text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Place chicken wings in the air fryer without overlapping and cook at 250F for 10 minutes. Turn the temperature to 400F and cook for 15 minutes. Optional: You can flip the wings halfway for even crispiness.... The use of a deep fryer insures that your chicken wings will have the crispy, brown skins that make them so tasty and desirable. Whether the wings are coated or simply spiced, a deep fryer will cook them to perfection.
Don't be afraid of overcooking these —the longer they stay in the air fryer, the more crispy goodness you'll get—add up to 10 more minutes to the cook time, if you like. Drumettes have more meat than wings; that's what we use, but either will work. how to make a good background for a drawing Generally, with these chicken wings, we invite a lot of oil/fat and the especially the deep fried wings that are way too heavy on the calories. Having a recipe to cook these wings inside the air fryer is really going to help you keep your weight on track and avoid unnecessary calories.
Place chicken wings in the air fryer without overlapping and cook at 250F for 10 minutes. Turn the temperature to 400F and cook for 15 minutes. Optional: You can flip the wings halfway for even crispiness.
Australian deep fried chicken wings allrecipes.com recipe. Learn how to cook great Australian deep fried chicken wings allrecipes.com . Crecipe.com deliver fine selection of quality Australian deep fried chicken wings allrecipes.com recipes equipped with ratings, reviews and mixing tips.
To ensure your wings are extra crispy put the lid on the oil-less fryer. Remove the basket from The Charbroil® Oil-less Turkey Fryer, remove wings from the their leg racks and place on a clean plate. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,429 |
Q: Laravel 5.4 How to responses JSON with Relationship? I want to response JSON with Laravel that I have CategoryModel belongTo SongModel.
This is my CategoryModel
class CategoryModel extends Model
{
protected $table = 'categories';
protected $fillable = [
'id',
'name',
];
public function song()
{
return $this->hasMany(SongModel::class);
}
}
And this is my SongModel
class SongModel extends Model
{
protected $table = 'songs';
protected $fillable = [
'id',
'name',
'url',
'categories_id',
'singers_id',
'types_id',
];
public function category()
{
return $this->belongsTo(CategoryModel::class);
}
}
I want to response JSON with the relationship. I wrote:
class SongController extends Controller
{
public function index(SongModel $songModel)
{
$song = $songModel->category()->get();
return response()->json(["DATA" => $song], 201);
}
}
A: To do this you will just need to load the relationship before you return the response:
public function index(SongModel $songModel)
{
$songModel->load('category');
return response()->json(["DATA" => $songModel], 201);
}
https://laravel.com/docs/5.4/eloquent-relationships#lazy-eager-loading
Hope this helps!
A: Try this. return response()->json(["DATA" => $songModel->load('category')->toArray()], 201);
A: OR, you can use with('song')
lass SongController extends Controller
{
public function index(SongModel $songModel)
{
$song = $songModel::with('song')->get();
return response()->json(["DATA" => $song], 201);
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,418 |
Epiblema banghaasi es una especie de polilla del género Epiblema, familia Tortricidae.
Fue descrita científicamente por Kennel en 1901.
Distribución
Se encuentra en China (Heilongjiang) y Rusia.
Referencias
Enlaces externos
Bisby F.A., Roskov Y.R., Orrell T.M., Nicolson D., Paglinawan L.E., Bailly N., Kirk P.M., Bourgoin T., Baillargeon G., Ouvrard D. (red.) (2011). Species 2000 & ITIS Catalogue of Life: 2011 Annual Checklist. Species 2000: Reading, UK.
banghaasi | {
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} | 5,502 |
Q: Oracle XMLTABLE - how to remove a node from XMLType? Assume we have following XML:
<root>
<item>
<a>a1</a>
<b>b1</b>
<c>c1</c>
<d>d1</d>
<e>e1</e>
</item>
<item>
<a>a2</a>
<b>b2</b>
<c>c2</c>
<d>d2</d>
<e>e2</e>
</item>
...
</root>
How to get following result using XMLTABLE and PATH?
A B ITEM_XML (excluding <d>)
a1 b1 <item><a>a1</a><b>b1</b><c>c1</c><e>e1</e></item>
a2 b2 <item><a>a2</a><b>b2</b><c>c2</c><e>e2</e></item>
No DELETEXML please as it is deprecated. I am particularly interested how to remove/exclude some node from XMLType. Mind that ITEM_XML should be pretty printed like in original, it was just more convenient to put it in the table this way.
A: Starting with Oracle Database 12c Release 1 (12.1.0.1) use XQuery Update to update XML data. (https://docs.oracle.com/en/database/oracle/oracle-database/12.2/adxdb/deprecated-functions-for-updating-XML-data.html). Example of deleting node:
UPDATE warehouses SET warehouse_spec =
XMLQuery('copy $tmp := . modify delete node
$tmp/Warehouse/VClearance return $tmp'
PASSING warehouse_spec RETURNING CONTENT)
WHERE warehouse_spec IS NOT NULL;
| {
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} | 5,273 |
{"url":"https:\/\/questions.examside.com\/past-years\/jee\/question\/the-value-of-the-integral-intlimits410-left-x-jee-main-mathematics-trigonometric-functions-and-equations-bfc3gtzp3xqighg8","text":"1\nJEE Main 2016 (Online) 10th April Morning Slot\n+4\n-1\nThe value of the integral\n\n$$\\int\\limits_4^{10} {{{\\left[ {{x^2}} \\right]dx} \\over {\\left[ {{x^2} - 28x + 196} \\right] + \\left[ {{x^2}} \\right]}}} ,$$\n\nwhere [x] denotes the greatest integer less than or equal to x, is :\nA\n6\nB\n3\nC\n7\nD\n$${1 \\over 3}$$\n2\nJEE Main 2016 (Online) 10th April Morning Slot\n+4\n-1\nFor x $$\\in$$ R, x $$\\ne$$ 0, if y(x) is a differentiable function such that\n\nx $$\\int\\limits_1^x y$$ (t) dt = (x + 1) $$\\int\\limits_1^x ty$$ (t) dt, \u00a0then y (x) equals :\n\n(where C is a constant.)\nA\n$${C \\over x}{e^{ - {1 \\over x}}}$$\nB\n$${C \\over {{x^2}}}{e^{ - {1 \\over x}}}$$\nC\n$${C \\over {{x^3}}}{e^{ - {1 \\over x}}}$$\nD\n$$C{x^3}\\,{1 \\over {{e^x}}}$$\n3\nJEE Main 2016 (Online) 9th April Morning Slot\n+4\n-1\nIf\u00a0\u00a0 $$2\\int\\limits_0^1 {{{\\tan }^{ - 1}}xdx = \\int\\limits_0^1 {{{\\cot }^{ - 1}}} } \\left( {1 - x + {x^2}} \\right)dx,$$\n\nthen $$\\int\\limits_0^1 {{{\\tan }^{ - 1}}} \\left( {1 - x + {x^2}} \\right)dx$$ is equalto :\nA\nlog4\nB\n$${\\pi \\over 2}$$ + log2\nC\nlog2\nD\n$${\\pi \\over 2}$$ $$-$$ log4\n4\nJEE Main 2016 (Online) 9th April Morning Slot\n+4\n-1\nThe area (in sq. units) of the region described by\n\nA= {(x, y) $$\\left| {} \\right.$$y$$\\ge$$ x2 $$-$$ 5x + 4, x + y $$\\ge$$ 1, y $$\\le$$ 0} is :\nA\n$${7 \\over 2}$$\nB\n$${{19} \\over 6}$$\nC\n$${{13} \\over 6}$$\nD\n$${{17} \\over 6}$$\nJEE Main Subjects\nPhysics\nMechanics\nElectricity\nOptics\nModern Physics\nChemistry\nPhysical Chemistry\nInorganic Chemistry\nOrganic Chemistry\nMathematics\nAlgebra\nTrigonometry\nCoordinate Geometry\nCalculus\nEXAM MAP\nJoint Entrance Examination","date":"2023-03-29 01:34:07","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\": 0.8790120482444763, \"perplexity\": 8279.947591384767}, \"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-2023-14\/segments\/1679296948900.50\/warc\/CC-MAIN-20230328232645-20230329022645-00351.warc.gz\"}"} | null | null |
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Consider a ...\n50 views\n\n53 views\n\n### Mutual authentication with Public Key and session Key\n\nI am trying to understand two protocols for mutual authentication and if they are secure or not. $K$ is the session key, and it's calculated $k=H(TimeStamp)$. Are the following both cases secure? ...\n130 views\n\n### Commutative Encryption with RSA scheme?\n\nI wanted to know how I could manage to do what I'm going to tell you next, with the RSA encryption\/decryption scheme. So Alice and Bob each have a public key $(n, e)$ and a private key $(p, q, d)$; ...\n49 views\n\n### Autocorrelation in Autokey\n\nI know that Vigen\u00e8re can broken using auto-correlation (as explained at cryptool-online.org). Now my question is: Why does this also work with autokey (tested it using crypttool)? To prove that I ...\n69 views\n\n### Bilinear pairing\n\nI am working on Efficient Construction of Pairings which are being realized by Miller's algorithm. In this algorithm the basic steps are point doubling and line function computation point addition ...\n178 views\n\n### Secure Secret sharing\n\nI'm trying to come up with a new way to do oblivious transfer that is faster and requires less computation than existing methods. The basis of my method is shamir secret sharing. Below is an example ...","date":"2014-10-26 08:59:25","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\": 0.806494414806366, \"perplexity\": 2406.9515618262226}, \"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\/1414119661285.56\/warc\/CC-MAIN-20141024030101-00095-ip-10-16-133-185.ec2.internal.warc.gz\"}"} | null | null |
Precious metals coins can offer a unique safe haven from market volatility, but they are also a difficult asset to own. Here's some expert advice
Precious Metals Coins Provide Refuge from Market Fallout
Precious metals coins can offer a unique safe haven from market volatility, but they are also a difficult asset to own. Here's some expert advice
Precious metals coins provide investors with diversification and a refuge from the current volatility of stocks.
With the prices of gold and silver falling during the past five years by 14.81 percent and 40.86 percent, respectively, the timing may be good to consider such purchases. It also would be in keeping with tradition for investors to use precious metals coins as a safe haven when the market slumps.
However, when venturing into the purchase of precious metals, investors need to tread carefully and use reputable coin dealers. For guidance, I contacted coin dealers who have rich experience in helping investors navigate the opportunities and pitfalls of buying and selling precious metals coins.
Silver or Gold?
???The trend for gold and silver has been for silver to appreciate or fall by a greater percentage than gold,??? said Patrick Heller, communications officer at Liberty Coin Service in Lansing, Michigan.
Gold and silver prices began climbing at the end of 1999 until they peaked in 2011, Heller said. The price of gold rose 563 percent by jumping from $288.25 an ounce in 1999 to $1,911 an ounce in September 2011. Silver soared 824 percent from $5.41 an ounce in 1999 to $50 an ounce at the end of April 2011, he added.
However, from those peaks to today, gold is down 30 percent (from $1,911 to $1,331.50) while silver has dropped 67 percent (from $50 to $16.34).
???Silver is a far smaller traded market than gold, so it tends to be more volatile,??? Heller said. ???Also, industrial demand for silver is a major driver for the price, unlike gold.???
Buying Silver and Gold
???From any particular point in time to another, either gold or silver could outperform the other,??? Heller said. ???Therefore, I recommend owning some of both metals. However, I do anticipate that silver???s price will outperform gold over the next five years, so skew my recommendation between the two toward silver.???
For buyers of both precious metals in the current market environment, Heller recommended an allocation of 60 percent to two-thirds silver and one-third to 40 percent gold.
???In silver, my current best recommendation is the U.S. 90 percent Silver Coins struck through 1964,??? Heller said. ???These can be purchased from many sources now at 1.5-3.0 percent above metal value if purchased in quantity.???
The 90 percent Silver Coins struck through 1964 have ???advantages??? other than price over the exact weight .999 fine ingots, Heller said. Those advantages, according to Heller, include: extreme divisibility, as one 90 percent silver dime has about 1/14 of an ounce of silver; familiarity, as the older general public can recall when these coins were in circulation; liquidity, in which the coins can be sold by almost any coin dealer and in the desired quantity (whereas a 100 ounce ingot would require the sale of an entire bar of silver); and reduced risk, as there is a small, but growing, problem of counterfeit products such as ingots and U.S. Silver Eagle Dollars.
When purchasing U.S. 90 percent silver coins in $1,000 face value bags (with about 715 ounces silver content), figure that the spread between the buy and sell price will run 8-12 percent, Heller said. The spread would be a bit wider for smaller quantities, he added.
Another Fan of Buying Silver
Van Simmons, president of David Hall Rare Coins in Newport Beach, California, said he buys and owns both silver and gold coins. He also recommends the 90 percent pre-1965 silver dimes and quarters. Simmons said they are about 50 cents above the spot price of silver. A bag of silver coins with a face value of $1,000, consisting either of 10,000 dimes or 4,000 quarters, has 715 ounces of silver.
A buyer would take the spot price of silver of approximately $16.35 an ounce, plus 50 cents, for a total of $16.85 times 715 ounces and pay roughly $12,050, Simmons explained. There also is a shipping and insurance charge of about $50, depending on where the bag of coins will be sent, he added.
???I like silver; it is very cheap,??? Simmons said. ???The average mining cost is about $5.75 an ounce.???
Silver not only is a collectible but an ???industrial metal,??? Simmons said.
The weight of buying a bag of silver can pose a problem arranging storage, Simmons cautioned.
???If you buy $10,000 worth of silver, you have about 50 pounds,??? Simmons said. ???If you buy $10,000 in gold you can put it in your pocket.???
???Some investors or readers want to preserve the purchasing power of their dollar and prefer gold,??? Simmons said. ???Others want to speculate in metals or commodities and swing for the fences and prefer silver.???
Assessing the Gold/Silver Ratio
???Gold and silver are currently both good buys, but they typically play a different role in your portfolio,??? said Rich Checkan, president and chief operating officer of Asset Strategies International, a full-service, tangible asset dealer in Rockville, Maryland, that offers precious metals, rare coins and foreign currencies.
Investors who are looking to invest in either gold or silver may want to check the Gold/Silver Ratio (GSR), which shows the amount of silver it takes to purchase one ounce of gold.
???Right now, the GSR hovers around 80.9, which means it takes approximately 81 ounces of silver to buy one ounce of gold, Checkan said. ???At times like these, when silver is highly undervalued compared to gold, there is typically a rush to take advantage of lower prices in the silver market.
???When you see us talk about the GSR, we typically mention a ???magic number??? of 80 for the ratio. The GSR has only reached 80 four times in the past two decades. Before today, the most recent occurrence was March 2016. Each of the three previous times it reached 80, the ratio responded by correcting downward roughly 40 percent to 60 percent! This effect was the result of silver outpacing gold as both gold and silver prices rose from the lows.???
Gold Eagle coins currently are available at ???ultra-low premiums,??? said Checken, who added investors still are buying and selling them at an equal pace.
Plus, gold prices tend to fluctuate less widely than silver, Checkan said. As a result, an investment in gold is more likely to hold onto its value in the long-term.
At today???s gold prices, the market is currently offering an opportunity that has not been seen in 40 years, Simmons said.
???The U.S. Twenty Dollar gold coins struck prior to 1933 are at extremely low premiums to the current price of gold,??? Simmons said. ???Over the last 40 years, these coins have sold at premiums ranging from 50 percent over the current spot price to 125 percent over the current sport price.???
Circulated grades of Extra Fine U.S. Twenty Dollar gold coins are priced at about 3 percent over the spot price of gold, Simmons said.
The Fast Money Alert trading service recommended the VanEck Vectors Gold Miners ETF (NYSE: GDX) in its April 2 report. This gold and metals mining ETF seeks to replicate the price and yield performance of the NYSE Arca Gold Miners Index and has an expense ratio of .53 percent, while offering an .80 percent yield.
Dr. Mark Skousen???s Forecasts & Strategies investment newsletter recommends each of the coin dealers quoted in this column. For investors who are looking to sidestep market volatility and invest in something they can hold in their hands, silver and gold coins offer such an opportunity.
Paul Dykewicz is an accomplished, award-winning journalist who has written for Dow Jones, the Wall Street Journal, Investor???s Business Daily, USA Today, the Journal of Commerce, Seeking Alpha, GuruFocus and other publications and websites. Paul is the editor of StockInvestor.com and DividendInvestor.com, a writer for both websites and a columnist. He further is the editorial director of Eagle Financial Publications in Washington, D.C., where he edits monthly investment newsletters, time-sensitive trading alerts, free e-letters and other investment reports. Paul previously served as business editor of a daily newspaper in Baltimore. Paul also is the author of an inspirational book, ???Holy Smokes! Golden Guidance from Notre Dame???s Championship Chaplain,??? with a foreword by former national championship-winning football coach Lou Holtz.
A Simple Fund for the Euro
Markets Holding up under Tariff Tantrum Conditions
Written By Paul Dykewicz
Paul Dykewicz is the editorial director of the Financial Publications Group at Eagle Publishing Inc., www.eaglepub.com, of Washington, D.C. Eagle publishes five free, e-letters, 10 weekly trading services and five monthly investment newsletters, Forecasts & Strategies, Successful Investing, Cash Machine, Growth & Dividend Report and The Alpha Investor Letter. He also is the editor of Eagle Daily Investor and the author of the inspirational book, "Holy Smokes! Golden Guidance from Notre Dame's Championship Chaplain." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,834 |
{"url":"https:\/\/www.gamedev.net\/blogs\/entry\/649572-of-mice-and-mittens\/","text":"\u2022 entries\n182\n235\n\u2022 views\n189502\n\n# Of Mice and Mittens\n\n202 views\n\nI've been spending the last few days with the the Torque Game Engine, and I must say, this is some of the most fun I've had programming in a very long time. Here's a quick run-through of the things I've been playing with over the last few days. Of note, though, is that I haven't stayed up this late in a while, so I may not be quite as witty (read: less than zero), verbose (no treatise on immersion :(), nor creative as I may [or may not] usually be.\n\nThe first thing I did with Torque was, naturally, play around with its terrain editor and particle editor. After about an hour of playing around with the editor, which has a built-in texture blender that can blend up to six 256x256 resolution textures (this 256x256 is a limit which made me angry) and having a lot of fun trying to figure out the particle editor without having so much as a clue as to what the various parameters did, I came up with this image:\n\nThe particle editor in Torque was surprisingly capable and powerful, though maybe not quite as powerful as the Max Payne 1\/2 particle editors, nor as some of the particle engines I've cooked up over the years, but this will certainly do. And it will certainly do better as I update it. One really nice thing is that the editor already has support for a relatively large number of animated textures for the particle sprites (if I remember what I read correctly, it can support up to 5^6 animated textures for particles).\n\nAfter playing with the particles and terrain, I set up an empty project in VS.NET 2k3 for all the various script files for the project I was working on at the time. It's a fairly large image (height-wise), so I don't want to post it here, but it's there if you want to take a peek at it.\n\nThroughout yesterday and today, I was working with the GUI system to create a slightly more aesthetically-pleasing look for the game's option screen, and though I haven't started work on fine-tuning the rest of the GUIs yet, they do operate off certain parameters\/images that I altered while playing with the look of the options window. I still have yet to finish my tweaking of this window, since I had gotten sidetracked, but eventually I'll be adding transparency to every GUI window as well. Here are a couple shots of the options window (left being the original look, middle being a rough draft, the third\/final image being the near-current state of the options GUI):\n\nWhat I got sidetracked with was the build of the Torque Shader Engine (I am now officially near-broke with just enough money to cover the first month's rent of my house for school this fall) that I decided to buy, going against my previous claim that I wanted to upgrade the engine with an entirely unique shader system of my very own (instead I'll just be working on a dynamic lighting and shadow system of my own). This decision came from the fact that I got an actual game idea in my head that I plan to flesh out and create a prototype level for all in the next six months. The screenshot below is the first build of the engine where I finally (only after about an hour of work) got the FPS demo from the first screen in this post working with the TSE:\n\nIf I didn't have class tomorrow, I would almost certainly be able to stay up all night just playing with the engine. I'm having an absolute blast with this. Another update tomorrow night.\n\nPlease keep posting about TGE, I haven't had a chance to dig into it much yet (although I am planning to) as I have been working on Torque2D. I bought TGE because I needed access to a lot of the engine docs that T2D owners don't get, yet they are the same engine.\n\nIf you do get some more money you might want to pick up T2D. Its a drop in compile for TGE and you can layer T2D gui elements over the TGE screen. Meaning all the fancy menu stuff you can think of with sprites and particle engines you can do.\n\nRight now I've been hacking up T2D to use as a client for my multiplayer game server I'm developing. I've been posting my exploits in my GDNet journal.\n\nI must reiterate how impressed I am with the Torque engine. Sure, it has its limitations, but getting the engine and a $250,000 or less profit license for$100 is pretty damn amazing.\n\nThe only issue I do have is how the GG site charged for things like the ShowTool Pro (something that should be part of the engine suite) and their soon-to-be-released map editor. I feel that the community provides so much for the engine that things like this should just be free (as of now, while it is a symbiotic relationship, it's a slightly unbalanced one, in my opinion).\n\n\"The only issue I do have is how the GG site charged for things like the ShowTool Pro (something that should be part of the engine suite) and their soon-to-be-released map editor. I feel that the community provides so much for the engine that things like this should just be free (as of now, while it is a symbiotic relationship, it's a slightly unbalanced one, in my opinion).\"\n\nTGE is amazing engine which IMO is worth a hell of a lot more than $100 (same with TSE and T2D) ShowTool Pro was made by a third party developer (nothing to do with garagegames) which is why it's on sale. And the soon to be released \"Torque Constructor\" is being made by GG with most of the work being contracted out to third party developers, and is looking pretty amazing... I don't think GG should supply this free because they sell the engines for such a cheap price, they deserve to make money off other products (they have make up for the low engine price and they have to pay the bills some how) No one has to buy the products\/addons... TGE\/TSE has it's own built in show tool type function, and there\u2019s lots of free software on the internet you can use to make content for TGE,TSE and T2D. I'm not trying to flame you or anything, the problem is GG gets too many people (mainly new people to the engine) saying the tools should be free.:( They say the tools should be free... I (and many others) say GG should charge much more for the engines :\\ But anyway, hope you continue to have fun with Torque... look forward to seeing your progress. -Joseph #### Share this comment ##### Link to comment Well, to be honest, I really think it would be nice if the GG site explicitly said that the engine cost doesn't include the cost of certain tools. I never was told that the engine specifically included the tools (but I was confused as to what the package included, to be honest), but for some reason the screenshots led me to believe otherwise. So I guess the problem is just that I felt misinformed. Though, while ShowTool Pro was developed by a third-party (which I knew), my point was that that particular program should be part of the engine suite. It seems to me that it is a fairly necessary (not necessary, but helpful to the point where it's almost a required buy), and should have been included by the Torque guys to begin with. And the engine is great, don't get me wrong, but it's not like you get an incredibly polished, perfectly-working product for your money. The$100 you pay for the indie license seems about right to me, considering that the graphics tech in the engine is a bit \"aged\" at this point in time; but if you consider you can get the shader engine for $150 more, the$250 value is fantastic for the whole engine (a bit of a steal for the developers, actually). Though I'd say the \\$100 for the Torque Game Engine certainly within reason.\n\n\"I never was told that the engine specifically included the tools (but I was confused as to what the package included, to be honest), but for some reason the screenshots led me to believe otherwise.\"\n\nTheres a feaure list on the product page that lists what you get (and states what they do)\n\nI do agree when you say \"the tech in the engine is a bit \"aged\" \"\nBut this is changing all the time with new versions being released, and it hasen't stopped people making amazing looking games such as the one being made at the moment called Poacher...\n\n(Pics are very early dev shots made in TGE)\n\n## Create an account\n\nRegister a new account","date":"2018-07-17 16:29:12","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\": 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\": 0.30452045798301697, \"perplexity\": 1169.6949657147875}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589752.56\/warc\/CC-MAIN-20180717144908-20180717164908-00169.warc.gz\"}"} | null | null |
{"url":"https:\/\/schoolportalng.com\/classes\/quantities\/","text":"Mon \u2014 Sat: 8.00 AM \u2014 8.00 PM\n\n# Tag: Quantities\n\n## FUNDAMENTAL QUANTITIES AND UNITS\n\nFundamental quantities are physical quantities whose dimensions and units are not usually derived from other physical quantities. Basically, there are three fundamental quantities in mechanics. They include:\n\nMass\nLength and\n\ni) Mass: This is a fundamental quantity with dimension \u2018M\u2019, usually written in capital letter. The S.I. unit of mass is kilogramme (kg). Mass can also be measured in gramme (g), tonne (t), etc.\nii) Length: This is another fundamental quantity with dimension \u2018L\u2019, written in capital letter. The S.I. unit of length is metre (m). Length can also be measured in kilometre (km), centimetre (cm), inches (inch), feet (ft), etc.\n\nerror: Content is protected !!","date":"2021-02-26 21:46:34","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8779233694076538, \"perplexity\": 6429.543759957511}, \"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-2021-10\/segments\/1614178357984.22\/warc\/CC-MAIN-20210226205107-20210226235107-00490.warc.gz\"}"} | null | null |
{"url":"http:\/\/openstudy.com\/updates\/5574b6dfe4b0edc5dbb8a1fc","text":"## anonymous one year ago Forty-two is what percent of 70? A. 21% B. 60% C. 74% D. 167%\n\n1. saylilbaby\n\nB.60%\n\n2. anonymous\n\nthank u so much\n\n3. anonymous\n\nwhat r u people doing\n\n4. Disco619\n\n$\\frac{ 42 }{ 70 } \\times 100$ $0.6 \\times 100 = 60$ 60% is the answer, hope you get it. |dw:1433712546556:dw|\n\n5. mathstudent55\n\nWe need to translate this word problem into a math equation and solve the equation. In math, \"is\" means an equal sign. \"What\" is an unknown, so we can use x. \"Percent\" means divide by 100. \"Of\" means times. Forty-two is what percent of 70? <--- English 42 = x \/100 * 70 <--- Math You have this equation: $$42 = \\dfrac{x}{100} \\times 70$$ $$42 = \\dfrac{70x}{100}$$ $$4200 = 70x$$ $$70x = 4200$$ $$x = 60$$ Answer: 42 is 60% of 70.\n\n6. Disco619\n\nOr that^ Haha","date":"2017-01-19 19:35:41","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\": 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\": 0.7892206907272339, \"perplexity\": 4338.057889413889}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560280730.27\/warc\/CC-MAIN-20170116095120-00013-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
Breakdown: In Your House was the 24th In Your House professional wrestling pay-per-view (PPV) event produced by the World Wrestling Federation (WWF; WWE as of 2002). It took place on September 27, 1998, at the Copps Coliseum in Hamilton, Ontario, Canada. Nine matches were held on the PPV, while three matches were held for the Sunday Night Heat pre-show. The promotional poster featured D-Generation X, whose entrance theme was titled "Break It Down." The event is notable for featuring the debut of Christian.
The main event was a Triple Threat match between Stone Cold Steve Austin, The Undertaker, and Kane for the WWF Championship, which ended in controversy when The Undertaker and Kane simultaneously pinned Austin. Another match at the event was a Triple Threat match in a steel cage between Ken Shamrock, Mankind, and The Rock to determine the number-one contender to the WWF Championship.
Production
Background
In Your House was a series of monthly pay-per-view (PPV) shows first produced by the World Wrestling Federation (WWF, now WWE) in May 1995. They aired when the promotion was not holding one of its then-five major PPVs (WrestleMania, King of the Ring, SummerSlam, Survivor Series, and Royal Rumble), and were sold at a lower cost. Breakdown: In Your House was the 24th In Your House event and took place on September 27, 1998, at the Copps Coliseum in Hamilton, Ontario, Canada.
Storylines
The event comprised twelve professional wrestling matches with outcomes predetermined by WWF script writers. The matches involved wrestlers portraying their characters in planned storylines that took place before, during and after the event.
The main feud heading into the event was between Stone Cold Steve Austin and Vince McMahon, who did not want Austin as the WWF Champion. On the September 5 episode of Raw Saturday Night McMahon criticized and insulted The Undertaker and Kane for failing to dethrone Austin. On the September 6 episode of Sunday Night Heat, McMahon apologized and scheduled Austin to defend his title at Breakdown against both in a Triple Threat match. On the September 14 episode of Raw, McMahon added a stipulation that The Undertaker and Kane could not pin each other. He then confronted Austin, who attacked him. The Undertaker and Kane broke up the attack and later that night interfered in Austin's match with Ken Shamrock. With the help of The Rock and Mankind, Austin fought off the attackers and chased after McMahon, who was watching from the aisle. On the September 21 episode of Raw, The Undertaker and Kane defeated Austin and Billy Gunn. After the match, Austin hit The Undertaker and Kane with a chair.
On the same night, in order to prevent Shamrock, The Rock and Mankind from teaming with Austin, McMahon gave them the opportunity to win a shot at the WWF Championship in a Triple Threat match. The match ended in a no-contest when The Undertaker and Kane attacked the three contestants. The Triple Threat match was rescheduled for Breakdown. to be contested inside a steel cage.
Lower down the card, Dustin Runnels clashed with Val Venis. Dustin's new gimmick was that of an evangelist who preached against the vulgarity of the WWF's Attitude Era, mainly its violence, bloodlust and sexual content, making Venis, whose gimmick was that of a porn star, a prime target. Venis responded the following evening on Raw with a film involving Runnels's wife, Terri, who had been absent for almost a year. On the September 13 episode of Sunday Night Heat, Dustin jumped through the crowd and attacked with a religious-themed sign, using the wood to choke him. The following week, Dustin, who was providing commentary, assaulted Venis, who was busy fighting Owen Hart, but Venis managed to tie him up in the ropes and deliver a promo further insinuating the details of his adulterous relationship with Terri.
Event
Despite featuring prominently on the promotional poster, Triple H did not participate in the event, due to a legitimate knee injury.
Before the event officially began, three matches were taped for Sunday Night Heat. Golga defeated Mosh, The Hardy Boyz (Matt Hardy and Jeff Hardy) defeated Kaientai (Men's Teioh and Funaki), and 8-Ball from The Disciples of Apocalypse defeated Billy Gunn and Skull from The Disciples of Apocalypse in a Triple Threat match.
The first match of the event was between Edge and Owen Hart. Edge gained the early advantage after a dropkick. After Edge sent Hart outside the ring, Edge attempted a move from the apron, which Hart countered into a powerslam. Hart gained the advantage afterwards until Edge performed an electric chair facebuster as Hart climbed the turnbuckles to regain the advantage. Hart eventually countered a suplex attempt into a bridging German suplex. Hart then missed an attack in the corner, and Edge attempted a superplex. Hart countered it into an inverted suplex slam from the turnbuckles. Hart then attempted the Sharpshooter, which Edge countered into a small package. After performing a spinning heel kick, Edge saw the debuting Christian at ringside. This distraction allowed Hart to pin Edge with a roll-up. After the match, Edge followed Christian into the crowd.
The second match was between Too Much (Brian Christopher and Scott Taylor), and Al Snow and Scorpio. After going back and forth, Taylor performed a pumphandle suplex. Snow and Christopher tagged in, and Snow delivered trapping headbutts. Scorpio and Taylor tagged in. As Scorpio delivered corner clotheslines, as Snow threw a steel chair into the ring. With Scorpio distracting the referee, Snow performed a calf kick with the chair. Scorpio then performed a dropkick with the chair to Christopher into Taylor. As Scorpio performed a splash to Taylor, Snow performed a moonsault on the outside from the security wall onto Christopher. As Scorpio was on the top turnbuckle, Christopher tripped him, and performed a suplex on the outside. Taylor followed with a springboard axe handle. Back inside, Too Much performed a back body drop, and Christopher performed a bulldog to Scorpio. Scorpio then countered a back body drop from Too Much into a double dropkick. Snow tagged in, and, as Scorpio distracted the referee, attacked Too Much with his mannequin head, also accidentally hitting Scorpio. Snow then performed a low blow to Taylor with the head. Christopher performed a Tennessee Jam onto Snow, Scorpio performed a flying clothesline to Christopher, and Snow finally pinned Taylor after a Snow Plow.
The third match was between Marc Mero (accompanied by Jacqueline) and Droz. Mero had the early advantage with punches and kicks. Droz came back with a flapjack and a dropkick. Mero left the ring and looked to leave as Droz came after him and attacked Mero with the guard rail and ringsteps. Back in the ring, Mero performed a knee lift, but Droz performed a flying shoulder block and a clothesline. Mero then performed a back body drop, sending Droz outside the ring. Mero followed with a suicide senton. Back inside, Droz blocked a slingshot splash attempt with his knees, and followed with a flying spinning back elbow, an inverted atomic drop, and a powerslam. As Droz went for the pin, Jacqueline placed Mero's foot on the bottom rope, stopping the count. Mero then came back with a clothesline, and choked Droz with the athletic tape around his wrist. Mero then distracted the referee as Jacqueline hit Droz with the heel of her shoe from the top rope. Mero then pinned Droz after a Marvelocity.
The fourth match was a Falls Count Anywhere match between Vader and Bradshaw. Bradshaw gained the early advantage with attacks in the corner until Vader performed a clothesline, and a big splash. Bradshaw fought back with a big boot, and performed a clothesline, sending Vader outside the ring. Outside, Bradshaw attacked Vader with the ringpost and the ringbell, but Vader fought back using the ringsteps. Back inside the ring, Bradshaw performed attacks in the corner, and a belly to back suplex. Vader then sent Bradshaw outside the ring, where Vader attacked Bradshaw using the guard rail, and delivered a low blow. Back inside, Vader performed a splash from the second rope, and a Vader Splash. Bradshaw fought back, performing a Clothesline from Hell, and pinned Vader after a neckbreaker slam.
The fifth match was between D'Lo Brown and Gangrel. The match went back and forth until Brown performed a low blow, and a running sitout powerbomb. Brown kept the advantage until Gangrel performed a flapjack onto the ropes. Mark Henry then came down to ringside as Gangrel performed a back body drop, and a flying clothesline. As Gangrel ran at the ropes, Henry pulled down the ropes, sending Gangrel outside the ring. Brown distracted the referee as Henry sent Gangrel into the ringpost, and back into the ring. Brown then pinned Gangrel after a Sky High. After the match, Gangrel spat "blood" into Henry's face, attacked Brown, and performed an Impaler on Brown.
The sixth match was a Triple Threat match in a steel cage between Ken Shamrock, Mankind and The Rock. The match went back and forth between all three, and each was double-teamed. As Mankind and Shamrock double-teamed The Rock, The Rock fought back with a DDT to Mankind and a float-over DDT to Shamrock. The Rock then performed a scoop slam and a People's Elbow to both before climbing the cage. Mankind and Shamrock stopped, and attacked The Rock. The Rock then performed a low blow to Shamrock, and a Rock Bottom to Mankind. Shamrock fought back with a belly to belly suplex, and applied the ankle lock. Mankind stopped Shamrock and climbed to the top of the cage. The Rock followed but Mankind threw him off. Mankind then performed a diving elbow drop from the top of the cage, but The Rock avoided it. Shamrock then attempted to climb out the door, but Mankind held onto his leg. Shamrock brought a chair into the ring, and attempted to hit Mankind. Mankind avoided it, performed a double-arm DDT on Shamrock, and hit him with the chair. As Mankind climbed the cage, The Rock pinned Shamrock to win the match, and become the number-one contender to the WWF Championship.
The seventh match was between Dustin Runnels and Val Venis (accompanied by Terri Runnels). Venis performed a spinebuster, but Runnels came back with a powerbomb. Venis knocked Runnels outside and attacked Runnels. As they re-entered the ring, Runnels performed a back body drop, but Venis fought back with a Russian legsweep, and applied the camel clutch. Runnels blocked a suplex into his own suplex, but Venis came back with a chinlock. Runnels fought back with a DDT and climbed the turnbuckles, but Venis threw him to the outside. They then went back and forth until Venis performed a powerslam, several elbow drops, and pinned Runnels after a Money Shot.
The eighth match was between Jeff Jarrett and Southern Justice (Mark Canterbury and Dennis Knight), and D-Generation X (Billy Gunn, Road Dogg, and X-Pac). Gunn and Dogg attacked Jarrett and Southern Justice from behind during X-Pac's entrance. X-Pac and Jarrett started, with X-Pac countering a hurricanrana attempt into a powerbomb. Road Dogg tagged in, but was dominated by Jarrett and Southern Justice after a cheap shot from Canterbury. X-Pac eventually tagged in after Road Dogg's head hit Jarrett's head. X-Pac was also dominated after Canterbury countered a spinning heel kick into a powerbomb. Jarrett applied the sleeper hold. X-Pac escaped, performed a belly to back suplex, and tagged in Gunn. Gunn fought off Jarrett and Southern Justice, and X-Pac performed the bronco buster on Jarrett. Jarrett tried to bring a guitar into the ring, but Gunn took it from him. The referee stopped Gunn using the guitar, and Southern Justice double-teamed Gunn. Outside, Jarrett hit X-Pac in the head with the guitar, as Gunn pinned Knight after a Fameasser. X-Pac left holding onto his right eye.
The main event was a Triple Threat match between Kane, The Undertaker and Steve Austin for the WWF Championship. The Undertaker and Kane were prohibited from pinning each other. Another stipulation added was that if anyone caused interference on Austin's behalf, Austin would immediately be stripped of the title. During The Undertaker's entrance, Austin attacked him with a chair. Austin then fought off Kane using the chair and the ringpost. Kane fought back, and threw Austin outside, where The Undertaker attacked him. Austin countered by sending The Undertaker into the ringsteps, and then pulled Kane's crotch into the ringpost. Austin then performed the Stone Cold Stunner to Kane, but Undertaker pulled him out, and sent him into the ringpost. Back in the ring, The Undertaker attacked Austin until Austin came back with a Lou Thesz press, and a swinging neckbreaker. Kane pulled Austin outside, and The Undertaker accidentally hit Kane. Austin then shoved The Undertaker into Kane, and choked Kane with cables. The Undertaker then choked Austin with cables, and The Undertaker and Kane dominated Austin. As Austin was kicked outside, he took the chair and hit Kane, but then Undertaker hit Austin with the chair. Austin was dominated, but The Undertaker and Kane stopped each other from pinning Austin. Undertaker then fought with Kane until Kane and Austin double-teamed him. Austin sent The Undertaker outside, and performed a Russian legsweep to Kane. Kane fought back with a headbutt, and Austin was dominated again. The Undertaker and Kane stopped each other from pinning Austin, and they fought, until they performed clotheslines to each other. Austin took advantage, and attacked both. Austin attempted a Stone Cold Stunner on Kane, but was shoved into The Undertaker, who performed a big boot. The Undertaker and Kane then performed a double chokeslam, and pinned Austin simultaneously. Austin lost the title, but no new champion was announced. Vince McMahon came to ringside, and took the title belt. McMahon left, and Austin followed him backstage. McMahon told Austin that he wasn't the champion anymore, and the title belt was his.
Reception
The event has received mixed reviews from critics.
In 2008, J.D. Dunn of 411Mania gave the event a rating of 6.0 [Average], stating, "The seeds for Rockamania were planted here, and it launched Edge as a singles star -- before they decided to de-push him. Like a lot of 1998 stuff that was heavy on the Russo, it worked at the time but had a short shelf life. It might be worth a look for nostalgia, and for some historical reasons, but the wrestling leaves a lot to be desired. I'll be generous, and call it a mild thumbs up. Don't take it as a ringing endorsement or anything, though.
Mildly recommended."
In 2018, Chris of Retro Pro Wrestling described the event as "odd," stating, "Though I can understand why not everybody was a fan of that main event, it was typical of the 'f**k the rules' brawling style that was so typical of the WWF main event scene at the time, and it actually worked pretty well. As for the show as a whole, this was an odd one. Outside of the last three matches, nothing much seemed to matter. Edge/Owen, MeroDroz, and D'Lo/Gangrel were as random as you could have asked for and didn't seem to have any point to them, whilst the best thing about Venis/Runnels was seeing Terri looking stunning. Not that I'm saying this was a bad show. Most of the matches were decent enough, just not the kind of classics that would make it must-see viewing 20 years later. If there is one reason to watch this show, it's The Rock. This wasn't the best he ever performed, but it was the first PPV where you saw that the former Rocky Maivia wasn't just over, but was Main Event Superstar over, and during the hottest period in the company's history, that's saying something."
In 2022, Paul Matthews of Classic Wrestling Review described the event as "average," stating, "This show was average. Nothing on it was bad, but much of it was dull and flat. The first half was throwaway matches with little build. They developed the matches for the second half well. But the action was lackluster, except the cage. Even the main event was disappointing. I didn't hate it, but it was forgettable. However, I love the closing scene of the PPV. Despite the rest of the show, that final shot is memorable."
Aftermath
The following night on Raw is War, McMahon attempted to announce a new WWF Champion. He held a presentation ceremony, and introduced The Undertaker and Kane. After saying that both deserved to be the WWF Champion, Austin drove a Zamboni into the arena, and attacked McMahon before police officers stopped him, and arrested him. Because The Undertaker and Kane both failed to defend McMahon from Austin, McMahon did not name a new champion, but instead made a match at Judgment Day between The Undertaker and Kane with Austin as the special referee. This prompted The Undertaker and Kane to attack Mr. McMahon, injuring his ankle because he gave them the finger behind their backs.
Due to the controversy with the WWF Championship, The Rock was not given a match for the title. The Nation of Domination disbanded shortly after Breakdown, and The Rock went on to feud with Nation members Mark Henry and D'Lo Brown before winning the WWF Championship at Survivor Series. Mankind and Shamrock feuded over the WWF Intercontinental Championship after Shamrock won the title in an eight-man tournament. Mankind and The Rock continued their feud after The Rock won the WWF Championship.
This would be the final Pay-per-view appearance for both Henry O. Godwinn and Vader, the former due to injury and the latter due to creative differences.
Results
References
External links
Breakdown Report at 411mania
Breakdown Report at SLAM! Sports
In Your House
Events in Hamilton, Ontario
1998 in Ontario
Professional wrestling in Ontario
1998 WWF pay-per-view events
September 1998 events in Canada
WWE in Canada | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/3-phase-system-electrical-question.772101\/","text":"# 3 phase system electrical question\n\n1. Sep 21, 2014\n\n### darwinharianto\n\n1. The problem statement, all variables and given\/known data\n\nhttps:\/\/www.physicsforums.com\/attachments\/question-jpg.73386\/ [Broken] the current go from a A N and n\nthe 2nd current go from b B N and n\n\nthe first thing, i make the matrix\n27-4j -3 i1 230 <0\n-3 27-4j x i2 = 230<0\n\nx means the matrix multiplied to the next\n< means the degree\nthen with determinant from the first matrix\ni find 767.04 <16.35\n\nthen to find i\ni1 = 230<0 \/767.04<16.35 27-4j -3 1\ni2 x -3 27-4j x 1\n\nis this right?\nhow to count 230<0 \/ 767.04<16.35\ni dont understand\n\n2. Relevant equations\n\n3. The attempt at a solution\n\nthe first thing, i make the matrix\n27-4j -3 i1 230 <0\n-3 27-4j x i2 = 230<0\n\nx means the matrix multiplied to the next\n< means the degree\nthen with determinant from the first matrix\ni find 767.04 <16.35\n\nthen to find i\ni1 = 230<0 \/767.04<16.35 27-4j -3 1\ni2 x -3 27-4j x 1\n\nis this right?\nhow to count 230<0 \/ 767.04<16.35\ni dont understand\n\nLast edited by a moderator: May 6, 2017\n2. Sep 21, 2014\n\n### Staff: Mentor\n\nYour matrix looks okay (although its formatting has been punished by the font!), but your solution doesn't look correct. Here it is rendered with LaTeX:\n\n$$\\left( \\begin{array}{cc} {27 - j4} & -3 \\\\ -3 & {27 - j4} \\end{array} \\right) \\left( \\begin{array}{c} I1 \\\\ I2 \\end{array} \\right) = \\left( \\begin{array}{c} 230 \\\\ 230 \\end{array} \\right)$$\n\nCan you check your determinant calculation? I see different magnitude and angle values.\n\n3. Sep 21, 2014\n\n### darwinharianto\n\nand thank you for the matrix format and the reply\n\nthe determinan is 704-j216?\nso it means 736.39 with -17.05 degree?\n\nso the matrix go like this?\n$$\\left( \\begin{array}{c} I1 \\\\ I2 \\end{array} \\right) = 230<0 \/ 736.39 < -17.05 \\left( \\begin{array}{cc} 27-j4 & -3 \\\\ -3 & 27-j4 \\end{array} \\right) \\left( \\begin{array}{c} 1 \\\\ 1 \\end{array} \\right)$$\n\nwhat is that $$\\left( \\begin{array}{c} 1 \\\\ 1 \\end{array} \\right)$$ mean?\ni get the formula from the ppt but i still dont understand why it is there\nand i dont understand how to calculate 230<0 \/ 736.39 < -17.05 manually\n\n4. Sep 21, 2014\n\n### Staff: Mentor\n\nYes, that's the correct value for the determinant.\nI'm not sure. I haven't seen that particular method used before. The $\\left( \\begin{array}{c} 1 \\\\ 1 \\end{array} \\right)$ is a column vector. I presume that it's meant to be multiplied with the impedance matrix in order to yield a new column vector. That would make sense since there's a column vector to the left of the equals also. But I don't see how that equation is meant to solve for the currents.\n\nI would use Cramer's Rule to solve the problem (look it up). It uses determinants, and you've already got the value of one of them.\nThat's just division of one complex number by another.\n\n5. Sep 22, 2014\n\n### darwinharianto\n\nOIC\nit is just an inverse\nmoved the left 1st matrix to the right side\nthen it make i1 = 7.59< 0.31\nand i2 = 7.59<0.31\nis this right?\nthen IaA = I1\nInN = -(I1+I2)\nIbB = I2\n\n6. Sep 22, 2014\n\n### Staff: Mentor\n\nPlease don't use text-speak abbreviations here on PF. They're not allowed.\nIt doesn't work for me; I can't see how that math could be correct. If I wanted to \"move\" the impedance matrix from the left side to the right side of the equation, I'd pre-multiply both sides with the inverse of the matrix. But then what is moved to the right side would be the inverse, not the original matrix. Take a generic equation of this type; let I be the unknown current column vector, V the known voltage column vector, and Z the impedance matrix. The equation is written:\n\n$Z \\; I = V$\n$Z^{-1}Z \\; I = Z^{-1}V$\n$I = Z^{-1}V$\n\nThe problem I have with the equation that you've quoted is that I don't see the inverse of the impedance matrix coming about by simply dividing the matrix by its determinant. What you'd want is to use Cramer's rule which divides the adjoint matrix by the determinant, and solve for the I's individually.\nYes, once you've solved for the mesh currents then then branch currents are obtained as you've shown.\n\n7. Sep 22, 2014\n\n### darwinharianto\n\nokay\nand the result is I1 = 9.44<9.45 and I1=I2 or I1=9.31+J1.55\nis this correct?\n\n8. Sep 22, 2014\n\n### Staff: Mentor\n\nYes, that's correct.\n\n9. Sep 22, 2014\n\n### darwinharianto\n\nand for the phasor diagram just draw a line with 9.44 and 9.45 degrees right?\n\n10. Sep 22, 2014\n\n### Staff: Mentor\n\nPresumably you'll want to show the source voltage phasors along with the phasors for all the currents that you were asked to find.\n\n11. Sep 22, 2014\n\n### darwinharianto\n\noohhh\nso there will be 3 line\nthx a lot","date":"2017-10-19 00:38:48","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\": 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\": 0.5395038723945618, \"perplexity\": 2074.507012272562}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187823168.74\/warc\/CC-MAIN-20171018233539-20171019013539-00266.warc.gz\"}"} | null | null |
Q: How to create Idle time? I am creating a batch file program where if you open a mspaint.exe, if its idle for 10 seconds the program will close automatically. I have created this:
@echo off
:Start
tasklist /FI "IMAGENAME eq mspaint.exe" | findstr "mspaint.exe" >nul
IF "%ERRORLEVEL%" == "0" GOTO Running
IF "%ERRORLEVEL%" == "1" GOTO NotRunning
:Running
***::Check if idle***
GOTO Terminate
:NotRunning
GOTO EOF
:Terminate
timeout 5
taskkill /im mspaint.exe /f
ECHO Paint has been terminated due to inactivity
PAUSE
:EOF
EXIT
I have trouble finding the idle syntax(seems it doesn't exist), is there another way to make the idle time work? And since this is my first time creating a batch file, I really need a helping hand here.
A: Not the answer you want to hear, but the one that you need. The only way to monitor another process for activity is to install a window hook, but that's going to require a much lower level language (C or C++).
If that doesn't put you off then start with the Microsoft Hooks Overview and post further questions. Good luck (you'll probably need it).
A: Thanks for the help but that seems complicated. I used SCHTASKS instead and it worked :)
| {
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Robert J. Shelley (born July 27, 1941) is an American attorney and former politician in the state of Florida.
Shelley was born in New York City. He came to Florida in 1969 and attended Broward Community College. He is a former fireman and fire chief. He served in the Florida House of Representatives from 1983 to 1990, as a Republican, representing the 92nd district.
References
Living people
1941 births
Republican Party members of the Florida House of Representatives | {
"redpajama_set_name": "RedPajamaWikipedia"
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An Initiative Of NASPA and The Suder Foundation
Why First-Gen?
Loyola New Orleans Celebrates First-generation Students
Brent McLemore, M.S., Loyola University New Orleans / The Center / December 16, 2020
At Loyola University New Orleans, we have an opt-in mentoring program for our first-year, first-generation students. This program, aptly called First in the PACK, features peer mentoring as its main competent, but it also includes faculty and staff allyship and numerous events throughout the academic year. In years past (those without the disruptions and restrictions brought on by a pandemic), we have been able to meet in-person with our new students for meet and greets, retreats, luncheon events, and, most importantly, one-on-one mentoring opportunities. Given the challenges of this year, delivering a similar experience for these new students became a digital endeavor. How we celebrated them this year for the National First-Generation College Celebration was no exception.
For this year's celebration, we wanted our students to be encouraged and inspired. We wanted them to know that being first-generation means they have already demonstrated a resiliency and fortitude that they can continue to flex and lean into during a uniquely different and challenging first semester. We wanted them to see living success stories of first-generation students and hear of their difficulties and triumphs along their journey. With this in the mind, we hosted a panel featuring our Vice President for Equity and Inclusion, Dr. Kedrick Perry, who is first-generation, as well as an alumnus of the program and current first-year med school student, Josh Quintana. This virtual panel included both insightful lessons on the power of embracing one's identity and practical advice on how to make the most of one's college experience. It was well received and much appreciated by all in attendance.
Stepping into our pre-2020 ways, we also hosted an in-person drop-in event the following day that featured prepackaged treats (because it is 2020). This event was open to all first-generation students. We also included an interactive element to this part of the celebration. With our student center windows as the canvas, our first-generation students were able to create an artful expression of what it is to be first-generation. Early on in the day, one student wrote "never thinking about quitting" and then expanded on this with those of us present at the time. For her, as with so many first-generation students, getting to college–and not just getting through college–was a challenging process, but like so many of her peers and those that came before her, quitting was not an option. Lacking the support experienced by other students while also bearing more responsibilities is a common path for our first-generation students. This was her path to that moment, and she was able to get there because she wouldn't quit. She wouldn't even think about it. The resilience and tenacity of first-generation students should never be underestimated. This is one of the many reasons we celebrate them at Loyola University New Orleans. Not even a pandemic would keep us from doing so, because we're not quitting either. They deserve that.
For more information on Loyola University New Orleans's approach, please visit their website here.
Against All Odds: First-generation Students and Noncognitive Barriers
What is North Carolina Central University Doing to Help First-generation students during COVID-19?
Roderick T. Heath, Ed.D. / November 11, 2020
A Q&A on Advancing Equity in Higher Education Amid Coronavirus
For some students, the COVID-19 pandemic did not start a basic needs crisis; it exacerbated one that was already there.
Center for American Progress / April 13, 2020
Supporting First-gen Students in Times of Need
Challenges facing first-generation students during COVID-19 at Southern Illinois University Carbondale
Lanel Love, Ph.D. / November 04, 2020
Building a Foundation for First-generation College Student Success
This blog highlights collaborative efforts at TCU to provide an asset-based approach to first-gen programming through empowerment.
Jamartae Jackson, M.Ed. & Cynthia Montes, M.Ed. / June 11, 2020
CCRI opens food pantry for students
To combat the issue of food insecurity often faced by first-generation college students on campus, the Community College of RI opened its first food pantry.
Providence Journal / March 25, 2019
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© 2020 NASPA, 111 K Street NE, 10th Floor, Washington, DC, 20002 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Detecting non-members in sandbox hierarchy in MKS Is there any way how to see the list of files in MKS Integrity Client which are in the project sandbox hierarchy but not added as a member in the MKS project ?
For example when I create new class inside existing package in the sandbox. What I do now is that I have to go through the sandbox hierarchy, find the correct folder and then manualy find and add new member through "New member wizard". This is very tricky when you create more files in one step - it is very easy to forget some file.
Is there some more convinient way for finding non-members ?
A: There is a command for that (at least for the command line).
si viewnonmembers --sandbox=
It takes the same universal options as the other si commands, try
si viewnonmembers --help
to see them
The same command is available in the GUI: Try Sandbox->Views->View Non-Members
| {
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The most experienced level of Cal Ripken Baseball and is now played on a 50/70 foot diamond. Recommended for 11 and 12 year old players. Players build and refine their fundamental skills while continuing to learn game strategy as well as progressing further with pitching fundamentals. Safety is the number one priority. Once the season starts, coaches are expected to perpetually evaluate each player's skills and developmental opportunities. As such, players will experience positions that are consistent with their progress to avoid unnecessary risk of injury.
Teams meet 1 time during the week for a 1-1.5 hour practice and 2x a week for a 6 inning or 2 hour max game. Typically 1 Saturday morning game and 1 weekday game.
Games and practices are scheduled at Ewald Park in Southbury & Quassy Field in Middlebury.
Practice day and time is scheduled in early September after registration is complete and teams are formed.
Team rosters are formed by a coaches draft facilitated by Majors President, team assignments will be communicated the week of Labor Day. | {
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<?xml version="1.0" encoding="utf-8"?>
<layout>
<LinearLayout
xmlns:android="http://schemas.android.com/apk/res/android"
android:orientation="horizontal"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:gravity="center_vertical" >
<EditText
android:id="@+id/editable_list_entry_title"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"
android:layout_marginLeft="-3dp"
android:layout_marginStart="-3dp"
android:layout_marginRight="1dp"
android:layout_marginEnd="1dp"
android:longClickable="false"
android:imeOptions="actionDone"
android:inputType="textMultiLine" />
<ImageView
android:id="@+id/editable_list_entry_options"
android:contentDescription="@string/editable_list_overflow_cd"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:src="@drawable/ic_overflow_vertical_white"
android:tint="@color/gray600" />
<ImageView
android:id="@+id/editable_list_entry_save"
android:contentDescription="@string/editable_list_save_cd"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:src="@drawable/ic_check_white_24dp"
android:tint="@color/gray600"
android:visibility="gone" />
</LinearLayout>
</layout>
| {
"redpajama_set_name": "RedPajamaGithub"
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package com.carlomicieli.jtrains.config;
import com.codahale.metrics.ConsoleReporter;
import com.codahale.metrics.MetricRegistry;
import com.ryantenney.metrics.spring.config.annotation.EnableMetrics;
import com.ryantenney.metrics.spring.config.annotation.MetricsConfigurerAdapter;
import org.springframework.context.annotation.Configuration;
import java.util.concurrent.TimeUnit;
/**
* @author Carlo Micieli
*/
@Configuration
@EnableMetrics
public class SpringMetricsConfig extends MetricsConfigurerAdapter {
@Override
public void configureReporters(MetricRegistry metricRegistry) {
ConsoleReporter
.forRegistry(metricRegistry)
.build()
.start(1, TimeUnit.MINUTES);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,862 |
Na terenie gminy Dragacz, w powiecie świeckim, w województwie kujawsko-pomorskim znajduje się 12 pomników przyrody. Wyróżniamy 7 pojedynczych drzew, 1 aleję, 1 głaz narzutowy, 2 grupy drzew i 1 fragment lasu o charakterze naturalnym.
W strukturze gatunkowej przeważa dąb szypułkowy. Na szczególną uwagę zasługuje dąb o obwodzie 684 cm (2012) rosnący w Dragaczu i lipa o obwodzie 400 cm w Wielkim Lubieniu.
Poniższa tabela przedstawia stan prawny (rzeczywisty może być inny) pomników przyrody na terenie gminy:
Przypisy
Pomniki przyrody w województwie kujawsko-pomorskim
Dragacz | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/2912080\/solution-verification-post-problem-12-2-mathematical-analysis-apostol","text":"# Solution verification post : Problem $12.2$, Mathematical Analysis, Apostol\n\nAs a continuation of this post, I'm posting my solutions (or attempts) to the exercise problems of Chapter 12 (Multivariable Differential Calculus), Mathematical Analysis by Apostol. Since I'm essentially self-studying, I'd really appreciate if anyone checks the solutions and let me know if there is any gap in my arguments or if there exists any cleverer solutions. Thank you.\n\nProblem $12.2.$ Calculate all first order partial derivatives and the directional derivative $f'(x;u)$ for each of the real-valued functions defined on $\\mathbb{R}^n$ as follows : \\begin{align*} &(a)\\,\\, f(x)=a \\boldsymbol{\\cdot} x\\\\ &(b)\\,\\, f(x)=\\left\\Vert x \\right\\Vert^4\\\\ &(c)\\,\\, f(x)=x \\boldsymbol{\\cdot} L(x), \\text{ where } L:\\mathbb{R}^n \\to \\mathbb{R}^n \\text{ is a linear function.}\\\\ &(d)\\,\\, f(x)=\\sum_{i=1}^n \\sum_{j=1}^n a_{ij}x_ix_j, \\text{ where } a_{ij}=a_{ji}. \\end{align*}\n\nSolution. $(a)$ Let $a=(a_1,\\dots,a_n)$, $x=(x_1,\\dots,x_n)$. Thus, $$f(x)=a \\boldsymbol{\\cdot} x=a_1x_1+\\dots+a_nx_n \\tag1$$ Then we have, $$D_kf(x)=\\frac{\\partial f}{\\partial x_k}(x)=a_k=a \\boldsymbol{\\cdot} e_k; \\,\\,k=1,\\dots,n \\tag2$$ $D_kf(x)$ is constant for all $k$ $\\Rightarrow$ $D_kf(x)$ is continuous for all $k$. We recall the following :\n\nTheorem. Assume that one of the partial derivatives $D_1f,\\dots,D_nf$ exists at $c$ and that the remaining $n\u22121$ partial derivatives exists in some $\\delta$-ball $B_{\\delta}(c)$, and are continuous at $c$. Then $f$ is differentiable at $c$.\n\nUsing the theorem, we conclude that $f$ is differentiable and hence directional derivative at any direction exists. Then,\n\n\\begin{align*} f'(x;u)&=\\lim_{h \\to 0} \\frac{f(x+hu)-f(x)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{a \\boldsymbol{\\cdot} (x+hu)-a \\boldsymbol{\\cdot} x}{h}\\\\ &=\\lim_{h \\to 0} \\frac{a \\boldsymbol{\\cdot} (hu)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{h \\sum_{i=1}^n a_iu_i}{h}\\\\ &=\\sum_{i=1}^n a_iu_i=a \\boldsymbol{\\cdot} u \\tag3 \\end{align*}\n\nSolution. $(b)$ \\begin{align*} f(x)&=\\left\\Vert x \\right\\Vert^4=\\left(\\sum_{i=1}^n x_i^2\\right)^2=\\sum_{i=1}^n x_i^4+\\sum_{i \\neq j} x_i^2x_j^2\\\\ &=x_k^4+\\sum_{i \\neq k} x_i^4+x_k^2\\left(\\sum_{i \\neq k} x_i^2\\right)+\\left(\\sum_{i \\neq k} x_i^2\\right)x_k^2+\\sum_{i \\neq k,\\,j \\neq k,\\,i \\neq j} x_i^2x_j^2\\\\ &=x_k^4+\\sum_{i \\neq k} x_i^4+2x_k^2\\left(\\sum_{i \\neq k} x_i^2\\right)+\\sum_{i \\neq k,\\,j \\neq k,\\,i \\neq j} x_i^2x_j^2 \\tag4 \\end{align*}\n\nLet $k \\in \\{1,\\dots,n\\}$. Then, $$D_kf(x)=4x_k^3+4x_k\\left(\\sum_{i \\neq k}x_i^2\\right)=4x_k\\left(\\sum_{i=1}^nx_i^2\\right)=4x_k\\left\\Vert x \\right\\Vert^2 \\tag5$$ Thus, $D_kf(x)$ exists and is continuous for all $k \\in \\{1,\\dots,n\\}$. By the statement stated above, $f$ is differentiable, and hence directional derivative exists in all directions. Let $u=(u_1,\\dots,u_n)=u_1e_1+\\dots+u_ne_n$. Now, \\begin{align*} f'(x;u)&=f'(x)(u)=f'(x)(u_1e_1+\\dots+u_ne_n)=\\sum_{k=1}^n u_kf'(x)(e_k)=\\sum_{k=1}^n u_kf'(x;e_k)\\\\ &=\\sum_{k=1}^n u_k\\frac{\\partial f}{\\partial x_k}(x)=\\sum_{k=1}^n u_k\\left(4x_k\\left\\Vert x \\right\\Vert^2\\right)=4\\left\\Vert x \\right\\Vert^2\\sum_{k=1}^n x_ku_k=4\\left\\Vert x \\right\\Vert^2 \\left(x \\boldsymbol{\\cdot} u\\right) \\tag6 \\end{align*}\n\n$(c)$ \\begin{align*} D_kf_(x)&=\\lim_{h \\to 0} \\frac{f(x+he_k)-f(x)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{(x+he_k) \\boldsymbol{\\cdot} L(x+he_k)-x \\boldsymbol{\\cdot} L(x)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{(x+he_k) \\boldsymbol{\\cdot} \\left(L(x)+L(he_k)\\right)-x \\boldsymbol{\\cdot} L(x)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{x \\boldsymbol{\\cdot} L(x)+x \\boldsymbol{\\cdot} L(he_k)+he_k \\boldsymbol{\\cdot} L(x)+he_k \\boldsymbol{\\cdot} L(he_k)-x \\boldsymbol{\\cdot} L(x)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{x \\boldsymbol{\\cdot} L(he_k)+he_k \\boldsymbol{\\cdot} L(x)+he_k \\boldsymbol{\\cdot} L(he_k)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{x \\boldsymbol{\\cdot} hL(e_k)+he_k \\boldsymbol{\\cdot} L(x)+he_k \\boldsymbol{\\cdot} hL(e_k)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{hx \\boldsymbol{\\cdot} L(e_k)+he_k \\boldsymbol{\\cdot} L(x)+h^2e_k \\boldsymbol{\\cdot} L(e_k)}{h}\\\\ &=\\lim_{h \\to 0} \\frac{h\\left(x \\boldsymbol{\\cdot} L(e_k)+e_k \\boldsymbol{\\cdot} L(x)+he_k \\boldsymbol{\\cdot} L(e_k)\\right)}{h}\\\\ &=\\lim_{h \\to 0} x \\boldsymbol{\\cdot} L(e_k)+e_k \\boldsymbol{\\cdot} L(x)+he_k \\boldsymbol{\\cdot} L(e_k)\\\\ &=x \\boldsymbol{\\cdot} L(e_k)+e_k \\boldsymbol{\\cdot} L(x) \\tag7 \\end{align*}\n\nBy continuity of $x$ and $L(x)$, we conclude that $D_kf_(x)$ is continuous for all $k \\in \\{1,\\dots,n\\}$. Thus, by the theorem stated above, $f$ is differentiable and hence, directional derivative of $f$ exists in all directions. Let $u=(u_1,\\dots,u_n)$. Then,\n\n\\begin{align*} f'(x;u)&=\\sum_{k=1}^n u_kD_kf_(x)\\\\ &=\\sum_{k=1}^n u_k\\left(x \\boldsymbol{\\cdot} L(e_k)+e_k \\boldsymbol{\\cdot} L(x)\\right)\\\\ &=\\sum_{k=1}^n \\left(x \\boldsymbol{\\cdot} u_kL(e_k)+u_ke_k \\boldsymbol{\\cdot} L(x)\\right)\\\\ &=\\sum_{k=1}^n \\left(x \\boldsymbol{\\cdot} L(u_ke_k)+u_ke_k \\boldsymbol{\\cdot} L(x)\\right)\\\\ &=x \\boldsymbol{\\cdot} \\sum_{k=1}^nL(u_ke_k)+\\sum_{k=1}^n\\left(u_ke_k\\right) \\boldsymbol{\\cdot} L(x)\\\\ &=x \\boldsymbol{\\cdot} L(\\sum_{k=1}^n u_ke_k)+\\sum_{k=1}^n\\left(u_ke_k\\right) \\boldsymbol{\\cdot} L(x)\\\\ &=x \\boldsymbol{\\cdot} L(u)+u \\boldsymbol{\\cdot} L(x) \\tag8 \\end{align*}\n\n$(d)$ \\begin{align*} f(x)&=\\sum_{i=1}^n\\sum_{j=1}^n a_{ij}x_ix_j\\\\ &=a_{kk}x_k^2+\\sum{i \\neq k}a_{ik}x_ix_k+\\sum{i \\neq k}a_{ki}x_kx_i+\\sum_{i \\neq k,\\,j \\neq k,\\,i \\neq j} a_{ij}x_ix_j \\tag9 \\end{align*}\n\nThen we have, \\begin{align*} D_kf(x)&=2a_{kk}x_k+\\sum_{i \\neq k} a_{ik}x_i+\\sum_{i \\neq k} a_{ki}x_i\\\\ &=2a_{kk}x_k+2\\sum_{i \\neq k} a_{ik}x_i \\,\\left(\\text{since } a_{ik}=a_{ki}\\right)\\\\ &=2\\sum_{i=1}^n a_{ik}x_i \\tag{10} \\end{align*}\n\nThus, $D_kf(x)$ exists and is continuous. Hence, $f$ is differnetiable and thus, it has directional derivative in every direction. Then, $$f'(x;u)=\\sum_{k=1}^n u_k \\frac{\\partial f}{\\partial x_k}(x)=2\\sum_{i=1}^n u_k\\left(\\sum_{i=1}^n a_{ik}x_i\\right)=2\\sum_{i=1}^n\\sum_{i=1}^n a_{ik}x_iu_k=2x^TAu, \\tag{11}$$ where $A={(a_{ij})_{i=1}^n}_{j=1}^n$.","date":"2019-07-23 03:11:33","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\": 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\": 0.9999810457229614, \"perplexity\": 9543.907260584658}, \"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-2019-30\/segments\/1563195528687.63\/warc\/CC-MAIN-20190723022935-20190723044935-00288.warc.gz\"}"} | null | null |
The Teocalli of the Sacred War (teocalli is Nahuatl for "temple;" Spanish: El Teocalli de la Guerra Sagrada) is the name given by archaeologist Alfonso Caso to a monolithic pre-Columbian miniature of an Aztec temple, thought by some to have served as a throne for Motecuhzoma II.
The sculpture was first discovered in 1831 in the foundations of the National Palace of Mexico, but was not removed until the 1920s.
It is now located in the Museo Nacional de Antropología, Mexico City.
Description and iconography
Generally speaking, the Teocalli is representative in form of Late Post-classic temple architecture and sculpture.
Two masses, a flat-roofed temple and a "truncated pyramid" complete with stairs up the front of the platform, compose the statue.
The upper part of the front of the sculpture contains a solar disk flanked by two figures; within the disk is the date Four Movement. The figures are identified as Huitzilopochtli (left) and Tepeyolotl, a form of Tezcatlipoca (right).
Below this, facing upwards, is a depiction of the earth deity Tlaltecuhtli, next to which are military equipment representing warfare.
The lower part of the front of the sculpture displays the dates One Rabbit (left) and Two Reed (right). The latter includes a rope representing the New Fire ceremony in that year. Together, the dates refer to both mythical events of the past as well as the beginning of a new cycle.
On the sides of the pyramid section of the Teocalli sit two pairs of figures carrying copal bags, maguey leaves, and tobacco containers. Caso identifies these four figures as Tlaloc, Tlahuiscalpantecuhtli, Xochipilli, and Xiuhtecuhtli.
The top of the sculpture displays the date Two House.
On the back of the sculpture is a depiction of an eagle on a prickly pear cactus (similar to the coat of arms of Mexico). The eagle, a representation of Huitzilopochtli, holds in its beak the glyph for war, atl-tlachinolli. Although part of the relief has eroded, the cactus appears to grow from a defeated Chalchiuhtlicue, goddess of lakes and streams.
Finally, on the Teocalli's platform sits a zoomorphic "earth monster."
Connection to Motecuhzoma II
Next to the figure of Tezcatlipoca is a hieroglyph containing the hair, ear plug, nose plug, and royal diadem of a ruler. This symbol is dubbed by Umberger as "The Headdress Glyph".
These elements are said to have represented the name 'Motecuhzoma' in Post-Conquest pictorial codices and on other Mexica sculptures, including the Hackmack Box in Hamburg and the Calendar Stone of the Museo Nacional de Antropología.
In early studies of the Calendar Stone, The Headdress Glyph has had many previous interpretations, including as a fire symbol.
Critical interpretations
In 1927, Alfonso Caso published a theory that the decoration on the sculpture "justified human sacrifice and warranted warfare as a means of procuring prisoners for immolation in the temples of Tenochtitlan." This myth was regarded as the foundation of a "mystical-military" ideology in which the Mexica believed themselves to be the chosen people of the sun.
Other scholars, such as Enrique Juan Palacios, hold a more historical and less allegorical interpretation––reading the symbols as cosmograms. He argued that these cosmograms did not represent mythology, but rather represent the sacred nature of the Mexica as a whole. Palacios' interpretation of this Teocall received wide acclaim from scholars, and served as the foundation of his broader assessments on Mexica religion.
According to historian Richard Townsend, the earth zoomorph featured on the platform of the Teocalli represents Mexica land held by force of arms.
References
External links
Teocalli de la Guerra Sagrada on Flickr — the back of the sculpture
Museo Nacional de Antropología, México record on Google Arts Project
Aztec artifacts
Mesoamerican stone sculpture | {
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Q: Using '?' as placeholder withFlask and SQLite3 yields OperationalError. The uncommented code below yields an OperationalError: near"?" while the commented lines works fine. What am I missing here?
g.db.execute('INSERT INTO ? (date,value) VALUES (?,?)', \
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#g.db.execute('INSERT INTO '+session['user']+' (date,value) VALUES (?,?)', \
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A: You can not use placeholders for column or table names. You can try this:
user = session['user']
data = request.form['date']
value = request.form['value']
g.db.execute('INSERT INTO {} (date,value) VALUES (?,?)'.format(user), (data, value))
A: You cannot parameterized table name. You have to use string format/concatenate for that.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 101 |
\section{Introduction}
\label{sec:intro}
Variational approaches based on tensor networks~\cite{CV} are a novel powerful numerical tool believed to be the key
ingredient to simulate efficiently quantum-many body systems. Although a detailed understanding of their potentialities and their
limitations is presently under scrutiny, there are already a number of encouraging results. Variational Tensor Networks (VTN) are free
of most of the problems of traditional numerical methods. Differently from quantum Monte Carlo methods, VTN do not suffer of the sign
problem. Compared to density matrix renormalization group~\cite{DMRG}, they are more versatile and allow to simulate efficiently critical
correlations, long-range interactions and two- and higher-dimensional quantum systems. Indeed the density matrix renormalization group can be
reformulated in terms of a particular class of tensor networks known as Matrix Product States~\cite{mps}. VTN include also projected
entangled pair states~\cite{peps} that generalize matrix product states in dimensions higher than one and weighted graph states~\cite{wgs}
designed to study systems with long-range interactions.
Among the proposed VTNs, a very promising one is the so-called Multiscale Entanglement Renormalization Ansatz (MERA)~\cite{mera}.
MERA has been already applied successfully to the study of a number of different physically relevant systems, like quantum models on a
two-dimensional lattices~\cite{cincio, Evenbly:2009p1759}, interacting fermions~\cite{corboz}, and critical
systems~\cite{jova08,pev-09,Giovannetti:2009p1563,Montangero:2008p1565}, only to mention a few of the most remarkable examples.
The capability of MERA to describe accurately critical systems derives directly from the scale-invariant self similarity of its tensor structure,
intimately related to a real-space renormalization procedure. The structure of the MERA state is designed~\cite{mera} in such a way
to reproduce scale-invariance and so, in one-dimensional systems, it naturally encoded several important features of the Conformal Field
Theory (CFT) underlying the critical lattice model~\cite{pev-09}.
The critical exponents can be computed directly from the spectrum of the MERA transfer matrix~\cite{jova08}.
Critical systems can however lack translational invariance, due to the presence of a physical boundary or to an impurity. The study of
boundary critical phenomena is, since many years, a very active field of research which ranges from the study of the critical magnets with
surfaces to quantum impurity problems (as e.g. Kondo) or the Casimir effects (for a review of the field see for example~\cite{bcs}).
The presence of the boundary does not spoil conformal invariance. Oppositely, boundary CFTs have a very rich structure and a deep
mathematical foundation (see e.g.~\cite{cardy-05}).
While in fact it started only as the study of critical two-dimensional systems in system with boundaries (surface critical behavior), it
found applications to open-string theory (D-branes), quantum impurity problems \cite{s-98},
quantum out-of-equilibrium studies (quantum quenches \cite{cc-q}) just to cite a few.
Furthermore, it attracted a large attention from the mathematical community for the recent
developments of stochastic Loewner evolution \cite{sle}.
In view of the connection between MERA and CFT, it is natural to wonder whether the MERA tensor-network can be employed to study
quantum systems with boundaries. In this work we introduce and analyze an entanglement renormalization tensor-network design which
takes into account the presence of a (critical) boundary, and we study its properties.
This is implemented by allowing the edge spin at the boundary of the system to interact at each step with an \emph{ancillary}
element, describing a fictitious degree of freedom.
Similarly to Refs.~\cite{jova08,Giovannetti:2009p1563,pev-09}, we will focus on
homogeneous configurations, where tensor elements of the same class are also identical to each other.
Interestingly enough this ansatz is able to capture some important properties predicted by boundary CFT.
Specifically, we show that the critical exponent associated to the decay of any
one-point function (as function of the distance from the boundary) is always half of the one of the bulk two-point correlation function
corresponding to the same scaling operator. We also compute the boundary corrections to the ground state energy.
\begin{figure*}[htb]
\hspace*{-0.6cm}
\includegraphics[width=280pt]{truebornbera} \hspace*{1.cm}
\includegraphics[width=140pt]{lambdabera}
\caption{ \label{fig:chibera}
Left panel - Entanglement renormalization network representation for $N=32$ sites. In the bulk the MERA structure consists of
isometries and disentanglers. The boundary is represented by an additional ancillary system indicated by the grey stripe.
Right panel - Alternative insertion of the boundary obtained by applying the ancilla interaction tensors (magenta crescent-moons) at the
same level of the disentanglers instead of the renormalizers, as done in the left panel. }
\end{figure*}
The paper is organized as follows. In Sec.~\ref{sec:ans} we introduce the tensor network and its main properties. In Sec.~\ref{sec:ex}
we discuss how the expectation values of local observable can be computed. Assuming a uniform tensor structure,
in the thermodynamic limit, these expectation values decay as power law. We relate the associated critical exponents to those of the
corresponding two-point correlations in the bulk. In Sec.~\ref{sec:tra} we discuss the boundary corrections to the ground state energy.
The conclusions of our work are summarized in Sec.~\ref{sec:con}.
\section{The tensor network}
\label{sec:ans}
Consider a 1D lattice of $N = 2^{n+2}$ spins (sites), of a given local dimension $d$, with open boundary conditions.
A generic pure state of such system can be expressed as
\begin{eqnarray}
|\psi^{(n)}\rangle = \sum_{\ell_1, \ldots , \ell_N = 1}^{d} \mathcal{T}_{\ell_1, \ldots \ell_N}
|\xi_{\ell_1} \ldots \xi_{\ell_N} \rangle \label{eq1}\;,
\end{eqnarray}
with $\{ |\xi_{i}\rangle \}_i$ a canonical basis for the single qudit and $\mathcal{T}$ a
type-$ \mbox{\tiny{$\left(\begin{array}{c} 0 \\N \end{array}\right)$}}$
tensor. Following the prescriptions of the MERA structure~\cite{mera}, we assume a formal decomposition of $\mathcal{T}$ which is
schematically sketched in Fig.~\ref{fig:chibera}. Here we use the standard graphical convention (see e.g. Ref.~\cite{mera}) for which
each node of the graph represents a tensor (the emerging legs of the node being its indices), while a link connecting any two nodes represents
the contraction of the corresponding indices.
The yellow element on the top of the figure describes a
type-$\mbox{\tiny{$\left(\begin{array}{c} 0 \\6 \end{array}\right)$}}$ tensor ${\cal C}$ of elements ${\cal C}_{a, \ell_1,\ell_2,\cdots, \ell_4,a'}$,
that we can call {\em hat} tensor.
The green triangles represent instead the same $d \times d^2$ renormalizer tensor $\lambda$ of
type-$ \mbox{\tiny{$\left(\begin{array}{c} 1 \\2 \end{array}\right)$}}$ of elements
${\lambda}^{u}_{\ell_1,\ell_2}$, and the blue circular elements represent the same $d^2\times d^2$ disentangler tensor $\chi$ of type-$
\mbox{\tiny{$\left(\begin{array}{c} 2 \\2 \end{array}\right)$}}$ of elements $\chi^{u_1,u_2}_{\ell_1,\ell_2}$.
At the boundary, we introduce extra tensors that couple the sites at the border with an ancillary degree of freedom represented in Fig.
\ref{fig:chibera} by the thick grey strip.
These new elements form the lateral edges of the network and describe the boundary at each level of the MERA, i.e. at each level of
the renormalization flow. As shown in the figure, each of them can be viewed as a matrix product state (yellow squares) whose bonding dimensions coincide with the coordinate space of the ancillas, which is coupled to the bulk via local coupling-elements (drawn as magenta
crescent-moons in the figure).
Via purification, we can always choose the dimension of such ancilla to be large enough so that the resulting interaction is described by a
unitary operator,
that we indicate as $\alpha$, a type-$
\mbox{\tiny{$\left(\begin{array}{c} 2 \\2 \end{array}\right)$}}$ tensor of elements $\alpha^{u_1,u_2}_{\ell_1,\ell_2}$.
Similarly to the case of the $\lambda$s and of the $\chi$s, we will also assume these elements to be uniform in the network (possibly allowing the
ones on the left-hand-side of the structure to differ from the ones on the right-hand-side \cite{footbcco}).
As customary with MERA-like configurations, to enforce efficient evaluation of local observables and correlation functions, the various elements
of the network are assumed to obey specific contraction rules (a detailed analysis of the efficiency requirements for MERA can be
found in Refs.~\cite{mera,EVE,Giovannetti:2009p1563}). In particular the renormalizers and the disentanglers obey isometric and unitary
constraints respectively, i.e.
\begin{eqnarray} \label{ffd}
\sum_{k_1,k_2} \lambda^{u}_{k_1,k_2} \bar{\lambda}_{\ell}^{k_1,k_2} = \delta^{u}_{\ell}, \quad
\sum_{k_1,k_2} \chi^{u_1,u_2}_{k_1,k_2} \bar{\chi}_{\ell_1,\ell_2}^{k_1,k_2} = \delta^{u_1}_{\ell_1}\; \delta^{u_2}_{\ell_2},
\end{eqnarray}
where $\delta^{u}_{\ell}$ is the Kronecker delta, while $\bar{\lambda}_{\ell}^{u_1,u_2} \equiv ({\lambda}^{\ell}_{u_1,u_2})^*$
and $\bar{\chi}_{\ell_1,\ell_2}^{u_1,u_2} \equiv ({\chi}^{\ell_1,\ell_2}_{u_1,u_2})^*$ are the adjoint counterparts of the $\lambda$ and $\chi$
tensors respectively, obtained by exchanging their lower and upper indices and taking the complex conjugate.
Similar conditions are imposed also for the edge tensors
\begin{eqnarray} \label{ffe}
\sum_{k_1,k_2} \alpha^{u_1,u_2}_{k_1,k_2} \bar{\alpha}_{\ell_1,\ell_2}^{k_1,k_2} = \delta^{u_1}_{\ell_1}\; \delta^{u_2}_{\ell_2}.
\end{eqnarray}
These rules are graphically represented in Fig.~\ref{fig:rulez}.
Finally to ensure proper state normalization, the tensor ${\cal C}$ is supposed to satisfy the
identity $\sum_{a, k_1, \cdots, k_4, a'}{\cal C}_{a, k_1, \cdots,k_4, a'} \bar{\cal C}^{a, k_1, \cdots, k_4, a'} =1$.
It is worth noticing that, by simply re-arranging the various tensorial components,
an entanglement renormalization configuration which differs from the one given in Fig.~\ref{fig:chibera} can be obtained.
In fact, the ancilla interaction tensors $\alpha$ (the magenta crescents) can be applied
at the same level of the disentanglers, instead of the renormalizers (see Fig.~\ref{fig:chibera} right panel).
Despite their different appearance, it can be shown that these two structures are formally equivalent to each other.
This can be verified by grouping together the edge-ancilla interaction with the nearest linked element belonging to the lower half-level.
Now, by just performing a polar decomposition~\cite{HORN}, we obtain a structure of the opposite type (although, the very first spin
of the chain is now taken out the system and put into the ancilla, while the second one becomes the edge spin).
Having acknowledged this equivalence, in the rest of the paper we will work with the structure of the left panel of Fig.~\ref{fig:chibera}.
\begin{figure}
\centering
\includegraphics[width=230pt]{berarules}
\caption{ \label{fig:rulez}
Rules for the elements of the tensor network of Fig.~\ref{fig:chibera}. The black marks drawn upon the tensors
are used to make distinction between top and bottom. The first two schemes correspond to the the contractions of Eq.~(\ref{ffd}),
while the last one represents the unitary constraint for the edge elements of the tensor.
}
\end{figure}
\section{Local averages in the presence of a boundary}
\label{sec:ex}
In the presence of the boundary, the average of any local observable depends on the distance from the boundary itself.
For a one-dimensional critical system, the case we consider here, the space-dependence will be a power-law characterized
by a set of critical exponents. In this Section we show how to compute local averages and how to extract these
exponents.
Consider a family ${\cal F}\equiv \{ |\psi^{(n)}\rangle ; n =2, 3, \cdots \}$ of states $|\psi^{(n)}\rangle$ of increasing sizes,
described by homogeneous networks of the form shown in Fig.~\ref{fig:chibera}, each sharing the same structural elements
(renormalizer, disentangler, edge-ancilla interaction, hat). For such family we want to calculate the expectation value
of a general observable acting on a small group of neighboring sites located at a given distance from the closest edge of the system,
say the left one. For instance, in the case of a three-sited observable $\Theta_\ell$ acting on the sites $\ell,\ell+1$, and $\ell+2$~\cite{MOTA}
we have
\begin{equation} \label{eq:prima}
\langle \Theta_\ell \rangle^{(n)} \equiv
\langle \psi^{(n)} | \Theta_\ell | \psi^{(n)} \rangle =
\text{Tr} [ \Theta_\ell \cdot \rho^{(n)}_{\ell,\ell+1,\ell+2}],
\end{equation}
where the site indices are counted starting from the leftmost spin
as the first one, and where $\rho^{(n)}_{\ell,\ell+1,\ell+2}$ is the reduced density matrix of $|\psi^{(n)}\rangle$ associated with the selected spins.
We assume a uniform MERA structure. This assumption may seem an over-simplification for a system which is not translational invariant, but
it turns out that it naturally accounts for the underlying (boundary) conformal invariance.
From the locality requirements imposed in Fig.~\ref{fig:rulez}, it is straightforward to verify that for all $1\leqslant \ell \leqslant 2^{n+1}-2$
and $n\geqslant 1$,
the following recursion rules apply:
\begin{equation} \label{eq:ricors}
\begin{array}{c}
\rho^{(n)}_{2\ell,2\ell+1,2\ell+2} = \mathcal{D}_L \left( \rho^{(n-1)}_{\ell,\ell+1,\ell+2} \right)\;, \\ \\
\rho^{(n)}_{2\ell+1,2\ell+2,2\ell+3} = \mathcal{D}_R \left( \rho^{(n-1)}_{\ell,\ell+1,\ell+2} \right)\;,
\end{array}
\end{equation}
where $\rho^{(n-1)}_{\ell,\ell+1,\ell+2}$ and $\rho^{(n-1)}_{\ell,\ell+1,\ell+2}$ are 3-sites reduced density matrices of
$|\psi^{(n-1)}\rangle \in {\cal F}$. In these expressions, $\mathcal{D}_L$ and $\mathcal{D}_R$ are completely positive trace preserving (CPT) maps
that depend only on the bulk elements of the network
(indeed they coincide with the $\mathcal{D}_L$ and $\mathcal{D}_R$ maps of an ordinary
homogenous MERA with the same $\lambda$ and $\chi$~\cite{jova08}) and whose
formal expression is graphically depicted in Fig.~\ref{fig:dielledierre}.
\begin{figure}[!b]
\centering
\includegraphics[width=230pt]{dielledierre}
\caption{ \label{fig:dielledierre}
Sketch of the CPT maps \emph{a)} $\mathcal{D}_L$ and \emph{b)} $\mathcal{D}_R$; the contracted tensors picture represents
their Kraus decomposition.}
\end{figure}
By means of the the renormalization procedure implied by Eq.~\eqref{eq:ricors}, at each application of the map the
site over which the average is performed approaches the boundary in an exponential fashion.
Correspondingly the network depth decreases linearly. Upon reaching the boundary one has to define further operations:
\begin{equation} \label{mappak}
\rho^{(n)}_{1,2,3} = \mathcal{K}_L \left( \rho^{(n-1)}_{A,1,2} \right)\;,
\end{equation}
where $A$ refers to the degree of freedom belonging to the ancillary
system, and $\mathcal{K}_L$ is again a CPT map, sketched in Fig.~\ref{fig:kel} (left).
At this point the causal cone of the ancilla jointed with the first two sites
is stable. Indeed one has
\begin{equation}
\rho^{(n)}_{A,1,2} = \mathcal{B}_L \left( \rho^{(n-1)}_{A,1,2} \right),
\label{bielledef}
\end{equation}
where $\mathcal{B}_L$ is the CPT map represented in Fig.~\ref{fig:kel} (right).
Analogously to $\mathcal{B}_L$ and $ \mathcal{K}_L$, we define the CPT maps at the right boundary from the mirror images of
Fig. \ref{fig:kel} and we call them $\mathcal{B}_R$ and $ \mathcal{K}_R$ respectively.
Because of the stability of the causal cone~\cite{mera,EVE,Giovannetti:2009p1563}, approaching the thermodynamical limit
we can determine the reduced density matrix in proximity of the boundary by
calculating the fixed point of $\mathcal{B}_L$.
This is unique provided that the CPT map is mixing~\cite{NJP,jova08}, i.e.
\begin{equation}
\lim_{n \to \infty}\rho^{(n)}_{A,1,2} = \rho^{f}_{A,1,2} = \mathcal{B}_L \left( \rho^{f}_{A,1,2} \right).
\end{equation}
We can now use this argument to obtain the expectation value in Eq. \eqref{eq:prima} for infinite volume.
The resulting expression becomes particularly simple when
$\mathcal{D}_L = \mathcal{D}_R = \mathcal{D}$. In this case indicating the integer part of $\log_2 \ell$ with $\lfloor \log_2 \ell\rfloor$,
we have
\begin{equation} \label{eq:sviluppo}
\langle \Theta_{\ell} \rangle^{(\infty)} =
\text{Tr} \left[ \Theta \cdot \mathcal{D}^{\lfloor \log_2 \ell\rfloor }\circ \mathcal{K}_L (\rho^{f}_{A,1,2}) \right]\;,
\end{equation}
where ``$\circ$" stands for super-operator composition and where
${\cal D}^\tau$
describe a $\tau$ reiterated applications of the map ${\cal D}$.
We can now exploit the Jordan block decomposition~\cite{HORN} to simplify further this expression.
\begin{figure}
\centering
\includegraphics[width=110pt]{kappaelle} \hspace{2mm}
\includegraphics[width=110pt]{bielle}
\caption{ \label{fig:kel}
Tensor graph representations of the CPT map $\mathcal{K}_L$ (left) and $\mathcal{B}_L$ (right) defined in Eq.~(\ref{mappak}) and (\ref{bielledef}) respectively.}
\end{figure}
Adapting the derivation for the bulk in Ref. ~\cite{Giovannetti:2009p1563} to the boundary case,
we easily get
\begin{equation} \label{ddd}
\langle \Theta_{\ell} \rangle^{(\infty)} =
\sum_{\kappa} \kappa^{\lfloor \log_2 \ell\rfloor}
\; {g}_\kappa(\lfloor \log_2 \ell\rfloor)
\;,
\end{equation}
where the sum spans over the eigenvalues $\kappa$ of $\mathcal{D}$ and ${g}_\kappa(\cdot)$
is a (finite degree) polynomial in its main argument with coefficients
which depends on $\rho^{f}_{A,1,2}$ and $\mathcal{D}$.
Since CPT maps are contractive, the values of $\kappa$ entering in Eq.~(\ref{ddd}) belong to the unit circle (i.e. $|\kappa|\leqslant 1$).
Furthermore, if $\mathcal{D}$ is mixing (which is a reasonable assumption~\cite{jova08}) then its spectrum admits a unique eigenvector (the fix point $\rho^f$ of the channel) associated with $\kappa=1$; all the remaining eigenvalues have modulus strictly smaller than $1$. Under these circumstances,
in the limit of large distance from the boundary, the quantity $\langle \Theta_{\ell} \rangle^{(\infty)}$ converges toward its bulk limit which
is obtained by computing the expectation value of $\Theta$ on the fix point of the channel, i.e. $\langle \Theta_{\text{bulk}} \rangle^{(\infty)} = \mbox{Tr}[ \Theta \rho_f]$.
The deviations from such limiting expression can be evaluated by keeping the largest contribution associated with the terms with $\kappa \neq 1$.
This yields an
exponential decay in $\log_2 \ell$ of the form
\begin{eqnarray} \label{befo}
\langle \Theta_{\ell
} \rangle^{(\infty)} &=& \langle \Theta_{\text{bulk}} \rangle^{(\infty)} +
\sum_{\kappa\neq 1} \kappa^{\lfloor \log_2 \ell\rfloor}
\; {g}_\kappa(\lfloor \log_2 \ell\rfloor) \nonumber \\
&=& \langle \Theta_{\text{bulk}} \rangle^{(\infty)}+ \ell^{\log_2 |\bar{\kappa} |} \; g'(\log_2 \ell) \;,
\end{eqnarray}
where $\bar{\kappa}$ is the eigenvalue of $\mathcal{D}$ which has the largest absolute value smaller than one and
which contributes not trivially to Eq.~(\ref{ddd}), and where $g'(\log_2 \ell)$ is instead
some complicated function which is dominated by a polynomial of $\log_2 \ell$.
In particular, if $\Theta$ is an eigenvector of the Heisenberg adjoint of the channel $\mathcal{D}$, then
Eq.~(\ref{befo}) yields an exact power-law decay (without polynomial corrections), i.e.
\begin{eqnarray} \label{befo1}
\langle \Theta_{\ell
} \rangle^{(\infty)} &=&c \; \kappa^{\lfloor \log_2 \ell\rfloor} \;,
\end{eqnarray}
where $c = \; \mbox{Tr} [ \Theta \; {\cal K}_L (\rho_{A,1,2}^f) ]$, and where $\kappa$ is the associated eigenvalue (notice that
for such a $\Theta$ one has $\langle \Theta_{\text{bulk}} \rangle^{(\infty)}=0$).
The above expressions
show that the quantities $- \log_2 |{\kappa}|$ play the role of the critical exponents of the system.
It is now evident that these critical exponents are the half of the corresponding ones for two-point correlation functions computed in the bulk,
a well-known result in conformal field theory \cite{cardy-05}.
For instance fixing the distance $\Delta \ell = 2^m$ of the two points, the bulk connected correlation function has been computed~\cite{jova08}, and it holds
\begin{eqnarray}
\mathfrak{C}^{(n)}_{\Delta \ell} \equiv \frac{1}{N} \sum_{\ell = 1}^{N}
[ \langle \Theta_{\ell} \, \Theta_{\ell + \Delta\ell} \rangle^{(n)} -
\langle \Theta_{\ell} \rangle^{(n)} \langle \Theta_{\ell + \Delta\ell}
\vphantom{\sum} \rangle^{(n)} ] \nonumber \\= \mbox{Tr} [ (\Theta \otimes \Theta) \; \cancel{\mathcal{D}}^{\log_2 \Delta \ell } (\sigma) ] = \sum_{\xi\neq 1} \xi^{\log_2 \Delta \ell} \; h_\xi(\log_2 \Delta\ell),
\end{eqnarray}
where the summation is performed over the eigenvalues $\xi$ of the CPT map
$\cancel{\mathcal{D}} = \frac{1}{2} \left(
\mathcal{D}_L \otimes \mathcal{D}_L + \mathcal{D}_R \otimes \mathcal{D}_R \right)$, and where $h_\xi(\cdot)$ is a polynomial function of its argument (in this expression
$\sigma$ stands for a traceless operator while finally $N=2^{n+2}$ is the size of the associated homogeneous MERA with periodic boundary conditions).
The result then follows by noticing that by construction $\cancel{\mathcal{D}} = \mathcal{D} \otimes\mathcal{D}$ so that the $\xi$ can be expressed as products $\kappa \kappa'$ of the eigenvalues of $\mathcal{D}$.
In particular if as in Eq.~(\ref{befo1}),
$\Theta$ is an eigenvector the Heisenberg adjoint of $\mathcal{D}$ we have that $\Theta \otimes \Theta$ is an eigenvector of the adjoint of $\cancel{\mathcal{D}}$ at the eigenvalue $\xi = \kappa^2$ and thus,
\begin{eqnarray}
\mathfrak{C}^{(n)}_{\Delta \ell} = c' \kappa^{2 \log_2 \Delta \ell} =
c'
\;(\Delta \ell)^{2 \log_2 \kappa } \;,
\end{eqnarray}
which proves the claim (here $c' = \mbox{Tr} [(\Theta \otimes \Theta) \sigma]$).
\section{Boundary contribution to the ground state energy}
\label{sec:tra}
In the presence of a boundary, the average of extensive observables (the ground state energy for example) does
contain a bulk and a boundary contribution (negligible in the thermodynamic limit).
In this section, we evaluate the {\it density} of the ground-state energy
for a local Hamiltonian (with interactions among sites at maximum distance $\nu$) of the form
\begin{equation}
\mathcal{H} = \frac{1}{L-\nu+1} \sum_{j = 1}^{L - \nu+1} H_{j \ldots j+\nu},
\end{equation}
where $\nu$ is the number of sites involved in the model interaction $H$.
While this problem is easily solved in a level-recursive manner for
a MERA structure (in which periodic boundary conditions hold), when
explicit conditions over a defined boundary are involved things become slightly
more complicated.
Suppose, for simplicity, that the interaction is again a $\nu =$ 3-body operator,
therefore
\begin{equation}
\langle \mathcal{H} \rangle = \text{Tr} [ H_3 \cdot \bar{\rho}_3^{(n)}]\;,
\end{equation}
where
\[
\bar{\rho}_3^{(n)} \equiv \frac{1}{2^{n+2}-2} \sum_{j = 1}^{2^n} \rho^{(n)}_{j,j+1,j+2}\;\; .
\]
We need to build a recursive function which relates this average density matrix
to the one belonging to the previous tensor level $\bar{\rho}_3^{(n-1)}$.
Of course, the boundaries will play some role too in this relation
\begin{multline} \label{eq:grossa}
\bar{\rho}_3^{(n)} =
\frac{1}{2^{n+2}-2} \left[ \mathcal{K}_L \left( \rho^{(n-1)}_{A,1,2} \right) +
\mathcal{K}_R \left( \rho^{(n-1)}_{2^{n+1}-1, 2^{n+1}, A'} \right) \right]
\\ +
\left(1 - \frac{1}{2^{n+1} -1}\right) \cdot \mathcal{D} \left(\bar{\rho}_3^{(n-1)} \right),
\end{multline}
(here $\mathcal{D}$ is the average of $\mathcal{D}_L$ and $\mathcal{D}_R$).
This equation shows the contributions of both bulk and edge terms; nevertheless,
when we approach the thermodynamical limit $n \to \infty$, the contribution
of the first two terms vanishes in every norm, because any density matrix
has trace norm bounded by one and CPT maps are contractive.
This means that the extensive influence of the boundary upon the lattice
grows \emph{slower} than the size of the system, a physical sounding and known property.
To quantitatively describe such behavior,
we compute the (total) energy associated with the block of the first $2^\tau-1$ spins near a boundary,
say the left one. In our notation this corresponds to
\begin{equation} \label{eq:estens}
E^{(n)}_{1 \ldots 2^{\tau}-1} =
\text{Tr}\left[ H_3 \cdot \sum_{j = 1}^{2^\tau-1} \rho^{(n)}_{j,j+1,j+2} \right].
\end{equation}
Exploiting the usual formalism of level-growing CPT maps, we can
rewrite the sum in~\eqref{eq:estens} as
\begin{equation}
\sum_{j = 1}^{2^\tau-1} \rho^{(n)}_{j,j+1,j+2}
= \sum_{p = 0}^{\tau - 1}
\, 2^p \cdot \mathcal{D}^p \circ \mathcal{K}_L \circ \mathcal{B}_L^{\tau - p -1} \left( \rho^{(n-\tau)}_{A,1,2}\right).
\end{equation}
Now, we can successfully approach the thermodynamical limit while keeping $\tau$ fixed.
Recalling that $\rho^{f}_{A,1,2}$ is the fixed point of $\mathcal{B}_L$, we obtain
\begin{equation}
E^{(\text{th})}_{1 \ldots 2^{\tau}-1} = \text{Tr} \left[ H_3 \cdot \sum_{p = 0}^{\tau - 1}
\, 2^p \; \mathcal{D}^p \circ \mathcal{K}_L \left( \rho^{f}_{A,1,2}\right) \right].
\end{equation}
As expected, the result diverges for $\tau \to \infty$ since the
series is made of terms growing in trace norm.
To explicitly estimate how this quantity \emph{deviates} from
its corresponding value in the bulk as $\tau$ grows we evaluate the following quantity
\begin{multline}
\Delta E^{(\text{th})}_{1 \ldots 2^{\tau}-1} = - (2^{\tau}- 1) \;\text{Tr} \left[ H_3 \cdot \rho^f_3 \right]
+ \\ +
\text{Tr} \left[ H_3 \cdot \sum_{p = 0}^{\tau - 1}
\, 2^p \; \mathcal{D}^p \circ \mathcal{K}_L \left( \rho^{f}_{A,1,2}\right) \right]
= \\ =
\text{Tr} \left[ H_3 \cdot \sum_{p = 0}^{\tau - 1}
\, 2^p \; \mathcal{D}^p \left\{ \mathcal{K}_L \left( \rho^{f}_{A,1,2} \right) - \rho^f_3 \right\} \right].
\end{multline}
Notice that in this case the map $\mathcal{D}$ applies to a traceless
operator, therefore if we decompose the argument in a basis of generalized eigenvectors
for $\mathcal{D}$, it must have null component over the unique state of eigenvalue one.
As a result the boundary contribution to the ground state energy has the form
\begin{equation} \label{eq:diverg}
\Delta E^{(\text{th})}_{1 \ldots \infty} = \sum_{\kappa_D \neq 1} \; \sum_{p=0}^{\infty}
( 2 \kappa_{\mathcal{D}} )^{p} \cdot g_{\kappa_{\mathcal{D}}}(p),
\end{equation}
where $g(\cdot)$ is a polynomial in its main argument.
Looking at Eq. \eqref{eq:diverg}, we notice that the inner sum diverges for any
eigenvalue $\kappa_{\mathcal{D}}$ of $\mathcal{D}$ greater or equal to $1/2$, unless the $g$ are identically zero
for such values of $\kappa_{\mathcal{D}}$.
In general this will happen
when $\rho^f_3 - \mathcal{K}_L(\rho^f_{A,1,2})$ has null component over any
generalized eigenspace whose $\kappa_{\mathcal{D}} \geq 1/2$.
Interestingly enough the capability of such deviation to diverge is another
manifestation that the MERA states of Fig.~\ref{fig:chibera} are critical: indeed only
the integral of a power-law function can diverge, while for an exponential decaying correlation function
the integral of the corresponding action is always finite.
\section{Conclusions}
\label{sec:con}
In this paper we exploited the properties of MERA to describe boundary critical phenomena. We considered the
case of a one-dimensional critical system with a boundary. To this end we modified the local structure of the MERA
at the boundary to account for more flexibility in its description.
Besides showing how to compute local observables, we achieved two main results.
First of all we showed, as predicted by boundary conformal field theory, that the critical
exponents associated to the decay of the one-point function (as a function of the distance from the boundary) is always
half of the one of the bulk two-point correlation function corresponding to the same scaling operator.
Secondly we compute the boundary corrections to the ground state energy and determined its scaling behavior.
As in the bulk case, also in the presence of the boundary, most of the critical properties are determined solely by the eigenvalues of the
MERA transfer matrix.
A remarkable feature of treating boundary critical phenomena with MERAs is that it is enough to consider uniform tensor
network. This is the result of the scale invariance of the underlying tensor network which holds also in the presence of
boundaries. In addition to the practical advantage in the numerical simulations, this observation further clarifies the
properties of the MERA.
It is worth noticing that such property (that is at the basis of the effectiveness of a bounded MERA) is physically equivalent to the
fact that, in boundary critical phenomena, the operator content of the bulk is not influenced by the boundary \cite{bcs}, suggesting
that maybe the connection between MERA and general renormalization group theory is deeper than what nowadays understood.
One dimensional systems display the peculiar feature that the boundary can be critical only when also the bulk is.
This is not the case in higher dimensions \cite{bcs}, where we can have a critical boundary in a gapped system, resulting in a
richer scenario for the boundary-bulk phase diagram.
This richness will reflect in the possibility to have different compositions of tensor structure.
In this paper we considered a matrix product state (at the border) connected to a MERA.
It is easy to imagine that, to describe critical surfaces in a non-critical bulk, different compositions of tensor
networks are required.
During the writing of this work a paper by G. Evenbly {\em et al.}
appeared on the archive discussing boundary critical phenomena using MERA, see Ref.~\cite{BOU}.
We acknowledge fruitful discussions with M. Campostrini, S. Montangero and M. Rizzi, and financial support from
FIRB-RBID08B3FM and the National Research Foundation and Ministry of Education
Singapore.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,237 |
\section{Introduction}\label{sec:Introduction}
\IEEEPARstart{N}{etwork} or graph is usually used to model the complex interaction relations in real-world data and systems~\cite{strogatz2001exploring,gao2016universal}, e.g., transportation, social pattern, cooperative behavior and metabolic phenomenon. By convention, the network (or graph) is usually abstracted as some nodes and their complicated and elusive links. To understand such data, network analysis can help to explore the organization, analyze the structure, predict the missing links and control the dynamics in complex systems. For a long time, researchers have proposed specially designed methods and models for different graph mining tasks, such as preference mechanism, hierarchical structure and latent space model for link predication~\cite{martinez2016survey}; grouping or aggregation, bit compression and influence based for network summarization~\cite{liu2018graph}; generalized threshold model, independent cascade model and linear Influence Model for information diffusion~\cite{zhang2016dynamics}. Among the core issues and applications of network analysis and graph mining, clustering~\cite{serrano2006clustering}, dividing the nodes into distinct or overlapping groups, has attracted attracted the most interest from different domains including machine learning and complex networks.
The field of network clustering has two main branches: \textbf{community detection}~\cite{girvan2002community,rosvall2007information,8531771} and \textbf{role discovery}~\cite{rossi2014role}. Community detection, the currently dominant clustering branch, is devoted to find common groups in which nodes interact more intensively than outside~\cite{fortunato2010community}. However, role discovery, which has a long research history in sociology~\cite{merton1968social} but had been inconspicuous in network science, groups the nodes based on the similarity of their structural patterns\cite{liu2012social}, such as the bridge or hub nodes~\cite{gilpin2013guided}. In general, nodes in the same community are likely to be connected to each other, while nodes in the same role may be unconnected and often far away form each other. Since their rules on dividing nodes are fundamentally different, the two branches are usually considered as orthogonal problems. A variety of algorithms and models are proposed for both of the two branches. For community detection, the modularity optimization~\cite{chen2014community,li2013multicomm}, statistical model~\cite{stanley2016clustering,7745890}, non-negative matrix factorization~\cite{yang2014unified,pei2015nonnegative,ma2017evolutionary,ma2018community} and deep learning methods~\cite{tu2018unified,liu2020deep} are developed and show crucial influence for other tasks and applications, such as recommendation\cite{feng2015personalized,zheng2019personalized} and identifying criminal gangs~\cite{lu2014algorithms}. Some surveys on community detection can be seen in~\cite{fortunato2010community,fortunato2016community,jin2021survey}. For role discovery, traditional methods are usually graph based and related to some equivalence, such as the structural~\cite{lorrain1971structural}, regular~\cite{white1983graph} and stochastic equivalence~\cite{holland1981exponential,nowicki2001estimation}. Blockmodels\cite{faust1992blockmodels} and mixed-membership stochastic blockmodels~\cite{airoldi2008mixed} are the important and influential methods are based on the graph. Besides, there are also some combinatorial or heuristic methods \cite{arockiasamy2016combinatorial} for this problem.
\begin{figure*}
\centering
\subfigure[Truth label]{
\includegraphics[width=.3\textwidth]{pic/brazil-flights-role.pdf}
}
\subfigure[Community detection]{
\includegraphics[width=.3\textwidth]{pic/brazil-flights-community.pdf}
}
\subfigure[Role discovery]{
\includegraphics[width=.3\textwidth]{pic/brazil-flights-rolx.pdf}
}
\caption{Air Brazil network for understanding the community detection and role discovery in network clustering. Nodes with same clustering label have same color. (a) Brazil network with ground-truth clustering label; (b) Community detection with Louvain~\cite{blondel2008fast}; (c) Role discovery with RolX~\cite{henderson2012rolx}.}
\label{fig:Brazil-sample}
\end{figure*}
Here we take the Brazilian air-traffic network as an example. As shown in Fig.~\ref{fig:Brazil-sample} (a), the nodes and edges denote the airports and their direct flights, respectively. The clustering labels are marked based on the activity of nodes~\cite{ribeiro2017struc2vec}. The size and color of the circle represent the degree and label of the node, respectively. It can be observed that, the nodes with the same label (color), i.e., they are structurally similar, are usually not connected.
To better illustrate these two clustering tasks, we choose two typical methods, Louvain~\cite{blondel2008fast} and RolX~\cite{henderson2012rolx}, which are specially designed for community detection and role discovery, respectively. The clustering results are presented in Fig.~\ref{fig:Brazil-sample} (b) and (c). Community detection divides this network into some tightly connected groups, which means the airports with more flights among them belong to the same community. However, the clustering result detected by the role discovery, usually related to the flow and scale of the airport, is closer to the truth.
In recent years, network embedding (NE) has become has become the focus of studying graph structure and been demonstrated to achieve promising performance in many downstream tasks, e.g., node classification and link prediction. The motivation of NE is to transform the network data into independently distributed representations in a latent space and these representations are capable of preserving the topological structure and properties of the original network. On the whole, current methods for NE can be categorized into two types: shallow and deep learning (here we focus on unsupervised NE approaches without explicit mentioning). The former includes the matrix factorization and random walk based methods. The goal of matrix factorization methods is to learn node embedding via the low-rank approximation and approaching the adjacency matrix or higher-order similarity of the network, such as the singular value decomposition, non-negative matrix factorization and NetSMF~\cite{qiu2019netsmf}. With the different random walk strategies, a series of methods have been proposed to optimize the co-occurrence probability of nodes and learn effective embeddings. The later is mainly rooted in autoencoder and graph convolutional networks. These methods generally consist of the encoder, similarity function and decoder. There are also some attributes, characteristics and constraints that can be combined to enhance the embedding, and VGAE, GAT and GraphSAGE are some representatives of such methods. There are also some other types of embedding approaches, e.g., the latent feature model, but discussion on general NE methods is out of our scope. We refer the interested readers to some recent survey papers on NE~\cite{cui2018survey,zhang2018network,zhang2020deep,wu2020comprehensive,goyal2018graph,cai2018comprehensive,hamilton2017representation,chen2018tutorial}.
