops being beneficial since the gain in performance does not repay the higher computational burden. This aspect is connected with the Richardson approximation applied for the polynomial representation of the dynamics.
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth]{order_good_blkv2.png}
\caption{Convergence analysis of the Halo orbit solution of the KO for different orders. }
\label{fig:cool}
\end{figure}
The Richardson's ODEs representation fails to provide an accurate approximation of the system for initial conditions far from the equilibrium point. Consequently, looking back at Figure~\ref{fig:halo}, it can be noted that for each set of curves, at different orders, there is a well-marked separation in the bottom part of the graph, while curves from different orders start to get closer as the orbit becomes larger. This aspect has been highlighted in Figure~\ref{fig:acc}, where the mean position error ($\bar{\epsilon}_p$) for each orbit is compared with the size of the Halo orbit ($r_0$) and the maximum order of the basis functions used to represent the KO. Thus, orbits with the furthest initial conditions from the $\mathcal L_1$ point generate mean position errors that range in value by up to one order of magnitude. However, as we get closer to $\mathcal L_1$, and thus reducing $r_0$, the mean position error for each KO solution order considered decreases with a different slope, making the accuracy gain more substantial as we increase the order of basis functions used. Additionally, it can also be observed that there is a region very close to $\mathcal L_1$ where the error seems to reach convergence and stabilize for 6th order basis functions. Thus, for Halo orbits close to $\mathcal L_1$, an increase of the KO order leads to a substantial improvement in accuracy and in the performance of the approximation of the system.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{orderbyorbitv2.png}
\caption{$\mathcal L_1$ Halo orbits convergence and accuracy analysis for different KO orders. }
\label{fig:acc}
\end{figure}
\section{Comparison with Extended Dynamic Mode Decomposition}
We have seen before that the Galerkin method can be used to approximate the Koopman Operator in closed-form. In this section, we obtain the Koopman Operator using Extended Dynamic Mode Decomposition (EDMD), which is a numerical technique that has been extensively used in non-linear engineering problems in the last decade. The aim of the comparison is to assess that the Galerkin methodology is a far superior technique compared to EDMD both in terms of accuracy and robustness of the approximation. Consequently, this chapter shows that a complete EDMD approximation analysis is not needed because of its approximation errors and the performance of its solution when compared to the Galerkin method. EDMD is related to the KO that we presented in previous sections and provides a relatively simple tool for solving nonlinear KO models. This section describes the EDMD algorithm at a high level; for more details please refer to the work by Williams~\cite{williams2015data}.
The EDMD approach attempts to estimate a finite-dimensional representation of the Koopman operator $\mathcal{K}$ using a linear mapping of the Koopman modes. The EDMD approach is restricted to discrete time systems in the form:
\begin{equation}
{\bf x}_{k+1}={\bf F}({\bf x}_k),
\end{equation}
where ${\bf x}_k$ are the discrete states and ${\bf F}$ is the nonlinear dynamic model.
In the same fashion as in the Galerkin method, eigenvalue decomposition is used to
find the $\mathcal{K}$ that minimizes a loss function related to the EDMD prediction error over the data samples (See Ref.~\cite{williams2015data} for full description). The EDMD solution is given by:
\begin{equation}\label{EDMD_solution}
\mathcal{K} = G^\dag A
\end{equation}
where $\dag $ denotes the pseudoinverse, $G$ and $A$ are defined as:
\begin{eqnarray}
G & = & \frac{1}{M}\sum_{k=0}^{M-1}\Psi^T({\bf x}_k)\Psi({\bf x}_k), \nonumber \\
A & = & \frac{1}{M}\sum_{k=0}^{M-1}\Psi^T({\bf x}_k)\Psi({\bf x}_{k+1}),
\end{eqnarray}
and $\mathcal{K} ,G,A\in \mathbb{C}^{K \times K}$. It is important to note that the solution for $\mathcal{K}$ given above
is a finite dimensional approximation of the KO. Furthermore, with the finite dimensional approximation of the KO given by Eq.~\eqref{EDMD_solution}, we can
determine an approximation of the eigenfunctions and eigenvalues of KO.
For the EDMD to work, we need to provide some numerical information about the dynamical system. To that end, initial conditions are drawn from a normal distribution with ${\bf x}\sim\mathcal{N}\left({\bf 0},\sigma^2 I_{4\times4}\right)$, where $\sigma$ is the scale used in the Gaussian weighting function when computing the Galerkin inner product. Note that due to its numerical nature, this method is not as accurate as the Galerkin approximation. Figure~\ref{EDMD} shows the eigenvalues computed for the Richardson third-order model using the EDMD approach. The linear approximation near $\mathcal L_1$ is known to have two real eigenvalues and four complex eigenvalues (in six dimensions). These linear eigenvalues, which define the shape of the spectrum, can be found analytically and they are matched by the EDMD. However, EDMD shows distortions on the higher-order eigenvalues that result from the linear ones, where the Galerkin method does not have this distortion. The predicted accuracy for the EDMD method is not shown since the error is extremely large and the EDMD solution does not provide a meaningful approximation of the KO for the three-body problem. The authors conclude that the three-body problem is poorly approximated by the EDMD method because this method relies on numerically approximating the KO and this approximation introduces too much error (in the eigenvalues and eigenvectors of the KO). In particular, the unstable eigenvalues of the RTBP make it challenging to numerically approximate the KO matrix form. However, using the closed-form computation of the KO avoids this approximation error and results in a far superior solution.
\begin{figure}[!ht]
\begin{center}
\psfrag{x1}[][]{\footnotesize{$x_1$}}
\psfrag{x2}[][]{\footnotesize{$x_2$}}
\psfrag{Time (min)}[][]{\footnotesize{Time (min)}}
\includegraphics[keepaspectratio,
width=0.65\textwidth]{eig_EDMD_L1_ES_BandW.pdf}\label{EDMD_eig}
\caption{\bf Extend Dynamic Model Decomposition}\label{EDMD}
\end{center}
\end{figure}
\section{Closed-Loop Station-Keeping for a Halo Orbit}
In previous sections, we show that the Koopman solution leads to accurate approximations of the dynamics of the RTBP while providing a linearized system of these dynamics. As such, these properties can now be used to generate an orbit-keeping application, where the goal of the approach is to keep a spacecraft orbiting in a specific Halo orbit.
We first assume that for the initial condition, the spacecraft has an offset with respect to the desired pathway. In addition, we consider the case where the state of the satellite is provided by some measurements from sensors or the ground stations. Therefore, the actual position and velocity vectors of the satellite need to be estimated and based on that, the spacecraft has to be redirected and controlled to achieve the desired behavior. Consequently, an estimation and control system based on the KO is proposed. Given GPS measurements from the true trajectory, the state of the spacecraft, in terms of its position and velocity, is estimated through a KO version of the particle filter. Afterward, reaching and keeping the desired orbit is achieved through a nonlinear model predictive controller (NMPC), which uses the KO solution for the integrative part of the control.
\subsection{The Koopman Operator Particle Filter - KOPF}
The particle filter (PF) is a well-known nonlinear estimator that predicts the state of a system by approximating probability density functions (PDFs) through a set of samples~\cite{ristic2003beyond}. The PF is a robust and accurate filter. However, its use in astrodynamics applications is limited by the high computational burden requested by the time propagation of every single particle, an aspect that makes the filter unsuitable for onboard applications. After initializing an ensemble of particles according to the given initial state PDF, each sample is integrated forward in time until a measurement becomes available. Thus, each particle is propagated numerically according to the selected numerical integrator. Once measurements are given, each particle receives an importance weight which is proportional to the likelihood given the measurements. After normalizing the weights such that they sum the unity, the estimated mean and covariance are evaluated as a weighted mean among all of the particles. Before starting the next step and integrating the ensemble forward in time, particle resampling is performed in order to keep the system tractable. The particles are therefore resampled according to their own weight, as in the Bootstrap Particle Filter (BPF)~\cite{douc2005comparison}, or using a Gaussian approximation of the \textit{a posteriori} PDF, given the posterior mean and covariance, as in the Gaussian Particle Filter (GPF) ~\cite{kotecha2003gaussian}.
The main drawback of any particle-based filtering technique is performing multiple integrations, which are time-consuming and drastically increase the machine workload. Thus, different techniques where the ensemble of particles of Monte Carlo filters are evaluated through a polynomial at a given time step have been developed~\cite{Servadio2021}. Therefore, a new particle filter based on the KO is proposed, where each integration is substituted by the mere evaluations of functions using the KO approximation of the solution for the dynamics. In particular, the Koopman Operator Particle Filter (KOPF) evaluates the propagated particles, each time step, by considering their different initial condition during the evaluation of the eigenfunctions. After the dynamics have been analyzed and linearized in the set of basis functions, the state of each particle, for any given time, is a function of its own initial conditions. Looking back at Eq.~\eqref{solution}, each particle is propagated through the evaluation of different ${\bf x}(t_0)$ into the eigenfunctions. Therefore, all the particles are integrated in the KO framework and the state PDF is propagated rapidly, making the KOPF suitable for on-board applications. The measurement update part of the filtering algorithm remains untouched and it continues normally according to the BPF or GPF mathematics, and following the selected resampling technique.
\subsection{The Nonlinear Model Predictive Controller - NMPC}
The Koopman theory is also integrated into the control segment of the station-keeping application. The KO solution is therefore merged inside a model predictive controller, where the control feedback input is optimized by predicting the future behavior of the state of the system given the current state.
The main idea of the Nonlinear Model Predictive Controller, NMPC, is that, at each sampling time step, the predicted future behavior of the system is optimized over a finite time horizon~\cite{grune2017nonlinear}. The optimal control sequence is then used as feedback control values to modify the state of the spacecraft at the current time step. The prediction of the future behavior of the state is compared to a given constant reference. The cost function to be used in the optimization should penalize the distance of the estimated state from the filter with respect to the reference trajectory. As expected, increasing the prediction horizon length directly relates to a more accurate feedback control input, which will lead the NMPC to achieve better performance. However, it is often convenient to penalize the control input $\bf u$ for modeling purposes, such as avoiding control values that are connected to high energy. For station-keeping applications, this aspect connects to fuel optimization in the particular case that the selected cost function analyzes the amount of fuel requested by the spacecraft. Constraints, both on the control, as well as on the states, can be easily taken into account for the classic NMPC algorithm~\cite{grune2017nonlinear}. The basic NMPC algorithm can be summarized in three major steps. At first, the state of the system is measured. This step is achieved through the state estimation performed by the KOPF. Then, an optimal control problem (OCP) is solved, with the goal of minimizing some well-defined cost function with respect to the control input $\bf u$, and subject to either control or state constraints. Lastly, the NMPC feedback value is selected and used as a control value for the next sampling period. Open-loop optimal control is also possible~\cite{patwardhan1990nonlinear}.
The Koopman operator is used inside the NMPC for the prediction horizon length, where the state of the system is propagated forward in time to assess the validity of the proposed control input value. Therefore, the newly proposed KO solution of the RTBP is embedded both in the estimator and in the controller part of the closed-loop station-keeping algorithm.
\subsection{Numerical Results}
The closed-loop station-keeping technique described before has been applied to the control of a spacecraft about a Halo orbit defined by Eq. (\ref{horbit}). After the coordinate transformation presented in Eq.~\eqref{coord}, which translates the origin of the system to the $\mathcal L_1$ point and scales the problem by the distance between the secondary body and $\mathcal L_1$ itself, the new initial conditions are:
\begin{align}
\bf{x}_{0,T}& =
\left[
-0.084775484328489 \quad 0.03 \quad 0.111161030000000 \quad 0.003 \quad 0.896068969160230 \quad 0.003
\right]^T, \\
\bf{x}_{0,D} &=
\left[
-0.114775484328489 \quad 0 \quad 0.081161030000000 \quad 0 \quad 0.893068969160230 \quad 0
\right]^T,
\end{align}
where $\bf{x}_{0,T}$ indicates the initial true position of the spacecraft. Note that this represents a large offset with respect to the desired initial position $\bf{x}_{0,D}$. The initial estimate is then selected randomly from a Gaussian PDF, with the true initial state as the mean value and a covariance matrix given by:
\begin{equation}
\bf{P}_{0} =
\begin{bmatrix}
10^{-6}\bf{I}_{3x3} & \bf{0}_{3x3} \\
\bf{0}_{3x3} & 10^{-7}\bf{I}_{3x3}
\end{bmatrix},
\end{equation}
where $\bf{I}_{3x3}$ indicates the identity matrix, and $\bf{0}_{3x3}$ the null matrix. The satellite is controlled such that the maneuver corrections and observations are performed every 4 days. Additionally, it is assumed that the thruster uses a negligible amount of fuel compared to the satellite's mass, meaning that the mass of the spacecraft remains constant during the simulation. Finally, the system is considered to be subject to Gaussian process noise $\mu \sim \mathcal{N}(\bf{0}_{6\times 1},10^{-5}\bf{I}_{6 \times 6})$, and the measurements are obtained considering that the spacecraft is equipped with GPS with a Gaussian observation noise $\nu \sim \mathcal{N}(\bf{0}_{3 \times 1},10^{-5}\bf{I}_{3 \times 3})$.
The NMPC optimizes the control input following the cost function $J = {\bf u^T}W{\bf u}$, where $W$ is a weighting matrix, selected as the identity in the current application. Constraints are given for the maximum feedback control input that the thruster can generate in the NMPC: a value of $\pm0.00474$ km/s is selected.
\begin{figure}[h!]
\centering
\includegraphics[width=.65\textwidth]{control.png}
\caption{Station-keeping application of a spacecraft orbiting in a Halo orbit around $\mathcal L_1$. }
\label{fig:keep}
\end{figure}
The station-keeping results using a KO with basis functions of order 3 are presented in Figure~\ref{fig:keep}, where the reference orbit, the true position of the satellite, and its estimated position, are shown respectively as dashed, continuous, and point-dashed lines. As can be observed from the figure, the KOPF and NMPC are able to control the spacecraft and bring it to the desired pathway after just a few time steps. Particularly, the initial offset can be noted at the beginning of the simulation, and soon after, the spacecraft is already in the trajectory of the Halo orbit. In this example, the NMPC prediction horizon takes into account the following 10-time steps, which avoid the overshooting during the approaching to the desired orbit, and provides a smooth approach to the desired state. This means that after the quick transient, the three lines overlap for the remaining part of the trajectory, assessing the validity of the proposed technique. The control horizon is set for 3 time steps.
\begin{figure}[h!]
\centering
\includegraphics[width=.7\textwidth]{error.png}
\caption{Tracking and estimation error of a spacecraft orbiting in a Halo orbit around $\mathcal L_1$. }
\label{fig:err}
\end{figure}
In this regard, it is also possible to assess the tracking error of the spacecraft. This is done by evaluating the difference between the reference orbit and the true position. Analogously, the estimation error is calculated as the difference between the true position of the satellite and the estimated state from the filter. Figure~\ref{fig:err} shows the tracking and the estimation errors for the presented application. After a quick transient where the system corrects the initial offset, the error levels settles down and it shows an accurate and robust estimate and control of the dynamics of the spacecraft.
\begin{figure}[h!]
\centering
\includegraphics[width=.7\textwidth]{uu.png}
\caption{Feedback control input for a spacecraft orbiting in a Halo orbit around $\mathcal L_1$. }
\label{fig:uu}
\end{figure}
The duration of the transient is directly connected to the maximum thrust available from the thrusters. Thus, in the first time steps, while compensating for the initial offset, the thrusters are saturated and reach the maximum available thrust. Having a more powerful thruster would reduce the transient time and achieve a quicker match between reference and target orbits. Figure~\ref{fig:uu} shows the feedback control input, in terms of change in the spacecraft's velocity, during the complete simulation.
\section{Conclusions}
The Koopman Operator methodology presented in this work is shown to provide an accurate transformation that embeds the nonlinear dynamics in a global linear representation. The selected application, the circular restricted three-body problem, is a highly nonlinear and highly chaotic system, whose periodic solution is hard to obtain. Nevertheless, the Koopman approximation provides an accurate analytical representation of the dynamics of the system, being able to accurately reproduce both Lyapunov and Halo orbits as a linear combination of well-selected eigenfunctions. As such, the dynamics of the system are transformed into a new framework, where each contribution has a linear dependence on the overall solution.
An analysis of the accuracy of the KO solution supports the validity of the newly proposed technique, especially for those initial conditions close to the equilibrium points, as shown by the study of the location of the minimum curves in the eigenfunctions.
In addition, the accuracy of the newly proposed techniques has been assessed through a convergence analysis, where the position error of different families of Halo orbits has been compared to classical numerical integration techniques for the different orders of the KO. The results show that the behavior of the KO framework is similar to the perturbation theory since the robustness of the KO approximation decreases the further the initial conditions are selected from the equilibrium points.
Moreover, the KO mathematics has been proven helpful to provide the spectral behavior of the system. Particularly, a study of the frequencies and modes of the dynamics is included with this work. As such, periodic orbits are found by studying the eigenfunctions of the system directly and their curves of minima.
Furthermore, the KO has been applied to control problems. In particular, a station-keeping application is proposed trhough a new filtering technique and a model predictive controller developed in the KO framework. The KOPF shows high accuracy levels for spacecraft state estimation, while the NMPC uses the KO solution to optimize the control feedback input for the satellite.
Lastly, a quick comparison of different methodologies for the evaluation of the Koopman matrix is presented. The analytical approach proposed in this paper is shown to better describe the spectral behavior of the dynamics of the system when compared to data-based methods such as the EDMD.
In conclusion, the KO provides a novel alternative to classical numerical integration techniques for solving Ordinary Differential Equations. The new methodology provides, at the same time, the ODE's solution and a deep understanding of its characteristics, in terms of stability and robustness, achieved thanks to the knowledge of the system eigenvalues and modes. Therefore, the new framework holds promise for a variety of future applications including onboard navigation applications (estimation and/or control), filtering problems, and Attitude Determination and Control Systems~\cite{chen2020koopman}.
\section*{Acknowledgments}
The authors wish to thank Hailee Hettrick at Massachusetts Institute of Technology for her help in reviewing and improving this manuscript. The authors want to acknowledge the support of this work by the Air Force's Office of Scientific Research under Contract Number FA9550-18-1-0115.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,665 |
Rostmantlad taggstjärt (Thripophaga berlepschi) är en fågel i familjen ugnsfåglar inom ordningen tättingar.
Utseende och läte
Rostmantlad taggstjärt är en 18 cm lång ugnfågel. Fågeln är rödbrun på det mesta av ovansidan samt på bröstet, mot buk och övergump olivbrun och gråbeige på hjässa oh haka. Näbben är blåaktigt hornfärgad och ögat hasselnöts- till orangefärgat. Lätet inleds vanligen med ett lågt, spinnande ljud som snabbas upp till ett fallande "cheecheecheecheecheechiddididrrrrrr". Duetter eller sång från flera individer samtidigt är vanligt förekommande.
Utbredning och status
Fågeln förekommer i Anderna i norra Peru (Amazonas till La Libertad).
Släktestillhörighet
Rostmantlad taggstjärt placeras traditionellt i släktet Thripophaga. Genetiska studier visar dock att den istället är en del av Cranioleuca och förs därför numera dit av bland andra IUCN och BirdLife International. De tongivande auktoriteterna Clements et al och International Ornithological Congress (IOC) behåller den dock ännu i Thripophaga och denna linje följs därför här.
Status
IUCN kategoriserar arten som nära hotad.
Namn
Fågelns vetenskapliga artnamn hedrar den tyske ornitologen Hans von Berlepsch (1850-1915). Den kallades tidigare perumjukstjärt men blev tilldelat ett nytt namn av BirdLife Sveriges taxonomikommitté 2022 för att bättre återspegla artens korrekta släktskap.
Referenser
Externa länkar
Läten på xeno-canto.org
Ugnfåglar
Fåglar i neotropiska regionen
Endemiska fågelarter i Peru | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,308 |
A Love Letter to Traditions Old & New, Our 2019 Celebration Issue Is Here
By Meghan Harlow| December 15, 2019
This article appears in Winter 2019: Issue No. 61 of Edible Manhattan.
The 2019 celebration issue of Edible Manhattan is finally here. Now, read editor Meghan Harlow's introduction to the magazine.
Our 2019 celebration issue is here, with: Christmas Nostalgia • Holiday Baking—Rugelach • New Year's Eve Recipes • "Odd Apples"—A Photo Essay • The Future of Local Food
My grandmother was born 90 years ago in a tenement house on 119th Street. "Not quite a manger," she says, "but close." Back then, December in New York used to chime like a monthlong ritual, as a time when aromas, and all the neighborhood's children, used to meet on the streets in knots. I've listened to my grandmother's stories of this time and place so often that it exists, dog-eared, in my memory; as if, through listening to her stories alone, I, too, have inhabited this small Italian corner of Old New York; as if I, too, have breathed its sweet, smoky air carrying chestnuts, incense from nearby churches, and oil, oil, oil from all the local families' frying.
"I just wish you could have seen it on Christmas Eve," said my grandmother back in August. We were sitting at her kitchen table, just the two of us, holding hands and time traveling. "It was magical. We did all the seven fishes back then; we did everything."
This, I knew, was my opening.
So why, I asked, couldn't we try to re-create it together? My husband and I had been thinking; and, yes, all seven fishes was too much for anyone alone to handle—much less a 90-year-old, however vibrant and spunky—but with the three of us, it would only work out to 2.3 fishes per person. And, no matter what, it would be better than eating out, as usual, at a restaurant. "Please," I said finally, "can't we at least try co-hosting?"
My grandmother smiled and squeezed my hand. "I think I'm finally ready."
And so now, in a few, short weeks, my grandmother and I will co-host our first Christmas Eve. And no matter how much we may bicker, or how much she may criticize my cooking, I know the experience will be the greatest gift she could have given me. Because here is perhaps the only thing I know with certainty: More than anything else, we are the hands we hold and the batons we choose to carry.
If I hadn't already been sure of this, the stories we've collected for you in this issue—our most loving, festive edition of the year—would have removed any lingering doubt. In the coming pages, you will meet a woman who honors her Jewish heritage by baking rugelach, using her own grandmother's beloved recipe; you will celebrate Christmas with a family of Coptic Christians who keep each other and their Egyptian customs close as they worship and feast and fast; you will cook arroz con dulce alongside a writer who has returned home to Puerto Rico, following the diaspora that brought his grandparents to New York, with culinary traditions so colossal they never could have fit in his carry-on. And, because batons are items we carry forward, you will learn about the people and organizations working to ensure the future of local food, as we near the point of no return for our planet.
In all of these stories—and even in this one, my own—so many batons are changing hands, but the staffs themselves are good and sturdy. Now it's our turn to carry them, to take them further or to an entirely different place, to honor everyone who held them before us and maybe, if we're lucky, finally finish the race. What a beautiful responsibility. What abundant cause to celebrate.
Wishing you and yours so much joy and wonder,
This New Year's Eve, Stay Home and Cook
Patadas Chafadas, Spanish Olive Oil Mashed Potatoes, Are a Simple Holiday Dish
Join Our Collective
Where to Find Edible Manhattan
June/July/August 2016: Issue No. 46
November 2015—January 2016: The Holiday Issue
The Travel Issue: September/October 2015, Issue No. 42
Edible Manhattan is published six times a year and available by subscription, for sale at selected retailers and at other distribution spots throughout Manhattan. Please visit our sister magazines, Edible Brooklyn, Edible East End, and the Edibles in New York State. And visit Edible Communities to find the publication nearest you.
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Subscribe to the Edible Manhattan newsletter to keep up with local food news, events and more! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 14 |
#include "ICM42688P.hpp"
#include <px4_platform_common/getopt.h>
#include <px4_platform_common/module.h>
void ICM42688P::print_usage()
{
PRINT_MODULE_USAGE_NAME("icm42688p", "driver");
PRINT_MODULE_USAGE_SUBCATEGORY("imu");
PRINT_MODULE_USAGE_COMMAND("start");
PRINT_MODULE_USAGE_PARAMS_I2C_SPI_DRIVER(false, true);
PRINT_MODULE_USAGE_PARAM_INT('R', 0, 0, 35, "Rotation", true);
PRINT_MODULE_USAGE_DEFAULT_COMMANDS();
}
I2CSPIDriverBase *ICM42688P::instantiate(const BusCLIArguments &cli, const BusInstanceIterator &iterator,
int runtime_instance)
{
ICM42688P *instance = new ICM42688P(iterator.configuredBusOption(), iterator.bus(), iterator.devid(), cli.rotation,
cli.bus_frequency, cli.spi_mode, iterator.DRDYGPIO());
if (!instance) {
PX4_ERR("alloc failed");
return nullptr;
}
if (OK != instance->init()) {
delete instance;
return nullptr;
}
return instance;
}
extern "C" int icm42688p_main(int argc, char *argv[])
{
int ch;
using ThisDriver = ICM42688P;
BusCLIArguments cli{false, true};
cli.default_spi_frequency = SPI_SPEED;
while ((ch = cli.getopt(argc, argv, "R:")) != EOF) {
switch (ch) {
case 'R':
cli.rotation = (enum Rotation)atoi(cli.optarg());
break;
}
}
const char *verb = cli.optarg();
if (!verb) {
ThisDriver::print_usage();
return -1;
}
BusInstanceIterator iterator(MODULE_NAME, cli, DRV_IMU_DEVTYPE_ICM42688P);
if (!strcmp(verb, "start")) {
return ThisDriver::module_start(cli, iterator);
}
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 169 |
vahidshirvani.se site which i work on in my spare time.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,528 |
\section{Introduction}
In recent years, we have witnessed the trend of using larger and larger neural network (NN) models to deliver improved accuracy and generalization
in various machine learning tasks~\cite{devlin2018bert, fedus2021switch}.
However, training these models requires a considerable amount of on-device GPU memory.
Unfortunately, the increase of GPU memory capacity has been relatively slow, leading to a fundamental barrier to the development of large NN models.
Activation Compressed Training (ACT) is a promising approach to reduce the memory footprint of models during training. As all layers' activations need to be kept in the memory for computing the gradients during training,
ACT reduces memory consumption by compressing these saved activations.
Prior work~\cite{chakrabarti2019backprop, fu2020don, chen2021actnn, evans2021ac} has shown the effectiveness of ACT by reducing activation footprint by up to $12\times$ with 2-bit activations.
Although ACT has already demonstrated impressive compression capabilities, previous work on ACT is restricted to specific NN architectures.
For example, ActNN~\cite{chen2021actnn} is a quantization framework for convolutional NNs only; Mesa~\cite{pan2021mesa} proposes a per head/layer quantization method for vision transformers; and AC-GC~\cite{evans2021ac} derives convergence error bound for different types of operators separately.
Developing a generic ACT framework is challenging. Theoretically, convergence guarantees must be made without assumptions on the network architecture. Algorithmically, the framework should find effective compression strategies for all kinds of networks automatically. From the system perspective, the framework should support arbitrary NN operations, including user-defined ones.
In this work, we propose GACT\xspace, a general framework for ACT that is agnostic to the NN architecture.
Neither specialized mathematical derivations nor customized implementation is needed to support different operators.
To enable this, we develop a general convergence theory by analyzing the stochastic gradient (SG) introduced by ACT. We show that the SG can be well approximated by a linearized version, which is unbiased to stochastic compressors. The variance of the linearized gradient has a particularly simple structure that allows a numerical algorithm to predict the variance given a compression strategy. Then, we generate the strategy by approximately solving an integer program.
We implement our method as a library based on PyTorch that can be quickly integrated into real-world machine learning systems.
The library also provides several optimization levels to explore the trade-off between memory and speed.
We demonstrate the flexibility and efficiency of GACT\xspace on various tasks, including image classification, object detection, text, and graph node classification. Our evaluation shows that GACT\xspace can reduce activation memory by up to 8.1$\times$, enabling training with a 24.7$\times$ larger batch size on the same GPU.
In sum, our main contributions are as follows:
\begin{itemize}
\itemsep0em
\item We propose a general convergence theory for ACT.
\item We develop an algorithm that automatically estimates the sensitivity of each compressed tensor and selects the optimal compression strategy.
\item We build efficient implementation of GACT\xspace in PyTorch with an easy-to-use API that can also be combined with other memory-saving techniques seamlessly.
\end{itemize}
\section{Related Work}\label{sec:related}
\textbf{Activation Compressed Training.}
ACT has been applied to convolutional NNs using different compressors, such as quantizers~\cite{chakrabarti2019backprop, fu2020don,chen2021actnn}, JPEG~\cite{evans2020jpeg}, or scientific data compression algorithms~\cite{jin2021novel,evans2021ac}.
ACT is also applied to transformers~\cite{pan2021mesa} and graph NNs~\cite{anonymous2022exact}.
However, the existing theory for ACT~\cite{chakrabarti2019backprop,fu2020don,chen2021actnn,evans2021ac} relies on the case-by-case analysis of specific network operators, such as convolution, ReLU, and batch normalization.
It also requires dedicated implementations for each operator.
On the contrary, GACT\xspace focuses on the generality of activation compressed
training, not a specific quantizer design, which is the main
topic of previous work. Instead of assuming that the network is a stack
of layers, GACT formulates the problem as a computational graph of operators.
This is general enough to cover transformers~\cite{vaswani2017attention}, graph NNs~\cite{kipf2016semi}, second-order
derivatives, and unknown future~architectures.
\noindent{\textbf{Reduced Precision Training.}}
Apart from ACT, reduced precision training~\cite{micikevicius2018mixed, wu2018training,wang2018training,banner2018scalable,chen2020statistical,sun2020ultra} performs calculations directly on low precision data, reducing the computation cost and memory footprint simultaneously. To achieve this, specialized kernels are used to calculate on low precision data. In contrast, ACT only considers storage, and it can thus use more flexible compression strategies and achieve a much better compression ratio with the same accuracy loss.
\noindent{\textbf{Memory-Efficient Training.}}
Gradient checkpointing~\cite{chen2016training, jain2019checkmate} trades computation for memory by dropping some of the activations in the forward pass from memory and recomputing them in the backward pass.
Swapping~\cite{kirisame2020dynamic, huang2020swapadvisor, wang2018superneurons, peng2020capuchin} offloads activation or model parameters to an external memory (e.g., CPU memory).
Recent work~\cite{beaumont2021efficient} explores the possibility of combining the gradient checkpointing and swapping.
All these methods save memory by storing fewer tensors on the GPU. In contrast,
GACT\xspace compresses the saved tensors and is complementary to these approaches. Moreover, the generality of GACT\xspace enables easy combination with these methods, which we explore in this paper.
\section{Formulation}
We first present the mathematical formulation of our activation compressed training (ACT) framework.
As we would like to develop a general ACT algorithm, applicable to a wide range of NN architectures, we make minimal assumptions on our formulation.
Throughout the paper, we define the variance of a vector $x$ as $\Var{x}=\E{\norm{x}^2}-\norm{\E{x}}^2$.
\subsection{Activation Compressed Training}
In this work, we abstract the forward propagation as two functions $\ell(x; \theta)$ and $h(x; \theta)$.
Both take a datum $x$ and the model parameter $\theta$ as the input. The loss function $\ell(x; \theta)$ outputs the loss of the network $\theta$ on datum $x$. The context function $h(x; \theta)$ outputs tensors to be stored in the memory for computing the gradients, which are referred as the \emph{context}. Assume that the context consists of $L$ tensors, where each tensor $h^{(l)}(x; \theta)$ is represented by a flattened $D_l$-dimensional vector.
Denote $h(x; \theta)=(h^{(l)}(x; \theta))_{l=1}^L$. Our notations are somewhat unconventional in the sense that we do not explicitly define each layer's activation.
We do not even assume that there is a NN. It could be any computational graph that saves context tensors.
Given a dataset $\mathcal X=\{x_n\}_{n=1}^N$, define the batch loss $\mathcal L(\theta):=\frac{1}{N}\sum_{n=1}^N \ell(x; \theta)$.
The dataset can be equivalently represented as an empirical data distribution $p_{\mathcal X}(x):=\frac{1}{N}\sum_{n=1}^N \delta(x-x_n)$, where $\delta$ is the Dirac delta function. The batch loss can be written as $\mathcal L(\theta)=\mathbb E_{\mathcal X}[\ell(x; \theta)]$, where $\mathbb E_{\mathcal X}$ denotes for taking expectation over $p_{\mathcal X}$.
The network is trained with stochastic gradient descent (SGD)~\cite{bottou2010large}. Starting from an initial model $\theta_0$, at the $t$-th iteration, SGD updates the model with:
\begin{align}
\theta_{t+1}\leftarrow \theta_t - \eta \nabla_{\theta} \ell(x; \theta_t),\label{eqn:sgd}
\end{align}
where $\eta$ is a learning rate, and the SG $\nabla_{\theta} \ell(x; \theta)$ is computed on a random datum $x\sim p_{\mathcal X}$. Notice that $\Es{\mathcal X}{\nabla_\theta \ell(x; \theta)}=\nabla_{\theta} \mathcal L(\theta)$, i.e., the SG is an unbiased estimator of the batch gradient $\nabla_{\theta} \mathcal L(\theta)$.
Crucially, the SG can be written in the form $\nabla_{\theta} \ell(x; \theta_t)=g(h(x; \theta_t); \theta_t)$. In other words, the back propagation only depends on the forward propagation through the context $h(x; \theta_t)$.
The entire context must be kept in memory for computing the gradients. The context dominates the memory consumption in many applications.
ACT reduces the training memory footprint by compressing the context.
Let $Q(h)$ be a compressor, which converts $h$ to compact formats while keeping $Q(h)\approx h$.
Then, ACT computes the gradient with compressed context: \begin{align}\label{eqn:ac}
\theta_{t+1}\leftarrow \theta_t - \eta g(Q(h(x; \theta_t)); \theta_t).
\end{align}
We refer to $g(Q(h(x; \theta_t); \theta_t)$ as the activation compressed (AC) gradient.
ACT is significantly more memory efficient then the plain SGD, Eq.~(\ref{eqn:sgd}), since it only needs to store a compressed version of the context. Suppose the original context $h(x; \theta_t)$ consists of 32-bit floating point tensors, and $Q(\cdot)$ is a compressor which quantizes tensors to 2-bit integers, ACT will reduce the context memory by $16\times$.
Fig.~\ref{fig:architecture} illustrates the computational graph of ACT with these notations.
In the following presentation, we might denote $h(x, \theta)$ simply by $h$ when there is no confusion.
\begin{figure}[t]
\centering
\includegraphics[width=\mywidth]{figures/architecture.pdf}
\ifisarxiv
\else
\vspace{-2em}
\fi
\caption{\small The architecture of GACT\xspace.}
\label{fig:architecture}
\end{figure}
\subsection{Convergence of ACT}\label{sec:convergence}
ACT is a lossy approximation of SGD, as it uses an approximate gradient $g(Q(h); \theta)$. Therefore, some kind of theoretical guarantee is required for ACT to be useful.
Fortunately, analyzing ACT is made significantly simpler by introducing an \emph{unbiased stochastic} compressor $Q(\cdot)$, such that $\Es{Q}{Q(x)}=x$ for any $x$. $\Es{Q}{\cdot}$ means taking expectation over the compressor.
In this way, $g(Q(h); \theta)$ can be viewed as a stochastic estimator of the batch gradient $\nabla \mathcal L(\theta)$, but the randomness comes not only from the datum $x$ but also the compressor $Q(\cdot)$. Therefore, ACT is still an SGD algorithm.
Standard analytical tools for SGD~\cite{bottou2018optimization} are applicable for studying ACT.
SGD algorithms have particular good properties when the SG is unbiased. In our case, this means $\Es{Q}{g(Q(h);\theta)}=g(h;\theta)$. However, the SG is biased general, even when the stochastic compressor itself is unbiased.\footnote{Consider the example $g(h)=\mathbb I(h\ge 0.5)$, where $h\in [0, 1]$ and its AC gradient $g(Q(h))=\mathbb I(Q(h)\ge 0.5)$ with the compressor $Q(h)\sim \mathrm{Bernoulli}(h)$. Then, $\E{g(Q(h))}=P(Q(h)=1)=h\ne g(h).$
}
The key technique in this work is to construct an unbiased approximation of the AC gradient by linearizing the gradient function $g(\cdot; \theta)$. Consider the first-order Taylor expansion of $g(\cdot; \theta)$ at $h$:
\begin{align}\label{eqn:first-order}
\hat g(Q(h); h, \theta):=
g(h; \theta) +
J(h, \theta)
\Delta h,
\end{align}
where $J(h, \theta):=\frac{\partial g(h;\theta)}{\partial h}$ is a Jacobian matrix, $\Delta h:=Q(h)-h$ is the compression error.
We further denote $\hat g_{x\theta}(Q(h); h):=\hat g(Q(h); h, \theta)\vert_{h=h(x; \theta)}$ and $J_{x\theta}(h):=J(h, \theta)\vert_{h=h(x; \theta)}$ for short.
Since $\E{\Delta h(x; \theta)}=0$,
$\hat g_{x\theta}(Q(h); h)$ is an unbiased SG, Furthermore, the approximation error is small:
\begin{proposition}\label{prop:bias-order}
Assuming that $g(h; \theta)$ is twice differentiable w.r.t. $h$, and the second order derivative is bounded, then
\begin{align*}&\E{\norm{g(Q(h); \theta) - \hat g_{x\theta}(Q(h); h)}_2}
= O(\Vars{Q}{\Delta h}).\end{align*}
\end{proposition}
Since $\Delta h$ itself is unbiased, $\Vars{Q}{\Delta h}=\Es{Q}{\norm{\Delta h}^2}$ is simply the expected compression error. Prop.~\ref{prop:bias-order} implies that the linearization error is bounded by the compression error. The linearized gradient $\hat g$ is accurate if the compression is accurate. Using $\hat g$ as a bridge, we arrive in the following convergence theorem:
\begin{theorem}\label{thm:convergence}
Assume that:\\
\noindent\textbf{A1.} $\mathcal L(\theta)$ is a continuous differentiable, $\nabla\mathcal L(\theta)$ is $\beta$-Lipschitz continuous.\\
\noindent\textbf{A2.} $\mathcal L(\theta)$ is bounded below by $\mathcal L_*$.\\
\noindent\textbf{A3.} $g(h; \theta)$ is differentiable w.r.t. $h$ and $\exists b>0$, s.t. $\forall \theta, \mathbb E\norm{g(Q(h(x; \theta)); \theta)-\hat g_{x\theta}(Q(h); h)}\le b$.\\
\noindent\textbf{A4.} $\exists \sigma^2>0$, s.t., $\forall\theta$, $\Var{\hat g_{x\theta}(Q(h); h)}\le \sigma^2$. \\
Then, for all $\eta < \frac{1}{2\beta}$, if we run ACT defined as Eq.~(\ref{eqn:ac}) for $T$ iterations, then we have
\begin{align*}
\min_{t=0, \dots, T-1}\E{\norm{\nabla \mathcal L(\theta_t)}^2}
\le \frac{4(\mathcal L(\theta_0)-\mathcal L_*)}{\eta T}+3b^2+ \eta\beta\sigma^2
\end{align*}
\end{theorem}
\paragraph{Remark: } The analytical technique used in Thm.~1 is rather standard, see Thm.~4.8 in~\citet{bottou2018optimization}. However, we consider the variance term $\sigma^2$ of the \emph{linearized gradient}, rather than the SG itself. This formulation brings better analytical properties and an adaptive algorithm for determining the compression scheme, as we shall see soon in Sec.~4.
The convergence of ACT is affected by both the linearization error (A3) and the variance of the unbiased gradient $\hat g(\cdot; \theta)$ (A4). The latter is characterized as:
\begin{proposition}\label{prop:var-order}
$
\Var{\hat g_{x\theta}(Q(h); h)}
=\Vars{\mathcal X}{g(h; \theta)} +
\mathbb E_{\mathcal X}\left[\Vars{Q}
{
\hat g_{x\theta}(Q(h); h)
}\right],
$
where the second term on the RHS equals to
$
\mathbb E_{\mathcal X}\left[\Vars{Q}
{J_{x\theta}(h)\Delta h}\right]=O\left(\Vars{Q}{\Delta h}\right).$
\end{proposition}
Prop.~\ref{prop:var-order} separates the variance from different noise sources.
$\Vars{\mathcal X}{g(h(x, \theta); \theta)}$ is the variance raised by random sampling of data (``sampling variance'').
$\mathbb E_{\mathcal X}\left[\Vars{Q}
{J_{x\theta}(h)\Delta h(x, \theta)}\right]$ is the variance raised by compression. Now, the convergence in Thm.~\ref{thm:convergence} is depicted by $3b^2+\eta\beta\sigma^2$. By Prop.~\ref{prop:bias-order}, $b^2=O(\Vars{Q}{\Delta h}^2)$. By Prop.~\ref{prop:var-order}, $\sigma^2 =O(1)+O(\Vars{Q}{\Delta h})$, since the sampling variance is not affected by compression. Therefore, when the compression is accurate ($\Delta h\rightarrow 0$), the impact of the linearization error is negligible, and the variance of the unbiased gradient dominates. ACT behaves as if the AC gradient is unbiased.
\section{Adapting the Compression Rate}
In a network, some context tensors (such as those stored for computing the cross entropy loss) are extremely sensitive, a small amount of compression would result in diverged training, while other tensors are quite robust to compression.
Therefore, we must apply different amounts of compression for each context tensor.
As a general framework, we have no prior knowledge of the users' model architecture, so we designed an algorithm to infer the sensitivity for each context tensor and determine their compression rate automatically.
There is a tradeoff between the compression error and the storage requirement.
We represent the storage requirement of the compressed context in \emph{bits per dimension}. We assume that $b_l$ bits/dim. are used for compression $h^{(l)}$, and $Q_{b_l}(h^{(l)})$ be the compression result. Let $b=(b_l)_{l=1}^L$ be a \emph{compression scheme}, $Q_b(h):=\{Q_{b_l}(h^{(l)})\}_{l=1}^L$, and $\Delta_b h = Q_b(h) - h$.
\subsection{Structure of Variance}
As discussed in Sec.~\ref{sec:convergence}, when the compression is relatively accurate, the variance plays the main role in determining the convergence. Therefore, we would like to investigate how the compression scheme would impact the variance. Formally, we are interested in:
\begin{align*}
V(b; h, \theta) := \Vars{Q}
{
\hat g(Q_b(h); h, \theta)
}.
\end{align*}
Once $V(b, h; \theta)$ is known, we can find the minimum variance compression scheme under a given total bits budget $B$, by solving the integer programming problem:
\begin{align}
\min_b V(b; h(x; \theta), \theta),~~~\mbox{s.t. }\sum_{i=1}^L b_l D_l \le B,\label{eqn:ilp}
\end{align}
where $D_l$ is the dimensionality of $h^{(l)}$.
To proceed, we need the following assumptions on the compressor $Q_b(\cdot)$:\\
\noindent\textbf{Assumption B1: } The compressed result is element-wise uncorrelated. That is, for any $i\ne j$, $\Cov{Q_b(h)_i}{Q_b(h)_j}=0$.\\
\noindent\textbf{Assumption B2: } For compressing $h^{(l)}(x; \theta)$ to $b_l$ bits/dim., the compression error can be written in the form $\Var{\Delta_{b_l} h^{(l)}(x; \theta)_j}\le R_{lj}(x; \theta)S(b_l)$, where $S(b_l)$ is a known function. This isolates the effect of $b_l$ through the unary factor $S(b_l)$.\\
Both assumptions can be achieved by a stochastic rounding quantizer~\cite{courbariaux2015binaryconnect}, where $R_{lj}(x; \theta)=\frac{1}{4}\left(\max_k h_k^{(l)}-\min_k h_k^{(l)}\right)^2$ and
$S(b) = (2^{b_l}-1)^{-2}$. See Appendix~\ref{sec:appendix-var-structure} for the derivations.
The following theorem reveals the structure of the variance:
\begin{theorem}\label{thm:variance-structure}
Under assumptions B1, B2, there exists a family of functions $\{c_l(h, \theta)\}_{l=1}^L$, such that the compression variance can be written in the form
\begin{align}\label{eqn:var-decomposition}
V(b; h, \theta)\le \sum_{l=1}^L c_l(h, \theta) S(b_l).
\end{align}
\end{theorem}
\subsection{Computing Sensitivity}
\label{sec:compute_sensitivity}
Thm.~\ref{thm:variance-structure} reveals two good properties of the variance: (1) the impact of compressing different context tensors simply sums up, without affecting each other; and (2) the compression scheme only impacts the variance through $S(b_l)$. Both properties are brought about by linearization. Since $S(\cdot)$ is a known function, we only need to know $c_l(h, \theta)$ to solve problem Eq.~(\ref{eqn:ilp}).
$c_l(h, \theta)$ can be understood as the sensitivity of the AC gradient to the compression of the $l$-th tensor.
We can compute $c_l(h, \theta)$ numerically
by leveraging the idempotence of compressing a tensor:
\\
\noindent\textbf{Assumption B3: } If $h=Q(h^\prime)$ for some $h^\prime$ with non-zero probability, then $Q(h)=h$ and $\Vars{Q}{Q(h)}=0$.
Let $Q^{\neg (l)}_b(h)=\{Q_{b_1}(h^{(1)}), \dots, h^{(l)}, \dots, Q_{b_L}(h^{(L)})\}$ be some tensors, where every tensor except $h^{(l)}$ is compressed. Plug $h=Q^{\neg (l)}_b(h)$ into Eq.~(\ref{eqn:var-decomposition}), and use B3, we have
\begin{align*}
V(b; Q^{\neg (l)}_b(h), \theta) \le c_l(Q^{\neg (l)}_b(h), \theta) S(b_l).
\end{align*}
The left hand side can be approximated by taking $\hat g(Q_b(h); h, \theta)\approx g(Q_b(h); \theta)$. Assume that $c_l(\cdot, \theta)$ is reasonably continuous, we have
\begin{align*}
c_l(h, \theta) \approx \Vars{Q}
{
g(Q_b(h); \theta)
}\vert_{h=Q_b^{\neg (l)}(h)} / S(b_l).
\end{align*}
The variance can be replaced by empirical variance.
\begin{algorithm}[t]
\caption{Numerical algorithm for computing $c_l(h, \theta)$.}\label{alg:autoprec}
\begin{algorithmic}
\REQUIRE A gradient evaluation function $g(\cdot; \theta)$
\REQUIRE A series of $L+1$ random seeds $(r_l)_{l=1}^{L+1}$.
\REQUIRE Any compression scheme $b=(b_l)_{l=1}^L$
\STATE $\forall l$, seed $Q^{(l)}$ with $r_l$
\STATE $g_0\leftarrow g(Q_b(h); \theta)$ \COMMENT{First iteration}
\STATE $\forall l$, seed $Q^{(l)}$ with $r_l$
\STATE seed $Q^{(l)}$ with $r_{L+1}$
\STATE $g_1\leftarrow g(Q_b(h); \theta)$ \COMMENT{Second iteration, with another seed}
\textbf{Return} $\frac{1}{2}\norm{g_0 - g_1}^2 / S(b_l)$
\end{algorithmic}
\end{algorithm}
Alg.~\ref{alg:autoprec} illustrates this idea. To compute $\Vars{Q}
{
g(Q_b(h); \theta)
}$ at $h=Q_b^{\neg (l)}(h)$, we keep the random seeds fixed for all the compressors except the $l$-th one. We compute the empirical variance by two evaluations of $g(Q_b(h); \theta)$, which are two NN iterations (forward + backward propagation).
Finally, we assume that $c(h, \theta)$ remains stable for different mini-batches $h$, and along the training trajectory $(\theta_t)$. Therefore, we maintain a $c_l$ for each tensor $l$, which is updated by periodically running Alg.~\ref{alg:autoprec}.
Eq.~(\ref{eqn:ilp}) is approximately solved by the $O(L\log_2 L)$ greedy algorithm~\cite{chen2021actnn}.
Another useful feature of this approach is predicting failure (in an \emph{a posteriori} manner). If the compression variance $V(b; h, \theta)$ is dominating the overall gradient variance $\Var{g(Q(h); \theta_t)}
$, compression is adding too much noise to the gradient, and the convergence might be affected. The overall gradient variance can be computed by maintaining a running mean of the gradient. If $V(b; \theta)/\Var{\hat g(Q(h); \theta_t)}$ is too large, we can raise an alert to the user to increase the storage budget.
\section{System Implementation}
We implemented GACT\xspace as a lightweight library in PyTorch. Users can use GACT\xspace for any NN architecture with several lines of code change.
GACT\xspace uses low-level PyTorch hooks to capture context tensors, so it supports arbitrary operators, including custom operators defined by users.
We implemented efficient CUDA kernels to infer tensor sensitivity and to perform compression during run time.
GACT\xspace uses the same per-group quantizer in ActNN~\cite{chen2021actnn} as the compressor. However, GACT\xspace differs from ActNN in several aspects.
ActNN relies on manual analytical deduction to compute the sensitivity for different operators, while GACT\xspace infers tensor sensitivity automatically, as described in Sec.~\ref{sec:compute_sensitivity}.
Moreover, ActNN performs layer-level quantization.
It has to implement an activation compressed version for each operator and substitute operators during the training (e.g., replace {\code torch.nn.Conv2d} with {\code actnn.Conv2d}).
In contrast, GACT\xspace runs at tensor level and uses a single hook interface to compress saved tensors for all operators.
\subsection{General API}
\ifisarxiv
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{figures/api.pdf}
\caption{Usage example of GACT\xspace}
\label{fig:act_layers}
\end{figure}
\else
\begin{figure}[t]
\centering
\includegraphics[width=0.92\linewidth]{figures/api.pdf}
\vspace{-1.4em}
\caption{\small Usage example of GACT\xspace}
\label{fig:act_layers}
\end{figure}
\fi
As shown in Fig.~\ref{fig:act_layers}, the interface of GACT\xspace is straightforward and intuitive, requiring the user to (i) initialize the GACT\xspace controller and specify an optimization level (Line 5); (ii) install hooks (Line 6); and (iii) instruct GACT\xspace
how to perform forward and backward propagation (Lines 13-17) and pass it as a function ({\code fwdbwdprop}) to the controller (Line 19). We require users to specify (iii) because GACT\xspace needs to numerically run the forward and backward pass to infer tensor sensitivity. Although {\code fwdbwdprop}
is passed to the controller every iteration, it is only called internally every {\code adapt\_interval} iterations when tensor sensitivity changes. As shown in Sec.~\ref{sec:cpr_strategy}, tensor sensitivity stabilizes quickly after the first several epochs, {\code adapt\_interval} can thus be set to a large number, introducing negligible impact on training speed.
\subsection{System Architecture}
Fig.~\ref{fig:architecture} shows an overview of GACT\xspace. The GACT\xspace controller has three modules: Adaptivate Algorithm; Compressor; and Decompressor. In the forward pass, the controller uses PyTorch {\code pack\_hook} to capture all context tensors. Then Adaptive Algorithm infers tensor sensitivity based on gradients and assigns higher bits to more sensitive tensors, as described in Sec.~\ref{sec:compute_sensitivity}. The bits information is used to instruct Compressor to perform quantization. In the backward pass, Decompressor dequantizes context tensors and uses {\code unpack\_hook} to send the dequantized results back to the PyTorch's auto differentiation engine. The controller is also responsible for swapping quantized tensors to the CPU and prefetching them back during the backward propagation if swapping is enabled.
\subsection{Identifying Tensors to Quantize} The {\code pack\_hook} and {\code unpack\_hook} process all types of context tensors, including activation, parameters trained by the optimizer, and training states such as running mean/variance used by batch normalization. To guarantee that only the activations are quantized, we filter out saved parameters by recording the data pointers of all the model parameters before training, and we skip quantization if the input tensor pointer exists in the parameter pointer set. Similarly, GACT\xspace does not quantize training states by checking if the input tensor requires gradients.
However, using hooks blindly disables some memory-saving optimization. For example,
in a transformer's self-attention layer,
the keys, query, value tensors are all calculated from the same input tensor. The saved objects of the three operations thus all refer to the same tensor.
In this case, PyTorch triggers the {\code pack\_hook} three times. If we perform quantization blindly, we waste computation resources and introduce extra memory consumption because the same underlying tensor is quantized and saved more than once. GACT\xspace avoids duplication by generating footprints for each input context tensor. We use the CUDA data pointer, sampled data points, and the tensor statistics (e.g., sum) as the footprint. GACT\xspace manages all quantized context tensors and uses the footprint to differentiate them. If a tensor is already quantized, GACT\xspace will skip quantization and return previous results directly.
\subsection{Parallel Swap and Prefetch}
To further reduce activation memory, we combine GACT\xspace with swapping. All compressed tensors are offloaded to the CPU during the forward pass and swapped back in the backward pass. Here, we replace the original tensor with quantized activation, as data movement is more expensive than computation. Swapping the original tensor saves the quantization overhead but adds more data movement cost between CPU and GPU. As shown in Sec.~\ref{sec:swap}, quantization overhead is much smaller than copying full-precision data to CPU in modern GPU architecture.
Furthermore, we create two new streams (swap in/out) to parallelize the computation and swapping operation to reduce the swap overhead. The forward computation and swap-out process happen in parallel during the forward pass. During the backward pass, in each layer the swap-in stream is responsible for prefetching the compressed activation of the previous layer to avoid synchronization overhead.
We leverage the CUDA event to ensure tasks in different streams are executed in the correct order.
\subsection{Other Memory Optimizations}
\textbf{Gradient checkpointing.}
Gradient checkpointing~\cite{chen2016training} works by dividing the NN into segments.
The algorithm only stores the inputs of each segment and recomputes the dropped activations segment by segment during backpropagation.
The memory consumption is thus the cost of storing the inputs of all segments plus the maximum memory cost to backpropagate each segment.
When combined with gradient checkpointing, GACT\xspace can reduce the memory consumption of both parts.
GACT\xspace reduces the memory consumption of the first part by quantizing the segment inputs. Moreover, the activations saved during the recompute phase are also quantized, reducing the memory cost of the second part.
Combining GACT\xspace with gradient checkpointing might introduce more training noise because the recompute starts from quantized segment inputs, making the forward pass of recompute phase not exact. However, in Sec.~\ref{sec:swap}, we show the noise introduced by forwarding from the quantized tensors is negligible.
\ifisarxiv
\noindent{\textbf{Memory efficient self-attention.}}
\else
\textbf{Memory efficient self-attention.}
\fi
When the batch size is very large, the single layer after dequantization occupies a large amount of memory and prevents the batch size from increasing further. We observe this problem in transformer-based models where self-attention has quadratic space complexity in terms of sequence length.
To reduce the memory footprint of the self-attention layer, we implement the algorithm introduced in~\cite{rabe2021self} that achieves linear space complexity, and combines it with GACT\xspace.
\subsection{Optimization level}
\begin{table}[t]
\centering
\vspace{-1em}
\caption{\small Optimization levels for GACT\xspace. \label{tab:opt_level}}
\resizebox{0.9\mywidth}{!}{
\begin{tabular}{ccc}
\toprule
Level & Compression Strategy & Bits\\
\midrule
L0 & Do not compress & 32\\
L1 & per-group quantization with auto-precision & 4\\
L2 & L1 + swapping/prefetching & 4 \\
CB1 & L1 + gradient checkpointing & 4\\
CB2 & CB1 + efficient self-attention & 4\\
\bottomrule
\end{tabular}}
\end{table}
To exploit the trade-off between memory saving and training speed, GACT\xspace provides several optimization levels.
Higher levels can save more memory but with more overhead.
Tab.~\ref{tab:opt_level} lists these optimization levels.
L1 uses per-group quantization with the adaptive algorithm. L2 combines per-group quantization with swapping and prefetching.
For transformer-based models, CB1 combines GACT\xspace with gradient checkpointing. CB2 further reduces the peak memory by adding efficient self-attention to CB1.
\section{Experiments}
We first demonstrate the effectiveness of the GACT\xspace adaptive algorithm. We further apply GACT\xspace to a wide range of machine learning tasks, including image classification, object detection, text, and graph node classification. We compare the training accuracy and activation compression rate for full precision, adaptive 4/3/2 (using GACT\xspace to adaptively decide quantization bits with an average of 4/3/2 bit) and fix-4 bit (quantizating all tensors uniformly with 4 bits).
Next, we study the trade-off between compression rate and training throughput and compare GACT\xspace with other state-of-the-art memory-saving methods. Lastly, we demonstrate the flexibility of GACT\xspace by exploring the possibility of combining it with other memory optimization methods (CB1, CB2 as listed in Table~\ref{tab:opt_level}). We use open-source model implementations for all tasks.
\subsection{Compression Strategy}
\label{sec:cpr_strategy}
\ifisarxiv
\begin{figure}[t]
\centering
\subfigure[Inferred per-tensor $c_l$ (line) and bits/dim. (bar) for VGG-11. Layers with * have a preceding ReLU layer with shared context. drop=dropout, loss=cross entropy~loss. ]{
\includegraphics[width=0.36\linewidth]{figures/bits.pdf}
}
\subfigure[Gradient variance.]{
\includegraphics[width=0.18\linewidth]{figures/var.pdf}
}
\subfigure[Evolution of the per-tensor sensitivity. Each line is $c_l$ for a tensor.]{
\includegraphics[width=0.18\linewidth]{figures/evo.pdf}
}
\caption{Effectiveness of the adaptive algorithm.}
\label{fig:sensitivity}
\end{figure}
\else
\begin{figure}[t]
\centering
\subfigure[Inferred per-tensor $c_l$ (line) and bits/dim. (bar) for VGG-11. Layers with * have a preceding ReLU layer with shared context. drop=dropout, loss=cross entropy~loss. ]{
\includegraphics[width=0.82\mywidth]{figures/bits.pdf}
}
\subfigure[Gradient variance.]{
\includegraphics[width=0.4\mywidth]{figures/var.pdf}
}
\subfigure[Evolution of the per-tensor sensitivity. Each line is $c_l$ for a tensor.]{
\includegraphics[width=0.4\mywidth]{figures/evo.pdf}
}
\vspace{-1em}
\caption{Effectiveness of the adaptive algorithm.}
\label{fig:sensitivity}
\end{figure}
\fi
We first test the effectiveness of our adaptive compression rate algorithm for training VGG-11~\cite{simonyan2014very} on ImageNet. Fig.~\ref{fig:sensitivity}(a) plots the inferred per-tensor sensitivity $c_l$ and the corresponding optimal bits/dim. GACT\xspace assigns more bits to more sensitive layers. The context tensor saved by the cross-entropy loss operator is most sensitive. A small amount of compression leads to a huge gradient variance. This makes sense since the loss is the first operator to back-propagate through, where the error accumulates. Therefore, GACT\xspace assigns 32 bits/dim. for the tensors in the classification head. With the adaptive algorithm, GACT\xspace with an average of 4 bits/dim. achieves smaller gradient variance than uniformly assigning 8 bits/dim. for all the tensors, as shown in Fig.~\ref{fig:sensitivity}(b). Finally, Fig.~\ref{fig:sensitivity}(c) shows that the sensitivity $c_l(h; \theta_t)$ remains stable during training.Therefore, periodically updating $c_l$ at a large interval is reasonable, and this introduces negligible impact on training speed.
\subsection{Optimization level}
\begin{table}[t]
\centering
\caption{For classification, we train VGG11~\cite{simonyan2014very}, ResNet-50~\cite{he2016deep}, and Swin-Tiny~\cite{liu2021Swin} on ImageNet~\cite{imagenet_cvpr09}.
For object detection, we train RetinaNet~\cite{lin2017focal}, Faster R-CNN~\cite{ren2015faster} on Coco~\cite{lin2014microsoft}. We report accuracy on validation sets (Div. indicates diverge) and the compression rate of context tensors (numbers in brackets) for both tasks. }
\resizebox{\mywidth}{!}{
\begin{tabular}{c|c|c|c|c}
\toprule
Task & Model & FP32 & \shortstack{GACT\xspace \\Adapt 4bit (L1)} & \shortstack{GACT\xspace \\Adapt 2bit} \\
\midrule
\multirow{2}{3em}{Cls.}
& VGG11 & 68.75 & 68.77 (2.84$\times$) & 68.49 (3.34$\times$) \\
& ResNet-50 & 77.29 & 76.96 (6.69$\times$) & 76.13 (11.39$\times$)\\
& Swin-tiny & 81.18 & 80.92 (7.44$\times$) & 77.91 (13.73$\times$)\\
\hline
\multirow{2}{3em}{Det.}
& Faster RCNN & 37.4 & 37.0 (4.86$\times$) & 36.1 (6.81 $\times$)\\
& RetinaNet & 36.5 & 36.3 (3.11$\times$) & Div. \\
\bottomrule
\end{tabular}}
\label{tab:vision_acc}
\end{table}
We apply GACT\xspace on various computer vision tasks, including image classification and object detection, as shown in Fig.~\ref{tab:vision_acc}. We also vary the average bits used by the adaptive algorithm to explore the memory accuracy trade-off. On both tasks, GACT\xspace L1 achieves comparable ($<0.5\%$ accuracy drop) or even better results than the full precision training, while reducing activation memory by up to 7.44$\times$. Here, we list the accuracy of FP32 as the strongest accuracy baseline. For other lossy methods we consider in Sec.~\ref{sec:mem-speed}, the accuracy is no better than FP32, and we list their training accuracy in Appendix~\ref{sec:baseline-acc}.
Notice that here GACT\xspace Adapt 2bit diverges on the detection task. This is because, as
shown in Sec.\ref{sec:convergence}, although ACT has unbiased gradients, the compression
error and learning rate affect the convergence. When using
2 bit, the compression error is large and the learning rate has to
be reduced accordingly to guarantee convergence. However,
we do not want to slow training by decreasing the learning
rate. All experiments are run with the same learning rate as
the full precision. Therefore when compression error is large,
the training diverges.
Furthermore, we observe that the memory reduction varies among networks because GACT\xspace does not quantize intermediate states, and the size of intermediate states differs between networks. For example, in VGG11, when the batch size is 128, GACT\xspace reduces the saved tensor size from 5889MB to 2080MB, among which 78\% (1494MB) is used to store the intermediate index for the max-pooling layer that is not quantized by GACT\xspace.
Next, we demonstrate the flexibility of GACT\xspace by applying it to a wider variety of natural language processing (NLP) and graph machine learning (Graph) tasks. We run multiple seeds for each task, and we report the mean$\pm$std of accuracy/F1 across runs as shown in Tab.~\ref{tab:nlp-graph-acc-cpr}. We include the detailed experimental setup in Appendix \ref{sec:setup}. For both NLP and Graph tasks, GACT\xspace L1 achieves comparable training results with FP32, introducing less than 0.3\% accuracy/F1-score drop, while reducing activation memory by 4.18$\times$ to 7.93$\times$. Moreover, the results are stable across runs, introducing similar accuracy variance as FP32.
We also show the training results of fix-4bit quantization, where all tensors are uniformly quantized with 4 bits. As shown in Tab.~\ref{tab:nlp-graph-acc-cpr}, fix-4 bit quantization causes significant accuracy/F1-score loss on various graph models. For Bert-large, fixed-4 bit quantization works fine because all the context tensors have similar sensitivity.
On the other hand, GACT\xspace L1, using a similar amount of memory as always quantizing each layer to 4 bits, still performs on par with full precision training on all the models.
This shows the necessity of using adaptive algorithms to assign bits based on tensor sensitivity for stabilized training.
Moreover, for Bert-large and three graph models (GCN/GAT/GCNII), GACT\xspace converges and gives lossless results with 3 bits. Remarkably, across all the graph models, training with 2-bit GACT\xspace causes little accuracy loss ($<1\%$).
This shows the robustness of our adaptive algorithm.
\begin{table*}[t]
\ifisarxiv
\else
\vspace{-1em}
\fi
\caption{Accuracy and activation compression rate for NLP and Graph tasks. Accuracy that drops $>$ 1\% is in italic font.}
\label{tab:nlp-graph-acc-cpr}
\centering
\resizebox{0.96\linewidth}{!}{
\begin{tabular}{ c|c|c|c|c|c|c }
\toprule
Model & Dataset & FP32 & Fix 4bit & GACT\xspace Adapt 4bit (L1) & GACT\xspace Adapt 3bit & GACT\xspace Adapt 2bit \\
\midrule
\multirow{4}{3em}{GCN}
& Flickr & 51.17 $\pm$ 0.19 & 50.93 $\pm$ 0.16 (7.56$\times$) & 51.08 $\pm$ 0.18 (7.93$\times$) & 51.14 $\pm$ 0.18 (11.34$\times$) & 51.20 $\pm$ 0.18 (17.56$\times$)\\
& Reddit & 95.33 $\pm$ 0.07 & 94.42 $\pm$ 0.11 (7.55$\times$) & 95.32 $\pm$ 0.07 (7.90$\times$) & 95.31 $\pm$ 0.07 (9.70$\times$) & 95.34 $\pm$ 0.06 (13.68$\times$)\\
& Yelp & 39.86 $\pm$ 0.94 & 39.85 $\pm$ 1.22 (5.94$\times$) & 40.06 $\pm$ 0.74 (6.42$\times$) & 40.21 $\pm$ 0.82 (7.46$\times$) & 39.89 $\pm$ 1.45 (9.00$\times$)\\
& ogbn-arxiv & 71.51 $\pm$ 0.65 & \textit{68.61 $\pm$ 0.77 (7.54$\times$)} & 71.35 $\pm$ 0.36 (8.09$\times$) & 70.82 $\pm$ 0.95 (10.45$\times$) & 70.87 $\pm$ 0.66 (13.75$\times$)\\
\multirow{4}{3em}{GAT}
& Flickr & 52.40 $\pm$ 0.28 & \textit{35.24 $\pm$ 11.90 (4.23$\times$)} & 52.26 $\pm$ 0.31 (4.34$\times$) & 51.68 $\pm$ 1.13 (5.04$\times$) & 51.62 $\pm$ 1.19 (5.46$\times$)\\
& Reddit & 95.95 $\pm$ 0.06 & \textit{59.37 $\pm$ 11.48 (4.12$\times$)} & 96.02 $\pm$ 0.09 (4.29$\times$) & 95.96 $\pm$ 0.06 (4.64$\times$) & 95.82 $\pm$ 0.06 (5.24$\times$)\\
& Yelp & 52.41 $\pm$ 0.69 & \textit{36.09 $\pm$ 13.70 (4.04$\times$)} & 52.18 $\pm$ 0.38 (4.18$\times$) & 51.63 $\pm$ 0.83 (4.53$\times$) & 51.15 $\pm$ 0.53 (5.24$\times$)\\
& ogbn-arxiv & 71.68 $\pm$ 0.54 & \textit{54.64 $\pm$ 5.62 (5.04$\times$)} & 71.80 $\pm$ 0.47 (5.09$\times$) & 71.47 $\pm$ 0.50 (6.14$\times$) & 71.21 $\pm$ 0.68 (6.98$\times$)\\
\multirow{4}{3em}{GCNII}
& Flickr & 52.37 $\pm$ 0.16 & 52.28 $\pm$ 0.16 (4.84$\times$) & 52.31 $\pm$ 0.16 (4.91$\times$) & 52.36 $\pm$ 0.16 (5.54$\times$) & 52.23 $\pm$ 0.15 (6.44$\times$)\\
& Reddit & 96.32 $\pm$ 0.24 & \textit{86.50 $\pm$ 1.08 (4.51$\times$)} & 96.11 $\pm$ 0.22 (4.52$\times$) & 96.01 $\pm$ 0.33 (5.16$\times$) & 95.54 $\pm$ 0.29 (5.92$\times$)\\
& Yelp & 62.33 $\pm$ 0.20 & 62.21 $\pm$ 0.22 (5.26$\times$) & 62.28 $\pm$ 0.26 (5.34$\times$) & 62.53 $\pm$ 0.36 (6.29$\times$) & 62.33 $\pm$ 0.37 (7.28$\times$)\\
& ogbn-arxiv & 72.52 $\pm$ 0.12 & \textit{44.57 $\pm$ 5.01 (6.54$\times$)} & 72.28 $\pm$ 0.35 (6.74$\times$) & 72.22 $\pm$ 0.28 (7.98$\times$) & 71.74 $\pm$ 0.26 (10.24$\times$)\\
\hline
\multirow{4}{3em}{Bert-large}
& MNLI & 86.74 $\pm$ 0.24 & 85.98 $\pm$ 0.16 (7.55$\times$) & 86.61 $\pm$ 0.11 (7.38$\times$) & 86.68 $\pm$ 0.08 (9.13$\times$) & \textit{ 84.24 $\pm$ 0.74 (12.87$\times$)}\\
& SST-2 & 93.69 $\pm$ 0.30 & 93.46 $\pm$ 0.23 (7.55$\times$) & 93.54 $\pm$ 0.52 (7.30$\times$) & 93.20 $\pm$ 0.37 (9.05$\times$) & \textit{ 91.90 $\pm$ 1.04 (12.91$\times$)} \\
& MRPC & 88.20 $\pm$ 0.02 & 87.36 $\pm$ 0.19 (7.55$\times$) & 87.90 $\pm$ 0.10 (7.40$\times$) & 87.69 $\pm$ 0.07 (9.19$\times$) & \textit{ 82.54 $\pm$ 0.38 (12.91$\times$)}\\
& QNLI & 92.29 $\pm$ 0.14 & 92.34 $\pm$ 0.07 (7.55$\times$) & 92.44 $\pm$ 0.07 (7.42$\times$) & 92.43 $\pm$ 0.31 (9.19$\times$) & \textit{90.74 $\pm$ 0.13 (12.95$\times$)}\\
\bottomrule
\end{tabular}
}
\ifisarxiv
\else
\vspace{-1em}
\fi
\end{table*}
\subsection{Memory Saving and Computational Overhead}
\label{sec:mem-speed}
\iffalse
\begin{figure}[t]
\centering
\subfigure{
\includegraphics[scale=0.74]{figures/mem_speed_resnet50.pdf}
}
\subfigure{
\includegraphics[scale=0.74]{figures/mem_speed_bert-large-cased.pdf}
}
\subfigure{
\includegraphics[scale=0.74]{figures/mem_speed_bert-base-cased.pdf}
}
\subfigure{
\includegraphics[scale=0.74]{figures/mem_speed_swin_tiny.pdf}
}
\caption{Training throughput vs batch size. Red cross mark means out-of-memory. The shaded yellow region denotes the possible batch sizes with full precision training given the memory budget. CKPT: Gradient checkpointing, L0/1/1: Different GACT\xspace optimization levels, CB1: Combine GACT\xspace with gradient checkpointing, CB2: Combine GACT\xspace with gradient checkpointing and efficient self-attention, ZeroOff: ZeRO-Offload.}
\label{fig:mem_speed}
\end{figure}
\fi
\begin{figure}[t]
\centering
\begingroup
\setlength{\tabcolsep}{0pt}
\renewcommand{\arraystretch}{1}
\scriptsize
\begin{tabular}{m{0.6cm}m{9cm}}
\begin{minipage}{\linewidth}(a)\end{minipage} &\includegraphics[width=.85\linewidth]{figures/mem_speed_resnet50.pdf}\\
\begin{minipage}{\linewidth}(b)\end{minipage}
&\includegraphics[width=.85\linewidth]{figures/mem_speed_bert-large-cased.pdf}\\
\begin{minipage}{\linewidth}(c)\end{minipage}
&\includegraphics[width=.85\linewidth]{figures/mem_speed_swin_tiny.pdf}\\
\end{tabular}
\endgroup
\vspace{-1.5em}
\caption{Training throughput vs batch size. Red cross mark means out-of-memory. The shaded yellow region denotes the batch sizes with full precision training given the memory budget. CKPT: Gradient checkpointing, ZeroOff: ZeRO-Offload.
}
\label{fig:throughput_batch_size}
\end{figure}
\textbf{Settings and baselines.}
We implement the benchmark with PyTorch 1.10 and measure the memory saving and overhead of GACT\xspace on an AWS g4dn.4xlarge instance, which has a 16GB NVIDIA T4 GPU and 64GB CPU memory.
On ResNet-50, we compare with ActNN~\cite{chen2021actnn}, a dedicated quantization framework for convolutional NNs, and DTR~\cite{kirisame2020dynamic}, a state-of-the-art rematerialization method for dynamic graphs. ``swap'' is a simple swapping strategy that swaps all activations to the CPU.
For Bert-large, we also show the results on Mesa ~\cite{pan2021mesa}, a memory-saving resource-efficient training framework for transformers, and ZeRO-Offload~\cite{ren2021zero}, a highly optimized system for training large-scale language models. Gradient checkpointing uses the default checkpointing policy provided by the transformer library~\cite{wolf-etal-2020-transformers}, where only the input to each transformer block is saved before the backward pass.
On Swin-tiny, we only include Mesa and swap because other baselines lack the support for this network.
\textbf{Results.}
We compare the training throughput of GACT\xspace against other memory saving systems in Fig.~\ref{fig:throughput_batch_size}.
On ResNet-50, GACT\xspace achieves similar throughput as ActNN (ActNN optimization L5 is not listed because it optimizes PyTorch memory allocation, which is unrelated to quantization and can also be applied to GACT\xspace), but ActNN enables training with a larger batch size. This is expected because ActNN implements efficient, customized layers for different operators in convolutional NNs.
For Bert-large, Zero-offload fails quickly because it only offloads optimizer states that occupy a small portion of total memory to CPU. GACT\xspace L1 outperforms Mesa because Mesa only compresses tensors to 8 bit. When the batch is bigger, the activation size of each segment becomes the memory bottleneck and prevents gradient checkpointing from increasing the batch size. Moreover, combining GACT\xspace with gradient checkpointing and efficient self-attention further reduces the peak memory, increasing the batch size by up to 24.7$\times$. Meanwhile, it introduces a small throughput overhead compared with the original gradient checkpointing. Across all the network architectures, GACT\xspace enables training with a 4.2$\times$ to 24.9$\times$ larger batch size under the same memory budget.
\textbf{Network scaling.}
With GACT\xspace, we can construct larger models or train with a higher image resolution. Tab.~\ref{tab:scale} compares the largest model we can train against full precision. With the same batch size and memory budget, GACT\xspace can scale a ResNet-152 to 7.0$\times$ deeper, 3.6$\times$ wider or 3.0$\times$ higher resolution. Similarly, Bert-large can be scaled to 2.0$\times$ deeper or 1.6$\times$ wider. In GCN, GACT\xspace enables training 10.0$\times$ deeper and 1.7$\times$ wider network. Overall, GACT\xspace maintains 75\% - 136\% original training throughput.
\begin{table}[t]
\caption{Largest models GACT\xspace can train with 16G GPU memory. In ResNet (batch size=64), D (depth): number of layers, W (width): base width of the bottleneck block, R (resolution): width and height of input images. In Bert-large (batch size=16) and GCN, D (depth): number of transformer/gcn blocks, W (width): hidden size.}
\label{tab:scale}
\centering
\resizebox{\mywidth}{!}{%
\begin{tabular}{cc|ccc|ccc}
\toprule
&\multirow{2}{3em}{Dim} & \multicolumn{3}{|c|}{Maximum Value}
& \multicolumn{3}{|c}{ Throughput (TFLOPS)} \\
& & FP & L1 & L2 & FP & L1 & L2 \\
\midrule
\multirow{3}{2em}{ResNet-152}
& D & 160 & 460 & 1124 & 0.43 & 0.47 & 0.41 \\
& W & 88 & 304 & 320 & 0.44 & 0.89 & 0.6 \\
& R & 232 & 548 & 716 & 0.41 & 0.39 & 0.44 \\
\hline
\multirow{2}{2em}{Bert-large}
& D & 32 & 56 & 64 & 0.67 & 0.56 & 0.53 \\
& W & 1280 & 1488 & 2032 & 0.68 & 0.61 & 0.60 \\
\hline
\multirow{2}{2em}{GCN}
& D & 24 & 152 & 240 & 0.20 & 0.14 & 0.15 \\
& W & 2464 & 3948 & 4244 & 0.36 & 0.38 & 0.40 \\
\bottomrule
\end{tabular}%
}
\end{table}
\vspace{-1em}
\subsection{Other Optimizations}
\label{sec:swap}
\begin{table}[t]
\caption{Swap and prefetch speed/memory on Bert-large.}
\label{tab:swap}
\centering
\resizebox{\mywidth}{!}{%
\begin{tabular}{c|c|c|c}
\toprule
Algorithm & \shortstack{Speed\\(sequence/s)} & \shortstack{Peak Mem.\\ (MB)} & \shortstack{Total Mem.\\ (MB)} \\
\midrule
FP32 & 16.41 & 9573 & 9527 \\
FP32 + swap & 6.02 & 5215 & 5093\\
GACT\xspace swap & 12.95 & 5426 & 5325 \\
GACT\xspace swap + prefetch & 14.02 & 5426 & 5324 \\
\bottomrule
\end{tabular}%
}
\end{table}
We evaluate the idea of combining GACT\xspace with swapping on Bert-large-cased. As shown in Tab.~\ref{tab:swap}, swapping compressed tensors is faster than swapping the original ones because communication between CPU and GPU is more time-consuming than computation. Combining GACT\xspace with swapping increases training speed by up to 2.3$\times$. Notice here that the peak memory use of ``GACT\xspace swap'' is slightly higher than ``FP32 + swap'' because GACT\xspace does not quantize and swap intermediate states such as running mean/var of BatchNorm layer. Moreover, prefetch increases the speed by about 7\% with negligible memory overhead.
\begin{table}
\caption{Accuracy of Bert-large-cased on SST-2 and QNLI datasets}
\label{tab:acc-ckpt}
\centering
\resizebox{0.9\mywidth}{!}{%
\begin{tabular}{c c c|c c c}
\toprule
Algorithm & SST-2 & QNLI & Algorithm & SST-2 & QNLI\\
\midrule
FP32 & 93.58 & 92.42 & CB1 & 93.81 & 92.26\\
\bottomrule
\end{tabular}%
}
\end{table}
\begin{table}
\caption{Memory use of different algorithms on Bert-large. AM1: Activation size before backward, AM2: Activation size after reforwading the first transformer block. When batch size = 288, L0 runs out of memory, and therefore it is not listed below. }
\label{tab:mem-ckpt}
\centering
\resizebox{0.8\mywidth}{!}{%
\begin{tabular}{c|c|c|c|c}
\toprule
Batch Size & Algorithm & \shortstack{AM1\\(MB)} & \shortstack{AM2\\(MB)} & \shortstack{Peak Mem.\\(MB)} \\
\midrule
\multirow{4}{3em}{16} & L0 & 4434 & - & 9573 \\
& FP32 + CKPT & 210 & 394 & 5541 \\
& CB1 & 37 & 99 & 5286 \\
& CB2 & 31 & 79 & 5269 \\
\hline
\multirow{4}{3em}{288}
& FP32 + CKPT & 3783 & 7092 & 12885 \\
& CB1 & 515 & 1497 & 8251 \\
& CB2 & 486 & 1307 & 8102 \\
\bottomrule
\end{tabular}%
}
\end{table}
We next demonstrate combining GACT\xspace with gradient checkpointing (CB1). Gradient checkpointing is performed at the beginning of each transformer block, thus avoiding saving tensors generated within the block. We then apply GACT\xspace with gradient checkpointing, where the saved tensors are quantized with 4 bits. As shown in Tab.~\ref{tab:acc-ckpt}, the accuracy is unaffected. We also compare the activation memory and peak memory of CB1 and CB2 in Tab.~\ref{tab:mem-ckpt}. AM2 denotes the peak activation memory, which is the size of saved tensors after reforwarding the first transformer block. When batch size = 288, compared with gradient checkpointing on full precision (FP32), CB1 and CB2 reduce the peak activation size by 4.7$\times$ and 5.4$\times$ respectively.
\section{Proof of Theorems}
\subsection{Theorem 1: Convergence of ACT}
Assume that:\\
\noindent\textbf{A1.} $\mathcal L(\theta)$ is a continuous differentiable, $\nabla\mathcal L(\theta)$ is $\beta$-Lipschitz continuous.
.\\
\noindent\textbf{A2.} $\mathcal L(\theta)$ is bounded below by $\mathcal L_*$.\\
\noindent\textbf{A3.} $g(h; \theta)$ is differentiable w.r.t. $h$ and $\exists b>0$, s.t. $\forall \theta, \mathbb E\norm{g(Q(h(x, \theta)); \theta)-\hat g(h(x, \theta); \theta)}\le b$.\\
\noindent\textbf{A4.} $\exists \sigma^2>0$, s.t., $\forall\theta$, $\Var{\hat g(h(x, \theta)}\le \sigma^2$. \\
Then, for all $\eta < \frac{1}{2\beta}$, if we run ACT defined as Eq.~(\ref{eqn:ac}) for $T$ iterations, then we have
\begin{align*}
\min_{t=0, \dots, T-1}\E{\norm{\nabla \mathcal L(\theta_t)}^2}
\le \frac{4(\mathcal L(\theta_0)-\mathcal L_*)}{\eta T}+3b^2+ \eta\beta\sigma^2
\end{align*}
\begin{proof}
Denote $m:=\nabla_\theta \mathcal L(\theta_t)$, $\epsilon:=\hat g(h(x, \theta_t); \theta_t)-m$, $d:=g(Q(h(x;\theta_t));\theta_t)-\hat g(h(x, \theta_t); \theta_t)$.
Then, by A3 and A4, we have
\begin{align}
\E{\epsilon}&=
\E{g(h(x, \theta_t); \theta_t) - \nabla_\theta \mathcal L(\theta_t)}+
\E{\iprod{J(x, \theta_t),
\Delta Q(h(x, \theta_t))} }\nonumber\\
&=\iprod{J(x, \theta_t),
\E{\Delta Q(h(x, \theta_t))}}=0. \label{eqn:epsilon}\\
\E{\norm{\epsilon}^2}&=\norm{\E{\epsilon}}^2
+\Var{\epsilon}=\Var{\hat g(h(x, \theta_t); \theta_t)}\le \sigma^2.\label{eqn:epsilon-sqr}\\
\E{\norm{d}}&\le b\label{eqn:d}.
\end{align}
By the definitions, the ACT dynamics can be written as
\begin{align*}
\theta_{t+1}\leftarrow \theta_t - \eta(m+d+\epsilon).
\end{align*}
By A1, we have
\begin{align}
\mathcal L(\theta_{t+1})\le \mathcal L(\theta_t) - \eta \iprod{m, m+d+\epsilon} + \frac{\beta\eta^2}{2}\norm{m+d+\epsilon}^2.\label{eqn:smoothness}
\end{align}
By Eq.s~(\ref{eqn:epsilon},\ref{eqn:d})
\begin{align}
\E{\iprod{m, m+d+\epsilon}}\ge\norm{m}^2 - \norm{m}\norm{d}+ \iprod{m, \E{\epsilon}}\ge\norm{m}^2-\norm{m}b
.\label{eqn:first-order-ineq}
\end{align}
By Eq.s~(\ref{eqn:epsilon},\ref{eqn:epsilon-sqr},\ref{eqn:d}), and $\norm{x+y}^2\le 2\norm{x}^2+2\norm{y}^2$,
\begin{align}
\E{\norm{m+d+\epsilon}^2}=\E{\norm{m+d}^2}+\Var{\epsilon}\le 2\E{\norm{m}}^2+2\E{\norm{d}}^2+\Var{\epsilon}=2\E{\norm{m}}^2+2b^2+\sigma^2.\label{eqn:second-order}
\end{align}
Taking expectation on both sides of Eq.~(\ref{eqn:smoothness}), plug in Eq.s~(\ref{eqn:first-order-ineq}, \ref{eqn:second-order}), and use $\eta<\frac{1}{2\beta}$, we have
\begin{align*}
\E{\mathcal L(\theta_{t+1})}\le& \mathcal L(\theta_t) - \eta (\norm{m}^2-\norm{m}b) + \frac{\beta\eta^2}{2}(2\E{\norm{m}}^2+2b^2+\sigma^2).\\
=&\mathcal L(\theta_t) - (\eta-\beta\eta^2)\norm{m}^2
+\eta \norm{m}b+\frac{\beta\eta^2}{2}(2b^2+\sigma^2)\\
=&\mathcal L(\theta_t) - \frac{\eta}{2}\norm{m}^2
+\eta \norm{m}b+\frac{\beta\eta^2}{2}(2b^2+\sigma^2).
\end{align*}
Completing the squares,
\begin{align*}
\E{\mathcal L(\theta_{t+1})}\le
\mathcal L(\theta_t) - \frac{\eta}{2}(\norm{m}-b)^2
+\frac{\beta\eta^2}{2}(2b^2+\sigma^2).
\end{align*}
Take expectation on both sides and sum up for $t={0, \dots, T-1}$,
\begin{align*}
\E{\mathcal L(\theta_T)} - \mathcal L(\theta_0) \le -\frac{\eta}{2}\sum_{t=0}^{T-1}\mathbb E\left(\norm{\nabla \mathcal L(\theta_t)}-b\right)^2 +\frac{\beta\eta^2T}{2}(2b^2+\sigma^2).
\end{align*}
Reorganize the terms,
\begin{align*}
\Es{t}{\mathbb E\left(\norm{\nabla \mathcal L(\theta_t)}-b\right)^2 }
\le \frac{2(\mathcal L(\theta_0)-\mathcal L(\theta_T))}{\eta T}+\eta\beta(2b^2+\sigma^2).
\end{align*}
Let $t_* = \operatornamewithlimits{argmin}_t \E{\norm{\nabla \mathcal L(\theta_t)}}$, and use A1, we have
\begin{align*}
\mathbb E\left(\norm{\nabla\mathcal L(\theta_{t_*})}-b\right)^2\le
\frac{2(\mathcal L(\theta_0)-\mathcal L_*)}{\eta T}+\eta\beta(2b^2+\sigma^2).
\end{align*}
Use $(a+b)^2\le 2a^2 + 2b^2$, we have
\begin{align*}
\E{\norm{\nabla\mathcal L(\theta_{t_*})}^2}\le
\frac{4(\mathcal L(\theta_0)-\mathcal L_*)}{\eta T}+(2\beta\eta+2)b^2+ \eta\beta\sigma^2
\le \frac{4(\mathcal L(\theta_0)-\mathcal L_*)}{\eta T}+3b^2+ \eta\beta\sigma^2.
\end{align*}
\end{proof}
\subsection{Proposition 1: The Linearization Error}
\begin{proof}
Consider the gradient function $g(Q(h(x; \theta); \theta))$, whose output is a $P$-dimensional vector.
Since it is twice differentiable, we construct the Taylor's expansion at $h(x; \theta)$ with Lagrange remainder:
\begin{align*}
\exists H_1, \dots, H_P, \mbox{s.t., }\forall i,~~
g_i(Q(h(x; \theta)); \theta) =
g_i(h(x, \theta); \theta) +
J_i(x, \theta)\Delta h(x, \theta)
+\Delta h(x, \theta)^\top H_i \Delta h(x, \theta),
\end{align*}
where $J_i(h(x; \theta), \theta):=\frac{\partial g_i(h(x; \theta); \theta)}{\partial h}$.
By the assumption, there exists $P>0$, such that the linearization error is
\begin{align*}
\norm{g(Q(h(x; \theta)); \theta) - \hat g(h(x; \theta); h(x; \theta), \theta)}_1
=\sum_{i=1}^P \Delta h(x, \theta)^\top H_i \Delta h(x, \theta)
\le \gamma P \norm{\Delta h(x, \theta)}^2.
\end{align*}
Taking expectation,
\begin{align*}
&\E{\norm{g(Q(h(x; \theta)); h(x; \theta), \theta) - \hat g(h(x; \theta); \theta)}_2}\le
\E{\norm{g(Q(h(x; \theta)); \theta) - \hat g(h(x; \theta); h(x; \theta), \theta)}_1}\\
&\le \gamma P \Var{\Delta h(x, \theta)}
=O(\Var{\Delta h(x, \theta)})
\end{align*}
\end{proof}
\subsection{Proposition 2: The Order of the Variance}
\newtheorem{innercustomthm}{Proposition}
\newenvironment{customthm}[1]
{\renewcommand\theinnercustomthm{#1}\innercustomthm}
{\endinnercustomthm}
The following proposition is convenient for isolating the different noise sources.
\begin{customthm}{A}
(Law of Total Variance)
$$\Var{X}=\E{\Varcond{X}{Y}} + \Var{\Econd{X}{Y}}.$$
\end{customthm}
\begin{proof}
By definition
\begin{align*}
\Var{\hat g(h(x; \theta_t); h(x; \theta), \theta_t)}
=\Var{g(h(x, \theta); \theta)} +
\Var{J(h(x; \theta), \theta)\Delta h(x, \theta)},
\end{align*}
where $\Var{g(h(x, \theta); \theta)}$ is the noise introduced by subsampling the data $x$. By law of total variance,
\begin{align*}
\Var{J(h(x; \theta), \theta)\Delta h(x, \theta)}= \mathbb E_{\mathcal X}\left[\Vars{Q}
{J(h(x; \theta); \theta_t)\Delta h(x, \theta)}
\right]+
\underbrace{
\Vars{\mathcal X}{\Es{Q}{J(h(x; \theta); \theta_t)\Delta h(x, \theta)}}}_{=0},
\end{align*}
where
\begin{align*}
\Vars{Q}{J(h(x; \theta); \theta_t)\Delta h(x, \theta)}
=&\Es{Q}{\norm{J(h(x; \theta); \theta_t)\Delta h(x, \theta)}^2}
\le \Es{Q}{\norm{J(h(x; \theta); \theta_t)}^2\norm{\Delta h(x, \theta)}^2}
\\=&
\norm{J(h(x; \theta); \theta_t)}^2
\Es{Q}{\norm{\Delta h(x, \theta)}^2}
= O\left(\Var{\Delta h(x, \theta)}\right).
\end{align*}
\end{proof}
\subsection{Proposition 3: The Structure of the Variance}\label{sec:appendix-var-structure}
Before investigating the structure of $\Vars{Q}
{J(x; \theta_t)\Delta h(x, \theta)}$, let's do some recap:
the parameter $\theta_t$ is a $P$-dimensional vector; the context difference $\Delta h(x, \theta)$ is a $D$-dimensional vector, and $J(x; \theta_t)$ is a $P\times D$ matrix. Recall that $\Delta h(x, \theta)$ is the concatenation of $L$-vectors, $\Delta h^{(l)}(x, \theta)$, and let $J^{(l)}(x, \theta):=\frac{\partial g}{\partial h^{(l)}}g\left((h^{(l)}(x; \theta))_{l=1}^L, \theta\right)$, which is a $P\times D_l$ matrix. Furthermore, let $h^{(l)}_j(x, \theta)$ be the $j$-th dimension, and $J^{(l)}_j(x, \theta)$ be its $j$-th column.
To proceed, we need to make the following assumptions to the compressor $Q(\cdot): \mathbb R^{D}\rightarrow \mathbb R^{D}$:\\
\noindent\textbf{B1: } The compressed result is element-wise uncorrelated. That is, for any $i\ne j$, $\Cov{Q(h)_i}{Q(h)_j}=0$.\\
\noindent\textbf{B2: } For compressing a vector $h$ to $b$ bits, the compression variance of each dimension can be written in the form $\Var{Q(h)_j}\le R_j(h)S(b)$, where $S(\cdot)$ is a known function.
Both assumptions can be achieved by a stochastic rounding~\cite{courbariaux2015binaryconnect} quantizer, where
\begin{align*}
Q(h)_j = \begin{cases}
T_{h, b}^{-1}\left(\ceil{T_{h, b}(h_j)}\right) & \mbox{w.p. } T_{h, b}(h_j)-\floor{T_{h, b}(h_j)} \\
T_{h, b}^{-1}\left(\floor{T_{h, b}(h_j)}\right) & \mbox{otherwise}\\
\end{cases},
\end{align*}
where $T_{h, b}(h_j)=(2^b-1)\frac{h_j - \min_j h}{\max_j h-\min_j h}$. Since each dimension is quantized independently, B1 is met. Moreover,
\begin{align*}
\Var{Q(h)_j}\le \frac{1}{4} \left(\frac{\max_j h-\min_j h}{(h_j - \min_j h)}\right)^2 (2^b-1)^{-2}=R_j(h) S(b),
\end{align*}
where
\begin{align*}
R_j(h) = \frac{1}{4} \left(\frac{\max_j h-\min_j h}{(h_j - \min_j h)}\right)^2,~~~ S(b) = (2^b-1)^{-2}.
\end{align*}
\begin{proof}
By definition,
\begin{align*}
J(h; \theta)\Delta h =
\sum_{l=1}^L \sum_{j=1}^{D_l} J^{(l)}_j(h; \theta_t) \Delta h^{(l)}_j.
\end{align*}
Using Assumption B1, we have
\begin{align*}
\Vars{Q}{J(h; \theta)\Delta h} &=
\Es{Q}{\norm{\sum_{l=1}^L \sum_{j=1}^{D_l} J^{(l)}_j(h; \theta_t) \Delta h^{(l)}_j}^2}\\
&=\sum_{l=1}^L \sum_{j=1}^{D_l} \Es{Q}{\norm{J^{(l)}_j(h; \theta_t) \Delta h^{(l)}_j}^2}.\\
&=\sum_{l=1}^L \sum_{j=1}^{D_l} \norm{J^{(l)}_j(h; \theta_t)}^2 \Vars{Q}{\Delta h^{(l)}_j}
\end{align*}
Using Assumption B2, we have
\begin{align*}
\Vars{Q}{J(h; \theta)\Delta h}
&\le\sum_{l=1}^L \sum_{j=1}^{D_l} \norm{J^{(l)}_j(h; \theta_t)}^2 R_l(h) S(b_l)
=\sum_{l=1}^L c_l(h, \theta) S(b_l),
\end{align*}
where $c_l(\theta, h) :=R_l(h) \norm{ J^{(l)}(h; \theta_t)}^2_F$.
\end{proof}
\section{Training Accuracy of Baselines}
\label{sec:baseline-acc}
For all the baselines we compared in Sec.~\ref{sec:mem-speed}, only ActNN, Mesa, and ZeRO-Offload are lossy methods. All other methods are lossless and have the same training accuracy as FP32. For ResNet-50 on ImageNet, the training accuracy for FP32, GACT\xspace, ActNN L2, and ActNN L3 are 77.3, 77.0, 77.4, and 76.9. For Bert-Large on SST-2, the accuracy for FP32, GACT\xspace, Mesa, and ZeRO-Offload are 93.7, 93.5, 93.8, and 93.3.
For Swin-tiny on ImageNet, the training accuracy for FP32, GACT\xspace, and Mesa are 81.2, 81.0, and 81.3 respectively.
\section{Experiment Setup}
\label{sec:setup}
\subsection{Node classification task on graphs} We conduct experiments on four node classification datasets with standard splits, including Flickr, Reddit, Yelp from GraphSAINT~\cite{zeng2019graphsaint}, and ogbn-arxiv from Open Graph Benchmark (OGB)~\cite{hu2020open}. The four datasets cover extensive downstream applications with different scales.
We use accuracy as the evaluation metric for multi-class classification and micro-F1 for multi-label classification. We run ten seeds (0 to 9) and report the average accuracy across~runs.
We evaluate GACT\xspace on three representative GNN models, including GCN~\cite{kipf2016semi}, GAT~\cite{velivckovic2017graph}, and GCNII~\cite{chen2020simple_gcnii} under the full-batch training setting.
All three models are implemented by CogDL~\cite{cen2021cogdl}, a toolkit for graph neural networks.
\subsection{Text classification task} We select four largest datasets, MNLI, QQP, SST-2, and QNLI, from the GLUE benchmark~\cite{wang2018glue}. The four datasets cover different aspects of natural language understanding, including sentiment classification, natural language inference and paraphrase detection. We use the mainstream transformer implementation~\cite{wolf-etal-2020-transformers} to train Bert-large~\cite{devlin2019bert}. We run three
seeds (42, 43, 44) and report F1 for QQP, accuracy for the others.
\section{Conclusion}
This paper presents GACT\xspace, an ACT framework for generic NN architectures. We prove the convergence of GACT\xspace without prior knowledge about operator type or network architecture by analyzing a linearized approximation of ATC's gradients. With the adaptive algorithm, GACT\xspace achieves negligible accuracy loss on various tasks, reducing activation memory by up to 8.1$\times$ and enabling training with up to 24.7$\times$ batch size compared with full precision training.
\section*{Acknowledgements}
This work was supported by the National Key Research and Development Project of China (No. 2021ZD0110502); NSF of China Project (No. 62106120), by the National Science Foundation through grants IIS-1955488, IIS-2027575, CCF-1723352, ARO W911NF2110339, ONR N00014-21-1-2724, and DOE award DE-SC0016260.
We would also like to acknowledge partial support from DARPA, IARPA, the Sloan Foundation, NSF, and ONR.
Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,032 |
{"url":"http:\/\/tex.stackexchange.com\/questions\/162682\/ps-to-pk-converter-has-stopped-working","text":"# PS- to-PK converter has stopped working [closed]\n\nI compile the latex file using miktex, during the preview in yap it shows PS- to-PK converter has stopped working error. There is no problem during compilation. Please suggest me the correct.\n\n-\n\n## closed as unclear what you're asking by Martin Schr\u00f6der, Svend Tvesk\u00e6g, Peter Jansson, Jubobs, JesseFeb 28 at 12:27\n\nPlease clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it\u2019s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.\n\nDo you really need to preview DVI files? \u2013\u00a0 egreg Feb 27 at 10:35\n@egreg Without DVI how can i create PS or PDF? so DVI preview is importent. \u2013\u00a0 PSK Feb 27 at 10:36\nIf you don't need PS, but only PDF, then DVI is not required, as pdflatex directly outputs PDF. \u2013\u00a0 egreg Feb 27 at 10:42\nNo I need PS. so i dont want the pdflatex \u2013\u00a0 PSK Feb 27 at 10:48\nWith given information noone can help you in the moment. Please add a minimal working example (MWE) that illustrates your problem. It will be much easier for us to reproduce your situation and find out what the issue is when we see compilable code, starting with \\documentclass{...} and ending with \\end{document}. \u2013\u00a0 Speravir Feb 27 at 22:35","date":"2014-10-24 10:36:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8743447661399841, \"perplexity\": 3670.1661497402993}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414119645845.57\/warc\/CC-MAIN-20141024030045-00286-ip-10-16-133-185.ec2.internal.warc.gz\"}"} | null | null |
"use strict";
import React from "react";
import { ContextMenuLayer } from "../../src";
import MenuTypes from "./constants";
const Container = React.createClass({
displayName: "Container",
render() {
return (
<div className="well">
right click to see the menu
</div>
);
}
});
export default ContextMenuLayer(MenuTypes.simple, (props) => {
return {};
})(Container);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,520 |
package org.apache.spark.internal.io.cloud
import java.nio.file.FileAlreadyExistsException
import java.util.EnumSet
import scala.util.control.NonFatal
import org.apache.hadoop.conf.Configuration
import org.apache.hadoop.fs._
import org.apache.spark.internal.Logging
import org.apache.spark.sql.execution.streaming.AbstractFileContextBasedCheckpointFileManager
import org.apache.spark.sql.execution.streaming.CheckpointFileManager.CancellableFSDataOutputStream
class AbortableStreamBasedCheckpointFileManager(path: Path, hadoopConf: Configuration)
extends AbstractFileContextBasedCheckpointFileManager(path, hadoopConf) with Logging {
if (!fc.hasPathCapability(path, CommonPathCapabilities.ABORTABLE_STREAM)) {
throw new UnsupportedFileSystemException("AbortableStreamBasedCheckpointFileManager requires" +
s" an fs (path: $path) with abortable stream support")
}
logInfo(s"Writing atomically to $path based on abortable stream")
class AbortableStreamBasedFSDataOutputStream(
fsDataOutputStream: FSDataOutputStream,
fc: FileContext,
path: Path,
overwriteIfPossible: Boolean) extends CancellableFSDataOutputStream(fsDataOutputStream) {
@volatile private var terminated = false
override def cancel(): Unit = synchronized {
if (terminated) return
try {
fsDataOutputStream.abort()
fsDataOutputStream.close()
} catch {
case NonFatal(e) =>
logWarning(s"Error cancelling write to $path (stream: $fsDataOutputStream)", e)
} finally {
terminated = true
}
}
override def close(): Unit = synchronized {
if (terminated) return
try {
if (!overwriteIfPossible && fc.util().exists(path)) {
fsDataOutputStream.abort()
throw new FileAlreadyExistsException(
s"Failed to close atomic stream $path (stream: " +
s"$fsDataOutputStream) as destination already exists")
}
fsDataOutputStream.close()
} catch {
case NonFatal(e) =>
logWarning(s"Error closing $path (stream: $fsDataOutputStream)", e)
} finally {
terminated = true
}
}
override def toString(): String = {
fsDataOutputStream.toString
}
}
override def createAtomic(
path: Path, overwriteIfPossible: Boolean): CancellableFSDataOutputStream = {
import CreateFlag._
val createFlag = if (overwriteIfPossible) {
EnumSet.of(CREATE, OVERWRITE)
} else {
EnumSet.of(CREATE)
}
new AbortableStreamBasedFSDataOutputStream(
fc.create(path, createFlag), fc, path, overwriteIfPossible)
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,860 |
{"url":"https:\/\/sv.overleaf.com\/latex\/examples\/luatex-parsing-typeset-paragraphs\/mzbpwxtjshsj","text":"SammanfattningThis example is a LuaTeX-based plain TeX project which implements a LuaTeX callback function to show the components of a typeset paragraph. It is designed to accompany a blog post and, for simplicity, it is not a full-blown \u201cparser\u201d\u2014for example, it is not recursive and ignores a number of node types. However, it offers a useful starting point for anyone wishing to explore callbacks in more detail.","date":"2022-01-21 07:31:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 1, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6327052712440491, \"perplexity\": 852.3506292972598}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320302740.94\/warc\/CC-MAIN-20220121071203-20220121101203-00210.warc.gz\"}"} | null | null |
Mogangina chubsugulicus är en insektsart som beskrevs av Dlabola 1967. Mogangina chubsugulicus ingår i släktet Mogangina och familjen dvärgstritar. Inga underarter finns listade i Catalogue of Life.
Källor
Dvärgstritar
chubsugulicus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,447 |
\section{Introduction}
\label{S:intro}
Although the solar corona is dominated by the Sun's magnetic field, accurately determining its strength continues to be a difficult task. Estimates may be obtained in a small number of long-wavelength (forbidden) emission lines by measuring their Zeeman splitting \citep[\emph{cf}.][]{Lin:2004ab} or by using the Hanle effect \citep[\emph{cf}.][]{Raouafi:2002ab}, but these are generally obtained near active regions where the magnetic-field strength is high and very strong lines in the near infrared may be used \citep[such as, \emph{e.g.,} \corr{\ion{Fe}{13}}~$\lambda$10\,747;][]{Lin:2000ab}. The strength of the coronal magnetic field can also be derived from the gyro-resonance emission in radio wavelengths \citep[\emph{e.g.,}][]{White:1997ab}. However, the optically thin emission lines and weak magnetic-field strength above quiet-Sun regions mean that these techniques are typically not suited to measuring magnetic field strength there. An alternative approach is to infer the coronal magnetic-field strength and other plasma parameters by examining how the properties of waves change with propagation: a technique called \emph{coronal seismology} \citep{Uchida:1970ab,Roberts:1984ab}.
From their initial discovery in images from the \corr{\emph{Extreme ultraviolet Imaging Telescope}} \citep[EIT:][]{Delaboudiniere:1995ab} \corr{onboard the \emph{SOlar and Heliospheric Observatory} (SOHO) spacecraft}, large-scale coronal disturbances \citep[commonly called ``EIT waves'':][]{Dere:1997ab,Moses:1997ab,Thompson:1998ab} have been suggested as possible probes for studying the plasma parameters of the low corona \citep[\emph{e.g.,}][]{Ballai:2003ab,Ballai:2004ab}. These pulses are quite fast, with typical velocities of \corr{$\approx$} 200\,--\,400~km~s$^{-1}$ \corr{measured using data from \corr{SOHO}/EIT} \citep{Thompson:2009ab} and have been observed to exhibit deceleration \citep{Long:2008ab,Warmuth:2004a} and pulse broadening \citep{Long:2011a,Muhr:2011ab} with propagation: features consistent with the propagation of a fast-mode magnetohydrodynamic (MHD) wave through a randomly structured medium \citep[\emph{e.g.,}][]{Murawski:2001ab}. The pulses appear to expand isotropically through \corr{quiet-Sun regions}, \corr{although they do tend to avoid regions of lower and higher density such as} coronal holes and active regions \citep{Thompson:1999ab}, a property which makes them ideal for determining the nature of the quiet corona.
However, the use of ``EIT waves'' to probe the quiet corona is predicated on the interpretation of these disturbances as MHD waves, which is not a simple issue. There are currently two competing physical interpretations for these disturbances. The first uses MHD wave theory to explain the phenomenon \citep[\emph{e.g.,}][]{Long:2008ab,Veronig:2010ab,Wang:2000ab,Kienreich:2012ab,Patsourakos:2009ab} with observations of wave properties such as reflection and refraction at coronal-hole boundaries \citep{Gopal:2009ab} supporting this approach. An alternative interpretation visualises the pulse as a ``pseudo-wave'' resulting from the restructuring of the global magnetic field during the eruption of a coronal mass ejection \citep[CME: \emph{e.g.,}][]{Delannee:2008ab,Schrijver:2011ab,Attrill:2007ab}. In this scenario, the bright feature observed as the ``EIT wave'' is due to Joule heating or small-scale magnetic reconnection as the erupting CME passes out of the low corona.
An alternative explanation for ``EIT waves'' combines both the wave and ``pseudo-wave'' theories to interpret this phenomenon as consisting of both a fast-mode wave initially driven by the erupting CME and a slower brightening due to reconfiguration of the magnetic field. This form was originally posited by \citet{Chen:2002ab} and has been expanded in simulations performed by \citet{Chen:2005ab,Cohen:2009ab} and \citet{Downs:2011ab,Downs:2012ab}. There has also been some evidence of two distinct fronts in observations \citep[\emph{e.g.,}][]{Zhukov:2004ab,Chen:2011ab}. Recent statistical analysis of a large sample of ``EIT waves'' performed by \citet{Warmuth:2011ab} suggests that there may be three distinct classes of ``EIT wave''. Class~1 pulses are initially fast waves that exhibit pronounced deceleration, Class~2 pulses are waves with moderate and almost constant velocities, while Class~3 pulses exhibit erratic kinematic behaviour and are most likely explained as pseudo-waves.
``EIT waves'' are traditionally identified and analysed using data from imagers such as \corr{SOHO}/EIT \citep{Thompson:1999ab}, the \emph{Transition Region And Coronal Explorer} \citep[TRACE, \emph{e.g.,}][]{Wills-Davey:1999ab}, the Extreme UltraViolet Imager (EUVI) onboard the \emph{Solar Terrestrial Relations Observatory} \citep[STEREO, \emph{e.g.,}][]{Long:2008ab}, and the Atmospheric Imaging Array (AIA) onboard the \emph{Solar Dynamics Observatory} \citep[SDO, \emph{e.g.,}][]{Liu:2010ab}, as these instruments allow easy identification of the disturbance within a large field-of-view. However, a more detailed understanding of these disturbances requires the use of spectroscopic instruments as these allow their true physical nature to be investigated. This approach is hindered by the fact that these instruments generally have a restricted field-of-view, making observations of ``EIT Waves'' rare. Despite this, several events have been observed using the \corr{\emph{Extreme ultraviolet Imaging Spectrometer}} \citep[EIS:][]{Culhane:2007ab} onboard the \emph{Hinode} spacecraft \citep{Kosugi:2007ab}.
As rare cases of ``EIT waves'' being observed by spectroscopic instruments, both of the events discussed here have previously been studied by other authors. The event from 12~June~2010 was analysed by \citet{Chen:2011bc}, who found a significant change in the magnetic topology as the nearly circular pulse passed over an upflow region near a magnetic bipole. This led the authors to suggest that the event may be best explained using the magnetic field-stretching model proposed by \citet{Chen:2005ab}. The 16~February~2011 event was one of a series of eruptions from AR 11158 over the course of several days and the active region was the subject of a specialist \emph{Hinode} Observing Plan (HOP) for studying ``EIT waves''. The eruption has been studied by both \citet{Harra:2011ab} and \citet{Veronig:2011ab} who found clear downward bulk motion towards the chromosphere at the pulse, followed by a later upward motion behind the pulse. The kinematics measured by \corr{\emph{Hinode}}/EIS also matched those using imagers, suggesting that ``EIT waves'' may be best interpreted as MHD waves propagating through the low corona.
In this article, we examine \corr{those} two ``EIT wave'' events observed by both \emph{Hinode}/EIS and \corr{SDO}/AIA. The spectroscopic observations from \corr{\emph{Hinode}}/EIS are used to determine the density of the plasma through which the pulse is propagating, and this information is combined with the kinematics obtained from \corr{SDO}/AIA, allowing the magnetic-field strength of the quiet corona to be estimated and compared with theoretical predictions from multiple extrapolated-field models. The observations of both events studied are outlined in Section~\ref{S:data}, with the analysis of these observations from both \corr{SDO}/AIA and \corr{\emph{Hinode}}/EIS discussed in Section~\ref{S:methods}. The results of this analysis are presented in Section~\ref{S:results} before being compared to extrapolated-field estimates in Section~\ref{S:mag_field}. Finally, some conclusions are outlined in Section~\ref{S:concs}.
\section{Observations and Data Analysis}
\label{S:data}
The events studied here were quite similar. The 12~June~2010 eruption from active region (AR) NOAA~11081 was associated with a GOES M2.0 flare, which began at 00:30~UT and peaked at 00:57~UT, while the eruption on 16~February~2011 from AR NOAA~11158 had an associated M1.6 flare (starting at 14:19~UT, peaking at 14:25~UT). Both events were also associated with Type~\Rmnum{2} radio bursts, while CMEs were also identified using the Coordinated Data Analysis Workshops (CDAW) catalogue (\href{http://cdaw.gsfc.nasa.gov/CME_list/}{http://cdaw.gsfc.nasa.gov/CME\_list/}) for the 12~June~2010 event and by the instrument team for the Cor-1 coronagraph (\href{http://cor1.gsfc.nasa.gov/catalog/}{http://cor1.gsfc.nasa.gov/catalog/}), part of the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) instrument package onboard the STEREO spacecraft for the 16~February~2011 event.
The \corr{\emph{Hinode}}/EIS data for these events were taken from two separate observation programmes, and consequently measured different sets of emission lines. The data for the 12~June~2010 event were taken using EIS study~387, which was designed to study the asymmetry of transition region emission lines. This produced a set of 12 observations of \corr{$\approx$}~five~minute duration taken in a region of the quiet Sun adjacent to the erupting active region (as shown in panels (a) and (b) of Figure~\ref{fig:overview}). The data for the 16~February~2011 event were taken using HOP~180, which was a co-ordinated programme designed to study the plasma properties of ``EIT waves''. This data-set consists of a single sit-and-stare raster observing both the edge of the erupting active region and the adjacent quiet Sun (see panels (c) and (d) of Figure~\ref{fig:overview}) and lasting \corr{$\approx$}~30~minutes.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.9\textwidth,clip=,trim=0 30 0 0]{overview.eps}
\caption{SDO/AIA 193~\AA\ images showing the events from 12~June~2010 (\corr{\emph{t}} = 00:58:30~UT; full-disk in panel (a) and zoomed-in in panel (b)) and 16~February~2011 (\corr{\emph{t}} = 14:25:31~UT; full-disk in panel (c) and zoomed-in in panel (d)). The great-circle arc sectors used to determine the kinematics of the pulses are shown bounded in white, while the \corr{\emph{Hinode}}/EIS slit positions are in black in all panels. \corr{AR~11159, which lay to the North of the erupting AR~11158 for the 16~February~2011 event, is indicated in panel (d).} The arcs in panel (b) are labelled \Rmnum{1}, \Rmnum{2}, and \Rmnum{3} for easier identification throughout the text.\corr{The variation of the zoomed-in regions with time is shown in the running-difference movies~1 and 2 in the electronic supplementary material.}}
\label{fig:overview}
\end{figure*}
Full-disk images (0.6\corr{\hbox{$^{\prime\prime}$}}~pixel$^{-1}$) from the AIA instrument \citep{Lemen:2012ab} onboard the SDO spacecraft \citep{Pesnell:2012ab} were used to determine the kinematics and morphological evolution of both ``EIT waves''. The \corr{SDO}/AIA data used here were taken from the 193~\AA\ passband (which is sensitive to coronal plasma at \corr{$\approx$}~1-2~MK) as it provided the clearest observations of the pulse. The kinematic properties of both pulses were determined using the semi-automated intensity-profile technique utilised by \citet{Long:2011a} and \citet{Long:2011b}. This is outlined in more detail in Section~\ref{SS:methods_aia}.
\section{Methods}
\label{S:methods}
Since our data are from two distinct instruments, each measuring different properties of the observed phenomenon, analysis of the observations from both \corr{\emph{Hinode}}/EIS and \corr{SDO}/AIA was a two-step process. The full-disk images from \corr{SDO}/AIA were primarily used to examine the kinematics of the disturbances, as well as to allow some comparison with the magnetic field extrapolations obtained from the \corr{\emph{Helioseismic and Magnetic Imager}} \citep[HMI:][]{Scherrer:2012ab} onboard \corr{SDO}. The spectra obtained from \corr{\emph{Hinode}}/EIS were used to determine the \corr{number} density of the quiet-Sun region being examined using several density-sensitive line ratios.
\subsection{\corr{SDO}/AIA analysis}
\label{SS:methods_aia}
The kinematics of the two pulses studied were determined using the semi-automated intensity-profile technique promulgated by \citet{Long:2011a} and \citet{Long:2011b}. This technique uses percentage base-difference intensity \citep[PBD:][]{Wills-Davey:1999ab} images to identify the disturbance, with each image derotated to the same pre-event time for each event. A series of 36 arcs of \corr{$\approx$}~10\corr{\hbox{$^{\circ}$}} width radiating from a source point are used to create intensity profiles by averaging the PBD intensity across each arc sector in annuli of increasing radii and 1\corr{\hbox{$^{\circ}$}} width on the spherical surface. \corr{As this algorithm is designed to operate automatically with minimal user input, the source point from which to measure the distance of the pulse is taken as the position of the flare defined by the Heliophysics Event Knowledgebase (\href{https://www.lmsal/hek/isolsearch/isolsearch.html}{https://www.lmsal/hek/isolsearch/isolsearch.html}; note that this is used to ensure consistency between events, but does not imply that the flare is the physical source of the wave).} This creates a set of 36 intensity profiles for each image, with the \corr{mean and standard deviation of the PBD intensity values across the annulus taken as the intensity and associated error for that point on the profile}.
\corr{Once the intensity profiles had been obtained, the intensity peak corresponding to the flare, the propagating pulse, and any associated ``stationary brightenings'' were identified automatically by the algorithm for each arc, with a Gaussian model used to fit the position of the peak intensity of each feature individually. The pulse was then identified through its motion, with the pulse position, height, and width given by the centroid, peak, and full width at half maximum (FWHM) respectively \citep[as ``EIT waves'' display a Gaussian cross-section, see][]{Wills-Davey:1999ab}. The errors associated with each parameter were obtained from the error on the fit to the intensity profile. This approach was used as it operates automatically, thus minimising user bias; there is no user input into the algorithm since the source of the arcs and start time of the analysis were both obtained from the identification of the flare defined by the HEK. In addition, the algorithm is designed to compare features between arc sectors, ensuring their accurate identification.} The pulse position and width were therefore obtained with respect to time for each arc, allowing the kinematics and temporal behaviour of the pulse to be determined. The kinematics of the pulse were derived by fitting the pulse position with time using a quadratic equation of the form
\begin{equation}
r(t) = r_0 + v_0 t + \frac{1}{2}a t^2,
\end{equation}
where $r_0$ is the initial pulse position, $v_0$ is the initial pulse velocity, and $a$ is the acceleration of the pulse. Broadening of the pulse was identified by fitting the variation in pulse width with time using a linear function. \corr{Note that to compare the \corr{SDO}/AIA kinematics for these events directly with the \corr{\emph{Hinode}}/EIS observations, only the three arcs that intersected the \corr{\emph{Hinode}}/EIS slit were used for the 12~June~2010 event, while just one arc was suitable for the 16~February~2011 event. This is shown in Figure~\ref{fig:overview}.}
\begin{figure*}[!t]
\centering
\includegraphics[width=0.95\textwidth,clip=,trim=20 20 0 0,angle = 0]{kinematics.eps}
\caption{Time-distance plots for the events of 12~June~2010 (top) and 16~February~2011 (bottom). The kinematics derived for each of the arcs \Rmnum{1}, \Rmnum{2}, and \Rmnum{3} for the 12~June~2010 and 16~February~2011 ``EIT waves'' are given in the respective legends.}\label{fig:kins}
\end{figure*}
The resulting distance-time plots produced by this analysis are shown in Figure~\ref{fig:kins}. The kinematics derived for each arc were obtained using a residual resampling bootstrapping approach \citep[see, \emph{e.g.},][for more details]{Long:2011a}. This technique was chosen to determine the pulse kinematics as it provides a statistically significant estimate of the errors associated with the derived kinematic values by allowing each parameter to be characterised by a distribution.
The 12~June~2010 pulse was observed by \corr{\emph{Hinode}}/EIS with its slit positioned approximately perpendicularly to the direction of the pulse's propagation (see Figure~\ref{fig:overview}). As a result, it was possible to measure the pulse kinematics where three different arc sectors cross the slit; these measurements are shown in the upper panel of Figure~\ref{fig:kins}. It is clear that while the distance travelled by the pulse in each arc sector is comparable, the resulting pulse kinematics are different for each arc sector. This indicates that the pulse does not propagate isotropically, and \corr{it is therefore of interest to examine whether the differences in kinematics are driven by differences in the local or global plasma conditions, where \corr{\emph{Hinode}}/EIS may be used to study the local variations and \corr{SDO}/AIA may be used to study the variations in global propagation.}
The geometry of the 16~February~2011 event was slightly different, in that the pulse propagation direction was almost parallel to the \corr{\emph{Hinode}}/EIS slit (see Figure~\ref{fig:overview}). Only one arc sector was therefore required to determine the kinematics of the pulse. The resulting kinematics, shown in the bottom panel of Figure~\ref{fig:kins}, indicate a slightly higher initial velocity and much stronger deceleration. This suggests that while the initial driver of the pulse may have been comparable to that of the 12~June~2010 event, the free propagation of the 16~February~2011 pulse was subject to more resistance.
\subsection{\corr{\emph{Hinode}}/EIS analysis}
\label{SS:methods_eis}
Analysis of the \corr{\emph{Hinode}}/EIS observations for the events studied was complicated by the fact that the data were taken from two distinct observing plans that were designed with different scientific goals in mind. Despite this, both sets of observations included density-sensitive line ratios that can be used to estimate the \corr{electron number-}density variation of the low solar corona through which each pulse propagated.
Two density-sensitive line ratios were included in the 12~June~2010 observations: the \corr{\ion{Si}{10}}~$\lambda$258.37/261.04 and \corr{\ion{Fe}{14}}~$\lambda$264.78/274.20 ratios, which are sensitive to \corr{electron number}-densities of \corr{log$_{10}(n_{\textrm{e}}$)~$\approx$}~(8\,--\,10) and \corr{log$_{10}(n_{\textrm{e}}$)~$\approx$}~(9\,--\,11)~cm$^{-3}$ respectively \citep{Young:2007ab}. This combination meant that the coronal density could be determined over a temperature range of \corr{log$_{10}(T)$ $\approx$} (6.1\,--\,6.3)~K \citep{Mazzotta:1998ab}, which covers the peak emission temperature of the 193~\AA\ passband observed by \corr{SDO}/AIA (\corr{log$_{10}(T)$~$\approx$}~6.1~K). The 16~February~2011 event was observed using a more targeted EIS observing plan, and therefore the data include four density-sensitive line ratios. These were the \corr{\ion{Fe}{12}}~$\lambda$186.88/195.12, \corr{\ion{Fe}{13}}~$\lambda$196.54/202.04, \corr{\ion{Fe}{13}~$\lambda$203.82/202.04} and \corr{\ion{Mg}{7}}~$\lambda$278.39/280.75 line ratios, corresponding to a \corr{range of densities} of \corr{log$_{10}(n_{\textrm{e}}$)~$\approx$}~(8\,--\,11)~cm$^{-3}$ and a temperature range of \corr{log$_{10}(T)~\approx$}~(5.8\,--\,6.2)~K.
For both events, the density-sensitive line ratios were obtained first by averaging the measured intensity in \corr{time} to obtain a one-dimensional intensity profile \corr{along the \corr{\emph{Hinode}}/EIS slit}. The \corr{number}-densities were then calculated using the $\sf{eis\_density.pro}$ routine contained within the SolarSoftWare (SSW) software package, producing a one-dimensional density profile along the slit for both events studied. Although multiple line ratios were available for each event, only the \corr{\ion{Si}{10}~$\lambda$258.37/261.04} and \corr{\ion{Fe}{13}~$\lambda$196.54/202.04} line ratios were used in this analysis \corr{for the 12~June~2010 and 16~February~2011 events respectively} as they were most sensitive to \corr{variations in the range log$_{10}(n_{\textrm{e}}$)~$\approx$~(8\,--\,10)~cm$^{-3}$. The variation in density with time is shown in Figure~\ref{fig:dens_imgs} for both events studied.} The resulting density profiles are shown \corr{in the upper panels of} Figures~\ref{fig:eis_20100612} and \ref{fig:eis_20110216} for each observed event.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.85\textwidth,clip=,trim=0 0 0 0,angle = 0]{dens_imgs.eps}
\caption{\corr{Variation in number density with time calculated for the 12~June~2010 event (\ion{Si}{10}~$\lambda$258.37/261.04 ratio; left panel) and 16~February~2011 event (\ion{Fe}{13}~$\lambda$196.54/202.04 ratio; right panel). The sections delineated by the horizontal lines in the left panel correspond to arc sectors (from bottom to top) \Rmnum{1}, \Rmnum{2}, and \Rmnum{3} respectively, with the vertical dashed lines indicating the time in each arc sector at which the pulse passed through the slit.}}
\label{fig:dens_imgs}
\end{figure*}
\section{Results}
\label{S:results}
The derived kinematic parameters of the pulses studied here are shown in Figure~\ref{fig:kins} to be quite high (with initial velocities of $\approx$~\corr{496}~km~s$^{-1}$ and \corr{658}~km~s$^{-1}$ respectively). These values are higher than the typically observed pulse velocities reported by \citet{Thompson:2009ab}, but are consistent with other measurements using data from \corr{SDO}/AIA reported by \corr{\emph{e.g.}, }\citet{Zheng:2012ab} and \citet{Olmedo:2012ab}. In addition, both events exhibited significant deceleration with values of $a = -$\corr{(136 to 635)}~m~s$^{-2}$ and $-$\corr{903}~m~s$^{-2}$ for the 12~June~2010 and 16~February~2012 events respectively. The higher pulse velocity and significant deceleration observed here are consistent with the Class~1 ``EIT wave'' classification proposed by \citet{Warmuth:2011ab}. According to this classification system, ``EIT waves'' exhibiting a high initial velocity \citep[i.e., $>$325~km~s$^{-1}$:][]{Warmuth:2011ab} and a resulting strong deceleration (such that the final pulse velocity is \corr{$\approx$}~200\,--\,300~km~s$^{-1}$) is thought to correspond to fast MHD wave modes, with the result that the events reported here are interpreted as such.
By interpreting these phenomena as fast-mode MHD waves \corr{\citep[\emph{e.g.},][]{Priest:1987}}, it is possible to examine their kinematics using the equation
\begin{equation}\label{eqn:v_fast_mode}
v_{\textrm{fm}} = \sqrt{v_\textrm{A}^2 + c_\textrm{s}^2},
\end{equation}
where the Alfv\'{e}n speed and sound speed are defined as $v_\textrm{A} = B/(4 \pi n m)^{1/2}$ and $c_\textrm{s} = (\gamma k T/m)^{1/2}$ respectively and the propagation of the pulse is approximately perpendicular to the coronal magnetic field. The magnetic-field strength is defined by $B$, $n$ is the particle \corr{number} density, $m$ is the proton mass, $\gamma$ is the adiabatic index (typically $5/3$), $k_\textrm{B}$ is the Boltzmann constant and $T$ refers to the \corr{peak emission temperature of the density sensitive lines used \citep[see Section~\ref{SS:methods_eis} and][]{Mazzotta:1998ab}}. This equation can then be rewritten in terms of the magnetic-field strength $B$ as,
\begin{equation}\label{eqn:b_field}
B = \sqrt{4 \pi n (m v^2_{\textrm{fm}} - \gamma k_\textrm{B} T)}.
\end{equation}
This approach is discussed in more detail by \citet{Long:2011b} and \citet{West:2011ab} and may be used to estimate the magnetic field strength, given the plasma density and pulse velocity.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.9\textwidth,clip=,trim=0 30 0 0,angle = 0]{eis_dens_magfield_20100612.eps}
\caption{Variation in \corr{number} density \corr{(top panel)} and magnetic-field strength \corr{(bottom panel)} with position for the 12~June~2010 event. The sections delineated by the vertical lines correspond to arc sectors (from left to right) \Rmnum{1}, \Rmnum{2}, and \Rmnum{3} respectively \corr{while the error associated with each parameter is indicated by the grey shaded region}. The density variation was determined using the \corr{\ion{Si}{10}~$\lambda$258.37/261.04} density-sensitive line ratio.}\label{fig:eis_20100612}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth,clip=,trim=20 0 0 0,angle = 0]{eis_dens_magfield_20110216.eps}
\caption{Variation in \corr{number} density \corr{(top row)}. Note that both panels are the same in order to calculate magnetic-field strength \corr{(bottom row)} with position for the 16~February~2011 event using a \corr{decelerating pulse velocity (left) and constant pulse velocity (right)}. \corr{The error associated with both the number density and magnetic field strength is indicated by the grey shaded region.} The density variation was determined using the \corr{\ion{Fe}{13}~$\lambda$196.54/202.04} density-sensitive line ratio.}\label{fig:eis_20110216}
\end{figure*}
Equation~(\ref{eqn:b_field}) was used to estimate the quiet coronal magnetic-field strength for both of the events presented here. For the 12~June~2010 event, the distance of the \corr{\emph{Hinode}}/EIS slit from the pulse source was measured for each of the arc sectors studied, and the pulse kinematics were determined for this distance. Resulting velocities of \corr{$\approx$~469}~km~s$^{-1}$, \corr{$\approx$~558}~km~s$^{-1}$, and \corr{$\approx$~400}~km~s$^{-1}$ were estimated for arcs~\Rmnum{1}, \Rmnum{2}, and \Rmnum{3} respectively. These values were combined with the \corr{number} density profile shown in \corr{the upper panel of} Figure~\ref{fig:eis_20100612} to produce the magnetic field strength in the range 2\,--\,6~G, shown in \corr{the lower panel} in the same figure. The sections delineated by the vertical black lines correspond to (from left to right) arc sectors \Rmnum{1}, \Rmnum{2} and \Rmnum{3} respectively.
The slightly different geometry of the 16~February~2011 event meant that the pulse propagated along the \corr{\emph{Hinode}}/EIS slit with a velocity shown in the bottom panel of Figure~\ref{fig:kins}. Two kinematic models for the pulse were examined for this event: a pulse that propagated at a constant velocity of \corr{$\approx$~472}~km~s$^{-1}$, obtained by taking the mean velocity of the pulse across the \corr{\emph{Hinode}}/EIS field-of-view; and a decelerating pulse with a velocity given by\corr{
\begin{equation}
v = 658 - 0.903t\textrm{~km~s$^{-1}$}.
\end{equation} }
These velocity values were combined with the \corr{number} density profile obtained for this event \corr{(top panels of Figure~\ref{fig:eis_20110216})} to produce the magnetic field strength profiles shown in \corr{the bottom panels of} Figure~\ref{fig:eis_20110216}, with the \corr{right panels} showing the results for constant velocity and the \corr{left panels} showing the results for variable velocity. It is clear that while both profiles are similar, the variation in pulse velocity does affect the derived magnetic-field strength. When a constant velocity is assumed, $B$ is in the range 2\,--\,6~G; where deceleration is considered, the range of $B$ increases to 1.5\,--\,10~G, and the difference in results is most easily seen in the upper part of the \corr{\emph{Hinode}}/EIS slit (100\,--\,220\corr{\hbox{$^{\prime\prime}$}}).
These observations suggest \corr{that the magnetic field strength of the quiet corona exhibits some variability with position, ranging here between $\approx$~0\,--\,10~G (within errors)}. Previous estimates of the quiet coronal magnetic-field strength \corr{using ``EIT Waves''} were constrained by the use of lower cadence images from \corr{STEREO}/EUVI or by the use of general estimates of the coronal \corr{number} density rather than precise measurements using \emph{Hinode}/EIS. This was reflected in the resulting general estimates, with values of 0.7$\pm$0.7~G and \corr{$\approx$}~1\,--\,2~G returned by \citet{West:2011ab} and \citet{Long:2011b} respectively. The higher values of magnetic field strength here may reflect the use of measured rather than canonical values of density, in combination with higher-cadence data that better capture the kinematic properties of the pulse.
\section{Magnetic Field Extrapolation}
\label{S:mag_field}
These measurements of the coronal magnetic-field strength invite further investigation and comparison with theoretical models. A detailed examination of the magnetic-field configuration in the vicinity of the \emph{Hinode}/EIS slit was therefore undertaken to try and understand them. Two different methods were used to examine the magnetic field; a potential field source surface (PFSS) model giving a potential-field configuration for the full Sun (discussed in Section~\ref{SS:pfss}), and a potential-field extrapolation in a Cartesian box above the quiet Sun (discussed in Section~\ref{SS:qs_model}). Given the magnetic-field strength estimated using the propagation of the ``EIT wave'' and shown in Figures~\ref{fig:eis_20100612} and \ref{fig:eis_20110216}, the methods below are used to derive the height at which this field strength is most probable.
\subsection{PFSS model}
\label{SS:pfss}
The PFSS model developed by \citet{alt69}, \cite{sch69}, and \citet{sch03} was used to investigate the magnetic field strength associated with the observed ``EIT waves''. This model has the advantage of requiring only the distribution of the radial magnetic field on the photosphere as a boundary condition. Synoptic maps from \corr{SDO}/HMI were used to determine the photospheric magnetic field for each event, using the maps at 00:04 UT for the 12~June~2010 event (\corr{Figure~\ref{fig:b_pfss}(a)}), and at 12:04 UT for the 16~February~2011 event (\corr{Figure~\ref{fig:b_pfss}(c)}). In both cases, the spatial resolution of the maps is \corr{1\hbox{$^{\circ}$}}, while the source surface is located at 2.5~R$_{\odot}$ and assumes that the magnetic field is radial above this height. \corr{The PFSS model provides the magnetic field for each pixel of the domain.}
A spherical-co-ordinate wedge \corr{($\approx23^{\circ}\times26^{\circ}$ for the 12~June~2010 event and $\approx19^{\circ}\times37^{\circ}$ for the 16~February~2011 event)} that included the \emph{Hinode}/EIS slit \corr{was then extracted. This} is shown by the white rectangle \corr{and the zoomed in section of panels (a) and (c) of Figure~\ref{fig:b_pfss}}.
Figure~\ref{fig:b_pfss}(b) shows the variation of the average $B$ field with height for the 12~June~2010 event. The grey area defines the distribution of $B$ delineated by the maximum of $B$ at a given height. The vertical dashed lines indicate the \corr{heights at which $B$ drops below 6~G and 2~G respectively, implying that the height range of this event lies} between 70 and 128~Mm. Similarly for the 16~February~2011 event, Figure~\ref{fig:b_pfss}(d) indicates that the heights at which $B$ is lower than 6~G and 2~G are 72~Mm and 131~Mm respectively.
\begin{figure*}
\centering
\includegraphics[width=1\linewidth,trim=0 25 0 0, clip=]{b_vary_pfss.eps}
\caption{\corr{Synoptic maps from 12~June~2010 (panel (a)) and 16~February~2011 (panel (c))} used as a boundary condition for the PFSS model. The white box \corr{and enlarged section indicate} the area in which the average magnetic field strength is computed. \corr{The corresponding average field strength as a function of height in the corona is shown in panel (b) for the event of 12~June~2010 and panel (d) for the event of 16~February~2011}. The grey area shows the spread of the magnetic-field strength. The vertical dashed lines indicate the height at which a maximum field strength of 6~G and 2~G is achieved (horizontal lines).}
\label{fig:b_pfss}
\end{figure*}
The PFSS extrapolation allows an examination of the variation in the large-scale magnetic field of the Sun. The coarse spatial resolution removes the small-scale magnetic field, and thus the complexity of the quiet Sun as well as part of the complexity of active regions. In the following section, the influence of the quiet-Sun magnetic field on the determination of the height of the feature is investigated.
\subsection{Quiet-Sun model}
\label{SS:qs_model}
The procedure described by \citet{reg08} is adopted here to compute the potential field above quiet-Sun regions. \corr{SDO}/HMI magnetograms are again employed, although in this case magnetograms closest in time to observations of the ``EIT wave'' are used rather than synoptic maps. The initial pixel size of 0.6\corr{\hbox{$^{\prime\prime}$}} for the \corr{SDO}/HMI magnetograms is increased by a factor of 2 for the extrapolation. A small field-of-view encompassing the \emph{Hinode}/EIS slit was extracted from the full-disk observations, with care taken to ensure that there is no contamination from the stronger magnetic field associated with nearby active regions. The field-of-view used for the potential field extrapolation is shown in Figure~\ref{fig:b_local}(a) for the 12~June~2010 event (480\corr{\hbox{$^{\prime\prime}$}}$\times$360\corr{\hbox{$^{\prime\prime}$}}) and Figure~\ref{fig:b_local}(c) for the 16~February~2011 event (330\corr{\hbox{$^{\prime\prime}$}}$\times$300\corr{\hbox{$^{\prime\prime}$}}). The vertical extension \corr{of the computational domain} is approximately 350\corr{\hbox{$^{\prime\prime}$}} (about 250~Mm) for both computations.
\begin{figure*}
\centering
\includegraphics[width=1\linewidth,trim=0 25 0 0, clip=]{b_vary_local.eps}
\caption{\corr{Quiet-Sun magnetograms from 12~June~2010 (panel (a); field-of-view of 480\hbox{$^{\prime\prime}$}$\times$360\hbox{$^{\prime\prime}$}) and 16~February~2011 (panel (c); field-of-view of 330\hbox{$^{\prime\prime}$}$\times$300\hbox{$^{\prime\prime}$})} used as a boundary condition for the local domain model. \corr{The corresponding average field strength as a function of height in the corona is shown in panel (b) for the event of 12~June~2010 and panel (d) for the event of 16~February~2011}. The grey area shows the spread of the magnetic-field strength. The vertical dashed lines indicate the height at which a maximum field strength of 6~G and 2~G is achieved (horizontal lines).}
\label{fig:b_local}
\end{figure*}
Similar to the approach followed for the PFSS model, the average magnetic-field strength was estimated at a given height, with the resulting variation with height then plotted (see \corr{Figure~\ref{fig:b_local}(b),(d)}). It is determined that the height from which the magnetic-field strength is above 6~G or 2~G is 13~Mm or 36~Mm for the 12~June~2010 event \corr{(panel (b))} and 9~Mm or 17~Mm for the 16~February~2011 event \corr{(panel (d))}.
The values obtained for the characteristic height of the ``EIT wave'' pulse using the quiet-Sun field extrapolation are quite low, and not consistent with previous estimates or the heights estimated here using density and temperature scale heights. However, they may be explained by the fact that, even if the model includes the small-scale connectivity and complexity of the magnetic field, the intermediate scale of the active-region magnetic field has been removed.
Given the two different extrapolation techniques outlined above, it may be observed that the PFSS model provides more appropriate results for a large-scale event such as the eruption and propagation of an ``EIT wave''. The active-region scale extrapolation cannot be tackled without removing the small-scale field or the large-scale component with \corr{present} limitations on the field-of-view and pixel size that are reasonably possible to use in extrapolation codes.
\section{Discussion and Conclusions}
\label{S:concs}
The results presented here indicate that the magnetic-field strength of the quiet-solar corona may be determined in-situ using ``EIT waves''. High-cadence images from \corr{SDO}/AIA have been analysed using a semi-automated technique for identifying and tracking ``EIT waves'', thus minimising user bias and allowing a more accurate estimation of the kinematic properties of the disturbance. This approach is complemented by high-resolution spectral observations from \emph{Hinode}/EIS, which allow the \corr{number} density of the corona, through which the pulse was propagating, to be determined.
Two events from 12~June~2010 and 16~February~2011 are examined here using this rare combination of data: they were both observed at high cadence by \corr{SDO}/AIA and \emph{Hinode}/EIS. However, the magnetic structure of the local corona varies between these cases. The pulse on 12~June~2010 \corr{originated} in a relatively isolated, simple-topology active region. The 16~February~2011 event instead erupted from a very complex active region and was tracked across a region of quiet Sun towards an extended active region of much simpler topology.
The combination of pulse kinematics and density measurements allows the coronal magnetic-field strength to be estimated for the region of the quiet corona through which the pulse propagated. Using this approach, some variability in \corr{number} density and magnetic-field strength is observed for the region of the quiet corona studied, showing a range of 2\,--\,6~G. This variability may play a role in \corr{terms of} the observed properties of the pulse, causing it to decelerate and broaden as it propagates through the randomly structured medium \citep[see, \emph{e.g.},][]{Murawski:2001ab}. These observations \corr{indicate that interpretation of the coronal magnetic-field strength, particularly in quiet coronal regions, is not trivial}.
The estimates derived using this combination of data from \corr{SDO}/AIA and \emph{Hinode}/EIS are then compared to magnetic-field extrapolations of the coronal magnetic field. Data from \corr{SDO}/HMI have been analysed using both a global-scale PFSS extrapolation and a local-scale quiet-Sun magnetic-field extrapolation technique outlined by \citet{reg08}. Using each of these techniques it is possible to estimate the height range corresponding to the magnetic-field strength estimated using the propagating ``EIT wave'' pulse. It is found that the values obtained using the quiet-Sun approach are too low to be considered a realistic estimate of the height range. This indicates that the magnetic-field at the height of the pulse is not dominated by the small-scale magnetic-field anchored in the photosphere below.
The PFSS extrapolation produces height estimates of \corr{$\approx$}~\corr{(}70\,--\,130\corr{)}~Mm, which are most consistent with the height range estimated by \citet{Patsourakos:2009bc} and \citet{Kienreich:2009ab} using quadrature observations of an ``EIT wave'' made by \corr{STEREO}/EUVI. This increasing \corr{number} of observations made using a variety of instruments and techniques suggests that this is the true height range at which ``EIT waves'' propagate. Furthermore, this consistency strongly indicates that ``EIT waves'' are a global phenomenon influenced by global-scale features.
Direct measurements of the coronal magnetic-field are particularly difficult to make, with efforts generally focussed on the variation of the magnetic-field strength in active regions rather than the quiet corona. These results indicate that it is possible to estimate the magnetic-field strength in the solar corona using ``EIT waves'' observed using a combination of broadband and spectroscopic imagers.
\begin{acks}
\corr{The authors wish to thank the anonymous referee whose comments improved the article.} \emph{Hinode} is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the \emph{Hinode} mission is conducted by the \emph{Hinode} science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and NSC (Norway). \corr{SDO}/AIA data are courtesy of NASA/\corr{SDO} and the AIA science team. \corr{The research leading to these results has received funding from the European Commission's Seventh Framework Programme under the grant agreement No. 284461 (eHEROES project).}
\end{acks}
\bibliographystyle{spr-mp-sola}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 794 |
What can stop propaganda on the Internet?
The Russian attempts to influence the US election campaign via social media were apparently on a larger scale than previously assumed. Facebook alone has admitted that 126 million US users received comments and other content published by Russian sources between 2015 and 2017. Commentators disagree over who should be called to account.
Die Welt (DE) / 02 November 2017
Internet users also responsible
Users must also stand up to political propaganda, Die Welt believes:
"The Internet places part of the responsibility on its users. Whether it's flight bookings, banking or publications on networks. This is one of its principles. We, as users of social networks, must assume and fulfil this responsibility. We must critically examine claims, check sources and above all not simply believe everything that appears on our timeline. If we don't then we're an easy target for all kinds of propaganda."
Frank Schmiechen
Tages-Anzeiger (CH) / 03 November 2017
Only laws can regulate social media
The technology giants won't stop the propaganda of their own volition, the Tages-Anzeiger fears:
"It's unlikely that the digital companies will start self-regulating on their own. Facebook is a listed company. As such it needs to expand its business model. It would pay dearly for any attempts to undermine the mechanisms for users to vent their anger. Would a car company voluntarily reduce the power of its engines? A sense of responsibility is a virtue. But on the stock markets it has no value. No, we can't expect any answers from the digital industry. Democratic society must find its own answers - and act on them. Even if the consequence is un-American: legal regulations."
Andrian Kreye
Neue Zürcher Zeitung (CH) / 02 November 2017
End special treatment for big tech firms
The US Congress must use the investigation into Russian electoral propaganda on Facebook, Twitter and Google to end special treatment for Internet companies, the Neue Zürcher Zeitung urges:
"The fact that the true extent of Russian influence is still unknown is a result of either these companies' inability to control their own platforms or of a lack of desire for transparency. Both are sufficient reason to subject the big tech companies to the same laws that already hold for their competitors. Traditional media companies, for example, must thoroughly document all political advertising they publish. If they don't, they're held to account. For too long the companies of Silicon Valley have enjoyed special treatment in Washington."
Marie-Astrid Langer
Irish Examiner (IE) / 01 November 2017
Irish Examiner
Landmark Media Investments
Internet giants only care about money
Facebook, Google and the like aren't taking their duty to help protect democracy seriously, the Irish Examiner criticises:
"This week's events in Washington raise questions relevant to the security of democracy. ... Today, Twitter, Facebook and Google will try to explain how and why they allowed foreign agencies to target American voters. In the plainest terms, commercial interests topped any fleeting sense of national responsibility in a way that makes our bankers look like paragons of virtue. This raises a fundamental question of our age - are democracies capable of withstanding the machinations of the data plutocrats?"
GlobalDemocracyMedia policyU.S.EconomyRussiaElectionsDomestic PolicyOnline media | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,701 |
The Aaron Murray Project plays East Coast flavoured Canadiana .
music reflects his folk-country roots. His warm vocals transports the listener to the heart of Acadie.
Rodrigue Dechaine on foot-stomping, syncopated hand percussion.
Bill has spent a lifetime immersing himself in the blues, starting his professional career at the age of 15.
Go to Bill's website for his music and his rich career history. | {
"redpajama_set_name": "RedPajamaC4"
} | 683 |
echo "building baviecaAPI for java"
# generate java files from the API header file
mkdir -p ./blt/bavieca
swig -c++ -java -package blt.bavieca -outdir ./blt/bavieca -I../src/api BaviecaAPI.i
# move the wrapper file to the C++ api folder
cp BaviecaAPI_wrap.cxx ../src/api/BaviecaAPI_wrap.cxx
# compile java files
javac ./blt/bavieca/*.java
# create a jar file containing all the classes
jar -cf baviecaAPI.jar ./blt/bavieca/*.class
echo "baviecaAPI for java was successfully built!"
echo "note: library for native API needs to be rebuilt with the wrapper"
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,853 |
The Millennium Software newsgroups are no more. We've replaced them with a community based forum that is integrated directly into our support ticket and knowlegebase system.
While the old NNTP newsgroups worked well, the new software will provide for better support and interaction between users and our support technicians. In addition, users can indicate up or down votes to existing posts, which will allow you to communicate your opinions on the direction of future development.
The new community forums are available at http://support.1000years.com/community . | {
"redpajama_set_name": "RedPajamaC4"
} | 3,522 |
Bristol Bits
Random Writing
Hidden Illnesses
School & SEN
PR and Contact
Famous, Rich and Homeless; View from a Cynic
How on earth do you pretend to be homeless? This was the question a lot of people were asking when Famous, Rich and Homeless arrived on our screens in aid of Sport Relief. This was a very sensitive subject and, with a title like that, this show had the potential to fall flat on its face as a patronising pat on the head to anyone currently grappling with the reality of this horrendous situation. Surely, anyone who goes into this knowing they have a warm bed to return to at the end of the 'experiment' can't really learn anything... can they?
There were some interesting stories behind the personalities who were featured on the show, particularly Kim Woodburn. I have been a huge fan of hers, since the 'How Clean is your House' days, and not just because she knows her way around a U-bend. I have read Kim's autobiography and it's a very, very difficult read. What she went through in her youth was horrific and to walk out the other side at all is nothing short of miraculous, but to do it with the determination and stoicism that she has, shows a phenomenal strength of character that it's impossible not to admire. I love her positive outlook on life and was really interested to see how this would influence the people she met in the course of the show. Although she was slated after the show for her harsh words to one rough sleeper, taking into account her background, and the editing of the show, things may not be all they seem. Willie Thorne has also battled a few demons in his past, and I wondered if that would help him relate to some of the people he met. Also featured were Nick Hancock, who I remembered from They Think It's All Over, but didn't know much about and Julia Bradbury, who is around my age and always seems so normal and down to earth, she could be anyone's mum and her approachability made her a good candidate to find the best in people.
What surprised me most was that, yes, they were followed by a film crew and, obviously (quite rightly), they were clearly kept safe, but they were also expected to fend for themselves, and it was really interesting to see how that worked out. Kim wanted to be visible and approachable and looked for somewhere to be where other homeless people would see her. Nick was the most sociable, seeking out people to talk to and asking the advice of those he met. Julia had me in stitches, with her reasoning for asking strangers for money, but not actually begging. Genius! Although I felt she embraced the experience in a totally different way. Her aim was to find a way not to sleep on the streets, which was brilliant because that is exactly what a homeless person probably would do. It also highlighted an issue that I had no idea about. It's widely assumed that people are homeless because they have no money, but Julia's attempts to secure a place in a hostel, with money she had managed to cadge off a generous public, failed because she had no I.D. So, even if you give enough money to a street sleeper to try to keep them safe, they may still end up sleeping in a doorway.
Willie Thorne. Hmmm. What about Willie? In the early stages of the show he struck me as a man who is very well looked after at home! Not a crime, but it did mean he had a much harder time than the others, because he didn't seem to get the memo about preparation for the night ahead. Where Kim, Julia and Nick were making friends, who would help them through the difficult bits and finding shelter and warmth as best they could, Willie was... not. I don't think it was his fault, he was just vastly underprepared for the task and the question we were all asking was... Willie make it? Unfortunately not, really. He managed to wangle two nights in a hotel which seemed to put him at a massive disadvantage to the others. All the participants who threw themselves into this show, who got to know the genuine street sleepers, listened to them, cared about them, all got something back. Kim was so touched when two young men gave her an extra sleeping bag and brought her something to eat. Julia, in her own unique, bossy-but-caring way, managed to reunite a lovely young man with his mum, so he could get the support he needed for a drug addiction. She did that just by listening to him. And then there's Nick. The unlikely hero of the piece. Faced with aggression, hostility and drug abuse, his compassion and caring nature turned every situation around, so that eventually people were talking, even laughing, and he was listening. He, too, was moved to tears (well, almost, he is a Northerner!) by the kind actions of a young homeless couple who couldn't do enough to help him, despite having nothing themselves.
King of the pillow fort!
I approached this programme with the utmost cynicism. Like Nick, I asked myself; 'What can (us) doing this change?' We know the homeless are there, we know that this needs to change. One of the things the show highlighted though, is that homelessness isn't always about drug use. Anyone can be homeless, from any walk of life. When Willie Thorne told a homeless man that he had to go to a hotel because; 'We've all come from a life at home.' The man quietly replied; 'Yeah, so have I'. In that moment, every stereotypical myth that has ever been created about the lives of the homeless disintegrated. Anyone, at any time, is at risk. Especially in today's economy. You can't judge a person by the clothes they are wearing, or the place they call home, they could have been your next door neighbour, or someone you sat next to on the commute to work. Which brings me to the point of this post. What I learned from this show. Something I hadn't even considered. Where Willie was getting cross with everyone and removing himself from the group, because he felt alone and vulnerable, the other celebrities involved, particularly Nick and Julia, put a huge amount into the project. They went out of their way to find out the truth and try to help and the rewards they got for that were clearly apparent. This really illustrated to me, that you get out of life what you put in. Even with no money, or resources, they still made a difference. I also learned that Nick Hancock is the most resourceful person to have on your side if you need shelter, as he emerged as the star of the show. His pillow fort, built from a supermarket cage and an old mattress, was phenomenal. Clearly a quiet, unassuming man, he stole the show with his down to earth manner and incredible survival skills. I really hope he returns to our screens soon. I wonder if Ray Mears needs a sidekick?
posted by Lucy D on day April 01, 2016
labels Commentary, Pinterest
Gtech Multi Review
My Sunday Photo: My Broken Heart
Win Prizes for You and Your Child!
Awesome Autism
Home Alone - How Young is too Young?
Dear Mr Nobody
The Run In. A Football Guest Post by Phil
Instant Ink from HP; One Less Thing to Remember!
My Sunday Photo: Crazy Cloud Formations!
Lumie Brightspark SAD Lightbox Review
© The Parent Game. All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,055 |
#pragma once
/*! \file numeric/netcdf.hh
* \brief Provides functions to save/load arrays from NetCDF files.
*/
#include "base.hh"
#include "ndarray.hh"
#include <netcdf>
#include <cstddef>
namespace netCDF
{
template <typename T>
struct type_traits;
template <>
struct type_traits<float>
{
static NcType nc_type;
};
template <>
struct type_traits<int>
{
static NcType nc_type;
};
template <>
struct type_traits<size_t>
{
static NcType nc_type;
};
NcType type_traits<float>::nc_type = ncFloat;
NcType type_traits<int>::nc_type = ncInt;
NcType type_traits<size_t>::nc_type = ncUint64;
}
namespace HyperCanny {
namespace numeric
{
template <typename F>
void variadic_for_each_helper(F func, unsigned idx) {}
template <typename F, typename Arg, typename ...Args>
void variadic_for_each_helper(F func, unsigned idx, Arg &&arg1, Args &&...args)
{
func(idx, std::forward<Arg>(arg1));
variadic_for_each_helper(func, idx + 1, std::forward<Args>(args)...);
}
template <typename F, typename ...Args>
void variadic_enumerated_for_each(F func, Args &&...args)
{
variadic_for_each_helper(func, 0, std::forward<Args>(args)...);
}
template <typename T_from, typename T_to>
void netcdf_convert_helper(
netCDF::NcVar &nc_var,
T_to *data,
size_t size)
{
T_from *temp = new T_from[size];
nc_var.getVar(temp);
std::copy(temp, temp + size, data);
}
template <typename T>
void netcdf_convert(
netCDF::NcVar &nc_var,
T *data,
size_t size)
{
using namespace netCDF;
switch (nc_var.getType().getTypeClass())
{
case NcType::nc_FLOAT:
netcdf_convert_helper<float, T>(nc_var, data, size);
break;
case NcType::nc_INT:
netcdf_convert_helper<int, T>(nc_var, data, size);
break;
case NcType::nc_UINT64:
netcdf_convert_helper<uint64_t, T>(nc_var, data, size);
break;
default:
throw Exception(format(
"Could not convert NetCDF data: ",
nc_var.getType().getTypeClassName()));
}
}
template <typename T, unsigned D, typename ...DimArgs>
void netcdf_add_array(
netCDF::NcFile &file,
NdArray<T,D> const &data,
std::string const &var_name,
DimArgs &&...dim_args)
{
using namespace netCDF;
static_assert(
sizeof...(DimArgs) == D,
"Number of dimension arguments should match dimension of array.");
std::vector<NcDim> dims;
variadic_enumerated_for_each(
[&file, &dims, &data] (unsigned idx, std::string const &dimname)
{
dims.push_back(file.addDim(dimname, data.shape()[D-idx-1]));
}, std::forward<DimArgs>(dim_args)...);
NcVar nc_data = file.addVar(var_name, type_traits<T>::nc_type, dims);
nc_data.putVar(data.const_container().data());
}
template <typename T, unsigned D>
std::unique_ptr<NdArray<T,D>> netcdf_read_array(
netCDF::NcVar &nc_var)
{
using namespace netCDF;
Console::Log console;
if (nc_var.getDimCount() != D)
throw Exception(format(
"NetCDF variable does not have the expected number of ",
D, " dimensions: got ", nc_var.getDimCount(), "."));
shape_t<D> shape;
for (unsigned i = 0; i < D; ++i)
shape[D-i-1] = nc_var.getDim(i).getSize();
auto data = std::make_unique<NdArray<T,D>>(shape);
if (nc_var.getType() == type_traits<T>::nc_type)
{
nc_var.getVar(data->container().data());
} else {
console.warning(
"converting data from ", nc_var.getType().getTypeClassName(),
" to ", type_traits<T>::nc_type.getTypeClassName());
netcdf_convert<T>(nc_var, data->container().data(), data->size());
}
return std::move(data);
}
template <typename T, unsigned D>
std::unique_ptr<NdArray<T,D>> netcdf_read_array(
netCDF::NcFile &file,
std::string const &var_name)
{
using namespace netCDF;
NcVar nc_var = file.getVar(var_name);
if (nc_var.isNull())
throw Exception(format(
"NetCDF file does not contain requested variable: ", var_name));
return netcdf_read_array<T, D>(nc_var);
}
}}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,152 |
id: lazyweb-2015
showid: lazyweb
layout: shows/series-index
collection: lazyweb
title: "Lazy Web: 2015"
description: ""
published: true
date: 2015-04-14
article:
written_on: 2015-04-15
updated_on: 2015-04-15
---
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,094 |
Q: Background image is not showing at login screen. 14.04 So I changed my background on my computer to an image I liked on the internet. However unlike on my laptop, the background image doesn't display for the login to my user account. Is there a way to fix this on 14.04?
A: Install dconf-editor either via the Ubuntu Software Center or CLI.
In dconf-editor go to: com > canonical > unity > unity-greeter
and then make sure that the tick box draw-user-backgrounds is set to True.
A: You can set it through var/lib/AccountsService/users/yourusername, under [User] section. I use this very same file to set a login background different from desktop.
Here's how mine looks like. Notice the Background= line
[com.canonical.indicator.sound.AccountsService]
PlayerName=''
Timestamp=0
Title=''
Artist=''
Album=''
ArtUrl=''
PlayerIcon=<('themed', <['application-default-icon', 'application-default', 'application']>)>
[User]
Language=en
FormatsLocale=en_US.UTF-8
XSession=gnome
Background=/usr/share/backgrounds/LOGIN-WARNING.png
Icon=/usr/share/pixmaps/faces/penguin.jpg
SystemAccount=false
[InputSource0]
ibus=pinyin
[InputSource1]
xkb=us
[InputSource2]
xkb=ru
A: Follow these steps to configure your background image:
*
*If your wallpaper is in another partition, kindly copy it to your home folder - like inside the pictures folder.
*Right click on it and choose Properties then select the permissions tab. Set the permissions as:
owner - Read and Write
Group - Read and Write
Others - Read only
*Open up your wallpaper in Image Viewer (EOG) then right click and choose Set as Wallpaper
(At this stage you can try and logout and check if the image has appeared as the login background image). If not, continue - though there would be no harm in continuing all the same.
*Install Ubuntu Tweak
*Open up Ubuntu Tweak and choose the Tweaks tab then the Login Settings section - see screenshot below:
*Unlock the settings using your password - see screenshot below:
*Scroll down and press the Set the same background as the current desktop background button - see screenshot below:
*And that's it, the wallpaper has been set as the login background.
Here's my setup:
Desktop:
Login Screen:
https://askubuntu.com/a/607095/163331
A: Another reason for this could be wrong permissions on the image file and/or the folder it is inside. This has happened to me.
The permissions for folders should be 775(rwxrwxr-x) and for image files 664(rw-rw-r--).
For example, if you have:
./wallpapers/animals/cat_0.jpg
wallpapers and animals must have permissions 775 and cat_0.jpg - 664.
You can set permissions using chmod:
chmod 775 [folder/file name]
A: if all else fails convert the graphic to a different format.. in the the past there have been graphics that would display as the wallpaper while logged in but would not work for the login screen until the format was changed.. forget what the valid format was but sure that you can figure that one out with a little experimentation.
oh.. and forgot about this.. but if you use a separate desktop environment instead of unity then you must change the background wallpaper in unity because unity greeter does not care what background that you use in other desktop environments. for myself there is lxde so always have to remember to change my wallpaper in unity to the same as my background in lxde.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,384 |
\section{Introduction}
Power Distribution Systems (PDS) have been historically designed to transport power from the bulk transmission system to end consumers of electricity. However, the recent increase in the adoption of Distributed Energy Resources (DERs) has started to shift this paradigm, with consumers and utilities leveraging these resources at increasing rates. This high penetration of DERs in PDS will create multiple technical challenges associated with new operating characteristics such as bidirectional power flows and voltage fluctuations due to the volatility of renewable power generation. Specifically, in a distribution system with Energy Storage Systems (ESS), renewable generation and advanced controls, traditional monitoring and protection schemes are not well suited for detecting faults, changes in topology, and malfunction of controllable devices. Furthermore, the methods that rely on accurate measurements and knowledge of parameters currently used in transmission systems, such as traditional power system state estimation, are inadequate for monitoring in PDS. Thus, it is essential to develop new methodologies to detect abnormities in PDS due to critical events (e.g., natural disasters, physical and cyber-attacks) and to mitigate the consequences of these abnormities.
Classical approaches for incident detection in the transmission grid rely on some knowledge of the system model. For example, the integer programming approach in \cite{tate}, compressive sensing based approach in \cite{zhu}, quickest change detection method in \cite{banerjee}, and the Gauss-Markov graphical model in \cite{he} all rely on some knowledge of grid parameters and of the assets connected to the grid. As compared to works that study the transmission system, the body of literature that considers incident detection in the distribution grid is not as large. Despite the differences between both grids, such as the fact that distribution grid models are little known and sensing is scarce, studies that aim to detect events in the distribution grid started by also assuming knowledge of the network topology, such as in graph-based techniques and methods based on traveling waves (see \cite{bahmanyar} for a review of methods for fault and outage area detection in distribution grids). With the introduction of PMUs, data-driven techniques started receiving more attention, as in the case with SVD-based approaches \cite{lim}. However, most of the focus has still been on the transmission grid, where sensing is more abundant and uniform. When it comes to distribution grids, an increase in the use of micro PMUs and AMIs has been observed, but the existence of heterogeneous sources of data has also posed challenges.
Our proposed approach draws on Koopman operator theory, which accounts for the causal relationship among multiple sensor data streams without prior knowledge of the dynamic model. The major feature of the proposed approach lies on the detection of events which produce sharp changes in the causal map of the system dynamics, while being robust to small variations such as measurement noise and load fluctuation. Koopman operators have been explored in the power systems literature for the identification of system dynamics, stability assessment, and other topics \cite{sinha,sinha2,susuki,susuki2,korda}. In this work, we propose an algorithmic approach that provides evidence of the effectiveness of this operator in incident detection tasks relevant to distribution systems. We also tackle the issue of sensor availability by proposing a way to augment the dataset through a transformation that computes the relationship between data streams. Our case studies explore the detection of incidents in ESS operation that relates both to active power dispatch and reactive power control, while accounting for the presence of measurement noise and load fluctuations.
The remainder of this paper is organized as follows. We introduce the Koopman operator theory in Section~\ref{sec:theory}, present the proposed algorithm for incident detection in Section~\ref{sec:algorithm}, and detail the steps in the methodology for incident detection in Section~\ref{sec:methodology}. Section~\ref{sec:cases} presents two representative case studies which exemplify the use of the proposed algorithm, and concluding remarks are presented in Section~\ref{sec:conclusion}.
\section{Koopman Operator Theory Preliminaries}\label{sec:theory}
Consider a nonlinear system with states $x_t \in \mathbb{R}^n$ whose dynamics can be characterized by
\begin{equation}
x_{t+1} = \bm{F}(x_t).
\end{equation}
The Koopman operator $\bm{K}$ is a linear infinite-dimensional operator that acts on the space of observables $g(x): \mathbb{R}^n \to \mathbb{R}$, i.e. the space of scalar-valued functions of the states of the system, as follows:
\begin{equation}
\bm{K}g(x) = g(x) \circ \bm{F}.
\end{equation}
The map $\bm{F}$ can be described as a linear combination of the eigenfunctions of the linear infinite-dimensional operator $\bm{K}$. Then, the evolution of the state becomes a linear combination of independent dynamics along each eigenfunction. Assume that the dominant eigenfunctions are roughly in the span of the $D$ dictionary functions
\begin{equation}
\bm{\Psi} = (\Psi_1,...,\Psi_D)^T.
\end{equation}
Then, the dynamics of $\bm{F}$ becomes roughly linear if lifted to a space in which the functions in $\bm{\Psi}$ are roughly taken as coordinates. This linear description is characterized by $\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D$, which is a finite-dimensional approximation of the Koopman operator. An observable $g(x)$ in the span of dictionary functions identified by weights $b$ is given by $g(x) = \bm{\Psi}(x)^Tb$. Under the action of the Koopman operator, this observable can be approximated as
\begin{equation}
g(\bm{F}(x)) \approx \bm{\Psi}(x)^T\mathcal{K}b.
\end{equation}
Techniques such as the Extended Dynamic Mode Decomposition (EDMD) algorithm proposed in \cite{williams} provide a way to calculate an approximation of the Koopman operator by minimizing $||\bm{\Psi}(\bm{F}(x))^Tb-\bm{\Psi}(x)^T\mathcal{K}b||^2_2$ for an arbitrary $b$. Let $(x_1,x_2),...,(x_M,x_{M+1})$ be a time-series data. The minimization problem of interest is equivalent to
\begin{equation}
\underset{\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}-\bm{G}\mathcal{K}||^2_F,
\end{equation}
where $||.||_F$ is the Frobenius norm, and $\bm{A}$ and $\bm{G}$ are defined~as
\begin{equation}
\bm{A} := \frac{1}{M}\sum_{j=1}^M \bm{\Psi}(x_j)\bm{\Psi}(x_{j+1})^T,~ \bm{G} := \frac{1}{M}\sum_{j=1}^M \bm{\Psi}(x_j)\bm{\Psi}(x_j)^T.
\end{equation}
\section{Algorithm for Incident Detection}\label{sec:algorithm}
In this work, we leverage a sparsity-promoting variant of the EDMD algorithm \cite{quade} for the detection of incidents in the distribution grid, given by
\begin{equation}\label{eq:ksparse}
\underset{\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}-\bm{G}\mathcal{K}||^2_F + \lambda ||\text{vec}(\mathcal{K})||_1.
\end{equation}
For our algorithm, we consider the following notation. For time-series data $x$ and $t_1\leq t_2$, define
\begin{equation}
x[t_1:t_2] := (x_{t_1}x_{t_1+1},...,x_{t_2}).
\end{equation}
At time $t$, let $\bm{A}[t-T:t]$ and $\bm{G}[t-T:t]$ be computed with $T+1$ observations and their one-step propagation in the observed data series $x[t-T:t]$.
Our goal is to maintain a sparse representation of the dynamic system, and detect incidents by detecting changes in this sparsity pattern. For that, we build on \eqref{eq:ksparse} to define our algorithm for incident detection as follows:
\begin{itemize}
\item At time $t=T+1$, compute $\mathcal{K}_{t}$ by solving
\begin{equation}
\underset{\mathcal{K}_t\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}[1:T+1]-\bm{G}[1:T+1]\mathcal{K}_t||^2_F + \alpha ||\text{vec}(\mathcal{K}_t)||_1.
\end{equation}
\item From $t=T+2$ onward, compute $\mathcal{K}_{t}$ by solving
\begin{equation}
\begin{split}
\underset{\mathcal{K}_t\in \mathbb{R}^D \times \mathbb{R}^D}{\min} &||\bm{A}[t-T:t]-\bm{G}[t-T:t]\mathcal{K}_t||^2_F \\ &+\alpha ||\text{vec}(\mathcal{K}_t)||_1+\beta ||\mathcal{K}_{t-1}-\mathcal{K}_t||_F.
\end{split}
\end{equation}
\item If the sparsity pattern of $\mathcal{K}_{t}$ differs significantly from that of $\mathcal{K}_{t-1}$, flag the occurrence of an incident that has altered the causal map between a state and its one-step propagation.
\end{itemize}
The sparsity pattern of each $\mathcal{K}$ encodes the causal relationship among observables, and the level of sparsity is adjusted by the choice of the hyper-parameter $\alpha$. Small values of $\alpha$ define more complete networks, while large values promote sparsity. The hyper-parameter $\beta$ is related to the smoothness of transitions. Thus, larger values of $\beta$ tend to stabilize $\mathcal{K}$ along a trajectory, making its less prone to change due to smaller variations which do not lead to major structural changes.
\section{Methodology for Incident Detection}\label{sec:methodology}
Our methodology can be divided into two distinct parts, as illustrated in Fig.~\ref{fig:method}. The first part consists of a data collection step, followed by a data transformation. The transformed data is then used to compute the $\mathcal{K}$ matrices using the sliding window approach introduced in Section~\ref{sec:algorithm}. The second part is a post-processing clustering task which performs a clustering analysis with the goal of grouping the $\mathcal{K}$ matrices calculated into clusters that are uniquely identified by one of the scenarios simulated in the data collection step. Each part of this approach is discussed in more detail in the next sections.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/methodology.png}
\caption{Two-part methodology developed for the detection of incidents in distribution grids.}
\label{fig:method}
\end{figure}
\subsection{Part 1: Koopman Operator}
\subsubsection{Data Collection}
We begin by selecting suitable simulation scenarios from which our data will be gathered. Since our focus is on incidents related to the operation of ESS in the distribution grid, the following two broad types of events were considered:
\begin{enumerate}
\item Changes in ESS charging/discharging rate: In distribution systems, ESS can be used for peak-shaving operations, absorption of excess local renewable production, and other tasks which involve a coordinated charging/discharging schedule. We aim to analyze whether changes in the charging/discharging rates of ESS can be identified with our proposed framework. Detecting such changes would allow us to recognize, for example, when a battery fleet is no longer following the command issued for their power supply/absorption operation.
\item Changes in ESS controller parameters: Voltage regulation is an important task in power distribution systems, especially with the increasing adoption of DERs. By considering a scenario in which batteries offer voltage regulation services, we model Volt/VAR curves to control the reactive power output of these devices. The goal is to detect changes in controller parameters, which could have been caused by attacks that aim to destabilize the grid.
\end{enumerate}
We assume the data collected comes from PMUs, and thus we use voltage magnitude and phase measurements. The data was collected with a 0.25s sampling time. Because sensor measurements are typically noisy, Gaussian noise was added to the voltage magnitude ($\pm 0.01\%$) and voltage angle ($\pm 0.01\degree$) measurements. Further, considering that sensing capability would be limited in the distribution grid, we assumed that only nodes with an ESS had PMUs. These scenarios were simulated using OpenDSS.
\subsubsection{Data Transformation}
We hypothesize that the relationship between different time-series data carries more information than individual data streams. In \cite{fujii}, this dependence among observations is explored in a data-driven spectral analysis using the Koopman operator with the objective of understanding complex biological network dynamics. For this purpose, using a ballgame as an example, the authors transformed the data using the Gaussian kernel
\begin{equation}\label{eq:gaussian}
g(x_i,x_j) = \exp\left(-\frac{||x_i-x_j||^2_2}{2\sigma}\right),
\end{equation}
where each $x_i$ is a stream of raw data, $||.||_2$ is the Euclidean norm, and $\sigma$ is an adjustment parameter.
We apply \eqref{eq:gaussian} to the distribution grid data to be used in our analysis. This nonlinear transformation lifts the data into a higher dimensional space by giving a measure of similarity between states, thus augmenting our dataset. The transformed data is then used to compute a sequence of $\mathcal{K}$ matrices through a sliding window.
\subsubsection{Approximate Koopman Operator}
In this step, the finite-dimensional approximation of the Koopman operator for the transformed data was performed numerically following the steps detailed in Section~\ref{sec:algorithm}. Significant changes in $\mathcal{K}$ indicates altered causality in states (i.e., occurrence of incidents). Radial basis functions of the form
\begin{equation}\label{eq:rbf}
\Psi_i(x) = ||x-c_i||^2\log_e(||x-c_i||),
\end{equation}
where $c_i$ are unique center points, have shown to be effective for our application. We face the following trade-off when choosing the amount of dictionary functions to be used: too few dictionary functions $\Psi_i(x)$ may lead to poorer distinguishability of the sparsity pattern of $\mathcal{K}$'s from different incidents; increasing the amount of functions, however, also increases computational time.
\subsection{Part 2: Offline Clustering}
The previous step was aimed at generating data for different case studies, transform the data collected, and compute the approximate Koopman operator for multiple time windows. The offline clustering task performs a clustering analysis in the $\mathcal{K}$ matrices calculated. Since the sparsity pattern of these matrices is expected to be similar if the system dynamics remains the same, this analysis is expected to cluster together the $\mathcal{K}$'s coming from the same simulated scenario. For example, the matrices calculated from a scenario in which a battery fleet is discharging at
25\% rate should be clustered together, while $\mathcal{K}$'s from a scenario where these batteries are discharging at 100\% should be together but in a different cluster.
We used the k-means clustering algorithm for this analysis. The idea behind k-means is that observations within a cluster are close to each other, while observations in different clusters are far apart. This method takes as inputs a distance measure, the number of clusters, the data to be clustered, and a random seed that is used to initialize the algorithm. For our application, the correlation distance was shown to perform better than other metrics. Further, prior to the clustering analysis, the $\mathcal{K}$ matrices were transformed into binary matrices, i.e. values below a certain threshold were set to zero and all others were set to one. The purpose behind this transformation is that we are only concerned with the sparsity pattern of these matrices, and not with the actual values of their elements. This step also avoids numerical issues that may arise when working with small numbers. Since the approximate Koopman operator is a representation of our original data in a high-dimensional space, it is common to see elements of $\mathcal{K}$ which have a really small order of magnitude. The effectiveness of our results is evaluated based on the misclassification rate achieved in this clustering analysis.
\section{Case Studies}\label{sec:cases}
We considered the IEEE 8500-node test feeder with 7 battery energy storage systems (BESS) added, implemented using OpenDSS. This feeder is an unbalanced radial network, which are typical characteristics of distribution grids. These BESS can take real power dispatch commands and perform Volt/VAR control. The data was collected with a 0.25s sampling time, and Gaussian noise was added to the voltage magnitude ($\pm 0.01\%$) and voltage angle ($\pm 0.01\degree$) measurements. Besides noise, fluctuations in real and reactive load were considered in the feeder model. Our aim is to evaluate the robustness of our algorithm when these variations are accounted for.
As previously discussed, changes in the sparsity pattern of the $\mathcal{K}$ matrices indicated the occurrence of an incident. To confirm the $\mathcal{K}$ matrices had unique sparsity patterns for each scenario in which the causality of the model was maintained, an offline analysis was performed using k-means clustering to cluster these matrices. For both case studies, the centers for the radial basis functions $c_i$ in \eqref{eq:rbf} were chosen in the computation of the first $\mathcal{K}$ matrix, and then kept constant throughout the experiment. The center points were selected using a randomly generated perturbation, so that they were of the same order of magnitude as the data itself. Further, we used $D=400$ dictionary functions, a time window of $T=100s$, and a new $\mathcal{K}$ was estimated every $25s$. We also assumed voltage magnitude and phase measurements from PMUs were available at the nodes of 3 of these BESS.
\subsection{Detecting changes in discharging rates}
We first consider a scenario in which the battery fleet is initially idle. After 5min, all the batteries start discharging at 50\% of their discharging rate, maintaining this rate for 5min. Fig.~\ref{fig:case1} shows the results for this case. For the Koopman operator approximation step, we let $\alpha=0.1$ and $\beta=0.4$.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/casestudy1.png}
\caption{Results for case study which aims to detect changes in the rate at which the battery fleet supplies active power to the grid.}
\label{fig:case1}
\end{figure}
The red numbers in Fig.~\ref{fig:case1} mark the time instants at which a new $\mathcal{K}$ matrix was estimated. The plots on top show a representative sparsity pattern for each of the two situations considered, and we can clearly observe that this pattern changes after the incident happens, allowing us to successfully detect its occurrence. To corroborate that the two patterns are indeed distinct, all the 20 $\mathcal{K}$ matrices computed were subjected to a clustering analysis using k-means. For this case study, we have two different clusters: the first representing our network dynamics when the batteries are idle, and the second related to the case in which the battery fleet discharges at 50\% of their rate. As a result, we found that 3 of these matrices were misclassified, i.e. they were classified to the cluster that does not correspond to the scenario they actually refer to. We note that the misclassified matrices, highlighted in Fig.~\ref{fig:case1}, were computed immediately after the event occurred. Due to the sliding window approach, their computation used data from both the first and the second scenarios, which impacts their sparsity pattern. However, the matrices after those were correctly classified. Thus, we can state that some time may be needed after the event for the new sparsity pattern to stabilize. We remark that the results were robust to measurement noise, as well as load variations, since these fluctuations did not lead our algorithm to erroneously identify the occurrence of nonexistent incidents.
\subsection{Detecting changes in controller parameters}
In this case study, we considered the battery fleet provides voltage regulation services. For this purpose, the hierarchical Volt/VAR (VV) control strategy proposed in \cite{quiroz} is used to control the reactive power supply/absorption of these devices so that the voltage magnitudes are maintained close to their nominal value. Each battery is assigned a VV curve, which is designed to adjust the reactive power of the device according to measured local voltage levels. An example of this curve is shown in Fig.~\ref{fig:vvcurve} \cite{quiroz}. The VV curve has a deadband corresponding to acceptable voltage levels which do not trigger control action for correction. When voltage levels become high, the battery absorbs reactive power to lower the voltage back to acceptable levels. The opposite happens when voltage levels are low, with the batteries injecting reactive power into the grid. This control layer is local, as each device does not have information about the overall system condition, and only responds to their local condition.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{figs/vvcurve2.png}
\caption{Example of VV curve and shifting logic \cite{quiroz}.}
\label{fig:vvcurve}
\end{figure}
To overcome this limitation and expand the control strategy to regulate voltage levels throughout the network, a centralized control layer is added. This layer has full information about the voltages over the entire feeder, and is able to dispatch new VV curves to the devices with the goal of regulating the overall voltage in the system, even when the device's local voltage magnitude is within acceptable ranges. Fig.~\ref{fig:vvcurve} shows an example of how the initial VV curve assigned to a battery can be shifted using the proposed centralized strategy. We note that shifting the curve to the left induces an increase in reactive power absorption, which helps decreasing voltage levels more significantly across the network. The reverse holds for right shifts in the curve.
Using this hierarchical VV control, we modeled two events for this case study, leading to a total amount of three different scenarios. For each event, the controller parameters are changed by adjusting the set-points for the VV curves assigned to the batteries. The dead band of the controller changes from 0.95 to 0.98 and then 0.99 pu, with saturation point shifting right accordingly at $t=200s$ and $t=400$, respectively. The results for this case study are presented in Fig.~\ref{fig:case2}, in which we observe three distinct sparsity patterns for the $\mathcal{K}$ matrices in each scenario. These patterns were achieved by using $\alpha=0.1$ and $\beta=0$ in the Koopman operator approximation. Similarly to the first case study, the differences in the $\mathcal{K}$ matrices were sufficient for us to identify the occurrence of the events considered. Further, in the offline classification task, 3 out of the 32 matrices were misclassified, all of which were computed in time instants immediately following an event.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/casestudy2.png}
\vspace{-5mm}
\caption{Results for the detection of changes in the controller parameters of batteries providing voltage regulation through Volt/VAR control.}
\label{fig:case2}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
We proposed a data-driven method that requires no prior knowledge of the network dynamic model for the detection of incidents in power distribution systems. Our case studies considered the occurrence of changes in BESS operations~that refer both to active power dispatch and reactive power control of these devices. The successful detection of the events modeled allows for proper mitigation strategies to be set forth, if needed. Our methodology introduces a data transformation step which augments the dataset while maintaining meaningful information of the system states. This strategy is particularly useful in situations with restricted sensor availability, which can be common in distribution grids. Further,
the algorithm proposed was shown to be robust to measurement noise and load fluctuations, and thus such variations do not trigger a~false detection. Future work includes
designing a more systematic way to choose the hyper-parameters for the Koopman operator approximation, and evaluating our methodology in the presence of data loss or data streams with different~sampling~time.
\section*{Acknowledgment}
This research was funded by the energy storage program at the U.S. Department of Energy under the guidance of Dr. Imre Gyuk. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy National Nuclear Security Administration under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Jamal Al Hourani helped with some numerical experiments. Hyungjin Choi helped with the initial code and explanation of Koopman operator calculations. The work of the first two authors was supported in part by Sandia National Lab under the grant PO 2079716.
\bibliographystyle{IEEEtran}
\section{Introduction}
Power Distribution Systems (PDS) have been historically designed to transport power from the bulk transmission system to end consumers of electricity. However, the recent increase in the adoption of Distributed Energy Resources (DERs) has started to shift this paradigm, with consumers and utilities leveraging these resources at increasing rates. This high penetration of DERs in PDS will create multiple technical challenges associated with new operating characteristics such as bidirectional power flows and voltage fluctuations due to the volatility of renewable power generation. Specifically, in a distribution system with Energy Storage Systems (ESS), renewable generation and advanced controls, traditional monitoring and protection schemes are not well suited for detecting faults, changes in topology, and malfunction of controllable devices. Furthermore, the methods that rely on accurate measurements and knowledge of parameters currently used in transmission systems, such as traditional power system state estimation, are inadequate for monitoring in PDS. Thus, it is essential to develop new methodologies to detect abnormities in PDS due to critical events (e.g., natural disasters, physical and cyber-attacks) and to mitigate the consequences of these abnormities.
Classical approaches for incident detection in the transmission grid rely on some knowledge of the system model. For example, the integer programming approach in \cite{tate}, compressive sensing based approach in \cite{zhu}, quickest change detection method in \cite{banerjee}, and the Gauss-Markov graphical model in \cite{he} all rely on some knowledge of grid parameters and of the assets connected to the grid. As compared to works that study the transmission system, the body of literature that considers incident detection in the distribution grid is not as large. Despite the differences between both grids, such as the fact that distribution grid models are little known and sensing is scarce, studies that aim to detect events in the distribution grid started by also assuming knowledge of the network topology, such as in graph-based techniques and methods based on traveling waves (see \cite{bahmanyar} for a review of methods for fault and outage area detection in distribution grids). With the introduction of PMUs, data-driven techniques started receiving more attention, as in the case with SVD-based approaches \cite{lim}. However, most of the focus has still been on the transmission grid, where sensing is more abundant and uniform. When it comes to distribution grids, an increase in the use of micro PMUs and AMIs has been observed, but the existence of heterogeneous sources of data has also posed challenges.
Our proposed approach draws on Koopman operator theory, which accounts for the causal relationship among multiple sensor data streams without prior knowledge of the dynamic model. The major feature of the proposed approach lies on the detection of events which produce sharp changes in the causal map of the system dynamics, while being robust to small variations such as measurement noise and load fluctuation. Koopman operators have been explored in the power systems literature for the identification of system dynamics, stability assessment, and other topics \cite{sinha,sinha2,susuki,susuki2,korda}. In this work, we propose an algorithmic approach that provides evidence of the effectiveness of this operator in incident detection tasks relevant to distribution systems. We also tackle the issue of sensor availability by proposing a way to augment the dataset through a transformation that computes the relationship between data streams. Our case studies explore the detection of incidents in ESS operation that relates both to active power dispatch and reactive power control, while accounting for the presence of measurement noise and load fluctuations.
The remainder of this paper is organized as follows. We introduce the Koopman operator theory in Section~\ref{sec:theory}, present the proposed algorithm for incident detection in Section~\ref{sec:algorithm}, and detail the steps in the methodology for incident detection in Section~\ref{sec:methodology}. Section~\ref{sec:cases} presents two representative case studies which exemplify the use of the proposed algorithm, and concluding remarks are presented in Section~\ref{sec:conclusion}.
\section{Koopman Operator Theory Preliminaries}\label{sec:theory}
Consider a nonlinear system with states $x_t \in \mathbb{R}^n$ whose dynamics can be characterized by
\begin{equation}
x_{t+1} = \bm{F}(x_t).
\end{equation}
The Koopman operator $\bm{K}$ is a linear infinite-dimensional operator that acts on the space of observables $g(x): \mathbb{R}^n \to \mathbb{R}$, i.e. the space of scalar-valued functions of the states of the system, as follows:
\begin{equation}
\bm{K}g(x) = g(x) \circ \bm{F}.
\end{equation}
The map $\bm{F}$ can be described as a linear combination of the eigenfunctions of the linear infinite-dimensional operator $\bm{K}$. Then, the evolution of the state becomes a linear combination of independent dynamics along each eigenfunction. Assume that the dominant eigenfunctions are roughly in the span of the $D$ dictionary functions
\begin{equation}
\bm{\Psi} = (\Psi_1,...,\Psi_D)^T.
\end{equation}
Then, the dynamics of $\bm{F}$ becomes roughly linear if lifted to a space in which the functions in $\bm{\Psi}$ are roughly taken as coordinates. This linear description is characterized by $\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D$, which is a finite-dimensional approximation of the Koopman operator. An observable $g(x)$ in the span of dictionary functions identified by weights $b$ is given by $g(x) = \bm{\Psi}(x)^Tb$. Under the action of the Koopman operator, this observable can be approximated as
\begin{equation}
g(\bm{F}(x)) \approx \bm{\Psi}(x)^T\mathcal{K}b.
\end{equation}
Techniques such as the Extended Dynamic Mode Decomposition (EDMD) algorithm proposed in \cite{williams} provide a way to calculate an approximation of the Koopman operator by minimizing $||\bm{\Psi}(\bm{F}(x))^Tb-\bm{\Psi}(x)^T\mathcal{K}b||^2_2$ for an arbitrary $b$. Let $(x_1,x_2),...,(x_M,x_{M+1})$ be a time-series data. The minimization problem of interest is equivalent to
\begin{equation}
\underset{\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}-\bm{G}\mathcal{K}||^2_F,
\end{equation}
where $||.||_F$ is the Frobenius norm, and $\bm{A}$ and $\bm{G}$ are defined~as
\begin{equation}
\bm{A} := \frac{1}{M}\sum_{j=1}^M \bm{\Psi}(x_j)\bm{\Psi}(x_{j+1})^T,~ \bm{G} := \frac{1}{M}\sum_{j=1}^M \bm{\Psi}(x_j)\bm{\Psi}(x_j)^T.
\end{equation}
\section{Algorithm for Incident Detection}\label{sec:algorithm}
In this work, we leverage a sparsity-promoting variant of the EDMD algorithm \cite{quade} for the detection of incidents in the distribution grid, given by
\begin{equation}\label{eq:ksparse}
\underset{\mathcal{K}\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}-\bm{G}\mathcal{K}||^2_F + \lambda ||\text{vec}(\mathcal{K})||_1.
\end{equation}
For our algorithm, we consider the following notation. For time-series data $x$ and $t_1\leq t_2$, define
\begin{equation}
x[t_1:t_2] := (x_{t_1}x_{t_1+1},...,x_{t_2}).
\end{equation}
At time $t$, let $\bm{A}[t-T:t]$ and $\bm{G}[t-T:t]$ be computed with $T+1$ observations and their one-step propagation in the observed data series $x[t-T:t]$.
Our goal is to maintain a sparse representation of the dynamic system, and detect incidents by detecting changes in this sparsity pattern. For that, we build on \eqref{eq:ksparse} to define our algorithm for incident detection as follows:
\begin{itemize}
\item At time $t=T+1$, compute $\mathcal{K}_{t}$ by solving
\begin{equation}
\underset{\mathcal{K}_t\in \mathbb{R}^D \times \mathbb{R}^D}{\min} ||\bm{A}[1:T+1]-\bm{G}[1:T+1]\mathcal{K}_t||^2_F + \alpha ||\text{vec}(\mathcal{K}_t)||_1.
\end{equation}
\item From $t=T+2$ onward, compute $\mathcal{K}_{t}$ by solving
\begin{equation}
\begin{split}
\underset{\mathcal{K}_t\in \mathbb{R}^D \times \mathbb{R}^D}{\min} &||\bm{A}[t-T:t]-\bm{G}[t-T:t]\mathcal{K}_t||^2_F \\ &+\alpha ||\text{vec}(\mathcal{K}_t)||_1+\beta ||\mathcal{K}_{t-1}-\mathcal{K}_t||_F.
\end{split}
\end{equation}
\item If the sparsity pattern of $\mathcal{K}_{t}$ differs significantly from that of $\mathcal{K}_{t-1}$, flag the occurrence of an incident that has altered the causal map between a state and its one-step propagation.
\end{itemize}
The sparsity pattern of each $\mathcal{K}$ encodes the causal relationship among observables, and the level of sparsity is adjusted by the choice of the hyper-parameter $\alpha$. Small values of $\alpha$ define more complete networks, while large values promote sparsity. The hyper-parameter $\beta$ is related to the smoothness of transitions. Thus, larger values of $\beta$ tend to stabilize $\mathcal{K}$ along a trajectory, making its less prone to change due to smaller variations which do not lead to major structural changes.
\section{Methodology for Incident Detection}\label{sec:methodology}
Our methodology can be divided into two distinct parts, as illustrated in Fig.~\ref{fig:method}. The first part consists of a data collection step, followed by a data transformation. The transformed data is then used to compute the $\mathcal{K}$ matrices using the sliding window approach introduced in Section~\ref{sec:algorithm}. The second part is a post-processing clustering task which performs a clustering analysis with the goal of grouping the $\mathcal{K}$ matrices calculated into clusters that are uniquely identified by one of the scenarios simulated in the data collection step. Each part of this approach is discussed in more detail in the next sections.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/methodology.png}
\caption{Two-part methodology developed for the detection of incidents in distribution grids.}
\label{fig:method}
\end{figure}
\subsection{Part 1: Koopman Operator}
\subsubsection{Data Collection}
We begin by selecting suitable simulation scenarios from which our data will be gathered. Since our focus is on incidents related to the operation of ESS in the distribution grid, the following two broad types of events were considered:
\begin{enumerate}
\item Changes in ESS charging/discharging rate: In distribution systems, ESS can be used for peak-shaving operations, absorption of excess local renewable production, and other tasks which involve a coordinated charging/discharging schedule. We aim to analyze whether changes in the charging/discharging rates of ESS can be identified with our proposed framework. Detecting such changes would allow us to recognize, for example, when a battery fleet is no longer following the command issued for their power supply/absorption operation.
\item Changes in ESS controller parameters: Voltage regulation is an important task in power distribution systems, especially with the increasing adoption of DERs. By considering a scenario in which batteries offer voltage regulation services, we model Volt/VAR curves to control the reactive power output of these devices. The goal is to detect changes in controller parameters, which could have been caused by attacks that aim to destabilize the grid.
\end{enumerate}
We assume the data collected comes from PMUs, and thus we use voltage magnitude and phase measurements. The data was collected with a 0.25s sampling time. Because sensor measurements are typically noisy, Gaussian noise was added to the voltage magnitude ($\pm 0.01\%$) and voltage angle ($\pm 0.01\degree$) measurements. Further, considering that sensing capability would be limited in the distribution grid, we assumed that only nodes with an ESS had PMUs. These scenarios were simulated using OpenDSS.
\subsubsection{Data Transformation}
We hypothesize that the relationship between different time-series data carries more information than individual data streams. In \cite{fujii}, this dependence among observations is explored in a data-driven spectral analysis using the Koopman operator with the objective of understanding complex biological network dynamics. For this purpose, using a ballgame as an example, the authors transformed the data using the Gaussian kernel
\begin{equation}\label{eq:gaussian}
g(x_i,x_j) = \exp\left(-\frac{||x_i-x_j||^2_2}{2\sigma}\right),
\end{equation}
where each $x_i$ is a stream of raw data, $||.||_2$ is the Euclidean norm, and $\sigma$ is an adjustment parameter.
We apply \eqref{eq:gaussian} to the distribution grid data to be used in our analysis. This nonlinear transformation lifts the data into a higher dimensional space by giving a measure of similarity between states, thus augmenting our dataset. The transformed data is then used to compute a sequence of $\mathcal{K}$ matrices through a sliding window.
\subsubsection{Approximate Koopman Operator}
In this step, the finite-dimensional approximation of the Koopman operator for the transformed data was performed numerically following the steps detailed in Section~\ref{sec:algorithm}. Significant changes in $\mathcal{K}$ indicates altered causality in states (i.e., occurrence of incidents). Radial basis functions of the form
\begin{equation}\label{eq:rbf}
\Psi_i(x) = ||x-c_i||^2\log_e(||x-c_i||),
\end{equation}
where $c_i$ are unique center points, have shown to be effective for our application. We face the following trade-off when choosing the amount of dictionary functions to be used: too few dictionary functions $\Psi_i(x)$ may lead to poorer distinguishability of the sparsity pattern of $\mathcal{K}$'s from different incidents; increasing the amount of functions, however, also increases computational time.
\subsection{Part 2: Offline Clustering}
The previous step was aimed at generating data for different case studies, transform the data collected, and compute the approximate Koopman operator for multiple time windows. The offline clustering task performs a clustering analysis in the $\mathcal{K}$ matrices calculated. Since the sparsity pattern of these matrices is expected to be similar if the system dynamics remains the same, this analysis is expected to cluster together the $\mathcal{K}$'s coming from the same simulated scenario. For example, the matrices calculated from a scenario in which a battery fleet is discharging at
25\% rate should be clustered together, while $\mathcal{K}$'s from a scenario where these batteries are discharging at 100\% should be together but in a different cluster.
We used the k-means clustering algorithm for this analysis. The idea behind k-means is that observations within a cluster are close to each other, while observations in different clusters are far apart. This method takes as inputs a distance measure, the number of clusters, the data to be clustered, and a random seed that is used to initialize the algorithm. For our application, the correlation distance was shown to perform better than other metrics. Further, prior to the clustering analysis, the $\mathcal{K}$ matrices were transformed into binary matrices, i.e. values below a certain threshold were set to zero and all others were set to one. The purpose behind this transformation is that we are only concerned with the sparsity pattern of these matrices, and not with the actual values of their elements. This step also avoids numerical issues that may arise when working with small numbers. Since the approximate Koopman operator is a representation of our original data in a high-dimensional space, it is common to see elements of $\mathcal{K}$ which have a really small order of magnitude. The effectiveness of our results is evaluated based on the misclassification rate achieved in this clustering analysis.
\section{Case Studies}\label{sec:cases}
We considered the IEEE 8500-node test feeder with 7 battery energy storage systems (BESS) added, implemented using OpenDSS. This feeder is an unbalanced radial network, which are typical characteristics of distribution grids. These BESS can take real power dispatch commands and perform Volt/VAR control. The data was collected with a 0.25s sampling time, and Gaussian noise was added to the voltage magnitude ($\pm 0.01\%$) and voltage angle ($\pm 0.01\degree$) measurements. Besides noise, fluctuations in real and reactive load were considered in the feeder model. Our aim is to evaluate the robustness of our algorithm when these variations are accounted for.
As previously discussed, changes in the sparsity pattern of the $\mathcal{K}$ matrices indicated the occurrence of an incident. To confirm the $\mathcal{K}$ matrices had unique sparsity patterns for each scenario in which the causality of the model was maintained, an offline analysis was performed using k-means clustering to cluster these matrices. For both case studies, the centers for the radial basis functions $c_i$ in \eqref{eq:rbf} were chosen in the computation of the first $\mathcal{K}$ matrix, and then kept constant throughout the experiment. The center points were selected using a randomly generated perturbation, so that they were of the same order of magnitude as the data itself. Further, we used $D=400$ dictionary functions, a time window of $T=100s$, and a new $\mathcal{K}$ was estimated every $25s$. We also assumed voltage magnitude and phase measurements from PMUs were available at the nodes of 3 of these BESS.
\subsection{Detecting changes in discharging rates}
We first consider a scenario in which the battery fleet is initially idle. After 5min, all the batteries start discharging at 50\% of their discharging rate, maintaining this rate for 5min. Fig.~\ref{fig:case1} shows the results for this case. For the Koopman operator approximation step, we let $\alpha=0.1$ and $\beta=0.4$.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/casestudy1.png}
\caption{Results for case study which aims to detect changes in the rate at which the battery fleet supplies active power to the grid.}
\label{fig:case1}
\end{figure}
The red numbers in Fig.~\ref{fig:case1} mark the time instants at which a new $\mathcal{K}$ matrix was estimated. The plots on top show a representative sparsity pattern for each of the two situations considered, and we can clearly observe that this pattern changes after the incident happens, allowing us to successfully detect its occurrence. To corroborate that the two patterns are indeed distinct, all the 20 $\mathcal{K}$ matrices computed were subjected to a clustering analysis using k-means. For this case study, we have two different clusters: the first representing our network dynamics when the batteries are idle, and the second related to the case in which the battery fleet discharges at 50\% of their rate. As a result, we found that 3 of these matrices were misclassified, i.e. they were classified to the cluster that does not correspond to the scenario they actually refer to. We note that the misclassified matrices, highlighted in Fig.~\ref{fig:case1}, were computed immediately after the event occurred. Due to the sliding window approach, their computation used data from both the first and the second scenarios, which impacts their sparsity pattern. However, the matrices after those were correctly classified. Thus, we can state that some time may be needed after the event for the new sparsity pattern to stabilize. We remark that the results were robust to measurement noise, as well as load variations, since these fluctuations did not lead our algorithm to erroneously identify the occurrence of nonexistent incidents.
\subsection{Detecting changes in controller parameters}
In this case study, we considered the battery fleet provides voltage regulation services. For this purpose, the hierarchical Volt/VAR (VV) control strategy proposed in \cite{quiroz} is used to control the reactive power supply/absorption of these devices so that the voltage magnitudes are maintained close to their nominal value. Each battery is assigned a VV curve, which is designed to adjust the reactive power of the device according to measured local voltage levels. An example of this curve is shown in Fig.~\ref{fig:vvcurve} \cite{quiroz}. The VV curve has a deadband corresponding to acceptable voltage levels which do not trigger control action for correction. When voltage levels become high, the battery absorbs reactive power to lower the voltage back to acceptable levels. The opposite happens when voltage levels are low, with the batteries injecting reactive power into the grid. This control layer is local, as each device does not have information about the overall system condition, and only responds to their local condition.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{figs/vvcurve2.png}
\caption{Example of VV curve and shifting logic \cite{quiroz}.}
\label{fig:vvcurve}
\end{figure}
To overcome this limitation and expand the control strategy to regulate voltage levels throughout the network, a centralized control layer is added. This layer has full information about the voltages over the entire feeder, and is able to dispatch new VV curves to the devices with the goal of regulating the overall voltage in the system, even when the device's local voltage magnitude is within acceptable ranges. Fig.~\ref{fig:vvcurve} shows an example of how the initial VV curve assigned to a battery can be shifted using the proposed centralized strategy. We note that shifting the curve to the left induces an increase in reactive power absorption, which helps decreasing voltage levels more significantly across the network. The reverse holds for right shifts in the curve.
Using this hierarchical VV control, we modeled two events for this case study, leading to a total amount of three different scenarios. For each event, the controller parameters are changed by adjusting the set-points for the VV curves assigned to the batteries. The dead band of the controller changes from 0.95 to 0.98 and then 0.99 pu, with saturation point shifting right accordingly at $t=200s$ and $t=400$, respectively. The results for this case study are presented in Fig.~\ref{fig:case2}, in which we observe three distinct sparsity patterns for the $\mathcal{K}$ matrices in each scenario. These patterns were achieved by using $\alpha=0.1$ and $\beta=0$ in the Koopman operator approximation. Similarly to the first case study, the differences in the $\mathcal{K}$ matrices were sufficient for us to identify the occurrence of the events considered. Further, in the offline classification task, 3 out of the 32 matrices were misclassified, all of which were computed in time instants immediately following an event.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figs/casestudy2.png}
\vspace{-5mm}
\caption{Results for the detection of changes in the controller parameters of batteries providing voltage regulation through Volt/VAR control.}
\label{fig:case2}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
We proposed a data-driven method that requires no prior knowledge of the network dynamic model for the detection of incidents in power distribution systems. Our case studies considered the occurrence of changes in BESS operations~that refer both to active power dispatch and reactive power control of these devices. The successful detection of the events modeled allows for proper mitigation strategies to be set forth, if needed. Our methodology introduces a data transformation step which augments the dataset while maintaining meaningful information of the system states. This strategy is particularly useful in situations with restricted sensor availability, which can be common in distribution grids. Further,
the algorithm proposed was shown to be robust to measurement noise and load fluctuations, and thus such variations do not trigger a~false detection. Future work includes
designing a more systematic way to choose the hyper-parameters for the Koopman operator approximation, and evaluating our methodology in the presence of data loss or data streams with different~sampling~time.
\section*{Acknowledgment}
This research was funded by the energy storage program at the U.S. Department of Energy under the guidance of Dr. Imre Gyuk. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy National Nuclear Security Administration under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Jamal Al Hourani helped with some numerical experiments. Hyungjin Choi helped with the initial code and explanation of Koopman operator calculations. The work of the first two authors was supported in part by Sandia National Lab under the grant PO 2079716.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,225 |
Waigeo liveaboards will usually visit here as part of a Raja Ampat itinerary. It is situated to the North of West Papua, Indonesia. The island is the largest in the famous Raja Ampat archipelago, renowned for its diverse flora and fauna both above and below the ocean. The region was discovered by Portuguese explorers in the 16th century and later studied by British explorer and naturalist Alfred Wallace, who studied some of the islands fauna in the 19th century. A wide selection of Indonesian liveaboard operate itineraries tthroughout Raja Ampat which will include Waigeo.
The island has a booming pearl farming trade in addition to being a popular tourist destination with a few resorts available for visitors. The pearl farms can be seen around the island and the one at Sele Pele is open to visits if they are booked in advance. This is a fascinating opportunity learn about the history of pearl fishing and to see how pearls are seeded and grown in farms nowadays. Waigeo is also home to the region's capital Waisai which is a modest city with some basic tourist facilities. Other than Waisai the island is, like much of the Raja Ampat islands quite isolated and untouched.
Liveaboards to Waigeo offer a diverse range of dive sites, for a variety of different interests and levels of diving experience. The sites are normally around 25-30 meters deep with excellent visibility, sometimes up to 30 meters. The currents on the South-west end of the island are the most severe, and dive sites on this side are not really suitable to absolute beginners.
The diving is mostly on coral reefs, some with deeper areas and drop-offs where you can see some sharks. The main reason to take a dive cruise to Waigeo however, is the sheer number and variety of fish and critters you can find in every corner of the reefs here. There are lion fish hiding in small caverns, intricately coloured juvenile emperor angel fish can be found close to the corals along with napoleon wrasse and fusiliers.
Waigeo is also a top location in Raja Ampat, for diving with cephalopods. There are numerous different kinds on the reefs including, squid, blue ringed and red octopus and cuttlefish. There are also occasionally turtles at some of the dive sites in Waigeo which can be seen grazing on the reefs. Macro lovers won't be disappointed either, with mantis shrimp and the stunning sea dragons to photograph.
Four Kings dive site is one of the most stunning sites. Named after the archipelago, this reef boasts some of the finest corals in the region. This place is a photographers dream, with a rainbow of colour, critters and reef fish galore. Bird wall to the North is also rich with fish life and the occasional green turtle too.
Sele Pele, and the jetty that is located near the pearl farm, are both critter heaven. There are frogfish, razor fish and cuttlefish and octopus to be found in these reefs. The advantage of these two sites is, that unlike much of this region the current and conditions are normally quite mild.
The Corner on the South-western side, offers up a more challenging dive with strong currents along the reef. The payoff is well worth the extra effort to stay on though, white and black tipped reef sharks are common here in deeper waters. If you keep a keen eye out on the bottom it is also possible to see the rare Wobbegong sharks resting on the slope.
The currents in parts of Waigeo can be milder than other areas and so these might make a better dive for those with less experience.
The water is warm but bear in mind that there are fire urchins and stone fish in this area so consider covering you skin.
Waigeo town has more facilities than most of the Raja Ampat islands, but this is a frontier destination so make sure you bring lots of spares of anything you need with you.
To reach Waigeo visitors must first reach West Papua, where the closest airport to the Raja Ampat islands is located. The airport of Sorong has flights from Jakarta and Singapore but not international flights. From Sorong there are two types of boats to Waigeo, the faster hydrofoil and the slower ferry boat which takes around two hours.
The most popular way for divers to reach this region however is a Indonesian diving liveaboard which normally depart from Sorong. Due to the isolated nature of the sites the majority of diving in Raja Ampat is done from dive cruises. There are various kinds of boats, to suit all budgets and tastes. There are customised motor vessels which offer comfort and top diving facilities. The most popular however are the Indonesian sailing boats, called phinisi, which offer a luxurious and traditional experience.
Beautiful reefs and lot of fish!
Sardine Reef, Cape Kri, Mike's Point in Dampier Straits near Waigeo all very healthy reefs. Good slope and wall dives. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,402 |
## TABLE OF CONTENTS
Introduction
NORTH AMERICA
* * *
New York, New York
Washington, DC
Freeport, Maine
Woodstock, Vermont
Weston, Vermont
Boston, Massachusetts
Los Angeles, California
Santa Barbara, California
Palm Springs, California
San Francisco, California
Seattle, Washington
Portland, Oregon
Santa Fe, New Mexico
Austin, Texas
Dallas & Fort Worth, Texas
Houston, Texas
Charlotte, North Carolina
Atlanta, Georgia
Palm Beach, Florida
Miami, Florida
New Orleans, Louisiana
Nashville, Tennessee
Chicago, Illinois
Detroit, Michigan
Toronto, Canada
Montréal, Canada
Vancouver, Canada
MEXICO & SOUTH AMERICA
* * *
Mexico City, Mexico
San Miguel De Allende, Mexico
Tulum, Mexico
Rio de Jainero, Brazil
Sao Paolo, Brazil
Buenos Aires, Argentina
Lima, Peru
Cusco, Peru
EUROPE
* * *
London, UK
Dublin, Ireland
Paris, France
Milan, Italy
Rome, Italy
Florence, Italy
Venice, Italy
Madrid, Spain
Barcelona, Spain
Berlin, Germany
Stockholm, Sweden
EURASIA
* * *
Istanbul, Turkey
Moscow, Russia
ASIA
* * *
Tokyo, Japan
Hong Kong, China
Shanghai, China
Beijing, China
Seoul, South Korea
Mumbai, India
New Delhi, India
MIDDLE EAST
* * *
Riyadh, Saudi Arabia
Jeddah, Saudi Arabia
Dubai, UAE
AFRICA
* * *
Marrakech, Morroco
Freetown, Sierra Leone
Lagos, Nigeria
Lamu, Kenya
Nairobi, Kenya
Accra, Ghana
Cape Town, South Africa
AUSTRALIA
* * *
Sydney
Melbourne
Contributors
Acknowledgments
About Booth Moore
Image Credits
## INTRODUCTION
### Why write a book about shopping now?
Every day, headlines herald the death of brick-and-mortar stores. There are so many abandoned shopping centers, we now have a name for them—ghost malls. Amazon.com gets bigger by the click, and as soon as there is drone delivery, we won't ever have to leave our homes to buy anything.
But then what?
I decided to write a book about shopping now because there is still so much joy in it. It's thrilling to discover that special find or bargain on your travels that no one else at home will have, a part of the experience of creating a style uniquely yours. Whether you are treating yourself, or someone else is treating you, there's nothing quite like it. I'll never forget my friend, Heather, who, upon leaving a store with a new pair of shoes, would put them in the passenger seat of her car, box top off, so she could cast loving glances at her new beauties all the way home.
Originally, I wanted to find out where stylists shop, because they are the gatekeepers to the best clothing and accessories worn by celebrities and models on red carpets and in magazines—they are the ultimate insiders.
But when I started to think about it, I realized that everyone in the fashion industry has shopping intel. Designers, editors, retailers, models, and bloggers have it, too; in fact, anyone who loves fashion and style has it.
Knowing the newest, the old schooliest, the hidden gems, and the unexpectedly chic is valuable currency in the fashion business. Every season that I've covered the runway shows, I stay up way too late figuring out where to shop, combining research from magazines (even the ones in French and Italian), blogs, and travel guides, with tips passed down the front row from editor to editor.
Between shows, I'd venture out and explore. Inevitably, I'd see other in-the-know colleagues treading the same path to the Marni outlet in a residential neighborhood in Milan, where you have to buzz an intercom for the creaking gate to be opened. Or, I'd spot them gawking at the stuffed animals in the sprawling, museumlike taxidermy mecca of the world in Paris.
What if you could have all the best places, and the industry secrets, in one handy guide? That's what this book is. I surveyed 175 people in the fashion industry, including celebrity stylists, designers, bloggers, editors, street-style stars, models, and store owners and asked them to share their favorite shopping destinations. I used their answers to compile a global guide with a special focus on fashion hubs New York, London, Milan, Paris, Los Angeles, and Tokyo, cutting-edge fashion destinations such as Cape Town and Seoul, and places that the style set frequents on holiday or for inspiration, such as Marrakech and Tulum.
The book tells you about the designer stores you have to visit to really understand the fashion climate of a country, as well as the stores you might never have heard about otherwise, like the best place in Paris to get Indian block-print tunics and tablecloths, or the best T.J. Maxx ($37 Marc Jacobs dresses!) outside Boston. It's the stores you have to see IRL because they are just fun, such as Bedrock in Tokyo, which can only be accessed through the Forbidden Fruit café; the Doota mall in Seoul where you can shop for the latest teeny-bopper fashion twenty-four hours a day; and Mall of the Emirates, which has its own indoor ski slope.
Although this guide is devoted largely to brick-and-mortar stores around the world, the thrill of the hunt and the find can be experienced online, too, if you know where to look, which is why I've included Internet tips—from the best Etsy shop to find traditional boro-style Japanese clothing to a fantastic resource for discovering the hottest new African designers.
Not a week goes by when I don't ask a total stranger, "Where did you get that?" Recently, it was the lady pouring my iced coffee on Third Street in Los Angeles who was wearing a fabulous pair of gold nameplate earrings that spelled out BONITA. She told me they came from the Slauson Super Mall. I went the next weekend and bought my own. They were made by a woman named Boss Lady for $100 in less than two hours, and packaged in a robin's-egg-blue bag as if they were from Tiffany & Co.
At its best, shopping is about discovery, and there's no better way to get to know a city than by hitting the stores. You can find out so much about a place through its local designers, its markets, and its favorite chains. The things you come home with create a scrapbook of your trip, which you can relive every time you wear them. Done over a lifetime, shopping is a collection of experiences that informs your fashion sense. My style has been influenced by trying to dress like TV character "Murphy Brown" when I was a teenager, using Ralph Lauren pieces scored at one of the label's first outlets near my parents' house in Manchester, Vermont; by the Native American jewelry I collected on multiple trips to the Southwest; by the sparkly socks amassed on visits to the Tabio store in Paris during fashion week; and by my desire to mimic the style of my California rock 'n' roll idols Stevie Nicks and Belinda Carlisle, first using treasures scored at vintage stores in New York City, where I grew up, and eventually in Los Angeles, where I live now.
Shopping is about the thrill of the hunt. I'll never forget, in the days before the Internet, calling Lord & Taylor store after Lord & Taylor store up and down the Eastern seaboard, looking for a pair of pink silk shantung Enzo Angiolini slides in my size. I finally found a pair, after an entire Saturday afternoon on the phone, and made the drive all the way to Paramus, New Jersey, just to pick them up. I couldn't wait to wear the shocking pink shoes and tell everyone the lengths I had gone to find them. Moral of the story? No serious shopping conquest is really over unless, and until, you have scored. And then, there's nothing better than showing it off.
Everyone has a shopping story.
Happy hunting.
## NEW YORK
### NEW YORK
LOVE ADORNED
#### THE MAJORS
#### ABC Carpet & Home
* * *
This mecca for home design, tabletop, and apothecary items bills itself as a "three-dimensional living magazine and interactive museum." The original New York location also carries quirky gifts, such as Macon & Lesquoy embroidered pins, flowy dresses by Ryan Roche, Giada Forte, and Vlas Blomme, metallic clutches by Tracey Tanner, delicate jewelry by Sweet Pea and Elisa Solomon, and chakra-balancing bracelets by Tiny Om. You might see fashion designer Mara Hoffman wandering the floor, searching for inspiration: "I feel like I'm shopping in a hundred different countries during just one visit," she says. Grab lunch at one of Jean Georges Vongerichten's on-site restaurants, ABC Kitchen or ABC Cocina.
888 Broadway, New York, NY 10003, +1-212-473-3000, abchome.com.
#### Barneys New York
* * *
The Madison Avenue flagship showcases the edgy boho mix that Barneys is known for. The collection includes bags by Goyard, Loewe, and Céline, jewelry from Irene Neuwirth, Jennifer Meyer, and Sidney Garber, and an unparalleled collection of clothes by Dries Van Noten, Azzedine Alaïa, Dolce & Gabbana, Narciso Rodriguez, as well as contemporary labels like Ulla Johnson, Isabel Marant, Raquel Allegra, and Pas de Calais. Don't miss lunch at Freds. The Palm Beach shrimp salad is designer Joseph Altuzarra's favorite, and jewelry designer Meyer splurges on the Robiola with truffle oil. There's also a new Chelsea location that's worth checking out.
660 Madison Ave., New York, NY 10065, +1-212-826-8900, barneys.com, and other locations.
#### Bergdorf Goodman
* * *
The grande dame of American department stores, Bergdorf Goodman sits on the site once occupied by the Cornelius Vanderbilt II mansion. Inside, you'll find every luxury label on the planet (Dries Van Noten, Gucci, Oscar de la Renta, Altuzarra, Libertine, to name just a few), one of the world's best women's shoe departments, and fab finds from emerging labels. Personal note: I bought my first pair of high heels here as a teenager, at the now-defunct Delman shoe salon. Today, the second-floor shoe department is not to be missed, with fancy footwear from Chanel, Aquazzura, Roger Vivier, Christian Louboutin, and more. "The label whore in me continues to go back to Bergdorf for my fancy footwear. Whether it's a fancy Valentino or a platform from Stella McCartney, it's always the best. And, if you can hold out, their sales are beyond," says fashion editor Sasha Charnin Morrison. Dozens of our stylish shoppers cited it as a favorite. "It's windowless and cozy, even a little bit cramped, but the edit of the shoes is flawless. I basically want to live there," says stylist Kemal Harris. Enjoy a Gotham salad, a glass of champagne, and views of Central Park at the Kelly Wearstler–designed BG Restaurant on the top floor, which also boasts an extensive collection of home accessories and gifts. "They have this really cool section of vintage objects and barware that I'm obsessed with, and don't even get me started on their room of ornaments during the holidays." says Empire costume designer Paolo Nieddu. The men's store is across the street.
754 Fifth Ave., New York, NY 10019, +1-888-774-2424, bergdorfgoodman.com.
#### Bloomingdale's
* * *
From hoopskirts (that's what the Bloomingdale brothers got their start selling in the 1860s) to the iconic Bloomie's bikini underwear that made the store a tourist attraction starting in the 1970s, Bloomingdale's is a classic. The 59th Street flagship can't be beat, especially for its window displays and in-store boutiques featuring pop culture merchandise, whether souvenirs from the movie Star Wars or the Broadway musical Hamilton. But the place deserves high-fashion props, too. "Often when I go to Forty Carrots for fro-yo, I get sucked into the designer floor; I've found that they tend to have the best Chanel shoe sales, because few people actually frequent Bloomingdale's for its selection of Chanel," says Man Repeller's Leandra Medine.
1000 Third Ave., New York, NY 10022, +1-212-705-2000, bloomingdales.com, and other locations.
#### Dover Street Market
* * *
Located in a seven-story Beaux Arts building sans sign on the corner of Lexington and 30th Street near New York's Curry Hill, this is the only stateside location of Japanese designer Rei Kawakubo's men's and women's avant-garde-fashion superstore. Inside, you'll find Comme des Garçons and other labels under its umbrella, including Play, Black, Shirt, and Junya Watanabe, as well as an edited mix of high-end labels (Simone Rocha, Gucci, J.W. Anderson, Rick Owens, Gosha Rubchinskiy, Vetements, Sacai) and streetwear (Supreme, Facetasm, Visvim, Hood by Air). The interior is equally inspiring, from the colorful installation by the mother of yarn bombing, Magda Sayeg, to the "Biotopological Scale-Juggling Escalator," a tunnel-shaped stairway designed by visionary architect Madeline Gins that connects the second and third floors. "One of the best curated stores in the world, DSM mixes young street-skate brands like Alltimers with the heavy hitters like Louis Vuitton," says Norstrom's head of creative projects, Olivia Kim. "I grab lunch at Rose Bakery on the ground floor, then take the elevator up and circle each of the seven floors," says accessories designer Sally Perrin. "It's small enough not to feel overwhelmed by the selection; the pieces are unique and avant-garde, so I always feel like I've come away with a piece of wearable art." "I love the museum-y feel of all the different floors," says retailer/designer Ann Mashburn, adding, "the staff is so incredibly pleasant and approachable, especially for such an over-the-top fashiony place, and I love the Japanese toilets."
160 Lexington Ave., New York, NY 10016, +1-646-837-7750, shop.doverstreetmarket.com/us, and other locations.
#### Saks Fifth Avenue
* * *
The first Saks opened in 1924 between 49th and 50th streets on upper Fifth Avenue in what was at the time primarily a residential neighborhood. Now, it has grown into an international empire of stores. The flagship is already costume designer Jenn Rogien's go-to for "a constantly evolving assortment," and come 2018, it will have even more to offer, after a major renovation that will move the beauty department to the second floor, and enhance it with a brow bar, 16 spa treatment rooms, and a flower shop. There will be a high jewelry vault in the basement, a replica of Coco Chanel's apartment upstairs, and the first US outpost of Paris' L'Avenue restaurant with terrace views of Rockefeller Center and St. Patrick's Cathedral. Add to that the already world-famous shoe department with its own zip code, 10022-SHOE, and Saks is shaping up to be the new luxury standard. Designer Mary Alice Haney praises the salespeople as tops for pulling special pieces, "if you are looking for pieces for an event and don't want to worry about wearing the same thing as someone else." 'Nuff said.
611 Fifth Ave., New York, NY 10022-SHOE, +1-212-753-4000, saksfifthavenue.com, and other locations.
### SOHO
#### Alice and Olivia
* * *
Stacey Bendet's New York–based contemporary clothing label is beloved by Gwyneth Paltrow, Katy Perry, Dakota Fanning, and other celebs for its girlish print dresses and blouses, art and pop cultural collabs, and charitable initiatives. Recent collections have been inspired by The Grateful Dead (Bendet's husband, Eric Eisner, is producing a Grateful Dead documentary with Martin Scorsese) and the artist Jean-Michel Basquiat.
98 Greene St., New York, NY 10012, +1-646-790-8030, aliceolivia.com, and additional locations.
#### De Vera
* * *
This store slash gallery boasts a finely curated selection of Venetian glass, ivory carvings, and jewelry—whether its Georgian brooches or Sardinian coral pendants. It's Tory Burch's go-to for "exquisite antique jewelry and the most interesting collection of objets d'art." "Everything from Japanese lacquer to Roman ephemera is available. It's truly a jewel box of a store," says shoe designer Paul Andrew. Joseph Altuzarra calls it "a treasure trove," and it is shoe designer Brian Atwood's favorite place for gifts.
1 Crosby St., New York, NY 10013, +1-212-625-0838, deveraobjects.com.
#### Gentle Monster
* * *
We challenge you to walk out of the flagship of this young Korean eyewear brand without a pair of sunglasses. The immersive space with rotating art installations is so calm and Zen-like, you'll want to stay and try on every pair of confetti-flecked acetate cat-eye, mirrored round, and goggle-style sunnies (and then take selfies). It doesn't hurt that the brand has collaborations with cool-kid labels such as Opening Ceremony, Tome, and Hood by Air, and most frames are less than $300.
79 Grand St., New York, NY 10013, +1-646-928-1694, en.gentlemonster.com, and other locations.
#### Journelle
* * *
It's rare to find a lingerie store that focuses on the tried-and-true brands (Chantelle, Eberjey, La Perla) as well as up-and-comers (Le Petit Trou, Paloma Casile), but Journelle is that store. "Part of the fun of being a woman is being able to wear something supersecret and sexy under a suit and feel great about it," says accessories designer Aurora James. "A new pair of undies or bra is a pick-me-up. Journelle has a great selection of everything from cute panties to seductive lingerie at a great cross section of prices."
125 Mercer St., New York, NY 10012, +1-212-255-7803, journelle.com, and additional locations.
JOURNELLE
#### Kirna Zabête
* * *
This boutique, started in 1999, helped pioneer high fashion in SoHo. Graphic displays, neon signs, and colorful chandeliers lend a pop art sensibility to the 10,000-square-foot store, which offers a whimsical edit of pieces both super-high-end (Céline, Givenchy, Gucci, Valentino, Fendi) and contemporary cool (Frame Denim, Zimmerman, Piamita, Mira Mikati, Ellery, Sarah's Bags, Yazbukey). "Owner Beth Buccini has an impeccable eye," says fashion insider Alina Cho. "She always seems to buy that dress, shoe, or bag I never knew I wanted but suddenly can't live without. . . . I can find established designers like Gucci and Chloé as easily as I can Rosie Assoulin and Isa Arfen."
477 Broome St., New York, NY 10013, +1-212-941-9656, kirnazabete.com.
#### Odin New York
* * *
Casual-cool menswear by Comme des Garçons Shirt, Duckie Brown, Acne Studios, Won Hundred, Engineered Garments, Common Projects, and more, plus basics and fragrance by Odin's in-house label. "A small shop but very well curated," says Sabah founder Mickey Ashmore. "Feels more like my closet than a store," says fashion editor/TV host Joe Zee.
161 Grand St., New York, NY 10013, +1-212-966-0026, odinnewyork.com, and an additional location in the East Village.
#### Patron of the New
* * *
"For a serious fashion fix, the style here is bold and brash with an eye for the avant-garde," says Hong Kong–based fashion journalist and blogger Divia Harilela. Inside this darkly romantic boutique with a Parisian vibe, you'll find second-skin minidresses by Balmain and Alexandre Vauthier, blinged-out sneakers by Buscemi, harem shorts by DRKSHDW, Fear of God bomber jackets, Unravel destroyed knits, Zilla handbags made out of artificial sponges, and other goodies. Men's and women's.
151 Franklin St., New York, NY 10013, +1-212-966-7144, patronofthenew.us.
#### Rachel Comey
* * *
Comey's namesake label, which began as menswear, grew out of the New York downtown arts scene in the early 2000s. Her collections are a favorite with Rashida Jones, Taylor Schilling, and others for their arty textiles (foam, linen, pebble, crochet), modern silhouettes, and covetable footwear. Comey and her partner Sean Carmody collaborated with Brooklyn-based architect Elizabeth Roberts and San Francisco–based interior designer Charles de Lisle on the spare design of her store with a wood beam ceiling and skylight to showcase denim culottes, drapey trench coats, and chunky wood-soled sandals and boots.
95 Crosby St., New York, NY 10012, +1-212-334-0455, rachelcomey.com, and an additional location in Los Angeles.
RACHEL COMEY
#### Rebecca Minkoff
* * *
In 2004, Rebecca Minkoff's first runway show was taped as a plotline for the TV shows Laguna Beach and The Hills, but it wasn't until 2005, when she debuted her "Morning After Bag," that things really took off. A satchel designed to be small enough for a night on the town but large enough to carry a change of clothes, the style attracted a celebrity following that included Jessica Alba and Reese Witherspoon. Now, Minkoff has a full-fledged collection of clothing and accessories, loosely inspired every season by rock 'n' roll. Several of her boutiques, including this one, are "wired" for millennial-friendly shopping, with touchable video walls and digitized dressing room experiences.
96 Greene St., New York, NY 10012, +1-212-677-7883, rebeccaminkoff.com, and additional locations.
RACHEL COMEY
NYC LOCAL FAVES
* * *
Rachel Comey is a New York–based fashion designer whose career kicked off with a collaboration with the band Gogol Bordello, earning her a place in the 2001 Whitney Biennial. Her namesake women's label, which launched in 2004, is sold at more than a hundred high-end boutiques and stores in sixteen countries as well as stand-alone stores in New York and Los Angeles.
PEARL RIVER in NYC was always a favorite for home items and party decorations (it's online now). HOUSING WORKS for old books. ESSEX CARD SHOP ON AVENUE A for Le Pens. There are some old bra and underwear shops on lower ORCHARD STREET that are still there and fun for surprising negligees or undergarments. For me, the search is as thrilling as the find.
I love the FLEA MARKET ON 26TH STREET. I know it's been reduced to one open lot, but there are still some great vendors there. I love our store on Crosby Street. Crosby Street has so many great stores, so if you are looking for locally designed and manufactured productions, then a visit to Crosby Street is a must. JILL PLATNER on Crosby north of Prince is a great jewelry designer; just below, is us and TESS GIBERSON. Then heading south you eventually get to BDDW, such beautiful furniture in an epic storefront.
My friend Leanne just turned me on to the tin plates at the FRICK COLLECTION. Such a beautiful alternative to disposable plates or carrying your heavy ceramics outside for a picnic. While uptown, a visit to the bookstore ALBERTINE at the French Embassy is a secret must-visit spot!
PEARL RIVER MART, PearlRiver.com.
HOUSING WORKS BOOKSTORE CAFÉ, 126 Crosby St, New York, NY 10012, +1-212-334-3324, housingworks.org.
ESSEX CARD SHOP, 39 Avenue A, New York, NY 10009, +1-212-228-7740.
CHELSEA FLEA MARKET, Uptown Side of West 25th Street between Broadway and Sixth Avenue, NY 10010.
RACHEL COMEY, 95 Crosby St, New York, NY 10012, +1-212-334-0455, rachelcomey.com.
JILL PLATNER, 113 Crosby St, New York, NY 10012, +1-212-324-1298, jillplatner.com.
TESS GIBERSON, 97 Crosby St, New York, NY 10012, +1-646-998-4391, tessgiberson.com.
BDDW, 5 Crosby St, New York, NY 10013, +1-212-625-1230, bddw.com.
THE FRICK COLLECTION, 1 East 70th Street New York, NY 10021, +1-212-288-0700, frick.org.
ALBERTINE, 972 Fifth Avenue (at 79th Street), New York, NY 10075, +1-212-650-0070, albertine.com.
ONLINE SHOPPING TIP
"I like Farfetch.com because you get a variety of styles and designer pieces that are not commonly found in other stores," says stylist/designer Maryam Malakpour. "They have about 400 boutiques all around the world that are subscribed to Farfetch.com, which gives more variety and extensive options." Raven Kauffman agrees: "I just love being able to shop every amazing boutique around the world at midnight from my iPad. It's such an indulgence."
#### Saturdays NYC
* * *
"For guys, this is such a cool spot," says actress, blogger, and model Christina Caradona of the New York–based board-short and surf lifestyle brand. "They have this little outdoor area and a coffee shop in the store—I used to get coffee there all the time because the guys were so attractive."
17 Perry Street, New York, New York 10014, +1-347-246-5830, saturdaysnyc.com, and additional locations.
#### Tibi
* * *
Flagship for Amy Smilovic's relaxed, feminine label, a favorite with street-style stars such as Leandra Medine. Smilovic's Instagram-friendly designs feature dramatic ties, back vents, and artful ruffles. You can also thank her for helping to kick off the off-the-shoulder trend.
120 Wooster St., New York, NY 10012, +1-212-226-5852, tibi.com.
#### Wendy Nichol
* * *
Silk dusters, lace jumpers, leather bullet bags, and pavé diamond "middle finger" rings for the chic Goth. "I'm completely obsessed with this designer and her entire existence," says stylist Kemal Harris. "One of my clients told me she's an actual witch! Her collections are very feminine and wearable, despite being mostly produced in all-black silk and leather. Her jewelry is to die for, and if you are in need of a crown made of patent leather, trust me, she's designed the perfect one."
147 Sullivan St., New York, NY 10012, +1-212-431-4171, wendynicholnyc.com, and an additional location in Nolita.
#### What Goes Around Comes Around
* * *
Collectible vintage from the 1890s to the 1990s, including Chanel bags galore, concert tees, Levi's, and more. "Always offers something I must have, whether for personal use or inspiration for my collections," says A.L.C. designer Andrea Lieberman. The original location in SoHo is the best.
351 West Broadway, New York, NY 10013, +1-212-343-1225, whatgoesaroundnyc.com, and additional locations.
### LOWER EAST SIDE, NOLITA, EAST VILLAGE
#### Anna Sheffield
* * *
Tucked away next door to a Chinese-menu printing shop on the Lower East Side is this hidden gem full of gems. "She basically invented the concept of stacking rings and then took it a step further with curved nesting rings that fit snugly against raw diamonds and triple-gem pyramid rings," says stylist Kemal Harris. "Where else can you get an eternity band of black diamonds, the ultimate example of inconspicuous luxury, for $500?"
47 Orchard St., New York, NY 10002, +1-212-925-7010, annasheffield.com.
#### DEAR: Rivington
* * *
"So, so inspiring," says Valery Demure. "Beautiful clothing, vintage Comme des Garçons, Junya Watanabe, Yohji Yamamoto, etc. Handbags, socks, jewelry, headpieces. They create the most beautiful collection downstairs with homewares, furniture, ceramics, handmade dolls, art, illustrations. It's owned by a supercool Korean couple with a gorgeous dog—an English bull terrier called Dumbo."
37 Great Jones St., New York, NY 10012, +1-212-673-3494, dearrivington.com.
#### Dö Kham
* * *
A Tibetan emporium in the heart of Nolita. "Very jet-set boho style," says Catherine Ryu, creative director of Citizens of Humanity. "They even sell bracelets blessed by the Dalai Lama. I've been frequenting this shop since the '90s. It's my go-to place for beachy tunics that I wear on vacation or in the heat of summer, in an array of gorgeous colors! The owner, Phelgye [Kelden], is a kind and delightful man filled with a wealth of knowledge about Tibet." "I always find something, whether it's a piece of jewelry or an unusual caftan that I never would have noticed outside the confines of the shop and am often resistant to purchase but always end up wearing in spades," says Leandra Medine.
51 Prince St., New York, NY 10012, +1-212-966-2404, dokham.com.
#### John Derian
* * *
Three boutiques in a row, this lifestyle fantasyland sells Derian's signature plates, plaques, and paperweights decoupaged with letters, insects, and flowers. You'll also find furniture, Moroccan poufs, blankets, scarves, Astier de Villatte ceramics, incense and candles, colorful paper lanterns, and more. "My go-to for gifts," says accessories designer Lizzie Fortunato. "There is always something to discover," says Moscow-based textile designer Olya Thompson. "An unusual light fixture, a fringed Moroccan bath towel in that perfect baby-coral color, a birthday hat with a pinup beauty on top. . . . Everything I have ever bought here I absolutely cherish."
6 E. 2nd St., New York, NY 10003, +1-212-677-3917, johnderian.com.
#### Love Adorned
* * *
In two decades, traveler and collector Lori Leven has built her small tattoo, piercing, and jewelry shop into a trio of lifestyle boutiques selling jewelry, crystals, and trinkets from India, Indonesia, Japan, and other countries. Also featured are shibori textiles, fringed moccasins, dream catchers, planters, and other housewares. She has pieces for men and women, and alternative bridal options, too. No surprise that boho designers Ulla Johnson and Mara Hoffman are devoted fans. The Elizabeth Street store is the flagship. "I die for all the jewelry in this store," says Olivia Kim. "She has such a great eye for mixing the precious with vintage and the store has a modern bohemian feel."
269 Elizabeth St., New York, NY 10012, +1-212-431-5683, loveadorned.com, and additional locations.
#### Maryam Nassir Zadeh
* * *
Launched in 2008, Zadeh's art-gallery-like boutique on the Lower East Side specializes in haute minimalism and fresh indie designers. "A true incubator of young and important talent—it's almost like you get to see the next generation of fashion hanging in a store," says Leandra Medine. You'll find conceptual designs by Jacquemus, Bless, Jesse Kamm, and Dusan alongside accessories from Rochas, jewelry by Sophie Buhai, and the popular in-house line of wide-leg pants, cotton dresses, block-heel pumps, sandals, and other basics.
123 Norfolk St., New York, NY 10002, +1-212-673-6405, mnzstore.com.
#### No Relation Vintage
* * *
This secondhand store with budget prices "is always our final stop at the end of a long workday in the city," says jewelry designer Gabriela Artigas. "We love to unwind by browsing through the endless racks of everything." The store is part of the L Train Vintage family.
204 First Ave., New York, NY 10009, +1-212-228-5201, ltrainvintage.com.
#### Opening Ceremony
* * *
Founded by California natives Carol Lim and Humberto Leon in 2002, Opening Ceremony was conceived as a shopping version of the Olympics, featuring emerging designers from a different country each year alongside up-and-coming American labels such as Proenza Schouler, Rodarte, and Alexander Wang. The idea was to bring the magic of travel and discovery to local customers. Now with stores in L.A., London, and Tokyo, Lim and Leon can be credited with fueling collaboration fever in fashion and helping a younger generation discover heritage brands such as Pendleton, Levi's, and Reyn Spooner. "All Opening Ceremony stores have a creative architectural solution to the use of space," says stylist/designer Maryam Malakpour. Lim and Leon, who design their own in-house label as well as LVMH-owned luxury label Kenzo, are known for collaborating with Spike Jonze, Terence Koh, Chloë Sevigny, and other creatives on fashion shows, installations, and product ranges.
35 Howard St., New York, NY 10013, +1-212-219-2688, openingceremony.com, and additional locations.
#### Ritual Vintage
* * *
Chloë Sevigny loves this space for "Victorian, Edwardian, 1970s Saint Laurent," and everything in between. Bonus? "They also have a trick mirror—a tall, skinny mirror," she says.
377 Broome St., New York, NY 10013, +1-212-966-4142, ritualvintage.com.
#### The Sabah Dealer
* * *
The place to get former hedge funder Mickey Ashmore's comfy-chic, handmade-in-Turkey leather espadrilles in a rainbow of colors. He'll even serve you a cup of tea while you find your perfect size. Call or email for hours.
Sabah House, 211 E. 12th St., New York, NY 10003, +1-646-864-0790, sabah.am.
#### Timbuktu
* * *
"One of my favorite stores in Manhattan is this Turkish and Moroccan importer," says menswear retailer Josh Peskowitz. "I've bought clothes, home goods and textiles from them constantly over the years. It's by no means a fashion store, but I've definitely worn a bunch of stuff I've bought there to Milan and Paris Fashion Week."
45 Second Ave., New York, NY 10003, +1-212-473-4955.
#### Tokio 7
* * *
"I'm a vintage hound, and this is probably my favorite place," Chloë Sevigny says of this secondhand spot on East 7th Street where she's bought and sold many a piece. Just don't go on a Saturday afternoon; it gets so crowded inside that it's hard to see anything!
83 E. 7th St, New York, NY 10003, +1-212-353-8443, tokio7.net.
#### United Nude
* * *
The only US store for the architectural footwear brand known for its injection-molded and carbon-fiber heels designed by creative director Rem D. Koolhaas (not to be confused with his architect uncle Rem Koolhaas). Lady Gaga, Kim Kardashian, and Solange Knowles are fans of the alien-looking styles created in collaboration with Dutch fashion designer Iris van Herpen. United Nude's core collection includes more down-to-earth elastic fabric booties and Möbius carved-out heels. The store's darkly lit interior, with shoes mounted in glowing cubbies, adds to the allure.
25 Bond St., New York, NY 10012, +1-212-420-6000, unitednude.com, and additional locations.
#### Warm
* * *
It's always summer at Warm, which was opened by the "urban hippie" husband-and-wife team Winnie Beattie and Rob Magnotta. Specializing in the "French girl goes to Costa Rica" look, they stock designers such as Isabel Marant, the Elder Statesman, Heidi Merrick, Roseanna, Giada Forte, Mara Hoffman, and NSF, as well as their own in-house label of breezy dresses and tunics, and a Warm fragrance that smells like a day at the beach. "Built on the premise of warm people, warm places, and warm things, this shop offers the best of the best for the refined, beach-loving, chic yet bohemian girl," says stylist Cristina Ehrlich. The store features men's, women's and children's clothing, photography, books, home accessories, and more.
181 Mott St., New York, NY 10012, +1-212-925-1200, warmny.com.
### MEATPACKING DISTRICT, CHELSEA
#### Artists & Fleas
* * *
"At Chelsea Market, I always make a pit stop at Pamela Barsky's booth for pouches with witty quotes and sayings on them," says Jacey Duprie, creator of Damsel in Dior, of this artist, designer, and vintage market held daily. "They make for wonderful gifts and I never leave without a smile on my face and pouch in my hand."
Chelsea Market, 88 Tenth Ave., New York, NY, 10011, artistsandfleas.com, and additional locations.
#### Jeffrey
* * *
One of the first to open a retail boutique in the Meatpacking District, Atlanta transplant Jeffrey Kalinsky helped turn the area into a trendy hot spot. Open since 1999, the boutique stocks men's and women's designer labels, like Prada, Gucci, Valentino, and Sacai, and one of the best selections of shoes in the city, always with plenty of exclusives. "Filled with inspiration," says Alice and Olivia designer Stacey Bendet.
449 W. 14th St., New York, NY 10014, +1-212-206-1272, jeffreynewyork.com.
#### Story
* * *
"Point of view of a magazine, changes like a gallery, sells things like a store." That's how founder Rachel Shechtman describes her ever-changing, experiential retail space. Every four to eight weeks, there's a complete overhaul of theme, design, and inventory, so "if you visit NYC a few times in an year, the store will be completely visually different each time," says stylist and Newbark designer Maryam Malakpour. "You can find clothing, decor, accessories, et cetera." Past "stories" have curated merchandise around themes of "love," "Made in America," and "wellness."
144 Tenth Ave., New York, NY 10011, +1-212-242-4853, thisisstory.com.
#### Urban Zen
* * *
Globally inspired clothing, accessories, and home furnishings supporting Donna Karan's charitable Urban Zen Foundation. "For cashmere, it is the perfect shop," says fashion industry consultant Fern Mallis. "Plus they also sell beautiful handmade jewelry and decor from artisans around the world."
705 Greenwich St., New York, NY 10014, +1-212-206-3999, urbanzen.com, and additional locations.
### SOUTH STREET SEAPORT
#### Brother Vellies
* * *
"Aurora James's eclectic and elegant store sets a new standard for luxury," says Studio 189 cofounder Abrima Erwiah. The store features her full range of magical shoes and handbags handmade in Kenya, Morocco, and South Africa with handwoven details from Burkina Faso.
4 Fulton St., New York, NY 10038, +1-212-480-8869.
### UNION SQUARE/FLATIRON/WEST 30s
#### AEO & Aerie
* * *
The lingerie sub-brand of casual clothing chain American Eagle Outfitters, which has more than a hundred locations (both stand-alone and inside AEO stores) throughout the US and Canada, is worth checking out. You can pick up cute underwear at very reasonable prices. "I feel very old when I go in there because there are tweens in the store," says designer Rebecca Minkoff, "but the underwear is great. They make the best boy shorts—I stock up on twelve pairs for like $35."
19 Union Square West, New York, NY 10003, +1-212-645-2086, ae.com, and additional locations.
#### Aritzia
* * *
Canadian clothing chain with its own in-house brands of minimalist and boho-inspired attire that can hold its own against the high-end Marni, Mansur Gavriel, Maryam Nassir Zadeh, and Céline accessories you may already have (or want to have) in your closet. Look for washed-indigo cross-back dresses; textural cotton culottes; wrap-front jumpsuits; elongated, split-side rib-knit sleeveless sweaters; fringed kimono jackets; and lace-inset festival blouses plus denim by Citizens of Humanity and Frame, leather jackets by Mackage, and more. Most pieces are less than $300. Although there are stores across the US and Canada, the 8,000-square-foot Flatiron flagship is one of their best.
89 Fifth Ave., New York, NY, 10003, +1-212-462-1095, aritzia.com, and additional locations.
#### Bandier
* * *
Workout wear too fashionable to stay in the gym by Sundry, Spiritual Gangster, Michi, Koral, Mackage, APL, and more. "I stop in every time I'm in New York City," says blogger/socialite NJ Goldston. "You gotta love the pulsing music and the knowledgeable fitness staff."
164 Fifth Ave., New York, NY 10011, +1-646-360-3345, bandier.com, and additional locations.
#### Elegance Hosiery
* * *
A true hidden gem, this hosiery and sock place is in Penn Station near the Amtrak. Great for metallic socks and patterned stockings from Europe. "Every time I pass through there, I stock up on fishnets and interesting hosiery," says designer/personal shopper Raven Kauffman. "Not white glove service, mind you, but a great selection that I have never seen anywhere else. I always buy the micro-fishnets, which are the absolute best."
One Penn Plaza, New York, NY 10119, +1-212-736-5306.
#### Marlene Wetherell
* * *
Vintage Yves Saint Laurent gypsy blouses, Dior turbans, and a Teal Traina serape-stripe halter dress are just a few of the fabulous finds to be snapped up here. "My best-kept secret . . . until now," says blogger Lainy Hedaya. Paris-based handbag designer Olympia Le-Tan shops the store online at www.1dibs.com.
40 W. 25th St., Gallery 210, New York, NY 10010, +1-917-225-0662, marlenewetherell.com.
#### New York Vintage
* * *
The hands-down favorite of stylish shoppers (stylist and designer Rachel Zoe, jewelry designer Kara Ross, fashion tech entrepreneur Alexandra Wilkis Wilson) for high-end vintage by Chanel, Emilio Pucci, Giorgio di Sant'Angelo, Holly Harp, Zandra Rhodes, and more. "I adore the owner, Shannon Hoey—she has a true passion for vintage and it shows," says Zoe. "I can spend a whole day in her store looking through evening gowns from all of my favorite designers." What do first lady Michelle Obama's vintage Norman Norell dress, Sarah Jessica Parker's Sex and the City bird-shaped wedding headpiece, and the 1920s showgirl costumes in Boardwalk Empire have in common? They all came from this place, which also has a rental-only archive for costume designers and stylists.
117 W. 25th St., New York, NY 10001, +1-212-647-1107, newyorkvintage.com.
#### The Strand
* * *
New York's most iconic bookstore, opened in 1927, boasts eighteen miles of new and used books, and is an endless source of inspiration for designers. "I never walk out without something in my hand," says Jeremy Scott. French designer Olympia Le-Tan, who reimagines classic book covers as handbags and accessories, also counts this as one of her favorite haunts. "My favorite place to get lost," says shoe designer Isa Tapia. "Also, the basement is full of hidden gems. . . . once I found Isaac Mizrahi comic books there."
828 Broadway, New York, NY 10003, +1-212-473-1452, strandbooks.com.
ONLINE SHOPPING TIP
"I love the Muzungu Sisters website (muzungusisters.com)," says Leandra Medine of the London-based, globally inspired collection. "I'm a big fan of their straw totes, velvet bolero jackets, and beaded jewelry and belts of which they have plenty. I have also recently really taken to an L.A. site called Lisa Says Gah (lisasaysgah.com)," she says of the multi-label online retailer. "Reminds me a lot of Maryam Nassir Zadeh, but has its own sort of subversive West Coast vibe of maximalism even though it is very minimalist."
### UPPER EAST SIDE and UPPER WEST SIDE
#### Aquazzura
* * *
The first US location for the Florentine shoe brand designed by Edgardo Osorio. The fringed "Wild Thing" sandals, lace-up suede "Belgravia" flats, and cutout "Sexy Thing" sandals have made fans of Moda Operandi founder Lauren Santo Domingo, Her Royal Highness Beatrice of York, and Leandra Medine.
939 Madison Ave., New York, NY 10021, +1-347-328-0080, aquazzura.com, and additional locations.
AQUAZZURA
#### Fivestory
* * *
This multi-brand boutique occupying a jewel box of a town house on East 69th Street has cocktail frocks, whimsical clutch purses, and strappy sandals for Park Avenue princesses, by Rosie Assoulin, Monse, Mary Katrantzou, Edie Parker, Charlotte Olympia, Paul Andrew, and more. There's also a fun selection of jewelry, including emoji-inspired studs by Alison Lou and silk thread Les Bonbons earrings by Rebecca de Ravenel. "An intimate shopping experience with a great point of view," says accessories designer Karen Erickson.
18 E. 69th St., New York, NY 10021, +1-212-288-1338, fivestoryny.com.
#### Gianvito Rossi
* * *
The only stateside boutique for the Italian footwear designer known for his sexy leather and PVC pumps, and velvety-soft suede sandals with comfortable block heels.
963 Madison Ave., New York, NY 10021, +1-646-869-0201, gianvitorossi.com, and additional locations.
#### Hayward House
* * *
You will want to move into the second-story Hayward flagship in the palatial Grosvenor Atterbury mansion. Here, Marin Hopper (daughter of Dennis) sells her line of chic, made-in–New York handbags and accessories inspired by Hollywood royalty such as agent Leland Hayward and fashion icon Slim Keith. The store's wood-paneled interior, gold-etched ceiling, and fretwork fireplace are not to be believed.
131 E. 70th St., New York, NY 10021, +1-212-585-1712, haywardluxury.com.
ONLINE SHOPPING TIP
"I love Net-A-Porter.com for the one-stop shopping and, of course, same day delivery in Manhattan," says accessories designer Lee Savage. "I also love SSENSE.com, which always has a cool edit of edgier styles."
#### Neue Galerie Gift Shop
* * *
"I adore the artists of the Viennese Secession and their influence on both fashion and decorative arts and textiles," says designer/personal shopper Raven Kauffman. "You can pick up a fabulous caftan here in the spirit of Koloman Moser, Klimt's painting smock, or even a pair of fingerless gloves and a beautiful cashmere scarf. Great for unique gifts."
1048 Fifth Ave., New York, NY 10028, +1-212-628-6200, neuegalerie.org.
#### Norma Kamali
* * *
The quintessential New York designer, Kamali has been in business nearly fifty years and is still going strong. She's an innovator for her jersey sportswear, sweatshirt dresses, sleeping bag coats, and sophisticated swimwear (she designed the red one-piece bathing suit worn by Farrah Fawcett in the iconic 1976 poster). Kamali's flagship boutique also has an organic café.
11 W. 56th St., New York, NY 10019, +1-212-957-9797, normakamali.com.
#### Proenza Schouler
* * *
Flagship store for designer darlings Jack McCollough and Lazaro Hernandez featuring two floors of clothing, shoes, and handbags. Steel shelves, exposed wood beams, and an indoor cactus garden add to the modern and architectural vibe.
822 Madison Ave., New York, NY 10065, +1-212-585-3200, proenzaschouler.com, and additional locations.
#### Sanjay Kasliwal
* * *
For nine generations, the Kasliwal family has been handcrafting jewelry, a lineage that can be traced back to India and the crafting of megawatt bling for the maharajas of the Mughal empire. The selection here blends the old (an enameled band ring) with the new (a yellow gold bubble ring).
971 Madison Ave., New York, NY 10021, +1-212-988-1511, sanjaykasliwal.com, and additional locations.
#### Tory Burch
* * *
The New York designer has transformed a small collection of tunics made at her kitchen table into a lifestyle empire, with this 8,000-square-foot Madison Avenue townhouse flagship as its crown jewel. Browse and buy Burch's entire retro-feminine collection.
797 Madison Ave, New York, NY 10065, +1-212-510-8371, and additional locations, toryburch.com.
### FINANCIAL DISTRICT
#### Century 21
* * *
A NYC icon for more than fifty years, this discount department store is the mack daddy of off-price designer retail. Although there are now several stores, the original in the Financial District is the one to hit for deals and steals from Rick Owens, Saint Laurent, Givenchy, Jimmy Choo, Giorgio Armani, and more. "You can find some insane bargains on designer pieces, especially during their semiannual sales. And the designer shoe game is serious," says costume designer Jenn Rogien,
22 Cortlandt St., New York, NY 10007, +1-212-227-9092, c21stores.com, and additional locations.
### BROOKLYN
#### Bird
* * *
Jen Mankins's boutiques are worth crossing the river for. You'll find printed dresses and coats by Dries Van Noten, Maria Cornejo, and Marni; pants and clogs by Rachel Comey; bags by Mansur Gavriel, Building Block, and Lizzie Fortunato; and more. The Williamsburg location is the largest; there is also one in Cobble Hill and one in Park Slope (the original, open since 2004).
203 Grand St., Brooklyn, NY 11211, +1-718-388–1655, shopbird.com, and additional locations.
#### Brooklyn Flea Market
* * *
Starting the first weekend in April and running through November, the market takes place outdoors: Saturdays in Fort Greene and Sundays in DUMBO. Our stylish shoppers prefer the Fort Greene location, in a schoolyard on leafy Lafayette Avenue, where 150 vendors sell a mix of old (vintage jean jackets, Bakelite jewelry, and Indian dresses) and only-in-Brooklyn new (palmistry-inspired stud earring sets, grandpa cardigans for babies, and bicycle paintings). "Great food and offbeat treasures," says fashion designer Ulla Johnson, who lives in the neighborhood. Local food offerings include Ramen Burger, Red Hook Lobster Pound, and the Good Batch cookies.
176 Lafayette Ave., Brooklyn, NY 11238, +1-718-928-6603, brooklynflea.com.
JENN ROGIEN
SHOP BROOKLYN LIKE HBO'S GIRLS
* * *
Emmy-nominated costume designer Jenn Rogien is behind the personality-print blouses and onesies, folksy maxi dresses, and jeweled barrettes worn by the characters on Lena Dunham's Girls. And when it comes to research, there's no place like home. (Jenn is also responsible for outfitting the prison population in the Netflix series Orange Is the New Black.)
There's a Tibetan store in Fort Greene that has the best jewelry: 21 TARA on Myrtle Avenue. It not only has a great selection of Tibetan and Nepalese pieces but also delicate gold and silver pieces that are very much contemporary. Cool textiles and clothes also make it worth stopping in.
I also love PEOPLE OF 2MORROW in Greenpoint. It's a hybrid of amazing things: vintage, gift, home goods, young jewelry designers, all pulled together in a bright, airy cool space. I've done quite a bit of shopping for the "Girls" here over the years.
I love OAK in Williamsburg for a bit of personal shopping or for dressing an aggressively casual character. Oak's pieces are cool, casual, and worth living in. I try to drop in whenever we shoot in the neighborhood.
URBAN JUNGLE might be my favorite thrift warehouse in Brooklyn. It's part of the L Train Vintage family. Urban Jungle is massive and full of amazing finds. We've shopped with them fairly regularly for flashbacks on Orange Is the New Black. Most recently my team and I bought bags and bags of '90s clothes for a pilot we were working on. I'm looking forward to stopping in when we head back for our final season of Girls.
21 TARA, 388 Myrtle Ave., Brooklyn, NY 11205, +1-347-916-0045.
PEOPLE OF 2MORROW, 65 Franklin St., Brooklyn, NY 11222, +1-718-383-4402, peopleof2morrow.com.
OAK, 55 Nassau Ave., Brooklyn, NY 11222, +1-718-782-0521, oaknyc.com, and additional locations.
URBAN JUNGLE, 118 Knickerbocker Ave., Brooklyn, NY 11237, +1-718-381-8510, ltrainvintage.com.
ONLINE SHOPPING TIP
"I like to visit New York's CapBeauty.com for nontoxic beauty products," says Shiva Rose, who has her own line of skincare at ShivaRose.com. "I also love Mountain Rose Herbs (mountainroseherbs.com) for teas and oils."
#### Catbird
* * *
Founded in 2004, this gem of a store features delicate jewelry such as first knuckle rings and ear climbers made in house, alongside paper items by Rifle, trays by John Derian, candles, and apothecary items.
219 Bedford Ave., Brooklyn, NY 11211, +1-718-599-3457, catbirdnyc.com.
#### Concrete & Water
* * *
Cute lifestyle boutique with a mix of local and international labels (Sea, A.P.C., Ganni), plus quirky accessories (taco-shaped lapel pins by Pintrill, cork-heel pumps by Anne Thomas, clutches by Lizzie Fortunato). A back patio with a ping-pong table invites hanging out in the warmer months.
485 Driggs Ave., Brooklyn, NY 11211, +1-917-909-1828, concreteandwater.com.
CONCRETE & WATER
#### Electric Feathers
* * *
"I always visit for pants or dresses by the visionary designer Leana Zuniga," says actress/blogger/beauty entrepreneur Shiva Rose of the Brooklyn-based designer's flagship. It features versatile garments such as her signature Infinite Rope dress, which can be rolled up, tucked in a knapsack, and unfurled for ease when traveling and her Infinite Bandeau, which can be worn as a top, scarf, belt, or headdress.
60 Broadway, Brooklyn, NY 11249, +1-347-227-7023, electricfeathers.com.
#### Oroboro
* * *
Boho duds by Ulla Johnson, Electric Feathers, Ryan Roche, 69 denim, Rachel Craven, Ace & Jig, and other designers, alongside shoes by Brother Vellies, A Détacher, and Dieppa Restrepo, apothecary items by Brooklyn Beach, Fig & Yarrow, plus textiles and wall hangings, and ceramics. "The buyer, April, has such incredible and original taste," says Shiva Rose. "It's sort of my home away from home," says Brother Vellies designer Aurora James, who suggests ducking into Sprout Home around the corner for flowers, and Devoción across the street for a latte.
326 Wythe Ave., Brooklyn, NY 11249, +1-718-388-4884, oroborostore.com.
OROBORO
## WASHINGTON
### DISTRICT OF COLUMBIA
#### Hu's Wear and Hu's Shoes
* * *
Within one block of each other, Hu's Shoes and Hu's Wear feature designer ready-to-wear, footwear, handbags, and accessories by Kenzo, Erdem, Chloé, Isabel Marant, Valentino, Jerome Dreyfuss, and more.
3005 and 2906 M St. NW, Washington, DC 20007, +1-202-342 0202, husonline.com.
#### National Museum of the American Indian
* * *
"My favorite gift shop in the world," says retailer Josh Peskowitz. "They work with so many artisans, and I've bought jewelry and home goods that I use all the time. Both as reference and just to wear. The food court there is also terrific. They have the stalls set up by region. Great Plains cooking here, Pacific Northwest cooking there."
4th St., and Independence Ave. SW, Washington, DC 20560, nmai.si.edu.
#### Relish
* * *
"An artfully chosen array of well-known labels such as Marni, Aspesi, and Marc Jacobs, but also lesser-known ones, eclectic brands such as Sacai and Simone Rocha," says Washington Post fashion editor Robin Givhan. "It's the very personal point of view of owner Nancy Pearlstein."
3312 Cady's Alley NW, Washington, DC 20007, +1-202-333-5343, relishdc.com.
RELISH
#### Salt and Sundry
* * *
"For housewares and hostess gifts, Salt and Sundry in Union Market is a delight. It's got craft cocktail mixers, Turkish towels, hand-carved furniture, and jewelry," Robin Givhan says.
1309 5th St. NE, Washington, DC 20002, +1-202-556-1866, shopsaltandsundry.com, and an additional location in Logan Circle.
## FREEPORT
### MAINE
#### L.L. Bean
* * *
"I don't think I would ever classify myself as a preppy guy, but I love the classic, all-American L.L. Bean flagship," says fashion editor/TV host Joe Zee. "First of all, it is open twenty-four hours(!), but I also think it's because where I grew up the must-have items in high school were always the L.L. Bean sweaters and 'duck boots.' And the fact that you can buy those here along with a shotgun case is so authentic."
95 Main St., Freeport, ME 04032, +1-877-755-2326, llbean.com, and additional locations.
L.L. BEAN
## WOODSTOCK
### VERMONT
#### Who Is Sylvia?
* * *
"Every summer we go to Woodstock, Vermont, to see family," says L.A.–based jewelry designer Sonia Boyajian. "It's a very charming, classic New England town with covered bridges and old-world charm. There is a tiny vintage shop there called Who is Sylvia? and it's by far my favorite destination to visit. You can get anything from beautiful vintage clothing to hand-embroidered tablecloths, stunning hats, and even some children's clothing."
26 Central St., Woodstock, VT 05091, +1-802-457-1110.
## WESTON
### VERMONT
#### The Vermont Country Store
* * *
Open since 1946, you can still spend hours browsing at this ultimate general store. There's everything from Lanz of Salzburg flannel nightgowns and made-in-Vermont Darn Tough socks to Tangee color-changing lipstick and Vermont maple syrup. The Weston location also has the Bryant House restaurant, which serves such tavern classics as Vermont cheddar cheese soup.
657 Main St., Weston, VT 05161, +1-802-824-3184, vermontcountrystore.com, and an additional location in Rockingham.
VERMONT COUNTRY STORE
## BOSTON
### MASSACHUSETTS
#### Ball and Buck
* * *
This menswear shop with hunting-inspired apparel is on lad-mag top lists. It stocks rugged-yet-refined apparel, all made in the USA, plus oyster knives, wallets, belts, and vintage Playboy magazines. There's a barbershop, too.
144B Newbury St., Boston, MA 02116, +1-617-262-1776, ballandbuck.com.
#### Bobby From Boston
* * *
Bobby Garnett was a vintage clothing pioneer when he founded his Provincetown store in 1974 (it moved to the South End in 1995). Over the years, he sold vintage menswear to the design teams at Marc Jacobs, Ralph Lauren, J. Crew, and more, and assembled a museum-worthy collection of pleated pants, two-tone shoes, and workwear, with corresponding, historically accurate props, dating back to the 1930s. Garnett, who also provided costumes for dozens of films, died in May 2016, but his legacy continues.
19 Thayer St., Boston, MA 02118, +1-617-423-9299, bobby-from-boston.com.
#### Bodega
* * *
"A store that left a great impact on me," says journalist and author Susana Martinez Vidal, about Bodega, "It's hidden behind an actual bodega that sells sodas, water, and snacks on a small street off the commercial thoroughfare. The store is only accessible to those who know its secret. The bodega looks old and unkempt, but a marvelous store appears suddenly behind a Snapple vending machine, which in fact is a sliding door." Inside, you'll find rare and stylish sneakers, plus amazing menswear from Garbstore, A.P.C., Norse Projects, Woolrich, and more.
6 Clearway St., Boston, MA 02115, shop.bdgastore.com.
BODEGA
#### Castanet
* * *
Building on the legacy of beloved Boston consignment store The Closet, Castanet took over the space in the spring of 2016, and is the place to go for gently used clothing, handbags, and shoes by Chanel, Isabel Marant, Balenciaga, The Row, Marni, and more.
175 Newbury St., second fl., Boston, MA 02116, +1-617-536-1919, shopcastanet.com.
#### December Thieves
* * *
Stop by this local fave for limited-run jewelry, including stackable rings and earthy, crystal-adorned necklaces, modern-looking bags, arty home decor, and fun men's accessories.
88 Charles St., Boston, MA 02114, +1-617-982-6802, decemberthieves.com.
#### Dress
* * *
The boutique is named after the one piece of clothing every woman probably has in her wardrobe, the little black dress, but it sells much more, including denim by M.i.h Jeans, Lizzie Fortunato jewelry, Rachel Comey tops, and Ulla Johnson wraps.
70 Charles St., Boston, MA 02114, +1-617- 248-9910, dressboston.com.
#### Riccardi
* * *
This edgy, bi-level Newbury Street boutique stocks all the major designers for men and women, including Kenzo, Moschino, Comme des Garçons Play, Neil Barrett, Thom Browne, Valentino, and Givenchy.
116 Newbury St., Boston, MA 02116, +1-617-266-3158, riccardiboston.com.
#### Serenella
* * *
Credited with introducing Emilio Pucci, Romeo Gigli, and other high-end designers to Boston back in the day, Serenella has been at the cutting edge of fashion since opening in 1980. The two boutiques (in Boston and Nantucket) feature an array of luxury designers, including old standbys Agnona, Bottega Veneta, Dolce & Gabbana, Loewe, and Roberto Cavalli, and members of the new guard, including Lisa Marie Fernandez, Rosie Assoulin, and Roksanda.
134 Newbury St., Boston, MA 02116, +1-617-262-5568, serenella-boston.com.
#### Twentieth Century Limited
* * *
A hidden gem full of vintage costume jewelry gems, this Beacon Hill basement store stocks sparklers by Miriam Haskell, Trifari, Dior, Georg Jensen, Kenneth Jay Lane, Oscar de la Renta, and more.
73 Charles St., Boston, MA 02114, +1-617-742-1031, boston-vintagejewelry.com.
FARAN KRENTCIL
BOSTON SECRETS
* * *
Boston-bred Faran Krentcil is a contributing editor at ELLE.com and Yahoo News. She is the founding editor of Fashionista.com and served as NYLON's first-ever digital director. Last year, she illustrated her first book, The Craft, by rock stylist Lou Teasdale.
In Boston, I always visit THE GARMENT DISTRICT, a bonkers clubhouse for club kids, vintage lovers, and deadstock denim junkies. Think: If Patricia Field had four floors of retail.
SUDO SHOES is the only cruelty-free shoe store in the Boston area, with good weatherproof options and a full selection of vegan Doc Martens!
Where college girls go to scope potential dates, CONCEPTS has hard-to-find collaborations from A Bathing Ape + Adidas, and Rihanna's Fenty line.
Now that Boston's beloved Filene's Basement has bit the dust, Framingham, Massachusetts–based TJ MAXX is the hometown discount store to visit for crazy deals on designer clothes. The one on Route 4/225 in Bedford is the one you want. It's the unofficial flagship with designer stuff.
I love to buy custom-made embroidered moccasins and loafers online at Boston–based ARTEMIS DESIGN CO.—cuter than Gucci, I swear.
THE GARMENT DISTRICT, 200 Broadway, Cambridge, MA 02139, +1-617- 876-5230, garmentdistrict.com.
SUDO SHOES, 1771 Massachusetts Ave., Cambridge, MA 02140, +1-617- 354-1771, sudoshoes.com.
CONCEPTS, The Atrium, 37 Brattle St., Cambridge, MA 02138, +1-617- 204-2104, cncpts.com.
TJ MAXX, 297 Great Rd, Bedford, MA 01730, (781) 275-8864, tjmaxx.com.