However, most of these methods, whether or not for network clustering, are designed for modeling the \textbf{proximity}, i.e. the embedding vectors are community oriented. They fail to capture the structural similarity, or the role information~\cite{rossi2020proximity}, Therefore, it raises several inherent challenges in the research of role-oriented network embedding. Firstly, the most and important is that two nodes with structural similarity have nothing to do with their distance, which makes it difficult to define the loss function effectively. Secondly, strict role definitions, such as some definitions based on equivalence, are difficult to be implemented in real-world networks especially large-scale networks. Thirdly, the distribution of nodes with same role in the network is very complex and interaction patterns between different roles are unknown.
\iffalse
In essence, there are still some scattered methods being proposed one after another recently years. The struc2vec~\cite{ribeiro2017struc2vec} constructs a multi-layer network based on structural similarity and takes the random walk algorithm. The REACT~\cite{pei2019joint} analyzes the community structure and role discovery under a unified framework and describes their relations via non-negative matrix factorization. The DRNE~\cite{tu2018deep} is the first deep learning method for role based embedding with the regular equivalence. Although some researchers have developed diverse approaches and achieved satisfactory results from distinct options, there is still a lack of systematic understanding of role-oriented network embedding. Besides, we also lack a taxonomy for deep thinking of this problem. Meanwhile, there is short of performance and efficiency comparison of currently methods. All these limit the development and applications of role embedding.
\fi
In essence, there are still some scattered methods being proposed one after another recently years. These methods uses various embedding mechanisms. Struc2vec~\cite{ribeiro2017struc2vec} leverages random walks on graphs in which edges are weighted based on structural distances. DRNE develops a deep learning framework with layer-normalized~\cite{tu2018deep} LSTM model to learn regular equivalence. REACT~\cite{pei2019joint}, generating embeddings via matrix factorization, focus on capturing both community and role properties. Though the number of diverse role-oriented embedding methods is gradually increasing, there is still a lack of systematic understanding of role-oriented network embedding. Besides, we also lack a taxonomy for deep thinking of this problem. Meanwhile, there is short of performance and efficiency comparison of currently methods. All these limit the development and applications of role embedding.
So in this survey, we systematically analyze role-oriented network embedding and the analysis can help to understand the internal mechanism of currently methods. First, we propose a two-level categorization scheme for existing methods which is based on embedding mechanism of currently methods and models. Further more, we evaluate selected embedding methods from the perspectives of both efficiency and effectiveness on different tasks related to role discovery. In specific, we conduct comprehensive experiments on some representative methods on running time (efficiency), node classification and clustering (role discovery), top-k similarity search and visualization with widely used benchmark networks. Last, we summarize the applications, challenges and future directions of role-oriented network embedding.
Some surveys on network embedding, community detection, role discovery, and deep learning on graph have been conducted. Our survey has several essential differences compared to these works.~\cite{fortunato2010community,fortunato2016community,jin2021survey} mainly study the problem of community detection with different focuses from the perspective of network analysis and machine learning. \cite{rossi2014role} is a seminal work in reviewing the development and methodology of role discovery. However, this survey is relatively outdated where more advanced methods, e.g., deep learning based methods, have not been discussed. Besides, these surveys focus on the methods specific for community or role task, while our work studies roles with a focus on network embedding approaches which can preserve the role information. The surveys~\cite{hamilton2017representation,cui2018survey,zhang2018network,chen2018tutorial} are influential works on network embedding from different principles. However, they all focus on community-oriented methodology. Similarly, some graph embedding\footnote{We do not distinguish the difference between network embedding and graph embedding.} reviews~\cite{goyal2018graph,cai2018comprehensive}, however, except for some technologies, have nothing to do with the role-oriented embedding. Meanwhile, surveys such as~\cite{wu2020comprehensive} and~\cite{zhang2020deep} introduce the effective deep learning framework and methods on graph or networks. They focus on general problems of how to use machine learning on networks and are less relevant to our problem. One relevant work is~\cite{rossi2020proximity}, it clarifies the difference between the community-oriented and role-oriented network embedding for the first time, and proposes the normal mechanisms which can help to understand if a method is designed for community or role. However, it does not systematically discuss the series of role-oriented NE methods: some advanced methods have been ignored, some introduced works are not used for role discovery or role related tasks. Moreover, it does not evaluate methods empirically by analyzing the relevant data, tasks and performance. Another recent work~\cite{jin2021towards}, which introduces some structural node embedding methods and evaluate them empirically, is the most relevant to our work. In analysis, they mainly focus on analyzing the relationships between NE methods and equivalence. In evaluation, they evaluate the discovered roles on direct tasks such as role classification and clustering. In contrast, we concentrate on analyzing advantages and disadvantages of different role-oriented approaches using a new two-level categorization from the analysis perspective. We conduct more comprehensive experiments to evaluate different methods w.r.t. both efficiency and effectiveness in role discovery and downstream tasks including running time, classification, clustering, visualization, and top-k similarity search.
To sum up, our survey has several contributions as follows.
\begin{itemize}
\item We first show the summary of role-oriented network embedding and discuss the relationship and differences of it and community oriented.
\item We propose a two levels categorization schema of currently role-oriented embedding methods and briefly describe their formalization, mechanism, task, connection and difference.
\item We provide full experiments of popular methods of each type on different role-oriented tasks and detailed comparison on effectiveness and efficiency.
\item We share all the open-source code and widely used network datasets on Github \url{} and point out the development and questions of role-oriented network embedding.
\end{itemize}
\section{Notations and Framework}\label{sec:definition}
In this section, we give formal definitions of basic graph concepts and role-oriented network embeddings. In Table~\ref{tab.notation}, we summarize the main notations used throughout this paper. Then, we propose a unified framework for understanding the process of role oriented network embedding.
\newtheorem{definition}{Definition}
\begin{definition}[Network]
A network is denoted as $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V} =\{v_1,...,v_n \}$ is the set of $n$ nodes and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of edges. An edge $e_{ij} = (v_i, v_j) \in \mathcal{E}$ denotes the link between node $v_i$ and $v_j$.
\end{definition}
In usual, a network $\mathcal{G}$ is represented by an weight matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$. If $e_{ij} \in \mathcal{E}$, $\mathbf{A}_{ij} > 0$ ($\mathbf{A}_{ij} = 1$ for an unweighted network and $\mathbf{A}_{ij} = \mathbf{A}_{ji}$ for an undirected network), otherwise $\mathbf{A}_{ij} = 0$. Some networks may have an attribute matrix $\mathbf{X} \in \mathbb{R}^{n \times x}$ whose $i$th row represents attributes of $v_i$.
For an undirected network, denote the degree of node $v_i$ as $d_i = \sum_{j}\mathbf{A}_{ij}
$, and we have the degree matrix $\mathbf{D}=\mathrm{Diag}(d_1,...,d_n)$. $\mathbf{L}=\mathbf{D}-\mathbf{A}$ is called the Laplacian matrix, it can be decomposed as $\mathbf{L}=\mathbf{U} \Lambda \mathbf{U}^\top$ where $\Lambda = \mathrm{Diag}(\lambda_1,...,\lambda_n)$ is the matrix of eigenvalues satisfying $\lambda_1 \le \lambda_2 \le ... \le \lambda_n$.
Denote the $k$-hop ($k>0$) reachable neighbor set of node $v_i$ as $\mathcal{N}^k_
i$ ($k$ is omitted when $k=1$), where the shortest path between $v_i$ and each node $v_j \in \mathcal{N}^k_
i$ is less than or equal to $k$. For a directed network, use $d_i^+$, $d_i^-$, $\mathcal{N}^{k+}_i$ and $\mathcal{N}^{k-}_i$ to represent the out/in-degree and $k$-hop reachable out/in-neighborhood of $v_i$ respectively. Unless otherwise stated, a model is discussed on unweighted undirected networks without attributes in later part of this paper.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.7}
\caption{Main Notations.}
\label{tab.notation}
\centering
\begin{tabular}{c|c}
\hline
\bfseries Notation & \bfseries Definition\\
\hline
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ & the network/graph with node set $\mathcal{V}$ and edge set $\mathcal{E}$\\
\hline
$\mathcal{N}^k_i$& the set of $v_i$'s $k$-hop reachable neighbors\\
\hline
$\mathcal{G}^k_i$& the subgraph induced by $v_i$ and $\mathcal{N}^k_
i$\\
\hline
$d_i$& the degree of node $v_i$\\
\hline
$s_{ij}$& the shortest path between $v_i$ and $v_j$\\
\hline
$\mathbf{X}$& attribute matrix \\
\hline
$\mathbf{I}$& identity matrix\\
\hline
$\mathbf{H}$& embedding matrix\\
\hline
$\mathbf{F}_{\rm m}$& the feature matrix extracted by or in method $\rm m$\\
\hline
$\mathbf{S}_{\rm m}$& the similarity matrix obtained by or in method $\rm m$\\
\hline
$\circ$, $\langle \langle \cdot \rangle \rangle$& the concatenation operator\\
\hline
\end{tabular}
\\\footnotesize{*For conveniece, method notation $\rm m$ is omitted in some descriptions.}
\end{table}
\begin{figure*}
\centering
{\includegraphics[width=0.9\linewidth]{./pic/paradigm3.pdf}}
\caption{The common framework of role-oriented network embedding methods includes two main step: structural property extraction and embedding. The former one can be accomplished via a variety of ways of which some are feature-based and some are similarity-based methods. On the extracted properties, role-oriented embedding methods then employ some specific embedding mechanisms to generate embeddings. Note that the discussed role-oriented methods are unsupervised. Thus, though the generated embeddings can be applied on some downstream tasks with ground truth, the whole process of embedding generation has no interaction with the target tasks.}
\label{fig:paradigm}
\end{figure*}
\begin{definition}[Motif/Graphlet]
A motif/graphlet $\mathcal{M} = (\mathcal{V}_{\mathcal{M}},\mathcal{E}_{\mathcal{M}})$ is a small connected subgraph representing particular patterns of edges on several nodes. The pattern can be repeated in or across networks, i.e., many subgraphs can be sampled from networks and isomorphic to it. Nodes automorphic to each other, i.e., having the same connectivity patterns, are in the same orbits.
\end{definition}
\begin{figure}
\centering
{\includegraphics[width=0.8\columnwidth]{./pic/motifs.pdf}}
\caption{Motifs and orbits (denoted by numbers) with size 2-4 nodes.}
\label{fig:motifs}
\end{figure}
For unweighted networks, there are 9 motifs and 15 orbits with size 2-4 nodes as shown in Fig.~\ref{fig:motifs}. Because of their ability to model the smallest but most fundamental connectivity patterns, motifs are wildly used for capturing structural similarities and discovering roles.
\begin{definition}[Network Clustering]
A clustering $\boldsymbol{\mathcal{R}}
=\{ \mathcal{R}_1,...,\mathcal{R}_k\}$ of network $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is a group of node sets satisfying $\forall 1 \le i \le k: \mathcal{R}_i \neq \emptyset, \mathcal{R}_i \subset \mathcal{V}$ and $\bigcup_1^{k} \mathcal{R}_i = \mathcal{V}$.
In this paper, we discuss about hard clustering, i.e., $\forall 1 \le i < j \le k: \mathcal{R}_i \cap \mathcal{R}_j = \emptyset$. If $\mathcal{R}_i \cup \mathcal{R}_j \neq \emptyset$, it is usually called overlapping or soft clustering.
\end{definition}
For the \textbf{community detection}, each set $\mathcal{R}_i$ is a tightly interconnected collection of nodes.
And for \textbf{role discovery}, it usually composed of unconnected nodes which have similar structural patterns or functions. So every network clustering algorithm is committed to achieve the clustering results under different goal constraints. However, there is no common understanding of role equivalence or similarity, which leads to multifarious definitions of equivalence and designs of similarity computation. For example, two nodes are automorphic equivalent~\cite{autoandsto-equi} as the subgraphs of their neighborhood are isomorphic, while regular equivalence~\cite{regular-equi} means that if two nodes have the same roles, there neighbors have the same roles.
\begin{definition}[Network Embedding]
Network Embedding is a process to map nodes of network $\mathcal{G}=(\mathcal{V},\mathcal{E})$ to low-dimensional embeddings $\mathbf{H} \in \mathbb{R}^{n \times r}$ so that $r \ll n$. In general, for nodes $v_i$ and $v_j$, if they are similar in the network, their embedding vectors $\mathbf{H}_i$ and $\mathbf{H}_j$ will be close in the low dimensional space.
\end{definition}
With the node embedding, we can take it for different network tasks. If we focus on the community detection or link predication, we want to discriminate by embeddings whether the nodes are connected or likely to be connected. However, for role discovery, the embeddings should reflect some structural patterns including local properties like subgraph isomorphism, global properties like regular equivalence and higher-order properties like motifs.
Based on the discussion and notations above, here we firstly propose a unified framework for understanding the role-oriented network embedding. To our knowledge, it can cover almost all the existing methods and models in a unified way.
The framework is illustrated in Fig. \ref{fig:paradigm}. Structure of networks is discrete, but embeddings are usually designed to lie in continuous space. Thus, role-oriented embedding methods always take two steps to capture structrual properties and generate embeddings respectively to bridge the gulf between two spaces:
\begin{itemize}
\item \textbf{Structure Property Extraction}. The ways to extract structural information are diverse. Some methods such as RolX~\cite{henderson2012rolx} and DRNE~\cite{tu2018deep} leverages some primary structural features including node degree, triangle numbers. Part of these methods such as SPINE~\cite{ijcai2019-333} will continue to transform the features into distances or similarities. There are also some methods captureing similarity between node-centric subgraphs. For example, struc2vec~\cite{ribeiro2017struc2vec} compute structural distances between $k$-hop subgraphs based on degree sequences of the subgraphs. SEGK~\cite{nikolentzos2019learning} employs one graph isomorphism test skill called graph kernel on subgraphs. As the result of these hand-craft process, these structural properties are contained in interim features or pair-wise similarities.
\item \textbf{Embedding}. The extracted properties then are mapped into embedding space via different mechanisms. In the process of embedding, the structural properties are used as inputs or training guidance. For example, RolX~\cite{henderson2012rolx} and SEGK~\cite{nikolentzos2019learning} apply low-rank matrix factorization on feature matrix and similarity matrix respectively, as they implicitly or explicitly reflect whether nodes are structurally similar. Struc2vec\cite{ribeiro2017struc2vec,ijcai2019-333} leverages word embedding methods to the similarity-biased random walks. DRNE~\cite{tu2018deep} utilizes LSTM~\cite{hochreiter1997long} on degree-ordered node embedding sequences to capture regular equivalence with a degree-guided regularizer.
\end{itemize}
As the embeddings capture crucial structural properties, they can be used on the downstream tasks such as role-based node classification and visualization. With this framework, we generalize the process of role-oriented embedding. However, as we can learn from Fig. \ref{fig:paradigm}, the core of designing role-oriented embedding methods is the way to extract structural properties. In contrast, the embedding mechanisms for mapping structural features/similarities into low-dimensional continuous vector space are much more regular. Thus, We introduce the popular methods in the next section from the perspective of embedding mechanisms.
\section{Algorithm Taxonomy}\label{sec:algorithm}
In this section, we introduce these approaches categorized according to their embedding mechanisms. In detail, we propose a two-level classification ontology for these popular methods. Similar to the taxonomy of community oriented network embedding, we divide these into three categories, low-rank matrix factorization, random walk based and deep learning methods from the first level. Further more, with there embedding mechanisms and constraint information, we give a more refined classification taxonomy as shown in TABLE~\ref{tab.methodlist}. At the same time, we also list the tasks which can be served by different methods. Next, we will introduce these methods in detail.
\newcommand{\tabincell}[2]{\begin{tabular}
{@{}#1@{}}#2\end{tabular}}
\begin{table*}[htbp]
\renewcommand{\arraystretch}{1.35}
\caption{A summary of role-oriented embedding methods. The abbreviations of tasks CLF, CLT, LP, ER, NA, SS and Vis denote node
classification/clustering, link prediction, entity resolution/Network alignment/Top-k similarity search and visualization, respectively. }
\label{tab.methodlist}
\centering
\begin{tabular}{|c|c|c|p{1.3cm}<{\centering}|p{1.3cm}<{\centering}|p{1.3cm}<{\centering}|p{1.3cm}<{\centering}|c|}
\hline
\multirow{2}{*}{\bfseries Method} & \multicolumn{2}{c|}{\multirow{2}{*}{\bfseries Embedding Mechanism}} & \multicolumn{4}{c|}{\bfseries Conducted Tasks} & \multirow{2}{*}{\bfseries Year}\\ \cline{4-7}
~ & \multicolumn{2}{c|}{~} & \bfseries Vis & \bfseries CLF/CLT & \bfseries ER/NA/SS & \bfseries LP & \\ \hline
RolX\cite{henderson2012rolx} & \multirow{10}{*}{\tabincell{c}{low-rank \\ matrix \\ factorization \\ (Sec.\ref{sec:lowrank})}}& \multirow{5}{*}{\tabincell{c}{on structural feature \\ matrix (Sec.\ref{sec:lowrank-feature}) }} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2012\\ \cline{1-1} \cline{4-8}
GLRD\cite{gilpin2013guided} & & & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2013\\ \cline{1-1} \cline{4-8}
RID$\large\boldsymbol{\varepsilon}$Rs\cite{gupte2017role} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2017\\ \cline{1-1}
\cline{4-8}
GraphWave\cite{donnat2018learning} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2018\\ \cline{1-1} \cline{4-8}
HONE\cite{rossi2020structural} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & 2020\\ \cline{1-1} \cline{4-8} \cline{3-3}
xNetMF\cite{heimann2018regal} & & \multirow{5}{*}{\tabincell{c}{on structural similarity \\ matrix (Sec.\ref{sec:lowrank-similarity}) }} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2018\\\cline{1-1} \cline{4-8}
EMBER\cite{jin2019smart} & & & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-1} \cline{4-8}
SEGK\cite{nikolentzos2019learning} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} &\textcolor{red}{\XSolidBold} & 2019\\ \cline{1-1} \cline{4-8}
REACT\cite{pei2019joint} & & & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-1} \cline{4-8}
SPaE\cite{shi2019unifying} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-3} \cline{4-8}
struc2vec\cite{ribeiro2017struc2vec} & \multirow{6}{*}{\tabincell{c}{random \\ walk-based \\ methods \\ (Sec.\ref{sec:randomwalk})}}& \multirow{3}{*}{\tabincell{c}{on similarity-biased \\ random walks (Sec.\ref{sec:randomwalk-similarity}) }} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2017\\ \cline{1-1} \cline{4-8}
SPINE\cite{ijcai2019-333} & & & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-1} \cline{4-8}
struc2gauss\cite{pei2020struc2gauss} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2020\\ \cline{1-1} \cline{4-8} \cline{3-3}
Role2Vec\cite{ahmed2019role2vec} & & \multirow{3}{*}{\tabincell{c}{on feature-based \\ random walks (Sec.\ref{sec:randomwalk-feature}) }} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} &\textcolor{blue}{\CheckmarkBold} & 2019\\ \cline{1-1} \cline{4-8}
RiWalk\cite{xuewei2019riwalk} & & & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-1} \cline{4-8}
NODE2BITS\cite{jin2019node2bits} & & & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2019\\ \cline{1-3} \cline{4-8}
DRNE\cite{tu2018deep} & \multirow{5}{*}{\tabincell{c}{deep \\ learning \\ (Sec.\ref{sec:deeplearning})}}& \multirow{5}{*}{\tabincell{c}{via structural information \\ reconstruction/guidance \\ (Sec.\ref{sec:structure-guidance}) }} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2018\\ \cline{1-1} \cline{4-8}
GAS\cite{guo2020role} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{red}{\XSolidBold} & 2020\\\cline{1-1} \cline{4-8}
RESD\cite{zhang2021role} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2021\\ \cline{1-1} \cline{4-8}
GraLSP\cite{jin2020gralsp} & & & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & 2020\\\cline{1-1} \cline{4-8}
GCC\cite{qiu2020gcc} & & & \textcolor{red}{\XSolidBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{blue}{\CheckmarkBold} & \textcolor{red}{\XSolidBold} & 2020\\ \cline{1-3} \cline{4-8} \cline{3-3}
\end{tabular}
\end{table*}
\subsection{Low-rank Matrix Factorization}
\label{sec:lowrank}
Low-rank matrix factorization is the most commonly used method for role-oriented embeddding methods. They generate embeddings by factorizing matrices preserving the role similarities between nodes implicitly (i.e. feature matrices) or explicitly (i.e. similarity matrices).
\subsubsection{Structural Feature Matrix Factorization}
\label{sec:lowrank-feature}
\noindent\textbf{RolX}~\cite{henderson2012rolx}. RolX takes the advantages of feature extraction method ReFeX~\cite{henderson2011s} by decomposing the ReFeX feature matrix $\mathbf{F}_{ReFeX} \in \mathbb{R}^{n \times f}$. ReFeX firstly computes some primary features such as degree and clustering coefficient for each node. Then it aggregates neighbors' features with sum- and mean-aggregator recursively. In $k$ recursive steps, it can capture very thorough features to express the structure of $(k+1)$-hop reachable neighborhood. Non-negative Matrix Factorization (NMF) is used for generating embeddings as it is efficient compared with other matrix decomposition methods. The non-negative constraints are adapted to interpretation of roles. Thus, RolX aims to obtain two low-rank matrices as follows:
\begin{equation}
\label{eq:rolx1}
\min_{\mathbf{H},\mathbf{M}} \Bigl| \Bigl| \mathbf{F}_{ReFeX}-\mathbf{HM} \Bigr| \Bigr|_{F}^2, \ s.t. \ \mathbf{H,M} \ge 0
\end{equation}
where $\mathbf{H} \in \mathbb{R}^{n \times r}$ is the embedding matrix (or role assignment matrix) and the matrix $\mathbf{M} \in \mathbb{R}^{r \times f}$ (role definition matrix) describes the contributions of each role to structural features. $r$ is the number of hidden roles which is determined by Minimum Description Length (MDL)~\cite{rissanen1978modeling}.
\noindent\textbf{GLRD}~\cite{gilpin2013guided}. GLRD extends RolX by adding different optional constraints to objective function (\ref{eq:rolx1}).
Sparsity constraint ($\forall i, \left \| \mathbf{H}_{\cdot i} \right \|_{1} \le \epsilon_\mathbf{H} \land \left \| \mathbf{M}_{i \cdot} \right \|_{1} \le \epsilon_\mathbf{M}$) is defined for more definitive role assignments and definitions while diversity constraint ($\forall i \ne j , \mathbf{H}_{\cdot i}^\top \mathbf{H}_{\cdot j} \le \epsilon_\mathbf{H} \land \mathbf{M}_{i \cdot}\mathbf{M}_{j \cdot}^\top \le \epsilon_\mathbf{M}$) is for reducing the overlapping. $\mathbf{H}^*$ and $\mathbf{M}^*$ are previously discovered role assignments and definitions with which alternativeness constraint ($\forall i \ne j , \mathbf{H}_{\cdot i}^{*\top}\mathbf{H}_{\cdot j} \le \epsilon_\mathbf{H} \land \mathbf{M}_{i \cdot}^*\mathbf{M}_{j \cdot}^\top \le \epsilon_\mathbf{M}$) can be used for mining roles unknown.
\iffalse
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.7}
\caption{Optional Constraints in GLRD.}
\label{tab.glrd_constraints}
\centering
\begin{tabular}{c|c}
\hline
\bfseries Constraint & \bfseries Formula\\
\hline
\multirow{2}{*} {Sparsity} & $\forall i \ \ \left \| \mathbf{H}_{\cdot i} \right \|_{1} \le \epsilon_\mathbf{H}$ \\
& $\forall i \ \ \left \| \mathbf{M}_{i \cdot} \right \|_{1} \le \epsilon_\mathbf{M}$ \\
\hline
\multirow{2}{*} {Diversity} & $\forall i,j \ \ \mathbf{H}_{\cdot i}^\top \mathbf{H}_{\cdot j} \le \epsilon_\mathbf{H} \ \ i \ne j $ \\
& $ \forall i,j \ \ \mathbf{M}_{i \cdot}\mathbf{M}_{j \cdot}^\top \le \epsilon_\mathbf{M} \ \ i \ne j $ \\
\hline
\multirow{2}{*} {Alternativeness} & $\forall i,j \ \ \mathbf{H}_{\cdot i}^{*\top}\mathbf{H}_{\cdot j} \le \epsilon_\mathbf{H} \ \ i \ne j $ \\
& $ \forall i,j \ \ \mathbf{M}_{i \cdot}^*\mathbf{M}_{j \cdot}^\top \le \epsilon_\mathbf{M} \ \ i \ne j $ \\
\hline
\end{tabular}
\end{table}
\fi
\noindent\textbf{RID$\large\boldsymbol{\varepsilon}$Rs}~\cite{gupte2017role}.
RID$\large{\varepsilon}$Rs uses $\varepsilon$-equitable refinement ($\varepsilon$ER) to partition nodes into different cells and compute graph-based features. An $\varepsilon$-equitable refinement partition $\pi=\{\mathcal{C}_1,\mathcal{C}_2,...,\mathcal{C}_K\}$ of $\mathcal{V}$ satisfies the following
rule:
\begin{equation}
\label{eq:riders1}
|\mathrm{deg}(u,\mathcal{C}_j)-\mathrm{deg}(v,\mathcal{C}_j)| \le \varepsilon, \forall u,v \in \mathcal{C}_j, \forall 1 \le i,j \le K
\end{equation}
where $\mathrm{deg}(u,\mathcal{C}_j) = |\{ u | (u,v) \in E \land v \in \mathcal{C}_j \}|$ denotes the number of nodes in cell $\mathcal{C}_j$ connected to node $v_i$. As nodes in the same cell have similar number of connections to the nodes in another cell, $\varepsilon$ERs could capture some connectivity patterns.
Based on the cells partitioned by $\varepsilon$ERs with an relaxation parameter $\varepsilon$, the feature matrix is defined as $(\mathbf{F}_{\varepsilon ER}^{\varepsilon})_{ij} = |\mathcal{N}_i \cap \mathcal{C}_j|$. After prunning and binning process, the feature matrices for all $1 \le \varepsilon \le \lfloor d_{avg} \rfloor$ are concatenated as the final feature matrix $\mathbf{F}_{\varepsilon ER}$. Finally, like RolX and GLRD, NMF is applied for embedding generation while right sparsity constraint (on $\mathbf{M}$) is optional for more definitive role representations.
\noindent\textbf{GraphWave}~\cite{donnat2018learning}. GraphWave treats graph diffusion kernels as probability distributions over networks and gets embeddings by using characteristic functions of the distributions. Specifically, take the heat kernel $g_\varsigma(\lambda) = e^{-\lambda \varsigma}$ with scaling parameter $\varsigma$ as an example, the spectral graph wavelets $\boldsymbol{\Psi} \in \mathbb{R}^{n \times n}$ are defined as:
\begin{equation}
\label{eq:graphwave1}
\boldsymbol{\Psi} = \boldsymbol{\mathcal{I}}\mathbf{U} \mathrm{Diag}(g_\varsigma(\lambda_1),...,g_\varsigma(\lambda_n)) \mathbf{U}^\top
\end{equation}
where $\boldsymbol{\mathcal{I}}$ is the one-hot encoding matrix on $\mathcal{V}$ and the scaling parameter $\varsigma$ is omitted. The $i$-th row $\boldsymbol{\Psi}_i$ represents the resulting signal from a Dirac signal around node $v_i$. Considering the empirical characteristic function:
\begin{equation}
\label{eq:graphwave2}
\varphi_i(t) = \frac{1}{n} \sum_{j=1}^{n} e^{\mathfrak{i}t\boldsymbol{\Psi}_{ij}}
\end{equation}
where $\mathfrak{i}$ denotes the imaginary number, $v_i$'s embedding vector $\mathbf{H}_i$ is generated by concatenating pairs of $\mathrm{Re}(\varphi_i(t))$ and $\mathrm{Im}(\varphi_i(t))$ at $r$ evenly spaced points $t_1,...,t_r$.
\noindent\textbf{HONE}~\cite{rossi2020structural}. HONE constructs weighted motif graphs in which the weight of an edge is the count of the co-occurrences of the two endpoints in a specific motif. For a motif represented by its weighted motif adjacency matrix $ \mathbf{A}_{\mathcal{M}_m}$, HONE characters the higher-order structure by deriving matrices from its k-step matrices $\mathbf{A}_{\mathcal{M}_m}^k$. These new matrices are designed by imitating some popular matrices based on normal adjacency matrix such as transition matrix $\mathbf{P} = \mathbf{D}^{-1}\mathbf{A}$ and Laplacian matrix $\mathbf{L} = \mathbf{D}-\mathbf{A}$. Here we use $(\mathbf{F}_{HONE})_{\mathcal{M}_m}^{(k)}$ to denote the derived matrices. Then the k-step embeddings can be learned as:
\begin{equation}
\mathop{\arg\min}_{\mathbf{H}_{\mathcal{M}_m}^{(k)},\mathbf{M}_{\mathcal{M}_m}^{(k)}} \mathbb{D}_{Breg}(\mathbf{F}^{(k)}_m | \Psi(\mathbf{H}_{\mathcal{M}_m}^{(k)}\mathbf{M}_{\mathcal{M}_m}^{(k)}))
\label{eq:hone1}
\end{equation}
where $\mathbb{D}_{Breg}$ is the Bregman divergence and $\Psi(\cdot)$ is a matching function. The global embeddings are generated by minimizing the following objective:
\begin{equation}
\min_{\mathbf{H},\mathbf{M}} \Bigl| \Bigl| \mathbf{F}_{HONE}-\mathbf{HM} \Bigr| \Bigr|_{F}^2
\label{eq:hone2}
\end{equation}
where $\mathbf{F}_{HONE}$ is obtained by concatenating the $\mathbf{H}_{\mathcal{M}_m}^{(k)}$ with all the considered motifs and steps. If necessary, attributes diffused by transition matrix based on different motifs and steps can be added into $\mathbf{F}_{HONE}$.
\noindent\textbf{\textit{Remark.}} Aforementioned methods assume that nodes in similar roles have similar structural features. Thus, they apply matrix factorization on the feature matrices to obtain role-based representations. RolX, GLRD and RID$\large{\varepsilon}$Rs directly get embeddings which give soft role assignment by factorizing feature matrices. As the feature matrices are usually lower dimension, these methods are quite efficient. GraphWave uses
eigen-decomposition and empirical characteristic function to characterize the structural patterns of each node, which leads to robust embeddings but high computation cost. The weighted motif adjacency matrices in HONE capture higher-order proximities actually, while they can obtain structural information because each matrix represents one motif.
\subsubsection{Structural Similarity Matrix Factorizaiton}
\label{sec:lowrank-similarity}
\noindent\textbf{xNetMF}~\cite{heimann2018regal}.
xNetMF is an embedding method designed for an embedding-based network alignment approach REGAL. It firstly obtains a node-to-node similarity matrix $\mathbf{S}_{REGAL}$ based on both structures and attributes:
\begin{equation}
\label{eq:xnetmf1}
\mathbf{S}_{ij} = \mathrm{exp}(- \gamma_{s} \mathrm{dist}_s(v_i,v_j) - \gamma_{a} \mathrm{dist}_a(v_i,v_j))
\end{equation}
where $\mathrm{dist}_s(v_i,v_j)$ and $\mathrm{dist}_a(v_i,v_j)$ are structure-based and attribute-based distance between node $v_i$ and node $v_j$ while $\gamma_s$ and $\gamma_a$ are balance parameters of the two distances. $\mathrm{dist}_s(v_i,v_j)$ is the Euclidean distance between node features. And $\mathrm{dist}_a(v_i,v_j)$ counts different attributes between nodes, i.e., $\mathrm{dist}_a(v_i,v_j) = |\{a | \mathbf{X}_{ia} \neq \mathbf{X}_{ja} \land 1\le a \le x\}|$. The feature matrix $\mathbf{F}_{REGAL}$ is defined by counting nodes with the same logarithmically binned degree in each node's $k$-hop reachable neighborhood as follows:
\begin{equation}
\label{eq:xnetmf2}
\begin{aligned}
\mathbf{F}_{ic}^k & = | \mathcal{D}_{i,c}^k| = | \{ v_j \in \mathcal{N}^k_i| \lfloor \mathrm{log}_2 d_j\rfloor= c \}|,\\
\mathbf{F}_{i} & = \sum_{k=1}^{K} \delta^k \mathbf{F}_{i}^k
\end{aligned}
\end{equation}
where $\delta^k \in (0,1]$ is a discount factor for lessening the importance of higher-hop neighbors.
Then on computed $\mathbf{S}$, matrix factorization methods can be applied for obtaining embedding matrix $\mathbf{H}$ satisfying $\mathbf{S} \approx \mathbf{H}\mathbf{M}^\top$. As the high dimension and rank of $\mathbf{S}$ lead to high computation, an implicit matrix factorization approach extending Nyström method~\cite{drineas2005nystrom} is proposed as follows:
\begin{enumerate}
\item Select $r\ll n$ nodes as landmarks randomly or based on node centralities.
\item Compute a node-to-landmark similarity matrix $\mathbf{C} \in \mathbb{R}^{n \times r}$ with Eq.(\ref{eq:xnetmf1}) and extract a landmark-to-landmark similarity matrix $\mathbf{B} \in \mathbb{R}^{r \times r}$ from $\mathbf{C}$.
\item Apply Singular Value Decomposition on the pseudoinverse of $\mathbf{B}$ so that $\mathbf{B}^\dagger = \mathbf{V}\mathbf{\Sigma}\mathbf{Y}^\top$.
\item Obtain embedding matrix $\mathbf{H}$ by computing and normailize $\mathbf{CV\Sigma}^{- \frac{1}{2}}$.
\end{enumerate}
With above method, embeddings are actually generated by factorizing a low-rank approximation of $\mathbf{S}$, i.e., $\widetilde{\mathbf{S}} = \mathbf{C(V \Sigma Y}^\top)\mathbf{C}^\top$.
Meanwhile, the computation can be reduced, as only a small matrix $\mathbf{B}^\dagger$ is decomposed.
\noindent\textbf{EMBER}~\cite{jin2019smart}. EMBER is designed for mining professional roles in weighted directed email networks. It defines node outgoing feature matrix $\mathbf{F}_{EMBER}^+$ as:
\begin{equation}
\label{eq:ember1}
\begin{aligned}
\mathbf{F}_{ic}^{k+} &= \sum_{v_j \in \mathcal{D}^{k+}_{i,c}} \mathrm{pw}(\mathcal{P}_{v_i \rightarrow v_j}^{k+}),\\
\mathbf{F}_{i}^{+} &= \sum_{k=1}^{K} \delta^k \mathbf{F}_{i}^{k+}
\end{aligned}
\end{equation}
where $\mathrm{pw}(\mathcal{P}_{v_i \rightarrow v_j}^{k+})$ denotes the product of all edge weights in a $k$-step shortest outgoing path $\mathcal{P}_{v_i \rightarrow v_j}^{k+}$. The incoming feature matrix $\mathbf{F}$ is defined similarly. By concatenating the incoming and outgoing feature matrices, the final feature matrix $\mathbf{F}_{EMBER} = [\mathbf{F}^+,\mathbf{F}^-]$ is obtained. The node-to-node similarities are computed through Eq.(\ref{eq:xnetmf1}) without attribute-based distance, i.e., $\mathbf{S}_{ij} = \mathrm{exp}(-\left \| \mathbf{F}_i - \mathbf{F}_j \right \|^2)$. EMBER uses the same implicit matrix factorization approach to generate embeddings. Note that if the feature extraction part of EMBER is applied on an undirected unweight network, EMBER
will be equivalent to xNetMF without attributes.
\noindent\textbf{SEGK}~\cite{nikolentzos2019learning}. SEGK leverages graph kernels to compute node structural similarities. To compare the structure more carefully, it computes node similarities with different scales of neighborhood as follows:
\begin{equation}
\label{eq:segk1}
\mathbf{S}_{ij} =\sum_{k=1}^K \hat{\mathcal{K}}(\mathcal{G}_i^k,\mathcal{G}_j^k) \hat{\mathcal{K}}(\mathcal{G}_i^{k-1},\mathcal{G}_j^{k-1})
\end{equation}
where $\hat{\mathcal{K}}(\mathcal{G}_i^{0},\mathcal{G}_j^{0}) = 1$ and $\hat{\mathcal{K}}$ denotes the normalized kernel which is defined as:
\begin{equation}
\label{eq:segk2}
\hat{\mathcal{K}}(\mathcal{G},\mathcal{G}') = \frac{\mathcal{K}(\mathcal{G},\mathcal{G}')}{\sqrt{\mathcal{K}(\mathcal{G},\mathcal{G}) \mathcal{K}(\mathcal{G}',\mathcal{G}')}}
\end{equation}
SEGK chooses the shortest path kernel, Weisfeiler-Lehman subtree kernel, or graphlet kernel for practical use of $\mathcal{K}(\cdot,\cdot)$. Then Nyström method~\cite{williams2001using} is employed on the factorization of $\mathbf{S}$ for efficient computation and low dimensions of embeddings as follows:
\begin{equation}
\label{eq:segk3}
\mathbf{H}=\mathbf{S}\mathbf{U}_{[r]}\mathbf{\Lambda}^{-\frac{1}{2}}_{[r]}
\end{equation}
where $\mathbf{U}_{[r]}$ denotes the matrix of first $r$ eigenvectors and $\mathbf{\Lambda}_{[r]}$ is the diagonal matrix of corresponding eigenvalues.
\noindent\textbf{REACT}~\cite{pei2019joint}. REACT aims to detect communities and discover roles by applying non-negative matrix tri-factorization on RoleSim~\cite{jin2011axiomatic} matrix $\mathbf{S}$ and adjacency matrix $\mathbf{A}$, respectively. RoleSim matrix is developed with the idea of regular equivalence and is a pair-wise similarity matrix computed by iteratively updating the following scores:
\begin{equation}
\mathbf{S}_{ij}=(1-\beta) \max _{\mathrm{M}(v_i, v_j)} \frac{\sum_{(v_{i'}, v_{j'}) \in \mathrm{M}(v_{i}, v_{j})} \mathbf{S}_{i'j'}}{d_{i}+d_{j}-|\mathrm{M}(v_i, v_j)|}+\beta
\label{eq:rolesim}
\end{equation}
where $\mathrm{M}(v_i, v_j)$ is a matching between the neighborhoods of $v_i$ and $v_j$, and $\beta$ ($0 < \beta <1$) is a decay factor. In addition, $L_{2,1}$ norm is leveraged as the regularization to make the distribution of roles within communities as diverse as possible.
Thus, the objective function of REACT is:
\begin{equation}
\label{eq:react1}
\begin{aligned}
\quad &\min_{\mathbf{H}_R,\mathbf{M}_R,\mathbf{H}_C,\mathbf{M}_C} \ \left \| \mathbf{S}-\mathbf{H}_R\mathbf{M}_R\mathbf{H}_R^\top \right \|_{F}^2 \\
\quad & \quad + \left \| \mathbf{A}-\mathbf{H}_C\mathbf{M}_C\mathbf{H}_C^\top \right \|_{F}^2 + \gamma_{2,1} \left \| \mathbf{H}_C^\top\mathbf{H}_R \right \|^{\qquad}_{2,1}, \\
s.t. \quad &\mathbf{H}_R,\mathbf{M}_R,\mathbf{H}_C,\mathbf{M}_C \ge 0, \mathbf{H}_R^\top \mathbf{H}_R = \mathbf{I},\mathbf{H}_C^\top \mathbf{H}_C = \mathbf{I}.
\end{aligned}
\end{equation}
where $\mathbf{H}_R$/$\mathbf{H}_C$ denotes the embedding matrix for roles/communities, and $\mathbf{M}_R$/$\mathbf{M}_C$ denotes the interaction between roles/communities. $\gamma_{2,1}$ is the weight of regularization. Orthogonal constraint on embedding matrices is added for increased interpretability.
\noindent\textbf{SPaE}~\cite{pei2019joint}. SPaE also tries to capture communities and roles simultaneously. For node structural similarity, it computes cosine similarity between the standardized Graphlet Degree Vectors of nodes, and generates role-based embeddings via Laplacian eigenmaps method as follows:
\begin{equation}
\label{eq:spae1}
\max_{\mathbf{H}_R} \mathcal{J}_R = \mathrm{Tr}(\mathbf{H}_R^\top \mathbf{L_S} \mathbf{H}_R), \
s.t. \ \mathbf{H}_R^\top \mathbf{H}_R = \mathbf{I}.
\end{equation}
where $\mathbf{L_S}$ is the symmetric normalized matrix of structural similarity matrix $\mathbf{S}_{SPaE}$. SPaE obtains community-based embeddings similarly as follows:
\begin{equation}
\label{eq:spae2}
\max_{\mathbf{H}_C} \mathcal{J}_C = \mathrm{Tr}(\mathbf{H}_C^\top \mathbf{L_\mathbf{A}} \mathbf{H}_C), \
s.t. \ \mathbf{H}_C^\top \mathbf{H}_C = \mathbf{I}.
\end{equation}
where $\mathbf{L_\mathbf{A}} = \mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}$ is the symmetric normalized adjacency matrix. To map $\mathbf{H}_R$ and $\mathbf{H}_C$ into a unified embedding space, SPaE generates hybrid embeddings by maximizing the following objective function:
\begin{equation}
\begin{aligned}
\label{eq:spae3}
&\max_{\mathbf{H}_R,\mathbf{H}_C,\mathbf{H}_H} \ \mathcal{J}_R + \mathrm{p}_R
+ \gamma (\mathcal{J}_C + \mathrm{p}_C), \\
&s.t. \ \mathbf{H}_R^\top \mathbf{H}_R = \mathbf{I}, \mathbf{H}_C^\top \mathbf{H}_C = \mathbf{I}, \mathbf{H}_H^\top \mathbf{H}_H = \mathbf{I}.
\end{aligned}
\end{equation}
where $\mathbf{H}_H$ denotes the hybrid embedding matrix and $\gamma$ is the balance parameter. $\mathrm{p}_R = \mathrm{Tr}(\mathbf{H}_R^\top\mathbf{H}_H\mathbf{H}_H^\top\mathbf{H}_R)$ and $\mathrm{p}_C = \mathrm{Tr}(\mathbf{H}_C^\top\mathbf{H}_H\mathbf{H}_H^\top\mathbf{H}_C)$.
\noindent\textbf{\textit{Remark.}} These methods all explicitly compute structural similarities based on features, e.g., graph kernels, role equivalence, and so on. Most of them have considered the similarities between multiple hops of neighborhoods. Their effectiveness on role discovery depends on the quality of the similarity matrices. One major problem of this kind of methods is the issue of efficiency: computing pair-wise similarity and factorizing the high-dimensional similarity matrix $\mathbf{S} \in \mathbb{R}^{n \times n}$ are time-consuming. So xNetMF, EMBER and SEGK apply Nyström method to improve the efficiency as their similarity matrices are Gram matrices~\cite{drineas2005nystrom}.
\subsection{Shallow Models Using Random Walks}
\label{sec:randomwalk}
Random walk is a common way to capture node proximity used by network embedding methods~\cite{perozzi2014deepwalk,grover2016node2vec}.
Recently, two strategies have been proposed to adapt random walks to role-oriented tasks: (1) structural similarity-biased random walks makes structurally similar nodes more likely to appear in the same sequence (as shown in Fig.~\ref{fig:randomwalks}(b)). (2) structural feature-based random walks, e.g., attributed random walks~\cite{ahmed2019role2vec}, map nodes with similar structural features to the same role indicator and replace ids in random walk sequences with the indicators (see Fig.~\ref{fig:randomwalks}(c)). The first way can preserve structural similarity into co-occurrence relations of nodes in the walks. While the second way preserves structural similarity into role indicators and may capture the proximity between roles through the co-occurrence relations of the indicators as well.
Usually, language models such as Skip-Gram~\cite{mikolov2013distributed} are applied on generated random walks to map the similarities into embedding vectors~\cite{perozzi2014deepwalk,grover2016node2vec,ribeiro2017struc2vec,ahmed2019role2vec}.
However, some different mapping mechanisms are also employed such as the SimHash~\cite{charikar2002similarity} used in NODE2BITS~\cite{jin2019node2bits}.
\begin{figure}
\centering
\subfigure[Normal Random Walks]{\includegraphics[width=0.8\columnwidth]{./pic/randomwalks.pdf}}
\subfigure[Structural Similarity-biased Random Walks]{
\includegraphics[width=0.8\columnwidth]{./pic/biasedrandomwalks.pdf}}
\subfigure[Structural Feature-based Random Walks]{
\includegraphics[width=0.8\columnwidth]{./pic/featurebasedrandomwalks.pdf}}
\caption{Different types of random walks. Note that $\mathbf{P}'$ is the biased transition matrix computed based on node structural similarities. $R_i$ is the role indicator mapped from node structural features.}
\label{fig:randomwalks}
\end{figure}
\subsubsection{Structural Similarity-biased Random Walks}
\label{sec:randomwalk-similarity}
\noindent\textbf{struc2vec}~\cite{ribeiro2017struc2vec}. Struc2vec generates structurally biased id-based contexts via random walks on a hierarchy of constructed complete graphs.
In detail, it firstly computes structural distances between a pair of nodes as follows:
\begin{equation}
\label{eq:struc2vec1}
\begin{aligned}
\mathrm{dist}_d^k(v_i,v_j) = & \mathrm{dist}_d^{k-1}(v_i,v_j) + \mathrm{DTW}(\mathcal{H}^k_i,\mathcal{H}^k_j), \\
& 0 \le k \le k^*
\end{aligned}
\end{equation}
where $\mathrm{DTW}(\cdot,\cdot)$ denotes Dynamic Time Warping (DTW). $\mathrm{d}(a,b) = \mathrm{max}(a,b) / \mathrm{min}(a,b) -1$ is adopted as the distance function for DTW. $k^*$ is the diameter of the $\mathcal{G}$. $\mathcal{H}^k_i$ is the ordered degree sequence of nodes at the exact distance $k$ from $v_i$. Note that $\mathcal{H}^0_i = \{ d_i \}$ and $\mathrm{dist}_d^{-1}(v_i,v_j)$ is set to the constant 0.
Then a multi-layer weighted context graph $\mathcal{G}_C = (\mathcal{V}_C,\mathcal{E}_C)$ is built. Each layer $k = 0,...,k^*$ is an undirected complete graph $\mathcal{G}_C^k
= (\mathcal{V}_C^k,\mathcal{E}_C^k)$. $\mathcal{V}_C^k = \{ v_1^k,...,v_n^k \}$
where the corresponding node of $v_i \in \mathcal{V}$ in $k$-layer is denoted as $v^k_i$. The weight $w_C^k(v_i^k,v_j^k)$ of edge $(v_i^k,v_j^k) \in \mathcal{E}_C^k$ is defined as follows:
\begin{equation}
\label{eq:struc2vec2}
w_C^k(v_i^k,v_j^k) =
\mathrm{exp}(-\mathrm{dist}_d^k(v_i^k,v_j^k)) , k = 0,...,k^*
\end{equation}
The neighboring layers are connected through directed edges between the corresponding nodes. Thus $\mathcal{V}_C = \bigcup_0^{k^*} \mathcal{V}_C^k$ and $\mathcal{E}_C = (\bigcup_0^{k^*} \mathcal{E}_C^k) \cup (\bigcup_1^{k^*} \bigcup_1^{n} \{ (v_i^k,v_i^{k-1}) \}) \cup (\bigcup_0^{k^*-1} \bigcup_1^{n} \{ (v_i^{k-1},v_i^{k}) \})$. The edge weights between layers are defined as follows:
\begin{equation}
\label{eq:struc2vec3}
\begin{aligned}
w_C(v_i^k,v_i^{k+1}) = &
\mathrm{log}(\Gamma(v_i^k) + \mathrm{e}), k = 0,...,k^*-1 \\
w_C(v_i^k,v_i^{k-1}) = &
1, k = 1,...,k^*
\end{aligned}
\end{equation}
where $\Gamma(v_i^k)$ counts the edge $(v_i^k,v_j^k) \in \mathcal{E}_C^k$ whose weight is larger than the average edge weight of $\mathcal{G}_C^k$. That is:
\begin{equation}
\label{eq:struc2vec4}
\begin{aligned}
\Gamma(v_i^k) = |\{ (v_i^k,v_j^k) | w_C^k(v_i^k,v_j^k) > \\ \frac{\sum_{(v^k_{i'},v^k_{j'})\in \mathcal{E}_C^k} w_C^k(v^k_{i'},v^k_{j'})}{\tbinom{n}{2}} \}|
\end{aligned}
\end{equation}
Then id-based random walks can be employed on $\mathcal{G}_C$ and started in layer $0$ for context generation of each node. In detail, the walk stays in the current layer with a given probability $q_b$. In this situation, the probability of a walk from $v_i^k$ to $v_j^k$ is:
\begin{equation}
\label{eq:struc2vec5}
(\mathbf{P}^k_{S2V})_{ij} = \frac{w_C^k(v_i^k,v_j^k))}{\sum_{(v_i^k,v_{j'}^k) \in \mathcal{E}^k_C}w_C^k(v_i^k,v_{j'}^k))}
\end{equation}
With probability $1-q_b$, the walk steps across layers with the following stepping probability:
\begin{equation}
\label{eq:struc2vec6}
\begin{aligned}
p(v_i^k,v_i^{k+1}) = & \frac{w_C(v_i^k,v_i^{k+1})}{w_C(v_i^k,v_i^{k+1})+w_C(v_i^k,v_i^{k-1})} \\
p(v_i^k,v_i^{k-1}) = & 1 - p(v_i^k,v_i^{k+1})
\end{aligned}
\end{equation}
Note that $v_i^k$ with different $k$ have the same id in the context.
On the structural context, struc2vec leverages Skip-Gram with Hierarchical Softmax to learn embeddings.
\noindent\textbf{SPINE}~\cite{ijcai2019-333}. SPINE uses largest $f$ values of $i$th row of Rooted PageRank Matrix $\boldsymbol{\Omega} = (1 - \beta_{RPR})(\mathbf{I} - \beta_{RPR}\mathbf{P})^{-1}$ as $v_i$'s feature $(\mathbf{F}_{RPR})_i$. $\beta_{RPR}$ is the probability of stepping to a neighbor, while with probability $1-\beta_{RPR}$, a walk steps back to the start node. For the inductive setting, SPINE computes $\mathbf{F}_i$ via a Monte Carlo approximation.
To simultaneously capture structural similarity and proximity, SPINE designs a biased random walk method. With probability $\varrho_{br}$, the walk steps to a structural similar node based on the following transition matrix:
\begin{equation}
\label{eq:spine1}
(\mathbf{P}^k_{SPINE})_{ij} = \frac{\mathrm{sim}(v_i,v_j)}{\sum_{v_k \in \mathcal{V},v_k \ne v_i} \mathrm{sim}(v_i,v_k)}
\end{equation}
Here $\mathrm{sim}(\cdot,\cdot)$ can be computed via DTW or other methods based on node features. With probability $1-\varrho_{br}$, normal random walks are applied. Thus, with larger $\varrho_{br}$, SPINE can be more role-oriented. The embeddings are learned through Skip-Gram with Negative Sampling (SGNS). To leverage attributes, the embeddings are generated as:
\begin{equation}
\label{eq:spine2}
\mathbf{H}_{i} = \sigma(\sum_{j=1}^{f} \mathbf{F}_{ij}\mathbf{X}^{<i,f>}_j\mathbf{W})
\end{equation}
where $\mathbf{X}^{<i,f>}$ represents the attribute matrix of which the rows correspond to $f$ largest values of $\boldsymbol{\Omega}_{i}$. $\mathbf{W}$ is the weight matrix of the multi-layer perceptron (MLP).
\noindent\textbf{struc2gauss}~\cite{pei2020struc2gauss}. For each node $v_i$, struc2gauss generates a Gaussian distribution: $\mathcal{Z}_i = \mathrm{Gauss}(\boldsymbol{\mu}_i, \boldsymbol{\Sigma}_i)$
to model both structural similarity and uncertainty. After calculating structural similarity via existing methods such as RoleSim~\cite{jin2011axiomatic}, it samples the top-$K$ most similar nodes for a node as its positive set $\mathcal{S}^+_i$. The positive sampling of struc2gauss could be regarded as special random walks with mandatory restart on a star-shaped graph where the star center is the target node and star edges are the most similar nodes. The negative sample set $\mathcal{S}^-_i$ has the same size of $\mathcal{S}^+_i$ and is generated as in the normal random-walk based methods. To push the Gaussian embeddings of similar nodes closer and those of dissimilar nodes farther, struc2gauss uses the following max-margin ranking objective:
\begin{equation}
\label{eq:struc2gauss1}
\mathcal{L}=\sum_{v_i \in \mathcal{V}} \sum_{v_j \in \mathcal{S}^+_i} \sum_{v_k \in \mathcal{S}^-_i} \max \left(0, m-\mathrm{sim}\left(\mathcal{Z}_{i}, \mathcal{Z}_{j}\right)+\mathrm{sim}\left(\mathcal{Z}_{i}, \mathcal{Z}_{k}\right)\right)
\end{equation}
where $m$ is the margin parameter to push dissimilar distributions apart, and $\mathrm{sim}\left(\mathcal{Z}_{i}, \mathcal{Z}_{j}\right)$ is the similarity measure between distributions of $v_i$ and $v_j$. There are different similarity measures that can be used such as logarithmic inner product and KL divergence. For normal tasks, the mean vectors of those Gaussian distributions can be treated as embeddings, i.e., $\mathbf{H}_i = \boldsymbol{\mu}_i$.
\noindent\textbf{\textit{Remark.}} These methods reconstruct the edges between nodes based on the structural similarities so that the context nodes obtained by random walks are structurally similar to the central nodes. Compared with SPINE and struc2gauss, struc2vec clearly construct edges that better represent role information in the multi-layer complete graphs, which leads to better embeddings but higher time and space complexities.
\subsubsection{Structural Feature-based Random Walks}
\label{sec:randomwalk-feature}
\noindent\textbf{Role2Vec}~\cite{ahmed2019role2vec}. Role2Vec firstly maps nodes into several disjoint roles. Logarithmically binning, K-means with low-rank factorization and other methods on features and attributes can be chosen for the role mapping $\phi: \mathcal{V} \rightarrow \mathcal{R}$. Motif-based features, such as Graphlet Degree Vectors, are recommended since motif can better capture the high-order structural information.
Then random walks are performed but the ids in generated sequences are replaced with role indicators. With feature-based role context, the language model CBOW can be used for obtaining embeddings of roles. Nodes partitioned into the same role have the same embeddings
\noindent\textbf{RiWalk}~\cite{xuewei2019riwalk}. RiWalk designs structural node indicators approximating graph kernels. In a given subgraph $\mathcal{G}_i^k$, the indicator approximating shortest path kernel for node $v_j \in \mathcal{N}_i^k$ is defined as the concatenation of the degrees of $v_i$ and $v_j$ and the shortest path length between them:
\begin{equation}
\label{eq:riwalk1}
\phi_{SP}^i(v_j) = \mathrm{b}(d_i) \circ \mathrm{b}(d_j) \circ s_{ij}, v_j \in \mathcal{N}_i^k
\end{equation}
where $\mathrm{b}(x) = \lfloor \mathrm{log}(x) + 1 \rfloor$ is a logarithmically binning function.
The indicator approximating Weisfeiler-Lehman sub-tree kernel is defined as:
\begin{equation}
\label{eq:riwalk2}
\phi_{WL}^i(v_j) = \mathrm{b}(\mathbf{l}^{<i,i>}) \circ \mathrm{b}(\mathbf{l}^{<i,j>}) \circ s_{ij}, v_j \in \mathcal{N}_i^k
\end{equation}
where $\mathbf{l}^{<i,j>}$ is a vector of length $k+1$ whose $l$-th element is the count of $v_j$'s neighbers at distance $l$ to $v_i$ in $\mathcal{G}_i^k$, i.e.:
\begin{equation}
\label{eq:riwalk3}
\mathbf{l}^{<i,j>}_l = |\{ v_{j'} \in \mathcal{N}_j | s_{ij'} = l \}|, l=0,...,k
\end{equation}
Then the random walks starting from $v_i$ are performed on each $\mathcal{G}_i^k$. The nodes are relabeled indicated by Eq.(\ref{eq:riwalk1}) or Eq.(\ref{eq:riwalk2}) while only $v_i$ is not relabeled. And embeddings are learned via SGNS on the generated sequences.
\noindent\textbf{NODE2BITS}~\cite{jin2019node2bits}. NODE2BITS is designed for entity resolution on temporal networks. Here we use $\tau_{ij}$ to denote the the timestamp of edge $e_{ij}$. To integrate temporal information, NODE2BITS utilizes temporal random walks in which edges are sampled with non-decreasing timestamps. The following stepping probability is defined to capture short-term transitions in temporal walks:
\begin{equation}
\label{eq:node2bits1}
p_s(v_i,v_j) = \frac{\mathrm{exp}(-\tau_{ij}/T)}{\sum_{(v_i,v_{j'})\in \mathcal{E}} \mathrm{exp}(-\tau_{ij'}/T)}
\end{equation}
where $T$ is the maximal duration between all timestamps. The stepping probability in long-term policy is defined similarly with positive signs. Multiple walks are generated for each edge and the temporal context of different hops $\Delta t$ for a node can be extracted from the walks. Then structual features and attributes are fused in temporal walks. For each node $v_i$ with a specific $\Delta t$, histograms are applied on multi-dimensional features (and node types if the network is heterogeneous) to aggregate information in the neighborhood and they are concatenated as a vector $(\mathbf{H}_{HIST})_i^{\Delta t}$. SimHash~\cite{charikar2002similarity} is applied by projecting the histogram $(\mathbf{H}_{HIST})_i^{\Delta t}$ to several random hyperplanes for generating binary hashcode $(H_{HASH})_i^{\Delta t}$. The final embeddings are obtained via concatenation on $(H_{HASH})_i^{\Delta t}$ across different $\Delta t$s.
\noindent\textbf{\textit{Remark.}} The above three methods are very different on their motivations of utilizing structural feature-based random walks. Role2vec assigns roles firstly and then employs random walks with role indicators. It essentially captures proximity between assigned roles. RiWalk relabels the walks in subgraphs to approximate graph kernels. NODE2BITS uses random walks as neighbor feature aggregators.
\subsection{Deep Learning Models}
\label{sec:deeplearning}
Recently, a few works focus on leveraging deep learning techniques to role-oriented network representation learning. Though deep learning can provide more varied and powerful mapping mechanisms, it needs to be trained with more carefully designed structural information guidance.
\subsubsection{Structural Information Reconstruction/Guidance}
\label{sec:structure-guidance}
\noindent\textbf{DRNE}~\cite{tu2018deep}. DRNE is proposed to capture regular equivalence in networks, so it learns node embeddings in a recursive way with the following loss function:
\begin{equation}
\label{eq:drne1}
\mathcal{L}_{equiv}=\sum_{v_i\in \mathcal{V}} \left \| \mathbf{H}_i - \breve{\mathbf{H}}_i \right \|_2^2
\end{equation}
where $\breve{\mathbf{H}}_i$ is the aggregation of the neighbors' embeddings via a layer normalized Long Short-Term Memory. To make the neighbor information available for LNLSTM, for each node $v_i$, it downsamples a fixed number of neighbors with large degrees and orders them based on the degrees. Denoting their embeddings as $\{\mathbf{H}_{(1)},...,\mathbf{H}_{(T)}\}$ the aggregating process is $\breve{\mathbf{H}}_{(t)} = \mathrm{LNLSTM}(\mathbf{H}_{(t)},\breve{\mathbf{H}}_{(t-1)})$ and finally $\breve{\mathbf{H}}_i = \breve{\mathbf{H}}_{(T)}$.
Additionally, DRNE proposes a degree-guided regularizer to avoid the trivial solution where all embeddings are $\mathbf{0}$. The regularizer is as follows:
\begin{equation}
\label{eq:drne2}
\mathcal{L}_{deg}=\sum_{v_i\in \mathcal{V}} (\mathrm{log}(d_i+1) - \mathrm{MLP}_{deg}(\breve{\mathbf{H}}_i))^2
\end{equation}
The regularizer with a parameter $\gamma_{deg}$ is weighed and the whole model is trained via the combined loss:
\begin{equation}
\label{eq:drne3}
\mathcal{L}=\mathcal{L}_{equiv}+\gamma_{deg}\mathcal{L}_{deg}
\end{equation}
\iffalse
\noindent\textbf{DMER}~\cite{ke2019deep}. DMER applies the structural feature extraction algorithm ReFeX and then use different encoders to learn different structural information. One of the encoders is a MLP model for preserving local structural information:
\begin{equation}
\label{eq:dmer1}
\mathbf{H}_{MLP} = \mathrm{MLP}_{enc}(\mathbf{F}_{ReFeX})
\end{equation}
where $\mathrm{tanh}(\cdot)$ is used as the activation function. And the other encoder is a multi-layer Graph Convolution Network model which passing local structural information among the network and obtain global structural information. Each layer of the GCN encoder is as follows:
\begin{equation}
\label{eq:dmer2}
\mathbf{H}_{GCN}^{(l)} = \mathrm{ReLU}(\Tilde{\mathbf{D}}^{-\frac{1}{2}}\Tilde{\mathbf{A}}\Tilde{\mathbf{D}}^{-\frac{1}{2}}\mathbf{H}_{GCN}^{(l)}\Theta^{(l-1)})
\end{equation}
where $\Tilde{\mathbf{A}} = \mathbf{A} + \mathbf{I}$, $\Tilde{\mathbf{D}}_{ii}= \sum_{j}\mathbf{A}_{ij}$ and $\Theta^{(l-1)}$ is the $l-th$ layer's parameter matrix. The $L$-layer GCN encoder takes $\mathbf{F}_{ReFeX}$ as input and outputs $\mathbf{H}_{GCN} = \mathbf{H}_{GCN}^{(L)}$. The two encoders share the same MLP decoder to reconstruct the feature and the adjacency matrices:
\begin{equation}
\label{eq:dmer3}
\begin{aligned}
\hat{\mathbf{F}} = \mathrm{MLP}_{dec}(\mathbf{H}_{MLP})\\
\hat{\mathbf{A}} = \mathrm{MLP}_{dec}(\mathbf{H}_{GCN})
\end{aligned}
\end{equation}
To achieve that the two parts of embeddings enhance each other mutually, the whole deep framework is trained via the following loss function:
\begin{equation}
\label{eq:dmer4}
\begin{aligned}
\mathcal{L} = &\left \| \mathbf{H}_{GCN} - \mathbf{H}_{MLP} \right \|^2_F + \gamma_{MLP}\left \| \mathbf{F} - \hat{\mathbf{F}} \right \|^2_F \\&+ \gamma_{GCN}\left \| \mathbf{F} - \hat{\mathbf{A}} \right \|^2_F
\end{aligned}
\end{equation}
where the first part is designed as mutual guidance and the other are reconstruction loss. The final embedings are obtained by concatenating the two parts, i.e., $\mathbf{H} = \mathbf{H}_{GCN} \circ \mathbf{H}_{MLP}$
\fi
\noindent\textbf{GAS}~\cite{guo2020role}. Graph Neural Networks have the power to capture structure as they are closely related to Weisfeiler-Lehman (WL) test in some ways~\cite{xu2018how}. GAS applies a $L$-layer graph convolutional encoder, in which each layer is :
\begin{equation}
\label{eq:gas1}
\begin{aligned}
\mathbf{H}^{(l)} = \sigma(\Tilde{\mathbf{A}}\mathbf{H}^{(l-1)}\Theta^{(l-1)})
\end{aligned}
\end{equation}
where $\Tilde{\mathbf{A}} = \mathbf{A} + \mathbf{I}$ and $\Theta^{(l-1)}$ is the parameter matrix in the $l$-th layer. The input $\mathbf{H}^{(0)}$ could be $\Tilde{\mathbf{A}}$ or an embedding lookup table. Here the sum-pooling propagation rule is applied instead of the original GCN~\cite{gcn} to better distinguish local structures. In fact, more powerful GNNs such as Graph Isomorphic Network~\cite{xu2018how} may further improve the performance. The key idea for GAS is that using a few critical structural features as the guidance information to train the model. The features are extracted in a similar way proposed in ReFeX but aggregated only once, normalized and not binned. With a MLP model as the decoder to approximate the features, i.e., $\hat{\mathbf{F}} = \mathrm{MLP}_{dec}(\mathbf{H})$. The loss function is:
\begin{equation}
\label{eq:gas2}
\begin{aligned}
\mathcal{L} = \left \| \mathbf{F} - \hat{\mathbf{F}} \right \|^2_F
\end{aligned}
\end{equation}
\noindent\textbf{RESD~\cite{zhang2021role}}. RESD also adopts ReFeX~\cite{henderson2011s} to extract appropriate features $\mathbf{F}_{ReFeX}$. It uses a Variational Auto-Encoder~\cite{Kingma2014} architecture to learn the low-noise and robust representations:
\begin{equation}
\begin{aligned}
\mathbf{Z}_i &= \mathrm{MLP}_{enc}(\mathbf{F}_{i}) \\
\boldsymbol{\mu}_i &= \mathbf{W}_{\boldsymbol{\mu}}\mathbf{Z}_i + \mathbf{b}_{\boldsymbol{\mu}} \\
\mathrm{log}(\boldsymbol{\sigma}_i) &= \mathbf{W}_{\boldsymbol{\sigma}} \mathbf{Z}_i + \mathbf{b}_{\boldsymbol{\sigma}} \\
\mathbf{H}_i &= \boldsymbol{\mu}_i + \boldsymbol{\sigma}_i \odot \boldsymbol{\epsilon}, \boldsymbol{\epsilon} \sim \mathrm{Gaussian}(\mathbf{0},\mathbf{I})
\\
\hat{\mathbf{F}}_i &= \mathrm{MLP}_{dec}(\mathbf{H}_{i})
\end{aligned}
\label{eq:resd1}
\end{equation}
The VAE model is trained via feature reconstruction. A degree-guided regularizer Eq.(\ref{eq:drne2}) designed in DRNE~\cite{tu2018deep} is introduced in RESD for preserving topological characteristics. The combined objective is as follows:
\begin{equation}
\mathcal{L} = \left \| \mathbf{F} - \hat{\mathbf{F}} \right \|^2_F + \gamma_{deg}\mathcal{L}_{deg}
\label{eq:resd2}
\end{equation}
\noindent\textbf{GraLSP}~\cite{jin2020gralsp}. GraLSP is a GNN framework integrating local structural patterns that can be employed on role-oriented tasks. For a node $v_i$, it captures structural patterns by generating $w$ random walks starting from $v_i$ with length $l_w$: $\mathcal{W}_i = \{\omega_{i1},...,\omega_{iw}\}$, and then anonymizes them $\mathrm{aw}(\omega)$~\cite{ivanov2018anonymous}. Each anonymous walk $\mathrm{aw}(\omega)$ is represented as an embedding lookup table $\mathbf{u}_{\mathrm{aw}(\omega)}$. Then the aggregation of neighborhood representation is designed as follows:
\begin{equation}
\label{eq:gralsp1}
\begin{aligned}
(\mathbf{H}_{nei})^{(l)}_i &= \mathrm{MEAN}_{\omega \in \mathcal{W}_i, j \in \left [1,\left\lfloor \frac{2l_w}{|\omega|} \right \rfloor\right]}(\alpha_{i,\omega}^{(l)}(\mathbf{a}_{i,\omega}^{(l)}\odot \mathbf{H}_{\omega_j}^{(l-1)}))\\
\mathbf{H}^{(l)}_i&=\mathrm{ReLU}(\mathbf{W}_{self}^{(l)}\mathbf{H}^{(l-1)}_i + \mathbf{W}_{nei}^{(l)}(\mathbf{H}_{nei})^{(l)}_i)
\end{aligned}
\end{equation}
where $\mathbf{W}_{self}$ and $\mathbf{W}_{nei}$ are trainable parameter matrices. $\alpha_{i,\omega}^{(l)}$ is learned attention values based on their local structure:
\begin{equation}
\label{eq:gralsp2}
\begin{aligned}
\alpha_{i,\omega}^{(l)} = \frac{\mathrm{exp}(\mathrm{SLP}_{att}(\mathbf{u}_{\mathrm{aw}(\omega)}))}{\sum_{\omega' \in \mathcal{W}_i}\mathrm{exp}(\mathrm{SLP}_{att}(\mathbf{u}_{\mathrm{aw}(\omega')}))}
\end{aligned}
\end{equation}
$\mathrm{SLP}(\cdot)$ denotes a single-layer perceptron. $\mathbf{a}_{i,\omega}^{(l)}$ is the amplification coefficients:
\begin{equation}
\label{eq:gralsp3}
\begin{aligned}
\mathbf{a}_{i,\omega}^{(l)} = \mathrm{SLP}_{amp}(\mathbf{u}_{\mathrm{aw}(\omega)})
\end{aligned}
\end{equation}
To preserve proximities between nodes, the loss function in DeepWalk~\cite{perozzi2014deepwalk} is leveraged:
\begin{equation}
\label{eq:gralsp4}
\begin{aligned}
\mathcal{L}_{prox} = - \sum_{v_i \in \mathcal{V}} \sum_{v_j \in \mathcal{N}_i} (\mathrm{log}\sigma(\mathbf{H}_i\mathbf{H}_i^\top)\\-\gamma_{neg}\mathbb{E}_{v_k \sim P_n(v)}\left[ \mathrm{log}\sigma(\mathbf{H}_i\mathbf{H}_k^\top) \right])
\end{aligned}
\end{equation}
After $L$ aggregations, the embeddings are $\mathbf{H} = \mathbf{H}^{(L)}$. To capture structural similarities between nodes, GraLSP designs the following loss:
\begin{equation}
\label{eq:gralsp5}
\begin{aligned}
\mathcal{L}_{struc} = &- \sum_{v_i \in \mathcal{V}, \omega_j,\omega_k,\omega_s \in \mathcal{W}_i} \mathrm{log}\sigma(\mathbf{u}_j^\top\mathbf{u}_k-\mathbf{u}_k^\top\mathbf{u}_s),\\
s.t. \;\; &\hat{p}(\omega_j|v_i) > \hat{p}(\omega_j|\mathcal{G}), \hat{p}(\omega_k|v_i) > \hat{p}(\omega_k|\mathcal{G}), \\
&\hat{p}(\omega_n|v_i) < \hat{p}(\omega_n|\mathcal{G}).&
\end{aligned}
\end{equation}
where $\hat{p}(\cdot)$ is the empirical distribution of anonymous walks:
\begin{equation}
\label{eq:gralsp6}
\begin{aligned}
\hat{p}(\omega_j|v_i) &= \frac{\sum_{\omega\in \mathcal{W}_i} \mathbb{I}(\mathrm{aw}(\omega)=\omega_j)}{w}\\
\hat{p}(\omega_j|\mathcal{G}) &= \frac{\sum_{i=1}^{n}\hat{p}(\omega_j|v_i) }{n}
\end{aligned}
\end{equation}
The objective is to combine the two losses with a trade-off parameter $\gamma_{struc}$:
\begin{equation}
\label{eq:gralsp7}
\begin{aligned}
\mathcal{L}= \mathcal{L}_{prox} +\gamma_{struc}\mathcal{L}_{struc}
\end{aligned}
\end{equation}
\begin{table*}[htbp]
\caption{Statistic of the networks for node classification and clustering.}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Dataset & \# Nodes & \# Edges & \# Classes & Density(\%) & Mean Degree & Average CC & Transitivity \\
\hline
Brazil & $131$ & $1,074$ & $4$ & $12.6130$ & $16.3969$ & $0.6364$ & $0.4497$ \\
Europe & $399$ & $5,995$ & $4$ & $7.5503$ & $30.0501$ & $0.5393$ & $0.3337$ \\
USA & $1,190$ & $13,599$ & $4$ & $1.9222$ & $22.8555$ & $0.5011$ & $0.4263$ \\
Reality-call & $6,809$ & $7,697$ &$3$ & $0.0332$ & $2.2608$ & $0.0178$ & $0.0024$ \\
Actor & $7,779$ & $26,733$ &$4$ & $0.0888$ & $6.8917$ & $0.0790$ & $0.0156$ \\
Film & $27,312$ & $122,706$ &$4$ & $0.0329$ & $8.9855$ & $0.1180$ & $0.0278$ \\
\hline
\end{tabular}
\label{tab.datasets1}
\end{table*}
\begin{table*}[htbp]
\caption{Statistic of the networks for top-k similarity search.}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Dataset & \# Nodes & \# Edges & \# Bots & \# Admins & Density(\%) & Mean Degree & Average CC & Transitivity \\
\hline
ht-wiki-talk & $446$ & $758$ &$24$ & $0$ & $0.7638$ & $3.3991$ & $0.0941$ & $0.0085$ \\
br-wiki-talk & $1,049$ & $2,330$ & $35$ & $8$ & $0.4239$ & $4.4423$ & $0.1998$ & $0.0410$ \\
cy-wiki-talk & $2,101$ & $3,610$ & $31$ & $16$ & $0.1636$ & $3.4365$ & $0.1579$ & $0.0090$ \\
oc-wiki-talk & $3,064$ & $4,098$ &$43$ & $4$ & $0.0873$ & $2.6749$ & $0.0994$ & $0.0023$ \\
eo-wiki-talk & $7,288$ & $14,266$ & $120$ & $21$ & $0.0537$ & $3.9149$ & $0.1206$ & $0.0085$ \\
gl-wiki-talk & $7,935$ & $19,887$ &$12$ & $14$ & $0.6318$ & $5.0125$ & $0.4913$ & $0.0037$ \\
\hline
\end{tabular}
\label{tab.datasets2}
\end{table*}
\noindent\textbf{GCC~\cite{qiu2020gcc}}. GCC is a pre-train Graph Neural Network model, which represents a node as $\mathbf{H}_i$ by encoding its node-centric subgraph and tries to distinguish the similar subgraphs from the dissimilar ones. It aims to leverage several large-scale networks to train the ability of a GNN-based model to discriminate node substructures in an unsupervised manner. When the scale of the pre-trained datasets is large enough, GCC has the power to recognize local connective patterns of nodes, and the representations generated by it can measure structural similarities. While other GNN methods (such as~\cite{You.2021.arXiv},~\cite{you2019position}) either concentrates on community and proximity, or trains model in a supervised manner. So we introduce GCC as role-oriented network embedding method. Specifically, for each node $v_i$ in a network, GCC extracts its $k$-hop reachable neighborhood as $\mathcal{G}_i^k$, and leverages the Graph Isomorphic Network~\cite{xu2018how} as the encoder whose convolutional layer is:
\begin{equation}
\mathbf{H}^{(l)} = \mathrm{MLP}_{GIN}((\mathbf{A} + (1+\epsilon)\cdot\mathbf{I})\mathbf{H}^{(l-1)})
\label{eq:gcc1}
\end{equation}
where $\epsilon$ could be a learnable or fixed parameter and the input attributes of GIN is initialized as the eigenvectors of $\mathcal{G}_i^k$. A similar subgraph instance of $v_i$ is induced based on the nodes in the random work with restart starting from $v_i$. $K$ dissimilar instances are subgraphs induced in the same way but starting from other nodes which could be in other networks. The representations $\{ \mathbf{x}_0, ...,\mathbf{x}_K \}$ of these $K+1$ instances are generated via another GIN encoder. The representation of the similar instance is denoted as $\mathbf{x}^+ \in \{ \mathbf{x}_0, ...,\mathbf{x}_K \}$, and GCC is pre-trained by a contrastive learning method called InfoNCE~\cite{oord2018representation}:
\begin{equation}
\mathcal{L} = \sum_{v_i \in \mathcal{V}} - \mathrm{log}\frac{\mathrm{exp}(\mathbf{H}_i \mathbf{x}^+/\iota)}{\sum_{j=0}^{K} \mathrm{exp}(\mathbf{H}_i \mathbf{x}_j/\iota) }
\label{eq:gcc2}
\end{equation}
where $\iota$ is a hyper-parameter.
\noindent\textbf{\textit{Remark.}} These deep methods incorporate model traditional role-related concepts used in shallow methods
with deep learning techniques to map structural information into non-linear latent representations. DRNE learns regular equivalence and reconstructs node degrees. GAS uses structural features to guide the training of graph convolutional networks. RESD combines variants of DRNE and ReFeX via a VAE architecture. GraLSP reconstructs the similarities based on anonymous walks. And GCC is a pre-train model which encodes node-centric subgraphs and is pre-trained via a constrastive learning manner.
\section{Experimental Evaluation}\label{sec:Experiment}
In this section, we show the comprehensive analysis of these popular role-oriented embedding methods on widely used benchmarks. The experimental evaluations are conducted from the perspectives of both efficiency and effectiveness. To analyze the efficiencies of these methods, we compare their running time for generating node representations on both real-world and synthetic networks with varying sizes. To evaluate the effectiveness of these methods, we select four tasks for the evaluation including (1) the classification experiment based on the ground-truth labels of datasets by comparing the Micro-F1 and Macro-F1 scores, (2) the clustering experiment by comparing some clustering indices with K-Means model in an unsupervised manner. (3) the visualization experiment by plotting the node representations in a $2$-$D$ space to observe the relationships between node embeddings and their roles. (4) the top-k similarity search experiment to see if nodes in the same role are mapped into close position in the embedding space.
It's worth noting that we do not select link prediction for the experimental study, mainly because (1) only limited previous role-oriented NE methods evaluate the performance of this task (see Table~\ref{tab.methodlist}). Thus, it is difficult to make a consistent and fair comparison for all these methods. (2) It has been demonstrated that global information, i.e., roles, is less useful than local information, i.e., proximity, in link prediction task~\cite{lyu2017enhancing}. So we believe that this task is more suitable for proximity-preserving rather than role-oriented node embeddings.
\subsection{Datasets}
We perform our experiments on the following datasets which form unweighted and undirected networks to illustrate the potential of role-oriented network embedding methods in capturing roles. Based on the type of conducted experiments, these networks are divided in two groups. One group of networks, which include \textbf{Brazil}, \textbf{Europe}, \textbf{USA}, \textbf{Reality-call}, \textbf{Actor} and \textbf{Film}, are for node classification and clustering in which the nodes are labeled based on some role-related rules. The other networks are all Wiki-talk networks in which some nodes have the same but very rare role. These Wiki-talk networks are for top-k similarity search. We show some statistical characteristics from various aspects of these networks in Table~\ref{tab.datasets1} and Table~\ref{tab.datasets2}, respectively. More details about the datasets can be found in the Supplementary Materials.
\iffalse
We report the network density, mean degree, average clustering coefficient (CC) and transitivity of these networks for simply depicting their degree of structural sparsity and heterogeneity. These characteristics are usually used to analyze roles in networks~\cite{bartal2019member}. Network density is the ratio of the actual number of edges in a network to the upper limit of the edges number that can be accommodated, which is used to describe the density of the connected edges between nodes in the network. Degree centrality is the most direct measure of node centrality in network analysis. The assumption behind this indicator is that important nodes have more connections. The clustering coefficient is a measure of the degree to which nodes in a network tend to cluster together. Transitivity is also called global clustering coefficient. It is the proportion of triangle structure in the network to all possible triangle structures.
We introduce the details of the networks for node classification and clustering as follows:
\begin{itemize}
\item Air-traffic networks~\cite{ribeiro2017struc2vec}. They are from the aviation network of Brazil, USA and Europe, respectively. In each network, nodes represent airports and edges represent existing aircraft routes between airports. As the labeled networks, the nodes are labeled according to the activity level of airports,
which forms $4$ classes in each network. In the following paper, we will simply denote the $3$ networks as Brazil, USA and Europe.
\item Reality-call network~\cite{guo2020role}. It records $52,050$ phone calls of $6,809$ users from September $2004$ to January $2005$. After processing the dataset, we get a static network with $6,809$ nodes and $7,697$ edges, in which nodes represent users and edges represent communication between users. The nodes are divided into different categories according to the call frequency of users.
\item Actor co-occurrence network~\cite{xuewei2019riwalk} (shortly as Actor). This is a subgraph of the Film network that only contains actors. The nodes are sorted in accordance with the number of words on their Wikipedia pages, and then are divided into $4$ labels which can be regarded as a measure of their influence.
\item English-language film network~\cite{xuewei2019riwalk} (shortly as Film). The network assigns a label to each node to represent the role attributes of the node. In other words, it indicates whether the node is a film, director, actor or writer. The edge denotes that the two nodes have appeared on the same Wikipedia page.
\end{itemize}
Details on the networks for top-k similarity search are shown as follows:
\begin{itemize}
\item Wiki-talk networks~\cite{sun2016predicting}. These networks are extracted from talk pages of Wikipedia in different languages. The nodes denote Wikipedia users and two nodes have an edge if their corresponding users have communications. Each network has a very small number of nodes which are bots and administrators in reality. Top-k similarity search on these bots and administrators requires role-oriented network embedding methods to have ability to capture more
significant or intrinsic structural patterns. Here we use $6$ of the $28$ networks because their scale is suitable for both employment and comparison of most baseline methods. For each used network, we denote it as "wiki-talk" shortly with the ISO 639 code of the language as the prefix (ht-Haitian, br-Breton , cy-Welsh, oc-Occitan, eo-Esperanto, gl-Galician).
\end{itemize}
\fi
\begin{figure*}[htbp]
\centering
{\includegraphics[width=0.99\linewidth]{./pic/linear_time.pdf}}
\caption{Running time of $12$ methods on four different types of synthetic networks with different sizes. }
\label{fig:linear time}
\end{figure*}
\begin{figure}[htbp]
\centering
{\includegraphics[width=0.8\linewidth]{./pic/time.pdf}}
\caption{Running time of $12$ methods on the six real networks. }
\label{fig:time}
\end{figure}
\subsection{Experimental Settings}
Although different methods with different embedding mechanism have been proposed, according to our proposed two-level classification taxonomy discussed in Section~\ref{sec:Definition}, for each class of role-oriented network embedding, we choose $4$ methods to analyze the performance on different role related tasks. In specific, RolX~\cite{henderson2012rolx}, RID$\large\boldsymbol{\varepsilon}$Rs~\cite{gupte2017role}, GraphWave~\cite{donnat2018learning} and SEGK~\cite{nikolentzos2019learning} belong to the low-rank matrix factorization.
struc2vec~\cite{ribeiro2017struc2vec}, struc2gauss~\cite{pei2020struc2gauss}, Role2Vec~\cite{ahmed2019role2vec} and NODE2BITS~\cite{jin2019node2bits} are all based on random walk.
DRNE~\cite{tu2018deep}, GraLSP~\cite{jin2020gralsp}, GAS~\cite{guo2020role} and RESD~\cite{zhang2021role} pertain to the scope of deep learning. In the subsequent experiments, all the parameters are fine-tuned. Note that for Role2vec, we use motif-count features as the default setting, but it causes the out-of-memory issue in large networks and we use node degrees to circumvent. We release the datasets and source code used in the experiments on Github \url{}.
\begin{figure*}[htbp]
\centering
{\includegraphics[width=\linewidth]{./pic/brazil-flights_tsne.pdf}}
\caption{Visualization results of $12$ methods using t-SNE on the Air Brazil network. The points denote the nodes and the same color indicates that they belong to the same label.Two nodes are more closer to each other if their embeddings are role similar.}
\label{fig:tsne}
\end{figure*}
\begin{table*}[htbp]
\centering
\caption{Node classification average F1 score on different networks.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{Brazil} & \multicolumn{2}{c|}{Europe} & \multicolumn{2}{c|}{USA} & \multicolumn{2}{c|}{Reality-call} & \multicolumn{2}{c|}{Actor} &
\multicolumn{2}{c|}{Film} \\
\cline{2-13}
& Micro & Macro & Micro & Macro & Micro & Macro & Micro & Macro & Micro & Macro & Micro & Macro \\
\hline
RolX & $0.749$ & $0.741$ & $0.556$ & $0.546$ & $0.623$ & $0.617$ & $0.593$ & $0.383$ & $0.465$ & $0.451$ & $\textbf{0.494}$ & $\textbf{0.396}$ \\
RID$\large\boldsymbol{\varepsilon}$Rs & $\textbf{0.790}$ & $\textbf{0.783}$ & $0.539$ & $0.519$ & $0.626$ & $0.618$ & $0.635$ & $0.396$ & $\textbf{0.470}$ & $0.448$ & $0.479$ & $0.380$ \\
GraphWave & $\textbf{0.762}$ & $\textbf{0.753}$ & $0.526$ & $0.491$ & $0.517$ & $0.469$ & $\textbf{0.839}$ & $\textbf{0.516}$ & $0.251$ & $0.180$ & OM & OM \\
SEGK & $0.723$ & $0.718$ & $0.536$ & $0.524$ & $0.615$ & $0.607$ & $\textbf{0.839}$ & $\textbf{0.514}$ & $\textbf{0.479}$ & $\textbf{0.460}$ & OM & OM \\
\hline
struc2vec & $\textbf{0.766}$ & $\textbf{0.757}$ & $\textbf{0.577}$ & $\textbf{0.572}$ & $0.599$ & $0.594$ & $0.593 $ & $0.376$ & $0.463 $ & $\textbf{0.456}$ & $0.474$ & $0.365$ \\
struc2gauss & $0.730$ & $0.715$ & $\textbf{0.585}^{\star}$ & $\textbf{0.580}^{\star}$ & $\textbf{0.641}$ & $\textbf{0.632}$ & $0.603$ & $0.391$ & $0.456$ & $0.449$ & OT & OT \\
Role2vec & $0.385$ & $0.358$ & $0.362$ & $0.347$ & $0.467$ & $0.455$ & $0.541$ & $0.354$ & $0.277\dag{}$ & $0.277\dag{}$ & $0.278\dag{}$ & $0.243\dag{}$ \\
NODE2BITS & $0.593$ & $0.577$ & $0.490$ & $0.477$ & $0.583$ & $0.573$ & $0.524$ & $0.354$ & $0.442$ & $0.424$ & $\textbf{0.542}^{\star}$ & $\textbf{0.428}^{\star}$ \\
\hline
DRNE & $0.716$ & $0.700$ & $0.542$ & $0.521$ & $0.601$ & $0.588$ & $0.630$ & $0.478$ & $0.462$ & $0.449$ & $0.453$ & $0.339$ \\
GAS & $0.757$ & $0.750$ & $\textbf{0.583}$ & $\textbf{0.574}$ & $\textbf{0.668}^{\star}$ & $\textbf{0.659}^{\star}$ & $\textbf{0.841}^{\star}$ & $\textbf{0.529}^{\star}$ & $\textbf{0.480}^{\star}$ & $\textbf{0.466}^{\star}$ & $\textbf{0.490}$ & $\textbf{0.380}$ \\
RESD & $\textbf{0.797}^{\star}$ & $\textbf{0.791}^{\star}$ & $0.557$ & $0.544$ & $\textbf{0.640}$ & $\textbf{0.634}$ & $0.607$ & $0.411$ & $\textbf{0.476}$ & $\textbf{0.463}$ & $\textbf{0.516}$ & $\textbf{0.415}$ \\
GraLSP & $0.510$ & $0.490$ & $0.455$ & $0.422$ & $0.535$ & $0.523$ & $0.454$ & $0.300$ & $0.341$ & $0.316$ & OM & OM \\
\hline
\end{tabular}
\label{tab.3}
\end{table*}
\subsection{Efficiency analysis}
All experiments are performed on a machine with Intel(R) Xeon(R) CPU E5-2680 v4 at 2.40GHz and 125GB RAM.
In this experiment, by ignoring the differences in some implementation details, we report the running time of these methods on the above six real networks. The results are shown in Fig~\ref{fig:time}. The y-axis represents the average logarithmic time (seconds) of $10$ times running of each method on one specific network and x-axis is the size (number of nodes) of the network. If the value is above the dotted line, it means that the cost of corresponding method is beyond our tolerance.
Generally speaking, the methods based on deep learning are relatively efficient compared to others, especially for the methods that need higher-order features. GAS~\cite{guo2020role}, RESD~\cite{zhang2021role} and NODE2BITS~\cite{jin2019node2bits} are the three most efficient methods and RolX~\cite{henderson2012rolx} also has competitive results. struc2gauss~\cite{pei2020struc2gauss}, SEGK~\cite{nikolentzos2019learning} and Role2vec~\cite{ahmed2019role2vec} cost more time to learn the node embeddings since they need to compute higher-order features, e.g., motif. In particular, as the most classical method proposed in the early stage, RolX still shows competitive in efficiency.
However, the numbers of nodes in these networks are quite different in a random way. In order to show the efficiency of different methods more fairly, we generate four different types of networks with linear variation of the size. The details are as following.
Regular network. It is a regular graph in which each node has the same number of neighbors and we set it as $3$.
ER network~\cite{erdos59a}. It is based the ER random graph and we set the probability of having a link between any two nodes as $0.1$.
Small world network. It comes from the popular Watts–Strogatz model~\cite{watts1998collective} and for each link, we set it rewiring probability as $0.5$.
Scale-free network. It is based on the Barabási–Albert model~\cite{barabasi1999emergence} with preferential attachment, for each new nodes, the number of its links is set as $3$.
For each type of network, we vary the number of nodes $N$ from $1,000$ to $10,000$. The running time of the baselines is shown in Fig~\ref{fig:linear time}. From these results, some conclusions can be drawn. Firstly, the ER network is relatively dense for its fixed link probability. The average degree of nodes in larger networks is larger correspondingly. In this type of network, NODE2BITS has significant efficiency advantage than all others, and it also achieves competitive results in other three types of networks. Secondly, SEGK is very efficient on the regular and small world networks but time-consuming on the ER and scale-free networks. Thirdly, GAS is the most effective on the scale-free networks and has second results only to the SEGK. Besides, RolX and RESD rank in the middle of this experiments for they all are based on ReFex~\cite{henderson2011s} and reconstruct the features with different methods. GAS is one of the most efficient on almost all the networks. GAS reconstructs a small number of primary features which leads to few parameters and epochs for loss convergence. NODE2BIT consumes little time on all the networks stably. Because its feature aggregation process, which is based on random walk and the Simhash mapping process, is almost linear to the number of edges.
\begin{figure*}[htbp]
\centering
{\includegraphics[width=0.8\linewidth]{./pic/classification.pdf}}
\caption{Classification results of the $12$ role-oriented network embedding methods on the six real world networks.The performance is based on the Micor-F1 and Macor-F1 with different percentages of training set, which is the average values of $20$ runs.}
\label{fig:classification}
\end{figure*}
\subsection{Visualization analysis}
Visualization can help to understand the performance of the different methods intuitively. It is also a common way for understanding the network structure and the methods. In this section, we show the t-SNE results of these methods based on the Brazil network in Fig~\ref{fig:tsne}. It can be observed that, RID$\large\boldsymbol{\varepsilon}$Rs~\cite{gupte2017role}, GraphWave~\cite{donnat2018learning}, SEGK~\cite{nikolentzos2019learning}, struc2vec~\cite{ribeiro2017struc2vec}, struc2gauss~\cite{pei2020struc2gauss}, DRNE~\cite{tu2018deep}, GAS~\cite{guo2020role} and RESD~\cite{zhang2021role} can be more accurate to identify the labels compared with others. The cyan nodes correspond to the airports with large passenger flow and all these methods can embed it effectively except NODE2BITS~\cite{jin2019node2bits}. The red nodes represent the marginal airports. These nodes are usually hard to be well identified for they are more scattered in the network. Therefore, these methods can not identify this kind of nodes well. Besides, although we expect to embed nodes with same role into close points in latent space, it is also still the demand nodes should also be slightly different even they are in the same role. From this view, struc2gauss, SEGK and GAS are better selections. GAS benefits from the low-dimensional features which leads to the same-role embeddings gathering. The embeddings of SEGK and struc2gauss shape some clumps since they are directly constrained by effective similarities.
\subsection{Classification results}
Here we report the performance of the classification experiment on the six networks.
In detail, with the learned node embeddings of each method, we randomly select $70\%$ of the nodes as the training set and the others as the test set and take the logistic regression classifier to learn the labels of nodes. The results are shown in Table~\ref{tab.3}. It represents the Micro-F1 and Macro-F1 value of classification results. To avoid losing of generality, we report the average results of $20$ times of independent runs on each network.
As shown in Table~\ref{tab.3}, for each column, we mark the values of methods with significant advantages, i.e. the top results of these methods. OM and OT mean that it cannot be calculated for fixed memory and limited time, and the \dag{} denotes that Role2vec uses degree features rather than motif features because it is very time-consuming.
\iffalse
\begin{itemize}
\item For the three air-traffic networks, RESD~\cite{zhang2021role} performs best on Brazil and RID$\large\boldsymbol{\varepsilon}$Rs~\cite{gupte2017role} follows because the ReFeX~\cite{henderson2011s} features used by them can effectively capture structural similarities in small networks. While on the other two networks, GAS~\cite{guo2020role} and struc2gauss~\cite{pei2020struc2gauss} perform well. For the rest three large networks, struc2guass cannot generate node embeddings within limited time, though it performs well in small networks. GAS outperforms others on Reality-call and Actor, while NODE2BITS~\cite{jin2019node2bits} gets the highest score on Film.
\item Among all the datasets, we observe that Role2vec~\cite{ahmed2019role2vec} obtains the worst performance. This may because that it leverages the features-based random walk and damages the original network structure. It can be applied to link prediction task, but is not proper for role discovery. Though GraLSP~\cite{jin2020gralsp} leverages the local structural patterns, it concentrates more on node proximity and fails to distinguish roles of nodes.
\item We divide these methods into three types: random walk, matrix factorization, and deep learning. In general, deep learning methods perform well on all the networks. For the three small air-traffic networks, random walks (struc2vec~\cite{ribeiro2017struc2vec} and struc2gauss) perform well, while matrix factorization methods get better results on large networks, even if some of them cannot obtain embeddings because of the out of memory error. Overall, deep learning methods are scale-well and can effectively discover roles.
\item For matrix factorization methods, RID$\large\boldsymbol{\varepsilon}$Rs is better than RolX~\cite{henderson2012rolx} except on Film, and they both can be applied to large-scale networks. GraphWave~\cite{donnat2018learning} and SEGK~\cite{nikolentzos2019learning} perform best on Reality-call but consume too much memory. For random walk methods, struc2vec and struc2gauss show their superiority on small networks, while NODE2BITS is more suitable for large-scale networks. As for deep learning methods, GAS and RESD stay ahead of other approaches on all networks.
\item As shown in Fig.~\ref{fig:time}, for RESD, NODE2BITS, and GAS, there is a linear relationship between their computational complexities and network sizes. In Table~\ref{tab.3}, GAS and RESD perform well on all networks, while NODE2BITS only gets the best score on Film. It is obvious that combining structural features and deep learning models has advantages in both time and classification.
\end{itemize}
\fi
\begin{figure*}[htbp]
\centering
{\includegraphics[width=0.8\linewidth]{./pic/clustering.pdf}}
\caption{Clustering results of the $12$ role-oriented network embedding methods on the six real world networks. The performance is based on the silhouette coefficient and NMI and evaluated on $10$ times run.}
\label{fig:clustering}
\end{figure*}
For the three air-traffic networks, RESD~\cite{zhang2021role} performs best on Brazil and RID$\large\boldsymbol{\varepsilon}$Rs~\cite{gupte2017role} follows because the ReFeX~\cite{henderson2011s} features used by them can effectively capture structural similarities in small networks. While on the other two networks, GAS~\cite{guo2020role} and struc2gauss~\cite{pei2020struc2gauss} perform well. For the rest three large networks, struc2guass cannot generate node embeddings within limited time, though it performs well in small networks. GAS outperforms others on Reality-call and Actor, while NODE2BITS~\cite{jin2019node2bits} gets the highest score on Film.
Among all the datasets, we observe that Role2vec~\cite{ahmed2019role2vec} obtains the worst performance. This may because that it leverages the features-based random walk and damages the original network structure. It can be applied to link prediction task, but is not proper for role discovery. Though GraLSP~\cite{jin2020gralsp} leverages the local structural patterns, it concentrates more on node proximity and fails to distinguish roles of nodes.
In Section~\ref{sec:algorithm}, we have divided these methods into three types: random walk, matrix factorization, and deep learning. In general, deep learning methods perform well on all the networks. For the three small air-traffic networks, random walks (struc2vec~\cite{ribeiro2017struc2vec} and struc2gauss) perform well, while matrix factorization methods get better results on large networks, even if some of them cannot obtain embeddings because of the out of memory error. Overall, deep learning methods are scale-well and can effectively discover roles.
For matrix factorization methods, RID$\large\boldsymbol{\varepsilon}$Rs is better than RolX~\cite{henderson2012rolx} except on Film, and they both can be applied to large-scale networks. GraphWave~\cite{donnat2018learning} and SEGK~\cite{nikolentzos2019learning} perform best on Reality-call but consume too much memory. For random walk methods, struc2vec and struc2gauss show their superiority on small networks, while NODE2BITS is more suitable for large-scale networks. As for deep learning methods, GAS and RESD stay ahead of other approaches on all networks.
As shown in Fig.~\ref{fig:time}, for RESD, NODE2BITS, and GAS, there is a linear relationship between their computational complexities and network sizes. In Table~\ref{tab.3}, GAS and RESD perform well on all networks, while NODE2BITS only gets the best score on Film. It is obvious that combining structural features and deep learning models has advantages in both time and classification.
Further more, in order to analyze the influence of different percentages of training set on classification results, we also show the performance of these methods with varying size of training (from $10\%$ to $90\%$). The results are shown in Fig~\ref{fig:classification}. The experimental settings are same introduced above.
In general, as the ratio of the training set increases, the F1 scores of classification results are improved. On the three air-traffic networks, we observe significant increase, but when we use too many training samples (more than $80\%$), the scores go down because of over-fitting. The scores on large networks are stable, because small percentage of training samples are sufficient enough to make the linear model constraint.
The scores on Reality-call network are different to others, because the distribution of labels is unbalance, and the Micro-F1 scores are higher than Macro-F1. In general, we observe the scores of Role2vec fluctuate, which means that Role2vec fails for the task of role classification. GraLSP also performs a litter better than Role2vec, but it gets worse on Reality-call. Other methods show similar tendencies, while deep learning methods (GAS and RESD) and NODE2BITS exhibit significant superiority.
\subsection{Clustering results}
In the social science, the role discovery is usually denoted as an unsupervised task which can be implemented by the unsupervised clustering in machine learning. In this experiment, we study the role clustering problem based on the embedding of different methods. Based on the embedding, we take the k-means algorithm to obtain the clusters, and report the silhouette coefficient~\cite{rousseeuw1987silhouettes} and Normalized Mutual Information (NMI)~\cite{danon2005comparing} in Fig.~\ref{fig:clustering}. The silhouette coefficient is a measure of how similar an object is to its own cluster compared to other clusters. The result only relies on the embeddings themselves rather than true labels. While the NMI measures the similarity between clustering results and real distribution.
We observe that GraphWave~\cite{donnat2018learning} obtains high scores on both measurements in air-traffic networks, but in larger networks (Actor and Reality-call), it gets low NMI scores. DRNE~\cite{tu2018deep} shows similar tendency because its aggregation methodology leads to obvious clustering, but may not fit the true distributions. The results of Role2vec~\cite{ahmed2019role2vec} and GraLSP~\cite{jin2020gralsp} are similar to classification, and they fail to cluster roles of nodes. The performances of other random walk and matrix factorization methods are moderate. In general, it can be observed that deep learning methods (GAS~\cite{guo2020role} and RESD~\cite{zhang2021role}) show overall superiority. They perform well on small networks but significantly outperform others in large-scale networks. This is also consistent with findings of deep learning methods in other domains that deep learning methods can better learn the patterns from larger-scale data.
\subsection{Top-k similarity search}
We conduct top-k similarity search experiments on the $6$ Wiki-talk networks to evaluate the ability to retrieve rare roles of different methods. In specific, after generating role-based embeddings for each node through a candidate method, we find the $k$ most similar users for each bot or administrator by computing the euclidean distance between the embeddings. The mean precision for both bots and administrators on different sizes of the retrieved node list, i.e., different $k$s, is reported in the Supplementary Materials.
In general, we assume that nodes literally having the same role (such as bots in Wiki-networks) in a network have similar structural patterns. However, there are more bots than administrators in most Wiki-talk networks, almost every method achieves higher precision on searching administrators than that on searching bots when the retrieved list size $k$ is fixed and smaller than or close to the number of administrators. This shows that the structures of bots are more irregular or these methods can not effectively capture the specific structural patterns of bots. The structural representation learning on literal roles is much more difficult than on function-based defined roles (such as the classes in Table~\ref{tab.datasets1}), since the latent patterns of literal roles are usually not reflected in the plain structural properties.
What's more, the inconsistency between literal roles and node structure is obvious across networks. We can come to this conclusion gradually through the following observations:
the proportion of the same literal role in different networks varies greatly as shown in Table~\ref{tab.datasets2}; none of the compared methods can produce top results on all the networks with $k$ fixed; although the proportion of bots in cy-wiki-talk and oc-wiki-talk networks are very close, the results of almost every method on cy-wiki-talk are better than those on oc-wiki-talk. Thus, the literal roles in different networks are less comparable and transferable to those roles (such as the classes in Table~\ref{tab.datasets1}) defined directly by node functions.
Though no one method achieves best performance on all the networks, the ranks of each method on a network with different retrieved list size $k$ are usually close. Some methods, such as RID$\large\boldsymbol{\varepsilon}$Rs~\cite{gupte2017role}, GraphWave~\cite{donnat2018learning} and NODE2BITS~\cite{jin2019node2bits} show their wider applicability across different networks. NODE2BITS surprisingly achieves good and stable results compared with its performance on node classification and clustering tasks. Its feature aggregation mechanism based on random walks makes it lose some elaborate information but capture more latent structural patterns. Similarly, RID$\large\boldsymbol{\varepsilon}$Rs and GraphWave can also capture more latent structural patterns via $\varepsilon$-equitable refinement and characterizing wavelet distributions, respectively. Role2vec still lacks competitiveness on this task, since it assigns roles based on features firstly and generates embeddings capturing proximities between roles as we argued above. The deep learning methods DRNE~\cite{tu2018deep}, GAS~\cite{guo2020role} and RESD~\cite{zhang2021role} show mediocre performance as they overfit some features due to the deep process of feature reconstruction. While the similarity captured by GraLSP~\cite{jin2020gralsp} based on anonymous walks is too coarse.
\subsection{Discussion on experiments}
In this section, we show a variety of experiments on some popular role-oriented network embedding methods. These methods cover all the three major categories and the five minor categories we propose. We evaluate their effectiveness on classification, clustering, visualization and top-k similarity search and efficiency on networks in different scales.
We find that no one method can outperform the others on all tasks. As we argued above, each task has a different focus and the evaluated methods are only suitable for parts of them. On balance, GAS~\cite{guo2020role} and RESD~\cite{zhang2021role} are good at visualization, node classification and clustering tasks. NODE2BITS~\cite{jin2019node2bits} show outstanding performance on both top-k similarity search and efficiency test. Role2vec~\cite{ahmed2019role2vec} ceases to be effective on these role-based tasks as it assigns roles based on features directly while the embeddings do not capture the structural similarities. It specializes in link prediction for non-direct role-based tasks. Similarity, the structural similarity objective of GraLSP~\cite{jin2020gralsp} is not powerful enough for role-based tasks. The performance of the other methods fluctuates a lot on different tasks and networks. Thus, it is still an open problem that how we can capture both fine and essential role information effectively and efficiently.
Though some deep learning methods seems good at many experiments, they have not yet addressed the nature of the role. The deep learning manners give them better capabilities on machine learning tasks such as classification and clustering. However, they utilize additional skills to extract structural information, and lack a suitable objective function to embed it without bias and impurities. Thus, deep role-base representation research, which is still in the beginning stage, is promising and needs more attention.
\section{Applications}\label{sec:applications}
Role discovery can complement community detection in network clustering and has always been the focus of social science and network science research. Role oriented network embedding has gradually become one of the focuses of graph machine learning. It can benefit to node ranking, community detection, information spreading and other problems in complex networks. It also contributes to the research and development of graph neural networks in machine learning. So we summarize some important applications of role oriented network embedding as follows.
First, it is beneficial to a variety of network mining tasks. Community detection and role discovery are two complementary tasks from local and global perspective of networks structures respectively~\cite{pei2020onlocal}. Thus, combining these two tasks can help to achieve better network generation and mutually improve each other. For instance, MMCR has been proposed to integrate community detection and role discovery in a unified model and detect both of them simultaneously for information networks~\cite{chen2016integrating}. The proposed method extends the Mixed Membership Stochastic Blockmodel (MMSB)~\cite{airoldi2008mixed} to combine the generative process of both community and role. REACT~\cite{pei2019joint} analyzes the community structure and role discovery under a unified framework and describes their relations via non-negative matrix factorization. Moreover, role-oriented embedding can also help other tasks including link prediction~\cite{lyu2017enhancing}, anomaly detection~\cite{henderson2012rolx}, and structural similarity search on network~\cite{rossi2020proximity}.
Second, it can help to analyze the network dynamics and evolution. As we know, nodes with different roles may have inequable influence in information diffusion. RAIN~\cite{wof} analyzes the effect of users with different roles on their reposting messages and models the generation of diffusion process under a unified probabilistic framework. Then, with a general representation learning on the network, it can effectively improve the performance of cascade~\cite{10.1145/3336191.3371811} and its popularity~\cite{10.1109/infocom41043.2020.9155349} prediction. Besides, analyzing the roles of nodes could be used to predict the network evolution and its dynamic behaviors~\cite{7303952,LI2020458} and detecting the varying of roles of node could capture the dynamic structural information~\cite{10.24963/ijcai.2018/531}.
Moreover, it has induced some new graph neural networks with more expressive ability. Although GNNs with the message passing are mainly used for supervised node or graph classification and link prediction, some new architectures have been developed to improve their expressive ability with the heterogeneity and roles in the network. Geom-GCN~\cite{pei2020geom-gcn:} has been proposed to use the role embedding method
to obtain the structural neighborhood and a bi-level aggregation in GNN. ID-GNN~\cite{You.2021.arXiv} extracts the ego network centered at one node of the network and takes rounds of heterogeneous message passing. It can compute the shortest path and clustering coefficient of the network. These features are are extremely relevant to the role. GCC~\cite{qiu2020gcc} also makes use of ego network and then employs pre-training for role-oriented network embedding.
Besides, it can shed light on new patterns discovery in specific networks. Role discovery and identification can be used to predict the social behaviors, identities and the temporal patterns~\cite{zhang.2017.qi}.~\cite{Zygmunt.2020.Demazeau} analyzes the user behaviors in the Blog social network and learn roles from the concepts of activity, influence and competition. It also can predict the sentiment and topics with the proposed definitions of roles. EMBER~\cite{jin2019smart} considers professional role inference in a large-scale email network, so it can capture and distinguish the behavior similarity of nodes automatically. SADE~\cite{pei2020subgraph} combines role information and subgraph embedding to detect subgraph-level anomalies from financial transaction networks.
\section{Future Directions}\label{sec:direction}
Although many creative and innovative methods have been proposed for role oriented network embedding, it still faces some problems and challenges to be solved in this field. In this section, we will discuss these challenges as future directions.
\textbf{Role-oriented embedding on dynamic networks.}
Most methods reviewed in this survey are for static networks. However, real-world networks are naturally dynamic and continuously streaming over time with evolving structures. Thus role-oriented NE methods for dynamic networks are of fundamentally practical and theoretical importance. There have only been a few approaches for role discovery on dynamic networks e.g., Role-Dynamics~\cite{10.1145/2187980.2188234} and DyNMF~\cite{10.24963/ijcai.2018/531}. However, some questions remain open. For example, how to separate the dynamic networks into different snapshots and how to model the deletion of nodes and edges. Thus, additional investigation is needed to extend current NE methods for dynamic scenarios.