ARTEMIS DESIGN, Artemisdesignco.com.
## LOS ANGELES
### CALIFORNIA
FREE CITY
#### THE MAJORS
#### Neiman Marcus
* * *
The Beverly Hills outpost of Neiman Marcus has a secret weapon, and that's Catherine Bloom, a personal shopper so renowned, her bosses built her "Bloom's Room" to cater to stylists and celebs. "My two big secrets in Los Angeles are the amazing personal shoppers Catherine Bloom at Neiman's and Tony Ferreira at Saks," says Haney designer Mary Alice Haney. "If I had the resources, Catherine is the only person I would shop with," says Rodarte's Kate Mulleavy. Our stylish shoppers also praise the store's Mariposa restaurant and its irresistible warm popovers.
9700 Wilshire Blvd., Beverly Hills, CA 90212, +1-310-550-5900, neimanmarcus.com, and additional locations.
#### Maxfield
* * *
Temple to dark and decadent high fashion in the land of sunshine since its opening in 1969, Maxfield combines the latest from Chanel, Givenchy, Balmain, Saint Laurent, Alaïa, and Isabel Marant, with accessories by Céline and Gucci, jewelry by Lisa Eisner, vintage Hermès bags, and Cartier baubles. Not only did founder Tommy Perse define a new, largely all-black aesthetic for L.A., he also gave many designers a launch pad in the US, including Rick Owens, Giorgio Armani, Yohji Yamamoto, and Comme des Garçons's Rei Kawakubo. Crazy-expensive jeans and leather jackets by Japanese designers abound, as do cashmere sweaters as thick as blankets, for warding off that California coastal chill. "Push open the glass doors and get ready to find anything from a taxidermy chicken to an overnight bag to the most beautiful dress you've ever seen," says interior designer Brigette Romanek, "and sometimes they have vintage Kelly bags or a Cartier bangle from the '70s. They have the best of everything." "My go-to source for special pieces since my days as a bicoastal stylist," says designer Andrea Lieberman. "Added bonus: Gracias Madre is on the same block." "The first time I shopped there, it was the late '80s. I came home flush from a modeling gig in Paris and went to Maxfield," says handbag designer Kendall Conrad. "They offered me champagne while I shopped and I thought that was so cool. I bought an incredible Thierry Mugler dress that had a sheer black front with flames covering the chest. I thought Maxfield was out of Scruples." You can also browse cases of antique erotica, skulls, and other offbeat ephemera.
8825 Melrose Ave., West Hollywood, CA 90069, +1-310-274-8800, maxfieldla.com, and additional locations.
### BEVERLY HILLS
#### Alo Yoga
* * *
Gigi and Bella Hadid, Kendall Jenner, and Hailey Baldwin wear this brand's moto and goddess leggings in the studio and on the street. "This is the new face of retail," says blogger/socialite NJ Goldston. "It's a wonderful mix of perfectly curated and spectacularly merchandised athletic apparel with everything from an open-air coffee and juice bar to a rooftop yoga studio and event space. You won't want to leave." Another reason to stop in? Stumptown coffee, cold-pressed Raw Juicery juice, and local, sustainable bites by Café Gratitude.
370 N. Canon Dr., Beverly Hills, CA 90210, +1-310-295-1860, aloyoga.com.
#### Beverly Hills Hotel Gift Shop
* * *
This fabulous hotel gift shop routinely collaborates with designers on stylish items inspired by the famed Pink Palace, including Charlotte Olympia's Charlotte Dellal and milliner Stephen Jones. "I could spend a day and $1,000 in the gift shop of the Beverly Hills Hotel just poring over all the pink-and-white striped and palm-tree-covered tchotchkes," says San Francisco–based fashion retailer Emily Holt. "I've always loved the gift shop at the Beverly Hills Hotel! #thinkpink!" says accessories designer Sally Perrin.
9641 Sunset Blvd., Beverly Hills, CA 90210, +1-310-276-2251, dorchestercollection.com.
#### Lily et Cie
* * *
A vintage store even an Oscar nominee would love, with rare pieces from the 1920s to the 1990s. This super-high-end spot is "a gem on epic levels," says A.L.C. designer Andrea Lieberman. "[Owner] Rita Watnick has the most incredible collection of vintage I have ever seen."
9044 Burton Way, Beverly Hills, CA 90211, +1-310-724-5759.
#### Mameg
* * *
Hidden behind the Maison Martin Margiela store, Sonia Eram's chic spot stocks arty clothing and accessories by Jil Sander, Hussein Chalayan, and more. "It's a very intimate shopping experience. On most afternoons clients are offered a glass of champagne and some Persian sweets while they look around," says jewelry designer Sonia Boyajian.
9970 S. Santa Monica Blvd., Beverly Hills, CA 90212, +1-310-556-2600, mameg.com.
#### Maxfield Bleu
* * *
Love to shop at Maxfield but don't have the budget? Our stylish shoppers' secret is Maxfield Bleu, the sale shop for Maxfield, which has past-season goodies by all the top brands at up to 70 percent off.
301 N. Canon Dr., Beverly Hills, CA 90210, +1-310-275-7007, maxfieldla.com.
#### Nadine Krakov Collection
* * *
An antique jewelry store in the heart of Beverly Hills. "The couple that owns the business have become friends," says Brigette Romanek. "I sit with them and try on antique and new jewels. They have every high-end jewelry brand and can give you the history of each piece. Nadine knows me better than I know myself. She calls me when she has a piece that she thinks I'd like. She hasn't missed once in the past ten years."
191 S. Beverly Dr., Beverly Hills, CA 90212, +1-310-860-9991, beverlyhillsjewelry.net, and an additional location on Canon Drive.
#### Tom Ford
* * *
The designer's namesake label may be based in London, but his spiritual home is in L.A., where his Rodeo Drive store is a hangout for friends and fans who come by to drop serious cash or to just say hi to their favorite salespeople. Ford has outfitted Daniel Craig as James Bond; dressed a constellation of stars, including Justin Timberlake, Gwyneth Paltrow, and Julianne Moore, for stage and red carpets; and been the subject of a Jay Z song. He's also directed two films. While his menswear may be deeply rooted in Savile Row tradition, his womenswear, with its myriad cinematic and pop culture references, looks very at home in L.A. In this two-level store, you'll find Ford's fragrances, accessories, and men's and women's collections, as well as the kind of over-the-top luxe pieces, like a $98,000 crocodile minidress, that make Beverly Hills what it is.
346 N. Rodeo Dr., Beverly Hills, CA 90210, +1-310-270-9440, tomford.com, and additional locations.
### WEST HOLLYWOOD/MELROSE PLACE/MELROSE
#### The Apartment by the Line
* * *
Vanessa Traina Snow, designer muse and stylish daughter of novelist Danielle Steel, opened her first home-as-store concept Apartment by the Line in New York's SoHo, but we prefer the airy, light-filled Los Angeles location on Melrose Place. "You want to move right in," Louise Roe says of the store, which takes the retail trend of curation to a new level, with absolutely everything for sale, from the $50,500 Helmut Newton photograph hanging on the wall to the $895 pair of exclusive alligator Alexandra Knight Birkenstocks in the walk-in closet to the $8 Morihata charcoal toothbrush on the bathroom sink. "The epitome of great taste," says fashion editor/TV host Joe Zee. "Like walking into one of the chicest apartments you have ever seen," says stylist and designer Maryam Malakpour.
8463 Melrose Place, second fl., Los Angeles, CA. 90069, +1-323-746-5056, theline.com, and an additional location in New York.
#### Curve
* * *
Nevena Borissova was a retail pioneer when she opened her first store on Robertson Boulevard in 1997, before the stretch became hot, and now her empire has extended to other locations in L.A. and the rest of the country. Throughout, she's catered to women looking for a well-edited selection of the newest labels, including J.W. Anderson, Baja East, Co, Ellery, Mugler, and more.
154 N. Robertson Blvd., Los Angeles, CA 90048, +1-310-360-3008, shopcurve.com, and additional locations.
ONLINE SHOPPING TIP
"I love sites like Eyebobs.com for their velvet readers, which I give out as gifts all the time, Zerouv.com for sunglasses, and APL.com for metallic knit sneakers (full disclosure, my sons run the company)," says NJ Goldston. "Yoox.com is the only one I know who carries French designer Marie Beltrami's mouse rings. You stack them on every finger. I met the designer at a party in Paris, she took a ring off her hand and insisted I take it—I'd never met her! I also love Ashlynd.com for clutches that are completely whimsical. At very social media-savvy parties, I always grab my chalkboard clutch from her and write the hashtag of the night. It's such an icebreaker."
#### Decades
* * *
Cameron Silver may have migrated to the East Coast to be fashion director of H by Halston for QVC, but back at home, Decades still sets the bar for upscale vintage in Los Angeles. The boutique opened in 1997, and helped introduce vintage to the red carpet, dressing stars such as Julia Roberts, Renée Zellweger, and Chloë Sevigny. Upstairs is the primo stuff; downstairs, you'll find more recent designer pieces, and the famous twice-annual shoe sale.
8214 Melrose Ave., Los Angeles, CA 90046, +1-323-655-1960, decadesinc.com.
#### The Elder Statesman
* * *
Located in a converted bungalow, the home base for L.A. designer Greg Chait's ultra-luxe boho cashmere label sells supersoft cashmere Baja hoodies, palm tree intarsia sweaters, tie-dyed sweatpants, striped blankets, teddy bears, and more.
607 Huntley Dr., West Hollywood, CA 90069, +1-424-288-4221, elder-statesman.com.
#### Elodie K.
* * *
Resorty, whimsical boutique located next to Alfred Coffee, with shoes by Aquazurra, Sophia Webster, and Ancient Greek Sandals; denim by Seafarer; beachwear by Lisa Marie Fernandez and Camilla; and more. "Every time I come in, I discover great new European labels, unavailable anywhere else in L.A.," says blogger Annabelle Fleur. "Elodie K. was probably one of the first places to carry Dodo Bar Or's stunning hand-embroidered separates and Paula Cademartori's statement-making bags." Owner Elodie Khayat has her own collection of jewelry, including body chains, hand chains, and earrings.
8428 Melrose Place, West Hollywood, CA 90069, +1-323-658-5060, elodiek.com.
#### Gabriela Artigas
* * *
Stop by this charming store to shop tusk earrings, chain rings, star-shaped barrettes, and other pieces by the L.A.–based sister act Gabriela and Tere Artigas, who started their brand in 2003. Since then, their everyday statement jewelry has found favor with the likes of Tyra Banks, Emma Roberts, Chelsea Handler, and Carey Mulligan.
370 N. La Cienega Blvd. #1, Los Angeles, CA 90048, +1-310-360-0796, gabrielaartigas.com.
#### Indigo Seas
* * *
"For the last thirty years or so I've loved this divine shop by Lynn Von Kersting, who also owns the two Ivy restaurants in Los Angeles," says Libertine fashion designer Johnson Hartig. "Lynn has one of the most sophisticated and clever eyes in the business and curates such a fantastic selection of her own upholstered furniture and quirky, divine objets in a delightful way. I remember when Michael Smith, the decorator, worked there as Lynn's assistant in the 1980s, and he has kind of adopted Lynn's best attributes in his own design work."
123 N. Robertson Blvd., Los Angeles, CA 90048, +1-310-550-8758, indigoseas.us.
#### Irene Neuwirth
* * *
The Venice, California–based jewelry designer is known for carefully balanced constructions of bold, rough-cut, and semiprecious stones. She upped the retail game with her first store, an impressive mix of whimsical chic and at-home L.A. ease. Jewelry is displayed in glass cases with dioramas designed by local artist Clare Crespo. Tiny pieces are draped on birds, flowers, and butterflies fashioned out of cashmere, snakeskin, and suede. A pink velvet couch and a pair of shearling chairs set up in front of a fireplace and bookshelves filled with volumes on everything from the artist Elizabeth Peyton to Audubon animals encourage shoppers to stay awhile.
8458 Melrose Place, West Hollywood, CA 90069, +1-323-285-2000, ireneneuwirth.com.
IRENE NEUWIRTH
#### Isabel Marant
* * *
Sure, the French designer has other stores—in Paris, New York, Tokyo, etc.—but the L.A. location is her most inspiring, thanks to the cactus garden out front and the artist-studio feel inside. Both her runway and Étoile brands are sold here, along with her coveted accessories.
8454 Melrose Place, Los Angeles, CA 90069, +1-323-651-1493, isabelmarant.com, and additional locations.
#### Jenni Kayne
* * *
The L.A. designer has earned fans near and far for her relaxed-yet-polished pieces that mix masculine and feminine elements: dotted swiss boyfriend shirts, crepe slim pants, twill military jackets, and beach-appropriate sweater coats. She has her own line of shoes, too, including her signature much-imitated d'Orsay flats. You'll also find handbags by Mansur Gavriel and jewelry by Sophie Bille Brahe. "So chic and well curated," says stylist Ilaria Urbinati. "She has beautiful taste." Check out Kayne's lifestyle blog Rip and Tan, which features recipes, entertaining tips, and interviews.
614 N. Almont Dr., West Hollywood, CA 90069, +1-310-860-0123, jennikayne.com, and additional locations.
#### Nasty Gal
* * *
Ground zero for #GirlBoss Sophia Amoruso, the Melrose flagship features a mix of trendy apparel, shoes, accessories, and intimates. You'll find lace-up tops, ruffled rompers, leather minis from the Nasty Gal brand alongside other labels, including Jeffrey Campbell, Vans, and For Love & Lemons, plus vintage bags and jewelry from Louis Vuitton and Chanel. There's a magazine nook with tech gifts and a few vinyl records for sale, including one by Betty Davis, whose funk song "Nasty Gal" inspired the brand's name.
8115 Melrose Ave., Los Angeles, CA, 90046, +1-323-658-1010, nastygal.com, and an additional location in Santa Monica.
#### The Office of Angela Scott
* * *
The Santa Barbara designer's menswear-inspired footwear for women includes black-and-navy striped oxfords with chunky heels, black patent leather platform kiltie loafers, chocolate suede cutout flat booties with perforated details, black-and-white deco-inspired derby oxfords, priced from $195 to $625. Scott's shoes have been worn by Cate Blanchett, Julia Roberts, Ellen DeGeneres, and Taylor Swift. There are select styles for men, too.
7975 Melrose Ave., Los Angeles, CA 90046, +1-323-424-7796, theofficeofangelascott.com.
#### Reformation
* * *
Founder Yael Aflalo has caused a sensation with her reworked vintage garments, which are not only eco-friendly but also sexy enough to make fans of the model crowd, including Lily Aldridge, Rosie Huntington-Whiteley, Karlie Kloss, and others. Popular styles include thigh-high-slit maxi dresses, open-back jumpsuits, and off-shoulder blouses, and most items top out at $300. "My favorite affordable shop," says stylist Ilaria Urbinati. "I shop there an embarrassing amount (and on their site, too). And, like, where in the hell was this store when I was in my 20s? I love that the clothes are all about being eco and ethically made, but they're actually sexy, cool-girl clothes, nothing granola about them." In addition to the in-house label, the store stocks vintage denim, accessories by Monserat De Lucca, B-Low the Belt, and more.
8253 Melrose Ave., Los Angeles, CA 90048, +1-323-852-0005, thereformation.com, and additional locations.
REFORMATION
#### Reservoir
* * *
The concept of New York City transplants Aliza Neidich and Alissa Jacob, Reservoir features a mix of clothing, accessories, and home goods with an easy sophistication made for L.A., including Ryan Roche hand-knit sweaters, Denis Colomb linen dresses, Ellery sleek crepe dresses and tops, Re/Done denim, MadeWorn tees, Newbark slides, and Wendy Nichol fringed leather bucket bags. Home furnishings and gift items have a handmade modern vibe.
154 S. Robertson Blvd., Los Angeles, CA 90048 +1-323-300-5309, reservoir-la.com.
#### Ron Herman at Fred Segal
* * *
As a young entrepreneur in the early 1960s, Fred Segal unleashed the first wave of designer denim mania (when jeans were retailing for $3.98, he priced his at $19.95) and promptly had customers lining up around the block. What made him a true retail visionary, though, was pioneering the shop-in-shop concept that is now standard practice in department stores. Segal served as the original savvy landlord, clustering a collection of cutting-edge, independent retailers inside his ivy-covered walls on Melrose, starting in 1965. With their ability to spot the next big thing, his deputized curators of cool—Ron Herman chief among them—helped establish the Fred Segal center as a retail must-visit (and as a de facto extension of the wardrobe departments for many a fashion-influencing TV show, Melrose Place and Friends among them). Ron Herman carries the torch for SoCal style at his men's and women's stores at Melrose Center (and in other locations around Southern California and in Japan), where you'll find everything for the L.A. look from must-have denim, rocker T-shirts, and swim trunks to breezy dresses; patchwork men's jackets by Junya Watanabe; hoodies by Solca; diver-print button-down shirts by Portuguese Flannel; and more.
8100 Melrose Ave., Los Angeles, CA 90046, +1-323-651-4129, ronherman.com, and additional locations.
#### Roseark
* * *
This converted West Hollywood bungalow is a gallery of must-have high-end jewelry, accessories, and art from around the world. It's full of unique discoveries such as a Dagmar Zaragoza hand-embroidered fabric collar and porcelain bead scarves from Guatemala, Amedeo's edgy cameo rings, Daniela Villegas's butterfly fish earrings, Elisabeth Bell Jewelry shark tooth ear climbers, Victorian estate pieces that have been tweaked by in-house artisans, and more. Prices start at $25.
1111 N. Crescent Heights Blvd., West Hollywood, CA 90046, +1-310-395-6706, roseark.com.
#### RTH
* * *
Designer René Holguin has two neighboring stores on La Cienega Boulevard devoted to his label RTH. With rotating art displays, talisman-like handcrafted leather jewelry, handmade apothecary items, and the scent of piñon incense in the air, they are like upscale souvenir shops. Holguin's unisex collection isn't sold online, so you have to go to his shops to procure his twist on the classic navy blazer, or the cult-favorite drop-crotch jeans. "Japan-meets–Ralph Lauren in Los Angeles," says stylist Amanda Ross. "Familiar yet wholly original," says Lori Leven, the founder of Love Adorned. "His refined eye brings you items that other retail stores will say they discovered a year later." "And the fragrance and the body oil . . . it is all made with care and so, so cool," says stylist Sarah Schussheim. Experiential retail at its best.
529 and 537 N. La Cienega Blvd., West Hollywood, CA, 90048, +1-310-289-7911, rthshop.com.
ONLINE SHOPPING TIP
Jeweler Irene Neuwirth recommends matchesfashion.com for clothes. "I feel like they have a better curated shopping experience than others," and ylang23.com for jewels (including her own). She likes boho–inspired L.A.–based brand Doen Collective, too, at shopdoen.com. Also based in L.A., dreslyn.com has a nice curation of contemporary labels. For vintage, Neuwirth, like most every other contributor in this book, singles out 1stdibs.com.
#### Sielian's Vintage Apparel
* * *
"Every stylist comes here," says stylist Jen Rade, which is why Miley Cyrus, Kim Kardashian, Katy Perry, and others have worn vintage sourced from this West Hollywood gem, which stocks cocktail frocks and more by Jean Paul Gaultier, Thierry Mugler, Oscar de la Renta, James Galanos, and Gianfranco Ferré.
8629½ Melrose Ave., West Hollywood, CA 90069, +1-310-246-9595, sieliansvintageapparel.com.
#### Ten Over Six
* * *
Accessory-obsessed cool girls head to this airy, all-white space for offbeat items by fashion-insider favorites like Rachel Comey, No. 6, Marlow Goods, and Thierry Lasry. "My wildcard store," says Brigette Romanek. "They carry really unique bags, clothes, shoes, and jewelry. I find things here that I won't find everywhere else."
8425 Melrose Ave., Los Angeles, CA, 90069, +1-323-330-9355, shop.tenover6.com.
#### The Row
* * *
First retail store for The Row, the stealth-wealth luxury brand created in 2006 by former child stars Mary-Kate and Ashley Olsen. The residential-style space features furniture sourced from local purveyors JF Chen, Galerie Half, and Thomas Hayes Gallery, paintings by artists John Tweddle, Sergej Jensen, and vintage jewelry and objects—all of it for sale. There are also Manolo Blahnik shoes and Sidney Garber jewelry. "It's so beautifully appointed and designed, it feels a little like being in a live-in museum," says Refinery29's Christene Barberich. "I want everything, all the time, including the furniture that's for sale," says interior designer Brigette Romanek. "The sweaters are 'forever' items; they get more cozy with time. The leather leggings, the dresses! When I wear their clothes, I feel empowered."
8440 Melrose Place, Los Angeles, CA 90069, +1-310-853-1900, therow.com, and an additional location in New York City.
#### Violet Grey
* * *
Expertly curated beauty boutique with the best products by Charlotte Tilbury, Tom Ford, Giorgio Armani, Le Mer, RGB, Root Vanish, and more, tested and approved by top Hollywood makeup artists, hairstylists, and celebrities. Check out the website's Violet Files section for celebrity profiles and beauty routines, expert beauty lessons, and how-tos.
8452 Melrose Place, Los Angeles, CA 90069, +1-323-782-9700, violetgrey.com.
#### Wasteland
* * *
It's Coachella year-round at this Melrose Avenue institution, which has a large, on-trend collection of vintage clothing for men and women that won't break the bank; denim cutoffs and overalls start at $32, Mexican blouses at $22, and Lurex hippie skirts at $28. There is more serious stuff, too, including I. Magnin cocktail coats and sequin cocktail dresses. "If you really look, you can find great stuff like '90s Helmut Lang, Prada, and Comme des Garçons," says designer Rosetta Getty. Bonus: You might run into Chloë Sevigny; she has said that this is her favorite store in L.A.
7428 Melrose Ave., Los Angeles, CA 90046, +1-323-653-3028, shopwasteland.com, and additional locations.
#### Zero Maria Cornejo
* * *
Maria Cornejo may be a New York–based designer (and have her original store there), but we prefer to shop her relaxed draped dresses, cocoon coats, and jumpsuits in geometric-inspired cuts at the light-filled L.A. outpost. Cornejo's demo is artsy and intellectual; her aesthetic, minimalist, feminine, and friendly. Is it any wonder that former First Lady Michelle Obama, actor Tilda Swinton, and artist Cindy Sherman are fans?
8408 Melrose Place, Los Angeles, CA 90069 +1-323-782-4915, zeromariacornejo.com, and an additional location in New York.
### BEVERLY BOULEVARD and THIRD STREET
#### Anine Bing
* * *
Why not go straight to the source for the army jackets, leather biker pants, tassel minidresses, vintage-looking "Los Angeles" tees, and lace bralets that have earned a cultlike following among the young Hollywood set, including Kendall Jenner, Rosie Huntington-Whiteley, Kate Bosworth, and Jessica Alba? This is Danish model-turned-designer (and L.A. transplant) Bing's hometown flagship.
8128 W. Third St., Los Angeles CA, 90048, +1-323-424-3165, aninebing.com, and additional locations.
#### Elaine Kim
* * *
The L.A. designer creates timeless classics (lace blazers, silk bombers, linen pants, pencil skirts) as well as custom-made garments. Her styling services are available on-site or via Skype. "I wear a lot of slouchy suits, and Elaine makes them for me in whatever color I want," says stylist Ilaria Urbinati. "It feels very old school to have my suits made."
8373 W. 3rd St., Los Angeles, CA 90048, +1-323-937-0355, elainekim.com.
#### George Esquivel
* * *
The L.A. shoemaker and his band of craftsmen have been hand-cobbling high-end shoes for a who's who of the well-heeled, including rock stars, NBA players, politicians, and Hollywood heavyweights, for more than a decade. His signature wing tips and boots are made from high-quality European leathers that are pebbled, painted, sunbaked, or pummeled into laid-back luxe. His appointment-only studio has plenty of pairs to try on and buy on the spot, and he takes custom orders, too. The shoes are made about an hour south of L.A. at Esquivel's Buena Park workshop.
8309 W. 3rd St., Los Angeles, CA 90048, +1-714-670-2200, esquivelshoes.com.
#### L.A. Eyeworks
* * *
An institution for creative peoples' creative frames since 1979, L.A. Eyeworks breaks the mold with its whimsical designs, and with its store windows that are a messaging platform for political discussion (and a good pun). Since 1981, the brand's portrait ad campaign, photographed by Greg Gorman, has featured more than 200 cultural provocateurs (Grace Jones, Andy Warhol, Joey Arias, and Debbie Harry, to name just a few), anchored by the brand's tagline: "A face is like a work of art. It deserves a great frame." "It's my candy store," fashion journalist/retailer Rose Apodaca says of the Beverly Boulevard space, designed by L.A. architect Neil Denari.
7386 Beverly Blvd., Los Angeles, CA 90036, +1-323-931-7795, laeyeworks.com, and an additional location on Melrose Ave.
DITA VON TEESE
HOW TO SHOP FOR RETRO FASHION ONLINE
* * *
The Los Angeles–based burlesque dancer and style icon is a vintage connoisseur and collector. Her high-end striptease act has been performed to sold-out crowds around the world.
I grew up watching classic films because my mother loved them, and that's where my obsession with retro glamour comes from. As a little girl, I especially loved those big Technicolor musicals starring Betty Grable, Carmen Miranda, and Rita Hayworth, and so 1940s era style has always been the dearest to me. I think it was the most glamorous time for both beauty and fashion, and many of the silhouettes are still chic and easy to wear today. I have a "code" that I rarely stray from, and it's all rooted in my childhood fantasies about growing up to be like the movie stars of the 1940s.
I like to get my swimsuits at PINUPGIRL-CLOTHING.COM They fit well and are made with nice luxurious satin Lycra that gives that glamorous retro shape, plus they're made in a broad range of sizes and look fantastic on every woman.
I like LENAHOSCHEK.COM for well-made retro style skirts and sweaters. I also love wearing pieces from their dirndl collection.
THE DEPOP MARKETPLACE APP, depop.com, is like Instagram for shopping, lots of glamour girls and fashion bloggers sell their things on it. I sell my own vintage clothes, jewelry, and samples from my lingerie collection, too.
WHEELSANDDOLLBABY.COM is great for cardigans and pencil skirts.
SECRETSINLACE.COM is a go-to for authentic fully fashioned seamed stockings with French and Euro heels. These are for the next-level stocking aficionado, made on the same machines as they were in the 1940s and '50s. Once you feel the difference of fully fashioned stockings, you can't go back to modern Lycra stockings. It's like becoming part of an elite club.
I like BARENECCESSITIES.COM for everything lingerie, they carry a vast selection of brands and a wide variety of sizes. I aways send fans there to get my lingerie brand because they stock all the larger sizes, whereas many online retailers don't.
#### Mister Freedom
* * *
A brick building houses the retail space and design studio created by French expat Christophe Loiron in 1990. Clothing spans the late 1800s to the 1970s from around the world, but Loiron specializes in men's rugged US and European vintage. "A denim lover's dream," says designer Andrea Lieberman. Loiron also designs a handful of his own vintage-inspired collections in the style of the American frontier, the Mexican Revolution, and 1950s California. This is one of French designer Isabel Marant's favorite L.A. haunts. Emily Current and Meritt Elliott are also devotees.
7161 Beverly Blvd, Los Angeles, CA 90036, +1-323-653-2014, misterfreedom.com.
#### Noodle Stories
* * *
This is L.A.'s top destination for clothing and accessories by interesting, intellectual designers such as Sacai, Martin Margiela, Issey Miyake, Antipast, Pas de Calais, Sofie D'Hoore, Y's, and more. Two doors down, the NS Temp store currently features men's and women's Comme des Garçons, plus selections from CDG Black, Tricot Comme des Garçons, and Junya Watanabe.
8323 W. 3rd St., Los Angeles, CA 90048, +1-323-651-1782, noodlestories.com.
#### Regency Jewelry Co.
* * *
The store offers expert jewelry repair and several cases of tempting vintage pieces, including Southwestern squash blossom necklaces, and Elsa Peretti originals. "Many years ago, I bought myself a garnet and gold princess ring from a 75 percent–off tray," says WhoWhatWear's Hillary Kerr. "The craftsmanship is amazing, and the price was almost criminally affordable. It makes me feel like a 1940s dame in all the right ways." Personal note: I, too, have found great vintage pieces there, including a 1950s-era aurora borealis cocktail bracelet.
8129 W. 3rd St., Los Angeles, CA 90048, +1-323-655-2573, regencyjewelry.com.
#### Re-Mix Vintage Shoes
* * *
Fashion insider Liz Goldwyn and other retronauts hit this store for deadstock vintage shoes and reproductions of styles from the 1920s to the 1970s, including ankle-strap platforms and perforated lace-ups. It has men's shoes, too.
7384 Beverly Blvd., Los Angeles, CA 90036, +1-323-936-6210, remixvintageshoes.com.
#### Scent Bar
* * *
This olfactory heaven is set up like a bar, allowing shoppers to try on fragrances while they enjoy coffee or prosecco and features hard-to-find brands from around the globe such as Escentric Molecules, Ex Nihilo, 10 Corso Como, Altaia, and Mancera. "Best in L.A. for niche fragrances and candles," says fashion journalist and consultant Melissa Magsaysay.
7405 Beverly Blvd., Los Angeles, CA 90036, +1-323-782-8300, luckyscent.com.
#### Scout
* * *
"Best selection of vintage clothing at affordable prices," says stylist and designer Maryam Malakpour. "Joey [Grana], the owner, has incredible taste and I have been following his handpicks for more than ten years. I found two Valentino Couture silk blouses there that I love." Other stylish shoppers praise the collection of '90s and minimalist clothing. "Exactly the type of vintage shopping I like to do," says retailer Lori Leven. "A supertight edit of pieces you aren't sure are contemporary or vintage when you pick them up."
8021 Melrose Ave., Los Angeles, CA 90046, +1-323-461-1530, scoutla.net.
#### Shabon
* * *
"One of my favorite vintage stores in L.A.," Nicole Richie says of this just-right-sized spot with wares spanning from the Victorian era to the '90s, where she nabbed one of her all-time-greatest finds, a vintage YSL oversized coat.
7617 Beverly Blvd, Los Angeles, CA 90036, +1-323-692-0061, shabonla.com.
### LA BREA
#### A + R Store
* * *
Global design, edited by fashion journalist and author Rose Apodaca and her husband, Andy Griffith (no, not that Andy Griffith). This inspirational showroom for modern furniture, lighting, and rug design also has plenty of grab-and-go gifts and toys, including safety pin earrings, animal head pins, digital wristwatches, and other tempting baubles.
171 S. La Brea Ave., Los Angeles, CA 90036, +1-323-692-0086, aplusrstore.com.
#### Aether
* * *
L.A.–based brand of men's and women's performance gear and off-duty athleisure for mountain, motorcycle, gym, or street. The flagship on La Brea Avenue has a walk-in freezer, chilled to 9 degrees Fahrenheit for coat testing, and a curated selection of "man toys," including a $19,850 Ducati motorcycle, a $1,500 electric skateboard, a $2,000 folding kayak, a $174 ax, and books on subjects such as subterranean London and container architecture.
161 S. La Brea Ave., Los Angeles, CA 90036, +1-323-746-5147, aetherapparel.com, and additional locations.
#### American Rag Cie
* * *
Southern California high-low style and culture in all its glory—vintage mixed with denim and workwear, contemporary clothing and streetwear, plus books, DVDs, and housewares, and a see-and-be-seen sidewalk café. American Rag is really three stores in one. The main shop features designer clothing and shoes alongside a superbly edited selection of vintage clothing and accessories. The denim bar has workwear and denim (Denham, Levi's Made & Crafted, Current Elliott, Mother), and the housewares store features outdoor furniture, glassware, and accessories with a French Mediterranean twist to match the restaurant's menu. "The best omelets," says Nicole Richie.
150 S. La Brea Ave., Los Angeles, CA 90036, +1-323-935-3154, americanrag.com, and an additional location in Newport Beach.
#### Rick Owens
* * *
The visonary designer who started his Goth luxe brand in L.A. in 1994, comes full circle with his L.A. store. Located on a seedy stretch of La Brea Avenue, it's just down the street from the Plaza Salon, a drag bar where he spent many a late night, including one when he met "a shriveled little man" on the dance floor who turned out to be his hero, Iggy Pop. The 5,200-square-foot concrete-floor space has a water tank that releases slow-motion bubbles and a wall that emits fog every five minutes. Featured inside? Owens's entire glamorous, gritty universe, including women's knitwear and T-shirts in his signature drapey silhouettes; sports-infused menswear, shoes, and accessories; the simple benches and tables he designs; and jewelry designed by his muse and wife, Michele Lamy.
819 N. La Brea Ave., Los Angeles, CA 90038, +1-323-931-4960, rickowens.eu, and additional locations.
#### The Way We Wore
* * *
Doris Raymond's vintage gold mine is the best combination of accessible (a $25 boho belt) and aspirational (a $9,500 1950s Givenchy gown), which is why it's a hit with designers, stylists, and tastemakers alike. Personal note: I once took fashion collector Iris Apfel on a shopping trip there and she had a field day. "You could stumble upon anything from a Madame Grès gown to original Yves Saint Laurent Rive Gauche culottes," says fashion journalist Christina Binkley. "If you're a design professional, she might even let you peek into her archives next door—that's really special." Raymond has been called "the godmother of fashion inspiration"; she even starred in her own docuseries on the Smithsonian Channel, L.A. Frock Stars.
334 S. La Brea Ave., Los Angeles, CA 90036, +1-323-937-0878, thewaywewore.com.
#### Union
* * *
A source for hard-to-find menswear labels from around the world, including Christopher Kane, Gosha Rubchinskiy, Raf Simons, Visvim, and Ganryu.
110 S. La Brea Ave., Los Angeles, CA 90036, +1-323-549-6950, store.unionlosangeles.com.
### HOLLYWOOD
#### Des Kohan
* * *
A trend forecaster for brands such as Prada and Gucci before opening her boutique in 2005, Desiree Kohan has built her business on her ability to teach clients how to incorporate clothing by avant-garde local and European designers—Juan Carlos Obando, Dusan, Damir Doma, Hussein Chalayan, etc.—into their everyday wardrobe. She also has a great eye for modern-yet-organic jewelry by Annie Costello Brown, Gabriela Artigas, and Aesa. "I know that I can walk in there at any time and emerge with a complete outfit that feels unique, elegant, timeless, and effortlessly chic," says designer/personal shopper Raven Kauffman. "I love the vibe there. You always feel like you are their most important client, no matter how often you go or how much you spend."
671 Cloverdale Ave., Los Angeles, CA 90036, +1-323-857-0200, deskohan.com.
#### Free City Supershop
* * *
Nina Garduno's hippie chic–meets–pop art–meets–commune store/gallery is L.A.'s antidote to cookie-cutter fashion. You'll browse unisex T-shirts, sweatshirts, sweatpants, and phone cases hand-printed with doves, rainbows, bicycles, and other feel-good graphics, and slogans such as COLOR AND SOUND, ARTISTS WANTED, and LIFE NATURE LOVE. There's a different theme each year, and the store also sells patchouli perfume, brown bread, ceramic pots, and books on subjects as diverse as Corita Kent and Yoko Ono.
1139 N. Highland Ave., Los Angeles, CA 90038 +1-323-461-2226, freecitysupershop.com, and additional locations.
#### Just One Eye
* * *
Part boutique, part art gallery, Just One Eye is the brainchild of Paola Russo, former fashion director of Maxfield. Expect the unexpected when you walk inside the art deco building in which tycoon Howard Hughes once ran his business empire, from a $65,000 Marilyn Minter print to a $12,500 Ulysses Tier 1 survival kit designed by the store and filled with supplies to get through a disaster (yes, really). In the men's and women's clothing departments, offerings run the gamut from minidresses by Anthony Vaccarello and rabbit ear headbands by Maison Michel to streetwear by Off-White and sneakers by Y-3. The store is also known for its quirky collaborations; one paired The Row and artist Damien Hirst, which resulted in Nile crocodile backpacks festooned with colorful pills and polka dots and priced at an astronomical $55,000. "Beautiful just to visit," says Hong Kong–based fashion journalist Divia Harilela.
"It's in an odd location and the storefront isn't a typical one. They needed to make it a destination, and they have, first of all by having an unusual space, and decor by Blackman Cruz, one of the best furniture stores ever created," says interior designer Brigette Romanek. "You'll have a unique shopping experience."
7000 Romaine St., Los Angeles, CA 90038, +1-323-969-9129, justoneeye.com.
#### Lost & Found
* * *
This string of five storefronts on Yucca Street in Hollywood is a one-stop shop for the laid-back L.A. lifestyle, with items for men, women, and kids; and an impeccable selection of boho dresses and blouses by Ulla Johnson, Nili Lotan, and Burning Torch, Local indigo jackets, Closed denim, B'Sbee shirts, furniture and other items for the home, such as maple cutting boards, Wallace Sewell throws, Studio One selvedge baskets, and more.
6320 Yucca St., Los Angeles, CA 90028, +1-323-856-5872, lostandfoundshop.com, and an additional location in Santa Monica.
#### Sonia Boyajian
* * *
L.A. jewelry designer known for kinetic-looking earrings and sculptural necklaces made of mismatched beads, candy-colored crystals, and fish-shaped pendants. Boyajian's studio/store is a jewelry fantasyland, with pieces hanging on the walls, spilling from drawers, and coming together on her worktable (she makes it all by hand, from setting stones to twisting wires). Boyajian has collaborated with several designers over the years on jewelry for runway collections, including Ulla Johnson, Peter Copping for Nina Ricci, and Bernhard Wilhelm, and will happily collaborate with customers to make custom pieces.
1947 N. Cahuenga Blvd., Ste. 102, Los Angeles, CA 90068, +1-323-325-1222, soniabstyle.com. By appointment.
#### The Supply Sergeant
* * *
Established in 1946, this regional chain is the go-to for Hollywood studios looking to outfit casts and backup dancers in military gear. You'll find American military uniforms from the Vietnam War, foreign uniforms, and the kind of camping and survival gear people snap up for the Coachella music festival. Costume designer Arianne Phillips singles out the Hollywood Boulevard location, across from the Musso & Frank Grill.
6664 Hollywood Blvd., Hollywood, CA 90028, +1-323-463-4730, supplysergeantstores.com, and an additional location in Burbank.
### SILVER LAKE/LOS FELIZ/ECHO PARK
#### Broome Street General Store
* * *
"Cozy, a little folksy, and has great stuff for your home," says Tere Artigas, who handles sales and press for her sister Gabriela Artigas's jewelry line. "Sells the kind of pieces that you would expect to find in a cold-weather beach house in Northern California, plus they make really good coffee and chocolate cookies."
2912 Rowena Ave., Los Angeles, CA 90039, +1-323-570-0405, broomestgeneral.com.
#### Clare V.
* * *
Hometown handbag designer Clare Vivier's first store helped put Silver Lake on the fashion map. Her colorful leather accessories marry French chic and minimalist L.A. cool. You'll find all her colorful bags here, including the Karlie Kloss–approved Sandrine duffle, totes, belt bags, and clutches, along with her tees and sweatshirts emblazoned with French phrases, sunglasses made in collaboration with Garrett Leight, and jewelry by L.A. local Annie Costello Brown. "One of those places I can't drive by without stopping," says fashion journalist Christina Binkley. "Her bags are great quality for the price, and she single-handedly transformed my evening-bag experience with her fold-over pockets. If I don't buy a bag, I can always grab one of her fun 'Merci Beaucoup' matchbooks—they're cheery and they're free!"
3339 W. Sunset Blvd., Los Angeles, CA 90026, +1-323-665-2476, clarev.com, and additional locations.
#### Lake
* * *
Cute boutique offering sunny, colorful boho styles from Ace & Jig, Maison Scotch, Matta, Humanoid, Isabel Marant Étoile, Iro, and Ulla Johnson, alongside Petit Bateau T-shirts and Jerome Dreyfuss bags, plus throw pillows, objets, and perfumes. Grab a coffee at L.A. Mill two doors down.
1618½ Silver Lake Blvd., Los Angeles, CA 90026, +1-323-664-6522, lakeboutique.com.
#### Mohawk General Store
* * *
Started by husband-and-wife team Kevin and Bo Carney, this spare but warm boutique is the cornerstone of the Sunset Junction retail scene, offering arty, minimalist clothing and accessories from a mix of local (Jesse Kamm, Raquel Allegra, Building Block, Black Crane), national (Apiece Apart, Engineered Garments, Rachel Comey, Mansur Gavriel), and international brands (Lemaire, Carven, A Détacher, Comme des Garçons, Dries Van Noten). The boutique also sells mid-century furniture, vinyl records, design books, and apothecary items. Stylist Jessica de Ruiter is a regular, and Nasty Gal's Sophia Amoruso likes to stop by after brunch at Cafe Stella. The men's store next door has elevated basics by Engineered Garments, Dries Van Noten, Gitman Bros., and more.
4011 W. Sunset Blvd., Los Angeles, CA 90029, +1-323-669-1601, mohawkgeneralstore.com., and an additional location in Pasadena.
#### Steven Alan Outpost
* * *
This is the permanent sale outpost for the chain of boutiques offering well-made, often American-manufactured men's and women's clothing from Apolis, RRL, Our Legacy, Rachel Comey, and more, with a smattering of Isabel Marant, A.P.C., Acne, and Apiece Apart. "A great place to buy all the women's and menswear of Steven Alan's regular stores, at discount," says fashion journalist Christina Binkley. "I never know what I'll find there—could be a snuggly sweater for my husband, or socks for myself, or a blouse."
1937½ Hillhurst Ave., Los Angeles, CA 90027, +1-323-667-9500, stevenalan.com.
#### Tavin
* * *
Opened in 2009 in Echo Park, Erin Tavin's vintage goldmine carries clothing from Victorian-era pieces to present day. There's also a strong collection of 1960s and '70s pieces, including Indian and Afghan dresses. Most items are in the $200-and-up range, though there is an "under $100" section.
1543 Echo Park Ave., Los Angeles, CA 90026, +1-213-482-5832, tavinboutique.com.
RODARTE SISTERS
L.A. NOSTALGIA
* * *
Few designers are as closely tied to California as Kate and Laura Mulleavy of Rodarte. The sisters started their label in 2005 out of their parents' Pasadena guesthouse. Since then, they've designed costumes for the film Black Swan, dressed celebrities for the red carpet (including Dakota Fanning, Kirsten Dunst, and Katy Perry), and won numerous awards for their multilayered runway collections, spun from memories of trips to the Redwood National Park, the Santa Cruz boardwalk, and Caffe Trieste in San Francisco's North Beach. More than a decade later, the Mulleavys still live together in Pasadena. Here are their shopping favorites in their 'hood.
Laura: For vintage, Carmen Hawk, who used to be the designer of Jovovich-Hawk. Her store, AVALON VINTAGE in Eagle Rock, is amazing. She also has the best Instagram page.
Kate: It was our dad's birthday, and up on the wall were these mushroom candles from the 1970s. We had to get them because (a) our dad is a mushroom guy, and (b) he loves candles. Basically, Carmen has great stuff that always has a story.
Laura: We grew up going to the HUNTINGTON LIBRARY GIFT SHOP because our grandmother was a docent. Over the years, we discovered how much better it is than other museum gift shops. The NORTON SIMON MUSEUM also has a great bookstore.
Kate: They sell you bottles of wine and you can go sit outside. It's like being in Europe. I got my favorite jean jacket at VIRGIL NORMAL, which is owned by our friend stylist Shirley Kurata. My favorite bookstore in L.A. is CARAVAN BOOK STORE. It's right near the Biltmore Hotel downtown. It's a little jewel box specializing in Western America books, from Jack Keruoac to westward expansion.
AVALON VINTAGE, 106 North Ave. 56, Los Angeles, CA 90042, +1-323-309-7717.
VIRGIL NORMAL, 4157 Normal Ave., Los Angeles, CA 90029, +1-323-741-8489, virgilnormal.com.
THE HUNTINGTON STORE, 1151 Oxford Rd., San Marino, CA 91108, +1-626-405-2142, thehuntingtonstore.org.
NORTON SIMON MUSEUM STORE, 411 W. Colorado Blvd., Pasadena, CA 91105, +1-626-449-6840, store.nortonsimon.org.
CARAVAN BOOK STORE, 550 S. Grand Ave., Los Angeles, CA 90071, +1-213- 626-9944.
### PASADENA/DOWNTOWN/EAST OF DOWNTOWN
#### Acne Studios
* * *
Acne (an acronym for Ambition to Create Novel Expression) was founded in 1996 in Stockholm by musician-turned-fashion designer Jonny Johansson. The brand has boutiques around the world—but the L.A. outpost is special. Opened in 2013 in the historic art deco Eastern Columbia Building on Broadway, it helped cement the gentrification of downtown L.A. and draw other high-end retailers to the neighborhood. "My go-to for everything," says blogger Chriselle Lim of this hipster haven, with denim, cool boots, and coats. "They are connected to iL caffe, which is a great spot to pick up my daily coffee and/or to have a meeting. They also have a very cute floral shop right across the street that just opened up. One-stop shop!"
855 S. Broadway, Los Angeles, +1-213-243-0960, acnestudios.com, and additional locations.
#### Alchemy Works
* * *
A retail, gallery, and events space with the added bonus of a Warby Parker showroom. You'll find everything from Will Adler photographs and boards by Almond Surfboards to Miansai bracelets and Martiniano shoes. Owned by Raan Parton (cofounder of Apolis, below) and his wife, Lindsay.
826 E. 3rd St., Los Angeles, CA 90013, +1-323-487-1497, alchemyworks.us.
#### Apolis
* * *
The new uniform of the creative class, Apolis's relaxed menswear includes indigo-dyed boiled-wool blazers, slim-cut cotton canvas utility pants, and washed oxford button-downs with do-gooder appeal. L.A.–based founders—and brothers—Raan and Shea Parton scour the globe for artisanal manufacturers to produce their designs using the highest social and environmental standards. "They have simple styles mainly for men, but much of it could be labeled unisex," says costume designer and author Tracy Tynan. "Some terrific indigo-dyed items like a soft, cozy alpaca cardigan." This is their flagship, located in downtown L.A.'s trendy Arts District.
806 E. 3rd St., Los Angeles, CA 90013, +1-213-613-9626, apolisglobal.com, and an additional location in New York City.
#### Building Block
* * *
The West Coast's cool-girl leathergoods brand opened its first store in downtown L.A.'s burgeoning Chinatown in 2016, showcasing its full range of industrial design–minded accessories. The store's design was inspired by the seminal 1934 Museum of Modern Art exhibition, "Machine Age," which put industrially designed objects like springs, ball bearings, and propellers in the context of an art museum. Bags hang from oversized springs on the walls, and sit on beds of foam balls in planters.
970 N. Broadway, Los Angeles 90012, +1-323-803-3420 building-block.com.
#### Casa Bernal
* * *
Located on Olvera Street, known as the birthplace of the city, where a Mexican marketplace re-creates old Los Angeles with structures, painted stalls, street vendors, cafés, restaurants, and gift shops.
Casa Bernal has been open since 1944, selling embroidered Mexican blouses, charro suits, handblown glassware, serapes, blankets, paper flowers, baskets, and more. This is accessories designer Sally Perrin's go-to for traditional Mexican clothing.
W-23 Olvera St., Los Angeles, CA 90012, +1-213-687-4568, casabernalstore.com.
#### Daiso Japan
* * *
Japanese version of a dollar store, with 8,000 square feet of goodies for the home, bath, and closet, many priced at about $1.50 each. "So fun to wander through and pick up everything from my favorite green tea to toilet scrubbers and makeup sponges," says fashion journalist Melissa Magsaysay. "The brightly lit environment and the shelves packed with colorful products in adorable packaging is a sensory overload in the best way imaginable." There are numerous locations in the L.A. area alone; this one in Little Tokyo is well worth browsing.
333 S. Alameda St., at the Little Tokyo Galleria, Los Angeles, CA 90013, +1-213-265-7821, daisoglobal.com, and additional locations.
#### Replika Vintage
* * *
"The guy that runs it has an incredible eye," says costume designer and author Tracy Tynan. "Not cheap but everything is in mint condition. Recently, I got a wonderful Dior chiffon camouflage skirt there, cut on the bias."
1812 N. Broadway, Los Angeles, CA 90031, by appointment at +1-323-251-7613, replikavintage.com.
#### Shareen
* * *
Chic downtown warehouse of vintage goodness curated by former fashion editor, model scout, and actor Shareen Mitchell, who got her start at the Melrose Trading Post. A treasure trove of restored and reconstructed dresses, leather coats, tutus, and sequin jackets on racks organized by decade (1940s through the 1990s) or theme (ethnic, pantsuits, etc.), with alterations available on-site.
"It feels like a secret society for cool girls and vintage lovers," says designer Erin Fetherston. "A great resource for design inspiration and cool vintage finds," says Mad Men costume designer Janie Bryant. Mitchell has since opened a second store in New York City, launched a line of bridal, evening, and ready-to-wear pieces, and starred in a reality TV series, Dresscue Me.
1721 N. Spring St., Los Angeles, CA 90012, +1-323-276-6226, shareen.com.
#### Slauson Super Mall
* * *
This indoor mall is an L.A. experience, featuring a variety of vendors selling trucker caps emblazoned with sports teams, neighborhoods, bejeweled mustaches, spikes, and horns; tube socks; lace thongs; hair extensions; sneakers shrink-wrapped for safe keeping; and screen print while-you-wait NWA and Cali T-shirts. Don't miss the lady with soaps that look like slices of cake, or TF Jewelry, where you should ask for Boss Lady. She can make you a custom gold nameplate necklace or earrings for about $100 in two to three hours, and will even wrap it in robin's egg blue packaging like that other Tiffany.
1600 W Slauson Ave, Los Angeles, CA 90047, +1-323-778-6055, slausonsupermallinc.com.
#### St. Vincent de Paul
* * *
"A huge thrift store with a mix of clothes, furniture, paintings, and appliances. I find weird and great things here—I might come out with a dress and a couch," says Brigette Romanek.
210 N. Avenue 21, Los Angeles, CA 90031, +1-323-224-6280, svdpla.org.
### SANTA MONICA/BRENTWOOD/PACIFIC PALISADES/MALIBU
#### Becker Surfboards
* * *
This iconic surf-shop chainlet is a favorite for bikinis, flip-flops, Baja hoodies, and beach cardigans. The Malibu outpost is a good place to stop after a morning riding the waves or lounging on the sand. "I can do some serious damage here," says Chrome Hearts cofounder Laurie Lynn Stark. "I buy weird socks and flip-flops, and squeeze into some cheap bathing suits. I love it."
23755 Malibu Rd., Malibu, CA 90265, +1-310-456-7155, beckersurf.com, and an additional location in Hermosa Beach.
#### Brentwood Country Mart
* * *
This outdoor shopping center is a quintessential L.A. experience. "One of my favorite places to spend an afternoon," says jewelry designer Jennifer Meyer. "My family and I love to eat at Farmshop, and it also has a delicious selection of groceries to bring home for dinner. Broken English has the most unique jewelry selection. Poppy Store has my favorite selection of children's clothes. I love stopping by Jenni Kayne for clothes, accessories, and home goods. Sugar Paper is my go-to for all things stationery, plus it has the sweetest little gifts. We usually finish the day with a treat from Sweet Rose Creamery—it has such fresh and creative flavors, and vegan ones, too!"
225 26th St., Santa Monica, CA 90402, +1-310-451-9877, brentwoodcountrymart.com.
#### Elu
* * *
"Elu has a clear, intentional point of view and understated, casual elegance," says Dreslyn.com owner Brooke Taylor Corcia of the Malibu outpost of this chainlet, featuring low-key luxe by Guidi, Uma Wang, Henry Beguilin, Rick Owens, and Vetements.
3824 Cross Creek Rd., Malibu, CA +1-310-317-6177, elubycn.com, and additional locations.
#### Elyse Walker
* * *
Beachy casual with a metropolitan edge, this 6,500-square-foot store has everything a woman needs to take her from carpooling (Isabel Marant jackets, Raquel Allegra sweaters, Frame Denim jeans) to a business meeting (Stella McCartney suits, Chloé separates) to a museum gala (Givenchy and J. Mendel gowns). Owner Elyse Walker, whose family was in the shoe business, puts a huge focus on accessories, including bags by Givenchy, Proenza Schouler, and Balenciaga and shoes by Gianvito Rossi and Golden Goose. "A little pocket of high fashion in the Palisades," says stylist Elizabeth Stewart.
15306 Antioch St., Pacific Palisades, CA, 90272, +1-310-230-8882, elysewalker.com, and an additional location in Newport Beach.
#### Hidden Treasures
* * *
There's nothing hidden about this place. Located in a house kitted out to resemble a pirate ship, it's complete with a steering wheel on the balcony and a mermaid statue and life preserver out front. The bounty inside ranges from colorful rave wigs to band T-shirts to Mexican embroidered dresses to an antique diving helmet. You can get lost for hours trying on things. A must-see, this is where Rodarte's Kate Mulleavy found her favorite vintage T-shirt of all time, with a picture of Barbra Streisand from Hello, Dolly! airbrushed on the back. "It's my favorite thing ever and it was under $50," she says.
154 S. Topanga Canyon Blvd., Topanga, CA 90290, +1-310-455-2998, hiddentreasurestopanga.com.
HIDDEN TREASURES
#### James Perse
* * *
Perhaps the most enjoyable of all of Perse's stores, the Malibu Lumber Yard outpost suits his upscale laid-back beach-luxe lifestyle brand to a tee, with its outdoor lounge chairs and a pool table. Inside, you'll find the full range of the brand's surf 'n' sand–inspired colored jersey and French terry separates, linen shorts and pants, plus furniture and home goods, bikes, and boards. Perse is the son of Maxfield founder Tommy Perse, who brought the all-black look to L.A. The two couldn't be more different, except when it comes to retail know-how.
3939 Cross Creek Rd. Ste. E-100, Malibu, CA 90265, +1-310-469-6030, jamesperse.com, and additional locations.
### CULVER CITY/VENICE
#### Arcana: Books on the Arts
* * *
This high-end purveyor of fine art, fashion, photography, and film books is a must-visit for creatives, including Bottega Veneta designer Tomas Maier. It's not cheap, but you can find lots of first editions, including Black Panthers 1968, Calder Jewelry, and other eclectic items, such as a 1992 edition of Paris Vogue edited by the Dalai Lama. "A great source of inspiration," says accessories designer Sally Perrin. The store will ship books for you, so when you arrive home, a gift will be waiting for you, Maier says.
8675 Washington Blvd., Culver City, CA 90232, +1-310-458-1499, arcanabooks.com.
ARI SETH COHEN
RETRO ROW IN LONG BEACH
* * *
Ari Seth Cohen is the street-style photographer and creator of Advanced Style, a blog that's been documenting the fashion and style of the over-60 set since 2008. He's also the author of two books based on the blog: Advanced Style (2012) and Advanced Style: Older & Wiser (2016) as well as a documentary released in 2014.
Kathleen Schaaf's 4th Street Shop, MEOW always has a rack of patterned tuxedo jackets ripe for the picking. IN RETROPSECT has great mod and midcentury furniture, clothing, and collectibles. I always find an old Charles Schulz book or '70s embroidered pillow. My favorite pin and sliver-flecked armchair come from there. One of the best gift stores I have ever seen stocked to the brim with collections of old glassware, trophies, kitschy elves and ornaments, globes, ceramic dogs, classic board games, and treasures is PAST AND PRESENT RETROADA. During Christmas they have stacks of Technicolor vintage ornaments in their original packing!
MEOW, 2210 E 4th St., Long Beach, CA 90814, +1-562-438-8990, Meowvintage.com.
IN RETROSPECT, 2122 E 4th St., Long Beach, CA 90814, +1-562-433-6600, inretrospect.com.
PAST AND PRESENT RETROADA, 2001 E 4th St., Long Beach, CA 90814-1001, +1-562-434-6464.
#### Garrett Leight California Optical
* * *
Leight, whose father, Larry, founded Oliver Peoples in 1987, struck out on his own in 2009, opening this store in Venice Beach. His geek-chic and retro-inspired frames (each one is named after a Venice street), are a hit with Kendall Jenner, Kristen Stewart, Emily Ratajkowski, Garance Doré, and Johnny Depp. He has collaborated on frames with indie labels Want Les Essentiels, and Mark McNairy. The brand also publishes a pretty cool biannual magazine, Spectacle, that covers music, food, and culture.
1423 Abbot Kinney Blvd., Venice, CA 90291, +1-310-392-3400, garrettleight.com, and additional locations.
#### General Store
* * *
Vintage clothes, textiles, handmade espadrilles, macramé market totes, handmade ceramic cactus sculptures, brass incense burners, wooden baby rattles, and more things you didn't know you needed, all with a modern sensibility. It's hard to come out of this place empty-handed. Personal note: Last time, I bought a wood surf-wave nesting puzzle that sits on the mantel in my living room. There's also a location in San Francisco.
1801 Lincoln Blvd., Venice, CA 90291, +1-310-751-6393, shop-generalstore.com, and an additional location in San Francisco.
#### Govinda's International Imports
* * *
"My happy place for a peaceful lunch and authentic wares is the vegetarian restaurant and general store located within the Hare Krishna Cultural Center," says fashion consultant Victoria Brynner. Affordably priced tunics, saris, wrap skirts, colorful clutch bags, bangles, incense, pillows, and more abound at this authentic Indian shopping emporium that's been drawing the fashion crowd and celebrities to the West L.A. Hare Krishna temple since 1982.
3764 Watseka Ave., Los Angeles, CA 90034, +1-310-204-3263.
#### Heist
* * *
Stylish boutique with a boho bent, featuring low-key clothing and accessories by Nili Lotan, Giada Forte, Isabel Marant, Jerome Dreyfuss, Ulla Johnson, Closed, Mes Demoiselles, K. Jacques, Golden Goose. "Our favorite designers and the friendliest staff," say stylists and designers Emily Current and Meritt Elliott.
1100 Abbot Kinney Blvd., Venice, CA 90291, +1-310-450-6531, shopheist.com.
#### Kendall Conrad
* * *
The former fashion model's down-to-earth, SoCal-Euro-inspired accessories line features handbags, leather bracelets, and sandals in luxe materials and muted colors, plus organic-shaped sterling silver and brass jewelry. One of her best-sellers is a a zip-around pouch with tassel details that can hold an iPhone and also has slots for credit cards or cash. It comes in leather, alligator, or snakeskin. "Everything she carries is artisanal, handmade, timeless, and made in the USA," says Figue designer Stephanie Von Watzdorf of the L.A.–based talent.
1121 Abbot Kinney Blvd., Venice, CA 90291, +1-310-399-1333, kendallconraddesign.com, and additional locations.
#### Le Magazyn
* * *
Bringing the best of Bossa to L.A., this is a treasure trove of bags, shoes, jewelry, clothing, furniture, home goods, and art, from a range of international designers, mostly from Brazil and made by hand. Think macramé sandals and laborodite earrings.
904 Pacific Ave., Venice, CA 90291, +1-310-450-5333, lemagazyn.com.
#### LFrank
* * *
Earthy, elegant jewelry and lingerie by California native Liseanne Frankfurt. Lapis feather earrings, tourmaline pendant necklaces, multicolored tourmaline cigar band rings, luscious silk kimonos, and slip dresses: This is the stuff boho boudoir dreams are made of.
226 Main St., Venice, CA 90291, +1-310-452-0771, lfrankjewelry.com.
#### Magasin
* * *
Josh Peskowitz, the former men's fashion director of Bloomingdale's, settled on L.A. for his first independent retail venture, a 1,500-square-foot store in Culver City's Platform development (where you'll also find Linda Farrow and Aesop). The boutique features a strong mix of upscale international labels selling Peskowitz's interpretation of the men's creative-casual wardrobe, including unstructured Camoshita jackets, garment-dyed polos from Massimo Alba, cashmere T-shirts from Naadam, handmade slipper-like leather shoes by Feit, bucket hats and totes by Monitaly.
8810 Washington Blvd., Culver City, CA 90232, +1-213-458-8424.
#### Strange Invisible Perfumes
* * *
Venice-based natural fragrance house that uses botanical essences to create transporting scents, such as Ojai and Magazine Street, plus perfumes inspired by your zodiac sign. "I am never without their incredible botanical perfume, or any number of the excellent brands of skincare that they offer, especially May Lindstrom's line, or a flower remedy from Alexis Smart Flower Remedies," says jewelry designer Liseanne Frankfurt.
1138 Abbot Kinney Blvd., Venice, CA 90291, +1-310-314.1505, siperfumes.com.
### NEWPORT BEACH/LONG BEACH
#### A'Maree's
* * *
A fashion force forty miles south of L.A., this waterfront Newport Harbor space invites comparisons to Peggy Guggenheim's palazzo in Venice, Italy. Boasting soaring arched windows and portholes in the floor that allow views of the fish swimming below, it even has its own boat slip for those who want to dock and shop. The modernist building was designed in 1961 by Pasadena architects Thornton Ladd and John Kelsey, who would go on to design the Norton Simon Museum. As yachts float by, sunlight pours over cases of sparkling jewelry. Racks are filled with jackets and shorts by The Row, fringed jeans by Chloé, lace-trimmed T-shirts by Sacai, oversized shirts by Junya Watanabe, and wispy Dosa dresses. The business began in 1976 as a labor of love for family matriarch Nancy Brown. Now, Brown's three daughters—Dawn Klohs, Denise Schaefer, and Apryl Schaefer—run the show. "One of the finest boutiques ever," says stylist Jessica de Ruiter. "The candles, the sunglasses, jewelry, of course," says jewelry designer Irene Neuwirth. "And the clothing and books and . . . the list goes on. I love Denise and Dawn! I feel like I'm on vacation when I'm sitting and having a deliciously healthy lunch and a coffee in their gorgeous kitchen." There's also a sale shop on Balboa Island.
2241 West Coast Hwy., Newport Beach, CA 92663, +1-949-642-4423, amarees.com.
### SPECIAL EVENTS/SERVICES
#### A Current Affair
* * *
This twice-yearly pop-up vintage marketplace is a must-shop for designers and vintage aficionados, including Alice and Olivia's Stacey Bendet, Figue's Stephanie von Watzdorf, and A.L.C.'s Andrea Lieberman, who says, "I wish they would pop up more often." "My girlfriends and I turn it into a half-day adventure," says Nicole Richie. More than sixty vendors, and events in L.A. and Brooklyn. Co-founded by Joey Grana, owner of vintage boutique Scout.
Itsacurrentaffair.com.
#### DTLA Custom
* * *
This group of artists will personalize any leather good in your closet, from a Louis Vuitton tote to a pair of Minnetonka baby moccasins. They'll hand-paint stripes, initials, palm trees, lipsticks, pet portraits, cartoon characters, or more on any leather surface, starting at $175.
+1-310-935-0477, dtlacustom.com.
HEATHER JOHN FOGARTY
WEEKEND SHOPPING IN OJAI
* * *
As a Bay Area native, and a freelance fashion and food writer for Los Angeles magazine, the Los Angeles Times, and other publications, Heather John Fogarty has shopped her way up and down the California coast.
I'm obsessed with IN THE FIELD! I want everything in the store, from the moose antlers to the home stuff—Rachel Craven linens, local ceramics, vintage surfboards. The clothing edit is heaven—chic and easy pieces from Ulla Johnson, Imogene & Willie, and Beatrice Valenzuela, and Agnes Baddoo handbags—exactly what you want to be wearing around town. Laid-back, modern boho at its best. There are great men's T-shirts, too.
I love SUMMER CAMP for housewares, and I kind of "need" some of the Bernscott Pottery ceramics by Ojai artist Karen Scott. Great vintage Southwestern rugs and throws, and reclaimed wood frames. They have a lot of vintage/flea market finds and soy candles, tongue-in-cheek Ojai Nightlife T-shirts, that kind of thing.
Another place that's good for home stuff is DEKOR & CO. Really beautiful selection of Swedish beauty products, Moroccan rugs, Danish modern furniture, and artisanal teas.
IN THE FIELD, 730 E. Ojai Ave., Ojai, CA 93023, +1-310-403-4292, inthefieldojai.com.
SUMMER CAMP, 1020 W. Ojai Ave., Ojai, CA 93023, +1-805-861-7109, shopsummercamp.com.
DEKOR & CO., 105 S. Montgomery St., Ojai, CA 93023, +1-805-272-8675, dekorandco.com.
#### Echo Park Craft Fair
* * *
Not your traditional craft fair. This twice-yearly indie marketplace founded by designers Beatrice Valenzuela and Rachel Craven in 2009 features vendors who also sell their wares at boutiques such as Mohawk General Store in Los Feliz and Bird in Brooklyn. Typically, you'll find jewelry by Annie Costello Brown, Dream Collective, and Gabriela Artigas, leather totes by Agnes Baddoo, embroidered jeans by Bliss and Mischief, and gauzy dresses by Doen. The event has helped shape the made-in-L.A. aesthetic associated with the creative communities of Los Feliz, Echo Park, and Silver Lake that has been embraced around the world.
Echoparkcraftfair.com.
#### Long Beach Antique Market
* * *
"Super-expansive, with hundreds of vendors and an incredibly eclectic mix of furniture, clothing, and collectibles," says Brother Vellies designer Aurora James. "Because it's a little ways outside of downtown L.A., it's slightly less frequented than the Rose Bowl, which ups your chances of scoring that hidden treasure." "The Rose Bowl Flea is more famous," says Tomboy Style author Lizzie Garrett Mettler, "but there are more deals to be had in Long Beach. I like to pop into Number Nine afterward for a great bowl of pho."
Third Sunday of every month. 4901 E. Conant St., Long Beach, CA 90808, +1-323-655-5703, longbeachantiquemarket.com.
#### Melrose Trading Post
* * *
Small but mighty, this weekly open-air flea market in the parking lot of Fairfax High School in the heart of L.A.'s Melrose shopping district offers a vintage fix for stylist/designer Maryam Malakpour, jewelry designer Daniela Villegas, and other stylish shoppers. "Small but good quality, great for finding cool dresses for every day, very accessible and easy," says Villegas. Canadian model-turned-blogger Elizabeth Minett hits the textile stand that sells "exquisite hand-dyed prints from Mali."
7850 Melrose Ave., Los Angeles, CA 90046, +1-323-655-7679, melrosetradingpost.org.
ONLINE SHOPPING TIP
"There is this website/blog called Love The Edit (lovetheedit.com) that makes navigating Amazon easy and actually chic!" says L.A.–based fashion journalist/consultant Melissa Magsaysay. "We all use Amazon for paper towels and dog food reorders, but there is a lot of fashion and accessories to be had and often at better prices. For me, the biggest draw for buying clothing or accessories on Amazon is how quickly it can arrive, because I am the most impatient person, particularly when having to wait for something I want to wear right away."
#### Rose Bowl Flea Market
* * *
One of the world's best, this monthly market with more than 2,500 vendors is regular stop for designers, stylists, and fashion lovers from as far away as China and Japan. "You have to get there at 7 a.m., and you couldn't go through the whole thing in a day if you tried," says blogger Shea Marie. "You can find everything from handmade furniture, furs, vintage jewelry, and amazing vintage designer finds. One of my favorite finds was a vintage Escada star-studded belt." "There are endless options for everything you need or don't need in your closet," says designer Anine Bing. Stylist Tara Swennen goes monthly, TV fashion personality Joe Zee shows up early for midcentury furniture, and The Great designers Emily Current and Meritt Elliott scour the vendors for vintage denim, tees and Victorian dresses. This is where stylist B Åkerlund scored her single best shopping find of all time: a life-size Andy Warhol doll.
Second Sunday monthly, 1001 Rose Bowl Dr., Pasadena, CA 91103, +1-626-577-3100, rgcshows.com/rosebowl.
ROSE BOWL FLEA MARKET
#### Vintage Fashion Expo
* * *
Twice-yearly haunt for vintage hounds in Los Angeles and San Francisco, featuring more than fifty dealers from around the country. Rachel Zoe, Who What Wear's Hillary Kerr, Liz Goldwyn, and Dita Von Teese are all devotees. "I have found so many unbelievable pieces, one being an amazing Oscar de la Renta, all black, chantilly lace couture cocktail dress," Zoe says. "I also found my Chanel briefcase there, which is one of my most treasured pieces."
Vintageexpo.com.
## CALIFORNIA
### SANTA BARBARA
#### Wendy Foster
* * *
Wendy Foster brought designer fashion to Santa Barbara in the 1970s and her lifestyle shop (near San Ysidro Ranch) is worth a stop anytime you're nearby for inspired, relaxed-yet-elegant pieces. Foster's retail empire, which she owns with her husband, at Pierre Lafond, also includes two vineyards, a bistro, a gourmet market, a home accessories store upstairs, and three other area clothing boutiques. (Designer Jenni Kayne singles out the Pierre Lafond store in the same complex, for "a really eclectic and inspiring assortment of home goods, such as well-designed baskets and cutting boards.")
516 San Ysidro Road, Santa Barbara CA 93108, +1-805-565-1506, wendyfoster.com.
### PALM SPRINGS/PALM DESERT
#### Desert Hills Premium Outlets
* * *
A must-stop on the way to or from the desert, this outlet mall has everything, from the tip-top (Prada, Saint Laurent, Gucci, Valentino, Jimmy Choo, Etro, Marni) to Barneys New York, James Perse, J. Crew, Maje, Rag & Bone, Wolford, and more. Personal note: Some of my best shopping scores of all time have come from here, including a pair of Manolo Blahnik pink crocodile slingbacks I got for less than $500, and a Prada peacock-feather-covered, straight-from-the-runway hat I snapped up for about $300.
48400 Seminole Dr., Cabazon, CA 92230, +1-951-849-6641, premiumoutlets.com/outlet/desert-hills.
#### The Fine Art of Design
* * *
When Louis Vuitton hosted its cruise show in Palm Springs in 2015, fashionable guests such as blogger Susie Bubble found their way to this Palm Desert vintage clothing must-see while they were in town. Bubble nabbed a vintage Courrèges jacket here, and I found a fabulous rainbow-striped dress. "I have found some true gems here, an Oscar de la Renta lace blouse, an amazing velvet and taffeta gown from the '80s," says jewelry designer Liseanne Frankfurt, who grew up in Palm Springs.
73717 Hwy. 111, Palm Desert, CA 92260, +1-760-565-7388, thefineartofdesign.com.
TRINA TURK
PALM SPRINGS
* * *
L.A. designer Trina Turk has been the unofficial fashion ambassador for Palm Springs for almost as long as she's been in the business (since 1995). She's a lover of the area's midcentury aesthetic, which inspires her California resort-chic label, and its architecture (her gleaming 1936 "Ship of the Desert" weekend home has been photographed numerous times for magazines and is the brand's spiritual home).
Turk opened her first store in Palm Springs in 2002. Housed in a low-slung, glass-walled 1960s Albert Frey building, it's become a destination for stylish travelers, and has expanded to three storefronts carrying womenswear, home goods, and the Mr. Turk collection.
There are so many local gems in Palm Springs. One of my favorites is BON VIVANT, which has decorative accessories for the home, great vintage jewelry and, occasionally, furniture. I love Patrick and James, the proprietors, their knowledge of their vintage finds, and their enthusiasm for seeing their finds go to a good home. I would not want to describe this store as cheap, because it's not. But chic, yes—due to Patrick's amazing eye for the beautiful and the whimsical. He scours flea markets, estate sales, and church bazaars for an unrivaled selection of decorative objects for your home at very reasonable prices. MODERNWAY has mostly midcentury furniture, with a special love for the outrageousness and color of the 1970s. HEDGE in Cathedral City is a serene, graphic, and modern point of view on how to decorate your home with mostly vintage furnishings and objects. And, of course, TRINA TURK and MR. TURK! A visit to Palm Springs is not complete without a stop here.
BON VIVANT, 766 N. Palm Canyon Dr., Palm Springs, CA 92262, +1-760-534-3197, gmcb.com.
MODERNWAY, 745 N. Palm Canyon Dr., Palm Springs, CA 92262, +1-760-320-5455, psmodernway.com.
HEDGE, 68-929 Perez Rd., Cathedral City, CA 92234, +1-760-770-0090, hedgepalmsprings.com.
TRINA TURK PALM SPRINGS, 891 N. Palm Canyon Dr., Palm Springs, CA 92262, +1-760-416-2856, trinaturk.com.
## SAN FRANCISCO
### HAYES VALLEY
METIER
#### Azalea
* * *
Denim-heavy store features trendy crop tops, high-low dresses, and culottes from Alexander Wang, Ksubi, and Iro, as well as in-house label Azalea—which is priced at less than $100—also menswear from John Elliott, Stampd, and more. Don't miss sister store Rand and Statler, next door to the Hayes Street location, offering edgier, higher-priced labels such as Acne, A.L.C., Comme des Garçons Play, and Anine Bing.
411 Hayes St., San Francisco, CA 94102, +1-415-861-9888, and 956 Valencia St., San Francisco, CA 94102, +1-415-682-6988, azaleasf.com.
#### Bell'occhio
* * *
The chicest general store you'll ever come across, Bell'occhio translates to "beautiful eye" in Italian. "From the eighteeth-century French jewelry in its original boxes to vintage commemorative French Liberation ribbons, owner Claudia Schwartz's shop has some amazing finds," San Francisco–based model, fashion journalist, and couture collector Tatiana Sorokko says. "Scented Armenian burning papers, unique handmade stationery items, and carrot-shaped paper packages of amandes au chocolat are just a few items I love." Also a favorite of Martha Stewart's, it's a one-stop shop for gifts, with many things under $100.
10 Brady St., San Francisco, CA 94103, +1-415-864-4048, bellocchio.com.
BELL'OCCHIO
#### Metier
* * *
Estate pieces mingle with gems by contemporary masters at Metier, a gem of a store, with midnight-blue walls artfully leafed with gold, and a curated collection of new and vintage jewelry and accessories. The owners seek out unique designers, such as Gabriella Kiss and Montréal-based Arielle de Pinto, who treats metal like fabric, crocheting sterling silver and gold vermeil into lacy chain-mail-like adornments. Pieces start at $50.
546 Laguna St., San Francisco, CA 94102, +1-415-590-2998, metiersf.com.
#### Modern Appealing Clothing (MAC)
* * *
"A San Francisco institution," says San Francisco–based retailer Emily Holt. "They've been around for thirty-five years and are the authority on avant-garde designers from places like Japan and Belgium. They're unafraid of wacky, which, in its own way, is very San Francisco." Men's and women's clothing by Dries Van Noten, Sofie D'Hoore, Tricot Comme des Garçons, Noir Kei Ninomiya, and Walter Van Beirendonck.
387 Grove St., San Francisco, CA 94102, +1-415-863-3011, modernappealingclothing.com, and an additional location in Dogpatch.
#### Reliquary
* * *
This is one of San Francisco designer Erica Tanov's local favorites, "a curiosity shop of sorts, featuring vintage and new women's clothing, jewelry, and folk art." You'll find comfy basics by CP Shades alongside vintage American, French, and Japanese workwear, Southwestern and Tuareg jewelry, and RTH leather totes.
544 Hayes St., San Francisco, CA 94102, +1-415-431-4000, reliquarysf.com.
### THE MISSION
#### Anaïse
* * *
Great selection of hard-to-find, cool-girl labels from the US (Jasmin Shokrian, Dosa, Co) and abroad (Carven, Hache, Chalayan, Vanessa Bruno), with anti-It bags by Eatable of Many Orders and Isaac Reina.
3686 20th St., San Francisco, CA 94110, +1-408-807-9379, shopanaise.com.
#### Gravel & Gold
* * *
Women-owned design collective that makes whimsical, print-centric housewares (boob-print towels, anyone?) and casual clothing. The store also stocks artsy fun jewelry by Dream Collective and Annie Costello Brown, notebooks, pottery, incense, and more by indie makers.
3266 21st St., San Francisco, CA 94110, +1-415-552-0112, gravelandgold.com.
#### Le Point
* * *
Spare concept space opened by stylist Pauline Montupet to showcase classic-cool labels such as Clare Vivier, Ryan Roche, Opening Ceremony, Lemlem, and Creatures of Comfort.
301 Valencia St., San Francisco, CA 94103, +1-415-400-4275, shoplepoint.com.
#### Mira Mira
* * *
Small boutique selling next-gen indie designers with a boho sensibility, including Polder (from France), Rodebjer (from New York), and Ottod'ame (from Italy). This is where you'll find kimono jackets, diaphanous dresses, and high-waist pants from the next Isabel Marants.
3292 22nd St., San Francisco, CA 94110, +1-415-648-6513, miramirasf.com.
#### Small Trade Company
* * *
Matt Dick's design atelier above the Heath Ceramics flagship in the Mission District produces Japanese-inspired aprons and other workwear, bags, and shibori-dyed apparel. He has collaborated with Levi's, Blue Bottle Coffee, and Commune Design on products.
550 Florida St., Ste. D, San Francisco, CA 94110, +1-415-570-1019, smalltradecompany.com.
#### Voyager
* * *
A concept store for men and women with a Japanese vibe, it stocks minimalist clothing by Priory, Shaina Mote, Black Crane, WayWay, Metier, and others; arty Scandinavian textile-driven styles by Dusen Dusen and Henrik Vibskov, plus Hobonichi notebooks, gifts, and decorative objects.
365 Valencia St., San Francisco, CA 94110, +1-415-795-1748, thevoyagershop.com, and an additional location in downtown Los Angeles.
### NORTH BEACH
#### Al's Attire
* * *
A taste of couture in North Beach.This mom-and-pop bespoke clothing and accessories store can make everything from patchwork leather boots to seersucker suits. "A favorite for custom-made footwear," says model and fashion journalist Tatiana Sorokko. "I first met [owner] Al Ribaya in 2008, when I had the idea to make flat sandals with burnt ostrich feathers, just like the adornment of Ralph Rucci's chiffon evening gown that I wanted to wear for the gala opening of the San Francisco Opera." Shoes are built around a wooden last made for each customer's feet.
1300 Grant Ave., San Francisco, CA 94133, +1-415-693-9900, alsattire.com.
#### City Lights
* * *
The independent bookstore founded in 1953 by Lawrence Ferlinghetti, published Allen Ginsberg's Howl in 1956, inspiring a generation of nonconformist youth and propelling the Beats into international consciousness. A copy of Howl from here is the consummate San Francisco souvenir.
261 Columbus Ave., San Francisco, CA 94133, +1-415-362-8193, citylights.com.
### PACIFIC HEIGHTS
#### March
* * *
"Next-level home store filled with the most wonderful pottery, enamelware, ridiculously luxurious knives and wooden utensils, hand-painted glassware, and beyond-beautiful packaged pantry goods like olive oil, marmalade, and almond butter," says Emily Holt. "Stocks some of the most amazing ceramics, wood, and leather products you could ever imagine," says fashion designer Mara Hoffman. Think spatterware, copper teakettles, and haute aprons.
3075 Sacramento St., San Francisco, CA 94115, +1-415-931-7433, marchsf.com.
### PRESIDIO HEIGHTS
#### Betty Lin
* * *
Chic and cheerful cropped flare pants by Suno and Marni, ruffled blouses by Ellery, jumpsuits by Rachel Comey and Nili Lotan, and more.
3625 Sacramento St., San Francisco, CA, 94118, +1-415-345-8688, shopbettylin.com.
### TENDERLOIN
#### Hero Shop
* * *
A highly curated fashion and lifestyle concept store created by former Vogue editor Emily Holt, it stocks women's ready-to-wear and accessories plus home, lifestyle, and tech items from established designers and local West Coast brands. Labels include Creatures of the Wind, Adam Lippes, Sophie Buhai, Myriam Schaefer, Paul Andrew, Vans, Stevie Howell, Future Glory, and Lola Fenhirst.
982 Post Street, San Francisco, California 94121, +1-917-539-1562, hero-shop-sf.myshopify.com.
HERO SHOP
#### JayJeffers
* * *
Vintage and made-to-order home furnishings and accessories housed in a 1910 San Francisco warehouse in the newly trending Tenderloin neighborhood. Penguin cocktail shakers, tuxedo-inspired lighters, design books, candles, acrylic trays, woven leather totes, and more.
1035 Post St., San Francisco, CA 94109, +1-415-440-7300, jayjeffers-thestore.com.
#### In Fiore
* * *
Old-world-style apothecary with a cultlike following. Since 1999, owner and founder Julie Elliott has drawn on the power of flowers to create her range of body balms (try the Bois de Rose), face serums, and solid perfumes.
2044 Fillmore St., San Francisco, CA 94115, +1-415-563-5000, infiore.net.
### FINANCIAL DISTRICT
#### Lang Antique & Estate Jewelry
* * *
Fine antique and estate jewelry, from Georgian period to contemporary styles. "I don't know why anyone would shop for fine jewelry anywhere else," says editor-turned-retailer Emily Holt. "They have amazing Victorian pieces and it's possible to find a ring for under $500."
309 Sutter St., San Francisco, CA 94108, +1-415-982-2213, langantiques.com.
### NOE VALLEY
#### MAAS & Stacks
* * *
Owner Otto Zoell stocks high-concept menswear by labels such as Craig Green, J.W. Anderson, and Gosha Rubchinskiy, as well as subversive streetwear brands like Brain Dead.
2128 Market St., San Francisco, CA 94114, +1-415-678-5629, maasandstacks.com.
#### Mill Mercantile
* * *
Founded in 2012 as the women's counterpart to San Francisco's menswear stalwart Unionmade, this store specializes in tomboy-chic jeans by Raleigh Denim, culottes by Black Crane, popover dresses by Samuji, striped shirts by Saint James, beauty items by Aesop, and a handful of exclusives, such as indigo-dyed Tretorn sneakers.
3751 24th St., San Francisco, CA 94114, +1-415-401-8920, millmercantile.com.
ONLINE SHOPPING TIP
"I buy Everlane religiously," says Emily Holt of the online retailer everlane.com. "It's such an easy purchase because it's reasonably priced ($18 to $200 for ready-to-wear), I know the clothes are responsibly made, the designs are basic enough to wear every day but not boring, and their user experience is easy and friendly. And, because things often sell out, there's that crucial sense of urgency. But most of all, it's a SF brand so I get a little jolt of hometown pride and satisfaction with each purchase." Stylist Jen Rade agrees. "Their modern point loafers, I wear every day. If they stopped making them, I don't know what I would do. I'd have to become a dairy farmer."
### HAIGHT-ASHBURY
#### Relic Vintage
* * *
A vintage store in the center of Haight-Ashbury with a focus on the 1920s to the 1960s. This is where vintage connoisseur Dita Von Teese found her best vintage score in twenty-five years of collecting, a 1954 Christian Dior New Look three-piece tweed skirt suit, with the Christian Dior Paris label bearing the red serial numbers. "It's silk-lined, with garters attached to the blouse. It was like a mirage when I saw it!" says Von Teese. "The owner said, 'I have something in the back you might be interested in.' " The lesson? Always ask what's in the back.
1605 Haight St., San Francisco, CA 94117, +1-415-255-7460, relicvintagesf.com.
### CASTRO
#### Sui Generis
* * *
Sui Generis—its name comes from the Latin expression meaning "of its own kind"—is a consignment store that puts as much emphasis on menswear as womenswear, with separate locations dedicated to each. "A treasure in the heart of the Castro," says famed dandy Patrick McDonald of the men's location. "You can find fabulous pieces from Jean Paul Gaultier, Alexander McQueen, and Ralph Lauren, gently worn."
Women's, 2147 Union St., San Francisco, CA 94123, +1-415-800-7584. Men's, 2231 Market St., San Francisco, CA 94114, +1-415-437-2231, suigenerisconsignment.com.
### PRESIDIO
#### Susan
* * *
Owner Susan Foslien is the grande dame of San Francisco fashion retail, known for her early championing of such labels as Comme des Garçons, Rodarte, and Barbara Tfank. The store doesn't have a website, so you have to visit to see what's new from Vetements, Yang Li, and Simone Rocha. Don't miss Foslien's contemporary fashion space Grocery Store next door.
Susan, 3685 Sacramento St., San Francisco, CA 94118, +1-415-922-3685. Grocery Store, 3687 Sacramento St., San Francisco, CA 94118, +1-415-928-3615.
ERICA TANOV
THE BAY AREA
* * *
Erica Tanov is the consummate earthy-chic Bay Area designer. A California native, Tanov launched her namesake collection in 1990. Her earliest pieces—seamed slips and chemises—reflected her dedication to fine fabrics, construction, and subtle details. Tanov opened her first retail store in 1994 in the Berkeley building where her grandfather once ran his laundry business. Her store sells raw-edge linen dresses, tie-dyed silk blouses, damask maxi skirts, and more.
Right down the street from my studio in Berkeley, FAR & FEW has an eclectic mix of vintage clothing, home furnishings, and jewelry. It's a good place to dig and find a treasure. TAIL OF THE YAK is a Berkeley institution and a must go-to when visiting the Bay Area. It has a fantastic assortment of bibelots and unusual objects, vintage and new, and a gorgeous selection of estate jewelry.
In Oakland, TEMESCAL ALLEYS is an indie shopping enclave where you'll find eighteen locally owned specialty shops, some with their own workspace. My favorite is HOMESTEAD APOTHECARY, which has the best selection of tinctures, herbs, fragrances for healing, and beauty products. On Piedmont Avenue, MERCY VINTAGE is a nicely edited, supercool high-end designer vintage clothing and accessories store.
My family has a cabin in Soda Springs and while my time spent there isn't usually for shopping, there is, surprisingly, a great store in nearby Truckee tucked among some touristy, not-so-appealing shops. BESPOKE has a nice "general store" feel. It's casual, earthy, and masculine. I especially like the pocketknife collection, both vintage and new.
ERICA TANOV, 1827 4th St., Berkeley, CA 94710, +1-510-849-3331, ericatanov.com, and an additional location in Marin.
FAR & FEW, 1643 San Pablo Ave., Berkeley, CA 94702, +1-707-490-6665.
TAIL OF THE YAK, 2632 Ashby Ave., Berkeley, CA 94705, +1-510-841-9891.
TEMESCAL ALLEYS, 49th St. between Telegraph Ave. and Clarke St., Oakland, CA 94609, temescalalleys.com.
HOMESTEAD APOTHECARY, 486 49th St. #C, Oakland, CA 94609, +1-510-541-5225, homesteadapothecary.com.
MERCY VINTAGE, 4188 Piedmont Ave., Oakland, CA 94611, +1-510-654-5599, mercyvintage.com.
BESPOKE, 10130 Donner Pass Rd., Truckee, CA 96161, +1-530-582-5500, bespoketruckee.com.
### FISHERMAN'S WHARF
#### Art Attack SF
* * *
"Such a cool place, the art is all created by talented local artists," says shoe designer Sophia Webster. "I bought a piece there titled Hello Frida, by an artist named María Motta. It's a print of Hello Kitty dressed as one of my favorite artists, Frida Kahlo. It makes me so happy to have it hanging next to my desk in my studio."
2722A Hyde St., San Francisco, CA 94109, +1-415-814-3136, artattacksf.com.
#### Helpers House of Couture
* * *
This 100 percent for-charity, by-appointment-only boutique is run by San Francisco style icon Joy Venturini Bianchi. It features several rooms of couture, men's apparel, and jewelry, plus one room where everything is less than $100. There's also a second, more affordable boutique, Helpers Homes Bazaar.
Art Attack SF, 2626 Fulton St., San Francisco, CA 94118, +1-415-609-0658. Helpers Homes Bazaar, 900 North Point St., San Francisco, CA 94109, +1-415-441-0779.
### INNER/OUTER RICHMOND
#### Legion of Honor Museum Store
* * *
"Museum shops are always a good idea," says designer Ulla Johnson, who praises the Legion of Honor's for unusual items and one-of-a kind gifts. "Related to an exhibition of royal Hawaiian featherwork, they had the most beautiful necklaces made from red and white swallow feathers, all sewn meticulously by hand," she says. "You would have to travel around the world and wait weeks for an opportunity to find such a rare and almost extinct art form."
Lincoln Park, 100 34th Ave., San Francisco, CA 94121, +1-415-750-3677, shop.famsf.org.
ONLINE SHOPPING TIP
"Totokaelo.com does a terrific job with their online store," says fashion consultant Libby Callaway. "They integrate customers' comments and questions in a way that I have never seen on another site; basically, they post and answer folks' questions in the info section beside the garment or accessory. I love that."
#### Park Life
* * *
Started in 2006 by Jamie Alexander and Derek Song, Park Life features art and design products culled from all over the world, including witty T-shirts, skateboard decks, tea towels, David Bowie tribute pins, books, paper goods, and original art.
220 Clement St., San Francisco, CA 94118, +1-415-386-7275, parklifestore-webstore.com.
### UNION SQUARE/CHINATOWN
#### Neiman Marcus
* * *
The Union Square flagship is worth a stop for its interior alone. The building was constructed in 1983 by architect Philip Johnson, who preserved the remarkable stained-glass rotunda from the old City of Paris department store that once stood on the same corner, turning it into a restaurant that's become a ladies-who-lunch, San Francisco tradition. "There's nothing like having a fancy little tea at the Rotunda inside of Neiman Marcus. The view is incredible," says Nasty Gal founder Sophia Amoruso. "There's definitely an old-school glamour to it," adds designer Erin Fetherston. "I love that it has a history and they maintained it," stylist George Kotsiopoulos adds.
150 Stockton St., San Francisco, CA 94108, +1-415-362-3900, neimanmarcus.com, and additional locations.
NEIMAN MARCUS
ONLINE SHOPPING TIP
"Flora and Henri (floraandhenri.com), a Seattle–based company, has been my favorite children's brand for years," says Olya Thompson. "They have a rare delicate palette that underscores children's natural, unaffected charm. Dresses of restrained, sweet beauty, perfect and extremely practical cashmere cardigans of camel, seafoam green, and buff shades. There are great paper-thin cotton long sleeve T-shirts. Pepe shoes and Derek Rose batiste pajamas are also found on the site."
#### Old Shanghai
* * *
This is a cut above other Chinatown stores, with sophisticated traditional Chinese wares, including floral cheongsam and beaded evening purses, hanging lanterns, silk fans, and a line of contemporary home and fashion goods (think bamboo dishware, silk pillows, printed scarves, and jewelry rolls), all at reasonable prices.
645 Grant Ave., San Francisco, CA 94108, +1-415-986-1222, oldshanghaionline.com.
#### Torso Vintages
* * *
This museum-like collection of vintage is beloved by stylish shoppers (Alice and Olivia's Stacey Bendet, jewelry designer Kara Ross) and well-heeled celebrities (Victoria Beckham, M.I.A., Lenny Kravitz). You'll find Dior pieces in the four figures, as well as old I. Magnin styles, colorful furs, and affordable sunglasses, turbans, beaded clutches, and enamel compacts.
272 Sutter St., San Francisco, CA 94108, +1-415-391-3166, torsovintages.com.
ONLINE SHOPPING TIP
With the launch of Olivia Kim's new shop-in-shop boutiques SPACE at select Nordstrom stores, the retail stalwart is introducing up-and-coming designers as well as more established brands that aren't necessarily traditional department store staples. SPACE also lives online at shop.nordstrom.com/c/space-shop-all, where you can find edgy items from Comme des Garçons, Simone Rocha, Vetements, Brother Vellies, and more.
#### Wilkes Bashford
* * *
Luxury clothier and menswear icon Wilkes Bashford changed San Francisco's style landscape when he opened his men's store in 1966. He brought Ralph Lauren, Giorgio Armani, and other designers to the area for the first time. Bashford promoted a "bold conservative" look, educating shoppers about how to wear Brioni and Versace suits; he eventually added a strong womenswear selection. In 2009, the company invested millions of dollars in a remodel of the store's seven-story town house on Sutter Street. A charming figure, Bashford lunched with legendary newspaper columnist Herb Caen and former San Francisco Mayor (and avowed fashion enthusiast) Willie Brown every Friday for years.
375 Sutter St., San Francisco, CA 94108, +1-415-986-4380, wilkesbashford.mitchellstores.com, and an additional location in Palo Alto.
## SEATTLE
### BALLARD
#### Horseshoe
* * *
Clothing and accessories with a nod to Americana style (dresses by Prairie Underground, earmuffs by Pendleton, boots by Frye).
5344 Ballard Ave. NW, Seattle, WA 98107, +1-206-547-9639, shophorseshoe.com.
#### Lucca Great Finds
* * *
The name says it all. Lucca carries ceramics by Astier de Villatte, embroidered pillows and accessories by Brooklyn's Coral & Tusk, Mariage Frères teas, and custom cards.
5332 Ballard Ave. NW, Seattle, WA 98107, +1-206-782-7337, luccagreatfinds.com.
#### Prism
* * *
Modern arty jewelry, clothing, and objets (marble necklaces by RillRill, Wonders of Washington patches, solar-powered watches, Lulu Organics hair powder, and the like).
5208 Ballard Ave. NW, Seattle, WA 98107, +1-206-402-4706, templeofcairo.com/prism.
### FREMONT
#### Atlas Clothing
* * *
"I've found amazing old Betsey Johnson dresses, Michael Jordan T-shirts, Levi's jean shorts, and other awesomeness here each and every time," says Nordstrom VP of Creative Projects Olivia Kim.
3509 Fremont Place N., Ste. 200, Seattle, WA 98103, +1-206-323-0960.
#### Essenza
* * *
Owned by Becky Buford, this jewelry and apothecary shop has delicate earrings by Rockstella and Tai and fragrances by Parfums DelRae, Miller et Bertaux, and others.
615 N. 35th St., Seattle, WA 98103, +1-206-547-4895, essenza-inc.com.
#### Fremont Vintage Mall
* * *
Dozens of vendors under one roof selling vintage clothing, accessories, home furnishings, vinyl, Pyrex, old sports jerseys, taxidermic animals, and more. Olivia Kim loves coming here on Sunday afternoons on her way to the Fremont flea market.
3419 Fremont Place N., Seattle, WA 98103, +1-206-329-4460, fremontvintagemall.com.
#### Les Amis
* * *
Also owned by Essenza's Becky Buford, this darling boutique has a French country vibe, with Isabel Marant, Rachel Comey, Dosa, and Giada Forte clothing.
3420 Evanston Ave. N., Seattle, WA 98103, +1-206-632-2877, lesamis-inc.com.
### DOWNTOWN/CAPITOL HILL
#### Nordstrom
* * *
The flagship of the department store chain has a wider selection than most Nordstrom stores, including top-of-the-line styles by Chanel, Chloé, and more.
500 Pine St., Seattle, WA 98101, +1-206-628-2111, shop.nordstrom.com, and additional locations.
#### Filson
* * *
"Seattle is home to heritage outdoor outfitter Filson. They still manufacture a large number of their products here in town," says Olivia Kim. The 6,000-square-foot-flagship features limited-edition items not available anywhere else.
1741 First Ave. S., Seattle, WA 98134, +1-206-622-3147, filson.com, and additional locations.
#### Totokaelo
* * *
This tranquil, all-white concept store dedicated to avant-garde designers—think Black Crane, Lemaire, Comme des Garçons, Dries Van Noten, Acne, Marni, and Rick Owens—is on the rise. It has expanded its floor space twice over the past thirteen years, and recently opened a New York City location. The men's, women's, and home collections are outstanding, and they dive deep into labels not always well represented in the US, including Haider Ackermann, Our Legacy, and Ann Demeulemeester.
1523 Tenth Ave., Seattle, WA 98122, +1-206-623-3582, totokaelo.com, and an additional location in New York City.
OLIVIA KIM
WASHINGTON
* * *
As director of creative projects for Nordstrom, it's Seattle-bred Olivia Kim's job to curate cool for the department store's themed Pop-In shops, whether she's discovering made-in-Africa antelope fur shoes by Brother Vellies in New York or calling on downtown L.A. design destination Poketo for quirky home products. Here are her favorite spots in her adopted hometown.
The FRYE ART MUSEUM SHOP has a unique and curated collection of gifts, objects, and art from local Seattle- and Washington-based artists with a superclean aesthetic.
I absolutely love PIKE PLACE MARKET, even if it is a tourist trap, with all their yummy food and fresh flower stands, local artist booths, amazing restaurants, and hidden gem shops—a shop dedicated to old movie posters, a Chinese cantina, hand-knit mukluks from local Native Americans, the best magazine stand with every magazine from around the world, an umbrella store, a map store, a Mexican hot sauce and spices store.
Elsewhere in Washington, the entire town of Edison is about the length of half a New York City block! But there's a taco stand, a great café, a bread store, and a couple of shops including HEDGEROW, a mix of eclectic finds and housewares; THE LUCKY DUMPSTER, a collective of local artists who make and sell their fun ceramics and homemade art objects; and SHOP CURATOR, a gallery and shop in one. They make their own soap and incense sticks from fresh pine oil, which apparently is seasonal so stock up when you can.
FRYE ART MUSEUM SHOP, 704 Terry Ave., Seattle, WA 98104, +1-206-622-9250, store.fryemuseum.org.
PIKE PLACE MARKET, 1st Ave and Pike St., Seattle, WA 98101, pikeplacemarket.org.
HEDGEROW, 5787 Cains Ct., Edison, WA 98232, +1-206-605-8639, christy-erickson.squarespace.com.
THE LUCKY DUMPSTER, 14011 Mactaggart Ave., Bow, WA 98232, (360) 766-4049
SHOP CURATOR, 14010 Mactaggart Ave., Bow, WA 98232, +1-360-820-9912, shopcurator.blogspot.com.
## PORTLAND
### OREGON
TENDER LOVING EMPIRE
#### Backtalk
* * *
Right around the corner from the Ace Hotel, Katie Freedle spotlights clothing and accessories from emerging designers across the US, including her own RillRill jewelry line, as well as vintage and ceramics.
421 SW Tenth Ave., Portland, OR, 97205, +1-503-477-7144, backtalkpdx.com.
#### Canoe
* * *
Highly curated design store where every object leaves a distinct aesthetic impression, from a Portland-made oiled leather phone holder for your bike to a stainless-steel condom dispenser.
1233 SW Tenth Ave., Portland, OR 97205, +1-503-889-8545, www.canoe-online.net.
#### Cargo
* * *
A 20,000-square-foot warehouse full of Asian imports and knickknacks, such as paper lanterns, blue-and-white porcelain, Buddha statues, embroidered totes from India, indigo scarves, and pens made from old electrical wires.
81 SE Yamhill St., Portland, OR 97214, +1-503-209-8349, cargoinc.com.
#### Field Trip
* * *
Cheery, modern-hippie lifestyle shop and community space that brings together small-batch products—moccasins, maple wood hair combs, THE FUTURE IS FEMALE T-shirts, and more—from makers in Portland, Brooklyn, Santa Monica, and beyond.
3725 SE Division St., Portland, OR 97202, +1-971-703-4523, shop-fieldtrip.com.
#### Frances May
* * *
A top destination for men's and women's clothing from independent labels such as A.P.C., Ace & Jig, Whit, Rachel Comey, Acne, Industry of All Nations, and Raquel Allegra.
1003 SW Washington St., Portland, OR 97205, +1-503-227-3402, francesmay.com.
#### House of Vintage
* * *
Vintage wonderland with more than sixty-five vendors under one roof. Room after room of inexpensive finds, including a $12 shocking-pink crochet knit top and $25 cowboy boots.
3315 SE Hawthorne Blvd., Portland, OR 97214, +1-503-236-1991, houseofvintagenw.com.
#### Lizard Lounge
* * *
Upscale workwear for men and women from Levi's Made & Crafted, Raleigh Denim, Norman Russell, Red Wing, and more in a historic building in Portland's Pearl District.
1323 NW Irving St., Portland, OR 97209, +1-503-416-7476, lizardloungepdx.com.
#### Mario's
* * *
The most respected name in specialty retailing in Portland, Mario's was founded by Mario Bisio Sr. as a family clothing store in 1960. It was transformed into a menswear store by his son Mario Jr. in the 1970s, when he brought Italian luxury labels Canali, Ermenegildo Zegna, and Diesel to the US for the first time. The first women's store opened in the 1980s, introducing Prada, Loro Piana, and Brunello Cucinelli to the area. A Seattle location followed. The best of the best is represented here, including Valentino, Stella McCartney, Gucci, Hermès, Lanvin, Chloé, and Oscar de la Renta, plus contemporary offerings from Ulla Johnson, L'Agence, Vince, and more.
833 SW Broadway, Portland, OR 97205, +1-503-227-3477, marios.mitchellstores.com, and additional locations.
#### Monograph Bookwerks
* * *
"A small, very special art bookstore for new, used, and rare books on contemporary art, architecture, and design, plus ephemera, unique objects, and artworks," says Berkeley-based designer Erica Tanov. "Carefully curated and handpicked."
5005 NE 27th Ave., Portland, OR 97211, +1-503-284-5005, monographbookwerks.com.
#### Powell's City of Books
* * *
An independent bookstore that takes up a whole city block and caters to every esoteric interest. It stocks used books, too. Personal note: My favorite find? A fashion history book titled Furs for Men.
1005 W. Burnside St., Portland, OR 97209, +1-503-228-4651, powells.com, and additional locations.
#### Pendleton Home Store
* * *
The family-owned Pendleton Woolen Mills, known both for surfer- and grunge-rocker-approved plaid shirts and colorful wool blankets, was born in Oregon in 1889, and has been rediscovered in recent years by American heritage–loving hipsters. This is the brand's "experimental" store, and it does feel younger and hipper, mixing housewares and vintage Native American jewelry and belts with Pendleton shirts, pants, and skirts.
210 NW Broadway, Portland, OR 97209, +1-503-535-5444, pendleton-usa.com, and additional locations.
#### Tender Loving Empire
* * *
The dream of the '90s is alive at this indie store and record label (for real). This is the source for amusing local goods, including CATS AGAINST CATCALLS totes, PBR duct tape flasks, and MY DAD'S BEARD IS BETTER THAN YOUR DAD'S BEARD baby bibs. Is it any wonder Carrie Brownstein has called this her favorite shop in Portland?
3541 SE Hawthorne Blvd., Portland, OR 97214, +1-503-548-2927, tenderlovingempire.com, and additional locations.
#### Una
* * *
Modern romanticism is the name of the game at this top-notch boutique, which features pretty dresses by Gary Graham and A Détacher; high-waist pants by Rodebjer; dusters by Pas de Calais; and a drool-worthy selection of minimal, modern jewelry by Monica Castiglioni, Quarry, and others.
922 SE Ankeny St., Portland, OR 97214, +1-503-235-2326, unanegozio.com.
#### Xtabay Vintage
* * *
Girlie-girl heaven, this boudoir-like shop full of old hatboxes and gilded mirrors specializes in vintage cocktail frocks and gowns from the 1930s to the 1960s, and is Carrie Brownstein–approved. It's also a treasure trove of vintage wedding gowns.
2515 SE Clinton St., Portland, OR 97202, +1-503-230-2899, xtabayvintage.com.
XTABAY VINTAGE
## SANTA FE
### NEW MEXICO
#### Five & Dime General Store
* * *
Set in the historic Santa Fe Plaza, this general store is a reminder of the Santa Fe of yesteryear. "Whether you need a tote bag, postcards (usually vintage dead stock), or a pair of Foster Grant sunglasses, this institution has it all," says designer Claude Morais. "Don't forget to try the Frito Pie at the lunch counter, the best in New Mexico!"
58 E. San Francisco St., Santa Fe, NM 87501, +1-505-992-1800, fiveanddimegs.com.
#### Santa Fe Double Take
* * *
You can always find something amazing at this vintage Valhalla, whether it's the perfect black velvet broomstick skirt, a pair of broken-in-cowboy boots, or a Native American turquoise treasure. There are several different rooms to explore, including a vintage designer space, and Hacienda (for vintage Western furniture, quilts, and the like).
321 S. Guadalupe St., Santa Fe, NM 87501, +1-505-989-8886, santafedoubletake.com.
#### Santa Fe Dry Goods
* * *
Located right on the plaza, SFDG has been a source for avant-garde design since 1979, when Santa Fe first arrived on the international art scene. Today, it's a favorite of LACMA Costume and Textiles department curator Sharon Takeda for browsing Rick Owens, Dries Van Noten, Issey Miyake, A.F. Vandevorst, and Monies.
53 Old Santa Fe Trail, Santa Fe, NM 87501, +1-505-992-8083, santafedrygoods.com.
SANTA FE DRY GOODS
#### Santa Fe Vintage
* * *
Santa Fe's best-kept fashion secret, this by-appointment spot is a destination for designers looking to source denim, cowboy boots, blankets, and antiques. "Every year I visit Scott Corey's vintage warehouse, Santa Fe Vintage, on the outskirts of Santa Fe, near the airport," says Sabah founder Mickey Ashmore. "It's full of so many gems, it's overwhelming. Scott is a prince of a person and a pleasure to spend time with."
7501 Avenger Way, Ste. B, Santa Fe, NM 87508, +1-505-690-1075, santafevintage.com
SANTA FE VINTAGE
#### Shiprock Santa Fe
* * *
"The chicest turquoise jewelry," says fashion consultant Amanda Ross of this high-end source for Southwestern jewelry, textiles, and arts and crafts, as well as Visvim clothing. The gallerylike space is also filled with collectible modern furniture.
53 Old Santa Fe Trail, Santa Fe, NM 87501, +1-505-982-8478, shiprocksantafe.com.
## AUSTIN
### TEXAS
ALLENS BOOTS
#### Allens Boots
* * *
You can't miss the giant red boot above the entrance to this cowboy boot paradise with thousands of men's, women's, and children's styles, from old-school Luccheses that run in the $700 range to more-contemporary fringed booties in the $200 to $300 range, plus snap-front shirts, denim, prairie dresses, and accessories. After you shop, boot-scoot across the street to the Continental Club to listen to some live music.
1522 South Congress Ave., Austin, TX 78704, +1-512-447-1413, allensboots.com.
#### By George
* * *
The location on North Lamar Boulevard is the city's go-to for men's and women's high-end designers (Chloé, Jacquemus, Suno, Lanvin, Protagonist, Tome, Céline, Dries Van Noten, Novis), while the South Congress shop is fantastic for more-contemporary pieces from The Great, Maison Kitsuné, and Simon Miller.
524 N. Lamar Blvd., Austin, TX 78703, +1-512-472-5951, bygeorgeaustin.com, and an additional location on South Congress Ave.
BY GEORGE
#### Feathers
* * *
"A gold mine for quirky vintage finds," says Austin-based jewelry designer Kendra Scott. With its casual, boho vibe and edited collection, this store has offered such finds as a Koos van den Akker patchwork duster coat, gauzy floral-print Indian dresses, 1960s fringed jackets, and a vintage Michael Jackson Thriller T-shirt.
1700 S. Congress Ave., Austin, TX 78704, +1-512-912-9779, feathersboutiquevintage.com.
#### Friends & Neighbors
* * *
Located in an actual home, this place stocks vintage clothing, arty jewelry, planters, pipes, and other housewares, with beer, wine, Stumptown coffee, and snacks served in the backyard. Trust us, you'll want to stay awhile.
2614 E. Cesar Chavez St., Austin, TX 78702, friendsaustin.com.
#### JM Drygoods
* * *
Spot-on chic collection of Mexican- and Spanish-inspired clothing, Oaxacan ceramics, wax skull candles, bison wallets, Marfa Brands soap, and more goodies with a Texas vibe, opened by former costume designer Michelle Teague.
215 S. Lamar Blvd., Ste. C, Austin, TX 78704, +1-512-579-0303, jmdrygoods.com.
#### Kendra Scott
* * *
The Austin-based jewelry designer started her company with $500 in 2002; today, she has boutiques nationwide. Her druzy drop earrings, colorful statement necklaces, and double rings make a perfect Austin souvenir. You can also design your own pieces at the store's Color Bar.
1400 S. Congress Ave., Ste. A-170, Austin, TX 78704, +1-512-354-4737, kendrascott.com, and additional locations.
#### Kick Pleat
* * *
Must-haves from modern minimalist labels including Sofie D'Hoore, Apiece Apart, Shaina Mote, Priory, and Creatures of Comfort. "A perfect edit of refined and cool fashion, from M. Martin to Lemaire to Isa Arfen," says accessories designer Lizzie Fortunato. "I never walk out empty-handed."
918 W. 12th St., Austin, TX 78703, +1-512-445-4500, kickpleat.com, and an additional location in Houston.
KICK PLEAT
#### Olive
* * *
"Artful independent design with everyday dressing in mind," is how photographer and stylist Laura Uhlir describes her store, which stocks elevated basics, fun prints, easy day dresses, classic jeans, and cool jewelry. Designers include Ace & Jig, Dusen Dusen, Samantha Pleet, Whit, Wray, and Maryam Nassir Zadeh.
1200 E. 11th St., No. 101, Austin, TX 78702, +1-512-522-9462, oliveaustin.com.
#### Stag Provisions For Men
* * *
Hands down, one of the best menswear boutiques in the US, Stag stocks relaxed and rough-hewn Americana-inspired suiting and workwear by Alex Mill, General Assembly, Rogue Territory, RRL, and Levi's Made & Crafted alongside one-of-a-kind vintage furniture pieces, artwork, taxidermic animals, bags, belts, vintage books, classic vinyl albums, and more. This is the original location.
1423 S. Congress Ave., Austin, TX 78704, +1-512-373-7824, stagprovisions.com, and additional locations.
#### Tesoros Trading Company
* * *
Folk art, furniture, ethnic clothing, colorful beads, paper roses, Frida Kahlo postcards, oilcloth bags, multicolored button bags, and much more from Mexico, Guatemala, India, China, Nepal, and elsewhere.
1500 S. Congress Ave., Austin, TX 78704, +1-512-447-7500, tesoros.com.
#### Uncommon Objects
* * *
A gallery of antique oddities, from Lone Star belt buckles to dollhouses. "You can find anything from vintage indigo fabrics to antique anatomical pieces. It's incredible," says designer Mara Hoffman. "I could spend hours there," says designer Brandon Maxwell. "They have rooms and rooms of antiques and interesting little objects, cases and cases of vintage jewelry. Everything is so special and unique."
1512 S. Congress Ave., Austin, TX 78704, +1-512-442-4000, uncommonobjects.com.
## DALLAS/FORT WORTH
### TEXAS
FORTY FIVE TEN
#### Cabana and Canary
* * *
This pair of Highland Park stores owned by Merry Vose caters to students at neighboring Southern Methodist University. Cabana is beachy, with K. Jacques sandals, Mother denim, and Frank & Eileen shirts, while Canary stocks more sophisticated duds from Veronica Beard, Stella Jean, Preen, Sacai, Giada Forte, and Roksanda Ilincic.
4609 W. Lovers Lane and 4711 W. Lovers Lane, Dallas, TX 75209, +1-214-351-4400.
#### Dolly Python
* * *
When Jean Paul Gaultier came to Dallas for the opening of his retrospective at the Dallas Museum of Art in 2011, he discovered this vintage gem and tweeted about it. Since then, it's become a favorite of stylish shoppers, including Mary-Kate and Ashley Olsen and Florence Welch, who come for vintage clothes from the 1940s to the 1980s, old cowboy boots, vinyl records, and amazingly cool kitsch, such as velvet paintings, and taxidermic animals. "You can find everything from amber bugs, to those old hair dryers women used to sit under in hair salons in the 1960s," says style influencer Billy Fong. We're sold.
1914-1916 N. Haskell Ave., Dallas, TX 75204, +1-214-887-3434, dollypythonvintage.com.
#### Fort Worth Stockyards
* * *
"If you want a cool saddlebag, you don't have to go to Céline, come here," says Billy Fong, referring to the historic district—and former livestock market—just north of the central business district in Fort Worth. The Stockyards has several stores that capitalize on the city's cow town image, including bootmaker M.L. Leddy's and the Maverick Fine Western Wear, where shoppers can down a beer while they browse.
130 E. Exchange Ave., Fort Worth, TX 76164, +1-817-624-4741, fortworthstockyards.org.
#### Forty Five Ten
* * *
With clothes and housewares in an 8,000-square-foot historic building surrounding a lush courtyard with its own tearoom, Forty Five Ten is a legend in these parts. "The best game in town," says Billy Fong. There's something for everyone, from tried-and-true labels like The Row, Céline, Chloé, and Dries Van Noten to edgy Vetements, Jeremy Scott, Maison Martin Margiela, and Sacai to up-and-comers Delpozo, Monse, Co, Brock Collection, and Rosie Assoulin. "Unlike many multi-brand stores, they still take risks on interesting, new designers," says designer Sally Perrin. "Great clothes, great food!" says fashion editor Karla Martinez de Salas. The store has attracted a clientele that includes Oprah Winfrey, Laura Bush, Gwyneth Paltrow, and Melissa Etheridge, and is a favorite of local celebrities Angie Harmon, Erykah Badu, and Jessica Simpson.
1615 Main St., Dallas, TX 75201, +1-214-559-4510, fortyfiveten.com, and additional locations.
#### Grange Hall
* * *
"A unique spot for gifts, including egg-shaped vases, Cire Trudon candles, Myriam Schaefer handbags, and Lydia Courteille chandelier earrings. There's also a florist. Every time I come, something ends up in my home!" says Cameron Silver. There's a wonderful restaurant, too, where "all the servers look like male models," says Billy Fong. "You have to get the Snob Sandwich, which has Petrossian smoked salmon, caviar, and crème fraîche."
4445 Travis St., Ste. 101, Dallas, TX 75205, +1-214-443-0600, ufgrangehall.com.
#### Gypsy Wagon
* * *
Cute, girlie clothing and accessories, plus home goods and gifts with a Texas/Western vibe. Come here for lacy florals and denim cutoffs.
2928 N. Henderson Ave., Dallas, TX 75206, +1-214-370-8010, the-gyspy-wagon.com.
#### Highland Park Village
* * *
The first self-contained shopping center in the US, this collection of shops opened in 1931 and became the model for the modern-day shopping mall. While planning the town-square-like design, original proprietors Edgar Flippen and Hugh Prather Sr. traveled to California, Mexico, and Spain to study architecture. Dior, Hermès, Chanel, Tom Ford, and the other usual luxury-brand suspects are all represented among the 100-plus stores. Both Gwyneth Paltrow's Goop and Reese Witherspoon's Draper James have hosted pop-ups here. And don't miss the great restaurants, including Mi Cocina, Bistro 31, and Café Pacific.
Mockingbird Lane at Preston Rd., Dallas, TX, 75205, hpvillage.com.
#### Neiman Marcus
* * *
This is the mother ship of the Dallas-based Neiman Marcus chain, located in the same spot since 1914. It's worth stopping in for the nostalgic thrill of remembering all the innovation that the store has wrought, from establishing its own fashion awards, which brought the likes of Coco Chanel and Bonnie Cashin to Texas, to its over-the-top Christmas catalogs. Check out the vintage photos on the walls of the Zodiac restaurant, especially the ones of late store patriarch Stanley Marcus, and be sure to order the mandarin orange soufflé.
1618 Main St., Dallas, TX 75201, +1-214-741-6911, neimanmarcus.com, and additional locations.
#### Stanley Korshak
* * *
Another Texas institution. "They sell Prada next to cowboy boots. You can buy a Nancy Gonzalez alligator-skin bag with ostrich-skin boots to go with your Roberto Cavalli dress," says Billy Fong.
500 Crescent Court, Ste. 100, Dallas, TX 75201, +1-214-871-3600, stanleykorshak.com.
STANLEY KORSHAK
#### Vintage Martini
* * *
A vintage and consignment boutique that's full of 1970s-era Saint Laurent and also has a fabulous costume jewelry selection.
2923 N. Henderson Ave., Ste. A, Dallas, TX 75206, +1-469-334-0584, vintagemartini.com.
#### V.O.D.
* * *
The emphasis here is on cutting-edge indie labels—think Ryan Roche, Tsumori Chisato, Isabel Marant, Roseanna, and Lisa Marie Fernandez—and a relaxed aesthetic.
2418 Victory Park Lane, Dallas, TX 75219, +1-214-754-0644, vodboutique.com.
#### Ylang 23
* * *
Dallas's best boutique for designer fine jewelry stocks sparklers by Irene Neuwirth, Jennifer Meyer, Ileana Makri, Cathy Waterman, Sharon Khazzam, and Sydney Evan. The range includes colorful rings, edgy ear adornments, and personalized pendants, from $175 to $75,000 and up.
The Plaza at Preston Center, 8300 Preston Rd., Ste. 700, Dallas, TX 75225, +1-866-952-6423, ylang23.com.
## HOUSTON
### TEXAS
À BIENTÔT
#### À Bientôt
* * *
Betty Newton and Cristiana Anderson's two-story treasure trove has amazing jewelry from around the globe, including horn necklaces and shell and coral earrings, beaded clutch purses, and monogrammed totes, plus fun paper goods, cashmere wraps, and sandals.
2501 River Oaks Blvd., Houston, TX 77019, +1-713-523-3997, shopabientot.com.
#### Cheeky Vintage
* * *
This bungalow turned closet is a girlie destination for vintage clothing from the 1950s to the 1980s.
2134 Richmond Ave., Houston, TX 77098, +1-713-533-1121, cheeky-vintage.myshopify.com.
#### Elaine Turner
* * *
Houston is home base for the designer, whose colorful cork, raffia, and faux python accessories are available at department stores around the country and at her namesake boutiques. She started in 2000 with handbags and has since expanded to shoes, jewelry, and clothing, for an empire in the making.
2439 University Blvd., Houston, TX +1-713-255-0052, elaineturner.com.
#### Katia
* * *
This by-appointment-only, three-room boutique takes over half of a shopping center on Westheimer Road. Owner Denise Lucia and her daughter Katia are known for their edgy East-meets-West-Coast edit, stocking pieces by Anine Bing, Barbara Bui, Drome, Helmut Lang, Isabel Marant, and more.
5634 Westheimer Rd., Houston, TX 77056, +1-713-621-1817, katiaboutique.com.
#### Kuhl Linscomb
* * *
This lifestyle/home store spanning four storefronts is a one-stop-shop for gifts, including fragrances by Carthusia and LAFCO; plates by Tom Dixon and Astier de Villate; linens by K Studio, Missoni Home; and jewelry and accessories from Alexis Bittar, Me and Ro, Ten Thousand Things, and more.
2418 W. Alabama St., Houston, TX 77098, +1-713-526-6000, kuhl-linscomb.com.
#### Laboratoria
* * *
This hidden River Oaks boutique stocks clothing and accessories from edgy designers not often found between the East and West Coasts, such as Ellery, Rodebjer, Holly Fulton, Preen, and Simone Rocha. A changing display of art by local artists is featured on the walls.
2803 Westheimer Rd., Houston, TX 77027, +1-832-407-2832, laboratoria-shop.com.
#### Little Bird
* * *
The place to find barely worn designer pieces consigned by Texas fashionistas. Hermès, Chanel, Céline, Gucci—it's all here.
1735 Post Oak Blvd., Houston, TX 77056, +1-832-767-4483, and an additional location in Aspen, littlebirdinc.com.
#### Myth and Symbol
* * *
Minimalist, cool clothing by Black Crane, Dusen Dusen, Apiece Apart, A.P.C., Samantha Pleet, and other arty brands, bags by Building Block, and delicate jewelry by Mociun.
2537 Times Blvd., Houston, TX 77005, +1-832-422-6984, mythandsymbol.com.
#### Pinto Ranch
* * *
Everything Texans need for that upscale-ranch lifestyle, including embroidered cowboy shirts, exotic leather boots, and trophy buckles. "I've gotten my fair share of cowboy boots here," says fashion tech entrepreneur Alexandra Wilkis Wilson. "I haven't figured out how to wear them yet in NYC, though. In Texas they look normal!"
1717 Post Oak Blvd., Houston, TX 77056, +1-713-333-7900, pintoranch.com.
#### Saint Cloud
* * *
Contemporary and globally sourced artisan-designed accessories and home goods from Clare Vivier, Creatures of Comfort, Lemlem, Gabriela Artigas, Brother Vellies, and Cold Picnic, plus embroidered Oaxacan pillows, Moroccan baskets, and more in a minimalist space.
5217 Kelvin Dr., Houston, TX, +1-713-522-0077, shopsaintcloud.com.
#### Sloan/Hall
* * *
Multi-category lifestyle store stocking avant-garde clothing and accessories by Christopher Kane, David Koma, Maison Martin Margiela, Sacai, Halleh, and Perrin Paris, alongside antiques, contemporary objects, fragrances, limited-edition books, and greeting cards.
2620 Westheimer Rd., Houston, TX 77098, +1-713-942-0202, sloanhall.com, and an additional location in San Antonio.
#### Tootsies
* * *
Houston's largest specialty store, Tootsies has 35,000 square feet of designer clothing by a lot of the majors, including Alexander McQueen, J. Mendel, Carolina Herrera, Rick Owens, Mugler, Moschino, and more, as well as contemporary offerings from Veronica Beard, Mara Hoffman, Mansur Gavriel, and others.
2601 Westheimer, Houston, TX 77098, +1-713-629-9990, tootsies.com.
## CHARLOTTE
### NORTH CAROLINA
#### Capitol and Poole
* * *
Laura Vinroot Poole created her trio of boutiques—Capitol for women's designer, Poole for women's contemporary, and Tabor for men—because she couldn't find any interesting retail options in her home state. And she brought it with her gorgeous courtyard space, punctuated by a two-story vertical garden by French landscape designer Patrick Blanc. She has an eye for what's new, as well as what works for Southern ladies, stocking Alexander McQueen, Barbara Tfank, Dolce & Gabbana, Dries Van Noten, Giambattista Valli, Rodarte, Altuzarra, Roksanda, fine jewelry, and fragrances in a space decorated with vintage furniture. "The most gorgeous dresses," says jewelry designer Irene Neuwirth. "Laura is one of my dearest friends and her shop is torture (in the best way)." "People must have thought she was completely crazy putting all that Giambattista Valli in a shopping center next to a Barnes & Noble," says Ann Mashburn. "But it is just the most fantastic space and assortment . . . and she has totally changed the way women in that city dress."
4010 Sharon Rd., Charlotte, NC, 28211, +1-704-553-8868, capitolcharlotte.com.
CAPITOL AND POOLE
#### Tabor
* * *
For men, Poole's converted bungalow offers Dries Van Noten, Mark McNairy, Rag & Bone, Steven Alan, and much more, plus on-site alteration, closet curation, and coffee!
421 Providence Rd., Charlotte, NC, 28207, +1-980-207-4860, capitolcharlotte.com.
## ATLANTA
### GEORGIA
#### Ann Mashburn
* * *
Pick up a pair of jeans or get measured for a custom suit, play ping-pong, listen to records, drink a Coke, and settle in for a while. Caran d'Ache pens and Musgo Real shaving creams are displayed below Sartorio suits and Isabel Marant jackets; Danish cattle horns and copies of The Secret Garden sit next to Laguiole knives and Mason Pearson hairbrushes; Levi's 501s and Saint James striped shirts hang alongside Sid and Ann Mashburn's preppy-with-a-global-punch clothing line.
1198 Howell Mill Rd., Atlanta, GA 30318, +1-404-350-7132, annmashburn.com, and additional locations.
ANN MASHBURN
#### Scott's Market
* * *
Monthly antiques show near the airport. "There is so much great stuff from dealers all over the South with a great mix of high and low . . . 100 percent worth making the trip," says boutique owner Ann Mashburn. "They also serve a great meat-and-three for lunch. The smartest dealers and designers go every month with a slew of new stuff."
3650 Jonesboro Rd. SE, Atlanta, GA 30354, +1-404-361-2000, scottantiquemarket.com.
## PALM BEACH
### FLORIDA
#### Tomas Maier
* * *
The designer, who is creative director of luxury powerhouse Bottega Veneta, also produces his own line, which has evolved from swimwear to a full lifestyle collection. He opened his first store in 2004, selling his clothing alongside art books and artisanal objects. "The best lifestyle brick-and-mortar I've ever been to," says author and filmmaker Liz Goldwyn.
38 Via Mizner, Palm Beach, FL 33480, +1-561-650-1221, tomasmaier.com, and additional locations.
VIA MIZNER, PALM BEACH
## MIAMI
### FLORIDA
THE WEBSTER
#### Alchemist
* * *
A jewel box in the sky, this arty concept shop on the ground and fifth floors of a parking garage was built by Swiss architecture firm Herzog and de Meuron. Stylist B. Åkerlund calls the Miami Design District stunner her favorite destination for "dark fashion" from Maison Martin Margiela, Junya Watanabe, Givenchy, Chrome Hearts, and more.
1111 Lincoln Rd., Miami Beach, FL 33139, +1-305-531-4815, shopalchemist.com.
#### Bal Harbour Shops
* * *
One of the most famous shopping malls in the world, this family-owned, open-air destination opened for business in 1965. Neiman Marcus and Saks Fifth Avenue are the anchor stores. "My Sunday ritual starts with brunch at the Zodiac Café, where they have the best popovers and strawberry butter," says fashion publicist Tara Solomon. There are more than 100 upscale stores here, including Balenciaga, Chanel, Gucci, Céline, Stella McCartney, Valentino, and Tom Ford.
9700 Collins Ave, Bal Harbour, FL 33154, +1-305-866-0311, balharbourshops.com.
#### Boho Hunter
* * *
Art District concept store that brings together designs from Mexico and South America including sexy cutout swimsuits by Suki Cohen, embroidered Pitusa beach bags, and Balkanica kimonos. "For unique gifts, great accessories, beach cover-ups, et cetera," says fashion editor Karla Martinez de Salas.
184 NW 27th St., Miami, FL 33127, +1-786-558-4486, bohohunter.com.
#### C Madeleine's
* * *
This 10,000-square-foot showroom located in North Miami Beach is a vintage lover's paradise with designer couture (Chanel, Versace) and other rare finds attracting the likes of Miley Cyrus and Lenny Kravitz. But don't let the prices scare you; there are things to be had for less than $100. "It's off the beaten path but well worth the visit, especially on rainy days," says designer Charlotte Ronson. "It's a huge space, but everything is very well organized into different eras and styled to inspire. Just the range and assortment make it overwhelming for all the right reasons."
13702 Biscayne Blvd., Miami, FL 33181, +1-305-945-7770, cmadeleines.com.
#### Chrome Hearts
* * *
The Los Angeles-based rock 'n' roll luxe lifestyle purveyor has stores around the world, but the 5,000-square-foot Miami flagship is the most impressive. The store features such delightfully decadent gifts as diamond-encrusted toilet plungers and 22K-gold toothpicks.
Miami Design District, 4025 NE 2nd Ave., Miami, FL 33137, +1-786-953-7384, chromehearts.com.
#### En Avance
* * *
Located in Miami's Design District, this multi-label boutique is always ahead of the curve in finding pieces from innovative young talent around the world, including ruffled silk dresses by Japan's Leur Logette and handwoven clutches by Turkey's ChuChuMai, Ellery, Sacai, Irene Neuwirth, Vilshenko, and Marques Almeida are also represented.
53 NE 40th St., Miami, FL 33137, +1-305-576-0056, enavance.co.
#### Fly Boutique
* * *
This consignment store is a must-visit for fab finds from Gucci, Jimmy Choo, Chloé, Chanel, and more, as well as funky rocker tees, ponchos, and kooky sunglasses in a Hollywood-worthy setting filled with chandeliers, vintage record players, love seats, and more.
7235 Biscayne Blvd, Miami, FL 33138, +1-305-604-8508, flyboutiquevintage.com.
#### Iniva Miami
* * *
Located in the trendy Arts District, Iniva is part art gallery, part concept store, featuring work by African artists, alongside African-made home furnishings, jewelry, and clothing. Look for fabric-covered glasses frames, clutch bags, shawls, and more.
Wynwood Block, Space 19-20, 2621 NW 2nd Avenue, Miami, FL 33127, +1-786-614-8380, inivaboutiques.com.
#### Malaquita
* * *
A showcase for contemporary Mexican designers making everything from cool knitwear to cowboy boots to woven mandalas and wicker coffee tables.
2613 NW Second Ave, Unit 13, Wynwood, Miami, FL 33127, +1-786-615-4917, malaquitadesign.com.
#### Miami Design District
* * *
Art, architecture, and fashion meet in a pedestrian-friendly retail neighborhood, located in what used to be Miami's furniture area. The district sprang up in the wake of the success of Art Basel Miami Beach, the American outpost of the famed Swiss art fair that draws a starry international crowd each December. The neighborhood is a who's who of luxury retailers, including Prada, Dior, Bulgari, Louis Vuitton, Maison Martin Margiela, Rick Owens, Hermès, Marni, and Van Cleef & Arpels; many stores have interesting architectural elements and their own cafés. Loewe has its only US store here. "It's outdoors, so you can see great art," says Laurie Lynn Stark, cofounder of Chrome Hearts, which has a stunning 5,000-square-foot boutique in the district with the largest living wall in the world, a gallery with rotating photo exhibitions, and a Ladurée café.
miamidesigndistrict.net.
#### Ramona La Rue
* * *
Hand-painted resort wear by Coconut Grove native Arianne Brown, whose one-of-a-kind silky bohemian blouses, pants, and dresses make a unique Miami souvenir.
3444 Main Highway, Miami FL 33133, +1-786-564-5286, ramonalarue.com, and additional locations.
#### The Rabbit Hole
* * *
No need to ask Alice, just jump in! The brick-and-mortar location of the beloved online vintage destination is now located in North Miami across the street from the Museum of Contemporary Art. You'll find everything from 1960s psychedelic floral house dresses to 1980s sunglasses.
791 NE 125th Street, North Miami, FL 33161, +1-305-892-0213, shoprabbithole.com.
#### The Webster
* * *
The boutique associated worldwide with Miami style, the 20,000-square-foot flagship (and first) location is in a historic art deco building designed in 1939 by famed architect Henry Hohauser. Founded by Laure Heriard Dubreuil in 2009, the store itself has become a brand, collaborating on capsule collections with Target in 2011 and Paris's Le Bon Marché in 2015. The store is an A-to-Z of fashion for men and women with an eye toward Miami's tropical vibes, from Alaïa to Yeezy, while not forgetting indie and up-and-coming names such as Vetements, Sofie D'Hoore, Creatures of the Wind, Mira Mikati, Hood by Air, and Thom Browne. "I love how they curate designers," says model-blogger Elizabeth Minett. "Awesome wallpaper and awesome clothes," says accessories designer Lizzie Fortunato. "The assortment is fantastic."
1220 Collins Ave., Miami Beach, FL 33139, +1-305-674-7899, thewebster.us, and additional locations.
## NEW ORLEANS
### LOUISIANA
#### Ashley Longshore Studio
* * *
A glittery, colorful gallery space bursting with energy, this is the home base for New Orleans local artist Ashley Longshore, whose bold, hilarious, pop artwork has made fans of celebrities like Blake Lively, Salma Hayek, and Penelope Cruz. It's the perfect place to pick up a (totally instagrammable) painting of a gorgeously bedazzled Frida Kahlo, Jesus holding a Louis Vuitton bag, or Anna Wintour surrounded by Care Bears.
4537 Magazine St., New Orleans, LA 70115, +1-504-333-6951, ashleylongshore.com.
#### Bambi Deville's Vintage Clothing
* * *
Swanky cocktail dresses, kimonos, Bakelite jewelry, and furs abound at this second-floor treasure trove of Blanche DuBois–worthy fashion finds embellished with plenty of beads and chiffon. Vintage Mardi Gras costumes and headdresses, too.
818 Royal St., New Orleans, LA 70116, +1-504-491-0824.
#### Bourbon French Parfums
* * *
This perfumery has been in the French Quarter for more than 170 years. They have their own Southern-inspired scents (Voodoo Love, Forever New Orleans, Magnolia) and also do custom blending. "One of our favorite olfactory haunts in the world," say designers Brian Wolk and Claude Morais. "We always look forward to getting lost in the dizzying array of intoxicating elixirs of love. For only $90—which includes a one-hour assessment of your likes and dislikes—the specialist will create an original essence for you, keep the formula on file, and mix and ship worldwide upon demand."
805 Royal St., New Orleans, LA 70116, +1-504-522-4480, store.neworleansperfume.com.
#### Clover
* * *
This newish Magazine Street boutique stocks sexy, minimalist-cool pieces by Anine Bing, Milly, Sir, L'Agence, Iro, and Frame Denim, plus slides by Avec Modération, scented candles by Maison La Bougie, and more.
2240 Magazine St. #101, New Orleans, LA 70130, +1-504-272-0792, boutiqueclover.com.
#### George Bass
* * *
After more than thirty years in the business, George Bass is an expert in the Southern gentleman's wardrobe. With a master tailor on the premises, alterations are available for everything from a new pair of jeans to a custom-tailored suit, and George will tell you how they're supposed to look, too. He stocks Isaia, Kiton, and Brunello Cucinelli clothing, Hamilton and Luciano Barbera shirts, Loro Piana knits, Alden shoes, and much more.
201 St Charles Ave. #103, New Orleans, LA 70170, +1-504-582-1180, georgebass.com.
#### Hazelnut
* * *
Old World meets modern at this fine gift and home accessories boutique owned by actor/decorator Bryan Batt (he played art director Salvatore Romano on HBO's Mad Men). In addition to ceramics, books, candles, New Orleans–inspired toile place mats and the like, Batt has several charming exclusives, including a brass lapel pin shaped like a king cake baby, and Laurel Wilder commemorative art-glass trays featuring New Orleans scenes.
5515 Magazine St., New Orleans, LA 70115, +1-504-891-2424, hazelnutneworleans.com.
#### Hove Parfumeur
* * *
Opened in 1931 by Mrs. Alvin Hovey-King, the daughter of a calvary officer and wife of a Navy commander. She spent much of her life traveling the world and expanding her love of making perfume, a craft learned from her French Creole mother. This 18th-century-style shop sells New Orleans–inspired blends (Carnaval, Creole Days, Kiss in the Dark), French milled soap (in Spanish moss and other scents), and handmade powder puffs, as well as gifts and antiques.
34 Chartres St., New Orleans, LA 70130, +1-504-525-7827, hoveparfumeur.com.
#### Lili Vintage
* * *
Located in a renovated Victorian house, this vintage haunt has elegant dresses, ball gowns, fur capelets, and more, with an eye toward the feminine. Great jewelry, too.
3329 Magazine St., New Orleans, LA 70115, +1-504-931-6848, lilivintage.com.
#### Meyer the Hatter
* * *
Venerable third-generation, family-run hat store, open since 1894. Some people consider it the best hat store in the South. Personal note: My hat-loving husband bought a straw boater, which was perfect for the heat.
120 St. Charles Ave., New Orleans, LA 70130, +1-504-525-1048.
#### Mignon Faget
* * *
One of New Orleans's beloved local jewelers, this is the place to go for a special pair of fleur-de-lis earrings (the fleur-de-lis is the symbol of Louisiana). In fact, Faget has a whole Louisiana-themed collection (crawfish tie pin, anyone?). But don't overlook the more abstract pieces either, such as pearl-scarf lariat necklaces and honeycomb rings.
Mignon Faget, 3801 Magazine St., New Orleans LA 70118, +1-504-891-2005, mignonfaget.com.
MIGNON FAGET
#### Mimi
* * *
Housed in a 5,000-square-foot 1930s building, Mimi is the city's go-to boutique for high-end designer clothing, including Michael Kors, Derek Lam, Proenza Schouler, and Monique Lhuillier, plus shoes by Aquazzura, Tabitha Simmons, and Paul Andrew, jewelry by Aurélie Bidermann, and more. Alterations, manicures, and hairstyling services on-site.
5500 Magazine St., New Orleans, LA 70115, +1-504-269 6464, miminola.com.
#### Marion Cage
* * *
Handcrafted contemporary jewelry, from leather "O" ID bracelets to sculptural rosewood disc earrings, and tabletop items made in New Orleans by designer and architect Marion Cage McCollam. Worth a look.
3807 Magazine St., New Orleans, LA, 70115, +1-504-891-8848, marioncage.com.
#### Pied Nu
* * *
Chic Magazine Street lifestyle shop with breezy dresses by Dosa, Megan Park, Mes Demoiselles, and other indie labels, ceramics by Mud Australia and John Derian, gorgeous books, jewelry, handmade espadrilles from Spain, and more.
5521 Magazine St., New Orleans, LA 70115, +1-504-899-4118, piednunola.com.
#### Rendezvous Linen and Lace
* * *
Calling all Southern belles, this charming, old-world store on Jackson Square has delicate linens and lace, from napkins, handkerchiefs, and table runners to christening gowns, plus antique perfume bottles and tea sets.
Jackson Square, 522 St. Peters St., New Orleans, LA, 70116, +1-504-522–0225.
#### Trashy Diva
* * *
A must-shop in New Orleans. What started in 1996 as a vintage clothing store has grown into a full-fledged brand of vintage-inspired clothing nodding to the 1930s through the 1950s, designed by owner Candice Gwinn to fit modern bodies (sizes range from 0 to 24). Personal note: For less than $200, I picked up a 1940s-inspired dress in a Mardi Gras print that I treasure. Gwinn also sells pinup-worthy swimwear, lingerie, shoes, and accessories.
537 Royal St., New Orleans, LA 70117, +1-504-522-4233, trashydiva.com, and an additional location uptown.
#### Weinstein
* * *
What started as a family-owned men's store now caters to women with Rick Owens, Self-Portrait, Dries Van Noten, Co, Avant Toi, and other edgy labels.
4011 Magazine St., New Orleans, LA 70115, +1-504-895-6278, weinsteinsinc.com.
## NASHVILLE
### TENNESSEE
DRAPER JAMES
#### Draper James
* * *
Flagship for Reese Witherspoon's line of Southern-inspired clothing, accessories, and home accents. The white-and-Wedgwood-blue 3,000-square-foot space was designed by Los Angeles interior designer Mark D. Sikes, who also designed Witherspoon's home in Music City.
2608 Twelfth Ave. S., Nashville, TN 37204, +1-615-997-3601, draperjames.com.
DRAPER JAMES
#### Fond Object Records
* * *
Filled with new and used vinyl, vintage clothing, boots, and oddities curated by local fashion designers Poni Silver and Leslie Stevens, this funky place exemplifies Nashville's hipster scene.
1313 McGavock Pike, Nashville, TN 37216, +1-615-499-4498, fondobjectrecords.com.
#### Goodbuy Girls
* * *
Vintage cowboy boots, clothing, and turquoise jewelry, screen-print T-shirts, and other items by local designers.
1108 Woodland St., Nashville, TN 37206, +1-615-281-9447, goodbuygirlsnashville.com.
#### H. Audrey
* * *
Lily Aldridge, Gwyneth Paltrow, and Sheryl Crow are fans of this boutique, owned by singer-songwriter Holly Williams (her grandfather was OG country legend Hank Williams Sr.), which stocks everyday pieces from Anine Bing, Helmut Lang, Rag & Bone, Raquel Allegra, Veronica Beard, Ulla Johnson, and more.
4027 Hillsboro Pike, Nashville, TN 37215, +1-615-760-5701, haudrey.com.
#### Imogene and Willie
* * *
"Nashville is known for having a lot of small companies with singular retail spaces," says Nashville-based fashion publicist Libby Callaway, and this is one of them. The top shop for premium denim occupies a former 1920s gas station, and musicians like Brooks and Dunn, Matt Wertz, and Kings of Leon stop in for the brand's famously flattering cuts. Carrie and Matt Eddmenson opened the shop in 2009 with a belief in made-in-the-USA quality, and they have built a community around their Supper and Song event, which opens up the courtyard to live music in the warmer months.
2601 12th Ave. South, Nashville, TN, +1-615-292-5005, imogeneandwillie.com.
LIBBY CALLAWAY
FAVORITE NASHVILLE VINTAGE
* * *
After building a career in fashion journalism in New York City, Libby Callaway now calls East Nashville home. Former marketing director for Alabama fashion house Billy Reid and former media director for Nashville denim company Imogene and Willie, Callaway is the owner of a marketing consulting firm and makes frequent presentations on the subjects of style and the fashion industry.
Callaway is chair of the board of directors of the Nashville Fashion Alliance. A former wardrobe stylist and vintage clothing dealer, she is also a noted secondhand-shopping expert. Her personal style, and professional advice on the subject, has been featured in InStyle, Elle, Lucky, Us Weekly, Nylon, and Vogue, and on Style.com, The Selby, and the Today show.
Open since the late '90s, Trisha Brantley's East Side store HIP ZIPPER is the OG on the Nashville vintage scene. Her prices are incredibly reasonable and her selection unbelievably deep. The '60s and '70s are especially well represented.
Expensive, but Beverly Chowning from SAVANT VINTAGE has the best and deepest selection in town of highly collectible Americana and dressy wear from the mid- to late 20th century.
Proprietress of HIGH CLASS HILLBILLY, Nikki Lane is known in music circles as "the Queen of Outlaw Country"; in the world of vintage, she's got a reputation for excellent vintage Western wear and motorcycle gear.
Don't miss PINK STAR VINTAGE. You can find Lynda Herdelin's vintage collection at Fanny's House of Music in East Nashville and at Pre to Post Modern on Eighth Avenue. She's got everything from Victorian lace slips to '90s leather.
Carmen Jaudon has an excellent eye for goods from the 1930s through the mid-'70s, all housed in her Wedgewood Houston boutique CLOSET CASE VINTAGE. Lots of calico dresses and excellent collectible denim.
Every few months, Andrew Clancey and Laura Citron get a shipment of vintage dresses from Japan. Those, and pieces from model Karen Elson's personal collection, are some of the highlights to be found in their East Nashville store ANY OLD IRON.
I am passionate about SOUTHERN THRIFT, a Nashville chain of thrift stores. I've found vintage gems from Pucci, Gucci, YSL, Stephen Sprouse, Versace, and Bill Blass on the racks there, which is plenty to keep me coming back.
HIP ZIPPER, 1008 Forrest Ave., Nashville, TN 37206, +1-615-228-1942, hipzipper.com.
SAVANT VINTAGE, 2302 Twelfth Ave. S., Nashville, TN 37204, +1-615-385-0856, savantvintage.com.
HIGH CLASS HILLBILLY, 4604 Galatin Pike (back of building), Nashville, TN 37216, highclasshillbilly.com.
PINK STAR VINTAGE, 2110 Eighth Ave. S., Nashville, TN, 37204, +1-615-414-4881.
CLOSET CASE VINTAGE, 1117 3rd Ave. S., Nashville, TN 37210, +1-615-649-8410, closetcasevintage.com.
ANY OLD IRON, 1629 Shelby Ave., Nashville, TN 37206, +1-615-953-2502.
SOUTHERN THRIFT, 2710 Old Lebanon Pike, Nashville, TN 37214, +1-615-872-0499, and additional locations.
#### Jamie
* * *
Where the society set shops, this is the destination in Nashville for high-end designer duds by The Row, Brunello Cucinelli, Lela Rose, and Rosetta Getty. Owner Jamie Stream brought designer labels to Nashville before anyone else, and now her daughter and granddaughters are carrying on the tradition.
4317 Harding Pike, Nashville, TN 37205, +1-615-292-4188, Jamie-nashville.com.
#### Manuel's Custom Clothiers
* * *
Nashville's resident rhinestone expert has made custom clothing for all the music greats, including Bob Dylan, Elvis, Johnny Cash, Elton John, Dwight Yoakum, and Jack White. "I fell in love with Manuel Cuevas's embroidered blazers," says showroom owner Valery Demure of this shop, where you can also buy rhinestone-embellished T-shirts and scarves.
800 Broadway, Nashville, TN 37203, +1-615-321-5444, manuelcouture.com.
#### Old Made Good
* * *
Fun selection of vintage and locally made goods, artwork, and accessories, including Naughty Needlepoint hoops and custom necklaces.
3701B Gallatin Pike, Nashville, TN 37216, +1-615-432-2882, oldmadegoodnashville.com.
#### Peter Nappi
* * *
This leather accessory company (designed locally, made in Italy) situated in part of a Victorian-era meat processing plant is one of fashion publicist Libby Callaway's Nashville faves for cool clogs and aged leather bags for women, and work boots and belts for men. The large, industrial-chic space serves as the brand's studio, store, and events space, and is filled with vintage decor.
1308 Adams St., Nashville, TN 37208, +1-615-248-3310, peternappi.com.
#### Two Old Hippies
* * *
Guitars, boho apparel, and gifts, plus live music five nights a week? What's not to love about this eclectic boutique in Nashville's trendy Gulch neighborhood? You'll find premium guitars, music memorabilia, western rock 'n' roll-inspired clothing, and gifts.
401 Twelfth Ave. S., Nashville, TN 37203, +1-615-254-7999, twooldhippies.com.
#### Two Son
* * *
This Vogue-approved East Nashville store is minimalist and bright, with an in-house design studio turning out denim, clothing, and accessories to augment collections by Ace & Jig, Norse Projects, Freenote Cloth, and other indie brands, and a photo booth for customers.
918 Main St., Nashville, TN 37206, +1-615-678-4953, twoson.com.
#### White's Mercantile
* * *
Owned by H. Audrey's Holly Williams, this general store stocks Williams's favorite cookbooks, candles, pillows, locally made bread mixes, Turkish towels, and other homey items, plus baby gifts and custom dog collars.
2908 Twelfth Ave S., Nashville, TN 37204, +1-615-750-5379, whitesmercantile.com.
#### Wilder
* * *
This design shop and arts space opened by a couple of New York transplants sells gifts and objects from near and far, including alpaca scarves by New York–based label A Peace Treaty, brooches by Azumi Sakata, jewelry by local artists, and more.
1212 Fourth Ave. N., Nashville, TN 37208, +1-615-679-0008, wilderlife.com.
JEREMY SCOTT
KANSAS CITY, MISSOURI
* * *
Since the late 1990s, fashion rabble-rouser Jeremy Scott has dressed nearly every celebrity in the pop culture pantheon (Miley! Kanye! Katy!) in cheeky designs from his namesake label, as well as in tracksuits and teddy-bear sneakers from his collection for Adidas Originals. In 2013, he was appointed creative director of Italian fashion house Moschino, where he has brought his witty commentary to the runway. His collections reflect on the Barbie beauty ideal and how fast food meets fast fashion. Before Scott began jet-setting around Milan, Los Angeles, and the rest of the world, he grew up in Kansas City, Missouri. As a teenager, he spent afternoons in thrift stores, digging through military uniforms and prom dresses to assemble outfits for his friends. Even now, he hasn't forgotten where he came from and visits home often.
Kansas City has one of the best spots for iconic furniture design at Rod Parks's RETRO INFERNO. You will find things that are out of books, things that you have never seen in real life before, and pieces that you won't know how you lived without until having discovered them at his amazing store.
When in Kansas City, stop by PEGGY NOLAND'S fun fashion boutique. She changes the displays and themes a few times a year; her passion and bravado make the boutique a real experience and something completely unique.
PANACHE CHOCOLATIER on the Country Club Plaza has my favorite guilty pleasure—Choco Poppo. It's chocolate-covered popcorn (I prefer the milk chocolate version). Their chocolate chip cookies with chunks of chocolate the size of half-dollars and their hot chocolate, made with real chocolate, are must-haves for those with a sweet tooth.
RETRO INFERNO, 1500 Grand Blvd., Kansas City, MO 64108, +1-816-842-4004, retroinferno.com.
PEGGY NOLAND, 124 W. 18th St., Kansas City, MO 64108, +1-816-221-7652, peggynoland.com.
PANACHE CHOCOLATIER, 418 Nichols Rd., Kansas City, MO 64112, +1-816-931-3191, chocolatekc.com.
## CHICAGO
### ILLINOIS
#### Azeeza
* * *
Chicago-based designer Azeeza Khan has made fans of the Blonde Salad's Chiara Ferragni, Sarah Jessica Parker, Sophia Bush, Gabrielle Union and Erica Pelosini with her voluminous, off-the-shoulder babydoll dresses, satin slips and wrap gowns in silk crepe de chine and velvet, her embellished chokers, and clutch bags.
900 N. Michigan Ave., Level 5, Chicago, IL 60611, +1-312-649-9373, azeeza.us.
#### Blake
* * *
This minimalist mainstay stocks everything from The Row to Rick Owens, in a pared-down space that used to be a post office. You'll also find Dries Van Noten, Saint Laurent, Balenciaga, Pierre Hardy, and more at this Chicago institution.
212 W. Chicago Ave., Chicago, IL 60654, +1-312-202-0047.
#### Ikram
* * *
"High fashion's ambassador from the Midwest," is how the New York Times described Chicago's powerful retailer Ikram Goldman. A longtime advocate for young designers, Goldman was among the first supporters of American labels Thakoon, Proenza Schouler, and Prabal Gurung, and counts Michelle Obama and Mellody Hobson among her well-heeled clients. Her 16,000-square-foot boutique features a mix of American, European, and Japanese labels, including the most over-the-top-gorgeous runway pieces from Comme des Garçons, Dolce & Gabbana, Givenchy, Junya Watanabe, Simone Rocha, Moschino, Sacai, Rodarte, and Monse, along with newer names you may not have heard of yet but will be happy to discover. "Ikram's singular eye never disappoints," says Karen Erickson, jewelry designer and cofounder of Erickson Beamon. The store also has a restaurant featuring farm-to-table cooking, and an art gallery, "so you can really make a day of it," Erickson says.
15 E. Huron St., Chicago, IL 60611, +1-312-587-1000, ikram.com.
#### Luxury Garage Sale
* * *
"I found them on eBay originally, and then discovered their brick-and-mortar store was in Chicago when I was getting ready to head out there for Empire," says Paolo Nieddu, the show's costume designer. "They are a designer consignment store that has a huge selection of clothing, shoes, bags, and accessories by the best designers, from seasons back to very current stuff. I have gotten amazing pieces for all of my ladies here."
1658 N. Wells St., Chicago, IL 60614, +1-312-291-9126, luxurygaragesale.com.
#### Madison Hall at Chicago Athletic Association Hotel
* * *
Occupying two storefronts along the hotel's Madison Street entrance, Madison Hall takes visitors for a trip in time with apothecary items and men's accessories that hark back to the old-school drugstores of the 1930s. There's also a women's lounge with accessories and a florist. "Somewhere great that I found without really looking for it," Tere Artigas says of the store, conceived by celebrated Chicago independent retailers Lance Lawson and Jim Wetzel, who also own Space 519.
71 E. Madison St., Chicago, IL 60602, +1-312-683-9586, madisonhallchicago.com.
MADISON HALL
#### Robin Richman
* * *
Robin Richman opened her namesake store in 1997, in Bucktown, laying the foundation for what would become an enclave of indie boutiques. She stocks an inspiring collection of under-the-radar, avant-garde fashion and home accessories brands from around the world, including Eckhaus Latta, Uma Wang, and Parts of Four.
2108 N. Damen Ave., Chicago, IL 60647, +1-773-278-6150, robinrichman.com.
#### Space 519
* * *
Lance Lawson and Jim Wetzel scour the globe for fab finds for this concept store, including Atea Oceanie dresses, Trademark basket-weave bucket bags, French cookbooks, handmade botanical prints, and more.
900 N. Michigan Ave., level five, Chicago, IL 60611, +1-312-751-1519, space519.com.
SPACE 519
#### Store B
* * *
"I love to rework vintage pieces for my actors," says Paolo Nieddu. "It's fun and keeps things interesting and inspiring. I've found great long dresses here that I cut into minis, cool jewelry and bags, too. They have clothing organized by designer, which is really helpful when shopping vintage."
1472 N. Milwaukee Ave., Chicago, IL 60654, +1-773-772 4296, storebvintage.com.
## DETROIT
### MICHIGAN
#### Linda Dresner
* * *
"Her store curation rivals any around the globe," says Karen Erickson of Detroit's landmark boutique, which stocks an avant-garde modern/streetwear mix, including Balenciaga, Hood by Air, Comme des Garçons, Vetements, and Zaid Affas. Brooke Taylor Corcia, owner of TheDreslyn.com, says, "She has a great curatorial eye and understanding of her customers, which explains her longevity in the space."
299 W. Maple Rd., Birmingham, MI 48009, +1-248-642-4999, lindadresner.com.
#### Pewabic Pottery
* * *
"Timeless handmade tiles, vases, bowls, et cetera," says fashion writer Robin Givhan. "It dates back to the Arts and Crafts Movement, and the tiles are featured in some of the grand, architecturally significant buildings and houses around the country." Visit the studio and take a self-guided tour of this nonprofit studio.
10125 E. Jefferson Ave., Detroit, MI 48214, +1-313-626-2100, pewabic.org.
#### Shinola
* * *
This is the flagship for the made-in-Detroit retro watch, leather accessory, and bicycle brand that has been a hometown success story, bringing jobs and hipster cachet back to Motor City. The store, which now spans 10,000 square feet, is outfitted with American white oak cabinetry, chairs, sofas, and tables, a lounge area, a reference library, and a café that serves fare from local purveyors. It's just a few doors down from Jack White's Third Man Records. "It also sparked a lot of retail in the area," says Robin Givhan. "Oh, and there's an adjacent Shinola dog park!"
441 W. Canfield St., Detroit, MI 48201, +1-313-285-2390, shinola.com, and additional locations.
SHINOLA
#### Tender Birmingham
* * *
The more feminine and sweet yin to Linda Dresner's yang, this boutique, owned by sisters Karen and Cheryl Daskas, carries Marni, Isabel Marant, Simone Rocha, Thakoon, Marques' Almeida, and Peter Pilotto.
271 W. Maple Rd., Birmingham, MI 48009, +1-248-258-0212, tenderbirmingham.com.
TENDER BIRMINGHAM
## TORONTO
### CANADA
LOST & FOUND
#### Chosen Vintage
* * *
One of the city's best-kept secrets, this store has clothing and accessories that aren't just chosen, they're "handpicked," according to the sign. "An amazing collection of cheap and chic vintage," says model-turned-blogger Elizabeth Minett.
1599 Dundas St. W., Toronto, ON M6K 1T9, +1-647-346-1993, chosen-vintage.com.
#### DAVIDS
* * *
"A must for shoes," says stylist Jessica de Ruiter of this 50-plus-year-old, family-owned Toronto institution. You'll find the hottest styles from Chloé, Christian Louboutin, Valentino, Sophia Webster, Charlotte Olympia, and more.
66 Bloor St. W, Toronto, ON M5S 1L9, +1-416-920-1000, davidsfootwear.com, and additional locations.
#### Ewanika
* * *
Designer Trish Ewanika is the mastermind of this überchic boutique, which sells her streamlined modern classics alongside items by local and international designers such as MM6 Maison Margiela, Samuji, and Hope Stockholm, plus Sophie Buhai jewelry and shoes by the Palatines and Martiniano.
1083 Bathurst St., Toronto, ON M5R 3G8, +1-416-927-9699, ewanika.ca.
#### Frank and Oak
* * *
Founded in 2012, this trailblazing millennial men's lifestyle brand continues to focus on tech-enabled service. It has an app with chat bots and stylists to make recommendations based on your personal preferences, and offers two-hour delivery in select cities. The Toronto flagship combines clothing with a barbershop, café, and community space. Customer accounts are synced online and in the brick-and-mortar stores for ease of shopping.
735 Queen St. W., Toronto, ON M61 1J1, +1-647-930-8711, frankandoak.com, and additional locations.
#### Holt Renfrew
* * *
Holt Renfrew started out in 1837 as a modest hat shop, and eventually expanded to become Canada's premier destination for luxury retail. In 1947, it hosted Christian Dior when he launched his New Look. The department store carries Tom Ford, Dior, Chloé, Gucci, Alexander McQueen, and Manolo Blahnik, as well as emerging Canadian luxury labels Greta Constantine and Sid Neigum. The flagship is in Toronto, and there are several other stores across Canada.
50 Bloor St. W., Toronto, ON, M4W 1A1, +1-416-922-2333, holtrenfrew.com, and additional locations.
#### Klaxon Howl
* * *
"Super hidden gem of an outpost in Toronto, where I grew up," says Citizens of Humanity women's creative director Catherine Ryu. "Cool vintage collectibles, specializing in military gear. They also produce their own line of Canadian-made men's workwear and military-inspired clothing. The entrance is located in the back alley of an old coach house."
694 B Queen St. W. (back alley entrance off Manning), Toronto, ON M6J 1E7, +1-647-436-6628, KlaxonHowl.com.
#### Lost & Found
* * *
Menswear and coffee shop that curates classic heritage brands including Alden, Champion, Dickies, Shinola, Steven Alan, Patagonia, and Levi's.
44 Ossington Ave., Toronto, ON, M6J 2Y7, +1-647-348-2810 shoplostfound.com.
#### Nomad
* * *
When Justin Bieber's Purpose tour rolled into Toronto, this is where he popped up with his tour merch, created by Fear of God designer Jerry Lorenzo. It's no surprise, then, that Nomad is the place to find cool international men's brands like Off-White by Virgil Abloh, Engineered Garments, and Gosha Rubchinskiy alongside Canada's own Wings and Horns, Naked & Famous Denim, and Reigning Champ.
819 Queen St. W., Toronto, ON M6J 1G1, +1-416-202-8777, nomadshop.net.
#### Nordstrom at the Toronto Eaton Centre
* * *
The US department store chain is slowly making inroads in Canada, and the new Eaton Centre store is worth checking out for the cocktail bar and in-store Madewell boutique (Canada's first) alone.
260 Yonge St., Toronto, ON M5B 2L9, +1-416-552-2900.
#### OVO
* * *
Canadian rapper Drake's flagship for his owl-festooned OVO streetwear line, much of which is made in Canada, including his collaboration with Canada Goose.
899 Dundas St. W., ON M6J 1W1, +1-416-276-0568, octobersveryown.com.
#### Pink Tartan
* * *
The flagship for Kimberley Newport- Mimran's Toronto-based label, known for elegant sportswear in understated, super-luxe fabrics, also carries a curated selection of clothing by other labels such as Mira Mikati, Rochas, Narciso Rodriguez, and Tome, and accessories and gifts by Tabitha Simmons, No. 21, Castañer, Want Les Essentials, and Mark Cross.
77 Yorkville Ave., Toronto, ON M5R 1C1, +1-416-967-7700, pinktartan.com.
SUSIE SHEFFMAN
TORONTO
* * *
Toronto-based Susie Sheffman has spent thirty years creating award-winning fashion editorials for Canada's leading magazines, as well as shaping the vision for Canadian brands and retailers, including Holt Renfrew, Joe Fresh, Roots, and Hudson's Bay. Her career started at age five, when she forced friends to play "store" in her bedroom. (No one left without a tissue-wrapped package and a receipt!) She's collaborated with international fashion icons, including Linda Evangelista, Jessica Stam, and Coco Rocha, and styled numerous celebrities (Rihanna, Britney Spears, Taylor Swift, and Ashley Olsen, to name just a few).
Since JONATHAN + OLIVIA rolled into town, I've found my shopping mecca, and built a wardrobe of all-time personal favorites along the way. Owner-buyer Jackie O'Brien is a retail risk-taker and pioneer who stocks her hip Ossington Avenue boutique with the kind of cult closet classics that cool girls (and guys) dream about. With a confident eye and wicked taste, she was the first to launch Topshop in Canada, and she carried cult favorites like Vetements and Jacquemus long before they were even whispered about abroad. Alexander Wang, Acne Studios, A.P.C., Isabel Marant, Étoile, and local designers like Horses round out the selection of clothing, shoes, bags, and jewelry that manages to hit everything on your list. If it's not there, you probably don't want it. Plus killer coats—this is Canada, after all.
The jam-packed shoe store GRAVITYPOPE (half a block from Jonathan + Olivia) hits all the right shoe notes, everything from Church's and Marni to Converse and Vans for men, women, and kids, plus a great private-label line. The second floor is full of clothing from Marni, Rachel Comey, Isabel Marant, Jil Sander Navy, and other killer lines whose labels you may not recognize and won't find elsewhere. Check out the all-white sneaker collection, the jewelry and leather goods plus fragrance and body care. The store originated in Calgary, and there's also an outpost in Vancouver. GRAVITYPOPE and JONATHAN + OLIVIA are the main retail anchors in this West End neighborhood that is rapidly gentrifying with a solid mix of restaurants, bars, and vintage shops.
OVER THE RAINBOW
In terms of local midtown hits (small independents within a block of each other), GEE BEAUTY is a one-stop beauty bar/day spa run by former makeup artist/beauty editor Miriam Gee and her three gorgeous and stylish daughters. (You can find them at their Bal Harbour outpost as well.) Entering 6 BY GEE BEAUTY, their teeny-tiny gift emporium, feels like you're stepping into a beach vacation. Grab Le Labo candles and fragrance, beachy caftans, books, sunglasses, and personal jewelry.
ADVICE FROM A CATERPILLAR has baby and kids' clothes—nothing cloying or cutesy, just the most perfect bite-size bits that you'll wish you could squeeze into yourself! Plus tepees, baby moccasins, vintage toys, and other delights in an übercool, airy space. ("My favorite kids' store of all time," says stylist Jessica de Ruiter.)
In the Bloor Yorkville area, which is like Toronto's Fifth Avenue, you'll find all the designer mono-brand boutiques. SPECCHIO has a roster of Marni, Dries Van Noten, Stella McCartney, and Fiorentini and Baker, so it's hard to go wrong—this small shoebox of a shop gets it right every season. Handsome owner Albert Moryoussef handpicks a highly personalized collection of lust-worthy, hard-to-resist shoes and boots that ramp up my heart rate before I've even walked in. (He's also known to offer hard-to-find small sizes!) Gotta love a shoe store that stocks killer over-the-knee suede boots as a basic.
At 119 CORBO, pieces from Dries Van Noten, Céline, Haider Ackermann, Stella McCartney, Sacai, and other lusted-after labels keep company with equally drool-worthy shoes and bags in this beautifully sophisticated Yorkville shop.
Family-run denim destination OVER THE RAINBOW has been going strong for forty years thanks to Dr. Denim himself, Joel Carman, who understands just how important it is to get your jeans right. With an enormous selection and a genuine, friendly spirit, Carman boasts all the best denim brands and still gets down on his hands and knees to check the fit. Need an inch here, a maternity panel there? Done! On-the-spot alterations and service with a smile are the key to his longevity and success. Plus, no charge for hugs!
A couple of randoms: HER MAJESTY'S PLEASURE downtown is a beauty emporium that goes way beyond the blowout! You can hang out for hours in this potent, pampering space that's also part cocktail lounge and part coffee/dessert bar, and elevate your mani to the next level!
For more than fifty years, trusted North Toronto family shoe store CIRCLE SHOES has been serving up the kind of unironic normcore footwear that fashion people freak for. All the major shoe trends are covered, from Birkenstocks and Sorels to Minnetonka moccasins. Look closely and you might find original versions of the stacked-heel loafers and crossover sandals that have inspired designers everywhere of late. The good old-fashioned service here not only encourages special orders but also writes them in an actual notebook with a pen.
JONATHAN + OLIVIA, 49 Ossington Ave., Toronto, ON M6J 2Y9, +1-416-849-5956, and also in Vancouver, jonathanandolivia.com.
GRAVITYPOPE, 1010 Queen St. W., Toronto, ON M6J 1H6, +1-647-748-5155, gravitypope.com.
GEE BEAUTY, 2 Roxborough St. W., Toronto, ON M5R 1T8, +1-416-486-0080, geebeauty.com, and an additional location in Miami.
ADVICE FROM A CATERPILLAR, 8 Price St., Toronto, ON M4W 1Z4, +1-416-960-2223, advicefromacaterpillar.ca.
SPECCHIO BOUTIQUE, 1240 Bay St., Toronto, ON M5R 2A7, +1-416-961-7989, specchioshoes.com.
119 CORBO, 119 Yorkville Ave., Toronto, ON M5R 1C4, +1-416-928-0954, 119corbo.com.
OVER THE RAINBOW, 101 Yorkville Ave., Toronto, ON M5R 1C1, +1-416-967-7448, rainbowjeans.com.
HER MAJESTY'S PLEASURE, 556 King St. W, Toronto, ON M5V 1M3, +1-416-546-4991, hermajestyspleasure.ca.
CIRCLE SHOES, 2597 Yonge St., Toronto, ON M4P 2J1, +1-416-489-4379, circleshoes.ca.
#### The Drake General Store
* * *
A traditional hotel gift shop, general store, flea market stand, and museum gift shop rolled into one is how fans describe this Toronto staple, which has since expanded to a few more stores around the city. You'll find the best of locally designed Canadian goods, including classic toque hats, "Canada creatures" kids' tees, a Canadian log-print "lumber" pillow, and more.
1151 Queen St West, Toronto, ON M6J 1J4 +1-647-346-0742, drakegeneralstore.ca, and additional locations.
#### The Narwhal
* * *
"A great little boutique for contemporary women's fashion," says stylist Jessica de Ruiter of this spot where Edie Parker acrylic clutches sit next to Golden Goose sneakers, and Rachel Comey linen blazers hang alongside Apiece Apart and Ulla Johnson skirts.
8 Price St., Toronto, ON M4W 1Z4, +1-647-351-5011, narwhalboutique.com.
#### TNT
* * *
Arguably the toniest fashion destination in town. Founders Arie Assaraf and Carrie Richmond have three locations in Toronto and one in Montréal offering men's and women's streetwear-inspired labels such as Yeezy, D. Gnak, and Vancouver-based Wings and Horns, alongside luxe mainstays Balmain, Brunello Cucinelli, and The Row.
87 Avenue Rd., Toronto, ON M5R 3R9 +1-416-975-1810, tntfashion.ca, and additional locations.
#### The Room at the Bay
* * *
This is Hudson's Bay Company department store's high-end section, with top-of-the-line ready-to-wear and accessories from Giambattista Valli, Balmain, Alaïa, Sonia Rykiel, and Canadian designer Marie Saint Pierre (a favorite of Sophie Grégoire Trudeau, wife of the Canadian prime minister) among others.
176 Yonge St., The Bay Queen Street, Toronto, ON M5C 2L7, +1-416-861-6251, www2.thebay.com/theroom.
#### VSP Consignment
* * *
Stylist and blogger favorite for designer consignment fashion finds.
1410 Dundas St West, Toronto, ON M6J 1Y5 +1-416-588-9821, vspconsignment.com.
## MONTRÉAL
### CANADA
#### Cahier d'Exercices
* * *
Sleek, industrial space with an eclectic array of women's clothing from Céline, Stella McCartney, Dries Van Noten, Junya Watanabe, and other well-known designers, as well as lesser-known names.
369 Rue Saint-Paul O., Montréal, QC H2Y 2A7, +1-514-439-5169, cahierdexercises.com.
CAHIER D'EXERCICES
#### Éditions de Robes
* * *
At the flagship for Julie Pesant's chic Montréal-based brand, you'll be hard-pressed not to find the perfect dress for work or cocktails—dresses are the sole focus. You'll find cape-back ones, trapeze silhouettes, floral lace designs, and more, and while they don't always come cheap, they are made to last.
178 Rue Saint Viateur O. Montréal, QC H2T 2L3, +1-514-271-7676, editionsderobes.com.
#### Espace Pepin
* * *
One-stop shop in Old Montréal for down-to-earth clothing, accessories, and rustic home design objects and furniture with an emphasis on Canadian artists and makers. There's also a lunch counter, Le Comptoir Végétarien, in the back of the store.
350 Rue Saint-Paul O., Montréal, QC H2Y 2A3, +1-514-844-0114, thepepinshop.com.
#### Eva B
* * *
This bistro, boutique, and fabulously cluttered used clothing store tucked inside a graffiti-covered building is unique, that's for sure. It's a junk-lover's paradise—you just can't be afraid to get your hands dirty.
2015 Boul. Saint-Laurent, Montréal, QC H2X 2T3, +1-514-849-8246, eva-b.ca.
#### General 54
* * *
Originally conceived as a gallery/general store (hence the name), General 54 evolved into a boutique selling women's clothing, accessories, and jewelry with a playful twist. More than sixty Canadian labels, including Birds of North America and Heidi Martens, are represented. Located in the highly shoppable, indie Mile End area (some call it the Brooklyn of Montréal!).
5145 Boul. Saint-Laurent, Montréal, QC H2T 1R9, +1-514-271-2129, general54.ca.
#### Harricana
* * *
Workshop and boutique for the Montréal-based brand that uses recycled fur to create cool scarves, mittens, hats, ski jackets, and even fringe necklaces and mules.
416 Rue McGill, Montréal, QC H2Y 261 +1-514-287-6517, harricana.qc.ca.
#### Henri Henri
* * *
One of the most renowned hatmakers in Canada, this old-school shop was founded as a family business in 1932. According to store lore, Henri Henri introduced the expression "hat trick" into hockey in the 1950s, by rewarding players who scored three goals or more in a single game with a custom-made hat.
189 Rue Sainte-Catherine E., Montréal, QC H2X 273, +1-514-288-0109, henrihenri.ca.
#### La Maison Simons
* * *
The Quebec City–based department store chain has locations throughout Canada, including two in Montréal. The downtown store carries international labels both well known (Philosophy, Acne, Dries Van Noten, McQ, Faith Connexion) and new (Paskal) alongside the best of Canadian designers, including Denis Gagnon, Smythe, and UNTTLD.
977 Rue Sainte-Catherine O., Montréal, QC H3B 4W3, +1-514-282-1840, simons.ca, and additional locations.
#### Les Etoffes
* * *
One of the city's most carefully curated boutiques, Les Etoffes has a fashion-forward point of view, stocking Samuji, Apiece Apart, Rodebjer, and other labels.
5253 Boul. Saint-Laurent, Montréal, QC H2T 154 +1-514-544-5500, lesetoffes.com.
#### Ssense
* * *
The brick-and-mortar outpost of the online luxury purveyor has an encyclopedic offering, with everything from Altuzarra and Dolce & Gabbana to Giuseppe Zanotti and Yang Li.
90 Rue Saint-Paul O., Montréal, QC H2Y 3S5, +1-514-289-1906, ssense.com.
#### Want Apothecary
* * *
Launched in Montréal in 2006 by twin brothers Byron and Dexter Peart, Want Les Essentials is known for its architectural-looking bags for men and women (Sophie Grégoire Trudeau, wife of the Canadian prime minister, carries one). This store also features clothing by like-minded labels such as Tomorrowland, Acne Studios, and Maison Kitsuné, plus apothecary items by Byredo and others.
4960 Sherbrooke St. W., Westmount, QC, H3Z 1H3, +1-514-484-3555, wantapothecary.com, and additional locations.
WANT APOTHECARY
#### Unicorn
* * *
Also in the Mile End neighborhood, Unicorn is dedicated to showcasing casual clothing, most of it by local designers such as Betina Lou and Melissa Nepton.
5135 Boul. Saint-Laurent, Montréal, QC H2T 1R9, +1-514-544-2828, boutiqueunicorn.com.
## VANCOUVER
### CANADA
SECRET LOCATION
#### A'hoy Goods
* * *
Launched by Jamie and Lyndon Cormack, founders of the popular Vancouver-based bag label Herschel Supply, this store features a nautical- themed apparel line inspired by the Deep Cove neighborhood and sporting such slogans as LOCALS ONLY and A'HOY OR HIGH WATER. The shop also carries classic threads and accessories by Levi's, Vans, Patagonia, Stussy, Hunter, and, of course, Herschel Supply.
4391 Gallant Ave., North Vancouver, BC V7G 1L2, +1-604-770-3100, ahoygoods.com.
#### Lululemon Lab
* * *
Opened in 2009 by the Vancouver athleisure giant, this concept store is an incubator for the brand's newest styles, such as bomber jackets, tunics, and onesies, designed to extend Lululemon's appeal from the gym to the street.
50 Powell St., Vancouver, BC V6A 1E7, +1-604-708-1126, lululemonlab.com, and additional locations.
#### Oak + Fort
* * *
Since its inception in 2010, the Vancouver-based Oak + Fort has curated collections of uncomplicated, thoughtfully designed clothing. You'll find androgynous pieces in languid silhouettes that hark back to the 1990s.
355 Water St., Vancouver, BC V6B 1B8, +1-604-566-9199, oakandfort.com, and other locations.
#### Secret Location
* * *
Located in the city's trendy Gastown neighborhood, this gallerylike concept store is one part retail, selling cutting-edge fashion and objets by David Koma, Ileana Makri, Osman, No. 21, Thierry Lasry, and Yazbukey, and one part modern café, where you can enjoy a six-course tasting menu.
1 Water St., Vancouver, BC, V6B 2H9, +1-604-685-0090, secretlocation.ca.
SECRET LOCATION
#### Roden Gray
* * *
Top-notch menswear store stocking everything from Thom Browne and Givenchy to Buscemi and John Elliott, from Carhartt to Engineered Garments.
8 Water St., Vancouver, BC V6B 2K8, +1-604-689-7302, rodengray.com.
#### Wings + Horns
* * *
Inspired by trips between Tokyo and Vancouver, this contemporary menswear label is one of Canada's best, integrating innovative fabrics with a meticulous approach to detail. Look for knit bombers, washed-linen blazers, and awesome collaborations, like deconstructed Wings and Horns x New Balance sneakers. This is the flagship.
133 West Fifth Ave., Vancouver, BC V5Y 1H9 +1-604-568-0140, wingsandhorns.com.
JENNIFER MORRISON
VANCOUVER
* * *
American actor/model/producer Jennifer Morrison calls Vancouver her "home away from home." She's been living on and off in the city often referred to as "the Hollywood of the North," since the 1990s, when she filmed the pilot of House, the medical drama in which Morrison played Dr. Allison Cameron alongside star Hugh Laurie. She's back in Vancouver again for her current role as Emma Swan on the fantasy-adventure series Once Upon a Time.
My favorite boutique is MISCH on Granville. The buyers are very consistent and always select good stuff (Isabel Marant, Forte Forte, Vanessa Bruno, Co, Protagonist).
There's a boutique called BLUEBIRD on Alberni that I also like a lot. They have a great jeans selection. (Plus contemporary labels like A.L.C., Derek Lam 10 Crosby, Mackage, Jonathan Simkhai, etc.)
NORDSTROM just opened in downtown Vancouver, and people are very excited about it. Nearby is the flagship of Vancouver-based ARITZIA. The vibe and feel of the brand is a great reflection of Vancouver style. The focus is on quality fabrics that last and affordable prices. I've taken a tour of their offices and warehouse, and find them a very impressive quality-driven and forward-thinking company.
For secondhand shops, COMMERCIAL DRIVE is the place to go. Store after store with vintage finds.
There are also a few great more-specialty boutiques in Gastown. I like THE BLOCK on Cordova Street. Consistently good finds and cool shoes and bags by A Détacher, No. 6, A.P.C., Wendy Nichol, etc.
MISCH, 2960 Granville St., Vancouver, BC V6H 3J7, +1-604-731-1017, misch.ca.
BLUEBIRD, 1055 Alberni St., Vancouver, BC V6E 1A1, +1-604-257-0700, also in Oakridge Centre, bluebird.ca.
NORDSTROM, Pacific Centre, 799 Robson St., Vancouver, BC V7Y 0A2, +1-604-699-2100.
ARITZIA, 1110 Robson St., Vancouver, BC V6E 1B2, +1-604-684-3251, aritzia.com, and additional locations.
THE BLOCK, 350 West Cordova St., Vancouver, BC, V6B 1E8 +1-604-685-8885, theblock.ca.
## MEXICO CITY
### MEXICO
#### 180°
* * *
This streetwear-oriented shop features locally designed T-shirts, art magazines, sunglasses, and bicycles. You can't miss the eye-catching black-and-white awning outside. Inside, make sure to take a look at the front counter made out of old VHS tapes.
Calle Colima 180, Roma Norte, 06700 Ciudad de México, D.F., +52-55-5525-5626, 180grados.mx.
#### Anatole 13
* * *
Fashion, food, tea, and design objets in a minimalist, gallerylike environment. Carries a variety of clothing from Mexico, the US, and Europe, and has an organic café.
Anatole France 13, Polanco III Secc, 11550 Ciudad de México, D.F., +52-55-5280-5267, anatole13.net.
#### Caballería
* * *
A stylish, multilevel men's marketplace (caballería translates to "chivalry") with an assortment of clothing and accessories, a barbershop, a tattoo parlor, and a pizza-and-beer restaurant.
Calle Havre 64, Col. Juárez, 06600 Ciudad de México, D.F., caballeria.mx.
#### Cañamiel
* * *
This store is dedicated to showcasing styles from independent Latin American designers, including shoes by Minhk, bags by Oriana Rodriguez, jewelry by Paula Mendoza, and clothing by Julia y Renata, and more.
Zedec Santa Fe, Av. Javier Barros Sierra 540, Col. Lomas de Santa Fé, 01219 Ciudad de México, D.F., +52-55-5292-3869, canamielmx.com.
#### Carla Fernández
* * *
Fernández launched her namesake fashion label in 2000, with the goal of working with indigenous communities in Mexico to preserve handweaving and dyeing, felting, and embroidery techniques. Her collection, priced from $50 to $600 or more for special orders, has a look that is more avant-garde than arts and crafts. "Worth a stop for her signature wrap-this-way-and-that-way Cobra dresses alone," says fashion journalist/retailer Rose Apodaca.
Isabel La Catolica 30, First Floor, Mexico City 06000 Ciudad de México, D.F., +52-55-5510-9624, carlafernandez.com, and additional locations.
#### Carmen Rion
* * *
Another fashion designer who often works with indigenous communities, Rion's blouses and wraps have a graphic sensibility. Her store also carries flat leather sandals and modern silver jewelry.
Av. México 135 A, Hipódromo, 06140 Ciudad de México, D.F., +52-55-5564-1666, carmenrion.com.
#### Chic by Accident
* * *
French expat Emmanuel Picault's gallery, workshop, and design studio juxtaposes traditional Mexican folk objects with midcentury antiques. The gallery was established in 2000; in 2010, a design studio was added. Picault has worked on commercial and residential projects in Mexico City and Paris.
Lago Texcoco 112, Anáhuac I Secc, 11320 Ciudad de México, D.F., +52-55-5511-1312, chicbyaccident.com.
#### Common People
* * *
This concept shop in a 1940s colonial-style mansion in Mexico City's high-end Polanco district has been called Mexico City's answer to Paris's colette. It has three floors of clothing, shoes, and decor items at all price points from well-established international and up-and-coming local brands, including Vivienne Westwood, Maison Martin Margiela, THVM, Opening Ceremony, Avocet, and Pays. On the second floor, recharge with a Hungarian coffee at Café Budapest.
Emilio Castelar 149, Polanco III Secc, 11560 Ciudad de México, D.F., +52-55-5281-0800, commonpeople.com.mx.
#### El Bazaar Sabádo
* * *
Open on Saturdays from 9 a.m. to 6 p.m., this pop-up flea market showcases a selection of handmade traditional and contemporary jewelry, ceramics, decorative items, artwork, and textiles by local artisans, many of whom also sell in local boutiques and galleries. Top contemporary Mexican fashion designer Carla Fernández has a stall, and our stylish shoppers also recommend Androna Textiles "for the best Mexican traditional clothes—the quality of the product is incredible," says jewelry designer Daniela Villegas.
San Jacinto 11, San Ángel, 01000 Ciudad de México, D.F, +52-55-5616-0082, elbazaarsabado.com/mx.
#### El Palacio de Hierro - Moliere
* * *
Mexico's biggest department store chain with more than a dozen locations (this is the granddaddy) has a wide selection of international and local designers. "Great shop-in-shops from Tiffany & Co. to Louis Vuitton," says fashion editor Karla Martinez de Salas. Don't miss the 20,000-square-foot food hall, a terrific spot for hot chocolate and churros, and pretty much everything else you'd ever want to eat.
Av. Moliere 222, Polanco III Secc, 11570 Ciudad de México, D.F., +52-55-5283-7200, elpalaciodehierro.com.
THE ARTIGAS SISTERS
MEXICO CITY FAVORITES
* * *
In 2003, while studying fabric design in her hometown of Mexico City, Gabriela Artigas fashioned her first cuff bracelet out of a supermarket toothbrush, minus the bristles. Now based in Los Angeles, she has a full-fledged label, Gabriela Artigas, designed and handcrafted in L.A., which counts Carey Mulligan, Emma Roberts, and Tyra Banks among its fans. Her sister, Tere Artigas, handles sales and press for the minimalist-yet-edgy brand, which is known for its signature gold tusks and shooting stars.
In Mexico City, there is an organic restaurant and grocery store called OJO DE AGUA. It reminds me of a traditional fruit-and-vegetable market stall. You can buy amazing avocados or pressed juices, pastries or marmalades and preserves. For hot chocolate and tortas after a long day shopping, ABARROTES DELIRIO is another great find. It has a beautiful wine and mescal selection—perfect for gifts! ONORA sells traditional crafts and textiles in nontraditional colors that you won't find anywhere else. You can buy ceramics made out of traditional materials but designed in a contemporary way. CASA BOSQUES is another gem, and I don't want to forget AVERY, which sells super-minimal-style clothes in black, gray, or white. It's one of my favorite stores in Mexico City.
If you want to experience something truly local, LA LAGUNILLA MARKET is a must-visit. It's a traditional Mexican market selling antiques, furniture, art, clothes—everything you could possibly imagine! A friend of mine once found a Knoll dining room table and chairs in brand-new condition. Go on a Sunday morning, order a michelada (cerveza with lime juice and assorted sauces, and spices on the rim of the glass), and take the time to hunt for treasures.
OJO DE AGUA, Calle Citlaltépetl 23C, Hipódromo, 06100 Ciudad de México, D.F., +52-55-6395-8000, grupoojodeagua.com.mx.
ABARROTES DELIRIO, Colima 114, Cuauhtemoc, Roma Norte, 06700 Ciudad de México, D.F., +52-55-5264-1468, abarrotes.delirio.mx.
ONORA, Lope de Vega 330, Polanco V Secc, 11560 Ciudad de México, D.F., +52-55-5203-0938, onoracasa.com.
CASA BOSQUES, Calle Córdoba 25, Roma Norte., 06700 Ciudad de México, D.F., +52-55-6378-2976, casabosques.net.
AVERY, Córdoba 25, Roma Norte, 06700 Ciudad de México, D.F., +52-55-6378-2976 Ext. 105, averyshop.com.
LA LAGUNILLA MARKET, La Lagunilla, 06020 Ciudad de México, D.F., +52-55-5529 8352.
#### Fábrica Social
* * *
This organization with a fair-trade philosophy helps train Mexican women who are already skilled in traditional textile work become more proficient in fashion design, so that they can produce more commercial pieces. Price tags include the artisan's name and the number of hours it took for her to make the garment.
Calle Isabel la Catolica 30, Centro Histórico, 06000 Ciudad de México, D.F., +52-55-5512-0730, fabricasocial.org.
#### Goodbye Folk
* * *
One of Mexico City's most beloved fashion haunts, this vintage store and hair salon is known for its handmade brogues, some of which incorporate serape blanket remnants or floral-print fabric. "Handmade shoes and redone vintage at great prices," says fashion editor Karla Martinez de Salas.
Calle Colima 198, Cuauhtémoc, 06700 Ciudad de México, D.F., +52-55-5525-4109, goodbyefolk.com.
#### Lago DF
* * *
This arty boutique by local designers Regina Barrios, of the jewelry brand Ishi, and Alessandro Cerutti, of Boca MMXII, features their favorites from Mexico and the rest of the world.
Calle Emilio Castelar 209, Polanco V Secc, 11560 Ciudad de México, D.F., +52-55-655-2059, lagodf.com.
#### Lemon Chic
* * *
Right next door to many of the city's luxury-brand stores on Masaryk, Lemon Chic has a great assortment of party dresses, rompers, and must-have denim by contemporary brands such as Alice and Olivia, Iro, Frame Denim, and Rebecca Taylor.
Masaryk 311, Third Floor N. 4 , Polanco, IV Secc, 11550 Ciudad de D. F., +52-55-9155-5318, lemonchic.com.
#### Onora
* * *
One of Mexico City's coolest design stores, Onora strikes a balance between traditional and contemporary in their shop, which stocks modern-looking black clay candleholders from Oaxaca, table runners from Chiapas, serving dishes from Puebla, and more.
Lope de Vega 330, Polanco V Secc, 11560 Ciudad de México, D.F., +52-55-5203-0938, onoracasa.com.
#### Pineda Covalin
* * *
This Mexico City–based brand has gone global with its Hermès-style silk ties, scarves, clothing, bags, and small accessories in colorful, artisan-designed prints inspired by Latin American culture, with sophisticated takes on sugar skulls, dream catchers, huipil embroidery, and more. This is the headquarters; there are also boutiques across Mexico and around the world.
Sinaloa 237, Roma Norte, 06700 Ciudad de México, D.F., +52-55-5256-3606, pinedacovalin.com.
#### Raquel Orozco
* * *
Caters to sophisticated Mexican ladies with sexy, cape-back Halston-esque cocktail dresses, jumpsuits, and sleek coats, plus bold, modern accessories, all at prices that won't break the bank.
Calle Emilio Castelar 2271-B, Polanco IV Secc, 11550 Ciudad de México, D.F., +52-55-5280-5081, raquelorozco.mx.
#### Roma Quince
* * *
Housed in a century-old mansion, this concept store features individual fashion boutiques (including the Mexico City–based Les Filles du Nord brand of cute resortwear and tote bags), antique furniture, luxury linens, a café, and a gelato stand.
Medellín 67, Roma Norte, Ciudad de México, D.F., +52-55-5207-8682.
#### Sandra Weil
* * *
One of Mexico City's most talented young designers, Weil founded her line in 2008 with a vision of creating contemporary, feminine clothing and gowns with subtle nods to her South American roots. Think colorful tweed bustiers, navy zigzag-stripe organza blouses, and lace maxi dresses with appliquéd sequins.
Calle Emilio Castelar 185, Polanco III Secc, 11560 Ciudad de México, D.F., +52-55-5280-7597, sandraweil.com.
#### Sangre de mi Sangre
* * *
Designer Mariana Villarreal creates edgy contemporary jewelry with skull, pyramid, and snake motifs.
Edificio Balmori L-F, Orizaba 101, Roma Norte, 06700 Ciudad de México, D.F., +52-55-551-8599, sdemis.com.
#### Silver Deer
* * *
This superb menswear store stocks a contemporary luxe mix of clothing, shoes, books, grooming products, accessories, and home furnishings, including Thom Browne, Isaia, Drakes, and exclusive collabs such as Alden for Silver Deer.
Av. Javier Barros Sierra 540, Zedec Sta Fé, 01219 Ciudad de México, D.F., +52-55-5292-9587, thesilverdeer.com.
#### Tane
* * *
Since the 1940s, this Mexico City–based luxury brand has been the go-to for sleek silver jewelry inspired by pre-Hispanic design and crafts. It continues to thrive today, thanks to collaborations with contemporary designers and style-setters such as Iris Apfel. You'll find chunky silver chains, silver-and-wood cuffs, and silver minaudières, as well as gold pieces, silk and leather accessories, and candles.
Av. Presidente Masaryk 430, Polanco III Secc, 11560 Ciudad de México, D.F., +52-55-5282-6200, tane.com.mx, and additional locations.
#### Taxonomía
* * *
Located on the first floor of Hotel Carlota, this shop curates the best of contemporary Mexican fashion, jewelry, and home design, including shirts by 1/8 Takamura, leather goods by Robin Archives, and mescal cups by Lagos del Mundo.
Hotel Carlota, Calle Río Amazonas 73, 06500 Ciudad de México, D.F, +52-55-116300, taxonomia.mx.
#### Yakampot
* * *
Francisco Cancino's line of feminine blouses, tunic dresses, and ponchos is a big hit among Mexico's stylish set, including fashion editor Karla Martinez de Salas. Cancino sources fabrics in Mexico or from specialty mills in Europe and collaborates with indigenous artisans. The first Yakampot shop in Polanco, designed by Emiliano Godoy and Tuux, was built with the same sustainable practices. Check out Yakampot's sister brand, Arroz con Leche, for adorable kids' clothes.
Calle Emilio Castelar 215B, Polanco III Secc, 11550 Ciudad de México, D.F., +52-55-6721-3324, yakampot.com.
YAKAMPOT
#### Void
* * *
This high-end vintage store, owned by former model and current Mexico City "It" girl Olympia de la Macorra, features everything from collectible Chanel to rocker T-shirts.
Calle Juan Escutia 89, Col. Condesa, 06140 Ciudad de México, D.F., +52-55-5211-7213, voidmx.com.
## SAN MIGUEL DE ALLENDE
### MEXICO
MIXTA
#### Abrazos
* * *
Shirts, napkins, aprons, baby bibs, tote bags, and more in Day of the Dead, lucha libre, cactus, and other colorful cotton prints that capture the spirit of Mexico.
Zacateros 24, Centro, 37700 San Miguel de Allende, GTO., +52-415-154-8580, sanmigueldesigns.com.
#### Artisan Alley
* * *
Three blocks of small stores and tented stalls selling embroidered pillowcases, tablecloths, purses, sugar skulls, jewelry, tin lamps, Catrina dolls, and more.
Lucas Balderas S/N, Centro, 37700 San Miguel de Allende, GTO.
#### Camino Silvestre
* * *
Gorgeous collection of hummingbird feeders, birdhouses, ceramics, glassware, and gifts in an enchanting setting.
Zacateros 46, Centro, 37700 San Miguel de Allende, GTO., +52-415-121-3359, caminosilvestre.com.
#### Casa Kiri
* * *
Hand-embroidered blouses, dresses, shawls, table runners, pillowcases, and more, all with a traditional-meets-boho modern vibe. One of the best spots in town to find something truly unique.
Calle Diez de Sollano 27, Centro, 37700 San Miguel de Allende, GTO., +52-415-152-3758, kiri.com.mx.
#### Ceramica Lopez
* * *
Colorful, hand-painted ceramics, including bowls, vases, dinnerware, and mugs, with a traditional-meets-contemporary feel.
Recreo 10, Centro, 37700 San Miguel de Allende, GTO., +52-415-154-4009.
#### Mixta
* * *
This beautiful 1700s courtyard building has several rooms full of eclectic clothing, housewares, accessories, furniture, and art, mostly by contemporary Mexican designers.
Pila Seca 3, Centro, 37700 San Miguel de Allende, GTO., +52-415-152-7343, mixtasanmiguel.com.
## TULUM
### MEXICO
#### Caravana
* * *
Jacopo Janniello Ravagnan's collection of artisan-made leather bags, choker necklaces, fringed shawls, ponchos, and more, with an earthy Mayan vibe and all housed in a giant tent.
5 miles along the main beach road in Tulum—Carretera Tulum Ruinas Boca Paila km 7.5, 77780 Tulum, Q.R., +52-985-856-0665, caravan.land.
CARAVANA
#### Josa
* * *
Cute caftans, kimonos, maxi dresses, and jumpsuits to take you from the beach to the dance floor, by New York transplant and photographer-turned-designer Joanne Salt and co-founder Ana Cabello.
5 miles along the main beach road in Tulum—Carretera Tulum Boca Paila km 7.5, 77780 Tulum, Q.R., +52-984-115-8441, josatulum.com.
#### KM33
* * *
Handmade luxury accessories produced in Mexico and South America, including colorful exotic leather backpacks, fringed clutches, chunky gold jewelry, straw hats, and more.
5 miles along the main beach road in Tulum—Carretera Tulum Boca Paila km 8.5, 77780 Tulum, Q.R., +52-55-4592-3594, km33tulum.com.
SHOPPING FOR THE FRIDA KAHLO LOOK IN MEXICO
* * *
Susana Martínez Vidal literally wrote the book on Frida Kahlo style. Based in Mexico City and Madrid, Vidal is a fashion and lifestyle journalist with more than twenty-five years of experience. Frida Kahlo inspired the first fashion editorial she produced as director of Elle Spain. Since then, she has passionately followed the artist's influence on fashion, music, and culture. After seeing the first-ever exhibition of Frida's clothing at La Casa Azul in 2012, Vidal was inspired to write Frida Kahlo: Fashion as the Art of Being. If you're looking for a more authentic take on Frida's look, one beyond the T-shirts, tote bags, and tchotchkes you can find on every corner in Mexico, these are Vidal's favorite places.
There are wonderful shawls in the store at the MUSEO DE ARTE POPULAR (MAP), at the CIUDADELA MARKET, and at the well-stocked EL BAZAAR SÁBADO in San Ángel.
Frida's jewelry, which was made from silver filigree and semiprecious stones, inspires the jewelry line SANGRE DE MI SANGRE, which has a store in Colonia Roma, and also ships all over the world.
There's an Argentine chain store, RAPSODIA, with several branches in Mexico and across South America that sometimes has Frida-inspired looks with embroidery and an ethnic air.
MUSEO DE ARTE POPULAR, Calle Revillagigedo 11, Centro, 06050 Ciudad de México, D.F., +52-55-5510-2201, map.cdmx.gob.mx.
LA CIUDADELA: MERCADO DE ARTESANÍAS, Av. Balderas Centro, 06040 Ciudad de México, D.F., +52-55-5510-1828, laciudadela.com.
EL BAZAAR SÁBADO, San Jacinto 11, San Ángel, 01000 Ciudad de México, D.F, +52-55-5616-0082, elbazaarsabado.com/mx.
SANGRE DE MI SANGRE, Edificio Balmori L-F, Orizaba 101, Roma Norte, 06700 Ciudad de México, D.F., +52-55-551-8599, sdemis.com.
RAPSODIA, Av. Tamaulipas 88, Condesa, Ciudad de México, D.F., +52-55-5211-6372, rapsodia.com, and additional locations.
ONLINE SHOPPING TIP
CooperativaShop.com is a terrific website founded by Araceli Graham, that brings top Latin American designers together (Carla Fernandez, Johanna Ortiz, Mercedes Salazar, Yakampot, and more) under one shoppable site that ships internationally.
Susana Martínez Vidal recommends Jalinedesign.com for "exquisite designs produced artisinally in southern Mexico." She also likes olgaprieto.es for "wonderful jewelry that illuminates any look, made by hand." And shoprarely.com is worth a look for "quality ready-to-wear sweatshirts and T-shirts."
For discovering new designers all over the world, Designer Carla Fernandez looks to NotJustALabel.com, a global e-commerce platform and organizer of a trade show called The Future Of Fashion, dedicated to sustainable design. Another site she likes is Yoox.com. "They showcase funky fashion that has folk spirit."
#### La Troupe
* * *
Breezy bohemian clothing, accessories, and home goods made locally by Mayan women, with lots of hand embroidery but also a contemporary look, thanks to soft, neutral colors.
5 miles along the main beach road in Tulum—Carretera Tulum Boca Paila km 7.5, 77780 Tulum, Q.R., +52-984-147-1178, latroupe.com.mx.
#### Mixik
* * *
Day of the Dead figurines, glassware, ceramics, scarves, and folk art abound at this gem for Mexican souvenirs. There is no street address, but the store is a block away from the bank in town, next to Charlie's restaurant.
Ave. Tulum, Centro, 77780 Tulum, Q.R.
#### Mr. Blackbird
* * *
Quartz necklaces, fringed and knotted leather jewelry, crochet bags, and sandals can be found at this tiny, sandy- floored boutique.
5 miles along the main beach road in Tulum—Carretera Tulum, Boca Paila km. 7.5, 77780 Tulum, Q.R., +52-948-114-3796
CARLA FERNANDEZ
SHOPPING FOR TRADITIONAL MEXICAN CLOTHING
* * *
Susana Carla Fernández is one of Mexico's best-known, and best-loved, fashion designers. Inspired by the architecture and textile traditions of her homeland, she launched her namesake fashion label in 2000, with the goal of working with indigenous communities around Mexico to preserve handweaving, dyeing, felting, and embroidery techniques.
For amazing folk clothes I like Oaxaca, a big city in the south of Mexico that is full of clothes woven or embroidered by hand and dyed with the most exquisite natural dyes. My favorite store in Oaxaca is TIENDA Q, where you can find new Mexican designs mixed with accessories from Toledo or traditional blouses dyed with indigo. Another store I love is LOS BAÚLES DE JUANA CATA, which is owned by Remigio Mestas Revilla, an advocate for the local textile industry.
I love traditional garments. I love to go to old bookstores and buy traditional garment patterns of the country I am visiting. If you are in downtown Mexico City, visit DONCELES STREET and you will see the kinds of bookstores I am talking about. I also like to go to local markets in little towns in Mexico. When you arrive in a new town, ask when the local market day is and don't miss it.
I love San Miguel de Allende, which is three hours from Mexico City. It has amazing Mexican design and delicious food. A new concept shopping center, DOCE18, recently opened in a historic San Miguel building. It has six restaurants and several little design boutiques. Another must is THE RESTAURANT, the place to eat in San Miguel.
TIENDA Q, Manuel Bravo 109, Centro, 68000 Oaxaca, OAX., +52-951-514-8855.
LOS BAÚLES JUANA CATA, Macedonio Alcala 403, Lazaro Cardenas 1ra Secc, 68140 Oaxaca, OAX., Mexico.
DÔCE 18, Calle Relox 18, Centro, 37700 San Miguel De Allende, GTO., +52-415-152-0215, doce-18.com.
THE RESTAURANT, Calle Sollano 16, Centro, 37700 San Miguel de Allende, GTO., +52-415-154-7877, therestaurantsanmiguel.com.
## RIO DE JANEIRO
### BRAZIL
#### Andrea Marques
* * *
Up-and-coming designer distinguished by her striking use of color and prints, and her classic yet sensual silhouettes, including printed bodysuits and midi skirts reminiscent of Diane von Furstenberg classics.
Rua Garcia d'Avil, 149, Sobreloja 201, Ipanema, Rio de Janeiro-RJ, 22421-010, +55-21-3202-2700.
#### Animale
* * *
One of the largest brands in Brazil, with more than eighty freestanding boutiques, the Rio-based Animale caters to professional women, with sexy, draped dresses, blouses, and pants that show off Brazilian bods, plus biker jackets, velvet slip dresses, and edgy accessories.
Av. Rio Branco 128, Loja B, Centro, Rio de Janeiro-RJ, 20040-002, +55-21-2224-4387, animale.com.br, and additional locations.
#### Antonio Bernardo
* * *
Rio-based fine jeweler whose designs are organic, sculptural, and glam (his gold folded-fan-shaped Scandal earrings are perfect for the disco). His flagship has a gallery next door that features contemporary art.
Rua Garcia d'Avila 121, Ipanema, Rio de Janeiro-RJ, 22421-010, +55-21-2512-7204, antoniobernardo.com.br, and additional locations.
#### Blue Man
* * *
This iconic Brazilian swimwear company has been in business since the 1970s, first becoming famous for its denim bikini, and later, for running afoul of the Roman Catholic Church when it produced a bikini with an image of Jesus on the seat. Today, it remains a fashion favorite, selling a range of colorful, itty-bitty women's styles as well as one-pieces, men's sungas (the Brazilian mankini), kids' bathing suits, and resort and workout wear.
Rua Visconde de Pirajá 351, Ipanema, Rio de Janeiro-RJ, 22410-003, +52-21-2247-4905, blueman.com.br, and additional locations.
#### Dona Coisa
* * *
This concept store (Brazil's colette) serves up the best of Brazilian and international designers, as well as beauty and lifestyle items. A must-stop for serious fashion fans.
Rua Lopes Quintas 153, Jardim Botânico, Rio de Janeiro-RJ, 22460-020, +55-21-2249-2336, donacoisa.com.br.
HELENA BORDON MAKES
A CASE FOR THE MALL
* * *
Who says malls are dead? Not Helena Bordon, the popular Brazilian entrepreneur, model, and fashion blogger based in São Paulo. Bordon began her fashion training earlier than most thanks to her mother, Donata Meirellescurrent style director of Vogue Brazil. When Bordon was seven, she started accompanying her mom on trips to Europe, attending Chanel shows and meeting designers like Valentino. Bordon spent her formative years learning buying tips and engaging with fashion teams and retailers. She cofounded Brazilian fast fashion chain 284 with childhood friends. An avid follower of fashion, Bordon regularly updates her website (helenabordon.com) with style, travel, and beauty tips. You can see her in ad campaigns for Cîroc Vodka, L'Oréal Professionnel, Tod's, and Salvatore Ferragamo.
In Brazil we tend to shop more in malls than in street shops. My two favorites are SHOPPING IGUATEMI and SHOPPING CIDADE JARDIM. They both carry all the international brands and the local must-have ones. The Cidade Jardim is very beautiful and open with lots of trees. It's very nice to go there just to walk around and have lunch.
At the Iguatemi, they recently opened MANI, which is my favorite restaurant in São Paulo. The food and the caipirinha (Brazil's traditional drink) are amazing. Ask for the Mani caipirinha—a must. Both of the malls have hair salons and a gym, so you can basically spend the entire day there.
SHOPPING IGUATEMI SAN PAOLO, Av. Brigadeiro Faria Lima 2232, Jardim Paulistano, São Paulo-SP, 01489-900, +55-11-3048-7344, iguatemi.com.br.
SHOPPING CIDADE JARDIM, Av. Magalhães de Castro 12000, Cidade Jardim, São Paulo-SP, 05502-001, +55-11-3552-3560, shoppingcidadejardim.com.
#### Farm
* * *
Dressing the fashionable girls of Rio since the late 1990s, Farm started as a stall at a flea market and has since grown into a lifestyle brand with festive, tropical prints as its signature.
Rua Visconde de Pirajá 365, Ipanema, Rio de Janeiro-RJ, 22410-003, +52-21-99834-4486, farmrio.com.br.
#### Fiera do Rio Antigo
* * *
Famous for its antiques, this market is "a local gem," says swimwear designer Lenny Niemeyer. There is also a selection of arts and crafts stalls. Open the first Saturday of the month, from 10 a.m. to 7 p.m.
Rua do Lavradio, Lapa, Rio de Janeiro-RJ, 20230-014.
#### Francesca Romana Diana
* * *
Colorful and chic costume jewelry featuring quartz, amethysts, citrines, and other stones, inspired by Rio's bossa lifestyle.
Rua Visconde de Pirajá 547, Ipanema, Rio de Janeiro-RJ, 22410-003, +55-21-2274-8511, francescaromanadiana.com.br.
#### H.Stern
* * *
Founded in 1945, Rio's very own luxury jewelry brand captures the spirit of Brazilian modernism in fine metals and stones. It's stayed relevant by collaborating on collections with international style-setters, including Diane von Furstenberg and architect Oscar Niemeyer. There are more than 160 H. Stern stores worldwide, but the flagship is in Ipanema, where you can take a self-guided workshop tour.
Rua Garcia d'Avila 113, Primeiro andar, Ipanema, Rio de Janeiro-RJ, 22410-002, +55-21-3204-1334, hstern.net, and additional locations.
#### Isabela Capeto
* * *
One of Brazil's buzziest contemporary fashion designers, Capeto is known for her hand-dyed, embroidered, boho dresses and skirts (think Isabel Marant, Brazilian-style), as well as kids' clothes and home accessories.
Rua Alberto Ribeiro 17, Horto, Rio de Janeiro-RJ, 22460-250, +55-21-2537-3331, isabelacapeto.com.br, and an additional location in São Paulo.
#### Lenny Niemeyer
* * *
With flattering cuts, sensual draping, and bold, nature-inspired prints, Niemeyer's swimwear and beachwear reflect her past experience as a landscape architect. (She's the niece of architect Oscar Niemeyer.)
Rua Garcia d'Avila 149, Loja A, Ipanema, Rio de Janeiro-RJ, 22421-010, +55-21-2227-5527, lennyniemeyer.com, and additional locations.
LENNY NIEMEYER
#### Rosa Chá
* * *
Another Brazilian brand elevating swimwear to high fashion is Rosa Chá. "I love seeing all the swimwear designs in Rio," says fashion tech entrepreneur Alexandra Wilkis Wilson. "I also love that the women there are confident no matter their shape."
Av. das Américas 7777, Barra da Tijuca, Rio de Janeiro-RJ, 22793-081, +55-21-4062-7953, rosacha.com.br.
#### Sobral
* * *
Carlos Sobral started his business in Rio in the 1960s, after coming across the material resin at the Hippie Fair in Ipanema. His Brazilian-made resin jewelry, including chunky necklaces and rings, makes colorful souvenirs.
Fórum de Ipanema, Rua Viscconde de Pirajá 351, Ipanema, Rio de Janeiro-RJ, 22410-003, +55-21-2267-0009, sobraldesign.com.b.
## SÃO PAULO
### BRAZIL
#### Adriana Degreas
* * *
With couture-like construction, luxury fabrics, and cheeky designs (such as a bikini bottom with trompe l'oeil lips), Degreas's swimwear and beachwear is stylish enough to wear to the club. No wonder that Alessandra Ambrosio, Poppy Delevingne, and Erica Pelosini are fans.
Shopping Iguatemi, Av. Brigadeiro Faria Lima 2232, Jardim Paulistano, São Paulo-SP, 01489-900, +55-11-3045-5646, adrianadegreas.com.br.
#### Herchcovitch Alexandre
* * *
One of Brazil's first designers to become internationally known in the 1990s, Herchcovitch has shown his pretty-punk collections in New York and Paris as well as Brazil.
Rua Melo Alves 561, Jardim Paulista, São Paulo-SP, 01417-010, +55-11-3063-2888, loja.herchcovitch.com.br.
#### Amir Slama
* * *
A history professor-turned-swimwear designer who was the designer/owner of beach brand Rosa Chá before launching under his own name, Slama now counts Naomi Campbell and many other stylish girls as devotees. "I can only buy my swimsuits from him now—and I also bring them back as gifts," says L.A. retailer Desiree Kohan. "He executes the perfect string bikini for all shapes and sizes as well as bodysuits that rival Wolford. I have a collection of minimal architectural suits as well as non-structured string bikinis. I'm really into great fabrics, especially for swimsuits and he does it best."
Rua Oscar Freire 977, Jardim Paulista, São Paulo-SP, 01426-001, Brasil, amirslama.br.
#### Cartel 011
* * *
"One of the coolest stores in the city," says stylist/designer Maryam Malakpour. "Right in the entrance there is an art gallery, in the back you will find a great restaurant, and they have a good selection of modern clothing in the store. A great mix of art, design, fashion, and gastronomy."
Rua Artur de Azevedo, 517 - Pinheiros, São Paulo-SP, 05404-001, +55-11-3081-4171, www.cartel011.com.br.
#### Casa Juisi
* * *
"Located in an old mansion in São Paulo's historical downtown that was converted into a boutique vintage retailer with an offering that is carefully curated by its eagle-eyed creators," says Helena Bordon of this off-the-beaten-path place, where she scored her best find ever, a vintage Lanvin coat.
Rua Roberto Símonsen, 108 - Sé, São Paulo-SP, 01017-020, +55-11-3063-5766.
UXUACASA
#### Fernanda Yamamoto
* * *
This Brazilian-born fashion designer draws on the local landscape and her Japanese ancestry to create conceptual, multitextured pieces in minimalist, architectural silhouettes. A bit Comme des Garçons–like.
Rua Aspicuelta 441, Pinheiros, São Paulo-SP, 05433-011, +55-11-3032-7979, fernandayamamoto.com.br.
FERNANDA YAMAMOTO
#### Gig Couture
* * *
Founded in 2002 by Gina Guerra and Patricia Schettino, Gig Couture has risen to success on its vividly patterned knitwear, created with jacquards and special yarns in Belo Horizonte. Beyoncé gave the label her stamp of approval when she chose a Gig Couture look from the spring/summer 2016 collection for her film Lemonade.
Rua Peixoto Gomide 1789, Jardim Paulista, São Paulo-SP, 01409-003, +55-11-2386-1500, gigcouture.com.
#### Havaianas
* * *
Situated on the high-end Rua Oscar Freire is the flagship for Brazil's native shoe. Behind the minimalist facade is an impressive space with greenery to accent the shoes, plus rotating art installations. You can also have your own designs made.
Rua Oscar Freire 1116, Cerqueira César, São Paulo-SP, 01426-000, +55-11-3079-3415, havaianas.com.br, and additional locations.
ANA KOZAK
THE BEST OF BOSSA DESIGN
* * *
Ana Kozak is so enthusiastic about Brazilian design, she brought it to Los Angeles when she relocated from São Paulo, opening Le Magazyn in Venice Beach in 2016. When she returns to Brazil twice a year to scout new designers and artisans for her store, these are the places she does not miss.
For swimsuits and cool bikinis, I go to LENNY NIEMEYER. Lenny is a "Carioca" brand, which means, "born in Rio." She has several stores in many Brazilian cities. In São Paulo, ADRIANA DEGREAS has the best swimsuits and the coolest lounge outfits. For amazing sandals you can't go wrong with PAULA FERBER. They are exquisite sandals and original creations; I'm very proud to exclusively represent her in the US at Le Magazyn.
For casual clothing with Brazilian DNA, I would recommend A. NIEMEYER in São Paulo. ADRIANA BARRA has the best colorful print long dresses in exotic patterns with tropical flowers and birds and I love Rio label OSKLEN for fashionable sneakers. Also in Rio is DONA COISA, a store that mixes fashion and home decor, and has a cafe inside the store. It is located in the Jardim Botanico neighborhood—a destination worth taking some time to explore.
A nice lifestyle store in São Paulo is AMOREIRA, where you can have a coffee and a piece of cake while shopping. I usually spend my Sunday mornings in São Paulo at the MUSEU BRASILEIRO DA ESCULTURA where they have a market. While mostly focused on antiques, you can also find some vintage clothing. Once there, take the opportunity to have lunch at CHEZ MIS restaurant, which is one block away from MuBE.
C&A is an international Dutch retail (like Topshop and H&M) that cannot be classified as chic but is definitely cheap and cool! In Brazil they often collaborate with local designers like Alexandre Herchcovitch, Lenny Niemeyer, and others. The collaborations are really good and things fly from the store.
One of my favorite vacation spots in Brazil is Trancoso, in Bahia; my family meets there once a year and I really try to not skip a single spring (the best season). Trancoso is a small little village full of charm and a great vibe. I love the woodwork of MARCENARIA TRANCOSO (I carry at Le Magazyn) and I always visit UXUACASA HOME. They are both located in the "quadrado" (central square).
LENNY NIEMEYER, Av. Ataulfo de Paiva, 270, 2F, Leblon, Rio de Janeiro – RJ, 22440-033, +55-21-3114-8887, lennyniemeyer.com, and additional locations.
ADRIANA DEGREAS, Rua Haddock Lobo, 1151, Cerqueira César, São Paulo - SP, 01414-002, +55-11-3331 1113, adrianadegreas.com.br, and additional locations.
PAULA FERBER, Shopping Iguatemi São Paulo, Av. Brigadeiro Faria Lima, 2232, Jardim Paulistano, São Paulo - SP, 01451-000, +55-11-3811 9810, paulaferber.com.
A. NIEMEYER, Shopping Iguatemi, Av. Brg. Faria Lima, 2232, Jardim Europa, São Paulo - SP, 01489-900, +55-11-3034-4411, aniemeyer.com.br, and additional locations.
ADRIANA BARRA, Alameda Franca, 1243, Jardim Paulista, São Paulo - SP, 01422-001, +55-11-2925-2300, adrianabarra.com.br, and an additional location in Leblon.
OSKLEN, Rua Maria Quitéria, 85, Ipanema, Rio de Janeiro - RJ, 22410-040, +55-21-2227-2911 osklen.com, and additional locations.
AMOREIRA, Rua dos Macunis, 510, Vila Madalena, São Paulo - SP, 05444-001, +55-11-3032-5346, amoreira.com.
DONA COISA, Rua Lopes Quintas, 153, Jardim Botânico, Rio De Janeiro-RJ, 22460-020, +55-21-2246-2336.
MUBE MUSEUM, Rua Alemanha, 221 - Jardim Europa, São Paulo - SP, 01448-010, +55-11-2594-2601, mube.com.br).
CHEZ MIS, Av. Europa, 158, Jardim Europa, São Paulo - SP, 01449-000, +55-11-3467-3441, chezmis.com.br.
C&A, for locations and online store see c-and-a.com.
MARCENARIA TRANCOSO, Square St. John, 12, Trancoso, Porto Seguro - BA, 45818-000, +55-73-3668-1179, mtrancoso.com, and additional locations.
UXUACASA, Porto Seguro - BA, 45810-000, Brazil, +55-73-3668-2277, uxua.com.
#### Lilly Sarti
* * *
Cool boho pieces with a sophisticated-city-girl twist. Think airy silk chiffon dresses, blouses, and asymmetrical skirts, and relaxed cotton jumpsuits, all in neutral tones.
Rua Peixoto Gomide 1749, Jardins, São Paulo-SP, 01409-003, +55-11-3083-4509, lillysarti.com.br.
#### Lolitta
* * *
Up-and-coming talent Lolita Zurita Hannud is making a mark with her sexy, intricately knit designs reminiscent of Herve Leger and Missoni.
Shopping Iguatemi, Av. Brigadeiro Faria Lima 2232, Piso Superior, São Paulo, 55-11-3034-0610, lolitta.com.br.
#### Melissa
* * *
This is the impressive flagship for Brazil's stylish plastic and rubber footwear brand, which has collaborated with a host of designers on exclusive designs, including Vivienne Westwood, Jason Wu, and Jeremy Scott. The store also features a series of rotating installations by local artists.
Rua Oscar Freire 827, Cerqueira César, São Paulo-SP, 01426-003, +55-11-3083-3612, melissa.com.br, and additional locations.
#### NK Store
* * *
Founded by Natalie Klein in 1997 in São Paulo, this concept store boasts a well-curated selection of designers. "One-stop shop for a great selection of brands including Isabel Marant, Stella McCartney, Balmain, and their own brand," says Brazilian style blogger Helena Bordon. Klein has two in-house labels: the NK collection and the diffusion line TalieNK.
Rua Sarandi 34, Jardim America, São Paulo-SP, 01414-010, +55-11-3897-2600, nkstore.com.br.
#### Osklen
* * *
Designer, environmental activist, and United Nations Goodwill Ambassador Oskar Metsavaht balances sustainability and luxury in his stylish collection, one of Brazilian fashion's biggest success stories with more than 50 stores worldwide. His designs are graphic and earthy with a slight edge, often using raffia and raw linen contrasted with colorful tropical prints.
Rua Oscar Freire 645, Cerqueira César, São Paulo-SP, 01412-100, +55-11-3083-7977, osklen.com, and additional locations.
#### Riachuelo
* * *
Owned by Brazil's largest fashion group, Riachuelo is the Brazilian H&M, with more than 200 stores. The fast-fashion giant has collaborated on collections with well-known Brazilian designers such as Oskar Metsavaht and Pedro Lourenço, as well as international names Versace and Karl Lagerfeld. This is the brand's concept store.
Rua Oscar Freire 777, Jardim Paulista, São Paulo-SP, 01426-003, +55-11-2739-1960, riachuelo.com.br, and additional locations.
#### Schutz
* * *
Designer Alexandre Birman's affordable Brazilian footwear chain has gone global with its sexy, boho sandals, wedges, and boots that feature fringe, woven accents, pom-poms, and beads.
Rua Oscar Freire 944, Jardim Paulista, São Paulo-SP, 01426-000, +55-11-4508-1409, schutz.com.br, and additional locations.
## BUENOS AIRES
### ARGENTINA
#### Arandú
* * *
This three-story town house brands itself as a talabartería, or equestrian store, but in addition to beautiful burnished-leather riding boots and saddles, it also sells fabulous leather goods, silver jewelry, needlepoint belts, and home accessories, as well as alpargatas, a traditional Argentine shoe, in every color and pattern. "Best selection for a weekend in an Argentine estancia," says model/blogger Sofia Sanchez de Betak. "Great hats and handwoven leather accessories," says Ulla Johnson.
Ayacucho 1924, C111211J Buenos Aires, +54-11-4800-1575, arandu.com.ar.
#### Ayres
* * *
Trendy chain with eclectic prints and casual pieces (bomber jackets, moto zip jeans, sharp-tailored jackets), most priced for less than $300. Look for the store in the Alto Palermo mall, along with other interesting Argentine brands such as Maria Cher and A.Y. Not Dead.
Alto Palermo Shopping Center, Primero piso, Av. Santa Fe 3253, C1425BGH Buenos Aires, +54-11-5777-8228, ayres.com.ar.
#### Benito Fernandez
* * *
Crazy colors and prints and unusual combinations of textures characterize the whimsical looks of local designer Benito Fernandez. You'll find geometric-print blazers, fringed sweaters, patchwork-print button-down shirts, and even fun socks.
El Salvador 4666, C1414BPJ Buenos Aires, +54-11-4833-0303, benitofernandez.com.ar.
#### Casa Fagliano
* * *
In this little shop in Hurlingham, right outside of Buenos Aires, the Fagliano family handcrafts the best riding boots in the world. Among their exclusive clientele are top polo players, the king of Spain, Prince Charles, and the sultan of Brunei. A bespoke pair of boots starts at $2,600. The workshop opened in 1892 and is still family-run.
Tambo Nuevo 1449, B1686EOU Hurlingham, Buenos Aires, +54-11-4665-0128, fagliano.com.ar.
#### Celedonio
* * *
Local designer Celedonio Lohidoy's nature-inspired jewelry makes a statement, particularly his massive necklaces constructed of giant amethyst floral bouquets, or faux pearls, crystal and glass beads, and butterflies.
Av. del Libertador 1774, C1425AAQ Buenos Aires, +54-11-4803-7598, celedonio.com.ar.
#### Comme Il Faut
* * *
This is the place to go for a pair of handmade tango shoes stylish enough for a night on the town or on the dance floor. Styles come in lizard, suede, or lace, with heart embellishments or sequin fringe on the heels. The store, tucked away in a courtyard, also carries slingbacks and pumps.
Arenales 1239, Cdad. Autónoma de Buenos Aires, +54-11-4815-5690, commeilfaut.com.ar.
COMME IL FAUT
#### EDITOR Market
* * *
Buenos Aires's hip, new lifestyle hub, created by Gabriel Brener, the man behind Maria Cher and A.Y. Not Dead. The eight-floor building, inspired by New York City's Dover Street Market and Paris's merci, features a carefully curated selection of Argentine design, including lighting by Federico Churba, Monochrome bicycles, Fueguia perfumes and candles, fashion, jewelry, and accessories. There's also a café with fresh juices and artfully decorated lattes.
Av. Corrientes 503, C1043AAF Buenos Aires, +54-11-5356-2575, editormarket.com.ar.
#### El Ateneo Grand Splendid
* * *
One of the oldest bookstores in Argentina, the building—a converted theater—is as much an attraction as the thousands of volumes inside.
Av. Santa Fe 1860, C1123AAA Buenos Aires, +54-11-4813-6052.
#### Elementos Argentinos
* * *
Design your own rug or choose from hundreds of colorful designs at this local gem, which specializes in handloomed llama wool rugs, sofa throws, table runners, colorful cushions, stools, and more.
Gurruchaga 1881, C1414DIK Buenos Aires, +54-11-4832-6899, also in Recoleto, elementosargentinos.com.ar.
#### Gil Antigüedades
* * *
High-end vintage clothing institution. The beaded dresses and jeweled minaudières will take you back to pre-revolution Argentina.
Humberto Primero 412, C1103ACJ Buenos Aires, +54-11-4361-5019, gilantiguedades.com.ar.
#### Jessica Kessel
* * *
This young designer's innovative color-block mules, ankle booties, and slingbacks are a real find.
Defensa 1009, C1065AAS Buenos Aires, +54-11-4362-2144, jkshoes.com.ar.
#### Kabinett
* * *
Expect collaborations with leading local and international designers and artists in this ever-changing concept boutique. You'll find colorful woven cuff bracelets, clutch bags, wooden sunglasses, hats, and whimsical home accessories.
Gurruchaga 1744, C1414DIJ Buenos Aires, +54-11-4833-7447, kabinett.us.
#### Lupe
* * *
Local designer Guadalupe Villar's selection in this Palermo shop is charming and feminine—think a cross between A.P.C. and rock 'n' roll. Melissa Magsaysay snagged her favorite fashion find of all time here: "Probably the most comfortable brown leather loafers ever. I wear them nonstop."
El Salvador 4666, C1414BPJ Buenos Aires, +54-11-4832-6743.
#### Mishka
* * *
This trendy shoe store chain has supercool platform loafers and chunky-heel ballet pumps, plus a rotating selection of clothing and denim, all at similar prices to fast fashion chain COS.
El Salvador 4673, C1414BPI Buenos Aires, +54-11-4833-6566, mishka.com.ar, and additional locations.
#### Feria de San Pedro San Telmo
* * *
Don't miss the Sunday antiques market in the oldest neighborhood in Buenos Aires, with hundreds of stalls of knickknacks and souvenirs, street performers, tango dancers, and more. Sofia Sanchez de Betak makes it a regular stop when she's in town.
Defensa 1098, C1103AAA Buenos Aires, +55-11-3181-5188, feriadesantelmo.com.
#### Rapsodia
* * *
This Argentine chain is boho paradise. You'll think you've entered bohemian babe Alessandra Ambrosio's closet when you see all the fab patchwork maxi skirts, embroidered velvet jackets, and fringed booties, at prices that won't break the bank.
Alto Palermo, Av. Santa Fe 3251, C1425 CABA, Buenos Aires, +54-11-5777-81111 rapsodia.com, and additional locations.
#### Tokonoma
* * *
Gallery owner Oli Martinez has curated a selection of handmade pottery, jewelry, glass, and tableware by local designers at this minimalist spot accessed by a metal bridge.
José Antonio Cabrera 5037, C1414BGQ Buenos Aires, +54-11-4831-8365.
#### Tramando
* * *
This luxury label, founded in Buenos Aires by textile designer and former photographer Martín Churba, is known for its experimental weaving, futuristic prints, and avant-garde silhouettes.
Rodríguez Peña 1973, C1021ABO Buenos Aires, +54-11-4811-0465, tramando.com.
#### Vasalissa
* * *
"The best chocolates and the chicest wrapping," says fashion consultant Amanda Ross. The red-and-white toile boxes are indeed gorgeous, and the tango-shoe-shaped confections a must.
Av. Callao 1940, C1024AAS Buenos Aires, +54-11-4806-4158, vasalissa.com.
#### Zapatos de Maria
* * *
Sophisticated handmade shoes in rich colors, including mesh ballet flats, loafers, ribbon-tie oxfords, and ankle-strap pumps.
Libertad 1655, C1016ABG Buenos Aires, +54-11-4815-5001, zapatosdemaria.com.
## LIMA
### PERU
#### Alessandra Petersen
* * *
The Peruvian designer, and local fashion industry advocate, is known for her beautifully edgy crochet work, including sweaters, dresses, and neckpieces. Her clothing has been featured in Vogue.
Calle Atahualpa 479, Miraflores 15074, Lima, Peru, +51-1-242-5378, alessandrapetersen.com.
#### Andrea Llosa
* * *
Drawing inspiration from her time living abroad in Spain and London and from her Peruvian heritage, this buzzy young designer creates cool contemporary clothing and silhouettes. Visit her showroom or look for her designs at Hanger.
Av. Mariscal La Mar 352 350 348, Miraflores 15074, Lima, Peru, +51-1-4400711, andreallosa.com.
#### Artesanías Las Pallas
* * *
"For incredible crafts," says designer Ulla Johnson of this Lima gem, which promotes the traditional art of the coast, highlands, and jungle by working directly with artisans and paying fair prices. It offers high-quality alpaca clothing, jewelry, retablos, carved gourds, masks, ceramics, and more. Owner Mari Solari displays it all in several rooms of her fine Barranco house.
Cajamarca 212, Barranco 15063, Lima, Peru, +51-1-477-4629.
#### Dédalo Arte
* * *
A contemporary art gallery and café in a restored colonial home that is also a store featuring gifts, accessories, and jewelry (check out the pieces by Hiro Yoshimoto) by local designers. If you don't make it to Barranco, there's another store in the Larcomar shopping mall.
Paseo Saenz Peña 265, Barranco 15063, Lima, Peru, +51-1-652-5400.
#### Galería Indigo
* * *
Great selection of high-end Peruvian art, pottery, and one-of-a-kind jewelry.
Av. El Bosque 260, San Isidro 15073, Lima, Peru, +51-1-441-2232, galeriaindigo.com.pe.
#### Jallpa Nina
* * *
Amazing pottery, from dinner sets to lamps, from all over Peru with a look more modern than traditional. Artisans here also make ceramics for Jonathan Adler.
Federico Villareal 290, Miraflores 15074, Lima, Peru, +51-1-430-1435, jallpaninaperu.com.
#### Kuna
* * *
Inspired by Incan textile heritage, this high-end alpaca and vicuña clothing brand has multiple stores in Lima and throughout Peru.
Av. Larco 671, Miraflores 15074, Lima, Peru, +51-1-447-1623, kuna.com.pe, and additional locations.
#### Museo Larco
* * *
This privately owned museum of pre-Columbian art has a beautiful garden-facing café and a boutique featuring Peruvian-designed clothing and gifts.
Av. Simón Bolivar 1515, Pueblo Libre 15084, Lima, Peru, +51-1-461-1312, museolarco.org.
#### Meche Correa
* * *
One of Peru's best-known designers uses native fabrics and techniques to create gorgeous embroidered shawls, chunky gilt necklaces, plastic handbags with see-through pockets full of religious charms and offerings, and horn jewelry. Visit her atelier by appointment, or look for her collection at Dedalo and Gallery Indigo.
Av. Aurelio Miroquesada 136, San Isidro 15073, Lima, Peru, +51-1-441-1441, mechecorrea.com.
#### The Hanger
* * *
Great selection of Instagram-worthy styles by young Peruvian designers, including colorful Capittana swimsuits, Karen Mitre fringe clutches, feminine striped dresses by Carolina Tola, streetwear-inspired denim and knits from Camote Soup, cool handmade wood platform shoes by Quimera, and more.
Calle Miguel Dasso 110, San Isidro 15073, Lima, Peru, +51-1-421-6462.
#### Verné
* * *
Hip art gallery, restaurant, and clothing store selling local designers.
Av. 269, Barranco 15063, Lima, Peru, +51-1-586-8677.
## CUSCO
### PERU
#### Mercado Central de San Pedro
* * *
"This large open-air market in Cusco is a great place to find all types of local Peruvian craftwork," says jewelry designer Monique Péan.
Cascaparo, Cusco 84.
## LONDON
### UNITED KINGDOM
ROKSANDA
#### THE MAJORS
#### Harrods
* * *
"Probably my favorite big-city store," jewelry designer Kara Ross says of this institution whose branded tote bags and teddy bears are beloved souvenirs. "It has everything you can imagine. People come from all over the world and it really has something for everyone, from very expensive luxury items to more affordable things. It is such a chic store, but very approachable—and don't get me started on the food court." Don't miss the fish-and-chips at Bentley's Sea Grill on the ground floor, some of the best in the city; the Georgian Restaurant is great for tea. The shoe department is pretty epic, as are the pet and toy departments.
87-135 Brompton Rd., London SW1X 7XL, +44 (0)20 3626-7020, harrods.com.
#### Harvey Nichols
* * *
"I love a spin through Harvey Nichols," says costume designer Jenn Rogien of this Barneys-like department store with a fifth-floor market (don't miss the Harvey Nichols–brand stem ginger biscuits), a champagne bar, a rooftop café, and a Yo! Sushi conveyer-belt sushi spot. "I bought a fantastic dress there when I was nominated for a Costume Designers Guild award a few years ago. Even on sale it involved some creative financing," Rogien says. "A friend and I kicked off that shopping day with tea and scones upstairs. When in London, right?"
109-125 Knightsbridge, London SW1X 7RJ, +44 (0)20 7235-5000, harveynichols.com.
#### Liberty London
* * *
This Regent Street department store, housed in several buildings (including an incredible Tudor Revival constructed from the timber of two ships), harks back to Britain's imperial past in the best possible way. Liberty opened in 1875 selling ornaments, fabric, and objets d'art from the East. Today, it features an eclectic selection of top designer fashion and accessories, plus furniture, home accessories, fabrics and trims. The store played a key role in shaping British style in the 1890s, by helping to popularize art nouveau and Arts and Crafts styles. These are still celebrated in the store brand's iconic Liberty silk scarves, button-down shirts, and notebooks, as well as in collaborative collections with Manolo Blahnik, Nike, Barbour, and Vans. "The haberdashery department is amazing," says Olympia Le-Tan. "I can also spend hours looking at their fabrics." "When I worked at Harper's Bazaar UK, the office was right around the corner from Liberty, perfect for lunch meetings—Cafe Liberty, on the second floor, is gorgeous!" says fashion editor Alison Edmond. "I loved that there was such a mix of clients, and I still aspire to grow into one of the store's signature chic older ladies, shopping in my '90s, still wearing black, still into fashion, with Chanel Rouge Noir lipstick and white hair."
Regent St., London W1B 5AH, +44 (0)20 7734-1234, libertylondon.com.
ONLINE SHOPPING TIP
"Hoodlondon.com presents the work of independent milliners from California to Colombia. I love even just 'window' shopping and imagining myself in all those incredible flights of crowning fancy," says fashion journalist/retailer Rose Apodaca.
#### Marks & Spencer
* * *
It may not be as fancy as Harrods or Selfridges, but it's beloved nonetheless for its no-frills fashion and food. (The underwear is a staple for most Brits, because it is so longwearing.) The clothing is basic and well priced, and now ships to the States!
458 Oxford St, London W1C 1AP, +44 (0)20 7935-7954, marksandspencer.com, and additional locations.
#### Selfridges
* * *
"There's not a lot you can't find within those walls," says shoe designer Sophia Webster. "My husband and I go there every Christmas Eve and do our holiday shopping in one big sweep." "I love a good stationery shop. The one inside Selfridges is pretty great and it's next to their chocolate selection, which is also dreamy," says tastemaker Liz Goldwyn. "Not only does the store proper have everything—everything—anyone could ever need, but the luxurious personal-shopping suites are divine," says designer Karen Erickson. "I can spend hours at the ground floor beauty workshop, which is essentially a playground of the newest trend brands and products," says fashion expert Tania Mohan."
400 Oxford St., London W1A 1AB, United Kingdom, +44 0(11) 3369-8040, selfridges.com, and additional locations.
### MAYFAIR/SOHO/COVENT GARDEN/PICCADILLY
#### Accessorize
* * *
A high-street mainstay since 1984, this tons-of-fun accessories chain owned by fashion retailer Monsoon has gone international with its irresistible mix of statement jewelry, hats, beach bags, embellished sandals, swimwear, iPhone covers, and more. Most items are less than $50.
55-59 Oxford St., London, W1C 2PX +44 (0)20 7287-0531, accessorize.com, and additional locations.
#### Alex Eagle
* * *
Located in a town house, this retail experience showcases "a great combo" of high-end fashion, accessories, art, design, houseware, beauty, and photography from around the globe, says accessories designer Lee Savage. Fashion industry vet Alex Eagle not only champions timeless pieces by Rosetta Getty, Blazé Milano, Lemaire, Vilshenko, Pallas, and Vita Kin but also produces an in-house line of luxury staples, as well as collaborations with specialists such as tailor New & Lingwood; customized Rolex and Cartier watches by MAD; jewelry by Fernando Jorge and Sophie Buhai; and hand-selected antique pieces by Catherine Noll, Hervé Van Der Straten, and Gucci.
6-10 Lexington St., London W1R OLB, +44 (0)20 7589-0588, alexeagle.co.uk
#### Alexander McQueen
* * *
As a provocateur, McQueen helped elevate British fashion to the international stage with his unconventional and sometimes macabre designs. He died in 2010, but his brand lives on under the direction of Sarah Burton, who has left her own indelible stamp on the style world, from designing Kate Middleton's royal wedding gown, to dressing former First Lady Michelle Obama. There are several McQueen stores in London, including the women's flagship on Bond Street, the men's store on Savile Row (McQueen the man got his start as an apprentice for a Savile Row tailor), and McQ, the less expensive contemporary collection.
4-5 Old Bond Street, London W1S PD, +44 (0)20 7355-0088, mcqueen.com, and additional locations.
#### Browns Fashion
* * *
Retail icon and Browns founder Joan Burstein is known for her early discovery of British designers John Galliano, Alexander McQueen, and Christopher Kane and for introducing labels such as Calvin Klein, Missoni, and Sonia Rykiel to the UK market. Open since the 1970s, the store is still a fashion mainstay in London for men's, women's, and bridal apparel, even though Burstein has since retired.
24-27 South Molton St., London. W1K 5RD, +44 (0)20 7514-0016, and an additional location on Sloane St., brownsfashion.com.
#### Burlington Arcade
* * *
Open since 1819, this beautiful covered shopping arcade that runs behind Bond Street, from Piccadilly to Burlington Gardens, is a true London experience, and one of the precursors to the modern shopping center. Tenants include N. Peal Cashmere, Penhaligon's, Manolo Blahnik, Church's, and Ladurée, the French patisserie.
51 Piccadilly, London W1J 0QJ, +44 (0)20 7493-1764, burlingtonarcade.com.
#### Charlotte Olympia
* * *
London-based shoe and bag designer Charlotte Dellal is known for such cheeky, feminine classics as her Kitty smoking slippers, Pandora clear plastic box clutches with fabric pouch inserts, and the iconic Dolly pumps in a rainbow of colors, with island-shaped exterior platforms.
56 Maddox St., London W1S 1AY, +44 (0)20 7499 0145, charlotteolympia.com, and additional locations.
#### Christopher Kane
* * *
Designed by British architect John Pawson, the London designer's first boutique is split over two floors, with womenswear on the ground floor and men's in the basement. The gallerylike space is sparsely furnished, allowing Kane's colorful clothing to shine on floating bars, and the handbags on floating shelves.
6 Mount St., London W1K 3EH, +44 20 7493 3111, christopherkane.com.
#### Coco de Mer
* * *
This tasteful erotica shop sells beautiful sex toys and apothecary items, and lingerie by Stella McCartney, Mimi Holliday, and more (including some risqué latex items from the in-house brand).
23 Monmouth St., London WC2H 9DD, +44 (0)20 7836-8882, cocodemer.com.
#### COS
* * *
The H&M-owned, minimalist-chic chain's first store is on Regent Street. A must for stylish shoppers, COS offers modern clothing with clean lines and utilitarian details. It's akin to Marni and Céline, but with most pieces priced at less than $300.
222 Regent St., London W1B 5BD, +44 (0)20 3139-2440, cosstores.com, and additional locations.
ONLINE SHOPPING TIPS FROM DESIGNER/PERSONAL SHOPPER RAVEN KAUFFMAN:
I really love the modaoperandi.com edit. I feel like I am introduced to a new designer every time I visit the site and I love getting first access with their trunk shows.
Gal Stern tights at etsy.com/shop/SternTights out of Tel Aviv: I own a few of her tights with metallic details and they are fabulous for dancing away the night at Giorgios. They look like something McQueen might do.
French consignment boutique valoisvintage-paris.com has an amazing Hermes selection. Like raiding your French best friend's closet.
The British version of theoutnet.com is the best secret spot for Alaïa shoes at an amazing discount. I'm an Alaïa junky and I stalk this site for amazing shoes in my size. I've been able to amass quite a collection.
#### Erdem
* * *
Known for exquisite fabrics and laces and moody floral prints, London designer Erdem Moralioglu established his collection in 2005 and quickly made fans of Keira Knightley, Michelle Dockery, and Emma Stone. His flagship sells the brand's luxury, ready-to-wear, and accessories collections, including exclusive scarves and leather pouches.
70 S Audley St., London W1K 2RA, United Kingdom, +44 (0)20 3653-0360, erdem.com.
#### Fortnum & Mason
* * *
"I carve out extra room in my luggage for a tin or two of tea from Fortnum and Mason," says costume designer Jenn Rogien. "I visited the store because a chef friend mentioned that it is the Queen's grocery store and I couldn't resist. Only on my first visit to all the floors did I find that it's also a fantastic department store complete with a millinery boutique."
181 Piccadilly, London, W1A 1ER, +44 (0)20 7734-8040, fortnumandmason.com.
#### Grenson
* * *
This British purveyor of traditional shoes for men and women has been at it since 1866, and now stocks classic styles, along with trendier wedge sole tassel loafers and derby boots and platform brogues.
40 Lamb's Conduit St., London WC1N 3LB, +44 (0)20 3689-2970, grenson.com.
#### Gray's Antiques
* * *
This labyrinthine Mayfair antiques center, which has more than 200 stalls, is a fave of Liz Goldwyn's for vintage costume jewelry and accessories. Interesting factoid: there's a lost tributary of the Tyburn river running underneath the Mews section of the building, which has been channeled into a water feature stocked with fish.
58 Davies St., London W1K 5AB, +44 (0)20 7629-7034, graysantiques.com.
#### Hussein Chalayan
* * *
One of fashion's biggest thinkers, Chalayan has designed remote-controlled dresses, collapsible coffee table skirts, and dresses covered in rows of fake fingernails. The designer's low-key, black-and-white flagship, designed by his long-term collaborator, architect Zoe Smith, is a blank canvas for all that. It features a tiled floor resembling a backgammon board, and a center table evocative of the hull of a boat.
2 Bourdon St., London W1K 3PA, +44 (0)20 7493-8157, chalayan.com.
#### Jimmy Choo
* * *
Designed by David Collins Studio, the 3-story flagship for this Brit brand of It shoes oozes luxury, from the raw crystal chandeliers to the bridal salon with fully stocked bar. The store showcases the brand's range, from glitter fabric pumps to sunglasses and fragrance.
27 New Bond Street, London, W1S2RH, +44 (0)20 07493-5858, jimmychoo.com, and additional locations.
#### Miller Harris
* * *
Lyn Harris launched her brand in 2000, which uses globally sourced ingredients, from Florentine iris to Tunisian orange flower. The result? Fragrances such as Rose en Noir (my favorite), Coeur d'Ete, and Feuilles de Tabac inspired by "Parisian elegance" and "London's eclectic style-driven streets," according to the brand. Harris has since moved on to create a new project, Perfumer H, but this brand remains a beloved classic.
21 Bruton St., London W1J 6QD, +44 (0)20 7629-7750, millerharris.com, and additional locations.
#### Mulberry
* * *
The British heritage leather goods brand was recently turned over to designer Johnny Coca, formerly of Céline, who has given the bags and ready-to-wear a newly streamlined, dare we say, French touch.
50 New Bond St., London W1S 1BJ, +44 (0)20 7491-3900, mulberry.com, and additional locations.
#### Nicholas Kirkwood
* * *
The award-winning shoe designer's London flagship features his architectural styles with dynamic, boomerang- shaped straps, mirrored triangular heels and pearl embellishments.
5 Mount St., London W1K 3NE, +44 20 7290 1404, nicholaskirkwood.com, and additional locations.
BRONWYN COSGRAVE
NEXT GEN DESIGNERS
* * *
A multidisciplinary fashion professional, Bronwyn's CV includes serving as features editor of British Vogue, penning six fashion history books, and curating the exhibition Designing 007, Fifty Years of Bond Style. She is currently a contributing editor at Vogue India.
I love Mount Street in London. A generation of designers whom I got to know when they were all just starting out have established boutiques in and around this luxury thoroughfare. MARC JACOBS was the first designer to open on Mount Street. Then came BALENCIAGA, CHRISTIAN LOUBOUTIN, and CÉLINE. But the very special aspect of this street is the boutiques that young British designers established. CHRISTOPHER KANE and ROKSANDA ILLINCIC commissioned the architects John Pawson and David Adjaye, respectively, to design retail spaces that are more like galleries than shops. Nearby on South Audley Street, ERDEM designed his own boutique with his partner, Philip Joseph.
MARC JACOBS, 24-25 Mount St., London W1K 2RR, +44 (0)20 7399-1690, marcjacobs.com.
BALENCIAGA, 12 Mount St., London W1K 2RD, +44 (0)20 7317-4400, balenciaga.com.
CHRISTIAN LOUBOUTIN, 17 Mount St., London W1K 2RD, +44 (0)20 843-227 4322, us.christianlouboutin.com.
CÉLINE, 103 Mount St., London W1K 2AP, +44 (0)20 7491-8200, celine.com.
CHRISTOPHER KANE, 6 Mount St., London W1K 3EH, +44 (0)20 7493-3111, christopherkane.com.
ROKSANDA ILLINCIC, 9 Mount St., London W1K 3NG, +44 (0)20 7613-6499, roksanda.com.
ERDEM, 70 S Audley St., London W1K 2RA, +44 (0)20 3653-0360, erdem.com.
#### Roksanda
* * *
Designer Roksanda Ilincic launched her collection at London Fashion Week in 2005, and has gained fame for her boldly colored, feminine, draped silhouettes that often feature cutouts and have been worn by the Duchess of Cambridge, Cate Blanchett, Daisy Ridley, Gwyneth Paltrow, and Keira Knightley. Ilincic's first store opened in 2014, in a space with a marble herringbone floor, rose gold rails, and concrete walls conceived by architect David Adjaye. "I never miss a chance to pop in to her exquisite boutique," says Refinery29's Christene Barberich. "It's a full sensory experience, for sure!"
9 Mount St., London W1K 3NG, +44 (0)20 7613-6499, roksanda.com.
#### Simone Rocha
* * *
Since her London Fashion Week debut in September 2010, the designer's star has risen quickly, thanks to her rebellious-yet-romantic take on femininity and fresh, craftsy approach to using fabrics and textures. In 2015, she opened her first store, on London's Mount Street, showcasing her ready-to-wear, bags, shoes, knitwear, and jewelry, next to artworks, including a glazed ceramic chandelier by Los Angeles artist Pae White and a set of three original lithographs by Francis Bacon from 1984.
93 Mount St., London W1K 2SY, +44 (0)20 7729-7004, simonerocha.com.
#### Sophia Webster
* * *
A former assistant to shoe designer Nicholas Kirkwood, Sophia Webster now has her own fabulous footwear line characterized by candylike colorways, graphic detailing, and whimsical embellishments. Her Mount Street store features the full upbeat collection of shoes and bags, plush lavender velvet seating, and a place for shoppers to take butterfly-wing selfies.
124 Mount St., London W1K 3NR, +44 (0)20 7729-70004, sophiawebstercom.
#### Stella McCartney
* * *
The daughter of Sir Paul McCartney, and a lifelong vegetarian, Stella McCartney has built a luxury brand from the ground up based on the principles of sustainability and "no leathers, feathers, or fur." A recent ad campaign, featuring that slogan graphically superimposed over photos of her collection in collaboration with the American artist Ed Ruscha. McCartney is a resource for sporty tailored pieces, printed T-shirts, Falaballa bags and those platform brogues that bloggers can't get enough of. She has stores around the world.
30 Bruton St., London W1J6QR,+44 (0)20 7518-3100, stellamccartney.com, and additional locations.
ONLINE SHOPPING TIP
Most big name London-based designers (Mulberry, Alexander McQueen, Mary Katrantzou, etc.) have their own international e-commerce businesses. If you want to discover more up-and-coming names, check out UK-based Netaporter.com, Matchesfashion.com, and Avenue32.com, all of which do a good job of introducing new talent, and which ship globally. Another good UK-based global e-comm site is WolfandBadger.com, for indie labels from London and beyond.
#### Topshop
* * *
This British retailer may be a fast-fashion juggernaut, with hundreds of stores around the globe and a collaboration with Beyoncé on her activewear collection, Ivy Park, but the Oxford Street store was the original. The 65,000-square-foot retail mecca has its own radio station and manicure bar, and receives multiple daily deliveries of on-trend merchandise to satisfy the throngs who come through the doors. "So much fun," says British TV presenter, blogger and model Louise Roe. "I make a beeline for it whenever I'm back in London. There's a DJ, a blowout bar, a candy bar, a café, and an unreal shoe section." "Can also be overwhelming just by its sheer size," says Charlotte Ronson. "Luckily there is a sweet shop inside to give you that much needed sugar rush." Topshop was early on to the fast-fashion–designer collaboration game, sponsoring London Fashion Week and partnering with emerging British designers as far back as the mid-1990s.
Oxford Circus, 214 Oxford St., London W1W 8LG, +44 (8)44 848-7487, topshop.com, and additional locations worldwide.
#### Victoria Beckham
* * *
Designed by female architect Farshid Moussavi, who also worked on the London 2012 Olympic Park, the conceptual space features the entire VB universe, including her sexy, sophisticated runway collection and the less expensive Victoria Victoria Beckham collection. And occasionally, the former Spice Girl has been known to drop by!
36 Dover St., London W1S 4NH, +44 (0)20 7042-0700, victoriabeckham.com.
VICTORIA BECKHAM
#### Vivienne Westwood
* * *
The one, the only, Dame Vivienne Westwood, queen of the punk past and present, who has dressed everyone from Naomi Campbell (remember when she tottered on the runway in those famously tall shoes?) to Pharrell Williams, whose Mountain hat caused a sensation a few years back. In the 1970s, Westwood and her then-husband Malcolm McLaren played a major role in the punk movement in London, believing they could wage a social revolution through fashion and rock 'n' roll from their Kings Road shop, which was alternatively named "Let It Rock" and "Too Fast to Live, Too Young to Die." In 1974 they changed the store's name to Sex to give a symbolic middle finger to the establishment that prosecuted them under British obscenity laws for making shirts with pornographic images. And the rest is history. She has stores all over the world, but shopping for her twisted tartans and T-shirts in London feels right.
44 Conduit St., London W1S 2YL, +44 (0)20 7439-1109, viviennewestwood.com, and additional locations.
#### Wolf and Badger
* * *
If you're interested in discovering something beyond London's well-known fashion brands, this is the place to find indie fashion, jewelry, and home accessories design from around the world, and at reasonable prices. Launched in 2009, the company has two brick-and-mortar stores, in Mayfair and Notting Hill, and an e-commerce site. It carries more than 600 independent labels, including Rixo, Lautem, and Latelita London. More than 75 percent of each purchase goes back to the designer, and the owners also nurture the brands they stock with business advice and product development. There's also a juice bar by Raw Press.
32 Dover St., London W1S 4NE, +44 (0)20 3627-3191, wolfandbadger.com, and an additional location in Notting Hill.
### MARYLEBONE
#### Alfie's Antique Market
* * *
One of the largest indoor antiques markets in the country (35,000 square feet over five floors), this spot is a must for interior decorators and fashion enthusiasts. Liz Goldwyn shops Alfie's for costume accessories and jewelry. There's also a rooftop restaurant.
13-25 Church St., London NW8 8DT, +44 (0)20 7723-6066, alfiesantiques.com.
#### Mouki Mou
* * *
"Feels like you've stepped back in time—in a good way—with their old-world service and curated offering of clothes, jewelry, and beauty," says Tere Artigas of this store, which features indigo clothing by 45rpm, sundresses by Rachel Comey, sandals by Ancient Greek Sandals, jewelry by Pippa Small, and more.
29 Chiltern St., London W1U 7PL, +44 (0)20 7224-4010, moukimou.com.
#### Perfumer H
* * *
Lyn Harris, founder of Miller Harris, launched this fragrance concept space in late 2015, offering bespoke and seasonal fragrances. You can create a custom fragrance with her from scratch (a collaborative process that can take up to six months), buy a one-of-a-kind Laboratory Edition (after the customer takes the bottle, the formula is reserved as exclusively his or hers), or choose one of her ready-to-wear seasonal fragrances. The midcentury spot offers a warm welcome—and serves tea.
106a Crawford St., London W1H 2HZ, +44 (0)20 7258-7859, perfumerh.com.
#### William Vintage Shop
* * *
"One of my favorite, most unforgettable shopping experiences was when I met William Banks-Blaney and visited his extraordinary store," says Rachel Zoe. "This was my first time really 'shopping' in over a year—my longest time without a purchase since my son, Skyler, was born. I ended up spending six hours in the store—Skyler got comfortable and took a nap! I found some unbelievable Christian Dior haute couture pieces and a Jean Patou vintage gown that I wore on the cover of my second book, Living in Style. It was hands down the best shopping day I have ever had."
2 Marylebone St., London W1G 8JQ, +44 (0)20 7487-4322, williamvintage.com.
### KNIGHTSBRIDGE/SLOANE STREET/BELGRAVIA
#### Anya Hindmarch
* * *
Handbag designer darling Anya Hindmarch founded her business in London in 1987 at age nineteen. Over the years, her hits have included beaded evening bags designed to look like British candy packets, and clutches modeled after Walkers potato chip bags. In 2001, Hindmarch launched the "Be a Bag" concept, which allowed shoppers to turn a favorite photograph into a bag. In 2007 her $15 canvas I AM NOT A PLASTIC BAG totes raised awareness about plastic bag waste and caused a worldwide frenzy. In recent years, in addition to bags, her sticker business has been booming.
157-158 Sloane St., London SW1X 9AB, +44 (0)20 7730-0961, anyahindmarch.com, and additional locations.
#### egg
* * *
Hidden in a mews house on a noncommercial street, egg sells luxurious, casual basics. "I live for their supersoft cashmere, which I can never get enough of," says fashion industry consultant Fern Mallis.
36 Kinnerton St., London SW1X 8ES, +44 (0)20 7235-9315, eggtrading.com.
#### Emilia Wickstead
* * *
A favorite of the Duchess of Cambridge, the British designer is known for her classy, feminine, wearable pieces in pretty pastels and florals that make high-profile dressing a breeze. Spanning two floors and 2,000 square feet, her Sloane Street store offers the complete ready-to-wear collection, while maintaining the services that launched her career: made-to-order, made-to-measure, and bespoke options.
162A Sloane St., London SW1X 9BS, +44 (0)20 7235-1104, emiliawickstead.com.
SOPHIA WEBSTER
EAST LONDON AND CAMDEN TOWN GEMS
* * *
Sophia Webster debuted her quirky, fun shoe collection for spring 2013 after working as design assistant to Nicholas Kirkwood. In just a few years, she's built a following that includes Katy Perry, Rita Ora, and Elizabeth Banks. Spring 2014 saw the launch of her iconic Butterfly collection, which included the signature Chiara sandals. Her collections have expanded to include a wide range of styles, from flats to espadrilles, and the classic Coco pump showcasing the flamingo detail at the back of the heel, an image now synonymous with the brand. She opened her first London store in 2016.
As a glasses wearer I am always on the lookout for unique frames, and in London there is no place better than GENERAL EYEWEAR in Stables Market, Camden. They have an incredible selection of one-of-a-kind vintage frames; they handle your prescription as well, so it's all really good value for something so individual. I love buying art from PRINT CLUB LONDON in Dalston, East London. They have affordable contemporary prints in limited runs. I have lots of their prints in my house, ranging from pink pineapples to huge toucans to my favorite: a multicolored Beyoncé print, by illustrator Hattie Stewart.
BOROUGH MARKET in South East London is amazing for fresh, good-quality food, and BROADWAY MARKET in Hackney, East London, has so many cute cafés and shops selling original clothing and arts and crafts as well as food stalls offering authentic street food.
GENERAL EYEWEAR, The Stables Market, Arch 67, Chalk Farm Rd., London NW1 8AH, +44 (0)20 7428-0123, generaleyewear.com.
PRINT CLUB LONDON, 10-28 Millers Av., Unit 3, London E8 2DS, +44 (0)20 7254-9028, printclublondon.com.
BOROUGH MARKET, 8 Southwark St., London SE1 1TL, +44 (0)20 7407-10002, boroughmarket.org.uk.
BROADWAY MARKET, Broadway Market—pedestrian route between London Fields Park and Regent's Canal, London E8 4QJ, broadwaymarket.co.uk.
#### Feathers
* * *
An icon of Swinging '60s London, Feathers was founded in 1968 by Jean and Willie Burstein, brother of Sidney Burstein, who founded luxury boutique Browns a year later with his wife, Joan. The Knightsbridge store, where Manolo Blahnik worked as a young lad, is still going strong with more than ninety designer collections, including Moschino, Junya Watanabe, Sacai, and Etro. "Supercool boutique right across the street from Harrods," says the Gem Palace's Shalini Kasliwal.
42 Hans Crescent, London SW1X 0LZ, +44 (0)20 7589-5802, feathersfashion.com.
#### Philip Treacy
* * *
Irish hat designer Philip Treacy has elevated millinery to an art form. Championed from a young age by the late fashion editor Isabella Blow, Treacy opened his shop on Elizabeth Street in 1994, next door to Blow's old house, where Treacy's business first began in the basement. His famously iconic pieces worn by the likes of Kate Middleton and Grace Jones include hats fashioned into boats and lobsters. There are handmade couture items, and more affordable floral headbands, too.
69 Elizabeth St., London SW1 W9PJ, +44 (0)20 7730-3992, philiptreacy.co.uk.
#### Victoria & Albert Museum store
* * *
The V&A might be the best museum bookstore in the world, with must-have fashion, art, and design books, killer jewelry, scarves, and other accessories, plus cool cards, wrapping paper, and more. Give yourself plenty of time. Costume designer Arianne Phillips counts it among her favorites, and so do I.
Cromwell Rd., London SW7 2RL, +44 (0)20 7942-2000, vam.ac.uk.
### KINGS ROAD
#### L.K. Bennett
* * *
Founded in London in 1990, the brand rose to international fame in 2011, when it was revealed to be Kate Middleton's go-to source for sensible kitten heels. Known for classic, feminine shoes, $300-to-$500, and a range of clothing to match, L.K. Bennett has stores across Europe, the US, and in the Middle East.
164-166 Kings Rd., London SW3 4UR, +44 (0)20 7351-9659, lkbennett.com, and additional locations.
#### Manolo Blahnik
* * *
With a career spanning forty years, Blahnik was a luxury shoe designer before it was a thing. His designs have been worn by every celebrity under the sun (and earned him the honor of having his shoes referred to simply as Manolos), and he even collaborated with Rihanna on a recent capsule collection. Born in the Canary Islands, Blahnik designed sets before shoes, and was "discovered" by iconic Vogue editor Diane Vreeland in 1970. He moved to London and his breakthrough came in 1971, when he was asked by the flamboyant British fashion designer Ossie Clark to create shoes for his runway show. Blahnik made a high-heeled sandal that entwined the ankles with straps of ivy and cherries, and he hasn't stopped creating since.
49-51 Old Church St., London SW3 5BS, +44 (0)20 7351-5822, manoloblahnik.com, and additional locations.
RICCARDO TORTATO
BESPOKE LONDON AND BEYOND
* * *
Riccardo Tortato was born in Venice and lives in New York, but he takes an airplane every ten days to discover the best fashion trends around the world. Men Fashion Director and Online Fashion Director at Tsum Moscow, he is a style expert and best friend and business partner of Anna Dello Russo. He is also famous for his long gray beard and hardly ever wearing socks . . . even during the winter in Moscow.
For cashmere sweaters I go to TURNBALL AND ASSER, one of the most historic English bespoke shirtmakers—it serves Prince Charles and the British Royal family. GEO TRUMPER is one of the oldest barber salons in London, and it's where I buy all my grooming products, except perfume, which I buy at FLORIS, the best perfume store in London. I order velvet slippers from FOSTER AND SON, customized with my intials.
TELERIE SPADARI in Milan is a great place to order custom-made pajamas and tablecloths. LARUSMIANI in Milan is a wonderful Italian gentleman's clothing store in the center of Milan fashion district. I used to go to CAMICERIA SAN MARCO in Venice when I was child to have my shirts custom made. The Duke of Windsor and Hemingway are also former clients. They can make ladies blouses and dresses, too.
MERCURY in Moscow is the best place to buy watches from Patek Philippe, because the store stocks the best collections. I visit CHARVET in Paris for pocket squares and silk scarves and AGLITITALY online for unique sneaker laces.
TURNBULL ASSER, 71-72 Jermyn St., London SW1Y 6PF, +44 (0)20 7808 3000, turnballandasser.com, and additional locations.
GEO TRUMPER, 9 Curzon St., London W1J 5HQ, +44 (0)20 7499-1850, trumpers.com, and an additional location in St. James.
FLORIS, 89 Jermyn St., London SW1Y 6JH, +44 (0)20 7747 3612, florislondon.com, and an additional location in Belgravia.
FOSTER AND SON, 83 Jermyn St., London SW1Y 6JD, +44 20 7930 5385, foster.co.uk.
TELERIE SPADARI, Via Spadari, 13, 20123 Milano +39 02 8646 0908.
CAMICERIA SAN MARCO, Sant'Angelo, Venice, 3627, 30124 VE, +39 041 522 1432.
MERCURY, Tretskayov Passage 1, Moscow, and additional locations, mercury.ru.
LARUSMIANI, Via Monte Napoleone, 7, 20121 Milano, +39 02 7600 6957, larusimiani.it.
CHARVET, 28 Place Vendôme, 75001 Paris, +33 1 42 60 30 70, charvet.com.
AGLITITALY, aglititaly.com/shop.
LIBERTY LONDON
#### The Shop at Bluebird
* * *
A cabinet of curiosities located in a stunning art deco building that used to be a car showroom, this concept store specializes in laid-back luxe lifestyle. Open since 2006, Bluebird stocks a great mix of Brit and international brands for men and women (Mary Katrantzou, Emilia Wickstead, Chloé, Preen, Isa Arfen, Missoni, Junya Watanabe, Gant). There's a real emphasis on new talent, too, with the shop hosting events and pop-up shops featuring exclusives from around the world. On-site cozy café and spa.
350 Kings Rd., London SW3 5UU, +44 (0)20 7351-3873.
#### Whistles
* * *
When Kate Middleton was dubbed "princess of the High Street," before she married Prince William, Whistles was one of her favorite brands. More sweet than sexy, the brand made the scalloped-edge ivory blouse Middleton wore in some of her engagement photos. Personal note: The Kings Road store is my favorite.
31 Kings Rd., London SW3 4RP, +44 (0)20 7730-2006, whistles.com, and additional locations.
### NOTTING HILL
NOTTING HILL
#### Couverture & The Grabstore
* * *
"A cool mix of housewares, kids' clothes, and jewelry you can't find anywhere else," says M.i.h Jeans' Chloe Lonsdale. There's an emphasis on indie designers like Rachel Comey, A Détacher, Apiece Apart, Martiniano, and Clare V.
188 Kennsington Park Rd., London W11 2ES, +44 (0)20 7229-2178, couvertureandthegrabstore.com.
#### Matches Fashion
* * *
Husband-and-wife duo Tom and Ruth Chapman opened their first boutique in Wimbledon Village in 1987 and were the first retailers in the UK to stock Prada. Over the past three decades, their business has grown to include four stores, all in London, selling Alexander McQueen, Mary Katrantzou, Isabel Marant, Delpozo, and other top brands, as well as an international online luxury site that reaches more than 100 countries. "The shop in Notting Hill looks like a showroom," says interior designer Brigette Romanek. "The clothes are displayed hanging at eye level, which makes it easy to shop. And the white walls as background makes everything pop. I have to control myself when I walk in that place."
60-64 Ledbury Rd., London W11 2AJ, +44 (0)20 7221-0255, matchesfashion.com, and additional locations.