\textbf{Other types of embedding spaces.} Beside to the Euclidean space, a variety of methods project the nodes into the Hyperbolic space~\cite{Peng.2021.Zhao} in proximity preserved network representation learning. For role oriented embedding, as we know, the only work is Hyperboloid~\cite{10.1145/3340531.3412102} which extends the struc2vec~\cite{ribeiro2017struc2vec} with the hyperboloid model. Considering the non-Euclidean space is quite suitable for network embedding because of the power-law distribution of network data, it is interesting and meaningful to have some in-depth analysis and understanding in utilizing hyperbolic neural network for this problem.
\textbf{Construction of larger-scale benchmarks.} All the methods for role oriented network embedding are evaluated on relatively small-scale networks data with thousands of nodes (seen in section~\ref{sec:Experiment}). However, real-world networks are often of a massive scale, e.g., there are billions of users in social networks. Constructing larger-scale benchmark datasets is very important to evaluate existing approaches in effectiveness, efficiency and robustness, and also beneficial for researchers to develop new models.
\textbf{Interpretation on roles and role-oriented embeddings.} In social science, roles often correspond to social identifications, e.g., students and teachers in a school. Real-world networks may not contain such information and thus are difficult to understand the discovered roles. The lack of interpretability of roles and role-oriented embeddings may significantly impact our ability to gain insights into these roles. However, so far, there have not been much work focusing on discovering roles that are interpretable. One exception is RolX~\cite{henderson2012rolx}, and it attempts to explain roles based on several structural measures such as centrality. More research is needed to discover roles and learn embeddings that are more interpretable.
\textbf{Deep theoretical analysis.} Important but not the final, network embedding for role discovery lacks theoretical analysis. Although some methods stem from certain type of equivalence relation, the theoretical analysis has been ignored. Moreover, current methods cannot capture universal node representations and provide the upper/lower bound of the expressive ability. It is expected to have solid theoretical analysis in this filed similar to that in graph neural networks~\cite{xu2018how}.
\section{Conclusions}\label{sec:conclusions}
Role-oriented network embedding can complement the category of network representation learning. It has gradually become one of most important research focuses in network embedding. In this survey, we have proposed a general understanding mechanism for role-oriented network embedding approaches and a two-level classification ontology based on different embedding principles. Using this ontology, we categorize a series of popular methods into different groups. Then, we review the principles and innovations of selected representative embedding methods in different categories. We further conduct comprehensive experiments to evaluate these representative methods, including role classification and clustering, top-k similarity search, visualization and their efficiency test. Last, we summarize the important applications of this problem and outline some future directions. We believe that this survey can help to understand and deepen role-oriented network embedding and it will attract more attention from network science and deep learning.
\section*{Acknowledgment}
This work is supported by the National Natural Science Foundation of China (61902278).
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CMP#84 Cultivating the Mind
"Mrs. Bennet... was a woman of mean understanding, little information, and uncertain temper."
-- Jane Austen describing Mrs. Bennet in Pride and Prejudice
In the previous posts, we have looked at the way novelists of Austen's day wrote about the natural dispositions and innate intelligence (or lack thereof) of their characters. Of course, merely being blessed with intelligence, aka "a good understanding," was not sufficient. That understanding had to be enriched and cultivated with knowledge, or "information."
In Pride & Prejudice, Mrs. Bennet has "little information." She's narrow-minded and ignorant.
Elizabeth Bennet lists Darcy's "information" as one of his attractive attributes: "She began now to comprehend that he was exactly the man who, in disposition and talents, would most suit her. His understanding and temper, though unlike her own, would have answered all her wishes... and from his judgement, information, and knowledge of the world, she must have received benefit of greater importance."
Fanny Price's father "did not want abilities but he had no curiosity, and no information beyond his profession; he read only the newspaper and the navy-list; he talked only of the dockyard."
Henry and Mary Crawford are intelligent and well-educated, as Austen makes clear. Mary quotes poetry and invents a satire of that poetry on the spot. Henry converses knowledgably about Shakespeare and even how best to conduct a church service, "shewing it to be a subject on which he had thought before, and thought with judgment." Mary and Henry are lacking, however, in moral judgement, a topic we'll turn to later in this series.
Rousseau's Emile, learning from nature
In Austen's time, many novels and stories centered around the problems created when parents neglected their children's education. This theme is found both in adult novels and in works aimed at children.
In Ashton Priory, (1792) "Eliza is endowed with a good natural understanding… but… from the character of her parents…. it can have derived no advantages from cultivation…"
In Albert, or the Wilds of Strathnavern, "Frederick and Gertrude St. Austyn joined to good persons understandings which, had they been properly cultivated, might have proved an honour to society; but the follies of their parents slurred the fair tablets which nature had rendered capable of receiving the noblest impression."
In Fanny, or the Deserted Daughter (1792), Fanny Vincent vainly attempts to be an effective governess for a spoilt girl: "Sophy Wigstead was not yet twelve; she was pretty, but was so much neglected that no other judgment could be formed of her; neglected indeed only as to her mind, for the ornaments of her person were peculiarly studied..."
In The Brothers, a Novel for Children, (1794) the author contrasts George, "who did not want natural understanding, though it was lost in idleness and neglect" with his brother Harry who "cultivate[s] that understanding which nature had given him."
There was no such thing as universal public school education at this time. Educating children was the responsibility of the parents so it's not surprising that there was much debate about how best to go about it. The French philosopher Jean Jacques Rousseau was credited with popularizing the idea that children should be allowed to study what interests them, so that their natural curiosity would be awakened.
Colonel Lorton, the father in Emily, a Moral Tale [1809], thinks that is so much hogwash: "He thought that Rousseau's plan of education, founded as it is upon no good principle, can be conducive to no good end. It shows great ignorance of human nature, because it is evident if a child is left to the unrestrained exercise of its own caprice and passions, it is likely to be more wild and mischievous than any other young animal... [he rejected] without hesitation all such fanciful schemes of education, [and] considered moral discipline as essentially conducive to the right conduct of the human mind…"
It seems Jane Austen also disagrees with Rousseau. When Catherine Morland refers to lessons as "a torment," Henry Tilney responds: "That little boys and girls should be tormented, is what no one at all acquainted with human nature in a civilized state can deny."
He acknowledges that learning how to read is a boring and laborious task, but "it is very well worth-while to be tormented for two or three years of one's life, for the sake of being able to read all the rest of it."
Edward Austen treated his sisters to lunch at an inn when they were at boarding school
Boys usually acquired their education at boarding schools, or occasionally with a tutor, as Edward Ferrars does in Sense and Sensibility. When Austen says "the young people were all at home" when young Fanny first came to Mansfield Park, she means that Fanny arrived during school holidays, when the teen-age Tom and Edmund were not away at school.
Emma is glad that the Westons have a daughter because: "It would be a great comfort to Mr. Weston… to have his fireside enlivened by the sports and the nonsense, the freaks and the fancies of a child never banished from home."
Of course, girls sometimes did go to boarding schools, including Cassandra and Jane Austen.
In I'll Consider Of It, a grandfather disputes with his daughter over the education of his granddaughter Emily: "this was a point they never could agree upon—he hated all boarding-schools; but was particularly averse to his grand-daughter being kept any longer, now she was grown a fine tall girl, in a place where he declared she would now learn nothing but deceit and vanity, without any kind of domestic economy."
In The Advantages of Education, the mother rescues her daughter from just such a fashionable boarding school and commences her upon a solid course of reading and moral reflection.
In Middlemarch, set in the 1830s, Rosamond Vincy learns how to get in and out of a carriage elegantly at her boarding school. More about "accomplishments" in another post.
In Emma, Austen is at pains to contrast Mrs. Goddard's boarding school with fashionable establishments: "not of a seminary, or an establishment, or any thing which professed, in long sentences of refined nonsense, to combine liberal acquirements with elegant morality, upon new principles and new systems—and where young ladies for enormous pay might be screwed out of health and into vanity—but a real, honest, old-fashioned Boarding-school, where a reasonable quantity of accomplishments were sold at a reasonable price, and where girls might be sent to be out of the way, and scramble themselves into a little education, without any danger of coming back prodigies."
Governesses and Home Schooling
"So much mischief may be done by a silly governess in a single quarter of an hour!" exclaims the narrator in Maria Edgeworth's short story, Madame Panache. Well, the governess is French, so what can you expect?
In I'll Consider of It, Miss Clarkson's grandfather's friend advises against hiring a French governess.
"And do not all young girls show a predilection for reading novels and romances? therefore, to form the mind of her pupil, Madame [the hypothetical French governess] will very good-naturedly, procure for her all the love-breathing, unshackled romances which France has produced; from the Eloise of Rousseau, to the Delphine of Madame de Stael and perhaps will not scruple to put into her hands the more loose and easy writings of the licentious and immoral Louvet!"
"Horrible!" exclaimed the Captain. [the grandfather]
"And yet," said the Major, "the picture is not too highly coloured; would to Heaven it were! But such are the beings to whom the education of children belonging to the first people in the land is entrusted."
Many novels featured the relationship between well-born girls and their governesses. A governess is a central character in this children's book about the importance of a thorough, moral education for girls. If governesses refrained from corrupting the morals of their charges with French novels, then they might be expected to teach biography and memoir, history, modern languages, astronomy and geography, in addition to enough arithmetic so girls could keep their personal or household accounts. Some girls also studied botany, but other sciences were not considered suitable for females. I quoted this passage last month in a book review but it is so apropos I'll quote it again here: In A Winter in London, two old curmudgeons talk about the new-fangled craze for female education:
"I don't approve of the present system, of making prattling philosophers in petticoats." [says Mr. Ogilvy] I see no good that is to result to society from having our wives or daughters discharging electric or Galvanic batteries at our heads, or of converting our cookmaids into chemical analysers of smoke and steam."
"But are not the scientific pursuits of the present day at least as beneficial to society as the old amusements of working carpets and chair bottoms?" doctor Hoare asks, but he is just kidding. He concurs that "it is impossible there should be a difference of opinion" on the subject of females studying the sciences.
The wealthy Bertram family hire Miss Lee to educate Maria and Julia. The Bertram girls have the advantage of "early information." (If not moral instruction), and they are proud of the education they have acquired.
Studying Latin and Greek was controversial for women. In Coelebs in Search of a Wife, it accidentally slips out that Lucilla, the candidate for wifehood, has been studying Latin. She blushes and leaves the room. Her father Mr. Stanley approves of her modesty, for "a discreet woman will never produce in company" her knowledge of "a learned language" as opposed to "those acquirements which are always in exhibition" such as music and drawing.
Ladies who made it known they could speak Latin and Greek were derided as female pedants. For boys, however, education in those days strongly emphasized Latin. Our Austen heroes with the possible exception of Captain Wentworth (who might have gone to sea as a boy, not to university) would have been able to read Latin and to quote and refer to classical literature in Latin. Our clergyman heroes, Edmund Bertram, Edward Ferrars, and Henry Tilney, would have also studied Greek.
Of course, not everybody profited by their classical education. We can't imagine Mr. Rushworth quoting Cicero. The long-standing joke was that the sum total of classical knowledge schoolboys took away was the answer to the question, "Who dragged whom around the wall of what?"
Because Jane Austen's father kept a small school in his home, I feel certain young Jane would have overheard the lessons which were probably held six mornings a week. She would have heard the daily rhythmic chanting of the boys construing their Latin grammar.
Dr. Helena Kelly points out that Austen "certainly knew a little Latin. One of the notebooks in which her youthful writing is preserved is inscribed, in Latin, "Ex doni mei patris" (Given to me by my father), and she includes Latin phrases in her letters on more than one occasion."
However, Austen disclaimed all knowledge of the higher branches of education when she turned down a suggestion that she write a serious novel about a clergyman: "Such a man's conversation must at times be on subjects of science and philosophy, of which I know nothing; or at least be occasionally abundant in quotations and allusions which a woman who, like me, knows only her own mother-tongue, and has read little in that, would be totally without the power of giving. A classical education, or at any rate a very extensive acquaintance with English literature, ancient and modern, appears to me quite indispensable for the person who would do any justice to your clergyman; and I think I may boast myself to be, with all possible vanity, the most unlearned and uninformed female who ever dared to be an authoress."
Educational fads and fashions
The Regency period saw many books published for home educational use. Mary Shelley's father, the philosopher William Godwin, operated a publishing business and bookstore with his second wife, specializing in juvenile literature such as Charles and Mary Lamb's Tales from Shakespeare. Moralizing novels, some of which are quoted in this series, were judged safer for children than fantasies and fairy tales. In Traits of Nature, Young Christina Cleveland remarks to her cousin Adela, "fairy-tales are forbidden pleasures in all modern school-rooms. Mrs. Barbauld, and Mrs. Trimmer, and Miss Edgeworth, and a hundred others, have written good books for children, which have thrown poor Mother Goose, and the Arabian Nights, quite out of favour;—at least, with papas and mamas."
Finally, I'll quote a passage from Coelebs in Search of a Wife [1809], which illustrates what many people thought about educational fads for girls, but also so we can appreciate the difference between a didactic novel and the more natural, skillful, and witty way Austen treats this same subject in Mansfield Park. In Coelebs, a girl named Amelia Rattle (hint, hint) is brought into the novel just to give this speech. She plays no role in the plot:
"'I have not been idle, if I must speak the truth,' [says Amelia]. 'One has so many things to learn, you know. I have gone on with my French and Italian of course, and I am beginning German. Then comes my drawing-master; he teaches me to paint flowers and shells, and to draw ruins and buildings, and to take views.' [She adds that he "begins" projects for her, which will then be passed off as her own work, of course.]
'And then,' pursued the young prattle, 'I learn varnishing, and gilding, and japaning. And next winter I shall learn modelling, and etching, and mezzotinto and aquatints…Then I have a dancing-master, who teaches me the Scotch and Irish steps, and another who teachers me attitudes, and I shall soon learn the waltz… Then I have a singing-master, and another who teaches me the harp, and another for the piano-forte. And what little time I can spare from these principal things, I give by odd minutes to ancient and modern history, and geography, and astronomy, and grammar, and botany. Then I attend lectures on chemistry, and experimental philosophy, for as I am not yet come out, I have not much to do in the evening… as soon as I come out, I shall give it all up, except music and dancing.'
All this time Lucilla sat listening with a smile, behind the complacency of which she tried to conceal her astonishment."
This leads us, of course, to a discussion of "accomplishments."
Jane Austen and her contemporaries also used the word "information" as we use it today. "Miss Bertram could now speak with decided information of what she had known nothing about when Mr. Rushworth had asked her opinion; and her spirits were in as happy a flutter as vanity and pride could furnish, when they drove up to the spacious stone steps before the principal entrance."
Jean-Jacques Rousseau may hold an important place in the history of pedagogical theory, but he himself had no experience with children. As Paul Johnson relates in Intellectuals, "it comes as a sickening shock to discover what Rousseau did to his own children." Rousseau abandoned each of his five children by his common-law wife at a foundling home, where he knew they would most likely die of malnutrition, disease or neglect. "How could I achieve the tranquility of mind necessary for my work, [if] my garret [were] filled with domestic cares and the noise of children?"
Fanny Price's brother John and his friend Prudence pay a visit to the Juvenile Library in A Different Kind of Woman, the third volume of my Mansfield Trilogy. Click here for more about my novels. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,879 |
Q: No certificate in X-ARR-ClientCert header when called from Azure API Management I have configured an Azure website (with one ApiController) to use client-certificate authentication using the instructions provided here. Summarizing: you set the clientCertEnabled flag to true and from then on your website starts asking for a client-authentication-certificate.
Works great, however, now I want to access the client certificate that is sent to the server. According to the MSDN article, it should be available in the X-ARR-ClientCert request header, except that it isn't!!
This means that anyone with a client-authentication-certificate can access my API, which is undesirable in my case.
So how do I retrieve the client-authentication-certificate that a client sends to my Web API?
UPDATE 1: I'm actually calling my API through Azure API Management. I configured APIM with my client-authentication-certificate and APIM calls my API without issues. However, when the API is called from APIM, no X-ARR-ClientCert header is set. When called directly via Fiddler, I do see the header. So APIM is calling my API in some different way?!?
UPDATE 2: I went through everything again and produced some logging. First the relevant part of the DelegatingHandler class I'm logging from:
protected override async Task<HttpResponseMessage>
SendAsync(HttpRequestMessage request, CancellationToken cancellationToken)
{
Trace.TraceInformation("Going to validate client certificate.");
var x509Certificate2 = request.GetClientCertificate();
Trace.TraceInformation("Client cert: {0}", x509Certificate2 == null
? "<null>"
: x509Certificate2.Subject);
try
{
var headerKeys = string.Join("|", request.Headers.Select(h => h.Key));
Trace.TraceInformation("Header keys: {0}", headerKeys);
...
And the resulting log output:
2015-12-07T08:08:24 PID[8464] Information Going to validate client certificate.
2015-12-07T08:08:24 PID[8464] Information Client cert: <null>
2015-12-07T08:08:24 PID[8464] Information Header keys:
Connection|Host|Max-Forwards|Conf-Organisation-Key|Ocp-Apim-Subscription-Key|
X-Forwarded-For|X-LiveUpgrade|X-ARR-LOG-ID|DISGUISED-HOST|X-SITE-DEPLOYMENT-ID|
X-Original-URL
So there's no client certificate and also no X-ARR-ClientCert header.
UPDATE 3: And here's the log that results when I go to my actual API directly with a client-authentication-certificate:
2015-12-07T09:16:45 PID[8464] Information Going to validate client certificate.
2015-12-07T09:16:45 PID[8464] Information Client cert: CN=rwildenberg@itq.nl
2015-12-07T09:16:45 PID[8464] Information Header keys:
Connection|Accept|Accept-Encoding|Accept-Language|Cookie|Host|Max-Forwards|
User-Agent|Upgrade-Insecure-Requests|DNT|X-LiveUpgrade|X-ARR-LOG-ID|DISGUISED-HOST|
X-SITE-DEPLOYMENT-ID|X-Original-URL|X-Forwarded-For|X-ARR-SSL|X-ARR-ClientCert
Both a client certificate directly from the request and the expected X-ARR-ClientCert header.
UPDATE 4: In the end this happened to be my own mistake (of course). I was convinced that the url for the backend was https while in fact it was http. Client certificate authentication only works over https so in hindsight it made perfect sense no certificate was found in the backend...
A: You also need to enable client certificate authentication for the Azure hosted API - this step will ensure Azure carries over any X-ARR-ClientCert headers coming on the incoming request.
Otherwise Azure removes the X-ARR-ClientCert header from Request.Headers before it reaches your API.
Note: this setup is only available for paid subscriptions
*
*Go to https://resources.azure.com/ and select your desired azure account
*Select subscriptions -> resource groups -> your_resource_group -> providers -> microsoft.web -> sites -> your_website
*Enter in Read/Write mode from top button, enter in Edit mode then set the property "clientCertEnabled": true under properties.
"properties": {
"name": "my-site",
"state": "Running",
"hostNames": [
"my-site.azurewebsites.net",
"(string)"
],
...
"clientCertEnabled": true,
...
}
Documentation: https://learn.microsoft.com/en-us/azure/app-service-web/app-service-web-configure-tls-mutual-auth
A: Do you really want to manage client certificates for all users of the API? I understand using a client-cert to ensure only APIM can talk directly to your backend API. Usually developers who expose APIs via API Management use API keys to control access to the API. Using this approach enables policies to be applied based on different configured "products".
If you have the infrastructure in place to manage cert creation and revocation then this might be the right choice, it is just an uncommon approach. Having said that I will investigate what options are available in API Management to be able to extract the certificate thumbprint on the APIM gateway.
A: I am contacting the engineers in APIM to figure out how this is intended to work and will get back to you. When you have APIM in front of your actual API (web app), my guess is that APIM is the proxy that will take care of all the authN / authZ for you. So requests would go to your API only when this succeeds. I am guessing that is why they probably just drop the client certificate instead of forwarding it onward. But I can definitely see why the client cert would still be useful in the web app.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,148 |
\section{The \bighess\ Telescope System}
The \hess\ detector for observation of $>100\;\rm GeV$ \gr s has been
operating since June, 2002 in the Khomas highlands of Namibia (23\dg
S, 15\dg E, $1.8\;\rm km$ a.s.l.). It captures the Cherenkov light
emitted by cascades of particles in the atmosphere initiated by a
\gr\ or charged cosmic ray incident on the atmosphere. The
Cherenkov pulses ($\lambda\sim 350\;\rm nm$) are brief (few ns),
faint, and illuminate a light-pool of diameter $\sim 250\;\rm m$ on
the ground for vertical cascades. The Cherenkov images of these
cascades, roughly cometary in shape with an angular extent of a few
mrad, can be seen by a detector anywhere in the light-pool equipped
with a sufficiently fast and sensitive camera. This permits the
estimation of the nature of the initiating particle (signal
\gr\ or background cosmic-ray) and the measurement of its angular
origin and energy. The Atmospheric Cherenkov technique intrinsically
has a large ($\sim 50000\;\rm m^2$) collection area, though with a
small field of view (few degrees). Observations must take place on
clear, moonless nights.
The detector, in its Phase-I, consists of four IACTs in a square of
side $120\;\rm m$. Each telescope mount has a tessellated mirror of
$107\;{\rm m}^2$ area
with a camera in the focal plane at $15\;{\rm m}$. The camera
contains 960 photo-multipliers (PMs) with 0.16\dg\ pixel-size,
5\dg\ field of view. The read-out electronics, all contained
within the camera, is triggered when the signal from a number of PMs
exceeds a trigger threshold in an effective $\sim 1.3\;\rm ns$ trigger
window. The PM signals, which are stored in an analogue
memory while awaiting the trigger, are then read out, digitized, and
integrated within a $16\; \rm ns$ window. The results are then
sent from the camera's data-acquisition system to the control room via
optical fibres.
Soon after the second telescope became operational in January 2003, a
\lq Stereo' central trigger was implemented (June 2003), by which
events are only retained if multiple telescopes see the same cascade.
This decreases the dead-time for the individual telescopes, allowing the
trigger threshold to be decreased (thus decreasing detector's energy
threshold), while the multiple images of each cascade provide a
increase in the background-rejection capability and the angular and
energy resolution of the system.
The Phase-I of \hess\ was completed in December, 2003, with the
addition of the fourth telescope, since which time the system has been
operating at its full sensitivity. The energy threshold of the system
is $\sim 120\;\rm GeV$ for sources close to Zenith after background
rejection cuts ($\sim 400\;\rm GeV$ for single-telescope mode) with an
angular resolution improved to 0.06\dg\ (from 0.1\dg) and allowing
spectral measurements with an energy resolution of $\simeq 15\%$.
Observations of the Crab nebula have confirmed the system's
performance, with a rate of $10.8\;\rm \gamma/minute$ and a detection
significance of $26.6\;\sigma/\rm\surd hr$, which when extrapolated
for a sources close to Zenith give a 1 Crab-level sensitivity
($5\sigma$ detection) in only 30 seconds (1\% Crab in 25 hrs). See
\cite{HESS_WH,HESS_PV} for further details.
\section{Observations of AGNs with \bighess }
Since the first operation of the \hess\ detector, many galactic and
extragalactic sources have been studied. The observation of AGNs at
the highest energies is a probe of the emission mechanisms in the jets
of these sources, and studies of the their multi-wavelength spectral
energy distributions (SEDs) and correlated variability over wavelength
enable emission models (leptonic or hadronic) to be tested. In
addition, as these VHE photons interact with the intergalactic
Infra-Red (IIR) background (to give an electron-positron pair) and are
thus absorbed, they can also serve as a probe of this background
(resulting mainly from early star formation) which is difficult to
measure by direct means. However, this absorption limits the distance
at which we can see AGNs to a redshift 0.5 at the \hess\ detector
threshold energy. The large detection area of \hess\ allows us to
measure spectral and temporal characteristics on hour timescales
(depending on the strength of flares) for the sources seen.
Among the extra-galactic targets (with observing time up to Summer,
2004 in parentheses) are: \pks\ (92h), {\small PKS}$\;2005$-489 (52h),
M87 (32h), {\small NGC}253 (34h). Here we present results from two
AGNs:
\pks\ and \mkn .
\subsection{The AGN \pks }
\pks\ is the brightest AGN in the Southern Hemisphere, and has been
well studied in many energy bands over the last 20 years. It has been
previously detected at VHE energies \cite{durham}. With a redshift of
$z=0.117$ it is one of the most distant VHE blazars, and therefore
of interest not only for studies of this class of object, but also for
IIR studies.
Initial observations were taken over all the installation phase of
\hess\ Phase-I from July 2002 to October 2003, with an evolving
detector threshold and sensitivity. Clear detections ($>5\sigma$) are
seen in each night's observations, and an overall signal of
$44.9\sigma$ in $63.1\;\rm h$ of this mixed data, with $\sim
1.2\;\gamma\rm /min$, 10-60\% Crab level, with variability on
time-scales of months, days, and hours. The energy spectra are
characterized by a steep power law with a time-averaged photon index
of $\alpha=-3.31\pm 0.06$.
Owing to a particularly high level seen by \hess\ in October, 2003,
we triggered our RXTE \lq\lq target of opportunity" proposal on this
source, enabling quasi-simultaneous observations to be taken between
the two instruments. Short-term variations ($<30\;\rm min$) are seen
in both these datasets, and multi-wavelength correlations will be
published in a forthcoming paper.
A \hess\ multi-wavelength campaign with the PCA instrument on board
the Rossi X-ray Timing Explorer (RXTE) has been successfully completed
in August, 2004, with the full four-telescope Phase-I array, and
therefore full sensitivity, and these data are under analysis. This
intense study of this source should yield insights into its inner
workings.
\subsection{The AGN \mkn }
\mkn\ was the first extra-galactic source detected at VHE energies
\cite{me}.
It is the closest
such source (at $z=0.03$) and so is little affected by IIR absorption.
With a declination $\delta \sim 38^\circ$, it is still accessible to
\hess , though culminating at a Zenith angle above 60\dg . Under
these conditions, observations with the \hess\ detector have a higher
threshold, but a compensatory larger effective area (as the light-pool
is geometrically larger for showers developing at a greater
atmospheric slant distance), and so gives access to the highest
energies of the spectrum.
In April of this year, a great increase in activity from this source
was seen by the all-sky monitor aboard RXTE, reaching an
historically-high level of $110\;\rm mCrab$ in mid-April. A
multi-wavelength campaign was therefore triggered on this source,
including other IACTs, radio and optical telescopes, and RXTE.
The \hess\ observations, at an average Zenith angle of 62\dg, provided
a very clear signal in April, with $66\sigma$ in $9.71\;\rm h$ of
data, yielding $\sim 5.1\;\gamma\rm /min$, and an estimated 1-2 Crab
level. The flux clearly increases from the January level ($6\sigma$
in $2.12\;\rm h$, $\sim 0.8\;\gamma\rm /min$, 10-50\% Crab level), and
was also seen by other IACTs in the Northern hemisphere (Whipple,
{\small MAGIC}). Shorter-term variations and correlations with other
energy domains are currently under study.
\section{Future Plans for expansion to \bighess\ Phase-II}
Plans for Phase-II of the experiment are comprised of a large
telescope in the centre of the current Phase-I providing a lowered
threshold and increased sensitivity. This will provide access to a
number of astrophysical phenomen\ae, such as the spectral cut-offs in
pulsars, microquasars, GRBs, and dark matter in the form of WIMPs. As
concerns this paper, AGNs can be observed up to redshift of 2-3 with
\hess\ Phase-2 (vs. 0.5 with \hess ), provided that they are
sufficiently bright, as the optical depth due to absorption in the
intergalactic infra-red background is smaller at lower energies.
With detections of a larger number of AGNs at varying redshifts,
the effect of IIR absorption may be disentangled from the intrinsic
spectra of the sources.
Technical plans for this very-large telescope are well advanced. The
mount and dish structure (30\rm m \O) are well within the capabilities
of industry, since much larger radio-telescopes have been built. The
camera, using the same technology as Phase-I, with some improvements
in order to decrease the dead-time and readout speed, will have $\sim
2000$ pixels of size 0.05\dg ($\sim 3$\dg field of view). An improved
Analogue Memory ASIC (Application-Specific Integrated Circuit) is
being prototyped, and the associated camera and read-out electronics
are being designed, based on the experience gained with the Phase-I.
In operation with the four telescopes of Phase-I, Monte Carlo
simulations indicate that, in coincidence mode the \lq 4+1' system
would have a detection threshold of $\sim 50\rm GeV$ with fine-grained
and photon-rich image in the central telescope providing improved
background rejection and angular and energy resolution. In
stand-alone mode, a threshold as low as $15-25\; \rm GeV$ may be
achieved, though with lower background-rejection capability.
\section{Conclusions}
Phase-I of \hess\ has already provided many interesting new results,
of which some of those from extra-galactic sources are presented here.
Based on the experience gained with \hess\ Phase-I, a Phase-II
extension consisting of a very large Cherenkov Imaging Telescope is
being designed, which will provide an unprecedentedly low threshold
IACT, while greatly increasing the sensitivity at current energies.
\hess\ Phase-I will continue to provide exciting new results
in the future, while the Phase-II is being designed and installed.
\begin{figure}[b]
\begin{center}
\includegraphics[width=.99\textwidth]{punchF1.eps}
\end{center}
\caption[]{Photo of the current four-telescope \hess\ Phase-I array,
with an artist's impression of the Phase-II 30m \O\ telescope in the
centre of the array superimposed.
}
\label{eps1}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,246 |
namespace net {
namespace testing {
MockFileStream::MockFileStream(net::NetLog* net_log)
: net::FileStream(net_log),
forced_error_(net::OK),
async_error_(false),
throttled_(false),
weak_factory_(this) {
}
MockFileStream::MockFileStream(base::PlatformFile file,
int flags,
net::NetLog* net_log)
: net::FileStream(file, flags, net_log),
forced_error_(net::OK),
async_error_(false),
throttled_(false),
weak_factory_(this) {
}
MockFileStream::MockFileStream(
base::PlatformFile file,
int flags,
net::NetLog* net_log,
const scoped_refptr<base::TaskRunner>& task_runner)
: net::FileStream(file, flags, net_log, task_runner),
forced_error_(net::OK),
async_error_(false),
throttled_(false),
weak_factory_(this) {
}
MockFileStream::~MockFileStream() {
}
int MockFileStream::OpenSync(const base::FilePath& path, int open_flags) {
path_ = path;
return ReturnError(FileStream::OpenSync(path, open_flags));
}
int MockFileStream::Seek(Whence whence, int64 offset,
const Int64CompletionCallback& callback) {
Int64CompletionCallback wrapped_callback =
base::Bind(&MockFileStream::DoCallback64,
weak_factory_.GetWeakPtr(), callback);
if (forced_error_ == net::OK)
return FileStream::Seek(whence, offset, wrapped_callback);
return ErrorCallback64(wrapped_callback);
}
int64 MockFileStream::SeekSync(Whence whence, int64 offset) {
return ReturnError64(FileStream::SeekSync(whence, offset));
}
int64 MockFileStream::Available() {
return ReturnError64(FileStream::Available());
}
int MockFileStream::Read(IOBuffer* buf,
int buf_len,
const CompletionCallback& callback) {
CompletionCallback wrapped_callback = base::Bind(&MockFileStream::DoCallback,
weak_factory_.GetWeakPtr(),
callback);
if (forced_error_ == net::OK)
return FileStream::Read(buf, buf_len, wrapped_callback);
return ErrorCallback(wrapped_callback);
}
int MockFileStream::ReadSync(char* buf, int buf_len) {
return ReturnError(FileStream::ReadSync(buf, buf_len));
}
int MockFileStream::ReadUntilComplete(char *buf, int buf_len) {
return ReturnError(FileStream::ReadUntilComplete(buf, buf_len));
}
int MockFileStream::Write(IOBuffer* buf,
int buf_len,
const CompletionCallback& callback) {
CompletionCallback wrapped_callback = base::Bind(&MockFileStream::DoCallback,
weak_factory_.GetWeakPtr(),
callback);
if (forced_error_ == net::OK)
return FileStream::Write(buf, buf_len, wrapped_callback);
return ErrorCallback(wrapped_callback);
}
int MockFileStream::WriteSync(const char* buf, int buf_len) {
return ReturnError(FileStream::WriteSync(buf, buf_len));
}
int64 MockFileStream::Truncate(int64 bytes) {
return ReturnError64(FileStream::Truncate(bytes));
}
int MockFileStream::Flush(const CompletionCallback& callback) {
CompletionCallback wrapped_callback = base::Bind(&MockFileStream::DoCallback,
weak_factory_.GetWeakPtr(),
callback);
if (forced_error_ == net::OK)
return FileStream::Flush(wrapped_callback);
return ErrorCallback(wrapped_callback);
}
int MockFileStream::FlushSync() {
return ReturnError(FileStream::FlushSync());
}
void MockFileStream::ThrottleCallbacks() {
CHECK(!throttled_);
throttled_ = true;
}
void MockFileStream::ReleaseCallbacks() {
CHECK(throttled_);
throttled_ = false;
if (!throttled_task_.is_null()) {
base::Closure throttled_task = throttled_task_;
throttled_task_.Reset();
base::MessageLoop::current()->PostTask(FROM_HERE, throttled_task);
}
}
void MockFileStream::DoCallback(const CompletionCallback& callback,
int result) {
if (!throttled_) {
callback.Run(result);
return;
}
CHECK(throttled_task_.is_null());
throttled_task_ = base::Bind(callback, result);
}
void MockFileStream::DoCallback64(const Int64CompletionCallback& callback,
int64 result) {
if (!throttled_) {
callback.Run(result);
return;
}
CHECK(throttled_task_.is_null());
throttled_task_ = base::Bind(callback, result);
}
int MockFileStream::ErrorCallback(const CompletionCallback& callback) {
CHECK_NE(net::OK, forced_error_);
if (async_error_) {
base::MessageLoop::current()->PostTask(
FROM_HERE, base::Bind(callback, forced_error_));
clear_forced_error();
return net::ERR_IO_PENDING;
}
int ret = forced_error_;
clear_forced_error();
return ret;
}
int64 MockFileStream::ErrorCallback64(const Int64CompletionCallback& callback) {
CHECK_NE(net::OK, forced_error_);
if (async_error_) {
base::MessageLoop::current()->PostTask(
FROM_HERE, base::Bind(callback, forced_error_));
clear_forced_error();
return net::ERR_IO_PENDING;
}
int64 ret = forced_error_;
clear_forced_error();
return ret;
}
} // namespace testing
} // namespace net
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,604 |
Military / US News
26JUL2013: U.S. Army Buying Millions of Rounds of Russian Ammo and Popular Civilian Firearms
(Infowars) – Compelling proof Dept. of Defense is also drying up firearms and ammo supply. The U.S. Army is now looking to stockpile nearly 3,000,000 live rounds of Soviet-era Russian ammo popular with civilian shooters.
A U.S. Army solicitation posted July 18 on the Federal Business Opportunities web site asks for "non-standard" ammunition from vendors which includes:
– 2,550,000 rounds of 7.62x39mm ball ammo
– 575,000 blank rounds of 7.62x39mm ammo and
– 425,000 rounds of 9x18mm Makarov ball ammo
The army intends to store all these rounds in ammo storage facilities at both Camp Stanley in Boerne, Texas and the Blue Grass Army Depot in Kentucky.
As the solicitation implies, the 7.62x39mm and the 9x18mm Makarov are notstandard-issue in the U.S. military or NATO.
Rather they are calibers developed by the former Soviet Union which are now commonly used by civilian shooters in the United States.
The 7.62x39mm in particular is extremely popular with private gun owners due to the wide availability and affordability of both military surplus ammo and firearms chambered for this round, such as the AK-47 and the SKS.
Handguns chambered for the 9x18mm Makarov, such as the FEG PA-63, are common, inexpensive imports.
The desired list of calibers attached to aprevious, related acquisition request also included oddball rounds such as the .303 British and the 7.62×25mm Tokarev.
In addition to this solicitation for nearly 3,000,000 live rounds of Russian calibers popular with the public, the army made a similar request last year for a long-term weapon supplier who can ship both foreign non-standard and obsolete U.S. military weapons anywhere in the world.
According to this 2012 request, the U.S. Army Armament Research, Development and Engineering Center (ARDEC) wanted to find a vendor who could "reach around the world at any given moment to gather and provide multiple types of weapons and weapon parts."
The extensive list of desired weapons included firearms popular with civilians such as the aforementioned AK-47, 1911s, M1903 Springfields, Walther PP/PPKs (another common import), and other "commercial and para-military weapons."
This solicitation also asked for "books, manuals, tools, and gauges" pertaining to the firearms.
Headquartered in New Jersey, ARDEC is primarily known for its research in advanced weapons such as lasers and nanotechnology.
These unusual requests prompt the question as to why the U.S. Army, and especially the army's advanced weapons research and development division, needs a vast quantity of non-NATO rounds and decades-old – sometimes even 100-year-old – firearms popular with civilians for worldwide deployment "at any given notice."
The ARDEC request in particular seems too broad.
Are World War I era M1903 Springfields really that common in today's battlefields, or even the popular CZ-52 imports which have been retired from Czechoslovakian service since 1982?
Are these obsolete weapons used that frequently in current world conflicts to warrant specific mention in an army acquisition request?
Do century-old firearms really need to be shipped all around the world for "research and development?"
What about the huge purchase of 425,000 9x18mm Makarov rounds?
Are they going to somehow end up in the sidearms of Obama-backed Syrian rebels, especially aftertwo congressional panels cleared the way for shipping small arms to Syria?
Handguns chambered in 9x18mm Makarov are still commonplace among Syrian militants becauseSyria received military aid from the Soviet Union for over 20 years.
These solicitations, with planned acquisitions ranging between $500,000 to $22,000,000, definitely forge fears of back door gun control by creating artificial scarcity that denies Americans access to a wide-range of firearms and ammo, especially in the wake of the U.N. arms trade treaty which was signed by Obama but rejected by the Senate.
Regardless of the army's intentions, these large-scale purchases will only further intensify firearm and ammo shortages for gun owners across the country.
Read Original: http://www.infowars.com/u-s-army-to-buy-millions-of-russian-rounds-of-ammo-and-popular-civilian-firearms/
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26JUL2013: Pentagon to deploy huge blimps over Washington, DC for 360-degree surveillance
26JUL2013 – Shocking 'Extermination' Fantasies By the People Running America's Empire on Full Display at Aspen Summit | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,564 |
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A love story and murder mystery based on the most notorious unsolved murder case in New York history. The original screenplay uses newly discovered facts, court records and speculation as the foundation for a story of family, obsession, love and loss.
George Tillman Jr.
Dwayne Johnson, Billy Bob Thornton, Maggie Grace .....
A lone man, recently released from prison, seeks out the people from his past who he believes betrayed him. While he forces each to atone for their sins, he must also face what meting out vengeance will do to his immortal soul. Meanwhile, hes not the only one looking to settle old scores. Hired by the man responsible for the events which lead him to prison,
France, USA, Spain, UK
Action, Drama, Thriller
Matt Damon, Jason Isaacs, Greg Kinnear .....
Following the American invasion of Iraq in 2003 Chief Warrant Officer Roy Miller and his men are charged with finding the so-called weapons of mass destruction, whose existence justified American involvement, according to the Pentagon and their man in Baghdad, Poundstone. Veteran CIA operative Marty tells Miller that there are no weapons, it is a deception
Biography, Drama, Music
Kristen Stewart, Dakota Fanning, Michael Shannon .....
In 1975, San Fernando Valley teen Joan Larkin reinvents herself as Joan Jett, a guitarist who wants to form an all-girl punk band. She pitches the idea to a sleazy manager, Kim Fowley, who pairs her with a drummer and then searches for a face: he finds Cherie Currie, at age 15, the perfect jailbait image for his purpose; by luck, she can sing. Two others
Helen Mirren, Sam Worthington, Tom Wilkinson .....
The espionage thriller begins in 1997, as shocking news reaches retired Mossad secret agents Rachel (Helen Mirren) and Stefan (Tom Wilkinson) about their former colleague David (Ciarán Hinds). All three have been venerated for decades by their country because of the mission that they undertook back in 1965, when the trio (portrayed, respectively, by
Comedy, Drama, Music
Joaquin Phoenix, Antony Langdon, Carey Perloff .....
In 2008 while rehearsing for a charity event, actor Joaquin Phoenix, with Casey Afflecks camera watching, tells people hes quitting to pursue a career in rap music. Over the next year, we watch the actor write, rehearse, and perform to an audience. He importunes Sean Combs in hopes hell produce the record. We see the actor in his home: he parties, smokes,
Lisa Cholodenko
Annette Bening, Julianne Moore, Mark Ruffalo .....
Nic and Jules are in a long term, committed, loving but by no means perfect same-sex relationship. Nic, a physician, needs to wield what she believes is control, whereas Jules, under that control, is less self-assured. During their relationship, Jules has floundered in her nine to five life, sometimes trying to start a business - always unsuccessfully - or
Isaac Florentine
Scott Adkins, Mykel Shannon Jenkins, Mark Ivanir .....
Russian inmate Boyka, now severely hobbled by the knee injury suffered at the end of Undisputed 2. No longer the feared prison fighter he was, he has declined so far that he is now good only for cleaning toilets. But when a new prison fight tournament begins - an international affair, matching the best fighters from prisons around the globe, enticing them
Action, Adventure, Drama
Neil Marshall
Michael Fassbender, Dominic West, Olga Kurylenko .....
Britain, A.D. 117. Quintus Dias, the sole survivor of a Pictish raid on a Roman frontier fort, marches north with General Virilus legendary Ninth Legion, under orders to wipe the Picts from the face of the Earth and destroy their leader, Gorlacon.
David Michod
James Frecheville, Guy Pearce, Joel Edgerton .....
Despite being no saint herself, Julia Cody has shielded her seventeen year old son, Joshua J Cody, from her Melbourne-based criminal relatives who they have not seen in years. After Julia dies in front of Js eyes from a self-inflicted heroin overdose, J, who is slightly detached from life, feels he has no choice but to contact his maternal grandmother,
Massy Tadjedin
Keira Knightley, Sam Worthington, Eva Mendes .....
Joanna accuses her husband, Michael, of being attracted to his co-worker. Wanting to make-up before he goes on a business trip, she assumes she over-reacted. But when they are both away from each other for one night, Michael ends up testing his loyalty to Joanna and his attraction to Laura. And Joanna tests her honesty to Michael when her ex-boyfriend Alex
Step Up 3D
Drama, Music, Romance
Jon M. Chu
Sharni Vinson, Rick Malambri, Adam G. Sevani .....
A tight-knit group of New York City street dancers, including Luke (Malambri) and Natalie (Vinson), team up with NYU freshman Moose (Sevani), and find themselves pitted against the worlds best hip hop dancers in a high-stakes showdown that will change their lives forever.
Mel Gibson, Ray Winstone, Danny Huston .....
Thomas Craven is a detective who has spent years working the streets of Boston. When his own daughter is killed outside his own home, Craven soon realizes that her death is only one piece of an intriguing puzzle filled with corruption and conspiracy, and it falls to him to discover who is behind the crime.
Rachel McAdams, Harrison Ford, Diane Keaton .....
Becky (Rachel McAdams) is a hard-working morning TV show producer, or at least she was until she got fired. Desperate to get a job, she finally gets an interview with Jerry (Jeff Goldblum) - who is desperate to hire a producer for the struggling show Daybreak. Becky accepts the job and it proves to be more difficult than even she might be able to handle.
Reese Witherspoon, Paul Rudd, Owen Wilson .....
Star softball player, Lisa, has just been cut from the national team; Scholarly business man, George, has just been indicted from his fathers company. With everything that they know in their lives taken from them, Lisa and George attempt to find romance. Lisas potential boyfriend, Matty, however, is as clueless and perpetually single as they come, and
Naomi Watts, Sean Penn, Sonya Davison .....
Plames status as a CIA agent was revealed by White House officials allegedly out to discredit her husband after he wrote a 2003 New York Times op-ed piece saying that the Bush administration had manipulated intelligence about weapons of mass destruction to justify the invasion of Iraq.
The Call of the Wild (Watch movies online)
Adventure, Drama, Family
Karen Gillan, Cara Gee, Harrison Ford, Dan Stevens, Jean Louisa Kelly, Bradley Whitford, Wes Brown, Colin Woodell, Omar Sy, Scott MacDonald .....
A sled dog struggles for survival in the Alaskan wild.
In the Heights (Watch movies online)
Drama, Music, Musical, Romance
Corey Hawkins, Lin-Manuel Miranda, Stephanie Beatriz, Anthony Ramos, Jimmy Smits, Dascha Polanco, Melissa Barrera, Leslie Grace, Ariana Greenblatt, Marc Anthony .....
In the Heights centers on a variety of characters living in the neighborhood of Washington Heights, on the northern tip of Manhattan. At the center of the show is Usnavi, a bodega owner who looks after the aging Cuban lady next door, pines for the gorgeous girl working in the neighboring beauty salon and dreams of winning the lottery and escaping to the
Ghostbusters: Afterlife (Watch movies online)
Action, Comedy, Drama, Fantasy, Horror, Sci-Fi
Ernie Hudson, Mckenna Grace, Carrie Coon, Paul Rudd, Bill Murray, Annie Potts, Finn Wolfhard, Sigourney Weaver, Dan Aykroyd, Bokeem Woodbine .....
When a single mom and her two kids arrive in a small town, they begin to discover their connection to the original Ghostbusters and the secret legacy their grandfather left behind.
The Other Lamb (Watch movies online)
Malgorzata Szumowska
Raffey Cassidy, Michiel Huisman, Denise Gough, Eve Connolly, Kelly Campbell, Isabelle Connolly, Irene Kelleher, Charlotte Moore, Jane Herbert, Ailbhe Cowley .....
Life with Shepherd (
Port Authority (Watch movies online)
Danielle Lessovitz
Fionn Whitehead, Leyna Bloom, McCaul Lombardi, Louisa Krause, Eddie Plaza, William Dufault, Stephen Cavalieri, Taliek Jeqon, Devon Carpenter, Azza Melton .....
Port Authority is a love story set in New Yorks kiki ballroom scene, and follows Paul, a 20 year old midwesterner, who arrives at the central bus station and quickly catches eyes with Wye, a 22 year old girl voguing on the sidewalk. After Paul seeks her out in secret, an intense love between them blossoms. But when Paul discovers Wye is trans, he is forced
Buoyancy (Watch movies online)
Rodd Rathjen
Sarm Heng, Thanawut Kasro, Mony Ros, Saichia Wongwirot, Yothin Udomsanti, Chan Visal, Chheung Vakhim, Terb Sophon, Phouen Nisa, Sareoun Sopheara .....
A 14-year-old Cambodian boy leaves home in search of a better life but is sold to a Thai broker and enslaved on a fishing trawler. As fellow slaves are tortured and murdered around him, he realises his only hope of freedom is to become as violent as his captors.
The Warrior Queen of Jhansi (Watch movies online)
Swati Bhise
Rupert Everett, Jodhi May, Derek Jacobi, Nathaniel Parker, Ben Lamb, Devika Bhise, Swati Bhise, R. Bhakti Klein, Glenn Webster, Omar Malik .....
The Warrior Queen of Jhansi tells the story of Lakshmibai, the historic Queen of Jhansi who fiercely led her army against the British East India Company in the infamous mutiny of 1857. From Queen Elizabeth to Queen Victoria, two-and-a half-centuries of East India Company rule will be reversed by its attempt to crush Indias Warrior Queen. Lakshmibai is known
Mickey and the Bear (Watch movies online)
Camila Morrone, James Badge Dale, Calvin Demba, Ben Rosenfield, Rebecca Henderson, Rob Grabow, Gabriel Vega .....
Faced with the responsibility to take care of her addict, veteran father, headstrong teen Mickey Peck keeps her household afloat. When she has the opportunity to leave for good, she must choose between familial obligation and personal fulfillment.
In the Tall Grass (Watch movies online)
Drama, Horror, Thriller
Laysla De Oliveira, Avery Whitted, Patrick Wilson, Will Buie Jr., Harrison Gilbertson, Tiffany Helm, Rachel Wilson .....
When siblings Becky and Cal hear the cries of a young boy lost within a field of tall grass, they venture in to rescue him, only to become ensnared themselves by a sinister force that quickly disorients and separates them. Cut off from the world and unable to escape the fields tightening grip, they soon discover that the only thing worse than getting lost
The Personal History of David Copperfield (Watch movies online)
Armando Iannucci
Dev Patel, Tilda Swinton, Sophie McShera, Gwendoline Christie, Hugh Laurie, Ben Whishaw, Aneurin Barnard, Anna Maxwell Martin, Benedict Wong, Peter Capaldi .....
Based on the novel by Charles Dickens.
Cuck (Watch movies online)
Zachary Ray Sherman, Sally Kirkland, Timothy V. Murphy, Monique Parent, David Diaan, Hugo Armstrong, Albert Abraham, Travis Hammer, Jessica Jade Andres, Patrick Malone .....
CUCK is a gripping fly-on-the wall character study and psychosexual thriller tracking an isolated and lonely young man as he is indoctrinated into the online echo chambers of Alt-Right ideology and hate. Touching on several topical nerves, CUCK gives a raw, unforgiving glimpse into a dark world that hides in plain sight.
The Birdcatcher (Watch movies online)
Drama, History, Thriller, War
Ross Clarke
August Diehl, Laura Birn, Sarah-Sofie Boussnina, Jakob Cedergren, Anders Baasmo Christiansen, Johannes Kuhnke, Jonas Hoff Oftebro, Arthur Hakalahti, Christopher Dane, Tibo Vandenborre .....
On her attempt to flee the Nazi round-up in Norway, Esther finds herself alone, on an occupied farm forced to conceal her identity; leading to a series of choices and consequences which shift the paths of those around her. This story uncovers a hidden slice of history that grips at the heart and inspires us all at the deepest level.
Earthquake Bird (Watch movies online)
Crime, Drama, Mystery, Romance, Thriller
Alicia Vikander, Riley Keough, Jack Huston, Kiki Sukezane, Ken Yamamura, Crystal Kay, Yurino, Naoki Kobayashi, Akiko Iwase, Kenta Ogawa .....
Lucy ([link=nm2539953)) is a buttoned-up, MUJI-clad translator who has been in Japan for five years and is desperate to blend in. Her reticent nature hides old scars. Theres also the fact that people around her keep dying - accidentally or otherwise. After a street encounter with hot photographer Teiji (
Ema (Watch movies online)
Pablo Larraín
Santiago Cabrera, Mariana Di Girolamo, Josefina Fiebelkorn, Giannina Fruttero, Gael García Bernal, Paola Giannini, Antonia Giesen, Susana Hidalgo, Paula Hofmann, Mariana Loyola .....
A couple deals with the aftermath of an adoption that goes awry as their household falls apart.
Swallow (Watch movies online)
Carlo Mirabella-Davis
Haley Bennett, Austin Stowell, Denis OHare, Elizabeth Marvel, David Rasche, Luna Lauren Velez, Zabryna Guevara, Laith Nakli, Babak Tafti, Nicole Kang .....
Hunter, a newly pregnant housewife, finds herself increasingly compelled to consume dangerous objects. As her husband and his family tighten their control over her life, she must confront the dark secret behind her new obsession.
The Cave (Watch movies online)
Thailand, Ireland
Adventure, Drama, Thriller
Tom Waller
Ron Smoorenburg, Kelly B. Jones, Bobby Gerrits, Eoin OBrien, Ting Sue, Lawrence de Stefano, Gigi Velicitat, James Edward Holley, Valerie Bentson, Zeb Moore .....
When a soccer team of 12 boys, aged 11-16, and their 25-year-old coach is trapped deep inside a cave in Northern Thailand, thousands of volunteers and soldiers from around the world unite in a race against time to find them. Once the boys are found alive ten days later, the only way out is an impossible five-hour dive-swim that only expert cave divers could
O que arde (Watch movies online)
Spain, France, Luxembourg
Oliver Laxe
Amador Arias, Benedicta Sánchez, Ivan Yañez, Inazio Abrao, Elena Mar Fernández, Alvaro de Bazal, Luis Manuel Guerrero Sánchez, Rubén Gómez Coelho, David de Peso, Nando Vázquez .....
Amador is a notorious Galician arsonist who has been accused of causing a new fire. Lois, a young firefighter, explores the depths of a forest on fire. Their destinies are linked by the power of a mysterious fire.
Watch Movies Online - Full Series
Stream Full Movies | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,953 |
Das Richardson Bluff ist ein steilwandiges Felsenkliff an der Pennell-Küste des ostantarktischen Viktorialands. In den Anare Mountains ragt es gegenüber dem Frecker Ridge an der Ostflanke des Kirkby-Gletschers auf.
Wissenschaftler der Australian National Antarctic Research Expeditions (ANARE) nahmen die Benennung vor. Namensgeber ist Sergeant Alan Keith Richardson (* 1930) von der Royal Australian Air Force, Besatzungsmitglied beim Antarktisflug im Rahmen der ANARE-Kampagne mit dem Schiff Thala Dan zur Erkundung der Oates-Küste im Jahr 1962.
Weblinks
(englisch)
Richardson Bluff auf geographic.org (englisch)
Geographie (Antarktika)
Kliff
Anare Mountains
Pennell-Küste | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,270 |
Q: javascript: Clear all timeouts? Is there a way to clear all time outs from a given window? I suppose the timeouts are stored somewhere in the window object but couldn't confirm that.
Any cross browser solution is welcome.
A: You have to rewrite the window.setTimeout method and save its timeout ID.
const timeouts = [];
const originalTimeoutFn = window.setTimeout;
window.setTimeout = function(fun, delay) { //this is over-writing the original method
const t = originalTimeoutFn(fn, delay);
timeouts.push(t);
}
function clearTimeouts(){
while(timeouts.length){
clearTimeout(timeouts.pop());
}
}
A: To clear all timeouts they must be "captured" first:
Place the below code before any other script and it will create a wrapper function for the original setTimeout & clearTimeout.
New clearTimeouts methods will be added to the window Object, which will allow clearing all (pending) timeouts (Gist link).
Other answers lack complete support for possible arguments setTimeout might receive.
// isolated layer wrapper (for the local variables)
(function(_W){
var cache = [], // will store all timeouts IDs
_set = _W.setTimeout, // save original reference
_clear = _W.clearTimeout // save original reference
// Wrap original setTimeout with a function
_W.setTimeout = function( CB, duration, arg ){
// also, wrap the callback, so the cache reference will be removed
// when the timeout has reached (fired the callback)
var id = _set(function(){
removeCacheItem(id)
CB.apply(null, arguments)
}, duration || 0, arg)
cache.push(id) // store reference in the cache array
// id reference must be returned to be able to clear it
return id
}
// Wrap original clearTimeout with a function
_W.clearTimeout = function( id ){
_clear(id)
removeCacheItem(id)
}
// Add a custom function named "clearTimeouts" to the "window" object
_W.clearTimeouts = function(){
console.log("Clearing " + cache.length + " timeouts")
cache.forEach(n => _clear(n))
cache.length = []
}
// removes a specific id from the cache array
function removeCacheItem( id ){
var idx = cache.indexOf(id)
if( idx > -1 )
cache = cache.filter(n => n != id )
}
})(window);
// lets define some timeouts
setTimeout(()=> console.log('1s passed'), 1000); // should run
setTimeout(()=> console.log('2s passed'), 2000); // should be cleared
setTimeout(()=> console.log('3s passed'), 3000); // should be cleared
// lets clear them all after 1 and a half second:
setTimeout(()=> {
window.clearTimeouts()
}, 1500)
See a nice piece of code which I named "better-timeout" which does a bit more than the above.
A: For completeness, I wanted to post a general solution that covers both setTimeout and setInterval.
It seems browsers might use the same pool of IDs for both, but from some of the answers to Are clearTimeout and clearInterval the same?, it's not clear whether it's safe to rely on clearTimeout and clearInterval performing the same function or only working on their respective timer types.
Therefore, when the goal is to kill all timeouts and intervals, here's an implementation that might be slightly more defensive across implementations when unable to test all of them:
function clearAll(windowObject) {
var id = Math.max(
windowObject.setInterval(noop, 1000),
windowObject.setTimeout(noop, 1000)
);
while (id--) {
windowObject.clearTimeout(id);
windowObject.clearInterval(id);
}
function noop(){}
}
You can use it to clear all timers in the current window:
clearAll(window);
Or you can use it to clear all timers in an iframe:
clearAll(document.querySelector("iframe").contentWindow);
A: They are not in the window object, but they have ids, which (afaik) are consecutive integers.
So you may clear all timeouts like so:
var id = window.setTimeout(function() {}, 0);
while (id--) {
window.clearTimeout(id); // will do nothing if no timeout with id is present
}
A: This is very late... but:
Basically, setTimeout/setInterval ID's go in consecutive order, so just create a dummy timeout function to get the highest ID, then clear interval on all the IDs lower than that.
const highestId = window.setTimeout(() => {
for (let i = highestId; i >= 0; i--) {
window.clearInterval(i);
}
}, 0);
A: I have an addition to Pumbaa80's answer that might be useful for someone developing for old IEs.
Yes, all major browsers implement timeout ids as consecutive integers (which is not required by specification). Althrough the starting number differs form browser to browser. It seems that Opera, Safari, Chrome and IE > 8 starts timeout ids from 1, Gecko-based browsers from 2 and IE <= 8 from some random number that is magically saved across tab refresh. You can discover it yourself.
All that meens that in IE <=8 the while (lastTimeoutId--) cycle may run over 8digits times and show the "A script on this page is causing Internet Explorer to run slowly" message. So if you can not save all you timeout id's or don't want to override window.setTimeout you may consider saving the first timeout id on a page and clearing timeouts until it.
Execute the code on early page load:
var clearAllTimeouts = (function () {
var noop = function () {},
firstId = window.setTimeout(noop, 0);
return function () {
var lastId = window.setTimeout(noop, 0);
console.log('Removing', lastId - firstId, 'timeout handlers');
while (firstId != lastId)
window.clearTimeout(++firstId);
};
});
And then clear all pending timeouts that probably was set by foreign code so many times you want
A: I think the easiest way to accomplish this would be to store all the setTimeout identifiers in one array, where you can easily iterate to clearTimeout() on all of them.
var timeouts = [];
timeouts.push(setTimeout(function(){alert(1);}, 200));
timeouts.push(setTimeout(function(){alert(2);}, 300));
timeouts.push(setTimeout(function(){alert(3);}, 400));
for (var i=0; i<timeouts.length; i++) {
clearTimeout(timeouts[i]);
}
A: How about store the timeout ids in a global array, and define a method to delegate the function call to the window's.
GLOBAL={
timeouts : [],//global timeout id arrays
setTimeout : function(code,number){
this.timeouts.push(setTimeout(code,number));
},
clearAllTimeout :function(){
for (var i=0; i<this.timeouts.length; i++) {
window.clearTimeout(this.timeouts[i]); // clear all the timeouts
}
this.timeouts= [];//empty the id array
}
};
A: I use Vue with Typescript.
private setTimeoutN;
private setTimeoutS = [];
public myTimeoutStart() {
this.myTimeoutStop();//stop All my timeouts
this.setTimeoutN = window.setTimeout( () => {
console.log('setTimeout');
}, 2000);
this.setTimeoutS.push(this.setTimeoutN)//add THIS timeout ID in array
}
public myTimeoutStop() {
if( this.setTimeoutS.length > 0 ) {
for (let id in this.setTimeoutS) {
console.log(this.setTimeoutS[id]);
clearTimeout(this.setTimeoutS[id]);
}
this.setTimeoutS = [];//clear IDs array
}
}
A: Use a global timeout which all of your other functions derive timing from. This will make everything run faster, and be easier to manage, although it will add some abstraction to your code.
A: We've just published a package solving this exact issue.
npm install time-events-manager
With that, you can view all timeouts and intervals via timeoutCollection & intervalCollection objects.
There's also a removeAll function which clears all timeouts/intervals both from the collection and the browser.
A: Inside the setTimeout after the definition of function f(...){} and timeout can be additional parameters.
For example: $setTimeout( function(a, b){ run(a); run(b); }, 100, a, b); )
...args solves that problem.
var $timeouts = new Array();
function $setTimeout(...args)
{
var t = window.setTimeout(...args);
$timeouts.push(t);
return t;
}
function $clearTimeout(id)
{
if( $timeouts.indexOf(id) > -1 )
$timeouts = $timeouts.filter(n => n != id )
window.clearTimeout(id);
}
function $clearTimeouts()
{
while($timeouts.length)
window.clearTimeout($timeouts.pop());
}
For old browsers you need to use another methods to pass parameters and to delete a value from the array $timeouts.
var $timeouts = new Array();
function $setTimeout()
{
var t = window.setTimeout.apply( this, arguments );
$timeouts.push(t);
return t;
}
function $clearTimeout(id)
{
var index = $timeouts.indexOf(id);
if (index > -1) {
$timeouts.splice(index, 1);
}
window.clearTimeout(id);
}
function $clearTimeouts()
{
while($timeouts.length)
window.clearTimeout($timeouts.pop());
}
A: The answers of the other are actually correct. But for me this is how approach this kind of stuff. Check my code bellow.
// example when I person types on search input
function typeOnSearch(time = 3000) {
// this will clear timeout
clearTimeout(window.typeOnSearchTimeOut);
// we create a timeout variable and added it to window, this way, we can access this in any function in our app.
window.typeOnSearchTimeOut = setTimeout( () => {
//enter what you like to do here, like fetch data
}, time);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,168 |
Q: What's the maximum size of Web.config? When I tried to open my SharePoint extended site, I received error 500. The error message was:
Cannot read configuration file because it exceeds the maximum file size
My web.config file was just about 260 KB, because it had so many <SafeControl> entries in it. After removing some of the old entries, and getting the file down to 186 KB, my site started to load correctly.
My question:
*
*What is the maximum size of web.config?
*What can be done to avoid such problems in the future?
A: Its not hard coded its defined in
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\InetStp\Configuration\MaxWebConfigFileSizeInKB
or on 64 Bit machines
HKEY_LOCAL_MACHINE\Software\Wow6432Node\Microsoft\InetStp\Configuration\MaxWebConfigFileSizeInKB
The "cutoff point" by default, unless you configure the MaxWebConfigFileSizeInKB appears to be 250Kb
Remember you can always split your web.config files into multiple little files to get around this issue,
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,202 |
Q: Is it possible to set css rules based on value of non-sibling elements? Say you have this code:
<div class="hello">
<div class="cat dog"/>
</div>
<div class="notASibling" />
is it possible to set a property of notASibling, based on the class value of an element that is not a sibling? For example, in the above code I may want to set the display property of notASibling to hidden if the hello > cat class is not a dog.
If notASibling was a sibling of "cat dog", I could simply say:
.cat:not(.dog) ~ .notASibling{ display:none; }
A: This isn't possible in CSS. Your only real option here would be to use Javascript. The jQuery library is based around the idea of using enhanced CSS selectors for selecting elements to manipulate them, and you could do that here:
prevWasADog = $(".notASibling").prev().children('.cat').eq(0).is('.dog');
A: The answer is no. Spontaneously, you might think about combining the child combinator and sibling combinator, i.e. like this:
.notASibling ~ .hello > .cat:not(.dog) { display: none }
But this is not possible since in the CSS 3 selectors definition (http://www.w3.org/TR/css3-selectors/) selectors may only use combinators with "simple selector sequences" as their operands, not other selectors. So if you follow the W3C's grammar rules you can only use one combinator within a selector.
A: You can't reach a non-sibling in pure CSS. You can do this in JavaScript. If you want a pure CSS solution, consider refactoring your DOM a bit. Maybe something like this:
<div class='dog'>
<div class='hello'>
<div />
</div>
<div class='notASibling' />
</div>
.cat div.notASibling {
/* special styles for cats */
}
.dog div.notASibling {
/* special styles for dogs */
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,309 |
Cepora lichenosa is een vlindersoort uit de familie van de Pieridae (witjes), onderfamilie Pierinae.
Cepora lichenosa werd in 1877 beschreven door Moore.
Witjes | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,147 |
Baden Baden, Germany, Nov. 9.
she should inherit her fortune.
position iu Washington as a lawyer.
and the child stayed several months.
Hau in the mentiine weut to Paris.
Both men are past 60 years of age.
homestead land near Three Lakes.
McCormick of Wippany road. Mrs.
and when Miss McCormick saw it,"
senatorial situation in South Dakota.
defeat of Senator Gamble is possible.
republican votes in the legislature.
port have been given him.
ing of a divorcewill, be handed down.
Washington, Nov. 9.Dr. Charles W.
tess did not intend to fly to America.
Fort Meade on the 14th.
child, who also had been Suffocated.
day. National Chairman Charles E.
Swallow for president in 1904 was 25S""
few hours before the accident.
ready:rcontrol the supply of beef, mut-"
washing soap, toilet soap'and perfumes.
and Claude Johnson would have figured..
Minnesota officials are looking for him.
admission, Johnson shot thru the" door"
his family. He is in excellent health.
slated to step into 'Mr. Lincoln's shoes.
one onMreaTnesday and the last on'.
fresh air, Is given, and nourishing food being supplied.
taining opiates, as narcotics-only stultify the lungs and render them more"
Whiskey, a half pint. Use ltt teaspoonfui does every/four hours.
stretch' out o'n the flour and go to sleep.
The State Will get more or leSS'Decauseithe ninth district by 1,376 majority.
sons werjB .doit*-, tine. rcKofesiat-riS&tomiSt.
Des Moines, Nov. 9.Edward B.
of new pasteurizers and "churns.
while a third supervises students in charge'bf!
county the republicans elected a straight ticket.
W. B. Johnson, Des Moines treasurer, F. M.
Lake company have been filed with the register.
auder Stewart and others, $61,075.
wolf before:-it could again attack his comrade.
in one of the shelves.
Fargo, N. D., Nov. 9.-~-The case bf Major J.
trial will probably continue three to four weeks.
is -to sympathize with the defendant.
found wandering north of Fargo by a farmer.
was dismissed before the expiration of ils term.
He sued for salary and was given a judgment.
front of the opponents of the canal.
the canal on better terms. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,767 |
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<h1 class="page-title">Index</h1>
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<section>
<article><h1>Amazon Mobile Analytics SDK for JavaScript</h1>
<p><strong>Developer Preview:</strong> We welcome developer feedback on this project. You can reach us by creating an issue on the
GitHub repository or posting to the Amazon Mobile Analytics forums:
<em> <a href="https://github.com/aws/aws-sdk-mobile-analytics-js">https://github.com/aws/aws-sdk-mobile-analytics-js</a>
</em> <a href="https://forums.aws.amazon.com/forum.jspa?forumID=174">https://forums.aws.amazon.com/forum.jspa?forumID=174</a></p>
<h1>Introduction</h1>
<p>The Mobile Analytics SDK for JavaScript allows JavaScript enabled applications to create and submit events for analysis in the AWS Console and through Auto Export to S3 and Redshift. The library uses the browser's local storage API to create a local cache for the data, allowing your web application to batch and record events even when the app is offline.</p>
<h2>Setup</h2>
<ol>
<li><p>Download and include the AWS JavaScript SDK (minimum version 2.1.18):</p>
<ul>
<li><a href="http://aws.amazon.com/sdk-for-browser/">http://aws.amazon.com/sdk-for-browser/</a></li>
</ul>
</li>
<li><p>Download and include the Amazon Mobile Analytics SDK for JavaScript:</p>
<ul>
<li><a href="https://raw.githubusercontent.com/aws/aws-sdk-mobile-analytics-js/master/dist/aws-sdk-mobile-analytics.min.js">/dist/aws-sdk-mobile-analytics.min.js</a></li>
</ul>
</li>
</ol>
<pre class="prettyprint">
<script src="/js/aws-sdk.min.js"></script>
<script src="/js/aws-sdk-mobile-analytics.min.js"></script>
</pre>
<h2>Usage</h2>
<p><strong>Step 1.</strong> Log in to the <a href="https://console.aws.amazon.com/mobileanalytics/home/?region=us-east-1">Amazon Mobile Analytics management console</a> and create a new app. Be sure to note your App Id and Cognito Identity Pool Id.
* <a href="https://console.aws.amazon.com/mobileanalytics/home/?region=us-east-1">https://console.aws.amazon.com/mobileanalytics/home/?region=us-east-1</a></p>
<p><strong>Step 2.</strong> Initialize the credentials provider using a Cognito Identity Pool ID. This is necessary for the AWS SDK to manage authentication to the Amazon Mobile Analytics REST API.</p>
<pre class="prettyprint">
AWS.config.region = 'us-east-1';
AWS.config.credentials = new AWS.CognitoIdentityCredentials({
IdentityPoolId: COGNITO_IDENTITY_POOL_ID //Required e.g. 'us-east-1:12345678-c1ab-4122-913b-22d16971337b'
});
</pre>
<p><strong>Step 3.</strong> Instantiate the Mobile Analytics Manager, including your App ID generated in Step 1, above. Session events will be automatically recorded and the client will batch and automatically submit events to Amazon Mobile Analytics every 10 seconds.</p>
<pre class="prettyprint">
var options = {
appId : MOBILE_ANALYTICS_APP_ID //Required e.g. 'c5d69c75a92646b8953126437d92c007'
};
mobileAnalyticsClient = new AMA.Manager(options);
</pre>
<p>To manually force an event submission you can call:</p>
<pre class="prettyprint">
mobileAnalyticsClient.submitEvents();
</pre>
<h2>Additional Options</h2>
<h3>Custom Events</h3>
<p>You can optionally add custom events to capture additional information you find valuable.</p>
<pre class="prettyprint">
mobileAnalyticsClient.recordEvent('CUSTOM EVENT NAME', {
'ATTRIBUTE_1_NAME': 'ATTRIBUTE_1_VALUE',
'ATTRIBUTE_2_NAME': 'ATTRIBUTE_2_VALUE'
/* ... */
}, {
'METRIC_1_NAME': 1,
'METRIC_2_NAME': 99.3
/* ... */
});
</pre>
<h3>Session Settings</h3>
<p>By default a session lasts 10 minutes. You can override this default setting when initializing the Mobile Analytics Manager by including "sessionLength" in the "options" object.</p>
<pre class="prettyprint">
var options = {
appId : MOBILE_ANALYTICS_APP_ID,
sessionLength: 300000 //Session Length in milliseconds. This will evaluate to 5min.
};
mobileAnalyticsClient = new AMA.Manager(options);
</pre>
<p>A session's timeout can also be updated to allow for continuation of a session.</p>
<pre class="prettyprint">
//This will set the current session to expire in 5 seconds from now.
mobileAnalyticsClient.resetSessionTimeout(5000);
//This will reset the current session's expiration time using the time specified during initialization.
//If the default setting was used (10 minutes) then the session will expire 10 minutes from now.
mobileAnalyticsClient.resetSessionTimeout();
</pre>
<h3>Record Monetization Event</h3>
<p>You can record monetization events to enable reports such as Average Revenue Per User (ARPU) and more.</p>
<pre class="prettyprint">
mobileAnalyticsClient.recordMonetizationEvent(
{
productId : PRODUCT_ID, //Required e.g. 'My Example Product'
price : PRICE, //Required e.g. 1.99
quantity : QUANTITY, //Required e.g. 1
currency : CURRENCY_CODE //Optional ISO currency code e.g. 'USD'
},
{/* Custom Attributes */},
{/* Custom Metrics */}
);
</pre>
<h3>Add App Details to Events</h3>
<p>Additional app and environment details can be added to the "options" object when initializing the SDK. These details will be captured and applied to all events and can be useful if using Auto Export for custom analysis of your data.</p>
<pre class="prettyprint">
var options = {
appId : MOBILE_ANALYTICS_APP_ID, //Required e.g. 'c5d69c75a92646b8953126437d92c007'
appTitle : APP_TITLE, //Optional e.g. 'Example App'
appVersionName : APP_VERSION_NAME, //Optional e.g. '1.4.1'
appVersionCode : APP_VERSION_CODE, //Optional e.g. '42'
appPackageName : APP_PACKAGE_NAME, //Optional e.g. 'com.amazon.example'
make : DEVICE_MAKE, //Optional e.g. 'Amazon'
model : DEVICE_MODEL, //Optional e.g. 'KFTT'
platform : DEVICE_PLATFORM, //Required valid values: 'Android', 'iPhoneOS', 'WindowsPhone', 'Blackberry', 'Windows', 'MacOS', 'Linux'
platformVersion : DEVICE_PLATFORM_VER //Optional e.g. '4.4'
};
mobileAnalyticsClient = new AMA.Manager(options);
</pre>
<p>Please note, if device details are not specified Amazon Mobile Analytics will make best efforts to determine these values based on the User-Agent header value. It is always better to specify these values during initialization if they are available. </p>
<h3>Further Documentation</h3>
<p>Further documentation and advanced configurations can be found here:</p>
<p><a href="https://aws.github.io/aws-sdk-mobile-analytics-js/doc/AMA.Manager.html">https://aws.github.io/aws-sdk-mobile-analytics-js/doc/AMA.Manager.html</a></p>
<h2>Network Configuration</h2>
<p>The Amazon Mobile Analytics JavaScript SDK will make requests to the following endpoints
<em> For Event Submission: "<a href="https://mobileanalytics.us-east-1.amazonaws.com">https://mobileanalytics.us-east-1.amazonaws.com</a>"
</em> For Cognito Authentication: "<a href="https://cognito-identity.us-east-1.amazonaws.com">https://cognito-identity.us-east-1.amazonaws.com</a>"
* This endpoint may change based on which region your Identity Pool was created in.</p>
<p>For most frameworks you can whitelist both domains by whitelisting all AWS endpoints with "*.amazonaws.com".</p>
<h2>Change Log</h2>
<p><strong>v0.9.2:</strong>
<em> Bug Fixes:
</em> Fixed data loss issue on migration from previous versions</p>
<p><strong>v0.9.1:</strong>
<em> Updated Dependency: aws-sdk-js v2.2.37
</em> Increase base delay between retries from 30ms to 3s
<em> Allow passing of configurations to the low level client via clientOptions attribute
</em> Local events from different apps are stored in different locations
<em> Improved retry strategies
</em> CorrectClockSkew is enabled by default for Sigv4 Signatures <a href="https://github.com/aws/aws-sdk-mobile-analytics-js/issues/7">per Issue#7</a>
<em> Bug Fixes:
</em> Fixed timer from being killed in cases where multiple submissions happened in under a second
<em> Fixed duplicate batch re-submission to the Mobile Analytics service
</em> Fixed delayed auto-submission of first _session.start event
* Fixed Safari throwing exception when in private browsing mode</p>
<p><strong>v0.9.0:</strong>
* Initial release. Developer preview.</p></article>
</section>
</div>
<nav>
<h2><a href="index.html">Index</a></h2><h3>Modules</h3><ul><li><a href="module-AMA.html">AMA</a></li></ul><h3>Classes</h3><ul><li><a href="AMA.Client.html">Client</a></li><li><a href="AMA.Manager.html">Manager</a></li><li><a href="AMA.Session.html">Session</a></li></ul>
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<p id="p">Move mouse pointer over me!</p>
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Q: Creating HTTP4 Endpoint dynamically is correct? I am using ProducerTemplate in my project and have some concern regarding performance
@Produce
protected ProducerTemplate template;
and use as
String response = template.requestBody("http4://hostToBeCalled/v1/dogs",MESSAGE_BODY,String.class);
In my case, the Endpoint including component(http4) is not known at the time of initialization and I am using producer template to get around it. I see the issue of using template this way because consumer(http4) will be created and destroyed each time requestBody is called.
Does anyone else also see the potential performance issue with it or could suggest the better solution?
A: As you correctly said, producertemplate may affect performance if it is not correctly handled.
you can use Recipient List which will derive endpoint at run time. You can have some logic to compute your endpoint and assign it to exchange header or property. That header/property can be used by Recipient List to derive endpoint at run time.
If you still stick with ProducerTemplate as your usecase may insisting, - for better performance, you can inject your producer template instance to a IOC container and let that container to handle its life cycle.
Ref: http://camel.apache.org/why-does-camel-use-too-many-threads-with-producertemplate.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,640 |
\section{Introduction}
Let $K$ be a field, $\bar{K}$ its separable closure,
$\mathrm{Gal}(K)=\mathrm{Gal}(\bar{K}/K)$ the (absolute) Galois group of $K$. Let
$X$ be an abelian variety over $K$, $\mathrm{End}_K(X)$ its ring of
$K$-endomorphisms and $\mathrm{End}_K^0(X):=\mathrm{End}_K(X)\otimes{\mathbf Q}$.
Let us consider the set $\mathrm{Isog}(X,K)$ of $K$-isomorphism classes of
abelian varieties $Y$ over $K$ such that there exists a
$K$-isogeny $Y \to X$.
If $\ell$ is a prime different from $\mathrm{char}(K)$ then let us
consider the set $\mathrm{Isog}(X,K,\ell)$ of $K$-isomorphism classes of
abelian varieties $Y$ over $K$ such that there exists an
$\ell$-isogeny $Y \to X$ that is defined over $K$. (Recall that
$\ell$-isogeny is an isogeny, whose degree is a power of $\ell$.)
If $m$ is a
positive integer then we write $\mathrm{Isog}_m(X,K,\ell)$ for the subset of
$\mathrm{Isog}(X,K,\ell)$ that consists of all (isomorphism) classes of $Y$
with a $K$-polarization of degree $m$.
When $K$ is finitely generated over its prime subfield, Tate
\cite{Tate} conjectured the finiteness of $\mathrm{Isog}_m(X,K,\ell)$.
This conjecture played a crucial role in the proof of Tate's
conjecture on homomorphisms and the semisimplicity of the Galois
representations in the Tate modules of abelian varieties
\cite{Tate,Serre,Parshin,ZarhinFA,ZarhinIz,ZarhinMZ1,Faltings1,Faltings2,MB,ZarhinP}.
If $K$ is finitely generated over the field ${\mathbf Q}$ of rational
numbers then it was proven by Faltings \cite{Faltings2} that
$\mathrm{Isog}(X,K)$ is finite. (See also \cite{Faltings1,ZarhinIn}.) When
$K$ is finite, the finiteness of $\mathrm{Isog}(X,K)$ is proven in
\cite{ZarhinMZ2}. (See
\cite{Parshin,ZarhinFA,ZarhinMZ1,ZarhinMZ2,MB} for a discussion of
the case when $K$ is infinite but finitely generated over a finite
field.)
The aim of this note is to discuss the situation when $K$ is an algebraic
function field in one variable over ${\mathbf C}$. (In other words, $K$ is finitely
generated
and has transcendence degree $1$ over ${\mathbf C}$.) In order to state our results, notice
that one may view $K$ as
the field of rational functions on a suitable irreducible smooth algebraic
complex curve $S$ such that $X$ becomes the generic fiber of an abelian scheme
$f:{\mathcal X}\to S$. Let $s\in S$ be a complex point of $S$, let ${\mathcal X}_s$ be the fiber of
$f$ at $s$ and $H_1({\mathcal X}_s,{\mathbf Q})$ its
first rational homology group. (Recall that ${\mathcal X}_s$ is a complex abelian variety, whose dimension
coincides with $\mathrm{dim}(X)$.) There is the natural global monodromy representation
\[\pi_1(S,s) \to \mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Q}))\]
of the fundamental group $\pi_1(S,s)$ of $S$. Deligne \cite{Deligne} proved that
this representation is completely reducible and therefore its centralizer
$D_f:=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))$ is a (finite-dimensional) semisimple ${\mathbf Q}$-algebra. On
the other hand, every $u\in \mathrm{End}_K(X)$ extends to an endomorphism of ${\mathcal X}$ and therefore induces
a certain endomorphism $u_s$ of ${\mathcal X}_s$. This gives rise to the embeddings
\[\mathrm{End}_K(X) \hookrightarrow \mathrm{End}({\mathcal X}_s) \hookrightarrow
\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q})),\]
whose composition extends by ${\mathbf Q}$-linearity to the embedding
\[\mathrm{End}_K^0(X)\hookrightarrow \mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q})),\]
whose image lies in $D_f$. Further, we identify $\mathrm{End}_K^0(X)$ with its image in $D_f$.
Our main result (Theorem \ref{main0}) may be restated as follows.
{\sl The set $\mathrm{Isog}(X,K)$ is infinite if and only if $D_f\ne \mathrm{End}_K^0(X)$}.
If $D_f\ne \mathrm{End}_K^0(X)$ then the set $\mathrm{Isog}(X,K,\ell)$ is infinite for all but finitely many
primes $\ell$ (see Theorem \ref{main}). In addition,
if $X$ is principally polarized over $K$ then the set
$\mathrm{Isog}_1(X^2,K,\ell)$ is infinite for all but finitely many primes $\ell$ congruent to $1$ modulo $4$
(see Corollary \ref{principal}).
In order to describe other results of this paper (Sect. \ref{dim4}), let us further assume that
every homomorphism over $\overline{{\mathbf C}(S)}$ between $X$ and
``constant" abelian varieties (defined over ${\mathbf C}$) is zero. If $\mathrm{dim}(X)\le 3$
then Deligne \cite{Deligne} proved that $D_f= \mathrm{End}_K^0(X)$: this implies that
in this case $\mathrm{Isog}(X,K)$ is finite. On the other hand, Faltings \cite{Faltings0}
constructed a four-dimensional $X$ with $D_f \ne \mathrm{End}_K^0(X)$; in his example(s)
$X$ is the generic fiber of an universal family of abelian fourfolds
with level $n\ge 3$ structure over a Shimura curve.
We prove that if $\mathrm{dim}(X)=4$, all endomorphisms of $X$ are defined over $K$ and $\mathrm{Isog}(X,K)$ is infinite
then $\mathrm{End}_{K}^0(X)$ is a CM-field of degree $4$ (see Theorem \ref{dim4s}).
Almost conversely, if $X$ is a fourfold, $\mathrm{End}_K^0(X)$ is a CM-field of degree $4$
and all endomorphisms of $X$ are defined over $K$ then there exists
a finite algebraic extension $L/K$ such
that $\mathrm{Isog}(X\times_{K}L, L)$ is infinite (see Theorem \ref{foursuf}).
A rather explicit example of the latter case (with infinite $\mathrm{Isog}(X\times_{K}L, L)$)
is provided by the field
$K={\mathbf C}(\lambda)$ of rational functions in independent variable $\lambda$,
the jacobian $X$ of the genus $4$ curve $y^5=x(x-1)(x-\lambda)$ and
the overfield $L={\mathbf C}(\sqrt[10]{\lambda},\sqrt[10]{\lambda-1})$ (see Example \ref{mainex}).
The paper is organized as follows. Section \ref{prelim} contains basic notation
and useful facts about abelian varieties.
In Section \ref{monod} we
discuss abelian schemes over curves and corresponding monodromy
representations.
In Section \ref{mr} we state the main results. In Sections
\ref{AS} and \ref{isab} we discuss non-isotrivial abelian schemes
and isogenies of abelian schemes respectively. The next two
sections contains the proofs of main results. Section \ref{append}
contains auxiliary results about quaternions. In Section
\ref{dim4} we deal with isogeny classes of four-dimensional
abelian varieties.
{\bf Acknowledgements}. This paper is based on beautiful ideas and
results of J. Tate \cite{Tate} and P. Deligne \cite{Deligne}. I
learned about the Tate conjecture in the early 1970s from a paper
of A. N. Parshin \cite{Parshin}. I am
grateful to Minhyong Kim for stimulating questions and to B. van
Geemen for useful comments. The final version of this paper was written during
my stay at the Steklov Mathematical Institute of the Russian Academy of
Sciences (Moscow); I am grateful to its Department of Algebra for the
hospitality.
\section{Abelian varieties}
\label{prelim}
Let $X$ be an abelian variety of positive dimension over a field $K$
of characteristic zero.
If $n$ is a
positive integer
then we
write $X_n$ for the kernel of multiplication by $n$ in $X(K_s)$. It is
well-known \cite{Mumford} that $X_n$ ia a free ${\mathbf Z}/n{\mathbf Z}$-module of rank
$2\mathrm{dim}(X)$; it is also a Galois submodule in $X(\bar{K})$. We write $K(X_n)$
for the field of definition of all points of order $n$; clearly, $K(X_n)/K$ is
a finite Galois extension, whose Galois group is canonically identified with
the image of $\mathrm{Gal}(K)$ in $\mathrm{Aut}(X_n)$. We write $\mathrm{Id}_X$ for the identity
automorphism of $X$. We write $\mathrm{End}(X)$ for the ring of all
$\bar{K}$-endomorphisms of $X$ and $\mathrm{End}^0(X)$ for the corresponding
${\mathbf Q}$-algebra $\mathrm{End}(X)\otimes{\mathbf Q}$. We have \[{\mathbf Z}\cdot \mathrm{Id}_X \subset \mathrm{End}(X)\subset
\mathrm{End}^0(X).\]
\begin{rem}
\label{isogeny} Let $X'$ be an abelian variety over $K$. If $X$
and $X'$ are $K$-isogenous then
every $K$-isogeny $X \to X'$ gives rise to
a bijection between
$\mathrm{Isog}(X,K$ and $\mathrm{Isog}(X,K')$. In
particular, the sets $\mathrm{Isog}(X,K)$ and $\mathrm{Isog}(X',K)$ are either both
finite or both infinite.
\end{rem}
Let $\ell$ be a prime.
\begin{lem}
\label{onemod4}
Suppose that $\ell$ is congruent to $1$ modulo $4$. If $X$ admits a principal
polarization over $K$ and
$\mathrm{Isog}(X,K,\ell)$ is infinite
then $\mathrm{Isog}_1(X^2,K,\ell)$ is also infinite.
\end{lem}
\begin{proof}
Clearly, every abelian
variety $Y$ over $K$ that admits an $\ell$-isogeny $Y\to X$
over ${\mathbf C}(S)$ also admits a $K$-polarization, whose degree is
a power of $\ell$. In addition, there exists an $\ell$-isogeny $X\to Y$
over $K$;
its {\sl dual} $Y^t\to X^t\cong X$ is an $\ell$-isogeny that is also defined over
$K$.
Since $\ell$ is congruent to $1$ modulo $4$ then
$\sqrt{-1}\in {\mathbf Z}_{\ell}$ and it follows from Remark 5.3.1 on pp.
314--315 of \cite{ZarhinIn} that $Y \times Y^{t}$ is principally
polarized over $K$. Clearly, there is an $\ell$-isogeny $Y \times Y^{t}\to X\times X=X^2$
that is defined over $K$. On the other hand, for any given abelian variety
$Z$ over $K$
there are, up to an isomorphism, only finitely many abelian
varieties over $K$
that are isomorphic over $K$ to an abelian subvariety of $Z$ \cite{LOZ}. It follows that
$\mathrm{Isog}_1(X^2,K,\ell)$ is {\sl infinite}.
\end{proof}
\
\begin{rem}
\label{twist}
Let $L/K$ be a finite Galois extension with Galois group
$\mathrm{Gal}(L/K)$. Let $Y$ be an abelian variety over $K$ and
$\mathrm{Aut}_L(Y)$ be its group of $L$-automorphisms. It is well-known
that $\mathrm{Aut}_L(Y)$ is an {\sl arithmetic} group; it follows from a
theorem of Borel-Serre \cite{BS} that the corresponding first noncommutative Galois cohomology set
$H^1(\mathrm{Gal}(L/K), \mathrm{Aut}_L(Y))$ is finite, i.e., the set of
($K$-isomorphism classes of)
$L/K$-forms of $Y$ is finite. This implies that if $\mathrm{Isog}(X\times_{K}L,L)$ (resp.
$\mathrm{Isog}(X\times_{K}L,L,\ell)$)
is finite then $\mathrm{Isog}(X,K)$ (resp. $\mathrm{Isog}(X,K,\ell)$) is also finite \cite{ZarhinP}.
\end{rem}
Abusing notation, we sometime write $\mathrm{Isog}(X,L,\ell)$, $\mathrm{Isog}(X,L,\ell)$ and
$\mathrm{Isog}_m(X,L,\ell)$) instead of $\mathrm{Isog}(X\times_{K}L,L,\ell)$,
$\mathrm{Isog}(X\times_{K}L,L,\ell)$ and $\mathrm{Isog}_m(X\times_{K}L,L,\ell)$ respectively.
\section{Abelian schemes and monodromy representations}
\label{monod}
Let $S$ be an irreducible smooth (but not necessarily projective) algebraic curve over
${\mathbf C}$. We write ${\mathbf C}(S)$ for the field of rational functions on $S$
and $\overline{{\mathbf C}(S)}$ for its algebraic closure. Let $f:{\mathcal X} \to S$
be a polarized abelian scheme of positive relative dimension $d$
over $ S$. Let $\eta$ be the generic point of $S$ and $X$ be the
generic fiber; it is a $d$-dimensional abelian variety over
$k(\eta)={\mathbf C}(S)$.
An abelian variety $Z$ over ${\mathbf C}(S)$ is called {\sl constant} if
there exists an abelian variety $W$ over ${\mathbf C}$ such that $Z$ is
isomorphic to $W\times_{{\mathbf C}}{\mathbf C}(S)$ over ${\mathbf C}(S)$.
Recall that ${\mathcal X}$ is called {\sl isotrivial} if there exists a
finite \'etale cover $S'\to S$ (with non-empty $S'$) such that
the pullback ${\mathcal X}_{S'}={\mathcal X}\times_S S'$ is a constant abelian scheme
over $S'$. We say that ${\mathcal X}$ is {\sl weakly isotrivial} if (under
the same assumptions on $S'\to S$) ${\mathcal X}_{S'}$ contains a non-zero
constant abelian subscheme.
\begin{thm}
\label{iso}
\begin{itemize}
\item[(i)] ${\mathcal X}$ is isotrivial if and only if there
exists an abelian variety $W$ over ${\mathbf C}$ such that the
${\mathbf C}(S)$-abelian varieties $X$ and $W\times_{{\mathbf C}}{\mathbf C}(S)$ are
isomorphic over $\overline{{\mathbf C}(S)}$. \item[(ii)]${\mathcal X}$ is weakly
isotrivial if and only if there exist an abelian variety $W$ of
positive dimension over ${\mathbf C}$ and an abelian subvariety $Z\subset
X\times_{{\mathbf C}(S)}\overline{{\mathbf C}(S)}$ such that the
$\overline{{\mathbf C}(S)}$-abelian varieties
$W\times_{{\mathbf C}}\overline{{\mathbf C}(S)}$ and $Z$ are isomorphic.
\end{itemize}
\end{thm}
We prove Theorem \ref{iso} in Section \ref{AS}.
We write $R_1 f_*{\mathbf Z}$ for the corresponding local system of the first integral
homology groups $H_1({\mathcal X}_s,{\mathbf Z})$ of the fibres ${\mathcal X}_s=f^{-1}(s)$ on $S$. If $g: {\mathcal Y}
\to S$ is another polarized abelian scheme over $S$ with generic fiber $f$ then
the natural map \[\mathrm{Hom}_S({\mathcal Y},{\mathcal X}) \to \mathrm{Hom}_{{\mathbf C}(S)}(Y,X)\] is bijective. We write
$R_1 f_*{\mathbf Q}$ for the corresponding local system $R_1 f_*{\mathbf Z}\otimes{\mathbf Q}$ of the
first rational homology groups $H_1({\mathcal X}_s,{\mathbf Q})=H_1({\mathcal X}_s,{\mathbf Z})\otimes{\mathbf Q}$ of the
fibres ${\mathcal X}_s=f^{-1}(s)$ on $S$ and $\pi_1(S,s) \to \mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z}))\subset
\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Q}))$ for the corresponding monodromy representation.
\begin{rem}
\label{kz} An abelian scheme $f:{\mathcal X}\to S$ is isotrivial if and only if the image
of the monodromy representation $\pi_1(S,s)\to\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z}))$ is finite
\cite{Katz} (see also \cite[Sect. 4.1.3.3]{Deligne}).
\end{rem}
\begin{sect} {\bf Rigidity and specialization}.
\label{rigid} It follows from the rigidity lemma \cite[Ch. 6, Sect. 1, Cor.
6.2]{GIT} that the natural homomorphism \[\mathrm{Hom}_S({\mathcal Y},{\mathcal X})\to\mathrm{Hom}({\mathcal Y}_s,{\mathcal X}_s),
\quad u \mapsto u_s\] is injective. In particular, the natural ring
homomorphism $\mathrm{End}_S({\mathcal X}) \to \mathrm{End}({\mathcal X}_s)$ is also injective. It is well-known
that the natural homomorphisms
\[\mathrm{Hom}({\mathcal Y}_s,{\mathcal X}_s)\to \mathrm{Hom}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z})), \ \mathrm{End}({\mathcal X}_s)\to
\mathrm{End}(H_1({\mathcal X}_s,{\mathbf Z}))\] are embeddings. Taking the compositions, we get the
embeddings \[\mathrm{Hom}_S({\mathcal Y},{\mathcal X})\to \mathrm{Hom}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z})), \mathrm{End}_S({\mathcal X}) \to
\mathrm{End}(H_1({\mathcal X}_s,{\mathbf Z}));\] the images of $\mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ and of $\mathrm{End}_S({\mathcal X})$ lie
in $\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z}))$ and
$\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))$ respectively. Further we will identify
$\mathrm{End}_S({\mathcal X})$ with its image in $\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))\subset
\mathrm{End}(H_1({\mathcal X}_s,{\mathbf Z}))$ and $\mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ with its image in
$\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z}))$
respectively.
We have
\[\mathrm{Hom}^0_S({\mathcal Y},{\mathcal X}):=\mathrm{Hom}_S({\mathcal Y},{\mathcal X})\otimes{\mathbf Q}\subset\]
\[\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z}))\otimes{\mathbf Q}=
\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Q}),H_1({\mathcal X}_s,{\mathbf Q})),\]
\[\mathrm{End}^0_S({\mathcal X}):=\mathrm{End}_S({\mathcal X})\otimes{\mathbf Q}\subset
\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))\otimes{\mathbf Q}= \mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q})).\]
Notice that a theorem of Grothendieck \cite[p. 60]{Grothendieck} (see also
\cite[Sect. 4.1.3.2] {Deligne}) implies that
\[\mathrm{Hom}_S({\mathcal Y},{\mathcal X})=\mathrm{Hom}({\mathcal Y}_s,{\mathcal X}_s)\bigcap
\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z})),\]
\[\mathrm{End}_S({\mathcal X})=\mathrm{End}({\mathcal X}_s)\bigcap\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z})).\] It follows that
\[\mathrm{Hom}^0_S({\mathcal Y},{\mathcal X})=\mathrm{Hom}^0({\mathcal Y}_s,{\mathcal X}_s)\bigcap
\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Q}),H_1({\mathcal X}_s,{\mathbf Q})).\] Here
$\mathrm{Hom}^0({\mathcal Y}_s,{\mathcal X}_s)=\mathrm{Hom}({\mathcal Y}_s,{\mathcal X}_s)\otimes{\mathbf Q} \subset
\mathrm{Hom}_{{\mathbf Q}}(H_1({\mathcal Y}_s,{\mathbf Q}),H_1({\mathcal X}_s,{\mathbf Q}))$. In particular,
\[\mathrm{End}_S^0(X)=\mathrm{End}^0({\mathcal X}_s)\bigcap\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q})).\]
\end{sect}
\begin{sect}
\label{deligneT}
Let $\Gamma_s$ be the image of $\pi_1(S,s)$ in
$\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Q}))$, let $G_s$ is the Zariski closure of
$\Gamma_s$ in $\mathrm{GL}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))$ and $G_s^0$ its identity
component. By a theorem of Deligne \cite[Cor. 4.2.9]{Deligne},
$G_s^0$ is a {\sl semisimple} algebraic ${\mathbf Q}$-group.
Recall \cite[Ch. 1, Sect. 1.2]{BorelLin} that $G_s^0$ has
finite index in $G_s$.
It
follows that the intersection $\Gamma_s^0:=\Gamma_s\bigcap G_s^0$ is a normal
subgroup of finite index in $\Gamma_s$ and this index divides $[G_s:G_s^0]$.
The groups $\Gamma_s$ and $G_s$ do not depend on $s$ in the
following sense. Recall that $S$ is arcwise connected with respect to the complex topology.
If $t$ is another point of $S$ then every path $\gamma$ in $S$ from $s$ to
$t$ defines an isomorphism $\gamma_{*}:H_1({\mathcal X}_s,{\mathbf Z})\cong
H_1({\mathcal X}_t,{\mathbf Z})$ such that
$\Gamma_t=\gamma_{*} \Gamma_s \gamma_{*}^{-1}$ and therefore
\[G_t=\gamma_{*}G_s \gamma_{*}^{-1}, \ G_t^0=\gamma_{*}G_s^0
\gamma_{*}^{-1}.\]
\end{sect}
Let $\mathrm{Hdg}({\mathcal X}_s) \subset
\mathrm{GL}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))$ be the {\sl Hodge group} of ${\mathcal X}_s$ \cite{Mumford} (see
also \cite{ZarhinIzv,MZ,MZ2}). Recall that $\mathrm{Hdg}({\mathcal X}_s)$ is a connected
reductive algebraic ${\mathbf Q}$-group and its centralizer
$\mathrm{End}_{\mathrm{Hdg}({\mathcal X}_s)}(H_1({\mathcal X}_s,{\mathbf Q}))$ in $\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))$ coincides with
$\mathrm{End}^0({\mathcal X}_s)$.
The following assertion is a special case of a theorem of Deligne
\cite[Th. 7.3]{ZarhinIzv} (see also \cite[Prop. 7.5]{DeligneK3},
\cite{Masser}).
\begin{thm}
\label{deligneH} For all $s \in S$ outside a countable set, $G_s^0$ is a normal
algebraic subgroup in $\mathrm{Hdg}({\mathcal X}_s)$.
\end{thm}
\begin{defn}
We say that a point $s \in S$ is {\sl in general position} (with respect to ${\mathcal X}
\to S$ if $G_s^0$ is a normal algebraic subgroup in $\mathrm{Hdg}({\mathcal X}_s)$. It follows
from Theorem \ref{deligneH} that every $s$ outside a countable set is in
general position. In particular, a point in general position always does exist.
\end{defn}
\begin{cor}
\label{endoS} Suppose that $s$ is in general position and $G_s$ is connected.
Then \[\mathrm{End}^0({\mathcal X}_s)=\mathrm{End}_S^0({\mathcal X})=\mathrm{End}^0_{{\mathbf C}(S)}(X).\] If, in addition, ${\mathcal X}$ is
not isotrivial and $\mathrm{Hdg}({\mathcal X}_s)$ is a ${\mathbf Q}$-simple algebraic group then
$\mathrm{End}^0_{{\mathbf C}(S)}(X)=\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))$.
\end{cor}
\begin{proof}
We have $G_s=G_s^0\subset \mathrm{Hdg}({\mathcal X}_s)$. Clearly,
\[\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}_{G_s}(H_1({\mathcal X}_s,{\mathbf Q}))\supset
\mathrm{End}_{\mathrm{Hdg}({\mathcal X}_s)}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}^0({\mathcal X}_s).\] It follows that
$\mathrm{End}_S^0({\mathcal X})=\mathrm{End}^0({\mathcal X}_s)\bigcap\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}^0({\mathcal X}_s)$.
Assume now that ${\mathcal X}$ is not isotrivial and $\mathrm{Hdg}({\mathcal X}_s)$ is ${\mathbf Q}$-simple. By
Remark \ref{kz}, $\Gamma_s$ is infinite and therefore $G_s$ has positive
dimension. Since $s$ is in general position, $G_s$ is normal in $\mathrm{Hdg}({\mathcal X}_s)$.
Now the simplicity of $\mathrm{Hdg}({\mathcal X}_s)$ implies that $\mathrm{Hdg}({\mathcal X}_s)=G_s$ and therefore
\[\mathrm{End}^0({\mathcal X}_s)=\mathrm{End}_{G_s}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))=
\mathrm{End}_S^0(X)=\mathrm{End}_{{\mathbf C}(S)}(X).\]
\end{proof}
\begin{sect} {\bf Base change}.
\label{cover} Let $S' \to S$ be a finite \'etale cover and ${\mathcal X}'=X\times_S S'$
the corresponding abelian $S$-scheme. If a point $s'\in S'$ lies above $s \in
S$ then (in obvious notations)
\[{\mathcal X}'_{s'}={\mathcal X}_s, \
H_1({\mathcal X}_s,{\mathbf Z})=H_1({\mathcal X}'_{s'},{\mathbf Z}),\ H_1({\mathcal X}_s,{\mathbf Q})=H_1({\mathcal X}'_{s'},{\mathbf Q}),\
\mathrm{Hdg}({\mathcal X}'_{s'})=\mathrm{Hdg}({\mathcal X}_s).\]
The fundamental group $\pi_1(S',s)$ of $S'$ is a
subgroup of finite index in $\pi_1(S,s)$ and therefore $\Gamma_{s'}$ is a
subgroup of finite index in $\Gamma_s$. It follows that $G_{s'}$ is a subgroup
of finite index in $G_s$ ; in particular, $G_{s'}^0=G_{s}^0$. This implies that
$s$ is in general position with respect to ${\mathcal X}$ if and only if $s'$ is in
general position with respect to ${\mathcal X}'$. It also follows that if $G_s$ is
connected then $G_{s'}=G_s$.
On the other hand, let $\Gamma^{0}\subset \Gamma_s$ be a subgroup
of finite index (e.g., $\Gamma^{0}=\Gamma_s^{0}$). Clearly, its
Zariski closure lies between $G_{s}^0$ and $G_{s}$; in particular,
if $\Gamma^{0}\subset\Gamma_s^{0}$ then this closure coincides
with $G_{s}^0$. Let $\pi^{0}\subset \pi_1(S,s)$ be the preimage of
$\Gamma^{0}$: it is a subgroup of finite index in $\pi_1(S,s)$.
Let $S^{0} \to S$ be a finite \'etale map of irreducible smooth
algebraic curves attached to $\pi^{0}$ with
$\pi_1(S^{0})=\pi^{0}$. Clearly, the degree of $S^{0} \to S$
coincides with the index $[\Gamma_s:\Gamma^0]$.
Notice that if $\Gamma^0$ is normal in $\Gamma_s$ then $S^0\to S$
is Galois with Galois group $\Gamma_s/\Gamma^0$.
Fix a point $s_0\in S^0$ that lies above $s$. Then the image of the monodromy
representation $\pi_1(S^0,s_0)=\pi^{0} \to \mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Q}))$ attached to the
abelian $S^{0}$-scheme ${\mathcal X}\times_S S^0$ coincides with $\Gamma^{0}$. In other
words, $\Gamma_{s_0}=\Gamma^{0}$ and therefore $G_{s_0}=G_s^0$ is connected if
$\Gamma^{0}\subset\Gamma_s^{0}$.
\end{sect}
\begin{ex}
\label{min}
Let $n \ge 3$ be an integer and $\Gamma^{n}\subset \Gamma_s$
the kernel of the reduction map modulo $n$
\[\Gamma_s \subset \mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z})) \twoheadrightarrow
\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z}/n{\mathbf Z})).\] Here $H_1({\mathcal X}_s,{\mathbf Z}/n{\mathbf Z})=H_1({\mathcal X}_s,{\mathbf Z})\otimes{\mathbf Z}/n{\mathbf Z}$ is
the first integral homology group of ${\mathcal X}_s$ with coefficients in ${\mathbf Z}/n{\mathbf Z}$.
Clearly, $\Gamma^{n}$ is a (normal) subgroup of finite index in $\Gamma_s$. On
the other hand, since $\Gamma^{n}\subset 1+ n\mathrm{End}(H_1({\mathcal X}_s,{\mathbf Z}))$, Zariski
closure of $\Gamma^{n}$ is connected \cite[Prop. 2.6 ]{SZcomp}. This implies
that this closure coincides with $G_{s}^0$.
\end{ex}
\begin{rem}
\label{endostab}
\begin{itemize}
\item[(i)] It is known \cite{Silverberg} that all endomorphisms of $X$ are
defined over $L:={\mathbf C}(S)(X_3)$. Clearly, $L/{\mathbf C}(S)$ is a finite Galois extension
and all points of order $3$ on $X$ are $L$-rational. It follows from
N\'eron-Ogg-Shafarevich criterion \cite{SerreTate} that $L/{\mathbf C}(S)$ is unramified
at all points of $S$. This implies that if $S'$ is the normalization of $S$ in
$K$ then $S'\to S$ is finite \'etale. Clearly, ${\mathbf C}(S')=L$. \item[(ii)] Suppose
that $G_s$ is connected. Then $\mathrm{End}^0(X)=\mathrm{End}_{{\mathbf C}(S)}^0(X)$, i.e., all
endomorphisms of $X$ are defined over ${\mathbf C}(S)$. Indeed, first,
\[\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}_{\Gamma_s}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}_{G_s}(H_1({\mathcal X}_s,{\mathbf Q}))\]
and therefore $\mathrm{End}_{{\mathbf C}(S)}^0(X)=\mathrm{End}^0({\mathcal X}_s)\bigcap \mathrm{End}_{G_s}(H_1({\mathcal X}_s,{\mathbf Q}))$.
Second, pick a point $s'\in S'$ that lies above $s$.
We have ${\mathcal X}'_{s'}={\mathcal X}_s,\ G_s=G_{s'}$ and
\[\mathrm{End}^0(X)=\mathrm{End}_{{\mathbf C}(S')}(X)=\mathrm{End}^0({\mathcal X}'_{s'})\bigcap\mathrm{End}_{\pi_1(S',s')}(H_1({\mathcal X}'_{s'},{\mathbf Q}))=\]
\[\mathrm{End}^0({\mathcal X}'_{s'})\bigcap\mathrm{End}_{G_{s'}}(H_1({\mathcal X}'_{s'},{\mathbf Q}))= \mathrm{End}^0({\mathcal X}_s)\bigcap
\mathrm{End}_{G_s}(H_1({\mathcal X}_s,{\mathbf Q}))=\mathrm{End}_{{\mathbf C}(S)}(X).\]
\end{itemize}
\end{rem}
\begin{sect} {\bf Isogenies}. \label{isogenySS} It is well-known that $u\in
\mathrm{Hom}_{{\mathbf C}(S)}(Y,X)$ is an isogeny (resp. an $\ell$-isogeny) if and
only if there exist $v\in \mathrm{Hom}_{{\mathbf C}(S)}(X,Y)$ and a positive
integer $n$ such that the compositions $uv$ and $vu$ are
multiplications by $n$ (resp. by $\ell^n$) in $X$ and $Y$
respectively.
We say that $u\in\mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ is an isogeny (resp. an
$\ell$-isogeny) of abelian schemes if the induced homomorphism of
generic fibers $u_{\eta}:Y\to X$ is an isogeny of abelian
varieties.
Clearly, $u$ is an isogeny (resp. an $\ell$-isogeny) of abelian
schemes if and only if there exist $v\in \mathrm{Hom}_S(X,Y)$ and a
positive integer $n$ such that the compositions $uv$ and $vu$ are
multiplications by $n$ (resp. by $\ell^n$) in ${\mathcal X}$ and ${\mathcal Y}$
respectively (see \cite[Sect. 7.3]{Neron}).
If $u \in \mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ is an isogeny (resp. an $\ell$-isogeny) of abelian
schemes then it is clear that the induced homomorphism $u_s:{\mathcal Y}_s\to {\mathcal X}_s$ is an
isogeny (resp. an $\ell$-isogeny) of the corresponding complex abelian
varieties \cite[Sect. 7.3]{Neron}.
Conversely, suppose that $u \in \mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ and assume that $u_s \in
\mathrm{Hom}({\mathcal Y}_s,{\mathcal X}_s)$ is an isogeny (resp. an $\ell$-isogeny). Then there exist an
isogeny $w_s:{\mathcal X}_s\to {\mathcal Y}_s$ (resp. an $\ell$-isogeny) and a positive integer
$n$ such that the compositions $u_s w_s$ and $w_s u_s$ are multiplications by
$n$ (resp. by $\ell^n$) in ${\mathcal X}_s$ and ${\mathcal Y}_s$ respectively. Since $u_s\in
\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal Y}_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z}))$ and therefore \[u_s^{-1}\in
\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z})\otimes{\mathbf Q},H_1({\mathcal Y}_s,{\mathbf Z}))\otimes{\mathbf Q}),\] it follows
that $w_s\in\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}),H_1({\mathcal Y}_s,{\mathbf Z}))$. This implies that
\[w_s\in \mathrm{Hom}({\mathcal X}_s,{\mathcal Y}_s)\bigcap \mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}),H_1({\mathcal Y}_s,{\mathbf Z}))\]
and therefore there exists $v\in \mathrm{Hom}_S({\mathcal X},{\mathcal Y})$ with $v_s=w_s$. It follows that
$(uv)_s$ and $(vu)_s$ are multiplications by $n$ (resp. by $\ell^n$) in ${\mathcal X}_s$
and ${\mathcal Y}_s$ respectively. By the rigidity lemma (Sect. \ref{rigid}), $uv$ and
$vu$ are multiplications by $n$ (resp. by $\ell^n$) in ${\mathcal X}$ and ${\mathcal Y}$
respectively. This implies that $u$ and $v$ are isogenies (resp.