#### Soler
* * *
"Where my friend Alex al-Badr makes the most charming and lovely dresses, blouses, fur vests, and leather skirts," says jewelry designer Liseanne Frank.
88 Bevington Rd, London W10 5TW, +44(0)20 8968-4694, soler.co.uk.
### East London
#### Goodhood Store
* * *
This 2,000-square-foot boutique for men and women specializes in elevated minimalist clothing and streetwear with a fun grab'n'go items mixed in (T-shirts, sneakers, buttons, and stickers). The space is filled with wooden fixtures, and there's even a log cabin. "If you journey downstairs, they have amazing housewares and a café," says Tere Artigas.
151 Curtain Rd., London EC2A 3QE, +44 (0)20 7729-3600, goodhoodstore.com.
#### Hostem
* * *
Hostem, located in East London, is Harry Styles's go-to for high-end runway and street labels Junya Watanabe, Raf Simons, and Visvim. The brainchild of menswear visionary James Brown, the store expanded to include womenswear in 2013, with a focus on up-and-coming designers.
28 Old Nichol St., London E2 7HR, +44 (0)20 7739-9733, hostem.co.uk.
#### House of Hackney
* * *
House of Hackney was founded in 2011 around a kitchen table by husband-and-wife team Javvy M Royle and Frieda Gormley. Sharing a passion for interiors and a background in fashion and product design, they decided to create a new brand in the great tradition of British print houses, to champion print, color, and texture, taking inspiration from the past and the present. The 2,000-square-foot town house that is home to the brand is a print paradise of home design (wallpaper, lampshades, porcelain, towels, bedding fabric), plus fashion and accessories.
131 Shoreditch High St., London E1 6JE, +44 (0)20 7613-5559, houseofhackney.com.
#### J.W. Anderson Workshops
* * *
Not a store, but rather an exhibition space that occasionally sells things . . . Got that? The young British designer took inspiration from the Bloomsbury Group's Omega Workshops to create the space next to the Ace Hotel, which hosts rotating events, some lasting weeks, others just a few days. Anderson collaborates with different artists (photographer Ian David Baker was one, A$AP Rocky was another) on the installations, with spin-off merchandise for sale.
100 Shoreditch High St., London E1 6JQ, +44 (0)20 3441-7030, j-w-anderson.com.
#### LNCC
* * *
Edgy concept store in Dalston, LNCC (short for Late Night Chameleon Café) features top designer names and young talent (Acne, Gucci, Proenza Schouler, J.W. Anderson), plus interesting eco-conscious brands (Blackyoto, Eckhaus Latta) in a groovy spot designed by Gary Card with a bar and club space. You'll also discover hard-to-find vinyl and first edition books.
18-24 Shacklewell Ln., basement level, London E8 2EZ, +44 (0)20 3174-0744, ln-cc.com.
#### YMC
* * *
You Must Create is ground zero for London's modern minimalists, with canvas culottes, workwear jackets and boilersuits galore, inviting shoppers to create their own style, plus nifty things like a collab with groundbreaking 1980s designer Katherine Hamnett, queen of the slogan tee.
23 Hanbury St., London E1 6QR, +44 (0)20 3432 3010, youmustcreate.com.
### PADDINGTON/ST. GEORGE'S FIELD
#### Merchant Archive
* * *
Sophie Merchant's original vintage shop served as a source of inspiration for designers, but has now developed into a full-fledged store and her own label. "She has vintage clothing and new clothing, in addition to beautiful old dishware, books, and paintings by local artists," says Brigette Romanek. "She brings in lots of interesting odds and ends and has a very keen eye for bohemia chic."
19 Kensington Park Rd., London W11 2EU, +44 (0)20 7229-9006, merchantarchive.com.
#### Pebble London
* * *
"For amazing jewelry," says designer Stephanie von Watzdorf of this appointment-only treasure chest where founder, designer, and curator Peter Adler makes his own jewelry using metals, stones, wood, glass, coral, and amber sourced from India, China, Africa, Thailand, and South America, as well as stocking traditional pieces from tribal groups in Africa, the Pacific, China, Afghanistan, Nepal, and India. Over the years, Adler has worked with or sold to many fashion designers, including Alexander McQueen, John Galliano, Roberto Cavalli, Issey Miyake, Matthew Williamson, Emilio Pucci, Allegra Hicks, and Donna Karan.
191 Sussex Gardens, London W2 2RH, +44 (0)20 7262-1775, pebblelondon.com.
#### Viola Boutique
* * *
Sara Lauchlan opened Viola after a career spent buying for London boutiques and moonlighting as a personal stylist. Tucked away on leafy Connaught Street since opening its doors in 2009, the store has acquired a loyal following of women who appreciate Lauchlan's laid-back approach to London style. The range of womenswear and jewelry designers includes Tsumori Chisato, Forte Forte, Roseanna, Stouls, Pedro Garcia and Raquel Allegra.
25 Connaught St., London W22AY, +44 (20) 7262-2722, violalondon.com.
### BERMONDSEY
#### Kerry Taylor Auctions
* * *
"At this auction house for fabulous vintage, I bought an insanely beautiful custom dress by the first female couturier, Jeanne Paquin," says jewelry designer Kara Ross. "I actually bought it sight unseen. It's the most beautiful dress; I'm hoping that one of my daughters will wear it as a wedding dress."
249-253 Long Lane, London SE1 4PR, +44 (0)20 3137-0112, kerrytaylorauctions.com.
### PORTOBELLO
#### Portobello Road Market
* * *
This world-famous antiques market features hundreds of vendors on any given day, but Fridays are when you might see Kate Moss or Sienna Miller wandering the stalls looking at vintage clothes, military jackets, cashmere sweaters, and belts. "Fridays at Portobello Market still hold unearthed treasures, and on any given day you will be able to spot a handful of key stylists and designers," says M.i.h Jeans's designer Chloe Lonsdale. "I always source the best vintage there even though I travel all over the world for research."
Portobello Rd., London W11 1LA, +44 (0)20 7727-7684, portobelloroad.co.uk.
PORTOBELLO ROAD MARKET
#### Rellik
* * *
Opened in 1999 by three partners, Rellik has high-end vintage from the 1930s onward by Vivienne Westwood, Comme des Garçons, Bill Gibb, Azzedine Alaïa, and many more. "The most amazing vintage shop," says jewelry designer Jordan Askill. "The owners—Fiona, Claire, and Steven—are all dear friends. I used to work there when I was about nineteen and living in London for the first time. Each owner keeps their own selection, and each has a specialty, but the store operates as one unit."
8 Golborne Rd., London W10 5NW, +44 (0)20 8962-0089, relliklondon.co.uk.
## DUBLIN
### IRELAND
IRISH DESIGN SHOP
#### Arnotts
* * *
The oldest department store in Dublin, Arnotts was founded in 1843, and acquired by Selfridges in 2015. In addition to top fashion and housewares brands, it also carries Irish craft brands, including glassware, pottery, textiles, and home accessories.
12 Henry St., Dublin 1, +353-1-805-0400, arnotts.ie.
#### Brown Thomas
* * *
The most prestigious department store in the city, Brown Thomas stocks exclusive brands, as well as collections by top Irish designers such as Paul Costelloe, Lainey Keogh, and Louise Kennedy. Part of the same retail family that includes Selfridges in the UK and Holt Renfrew in Canada. The Grafton Street flagship opened in 1849.
88-95 Grafton St., Dublin 2, +353-1-605-6666, brownthomas.com, and additional locations.
#### Cleo
* * *
High-end, family-run source for incredible knitwear handmade throughout Ireland (Donegal, Kerry, the Aran Islands, etc.). Features gorgeous sweaters and shawls, linen shirts, and tweed jackets.
18 Kildare St., Dublin, +353-1-676-1421.
#### Costume
* * *
Launched in 1997 by three sisters, Costume is a Dublin fashion institution, with designs from Isabel Marant, Roland Mouret, Cédric Charlier, MSGM, Jérôme Dreyfuss, and more, plus top-notch customer service.
10 Castle Market, Dublin Southside, Dublin, +353-1-679-4188, costumedublin.ie.
#### Folkster
* * *
Owned by Irish stylist Blanaid Hennessy, this successful independent retail store specializes in the boho-romantic look with a mix of vintage and contemporary clothing and accessories, as well as bridal and housewares.
9 Eustace St. Temple Bar, Dublin 2, +353-1-679-4188, folkster.com
FOLKSTER
#### Havana
* * *
Destination for high-end, avant-garde fashion by Irish labels Simone Rocha, Sphere One, and Sian Jacobs, alongside international names like Rick Owens, Stella Jean, and Junya Watanabe. Nearly all of owner Nikki Creedon's designer stock is exclusive to her in Ireland.
2 Anglesea House, Donnybrook Road, Donnybrook, Dublin 4, +353-1-260-2707 havanaboutique.ie.
#### Harlequin
* * *
You'll want to play dress-up for hours in this chicly cluttered shop, with fabulous vintage finds from the 1920s onward, including evening gloves, top hats, and beaded choker necklaces.
13 Castle Market, Dublin 2, Ireland, +353-1-671-0202.
#### Indigo & Cloth
* * *
One of Ireland's top menswear boutiques, Indigo & Cloth opened in 2007. It features leading menswear labels, both international and Irish, alongside a ground-floor Clement & Pekoe café.
9 Essex St. E, Temple Bar, Dublin 2, Ireland, +353-1-670-6403, indigoandcloth.com.
#### Irish Design Shop
* * *
Established in 2008 by jewelry designers Clare Grennan and Laura Caffrey, this place promotes the work of Ireland's contemporary makers, including crockery, candles, knit hats, and modern-looking jewelry.
41 Drury St., Dublin 2, +353-1-679-8871, irishdesignshop.com.
ONLINE SHOPPING TIP
There are lots of places online to buy Irish knitwear and crafts, but I like the fashion-forward eye of AranSweaterMarket.com and TheIrishWorshop.com which ship worldwide.
#### Jenny Vander
* * *
Considered by many to be Ireland's best vintage store, Jenny Vander has been open since the 1970s, and carries an eclectic mix of clothing, accessories, flamboyant hats, and jewelry that dates back to the 1800s.
50 Drury St., Dublin, Ireland, +353-1-677-4006.
#### Om Diva
* * *
What started as a stall on Georges Street Arcade in 1999 has since become a five-floor boutique in the creative quarter of Dublin. Owner Ruth Ni Loinsigh is a champion of Irish design and fashion who mixes clothing by cutting-edge talents with carefully curated vintage from the 1950s to the 1980s, plus jewelry and accessories. A truly funky shopping experience.
27 Drury St., Dublin 2, Ireland, +353-1-679-1211, omdivaboutique.com.
#### Penneys/Primark
* * *
The international fast-fashion giant (it's known as Penneys in Ireland and Primark in the rest of the world) is the biggest retail success story to come out of the country, and this is the mother ship. Open since 1969, the store now occupies nearly 100,000 square feet, filled with trendy casual wear for men, women, and kids. "We watch how real people are wearing clothes; we watch celebrities, bloggers, and freestylers," a Primark buyer told the Irish Times.
47 Mary St., Dublin 1, Ireland, +353-1-888-0500, primark.com, and additional locations.
## PARIS
### FRANCE
DEYROLLE
#### THE MAJORS
#### colette
* * *
There's only one colette in the world, which is by owner Sarah Andelman's design. In other words, you have to visit Paris to experience the 7,500-square-foot temple of cool. Established in 1997, you'll find everything from Valentino party dresses to Vans sneakers, plus plenty of limited-edition art and fashion collabs (a Gary Baseman x Doc Martens pair of boots, oui), a great selection of music, magazines, and knickknacks, and a restaurant that serves more than 100 kinds of water. "I may actually love their displays even more than the shopping," says fashion editor/TV personality Joe Zee. "Everything is perfectly curated and merchandised and styled. Sometimes, I'm not even sure if I can buy it, but it does look meticulous." "A pop culture museum," adds designer Karen Erickson. "I love having lunch at the Water Bar," Olympia Le-Tan remarks. "I stole a lighter once!" admits Rodarte designer Laura Mulleavy. "We had an opening party there and [the photographer] Nan Goldin came, and she needed a lighter for her cigarette. They wouldn't sell it to her because the store was closed, so we took it. I have it in a shrine in our office!"
213 rue Saint-Honoré, 75001 Paris, +33 (01)-55-35-33-90, colette.fr.
#### Le Bon Marché
* * *
Founded in 1852, Le Bon Marché was the first modern department store, but that's just one reason to visit this remarkable place, now owned by luxury conglomerate LVMH. The fashion edit is top-notch, with all the best European and Japanese designers represented, a terrific contemporary mix (Mira Mikati, Tibi), an épicerie (grocery store), top-floor shoe heaven, inspirational home furnishings, themed pop-up shops, and more. "They often have special items created just for the season by a particular designer," says fashion editor Robin Givhan. "They did a special Vanessa Bruno Le Cabas in gray boiled wool and silver sequins for winter 2015 that became my regular work/gym bag. I also love the housewares and the helpful sales clerks who can translate the sizes from centimeters into inches or what have you." "By far the most elegant, beautiful department store," says stylist and designer Maryam Malakpour. "You can find everything you need in one stop. I particularly love their displays." When in Paris, accessories designer Sally Perrin likes to say, "If I go missing, you'll find me at the Bon Marché!" "Aside from clothing, shoes, and accessories, there are several charming cafés, a bookstore, an amazing perfume and cosmetics section, a gourmet grocery store, and, of course, a fantastic hat department. The salesgirls still know me by name—this is where I began collecting hats more than a decade ago."
24 rue de Sèvres, 75007 Paris, +33 (01)-44-39-80-00, lebonmarche.com.
#### L'Eclaireur
* * *
Since 1980, when founders Martine and Armand Hadida opened their first space in an art gallery on the Champs-Élysées, L'Eclaireur has been a visionary retailer of fashion and design. Today, there are multiple stores, selling avant-garde styles by Fendi, Sacai, Saint Laurent, Rick Owens, Margiela, Marni, Giambattista Valli, and names that you probably haven't heard of—yet. (The Hadidas are the ones who introduced the Antwerp Six to Paris. They also own the Paris trade show Tranoï.) "A wide and eclectic mix of merchandise in a very cool atmosphere," says fashion editor Robin Givhan. "I've bought everything from a ski cap with a strip of Swarovski crystals down the front—it was practically glued to my head last winter—to the softest cashmere scarf ever to a Dries Van Noten jacket with a tulle front." All the locations are different; the Boissy is close to shopping on rue de Faubourg Saint-Honoré. "They have the largest collection of Fornasetti objects in the world," famed dandy Patrick McDonald says of this location. "I adore the men's area upstairs, and they always have a fabulous chapeau for me."
10 rue Boissy d'Anglas, 75008 Paris, +33 (01)-53-43-03-70, leclaireur.com, and additional locations.
#### MONOPRIX
* * *
This clothing, housewares, and grocery store chain (think Target, but chicer) is a fashion week insider go-to for cute cotton underwear, socks and sleepwear, kids' clothes, and such everyday necessities as notebooks, bottled water, warm croissants, and snacks. "My friends joke that my favorite place to shop in Paris is Monoprix on the Champs-Élysées," says Alina Cho. "They're not wrong. I stock up on Nuxe hand cream, La Roche-Posay skin-care products, and small umbrellas, which are sturdier than the $5 versions you buy on the street in New York." "I wear their black or white denim almost exclusively," says fashion journalist Christina Binkley. "I must have eight pairs. Also, their socks are the best. And it's worth checking out their seasonal fashions as sometimes they can be quite good in design and fabrication (though they can also be quite boring—I don't quite understand their fashion strategy). Monoprix has stores all over, but I favor the one just off boulevard Saint-Germain in the Sixth Arrondissement." "Beautiful kids' clothing at a steal," adds blogger/model Elizabeth Minett.
52 rue de Rennes, 75006 Paris, +33 (01)-45-48-18-08, monoprix.fr, and additional locations.
### ST. GERMAIN DES PRES/LATIN QUARTER/7th ARR.
#### Alexandra Sojfer
* * *
An entire store dedicated to handmade umbrellas? Only in Paris, mes amis. Textile designer Olya Thompson calls out this place as "a hidden gem." The family has been in the business since the 1930s, and also handcrafts walking sticks. Materials are super-luxe. Think ebony, ostrich skin, organza, and silk satin. Almost too good for the rain.
218 bvld. Saint-Germain, 75007 Paris, +33 (01) 42-22-17-02, alexandrasojfer.com.
#### A.P.C.
* * *
Founded in 1987 by Jean Touitou in Paris, A.P.C. stands for Atelier de Production et de Création. The brand's vision of minimalist cool, in slim cuts, (selvedge jeans, military-inspired jackets, streamlined basics, simple boots) has gone global, but this is the flagship—and the headquarters.
35 (men) & 38 (women) rue Madame, 75006 Paris, France, +33 (01) 42-22-12-77, apc.fr, and additional locations.
#### Aurélie Bidermann
* * *
A mix of casual and glam is what has made Paris-based jewelry designer Aurélie Bidermann a fashion-world favorite. She's created modern-day heirlooms out of one-of-a-kind pieces of vintage cotton lace, dipping them in 18-karat gold and molding them into cuffs with a weathered patina. She's also elevated the humble friendship bracelet to must-have status, using gold hardware to anchor the braided threads and give the pieces strength and permanence. She is influenced by pre-Columbian art, Brazilian handicrafts, Santa Fe style, and the architecture of Miami Beach. Beyoncé, Naomi Watts, and Keira Knightley are fans. This jewel box of a shop features her casual (most pieces less than $2,000) and fine collections.
55 bis rue des Saints-Pères, 75006 Paris, +33 (01)-45-48-43-14, aureliebidermann.com, and additional locations.
#### Bellerose
* * *
A chain store that doesn't feel like one. Bellerose started in Belgium in 1990 and has since expanded to France. The stores are homey, like walking into a close friend's apartment, with wood floors and calm music. On the rails you'll find the kind of understated chic clothing you'll want to wear every day, but that still feels special and fashionable, for example a great silk pajama blouse, a reversible Chinese patterned bomber jacket, a metallic rose gold sweatshirt and slouchy khaki pants. You'll also find select pieces from other brands, such as Golden Goose sneakers, Birkenstock sandals and more. Most pieces are less than $300.
3 rue Jacob, 75006 Paris, +33 (01)- 43-54-77-90, bellerose.be, and additional locations.
#### Bonton
* * *
Where else but in Paris would you find a concept store dedicated to young people? Created in 2001 by Thomas Cohen (his parents created the popular children's brand Bonpoint), Bonton is casual, timeless, and fun. The brand quickly went international, selling clothing, furniture, toiletries, and gifts, and has done collaborations with the likes of A.P.C. and Peanuts. Some stores have hairdressers and host workshops and cooking classes. "It's a kids' store, but it's the best," says stylist Kate Young. "I love the bed linens, the party favors, the kids' toys and clothes. It's got the most fun stuff ever." Other fans include Victoria Beckham, Sofia Coppola, Kate Winslet, Stella McCartney, and Charlotte Gainsbourg.
82 rue de Grenelle, 75007 Paris, +33 (01)-44-39-09-20, bonton.fr, and additional locations.
#### Buly 1803
* * *
"The brand has been around for something like 200 years and makes beautiful soaps and lotions," says Washington Post fashion editor Robin Givhan of this old school apothecary where they have a calligrapher on hand to write gift messages. "I'm in love with their candles. They're in marble containers; my favorite is one that smells of cypress, like a Japanese temple." You'll also find facial oil, hand cream, fragrances and more in packaging so beautiful, you'll want to display it on your vanity long after the product is gone.
6 rue Bonaparte, 75006 Paris, +33 (01)-43-29-02-50, buly1805.com.
#### Des Petits Hauts
* * *
This French contemporary clothing brand, started by two sisters, is a terrific source for sweet lace-inset silk blouses, angora boyfriend cardigans, shiny Lurex jumpsuits, star-print miniskirts, suede ankle booties, and other cool-girl staples with that certain Parisian je ne sais quoi. Nothing is more than $400.
70 rue Bonaparte, 75006 Paris, +33 (01)-75-44-05-82, despetitshauts.com, and additional locations.
#### Deyrolle
* * *
"If you can't find a gift here for your crazy aunt, you can't find one anywhere," fashion journalist Christina Binkley says of this taxidermy place, which isn't a just a cabinet of curiosities, it's a whole store full. "From books to taxidermic bugs, it's totally exotic. Vanessa Friedman of the New York Times first took me to Deyrolle years ago, and now I go all the time with my daughter."
46 rue du Bac, 75007 Paris, +33 (01)-42-22-30-07, deyrolle.com.
#### Dries Van Noten
* * *
The Belgian designer's jewel box of a boutique in Saint Germain is a must-stop for the fashion flock looking for his artful coats and dresses, versatile knitwear, and fun shoes, especially since his collection is distributed through only a few international retailers. "I go to Dries because he's a personal favorite and there's usually something from the previous season's show that has stuck in my memory that I'm keen to try on," says fashion editor Robin Givhan. "I have done serious Amex damage." "Probably my favorite place to shop in the world," says accessories designer Lizzie Fortunato, while accessories designer Sally Perrin loves "the surge of colors and textures." "I love the prints," says jewelry designer Aurélie Bidermann. "Moreover, the interior design of this store is really amazing." There are men's and women's boutiques, both on the Quai Malaquais.
7 and 9 Quai Malaquais, 75006 Paris, +33 (01)-44-27-00-40, driesvannoten.be, and additional locations.
#### Editions de Parfums by Frédéric Malle
* * *
Paris-based fragrance nose Frédéric Malle grew up in the perfume industry; his grandfather founded Parfums Christian Dior. After working as a consultant in the industry, Malle founded Editions de Parfums Frédéric Malle in 2000 to support more-artisanal perfume creation as an alternative to marketing-driven perfume companies. Since then, he's had a number of hits, including Carnal Flowers and Portrait of a Lady. The fragrances' packaging is simple, and the labels bear the signature of the perfume creator just as a painting would bear the signature of an artist. Malle's original store feels like a fragrance lab inside a Parisian apartment.
37 rue de Grenelle, 75007 Paris, +33 (01)-42-22-76-40, fredericamalle.com.
EDITIONS DE PARFUMS
#### Hermès
* * *
Founded as a saddlery in 1880, the original store at 24 rue Faubourg de Saint-Honoré may be better known than the Left Bank boutique, which opened in 2010. But the Left Bank shop surpasses the original with its cool concept. French architecture firm RDAI transformed the 1930s art deco building and swimming pool into a modern marvel with otherworldly braided wood pods defining the accessories, fashion, tableware, and furniture departments. There's an in-house florist, a bookshop, and a café that serves tea and light lunch. Scattered throughout are items from the Petit H collection made entirely of production remnants, including bracelets fashioned from pieces of silk scarves, and a lamp made of coffee cup. "It's such an inspiration to see a luxury brand not take themselves so seriously," says Olivia Kim. "Best known for their leather craftsmanship, I love how they take that and create objects that wouldn't be expected—pinwheels, standing animals, rocking horses, even birdhouses."
17 rue de Sèvres, 75006 Paris, +33 (01)-42-22-80-83, hermes.com.
WHERE TO SHOP ON SUNDAYS IN PARIS
* * *
You're in the capital of couture on a Sunday and everything is closed (and we mean everything, even Monoprix), due to state regulations rooted in religious tradition. What's a shopaholic to do?
Three vintage stores in the Marais:
VINTAGE DÉSIR might try to fool you with the name Coiffure on the sign outside, but inside, it's a vintage bonanza—and not too expensive.
FREE'P'STAR has some vintage and some secondhand, meaning you're just as likely to find a sweater from Zara as an old pair of Levi's.
KILO SHOP sells clothing by the pound, er, kilo.
The Marais branch of L'ECLAIREUR, on Rue de Sévigné, is a work of art in and of itself, designed in collaboration with Belgian artist Arne Quinze and featuring two tons of wooden planks and 147 video screens. Menswear and womenswear collections, including Yohji Yamamoto, 08 Sircus, Haider Ackermann, and Boris Bidjan Saberi, are alternately concealed and revealed by moving walls.
With more than 150,000 objects, the MUSÉE DES ARTS DÉCORATIFS is dedicated to the French art of living, including the art of dressing. Not only is this a fabulous place to catch a fashion exhibition (past shows have been dedicated to everyone from Dries Van Noten to Barbie), the shop is top-notch. "An exceptional selection of exclusive objects, mostly related to current and past exhibitions," says Tatiana Sorokko. "You can find things like melamine René Magritte motif plates and a 5-inch square Jean Paul Gaultier microfiber navy stripe fabric swatch."
In 2009, stores in France's 569 existing tourist zones, including the Champs-Élysées, were allowed to be open on Sundays. (Before then, only stores having to do with sports, culture, or tourism were allowed.) So you can shop the headquarters of LOUIS VUITTON every day of the week. The store includes an art gallery accessible from a separate entrance. Also on the Champs-Élysées and open on Sundays are Tiffany & Co., Swarovski, Sephora, Montblanc, Longchamp, and the Publicis Drugstore (which is more like a high-end department store, food emporium, and restaurant than a drugstore).
Paris's largest flea market, MARCHÉ AUX PUCES DE SAINT-OUEN is actually fifteen markets in one. Stylist Cristina Ehrlich likes Marché Dauphine for "antiques, amazing vintage pieces, old photos, and postcards all housed under this amazing greenhouse-like space." Accessories designer Sally Perrin's favorite is Marché Paul Bert Serpette, where she has found beautiful vintage kimonos in the back alley. Shoe designer Isa Tapia prefers to wander around for "the best 1940s, '50s, and '60s exotic handbag selections. Also good for cosmetic packaging and old graphic rubber stamps and furniture." "You will find so many great little shops at Les Puces selling everything from antiques to junk to vintage clothes," says designer Marie France Van Damme. "I have found everything from drums to cacti to medieval jousting armor," says stylist Tara Swennen. "I found a beautiful vintage leather doctor's bag that I vowed would be an integral part of my collection should I start a handbag company. And it has been!" adds accessories designer Fiona Kotur Marin.
VINTAGE DÉSIR, 32 rue des Rosiers, 75004, Open every day 11:30 a.m.–9:00 p.m.
FREE'P'STAR, 8 rue Sainte-croix de la Bretonnerie, 75004, Mon–Fri: 11:00 a.m.–9:00 p.m.; Sat–Sun: 12:00 p.m.–2:30 p.m., freepstar.fr.
KILO SHOP, 69-71 rue de la Verrerie, 75004, Mon & Sun: 2:00 p.m.–7:45 p.m.; Tue–Sat: 11:00 a.m.–7:45 p.m., kilo-shop.fr.
L'ECLAIREUR, 40 rue de Sévigné 75003 Paris, +33 (01)-48-87-1022, open Mon-Sat: 11 a.m.–7 p.m.; Sun: 3 p.m.–7 p.m.
MUSÉE DES ARTS DÉCORATIFS, 107 rue de Rivoli, 75001, open Tue-Sun: 11 a.m.–6 p.m., Thurs: 11a.m.–9 p.m.
LOUIS VUITTON FLAGSHIP ON THE CHAMPS-ÉLYSÉES, 101 Ave. des Champs-Élysées, 75008 Paris, +33 (01)-53-57-52-00, louisvuitton.com.
MARCHÉ AUX PUCES DE SAINT-OUEN, Located at the Porte de Cligman Court metro, 75018 Paris, marcheauxpuces-saintouen.com. open Sat-Mon: 10a.m.–1p.m. and 2p.m.–5:30p.m.
#### Ines de la Fressange
* * *
Model/designer/author Fressange has added retailer to her considerable list of talents, with a new lifestyle boutique on the Left Bank showcasing her own clothing and accessories made in the atelier in back (think silk pajamas, logo T-shirts, suede miniskirts, vintage silk dresses, and ballet flats), alongside fun housewares, notebooks, iPhone cases, tea, and trinkets.
24 rue de Grenelle, 75007 Paris, +33 (01)-45-48-19-06, inesdelafressange.fr.
#### Jérôme Dreyfuss
* * *
Paris-based Dreyfuss shares his laid-back, bohemian design sensibility, and his home, with designer Isabel Marant. Dreyfuss's under-the-radar bags are logo-less and, in keeping with their tomboyish look, have boys' names. They also have clever design details, such as multiple straps to allow carrying by hand or over the shoulder, removable wallets, attached key rings, and mini-flashlights. Dreyfuss also designs cool shoes and boots. He launched his accessory collection in 2002 with the goal of creating practical, light, soft bags that feel as good next to your body as a teddy bear. Prices range from $115 for a small pouch to $2,515 for an oversized hobo. Dreyfuss's shoe store is across the street.
1 rue Jacob, 75006 Paris, +33 (01)-43-54-70-93, jerome-dreyfuss.com, and additional locations.
#### Karry O
* * *
Fashion journalist and blogger Divia Harilela recommends this as a source for vintage jewelry by Chopard, Van Cleef & Arpels, Dior, and more, plus original pieces designed by owner Karine Berrebi.
41 rue de Saint-Honoré, 750008 Paris, +33 (01)-45-48-94-67, karryo.com.
#### Les 3 Marches de Catherine B
* * *
Vintage expert Catherine B opened her first store in 1994, selling timeless, pristine-quality bags, belts, and scarves by Hermès and Chanel. Next to the original store is a newer one dedicated to shoes, clothing, and travel bags. Some pieces are so rare, they are not for sale, including a Chanel bicycle featuring two sidesaddle "handbags" and a seat with signature Chanel quilting. Catherine B also owns an original Birkin bag with the initials JB, for Jane Birkin. "An amazing vintage selection of Chanel and Hermès," says jewelry designer Aurélie Bidermann. "I always find something."
1 rue Guisarde, 75006 Paris, +33 (01)-43- 54-74-18, les3marchesdescatherineb.com.
LES 3 MARCHES DE CATHERINE B
#### Lefranc-Ferrant
* * *
"The proprietress and designer, Béatrice Ferrant, used to show her collections at Paris Haute Couture Week, but she has downsized and now sells her modern, architectural designs from her shop," says fashion journalist Christina Binkley. "She will do custom looks and sizes in addition to her ready-to-wear. On any given day, you're likely to find her in her shop serving clients."
22 rue de l'Echaudé, 76006 Paris, +33 (01)-40-21-03-29, lefranc-ferrant.fr.
#### Leon and Harper
* * *
After twenty-five years at Et Vous, Philippe Corbin launched Leon and Harper in 2011. The casual-boho chain sells distressed army jackets, embroidered tunics, frilly blouses, high-waist jeans, fringed moccasins, clog sandals, and more fun stuff.
46 rue du Four, 75006 Paris, +33 (01)-45-44-17-27, leonandharper.com, and additional locations.
#### L/Uniform
* * *
Why buy a designer bag when you can create your own? That's the concept behind Jeanne Signoles's preppy-utilitarian brand L/Uniform, launched in 2014. A veteran of luxury leather goods brand Goyard (which is owned by her father-in-law), Singoles opened her first L/Uniform store in 2015 with a sleek interior designed by Masamichi Katayama. Choose your bag shape (backpack, messenger, tote, pencil case, and more) and the color of canvas and trim; you can have it monogrammed, too. Most styles are less than $500.
21 quai Malaquais, 75006 Paris, +33 (01)-42-61-76-27, luniform.com.
#### Maison Kitsune
* * *
All the classicism of A.P.C., but with more whimsy and color, this contemporary French label, launched in 2002 by Gildas Loaëc and Masaya Kuroki, channels preppy Paris-meets-Tokyo style. You'll find intarsia fox head sweaters, French slogan tees, flannel sweatpants, and shrunken blazers. Maison Kitsune also has its own music label, and a couple of cafes, including one at the Palais Royale.
38 rue Madame, 75006 Paris, +33 (01)- 53-71-76-62, maisonkitsune.com, and additional locations.
#### Mes Desmoiselles
* * *
This vintage-boho French contemporary brand's boutiques are Aladdin's caves of maxi dresses and chunky knits, Lurex stripe caftans, silk bed jackets, and embroidered slippers. The collection is carried in a ton of multi-label stores around the world, but going to the source is a different feel altogether.
57 rue des Saints-Pères, 75006 Paris, +33 (01)-43-35-12-52, mesdesmoiselles.com, and additional locations.
#### Petit Bateau
* * *
"I love the utilitarian chic of the small but ubiquitous Petit Bateau shops in Paris," says textile designer Olya Thompson of this French chain store, which has been around for more than 100 years. "The best-cut kids' footies with snaps that actually work, great pj's and underwear, and T-shirts with a great price-quality ratio."
53 bis rue de Sèvres, 75006 Paris, +33 (01)-45-49-48-38, petit-bateau.fr, and additional locations.
#### Robert Clergerie
* * *
Sturdy, comfortable, untrendy, and 1990s-nostalgic, Robert Clergerie shoes have been embraced by a bevy of celebs, including Shailene Woodley, Scarlett Johansson, and Rihanna. The company has been around since 1895, when founder Joseph Fenestrier took over a small men's shoe manufacturing company. It chugged along through the 20th century, producing 1,200 shoes a day in 1926, and opening stores in France. Then, in 1978, Robert Clergerie purchased the company. He continued to produce men's styles until he hit on the idea, in 1981, of making women's shoes built on a men's last. His first collection of oxfords was an instant success. In 1987, his legacy was sealed when he was named designer of the year by Footwear News. He went on to win the award twice more. Today, the brand has more than two dozen boutiques worldwide. In 2011 French designer Roland Mouret, known for his formfitting dresses, was appointed creative director, and he has introduced sexier stiletto styles. Still, at Clergerie, simplicity will always be the guiding principle. These are shoes made for walking; key styles include mules and lace-ups. This is the flagship.
5 rue du Cherche-Midi, Paris 75006, +33 (01)-42-84-03-14, robertclergerie.com, and additional locations.
#### Sabbia Rosa
* * *
"For feminine, exquisite silk lingerie, and available in sooo many colors," says designer Marie France Van Damme of this upscale lingerie shop, opened in 1976 by owner/designer Monette Moati. Legend has it that one month after the opening, French singer Serge Gainsbourg stopped by to purchase a gift for his then-partner Jane Birkin, and the experience inspired his song "Les Dessous Chics." Whether or not the story is true, it helped secure Sabbia Rosa's enduring popularity.
73 rue des Saints-Pères, 75006 Paris, +33 (01)45-48-88-37.
VANESSA SEWARD'S
ADDRESS BOOK
* * *
Paris-based designer Vanessa Seward founded her namesake brand in 2014, after two years of creating capsule collections for French denim and casual wear label A.P.C., and nearly a decade designing for the high-end house of Azzaro before that. Her personal style is influenced by Ali MacGraw (preppy chic to perfection), a teenage Brooke Shields, and Caroline de Monaco.
I love LE BON MARCHÉ for its very chic selection of clothes and its beauty floor where there's lots of exclusive international brands to discover. I would cross Paris to do my grocery shopping at their food parlor the GRANDE ÉPICERIE, but luckily it's just across the street! I also like SPREE, a multi-label boutique located in Montmartre. I've been friends with the owner Roberta for years and I've always admired her impeccable taste. Bruno Hadjadj her husband is an artist and they also host art and design exhibitions.
For vintage, THE DRESSING FACTORY is small, but I could buy half the shop each time I go and they are very well priced.
GANTS MURIEL, this little glove shop is great, they propose the chicest glove selection in the most exquisite leathers and colors, all made in France. Very good for presents.
The French chain store PRINCESSE TAM TAM is very good for lingerie and home wear.
In Paris, they have great garage sales in summer and the best one was one on the very chic Rue Trochet. I bought a Burberry trench (which I wear practically every day), a '70s Céline blue suede trench and a few gorgeous YSL print scarves for around 200€.
LE BON MARCHÉ, 24 rue de Sèvres, 75007, +33 (01)-44-39-80-00, lebonmarche.com.
LE GRANDE ÉPICERIE, 38 rue de Sèvres, 75007, +33 (01)-44-39-81-00, lebgrandeepicerie.com.
SPREE, 16 rue La Vieuville, 75018, +33 (01)-42-23-41-40, spree.fr.
THE DRESSING FACTORY, 18 rue Saint Simon, Paris 75007, dressing-factory.com.
GANTS MURIEL, 4 Rue des Saussaies, Paris 75008, +33 (01)-42-65-95-34.
PRINCESSE TAM TAM, 109 Ave. Victor Hugo, 75116 Paris, +33 (01)-47-27-77-53 princesstamtam.eu, and additional locations.
#### Saint James
* * *
Saint James is the maker of the nautical Breton shirts in combed cotton jersey that have been part of the official French naval uniform since 1858. According to lore, each of the 21 stripes corresponds to a naval victory of Napoleon's fleet against the British. The brand has expanded to a full range of casual wear for men, women, and children and is sold around the world.
66 rue de Rennes, 75006 Paris, +33 (01)-53-63-09-82, saintjamesboutique.com, and additional locations.
#### Saint Laurent
* * *
In 1966, Yves Saint Laurent opened a ready-to-wear boutique in Paris called Saint Laurent Rive Gauche, the first time a couturier successfully launched a ready-to-wear line in France. (You could call it the original cheap chic.) Located in a former antiques store in the student-dominated Left Bank, the store was a complete 180 from the gilded interior of Saint Laurent's haute couture salon. Rive Gauche was such a success, some customers waited up to three hours to purchase items, and the store became a hub of youth culture. Although there are several Saint Laurent boutiques in Paris (and more around the world), the rive gauche spirit is particularly potent at this location, near Saint-Sulpice, inside a 17th-century building.
6 place Saint-Sulpice, 75006 Paris, +33 (01)-43-29-43-00, ysl.com, and additional locations.
#### Sandra Serraf
* * *
I always find something at this low-key multi-brand boutique. The look is Parisian boho, with offerings from Isabel Marant Étoile, Forte Forte, Masscob, Humanoid and Jacquie Aiche.
18 rue Mabillon, 75006 Paris, +33 (01)-43-25-21-24, sandraserraf.fr.
#### Simrane
* * *
"I buy something here every time I'm in Paris," says fashion journalist Christina Binkley. "It carries all sorts of clothing and household goods from India, and the quality is higher than you typically find in bargain boutiques. I have a half-dozen blouses with such beautiful prints from here, and a couple of dresses." Also carries beautiful napkins, scarves, pillows, robes, and doorknobs.
23 and 25 rue Bonaparte, 75006 Paris, +33 (01)-43-54-90-73, simrane.com.
#### Sonia Rykiel
* * *
In her heyday in the 1960s and '70s, Sonia Rykiel was synonymous with French chic, from the shrunken "poor boy" sweater that landed on the cover of Elle magazine and launched her career in 1968 to endless interpretations of her iconic striped sailor sweater, to the safety pin brooch that was a favorite fashionable Paris souvenir. The Rykiel store in the bohemian Saint Germain-des-Prés neighborhood, across the street from the Café de Flore, is the brand's spiritual home, filled with thousands and thousands of paperback books nestled into bookcases, with whimsical lip-printed carpeting on the floor. Julie de Libran took over as artistic director in 2014, and is earning rave reviews and fans in Miranda Kerr, Suki Waterhouse, Leighton Meester, and Selena Gomez.
175 blvd. Saint-Germain, 75006 Paris, +33 (01)-49-54-60-60, soniarykiel.com, and additional locations.
SONIA RYKIEL
#### Tabio
* * *
This made-in-Japan hosiery brand's Paris store is "a must-stop for sparkly socks," says accessories designer Lizzie Fortunato. "They will monogram them for an extra three euros in 15 minutes" "I try to visit whenever I'm in Paris, because who doesn't love fun socks?" says stylist Cristina Ehrlich.
32 rue Saint-Sulpice, 75006 Paris, +33 (01)-43-26-28-12, tabio.com.fr, and additional locations.
#### Tara Jarmon
* * *
"I discovered this brand in Paris," says author Aliza Licht of Canadian-born, Paris-based Jarmon's feminine label of metallic pleated skirts, printed minidresses, and lacy blouses. "It's my favorite place to shop when I'm there. The store is filled with clothing that I haven't seen before. The prices are reasonable, and the quality is amazing."
18 rue du Four, 75006 Paris, +33 (01)-46-33-26-60, tarajarmon.com, and additional locations.
#### Thierry Lasry
* * *
The French eyewear designer favorite of Rihanna and Gigi Hadid opened his first flagship in Paris in 2016 as a showcase for his colorful frames, including vintage acetate styles. Designed by artistic multi-hyphenate Vincnt Darré, the storefront beckons passersby to gaze at themselves through a reflective pupil and blue iris surrounded by rows of shelves. Stocked inside, the full collection of Thierry Lasry sunglasses, plus Harry Lary's frames, his other optical label, and his eyewear collaborations with Fendi and tattoo artist Brian Woo (aka Dr. Woo).
40 rue du Four, 75006 Paris, +33 (01)-43-27-44-65, thierrylasry.com.
#### Vanessa Bruno
* * *
Bruno started her namesake brand in 1996, to create a modern everyday wardrobe. She opened her first store on the Left Bank in 1998, the same year she launched the Cabas tote bag. With some sequins on cotton canvas, she created a new classic, and it became an international best-seller. The contemporary brand is a great source for sweet blouses, pointelle knits, laid-back culottes, and print dresses.
25 rue Saint Sulpice, 75006 Paris, +33 (01)-43-54-41-04, vanessabruno.fr, and additional locations.
### BASTILLE
#### Come on Eileen
* * *
"A thrift store that features super-high-end vintage," says Gabriela Artigas. "My sister, Tere, bought two Céline coats there at an amazing price. One of them is a floor-length, mustard-yellow suede coat with leather buttons that she still wears to this day. We love this store because they always remember us; they make us feel like part of their family, which is so important and keeps us going back time and again."
16/18 rue des Taillandiers, 75011 Paris, +33 (01)-43-38-12-11.
#### Isabel Marant
* * *
Marant is the master of Paris-meets-L.A. boho chic (greatest hits include the hidden platform sneaker, the Dicker Western-style boot, and the Kady washed-leather biker jacket). Even though the designer has stores around the world, it feels right shopping her collection in Paris. This store, her first, opened in 1998.
16 rue de Charonne, 75011 Paris, +33 (01)-49-29-71-55, isabelmarant.fr, and additional locations.
#### Sessún
* * *
Started in 1996 after owner Emma François took a trip to South America, this French brand's vision of boho clothing features contemporary basics (drapey trench coats, poet blouses, loose-fit trousers) in natural fabrics with the occasional handicraft-inspired tassel or fringe.
34 rue de Charonne, 75011 Paris, +33 (01)-48-06-55-66; sessun.com, and additional locations.
### MARAIS
#### Azzedine Alaïa Outlet
* * *
Easily missed, because it's difficult to spot, the outlet is accessed around the corner from the rue de Moussy boutique, on a side street under an archway. Inside, you'll find previous seasons' clothing, shoes, and accessories discounted at least 50 percent.
18 rue de la Verrerie, 75004, Paris, +33 (01)–42–72–19–19, alaia.com.
#### The Broken Arm
* * *
Paris's latest concept store, the Broken Arm was opened in 2013 by the team behind De Jeunes Gens Modernes. The shop stocks avant-garde and streetwear brands for men and women, including Vetements, Jacquemus, Lemaire, Nike, Gosha Rubchinskiy, Raf Simons, and Our Legacy, and also has a café with its own blend of coffee. "An incredible modern concept store," says costume designer Janie Bryant. "There's a great café attached," says Malone Souliers' co-designer Ma Malone. There's even a Broken Arm fragrance designed to evoke the materials used to construct the store.
12 Rue Perrée, 75003 Paris, +33 (01)-44-61-53-60, the-broken-arm.com.
#### Hod
* * *
"The girls who run this store travel all over the world and bring fine jewelry treasures back to their beautiful little boutique," says stylist Cristina Ehrlich. "It exudes a chic, young approach to jewelry that is distinctly French while showcasing jewels that reflect the far corners of the world."
104 rue Vieille du Temple, 75003 Paris, +33 (09)-53-15-83-34, hod-boutique.com.
#### Merci
* * *
Occupying a former wallpaper factory, Merci is the laid-back, boho alternative to colette, a concept store with designer furniture and vintage items, tableware, men's, women's, and kids' clothing, and more. There's a restaurant in the basement ("Just the right place to sip tea," says handbag designer Clare Vivier), and a bookstore, and some of the profits go to charity. "I love the all-encompassing experience with the home stuff and the restaurant," says designer Rebecca Minkoff. "The way they merchandise makes me want to buy things I don't even need. During fashion week, I fell in love with all these African textiles."
111 blvd. Beaumarchais, 75003 Paris, +33 (01)-42-77-00-33, merci-merci.com.
MERCI
#### Tom Greyhound
* * *
"Hipster haven," designer Bridget Romanek says of the Korean multi-brand store, which also has this branch in Paris. "Curated with items you haven't seen before. They carry some collections you know, but they pick the pieces that other stores don't. It's fun in there." Behind a wall of rotating wood slats, you'll find sections dedicated to young, independent labels.
19 Rue de Saintonge, 75003 Paris, +33 (01)-44-61-36-59, tomgreyhound.fr.
#### Vanessa Seward
* * *
Seward founded her namesake brand in 2014 and her vision of comfortable Parisian glamour is gaining steam on both sides of the Atlantic. French stars such as actor Marion Cotillard and actor-model Farida Khelfa have been wearing her retro print dresses, while model Karlie Kloss likes Seward's French-phrase tee (UNE FEMME FRANÇAISE). Seward makes great denim, too, and you can personalize the back pocket with your name. Prices range from about $150 to $2,100.
7 blvd. des Filles du Calvaire, 75003 Paris, +33 (01)-70-36-06-11, vanessaseward.com.
#### Odetta
* * *
"One of my favorite consignment stores," says jewelry designer Gabriela Artigas. "It's not super cheap but they have every designer you can imagine: Azzaro, Leonard, Yves Saint Laurent, Lanvin. Nearby is a great little coffee shop called Fragments, which is a must-visit."
76 rue des Tournelles, 75003 Paris, +33 (01)-48-87-08-61, odettavintage.com.
### RUE DE FAUBOURG SAINT-HONORÉ/OPERA/PLACE VENDÔME
#### Anouschka
* * *
"By appointment only, but such divine frocks and accessories," says designer and personal shopper Raven Kauffman of this insiders' spot, owned by a former model. "I love the secret feel of the space—it's like you are rifling through the private closets of one of the most chic women in the world."
6 ave. du Coq, 75009 Paris, +33 (01)-48-74-37-00.
#### Astier de Villatte
* * *
Known around the world since 1996 for handmade black terra-cotta ceramics and tableware finished in white paint, city-inspired incense, and candles. The ceramics are made in the Bastille, and deliberate surface imperfections add to their charm. "Everything they make is gorgeous," says stylist Kate Young. Personal note: My favorite piece is the Antoinette incense burner, a cheeky ode to Marie Antoinette's end that features her head on a plate.
173 rue Saint-Honoré, 75001 Paris, +33 (01)-42-60-74-13, astierdevillate.com, and an additional location on the Left Bank.
#### Chanel
* * *
Coco Chanel's first independent boutique, Chanel Modes, opened in 1910 on this spot and sold hats, which were her specialty at the time. In 1913, she expanded to the resort towns of Deauville and Biarritz selling casual sports clothes that appealed to women's increasingly active lifestyles. By 1919, Chanel had established her couture house (also on this site), near her home at the Ritz Hotel. The rest is fashion history. Chief designer Karl Lagerfeld has not only kept the Chanel flame burning since 1983 but also fueled incredible growth, bringing the house codes (the suit, the pearls, the chain-handled bag) into the 21st century with over-the-top shows. There are many Chanel stores around the world (and around Paris), but this is the one that started it all.
31 rue Cambon, 75001 Paris, +33 (01)-44-50-66-00, chanel.com, and additional locations.
#### Chantal Thomass
* * *
"Important to any [lingerie] aficionado," says filmmaker and author Liz Goldwyn of this designer shop, where lingerie is treated as fashion. The modern boudoir carries Thomass's collections of bras, merry widows, garter belts, thongs, bloomers, bikinis, bustiers, and stockings.
211 rue Saint-Honoré, 75001 Paris, +33 (01)-42-60-40-56, chantalthomass.com.
#### Charvet
* * *
High-end purveyor of bespoke and ready-to-wear men's shirts, neckties, pajamas, and suits, founded in 1838 by the son of the "wardrobe curator" to Napoléon Bonaparte, Charvet has outfitted kings, princes, heads of state, and creatives such as Charles Baudelaire, George Sand, Jean Cocteau, the Duke of Windsor, Yves Saint Laurent, Charles de Gaulle, Winston Churchill, Henry Kissinger, and John F. Kennedy. A great place to pick up a linen pocket square or a pair of passementerie-knot cuff links (which were invented by Charvet) for a gift.
28 place Vendôme, 75001 Paris, +33 (01)-42-60-30-70, charvet.com.
#### Goyard
* * *
The understated yin to Louis Vuitton's yang, Goyard has been making its signature chevron-patterned canvas trunks and bags since 1853, and is older than LV by one year. The brand is known for eschewing advertising and e-commerce, and limiting distribution, which has protected the allure. Pablo Picasso, Jacques Cartier, the Agnellis and the Rockefellers, the Romanovs and the Grimaldis, Estée Lauder and Barbara Hutton, Coco Chanel and Jeanne Lanvin, Edith Piaf and Karl Lagerfeld have all been customers, and chef Alain Ducasse had a Goyard culinary trunk custom made at this historic store. In 2008 Le Chic du Chien (Canine Chic) boutique dedicated entirely to pet accessories opened at 352 rue Saint-Honoré, right across the street.
233 rue Saint-Honoré, 75001 Paris, +33 (01)-42-60-57-04, goyard.com.
#### Hervé Chapelier
* * *
The French fashion company is known for its made-for-travel tote bags that come in a range of sizes and materials, including cotton and nylon, with braided handles made from seat belts. A great souvenir or solution for carting extra purchases home. Made in France.
390 rue Saint-Honoré, 75001 Paris, +33 (01)-42-96-38-04, hervechapelier.com, and an additional location on the Left Bank.
#### Opéra National de Paris Gift Shop
* * *
"I recently came across this shop and I was blown away by its great selection, from kids' storybooks to ballet flats, that I have never seen before!" says stylist and designer Maryam Malkapour.
La Galerie de l'Opéra de Paris, Palais Garnier, 8 rue Scribe, 75009 Paris, +33 (01)-53-43-03-97, boutique.operadeparis.fr.
#### Longchamp
* * *
Founded in 1948 as a pipe company (leather-covered pipes, natch), Longchamp expanded into accessories in the 1950s, and grew into a global brand on the success of its foldable Pliage nylon tote, first designed in 1993. More than 30 million Le Pliage bags have been sold worldwide, and it takes more than 100 steps to make each one. The brand has enlisted several designers to collaborate on designs, including Jeremy Scott, and it also has a modest ready-to-wear collection to accessorize the bags and small leather goods.
404 rue Saint-Honoré, 75001 Paris, +33 (01)-43-16-00-16, Longchamp.com, and additional locations.
#### Olympia Le-Tan
* * *
"A tiny little shop near Palais Royale with the most charming clothing and accessories by the designer herself," says jewelry designer Sonia Boyajian. Best known for clutch bags designed to look like books with fanciful artwork, Olympia Le-Tan's boutique also features her girlish print dresses and accessories.
Passage des Deux-Pavillons, 5 rue Des Petites Champs, Paris 75001, +33 (01)-42-36-42-92, olympialetan.com.
#### Perrin
* * *
Founded in 1893 in central France, Rigaudy-Perrin began a century-long tradition of glove making. In 2006, the family resurrected the brand and added luxury handbags, many of which nod to the brand's origins with their unusual grips (glove-like handles on clutch bags, for example). Other bags attach to the wrist with fur cuff bracelets, and others look to architecture for inspiration, such as the basket-weave Ball bag. The Paris flagship was designed by Chahan Minassian, with shelves displaying the items if they were rare specimens.
3 rue D'Alger, 75001 Paris, +33 (01)-42-53-54, perrinparis.com, and additional locations.
#### Repetto
* * *
Rose Repetto founded the French ballet shoe company in 1947, after her son, choreographer Roland Petit, came home from class complaining of sore feet. The company gained international attention for creating the Cendrillon ballet flat for Brigitte Bardot's 1956 film And God Created Woman. In 1959 Repetto opened the store at 22 rue de la Paix near the Paris Opera, and it remains open today. The brand has since gone global, with boutiques around the world selling professional dancewear, plus the Cendrillon flat, oxfords, Mary Janes, loafers, and bootie-style shoes that work as well on the street as on the stage.
22 rue de la Paix, 75002 Paris, +33 (01)-44-71-83-12, Repetto.com, and additional locations.
#### Roger Vivier
* * *
French designer Roger Vivier was the most famous shoemaker in the world in the 1950s. He invented the stiletto (depending on whom you ask), created the curved "comma" heel, was the first to use plastic in footwear, and designed the Belle Vivier chrome buckle-front pump that Catherine Deneuve wore in the 1967 film Belle de Jour. Since 2004, designer Bruno Frisoni has been at the helm of the brand, making exquisite creations, including red-carpet-worthy shoes and handbags, as well as the everyday Gommette flats.
29 rue du Faubourg Saint-Honoré, 75008 Paris, +33 (01)-53-43-00-85, rogervivier.com, and additional locations.
### AVENUE MONTAIGNE/AVENUE GEORGE V
#### Azzedine Alaïa
* * *
The revered designer has been going his own way his whole career, eschewing fashion week and trends in favor of his own timeline. After thirty years in business, he opened this maison in 2013. The renovated three-story 18th-century town house contains his showroom, offices, a workshop, and a boutique featuring an amazing chandelier as well as ready-to-wear and accessories. "The display, the clothes, and the accessories are always exquisite," says designer Zainab Sumu.
5 rue de Marignan, Paris 75008, +33 (01)-76-72-91-11, alaia.fr.
#### Balenciaga
* * *
Demna Gvasalia, the product of war-torn Georgia in the former Soviet Union, has redefined fashion at the vaunted house of Balenciaga with an off-kilter sense of cool that expresses itself in lopsided folksy floral dresses, trench coats designed to sit back on the shoulders, asymmetrical sweaters, and "broken heel" boots. Two side-by-side stores in the bottom of the iconic maison sell the clothing and accessories.
10 ave. George V, 75008 Paris, +33 (01)-47-20-21-11, Balenciaga.com, and additional locations.
#### Céline
* * *
Danish artist FOS created custom furnishings for the brand's divine, light-filled Avenue Montaigne flagship, including modern lamps, seating, and chandeliers that mimic the clean lines and raw materials that characterize the clothing and must-have shoes and handbags. Between the palm plants and pink marble tables, you'll get as much interiors inspiration here as fashion inspiration.
53 ave. Montaigne, 75008 Paris, +33 (01)-40-70-07-03, celine.com, and additional locations.
#### Dior
* * *
This maison is where Monsieur Dior himself used to fit his clients, including Marlene Dietrich, Ava Gardner, and Eva Perón, and where the haute couture creations are still made on the top floor. Dior presented his legendary New Look collection within these walls in 1947, and Carrie Bradshaw fell on the store's marble floor during the final episode of Sex and the City. The flagship offers the full collection of men's, women's, and children's clothing (the first Baby Dior store was established here), along with accessories, makeup, and perfume.
30 ave. Montaigne, Paris 75008, +33 (01)-40-73-73-73, dior.com, and additional locations.
#### Les Suites
* * *
Combining couture and ready-to-wear, the salon-like store is the brainchild of Ukrainian-born Eka Iukuridze, who was exposed to couture at an early age thanks to her parents. Pieces from Giambattista Valli, Haider Ackermann, and Rochas, as well as lesser-known designers from around the world, sit alongside made-to-order eveningwear from Copenhagen's Trash Couture, furs from Terzakou, and tailoring from Tailleur Premium Paris. "I work in interiors, so I'm entranced from the moment I walk in," says Brigette Romanek. "It's as if I'm walking into a palace and borrowing some of the queen's clothes. The dressing rooms, or 'suites,' are the most beautiful; the brands are elegant and sophisticated, like the store itself."
47 rue Pierre Charron, 75008 Paris, +33 (01)-56-59-11-11, boutiquelessuites.com.
#### Montaigne Market
* * *
"Always has all of my favorite pieces from every current designer," Rachel Zoe says of this multi-label boutique with more than 100 international brands, from haute to street, resortwear to denim, including Chloé, Altuzarra, Off-White, Victoria Beckham, Vetements, Frame Denim, Pitusa, Rochas, Rosetta Getty, and more. "The best-curated selection of must-have and up-and-coming designers," adds Shea Marie of Peace Love Shea. "It's also right by one of my favorite lunch spots, L'Avenue. I love sitting on the patio and people/fashion-watching. You can spot the chicest women (and men) in the world."
57 ave. Montaigne, 75008 Paris, +33 (01)-42-56-58-58, Montaignemarket.com.
### PALAIS ROYAL/LES HALLES
#### agnès b.
* * *
Synonymous with French chic since 1975, agnès b. sells the look defined by such icons as Jean Seberg and Françoise Hardy—subtle, everyday cool. There are more than 300 agnès b. stores worldwide selling simple tailored suits, long-sleeved striped T-shirts, and snap-front fleece sweatshirt cardigans. Founder Agnès Andrée Marguerite Troublé is an interesting character who has plenty of pursuits outside of fashion: She opened a gallery to exhibit graffiti artists in the 1980s, has dabbled in photography, and made a feature film in 2012, My Name Is Hmmm. It wasn't that much of a leap; an early agnès b. T-shirt read J'AIME LE CINEMA (the style is still available today), and it was an iconic agnès b white button-down shirt that Uma Thurman's character Mia Wallace wears in the 1994 film Pulp Fiction. Fans include Yoko Ono, Helena Bonham Carter, Natalie Portman, Philippe Starck, and Gérard Depardieu. She manufactures almost entirely in France. This is her original boutique.
6 rue du Jour, 75001 Paris, France, +33 (01)-42-33-04-13, agnesb.com, and additional locations.
#### Christian Louboutin
* * *
The Parisian shoemaker struck gold when he introduced his signature red soles, which have been worn by the most stylish ladies in the world. If there were a capital of Louboutin World, it would be here, in the storied retail arcade, Passage Véro-Dodat, where he opened his first store in 1991. The spot has such sentimental value, Louboutin designed the Passage bag with arch-shaped metal handles in honor of it. There is a men's store nearby.
19 rue Jean-Jacques Rousseau, 75001 Paris, +33 (08)-00-94-58-04, christianlouboutin.com, and additional locations.
ONLINE SHOPPING TIP
Famed Paris boutique Merci offers a nice selection of clothing, accessories, and home items that ship internationally at merci-merci.com. Another fun thing to do online is design your own Longchamp nylon Pliage tote at Longchamp.com. The French brand allows you to customize your own color, trim, and monogram, and ship it any place in the world. Jewelry designer Liseanne Frankfurt also recommends Poilane.fr, where you can order the best bread and have it sent from Paris. "Not the best for your carbon footprint, but really a treat once in a while and amazing to send to friends," she says.
#### Didier Ludot
* * *
The destination for museum-quality vintage clothing in Paris. This is where Reese Witherspoon's stylist and Haney designer Mary Alice Haney found the 1950 Christian Dior gown with rose garland embellishment, which Witherspoon wore to accept the Best Actress Oscar in 2006. "A regular haunt for my design team," says designer Tory Burch. "Like stepping back into a time of elegance and haute couture," says famous street style dandy Patrick McDonald.
Jardin du Palais Royal, 24 Galerie Montpensier, 75001 Paris, +33 (01)-42-96-56, didierludot.fr.
#### Pierre Hardy
* * *
The accessories designer, who founded his brand in 1999, is known for his quirky, architectural styles, including sneakers and messenger bags. He has also been responsible for Hermès's shoe design since 1990, and its fine jewelry design since 2001. In 2016, Hermès took a minority stake in Hardy's business.
Jardin du Palais Royal, 156 Galerie de Valois, 75001 Paris, +33 (01)-42-60-59-75, pierrehardy.com, and other locations.
### MONTMARTRE
#### Spree
* * *
Roberta Oprandi's version of the Paris concept store includes clothing and accessories by Acne, Marc by Marc Jacobs, Carven, Helmut Lang, and MM6 Maison Margiela casually draped on pieces of classic '50s designer furniture by Eames, Pierre Paulin, and others (the furniture is also for sale). Rotating artwork is featured on the walls. Across the street, you'll find more vintage furniture and art exhibitions in the gallery run by Roberta's husband, artist-designer Bruno Hadjadj, in an old DIY shop still marked Papiers Peints.
16 rue La Vieuville, Paris 75018, +33 (01)-42-23-41-40, spree.fr.
### REPUBLIQUE
#### Centre Commercial
* * *
Industrial space featuring artisan-minded clothing and accessories for men, women, and kids, plus lifestyle items. Brands include A Peace Treaty, Atlanta Mocassin, Birkenstock, Bodkin, Etudes, Filson, Norse Projects, Patagonia, and more.
2 rue de Marseille, 75010 Paris, +33 (01)-42-02-26-08, centrecommercial.cc.
#### Thanx God I'm A VIP
* * *
"Hands down the best vintage store in Paris," says Christene Barberich, editor in chief and cofounder of Refinery29.com of this spot, featuring racks of clothes arranged by color. It was launched in 1994 by Sylvie Chateigner, who is also a player in the Parisian nightlife scene, hosting regular house music parties at different locations around the city where the worlds of fashion, music, and media meet.
12 rue de lancry 75010 Paris, +33 (01)-42-03-02-09, thanxgod.com.
### TROCODERO
#### Reciproque
* * *
"Over the years, I accumulated a lot of amazing ball gowns and special finds from this secondhand store," says fashion tech entrepreneur Alexandra Wilkis Wilson of the Paris institution, which is actually six boutiques in a row selling tens of thousands of items from every brand in the fashion universe, from Alaïa to Yohji Yamamoto and everything in between.
89-101 rue de la Pompe, 75116 Paris, +33 (01)-47-04-30-28, reciproque.fr.
## MILAN
### ITALY
#### THE MAJORS
#### 10 Corso Como
* * *
Many consider this to be the first concept store in the world, founded in 1990 by former magazine editor Carla Sozzani (the sister of Vogue Italia editor in chief Franca Sozzani). "One of the first stores to do a cultural mix in the same space," says designer and stylist Maryam Malakpour. "It unites shopping (great brands, from Converse to Azzedine Alaïa), with a gallery, café, and bookshop." "I could spend hours, and I have, wandering through," adds retailer Emily Holt. Over the years, Sozzani has boosted the careers of everyone from Alaïa to Rei Kawakubo. The store's fragrance collection is top-notch, as is the accessories selection, and you'll always find interesting exclusives and collaborations, with everyone from Balmain to Swatch. In 2002, a 10 Corso Como branch opened in Tokyo, followed by Seoul, Shanghai, and Beijing. The brand's recognizable black-and-white logo, designed by American artist Kris Ruhs, as well as other artist-created graphics, has been splashed on fragrances, T-shirts, sneakers, and tote bags, making great souvenirs.
Corso Como, 10, 20124 Milano, +39 02 2900 2674, 10corsocomo.com.
10 CORSO COMO
#### Galleria Vittorio Emanuele II
* * *
One of the world's oldest shopping malls, this marble arcade with a stunning iron and vaulted glass roof dates back to the 19th century. Known as Milan's living room, it's the perfect place for an afternoon stroll, with plenty of cafés (have an apertivo at Bar Zucca, which Verdi and Toscanini favored) and, of course, shopping (Prada, Louis Vuitton, Tod's, Gucci, and more).
Piazza del Duomo, 20123 Milano, +39 02 8845 5555.
GALLERIA VITTORIO EMANUELE II
#### la Rinascente
* * *
The department store where Giorgio Armani got his start as a photographer and a buyer, this is the place to rub elbows with in-the-know Italians. Shop here for upscale clothing and accessories, plus sneakers, stockings (always better in Europe), and lingerie (who doesn't need a leopard-print Dolce & Gabbana undershirt?). The home section is full of colorful ceramics, tablecloths, and napkins, and the food hall on the top floor is a great resource for made-in-Italy delicacies. "A multi-brand department store with a view," says stylist Susie Sheffman. "Don't miss a post-shopping glass of prosecco at the rooftop bar, where you can practically reach out and touch Milan's magnificent Duomo." "I love it because they have such a beautiful and varied lingerie and hosiery department," adds costume designer Janie Bryant.
Piazza del Duomo, 20121 Milano, +39 02 88521, rinascente.it, and additional locations.
#### Prada at Galleria Vittorio Emanuele II
* * *
A mecca for fashion pilgrims, it has an old-world feel, with built-in wood shelves, a black-and-white checkerboard floor, and even the original cashier's desk and sign. The upstairs is packed with handbags; down the marble staircase in the basement is a treasure trove of shoes and clothing for men and women, as well as the brand's signature nylon totes and cosmetic bags displayed in antique steamer trunks—a nod to Prada's origins in the luggage business.
Galleria Vittorio Emanuele II, 63-65, 20121 Milano, Italy, +39 02 876979, prada.com, and additional locations.
#### Golden Quadrangle
* * *
Italy's biggest names are all represented on the cobblestone streets of the city's luxury shopping district—Versace, Valentino, Etro, Fendi, Tod's, Marni, Moncler, La Perla, Moschino, Miu Miu, Missoni, Emilio Pucci, Brunello Cucinelli, Roberto Cavalli, Borsalino, Tod's, and Gianvito Rossi—along with most international brands. "It's like the biggest mall in the world, only outdoors," says street-style star Anna Dello Russo. "You can go from one shop to the other one easily, and you can have fresh clothes right away."
Bordered by Via Manzoni (NW), Via della Spiga (NE), Corso Venezia (SE), and Via Montenapoleone (SW), 20123, Milano.
### Golden Quadrangle
#### Alan Journo
* * *
"Has a great selection of hats," says Anna Dello Russo, of this whimsical, high-end accessories boutique, where she picked up both her legendary cherry headpiece and her Peruvian doll headpiece, among many others. "For me, there's no vintage and no online. I like to spend time in the shops, talking to people."
Via della Spiga, 36, 20121 Milano, +39 02 7600 1309.
#### Aspesi
* * *
Understated sportswear and outerwear (including down-filled shirts) beloved by the Milanese. "I always stock up on their cashmere-and-silk scarves, which are amazing quality but won't break the bank," says fashion consultant Amanda Ross.
Via Monte Napoleone, 13, 20121 Milano, +39 02 7602 2478, aspesi.com.
#### Armani
* * *
Giorgio Armani's lifestyle emporium occupies an entire block on Via Manzoni. This minimalist palace features the designer's main line of suits and red-carpet gowns, as well as Emporio Armani, Armani Jeans, Armani Dolci (the best chocolates!), a florist, and a bookshop, as well as Milan's very own Nobu restaurant.
Via Alessandro Manzoni, 31, 20121 Milano, +39 02 7231 8600, armani.com.
#### Bottega Veneta
* * *
The first of the Italian luxury brand's "maison" concept stores is a massive residential-style space across two floors of an 18th-century building, with a curved staircase and windows opening onto a lush green courtyard. Creative director Tomas Maier designed the space, and the furniture inside. Featured is the entire product range of understated luxe ready-to-wear, accessories, and jewelry.
Via Sant'Andrea, 15, 20121 Milano, +39 02 7787 8116, bottegaveneta.com, and additional locations.
#### Brunello Cucinelli
* * *
Cashmere king Brunello Cucinelli's brand of casual super-luxe has made fans of Leonardo DiCaprio, Bradley Cooper, Jennifer Garner, and Girls Allison Williams, as well as one-percenters the world over. He has his headquarters in the medieval hilltop village of Solomeo in Umbria, Italy, but Milan is where his showroom is. His collections are well-represented in his boutique on Via della Spiga, which features made-to-measure suiting, plush cashmere sweaters and shawls, relaxed tailored separates, and the label's distinctive chain mesh embellished shoes and accessories.
Via della Spiga, 5, 20121 Milano, +39 02 7601 4448, brunellocucinelli.com, and additional locations.
#### DMAG Outlet
* * *
Nestled right in the high-end heart of Milan is this gem of an outlet (there are three locations), in which you can find overstock and past-season items from all the top brands, including Lanvin, Proenza Schouler, Tory Burch, and more.
Via Bigli, 4, 20121 Milano, +39 02 3664 3888, dmag.eu.
#### Dolce & Gabbana, Boutique Sartoria and Bar Martini
* * *
Housed in the 16th-century palazzo that has been home to the men's collections for years, Dolce & Gabbana Sartoria shares space with the men's shoe boutique, plus its Barbiere barbershop, Beauty Farm spa, and Martini Bar and Bistrot (in partnership with the Italian spirits brand Martini). The store opens onto a courtyard garden of citrus trees, palms, prickly pears, and other plants common to Sicily, the birthplace of Domenico Dolce. The women's boutique is just steps away at 2 Via Spiga.
Boutique Sartoria Man, Corso Venezia, 13, 20121 Milano +39 02 799135.
#### Fausto Puglisi
* * *
For Puglisi, who has dressed Katy Perry and Madonna on their tours, there was no question that Milan would be the location for his first store. Although he's known for a sexy, rock'n'roll aesthetic, the store also stocks tailored jackets and pleated skirts. The space is designed to resemble a Roman domus, complete with a geometric marble mosaic at the entrance that represents Puglisi's signature symbol, the sun.
Via della Spiga, 1, 20121 Milano, +39 02 7634 0454, faustopuglisi.com.
#### Gaimbattista Valli
* * *
Located in the courtyard of one of the historic buildings of Via Sant'Andrea, the Italian designer's Milan flagship showcases his entire frothy, flowery, elegant universe, including ready-to-wear and accessories, bags, shoes, jewels, and furs—even his signature sweet-smelling candle.
Via Sant'Andrea, 12, 20121 Milano, +39 02 780 218.
#### Gio Moretti
* * *
"Located on the famous Via della Spiga, it reminds me of Intermix in New York. I never leave without a pair of shoes. They have a great selection!" says tastemaker Shalini Kasliwal.
Via Della Spiga, 4, Milano, +39 02 7600 3186, giomoretti.com.
#### Gucci
* * *
The Montenapoleone flagship store is ground zero for the new, maximalist Gucci, as envisioned by creative director Alessandro Michele. Rihanna even stopped by to browse the Florentine brand's colorful handbags and clothes. This store sets the tone for others around the world and was the first to introduce the Gucci DIY service, with bags, denim jackets, silk bombers, tailored jackets, and more that can be customized in-store with patterns, patches, embroideries, and monograms. Head downstairs to the basement for the latest women's shoes and clothes.
Via Monte Napoleone, 5/7, 20121 Milano, +39 02 771271, Gucci.com, and additional locations.
#### La Sermoneta
* * *
The Rome-based glove maker has been in operation as a family-run business since 1960, offering gloves in every color and fabric imaginable.
Via della Spiga, 46, 20121 Milano, +39 02 7631 8303, sermonetagloves.com, and additional locations.
#### Madina Milano
* * *
This makeup store is a fashion insider favorite for must-have products like its Crystal Bronzing Powder and the Chic & Shine Highlighter Stick. It's great stuff and the prices are drug store–cheap.
Corso Venezia, 23, 20100 Milano, +39 02 7601 1692, madina.it.
#### Moschino
* * *
The Italian brand's biggest flagship in Italy, this boutique is a supersized showcase for creative director Jeremy Scott's wonderfully wacky vision. Feast your eyes on a giant mannequin that stretches across the entire store window, an XL-size leather-jacket- shaped handbag that serves as a display counter, a pair of huge high-heel pumps that act as shelves for actual shoes, and a gigantic teddy bear in the children's area.
Via Sant'Andrea, 12, 20121 Milano, +39 02 7600 0832, Moschino.com, and additional locations.
#### Ottod'ame
* * *
This Italian casual chic line with a girlie touch has fun frilly blouses, Lurex pleated skirts, faux fur coats, velvet sandals, and more, all at a contemporary price point.
Via Alessandro Manzoni, 39, 20121 Milano, +39 02 6556 0409, ottodame.it.
### DUOMO/CORSO VITTORIO EMMANUEL
#### Dixie
* * *
This Italian fast fashion chain is somewhere between H&M and & Other Stories, both in price point and aesthetic. In addition to trendy jogger pants and slip dresses, there are some surprisingly elegant pieces in the shops, including an oversized cable-knit sweater with pearl detail that was reminiscent of Dries Van Noten, Lurex pleated skirts in a range of colors, and retro-print pussy bow blouses. It's worth a look.
Via San Maurillo, 2, 20100 Milano, +39 02 7209 4475, shop.dixie.it, and additional locations.
#### Excelsior Milano
* * *
This luxury department store, designed by Jean Nouvel and Vincenzo de Cotiis, occupies a former Milanese cinema. The store spans 4,000 square feet and seven floors, with a perfume gallery, a Tiffany & Co. shop-in-shop, and clothing for men and women with denim and high-end streetwear (Enfants Riches Déprimés, K-Way, Yeezy, Visvim), ready-to-wear (Cédric Charlier, Chloé, Stella Jean, Versace, Facetasm), plus must-have accessories from Sophia Webster, Sara Battaglia, Aquazzura, Manolo Blahnik, and more. "I love the food hall downstairs," says fashion journalist Divia Harilela of the underground gourmet grocery, which boasts what was the most beautiful selection of fresh tomatoes I've ever seen. Only in Italy!
Galleria del Corso, 4, 20122 Milano, +39 02 7630 7301, excelsiormilano.com.
#### Funky Table
* * *
This fun home décor store across the street from Wait and See carries mismatched dishes, lace-like baskets, furry totes and more fun stuff.
Via Santa Marta, 19, 20123 Milano, +39 02 3674 8619, funkytable.it.
#### Wait and See
* * *
"Carries lots of great under-the-radar labels, often from Belgium and Holland," says Christina Binkley. "I buy clothes there, and sometimes other things, like a bunch of silver spoons engraved with YOU ARE LOVED that I gave my kids, husband, and mom."
Via Santa Marta, 14, 20123 Milano, +39 02 7208 0195, waitandsee.com.
### NAVIGLIA/PORTE TICINESE
#### Antonioli
* * *
Dark and edgy men's and women's boutique housed in a 1920s silent- movie theater, Antonioli stocks Rick Owens, Rodarte, Givenchy, Vetements, and more.
Via Pasquale Paoli, 1, 20143 Milano, +39 02 3656 6494, antonioli.eu.
#### Biffi
* * *
A cornerstone of the Milan boutique scene since the 1960s, when the Biffi sisters, Rosy and Adele, opened two stores in Milan. They've stocked such international designer labels as Kenzo and Yohji Yamamoto since the beginning. You'll also find Marni, Stella McCartney, MSGM, Stella Jean, Sacai, Tod's, Alexander McQueen, and Thom Browne, both men's and women's. The sister store, Banner, is in the Golden Quadrangle at Via Sant'Andrea 8.
Corso Genova, 6, 20123 Milano, +39 02 8311 1601, biffiboutiques.com.
#### LADoubleJ
* * *
Fashion journalist, consultant, street style favorite, and L.A.-to-Italy transplant JJ Martin opened this gorgeous showroom for her überchic collection of clothing (boy shirts, ball skirts, swing dresses, scarves, and more) and furniture made from reissued vintage prints from Italian textile company Mantero. Designed by architect Luca Cipoletti, the space features a white exhibition grid, called Wunderwall, which displays clothing, vintage jewelry that once belonged to legendary Italian fashion editor Anna Piaggi, design objects, and ceramics. Email for an appointment. And be sure to check out Martin's website for all things chic in Milan (and a killer selection of vintage clothing and jewelry that ships worldwide).
Piazza Arcole 4, 20143 Milano, ladoublej.com.
#### Navigli Antiques Market
* * *
Held on the last Sunday of the month, this market hosts 400 antiques dealers along the canals. "I always find great vintage finds and new treasures here," says shoe designer Brian Atwood. "Italy isn't big on vintage, but in this area you can find some interesting shops and cafés."
Alzaia Naviglio Grande, starting at the Via Valenza bridge on Viale Gorzia, 20144, Milano.
#### Pupi Solari
* * *
A Milan institution, the silver-haired Solari started with a children's shop in 1969, expanding her store over the years to include women's clothing and accessories (there's also a men's store, Host, next door). She is the arbiter of the classic, non-flashy Italian look, with understated clothing and accessories from Dusan, Emma Hope, Mantu, Sofie D'Hoore, Tomas Meier, George Esquivel, Massimo Alba, Massimo Piombo, Alberto Biani, and Aspesi.
Piazza Tommaseo, Via Lorenzo Mascheroni, 20123 Milano, +34 02 463 325, pupisolari.com.
### PORTO VENEZIA
#### Imarika
* * *
This small boutique packs a big punch with elegant, but sparkly frocks from Rochas, Simone Rocha, shoes by Elena Iachi, and more. It also has its own private label.
Via Giovanni Morelli, 1, 20129 Milano, +39 02 7600 5268, imarika.it.
### PORTO ROMANA
#### Etro Outlet
* * *
One of the few true outlet experiences left, meaning you'll still experience the thrill of the hunt in this small shop, where you can dig through bins of paisley scarves, a basement full of racks of last season's patchwork blazers and silk patterned dresses, and shelves of fringed leather goods from the Italian luxury stalwart. Its recent boho turn has found fans with Florence Welch, Rosie Huntington-Whiteley, and other stars. If you're a fan of the label, it's worth a stop.
Via Spartaco, 6, Milano, +39 02 5502 0218, etro.com, and other locations.
#### Rose's Roses Shoes
* * *
"Rosa Aiuto sells her designs—the brand is Rose's Roses—mainly in Italy," says fashion journalist Christina Binkley. "She used to design shoes for Marni and for Vera Wang. Her own style is more modern-classic, and she's always way ahead of the pack. I bought oxfords from her two years before they became a 'thing.' She recently launched online sales, but I go to her studio in Milan's Porta Romana neighborhood."
Corso Lodi, 5, 20135 Milano, +39 02 5412 1998, rosesroses.it.
### BRERA/CASTELLO/CORSO COMO
#### Antonia
* * *
The multi-brand boutique began with accessories in 1999, followed by a menswear location. In 2013 the two stores were combined in a renovated 17th-century palazzo. The menswear is street-casual (think Riccardo Tisci for Nike, Hood by Air, Balmain, Saint Laurent, Sacai), while the women's combines streetwear and denim (Fenty x Rihanna, Frame Denim, Isabel Marant) with haute labels (Gucci, Fendi, Anthony Vaccarello). Whenever she's in Milan, Amanda Ross hits this spot to shop the "design collaborations and exclusives" she can't get anywhere else.
Via Cusani, 5, 20121 Milano, +39 02 8699 8340, antonia.it.
#### Cavalli e Nastri
* * *
Italian designers head to these cheery vintage boutiques, which feature a masterful mix of high and low from Kenzo, Moschino, Gucci, and more. Each location has a different twist: At Via Brera, 2, you'll find dresses and accessories from 1940 to today as well as pieces by young new designers; on Via Gian Giacamo Mora, 12, you can pick up more unique pieces; while Via Gian Giacomo Mora, 3, stocks vintage menswear, as well as interior and housewares.
Via Brera and Via Gian Giacomo Moro, 20121 Milano, +39 02 7200 0449, cavallienastri.com.
#### Corso Como Outlet
* * *
If your wallet is on the light side, this outlet offers past-season merch from the big store, in addition to the odd pair of Prada or Manolo Blahnik shoes.
In the courtyard at Via Tazzoli, 3, 20154 Milano, +39 02 2901 5130.
#### Elena Iachi
* * *
This made-in-Italy shoe brand founded in 2011 is known for embellished cowboy booties and clogs, but what's kept us coming back lately is the elegant slip-on sneakers in floral velvet or sequined fabric, with crystal or fur embellishment.
Piazza Venticinque Aprile, 9, 20154 Milano, +39 02 6556 0413, elenaiachi.it.
#### ERAL 55
* * *
"I always appreciate a very singular point of view, and Ermanno Lazzarin really pulls it off," says Ann Mashburn of this menswear destination where you'll find everything from Japanese and American workwear to Italian tailoring, from ascots to cowboy boots. "He is not following trends, but driving trends himself."
Piazza Venticinque Aprile, 14, 20124 Milano, +39 02 659 8829, eral55.com.
#### Golden Goose Deluxe Brand
* * *
Founded in 2000, this Italian brand, which is responsible for lived-in luxury sneakers that have become a fashion cult classic, opened its first boutique in Milan in 2013. The store also features the leather jackets, classic shirts, and distressed boots that round out the collection.
Via Ponte Vetero, 1, 20121 Milano, +39 02 8901 1921, goldengoosedeluxebrand.com, and additional locations.
#### Madame Pauline Vintage
* * *
"I found this place by accident," Christene Barberich, editor in chief and cofounder of Refinery29, says of this vintage store. "I was running from lunch to a show and spotted all the bright colors and textiles out of the corner of my eye. Even though I was running late, I did a crazy speed round, and returned the next day when I had more time. I found a gorgeous lipstick-red topcoat that people always ask me about. The tailoring is next level."
Foro Buonaparte, 74, 20121 Milano, +39 02 4943 1201, madamepaulinevintage.it.
#### Massimo Alba
* * *
Based in Milan, Alba creates luxury T-shirts, chinos, and linen jackets that exemplify the new Italian elegance. Cashmere is a speciality.
Via Brera, 8, 20121 Milano, +39 02 7209 3420, massimoalba.com.
#### Moleskine Flagship and Cafe
* * *
The Milan-based brand of notebooks and planners opened this hybrid café–art gallery in 2016 in the heart of the Brera Design District. Milanese coffee roaster Sevengrams serves single-origin brew, notebooks are for sale, and hanging out is encouraged. Breakfast learning sessions and exhibitions are part of the concept.
Corso Garibaldi, 65, 20121 Milano, +39 02 7200 0608, moleskine.com.
#### Momoni
* * *
What started in 2009 as a lingerie brand has grown into a full line of casual, colorful ready-to-wear, including dreamy garden-floral silk blouses, Lurex knits, softly tailored jackets, and boyfriend jeans with silver embroidered details.
Corso Como, 3, 20154 Milano, +39 02 6379 3466, momoni.it, and additional locations.
#### MSGM
* * *
Launched by former DJ Massimo Giorgetti in 2009, this Italian contemporary brand specializes in bright hues, digital prints, and fun, Insta-friendly shapes.
Via Ponte Vetero, 9, 20121 Milano, +39 02 36595680, msgm.it.
#### No° 21
* * *
After parting ways with his namesake brand in 2009, Italian designer Alessandro Dell'Acqua founded Noº 21—named after his birthdate and lucky number. His Milan flagship features sumptuous marble floors and an accessories room with walls and floor made of the same stones used to build the Duomo. You'll find clothes and accessories with lots of romantic folds and flourishes that still manage to be sporty. Dell'Acqua also designs for the French house of Rochas.
Via Santo Spirito, 14, 20121 Milano, numeroventuno.com.
#### Piazza Gae Aulenti
* * *
Milan's newest business hub boasts the highest skyscraper in Italy (the Unicredit Tower) and is surrounded by arcades with numerous upmarket boutiques, including funky Italian design brand Toiletpaper. Opened in 2012 adjacent to Milan's main train station, the futuristic-looking square has become a tourist attraction with its vast reflecting pool, colosseum-style outdoor seating and pedestrian walks.
Piazza Gae Aulenti, 20154 Milano.
#### Richard Ginori 1735
* * *
For more than two centuries, this Florentine porcelainmaker has created tableware in cheery colors with Italian flair. "The most beautiful porcelain . . . it and the shop feels like a sanctuary," says retailer Ann Mashburn. "I love the little painted insects."
Piazza San Marco, 3, Milano, +39 02 8901 1646, richardginori1735.com.
#### Vintage Delirium
* * *
One of the world's best vintage stores, Vintage Delirium has museum-quality pieces by Thierry Mugler, Versace, Pucci, Pierre Cardin, and Callot Soeurs.
Via Giuseppe Sacchi, 3, 20121 Milano, +39 02 8646 2076, vintagedeliriumfj.com.
### SEMPIONE
#### Riedizioni
* * *
Milan-born textile designer Cevese's charming collection of bags, pouches, and clutches made out of feathers, fabric scraps, and industrial waste pressed in plastic is available by appointment at her showroom and store.
Via Pallavicino, 22, 20145 Milano, by appointment only, +39 02 3982 6647, riedizioni.com.
### ORTICA
#### Outlet Marni
* * *
Tucked away on a residential street, this fashion insider favorite is worth the trip. "Knowing editors and buyers discreetly slip behind the iron gate that swings open and shuts with a jarring clang. If you've come this far, rejoice! Once inside, count your blessings and gawk at heaps of whimsical, marked-down Marni print dresses, coats, jewelry, furs, leathers, bags, and shoes, and at the squealing editors paying respects to the church of Marni," says stylist Susie Sheffman. You'll find an array of outerwear, fur, dresses, sweaters, bags, jewelry, and sunglasses for men, women, and children starting at 50 percent off.
Via Giancarlo Sismondi, 65, 20133 Milano, +39 02 7104 0332, marni.com.
## ROME
### ITALY
#### Battistoni
* * *
The classic Rome tailor has made bespoke pieces for everyone from Marlon Brando to Sophia Loren. "A real treat to visit," says retailer Ann Mashburn. "The chicest men's shop, and amazing art on every wall."
Via dei Condotti, 61A, 00187 Roma, +39 06 697 6111, battistoni.com.
#### Eleuteri
* * *
Dates back to 1897, when it opened as a café, just off the Spanish Steps. Owner Pietro Eleuteri's son, Carlo, moved from the restaurant business into collecting antiques, specializing in vintage jewels. Fast-forward to today, and Eleuteri is one of Italy's premier vintage and antique jewelry destinations, attracting an international jet set that browses pieces by Bulgari, Cartier, and Van Cleef & Arpels alongside 18th-century Southern Italian coral items and rare silverware. There is also a branch of the store in New York.
Via dei Condotti, 69, 00187 Roma, +39 06 678 1078, eleuteri.nyc.
#### Fendi
* * *
After becoming the benefactor of the Trevi Fountain's restoration and resurrecting the Palazzo della Civiltà, an abandoned Fascist gem that is now the Rome-based luxury brand's headquarters, Fendi opened its largest flagship in the world right by the Spanish Steps. The Palazzo Fendi includes two floors of retail space designed by architect Gwenael Nicolas, a boutique hotel, a VIP apartment, a restaurant, and a rooftop bar. Most of the furniture is by Fendi Casa, and the Fendi family's art collection is on display. Inside the store, you'll find everything from fashion's most over-the-top furs to fur bag charms.
Via della Fontanella di Borghese, 48, 00187 Roma, +39 06 33 4501, fendi.com, and additional locations.
FENDI
ONLINE SHOPPING TIP
For Italian style, Luisaviaroma.com can't be beat. The site (which also has bricks-and-mortar spaces in Florence) features the best of the best of Italian (and European, Asian, and American) designers, including their most over-the-top pieces, as well as showcasing new talents, and it also ships internationally. Another must-shop site is LaDoubleJ.com, which has an expertly curated selection of vintage fashion and accessories come to life within the worlds of high style and interior design in Milan, plus the latest information about the city's creative talent, hidden shops, tantalizing food, and best-kept secrets. For made-in-Capri sandals online, delivered to your doorstep, try Canfora.com.
#### Saddlers Union
* * *
Launched in 1957, Rome's heritage leather-goods-maker gained a cult following, including Jackie Kennedy, who sported the shop's saddlebag. You can find all kinds of leather bags here, from bucket shapes to briefcases.
Via Margutta 11, 00186 Roma, 00187, +39 06 3212 0237, saddlersunion.com.
#### Valentino
* * *
It's fitting that the Roman fashion house's biggest flagship would be here, neighboring the 16th-century Palazzo Mignanelli, where the brand is headquartered. Opened in 2015, it was designed by architect David Chipperfield and houses the men's and women's collections, as well as accessories and made-to-measure denim for men.
Piazza di Spagna, 35/38, 00187 Roma, +39 06 9451 5710, valentino.com, and additional locations.
JJ MARTIN
ITALIAN INSIDER'S GUIDE
* * *
One of Milan's most stylish women, JJ Martin's ladoublej.com is dedicated to the best of Italian style, talent, and fashion.
My favorite big-city shop in Milan is PRADA in Montenapoleone. It sounds banal, but I associate the brand so closely with the Milanese aesthetic that it comforts me to go in there and just breathe in the style of the city that I admire. I also know most of the staff, which, for me, counts more than anything—I walk in and we chat, we play, they bring out a bunch of shoes no one else likes that they know I will appreciate. It's a very personal, homey environment, one that feels more and more essential to where I go and how I spend my time. (Be sure to stop by the Prada-owned HISTORIC PASTICCERIA MARCHESI next door for a snack.)
I love the upstairs emporium of ROSSANA ORLANDI'S design gallery in Milan. She's got an amazing selection of colorful plates, plastic glasses, and tabletop items you never knew you needed. When you go in the springtime you feel like you've entered an enchanted garden. I never miss going to the vintage furniture shop owned by RAIMONDO GARAU. He picks pieces no one else would ever consider and they look incredible together—plus, I love sitting and discussing decorating ideas with him. I also make sure to browse the colorful combination of unusual and not-that-expensive fashion at Uberta Zambeletti's well-edited shop WAIT AND SEE. These are creative Italian people who embody the qualities that I admire.
On the weekends we often travel to my husband's hometown of Pesaro, a small town on the Adriatic coast that no one cool ever visits and is not on any tourist hit list. This seaside town is a quintessential example of the Italian provinces. What was amazing to me was discovering two important shops here. One is an impressive multi-brand store called RATTI, which has been around for 100 years, and has an Hermès shop-in-shop in addition to edgy clothes from Christopher Kane and Erdem and an amazing selection of Valentino. I love finding these shrines dedicated to beauty in the middle of nowhere, where women come out like ants after the rain and you're like, "Where do all of these people live?" The other shop in Pesaro is ZUCCA, which has a very small but perfectly edited collection of vintage jewelry and furniture. No one in Pesaro shops there, so I'm not sure how it stays in business. I met the owners years ago and they have impeccable taste—whenever I go, we sit for hours, have an aperitivo, and recount vintage stories.
PRADA, Via Monte Napoleone, 6, 20121 Milano, +39 02 7602 0273, prada.com.
HISTORIC PASTICCERIA MARCHESI, Via Monte Napoleone, 6, 20121 Milano, +39 02 76008238, pasticceriamarchesi.it. Rossana Orlandi, Via Matteo Bandello, 14/16, 20123 Milano, +39 02 467 4471, rossanaorlandi.com.
RIAMONDO GARAU, Viale Francesco Crispi, 5b, 20121 Milano, Italy, +39 02 659 9913, riamondogarau.com.
WAIT N' SEE, Via Santa Marta, 14, 20123 Milano, +39 02 7208 0195, waitandsee.it.
RATTI, Via Gioacchino Rossini, 71/74, Pesaro, +39 07 213 1031, rattiboutique.com.
ZUCCA, Via Sabbatini, 12, Pesaro, +39 33 5597 5176, zuccadesign.it.
## FLORENCE
### ITALY
#### Elio Ferraro
* * *
Ferraro is known as Italy's maestro of vintage, and is a legendary figure on the fashion scene who has worked as a consultant and curated luxury vintage shops around the world. McQueen, Missoni, Mugler, and more, he's got the most important pieces, plus vintage furnishings. "Fabulous vintage emporium," says designer/personal shopper Raven Kauffman. "From vintage Pucci to rare Fornasetti housewares, there is always something wonderful to bring home."
Via Pavione, 47/R, 50123 Firenze, +39 055 290 425, ElioFerraro.com.
#### Gucci Museo and Icon Store
* * *
When Guccio Gucci opened his leather goods store in Florence in 1921, he couldn't have dreamed that his name would go global, and the brand grow big enough to warrant this three-story museum, full of archival clothing, accessories, and one-offs, such as a Cadillac upholstered in GGs. To round out the Gucci experience, there's a café, bookstore, gift shop, and Icon store dedicated to limited-edition products available only here.
Piazza della Signoria 10; +39-055-7592-3302, guccimuseo.com.
#### Lorenzo Villoresi
* * *
"Bypass his more available scents that you might see in boutiques and go directly to his atelier," says Raven Kauffman. "It's like entering a medieval alchemist's workshop. He takes only a few appointments each week, and it's a totally divine indulgence."
Via de'Bardi, 12, 50125 Firenze, +39 055 234 1187, lorenzovilloresi.it.
#### Loretta Caponi
* * *
"For the most beautiful nightgowns and custom linens," says Raven Kauffman. "Their embroidery is exquisite and you can work with the amazing artisans to create your own patterns. The lingerie is frothy and delicate, and their cotton pajamas are the best."
Piazza degli Antinori, 4/red, 50123 Firenze, +39 055 211 074, lorettacaponi.com.
#### Luisaviaroma
* * *
The three-floor, eighty-year-old Florentine fashion institution was early to the e-commerce game, launching its site in 1999, and building a loyal community of online shoppers with an edgy edit of more than 600 designers. The brick-and-mortar flagship doesn't disappoint, with all the top designers represented, from Alexander McQueen to Zuhair Murad, and a special emphasis on the Italians. "I could spend all day there," says tastemaker Shalini Kasliwal.
Via Roma, 19-21r; +39 055 906 4116, luisaviaroma.com.
LUISAVIAROMA
#### Luisaviaroma Contemporary
* * *
This outlet store sells clothing and accessories from previous seasons starting at 30 percent off. Canadian model-blogger Elizabeth Minett found her best score of all time here, a YSL clutch.
Via Silvio Pellico, 9, 50121 Firenze, +39 05 521 782, luisaviaroma.com.
#### Playground
* * *
This two-story men's and women's boutique is young and fun all right, with wear-it-out-the-door pieces by tons of modern contemporary brands, including Masscob, No. 21, Derek Lam 10 Crosby, Seafarer, and Iro, plus Golden Goose and Common Projects sneakers, Herschel Supply bags, and more.
Viale Don Giovanni Minzoni, 31/A, 50129 Firenze, +39 055 575 654, playgroundshop.com.
#### Salvatore Ferragamo Museo and Flagship
* * *
This 700-year-old Palazzo Spini Feroni, purchased by the shoemaker to the stars in 1938, is the spiritual home of the brand, with a fresco ceiling, arched entrances, many archival photographs, and shoes made from 1927 to 1960, the year Ferragamo died.
Flagship: Via dei Tornabuoni, 4r-14r, +39 055 292123. Museum: entrance on Palazzo Spini Feroni, Piazza Santa Trinita, 5r, 50123 Firenze, +39 055 3360 456/455, ferragamo.com.
ERICA PELOSINI
FLORENCE & CAPRI
* * *
A stylist, art director, and magazine contributor, Leeman is a jet-setting street-style star with a look that's equal parts rock-and-roll and la dolce vita. She hails from Florence and is married to Dutch shoe designer Louis Leeman. The couple make their home in Los Angeles, though they return to Italy as often as possible.
I love to shop at LUISAVIAROMA in Florence, maybe because it is in the city where I grew up and I remember going with my mum when I was little! It feels like home to me, with the best clothes ever.
Everything in Florence is within walking distance, and while shopping, I love to get a cappuccino at GIACOSA, the historic bar in the ROBERTO CAVALLI boutique, or a prosecco and a truffle sandwich at a classic winery on via Tornabuoni.
What is magical about Florence? Well, besides walking around some of the world's most incredible monuments, you can find yourself in hidden shops or restaurants inside some majestic Renaissance palazzo filled with affresco walls and precious chandeliers. It's art meets fashion, in the best way.
CAFFÉ GIACOSA
A secret local gem is IL RICCIO in Capri. It's an adorable beach club hidden on a top of a hill in Capri. Not easy to reach! The best way to get there is with a little old gozzo, the typical wooden Capri boat. Once you arrive you will be enchanted by all the blue-and-white décor and the most delicious cuisine you've ever had. It's no wonder I chose them to cater my wedding (even though it was on the opposite part of the island). To finish you can visit their "Temptation Room," a place filled with beautiful desserts, and even get your new beach outfit at their little boutique located in a rock on the cliffside.
One of my favorite things about vacationing in Capri in the summer is my shopping routine. There is a little sunglasses store called CAPRI PEOPLE, which makes the cutest sunglasses. You can even choose your lenses to go with them. LA BOTTEGA CAPRESE, a few steps away, is where you can make your own Capri sandals. By the time you are finished with your ice cream, your sunglasses and sandals will be ready! Of course, after all these years spent in Capri, I have a huge collection—but I just can't resist!
LUISA VIA ROMA, Via Roma, 19/21, 50123 Firenze, +39 055 217 826, luisaviaroma.com.
CAFFÈ GIACOSA, Via della Spada 10/r 50123 Firenze, +39 055 277 6328, caffegiacosa.com.
ROBERTO CAVALLI, ia dè Tornabuoni, 83/r, 50123 Firenze, +39 055 239 6226, robertocavalli.com.
ILL RICCIO, Via Gradola, 4, Anacapri, +39 081 837 1380, capripalace.com.
CAPRI PEOPLE, Via Le Botteghe, 27, Anacapri, +39 081 837 8664, capripeople.it.
LA BOTTEGA CAPRESE, Via G. Orlandi, 26, Anacapri.
## VENICE
### ITALY
#### Attilio Codognato
* * *
"When I first came across this extraordinary jewelry store, I stood mesmerized in front of the shop window—the jewelry pieces were like nothing I had ever seen before," says model-muse Tatiana Sorokko. "Enormous snake bracelets in gold and diamonds, stunning memento mori scull rings and necklaces, intricate blackamoor brooches, all exquisite works of art, and elaborately crafted. I knew instantly that I was hooked. I learned later that the Codognato jewelry house was Venice's best-kept secret. It was founded in 1866 by the great-grandfather of the present owner, Attilio Codognato. It soon became a go-to for European royalty as well as such tastemakers as Coco Chanel, Barbara Hutton, Maria Callas, and Elizabeth Taylor. Attilio has become my good friend, and I visit with him in Venice at least once a year; each time I come to his store, I am struck by the unique and inimitable elegance and beauty of his creations." Designer and personal shopper Raven Kauffman is also a devotee. "It's for the decadent Victorian Goth that lurks inside me," she says. "One of his snake cuffs and delicate cameo rings is on my manifestation board for 'One day you will be mine.' "
San Marco, 1295, 30124 Venezia, +39 041 522 5042, attiliocodognato.com.
VENICE
#### Dittura Giani
* * *
For velvet furlane shoes like the gondoliers wear, Venice's version of the espadrille, this is the place.
San Vio, 871, 30123 Venezia, +39 041 522 3502.
HOW TO HOLIDAY IN CAPRI LIKE MARY ALICE HANEY
* * *
A twenty-year veteran of the fashion industry in L.A., Haney worked as an editor at Allure, Marie Claire, and GQ, a stylist for A-listers such as Blake Lively and Sarah Jessica Parker, and a TV host before launching her collection of luxury eveningwear in 2013. Since then, she's dressed Taylor Swift, Reese Witherspoon, Kate Hudson, and many more. The Chatanooga, Tennessee, native travels to Capri almost every summer, and was even married there.
The tiny SUD CAPRI swim shop, squeezed into the island's main luxury drag, has a sexy assortment of clothes that are not just swimsuits. You have to dig and the store gets super hot in the summer, but I love it.
LA PARISIENNE is the store where Jackie Kennedy first bought her capri pants. The same family has owned it since it opened, and it's still the best place to get the original Jackie O look. A made-to-measure pair of capris takes only one day to make. The boutique also sells flamboyant designs by Roberto Cavalli and the latest Fendi bags.
My favorite sandal store is DA COSTANZO. Founded by Costanzo Ruocco in the 1960s, the small workshop is now run by Costanzo's son Antonio, who can whip up a new set of sandals in a matter of hours (or the next day for unusual requests). Prices start around $95.
SUD CAPRI, Via Camerelle 61, 80076 Capri, +39 08 1838 9445.
LA PARISIENNE, Piazza Umberto I/7, 80076 Capri, +39 08 1837 0283, laparisiennecapri.it.
DA COSTANZO, Via Roma 49, 80076 Capri, +39 06 697 6111.
## MADRID
### SPAIN
#### Adolfo Domínguez
* * *
The Galician designer's beautifully cut suits, asymmetrical dresses, and quirky footwear are mainstays in Spain, where he is known for a certain urban sophistication. He recently launched an eco-chic line of clothing made of sustainable materials. There's also a coffee shop that serves organic food and smoothies.
Calle de Serrano, 5, 28001 Madrid, +34 914 36 26 00, adolfodominguez.com.
#### Agatha Ruiz de la Prada
* * *
One of Spain's most colorful designers (think, a Spanish Betsey Johnson), the Madrid-born Ruiz specializes in fun, fearless designs for men, women, children, and the home, with lots of rainbows and hearts.
Calle de Serrano, 27, 28001 Madrid, +34 913 19 05 01, agatharuizdelaprada.com.
#### Beni Room
* * *
A cute, contemporary boutique founded by fashion industry vet Beatriz Nicolás, who has an eye for spotting new international brands.
Calle de Claudio Coello, 14, 28006 Madrid, +34 917 37 37 80, beniroom.com.
#### Bimba y Lola
* * *
Established in 2005 by two nieces of designer Adolfo Domínguez, the Vigo-based Bimba y Lola has nearly 200 stores across the world. The emphasis is on jackets, blouses, dresses, and pants in playful, fashion-forward shapes and cool prints. Great statement jewelry, too.
Calle de Serrano, 22, 28001 Madrid, +34 915 76 11 03, bimbaylola.com, and additional locations.
#### Capas Seseña
* * *
Founded in 1901, this famous, capes-only store has sold capes to Picasso, Michael Jackson, and Hillary Clinton. The velvet-lined style that designer Carolina Herrera gave to fashion editor André Leon Talley? It came from this Spanish treasure, which has styles from the traditional (an elegant cloak with frog closures, for example) to the contemporary (bomber capes) for men and women.
Calle de la Cruz, 23, 28012 Madrid, +34 915 31 68 40, sesena.com.
GRAN VÍA, MADRID
CAPAS SESEÑA
#### Casa Hernanz
* * *
There are countless stores selling rope-soled espadrilles in Spain (including Casa Crespo), but only one Casa Hernanz, which dates back more than 150 years, when Toribio Hernanz, the great-grandfather of the current owners, opened it in 1840. You'll find dozens of colors, seasonal prints, and styles, in cotton, canvas, and silk, at prices that encourage buying multiple pairs.
Calle de Toledo, 18, 28005 Madrid, +34 913 66 54 50, alpargatashernanz.com.
#### Antigua Casa Crespo
* * *
This rival espadrille shop has also been around for a century and a half, is family owned, and is packed ceiling to floor with alpargatas, as they are known in Spain, in every silhouette imaginable. Among the many customers is Queen Sofia of Spain.
Calle del Divino Pastor, 29, 28004 Madrid, +34 915 21 56 54, alpargateriacrespo.com.
#### Delitto E Castigo
* * *
Ranked by Vogue as one of the world's top twenty-five stores, with an exquisite selection of labels for men and women, including Emilio Pucci, Dolce & Gabbana, Balmain, Comme des Garçons, Givenchy, and many others.
Calle de Claudio Coello, 26, 28001 Madrid, +34 915 77 77 29, delittoecastigo.com.
#### Delpozo
* * *
Jesús del Pozo founded his Madrid fashion house in 1974. Since his death, Delpozo has found new life under the direction of Josep Font. The architect-turned-designer's dresses and tops with playful volumes and couture-level embroideries are a hit with the social set and with young Hollywood celebs, including Kiernan Shipka. Font opened this flagship in 2013.
Calle de Lagasca, 19, 28001 Madrid, +34 912 19 40 38, delpozo.com
#### El Corte Inglés
* * *
Spain's most prominent department store carries everything from auto parts to groceries, lingerie to designer fashion. Madrid's largest branch is on the corner of Calle Raimundo Fernández Villaverde and Castellana. "You can literally get your teeth cleaned, your hair done, insure your car, buy a Carolina Herrera gown, and a Chanel lipstick," says jewelry designer Liseanne Frankfurt. "If they don't have it, it doesn't exist!" The store's Gourmet Experience offers food products from all over Spain and features the creative cuisines of notable Spanish and international chefs, including Michelin-star chefs David Muñoz, Jordi Roca, and Roberto Ruiz. "I like to eat at StreetXO," says author and journalist Susana Martínez Vidal. "It's the less expensive offshoot of the three-star DiverXO. David Muñoz delights with exquisite plates, prepared quickly and at incredibly low prices."
Calle de Raimundo Fernández Villaverde, 79, 28003 Madrid, +34 914 18 88 00, elcorteingles.eu, additional locations.
#### Ekseption
* * *
"The most important multi-brand store in Madrid," says Susana Martínez Vidal. This gallerylike space stocks top designers, including Alaïa, Balenciaga, Prada, Givenchy, and Marc Jacobs. Next door, Eks targets the younger crowd with Alexander Wang, Acne, and the like.
Calle de Velázquez, 28, 28001 Madrid, +34 915 77 43 53, ekseption.es.
#### Felipe Varela
* * *
Hailing from Madrid, Varela is Queen Letizia's go-to for understated, anti-flash fashion. Before she helped put his name on the international stage, other members of the Spanish monarchy tapped him for special-occasion wear. Varela founded his namesake brand in 1996, opening his first store two years later.
Calle de José Ortega y Gasset, 30, 28006 Madrid, +34 915 77 92 20, felipevarela.com, and additional locations.
#### Intropia
* * *
This Madrid-born contemporary brand was started in 1994 by a team of female designers managed by Paloma Vázquez de Castro, former costume designer for the National Ballet of Spain. The clothing—embroidered maxi dresses, beaded blouses, silk jumpsuits—has a Mediterranean glam that has carried it to stores overseas and to websites such as Revolve.com.
Calle de Serrano, 18, 28001 Madrid, +34 917 81 06 12, intropia.com, and additional locations.
#### Loewe
* * *
What started in Madrid in 1846 as a small, artisan-run leather goods shop has turned into an international luxury fashion brand, owned by LVMH Möet Hennessy Louis Vuitton and under the creative direction of British designer Jonathan Anderson. The Madrid flagship remains a must-stop for colorful Amazonia bags, wallets, shoes, and ready-to-wear. The Gran Via shop has pop-up art installations.
Calle de Serrano, 26, 28001 Madrid, +34 915 77 60 56, and Calle Gran Vía 8, 28013 Madrid, +34 91 522 6815, loewe.com, and additional locations.
SUSANA MARTÍNEZ VIDAL
SECRET MADRID SPOTS
* * *
The fashion journalist and author lives in Mexico City and Madrid.
JOYERÍA MARQUISE has the most fabulous vintage jewelry. Each piece is a discovery. DO DESIGN mixes the latest trends with the more rustic and traditional, the organic and artisanal. Everything appears to be an object found during a long journey around the world. It's a place not only for shopping but also inspiration. GALIPÓ VINTAGE on Velarde Street specializes in customizing Levi's 501s. Ask for what you want and they will make it happen.
JOYERÍA MARQUISE, Calle Ayala, 10, 28001 Madrid, +34 914 35 79 06.
DO DESIGN, Calle Fernando VI, 13, 28004 Madrid, +34 913 10 62 17, dodesign.es.
GALIPÓ VINTAGE, Calle Velarde, 28004 Madrid.
#### Malababa
* * *
Designer Ana Carrasco produces her simple, chic bags and shoes in Spain, using local artisans for every step of the process, from tanning the leather to sewing the bags. You'll find chunky-soled espadrille sandals, leather collars decorated with geodes, and weathered, woven leather tote bags, most for less than $500.
Calle Lagasca, 68, 28001 Madrid, +34 912 03 59 90, malibaba.com, and additional locations.
#### Masscob
* * *
Spanish designers—and life partners—Marga Massanet and Jacobo Cobián take inspiration from their seaside hometown of La Coruña to create the kind of sophisticated, beachy boho tunics, frilly blouses, kimono jackets, and loose dresses you'll want to live in all year long.
Calle Puigcerdà, 2, 28001 Madrid, +34 914 35 85 96, and an additional location in La Coruña, masscob.com.
#### Mercado Central de Diseno
* * *
Held the first weekend of every month, this market showcases emerging Spanish designers and artists selling everything from wood sunglasses to cool ceramic jewelry to modern-looking chairs.
Centro Cultural Matadero, Paseo de la Chopera, 14, 28045 Madrid, +34 918 05 05 76, mercadodediseno.es.
#### Pedro del Hierro
* * *
In business since 1974, this Madrid designer creates simple, wearable women's pieces that suggest a bit of Spanish flair; think off-the-shoulder, ruffled maxi dresses, bow-tie lace-paneled blouses, and scallop-hem shorts, all at moderate prices.
Calle de Serrano, 29, 28001 Madrid, +34 914 35 22 10, pedrodelhierro.com.
#### Pez
* * *
Opened in 2004 in a former pharmacy, Pez is a concept store that sells the best of European and Spanish designs (Humanoid, Forte Forte, Golden Goose, Masscob), along with home goods, Danish, and industrial furniture, and more. "Where you find trends before they arrive elsewhere," Susana Martínez Vidal says. Next door, Pez Chico carries menswear.
Calle de Regueros, 15, 28004 Madrid, +34 913 10 66 77, pez-pez.es.
#### Sita Murt
* * *
Bohemian-chic tunics, shorts, and skirts à la Isabel Marant Étoile are the specialty of this Catalan designer, whose flagship is a whitewashed wood oasis of cool.
Calle Claudio Coello, 88, 28006 Madrid, +34 915 77 50 38, sitamurt.com.
#### Zara
* * *
The fast-fashion giant was born in the north of Spain, and succeeds on its low prices and its ability to turn around runway trends in a matter of weeks. The Gran Via store is the largest in Madrid; the Calle de Serrano store, which opened in 2014, is spread over six floors and houses the brand's showroom on top. "They nail it with making affordable fashion look very expensive," says model/TV presenter Louise Roe. "From simple tanks to more ornate coats and dresses, I always buy a few pieces from Zara every season." Adds costume designer Janie Bryant, "Zara is inexpensive but if you buy smart, it doesn't have to be throw-away fashion. I've had pieces from there that I've had for many years and still wear." Zara "covers the basic closet builders without breaking the bank and always has a fun selection of classic yet fun pieces," says stylist Tara Swennen.
Calle Gran Vía, 34, 28013 Madrid, +34 915 21 12 83, zara.com, and additional locations.
## BARCELONA
### SPAIN
LA COMERCIAL
#### Ailanto
* * *
Created by brothers Iñaki and Aitor Muñoz, Ailanto is one of Spain's big designer success stories in recent years. Known for their prints and patterns inspired by art and artists (recent collections have been inspired by artist David Hockney and Canadian painter Lawren Harris), Iñaki and Aitor are a regular presence on the runways in Madrid. They've collaborated with Campari, Lladró, Nespresso, Chupa Chups, W Hotels, and L'Oréal; in 2013, they designed the new uniform for the staff of the Guggenheim Museum Bilbao.
Carrer Enrique Granados, 46, 08008 Barcelona, +34 934 51 31 06, ailanto.com, and additional locations.
#### Beatriz Furest
* * *
Located just steps from the Basílica de Santa Maria del Mar, as well as lots of other must-see boutiques in the trendy El Born district, designer Beatriz Furest's store features her entire range of contemporary classic bags, sandals, woven bracelets, and belts, all of which make perfectly chic Barcelona souvenirs.
Carrer de l'Esparteria, 1, 08003 Barcelona, +34 932 68 37 96, beatrizfurest.com, and additional locations.
#### Boo
* * *
Named after To Kill a Mockingbird's Boo Radley, this store with a Silver Lake vibe features a mix of lovable labels from near (Barcelona-based geometric-jewelry-makers Après Ski and resortwear-focused Med Winds) and far (Saint James and Petit Bateau) along with plenty of exclusives, and a warm, inviting interior that includes a telephone booth repurposed as a dressing room.
Bonavista, 2, 08012 Barcelona, +34 933 68 14 58, boocn.com.
#### Colmado
* * *
The clothing and accessories at this El Born boutique have a candylike quality. It specializes in locally produced wares as well as ones from London, Milan, and beyond with a modern, arty edge, including geometric jewelry and handbags and fuzzy loafers.
Brosoli 5 (corner of Mirallers), 08003 Barcelona, +34 931 722 966, colmadoshop.tumblr.com.
#### Colmillo de Morsa
* * *
Designer Isabel Vallecillo and Javier Blanco have turned Colmillo de Morsa into a minimalist-chic cult brand reminiscent of New York's Creatures of Comfort. The duo shows their collection at Madrid Fashion Week and have opened outlets in Taipei and Moscow. Their showroom/store features not only their collection but also those of young designers with similar aesthetic.
Carrer de Vic, 15, 08006 Barcelona, +34 645 20 63 65, colmilliodemorsa.com, and additional locations.
#### Coquette
* * *
"The multi-brand store of reference in Barcelona," author/journalist Susana Martínez Vidal says of this trio of boutiques that stocks feminine frocks, blouses, and shoes by Mes Demoiselles, Masscob, Isabel Marant, Swildens, and more in an industrial-yet-warm space. The Calle Rec location is the original.
Calle Rec, 65, 08003 Barcelona, +34 93 319 29 76, coquettebcn.com, and additional locations.
#### Castañer
* * *
Perhaps the most internationally famous espadrille brand to come out of Spain, Castañer has Yves Saint Laurent to thank for its success. When the designer was walking the stalls at a Paris trade show in the early 1970s looking for someone to make espadrille wedges for his collection, the shoemaking family scion Lorenzo Castañer was happy to oblige. Since then, Castañer has been the high-fashion answer to Spain's less expensive espadrille shops. In addition to manufacturing shoes for brands such as Hermès, Louis Vuitton, and Christian Louboutin, Castañer has its own label, which is sold around the world and in its Barcelona flagship. (The brand is headquartered about 60 miles away, in Girona.) Prices are in the $100 to $300 range.
Carrer del Rosselló, 230, 08008 Barcelona, +34 934 14 24 28, castaner.com, and additional locations.
#### QK BCN
* * *
"A concept store with items from all over the world, including furniture, perfumes, and music," says author/journalist Susana Martínez Vidal. "An experience for all five senses." Look for boho-chic scarves and dresses by M Missoni, Antik Batik, Schutz, and more.
Cavrer de Calvet 29, 080021 Barcelona, +34 932 09 55 18, qkbcn.com.
#### Iriarte Iriarte
* * *
Carolina Iriarte's handmade leather designs resemble schoolgirl satchels and briefcases in the best possible way. Her showroom/studio is on the picturesque Plaza Real; the second location is on Cotoners.
Cabrer de Cotoners, 12, 08003 Barcelona, +34 933 19 81 75, iriateiriate.com.
#### L'Arca
* * *
A treasure trove of antique linens, vintage dresses by international designers (Dior, Chanel, Yves Saint Laurent) and Spanish ones (Balenciaga, Pertegaz), wedding gowns, and period turbans, lace gloves, and Spanish fans. They don't come cheap, though.
Banys Nous 20, 08002 Barcelona, +39 93 302 15 98, larca.es.
#### La Comercial
* * *
Located on Carrer de Rec in El Born, the wildly successful La Comercial is actually six stores in close proximity, each one featuring a different focus on international labels for men and women, for example, accessories, fragrances, home furnishings, and objects. Some of the brands you'll find along the shopping stretch are Aesop, Astier de Villatte, Jonathan Adler, Kenzo, Paul Smith, Carven, and Surface to Air, but you'll also find cheap and cheerful tchotchkes. The W Hotel in Barcelona recently tapped La Comercial to create its hotel shop.
MAN, Rec 73, +34 93 319 24 35. and Rec 75, +34 93 310 49 68.
WOMAN, Rec 52, +34 93 319 34 63.
CONCEPT STORE, Paul Smith Shoe, Rec 77 +34 93 319 86 01.
SELECTION, Paolo Pecora, Bonaire 7, +34 93 160 01 14.
HOME, Bonaire 4, +34 93 295 46 30
08003 Barcelona, LaComercial.com.
LA COMERCIAL
#### La Manual Alpargatera
* * *
Yet another espadrille store, this one claiming to cater to Penelope Cruz, Catherine Zeta-Jones, Jack Nicholson, and others. Like Antigua Casa Crespo in Madrid, La Manual features inexpensive handmade slip-ons, slides, lace-ups, and wedges.
Carrer Avinyó, 7, 08002 Barcelona, +34 933 01 01 72, lamanualalpargatera.com.
#### Le Swing
* * *
This well-curated vintage store with a funky 1960s vibe has "unique pieces," says Susana Martínez Vidal, including clothing in whimsical prints, chunky jewelry, and colorful handbags galore.
Rec, 16, 08003 Barcelona +34 933 101 449, leswingvintage.com. Sister store Blow: Carrer Bonaire, 6, 08003 Barcelona, +34 93 302 36 98.
#### Lupo
* * *
Barcelona's heritage leather brand since 1920, Lupo is known around the world for its architecturally inspired bags. Many of them echo Gaudí designs, including the fan-shaped Abanico, with pleated leather details. In 2015 the brand launched its first ready-to-wear collection.
Carrer de Mallorca, 257, 08008 Barcelona, +34 934 87 80 50, lupo.es, and additional locations.
#### Majoral
* * *
Enric Majoral's beautiful, organic- form jewelry is inspired by the island of Formentera, the Mediterranean Sea, popular culture, and nature.
Carrer del Consell de Cent, 308, 08007 Barcelona, +34 934 67 72 09, majoral.com.
#### MANGO
* * *
This Barcelona-based fashion chain is another one of the major players in the fast-fashion game. Over the years, the brand has tapped Kate Moss, Penélope Cruz, and her sister Mónica Cruz to design special collections. The Gràcia flagship showcases all the brand's lines (women's, men's, kids', sport, intimates, and accessories) in a spacious interior.
Passeig de Gràcia, 36, 08008 Barcelona, +34 901 15 05 43, mango.com, and additional locations.
#### Med Winds
* * *
A budding Mediterranean-inspired lifestyle collection of men's and women's resort-ready pieces, Med Winds began in Barcelona in 2011. It's distinguished by watercolor-like prints, seaside-inspired jewelry, chic sandals, and curated home accessories and gifts. Most items are less than $250, and most everything is made in Spain or Italy.
Elisabets 7, 08001 Barcelona, +34 935 210 056, medwinds.com.
#### Mushi Mushi
* * *
A Parisian-style boutique in the trendy Gràcia district, where there are lots of cute shops to browse (including Boo, Pinc, and the Outpost), Mushi Mushi features casual-cool T-shirts by Sessùn, feminine blouses by Des Petits Hauts, Saltwater sandals, jewelry by Barcelona-based Mimi Scholer, and more.
Carrer de Bonavista, 12, 08012 Barcelona, +34 932 92 29 74, mushimushicollection.com.
#### Pinc
* * *
The studio/store of Ana Tichy, a Barcelona-based designer who creates wonderfully whimsical prints and casual pieces with a quirky twist.
Encarnació, 39, 08024 Barcelona, +34 93 200 97 71, anatichy.com.
#### Santa Eulalia
* * *
Founded in 1843, Santa Eulalia is the city's oldest design house and also the grande dame of multi-label luxury shopping in Barcelona, featuring Tom Ford, Dior, Maison Martin Margiela, Proenza Schouler, Céline, Etro, and Saint Laurent, alongside its own line of ready-to-wear for women and tailoring and shirts for men. The William Sofield–designed store sits near the designer flagship stores on Passeig de Gràcia. To keep things interesting, Santa Eulalia has a rotating pop-up space and a lovely outdoor terrace bistro and champagne bar.
Passeig de Gràcia, 93, 08008 Barcelona, +34 932 15 06 74, santaeulalia.com.
#### Stradivarius
* * *
Founded in 1994 in Barcelona, Stradivarius is part of Inditex, which also owns Zara. The fast-fashion brand, which has a slightly younger, more casual point of view than Zara (lots of graphic tees, cutoff shorts, and varsity jackets), has expanded to hundreds of stores worldwide.
Avinguida Portal del Ángel, 24, 08002 Barcelona, +34 934 15 42 67, stradivarius.com, and additional locations.
#### Studiostore
* * *
This El Born concept shop and events space is the brainchild of Federica Sandretti (aka Lafede), an architect who started her career in window design for print-heavy fashion brand Custo Barcelona. Sandretti works on retail, furniture, lighting, and graphic design projects, and runs Studiostore, a Fiorucci-like showcase for whimsical furniture, pillows, tabletop accessories, objets, sunglasses, and gifts by designers from Barcelona and beyond. Many items are accessibly priced.
Calle Comerç, 17, 08003 Barcelona, +34 93 222 50 75, studiostore.es.
ONLINE SHOPPING TIP
One of my favorite Spanish brands, Masscob (think, Isabel Marant but not everywhere), ships to some countries from Masscob.com. Select pieces are also carried at Barneys.com and Matches.com.
#### The Outpost
* * *
Wood-paneled destination for haute men's shoes and accessories from top labels such as Want Les Essentiels, Marsèll, Mark McNairy, Thom Browne, Tomas Maier, and more, lovingly chosen by the stylish owner, Pep Esteva.
Rosselló 281 bis, 08037 Barcelona, +34 93 457 71 37, theoutpostbcn.com.
#### The Perfumery
* * *
"The most magical perfume shop," says costume designer Janie Bryant. "It's located in the heart of the Jewish quarter, the Gothic section of the city. I stumbled across it when I was traveling around Spain a couple of years ago. There are fragrances there that were worn by kings and queens. My favorite being Rance 1795, Eau Superbe. Worn by Napoleon himself."
Baixada de Santa Eulàlia, 3, 08002 Barcelona, +34 930 08 37 51, theperfumery.es.
## BERLIN
### GERMANY
#### KaDeWe
* * *
The Kaufhaus des Westens ("Department Store of the West"), usually abbreviated to KaDeWe, is the best department store in Berlin, with a food hall and a rooftop buffet that have to be seen to be believed (lobster tails, champagne, schnitzel, it's all here). Marlene Dietrich used to buy her lingerie in the store, and you can find everything from the top fashion labels to German magazines and tobacco. The food hall has concessions from all over the world, too. The US selection is a hoot (who knew there was an appetite in Germany for Marshmallow Fluff?).
Tauentzienstraße 21-24, 10789 Berlin, +49 30 21210, kadewe.de.
#### Bikini Berlin
* * *
Berlin's concept mall runs alongside the city zoo, so you can sit and watch the monkey enclosure through a big picture window, or from above, at 25hours Hotel's Monkey Bar. Inside the concrete-and-steel mall, which gets its name from the design of the two-building structure with exposed midsection, you won't find many chain stores, apart from a few sneaker and streetwear outposts. Instead, you'll find wood packing crates housing a rotating selection of designer pop-ups, plus spaces such as LNFA, a public relations firm/store featuring the work of many young Berlin designers who show at Berlin Fashion Week.
Budapester Straße 38-50, 10787 Berlin, +49 30 55496454, bikiniberlin.de.
#### Andreas Merkudis
* * *
Perhaps the best known boutique in Berlin, this concept store was founded in 2003, and recently moved north of Potsdamer Platz to an industrial space in the former Tagesspiegel newspaper building. It's out of the way, but an interesting stop as a reference point for Berlin's fashion scene. Designers featured include Dries Van Noten, Céline, Marni, The Row, and Aspesi for women; Kolor, Martin Margiela, and Sunspel. Founder Andreas Murkudis comes from the art scene, and his curation has a similar focus, with objects and furnishings rounding out the vast space. "Understated, simple, chic," says designer Zainab Sumu.
Potsdamer Straße 81e, 10785 Berlin, +49 30 6807 98306, andreasmerkudis.com.
#### Annette Görtz
* * *
Flagship for the German designer, whose understated, sophisticated clothes are a mix of Rick Owens and Donna Karan. Wearable and utilitarian but with an edge.
Markgrafenstraße 42, 10117 Berlin, +49 30 2007 4613, Annettegoertz.com.
#### Frau Tonis Parfum
* * *
The Berlin-based perfumer's signature scents make the perfect souvenir (there's even one named Berlin Summer that captures the essence of the city, with lime and lemon balm, peppermint, and orange).
Zimmerstraßee 13, 10969 Berlin,+49 30 20215310, frau-tonis-parfum.com.
FRAU TONIS PARFUM
#### Kauf Dich Glücklich
* * *
This chainlet of indie concept boutiques started as a waffle bar/vintage shop. The stores (visualize a cross between Aritzia and Urban Outfitters) have a calm, cool vibe, and stock the men's and women's KDG collections (drapey coats, tunic dresses, etc.) as well as pieces by Cheap Monday, Levi's, Sessùn, Jeffrey Campbell, Adidas, and more. Also shop for books, music, and accessories. This is the flagship.
Rosenthaler Straße 17, 10119 Berlin, +49 30 2887 8817, kaufdichgluecklich-shop.de, and additional locations.
#### Konk
* * *
Founded by Ettina Berrios-Negrón, this Mitte boutique features a spot-on, elevated edit of up-and-coming Berlin designers making all kinds of things, from sophisticated knits to 3-D printed jewelry to handmade hats. Look out for designer Tata Christiane's collagey pop art–inspired clothing.
Kleine Hamburger Straße, 15 10117 Berlin, +49 30 2809 7839, konk-berlin.de.
#### Lala Berlin
* * *
Designer Leyla Piedayesh's knitwear and print–based designs with a glam-rock twist have managed to win the hearts of celebrities including Heidi Klum, Claudia Schiffer, and Mischa Barton. Don't miss her signature cashmere triangle scarves.
Alte Schönhauser Straße 3, 10119 Berlin,+49 30 2009 5363, lalaberlin.com.
#### Mauerpark Flea Market
* * *
This Sunday flea market near the Berlin Wall Memorial is a must-see to experience local culture, from skateboarding kids to the crowd-pleasing karaoke sing-alongs that spring up. You'll find everything from vintage leather jackets to Trabant-shaped plastic earrings and other curiosities.
Bernauer Straße 63-64, 13355 Berlin, +49 30 2977 2486, flohmarktimmauerpark.de.
TOP BERLIN TIPS FROM STEFANIE HANSSEN
CREATIVE DIRECTOR OF FRAU TONIS PARFUM
* * *
Shop at BIKINI BERLIN. Not only can you buy Frau Tonis perfumes there but you can also have lunch with an incredible view of the city at the Asian-fusion restaurant NENI. You'll find exciting concept stores like LNFA (which features lots of Berlin designers), SUPER (a restaurant and home accessories concept space), and, on the top floor, ANDREAS MURKUDIS. Next to the concept mall is one of Berlin's most stylish movie theaters: THE ZOO PALAST.
I love to visit the WINTERFELDTMARKT on Saturdays, Berlin´s biggest and best-known farmers' market. Don't miss the incredible mix of edible flowers, locally made grilled sausages, escargot soup, and organic bread! The space is packed with more than 250 stalls. Sometimes, if you're lucky, you may run into Academy Award–winner Christoph Waltz there!
Stroll around Berlin's flea markets. I often find beautiful vintage clothes on Sundays at Fehrbelliner Platz or at Rathnas Schöneberg. It's a paradise for those who love cashmere knitwear and silk dresses!
LNFA, Neni Super and Andreas Murkudis at Bikini Berlin, Budapester Straße 38-50, 10787 Berlin, +49 30 5549 6454, Infa.de, neniberlin.de, super-space.de, andreasmurkdis.com.
NENI, Budapester Str. 40, 10787 Berlin, +49 30 1202 21200, neniberlin.de.
WINTERFELDTSTRASSE, 10777 Berlin, +49 17 5437 4303.
FLEA MARKET AT FEHRBELLINER SATURDAYS AND SUNDAYS, Platz 1, 10707 Berlin.
FLEA MARKET AT RATHAUS SCHONBERG SATURDAYS AND SUNDAYS, John F. Kennedy Platz 1, 10825 Berlin.
THE ZOO PALAST, Hardenbergstraße 29A 10623, Berlin.
#### MDC
* * *
Opened by Melanie Dal Canton, former manager of the concept store Andreas Murkudis, this cosmetics store carries products by Australia-based Aesop, Italy's Santa Maria Novella, and other coveted brands. "A lovely, small perfume and cosmetic store in Prenzlauer Berg, they stock little-known perfume brands and also offer great beauty treatments," says Valery Demure.
Knaackstrasse 26, 10405 Berlin, +49 30 40056339, mdc-cosmetics.de.
#### Rianna and Nina
* * *
Amazing collection of caftans, pillows, lamps, and kimonos, all made from vibrant vintage scarf fabrics, Rianna and Nina was created by two friends, former high-end-vintage dealer Rianna Kounou and former fashion publicist Nina Kuhn. Each piece is unique and ever so chic.
Steinstraße 4, 10119 Berlin, +49 30 28879122, riannaandnina.com.
#### Ritter Sport
* * *
You can make your own version of Germany's favorite chocolate bar here in thirty minutes or less. Choose from ingredients such as coconut, Pop Rocks, and pink peppercorn.
Französische Straße 24, 10117 Berlin, +49 30 2009 50810, ritter-sport.de.
#### Sabrina Dehoff
* * *
This Berlin-based jewelry designer makes superchic treasures, including modernist abstract cocktail rings and chokers, tiny bar earrings, and bold, Bauhaus-looking cuffs. Dehoff has boutiques around the world, but the flagship is in Mitte.