$\ell$-isogenies) of abelian schemes.
\end{sect}
\begin{sect} {\bf Semisimplicity}.
\label{semisimple}
Recall \cite[Sect. 4.2]{Deligne} that the
{\sl monodromy} representation \[\pi_1(S,s)\to
\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z})\subset\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Z})\otimes{\mathbf Q}))=\mathrm{Aut}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))\]
is completely reducible and therefore its centralizer
\[D=D_f:=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))\subset\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))\] is a
finite-dimensional {\sl semisimple} ${\mathbf Q}$-algebra. In addition, the {\sl center}
$E$ of $D$ lies in $\mathrm{End}_S({\mathcal X})\otimes{\mathbf Q}$ \cite[Sect. 4.4.7]{Deligne}. It
follows that $E$ lies in the {\sl center} $C$ of $\mathrm{End}^0_S({\mathcal X})$. Notice that
$\mathrm{End}_S({\mathcal X})\otimes{\mathbf Q}=\mathrm{End}_{{\mathbf C}(S)}(X)\otimes{\mathbf Q}$. So, we have the inclusion of
finite-dimensional semisimple ${\mathbf Q}$-algebras
\[\mathrm{End}_{{\mathbf C}(S)}(X)\otimes{\mathbf Q}=:\mathrm{End}^0_S({\mathcal X})\subset D \subset
\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}));\] in addition, $E$ lies in $C$. It follows easily that
if $D$ is commutative then $\mathrm{End}^0_S({\mathcal X})= D$.
\end{sect}
\begin{rem}
\label{isotype} Assume that $X$ is {\sl simple}. Then:
\begin{itemize}
\item[(i)] $\mathrm{End}^0_{{\mathbf C}(S)}(X)$ is a division algebra. Since the center $E$ of
$D_f$ lies in $\mathrm{End}^0_{{\mathbf C}(S)}(X)$ (Sect. \ref{semisimple}), it has no zero
divisors. This implies that $E$ is a field and therefore $D$ is a central
simple $E$-algebra. It follows that the $H_1({\mathcal X}_s,{\mathbf Q})$ is an isotypic
$\pi_1(S,s)$-module, i.e., is either simple or isomorphic to a direct sum of
several copies of a simple module. \item[(ii)] Notice that $\mathrm{End}^0_S({\mathcal X})= D$ if
and only if $\mathrm{End}_S({\mathcal X})=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))$. It is also known
\cite[Cor. 4.4.13]{Deligne} that $\mathrm{End}^0_S({\mathcal X})= D$ when
$d \le 3$ and ${\mathcal X}$ is not weakly isotrivial.
\end{itemize}
\end{rem}
\begin{rem}
\label{fal}
Faltings \cite[Sect. 5]{Faltings0}
has constructed a principally polarized abelian scheme $f:{\mathcal X}\to S$ with $d=4$ that is
{\sl not} weakly isotrivial, $\Gamma_s=\Gamma^{n}$ for a
certain integer $n \ge 3$ (in notations of Example \ref{min}) and $\mathrm{End}^0_S({\mathcal X})\ne D$.
It follows from arguments in Example \ref{min} that in Faltings'
example $G_s$ is connected.
\end{rem}
\section{Main results}
\label{mr} Our main result is the following statement.
\begin{thm}
\label{main0} Let $f:{\mathcal X}\to S$ be a polarized abelian scheme of
positive relative dimension $d$. Then the following conditions are
equivalent:
\begin{itemize}
\item [(i)] $D_f \ne \mathrm{End}^0_S({\mathcal X})$. \item [(ii)] The set
$\mathrm{Isog}(X,{\mathbf C}(S))$ is infinite.
\end{itemize}
\end{thm}
Theorem \ref{main0} is an immediate corollary of the following two
statements.
\begin{thm}
\label{finite} Let $f:{\mathcal X}\to S$ be a polarized abelian scheme of
positive relative dimension $d$. If $D_f = \mathrm{End}^0_S({\mathcal X})$ then
$\mathrm{Isog}(X,{\mathbf C}(S))$ is finite.
\end{thm}
\begin{thm}
\label{main} Let $f:{\mathcal X}\to S$ be a polarized abelian scheme of
positive relative dimension $d$. Suppose that $D=D_f \ne
\mathrm{End}^0_S({\mathcal X})$ (in particular, $D$ is noncommutative). If $\ell$ is
a prime such that $D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ is isomorphic to a
direct sum of matrix algebras over fields then there exist a
positive integer $d_0<2d$ and a sequence $\{Z(m)\}_{m=1}^{\infty}$
of ${\mathbf C}(S)$-abelian varieties $Z(m)$ that enjoy the following
properties:
\begin{itemize}
\item[(i)] For each positive integer $m$ there exists a
${\mathbf C}(S)$-isogeny $Z(m)\to X$ of degree $\ell^{(2d-d_0) m}$.
\item[(ii)] Let $Y$ be an abelian variety over ${\mathbf C}(S)$ and
let $M_Y$ be the set of positive integers $m$ such that $Z(m)$
is isomorphic to $Y$ over ${\mathbf C}(S)$. Then $M_Y$ is either empty or
finite. In other words, if
$M$ is an arbitrary infinite set of positive integers
then there exists an infinite subset $M_0\subset M$ such that for
$m \in M_0$ all $Z(m)$ are mutually non-isomorphic over ${\mathbf C}(S)$.
\end{itemize}
\end{thm}
\begin{rem}
\label{splitD}
Clearly, $D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ is isomorphic to a direct sum of
matrix algebras over fields for all but finitely many primes
$\ell$.
\end{rem}
We prove Theorems \ref{main} and \ref{finite} in Sections
\ref{inf} and \ref{fin} respectively.
\begin{cor}
Suppose that a point $s\in S$ is in general position and $\mathrm{Hdg}({\mathcal X}_s)$ is
${\mathbf Q}$-simple. If ${\mathcal X}$ is not weakly isotrivial then for all finite algebraic
extension $L/{\mathbf C}(S)$ the set $\mathrm{Isog}(X,L)$ is finite.
\end{cor}
\begin{proof}
Replacing $S$ by its normalization in $L$ and using (\ref{cover}), we may
assume without loss of generality that $L=K$. Choosing an integer $n \ge 3$,
replacing $S$ by $S_n$ and applying (\ref{cover}) and Remark \ref{twist}, we
may assume without loss of generality that $G_s$ is connected. It follows from
Corollary \ref{endoS} that
\[\mathrm{End}_{{\mathbf C}(S)}(X)=\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q})).\] Now the result
follows from Theorem \ref{main0}.
\end{proof}
\begin{cor}
\label{principal} Suppose that $D_f \ne \mathrm{End}^0_S({\mathcal X})$. If $X$
admits a principal polarization over ${\mathbf C}(S)$ then
$\mathrm{Isog}_1(X^2,{\mathbf C}(S),\ell)$ is {\sl infinite} for all but finitely
many primes $\ell$ congruent to $1$ modulo $4$.
\end{cor}
\begin{proof}
Let us pick a prime $\ell$ such that $D_f\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ is
a direct sum of
matrix algebras over fields. It follows from Theorem \ref{main}
that $\mathrm{Isog}(X,{\mathbf C}(S),\ell)$ is {\sl infinite}. Applying Lemma \ref{onemod4},
we conclude that $\mathrm{Isog}_1(X^2,{\mathbf C}(S),\ell)$ is {\sl infinite} if
$\ell$ is congruent to $1$ modulo $4$. Now the result follows from Remark \ref{splitD}.
\end{proof}
Till the end of this Section we assume that $K$ is a field of
algebraic functions in one variable over ${\mathbf C}$, i.e., $K$ is
finitely generated and of degree of transcendency $1$ over ${\mathbf C}$.
\begin{thm}
\label{dim3} Let $X$ be an abelian variety over $K$ of positive
dimension $d$. Suppose that the $\bar{K}/{\mathbf C}$-trace of $X$ is zero.
If $d\le 3$ then $\mathrm{Isog}(X,K)$ is finite.
\end{thm}
\begin{proof}[Proof of Theorem \ref{dim3}]
There exists a smooth irreducible algebraic ${\mathbf C}$-curve $S$ such that $K={\mathbf C}(S)$
and there exists a polarized abelian scheme ${\mathcal X} \to S$ of relative dimension
$d\le 3$, whose generic fiber coincides with $X$ \cite[Sect.1.4, p.20]{Neron}.
Pick a point $s\in S({\mathbf C})$. Recall \cite[Ch. VIII, Sect. 3]{Lang} that the
vanishing of the $\overline{{\mathbf C}(S)}/{\mathbf C}$-trace means that $X$ does not contain
over $\overline{{\mathbf C}(S)}$ a non-zero abelian subvariety that is isomorphic over
$\overline{{\mathbf C}(S)}$ to a (constant) abelian variety of the form
$W\times_{{\mathbf C}}\overline{{\mathbf C}(S)}$ where $W$ is a complex abelian variety. By
Theorem \ref{iso}, this means $X$ is not weakly isotrivial. It follows from
results of Deligne (Remark \ref{isotype}(ii)) that
$\mathrm{End}_S({\mathcal X})=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))$ and therefore $D_f:
=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))= \mathrm{End}^0_S({\mathcal X})$. Now the assertion follows from
Theorem \ref{main}.
\end{proof}
\begin{cor}
\label{prod3} Let $X$ be an abelian variety over $K$ that is
isogenous over $\bar{K}$ to a product of abelian varieties of
dimension $\le 3$. If $\bar{K}/{\mathbf C}$-trace of $X$ is zero then
$\mathrm{Isog}(X,K)$ is finite.
\end{cor}
\begin{proof}
Replacing if necessary, $K$ by its finite Galois extention and
using Remarks \ref{twist} anad \ref{isogeny}, we may assume
without loss of generality that $X$ is isomorphic over $K$ to a
product $Y_1\times \cdots \times Y_r$ of abelian varieties $Y_i$'s
of dimension $\le 3$ over $K$.
There exists a smooth irreducible algebraic ${\mathbf C}$-curve $S$ such
that $K={\mathbf C}(S)$ and there exist polarized abelian schemes ${\mathcal Y}_i \to
S$, whose generic fiber coincides with $Y_i$ \cite[Sect.1.4,
p.20]{Neron}. If $f: {\mathcal X}\to S$ is the fiber product of ${\mathcal Y}_i$ then
the generic fiber of the abelian $S$-scheme ${\mathcal X}$ coincides with
$X$.
Pick a point $s\in S({\mathbf C})$. Since all $\mathrm{dim}(Y_i)\le 3$, it follows from results
of Deligne \cite[Cor. 4.4.13]{Deligne} applied to all pairs $({\mathcal Y}_i,{\mathcal Y}_j)$ that
$\mathrm{End}_S({\mathcal X})=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}))$ and therefore $D_f:
=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))= \mathrm{End}^0_S({\mathcal X})$. Now the assertion follows from
Theorem \ref{main}.
\end{proof}
\begin{cor}
\label{dim4ns} Let $X$ be a four-dimensional abelian variety over
$K$. Suppose that the $\bar{K}/{\mathbf C}$-trace of $X$ is zero. If $X$ is
not absolutely simple then
$\mathrm{Isog}(X,K)$ is finite.
\end{cor}
\begin{proof}
Replacing if necessary, $K$ by its finite Galois extention and
using Remarks \ref{twist} anad \ref{isogeny}, we may assume
without loss of generality that $X$ is isomorphic over $K$ to a
product $Y\times T$ of abelian varieties $Y$ and $T$ of positive
dimension over $K$. Since $4= \mathrm{dim}(X)=\mathrm{dim}(Y)+\mathrm{dim}(T)$, we
conclude that both $\mathrm{dim}(Y)$ and $\mathrm{dim}(T)$ do not exceed $3$. One
has only to apply Corollary \ref{prod3}.
\end{proof}
\begin{ex}
\label{fl}
Let $f:{\mathcal X} \to S$ be the Faltings' example (Remark \ref{fal})
and let $X$ be its generic fiber, which is a principally
polarized four-dimensional abelian variety over ${\mathbf C}(S)$. Since
${\mathcal X}$ is not weakly isotrivial,
it follows from Theorem \ref{iso} that the
$\overline{{\mathbf C}(S)}/{\mathbf C}$-trace of
$X$ is zero. Since $D_f \ne
\mathrm{End}^0_S({\mathcal X})$, it follows that $\mathrm{Isog}(X,{\mathbf C}(S))$ is {\sl infinite} and $\mathrm{Isog}_1(X^2,{\mathbf C}(S),\ell)$ is
also {\sl infinite} for all but finitely many primes $\ell$ congruent to $1$ modulo $4$.
Recall that $G_s$ is connected (\ref{fal}) and therefore all
endomorphisms of $X$ are defined over ${\mathbf C}(S)$, thanks to Remark
\ref{endostab}. It follows from Corollary \ref{dim4ns} that $X$ is
absolutely simple.
\end{ex}
We discuss isogeny classes of absolutely simple abelian fourfolds
in Section \ref{dim4}.
\section{Non-isotrivial Abelian schemes}
\label{AS}
\begin{proof}[Proof of Theorem \ref{iso}]
In one direction the assertion is almost obvious. Indeed, if
${\mathcal X}_{S'}$ is a constant abelian scheme (resp. contains a non-zero
constant abelian subscheme) $W \times_ {{\mathbf C}} S'$ where $W$ is an
abelian variety over ${\mathbf C}$ then $X\times_{{\mathbf C}(S)} {\mathbf C}(S')$ is
isomorphic to $W\times_ {{\mathbf C}}{\mathbf C}(S')=(W\times_
{{\mathbf C}}{\mathbf C}(S))\times_{{\mathbf C}(S)}{\mathbf C}(S')$ (resp. contains an abelian
subvariety that is defined over ${\mathbf C}(S')$ and is isomorphic to
$W\times_ {{\mathbf C}}{\mathbf C}(S')$ over ${\mathbf C}(S')$. (By the way, we did not use
an assumption that $S'\to S$ is \'etale.)
In the opposite direction, let $W$ be an abelian variety over
${\mathbf C}$, let $W_{\eta}$ be the constant abelian variety $W\times_
{{\mathbf C}}{\mathbf C}(S)$ and $\bar{u}: W_{\eta}\times_{{\mathbf C}(S)}\overline{{\mathbf C}(S)}
\hookrightarrow X\times_{{\mathbf C}(S)}\overline{{\mathbf C}(S)}$ an embedding of
abelian varieties over $\overline{{\mathbf C}(S)}$. Let $L={\mathbf C}(S)(X_3)$ be
the field of definition of all points of order $3$ on $X$ and $S'$
is the normalization of $S$ in $L$. By Remark \ref{endostab}(i),
$S'\to S$ is a finite \'etale map of smooth irreducible curves,
${\mathbf C}(S')=L$ and all points of order $3$ on $X$ are $L$-rational.
Clearly, all torsion points of $W_{\eta}$ are ${\mathbf C}(S)$-rational; in
particular, all points of order $3$ on $W_{\eta}$ are defined
over $L$. It follows from results of \cite{Silverberg} that all
$\overline{{\mathbf C}(S)}$-homomorphisms between $W_{\eta}$ and $X$ are
defined over ${\mathbf C}(S')$; in particular, $\bar{u}$ is defined over
${\mathbf C}(S')$, i.e., there exists an embedding
$u:W_{\eta}\times_{{\mathbf C}(S)}{\mathbf C}(S')\hookrightarrow
X\times_{{\mathbf C}(S)}{\mathbf C}(S')$ of abelian varieties over ${\mathbf C}(S')$ such
that $\bar{u}$ is obtained from $u$ by extension of scalars from
${\mathbf C}(S')$ to $\overline{{\mathbf C}(S)}$. Notice that the ${\mathbf C}(S')$-abelian
varieties $X\times_{{\mathbf C}(S)}{\mathbf C}(S')$ and
$W_{\eta}\times_{{\mathbf C}(S)}{\mathbf C}(S')=W\times_ {{\mathbf C}}{\mathbf C}(S')$ are generic
fibers of abelian $S'$-schemes ${\mathcal X}_{S'}$ and $W\times_{{\mathbf C}} S'$
respectively.
It follows that $u$ extends
to a certain homomorphism of abelian $S'$-schemes $W\times_{{\mathbf C}}
S'\to {\mathcal X}_{S'}$, which we denote by $u_{S'}$. If $u$ is an
isomorphism of generic fibers (the isotrivial case) then $u_{S'}$
is an isomorphism of the corresponding abelian schemes; in
particular, ${\mathcal X}_{S'}$ is a constant abelian scheme. If $u$ is not
an isomorphism, it is still a closed emdedding; in particular, the
image $Y$ of $u:W\times_ {{\mathbf C}}{\mathbf C}(S')\hookrightarrow
X\times_{{\mathbf C}(S)}{\mathbf C}(S')$ is an abelian subvariety in
$X\times_{{\mathbf C}(S)}{\mathbf C}(S')$ and the abelian varieties $Y$ and
$W\times_ {{\mathbf C}}{\mathbf C}(S')$ are isomorphic over ${\mathbf C}(S')$. Let ${\mathcal Y}$ be
the schematic closure of the image $Y$ in ${\mathcal X}_{S'}$. It follows
from Corollary 6 on p. 175 of \cite{Neron} that ${\mathcal Y}$ is the
N\'eron model of $Y$ over $S'$. Since $W\times_ {{\mathbf C}}{\mathbf C}(S')\cong Y$
over ${\mathbf C}(S')$ and the N\'eron model of $W\times_ {{\mathbf C}}{\mathbf C}(S')$ over
$S'$ is $W\times S'$, we conclude that ${\mathcal X}_{S'}$ contains an
abelian subscheme isomorphic to the constant abelian scheme
$W\times_{{\mathbf C}}S'$.
\end{proof}
\section{Isogeny classes of abelians schemes}
\label{isab} Let ${\mathcal X} \to S$ be a polarized abelian scheme of positive relative
dimension $d$. Let us consider the category $Is({\mathcal X})$, whose objects are pairs
$({\mathcal Y},\alpha)$ that consist of an abelian scheme $g:{\mathcal Y}\to S$ and an isogeny
$\alpha:{\mathcal Y} \to {\mathcal X}$ of abelian schemes and the set of morphisms
$\mathrm{Mor}(({\mathcal Y}_1,\alpha_1), ({\mathcal Y}_2,\alpha_2)):=\mathrm{Hom}_S({\mathcal Y}_1,{\mathcal Y}_2)$ for any pair of
objects $({\mathcal Y}_1,\alpha_1)$ and $({\mathcal Y}_2,\alpha_2)$. Let us consider the category
$Is_s({\mathcal X})$, whose objects are pairs $(\Lambda, i)$ that consist of a
$\pi_1(S,s)$-module
$\Lambda$, whose additive group is isomorphic to ${\mathbf Z}^{2d}$ and an
embedding $i:\Lambda\hookrightarrow H_1({\mathcal X}_s,{\mathbf Z})$ of
$\pi_1(S,s)$-modules
and the set of morphisms $\mathrm{Mor}_{Is_s({\mathcal X})}((\Lambda_1,i_1), (\Lambda_2,i_2))$ is the
set
\[\{a \in \mathrm{Hom}_{\pi_1(S,s)}(\Lambda_1,\Lambda_2)\mid \exists u\in
\mathrm{End}^0({\mathcal X}_s) \text{ such that } i_2 a= u i_1\}.\]
\begin{thm}
\label{fiber} The functor \[\Psi_s:Is({\mathcal X}) \to Is_s({\mathcal X}), \ ({\mathcal Y},\alpha) \mapsto
(H_1({\mathcal Y}_s,{\mathbf Z}),\ \alpha_s:H_1({\mathcal Y}_s,{\mathbf Z}) \to H_1({\mathcal X}_s,{\mathbf Z})),\] \[(\gamma: {\mathcal Y}_1 \to
{\mathcal Y}_2) \mapsto \gamma_s\in
\mathrm{Hom}({{\mathcal Y}_1}_s,{{\mathcal Y}_2}_s)\bigcap
\mathrm{Hom}_{\pi_1(S,s)}(H_1({{\mathcal Y}_1}_s,{\mathbf Z}),H_1({{\mathcal Y}_2}_s,{\mathbf Z}))\] is an equivalence of
categories.
\end{thm}
\begin{proof}
First, we need to check that $\gamma_s \in
\mathrm{Mor}_{Is_s({\mathcal X})}((H_1({{\mathcal Y}_1}_s,{\mathbf Z}),H_1({{\mathcal Y}_2}_s,{\mathbf Z}))$, i.e. there exists $u\in
\mathrm{End}^0({\mathcal X}_s)$ such that ${\alpha_2}_s \gamma_s=u {\alpha_1}_s$. (Recall that
$\alpha_1: {\mathcal Y}_1 \to {\mathcal X}, \alpha_2:Y_2\to {\mathcal X}$ are isogenies of abelian schemes
and $\gamma_s \in \mathrm{Hom}({{\mathcal Y}_1}_s,{{\mathcal Y}_2}_s)$.) Clearly, both ${\alpha_1}_s:
{{\mathcal Y}_1}_s \to {\mathcal X}_s,\ {\alpha_2}_s:{\mathcal Y}_2\to {\mathcal X}_s$ are isogenies of abelian
varieties. Then $u:={\alpha_2}_s \gamma_s {\alpha_1}_s^{-1}\in \mathrm{End}^0({\mathcal X}_s)$
satisfies ${\alpha_2}_s \gamma_s=u {\alpha_1}_s$. Second, the injectiveness and
surjectiveness of $\mathrm{Hom}_S({\mathcal Y}_1,{\mathcal Y}_2)\to \mathrm{Hom}({{\mathcal Y}_1}_s,{{\mathcal Y}_2}_s)\bigcap
\mathrm{Hom}_{\pi_1(S,s)}(H_1({{\mathcal Y}_1}_s,{\mathbf Z}),H_1({{\mathcal Y}_2}_s,{\mathbf Z}))$ follow from the rigidity
lemma and Grothendieck's theorem (Sect. \ref{rigid}) respectively.
Let $(\Lambda,i_s)$ be an object of $Is_s({\mathcal X})$, i.e., a
$\pi_1(S,s)$-module
$\Lambda$, whose additive group is isomorphic to ${\mathbf Z}^{2d}$ and
an embedding $i_s:\Lambda\hookrightarrow H_1({\mathcal X}_s,{\mathbf Z})$ of
$\pi_1(S,s)$-modules. In order to check the {\sl essential surjectiveness} of $\Psi_s$,
we need to construct an abelian scheme
$g:{\mathcal Y}\to S$, an isogeny $\alpha:{\mathcal Y}\to {\mathcal X}$ and an isomorphism
$\phi_s:\Lambda\cong H_1({\mathcal Y}_s,{\mathbf Z})$ of
$\pi_1(S,s)$-modules such that $\phi_s=\alpha_s i_s$. In order to do
that, recall that the $\pi_1(S,s)$-module $\Lambda$ defines a
certain local system ${\mathcal U}$ of free ${\mathbf Z}$-modules of rank $2d$ on $S$,
whose fiber at $s$
coincides with $\Lambda$.
In addition, $i_s$ defines an embedding of local systems $i:{\mathcal U}
\hookrightarrow
R_1 f_*{\mathbf Z}$, whose fiber at $s$ coincides with our ``original" $i_s$. Rank arguments
imply that the corresponding embedding \[i:{\mathcal U}\otimes {{\mathbf Q}} \to
R_1 f_*{\mathbf Z}\otimes {{\mathbf Q}}=R_1f_*{\mathbf Q}\] is, in fact, an isomorphism. This
allows us to provide ${\mathcal U}$ with the structure (induced by $R_1 f_*{\mathbf Z}$) of the holomorphic family of
polarized Hodge structures of type $(-1,0)+(0,-1)$ \cite[Sect.
4.4]{Deligne}. The equivalence of the category of polarized abelian
schemes and the category of holomorphic families of
polarized Hodge structures of type $(-1,0)+(0,-1)$ over $S$ \cite[Sect.
4.4.2 and 4.4.3]{Deligne} (based on results of \cite{Borel}) implies that there exist an abelian
scheme $g:{\mathcal Y} \to S$, a homomorphism $\alpha\in \mathrm{Hom}_S({\mathcal Y},{\mathcal X})$ and an
isomorphism of local systems $\phi_S:{\mathcal U} \cong R_1 g_*{\mathbf Z}$ such that
$\alpha\psi_S=i$. Taking the fiber of the latter equality at $s$, we
get the desired $\alpha_s(\psi_S)_s=i_s$. Clearly,
$\alpha_s\in \mathrm{Hom}(Y_s,X_s)$ induces an isomorphism
$H_1({\mathcal Y}_s,{\mathbf Q})\cong H_1({\mathcal X}_s,{\mathbf Q})$ and therefore is an isogeny.
Applying results of Section \ref{isogenySS}, we conclude that $\alpha$ is an isogeny of
abelian schemes.
\end{proof}
\begin{rem}
The degree of $\alpha$ coincides with the index $[H_1({\mathcal X}_s,{\mathbf Z}):i_s(\Lambda)]$.
\end{rem}
\section{Finite isogeny classes}
\label{fin}
Let $L$ (resp. $L_{{\mathbf Q}}$) be the image of
the group algebra ${\mathbf Z}[\pi_1(S,s)]$ (resp. of ${\mathbf Q}[\pi_1(S,s)]$) in
$\mathrm{End}(H_1({\mathcal X}_s,{\mathbf Z}))$ (resp. in $\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))$) induced by the
monodromy representation. Clearly, $L$ is an order in the ${\mathbf Q}$-algebra
$L_{{\mathbf Q}}$.
It follows from Jackobson's density theorem and the semisimplicity of the
monodromy representation (over ${\mathbf Q}$) that $L_{{\mathbf Q}}$ is a semisimple ${\mathbf Q}$-algebra
that coincides with the centralizer $\mathrm{End}_{D_f}(H_1({\mathcal X}_s,{\mathbf Q}))$ of $D_f$ and
\[\mathrm{End}_{L_{{\mathbf Q}}}(H_1({\mathcal X}_s,{\mathbf Q}))=D_f.\] Clearly, each $\pi_1(S,s)$-stable
${\mathbf Z}$-lattice in $H_1({\mathcal X}_s,{\mathbf Q})$ is an $L$-module, whose additive group is
isomorphic to ${\mathbf Z}^{2d}$.
\begin{sect}
\label{ZS} It follows from the Jordan-Zassenhaus theorem
\cite[Theorem 26.4]{Reiner} that there are, up to an isomorphism,
only finitely many $L$-modules, whose additive group is isomorphic
to ${\mathbf Z}^{2d}$.
\end{sect}
\begin{proof}[Proof of Theorem \ref{finite}]
Since $D_f=\mathrm{End}^0_S({\mathcal X})$, we have
\[D_f=\mathrm{End}^0_S({\mathcal X})\subset \mathrm{End}^0({\mathcal X}_s)\subset \mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}).\]
Thanks to Theorem \ref{fiber}, it suffices to check that the set of isomorphism
classes of objects in $Is_s({\mathcal X})$ is finite. Thanks to the Jordan-Zassenhaus
theorem (Sect. \ref{ZS}), it becomes an immediate corollary of the following
statement.
\begin{lem}
\label{categ} Suppose that $(\Lambda_1,i_1)$ and $(\Lambda_2,i_2)$
are objects in $Is_s({\mathcal X})$.
Then $i_1(\Lambda_1)$ and $i_2(\Lambda_2)$ are $L$-submodules in
$H_1({\mathcal X}_s,{\mathbf Z})$, whose additive group is isomorphic to ${\mathbf Z}^{2d}$. If the
$L$-modules $i_1(\Lambda_1)$ and $i_2(\Lambda_2)$ are isomorphic then
$(\Lambda_1,i_1)$ and $(\Lambda_2,i_2)$ are isomorphic.
\end{lem}
\begin{proof}[Proof of Lemma \ref{categ}]
Recall that $\Lambda_1$ and $\Lambda_2$ are $\pi_1(S,s)$-modules, whose
additive groups are isomorphic to ${\mathbf Z}^{2d}$ and $i_1:\Lambda_1 \hookrightarrow
H_1({\mathcal X}_s,{\mathbf Z}), \ i_2:\Lambda_2 \hookrightarrow H_1(X_s,{\mathbf Z})$ are embeddings of
$\pi_1(S,s)$-modules. Clearly, $i_1:\Lambda_1\cong i_1(\Lambda_1), \
i_2:\Lambda_2\cong i_2(\Lambda_2)$ are isomorphisms of $\pi_1(S,s)$-modules.
Both $i_1(\Lambda_1)$ and $i_2(\Lambda_2)$ are $L$-submodules in
$H_1({\mathcal X}_s,{\mathbf Z})$,
Assume that there exists an isomorphism of $L$-modules $:\alpha:
i_1(\Lambda_1)\cong i_2(\Lambda_2)$. Clearly, $\alpha$ is an isomorphism of
$\pi_1(S,s)$-modules. Extending $\alpha$ by ${\mathbf Q}$-linearity, we obtain an
isomorphism \[u:i_1(\Lambda_1)\otimes{\mathbf Q}\cong i_2(\Lambda_2)\otimes{\mathbf Q}\] of
$L\otimes{\mathbf Q}$-modules. Recall that $L\otimes{\mathbf Q}=L_{{\mathbf Q}}, \
i_1(\Lambda_1)\otimes{\mathbf Q}= i_2(\Lambda_2)\otimes{\mathbf Q}=H_1({\mathcal X}_s,{\mathbf Q})$. We have \[u \in
\mathrm{End}_{L_{{\mathbf Q}}}(H_1({\mathcal X}_s,{\mathbf Q}))\bigcap \mathrm{Aut}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))= D_f\bigcap
\mathrm{Aut}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q})).\] Since $D_f\subset\mathrm{End}^0({\mathcal X}_s)$, we obtain that $u
\in \mathrm{End}^0({\mathcal X}_s)$. Since $u\in\mathrm{End}^0({\mathcal X}_s)\subset\mathrm{End}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))$ is an
automorphism of the finite-dimensional ${\mathbf Q}$-vector space $H_1({\mathcal X}_s,{\mathbf Q})$, we
have $u^{-1}\in \mathrm{End}^0({\mathcal X}_s)$, i.e., $u \in\mathrm{End}^0({\mathcal X}_s)^{*}$. Now if we put
\[a:=i_2^{-1}\alpha i_1: \Lambda_1 \to i_1(\Lambda_1)\to i_2(\Lambda_2)\to
\Lambda_2\] then $a \in \mathrm{Hom}_{\pi_1(S,s)}(\Lambda_1,\Lambda_2)$ is an
isomorphism of $\pi_1(S,s)$-modules and $i_2a=u i_1,\ i_1 a^{-1}=u^{-1} i_2$.
This implies that $a$ is an isomorphism of $(\Lambda_1,i_1)$ and
$(\Lambda_2,i_2)$.
\end{proof}
\end{proof}
\section{Infinite isogeny classes}
\label{inf}
We deduce Theorem \ref{main} from the two following
auxiliary statements.
\begin{lem}
\label{linalg} Let $V$ be a finite-dimensional vector space over a
field $k$, let $\mathrm{Id}_V$ be the identity automorphism of $V$, let
$A\subset \mathrm{End}_k(V)$ a $k$-subalgebra that contains $k\cdot\mathrm{Id}_V$
and is isomorphic to a direct sum of matrix algebras over fields.
Let $B\subset A$ be a semisimple $k$-subalgebra that contains the
center of $A$ but does not coincide with $A$.
Then there exists a proper subspace $W\subset V$ that enjoys the
following properties:
\begin{itemize}
\item[(i)] There does exist $a\in A$ such that $a(V)=W$.
\item[(ii)] There does not exist $b\in B$ such that $b(V)=W$.
\end{itemize}
\end{lem}
In order to state the next lemma, let us choose a prime $\ell$ and consider the
natural ${\mathbf Q}_{\ell}$-linear representation \[\pi_1(S,s)\to
\mathrm{Aut}(H_1({\mathcal X}_s,{\mathbf Q}))\subset \mathrm{Aut}_{{\mathbf Q}_{\ell}}(H_1({\mathcal X}_s,{\mathbf Q})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell})
\subset\mathrm{Aut}_{{\mathbf Q}_{\ell}}(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}));\]
here $H_1({\mathcal X}_s,{\mathbf Q}_{\ell})=H_1({\mathcal X}_s,{\mathbf Q})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$. Clearly, this
representation remains semisimple and the centralizer of $\pi_1(S,s)$ in
$\mathrm{End}_{{\mathbf Q}_{\ell}}(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}))$ coincides with
$D_f\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$. Let us put $D=D_f$. We have
\[\mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}\subset D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}\subset
\mathrm{End}_{{\mathbf Q}_{\ell}}(H_1({\mathcal X}_s,{\mathbf Q}_{\ell})).\] Clearly, $E\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ is
the center of $D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ and lies in $C\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$,
which is the center of $\mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$. It is also clear
that both ${\mathbf Q}_{\ell}$-algebras $D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ and
$\mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ are semisimple
finite-dimensional.
\begin{lem}
\label{tate} Let $\ell$ be a prime, $W$ a proper $\pi_1(S,s)$-stable
${\mathbf Q}_{\ell}$-vector space of $H_1({\mathcal X}_s,{\mathbf Q}_{\ell})$ and
$d_0=\mathrm{dim}_{{\mathbf Q}_{\ell}}(W)$. Then one of the following two conditions holds:
\begin{enumerate}
\item There exists $u \in D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ with
$u(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}))=W$. \item
There exists a sequence $\{Z(m)\}_{m=1}^{\infty}$
of ${\mathbf C}(S)$-abelian varieties $Z(m)$ that enjoy the following
properties:
\begin{itemize}
\item[(i)] For each positive integer $m$ there exists a
${\mathbf C}(S)$-isogeny $Z(m)\to X$ of degree $\ell^{(2d-d_0) m}$.
\item[(ii)] Let $Y$ be an abelian variety over ${\mathbf C}(S)$ and
let $M_Y$ be the set of positive integers $m$ such that $Z(m)$
is isomorphic to $Y$ over ${\mathbf C}(S)$. Then $M_Y$ is either empty or
finite.
\end{itemize}
\end{enumerate}
\end{lem}
\begin{rem}
The statement (and the proof) of Lemma \ref{tate} is inspired by
\cite[Prop. 1, pp. 136--137]{Tate}.
\end{rem}
\begin{proof}[Proof of Theorem \ref{main}(modulo Lemmas
\ref{linalg} and \ref{tate})] Let us apply Lemma \ref{linalg} to \[k={\mathbf Q}_{\ell},
V=H_1({\mathcal X}_s,{\mathbf Q}_{\ell}), A=D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell},
B=\mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}.\] We conclude that there exist a proper
${\mathbf Q}_{\ell}$-vector subspace $W$ in $H_1({\mathcal X}_s,{\mathbf Q}_{\ell})$, an element $v\in
D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ with $v(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}))=W$ but there does {\sl
not} exist an element $u\in \mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$ with
$u(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}))=W$. Since $v$ lies in $D\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$, it
commutes with $\pi_1(S,s)$ and therefore $W$ is $\pi_1(S,s)$-stable. Now the
result follows from Lemma \ref{tate}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{tate}]
Let us consider the $\pi_1(S,s)$-stable ${\mathbf Z}_{\ell}$-lattice
$H_1({\mathcal X}_s,{\mathbf Z}_{\ell})=H_1({\mathcal X}_s,{\mathbf Z})\otimes{\mathbf Z}_{\ell}$ in the ${\mathbf Q}_{\ell}$-vector
space $H_1({\mathcal X}_s,{\mathbf Q}_{\ell})$. Notice that for each positive integer $m$ we have
a canonical isomorphism of free ${\mathbf Z}/\ell^m{\mathbf Z}$-modules
\[H_1({\mathcal X}_s,{\mathbf Z}_{\ell})/\ell^m H_1({\mathcal X}_s,{\mathbf Z}_{\ell})=H_1({\mathcal X}_s,{\mathbf Z})/\ell^m
H_1({\mathcal X}_s,{\mathbf Z})=H_1({\mathcal X}_s,{\mathbf Z}/\ell^m{\mathbf Z})\] (induced by the canonical isomorphism
$Z/\ell^m{\mathbf Z}= {\mathbf Z}_{\ell}/\ell^m {\mathbf Z}_{\ell}$) that commutes with the actions of
$\pi_1(S,s)$.
The intersection $T:=W\bigcap H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ is a $\pi_1(S,s)$-stable
free pure ${\mathbf Z}_{\ell}$-submodule of rank $d_0$
in $H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$. Clearly,
$T\subset W$ and the natural map $T\otimes_{{\mathbf Z}_{\ell}}{\mathbf Q}_{\ell}\to W$ is an
isomorphism of ${\mathbf Q}_{\ell}$-vector spaces. The image $T_m$ of $T$ in
$H_1({\mathcal X}_s,{\mathbf Z}/\ell^m{\mathbf Z})$ is a free $Z/\ell^m{\mathbf Z}$-submodule of rank $d_0$. The
preimage of $T_m$ in $H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ coincides with
$T+\ell^mH_1({\mathcal X}_s,{\mathbf Z}_{\ell})$.
We write $\Lambda_m$ for the preimage of $T_m$ in $H_1({\mathcal X}_s,{\mathbf Z})$. Clearly,
$T+\ell^mH_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ contains $\Lambda_m$ and the natural map
$\Lambda_m\otimes{\mathbf Z}_{\ell} \to T+\ell^mH_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ is an isomorphism
of free ${\mathbf Z}_{\ell}$-modules. Notice that $\Lambda_m$ is a $\pi_1(S,s)$-stable
subgroup of index $\ell^{m(2d-d_0)}$ in $H_1({\mathcal X}_s,{\mathbf Z})$ and contains $\ell^m
H_1({\mathcal X}_s,{\mathbf Z})$. It follows from Theorem \ref{fiber} that there exists an
abelian scheme $h_m: \mathcal{Z}(m) \to S$ and an isogeny $\gamma(m): \mathcal{Z}(m)\to {\mathcal X}$ of
abelian schemes of degree $\ell^{m(2d-d_0)}$ such that
$\gamma(m)_s(H_1(\mathcal{Z}(m)_s,{\mathbf Z}))=\Lambda_m$. (Here $\mathcal{Z}(m)_s$ is the fiber of
$\mathcal{Z}(m)$ over $s$.) Since $\Lambda_m$ contains $\ell^m H_1({\mathcal X}_s,{\mathbf Z})$, there
exists an isogeny $\gamma(m)_s':{\mathcal X}_s \to \mathcal{Z}(m)_s$ of degree $\ell^{m d_0}$
such that the compositions
\[\gamma(m)_s\gamma(m)_s':{\mathcal X}_s \to \mathcal{Z}(m)_s \to
{\mathcal X}_s, \quad \gamma(m)_s'\gamma(m)_s: Z(m)_s \to {\mathcal X}_s\to \mathcal{Z}(m)_s\] coincide
with multiplication(s) by $\ell^m$. Clearly,
\[\gamma(m)_s(H_1(\mathcal{Z}(m)_s,{\mathbf Z}_{\ell}))=\gamma(m)_s(H_1(\mathcal{Z}(m)_s,{\mathbf Z}))\otimes{\mathbf Z}_{\ell}=
\Lambda_m\otimes{\mathbf Z}_{\ell}=T+\ell^m H_1({\mathcal X}_s,{\mathbf Z}_{\ell}).\] We also have
$H_1(Z(m)_s,{\mathbf Z})\supset\gamma(m)'_s(H_1({\mathcal X}_s,{\mathbf Z}))\supset \ell^m
H_1(\mathcal{Z}(m)_s,{\mathbf Z})$ and therefore
\[H_1(\mathcal{Z}(m)_s,{\mathbf Z}_{\ell})\supset\gamma(m)'_s(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\supset
\gamma(m)'_s \gamma(m)_s(H_1(\mathcal{Z}(m)_s,{\mathbf Z}_{\ell}))=\]
\[ \ell^m H_1(\mathcal{Z}(m)_s,{\mathbf Z}_{\ell}).\]
Since $\gamma(m)_s \in \mathrm{Hom}_{\pi_1(S,s)} (H_1(\mathcal{Z}(m)_s,{\mathbf Z}),H_1({\mathcal X}_s,{\mathbf Z}))$, we
have \[\gamma(m)_s^{-1} \in \mathrm{Hom}_{\pi_1(S,s)} (H_1({\mathcal X}_s,{\mathbf Q}),H_1(Z(m)_s,{\mathbf Q})).\]
It follows that $\gamma(m)_s'=\ell^m \gamma(m)_s^{-1}$ lies in
\[\mathrm{Hom}_{\pi_1(S,s)} (H_1({\mathcal X}_s,{\mathbf Q}),H_1(\mathcal{Z}(m)_s,{\mathbf Q}))\bigcap \mathrm{Hom}
(H_1({\mathcal X}_s,{\mathbf Z}),H_1(\mathcal{Z}(m)_s,{\mathbf Z}))=\]
\[\mathrm{Hom}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Z}),H_1(\mathcal{Z}(m)_s,{\mathbf Z})).\] This implies that
$\gamma(m)_s'$ coincides with the ``fiber over" $s$ of a certain isogeny of
abelian schemes $\gamma(m)':{\mathcal X} \to \mathcal{Z}(m)$ of degree $\ell^{m d_0}$. The
rigidity lemma implies that the compositions \[\gamma(m)\gamma(m)':{\mathcal X} \to
\mathcal{Z}(m) \to {\mathcal X}, \quad \gamma(m)'\gamma(m):\mathcal{Z}(m) \to {\mathcal X}\to \mathcal{Z}(m) \] coincide
with multiplication(s) by $\ell^m$.
The generic fiber $Z(m)$ of $\mathcal{Z}(m)$ is a ${\mathbf C}(S)$-abelian variety and
$\gamma(m)$ and $\gamma(m)'$ induce ${\mathbf C}(S)$-isogenies $\gamma(m)_{\eta}:Z(m)
\to X, \ \gamma(m)'_{\eta}:X \to Z(m)$. Their degrees are $\ell^{m(2d-d_0)}$
and $\ell^{m d_0}$ respectively. Their composition(s)
\[\gamma(m)_{\eta}\gamma(m)'_{\eta}: X \to Z(m) \to X, \
\gamma(m)'_{\eta}\gamma(m)_{\eta}:Z(m) \to X\to Z(m)\] coincide with
multiplication(s) by $\ell^m$.
Suppose that the condition (2) is not fulfilled. Then there exist an infinite
set $I$ of positive integers and an abelian variety $Y$ over ${\mathbf C}(S)$ such that
$Y$ and $Z(i)$ are isomorphic over ${\mathbf C}(S)$ for all $i \in I$. Let $n$ be the
smallest element of $I$. For each $i\in I$ let us fix a ${\mathbf C}(S)$-isomorphism
$v_{i,\eta}:Z(n) \to Z(i)$. Clearly, $v_{i,\eta}$ extends to an isomorphism of
abelian schemes $\mathcal{Z}(n)\to\mathcal{Z}(i)$, which we denote by $v_i$. Since $v_i$ is an
isomorphism, its specialization $v_{i,s}$ at $s$ satisfies
\[v_{i,s}(H_1(\mathcal{Z}(n)_s,{\mathbf Z}_{\ell}))=H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell}).\] Let us consider
the composition \[u_i:=\gamma(i)v_i\gamma(n)' : {\mathcal X} \to \mathcal{Z}(n)\to \mathcal{Z}(i) \to
{\mathcal X}.\] We have \[u_i \in \mathrm{End}_S({\mathcal X})\subset \mathrm{End}_S({\mathcal X})\otimes {\mathbf Z}_{\ell}\subset
\mathrm{End}_S({\mathcal X})\otimes {\mathbf Q}_{\ell} =\mathrm{End}^0_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}.\] Since
\[H_1(\mathcal{Z}(n)_s,{\mathbf Z}_{\ell})\supset \gamma(n)_s'(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\supset \ell^n
H_1(\mathcal{Z}(n)_s,{\mathbf Z}_{\ell}),\] we conclude that \[H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell})\supset
v_i\gamma(n)_s'(H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell}))\supset \ell^n
H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell})\] and therefore \[\ell^i
H_1({\mathcal X}_s,{\mathbf Z}_{\ell})+T=\gamma(i)(H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell}))\supset
\gamma(i)v_i\gamma(n)'(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\supset \] \[\ell^n
\gamma(i)(H_1(\mathcal{Z}(i)_s,{\mathbf Z}_{\ell}))=\ell^n(\ell^i
H_1({\mathcal X}_s,{\mathbf Z}_{\ell})+T)=\ell^{n+i}H_1({\mathcal X}_s,{\mathbf Z}_{\ell})+\ell^n T.\] It follows
that for all $i\in I$, \[\ell^i H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))+T\supset
\gamma(i)v_i\gamma(n)'(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))=u_i(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\supset\]
\[\ell^{n+i}H_1({\mathcal X}_s,{\mathbf Z}_{\ell})+\ell^n T.\] Recall that
$\mathrm{End}_S({\mathcal X})=\mathrm{End}_{{\mathbf C}(S)}(X)$ is a free commutative
group of finite rank and therefore the ${\mathbf Z}_{\ell}$-lattice
$\mathrm{End}_S({\mathcal X})\otimes{\mathbf Z}_{\ell}$ in $\mathrm{End}_S({\mathcal X})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$
is a compact metric space with respect to the $\ell$-adic topology. So, we can extract from $\{u_i\}_{i\in I}$ a
subsequence $\{u_j\}_{j\in J}$ that converges to a limit
\[u\in\mathrm{End}_S({\mathcal X})\otimes{\mathbf Z}_{\ell} \subset
\mathrm{End}_{{\mathbf Z}_{\ell}}(H_1({\mathcal X}_s,{\mathbf Z}_{\ell})).\]
We may assume that there is a sequence of nonnegative integers $\{m_j\}_{j\in
J}$ that tends to infinity and
such that for all $j$,
$u-u_j\in \ell^{m_j}\cdot \mathrm{End}_S({\mathcal X})\otimes{\mathbf Z}_{\ell}$.
In particular,
$(u-u_j)(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\subset \ell^{m_j}\cdot H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$.
It follows that
\[u(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))=\{\lim u_j(c)\mid c \in
H_1({\mathcal X}_s,{\mathbf Z}_{\ell})\}.\]
This implies easily that $u(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\subset T$.
On the other hand, if $t \in \ell^n T$ then for each $j\in J$
there exists $c_j \in H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ with $u_j(c_j)=t$.
Since $H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$ is a compact metric space with respect to the $\ell$-adic topology,
we can extract from $\{c_j\}_{j\in J}$ a
subsequence $\{c_k\}_{k\in K}$ that converges to a limit $c \in
H_1({\mathcal X}_s,{\mathbf Z}_{\ell})$. Then
$u(c)=\lim u_k(c) =\lim u_k(c_k)=t$.
It follows that
$\ell^n T \subset u(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\subset T$.
This implies that
$u(H_1({\mathcal X}_s,{\mathbf Q}_{\ell}))=u(H_1({\mathcal X}_s,{\mathbf Z}_{\ell}))\otimes_{{\mathbf Z}_{\ell}}{\mathbf Q}_{\ell}=
T\otimes_{{\mathbf Z}_{\ell}}{\mathbf Q}_{\ell}=W$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{linalg}]
{\bf Step 1}. {\sl Reduction to the case of simple} $A$. Suppose that the
semisimple $k$-algebra $A$ splits into a direct sum $A=A_1\oplus A_2$ of
non-zero semisimple $k$-algebras $A_1$ and $A_2$. Let $e_1$ and $e_2$ be the
identity elements of $A_1$ and $A_2$ respectively. Clearly, $\mathrm{Id}_V=e_1+e_2, e_1
e_2=e_2e_1=0, e_1^2=e_1, e_2^2=e_2$. It is also clear that both $e_1$ and $e_2$
lie in the center of $A$ and therefore in the center of $B$. Let us put
\[V_1=e_1 V, V_2=e_2 V, \ B_1=e_1 B= B e_1\subset A_1, B_2=e_2 B= B e_2\subset
A_2;\] we have $V=V_1\oplus V_2, \ B = B_1 \oplus B_2$. In addition, $A_1$
acts trivially on $V_2$ and $A_2$ acts trivially on $V_1$. So, we may view
$A_1$ as a subalgebra of $\mathrm{End}_k(V_1)$ and $A_2$ as a subalgebra of
$\mathrm{End}_k(V_2)$ respectively. Obviously, the center of $A_i$ lies in the center
of $B_i$ for both $i=1,2$. Clearly, either $B_1 \ne A_1$ or $B_2 \ne A_2$. It
is also clear that the validity of the assertion of Lemma \ref{linalg} for
$(V_i,A_i,B_i)$ with $A_i\ne B_i$ implies its validity for $(V,A,B)$. It
follows that it suffices to prove Lemma \ref{linalg} under an additional
assumption that $A$ is simple. So, further we assume that $A$ is a simple
$k$-algebra and therefore is isomorphic to a matrix algebra $\mathrm{M}_r(E)$ of size
$r$ over a field $E$. Since $E$ is the center of $\mathrm{M}_r(E)$, it is a finite
algebraic extension of $k$. The field $E$ lies in the center of $D$, i.e., $D$
is a $E$-algebra. It follows easily that a semisimple $k$-algebra $D$ is a also
a semisimple $E$-algebra.
{\bf Step 2}. {\sl Reduction to the case of simple} $V=E^r$. Since all
$\mathrm{M}_r(E)$-modules of finite $k$-dimension are isomorphic to direct sums of
finite number of copies of the standard module $E^r$, we may assume that
$V=E^r, A=\mathrm{M}_r(E)=\mathrm{End}_E(V)$. Clearly, if $W$ is $k$-vector subspace in $V$
then one can find $a \in \mathrm{End}(V)$ with $u(V)=W$ if and only if $W$ is a
$E$-vector subspace in $V$. So, in order to prove Lemma \ref{linalg}, it
suffices to prove the following statement.
\begin{prop}
\label{mat}
Let $V$ be a vector space of finite dimension $r$ over a
field $E$. Let $B\subset \mathrm{End}_E(V)$ be a semisimple $E$-subalgebra
that contains $\mathrm{Id}_V$. If $B \ne \mathrm{End}_E(V)$ then there exists a
proper $E$-vector subspace $W$ of $V$ that enjoys the following
property: there does not not exist $b \in B$ with $b(V)=V$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{mat}]
Suppose that $B$ is not simple, i.e. $B$ splits into a direct sum $B=B_1\oplus
B_2$ of non-zero summands $B_1$ and $B_2$. Let $e_1$ and $e_2$ be the identity
elements of $A_1$ and $A_2$ respectively. Clearly $V$ splits into a direct sum
$V=e_1 V\oplus e_2 V$ of non-zero $B$-stable subspaces $e_1 V$ and $e_2 V$. It
is also clear that for each $b\in B$ the image \[bV=bV_1\oplus bV_2, \
bV_1\subset V_1, bV_2\subset V_2.\] In particular, if $W$ is {\sl not} a direct
sum of a subspace of
$e_1 V$ and a subspace of $e_2 V$ then it does not coincide with
$bV$ for any choice of $b \in B$. This proves Proposition
\ref{mat}in the case of non-simple $B$. So further we assume that
$B$ is a simple $E$-algebra. Let $F$ be the center of $B$.
Clearly, $F$ is a field that contains $E\cdot\mathrm{Id}_V$. If $F \ne E\cdot\mathrm{Id}_V$
then every $b(V)$ is an $F$-vector space; in particular, its
$k$-dimension is divisible by the degree $[F: E\cdot\mathrm{Id}_V]$, which is
greater than $1$. So, none of $E$-subspaces $W$ of $E$-dimension
$1$ is of the form $b(V)$. So, without loss of generality, we may
assume that $F = E\cdot\mathrm{Id}_V$, i.e., $B$ is a central simple
$F$-algebra. Then there exist a division algebra $H$ of finite
dimension over its center $E$ and a positive integer $m$ such
that $B$ is isomorphic to the matrix algebra $\mathrm{M}_m(H)$. It follows from the
classification of modules over central simple algebras that the
left
$B=\mathrm{M}_m(H)$-module $V$ is isomorphic to a direct sum of finitely
many copies of the standard module $H^m$. Clearly, $V$ carries a
natural structure of right $H$-module in such a way that every
$b(V)$ is a right $H$-submodule. In particular, $\mathrm{dim}_E(bV)$ is
divisible by $\mathrm{dim}_E(H)$. This implies that if $H \ne E$ then
none of $E$-subspaces $W$ of $E$-dimension
$1$ is of the form $bV$. So, without loss of generality, we may
assume that $H=E$ and $B \cong M_m(E)$. Since $B \subset
\mathrm{End}_E(V)$ but $B \ne\mathrm{End}_E(V)$, we conclude that $m<r$.
Now dimension arguments
imply that the $B=M_m(E)$-module $V=E^r$ is isomorphic to the
direct sum of $r/m$ copies of the standard module $E^m$; in
particular, $m\mid r$. This implies that $\mathrm{dim}_E(bV)$ is
divisible by $r/m$. Since $m<r$, none of $E$-subspaces $W$ of $E$-dimension
$1$ is of the form $bV$.
\end{proof}
\end{proof}
\section{Quaternions and SU(2)}
\label{append}
This stuff is (or should be) well-known. However, I was unable to
find a proper reference. (However, see \cite[Lecture 1, Example
12]{Postnikov}.)
Let $E$ be a field of characteristic zero, $F/E$ its quadratic extension. We
write $\sigma:z\mapsto \bar{z}$ for the only nontrivial $E$-linear automorphism
of $F$, which is an involution. Fix a non-zero element ${\mathbf i}\in E$ such that
$\sigma({\mathbf i})=-{\mathbf i}$. Clearly, \[F=E+E\cdot {\mathbf i},\ 0\ne -a:={\mathbf i}^2\in E.\] Let $V$ be
a two-dimensional $E$-vector space. We write $\mathrm{Aut}_{F}^{1}(V)$ for the group of
$F$-linear automorphisms $u$ of $V$ with ${\det_F}(u)=1$. Here
${\det}_F:\mathrm{Aut}_F(V) \to F^{*}$ is the determinant map. If $\omega: V \times V
\to F$ is a map then we write $\mathrm{Aut}_{F}(V,\omega)$ for the group of all
$F$-linear automorphisms $u$ of $V$ such that $\omega(ux,uy)=\omega(x,y) \
\forall \ x,y \in V$ and put
\[\mathrm{Aut}^{1}_{F}(V,\omega):=\mathrm{Aut}_{F}(V,\omega)\bigcap \mathrm{Aut}_{F}^{1}(V).\] Let
$\psi: V \times V \to F$ be a non-degenerate $F$-sesquilinear Hermitian form.
Let us reduce $H$ to a diagonal form, i.e., pick $e_1 \in V$ with
$\psi(e_1,e_1)\ne 0$ and let $V_2$ be the orthogonal complement in $V$ to $e_1$
with respect to $\psi$. The non-degeneracy of $\psi$ implies that $V_2$ is
one-dimensional and the restriction of $\psi$ to $V_2$ is also non-degenerate,
i.e., $\psi$ does not vanish on non-zero elements of $V_2$. It follows that if
$e_2$ is a non-zero element of $V_2$ then $\{e_1,e_2\}$ is an orthogonal basis
of $V$ then \[\psi(z_1 e_1+z_2 e_2, w_1 e_1+w_2 e_2)= b_1 z_1\overline{w_1}+b_2
z_2\overline{w_2} \ \forall \ z_1,z_2,w_1,w_2\in F\] where $0 \ne
b_1:=\psi(e_1,e_1)\in E, \ 0 \ne b_2:=\psi(e_2,e_2)\in E$. Let us put
$\mathrm{discr}(\psi)=b_1 b_2$.
It is known \cite[Ch. 9, \S 2]{Bourbaki} that the (multiplicative) class of
$\mathrm{discr}(\psi)$ in $E^{*}/N_{F/E}(F^{*})$ does not depend on the choice of
basis. Here $N_{F/E}: F \to E$ is the norm map. Let us put $b=b_2/b_1 \in
F^{*}$ and consider the Hermitian form \[\psi':=\frac{1}{b_1}\psi: V \times V
\to F, \] \[\psi'(z_1 e_1+z_2 e_2, w_1 e_1+w_2 e_2)= z_1\overline{w_1}+b
z_2\overline{w_2} \ \forall \ z_1,z_2,w_1,w_2\in F.\] Clearly, the classes of
$b$ and $\mathrm{discr}(\psi)$ in $E^{*}/N_{F/E}(F^{*})$ do coincide and
\[\mathrm{Aut}_{F}^{1}(V,\psi')=\mathrm{Aut}_{F}^{1}(V,\psi).\] Let us consider the {\sl cyclic}
algebra \cite[Sect. 15.1]{Pierce} \[D=(F,\sigma,-b)\cong
(F,\sigma,-\mathrm{discr}(\psi)).\] Recall that $D$ is the four-dimensional central
simple $E$-algebra that contains $F$ and coincides as (left) $F$-vector space
with $F\oplus F\cdot {\mathbf j}$ where ${\mathbf j}$ is an element of $A^{*}$ such that
${\mathbf j}^2=-b, \ {\mathbf j} z {\mathbf j} ^{-1}=\sigma(z)=\bar{z}\ \forall z \in F$. If we put
${\mathbf k}:={\mathbf i}{\mathbf j}$ then \[A=F\oplus F\cdot {\mathbf j}=E\cdot 1\oplus E\cdot {\mathbf i}\oplus E{\mathbf j}\oplus
E\cdot {\mathbf k}\] with \[{\mathbf i}^2=-a,{\mathbf j}^2=-b, {\mathbf k}^2=-ab, {\mathbf k}={\mathbf i}{\mathbf j}=-{\mathbf j}{\mathbf i}, {\mathbf i}{\mathbf k}=-a{\mathbf j}=-{\mathbf k}{\mathbf i},
{\mathbf k}{\mathbf j}=-b{\mathbf i}=-{\mathbf j}{\mathbf k}.\] Clearly, $D=\left(\frac{-a,-b}{E}\right)$ (see \cite[Sect.
15.4]{Pierce}). Let us consider the standard $F$-linear involution $D \to D, \
q \mapsto \bar{q}$ that sends ${\mathbf i},{\mathbf j},{\mathbf k}$ to $-{\mathbf i},-{\mathbf j},-{\mathbf k}$ respectively.
Clearly,
\[\bar{q}q=q\bar{q}=x^2+ay^2+bs^2+abt^2\subset E \ \forall \ q=x\cdot
1+y\cdot{\mathbf i}+s\cdot{\mathbf j}+t\cdot{\mathbf k}; \ x,y,s,t\in E.\]
Let us consider the
``quaternionic" Hermitian $D$-sesquilinear form
\[\phi_D:D \times D \to D, (q_1,q_2) \mapsto q_1\overline{q_2}.\]
Taking the compositions of
$\phi_D:D\times D \to D$ with the projection maps
\[D=F\oplus F\cdot {\mathbf j} \twoheadrightarrow F, \ D=F\oplus F\cdot {\mathbf j} \twoheadrightarrow F {\mathbf j},\]
we get
$E$-bilinear forms
\[H_D:D\times D \to F, \ A_D:D\times D \to F\]
defined by
\[\phi_D(q_1,q_2)=H_D(q_1,q_2)\cdot 1 +A_D(q_1,q_2)\cdot{\mathbf j}.\]
Clearly, both $H_D$
and $A_D$ are $F$-linear with respect to first argument. Since
\[\phi_D(q_1,q_2)=\overline{\phi_D(q_2,q_1)}, \ \phi_D(q,q)=q\bar{q}\in E\subset
F,\]
we have
\[H_D(q_1,q_2)=\overline{H_D(q_2,q_1)}, \ A_D(q,q)=0 \ \forall
q_1,q_2,q \in D.\]
This means that
$H_D$ is an Hermitian
$F$-sesquilinear form on $D$ and $A_D$ is an alternating
$F$-bilinear form on $D$.
Clearly, $A_D$ is not identically zero and therefore is
non-degenerate, since the $F$-dimension of $D$ is $2$. This
implies that the {\sl symplectic} group $\mathrm{Aut}_F(D,A_D)$ coincides
with $\mathrm{Aut}_{F}^{1}(V)$
and therefore
\[\mathrm{Aut}_F(D,\phi_D)=\mathrm{Aut}_F(D,H_D)\bigcap
\mathrm{Aut}_F(D,A_D)=\mathrm{Aut}_F(D,H_D)\bigcap\mathrm{Aut}_{F}^{1}(V)=\] \[\mathrm{Aut}_{F}^{1}(D,H_D).\]
Clearly, $F\cdot 1$ and $F\cdot{\mathbf j}$ are mutually orthogonal with respect to
$H_D$. On the other hand \[H_D(z_1\cdot 1,z_2\cdot 1)=z_1\overline{z_1},\
H_D(z_1\cdot {\mathbf j},z_2\cdot {\mathbf j})=z_1{\mathbf j}\overline{z_2{\mathbf j}}=z_1{\mathbf j}(-{\mathbf j})\overline{z_2}= b
z_1 \overline{z_2}\ \forall \ z_1,z_2\in F.\] This implies that the isomorphism
of $F$-vector spaces \[\kappa:V\cong D, z e_1+w e_2 \mapsto z\cdot 1+ w \cdot
j\] is an isomorphism of {\sl Hermitian} $F$-vector spaces $(V,\psi')$ and
$(D,H_D)$, i.e., \[H_D(\kappa(v_1),\kappa(v_2))=\psi'(v_1,v_2)\ \forall
v_1,v_2\in V.\] For every $q\in D$ we denote by $R(q)$ the $F$-linear operator
$D \to D, \ d \mapsto d\cdot q$. Clearly, \[R(1)=\mathrm{Id}_{D},\
R(q_1q_2)=R(q_2)R(q_1), \ R(x q_1+y q_2)=x R(q_1)+y R(q_2)\] for all
$q_1,q_2\in D;\ x,y\in E$.
I claim that
\[\mathrm{Aut}_F(D,\phi_D)=
\{R(q)\mid q\bar{q}=1\}.\] Indeed, if $q\bar{q}=1$ then
\[\phi_D(R(q)q_1,R(q)q_2)=\phi_D(q_1q,q_2q)=q_1q\overline{q_2q}=q_1q\bar{q}\
\overline{q_2} =q_1\overline{q_2}=\phi_D(q_1,q_2),\] i.e., $R(q)$ preserves
$\phi_D$.
On the other hand, if a
$E$-linear automorphism $u$ of $D$ preserves $\phi_D$ then
$u(1)\overline{u(1)}=\phi_D(u(1),u(1))=\phi_D(1,1)=1\bar{1}=1\cdot
1=1$ and
$u'=R(u(1))^{-1} u$ also preserves $\phi_D$ and
satisfies $u'(1)=1$. This implies that for all $q\in D$, \[q=q\cdot 1= q=q\cdot
\bar{1}= \phi_D(q,1)=\phi_D(u'(q),u'(1))= u'(q)\cdot \bar{1}=u'(q),\] i.e.,
$u'$ is the identity map $\mathrm{Id}_D$ and therefore $u=R(u(1))$. Since
$\mathrm{Aut}_F(D,\phi_D)=\mathrm{Aut}_{F}^{1}(D,H_D)$, we conclude that
$\mathrm{Aut}_{F}^{1}(D,H_D)=\{R(q)\mid q\bar{q}=1\}$.
Viewing $D$ as the
left $D$-module, we get an embedding $D \subset \mathrm{End}_{E}(D)$.
Clearly, (the algebra) $D$ (of left multiplications) commutes with
all {right multiplications} $R(q)$ and therefore with
$\mathrm{Aut}_{F}^{1}(D,H_D)$ in $\mathrm{End}_{E}(D)$.
\begin{lem}
\label{SUD} The centralizer ${\mathcal D}$ of
$\mathrm{Aut}_{F}^{1}(D,H_D)$ in $\mathrm{End}_{E}(D)$ coincides with $D$.
\end{lem}
\begin{proof}
Clearly, ${\mathcal D}$ contains $D$. Let us pick {\sl non-zero} integers $n$
and $m$ such that $n^2+a\ne 0,\ m^2+b \ne 0$ and put \[q_1=\frac{(n^2-a)\cdot
1+ 2n\cdot {\mathbf i}}{n^2+a}, \ q_2=\frac{(m^2-b)\cdot 1+ 2m\cdot {\mathbf j}}{m^2+b}.\] One
may easily check that $q_1\overline{q_1}=1=q_2\overline{q_2}$. Clearly,
\[R(q_1)==\frac{n^2-a}{n^2+a}\mathrm{Id}_V+\frac{2n}{n^2+a}R({\mathbf i}),
R(q_2)==\frac{m^2-b}{m^2+b}\mathrm{Id}_V+\frac{2m}{m^2+b}R({\mathbf j})\]
and therefore ${\mathcal D}$ commutes with $R(q_1)$ and
$R(q_2)$. Since $n$ and $m$ do not vanish, we conclude that
${\mathcal D}$ commutes with right multiplications $R({\mathbf i})$ and
$R({\mathbf j})$. Clearly, $R({\mathbf j}(R({\mathbf i})=R({\mathbf k})$. This implies that ${\mathcal
D}$ commutes with $R({\mathbf k})$ and therefore commutes with all $R(q)$
($q\in D$). Now if $u\in {\mathcal D}$ then let us put $z:=u(1)\in
D$. For all $q\in D$ we have
$u(q)=u(R(q)(1))=R(q)(u(1))=R(q)(z)=zq$, i.e., $u=z\in
D\subset\mathrm{End}_E(D)$.
\end{proof}
Recall that the Hermitian $F$-vector spaces $(V,\phi')$ and
$(D,H_D)$ are isomorphic. Taking into account that
$\mathrm{Aut}_{F}^{1}(V,\psi)=\mathrm{Aut}_{F}^{1}(V,\psi')$ and applying Lemma
\ref{SUD}, we obtain the following statement.
\begin{thm}
\label{cyclic} Let us view $V$ as a four-dimensional $E$-vector
space. Then the centralizer $D$ of $\mathrm{Aut}_{F}^{1}(V,\psi)$ in
$\mathrm{End}_E(V)$ is a central simple four-dimensional $E$-algebra
isomorphic to the four-dimensional central simple $E$-algebra
$D=\left(\frac{-a,-b}{E}\right)$.
\end{thm}
\section{Abelian fourfolds}
\label{dim4}
\begin{thm}
\label{dim4s} Let $S$ be a smooth irreducible algebraic curve over
${\mathbf C}$ and $f:{\mathcal X} \to S$ a polarized abelian scheme of relative
dimension $4$ with generic fiber $X$.
Suppose that ${\mathcal X}$ is not weakly isotrivial and $\mathrm{Isog}(X,{\mathbf C}(S))$ is
infinite. Then:
\begin{itemize}
\item[(i)] $X$ is absolutely simple.
\item[(ii)] The center $E$ of
$D_f$ is a real quadratic field.
\item[(iii)] $D_f$ is a quaternion division $E$-algebra that is
unramified at one infinite place of $E$ and ramified at the other
infinite place.
\item[(iv)] $\mathrm{End}_{{\mathbf C}(S)}^0(X)$ is either $E$ or a CM-field of
degree $4$.
\item[(v)]
$\mathrm{End}^0(X)$ is a CM-field of degree $4$.
\item[(vi)] Let $s\in S$ and assume that $G_s$ is connected. Then
$\mathrm{End}_{{\mathbf C}(S)}^0(X)=\mathrm{End}^0(X)$ is a CM-field of degree $4$.
\end{itemize}
\end{thm}
\begin{proof}
The absolute simplicity of $X$ follows from Corollary \ref{dim4ns}, which
proves (i). This implies that $\mathrm{End}_{{\mathbf C}(S)}^0(X)$ has no zero divisors. Since
$E$ is isomorphic to a subalgebra of $\mathrm{End}_{{\mathbf C}(S)}^0(X)$, we conclude that $E$
is a number field and $[E:{\mathbf Q}]$ divides $2\mathrm{dim}(X)=8$. Since $\mathrm{Isog}(X,{\mathbf C}(S))$ is
infinite, it follows from Theorem \ref{main0} that $D_f \ne \mathrm{End}_S^0({\mathcal X})$. By
the last sentence of \ref{semisimple}, $D_f \ne E$, i.e., $D_f$ is a
non-commutative central simple $E$-algebra. Since $\mathrm{dim}_E(D_f)$ divides
\[\mathrm{dim}_{E}(H_1({\mathcal X}_s,{\mathbf Q}))=\frac{\mathrm{dim}_{{\mathbf Q}}(H_1({\mathcal X}_s,{\mathbf Q}))}{[E:{\mathbf Q}]}=\frac{8}{[E:{\mathbf Q}]},\]
we conclude that $8/[E:{\mathbf Q}]$ is {\sl not} square-free. It follows that $E$ is
either ${\mathbf Q}$ or a quadratic field. On the other hand, Deligne \cite[Prop.
4.4.11]{Deligne} proved that if $E$ is either ${\mathbf Q}$ or an imaginary quadratic
field then $D_f = \mathrm{End}_S^0({\mathcal X})$. It follows that $E$ is a real quadratic field
(which proves (ii)) and $\mathrm{dim}_E(D_f)=4$. It follows from \cite[Prop.
4.4.11]{Deligne} combined with the inequality $D_f \ne \mathrm{End}_S^0({\mathcal X})$ that $D_f$
is unramified at one infinite place of $E$ and ramified at the another one.
This rules out the possibility that $D_f$ is a matrix algebra over $E$. It
follows that $D_f$ is a quaternion division $E$-algebra, which proves (iii).
We have \[E \subset \mathrm{End}_S^0({\mathcal X})\subset D_f, \ \mathrm{End}_S^0({\mathcal X})\ne D_f.\] Since
$\mathrm{dim}_E(D_f)=4$, we conclude that either $\mathrm{End}_S^0({\mathcal X})=E$ or $\mathrm{End}_S^0({\mathcal X})$ is
a field of degree $4$ that contains $E$. Since there is an embedding $E
\hookrightarrow {\mathbf R}$ such that $D_f\otimes_E{\mathbf R}$ is the standard quaternion
${\mathbf R}$-algebra, either $E=\mathrm{End}_S^0({\mathcal X})$ or $\mathrm{End}_S^0({\mathcal X})$ is a degree four field
that is {\sl not} totally real. Since $\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{{\mathbf C}(S)}^0(X)$, we
conclude that either $E=\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{{\mathbf C}(S)}^0(X)$ or
$\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{{\mathbf C}(S)}^0(X)$ is a CM-field of degree $4$, which proves
(iv).
Let $s\in S$ be a point in general position and assume that $G_s$ is connected.
Suppose that $\mathrm{End}_{{\mathbf C}(S)}^0(X)=E$. We need to arrive to a contradiction. It
follows from the first assertion of Corollary \ref{endoS} that
\[\mathrm{End}^0({\mathcal X}_s)=\mathrm{End}_S^0({\mathcal X})=\mathrm{End}_{{\mathbf C}(S)}(X).\] This implies that $\mathrm{End}^0({\mathcal X}_s)=E$
is a real quadratic field. It follows from \cite[4.2, p. 566]{MZ} that
$\mathrm{Hdg}({\mathcal X}_s)$ is ${\mathbf Q}$-simple. It follows from the first assertion of Corollary
\ref{endoS} that $\mathrm{End}_{{\mathbf C}(S)}(X)=\mathrm{End}_S^0(X)=\mathrm{End}_{\pi_1(S,s)}(H_1({\mathcal X}_s,{\mathbf Q}))$.
Applying Theorem \ref{main0}, we conclude that $\mathrm{Isog}(X,K)$ is finite. The
obtained contradiction proves that $\mathrm{End}_{{\mathbf C}(S)}^0(X)$ is a CM-field of degree
$4$. It follows from Remark \ref{endostab} that $\mathrm{End}_{{\mathbf C}(S)}^0(X)=\mathrm{End}^0(X)$.
This proves (vi).
We still have to prove that $\mathrm{End}^0(X)$ is a CM-field of degree $4$
without assuming the connectedness of $G_s$. However, there exists a
(connected) finite \'etale cover $S'\to S$ such that for $s'\in S'$ the group
$G_{s'}$ attached to the abelian scheme ${\mathcal X}'={\mathcal X}\times_{S}S'$ is connected
and all endomorphisms of $X$ are defined over
the field $L={\mathbf C}(S')$ (see Subsect. \ref{cover}, Example \ref{min} and Remark \ref{endostab}).
By Remark \ref{twist}, $\mathrm{Isog}(X\times_{K}L,L)$ is also infinite. Applying
already proven (vi) to the generic fiber $X\times_{K}L$ of ${\mathcal X}'$, we conclude that
$\mathrm{End}^0(X)=\mathrm{End}^0(X\times_{K}L)$ is a CM-field of degree $4$. This proves (v).
\end{proof}
\begin{ex}
Let $f:{\mathcal X} \to S$ be the Faltings' example (Sect. \ref{fal}) and
\ref{fl}) and let $X$ be its generic fiber. It follows from
Theorem \ref{dim4s} combined with arguments in Example \ref{fl}
that $\mathrm{End}_{{\mathbf C}(S)}^0(X)=\mathrm{End}^0(X)$ is a CM-field of degree $4$
that is a purely imaginary quadratic extension of a totally real
quadratic field $E$ and $D_f$ is a quaternion division $E$-algebra
that is unramified at one infinite place of $E$ and ramified at
the other infinite place. In fact, the quaternion algebra $D_f$
was the starting point of Faltings' construction \cite[Sect.
5]{Faltings0}.
\end{ex}
\begin{thm}
\label{foursuf} Let $S$ be a smooth irreducible algebraic curve
over ${\mathbf C}$ and $f:{\mathcal X} \to S$ a polarized abelian scheme of relative
dimension $4$ with generic fiber $X$. Suppose that ${\mathcal X}$ is not
isotrivial and $\mathrm{End}^0(X)=\mathrm{End}_{{\mathbf C}(S)}^0(X)$. Assume, in addition,
that $F:=\mathrm{End}^0(X)$ is a CM-field of degree $4$. Let $\mu_F$ be
the (multiplicative) group of roots of unity in $F$ and $r_F$ its
order. Then:
\begin{itemize}
\item[(0)] ${\mathcal X}$ is not weakly isotrivial.
\item[(i)] There exists a finite \'etale cover $S^0\to
S$ such that ${\mathbf C}(S')/{\mathbf C}(S)$ is a cyclic extension of degree
dividing $r_F$ and the set $\mathrm{Isog}(X,{\mathbf C}(S^0))$ is infinite.
\item[(ii)] Let $s\in S$ and assume that $G_s$ is connected. Then
$\mathrm{Isog}(X,{\mathbf C}(S))$ is infinite.
\end{itemize}
\end{thm}
\begin{proof}
Clearly, $X$ is an absolutely simple abelian variety. It follows
easily that ${\mathcal X}$ is not weakly isotrviial.
Let us put $F:=\mathrm{End}_{{\mathbf C}(S)}^0(X)$. By assumption, it is
a purely imaginary quadratic extension of a real quadratic field $E$. Pick a
point $s$ in general position. By Corollary \ref{endoS}, $\mathrm{End}^0({\mathcal X}_s)=F$. Let
us put $V:=H_1({\mathcal X}_s,{\mathbf Q})$; recall that $V$ is an $8$-dimensional ${\mathbf Q}$-vector
space. Clearly, $V$ carries a a natural structure of two-dimensional $F$-vector
space and $\Gamma_s \subset \mathrm{Aut}_F(V)$. We can do better, using the
polarization on ${\mathcal X}$ that induces a polarization on ${\mathcal X}_s$, whose Riemann form
gives rise to a non-degenerate alternating $\pi_1(S,s)$-invariant ${\mathbf Q}$-bilinear
form $\phi: V \times V \to {\mathbf Q}$. Since the {\sl complex conjugation} $e \mapsto
e'$ on $F$ is the only positive involution on the CM-field $F=\mathrm{End}^0({\mathcal X}_s)$,
all Rosati involutions on $\mathrm{End}^0({\mathcal X}_s)=F$ coincide with the complex
conjugation. This implies that $\phi(ex,y)=\phi(x,e'y) \ \forall x,y\in V,
e\in F$ where $e \mapsto e'$ is the {\sl complex conjugation} on $F$. Pick a
non-zero element $\alpha \in E$ with $\alpha'=-\alpha$. Then there exists a
unique (non-degenerate) $F$-Hermitian form \[\psi: V \times V \to F\] such that
$\phi(x,y)=\mathrm{tr}_{F/{\mathbf Q}}(\alpha \psi(x,y)) \ \forall x,y\in V$ \cite[Sect.
9]{Deligne900}. Here $\mathrm{tr}_{F/{\mathbf Q}}: F \to {\mathbf Q}$ is the trace map.
Since $\phi$ is $\pi_1(S,s)$-invariant, $\psi$ is also
$\pi_1(S,s)$-invariant and therefore is $\Gamma_s$-invariant. Let us consider
the unitary group \[\mathrm{U}(V,\psi)\subset \mathrm{GL}(V)\] of the $F$-vector space $V$
relative to $\psi$. {\sl A priori} $U(V,\psi)$ is an algebraic group over $E$,
but we regard it as an algebraic ${\mathbf Q}$-group, i.e., take its Weil restriction
over ${\mathbf Q}$. In particular, $U(V,\psi)({\mathbf Q})=\mathrm{Aut}_{F}(V,\psi)$. Clearly, $\Gamma_s
\subset \mathrm{Aut}_{F}(V,\psi)=U(V,\psi)({\mathbf Q})$ and therefore $G_s \subset
\mathrm{U}(V,\psi)$. The semisimplicity of $G_s^0$ implies that $G_s^0\subset
\mathrm{SU}(V,\psi)$ where $\mathrm{SU}(V,\psi)$ is the special unitary group of the
$F$-vector space $V$ relative to $\psi$. As above, we view $\mathrm{SU}(V,\psi)$ as an
algebraic ${\mathbf Q}$-(sub)group (of $\mathrm{U}(V,\psi)$). Clearly,
\[\mathrm{SU}(V,\psi)({\mathbf Q})=\mathrm{Aut}_{F}^{1}(V,\psi):=\{u\in \mathrm{Aut}_{F}(V,\psi)\mid
{\det}_F(u)=1\}.\]
Since $\mathrm{dim}_F(V)=2$, the centralizer $\mathrm{End}_{\mathrm{SU}(V,\psi)({\mathbf Q})}(V)$ of
$SU(V,\psi)({\mathbf Q})$ in $\mathrm{End}_{{\mathbf Q}}(V)$ (and even in $\mathrm{End}_E(V)$) is {\sl strictly
greater} than $F$ (see Theorem \ref{cyclic}). Since the group $\mathrm{SU}(V,\psi)({\mathbf Q})$
is dense in $\mathrm{SU}(V,\psi)$ with respect to Zariski topology \cite[Ch. 5, Sect.
18, Cor. 18.3]{BorelLin}, the centralizer $\mathrm{End}_{\mathrm{SU}(V,\psi)}(V)$ of
$\mathrm{SU}(V,\psi)$ in $\mathrm{End}_{{\mathbf Q}}(V)$ coincides with $\mathrm{End}_{\mathrm{SU}(V,\psi)({\mathbf Q})}(V)$ and
therefore is also {\sl strictly greater} than $F$. This implies that if
$G_s\subset \mathrm{SU}(V,\psi)$ then $D_f \ne \mathrm{End}^0(X)$. Applying Theorem
\ref{main0}, we obtain the following statement.
\begin{lem}
\label{su2} If $G_s\subset \mathrm{SU}(V,\psi)$ then $\mathrm{Isog}(X,{\mathbf C}(S))$ is infinite.
\end{lem}
{\sl End of the proof of Theorem} \ref{foursuf}. Since $G_s^0\subset
\mathrm{SU}(V,\psi)$, the assertion (ii) follows readily. In order to prove the
assertion (i), recall (Sect. \ref{deligneT}) that $\Gamma_s^0:=\Gamma_s\bigcap
G_s^0$ is a normal subgroup of finite index in $\Gamma_s$. Since $G_s^0\subset
\mathrm{SU}(V,\psi)$, it follows that
${\det}_F(\Gamma_s)\subset \mu_F$. This implies that if we put
$\Gamma^0:=\{g\in \Gamma_s\mid {\det}_F(g)=1\}$ then $\Gamma^0$ is a normal
subgroup in $\Gamma_s$ and the quotient $\Gamma_s/\Gamma^0$ is a finite cyclic
subgroup, whose order divides $r_F$. Using the construction of Section
\ref{cover} applied to $\Gamma^0$, we get a finite \'etale Galois cover $S^0
\to S$ with Galois group $\Gamma_s/\Gamma^0$ and abelian $S^0$-scheme
${\mathcal X}^0={\mathcal X}\times_S S^0$ such that if $s_0\in S^0$ lies above $s$ then the image
$\Gamma_{s_0}$ of the corresponding monodromy representation coincides with
$\Gamma^0$. (Here we identify ${\mathcal X}_s$ with ${\mathcal X}^0_{s_0}$ and $H_1({\mathcal X}^0_{s_0},{\mathbf Q})$
with $H_1({\mathcal X}_{s},{\mathbf Q})=V$.) Clearly, $\Gamma_{s_0}=\Gamma^0\subset \mathrm{SU}(V,\psi)$.
Applying Lemma \ref{su2} to
the abelian $S^0$-scheme
${\mathcal X}^0 ={\mathcal X}\times_S S^0$, we conclude that $\mathrm{Isog}(X,{\mathbf C}(S^0))$ is infinite. In order
to finish the proof of (i), notice that the field extension ${\mathbf C}(S^0)/{\mathbf C}(S)$ is
normal and its Galois group coincides with $\Gamma_s/\Gamma^0$.
\end{proof}
\begin{rem} We keep the notation and assumptions of Theorem \ref{foursuf}.
\begin{enumerate}
\item
$\mathrm{Hdg}({\mathcal X}_s)=\mathrm{U}(V,\psi)$ \cite[Sect. 7.5]{MZ}. It follows from Deligne's Theorem \ref{deligneH}
that $G_s^0=\mathrm{SU}(V,\psi)$.
\item It follows from Theorem \ref{cyclic} that the centralizer $\mathrm{End}_{\mathrm{SU}(V,\psi)}(V)$
of $\mathrm{SU}(V,\psi)$ in $\mathrm{End}_{{\mathbf Q}}(V)$ is a four-dimensional central simple
$E$-algebra $D'$ containing $F$. \item Suppose
that $G_s=G_s^0$. It follows that $D_f$ contains $D'$ and
therefore does not coincides with $F=\mathrm{End}_S^0({\mathcal X})$. However, the
center ${\mathcal E}$ of $D_f$ lies in $F$. Since $F\subset
D'\subset D_f$ and the center of $D'$ is $E$, we conclude that
${\mathcal E}\subset E$. This means that either ${\mathcal E}={\mathbf Q}$
or ${\mathcal E}=E$, because $E$ is a quadratic field. In both
cases $D_f$ is a central simple ${\mathcal E}$-algebra. Applying
Proposition 4.4.11 of \cite{Deligne}, we conclude that ${\mathcal
E}\ne {\mathbf Q}$, i.e., ${\mathcal E}=E$. Since $\mathrm{dim}_{{\mathbf Q}}(D_f)$ must
divide $8$ and $\mathrm{dim}_{{\mathbf Q}}(D')=8$, we conclude that $D_f=D'$.
Applying again the same Proposition, we conclude that $D_f$ is
ramified at one infinite place of $E$ and unramified at another
one. In particular, $D_f$ is not isomorphic to the matrix algebra
$\mathrm{M}_2(E)$, i.e., $D_f$ is a {\sl quaternion} (division)
$E$-algebra.
\end{enumerate}
\end{rem}
\begin{ex}
\label{mainex} Recall that the $5$th cyclotomic field $F:={\mathbf Q}(\mu_5)$ is a
CM-field of degree $4$. Notice that $\mu_F$ is a cyclic group of order $10$,
i.e., $r_F=10$. Let us consider a smooth genus zero affine curve
\[S={\mathbf A}^1\setminus \{0,1\}={\mathbf P}^1\setminus \{0,1,\infty\}\] with coordinate
$\lambda$ and a family ${\mathfrak C}\to S$ of genus four smooth projective curves over
$S$ defined by the equation \[y^5=x(x-1)(x-\lambda)\] and the corresponding
family of jacobians $f:{\mathcal J}\to S$. Clearly, ${\mathcal J}$ is a (principally) polarized
abelian $S$-scheme of relative dimension four, ${\mathbf C}(S)={\mathbf C}(\lambda)$ is the field
of rational functions in one variable over ${\mathbf C}$ and the generic fiber $J$ of
$f$ is the jacobian of the ${\mathbf C}(\lambda)$-curve $y^5=x(x-1)(x-\lambda)$ of genus
four. It follows from \cite[Lemma 2.2 and Prop. 2.7]{JN} that ${\mathcal J}$ is {\sl not}
isotrivial and \[\mathrm{End}^0(J)=\mathrm{End}_{{\mathbf C}(\lambda)}^0(J)=F.\] This implies that $J$
is absolutely simple and therefore ${\mathcal J}$ is {\sl not} weakly isotrivial. By
Theorem \ref{foursuf}(i), there exists a cyclic extension $L/{\mathbf C}(\lambda)$ that
has degree $r$ dividing $10$, is unramified outside $\{0,1,\infty\}$ and such
that $\mathrm{Isog}(J,L)$ is {\sl infinite}. Using Kummer theory, we obtain easily that
there are nonnegative integers $a$ and $b$ such that \[0\le a<r\le 10, \ 0\le
b<r\le 10\] and $L={\mathbf C}(\lambda)(\sqrt[r]{\lambda^a (\lambda-1)^b})$. Clearly,
${\mathbf C}(\lambda)\subset L \subset {\mathbf C}(\sqrt[10]{\lambda},\sqrt[10]{\lambda-1})$. It
follows from Remark \ref{twist} that
$\mathrm{Isog}(J,{\mathbf C}(\sqrt[10]{\lambda},\sqrt[10]{\lambda-1}))$ is infinite. Now Corollary
\ref{principal} implies that
$\mathrm{Isog}_1(J^2,{\mathbf C}(\sqrt[10]{\lambda},\sqrt[10]{\lambda-1}),\ell)$ is {\sl infinite} for all but finitely
many primes $\ell$ congruent to $1$ modulo $4$.
Notice that ${\mathbf C}(\sqrt[10]{\lambda},\sqrt[10]{\lambda-1})$ is the
field of the rational functions on the affine Fermat curve $S':
u^{10}-v^{10}=1, \ u \ne 0, v \ne 0$ with
$u=\sqrt[10]{\lambda},v=\sqrt[10]{\lambda-1}$ and $S' \to S, \
\lambda=u^{10}$ is the corresponding finite \'etale cover.
\end{ex}
| {
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Blog: http://zhangwenli.com/blog.
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{"url":"https:\/\/stats.libretexts.org\/Courses\/Luther_College\/Psyc_350%3ABehavioral_Statistics_(Toussaint)\/13%3A_Analysis_of_Variance\/13.09%3A_Power_of_Within-Subjects_Designs_Demo","text":"# 13.9: Power of Within-Subjects Designs Demo\n\n\nLearning Objectives\n\n\u2022 State the relationship between variance and power\n\u2022 State the effect of using a one-tailed test on power\n\n## Instructions\n\nThis simulation demonstrates the effect of the correlation between measures in a one-way within-subjects ANOVA with two levels. This test is equivalent to a correlated t test. The default values for this demonstration are for an experiment with $$10$$ subjects each measured under two conditions. The population difference for the two conditions is $$1.85$$ and the variance in each of the conditions is $$4.0$$. The graph shows the power of the test as a function of the population correlation between the two scores for the $$0.10$$, $$0.05$$, and $$0.01$$ significance levels. The power of an independent-groups $$t$$ test (which assumes the correlation is $$0$$) is shown by the $$x's$$.\n\nExperiment with different combinations of the parameters. Is the correlation an important factor in power?\n\n## Illustrated Instructions\n\nVideo Demo\n\nThe video begins by changing the population variance to $$8$$ and increases the sample size to $$25$$ and then reduces it to $$5$$. Notice the impact that these changes have on the relationship between power and population correlation. The video concludes by changing the mean difference to $$4$$.\n\nThis page titled 13.9: Power of Within-Subjects Designs Demo is shared under a Public Domain license and was authored, remixed, and\/or curated by David Lane via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.","date":"2023-02-09 02:20:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.92757648229599, \"perplexity\": 571.7152901194353}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2023-06\/segments\/1674764501066.53\/warc\/CC-MAIN-20230209014102-20230209044102-00414.warc.gz\"}"} | null | null |
The following problems require the algebraic computation of limits of functions as x approaches plus or minus infinity. Most problems are average. A few are somewhat challenging. All of the solutions are given WITHOUT the use of L'Hopital's Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by giving careful consideration to the forms and during the computations of these limits. Initially, many students INCORRECTLY conclude that is equal to 1 , or that the limit does not exist, or is or . Many also conclude that is equal to 0 . In fact, the forms and are examples of indeterminate forms. This simply means that you have not yet determined an answer. Usually, these indeterminate forms can be circumvented by using algebraic manipulation. Such tools as algebraic simplification and conjugates can easily be used to circumvent the forms and so that the limit can be calculated.
PROBLEM 14 : Compute .
PROBLEM 15 : Compute .
PROBLEM 16 : Compute .
PROBLEM 17 : Compute .
PROBLEM 18 : Compute .
PROBLEM 19 : Compute . | {
"redpajama_set_name": "RedPajamaC4"
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Ти́нна — село в Україні, в Дунаєвецькій селищній територіальній громаді Кам'янець-Подільського району Хмельницької області. Населення становить 675 осіб.
Географія
Розташування
Тинна розташоване на південних відрогах Волино–Подільської височини. Та місцевість являє собою погорбовану хвилясту рівнину, в окремих місцях перерізану пересохлими річковими руслами і балками. Загальна площа території Тинни становить 20,5 км².
Тинна межує з такими селами: на заході з Малий Карабчіїв, на сході з Терновою, на півночі з Малою Тернівкою, на півдні з Лисогіркою. Найближча від Тинни залізнична станція знаходиться на відстані 9 км.
Клімат
Клімат в Тинні помірно континентальний. Середньомісячна температура січня -6,5 °С,
липня +24 °С. Середня тривалість без морозного періоду становить 125 днів на рік.
Водойми
На захід від Тинни, на відстані 11 км тече Смотрич, притока р. Дністер. Через Тинну протікає невеличка річка, права притока Тернавки, яка впадає у неї в селі Тернова. На західній околиці Тинни знаходиться урочище «Кринички», звідкіля витікає не великий струмок, що живить два ставки.
Ґрунти
Ґрунт в Тинні — звичайний чорнозем, лише в урочищі «На зрубі» опідзолений.
Природна зона
Тинна знаходиться в зоні лісостепу.
Рослинний світ
На південь від Тинни знаходиться лісовий масив площею 400 га. Цей ліс складають переважно листяні породи: дуб, граб, ясен, липа, клен. З ними ростуть і кущі шипшини, ліщини, бузини, глоду. В цьому лісі та в околицях Тинни зустрічається чимало лікарських і медоносних рослин. Різних видів рослин тут нараховується кілька сотень. Найпоширенішим деревом в Тинні є біла акація.
Тваринний світ
Серед найпоширеніших птахів в Тинні та її околицях трапляються такі: ворона сіра, грак, сорока, сова, шпак, голуб, яструб, дятел та інші.
Людність
Загальна кількість населення, що проживає в Тинні, за останнім переписом 2001 року, становить 734 особи, за національністю переважна більшість — українці.
Минувшина
Порівняно невелике українське село Тинни, яке розкинулось обабіч старого Подільського шляху, що вів від Проскурова до Кам'янець-Подільського, багате історичним минулим. Столітні липи, вздовж цього шляху, є німими свідками бурхливих історичних подій. За давніми переказами, під однією з них на відпочинок зупинявся сам Богдан Хмельницький зі своїм товариством, визволяючи Поділля від польської шляхти. Вздовж них гримів своїми кайданами народний месник Устим Кармелюк, їдучи на заслання у Сибір, і проїздив Тарас Шевченко, а також багато інших видатних і знаменитих людей, які побували у Тинні.
Кінець XIV — початок XV століття — ймовірний час заснування Тинни.
1607 рік — перша відома згадка у костельному архіві. Тинна разом з селами Тернова, Тарнавка та Томашів згадується там, як спадок подільського ловчого Томаша Гумецького. До того часу протягом 200 років це були королівські землі. Будучи тоді немалим містечком, що лежало на обох берегах річки Тернавки, Тинна була відомою завдяки своїм ярмаркам.
У 1736 році після смерті подільського воєводи Стефана Гумецького, Тинна перейшла у власність його сина Ігнація, коронного стольникa.
Під час Голодомору 1932—1933 років померло щонайменше 43 жителі села.
Про костел
Близько 1590-го року коштами Томаша Гумецького зводиться новий мурований храм. Точне місце розташування цього першого дерев'яного костелу св. Катерини в Тинні нині не відоме.
Початок XVII ст. — папа Урбан VIII видав привілей місцевому костелу, з відпустом на всі марійні (присвячені діві Марії) свята. В цьому документі храм називають ecclesia primae fundationis ловчого Гумецького.
У 1706 році костел, який був вже зовсім у руїні, зруйнували до фундаменту, на якому його відновлено Катериною та подільським воєводою Стефаном Гумецькими.
1717 рік, 10 листопада кам'янецький єпископ РКЦ Стефан Рупнєвський храм освячує, про що свідчить меморіальний камінь з написом, який прикрашав колись захристію.
Після 1736 року будівництво нового костелу, виконуючи останню волю батька, розпочав Ігнацій Гумецький. Процес зведення храму завершила вже його вдова, Тереза.
Колись у костелі було 9 вівтарів, головний прикрашав славетний дерев'яний образ Богородиці. Був тут і гарний різьблений образ вознесіння Марії.
У 1956 році костел у Тинні зруйновано і відбудовано на тому ж місті у 90-ті.
Нині храм стоїть на невеликому горбку, він прямокутний у плані. Раніше його фасад прикрашали скульптури святих Ігнація та Терези, діви Марії, Петра та Павла, та до нашого часу ці скульптури не збереглись.
Образ Богородиці
Можливо, від Гумецького костел у Тинні отримав у дарунок образ Богородиці, який прославився багатьма дивами.
У 1656 році татари пограбували містечко та костел, а чудодійний образ розітнули палашем над лівим оком Марії, пошрамувавши обличчя Богородиці.
У 1672 році на початку турецької навали, образ Богородиці перенесено у Кам'янецьку Катедру. Після здачі Кам'янця туркам, пані Гумецька перевезла образ до Львова.
У 1730 році, образ Богородиці повернули у Тинну.
У 1734 році тинська святиня пережила напад волохів та циган.
1743 рік, 2 липня поклонитися образу Богородиці в Тинну прийшло 30 000 вірних.
Останні володарі
Катерина Гумецька одружилася вдруге з подільським воєводою — Бернардом Годзьким, і від цього шлюбу народила ще одну дочку, теж Катерину, яка також була двічі одружена. Її другим чоловіком був принц Кароль (Шарль) де Нассау-Зіген. Не маючи нащадків в Тинні від цього шлюбу, він відписав Тинну у спадок Гречинці Екомон, пізніше — Бутягіновій, а вже від її дочки Тинна відійшла до Потоцьких.
Туристичні об'єкти
Церква Різдва Богородиці, 1869 рік
Польський цвинтар
Особистості
У 1808 році в Тинні помер і похований Шарль Анрі Насау-Зіген, флотоводець Катерини ІІ.
Ігнатенко Микола Андрійович. Літературознавець. Доктор філологічних наук. Родом (16.05.1939) з с.Калинки, тепер Тинна Дунаєвецького району Хмельницької області. Після закінчення в 1961 році філологічного факультету Чернівецького держуніверситету вчителював, був кореспондентом Чернівецького обласного радіо, працював у районних газетах. З 1972 р. – науковий співробітник Інституту літератури ім. Т. Г. Шевченка. Автор книг з питань теорії літератури та історії художньої культури "Читач як учасник літературного процесу", "Генезис сучасного наукового мислення" та низки статей у колективних збірниках і розділів у монографіях. М. І. Ігнатенко переклав з німецької мови романи "Брати-віталійці" В. Бределя, "Загибель воїнів –ягуарів" В. Мейнка, окремі поезії В. Гете; з румунської – повісті "Щасливий млин" і "Скарб" І. Славича, новели Х.Зінке, ряд віршів М. Емінеску. Загинув за нез'ясованих обставин. Юхим Гусар.
Галерея
Див. також
Поділля
Подоляни
Подільський говір
Децентралізація
Примітки
Посилання
Погода в селі Тинна
Сайт Тиннянської загальноосвітньої школи
Стаття Тинна на сайті «Замки та храми України»
Тинна на сайті «Старі мапи України»
Тинна на сайті «Поштові індекси України»
Тинна на сайті Верховної Ради України
Тинна на сайті «Енциклопедія пам'яток»
.— S. 714—716.
Інформація про наявність проідентифікованих та зареєстрованих в Реєстрі тварин ВРХ станом на 29.08.2010 (с. Тинна)
Села Хмельницької області
Населені пункти Кам'янець-Подільського району | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,837 |
module SchemaPlus
module ForeignKeys
VERSION = "1.1.0"
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,137 |
\section{Introduction}
Research on computational paralinguistics advances with the advent of artificial intelligence, big data, and speed up internet access. Among many issues, computational paralinguistics research still relies on limited small data for experiments \cite{Batliner2020}. The need to conduct research on big data is needed for generalization and performance improvements.
A large-scale dataset was made for accelerating research on computational paralinguistics, the HUME-VB dataset \cite{Baird2022}. The dataset contains more than 36 hours of audio data with a 48 kHz of the sampling rate. The samples in the dataset were labeled with three pieces of information: intensity rating for ten different expressed emotions (float in ranges [0, 1]), age (integer), and country (string). This dataset was created to support the following tasks: multitask learning, emotion generation, and few-shot learning. Multitask learning includes emotion recognition from vocalizations.
Speech emotion recognition is a branch of computational paralinguistics that deal with the accurate prediction of emotional score/class from speech. Since the expression of speakers' emotions can be perceived by the human ears of listeners, it is also possible for the computer to have the same ability: recognize emotion from the sound. Instead of speech, which has semantic or linguistic meanings, the burst of brief vocalizations is an interesting source for predicting expressed emotion in human social life since it contains rich information on emotion \cite{Cowen2019}.
Humans communicate emotion through two different kinds of vocalizations, prosody and vocal burst \cite{Scherer1986}. Prosody interacts with the words (linguistic) to convey feelings and attitudes via speech, while vocal burst just occurs without linguistic meanings. Examples are laughs, cries, sighs, shrieks, growls, hollers, roars, and oohs. While the study of the prosody of speech for speech emotion recognition is widely conducted (indicated by the number of available datasets), the study of the vocal burst emotion recognition is currently undergoing with support of new vocalization datasets, e.g., \cite{ Schuller2022,Baird2022}.
It is not the only emotion that can be recognized in the human's voice. Age, gender, and nationality could also be detected from voices. The change of voice in age is recognizable by both humans and computers, although it is the hardest among emotion and gender \cite{Kaya2017}. By transferring information about age and gender, the recognition of dimensional could be improved \cite{Zhao2018a}. The task of combining several tasks together based on the same or different inputs is known as multitask learning.
At the ICML Expressive Vocalizations (ExVo) Workshop and Competition, a multitask learning task is held to predict the average intensity of each of 10 emotions perceived in vocal bursts, the speaker's age, and the speaker's Native-Country. The challenge utilized a large vocalization dataset \cite{Cowen2019} which was shared across three different tasks. The emotion and predictions are regression problems; native-country prediction is a classification problem. The participants were evaluated by a single score of the harmonic mean from these three problems.
This paper contributes to evaluating the pre-trained speech embeddings extractor, which was trained specifically on an affective speech dataset, to jointly predict emotion, age, and the country as in the ExVo challenge. We hypothesize that using this kind of acoustic feature extractor will lead to better results than traditional feature extractors, which extract physical information of audio signals. The rest of this paper explains related work, methods, results and discussion, and conclusion.
\section{Related work}
Research on multitask learning, predicting several tasks simultaneously using unified architecture, progressively increased due to its effectiveness in predicting several outputs with the same or similar inputs. Speech processing is an ideal test bed for multitask learning. Several pieces of information could be extracted from the same speech input, whether it is text (automatic speech recognition, ASR), gender, age, nationality, emotion, language, and disease. The following is a resume of the work of multitask learning done in the past with key differences from this study presented at the end of this section.
Parthasarathy and Busso \cite{parthasarathy2017jointly} proposed two schemes of multitask learning architectures for evaluating valence, arousal, and dominance simultaneously. The first architecture is without an independent layer for each task (shared layers only), while the second architecture is with independent layers for each task. They found the second architecture with independent layers performed better than the first one without independent layers. The scores (measured in concordance correlation coefficient) also showed improvements from the baseline architecture with single-task learnings.
Lee \cite{Lee2019} also evaluated two architectures of multitask learning optimized for emotion recognition. The first architecture is composed of three task-specific softmax layers to predict gender, emotion, and language. The second architecture is composed of one softmax layer containing two tasks, language and emotion. The results revealed a better generalization of the second architecture to predict emotion categories.
Kim et al. \cite{Kim2017} proposed to use gender and naturalness information to minimize the large mismatch between the training and test data in speech emotion recognition in the wild. The method employed traditional acoustic features ($f_{o}$, voice probability, zero-crossing-rate, MFCC with their energies and first-time derivatives) extracted on frame-level and calculated the high-level features on top of it. The high-level features then were fed into an extreme learning machine to predict the categories of emotion. They obtain significant improvement over the baseline with single-task learning. The method was evaluated on five different datasets for generalization.
Atmaja and Akagi \cite{Atmaja2020b} evaluated different loss functions -- CCC, MSE, and MAE -- for multitask learning dimensional emotion recognition. The traditional approach of deep learning commonly employed MSE loss to minimize the error and training stage. Since the goal is to maximize CCC, they proposed to use CCC loss as the loss function to replace MSE. The results on two datasets (IEMOCAP and MSP-IMPROV) and two acoustic features (GeMAPS and python Audio Analysis) showed the consistency that CCC loss is superior to MSE and MAE.
Li et al. \cite{Li2020} proposed additional information on age, gender, and emotion for speaker verification using multitask learning and domain adversarial training. The multitask learning part minimizes losses of three variables (speaker, gender, and nationality). The domain adversarial training, which also employs multitask learning, minimizes losses of speaker and emotion. The results showed that multitask improved the performance from the baseline by about 16\% while the domain adversarial training improved the performance from the baseline by about 22\%. The baseline used ResNet networks.
Cai et al. \cite{Cai2021} employed multitask learning by predicting text characters and predicting emotion in the training phase. The model is trained to minimize the loss of categorical emotion (cross-entropy) and loss of character recognition (connectionist temporal classification). The inference phase removes the character recognition path to predicting emotion categories only. The proposed multitask learning achieved an accuracy of 78\% compared to 72\% of the baseline method with capsule networks.
Atmaja et al. \cite{Atmaja2022a} evaluated multitask learning of emotion recognition (dimensional) naturalness scores from speech. They evaluated two different architectures with and without independent layers. The architecture without independent layers (shared layers only) exhibits the best performance in predicting valence, arousal, and dominance scores. The shared layers have been built using three layers of fully connected networks with nodes of 512, 256, and 128. However, the scores for naturalness recognition in multitask learning is lower than in single-task learning.
Instead of focusing on the single task evaluation on multitask learning (e.g., only predicting emotion in multitask emotion and transcription \cite{Cai2021}, or emotion and language \cite{Lee2019}), this study focused on the all tasks evaluated on the multitask learning. We employed the harmonic mean evaluation from three metrics for three tasks and used this harmonic mean as the final evaluation. This multitask learning evaluation using all tasks was not evaluated on the previous tasks, where the authors only focused on emotion recognition or speaker verification.
\section{Methods}
\subsection{Datasets}
This study relies on the HUME-VB dataset, which is used at the ICML Expressive Vocalization (ExVo) Competition 2022. The dataset is a large-scale emotional non-linguistic vocalization known as vocal burst. An example of this vocal burst is ``argh!'' to express distress emotion. There are ten emotions rated in continuous scores. These emotions are Amusement, Awe, Awkwardness, Distress, Excitement, Fear, Horror, Sadness, Surprise, and Triumph. The data were collected from 1702 speakers aged 20-39 years old. The collection locations are China, South Africa, Venezuela, and the US. The total duration of the dataset is almost 37 hours (36 hours 47 minutes). Although the data were split into train, validation, and test, the test set was closed by the organizer of the competition. Hence, we evaluated our methods mostly on the validation set (except in the last part, where test results are reported). More details about the dataset can be found in the \cite{Baird2022}.
\subsection{Pre-trained Acoustic Embedding}
We evaluated a pre-trained model finetuned on an affective speech dataset. The model \cite{Wagner2022,Wagner2022a} is based on wav2vec2-large-robust model \cite{Hsu2021a}. The model is trained on MSP-Podcasts dataset \cite{Lotfian2019}, a large affective speech corpus derived from YouTube with valence, arousal, and dominance scores. For this finetuning, the combined samples on the MSP-Podcasts dataset have a combined length of roughly 21 hours. The model extracted the speech embedding from the dataset (HUME-VB) with a size of 1024 dimensions for each utterance. The output layers of the model, i.e., the logits, are the scores of arousal, dominance, and valence, in ranges [0, 1]. We experimented with two variants of speech embedding with this model. The first embedding is the hidden states of the last layer before the output layer (1024-dim), and the second embedding is the concatenation of hidden states with the logits (1027-dim). We named the first embedding ``w2v2-R-er" (wav2vec 2.0 robust emotion recognition), and the second embedding``w2v2-R-vad" (wav2vec 2.0 robust emotion recognition with valence, arousal, and dominance). Notice the term of acoustic embedding (extracted from vocal bursts) here is used instead of (traditonal) acoustic features or speech embedding (extracted from speech).
\subsection{Classifier}
We employed a model of multitask learning adopted from \cite{Baird2022}. The architecture of the model is shown in Figure \ref{fig:arch}. The model accepts the input of acoustic features depending on the size of the features. For instance, the dimension for w2v2-R-er is 1024. Then, two linear layers are stacked as shared layers. The number of nodes for each layer is 128 and 64, respectively. We applied layer normalization \cite{Ba2015} for each linear layer, followed by LeakyReLU activation functions. A group of layers builds up the independent layers for each task. For emotion and country prediction, there is only a layer followed by output layers. For age prediction, we used two independent layers to bridge the gap in the big number of nodes from the shared layer (64) to the single-node output layer. The nodes for these independent layers are 32 and 16, respectively.
The loss function minimizes losses of three tasks: MSE for emotion recognition age prediction, cross-entropy (CE) for country prediction. The total loss function ($\mathcal{L}_T$) is weighting sum of three losses given by the following formula,
\begin{equation}
\mathcal{L}_T = \sum_{i=1}^3 \left( \frac{\mathcal{L}_i}{2e^{\alpha}} + \frac{\alpha}{2} \right).
\end{equation}
The coefficient of $\alpha$ is set to be 0.34, 0.33, and 0.33 for emotion, country, and age, respectively.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{mtl-crop.pdf}
\caption{Architecture of multitask learning (MTL) for predicting emotion, age, and country. Emotion and age tasks are regressions; country prediction is a classification task; the number within the circle denotes units/nodes.}
\label{fig:arch}
\end{figure*}
\subsection{Evaluation Metrics}
The evaluation of three paralinguistic tasks in this study used specific metric for each task. The emotion recognition is evaluated in concordance correlation coefficient (CCC), age recognition is evaluated in mean absolute error (MAE), and nationality prediction is evaluated in unweighted average recall (UAR). These metrics are described below,
\begin{equation}
\mathcal{CCC}=\frac{2 \sigma_{x y}^{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+\left(\mu_{x}-\mu_{y}\right)^{2}}.