Torstraße 175, 10115 Berlin, +49 30 9362 4680, sabrinadehoff.com, and additional locations.
ONLINE SHOPPING TIP
Frau Tonis can deliver the scent of Berlin to your doorstep. Check out frau-tonis-parfum.com. If you're looking to discover new Berlin design talent, the Konk boutique website is a good place to start, konk-berlin.de. Another good curation of talent (many of whom show at Berlin Fashion Week) can be seen at LNFA-shop.de, the online site for the store/PR agency, though unfortunately neither site ships internationally.
#### The Store
* * *
"A really nice, curated shopping space," says designer Rosetta Getty of this high-end boutique and industrial-yet-warm hangout at Soho House Berlin that is curated by retailer Alex Eagle. "Everything is for sale and they have a great café with pressed juices and a bookstore." Designers represented include Balenciaga, The Row, Jil Sander, Christophe Lemaire, Proenza Schouler, Issey Miyake, and Junya Watanabe.
Torstraße 1, 10119 Berlin, +49 30 4050 44550, thestores.com.
#### The Corner
* * *
Luxury lifestyle boutique selling the best from established labels, including Céline, Givenchy, Isabel Marant, Valentino, Nike, and Apple Watch, and up-and-comers such as Stella Jean, MSGM, and Sara Battaglia. There is also a branch in West Berlin, as well as a men's store.
Französische Straße 40, 10117 Berlin, +49 30 2067 0940, thecornerberlin.de.
#### Voo
* * *
Off-the-beaten shopping path in Berlin's Kreuzberg district, Voo is the city's edgy streetwear-meets-runway purveyor. It carries Henrik Vibskov, Surface to Air, Acne, 3.1 Phillip Lim, Raf Simons, J.W. Anderson, Nike x Kim Jones, and more, plus indie magazines, cosmetics, and coffee.
Oranienstraße 24, 10999 Berlin, +49 30 6957 972710, vooberlin.com.
## STOCKHOLM
### SWEDEN
& OTHER STORIES
#### & Other Stories
* * *
Swedish fast-fashion giant H&M's premium lifestyle brand of fun, funky clothing, accessories, and beauty products is built on the concept of personal style, and designed to appeal to all ages. (The Stockholm-based label launched in 2013 with an ad campaign that included, among others, nonagenarian Iris Apfel.) "I frequent & Other Stories for trousers, T-shirts, and knitwear because it is young, funky, and inexpensive," says author/fashion editor Bronwyn Cosgrave. Managing director Samuel Fernström, who studied at the Stockholm School of Economics before starting at H&M's buying office, created the concept. & Other Stories has collaborated with several designer brands and tastemakers, including Rodarte, Clare V., and Garance Doré. H&M also owns the Cheap Monday, Monki, and Cos brands.
Biblioteksgatan 11, 111 46 Stockholm, +46 8 440 5290, stories.com, and additional locations.
#### Acne Studios
* * *
This quintessentially Swedish, high-end fashion and denim brand for men and women is headquartered in the bank building where the infamous "Stockholm syndrome" heist took place. Check out the Acne Archive store for past-season merchandise at discounted prices.
Norrmalmstorg 2, 111 46 Stockholm, +46 8 611 64 11, and Acne Archive, Torsgatan 53, 113 37 Stockholm, +46 8 30 2723, acnestudios.com.
#### Aplace
* * *
A local retail chainlet and magazine founded in 2007, Aplace is a gem that aims to help shoppers discover the best fashion from Scandinavia and beyond, including Wood Wood, Back, BLK DNM, Dagmar, Henrik Vibskov, and Sandqvist.
Norrlandsgatan 11, 111 43 Stockholm, +46 8 643 3230, aplace.com, and additional locations.
#### Army of Me
* * *
Samuel Fernström, managing director of & Other Stories, advises checking out this local brand of menswear, which has an edgy, cyberpunk vibe. "Go down to their basement to try on the latest long-lasting pieces," he says.
Bergsgatan 49, 112 31 Stockholm, armyofmedesign.com.
#### Byredo
* * *
Former pro basketball player Ben Gorham's shop has a chic, spare Scandi flair, befitting the unisex fragrance line he launched in 2006 with the concept of creating understated scents with simple compositions that translate memories into smells. Sample fragrances include Mojave Ghost and Rose of No Man's Land. Gorham also helped develop scents for & Other Stories.
Mäster Samuelsgatan 10, 111 44 Stockholm +46 8 525 026 10, byredo.com.
#### Dagmar
* * *
This Stockholm-based brand started by three sisters has a sophisticated point of view and interesting clothes, including inventive knits and faux fur outerwear.
Sturegallerian Stureplan 4, 114 35 Stockholm, +46 7 287 230 39, houseofdagmar.com.
#### Filippa K
* * *
Based in Stockholm, with stores throughout Europe, this contemporary brand was founded by Filippa Knutsson in 1993, and specializes in minimalist basics with subtle flourishes (jersey dresses with asymmetrical ties, sweaters with reverse seams).
Grev Turegatan 18, 114 46 Stockholm, +46 8 545 888 88, filippa-k.com, and additional locations.
#### Fran O Till A
* * *
Retail concept started in 2013 by denim designer Orjan Andersson, the store is four floors of cavelike rooms and a journey-of-discovery feeling. Andersson mixes established fashion brands with young designers in a playful way, combining them with second hand, vintage, and recycled products. In the store you can find designers such as as Marques Almeida, Acne, Altewaisaome, Haal, and Örjan Andersson.
Götgatan 105, 116 62 Stockholm, +46 8 644 1030, franotilla.com.
#### Grandpa
* * *
Flagship for the local chainlet of multi-label boutiques stocking the best of Swedish and Scandinavian fashion and design for men and women by Rodebjer, Hope Stockholm, Dagmar, Stylein, Back, Minimarket, and other brands (the edit is about clean lines, sporty details, and prices that won't cause too much pain). At this location, there's also a café, Sixten & Frans, with cocktails and seasonal dishes.
Fridhemsgatan 43, 112 46 Stockholm, +46 8 643 6081, grandpa.se, and additonal locations.
#### H&M
* * *
This fast-fashion gorilla is Sweden's biggest fashion import, and the world's second-largest clothing retailer, behind Spain's Inditex, with stores in more than sixty countries selling trendy items at rock-bottom prices. And it all started here, in Stockholm. The retailer has been around since the 1940s, and began to achieve international fame in 2004, when it collaborated with Karl Lagerfeld on a collection that sold out in some cities in less than one hour, helping to kick off the cheap-chic/designer collaboration craze. H&M has since collaborated with Stella McCartney, Jimmy Choo, Marni, Alexander Wang, Balmain, and others. This is the flagship and the head office.
Drottninggatan 53, 111 21 Stockholm, +46 3 314 0000, hm.com, and additional locations.
B. ÅKERLUND
STOCKHOLM SECRETS
* * *
She's worked with Lady Gaga, Madonna, Rihanna, and Britney Spears. Beyoncé's yellow Roberto Cavalli Lemonade dress? That was all Åkerlund. The costume designer, stylist, and Stockholm native moved to Los Angeles at age fourteen to pursue her dream of working in film, fashion, and music. In addition to Beyoncé's Lemonade looks, Åkerlund cites Gaga's madcap fashion in the Paparazzi video as one of her proudest moments. Oh, and Åkerlund's personal style is pretty madcap, too.
My passion for shopping drove me to the profession of styling. It feeds me with the instant gratification of making a purchase and discovering unique and rare things. In my hometown, Stockholm, I love the hidden gem SAKER & TING, a one-of-a-kind vintage shop where you can find anything from the turn-of-the-century to modern times. For interiors, I also love SVENSKT TENN, which carries traditional Swedish décor with Josef Frank prints. If you are looking for modern Swedish fashion, ACNE's flagship store is in the middle of the mecca of Stockholm. The best spot for home-cooked Swedish food is PA & CO. Its menu never fails, and it's the best spot for hanging out with locals. I also love MATTHIAS DALHGREN. It's located next to the Grand Hotel, which happens to have the best spa and massage therapists anywhere.
SAKER & TING, Sturegatan 28, 114 36 Stockholm.
SVENSKT TENN, Strandvägen 5, 114 51 Stockholm, +46 8 670 16 00, svenskttenn.se.
ACNE, Norrmalmstorg 2, 111 46 Stockholm, +46 8 611 64 11, acnestudios.com.
PA & CO, Riddargatan 8, 114 35 Stockholm, +46 8 611 08 45, paco.se.
MATTIAS DALHGREN, Södra Blasieholmshamnen 6, 111 48 Stockholm, +46 8 679 35 84, mdghs.se.
GRAND HOTEL, Södra Blasieholmshamnen 8, 103 27 Stockholm +46 8 679 35 00, grandhotel.se.
#### Herr Judit
* * *
A reference point for people anywhere in the world interested in men's vintage fashion, this store mixes gently used styles from the 1950s on with newer brands, plus interesting vintage furniture, toys, books, and watches. At 75 Hornsgatan, you'll find Judits, a women's secondhand store.
Hornsgatan 65, 118 49 Stockholm, +46 8 658 3037, herrjudit.se.
#### Jus
* * *
Avant-garde-meets-streetwear is the focus at this multi-brand boutique selling Ann Demeulemeester, Dries Van Noten, Rick Owens, Comme des Garçons, Maison Martin Margiela, and more for men and women.
Brunnsgatan 7, 111 38 Stockholm, +46 8 20 67 77, jus.se.
#### Monki
* * *
This H&M-owned brand merges Swedish design and Japanese street style. Monki customers—mostly members of Gen Z, who are referred to by staff as "our Monki friends"—are presented with a fable based around a cast of mischievous fuzz balls, the Monkis, who inhabit the colorful shops. The clothes are playful—think striped knit tops, snap-front suede miniskirts, and silk bomber jackets—and so are the fun phone cases and accessories.
Götgatan 19, 116 46 Stockholm, +46 8 640 0841, monki.com, and additional locations.
#### Nordiska Kompaniet (NK)
* * *
Swedish architect Ferdinand Boberg designed the art nouveau building for this luxury department store, established in 1915, and built around an atrium. Inside, you'll find Swedish and international fashion labels, cosmetics, design items, and food. There are ten restaurants on-site, including Korv & Glass for Swedish hot dogs, and a food hall in the basement.
Hamngatan 18-20, 111 47 Stockholm, +46 8 762 80 00, nk.se.
ONLINE SHOPPING TIP
Thanks to H&M, Swedish design has conquered the globe. Those sites, as well as Tiger of Sweden, Swedish Hasbeens, Rodejer, Dagmar, and Filippa K have comprehensive e-commerce sites with wide selections of merchandise shipping to many places in the world. To discover emerging designers, though, Aplace.com is both an online magazine and the website for a chainlet of brick-and-mortar stores in Stockholm dedicated to showcasing up-and-coming Scandinavian fashion designers. International shipping is available.
#### Rodebjer
* * *
Founded in New York in 1999 by native Swede Carin Rodebjer. After she was spotted on the streets of Manhattan wearing her handmade designs, she started selling to friends and stores in New York and Stockholm. Since then, Rodebjer has grown into a full-fledged, cool-girl contemporary collection available in the brand's two Stockholm boutiques, as well as in key stores around the world. Signature silhouettes include slouchy suits, drapey caftans, and easy-to-wear dresses and kimonos. The shoes are pretty rad, too.
Regeringsgatan 50, 111 56 Stockholm, +46 8 20 6614, rodebjer.com.
#### Swedish Hasbeens
* * *
This brand has taken a 1970s has-been, the clog, and elevated it to new fashion heights, with heels tall and small, Mary Jane, slip-on, sandal, and bootie styles, and in every color imaginable. The flagship is in Stockholm's trendy SoFo neighborhood.
Nytorgsgatan 36A, 116 40 Stockholm, swedishhasbeens.com, and an additional location in Bibliotekstan.
#### Tiger of Sweden
* * *
Founded in 1903 in the Swedish town of Uddevalla, by tailors Markus Schwarmann and Hjalmar Nordström as a classical menswear brand, Tiger of Sweden began to reposition itself in 1993 as younger and hipper, producing razor-sharp, mod-inspired suits in bold check prints. The slim-cut suits became a signature, amassing a loyal following of stylish men; womenswear was introduced in 2006. The brand has a slight rock 'n' roll vibe, through a pared-down, Scandi lens. Sold across Europe, Canada, and South Africa.
Biblioteksgatan 12, 111 46 Stockholm, tigerofsweden.com +46 8 200 3560, and addtional locations.
## ISTANBUL
### TURKEY
ADNAN & HASAN
#### Aida Pekin
* * *
Istanbul-based jewelry designer Bihter Ayda Pekin launched her collection in 2005 and opened her store in Galata three years later. She has earned a following for her reasonably affordable, whimsically modern wire creations, which are crafted by artisans in the Grand Bazaar. Her designs are also sold at art museums around the world.
Serdar-i Ekrem 44/A, Galata, Istanbul, +90 212 243 1211, aidapekin.com.
#### Atelier 55
* * *
One of Galata's chicest boutiques, Atelier 55 stocks clothing, accessories, and jewelry by Turkish and international designers, including Bora Aksu, Nathalie Trad, David Koma, and VPL. It also has its own label and an espresso bar for refueling.
Şahkulu Mahallesi Serdar-i Ekrem Sokak Seraskerci Çıkmazı No: 55, Galata, Istanbul, +90 212 245 3255, atelier-55.com.
#### Au Vintage
* * *
Au Vintage started as an online shop, but the basement apartment is now a wonderland of perfect pieces from the 1920s to 1990s, mixed with a selection of items by contemporary designers, such as Mila Biju's whimsical throwback jewelry (pineapple earrings) and Burlap's handbags with beaded honeybees.
Küçükbebek Bebeck Cad. No: 34 apt. D:1, Bebek, +90 541 252 6836 auvintage.com.
#### Baston Vintage
* * *
"In the Galata neighborhood, on a small street across from the Crimean church, this store sells beautiful vintage leather luggage at reasonable prices," says Mickey Ashmore, founder of Sabah.
Şahkulu mah. Serdar-ı ekrem Cad No: 61/A, Beyoğlu, Istanbul.
#### Dilek Hanif
* * *
On the scene since the 1990s, this Turkish designer is known for her elaborate gowns and sophisticated ready-to-wear pieces incorporating Ottoman motifs and embroidery.
Macka Cad. Rally Apt. No: 37, Floor Tesvikiye, Istanbul, +90 212 219 3730, dilekhanif.com.
DILEK HANIF
#### Fey
* * *
Owned by former fashion editor Fatoş Yalin Arkun, this boutique is a must-shop for its dreamy combination of vintage and contemporary clothing, accessories, and objets. You'll find circle skirts, classic shirts, bold vintage earrings, and more.
Mim Kemal Öke Cad. No: 9, Nişantaşı, Istanbul, +90 212 219 8724, fey.com.tr.
#### Gönül Paksoy
* * *
An Instanbul fashion stalwart, Paksoy reinterprets Ottoman designs for modern wardrobes, using vintage textiles to create breathtaking clothing and accessories that blur the line between fashion and art.
Teşvikiye Mah., Akkavak Sok. Demet Apt. 4/A, Nişantaşı, Istanbul, +90 212 236 0209.
#### Grand Bazaar
* * *
The centuries-old, labyrinthine complex with sixty-one covered streets and some 3,125 shops draws nearly 100 million visitors every year. Built approximately 554 years ago after Sultan Mehmed the Conqueror took Constantinople, the world's oldest shopping mall has been featured in several Hollywood films, including Skyfall, in which James Bond engages in a high-speed motorcycle chase on the Grand Bazaar rooftop. There are several stores that are musts, including Kafkas (No: 4-6) for fine jewelry, Adnan & Hasan for kilims (No: 89-92), Ottoamano for scarves and pashminas (No: 10), and Sivasli Istanbul Yazmacisi for fabric (No: 57). Shoe designer Paul Andrew calls the latter "one of the most inspiring stores for color and print that I've been to. The walls are stacked to the ceiling with bolts of fabrics in every color imaginable."
GRAND BAZAAR, Beyazıt Mh., Fatih, İstanbul, +90 212 519 1248, grandbazaar.org.
#### Haremlique
* * *
Luxury linens, towels, tunics, and pouches in exquisite fabrics. You can even order made-to-measure linens "for your home or yacht," as the website cheekily suggests.
Zorlu Center, Koru Sok. No: 2/194, Beşiktaş, İstanbul, +90 212 236 3843, haremlique.com, and additional locations.
#### Midnight Express
* * *
Owned by fashion designer Banu Bora and architect Tayfun Mumcu, this boutique offers a tight edit of clothing and accessories by regional designers including Nazli Bozdag and Selim Mouzzanar, alongside pieces from Masscob, Zadig & Voltaire, and Jupe by Jackie, plus beautiful objects and books. Both Bora and Mumcu design pieces for the shop.
Küçük Bebek Cad. No: 3/A, Bebek, Istanbul, and Boorum, +90 212 263 2111, midnightexpress.com.tr.
EVREN DOGANCAY
HOMETOWN FAVES
* * *
Evren Dogancay is an Istanbul local and the buying manager for Beymen, one of Turkey's best luxury department stores.
My favorite department store in my hometown is BEYMEN at the Zorlu Center mall. It's a key location with a big variety of departments—home, shoes, bags, ready-to-wear, etc.—so it's very convenient. (Beymen carries international as well as Turkish designers, including Hakaan Yildirim, who has dressed Rihanna, Jennifer Lopez, and Lady Gaga in his couture-like creations.)
My favorite independent store is SANAYI 313. It's an architecture office in the middle of an auto repair shop that has a healthy fusion kitchen, the best coffee in town by Petra, and a lifestyle shop that sells everything stylish, from toothpaste, shoe polish, scented candles, and books to handbags and jewelry. They also make their own supercool slipper and pouch line.
SOUQ KARAKOY is a pop-up market started by Vogue Turkey editor Yaprak Aras. I love it because it has lots of personality. The vendors vary each time but are all local, up-and-coming, independent creatives. You can find literally anything here. Handmade bicycles, records, robes, activewear, sunglasses, dolls, and homemade healthy dishes are only a few items on the list.
Another Istanbul gem is the GRAND BAZAAR. The best part is the antique section. You can score rare vintage watches, silverware, and antique fabrics at great prices. I also love the variety of ikat fabrics and Turkish pottery.
BEYMEN, Zorlu AVM Levazım Mah. Koru Sok. No: 2, Şişli, İstanbul, +90 212 306 3300, beymen.com.
SANAYI 313, Maslak, 10. Sk. No: 313, Şişli, Istanbul, +90 212 286 3857, sanayi313.com.
SOUQ KARAKOY, Murakıp Sokak No.12, Karaköy, Istanbul.
GRAND BAZAAR, Beyazıt Mh., Fatih, İstanbul, +90 212 519 1248, grandbazaar.org.
BESTE & MERVE MANASTIR
ISTANBUL SHOPPING SECRETS
* * *
Inspired by their father's work as a revered leather artisan in Istanbul, Beste and Merve Manastir launched their handbag line in 2014, and gained worldwide attention after fashion editor Eva Chen posted several photos of the bags on her Instagram page. Manu Atelier is now one of the buzziest contemporary accessories brands on the market, following in the footsteps of Mansur Gavriel. The sisters combine time-honored leather-making traditions with pared-down silhouettes, creating such styles as the signature Pristine square flap-over bag, a favorite with girls in the know.
We really love V2K DESIGNERS, which is the coolest concept store in Istanbul and a part of the Vakko department store chain. This store is not only our first and only exclusive retailer in Turkey but also the very first fashion house in Turkey, founded in 1934. As for new attractions in Istanbul, we love MAVRA CAFÉ DESIGN WORKSHOP in Galata, which is located in old-city Istanbul, a historic place with traditional narrow cobblestone streets. You can have great coffee, breakfast, lunch, and dinner here. Recently, we discovered that DEVELI RESTAURANT, which serves traditional Turkish cuisine, just opened a location near our showroom in the Nişantaşı area. The interior is much more fine dining than traditional kebab restaurant. As for coffee, we consume a lot during the day and we prefer to go to KRUVASAN BAKERY & COFFEE SHOP or MOC (Ministry of Coffee), which is in the Nişantaşı area.
We love FERIKOY FLEA MARKET on Sundays. It is an antiques market, so the selection of antiques and gadgets is huge. It has enough stalls with mysterious objects to spend an entire day there. You can find clothes, jewelry, watches, books, cassettes, decorations, vintage tea sets, cameras and lenses, everything. On one side of the market are stalls with traditional Turkish pancakes, fresh fruit juice, and tea, so you can enjoy great food while you shop. In the Cukurcuma/Beyoglu area, there's TURNACIBASI STREET, where you can find lots of secondhand and antique shops, bookstores, music shops, and stores with all kinds of decorations. There are also some nice cafés such as 49 CUKURCUMA, for example.
At SOFA ART & ANTIQUES you can find very unique art pieces, jewels, and gift alternatives for friends and family. You can also check their website (kashifsofa.com).
For cheap chic, we love SENTETIK SEZAR. They have two branches in Istanbul: one in the Europe side (Taksim/Beyoglu), one in the Asia side (Kadikoy). Taking into account the range of products and how amazing they are, the prices are cheap. If you love vintage, this place will become your favorite store in Istanbul.
V2K DESIGNERS, Abdi İpekçi Cad. No: 29, Şişli, Istanbul, +90 212 219 9487, vakko.com.
MAVRA CAFÉ DESIGN WORKSHOP, Hacımimi, Serdarı Ekrem Sok. No: 31/A, Beyoğlu, İstanbul, +90 212 252 7488.
DEVELI, A, Harbiye, Abdi İpekçi Cad. No: 61D, Blok. 3/4/5, Şişli, Istanbul, +90 212 514 8383, develikebap.com.
KRUVASAN BAKERY, Teşvikiye, Av. Süreyya Ağaoğlu Sok. No: 10, Şişli, Istanbul, +90 212 296 8656.
MOC ISTANBUL, Şakayık Sok. No: 4/A, Şişli, Istanbul, +90 212 234 4465, mocistanbul.com.
FERIKOY FLEA MARKET, Semt Pazarı No: 8, Cumhuriyet Mahallesi, Şişli, Istanbul.
49 ÇUKURCUMA, Kuloğlu Mh., Turnacıbaşı Cad. No: 49, Beyoğlu, Istanbul, +90 212 249 0048.
SOFA ART & ANTIQUES, NuruOsmaniye Cad. No: 53/A, Cagaloglu, Istanbul, +90 212 520 2850, kashifsofa.com.
SENTETIK SEZAR, Moda Cad., Leylak Sok. No: 20/aA, Caferağa, Kadıköy, İstanbul, +90 536 488 6013.
#### Sevan Biçakçi
* * *
This master fine jeweler is known for being a Michelangelo of miniatures, creating incredible dome rings that celebrate the city's historical Byzantine and Ottoman sites, such as the Hagia Sophia encased in rose quartz. American designer Tory Burch is a collector. His atelier produces fewer than 1,000 pieces a year, which are priced from $3,000.
Sevan Biçakçi, Molla Fenari Mah., Gazi Sinanpaşa Sk. No: 16 Eminönü, İstanbul + 90 212 520 4516, www.sevanbicakci.com.
#### Ümit Ünal
* * *
The Istanbul-based designer sells his drapey, neutral-colored unisex creations worldwide. His atelier is on-site.
Nişancı Mah, Eyüp Nişanca Cad. No: 17, Eyüp, Istanbul, umitunal.com.
#### Vakko
* * *
One of Turkey's oldest fashion houses, Vakko started selling hats in the 1930s, and is now a successful manufacturer (clothing, accessories, swimwear, and more) and a retailer with a string of stores. It operates the edgy V2K, Vakkorama (which carries sportier brands such as Adidas, Swims, and Herschel Supply), Vakko Wedding, and Vakko Chocolate.
Abdi İpekçi Mah. 33, NİŞANTAŞI, Istanbul, +90 212 248 5011, vakko.com, and other locations.
#### V2K
* * *
This offshoot of Vakko next door sells well-known contemporary brands such as Alexander Wang, Sea, and Opening Ceremony, plus cool-girl Turkish handbag label Manu Atelier.
Harbiye, Abdi İpekçi Cad. No: 29, Şişli, İstanbul, +90 212 219 9487, vakko.com.
#### Yastik by Rifat Özbek
* * *
The cushion store to end all cushion stores, spearheaded by renowned Istanbul-born fashion designer Rifat Özbek, who had success on the runways in London and Paris in the 1980s.
Şakayık Sokak, No: 13/1, Teşvikiye, Şişli, Istanbul, +90 212 240 8731, yastikbyrifatozbek.com.
#### Yuka Studio
* * *
This local designer's simple, modern geometric jewelry shapes in bronze and brass make the perfect fashion souvenir.
Maçka Meydanı Sok., No: 18 /A Beşiktaş, +90 212 259 1099, yuka-studio.myshopify.com.
## MOSCOW
### RUSSIA
#### Air Moscow
* * *
This concept shop features cutting-edge clothing and streetwear with an emphasis on Japanese brands (all the Comme des Garçons labels, Undercover, and Bao Bao Issey Miyake are represented). Also present are Craig Green, Walter Van Bierendonck, Off-White, KTZ, Christopher Ræburn, ZDDZ, and the newest artist-fashion streetwear collabs from Adidas, Nike, and the like.
Teatralnyy pr-d, 3/4, Moscow, 109012, +7 (495) 621-78-91, air-moscow.com.
#### Aizel
* * *
Street-style star Aizel Trudel, whose social circle includes Christian Louboutin and Natalia Vodianova, is the proprietor of this must-shop, multi-label, three-story boutique, which shut down for Jennifer Lopez to enjoy a private shopping spree when she was in Moscow in 2016. You'll find top international labels, including Valentino, Gucci, Balenciaga, Dolce & Gabbana, and Delpozo, plus the very best of Russian fashion and accessories (Alena Akhmadullina, Katya Dobryakova, Walk of Shame), and an inspired selection of vintage.
Stoleshnikov per., 10/3, Moscow, 125009, +7 (495) 629-95-01, aizel.ru.
#### Alexander Terekhov
* * *
This Moscow-based designer is a go-to for cool, everyday pieces, including print shirtdresses, leather track pants, patchwork knit dresses, and liquid metallic trench coats.
ul. Petrovka, 24/2, Moscow, 127051, +7 (903) 721-66-07, alexanderterekhov.com/en, and additional locations.
#### Flacon
* * *
A former glass factory turned into a colorful urban space for creative businesses. One part functions as office space for media groups, PR agencies, design bureaus, and art workshops. The rest is filled with boutiques selling unique clothes and accessories by local designers, cafés, and bars, and temporary exhibition spaces.
ul. Bolshaya Novodmitrovskaya, 36, Moscow, 127051, +7 (495) 150-03-58, flacon.ru.
#### Freak Frak
* * *
One of the best vintage stores in town. In fact, this place may have introduced the concept of vintage to Moscow when it opened in 1997. It has the makings for any retro look you could dream up, including Soviet-era pleated skirts and schoolmarmish sweaters.
uI. Shabolovka, d. 25, k.1, Moscow, 119049, +7 (926) 251-52-78, freakfrak.ru.
#### KM20
* * *
Moscow's most famous concept store opened in 2009, and was the first in Russia to introduce Raf Simons, Lemaire, J.W. Anderson, Marques'Almeida, and Yeezy. The multi-label boutique is home to a ton of avant-garde brands, including Off-White, ZDDZ, and Commes des Garçons. Among the Russian designers showcased here are A.W.A.K.E., Gosha Rubchinskiy, Nina Donis, Tigran Avetisyan, and Walk of Shame. The space also functions as a gallery, and the greenhouse café is a popular hangout.
Kuznetsky Most, 20, Moscow, 107031, +7 (495) 623-78-88, km20.ru.
#### Leform
* * *
Opened in 1997, this may have been the first avant-garde designer store in Moscow. It's the place to find the latest from Dries Van Noten, Comme des Garçons, Maison Martin Margiela, and Simone Rocha as well as up-and-comers Nili Lotan and Simon Miller, plus edgy home furnishings, accessories, and apothecary items.
ul. Povarskay, 35/28, Moscow, 121069, +7 (495) 691-82-20, leform.ru.
#### Moussa Project
* * *
Founded by Elena and Victoria Moussa, this retail project lets shoppers rent clothing and accessories for a fraction of the retail cost, with the option of buying them later. Unlike similar services in the US, which focus primarily on formal wear, the Moussa showroom offers a range of dark and arty brands, including Haider Ackermann, Iris van Herpen, and Piers Atkinson.
Bol'shoi Patriarshy per., 8, Moscow, 123001, +7 (916) 166-19-03, facebook.com/moussaproject.
#### SV Moscow
* * *
In addition to stocking Comme des Garçons, Junya Watanabe, Ann Demeulemeester, Yohji Yamamoto, Damir Doma, Rick Owens, and Maison Martin Margiela, this directional multi-label boutique was the first in Russia to carry Ashish, Haider Ackermann, and Vetements. SV has also added Raf Simons, Marni, The Row, Yang Li, and Undercover to the selection. Recently, it was the exclusive vendor for a Vetements hoodie dedicated to Zemfira Vetements and Balenciaga designer Demna Gvasalia's favorite Russian singer.
Malaya Molchanovka 6, Moscow, 121069, +7 (495) 215-53-51, svmoscow.com.
SV MOSCOW
ONLINE SHOPPING TIP
Moscow multi-label concept boutique km20.ru ships worldwide for a $50 flat fee, and stocks Russian designers, including Nina Donis, Tigran Avetisyan, and Walk of Shame.
#### TSUM
* * *
A rival of GUM, the nearby TSUM department store has upped its fashion quotient in recent years, with a floor curated by street-style star Natasha Goldenberg.
ul. Petrovka, d.2, Moscow, 125009, +8 (495) 933-73-00, tsum.ru.
#### Tsvetnoy Central Market
* * *
A six-floor department store selling fashion, food, and interior design items, with high-end brands such as Self-Portrait, Marc Jacobs, and Peter Pilotto featured alongside Adidas, Reebok, and Maje, plus up-and-coming talents such as Yulia Yefimtchuk, a Ukrainian designer known for creating clothing inspired by her Soviet-era youth.
Tsvetnoy bulvar, 15/1, Moscow, 127051, +7 (495) 737-77-73, tsvetnoy.com.
#### Vintage Voyage
* * *
Moscow's premier vintage destination is full of treasures, including pristine Hermès bags, Chanel chain belts, 1980s Yves Saint Laurent dresses, and more.
ul. Neglinnaya, 9, Moscow, 107031, +7 (495) 968-11-79, vintagevoyage.ru.
OLYA THOMPSON
OLD-WORLD WONDERS OF MOSCOW
* * *
Olya Thompson textiles are sold through John Derian in New York and the Jules et Jim showroom in Paris. In addition to being an interior decorator and fabric designer, the Moscow resident (and Vogue magazine favorite) recently launched a line of dresses inspired by Russian folkloric designs and '70s hippie looks that combine embroideries, handwoven silks, and vintage Uzbeki floral silks for a kind of "Russian Thea Porter" look.
GUM (translates as the State Universal Shop) is the big city shop. The late-19th-century building, reminiscent of the Russian medieval boyar palace, is an architectural landmark, a multistory glass-covered arcade right on Red Square. There is a cozy art-house movie theater on the third floor. There is a historic wood-paneled powder room. There is a shoe repair place that will restore even the most weathered heels. It is a Moscow tradition to stroll the arcade, looking at the free exhibits, often featuring vintage fashion collections while having an ice cream cone. The best gelato-style ice cream comes from the BOSCO CAFE CART. Or head straight to BOSCO CAFE, grab a table on its sunlit terrace with a view of St. Basil's Cathedral, and enjoy a perfect porcini risotto. It can only compare to Caffé Florian in Venice with a view of San Marco! Among the shops, my favorite is the IMPERIAL PORCELAIN FACTORY'S (Lomonosov), where one can always find a lovely cup or teapot at a reasonable price.
I have become a big fan of the YULIA YANINA COUTURE atelier, hidden in the back of Tverskaya Street off Kamergersky Lane. These are truly red-carpet-worthy hand-embroidered gowns and cocktail dresses with the most delicate lace and layers and layers of tulle. Lady Gaga has discovered Yulia Yanina, but somehow it is still a secret to the rest of the world!
GRISHKO is the place for Russian-made soft canvas ballet and pointe shoes of all shapes. It is the main Russian emporium for all things ballet. Tutus and costumes from many ballets are made to order here.
GUM
Located in a historic mansion featured in Tolstoy's War and Peace, AXENOFF JEWELLERY HOUSE is a gorgeously appointed showroom where one can find earrings, brooches, and tiaras in silver and gold and adorned with pearls, precious and semiprecious stones, and enamels inspired by the splendor of the Russian past.
The hat department in TSUM has a great selection of fur hats from Furland. They make perfect souvenirs from Moscow.
GUM, Red Square, 3, Moscow, 101000, +7 (495) 788-43-43, gum.ru/en.
BOSCO CAFÉ, GUM, +7 (495) 660-05-50, boscofamily.ru/en/restoration/bosco_cafe, and additional locations.
IMPERIAL CHINA, GUM, 1st line, 3rd Floor, +7 (495) 623-26-04, ipm.ru, and additional locations.
YULIA YANINA, ul. Tverskaya, 6 bldg. 6/1, Moscow, 125009, by appointment at +7 (495) 629-51-06, yaninafashion.com/en.
GRISHKO, 3-y Krutitskiy per., 11, Moscow, 109044, +7 (495) 287-45-77, grishkoshop.com, and additional locations.
AXENOFF JEWELLERY, ul. Povarskaya, 52/55-1, Moscow, 121069, +7 (925) 596-53-22, axenoffjewellery.com, and additional locations.
TSUM, ul. Petrovka, d.2, (across the street from the Bolshoi Theatre) Moscow, 125009, +8 (495) 933-73-00, tsum.ru.
LIANA SATENSTEIN
NEW WORLD WONDERS OF MOSCOW
* * *
New York–based Vogue.com news writer Liana Satenstein has been one of the magazine's resident experts on Russia for several years, using her experiences living abroad in Russia and Ukraine as an exchange student as a jumping-off point.
The concept store KM20 is a must. They have Western-based labels like Hood by Air and Raf Simons, but they also offer up a great selection of Russian designers like Tigran Avetisyan and Nina Donis. Plus, they have a stellar vegan café with an amazing selection of fresh juices, kombucha, and salads.
I absolutely love the secondhand store MEGASTYLE (or Megastil). It's very sterile and somewhat eerie, but amazing. It's located about fifteen minutes outside of Moscow. There is so much bizarre deadstock shipped in from Turkey and Greece, as well as oddball bazaar-plucked pieces that look straight out of a provincial Russian town from the '90s. My biggest shopping score of all time was from Megastyle and it was less than $5. It was a semisheer shirt with a tiger printed on it. It's heinous and dated but weirdly chic in a cheeky-throwback way. I live in it.
My off-the-beaten-path shopping location is actually very commercial. I love the famed ARBAT STREET, which is a pedestrian tourist street in the heart of Moscow. There are so many soccer scarves and handmade linens in the souvenir stores.
I love the label NINA DONIS. It is one of the most established but low-key labels in Russia, as well as one of the most creative, mixing Soviet and Western motifs together. (It's available at KM20.) Another brand to watch is AXENOFF by Petr Axenoff. The jewelry is opulent and classic, right out of a Pushkin story. What makes it extra special is they use an old enamel practice called "Rostov finift" in their pieces even today.
I'd have to say living in Russia and Ukraine had a huge influence on how I dress. I came from a mall-rat American town (Amesbury, Mass.) where it was totally fine to roll up to class in a pair of pajamas. My senior year in high school, I decided to do an exchange year in Crimea. Though people there may not have had a lot of money, they still took great care of their appearance and clothes. Crimean women are perpetually well-heeled and have impeccable manicures.
KM20, Kuznetsky Most, 20, Moscow, 10703, +7 (495) 623-78-88, km20.ru.
MEGASTYLE, ul. Skladochnaya, 1, Moscow, 127018, megagiper.ru, and other locations.
UL. ARBAT, Moscow, 119002.
NINA DONIS, available at KM20, ninadonis.com.
AXENOFF JEWELLERY, ul. Povarskaya, 52/55-1, Moscow, 121069, +7 (925) 596-53-22, axenoffjewellery.com, and additional locations.
## TOKYO
### AOYAMA/OMOTESANDO AREA
SHIBUYA CROSSING
#### 1LDK
* * *
"Almost all the brands they sell are available only there," says retailer Josh Peskowitz of this stylish but approachable place that redefines the workwear, military, and sportswear traditions in menswear. "They just opened a small store in Paris up the street from colette, so that's also a must for me." Across the street is the more spacious 1LDK Apartments location, with the Me boutique and editeD/Found café. Here you'll find women's clothing and decorative goods for the home alongside books and jewelry. Also worth mentioning: 1LDK's Depot in Jingumae and 1LDK Hotel in Minami-Aoyama.
#1A 1-8-28, Kamimeguro, Meguro-ku, Tokyo 153-0051, +81 3-5728-7140, 1ldk.com.
#### 45RPM
* * *
The flagship for the cult Japanese denim-and-more brand is located inside a traditional Japanese house.
7-7-21 Minamiaoyama, Minato-ku, Tokyo 07-0062, +81 3-5778-0045, 45rpm.jp, and additional locations.
#### Bedrock
* * *
When in Tokyo, stylist and editor Lori Goldstein heads to this "high-end punk-rock gem," which is famously difficult to find (down a set of stairs in the back of the Forbidden Fruit café/smoothie shop). Thanks to chandeliers and cages, Bedrock resembles a rocker dungeon, and it stocks everything from vintage Givenchy to the owners' own brands, If Six Was Nine and Le Grand Bleu. Stylish Hong Kong restaurateur Bonnae Gokson is a fan, too.
Omotesando Hills, 4-12-10 Jingumae, Shibuya, Tokyo 150-0001, +81 3-3423-6969, maniac-co.jp.
#### Comme des Garçons
* * *
The granddaddy of all CDG stores, this one stocks all of the brands under Rei Kawakubo's umbrella. It also has the biggest range of sizes and styles in the city.
5-2-1 Minami-Aoyama, Minato-ku, Tokyo 107-0062, +81 3-3406-3951, comme-des-garcons.com, and additional locations.
#### Commes des Garçons Design Shop Good
* * *
"Rei Kawakubo's postmodern take on a dry goods store is stocked with all of the iconic designer's favorite things," says designer Brian Wolk. "Products range from the practical to practically luxurious spanning the gamut from Ms. Kawakubo's preferred flair-tip pen to a grandma-worthy metal-frame leather coin purse and everything else in between." In the same upmarket Gyre mall, you'll find the Comme des Garçons Play, Maison Martin Margiela, Visvim, and MoMA Design stores.
5-10-1 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-3406-2323, comme-des-garcons.com, and additional locations.
#### Gallery Muveil
* * *
This contemporary womenswear line from Tokyo-born designer Michiko Nakayama launched in 2007, and is earning a reputation among those in the know for its quirky but feminine aesthetic, blending playful prints with whimsical details. Standouts include embroidered knitwear, beaded taffeta bomber jackets, romantic floral blouses, foil-print pleated skirts, and pearl-festooned ankle booties. Filled with vintage furniture and drawers of jewelry and accessories to explore, the store has the feel of a European apartment. Nakayama also designed the Muveil Work label and collaborated with Wes Anderson on a collection geared to the release of the film Moonrise Kingdom.
5-12-24 Minami-Aoyama, Minato-ku, Tokyo 107-0062, +81 3-6427-2162, gallerymuveil.com.
#### Garden
* * *
"An incredible store for underground-ish men's fashion," says retailer Josh Peskowitz. "It's not showy stuff, but progressive when it comes to silhouette and fabric. I hadn't heard of half the brands in there the first time I went, but I was blown away by the design."
4-8-12 Jingumae, Shibuya-ku, Tokyo 150-001, +81 3-3405-5075, gardenxxx.com.
#### Head Porter Plus
* * *
"Porter is a Japanese bag brand that is very famous locally and not really sold in the States," says designer Scott Sternberg. "They work with nylon and have stores throughout Japan; the Head Porter store in the Shibuya district is the most luxurious take on the brand."
3-21-112 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-5771-2621 headporter.co.jp.
#### Issey Miyake
* * *
The sculptural creations of the iconic Japanese designer are as relevant now as at any other time. Keep walking on the same street, away from Omotesando Station, and you will come across several of the designer's other boutiques, including Issey Miyake Men and Pleats Please. At the end of the street is Reality Lab, which carries Miyake's most experimental collections, Bao Bao and In-Ei; and Me, featuring Miyake's collection of lightweight, compact, easy-to-wear, and easy-to-care-for clothing.
3–18–11 Minami-Aoyama, Minato-ku, Tokyo 107-0062, +81 3-3423-1408, isseymiyake.com, and additional locations.
HOW OLIVIA KIM DOES
TOKYO'S NARITA INTERNATIONAL AIRPORT
* * *
An Opening Ceremony buying department veteran, Kim helps Nordstrom stay on the cutting edge as the store's Director of Creative Projects.
My favorite magazine store is in Narita International Airport. Japanese magazines are my favorite—I learned how to apply makeup from them, how to take the best photos of food for social media, how to wear slouchy socks. I buy as many as I can fit in my carry-on! Also in the airport, there's an outpost of my favorite sock store, Tabio. I get all my socks there and all my last-minute presents and gifts for friends and family. I buy the same style of socks in navy and black, and I ask them for all the pairs they have and clean house on them. And then I throw in a few fun novelty pairs—some with sushi pieces or crocodiles or colored checks and lace.
NARITA INTERNATIONAL AIRPORT, Tokyo 282-004.
#### Kigure
* * *
Celebrate nature in the heart of the city at this boutique featuring environmentally conscious clothing made of natural materials, vintage handmade pottery, and home workshop tools (mingu). "A really special store specializing in all sorts of Japanese folk crafts," says designer Kimberly Wu.
4-25-12 Jingu-mae, Shibuya-ku, 150 0001, +81 3-5414-5737, also in Marunouchi, kigure.com.jp.
#### North Face
* * *
"North Face Japan is a separate company, with its own design team and production," says designer Scott Sternberg. "The product is more specific, more interesting, and just cooler than the US version. They have two stores on the main drag of Aoyama; the smaller one houses the more interesting styles."
6-10-11, Jigumae, Shibuya-ku, Tokyo 150-0001, +81 3-5466-9278, goldwin.co.jp, and additional locations.
#### Quico
* * *
This design-oriented boutique around the corner from the MoMA Design Store stocks simple dresses, smocks, and work pants, sculptural jewelry, straw basket bags, kung fu shoes, beautiful pots and pans, organic towels, and more. Just try not to be charmed.
5-16-15 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-5464-0912, quico.jp.
#### Sacai
* * *
If you've worn a T-shirt with a lace panel in the back, or a sweater with shirt cuffs, it owes a debt to Japanese designer Chitose Abe, who has made her haute hybrid designs into a booming business. The entire collection, plus the less expensive offshoot Sacai Luck, is stocked in this two-level boutique.
Minamiaoyama City House 5-4-44 Minamiaoyama, Minato-ku, Tokyo 107-0062, +81 3 6418-5977, sacai.jp.
#### Sou Sou
* * *
Started by a textile designer who used to work for Marimekko, this brand has a similar aesthetic, with whimsical, colorful Ise-momen fabric made into shirts, baby clothes, kimonos, tabi shoes, tote bags, and phone covers.
5-4-24 Minami Aoyama, Minato-ku, Tokyo 107-0062, +81 3-3407-7887, sousou.com.
JENN ROGIEN
SHOPPING VINTAGE LIKE A BROOKLYNITE IN TOKYO
* * *
When Girls shot in Japan for season five, I got to do some serious shopping in Tokyo accompanied by Tony Crosbie, my Tokyo costume design counterpart and personal shopping guide.
KOMEHYO (which is really a secondhand department store) specializes in designer accessories. Between the insanely good prices and the exchange rate, I bought myself my first Chanel bag (the small classic) and a red Louis Vuitton Epi weekender. The weekender was $200 because of a small blemish on the interior. Komehyo is serious about the authenticity and condition of its items.
CHICAGO, which operates several branches, felt like a Tokyo version of a Brooklyn thrift store but with kimonos. I bought amazing vintage yukatas (a cotton summer-weight version of a kimono) for all the ladies in my family as Christmas gifts. And I bought myself a killer pair of '90s vintage Levi's that fit like a dream. The rack was labeled "crashed denim" as all the jeans were "pre-loved." My pair has the best holes in the knees and wear and tear all over that make them brilliantly broken in. They were twenty bucks max.
RAGTAG is a chain that focuses on secondhand designer clothing and accessories. There were some amazing pieces at pretty great prices. The stores are basically thrift stores that feel like lovely boutiques. My co-designer and I shopped several pieces for the Japanese characters on Girls from Ragtags around Tokyo.
KOMEHYO, 3-5-6, Shinjuku, Shinjuku-ku, Tokyo 160-0022, +81 3-5363-9188, komehyo.co.jp, and additional locations.
CHICAGO, B1F Olympia Annex Bldg, 6-31-21 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-3409-5017, chicago.co.jp, and additional locations.
RAGTAG, 1-17-7 Jinnan, Shibuya-ku, Tokyo 150-0041, +81 3-3476-6848, ragtag.jp.
#### Super A Market
* * *
Super A's tagline is "Fun with clothes," and you'll certainly have that here among the colorful styles by Dries Van Noten, J.W. Anderson, Tome, and Bless. "A concept store owned by the Japanese powerhouse Tomorrowland," says designer Lizzie Fortunato. "The store carries the best high-end fashion merchandised alongside playful jewelry, fragrances, housewares, menswear, and even buttons and rolls of ribbon. Another highlight of the store is the Bar & Grill restaurant at the back of the second floor—hidden and so cool."
3-18-9 Minami-Aoyama, Minato-ku, Tokyo, 107-0062, +81 3-3423-8428, superamarket.jp.
#### Tsumori Chisato
* * *
Popular both in Tokyo, where she has three stores, and beyond, Tsumori Chisato launched her label in 1990 and has earned a loyal fan base for her whimsical clothing and accessories that incorporate arty prints made from original illustrations, fun embroideries, and appliqués.
4-21-25 Minami Aoyama, Minato-ku, Tokyo, +81 3-3423-5170, tsumorichisato.com, and additional locations.
#### Undercover
* * *
The men's and women's flagship for the cult Japanese brand designed by Jun Takahashi, who creates edgy, streetwise clothing inspired by twisted fairy tales, lost innocence, and pretty punks.
5-3-22 Minami Aoyama, Minato-ku, Tokyo 107-0062, +81 3-3407-1232, undercoverism.com.
#### United Arrows
* * *
Founded in 1989, this multi-brand powerhouse has stores across Japan representing a diverse sensibility, from high-end to street, along with its own private label. It's worth stopping by to take a look at the latest pop-up (recently, it was Kith) or artist collab (Larry Clark, for example).
2-31-12 Jingumae, Shibuya-ku, Tokyo 150-0001, +61 3-3479-817, united-arrows.co.jp, and additional locations.
### SHIBUYA
#### Journal Standard
* * *
Many Japanese brands take inspiration from American heritage brands and workwear, none more so than Journal Standard, which has four fashion lines and thirty stores around the world.
1-5-6 Jinnan, Shibuya-ku, Tokyo 150-0041, +81 3-5457-0700, journal-standard.jp, and additional locations.
#### Laila Tokio
* * *
The impressive all-white interior, with walls covered in 8,000 pyramid studs, is the canvas for hard-to-find designer archives, as well as up-and-coming labels, art books, and more.
2F, 1-5-11 Shibuya, Shibuya-ku, Tokyo 150-0041, +81 3-6427-6325, laila-tokio.com.
NINA GARDUNO
GETS INSPIRED BY TOKYO
* * *
A retail visionary, Nina Garduno is the founder and creator of L.A.'s pop-art-meets-commune Freecity store, which changes theme every few months. Her brand of sweats and T-shirts with feel-good sayings has gone global, with a store in Tokyo. And Tokyo, as it turns out, is one of the places that always inspires Garduno.
THE DAIKANYAMA TSUTAYA BOOKSTORE in Tokyo is completely obsessive-compulsive. So if it's race cars they're into, there's every video and small model car, books on vintage Mercedes and Porsches, and video footage of races. Or if it's Star Wars, they have every movie on VHS, Beta, and DVD, plus signed pieces of C-3PO. It's mixing media—old, new, vintage, cutting edge. And they have every magazine, and every issue of W and National Geographic ever printed. It's a library, too.
I love the GOLDEN GAI, which has three tiny roads of bars, each one a different genre. So you'll have one themed around 1930s geisha, and another one that's about samurai movies. These places are tiny, like four square feet, and you have to inherit spaces when others leave. Quentin Tarantino has a place he frequents that's inspired by French New Wave cinema. And they serve drinks that are of the place, so you feel like you are living it.
Then there's TOYKU HANDS, which has different worlds of craft, from letter making to stickers to candle making. It's floor after floor of dense, intense worlds and genres.
It's very real to step into these worlds, and it's very inspiring to me because it's what I do, too. My favorite purchase isn't anything I have bought—it's the experiences I have taken with me.
TSUTAYA BOOKS, 17–5 Sarugaku-cho, Meguro-ku, Tokyo 150-0033, +81 3-3770-2525, tsite.jp/daikanyama.
GOLDEN GAI, 1-1 Kabukicho, Shinjuku-ku, Tokyo 160-0021.
TOYKU HANDS, 12-18 Udagawacho, Tokyo 150-0042, +81 3-5489-5111, tokyu-hands.jp.
#### Seibu
* * *
One of Tokyo's biggest department stores, Seibu is spread out over two facing buildings and fourteen floors, two of them dedicated to an incredible food hall (a favorite of designer Trina Turk). Prada, Iris van Herpen, Muveil, and Yohji Yamamoto are just a few of the brands represented here.
21-1 Udagawacho, Shibuya-ku, Tokyo 150-0042, +81 3-3462-0111, sogo-seibu.jp.
#### Shibuya 109
* * *
This shopping mall may be to Tokyo what the Sherman Oaks Galleria was to Los Angeles. In the '90s, the Shibuya gyarus (gals) brought 109 into popular culture with their dyed blond hair, fake tans, miniskirts, and makeup. Nowadays, the mall is still a place to shop for fun fashion, including cutesy clothing (most in very small sizes), bedazzled phone cases, wigs, colored contact lenses, and more.
2-29-1 Dogenzaka, Shibuya, Tokyo 150-0043, +81 3-3477-5111, shibuya109.jp.
#### Tokyu Hands
* * *
"A department store like none you've ever seen," says designer Scott Sternberg of this eight-floor DIY heaven. "They carry everything you can think of, from bags and T-shirts to watches and appliances. The mix can be overwhelming, but if you step back and take your time, you'll find some of the best local oddities and deals." "Love all the gadgets and knickknacks. I always buy lots of stuff there: brushes, stickers, stationery, black earbuds, sushi erasers, shiso seeds, et cetera," adds publicist and retailer Valery Demure. Designer Erica Tanov describes Tokyu Hands this way: "An awesome living-ware store carrying items from traditional Japanese stationery, paper clips and perfect pens to fun bath and beauty products. I could shop there for hours. Cheap and cheerful, for sure."
12-18 Udagawacho, Tokyo 150-0042, +81 3-5489-5111, tokyu-hands.jp, and additional locations.
#### Tomorrowland
* * *
This elegantly preppy brand is the namesake label of the Tomorrowland retail and manufacturing group, but you'll also find other international fashion and accessories labels at the beautiful flagship, including Loewe, Isabel Marant, and Dries Van Noten.
F B1F, 1-23-16, Shibuya, Shibuya-ku, Tokyo 150-0002, +81 3-5774-1711, tomorrowland.jp, and additional locations.
ERICA TANOV
BOUTIQUE TREASURES IN TOKYO
* * *
MINÄ PERHONEN is the best-designed store I've ever been in (especially the dressing rooms). I love his work, he's a true original. Beautiful clothing, accessories, and fabrics. A feast for the eyes and heart.
FOG LINEN has a thoughtful mix of clothing and home goods in a natural, soothing color palette.
ARTS & SCIENCE is a beautifully curated clothing, accessories, apothecary, and housewares store. There are several locations, with each offering something unique. I admire how many of the stores in Tokyo require a map (which they print on their business cards) to locate them—you feel like you're on a treasure hunt.
MINA PERHONEN, Hillside Terrace G-1F 18-12 Sarugakucho, Shibuya-ku, Tokyo 105-0033, +81 3-6826-3770, mina-perhonen.jp, and additional locations.
FOG LINEN, 5-35-1 Daita, Setagaya, Tokyo 155-0033, +81 3-5432-5610, foglinenwork.com.
ARTS & SCIENCE, 6-6-20, Pacific Aoyama, Minami Aoyama, Minato-ku, Tokyo 107-0062, +81 3-3498-1091, arts-science.com, and additional locations.
### HARAJUKU
#### 6% Doki Doki
* * *
The capital of kuwaii (cute) Harajuku fashion, this place is a can't-miss for colorful dolly dresses, cosplay outfits, and candy-like jewelry. The salespeople dress the part and are worth the stop alone.
4-28-16 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-3479-6116, dokidoki6.com.
#### Beams
* * *
Of all the stores owned by the Beams fashion empire (the chain has more than 100), the most luxurious is International Gallery Beams, which features the latest from Proenza Schouler, Jaquemus, and many other international labels, plus timely pop-up shops and limited-edition artist collaborations.
3-25-15 Jingumae, Shibuya-ku, Tokyo, +81 3-3470-3948, beams.co.jp, and additional locations.
#### Dog
* * *
"The first place I go to whenever I'm in Japan," says designer Pip Edwards of this hidden spot, a longtime favorite of Harajuku hipsters for reworked vintage. "You follow this winding path, then through a door and down a set of stairs. The store is all black, and it has the most amazing things, from vintage Mugler to new stuff."
3-23-3 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-3746-8110.
#### Facetasm
* * *
Facetasm is Japan's latest fashion export. It was founded by Hiromichi Ochiai, who showed his collection in Paris for the first time in 2016 and picked up several international accounts, including Net-a-Porter. Inspired by pop culture and the Antwerp Six, the collection includes idiosyncratic takes on utilitarian styles (pleated hybrid skirt-pants and bow-embellished bomber jackets, for example).
#601 2-31-9 Jingumae Shibuya-ku, Tokyo 150-0001, +81 3-6434-9893, facetasm.jp.
#### Laforet
* * *
Tokyo's trend-setting teenage pop palace. Looking for the next Goth Lolita or Ganguro style? You're likely to find it here, where a lot of the Harajuku kids shop. "I walk each floor looking for anything that I can squeeze into," says author and street-style photographer Ari Seth Cohen. "I am totally Harajuku at heart. Laforet has floors and floors of independent little shops and designers with styles ranging from rainbow-streaked club-kid wear to French floral country grandmother. The whole place is a feast for the eyes."
1-11-6 Jingumae, Shibuya-ku, Tokyo 150-0001, +81 3-3475-0411, laforet.ne.jp.
#### Toga
* * *
Former costume designer and stylist Yasuko Furuta is a go-to at home and abroad (Katy Perry is a fan) for her avant-garde clothes and shoes, which mix feminine shapes and Western touches such as oversized silver buckles and silk fringe.
6-31-10 Jingumae, Shibuya-Ku, Tokyo 150-0001, +81 3-6419-8136, toga.jp.
#### Visvim
* * *
"The way that designer Hiroki Nakamura approaches his brand is as an archivist, looking at the past and vintage to create something modern and authentic," says Nordstrom Director of Creative Projects Olivia Kim. "His flagship store in Harajuku is one of my favorite places to discover fashion—not only are the clothes there but his inspiration pieces are also on display—his photos of him and his family on the road in search of that inspiration. He also roasts his own coffee beans, Little Cloud Coffee, and brews one cup at a time. And on top of it all, Visvim makes the most delicious scented candles and room diffusers."
5-10-1 Gyre 2F Jingumae, Shibuya-Ku, Tokyo 150-0001, +81 3-5468-5424, visvim.tv.
HARAJUKU GIRLS
### GINZA
#### Dover Street Market
* * *
Rei Kawakubo's hometown fashion fun house doesn't disappoint. There are seven floors of goodness, it's connected to the Uniqlo flagship, has a rooftop shrine, a branch of Rose Bakery, and Komatsu Bar.
6-9-5 Ginza, Chuo-ku, Tokyo 104-0061, +81 3-6228-5080, Ginza.doverstreetmarket.com.
#### Muji
* * *
The mother of all Mujis, this flagship has everything you need to lead a serene and simple lifestyle, including apparel, furniture, home accessories, food, even a Muji prefab house you can tour and buy to build in Japan. There are aromatherapy and monogramming stations on site, plus a café and a children's play area. In the same building, you'll find Loft, "a famous store for stationery, office, and art supplies, which is boring in the US but like Disneyland in Tokyo," says designer Scott Sternberg.
3-8-3 Marunouchi, Chiyoda-ku, Tokyo 100-0005, +81 3-5208-8241, muji.com, and additional locations.
#### Matsuya
* * *
"One of my favorite department stores because of its smaller size, design shop, and designer selections," says museum curator Sharon Takeda. On the third floor is "Le Place," which sells only local designer fashion; the two basement floors have an impressive selection of delicacies.
3-6-1 Ginza, Chuo-ku, Tokyo 104-0061, +81 3-3562–1111.
#### Oedo Antique Market
* * *
The largest outdoor antique market in Japan, Oedo is held the first and third Sundays of the month. "A wondrous array of Japanese antiques at reasonable prices," says designer Erica Tanov. "I've found cherished items such as a favorite oil painting, handmade kitchen utensils, exquisite kimono fabric, and rustic ceramic vessels."
3-5-1 Marunouchi, Chiyoda-ku, Tokyo 100-0005.
ONLINE SHOPPING TIP
Tokyu Hands, perhaps one of the most referenced stores by contributors to this book, ships internationally to many countries via BuySmartJapan. Look for the icon next to items on the tokyu-hands.co.jp website.
For Japanese design goodies in North America, there are Daiso dollar stores in Canada and the US, including California, Texas, and Washington, or order online at daisojapan.com. Websites for the popular Japanese brands Muji (muji.com) and Uniqlo (Uniqlo.com) also ship worldwide.
#### Tansu-ya
* * *
This chain of vintage kimono stores is a hit with designer Elizabeth Kennedy, who says, "They sell beautiful vintage silk, shibori, and hand-painted kimonos. With a bit of luck, you can find a kimono on sale for as little as $35. Most start around $300 to 500, but they can cost up to $2,500!" The staff will teach you the traditional method of wearing a kimono, and there are stores all over the city.
3-4-5 Ginza, Chuo-ku, Tokyo 104-0061, +81 3-3561-8529, tansuya.jp, and additional locations.
#### Uniqlo
* * *
Standing twelve stories high, the Ginza flagship of the popular cheap-chic basics brand has a stunning glass facade and compelling displays inside, too, thanks to Wonderwall's Masamichi Katayama, famous for his innovative work with BAPE and colette in Paris.
6-9-5 Ginza, Chuo-ku, Tokyo 104-0061, +81 3-6252-5181, uniqlo.com, and additional locations.
### SHINJUKU
#### Don Quijote
* * *
Designer Brandon Maxwell names Donki (as it's known) as his favorite store in the world. The discount chain is famous for its extended hours (this location is open twenty-four hours) and its distinctive jingle, it's "miracle shopping." "You can literally buy everything you ever imagined there. You can get a Kelly bag and a Rolex on the eighth floor and laundry detergent on the first floor!" Maxwell says.
1-12-6 Okubo, Tokyo 169-0072, +81 3-5292-7411, and additional locations.
DON QUIJOTE
#### Isetan
* * *
"Hands down my favorite department store in the world," says men's retailer Josh Peskowitz of this not-to-be-missed Tokyo hot spot that carries all the best fashion brands, local and international. "Honestly, there's nothing like it. Their windows at holiday time are the illest, most surreal examples of the craft that I have ever seen, and it extends to every point of the store displays." The champagne bars and basement food court come highly recommended, too, and the store always seems to be hosting a special event, designer appearance, or pop-up. "I have never been more impressed with a more manicured department store," adds fashion editor Karla Martinez de Salas. "The fruit department with $200 muskmelons is unforgettable."
3-14-1 Shinjuku, Shinjuku-ku, Tokyo, 160-0022, +81 3-3225–2514, isetan.mistore.jp/store/shinjuku.
#### Mannen-Ya
* * *
"A really special store specializing in all sorts of Japanese workwear," says designer Kimberly Wu of this place, which is like an Army-Navy store for the hard-hat set.
3-8 Nishi Shinjuku, Shinjuku, Tokyo 160-0023, +81 3-3373-1111, mannen-ya.co.jp.
### EBISU
#### Biotop
* * *
"Boutique, café, tree house, and nursery rolled into one" is how designer Kimberly Wu describes this innovative retail space, where you can stock up on emerging and established labels such as The Row and Rosetta Getty, and grab lunch at the Irving Place café upstairs, with views of the tree house outside.
4-6-44 Shirokanedai, Minato-ku, Tokyo, 108-0071, +81 3-3444-2421, biotop.jp.
#### Kapital
* * *
This cult brand of folksy, Japanese- meets-Americana denim and workwear cut, patched, and distressed to imperfect perfection is so beloved that David Sedaris wrote a New Yorker piece about it. There are several stores in Tokyo and they are all equal parts clothing and curiosities. "Best merchandising ever," says designer Ulla Johnson.
2-20-2 Ebisu 1F, Shibuya-ku, Tokyo 150-0022, +81 3-5725-3923, kapital.jp, and additional locations.
### DAIKANYAMA
#### Okura
* * *
A treasure trove of all things indigo set up in a rustic house, Okura offers a glimpse into Japan's traditional hand-dye, batik, and block-print techniques on everything from sweatshirts to scarves. "It's all simple, classic style with wonderful Japanese tailoring," says Sabah founder Mickey Ashmore. "As at most Japanese institutions, the staff is very caring and attentive. I bought a double-breasted indigo suit here that I love." Downstairs, you can grab lunch and lotus baguettes at Bombay Bazar. Okura is owned by Seilin & Co., which operates several fashion stores and cafés around Japan, including the nearby Hollywood Ranch Market and High Standard.
20-11 Sarugakucho, Shibuya, Tokyo 150-0033, +81 3-3461-8511, hrm.co.jp/okura.
#### Tsutaya Books
* * *
Perhaps the most beautiful bookstore in the world, Tsutaya sells new and used vintage books, DVDs, music, magazines, and more. It also has its own outdoor eatery and gallery. The two-story building complex was designed by Tokyo architecture firm Klein Dytham. "I could easily spend an entire day there browsing the art books and back issues of magazines and interacting with robots before sitting down in one of the swankiest lounges on the planet for a Manhattan made by a true cocktail master," says writer and retailer Lizzie Garrett Mettler.
17-5 Sarugaku-cho, Meguro-ku, Tokyo 150-0033, +81 3-3770-2525, tsite.jp/daikanyama.
TSUTAYA BOOKS
#### White Mountaineering
* * *
Japanese designer Yosuke Aizawa's menswear and womenswear label is white hot. The premium brand may be inspired by the outdoors, but it has serious fashion cred, with reconstructed workwear and cozy, cool patterned knits.
2-7 1F, Sarugaku-cho, Shibuya-ku, Tokyo 150-0033, +81 3-6416-9110, whitemountaineering.com, and additional locations.
### NIHANBASHI
#### Starnet
* * *
"The most beautiful ceramics," says handbag designer Kimberly Wu. "The Tokyo store is out of the way, though. Go to the Mashiko location for a really special experience—it's about a two-hour drive from Tokyo."
1-3-9 Higashi-Kanda, Chiyoda-ku, Tokyo 101-0031, +81 2-85-72-9661, starnet-bkds.com, and additional locations.
### NAKAMEGURO DI
#### Cow Books
* * *
"A tiny, amazing rare bookstore in my favorite little neighborhood that is centered around a canal," says designer Scott Sternberg. "In April and May, it's cherry blossom season and the entire run of the canal is bright pink. It's stunning and charming."
1-14-11 Aobadai, Meguro-ku, Tokyo 153-0042, +81 3-5459-1747, cowbooks.jp.
#### Ganso Shokuhin Sample-ya
* * *
Plastic-food nirvana—you know, the stuff you see outside every Asian restaurant.
3-7-6 Nishi-Asakusa, Taito-ku, Tokyo 111-003, +81 120-171-839, ganso-sample.com.
SHOPPING THE WORLD FROM ETSY
For authentic Japanese workwear-style clothing, stylist Sarah Schussheim recommends the Etsy store Sasakiyohinten, located in Gunma, Japan. "It is run by the most lovely man, Mitsugu Sasaki. His pieces are a hodgepodge of techniques; the best way to describe them would be as remade from antique linen, Japanese cotton, sometimes dyed, sometimes not. Lots of boro-stitching and patchwork. My favorite pieces from here are a patched indigo chore jacket and a light-blue patched sweatshirt."
"One of my favorite shops on Etsy is Embroidered Dream," says Brigette Romanek. "It carries Ukrainian tunics and dresses. They're colorful garb with billowy oversize sleeves. I wear them day and night. I love them."
## HONG KONG
### CHINA
JOHANNA HO
#### Barney Cheng
* * *
"Great for a modernized cheongsam," says art dealer Pearl Lam of Hong Kong's premier designer for the social set. "Barney is greatly counterintuitive."
12/F, Worldwide Commercial Building, 34 Wyndham St., Central, Hong Kong, +852 2530 2829, barneycheng.com.
#### Cha Ling
* * *
This new LVMH-backed, tea-infused range of beauty products and fragrances includes cleansing powders, masks, steam tablets, and more.
Shop 3306A, 3/F, Gateway Arcade, Harbour City, Hong Kong, +852 3101 0708, cha-ling.com.
#### DelStore
* * *
"One of my favorite small-scale stores in the world," says retailer Josh Peskowitz of this multi-label menswear store, featuring indie brands such as Arts & Science, Kapital, Blue Blue, Yuketen, and Comoli. "The former menswear buyer at Lane Crawford opened it in a very out-of-the-way alley in the most fashionable neighborhood in Hong Kong and it's got such a well-thought-out aesthetic. Everything in the store from every brand could almost be made specifically for the store, they all meld so well into one whole."
3 Schooner St., Wan Chai, Hong Kong, +852 2528 1770, www.delstore.co.
#### Fang Fong Projects
* * *
Hong Kong designer Wu Lai-Fan makes contemporary-looking dresses in Asian-inspired fabrics.
69A Peel St., Central, Hong Kong, +852 3105 5557.
#### Initial
* * *
Founded in 2000, this chain of lifestyle stores with earthy furnishings and lulling soundtracks features its own romantic, vintage-inspired clothing for men and women. Inital has carved out an empire, opening several branches in China and even expanding into home (Initial Living) and dining (Initial Café).
Shop B13-15, Lab Concept, 93 Queensway, Admiralty, Hong Kong, +852 2623 2632, initialfashion.com, and additional locations.
#### Island Beverley Center
* * *
"An often-overlooked treasure of the Hong Kong shopping experience," local fashion expert Tania Mohan says of this trendy mall full of well-priced stores. "It has been a fashion-forward shopping experience since my late teens, and it remains a treasure with its four floors of quirky, independent street-forward fashion shops with a strong Japanese and Korean influence. You'll find everything from gorgeous dresses and well-cut blazers to laser-cut neon and PVC garments."
1 Great George St., East Point, Hong Kong, +852 2890 6823.
#### IT
* * *
"My husband and I stumbled upon this amazing store when we were in Hong Kong on our honeymoon," blogger Lainy Hedaya says of this multi-label monolith, which has eight locations in HK selling international and in-house brands. "We spent about five hours there and left with so many bags (it was also sale season). The curation of local designers and the quintessential luxury brands made for an amazing shopping experience."
2 Kingston St., Causeway Bay, Hong Kong, +852 2881 6102, ithk.com, and additional locations.
#### Jardine Crescent
* * *
"You can find anything from women's underwear to kids' clothing to cartoon pajamas," says fashion consultant and stylist Grace Lam of this narrow street market. "If you are looking for some essentials and fun local goodies, this is definitely the place to go."
Jardine's Bazaar, Causeway Bay, Hong Kong.
#### Johanna Ho
* * *
Hong Kong fashion designer Ho launched her brand in 1998 in London, and her first collection was quickly snapped up by Barneys New York. After continuing to build her business from Japan, she returned to Hong Kong a few years ago; in 2014, Ho opened her first flagship here, offering the quirky women's knitwear and whimsical men's tailoring that have made her a favorite of celebs such as Cantopop singers Eason Chan and Sammi Cheng and British actor Rosamund Pike. Popular in China, Ho has collaborated with numerous brands, including Birkenstock, Vans, and MAC.
Shop 315, 3/F, Windsor House, 311 Gloucester Rd., Causeway Bay, Hong Kong, +852 2722 6776, johannaho.com.
#### Joyce
* * *
"My all-time favorite boutique," says Hong Kong–based luxury marketing consultant Peter Cheung. "I have been shopping there since the '80s, and the founder, Joyce Ma, is truly the fashion and luxury pioneer of Asia. The boutique has been on the cutting edge of bringing the best in designer fashion to Hong Kong, as well as searching the globe for young, up-and-coming designers to introduce to the Hong Kong market. One young Chinese designer that I love to purchase at Joyce is Xander Zhou. The store is such a beautiful haven in the busy Central district, it transports you to a world of calm and elegance that I don't think any other shop in town does." Designer Karen Erickson agrees: "Joyce Ma's legacy of avant-garde luxury is subliminal and perfect."
G/F, New World Tower, 232 Pacific Place, Central, Hong Kong +852 2810 1120, joyce.com, and additional locations.
DIVIA HARILELA'S
FAVORITE HONG KONG BOUTIQUES
* * *
Harilela is a fashion media vet, having served as the fashion editor for Hong Kong's leading English-language newspaper, the South China Morning Post, before founding her own luxury and fashion website the D'Vine in 2011. She is also a regular contributor to Vogue China, Jing Daily, The Business of Fashion, and Departures, to name just a few. Born and raised in Hong Kong, Harilela is often sought out as an authority on fashion in Asia.
Hong Kong is full of big brand stores, but I prefer small, curated boutiques. One of my favorites is VEIN, which stocks hard-to-find Scandinavian brands (Ganni, Humanoid, Rodebjer, Wood Wood). TABLA is run by Tania Mohan and everything is handcrafted in India—think Western silhouettes with exquisite embroideries. You feel like you're buying an heirloom that you'll wear time and time again.
GRANA is relatively new; it started online à la Warby Parker. It's my go-to for everyday clothes. They cut out the middleman and offer great-quality basics such as T-shirts (comparable to Petit Bateau), denim made in Japan (the fit is almost as good as Uniqlo), and even cashmere. The prices are super reasonable as everything is direct-to-consumer. They just opened their first freestanding boutique in Hong Kong.
VEIN, 2 St. Francis Yard, Wan Chai, Hong Kong, +852 2804 1038, bvein.com.
TABLA, Princes Building, Central, Hong Kong, +852 2525 5590, houseoftabla.com.
GRANA, 108 Hollywood Rd., Sheung Wan, Hong Kong, +852 2755 8744, grana.com.
#### Kapok
* * *
Founded in 2006 by Frenchman Arnault Castel with the goal of bringing future classics to the Asian market, Kapok has grown into a chainlet of lifestyle shops carrying Bensimon, Building Block, Clare V., Maison Kitsuné, Martiniano, Rains, and other pared-down basics, as well as its own unpretentious, casual Future Classics label.
Kapok, G/F, 5 St Francis Yard, Wanchai, Hong Kong, +852 2549 9254, ka-pok.com, and additional locations.
#### Landmark
* * *
One of HK's most prestigious malls, with Harvey Nichols spanning five floors, a wide variety of luxury brand boutiques, including Louis Vuitton, Céline, and Marni, and Michelin-star restaurants such as L'Atelier de Joël Robuchon. "I love combing through their edit whenever I am in Hong Kong," says fashion editor Joe Zee. "There are always items that are available only in Asia, and I particularly love their home décor department."
The Landmark, 15 Queen's Rd., Central, Hong Kong, +852 3695 3388, landmark.hk.
#### Lane Crawford
* * *
Founded in 1850, Lane Crawford is still Hong Kong's leading luxury fashion and lifestyle specialty store. "One of my favorite department stores," says shoe designer Paul Andrew. "They have a very progressive, well-curated assortment of collections in all departments, and they spend a great deal of energy supporting and nurturing emerging talent. It's really unparalleled in China. Beyond the selection, the store is beautifully designed and filled with great rest areas, including a café opposite the designer shoe salon that serves the best coffee and delicious cookies and pastries (traditional French recipes mixed with Eastern flavors and influences)." "Their selection is really well curated," says shoe designer Isa Tapia. "Especially the shoe and vintage shop. The vintage collection at the ICF mall is the best I've ever seen. This is the best place to buy vintage Chanel that you will wear forever."
IFC Mall, Podium 3, 8 Finance St., Central, +852 2118 2288, lanecrawford.com, and additional locations.
WHERE DESIGNER MARIE FRANCE VAN DAMME
FINDS INSPIRATION IN HONG HONG
* * *
Van Damme is the Montréal-born, Hong Kong–based founder of the travel-inspired luxury resort line that bears her name. Van Damme has five boutiques, and her collection is at upscale department stores worldwide.
Hong Kong is famous for its street markets. CAT STREET MARKET on Upper Lascar Row is antiques central—it's where I buy jade rings that I use as Chinese spoon rests.
A chef's dream, SHANGHAI STREET is fantastic for understated porcelain pieces that are great for dinner parties or decorating the home. I love to get little trinkets here. Many of these pieces, when situated correctly, can add an understated elegance to any space.
The HONG KONG FLOWER MARKET in Mongkok is a little oasis, with rows upon rows of stalls, shops, and stands brimming with fresh-cut flowers, exotic plants, and an abundance of foliage and color that is truly a treat for the senses. It can get very crowded, especially during Chinese New Year, when families come and buy auspicious plants that will bring them good luck and fortune.
SHAM SHUI PO, Hong Kong's material and trimmings bazaar, began in the mid-1970s; to this day, it still offers everything. It has street upon street of wholesalers selling fabrics in every color and pattern imaginable.
My favorite place to stop for tea is TEAKHA, located in Sheung Wan. It's a small, hidden spot where you can sit on the terrace. It has good pastries and treats, too.
CAT STREET MARKET, Upper Lascar Row, Tai Ping, Hong Kong, cat-street.hk.
SHANGHAI STREET, Yau Ma Tei, Hong Kong.
HONG KONG FLOWER MARKET, Flower Market Rd., Mong Kok, Hong Kong, flower-market.hk.
SHAM SHUI PO, Ki Lung, Nam Cheong St. and Yu Chau St., sham-shui-po.com.
TEAKHA, 18 Tai Ping Shan St., Hong Kong, +852 2858 9185, teakha.com
#### Liger
* * *
Local fashion icon Hilary Tsui and her business partner have three Liger stores in HK carrying a fun range of international and local brands, such as Alice McCall, Holly Fulton, Shrimps, Fenty, Stella Jean, and Johanna Ho, as well as the in-house label, Oh My God.
Shop A & C Vienna Mansion, 55 Paterson St., Fashion Walk, Causeway Bay; +852 2503 5308, ligerstore.com, and additional locations.
#### Lii Lii
* * *
"With the help of the fabulous staff, it is possible to have customized dream shoes handcrafted," says Hong Kong fashion expert Tania Mohan. "It's quite a phenomenal shopping experience and so easy to get lost in the choices of skins, colors, textures, finishes, and heel heights." Orders take ten to fourteen days.
Shop 75, 1/F, Admiralty Center, 18 Harcourt Rd., Queensway, Admiralty, Hong Kong, +852 2865 3989.
#### PageOne
* * *
"You can find the widest selection of fashion and travel magazines plus back issues," shoe designer Isa Tapia says of this local chain that has been struggling in recent years. "I found some issues of Vitals magazine there that I treasure to this day."
Shop LG 1-30, Festival Walk, 80 Tat Chee Ave., Kowloon, Hong Kong, +852 2778 2808, pageonegroup.com.
#### PMQ
* * *
"A wonderful place for discovering young and local designers," says luxury marketing consultant Peter Cheung of this creative complex (it used to be a school, then served as living quarters for young married police officers). "PMQ offers subsidized rental for local brands to have workshop and retail space. In the two building blocks, there are different shops, brands, and pop-ups. There's always something new."
35 Aberdeen St., Central, Hong Kong. +852 2870 2335, pmq.org.hk.
BONNAE GOKSON
HONG KONG'S CHICEST RESTAURATEUR, TAKES FIVE
* * *
The creative force behind Hong Kong's most fashionable restaurants and bakeries, Bonnae Gokson is also a fashion and style icon to boot!
My restaurant, SEVVA, at Prince's Building, has the best afternoon-tea experience, and C'EST LA B CAFE at Pacific Place has the best sweet treats in town. Book Samuel So at MANDARIN ORIENTAL'S SALON for the best pedicure you've ever had. It's a dying art, and So was trained by his famous father to use the sharpest knives and blades for the prettiest feet!
SEVVA, Prince's Building, 25/F, 10 Chater Rd., Central, Hong Kong, +852 2537 1388, sevva.hk.
C'EST LA B CAFÉ, Shop 202, 2/F, Pacific Place, 88 Queensway, Admiralty, Hong Kong, +852 2536 0173, msbscakery.hk.
MANDARIN ORIENTAL HONG KONG SALON, 5 Connaught Rd., Central, Central, Hong Kong, +852 2825 4888, mandarinoriental.com/hongkong/luxury-spa.
ONLINE SHOPPING TIP
"I love Hong Kong designer Johanna Ho's online store johannaho.com. She has mega-cute and wearable womenswear and I like to support local designers," says Grace Lam. "Her knitwear collection is to die for and she ships internationally."
#### Qeelin
* * *
Owned by Kering, this Hong Kong–based luxury jewelry brand inspired by Chinese culture first caught the world's attention in 2004 when actor Maggie Cheung wore its curved, gourd-shaped Wulu earrings on the red carpet at Cannes. Since then, the brand has opened boutiques around the globe, including Paris, Beijing, and Shanghai, as well as wholesaling to stores like Neiman Marcus. The name Qeelin comes from the Chinese ideograms for male (qi) and female (lin), and symbolizes harmony, balance, and peace.
Festival Walk, 80 Tat Chee Ave., Hong Kong, +852 2787 2268, qeelin.com, and additional locations.
#### Marie France Van Damme
* * *
Clothes for the woman for whom every day is a holiday. The designer launched her namesake luxury resortwear collection in 2011, and it has since grown into a global lifestyle brand selling elegant yet relaxed swimwear, caftans, maxi dresses, and more that can take you from the beach to dinner.
IFC Mall, Podium 3, 8 Finance St., Central, +852 2882 9088, mariefrancevandamme.com, and additional locations.
#### Sasa
* * *
"You can find obscure Korean and Japanese skin care products and makeup here," accessories designer Fiona Kotur Marin says of this chain with more than a dozen locations. "I recently bought tattoos that you apply on your eyelids to resemble ornate makeup. It's great for unique beauty products."
G/F&2/F, Leighton Centre, 77 Leighton Rd., Causeway Bay, Hong Kong, +852 2555 0806, sasa.com, and additional locations.
#### Shanghai Tang
* * *
This international clothing company was founded in Hong Kong in 1994, and purchased by luxury conglomerate Richemont in 1998. There are forty-eight boutiques worldwide, selling brilliantly hued silk and cashmere ready-to-wear with a Chinese touch, cheongsams and mandarin suits, cardigans with silk linings, silk pajamas, lacquered boxes, and more. The stores also offer an Imperial Tailoring Service.
1 Duddell St., Central, Hong Kong, +852 2525 7333, shanghaitang.com, and additional locations.
## SHANGHAI
### CHINA
#### 10 Corso Como
* * *
In 2013 the Milan-based concept store opened a Shanghai branch with four floors of art, design, and fashion, plus a bookstore, exhibition space, and an Italian restaurant.
1717 West Nanjing Rd., Jing'An District, Shanghai 200040, +86 21 6286-1011, 10corsocomo.com.
#### Alter
* * *
This multi-brand and lifestyle concept store opened in 2010, offering an alternative to luxury megabrands—its name is short for alternative—and has become a cornerstone of the city's new fashion scene. The stock is a cross section of eclectic women's fashion from Europe and Asia (Thakoon, Jason Wu, Simone Rocha, Charlotte Olympia), as well as apothecary items, all in a stunning space inspired by M.C. Escher's descending and ascending stairs.
Xintiandi Style, 245 Madang Rd., Lu Wan District, Shanghai 200021, +86 21 6302-9889, alterstyle.com
#### Dongliang
* * *
This is the place to get schooled about the best new Chinese designers. Located in a three-story lane house with lots of nooks and crannies to explore, Dongliang promotes homegrown talent, including Uma Wang, Yirantian Guo, Deep Moss, and He Yang, by supporting brands with marketing and mentorship, and organizing contingents of Chinese designers to travel to fashion weeks around the world. There's a second store in Beijing.
184 Fumin Rd., Jing'An District, Shanghai 200000, +86 21 3469-6926, dongliangchina.com.
#### Erdos 1436
* * *
China's answer to Loro Piana, Erdos 1436 is a luxury cashmere purveyor with a pared-back collection of ready-to-wear for men and women. It's no coincidence that Erdos lent its cashmere production supply services to the likes of Hermès, Burberry, and, yes, Loro Piana before launching its own brand in 2007. The company recently hired Graeme Black, a designer with luxury experience, to give it a more fashionable spin.
Shanghai Gulf, 1118 Pudong South Rd., Pudong New District, Shanghai 200120, +86 21 6888-1436, 1436erdos.com, and additional locations.
#### Grace Chen
* * *
Chen was hailed by the South China Morning Post in 2015 as "the power dresser . . . whom China's top-achieving women turn to." She worked as a designer in New York and Los Angeles before returning to Shanghai in 2009 to establish her own brand. Her made-to-order collection includes elegant evening gowns and cocktail dresses, chic day and business wear with an Asian spin. Her first boutique is located in Shanghai's former French Concession building, which also includes a gallery, library, movie theater, and fashion lounge to cater to China's high rollers, including the country's first lady.
1515 Hua Shan Rd. (French Concession), Changning District, Shanghai 200052, gracechenstyle.com
GRACE CHEN
#### Guo Pei Xi
* * *
China's most famous couturier, Guo Pei, shot to international stardom when she dressed Rihanna for the 2015 Met Gala in a regal yellow cape gown with a train that went on for days. She opened this store for her (slightly) more accessible demi-couture wedding gown collection. Instead of a three-month lead-time, dresses can be made within two weeks. Prices are still high, but not as high as for her couture gowns, which can run in the $200,000 to $500,000 range, and are sold through her bespoke Rose Studio in Beijing or at her store in Paris.
South Bund 22, 22 Zhongshan Dong Er Rd., Huangpu District, Shanghai 200000, +86 21 5352-0796, guo-pei.fr.
#### Maison Charles Philip
* * *
The story of this made-in-China shoe brand started when founder Charles Philip had one of his blue-and-white-striped shirts crafted into a loafer by a Shanghai cobbler on Mao Ming Road. Now an international business, the brand produces chic loafers, smoking slippers, and sneakers in every color and pattern imaginable.
101 Gao An Rd., Suite B-2, Xuhui District, Shanghai 200031, +86 21 6422-6982, charlesphilipshanghai.com.
#### OOAK
* * *
This jewel box of a retail space showcases indie designers of costume jewelry, accessories, gifts, housewares, art pieces, and books, including Sharouk, Gentle Monster, Celestine, Delfina Delettrez, and Lizzie Fortunato.
No.30, 820 Nong, JingAn District, Shanghai 200000, +86 21 5424-8929, theooak.com.
#### One by One
* * *
Founded in 2003, One by One claims to be the first independent designer collection store in China. Designers Qiu Hao (winner of the 2008 Woolmark Prize) and Wang Chuqiao (her collection is Neither Nor) opened the store on Changle Road to showcase their minimalist avant-garde designs, alongside other top Chinese designers, including Masha Ma and Yun Linn. There are several branches in China, but the Shanghai store is the flagship.
Xintiandi Style, 245 Ma Dang Rd., Huangpu District, Shanghai 200000, +86 21 3331-2627, onebyonestudio.com, and additional locations.
#### Project Aegis
* * *
This two-story menswear temple combines contemporary labels (Kapital, Rag & Bone, Public School) with its own understated tailored brand. There's also a café. A must-see.
1 Taojiang Rd., Xuhui District, Shanghai 200000, +86 21 5403-9869, projectaegis.com and additional locations.
#### Shang Xia
* * *
China's first homegrown luxury brand, Shang Xia is backed by Hermès, and features furniture, decorative objects, clothing, accessories, and rotating exhibitions. The spare, blond-wood interior was designed by Japanese architect Kengo Kuma. Some of the items are reminiscent of Hermès products, including printed silk scarves, enamel bracelets, and bamboo pendants, but with an emphasis on materials and crafts native to China.
233 Middle Huai Hai Rd., Huangpu District, Shanghai 200000, +86 21 8017-9777, shang-xia.com, and additional locations.
#### Spin Ceramics
* * *
"A very chic ceramic store that I always find myself at to purchase tableware and kitchen accessories," says designer Tadashi Shoji of this destination for handmade vases, chopstick rests, and dim sum paperweights packaged in bamboo steamers. "It's not always cheap, but you can find great little treasures or bits of inspiration when visiting."
360 Kangding Rd., Jing'An District, Shanghai 200000, +86 21 6279-2545, spinceramics.com, and additional locations.
#### Triple Major
* * *
This destination for playful, indie fashion and design opened on Shaoxing Lu, a historic publishing street, in 2012, and eventually expanded from the ground floor to occupy the whole four-story building. In addition to Henrik Vibskov, Bless, Eatable of Many Orders, and other labels, you'll find home accessories, indie magazines, and artisan-made crafts.
25 Shaoxing Rd., Huangpu District, Shanghai 200000, +86 180 2100-2509, triple-major.com/shanghai, and additional locations.
#### Uma Wang
* * *
China's biggest international fashion success story, Wang is known for her oversized cuts, avant-garde draping techniques, and texture combinations. She studied in London at the prestigious Central Saint Martins and has shown her collections around the world. In 2012, Wang was selected to be the first Chinese designer to take part in the inaugural CFDA/China Exchange Program.
299 Fuxing West Rd., Xuhui District, Shanghai 200000, +86 21 6262-2032, umawang.com, and additional locations.
#### Xinlelu
* * *
Xinlelu highlights indie designers with the goal of creating what the website describes as a "new modern Shanghai girl." You'll find an edited collection of clothing, accessories, and gifts from China (Kun Man Ma, Sefhyir) and abroad (Alice & Trixie, Garance Doré for Rifle Paper).
414 Shaan Xi Bei Rd. (close to Plaza 66), Jing'An District, Shanghai 20000, +86 21 5213-3301, xinlelu.com.
#### Ziggy Chen
* * *
Shanghai-based designer Ziggy Chen established his namesake label in 2012, and has attracted international stockists based on his edgy but wearable menswear designs that incorporate prints on century-old Chinese and European silhouettes.
204/2F No. 6 Building, 123 Xing Ye Rd., Luwan District, Shanghai 200000, +86 21 3331-2968, ziggychen.com, and additional location in Beijing.
## BEIJING
### CHINA
#### Anchoret
* * *
This Beijing-based fashion boutique founded in 2011 is located in the heart of the Sanlitun shopping district. A multi-label boutique, it's been dedicated to bringing fashion to Beijing from the beginning. Yohji Yamamoto, Vetements, Yang Li, Lemaire, Undercover, Taichi Murakami, and more are represented.
Sanlitun Bei 25 Hao Lou, Beijing, +86 10 8400 2677, anchoret-beijing.com.
#### Brand New China
* * *
Opened by Chinese media veteran Hung Huang (known as "the Oprah of China"), and located in the basement of Tai Koo Li North, Brand New China brings together inventive clothing, shoes, bags, and accessories from a variety of local designers. A must-shop destination for cutting-edge talent.
NLG-09A, B/1F, Taikooli North, 11 Sanlitun Rd., Chaoyang District, Beijing, 100027, +86 10 6416 9045, brandnewchina.cn.
ONLINE SHOPPING TIP
Maison Charles Phillip Shanghai (charlesphilipshanghai.com) ships slippers internationally. Spin Ceramics (spinceramics.com) also has international shipping, as well as a brick-and-mortar store in New York City. LaneCrawford.com stocks a number of fabulous Chinese labels, including Shushu/Tong, Xiao Li, and Helen Lee, and ships internationally. Modaoperandi.com stocks several Chinese designers on the rise, including Huishan Zhang, known for his elegant, East-meets-West style.
Designer Tadashi Shoji, who splits his time between Los Angeles and China, recommends world.Taobao.com. "It's the largest online shopping market place in China. It's similar to eBay, but it is much more efficient, with extremely low prices."
## SEOUL
### SOUTH KOREA
#### Aland
* * *
Sometimes referred to as the Korean Urban Outfitters, this funky chainlet sells everything from new and emerging contemporary pop-culture-influenced Korean collections (SJYP, Stereo Vinyl Collection, Salad Bowls, Margarin Fingers, Fleamadonna) to vintage clothing and even iron-on patches, all at reasonable prices. There are several branches in Seoul and across Asia.
53-6, Myeongdong 2-ga, Jung-gu, Seoul, +82 2-3210-5890, and additional locations, alandglobal.com.
#### Beaker
* * *
This multi-brand shop is part of Samsung's Cheil Industries and stocks a wide range of contemporary brands, including Opening Ceremony, Our Legacy, Wood Wood, Rag & Bone, Melissa, Karen Walker, Building Block, and Common Projects, as well as hosting shop-in-shops for Korean brands.
78-6, Cheongdam-dong, Gangnam-gu, +82 2-543-1270, and additional locations, beakerstore.com.
#### Boon The Shop
* * *
"A fashion girl's mecca," says Olivia Kim of this Peter Marino–designed crown jewel of Seoul multi-label boutiques, which sells Céline, Rochas, Martin Margiela, Vetements, Balenciaga, and other top international labels. It also peddles Korea's own street-style-driven fun furs (KYE) and cheeky graphic separates (Kiok). "Located in the überchic neighborhood of Gangnam, Boon showcases the biggest brands in fashion as well as the coolest young, emerging brands to come out of Seoul, fashion's newest darling city. The level of service, the visual merchandising, and the great selection make this one of my faves," says Kim. Also has a café and a Japanese restaurant.
21 Apgujeong-ro 60-gil, Gangnam-gu, Seoul, +82 2-2056-1234, boontheshop.com.
#### Doota
* * *
In Dongdaemun, the retail and wholesale district that never sleeps, are several high-rise malls selling trendy clothing and accessories. The best is Doota, which has about 500 boutiques across a labyrinthine eight floors and devotes the basement to up-and-coming talent, some of whom eventually graduate to the Seoul Fashion Week runways. Open twenty-four hours, it's easy to lose track of time here. Consider yourself warned.
Doosan Tower, 18-12 Euljiro 6-ga, Jung-gu, +82 2-3398-3034, and additional locations, doota.com.
#### Ecru
* * *
"Located in the basement, it's not easy to find even with the small signs," says Minyeong Kim, brand director for Gentle Monster, of this boutique that stocks European and Japanese brands, including Acne, WTAPS, MSGM, Undercover, and its own in-house line. "It is quite like LNCC in London, but they carry uncommon things for Korea, so it's fun."
656-10 Sinsa-dong, Gangnam-gu, Seoul, South Korea, +82 2-545-7780, ecru.co.k.
#### Galleria
* * *
This department store has been described as the Korean version of Bergdorf Goodman. You'll find the crème de la crème of luxe labels, including Goyard bags, Tom Ford clothing, and Fred jewelry, alongside a great selection of contemporary Korean labels such as Low Classic, Pushbutton, Wooyoungme, and Lucky Chouette.
343 Apgujeong-ro, Gangnam-gu, Seoul, dept.galleria.co.kr.
#### Gentle Monster
* * *
This was the first flagship for the cool Korean eyewear brand, which has now expanded to several locations in Seoul and internationally. Set on two levels, one of which has a cactus garden, the store doesn't disappoint. Each Gentle Monster location has a different vibe.
192-10 Nonhyeon-Dong, Gannam-Gu, Seoul, +82 2-3443-2126, gentlemonster.com, and additional locations.
#### Goen J
* * *
Born in Korea, Goen Jong spent seven years in Paris before launching her own line in 2012. Her sculptural yet romantic pieces (ruffles, lace, and eyelet are mainstays, used in modern ways) have earned her an impressive roster of international retailers, including Barneys New York, Opening Ceremony, and Shopbop.com. This is her flagship.
546-18, Sinsa-dong, Gangnam-gu, Seoul, +82 70-4408-1754, goenj.com.
#### Gwangjang Market
* * *
While most people visit this traditional street market for food, it also has a bounty of textiles, traditional Korean hanbok outfits, used clothing, and accessories (if you have the patience to sift through the piles).
88 Changgyeonggung-ro, Jongno-gu, Seoul, +82 2-2267-0291.
#### Insa-dong
* * *
"One of the most memorable attractions in Seoul, it represents the focal point of Korean traditional culture and crafts," says fashion publicist Natalie Choy of this shopping district anchored by the main street of Insadong-gil, which connects to a multitude of alleys full of modern galleries and tea shops. "Stores in Insa-dong specialize in a wide variety of goods that can be purchased only in Korea such as hanbok [traditional clothing], hanji [traditional paper], traditional teas, pottery, and folk crafts." The street is closed to traffic on weekends.
62, Insadong-gil, Jongno-gu, Seoul.
#### Jardin de Chouette
* * *
Designer Kim Jae Hyun's young, fun collections (think, denim culottes, mixed plaid shirtdresses) have made fans of model Irene Kim and other Seoul tastemakers. Her lower priced Lucky Chouette label is also a hit, with printed backpacks, striped sailor shirts, velvet high top sneakers, and plenty of items featuring the brand's owl logo.
21-15 Cheongdam-dong, Gangnam-gu, Seoul, and other locations, +82 2-3444-1002, jdchouette.com.
#### Juun.J
* * *
One of the most important figures in Korean men's fashion, Juun.J debuted his first collection under the name Lone Costume in 1991, and launched his namesake label in 2007. Known for his directional collections, he describes his look as "street tailoring" (drapey trenches, reinvented zoot suits, and other oversized silhouettes). His collections are stocked at Dover Street Market, L'Eclaireur, and other influential retailers around the world. He has also collaborated with other brands, Adidas and Reebok among them.
4f, 343 Apgujeong Ro, Gangnam Gu, Seoul, +82 2-3449-4144, juunj.com.
#### Koon With a View
* * *
This Korean brand and high-end boutique sells J.W. Anderson, Alexander Wang, and other contemporary labels in a dynamic-looking space. A bit like Intermix but edgier.
21-12 Cheong-dam dong, Gangnam-gu, Seoul, +82 2-548-450, koon-korea.com, and additional locations.
#### Low Classic
* * *
One of the coolest contemporary brands to come out of Seoul, Low Classic is minimalist but whimsical (think Céline meets Tibi), with lots of asymmetrical wrap skirts, slip dresses, and sliced knits. For a recent collection, designer Lee Myeong Sin reflected on women's bodies by showing crisp white suits and slip dresses featuring ink sketches of female forms. Accessories are also strong, including square-toe mules and fold-detail bags.
554 Sinsa-dong, Gangnam-gu, Seoul, +82 2-516-2004, lowclassic.com.
IRENE KIM
NEW DESIGNER FAVORITES
* * *
Dubbed by Vogue as an "ambassador to the buzzing South Korean fashion scene" and Women's Wear Daily as fashion's "It Girl," Irene Kim is the host of popular Korean fashion and beauty programs such as K-Style. She is at the forefront of the Korean entertainment establishment, and she represents the emergence of Korean talents who are influencing the marketplaces of East Asia and beyond.
A great place to discover new talent is COMMON GROUND, a concept shopping venue made out of shipping containers with emerging new Korean designers, lifestyle items, accessories and Korean brands.
My favorite only-in-Seoul experiences are twenty-four-hour food delivery, Dongdaemun night shopping and the best fried chicken and beer in the world!
COMMON GROUND, 17-1 Jayang-dong, Gwangjin-gu, Seoul, +82 2-467-2747, common-ground.co.kr.
#### MIK247
* * *
Owned by stylist Chae Han-Suk, who works with some of K-pop's leading stars, MIK stands for Made in Korea. Suk exclusively stocks homegrown designers and collaborates with them on unique pieces, some of which are relatively affordable.
2/F, Shinwoo Building, 664-19 Sinsa-dong, Gangnam-gu, Seoul, +82 2-3446-8556, facebook.com/mik247.
#### MMMG
* * *
MMMG (Millimeter Milligram) puts the focus on the simple things in life by selling books and stationery and small quantities of clothing and accessories with a distinct point of view. A favorite of handbag designer Clare Vivier's.
240 Itaewon-ro, Yongsan-gu, Seoul, +82 2-549-1520, mmmg.net.
#### Monday Edition
* * *
Model/tastemaker Irene Kim loves the idiosyncratic jewelry from this local label, including ball-and-stick bracelets, twisted leather earrings, shell flower necklaces, and more.
1F, 57, Daesagwan-ro 11-gil, Yongsan-gu, Seoul, +82 70-4284-5923, Monday-edition.co.kr.
#### My Boon
* * *
Younger sibling of the Boon store, this space focuses on edgier brands, including Chrome Hearts, Visvim, Linda Farrow, Grenson, and Garret Leight, and stocks lots of gifts. It's inside the Shinsegae department store.
1F, SSG, Pie'N Polus, 4-1, Cheongdam-Dong, Kangnam-Gu, Seoul, +82 2-6947-1270, myboon.co.kr
#### Playnomore
* * *
This whimsical accessories brand turns out tasty treasures such as classic frame bags adorned with winking eye or red lip patterns.
60, Namdaemunggo, Jung-gu, Seoul, +82 2-3789-5526, playnomore.co.kr, and additional locations.
#### Queenmama Market
* * *
This seven-story concept store brings a bit of Los Angeles to Seoul. It was founded by Korean designers Kang Jin-Young and Yoon Han-Hee, who worked in New York and L.A. before returning to Korea to bring a taste of the laid-back West Coast lifestyle to the fashion retail scene. The building mixes nature and shopping, with clothes nestled among gardening tools and plants, housewares, stationery and candles, and a beautiful rooftop café, Manufact.
50, Apgujeong-ro 46-gil, Gangnam-gu, Seoul, queenmamamarket.com.
#### Rare Market
* * *
Opened in 2014 by lifelong friends Dami Kwon, sister of K-pop star G-Dragon, and Jessica Jung, Rare Market is a regular stop on the international fashion circuit for street-style stars (Miroslava Duma) and celebs (Rihanna) alike. The store specializes in the newest, coolest brands, whether that's Jacquemus, Marques'Almeida, and Faith Connexion from Europe, or KYE from Korea (a Rihanna favorite). It also has its own in-house label that launched in 2015 with a lookbook starring Leandra Medine.
95-5 Cheongdam-dong, Gangnam-gu, Seoul, raremarket.com.
GENTLE MONSTER
#### Shinsegae
* * *
"You can spend the day there, it's a whole world," says Chrome Hearts designer Laurie Lynn Stark of Shinsegae, one of the best department stores in the world. Shinsegae carries every brand imaginable, has a rooftop park and restaurant, upscale food courts, and a cinema. "You are sitting on the roof having lunch, looking at the mountains with a trolley going up the mountain, and then there's a Jeff Koons Sacred Heart sculpture next to you. It's beautiful," she says.
52-50 Chungmuro 1ga, Seoul, +82 2-1588-1234, shinsegae.com.
#### SJYP
* * *
Husband-and-wife team Steve J and Yoni P are stalwarts of Seoul Fashion Week. The duo met in London at Central Saint Martins, and founded their label there in 2006. Now based in Seoul, they specialize in edgy denim and utilitarian pieces at contemporary prices. This is their flagship store.
1F, 685-12 Hannam-dong, Seoul, +82 70-7730-9988, steveyoniworld.com.
HOW AIMEE SONG DOES SEOUL
* * *
Instagram star Aimee Song, creator of the popular fashion blog SONG OF STYLE, has more than 4 million online fans who follow her tips on fashion, design, food, and travel. A frequent traveler to fashion weeks around the world, she is also the author of the book Capture My Style.
BOON THE SHOP is like the Korean version of Barneys, but done even better. It's a super well-curated, multi-brand high-end fashion store with amazing interiors and futuristic archi-tecture spanned across many levels. It's easy to get lost here and for any design lover, it's the best place to be lost! Even the elevators are super cool and well designed.
I actually found MSMR SOCK STORE (also called Ladies and Gentlemen) through a graphic designer I was following on instagram, @venusmansion. He did all of the labeling and design and I'm so happy I discovered it. They make the best customizable gift boxes. The whole experience is really fun—choosing the box and stickers to decorate it with. Not to mention all the unique socks. I never thought socks would make such a great gift until I came here!
There are a bunch of GENTLE MONSTER stores in Seoul and the surrounding cities. They're one of my favorite sunglasses brands and each store has a distinct concept. One I went to has a butcher shop theme. Another revolved around a traditional Korean spa with sinks, tubs, and amazing tile work. I love to see how they showcase the sunglasses and create unique experiences.
DONGDAEMUN DESIGN PLAZA
DONGDAEMUN DESIGN PLAZA is so stunning. It was designed by Zaha Hadid who is one of my idols. It's part museum, part exhibition space, and also where they hold Seoul Fashion Week.
INNISFREE is a great store for any K-beauty beginner. For the most part, getting good Korean beauty products in Korea is easy but for someone who wants a one-stop, this is a good place to start. They have great products for both men and women and what I like about this wallet friendly beauty store is that they carry products with ingredients sourced from Jeju island which is very unique to Korea.
BOON THE SHOP, 17 Apgujeong-ro 60-gil, Gangnam-gu, Seoul, +82 2-2056-1234, boontheshop.com.
MSMR SOCK STORE, 247-1, Itaewon-2-dong, Yongsan-gu, Seoul.
GENTLE MONSTER, , 192-10 Nonhyeon-Dong, Gannam-Gu, Seoul, +82 2-3443-2126, and additional locations, gentlemonster.com.
DONGDAEMUN DESIGN PLAZA, 281 Eulji-ro, Jung-gu, Seoul, ddp.or.kr.
INNISFREE, Myeong-dong Store, 64-2 Myeong-dong Il(1)-ga Jung-gu, Seoul, +82 2- 776-0117, and additional locations, innisfreeworld.com.
ONLINE SHOPPING TIP
Aland's website ships internationally (alandglobal.com). If you want to purchase merchandise from a Korean site that does not support overseas delivery, momokorea.com can help. An online middleman, it works with top Korean labels, including SJYP, Steve J and Yoni P, Stereo Vinyls, and more. Enter payment info on MomoKorea's website and it will do the rest. Another site of note: Peach & Lily (peachandlily.com). Founded by K-beauty star Alicia Yoon, it's a great resource for Korean and Japanese beauty brands.
#### Space Mue
* * *
This menswear concept store (a must for US avant-street designer Chris Stamp when he's in Seoul) caters to the K-pop set with trendy items from Lanvin, Valentino, and others, in an imposing gallerylike space.
93-6 Chungdam-dong, Seoul, +82 2-3446-8074, spacemue.com.
#### Slow Steady Club & Café
* * *
Dedicated to the values of "slow and steady," this store specializes in apron dresses, pullovers, simple coats, and other pared-down basics built to stand the test of time (think, brands like Champion, Common Projects, and Still by Hand). It also boasts a café and gallery.
130-1 Palpan Jongno-gu, Seoul, +82 2-725-1301.
#### Tom Greyhound
* * *
This multi-brand store, which has an outpost in Paris, stocks the best international designers alongside up-and-coming Korean labels. Interior designer Brigette Romanek calls it a "hipster haven curated with items you haven't seen before. They carry some collections you know, but they pick the pieces that other stores don't. It's fun in there."
10-3, Dosan-daero 45-gil, Gangnam-gu, Seoul, +82 2-3442-3696.
#### Worksout
* * *
With a spaceship-like silver facade, this three-story streetwear destination features more than forty labels from the US, Japan, and Korea, including Carhartt, Reigning Champ, Obey, and Vans.
20-4 Seolleung-ro 157-gil, Gangnam-gu, Seoul.
## MUMBAI
### INDIA
#### Tribe by Amrapali
* * *
One of India's most well-known jewelry brands, Amrapali was founded in 1978 by Rajesh Ajmera and Rajiv Arora. The brand is influenced by the country's varied cultural traditions, as interpreted by 1,200 craftsmen. Amrapali jewelry is available in more than 36 outlets. You can find everything from a pair of gold-plate floral jaali hoop earrings for less than $50 to a multicolored sapphire bangle for $10,000.
Juhu Rd., Mumbai 400049, + 91 22 26125001/5002, tribebyamrapali.com, and additional locations.
#### Anita Dongre
* * *
Inspired by Rajasthan's rich heritage, this designer brand showcasing Indian aesthetics with a modern sensibility was thrust into the international spotlight in 2016 when Kate Middleton wore one of Dongre's breezy, bohemian print dresses during the royal visit to India and Bhutan. (Demand for the style crashed the designer's website.)
Everest Classic, Linking Rd., Khar (West), Mumbai 400052, +91 22 67411504, anitadongre.com, and additional locations.
#### Anokhi
* * *
Beloved around the world for preserving the tradition of Indian hand-block printing. Through Faith Singh, Anokhi's British designer, this ancient textile art has been revived in modern designs and exotic patterns ranging from pastel florals to dramatic geometrics. Anokhi is involved with educational and social projects in Rajasthan, where the company is based and its products produced. Anokhi first appeared in London in 1970. Printing is done twice yearly in limited collections. At the shops, you can find everything from tunics and robes to pillowcases, scarves, place mats, and cosmetic pouches.
Rasik Niwas, Metro Motors Lane, Dr. A. R. Rangnekar Marg., Mumbai 400007, +91 22 23685761, +91 22 23685308, anohki.com, and additional locations.
#### Bombay Electric
* * *
"For eclectic fashion by local and regional Indian designers," says costume designer Arianne Phillips of this Indian concept store, which is also a favorite of fashion consultant/author Fern Mallis. Opposite the five-star Taj Mahal Palace and Towers and Gateway of India on "the Bond Street of Mumbai," and a stone's throw from a busy street filled with food stalls, Bombay Electric is housed in a historic heritage building. Restoration and redesign exposed antique interiors featuring original tree-size teak beams that were a gift from the king of Burma, coupled with sculptural white racks and industrial trunks. This is the place to find everything from Manish Arora's kaleidoscopic pop clothing to earrings shaped like Valium capsules, whimsical clutches, coffee-table books, and vintage Bollywood posters.
1 Reay House, Best Marg., Mumbai, 400001, +91 22 22876276, facebook.com/bombayelectric.
#### Bungalow 8
* * *
Mumbai's first concept store, Bungalow 8 specializes in unconventional Indian- inspired antiques and fashion finds, and is a must-shop in Mumbai for Fern Mallis, Divia Harilela (and Madonna and Sting). Maithili Ahluwalia's bungalow is devoted to spotlighting the best of Indian design, from handmade ceramics to salvaged factory lights and baroque picture frames. Think ABC Carpet & Home, only in India and underneath a cricket stadium.
Inside Wankhede Stadium, North Stand, E & F Block, D Rd., Churchgate, Mumbai 400020, +91 22 22819880/1/2, bungaloweight.com.
#### D Popli & Sons
* * *
Jewelry for all price points and occasions, all of it irresistible. In the front section of this unassuming store you'll find costume jewelry and silver pieces starting at less than $50. In the back is the finer stuff, contemporary-looking pieces set with diamonds, rubies, and semiprecious stones, including basic stud earrings and edgier three-finger rings.
Chhatrapati Shivaji Maharaj Marg., Apollo Bandar, Colaba, Mumbai 400001, +91 22 22021694.
#### Good Earth
* * *
"Love shopping for linens and china here," says Arianne Phillips of India's most famous home design company, founded in Mumbai in 1996. Good Earth is known for its crafts-focused approach to luxury and for reviving the authentic skills of the crafts communities of India, by producing handmade Kansa tableware, wallpaper, quilts, and more. The brand also has a line of clothing made of chintz textiles.
2 Reay House, Colaba, Mumbai, 400001, +91 22 22021030, goodearth.in, and additional locations.
#### Le Mill
* * *
"The colette of Mumbai," says Arianne Phillips of this iconic store, where global fashion brands (Balenciaga, Peter Pilotto, Sacai) sit alongside the new Indian voices of fashion (Dhruv Kapoor, Nimish Shah). "They have a great café and are upstairs from a delicious restaurant." (The restaurant is a branch of Indigo Deli, mentioned by Kavita Daswani.) The store is housed in a 1930s warehouse that was originally used as a rice mill. The founders have international experience at Bergdorf Goodman, L'Oréal, and Condé Nast.
1st Floor, Pheroze Building, Colaba, Mumbai, 400005, +91 22 22041926/27, lemillindia.com.
## NEW DELHI
### INDIA
#### Dili Haat Market
* * *
"A focal point for artisans from every region of India to come and showcase their beautiful handicrafts," fashion editor/designer Tania Mohan says of this open-air crafts bazaar. "You'll find everything ranging from exquisite silk embroideries to mirror works to reinterpreted threadwork to pashminas. The handicrafts are always changing, and the prices very fair. For my brand Tabla, which is wholly inspired by and made in India, this hub is a rich source of inspiration and the reason why I visit it at least a few times a year. It allows me to research such things as new weaves, techniques, and shawl compositions. I'd recommend a visit to everyone."
Sri Aurobindo Marg, Laxmi Bai Nagar, New Delhi 110023, +91 98 480 22338.
ONLINE SHOPPING TIP
PerniasPopUpShop.com is a comprehensive e-commerce site for jewelry, accessories, saris, tunics, and gowns by top Indian designers, including Sabyasachi, Manish Arora, Tarun Tahiliani, Amrapali, Anita Dongre, and more. It's curated by leading stylist and Bollywood costume designer Pernia Qureshi. Offers worldwide shipping.
KAVITA DASWANI
AN INSIDER'S GUIDE TO INDIA
* * *
Kavita Daswani is an Indian-American author who has published eight novels, which have been translated into seventeen languages. She is also a journalist covering fashion, beauty, travel, design, and celebrities for a range of global publications. A former fashion editor for the South China Morning Post in Hong Kong and an Asian correspondent for Women's Wear Daily, both of which she still contributes to, she currently writes for the Los Angeles Times, and the international editions of Vogue, Condé Nast Traveler, and Grazia.
In Delhi, India's top designers have made their homes in EMPORIO MALL. Next door to anchor brands like LOUIS VUITTON and DIOR, shoppers will find homegrown couturiers like MONISHA JAISING (shop number 331, monishajaising.com), known for svelte evening gowns, and TARUN TAHILIANi (shop number 351, taruntahiliani.com), who dresses Bollywood's biggest stars and is notable for his avant-garde approach to the traditional sari. While stoles and shawls are a dime a dozen in India, Delhi boutique JANAVI (shop 323, janavi.com) specializes in them: look for cashmere and merino wool shawls featuring modern motifs. Nestled amid all this beauty is ZEST RESTAURANT, popular for its multiple cuisines (Indian, Asian, European) and expansive terrace for a day when the smog is bearable.
Outside of Emporio, OGAAN is a multi-designer boutique chain with several outposts throughout the city. It's beloved by the fashion set for its curated offerings: stop in and find crisp linen shirts, flowing tunics, wide-leg pants, and elegant ethnic wear. End the day with a visit to the AMBIKA PILLAI salon, where Delhi's beautiful people go to be fussed over before a night out. You can get a blow-dry anywhere, but India is known for its lush, luxurious head massages using coconut, olive, or mustard oil. No matter what, book one.
In Mumbai, stroll down COLABA CAUSEWAY for casual, locally made garments (cotton tunics, woven leather sandals). At PALLADIUM MALL, drop by GLOBAL DESI (second level), an India-based brand that fuses local fabrics and traditional patterns with a Western aesthetic and very appealing prices: breezy maxi dresses, cute dungarees with elephant prints, long tunics to be paired with jeans or belted and worn as a dress. Most items cost less than $50.
Similarly, FAB INDIA, with a flagship store in the trendy Kala Ghoda district, features clothing made by local craftspeople and modeled on Western wardrobe staples.
At the higher end, the city's socialites flock to the studio of SABYASACHI MUKHERJEE, whose extravagant pieces recall the nation's royal pedigree. Designer Tahrun Tahiliani's multi-brand boutique ENSEMBLE has long been a draw for brides-to-be compiling their trousseaux (even in modern India, this is a thing). There, seek out a hot favorite: ANAMIKA KHANNA, whose modern, almost architectural aesthetic has elevated Indian fashion to another level.
After a spot of shopping at ENSEMBLE, stop for lunch at INDIGO DELI for fusion offerings like pepper chicken stuffed into a paper-thin chapati-style wrap, or hop into a cab and adjourn to THE DOME, the rooftop bar at the Intercontinental overlooking the glittering lights of Marine Bay and the Indian Ocean.
DLF EMPORIO, 4, Nelson Mandela Marg, Vasant Kunj, New Delhi 110070, +91 11 461 6666, dlfemporio.com.
OGAAN FLAGSHIP STORE, H-2 Hauz Khas Village, New Delhi 110016, + 91 97 119 91998, +91 11 269 67595, ogaan.com.
AMBIKA PILLAI, South Extension, D-16 South Extension Part 2, 3rd floor, New Delhi 110049, +91 11 400 10000/400 10200, ambikapillai.com, and additional locations.
COLABA CAUSEWAY, Maharashtra, Mumbai 400005.
PALLADIUM MALL, 462, High St. Phoenix, Senapati Bapat Marg, Lower Parel, Mumbai 400013, +91 22 43339994, palladiummumbai.com.
FAB INDIA, Jeroo Building, 137, M.G. Rd., Kala Ghoda, Mumbai 400001, +91 22 22626539/40, fabindia.com.
SABYASACHI MUKHERJEE, Ador House, 6K Dubash Marg, Mumbai 400001, +91 22 22044774/75, sabyasachi.com.
ENSEMBLE, Great Western Building, 130/132 Shahid Bhagat Singh Rd., Mumbai 400023, +91 22 40564825/40564800, ensembleindia.com.
INDIGO DELI, Palladium Mall, Level 1, Senapati Bapat Marg, Lower Parel, Mumbai 400013, +91 22 4366 6666, and additional locations.
THE DOME, InterContinental Hotel, 135, Marine Drive, B Rd., Church Gate, Mumbai 400020, +91 22 39879999, ihg.com/intercontinental/hotels/us/en/mumbai/bomhb/hoteldetaili.
## RIYADH
### SAUDI ARABIA
#### O100°
* * *
"As you enter, you are taken through a tunnel of hanging Saudi roses, and your olfactory journey begins," says Marriam Mossalli, a Jeddah-based fashion consultant and editor. "Then a team of expert noses gives you a guided tour of the different offerings, from regional brands such as Tola to Histoires de Parfums."
Boutique # 10, Akaria Plaza, Olaya Street, Olaya, Riyadh 12244, +966 11 293 1119, O-100.com.
#### D'NA
* * *
In the heart of Riyadh, D'NA is an independent voice in the Middle East fashion scene. The space is a beautifully curated magazine come to life. Within its loftlike gallery space, you'll find specially commissioned pieces by designers and artists, including Jason Wu, Rodarte, Delpozo, The Row, Lisa Marie Fernandez, and Paul Andrew. "The best women's buy in the world," says Cameron Silver. D'NA is also a meeting place, where art exhibitions and film screenings are often held.
6611 Tukassisi St., Al Rehmania, Riyadh 12341, +966 11 419 9966, and an additional location in Doha, dnariyadh.com.
## JEDDAH
### SAUDI ARABIA
#### Life
* * *
This eclectic streetwear-inspired store stocks Yeezy, Comme des Garçons, Jeremy Scott, KTZ, Wildfox, and more.
Prince Sultan Rd., Al Andalus, Roshana Mall, Jeddah 23433, +966 12 284 4266, and an additional location in Riyadh.
#### Domvs
* * *
This lifestyle boutique (think Anthropologie), which includes a gallery and café, plays host to Homegrown Jeddah, a pop-up market for Middle Eastern designers on the first floor.
Al-Rawdah St., Jeddah 23432, +966 12 661 1447, domvs-ksa.com.
MARRIAM MOSSALLI
BUYING TRADITIONAL ARAB CLOTHING
* * *
Marriam Mossalli is the Jeddah-based owner of fashion consulting firm Niche Arabia and creator of the blog ShoesandDrama.com. In recent years, she has emerged as one of the Middle East's most revered fashion experts. As a fashion editor, she has fused cultural heritage with a modern edge, helping to define a new Arabic style. Since 2011, her consultancy firm has catered to high-profile clients, including Burberry, Carolina Herrera, TAG Heuer, and Harvey Nichols, helping them understand and thrive in the Middle Eastern market.
I love traditional clothing. Even as a youth, I would cross-dress and wear the men's thobe (a long, crisp white shirtdress that Arab men wear—it's the Bedouin equivalent to the Western business suit). TOBY by Hatem Alakeel is one of my favorite Khaleeji menswear brands; he's been called the "Tom Ford of Thobes" because his tailoring is so impeccable.
THAMANYAH from the Emirates is another chic brand; he brings an avant-garde touch to traditional wear—if Hatem is Ford, then Thamanyah's creative director, Ahmed Abdelrahman, is Rick Owens.
I also love midas, the traditional men's sandals. You've seen a lot of references to midas sandals from Christian Louboutin to Alexander Wang! My favorite place is downtown, in the old souk, AL-BALAD, where you can get custom leather ones made for less than $50.
I love going to downtown Jeddah. Historically, Jeddah was a major port city for ships traveling the Red Sea, which is why even today the traditional souks reflect a history of maritime trade and exchange. Jeddah continues to act as a vital gateway to Mecca (the holy city where Muslims come for pilgrimage), which is why you'll always find visiting Muslims selling their wares, from Russian dolls to Indian herbs and incense.
TOBY, Ameer SultanSt., after Naft Petrol Station beside Gardenia Flower Shop, Jeddah, +966546234838, tobybyhatem.com.
THAMANYA, stocked online at farfetch.com.
AL-BALAD MARKET, Jeddah.
## DUBAI
### UNITED ARAB EMIRATES
CARTEL
#### Bambah
* * *
For three years, Bambah was Dubai's only high-end vintage boutique. It served as an inspirational spot for fashion influencers from all corners of the world. Rasheed has since debuted a vintage-inspired ready-to-wear collection.
The Boulevard, City Walk 2 in Building 13, Dubai, +971 55 224 1538, bambah.com.
#### Boom and Mellow
* * *
This high-end accessories boutique opened its doors in 2005, and has been a destination since for fine and costume jewelry, clutches, hats, bags, belts, shawls, small gift items, and lounge/beach dresses. Designers include Alexis Bittar, Goddis, Jennifer Behr, and Jacquie Aiche.
Town Centre Jumeirah, Ground Floor Shop 21, Jumeirah Beach Rd., Dubai, +971 4 344 4512, shopboomandmellow.com.
#### Boutique 1
* * *
"Dubai has become the Middle East's shopping hub, and while designer flagships are popping up in every corner of every mega-mall, Boutique 1 is a feminine, multi-brand boutique that gives the local fashionistas access to all the brands from London and Paris fashion weeks," says Marriam Mossalli of this destination with locations at Jumeirah Beach Resort and Mall of the Emirates. You'll find designs by Elie Saab, Alexander Wang, Rosie Assoulin, Proenza Schouler, The Row, and Victoria Beckham.
The Walk at Jumeriah Beach Resort, Dubai, +971 4 425 7888, boutique1.com, and additional locations.
#### By Symphony
* * *
Curated by Emirati entrepreneur Salama Alabbar, Symphony showcases designer brands from around the world alongside new and exciting rising stars. By Symphony aims to be a "home away from home," so you can take a load off and stay a while. Clothing by Gucci, Bottega Veneta, Delpozo, Simone Rocha, and Peter Pilotto is stocked alongside items by Dubai-based designer Madiyah Al Sharqi and Saudi designer Razan Alazzouni.
The Dubai Mall, Ground Floor Fashion Ave., Dubai, +971 4 330 8050, bysymphony.com.
#### cARTel
* * *
"Positioning itself as a purveyor of wearable art and showcasing regional designers, cARTel sits among Dubai's leading art galleries in Alserkal Avenue. It hosts pop-ups and screenings while producing its own in-house concept magazine," says Eileen Wallis.
6 Alserkal Ave., Al Quoz 1, St. 17, Dubai, +971 4 388 4341, thecartel.me.
#### Comptoir 102
* * *
"Created by two Parisiennes, this is a one-stop design destination housed in a traditional Dubai villa with an organic kitchen and café as well as housewares, jewelry, and fashion," says Eileen Wallis. Look for Raquel Allegra, Swildens, Jérôme Dreyfuss, Ileana Makri, Pippa Small, and more.
102 Beach Rd., Jumeirah 1, Dubai, +971 4 385 4555, comptoir102.com.
#### Dubai Mall
* * *
Dubai is a city with more than 100 shopping malls, a monthlong shopping festival, and a chain of luxury boutiques named Rodeo Drive (yes, really). The fashion capital of the region features the largest mall on the planet by area, and it has some off-the-hook attractions and amenities, including an indoor theme park, an Olympic-sized ice rink, the Dubai Aquarium and Underwater Zoo, an Armani hotel, a Shake Shack, a personal porter and private locker service for purchases, a walkway to Burj Khalifa (the tallest skyscraper in the world), and, oh yeah, some pretty great shopping, too, with 1,200 retail stores anchored by Galeries Lafayette and Bloomingdale's. There's also an area called the Souk that features gold jewelry, accessories, traditional Arab clothing, and handicraft stores, so you can experience an "authentic" souk in the comfort of a mall. "From glamour gowns to caviar to a huge variety of nice shops, just being in this mall is enough," says Hong Kong restaurateur Bonnae Gokson.
Financial Centre Rd., Downtown Dubai, +971 800 3822 46255, thedubaimall.com.
#### Etoile
* * *
"With a median age of twenty-eight years old, Saudi is a pretty young country in terms of demographics—and Etoile is the multi-brand boutique made for the millennial!" says Marriam Mossalli. "From Anya Hindmarch to Chloé, everything in there is an Instagram-perfect item ready to be hashtagged." Other labels stocked include Camilla, Dior, Nicholas Kirkwood, Prabal Gurung, and Roksanda.
Mall of the Emirates, Dubai, +971 4 341 4166, etoilelaboutique.com, and additional locations.
#### Global Shopping Village
* * *
The Disney World of shopping gives you the chance to shop the entire world in one place, with pavilions themed around all corners of the globe, from the Americas to the Far East. You can find everything from Japanese kimonos to Syrian baklava, and at relatively affordable prices, too. Open seasonally from November to March.
Sheikh Mohammed Bin Zayed Rd., Dubai, +971 4 362 4114, globalvillage.ae.
#### Gold Souk
* * *
Dubai's most famous and most visited souk is all about gold, and you're expected to haggle for it. Also on offer are beautiful textiles and Ali Baba shoes.
54 Al Khor St., Dubai.
#### iF Boutique
* * *
A villa tucked away near the beach in Umm Suqeim has one of Dubai's edgiest selection of designers: Yohji Yamamoto, Rick Owens, Limi Feu, Uma Wang, and Ivan Grundahl. "A true stylist's destination," says Dubai fashion expert Eileen Wallis.
26 Umm Al Sheif Rd., Dubai, +971 4 394 7260, ifboutiquedubai.com.
#### Level Shoe District
* * *
This may be the best shoe store in the world (sorry, Saks Fifth Avenue 1002Shoe), with 96,000 square feet and 400 in-store boutiques, stocking everything from Alaïa and Aquazzura to Sophia Webster, as well as offering shoe styling and restoration services and an on-site foot clinic. "The most comprehensive collection of fashion-forward shoes I have ever seen," says author and journalist Bronwyn Cosgrave. "There are also extremely opulent variations of every style of ultra-fine footwear: sneakers, flats, stilettos. I was particularly taken with the rainbow array of Manolo Blahnik Hangisi flats, which I swear by. Dubai is the only place I have seen this shoe with pearl-embellished buckles."
The Dubai Mall, Ground Floor, Dubai, +971 800 538 3573, levelshoes.com.
#### Mall of the Emirates
* * *
Features nearly 700 stores anchored by the British department store Harvey Nichols, plus Ski Dubai, the famous indoor ski slope, complete with chair lifts.
Mall of the Emirates, Dubai, +971 4 409 9000, malloftheemirates.com.
#### Mochi
* * *
As worn by Gigi Hadid, Kourtney Kardashian, and Chiara Ferragni, this Dubai-based brand is worth getting to know for its fun multiculti-inspired, colorfully embroidered jackets, off-shoulder tops, and fringed clutches. Founder Ayah Tabari, who was born in Palestine, raised in Riyadh and Amman, and studied in London before settling in Dubai, works with artisans all over the world, supporting the communities with job creation and incorporating stylish flourishes from Africa, India, and beyond into her designs.
Dubai Design District Building 9, Office 103, Dubai, +971 4 443 3451, allthingsmochi.com.
#### O de Rose
* * *
Set in a gorgeous residential-style villa far from the mall madness, this concept shop features fashion, accessories, and home décor items from regional designers and artisans.
999 Al Wasl Rd., Umm Suqeim 2, Dubai, +971 4 348 7990, o-derose.com.
#### Sauce
* * *
The trendiest concept boutique in Dubai, Sauce is a must-shop for its whimsical edit of local and international labels, including Zimmermann, Sea, Self-Portrait, Joshua Sanders, Lucy Folk, Sarah's Bags, and its own Made by Sauce collection. Look for banana-embroidered beach bags, flower-festooned sunglasses, and denim skirts with floral and bird appliqués. There are nine stores total, including locations in the Dubai Mall and Village Mall.
The Dubai Mall, Fashion Section, 1st Floor, Dubai, +971 4 339 9696, shopatsauce.com, and additional locations.
#### Souk Madinat Jumeirah
* * *
A re-creation of a traditional Middle Eastern souk, with craftsmen at work, rugs, cushions, Aladdin lamps, and Arabian coffee sets spilling out onto the walkways—only it's indoors and air-conditioned, bonus! This is the place for quality souvenirs.
Madinat Jumeirah Hotel, 6 Al Sufouh Rd., Dubai, +971 4 366 8888, jumeirah.com.
#### Studio 8
* * *
Launched in May 2009, Studio 8 features an edited collection of designers from the Middle East and Asia, and is the brainchild of entrepreneur and fashion promoter Sarah Belhasa. If you are looking for the latest jalabya and abaya trends, Studio 8 offers a range by Khulood Bin Arab, Das Collection, Julea Domani, Sumaya Abdul Razak, and Amira Haroon, among others. You'll also find international designers, including Suneet Varma, Nikasha, Deepika Anand, Rahul and Anushka, Shaila Khan, Zeeshan Bariwala, and Rizwan Baig.
Villa 3, Jumeira Beach Rd., Dubai, +971 4 325 3258, studio8.ae.
ONLINE SHOPPING TIP
Up-and-coming Dubai-based brand Mochi's e-commerce site features a range of mirrored clutch bags, embroidered jackets, skirts, and shorts that can be shipped around the world (allthingsmochi.com). Zayan the Label also ships worldwide and has a comprehensive selection of items (zayanthelabel.com). Sauce also ships internationally and specializes in fearless, fun, Insta-friendly items (shopatsauce.com).
#### Ush Boutique
* * *
Founded by Fashion Arabia, Ush Boutique gave voice to a collection of avant-garde designers and created a welcoming space for shoppers looking for originality and craftsmanship. Ush became the vehicle for renowned Emirati designer Abeer Al Suwaidi, and her punk rock and contemporary takes on the traditional abaya. The store offers curated collections of the best abayas, jalabyas, caftans, dress shoes, bags, and accessories from some of the region's best-known and in-demand designers. The 3,000-square-foot villa also serves as a gathering place for events and fashion shows.
Villa C-125, 32B St., Al Wasl Rd., Jumeirah 1, Dubai, +971 4 344 0735, ushboutique.com, and an additional location in Abu Dhabi.
#### Zayan the Label
* * *
This decidedly girlie label was created by Zayan Ghandour, cofounder of the multi-brand concept boutique Sauce, where she serves as creative director and head buyer. Following the success of Sauce, Zayan began developing her namesake brand, which debuted at Paris Fashion Week in 2011 to wide acclaim; she recently opened her first stand-alone in Dubai. The collection includes flutter-sleeve dresses, lace-edged joggers, and scallop-edged crop tops.
Galleria Mall, Shop 10, Al Wasl Rd., Dubai, +971 4 344 0104, zayanthelabel.com.
## MARRAKECH
### MOROCCO
AKBAR DELIGHTS
#### Atika
* * *
Loafers, drivers and sandals in every color of the rainbow await at this footwear feast for the senses.
34, rue de la Liberté, Marrakesh, Marrakech-Tensift-Al Haouz, +212 5244-36409.