\end{equation}
The average CCC scores for ten emotion categories is the mean values,
\begin{equation}
\hat{\mathcal{C}}=\sum_{i=1}^{10} \mathcal{CCC}_{i} / 10.
\end{equation}
CCC is in range [-1, 1] with -1 for perfect disconcordance, 0 for absence of concordance/disconcordance, and 1 for perfect concordance.
Next is metric to evaluate the performance of nationality/country prediction that is unweighted accuracy, also known as unweighted average recall (UAR) and balanced accuracy. Unweighted accuracy is formulated as,
\begin{equation}
\hat{\mathcal{U}} = \frac{1}{4} \sum_{i=1}^{4}Recall_i
\end{equation}
\noindent
where $i$ is the corresponding country class, and $4$ is the number of countries (USA, China, South Africa, and Venezuela). UAR ranges in 0-100 in \% or 0-1 in normalized score.
The metric to evaluate the last task, age prediction, is mean absolute error (MAE). MAE is a common metric for evaluating regression, and it is scale-dependent. The lower the scores, the better the age prediction. The formulation of MAE is given by
\begin{equation}
MAE=\frac{1}{n} \sum_{i=1}^{n}\left|x_{i}-y_{i}\right|,
\end{equation}
where $n$ is the number of samples in the evaluation or test sets (which one is used to calculate the score).
Since MAE is scale-dependent, we inverted the MAE scores for consistency with the previous two metrics,
\begin{equation}
\hat{\mathcal{M}}=1 / MAE.
\end{equation}
Now, for all metrics (CCC, UAR, 1/MAE), the higher scores, the better predictions of emotion, country, and age. Finally, for calculating overall performance, we used the harmonic mean of three metrics above \cite{Baird2022},
\begin{equation}
S_{MTL} = \dfrac{3}{(1 / \hat{\mathcal{C} } + 1 / \hat{\mathcal{M} } + 1/ \hat{\mathcal{U} })}.
\end{equation}
$S_{MTL}$ is our main metric to judge the performance of the evaluated methods (hyperparameters, features, normalization). Other previous metrics are used to determine the performance of the corresponding method for individual tasks in multitask learning.
\section{Results and Discussion}
We presented our results in different ablation studies: choosing the right seed, comparing different acoustic features, effects of normalization, and test results. For each study, we run the experiment five times for each setting. For instance, in choosing the right seed, we run the experiment five times on seed "101". The reported results are the average scores, except for comparing acoustic features. The reported scores for w2v2-R-er and w2v2-R-vad are chosen from the best from five different runs, similar to the baseline. For the baseline results, we quote the scores from the source \cite{Baird2022}.
\subsection{Choosing the right seed}
Seed initialization is an important step in deep learning methods \cite{Pepino2020,Macary2020}. Here, as a first step, we choose the seven different seed values to choose the best one. We evaluated seed values 42, 101, 102, 103, 104, 105, 106. The reason for adding seed 42 to the other six seed values given in the baseline is due to its common use in the deep learning community.
Table \ref{tab:seed} shows the results of using different seed values for multitask learning of emotion, age, and country on the validation set. The results are average scores of five trials with their standard deviation. Since the values of STD are similar in other ablation tests in this study, we only report these STD scores in Table \ref{tab:seed}. The following reported scores, except stated, used the seed values of "106" which obtained the highest $S_{MTL}$ score from seed values evaluation.
\begin{table*}[htbp]
\caption{Evaluation of different seed values for initialization on validation set (mean values $\pm$ standard deviation); Feature: w2v2-R-er}
\centering\begin{tabular}{c c c c c}
\hline
Seed value & Emo-CCC & Cou-UAR & Age-1/MAE & $S_{MTL}$ \\
\hline
42 & 0.534 $\pm$ 0.010 & 0.520 $\pm$ 0.006 & 0.245 $\pm$ 0.005 & 0.3806 $\pm$ 0.003 \\
101 & 0.543 $\pm$ 0.006 & 0.520 $\pm$ 0.004 & 0.242 $\pm$ 0.003 & 0.3798 $\pm$ 0.004 \\
102 & 0.529 $\pm$ 0.002 & 0.531 $\pm$ 0.003 & 0.239 $\pm$ 0.002 & 0.3773 $\pm$ 0.002 \\
103 & 0.539 $\pm$ 0.004 & 0.512 $\pm$ 0.005 & 0.236 $\pm$ 0.003 & 0.3728 $\pm$ 0.003 \\
104 & 0.537 $\pm$ 0.004 & 0.527 $\pm$ 0.006 & 0.247 $\pm$ 0.005 & 0.3838 $\pm$ 0.004 \\
105 & 0.541 $\pm$ 0.005 & 0.518 $\pm$ 0.006 & 0.234 $\pm$ 0.008 & 0.3725 $\pm$ 0.008 \\
106 & 0.537 $\pm$ 0.010 & 0.527 $\pm$ 0.006 & 0.247 $\pm$ 0.005 & \textbf{0.3844 $\pm$ 0.003} \\
\hline
\end{tabular}
\label{tab:seed}
\end{table*}
\subsection{Comparison of different features}
The main focus of this study is to evaluate the acoustic embedding extracted using pre-trained models trained on the emotional speech dataset. Most speech processing tasks, especially speech emotion recognition, are trained using handcrafted acoustic feature extractors (e.g., MFCC, spectrogram, or mel-spectrogram) \cite{Atmaja2022}. Others are using pre-trained models but trained on neutral speech (e.g., wav2vec2 2.0, HuBERT, WavLM). In this study, we utilized a pre-trained model \cite{Wagner2022a,Wagner2022a} built using wav2vec 2.0 Robust on affective speech dataset.
Table \ref{tab:feat} proves our presumption that our model will surpass the baseline scores. Both w2v2-R-er and w2v2-R-vad obtain higher $S_{MTL}$ scores than the baseline scores on the same configuration (batch size, seed, and other hyperparameters). Specifically, these scores obtained by two acoustic embeddings show the most remarkable improvement in emotion recognition score, in which the model to extract the acoustic embedding is trained. Not only for the emotion recognition task but both scores for age and country predictions were also improved. The pre-trained model is shown to be helpful on other tasks probably due to the similarity of the task (paralinguistics and non-linguistic tasks), and the data trained to build the model contains the age and country information embedded on the extracted acoustic embedding. The data to train the model is MSP-Podcast in the English language. Although the dataset contains English only language, the pre-trained model may be able to discriminate between English with non-English language (related to country prediction) as in anomaly detection problems.
\begin{table}[htbp]
\caption{Comparison of best scores from different acoustic features on validation set; scores from ComParE to DeepSpec are obtained from \cite{Baird2022}}
\centering\begin{tabular}{l c c c c c}
\hline
Feature & Dims. & CCC & UAR & 1/MAE & $S_{MTL}$ \\
\hline
ComParE & 6373 &0.416 & 0.506 & 0.237 & 0.349 \\
eGeMAPS & 88 &0.353 & 0.423 & 0.249 & 0.324 \\
BoAW & 125 & 0.335 & 0.417 & 0.234 & 0.311 \\
& 250 & 0.354 & 0.423 & 0.238 & 0.319 \\
& 500 & 0.374 & 0.432 & 0.218 & 0.314 \\
& 1000 & 0.384 & 0.438 & 0.225 & 0.321 \\
DeepSpec & 4096 & 0.369 & 0.456 & 0.227 & 0.322 \\
w2v2-R-er & 1024 & 0.533 & 0.523 & 0.252 & 0.386 \\
w2v2-R-vad & 1027 & 0.534 & 0.525 & 0.253 & \textbf{0.388} \\
\hline
\end{tabular}
\label{tab:feat}
\end{table}
\subsection{Evaluation of different batch sizes}
We evaluated five different batch sizes since there is evidence the influence of batch size, particularly for emotion recognition task \cite{Wu2018}. As found in the \cite{Wu2018}, we also found that the smallest evaluated batch size in this study resulted in the best performance. As the batch size increases, the performance decreases. For the best performance on the use of batch size = 2 (scores in Table \ref{tab:batch} are average of 5 runs), we obtained the following scores: 0.534, 0.533, 0.267, and 0.401 for Emo-UAR, Age-CCC, Cou-1/MAE, and $S_{MTL}$, respectively. This is the highest $S_{MTL}$ score obtained in this study.
\begin{table}[htbp]
\caption{Average scores of different batch sizes from 5 runs on validation set; Feature: w2v2-R-vad}
\centering\begin{tabular}{c c c c c}
\hline
Batch size & CCC & UAR & 1/MAE & $S_{MTL}$ \\
\hline
2 & 0.532 & 0.508 & 0.264 & \textbf{0.393} \\
4 & 0.535 & 0.502 & 0.241 & 0.374 \\
8 & 0.534 & 0.525 & 0.252 & 0.387 \\
16 & 0.535 & 0.511 & 0.236 & 0.372 \\
32 & 0.516 & 0.505 & 0.228 & 0.361 \\
\hline
\end{tabular}
\label{tab:batch}
\end{table}
\subsection{Effect of waveform normalization}
Normalization may affect the performance of deep learning, particularly in speech processing. This effect is due to the model usually only working on the standard input and being prone to high deviation. It has been proved that such normalizations are effective for deep learning, e.g., batch normalization \cite{Ioffe2015}, group normalization \cite{Wu2018}, and layer normalization \cite{Ba2015}. Aside from layer normalization, we also evaluated the model with waveform normalization. In this case, we utilize librosa toolkit \cite{McFee2020} to normalize the audio array (the amplitude of waveform), i.e., the output is in the range [-1, 1]. Then, two acoustic embeddings are created with these normalizations, namely w2v2-R-er-norm and w2v2-R-vad-norm. The results are shown in Table \ref{tab:norm}.
As shown in Table \ref{tab:norm}, there are no such improvements by normalizing the waveform of speech. This finding may be explained by the fact that emotion, age, and gender are related to the loudness of the waveform. Hence, such normalization of the waveform will remove important information that discriminates these paralinguistics labels. The results for two acoustic emebddings, w2v2-R-er and w2v2-R-vad, are consistent with and without normalization, highlighting the unnecessary processing of normalizing the speech waveform.
\begin{table}[htbp]
\caption{Comparison of the acoustic embeddings with and without normalizations on validation set}
\centering\begin{tabular}{l c c c c c}
\hline
Feature & Batch size & CCC & UAR & 1/MAE & $S_{MTL}$ \\
\hline
w2v2-R-er & 8 & 0.537 & 0.527 & 0.247 & 0.384 \\
w2v2-R-er-norm & 8 & 0.542 & 0.524 & 0.242 & 0.381 \\
w2v2-R-vad & 2 & 0.532 & 0.508 & 0.264 & \textbf{0.393} \\
w2v2-R-vad-norm & 2 & 0.528 & 0.519 & 0.260 & 0.391 \\
\hline
\end{tabular}
\label{tab:norm}
\end{table}
\subsection{Test results}
We presented the results of our predictions from the previous validation evaluation for the closed test set evaluation. Note that the best result from the validation set reported in this study was not submitted to obtain the test scores due to time limitations (for obtaining the test scores as part of The ICML 2022 Expressive Vocalizations Workshop and Competition). In the submitted results, we evaluated two predictions from both w2v2-R-er features set with different seed and batch sizes. The first prediction is with a batch size of 8 and a seed of 42. The second prediction is with a batch size of 4 and a seed of 106. Note that this last prediction is submitted with the original architecture \cite{Baird2022} with one hidden layer for all independent layers.
Table \ref{tab:test} shows the score of harmonic mean ($S_{MTL}$) from our predictions (last two rows) and the baseline \cite{Baird2022}. Although we did not use the best-reported validation scores in this study, our submitted results are still higher than the baseline results. These scores reveal the effectiveness of the evaluated acoustic embedding, w2v2-R-er, which were extracted using the robust version of wav2vec 2.0 \cite{Baevski2020a} on the MSP-Podcast emotional speech dataset \cite{Lotfian2019}. We believe that the test score will be even higher for w2v2-R-vad with batch size 2 since there are remarkable improvements on the validation set by using this configuration (the best validation $S_{MTL} = 0.401$).
\begin{table}[htbp]
\caption{Results of the submitted predictions on the test set; 'best' in seed column is obtained from values in range [101, 102, 103, 104, 105, 106].}
\centering\begin{tabular}{l c c c}
\hline
Feature & Batch size & Seed & $S_{MTL}$ \\
\hline
ComParE & 8 & best & 0.335 \\
eGeMAPS & 8 & best & 0.214 \\
BoAW-125 & 8 & best & 0.299 \\
BoAW-250 & 8 & best & 0.305 \\
BoAW-500 & 8 & best & 0.302 \\
BoAW-1000 & 8 & best & 0.307 \\
BoAW-2000 & 8 & best & 0.303 \\
DeepSpec & 8 & best & 0.305 \\
w2v2-R-er & 8 & 42 & 0.358 \\
w2v2-R-er & 4 & 106 & \textbf{0.378} \\
\hline
\end{tabular}
\label{tab:test}
\end{table}
\section{Conclusions}
In this paper, we reported evaluations of multitask learning to jointly predict emotion, age, and country by using acoustic emebdding extracted from a pre-trained model. The model is finetuned on an affective speech dataset. The extracted acoustic emebddings were fed to an architecture consisting of shared layers and independent layers for these three tasks. The results showed improvements over the baseline methods with the common speech representations (ComParE, eGeMAPS, BoAW, and DeepSpectrum). Two variants of acoustic embeddings are evaluated with original hidden states and concatenation of hidden states with logits. The latter performed best on the validation set. We conducted ablation studies on different seeds, batch sizes, and normalization. The study finds the optimum seed and batch size of the evaluated ranges and finds no improvement in performing waveform normalizations. While this study treated all emotions in the same weights, future studies may be directed to adjust these weights for optimum emotion recognition, as well as to improve overall harmonic mean evaluation for all tasks in Multitask learning.
\section*{Acknowledgment}
This paper is based on results obtained from a project, JPNP20006, commissioned by the New Energy and Industrial Technology Development Organization (NEDO), Japan.
| {
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Silicon dioxide nanoparticles coated with a high dielectric constant polymer provides improved cooling for increasingly power-hungry electronic devices.
"We have shown for the first time that you can take a packed nanoparticle bed that would typically act as an insulator, and by causing light to couple strongly into the material by engineering a high dielectric constant medium like water or ethylene glycol at the surfaces, you can turn the nanoparticle bed into a conductor," said Cola, an associate professor in the George W. Woodruff School of Mechanical Engineering at the Georgia Institute of Technology. "Using the collective surface electromagnetic effect of the nanoparticles, the thermal conductivity can increase 20-fold, allowing it to dissipate heat."
In the last several years, theoretical papers have predicted the ability of surface phonon polaritons to increase thermal conduction in nanomaterials made from polar materials like silicon dioxide. Polaritons are quantum quasiparticles produced by strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. In the specific case of surface phonon polaritons, the electromagnetic waves are coupled to a certain frequency and polarisation of vibrating atoms in the material known as optical phonons. When materials are reduced to sizes below 100 nanometres, the surface properties of the material dominate over bulk properties, allowing phonons of heat to flow from particle to particle in the closely packed bed with the assistance of the coupled electromagnetic waves.
The researchers decided to experiment with those special properties, first using water to coat the nanoparticles and turn the silicon dioxide nanoparticle bed into a conductor. But the water coating was not robust, so the researchers switched to ethylene glycol, a fluid commonly used in vehicle antifreeze. The new combination increased the heat transfer by a factor of 20 to approximately 1W per meter-kelvin, which is higher than the value ethylene glycol or silicon dioxide nanoparticles could produce alone, and competitive with expensive polymer composites used for heat dissipation.
The research, which involved both theory and experiment, has been published online in the journal Materials Horizons, and was highlighted in the journal Science. The work was supported by the Air Force Research Laboratory and the U.S. Air Force. Co-authors include Professor James Hammonds at Howard University, and graduate students Eric Tervo from Georgia Tech and Olalekan Adewuyi from Howard University. | {
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Jan de Lichte (baptisé le 7 avril 1723 à Velzeke et mort à Alost, le 14 novembre 1748) était un criminel belge et un chef de bande. Durant la guerre de Succession d'Autriche, sa bande et lui-même, poussés par la pauvreté et la misère, ont commis des vols puis des meurtres. Sa tête a été mise à prix, et il fut arrêté par les troupes d'occupation françaises et exécuté en place publique par des bourreaux.
La bande de Jan de Lichte et sa fin ont donné vie à des légendes et un folklore en Flandre. De nos jours, Jan de Lichte est également connu pour l'œuvre littéraire de Louis Paul Boon dans son livre ''La bande de Jan de Lichte'' et par la diffusion d'une mini série télévisée, Bandits des Bois, sur Netflix à partir du 20 janvier 2020.
Biographie
La jeunesse
Jan de Lichte naît en 1723 près du Driesmolen à Velzeke en Belgique dans la famille de Joël ou Joseph (latinisé en Jodocus dans les registres catholiques de baptême) de Lichte et d'Elisabeth de Schepper. L'arrêt de mort de 1748 mentionne un âge de 23 ans, mais le registre de la paroisse de Velzeke mentionne son baptême en 1723. Ses parents étaient alors déjà d'âge mûr : en 1748, Joseph de Lichte avait 69 ans et Elisabeth de Schepper 70.
Pour gagner leur vie, ils ont dû faire appel à la "tablette du Saint-Esprit", une œuvre d'aide charitable pour les plus nécessiteux. Les parents de Jan et d'autres membres de la famille, comme son frère aîné Pieter de Lichte et son oncle Joannes de Schepper, ont été reconnus coupables de vol.
jeune, Jan est fanfaron, passionné et bravache.
Pour sortir de la pauvreté, Jan s'enrôle pour l'armée autrichienne, puis dans l' armée hollandaise, mais déserte à chaque fois peu de temps après. En 1740, il commet ses premiers larcins à Dikkele et Strijpen. Il erre alors avec d'autres mendiants et vagabonds, rencontrés au cours des errances sur les routes belges.
Le premier acte violent signalé de Jan de Lichte date de 1743 lorsqu'il tire un pistolet sur des pèlerins à la chapelle Onze-Lieve-Vrouw van Deinsbeke à Zottegem.
Le soldat
À la suite de la guerre de Succession d'Autriche, les troupes du royaume de France occupent les Pays-Bas autrichiens en 1745. L'obligation de nourrir les de l'armée française rend les conditions de vie encore plus difficiles pour les quelque d'Alost et des villages environnants.
La bande
La bande n'est pas une organisation structurée, mais une alliance tacite entre voleurs, gitans et d'autres personnes déclassées qui s'associent au gré des circonstances.
Les maraudeurs volent dans des fermes isolées, des moulins à vent, des forges, chez des personnes ordinaires. Le butin consiste principalement en vêtements, qui sont revendus aux guérisseurs, et en nourriture pour l'usage propre des membres de la bande. L'argent est rare et donc peu volé. Les délits ne sont souvent pas des exploits. ; Jan de Lichte et ses complices sont notamment chassés avec succès en 1747 lors d'une tentative de vol.
Les membres de la bande les plus durs suivent souvent Jan de Lichte.
La bande sévit essentiellement entre Zottegem et Grammont et est surtout active à partir de 1747. Le groupe comprend Francis van der Geenst ("Tincke"), Francis van den Haute ("Abeel"), Anthone van der Gucht ("Tone den Breteur"), Jan Savoye ("Klein Janneken"), Francis Meulenaere, Adriaan Vagenende, les frères Jean et Jacques Couvreur, Lieven Faviel, Jan de Vrieze, Gillis van der Elst et Simon Ysenbaert. Il n'y a aucune indication dans les sources historiques que Jan de Lichte fut formellement le chef.
Les bandits se cachent dans les bois et dissimulent leur butin dans des auberges comme De Honger (La famine) à Audenhove-Sainte-Marie et De Ealing à Aspelare.
Cependant, des conflits surgissent souvent entre bandits, à propos du partage du butin ou des réclamations au sujet des femmes avec lesquelles ils vivent ensemble sans être mariés.
Une bande violente
Le troisième jour de la Pentecôte 1748 (le 2 juin), alors que Jan de Lichte et ses accolites jouaient krulbol dans une auberge de Scheldewindeke, un différend éclate avec Jan de Vrieze, et dégénère en rixe. Jan de Lichte poignarde De Vrieze à l'estomac.
Les compagnons de Lichte, Vagenende et Meulenaere traînent le corps sur une centaine de pas et le jettent dans l'eau, non sans l'avoir d'abord déshabillé. Lors de son arrestation, Meulenaere portait toujours les vêtements de la victime.
C'est alors qu'un changement se produit chez Jan de Lichte : la violence monte contre ses anciens camarades. Gillis van der Elst est assassiné à Bavegem. À coups de pistolet dans le dos, De Lichte tente de se débarrasser de ses rivaux Tincke et Pieter van de Putte. Dans la nuit du 30 au 31 juillet il tue Jan Dossche, qui n'avait pour tort que de l'avoir surpris, d'un coup de pistolet. Dans la nuit du 15 au 16 août, à Grammene, Marie Anne de Smet,épouse de Pieter Bovjin est assassinée par De Lichte et Vagenende à coups de couteau et coups de pistolet.
Fin de la bande
Après le Chute de Maastricht le 7 mai 1748, un armistice intervint dans la guerre de Succession d'Autriche, permettant aux forces d'occupation françaises de porter leur attention vers les brigands qui infestent le pays. Le 28 septembre 1748, une chasse à l'homme a été organisée au cours de laquelle tous les résidents de la castellany et de Courtrai, Alost, Termonde et Audenarde ont été convoqués pour arrêter quiconque était sans papiers valides ni résidence permanente. Environ 130 personnes ont été arrêtées. Certains ont été emprisonnés au Beffroi d'Alost, mais en raison du grand nombre de personnes arrêtées, ils ont également été emprisonnés dans les locaux des rhéteurs de Sainte Barbara et de Sainte Catherine.
Du 7 octobre au 14 décembre 1748, un peu plus de 100 personnes ont été condamnées. Les peines ont été exécutées le jour même ou le lendemain.
Le 14 novembre 1748 à onze heures et demie, Jan de Lichte est exécuté par les bourreaux. Ce châtiment, rarement vu aux Pays-Bas, visait à priver le condamné de toute chance de résurrection le jour du jugement dernier. Quatre autres meurtriers (Simon Ysenbaert, Lieven Faviel, Augustijn Hendricx, Jan de Priester) ont subi le même sort. 17 autres ont été pendus; les femmes et les guérisseurs étaient généralement fouettés et bannis. 11 membres de la bande qui ont réussi à s'échapper à temps ont été condamnés à mort par contumace.
Hommages
Héritage littéraire
Jan de Lichte est devenu un capitaine légendaire, voleur, terrifiant et la bande est devenue «sa» bande. Dans le folklore flamand, le personnage de Jan de Lichte est mêlé à ceux d'autres bandits, comme les Bokkenrijders (actifs dans la vallée de la Meuse), et les relieurs et brûleurs de pieds (actif pendant la Révolution française).
Selon une étude achevée en 1967, des histoires mettant en scène Jan de Lichte et sa bande sont contées dans au moins 35 communes de Flandre orientale.
Certaines histoires illustrent la cruauté du chef de bande : dans une histoire récurrente, par exemple, il aurait versé une coulée d'étain bouillant dans la bouche d'une personne endormie. Dans d'autres histoires, l'arrestation de Jan de Lichte est évoquée: lors de la chasse à l'homme, il se serait caché dans un arbre creux et aurait été trahi par un chien qui aboyait. Enfin, il y a eu plusieurs versions disparates de ses derniers mots lors de son exécution
En 1874, le greffier de la ville de Wetteren, E. Ternest, publia «Jan de Lichte et sa bande». Présenté comme un «livre folklorique» qui, avec son contenu sensationnel et écrit dans une langue vernaculaire (en flamand) accessible, connut un certain succès. Le travail de Ternest deviendra l'inspiration pour les versions suivantes.
L'écrivain régionaliste Abraham Hans a publié en 1908 (sous le pseudonyme de Hans Van Horenbeek) le roman «Jan de Lichte et sa bande noire de voleurs».
Statuaire
Après la mort de Boon en 1979, la société Louis Paul Boon a voulu rendre hommage à l'écrivain et a chargé l'artiste Roel D'Haese de réaliser une statue. Cependant, au lieu de concevoir une représentation figurative de l'écrivain, il préféra une statue de trois mètres de haut représentant Jan de Lichte, qui était selon lui le personnage principal du principal roman de Boon. L'intention de D'Haese et d'Hugo Claus était que cette statue fut érigée sur la grand place (Grote Markt) d'Alost. Cependant, le conseil municipal d'Alost n'a pas souhaité qu'un tel hommage fut rendu à un bandit. La statue a été offerte à la commune de Zottegem, avec l'intention de l'ériger dans le village natal de Jan de Lichtes, Velzeke. Pour les mêmes raisons qu'à Alost, la requête y fut également rejetée. Finalement, à la fin de l'année 2009, la statue s'est retrouvée dans le musée de sculpture de plein air du parc Middelheim, au sud d'Anvers.
Depuis 1988, Le panneau Jan de Lichte pad marque des lieux où serait né et aurait vécu jan de Lichte.
Cinéma et télévision
Depuis janvier 2020, la plateforme de vidéo à la demande Netflix diffuse une série historique en costumes en une saison de 10 épisodes "Bandits des Bois" en version française et "De Bende van Jan de Lichte" en version originale.
Autres hommages
En 2005, la brasserie belge De Glazen Toren a créé la bière Jan de Lichte.
En 2012, dans le cadre des journées du patrimoine, à l'occasion de l'année Boon, le procès et l'exécution de Jan de Lichte ont été rejoués à Alost, sur une mise en scène de Anton Cogen. L'avocat Jef Versassen a joué le défenseur. Le spectacle a ensuite été rejoué à Velzeke.
Notes et références
Naissance en avril 1723
Naissance à Zottegem
Décès en novembre 1748
Décès à Alost
Personnalité de la guerre de Succession d'Autriche
Personnalité belge condamnée pour crime
Décès à 25 ans
Mort sous la torture
Condamné à mort exécuté au XVIIIe siècle
Condamné à mort exécuté en Belgique | {
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Guyers House near Corsham, Wiltshire is a building of historical significance and is listed on the English Heritage Register. It commenced modestly as a farmhouse in about 1670 and sections were added to it progressively through the centuries. It is now a hotel and restaurant and caters for special events, particularly weddings.
Early residents
The first residents of the house were the Snelling family who built a small farmhouse. In 1678 they sold it to Edward Bayley who transformed it into a very substantial building. A sketch of this house is shown. Edward was a member of the Quakers who were very prominent in Pickwick at this time and he is listed in the Quaker burials. Sometime after his death in 1707 it was sold to the Bennett family. The most likely Bennett would be Thomas Bennett who opened a Quaker Boarding School in Pickwick in about 1725. After Thomas died in 1764 his son John Bennett took over teaching at the school. A map of 1773 (shown at left) shows that John Bennett owns a property called Guyers at this time. John died in 1802 and by 1813 Thomas Pycroft (1774-1849) owned the house.
Thomas Pycroft was barrister who practiced in Bath. He was born in 1774 in Essex and gained his qualifications in 1811. In 1805 he married Mary Collinson. The couple lived in Hempstead, Essex for several years then in about 1813 moved to Guyers House. They had seven children two of whom two became notable as they have entries in the Dictionary of National Biography. In about 1830 Thomas and his wife moved to Bath. He died in 1849 and is buried in St Bartholomew's churchyard, Corsham. His tombstone is well preserved and states that he lived at one time in Guyers House. Reverend Timothy Conyers moved to Guyers House with his mother in about 1831.
Reverend Timothy Conyers was born in about 1790. After his father died his mother Susan married in 1812 Reverend Henry Brindley who was famous for his speeches against cruelty to animals. Brindley died in 1819 and Timothy Conyers went to live with his widowed mother. They moved to Guyers House in about 1830 and in 1835 his mother died. After this Timothy married Eliza Richards who was 30 years his junior and the couple remained at Guyers until 1850. When Timothy died in 1850 the house was put on the market for sale. The sale notice is shown.
Captain William Wallace Rooke (1816-1864) was the next resident and he remained there until about 1858.
Residents after 1860
Nugent Chichester (1828-1908) bought Guyers House in about 1860 and remained there for the next twenty years. He was born in Geneva in 1828 and was the eldest son of Mr. Joseph Chichester Nagle, and Lady Henrietta Caroline Fellowes, a daughter of the 4th Earl of Portsmouth. Nugent served in the 7th Dragoon Guards until he was about 32 and then moved to Guyer House. In 1856 he married Amelia Mary Lamb daughter of Joseph Lamb of Axwell Park, County Durham and the couple had nine children – six sons and three daughters. In 1880 Nugent inherited his father's property Calverleigh Court at Tiverton and his family moved from Guyers.
From about 1881 until 1895 Captain Lancelot Charles Brown (1825-1895) resided with his wife Sarah Maria at Guyers House. He was the great grandson of Lancelot "Capability" Brown (1716-1783) the famous British landscape architect. After he died in 1895 Henry Stratton Coles bought the house.
Henry Stratton Coles (1864-1914) was a retired banker in Bath. In 1888 he married Edith Maude Symons daughter of Thomas George Symons of Mynde Park in Hertfordshire. They had one daughter Bridgett Stratton Coles.
In about 1910 Knightley Stalker Dunsterville bought Gurneys House. His grandson Hugh gave an account of this part of his life. He said that after Knightley bought the house his two sons who were in the Army came home on leave and stayed with their parents. It was customary for the family to go to Corsham Church and it was here that his son Graham Eadley Dunsterville met his future wife Eviline Goldney. In 1912 the couple were married and a year later had their first child. Graham went to join his regiment and Eviline came to Guyers House to live. Unfortunately Graham was killed soon after the war started in 1914. He was trying to rescue a wounded soldier. A photo of Graham is shown.
In 1921 there was a for sale notice in the newspapers and Lieutenant Colonel Donald James Handford bought the house. He had married in 1916 Elinor D'Arcy who was the daughter of a Charles D'Arcy a Bishop in Ireland. The couple lived there for over fifty years.
Guyers House today
Today Guyers House is a hotel and restaurant.
References
External links
Country houses in Wiltshire
Hotels in Wiltshire
Corsham
Grade II listed buildings in Wiltshire
Country house hotels | {
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Moritz Leuenberger (Biel, 21 september 1946) is een Zwitsers politicus. Hij is lid van de socialistische partij. Hij is een advocaat en woont in Zürich.
Hij maakte vanaf zijn verkiezing op 27 september 1995 tot 31 oktober 2010 deel uit van de Zwitserse regering (Bondsraad). Hij was bondspresident in 2001 en 2006. Hij was plaatsvervangend bondspresident in 2000 en 2005. Op 2 december 2009 werd hij ten derden male gekozen tot vicepresident, voor het jaar 2010, echter door zijn aftreden in 2010, werd hij niet voor de derde keer bondspresident.
Moritz Leuenberger leidde gedurende zijn gehele regeerperiode het ministerie van Verkeer, Milieu, Energie en Communicatie.
Einde 2004 raakte hij in zwaar water, omdat verschillende verkeersprojecten met budgetoverschrijdingen en vertragingen te maken kregen. Ook het staatsverdrag met Duitsland voor het aanvliegen van luchthaven Zürich-Kloten over Duits grondgebied werd als veel te mager beschouwd.
Leuenberger trad af in oktober 2010 en werd opgevolgd door zijn partijgenoot Simonetta Sommaruga.
Lid van de Zwitserse Bondsraad
SP-politicus (Zwitserland) | {
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{"url":"https:\/\/www.normalabnormal.com\/connecting-to-sourcetv-with-password.html","text":"In order to connect to SourceTV with a password set the following syntax is used in console\u2026\n\nconnect SERVER_IP; password PASSWORD_HERE\n\n\nFor example\u2026\n\nconnect 192.168.0.1:27016; password mypassword\n\n\nHope this helps.","date":"2022-06-28 06:18:13","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\": 0.18276149034500122, \"perplexity\": 10854.128034648638}, \"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-2022-27\/segments\/1656103355949.26\/warc\/CC-MAIN-20220628050721-20220628080721-00612.warc.gz\"}"} | null | null |
The Florence Rideout Elementary School Chorus sings Dharma's song. Updated story here June 3, 2018.
Wow, so much has happened since Christmas.
Florence Rideout Elementary School in Wilton, New Hampshire.
Mama Chell and Papa Sommese told me all about it.
Being a dog, I can't really go to school, so I kind of live through them. And that is okay by me. The whole thing has been SO MUCH FUN!
Anyway, it began with Mama meeting with Nancy Tong, retired music teacher. DID YOU KNOW that Mama woke up one morning, singing the tune to my song. It was so COOL. I mean like, it would not go away. Da da da da da da da da da. I mean over, and over and over. She was still teaching then and since she was NOT GETTING ANY SLEEP, she decided to go and ask Mrs. Tong if she could put the music and words down, ON PAPER, because well, she was NOT GETTING ANY SLEEP until this was done. So kind Mrs. Tong did this. Then student Miski came and volunteered to sing it. Whew, Mama said, it was good to get it out of her head because clearly, IT NEEDED TO COME OUT. So, that was about four years ago, and all the music sat in a binder, on the shelf, until just this year. So Mama, went back to Mrs. Tong and asked her to help her to finish the whole, GETTING IT ALL TOGETHER part of the process which she did. Once that was done, Mama was looking for kids that might like to sing it. To the rescue again, Mrs. Tong suggested that it might work better if we found a group of kids, like a choir, and got them to sing the song, I think mostly so that they would ALL BE IN TUNE. I guess that makes sense. Any way, she said she knew a music teacher, Mrs. Aparo, who taught at the Florence Rideout Elementary School and would see if she had interest. This was SSSOOOOOOOOOOOO great because this is where Dharma Doxie began!!!.
People don't know this but six years ago Mama took the simple pictures that artist Judi Wing drew for the book and Mama added the story and read it to three classrooms at the time. Kids loved it so Mama decided to go on and see where we would go. I am sure you can see how wonderful the pictures are from the book samples on the front page, so here we are. We were so happy for the opportunity to go back to FRES and have students from there sing my song.
That is them singing my song on the HOME page.
Mrs. Aparo, the wonderful music teacher there took my song and her team of over forty students practiced it every Thursday during chorus, and also on their recess time because Thursday was just not enough time. They VOLUNTEERED to do this. How wonderful. Anyway they only had chorus on Thursday, which Mama Chell would go there and hear them practice. It warmed her heart. What dedication!! Thank you guys.
Then again, how to get it recorded. Mrs. Tong, to the rescue again. She suggested Mr. Jim McClure, from Betsy's Folly Studios in Lyndeborought N.H. and with Mrs. Tong, a very accomplished key board musician accompanying them as they sang the song, with Mrs. Aparo's direction. They were good.
It all came together and Jim recorded my song on March 29th.
Thank you Mrs. Tong and Mr. Jim. You can see their pictures below.
I was so proud. And excited, I almost peed the floor.
Our adventure continued because for two days before the spring concert, Mama and Papa went to FRES and read my book, teaching the little kids the sight words and the tune for the refrain (she is such the teacher) and then Mama sang the song to the Little Kids, with them singing my refrain. What fun!!!
Then they got to see my Dharma the Bee video, up on You Tube.
Mama and Papa taught every student how to give Dharma Hugs as they were leaving the classroom.
I give Dharma Hugs by putting my paws up, you put up your paws (hands) and we (high five) hug each other.
Then each student got a free Dharma Hugs sticker. See the pictures below.
I will say, sssssssshhhhhhhhhh, don't tell Mama and Papa, but any student who sends me a Self Addressed Stamped Envelop, I will send you a Dharma Hugs sticker. ABSOLUTELY FREE. Write me back. Let me know how you like my BLOG.
Dharma Doxie song book, just so they will remember this adventure.
The whole night was such fun.
The school's mascot is a fox so Mama Chell put it on this log entry so you could see it.
She loves birches, so she always stopped an admired the mosaic work that student in the school did themselves.
Anyway, did I say all of this was way too COOL? | {
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} | 2,168 |
{"url":"https:\/\/studydaddy.com\/question\/i-understand-what-its-asking-me-to-do","text":"QUESTION\n\n# I understand what its asking me to do.\n\nI need help symbolizing these statements.\u00a0I understand what its asking me to do. I just need help translating them into a symbolic expression and I get confused on where exactly which quantifier goes where, whether they're apart or together. Like for example: \u00a0Someone is having her cake and eating it too. (Hxy: x is having y; Px: x is a person; Cx: x is a cake; Exy: x is eating y) can be translated as ($x)($y)[(Px \u2022 Cy) \u2022 (Hxy \u2022 Exy)] where \\$ is the existential quantifier. Thanks very much again for all the help.\n\nYou\u00a0can\u00a0fool\u00a0some\u00a0of\u00a0the\u00a0people\u00a0all\u00a0of\u00a0the\u00a0time\u00a0and\u00a0all\u00a0of\u00a0the\u00a0people\u00a0some\u00a0of\u00a0the\u00a0time,\u00a0but\u00a0you\u00a0can't\u00a0fool\u00a0all\u00a0of\u00a0the\u00a0people\u00a0all\u00a0of\u00a0the\u00a0time.\u00a0(Fxyz:\u00a0x\u00a0can\u00a0fool\u00a0y\u00a0at\u00a0z;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Px:\u00a0x\u00a0is\u00a0a\u00a0person;\u00a0Tx:\u00a0x\u00a0is\u00a0a\u00a0time)\n\n1. An\u00a0Italian's\u00a0dinner\u00a0is\u00a0spicier\u00a0than\u00a0a\u00a0Norwegian's.\u00a0(Ix:\u00a0x\u00a0is\u00a0an\u00a0Italian;\u00a0Dxy:\u00a0x\u00a0is\u00a0a\u00a0dinner\u00a0of\u00a0y;\u00a0Sxy:\u00a0x\u00a0is\u00a0spicier\u00a0than\u00a0y;\u00a0Nx:\u00a0x\u00a0is\u00a0a\u00a0Norwegian)\n1. \u00a0If anything can kill a fast-moving mammal, the cheetah can. (Kxy: x can kill y; Fx: x is fast-moving; Mx: x is a mammal; Cx: x is a cheetah)","date":"2019-02-16 22:41:12","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\": 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\": 0.5786207318305969, \"perplexity\": 208.21511322546576}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247481122.31\/warc\/CC-MAIN-20190216210606-20190216232606-00313.warc.gz\"}"} | null | null |
<?php
namespace MushroomFramework\Facades;
use \MushroomFramework\Pattern\Facade;
class Session extends Facade {
protected static $locClassName = '\\MushroomFramework\\InputOutput\\Session';
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 333 |
Stratford fest: The dancing is dazzling in 'Guys and Dolls'
Director-choreographer Donna Feore delivers some unforgettable sequences
Stratford fest: The dancing is dazzling in 'Guys and Dolls' Director-choreographer Donna Feore delivers some unforgettable sequences Check out this story on Freep.com: http://on.freep.com/2xnXmKo
Special to the Detroit Free Press Published 6:02 a.m. ET Aug. 27, 2017 | Updated 6:47 a.m. ET Aug. 27, 2017
Blythe Wilson as Miss Adelaide and Sean Arbuckle as Nathan Detroit in the Stratford Festival's production of "Guys and Dolls." (Photo: Lynda Churilla)
Lady Luck plays a major role in "Guys and Dolls," the musical chestnut about tough-talking New York gamblers and the women (or dolls) who love them. But chance has little to do with the success of the production at Stratford, which employs some winning performances and gravity-defying dancing in its revival of a play you only think you know.
The musical with a memorable score by Frank Loesser arrived in 1950 and is a go-to show for high school and community theater groups, but until now, I'd never seen it done on a big, professional stage. The Stratford production gave me a sense of what what it must have been like to experience "Guys and Dolls" fresh 67 years ago.
The plot, based on a couple of Damon Runyan stories but updated to the early days of television, could fit on a matchbook cover. Manhattan hustler Nathan Detroit (Sean Arbuckle) is desperate for a spot to hold his floating crap game. He needs $1,000 to make it happen, so turns to fellow gambler Sky Masterson (Evan Buliung) with a unique bet that ladies man Sky can't convince pretty but naive young missionary Sarah Brown (Alexis Gordon) to fly with Sky to Cuba.
Stratford fest: 'Tartuffe' is amusing but heavy-handed
Stratford fest: An admirable 'Madwoman of Chaillot'
At the Stratford Festival, autumn is prime time
Meanwhile Nathan courts Adelaide (Blythe Wilson), a featured dancer at the Hot Box cabaret. She wants to get married and settle down, but Nathan can't think of anything, or anyone, else until he gets his dice problem fixed.
Director and choreographer Donna Feore has amplified the dance sequences for "Guys and Dolls" with unforgettable results. The 11 o'clock number, "Sit Down, You're Rocking the Boat" remains a standout, its acrobatic moves rivaled only by "Luck Be a Lady" earlier in the second half.
The entire ensemble has a blast turning the wise guy stereotypes into people with genuine emotions. This works in spades for Wilson's Adelaide, a nice girl with a genuine longing and extreme loyalty to Nathan.
Loyalty, honor and keeping your word are major themes in the show. Nathan may be a flake in his relationship, but his word is his bond. For Runyonesque guys like him, reneging on a bet, disrespecting a marker, is unthinkable.
Michael Gianfrancesco's set, which uses a map of New York City streets on the floor of the Festival Theatre's thrust stage, is a highlight of the show. It also gives the dancers a grid on which to perform the complicated dance routines. On a stage this small, you marvel at how they manage to perform their big moves while avoiding a Times Square pileup.
'Guys and Dolls'
out of four stars
Through Oct. 29
Read or Share this story: http://on.freep.com/2xnXmKo
Ex-WDIV reporter Kevin Dietz back on TV following controversy
New Eminem album just dropped — with strong message about gun control
One of metro Detroit's favorite spots for sushi is now double the size
'Shake Shack on acid' opening in Detroit this spring
Steely Dan with Steve Winwood coming to Detroit
Robert Downey Jr. disappoints in 'Dolittle' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,470 |
{"url":"http:\/\/tutorialspointexamples.com\/flingster-evaluation","text":"# Flingster Evaluation\n\nFlingster Reviews reveal that the web site version\u2019s design is optimized for any gadget users are comfortable with to log in. The solely requirement that the platform poses is that users ought to log in utilizing a web browser. The website on a phone or a tablet reveals the same interface with the same options and the same velocity, thus not letting the users miss anything. With a little bit of a humorous name, CooMeet is a spot where you\u2019ll be able to meet girls in random chat matches. It promotes itself as an internet dating web site, with their ladies all verified and exquisite, as the positioning itself declares. CooMeet comes with a moderation system, HD streams, and a straightforward method to meet new and interesting folks.\n\nWhen it comes to your account settings, you can share as a lot as you need with other users. After that, you\u2019ll have the ability to navigate that user\u2019s account, however solely whereas you\u2019re speaking. You already know that there are three options to choose from in relation to accessing Flingster.\n\n## In Style User Questions:\n\nThe largest improve you get with Flingster\u2019s premium service is the choice of verifying your id for a badge that shall be seen to other users. Even with a paid subscription, although, you can\u2019t store connections for later or revisit old conversations. Perhaps that\u2019s part of the site\u2019s attraction for some \u2014 these are real, fleeting connections. Once you\u2019re a Flingster user, a very simple process, by the way, you presumably can select to connect with males, girls, or couples. You can send as many messages as you want, whenever you like, to as many individuals as you want. Flingster was obviously created with sure randomness in mind, so you can\u2019t revisit matches or narrow down your wishes very far.\n\nBelow are a number of the incessantly requested questions by present and new customers on Flingster. If you enjoy communication on Flingster and surprise if they\u2019re similar platforms which are value trying out. Yes, we are going to record a couple of of them that work virtually with the same rules. Well, the statistics don\u2019t generally imply everybody will not get what they\u2019re looking for. As we all know, this website allows courting throughout sexual orientation. As lengthy as you are not lower than 18 years old, you might be free to mingle. Everything is determined by your expectations and desires.\n\n### How Will I Be Connected To A Different Flingster Member?\n\nYou will find some couples to talk with, and that I cherish browsing varieties. There are plenty horny visitors and intriguing personalities on this website! I like each immediate of being right here and anticipate to see easy finest complement. Whenever you possibly can look for a fantastic platform without jerks, make me aware. Continue to, I\u2019m into web site with all of its alternative and customers. It\u2019s a decent and secure location to satisfy stunning guests and intriguing personalities. Right after I view people which are uncertain or disagreeable, we keep away from these people and proceed.\n\n\u2022 Anytime I accompanied this group, I produced the perfect alternative, and I perceive this software isn\u2019t just a little bit of slap and tickle.\n\u2022 Don\u2019t be idle and seek for their chance a lot past your comfort zone, plus the location is great for a person.\n\u2022 I\u2019m solitary and possess neither second nor hope to roam the pubs, trying to find adore actions.\n\u2022 This incredible web site is useful, but, as you\u2019ll have the ability to think about, it is not magic medication.\n\u2022 Generally, I\u2019ve had gotten very valid fits that let me to make numerous friends.\n\nIf you could have any questions, please write a evaluate on this page. You can watch a limited variety of the randomly generated chat rooms. 6 months membership would go for $101.ninety four, whereby in 1 month would price you$14.ninety nine. Enable your webcam and broadcast your self and text the others.\n\n## We\u2019re Checking Your Browser Flingstercom\n\nThe coverage of filtering out minorities could be very strict. The relationship standing or intercourse is not any bar for signing up to this site. The platform gets full registration from the US; however, it is turning into extremely popular throughout the globe today. The new logins have been reported to be in millions. This is your ideal place if you love talking to people and studying extra about other cultures and ethnicities. I get pleasure from this app given that it should not make an effort me personally with frightening exams. In all honesty, we don\u2019t have confidence in being fully suitable based mostly around varied reviews since men and women all the time lay very usually.\n\nToday, I\u2019m content material with our account whereas the customers around me of the software program. Excellent place to unwind, have a great time, and switch enchanting. I tried to uncover the proper forms of enterprise companions by wanting these folks in cabaret and bars.\n\n### Are The Chats Encrypted On Flingster?\n\nI enrolled with it a 12 months in the past as a end result of subsequently met some close associates with features. Also, we speak with a number of customers from my private favourite identify. Talking is excellent, as a chat display screen comes in handy.\n\nFirst of all, the cool factor is that you could join free. Moreover, you need to use a trial account for any period. All essential data to affix this membership of strangers is your e-mail address and gender.\n\n### Messaging And Chatting\n\nSometimes, you\u2019ll find a way to encounter that a variety of the regions wouldn\u2019t have energetic users, so you\u2019d never watch them online. So, you presumably can see that the Flingster service is inexpensive for everyone! What\u2019s more, you can all the time take a look at the system by purchasing only a weekly membership. Now on, let\u2019s shortly come across the features you get. Discovering the right courting site\/app increases your odds of assembly somebody online by 73% based mostly on a 2021 survey. The solely downside to that\u2019s that you can only receive messages when you\u2019re on-line. Also, it\u2019s attainable to add your age, preferences, sexual orientation, as properly as hobbies and pursuits.\n\nRegistration and a lot of the features of the site are free. Registration is free so there is no want to fret if you are not willing to pay till you would possibly be satisfied with the trial interval. You can meet all kinds flingsyer of individuals here, however definitely not minors. Currently, the United States has the very best percentage of members. Two hundred thousand isn\u2019t bad particularly if you\u2019re aiming for a meetup.","date":"2022-05-19 02:44:50","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\": 0.1928998976945877, \"perplexity\": 2173.7862705861503}, \"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-2022-21\/segments\/1652662522741.25\/warc\/CC-MAIN-20220519010618-20220519040618-00006.warc.gz\"}"} | null | null |
YAASSS: 'How To Get Away With Murder' Renewed For Season 2
By: Sonya Eskridge
Theo Wargo
Viola Davis and her students will be back to slay your Thursday nights this fall!
The season finale of "How To Get Away With Murder" left us absolutely breathless yesterday. In case you were still gathering up the pieces of your spirit after the show faded to black, ABC slyly announced that the show has been renewed for season 2!
In a promo following the two-hour season finale, a voiceover stated, "'How To Get Away With Murder' returns this fall on ABC!"
MUST READ: Can We Talk About How Black 'How To Get Away With Murder' Was Last Night?
Season 2 will likely center on solving a whole new murder as one of the central characters met their untimely end before credits rolled last night, but we won't spoil it for you right now. Hit up Hulu to see which character didn't make it to the end and why. All we'll say is that there may be even more blood on the hands of Annalise Keating's students, which should also give you an idea of who their latest victim was not.
ABC has yet to make an official announcement about the fate of Shonda's other shows, "Grey's Anatomy" and "Scandal," but we doubt that the network will be dropping either series. They've become staples in the ABC lineup and ratings gold for Thursday nights.
Viola Davis Remembers How She Tried To Be The '90-Pound White Girl' Before 'How To Get Away With Murder'
GET THE LOOK: Sizzle In Winter White Like Aja Naomi King From "How To Get Away With Murder"
PM BUZZ: Keyshia Cole Gets Sued; Viola Davis Talks About That Time She Pulled Off Her Wig On 'HTGAWM' & More
YAASSS: 'How To Get Away With Murder' Renewed For Season 2 was originally published on hellobeautiful.com
How To Get Away With Murder , Shonda Rhimes , Viola Davis | {
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{"url":"https:\/\/en.wikisource.org\/wiki\/Page:SearleEllipsoid.djvu\/13","text":"Page:SearleEllipsoid.djvu\/13\n\nthat as far as terms in ${\\displaystyle u^{2}\/v^{2}}$ the electric part of the energy is unaltered by the motion.\n\n(C) Energy of a very slender Ellipsoid. When the ellipsoid is so slender that ${\\displaystyle b^{2}\/a^{2}}$ may be neglected in comparison with unity we have\n\n ${\\displaystyle \\mathrm {W} ={\\frac {q^{2}}{2\\mathrm {K} a}}\\left\\{\\left(1+{\\frac {u^{2}}{v^{2}}}\\right)\\log {\\frac {2a}{b{\\sqrt {1-{\\frac {u^{2}}{v^{2}}}}}}}-{\\frac {u^{2}}{v^{2}}}\\right\\}}$. (26)\n\nWhen ${\\displaystyle u\/v}$ is small, this becomes\n\n${\\displaystyle \\mathrm {W} ={\\frac {q^{2}}{2\\mathrm {K} a}}\\left\\{\\left(1+{\\frac {u^{2}}{v^{2}}}\\right)\\log {\\frac {2a}{b}}+{\\frac {1}{2}}{\\frac {u^{2}}{v^{2}}}\\right\\}}$.\n\n(D) Energy of a Disk.\n\nWhen ${\\displaystyle a^{2}<\\alpha b^{2}}$ the ellipsoid is more oblate than Heaviside's, and ${\\displaystyle l^{2}}$ becomes negative. In this case let us write\n\n${\\displaystyle r^{2}=b^{2}-{\\frac {a^{2}}{\\alpha }}}$,\n\nso that ${\\displaystyle r}$ is the radius of the disk which is the \"image\" of the ellipsoid ${\\displaystyle a,b}$. Then writing ${\\displaystyle {\\sqrt {-1}}=i}$ we have from (23)\n\n${\\displaystyle \\mathrm {W} ={\\frac {q^{2}}{4\\mathrm {K} ir{\\sqrt {\\alpha }}}}\\left(1-{\\frac {u^{2}a^{2}}{v^{2}r^{2}\\alpha }}\\right)\\log {\\frac {1+i{\\sqrt {a}}r\/a}{1-i{\\sqrt {\\alpha }}r\/a}}+{\\frac {q^{2}u^{2}a}{2Kv^{2}r^{2}\\alpha }}}$.\n\nBut\n\n${\\displaystyle {\\frac {1}{i}}\\log {\\frac {1+xi}{1-xi}}=2\\left(x-{\\frac {x^{3}}{3}}+{\\frac {x^{5}}{5}}\\dots \\right)=2\\tan ^{-1}x}$,\n\nso that (23) becomes\n\n ${\\displaystyle \\mathrm {W} ={\\frac {q^{2}}{4\\mathrm {K} r{\\sqrt {\\alpha }}}}\\left\\{\\left(1-{\\frac {u^{2}a^{2}}{v^{2}r^{2}\\alpha }}\\right)\\tan ^{-1}{\\frac {r{\\sqrt {\\alpha }}}{a}}+{\\frac {u^{2}a}{v^{2}r{\\sqrt {\\alpha }}}}\\right\\}}$. (27)\n\nWhen ${\\displaystyle a=0}$ we find for the energy of a disk of radius ${\\displaystyle r}$ moving along its axis\n\n ${\\displaystyle \\mathrm {W} ={\\frac {q^{2}\\pi }{4\\mathrm {K} r{\\sqrt {\\alpha }}}}}$. (28)\n\nIn all these cases it will be found that when ${\\displaystyle u=v}$ the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.","date":"2016-10-22 02:04:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 18, \"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\": 0.7728650569915771, \"perplexity\": 292.38496159823495}, \"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-2016-44\/segments\/1476988718423.28\/warc\/CC-MAIN-20161020183838-00451-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
**PITCH ANYTHING**
# **PITCH ANYTHING**
_An Innovative Method for_
**PRESENTING, PERSUADING,**
**AND WINNING THE DEAL**
**OREN KLAFF**
Copyright © 2011 by Oren Klaff. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.
ISBN: 978-0-07-175976-2
MHID: 0-07-175976-X
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-175285-5, MHID: 0-07-175285-4.
eBook conversion by codeMantra
Version 2.0
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps.
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— _From a Declaration of Principles Jointly Adopted by a Committee of the
American Bar Association and a Committee of Publishers and Associations_
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For Dad, true north
## **Contents**
**Chapter 1** The Method
**Chapter 2** Frame Control
**Chapter 3** Status
**Chapter 4** Pitching Your Big Idea
**Chapter 5** Frame Stacking and Hot Cognitions
**Chapter 6** Eradicating Neediness
**Chapter 7** Case Study: The Airport Deal
**Chapter 8** Get in the Game
Index
**PITCH ANYTHING**
## **Chapter 1**
**The Method**
Here's the "big idea" in 76 words: There is a fundamental disconnect between the way we pitch anything and the way it is received by our audience. As a result, at the crucial moment, when it is most important to be convincing, nine out of ten times we are not. Our most important messages have a surprisingly low chance of getting through.
You need to understand why this disconnect occurs in order to overcome it, succeed, and profit. This book tells you how.
### **I Am Not a Natural**
I pitch deals for a living. My job is to raise capital for businesses looking to expand rapidly or go public. I am good at it. When companies need money, I get it for them. I have raised millions for deals involving Marriott, Hershey's, Citigroup, and many other household names—and I continue to do so at a rate of about $2 million per week. From the outside, the reasons for my success seem simple: I offer wealthy investors profitable deals that involve Wall Street banks. But others do that, too. Yet I raise a lot more money than they do. They compete in the same market. Do the same types of deals. Pitch the same kinds of facts and figures. But the numbers show I am consistently one of the best. The difference isn't luck. It is not a special gift. And I have no background in sales. What I do have is a good method.
As it turns out, pitching is one of those business skills that depends heavily on the method you use and not how hard you try. Better method, more money. Much better method, much more money. It's no different for you. The better you are at advocating your position, the more successful you will be. Maybe you want to sell an idea to investors, convince a client to choose you over the other guy, or even explain to your boss why you should be paid more. I can help you get better at it using the five methods in this book.
#### **Pitching a Master of the Universe**
Over the years, I've pitched to—and closed deals with—some of the iconic businesspeople of our time, including founding members of Yahoo!, Google, and Qualcomm. But the story of what I can offer you cannot really be told without my explaining the day I went to pitch one of the guys Tom Wolfe would describe as a "master of the universe."
"Jonathan" (never Johnny or even John) is an investment banker who controls vast sums of capital. He gets between 600 and 800 pitches a year; that's three to four every business day. He often makes multimillion-dollar investment decisions based on no more information than a few e-mails on his BlackBerry.
As a dealmaker, this guy—and I have absolutely no intention of giving you his name; he sues everyone and anyone at a moment's notice—is the real deal.
There are three things you must know about Jonathan. First, he's a math phenom who can calculate yield curves in his head. He doesn't need spreadsheets. He can instantly analyze what you are pitching him. Second, he's seen more than 10,000 deals and can detect any kind of flaw or BS no matter how well hidden. Third, he's tough talking and, at the same time, witty and charismatic. The upshot: When he's pitching you, _his_ chances are good. When you're pitching him, _yours_ aren't. Yet, if you want to be taken seriously in venture capital, you need to have done a deal with this guy. And so, some years ago, when I was working to raise money for a software company, I arranged to pitch Jonathan and his investment team. Given their reputation, I knew if I got them on board, it would be a lot easier to raise money from other investors who were still undecided. They'd say, "Hey, if Jonathan signed off on this, then I'm in too." But Jonathan knew the power of his endorsement—and he wasn't going to give me an easy win.
As my pitch got underway, he made things difficult. Maybe it was for sport. Maybe he was having a bad day. But it was clear he wanted to take—and keep—control of the whole presentation. I didn't realize this at the start, however, so, I began, as I always do, by _framing_ (frames create context and relevance; as we will see, the person who owns the frame owns the conversation). I explained exactly what I would—and would not—be talking about, and Jonathan immediately started giving me a type of resistance called _deframing_ , which is exactly like it sounds.
For example, when I said, "We expect revenues to be $10 million next year," he cut me off and changed the frame with, "Who cares about your made-up revenue projections. Tell me what your _expenses_ are going to be."
A minute later, I was explaining, "Our secret sauce is such-and-such advanced technology."
And he said, "No, that's not a secret sauce. That's just ketchup."
I knew not to react to these comments. I pressed on.
"We have a Fortune 50 company as our largest customer."
He interrupted with, "Look, I'm done here in nine minutes, so can you get to the point?"
He was really making it difficult. You can imagine how hard it was to use all the right techniques: _setting the frame, telling the story, revealing the intrigue, offering the prize, nailing the hookpoint, and getting the decision._
Collectively, I call these the _STRONG method_ (you will learn about these soon).
Some 12 minutes after I began, what I had hoped was going to be my best pitch ever instead showed all the signs of being my one of my worst.
Put yourself in my situation. After just 12 minutes of your presentation, you've been told that your secret sauce is ketchup. Told that your projections are made-up numbers. And that you have nine minutes left to actually make a point.
I was faced with the _presenter's problem_ : You can have incredible knowledge about your subject. You can make your most important points clearly, even with passion, and you can be very well organized. You can do all those things as well as they can be done—and still not be convincing. That's because a great pitch is not about procedure. It's about getting and keeping attention. And that means you have to own the room with _frame control_ , drive emotions with _intrigue pings_ , and get to a _hookpoint_ fairly quickly. (Details on those last two in a second.)
I reminded myself of these steps in the face of Jonathan's interruptions. Then I swallowed hard and hoped my nervousness wasn't showing. I went back to my pitch, concentrating on my three objectives. I was determined. When he deframed, I reframed. When he looked disinterested, I delivered an _intrigue ping_ (this is a short but provocative piece of information that arouses curiosity): "By the way, an NFL quarterback is also an investor." And finally, I got him to the _hookpoint_ , the place in the presentation where your listeners become emotionally engaged. Instead of you giving them information, they are asking you for more on their own. At the hookpoint, they go beyond interested to being involved and then committed.
At the end of the 21 minutes, my pitch was complete. I knew Jonathan was _in_. He leaned forward and whispered, "Forget the deal for a moment. What in the hell was that? Nobody pitches like that but _me_."
I tried to show no emotion as I told him, " _That_ , in general terms, is called _neurofinance_ , an idea that combines neuroscience—how the brain works—with economics. I have taken it a step further and have broken it down into five parts" (the method we talked about above).
Now, even though Jonathan has MENSA-level intelligence, he doesn't have much interest in concepts like neuroscience. He—maybe like you—had always believed that the ability to pitch was a natural talent. But given what he had just seen me do in 21 minutes—it changed his mind. It was clear my pitching was a learned skill and not naked, natural talent like his.
"You can do that all the time?" he asked.
"Yes," I said. "It's based on research about how the brain receives new ideas. And I'm raising a lot of money with it."
Jonathan hears a lot of big claims. When you listen to three or four pitches a day, your "BS detector" becomes finely tuned. So he asked, "How many hours do you have working on this neuro-whatever-it's-called?"
He was sure my answer was going to be 20 hours. Maybe 50.
I shocked him when I said, "Over 10,000 hours."
He looked at me with a wry half-smile. Giving up all pretense of being disinterested, he said, "I need you on my team. Come do this for my deals, and you'll make a lot of money."
I had never been more flattered. Not only had Jonathan, a guy who had been on magazine covers, offered me a partnership, he had given me an even higher compliment—validation that my method worked in high-stakes situations.
I turned him down. He had a reputation for being difficult to work for, and no amount of money is worth that. But his reaction persuaded me to try my approach as part of an investment company. I joined Geyser Holdings in Beverly Hills, the most profitable venture firm you have never heard of. Even as the economy cooled down (and then frosted over), I helped take Geyser from $100 million to $400 million in about four years. How I did that can serve as your blueprint for success. As you will see, it's possible to use the PITCH method in any presentation where you need to be truly convincing. What worked for me will work for you—no matter what you do for a living.
### **The Need for a New Method**
If ever there is a time to learn to pitch effectively, it is now. Funding is tight. Competition is more aggressive. On a good day, your customers are distracted by text messages, e-mails, and phone calls, and on a bad day, they are impossible to reach. If you've been in business for more than 10 minutes, you have figured this much out: The better you are at keeping someone's attention, the more likely that person will be to go for your idea.
But what kind of advice is this really? Telling someone, "Keep the audience's attention" is like telling someone learning to play tennis to "hit the ball with topspin when it comes." _They know that_! What they don't know is how to do it. But it's worth figuring out. If you have to sell anything as part of your job—a product, a service, an idea, and we all do at some point—you know how the right pitch can make a project go forward and the wrong pitch can kill it. You also understand how difficult it can be to pitch to a skeptical audience that is paying attention to you one minute and distracted by a phone call the next. But we all have to go through this because we all have to pitch if we need something. And though most of us spend less than 1 percent of our time doing it, pitching may be the most important thing we do. When we have to raise money, or sell a complicated idea, or get a promotion, we have to do it. And yet we do it incredibly badly.
One reason is that we are our own worst coach. We know way too much about our own subject to be able to understand how another person will experience it in our pitch, so we tend to overwhelm that person. (We will deal with this in Chapter 4.) But the biggest reason we fail is not our fault. As you will see in the pages that follow, we don't pitch well because there is an evolutionary flaw in our brain—a wiring kluge in our hardware—that we must understand and learn to deal with if we are ever going to pitch successfully.
### **Dealing with the Crocodile Brain**
A brief history of how the brain developed will show
1. How the kluge got there.
2. Why pitching is so much more complicated than we first thought.
3. Why, as with any high-order skill, such as physics, mathematics, or medicine, pitching must be learned.
The three basic parts of the brain are shown in Figure 1.1.
First, the history. Recent breakthroughs in neuroscience show that our brain developed in three separate stages. First come the old brain, or "crocodile brain"—we'll call it the "croc brain" for short. It's responsible for the initial filtering of all incoming messages, it generates most survival fight-or-flight responses, and it produces strong, basic emotions, too. But when it comes to decision making, the croc brain's reasoning power is . . . well, primitive. It simply doesn't have a lot of capacity, and most of what it does have is devoted primarily to the things it takes to keep us alive. When I am referring to the croc brain, I am referring to this level.
The midbrain, which come next, determines the meaning of things and social situations. And finally, the neocortex evolved with a problem-solving ability and is able to think about complex issues and produce answers using reason.
**Figure 1.1** Three parts of the brain.
#### **The Disconnect Between Message and Receiver**
I learned from molecular biologist Craig Smucker that when we pitch something—an idea, product, deal, or whatever—the highest level of our brain, the neocortex, is doing the work. It's the neocortex that is forming ideas, putting them into language, and presenting them. This is fairly intuitive.
* * *
**Three Brains Working Independently and Together**
You can actually sense how the three parts of your brain work separately from each other.
When you are walking to your car and are surprised by someone shouting, you will first act reflexively with some fear. (This is the old crocodile/survival brain at work.)
Then, you will try to make meaning from the situation by identifying the person doing the yelling and placing him or her in a social context. This is your midbrain trying to determine if it is a friendly coworker, an angry parking attendant, or something worse.
Finally, you will process the situation in the neocortex, the problem-solving brain (which figures it out: "It's okay. It's just some guy yelling out to his buddy across the street.")
Our thought process exactly matches our evolution: First, survival. Then, social relationships. Finally, problem solving.
* * *
Pitching anything means explaining abstract concepts—so it didn't surprise me that ideas would be formed by the most modern, problem-solving part of the brain.
_But this is exactly where my thinking—and probably yours—went off track. I assumed that if my idea-making abilities were_ _located in the neocortex (as they are), then that's where the people listening to my pitch were processing what I had to say._
_It's not._
Messages that are composed and sent by your young neocortex are received and processed by the other person's old crocodile brain.
You may be where I was about 10 years ago. Back then, I subscribed to "the brain is like a computer" metaphor. With a computer, if I send you an Excel spreadsheet file, you open it and read it in Excel. This is how I thought the brain worked. If I created a message in _my_ smart neocortex and "sent" it over to you (by telling you about it), I figured that you'd be opening that message in _your_ neocortex.
But no pitch or message is going to get to the logic center of the other person's brain without passing through the survival filters of the crocodile brain system first. And because of the way we evolved, those filters make pitching anything extremely difficult.
So instead of communicating with people, my best ideas were bouncing off their croc brains and crashing back into my face in the form of objections, disruptive behaviors, and lack of interest.
Ultimately, if they are successful, your pitches _do_ work their way up to their neocortex eventually. And certainly by the time the other person is ready to say "Yes, we have a deal," he is dealing with the information at the highest logic center of his brain. But that is not where the other person initially hears what you have to say.
Let me explain further. Because we are a soft, weak, slow species compared with just about everything else out there, we survived for millions of years by viewing everything in the universe as potentially dangerous. And because very few situations we faced back then were safe, we learned to err on the side of extreme caution. And that continues (unconsciously) to this day every time we encounter something new. It happens whenever we encounter a pitch from someone who wants us to do something.
* * *
We are hardwired to be bad at pitching. It is caused by the way our brains have evolved.
* * *
The fact that you are pitching your idea from the neocortex but it is being received by the other person's croc brain is a serious problem.
It's the kluge we talked about earlier. The gap between the lower and upper brain is not measured in the two inches that separate them physically. It must be measured in millions of years (the five million years or so that it took for the neocortex to evolve, to be more precise). Why? Because while you are talking about "profit potential," "project synergy," "return on investment," and "why we should move forward now"—concepts your upper brain is comfortable with—the brain of the person on the other side of the desk isn't reacting to any of those highly evolved, relatively complicated ideas. It is reacting exactly as it should. It is trying to determine whether the information coming in is a threat to the person's immediate survival and, if it isn't, whether it can be ignored without consequence.
* * *
**The Croc Brain at Work**
As you are pitching your idea, the croc brain of the person sitting across from you isn't "listening" and thinking, "Hmmm, is this a good deal or not?" Its reaction to your pitch basically goes like this: "Since this is not an emergency, how can I ignore this or spend the least amount of time possible on it?"
This filtering system of the crocodile brain has a very short-sighted view of the world. Anything that is not a crisis it tries to mark as "spam."
If you got a chance to look at the croc brain's filtering instructions, it would look something like this:
1. If it's not dangerous, ignore it.
2. If it's not new and exciting, ignore it.
3. If it is new, summarize it as quickly as possible—and forget about the details.
And finally there is this specific instruction:
4. _Do not_ send anything up to the neocortex for problem solving unless you have a situation that is really unexpected and out of the ordinary.
These are the basic operating policies and procedures of our brains. No wonder pitching is so difficult.
* * *
Sure, after initial filtering, parts of your message move quickly through the midbrain and on to the neocortex—business meetings would be very odd otherwise—but the damage to your message and your pitch has already been done.
First, given the limited focus and capacity of the croc brain, up to 90 percent of your message is discarded before it's passed on up to the midbrain and then on to the neocortex. The crocodile brain just doesn't process details well, and it only passes along big, obvious chunks of concrete data.
Second, unless your message is presented in such a way that the crocodile brain views it to be new and exciting— _it is going to be ignored._
Third, if your pitch is complicated—if it contains abstract language and lacks visual cues—then it is perceived as a threat. Not a threat in the sense that the person listening to your pitch fears he is going to be attacked, but a threat because without cues and context, the croc brain concludes that your pitch has the potential to absorb massive amounts of brain power to comprehend. And that is a major threat because there just isn't enough brain power to handle survival needs, the problems of day-to-day life, and existing work problems plus whatever unclear thing you are asking it to do. Presented with this kind of situation, a circuit breaker in your brain is tripped. The result? A neurotoxin gets attached to the potentially threatening message (your pitch). This is like a FedEx tracking number, which, in turn, routes your message to the amygdala for processing—and destruction.
Now, if there is one place in the brain you do not want your pitch to end up, it is the amygdala. This is the fear circuitry of the brain. The amygdala turns messages into physical sensations like a faster heart rate, sweating, increased breathing, and increased anxiety. And it produces a feeling that makes the person want to escape from the presentation.
* * *
Pitches are sent from the modern—and smart—part of the brain: the neocortex. But they are received by a part of the brain that is 5 million years older (and not as bright.)
This is a serious problem if you are trying to pitch anything.
* * *
Again, this is part of the hardwiring that has allowed us to survive. A lion is chasing you, and without needing to kick it up to your highly evolved neocortex (which would spend a lot of time trying to solve the problem), the _danger_ switch in the amygdala is flipped _on_ , and it sends the alarm to the rest of the brain to start spitting out chemical and electrical messages that get you to _Run!_ before you even have a chance to think. And while we don't live in the wild any longer, our brains are still wired to work this way.
Everything in the recent research points to the same conclusion: Nine out of 10 messages that enter the crocodile brain—and remember, every single pitch starts by going through the crocodile brain—end up being coded.
• Boring: Ignore it.
• Dangerous: Fight/run.
• Complicated: Radically summarize (invariably causing a lot to be lost in the process) and pass it in severely truncated form.
We've been thinking about this all wrong for years. Clearly, we need a new way of pitching.
#### **Rules of Engagement**
There are the two questions we always ask ourselves after we have made a presentation or pitch:
1. Did I get through?
2. Was my message well received?
We assume that our audience will do what we want them to do if our idea is good, if we didn't stumble through the pitch, and if we showed a winning personality. Turns out, it doesn't work that way. What is vitally important is making sure your message fulfills two objectives: First, you don't want your message to trigger fear alarms. And second, you want to make sure it gets recognized as something positive, unexpected, and out of the ordinary—a pleasant novelty.
Bypassing those fear alarm sensors can be extremely difficult. Creating novelty in the message can be tricky, too. But it is the only way our pitch stands any chance whatsoever because the crocodile brain wants information a certain way—simple, clear, nonthreatening, and above all, intriguing and novel. You need to communicate in these ways, or you are never going to capture people's attention.
* * *
The croc brain is picky and a cognitive miser whose primary interest is survival. It doesn't like to do a lot of work and is high maintenance when it is forced to perform. It requires concrete evidence—presented simply in black and white—to make a decision. Minor points of differentiation don't interest it. And this is the brain to which you are pitching.
* * *
As the principal gateway to the mind, the croc brain doesn't have a lot of time to devote to new projects. It's overseeing a big, complex operation (taking care of survival) and can't get bogged down in nuance and details. It likes facts clearly explained. It wants to choose between just two clearly explained options. And it needs you to get to the point fast. It goes to sleep during PowerPoint presentations, and it needs strong summarizing points to keep its attention.
If it gets really excited about some new project you have presented, then it approves it. Otherwise, it gives up on it, doesn't give it another thought, and goes on to the next issue.
The harsh but true reality is that the croc brain—the source of your target's first reaction to your pitch—is
• Going to ignore you if possible.
• Only focused on the big picture (and needs high-contrast and well-differentiated options to choose between).
• Emotional, in the sense it will respond emotionally to what it sees and hears, but most of the time that emotional response is fear.
• Focused on the here and now with a short attention span that craves novelty.
• In need of concrete facts—it looks for verified evidence and doesn't like abstract concepts.
When I learned these rules of engagement for dealing with the crocodile brain, I had my big "Aha" moment. I understood two very important things: First, I finally _got_ the fundamental problem you and I have when we pitch something: We have our highly evolved neocortex, which is full of details and abstract concepts, trying to persuade the crocodile brain, which is afraid of almost everything and needs very simple, clear, direct, and nonthreatening ideas to decide in our favor. Second, I realized that when my pitches had gone well, I had inadvertently adhered to the five rules of engagement contained in the bullet points above. I had made the crocodile brain feel safe; I was feeding it short vignettes of clear, visual, and novel information; and I wasn't making it do much work. (I also understood that when I didn't stick to those rules of engagement, I usually failed.)
Why do these rules of engagement matter for pitching? Sometimes they don't. If you're pitching the Google Android phone, 3D television, or a Ferrari coupe, the brain becomes so flooded with dopamine—a chemical in your brain that sends messages about pleasure and rewards—that any old pitch will work. But short of having a product that's so sexy it's irresistible—you have to observe the rules about how the brain works. How to do this makes up the heart of this book.
### **What Comes Next**
What became clear to me after my big "Aha" moment was that I needed to bridge the gap between the way the neocortex and crocodile brain see the world. More specifically, if I wanted my pitch to get through, I needed to be able to translate all the complex ideas coming out of my neocortex and present them in a way that the crocodile brain of the person I was pitching could easily accept and pay attention to.
It took me countless efforts to come up with a formula that worked. Now you are going to learn that formula.
As you will see, it begins by _setting the frame_ for your pitch, putting your big idea into an easily understood context. And then, once the frame is established, you must seize _high social status_ so that you have a solid platform from which to pitch. Then you must create messages that are full of _intrigue_ and _novelty_.
To make this process easier to remember, I use the acronym _STRONG_ :
_S_ etting the frame
_T_ elling the story
_R_ evealing the intrigue
_O_ ffering the prize
_N_ ailing the hookpoint
_G_ etting a decision
Over the years, I've used this formula—which we will be exploring in detail—in deal after deal with executives from Bear Stearns, Boeing, Disney, Honda, LinkedIn, Texas Instruments, and Yamaha. Each time I pitched, I learned more about the behaviors of the croc brain, and I eventually come to the understanding that there are five separate places where you can stumble in a pitch. Each step in the process represents one of these points where missteps can be fatal. When the other person's croc brain becomes either bored, confused, or threatened, your pitch is in trouble.
In the pages ahead I will discuss how to avoid those problems and create the perfect pitch, one that gains the full endorsement of the croc brain and increases your chances of success dramatically.
## **Chapter 2**
**Frame Control**
It was July 2001, and I stood in front of a towering office in the heart of Beverly Hills. This was a corridor of power, both in Hollywood and the financial world, a place where careers were made, a place where deals got done.
And here I was, headed to the office of a guy who controlled close to $1 billion in assets. It's not every day you pitch someone with this level of influence. If you think I was nervous, think again. For once, I wasn't pitching. Instead, a colleague, Tom Davis, would be pitching to this icon of corporate finance, Bill Belzberg, one of the three billionaire Belzberg brothers.
You might have heard of the Belzbergs if you follow the business press. They rose to prominence as corporate raiders in the 1980s. Merely observing one of them in the boardroom was a master class in finance, so I was looking forward to what would materialize in the next hour.
Tom was 31, charismatic, a likeable CEO type. He had a nice company in place, but he lacked the money to grow. To get that money, he was willing to try the impossible—impress Belzberg.
I smiled to myself. This was going to be interesting. I'd watched Tom rehearse his presentation, and he had good instincts.
"My pitch is totally bulletproof, I have nerves of steel, and I'm bringing my A-game," he had said while we were waiting in Belzberg's lobby. His confidence was inspiring.
"We'll see," I said. "Just relax."
Soon after that, we were moved to the conference room. After more than 30 minutes of waiting, we watched as the double doors swung open. Bill Belzberg strode through as if entering a saloon. At 69 years old, he was tall and lanky. He waved his arm at Tom, motioning him to get started. Tom looked at me, and I nodded the go-ahead. Belzberg remained standing and almost immediately cut Tom off, "Look, I only want to know two things from you. What are monthly expenses, and how much are you paying yourself?"
Not what Tom wanted to hear. He had a different pitch planned, and now he was looking foolish, searching his bag to find expense charts. Where were the confidence and nerves of hardened titanium? He dropped his papers and stuttered a bit. He was lost.
Belzberg had said only 20 words. As you will see, it's possible for a 20-word disruption to control the fate of any deal. Why is that? An analogy, like the one below, might help explain all this.
Imagine for a moment that there is some kind of powerful energy field that surrounds all of us, silently transmitting from the depths of our subconscious. This invisible defense shield is genetically designed to protect our conscious minds from sudden intrusion by ideas and perspectives that are not our own.
When that energy field is overwhelmed, however, it collapses. Our mental defenses fail, and we become subject to another person's ideas, desires, and commands. That person can impose his will.
No one really knows whether there are human energy fields or not, but perhaps this is the best way to think about the mental structures that shape the way we see the world, which I call _frames._ And in a moment, you will begin to understand what happened when Tom's frame come into contact with—and collapsed under—Bill Belzberg's power frame.
Imagine looking at the world through a window frame that you hold in your hands. As you move the frame around, the sounds and images you encounter are interpreted by your brain in ways that are consistent with your intelligence, values, and ethics. This is your _point of view_.
Another person can look at the same thing through his own frame, and what he hears and sees may differ—by a little or a lot.
The common label given to this is _perspective_. I might perceive and interpret things differently than you do—which is a good thing. Another perspective is often what we need as we nurture our ideas and values.
Yet, as we interpret the world through our frames, something else happens. Our brains process what our senses tell us and quickly react with a series of questions: Is it dangerous? Should I eat it or mate with it? This is the croc brain at work, doing what it does best—detecting frames, protecting us from threats, and using dominance and aggression to deflect attacking ideas and information.
There are millions of people in the business world, and each brings a frame to his or her social encounters. Whenever two or more people come together to communicate in a business setting, their frames square off and then come into contact, but not in a cooperative or friendly manner. Frames are extremely competitive—remember, they are rooted in our survival instincts—and they seek to sustain dominance.
When frames come together, the first thing they do is collide. And this isn't a friendly competition—it's a _death match_. Frames don't merge. They don't blend. And they don't intermingle. _They collide, and the stronger frame absorbs the weaker._
Only one frame will dominate after the exchange, and the other frames will be subordinate to the winner. _This is what happens below the surface of every business meeting you attend, every sales call you make, and every person-to-person business communication you have._
The moment your frame makes contact with the frame of the person you are calling on, they clash, battle, and grapple for dominance. If your frame wins, you will enjoy _frame control_ , where your ideas are accepted (and followed) by the others. But if your frame loses, though, you will be at the mercy of your customer, and your success will depend on that customer's charity.
Understanding how to harness and apply the power of frames is the most important thing you will ever learn.
### **Frame-Based Business**
One of the many benefits of using a frame-based approach to doing business is that it does not require a lot of technique, tactics, or smooth talk. In fact, as you will soon see, the less you say, the more effective you will be.
Sales techniques were created for people who have already lost the frame collision and are struggling to do business from a subordinated or low-status position. The sad fact is, these methods are typically ineffective and usually end up offending people instead of promoting pleasant, mutually beneficial business.
For decades, there have been many books and seminars—there are more than 35,000 on Amazon—promoting methods to persuade, influence, cajole, and browbeat customers into making rapid purchase decisions. Many years ago, when the promoters of these programs realized how inefficient their methods were, they explained it away with the _law of large numbers_. Their typical promise is "Make 100 sales calls using our sales technique, and you will land two sales." In other words, work much, much harder than everyone else, and you will get a 2 percent success rate. But really, _what kind of success is that_?
What these sales gurus are missing is this: When you fail to control the _social frame_ , you probably have already lost. All you can do then is fight for survival by fast talking, spin selling, trial closing, and a myriad of equally ineffective and annoying tactics that signal to the customer that you are needy and desperate—and defeated.
By preaching the law of large numbers, the purveyors of sales techniques are asking you to work longer and harder, with no real competitive advantage. They are forcing you to compensate for your weak position with a Herculean effort to win new business, claiming that _it's just a numbers game_. It's rather rude of them to give away so much of your life this way, isn't it?
Frame-based business takes the opposite approach. It promotes the use of social dynamics, stacking things in your favor before the game even begins.
When we think back about why this pitch or that pitch failed, we usually arrive at the fact that the terms of the deal weren't right for the buyer. Or we had a bad day and didn't position things correctly. Or the potential buyer found something he or she liked better. The reality is, however, that a pitch will fail for reasons that are far less obvious. And that's because frame control is won or lost even before the pitch starts.
When you own the frame, you are positioned to reach an agreement with your buyer. And you are also in a position to decide which deals, orders, or projects you want to work on instead of taking what you can get.
Think it's not possible? I do it every day and for the simple reason that I want to serve my buyers well. I can't do that if I'm continuously engaged in a frantic chase for new business.
Instead of flogging yourself to the point of exhaustion by making dozens of mind-numbingly unproductive sales calls and presentations, I'm going to show you how to get, and keep, frame control. And you are more likely to find yourself pitching five deals, tossing out the two deals you don't like, and keeping the three that interest you. How do you like my law of large numbers? This is what I do and what I have been doing for years.
### **Own the Frame, Win the Game**
Let's do a quick review: A frame is the instrument you use to package your power, authority, strength, information, and status.
1. Everyone uses frames whether they realize it or not.
2. Every social encounter brings different frames together.
3. Frames do not coexist in the same time and place for long. They crash into each other, and one or the other gains control.
4. Only one frame survives. The others break and are absorbed. Stronger frames always absorb weaker frames.
5. The winning frame governs the social interaction. It is said to have frame control.
#### **The Cop Frame: An Introduction to How Frames Work**
So that you can become familiar with the terminology of frames and the basic function of frames in social encounters, here is an example of a dominant frame that you already know about—an almost textbook example of frame control.
Imagine you are driving along California's Highway 101 north of Santa Cruz. The weather and the scenery are intoxicating, as is the rush of speed you feel as you take the fast lane at 80 mph in your pursuit of the setting sun. The moment is perfect—until you see those flashing lights in your rearview mirror. It's a police interceptor. The _whoop-whoop_ from a piercing siren and the Technicolor strobing of the light bar alert your croc brain that danger is imminent. _Dammit, where did he come from? How fast was I going?_ These are the last few thoughts going through your neo-cortex before fear (a basic and primal emotion) sets in, and your croc brain seizes control of your actions. You are now "pulled over." As you reach for your license and registration, you see the cop approaching in the driver-side mirror.
As you will see from this example, frames make human communication simple because they package a particular perspective and all the information that goes with it.
You roll down the window. In this moment, two frames are about to collide: the cop's frame and yours.
Quick! What is your frame made of? "I was going with the flow of traffic" or "I thought the speed limit was higher out this way."
You settle on the "nice guy" frame: "Officer, I'm usually a good driver. How about cutting me a break this one time?"
But the cop frame is nearly invincible. It's reinforced—morally, socially, and politically. Oh yeah, he's got you on a speed gun, too.
You meekly smile as you hand him your license and registration. He pauses, scowling at you through his mirrored aviators. Now, your "nice guy" frame is about to be disrupted. "Do you know why I pulled you over?" he asks.
You know you were speeding. Because you do not have any higher moral authority to bring to the frame game, your frame will be destroyed. This is the key to frame control. _When you are responding ineffectively to things the other person is saying and doing, that person owns the frame, and you are being frame-controlled._
Of course, there's no mystery about the outcome here. The officer has the stronger frame. Your two frames collided, and the cop frame won.
I chose this example so you could see how lesser frames literally crumble under a frame built from authority, status, and power. In this example, the officer had every form of power possible: physical, political, and moral power (you broke the law, and you knew it).
Let's explore the officer's frame on a deeper level so as to understand what really happened. The silhouette of his cruiser in your rearview mirror and the flashing lights pulled your primal levers of fear, anxiety, and obedience. Your croc brain went into defense mode. Your stomach tightened. Your breathing accelerated, along with your heart rate, and blood rushed to your face. All this happened the moment your croc brain was alarmed. You couldn't come up with any frame, any perspective, any way to view the situation that would be strong enough to break the officer's frame.
The lesson of the cop frame is an essential one: _If you have to explain your authority, power, position, leverage, and advantage, you do not hold the stronger frame._ Rational appeals to higher order, logical thinking never win frame collisions or gain frame control. Notice, the officer does not need to pitch you on why he is going to issue you a citation. He does not need to rationalize with you. He doesn't have to explain his power, he doesn't need to rest a hand on his gun, and he doesn't need to describe to you what will happen if you decide to resist. He feels no need to explain how critical it is that you remain calm and obedient. He doesn't suggest that you have fear and anxiety. Your croc brain instantly and naturally has these reactions to the cop frame. You are reacting; your croc brain is in control. Your actions are automatic, primal, and beyond your grasp.
In the final moments of the social encounter, the officer hands you the ticket. This roadside meeting is over. The only other thing he says to you is: "Sign here. Press hard. Fifth copy is yours."
Finally, not quite an afterthought but perhaps intended as a reward for your calm obedience, he says, "Slow down, and have a nice day," crowning your defeat with shame. _Every social interaction is a collision of frames, and the stronger frame always wins. Frame collisions are primal. They freeze out the neocortex and bring the crocodile brain in to make decisions and determine actions._
Strong frames are impervious to rational arguments. Weak arguments, made up of logical discussions and facts, just bounce off strong frames.
Over the years, I observed that a successful pitch depends on your ability to build strong frames that are impervious to rational arguments. These strong frames can break weak frames and then absorb them. Is there a formula for creating such a frame and using it? Turns out, there is.
#### **Choosing a Frame**
Whenever you are entering a business situation, the first question you must ask is, "What kind of frame am I up against?" The answer will depend on several factors, including the relative importance of your offering to the business interests of your buyer. But know this: Frames mainly involve basic desires. These are the domain of the croc brain. It would be fair to say that strong frames _activate basic desires._
One way to think about this is that there are only a few basic approaches that the buyer's croc brain reacts to, so you don't need to carefully _tune_ each frame to individual personalities. If you were a mechanic reaching into your toolbox, then a frame would be more like a rubber mallet than a screwdriver.
I think of these things before I take a meeting: What are the basic primal attitudes and emotions that will be at play? Then I make simple decisions about the kind of frame I want to go in with. For many years, I used just four frames that would cover every business situation. For example, if I know the person I'm meeting is a hard-charging, type A personality, I will go in with a _power-busting frame._ If that person is an analytical, dollars-and-cents type, I will choose an _intrigue frame._ If I'm outnumbered and outgunned and the deck is stacked against me, _time frames_ and _prize frames_ are essential.
I am also ready and willing to switch to a different frame as the social interaction develops or changes.
Going into most business situations, there are three major types of opposing frames that you will encounter:
1. Power frame
2. Time frame
3. Analyst frame
You have three major response frame types that you can use to meet these oncoming frames, win the initial collision, and control the agenda:
1. Power-busting frame
2. Time constraining frame
3. Intrigue frame
There is a fourth frame you can deploy. It's useful against all three of the opposing frames and many others you will encounter:
4. Prize frame
What follows is a discussion of how you can recognize opposing frames and defeat them.
### **The Power Frame**
The most common opposing frame you will encounter in a business setting is the _power frame._ The power frame comes from the individual who has a massive ego. His power is rooted in his status—a status derived from the fact that others give this person honor and respect. You will know that you are facing a power frame when you encounter arrogance, lack of interest (a vibe that conveys "I'm more important than you"), rudeness, and similar imperial behaviors.
Power frame types (a.k.a. big shots, egomaniacs—whatever you want to call them) tend to be oblivious to what others think. They are more likely to pursue the satisfaction of their own appetites. They are often poor judges of the reactions of others. They are more likely to hold stereotypes. They can be overly optimistic. And they are more likely to take unmeasured risks.
They are also the most vulnerable to your power-busting frame because they do not expect it. They expect your fawning deference and obedience. They expect you to laugh at their bad jokes. They expect you to value their feelings above your own. They expect you to adopt their frame. Therein lies their weakness. Not for a moment do they think that your frame is going to take control. You will almost always take them by surprise.
When you approach an opposing power frame, your first and most important objective is to avoid falling into the other person's frame by reacting to it. And make absolutely certain that you do nothing that strengthens the other person's frame before your frames collide.
Observing power rituals in business situations—such as acting deferential, engaging in meaningless small talk, or letting yourself be told what to do—reinforces the alpha status of your target and confirms your subordinate position. _Do not do this_!
As the opposing power frame approaches, when you first encounter the person you are meeting, you must be prepared for the frame collision to happen at any moment.
Prepare well and your frame will disrupt his, causing a momentary equilibrium in the social forces in the room, and then your frame will overtake and absorb his.
This all sounds like high drama, but in practice, it is often swift and tranquil. Before your target realizes what has occurred, control of the frame has shifted. Once you get used to establishing the dominant frame, it will become second nature. And when it does, you are going to have the time of your life.
#### **Encountering the Power Frame**
Several years ago, I had a meeting at a large money center bank whose name you would recognize in an instant. This was supposed to be a one-hour meeting, and it was made clear by the guy we were meeting that he would give us precisely _one_ hour. This is classic power framing with hard time pressure thrown in.
The cost of getting our team to Washington for this pitch was more than $20,000. But the meeting could be worth millions if we pitched it right.
After my team and I were escorted through security, we rode the elevator to the nineteenth floor, where more than _$1 trillion_ worth of business was traded each year. We felt like we were about to take a place among the nation's most powerful and elite financial traders.
Thirty-five traders moved billions of dollars a month here, and we were one hour away from being part of the game. I had contacted all my investors, and together we had pooled about $60 million in investment money that I was bringing to the table.
My contact, a trader named Steve, was meeting us, and I would be pitching him and two analysts. After a long wait, an impeccably dressed young woman led us to the largest conference room I've ever seen, about half the size of a basketball court. Steve and his entourage come in and exchanged the standard pleasantries. Steve was one of the bigger volume traders on the floor. He showed up several minutes late and then spent 15 minutes talking about himself. A precious 22 minutes had been burned. Finally, I was able to hand out our materials and begin the pitch.
During the economic boom of the time, Steve had become accustomed to doing $100 million deals that would close in a single day; by contrast, we had a $60 million deal that would take at least 30 days to close. So he didn't seem terribly interested.
I talked about the types of assets we wanted to buy and what we would pay. During a moment of pause, I looked over at Steve. He had taken our pitch book, flipped it over, and was absent-mindedly tracing his hand on the back of it with a pen.
_How significant is this lack of attention?_ Well, it's pretty bad. However, if you view the world through the lens of traditional sales techniques, you would think there's something wrong with my information or my deal. But instead, if you view the world through frames and social dynamics, then you would understand that the _deal was fine._ This is just the power frame coming at you, and in the collision of frames, you've just lost.
I first thought, _Ouch, how could this be happening?_ I had burned a lot of time and money getting to this meeting, and I could see our opportunity slipping away. The guy was _tracing his hand_ on _my_ executive summary. I felt two inches tall. My crocodile brain became overwhelmed with basic, primal emotions. I was frame-controlled. My simple, emotional, reactive croc brain told me to run, and I considered it.
_When you abide by the rituals of power instead of establishing your own, you reinforce the opposing power frame._
I soon recovered my poise, and here is what followed:
"Steve, gimme that," I said, pulling the pitch book away from him.
_That's a power frame disrupter._
Dramatic pause . . .
I looked at Steve's drawing intently. "Hold on, wait a sec. Now I see what's going on. This drawing is pretty damn good. Forget the big deal for a minute. How about you sell this to me. Name a price."
This is an extreme example of high-stakes power frames. But you can do this in everyday meetings in a far less dramatic way to change and refocus the frame to a totally different subject. If a guy is going to dominate you, let him dominate you on the price of something like a hand drawing in this case, something that doesn't matter. If you find yourself in a similar situation (the day will come when this happens to you, too), then pick something abstract and start an intense price negotiation over it—and it doesn't matter if you win or lose. The power of the person's frame is rendered trivial, and the focus is back to you and what you want to do with the meeting.
Steve didn't expect this, and the concussion from the force of my frame-busting move completely changed the dynamic of that moment and the remainder of the meeting. I got another chance to get the focus back on the real subject—the $60 million I was there to spend. And now I had Steve's complete attention.
_To instigate a power frame collision, use a mildly shocking but not unfriendly act to cause it. Use defiance and light humor. This captures attention and elevates your status by creating something called "local star power." (You will read about creating status and local star power inChapter 3.)_
#### **Taking the Frame**
Here are some subtler examples of taking the power frame away. As soon as you come in contact with your target, look for the first opportunity to
1. Perpetrate a small denial, or
2. Act out some type of defiance.
#### **Examples**
You place a folder on the conference table that is labeled "Confidential—John Smith." When the target reaches for the file, you grab it and say, "Uh-uh, not yet. You have to wait for this."
If you deal in creative work and you brought visuals, let the target sneak a peek and then, when you see him curiously looking, turn it over, take it away, and deliver a soft reprimand that says, _not until I say you're ready_.
This is a quick tease followed by a strong denial, and it is massively disruptive to the target's croc brain. What you are doing is not offensive, and it's not mean. It's playful, and it tells the target subconsciously, "I'm the one in charge here, not you, my friend."
The key to taking the frame is to perpetrate the denial and make it clear: Not yet. This is my meeting, we're following my agenda, and everything that happens will be on my timeline.
Another way to control the frame is to respond to a comment with a small but forceful act of defiance.
TARGET: "Thanks for coming over. I only have 15 minutes this afternoon."
YOU: "That's okay, I only have 12." _You smile. But you are serious, too._
With this simple remark, you have just snatched the power frame away from your target. This can easily become a frame game. I've had meetings get cut down to just two minutes this way. They will say, you only have 12 minutes? I forgot, I only have 10. Then I will come back with 8. And so on. As you'll find out, these kinds of frame games are good for relationships. They are a way of _prizing_ (which you will read about next) and can be entertaining for both parties. It can be that simple. The better you are at giving and taking frame control, the more successful you will be.
Think of how many ways you can use small acts of denial and defiance in the opening moments of meetings. The possibilities are only limited by your imagination. Defiance and light humor are the keys to seizing power and frame control. Keep it fun, do it with a grin on your face, and the moment the power shifts to you, move the meeting forward in the direction you want. This is the foundation of frame control. You'll be seizing more power and status as the pitch continues.
Power shifts and frame grabs start small and escalate quickly. When this first power transfer takes place, when your target loses the frame, he knows it—he can feel that something just happened. His cognition is hot, which means that his basic desires have been activated. Now, he is paying close attention and is fully engaged. He is thinking, _Whoa, what do we have here?_
He might be feeling a little buzz from what you've just done but is not offended because you were not rude or mean. _When you are defiant and funny at the same time, he is pleasantly challenged by you and instinctively knows that he is in the presence of a pro._ This is the moment when he realizes that this is a game, that the game is now _on_ , and that you are both about to have a lot of fun playing it.
Once started, the game has its own inertia, and you can use it to your advantage. Don't be afraid to play with your power by engaging in a little give and take to keep his attention in the moment because that is the entire purpose of this game— _to capture and keep attention until your pitch is complete._
You must also take care not to abuse the power you now hold. The _frame master_ , which is what you will be when you get good at this, knows that dominating the frame is not _how_ you win the game but rather a _means_ to win the game. No one likes to be dominated, so once you own the frame, use this power in ways that are fun and mutually exciting.
Small acts of denial and defiance are enormously powerful frame disrupters. They equalize the social power structure and then transfer all that power to you. Then, all you need to do is hold on to the power and use it wisely.
### **The Prize Frame**
Another common situation occurs when the key decision maker does not attend the meeting as was agreed to. This situation requires a special kind of response that not only will reaffirm your control of the frame but also will establish you as someone unlike anyone else they have dealt with.
Let's say that you've done everything right so far. You've come into the business interaction and quickly asserted strong frames and, hopefully, frame control with the people you've just met. You're ready to start your pitch and are waiting for "Mr. Big" to come in, when his assistant steps in to announce, "I am so sorry. Mr. Big just called. He can't make the meeting for another hour. He says to start without him." She turns to leave.
This is a defining moment for you. You have just lost the frame, and there is nothing you can do about it. However, this does not mean that you do not have choices. Your options are
1. Go ahead with your presentation, even though you know you've lost the frame, hope for the best, and hope that maybe Mr. Big will join the group toward the end of the meeting. I would not recommend this.
2. _Stop everything._ Reframe using power, time, or prize frames (which are covered in this chapter) or perhaps all three. Immediately take the power back.
You've traveled to this meeting, prepared for it, and have an established goal. Are you willing to throw that away?
No one can tell your story as well as you can. If you trust your presentation to subordinates and expect them to pass it on to the decision maker with the same force and qualities of persuasion that you have, then you are not being honest with yourself. Again, no one can tell your story as well as you can. Mr. Big must hear it. _He must hear it from you._
This is what I usually say in this situation:
"So you guys are asking me to delay the start? Okay. I can give you 15 minutes to get organized. But if we can't start by then, _then let's just call it a day._ "
Usually someone will volunteer to track down Mr. Big, and that person will try as hard as he or she can to find him and request that he join the meeting.
Or someone will say, "Let's go ahead with the presentation, and we'll make sure that Mr. Big is briefed." You can't let your frame get absorbed by this. Your response? "No, we're not going to follow your agenda. This meeting is going to start when I say _start_ , and it will end when I say _stop_. You're going to make sure that all the right people come to the meeting on time. Then we're only going to cover the items on my agenda, and you're going pay attention to every minute of my presentation."
You only _think_ this way, of course. What you actually say is, "I can wait 15 minutes, but then I have to leave." That's enough to get the message through.
The first time you think this way and say these words, you'll be uncomfortable—no, make that terrified—and you'll wonder if you are doing the right thing. Your heart will race, and you'll fear the consequences of your boldness, afraid of having offended your audience. You'll second-guess yourself and think you've just made an awful mistake.
And then something awesome will happen. The people in the room will scramble, doing their best to prevent you from being offended, doing their best to keep you from leaving. They are worried about _you._
When you own the frame, others react to _you._
Like Peter Parker's transformation into Spiderman, you will suddenly be empowered by an internal change state that is felt by everyone in the room. Be judicious with this power as you are now in complete control of the situation. If you stand, pack up your things, and leave, it will be a social disaster for Mr. Big and his staff. So be benevolent, give Mr. Big the promised 15 minutes to arrive, and act politely but true to your frame.
And if he does not show at that point, you leave. You do not deliver your presentation, you do not leave brochures, and you do not apologize. _Your_ time has been wasted, and you don't even need to say it. They know.
If it seems appropriate, and if this is a company with which you want to do business, tell the most important person in the room that you are willing to reschedule—on your turf. That's right, you offer to reschedule and acknowledge that these things happen (we have all missed meetings before), but for the next meeting, they must come to you.
_This is a subtle framing technique known as_ prizing. What you do is reframe everything your audience does and says as if they are trying to win you over.
A few moments earlier, you learned that Mr. Big wasn't coming to your meeting and apparently you were just the morning entertainment. Now, however, you are communicating to your buyers that _they are here to entertain you_. What prizing subconsciously says to your audience is, "You are trying to win my attention. I am the prize, not you. I can find a thousand buyers (audiences, investors, or clients) like you. There is only one me."
It also conveys to your audience that if they wish to get any further information from you, they will first have to do something to earn it.
#### **Prizing 101**
To solidify the prize frame, you make the buyer qualify himself to you. "Can you tell me more about yourself? I'm picky about who I work with." At a primal, croc brain level, you have just issued a challenge: _Why do I want to do business with you?_
This is a powerful and unspoken expression of your high status and your frame dominance. It forces your audience to qualify themselves by telling you exactly how interested they really are.
Sound outrageous? It's not, I promise you. _When you rotate the circle of social power 180 degrees, it changes everything_. The predator becomes the prey. In this instance, what your target is feeling is a kind of moral shame—they have wronged you—and they feel obligated to make things right.
Initially, you walked in with low status. Just another pitch in a long string of pitches. Over many experiences, these people have learned how to have their way with salespeople and presenters like you. But now, you've broken their power play. They will apologize, appease, and try to correct for the social gaffe, and in most cases, if Mr. Big is in the building, they will find a way to get him in front of you.
In a moment, I'm going to address what happens when you encounter _time frames_ and _analyst frames_. Before going into these aspects of framing in greater detail, though, I think it might help to prepare the ground if I recount how I come to develop and use frames over the years. As you will see, the practical side of frames grew out of my personal experiences, sometimes in high-stakes situations where there was much to be gained and lost.
Remember, when you own the frame, people respond to you. Let me share an example from my own experience.
### **The Avocado Farmer's Money**
I looked down at my phone. Fourteen missed calls, all from the same person, D. WALTER. I'd turned off my ringer for less than 30 minutes, and the phone blew up. I listened to one of his messages:
"Oren, I have a serious problem," he started.
His serious problem was a deal that had already gone bad, and now it was my job to help.
Dennis Walter was an avocado farmer, a guy who got his overalls dirty, a guy who put in long days in the hot sun. After 35 years, he was ready for retirement. He had money saved, but a good portion of it ($640,000) was in an escrow account, controlled by a man named Donald McGhan.
Dennis wanted his money now, and it was his, legally. But he was unable to get it back despite repeated attempts. This was now affecting an $18 million deal that both Dennis and I were in. If Dennis couldn't wire his money to me, all $640,000, then the deal—a large property we were buying in Hawaii—would start to unwind. So his problem was now my problem, too.
To retrieve Dennis's money, I would have to sit down with McGhan and make an appeal to have the money returned. This is how I was thrust into a pitch that clearly was doomed to fail. It wasn't life and death, but it was close. This was a man's life savings.
I knew a little bit about McGhan. He had a reputation as a successful businessman, primarily in the field of medical devices. Intriguingly, while at Dow Corning in the 1960s, he helped to invent the first generation of silicone breast implants. Today, he owned two companies: MediCor and Southwest Exchange.
MediCor's breast implant business had looked promising for a while. But the success enjoyed there was short-lived, and McGhan turned desperate. To keep MediCor solvent, McGhan began siphoning money from Southwest Exchange.
Southwest Exchange, which McGhan bought in 2004, gave him instant access to over $100 million in escrow accounts. Real estate investors, like Dennis, had used Southwest Exchange to hold their money while looking for new investments.
Soon after acquiring Southwest Exchange, McGhan, according to federal investigators, transferred $47.3 million from Southwest Exchange to MediCor. Just like that. Including $640,000 from Dennis, the avocado farmer.
Now I was on our corporate jet, en route to Las Vegas, on my way to help Dennis attempt the impossible.
I thought about McGhan and what it might be like to confront him face to face. At the time, I had no idea I was walking into a $100 million problem involving hundreds of investors. Or that McGhan was a bad guy, a criminal, presiding over a large-scale Ponzi scheme. All I knew was that this wasn't going to be pleasant.
As I drove to Henderson, a Las Vegas suburb, I had a strong sense of purpose. Not only was McGhan harming Dennis, not only was McGhan in the wrong, but the lost $640,000 also was holding up my Hawaii deal.
I pulled into the Southwest Exchange parking lot, and I met Dennis for the first time in person. He was a nice guy, looked like your typical farmer, and looked like a guy who really needed my help.
I was clearly nervous. Although I almost always enjoy pitching deals, it's usually for new business. Making this kind of pitch, to get money back—a lot of money—from a bad deal, is mentally and emotionally tough.
To calm myself, I thought about frame control and all the other methods that I had spent countless hours learning, and trying to master. As I mentioned before, no situation has real _meaning_ until you frame it. _Frames_ are mental structures that shape the way we see the world and put relationships in context. The frame you put around a situation completely and totally controls its meaning. But you aren't the only one framing. People are always trying to impose frames on each other. The frame is like a picture of what you want the interaction to be about. And the most powerful thing about frames? There can be only one dominant frame during any interaction between two people.
When two frames come together, the stronger frame absorbs the weaker frame. Then weak arguments and rational facts just bounce off the winning frame.
Dennis and I spoke for a few minutes in the parking lot. I prepared my frame. Then, just like that, I was ready, so we walked into the building together, and I went looking for the one guy who had caused all these problems: Donald McGhan.
It was 9 a.m. when we walked into the building. It was a generic looking office with a black leather couch and magazines spread neatly on the coffee table.
"Good morning. How can I help you?" a receptionist asked.
"I don't need help," I said. "Just tell me which office Don McGhan is in."
She began her gatekeeper script: "I'll see if he's in."
Rituals like these are meant to reinforce status hierarchies. But I was there to establish my own status and frame control and certainly not to supplicate a gatekeeper.
I strode past the front desk and down the hallway, the gatekeeper chasing behind me. She tried to keep me out of the office, to keep me from finding Dennis's money, so I had no choice but to start throwing open doors to various offices, interrogating anyone and everyone. What were they going to do, call the cops? Back at the office, my partner already had the local police and the FBI on speed dial.
"Where's Don McGhan?" I bellowed!
There were plenty of people now trying to stop me, but I wasn't going to stop until I'd spoken to McGhan. And I wasn't going to leave without Dennis's $640,000.
As I made my way through the building, office by office, Don McGhan hustled himself out the back door, not wanting to deal with me. Instead, he sent his son, Jim, who finally come out to "handle" things.
Jim McGhan, in his early 40s, was dressed in an Armani suit and had a confident, arrogant way about him. He was tall, and he looked down at me. We sat in a conference room, and right away, he was trying to take control, trying to frame things by saying, _There's a rational explanation for all of this._
So _that_ was his game; he was playing with the _analyst frame_ , which relies on facts, figures, and logic.
I had a better frame prepared, the _moral authority_ frame, and it's an analyst disruptor.
"Jim, you cannot hold Dennis's money," I told him. "We've requested it properly, and you're going to get it for him right now."
Jim was a player. I saw it in his eyes. But he knew that his scheme was falling apart, and he wasn't interested in giving Dennis the money. Instead, the money probably would be wired to McGhan's attorney by day's end, and then, we'd never get it. He knew what he was doing. He was using his status and authority to confidently explain the so-called facts.
I give him credit for one thing: Jim pulled off a beautiful analyst frame. He was completely unfazed, arrogant, and acting puzzled as to why we were there. Then he began with a rational, highly detailed, and analytical explanation of why the money couldn't be transferred right away.
This was the squaring-off phase.
He was trying to spin. He thought he could put us off and have us leave empty-handed.
Of course, I wasn't having that. I come in with a moral authority frame—that we were right and he was wrong—a nearly unshakeable frame when used correctly. The game was on. He knew my frame, and I knew his.
Next come the moment of first contact. It's that moment when two opposing frames are about to collide with full force. You can feel it—usually as a pang of anxiety in the pit of your stomach. It is at this moment when you need to strengthen your resolve and commit completely to your frame. No matter what happens, no matter how much social pressure and discomfort you suffer, you must stay composed and stick to your frame. This is called _plowing_. So you prepare yourself to plow, as an ox might plow a field. _Always moving forward. Never stopping. Never any self-doubt._ And, as you are about to see, when two frames collide, the stronger one always wins.
The niceties didn't last long. I spoke plainly and looked Jim right in the eyes. "We want Dennis's $640,000, and we are getting all of it back from you, today, right now."
He hemmed and hawed. He threw out a bunch of promises, half-truths, and MBA double talk. But I saw through the jibberish. And I had the stronger frame: _moral authority_.
I plowed.
"Look," I said. "Your lips are moving, but I'm not listening to a single word. Your words have no meaning. Stop talking. Start transferring money." He blinked. He tried one more time to explain, to argue, to rationalize why the money hadn't been transferred to Dennis, something about misplaced wire-transfer numbers. But rational explanations will never override a moral authority frame.
At one point, I saw the realization cross his face. He knew that he had picked the weaker frame. In fact, he tried the moral authority frame out for himself, "You know what, I've had enough of this. Get out of here now, or I'm going to call the cops."
But it was too late for him. He had already picked a weak analyst frame and had overcommitted to it—and was about to pay for doing so. It was time for _frame disruption_. I was ready to pulverize his frame into a puff of fine mist.