#### Beldi
* * *
"A nicely curated market of shops, and a welcome break from the hustle of the main souks," says designer Stephanie von Watzdorf.
9-11 Soukiat Laksour, Bab Fteuh, Marrakech 40000, +212 5244-41076.
#### Lahandira
* * *
"My favorite rug source. I lose my mind every time I visit here," says designer Mara Hoffman, "you can witness this in my office, which is covered in beautiful carpets." "I love the baskets, the embroidered basket bags, the colorful vintage beaded bags with tassels, the babouches in every color. A feast for the eyes, with so much visual deliciousness to take in. I always buy hamsa hands, in metal, wood, glass, any material," says Stephanie von Watzdorf.
Foundouk Namouss ''Draz'' Sidi Ishaq Num 100 Errahba El Qadima, Marrakech, +212 6774 23353; lahandira.com.
ONLINE SHOPPING TIP
Maryam Montague's website (mmontague.com) has a beautifully curated selection of Moroccan goodies, including woven rugs, poufs, shell necklaces, baskets, Tuareg leather bags, indigo leather scarves, and vintage clothing, with items at all price points shipping to fifteen countries. A portion of proceeds benefits Project Soar, a nonprofit dedicated to helping prevent Moroccan girls from becoming child brides.
MARYAM MONTAGUE
MARRAKECH MUST-DOS
* * *
Designer, hotelier, and humanitarian Montague is also the founder of tribal-chic lifestyle brand M. Montague. Born in Cairo, raised in Tunis and New York, Maryam Montague has lived in Dakar, Kathmandu, and Windhoek. She and her architect husband moved to Morocco, where they pitched a tent on a working olive grove on the outskirts of Marrakech. There they designed, built, and decorated boutique hotel Peacock Pavilions, a favorite spot for travelers, and a shoot location for brands like J. Crew and Tory Burch.
I hang out at 33 MAJORELLE, a big, beautiful concept store that carries everything from housewares to fashion—kind of like the Merci of Marrakech. The Technicolor Majorelle Garden, once owned by Yves Saint Laurent, is right across the street, making this an überstylish address.
For loungy fashions that channel Marrakech in the '60s and '70s, I'm a fan of MAX & JAN. I'm obsessed with their sarwal/harem pants, which are very cool without making you look like you're wearing a diaper, if you know what I mean.
For the very prettiest handblown Moroccan tea glasses, I head to LE VERRE BELDI at the Beldi Country Club. The glasses come in sets of twelve and are wrapped in brown paper packages tied up with string. No singing necessary but definitely one of my favorite things.
If I could afford it, I would buy most things at AKBAR DELIGHTS. This tiny jewel of a store in the Marrakech medina is filled with over-the-top hand-beaded tunics and coats—think, Marni meets Maasai warriors.
For color and pattern addicts, there is POPHAM DESIGN—the last word in Moroccan patterned cement tiles. My husband gets a look of fear when I tell him I am making a trip there. For those not headed to Marrakech, the tiles are carried at Ann Sacks (US and London).
AKBAR DELIGHTS
I also love the SPICE SOUK. Hidden among the spices are all kinds of local beauty concoctions. I recommend the supercheap rose cream, the homemade black olive soft soap, and argan oil for hair, skin, and nails. (Note that local chameleon sellers can also be found in the spice souk. Chameleons might be the coolest reptiles ever.)
I also have a tiny weird-jewelry obsession. In Morocco, people take genies and the evil eye seriously. Thankfully, the antidote is jewelry, especially amulets and talismans. For the real thing, there is no better place to shop than JEWELS ARTS. I have a single necklace made from fifty-two talismans—I think of it as supplemental life insurance.
RUE MARJORELLE, 33 Rue Yves Saint Laurent, Marrakech 40000, +212 5243-14195, 33ruemarjorelle.com.
MAX & JAN, 14 Rue Amsefah, Sidi Abdelaziz, Medina, Marrakesh 44000 +212 5243-75570, maxandjan.ma.
LE VERRE BELDI, Km 6 route d'Amizmiz, Chrifia, Marrakech 40000, +212 5243-83950, verrebeldi.com.
ABKAR DELIGHTS, Square Bab Fteuh, Médina, Marrakech 40000, +212 671-661307, akbardelightscollections.com.
POPHAM DESIGNS, 7 Km Route d'ourika, Tassoultante, Marrakech 40065, +212 5243-78022, pophamdesign.com.
SPICE SOUK, Medina, Marrakech 40000.
JEWELS ARTS, 15 Souk Cherifia, Medina, Marrakech, +212 5243-85066, jewelsarts.com.
## FREETOWN
### SIERRA LEONE
#### Big Market
* * *
One of the oldest markets in Sierra Leone. "Great artisanal goodies: art, jewelry, textiles, et cetera," says designer Zainab Sumu. Vendors have been trading here since the days when slave ships docked in the nearby harbor, and the market's location eventually helped establish the city as a refuge for freed slaves.
Wallace Johnson St., Freetown.
## LAGOS
### NIGERIA
#### Temple Muse
* * *
Along with Alara, mentioned by Abrima Erwiah, Temple Muse is a leader in Nigeria's emerging high fashion scene. This luxury lifestyle concept store stocks top African labels such as Lisa Folawiyo, Pop Caven, and Ituen Basi, alongside European and American brands like Alexander Wang, Carven, Manish Arora, and Victoria Beckham, all in a gallerylike space featuring rotating art exhibitions and an in-house champagne bar.
21 Amodu Tijani Close, Victoria Island, Lagos, +23 708 726 4853, temple-muse.com.
TEMPLE MUSE
## LAMU
### KENYA
#### African Crafts
* * *
Sells art, crafts, ornaments, home décor, and jewelry made mostly from recycled materials, from throughout Africa.
+254 706 718562, facebook.com/Africancornerexportcom.
#### Aman Lamu
* * *
"Sandy Bornman's shop has a nice selection of her designs and a curated selection of items from smaller artisans from Southeast Asia," says designer Stephanie von Watzdorf of this spot on the island that's been referred to as "Africa's St. Barts." It stocks kurtas, Egyptian cotton shirts, African print pants, and more.
+254 733 455821, facebook.com/pages/Aman-Lamu.
AMAN LAMU
#### Ali Lamu
* * *
"Goods made out of recycled materials, mostly sails, by local craftspeople," says Stephanie von Watzdorf. "They have great pillows, pouches, and totes. My best shopping score of late has been from Ali Lamu: an old canvas sail embroidered with a giant pink heart made out of recycled plastic sequins and Masai beading. It makes me think, My heart is in Africa, and it's just one of many amazing things that I have found while traveling."
+254 722 702510, alilamu.com.
#### Gallery Baraka
* * *
Well-curated selection of masks, jewelry, and crafts from Africa and the Far East.
+254 424 633399, +254 722 379934.
#### Slim the Silversmith
* * *
A true hidden gem. This guy makes beautiful jewelry, including his signature pieces: pendants fashioned from remnants of Chinese porcelain found embedded in Swahili houses in Lamu.
P.O. Box 58, Lamu 80500, +254 722 478878
ABRIMA ERWIAH
AFRICAN STYLE
* * *
In 2011, after a decade at Bottega Veneta, Abrima Erwiah partnered with actor Rosario Dawson to start Studio 189 (studiooneeightynine.com), a Ghana-based ethical-fashion label focused on creating jobs and supporting education and skills training. Erwiah also produces a private-label brand of breezy separates that highlights traditional African dyeing techniques. The two friends travel often in Africa to work directly with artisans and designers.
ALARA in Nigeria is the most beautiful store in Africa and one of the most beautiful stores in the world. Designed by famous architect David Adjaye and curated by Reni Folawiyo, the store looks like a contemporary art museum. It carries luxury brands from around the world mixed with innovative new African brands. It's a true lifestyle store including ready-to-wear, home, jewelry, accessories, bags, books, and even a stunning garden and restaurant.
In Ghana, DUABA SERWA is one of the most popular brands, worn by Ghana's fashion-savvy crowd. The designer, Nelly Hagan-Aboagye, creates refined, fitted, sexy dresses perfect for cocktail parties, weddings, the red carpet, or a fabulous dinner date. AAKS incorporates Ghanian weaving techniques into chic raffia bags in bright colors, available at ELLE LOKKO.
ZAINA LODGE, Ghana's first real eco-safari lodge, sets a new standard of luxury. Super relaxing, it's the only safari lodge in Mole National Park. It carries exquisite one-of-kind, locally made artisanal goods, like natural shea butter made by a woman in Larabanga village, black soap, and mud cloth pillowcases. You also get to enjoy incredible food and discover Ghana's wildlife—elephants, baboons, and more—from your cabin.
One of the largest festivals in Africa is CHALE WOTE, started by Mantse Aryeequaye and Sionne Neely a few years ago. It brings together Ghana's (and Africa's) most creative people. You can find one-of-a-kind clothing and art by many amazing young artists.
In Dworzulu, HEEL THE WORLD is a brand and social enterprise that works to empower Ghanian startup companies. You can order bespoke leather shoes, handmade by expert cobblers.
The Makola Market, known to locals as THE BEND DOWN BOUTIQUE, is where secondhand international wares end up. You can find all kinds of goodies and give them a second life.
Based in Senegal and with stores in Cape Town, South Africa, and Agadir, Morocco, BANTU is a swim, surf, and beachwear brand started by Yodit Eklund. The brand caters to Africa's fast-growing surfing and beach-going population, with swimwear, surfboards, and beach bags in prints that reference African art and textiles.
ALARA, 12A Akin Olugbade St., Lagos, Nigeria, +234 909 685 2076.
DUABA SERWA, Accra, Ghana, +233 26 210 1656.
A A K S, aaksonline.com.
ELLE LOKKO, N. F604/1, Lokko Rd., Accra, Ghana, +233 24 644 9944, ellelokko.com.
ZAINA LODGE, Mole National Park, Northern Region, Ghana, +233 54 011 1511, zainalodge.com.
CHALE WOTE STREET ART FESTIVAL, High St., James Town, Ghana, accradotaltradio.com/chale-wote-street-art-festival.
HEEL THE WORLD, Oxford St., Osu, Accra, Ghana +233 24 658 9160, +233 26 210 1656, htwshoes.com.
MAKOLA MARKET, Kojo Thompson Rd., Accra, Ghana.
BANTU, 368 Albert Rd Woodstock, Cape Town, 7295, South Africa, and KM 16 Route d'Essaouira, Agadir, Morocco, bantuwax.com.
## NAIROBI
### KENYA
#### Mille Collines
* * *
Founded by Spaniards Inés Cuatrecasas and Marc Oliver following a 2008 trip to Rwanda and an introduction to a local dressmaker, this fashion atelier focuses on both design and retail with an African spirit. Designs combine edgy tailoring and cool, African-inspired prints.
Village Market, Limru Rd., Gigiri, Nairobi 00621, +254 703 216541, and additional locations, millecollines.com.
MILLE COLLINES
#### Ngong House
* * *
"Home to the most amazing jewelry store," says designer Aurora James of this boutique inside the intimate luxury wildlife lodge Ngong House. "Penny Winter, the cofounder of Ngong House, has been designing jewelry for decades and has it all handcrafted in Kenya. Beautiful gems are fashioned into an eclectic and exciting mix of jewelry," such as horn cuffs and other pieces.
60 Ndovu Rd., Nairobi 24963, +254 422 434965, Ngonghouse.com.
#### West African Market
* * *
"It has no address and no sign," says Stephanie von Watzdorf. "You can find all kinds of African beads, beaded furniture, and shells. It's a real treasure trove."
## ACCRA
### GHANA
#### Christie Brown
* * *
One of the most well-known designers in Ghana, Aisha Obuobi learned her craft from watching her seamstress grandmother Christie Brown. Her women's apparel brand ranges from bespoke gowns to accessories, inspired by African culture and art. Think sophisticated, architecturally inspired sheath dresses, tailored tailcoats, and full skirts with hand-embroidered pockets, all making use of muted African prints. Obuobi has dressed everyone from Beyoncé's backup dancers on the Mrs. Carter Show World Tour to the stars of the Web series An African City, (the show has been dubbed the African Sex and the City).
809 Eleventh Lane, Viva Court, Osu, Accra, +233 24 441 8477, christiebrownonline.com.
#### Elle Lokko
* * *
A terrific boutique championing young African labels, including Osei-Duro, Raffia, Iamisigo, Monaa, Aaks, and Anago. Carries batik and batakari skirts, cool cross-body straw bags, beaded necklaces, sleek leather sandals, and more.
N. F604/1, Lokko Rd., Osu, Accra, +233 24 644 9944, ellelokko.com.
ELLE LOKKO
#### Kiki Clothing
* * *
Asymmetrical dresses, jumpsuits, off-the-shoulder tops, and other contemporary silhouettes in incredibly vibrant African fabrics.
Accra Mall, Shop G39A, Accra, +233 20 817 4362, kikiclothing.com.
#### La Maison
* * *
A superbly edited selection of art, home accessories, furniture, and decorative items from more than forty brands, many from Africa, plus jewelry and candles. La Maison was opened nearly two decades ago by Nada Moukarzel, who designed the interiors of Coco Lounge and Urban Grill, two trendy eateries near the store's second location, in the Icon House development in Stanbic Heights. La Maison is a part of the West African luxury group Yolo Experiences.
#F881/1 Annexe, 6th St., Osu R.E., Accra, + 233 30 278 9031, lamaisonghana.com.
#### Mina Evans
* * *
This up-and-coming Ghanian designer creates feminine pieces, from circle skirts to maxi dresses, in vibrant colors with print accents. They're a hit with the local celebrity and style-blogging set. In 2015 Mina Evans won brand of the year at the Ghana Fashion Awards.
Ringway Link, Ringway Estates, Accra, +233 24 422 8120, mina-evans.com.
#### Poqua Poqu
* * *
A Ghanian fashion scene mainstay, this label specializes in cropped blouses, high-low dresses, and fluid gowns featuring combinations of textures, vivid colors, and African prints.
Osu Forico Mall, Mission St., Osu, Accra, +233 30 296 3909, +233 27 023 1261, poquapoqu.com.
#### Viva
* * *
Accra's top luxury fashion multi-label boutique sells a colorful selection of Dolce & Gabbana, Valentino, Antonio Marras, No. 21, MSGM, Self-Portrait, and more.
809, 11th Lane, Osu, Accra, +233 02 077 6629.
## CAPE TOWN
### SOUTH AFRICA
#### AKJP Collective
* * *
This concept store features the most exciting, cutting-edge designers on the men's and women's South African fashion scene, including Lukhanyo Mdingi, Rich Mnisi, Selfi's Celeste Arendse, and Drotsky's Elaine du Plessis.
73 Kloof St., Cape Town, 8001, +27 21 424 5502, www.adriaankuiters.com.
#### Chandler House
* * *
This charming shop is housed in an 18th-century Cape Georgian building that is also home to the Voorkamer Gallery. Walk inside and feast your eyes on gorgeous, locally made housewares, artisanal gifts, jewelry, antiques, textiles, and fine art. Michael Chandler opened the doors in December 2013 to showcase his own designs, and soon included other Cape artists and designers as well.
53 Church St., Cape Town, 8001, +27 21 424 4810, chandlerhouse.co.za.
CHANDLER HOUSE
#### Dear Rae
* * *
Dear Rae jewelry focuses on natural lines and simple forms translated into gold, silver, brass, and copper, with the occasional gemstone. Drawing on inspiration from her South African and German heritage, Karin Rae Matthee launched Dear Rae in 2010. Each Dear Rae piece—precious gold chain bracelets with palm-shaped charms, delicate half-sun brass bangles, rose gold heart rings—is designed and made by Matthee and her team at her Cape Town studio.
Shop 3, The Woodstock Foundry, 160 Albert Rd., Cape Town, 7925, dearrae.co.za, and other locations.
#### Glitteratti
* * *
This blogger-approved vintage emporium specializes in wares from the 1950s to the 1980s. There are several other vintage stores in the same complex.
Antique Arcade, 127 Long St., Cape Town, 8000, +27 82 819 5247, theantiquearcade.co.za/stores/glitterati.
#### Haas Collective
* * *
Haas (the Afrikaans word for rabbit) was created by Glynn Venter, Francois Irvine, and Vanessa Berlein. Their dream was to create a hub for edgy collectibles and one-of-a-kind quirky art pieces. They also operate Haas Coffee next door, which serves kopi luwak, the most expensive coffee in the world, and an ad agency upstairs.
19 Buitenkant St., Cape Town City Centre, Cape Town, 8001, +27 21 461 1812 haascollective.com.
#### Klûk CDGT
* * *
A collaboration between Malcolm Klûk and Christiaan Gabriël Du Toit, this is one of South Africa's best-known fashion labels. Their glamorous dresses and coats emphasize lush embellishments and prints, making them a favorite with local celebs. Based in Cape Town, the designers also have a shop in Johannesburg and have showcased their designer collections as far away as New York and Paris. Their stores feature accessories and clothing by like-minded local designers alongside their collections.
43-45 Bree St., Cape Town, 8001, +27 83 377 7780, and 46 6th St., Parkhurst, Johannesburg, 2193, +27 83 377 7730, kluk.co.za.
KLÛK CDGT
#### Mememe
* * *
Owned by Doreen Southwood, another of Cape Town's most well-known designers, who shows at South Africa Fashion Week, this is the place to check out the best local talent, as well as Southwood's own collection of nostalgic, feminine designs.
117A Long St., Cape Town, 8000, and 54 6th St., Johannesburg, 2193, +27 21 424 0001, mememe.co.za.
#### Maison Mara
* * *
Upmarket concept store selling clothing and accessories from international labels, including J.W. Anderson (handbags), Acne (jeans), Balenciaga (coats), Petit Bateau (cotton T-shirts), and Costes (candles).
5 Jarvis St., Cape Town, 8001, +27 21 418 1600, maisonmara.co.za.
#### Merchants on Long
* * *
Amazing concept store housed in a striking building that was built in Edinburgh and shipped to Cape Town in 1903. Carefully curated selection of luxury and handmade brands, all sourced and made in Africa, including Bantu, Brother Vellies, Lalesso, MaXhosa by Laduma. Look for antelope-fur cuff bracelets, ostrich-feather bag charms, khanga-print beachwear, Xhosa-inspired knitwear, and men's shirts made of West African textiles, at this must-visit. Merchants is also home to the ethical African luxury handbag and accessories line Okapi.
34 Long St., Cape Town, 8001, +27 21 422 2828, merchantsonlong.com.
#### Missibaba
* * *
This colorful and playful luxury accessory label was established by designer Chloe Townsend, who is passionate about South Africa and supporting local industry. Townsend works from a studio in Woodstock, where she experiments with texture and color to craft boldly patterned bags and accessories with an exotic flair—leather appliqués in the shape of bananas and palm leaves, fun fringe, and patchwork details—which are sold at a flagship boutique on Bree Street, shared with jewelry designer Kirsten Goss.
229 Bree St., Cape Town, 8001, +27 21 424 8127, missibaba.com, and other locations in Johannesburg.
#### Mr. Price
* * *
The South African version of H&M, this chain of more than 400 stores offers value-priced trendy fashions for men, women, and kids, including off-shoulder tops, light-up sneakers, and printed varsity jackets. The retail group also includes Mr. Price Home, Mr. Price Sport, and Miladys.
Shop 146A, Lower Level, Victoria Wharf, Cape Town, 8001, +27 21 418 1499, mrp.com, and other locations.
#### Mungo & Jemima
* * *
Designers Kirsty Bannerman of Coppelia and Marian Park-Ross of Good opened Mungo & Jemima in 2008 to showcase the work of the emerging South African fashion scene. The flagship store in Cape Town's epicenter, Long Street, a second store in the Watershed at the V&A Waterfront, and the online shop, feature items from Coppelia, Good, I Love Leroy, Selfi, Margot Molyneux, and more.
108 Long St., Cape Town City Centre, Cape Town, 8000, +27 21 424 5016, and B1&2 Watershed, V&A Waterfront, Cape Town, +27 21 418 1180, mungoandjemima.com.
#### Nap
* * *
This lovely store and South African chainlet was created by Carina Marescia and daughter Tamryn, who share a love for beautiful, and simple design, preferably raw-edged and neutral in color. Their motto is "believe in homegrown," and you'll find lots of locally produced goods, including resort clothing and subtle home accessories. Check the website for locations, since different stores have different specialties.
Nap Lifestyle & Apparel, Shop B3, Mainstream Centre, Hout Bay, Cape Town, 7806, +27 21 421 6482, napliving.co.za.
#### Neighbourgoods Market
* * *
Founded by entrepreneurs whose aim is to revive the public market as a civic institution, Neighborgoods features 100 specialty traders every Saturday, including local farmers, fine-food purveyors, artisan producers, and local designers such as Selfi (artsy, Marni-esque togs), and Chiumbo (cool printed harem pants).
Every Saturday 9 a.m.-2 p.m. at the Old Biscuit Mill, 373 Albert Rd., Woodstock, Cape Town, 7915, and Johannesburg, neighbourgoodsmarket.co.za.
#### Old Biscuit Mill
* * *
"One of my favorite places in the world," Fashionkind founder Nina Farran says of this mixed-use retail, fashion, and food destination. "From drinking fresh machete-cut coconuts to sampling food stands and discovering artisan talent in every nook and cranny, it is a must-go when in Cape Town. I discovered one of my all-time favorite artists there—Andile Dyalvane —who is now a dear friend. His ceramics tell the story of his country, his roots, and his soul. They are masterful demonstrations of the talent that exists around the world just waiting to be discovered. To me, they are the true definition of luxury."
373-375 Albert Rd., Woodstock, Cape Town, 7915, +27 21 447 9120, theoldbiscuitmill.co.za.
ONLINE SHOPPING TIP
Luxury e-commerce is a relatively new frontier in Africa. Kisua.com and Oxosi.com are both great places to discover and order clothing and accessories by emerging African designers. Based in New York, Oxosi.com is a design-centered marketplace for African brands and enthusiasts, as well as a discovery platform for fashion, art, and design products made in Africa. South Africa–based kisua.com counts renowned Savile Row tailor Ozwald Boateng among its advisory board members.
#### Pichulik
* * *
Incredibly cool sculptural jewelry made of colorful cord and leather nods to traditional Masai and other African designs, but with a contemporary eye. Everything is made in South Africa by designer Katherine-Mary Pichulik, who has also collaborated with other labels on bags.
F36/F37 Watershed, V&A Waterfront, Dock Rd., Cape Town, 8001, and Johannesburg, +27 74 869 7979, pichulik.com.
#### SAM
* * *
Frustrated by the lack of a single beautiful space to find a variety of cool fashion and accessories, designers Berlinda White and Naomi Bossert created SAM (South African Market). It carries stenciled leather-detail clutches by Ilundi, jewelry by Pichulik, Bamboo Revolution watches, housewares, stationery, and art, too.
67 & 69 Shortmarket St., Cape Town, 8001, +27 83 690 6476, facebook.com/SouthAfricanMarket.
#### The Space
* * *
In 2000, the Space launched in Durban to partner with and showcase established designers and to provide a platform for promising young designers, giving them a chance to break into the fashion business. There are now ten Space stores in South Africa.
Shop L69, Cavendish Square, Claremont, Cape Town, +27 21 674 6643, thespace.co.za, and other locations.
THITHI NTETA
A PRIMER ON THE JO'BURG SHOPPING SCENE
* * *
Thithi Nteta is a PR consultant and creator of style blog teeteeiswithme.com, which focuses on fashion in her home country of South Africa.
In the past five years or so, a variety of international retailers have landed on our shores. It started with Zara, then Topshop, H&M, and Forever 21 opened shortly after. When the stores opened, South African consumers were excited. I remember vividly how Zara was full every single day for six months after it opened. At the risk of sounding like an awful person, as someone who travels, the opening of these stores didn't excite me one bit. It kind of upset me because it meant South African designers and the boutiques housing South African designers had more competition.
I made a conscious decision years ago to spend my money buying from local designers and boutiques. If someone was traveling to South Africa and wanted somewhere to shop, have lunch, get a manicure, and have drinks, I would say go to SANDTON CITY SHOPPING CENTRE in Johannesburg, which is connected to Nelson Mandela Square.
It has everything in that respect, and is very convenient, but it's a mall, so no real ambience. A drink and sushi at THE BIG MOUTH or TASHA'S in the mall isn't a bad day out.
As someone who is passionate about local South African fashion, and it becoming an actual industry, I like to discover places where I can buy local. One favorite is SUPERELLA, which is South African fashion designer Ella Buter's stand-alone store/studio. She creates comfortable, everyday clothes, including geometric tops and apron dresses. CONVOY is owned by a group of South African designers. They stock wearable pieces that are unique and locally manufactured.
MAXHOSA BY LADUMA (for knitwear), PICHULIK (for jewelry), and BLACK COFFEE (for hand-dyed dresses) are some of my favorite stores to frequent in WORK SHOP NEWTOWN, a fashion, design, and lifestyle concept center.
It isn't new, but the JOHANNESBURG ART GALLERY is one of my favorite spaces in the city. I think a lot of people are still quite apprehensive to go into Johannesburg and explore, but it's a beautiful space with beautiful works of art.
Re cheap chic, I opt to shop for high-quality, unique pieces over cheap and mass-produced ones. We have a South African retailer called MR. PRICE. I always shop from their collaborations with local designers because that means money back in the designers' pockets.
My biggest shopping score is an Egyptian cotton shirt from RICH MNISI that can be worn two ways. The other day I discovered that I could also wear it as lightweight outwear, so that's three ways.
SANDTON CITY SHOPPING CENTRE, 0 2031 South, 163 5th St., Sandton, 2031, +27 11 217 6000, sandtoncity.com.
THE BIG MOUTH, Shop No. 13 & 14, Nelson Mandela Square, Johannesburg, 2031, +27 63 293 8869, thebigmouth.co.za.
TASHA'S, Shop No. 16, Atholl Square, Sandton Johannesburg, 2196, +27 11 884 0365, tashascafe.com.
SUPERELLA, 7th Ave. and 9th St., Melville, Johannesburg, 2092, +27 82 845 5551.
CONVOY, The Bamboo Lifestyle Centre, 9th St., Johannesburg, 2109, +27 83 548 5321.
MAXHOSA BY LADUMA, Newtown Junction, Lilian Ngoyi St., Johannesburg, 2000, maxhosa.co.za.
PICHULK, Unit S16, Newtown Junction, Newtown, Johannesburg, 2001, and Cape Town, +27 74 869 7979, pichulik.com.
BLACK COFFEE, 44 Stanley Ave., Milpark, Johannesburg, 2092, +27 11 482 9148, and 100 Carr St., Newtown, Johannesburg, 2001, +27 72 943 6192, blackcoffee.co.za.
WORK SHOP NEWTOWN, 100 Carr St., Newtown, Johannesburg, 2001, +27 79 676 1826, workshopnewtown.com.
JOHANNESBURG ART GALLERY, Cnr Klein and King George Streets, Joubert Park, Johannesburg, 2044, +27 11 725 3130, friendsofjag.org.
MR. PRICE, Shop 223 Carlton Centre, Commissioner St., Johannesburg, 2001, +27 11 331 9840, mrp.com, and additional locations.
RICH MNISI, +27 79 066 9515, richmnisi.com.
## SYDNEY
### AUSTRALIA
ZIMMERMANN
#### Akira
* * *
One of Australia's most important designers, Akira Isogawa draws inspiration from his Japanese heritage, creating soft and romantic silhouettes with natural fabrics in whimsical prints or distressed for a lived-in feel. He has received numerous awards, exhibited his work in museums, made costumes for the Australian ballet, and even had his face on a commemorative postage stamp. He has four stores in Australia.
The Strand Arcade, Shop 110, Level 2, 412-414 George St., Sydney NSW 2000, +61 2 9232 1078, akira.com.au, and additional locations.
#### bassike
* * *
Basic T-shirts that don't lose their shape after washing, in styles elevated by asymmetrical hems, contrast topstitching, and raw edges. Although there are several stores in Australia (and one in Venice, California), the two-story layout of the Paddington store makes it a top spot to see the breadth of Deborah Sams and Mary Lou Ryan's collection.
26 Glenmore Rd., Paddington NSW 2021, +61 2 8457 6882, bassike.com, and additional locations.
#### Berkelouw Bookshop
* * *
"You can have a herbal tea while you look at books and enjoy the view of Oxford Street in Paddington," says jewelry designer Jordan Askill. "They specialize in both new and vintage books."
19 Oxford St., Paddington NSW 2021, +61 2 9360 3200, and additional locations, berkelouw.com.au.
#### Camilla
* * *
Designer and artist Camilla Franks is synonymous with Australian fashion and resort lifestyle. She specializes in caftans, bathing suits, and maxi skirts characterized by vibrant-colored digital prints and delicate crystal details. In 2004 Franks opened her first boutique at Bondi Beach.
132a Warners Ave., Bondi Beach, NSW 2026, +61 02 9130 1430, Camilla.com.au.
#### Camilla and Marc
* * *
One of Australia's most successful women's fashion labels was launched in 2003 at Australian Fashion Week by Sydney-based brother-and-sister duo Camilla Freeman-Topper and Marc Freeman. The label mixes masculine tailoring with feminine draped silhouettes.
217 Glenmore Rd., Paddington NSW 2021, +612 9357 5822, camillaandmarc.com, and additional locations.
#### Cream on Crown
* * *
The Crown Street location of this vintage and repurposed vintage chainlet is beloved by locals, as much for the people-watching opportunities as for the shopping.
32/277-285 Crown St., Surry Hills NSW 2010, +61 02 9331 5228, creamonline.com.au, and additional locations.
#### David Jones
* * *
Australia's upscale department store, owned by South African retail group Woolworths Holdings Limited. At the Sydney flagship, you'll find a wide selection of Australian and international brands (Dries Van Noten, Dion Lee, Erdem, Phillip Lim, Paul Smith), plus a food hall.
David Jones Elizabeth St., 86-108 Castlereagh St., Sydney NSW 2000, davidjones.com.au, and additional locations.
#### Dinosaur Designs
* * *
Iconic resin jewelry and housewares line, founded in Sydney. "Working with resin is like working with paint," says Louise Olsen, who started the collection with her partner, Stephen Ormandy, when they were both art-school students. "My go-to place for a little accessory or gift," says Alyce Tran of The Daily Edited.
Shop 77, Strand Arcade, 412–414 George St., Sydney NSW 2000, +61 02 9223 2953, dinosaurdesigns.com.au, and additional locations.
#### Dion Lee
* * *
Since Lee established his brand in Sydney in 2009, he has been building a serious Hollywood fan club of leading ladies (Selena Gomez, Diane Kruger, Priyanka Chopra) who appreciate his sexy, red-carpet-ready clothing, which marries innovative construction and cuts with respect for the female form. Think fluid silk suiting and loop-woven dresses designed to enhance movement. After debuting at Australia Fashion Week, Lee presented collections in London before relocating to New York, where the brand has become a fixture on the New York Fashion Week schedule. He has six stores in Australia. "I'm a huge fan. Part of my uniform is a pair of Dion Lee cropped trousers (I have three of the exact same pair), and this is my favorite of his stores, with his full collection on display. The Strand Arcade is pretty great itself!" says Alyce Tran.
Shops 62–66, The Strand Arcade, 412–414 George St., Sydney NSW 2000, dionlee.com, and additional locations.
PIP EDWARDS
BONDI BEACH AND BEYOND
* * *
Aussie style star Pip Edwards is a designer and stylist who had roles at Sass & Bide and Ksubi before becoming design director of Australian brand General Pants. Based in Bondi Beach, she also has her own brand, P.E. Nation, which is a key part of the activewear stable at department store David Jones, and has been worn by Kendall Jenner, Bella Hadid, and Ruby Rose.
GENERAL PANTS is my love. I worked there at age eighteen, and so many people worked there when they were starting out. In Australian culture, we live in denim and this is where you go. General Pants is a partner in David Jones, too, so we have concept spaces within their stores.
In Bondi, TUCHUZY is amazing. You can get everything from Céline to Sir the Label, which is by two Aussie girls. Georgia Alice is another label to watch. Tuchuzy has its own in-house label, too. SATURDAYS SURF is also iconic.
I hardly ever shop in the city, but the CORNER SHOP at the Strand Arcade has all the brands, including Dion Lee and Romance Was Born, which is another favorite of mine.
GLEBE MARKETS, Saturdays 10 a.m.–4 p.m., and BONDI MARKETS, Sunday 10 a.m.–4 p.m., are fun, too if you're looking for a flea market experience.
GENERAL PANTS, Westfield Shopping Centre, 500 Oxford St., Bondi Junction NSW 2022, +61 2 8275 5145, generalpants.com.au, and additional locations.
TUCHUZY, Shop 11, The Beach House, 178 Campbell Parade, Bondi Beach.
NSW 2026, +61 2 9365 7775, tuchuzy.com.
SATURDAYS NYC, L08/ Bondi Beach (entry via Gould St.), 180-186 Campbell PARADE, Sydney NSW 2026, +61 2 8316 4518, saturdaysnyc.com.
CORNER SHOP, The Strand Arcade, 1/412-414 George St., +61 2 9221 1788.
GLEBE MARKETS, Glebe Public School, Corner of Derby Place and Glebe Pt. Rd., Glebe NSW 2037, glebemarkets.com.au.
BONDI MARKETS, Bondi Beach Public School, Campbell Parade, Bondi Beach NSW 2026, +61 2 9315 7011, bondimarkets.com.au.
#### Elleryland
* * *
Before making waves on the international fashion scene (she now shows at Paris Fashion Week), Australian designer Kym Ellery worked as an editor and stylist at Australia's Russh magazine. In 2007, she launched her own label, which features sophisticated clothing with dramatic slits, bell sleeves, and ruffled edges. Fans include Rihanna, Solange Knowles, Miranda Kerr, and Cate Blanchett. Her store is dubbed Elleryland.
2-16 Glenmore Rd., Sydney NSW 2021, +612 8068 2361, elleryland.com.
#### Gould's Book Arcade
* * *
An archive of used books, established in 1967. "A place of discovery where you could easily spend an entire day," says fashion designer Dion Lee.
32 King St., Newtown NSW 2042, +61 2 9519 8947, gouldsbooks.com.
#### Harrolds
* * *
"Amazing service. A lovely small-format store with lots of shop-in-shops with an interesting edit of great labels, Victoria Beckham, J.W. Anderson, and Vetements, to name a few," says Alyce Tran, cofounder of The Daily Edited, of this family-run department store, which started in menswear and expanded to women's in 2015.
Westfield Sydney, 188 Pitt St., Sydney NSW 2000, harrolds.com.au.
#### Hatmaker
* * *
"One of my favourites little stores in Sydney, located in a small shop front in the leafy streets of trendy Surry Hills," says Gary Pepper Girl blogger Nicole Warne. "The milliner Jonathon is a genius and creates the most spectacular hats for every season."
284 S Dowling St., Paddington NSW 2021, +61 2 9360 0041, hatmaker.com.au.
#### Incu
* * *
Established in 2002 by twin brothers Brian and Vincent Wu, Incu has twelve retail stores located across Sydney, Melbourne, and the Gold Coast, as well as a successful online business, selling understated clothing and accessories by Isabel Marant, Acne, Rag & Bone, Want Les Essentiels, Paul Smith, Outerknown, and the like. In 2014, Incu announced a retail partnership with French designer label A.P.C. and shortly after launched the brand's first Australian store, in Melbourne.
Shop RG 23-24 (women's) and RG 19-20 (men's), The Galeries, Sydney NSW 2000, +61 2 9266 0244, incu.com, and additional locations.
#### Jac and Jack
* * *
Originally established in 2004 as a knitwear business—with a focus on cashmere and luxurious yarns—the label has grown to encompass all aspects of the contemporary wardrobe for men and women. Designers and founders Jacqueline "Jac" Hunt and Lisa "Jack" Dempsey focus on creating effortless clothing and accessories, including incredibly chic, drapey silk "Hef" robes, polished T-shirts, ribbed knit flute skirts, and other modern essentials.
Shop 3, 82-92 Gould St., Bondi Beach NSW 2026, +61 2 8384 8062, jacandcak.com, and additional locations.
#### Lee Mathews
* * *
Established in 2000, contemporary Australian brand Lee Mathews creates collections with a quiet elegance: classic in design, considered in fabrication. Prints and textiles come from the namesake designer's interest in art, silhouettes from her passion for architecture, and color palettes from Australia's unique flora and fauna.
18 Glenmore Rd, Paddington NSW 2021, and additional locations, +6 12 9331 1699, leemathews.com.au.
#### Lover
* * *
Nic Briand and Susien Chong have seen their label, Lover, go from a humble stall at the Bondi Markets to a critically acclaimed ten-year anniversary runway show at the Sydney Opera House in 2011. They are a local fave for their beautiful lacy dresses and feminine floral prints.
Shop 69-71, The Strand Arcade, 412-414 George St., Sydney NSW 2000, +61 02 9232 7289, loverlabel.com.
#### Mode Sportif
* * *
"I don't like wearing an actual gym kit around, but I do love a relaxed luxe-sport look, and Mode Sportif does a great edit of relaxed designer pieces, highs and lows," says Alyce Tran of this store, which stocks Nike, Iro, Adidas by Stella McCartney, and other lifestyle collections.
24 Glenmore Rd., Paddington NSW 2021, +61 2 9331 7222, modesportif.com.
#### MUD
* * *
Australian fashion designer Dion Lee loves these handmade porcelain ceramics with a minimalist aesthetic, which come in a rainbow of glazes.
Shop 4, 1 Kiaora Ln., Double Bay, NSW 2028, mudaustralia.com, and additional locations.
#### Poepke
* * *
This upmarket boutique stocks Dries Van Noten, Comme des Garçons Girl, Marques'Almeida, Simone Rocha, and other arty-girl faves, alongside accessories from Maryam Nassir Zadeh, Martiniano, and Linda Farrow.
47 William St., Paddington NSW, 2021, +61 2 9380 7611, poepke.myshopify.com.
ERIN WEINGER
SYDNEY FASHION ESSENTIALS
* * *
Erin Weinger is a journalist and entrepreneur currently serving as the digital commercial editor of Vogue Australia, where she's responsible for editorial and commercial content strategy across Vogue.com.au and its multi-platform subsidiaries. She previously served as digital style director of The Hollywood Reporter, where she launched and helmed the brand's digital style and beauty destination, Pret-a-Reporter.com.
Shoebox-size boutique, BLOODORANGE has a selection of US and French labels I love—A.P.C., Alexander Wang, and Vanessa Bruno—along with my beloved Aesop hand soap and body oil.
DAISO is an amazing Japanese dollar store inside the Jean Nouvel–designed Central Park complex, which has amazing $2 collagen sheet masks and made-in-Japan pottery for exactly the same price.
I also adore LOVEDUCK, a jewel box of a shop in tony Paddington that's filled with a genius supply of vintage and vintage-inspired bohemian dresses of every variety—caftans, off-the-shoulder, and linen. Pay particular attention to pieces by Paddo to Palmy, the dreamy dress line started by former interior designer Heidi Carter in 2011.
The colette of Sydney if you will, owner Eva Galambos' high-end concept store PARLOUR X calls a circa-1845 Paddington church home and is filled to the, uh, pews with Valentino gowns, Céline bags, Balenciaga blouses, and Isabel Marant everything. This is also the place in Sydney to find Jennifer Meyer jewels and Vetements' latest creations—not to mention goods from Australian labels, including Ellery and Christopher Esber—as well as the chic-meets-utilitarian signature prints from Cecilie Copenhagen, a line every stylish Sydneysider seems to swear by.
In the heart of the Silver Lake–meets–Brooklyn enclave of Newtown, PENTIMENTO PAPERIE is the perfect place to pick up a Life notebook from Japan, Neil Perry's latest coffee-table-worthy cookbook, or a sleek, matte porcelain vessel from Mud Australia, the famed twenty-two-year-old Sydney-produced ceramics brand to which Pentimento has devoted an entire room.
A fairly obvious inclusion, but no stylish Australian—from Adelaide to Tasmania—would be caught dead without a pair of R.M. WILLIAMS bush-inspired, Chelsea-esque boots from the ninety-year-old homegrown brand, which sold a minority stake to one of LVMH's investment groups in 2013. Go for the Adelaide in calfskin, suede, or—for the more daring—kangaroo.
Walking into THE SLEEVELESS SOCIETY, a small but mighty vintage shop, I feel as though I've been transported to Los Angeles in the best possible way. That makes sense, as owner Jaimi Kost sources a bulk of her bounty Stateside. The perfect mix of mostly '80s and '90s label-less capes, caftans, and shirtdresses that you'd never want to lose combined with the odd Escada blazer and pair of Hermès trousers—at fairly reasonable prices, no less—makes this a must-visit for anyone who adores perfectly curated, unstuffy treasure.
Beachside contemporary boutique TUCHUZY mixes Australian designers—think, Kit Willow's KITX, Camilla and Marc, and Bassike—with the likes of Helmut Lang, Acne, and Common Projects. Because of its waterside location, the styling has that coveted, undone beachy vibe with a decidedly casual yet polished Australian aesthetic. You could walk in here shoeless and buy a $1,000 Sophie Hulme shoulder bag and no one would bat an eyelash.
Attached to the Rosebery headquarters and design studio is ZIMMERMANN'S famed outlet store, where everything from bikinis to children's clothes to garden-party-appropriate rompers to potential bridesmaid dresses can be found at 40 to 60 percent off retail. On my last visit, I spotted last-season off-the-shoulder floral dresses for $450 and silk jumpsuits for $190.
BLOODORANGE, 35 Elizabeth Bay Rd., Elizabeth Bay NSW 2011, +61 2 9357 2424, bloodorange.com.au.
DAISO JAPAN, 727 George St., Haymarket NSW 2000, +61 2 9212 6888, daisojapan.com, and additional locations.
LOVEDUCK, 222 Glenmore Rd., Paddington NSW 2021, +61 2 9361 4427, loveduck.com.au.
PARLOUR X, 261 Oxford St., Paddington NSW 2021, +61 2 9331 0999, parlourx.com.
PENTIMENTO PAPERIE, 249 King St., Newtown NSW 2042, +61 2 9565 5591, pentimentonewtown.com.au.
R.M. WILLIAMS, Westfield, Shop 4020, Level 4, 188 Pitt St., Sydney NSW 2000, rmwilliams.com.au.
THE SLEEVELESS SOCIETY, Shop 2, 181c Edgecliff Rd., Woollahra NSW 2025, thesleevelesssociety.com.au.
TUCHUZY, Shop 11, The Beach House, 178 Campbell Parade, Bondi Beach.
NSW 2026, +61 2 9365 7775, tuchuzy.com.
ZIMMERMANN OUTLET, 2E Hayes Rd., Rosebery NSW 2018, +61 2 8324 7323.
BASSIKE
#### Reverse Garbage
* * *
Sydney's recycling institution, established in 1974 by a collective of teachers looking for inexpensive classroom materials, is dedicated to reusing stuff headed for the landfill. This place offers recycled materials for DIY, costume, and art projects, some of them donated by local theater companies. If you need sequin fabric, a road sign, a mannequin, or a flag that says SYDNEY, this is your place.
8/142 Addison Rd., Marrickville NSW 2204, +61 02 9569 3132, reversegarbage.org.au.
#### Sarah & Sebastian
* * *
Founded in 2011 by Sarah Gittoes and Robert Sebastian Grynkofki, this Australian jewelry brand quickly achieved fashion status. Designed and handmade in their Sydney studio, each piece is simultaneously modern and classic. Look for tiny bar and chain earrings, delicate multistone rings, and more.
12B, 32 Ralph St., Alexandria NSW, 2015, sarahandsebastian.com.
#### Scanlan & Theodore
* * *
Locals compare this chain to Club Monaco in the US, because of its mix of everyday basics with going-out dresses in lace, linen, or floral stripes, but at slightly higher prices.
122 Oxford St., Paddington NSW 2021, scanlantheodore.com, and additional locations.
#### Sneakerboy
* * *
Sneakerboy carries the best luxury streetwear and sneaker brands, including Raf Simons for Adidas, Buscemi, Fendi, Fear of God, DRKSHDW, Yeezy, and more. "A brick-and mortar/online hybrid. You go into one of their Australian stores to try the sneakers on, then they send them out to you," says Dion Lee, who counts sneakers as part of his daily uniform.
3 Temperance Lane, Sydney NSW 2000, +61 2 9279 4066, sneakerboy.com, and additional locations.
#### The Standard Store
* * *
Established in 2011 by Sydney-based husband-and-wife team Nicola and Orlando Reindorf, the Standard Store is their vision for fashion retail. With its unique mix of small but interesting and wearable labels from Europe, Scandinavia, and the US (Sea, Bellerose, Carven, Clare V., Humanoid, Rachel Comey), the Standard Stores in Surry Hills and Melbourne's Fitzroy feature a tightly curated selection of clothing, accessories, and lifestyle must-haves for men and women.
503 Crown St., Surry Hills NSW 2010, +61 02 9310 1550, thestandardstore.com.au.
#### Supply
* * *
"Has a great selection of skate brands like Richardson and Palace," says jewelry designer Jordan Askill. "It is right off the Main Street, but it is down a set of stairs, and a bit hidden."
Burton St & Riley St., Darlinghurst NSW 2010, +61 2 9361 0188, supplystore.com.au.
#### The Vintage Clothing Shop
* * *
"The store opened in the '70s, and at the time, there was nothing else of its kind. The owners have been collecting vintage for over forty years and are the main supplier for period film and television production in Australia, as well as loaning to fashion editors for editorial shoots," says Gary Pepper Girl founder Nicole Warne. "The store offers a wonderfully curated selection of good-quality vintage pieces at reasonable prices. It's my favorite place to pick up something that I know I won't see anyone else in."
St. James Arcade, 7/80 Castlereagh St., Sydney NSW 2000, +61 2 9238 0090.
#### Zimmermann
* * *
Sisters Nicky and Simone Zimmermann are behind the Zimmermann label, which is known for sophisticated feminine dresses in delicate laces and prints, swimwear, and resortwear worn by everyone from Beyoncé to Miranda Kerr. The Zimmermanns started their brand in 1991, and opened their first store in Sydney in 1992. Their airy, high-spirited pieces have made them synonymous with Australian style.
2/2 - 16 Glenmore Rd., Paddington NSW 2021, +61 02 9357 4700, zimmermann.com, and additional locations.
## MELBOURNE
### AUSTRALIA
THIRD DRAWER DOWN
#### Aesop
* * *
First established in Melbourne in 1987, Aesop is a beauty-turned-lifestyle brand of hair care, skin care, soaps, and fragrances infused with parsley seed, clove, rose, cardamom, and chai scents. Each store has its own version of a minimalist aesthetic (the details of which are broken down in a special "design taxonomy" section on the website), and the brand is into supporting the arts, collaborating with the Paris Review and publishing its own literary journal. There are stores around the world, but this is the brand's Australian flagship.
87 Collins St., Melbourne VIC 3000, +61 3 9650 3027, and additional locations, aesop.com.
#### Alpha 60
* * *
If there was ever a store that represented the "Melbourne look"—black, simple, chic silhouettes, more black—it is Alpha 60. Known for its fresh take on classic styles, visit for a piece of the brand's signature "sophisticated quirk."
201 Finders Ln., Melbourne VIC 3000, +61 3 9663 3002, alpha60.com, and additional locations.
#### Assin
* * *
Dark and minimalist, Assin houses a curated selection of men's and women's apparel and accessories by Comme des Garçons, Boris Bidjan Saberi, Daniel Andresen, Junya Watanabe, Nude:Masahiko Maruyama, Pearls Before Swine, and others, in addition to its own in-house label. The polished concrete interior matches the cooler-than-thou mood.
Basement 138 Little Collins St., Melbourne VIC 3000, +61 3 9654 0158, assin.com.au.
#### Bul
* * *
"Supersculptural, minimalist, and modern," Erin Weinger says of this label started by Virginia Martin, a designer who worked for Proenza Schouler and Cynthia Rowley in the US before moving home to Melbourne to start her business in 2010. There are seven stores across Australia.
241 Brunswick St., Fitzroy VIC 3065, +61 3 9415 8844, and additional locations, bul.com.au.
#### Christine
* * *
A treasure chest of accessories in a Flinders Lane basement. Blogger Nicole Warne is a fan of owner Christine Barro's careful edits of elegant shoes, bags, gloves, and scarves.
Basement, 181 Flinders Lane, Melbourne VIC 3000, +61 3 9654 2011, christineaccessories.com.
#### Camberwell Market
* * *
Every Sunday, rain or shine, this market is the place to find secondhand designer labels and vintage furniture.
Market Place, Camberwell Victoria 3124, camberwellsundaymarket.org.
#### Cose Ipanema
* * *
This Melbourne institution stocks Dries Van Noten, Dolce & Gabbana, Jean Paul Gaultier, Comme des Garçons, Pauw, and more. From the people behind Assin.
113 Collins St., Melbourne, +61 03 9650 3457, www.coseipanema.com.
#### Country Road
* * *
A classic Australian chain with a simple, easy-to-wear aesthetic and a very affordable price point. "They do the best basics with a twist and use lots of natural fabrications so the clothing breathes and wears really well," says blogger Nicole Warne.
252 Toorak Rd., South Yarra VIC 3141, +61 3 9824 0133, and additional locations, countryroad.com.au.
#### Dejour
* * *
Nam Huynh, the man behind this cult custom-denim operation, has been making jeans for more than twenty years. His hole-in-the-wall shop has floor-to-ceiling shelves stocked with jeans in every size, shape, color, and design. They're made using wholesale denim from Bradmill, an Australian textile mill that's been operating since 1927. On weekends, expect to wait several hours to be helped. But with jeans at less than $100, including on-site alterations, it's worth it. While you're in Brunswick, try some Lebanese pastries, rose water, and pomegranate molasses from the Middle Eastern cafés on Sydney Road.
542 Sydney Rd., Brunswick VIC 3056, +61 3 9380 4884.
#### Fiorina
* * *
Melbourne-based silversmith and fine jeweler Fiorina Golotta specializes in a "modern tribal" aesthetic, creating pieces that cross historical periods, referencing everything from the Byzantine to the Victorian periods. Her chunky skull rings are particularly popular, as are her bracelets embellished with antique coins and tassels.
897 High St., Armadale VIC 3143, +61 3 9509 8883, fiorinajewellery.com.
#### Gorman
* * *
In 1999 Lisa Gorman launched her namesake label in Melbourne, with a collection titled Less than 12 Degrees. The success of that first offering led to the Gorman label becoming an iconic part of the first Australian designer wave in the early aughts. Her stores are eco-friendly and cool, selling her funky gem-print dresses and shirts, smart knitwear, and clogs. She recently collaborated with Melbourne artist Mirka Mora on a collection featuring her famous snakes and birds. There are fifteen Gorman stores in Australia.
235 Brunswick St., Fitzroy VIC 3065, +61 3 9419 5999, gormanshop.com.au, and additional locations.
GORMAN
#### Kleins Perfumery
* * *
Established in 1993, Kleins Perfumery remains a favorite with locals and visitors alike. Located in Fitzroy, this beauty boudoir is decorated with vintage Australian wallpaper and Florentine chandeliers, and has a store cat. Kleins houses more than eighty different brands from Australia and beyond. The fig perfume, in particular, is a must.
313 Brunswick St., Fitzroy VIC 3065, +61 3 9416 1221, kleinsperfumery.com.au.
#### Kuwaii
* * *
Established in 2008 by designer Kristy Barber, Kuwaii makes beautiful, intelligently designed clothing and footwear for women. An alternative to mass-produced fashion, all Kuwaii products are made in Australia. Refined and romantic, the slip dresses are a must-have, as are the shoes made from stock leather remnants and the selection of vintage and locally designed jewelry.
37-39 Glenlyon Rd., Brunswick VIC 3056, +61 3 9380 5731, kuwai.com.au.
#### Le Louvre
* * *
This salon-like store with gilt mirrors and a glass staircase began more than ninety years ago as a place for Melbourne's society ladies to shop. Today it's a destination for the latest from Chloé, Saint Laurent, Lanvin, Gucci, Valentino, Roland Mouret, and other high-end designers, with an ample selection of eveningwear and bridalwear, too.
2 Daly St., South Yarra VIC 3142, +61 03 9560 1300; lelouvre.com.au.
#### Lost and Found Market
* * *
Vintage furniture, clothing, and bric-a-brac market on Saturdays and Sundays.
499-511 Lygon St., East Brunswick VIC 3057, +61 3 9383 1883, lostandfoundmarket.com.au.
#### Marais
* * *
"I don't think I'll ever get over their amazing windows. They do a serious sale at the end of each season, too," says fashion editor Alyce Tran of this multi-brand boutique that stocks Céline, Chloé, Lanvin, and more.
79/87-91 Bourke St., Melbourne VIC 3000, +61 3 8658 9555, marais.com.au.
#### Monk House Design
* * *
A local favorite that stocks a select range of creative items by established and up-and-coming designers. Check out the Witu bags—these neoprene totes are an Australian fashion essential.
102 Lygon St., East Brunswick VIC, Australia 3057, +61 3 9381 1191, monkhousedesign.com.
#### P.A.M.
* * *
This Melbourne streetwear brand known for putting its own high-fashion spins on track pants, jogging shorts, and bomber jackets for men and women, often using cool graphics, is housed in the historic Curtin House, a 1922 art nouveau building named after former Labor Prime Minister John Curtin. Also in the building: a live music venue on level two, and the Rooftop Cinema on level six.
Level 3, 252 Swanston St., Melbourne VIC 3000, +61 3 9654 6458, perksandmini.com.
#### Peter Alexander
* * *
Many a last-minute gift crisis has been solved inside the iconic Peter Alexander pink doors—an Australian institution for sleepwear. There is no better place in the world to buy soft men's-style flannel pajamas, colorful cotton sleep shorts, silky nighties, and the most coveted Australian fashion accessory of all—the wool-lined Ugg boot.
Shop G045 Emporium 269 321 Lonsdale St., Melbourne, VIC 3000, +61 03 966 28024, peteralexander.com.au, and additional locations.
VIRGINIA MARTIN
MELBOURNE
* * *
Martin is the owner and designer of Melbourne-based contemporary women's fashion label Búl. Her love of travel is the inspiration behind the label, with each collection based on a different world destination and referencing the individual landscapes, cultures, and architecture.
BÚL is my favorite, of course. The stores are unassuming and inviting, oases you can escape to from the hustle and bustle of the streets. CRAFT VICTORIA GALLERY STORE supports local artisans with exhibitions and retail programs. There's an outlet in the city where you'll find beautiful one-of-a kind pieces.
METROPOLIS BOOKSHOP is tucked away on the third floor of the iconic Curtin House building. It boasts an amazing range of specialist art and design books and hard-to-find editions. Explore the whole building to find great stores, bars, and restaurants. Make your way to the sixth floor to enjoy some cocktails and the beautiful view of the Melbourne skyline at ROOFTOP BAR.
NGV STORE (National Gallery of Victoria) is a design store that offers everything from books to fashion to housewares. They often have interesting collaborations with local artists, and editions relevant to the current exhibitions.
BÚL, Melbourne Central, Level 2, 211 Latrobe St., Melbourne VIC 3000, +61 3 9663 0139, bul.com.au, and additional locations.
CRAFT VICTORIA GALLERY, 31 Flinders Ln, Melbourne VIC 3000, +61 3 9650 7775, craft.org.au.
METROPOLIS BOOKSHOP, Level 3 Curtin House, 252 Swanston St., Melbourne, Victoria 3000, +61 3 9663 2015, metropolisbookshop.com.au, curtinhouse.com.
ROOFTOP BAR, Level 6 Curtin House, 252 Swanston St., Melbourne VIC 3000, +61 3 9654 5394, rooftopcinema.com.au.
NGV STORE (NATIONAL GALLERY OF VICTORIA), 180 St Kilda Rd., Melbourne VIC 3006, +61 3 8620 2243, ngv.vic.gov.au.
#### Shag
* * *
Showcases treasures from every era in a funky environment.
130 Chapel St., Windsor VIC 3181, +61 3 9510 8817, shagmelbourne.com.
#### Shifting Worlds
* * *
A luxury store located in the heart of the city, stocking clothing, accessories, and home goods from names such as Pleats Please, Bao Bao, and Maryam Nassir Zadeh—the shopping destination for obscure-label nerds!
187-193 Elizabeth St., Melbourne VIC 3000, +61 3 9600 0459, shifting-worlds.com.
ONLINE SHOPPING TIP
Tuchuzy.com offers international shipping and a great selection of Aussie labels, including KITX, Georgia Alice, and Sir the Label (a favorite of Sydney-born model and Instagram star Mimi Elashiry). Top designer sites such as elleryland.com, zimmermann.com, and dionlee.com also include international shipping options.
#### Third Drawer Down
* * *
Curated objects from around the world, including home accessories, books, and jewelry, as well as items produced from the Third Drawer Down Studio. You'll find scarves and tote bags by artists, products from Tom Dixon, Lucy Folk, and other well-known interior designers, and more.
93 George St., Fitzroy VIC 3065 +61 03 9534 4088, Prahran, thirddrawerdown.com, and an additional location in Prahran.
THRID DRAWER DOWN
#### Verner
* * *
Ingrid Verner first emerged on the Australian fashion scene as a TV star, before launching her namesake label in 2012. The brand has an upscale streetwear vibe; think, sweatshirts and sweatpants with artful cutouts. Sharing the space with the Verner store is Slow Waves, a concept store that collects small, independent labels. It's curated by Matli Atterton, who is focused on supporting the designers and getting the low-high balance right.
358 Lonsdale St., Melbourne VIC 3000, +61 3 9088 0760, verner.com..
## CONTRIBUTORS
B. ÅKERLUND is a costume designer, stylist, and Stockholm-to–Los Angeles transplant whose hand can be seen in everything: Television commercials for Dior, Absolut, and Nike; album covers for Lady Gaga and the Red Hot Chili Peppers; feature films (Spun, Horsemen); music videos ("Hold Up" from Beyoncé's visual album Lemonade); and Super Bowl halftime shows with the Black Eyed Peas (2011) and Madonna (2012).
Biggest score: A life-size Andy Warhol doll that I found at the Rose Bowl Flea Market.
JOSEPH ALTUZARRA launched the luxury women's line that bears his name in 2008, and has been raking in fashion industry accolades ever since. Recognitions include the CFDA/Vogue Fashion Fund award (2011), the CFDA Swarovski Award for Womenswear (2012), and the CFDA Womenswear Designer of the Year (2014). His strong, feminine dresses are worn by the likes of Lupita Nyong'o, Rihanna, and Zoe Saldana.
Style inspiration: Marlon Brando in The Wild One.
Words of wisdom:
If you're still thinking about a potential buy the next day, it's probably worth the purchase.
SOPHIA AMORUSO used a laptop computer and a love of vintage to build Nasty Gal into a retail empire. Her business memoir #GIRLBOSS hit the New York Times bestseller list in 2014, the same year she launched the GIRLBOSS Foundation, which awards financial grants to female creatives pursuing entrepreneurial endeavors. Her second book, Nasty Galaxy, was published in 2016.
Style icon: Bianca Jagger in her white suit.
Biggest score: A vintage Chanel jacket bought for $8 at the Salvation Army—and sold on eBay for $1,200.
PAUL ANDREW is a British-born, New York–based shoe designer who launched his namesake collection in 2013. In 2014, he became the first footwear designer to win the CFDA/Vogue Fashion Fund award; in 2016, he took home the CFDA Swarovski Award for Accessory Design. Among his celebrity fans are Cate Blanchett, Jessica Chastain, and Lupita Nyong'o.
Biggest score: A vintage Hermès Garden Party PM bag in navy-blue canvas and tan leather, purchased for $40 from Housing Works, a secondhand store in New York's West Village.
Words of wisdom:
Everything in life is about moderation, including the fun, frivolous, and impractical!
ROSE APODACA, the godmother of the L.A. fashion journalist set, has uncapped her pen on behalf of the Los Angeles Times, Harper's Bazaar, and Women's Wear Daily over the past several decades. She's penned a string of style-centric books about or with some of the biggest names in the fashion business, including retailer Fred Hayman, stylist and designer Rachel Zoe, and burlesque queen Dita Von Teese. She also co-owns and operates the home goods emporium A+R with her husband, Andy Griffith.
Style icon: Nancy Cunard, and not only for her turbans, kohl-rimmed peepers, and limbs ringed in bangles. Those things only served to accessorize a maverick spirit of activism, intelligence, feminism, and conviction—and that is true lifestyle.
Words of wisdom:
Shopping is like a one-night stand: if there's any inkling you're going to end up regretting some part of it, then move on!
GABRIELA ARTIGAS AND TERESITA ARTIGAS, who originally hail from Mexico City, are the sister act behind Gabriela Artigas & Co., an L.A.-based jewelry business. Since the brand's launch in 2003, its everyday statement pieces have found favor with the likes of Tyra Banks, Emma Roberts, Chelsea Handler, and Carey Mulligan.
Words of wisdom:
For everything that comes into your wardrobe, something must always come out. Extend the life of items you don't want anymore by reselling them. Someone, somewhere is going to appreciate the thing that you once appreciated. (Teresita)
Style inspiration: White clothes—tees, dresses, pants, shirts, everything—and red lips. There's a certain understated elegance when you have a white canvas. (Gabriela)
MICKEY ASHMORE, known to comfortably shod clients around the globe as "the Sabah Dealer," is a New York City–based former hedge fund manager who leveraged a six-month stint in Istanbul while working for Microsoft into a burgeoning business selling handmade Turkish slippers.
Words of wisdom:
Keep it simple. A few great items go a long way. I think one's choice of what to wear should take up minimal time each day.
Style icon: René Holguin, owner of RTH in Los Angeles.
JORDAN ASKILL is a New York City–based jewelry designer and sculptor who was born and raised in Sydney, Australia. He launched his namesake jewelry brand in London in 2010. In 2015 he won the British Fashion Award for Emerging Accessory Design—the first jewelry designer to win a BFA in more than five years.
Biggest score: A Tibetan bamboo cuff found at the gem market in Taipei.
Style icon: Elizabeth Taylor, because of her incredible collection of jewelry.
BRIAN ATWOOD is a New York– and Milan-based FIT-educated model turned designer who holds the distinction of being the first American ever hired by Gianni Versace. He launched his namesake footwear collection in 2001. Atwood's fantastical footwear has appeared on the feet of such luminaries as Lady Gaga, Taylor Swift, Victoria Beckham, Jessica Alba, and Kate Hudson.
Biggest score: A vintage print of Chris von Wangenheim's Dior ad with Liz Taylor and a Doberman.
Words of wisdom:
Don't think an item will be there when you return. If you love it, buy it.
CHRISTENE BARBERICH is the global editor-in-chief and cofounder of the Webby award-winning lifestyle media company Refinery29. Barberich is also the New York Times bestselling co-author of Style Stalking. Prior to cofounding Refinery29 in 2005, she held posts at Gourmet magazine, The Daily, The New Yorker, and was the founding editor of the ASME-winning CITY magazine. Her fashion and design writing have appeared in the New York Times, Travel & Leisure, and New York magazine among other titles. She lives in Brooklyn.
VANESSA BELL's insatiable curiosity about her adopted hometown Buenos Aires led her to investigate its sleepy backstreets and unearth its most tucked away spots, emerging trends, and up-and-coming talent to create her blog cremedelacremeba.com. As a freelance writer, she is a regular contributor to Wallpaper* and Monocle, and her work has appeared in The Independent, Departures, Fathom, Traveldo.se, and Time Out. She also operates the Crème de la Crème bespoke shopping service, which designs personalized itineraries in Buenos Aires for each client's shopping needs.
STACEY BENDET is creative director and CEO of the contemporary label Alice and Olivia, which she founded in 2002. Her ornamental fabrics and whimsical prints are a go-to for a stylish celebrity set that includes Jessica Alba, Eva Longoria, Anne Hathaway, and Heidi Klum. Bendet was named to Vanity Fair's International Best-Dressed List Hall of Fame in 2014.
Style inspiration: A photo of my mother on her wedding day wearing this amazing Fred Leighton wedding dress with a pair of super-flare bell-bottom jeans underneath.
Words of wisdom:
Shopping is cheaper than a psychiatrist!
SOFÍA SANCHEZ DE BETAK is a Buenos Aires–born, New York–based fashion consultant, art director, and all-around fashion darling who has worked with Marni, Chloé, Tiffany & Co., Jimmy Choo, Belstaff, and Ermenegildo Zegna among others. In 2012 she turned her focus toward her home country and launched the website UnderOurSky.com to showcase Argentine design with a focus on handmade and one-of-a-kind accessories. In early 2016, the world traveler—she shares her globe-trotting adventures with her 92,000-plus Instagram followers (@chufy)—collaborated on a line of luggage with Globe-Trotter and the Luxury Collection Hotels.
Words of wisdom:
If you love it, get two!
AURÉLIE BIDERMANN is a Paris-based jewelry designer who launched her namesake collection in 2004. Her travels around the globe—to California, Bali, and India—inspire her collections, which are available at hundreds of boutiques worldwide. Celebrity fans of her baubles include Céline Dion, Beyoncé, and Keira Knightley.
Style inspiration: I adore Charlotte Rampling. Her style is so timeless and elegant; she looks chic in anything she puts on and it always seems so effortless.
Biggest score: A vintage piece from Yves Saint Laurent's '70s-era Rive Gauche collection.
Words of wisdom:
Shop when you feel good.
ANINE BING is the Denmark-born, L.A.-based founder and creative director of the namesake Scandinavian-meets-American fashion line, which launched in 2012 and is currently sold in more than 300 boutiques around the globe. Her off-duty-model styles, including leather moto pants, studded boots, and lace bralettes, are faves with Kendall Jenner and Rosie Huntington-Whiteley.
Style inspiration: Jane Birkin.
Words of wisdom:
Less is more. That goes with everything I do in life, including shopping. I'd rather save up for one very nice thing than buy a bunch of stuff I'll only use one time.
CHRISTINA BINKLEY is the Wall Street Journal's fashion and style columnist and author of the New York Times bestseller Winner Takes All: Steve Wynn, Kirk Kerkorian, Gary Loveman, and the Race to Own Las Vegas. Based in Los Angeles, she travels the globe, from New York to Milan and Paris, covering the twice-yearly fashion week circuit. She was part of a team that won a Pulitzer Prize for its coverage of 9/11.
Style influence: I don't even think of fashion that way. I just know what I like.
Words of wisdom:
Shop like a man. Demand great fit, and if the fabric doesn't feel good, walk away.
HELENA BORDON is a São Paulo–born blogger, entrepreneur, and designer who cofounded the Brazilian fast-fashion brand 284 in 2008. The daughter of Donata Meirelles, former creative director of Brazilian high-fashion boutique Daslu, Bordon documents her globe-trotting, style-centric adventures online at helenabordon.com. In 2013, she collaborated with LOOL on a limited-edition jewelry collection.
Biggest score: A vintage Lanvin coat from Casa Juisi in São Paulo.
Words of wisdom:
Go shopping with the same discerning eye a gallerist has at an art fair.
SONIA BOYAJIAN is an Antwerp-trained, Los Angeles–based designer of artful, eye-catching statement jewelry. She launched her namesake line in 2003 and opened an atelier/boutique in Hollywood in 2014. Her baubles have found favor with the likes of Scarlett Johansson and Natalie Portman.
Style inspiration: Women who look at ease in their clothing.
Biggest score: A Valentino bustier with a massive bow on it for $50 at a resale boutique.
Words of wisdom:
In the words of Edith Head, "A dress should be tight enough to show you're a woman and loose enough to prove you're a lady."
JANIE BRYANT is a costume designer best known for her work on the AMC television series Mad Men, which earned her multiple Emmy Award nominations, two Costume Designers Guild Awards, a clothing collaboration with Banana Republic, and a book, The Fashion File: Advice, Tips, and Inspiration From the Costume Designer of Mad Men (co-authored with Monica Corcoran Harel). She also has an Emmy Award on her mantel for her work on the HBO series Deadwood.
Style inspirations: The rococo period, Jean Shrimpton, and Vivien Leigh as Scarlett O'Hara.
Biggest score: A 1960s white gold, mother-of-pearl, and diamond domed cocktail ring bought at Treasure Isle Flea Market in Naples, Florida.
Words of wisdom:
Buy pieces you are passionate about and always try on.
VICTORIA BRYNNER is a Swiss-born, Beverly Hills–based international creative consultant and the founder and president of Stardust Visions, a photo production company whose client list reads like a who's who of the luxury fashion world (Balenciaga, Prada, Giorgio Armani, and Valentino, among others). She is the daughter of Academy Award–winning actor Yul Brynner.
Style inspirations: My mother, Doris Brynner, Audrey Hepburn, early Balenciaga, and Valentino.
Biggest score: An original Arthur Elgort print of jazz tenor saxophonist Dexter Gordon—bought at a garage sale for $20.
TORY BURCH is the chairman, CEO, and designer of her namesake label. Launched in 2004 with a single NoLIta boutique, her company has since grown into a global bohemian-prep lifestyle brand that is a go-to for everyone from Kerry Washington, Jessica Alba, and Kiernan Shipka to the Duchess of Cambridge. In 2009 Burch launched the Tory Burch Foundation with the goal of empowering women and families through mentoring and microfinancing.
Style inspiration: My mom in a head-to-toe gold lamé outfit by Zoran.
Biggest score: A green tunic found at a Paris flea market. It would go on to inspire one of my first designs.
LIBBY CALLAWAY has contributed as a fashion journalist to the New York Post, Elle, the New York Times's T Magazine, Style.com, Travel + Leisure, and Glamour, where she had her own fashion advice column. She is the media director for Nashville denim company Imogene + Willie. As a former wardrobe stylist and vintage clothing dealer, Libby is a noted secondhand shopping expert.
Style inspiration: I am a massive fan of '40s and '50s style, and I'd also have to say the wardrobe of Gialo horror films Suspiria and Tenebre by Dario Argento, and postmodernist architecture and design.
CHRISTINA CARADONA was born in New York and raised in France by a mother who was a professional model and a father who loved to travel. Christina is the model/actress who created the popular blog Trop Rouge in 2010. She has been featured in Elle, Glamour, Cosmopolitan, Nylon, and Seventeen magazines, and her signature curly locks have appeared in campaigns for Paul Mitchell, H&M, Joe Fresh, OPI, and many other brands. She lives in New York City.
PETER CHEUNG is the Hong Kong–based founder and chief executive officer of Peter Cheung Asia Limited, a strategic branding and marketing firm. Cheung works with a range of luxury brands and celebrities, including Van Cleef & Arpels, Parisian couturier Gaspard Yurkievich, and model Philip Huang.
Style influence: Hong Kong in the 1980s. Luxury and fashion brands were just being introduced and the pioneer was Joyce Ma. Everyone dressed so perfectly for every occasion, at every destination—it was high-octane, put-together fashion!
Biggest score: A one-of-a-kind, full-length python trench coat purchased at the John Galliano boutique in Paris in 2005.
Words of wisdom:
Don't bother squeezing into it.
ALINA CHO is a New York–based journalist and editor-at-large at Ballantine Bantam Dell, where she develops fashion and lifestyle books. She is the host of the Metropolitan Museum of Art's lecture series "The Atelier With Alina Cho," for which she has interviewed fashion luminaries, including Anna Wintour, Donatella Versace, Alber Elbaz, Diane von Furstenberg, and Alexander Wang.
Style inspiration: The 1958 Willy Rizzo photograph of Yves Saint Laurent with models wearing clothes from his first collection for Christian Dior. It hangs in my bedroom and I look at it every day. It reminds me of a beautiful, feminine, formal style of a bygone era.
Biggest score: A floor-length ivory feather Lanvin coat found at Frock, a now-defunct vintage store in SoHo.
NATALIE CHOY was born in Providence, Rhode Island and moved to Korea at a very young age. As an adult, she moved to New York to study Photography and English Literature at the Parsons & Eugene Lang College, then Paris to pursue her degree in Fashion Styling at Istituto Marangoni. She has worked in event planning, styling and PR in London, Moscow and New York.
ARI SETH COHEN is the street-style photographer and creator of Advanced Style, a blog that's been documenting the fashion and style of the over-sixty set since 2008. He's also the creator of a documentary and two books based on the blog, Advanced Style (2012) and Advanced Style: Older & Wiser (2016).
Style inspirations: Nancy Cunard, Cecil Beaton's images of Edith Sitwell, the Marchesa Casati, the Seeberger Brothers street-style photos from the early 1900s, Liberace, Nudie's Tailors, Cary Grant, and Quentin Crisp.
Biggest score: A multicolored sequin kimono jacket from Etsy—a surprise gift from my boyfriend.
Words of wisdom:
You'll never regret buying a vintage treasure, but you may regret leaving it behind.
RACHEL COMEY is a New York–based fashion designer whose career kicked off with a collaboration with the band Gogol Bordello that earned her a place in the 2001 Whitney Biennial. Her namesake women's label launched in 2004. Known for arty textiles (foam, linen, pebble, crochet), modern silhouettes, and covetable shoes, it is sold at more than 100 high-end boutiques and department stores in sixteen countries as well as stand-alone stores in New York and Los Angeles.
KENDALL CONRAD is a former model whose accessories design career began in 2000 when she couldn't find a handbag she liked. Today her Los Angeles–based namesake collection is available online and at brick-and-mortar boutiques in California, where the Santa Barbara native lives.
Style inspiration: California beach girl Lauren Hutton and Candice Bergen and Ali MacGraw mixed with a touch of European influence (Charlotte Rampling, Anouk Aimée, and Julie Christie).
Biggest score: Black satin YSL Rive Gauche sailor pants from the '70s purchased in the '90s from L.A. vintage store the Paper Bag Princess.
Words of wisdom:
Make sure it fits. Better to buy big, and tailor.
BROOKE TAYLOR CORCIA is the founder and CEO of Los Angeles–based online retail site The Dreslyn, which launched in 2012. She also served as the senior womenswear and accessories buyer for ssense.com, prior to which she was the associate fashion editor at C Magazine.
Style inspiration: Nineties minimalism was at its height during my formative years, so that inevitably had an impact on my way of imagining style. I still reference the timeless and understated costume design by Ellen Mirojnick for the film A Perfect Murder. It holds up after nearly twenty years.
Words of wisdom:
Less is more.
BRONWYN COSGRAVE is a multidisciplinary fashion professional whose diverse career combines journalism, broadcasting, consulting, public speaking and producing feature-length documentaries exploring the world of luxury. Her fashion CV includes serving as features editor of British Vogue, penning six fashion history books, serving as guest curator of the exhibition "Designing 007: Fifty Years of Bond Style" at the Barbican Centre, and contributing to a host of international fashion newspapers and magazines, including Vogue India, where she is a contributing editor.
EMILY CURRENT AND MERITT ELLIOTT are L.A.-based stylists, consultants, authors, and the design duo behind the Current Elliott denim brand, which they founded in 2008 and left in 2012. They are the creative directors of contemporary women's collection the Great.
Style influence: The Boxcar Children was our favorite childhood book. It's about children who are forced to be resourceful and imaginative, and create magical worlds in their minds. We love that childlike sense of discovery, and see it affect the way we create in our jobs.
Biggest score: A vintage sign that says SHIRTS AND TIES found at a vintage store in Ventura County that hangs in our office.
Words of wisdom:
Shopping can be a personal and emotional journey, and you should be led not by what you need or should get but what makes you happy.
KAVITA DASWANI is an international journalist and author, and the former fashion editor for the South China Morning Post in Hong Kong and Asian correspondent for Women's Wear Daily, both of which she still contributes to. She also currently writes for the Los Angeles Times and international editions of Vogue, Condé Nast Traveler, and Grazia Italia. She has published seven books, including the 2012 Harper Collins' India title Bombay Girl, set in the high-society world of one of India's most sophisticated cities.
JESSICA DE RUITER started her career more than decade ago working as a fashion editor with a range of Condé Nast titles, including Vogue, Teen Vogue, W, and WWD. She works as a stylist to the stars, and her list of A-list clients includes Jennifer Lawrence, Michael Fassbender, Emily Blunt, Robert Pattinson, Diane Kruger, and Angelica Huston. She's also worked with a range of commercial clients, including Ferragamo, James Perse, Bebe, and Madewell. She lives in Ojai, California.
Style influence: Carolyn Bessette Kennedy, for her easy elegance and classic, clean style.
Words of wisdom:
Sleep on it.
ANNA DELLO RUSSO is a Milan-based fashion journalist, fashion-show front-row fixture, and enthusiastic collector of apparel and jewelry. She spent nearly two decades at Condé Nast Italia, including roles as fashion editor at Vogue Italia and editor of L'Uomo Vogue. She is editor-at-large and creative consultant for Vogue Japan. In 2012, she collaborated with Swedish fast-fashion brand H&M on a line of accessories and jewelry.
VALERY DEMURE founded her namesake jewelry-focused PR agency in London in 2005. In 2012 she launched e-commerce site valerydemure.com, which curates jewelry and accessories from around the globe. In addition, she serves as an industry consultant to a range of brands and is an adviser for the fashion jewelry and accessories curriculum at London College of Fashion.
Biggest score: A pair of black suede and gold metallic open-toe Chanel booties, lusted after for years and eventually scored during a 2 a.m. eBay session.
Words of wisdom:
It's not about needing it.
EVREN DOGANCAY is the chief buyer for the department store Beymen and former head buyer for the Kirna Zabête boutique in SoHo. Based in Istanbul, she can be found trotting the globe, from Milan to Paris and beyond, to find the right mix of merchandise to stock the store.
Style influences: Lee Radziwill and Catherine Deneuve.
Biggest score: Thigh-high Alaïa boots that I got at Kirna Zabête in New York. I was working as the buying director at the time. The boots were from the previous season and there was only one pair left.
JACEY DUPRIE is a style blogger and the editor of Damsel in Dior, a website she launched in 2009 that encompasses fashion, lifestyle, travel, and home décor. The website has been featured in the Huffington Post, Vanity Fair, WhoWhatWear, Forbes, and E! News, and Duprie has appeared in ad campaigns for Saks, GAP, and Nordstrom. She lives in Los Angeles.
ALISON EDMOND is one of the most highly regarded editorial stylists in the fashion industry and has worked with the who's who of Hollywood, including Beyoncé, Gwyneth Paltrow, and Angelina Jolie. A native Brit, she has worked all over the world styling, art directing, and consulting for magazines, advertising, TV commercials, and design houses. She is currently fashion director at C Magazine in Los Angeles, also covering C Men, C Home, and C Weddings.
PIP EDWARDS is a stylist and veteran of the Australian fashion scene. She serves as design director for the multi-brand retailer General Pants Co. She is the cofounder of the retro-inspired activewear label that bears her initials—P.E. Nation—which launched in early 2016. She lives a stone's toss from Sydney's Bondi Beach.
Style inspiration: Seventies denim and '90s street.
Biggest score: At One of a Kind, the London vintage shop where Kate Moss shops, I found an original Vivienne Westwood Pirate collection dress. I can wear it, but I don't, it hangs on my wall. It cost a fortune, and my baby daddy bought it for me. I buy not necessarily to wear but to appreciate.
CRISTINA EHRLICH is a former professional ballet dancer who works as a New York–based stylist. Her list of A-list clients includes Margot Robbie, Priyanka Chopra, Anna Kendrick, Uzo Aduba, Gugu Mbatha-Raw, Julia Louis-Dreyfus, and Tina Fey. She has worked with photographers such as Patrick Demarchelier, Mert Alas & Marcus Piggott, Peter Lindbergh, and Peggy Sirota. She was awarded "Celebrity Stylist of the Year" at the 2012 Style Awards in New York City and has partnered with Pandora to serve as a brand ambassador.
Style inspiration: Jean Seberg in Breathless. It was all about her effortless street style, her flats, jeans, T-shirts. She was the absolute French gamine.
Words of wisdom:
Take risks but be realistic.
MIMI ELASHIRY is a Sydney-born model and fashion blogger who leveraged a cult Instagram following into a career working with brands like Diesel, Free People, and Glue Store. In 2014, she was tapped to be the official style ambassador of MTV Australia. She is currently based in Los Angeles.
Style inspiration: Cher in the '70s had an incredibly eclectic style—from the neon mesh to the heavily beaded cutout gowns, Indian-inspired two-pieces, and very little makeup or accessories—that has always influenced and inspired me.
MERITT ELLIOTT (see Emily Current).
KAREN ERICKSON is cofounder and designer of New York–based Erickson Beamon, a vintage-inspired jewelry brand founded in the early '80s. She has accessorized high-profile A-list clients from Lady Gaga to Michelle Obama, with Gwyneth Paltrow, Beyoncé, Madonna, and Nicole Kidman in between. A go-to runway collaborator for luxury fashion labels such as Alexander McQueen, Givenchy, Donna Karen, and Zac Posen, Erickson launched a collaboration with Target in 2009.
Style inspiration: I became fashion-aware in the '70s, and I thank my lucky stars the '70s remain fashion insiders' most referenced decade. The designers in the '70s were inspired by the '20s and the '40s, so with one decade I get three. My fashion icon is a decade.
ABRIMA ERWIAH is cofounder and co-creative director of the Ghana-based Studio One Eighty Nine, which she created in 2013 with actress and activist, Rosario Dawson. Their social enterprise helps promote and curate African and African-inspired content and brands through worldwide distribution and manufacturing of their private label, Studio One Eighty Nine, as well as through support of other brands. Erwiah is a veteran of the fashion and luxury industry, with more than sixteen years' experience at Bottega Veneta, Hermes, John Lobb, and other brands.
NINA FARRAN is the founder of e-commerce site Fashionkind.com, which features a curated mix of ethically produced and sustainable luxury products from labels such as Coclico, A Peace Treaty, Ryan Roche, and Rialto Jean Project. Her background is in developing new opportunities that make a positive impact. While at the University of Pennsylvania, she launched a humanitarian fashion effort that built schools in sub-Saharan Africa and, more recently, she initiated and successfully built an impact-investing program at a national investment and wealth management firm. She's based in New York.
CARLA FERNÁNDEZ is creative director of the Mexico City–based fashion label that bears her name. Launched in 2000, her ready-to-wear collection takes inspiration from traditional Mexican textiles and geometric patterns, and is created in collaboration with artisans across Mexico.
Words of wisdom:
I always tell my kids that shopping is like the warning they give you on an airplane. Place the oxygen mask on yourself before helping others; shop for yourself first and then shop for the rest of the family and friends!
SAMUEL FERNSTRÖM is the managing director of fashion brand & Other Stories, and works out of ateliers in Paris and Stockholm. He's been on board since day one, cofounding the brand together with a small group of creatives. Samuel worked at H&M for ten years before he started to develop & Other Stories. Most of his years at H&M were spent in the concept development group at the new business department in Stockholm.
ERIN FETHERSTON was raised in the San Francisco Bay Area, and studied fashion design in Paris, where she first launched her namesake women's collection in 2005. She relocated to Los Angeles in 2016, and launched a home collection.
Words of wisdom:
Before I make a purchase, I always ask myself, "Will I love it years from now?"
ANNABELLE FLEUR is a Latvia-to-Los Angeles transplant who helms the fashion blog Viva Luxury. The Victoria's Secret ambassador has worked with a range of beauty and fashion brands, including Olay for the Grammys, RED Valentino, Stuart Weitzman, Bergdorf Goodman, 7 for All Mankind, Pandora, and Express.
Biggest score: An in-season Marc Jacobs Trouble bag bought at a 70 percent discount from the sale section of TheCorner.com.
Words of wisdom:
Regardless of how beautiful something looks on a hanger, the only thing that really matters is how good it looks on you.
HEATHER JOHN FOGARTY writes about the things she loves, whether that's a well-executed gin and tonic, a Case Study house, a Rodarte chiffon coat, her favorite hotel swimming pools, or a Jambon-Beurre sandwich slathered in buckwheat honey butter. Her work has appeared in publications such as Australian Vogue, Marie Claire, Los Angeles Magazine, C Magazine, Playboy, Robb Report, the Los Angeles Times, and Bon Appétit, where she was wine and spirits editor from 2004 to 2011. She is currently working on her first novel.
BILLY FONG has served as executive director of the Texas Association of Museums since 2016, prior to which he held posts at prestigious museums around the country. He's also a stylist and writer, and penned a style column for PaperCity magazine for nearly two years. He lives in the Dallas/Fort Worth area.
Biggest score: My first-ever pair of Gucci loafers purchased in 1995 for $395 from the brand's Boca Raton boutique. Twenty-five pairs later, they're still displayed in a place of honor—and still in the original box.
LIZZIE FORTUNATO is the designer and cofounder (with twin sister Kathryn) of the Lizzie Fortunato jewelry line, which launched in 2008 and added leather goods in 2011. Her artful modernist accessories with an exotic flair have adorned the fashion world ever since. The sisters have expanded their line, offering a range of artisanal home goods discovered on their travels to India, Morocco, Mexico, and Peru.
Style influence: Angelica Huston modeling Alexander Calder's curvy "feathered" brass necklace with a high chignon and black turtleneck. Her look is so classically chic and the jewelry makes such a statement. It's an image I think of a lot when getting dressed.
LISEANNE FRANKFURT is native Californian whose career as a jewelry designer began when she started selling her work during her sophomore year at UCLA. She launched LFrank Jewelry and opened a retail store in Venice, California, in 2007. In 2015, Frankfurt expanded the line to include a lingerie collection.
Biggest score: An incredible red leather trench coat from Alber Elbaz's last collection for YSL at the Barneys Warehouse sale. It was originally around $10,000. I was pregnant with my daughter at the time, with a toddler running around and a renovation at home, so it was far too indulgent. I found it at the sale for $199. I felt faint, and as it turned out, went into the hospital the next day in premature labor!
NINA GARDUNO launched the L.A. brand FREECITY in 2002 with a simple T-shirt that read FREECITY NEIGHBORHOOD. The idea blossomed into a lifestyle movement, which is brought to life in the brand's 3,000-square-foot Highland Avenue store in Los Angeles. The inspirational retail environment has a different theme each season, which translates to the artwork on the walls, the tunes played in the store, and the words on the clothing.
LIZZIE GARRETT METTLER is a Los Angeles–based freelance writer and creator of the-reed.com, an online destination that's part blog, part travelogue, and part carefully curated e-commerce experience. Her first book, Tomboy Style—based on her blog of the same name—was published in 2012.
Style influences: Coco Chanel's rebellious attitude, Diane Keaton's nonconformist nonchalance, Katharine Hepburn's "I don't give a damn, I'm wearing pants anyway" mantra all help me out on days when I don't know what to wear.
ROSETTA GETTY is a former model turned designer whose namesake collection of modern, sophisticated, and uncomplicated clothing launched in 2014, and is carried by some of the top boutiques and department stores worldwide including A'Maree's in Newport Beach, Ikram in Chicago, and Harvey Nichols and Selfridges in London. She lives in Los Angeles.
Style influences: Louise Bourgeois, Betty Parsons, '90s Helmut Lang, Prada, and Jil Sander.
Biggest score: A Wolfgang Tillmans photograph—one of the first purchases in my collection. I bought it from Shaun Regen.
ROBIN GIVHAN is the Pulitzer Prize–winning fashion editor of the Washington Post and the co-author of Michelle: Her First Year as First Lady. Her first solo book, The Battle of Versailles (2015), is being made into a movie for HBO Films co-written and directed by Ava DuVernay. In 2006, she won a Pulitzer Prize in criticism for her fashion coverage. She is based in Washington, DC.
Style influence: My mother from her heyday in the 1960s and '70s. She was always polished. Never fancy, but pulled together. I hear her voice telling me not to go out in wrinkled clothing. It's because of my mother that I am incapable of traveling in sweatpants. She would be appalled.
BONNAE GOKSON is the Hong Kong–based style icon, restaurateur, and creative force behind Sevva, C'est La B, and Mrs. B's Cakery. Prior to focusing on foodstuffs, the "queen of cakes" worked in the fashion space helping luxury brands Prada, Giorgio Armani, and Chanel build their presence in Asia. Her edible works of art are the subject of an award-winning coffee-table book Butterflies and All Sweet Things (2014)
Style influence: Grace Kelly.
Words of wisdom:
Quality over quantity always.
LORI GOLDSTEIN is a New York–based stylist who enthusiastically embraces the philosophy that "everything goes with anything." She's worked for a who's who of the celebrity set on both sides of the camera. In 2009, she added designer to her résumé with the launch of LOGO by Lori Goldstein exclusively for QVC. She's also the fashion editor-at-large for Elle magazine and the author of Lori Goldstein: Style Is Instinct (2013).
Style influences: Everyone from my grandmother GaGa Gladys, who wore head-to-toe leopard, to Marilyn Monroe. Everything Liz Taylor did and wore was heaven. Cher, Bowie, Barbra in every movie, The Thomas Crown Affair, Valley of the Dolls . . . I'm stopping now.
Words of wisdom:
If you can't stop thinking about it, get it.
NJ GOLDSTON is the Beverly Hills–based founder and editor-in-chief (and The Brunette) of the Webby-award-winning fashion and style blog The Blonde & The Brunette. She's also the chief creative officer of advertising, branding, and digital agency, The UXB, and a managing partner of her twin sons' L.A.-based footwear brand Athletic Propulsion Labs.
Biggest score: A Sergeant Pepper–style jacket from the Christophe Decarnin days at Balmain. It was hidden in the back at Browns in London. I still wear it at least once a year, on Michael Jackson's birthday.
Words of wisdom:
Don't buy something with an occasion in mind. Buy something because you love it, and it will be ready to go when the moment arrives.
LIZ GOLDWYN is a Los Angeles–based writer, filmmaker, artist, and an enthusiastic collector of vintage clothing. Her works include the nonfiction book Pretty Things: The Last Generation of American Burlesque Queens, the HBO documentary based on it (which she also directed), and the novel Sporting Guide: Los Angeles, 1897 (2015).
Biggest score: A Rudi Gernreich kabuki dress bought at a serious discount from Cameron Silver the week he opened Decades.
Words of wisdom:
I crib Janis Joplin's ethos—"If you buy it today, you don't wear it tomorrow." Preach.
MARY ALICE HANEY is a celebrity stylist and fashion editor (Marie Claire, GQ) turned designer who launched her namesake Los Angeles–based luxury women's label Haney in 2013. Celebrity fans of her sexy, red-carpet-ready creations include Jennifer Lawrence, Chrissy Teigen, Kate Hudson, and Taylor Swift.
Style inspiration: Old Hollywood. It's the inspiration for my collections, too.
STEFANIE HANSSEN is the founder and creative director of Berlin-based perfume brand Frau Tonis Parfum. In addition to creating a suite of fragrances that use the city of Berlin as a theme, she has partnered with other brands, from the Ritz-Carlton Hotel to concept stores and fashion magazines, to create bespoke scents.
Style influence: Diana Vreeland! I love her style and her idea of beauty. Her book The Eye Has to Travel is my bible.
DIVIA HARILELA is a thirteen-year veteran of the fashion media, having served as the fashion editor for Hong Kong's leading English-language newspaper, the South China Morning Post, before founding the luxury and fashion website the D'Vine in 2011. Born and raised in Hong Kong, she's often sought out as an authority on fashion in Asia.
Biggest score: An incredible jacket I found in Berlin made from a hand-painted vintage kimono. It's been deconstructed and put back together again all by hand—even the stitching is a detail in itself. The back looks like a piece of art, and people always stop me when I wear it.
KEMAL HARRIS is the New York–based half of the bicoastal styling team known as Kemal and Karla (the other half, Karla Welch, is based in L.A.), who have created red-carpet looks for the likes of Olivia Wilde, Felicity Jones, Hailee Steinfeld, and Zooey Deschanel, as well as tour wardrobes for Justin Bieber, Pink, and Idina Menzel. Harris added costume designer to her CV in 2015 when she was tapped to design wardrobe pieces for longtime client Robin Wright's character Claire Underwood on the Netflix series House of Cards.
Words of wisdom:
Trust your gut but also watch out for repeat offenders. I own approximately twenty-five biker jackets—nobody needs that many.
JOHNSON HARTIG is the cofounder and designer of the Los Angeles–based Libertine label that launched in 2001. His book, Libertine: The Creative Beauty, Humor and Inspiration Behind the Cult Label, was published in 2015.
Style influences: I'm a "more is more" guy: Tony Duquette, Mario Buatta, Mark Hampton, Madeleine Castaing, and Jean Cocteau. Last year I visited Cocteau's house in Milly-la-Forêt, outside Paris. Much of his place is still as it was when he died, and I continually reference it as inspiration. It's well worth the day trip; I recommend it if one has an extra day in Paris.
Biggest score: I just found a Birkin and a classic '70s Gucci bag for $24 each. I'm not going to tell you at which Goodwill I found them, because then I'd have to kill you.
LAINY HEDAYA is a model, writer, brand consultant, and the founder/creative director of the New York City fashion blog the Haute Inhabit, which she launched in 2011. She's worked with a range of fashion- and style-focused brands, including Chanel, Tiffany & Co., H&M, and W Hotels, and has appeared in the pages of fashion magazines (Marie Claire, Elle, and W, to name but a few).
Biggest score: A J. Mendel gown for $100 at a sample sale. It had a bloodstain on it.
Words of wisdom:
Unless you know you're going to wear it until it's destroyed, it's not worth a penny to you. Clothes are meant to be a part of your memories, not stuck in a closet collecting dust.
BRETT HEYMAN is the founder of Edie Parker, a brand of vintage-inspired clutches and handbags based in New York City. The aesthetic of the brand and its signature acrylic clutches, which have been carried by nearly every star in Hollywood, sprang from Heyman's love of midcentury style.
MARA HOFFMAN is a New York–based fashion designer who launched her namesake label in 2008 with a women's swimwear collection. In the years since, it has expanded to include a colorful kaleidoscopic range of colorful women's ready-to-wear, bridal, activewear, accessories, and a kids' collection.
Biggest score: My vintage Levi's boys jacket, from Narnia Vintage near my place in Williamsburg, that has drawings of all The Simpsons characters that (I assume) were done by a guy named Nick R. Ferrara, which is the name scribbled inside the jacket.
Words of wisdom:
Pay attention to where the things you're buying were made and what they were made from . . . and don't over-consume.
EMILY HOLT is a fashion and lifestyle writer turned retail entrepreneur based in San Francisco. She launched her latest project—a high-end boutique called Hero Shop—in 2016. She's also the creative director in residence for Fab.com, and her previous fashion industry experience includes a four-year stint as the fashion news editor at Vogue and a five-year run as the editor of W magazine's party pages.
Style influences: Sophia Loren and Monica Bellucci, two of those great Italian film stars who, to me, epitomize what it is to look like a woman. I can't say I dress like them except for the rare occasion, but I do think about them a lot, so the intention's there.
AURORA JAMES's fashion industry career included stints as a model and a creative consultant before she founded footwear brand Brother Vellies in 2013 with the goal of preserving the shoemaking craft in Africa. As the label's creative director, she travels to Africa every two to three months to work with artisans to develop the collection. She lives in Brooklyn.
Style inspiration: Donyale Luna was a magical and inspiring soul. I try to channel her creativity and spirit into my collections every season. I am also profoundly influenced by traditional African attire and the artisanal skills passed down through centuries of the people in my workshops in Africa.
Words of wisdom:
It's on sale for a reason.
ULLA JOHNSON is a Manhattan fashion designer who founded her namesake women's label in 2000 just out of college. Celebrity fans of the brand include Jennifer Lawrence, Mila Kunis, Kate Hudson, Anne Hathaway, Amber Heard, and Kate Bosworth. Johnson lives in Brooklyn.
Style influence: Georgia O'Keeffe.
Biggest score: An antique hand-loomed Moroccan vest covered in sequins and fringe and silk tassels that I picked up in Marrakech. It's just beyond.
MINYEONG KANG is the creative manager of the Korean eyewear brand Gentle Monster. The brand has already amassed a huge following in Asia, but is quickly finding fans in the US. The flagship stores feature a gallery showcasing the work of local artists, which Kim herself curates.
Style influence: The Beatles.
SHALINI KASLIWAL is the president and CEO of Sanjay Kasliwal Jaipur-New York, the Stateside outpost of India's famed jewelry business, making her the ninth generation of the Kasliwal family involved in the gem trade, a lineage that can be traced back to the crafting of megawatt bling for the maharajas of the Mughal empire.
Style influence: Carolina Herrera is so very elegant. She visits my shop and I love what she is wearing every single time.
Words of wisdom:
Don't think too hard about something that makes you happy.
RAVEN KAUFFMAN is a luxury lifestyle consultant and accessories designer who launched her namesake line of fantastical clutches, handbags, and jewelry in Los Angeles in 2007. Fans Jennifer Lopez and Sarah Jessica Parker, among others, love Kauffman's one-of-a-kind custom pieces and limited-edition series, which incorporate all things organic, from peacock feathers to laser-cut metal snakes on clutches.
Biggest score: Paul Smith used to have a beautifully curated collection of vintage in his Los Angeles store. I got the most divine 1930s brocade tapestry coat with Middle Eastern motifs in gold bullion. I adore this coat—it instantly makes me feel like an eclectic marchesa, even when I'm wearing it with jeans.
ELIZABETH KENNEDY launched her namesake exclusive-to-Bergdorf-Goodman luxury collection of couture, ready-to-wear, and eveningwear in 2011 after a decade-long fashion industry career that included stints at Isaac Mizrahi, J. Mendel, Donna Karan, and Max Mara. Her list of A-list clientele includes Mariah Carey, Molly Sims, Sharon Osbourne, and Mindy Kaling, the latter of whom wore one of Kennedy's gowns on the 2016 Academy Awards red carpet. The collection is produced in New York City's Garment District.
Style influence: For personal dress, the '90s; as an eveningwear designer, the '50s and '60s and Charles James.
Biggest score: A squash blossom necklace I found on eBay.
HILLARY KERR is the cofounder and co-CEO of content and commerce company Clique Media Group, parent of the celebrity-focused fashion and shopping website Who What Wear, which she and Katherine Power launched in 2006. The site was so successful that in 2013, the duo added sister sites focused on beauty and home (Byrdie.com and MyDomaine.com, respectively). Kerr is also co-author (with Power) of three books, most recently The Career Code: Must-Know Rules for a Strategic, Stylish, and Self-Made Career (2016).
Words of wisdom:
Know your measurements. I do 95 percent of my shopping online and it makes all the difference if you can find the garment's measurements.
IRENE KIM is a Seattle-born, Seoul-raised, FIT-educated model, YouTube sensation, TV star, and social media standout who has served as a global beauty contributor for the Estée Lauder cosmetics brand since 2015.
OLIVIA KIM is the Seattle-based vice president of creative projects for Nordstrom, where, since 2013, she's been switching up the traditional brick-and-mortar shopping experience by staging a series of themed, curated pop-up shops. Prior to joining Nordstrom, she spent a decade at Opening Ceremony.
Style influences: Esprit, Benetton, Generra, Contempo Casuals, anything '80s that moves into early-'90s grunge.
Biggest score: My favorite pair of vintage jeans from Metropolis on Third Avenue in NYC. On my way out, I saw the jeans hanging there and was all, "WHOA WHOA WHOA. What are those?" I had never dreamed that a pair of vintage jeans could look like those did. After I saw them, I couldn't imagine a day in my life without them.
DESIREE KOHAN opened the doors of her namesake Los Angeles boutique in 2005. In the decade since, the converted 1920s Miracle Mile warehouse on Cloverdale Avenue has become a must-visit retail destination for the stylish set thanks to its highly curated assortment of apparel, footwear, accessories, gifts, and home goods representing established luxury brands and up-and-coming designers, as well as a range of vintage Chanel, Gucci, and Hermès pieces.
GEORGE KOTSIOPOULOUS is a Los Angeles–based fashion editor, consultant, stylist, and former co-host of E!'s Fashion Police. He's also the author of Glamorous by George: The Key to Creating Movie-Star Style (2014).
Style influence: Late '50s, early '60s, when Steve McQueen ruled with his style and swagger.
Words of wisdom:
If it's clothing, one should ask, "Where am I wearing this?" If there is no answer, then don't buy it. If it's furniture or décor, one should ask, "Where am I putting this?" Again, if there's no answer, don't buy it.
FARAN KRENTCIL is a contributing editor at ELLE.com and Yahoo News. She is the founding editor of Fashionista.com and served as Nylon's first-ever digital director. In 2015, she illustrated her first book, The Craft, by rock stylist Lou Teasdale.
GRACE LAM is a Hong Kong–based fashion director, stylist, and consultant. In 2005, she moved to Shanghai as part of the team that launched Vogue China.
Style influence: I still like singer Faye Wong's style. She has a model's figure, therefore she can carry lots of different and diverse styles. She is not afraid to try new looks. She doesn't care what other people think of her, which is extremely rare in Chinese entertainment culture.
Words of wisdom:
Don't wear heels unless you can walk gracefully in them.
PEARL LAM is a Hong Kong–born art dealer and collector and is the owner of Pearl Lam Galleries. A pioneer in the Chinese art world, Lam opened her first permanent gallery space in Shanghai in 2005, in the early years of its rise as a center for contemporary art. Focusing on design, Lam used the gallery to introduce Chinese collectors to the international art market. Lam also manages a design workshop that produces custom furnishings and lighting for her galleries and apartments. She lives between Hong Kong, Shanghai, and London.
OLYMPIA LE-TAN founded her Paris-based namesake collection of storybook-inspired handbags and minaudière in 2009 and expanded to include women's ready-to-wear in 2013. Her collaborative projects include key chains for Uniqlo (which was also a partnership with the United Nations' refugee agency UNHCR), candles with Diptyque, and a range of clutch purses featuring the artwork of Keith Haring.
Style inspiration: Bettie Page always and forever.
Biggest score: I once bought a pink-and-white Alaïa coat with the Tati print from Etsy, it was super cheap and in perfect condition. I love it and wear it all the time.
DION LEE is a designer hailing from Sydney, Australia, and launched his eponymous label in 2009. Known for sensual silhouettes, dissected to enhance the body's movement and shape, Lee's designs have been worn by Selena Gomez, Miranda Kerr, Cara Delevingne, and many other celebrities. After debuting at Australia Fashion Week, Lee presented collections in London before relocating to New York, where the brand has become a fixture on the seasonal schedule.
LORI LEVEN is a veteran retailer and the director/designer of Love Adorned, a fine jewelry and lifestyle store that focuses on vintage, one-of-a-kind pieces and artisanal goods informed by years of traveling and collecting. It has attracted a celebrity following that includes Leonardo DiCaprio, Christy Turlington, and Paul McCartney.
Style inspiration: Carol Beckwith and Angela Fisher's "African Wedding Rituals," published in 1999 by National Geographic. These ladies' photos influenced me to open New York Adorned.
Words of wisdom:
Always ask if there are other things not on the sales floor.
ALIZA LICHT is a fashion industry veteran and founder/president of Leave Your Mark, where she works as a strategic consultant to brands at the intersection of fashion and technology. Her fashion CV includes serving as the longtime SVP of global communications for Donna Karan International (where she created the popular DKNY PR GIRL social media presence). She's also the author of the bestselling book Leave Your Mark: Land Your Dream Job. Kill It in Your Career. Rock Social Media (2015).
Style inspiration: A fashion editorial styled by Carlyne Cerf de Dudzeele for American Vogue. She had supermodels in Chanel ball gowns on motorcycles. I love the juxtaposition of feminine and tough.
Words of wisdom:
Don't buy for a future size.
ANDREA LIEBERMAN was born and raised in New York City and her career has included styling stars like Gwen Stefani and Jennifer Lopez (including the memorable plunging green Versace number Lopez wore to the 2000 Grammys). She launched her L.A.-based A.L.C. women's clothing label in 2009.
Style inspiration: The iconic images of Veruschka in the desert styled by Giorgio di Sant'Angelo. They were hanging on the wall of his studio when I worked there in my first job out of college.
Words of wisdom:
Never try to rationalize an emotional purchase. If you are completely swept away by something and it brings you joy, go for it!
CHRISELLE LIM is a stylist, digital influencer, and founder/creative director of Chriselle. In 2011, she launched the Chriselle Factor, a blog and YouTube channel where she chronicles her personal style. The luxury brands she's worked with include Dior, Gucci, Fendi, Louis Vuitton, Valentino, Cartier, Elie Saab, Miu Miu, Maison Margiela, and Stella McCartney. Her influence—and more than 2.2 million social media followers—has earned her the title of one of Fashionista.com's eight most successful bloggers for 2016.
Words of wisdom:
Invest in the classic pieces and experiment with trends wisely.
CHLOE LONSDALE is the founder and chief creative officer of London-based M.i.h. jeans, which launched in 2006 and has celebrity fans in Rachel Weisz, Jennifer Lawrence, and Julianne Moore. Lonsdale's parents were influential in the denim revolution of the '70s, which affords her the ability to draw on a personal design archive that spans nearly half a century.
Words of wisdom:
Pick things up on your travels. It's far more special to have something unique to a place and time, to a memory, than something that was bought in an anonymous environment. It's always worth it. I drove home with an eight-foot surfboard from Cap Ferret last weekend!
MELISSA MAGSAYSAY is a Los Angeles–based style and beauty writer and creative consultant whose work currently appears in a range of publications, including the Hollywood Reporter and the Los Angeles Times. She is also author of City of Style: Exploring Los Angeles Fashion From Bohemian to Rock (2012).
Words of wisdom:
Never shop while in a hurry or after having consumed more than two glasses of wine.
MARYAM MALAKPOUR is an Iranian-born, Los Angeles–based fashion editor, stylist, and shoe designer who has worked with a celebrity set that includes Rolling Stones frontman Mick Jagger (as well as the band itself for several world tours) In 2009 she and sister stylist Marjan Malakpour launched the minimalist footwear line Newbark.
Biggest score: At Scout LA I found three Valentino couture silk blouses [in] beautiful prints. All three were the perfect size, no alterations needed.
Words of wisdom:
If you're not having fun, then you're doing it wrong.
MARY ALICE MALONE is the creative director and ROY LUWOLT is the managing director of the London-based women's luxury shoe brand Malone Souliers, which they cofounded in 2014. Since then, their shoes, handmade in Italy, have shod the feet of such famous folks as Lupita Nyong'o, Solange Knowles, Kylie Jenner, and Kim Kardashian, who was spotted sporting a pair during her honeymoon.
Words of wisdom:
Fussy, uncomfortable, or wrong for you should never make the cut.
FERN MALLIS is a New York–based fashion consultant, author and the creator/organizer of New York Fashion Week during her decade-long tenure as executive director of the Council of Fashion Designers of America (CFDA) from 1991 to 2001. She has also served as senior vice president of fashion for IMG and is currently president of her own international consulting firm. Her book Fashion Lives: Fashion Icons with Fern Mallis, was published in 2015.
Words of wisdom:
The last thing you want is to spend years regretting not buying something.
BESTE AND MERVE MANASTIR, daughters of a leather craftsman, founded their handbag label Manu Atelier in Istanbul in February 2014. Their distinctive designs, handmade in Turkey, including the rectangular Pristine bag with a flap top, have been featured in Vogue, Business of Fashion, and Women's Wear Daily, and on the arms of such fashionable women as Eva Chen and Bella Hadid. Manu Atelier is available at Net-a-Porter, Selfridges, and other stores around the world, as well as at ManuAtelier.com.
SHEA MARIE is a globe-trotting fashion influencer, designer, stylist, creative consultant, and editor of PeaceLoveShea.com. Her website and various social media accounts generate more than 8 million impressions a day. She's appeared in the pages of fashion glossies including Glamour, Marie Claire, Harper's Bazaar, and Vogue. In 2015 she launched the luxury swimwear label Same Swim.
Biggest score: I found a pair of zippered Balmain booties in a teeny vintage store in London for a couple hundred dollars. They had been worn maybe once. Definitely the best vintage piece I ever found for the price.
Words of wisdom:
Work hard so you can shop harder.
FIONA KOTUR MARIN launched her accessories label Kotur in 2005. Celebrities seen clutching Kotur clutches over the years include Iman, Ginnifer Goodwin, Jennifer Lopez, Emma Roberts, Kendall Jenner, and Selena Gomez. Martin splits her time between New York City—where she was born and raised—and Hong Kong where her business is based.
Biggest score: When I was a student visiting Paris's Porte de Clignancourt flea market, I found a beautiful vintage leather doctor's bag that I vowed would be an integral part of my collection should I start a handbag company—and it has been!
JJ MARTIN is the founder and editor of the shoppable online magazine and creative agency LaDoubleJ.com. The site sells vintage clothing, makes new clothing with vintage prints from hidden Italian archives, promotes Italian talent and style, and wraps it all up in a colorful, fun, humorous package. Martin grew up in Los Angeles and now lives in Milan, where she's one of the city's most stylish women. She has been chronicling the fashion and design industries for more than fifteen years as an editor and writer at publications including Harper's Bazaar and Wallpaper magazine, where she is currently editor at large, Italy.
VIRGINIA MARTIN's passion for clothing was nurtured by her mother, who taught her to sew. After finishing high school Virginia threw herself into design, and at nineteen years old, stores throughout Australia took up her first collection. Virginia eventually took a break and moved to New York, further growing as a designer while interning at Proenza Schouler, Cynthia Rowley, and Heatherette. She returned to Melbourne eager to establish her own label. The ocean and the coast acted as inspiration and motivation for launching her line of sculptural basics, búl, in 2010.
KARLA MARTINEZ SALAS is a longtime fashion editor who currently serves as editorial director at Vogue Mexico and Latin America. She grew up in El Paso, Texas, spent a dozen years in New York City, and currently lives in Mexico City. In addition, she's the cofounder of luxury loungewear label Piamita (with designer Cecilia de Sola), which launched in 2011.
Style influence: Dolores Guinness and her old-world glamour.
Biggest score: Mink coat by Revillon from the '70s from a store in El Paso called Vintage Mode.
ANN MASHBURN, a former assistant and editor at Vogue and Glamour who, along with her equally fashionable husband, Sid, opened their first Sid Mashburn men's store in Atlanta in 2007, followed by an Ann Mashburn women's store in 2010. Today they sell apparel and accessories bearing their names at four sets of connected men's and women's shops in Atlanta, Houston, Dallas, and Washington, DC as well as at a stand-alone Sid Mashburn store in Los Angeles. She lives in Atlanta.
Style influence: Every single photo of Jackie Kennedy wearing short, flared white jeans and a black crewneck T-shirt. I wore this outfit for like ten years straight as a young housewife in paradise.
Words of wisdom:
If you really love it, buy a backup.
BRANDON MAXWELL is a stylist turned designer who earned early name recognition for his work with Lady Gaga, for whom he's been fashion director since 2012. He launched his namesake collection of luxury eveningwear in 2015, which finds favor with Jennifer Lawrence, Reese Witherspoon, Kate Hudson, Uma Thurman, and Michelle Obama. His industry recognitions include the Fashion Group International Rising Star award for Womenswear and the CFDA Swarovski award for Womenswear, both in 2016.
Style influence: Everything Jackie Kennedy, always.
Words of wisdom:
If the shoe fits, buy it in every color.
PATRICK MCDONALD's sartorial flair—spotted in the front rows of New York Fashion Week for decades—has earned him a reputation as "the Dandy of New York" as well as countless street-style photo ops. His adventures in high style include modeling and stints at Fiorucci and Barneys as well as penning Paper magazine's Highbrow column for several years. Formerly a fixture on the New York fashion scene, he currently hangs his distinctive top hat in San Francisco.
Style influence: Silent movies from the 1920s with Rudolph Valentino.
Words of wisdom:
Fashion finds are everywhere—explore, find, create your own style. Be original!
LEANDRA MEDINE is the founder of the Man Repeller, a fashion-meets-humor website launched in 2010. It has since grown into a multimedia business that has earned Medine spots on all kinds of power lists including Forbes' "Top 30 Under 30," Time's "25 best blogs," and Adweek's Fashion Power 25." In 2013 she published her first book Man Repeller: Seeking Love, Finding Overalls. She lives in New York City.
Biggest score: I got a yellow Valentino military jacket for $2 from an Outnet.com sale. When they were just starting, they used to host these ridiculous $1 and $2 sales once a year and I scored access to the $2 sale, et voilà.
Style influence: A photograph of Jane Birkin in the South of France holding a straw basket and wearing a T-shirt and high-rise jeans that hit just below the ankle—this photo is an important reference for my frame of mind. No matter the time of year, or what I'm doing, I always want to feel like I'm carrying a straw basket in the South of France.
Words of wisdom:
If you wouldn't buy it at retail, don't buy it on sale.
JENNIFER MEYER is a Los Angeles native who worked in communications for high-profile fashion brands including Ralph Lauren and Giorgio Armani before launching her namesake jewelry collection in 2005. In 2012 she was awarded runner-up in the CFDA/Vogue Fashion Fund; the following year she was nominated for the CFDA's Swarovski Award for Accessories Design. Her delicate, personalized pendants have become a Hollywood staple, spotted on the likes of Amy Adams, Claire Danes, Alicia Vikander, Kerry Washington, and Emma Roberts.
Style influence: Jane Birkin really paved the way for the chic casual look that works so well living in California. Diane von Furstenburg has also been an inspiring style icon for me, in every way.
ELIZABETH MINETT is a Canadian model turned lifestyle influencer and creator of the website HautAppetit.com—a guide to satisfy an "It" girl's appetite for life—which she launched in 2011. A frequent fixture at fashion weeks around the globe, she has collaborated with a range of international fashion and beauty brands. She splits her time between Los Angeles and Paris.
Style influence: Masculine power dressing from the '80s (think Grace Jones), juxtaposed with the feminine bombshell vibe of the '60s (think, Brigitte Bardot).
Words of wisdom:
Retail therapy does, in fact, mend a heavy heart.
REBECCA MINKOFF is a New York–based fashion designer who launched her namesake collection in 2005. Today she helms a global lifestyle brand that also includes footwear and jewelry as well as men's clothing and accessories under the Uri Minkoff label (named after her label cofounder and brother), sold globally through 900 retail doors as well as stand-alone stores in the US and internationally. Her industry recognitions include the 2011 Breakthrough Designer Award bestowed by the Accessories Council.
Style influence: Anything Bianca Jagger wore at Studio 54.
Words of wisdom:
My best pieces are the ones that made me a little uncomfortable. Push yourself into something uncomfortable.
TANIA MOHAN is a Hong Kong–based fashion expert who has retail in her blood—her family founded the Mohan's department stores in India in the 1950s—who served as the first fashion editor of the Hong Kong Standard newspaper in the 1990s. In 1999 she launched Tabla, an India-inspired luxury brand.
Style influence: The regal elegance and grace of Maharani Gayatri Devi fused with the timeless essence of Coco Chanel. Indian majestic regal charm, polo, tiaras, maharanis, and jodhpurs.
Biggest score: My first designer bag, the Louis Vuitton Speedy, was purchased during my first international trip to Paris. The bag is forever linked to my memories of sitting in Les Deux Magots relishing in all the beauty and romance of the City of Light. The bag is now almost twenty years old, I use it regularly.
MARYAM MONTAGUE is about as much of a global citizen as you can be: born in Cairo, raised in Tunis and New York, she did humanitarian work in Dakar, Kathmandu, and Windhoek before alighting in Marrakech, Morocco, where she and her husband designed, built, and decorated a boutique hotel, Peacock Pavilions, that's been the backdrop for fashion shoots for a range of brands, including J. Crew and Tory Burch. Her blog, My Marrakesh, launched in 2007, led to the creation of the M. Montague Souk, an online shop focusing on tribal-inspired and Moroccan home décor. She's also the author of Marrakesh by Design (2012).
Biggest score: I was in Yemen on a humanitarian aid assignment, when I found in the souk an antique hand-embroidered dress, once worn by a Yemeni woman. More than 100 years old and less than $199, the dress fits like it was made for me.
CLAUDE MORAIS AND BRIAN WOLK are the globe-trotting fashion designers behind the L.A.-based Wolk Morais fashion label. Before launching their namesake label in 2015, they made a name for themselves with the critically acclaimed New York cult label Ruffian.
Biggest score: A Napoléon-era letter-carrying bag (portefeuille) from the legendary (and since-shuttered) Old England department store in Paris.
JENNIFER MORRISON is an actress/director/producer/model and lover of fashion best known for her roles as Dr. Allison Cameron on House, and as Emma Swan on the adventure fantasy Once Upon a Time. She catalogs the brands she wears and riffs on the labels she likes via the website jennifermorrisonstyle.com. Her directorial feature film debut, Sun Dogs, is scheduled for a 2017 release.
SASHA CHARNIN MORRISON is a freelance stylist and editor but is probably best known for a nine-year run as the fashion director at Us Weekly magazine (2006–2015). She is also the author of Secrets of Stylists: An Insider's Guide to Styling the Stars. She is based in New York City.
Style influences: My parents, Genii and Martin Charnin—they are the bomb and my first stylists and style icons. We wore the greatest looks—yes, we wore "looks" as a family. I also love Theda Bara. Look at her. Her style in 1917 was so ahead of its time. The hair, the lined eyes, and the clothing.
Words of wisdom:
Buy the shoes.
MINNIE MORTIMER is a New York City to SoCal transplant and cofounder of the members-only cosmetics website Vainité, which launched in 2016. Her other adventures in the fashion space include launching a namesake women's sportswear line in 2009, a 2010 collaboration with W Hotels, a 2013 capsule collection with Three Dots, and a consultation/collaboration partnership with Boast USA.
MARRIAM MOSSALLI is the Jeddah, Saudi Arabia–based founder of luxury consultancy firm Niche Arabia, which she launched in 2011, and has a client list that includes Burberry, Versace, and Prada. That's the same year she launched her blog, Shoes & Drama. Her fashion CV includes stints as executive editor at Style.com Arabia, fashion editor at Destination Jeddah, and editor-in-chief at Design magazine.
Style influence: Catherine Deneuve—I was lucky enough to be on set with her in Malaysia (my mom and sister both had small parts in the 1992 film IndoChine) and I remember being memorized by her elegance and dominance. She was assertive, unfriendly, and unapologetic. It was her confidence and diva-ness that drew me in, and I memorized everything about her. Her wardrobe on set was very Coco Chanel and Jeanne Lanvin. Off set, she wore crisp white men's shirts with long maxi dresses, her hair in a messy bun. I recall seeing her Louis Vuitton trunks (which encouraged me to call dibs on all of my mother's trunks and luggage as soon as we returned home).
Words of wisdom:
"Charge it!" I remember watching Troop Beverly Hills (1989), and there was this scene in the beginning where Shelley Long is shoe shopping and says, "That one, and that one . . . out! I'll take the rest!" I would mimic that scene over and over. As a result I was the most obnoxious five-year-old ever!
KATE AND LAURA MULLEAVY are the cofounding sisters of the Los Angeles–based luxury women's label Rodarte, which they launched in 2005. Their collections have earned them numerous industry recognitions including multiple CFDA award nominations and wins. The costumes they designed for the film Black Swan earned them a Broadcast Film Critics award nomination. Enthusiastic collaborators, they've worked on special projects with Frank Gehry, Gustavo Dudamel, Benjamin Millepied, Target, and the Gap among others. Their pieces are in the permanent collections of the Costume Institute of the Metropolitan Museum of Art, and the Los Angeles County Museum of Art, among others.
IRENE NEUWIRTH is a Los Angeles–based jewelry designer who launched her namesake collection of fine jewelry in 2003. Her industry recognitions include the CFDA's Swarovski Award for Accessory Design in 2014 and her list of celebrity clientele includes Reese Witherspoon, Claire Danes, and Scarlett Johansson. She opened a stand-alone flagship boutique on Melrose Place in West Hollywood in 2014.
Style influence: Frida Kahlo, need I say more?
Words of wisdom:
Invest in something you know is colorful, fun, unusual, and will separate you from everyone else.
PAOLO NIEDDU is the two-time-Emmy-nominated costume designer for the television show Empire. He began his costume design career as an assistant costume designer to Pat Field on the 2008 big-screen version of Sex and the City, and his other screen credits include Sex and the City 2 and Confes-sions of a Shopaholic. He grew up in the suburbs of Detroit and currently calls the New York City area home.
Style influence: I'm obsessed with the book Those Glorious Glamour Years: Classic Hollywood Costume Design of the 1930s, by Margret J. Bailey. It inspired me as a kid and still does today.
Biggest score: My superthin Lanvin jacket. I saw it in the Paris men's store—I didn't buy it 'cause it was 2,700 euros. I saw it in Bergdorf's later that season and it was on sale for $1,000, but they didn't have my size. The salesperson took my info and believe it or not, someone returned my size!
LENNY NIEMEYER is a Rio de Janeiro–based landscape architect turned swimwear designer who launched her namesake label in 1993 out of an Ipanema storefront, and went on to become Brazil's bikini queen—now her collection is stocked in stores around the globe. Among the high-profile folks who've worn her wares are Nicole Kidman and Lady Gaga.
Words of wisdom:
Learn how to appreciate a piece and not necessarily consume it.
THITHI NTETA is the creator of the popular style blog TeeTeeIsWithMe.com. Nteta works as a freelance stylist, consultant, publicist, and graphic designer in Cape Town, South Africa. She posts regularly about fashion, beauty, lifestyle, and events in South Africa and beyond.
MONIQUE PÉAN is a former investment banker turned fine jewelry designer who launched her namesake, socially responsible and sustainable fine jewelry line in 2006. She has guest lectured at Harvard Business School, the University of Pennsylvania, the Smithsonian Institution and Parsons the New School for Design. She is based in New York City.
ERICA PELOSINI is a jet-setting, globe-trotting, Florence-born model, stylist, art director, and contributor to Vogue and L'Officiel Paris. She is also the cofounder (in 2012) of the shoe business that bears the name of her husband, Louis Leeman. A recent Paris-to-Los Angeles transplant, she can frequently be found at fashion weeks around the world.
Style influence: Marchesa Casati was an early-20th-century Italian muse and patron of the arts. It's always intrigued me how she used to walk through the streets of Venice with her cheetahs and their diamond-studded leashes wearing her furs and piled on jewelry.
Biggest score: My monogrammed Louis Vuitton luggage.
SALLY PERRIN is a Minnesota-born, Seattle-raised model turned handbag designer for the French luxury leather goods brand Perrin Paris, which dates to 1893. The brand has stand-alone stores in New York, Beverly Hills, Hong Kong, Kuwait, and Paris, the last of which is also home to the design studio and showroom. The Perrins (her husband, Michel, is the family-owned company's CEO) call Los Angeles home.
Biggest score: A black tulle Yves Saint Laurent dress from the '80s with bright color pom-poms. I mentioned to vintage guru Cameron Silver of Decades at a dinner an article I had read about him in Elle, with a photo of this very dress, and how it caught my eye. The very next day he had it delivered to my house. It fit as if it were custom-made for me. I have two daughters, they will have to work out a time share.
JOSH PESKOWITZ is the cofounder of Magasin, a multi-brand men's concept store in Culver City, California that opened its doors in 2016. His style-centric CV also includes stints as men's fashion director at Bloomingdale's, men's style director for Gilt Groupe, fashion editor for men.style.com and market editor for Vibe magazine. He currently hangs his knit cap in Los Angeles.
Biggest score: Probably the thing that I've worn the most over the years is a cable-knit shawl collared cardigan from RRL. I found it at the Polo Ralph Lauren outlet in Las Vegas (hey, seemed better than losing money at the craps table). This was in, like, 2004, so no one in Las Vegas was checking for RRL, and I got the sweater for $18. Retail on something like that is north of $700, so I was feeling like I was the only person to beat the house in Sin City that day.
Words of wisdom:
If it doesn't fit in the store, it's never going to. Keep it moving.
Also, if you're still thinking about it thirty minutes later, go back and buy it. You may never see it again. Especially if you're traveling. Which brings me to my most important point: Unless the deal is so good you can't pass it up, never buy something abroad that you could buy at home. My clothing is sort of like travel slide show, and I'm far more interested in the local market than the mall. Leave extra room in your suitcase and budget to go shopping a little bit. Nothing's worse than showing up back at the hotel room from Istanbul's Grand Bazaar and realizing you've got no room for that robe you just bought.
ARIANNE PHILLIPS is a stylist whose client list includes Madonna, Lenny Kravitz, Courtney Love, and Justin Timberlake, and a costume designer whose film work earned her a pair of Academy Award nominations for Walk the Line and W.E. She also designed costumes for Tom Ford's two films, A Single Man and Nocturnal Animals.
Style influence: Does film history count? Does Yentl count? David Bowie, Charlotte Rampling in The Night Porter, the Duke and Duchess of Windsor, Sally Bowles from Cabaret, Amelia Earhart, Peggy Guggenheim, Lee Miller, Marie Antoinette, and more.
Biggest score: I found a rare 1960s Rudi Gernreich dress in perfect condition at a thrift store in St. Louis for $3 and sold it at Decades for $1,200.
KATHERINE POWER (see Hillary Kerr).
JEN RADE is a costume designer, stylist, and self-described "wicked witch of the wardrobe" whose list of A-list clientele stretches nearly as long as an awards-show red carpet and includes Angelina Jolie, Abbie Cornish, Cher, Jenna Fischer, Pink, Marilyn Manson, Dave Matthews, and Tina Turner. Her industry recognitions include a pair of Costume Designers Guild awards for excellence in commercial costume design for her work with Apple and Target.
Words of wisdom:
Something has to grab you.
NICOLE RICHIE is a Los Angeles–based former television star turned designer whose House of Harlow 1960 brand launched in 2009 as a collection of jewelry, expanding over the years to include ready-to-wear, eyewear, footwear, and handbags, sold at high-end department and specialty stores around the world.
Style inspiration: I love moon manicures from the 1920s, the dark lipstick and glamour of the 1940s, the freedom of the 1960s and '70s. The '80s . . . not so into the '80s.
Biggest score: This year it was a vintage YSL oversized coat. I found it at Shabon, one of my favorite vintage stores in L.A.
LOUISE ROE is a model, digital influencer, writer, and television host whose fashion-focused résumé includes serving as the news editor for Vogue.com in London, writing for Elle UK, InStyle UK, and People Stylewatch. Her first nonfiction book, Front Roe: How to Be the Leading Lady in Your Own Life, was published in 2015.
Biggest score: A bright-red printed Gucci silk scarf from the '80s from Ebth.com (Everything but the House). They auction estate sales online and have an incredible selection.
Words of wisdom:
Imagine it worked into three outfits you already own before buying it. Then you know you'll wear it.
JENN ROGIEN is a Brooklyn-based costume designer whose work can be seen on Netflix's Orange is the New Black, and HBO's Girls, the latter of which earned her a 2013 Costume Designers Guild Award nomination. She was also nominated for an Emmy for her work on The Good Wife. In addition, she's collaborated with a range of brands and retailers including Sorel Footwear, TJ Maxx, the Gap, and Aerie Lingerie.
Words of wisdom:
Try. It. On. Hanger appeal is a real thing. Some garments are a disaster on the hanger but will be the best thing you've ever had on. Some things look great on a hanger and are an utter disaster once you've got them on.
BRIGETTE ROMANEK is an L.A.-based former actress/singer turned handbag designer turned interior designer. Her line of luxury handbags has graced the shelves of Barneys, the pages of Vogue, and the arms of celebrities like Kate Moss and Gwyneth Paltrow.
Style influence: My style icon hands down, is Nina Simone. Almost any image I find of her is strong and powerful. She always looked like a warrior. She was an African American doing bold things during a time that wasn't easy. She's my icon on many levels.
Words of wisdom:
Know yourself, what makes someone else look cool, may not be the look for you.
CHARLOTTE RONSON is a London-born, New York City–raised designer whose fashion career began with the launch of C. Ronson women's apparel in 2000, a label that would evolve, five years later, into the Charlotte Ronson brand. A frequent collaborator, she's partnered with a range of brands including Urban Outfitters, Uniqlo, JC Penney, Sephora, Luxottica, and even Starbucks.
Biggest score: A Hermès Kelly bag, it's less a shopping score and more of a "lucky me" loving gift score from my mother and stepfather. But, I am too scared to wear it, so it remains in the box in my closet.
Words of wisdom:
In the words of my mother: "Do you love it? Now, are you sure you love it?"
SHIVA ROSE is an actress and activist whose holistic lifestyle blog, The Local Rose, eventually resulted in the launch of a namesake line of toxic-free products and eventually expanding to include body scrubs, eye creams, body oils, and candles. She lives in Pacific Palisades.
Style influences: Georgia O'Keeffe and Frida Kahlo, because they were able to look chic and original no matter what the style of the day was.
Words of wisdom:
For me, dressing is a bit like storytelling. I feel I can be whatever character I want in whatever script I choose.
AMANDA ROSS is a fashion director, stylist, costume designer, and consultant who muses on all things life and style through the website ARossGirl.com, which also serves as a platform for see now, buy now seasonal designer collaborations. Her prior fashion industry CV includes serving as the fashion director for Departures magazine, fashion market director for Harper's Bazaar, and global fashion director to W Hotels. She is currently based in New York City.
Style influence: My favorite style icons are my grandmothers and my mother and stepmother.
Words of wisdom:
Shop to explore. Be curious. Only buy what you love.
KARA ROSS is a New York–based jewelry designer whose latest project is Diamonds Unleashed by Kara Ross, a jewelry brand with the goal of promoting and supporting women's empowerment that launched in 2015. She launched her namesake jewelry collection, Kara Ross New York, in 2003 with fine jewelry that incorporated materials like jet, lava, wood, and titanium, later expanding to include handbags made from exotic skins. Industry recognitions include Fashion Group International's "Rising Star" Award for Jewelry Design in 2008 and the GEM Award for Design from the Jewelry Information Center in 2014.