I pulled out my phone and dialed a colleague, Sam Greenberg. I put him on speaker and discussed the logistics of getting the FBI involved. Dramatic? Yes. But Jim McGhan knew at that moment we were 100 percent committed to following through. I was activating the primal fears in his croc brain. As soon as he became afraid, my frame would crush his, and he would bend to my will.
"Let me paint a picture for you, Jim," I told him. "You've seen SWAT teams in the movies. It happens _just like that_. They are going to swarm through this door, FBI accountants wearing Kevlar vests and Glock 22s. And the sheriff will be blasting pepper spray at anything that moves, dogs will be barking, and they'll be fastening your hands behind your back with zip ties. Is that how you want today to end, hog-tied, pepper sprayed, lying in the back of a black van with no windows? The other option is—you starting transferring money to us."
SMASH! That was the moral authority frame, delivered with emotional realism, and here, I achieved the _hookpoint_. Our frames had collided. My frame had absorbed his. The only options were my options. There's a moment in games of strategy when the other side realizes that no matter what moves it makes, the game is lost. This was _that moment._
I now had his full attention. Although it was his office and his domain, I had the seized the high-status position. Although he still had our money, $640,000, I had the _frame control_.
"Jim, starting right now, every 15 minutes, you're going to give me a deliverable. That means—just so you understand me perfectly—every 15 minutes something happens that benefits _me_. Cancel your schedule, do not leave this room, pick up the phone, and start finding our money."
He was listening, and I continued.
"I need the money wired to Dennis's account, right now."
Just because you have frame control doesn't mean that someone won't push back. You _just stay committed to your frame and keep it strong. You plow._ Jim started with more MBA doublespeak, returning to rationalization mode. So I expanded the frame to include new characters and new consequences.
"Listen, Jim. _Stop_ ," I told him. "Get your friends and family and investor's list and start dialing for money. Every 15 minutes you need to hand me a wire-transfer confirmation."
This was the point of consolidation. Because I had done everything right, up to this point, there was no need to make threats or create drama. _The frame was set. The agenda was my agenda. Because the social interaction was being governed by my frame, these were the rules Jim had to follow:_
Rule 1: Everything happening must involve Dennis's money.
Rule 2: Something good must happen every 15 minutes.
Rule 3: The meeting isn't over until all $640,000 is wired.
I sat with Jim for six long hours as he dialed associates, family members, and friends. The money come rolling in, in small increments ($10,000 here, $15,000 here).
As I mentioned earlier, when two mental frames come together, when they collide, the stronger frame disrupts and absorbs the weaker frame. I'd controlled the frame, started it small, and expanded it, and Jim's weaker frame collapsed. His internal state went from nonchalance and arrogance to panic and desperation. His status went from high to low. Responding to my frame, he raised Dennis's money, and we walked out with all the $640,000—irrevocably secured via wire transfers. Mission accomplished.
Over the next few days, Dennis and I and some other victims worked with the authorities and Southwest Exchange was raided. I got Dennis's $640,000 out just in time, thanks to my knowledge of framing. Not for a moment was it about threats or power plays.
Although it was Dennis's money legally, perhaps Jim and Don McGhan never should have given back that $640,000. It wasn't in their best interests. If Jim McGhan really thought I was going to call the FBI, he should have wired that money to his attorney. It was clearly the last bit of cash Jim and Don could scratch together.
I had always respected the nature of frame control. But now, with Dennis's $640,000 back in my escrow account, I was learning to rely on it more and more often.
All told, the McGhans had bilked more than 130 investors out of more than $180 million. Several people lost their life savings, and the case spawned numerous lawsuits. In 2009, Don McGhan, age 75, was sentenced to a 10-year prison sentence for wire fraud.
This is an example of owning the frame. There are still more frames that you will encounter that I haven't discussed yet. Let's take a look at _time-based frames_ and how to respond to them.
### **The Time Frame**
Frames involving _time_ tend to occur later in the social exchange, after someone has already established frame control. Again, if you want to know who has the frame, it's easy to observe. _When you are reacting to the other person, that person owns the frame. When the other person is reacting to what you do and say, you own the frame._
Time frames are often used by your Target to rechallenge your frame by disrupting you and, in the moment of confusion, unwittingly take back control. As long as you are alert, time frames are easy to defeat.
You will know that a time-frame collision is about to occur when you see attention begin to wane. You've been pitching for a few minutes, and the temperature in the room is noticeably cooler. The game you initiated was fun at the beginning, and now the audience has cooled and might be a little bored. There are limits to the human attention span, which is why a pitch must be brief, concise, and interesting, as you will read about in Chapter 4.
If you wait for someone in the audience to say (or give body language to the effect), "We only have a few minutes left, so let's wrap this up," you will lose the frame because you now have to react to that person.
Instead, when you see attention begin to bottom out and expire, that's it. You're done. Stay in control of time, and start wrapping up. _Running long or beyond the point of attention shows weakness, neediness, and desperation._
In Chapter 4, I explore attention extensively and you'll begin to understand that attention is an extremely rare cognitive phenomenon that is exceedingly difficult to create and manage. When attention is lacking, set your own time constraint, and bounce out of there:
"Hey, looks like time's up. I've got to wrap this up and get to my next meeting." If they are interested in you, they will agree to a follow-up.
Ironically, the mistake most people make when they see their audience becoming fatigued is to talk faster, to try to force their way through the rest of the pitch. _Instead of imparting more valuable information faster, however, they only succeed in helping the audience retain less of their message._ Here is another example of an opposing time frame and how to respond to it. If you visit customers' offices, you will recognize this situation:
CUSTOMER: "Hi, yes, um, well, I only have about 10 minutes to meet with you, but come on in."
SALESPERSON: "I really appreciate your time. Thanks for fitting me into your busy schedule."
This is a common dialogue and form of business etiquette— _and it is exactly the wrong thing to do._ You are reinforcing your target's power over you and confirming your target's higher status. You are essentially handing your target your frame and saying, "Here, please, crush my frame, control me, and waste my time."
When you encounter a time frame like this, quickly break it with a stronger prize frame of your own. Qualify your target on the spot.
YOU: "No. I don't work like that. There's no sense in rescheduling unless we like each other and trust each other. I need to know, are you good to work with, can you keep appointments, and stick to a schedule?"
YOUR TARGET: "Okay, you're right about that. Yeah, sure I can. Let's do this now. I have 30 minutes. That's no problem. Come on in."
You have just broken your target's time frame, established that your time is important, and he is now giving you focused attention instead of treating your visit like an annoyance.
Another frame that you will encounter is called the _analyst frame_. Like the time frame, the analyst frame usually appears after the initial frame collision and can derail you just when you are about to reach a decision. It is a deadly frame that you must know how to repel using the _intrigue frame_.
### **The Intrigue Frame**
How many times have you been giving a presentation when suddenly one or more people in the room take a deep dive into technical details? That's the _analyst frame_ coming at you. This is especially common in industries that involve engineers and financial analysts. This frame will kill your pitch.
The moment your audience does a "deep drill-down" into the minute details, you are losing control. The cognitive temperature of the audience, which was hot when things got started, naturally will cool as audience members listen to your pitch. But once you give their neocortex(es) something to calculate, they will go cold. _Problem solving, numerical calculations, statistics, and any sort of geometry are called_ cold cognitions. Nothing will freeze your pitch faster than allowing your audience to grind numbers or study details during the pitch.
As you will learn in Chapter 4, the key to preventing this is to control access to details. Sometimes, however, a drill-down will happen anyway, and you have to act—fast.
It is important to realize that _human beings are unable to have hot cognitions and cold cognitions simultaneously._ The brain is not wired that way. _Hot cognitions_ are feelings like wanting or desire or excitement, and _cold cognitions_ come from "cold" processes like analysis and problem solving. To maintain frame control and momentum, you must force your audience to be analytical on its own time. You do this by separating the technical and detailed material from your presentation.
Oh, for sure, audience members will ask for details. They believe that they need them. So what should you do if someone demands details? You respond with summary data that you have prepared for this specific purpose.
You answer the question directly and with the highest-level information possible. Then you redirect their attention back to your pitch.
In financial deals, I respond with something like this:
"The revenue is $80 million, expenses are $62 million, the net is $18 million. These and other facts you can verify later, but right now, what we need to focus on is this: Are we a good fit? Should we be doing business together? This is what I come here to work on."
If you're pitching a product and the drill-down is on price, don't chase this conversation thread. Do answer fast, answer directly with high-level details only, and go straight back to the relationship question.
What this tells the audience is that (1) I'm trying to decide if you are right for me; (2) if I decide to work with you, the numbers will back up what I'm telling you, so let's not worry about that now; and (3) I care about who I work with.
Keep the target focused on the business relationship at all times. Analysis comes later. This is the best and most reliable way to deal with a target who suddenly becomes bored and tries to entertain himself with the details of your deal.
_Remember, when you own the frame, you control the agenda, and you determine the rules under which the game is played._
There will be times when you are doing everything right, but for reasons beyond your understanding or control, the other person stops responding to you. The personal connection you had at one point seems to be fading.
When it no longer seems that communication is flowing back and forth, the other person is in something called a _nonreactive state_. It's like the other person's mind is wandering or thinking about something else. This is a state of disinterest that you can correct for if you recognize it in time and act quickly.
You can tell that this is starting to happen when you notice remarks or body language that indicate that your presentation is not intriguing—when the target thinks he can easily predict what your idea is before you even explain it or when he feels that he can anticipate what you are going to say and how you're going to say it.
_Most intelligent people take great pleasure in being confronted with something new, novel, and intriguing._ Being able to figure it out is a form of entertainment, like solving the Sunday puzzle. Our brains are wired to look for these kinds of pleasurable challenges.
When you described your idea initially to your target, you were pulling on a primal lever. When the target agreed to the meeting with you, what he or she really was saying was, "This is a puzzle I am interested in solving."
_No one takes a meeting to hear about something they already know and understand._ It's a fundamental concept driving every single presentation—it's the hook that allows you as the presenter to grab and hold attention by subconsciously saying, "I have a solution to one of your problems. I know something that you don't." This is why people agree to take meetings and to hear a pitch.
At the start of the meeting, you have the audience's attention. It's a rare moment, but not for the reason you may think. Audience members are, with full concentration and at the most basic and primal level, trying to figure out the answer to this question:
"How similar is your idea to something I already know about or to a problem I have already solved?"
If audience members discover that the answer is close to what they had earlier guessed, they will mentally check out on you. They will experience a quick ping of self-satisfaction at the moment of realization, just before they mentally _check out_.
But _checking out_ is not just a catch phrase to describe drifting attention or wandering minds. Checking out, in this context, refers to something very specific: an _extreme and nearly total loss of alertness_ , and this is exactly what you need to avoid.
As your pitch moves along, at any time, some or all members of your audience will solve the puzzle, see the solution, and _get_ the whole story. Then they check out. This is why you see presenters lose more and more of the audience as time goes on— _those who solve the puzzle drop out._
We generalize by saying, "Oh, they lost interest." But what really happened is that they learned enough about our idea to feel secure that they understand it—and there is nothing more to be gained by continuing to pay attention. They determined that there was no more value to be had by engaging with us on any level.
_As I've said before, the brain is a cognitive miser. Unless it can get value for itself, it stops paying attention._ The analyst frame can devastate your pitch because it only values hard data and ignores the value of relationships and ideas. This frame is completely lacking in any kind of emotion or connection to the people in the room.
The most effective way to overcome the analyst frame is with an _intrigue frame_. Of the four frame types at your disposal, intrigue is the most powerful because it hijacks higher cognitive function to arouse the more primitive systems of the target's brain.
_Narrative and analytical information does not coexist._ It cannot; that's simply impossible. The human brain is unable to be coldly analytical and warmly engaged in a narrative at the same time. This is the secret power of the intrigue frame.
When your target drills down into technical material, you break that frame by telling a brief but relevant story that involves you. This is not a story that you make up on the spot; this is a personal story that you have prepared in advance and that you take to every meeting you have. Since all croc brains are pretty similar, you will not need more than one story because the intrigue it will contain will have the same impact on every audience.
You need to be at the center of the story, which immediately redirects attention back to you. People will pause, look up, and listen because you are sharing something personal.
As you share your story, there has to be some suspense to it because you are going to create intrigue in the telling of the story by _telling only part of the story_. That's right, you break the analyst frame by capturing audience attention with a provocative story of something that happened to you, and then you keep their attention by not telling them how it ends until you are ready.
This is much more powerful than you may imagine. Now I can't give you a story to tell; that has to come from you. But what I can do is tell you what your story should contain and then tell you my personal analyst frame crusher so you can see how the elements come together to recapture and hold audience attention.
#### **The Intrigue Story**
Your intrigue story needs the following elements:
1. It must be brief, and the subject must be relevant to your pitch.
2. You need to be at the center of the story.
3. There should be risk, danger, and uncertainty.
4. There should be time pressure—a clock is ticking somewhere, and there are ominous consequences if action is not taken quickly.
5. There should be tension—you are trying to do something but are being blocked by some force.
6. There should be serious consequences—failure will not be pretty.
What's new here is not that you should tell some kind of story to your target. What's new and important is _when_ to use it—as soon as you recognize that the target is coming at you with an analyst frame. Then use it to nudge him out of analytical thinking. There are half a dozen other ways to disrupt the analyst frame—anger and extreme surprise are two. But in most social situations they are impractical. The intrigue frame does it better and does it fast.
Here is my intrigue story, which I will tell you first, and then I will show you how I tell this story to my audience.
_**My Intrigue Story: The Porterville Incident.**_ Recently, I was traveling in our company plane with my business partner and our attorney. We were at an airstrip in Porterville, a small California town about 300 miles from San Francisco. While this tiny airstrip served mostly small local aircraft, jet traffic in the air was heavy because of the many commercial planes going in and out of San Francisco. A jet must make a rapid and steep ascent after takeoff to join in with the busy traffic pattern.
In a pitch setting, I do not tell this story the way I just relayed it to you. When I was meeting with officials from a local airport, I told this story much differently. Knowing that my audience was made up of aviators, engineers, and guys interested in jets, I come to the meeting with this story prepared and ready to deploy if needed. As it happened, I did encounter an attack from an opposing analyst frame, and this story easily brought the meeting back under my control.
As the audience's attention began to shift to analytical questions, this is what I said:
"This reminds me of the Porterville incident. A while ago, my partner and I flew to Porterville to look at two deals. You guys know, they have a tiny airfield; it's visual-flight-rules-only and has no control tower."
"Mostly they get single-engine traffic—Cessna Skycatchers and Beechcraft Bonanzas—and maybe a few small jets. So when we got there, our big Legacy 600 skidded to a stop at the far edge of the runway. But the landing was nothing compared with the takeoff.
"Since Porterville airspace is under San Francisco air traffic control, 260 miles away, the trick to getting out of there is to climb fast and merge quickly into the traffic pattern. We expected an aggressive takeoff. It was no big deal when we found ourselves accelerating hard into a steep climb.
"The Legacy 600 is a 'muscle car' of a jet. When it's under full power, you feel it. So we're heavy and deep in this full-power ascent, we're having casual business conversation, and I would estimate that our altitude was 9,000 feet when abruptly the jet surges and then _nosedives_.
"We dropped 1,000 feet in a few seconds.
"My seat is facing forward, toward the cockpit. The door is open, and I can see the pilots.
"We are all clinging to our seats and cursing, a Klaxon is howling, and one of the pilots is saying, 'It's the TCAS! It's the TCAS!' But I didn't even know at the time what a traffic collision-avoidance system was.
"I'm trying to figure this all out, and I'm thinking this is it—I'm done, _Soy un perdedor. . . ._
"As we're plummeting in this nosedive, I look through the door into the cockpit and see both pilots with their hands on the throttle. _Then_ the plane rolls into a steep climb, _and I see the pilots fighting_ , literally slapping each other's hands off the throttle. The climb is short—just five seconds—and then the plane goes into a _nosedive again_.
"Anyway . . ."
And I go right back into my pitch. Why does this strategy work so well? The most extreme explanation is that the audience becomes immersed in the narrative. They take the emotional ride with me. Sure, they know that we obviously survived, but I've piqued their curiosity— _why were the pilots fighting?_ They want to know. When I do not tell them, the intrigue spikes high enough to shock them out of the analyst frame.
_In my experience with this approach, the opposing analyst frame gets crushed by emotional, engaging, and relevant narratives like this. Attention redirects back to me, allowing me to finish my pitch on my agenda, my timeline, and my topics._
After I finish the pitch, I complete the narrative arc by explaining the whole story:
"It turns out that the sudden dive was caused by the traffic collision-avoidance software built into the autopilot system. It had detected another airplane flying into our ascent path, and the computer had taken evasive measures just in time to avoid a crash. This was a very close call, and I am fortunate to be able to share this story with you now.
"The reason the pilots were fighting over the controls was because the copilot did not know the computer had taken over. But the pilot, older and more experienced, knew this and was pulling the copilot's hands off the controls. The collision-avoidance software was doing its job."
This true story has everything I need in an intrigue story—it's brief; it has a tight timeline; it has danger, suspense, and intrigue (what were those pilots doing?)—and it happened to be perfectly relevant to a pitch I was making to the operators of an airport, which you will read about later.
_Perhaps, in a broader sense, this is why we tell each other intriguing narratives—to participate in powerful emotional experiences involving high-stakes situations that we hope we will never have to face ourselves._ A short, personal narrative like this is important to your audience because it reveals something about you, your character, and your life. As you think about your intrigue story, don't be afraid to make it very personal. As long as it's relevant to your business and has the six elements described earlier, it will serve you well.
#### **Stop the Analyst Frame Cold**
The key to using an intrigue frame is to trust in its power to stop the analyst frame cold. Remember, the person using the analyst frame will break your pitch into pieces and ultimately crush it if unchecked. The analyst frame filters your deal like this:
1. It focuses on hard facts only.
2. It says that aesthetic or creative features have no value.
3. It requires that everything must be supported by a number or statistic.
4. It holds that ideas and human relationships have no value.
Do not let your audience go there—keep audience members focused on the relationship they are building with you. Your intrigue story breaks this analyst rule set in an entertaining way and replaces analytical thinking with narrative discourse.
#### **Breaking the Analyst Frame with Suspense**
Consider the movie _Jaws_ for a moment. This 1975 Steven Spielberg film is a classic, and decades later, it is still doing a brisk business on DVD. Why does this story work so well? In the first part of the film, Spielberg doesn't show you the shark. The great white lurks below the surface, creating a sense of terror and suspense. Where is it? When will it strike next? How big is it?
We see someone in the water, minding her own business. Then we see her as a victim, screaming, kicking, getting pulled under, and eventually disappearing in a froth of red water. This predator is unseen, and we have no idea when it's going to strike next. This creates great tension, and we are riveted to the action.
Now let's reimagine _Jaws_. Let's assume that the shark is fitted with a GPS transponder and that we know its exact location at all times. We know where the shark is going, where it's been, and what it looks like. When it comes time to hunt the shark, Police Chief Martin Brody and the crazy shark hunter, Quint, know exactly where to go and what they're up against.
Strapping a GPS transponder on the shark strips away the mystery and the intrigue. Telling the story this way would have wiped out nearly a billion dollars in box office revenue. _If you know where the shark is at all times, you have no tension, no suspense, no blockbuster._ The same can be said for your narrative.
Use the elements of surprise and tension, and as you approach the most interesting part of the story, move away from it and leave the audience _intrigued_ —until _you_ are ready to reveal. Clearly, this technique made Spielberg one of the most successful directors in history. It works for me in business settings, and it will work for you.
### **The Prizing Frame: Reloaded**
_Prizing is a way to deal with threatening and fast-approaching frames that are likely to push you into a low-status position._ When you prize, you frame yourself as high value in the eyes of your target. Prize correctly, and your target will be chasing you.
Establishing a prize frame is the very first thing you need to do when you are on someone else's turf, ready to begin your pitch. When you get to the end of your pitch and it's time to get a deal, your success depends on how well you establish your frames at the beginning and how strong those frames actually are.
For a moment, think of the alternative to having strong frames. One is to sell harder by making more calls and being more pushy. In fact, our business culture has a fascination with the idea that a salesperson should never take no for an answer. There's pressure from the top. Always be chasing. Always be closing.
Everyone has heard a friend tell a version of this story: "The buyer didn't want my product, but I wouldn't take no for an answer. I just kept pounding away, until he finally signed up."
Such stories dramatize the myth that you can succeed by badgering your clients into buying something. The truth is, this rarely works, and when it does, you are sure to encounter buyer's remorse.
The same is true with pitching. If you think you can browbeat your target until finally he relents, you have it backwards.
Whenever we chase someone or value someone else more than ourselves, we assume the subordinate position and put ourselves at a disadvantage. Although we talked briefly about this before, I'll now introduce you in greater depth to the concept of prizing and the prize frame.
Who is the prize, or who is chasing whom, is one of the underlying social dynamics that influences most meetings. The answer establishes a person's motivation and tells us how they will probably behave in the meeting. The basics:
• If _you_ are trying to win your target's respect, attention, and money, he becomes the prize.
• When _your target_ is trying to win your attention and respect, you are the prize. (This, of course, is what you want.)
_Prizing_ is the sum of the actions you take to get to your target to understand that he is a commodity and you are the prize. Successful prizing results in your target chasing you, asking to be involved in your deal.
#### **Why Is Prizing Important?**
_Successful prizing restores calm and poise to the social interaction._ You won't have to chase as hard or worry so much about impressing your target. But there's another important benefit. It reduces your feeling of needing to perform to get a reward. Consider the way people talk about a presentation—they often call it a "dog and pony show." This label evokes self-defeating imagery of you riding around in a circle on a pony. The only thing missing is rainbow suspenders and a clown nose.
Getting rid of those negative labels and ideas is an important step. When you are no longer performing for the money, the frame changes drastically.
Sure, it's easy to think that you have to earn the buyer/investor's approval to win the money. Especially when you're in the investor's office, on his turf, giving your pitch. The prize frame is the window through which you look at the world that allows you to see yourself as the prize: The money has to earn you, not the other way around. You're flipping the script.
#### **Why Does Prizing Work?**
Your pitch is first going to register in the target's croc brain. And as we discussed in Chapter 1, the croc brain would like to ignore you. But if you are dynamic enough—giving new and novel information—you will capture the croc's attention. Once that happens, the croc is going to have one of two primal reactions:
• Curiosity and desire, or
• Fear and dislike.
Breaking it down into such simple terms helped me to understand a crucial concept: If you trigger curiosity and desire, the croc sees you as something it wants to chase. You become the _prize_.
Let's consider three of the most fundamental behaviors of human beings:
1. We chase that which moves away from us.
2. We want what we cannot have.
3. We only place value on things that are difficult to obtain.
Are these universally valid laws that can be relied on in all social interactions? I think they are. And by now, you can see where I'm going with this. If you pitch in front of strangers, you know how easy it is to come across as a little too eager to do business. At the same time, you might make it seem too easy to get what you have. All your audience has to do is nod, and you will do everything else—it's too obvious that you're willing to do anything at all to make them happy.
_The problem with this approach is that if it is true that people only value things that are hard to get, you are not hard to get. There's no challenge. Behaving this way means that you are failing to prize._
And if you're pitching for money, your problems can multiply. Framing money as the prize is a common error—and often a fatal one. Money is never a prize; it's a commodity, a means for getting things done. Money simply transfers economic value from place to place so that people are able to work together.
### **Prizing 201: Avoiding the Mistakes**
The prize frame works only if certain conditions are fulfilled. In Prizing 101, you learned two basic ideas:
1. _Make the buyer qualify himself back to you._ Do this by asking such questions as, "Why do I want to do business with you?"
2. _Protect your status._ Don't let the buyer change the agenda, the meeting time, or who will attend. Withdraw if the buyer wants to force this kind of change.
Prizing 201 offers additional lessons beyond the previous learning:
1. There is a great temptation to use trial closes because we've all been told this is how selling should be done: "So, are we in the ballpark?" or "What do you think so far?" _Don't do it. This shows you as being too eager to get a deal done._ Anyway, trial closes are crude and ineffective.
2. Instead, take the time to step back, to withdraw. Work to control the underlying prize frame—then you don't have to push your ideas so hard. Instead of a trial close, you might issue a challenge (do it with humor or it will feel forced): "So many buyers, yet only one of me. How are you going to compete for my attention." I left out the question mark for a reason—because you are not seeking validation from the target. You don't have to ask it as a question; just issue it as a statement. It's important to get used to making statements instead of asking questions. Doing it this way shows that you aren't constantly seeking validation.
3. Make the target perform a legitimate task to earn the deal. For example, BMW has a special-edition M3 that requires you to sign a contract promising you'll keep it clean and take care of the special paint. The company won't even let you buy one until you promise this in writing.
4. What follows might sound like advice from the positive-mental-attitude crowd, but it's an important part of the learning: The prize frame works best when you change your attitude about money—fully realizing that money is almost useless to any buyer/investor until it purchases what you have. Oh sure, the investor's money can earn a few bucks in Treasury bills or corporate bonds. But that's not what money wants to do. It wants to go to work by investing in deals and buying products. How does this work in the real world? This can seem a little abstract until you fully internalize the following fact: Money cannot do anything without you. The money needs _you_.
When you combine the elements in Prizing 101 and 201, at first it feels like you are walking up the down escalator. This is a natural reaction. Don't worry, prizing does not mean that we have given up the pursuit of buyers—that would be an absurd notion. It means we must give up the concept of ABC, or "Always Be Closing," a phrase popularized by the sales gurus of the 1980s. Instead, you must embrace the idea that money is a commodity, that it is available in a thousand places, and that it's all the same no matter where it comes from. Knowing this, it is more likely you will embrace ABL—"Always Be Leaving." And it's also likely that you'll embrace the money that comes with it.
Money is a commodity. Every investment banker and economist you'll meet will confirm this. Imagine that—Investors reframed as a commodity, a vending machine for money. When you think about it, this makes perfect sense because there are many places to source money, but there is only one you. Your deal is unique among all others. If you think of yourself and your deal in this way and build frames around this idea, you will be pleased at how it will change the social dynamics in your meetings with investors.
If you want to get started with this, in a simple, low-risk way, here is a phrase I often use to set the prize frame firmly in place: "I'm glad I could find the time to meet with you today. And I do have another meeting right after this. Let's get started." This is always a good start because it tells the audience that there are many like them but only one of you.
As you move into your pitch, find moments to reinforce the other frames you hold. For example, make appropriate comments about the value of your time to strengthen both your time frame and your prize frame.
If someone asks a question that is relevant yet veers toward an analytical tangent, let the question just bounce off your stronger power frame. Save the discussion of details for later, after you have said what you want to say.
Remember, small acts of defiance and denial, combined with humor, are extremely powerful in maintaining your frame control and in reinforcing your high status. Humor is important here—don't leave it out, or I guarantee that you will encounter unpredictable responses.
## **Chapter 3**
**Status**
Status plays an important role in frame control. How others view you is critical to your ability to establish the dominant frame and hold onto the power you take when you win the frame collision. But most people in business and social interactions view status incorrectly. You don't earn status by being polite, by obeying the established power rituals of business, or by engaging in friendly small talk before a meeting starts. What these behaviors might earn you is a reputation for being "nice." They do nothing for your social position—except _reduce_ it.
Another common mistake is underestimating the value of status. People confuse status with charisma or ego, which are entirely different things. And they mistakenly believe that working to raise one's social value is foolish or just an act of peacocking. Nothing could be further from the truth.
Unless you are a celebrity, a tycoon, or the guy who just landed your company the largest deal it has ever done, in most cases you enter a new business setting with a low social position. The harder you try to fit into this social scene, the lower your perceived social value becomes.
Yet fitting in and having high social status are essential. Every interaction is affected by pecking order—who is the dominant group member and who are the subordinates. And the moment you enter a room to pitch is a beautiful example of how the social animal inside you works. In those first moments, the alpha and beta social positions are up for grabs. But it's not a physical skirmish—it's the rapid and sometimes instant assessment of each other's social position. When it comes down to finding the alpha, nobody takes the time to draft a balance sheet of who owns the most assets, who commands the most wealth, and who is the most popular. It's a subconscious and instant recognition of status.
Within seconds, we each need to decide, for the sake of our own self-preservation, _who in this room is the dominant alpha?_ And if it turns out that someone else is the dominant alpha and we are the beta, there is a second, even more valuable question: In the short amount of time we have to orient ourselves in this social interaction, _can we switch out of the beta position and take the alpha?_
People will judge your social status almost immediately, and changing their perception is not easy. But it's important because your social status is the platform from which you must pitch.
If you are pitching from a lower-level platform, or low social status, your ability to persuade others will be diminished, and your pitch will be difficult, no matter how great your idea or product. However, if you hold high social status, even on a temporary basis, your power to convince others will be strong, and your pitch will go easily.
What I am saying—and what I have proven to myself and to others—is that you can alter the way people think about you by creating _situational status._ Let's look at how situational status plays out in a familiar social structure, one we have all encountered at one time or another.
### **The French Waiter**
French waiters are respected throughout the world for their skill in controlling social dynamics. From the moment you enter their world, they set the frame and control the timing and sequence events according to their wishes. They wipe your status instantly, redistribute it as they choose, and control the frame throughout the exchange. You regain control only after the check has been paid, the tip has been left, and you're ushered out the front door.
I watched the waiters work their frame magic a few years ago on a bustling boulevard in Paris. I stopped in at Brasserie Lipp on the Boulevard Saint-Germain-des-Pres. My waiter was Benoit, who started there busing tables and washing dishes and moved his way up to head waiter. His father worked at this famous Left Bank _boîfite_ before and after World War II, and today, there is nothing about the history of this place that Benoit does not know.
Benoit can show you the table where Ernest Hemingway did much of his writing during the 1920s and can seat you there if he is feeling generous (and senses that you will be generous in return).
There is nothing Benoit cannot tell you about the menu—every dish, every ingredient, every method of preparation. But to ask questions about the menu is a mild insult. Instead, it's better to ask him to recommend something. The same goes for the wine list, which is even longer than the menu. This is his job. He is the expert within the walls of his restaurant.
I entered Lipp's with some friends I'd invited out for dinner. I was the host, so I carried myself with authority and high status. After all, I was the paying customer about to drop a big wad of cash. I wanted the maître d' and waitstaff to understand my status and give me the best they had to offer. The maître d' gave me a practiced look that said, _Yes, I know your kind. You're all the same to me._
The restaurant was starting to get busy, but it was not full. We wouldn't have to wait long. The maître d' looked down at his schedule and intoned, "It will be a few minutes while we prepare your table, monsieur. Please wait here." Yet he didn't move. He looked down, scribbled a note on his seating chart, and began to ignore me.
Fifteen minutes passed. I watched as the best tables began to fill. I looked at the maître d', anxious, and he held up his index finger indicating _only a minute more._ I returned to my guests, defending my choice of restaurant and commenting on how good the food is.
"I promise you, it will be worth the wait," I told them.
Finally, when the right amount of time had passed, the maître d' stepped away from his podium and said, "Madames et messieurs, your table is ready," waving us toward our table with an open palm and outstretched arm.
He seated us, handed us menus, and told us that Benoit would soon arrive to take our orders. A trainee brought water and bread, smiled, and then disappeared.
Another 15 minutes ticked by before Benoit appeared, and the first thing he did was flash me a rebellious look. "Do you know what you would like to drink?" he asked, looking at the hand-tooled leather-bound _carte du vin_ (wine list) resting near my left hand. I didn't recognize many wines on the list, so I played the part of good host and ordered an expensive bottle for the table.
This was Benoit's opportunity to perform a small but defiant act and to seize control of the high-status position, taking it away from me. You almost could hear the power transfer, clean and smooth, like the flip of switch:
"Hmm, monsieur, I do not think this wine is the best choice," he grimaced, taking the wine list away from me.
Benoit turned the page and paused. I was embarrassed, and my face turned red. "While all the wines in our cellar are fine wines, you must select a better pairing for this evening's meal," he said. He scanned our table, making eye contact with my guests, ignoring me.
He suggested various meals for my guests and, after several minutes, finally returned his attention to me. He flipped open the _carte du vin_ , stabbing his index finger at a wine that would meet his standard. His recommendation was less expensive than the wine I had chosen. So I gave up on my selection and gave the nod to his.
"An excellent choice, monsieur," he announced to the table, pretending to the group that it was my knowledge—and not his—that resulted in the best choice. I was the butt of the joke, and my guests had a fun laugh.
Benoit flashed me a look that said, _This table is mine!_
The wine arrived, and Benoit carried out the time-honored ritual of corking, testing, and decanting. He executed the steps with precision, tradition, and respect for his craft. My guests were in awe. Only when it had been established that the wine would meet his exacting standards did he offer me, the host, the first taste.
At this point, he could have served me stale vinegar, and I would have said that it was heavenly, just to save face.
I wasn't sure whether I was angry with Benoit or just amused that he had turned me into a low-status dork. Benoit had simply and effectively grabbed _local star power._
He had captivated the attention of the table, and now, in full possession of the social power that I once had, he decided to redistribute some of this power to further strengthen his position with my group.
As Benoit settled into a smooth rhythm of frame controls and status moves, I could see the game unfolding perfectly: acts of small defiance; seizing status, redistributing it; taunting me to behave like a beta. I was at the center of a master class on frame control.
As the wine was poured around the table, one of my guests smelled her wine and asked, "Is this a Bordeaux?" Benoit stood tall, placed a hand on her shoulder, and said, "Madame clearly knows French wine. This Bordeaux is from a small _terroir_ that most people mistake for Langedoc. Your palate is very sophisticated." This comment absolutely melted her, and her eyes were sparkling with emotional pyrotechnics. The table was smiling, and again, I was ignored.
Let's pause to review what Benoit, this practiced master of social frame control, had accomplished. First, he seized local star power by using simple, seemingly innocent and benign acts. And he isolated me by making me wait.
As I discussed earlier, the croc brain is a social organ that craves acceptance and belonging. No one likes being made an outsider, especially when there are guests to impress.
Then, after ostracizing me, Benoit swooped in, using his superior domain knowledge, and made me look like a fool. He then quickly rescued me from a "mistake" he had allowed me to make.
He knew that without first understanding what kind of food the table would be ordering, it would be impossible to order the wine. And yet he asked me to make a wine selection first. No matter what I ordered, my order would be wrong. _Thanks, Benoit!_
He called out my mistake and then quickly polled the table for the information needed to make the correct decision. He chose the correct wine, making sure it was a better wine at a lower price than I had chosen, and then gave me credit for making the proper choice. This was an early masterstroke that secured his control of the social power he had taken from me two minutes earlier.
His next move was to strengthen his position by co-opting one of my guests into his frame, making it impossible for me to attack him without simultaneously attacking her.
He waited for someone to comment on the wine—anyone, any comment—and lavished praise on them to distribute some of his social power to that person. When one person joins his frame, the others will follow. And now the table was his. _Voilà!_
Back to the dinner. As expected, the entrees were superb, and Benoit suggested a second bottle of wine, something a little different to capture the flavors coming from the progressing meal. Benoit appeared more regularly now, floating around the table, collecting information, making suggestions, and basically doing the hard work of protecting his superior social position. My guests told me that this was one of the best meals they'd ever had. I thanked them for joining me and then gave Benoit an appreciative nod. At first, I wanted to smack Benoit, but now I was really starting to like him.
As the plates were cleared, Benoit disappeared. I had an expectation that something more was coming, but what? Ten minutes went by. Where was Benoit? I knew something was up.
I was right about that. Benoit had chosen the dessert. Moments later, a gleaming silver cart was rolled to our table, followed by a cart lined with brandy and cigars. Behind all this was the coffee cart—the attendant filled individual French press cylinders with freshly ground coffee.
"Madames et messieurs, for this evening's dessert, I have taken the liberty of making something special for you," Benoit announced. What he meant was, "I have taken over your host's mind and bank account."
" _Baba de rhum_ ," he continued, "our most famous dessert, a light and delicious cake made with cream, rum, and a little sugar. Please enjoy."
The table applauded, and Benoit cut the cake with flair. I was so owned by Benoit at this point that it really no longer mattered. I smiled, relaxed, and decided that Benoit was going to get the largest tip he'd ever seen. In fact, the tip is the only power I had left in this situation.
My guests were delighted, and now, as the evening closed over coffee and brandy, Benoit slowly released some social power back to me. For a very good reason: The check was coming, and I was sure it would melt my neocortex.
"Madames et messieurs, it has been our pleasure to serve you this evening," Benoit gushed. He deftly placed a small silver tray near my left arm. On the tray was a tiny slip of paper turned face down, held in place by a small silver fleur-de-lis paperweight. No itemized bill could possibly fit on this miniscule slice of paper, just a single number would be on it. As my guests warmly thanked Benoit with hugs and handshakes, I managed a peek at the bill, flipping up the corner, like a poker player, not wanting to show any reaction.
It wasn't as bad as I thought.
With the performance Benoit gave, and the control he held throughout the evening, I was expecting him to take advantage. He had the power to do so, but in a final display of total frame control, he chose moderation over self-indulgence. Now I was absolutely delighted, and the large tip I was thinking of leaving him was just raised.
### **Alpha and Beta**
It doesn't matter how well you argue, the way your points are crafted, or how elegant your flow and logic. If you do not have high status, you will not command the attention necessary to make your pitch heard. You will not persuade, and you will not easily get a deal done.
As you are now starting to realize, pitching any kind of idea or deal involves playing a complex and tricky status game. And before I talk about the ways to win the game, and possible ways to lose it as well, I should cover the real advantages held by the person with the highest status, the _alpha_. The alpha enjoys most of the attention in a social interaction, even when he's not demanding it. And when he _does_ demand it, the alpha captures the group's attention immediately. When he makes a statement, it's regarded as true, and the claims go unchallenged. There's plenty of evidence to suggest that the alpha in a group is trusted and followed without question. To illustrate this, researchers have set up many tests in which men, dressed in high-status business suits, jaywalk across a busy street when it's unsafe to cross. Lower-status pedestrians tend to follow the high-status decoy into the danger zone. They will not follow someone who is dressed badly, however.
When you take the high-status position in a social interaction, you _feel it,_ and it is also felt by your audience. Do not underestimate the importance and value of status to your overall success.
For more than 40 years, sales trainers have been teaching techniques and methods that help "situationally disadvantaged" salespeople (read: those with low social value) get an appointment, establish a temporary relationship (called _building rapport_ , which contributes absolutely nothing to your social rank), package a business transaction in a thin and fragile emotional wrapper, and sometimes, if they're either lucky or doggedly persistent, close a sale.
During the 1970s and 1980s, these techniques worked, but even then, only for the most tenacious, hard-driving, type A personalities. Still, the purveyors of process-based sales methods continue training millions of ambitious salespeople in seminars from coast to coast. Today it's difficult to find an executive who isn't familiar with techniques like building rapport, pushing features and benefits, overcoming objections, and trial closing.
As a consequence, after generations of this process-driven business behavior, targets know how the sales game is played. _Use this stuff, and they will see it coming._ Even the best of these techniques are now so commonplace that clients have developed strong defenses and barriers to block them. These defenses are called _beta traps_ , meaning that you are held in a subordinate position to the target (or buyer) at all times. You are _PWNED_ from start to finish.
There is no substitute for holding a position of high status. The good news is that you need not be a celebrity or a billionaire to enjoy high levels of social status. There are ways you can create it instantly. And with it, you can capture and hold the attention of any audience or target.
The first step toward elevating your social status is to avoid the beta traps.
#### **Beta Traps**
In social interactions and business meetings as in nature, those who hold the dominant alpha rank are able to accomplish more than those holding a lesser rank. Alphas call the shots, give the orders, and create the outcomes they want with a minimum of effort. It's important to them emotionally and economically to remain the highest-ranking person in their social group.
Because they occupy a coveted rank, alphas have to constantly fight to maintain and protect their position. As top dog, their rank is under constant threat, and alphas protect themselves by asserting their authority over their employees and coworkers. They ask subordinates to run their errands, bring them coffee, and deal with matters that are disinteresting to them or are deemed to be below their rank. These are the nicer forms of dominant turf-protecting behaviors; many who hold alpha rank behave in far worse ways.
To shield themselves from people of higher social rank who visit them in their work environment, they erect a protective ring of social barriers intended to deflect and demote any threatening alphas.
A beta trap is a subtle but effective social ritual that puts you in the low-status position and works to keep you there, beneath the decision maker you have come to visit, for the entire duration of the social interaction. Most business environments are surrounded by a moat of beta traps that you already recognize and know: the reception desk, the lobby, the conference room, and any public meeting space in or near the office.
The first beta trap you encounter is the lobby. It's a venue created to welcome visitors, right? In fact, the lobby serves to demote you from the moment you arrive and keep you demoted throughout your visit.
You know the drill; how many times have you experienced this scenario?
You enter the lobby of the office where you will be meeting your target. You approach the reception desk. The receptionist looks up—"Hi, can I help you?"—then takes a call before you can answer. You stand, wait, and take a business card from the tray on counter. The receptionist transfers a call and then looks at you. "Yes? Can I help you?"
You say, "I'm here to see Bill Jones for a 2 o'clock. I talked to you earlier, I think, and you confirmed...."
The receptionist looks past you. "Sign the visitor's book, please. Here's your visitor's pass. Keep it with you at all times. Please take a seat. Bill's assistant will come get you in a few minutes." She then turns to finish a text message. You take a seat in the lobby. A table filled with dog-eared trade magazines and week-old newspapers indicate that others like you have been here before.
This sequence, in translation, reads as follows: Be a well-behaved salesperson, do as you're told, and you shall be rewarded with a bottle of water, a short visit, and a vague promise to "review your materials and information" after you leave. When you observe office power rituals, you are signaling to your target that you are a beta.
At 2:10 p.m., a young aide approaches you. "Hi. Yes, Bob is running a little late, shouldn't be more than another 10 minutes. Water and coffee are over there. Help yourself." You blink, she's gone.
Your target arrives late, offering a mock apology for his impossible schedule, telling you that he now has only a few minutes and still hasn't had a chance to review your materials. And now the decision maker, "Mr. Big," won't be able to attend the meeting as planned. Sorry. _At this point, you have been beta trapped and are completely and utterly defeated. You may as well go home._
What a demoralizing way to do business. Yet this is how millions of people set and conduct business meetings. It's a waste of time because the behaviors and outcomes are so predictable and so unproductive.
Another common beta trap is the conference room. If it's empty when you arrive, you are usually left alone for several minutes, cooling your heels while you wait for your targets to arrive. When they arrive, the mood is often jovial, with lots of light social chatter, smiles, and handshakes. They are happy because they are now taking a break from their daily work to come into a nice, larger room to see today's entertainment—that's you. Who isn't happy when they step into a circus tent and take a seat at ringside? They know a show is about to start, and they're looking forward to relaxing and having a good time.
As you wait for the latecomers—the decision makers you really need in the room—conversations are now taking place that do not include you. Others talk to each other as if you were not in the room, which is not only annoying, but it's also one of the most degrading things one person can do to another. In this situation, you are the jester in another king's court, and your value is purely based on the quality of your entertainment. You have no status whatsoever.
Then there are public spaces where customers sometimes decide to take a meeting. "Let's get a coffee and talk," they say, leading you into a cafeteria or close-by café where you exchange small talk in the queue and manage an awkward moment over who should pay for the drinks. You take a place at a nearby table, within earshot of a dozen strangers. This is no place for a pitch.
Your status level is zero. You are owned, processed, and now are nothing but a pleasant social interlude in an otherwise boring day. But you press on, believing in yourself, and your offering. You open your pitch and are moving along nicely when suddenly someone walks up to your customer and starts talking to him as if you do not exist. "Hey, Jim, how's it going?" the intruder says, shaking his hand and ignoring you. "Did you get my e-mail about the shipping delays in Dallas?" They continue their conversation for a while as you can do nothing but stare.
Eventually, when he decides he needs to go bother someone else, the intruder leaves, and your customer turns back to you. His face is blank, his eyes are empty, and his brain has stopped functioning. "Where were we?" he asks.
Need I go on?
There have been many frame collisions in this interaction—but you didn't win a single one of them. You have no control over the situation.
In general, public spaces are the most deadly beta traps and should be avoided. For a real pitch, coffee shops are an absolute last resort. I will mention one more public beta trap because it's common: trade shows and conventions.
If you exhibit at trade shows, you know that the absolute worst possible venue for pitching a customer is in a tiny booth or even on the convention floor. There are so many distractions that not even a frame-control ninja could hold an audience's attention for more than a few minutes without being interrupted by noise, announcements, or throngs of bag-carrying conventioneers mindlessly gathering free items to fill their brightly colored sacks.
If you need to pitch someone attending a conference, rent a hospitality suite or a hotel conference space or borrow someone's office conference room—pitch anywhere but on the floor of the convention hall.
A person standing in a trade show booth may as well erect a neon sign above his or her head that reads, _"I Am Needy!"_ Like a caged pet-shop puppy or a late-night infomercial host, you try to draw them into your 8- by 10-foot cube and hope to wow them with your pitch. It's sad.
_**Beta Trapping in Bentonville.**_ In the town of Bentonville, Arkansas, the art of beta trapping has been taken to an unparalleled level. You might call it the "Frame Supercollider."
The world leader in the design, construction, and operation of beta traps is Walmart. At its headquarters in Bentonville is the world's most efficient salesperson-grinding apparatus you will ever see. No matter what you have to offer the company, no matter how great its value, to do business with Walmart, you must submit to a process that is designed to beat you down and wipe out your status, all in the name of lower prices.
Think I'm exaggerating? Go to 702 Southwest Eighth Street in Bentonville. Walk into the lobby. There you will find two enormous reception desks, one on each side of the room, with a hospitality area on the far right filled with grade school–style chairs with writing desks attached to them for those who need to fill out forms. The perimeter of the room is lined with junk-food vending machines for those who need a quick energy boost to endure what is coming.
Between the two reception stations is a gleaming blue hallway marked with the Walmart logo that leads to another long hallway lined with dozens of six- by eight-foot meeting rooms. These meeting rooms are equipped with a door, one window, one small table, and four small plastic chairs. These rooms are where Walmart buyers meet with vendors.
Let's take a look at the company's process. First, you sign in, receive a visitor's badge, and are told to wait in the lobby. You are welcome to enter the company's hospitality room, and you can purchase candy and Walmart-branded soft drinks from the vending machines. The person you are visiting receives a message that you are in the lobby. When the buyer is ready to meet, you are paged to the reception desk and walked back to an assigned meeting room, where you are instructed to wait for your buyer to appear. As you are escorted to your assigned meeting room, you are allowed to see other vendors through the small glass windows of their cells. When you reach your cell, you are instructed to remain in the room until you are escorted out. Finally, the door is closed.
Eventually, one or two buyers will enter the cell, and your meeting will begin. The meetings are short and focused on price, volume, logistics, your financial ability to support the Walmart account, and then price again. Price is methodically and systematically driven down, whereas your logistical and product-support responsibilities are increased until you can no longer negotiate. When this point is reached, the Walmart buyers make a decision (buy or not) and move on to the next item in the product category.
The frame is so tightly controlled that even the most successful selling techniques do you absolutely no good. Walmart turns everything into a commodity, and every commodity is acquired through this process. Using scale, magnitude, and domination psychology for purchasing, Walmart has created the most effective frame supercollider in the history of free enterprise.
This is an extreme example of how beta trapping strips you of your power and ability to do good business. Old-fashioned sales techniques can help, but you are disadvantaged, you do not control the frame, and you are at the mercy of the buyer.
To compensate, you would need an enormous amount of self-confidence and self-belief to be convincing enough to succeed. You are forced to browbeat, manipulate, and cajole targets into buying decisions, and this is precisely why conventional sales methods focus on pressure closing.
Most of us don't have the required stamina and chutzpah—I certainly do not—and it's emotionally draining to have to make 100 sales calls to win an order or two.
When you are held down in beta position, the only tool you have at your disposal is emotional manipulation. At best, it works in the moment, and maybe you can land a deal. But your success is random, and it's not satisfying because the buyer really does not want to buy. He is doing so to please you now and will regret it later (buyer's remorse).
There is a much better and more natural way to attract business opportunities. You simply elevate your social value, and it's easier to do than you might think.
_**The Cardiac Surgeon and the Golf Pro.**_ Most golf professionals make their living by teaching the game, running golf clubs, operating golf courses, and dealing in golf equipment—not by caddying for touring professionals, like Phil Mickelson. In the United States, the label _golf pro_ means an experienced golfer who helps other golfers with their game. It's a fun job. In many ways, being a golf pro is a dream job. You're working outside, teaching people a sport, and getting paid for it. The catch? It doesn't pay well. And being a golf pro does not bring you much global status. That is to say, when someone asks you what you do, and you say, "I'm a golf pro," it's not like saying, "I'm a CEO" or "I'm a doctor" or "I'm a professor." In fact, it's more like saying, "I'm chronically underemployed," which doesn't carry much cachet.
What should we make of this? Is the golf pro any less intelligent, sociable, or reputable than, say, a cardiac surgeon from the regional hospital?
Of course not. The only difference is that the golf pro holds a lower position on the social hierarchy. Your position in the social hierarchy is an artificial measure of your worth to others, a construct based on your wealth, your popularity within society at large, and the power of the position you hold. This is not my formula. This is just the way we measure each other. The golf pro doesn't make a lot of money compared with a cardiac surgeon. That makes his social status even lower.
Or is it? When the surgeon takes a lesson with the golf pro, the status of each suddenly becomes fluid. Something called _situational status_ takes over. Out on the golf course, whatever wealth, power, and popularity the surgeon has become irrelevant. The domain has changed, and with that comes instant and significant changes in social value. Once the surgeon stepped into the realm of the golf pro, his status fell, and the golf pro's rose, and this change in social elevation remains in effect for as long as the surgeon is in the golf pro's domain.
The golf pro's situational status is suddenly very high—now much higher than it was back in the parking lot. Now the golf pro tells the surgeon what to do, when to do it, and how to do it. If the surgeon doesn't comply, the golf pro reprimands him. Through a simple domain change, the roles have reversed. _And it is in this role reversal that we begin to see the incredible power of situational status._
Think about this for a moment: Your social value is fluid and changes with the environment you are in—or the environment you create. _If you wish to elevate your social value in any given situation,_ _you can do so by redirecting people into a domain where you are in charge._ This is easier to do than you might think.
Our position in the social hierarchy is not locked in place. While the global status of our social standing may remain whatever it is, our situational status can be mobilized in order to temporarily create a high-status position whenever that is needed. While we temporarily hold high power, we can get a lot of things done just as effectively as those with financial or political means.
This is called creating _local star power_. This is critically important. With local star power, you'll be able to succeed in pitching audiences who don't know you; the ability to create and sustain local star power literally is going to mean the difference between success and failure.
The first impression we make on another person is based on that person's automatic calculation of our social value. As a survival mechanism, the other person's brain is making it a priority to understand where you fit in the social structure. The person makes a hasty judgment using three measurable criteria: your wealth, your power, and your popularity. Based on some quick mental shorthand, the person is going to assign you a social status level, and from that calculation, a frame will be fixed. The person will not necessarily even consciously think about this. The people jaywalking behind the man in the nice suit did not deliberately pause to consider his status or think about whether it meant he was likely to cross streets safely. They just automatically calculated his likely status and behaved accordingly.
### **Elevating Your Social Status**
The first thing you do when you meet with a target is to establish local star power. If your meeting happens to be on your turf, like the golf pro or the French waiter, use your domain expertise and locational knowledge to quickly take the high-status position.
If you are meeting in the target's domain—his (or her) office or at an off-site location—you must neutralize the person holding high status, temporarily capture his star power, and redistribute some of his status to others in the room who will support your frame.
I have given you two examples of situational status and how to capture local star power. Now let's look at how you can elevate your status when your target is coming at you with the power frame.
#### **The Hedge Fund Manager**
A couple of years ago, I took a meeting with Bill Garr, a hedge fund manager. The meeting was arranged by a mutual friend, Dan. Arriving a few minutes early, I checked in with the receptionist and immediately recognized the lobby beta traps. Sign the guest book, here's your visitor badge, take a seat, and how about a stale cup of coffee? Someone will attend to you shortly.
Looking around the lobby, I made a quick read of the situation. Green marble floors, modern chrome and leather furniture, rich accents, all designed to convey a single message—I'm rich, I'm powerful, fear me, _revere_ me. I knew what this was. I was on a conveyor belt headed straight for a status-crushing machine. Soon, my forehead would be stamped "Beta," and I'd have a 15-minute meeting with Bill and then would be shown the door. I knew instinctively that our first frame collision would not yield power to me. While I waited for Bill, I began to think of another way to acquire high status and take control of the frame.
Eventually, an assistant ushered me back to Bill's corner office. The level of luxury had been taken up a notch. His private office made the lobby look like a construction trailer. Teak furniture, Persian rugs, titanium and glass fixtures, a couple of dozen framed pictures of Bill with various politicians and celebrities, and from his windows, a panoramic view of Beverly Hills that rivaled the view from Mulholland Drive.
"Have a seat," Bill said, without looking up from a document he was reviewing at his desk. I took a seat at the conference table by the window. "No, come over here," he said, pointing to a low-slung Eames chair in front of his desk. _The secretary's chair_ , I thought to myself as I took the seat.
Bill was old school and enjoyed using classic power rituals like seating people below him to confirm his position as Lord William. I began to feel excitement because I have learned that the bigger they think they are, the harder they fall when my hookpoint is set. But I could sense that it was going to be a real challenge to get there.
Bill pressed a button on his phone and said, "Gloria, please ask Martin and Jacob to step in." A moment later, two smart-looking Ivy League MBAs come trotting in, taking subordinate seats on either side of mine. _I'm surrounded_ , I thought. _Points for style, Bill_.
Bill reached into a rare Jean Cocteau ceramic bowl sitting on his credenza and plucked out a large red apple. As he did, he asked me to hold on another moment while he asked Gloria to e-mail someone he had forgotten to call. Turning back to me and his underlings, he propped one foot against a desk drawer and took a large bite from his apple. He set it on his desk while he searched for a napkin, and that was when I saw my first opportunity.
While he was chewing on his bite of fruit, I tried to get some kind of frame control. "Look, guys, I only have 15 minutes, so I'm going to get right to it. This is the deal I'm working on," and I quickly briefed the group on the project. But this frail attempt didn't accomplish much. The status gap between us was too big to overcome with just frame control. I could see that Bill was hearing every third word—he was more interested in his apple than in the opportunity I'd come to offer. I made a good opening and my pitch was progressing, but my status was still too low to have any chance at getting this deal to the closing table.
_I'm good at this stuff_ , I said to myself. _Don't force an error. Wait for it._
That was when I saw the golden opportunity. After years of dealing with similar—but not as difficult—social situations, the idea formed in my head, and I knew how to crush his frame, captivate his attention, and establish high status for myself with one simple move.
I said, "I need a glass of water. Excuse me" and raced for the kitchenette I had seen on the way in. There I grabbed a glass of water, a paper towel, and a plastic knife. I thought, _If this doesn't work, Bill is going to grab this knife and kill me with it._
I walked back in but didn't sit. I said, "Listen, Bill, I hope that isn't how you do deals," nodding toward the apple that already had a bite out of it.
"In a real deal, everyone needs a piece. I'll show you what _my_ deals look like."
I reached for the apple on the desk. "May I?" Not waiting for an answer, I took the apple, cut it in two, and took half for myself.
As I returned the half-apple to the place on the desk where Bill had set it, you could hear the roar of silence. The do-boys Martin and Jacob were stunned, and Bill was staring at me with mean and squinty eyes. I took a bite of apple, chewed it quickly, complimented its flavor, and commented a little more on how our deals always have been split fairly with investors. Then I finished the pitch, acting as natural and as informal as if I were having a conversation in my living room with friends.
The three of them listened to every single word from that moment on. I focused intently on the parts of the deal in which I had expertise. Like Benoit or the golf pro, I was working hard to establish local star power.
When I had finished, and before Bill could speak, I quickly began to pull away. "Whoa, look at the time," I said a little comically, glancing at my watch, "I've got to run. Listen, guys, thanks for your time today. If this works for you, let me know."
As I reached for my folio and began to slide out of the chair, Bill waved his arm in the air and said, "Wait, wait. Wait a minute, Oren." And then he started hysterically laughing. This relieved the tension for Martin and Jacob. They smiled and laughed nervously with their boss, and I sat as straight-faced as I could while Bill got the huge laugh out of his system.
"I can definitely see why Dan said I should meet with you. Listen, tell me again who else you have in this deal."
The hookpoint was made. For the next 20 minutes, I answered questions and exchanged information with Martin and Jacob, who were tasked with the due diligence, and I continued pulling away, looking at my watch, worried about arriving late to my next meeting.
Finally, I stood up to leave. As I was shaking Bill's hand, he said, "If Marty and Jake tell me the numbers check out, I'm in."
What this example demonstrates is that a well-chosen, well-timed friendly but disruptive act will dethrone the king in a single stroke. In that brief, shocking moment when no one is quite sure what you've just done, that is when your frame takes over and when high status transfers to you.
To keep my frame strong, after the apple thing, I just ignored anything that didn't advance the pitch. This is an important lesson. In general, just ignore conversation threads that don't support your deal, and magnify ones that do. I kept talking about the deal—you'll see specifically some of the things I talked about in Chapter 4.
Here is a quick review of what happened in Bill's office that day:
1. I found myself in Bill's office with no frame control and in the beta position.
2. I perpetrated a mildly shocking but not unfriendly act that caused a new frame collision.
3. As the shock of my action wore off, the attention of the targets did not waver—believe me, it never does when you do something like this—and I continued accumulating status like a video gamer collecting power stars as each of the targets advances to new levels. The faster you grab status, the more is available for you to take.
4. As I captured attention, I then shifted my focus to acquiring local star power and the alpha status.
5. I got local star power by using information dominance to quickly shrink the frame around my area of specialization, making me unassailable. Because I was the expert, no one could undermine my deal points.
6. Using my newly acquired local star power, I quickly moved the discussion to a level where I could not be challenged by using the primary core values of hard work, domain expertise, and moral authority—which we will discuss in a moment.
7. The moment I was done with my pitch, I began to pull away and kept pulling away until I finally left the office—but not before I had set the hookpoint and received a decision.
These rules are applicable to any situation where you are pitching on someone else's territory.
Here are some other important things to remember:
• If you think you'll start a meeting from the beta position, always be on time for the appointment. When you are late, you are giving away power. It's difficult to establish strong frames when you can't play the game of business by its most basic rules.
• Momentum is key. Create high status immediately. Do not hesitate. Choose a frame, and force a collision at the most opportune moment—and do it early. The longer you wait, the more you reinforce the status of your target.
• Avoid social rituals that reinforce the status of others. Idle social banter diminishes your status.
• Have fun. Be popular. Enjoy your work. There is nothing as attractive as someone who is enjoying what he or she does. It attracts the group to you and allows you to build stronger frames and hold them longer.
As we have been saying, when you are the high-status person in a social interaction, you get all the good stuff. When you're the alpha, life is easy. The statements you make are trusted. The emotions you show will set the overall mood in the room. And most important, when you speak or gesture or even look like you _might_ speak or gesture, people turn their attention toward you.
Just remember, this process is geared to build and seize situational status, which is temporary. Once you leave the social encounter, it's wiped out. Gone. And if you come back later, you'll have to start over and build it again. Even if that's only five minutes later.
And you won't be able to seize global status, which is the honor or prestige attached to a person's position in society. It's the sum of the person's wealth, popularity, and power. For example, you're not going to sit down with a billionaire and have him believe that you're somehow a triple billionaire. Global status is fixed. It's only situational status that you can grab and control.
Fortunately, you do not need to be rich, famous, or powerful to enjoy status in your business encounters and social situations. If you do not have high status, you can create it temporarily.
#### **Seizing Situational Status**
Here are the steps involved in elevating your status in any situation. You will recognize some of these actions from framing, and for good reason. Frame control and status are closely related, as are the pitch techniques you will learn in Chapter 4.
1. Politely ignore power rituals and avoid beta traps.
2. Be unaffected by your customer's global status (meaning the customer's status inside and outside the business environment).
3. Look for opportunities to perpetrate small denials and defiances that strengthen your frame and elevate your status.
4. As soon as you take power, quickly move the discussion into an area where you are the domain expert, where your knowledge and information are unassailable by your audience.
5. Apply a prize frame by positioning yourself as the reward for making the decision to do business with you.
6. Confirm your alpha status by making your customer, who now temporarily occupies a beta position, make a statement that qualifies your higher status.
The last step in this sequence is of vital importance, and it's not as scary as it sounds. As I pointed out earlier in this book, I do not abuse the power I am holding by committing overt acts of dominance. Instead, I am playful, with lots of give and take that makes doing business fun.
One of the best ways to get a customer to confirm your alpha status is to make him defend himself in a light-hearted way. Not only does this let you know that you are still in control, but more important, it also reminds the customer that he holds a subordinate position. The customer then will defer to you, even in front of his underlings.
I may say something like, "Remind me again why in the world I want to do business with you?"
This usually elicits a few guffaws—and a serious response amid the laughter: "Because we're the largest bank in California, Oren."
To which I say, "Yeah, that's good, I'll keep that in mind."
It needs to be playful and interesting, with just a little edge to it. Keep the customer qualifying back to you as long as you can. Do it as much as possible right up to the point where it becomes a little awkward—or is just taking too much time. Ask another qualifying question: "Have you ever done a deal this large before?" This is the best way I've found to get an audience to qualify my dominant frame.
Now that you have an understanding of frames and how to create and use status to support frame control, let's move on to the heart and soul of my method—the pitch.
## **Chapter 4**
**Pitching Your Big Idea**
In 1953, molecular biologists James Watson and Francis Crick introduced the world to the double-helix DNA structure, the so-called secret of life, widely considered the most important scientific discovery of the twentieth century. The presentation earned Watson and Crick the Nobel Prize. And what is most striking about this accomplishment is that the full presentation takes just five minutes to read aloud. That's the _complete presentation_ —introducing the secret of life, explaining it in detail, and showing how it works.
Pause and consider this for a moment: _The most important scientific discovery of the twentieth century can be pitched in five minutes._ Yet nearly every pitch that I've seen—and I see hundreds every year—takes at least 45 minutes and usually an hour, _a ridiculous amount of time!_ No company in America should let its executives pitch for an hour. In a moment, you'll see why.
### **Pitching the Big Idea**
So far we have worked in the realm of frames and status, which are abstract notions. Now, however, lace up your shoes and tuck in your shirt—it's time to get in front of someone and deliver a pitch.
And if you're the front man, the guy who has to take a big idea on the road and pitch it, you need to know exactly how to give a complete presentation in a much shorter time frame than most. But as you'll see in a moment, short time frames are not a choice. You can't afford to run longer. The audience's brain won't give you more time. And worse, when attention runs dry—after about 20 minutes—the brain starts forgetting things it has already learned. Talk about going in reverse.
As soon as the pitch or presentation begins, one critical thing must happen: The target must feel at ease. In the vast majority of cases, they don't _because they don't know how long they're going to be stuck listening to you_ , and you're a stranger. Most people just don't want to sit through an hour-long pitch. To put them at ease, I have a simple solution: It's called the _time-constraint pattern_. This is what you say, exactly, to let the target know he isn't trapped in the typical hour-long-meeting: "Guys, let's get started. I've only got about 20 minutes to give you the big idea, which will leave us some time to talk it over before I have to get out of here."
Doing it this way puts the target at ease. It shows that you know what you're doing and that you're a pro. _Anything_ can be pitched in 20 minutes by a pro. It also shows that you're busy because you have a strong idea and you can't hang out too long in a single meeting.
What's important here is not your mastery over the details but your mastery over attention and time. Instead of trying to achieve what is virtually impossible—holding the target's attention for longer than 20 minutes—we need to observe the limits of the human attention span.
You're going to make the pitch in four sections or phases:
1. Introduce yourself and the big idea: 5 minutes.
2. Explain the budget and secret sauce: 10 minutes.
3. Offer the deal: 2 minutes.
4. Stack frames for a hot cognition: 3 minutes.
### **Phase 1: Introduce Yourself and the Big Idea**
Following this formula, the very first thing you need to do—even before you think about explaining your idea—is to give people your background. But you have to do this in a specific way; your success depends on how well (and how fast) you do it. After the introductory chitchat, where you establish status and use frame control, it's natural for the target to ask, "What's your background?" or "How'd you get started in this?" At this point, you can begin the pitch, _starting with your track record of successes. Not_ a long rundown of all the places you worked. _Not_ all the projects you were tangentially involved with. _Not_ your whole life story. The key to success here is making it about your track record. Things you built. Projects that actually worked out. Successes. Spend less than two minutes on it and definitely not more—and don't worry. Before your pitch is over, the target is going to know a whole lot more about you.
When a friend, Joe, was getting funding from Boeing, here's how he did it:
1. "My degree is from Berkeley. I did my MBA at UCLA.
2. After that I was at McKinsey for four years, but really, my only homerun there was the sales program I did for Lexus. Saved them about $15 million, and they still use it today.
3. I left consulting six months ago to work on the 'big idea.'"
Yes, Joe has done a lot more than that over the years, but so what. Only his big wins are worth talking about at this point. Is there a lot more to _your_ background than this? Of course there is. But in the pitch, time and attention are not infinite. In fact, they're _extremely scarce_. And you're going to need all the time available to get (and keep) frame control. Plan to stroke your own ego later—when the deal is more likely to go through.
Many times I've seen people spend 15 minutes or longer on their background. Absurd. No one is that fantastic. Yet people often think that if some background biography is good, then more is better. _But people's brains do not work that way._ Research has shown that your impression of someone is generally based on the average of the available information about them, not the sum. So telling people one great thing about yourself will leave them with a better impression of you than telling them one great thing and one pretty good one. And it gets worse if you tell them one great thing, one pretty good thing, and two mediocre things. _Stop with one great thing._ Get your track record on the table, and do it fast, clean, and problem-free. This is not the place to get hung up with questions, deep conversations, and analysis— _there's still a lot to do._
Is this different from what you're used to? Is framing this way a completely new way of looking at pitching? Yes to both. But if you don't feel like changing to frame-based pitching, you can always console yourself with the fact that you are not alone. Wasting time and focusing on the wrong things are problems that exist at the highest levels of business.
### **The "Why Now?" Frame**
You're almost ready to pitch the "big idea." But first, a reminder of the obvious: Nobody wants to invest time or money into an old deal that has been sitting around. _This is why you need to introduce a "Why now?" frame. It's vitally important that the target knows that your idea is new, emerging from current market opportunities and that it's not some relic left over from bygone days._ The target needs to know that you are pitching a new idea that came to life from a pattern of forces that you recognized, seized, and are now taking advantage of. And the target needs to know that you have more knowledge about these things than anyone else.
_There are unspoken questions in the target's mind as to why your idea is relevant and important and why it should be considered as important now._ By anticipating these questions and definitively answering them before they are verbalized, you will tick an important checkbox in the target's mind and put the target more at ease. Everything you say from that point forward will have context, greater meaning, and more urgency, reinforcing its scarcity.
What I have discovered over time is that in every business there are three market forces that together triangulate to answer the "Why?" question, and you can use these forces to create a strong "Why now?" frame.
#### **Three-Market-Forces Pattern: Trendcasting**
When you describe your idea, project, or product, first give it context by framing it against these three market forces or trending patterns that you believe are important.
1. _Economic forces._ Briefly describe what has changed _financially_ in the market for your big idea. For example, are customers wealthier, is credit more available, is financial optimism higher? Increases or decreases in interest rates, inflation, and the value of the dollar are considered as prime examples of forces that have significant impact on business opportunities.
2. _Social forces._ Highlight what emerging changes in people's behavior patterns exist for your big idea. An obvious example in the market for automobiles, concern over the environment—a social force—is driving demand for electric vehicles.
3. _Technology forces._ Technological change can flatten existing business models and even entire industries because demand shifts from one product to another. In electronics, for example, change is rapid and constant, but in furniture manufacturing, change is more gradual.
Describe the genesis of your idea, how it evolved, and the opportunity you saw as it was emerging. _The backstory of the idea is always interesting to the target._ Once this story is told, everything you say in your pitch will be legitimized by it.
As you craft your backstory, think in terms of how it come to be where it is today and how you found it. Describe the steps in its evolution, and show how it evolved—how it moved—to finally become the opportunity you have now identified and captured.
The three basic steps are:
1. Explain the most important changes in our business. Forecast the trends. Identify important developments—both in your market and beyond.
2. Talk about the impact of these developments on costs and customer demand.
3. Explain how these trends have briefly opened a market window.
Here is an example that ties the three market forces together into a tight pattern that supports the "Why now?" frame for a product called _UpRight_ , a device you wear on your wrist that wakes you up slowly, at exactly the right time, so that you feel well rested.
_Economic force._ The cost of making this product has just gone below the $10 mark. This means that the retail price can be $69. We've been waiting two years to hit this price point.
_Social force._ One of the changes in our society is that people don't get enough sleep or even the right kind of sleep. While this problem is growing only 1.8 percent a year, awareness of it is skyrocketing. People know that they need better sleep; it is a hot topic at all levels of society.
Against this backdrop, your idea begins to occupy the foreground. To continue with the example:
_Technological force._ This device requires a controlling chip and solenoid that now can be manufactured small enough and at a controllable price, allowing us mass-market capabilities.
By starting your pitch with the three market forces, your idea now enjoys a prominence that it did not have before. Now your idea has a history, an exciting evolutionary path to the present time, and credibility. The idea is displayed against the economic, historical, and sociologic changes that made it emerge from the shadows—but it has just barely emerged. You were alert and saw the potential and are now developing it. (This is a nice point to strengthen your prize frame.)
It does not matter what your idea, project, or product is—they all have a history and legitimacy when framed within the three-market-forces pattern. With the three-market-forces pattern, everything has a story.
As you write the "Why now?" frame for your idea, start by thinking in the broadest terms possible, going back as far as you need to go to understand and explain how it come from the past into the present and why it is special. Remember, it doesn't matter if you're pitching jet fighters, securities, real estate, software, or cotton balls, you need to frame your deal in this way because it explains the force behind its evolution.
_Movement is a critical element in the "Why now?" frame._ Your target needs to understand the forces that are pushing your deal and to understand that your success is inevitable and imminent as a consequence of these greater forces.
This is another area where you need to understand how the crocodile brain of your audience is working. _A huge part of the brain is devoted to detecting movement._ This is what makes it so hard to find things you have lost. Your keys, cellphone, or pencil do not move. You can look right at something and not see it because it is still. This is why animals, when they are frightened, tend to freeze. _Your brain grows accustomed to things that are not changing, and they effectively vanish._ A bird, holding its head very still, can hunt efficiently because the only thing it can see is a wiggling worm. If you injected your eye muscles with curare—a paralytic poison—the same thing would happen to you. Even without such injections, though, it is clear that movement captures your attention. And this is what you want to take advantage of with your pitch. You don't show people a static picture of how the world would be if your plan were implemented, but instead you show them how your idea is moving away from the current standard to a new way of doing things.
There is one more detail about how the brain works that you need to know, and it's called _change blindness._ It is surprising, but if you show people two pictures in rapid alternation, and one of them has some change in it—even a relatively major one—people will not see it. You can replace grandma with a tree. _It does not count as movement, and the brain ignores it._ You can hunt and hunt for what is different between the two pictures as they go back and forth and think they are identical. It is only when your attention is focused deliberately on the thing that is changing that you can finally "see" it. Once you know this fact about how your audience's mind works, _you realize that you cannot just show audience members two possible states and hope that the difference captures their attention._ You need to show them the movement from one to the other.
We are not wired to see or hear a static pitch: " _That_ was the old way, but _this_ is the new way." _That can trigger change blindness, where the target won't_ get _your deal at all._ The formula I've just given you, the three changing market forces, overcomes the potential for change blindness. With three market forces coming into alignment, you are literally showing the mind's eye how the market is moving to benefit your big idea.
Here's an example of my colleague Joe giving a pitch:
"In recent years, there hasn't been much going on in the business of building new airports. In fact, it would be fair to say that the market has been dead. But now things are heating up. _Three major forces are changing the market. First,_ banks have started lending to aviation projects. _Second,_ the Federal Aviation Authority (FAA) is now issuing building permits. _Third,_ our main competitor is excluded from bidding on this deal because of a conflict of interest."
The target sees that there's a market rationale for Joe's airport deal. _It makes sense._ The combination of the bank's new willingness to lend, the new FAA pressure on the airport, and the lack of competition have created the opportunity for this deal to happen now.
One of the most important things I've learned that has made every one of my deals possible is that targets simply do not like old deals. They want to see movement, and they don't like deals that have been sitting around, ignored by other investors or partners. It would be like the copier salesperson saying, "Hey, how would you like the Model T100? We've had 50 of them in the warehouse forever."
### **Introducing the Big Idea**
This does not take 15 minutes. It takes 1 minute. You don't have to explain the big idea in great detail. Oh, I know you want to. It's instinctive: First, introduce yourself; then dive into details. I get the same urges. And this seems like the perfect time to do it. _But it's not time for details._ Your target doesn't want the deal yet. So the pitch temperature is cool. Lots of details will turn it cold. The details can come later. First, you will establish the big idea using an _idea introduction pattern_. The venture capitalist Geoff Moore developed this pattern in 1999, and it still works today.
#### **The Idea Introduction Pattern**
This idea introduction pattern goes like this:
"For [target customers]
Who are dissatisfied with [the current offerings in the market].
My idea/product is a [new idea or product category]
That provides [key problem/solution features].
Unlike [the competing product].
My idea/product is [describe key features]."
Here's an example of a quick introduction for a big idea called the "EnergyTech 1000."
#### **Example 1**
"For companies with large buildings in California and Arizona
Who are dissatisfied with their aging solar panels.
My product is a plug-and-play _solar accelerator_
That provides 35 percent more energy from old panels.
And unlike the cost of replacing panels,
My product is inexpensive and has no moving parts."
That's it. The big idea can be introduced with this pattern in about one minute.
Here's another example of the idea introduction pattern.
#### **Example 2**
"For busy executives
Who don't have enough work space on their computer monitor.
My product is a visual array
That provides eight flat-screen monitors, linked together, that can fit on any desk.
Unlike the common do-it-yourself solutions, having just two or three monitors,
My visual array lets executives use Excel, Firefox, Word, Gmail, Skype, Photoshop, Explorer, and TradingDesk at the same time with no confusing windows."
Here's how Joe used the idea introduction pattern to introduce his airport deal.
#### **Example 3**
"For investors needing a 10 percent cash yield or better
Who are dissatisfied with risky investments such as stocks.
My airport deal is a project with low risk and lots of protection
That provides a current cash flow.
And unlike most development projects, you can cash out any time you want."
Certainly this does a lot to capture the target's attention. _It is important to realize, however, that capturing the target's attention_ _doesn't mean that you are commanding attention._ You will soon be skeptical that attention can ever be _commanded._ What's worse, you'll appreciate how it can be lost in a few seconds by making the wrong moves. The rudimentary model of how attention works goes like this: _We notice things that have movement through space and time because they are likely to be important._ But there's a catch—a lot of the time things that move are also things we have to run away from. Starting from this premise, in the pitch we want to create attention without threat. This is why I have come to believe in—and rely on—the idea introduction pattern, because, of all the ways to introduce an idea, _it does the least to trigger threat avoidance in the croc brain._
Neuroscientist Evian Gordon is convinced that minimizing danger and threats around us is "the fundamental organizing principle of the brain." As I've said before, _the croc brain doesn't think about threats too deeply._ It just reacts. It doesn't stop to research whether the snake coming toward us is a copperhead or a cottonmouth.
Although this natural defense mechanism is beneficial in evolutionary terms, researchers believe that when we enter social situations (like a boardroom where we are expected to pitch), there's one undeniable fact—we sense a potential threat to our own well-being. For example, we could get rejected. We could embarrass ourselves. We could lose the deal or lose face. When these social threats appear, the threat-avoidance system in our own brain starts pumping adrenaline and other neurotransmitters. Anxiety kicks in. We've all felt it—that moment when we are standing in front of an audience and it feels like they're not paying attention. Our heart rate increases, our face turns flush, and sweat pores open up. We are responding to a social threat.
It's important to recognize that humans are hardwired for social interaction. So, if you haven't thought of social situations as potential threats, it's might be time to start thinking that way now. In one study, researchers had subjects play a computer game where they thought they were throwing a digital "ball" with a few other participants. After a while of playing the game, the subject's online partners started throwing the ball _only_ to each other, leaving the subject as the odd-man out. Ouch! The researchers measured the subject's reaction through brain scans. _What they found is that social threats engage the same threat-response system in the brain as physical threats do._ And to make matters worse, the brain can trigger threat responses far ahead of when you consciously become aware of the threat.
If you don't use the idea introduction pattern to deliver the big idea (or some other highly controlled way to encapsulate the big idea), here's how trouble can start brewing. First, the target picks up on your anxiety. Second, when you see the target get uncomfortable, you get more tense; you _look_ tense. Third, an endless feedback loop starts: The targets senses your anxiety, and a similar threat response triggers in his system. You have lots more to do in this pitch, you've barely started, and you don't want to get caught up in a negative-feedback loop. It's too early in the pitch to start dealing with serious malfunctions.
The idea introduction pattern breaks the idea down to the essential basics: Here's what it is; here's who it's for; and here's who I compete with. No anxiety, no fear, no drama.
Let's review the actions to take in phase 1 of the pitch:
• First, you put the target at ease by telling him _in advance_ that the pitch is going to be short, just about 20 minutes, and that you're not going to be hanging around too long afterward. This keeps the target's croc brain focused on the here and now and feeling safe.
• Then, you give your background in terms of a track record of successes, not a long list of places and institutions where you simply "punched the clock." There's plenty of evidence to suggest that the more you talk about your background, the more average it becomes because the target is hardwired to average information about you, not add it up.
• Next, you show that your idea is not a static flash of genius. Rather, there are market forces driving the idea, and you are taking advantage of a brief market window that has opened. (And you've admitted that there will be competition, showing that you're not naive about business realities.)
• Because the brain pays attention to things that are in motion, you paint a picture of the idea moving out of an old market into a new one. Doing it this way, you don't trigger change blindness, which would make your deal easy to neglect.
• Last, you bring the big idea into play using the idea introduction pattern. Now the target knows exactly what it is, who it's for, who you compete with, and what your idea does better than the competition's. This simple pattern makes sure that your idea is easy to grasp and focuses on what is real. This strategy works so well because it avoids triggering a threat response.
However, this method should not imply that everything in your pitch must be simplified and reduced—you'll be delivering plenty of complex and detail-oriented information soon.
### **Phase 2: Explain the Budget and Secret Sauce**
It's been easy to maintain the target's attention so far. In phase 1, all you had to do was introduce yourself and the big idea in about 5 minutes (or less.) In phase 2, it gets harder to hold the target's attention. Now you have to explain what problems the big idea really solves and how it actually works. _The opportunities to scare the croc brain seriously multiply when you start to explain how stuff works._
Over the years, an enormous amount of pressure has been put on businesspeople to make their complex ideas more simple—but few gurus have come up with methods to do this that translate into real-world success. At least I haven't seen them—and I've been looking for 10 years.
The frustrating thing about simplicity is that it's _supposed_ to work wonders in a presentation. Summarize your information, make it supereasy to understand, roll the concepts into an "executive summary"—the target will love you for it.
I realize that what comes next goes against conventional wisdom, but what I've discovered is that _simplicity doesn't really matter._ If it really worked, everyone would be doing it. But it doesn't, and they aren't. Simplicity can make you seem naive or unsophisticated. You can underwhelm the target with too little information just as easily as you can overwhelm him with too much information.
_What you really want to do is tune the message to the mind of the target._
Think about the way you have to talk to a child. You don't just make the thing you want to say simple. For example, if you want to say, "There's no dessert being served before dinner is eaten," you don't make it _more_ simple and say, "No dessert before dinner." In fact, you might have to make it even longer and more complex, explaining the reasons. Again, what's important is that a child's mind is different from your own, and you have to understand how that mind reasons. This is why it's so important to understand how the croc brain reasons. I've concluded that _ideas you come up with_ _using your problem-solving brain—the neocortex—must be intentionally retuned for the croc brain that will receive them._
Early in my work to figure this stuff out, I stumbled onto something cognitive psychologists called "theory of mind" that supports this. _When you have a working theory of mind, you are able to understand how thoughts, desires, and intentions of others cause them to act._ When someone can only see a situation one way, their theory of mind is weak. When you have a strong theory of mind, you recognize how other people have different perspectives—and that they know different things about the situation, and that their desires are not always the same as your desires. A strong theory of mind also will let you know that anything involving statistics needs to be highly simplified. _The croc brain hates thinking about probabilities._ Our advanced society had to invent complex formulas and equations for statistics exactly because our brains are not wired to think about statistics on their own. While there's ongoing debate about what "complexity" the average audience will like and dislike, one thing is certain: _If you're describing relationships between people, you can provide plenty of detail._ The brain is really good at understanding complex human relationships.
As I look back on my experiences, two giant realizations tower above all others:
_Realization 1:_ It doesn't matter how much information you give, a lot or a little, but instead how good your theory of mind is. In other words, it's important how well you can tune your information to the other person's mind.
_Realization 2:_ All the important stuff must fit into the audience's limits of attention, which for most people is about 20 minutes.
### **Get Their Attention**
Earlier I said that one of the things that can go wrong is that your pitch is boring. In a large majority of presentations, this is the problem. In fact, virtually everyone is long-winded when they present. Yet there is absolutely no doubt among either executives or academics that audience attention fades out fast once a presentation has started. Studies of vigilance show that the targets generally can't focus on an idea for more than a few minutes. Some believe that it is a few seconds. Either way, why quibble? Attention wavers almost uncontrollably. People's minds wander. Distractions, from inside the person and the outside world, are constantly competing with your pitch. And anyway, even if there were no distractions, _the brain is still a cognitive miser—it wants to exert as little energy as possible figuring out you and your idea._
What grabs the target's attention, and once attention is grabbed, what holds it there?
_Attention will be given when information novelty is high and will drift away when information novelty is low._ You already know this. If your stuff looks boring, if it has no visual stimulus, is a bunch of cold, hard facts and involves spaghetti-like complexity—no one is going to offer you much attention.
Yet there's nothing more important than attention. Oh, we can vigorously debate whether attention accounts for 70 percent of the reason someone succeeds with a pitch or 50 percent, but no one can seriously question that getting and holding attention are the biggest reasons a pitch either connects with the target and succeeds or misses the target and fails.
Looking at it another way, if the target were willing to pay attention to you for a few hours, then just about any pitch—good or bad—would work. _But you don't have hours._ More likely, you've got those 20 minutes we've been talking about. And maybe, if you're just winging it, you only have five minutes before your pitch wanders into a mental no-man's land.
#### **What Is Attention?**
_To control attention, I have always felt that it's important to know what it's made of. Attention_ is a sort of vague, all-encompassing term that seems to just _define itself_. But who would ever try to make a martini without knowing what's in it first? I phrase the question this way because you'll see in a moment that attention is just that—a cocktail of chemicals served up to the brain as a lubricant for social interaction. You need to know how to blend this perfect cocktail and when to serve it.
How did I find what the ingredients are? I didn't have to. Researchers with brain scanners and hardcore neuroscience chops did the work. What they've worked out is that _when a person is feeling both desire and tension, that person is paying serious attention to what's in front of him or her._
The critical lesson of brain scans is that attention is always a delicate and unstable balancing act between desire and tension. It comes down to the presence of two neurotransmitters: dopamine and norepinephrine.
_Dopamine_ is the neurotransmitter of _desire_.
_Norepinephrine_ is the neurotransmitter of _tension_.
Together they add up to _attention_.
If you want someone's undivided, fully engaged attention, you have to provide these two neurotransmitters. These two chemicals work together—you need them both to be coursing through the crocodile brain of the target. But each has a different triggering mechanism.
To give a dopamine kick and create desire, _offer a reward._
To give a norepinephrine kick and create tension, _take something away_.
You're going to learn the patterns for triggering the desire and tension right now.
_**What Does Dopamine Do?**_ Dopamine is the chemical in the brain that chases rewards. It takes about 1/20 of a second for dopamine to guide humans toward some kind of action. Dopamine levels rise in the brain when you see or hear about something you want. When you see a person acting curious, open-minded, and interested in something—it's dopamine that's motivating them. A strong cup of coffee, Yohimbe root, cocaine, and the cold medication sudafed all increase dopamine levels in the brain. In most people, so does the thought of winning a large gamble or of even buying what's known as an _ornament_ —like a Rolex watch or some other status-enhancing product.
Dopamine release in the brain is connected to _pleasure activities_ , such things as food, sex, and drugs. But now brain scans show that dopamine isn't exactly the chemical of experiencing pleasure. Instead, it's the chemical of _anticipating a reward._ In his book, _Satisfaction_ , Dr. Greg Berns explains this: "How do you get more dopamine flowing in your brain? NOVELTY. A raft of brain imaging experiments has demonstrated that novel events... are highly effective at releasing dopamine. Your brain is stimulated by surprise because our world is fundamentally unpredictable." He adds, "You may not always like novelty, but your brain does."
_You create novelty by violating the target's expectations in a pleasing way._
Let's review. When you introduce something novel to the target's brain, a release of dopamine occurs. This triggers desire. For example:
A short product demo provides novelty.
A new idea provides novelty.
Good metaphors for otherwise complex subjects provide novelty.
Bright objects, moving objects, and unique shapes, sizes, and configurations all provide novelty.
You want the audience's full attention, and you want to erase everything else audience members are paying attention to, so introduce novelty.
_**How Dopamine Leads the Feeling of Novelty.**_ Until now, I've talked about the raw amount of information that comes into the brain and how it all can't be processed at once. All this information and data from the senses collect in one small part of the brain. There has to be some way of selecting what to ignore and what to act on. Dopamine motivates the human body to act on some things and ignore others.
Research done at the University College London, written about by _Wall Street Journal_ reporter Jason Zweig, suggests that getting what you expected to get produces no dopamine kick, but a novelty in the form of an unexpected gain gives the brain a blast of dopamine. On the other hand, if a reward you expected fails to materialize, then dopamine dries up, and negative feelings start happening.
Just like the martini making we talked about earlier, the amount of dopamine in the cocktail has to be just right. _Not enough, and there is no interest in your or your ideas; too much, and there is fear or anxiety._
Earlier we also talked about the importance of a simple introduction to your big idea but that simple is not always better. The dopamine kick explains why: _People enjoy some intermediate level_ _of intellectual complexity._ It has been argued that people are curious about things they cannot explain but that seem explainable—mystery stories work this way. And this, of course, is why novelty is so important in the pitch. Curiosity is the croc brain becoming interested—feeling like it's safe to learn more. Curiosity derives from an information gap—the difference between what you know and what you want to know. This is the addictive quality of curiosity—and what you are trying to create for the target: curiosity about the big idea.
It's only when the target feels that he knows enough to fully understand your big idea that the curiosity ends—and he becomes satiated. _At that point of satiation, whether you recognize it or not, the pitch is over._
Novel information has the potential to trigger one of two responses—retreat or exploration. Curiosity is a feeling of novel information taking on the second, exploratory path, which is the first step toward a satisfying intellectual experience.
_When a signal from the pitch tells the target there is something new to be discovered, dopamine is released in the brain._ Unexpected (and pleasant) rewards release more dopamine than expected ones. But dopamine has a dark side, too; if the target is expecting a reward and don't get it, dopamine levels fall off fast. And when dopamine levels drop that fast, the feeling of stress is just around the corner. Not only does the target stop taking in new information from you, but he starts forgetting the information you've already delivered.
In summary, expecting rewards generates dopamine. Dopamine is the buzz of novelty. Alone, however, it's not enough to create attention. While dopamine is the chemical of curiosity, interest, and desire, it can't generate attention without norepinephrine, which tends to create tension, and this is why I call it the _chemical of alertness_.
### **Tension**
In the earlier discussion of novelty and desire, I talked about just half the formula for creating attention. The other half of that formula is _tension_. First, let's start with some definitions: _Tension_ is the introduction of some real consequences to the social encounter. It's the response to a clear and unequivocal realization that something will be gained or lost. It is letting the target know that there are high stakes. _Tension indicates consequences and therefore importance._
There's no reason for the target to pay attention when there are no stakes—when tension is absent. A few words about the purpose of tension will help. Here we are interested in the interplay between pushing the target away and pulling him toward us. Not as a point of manipulation—at no point in the pitch are we ever interested in that—but as a way to keep the target alert. If you want your target focused and energized—to really pay full attention—you also must get him to stay alert. Tension accomplishes this by injecting a shot of norepinephrine into the target's brain.
This brings us to an examination of the relationship between novelty and tension. Without both, Dennis, the avocado farmer, loses $640,000; _Jaws_ becomes one of the worst movies of all time; and Benoit, the French waiter, can barely make a living.
We may never have thought about attention this way before, as a cocktail of neurotransmitters, and honestly, why bother now? We certainly don't know what neurotransmitters are, and if we're being honest with ourselves, this is something we don't really care about at a deep level. To most of us, it's worthwhile only in that it illustrates the following:
* * *
_The two parts of the attention cocktail are novelty and tension, which in a pitch work together in a feedback loop for about 20 minutes until—no matter what you do or how hard you try—they get out of balance and then stop working altogether._
* * *
Tension comes from conflict. Some beginning presenters want to rely on their charisma (a pure form of novelty) and try to avoid all conflict in their pitch narrative. They want everyone to play nice. Only smiles, no grimaces. Why? Because in regular life, outside the pitch, confrontation can be stressful and nerve wracking, so it makes sense that we would try to avoid it everywhere. _But in narrative- and frame-based pitching, you can't be afraid of tension. In fact, you have to create it._
The rudimentary patterns I have outlined below have proved extremely rewarding in my career. This might seem surprising—and not because patterns are so simple and basic, but because they are intended to build tension. This is what has given me an edge.
There are three tension patterns, each with an increasing level of intensity. These are conversational patterns you can use at any point in a presentation when you sense the target's attention dropping.
#### **Low-Key, Low-Intensity Push/Pull Pattern.**
PUSH: "There's a real possibility that we might not be right for each other."
_[Pause. Allow the push to sink in. It must be authentic.]_
PULL: "But then again, if this did work out, our forces could combine to become something great."
#### **Medium-Intensity Push/Pull Pattern.**
PUSH: "There's so much more to a deal than just the idea. I mean, there's a venture-capital group in San Francisco that doesn't even care what the idea is—they don't even look at it when a deal comes in. The only thing they care about is _who_ the people are behind the deal. That makes sense. I've learned that ideas are common, a dime a dozen. What really counts is having someone in charge who has passion and experience and integrity. So if you and I don't have that view in common, it would never work between us."
[ _Pause._ ]
PULL: "But that's crazy to think. _Obviously_ you value people over smart ideas. I've met corporate robots before that only care about numbers—and you are definitely not a robot."
#### **High-Intensity Push/Pull Pattern.**
PUSH: "Based on the couple of reactions I'm getting from you— _it seems like this isn't a good fit._ I think that you should only do deals where there is trust and deals you strongly believe in. So let's just wrap this up for now and agree to get together on the next one."
[ _Pause. Wait for a response. Start packing up your stuff. Be willing to leave if the target doesn't stop you.]_
_There's a two-way connection between pushing and pulling that, when it operates simultaneously, introduces enough tension to create alertness._ If you always pull the target toward you, he or she becomes cautious and anxious. Constantly pulling someone in, also known as _selling hard_ , is a signal of neediness. It's a balancing act, of course, because if you are constantly pushing them away, they will take the hint and leave.
One of the most celebrated examples of this push/pull in the Pitch community involves _Mad Men_ 's Don Draper, a pitchman for a fictional ad agency who gets a negative reaction from a client during a pitch. He pushes.
"Looks like there's not much else to do here. Let's call it a day," he tells the client, extending a handshake. "Gentlemen, thank you for your time." _Draper stands up to leave._
I've watched the clip on many occasions, and the result always leaves me with a greater appreciation for the perfect push/pull delivery, one that creates a blast of norepinephrine in the client's brain.
In the clip, the tension grows as the client, surprised, asks, "Is that all?"
Draper replies, "You're a nonbeliever. Why should we waste time on Kabuki [theater]?"
The client responds to the push—suddenly interested in Draper's ideas and paying attention, he asks Draper to sit back down.
Perhaps the most outstanding example of a push/pull pattern that I've encountered occurred some years ago when an audience tried to impose its will on me at a conference.
I was offered an opportunity to pitch my deals to a handful of the most influential investors in a $10 billion market. How could I resist? I would meet them at an upcoming conference in a closed-room, one-on-one experience. The conference organizers charged me $18,000 for the privilege, and they set it all up.
I gladly paid the fee, and I jumped my company jet to Denver, looking forward to a great opportunity to generate new business. After breakfast, I went up to the conference room, ready to rock. I walked in and was shocked by what I saw.
Inside the room were 25 people—more than I expected—and here was the kicker: They were not investors or buyers. They were _due-diligence analysts_. I shook my head. I couldn't believe it.
A _due-diligence analyst_ , as the name suggests, is someone whose job is to analyze and evaluate a deal based on facts and figures. These are neocortex people, and they are tough to pitch because they are all about numbers and are trained to avoid emotion. Imagine a bunch of nicely dressed robots looking for flaws in everything you do and say. Dealing with just one of these people is hard enough, but now I stood in front of 25 of them. And none of them could pull the trigger on a deal, anyhow. This was the worst possible audience for my deal.
The tables and chairs were set up in a U-shape, and I stepped up in the horseshoe, despite my misgivings. I started my pitch by passing out our marketing materials—a beautiful 56-page deal book.
The book outlined the math behind a new kind of financial technique called _bifurcation_ , which can really amplify profits. As the audience studied it page by page, I launched into my pitch. And I absolutely nailed it—or so I thought. But my pitch was all dopamine and no epinephrine—that is to say, _all promise of reward and no tension._ I looked up at one point, hoping to see smiling faces, expecting to hear a barrage of questions from these due-diligence guys. Instead, I got stone cold faces looking back at me. Silence. Not a single question. Imagine looking out at 25 concrete garden gnomes. I can't remember such a nonresponse to one of my pitches. Ever! This doesn't mean that the targets were unreachable. It just means that they had strong analyst frames that were not easily disrupted.
I said, "Guys, since you can't think of any questions to throw at me, let me get those bifurcation books back from you." I started walking around the horseshoe, taking them gently out of their hands. In some cases, I had to pull more aggressively. That's when I knew: _I had the prize frame._ Now they had something to lose—and questions started flowing. Over the next two years, I closed over $5 million with these targets.
_To hold your target's attention, there must be tension—a form of low-level conflict—guiding the interaction._ If there's no conflict, the target may be politely "listening," but there's no real connection. The target is thinking, "He seems like a nice guy, and his idea seems good, but I have other things to worry about right now."
This is a confidence problem. I used to be afraid of creating tension. I was afraid to do anything that might upset the target in any way. Sure, when you and the target are each nodding in happy agreement, it _feels_ great in the moment. You think to yourself, _it's a lovefest._ But when it goes on too long with no counterbalance—it's boring. At the end, the target gets up and says, "That was really nice," and then walk away. Targets want a challenge of some sort. They don't want the easy answers.
If there's a single reason why some of my most important pitches failed, it's because I was nice and the audience was nice, and we were all very polite with each other. There was no tension or conflict. Conflict is the basis of interesting human connections.
_As businesspeople, we come together to find solutions to problems—not to admire problems that have already been solved for us._ If you don't have a series of challenges for the target to over-come—with pushes and pulls and tension loops—then you don't have a _pitch narrative_.
A pitch narrative can be thought of as a series of tension loops. Push then pull. Create tension. Then resolve it.
When there's no tension between you and the target, there's no interest in what happens. The target also has no emotional involvement in what's going on. In other words, the target doesn't much care about what you do, why you're doing it, or what happens to you after you leave. Without tension loops, nothing is compelling the reader to stay with the pitch storyline.
### **The Heart of the Pitch**
Once you have attention by creating desire and tension—you're ready to deliver the heart of your pitch. But keep moving fast because this cocktail of dopamine and norepinephrine you're serving is sloshing around in the brain of the target in the right combination for just a few minutes. And as we discussed earlier, no matter how hard you work, eventually the target's desire is going to become fear, and the tension in the room is going to turn into anxiety.
The greatest problem in short-form pitching is deciding what details and specifics to single out for attention—what to leave in and what to leave out. And since I'll need to give you specific examples, I think that selling a company or raising money is a good framework in which to discuss this subject. For one thing, it is the market with which I am most familiar. I have been working in the capital markets for about 15 years. For another, these markets have provided a fantastic test bed for me because hard data and fast feedback have been readily available. I've got this down to a method.
Actually delivering the core of the pitch is very straightforward stuff. The main requirement is that you understand that what's happening in your mind is not what's happening in the target's mind. Package the information for the croc brain, as I described in Chapter 1. Big picture. High contrast. Visual. Novel. With verified evidence.
Before you decide to spend too much time on this part of the presentation, remember that the following items are a simple punchlist of issues that most pitches have to cover. _These are the prerequisites._ The stuff you _have_ to have no matter what. It's the minimum information you need to show up and be relevant.
Recognize that you can be incredible at turning a business plan into an executive summary or any other kind of elevator pitch and still have that pitch fail miserably. Doing a good job here is not about some genius new way of organizing and presenting information. Would you argue? We don't need yet another organizing theory for information. The basics work fine. What we need is a way to present this material without the target becoming too analytical about it.
When it comes to a choice of what to focus on when pitching the plan that will make the big idea work, I would start by presenting the budget because most people screw this up. It's your chance to be different.
#### **Pitching Numbers and Projections**
In his book _High Tech Ventures_ , Gordon Bell writes, "Start-ups often prepare absurdly aggressive and optimistic plans, which have a very low likelihood of success, just to maximize the company's perceived dollar value." Your financial projections, whether for a product or a company, are supposed to answer such basic questions as, How strong is the company? What if plans go awry, does the company have enough cash to last a few bad quarters? Do you know how to budget well?
A word of caution, however, as you approach these topics: Every experienced buyer and investor knows that you will be doing these two things:
1. Saying that your budgets are "conservative"
2. Preparing _absurdly aggressive and optimistic plans_
To the investor, for example, every pro forma looks the same, a hockey stick chart that shows the following: _We need lots of_ _money today, and way down the road we'll make it back_ (sometimes it works out that way; usually it doesn't).
_Unrealistic budgets and miscalculating costs are the greatest risks to a growing company,_ especially startups. How do you get around the skepticism that surely will fall on your plans? _Focus on demonstrating your skill at budgeting, which is a difficult and highly regarded executive talent._ Spend almost no time on your skills at projecting revenue—a task any simpleton can perform.
#### **Competition**
The act of introducing the budgets to the target will lead him or her to wonder, Who does the big idea compete with? This is a valid question that you cannot ignore. The attractiveness of an idea is based on the industry it's in and how much competition there is. Yet almost no one describes the competition they face in adequate terms. Let's do it right in the pitch. Here are the two major elements of competition:
1. How easy it is for new competitors to jump in the game?
2. How easy it is for customers to switch out your product with another?
#### **Secret Sauce**
To avoid the impression that you are a come-and-go idea that will shine brightly in the market one day and be forgotten soon after, you'll need to show what your competitive advantage is based on. This one thing will give you staying power against competition. In almost every pitch situation, you need something special. Briefly describe it as your "secret sauce"—the _unfair advantage_ you have over others.
You don't have to get too fancy here—just don't take longer than 10 minutes to describe the fundamental workings of your big idea—because you're going to need the last five minutes to offer the deal and stack the frames.
Think the need to move fast doesn't apply to you? _You want to leisurely take an hour to do all this?_ I've met many people who don't believe in the limits of human attention and feel exempt from the consequences of running long. An investment banker I know brags, "I can read them the phone book for an hour, and they'll pay attention." Is the science wrong—should we forget about dopamine and epinephrine cocktails entirely?
Consider for a minute the actor Jerry Seinfeld. His movie, _Comedian_ , is a behind-the-scenes look at the business of performing comedy. In it, Seinfeld reveals the difficulty of being in front of an audience. He is one of the most recognized personalities anywhere in the world. Probably the best-known comedian on the planet. Sure, there's Chris Rock, Dave Chappelle, and Robin Williams, but really, when you think about it, Jerry Seinfeld is as big as it gets.
When he decides to go on the road to test new material, Jerry says that it's not as easy as you might think. He can walk on the stage anywhere, even a small town, and it's clear that the audience knows he's one of the most accomplished performers of modern times, with over $1 billion of television revenues. They're thrilled to be in the presence of a man so popular and funny. _But the thrill doesn't last long._
"I have about three minutes where they will just listen to whatever I have to say," Seinfeld says. "But after that—it can fall apart fast. I get no credit. After three minutes, I have to be just as funny as any other comedian. That's it—three minutes."
And there's more to the story. Seinfeld became aware of the three-minute mark because it takes him as long as a month of full-time work to build up just three minutes of quality content. When he first goes out on tour, that's about all the material he starts with. Three minutes. It takes him months more of steady work to build up 20 minutes of material that can hold an audience's attention. That's worth thinking about. One of the most well-known performers and presenters in the world has to put in months of hard work to build up 20 minutes of material—and when he eventually goes on stage, the average audience will cut him slack for only three minutes. After that, the material had better be really good, or the audience will turn on him.
So when we frame the issue of how long a pitch should be, with the Jerry Seinfeld story in mind, it becomes easier to understand why time is so precious in the pitch: How long can you really be interesting to listen to? Perhaps there is someone who can pitch a deal as dull as the phone book for an hour, which is up to three times the basic limits of human attention, but if so, he or she is a lot smarter and more charismatic than Seinfeld or any other entertainer.
### **Phase 3: Offer the Deal**
In the third phase of the Pitch process, you need to do one thing and do it well: Describe to your audience what they are going to receive when they decide to do business with you. You'll want to push through this quickly for the sake of time—and get back to framing.
In clear and concise terms, tell the audience exactly what you will be delivering to them, when it will be delivered, and how. If they play a part in this process, explain what their roles and responsibilities will be. Don't drill down into a lot of detail; just provide summarized facts that they need to know so that their mental picture of your offering is complete.
It does not matter if you are offering a product, a service, an investment, or an intangible—there will be a fulfillment process involved, and that is what you must explain.
Keep it brief but rich in high-level details so there is no question as to what the audience is going to get. And remember, the most important deliverable in your deal is you.
## **Chapter 5**
**Frame Stacking and Hot Cognitions**
In Chapter 4, I showed you the first three phases of a pitch. By this point, you have held the target's attention for a while. The target knows the essentials: who you are, why this idea is important, how it works, what the "secret sauce" is—and what the target gets when he or she buys. But you're here to do more than just show and tell; this is a pitch, and you're here to make a deal happen. Now you have about five minutes left to propose something concrete and actionable—something so compelling that it will cause your target to chase you to get what you have.
Welcome to the next phase.
### **Phase 4: Frame Stacking and Hot Cognitions**
In the course of my activities seeking out money for deals, I discovered that investors do not operate only on cold, rational calculation. Do you think that the guy sitting across the table from you is an analytical machine?
* * *
The target can like your deal (or be afraid of it) before he knows much detail about it—and the target probably can decide "Yes" or "No" without even knowing what it is. This is _hot cognition_ at work. _Deciding that you like something before you fully understand it—that's a hot cognition._
* * *
We have been led to believe over time by managers, consultants, bankers, and professors of finance that business is analytical. That it's rational. That there are three very well-ordered stages in each business decision: Identify the problem, examine solutions, and make judgments. This makes sense, and this is how it should be in a perfect economic world. In fact, if you took out a blank sheet of paper and asked yourself, "How _should_ I make this decision?" that's how you probably _would_ do it. Research. Analyze. Decide. And if we were all computer-like or even behaved like rational economists think we do, it _would_ work this way. But we aren't, and it doesn't. What's intriguing here is that when we decide on something, we believe that it's because we really "thought it through" or we "used a decision matrix." We think that we are smart, careful, and rational decision makers.
In decision making, however, we don't do much analysis, if any at all. We go with our gut. When Jack Welch eventually wrote his biography, it wasn't called _Intense Analysis_ ; it was titled, _Straight from the Gut_. And when George Soros updates his next edition of _The Alchemy of Finance_ , he's going to include the research of Dr. Flavia Cymbalista, who believes that we feel decisions in our body, not our mind.
_There's a whole side to us that computers don't have and the "rational economic man" economists like to talk about doesn't have either. Our bodies "know" the situations we meet in life and how we should respond._
* * *
"Brain Scanners Can See Your Decisions Before You Make Them" is the title of a provocative article that appeared in _Wired._ The first line in the article reads, "You may think you decided to read this story—but in fact, your brain made the decision long before you knew about it," referring to a study by John-Dylan Haynes, a Max Planck Institute neuroscientist. Haynes says, "Your decisions are strongly prepared by brain activity. By the time consciousness kicks in, most of the work has already been done."
The patterns he found in the brain consistently predicted whether test subjects eventually pushed a button with their left or right hand—about seven seconds before they felt they had made a conscious choice to do it. Do you still think that your decisions are postconscious, in other words, that you rationally think about things and make decisions afterwards? The peer group is shrinking of people who think like that.
* * *
#### **We Tend to Like (or Dislike) Things Before We Know Much About Them**
People do not become friends with each other, choose one career over another, or choose what sport to watch on the weekend based on a detailed cognitive analysis of the pros and cons of each situation. If we stop to think about it, most major decisions are _not_ made by cold cognitive processes such as evaluation and analysis but instead by _hot cognition_. We quickly realize that there probably are very few decisions in our lives that aren't "hot."
Most of the time, the data we have collected about choices and alternatives and options aren't used to make a decision anyway. They are used _to justify decisions after the fact._ We buy the cars we "like," choose the jobs and houses that we find "attractive," and then justify those choices to other people with any number of facts and explanations. "Why this deal?" or "Why this investment?" We don't need facts and explanations to convince ourselves. _We_ know what we like. _Even when we try the rational approach—making lists of pros and cons—if it does not come out how we like, we go back and redo the list until it does._
If you had invested $1,000 with George Soros when he opened his Quantum Fund, you would have about $4 million today. Yet he is known for whimsically changing his investment tactics—we're talking market positions of hundreds of millions of dollars—on a feeling in his back or some other physical signal.
Cymbalista, who has studied Soros and financial decision making, writes, "This might sound mysterious but, in fact, human thinking is constantly guided by subtle bodily tensions. Traders need to learn how to isolate and identify these bodily tensions and relate them to the analysis of the market problem at hand. Certainly, Soros has learned how to combine theory and instinct to make money."
George Soros' backache decision-making is consistent with the research of Dr. Jerome Bruner. According to Bruner, "There are two modes of cognitive functioning, two modes of thought, each providing distinctive ways of ordering experience and constructing reality." Bruner says that one mode of "constructing reality" is called the _paradigmatic mode_ (one can think of this as the detective mode). In paradigmatic mode, the target takes the content of your pitch and analyzes it in terms of "tightly reasoned analysis, logical proof and empirical observation." In other words, the information you're providing is getting analyzed. If you push your listener into this mode, he or she is looking to find a formula that explains you. _Your audience/target will be doing only one thing in paradigmatic mode—trying to analyze. All your creative concepts, future projections, and human inferences are going to be ignored by the analytical/paradigmatic thinker. The only thing that will count are cold, hard facts._
In our pitch, we are _not_ looking to engage with the cold neo-cortex. We are _not_ going to push the target into paradigmatic mode. And we don't need a quantitative analysis of our numbers done by the target right now. Sure, our numbers will stand up to scrutiny and we're not afraid of a stress test—but there will be a time for cold analysis later once the relationship is firmly set in place.
#### **Creating Hot Cognition**
To avoid cold, reasoned analysis, unemotional judgment of ourselves and our idea, _we are going to create hot cognition by stacking frames._ I come to learn about stacking frames by accident.
### **How to Stack Frames**
Part of my work involves buying "defaulted debt" with a partner—specifically high-risk debt. It's a hard-knocks game where it seems that you either do very well or you get _crushed_. My partner did about $250 million in transactions from 2008 through 2010. In this field, however, $250 million is small time among the multibillion-dollar hedge funds and the giants of Wall Street with whom we work—Citi, Goldman Sachs, JP Morgan. Our success is based on keeping alert and making quick, small trades of $20 million to $50 million. We are like gazelles running among elephants. In this game, if you sense a giant foot coming down, you take off running because if JP Morgan, Chase Manhattan, or Goldman Sachs steps on you, they wouldn't even notice. But you would.
In June 2008, the market was in the early stages of a screaming freefall that bottomed out in March 2009. And we knew it was bad, but we didn't know how bad it was going to get: At one point, the housing market in Phoenix dropped 9 percent in a single month. Movements of the Dow Jones Industrial Index of 100 points a day was the norm. Any Wall Street trader knows that you want to be careful trading in this kind of volatility because a single mistake can cause you to blow up.
Buying distressed assets sounds like it should be easy. After all, the market was crumbling and everyone needed to sell their toxic assets.
At the time, I was doing a deal with a trading desk at one of the largest money-center banks. I needed another point of view on the deal, so I called a colleague, Mike. He had a lot of experience in this kind of deal—and a second opinion is always good. I thought that the price was high, and I wasn't sure about the deal.
Now, as I stayed late at work on a Wednesday night, the phone rang. It was a trader at this large money-center Wall Street bank, and this was his fifth call to me. In the back of my croc brain, threat alerts started going off. Why was this $300 billion bank courting me so hard? There had to be better buyers than me in the market? But once I started talking to the trader, I realized that he wasn't selling me in the traditional ways.
"Oren, once we get through this deal, and we know you can close deals, I'm going to introduce you to our senior trader, John Kincaid," the seller told me. "He's a wildman, just like you. It's going to be a total love connection, and he'll get you into the big deals that don't come to my desk."
This was _hot cognition 1—_ intrigue. I wanted to meet the senior trader and get introduced to these bigger deals.
The bank trader continued: "You know the market is on fire right now, and I have the French, English, and South Africans begging me for this package, but if you work hard and don't play any funny retrade games, you can earn your way in." It was true, the market was hot, and those were all players.
This was _hot cognition 2_ —prizing. Although I was the buyer, he was asking me to prove myself. I wanted to impress him so that I could earn my way into the deal.
He continued: "I'd love to give you until next week, but this market is not letting me, and you have to make up your mind by Friday." He said, "I'm totally okay with a 'No'; there's no pressure. But Friday is D-day."
This was _hot cognition 3—_ time frame. He gave me just enough time that I felt I had free will. This wasn't time pressure, just a reasonable time constraint. In the end, the decision was mine to make.
He continued: "And I don't need to tell you, we've done $150 billion in trades this year without a single SEC [Securities and Exchange Commission] sanction; right now we're very particular about our reputation and counterparties. We do things right over here, so no games, no missing wire numbers, just clean paper. We give you a fair price, and that's the deal. Can you play by the rules?"
This was _hot cognition 4—_ moral authority. I assured him that even though my company was small, just a $250 million blip on the outskirts of San Diego, I knew the rules and could do things right.
From the beginning, I never felt like he was selling me. My normal deal-making processes were disrupted by this four-frame stack. The Wall Street trader ran this stack on me _perfectly_ : I was intrigued, I was trying to impress him so I could have a chance to buy the deal, he boxed me into a very tight time frame and yet I felt no pressure, and I was trying hard to prove that I had a good moral values. I was a puppet. My cold analytical decision processes weren't just disrupted; they were shut down and turned off. My neocortex was short-circuited, and my croc brain was rolling around in cognitive mud, happy and calm. I was going to do the deal. And the next day I called the Wall Street trader back and said, "Send me the docs. I'm doing it!"
There's no question that the pitch worked on me. Fortunately, it also worked on Mike.
Mike Hanson soon called me, bragging that he had taken down the deal— _stolen it right out of my eager hands._ Thank God for that. Two years later, he was still down 15 percent in that trade and losing money daily. He was miserable with the deal. For me, though, it had worked out fine.
I had learned the _four-frame hot cognition stack_.
Now, a few years later, I've found that this stack works in many kinds of pitches. And it's a lot of fun for both me and the targets. It's hot, emotional, and fast-paced. When the target starts getting analytical and cold, it's time for the four-frame hot cognition stack to enter the pitch.
To make it work, you just implement the frames you are already familiar with from Chapter 2. All you have to do is learn how to stack them up one after the other to generate a _hot cognition_ —in other words, to create what cognitive scientists call a _wanting._
Specifically, we're _not_ trying to get the target to "like" us because learning to "like" something is the slow and intellectual business of the neocortex. This is not the business we're in. We want high-temperature frames that create hot cognitions. And we're using frame stacks to make sure that the target's croc brain wants us and moves toward us—even ends up chasing us to get the deal. Let's do it now.
Here are the four frames we're going to stack in quick succession. (Doing this correctly will move you quickly into the last part of the pitch—the _hookpoint_.)
_Hot cognition 1:_ the intrigue frame.
_Hot cognition 2:_ the prize frame.
_Hot cognition 3:_ the time frame.
_Hot cognition 4:_ the moral authority frame.
#### **Hot Cognition 1: The Intrigue Frame**
In Chapter 2, we talked about the theory of _frame-based pitching_ , and I introduced the _intrigue frame. Now we're going to use it._ The purpose is to get a large dopamine dump into the croc brain of the target—and build desire. I do this by introducing something the target is sure to want—but cannot get right now.
Here's the intrigue frame I used recently in a situation where I had given the pitch, offered up the deal, but, before I could blink, was getting hammered with technical questions:
"Guys, before we spend our last few minutes on financial details, let's decide first if you love me and you love the basic deal. And look, if you decide that you do love the deal, you're obviously going to meet my partner Joshua," I tell them. "He's a very interesting guy, a great guy, but a little eccentric."
I see if I have their attention, which I usually do. People like to hear stories about interesting and slightly eccentric characters.
"Last year, when the markets were volatile, I had this little deal, about $10 million," I tell them. "It seemed easy because it was such a small deal, and I was the only one working it. Things were going like clockwork until the bank called and at the eleventh hour and backed out. No explanation; they just pulled out. That left a $3 million hole in the plan—the deal was falling apart fast. This had come out of the blue, and I was sure that the board of directors of my company would fire me when they found out about the screw-up. I knew I had to go to Joshua with this."
My audience leans in. They want to find out how the problem was solved. And who is this Joshua? They are intrigued.
"Joshua asked me, 'Oren, is this a good deal?' I said, 'Yes, it's good. Let me tell you all about it." But he didn't stick around to listen. Instead, he went to lunch, without even giving me time to grovel. What could I do? I had to save the investors—and myself too. I wanted to pitch Joshua, to do anything to convince him to save the deal. He, it seemed, just wanted to eat lunch. I was preparing for my funeral when I got the call from the board. They had mysteriously gotten the $3 million. Joshua had wired it in, from his BlackBerry, while having sushi. He didn't ask me to sign any guarantees. He didn't even ask to see the file. If he hadn't made that wire, my investors would have lost a lot of money, and my reputation would have taken a hit. The thing about it is that he does this kind of stuff all the time. Wait until you meet him!"
This is the kind of narrative that targets truly enjoy. Who is this mystery man, Joshua, and how do we meet him? This works because it is not about _what_ happened. That's actually a boring story. What's important is _who it happened to and how the characters reacted to their situation._ Nobody cares about narratives where you witnessed something. They want to see someone forced into action and positively overcoming obstacles. This new notion of building an intrigue frame makes a lot of sense when you think about _why_ the target is sitting there listening to _your_ pitch.
* * *
_The targets have given you their time because they want to visit a new world to learn about new things and interesting ideas and become involved in the lives of unique, interesting, and talented people._
No one is seriously going to go into business with you until they know something about how you conduct yourself in tough situations. And it's no use telling anyone, "I'm a good person." That's a useless analytical fact that has no narrative to support it.
People want to know how you have faced obstacles and overcome them. They want to see you in situations that reveal your character. They want to know that you are someone who rises to whatever level necessary to overcome obstacles and someone who travels in the company of interesting people who are players in whatever game you are playing.
* * *
This kind of story, according to Dr. Bruner, puts the target into narrative thinking mode. In narrative mode, we seek to understand reality from events in terms of "human actors striving to do physical things over time." And from this simple statement, we draw an important insight: Your big idea is probably an abstract notion. If you are honest with yourself, what is it _really_? A pile of financials, a bunch of timelines, some customer orders, a marketing plan, an Internet site, and some smart new ideas. You probably have projections, information technology, competitive analysis, and market timing. This kind of info is much too abstract.
* * *
_The target's brain does not love abstract concepts—every abstract concept has to be kicked up to the neocortex to be worked on, slowly and painstakingly._
* * *
This is why you need analog human narratives to do your explaining. They don't have to be sent to the neocortex for processing. For example, the Joshua story is anchored in the real world with human characters: The target can relate to the human elements in it.
Why is the intrigue frame best performed as a narrative? Here's how the target's brain interprets narratives at the crocodile level:
In narrative mode, the croc brain sees human characters confronting real-world obstacles in time scales that make sense. The croc brain can sort of verify events in time because they are easy to relate to our previous experiences and understanding of how the world works. If the unfolding of events, as told by you, feels accurate to the target, then a truth is assigned to it. _A narrative that feels correct in time will convey a strong sense of truth and accuracy._
In stark contrast, facts and figures have no built-in internal mechanism for feeling the truth. If we present just facts and figures, then we trigger a paradigmatic mode that encourages the target to use logical rigor over imagination, intellect over feeling, and theories over stories. Clearly, you don't want your target in this paradigmatic thinking. _Short and strong narratives that introduce characters who are overcoming real-world obstacles can ignite hot cognitions—which, in turn, push the target out of paradigmatic and analytical thinking mode._
There's a basic formula for developing this kind of intrigue frame.
_**Narrative Pattern for Building an Intrigue Frame.**_ Intrigue frames, like all narratives, whether fiction or nonfiction, need structure. Without structure, a story wanders around without purpose and becomes boring. Here's a pattern that will give any of your stories a dramatic arc that ends with intrigue:
• Put a man in the jungle.
• Have beasts attack him.
• Will he get to safety?
Clearly, being stuck in the jungle is a metaphor for being in a difficult situation. The attacking beasts are the conflict and tension. These are the problems being faced by the man and the motivation for him to start moving toward safety. Once he is out of the jungle, the tension is resolved and the narrative arc is complete, so hold the man just short of safety as long as you want to use the intrigue frame.
This is going to give you a narrative arc that takes the target, quickly, on a strong emotional ride that has conflict and tension, and doing it this way will fire the hot cognition.
Things don't always need to be told in terms of extreme events—but they always should be extreme in terms of the character's emotional experience. This is what makes a good narrative.
Why use this pattern? The man-in-the-jungle formula forces you to deliver a narrative in a human, active way where you do something in the real world that shows drive and tenacity, self-confidence, and a connection to reality.
_When we listen to your narrative, it's not what happens to you that makes you interesting, but it's what you do about the situations you are in._ The emotional power in a narrative comes from a character that engages difficult obstacles and finds ways to overcome them.
Here's another example of the pattern, where I built an intrigue frame with a compelling narrative. I was taught this pattern by a Hollywood screenwriter. It's called the "ticking time bomb":
_Put a man in the jungle._ I had an $18 million deal some time ago in which I was responsible for finding $6.4 million from investors (the bank would provide the rest of the money). It took about 10 days, and I had commitments on for all $6.4 million. Then, less than 72 hours before closing, something unwanted happened.
_The beasts attack._ One of the investors, Jeff Jacobs, went AWOL. His bank would not wire the money without his signature, and I could not close the deal. That was a Friday afternoon. The entire $18 million deal was in jeopardy. I was imagining the worst: Maybe he was at the bottom of his Malibu swimming pool clutching a bag of bricks and a "goodbye cruel world" note. I spent all weekend looking for him—but he could not be found. By Monday morning, I had less than eight hours to come up with either Jeff Jacobs or his money. All my phones were ringing. On the other end of the line: the other investors, the bank, the seller, and my partners. Each caller was madder than the last.
_Will he get out of the jungle?_ I sat down at the computer and started e-mailing consultants and sales types in our industry. I offered $1,000 for any kind of information on Jacobs. One of them found me a second address and a phone number, which I called. A woman answered. Luckily, it was Mrs. Jacobs.
"You're his wife?" I asked.
"Yes sir, I am," she said.
I was thrilled. "Mrs. Jacobs, I am so glad I found you," I told her. "Can you please sign these closing documents on behalf of your husband—they allow for the wife to sign. I would appreciate it so much. I'll even drive out there [to Palm Springs]."
"Oh, you said this is to help Jeff out?" she asked nicely.
"Yes!" I said.
"Well. You know I would love to do that for you..."
I cut her short, "Great!"
Then she cut _me_ off. " _But_ I've been separated from that snake-eyed sonofabitch for 11 years," she told me, "and I'll burn in hell before I sign anything to help him out."
The minute I heard that, I dropped everything and jumped the jet for Palm Springs.
That's the important fourth step to the narrative pattern building the intrigue frame: _Get the man to the edge of the jungle, but don't get him out of it._ In other words, the intrigue is created by the fact that there's no final resolution.
To hold intrigue and make it work as an emotional event, a hot cognition, I don't complete the story (although it has a truly great ending, and it will always come up later) and instead move to the next frame in the stack: the prize frame.
#### **Hot Cognition 2: The Prize Frame**
As I mentioned in Chapter 2, the prize frame—or _prizing_ —let's you position yourself as the most important party in the deal, not the people on the other side of the table. Successful prizing _flips the frame._ Even though you are pitching the deal—it results in the target chasing you, trying to win _your_ attention.
A simple example of the prize frame comes from my trip to the Helen Woodward Animal Shelter in Rancho Santa Fe, California—one of the first times I recognized my own power disrupted and status seized and watched as the frame _flipped on me._ On entering the animal shelter, I had the classic hero's frame, announcing, "I'm here to rescue a homeless, abandoned dog." It was true. The dog that impressed me the most would win the prize—moving in with me and getting free food and medical care for life. Soon I found the dog I liked and was ready to pay the fee and complete the rescue. Spot seemed like a good name. But wait!
"Excuse me, sir."
It was the "Adoption Counselor." She was in her early 20s, and if she was your sister, you'd tell her to use less hair gel and lose the purple glitter eye shadow.
"What kind of home do you run?" she asked. "Do you have young children? What kind of job do you have? If you're backyard isn't big enough, we don't recommend a pet like this. And who will take care of the animal when you are at work? What's their number, and what's your income level?"
This was wild. A 23-year-old volunteer with pink streaks in her hair was telling me I might not be a suitable rescuer of this homeless mutt. My hero frame was smashed. Now I was busy defending myself. Giving examples of how I really was a good person.
I answered her questions. Once she gave the nod, I was ready to pay the fee and rescue the animal. Wait! Next, I had to fill out an application. Then I was told to come back in a few hours to see whether I was approved. I come in the hero, and the shelter turned me into a supplicant. Now I was begging to be considered a good enough citizen to rescue a homeless, abandoned animal, of unknown potty skills. I became the commodity, and Spot became the prize. The shelter had flipped the frame.
Let's continue with an example of using the prize frame in a pitch. Because it's specific to my business, pitching deals, I'll start with a detailed example and follow up with a more generic pattern that you can use to develop your own prize frame. What follows can be used toward the end of the pitch session:
"Guys, I'm glad I was able to find some free time to come here and show you my deal. I don't always get to meet the buyers. I know we're having fun here, but I have to wrap up. I have another meeting. We are busy, and there just aren't many deals like this—and obviously none that include me—and I'm fortunate to be in demand. Getting serious for a moment, I do have to choose which investors to let in and which to turn away. Before things go any further here, I need to figure out who you people really are. Yeah, we have your bios and know your reputation. But we have to be cautious about who we bring on board. And I have to sell you to my partner, Joshua—who is going to want to know why I think you would be good partners. _Can you give me that—can you tell me why we would enjoy working with you?"_
So what have I done in such a statement? I've delivered the prize frame, and the basic elements include
1. I have one of the better deals in the market.
2. I am choosy about who I work with.
3. It seems like I could work with you, but really, I need to know more.
4. Please start giving me some materials on yourself.
5. I still need to figure out if we would work well together and be good partners.
6. What did your last business partners say about you?
7. When things go sideways in a deal, how do you handle it?
8. My existing partners are choosy.
The prize frame is a hot cognition that signals the target's croc brain that you are strong, you are not needy, and you are not going to supplicate for a deal.
Dr. Robert Zajonc, writing in _The American Psychologist_ , describes the importance of these hot cognitions and the importance of these emotional processes. He suggests, for example, that it's not really important for us to know if someone has just said, "You are a friend" or "You are a fiend." What you really need to know is whether the statement was made with _affection_ or _contempt_. Whether the word was _friend_ or _fiend_ is the cold part of the message. It doesn't matter. Affection or contempt is the hot part. Researchers found that 22 times as much information is given in the hot part of the message.
Unlike some of the other frames, the prize frame relies a great deal on how strong your conviction is. In the pattern noted earlier, I've given you the external formula for the prize frame—which is what you say to the target. However, the prize frame doesn't come only from words that you say. It's how you're organized internally. Here's the internal pattern, the words you say to yourself to fully activate and deploy the prize frame:
I am the prize.
You are trying to impress me.
You are trying to win my approval.
Over time, as you get good at this stuff, you'll begin to see that the prize frame does not rely on words and explanations. It's more about the strength of your convictions about who or what is the prize.
#### **Hot Cognition 3: The Time Frame**
When I was selling a deal called _Geomark_ to Boeing, I used this version of the time frame:
"Guys, my company, Geomark, is a great deal, and you can't bluff me about what you are thinking; I know you agree. Consider the situation we're in. We are here for a third meeting at your corporate headquarters. Right now I'm looking at your team: four Boeing executives, three engineers, and two of your consultants. Why are you here in force? Because you love the deal. And you should love it. The deal is hot, that's no secret, and I've never used this fact to pressure you, but we can't ignore it either. For this reason, we have all got to make a decision about the deal in the next week. Why one week? This time constraint is not under my control; it's the market working. It's harsh but true: We have to decide by July 18 if you're in or out."
The effect of time on decision making has been researched for 100 years, and nothing has changed about human nature in that time: In nearly all instances, _the addition of time pressure to a decision-making event reduces decision quality._ It is true, for instance, that you can get someone to buy a car more easily if you tell him that the sale ends at the end of the day. Why does this strategy work so well? _There's a scarcity bias in the brain, and potential loss of a deal triggers fear._ But just because imposing scarcity works well isn't a recommendation to use it—we don't want to taint our deal with the whiff of cheap 1980's sales tactics. We want the target to see us as a professional agent. To trust us. So I tend not to use much time pressure at all. Extreme time pressure feels forced and cutrate. But the truth is that time is a factor in every deal. You have to find the right balance between fairness and pressure and set a real time constraint.
Here's the time frame pattern you can use and follow:
"Guys, nobody likes time pressure. I don't like it, and you don't like it. No one does. But good deals with strong fundamentals are like an Amtrak train, or more like a _deal train_. They stop at the station, pick up investors, and have a set departure time. And when it's time—the train has to leave the station.
"You have plenty of time to decide if you like me—and if you want this deal. If you don't love it, there's no way you should do it; we all know that.
"But this deal is bigger than me, or you or any one person; the deal is going ahead. There's a critical path, a real timeline that everyone has to work with. So we need to decide by the 15th."
That's it. You don't have to do any more. With just that simple pattern, the time constraint is set. You don't have to be overt or aggressive with time pressure. Every single person knows what you're talking about when you say the train is leaving the station at such and such date and time.
#### **Hot Cognition 4: The Moral Authority Frame**
Robert Zajonc, the thought leader on hot cognitions, once wrote, "We evaluate each other constantly, we evaluate each others' behaviors and we evaluate the motives and consequences of that behavior." And this, of course, is the key to the reason we stack frames. Because we are going to be evaluated no matter what happens and what we do, let's get the evaluation we want, something Zajonc calls _wanting_.
So, while it's tempting to get caught up in the best way to explain the financial mumbo-jumbo and how to best demo our products, the heart of the matter is that you have to do what it takes to create this wanting. There may be other factors that contribute to our effectiveness in the pitch, but certainly one of the most important is getting this wanting to happen. How is it done? To create a desire in the target's mind and to go on from the pitch to the hookpoint, every presenter has to use hot cognitions to create wanting and desire.
Until a wanting evaluation is in the target's croc brain, the information you are giving is largely being ignored or at least not making a big impression.
Now that we've gone through prize frames, intrigue frames, and time frames and their uses, here is another example that will deepen your understanding of how to use frames to create hot cognitions and wanting.
_**The Morality Frame in Practice**_ The most powerful politicians in the world have people underneath them who will do exactly what they are told. Each has phalanx of subordinates who do his or her bidding.
Take the president of the United States. If he orders a precision air strike on a clandestine enemy stronghold, a succession of people underneath him will execute his order, all the way down to the pilot in the F-22. The president can lead us into a war or, with a few pen strokes, sign a bill that will affect millions. His frame, in most of his encounters, is stronger than that of nearly any opposition.
The president, like many other world leaders, isn't used to being told what to do. Think about what one has to go through just to become president, how many personal attacks you suffer, and the constant political reframing of things you have said. Once president, though, you may have one of the most sophisticated and strongest frames in modern history. Yet there is one person whose direction the president will follow, almost blindly. When David Scheiner says to Barack Obama, turn around and take your clothes off—he does it without question.
In every social situation, there are basic, human, hardwired functions. Let's call them _ritual elements_ of social interaction. Every person navigates the world through social encounters. In each further contact with other people, as we have been discussing, each person brings a frame, which is a viewpoint or perspective. It doesn't matter if the person intends to bring a frame, he always will find that he has done so. All social encounters are framed. In light of this, when we think about it, not only do physicians such as Dr. David Scheiner tell us what to do, their frame is so powerful that we are nearly helpless in its presence.
In fact, the doctor frame may be the most powerful frame in the world. _Really?_ The most powerful frame in the world? Let's explore.
If we want to stay alive, we all follow our doctor's orders. And we have deep respect for the medical profession. Physicians, cardiologists, radiologists, internists, and surgeons—these are people who can save our life or the life of a loved one. So we have an embedded script that we follow when we deal with medical professionals. When the surgeon stands, we sit. If the surgeon waves his hand toward the table, we go sit on it, uncomfortably trying to cover up private areas.
Literally, when the surgeon does anything, we are programmed to react. We react to the surgeon automatically. He doesn't react to us. We obey him. The surgeon only nods appreciatively when we say something but doesn't react.
The consulting surgeon wears whatever he wants, sometimes a nice suit, other times comfortable-looking casual clothes, whereas at the same time we are dressed in a generic green gown, sans underwear, that signifies our low situational status. If we were seen anywhere in public in this ridiculous gown, it would cause us emotional scarring. The surgeon is wealthy, and has all the accoutrements of status: a prestigious degree, a respected position, and an expertise that took nearly 20 years to acquire, and he literally has the power to determine life or death.
We are hardwired to follow this script. Most of us, but not all of us. Certainly not Mother Teresa.
In December 1991, Mother Teresa entered Scripps Clinic and Research Foundation in La Jolla, California, where she was treated for bacterial pneumonia and heart problems. With such a notable world figure under their care, doctors and surgeons rushed in to meet her. And frames collided.
The doctor's frame has three rules:
Rule 1: Do what I say.
Rule 2: Defer to my expertise.
Rule 3: Accept my conclusions about life (and death).
However, when encountering Mother Teresa, the doctors found a person who did not follow the script or fall into the frame.
Here was Mother Teresa's frame:
1. Material wealth is worth nothing.
2. Life and death isn't critical.
3. Help the downtrodden.
4. A rich man is less likely to enter the kingdom of heaven than a camel is to pass through the eye of a needle.
Her frame is not powered by wealth or expertise—instead, by a high moral authority: Help the downtrodden! Life and death isn't critical!
As doctor after doctor met with Mother Teresa, their strong frames collapsed like a series of dominoes. She did not react to their status or their control over life and death. After all, even death was not one of her primary concerns, and she had often ignored doctors' orders in the past. And as they fell into her frame, something remarkable happened: These doctors could not impress her. Their power frame was disrupted.
This is why she so easily convinced American doctors to do something that they hadn't previously considered. Before coming to La Jolla—an affluent seaside community in north San Diego, Mother Teresa had visited Tijuana, a city across the U.S.-Mexican border where poverty is severe. It was there she learned of the huge disparity between the United States and Mexico, between Tijuana and La Jolla, between the haves and the have-nots. So, as these doctors fell over themselves to visit her, she sensed a tremendous opportunity. She asked the doctors what they were doing to give back. And then she asked them if they had ever seen the medical facilities, a mere 25 miles away, in Tijuana. Most of them said no.
She then asked each doctor to add his or her name to a signup sheet outside her room, pledging to donate time and resources to help mobile medical clinics in Tijuana.
The doctors, who were used to having the dominant frame, couldn't impress Mother Teresa with any of the usual trappings of success. They could only do it by pledging their time and expertise to her cause.
When it was time for her to leave the hospital, after 20 days at Scripps, Mother Teresa had imposed her frame on the richest, smartest, best-educated, and highest social class in southern California, and this was evidenced by the sheer number of doctors who had pledged their time to help out in Tijuana. And the doctors? Without any noticeable effort, their frames had been shaken, broken, and disrupted. With the high-status surgeon power frame disrupted, Mother Teresa's frame took over. According to the _Los Angeles Times_ , "On January 16, 1992, Mother Teresa of Calcutta was released from the Scripps Clinic and Research Foundation hospital, after securing a pledge from doctors and nurses to set up a volunteer network of mobile medical clinics to serve the poor of Tijuana."
When asked whether she was going to take better care of herself, she replied, "Oh, sure."
### **Reality Is Waiting to Be Framed**
A hot cognition—or a series of hot cognitions—is a fast method for getting the target's croc brain to want you and your big idea.
But this is not a 'sales technique. Stacking frames is not going to work, in my opinion, if you view it as just another type of sales tactic. Those old-fashioned sales techniques are about chasing the neocortex with features and benefits and rational explanations. "Selling" tempts you to do the three things I dislike the most: (1) supplicate; (2) make rational appeals to the neocortex, and (3) ask invasive questions. Hot cognitions, on the other hand, do not hit the target like a sales technique.
• _Hot cognitions are primal._ Whenever there's a rush of excitement, it's hard to get the neocortex to do any work at all. To protect us from potential physical or social threat, the croc brain hijacks brain function. No analysis gets done. As a result, it's much easier and natural to react to what's hot and vivid and moving right in front of us.
• _Hot cognitions are unavoidable._ You might be able to control the _expression_ of emotion, but there's no way you can get out of the path of having and experiencing it.
• _Hot cognitions tend to be instant and enduring._ Do you like the movie you just watched? Do you like the new model of Ford Mustang? Do you like eating snails? You never sat down to analyze these things; they're obviously hot cognitions—you got a sense of these things the moment you encountered them.
#### **Hot versus Cold Cognition**
Maybe the best way to define hot versus cold cognition is to compare it to chocolate and spinach. You know the cold, hard facts. Spinach is good for you, it has lots of nutrients, and you should be eating more of it. But when offered a piece of chocolate instead, you go for it.
The acid test of whether your pitch goes well will be: Does the target want to buy your stuff, be a part of your team, or invest in your idea?
How much thinking about your presentation does your target need before he or she forms a preference about it? How "fully and completely" must an idea be presented and thought about, and how much rational analysis needs to be completed before the target decides: "good" or "bad"? I argue here that as you approach the end of your pitch, you don't need to wait for an evaluation, or the target will wander off into a cold cognition process and think about you: did we like him, did we like his deal? Instead, stack the four frames, trigger hot cognitions, and create the instant evaluation that ought to be _wanting._
If hot cognitions targeted at the croc brain are so powerful, why do most people make presentations in the cold cognition style targeted to the neocortex? Here's why I think people go this way: Our faculties of reason tell us that the neocortex is way smarter than the croc brain. We think that if we create a message in our own smart neocortex, it should be sent to the target's neocortex, which will do a better job of understanding the pitch. It makes sense to think this way because the neocortex really _is_ an insanely capable problem solver. It has awesome language and math and creative abilities. It's a Swiss Army knife of mental ability.
By contrast, if the neocortex is like a Swiss Army knife, then the croc brain is a like a rubber mallet—best for simple jobs only. It works on only a few emotions, and those are very limited in scope and range. The croc brain almost seems too simple to "get" our smart ideas. We think, _Who are we going to trust this decision to, the target's infinitely capable neocortex or the emotional and simplistic croc brain?_ Our intuition tells us to trust the neocortex. But that's not the right choice. Let's return to the essential idea from Chapter 1: _No pitch or message is going to get to the logic center of the other person's brain without passing through the survival filters of the crocodile brain system first. And because of the way we evolved, those filters make pitching anything extremely difficult._
By this point, you know what I'm going to say next: Focus your energy on getting the target's croc brain to want your product. Because, at best, no matter how much you try to sell to the neo-cortex— _it can only "like" your idea._
A _hot cognition_ is the inner certainty of "knowing" something that comes through feeling it. A _cold cognition_ is the certainty of "knowing" something is good or bad by having evaluated it.
As we've said before, _hot cognitions are extremely fast._ Hot cognitions develop through the ancient brain structures—the brain stem and the midbrain—our croc brain. Cold cognitions are analytical and develop in the neocortex. Cold cognitions are calculated and take time to suggest a solution—this is how the neocortex goes about its business—it aggregates information over time and solves problems. You've heard the expression, "Just give me the cold, hard facts?" This is what is meant by a cold cognition, the labor-intensive processing of facts through a decision matrix.
_You can trigger a hot cognition instantly, but cold cognition can take hours or days._ Most presentations are set up to take the target down the path of a cold cognition. They try to justify the big idea with facts and information.
Hot cognitions encode value. It's the anticipation of a large financial gain that is emotionally compelling to the target. Actually receiving it is not nearly as exciting. As one researcher noted, "The human brain acquired its reward-reinforcement system for food, drink, ornaments, and other items of cultural value long before money was discovered." The brain thinks of money as it does of food, ornaments, and drugs and records the utility that can be collected by using it only indirectly. There's no cash register or balance sheet up there.
George Soros once wrote: "The philosophers of the Enlightenment put their faith in reason;... and they expected reason to provide a full and accurate picture of reality. Reason was supposed to work like a searchlight, illuminating a reality that lay there, passively awaiting discovery."
As we have been discussing, reality isn't waiting to be discovered—it's waiting to be framed. By stacking four frames quickly one after the other, you can achieve the hot cognition in the target—helping the target to discover a _wanting_. Yet, once the frame stacking is complete, we've got the target's attention for about another 30 seconds. And it still can all go wrong. We have to find, in that brief time, a way to translate the target's desire into action. But how? What do you do now?
## **Chapter 6**
**Eradicating Neediness**
Over the years, I have faced a lot of rejection. And the disturbing thing about rejection is that you never really get used to it. It's natural and even unavoidable to feel disappointment when you get a "No." We all do. What's certain is that none of us like being rejected. We want to avoid it. In high-stakes situations, we're nearly always anxious about it.
As businesspeople, friends, neighbors and citizens, we believe that when we need something from another person, there is a thin line of empathy that runs through all human hearts. We believe that we'll be treated well by others _just because_. But there isn't, and we won't. So we nearly always become anxious and needy.
Showing signs of neediness is about the worst thing you can do to your pitch. It's incredibly bad for frame control. It erodes status. It freezes your hot cognitions. It topples your frame stacks.
If you talk to investment bankers, the pros that make million dollar decisions almost daily, they'll tell you—validation-seeking behavior (neediness) is the number one deal killer.
### **Four Pitches, No Room for Mistakes**
Twelve years ago, I was trying to raise money for a technology company that I had also invested in. The company was quickly burning through cash. I needed to find a big-time investor, so each day I'd make up to 50 calls to leading venture-capital (VC) firms. I talked to a lot of receptionists and secretaries, and I got a lot of voice-mail action. But nobody was excited enough to return my call.
My company had a great idea, but it was difficult to explain over the phone. I needed face time to explain it, so I was desperate to simply land a meeting. I continued to be persistent, and the following week, I got a few people to pick up and tried to pitch them by phone. That didn't go well. Bill Reichert at Garage Technology Ventures told me, "I have no idea why anyone would want to build it, use it, or invest in it." Ron Fisher, at Softbank VC said, "Do yourself a favor, son. Try a different idea."
I reached Vinod Khosla at VC firm Kleiner Perkins, and he quickly transferred me to his analyst; an obvious dead end. I was getting nothing but rejections from the largest VC firms in North America. I questioned whether I should continue. It was a constant battle in my mind, press on or give up? But I'd passed the point of no return.
It sounds simple, but no greater truth exists in business today: Persistence pays. So I persisted. Eventually, I was able to get four big pitches lined up with top-tier VC firms. But setting a meeting is just step one of a two-step process. You have to impress and persuade, or you go home with nothing, which I was getting good at—the _going home with nothing part_ , that is.
This was a confidence killer—I knew I was good at pitching, but for some reason, my pitches were failing. And now, I was in big trouble. I'm embarrassed to say it, but it's a matter of public record (UCLA's Anderson School of Management uses a case study on this deal in its MBA program), I had less than $1,000 in my account, and my company was down to one final pitch, one last chance with a major VC firm, before the cash ran out and our doors closed for good.
I tried to develop a theory to explain what was going wrong—but I come up with nothing. I had a lot of self-doubt. _What was I doing wrong? There must be something._ Rather than make the same mistakes again in my final pitch, I decided to regroup and reorganize myself. A bit humbled, I went back to my former employer to speak with the senior partner, Peter.
Peter was another master of the universe type, and I had helped him put together a number of big deals that had made him a lot of money. If anyone would help me, it would be Peter.
I shut his office door and sat down, unsure if he was going to help or just lecture me. After I thanked him for seeing me, he said, "Oren, I've watched your career very carefully, and while I see flashes of brilliance, you know I've also identified plenty of problems over the years."
"Right," I said, bracing for the lecture that was to come.
"You're just inconsistent," he went on. "Sometimes you're fantastic. Other times you let us down. We were always unsure which Oren we were going to get on what day."
I really wanted to defend myself, but I knew that nodding my head and staying quiet was the best approach—for now.
"Let's go back," he continued. "Two years ago, you seemed invincible. You had just helped us close Somatex, the most profitable deal in our history."
How could I forget? Without me, that deal would have died. Guys got rich on that deal, thanks to me. "I remember it," I said.
"You were a few deals away from becoming a partner."
Becoming a partner, the most coveted position at any investment bank, had indeed been within my grasp. I'd also landed a sweet deal with Hershey's that made the firm over $1 million. And my work helped the firm to close several other big deals. The competition could only watch in awe and envy as we put together a great run.
"You were strong and confident and acing deals that we really liked," Peter said, "But...." His voice trailed off.
I'm sure he was still disappointed and hurt that I had abruptly left the firm, seduced by the promise of the Internet.
"I'm sorry," I told him. "But I'm here because I'm in dire straits. I have one more chance, or this whole thing is going to implode."
He looked at me and nodded.
For the next hour, I replayed the three failed VC meetings while Peter asked me questions. Eventually, his eyebrows lifted, he smiled, and he broke into a laugh. He stopped me midsentence.
"I know why you seem to have lost your edge," he said.
"Why?" I asked.
He allowed the moment to build and finally said, "Because you're out there, on your own, and you know there's no safety net. Son, you're going to these meetings... _needy_."
I snapped to attention. Of course. Classic validation-seeking behavior. Signals of desperation. No investor wants to work with a needy company run by an entrepreneur who is almost out of cash! Sure, the investor knows you need money, but giving a hint of neediness or any sign of desperation, plainly put, is like saying, "I'm holding a bomb that could go off at any minute." Everyone will respond by going on the defensive. Their first reaction is—Run!
_Self-protection is an unconscious reaction that comes from the crocodile brain._ This is the critical lesson of my failed pitches and the key to understanding why otherwise good deals can fail to impress the target. Neediness triggers fear and uncertainty, causing the target's croc brain to take over—but not in a good way. Its goal is to prevent further threat by effectively blocking out the higher-level brain, which likes to debate and consider and analyze. No time for that. Threats require immediate action.
_Neediness is a signal of threat._ If you display neediness, it is perceived as just the kind of threat that the crocodile brain wants to avoid. Neediness results in avoidance.
I knew that Peter was right, and I listened carefully as he gave me advice on eliminating neediness. And although my situation was desperate, he told me to "Get some game." In other words, find some source of inner strength, confidence, and poise. Easy to say. Hard to do.
When pitches work well, we tend to believe that it was our great idea that impressed the target. Or that our sublime explanation of the big idea was impressive and convincing. However, when pitches fail, we see it through a different lens. In such cases, we believe that the problem lies with the target, not us. We believe that the target somehow couldn't see the value of the big idea or that he was the wrong target. But pitches can fail for reasons that are hidden from view.
I thought back to my previous pitch, the third of the three failures. A VC group in Silicon Valley was interested in my deal. During my call, the representative said, "Your executive summary was good, we like your idea, and under the right leadership, this could become a large company and eventually go public. We'd like you to come up Tuesday to pitch the partnership."
I made plans to fly north and thought that this was my big break. When we got there, though, the circumstances felt familiar. I'd been in this position before. It was just another one-hour meeting in an anonymous office building overlooking a freeway. The conference room looked virtually the same as every other one I had pitched in: black leather chairs, a long conference table, a whiteboard, and an easel.
Thinking back on those days, if there's anything that reminds me of deal making, it's the acrid whiff of dry-erase markers. In 1999, no deal could be pitched properly without extensive—and highly abstract—whiteboard diagrams.
Now, when I was pitching, I spoke clearly and made my points elegantly. I maintained eye contact with the targets, exuding calm confidence. My voice rose and fell with dramatic timbre as I drew a chart on their whiteboard so artistic that, had it been preserved, it would today be hanging in the modern art hall of the Getty Museum.
Before I knew it, 30 minutes had gone by. Although I felt that there was much more to say, the targets were glancing at their watches. I knew it was time to wrap up, and with nearly perfect comedic timing, I delivered a joke, and laughter filled the room. I'd nailed the pitch.
Now I found myself in that awkward two-minute period that every presenter must face after finishing his or her pitch. This is a dangerous _beta trap_ and the easiest time to screw things up. Here is where small mistakes are amplified into deal killers. Slight missteps can erase all the good work you've done in the last 20 minutes. And it's unnerving because there's always an unspoken directive: You need something from the target. Money.
So, during the end of that third pitch, in front of some of the best venture capitalists in Silicon Valley, I became anxious and needy while I explained:We need a lot of your money, and we need it fast. At that point, I realized how high the stakes were. If these people said no, it would be my third rejection in a row. And the company would have almost no options left. I felt scared and anxious. I said things like:
• "Do you still think it's a good deal?"
• "So, what do think?"
• "We can sign a deal right away if you want us to."
This is the purest form of validation seeking and the most lethal form of neediness. And that was the end of that opportunity. Just that fast, the target's excitement turned to fear and anxiety. And of course, I failed to get a term sheet or an investment offer.
### **Why It's Important to Eradicate Neediness**
Plain and simple, neediness equals weakness. Broadcasting weakness by seeking validation is often a death sentence. This may sound harsh, but it is true. Neediness—displaying so-called validation-seeking behaviors—will affect all social interactions dramatically.
It is almost redundant to criticize the concept of validation seeking any further. Simply put, there's almost no way to get through the postpitch time period if you are needy. Let's define validation seeking and neediness and talk about how to get through the two-minute beta trap or any other point where you might be showing neediness.
#### **What Causes Neediness?**
You can tell when you are losing your audience because their growing discomfort is easy to read. They glance at their watches, turn their bodies away from you, cough nervously, and/or close the folder they had been leafing through. There are lots of outward signs.
When you notice that audience members are uncomfortable, you feel yourself losing the deal. Your anxiety and insecurity start turning into fear, and you begin falling into acceptance-seeking behaviors.
The experience of feeling disappointment creates problems that need to be considered carefully. When we feel even a touch of disappointment, our first reflexive reaction is to cure it by seeking validation, which, of course, broadcasts neediness to the audience.
Now our brain is subconsciously thinking, "If I can just get them to agree to do the deal with me, everything will be okay." This is what our brain wants to relieve the stress and fear caused by rejection. If we are lucky and the target decides to give us the deal, all is well again. We instantly feel better, our anxiety fades, our heart-rate returns to normal, and we feel in control.
However, in the panic moment caused by that flash of disappointment, we couldn't help but signal neediness to our target. Chances are good that he noticed it and will not respond by giving us what we want. What happens then? Further rejection poses the threat for an emotional catastrophe.
In practice, here is how we fall into validation-seeking behaviors:
1. When we want something that only the target can give us (money, an order, a job,) we set the stage for neediness.
2. When we need cooperation from the target and can't get it, it's frustrating and causes us anxiety. And audiences, at some point, always become uncooperative. Audience members turn their attention elsewhere—usually by texting, scanning e-mail, or taking phone calls. They allow interruptions by people coming in and out of the meeting room. Or they cut us short—before we've made our key points.
3. Neediness is created inside of us when we firmly believe that the target can make us feel good by accepting our pitch and by saying "Yes." When we set ourselves up to need the target to accept us, we have trapped ourselves. The more we want the target's desired behavior to occur, the more neediness we broadcast, and the less likely the target is to give us what we want. It's a downward spiral.
4. Finally, validation-seeking behaviors are triggered in us when the target seems uninterested in our pitch, begins to withdraw, or shifts his or her attention to something else. At that moment, we have a natural fear response, and the potential for an involuntary expression of neediness is high. Fear and anxiety are emotions that are both natural and reflexive, and they are very difficult to manage. Even the most common social rituals are loaded with situations that can trip us up, so you really need to pay attention to avoid outward signs of neediness—a reaction that surrenders your status and frame dominance.
### **Counteract Your Validation-Seeking Behaviors**
One dramatic way to eradicate neediness involves going into every social interaction with a strong time frame that you are prepared to use at any moment. This frame communicates, loudly and clearly, that you are needed somewhere else.
But this is just a part of a broader, more comprehensive solution to eradicating neediness. Here's the basic formula:
1. Want nothing.
2. Focus only on things you do well.
3. Announce your intention to leave the social encounter.
Executing these three steps will calm down the fear circuitry in your own brain. Your increased heart rate, perspiration, rapid breathing, and anxiety will subside slowly. Once under control, you'll impress others and make them actually come after you. Most important, the willingness to withdraw demonstrates a self-control, strength, and confidence that most targets will greatly admire.
To better appreciate the importance of controlling validation-seeking behaviors, let's take a closer look at the way others have solved this problem.
In the film _The Tao of Steve_ , the lead character, Dex, also employs the withdrawal technique to maintain a high status in social situations. This technique is part of a larger Taoist philosophy that Dex uses to guide his life.
Dex is an elementary school teacher's assistant who lives in Santa Fe, New Mexico, with several roommates. He has a huge belly, he smokes pot, and his wild, unkempt look is the opposite of Madison Avenue's image of the American male. Looking at Dex from the outside, you'd assume that he is _not_ a social success. But surprisingly, the opposite is true.
Dex is a practitioner of the Taoist philosophy, but with an ingenious twist (which I'll explain in a moment). Taoism is an Eastern philosophical and religious tradition that originated in China around 500 B.C. It is based on the writings of the philosopher Lao-Tsu. As in Buddhism, there is a heavy emphasis on connecting with the universe and controlling your desires.
In the movie, Dex has developed his own derivative version of the Tao that draws its inspiration from popular culture. He calls it the "Tao of Steve"—named for three guys named Steve who embody all the coolness and social poise that Dex is seeking. The three Steves are Steve McQueen, Steve Austin, and Steve McGarrett.
McQueen, the so-called King of Cool, rode an antihero persona to become the highest-paid actor of his generation. His roles in movies such as the _Magnificent Seven, The Great Escape_ , and _Bullitt_ won him a legion of female admirers and also cemented his reputation as a "man's man."
Steve McGarrett (Jack Lord) was the detective in charge of the elite crime unit on the popular 1960s show _Hawaii Five-O_. He was a no-nonsense, dedicated investigator who always stayed one step ahead of the bad guys.
Steve Austin (Lee Majors) was the bionic man in the 1970s hit, _The Six Million Dollar Man_. Austin was a former astronaut who survived an experimental plane crash. He was rebuilt with bionic parts and became an operative for a covert government agency.
All three were noted for being cool without really trying. The actors helped convey this feeling by maintaining poise in pressure situations. They never got flustered, even with the bad guys closing in.
Dex believed that the three Steves attracted admirers not because they had cool cars or bionic legs, but rather because they understood the three main rules of the Tao:
_Eliminate your desires._ It's not necessary to want things. Sometimes you have to let them come to you.
_Be excellent in the presence of others._ Show people one thing that you are very good at.
_Withdraw._ At a crucial moment, when people are expecting you to come after them, pull away.
The Tao of Steve is the perfect philosophy to use when you finish your pitch. Use it to stifle the urge to seek approval from your audience. People want what they can't have. So, when you finish your pitch, deny your audience. Start to pull away. In so doing, you banish insecurity and trigger a powerful prizing effect on your audience. They will come to you.
### **The Final Pitch**
Now, finally, it was the day of my fourth and final pitch. Six months ago, I had hundreds of thousands in my bank account. Now I had $468, enough for half of a payment on my Porsche, and that was if I didn't pay rent or eat. Nevertheless, in my head, I ran a constant audio loop: _I don't need these people; they need me. I am the prize._
I stood in front of Enterprise Partners, the largest VC firm south of Silicon Valley. It was nearly 4 p.m., and the people inside were ready to go home. I needed to electrify them, light up the room, and most important, I needed to forget that this was _do or die._ If they sensed any desperation, my partner and I would go home busted. This was my last chance.
I walked into a room of tired-looking investors. They'd been hearing pitches all day; none of them even bothered to say hello. It was unnerving.
They asked me a series of hostile questions. They wondered aloud if the market was big enough or if there was too much competition already. One of them actually laughed at our idea. And to make matters worse, the partner who brought us through the door ( _sponsored us_ , so to speak) questioned whether we should even be an Internet company, instead, suggesting that we settle on making PC software, which was the equivalent of saying, I no longer support this deal.
But none of that mattered. I delivered the knock-down, drag-out, bare-knuckled superpitch. Exactly 20 minutes of hardcore finance in a sexy Internet wrapper. At the end, I continued with my plan to run the prize frame and qualify them.
I made three points. Here's what I told the Enterprise investors:
1. This deal will be fully subscribed in the next 14 days.
2. We don't need VC money, but we want a big name on our cap sheet that will strengthen our initial public offering (IPO) registration.
3. I think you guys are interesting, but are you really the right investor? We need to know more about you and the relationships and brand value your firm can bring to our deal.
That was it. I was totally spent. At that moment, I couldn't muster enough mental power or energy to do anything but sit still and be silent. Now it was my turn to be the stone-faced gnome. I just sat there, with no trace of validation-seeking behavior, no neediness, waiting for them to react to me.
To my surprise, they fully engaged with me and in minutes had committed to the deal. They wanted what I had. I'd redeemed myself with a great opening and finished the pitch with confidence and poise, and the whole thing come together perfectly.
We sat in the room for another hour and discussed all the variables. We come up with a huge $14 million valuation—$6 million more than I had expected.
Yes, my bank account had dwindled to $468, but the next day, Enterprise Partners officially agreed to invest. And a few days later, they wired $2.1 million into my account. I remember going to the ATM and getting a balance slip so that I could see the $2.1 million printed on it. Thirty days later, Enterprise completed the financing with an additional $4 million.
Despite the difficulties and hard work, going after a deal is a thrill. Sure, I do it because I want the results—the loan, the job, the money, the deal... whatever. But that's not the only reason. I chase these things because the adventure excites me. All deal makers know: There's a king-of-the-world moment when we close a deal that is never forgotten. It's the satisfaction of preparing for a high-stakes meeting, giving a mouth-watering pitch, and with a chest full of confidence, getting the deal.
Over time, I would have had many fewer of these moments if I had not learned: _Never be needy!_
## **Chapter 7**
**Case Study: The Airport Deal**
This is the story of a pitch so massive and so high stakes that in order to tell it properly, I'll need to tell you what happened in the months leading up to the day of the pitch. There is perhaps no other single deal I've ever been involved with that better demonstrates the ideas at the core of my method and why it works. If you find that the story borders on the cinematic at times, well, that's what it was like to be there, and that's what it's like when you're playing with hundreds of millions and there's so much at stake. Let's begin.
I was 100 miles away from Los Angeles and in no hurry to return to the city. I had a very good reason, too: Business was dead. The collapse of the U.S. economy had seized the credit markets, freezing deals for more than a year. After pitching nearly nonstop for more than five years, I found myself with nothing to pitch—literally nothing. This lack of action was frustrating because at heart, I like the adrenaline rush that comes from playing this game. That's why I do it. I get the same adrenaline rush from action sports as I do in a boardroom—where I've raised more than $400 million.
Then it all went away— _Poof!_ —as the economy soured and the deal window shut. It was clear, after a full year, that it wasn't going to return to normal—for a long time. That's why I'd decided to get away—from all of it. The deal business had been very good to me, and I was in a position to do anything I wanted. I told friends, colleagues, everyone that I was done. _"I'm hanging up my guns,"_ I said.
And now I was due for some serious downtime in the middle of nowhere to figure out what I'd be doing next in life. A one-week vacation to the Anza-Borrego Desert had turned into two, and I was enjoying all that the tiny town of Ocotillio Wells had to offer—which wasn't much. It was a Monday in December, a dry 78 degrees, and most of the weekend guests had now retreated from this outpost in southern California on the east end of the Salton Sea. The solitude felt good. Most of my day was spent mountain biking and relaxing on a mesa. I snapped pictures of the sun as it settled behind a bluff. Life was good. And I was oblivious to the fact that someone was desperately trying to reach me.
### **A Monster Deal**
Across the mountain range that separates the high desert from the coast, in Los Angeles, Sam Greenberg was that someone. He sat in his office, frustrated, wondering what it would take to get in touch with me. I had pretty much made myself unreachable—my phone had been switched off, my e-mail autoresponder was switched on, and almost no one knew where I was staying.
Greenberg had a deal on the table, a big deal—10 figures, $1 billion. He wanted me to be his point man, just like the way things were before the economy collapsed. Greenberg was a man of action. He kept a private jet at Palomar Airport—not as a luxury item but as a tool to get to deals faster than the next guy. He would find me, get the plane in the air, and get me back to work. At least that was his plan, and it served his purpose. An acquaintance finally spilled my hotel information, and Greenberg left a short message on the hotel answering machine.
"I've got a monster deal," he said in a loud voice. "Call me, today. ASAP."
Three hours later, I entered my hotel room and saw the red light flashing—so much for privacy. _Who needs to reach me that badly?_ Only a handful of people knew how to get in touch. I listened to Greenberg's message, alternately ignoring it and then wondering about it. _No_. After 10 years of nonstop deal making, I'd sworn off deals. I'd sworn off the industry.
_No._
I was resolute, but I owed Greenberg enough to call him back and tell him as much. When he answered the phone, my first word was "No."
"I haven't even told you the deal," Greenberg said.
Greenberg had a natural gift of persuasion and typically got what he wanted, but this time I was sure I wanted no part of it.
"My answer is still no," I said.
"Just listen to me for one minute," Greenberg demanded.
"It's not going to matter," I said. "I've seen about every deal out there—it's a wasteland."
Greenberg told me about a deal right in our backyard. Davis Field, a regional airport on the outskirts of Los Angeles, was being rechristened as a large private fixed-base operator (FBO) as a relief valve for the overflowing Los Angeles air space and would need a whopping $1 billion raised to finance the effort.
"You _can't_ turn this one down," Greenberg bellowed. "This is not some shopping center. This is a _friggin' airport!_ "
"Sounds great, but I can't do it," I said. "I'm out of the game."
After three or four expletives, Greenberg tried another approach.
"That's fine. I got someone else lined up anyway, so..." His voice trailed off.
After a long pause, I took the bait. "Who?"
"Barnes."
"Oh, please."
"He can pull it off," Greenberg said. "And on your end, I guess we'll just chalk it up to a missed opportunity. You're gonna kick yourself."
Greenberg indeed had another guy lined up, but everyone knew that Paul Barnes was not exactly "the best."
Greenberg decided to stop his pursuit—for now.
"Have fun sticking your head in the sand while we're doing a monster deal," he added in closing. Then he hung up, not even waiting for a response.
I laughed and thought about the call. Over the years, I'd become hardwired to analyze every pitch to find out what worked and what didn't. Greenberg's pitch meant there was just one deal in a dead market, and that was no reason to unretire. I called the front desk and asked the clerk to hold all calls.
Two weeks went by, and sure enough—Paul Barnes did not perform as expected. Sure, he was confident and had a great analytical mind, but he couldn't think on his feet. If circumstances changed or things got off plan, he spooked easily. When Greenberg suddenly asked him to fly to Chicago on a red-eye for a meeting, Barnes begged off. "I need a few days to prepare," he said.
One of the things my business associates know about me is this: I would go anywhere, anytime to make a deal happen. If a flight was canceled, I'd drive. No cars? I'd take a bus. And after all these years, Greenberg trusted me. He wouldn't think twice about making a strategy change based on my recommendations.
So Greenberg was worried. He had two months to prepare the pitch for the airport selection committee—not just a presentation, but a full-pitch package, complete with a detailed financial analysis and strategy. He knew I could deliver a real face melter of a pitch, a compelling and dramatic story that explained everything. _Why now, why us, critical path, upside, downside, and competitive advantage_. All stuff that I'd done hundreds of times.
After Barnes proved unsuitable for the situation, Greenberg realized that he needed me.
### **Greenberg Goes to Borrego Springs**
Greenberg's Legacy 600 landed at the alarmingly tiny runway in Borrego Springs before 10 a.m., and within 15 minutes, he was in a car, headed to the Borrego Valley Inn. I'd agreed to meet him at noon in the hotel lobby.
It was nearly 10 years earlier that Greenberg had listened to one of my pitches and tried to hire me on the spot. I turned him down, saying I wasn't for hire, but we went on to partner up eventually. Together we made a lot of deals that propelled Greenberg Capital into a premiere position in the financial markets.
Now, the team had disbanded, and no deals were happening.
At 11:54 a.m., he approached the lobby. This was classic Sam Greenberg, who, like the legendary Carthaginian commander Hannibal, would either find a way or make one. This guy would not fly back without pushing our relationship to the brink. This was just the way he worked.
Shortly after 11:30 a.m., I thought about possible scenarios. Was there anything Sam could say to sway me? Not likely. Was the deal economy coming back any time soon? Also not likely.
There was a certain peace I had now, away from the rat race, not tied down by phones and e-mails. And I didn't miss the grind. Not at all.
It was getting close to noon. I headed to the lobby. Sam was there, dressed casually, sitting on a blue vinyl couch. Both of us were uncomfortable at first—as if meeting for the first time—we both had our game faces on.
"You know I can't let the plane idle on the runway for too long, so pack up and let's head back," Greenberg said with a straight face.
Ahhh, the time frame—it would need to be dealt with.
"At this altitude, after your 25-minute short hop, that jet can idle for three and a half hours in 98 degree heat with a half tank of fuel," I said, cracking Greenberg's power frame in half with the stronger expert frame. "And I'm hungry. It's your turn to buy lunch."
Greenberg countered the expert frame with a prize frame. "You've never made as much money as you have working with me," he calmly replied. "Look at you—you're leaning forward, hanging on every word I say. You're practicality drooling. That's how much you want this deal. It's almost sad."
"Ah, you're throwing a prize frame at me! I taught you too much!" I said, easily handling Greenberg's reframe. "Look, we could do this all afternoon. Forget it. Let's go eat. You're still buying. Follow me."
On the way to the sandwich shop, we continued to frame and reframe and deframe about who would buy a $20 lunch, even though Greenberg's fueled, piloted, and fully staffed jet was on the runway burning $8,400 an hour.
"Let's get back to doing what we always do," Greenberg said, continuing the conversation at the restaurant.
"Market says that's a bad idea," I replied.
This was probably an anticipated objection. He decided to challenge me. Chide me a bit.
"Why exactly did you come here?" he asked. "You just ran away. _You left because you lost your edge_ ," Greenberg continued.
"There was a point where I thought you might be one of the best," he said. "Now, the only thing next to your name will be... _quitter_."
That one made me clench my teeth. I stood up and gave him a long stare. I was ready to walk away right there.
"You're mad because I'm right," he said. "Maybe you'll come back with me and prove me wrong. You can't hang out and be a hippie—or whatever you call this—the rest of your life."
I could feel his neediness, and it was clear that I had the upper hand, for the moment. But you never have that upper hand long with Greenberg.
"Give me a minute to think about it," I told him.
The $1 billion figure for this deal was off the charts—I'd been accustomed to pitching deals in the $30 million range. This would be the biggest deal I'd ever put together. And it was only falling into my wheelhouse because the bad economy had sent everyone else scrambling. The fact that Greenberg had flown out to the middle of the desert also gave me a sense of importance—and a notion that loyalty still existed in the cutthroat world of investment banking.
"Let's talk about this," I said finally, looking across at Greenberg. "What exactly are you offering if I get back in the game?"
Greenberg's eyes lit up. The bullying had worked. He now had a hookpoint. It was time to take advantage. For the next 15 minutes, he explained the deal, pitched it actually, and laid out my role in the big picture. Then he closed the deal and sealed it, giving me some reasonable terms and a rare handshake.
Little did I know that Greenberg had withheld a key piece of information—there was a competitor in this pitch—a _big competitor_.
### **Preparing the Big Pitch**
The next day, back in Los Angeles, Greenberg and I got together at Greenberg's high-rise office to discuss an action plan. It was Wednesday. Greenberg come to the meeting knowing that he'd have to mention the competitor. But he waited until the end of the meeting and then sprung it on me.
"Did I mention that Goldhammer is pitching the deal, too?" Greenberg asked.
I almost spit up my coffee.
"What?!"
"Yeah, they have a team on it," Greenberg said. "Relax. It's nobody we can't take down."
"You knew that when you flew out, and you didn't tell me?"
"What does it matter?" Greenberg said, trying to play it down.
"It matters because they beat us most of the time," I said. "Goldhammer is Goldhammer. They have 10 times the resources, and they have the Goldhammer pedigree."
"That's why I got you, so we could beat them," Greenberg said.
"No," I said. "You didn't disclose this, and if you had, I wouldn't have come. I'll go toe to toe with anyone, but this is not a fair fight."
I was seething. Greenberg had purposely withheld this information, and now I felt betrayed. We were up against Goldhammer, who would throw 12 to 15 people at the effort against our 6. But the potential rewards certainly made the effort worth it. If we won the pitch and actually raised the $1 billion for the airport over a period of five years, Greenberg Capital would reap more than $25 million in fees. Of that, I would take 30 percent. So I had a lot to prove, a lot at stake, and a lot to gain. All I had to do was take down Goldhammer and my old arch-rival Timothy Chance.
#### **Meanwhile at Goldhammer**
The Goldhammer office in Los Angeles occupies the entire twelfth floor of a gorgeous downtown skyscraper with a view all the way across Hollywood to the Pacific. The floor is decorated throughout with touches from the Orient—a jade dragon, ornate vases, and Japanese flower arrangements. I'd tangled with Goldhammer before, so I'd been in that office. I could picture in my head _exactly_ how the company's planning session against my team was going. This is how I envisioned it:
Inside the main conference room, seven people would be gathered to discuss a big deal—a monster deal, the airport deal. The discussion would be led by Bill Miner, a second-generation investment banker who had been hand picked by Goldhammer corporate to lead the Los Angeles office. It was Miner who had chosen the Far East motif for the office, and Miner often quoted his favorite book, Sun Tzu's _Art of War_. After briefing everyone on the background of the airport, he would turn his attention to the competitors—us.
"There are three or four of the usual suspects involved," he would say. "But the wildcard is Greenberg Capital."
This name would be familiar to all of them—we were local, and occasionally, Goldhammer and Greenberg would cross paths.
Goldhammer's top pitch guy, Timothy Chance, would be quietly listening. He knew us and had worked with me for a few months in the 1990s. Since then, we had clashed several times. Three years ago, at a funding conference, Chance and I had exchanged words and had to be separated.
In the end, I imagined that Chance and Miner would look at each other, both knowing that this was the biggest pitch of the year—the one that bonuses would rest on. And it would be the one deal that would let the industry know who was best.
#### **Strategy Sessions and Research**
The first meeting at Greenberg Capital was a strategy session. Sam Greenberg, Rob McFarlen, and I sat in a conference room, and Greenberg delegated the various responsibilities. The pitch was a risk for all parties—the legal fees alone would be nearly $40,000. And the final bill for all the soft costs would be close to $100,000.
McFarlen and I began the long and arduous task of running numbers on the deal. McFarlen was a quantitative analyst, and he knew the types of financial models that were needed.
I had the role of putting together the big picture and the storyline—and I would be the one to actually make the presentation.
Greenberg would pay the bills and make sure that everything was tracking to his liking.
A few hours later, we took a break for lunch and started talking about Goldhammer.
"I'm wondering if they are gonna bring in Timothy Chance." Greenberg said.
"I hope so," I said. "He's the best they have, and the good news is that I know how his mind works."
"What do you think, Rob?" Greenberg asked.
McFarlen had a conflict of interest because he freelanced for Goldhammer as well. He shrugged. "My job is just to run numbers, not to pick sides."
"Go ahead. You can say it," I told him.
"Timothy is as good as they get," McFarlen said. "Superpolished. But I'm sure you are, too."
Greenberg watched me wince and smiled to himself. I'd taught him frame control, and now he was using it on me. Nice. This was typical Greenberg, always playing the game, trying to get people to work beyond their limits.
"You know what I think," I said. "If they do bring in Timothy, it will make things difficult. He's good."
We were making progress on the financing structure of the deal. We'd done research on comparable deals worldwide and felt that we had a good finance strategy in place with a track record to back it up.
We also confirmed that, indeed, Timothy Chance would be making the pitch on behalf of Goldhammer. This provided a jolt of adrenaline that helped me through the late nights of preparation.
Over the years, I'd been obsessed with putting together a method to the madness of presenting and closing deals. I'd developed the concept of neurofinance, perused scholarly journals, interviewed professors and researchers, and even set up experiments with executives to gauge their reactions to various pitching styles. However, all this research, as well as the 10,000 hours I spent on the subject, would do nobody any good unless it really worked. And it was clear that this $1 billion airport pitch would be the ultimate test of what I had learned and, if Greenberg Capital won, the ultimate confirmation of my methods.
#### **Middle of January**
The daily rhythm of work, the rhapsody of ticking things off the to-do list, the grind of crunching numbers—these things were giving me a sense of purpose and a daily jolt of exhilaration. I didn't tell Greenberg, but I'd been bored out there in the desert. Yeah, hot sand feels good on your toes, but if you're used to being a gunslinger, there's _nothing_ like being in a live deal.
McFarlen had come over to my home to discuss the latest numbers, and I was explaining this very thing—the thrill of the deal.
"In a sense, it's very gladiator-oriented," I said. "You have to kill—or be killed. And if you do go down if you fail, the spectators, the people you're pitching, often experience perverse joy."
McFarlen nodded. I was always giving him vivid and clarifying metaphors, but McFarlen wasn't wired to really pay attention to all that. He was an introvert, who only broke out of quiet mode to defend his financial analysis. He was a numbers guy with a low-key personality.
"Who is Goldhammer using to run these numbers?" I asked. "If you know."
"They are doing it in-house, so it'll be Brandon Caldwell," McFarlen replied.
"Can he do what you do?" I asked.
"What do I do?" McFarlen wondered.
"You make magic out of the mundane," I said.
"No," McFarlen said. "Caldwell can't do... what I do. Not with the short timeline." And this was the most bravado that I could pull out of McFarlen.
In McFarlen and me, Greenberg had indeed lined up two very skilled professionals. But Greenberg was no slouch either. He was a math whiz, and when I was just starting in the business, I looked up to Greenberg as my pitching mentor. This little team certainly had the talent and the experience. We just had a lot to overcome to bring the airport deal home.
#### **The Client**
Simon Jeffries owned the airport deal. He had worked for years to get all the pieces in place. Jeffries and Greenberg had known each other for more than a decade, and from time to time, they'd see each other in development circles. Now Jeffries was in an alpha status position. He was the guy who would hear the Goldhammer and Greenberg pitches and ultimately decide who would get a contract to raise $1 billion for the new airport.
Interestingly, while Chance was probably doing tons of research on Jeffries, I did none. I was not interested in building so-called deep rapport—a personal connection with my audience.
My research had shown that the small talk at the beginning of a pitch typically was fruitless. People who make million- and billion-dollar decisions don't care where you play golf or whether you had trouble finding a parking spot. I had learned this early on and avoided the deep-rapport trap that many pitchmen step into. I would be focused, instead, on a unique theme and storyline. A compelling human drama.
On paper, the proposed JetPark Airport is a beauty. A renowned architect had designed a 1,000-acre metropolis built around the famous Davis Field runway, which extends nearly 7,000 feet. The plans included restaurants, shopping, and amenities. Most of the buildings—now just renderings—would be multistory glass-and-steel monoliths. No detail had been spared.
The revamped airport was expected to help southern California deal with a staggering rise in air traffic (an estimated 30 million passengers will fly out of LAX in 2010 alone). The airport also would service smaller aircraft and provide quality office space for businesses that support the aviation industry. All told, the airport was expected to bring 10,000 jobs and and to have an economic impact of $2.2 billion.
Getting funding, therefore—this was serious business. Southern California _needed_ the new airport. The new airport needed money. And both Greenberg and Goldhammer needed to win.
#### **Nine Days Before the Big Pitch**
McFarlen was working 16-hour days reconfiguring the deal structure. Today, I was working with a graphic designer on creating a visual "face-melter" to accompany my pitch. I wanted the visuals for the presentation to pop and shock.
I also was working on the elements of "the story." Some of my friends were Hollywood screenwriters, and they had beat it into me: _Every pitch should tell a story._
"There's gotta be an intrigue hook," I told McFarlen. "If the shark in _Jaws_ has a GPS beacon on it, and you know where it is all the time, then there's no drama, and the story is not interesting."
I was now reframing the entire campaign to include a human element. Sometimes deals are done strictly on numbers—but not this time. This time, it was about people.
McFarlen nodded and got back to his numbers.
#### **The Day of the Pitch Arrives**
At 2:52 p.m. on pitch day, I saw Tim Chance entering the building a few steps ahead of me. In the main building lobby, I went over the pitch again in my head. The plan was to focus on the Hollywood story aspect—to talk about the people I'd met in Spring Hill, the community where the airport was located. I felt certain that neither Tim Chance nor anyone else at his firm had ever set foot in Spring Hill before. As I made my way up the elevator to the ninth floor, I had a quiet confidence in my pitch strategy. The financial structure was solid, and Greenberg Capital had a strong track record to stand on, but the story I'd prepared had something stronger. A human angle that was compelling.
Simon Jeffries' office covers 3,000 square feet of prime Los Angeles office space. I walked into the reception area and saw Chance tapping out a text. We made eye contact, and I arched my eyebrows—my way of saying hello—and turned to the receptionist.
"The Greenberg team is here," I said, grinning.
"Go ahead and have a seat," she said.
There were six chairs in this beta-trapped lobby. Rather than sit, I tried to provoke Chance a little ("You texting headquarters for last minute advice?"), but he wasn't in the mood to talk. He knew this was frame control in action. Anything he said would be framed, deframed, reframed, and flipped.
"Good luck," Chance said, and looked back down at his iPhone.
Jeffries eventually entered the reception area and shook hands with me and then Chance.
"Okay, gentleman, come this way," he said.
He led us down a long hallway into a conference room.
"Have a seat," he said.
Chance and I exchanged an uneasy look.
Jeffries excused himself, and as soon as he was out of earshot, Chance said, "We're pitching in front of each other? You've gotta be kidding me."
Perfect. "This happens all the time," I said. "You should get out and pitch more."
What Jeffries probably knew was that if he listened to both of us pitch, on our terms, he'd get our rehearsed efforts. If he was going to trust either of us to raise $1 billion, he wanted to see how we reacted when things didn't go as planned.
Just then, a third pitchman entered. He was from a firm in London. This thing was going to be even more competitive than I'd anticipated.
### **Anatomy of a Pitch**
Two months ago, I'd begun sizing up Goldhammer and wondering how I could get an advantage. Both teams would be working from the same information, so what would make the difference? It was a puzzle to solve, with the winner earning the opportunity to raise $1 billion and getting a $25 million payday.
Despite the stakes, I'd rewired my mind to think about it as "just another pitch" so I wouldn't feel all the pressure or do anything desperate when the moment came. Easy to say, but difficult to do. How do you spend weeks or months working on a big presentation and _not_ be anxious about it? I had to invert my own psychology because, as human beings, we are all hardwired to be emotional about important social encounters. What worked for me were the three rules of _eradicating neediness_ :
1. _Eliminate your desires._ It's not necessary to want things. Sometimes you have to let them come to you.
2. _Be excellent in the presence of others._ Show people one thing that you are very good at.
3. _Withdraw._ At a crucial moment, when people are expecting you to come after them, pull away.
If I could not eliminate my desire to win while preparing, then the team likely would seem needy and desperate on the day of the pitch. If I could not be excellent at pitching just one simple idea, then the competition would win because they were, on average, stronger. If I didn't have the nerve to withdraw at the right time, then I would just end up chasing the deal—and therefore losing it.
I knew that this had become a simple game involving the four pitch phases, and I should just have fun playing it. Toward that end, the first task for the team was to understand the mind-set of Simon Jeffries. I had to tune my pitch message to Jeffries' crocodile brain.
_First, I had to hit the right tone._ In fact, this would be a formal affair. Jeffries had been dealing with the Federal Aviation Administration (FAA) for several years. That experience surely would have set the "fun-dial" to low—these guys wouldn't be used to free-wheeling humor and high energy. Jeffries also was working with city, state, and federal agencies, so I would have to show a certain serious, respectful tone. But a "serious" tone, doesn't mean somber. Having fun with the pitch was absolutely critical. If the presentation isn't fun for the person giving it, then everyone else becomes anxious. And because there's no way to fake "having fun," I would really have to be enjoying myself. That in itself would eliminate desire.
_Second, I had to get the frame right._ This can be explained quite simply: The competition would make this about money and profits. They were sure to frame this deal as a "money-making opportunity." That's what they always did. What was being overlooked by these Wall Street types is that Simon Jeffries is not the head of just any old development company. He was head of a company that was about to redevelop one of the most historically significant airport runways in southern California. Jeffries would want to be known as the man who _saved_ Davis Field when others couldn't. This redevelopment plan had been tried many times before—and failed. Jeffries would be building an airport on 1,000 acres of southern California land, land with a history dating back to the 1920s. This didn't have to be a deal about money. Instead, it could be about something bigger, something that tapped into the human desire to be the alpha in a social situation. The brain is wired to do things to achieve status, not money. And within that notion, the big idea was born. This deal was about _legacy_.
This deal was about building a legacy from a piece of American history. Simon Jeffries wanted to be remembered for doing something important. That is desire working, not greed. In this realization, the hard work was done. All I had to do was tune my pitch to this desire and be part of the plan by which Jefferies could secure his legacy.
_Third, I had to hit the buttons that produce hot cognitions with a sledgehammer._ All the hours logged on Sam Greenberg's plane had made an impression. Jets are intoxicating, plain and simple. They are pure hot cognition. Jeffries and the committee were deeply involved in the aviation business. They either owned planes or worked with them. Two were pilots themselves. When presenting to someone who loves jet aircraft, hitting hot buttons is almost too easy. You just show lots of visuals of jets.
Any product that your target consciously or subconsciously believes will enhance his social image _will get his brain hot with desire._ Show the brain something that society values, and you won't just be hitting hot buttons, you'll be _stomping on them._ Dopamine will pour into the reward structure of the brain, and the emotion of pleasure will rise fast. It's the same reward structure that's involved in the response to recreational drugs. When most people enter a social interaction that involves something like a Ferrari; a Rolex; beautiful ornaments; a Renoir, a Cezanne, a Titian, or a de Kooning; a pedigreed Rottweiler; a beachfront mansion; or as in this case, a private jet, their hot cognitions fire like crazy. They anticipate desire and rewards, and it feels good to them.
This is why I planned to show Jeffries and the committee large poster boards covered with "aviation porn." Every few minutes, I'd flip around a new posterboard—each with an increasingly provocative photo of a beautiful jet taking off, landing, or doing a high-bank turn.
In other presentations, when pitching derivatives or abstract financial instruments, it was much harder to hit the target's hot buttons visually, but with this deal, the jet narrative would make it _easy_.
I knew if I could just make it to the final selection, it would come down to Goldhammer and us. They had the deepest bench, the best track record, and the most influence. The inventory of their strengths was awesome. In the preceding 10 years, they had trounced us whenever they went after the same target. And there is no doubt that they would be discounting us because of our small size. I'd personally raised $400 million. But those guys? Goldhammer was doing billions.
As the presentation began to take shape, I built it around the four phases:
_Phase 1:_ Frame control, grab status, introduce the big idea.
_Phase 2:_ Explain the problem/solution and our special advantage.
_Phase 3:_ Offer the deal.
_Phase 4:_ Stack frames for hot cognition.
There was a deeper problem than just the competition. Simon Jeffries and the committee were natural alphas. During the pitch, they would be vocal, disruptive, and distracting. If I didn't seize the status and control the frame, these guys certainly would. At any point, if my pitch slowed to a crawl, they would jump in and try to assert their alpha status. They'd demand to know, "How do you plan to do that specifically?" or "Where'd you get those figures?" or "How much will such-and-such cost?"
To keep them out of this detective mode, I would need to constantly deploy push/pulls. This would keep them either rocked back on their heels or leaning forward and intrigued to know more. They wouldn't have time to become cold and analytical. They wouldn't be able to disrupt the frame. They would always be in _my_ frame, reacting to _me_. I would have to seize star power status early.
#### **Prepitch Thoughts**
Here's what was running through my head at the last minute:
1. Get the tone right, frame myself as the alpha, seize status, and hit their hot buttons.
2. Deploy a big idea that is human and captures the theme of "building a legacy."
3. Keep it captivating with visuals that resonate.
4. Create hot cognitions. Make Jeffries and the committee _want_ the idea before they even know the details.
The plan was to have a 20-minute high-temperature conversation with the targets' croc brains. The goal was to achieve a pitch full of hot cognitions. I believe that if two equally skilled people pitch the same idea and one tunes it to the neocortex and the other tunes it to the crocodile brain, you have two very different results. I had tuned my pitch for the targets' croc brains and was ready to start.
### **The Presentation**
I'd been preparing for over two months, and now, with Chance watching, I stood up to address Jeffries and his committee. Speaking slowly, I began:
"There is a tremendous responsibility for all of us today. This is a decision not about who is the most charming or the most skilled in finance but instead about who has the _right ideas_ that can raise $1 billion for Davis Field. Others have tried this kind of thing before and failed, so it's not that the best man should win or that the best team should win, but instead, the best ideas should triumph today. This runway has served the United States of America in World War II and hosted squadrons of B-17 bombers and other fighter planes that took part in the Pacific campaigns. Today, we are not talking about building a shopping center or strip mall or motel. We are building an airport, and we are doing it on hallowed ground. This has to be done right."
It was critical to get the frame right at the beginning. And because Goldhammer definitely would open by highlighting their size, experience, and track record, I had to pick a frame that minimized their strengths and focused the lens of attention on ours. This is why I chose the _best idea frame_. In other words, I was telling the targets that they should forget about picking a bidder with size and power. Instead, they should focus on the quality of ideas. We could not compete on size and power against the others, but if I changed the frame, if my big idea frame was stronger than Goldhammer's, then we could still win.
I'd also cranked up the tension, injecting norepinephrine into their brains by saying, "This is hallowed ground." In other words, a screw-up would have serious consequences.
If had I hit the tone just right, then a strong frame would be in place. The next task was to reframe the competition:
"We are honored to be competing against two other great firms today. I know each of them could serve you well because they have large teams, multiple offices, legions of young, energetic researchers, and the best-paid analysts in the world, and when working a deal, these firms spare no expense to tackle the job."
This was my way of saying that Goldhammer and the group from London were big, bloated corporations with too many people, many of them young and inexperienced. With this statement, I had reframed the competition as young, overstaffed, worried about their fees, and generally overweight. This fit the general image that dogged Wall Street banks in the media and would be an easy idea to convey. I knew that Goldhammer's Tim Chance would have to invest much of his precious time digging the company out of that frame. Chance recognized what I'd just done, and it's no wonder he was scowling. Things had started well for us.
"Simon, the vast majority of people you have met in the last three months have told you that the market is flat and that nothing has changed for some time. But if you start challenging the way these folks are thinking, you can start to see this market through a different lens—the lens we use. Let me explain. Three market forces that we follow very carefully have formed an important market window that we can step through—if we time it right. We don't think the window stays open for long, but if we do this now, we can go get $1 billion from investors faster than any other deal out there. Here's our analysis of the way markets are moving:
_"Social factors._ Everyone is tired of investment bankers getting fat off these deals with no risk, so we have to be more transparent on fees.
_"Economic factors._ For deals that are transparent, where the bankers and consultants are willing to risk alongside investors, there's a glut of investor money in the market now. $5 billion more than last quarter.
_"Technology factors._ If we go green and make the buildings LEED-certified, I know a government agency that will get us a 10 percent reduction in taxes.
"I know this is news to you, but this is how the market is moving, and these three forces are important to our strategy. Again, we have a short market window to step through. If we fight against these forces, we will struggle. If we hit it right, though, we'll be one of the few deals that gets through this small market window."
This was the three-market-forces pattern that I had long endorsed. Simple and straightforward. I wasn't going to start selling and promoting before I was done framing. That would be a mistake. By doing it this way, I would show the targets how the market was moving. This would work because the mind is not a camera; it is a machine dedicated to observing motion and predicting what will happen next. I continued:
"Before going into my plan in any detail, let me tell you what we recognized some time ago. This project is more than an airport upgrade or an airport relaunch. _It is a legacy you will leave behind._ You will have your names written in history, and you we will be judged by generations to come by what you build here."
I had issued a challenge to the committee. Done this way, it creates a dopamine and norepinephrine kick at the same time. In other words, they would feel the twin pull of desire and tension.
"Simon, Jeff, Jim [I addressed the committee by their first names], I know you need to find investors for this airport as fast as possible, and I appreciate how questioning conventional wisdom is hard to do when time is tight. But today we are going to ask you to question how things have always been done because lately, the 'usual' way of doing these deals turns out to be wrong or wide off the mark or both.
"There are too many similar, me-too deals in this market. Right now, unless you are different in your approach from all the others, you will be wasting time and money.
"That's why our big idea is different from all the other plans, as you'll see.
[I flipped around a few big posterboards with the theme and logotype boldly printed.]
"As you can see, our theme is 'Invest in an American Legacy.'
"Our plan gives plenty of profit to investors but also gives them a chance to be part of an amazing story. Unlike the other bidders here today, who will just be telling the potential investors a profit-and-loss story, we plan to tell the investors a wonderful story about an airport that has a rich aviation history.
"The combination of the 'American Legacy' theme with our financial plan will work best in the market. Our approach will raise $1 billion faster and more easily. Our big idea delivers a better velocity of capital and more certainty that you will get the money you need. We are going to set our sights high, to literally become heroes by protecting—not destroying—a piece of aviation history and get $1 billion at the same time."
This was my classic _big idea introduction pattern_. Why would this kind of introduction for the big idea work here? There are three basic truths about the brain and decision making that went into it. _First,_ the most basic working principle of the brain is: Decisions of _wanting something_ are not conscious. _Second,_ the opportunity to gain a social reward, such as becoming a "hero," is _even more enticing than making money. Third,_ you can flood the target's brain with dopamine by focusing on three ideas: (1) the idea of social rewards, (2) the idea of becoming a "hero," and (3) the idea of making a lot of money. The purpose? Ignite desire.
Now, at this point of the pitch, I'd served the committee's crocodile brains the right cocktail of dopamine (desire) and norepinephrine (tension), and I was free to proceed with the boring stuff: the numbers.
By creating a frame that focused the attention of the targets on what we were good at, I'd put some distance between ourselves and Goldhammer. Being different also creates novelty. And that gets the dopamine injectors in the brain flowing. Traditional pitches often start out, "We worked really hard to come up with a great plan...."
But my approach takes two steps back and then goes three steps forward. It begins by saying: "Things have changed out there in the market [and the normal approach isn't going to work]," and it ends up by saying, "There is a better way that is different from the others," and "It's different because it isn't just cold, hard numbers. It has a human story to it."
The other groups probably would do precisely the wrong thing: Focus a huge amount of time on their wonderful résumés. They would all be using the same theme, only the details would be different, relying on clichés, framing themselves as a "full-service firm" that caters to clients and provides the "highest levels of integrity, service, and quality." There is nothing gained with that old, ineffective way, so why waste the time? Doesn't everyone just assume that you're going to provide service and quality?
Over the next five minutes, I gave highlights of the budget and what timeline I could deliver on. If I could not pitch the full plan in five minutes, then the last two months would become a very expensive waste of time and money.
In preparing for this day, the difficulty was knowing what to take out of the pitch without stripping away the richness and complexity of the idea. But I felt that I would have a better chance of success with fewer cold, hard details, the kind that switch the brains of the targets into analytical mode.
The length of the Pitch also was of paramount importance. During a rehearsal a month earlier, the pitch come in at more than 55 minutes. Too long. So I started hacking. I edited three minutes from the total, then another two minutes, and so on. With each new rehearsal, I removed details that lacked intrigue or hot cognition. As recently as one week before the presentation, I was still working on making the pitch hotter, shaving away distracting details while keeping the core message.
So now, here I was, pushing through the budget and the financing details in about five minutes. It was the coldest part of the pitch. Soon, I would deliver the _four-frame hot cognition stack—_ which would heat things up. But first a quick push/pull:
"Is this plan bold? Well, we can certainly debate if my numbers are 5 percent too high or 3 percent too low, but there is no doubt the big idea _is_ bold. We think that boldness is important. And if you don't like bold plans, then there's a real possibility that we are not right for each other because my team would always be working quickly in an entrepreneurial way, and you always would be responding like a big corporation—slow and methodical. And how could that ever work? So I'm okay with the notion that our plan is too bold and that we aren't right for each other."
I was employing the classic push part of the push/pull pattern, which challenged the targets and amplified tension. Now it was time to back off. I had a lot going on here, drawing on the techniques and research that had consumed me for more than a decade. No matter what kind of soft touch I had, this was still a form of selling, I was trying to get the targets to decide in my favor, trying to take control. And to my targets, this was a form of stress and pressure. Humans behave a certain way when they are put under this kind of pressure. At a basic level, in a target's croc brain, there's a feeling that you are taking away his or her autonomy. A threat response could be triggered.
The push would counter this problem, giving the targets the opportunity to make a pressure-free decision.
Because the human brain evolved in response to stressors over thousands of years, humans are constantly attuned, at the level of the croc brain, to the ways social encounters threaten their capacity for choice. This is one of my guiding theories: The slightest perception that you are taking away free will (scientists call this _reducing the autonomy of choice_ ) will trigger a threat response.
After letting the croc brain know that it didn't have its back against the wall, it was time to complete the other half of the push/pull pattern:
"But then again, if this did work out, our forces could combine to become something great. Imagine, your aviation experience and passion combined with our strategy and financial know-how. Almost like some kind of superpower, when we focus our gaze on any individual investor, he would just explode with desire!"
I then returned to the idea of status. The brain is always assessing how social encounters either enhance or diminish its status. Yet, at this point, all the competitors had higher global status than we did. There was no way around that. They had more wealth, more popularity, and more power—the three measures of status. So I needed some local star power, and I needed it fast.
"Look, in all seriousness, we love this project."
I started flipping over heavy posterboards that were set around the room. These were big, physical, real-world boards, about ½ inch thick each. Unlike a PowerPoint slide, which would just disappear, these would remain, adding a certain concrete feeling of reality to the whole pitch.
"And I know how difficult it is not to choose Goldhammer or the London team, sitting here with us. How great are those guys? Is there anything they can't do with all that youthful energy and those amazing bespoke suits? But one thing I would have to ask them: How much do they know about Spring Hill pickup football?"
This was a novelty play that would keep attention high. But it was risky, because if you are going to go off on a tangent—it had better be good.
"There's a reason I ask, because the full story of Davis Field and why the previous attempts to build a new airport here failed cannot be fully told until you meet Joe Ramirez."
I had indeed met an auto mechanic named Joe Ramirez while doing my research on the airport deal. He was tall, with wavy hair and a prematurely graying goatee. Now, here he was, striding across the room dressed as if he was going to church. You can imagine how a mechanic would have been received by the committee, in the middle of a pitch for a $1 billion contract. Nobody expected this. Clearly, he was not here to explain financial plans or flight intervals. I urged him to take his time and speak from the heart.
The clock was ticking, but this moment was too rich to be rushed. Joe moved in front of the lectern and pulled from his pocket a folded piece of notebook paper. He read aloud his prepared comments:
"I grew up in Spring Hill. It's been my home since my father come here from Dallas, Texas. As a kid, there wasn't much to do. We didn't have the mall and the theaters and the skateboard park. But what we did have was a football field. It was at the airport, here [he pointed to a place on the map, just adjacent the runway]. Every Saturday and Sunday we played football there. Two or three games would be going all the time. This place was easy for everyone to get to, and some of my best memories growing up happened right here. But in 1997, nobody knows why, the city paved the field. It's been an empty parking lot since that day. If there's anything you can do...."
You could clearly see the emotion in Joe's face; you'd have to have been either a robot or from an alien planet not to be moved yourself. When Joe said that the city had paved his football field into an unused parking lot, there was a certain heaviness in the room.
Heightened states of emotion create strong memories. _Where were you when such-and-such famous person died?_ It's easy to remember. The parts of the brain where memories are stored needs to distinguish between significant experiences and those which carry less importance. This was one such moment. While it is hard to define emotion specifically, what is not hard is to show its effect on cognition and decision making. Emotions are how we encode things of value and how we link events to our memory. If it is true that emotional intensity creates a moment where attention is high and encoding is high and where desire could be created, _I would not get a better opportunity to create "wanting" from the committee._
I thanked Ramirez and stepped back to the front of the room.
"Simon, committee... You see, we can look at the numbers behind this deal all day long: 24 percent of this, 15 percent of that, $100 million for solar panels, $100 million for construction of just one terminal. One billion dollars. It's all just numbers. We have all been thinking of this airport as if it is a purely financial transaction. As if this is a 7,000-foot runway in some kind of cyberspace. What I realized just 30 days ago, and what has been forgotten in all our desire to design it and build it and profit from it, is that this airport isn't in cyberspace and it's no longer 1948 when this runway was in the middle of nowhere. What I'm saying is, 50 years ago, Spring Hill was populated by fewer than 1,000 people, who were probably outnumbered by the jackrabbits. Now, 115,000 people live here. Think about that. We are deciding in this room, 50 miles away from the site, what will be done with 1,000 acres of land in the middle of a community."
With this, I put the morality frame into play. It would be hard for the other groups to find a higher set of values to champion than protecting a community of more than 100,000 souls. This frame is so basic, so tied to the workings of the social animal, that it had to be deployed. This was the right moment. Next, the time frame would be used:
"I have to finish in five minutes, so I don't have the time right now to introduce you to the 37 other friends of mine who live in Spring Hill. You see, I've been living there for the last few weeks in a cool little hotel on Main and 19th Street. I've played pickup football in a dirt field outside town. That's where I met Joe. So I can tell you that this is an amazing community that will support us if we play fair and support them as well."
Emotion in the room was peaking.
Simon Jeffries couldn't hold off any longer. He was already leaning forward in his seat almost past the edge. "You lived there? You know all these people?" Jeffries asked. What had been a formal pitch was now becoming a casual conversation. "They're your friends, and you know their names?"
"I'm good with names," I said. "And those people all had something important to say about this project."
"This is why our plan includes an athletic park to return to Joe and the community the football field he loved so much growing up. We also want a young aviators' center added to the build. With $1 billion, this is a rich project, and we can afford it. We'll pay for it ourselves. Here are the plans."
I flipped around another posterboard. Now it was becoming like a game show.
"Are you sure that you want to commit to these things?" Jeffries asked.
"How can we _not_ do it?" I said. "You can't just pull value out of a community. You have to put value back in."
Following the principles of seized status, I would now redistribute some of the alpha status and frame control I was holding to some of the other players:
"And the park plans with the restoration of the historic football field are complete. They're not just part of my theory. These are real plans. We completed the engineering specifications, and I want them to be part of any plan that goes forward. Five minutes ago we e-mailed these plans to everyone in this room. No matter what happens with our bid, whether you choose us or not, we want the football field to be restored."
I flipped around the final display boards with their beautiful, evocative visuals of aircraft overhead and kids playing football and proud community members holding their arms in an open embrace. These were all big-picture visuals meant to stoke the fires of hot cognitions. For a closing statement, I would bring it all together. _Time frame. Prize frame. Intrigue. Morality frame. Push. Pull. Desire. Tension_. It was a fireworks finale of frame collisions:
"Committee, the only thing worse than an idea you hate is an idea you just 'like.' When you only 'like' an idea, then you are still unsure about it. Imagine getting married to someone you only 'liked.' It would seem cold. If I were sitting where you are, what would be important to me? I might think, if we don't love this idea of 'American Legacy,' then we have to throw these guys out of here right now.
"And that would be okay with me because it would be the right thing to do. And by the same measure, if you only 'like' us, then you also must throw us out. And I'm totally okay with that, too. Because we could not possible work with you if you didn't love our big idea. We believe in the big idea that strongly.
"Right now, as we are sitting here, the terminal paint at Davis Field is peeling, the old observation deck is rotting, and a local park has been paved over. Almost everything about Davis Field Municipal Airport shows age and neglect. It is a place that by almost every appearance has been left behind in time.
"But time should not leave this place behind. The war in the Pacific was fought from this field. Bomber squadrons had thousands of movements here. Men left this field to fight for our country, and for some, it was the last patch of American soil they ever touched.
"So if you love the idea of an American Legacy and you want Joe Ramirez's kids to be able to play on that field, and you want to be known as the capitalists who built a legacy for the ages, then we are the right team to pick today because we know how to do this better than anyone else. But we are not going to do this _for_ you. We will have to do it _together_ with you. When you feel that the time is right, _I encourage you to come to our office and talk over how we can make that happen."_
The prize frame can be boiled down to one thing: Withdrawal. At a crucial moment, when the committee was expecting me to come after them, I pulled away.
I remembered what was once written in the _US Air Force Training Manual_ , "It is generally inadvisable to eject directly over the area you have just bombed." Strictly following that advice, it was time to leave.
In the course of my many pitches, I discovered that people won't do what you tell them to. They must feel as if they have free will to make their own decisions. They won't even know what to do unless you have created primary basic and inescapable emotions for them to react to. They can't encode your pitch into their memory without strong rushes of dopamine and norepinephrine, resulting in the twin forces of desire and tension.
In that moment, everyone realized that Greenberg's little company of six people (plus seven consultants) had a chance to win against the biggest and best in the financial industry. I had built from scratch a pitch that worked in a market where nothing worked. That's when I realized that this was the most exciting 20 minutes in my career of pitching deals.
### **The Competition Strikes Back**
Tim Chance was up next, and as expected, his presentation was polished, practiced, and predictable. He opened with a long explanation of the many large deals his firm, Goldhammer, had done in recent years, the amazing capability of his organization, and the respected name it held. The logo on his business card was known around the world, and he was using it to his best advantage.
There was a moment of unintentional comedy right away: While Tim was making his opening remarks, his team was fumbling with the laptop they had brought, trying to connect it to the conference room projector. Although we've all seen this happen before—it still makes me laugh. With all that was at stake— _how could they burn five minutes this way?_ We had worked for _two days_ to cut just _three minutes_ from our presentation. My question was soon answered: As his slide deck come into focus, I and several others in the room noticed the little number in the lower right corner of the screen: 42. Oh boy! There were 42 slides in his presentation. _This was going to take a while._
After the review of Goldhammer's bona fides, information we all knew anyway, Chance launched into a lengthy and detailed opinion of current market conditions. I could feel the temperature of the room falling toward subzero as the cold cognitions he was flashing on the screen were sending our brains into a deep freeze. While he certainly looked good and sounded good from the podium, he was talking about data and not about what really mattered, which should have been Why now? How? and the critical path for doing it.
Chance was banking on a tried-and-true method that Goliath-sized companies often use. Because they're big and successful, they think that it implies that they are also capable, and therefore, they often do not speak directly to how they will accomplish an objective. They believe that their audience assumes "it will get done," but _will it really?_ We are left to wonder. In large companies, deal makers like Chance are rewarded for the business they bring to their firms, not necessarily the results.
Each of us had been granted up to one hour to present, and unbelievably, Chance used every minute he was allotted. After 40 minutes of this financial mumbo-jumbo, he was inducing a coma in me. Tim was the only member of his team to speak, and he dragged the audience through every single one of the words in every single one of his densely worded slides. _This is doing them no favors_ , I thought, but it's good for me.
Next up was the team from London, who, _mercifully_ , did not speak for an hour. On the contrary, their presentation was the model of European efficiency—clean, slick, and heavy with emphasis on financial models. This team had the "Wow factor" on their side, too, using an animated three-dimensional digital presentation of their previous aviation projects. I was impressed. They had more experience in aviation than the rest of us combined.
Like Goldhammer, they eventually succumbed to the temptation every presenter faces: taking a deep dive into complicated financial figures, and as their team commented on how they would manage the deal, it became clear that this was merely production work to them—no different than any other airport project they've ever worked on. If awarded, they'd stamp this project out cookie-cutter style. They expressed no interest in the local community and seemed to have no concern with the economic impact. They focused only on the project funding—how quickly they could get in, get it done, and get out.
I could not help but be impressed by their confidence and felt certain that they could do this deal—and do a great job. These guys could win it easily.
As they finished with a rich European flourish—puffed chests, thick Oxford accents, and broad smiles—the last thing they said was, "So we would be proud to work on this prestigious project and will eagerly wait to hear your decision."
_Beta trapped!_ After all that work, showing neediness was the wrong move. Simon Jeffries approached the podium to make some final remarks. Simon is a class act. He graciously thanked the teams for their presentations, described the week-long deliberation process he would now preside over, and quickly brought the meeting to a close.
### **The Hour of Judgment**
After it was all said and done, I sat there, looking out the window of Greenberg's Los Angeles office. There were five others with me, waiting for a phone call. _The_ phone call. Jeffries, meanwhile, had gathered several members of his selection committee at his office, presumably to discuss last-minute details.
As I looked out at Los Angeles, I reflected on the last few months, this pitch, and its repercussions. I'd boiled down two months of work into a lean and elegant pitch lasting 20 minutes and 52 seconds. Now it would all come down to this call. This moment. The decision. The phone rang. I sat down at the table in the conference room, and Jeffries was put on speaker.
Jeffries began, "If you go down to Davis Field right now, as you pointed out to me, the terminal paint has worn off and the old observation deck is just rotting away. Almost everything has a hole in it, including important parts of the runway. Who would want to land a jet there, have it serviced there, or take a meeting there? Nobody." This was becoming melodramatic. We just wanted _the decision_.
"This is why I am so passionate about the new Davis Field and excited about building a new entrance to the airport and all new facilities. This can be one of the best private airports in the world. But I have got to pick the right team. The Greenberg team was fantastic in last week's pitch. There were some things that we felt you didn't get right, but we really enjoyed it. This is a hard decision, and of course, we can only pick one winner. . . ."
Jeffries let the moment linger for a torturous amount of time, and then he cleared his throat and said the word, " _Congratulations!_ "
The office erupted into celebration.
For me, this journey back from the desert was complete, and my methods were validated. They weren't just a personal collection of pitching notes stuffed in my notebooks. It wasn't just thousands of index cards in my office. It wasn't just a bunch of academic notions or theories. And it wasn't just a checklist of dos and don'ts. Much in the way that calculus is a system for solving math problems, or civil engineering is a system for building bridges, my STRONG method was now a system to get deals done, especially when the stakes are high. It worked.
## **Chapter 8**
**Get in the Game**
Learning to manage social dynamics is not an intuitive undertaking. Ten years ago, I found myself in many situations where I was the beta. I thought I had to accept this low social status and thought that there was almost nothing I could do to control the frame. I didn't even know what a frame was. And I cannot rationally describe to you why in those early days I disliked—even hated—traditional sales techniques.
I just knew what I wanted—a method that requires no blunt-force trauma. No beg-or-bash modalities that aggravate people and make them regret doing business with you. I wanted nothing to do with the anxiety and fear that accompany those beta-trapped, pushy methods.
Beta methods do not exist in my approach for the simple reason that you are not pushing—you are _interacting_ with people using basic rules of social dynamics.
For years I've been doing this stuff around the country and the world, and part of what I've learned is that the crocodile brain is the same everywhere. There aren't New York croc brains and California editions and special French ones. Every croc brain responds the same:
• When something is boring: Ignore it.
• When something seems dangerous: Fight/run.
• When something is complicated: Radically summarize (causing information loss) and pass it on in severely truncated form.
With my approach, you are respecting the croc brain by introducing a game, and you are inviting others to play with you. It will feel new and different to everyone involved—because _it is_. Instead of flogging people with canned responses and pressure tactics, frame-based interactions excite the senses and engage people in a much more social way. In a world of robotic sameness, this approach will distinguish you from others.
I learned this alone, and it took me more than 10,000 hours of trial and error (and many patient and forgiving clients) to get it right. In the beginning, I seriously screwed up some important deals. I should have worked with a partner or a small group, but everyone I talked to about my method was afraid of it. Most thought it was chaotic and unpredictable because I didn't have the model worked out at that point. Today, the method isn't chaotic or unpredictable at all. Frames are now easy to control, and local star power can be created in every situation.
In the most basic sense, what are the frames I have been talking about here? Frames are psychological referencing systems that all people use to gain a perspective and relevance on issues. Frames influence judgment. Frames change the meaning of human behavior. If a friend rapidly closes and opens her eyes, we will respond differently depending on whether we think this is a physical frame (she blinked) or a social frame (she winked). Consider the words: _hit, bumped, collided_ , and _smashed_. These words tell you the severity of an automobile accident. _Frames shape the underlying meaning of every social interaction._
It is true, for instance, when we get together for a presentation, meeting, or pitch, that we can't just conduct a wholesale drop and transfer of information. You don't send a cargo container full of information to your customers or potential investors and say, "Here, look through this stuff. See what you can make out of it." They can't absorb it, and if they could, they don't have the time. This is a part of the presenter's problem: Deciding what to pitch and how is not like a math or engineering problem that can be worked out by having more and more information. It's about figuring out what parts of the information to use—which parts of your deal will trigger cold, analytical processing by the neocortex and which parts will engage the hot and vibrant processes of the crocodile brain.
This is exactly why frame control is so crucial. It serves to filter information and provide meaning, bridging the natural disconnect between the you and the target. It is always the case that frames can simplify complex issues by putting greater emphasis on one interpretation over others. In the process, frames construct a point of view.
And when you set the frame correctly, you control the agenda, which, of course, is important to do because every situation can be seen from many different angles. Frame control is about controlling which angle your deal is seen _from_. A frame helps to package a deal in a way that encourages certain interpretations and discourages others.
For instance, during the 1984 presidential campaign, there was considerable concern about Ronald Reagan's age. Speaking during the presidential debate with Walter Mondale, Reagan said, "I will not make age an issue of this campaign. I am not going to exploit, for political purposes, my opponent's youth and inexperience."
In one sense, this was a beautiful example of frame control in action. Reagan changed the underlying meaning of the social encounter and seized the alpha status, building a strong, unassailable point of view for the rest of the audience to follow and get behind. There is a second and equally valuable lesson from this example, and perhaps it is the critical learning of social dynamics: Humor, fun, and light-heartedness are crucial components of every pitch.
As I discussed in Chapter 1, in recent years I finally _got_ the fundamental problem you and I have when we pitch something. We have our highly evolved neocortex, which is full of details and abstract concepts, trying to persuade the croc brain, which is afraid of almost everything and needs very simple, clear, direct, and non-threatening ideas, to decide in our favor. This realization guided me into the world of frames and status.
Since the beginning of this book, I have offered you two principal insights into social dynamics. The first is structural—you have to package ideas for the croc brain in such a way that you are generating _hot cognitions_. In other words, you avoid the kind of cold analysis that is done by the neocortex. Instead, you use visual and emotional stimuli to push your target's primal hot buttons—to create _wanting_.
The second insight is procedural: You always have to be on the watch for opposing power frames and then win the ensuing frame collisions with better, stronger frames. And then you must further your frame control by perpetrating small denials and showing defiance.
But now, there's a third element that I believe is fundamental to successful frame control, to seizing alpha status, and to social dynamics in general: _humor and having fun._
Certainly, the purpose of perpetrating denials and defiance is to reframe social situations and to prize—that perhaps you aren't really selling them but that _they need to sell you_ , that your own time is even _more valuable than theirs_ , that under the circumstances, if they try to beta-trap you, you're _willing to withdraw_ , that you aren't automatically willing to take the beta position you're being led into. _But in all this you must use humor as well._
Importantly, the humor is not there to _relieve_ tension. Instead, it's there to signal that although the tension is real, you are so confident that you can play around a little. Perhaps it's best to think about it this way: People who have lots of options are not uptight, and they don't take themselves too seriously.
It also signals that framing is a game, and it invites others to join in. If you take the power frame from a "master of the universe" and he or she takes it back from you, is that not a challenge to improve your game? If you talk to _frame masters_ , they'll tell you that the secret of success is to create tension in a fun way that invites people to join in the frame game.
I mention this because most buyers/customers/investors will try to use the power frame on you. You'll see it frequently. Don't worry. It's a clumsy frame that is easy to disrupt using power-busting, intrigue, prizing, and time frames.
As easy as the power frame is for _you_ to break, absorb, and control, many others can't, so buyers will be shell-shocked. Take it easy on them, and don't take advantage. In the buyer's experience, most salespeople submit to their every whim and command: _Meet me at such-and-such inconvenient place; wait for me; start now; no, wait, stop; do this; send me more info;_ and so on. Once they come across someone like you who doesn't submit to these whims, they take notice, thinking, _This person is interesting. He isn't falling over himself to impress me like all the others. What's going on here?_
It's one thing to acknowledge the power of frames and seizing status but quite another to put the method to use. Becoming a frame master isn't easy. It takes thought, effort, and will, but the rewards are substantial. The good news—this is a journey that is fun from the very start, and if you are doing it right, it remains fun. In fact, if at any point you find that you are not having fun, something has gone wrong. Get with a colleague or friend who also knows this stuff and backtrack. See where the train hopped the tracks. I've had to do this many times. It's humbling, but what's the alternative? Go back to common industry practices like interview the customer and trial closing?
There is another benefit to becoming a frame master that isn't immediately apparent but will make a remarkable difference in your life. Over time, you will begin to notice an increase in the velocity of your work and leisure activities. This is so because strong frames allow you to selectively ignore things that do not move you forward toward your goals, and such a recognition amplifies your focus on the things that do.
In a natural way, framing keeps you focused on what is most important—human relationships—and prevents you from becoming distracted or burdened by unimportant matters when you are in social situations. Weak frames and nonessential details bounce off strong frames. Your abilities to discern, judge, decide, and act will vastly improve because the frame guides you.
Throughout this book I have talked about the structure of frames and the methods of seizing status. But mastery comes from actually doing it. This book will serve as your guide for a while, but as soon as you can, leave it behind. Your skills will come from practice, not from being a desk jockey, reading, or browsing the Internet for more learning. And I'm going to insist that you learn this stuff with a colleague or with a group because, as I said before, _learning to manage social dynamics is not an intuitive undertaking._
Fortunately, frame control method comes naturally for most people who can follow the blueprint here and have a good sense of humor and a positive outlook on life. If this describes you, you should have no difficulty getting started.
### **Getting Started**
When somebody wants to learn frames, social dynamics, and the overall method from me, I always begin with this warning: Frame-based social dynamics is strong medicine. Instead of reciting the same old business clichés that your audience expects to hear, you are jacking into the wetware that controls their brains, their primordial programming. You are simultaneously communicating with them on the surface and below the surface of their consciousness. If you do it wrong, for example, without humor, poise, and grace, I guarantee that security will be called, and you will be escorted from the building. I don't want to receive an angry e-mail from you telling me that you've just been fired, so please pay attention to the advice I'm about to give you.
Here are the progressive steps to learning the method:
_Step 1:_ Learn to recognize beta traps and how to step around them. This is a low-risk way to train your mind to begin thinking in a frame-based way. As you go about the business of life, look for the beta traps. Identify anything that is designed to control your behavior, and think of how you would step around it. The key at this stage is to get good at seeing the traps (they are everywhere).
While there is no immediate harm in doing nothing, when you are told to wait in the lobby until called, it's a test. Remind yourself that if you step into this beta trap, the next one will be even larger and more difficult to overcome.
_Step 2:_ In a gradual way, start stepping around beta traps. It will feel uncomfortable at first, of course, but it will push you forward to the place where it becomes natural and hardly noticeable to you. Work with a partner to practice beta-trap avoidance.
As I said at the start of this book, this method is powered by its simplicity. I've been practicing it for over 10 years, and I've survived and prospered using only four basic frames and the ability to avoid beta traps. So don't overcomplicate this or worry over your lack of technique. It will come naturally to you. Just be sure to have fun at it—that's the secret to success.
_Step 3:_ Identify and label social frames. Notice the frames that are coming at you on every level of your life. Power frames, time frames, and analyst frames are everywhere, and they crash into you on a daily basis. Develop your ability to see them coming, describe them, and discuss them with your partner. Become very good at identifying frames using the unique language of framing.
_Step 4:_ Begin to initiate frame collisions with safe targets—those who pose no major career risk to you. What I'm saying is, tomorrow, don't stride into the CEO's office, grab a sandwich out of his hand, and put your feet up on his desk, telling him that it's time you and he had a talk about your bonus.
Working with a partner, begin to overtake opposing frames in a fun, light-hearted way. I'm repeating this because it's so critical: Remember that humor and a soft touch are absolutely necessary. Without it, you will appear rude and arrogant and will trigger croc brain defense responses instead of engaging your target in a fun and spirited social exchange.
_Step 5:_ The small acts of defiance and denial you use to take control of a social frame create a certain amount of conflict and tension. This is the point. Push. Pull. Delivering these acts with a soft touch reassures the target's croc brain that everything is okay—that there is no clear and present danger. If you are having difficulty at this stage, it is because you are triggering defensive responses, which means you are coming on too strong. If this is the case, _pause_. Do not press forward if you are struggling because that means that something is wrong. Find another partner to do this with, choose a different social environment, practice in another venue, or just punch "Reset" and start over.
_Step 6:_ Frame control cannot be forced because this takes the fun out of it. This is not theater for someone else to enjoy. It's not a dog and pony show. It's a game for your own personal enjoyment—and for a moment, consider why we play games—to enjoy ourselves in a challenging but fair way where we can rack up a win.
If you find yourself forcing the method, fortunately, this is an easy problem to fix. Simply lighten up a little. When you say something that causes a frame collision, do it with a twinkle in your eye and a smile in your heart. Your target will feel your good will and good humor and respond in a positive way.
Above all, remember that this is not a conventional sales technique. You need not be a back-slapping, guffawing blow-hard to win business from your customers. There is no pressure here, no brute force, and no anxiety. Instead, this is a fun game that you bring to every target with whom you meet. Simply enjoy every moment, and others will enjoy it with you. It's nice to know that your continued happiness is what will make you successful. What could be easier?
_Step 7:_ Work with other frame masters. Now that you have developed a basic level of skill, seek out others who are better than you. As with any other artistic or athletic endeavor, apprenticeship leads to mastery faster than going it alone. Continue to work with others. Like a 10th dan black belt, you never stop refining your technique and honing your mastery. Keep it simple, stick to a few frames that work for you, and avoid complication. In the PITCH method, less really is more. As you advance, teach others.
When you become a frame master—and even on your journey to becoming one—you will have the most fun you've ever had. I find myself cracking up sometimes in the middle of a pitch, even when the business I'm doing involves millions of dollars. Why not? _This is a game where you set the rules and then change the rules as needed to maintain your continuous advantage without ever upsetting your opponent._ Imagine that.
The only rule is that you make the rules that the others follow. Because you set the agenda and control the frame, this is a game you can never lose. How could that not be fun?