Words of wisdom:
You need to love it, not just like it.
CATHERINE RYU is the South Korea–born, Canadian-raised, L.A.-based creative director for the Citizens of Humanity brand, which she joined in 2012 after design stints at AG Adriano Goldschmeid, Gap, Calvin Klein, Urban Outfitters, and Club Monaco.
Style influence: Françoise Hardy and Carolyn Besette Kennedy.
LIANA SATENSTEIN is a news writer at Vogue.com, where she specializes in the Eastern European and Central Asian markets, as well as tracksuits (for real). She fell in love with all things Eastern European while on a high school exchange program in the Ukraine. She has written for Vogue, Vogue Ukraine, British Vogue, Elle.com, and Fashionista, and is based in New York City.
LEE SAVAGE is a Savannah-to-New York City transplant who transitioned from a career in interior design to handbag design with the launch of her namesake line of art-and-architecture-inspired clutches in 2013. Designed in New York and manufactured in Italy, her bags have become a go-to red carpet accessory found in the hands of famous ladies including Naomi Watts, Julie Bowen, Kate Hudson, Diane Kruger, and Jessica Alba.
Style influence: Lee Radziwill. One image of her that stands out is a Mark Shaw photograph from the 1960s in which she is in a mustard yellow Nina Ricci gown and blue cape. It's such a stunning simple gown, but impeccably tailored and chic.
Words of wisdom:
Always ask, "How long do I see myself wearing this?"
SARAH SCHUSSHEIM is a stylist, consultant, and contributing West Coast fashion editor at Elle magazine who has lent her considerable talents to the betterment of Penelope Cruz, Octavia Spencer, and Uma Thurman (all for Elle covers), and to glossy editorial spreads in Details, DuJour, Teen Vogue, and Vanity Fair. She lives and works between Los Angeles and New York City.
Style influence: I like the Town & Country look of Ralph Lauren, Céline Dion in Dior at the 1999 Oscars, Barbra Streisand's frumpy/sexy style in The Mirror Has Two Faces. Kanye's style over Kim for sure.
Words of wisdom:
If it feels right, it probably is.
JEREMY SCOTT is a Los Angeles–based fashion designer who launched his namesake line in Paris in 1997 and was quickly embraced by a celebrity set that included Björk, Madonna, Kylie Minogue, and Britney Spears and later, Miley Cyrus, Rihanna, Nicki Minaj, and Katy Perry. In 2013, he was tapped to serve as the creative director of Italian luxury label Moschino, injecting a new energy into the brand. He's the subject of the documentary film Jeremy Scott: The People's Designer, released in 2015—the same year he became the first fashion designer to be immortalized with a pair of handprints pressed into the wet cement in front of the TCL Chinese Theatre on Hollywood Boulevard.
Style influence: The film Blade Runner for fashion, architecture, and design. The mix of futuristic and vintage, the old and the starkly new mixing together to create a volatile cocktail.
Words of wisdom:
If you think about it for more than five minutes, you should probably just get it.
KENDRA SCOTT is the Austin, Texas–based designer and CEO of Kendra Scott Designs, the jewelry and accessories company, which she launched in 2002 with $500 and has since grown into a multi-million dollar business, known for drop earrings and Color Bars, which allow customers to customize designs.
Style inspiration: In 2005, I had the honor of meeting Oscar de la Renta and collaborating with him to design the jewelry for his Spring 2006 runway show. The detail and elegance with which Mr. de la Renta dressed women is an incredible inspiration to me.
Words of wisdom:
Every woman needs a great statement piece.
CHLOË SEVIGNY is an award-winning American film actress, fashion designer, and former model. Some of her most notable roles were in Kids (1995), Boys Don't Cry (1999), American Psycho (2000), and the HBO television series Big Love, for which she received a Golden Globe in 2010. Recently, she's had roles in Hit & Miss (2012) and American Horror Story: Hotel (2015–2016). She has designed several fashion collections with Opening Ceremony, and in April 2015 published a self-titled book about her style with Rizzoli.
VANESSA SEWARD was born in Argentina, and grew up in London before moving to Paris at age twelve. After studying at Studio Berçot, she worked for nine years as an accessory designer for Chanel and Tom Ford at Yves Saint Laurent before becoming second-in-command at the house of Loris Azzaro. Following Azzaro's death in 2003, Seward became artistic director. She left Azzaro in 2011, and for two years created capsule lines for French contemporary brand A.P.C. before launching namesake fashion label in 2014, which is known for its Charlie's Angels-meets-Parisian sense of cool.
DARIA SHAPOVALOVA is founder and creative director of Mercedes-Benz Kiev Fashion Days, founder of the Russian-language fashion website Fashion Week Daily (fw-daily.com), and an influential champion of up-and-coming designers in her native Kiev.
Biggest score: We could dedicate an entire book to my best scores. I have an amazing dress bought from the L.A. vintage store the Way We Wore, made by Karl Lagerfeld for Chloé. I was wearing it at a Chanel runway show in Paris, and was approached by the seamstress who made it. She remembered it from all those years ago.
Words of wisdom:
Work harder to shop harder.
SUSIE SHEFFMAN is a Toronto-based stylist and contributing fashion editor at Fashion magazine. The list of celebrities she's worked with over her three-decade career includes Ashley Olsen, Jennifer Hudson, Claire Danes, Taylor Swift, Britney Spears, and Rihanna. She's also served as fashion director for a range of brands, including Joe Fresh and Hudson's Bay.
Style influence: I'm a bit of a tomboy, so Steve McQueen and Paul Newman in American classics, and Pablo Picasso in a Breton shirt.
Biggest score: I became obsessed with finding a horn necklace like the one worn by Diana Vreeland. As I sat at Brioni's spring 2011 Milan show, there it was, around the necks of the models marching down the runway before me! I hunted down the sample and styled it as the signature piece on one of my favorite fashion magazine stories. To my delight, I was later gifted it by Brioni as a thank-you for the shoot. It's become my own signature and I'm rarely without it.
Words of wisdom:
If you love it, buy two!
And whether it's a tux, a watch, a brogue, or a cashmere V-neck, it's always better from the boy's department.
TADASHI SHOJI is the founder and chief designer of the women's apparel line that bears his name. His gowns have been worn on the red carpet by Florence Welch and Octavia Spencer, but the bulk of Shoji's business is in sales of tasteful, figure-flattering cocktail dresses and evening gowns for women who want to feel like celebrities in their own lives. He was born and raised in Sendai, Japan, moved to Los Angeles in the 1970s and launched his label there in 1982.
Style influence: Joan of Arc because of her iconic strength and dedication to what she believed in.
Biggest score: My Tibetan bamboo cuff that I got at the gem market in Taipei. The cuff has a historical importance in Taiwan and the legend is that it provides both safety and protection.
CAMERON SILVER is the co-owner of the Decades luxury vintage boutique he opened in Los Angeles in 1997. Since that time, he has enthusiastically worn, sold pieces from, consulted for, and partnered with enough big-name brands to stock a store sixteen times over. His book Decades: A Century of Fashion was published in 2012. The latest feather in the cap of the globe-trotting vintage clothing impresario came in 2015 when he was tapped to serve as the fashion director of the H by Halston and H Halston brands.
Biggest score: In the late 1990s, Paris flea markets and auctions were amazing and I have the Hermès collection to prove it!
Words of wisdom:
Would Cary Grant or Liberace wear this today?
TARA SOLOMON is a journalist-turned-PR guru and the founder of Tara, Ink., a boutique public relations, marketing, and special events firm in Miami Beach with clients including Dior, Chanel, Louis Vuitton, Vanity Fair, Art Basel, The Delano, Shore Club, STK, Soho Beach House, and W Hotels. In the 1990s, she chronicled Miami's nightlife scene as the Queen of the Night columnist at the Miami Herald, before going on to become a syndicated advice columnist for the publication.
TATIANA SOROKKO is a Russian-born, San Francisco–based model, fashion journalist, and collector of haute couture and antique jewelry whose treasures were featured in the 2010 Moscow Art Museum exhibition, "Extending the Runway: Tatiana Sorokko Style." Her fashion CV also includes styling photo shoots featuring the likes of Elizabeth Taylor and Joan Collins and serving as a contributing editor for Harper's Bazaar.
Style influence: Virginia Oldoini, Countess of Castiglione, is one of my most important style influences from history. La Castiglione, as she was known, was a mistress of Emperor Napoléon III and renowned for her immense beauty and elaborate court style.
Biggest score: A haute couture "sari" dress from Cristobal Balenciaga's spring 1964 collection, found at a vintage store in Paris when I was working there as a model. The brocaded lamé dress, with Lesage bead embroidery, was photographed by Richard Avedon for the October 1963 issue of Vogue. I had only ever seen one other, but it was in pale green, with a slightly different construction. In 2011, the Kerry Taylor Auction house in London held a sale of haute couture that included a version of my dress, owned by Elizabeth Taylor, which she wore to the New Review Lido premiere in Paris, in December 1964. Identical to mine in every way, the dress fetched £30,000.
Words of wisdom:
No cheap booze or cheap shoes.
LAURIE LYNN STARK is the co-owner and co-designer of the Los Angeles–based Chrome Hearts luxury line of jewelry, leather goods, and home furnishings that counts Cher, Steven Tyler, and Elton John among its clientele. She's also an accomplished photographer, art director, and creative director of photography of Chrome Hearts Magazine published in Japan and distributed internationally. She lives in New York and Los Angeles.
Words of wisdom:
Don't buy things just because everyone else is buying them.
SCOTT STERNBERG founded and served as creative director of the Los Angeles–based Band of Outsiders brand in 2003 and expanded it over the course of a decade to include complete men's and women's apparel and accessories collections and bricks-and-mortar retail stores in Tokyo and New York City. The cookie-and Polaroid-film obsessed designer left the label in 2015 and has since served as a much sought after consultant to a variety of brands and businesses in the fashion space.
ELIZABETH STEWART is a high-profile Hollywood stylist whose clients include Cate Blanchett, Jennifer Lawrence, Jessica Chastain, Julia Roberts, and Salma Hayek, to name just a few.
Style inspiration: The excess of Marie Antoinette. Also, I remember the first image that influenced me, a Vogue Patterns picture of a model in a wrap skirt on a tarmac. The wrap skirt was very easy and effortlessly glamorous. I felt the love!
ZAINAB SUMU is the Boston-based designer of the Primitive Modern label, which she launched in 2015 with a capsule collection of Mali-inspired, African-made scarves (singer Alicia Keys is a fan). Originally from Sierra Leone and educated in the UK, France, and the US, Sumu's fashion industry experience included a stint at Comme des Garçons.
TARA SWENNEN started her career working for Andrea Lieberman and Rachel Zoe before establishing herself as a go-to celebrity wardrobe stylist whose A-list clientele has included Kristen Stewart, Heather Graham, Kate Beckinsale, Mila Kunis, and Kaley Cuoco. Her work has also appeared in the pages of InStyle, Nylon, Teen Vogue, and Esquire UK. She lives in Los Angeles.
Style inspiration: The Dior post-war era is still truly one of the most influential times for me! The designs were structural, architectural, and the tailoring was impeccable.
Words of wisdom:
If you're on the fence about something, put it on hold.
SHARON TAKEDA is the senior curator and head of the Department of Costume and Textiles at the Los Angeles County Museum of Art (LACMA). Sharon has curated a number of exhibitions including Reigning Men: Fashion in Menswear 1715–2015; Kimono for a Modern Age; RODARTE: Fra Angelico Collection; and Breaking the Mode: Contemporary Fashion for the Los Angeles County Museum of Art.
ERICA TANOV has her name on a collection of women's apparel, accessories, and home goods—launched in 1990 in New York City—and a quartet of multi-brand boutiques located in San Francisco, New York City, Berkeley, and Marin. She currently resides in the San Francisco Bay area.
Style inspiration: I do love sculptor Louise Nevelson's style. She had such a raw, fearless glamour that I find timeless, ageless, and beautiful.
Biggest score: An antique, ornately embroidered, fur-trimmed kimono dress that I bought at a tiny (and, unfortunately, since closed) antiques store in Winters, California.
ISA TAPIA is a Puerto Rico–born, Manhattan-based designer of women's footwear who launched her namesake line in 2012. A member of the 2014–16 CFDA Fashion Incubator program, she spends most of her time traveling throughout South America, Europe, and Asia.
Words of wisdom:
From Carrie Bradshaw, "I have this little substance abuse problem. Expensive footwear."
OLYA THOMPSON is a Moscow-based textile designer whose style sense—and fabric designs inspired by everything from Ballets Russes costumes to Tartar robes—has earned her a reputation as Moscow's most stylish woman.
Style inspirations: Paris of 1913, Paul Poiret caftans, and Leon Bakst costumes for the Ballets Russes. Russian ballerina Tamara Karsavina in a kokoshnik, an ornate folkloric headdress.
Words of wisdom:
Support rare artisanal and unique individual brands. Be consistent with your style and you will be rewarded not just with purchases.
RICCARDO TORTATO is the New York–based fashion director for TSUM Moscow. Tortato can be found dressed in a bespoke suit, sitting in the front rows of fashion shows around the world.
ALYCE TRAN is the cofounder and creative director of (TheDailyEdited.com). Tran established the Australian online store and corresponding collection of customizable and personalized leather accessories collection with Tania Liu in 2014. The business has grown enormously in two short years, and in January 2016, they launched their first permanent retail space in David Jones Sydney, where their accessories are now monogrammed on the spot.
TRINA TURK launched her namesake women's contemporary clothing line in 1995, and, in the decades since, she and husband Jonathan Skow have built the L.A.-based label into a full-fledged California-flavored lifestyle brand that include accessories, swimwear, jewelry, home goods, textiles, a Mr. Turk menswear line, and a dozen stand-alone stores across the country. When not traveling the globe, Turk splits her time between Palm Springs and L.A.'s Silver Lake neighborhood.
Biggest scores: I love the macramé lion and rhino "trophies" we found at Studio 111 in Palm Springs (the gallery has since moved to Cathedral City).
TRACY TYNAN is a costume designer whose career has spanned more than thirty years and includes films (The Big Easy, Great Balls of Fire!, Blind Date) and TV movies (Tuesdays With Morrie, The Witches of Eastwick). In 2010 she was awarded the Women's Film & Television Showcase International Visionary award for her contributions to costume design. Her memoir, Wear & Tear: The Threads of My Life, was published in 2016. Originally hailing from the UK, she now calls downtown Los Angeles home.
Words of wisdom:
Check the return policy just in case.
ILARIA URBINATI is a Rome-born, Paris-raised, Los Angeles–based stylist whose client list has included Bradley Cooper, Ryan Reynolds, Ty Burrell, Ben Affleck, Shailene Woodley, Marisa Tomei, Lizzy Caplan, and Laura Dern, among others.
Style inspiration:
People often refer to Annie Hall, but I've always preferred Diane Keaton in Manhattan—all those little shirts buttoned to the top, tucked into those high-rise pants. She's perfect.
Words of wisdom:
Shopping is good for the economy.
MARY FRANCE VAN DAMME is the Montréal-born, Hong Kong–based founder of the travel-inspired luxury resort line that bears her name, which she launched in 2011. Her three decades of experience in the fashion industry include stints heading up a company that created private label brands for Macy's and Lord & Taylor and starting a manufacturing company (in Hong Kong) that developed and manufactured apparel for retail brands that include Monoprix, Marks & Spencer, and Saks Fifth Avenue.
Words of wisdom:
Only buy the best.
SUSANA MARTÍNEZ VIDAL is a fashion journalist based in Madrid and Mexico City. Over the past twenty-five years she's held posts as director of Elle Spain and Elle Décor Spain, founder of Ragazza magazine, and, most recently, contributor to the Huffington Post. Her coffee-table book Frida Kahlo: Fashion as the Art of Being was published by Assouline in 2016.
Words of wisdom:
To shop well, you must be prepared to return home empty-handed. If not, you will spend money on things you don't really want and won't end up wearing. And finally, as Frida Kahlo demonstrated, how one wears something is more important than what one wears.
DANIELA VILLEGAS is a Los Angeles–based jewelry designer known for translating her love of nature into wearable creations that combine 18-karat gold, gemstones, and a range of organic elements that include (but aren't limited to) beetles, porcupine quills, and feathers. Since launching her namesake jewelry collection in 2008, those spotted wearing her wares have included Demi Moore, Katy Perry, Salma Hayak, Christina Aguilera, Halle Berry, and Miley Cyrus.
Words of wisdom:
It's not about how much you pay, it's about how happy it makes you feel.
CLARE VIVIER was inspired in 2008 to launch her namesake Clare V. line of handbags and accessories after being unable to find a stylish and functional laptop case. Founded in 2008, the Los Angeles–based label that melds SoCal cool and Parisian chic currently sells a range of leather goods that includes clutches, totes, wallets, and luggage tags—and the occasional roll-up backgammon set—through retailers around the globe as well as a half-dozen US stand-alone stores.
Biggest score: My friend found white suede Prada platform brogues on the side of the road in Big Sur. She grabbed them and realized they weren't her size—but they were mine!
Words of wisdom:
It's okay to shop to make yourself feel better. And pay attention to where things are made!
DITA VON TEESE is a Los Angeles–based burlesque dancer whose high-end striptease act—which often involves Von Teese performing in a giant martini glass—has been performed to sold-out crowds around the world. She is also a designer, with her own line of lingerie, eyewear, and a fragrance collection. She has published three books, the most recent of which is Your Beauty Mark: The Ultimate Guide to Eccentric Glamour in 2015.
Biggest score: A Christian Dior New Look 1954 three-piece tweed circle skirt suit, with the Christian Dior Paris label bearing the red serial numbers. It has a silk lining and garters attached to the blouse, from Relic Vintage in San Francisco.
Words of wisdom:
I always dress well to shop and never buy anything that isn't better than what I have on.
STEPHANIE VON WATZDORF is the founder and creative director of the New York–based women's luxury label Figue (pronounced fig), which she launched in 2012. The collection of ready-to-wear, handbags, shoes, and jewelry has a gypsy vibe inspired by both her passion for travel and her upbringing; she was born on the outskirts of Paris to a German-French father and a Russian mother and her grandfather Leonide Massine was a renowned choreographer of the Ballets Russes.
Style influence: Anita Pallenberg and that effortless, very cool style of the late '60's and early '70's. Anouk Aimée has exquisite style, that perfect French look.
EILEEN WALLIS is the managing partner of the Portsmouth Group in Dubai, and has more than seventeen years of PR experience in the Middle East. Under her direction, The Portsmouth Group develops and manages regional communications strategies for clients in the retail, luxury, arts, entertainment, and hospitality sectors. The agency has worked with a number of brands, including Christie's, Rocco Forte Hotels, Swarovski, and Van Cleef & Arpels.
NICOLE WARNE started her Gary Pepper Girl website in 2009 as a hobby to sell clothes, and it has since grown into one of Australia's most successful lifestyle blogs. She became the first digital influencer to sign with IMG's Talent Division, and has graced the covers of Harper's Bazaar Singapore and Malaysia, Elle Australia, Miss Vogue Australia, Lucky, Nylon, and Net-A-Porter.com's The Edit. She counts premium brands Chanel, Valentino, Chopard, Dolce & Gabbana, L'Oreal Luxe, and Net-a-Porter.com among her list of clients.
SOPHIA WEBSTER is the London-based creative director of the namesake footwear and accessories label she founded in 2012 and which is now stocked by more than 200 retailers around the globe.
Style influence: Gwen Stefani. She has always stayed true to her unique sense of style and I love that about her. My favorite look of all time has to be from the "Just a Girl" video, classic Gwen.
Words of wisdom:
You can never have too many shoes!
ERIN WEINGER is a journalist and entrepreneur currently serving as the digital commercial editor of Vogue Australia. She previously served as digital style director of the Hollywood Reporter, where she launched and helmed the brand's digital style and beauty destination, Pret-a-Reporter.com. Her career in fashion writing began as a staff writer for the Los Angeles Times Image section. Her editorial contributions have appeared in Entrepreneur, Vanity Fair, Elle.com, InStyle.com, Paper, Departures, and other publications. She lives in Sydney.
ALEXANDRA WILKIS WILSON is a serial entrepreneur who cofounded luxury, members-only e-commerce site Gilt.com in 2007 and home-blowout business Glamsquad in 2014. She and Gilt cofounder Alexis Maybank are the authors of the New York Times bestseller By Invitation Only: How We Built Gilt and Changed the Way Millions Shop (2012).
Biggest score: A Valentino coat that I bought through Gilt. We didn't even photograph it, I paid $500, it was a $10,000, black suede with a Swarovski belt, a little bling, lined in fur.
Words of wisdom:
When you're traveling, you can be impulsive. Carpe diem!
BRIAN WOLK
(see Claude Morais).
KIMBERLY WU is one half of the Building Block sister act. Kimberly and her sister Nancy Wu launched their L.A. brand of industrial design–minded leather accessories in 2011. After graduating from Pasadena's Art Center College of Design, where she studied industrial design, Kimberly went to work for Honda's Advanced Studio in Tokyo, where she also discovered the hardware store to end all hardware stores, Tokyu Hands. There, she started to collect electrical sockets and cords that inspired the bags of Building Block, which are sold in stores around the world, including their own in downtown Los Angeles.
KATE YOUNG is a New York–based stylist who started her career as an assistant to Vogue magazine's Anna Wintour and Tonne Goodman, a job that led to the honor of styling the first-ever cover of Teen Vogue in 2003. Her list of red-carpet clients has includes Margot Robbie, Natalie Portman, Michelle Williams, Rachel Weisz, Dakota Johnson, Sienna Miller, and Selena Gomez. She's also worked with A-list designers and has styled ad campaigns for Dior, Joie, Guerlain, and Carolina Herrera, among many others.
Biggest score: I bought tons and tons of vintage YSL from Resee.com when it first launched. My best score ever.
JOE ZEE is the Los Angeles–based editor-in-chief and executive creative officer at Yahoo Style, prior to which he logged seven years as the creative director of Elle magazine. He is the author of That's What Fashion Is: Lessons and Stories From My Nonstop, Mostly Glamorous Life in Style (2015).
Style influences: Kurt Cobain and John F. Kennedy Jr. A bit of an oxymoron when it comes to style, but they represent two very distinct sides of me.
Words of wisdom:
Wear it now. If you love it enough to splurge on it, then use it. Before it's too late.
RACHEL ZOE is a Los Angeles–based stylist, fashion designer, TV personality and editor whose list of A-list clients over the years has included Nicole Richie, Jennifer Garner, and Jennifer Lawrence. She was the focus of Bravo's reality TV series The Rachel Zoe Project, which debuted in 2008 and aired for five seasons. The debut of her namesake collection of ready-to-wear, jewelry, and footwear, and the launch of her media company Zoe Report was in 2011. She has two books to her credit, the New York Times bestseller Style A to Zoe (2007) and Living in Style (2014).
Style influences: Brigitte Bardot, Jane Birkin, Coco Chanel, and countless others. I'm so influenced by the '60s and '70s in terms of my own taste and the aesthetic of my collection, and I will forever be inspired by these amazing and beautiful women.
Biggest score: One of my favorite most unforgettable shopping experiences was when I first met William Blanks Bailey, of William Vintage Shop in London, and visited his extraordinary store. This was my first time really "shopping" in over a year—my longest time without a purchase since my son Skyler was born. I ended up spending six hours in the store—Skyler got comfortable and took a nap! I found some unbelievable Christian Dior haute couture pieces and a Jean Patou vintage gown that I wore on the cover of my second book. It was hands down the best shopping day I have ever had.
Words of wisdom:
This took me about twenty years to learn, but I promise these are words to live by:
Shop with a purpose. Try to not always be driven by trends and only buy something if you can picture a moment in your life that you will wear it.
## ACKNOWLEDGMENTS
I would like to thank my friends near and far for helping me compile this guide, particularly Kelly Cutrone, Heather John Fogarty, Lizzie and Kathryn Fortunato, Jane Lim, Fiona Marin, Ann Mashburn, Lisa Parrish, Frances Pennington, Bryan Smith and Erin Weinger for being such generous connectors, and Ruth O'Neill for her extraordinary research skills. Thank you to my husband, Adam Tschorn, for his clever turns of phrase, and for all the long hours (years?) spent patiently waiting for me outside stores; to my parents Richard and Susan Moore for supporting me in every way; and to my sister Kathleen Moore for always being willing to tag along on a shopping excursion. Thanks to Missy Anderson and Billie Huff, two world-class shoppers in their own very different ways who taught me the thrill of the hunt, and to Judith Regan and Mia Abrahams for their guidance and determination.
## BOOTH MOORE
has been an Olympic-level shopper since she was a schoolgirl in New York City. After graduating from Duke University, she began her career in that city of the sensible suit, Washington, DC, at The Washington Post. Moore joined the Los Angeles Times in 1996, where she logged tens of thousands of miles as the paper's fashion critic, covering the runways in New York, London, Milan, and Paris (and stopping to shop along the way). She left the Times in 2015, is now the senior fashion editor at The Hollywood Reporter, and has contributed to the New York Times, Wall Street Journal, Town & Country, and Condé Nast Traveler, among other publications.
She lives in Los Angeles with her husband, two cats, and an overflowing closet room that could one day be a pop-up store.
## PHOTO CREDITS
#### NORTH AMERICA
* * *
Tommy Boudreau/Bodega, (top left), ; Chosen Vintage, (center left); FREECITY workshop, (bottom left), ; Ari Becker Photo, (bottom right), ; Want Apothecary, (center right), ; © 2016 Madison Hall, (top right), ; Draper James, (top right), , ; : Photo courtesy of Trashy Diva/Brittney Werner, (top center); Gus Powell, (top right), ; Billy Farrell/BFA, (bottom right); Courtesy of Ann Mashburn, (bottom center) ; Courtesy of Secret Location, (bottom left), , ; Love Adorned, ; Courtesy of Journelle, ; Courtesy of the author, , , ; Amy Dickerson, , ; Courtesy of Concrete & Water, ; Lucas Allen, ; Nancy Pearlsteen, ; John Phelan, Wiki Commons, Creative Commons License, ; Courtesy of the Vermont Country Store, ; Courtesy of Faran Krentcil, ; Courtesy of Irene Neuwirth, ; Courtesy of Reformation, ; Ali Mahdavi, ; Autumn de Wilde, ; Ben Ritter, ; Sharon Suh, ; Xander Bennett, Flickr, Creative Commons License, ; Courtesy of Trina Turk, ; Patrick Aguilar, ; Tom Schwartz, ; Leslie Santarina, ; Alana Hale, , ; Franco Folini Flickr, Creative Commons License, ; Courtesy of Nordstrom, , ; Jaclyn Campanaro, ; Liz Gross, ; Avant Toi, ; Jay Carroll, ; Courtesy of Allens Boots, ; Photo courtesy of ByGeorge, ; Courtesy of Kick Pleat, ; Nathan Schroder, ; Courtesy of Stanley Korshak, ; Betty Newton, ; Ben Gately Williams, ; Christopher Ziemnowicz, Flickr, Creative Commons License, ; Courtesy of the Webster, ; J Stephen Young, ; Caroline Allison, ; Giampaolo Sgura, ; Courtesy of Space 519, ; Tender, ; Shinola, ; Luke Truman, ; Courtesy of Susie Sheffman, ; Sarui Ghah Remanpour, ; Marc Cramer and Saucier + Perrotte Architects, ; Brian Bowen Smith, ;
#### MEXICO & SOUTH AMERICA
* * *
Courtesy of Comme il Faut, (top left), (center right), ; Courtesy of Uxuacasa, (center left), (bottom left), ; Holly Wilmeth, (bottom right), ; Courtesy of Fernanda Yamamoto, (top right), ; Courtesy of Gabriela and Tere Artigas, ; Courtesy of Yakampot, ; © 2016 Caravana Tulum, ; Alejandra Photography, , ; Gustavo Garcia Villa, ; Paulo Troya and Renan Teles, ; Courtesy of Lenny Neimeyer, ; Courtesy of Ana Kozak, ; Michell Zappa, Flickr, Creative Commons License, ;
#### EUROPE
* * *
Courtesy of La Commercial, (top left), , ; Courtesy of Caffé Giacosa, (top right), ; © & Other Stories, (center right), ; Courtesy of 10 Corso Como, (bottom right), ; Betty Newton, (bottom left); Courtesy of Prada, (center left), ; Al Higgins, (top left), ; Courtesy of Frau Tonis Parfum, (center top), , ; Pablo Recio, Flickr, Creative Commons License, (top right), ; Marc Dantan, (bottom right), ; Luisaviaroma – Luxury Shopping Worldwide Shipping, (bottom center), ; Jaclyn Campanaro, (bottom left), ; Courtesy of Roksanda, ; Jamie Kingham, ; Anna Bauer, ; Carmen De Witt, ; Alikhan & Radik, ; Courtesy of Liberty London, ; Stella Gehrckens, ; Blanaid Hennessy, ; Roland Halbe, ; Les 3 Marches de Catherine B, ; Pierre Bailly, ; Sunshinecity, Flickr, Creative Commons License, ; Alberto Zanetti, ; Courtesy of Erica Pelosini, ; Zoltán Vörös, Flickr, Creative Commons License, ; Doug Inglish, ; Eric Titcombe, Flickr, Creative Commons License, ; ANNA-KARIN KARLSSON Campaign shoot "DECADENCE" by Ekaterina Belinskaya, Head piece Agnieszka Osipa/ Sunglasses ANNA-KARIN KARLSSON/ Rings Loree Rorkin, Lynn Ban & Yeprem/ Fur collar Jh Nocturnal/ Outfit Patrik Guggenberger, ;
#### EURASIA
* * *
Courtesy of Axenoff, (top right), ; Courtesy of Dilek Hanif, (top right), ; Adnan & Hasan, (center right), ; Lelis Noroditky, (bottom right), (center left), ; Kemal Olca, (bottom left); Courtesy of Evren Dogancay, ; Cemre Mert, ; Charles Thompson, ; Martin de Lusenet, Flickr, Creative Commons License, ; Kathryn Wirsing, ;
#### ASIA
* * *
Courtesy of Johanna Ho, (top left), ; Courtesy of Marie France Van Damme, (top right), (bottom middle); Kate Shanasy, (center right), ; Elise Bergerson, (bottom right), ; Nestor Lacle, Flickr, Creative Commons License, (bottom left), ; Courtesy of Daikanyama Tsutaya Books, (center left), (left top), (center top), ; Seth Powers/KolcaiStudios, (top right), (bottom left), ; Lisa Eisner, ; Dick Thomas Johnson, Flickr, Creative Commons License, ; Courtesy of Divia Harilela, ; Sabrina Sikora, ; Courtesy of Bonnae Gokson, ; Justin Poland / The Society Management, ; Courtesy of Gentle Monster, ; Aimee Song, ;
#### MIDDLE EAST
* * *
Courtesy of Bambah, (top left), (center); The cARTel Dubai/Peter Richweisz, , (top right), (bottom right), (bottom left), ; Roy Ghattas, ;
#### AFRICA
* * *
Sandy Bornman, (top left), ; Courtesy of Chandler House, (top right), ; Akbar Delights, (center right), , ; Amfo-Akonnor Kwadwo (Amfo Connolly), (bottom right), (bottom left), ; Toby Murphy, (center left), ; Delphine Warin, ; Courtesy of Temple Muse, ; Emmanuel Andre @TBWA, ; Ines Cuatrecasas, ; Keagan Kingsley Carlin, ;
#### AUSTRALIA
* * *
Courtesy of Saturdays NYC, (top left), (center); Alpha60, (top right); Gorman, (bottom left), ; Camille Walala, (bottom right); Andrew Baker Photographer (top row); Courtesy of Peter Alexander, (bottom right); Shantanu Starick, (bottom left), , ; Courtesy of Zimmermann, ; Courtesy of Pip Edwards, ; Courtesy of Erin Weinger, ; Terence Chin, ; Kate Ballis, ;
65 Bleecker Street
New York, NY 10012
Copyright © 2017 by Booth Moore
All rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information, address Regan Arts Subsidiary Rights Department, 65 Bleecker Street, New York, NY 10012.
The information provided in this book is designed to provide helpful information on the subjects discussed. References are provided for information purposes only and do not constitute endorsement of any websites or other sources. Readers should be aware that the websites, addresses, and other store information listed in this book may change.
First Regan Arts paperback edition, January 2017
Library of Congress Control Number: 2016939698
ISBN 978-1-68245-032-1
ISBN 978-1-68245-033-8 (ebook)
Interior design by Catherine Casalino
Cover design by Richard Ljoenes
Image credits, which constitute an extension of this copyright page, appear on page 462
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The Pope arrives in Turkey, a land able to promote an encounter of civilisations and identify viable paths of peace and authentic progress
Vatican City, 28 November 2014 (VIS) – This morning Pope Francis began the sixth apostolic trip of his pontificate. His visit to Turkey is essentially of an ecumenical nature, like those of his predecessors; the country has occupied a privileged position in the geography of papal trips ever since the visit of the Apostolic Delegate Angelo Roncalli, referred to by the Turkish authorities as "the first Turkish pope in history" following his election as Pope John XXIII. Turkey was also the destination of Paul VI's fifth apostolic trip in 1967, a corollary of his pilgrimage to the Holy Land and his historic embrace with the Ecumenical Patriarch Atenagoras in Jerusalem. John Paul II continued the tradition with his fourth trip (1979) as did Benedict XVI with his fifth apostolic trip, in 2006.
The Holy Father departed from Rome's Fiumicino airport at 9 a.m., and reached the Turkish capital Ankara at 1 pm (local time) where he was received by the civil and religious authorities. He transferred by car to the mausoleum of Mustafa Kemal Ataturk, founder and first president of the Turkish Republic, the "Father of the Turks" who guided Turkey's radical rupture with its Ottoman past, laying down the foundations of the modern secular state in its 1937 Constitution. Upon arrival the Pope was received by the Commander of the Guard, ascended the Steps of Honour, left a floral tribute and prayed for a moment. He was then accompanied to the nearby "Tower of National Pact", where he signed the guest book.
He then paid a visit to the Presidential Palace or "Ak Saray" (White Palace), inaugurated just two months ago by President Recep Tayyip Erdogan and substituting the historic Cankaya Palace. The Holy Father was received by the president, and the two spoke in private for a few minutes, after which Pope Francis gave his first public address on Turkish soil, addressed to the authorities gathered in the Ak Saray.
"I am pleased to visit your country so rich in natural beauty and history, and filled with vestiges of ancient civilisations. It is a natural bridge between two continents and diverse cultures", the Pope began. "This land is precious to every Christian for being the birthplace of Saint Paul, who founded various Christian communities here, and for hosting the first seven Councils of the Church. It is also renowned for the site near Ephesus which a venerable tradition holds to be the 'Home of Mary',the place where the Mother of Jesus lived for some years. It is now a place of devotion for innumerable pilgrims from all over the world, not only for Christians, but also for Muslims.
"Yet, the reasons why Turkey is held with such regard and appreciation are not only linked to its past and ancient monuments, but also have to do with the vitality of its present, the hard work and generosity of its people, and its role in the concert of nations. It brings me great joy to have this opportunity to pursue with you a dialogue of friendship, esteem and respect, in the footsteps of my predecessors Blessed Paul VI, Saint John Paul II and Benedict XVI. This dialogue was prepared for and supported by the work of the then apostolic delegate, Angelo Giuseppe Roncalli, who went on to become Saint John XXIII, and by the Second Vatican Council".
The Pope reiterated the need for a dialogue to "deepen the understanding and appreciation of the many things which we hold in common. Such a dialogue will allow us to reflect sensibly and serenely on our differences,and to learn from them. There is a need to move forward patiently in the task of building a lasting peace, one founded on respect for the fundamental rights and duties rooted in the dignity of each person. In this way, we can overcome prejudices and unwarranted fears, leaving room for respect, encounter, and the release of more positive energies for the good of all".
Therefore, "it is essential that all citizens – Muslim, Jewish and Christian – both in the provision and practice of the law, enjoy the same rights and respect the same duties. They will then find it easier to see each other as brothers and sisters who are travelling the same path, seeking always to reject misunderstandings while promoting cooperation and concord. Freedom of religion and freedom of expression, when truly guaranteed to each person, will help friendship to flourish and thus become an eloquent sign of peace.
"The Middle East, Europe and the world all await this maturing of friendship. The Middle East, in particular, has for too long been a theatre of fratricidal wars, one born of the other, as if the only possible response to war and violence must be new wars and further acts of violence. How much longer must the Middle East suffer the consequences of this lack of peace? We must not resign ourselves to ongoing conflicts as if the situation can never change for the better! With the help of God, we can and we must renew the courage of peace! Such courage will lead to a just, patient and determined use of all available means of negotiation, and in this way achieve the concrete goals of peace and sustainable development".
Addressing the president, the Pope reaffirmed that "interreligious and intercultural dialogue can make an important contribution to attaining this lofty and urgent goal, so that there will be an end to all forms of fundamentalism and terrorism which gravely demean the dignity of every man and woman and exploit religion. Fanaticism and fundamentalism, as well as irrational fears which foster misunderstanding and discrimination, need to be countered by the solidarity of all believers. This solidarity must rest on the following pillars: respect for human life and for religious freedom, that is the freedom to worship and to live according to the moral teachings of one's religion; commitment to ensuring what each person requires for a dignified life; and care for the natural environment. The peoples and the states of the Middle East stand in urgent need of such solidarity, so that they can 'reverse the trend' and successfully advance a peace process, repudiating war and violence and pursuing dialogue, the rule of law, and justice.
"Sadly, to date, we are still witnessing grave conflicts. In Syria and Iraq, particularly, terrorist violence shows no signs of abating. Prisoners and entire ethnic populations are experiencing the violation of the most basic humanitarian laws. Grave persecutions have taken place in the past and still continue today to the detriment of minorities, especially – though not only – Christians and Yazidis. Hundreds of thousands of persons have been forced to abandon their homes and countries in order to survive and remain faithful to their religious beliefs.
Turkey, which has generously welcomed a great number of refugees, is directly affected by this tragic situation on its borders; the international community has the moral obligation to assist Turkey in taking care of these refugees. In addition to providing much needed assistance and humanitarian aid, we cannot remain indifferent to the causes of these tragedies. In reaffirming that it is licit, while always respecting international law, to stop an unjust aggressor, I wish to reiterate, moreover, that the problem cannot be resolved solely through a military response. What is required is a concerted commitment on the part of all, based on mutual trust, which can pave the way to lasting peace, and enable resources to be directed, not to weaponry, but to the other noble battles worthy of man: the fight against hunger and sickness, the promotion of sustainable development and the protection of creation, and the relief of the many forms of poverty and marginalisation of which there is no shortage in the world today".
The Pope concluded, "Turkey, by virtue of its history, geographical position and regional influence, has a great responsibility: the choices which Turkey makes and its example are especially significant and can be of considerable help in promoting an encounter of civilisations and in identifying viable paths of peace and authentic progress. May the Most High bless and protect Turkey, and help the nation to be a strong and fervent peacemaker".
Published by VISarchive 02 - Friday, November 28, 2014
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Indulgences for the Year of Consecrated Life | {
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{% extends "layout.html" %} {% block page_title %} GOV.UK prototype kit
{% endblock %} {% block content %}
<main id="content" role="main">
{% include "../includes/phase_banner_alpha.html" %}
<div class="column-three-thirds">
<h1 class="heading-large">
Register this S1<span class="heading-secondary"></span>
</h1>
The deadline for this registration is 14 May 2017.<br><br>
{% include "includes/S1InfoDetailsReg.html" %}
</div>
<br>
<form action="S1RegistrationConfirmation.html">
<input class="button" type="submit" value="Register S1">
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<a href="#">Dispute this registration</a>
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Q: What is wrong with the WHERE clause of my OUTER JOIN Query? I am joining tables: OLD and NEW
Table: OLD
Employee_Name Status
Ann Y
Bob Y
Carol
Doug Y
Ellen
Frank
Table: NEW
Employee_Name Status
Ann Y
Bob
Carol
Ellen
Frank
I can get the output I want with this join:
SELECT Old.Employee_Name AS [OldName]
,New.Employee_Name AS [NewName]
,Old.Status AS [OldStatus]
,New.Status AS [NewStatus]
FROM Old
FULL OUTER JOIN New ON Old.Employee_Name = New.Employee_Name
Which is:
OldName NewName OldStatus NewStatus
Ann Ann Y Y
Bob Bob Y
Carol Carol
Doug Y
Ellen Ellen
Frank Frank
Now I want to add a filter to the query so that it only keeps rows where the new and old status are different.
If I try:
SELECT Old.Employee_Name AS [OldName]
,New.Employee_Name AS [NewName]
,Old.Status AS [OldStatus]
,New.Status AS [NewStatus]
FROM Old
FULL OUTER JOIN New ON Old.Employee_Name = New.Employee_Name
WHERE Old.Status <> New.Status
...it doesn't give the proper output.
For this example I think it would give the output of:
OldName NewName NewStatus OldStatus
Bob Bob Y
...and fail to return the line with Doug.
A: When you do an outer join, the non-matching rows have NULL values. This is why you move conditions to the on clause from the where clause:
SELECT Old.Employee_Name AS [OldName],
New.Employee_Name AS [NewName],
Old.Status AS [OldStatus],
New.Status AS [NewStatus]
FROM Old FULL OUTER JOIN
New
ON Old.Employee_Name = New.Employee_Name AND
Old.Status <> New.Status;
A: If either operator is null in a SQL statement, it will return null (no rows) for that row. For example, if you do a <> b where a = 1 and b = null, it will always return no rows, even though it a and b are not equivalent.
You need to change your query to something like this:
WHERE COALESCE(Old.Status, '') <> COALESCE(New.Status, '')
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\section{Introduction}
\label{sec:Intro}
Let us give a few formal definitions and notations that we will use throughout this article.
\begin{notation}
For any natural number $p$ we will denote by $x^{[p]}$ the \textbf{``falling power'' polynomial}\footnote{also often called ``falling factorial''} $x(x-1)\cdots(x-p+1)$. That is,
$$
x^{[p]} = \prod_{m=0}^{p-1}(x-m) = p! \binom{x}{p} \pp
$$
Another common notation for this polynomial that you often see in texts on combinatorics is \textbf{Pochhammer symbol} $(x)_p$.
\end{notation}
\begin{notation}
We will denote by $\eulerian{n}{m}$ the so-called Eulerian number---the number of permutations of order $n$ with exactly $m$ ascents; an ascent in permutation $\pi = \{\pi_1, \pi_2, \ldots, \pi_n\}$ is index $1 \leqslant j < n$ such that $\pi_j < \pi_{j+1}$. In some texts the notation $\genfrac\langle\rangle{0pt}{}{n}{\raisebox{2pt}{$\scriptstyle m$}}$ is used.
Clearly, $\eulerian{n}{m} = 0$ if $m < 0$ or $m \geqslant n$. Therefore, skipping the zeros, all the Eulerian numbers can be arranged in the triangular shape constituting the Eulerian triangle, where the $m$th number in the $n$th row is $\eulerian{n}{m}$.
\end{notation}
The polynomial with coefficients taken from $n$th row of this triangle is called \textit{Eulerian polynomial} and denoted as $\mca_n(x)$, that is,
$$
\mca_n(x) = \sum_{m=0}^{n-1}\eulerian{n}{m}x^m \pp
$$
Below is the table with the first eight rows of the Eulerian triangle:
\begin{table}[H]
\begin{tabular}{|m{1cm}|C{1cm}|C{1cm}|C{1cm}|C{1cm}|C{1cm}|C{1cm}|C{1cm}|m{1cm}|}
\hline \vspace{1mm}
$n \backslash m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & \quad 7 \\
\hline \vspace{1mm}
1 & \textbf{1} & & & & & & & \\
\hline \vspace{1mm}
2 & \textbf{1} & \textbf{1} & & & & & & \\
\hline \vspace{1mm}
3 & \textbf{1} & \textbf{4} & \textbf{1} & & & & & \\
\hline \vspace{1mm}
4 & \textbf{1} & \textbf{11} & \textbf{11} & \textbf{1} & & & & \\
\hline \vspace{1mm}
5 & \textbf{1} & \textbf{26} & \textbf{66} & \textbf{26} & \textbf{1} & & & \\
\hline \vspace{1mm}
6 & \textbf{1} & \textbf{57} & \textbf{302} & \textbf{302} & \textbf{57} & \textbf{1} & & \\
\hline \vspace{1mm}
7 & \textbf{1} & \textbf{120} & \textbf{1191} & \textbf{2416} & \textbf{1191} & \textbf{120} & \textbf{1} & \\
\hline \vspace{1mm}
8 & \textbf{1} & \textbf{247} & \textbf{4293} & \textbf{15619} & \textbf{15619} & \textbf{4293} & \textbf{247} & \quad \textbf{1} \\
\hline
\end{tabular}
\end{table}
Various properties and formulas for Eulerian numbers and polynomials can be found in \cite{Pet}.
\begin{definition}
For any natural numbers $k \leqslant n$ the following two symmetric polynomials---$\mfp_{k,n}$ and $\mfe_{k,n}$---in $n$ variables $x_1$, \ldots, $x_n$, are defined by the formulas
\begin{align}
\mfp_{k,n}(x_1,\ldots,x_n) &= x_1^k + \ldots + x_n^k = \sum_{i=1}^n x_i^k \ ; \nc
\mfe_{k,n}(x_1,\ldots,x_n) &= \sum_{1\leqslant \alpha_1 < \ldots < \alpha_k \leqslant n}x_{\alpha_1}x_{\alpha_2}\ldots x_{\alpha_k}
\pp \nn
\end{align}
Both are, obviously, homogeneous symmetric polynomials of degree $k$. They are called, respectively, a \textbf{power-sum} polynomial and an \textbf{elementary symmetric} polynomial of $k$th order. When the set of variables is fixed, we will often denote these polynomials simply by $\mfp_k$ and $\mfe_k$.
\end{definition}
It is well-known that for any subfield $\mbf$ of complex numbers (such as $\mbq$, $\mbr$, or $\mbc$) both sets of polynomials $\{\mfe_k\}$ and $\{\mfp_k\}$ constitute a basis in the ring $\Lambda_\mbf[x_1,\ldots, x_n]$ of symmetric polynomials in $n$ variables.
\begin{definition}
For any natural number $k$ and any $n$-multiset $A = \{a_1,\ldots,a_n\}$ we define $\mfp_k(A)$, the power-sum of $k$th order of multiset $A$, as $\mfp_{k,n}(a_1,\ldots,a_n)$.
\end{definition}
\begin{notation}
For any two natural numbers $n$ and $s$ such that $n\geqslant s$, and an arbitrary $n$-multiset $A = \{a_1,\ldots,a_n\}$ we define the multiset $\sms As$ of its $s$-sums, i.e., the collection of all sums of the form
$$
a_{i_1}+a_{i_2}+\ldots+a_{i_s}\pc
$$
where $1\leqslant i_1 < i_2 < \ldots < i_s \leqslant n$.
\end{notation}
Obviously, power-sum $\mfp_k(\sms As)$ of multiset $\sms As$ is a symmetric homogeneous polynomial of degree $k$ in $a_1$, \ldots, $a_n$. Therefore if $k \leqslant n$, then this power-sum can be uniquely represented as
\begin{align}
\label{eq:fsk_sigma}
\mfp_k(\sms As)
&= \qskn skn(\mfp_1(A), \mfp_2(A), \ldots, \mfp_{k-1}(A), \mfp_{k}(A)) \nc
&= \mskn skn\, \mfp_k(A) + \tqskn skn(\mfp_1(A), \mfp_2(A), \ldots, \mfp_{k-1}(A)) \pc
\end{align}
where $\qskn skn$ and $\tqskn skn$ are polynomials in variables $\mfp_1$, \ldots,$\mfp_{k-1}$, $\mfp_k$, and $\mfp_1$, \ldots, $\mfp_{k-1}$ respectively, and the coefficient $\mskn skn$ is a constant (in terms of variables $a_i$) which depends only on $s$, $k$, and $n$.
Representation of symmetric polynomials of $\sms As$ via $\mfp_i(A)$ is of interest not only for purely algebraic or combinatorial reasons. One well-known example from topology is computation of Chern classes for exterior powers of a vector bundle. Let $\omega:E\rightarrow M$ be a vector bundle of rank $n$; consider computation of Chern classes of its $s$th exterior power $\bigwedge^s\omega$ via Chern classes $c_i(\omega)$ of the original bundle. The result will be the formula (very similar to \eqref{eq:fsk_sigma}) which expresses elementary symmetric polynomials $\mfe_k(\sms As)$ of multiset $\sms As$ via elementary symmetric polynomials $\mfe_k(A)$ of multiset $A$.
While that formula is clearly different from \eqref{eq:fsk_sigma}, the ``top'' coefficient (at $c_k(\omega)$) in that formula is the same number $\mskn skn$ from Equation \eqref{eq:fsk_sigma}. Namely, we have
$$
c_k(\bigwedge^s\omega) = \mskn skn\, c_k(\omega) + \tilde{\mathcal R}_{s,k,n}(c_1(\omega), c_2(\omega), \ldots, c_{k-1}(\omega)) \pp
$$
(see the proof below in Proposition \ref{thm:eskn_coeff}.)
We will also show how formula \eqref{eq:fsk_sigma} and polynomials $\qskn skn$ can be used to solve the so-called (Generalized) Moser Problem, or the Multiset Recovery Problem. The Moser Problem asks whether, given the multiset $\sms As$, it is always possible to uniquely restore (recover) the original multiset $A$. This question was originally posed by Leo Moser in 1957 as a problem in \textsc{American Mathematical Monthly} for $s=2$ and $n=4, 5$ (see \cite{Mos}.)
In article \cite{Fom} the reader can find a comprehensive survey of results and methods on this problem, circa 2017. In the next two sections we will compute $\mskn skn$ and show how it can be used in the Moser Problem.
\newpage\ \medskip
\section{Explicit formula for \texorpdfstring{$\qskn skn$}{Q s,k,n}}
\label{sec:Qformula}
In this section we prove the explicit formula for polynomials $\qskn skn$ and present some of its corollaries.
Consider an integer partition $\lambda$ of $k$, that is, $\lambda = \{\lambda_1, \ldots, \lambda_d\}$, where $\{\lambda_i\}$ is the non-increasing sequence of $d$ positive integers such that their sum equals $k$. Then $\mfp_\lambda$ will denote monomial $\mfp_{\lambda_1}\cdots \mfp_{\lambda_d}$ in variables $\mfp_i$.
Let $\delta = \{\delta_1, \ldots, \delta_q\}$ denote the sequence of of partition $\lambda$'s multiplicities---meaning that there are exactly $q$ different numbers among $\lambda_i$ with $i$th of these numbers occurring $\delta_i$ times. Obviously, the sum of these multiplicities equals $d$.
Since $\qskn skn$ is a polynomial in $\mfp_i$, it can be uniquely written in the following form
$$
\qskn skn = \sum_{\lambda\in\mcp(k)} c_{\lambda}\mfp_{\lambda} \pc
$$
with rational coefficients $c_\lambda$, where $\mcp(k)$ is the set of integer partitions of $k$. We cannot immediately claim that these coefficients are integers, as would be the case with elementary symmetric polynomials ($\mfe_k$ form a $\mbz$-basis of the ring of symmetric polynomials with integer coefficients $\Lambda_{\mbz}$, while $\mfp_k$ do not.)
\medskip
Our main objective now is to find an explicit formula for coefficient $c_\lambda$.
\begin{theorem}
\label{thm:qskn_coeff}
\begin{equation}
\label{eq:clambda}
c_{\lambda} =
\frac{(-1)^{s+d}k!}{\lambda_1!\cdots \lambda_d!\delta_1!\cdots \delta_q!}
\sum_{p=0}^s
\sum_{\mtop{m_1,\ldots,m_d\geqslant 1}{m_1+\cdots+m_d = s-p}}
(-1)^p \binom np
m_1^{\lambda_1-1} \cdots m_d^{\lambda_d-1}
\pc
\end{equation}
where the second summation is done over all length $d$ compositions of $s-p$; that is, the sequences of $d$ positive integers $\{m_1,\ldots,m_d\}$ such that the sum of these numbers equals $s-p$.
\end{theorem}
\begin{proof}
Given an arbitrary multiset $A = \{a_1, \ldots, a_n\}$ of $n$ (complex or rational) numbers $a_i$, let us consider the two functions
\begin{equation}
\label{eq:fx_sum}
f(x) = \sum_{i=1}^n e^{a_i x}
\end{equation}
and
\begin{equation}
\label{eq:gxy_sum}
g(x, y) = \prod_{i=1}^n (1 + ye^{a_i x}) \pp
\end{equation}
Now, using Taylor series expansion
$$
e^{ax} = \sum_{j=0}^\infty \frac{a^j x^j}{j!} \pc
$$
we can rewrite formula \eqref{eq:fx_sum} as follows
$$
f(x) = \sum_{i=1}^n \sum_{j=0}^\infty \frac{a_i^j x^j}{j!}
= \sum_{j=0}^\infty x^j \left(\frac{1}{j!}\sum_{i=1}^n a_i^j\right)
= \sum_{j=0}^\infty x^j \frac{\mfp_j(A)}{j!} \pc
$$
and formula \eqref{eq:gxy_sum} as
\begin{align}
g(x, y) &= \sum_{m=0}^n y^m \sum_{i_1 < i_2 < \ldots < i_m} e^{a_{i_1}x}e^{a_{i_2}x}\ldots e^{a_{i_m}x} \nc
&= \sum_{m=0}^n y^m \sum_{i_1 < i_2 < \ldots < i_m} e^{(a_{i_1} + a_{i_2} + \ldots + a_{i_m})x} \nc
&= \sum_{m=0}^n y^m \sum_{i_1 < i_2 < \ldots < i_m} \sum_{j=0}^\infty \frac{\left(a_{i_1} + a_{i_2} + \ldots + a_{i_m}\right)^j x^j}{j!} \nc
&= \sum_{m=0}^n y^m \sum_{j=0}^\infty x^j \sum_{i_1 < i_2 < \ldots < i_m} \frac{\left(a_{i_1} + a_{i_2} + \ldots + a_{i_m}\right)^j}{j!} \nc
&= \sum_{m=0}^n \sum_{j=0}^\infty y^m x^j \frac{\mfp_j(\sms Am)}{j!}
= \sum_{j,m\geqslant 0} x^j y^m \frac{\mfp_j(\sms Am)}{j!} \nn
\end{align}
where $\mfp_j(\sms Am)$ is defined as zero if $m = 0$, except for $\mfp_0(\sms A0) = 1$. Also, obviously, $\mfp_j(\sms Am) = 0$ if $m > n$.
For convenience sake we will use---in this proof only---the following notations.
$$
\lsp_j = \frac{\mfp_j(A)}{j!} \pc \quad
\Lsp_{j,m} = \frac{\mfp_j(\sms Am)}{j!} \pp
$$
\noindent Clearly, $f(x)$ and $g(x,y)$ are generating functions for sequences $\lsp_j$ and $\Lsp_{j,m}$. The following formula ties these two functions together.
\begin{lemma}
\begin{equation}
\label{eq:gxy_expfkt}
g(x,y)=\exp\left(\sum_{j=1}^\infty (-1)^{j-1}f(j x)\frac{y^j}j \right)\pp
\end{equation}
\end{lemma}
\begin{proof}
\begin{align}
\frac d{dy}\ln g(x,y) &= \frac d{dy}\ln \prod_{i=1}^n\left(1+ye^{a_i x}\right) = \sum_{i=1}^n\frac1y\;
\frac {ye^{a_i x}}{1+ye^{a_i x}} = \nc
&= \sum_{i=1}^n \frac1y \sum_{j=1}^\infty (-1)^{j-1}\left(ye^{a_i x}\right)^j=
\sum_{j=1}^\infty (-y)^{j-1}\sum_{i=1}^n e^{j a_i x}=\nc
&= \sum_{j=1}^\infty (-y)^{j-1}f(j x)\pp \nn
\end{align}
Since $g(x,0) = 1 = \exp(0)$, we have
$$
g(x,y) = \exp\left(\int\limits_0^y \sum_{j=1}^\infty (-y)^{j-1}f(j x) \,dy \right)
= \exp\left(\sum_{j=1}^\infty (-1)^{j-1}f(j x)\frac{y^j}j \right)\pp
$$
\end{proof}
\begin{lemma}
\begin{equation}
\label{eq:pkas_pka}
\sum_{j,m \geqslant 0}\Lsp_{j,m} x^j y^m = (1+y)^n\exp
\left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right)\pp
\end{equation}
\end{lemma}
\begin{proof}
Using equation \eqref{eq:gxy_expfkt} as well as formulas
$$
g(x,y)=\sum_{j,m \geqslant 0}\Lsp_{j,m} x^j y^m \ ; \quad
f(x)=\sum_{j=0}^\infty \lsp_j x^j =n + \sum_{j=1}^\infty \lsp_j x^j \pc
$$
we obtain
\begin{align}
\sum_{j,m \geqslant 0} \Lsp_{j,m} x^j y^m
&= \exp\left(\sum_{m=1}^\infty(-1)^{m-1}
\left(n + \sum_{j=1}^\infty \lsp_j(mx)^j\right) \frac{y^m}m\right) \nc
&= \exp\left (\sum_{m=1}^\infty (-1)^{m-1}n\frac{y^m}m\right)\cdot\exp
\left( \sum_{m=1}^\infty (-1)^{m-1}\sum_{j=1}^\infty \lsp_j(m x)^j
\frac{y^m}m\right) \nc
&= \left(\exp\left( \sum_{m=1}^\infty (-1)^{m-1}\frac{y^m}m \right)\right)^n
\cdot \exp\left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m
\right)\pp \nn
\end{align}
Now it suffices to note that
$$
\exp\left(\sum_{m=1}^\infty (-1)^{m-1}\frac{y^m}m\right) = \exp\bigl(\ln(1+y)\bigr)=1+y\pp
$$
\end{proof}
Let us go back to equation \eqref{eq:pkas_pka} and consider coefficients at the term $x^k y^s$ on both sides of it. Since they must be the same, the following equality holds.
\begin{equation}
\label{eq:lsp_binom}
\Lsp_{k,s} = \sum_{p=0}^s \binom{n}{p} c_{k,s-p} \pc
\end{equation}
where $c_{k,s-p}$ is the coefficient at the term $x^k y^{s-p}$ in
\begin{equation}
\label{eq:c_ksp}
\exp\left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right) \pp
\end{equation}
Now in order to express coefficient $c_{k,s-p}$ through $\lsp_k$ we rewrite and expand the expression \eqref{eq:c_ksp} as follows.
\begin{align}
\exp\left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right) =
1 &+ \frac{1}{1!}\left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right) \nc
&+ \frac{1}{2!} \left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right)^2 \nc
&+ \frac{1}{3!} \left(\sum_{j,m \geqslant 1}(-1)^{m-1}\lsp_j m^{j-1} x^j y^m \right)^3 + \ldots \nn
\end{align}
Thus the coefficient at $x^ky^{s-p}$ equals
$$
\sum_{d=1}^{s-p}
\frac{(-1)^{s-p-d}}{d!}
\sum_{\mtop{J, M}{\|J\| = k, \|M\| = s-p}}
\prod_{i=1}^d \lsp_{j_i}m_i^{j_i-1} \pc
$$
where summation is done over all length $d$ compositions $J = \{j_1, \ldots, j_d\}$ of number $k$, and all length $d$ compositions $M = \{m_1, \ldots, m_d\}$ of number $s-p$.
Finally, from this, using the formula \eqref{eq:lsp_binom} and definitions of $\Lsp_{j,m}$ and $\lsp_j$ we obtain
$$
\qskn skn(\mfp_1,\ldots,\mfp_k) = \sum_{p=0}^s \binom np
\sum_{d=1}^{s-p} (-1)^{s-p-d}\frac1{d!}
\sum_{\mtop{J, M}{\|J\| = k, \|M\| = s-p}}
\frac{k!}{j_1!\cdots j_d!}
\prod_{i=1}^d m_i^{j_i-1} \mfp_{j_i}
\pp
$$
If we combine the like terms, then each term that contains monomial $\mfp_\lambda = \mfp_{\lambda_1}\mfp_{\lambda_2}\cdots\mfp_{\lambda_d}$ occurs in the sum above exactly $\binom{d}{\delta_1,\ldots,\delta_q}$ times; therefore the coefficient $c_\lambda$ at $\mfp_\lambda$ equals
\begin{align}
\frac{k!}{\lambda_1!\cdots \lambda_d!}
\frac{d!}{\delta_1!\cdots \delta_q!}
&\sum_{p=0}^s \binom np
(-1)^{s-p-d}\frac1{d!}
\sum_{\mtop{m_1,\ldots,m_d\geqslant 1}{m_1+\cdots+m_d = s-p}}
\prod_{i=1}^d m_i^{\lambda_i-1}
\nc
&=
\frac{(-1)^{s+d}k!}{\lambda_1!\cdots \lambda_d!\delta_1!\cdots \delta_q!}
\sum_{p=0}^s
\sum_{\mtop{m_1,\ldots,m_d\geqslant 1}{m_1+\cdots+m_d = s-p}}
(-1)^p \binom np
m_1^{\lambda_1-1} \cdots m_d^{\lambda_d-1}
\pc \nn
\end{align}
which concludes the proof.
\end{proof}
\begin{remark}
Here is another, slightly different, way to present the same expression.
$$
c_\lambda =
\frac{(-1)^{d}k!}{\lambda_1!\cdots \lambda_d!\delta_1!\cdots \delta_q!}
\sum_{\|M\| \leqslant s}
(-1)^{\|M\|} \binom n{s-\|M\|}
m_1^{\lambda_1-1} \cdots m_d^{\lambda_d-1}
\pc
$$
where summation is done over all length $d$ compositions $M = \{m_1, \ldots, m_d\}$ of a positive integer not greater than $s$.
\end{remark}
The following are immediate corollaries of the formula \eqref{eq:clambda}.
\begin{corollary}
\label{thm:qskn_int}
All coefficients $c_\lambda$ of polynomial $\qskn skn$ are integers.
\end{corollary}
\begin{proof}
It is sufficient to show that the coefficient
$$
\frac{k!}{\lambda_1!\cdots \lambda_d!\delta_1!\cdots \delta_q!}
$$
on the right-hand side of the formula \eqref{eq:clambda} is an integer. But this is the number of ways to dissect the set with $k$ elements into $d$ subsets containing $\lambda_1$, $\lambda_1$, \ldots, and $\lambda_d$ elements, and therefore we are done.
\end{proof}
\begin{corollary}
\label{thm:qskn_zero}
Coefficient $c_\lambda$ is nonzero only if length $d$ of partition $\lambda$ does not exceed $s$.
\end{corollary}
\begin{proof}
Indeed, from formula \eqref{eq:clambda} it follows that $d \leqslant s-p \leqslant s$.
\end{proof}
\medskip
Finally, a short and easy proof of the fact we have mentioned at the end of Introduction section.
\begin{prop}
\label{thm:eskn_coeff}
$$
\mfe_k(\sms As) = \mskn skn \, \mfe_k(A) + \mathcal W(\mfe_1(A), \ldots, \mfe_{k-1}(A)) \pc
$$
where $\mathcal W(e_1, \ldots, e_{k-1})$ is some integer polynomial in $e_i$.
\end{prop}
\begin{proof}
From Newton-Girard identities (see Theorems 2.9--2.14 in \cite{MenRem}) we have
\begin{align}
\mfe_k(a_1, \ldots, a_n) = \frac{(-1)^{k-1}}{k}\mfp_k(a_1, \ldots, a_n) + \mathcal E(\mfp_1, \ldots, \mfp_{k-1}) \pc
\nc
\mfp_k(a_1, \ldots, a_n) = (-1)^{k-1}k\mfe_k(a_1, \ldots, a_n) + \mathcal P(\mfe_1, \ldots, \mfe_{k-1}) \pc
\nn
\end{align}
where $\mathcal E$, $\mathcal P$ are polynomials with rational coefficients. Notice that the coefficients $\frac{(-1)^{k-1}}{k}$ and $(-1)^{k-1}k$ are dependent only on $k$, not on $n$. Thus we have
\begin{align}
\mfe_k(\sms As) &= \frac{(-1)^{k-1}}{k}\mfp_k(\sms As) + \mathcal E (\mfp_1(\sms As), \ldots, \mfp_{k-1}(\sms As)) \nc
&= \frac{(-1)^{k-1}}{k}\mskn skn\mfp_k(A) + \mathcal V(\mfp_1(A), \ldots, \mfp_{k-1}(A)) \nc
&= \mskn skn \mfe_k(A) + \mathcal W(\mfe_1(A), \ldots, \mfe_{k-1}(A)) \pc \nn
\end{align}
using the fact that $\mfp_i(\sms As)$ ($i = 1, \ldots, k-1$) are polynomials in $\mfp_1(A)$, \ldots, $\mfp_{k-1}(A)$.
\end{proof}
\newpage\ \medskip
\section{Moser polynomials and their properties}
\label{sec:MoserPoly}
Theorem \ref{thm:qskn_coeff} provides us with another proof of an important formula, which was originally obtained in 1962 by Gordon, Fraenkel, and Straus (\cite{GorFraStr}) specifically for the purpose of solving the Moser's Problem.
\begin{theorem}[Gordon-Fraenkel-Straus Theorem]
\label{thm:fskn}
For any natural numbers $s$, $k$, $n$ such that $s, k \leqslant n$ we have
\begin{equation}
\label{eq:fsk_main}
\mskn skn = \sum_{p=1}^s (-1)^{p-1}\binom{n}{s-p}p^{k-1} \pp
\end{equation}
\end{theorem}
\begin{proof}
By definition, $\mskn skn = c_\lambda$, where $\lambda$ is the 1-part partition $\{k\}$. Thus $d=1$, $\lambda_1=d$, and $\delta_1=1$. There is only one length 1 composition $M$ of $s-p$, and therefore equality \eqref{eq:clambda} is reduced to the following
$$
c_\lambda = \sum_{p=0}^s (-1)^{s-p-1} \binom np (s-p)^{k-1} \pc
$$
which is equivalent to formula \eqref{eq:fsk_main}.
\end{proof}
Another way of deducing the last theorem from Theorem \ref{thm:qskn_coeff} is to use formulas \eqref{eq:lsp_binom} and \eqref{eq:c_ksp} for one specific $n$-multiset.
\begin{notation}
\label{def:znk}
For natural numbers $n\geqslant k$ we define $\Znk$ as $n$-multiset that consists of $n-k$ zeros and all complex $k$th roots of unity; that is,
\begin{equation}
\label{eq:znk}
\Znk = \Big\{e^{2\pi i \cdot m/k}: m=0,1,\ldots, k-1 \Big\} \msu \Big\{ \underbrace{0,\ldots,0}_{n-k} \Big\} \pp
\end{equation}
\end{notation}
\begin{lemma}
\label{thm:fskn_znk} For $n\geqslant k$ and $n\geqslant s$ we have $\mskn skn = \mfp_k(\sms{\Znk}s)/k$.
\end{lemma}
\begin{proof} Obviously, if $k \mid j$, then $\mfp_j(\Znk) = k$, otherwise $\mfp_j(\Znk) = 0$. Therefore, in equation \eqref{eq:fsk_sigma} for this multiset the last summand on the right-hand side $\tqskn skn(\mfp_1(\Znk), \mfp_2(\Znk), \ldots, \mfp_{k-1}(\Znk))$ is zero. Hence, we have $\mfp_k\left(\sms{\Znk}s\right) = \mskn skn\, \mfp_k(\Znk)$.
\end{proof}
We will leave finalizing this slightly different approach to the reader.
\medskip\begin{center}*\quad *\quad *\end{center}\medskip
Formula \eqref{eq:fsk_main} shows us that $\mskn skn$ is a polynomial in $n$ of degree $s-1$, which leads us to the following.
\begin{definition}
For any natural numbers $s$ and $k$ we will define \textbf{Moser polynomial} $\fsk sk(x)$ by the formula
\begin{equation}
\label{eq:fsk_psum}
\fsk sk(x) = \sum_{j=1}^s (-1)^{j-1} j^{k-1} \binom{x}{s-j} \pp
\end{equation}
The \textbf{normalized Moser polynomial} $(s-1)!\fsk sk(x)$ has integer coefficients and will be denoted by $\nfsk sk(x)$.
\end{definition}
This means that for natural numbers $s$, $k$ and $n$ such that $n\geqslant s$ and $n\geqslant k$ we have
$$
\mskn skn = \fsk sk(n) \pp
$$
Formula \eqref{eq:fsk_psum} can be rewritten to explicitly show the Moser polynomial's coefficients:
$$
\fsk sk(x) = \sum_{j=0}^s x^j \left( (-1)^{s+j-1} \sum_{i=j}^s \frac{(s-i)^{k-1} }{i!}
\stirling{i}{j} \right) \pc
$$
where $\stirling{i}{j}$ denotes the unsigned (positive) Stirling number of the first kind (see Chapter 6 in \cite{GraKnuPat}). Thus for any Moser polynomial signs of its coefficients alternate.
\medskip
The following theorem (proved in \cite{GorFraStr}) follows directly from Theorem \ref{thm:fskn} combined with the definition of the Moser polynomials.
\begin{theorem}
\label{thm:moser_problem}
Given natural numbers $n$ and $s$ such that $n\geqslant s$, consider sequence $\fsk sk(n)$, $k = 1, \ldots, n$. If none of these values vanish, then the answer to the Moser problem is positive---in other words, any $n$-multiset $A$ can be uniquely recovered from multiset $\sms As$ of its $s$-sums.
\end{theorem}
\begin{proof}
To begin with, $\mfp_1(\sms As) = \mskn s1n \mfp_1(A)$. Since $\mskn s1n = \fsk s1(n) \neq 0$, $\mfp_1(A)$ is fully determined by $\mfp_1(\sms As)$, and, therefore, by the multiset $\sms As$.
Now easy induction by $k$, using formula \eqref{eq:fsk_sigma} and the fact that $\mskn skn = \fsk sk(n)$ is nonzero, shows that for any $k \leqslant n$ the power-sum $\mfp_k(A)$ is determined by the power-sums $\mfp_1(\sms As)$, \ldots, $\mfp_k(\sms As)$, and therefore, by the multiset $\sms As$.
Finally, a multiset of $n$ numbers is fully determined by the sequence of its first $n$ power-sums, which concludes the proof.
\end{proof}
In this section we will prove several important properties of the Moser polynomials and their values.
\begin{prop}
\label{thm:fsk_nduality}
For any natural numbers $s$, $k > 1$, and $n\geqslant k$, the equality $\fsk sk(n) = (-1)^k\fsk {n-s}k(n)$ holds true.
\end{prop}
\begin{proof}
If $s \leqslant 0$ or $s\geqslant n$ then our equation follows from definition of Moser polynomials and from the previous item. Hence we can assume that $0 < s < n$.
Now let us again employ the $n$-multiset $A = \Znk$ from Lemma \ref{thm:fskn_znk}; we know that $\mfp_k(\sms As) = k \fsk sk(n)$.
Let multiset $B$ be a reflection of $A$ in complex plane with respect to zero; in other words, $B = -A$. Then, obviously, $\sms A{n-s} = \sms Bs$ and $\mfp_k(B) = (-1)^k\mfp_k(A)$.
Thus we have
$$
k \fsk {n-s}k(n) \mfp_k(A) = \mfp_k(\sms A{n-s}) = \mfp_k(\sms Bs) = (-1)^k k \fsk sk(n) \pp
$$
Divide this equality by $k$ and we are done.
\end{proof}
\begin{prop} For any natural numbers $s$, $k$, and $n$ the following recurrency equations hold.
\label{thm:MoserPolyRecs}
\begin{enumerate}
\item\label{itm:mskn_recur_sx} $\vfs$
$\mskn sk{n+1} = \mskn skn + \mskn {s-1}kn$.
\item\label{itm:mskn_recur_ksx} $\vfs$
$\mskn s{k+1}n = s\cdot \mskn skn - n\cdot \mskn {s-1}k{n-1}$.
\end{enumerate}
\end{prop}
\begin{proof} We begin with an easy but useful lemma.
\begin{lemma}
\label{thm:sk_tz}
Given $n$-multiset $A$ and number $z$ we construct $n$-multiset $A' = T_z(A)$, where the translation function $T_z:\mbr \rightarrow \mbr$ is defined by formula $T_z(x) = x+z$. Then the following equality holds true
$$
\mfp_k(A') = \sum_{i=0}^k \binom{k}{i} \mfp_{k-i}(A)\, z^i \pp
$$
\end{lemma}
\begin{proof}
Let us assume that $A = \{a_1, \ldots, a_n\}$. Then
\begin{multline}
\mfp_k(A') = \sum_{j=1}^n (z+a_j)^k =
\sum_{j=1}^n\sum_{i=0}^k \binom{k}{i} z^i a_j^{k-i} = \\
= \sum_{i=0}^k \sum_{j=1}^n \binom{k}{i} z^i a_j^{k-i} =
\sum_{i=0}^k \binom{k}{i} z^i \sum_{j=1}^n a_j^{k-i} =
\sum_{i=0}^k \binom{k}{i} z^i \mfp_{k-i}(A) \nn
\end{multline}
\end{proof}
Item \ref{itm:mskn_recur_sx} immediately follows from formula \eqref{eq:fsk_psum}. However, for variety sake, we will present here a proof which relies only on definition of $\mskn skn$ given in \eqref{eq:fsk_sigma}.
First, the case $s\leqslant 1$ is obvious. Therefore we can assume that $s\geqslant 2$.
Second, we can assume that $n\geqslant k$ and $n\geqslant s$.
Now let us consider some $n$-multiset $A = \{a_1, \ldots, a_n\}$, arbitrary number $z$ and $n+1$-multiset $B = A \msu \{z\}$. Then multiset $\sms Bs$ is, obviously, equal to $\sms As \msu T_z(\sms A{s-1})$.
Let us consider all the expressions below as polynomials in $z$. For instance, power-sum $\mfp_k(\sms Bs)$ taken as such polynomial has degree $k$.
By definition of $\mskn skn$ we have
\begin{align}
\mfp_k(\sms Bs) &= \mskn sk{n+1} \mfp_k(B) +
\tqskn sk{n+1}(\mfp_1(B), \mfp_2(B), \ldots, \mfp_{k-1}(B)) \nc
\mfp_k(\sms{A}s) &= \mskn skn \mfp_k(A) +
\tqskn skn(\mfp_1(A), \mfp_2(A), \ldots, \mfp_{k-1}(A)) \nn
\end{align}
and we also know that
\begin{equation}
\label{eq:skbs_skas}
\mfp_k(\sms Bs) = \mfp_k(\sms{A}s) + \mfp_k(T_z(\sms A{s-1})) \pp
\end{equation}
Using lemma \ref{thm:sk_tz} we get
\begin{multline}
\mfp_k(T_z(\sms{A}{s-1})) = \sum_{i=0}^k \binom{k}{i} \mfp_{k-i}(\sms A{s-1})\, z^i = \\
= \sum_{i=0}^k \binom{k}{i}\left( \mskn {s-1}{k-i}n \mfp_{k-i}(A) + \tqskn {s-1}{k-i}n(\mfp_1(A),\mfp_2(A),\ldots,\mfp_{k-i-1}(A)) \right)z^i \nn
\end{multline}
Now, let us compute the constant term of polynomials on both sides in \eqref{eq:skbs_skas}---for that we set $z=0$. With $z=0$ and $k>0$ we have $\mfp_k(B) = \mfp_k(A)$. Thus, the left-hand side of \eqref{eq:skbs_skas} evaluated at $z=0$ equals
$$
\mfp_k(\sms Bs) = \mskn sk{n+1} \mfp_k(A) +
\tqskn sk{n+1}(\mfp_1(A), \mfp_2(A), \ldots, \mfp_{k-1}(A))\pc
$$
and doing the same for the right-hand side of \eqref{eq:skbs_skas} we obtain
$$
\mskn skn \mfp_k(A) + \tqskn skn(\mfp_1(A), \mfp_2(A), \ldots, \mfp_{k-1}(A)) +
\mskn {s-1}kn \mfp_{k}(A) + \tqskn{s-1}kn(\mfp_1(A),\mfp_2(A),\ldots,\mfp_{k-1}(A)) \pp \nn
$$
Coefficients at terms $\mfp_k$ in these two expressions must coincide, Q.E.D. We have just proved the required equality for infinitely many values of $n$---namely, for all sufficiently large natural numbers. Thus, we have also proved the following polynomial identity
\begin{equation}
\label{eq:fsk_recur_sx}
\fsk sk(x+1) = \fsk sk(x) + \fsk {s-1}k(x) \pp
\end{equation}
\smallskip
The most straightforward way to prove item~\ref{itm:mskn_recur_ksx} is to use Theorem \eqref{thm:fskn}. So instead of item~\ref{itm:mskn_recur_ksx} we have to prove polynomial identity
\begin{equation}
\label{eq:fsk_recur_ksx}
\fsk s{k+1}(x) = s \cdot \fsk sk(x) - x \cdot \fsk {s-1}k(x-1) \pp
\end{equation}
To demonstrate that some two polynomials are identical it is sufficient to prove that sequences of their coefficients are identical. Here instead of representing each one of these polynomials in the regular way---as a sum of power monomials $x^j$---and then comparing their coefficients, we will use another basis of the polynomial ring $\mbr[x]$, namely the basis of the ``falling power'' polynomials. Thus, we express each polynomial as a sum $\sum_{j=0}^m\beta_jx^{[j]}$ and then show that their ``falling-power'' coefficients $\beta_j$ coincide.
For the the left-hand side of our recurrence equation we have
$$
\fsk s{k+1}(x) = \sum_{j=0}^s (-1)^{j-1} j^{k} \binom{x}{s-j} =
\sum_{j=0}^s x^{[s-j]} \frac{(-1)^{j-1} j^{k}}{(s-j)!}
$$
and for the right-hand side, using equality $a^{[b]} = a\cdot (a-1)^{[b-1]}$,
\begin{align}
s \cdot \fsk s{k}(x) - x \cdot \fsk {s-1}{k}(x-1) &=
s \sum_{j=0}^s (-1)^{j-1} j^{k-1} \binom{x}{s-j} - x \sum_{j=0}^{s-1} (-1)^{j-1} j^{k-1} \binom{x-1}{s-1-j} =
\nc
&= \sum_{j=0}^s x^{[s-j]} \frac{(-1)^{j-1} s j^{k-1}}{(s-j)!} -
x \sum_{j=0}^{s-1} (x-1)^{[s-1-j]} \frac{(-1)^{j-1} j^{k-1}}{(s-1-j)!}
\nc
&= \sum_{j=0}^s x^{[s-j]}\frac{(-1)^{j-1} j^{k-1}}{(s-j)!}
\left( s - (s-j) \right) = \sum_{j=0}^s x^{[s-j]} \frac{(-1)^{j-1} j^{k}}{(s-j)!} \pp
\nn
\end{align}
\end{proof}
\medskip
Applying recurrency equation \eqref{eq:fsk_recur_sx} multiple times gives us the following.
\begin{prop}
\label{thm:fsk_recurrs_d}
For any number $x$ and any natural numbers $d$, $s$ and $k$ equalities
\begin{align}
\label{eq:fsk_recurr_d}
\fsk {s+d}k(x+d) &= \sum_{j=0}^d \binom{d}{j} \fsk {s+j}k(x) \nc
\fsk sk(x+d) &= \fsk sk(x) + \sum_{j=0}^{d-1} \fsk {s-1}k(x+j) \pp \nn
\end{align}
hold true.
\end{prop}
Our next corollary shows how the formula for the Moser polynomials can be rewritten using the Eulerian numbers.
\begin{prop}
\label{thm:fsk_moser_eul}
\begin{equation}
\label{eq:fsk_moser_eul}
\fsk sk(x) = (-1)^{s-1}\sum_{j=0}^{s-1} (-1)^j \eulerian{k-1}{s-j-1} \binom{x-k}{j} \pp
\end{equation}
\end{prop}
\begin{proof}
This can be proved using the recurrency equations \ref{thm:MoserPolyRecs} but there is a more straightforward way. Regardless, we will need the explicit formula for Eulerian numbers (see \cite{GraKnuPat})
\begin{equation}
\label{eq:eulerian}
\eulerian{n}{m} = \sum_{j=0}^m(-1)^j\binom{n+1}j (m+1-j)^n
\end{equation}
(which, by the way, immediately allows us to see that $\fsk sk(k) = (-1)^{s-1}\eulerian{k-1}{s-1}$), and the well-known summation property of binomial coefficients
$$
\binom{x}{a} = \sum_{j=0}^{\infty} \binom{x-b}{j} \binom{b}{a-j} \pc
$$
which holds true for any integers $a$ and $b$.
Now,
\begin{align}
(-1)^{s-1}\fsk sk(x) &= \sum_{p=0}^s(-1)^p \binom{x}{p} (s-p)^{k-1} \nc
&= \sum_{p=0}^s(-1)^p (s-p)^{k-1} \sum_{j=0}^{\infty} \binom{x-k}{j}\binom{k}{p-j} \nc
&= \sum_{j=0}^{\infty} \binom{x-k}{j} \sum_{p=0}^s (-1)^p (s-p)^{k-1} \binom{k}{p-j} \nc
&= \sum_{j=0}^{\infty} \binom{x-k}{j} \sum_{p=-j}^{s-j} (-1)^{p+j} (s-p-j)^{k-1} \binom{k}{p} \nc
&= \sum_{j=0}^{\infty} \binom{x-k}{j} (-1)^{j} \sum_{p=0}^{s-j} (-1)^{p} \binom{k}{p} (s-p-j)^{k-1} \nc
&= \sum_{j=0}^{\infty} \binom{x-k}{j} (-1)^j \eulerian{k-1}{s-j-1} \nc
&= \sum_{j=0}^{s-1} (-1)^j \binom{x-k}{j} \eulerian{k-1}{s-j-1} \pp \nn
\end{align}
\end{proof}
When $x=n$ is an integer, this can be rewritten using the backward difference operator $\nabla$. Namely, in this case the formula \eqref{eq:fsk_moser_eul} is equivalent to
$$
\fsk sk(n) = (-1)^{s-1} \, \nabla^{n-k}_{s-1} \mca^{(k-1)} \pc
$$
where $\mca^{(k-1)}$ is the sequence of numbers $\eulerian{k-1}{m}$ constituting the $(k-1)$th row of the Eulerian triangle. That is, the Moser polynomial's value at $n$ is the $(n-k)$th discrete backward derivative (or corresponding discrete ``integral'', if $n<k$) of sequence $\mca^{(k-1)}$, computed at its $(s-1)$th term. Or, equivalently,
$$
(x-1)^{n-k}\,\mca_{k-1}(x) = \sum_{s=0}^n (-1)^{s-1}\fsk sk(n)x^s
$$
where $\mca_{k-1}(x)$ is the $(k-1)$th Eulerian polynomial.
\smallskip
One interesting corollary of the previous Proposition and Lemma \ref{thm:fskn_znk}.
\begin{corollary}
\label{thm:zskn_eulerian}
Consider set $\Zkk$ (see \eqref{eq:znk}) which consists of all complex roots of unity of $k$th order. Then eulerian number $\eulerian{k-1}{s-1}$ equals the sum of $k$th powers of all $s$-sums of $\Zkk$ multiplied by $(-1)^{s-1}/k$. In other words,
$$
\eulerian{k-1}{s-1} = (-1)^{s-1} \, \mfp_k(\sms{\Zkk}{s}) / k \pp
$$
\end{corollary}
\begin{proof}
Since $k = \mfp_k(\Zkk)$ both these numbers are equal to $(-1)^{s-1} \fsk sk(k) $.
\end{proof}
Finally, two more formulas. They express Moser polynomials using Stirling numbers of the second kind.
\begin{prop}
\label{thm:fsk_moser_stirling}
\begin{align}
\label{eq:fsk_via_stirling1}
\fsk sk(x) &= \sum_{i=1}^k (-1)^{i-1}(i-1)! \Stirling{k}{i} \binom{x-i}{s-i} \\
\label{eq:fsk_via_stirling2}
\fsk sk(x) &= \sum_{i=1}^k (-1)^{i+k-1} i! \Stirling{k-1}{i} \binom{x-i-1}{s-1}
\end{align}
(obviously, all the summands with index $i > k$ are zeros so the upper summation limit could be, if necessary, changed to infinity; similarly, the lower summation limit can be changed to zero or even to $-\infty$.)
\end{prop}
\begin{proof}
We can assume without loss of generality that $x = n$ is a natural number. Then, using the explicit formula for the Stirling numbers of the second kind (see identity \textbf{6.19} in \cite{GraKnuPat})
$$
\Stirling{n}{m} = \frac 1{m!}\sum_{j=0}^m (-1)^{m-j}\binom{m}{j} j^n \pc
$$
we turn the right-hand side of \eqref{eq:fsk_via_stirling1} into
\begin{align}
\sum_{i=1}^k (-1)^{i-1}(i-1)! \Stirling{k}{i} \binom{n-i}{n-s}
&=
\sum_{i=1}^s \sum_{j=1}^s (-1)^{i-1}(i-1)! \frac 1{i!} (-1)^{i-j} \binom ij j^k \binom{n-i}{n-s}
\nc
&=
\sum_{i=1}^s \sum_{j=1}^s (-1)^{j-1} j^{k-1} \binom{n-i}{n-s} \binom{i-1}{j-1}
\nc
&=
\sum_{j=1}^s (-1)^{j-1} j^{k-1} \sum_{i=1}^s \binom{i-1}{j-1} \binom{n-i}{n-s}
\nn
\end{align}
(notice that the limits' adjustments done here do not affect the sums.) Now we see that it is sufficient to prove that for any $s$, $j$, and $n$ the equality
\begin{equation}
\label{eq:ij_nis}
\sum_{i=0}^s \binom{i}{j} \binom{n-i}{n-s} = \binom{n+1}{s-j}
\end{equation}
holds true. To do that, consider the generating function
$$
\frac{t^j}{(1-t)^{j+1}} = \sum_{i=0}^{\infty} \binom{i}{j} t^i
$$
and compare coefficients at $t^n$ on the both sides of the equality
$$
\frac{t^j}{(1-t)^{j+1}} \cdot \frac{t^{n-s}}{(1-t)^{n-s+1}} = \frac{t^{n-s+j}}{(1-t)^{n-s+j+2}} =
\frac 1t \cdot \frac{t^{n-s+j}}{(1-t)^{n-s+j+1}} \pp
$$
Note: Formula \eqref{eq:ij_nis} is basically the same as a slightly more general identity \textbf{5.26} in \cite{GraKnuPat}, which can be proved in the exactly same manner.
To prove \eqref{eq:fsk_via_stirling2}, we will start with equality
$$
\eulerian{n}{k} = \sum_{i=k}^{n-1} (-1)^{i-k} \binom{i}{k}(n-i)! \Stirling{n}{n-i}
$$
proved in \cite{Kno}. Substituting that into \eqref{eq:fsk_moser_eul} we obtain
\begin{align}
\fsk sk(n) &= (-1)^{s-1}\sum_{j=0}^{s-1} (-1)^j \eulerian{k-1}{s-j-1} \binom{x-k}{j} \nc
&= (-1)^{s-1}\sum_{j=0}^{s-1} \sum_{i=s-j-1}^{k-2} (-1)^j \binom{n-k}{j} (-1)^{i-(s-j-1)} \binom{i}{s-j-1}(k-1-i)! \Stirling{k-1}{k-1-i} \nc
&= (-1)^{s-1}\sum_{i=0}^{k-1} (-1)^{i-s+1} (k-1-i)! \Stirling{k-1}{k-1-i} \sum_{j=0}^{s-1} \binom{n-k}{j} \binom{i}{s-j-1} \nc
&= \sum_{i=0}^{k-1} (-1)^i (k-1-i)! \Stirling{k-1}{k-1-i} \binom{n-k+i}{s-1} \pp
\nn
\end{align}
and making substitution $i \rightarrow k-1-i$, we have formula \eqref{eq:fsk_via_stirling2} as well.
\end{proof}
\newpage\ \medskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,308 |
Q: Perl: match symbols in a file, print data from a previous line I have a small data set in an XML format:
<symbolgroupdef id="bin_11-QQQQ">
<symbol>QQQ</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_6-AAPL">
<symbol>AAPL</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7-BIDU">
<symbol>BIDU</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7">
<symbol>AAPL</symbol>
<symbol>IBM</symbol>
</symbolgroupdef>
I want to print out the symbolgroupdef and the symbol where a certain symbol exists. The symbol may appear under several symbolgroupdef groups.
Here is the code I have written so far:
#!/usr/bin/perl
use warnings;
use strict ;
$symbol = $ARGV[0] ;
my $sym_file = "/data/xmlconfig/config.xml";
open my $sym_fh, '<', $sym_file or die $!;
while($line = <$sym_fh>) {
if (my $line =~ /\<symbolgroupdef id=\".*\"\>/) {
print $line ;
sleep 1;
}
}
Basically what I want is something with will find the symbolsgroupdef id line, look for the specified symbol under it, and if it finds it, print the symbolgroupdef is line and the symbol under it. The symbol will be a command line entry and specified by $ARGV[0]
in the above case theses two lines should be printed
<symbolgroupdef id="bin_6-AAPL">
<symbol>AAPL</symbol>
<symbolgroupdef id="bin_7">
<symbol>AAPL</symbol>
I don't have any modules on this machine, and can't install any on this machine. Please forgive me for parsing XML without a module.
A: Here's a solution based on the idea of keeping a record of the most recent <symbolgroupdef> attribute. It stores the id in $sgline, although you can store the whole line if you want. When a line turns up with the correct value in the symbol element, you can print out $sgline.
#!/usr/bin/perl
use warnings;
use strict;
my $id = $ARGV[0];
# uncomment these to use your file
#my $sym_file = "/data/xmlconfig/config.xml";
#open my $sym_fh, '<', $sym_file or die $!;
my $sgline = '';
# change DATA to $sym_fh to use your file
while (<DATA>) {
# match the symbolgroupdef element
if (m#<symbolgroupdef id="(.+?)">#) {
$sgline = $1; # or store the whole line using $sgline = $_;
}
# match the symbol element with the appropriate ID
elsif (m#<symbol>$id</symbol>#) {
print "$sgline\n";
}
}
__DATA__
<symbolgroupdef id="bin_11-QQQQ">
<symbol>QQQ</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_6-AAPL">
<symbol>AAPL</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7-BIDU">
<symbol>BIDU</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7">
<symbol>AAPL</symbol>
<symbol>IBM</symbol>
</symbolgroupdef>
Output:
bin_6-AAPL
bin_7
A: Don't use a regex to parse XML. Instead use an actual XML Parser.
I'd recommend using XML::LibXML:
use strict;
use warnings;
use XML::LibXML;
my $xml = XML::LibXML->load_xml(IO => \*DATA);
for my $group ($xml->findnodes(q{//symbolgroupdef/symbol[text()='BIDU']/..})) {
print $group->getAttribute('id'), "\n";
}
__DATA__
<root>
<symbolgroupdef id="bin_11-QQQQ">
<symbol>QQQ</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_6-AAPL">
<symbol>AAPL</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7-BIDU">
<symbol>BIDU</symbol>
</symbolgroupdef>
<symbolgroupdef id="bin_7">
<symbol>AAPL</symbol>
<symbol>IBM</symbol>
</symbolgroupdef>
</root>
Outputs:
bin_7-BIDU
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
Eddie Mair 4pm - 7pm
Nick Ferrari Rows With Guest Who Wants To Ban Short Prison Sentences
When this guest proposed banning prison sentences of less than six months, Nick Ferrari insisted he wanted these criminals in prison for longer.
Justice Secretary David Gauke suggested that short-term prison sentences should be scrapped as they have a poor record in stopping re-offending.
Peter Dawson, the Director of the Prison Reform Trust, agreed and suggested we need to help the perpetrators instead.
But Nick did not agree with that idea, saying: "We should put these people in for longer, shouldn't we?"
Mr Dawson responded: "The moment they come out of prison, they will start doing that again.
"When you send someone to prison for a short period, you postpone that issue. You don't save people from shoplifting, you make it worse."
Nick Ferrari wants criminals locked up for longer, not given shorter sentences. Picture: PA / LBC
Nick asked what the alternative to jail was and Mr Dawson said: "You've got to understand why a person is shoplifting. For some people, it will just be pure wickedness, but for other people, it is likely to be things that go much further back in their life.
"They need help. If you haven't got a job, if you haven't got an education, if you haven't got somewhere to live, then you're in a pretty poor place. You have to get alongside them."
But Nick asked: "Get alongside them? What does that mean? I don't understand.
"What are we going to do? There are 11,500 people behind bars for shoplifting offences, are they all going to get their own council flat?
"What are we going to do if someone keeps nicking stuff from Sainsbury's?"
They then went on to do some role-playing about how to deal with a shop-lifter, which became very entertaining.
How to watch Joe Biden's Inauguration Day in the UK from start time to TV schedule
Storm Christoph 'danger to life' weather warning triggers emergency response in England | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,128 |
The objective of this Competence Platform on Energy Crop and Agroforestry Systems for Arid and Semi-arid Ecosystems - Africa (COMPETE) is to stimulate bioenergy implementation in arid and semi-arid regions in Africa.
WIP is the coordinator of the COMPETE project. The COMPETE partnership comprises 18 European, 3 international (African Development Bank; Food and Agriculture Organisation of the United Nations, Conservation International) and 23 non-European partners. 13 partners are from African countries. COMPETE was co-financed by the European Commission in the 6th Framework Programme - Specific Measures in Support of International Cooperation (INCO).
Exploit the benefits of innovative bioenergy solutions in Africa with respect to sustainable rural development and improved livelihoods, increased energy access and income generation, alternative markets for agricultural products, security of energy supply, and diversification of energy sources.
Avoid the dangers of negative social and environmental implications, with regards to biodiversity, water availability, land competition, land ownership, insufficient value creation for local farmers, and the 'fuel versus food' debate.
In December 2010 the partnership of the COMPETE Bioenergy Competence Platform for Africa consist of 230 stakeholders from Africa, Asia, Europe, and Latin America. 140 stakeholders come from 28 African countries. | {
"redpajama_set_name": "RedPajamaC4"
} | 144 |
Q: NUnit Test Project - Unable to create Xamarin MockContext - java-interop could not be found I have an NUnit test project which works fine on its own until I start trying to bring Xamarin into the mix. Here is what my test looks like just to start:
[Test]
public void Test1()
{
//ARRANGE
var mockContext = new Android.Test.Mock.MockContext();
var textView = new Android.Widget.TextView(mockContext);
//ASSERT
var response = 1;
Assert.AreEqual(1, response);
}
This compiles fine, but when I run it, I get an error from the first line:
Message: System.DllNotFoundException : Unable to load DLL 'java-interop' or one of its dependencies: The specified module could not be found. (Exception from HRESULT: 0x8007007E)
I have verified that this reference java.interop is included in the project and also confirmed that the .dll is in the location on the drive that the reference is pointing to:
Is it just not possible to create instances of Xamarin objects in the NUnit test project?
I've seen this brought up numerous times from my research but the conclusion is usually along the lines of "Not sure if its possible, just avoid referencing Xamarin" Does anyone know for sure?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,078 |
Pseudacraea conradti is een vlinder uit de onderfamilie Limenitidinae van de familie Nymphalidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1893 door Charles Oberthür.
Limenitidinae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 303 |
Хрістіан Мозер (, 20 грудня 1972) — австрійський стрибун з трампліна, олімпійський медаліст.
Виступи на Олімпіадах
Зовнішні посилання
Досьє на sport.references.com
Австрійські стрибуни з трампліна
Стрибуни з трампліна на зимових Олімпійських іграх 1994
Австрійські бронзові олімпійські медалісти | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,773 |
<Record>
<Term>Epidural Abscess</Term>
<SemanticType>Disease or Syndrome</SemanticType>
<ParentTerm>Abscess</ParentTerm>
<ParentTerm>Central Nervous System Bacterial Infections</ParentTerm>
<ParentTerm>Perimeningeal Infections</ParentTerm>
<ParentTerm>Central Nervous System Infections</ParentTerm>
<ParentTerm>Spinal Cord Diseases</ParentTerm>
<ClassificationPath>Nervous System Diseases/Central Nervous System Diseases/Central Nervous System Infections/Perimeningeal Infections/Epidural Abscess</ClassificationPath>
<ClassificationPath>Nervous System Diseases/Central Nervous System Diseases/Central Nervous System Infections/Central Nervous System Bacterial Infections/Epidural Abscess</ClassificationPath>
<ClassificationPath>Nervous System Diseases/Central Nervous System Diseases/Spinal Cord Diseases/Epidural Abscess</ClassificationPath>
<ClassificationPath>Bacterial Infections and Mycoses/Infection/Suppuration/Abscess/Epidural Abscess</ClassificationPath>
<ClassificationPath>Bacterial Infections and Mycoses/Bacterial Infections/Central Nervous System Bacterial Infections/Epidural Abscess</ClassificationPath>
<ClassificationPath>Nervous System Diseases/Central Nervous System Diseases/Central Nervous System Infections/Epidural Abscess</ClassificationPath>
<BroaderTerm>Abscess</BroaderTerm>
<BroaderTerm>Infection</BroaderTerm>
<BroaderTerm>Central Nervous System Diseases</BroaderTerm>
<BroaderTerm>Central Nervous System Bacterial Infections</BroaderTerm>
<BroaderTerm>Perimeningeal Infections</BroaderTerm>
<BroaderTerm>Bacterial Infections</BroaderTerm>
<BroaderTerm>Suppuration</BroaderTerm>
<BroaderTerm>Epidural Abscess</BroaderTerm>
<BroaderTerm>Central Nervous System Infections</BroaderTerm>
<BroaderTerm>Nervous System Diseases</BroaderTerm>
<BroaderTerm>Bacterial Infections and Mycoses</BroaderTerm>
<BroaderTerm>Spinal Cord Diseases</BroaderTerm>
<Synonym>Cranial Epidural Abscess</Synonym>
<Synonym>Cranial Extradural Abscess</Synonym>
<Synonym>Intracranial Extradural Abscess</Synonym>
<Synonym>Spinal Extradural Abscess</Synonym>
<Synonym>Intracranial Epidural Abscess</Synonym>
<Synonym>Spinal Epidural Abscess</Synonym>
<Synonym>Extradural Abscess</Synonym>
<Synonym>Epidural Abscess</Synonym>
<Synonym>Intracranial Extradural Abscesses</Synonym>
<Description>Circumscribed collections of suppurative material occurring in the spinal or intracranial EPIDURAL SPACE. The majority of epidural abscesses occur in the spinal canal and are associated with OSTEOMYELITIS of a vertebral body; ANALGESIA, EPIDURAL; and other conditions. Clinical manifestations include local and radicular pain, weakness, sensory loss, URINARY INCONTINENCE, and FECAL INCONTINENCE. Cranial epidural abscesses are usually associated with OSTEOMYELITIS of a cranial bone, SINUSITIS, or OTITIS MEDIA. (From Adams et al., Principles of Neurology, 6th ed, p710 and pp1240-1; J Neurol Neurosurg Psychiatry 1998 Aug;65(2):209-12)</Description>
<Source>MeSH</Source>
</Record>
| {
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{"url":"https:\/\/brilliant.org\/problems\/an-easy-one-for-you\/","text":"# An easy one for you\n\nAlgebra Level 2\n\nIf the roots of the quadratic equation $$x^{2} - px + q = 0$$ differ by unity,then\n\n\u00d7","date":"2018-04-22 06:46:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7478647828102112, \"perplexity\": 1347.2906270417252}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125945497.22\/warc\/CC-MAIN-20180422061121-20180422081121-00399.warc.gz\"}"} | null | null |
using System;
using System.Numerics;
using Xunit;
namespace NeinMath.Tests
{
public class IntegerFunctionsData : IntegerDataSource
{
public TheoryData<Integer, int> Sgn()
{
var data = new TheoryData<Integer, int>();
foreach (var i in BigIntegers())
data.Add(ToInteger(i), i.Sign);
return data;
}
public TheoryData<Integer, Integer> Abs()
{
var data = new TheoryData<Integer, Integer>();
foreach (var i in BigIntegers())
data.Add(ToInteger(i), ToInteger(BigInteger.Abs(i)));
return data;
}
public TheoryData<Integer, Integer, Integer> Min()
{
var data = new TheoryData<Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers()))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2),
ToInteger(BigInteger.Min(i.Item1, i.Item2)));
return data;
}
public TheoryData<Integer, Integer, Integer> Max()
{
var data = new TheoryData<Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers()))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2),
ToInteger(BigInteger.Max(i.Item1, i.Item2)));
return data;
}
public TheoryData<Integer, Integer, Integer> Gcd()
{
var data = new TheoryData<Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers(x => x != 0)))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2),
ToInteger(BigInteger.GreatestCommonDivisor(i.Item1, i.Item2)));
return data;
}
public TheoryData<Integer, Integer, Integer> Lcm()
{
var data = new TheoryData<Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers(x => x != 0)))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2),
ToInteger(BigInteger.Abs(i.Item1 * i.Item2)
/ BigInteger.GreatestCommonDivisor(i.Item1, i.Item2)));
return data;
}
public TheoryData<Integer, Integer> ModInv()
{
var data = new TheoryData<Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers(x => x != 0,
BigInteger.Abs)))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2));
return data;
}
public TheoryData<Integer, int, Integer> Pow()
{
var data = new TheoryData<Integer, int, Integer>();
foreach (var i in Items(BigIntegers(), Integers(x => Math.Abs(x % 10))))
data.Add(ToInteger(i.Item1), i.Item2,
ToInteger(BigInteger.Pow(i.Item1, i.Item2)));
return data;
}
public TheoryData<Integer, double, double> Log()
{
var data = new TheoryData<Integer, double, double>();
foreach (var i in Items(BigIntegers(), Floats()))
data.Add(ToInteger(i.Item1), i.Item2,
BigInteger.Log(i.Item1, i.Item2));
return data;
}
public TheoryData<Integer, Integer, Integer, Integer> ModPow()
{
var data = new TheoryData<Integer, Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(),
BigIntegers(x => BigInteger.Abs(x)),
BigIntegers(x => x != 0, BigInteger.Abs)))
data.Add(ToInteger(i.Item1), ToInteger(i.Item2), ToInteger(i.Item3),
ToInteger(BigInteger.ModPow(i.Item1, i.Item2, i.Item3)));
return data;
}
public TheoryData<Integer, Integer, Integer, Integer> DivRem()
{
var data = new TheoryData<Integer, Integer, Integer, Integer>();
foreach (var i in Items(BigIntegers(), BigIntegers(x => x != 0)))
{
var remainder = default(BigInteger);
var expected = BigInteger.DivRem(i.Item1, i.Item2, out remainder);
data.Add(ToInteger(i.Item1), ToInteger(i.Item2), ToInteger(expected),
ToInteger(remainder));
}
return data;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: Sonarqube gives me issues of deleted codes and can't filter issues I'm trying to get Issues from a fileName on sonarqube with this code
public class SonarqubeServiceImpl implements SonarqubeService {
private final String SONAR_URL = "http://localhost:9000/";
private final String PROJECT_KEY = "refactor2";
private final String SRC_FOLDER = "src/main/java/com/uca/refactor2/activities";
private String EXECUTE_SONAR;
private String GET_ISSUES_URL = SONAR_URL + "api/issues/search?q=";
private static String POM_PATH = "pom.xml";
private RestTemplate restTemplate = new RestTemplate();
private InvocationRequest request;
private Invoker invoker;
@PostConstruct
private void init() {
System.out.println("PostConstruct");
buildSonarCommand();
configureMavenWithSonar();
}
@Override
public void runAnalysis() throws MavenInvocationException {
// Assuming everything is set up (done in init)
invoker.execute(request);
}
@Override
public IssuesResponse getIssuesFromFileName(String fileName) {
String URL = GET_ISSUES_URL + fileName;
return restTemplate.getForObject(URL, IssuesResponse.class);
}
private void configureMavenWithSonar() {
request = new DefaultInvocationRequest();
request.setPomFile(new File(POM_PATH));
request.setGoals(Collections.singletonList(EXECUTE_SONAR));
invoker = new DefaultInvoker();
// Set maven home in case env variables are not set
// (and using own installation)
invoker.setMavenHome(new File("apache-maven-3.5.2"));
}
private void buildSonarCommand() {
StringBuilder builder = new StringBuilder();
builder.append("sonar:sonar ");
builder.append(String.format("-Dsonar.host.url=%s ", SONAR_URL));
builder.append(String.format("-Dsonar.projectKey=%s ", PROJECT_KEY));
builder.append(String.format("-Dsonar.sources=%s ", SRC_FOLDER));
EXECUTE_SONAR = builder.toString();
}
}
I'm having two problem with this code. First one is that this is retrieving issues from filenames of source folder that were deleted long time ago and I don't understand why, because if I enter to sonarqube local web api on localhost/9000 that code is not there
and second and most important, I'm trying to get Issues from a filename but sonarqube is giving me issues from all the project(even projects when I had more than one), including deleted codes from the first problem I said above.
I'm fetching issues with this URL
http://localhost:9000/api/issues/search?q=" + fileName;
I'm using SonarQube version: 6.7.1
this is my first time with sonarqube so maybe I'm missing something
A: Where did you see that q is a valid parameter? It's not in the list I'm looking at, which is probably why you're getting random results.
You need to be using componentKeys instead, altho the right-hand side is going to be more than just the file name; instead it will be something like projectKey:path/to/file so you'll need to work out the details of that.
Regarding the issues you're getting back on deleted files, closed issues are cleaned out of the database 30 days (default value) later. Since your query is currently pulling a random set of issues, unconstrained by status, that explains seeing these "ghost" issues.
| {
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} | 5,703 |
Dry Fork is a rural unincorporated community in southern Barren County, Kentucky, United States.
References
Unincorporated communities in Barren County, Kentucky
Unincorporated communities in Kentucky | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 259 |
The Women's TRN Thermo sets a new standard for Orca in the entry level triathlon and open water swim wetsuit range. Designed with all the expertise and experience you would expect from 20 years experience. Orca and Wiggle have worked hard to give you a great value entry level/training wetsuit. The Orca TRN Thermo triathlon wetsuit offers 2-4mm Smoothskin Yamamoto neoprene for durability and flexibility, hidden seams for speed, and also thermal protection. The suit offers excellent value for those wanting to have a go at the fastest growing multi-sport or have a spare training suit without breaking the bank. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,114 |
\section{Introduction\label{sec:intro}}
Our current understanding of the nucleon's spin is that it originates from the sum of the spin and orbital angular momentum of the partons confined in the nucleon
\begin{equation}
\displaystyle S_{n} = \nicefrac{1}{2} = \nicefrac{1}{2} \Delta \Sigma + \Delta G + L_{q,\bar{q},g}
\label{eq:spinsum}
\end{equation}
\noindent where $\Delta \Sigma$ is the net quark spin from all flavors, $\Delta G$ is the net gluon spin, and $L_{q,\bar{q},g}$ is the orbital angular momentum contribution from the respective quarks, anti-quarks, and gluons. The net quark spin has been highly constrained by various experiments to be $\Delta \Sigma$\,$\approx$\,30$\%$ \cite{deltasigma}. Recent results from RHIC indicate that the gluon spin contribution is possibly non-zero, $\Delta G$\,=\,20\,$\pm$\,12$\%$ \cite{rhicspin0}. To fully understand the nucleon spin, the orbital angular momentum contributions of the quarks and gluons must be considered as well.
Measurements accessing the orbital angular momentum of the valence quarks have been done by Semi-Inclusive Deep Inelastic Scattering experiments (SIDIS) at HERMES \cite{hermes0}, COMPASS \cite{compass0}, and JLAB \cite{jlab0}. They have indicated that the angular momentum of up quarks is positive ($L_u$\,$>$\,0), while the down quark is negative ($L_d$\,$<$\,0), such that $L_{q}$\,=\,$L_{u}$\,+\,$L_{d}$\,$\approx$\,0 \cite{anselmino0}.
Measurements of the sea quark orbital angular momentum ($L_{\bar{q}}$) remain largely unexplored, as current SIDIS data have poor sensitivity to the sea quarks.
However, Lattice QCD calculations which predict the observed $L_{q}$\,$\approx$\,0, also predict up to 50$\%$ of the nucleon's spin to come from $L_{\bar{q}}$ \cite{lattice0,lattice1}. In addition to this theoretical prediction, indirect hints of a large non-zero $L_{\bar{q}}$ have been observed. Using proton induced Drell-Yan, the E866 experiment at Fermilab found an excess of $\bar{d}$ over $\bar{u}$ quarks, as seen in the $\bar{d}(x)/\bar{u}(x)$ ratio shown in Fig. \ref{fig:e866} \cite{e866}. Several models can explain this $\bar{d}(x)/\bar{u}(x)$ ratio, among them the pion cloud model \cite{pioncloud0}. The pion cloud model describes the proton as a linear combination of a bare proton plus pion-baryon states.
\begin{equation}
\displaystyle |p \rangle = |p \rangle + | N^{0}{\pi}^{+} \rangle + | {\Delta}^{++}{\pi}^{-} \rangle + ...
\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[width=25pc]{e866.pdf}
\end{center}
\caption{\label{fig:e866} The $\bar{d}(x)/\bar{u}(x)$ ratio measured by the E866 collaboration \cite{e866}, compared to predictions from the pion cloud model \cite{pioncloud0}.}
\end{figure}
Since the proton is more likely to be in the pion-nucleon state than the pion-delta state, this leads to an excess of $\bar{d}$ versus $\bar{u}$ quarks. Since the baryon has positive parity and the pion has negative parity, the pion cloud must be in an odd angular momentum state with respect to the baryon, \textit{i.e.} its substituent anti-quark will carry non-zero angular momentum, $L_{\bar{q}}$.
\section{Accessing the Sea Quark Orbital Angular Momentum}
The nucleon's structure is studied and understood by measuring it's parton distribution functions (PDFs). PDFs represent the probability that a given parton flavor carries a certain momentum fraction of the nucleon's momentum. By taking into account the spin of the proton with respect to the longitudinal momentum the proton, as well as spin-orbital properties of the partons, eight leading-twist transverse momentum dependent PDFs (TMDs) can be measured, which are summarized in \cite{jj_scheme, m_scheme}.
The different TMDs probe different properties of the proton. Our focus will be upon the Sivers TMD function, as it directly correlates to the orbital angular momentum of the partons mentioned in Eq. \ref{eq:spinsum}. It describes correlations between the transverse spin of the polarized nucleon and the transverse momentum of the parton with respect to the momentum of the nucleon \cite{sivers0,sivers1}.
The Sivers function was originally formulated to explain the large transverse single spin asymmetries (SSAs) measured for inclusive hadron production in $p+p^{\uparrow}$ collisions \cite{e704}. The quark Sivers function can be directly accessed from measured transverse SSAs in Polarized SIDIS $\left( e+p^{\uparrow} \rightarrow e+h+X \right)$ or Polarized Drell-Yan $\left( p+p^{\uparrow} \rightarrow {\gamma}^{*} \rightarrow \ell^{+} \ell^{-} \right)$. The latter of these has never been carried out experimentally. The same Sivers function is involved in both processes. However, transverse SSAs in SIDIS are the result of an attractive final state interaction whilst in Drell-Yan they are the result of a repulsive initial state interaction. This leads to a predicted sign change in the observed Sivers function
\begin{equation}
\displaystyle f^{\perp q}_{1T}|_{SIDIS} = -f^{\perp q}_{1T}|_{DY}
\end{equation}
Measured transverse SSAs in polarized SIDIS have been used to measure the quark Sivers function, which as already mentioned in Section \ref{sec:intro} has been used to show $L_{q}$\,=\,$L_{u}$\,+\,$L_{d}$\,$\approx$\,0 \cite{anselmino0}. However since valence and sea quarks cannot be isolated from each other in the SIDIS process, the sea quark Sivers function remains poorly constrained.
In measuring the transverse SSA from the polarized Drell-Yan process, the valence and sea quarks can be isolated from each other. In addition, there are no fragmentation functions involved, so the transverse SSA of polarized Drell-Yan, or \textit{Sivers asymmetry}, is directly related to the Sivers function. The E1039 experiment, described in the next section, will be the first to measure the sea quark Sivers function.
\section{The E1039 Experiment}
The E1039 experiment at Fermilab \cite{e1039} will measure the Sivers asymmetry of the Drell-Yan process from an unpolarized proton beam on a transversely polarized proton target via its decay into muons
\begin{equation}
\displaystyle A^{Sivers}_{N} \left( p_{beam} + p^{\uparrow}_{target} \rightarrow \gamma{*}\rightarrow \mu^{+}\mu^{-} \right) \propto \frac{1}{P_{target}}\frac{N^{DY}_L - N^{DY}_R}{N^{DY}_L + N^{DY}_R}
\label{eq:sivan}
\end{equation}
The dimuon pairs will be measured using the existing muon spectrometer used in the SeaQuest Drell-Yan experiment at Fermilab \cite{e906}. The 120 GeV Main Injector beam used for the SeaQuest experiment, with upgraded beam focusing, will also be used for E1039. The new E1039 target will be set upstream from the spectrometer to optimize measurement of DY pair production from annihilation of a high momentum fraction valence quark from the beam, and a sea quark from the target with momentum fraction 0.1\,$<$\,$x_{\bar{q}}$\,$<$0.35.
In order to create the transversely polarized target, it must consist of some ensemble of polarizable protons in a high magnetic field ($B$) and cooled to low temperature ($T$). The \textit{thermal equilibrium} (TE) polarization of spin-$\nicefrac{1}{2}$ particles is described by
\begin{equation}
\displaystyle P_{i} = tanh \left(\frac{g_i \mu_i B}{2 k_B T} \right)
\label{eq:tep}
\end{equation}
\noindent which is derived from Boltzmann's law and the Zeeman interaction of the particle's magnetic moment with the magnetic field using statistical mechanics. Due to the small magnetic moment ($\mu_p$) of the proton, the TE polarization of protons at a low temperature of $T$\,=\,1~Kelvin and a high magnetic field of $B$\,=\,5~Tesla is only $P_{p}$\,$=$\,0.5$\%$. However, electrons have a much higher magnetic moment ($\mu_e$\,$=$\,660$\mu_p$) and can be polarized up to $P_{e}$\,$=$\,99.8$\%$.
This high electron polarization can be transferred to the protons in a paramagnetic material utilizing \textit{Dynamic Nuclear Polarization} (DNP) \cite{dnp0}. The dipole-dipole interaction between the proton and electron leads to hyperfine splitting, as shown in Fig. \ref{fig:dnp}. By providing an RF field equal to the sum (difference) of the electron and proton Larmor frequencies ($\nu_e \pm \nu_p$), the high polarization of the electron is transferred to the proton anti-aligned (aligned) with the magnetic field $B$ (see Fig. \ref{fig:dnp}). Due to the high relaxation time of the proton spin ($\tau_p$\,$>>$\,$\tau_e$), a polarization of the proton can be achieved comparable to the electron polarization.
\begin{figure}[h]
\begin{center}
\includegraphics[width=30pc]{dnp.pdf}
\end{center}
\caption{\label{fig:dnp} Diagram indicating how polarization is passed from electron to nucleon via hyperfine splitting of the energy states \cite{Crabb}.}
\end{figure}
For E1039, the polarized target will be made of irradiated ammonia $\left( NH_{3} \right)$. Irradiation of $NH_{3}$ is done to create paramagnetic centers, and DNP is used to polarize the protons in the hydrogen atoms of $NH_{3}$ \cite{nh30}. When induced at a low temperature of $T$\,=\,1~Kelvin, and a high magnetic field of $B$\,=\,5~Tesla, polarization of the protons in $NH_{3}$ can reach up to $P$\,=\,92$\%$, which is measured using a \textit{Nuclear Magnetic Resonance} (NMR) technique \cite{NIM}.
NMR works by applying an RF at the proton Larmor frequency ($\nu_p$) to a series RLC circuit, where the inductor is within the target material. The polarization is measured by the absorption (emission) of RF in the circuit, which indicates that the polarization of proton's is aligned (anti-aligned) with the magnetic field. The RF Voltage across the circuit increases (decreases) for absorption (emission). The polarization can be measured as DNP is introduced to the target based on the known TE polarization
\begin{equation}
\displaystyle P_{DNP} = \frac{P_{TE}}{V_{TE}} V_{DNP}
\end{equation}
A liquid helium bath is used to cool superconducting coils which provide the 5-$T$ magnetic field to the $NH_{3}$ target. Liquid helium is also used in the refrigerator designed to cool the $NH_{3}$ target chamber. In addition, pumping on the liquid helium vapor in the target chamber lowers its vapor pressure, thus reducing the temperature of the $NH_{3}$ target to $\sim$1-K. A full description of this type of magnet-refrigerator system to be used for E1039 can be found in \cite{Crabb}.
\section{Conclusion}
The measurement of the Sivers asymmetry by the E1039 experiment will provide a first look into the $\bar{u}$-quark Sivers function.
\begin{equation}
\displaystyle A^{Sivers}_{N} \left( p_{beam} + p^{\uparrow}_{target} \rightarrow \gamma{*} \rightarrow \mu^{+}\mu^{-} \right) \propto \frac{f^{u}_{1} \left(x_{beam} \right) \cdot f^{\perp ,\bar{u}}_{1T} \left(x_{target} \right)}{f^{u}_{1}\left( x_{beam} \right) \cdot f^{\bar{u}}_{1}\left( x_{target}\right)}
\end{equation}
Based on the projected beam luminosity and the current target, accelerator, and muon spectrometer efficiencies of SeaQuest, one year of running at E1039 will provide $\sim$2.6\,$\times$\,$10^{18}$ Protons on Target (POT). Using this value and assuming an average polarization of $P$\,=\,80$\%$, Fig. \ref{fig:proj} gives the projected error on the Sivers asymmetry measurement. The current plan is that E1039 will run for two years.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=24pc]{projected.pdf}
\end{center}
\caption{\label{fig:proj} The projected error on the measurement of $A^{Sivers}_N$ vs. Bjorken-$x$ for one year of running based on annual POT\,=\,2.6\,$\times$\,$10^{18}$, compared to two predictions \cite{anselmino1,yuan0}.}
\end{figure}
Combined with QCD theory, this experiment will provide the first measurement of the sign and magnitude of the sea-quark orbital angular momentum within the proton. If $A^{Sivers}_N$\,$\neq$\,0, it will be the first experimental evidence to show that $L_{sea}$\,$\neq$\,0, a crucial piece of the nucleon spin puzzle. Equally interesting is if $A^{Sivers}_N$ is consistent with zero. If $A^{Sivers}_N$\,=\,0, the observed $\bar{d}\left(x\right) /\bar{u}\left(x\right)$ flavor asymmetry seen by E866 \cite{e866}, and the origin of the nucleon spin, will remain a mystery. Regardless of the findings, the measurement of $A^{Sivers}_N$ at E1039 is sure to provide new insights into nature of the nucleon.
\bibliographystyle{aipprocl}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,891 |
← At Last! Broom 3 is on the American Flying Broomstick site!
Norm is correct. I might also note that in researching this, Wikipedia states that 2710 of these handy, easy-to-build cargo ships were built during the war years, not quite measuring up to the "over 3000" on page 90. That still a lot of ships! Out of all these ships, only two are still seaworthy, the SS John W. Brown (pictured) and the SS Jeremiah O'Brien, the latter being preserved as a memorial in San Francisco with occasional trips at sea. The SS Otto Mears itself was launched in 1943, sold private in 1947, and wrecked and scrapped in 1967.
How's that for information in a book about railroads! Check out Narrow Gauge Railroading in the San Juan Triangle: The Rio Grande Southern, the Ouray Branch of the D&RG and Otto Mears' Silverton RR, in our catalog for $19.95 + shipping. You can be assured that Otto Mears was a railroad guy; it's interesting to learn of the ship built twelve years after his death that was finally scrapped when Mears would have been 128 years old.
This entry was posted in Railroad, Reader Comments. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,060 |
Entre las dos orillas corre el río es un drama teatral de corte proletario escrito por el escritor peruano César Vallejo por los años de 1930, pero que no pudo ser publicada en vida del autor al ser rechazada por los editores de ese tiempo, por su claro contenido marxista y revolucionario.
Tras publicarse algunos fragmentos en revistas y diversas publicaciones, la obra completa fue publicada al fin dentro de la recopilación: César Vallejo. Teatro completo (en dos tomos), editado por Enrique Ballón Aguirre y con la revisión y comentarios de Georgette Vallejo, viuda del escritor (Lima, Fondo Editorial de la PUCP, 1979). Colección ésta que reúne además otras tres piezas teatrales del poeta: "Lock-out", "Colacho Hermanos o presidentes de América" y "La piedra cansada".
Contexto
César Vallejo empezó a componer sus dramas de temática social tras su entusiasmo por el marxismo y sus primeros viajes a Rusia (1928 y 1929); es indudable pues, que influenció en él la literatura comprometida de la Rusia soviética, puesta al servicio de la denuncia social y los intereses del proletariado. Su primer drama de esas características fue "Lock-out", escrita en francés hacia el año 1930, pero fue a todas luces un fallido intento dramático. El autor, ya reconocido poeta, se hallaba recién haciendo sus tanteos en un género literario distinto y al parecer le costaba mucho acertar. Su siguiente creación del mismo tipo fue concebida tras un largo y tedioso proceso, lo que se vislumbra más en los cambios de título que sufrió: Vera Polianova a El juego del amor y del odio, El juego del amor, del odio y de la muerte y El juego de la vida y la muerte (inspirados estos dos últimos probablemente por Le jeu de l'amour et de la mort (1925) de Romain Rolland); luego, Moscú contra Moscú para, finalmente, llegar al título de Entre las dos orillas corre el río.
Argumento
El título del drama es un reflejo magistral del argumento de la obra: el antagonismo generacional establecido en dos riberas opuestas: los padres, representantes de la aristocracia, y los hijos, representantes del nuevo orden social. Entre ambos corre, incontenible, el río de la revolución, entre orillas que no podrán ya comunicarse.
Crítica
Se ha criticado a esta obra por las fallas en su estructura dramática: se denota la dificultad que tenía Vallejo en desarrollar dramáticamente un argumento. No obstante, el autor, con buen criterio extrajo de ella dos secciones y las convirtió en pequeñas piezas teatrales, con los títulos de El juicio final y La mort (esta última escrita en francés y traducida al castellano por Georgette de Vallejo), que a decir de algunos críticos son las más logradas de sus obras dramáticas.
Referencias
Vallejo, César: Teatro completo Prólogo, traducción y notas de Enrique Ballón Aguirre (Lima, Fondo Editorial de la PUCP, 1979).
Vallejo, César: Obras Completas. Prólogo de Ricardo González Vigil (Lima, Editora Perú, 1992)
Obras de César Vallejo
Obras de teatro de Perú | {
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Sonny Jurgensen • The Daily Dose
Sonny Jurgensen
In tackle football, a "gunslinger" is a strong-armed quarterback who plays in an aggressive and decisive manner by throwing deep and with risk. Sonny Jurgensen—who could throw a football 80 yards and hit his target—was a gunslinger.
Christian Adolph Jurgensen III was born in Wilmington, North Carolina, on this date in 1934. Always a great athlete, he led his grammar school team to titles in basketball and baseball, captured the boy's tennis championship of Wilmington and pitched his team to the city baseball crown. Jurgensen attended New Hanover High School–alma mater of 1969 NFL MVP Roman Gabriel and legendary newscaster David Brinkley—where as a senior he led the Wildcats to the state title game in basketball and was named his home state's starting quarterback in the North Carolina-South Carolina Shrine Bowl High School All-Star Game. Jurgensen played football and baseball at Duke University but turned down an invitation to try out for the basketball team. The three-year letterman led the Blue Devils to two ACC championships, a 34-7 win over Nebraska in the 1955 Orange Bowl and, as a defensive player, intercepted a pass in four straight games, still a school record. In December 1957, the Philadelphia Eagles selected Jurgensen in the 4th round of the NFL Draft. For three seasons, he served as backup to the great Norm Van Brocklin, who retired after the Eagles won the NFL Championship in 1960. The following year, Sonny got his chance.
In his first season as a starting quarterback, Sonny Jurgensen threw for an NFL-record 3,723 yards and 32 touchdowns. Three years later, he was traded to the Washington Redskins for quarterback Norm Snead. In October 1964, Jurgensen faced his former team for the first time, throwing for five touchdowns and 385 yards while compiling a passer rating of 120.6. After the trade, he went 13-3—with 33 touchdown passes against only 16 interceptions—against the Eagles during his career. In 1967, he broke his own record by throwing for 3,747 yards. Two years later, Vince Lombardi took over as the Redskins head coach. That season, Jurgensen—a classic drop back passer–led the NFL in completion percentage and passing yards. After going 7-5-1 and leading Washington to their first winning season in 15 years, Coach Lombardi died of cancer prior to the 1970 season. The defensive-minded George Allen took over for Lombardi and, after the "Red Roach" went down with a shoulder injury during the 1971 preseason, Allen replaced him with Billy Kilmer. The following year, the Redskins won the NFC Championship and earned a trip to Super Bowl VII to play the undefeated Miami Dolphins [Daily Dose, January 14]. Jurgensen, who sat out the second half of the season with a ruptured Achilles tendon, was banned from the team bench by Allen, a stern taskmaster who feared the fun-loving Jurgensen would disrupt the team. Allen relegated his veteran quarterback to watch the game alone from a 12-person suite inside the Los Angeles Coliseum. In 1974, at age 40 and in his final season, Jurgensen won his third NFL passing title despite splitting time with Kilmer. His final NFL game was his only postseason appearance, relieving Kilmer in a 19-10 loss to the Los Angeles Rams during the first round of the playoffs.
In 149 starts over 18 NFL seasons, Sonny Jurgensen had a record of 69-73-7. He only appeared in one playoff game—yet he is one of the greatest pure passers in football history. The Red Roach fell through the cracks because he never won a championship, but neither did Dan Marino who, along with Joe Namath [Daily Dose, May 31] join Jurgensen as the three greatest throwers of a football ever to play the game. Sonny was a true genius at work. He dropped passes into the hands of his receivers like feathers falling from the sky. Nobody ever dropped a pass from Jurgensen. His deep passes—and Jurgensen threw it down the field—were like handoffs. "Jurgensen may be the best the league has ever seen. He is the best I have ever seen," said Vince Lombardi, who won five NFL titles with the Packers. "If we would have had Sonny Jurgensen in Green Bay, we'd never have lost a game." The feeling was mutual. Jurgensen played for nine head coaches during his career and Lombardi was his favorite. "I learned more in five days from Lombardi than I had in 12 years in the NFL," said Sonny after Lombardi's death. Jurgensen was unconventional. He approached the line of scrimmage like a guy walking up to a bar to order a drink–pot-bellied, hands on his hips, scanning the scene. And he enjoyed the nightlife. When Jurgensen was traded from the Eagles to the Redskins in 1964, Philadelphia bartenders wore black armbands. He once said of Washington head coach Otto Graham, with whom he did not have a good relationship, "He likes candy bars and milkshakes. I like women and scotch."
Due to rule changes favoring offense, it is easier than ever to pass the ball in today's NFL. Last season, eight of the top nine highest rated passers in history were playing in the league. It is unfair to compare passers from the "Dead Ball Era"—pre 1978—against those in today's game, but Mr. Jurgensen's 82.6 passer rating is the highest from the Dead Ball Era. Sonny was a honey, throwing for over 32,000 yards and 255 touchdowns in the NFL. He played seven seasons in Philadelphia, eleven more in Washington, and is in the Ring of Honor of both teams. He passed for over 300 yards in 25 NFL games and five times threw for over 400 yards. The Redskins retired his "9" and he was named to the NFL 1960s All-Decade Team. Jurgensen won three league passing titles and played in five Pro Bowls. He is beloved inside The Beltway. For the past 35 years, the immensely popular Jurgensen has served as color analyst on Washington Redskins radio broadcasts. In 1983, Christian Adolph Jurgenson III—whose pregame ritual included sharing a bologna sandwich with Redskin trainer Tommy McVean—achieved his biggest accomplishment when he was inducted into the Pro Football Hall of Fame.
In 1968, famed Baltimore Colts quarterback John Unitas invited Sonny Jurgensen to the grand opening of his new restaurant, the Golden Arm. Upon entering, Jurgenson turned to Unitas and said, "Thanks for naming it after me."
Diana Nyad
Jahangir Khan
webmaster - CMS | {
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import os, sys, matplotlib
#import faulthandler; faulthandler.enable()
mpl_v = 'MPL-11' # change with every MPL upgrade
daptype = 'SPX-MILESHC-MASTARSSP' # change with every DAP type code
pcay_ver = os.environ['PCAY_VER']
csp_basedir = os.environ['PCAY_CSPBASE']
manga_results_basedir = os.environ['PCAY_RESULTSDIR']
mocks_results_basedir = os.path.join(
os.environ['PCAY_RESULTSDIR'], 'mocks')
from astropy.cosmology import WMAP9
cosmo = WMAP9
matplotlib.rcParams['font.family'] = 'serif'
matplotlib.rcParams['text.usetex'] = True
if 'DISPLAY' not in os.environ:
matplotlib.use('agg')
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,793 |
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