Most of the difficulties I've encountered in learning this method come from the fact that there was no common language to discuss these methods with people. Instead of saying, "Watch out! Here comes a power frame. We need to use strong moral authority and a power-busting frame to win this frame collision," I had to use long explanations, and in so doing, the moment and the opportunity were lost. This is why it's important to learn and use the _lingua franca_ of frame control. Future conversations with your partner or group should sound like this:
"These guys set beta traps from the lobby all the way to the conference room. You have to time frame them immediately and withdraw. After that, they just hit you with power frames. Just break it with a prize frame. And then frame stack a few push/pull patterns."
Or, "Here comes the analyst frame. Let's hard-core intrigue frame, seize local star power, and withdraw."
_Pitch Anything_ gives you a common vocabulary that will deepen your mastery of the method and ingrain frame-based thinking into your DNA. Here are the most important terms for you to know and to own personally:
Frame control
Power-busting frame
Frame collisions
Prizing
Beta traps
Seizing status
Local star power
Push/pull
Alpha
Hot cognition
Crocodile brain
Neocortex
These terms label the social phenomena that are invisible to others and that used to be invisible to you.
As you advance in your life and career, the challenges you will face will increase in proportion to the responsibilities you bear. Being a frame master will make those burdens seem lighter, will help others to see you as a sage and trusted leader, and will maintain your social value at a much higher level, even when you are not consciously practicing frame control.
By helping others to view situations and opportunities through your frame, it will make interacting with you seem effortless. It's easy for others to spend time with a person they agree with, in any setting, and that's how they will feel about you.
So go forth, learn frame control, practice it at every opportunity, and have fun doing so. I wish you much success, and I hope that it serves you as well as it continues to serve me. I invite you to learn more about frame control at pitchanything.net.
## **Index**
_Please note that index links point to page beginnings from the print edition. Locations are approximate in e-readers, and you may need to page down one or more times after clicking or tapping a link to get to the indexed material._
Abstract concepts, ,
Adrenaline,
Adrenaline rush,
Adventure,
"Aha" moment, ,
Airport deal. _See_ Case study (airport deal)
_Alchemy of Finance, The_ (Soros),
Alpha, ,
Always be closing (ABC),
Always be leaving (ABL),
Amygdala,
Analog human narratives,
Analyst frame, , , ,
Analyst frame disruption, –57, –61
Analytical tangent,
Anger,
Anxiety,
Apprenticeship,
Attention, –50,
Attention getting, –116
Austin, Steve _(Six Million Dollar Man),_
Avocado farmer's money, –48
Backstory,
Badgering,
"Being nice,"
Bell, Gordon,
Belzberg, Bill, –20
Berns, Greg,
Best idea frame,
Beta trap, –84, ,
Beta-trap avoidance,
Bifurcation,
Big idea introduction pattern,
Body language,
Boeing,
Brain
abstract concepts,
amygdala,
basic truths,
cognitive, as miser, ,
croc. _See_ Crocodile (croc) brain
decision making,
development of,
fundamental organizing principle,
how it functions, , –14,
memories,
movement, ,
narrative and analytical information,
narratives,
neocortex, , , , ,
parts,
pleasurable challenges,
probabilities,
scarcity bias,
threat-avoidance system,
threat responses,
"Brain is like a computer" metaphor,
"Brain Scanners Can _See_ Your Decisions Before You Make Them,"
Browbeat,
Bruner, Jerome, ,
Budget, ,
Building rapport,
Cardiac surgeon and golf pro, –85
Case study (airport deal), –206
American Legacy theme,
best idea frame,
beta trap,
big idea introduction pattern,
competition (Goldhammer), –179
competitor's pitch, –205
eradicating neediness,
hour of judgment, –206
how to pitch, –188
Klaff accepts to come on board, –178
length of the pitch, –196,
prepitch thoughts,
presentation, phases,
push/pull, –197
strategy sessions and research, –181
three-market-forces pattern, –193
Change blindness, –103
Chase Manhattan,
Checking out, ,
Chemical of alertness,
Chocolate vs. spinach,
Coffee shops,
Cognitive functioning,
Cold, reasoned analysis, , ,
Cold cognitions, ,
_Comedian_ (Seinfeld),
Competition,
Competitive advantage (secret sauce), –127
Complexity,
Conference room,
Conflict, ,
Convention floor,
Conviction,
Cop frame, –26
Crick, Francis,
Crocodile (croc) brain, , . _See also_ Brain
basic rules,
characteristics, –16
choice,
detailed explanation,
filtering system, –12
rubber mallet,
self-protection, –161
survival, ,
Curiosity,
Cymbalista, Flavia, ,
Death match,
Defaulted debt, –136
Deframing,
Denial and defiance, –36, ,
Disappointment,
Disconnect, , –11
Distractions,
Doctor frame, –150
Dopamine, , –116,
Double-helix DNA structure,
Draper, Don _(Mad Men),_
Due-diligence analyst,
Economic forces, –100
Emotional manipulation,
Emotions,
Empathy,
Enjoy your work,
Enterprise Partners, –169
Eradicating neediness, –170
basic formula,
beta trap,
causes of neediness, –165
disappointment,
example (Enterprise Partners), –169
rules,
self-protection, –161
_Tao of Steve,_ –167
threat,
time frame,
weakness,
Example. _See_ Case study (airport deal)
Eye contact,
Fear alarms,
Fisher, Ron,
Focus,
Four-frame hot cognition stack, –137
Frame, , , , –209
Frame collision,
Frame control, –68,
analyst frame, , , ,
denial and defiance, –36
example (avocado farmer's money), –48
example (cop frame), –26
example (money center bank), –33
frame collision,
frame disruption, ,
game, as a,
guiding principles, –25
intrigue frame, –61
intrigue story, –59,
plowing,
possible opening phrase,
power frame, –36
prize frame (prizing), –40, –68
rational arguments,
social dynamics,
time frame, –51
what is it,
who owns the frame,
Frame disruption, ,
Frame master,
Frame stacking and hot cognitions, –156
cold cognitions,
decision making, –131, ,
example (defaulted debt), –136
example (Helen Woodward
Animal Shelter), –144
example (Mother Teresa), –152
four-frame hot cognition stack, –137
hot cognitions, , ,
intrigue frame, –143
moral authority frame, –152
prize frame, –146
time frame, –148
wanting, ,
Framing,
French waiter, –76
Fundamental human behaviors,
Geyser Holdings,
Global status, –93
Goldhammer, –206
Goldman Sachs,
Golf pro and cardiac surgeon, –85
Gordon, Evian,
Greenberg, Sam, , –183
Guiding principles, –216
Haynes, John-Dylan,
Heart of the pitch,
Hedge fund manager, –90
Heightened states of emotion,
High-intensity push/pull pattern,
_High Tech Ventures_ (Bell),
Hookpoint, , ,
Hot cognitions, , , , . _See also_ Frame stacking and hot cognitions
Human behaviors,
Humor, , , , ,
Idea introduction pattern, –108
Idle social banter,
Impression of someone,
Information gap,
Intrigue frame, –61, –143
Intrigue ping,
Intrigue story, –59,
Investors,
_Jaws,_
Jaywalking, ,
Khosla, Vinod,
Kincaid, John,
King-of-the-world moment, –170
Law of large numbers,
Lao-Tsu,
Light-heartedness,
Likeability, –137
_Lingua franca,_
Local star power, , , ,
Lord, Jack,
Low-key, low-intensity push/pull pattern,
_Mad Men,_
Majors, Lee,
Man-in-the-jungle formula,
Martini making,
McGarrett, Steve _(Hawaii Five-O),_
McGhan, Donald, –44,
McGhan, Jim, –48
McQueen, Steve, –167
MediCor,
Medium-intensity push/pull pattern,
Midbrain,
Momentum,
Money,
Money center bank, –33
Moore, Geoff,
Moral authority frame, , , –152
Mother Teresa, –152
Movement, –104
Mystery stories,
Narrative pattern (intrigue frame), –143
Narrative thinking mode,
Neediness. _See_ Eradicating neediness
Neocortex, , , , ,
Neurofinance,
Neuroscience,
Neurotransmitters, ,
Nonreactive state,
Norepinephrine, , ,
Novelty, –116
Obama, Barack,
Old-fashioned sales techniques, –153
Paradigmatic mode,
Pecking order,
Persistence,
Personal story, –56
Perspective,
Pitch techniques, –128
attention getting, –116
backstory of the idea,
budget, ,
change blindness, –103
competition,
heart of the pitch,
idea introduction pattern, –108
movement, –104
novelty, –116
overview,
phase 1 (introduce yourself/big idea), –109
phase 2 (explain the budget/secret sauce), –127
phase 3 (offer the deal), –128
phase 4 (stack frames and hot cognitions). _See_ Stack frames and hot cognitions
push/pull, –120
secret sauce, –127
tension, –122
three-market-forces pattern, –101
time-constraint pattern,
time constraints, , , –127
"Why now?" frame, –105
Playfulness,
Plowing,
Point of view,
Porterville incident, –59
Power-busting frame, ,
Power frame, –36,
Power frame collision,
Power frame disrupter,
Practice,
Prepitch thoughts,
Presenter's problem,
Presidential campaign (1984),
Primal lever,
Prize frame (prizing), –40, –68, –146
Public spaces,
Push/pull, –120
Qualifying question,
Quantum Fund,
Rational arguments,
Rational decision making, , ,
Reagan, Ronald,
Real life example. _See_ Case study (airport deal)
Reducing the autonomy of choice,
Rules of engagement, –16
_Satisfaction_ (Berns),
Scarcity bias,
Scheiner, David,
Secret sauce, –127
Seinfeld, Jerry, –127
Seized status,
Self-protection, –161
"Selling,"
Selling hard,
Simplicity, ,
Situational status, , ,
_Six Million Dollar Man,_
Smucker, Craig,
Social barriers, –82
Social dynamics, , ,
Social encounters,
Social forces,
Social frame,
Social position. _See_ Status
Social threat, –108
Soft touch, ,
Soros, George, , ,
Southwest Exchange,
Spielberg, Steven,
Spinach vs. chocolate,
Stacking frames. _See_ Frame stacking and hot cognitions
Starting up, –216
Status, –94
alpha, ,
"being nice,"
beta trap, –84
charisma/ego, contrasted,
example (French waiter), –76
example (golf pro and cardiac surgeon), –85
example (hedge fund manager), –90
example (Walmart), –84
fluidity, –86
global, –93
guiding principles,
how it works,
jaywalking, ,
local star power,
situational, , ,
your turf vs. target's domain, –87
_Straight from the Gut_ (Welch),
Surprise, ,
_Tao of Steve, The,_ –167
Taoism,
Technical details,
Technical material,
Technology forces,
Tension, , –122
Terminology,
Theory of mind,
Threat, ,
Three-market-forces pattern, –101, –193
Ticking time bomb,
Time,
Time-constraint pattern,
Time constraints, , , –127
Time frame, –51, –148
Time frame collision,
Time pressure,
Timeliness,
Trade shows,
Trendcasting,
_U.S. Air Force Training Manual,_
Validation-seeking behavior. _See_ Eradicating neediness
Vocabulary,
Walmart, –84
Wanting, ,
Wasting time,
Watson, James,
Weakness,
Welch, Jack,
"Why now?" frame, –105
Wolfe, Tom,
Zajonc, Robert, ,
## **About the Author**
**Oren Klaff** is director of Capital Markets for the investment bank Intersection Capital, where he raises tens of millions of dollars from investors and institutions. Intersection Capital has grown to $250 million of assets under management by using Klaff's pioneering approaches to raising capital and incorporating neuro-science into its capital markets programs. Klaff is a specialist in financial modeling and the codeveloper of Velocity, a capital markets product that has raised over $100 million of private equity and venture capital. He lives in Beverly Hills, California.
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Everhard von Diest († 5. April 1301) war von 1275 bis 1301 der 31. Bischof von Münster. In seine Zeit fällt einerseits der Ausbau der Landesherrschaft, andererseits wurde die Stadt Münster zu einem eigenständigen politischen Faktor.
Frühe Jahre
Er entstammte der brabantischen Familie Diest. Verwandt war diese unter anderem mit der jüngeren Linie der Familie von Cuyk, dem Grafen von Rietberg aber auch mit Erzbischof Siegfried von Westerburg. Everhard war Sohn von Gerhard von Diest, der im Stedingerkrieg gefallen war. Seine Mutter ist nicht mit Namen bekannt. Große Bedeutung für seinen Aufstieg spielte sein Onkel. Johann von Diest war Minorit und stieg später zum Bischof von Samland und dann Lübeck auf. Er unterstützte den Papst und stand König Wilhelm von Holland nahe. Auch Everhard war Gefolgsmann des Papstes und König Wilhelms.
Urkundlich wird Everhard erstmals 1248 erwähnt. Damals war er als Propst des Klosters Hünfeld am königlichen Hof in Aachen. Er diente dem König lange auch als Notar. Im Jahr 1251 trat er zusammen mit dem späteren Papst Urban IV. als Zeuge in einer königlichen Urkunde auf. Im Jahr 1252 bezeichnete er sich selber in einer Königsurkunde als Propst von Fulda. Er begleitete Wilhelm, als dieser im selben Jahr zu Braunschweig als deutscher König anerkannt wurde. Everhard hatte auch die Propstei von St. Gereon in Köln inne. Im Jahr 1253 wurde er bereits vom Papst für eine Bischofsposition vorgesehen.
Schwierige Anfänge
Aber erst 1272 bot sich mit dem Tod des münsteraner Bischofs Gerhard von der Mark eine Gelegenheit für Everhard. Die Wahl erwies sich indes als schwierig. Es gab mit Wigbold von Holte einen ernsthaften Mitbewerber. Nachdem dieser im Domkapitel keine Stimmenmehrheit erzielen konnte, kam es im Bistum Münster zu Unruhen. Der Hintergrund ist nicht ganz klar. Möglicherweise beanspruchten Bürger und Ministeriale ebenfalls das Recht, an der Wahl teilzunehmen. Nur mit Unterstützung der Grafen von der Mark, der Edelherren zur Lippe sowie zahlreicher anderer hoher weltlicher und geistlicher Herren gelang es, Graf Otto von Tecklenburg als Stiftsverweser einzusetzen, bis die Bischofswahl vollzogen war. Erst als König Rudolf von Habsburg den Papst um eine Entscheidung bat, konnte Everhard gewählt werden. Wigbold von Holte wurde später Erzbischof von Köln.
Im Jahr 1275, nach einer mehrjährigen Vakanz, übernahm Everhard die Regentschaft im Bistum und Hochstift Münster.
Förderung des geistlichen Lebens
In Hinblick auf das geistige Leben versuchte Everhard, gegen Missstände vorzugehen. Dazu zählten 1282 die Verkündung von Synodalbeschlüssen gegen Konkubinate von Klerikern, ihr Handeltreiben, Simonie und der Mangel an Bildung. Außerdem bemühte sich Everhard, die päpstlichen Liturgiereformen umzusetzen.
Das Verhältnis des Bischofs zum Domkapitel war anfangs gut. Everhard versuchte auch, die wirtschaftliche Lage des Domstifts zu verbessern. Er ließ verpfändete Güter zurückkaufen und hat andere Besitzungen, die nicht genug einbrachten, verkauft. Weil der Bischof zu eigenmächtig agierte und das Domstift nicht genügend beteiligte, kam es in diesem zu einer Oppositionsbewegung. Im Jahr 1301 bildete sich ein Bündnis der Kapitulare gegen Everhard.
Besonders gefördert hat er im monastischen Bereich die Minoriten. Er bemühte sich aber auch, die weltlichen Vogteien über Klöster und Stift zu beseitigen. In seine Amtszeit fällt mit dem Nordenhospital bei Hamm nur eine Klostergründung.
Everhard ließ zahlreiche Ablässe ausstellen, insbesondere mit dem Ziel, den Bau von Kirchen zu fördern. In seiner Zeit tauchten die Pfarrkirchen im Bistum erstmals als eigene Rechtspersonen auf.
Sein Verhältnis zur römischen Kurie war nicht besonders eng. Dem Aufruf von Papst Gregor X. 1274 zu einem Kreuzzug etwa folgte die Diözese nicht.
Innere Politik
Im politischen Bereich erbte er die Verstrickung in die friesischen Wirren. Im Jahr 1276 gelang es ihm, zu einem Ausgleich mit den Friesen zu kommen und entsprechende Verträge abzuschließen. Es kam 1283 sogar zu einer Visitationsreise durch das zu Münster gehörende Friesland. Dabei zeigte sich, dass es in weiten Teilen des Landes kaum noch eine Landesherrschaft gab. Dies galt neben Friesland auch für das Niederstift Münster.
Auch in der Stadt Münster gestalteten sich die Verhältnisse schwierig. Die Bürger wollten mehr Unabhängigkeit von der bischöflichen Herrschaft und kritisierten vor allem den befestigten Bispinghof in der Stadt. Die folgenden Ereignisse sind nicht ganz klar. Möglicherweise wollte der Bischof mit Gewalt gegen Münster vorgehen, scheiterte aber an der fehlenden Unterstützung des Domkapitels. Klar ist, dass die Bürger, verbündet seit 1277 mit Soest, Osnabrück und Dortmund, stärker waren. Die Folge war, dass Everhard das halbe Stadtgericht und die Akzise an die Bürger verlor. Seit 1278 waren die Stadtrichter nicht mehr Ministeriale des Bischofs, sondern städtische Schöffen. Die Bürger erhielten auch die Aufsicht über die Befestigungsanlagen des Bispinghofes. Der Bischof verzichtete auch für die Zukunft auf jede gewaltsame Maßnahme gegen die Stadt. Damit verlor sein direkter Herrschaftsanspruch über Münster fast völlig an Substanz. Die Stadt stieg in seiner Zeit zu einer gleichberechtigten Macht auf. Durch ihre Teilnahme an den Münzproben drang sie sogar in den Kernbereich der landesherrlichen Rechte vor. Im selben Jahr nahm die Stadt Münster erstmals an einem Landtag teil. Damit begann sich die Struktur der Landstände der folgenden Zeit herauszubilden.
Auch anderen Städten und Siedlungen musste er entgegenkommen. Dazu zählten das Privileg von Beckum aus dem Jahr 1278 oder die Sühnebriefe für Borken und Coesfeld. Ahlen verlieh er die Zollfreiheit und Haltern bekam Stadtrecht.
Aus unbekanntem Anlass kam es 1287 in der Amtszeit von Everhard zur ersten Judenverfolgung in Münster. Dabei wurden mehr als 90 Juden getötet.
Wegen seines Machtverlustes in der Stadt Münster gab Everhard den alten Bischofspalast in der Nähe des Domes auf. Er residierte meist auf der Burg Wolbeck. Everhard begann damit, neben den Landständen Beamte einzusetzen, um die Landesregierung zu stärken.
Außenpolitik
Everhard war zu Beginn seiner Herrschaft mit den Kölner Erzbischöfen verbündet und ging zusammen mit diesen gegen nach Unabhängigkeit strebende Vasallen vor. Mit Köln schloss er auch ein Bündnis gegen die Herren zur Lippe. Aus nicht klaren Gründen kam es zu einer Fehde mit den Grafen von Limburg, die 1282 mit einem Vergleich endete. Zu einem Bündnis kam es auch mit dem Bischof von Osnabrück, das später um weitere Mitglieder erweitert wurde. Dieses richtete sich insbesondere gegen den Grafen von Tecklenburg. Dieser soll 1299 versucht haben, Everhard möglicherweise mit Unterstützung der Bürger von Münster gefangen zu nehmen. Everhard reagierte mit Zerstörungen in der Grafschaft Tecklenburg. Dies nutzte der Graf von der Mark für einen Einfall in das münstersche Gebiet. Erst die Vermittlung der Bischöfe von Osnabrück und Paderborn beendete den Konflikt.
Trotz seiner zeitweiligen Bündnisse mit dem Erzbischof von Köln stand er dem Versuch von Siegfried von Westerburg, seine Macht in Westfalen auszudehnen, nicht positiv gegenüber. Wohl sein verwandtschaftliches Verhältnis trug aber dazu bei, dass sich Everhard nicht dem Bündnis anderer westfälischer Großer anschloss, das mit der Niederlage des Erzbischofs in der Schlacht von Worringen 1288 endete.
Eine Rolle außerhalb seines Bistums spielte Everhard kaum. Insgesamt gelang es Everhard, seine Herrschaft im zentralen Münsterland (Oberstift) auszubauen. Misslungen war dies in Friesland, und in der Stadt Münster hatte er stark an Boden verloren. In seiner Zeit zeigte sich, dass der Bischof und das Domkapitel nicht mehr alleine herrschten, sondern auch Ministeriale und das Stadtbürgertum in Münster zu Herrschaftsfaktoren wurden.
Bestattet wurde er im Dom zu Münster.
Literatur
Wilhelm Kohl: Die Bistümer der Kirchenprovinz Köln. Das Bistum Münster 7,3: Die Diözese (= Germania Sacra NF. Bd. 37,3). de Gruyter, Berlin 2003, ISBN 3-11-017592-4, S. 340 ff. ().
Wilhelm Kohl: Die Bistümer der Kirchenprovinz Köln. Das Bistum Münster 7,1: Die Diözese (= Germania Sacra NF Bd. 37,1). de Gruyter, Berlin 1999, , S. 133 ff. ().
Weblinks
Everhard von Diest im Personenregister der Germania Sacra online
Römisch-katholischer Bischof (13. Jahrhundert)
Römisch-katholischer Bischof (14. Jahrhundert)
Bischof von Münster
Geboren im 13. Jahrhundert
Gestorben 1301
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,202 |
Louis Guy Marie Jean de Lubersac (1878-1932) was a French aristocrat, landowner and politician. He served as a member of French Senate from 1920 to 1932, representing Aisne.
Early life
Guy de Lubersac was born on 20 January 1878 in Paris, France. He studied the Law and graduated from Sciences Po.
Career
De Lubersac inherited the marquisate of Lubersac, including land in Aisne. He became a member of the Jockey-Club de Paris. He was elected as the mayor of Faverolles in 1904. He served in the French Army during World War I.
De Lubersac served as a member of French Senate from 1920 to 1932, representing Aisne.
Death
De Lubersac died on 15 April 1932 in Paris, France.
References
1878 births
1932 deaths
Politicians from Paris
Sciences Po alumni
French military personnel of World War I
French Senators of the Third Republic
Marquesses
Senators of Aisne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,593 |
\section{INTRODUCTION}
Driven diffusive systems have been of interest to a wide community of researchers since the first models were introduced \cite{Katz1,Katz2}. Mostly motivated by their ability to demonstrate the curious phenomena of non-equilibrium systems despite their theoretical simplicity, various different systems of such are proposed \cite{solversguide,rep1,rep2,rep3,rep4,rep5,schutz2part}. Asymmetric simple exclusion processes (ASEP) being one of the simplest of those, with particles interacting exclusively, hopping to both directions or in and out of the system with certain probability rates on a one dimensional lattice, can demonstrate the interesting theoretical phenomena such as phase separation \cite{Krug}, spontaneous symmetry breaking \cite{spontaneous}, phase coexistence \cite{spontaneous}, and shock formation \cite{shocks}. Furthermore, they can be utilised in modelling various real life problems such as transport of inter-cellular motor proteins \cite{solversguide}, traffic jams \cite{rep4}, surface growth \cite{surfacegrowth}.
One dimensional asymmetrical simple exclusion process (ASEP) systems with open boundaries are bounded to particle baths of constant densities at both ends, which can be modelled with constant boundary crossing rates for particles which enter or leave the system. Although considerable amount of work exists on the time-independent steady state properties, there are only a few examples which study the effect of applying time-dependent or oscillatory boundary rates to these systems.
For instance Popkov {\em et al.} apply an on and off boundary condition to the single-species, semi-infinite ASEP, such that the oscillating exit probability rates can be thought as red and green traffic lights \cite{schutz}. A significance of their result which is relevant to our work is the observation that the density fluctuations propagate with a typical velocity into the lattice from the boundary. The fluctuation although weakening as it propagates, preserves its characteristic shape within the bulk. They observe the density response of the system has a sawtooth-like
characteristics with periodic pileups related to the red-green light periods of the system independent of the initial conditions. They also showed same behaviour exists in the hydrodynamic limit. In another work, Basu {\em et al.}, applied a sinusoidal drive to the boundaries of single-species simple exclusion process (SEP) and ASEP models, in which particles are allowed to move to the in both directions in symmetric and asymmetric rates correspondingly. They performed Fourier analysis of the response of both systems. They found that the structure functions have bimodality, which they claim, indicate the modes of transportation in diffusive systems \cite{indians}.
In this present work, we carry out a Monte Carlo study of a two-species totally asymmetric simple exclusion process (TASEP) such that the boundary conditions (BC) are abruptly oscillating with relatively small amplitudes around a phase transition point. We show that the system responds in qualitatively different forms, depending on the frequency of the perturbation. In particular, there appears to be a dynamical transition where the response of the system changes from that of a symmetric state to that of an near-symmetric one abruptly, as a function of the frequency of the perturbation. This transition is independent of the size of the perturbation.
The phases in the phase diagram of the time-independent model were first reported by Evans {\em et al.} \cite{evans}. They identified through a mean field analysis, four different phases of the order parameter density, for all symmetric parameters of the two-species. One of the phases surprisingly display broken symmetry. Between the symmetric and asymmetric phases they report a tiny regime in which particle densities are low but not symmetric. (We will label three of the phases of interest to us as $LL$ [symmetric low density-low density], $HL$[the broken symmetry high density-low density], and $TR$ [tiny regime].) We will give a precise definition of the model in the next section. For our general discussion at this point, we demonstrate how the joint density
function $p(n_1,n_2)$ behaves near $TR$ as a function of the boundary exit rates $\beta_1$ and $\beta_2$ for the two types of particles. (Arndt {\em et al.} discuss the structure of these phases in detail in \cite{Arndt}.)
\begin{figure}[h]
\includegraphics[width=8.6cm]{bitisik_contours_v2.eps}
\caption{\small (Color online) Joint probability densities $p(n_1, n_2)$ for number of particles for various paramaters under constant BC. For a lattice size of $N=200$, parameters for the plot $(a)$ are $\beta_1=\beta_2=0.285$, for the plot $(b)$ are $\beta_1=\beta_2=0.275$ and for the plot $(c)$ are $\beta_1=\beta_2=0.265$ and for the plot $(d)$ they are asymmetric as $\beta_1=0.265$ and $\beta_2=0.285$. Contours enclose approximately 75 and 25 percent of the total probability. For all of the plots all other transition rates are equal to $1$. \label{fig:combined}}
\end{figure}
$TR$ was shown to be a finite size effect by Erickson {\em et al.} \cite{pruessner}. Through a Monte Carlo analysis they showed that the size of this phase decays exponentially with respect to lattice size. Detailed analysis of the joint density distributions of two types of particles for this regime reveals that the density is a superposition of ``shock profiles" along the length of the system \cite{schutz, Mallick, Arndt}. Each profile, which corresponds to a particular number of type I particles in the system, has an error-function like shape, whose midpoint is carrying out a random walk across the lattice \cite{Mallick}. The random walk is constrained when the shock approaches a boundary, if it gets too near the particle entry (exit) boundary, the increase (decrease) in the density near the boundary has a compensating effect on the position of the shock, pushing it away from the boundary. The entry and exit rates as a function of the position of the shock may then be interpreted as a ``force" on the shocks, with a corresponding ``potential", in which the random walk is carried out\cite{Arndt}.
Fig.~\ref{fig:equi_shocks} shows these profiles corresponding to several values of occupation of the lattice at time-independent steady state. The plots show the average density of first type of particles as a function of position, when the lattice contains a total of $n_1$ such particles with $n_1>n_2$. This last constraint limits the averaging to one leg of the boomerang-shaped probability density.
\begin{figure}[h]
\includegraphics[width=8.6cm]{equi_lat_dist.eps}
\caption{\small (Color online) Shock profiles in a system of size $N=100$, for various occupation numbers $n_1$. The boundary rates are kept fixed at $\alpha_1=\alpha_2=1$ and $\beta=0.2675$. It can be observed that the profiles for $n_1<40$ and $n_1>70$ deviate in character from others due to boundary effect. \label{fig:equi_shocks}}
\end{figure}
The discussion above points out two different features for the motion of the shock profile: The first corresponds to diffusive, damped motion in an effective potential. The second is the mechanism of application of an effective force through the manipulation of the boundary conditions, which will have a retarded effect dependent on the position in the lattice. We have looked into possibility of the production of interesting effects through the interplay of these two features. We investigate whether it is possible to force the shocks in the system by simply oscillating the boundary conditions. We observe significant frequency dependence which is unusual for a diffusive system.
We have also observed hysteresis in the density function of the system. Such behaviour was observed earlier in similar systems by Rakos {\em et al.}\cite{Rakos}. Hysteresis in our model appears abruptly as perturbation frequency is decreased,
associated with a typical velocity in the system.
In the following sections we first introduce the model we are studying, and how we apply the oscillatory boundary conditions. We then move on to the discussion of the response of the system to the boundary conditions.
\section{TASEP under Periodically Driven BC}
The system which is studied in this paper is TASEP on a finite, one-dimensional lattice with open boundaries and two species. Particles of type 1 (2) are allowed to enter the system from the left (right) with probability rate $\alpha_{1(2)}$, move only forward with probability rate $\gamma_{1(2)}$ if the following site is empty, and leave the system from the opposite end with probability rate $\beta_{1(2)}$. Different types of particles are allowed to switch places with rate $\delta$ when they come face to face. In our simulations all probability rates except the exit rates were taken to be equal to $1$. (These unitless quantities define a unitless time scale for the problem.) In comparison to the on and off exit rates of Popkov {\em et al.} \cite{schutz}, relatively small oscillations of the exit rates were applied to the system. We let the exit rate to oscillate around the $TR$ phase boundary point $\beta_o=0.275$ with an amplitude $\Delta\beta$:
\begin{eqnarray}
\beta_1 &=& \beta_0-\Delta\beta s(t) \\
\beta_2 &=& \beta_0+\Delta\beta s(t) \nonumber
\end{eqnarray}
where $s(t) = \mbox{sgn} (\sin(2\pi t/\tau))$ for time $t$ within a period of oscillation $\tau$.
To study the system we use Kinetic Monte Carlo simulation \cite{kinetic}, with Poissonian time dynamics. We maintain a list of all possible events $e$ (possible particle jumps within the system and motion through
the boundaries) and rates $\omega_e$ associated with them. The total rate for any one of these events happening is then given by $\Omega = \sum_e \omega_e$. A random variable $\Delta t$ which corresponds to the time increment for the next event is then given by $\Delta t = -\log(r)/\Omega$ with $r$ a random number uniformly distributed between $0$ and $1$. If $\Delta t$ implies a time increase past the next BC change time $t_t$ given by eqn 1, no changes are made to the system and the time is set to $t_t$. Otherwise we select the
particular type of change $e$ that takes place at that time randomly, with the probability $\omega_e /\Omega$. The procedure described in this paragraph is then repeated, and statistical averages evaluated, weighing the influence of each state with $\Delta t$. To produce such random numbers we use the Mersenne Twister pseudo-random number generator \cite{Mersenne1, Mersenne2}.
In the simulations a Monte Carlo step (MCS) was taken to be $N \times N$ time increments. This definition of MCS may be associated with the maximum lattice transit time of the particles in the system. (A particle has to make $N$ jumps to transit the system. For a ``typical" distribution of particles, each jump will take $\mathcal{O}(1)$ time unit, and $\Delta t$ is $\mathcal{O}(1/N)$.) Note that we also have a continuous time variable $t$ associated with time increments $\Delta t$, but large MCS is used to ensure good statistics. In each simulation, averages are calculated over $10^5$ MCS. Period dependent averages are calculated by obtaining time dependent averages within each period and averaging over the periods.
\section{Variations in the character of Frequency Dependence}
We change the boundary conditions in a way to break the symmetry between the two types of particles. We choose $TR$ as the unperturbed state which is associated with the presence of shock fronts in the widest region. Note that at very high frequencies, one obtains the unperturbed state, while at very low frequencies, the system moves from one constant asymmetric BC state to the other. To discuss these varying responses we focus on the joint density distributions for some values of period of oscillations. Depending on the frequency of oscillation, we observe very different types of responses.\\
Fig.~\ref{fig:contours_bitisik} displays the change in the joint distribution function for a system of size $N=200$ and $\Delta\beta=0.1$, for various frequencies. Note that the boomerang shaped profile (similar to those in Fig.~\ref{fig:combined}) disappears and re-appears as a function of frequency. At high frequencies of oscillation (low values of $\tau$) the density distribution preserves the boomerang shaped nature (similar to those in Fig.~\ref{fig:combined}) but the distribution tends to move as a whole in response to the changing boundary condition. We use the terminology ``near-symmetric"
states in association with such density functions, which although not preserving perfect symmetry between the two types of particles, maintain a shape which is a perturbation of the symmetric
time-independent version. This shape itself varies as the oscillation frequency is changed, resembling the time independent density distributions for different values of the parameter $\beta$ in Fig.~\ref{fig:combined}. (All figures display results for system size $N=200$, except where $N$ dependence is stated.) However note that in order to observe a time-independent distribution similar to that for $\tau=300$ in Fig.~\ref{fig:contours_bitisik}, one would need to go deeper into the $LL$ phase than the range of parameters used in oscillating BC (See Fig.~\ref{fig:combined}.~a). This is an indication of the resonance-like behaviour in the system; driving the density fluctuations much higher than the values one can obtain from the static values in the same range. Although the joint density is confined to a very small range $\tau=300$, smaller and longer values of $\tau$ result in densities which are still boomerang shaped.
\begin{figure}[h!]
\includegraphics[width=8.9cm]{contours_bitisik_definite2_v3_clr.eps}
\caption{\small (Color online) Time dependence of the joint density distribution $\rho(n_1,n_2)$ corresponding to the significant points of the Fig.~\ref{fig:gezinti_harfli}. In each column, the density at the indicated time as well as at the next quarter cycle (dashed lines) are shown to display the change. \label{fig:contours_bitisik}}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=8.9cm]{gezinti_harfli_clr.eps}
\caption{\small Average density spread ($\bar{\Delta}$) graph with respect to different period values. Inset shows the average spread for higher values of period. Interesting points are labeled with letters. Periods corresponding to those letters are: $\tau=140$ for $A$, $\tau=190$ for $B$, $\tau=300$ for $C$, $\tau=610$ for $D$ and $\tau=2900$ for $E$. Density distribution for these points are give in Fig.~\ref{fig:contours_bitisik}. \label{fig:gezinti_harfli}}
\end{figure}
To quantify this behaviour we introduce a parameter, which we call ``spread'', defined as follows: We calculate the averages below at 100 time values $t_i = i \tau/100$ within each period $\tau$:
\begin{eqnarray}
{\langle n_1^m\rangle}_{t_i} = \sum_{n_1,n_2} n_1^m~p(n_1,n_2,t_i) \nonumber \\
{\langle n_2^m\rangle}_{t_i} = \sum_{n_1,n_2} n_2^m~p(n_1,n_2,t_i) \nonumber \\
\Delta_1^2(t_i) = {\langle n_1^2\rangle}_{t_i} -{{\langle n_1\rangle}^2_{t_i}} \nonumber \\
\Delta_2^2(t_i) = {\langle n_2^2\rangle}_{t_i} -{{\langle n_2\rangle}^2_{t_i}} \nonumber
\end{eqnarray}
Then average spread is:
\begin{equation}
\bar{\Delta} = \sqrt{\frac{\sum_i(\Delta_1^2(t_i)+\Delta_2^2(t_i))}{100}} \nonumber
\end{equation}
This is then an average of the fluctuation in the number density during a period of oscillation.
Fig.~\ref{fig:gezinti_harfli} is a plot of this parameter as a function of oscillation period and indicates that system is going through a resonance-like behaviour at various frequencies. Extrema on this plot are identified with letters $A-E$ and correspond to the distributions in Fig.~\ref{fig:contours_bitisik}. For instance for $\tau=140$ (point A on Fig.~\ref{fig:gezinti_harfli}), density is mainly distributed around the $LL$ region with some tails into symmetry broken states. When $\tau=190$ (point B) joint density closely resembles the equilibrium density. The minimum at $C$ corresponds to the very compact distribution mentioned above. On the other hand, for low frequencies, {\em e.g.} for when $\tau=2900$ (point E), the system is in the broken symmetry state at all times within the period. The appearance of large scale hysteresis is apparent in this case. We discuss below the abrupt appearance of this hysteresis effect.
For even lower frequencies, the system is driven deeper into the symmetry broken phase at each half cycle, resulting in an even smaller spread as the inset to Fig.~\ref{fig:gezinti_harfli} displays.
The effect of the amplitude $\Delta \beta$ of the perturbation on the spread parameter is shown in Fig.~\ref{fig:amplitude_gezinti}. The structure of the response is preserved, but the magnitude dependence is apparent. Smaller perturbation leads to smaller variation in spread at higher frequencies. However, the spread
diminishes less slowly at longer periods because it takes a longer time to push the system into the asymmetric phase with a smaller perturbation. Fig.~\ref{fig:coklu_gezinti} displays the effect of the system size on the
response. Existence of a ``typical velocity" in the system would lead to an expectation of scaling of all characteristic time constants by $N$. Fig.~\ref{fig:coklu_gezinti} indicates that this is indeed the case. However characteristic times (such as response extrema) are not simply related to one another, indicating that the size of the boundary regions (which should be excluded from $N$) may be different for mechanisms which are responsible for various extrema. \\
\begin{figure}[h!]
\includegraphics[width=8.9cm]{amplitude.eps}
\caption{\small (Color online) Average density spread ($\bar{\Delta}$) responses of the system for different magnitudes of perturbation, $\Delta \beta = 0.05$ and $\Delta \beta = 0.1$. Note that the $\tau$ values for the extrema of the response is independent of the size of the periodic drive. \label{fig:amplitude_gezinti}}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=8.9cm]{coklu_gezinti_clr.eps}
\caption{\small (Color online) Average density spread ($\bar{\Delta}$) with respect to oscillation period $\tau$ for various values of lattice sizes. Both axes are scaled by N. \label{fig:coklu_gezinti}}
\end{figure}
We have looked at the hysteresis in the average values ${\langle n_1 \rangle}_t$ vs ${\langle n_2\rangle}_t$ of the joint probability distribution function $p(n_1,n_2,t)$ in some detail. Fig.~\ref{fig:kelebekler} displays this effect for various values of the oscillation period.
We calculate the area of the hysteresis curve
\begin{equation}
A=\sum_{t_i} {\langle n_2\rangle}_{t_i} \Delta {\langle n_1\rangle}_{t_i} \nonumber \\
\end{equation}
where $ \Delta {\langle n_1\rangle}_{t_i} = {\langle n_1\rangle}_{t_i}-{\langle n_1\rangle}_{t_{i-1}}$. Here the summation is over one lobe of the ($8$-shaped) hysteresis loop. Fig.~\ref{fig:hist_area} displays the result. Although some amount of hysteresis (not visible at the scale of Fig.~\ref{fig:hist_area}) exists at all frequencies, we find that a large-scale hysteresis starts at $\tau \sim 5N$, independent of $\Delta\beta$ or $N$. This may be interpreted as the onset of large scale motion of the probability density associated with symmetry broken phase. We then identify the value $N/\tau\sim 0.2$ as a typical velocity in the system. Hysteresis is not present when changes to the system are faster than that implied by this characteristic velocity. The inset to Fig.~\ref{fig:hist_area} shows that there is some structure associated with the break away point of the hysteresis magnitude. We identify this point as a dynamical phase transition point as a function of frequency. It is interesting to note that the values for which $\tau/N < 5$ correspond to the range in Fig.~\ref{fig:coklu_gezinti} where $\bar{\Delta}$ displays richer structure.
\begin{figure}[h!]
\includegraphics[width=8.9cm]{hysteresis_clr.eps}
\caption{\small (Color online) The trajectory for ${\langle n_2 \rangle}_{t_i}$ vs ${\langle n_1 \rangle}_{t_i}$ for values of $t_i$ within a period. The trajectories collapse to their limiting forms for $\tau \leq 5$ and $\tau \geq 5000$. Above $\tau =5000$ hysteresis area saturates, all overlap with $\tau =5000$. \label{fig:kelebekler}}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=8.9cm]{hist_area_clr.eps}
\caption{\small (Color online) Hysteresis area for two different perturbation amplitudes, $\Delta \beta =0.05$ and $\Delta \beta =0.1$ and $N=200$. The inset shows the detail near the break-away point. \label{fig:hist_area}}
\end{figure}
\section{Pulse Response}
To better understand the nature of the frequency dependence of the system, we further study the ``pulse response'': We have applied a constant perturbation, only to the exit rate of the first type of particles, $\beta_1=0.535$, for a duration of $\Delta t=100$ over a period of $\tau=10000$ with a repetition for $10^7$ MCS. When time reaches the end of the period, the system is relaxed to a near time-independent steady state. We have thus obtained the time-dependent shock profiles and average occupation values, which again show surprising oscillatory behaviour.
Fig.~\ref{fig:prob_dist} shows the variation of the occupation number probability of first type of particles in one leg of the boomerang-shaped probability density:
\begin{equation}
P(n_1,t) = A \sum_{n_2=0}^{n_2=n_1-1} p(n_1,n_2,t)\\ \nonumber
\end{equation}
where $A$ is a normalization constant. Note that, probability is reduced at early times for smaller and larger values of $n_1$. Figures \ref{fig:kucukn} and \ref{fig:buyukn} show the shock
profiles at various times after the pulse. It can be observed that the profiles for small $n_1$ show less of a distortion compared to those for larger $n_1$. The profiles for larger $n_1$ are distorted due to the exit of particles during the pulse. The change in $P(n_1,t)$ for small times is then due to two different mechanisms: The small $n_1$ shocks simply leave the system during the pulse, while the large $n_1$ shocks are deformed into smaller $n_1$ forms. The recovery of the system from these two effects seem to be qualitatively different. The statistics for large $n_1$ shock recover exponentially with a dynamics consistent with a diffusive system. Recovery of small $n_1$ shock statistics seem to be a contribution of multiple effects, resulting in an oscillatory damping.
\begin{figure}[h]
\includegraphics[width=8.6cm]{prob_dist.eps}
\caption{\small (Color online) $P(n_1,t)$ for various values of $t$. $P(n_1,10000)$ is a near steady-state distribution. \label{fig:prob_dist}}
\end{figure}
\begin{figure}[h]
\includegraphics[width=8.6cm]{kucukn_lat_dist.eps}
\caption{\small (Color online) Shock densities corresponding to $n_1=30$ for various values of $t$.\label{fig:kucukn}}
\end{figure}
\begin{figure}[h]
\includegraphics[width=8.6cm]{buyukn_lat_dist.eps}
\caption{\small (Color online) Shock densities corresponding to $n_1=65$ for various values of $t$. At early times, the profile has a distortion which indicates loss of particles on the right hand (exit) side due to the pulse. The entry side resembles a steady-state profile for a larger $n_1$ in Fig.\ref{fig:equi_shocks}. The perturbation pulse acts to shift the densities for large $n_1$ shocks to lower $n_1$ values. \label{fig:buyukn}}
\end{figure}
The inset to the Fig.~\ref{fig:diff_fr} displays the relation of $n_1$ to its steady state values as a function of time:
\begin{equation}
\delta_1(t)=\sum_{n_1}(P(n_1,t)-P(n_1,\infty))^2.
\end{equation}
To separate the two mechanisms discussed above, Fig.~\ref{fig:diff_fr} shows the contributions to this summation for values of $n_1<N/2$ and $n_1>N/2$. Note that for both cases deviation from the steady state increases for a period of time even after the perturbation pulse has ended. However, smaller $n_1$ statistics relax to the steady state with shorter time scale oscillations suggesting that the process may be associated with boundary events rather than the bulk. The oscillatory nature of the relaxation to steady state is also apparent in Fig.~\ref{fig:diff_fr}. This unusual behaviour forms the basis of the different type of response we report for the sinusoidal drive.
One does not expect to find an oscillatory response in a diffusive system. The effect seems to be a superposition of a number of recovery processes with different time scales dominated by the statistics of states with smaller number of particles. The time scale of the oscillations is consistent with our report of $N/\tau \sim 0.2$ for the sinusoidal drive. More work may be necessary to identify the details of the mechanisms involved in this interesting phenomena.
\begin{figure}[h]
\includegraphics[width=8.6cm]{diff_fr_equi.eps}
\caption{\small (Color online) Relaxation of the deviation $\delta_1(t)$ for smaller and larger values of $n_1$ for $N=100$. Inset shows $\delta_1(t)$ for all values of $n_1$. \label{fig:diff_fr}}
\end{figure}
\section{CONCLUSION}
We report the response of the TASEP model as a function of the perturbation
frequency of the boundary condition. We find that the response is qualitatively different for various ranges of the perturbing frequency. One type of change involves significant
modifications in the shape of the joint distribution function, which alternates between compact
and extended forms. Variation of this behaviour as a function of frequency
contains considerable structure which does not depend on the size of the periodic drive, and scales with the size of the system. This implies that the response is associated with motion of features through the system, in the form of shock fronts. A second type of change that was observed is the abrupt appearance of the hysteresis as the frequency of the perturbation is lowered. This also indicates a velocity threshold under which the the density distribution cycles from one phase (associated with that particular value of the BC) to the other, with significant changes during the cycle. We identify a characteristic velocity value of $\sim 0.2$ lattice sites per unit time. Higher frequencies correspond to near-symmetric states where the probability distribution moves more or less rigidly during the cycle,
albeit with a probability density profile which changes appreciably as a function of frequency.
We have reported the response of the system at a phase transition point, which we thought would be most interesting. Analysis of other special points on the phase diagram could also shed light on the dynamical mechanisms of interest in this system.
Authors acknowledge support from Turkish Academy of Sciences (TUBA).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,791 |
The Jacob Regez Sr. House is a historic house built in 1901 by an important local cheesemaker in Monroe, Wisconsin, United States. It was added to the National Register of Historic Places on January 17, 1980.
Description and history
Jacob Regez was born in 1849 in the Canton of Bern, Switzerland. His father made cheese, and Jacob learned cheesemaking at a factory scale. In 1872 he emigrated and in 1875 set up a cheesemaking business in Monroe. Soon he had an interest in twenty local cheese factories and operated the large Legler Cheese Factory. Later he was involved in the Monroe Milk Condensing Company.
In 1884 Regez had a Queen Anne-styled house built on the site of the current house, but it was hit by lightning and burned in 1901. Within a week of the fire, Regez hired Fridolin Heer & Son, Swiss architects of Dubuque, to design a replacement. It was finished within six months.
The 1901 house sits on the same limestone foundation as the 1884 house, but the frame structure is a different Queen Anne style design, 2.5 stories with bay windows, broad eaves, pedimented dormers, and a complex roof. On the southeast corner is a large one-story round veranda with its roof supported by seven Doric columns. The walls are clad in narrow clapboard, with bands of sawtooth shingles. Inside are parquet floors, a built-in china cabinet, a tiled fireplace, a mural, stenciling, and an ornate main staircase.
The NRHP considers the house significant as one of the finest historic houses in Monroe, and as an intact example of late Queen Anne architecture. Regez himself was "one of the dominant figures in the development of Monroe's Swiss cheese industry." And the design is an important work of Fridolin Heer. The house has also been listed on the Wisconsin State Register of Historic Places and designated a Monroe Landmark.
References
Houses in Green County, Wisconsin
Houses completed in 1901
Houses on the National Register of Historic Places in Wisconsin
Queen Anne architecture in Wisconsin
National Register of Historic Places in Green County, Wisconsin
Swiss culture | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,187 |
\section{Introduction}\label{sec:intro}
Quality control is typically one of the greatest challenges in the chain from
raw sensor data to scientific papers. This includes not only limited
observations for an individual scientist such as subsets of archival WFI data,
but also bulk observations of large astronomical surveys, such as those taken
with OmegaCAM on the VST (VLT Survey Telescope). In such surveys, the human
and financial resources required often dictate that not only the large survey
teams are spread over many institutes in many countries, but also the required
data storage and the parallel computing resources. Such a situation requires
an environment in which all non-manual qualifications are automated and the
scientist can graphically inspect where needed. This is easily achieved by going back and forth
through the data and metadata of the whole processing chain for large numbers of
data products, and for only those data products where it is necessary. Such
efficiency is clearly as beneficial to individual scientists as it is to large
survey teams.
These requirements force survey teams beyond the era of science on a desktop
and dictate a paradigm in which astronomers, calibration scientists, and
computer scientists spread over geographically distant locations in many countries
share their work and latest results in a single environment that allows the
optimized processing, quality control, and archiving of large data sets. This
means a federated system of humans, databases, computing resources, and data
storage yielding an \textit{integrated information system} \citep{adass}.
This integrated information system, \textsf{Astro-WISE}, is introduced and
described in detail in \citet{begeman}. It is assumed that the reader is
familiar with the fundamental concepts described in these papers as only the
most relevant concepts will be dealt with here.
\subsection{Traditional quality control}
The quality control of astronomical data is a key to success in obtaining
necessary data for scientific use cases. Quality control allows scientists to
verify observations, to improve observational plans, to correct the regime of
observations, to check the data processing and, finally, to distinguish between
an artifact and a real event detected during the observations.
Present day observations, especially the vast amounts in the case of large
astronomical surveys, require complicated processing systems involving a number
of data processing levels and programming efforts from many scientists and
programmers, usually distributed over a number of institutions. Tracing
data quality through the processing chain given the involvement of many scientists
and institutions becomes a non-trivial but crucial task.
There are many efforts invested in checking the quality of data delivered by an
instrument, but this quality control remains at the observation/reduction site and
comes to the scientific user as a reduced set of parameters describing the quality
of the observations \citep{hummel,VISIR}. There is no way for the user to return
to the raw observational material and check the quality of a particular
observation. In the case when the user does not process the data her/himself, but
accesses only the final product, she/he has to rely on the model of the quality
control chosen by the people behind the data processing. There is a general
understanding that the quality control should be shared by the observers and
scientists responsible for the data processing \citep{VLT}. Nevertheless, this
does not relieve the user from the task of making decision about the data quality
based on incomplete and non-reproducible information provided with the end product.
One mechanism bulk data providers employ to describe the quality of data products
is to introduce a number of attributes in the data model which will hold
information related to the quality control. For example, in the case of 2MASS
data products the quality control was performed during the observations and the
data processing, and the final catalog was formed according to the algorithm
described in \citet{2MASS}. From 60 attributes of the Two Micron All Sky Survey
Point Source Catalog (2MASS PSC), 31 attributes are related to data quality. This
allows the user to create a subset according to his/her preferences for the
quality of the data, but limits the user to the \textit{good quality} data. The
criteria for the data to be considered as \textit{good} are defined for a survey,
not for a user of its data. Similar approaches were used by SDSS and UKIDSS
surveys. In all these cases, data are delivered in a catalog with uniform quality
rather than optimizing quality for particular data subsets \citep{SDSS,UKIDSS}.
This is contrary to the typical goal of an individual scientist using the final
data products.
To make a sound decision about data quality, the user should be able to access
quality control algorithms at any point from the observation to the creation of
the end product. Thus, ideally, quality control should be performed on and
reviewed at each processing step. As a result, the user can trace the origin of
any problem associated with quality parameters back to the specific processing
step and/or the data entity responsible for it.
\subsection{\textsf{Astro-WISE}\ quality control}
\begin{figure}
\centering
\includegraphics[angle=0,width=118mm]{Fig01}
\caption{
The quality control model in the \textsf{Astro-WISE Information System}. Shown is a schematic of the
full data lineage with quality control at each processing step.
}\label{fig:quality}
\end{figure}
The core difference between this ``traditional'' quality control and the \textsf{Astro-WISE}\
approach to quality control is that the latter one uses features of \textsf{Astro-WISE}\ as an
\textit{integrated information system} to trace the quality at all stages of
data production. These features are: data processing and quality control
within the same system, an object-oriented framework, and full data
lineage with both forward and backward chaining. Together, they allow testing
of the quality of any data product, intermediate or final, from any other data
product at any stage of processing or analysis. The advantages to this approach
include allowing survey teams or individual scientists to inspect the quality of
any data product, allowing reprocessing of all or only part of one or multiple
data products in the most efficient way possible. In this way, the user knows
exactly what the final quality means and can even reprocess any set of data to
her/his needs.
Figure~\ref{fig:quality} shows an integral approach of quality control supported
by the \textsf{Astro-WISE Information System}. There are two types of quality control at each stage of the
data processing: automatic (default) and manual (optional). The user can visually
inspect each data item and validate/invalidate it. All the information about
the quality at every stage of data processing is saved in the database.
The object-oriented framework includes a set of parameters that are assigned
to each data class, and forms a built-in system of general quality estimators. The
following section describes these quality parameters used in the
\aw\ \textsf{Environment}\ (\texttt{AWE}) and how they are connected between different types of data.
Section~\ref{sec:quali} describes the quality control mechanisms built into \texttt{AWE}.
Section~\ref{sec:trend} gives examples of how trends in any aspect of the data
can be isolated using the command-line (\texttt{awe}-prompt). Finally,
Section~\ref{sec:qwise} describes the graphical interface for quality control in
\texttt{AWE}.
\section{Quality parameters}\label{sec:param}
\subsection{Data visibility}\label{sec:visib}
Visibility of data meeting the minimum level of quality to be processed
in \texttt{AWE}\ is governed by privilege level and by validity (i.e., privileged
data and data flagged as poor quality is hidden). Privileges in \texttt{AWE}\
are levels of accessibility for different groups, similar to permissions
levels on a UNIX file system.
All data entities in \texttt{AWE}\ are instances of Object-Oriented Programming (OOP)
\textit{objects}. Validity, and thus the processability, is
indicated by setting any or all of the following flag attributes of a given object:
\begin{enumerate}
\item
\texttt{is\_valid} -- manual validity flag
\item
\texttt{quality\_flags} -- automatic validity flag
\item
\texttt{timestamp\_start/end} -- validity ranges in time (for calibrations only)
\item
\texttt{creation\_date} -- the most recent valid data is the best
\end{enumerate}
For instance, obviously poor quality data can be flagged by setting its
\texttt{is\_valid} attribute to 0, preventing it from ever being processed
automatically. The calibrations used are determined by their
\texttt{timestamp\_start}, \texttt{timestamp\_end}, and \texttt{creation\_date} attributes
(Which calibrations are valid for the given data?), and the quality of processed
data by the automatic setting of its \texttt{quality\_flag} attribute (Is the
given data good enough?). Good quality data can then be flagged for promotion
(\texttt{is\_valid} $>1$) and eventually promoted in \textit{privilege} by its
creator (published from level 1 to 2) so it can be seen by the project manager
who will decide if it is worthy to be promoted once again (published from level
2 to 3 or higher) to be seen by the greater community. In the end, publishing
of data and results can be done by the manual setting of a single flag
attribute\footnote{All of these attributes can be modified via the command-line
\texttt{awe}-prompt\ or via one or more web services (see Sect.~\ref{sec:qwise}).}.
The example below shows how the user can invalidate a particular bias frame for
a particular instrument, detector and date using \texttt{AWE}.
\begin{verbatim}
awe> bias = BiasFrame.select(instrument='WFI', chip='ccd57',
.... date='2003-10-05')
awe> print bias.is_valid
1
awe> context.update_is_valid(bias,0)
awe> print bias.is_valid
0
\end{verbatim}
Note that the query returns the most recent, valid master bias object for the
given criteria. This same mechanism is used to query for objects
during processing.
\subsection{Provenance: full dependency linking and data lineage}\label{sec:prove}
The \aw\ \textsf{Environment}\ uses its federated database \citep{begeman,adass} to link all data products to their
progenitors (dependencies), creating a full data lineage of the entire
processing chain. This allows quick and simple troubleshooting of data results
by looking at processing settings, calibrations and more. It also allows for
direct monitoring of the progress of survey or individual observations, thus
simplifying observation management. This data lineage
also provides the ability to analyze trends in dependencies to aid in
troubleshooting (see Sect.~\ref{sec:fivel}).
Raw data is linked to the final data product via database links within the
\textit{data object}, allowing all information about any piece of data to be
accessed instantly. See \citet{mwebaze} for a detailed description of \texttt{AWE}'s
data lineage implementation. This data linking uses the power of OOP to
create this framework in a natural and transparent way.
\section{Built-in quality control mechanisms}\label{sec:quali}
In the \aw\ \textsf{Environment}, quality control permeates all aspects of the data reduction process.
From the moment data enters the system, through all processing steps, to the
final data product, data quality is retained and can be accessed transparently.
This is accomplished by integrating quality control concepts at the lowest
levels of the system.
\subsection{Integrated quality control}\label{sec:integ}
Quality control of the reduction process in \texttt{AWE}\ is integrated directly into
the objects. Three methods exist on all {\texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}}s (the afore mentioned OOP
\textit{objects} that describe data entities undergoing some level of
processing):
\begin{itemize}
\item{\texttt{verify()}}
compares values derived from the current \texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}\ instance to known acceptable
limits (e.g., image statistics) and automatically raises
\texttt{quality\_flags} if the limits are exceeded
\item{\texttt{compare()}}
compares values derived from the current \texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}\ instance to those of the
previous version and automatically raises \linebreak \texttt{quality\_flags} if
the values are worse
\item{\texttt{inspect()}}
provides an interface for manual inspection of the current \linebreak \texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}\
instance (e.g., viewing the image pixels)
\end{itemize}
The quality control parameters are stored in two persistent properties of the
object, \texttt{is\_valid} and \texttt{quality\_flags}. As mentioned before,
the \texttt{is\_valid} property is the manual flag used to validate or
invalidate any \texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}, and the \texttt{quality\_flags} property stores the
results of the automatic verification routines. This model shares similarities
with other quality control ``scoring'' models (e.g., \citet{hanuschik}) and is discussed in the processing context in Sect.~\ref{sec:proce}.
To give examples in contrast to this model, the Sloan survey uses automated
pipelines (e.g., \textit{runQA} and \textit{matchQA}) run separately from the
processing pipeline to assess and report the quality of the data \citep{SDSS}, and
the UKIDSS survey employs the metadata storage of FITS images to convey quality
parameters to the QC procedures (\citet{UKIDSS} and reference D06 therein). The
integrated nature of the quality parameters and procedures in \texttt{AWE}\ has clear
advantages over these other models because the quality parameters are directly
part of the \texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}.
This integrated quality control is one of the simplest, yet most powerful
aspects of \texttt{AWE}\ for survey operator and individual scientist alike. Both high
and low quality data can be accessed via a simple query and the cause of the
low quality can be known directly via the bit-masked value of its
\texttt{quality\_flags} attribute. Also, the nature of the queries in the
processing recipes guarantees that low quality data is never processed unless it
is manually specified.
This paradigm for quality control allows for construction of tools such as
\textsf{Quality-WISE}\footnote{\url{http://quality.astro-wise.org/}} that can act as the
QC front-end of the entire system. Data quality (of both pixel data and its
metadata) can be viewed through a simple interface. This interface allows access to flagging of
data (triggering automatic reprocessing), to direct reprocessing of data and
even to the quality of linked objects. This all exists within the information
system allowing effective sharing of human resources.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|r|r|}
\hline
\textbf{Class} & \textbf{process\_param} & \textbf{value} & \textbf{units}\\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.RawBiasFrame.html}{Raw\-Bias\-Frame}} & max\_bias\_stdev & 100.0 & ADU \\
& max\_bias\_level & 500.0 & ADU \\
& max\_bias\_flatness & 10.0 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.RawDomeFlatFrame.html}{Raw\-Dome\-Flat\-Frame}} & min\_flat\_mean & 5000 & ADU \\
& max\_flat\_mean & 55000 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.RawTwilightFlatFrame.html}{Raw\-Twilight\-Flat\-Frame}} & min\_flat\_mean & 5000 & ADU \\
& max\_flat\_mean & 55000 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.ReadNoise.html}{Read\-Noise}} & maximum\_readnoise & 5.0 & ADU \\
& maximum\_bias\_difference & 1.0 & ADU \\
& maximum\_readnoise\_difference & 0.5 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.GainLinearity.html}{Gain\-Linearity}} & maximum\_gain\_difference & 0.1 & $e^-/$ADU \\
& minimum\_gain & 2.0 & $e^-/$ADU \\
& maximum\_gain & 5.0 & $e^-/$ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.BiasFrame.html}{Bias\-Frame}} & maximum\_stdev & 10.0 & ADU \\
& maximum\_stdev\_differ & 10.0 & ADU \\
& maximum\_subwin\_flatn & 100000.0 & ADU \\
& maximum\_subwin\_stdev & 100000.0 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.HotPixelMap.html}{Hot\-Pixel\-Map}} & maximum\_hotpixelcount & 50000 & \\
& maximum\_hotpixelcount\_difference & 100 & \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.ColdPixelMap.html}{Cold\-Pixel\-Map}} & maximum\_coldpixelcount & 80000 & \\
& maximum\_coldpixelcount\_difference & 100 & \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.DomeFlatFlatFrame.html}{Dome\-Flat\-Frame}} & maximum\_subwin\_flatness & 1000.0 & ADU \\
& maximum\_subwin\_diff & 1000.0 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.TwilightFFlatFrame.html}{Twilight\-Flat\-Frame}} & maximum\_subwin\_flatness & 1000.0 & ADU \\
& maximum\_subwin\_diff & 1000.0 & ADU \\
& maximum\_number\_of\_outliers & 10000 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.MasterFlatFrame.html}{Master\-Flat\-Frame}} & maximum\_subwin\_diff & 1000.0 & ADU \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.PhotometricParameters.html}{Photo\-metric\-Parameters}} & max\_error & 0.03 & mag \\
\hline
\texttt{\href{http://doc.astro-wise.org/astro.main.AstrometricParameters.html}{Astro\-metric\-Parameters}} & min\_nref & 15 & \\
& max\_nref & 1200 & \\
& max\_sigma & 1.0 & arcsec \\
& max\_rms & 1.0 & arcsec \\
& min\_n\_overlap & 20 & \\
& max\_n\_overlap & 20000 & \\
& max\_sigma\_overlap & 0.1 & arcsec \\
& max\_rms\_overlap & 0.1 & arcsec \\
\hline
\end{tabular}
\end{center}
\caption{
Representative examples of QC limits used by the automated \texttt{verify()}
and \texttt{compare()} methods on the given class instances (objects). These
examples are limited to calibration data and are derived from the requirements
for the OmegaCAM instrument and updated based on experience with archive data of
the WFI instrument. See the document page linked from the class name of
appropriate links on \url{http://doc.astro-wise.org/astro.main.html} for more
details.
}\label{tab:proce}
\end{table}
\subsection{Quality control during ingestion}
A number of automatic, simple quality control procedures are executed at
the lowest level of data interaction--ingestion into the system. These
procedures are used to flag poor-quality data so they are excluded from further use.
The procedures include checks on the median and standard deviation of the
pixel values in bias exposures, and the exposure level of flat-fields.
The levels at which flags are raised are instrument and detector chip
dependent, as needed.
\subsection{Quality control during processing}\label{sec:proce}
Quality control at the processing stage starts well before any actual
processing is done. The selection of data to be processed is subject to the
visibility mechanism (see Sect.~\ref{sec:visib}). All processing tasks
first check the validity and quality of candidate science data, and the
validity, quality and timestamp ranges of applicable calibration data.
This guarantees that only the highest quality data is considered for
processing.
Once data processing is complete, the quality methods of data product object are
run to verify that this is the highest quality product
possible (see Sect.~\ref{sec:integ}). The \texttt{verify()} and \texttt{compare()} methods are
automatically run to check the data product against the accepted limits and to
make sure the quality is higher than the previous version if one exists. If
either test fails, one or more \texttt{quality\_flags} are raised.
Table~\ref{tab:proce} gives a representative sample of the limits tested via the
\texttt{verify()} and \texttt{compare()} methods. Optionally, the
\texttt{inspect()} method can be run manually to interactively check the data
product. A non-interactive version of this method is always run to create and
store a static version of the inspection plot for later perusal via the
command-line or through the \textsf{Quality-WISE}\ service (see Sect.~\ref{sec:qwise}).
\subsection{Inspection plots}\label{sec:inspe}
During processing, quality control inspection plots are made as a matter of
course. These can be viewed interactively during processing or saved for later
viewing. As most processing is done in a parallel environment, these
inspection plots tend to have a very low creation cost.
Inspection plots exist for many of the object types in \texttt{AWE}, particularly
those critical for assessing the quality of major data products (e.g., science
data quality, end-to-end detrending quality, astrometric and photometric
calibration quality). See Fig.~\ref{fig:ithum}~through~\ref{fig:iillu} for
examples of such plots.
These static plots are simple snapshots of the most useful information to be
inspected. In \texttt{AWE}, there exists the ability in most cases to interact with
the inspection plot. This is done using the PyLab interface to MatPlotLib.
This interface is integrated into \texttt{AWE}, and forms the backbone of all types of
plotting, including post-processing analysis.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=45mm]{Fig02a}
\hspace{8mm}
\includegraphics[angle=0,width=45mm]{Fig02b}
\caption{
(left panel) A thumbnail representation of a WFI \texttt{\href{http://doc.astro-wise.org/astro.main.ReducedScienceFrame.html}{Reduced\-Science\-Frame}}\ created by STIFF. The optimized
intensity cuts and binning allow a quick assessment of the quality. This
particular example shows an intensity gradient caused by either poor flat
fielding, nebulosity from a galaxy at the center of the mosaic field (to the
upper left), or simply a non-uniform illumination of the focal plane. The
intensity values are inverted.
(right panel) A thumbnail representation of a WFI \texttt{\href{http://doc.astro-wise.org/astro.main.WeightFrame.html}{Weight\-Frame}}\ created by STIFF. The optimized
intensity cuts and binning allow a quick assessment of the quality. This
particular example is associated with the thumbnail in the left panel.
Saturated stellar peaks and bad columns are clearly visible in addition to
``doughnuts'' of the primary mirror of the telescope that are part of the flat
field foundation of the \texttt{\href{http://doc.astro-wise.org/astro.main.WeightFrame.html}{Weight\-Frame}}. White pixels have values near 1, black pixels
have values at or near 0. The horizontal lines are artifacts of the CCD
manufacturing process. The higher weight of the pixels near some of the bad
columns is an artifact caused by Fourier processing of input flat frames
without properly taking into account bad pixels. It is possible to identify
some of these defects with pixel statistics \textit{a priori}, but these
unusual cases are generally only identified through this type of inspection plot.
}\label{fig:ithum}
\end{center}
\end{figure}
\begin{sidewaysfigure}
\begin{center}
\includegraphics[angle=270,width=165mm]{Fig03}
\caption{
A PSF Anisotropy plot of the \texttt{\href{http://doc.astro-wise.org/astro.main.ReducedScienceFrame.html}{Reduced\-Science\-Frame}}\ whose thumbnail is seen in
Fig.~\ref{fig:ithum}.
The left panel
shows the two-dimensional anisotropy in the PSF (in both FWHM and in
ellipticity) of the sources. The top-middle panel shows source magnitudes
on an arbitrary scale versus flux radius and gives an indication of how
``stellar'' the sample is. The remaining panels plot FWHM of the sources
versus ellipticity position angle, horizontal position and vertical
position, respectively, clockwise from top-right.
}\label{fig:ipsfa}
\end{center}
\end{sidewaysfigure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=80mm]{Fig04}
\caption{
An \texttt{\href{http://doc.astro-wise.org/astro.main.AstrometricParameters.html}{Astro\-metric\-Parameters}}\ inspection plot for a recent solution of the \texttt{\href{http://doc.astro-wise.org/astro.main.ReducedScienceFrame.html}{Reduced\-Science\-Frame}}\
associated with the thumbnail in Fig.~\ref{fig:ithum}. The plot displays the
statistics of the residuals (DRA and DDEC) between the RA and DEC of sources in
a source catalog to which the local astrometric solution has been applied and
the RA and DEC of those sources as listed in the reference catalog of
astrometric standards, USNO-A2.0 in this case. The text in the top of the figure lists the observation
date (DATE\_OBS), the number (N) of sources pairs plotted, their average RA
($<$RA$>$) and DEC ($<$DEC$>$) in degrees, the average RA and DEC residuals
($<$DRA$>$ and $<$DDEC$>$) and their standard deviations in arcsec, and finally
the root-mean-square (RMS) of the two-dimensional residual and the maximum
two-dimensional residual (Max) in arcsec. The large upper panel plots DRA
versus DDEC. The four panels below it show DRA and DDEC with respect to RA
(with a constant offset of 203.9 degrees) and then to DEC.
}\label{fig:iastr}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig05}
\caption{
A \texttt{\href{http://doc.astro-wise.org/astro.main.PhotometricParameters.html}{Photo\-metric\-Parameters}}\ inspection plot for a photometric observation comprising one WFI
detector. A graphical representation of the data used to calculate the
photometric zeropoint. In this plot, three photometric reference catalogs can
be seen: Stetson (blue points), \textsf{Astro-WISE}\ secondary standards (red points) and Sloan
Digital Sky Survey data release 5 (black points).
}\label{fig:iphot}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig06}
\caption{
An \texttt{IlluminationCorrection} inspection plot for WFI data. This plot is
a schematic representation of the illumination variations across the region of
consideration, usually the entire field-of-view, eight 2k$\times$4k detectors
in this case.
}\label{fig:iillu}
\end{center}
\end{figure}
\section{Trend analysis}\label{sec:trend}
Many powerful ways exist in the \aw\ \textsf{Environment}\ to examine both pixel data and
metadata. One of these ways is through the use of the command-line interface,
the \texttt{awe}-prompt. Through this interface, one can examine individual quality
parameters and processing parameters of any object or linked object
transparently.
\subsection{Five-line script}\label{sec:fivel}
\texttt{AWE}\ consists of Python classes representing {\texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}}s that can be created by
scripts (called recipes or Tasks). The Tasks are simply sophisticated versions
of what are termed five-line scripts\footnote{The term file-line script derives
from the observation that most simple tasks in \texttt{AWE}\ can be achieved in about
five lines of code.} (5LS). It is these 5LSs that do the bulk of the work of
the data reduction and analysis for the user. The 5LS is also a powerful tool
for quality control as atypical objects can be isolated easily.
This 5LS concept is a very simple and powerful way for users to interact with
the data contained in the system. They can be ``one-off'', ``on-the-fly'', or
``throw-away'' scripts used to locate some interesting aspect of the data, can
be written down in a source file for potential use at a later time, or can be
integrated into an existing or future Task for the benefit of the system. One
set of examples of 5LSs focuses on seeing how aspects of raw data in the system
change over time, another gathers statistical data for comparison and outlier
detection, and the last quickly investigates a scientific aspect of existing
data in the system.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig07}
\caption{
Plot of the bias level (median value of the science region) of ccd50 of the WFI
instrument from May 1999 to June 2005.
}\label{fig:biast}
\end{center}
\end{figure}
\subsection{Bias levels}
Display the bias level as a function of time for chip \textsf{ccd50} of the WFI camera:
\begin{verbatim}
awe> q = (RawBiasFrame.chip.name == 'ccd50') &\
.... (RawBiasFrame.quality_flags == 0) &\
.... (RawBiasFrame.is_valid > 0)
....
awe> biases = list(q) # instantiate all biases
awe> x = [b.MJD_OBS for b in biases]
awe> y = [b.imstat.median for b in biases]
awe> pylab.scatter(x,y,s=0.5)
\end{verbatim}
This script will result in a plot similar to that seen in Fig.~\ref{fig:biast}.
It is important to note how the query is done. Not only are the
objects of the desired detector queried for, the quality and validity (see
Sect.~\ref{sec:visib}) are also checked. This prevents any data that are out of
specified ranges from being plotted, thus removing the worst outliers in the
resulting plot before the data is even compiled. This lends significant
efficiency to this method of visualization.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig08}
\caption{
Plot of the dome flat exposure level (median value of the science region minus
the median value of the X overscan region) versus exposure time of
ESO\_CCD\_\#65 of the OmegaCAM instrument from data taken in 2011. This plot
gives a quick indication of how linear this detector is. The dashed red line
is only an indication of the slope in the data. The cluster of points at
EXPTIME$=$3 sec is from heavier sampling for diagnostic and detector health
purposes.
}\label{fig:media}
\end{center}
\end{figure}
\subsection{Exposure levels}
Not only can simple values be plotted over time as in the previous section, but more complex
investigations of object attributes can be performed easily. In this set of
examples, the linearity of an OmegaCAM detector is investigated:
\begin{verbatim}
awe> q = list((RawDomeFlatFrame.chip.name == 'ESO_CCD_#65') &
.... (RawDomeFlatFrame.filter.name == 'OCAM_g_SDSS') &
.... (RawDomeFlatFrame.quality_flags == 0) &
.... (RawDomeFlatFrame.is_valid > 0))
....
awe> exptime = [d.EXPTIME for d in q]
awe> med = [d.imstat.median-d.overscan_x_stat.median for d in q]
awe> pylab.plot(exptime, med, 'k.')
awe> pylab.plot([0,4], [0,60000], 'r--')
\end{verbatim}
This first example gives a plot similar to that shown in Fig.~\ref{fig:media}.
It is the overscan-corrected counts compared to the exposure time for
one detector of the OmegaCAM mosaic. Simple arithmetic is seen in the
\textit{list comprehension} that creates the \texttt{med} list. The second
example uses the data from the first, but adds the ability to perform array
arithmetic using NumPy\footnote{\url{http://numpy.scipy.org/}} to plot the
desired result (Fig.~\ref{fig:expos}).
\begin{verbatim}
awe> med = numpy.array(med)
awe> exptime = numpy.array(exptime)
awe> pylab.plot(med, med/exptime, 'k.')
awe> pylab.plot([0,60000], [15000,15000], 'r--')
\end{verbatim}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig09}
\caption{
Plot of the dome flat exposure (median value of the science region minus the
median value of the X overscan region divided by the exposure time) versus the
dome flat exposure level (median value of the science region minus the median
value of the X overscan region) of ESO\_CCD\_\#65 of the OmegaCAM instrument
from data taken in 2011. This plot quickly gives a different view of how
linear this detector is. The dashed red line is only an indication of the
mean detector exposure.
}\label{fig:expos}
\end{center}
\end{figure}
\noindent
This second example gives a quick exposure time-independent view of the same
data. As in the result of the previous script, outliers can easily be seen.
It is now easy to isolate these outliers with NumPy methods using visually
chosen limits:
\begin{verbatim}
awe> outlier_mask = (med/exptime < 10000)
awe> outlier_mask |= (med/exptime > 20000)
awe> outliers = med[outlier_mask], exptime[outlier_mask]
awe> good_data = med[~outlier_mask], exptime[~outlier_mask]
\end{verbatim}
\subsection{Twenty thousand light curves}
In the Fall of 2006, an investigation of light curves of the stars in the region of Centaurus-A\footnote{See \url{http://www.astro-wise.org/Presentations/LCnov06/CenA\_5LS\_valentijn/ for the details of the investigation and the various scripts used.}} was undertaken using
pre-reduced data in the \textsf{Astro-WISE}\ system.
The data was originally observed in the first half of 2005 with the WFI
instrument. Only example scripts and resulting plots are reproduced here. The
scripts have been updated and reformatted for inclusion.
The first example takes data from an association of two coadded frames. These
data exist in the system as an \texttt{\href{http://doc.astro-wise.org/astro.main.AssociateList.html}{Associate\-List}}\ object. Some astrometric and
photometric parameters are \textit{mined} from the association data. This is
plotted in such a way to test the astrometric accuracy of fainter sources (see
Fig.~\ref{fig:delta}). The plot clearly shows a slight degradation in this
accuracy, but also shows that it is not a source of concern as the position of
faintest sources is still generally well known.
\begin{verbatim}
awe> Al = (AssociateList.ALID == 1431)[0]
awe> arlist = ['RA', 'DEC', 'MAG_ISO', 'MAG_AUTO', 'MAG_APER']
awe> r = Al.get_data_on_associates(arlist,mask=3,mode='ALL')
awe> mag, dmag, ddec = [], [], []
awe> for aid in r.keys():
.... # index 0 = SLID, 1 = SID, # added automatically
.... # index 3 = DEC, 5 = MAG_AUTO
.... mag.append(r[aid][0][5])
.... dmag.append(r[aid][0][5] - r[aid][1][5])
.... ddec.append((r[aid][0][3] - r[aid][1][3])*3600)
....
awe> pylab.plot(mag, dmag, 'b.', ms=0.5)
awe> pylab.plot(mag, ddec, 'r.', ms=0.2)
awe> pylab.ylim([-2,2])
\end{verbatim}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig10}
\caption{
A plot of delta MAG\_AUTO (blue points) over-plotted with delta DEC (red points)
versus MAG\_AUTO. The increase in scatter of the astrometric residuals is far
lower than that of the photometric residual, a qualitative indication that
astrometry for faint sources is at acceptable levels.
}\label{fig:delta}
\end{center}
\end{figure}
The next example mines data and creates a plot of light curves for
approximately 7500 of the 20000 stars associated with at least one other star
in one of the other observations. These 7500 are the stars that were
associated for all 12 observations (i.e., where photometric data exists for all
12 observations). For brevity and clarity, only the first 100 of these are
plotted by the script and shown in the accompanying plot (see
Fig.~\ref{fig:light}).
\begin{verbatim}
awe> Al = (AssociateList.ALID == 1534)[0]
awe> sls = Al.sourcelists
awe> dates = [sl.frame.observing_block.start for sl in sls]
awe> arlist = ['RA', 'DEC', 'MAG_ISO', 'MAG_AUTO', 'MAG_APER']
awe> r = Al.get_data_on_associates(arlist, count=len(dates))
awe> #for aid in r.keys(): # plots eveything
awe> for aid in r.keys()[:100]: # plots only first 100 stars
.... # index 5 = MAG_AUTO
.... mags = [r[aid][i][5] for i in range(len(r[aid]))]
.... datesmags = zip(dates,mags) # sort by obsdate
.... datesmags.sort()
.... date = [datemag[0] for datemag in datesmags]
.... mag = [datemag[1] for datemag in datesmags]
.... l = pylab.plot(date, mag ,'k.', date, mag, '-')
....
awe> dt1 = datetime.datetime(2005,3,1)
awe> dt2 = datetime.datetime(2005,6,15)
awe> pylab.xlim(dt1, dt2)
\end{verbatim}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig11}
\caption{
A plot of MAG\_AUTO versus the date for 100 of approximately 7500 light-curves
containing 12 photometric data points. It is obvious that there remain
systematic offsets in the zeropoints.
}\label{fig:light}
\end{center}
\end{figure}
In this last example, the zeropoint of each chip is compared
over time with the zeropoints of all the other chips. The results can be seen in
Fig.~\ref{fig:zerop}.
\begin{verbatim}
awe> for chip in context.get_chips_for_instrument('WFI'):
.... zeropnts = []
.... for sl in sls:
.... for reg in sl.frame.regridded_frames:
.... if reg.chip.name == chip:
.... red = reg.reduced
.... break
.... pht = PhotometricParameters.select_for_reduced(red)
.... zeropnts.append(pht.zeropnt.value)
.... dateszps = zip(dates, zeropnts)
.... dateszps.sort()
.... date = [datezp[0] for datezp in dateszps]
.... zeropnt = [datezp[1] for datezp in dateszps]
.... pylab.plot(date, zeropnt, 'k.', date, zeropnt, '-')
....
awe> pylab.xlim(dt1, dt2)
\end{verbatim}
Zeropoint residuals with respect to that of any chip or to the mean zeropoint
per day can easily be obtained with only slight additions to the example code
presented above. This can give a clearer view of how the zeropoint of the set of chips
evolves over time.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=90mm]{Fig12}
\caption{
A plot of zeropoint versus the date for all 8 WFI detectors. The systematic
offsets in zeropoint from night to night is clearly seen.
}\label{fig:zerop}
\end{center}
\end{figure}
\section{Quality-WISE web service}\label{sec:qwise}
All objects stored in the \textsf{Astro-WISE}\ database are stored with their processing and quality
parameters. These parameters can be accessed in many ways: from the
command-line interface queries, from direct access to the database, or from web
services such as CalTS (\url{calts.astro-wise.org}) or DBView
(\url{dbview.astro-wise.org}). In \aw\ \textsf{Environment}, we have implemented a quality web
service that combines all three methods and collects the most
relevant metadata for the purpose of quality control:
\url{quality.astro-wise.org}.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=118mm]{Fig13}
\caption{
Screen-shot of the upper part of a \textsf{Quality-WISE}\ page. This view shows the quality of
an OmegaCAM coadded frame. At the very top is the type of object and a link to
the file on the dataserver (a unique hash value in the filename link is
purposely obscured for security reasons). Directly below the banner is the top
bar with links and basic actions. Below this is tabular information about the
object and graphical inspection plots (a thumbnail of the image on the left and
its weights on the right, cf. Fig.~\ref{fig:ithum}). Note that green fields
indicate values within specified ranges that will be red when out of specified
ranges.
}\label{fig:qwist}
\end{center}
\end{figure}
The \textsf{Quality-WISE}\ interface is accessed primarily through the DBView service by
clicking on the \textsf{\underline{quality}} links associated with science data
objects. The linked quality pages summarize observational and statistical
details and add a schematic representation of the detector, thumbnails of pixel
data, and various derived inspection plots (see Sect.~\ref{sec:inspe}).
A basic interface is also included to flag or to publish data directly.
Links to the quality pages of associated objects (e.g., progenitor
or derived data products) also exist. Details of how the \textsf{Quality-WISE}\ service can
be applied to real-world applications can be found in \citet{verdoes}.
\subsection{\textsf{Quality-WISE}\ top bar}
At the top of every \textsf{Quality-WISE}\ page is the class name of the object and a link to
the associated data file on a data server (see Fig.~\ref{fig:qwist}). There is
a bar below the banner image with links on the left to the \textsf{Astro-WISE}\ homepage and to
the database viewer, calibration timestamps and target processor web services.
On the right is the currently logged-in user and project name. These link to
interfaces to change the user and/or the project via browser cookies. In the
center, there is an indication of comments associated with the object and an
interface to add comments. This is typically done when the validity of the
object is changed using the \texttt{is\_valid} interface. This interface
allows one of 3 levels of validity to be assigned: $0
= invalid$, $1 = valid$ or $2 = publishable$ (see Sect.~\ref{sec:visib}). Pressing the
\textsf{\underline{Submit}} button stores the validity value and comment, where
applicable, prior to reloading the quality page. For special purposes such as
surveys, the validity choices can be expanded and the comment interface can have
pre-specified strings included for efficiency.
\subsection{Observational details}\label{sec:obser}
The observational details for the object being inspected are directly below the top bar of a \textsf{Quality-WISE}\ page (see Fig.~\ref{fig:qwist}). The values are taken
directly from the object stored in the database and include: date of the
observation in human readable and modified Julian date (\texttt{DATE\_OBS} and
\texttt{MJD\_OBS}, respectively), the name of the object observed
(\texttt{OBJECT}), right ascension and declination coordinates (R.A. and Dec.,
respectively), the observer responsible for the observation
(\texttt{OBSERVER}), the exposure time (\texttt{EXPTIME}), the airmass at the
start and end of the observation (\texttt{AIRMSTRT} and \texttt{AIRMEND},
respectively), the filter used for the observation (Filter), and the magnitude
identifier of the filter, i.e., the photometric system (\texttt{mag\_id}).
To the right of the observational details table is a graphical representation
of the detector-plane layout for the individual detectors. The detectors
highlighted in light blue are those that participated in the current data
object. In the example of a \texttt{\href{http://doc.astro-wise.org/astro.main.CoaddedRegriddedFrame.html}{Coadded\-Regridded\-Frame}}\ here, all detectors are highlighted as all
detectors are represented in the data.
\subsection{Processing and statistical details}
On the left side of every \textsf{Quality-WISE}\ page are processing details and
statistics of the main and associated objects (see Fig.~\ref{fig:qwist}).
The main characteristic of this side bar is the highlighting of important
quality parameters (see Table~\ref{tab:proce}). When a parameter
is within a specified range indicating good quality, the entire cell is colored
green, when the parameter is outside this range, the entire cell is colored
red. In addition, when the cursor is positioned over any of these cells, the reason for the indicated quality is displayed.
Processing details show when the object was created (\texttt{creation\_date}),
its validity (\texttt{is\_valid}), if any quality flags have been set
(\texttt{quality\_flags}), and to what level it has been published
(Privileges). See Sect.~\ref{sec:visib} for more on these last three parameters.
Furthermore, statistics of the main object and associated astrometric and photometric objects, if any, are also listed (see also Fig.~\ref{fig:qwism}).
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=118mm]{Fig14}
\caption{
Screen-shot of the middle part of the \textsf{Quality-WISE}\ page shown in Fig.~\ref{fig:qwist}.
The remainder of the statistical information of the combined global astrometric
solution can be seen on the left. The astrometric residuals plots representing
the quality of the solutions used to make the coadded frame are on the top-right.
The PSF anisotropy plot for the coadded frame is at the bottom.
}\label{fig:qwism}
\end{center}
\end{figure}
\subsection{Inspection plots}
The main body of each \textsf{Quality-WISE}\ page is dominated by the inspection plots. These
plots are of the sort described in Sect.~\ref{sec:inspe}. They always start
with an image thumbnail (with reverse pixel values) and a weight thumbnail
(when applicable) showing lower weights as darker values (see
Fig.~\ref{fig:qwist}). Below this is the astrometric reference residuals plot
of the individual \textit{reduced frame} local solution, or the astrometric reference and
overlap residuals plots of the composite global solution for \textit{coadded frames} (see Fig.~\ref{fig:qwism}).
In this latter case, the additional plot shows the internal accuracy of the
global solution. Below the astrometric plots can be the photometric plots showing
the data used to derive the zero point and the results of the illumination
correction derivation (see Figures~\ref{fig:iphot}~and~\ref{fig:iillu}). These are only shown for non-coadded objects. The last plot shown is the PSF anisotropy of the
sources in the observation shown at the bottom of
Fig.~\ref{fig:qwism}.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=118mm]{Fig15}
\caption{
Screen-shot of the lower part of the \textsf{Quality-WISE}\ page shown in Figures~\ref{fig:qwist}~and~\ref{fig:qwism}.
Near the top is the list of progenitor frames'
information. This list contains 160 entries and is truncated here. Basic information about the progenitor frames is provided in this
list along with links
to their quality pages (far right). Page
creation information is presented at the very bottom including a breakdown of creation times into three bins:
database time, processing time and web server time.
}\label{fig:qwisb}
\end{center}
\end{figure}
\subsection{Progenitor/derived quality}
For science data, each data product has progenitor data and derived data. The
quality pages for these data are linked near the bottom. In the case of the
\texttt{\href{http://doc.astro-wise.org/astro.main.CoaddedRegriddedFrame.html}{Coadded\-Regridded\-Frame}}\ quality page in Fig.~\ref{fig:qwisb}, there is only progenitor data.
This consists of a list of 160 {\texttt{\href{http://doc.astro-wise.org/astro.main.RegriddedFrame.html}{Regridded\-Frame}}}s. The information listed is nearly
identical to that described in the observational details table (see
Sect.~\ref{sec:obser}). At the far right of each entry is the link to the
quality page of the progenitor object.
\section{Summary}\label{sec:summa}
The approach for quality control of astronomical data in the \textsf{Astro-WISE Information System}\ has been described. The comparison to quality control techniques used in other systems has been presented. It was shown that
the \textsf{Astro-WISE}\ approach has advantages for any individual user or group of users in
that it allows the quality to be assessed for not only the final data product,
but also any other progenitor data product in a simple and transparent way
through database linking of all data objects ({\texttt{\href{http://doc.astro-wise.org/astro.main.ProcessTarget.html}{Process\-Target}}}s).
This quality control is built into all aspects of the \textsf{Astro-WISE}\ information system.
From the point where raw data enters the system, through all processing steps
to the final data product, quality control mechanisms permeate throughout.
Moreover, the quality of any stage of
data processing can be assessed with quality parameters and inspection plots.
Using metadata (quality- or non-quality-related) stored in all linked objects,
diagnostic plots can be created quickly using a relatively small
amount of command-line code. This has been shown with examples using archive data from the WFI instrument at La Silla Observatory and (pre-)survey data from the newly commissioned
OmegaCAM instrument at the Paranal Observatory. The code can be added to
simple scripts for the benefit of the individual user, or eventually find its
way into the core of the system benefiting all users alike.
All the quality control aspects of the \aw\ \textsf{Environment}\ have been gathered into a
webservice called \textsf{Quality-WISE}. This service allows quick viewing of the metadata
and inspection plots of the data in question and of any progenitor or derived
data. It also provides a simple interface for a user or group of users to
validate data and comment on its quality.
Taken as a whole, the \textsf{Astro-WISE}\ approach to quality control is a comprehensive and
efficient method to perform quality checks on individual users' data or on the
data from large astronomical surveys. It is constantly being updated as newer,
better quality control methods are discovered or derived, and will always stay
on the cutting edge to maintain its advantages.
\begin{acknowledgements}
\textsf{Astro-WISE}\ is an on-going project which started from a FP5 RTD programme funded
by the EC Action ``Enhancing Access to Research Infrastructures''. This work is
supported by FP7 specific programme ``Capacities - Optimising the use and
development of research infrastructures''. Special thanks to Philippe
H\'{e}raudeau and Ivona Kostadinova for their constructive comments.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,012 |
{"url":"https:\/\/openpiv.readthedocs.io\/en\/latest\/src\/tutorial1.html","text":"# OpenPIV tutorial\u00b6\n\nIn this tutorial we read a pair of images and perform the PIV using a standard algorithm. At the end, the velocity vector field is plotted.\n\n[21]:\n\nfrom openpiv import tools, pyprocess, validation, filters, scaling\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nimport imageio\nimport importlib_resources\nimport pathlib\n\n\n## Reading images:\u00b6\n\nThe images can be read using the imread function, and diplayed with matplotlib.\n\n[22]:\n\npath = importlib_resources.files('openpiv')\n\n[23]:\n\nframe_a = tools.imread( path \/ 'data\/test1\/exp1_001_a.bmp' )\nframe_b = tools.imread( path \/ 'data\/test1\/exp1_001_b.bmp' )\n\nfig,ax = plt.subplots(1,2,figsize=(12,10))\nax[0].imshow(frame_a,cmap=plt.cm.gray);\nax[1].imshow(frame_b,cmap=plt.cm.gray);\n\n\n## Processing\u00b6\n\nIn this tutorial, we are going to use the extended_search_area_piv function, wich is a standard PIV cross-correlation algorithm.\n\nThis function allows the search area (search_area_size) in the second frame to be larger than the interrogation window in the first frame (window_size). Also, the search areas can overlap (overlap).\n\nThe extended_search_area_piv function will return three arrays. 1. The u component of the velocity vectors 2. The v component of the velocity vectors 3. The signal-to-noise ratio (S2N) of the cross-correlation map of each vector. The higher the S2N of a vector, the higher the probability that its magnitude and direction are correct.\n\n[24]:\n\nwinsize = 32 # pixels, interrogation window size in frame A\nsearchsize = 38 # pixels, search area size in frame B\noverlap = 17 # pixels, 50% overlap\ndt = 0.02 # sec, time interval between the two frames\n\nu0, v0, sig2noise = pyprocess.extended_search_area_piv(\nframe_a.astype(np.int32),\nframe_b.astype(np.int32),\nwindow_size=winsize,\noverlap=overlap,\ndt=dt,\nsearch_area_size=searchsize,\nsig2noise_method='peak2peak',\n)\n\n\nThe function get_coordinates finds the center of each interrogation window. This will be useful later on when plotting the vector field.\n\n[25]:\n\nx, y = pyprocess.get_coordinates(\nimage_size=frame_a.shape,\nsearch_area_size=searchsize,\noverlap=overlap,\n)\n\n\n## Post-processing\u00b6\n\nStrictly speaking, we are ready to plot the vector field. But before we do that, we can perform some convenient pos-processing.\n\nTo start, lets use the function sig2noise_val to get a mask indicating which vectors have a minimum amount of S2N. Vectors below a certain threshold are substituted by NaN. If you are not sure about which threshold value to use, try taking a look at the S2N histogram with:\n\nplt.hist(sig2noise.flatten())\n\n[26]:\n\ninvalid_mask = validation.sig2noise_val(\nsig2noise,\nthreshold = 1.05,\n)\n\n\nAnother useful function is replace_outliers, which will find outlier vectors, and substitute them by an average of neighboring vectors. The larger the kernel_size the larger is the considered neighborhood. This function uses an iterative image inpainting algorithm. The amount of iterations can be chosen via max_iter.\n\n[27]:\n\nu2, v2 = filters.replace_outliers(\nu0, v0,\ninvalid_mask,\nmethod='localmean',\nmax_iter=3,\nkernel_size=3,\n)\n\n\nNext, we are going to convert pixels to millimeters, and flip the coordinate system such that the origin becomes the bottom left corner of the image.\n\n[28]:\n\n# convert x,y to mm\n# convert u,v to mm\/sec\n\nx, y, u3, v3 = scaling.uniform(\nx, y, u2, v2,\nscaling_factor = 96.52, # 96.52 pixels\/millimeter\n)\n\n# 0,0 shall be bottom left, positive rotation rate is counterclockwise\nx, y, u3, v3 = tools.transform_coordinates(x, y, u3, v3)\n\n\n## Results\u00b6\n\nThe function save is used to save the vector field to a ASCII tabular file. The coordinates and S2N mask are also saved.\n\n[29]:\n\ntools.save('exp1_001.txt' , x, y, u3, v3, invalid_mask)\n\n\nFinally, the vector field can be plotted with display_vector_field.\n\nVectors with S2N bellow the threshold are displayed in red.\n\n[30]:\n\nfig, ax = plt.subplots(figsize=(8,8))\ntools.display_vector_field(\npathlib.Path('exp1_001.txt'),\nax=ax, scaling_factor=96.52,\nscale=50, # scale defines here the arrow length\nwidth=0.0035, # width is the thickness of the arrow\non_img=True, # overlay on the image\nimage_name= str(path \/ 'data'\/'test1'\/'exp1_001_a.bmp'),\n);","date":"2023-01-30 01:35:23","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\": 0.5560453534126282, \"perplexity\": 3486.40701861681}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2023-06\/segments\/1674764499790.41\/warc\/CC-MAIN-20230130003215-20230130033215-00473.warc.gz\"}"} | null | null |
Revised Building & Construction Industry Forecasts
the impact of the COVID-19 economic shock
Prior to the onset of the COVID-19 crisis, Master Builders forecasted that the performance of the industry was set to be mixed over the next five years with infrastructure-related activity performing well and residential buiding recovering to stronger levels. However commercial building activity was set to fare less favourably and was forecast to lose considerable ground over the next few years.
Key forecasts were below:
With 1.2 million people employed directly by the construction industry and with research suggesting that a further 440,000 jobs in other sectors of the economy are dependent on construction work, our high impact scenario envisages that 464,300 construction jobs could be lost as a result of COVID-19 crisis, and an additional 171,600 jobs lost in the sectors which depend on construction – a total loss in employment of 635,900.
Unfortunately the high impact scenario may be playing out with key data as follows:
ABS data for late March/early April identifying employment nationally down by 6.3% and in the building and construction industry by 5.3%.
Master Builders April COVID-19 Survey results indicating an average reduction of forward contracts across all sectors of 40%.
Master Builders has subsequently undertaken a root and branch revision to its February 2020 forecasts.
These forecasts represent a major update to the set of projections to 2024–25 which were published in February 2020. The update is necessitated by the huge change in economic conditions which has unfolded since the beginning of March amid the global spread of coronavirus. It must be stressed from the outset that the economic environment is highly uncertain and that there is a strong possibility that the ultimate performance of construction could differ significantly from these forecasts.
The following tables summarise the significant downgrade to our forecasts for new home building and commercial building activity in light of COVID-19.
Updated Home Building Forecasts
With the COVID-19 crisis breaking towards the end of the 2019–20 financial year, we anticipate that the sudden and sharp nature of its impacts mean that new home building starts will hit a low of just under 116,000 in 2020–21 (compared with about 159,000) in our original set of forecasts. As stimulus unfolds and restrictions are lifted, activity is expected to recover during 2021–22. The tapering off of stimulus means that the recovery is likely to lose momentum during 2022–23 but thereafter the return to robust population growth and gains in employment will help propel new home building considerably higher and by 2024–25 activity will be close to reverting to the long-term home building trajectory set out in the February 2020 set of forecasts.
A comparison of the updated and previous set of forecasts for new home building to 2024–25 are summarised in the chart and table below.
Updated Commercial Building Forecasts
The commercial building landscape is changed hugely by the crisis: private sector-led projects are likely to suffer due to the very uncertain economic backdrop. In particular, activities like tourism, recreation and entertainment have particularly poor prospects given the combination of enforced closures, restrictions on travel as well as the reduced willingness of large numbers of consumers to engage in non-essential expenditure.
On the other hand, the public sector dominates important areas of commercial building like health and education and the short term prospects for these are much improved given that government stimulus is likely to be heavy here.
The updated forecasts for commercial building are shown below. Overall, the huge reverses in private sector investment will outweigh the possible expansions in government driven projects and the commercial building volumes are likely to be quite depressed over the three-year period between 2020–21 and 2022–23. Thereafter, a reasonably emphatic recovery is likely to resume as the economy returns to its long-run trajectory. | {
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Das Folklorefest Đakovački vezovi, bzw. Stickereikunst im kroatischen Đakovo findet jedes Jahr seit 1967 Anfang Juli statt. Dort werden Trachten gezeigt, Folkloregruppen führen ihre Tänze und Lieder auf, Chöre und Opernsänger treten in der Kathedrale zu Đakovo auf, bildende Künstler stellen ihre Werke aus. Es findet auch eine Schau der Pferde- und Hochzeitsgespanne statt. An der Prachtveranstaltung nehmen rund 50 Folklorensembles mit über 2000 Mitspielenden teil, wobei sich jedes Ensemble durch eine andere Tracht auszeichnet und jede Tracht mit einer anderen Stickerei verziert ist. Das Folklorefest Đakovački vezovi fand 2007 zum 40. Mal statt.
Geschichte
Das erste Folklorefest
Man kann behaupten, dass Đakovački vezovi heutzutage nicht nur ein anerkanntes Fest der Volkstrachten, der Stickereien und der Musik Slawoniens (Slawonien) und Baranjas (Baranja) ist, sondern des ganzen Landes (Kroatien). Aber dabei darf man nicht vergessen, dass man viel Zeit und Mühe brauchte, um dieses Image, die dieses Fest heute besitzt, zu schaffen.
Das erste Folklorefest Đakovački vezovi fand am 2. und 3. Juli 1967 als ein passendes Fest anlässlich des internationalen Jahres des Tourismus statt. Damals war es nur als eine einmalige Veranstaltung geplant. Wegen des unerwarteten Erfolges, einer großen Zahl der Teilnehmern und des positiven Anklang, die diese Veranstaltung in den Medien fand, entwickelte es sich aber zu einer Tradition. Schon 40 Jahre lang findet das Folklorefest Đakovački vezovi Anfang Juli an den Straßen von Đakovo statt.
Besuch der Königin Elisabeth II.
Das Folklorefest Đakovački vezovi wurde mit jedem Jahr immer interessanter und bekannter. Der Beweis dafür ist auch der Besuch der Königin Elisabeth II., der am 20. Oktober 1972 stattfand. Der Vierspänner mit Lipizzanern nahm an einem Wettbewerb in Großbritannien teil, wo auch die Königin Elisabeth II. war und auf das Gestüt und das Folklorefest Đakovački vezovi aufmerksam gemacht wurde. Mit ihrem Mann Philip Mountbatten, Duke of Edinburgh und ihrer Tochter Prinzessin Anne Mountbatten-Windsor, Princess Royal, besuchte sie 1972 Đakovo.
Daten
Das Folklorefest Đakovački vezovi dauert mit allen Festlichkeiten ungefähr 14 Tage. Es beginnt Ende Juni und endet Mitte Juli. Der Höhepunkt mit dem Festzug, den Folkloregruppen und Pferderennen dauert nur ein Wochenende. Gewöhnlich ist es das erste Juliwochenende. Das erste Fest, das stattfindet und der dann alle anderen folgen ist Bonavita. Danach geschehen alle anderen Feste der Reihenfolge, die durch ein Programm jedes Jahr geregelt wird.
Attraktionen
Es gibt mehrere Attraktionen, die das Folklorefest "Đakovački vezovi" bieten kann:
Die kleinen Đakovački vezovi
Gastrofest
Bonavita
Die Wahl der schönsten Tracht
Die sakrale Musik
Pferderennen
Die Ausstellung der kroatischen Schäfer (hrvatski ovčar)
Eine der interessantesten Attraktionen des Festes sind die kleinen Đakovački vezovi. Das ist das Fest der Volkstrachten der kleinen Kinder und der Jugendlichen. Es findet ein Wochenende vor dem Festzug statt. Das Gastrofest ist die Ausstellung und Degustationen der traditionellen slawonischen Küche mit den bekannten Fleischspezialitäten (Kulen = gefüllter Schweinemagen, Würste, Schinken), Speisen aus Wildbret und Süßwasserfischen. Eine weitere Attraktion ist Bonavita, der Wettbewerb, die Ausstellung und Degustation der ausgezeichneten Weine aus Đakovo und seiner Umgebung (Weißer Burgunder, Traminer, Riesling). Das Folklorefest Đakovački vezovi bietet als eine Attraktion die Wahl der schönsten Tracht für Mädchen und junge Frauen. Stickereikunst in Đakovo ist nicht nur eine Trachtenschau. In der Kathedrale treten unter der Orgelbegleitung die berühmte Opernsänger (Ivanka Boljkovac u. a.) oder Sängerchöre auf. Ein Teil des Festes sind auch die Pferderennen. Pferde werden am Gestüt Ivandvor seit 1502 gezüchtet. Das sportliche Programm verläuft auf dem Hippodrom. Springen über Hindernisse, Zwei- und Vierspännerfahrten bieten eine gute Gelegenheit dazu an, die Pferde in ihrer vollen Pracht ansehen zu können. Nach dem Defilee der kroatischen Schäferhunde (hrvatski ovčar) in dem Festzug gibt es auch eine Ausstellung derselben.
Festplatz
Man kann behaupten, dass die ganze Stadt zu einem großen Festplatz wird, weil es im Zentrum keinen Platz gibt, an dem keine Veranstaltung des Festes stattfindet. Die Eröffnung spielt sich vor der Kathedrale St. Peter und Paul ab. Als Zuschauerraum dienen die Straße und der Strossmayer-Hauptplatz. Die Teilnehmer des Festes gehen nach dem Festzug zusammen mit den Zuschauern in den "Strossmayer-Park". Die Sommerbühne im Strossmayer-Park steht im Mittelpunkt dieses Festes. Es wechseln sich Gruppen, Lieder und Tänze einander ab. Das Programm dauert bis in die Abendstunden hinein. Die Hochzeitsgespanne fahren nach dem Festzug zu dem Hippodrom des Gestüts, wo alle Pferderennen stattfinden. Alle Konzerte finden in dem großen Zelt auf dem Parkplatz hinter der Allerheiligenkirche statt. Im Fall von schlechtem Wetter spielen sich dort alle Festlichkeiten ab.
Literatur
Dean Zdravka / dr. Ive Mažuran; Đakovo. 2003.
40. Đakovački vezovi / Revija 2006.
Kultur (Kroatien)
Đakovo
Volksfest in Kroatien | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,964 |
promises a unified way to identify authors of scientific publications.
> The ORCID iD should always be expressed and stored as a URI: http://orcid.org/xxxx-xxxx-xxxx-xxxx (with the protocol (http://), and with hyphens in the number xxxx-xxxx-xxxx-xxxx).
Another issue is that there is actually no RDF exposed from orcid.org .
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
The 5d iridates have attracted much recent attention, because the comparable spin-orbit coupling and electronic correlation energy scales in these materials are expected to be conducive to exotic phases and phenomena, such as superconductivity,\cite{Wang2011, You2012} spin-liquid states,\cite{Okamoto2007} a J$_{eff}$=1/2 Mott state,\cite{Kim2008,Kim2009} topological phases,\cite{Pesin2010,Shitade2009,Kim2012c} and magneto-electric behavior.\cite{Chikara2009} The antiferromagnetic insulator, Sr$_{2}$IrO$_{4}$ (T$_{N}$$\sim$240 K),\cite{Cao1998,Ge2011,Ye2013} is of particular interest, because the strong spin-orbit coupling in this iridate contributes significantly to its insulating behavior:\cite{Kim2008,Kim2009} The Ir$^{4+}$ (5d$^{5}$) ions in Sr$_{2}$IrO$_{4}$ provide 5 d-electrons that occupy the t$_{2g}$ states, which are well separated from higher-energy e$_{g}$ bands by crystal-field effects. The spin-orbit interaction $\lambda\sim$0.4 eV further splits the t$_{2g}$ levels into half-filled J$_{eff}$=1/2 and filled J$_{eff}$=3/2 bands. Insulating behavior is thought to arise because of on-site Coulomb interactions, which split the J$_{eff}$=1/2 level into upper and lower Hubbard bands. This general picture is supported by angle-resolved photoemission, optical conductivity, and x-ray absorption measurements.\cite{Moon2006,Kim2008}
An important unresolved issue concerns the extent to which the strongly entangled spin-orbit states comprising the J$_{eff}$=1/2 Ir 5d moments influence both the magnetic properties and the magnetic excitation spectrum of Sr$_{2}$IrO$_{4}$.\cite{Kim2008,Kim2009,Kim2012b} Neutron scattering studies \cite{Crawford1994,Ye2013} have shown that Sr$_{2}$IrO$_{4}$ has an antiferromagnetic configuration with a $\sim$$11^{\circ}$ canting of the AF spins in the ab-plane, associated with the antisymmetric Dzyaloshinskii-Moriya (DM) spin exchange anisotropy in Sr$_{2}$IrO$_{4}$. This in-plane canting leads to net ferromagnetic in-plane moments (see Fig. \ref{Fig1}(c)), m$_{FM} \approx$0.06$\mu_{B}$,\cite{Fujiyama2012,Bahr2014} that are antiferromagnetically coupled along the c-axis in a ($\uparrow\downarrow\downarrow\uparrow$) pattern.\cite{Kim2009}
However, while the static magnetic configuration of Sr$_{2}$IrO$_{4}$ has been well established, the magnetic excitation spectrum of this strongly spin-orbit-coupled system has not been so well characterized. In particular, it is not clear whether the magnetic excitations associated with the spin-orbit entangled J$_{eff}$=1/2 moments in Sr$_{2}$IrO$_{4}$ can be described by the predictions of an isotropic S=1/2 Heisenberg model.\cite{Fujiyama2012,Igarashi2013,Igarashi2014} Unfortunately, inelastic neutron scattering studies of the magnetic excitations in Sr$_{2}$IrO$_{4}$ are hampered by the strong absorption of neutrons by Ir.\cite{Igarashi2014} Resonant inelastic x-ray scattering (RIXS) studies of Sr$_{2}$IrO$_{4}$ have probed both low-energy charge \cite{Ishii2011} and magnetic \cite{Kim2012b,Liu2015} excitations, but the relatively low resolution associated with RIXS measurements have not allowed a detailed study of the low energy magnetic excitations that would reveal deviations from Heisenberg model predictions. Interestingly, recent resonant magnetic diffuse x-ray \cite{Fujiyama2012} and field-dependent electron spin resonance \cite{Bahr2014} studies of Sr$_{2}$IrO$_{4}$ have offered evidence that the magnetic correlations and excitations are well-described by the two-dimensional S=1/2 Heisenberg model, in spite of the strong spin-orbital coupling associated with the Ir 5d moments.
In this paper, we present an inelastic light (Raman) scattering study of the low energy magnetic excitation spectrum of Sr$_{2}$IrO$_{4}$ and doped Eu-doped Sr$_{2}$IrO$_{4}$ as functions of temperature, applied magnetic field, and magnetic field orientation. Inelastic light scattering is a valuable probe for studying the spin-dynamics of Sr$_{2}$IrO$_{4}$: This technique is a very high resolution probe of the \textbf{q}=0 magnetic excitation energies, which are influenced by small anisotropy and interlayer coupling interactions that can uncover physics beyond the isotropic, two-dimensional S=1/2 Heisenberg model description.\cite{Chovan2000, Gozar2004,Benfatto2006a,Benfatto2006b} Additionally, Raman scattering can probe the spin dynamics both with and without an applied field. Consequently, this technique is useful for studies of spin dynamics in the interesting low-field region of Sr$_{2}$IrO$_{4}$, particularly through the field-induced antiferromagnetic (AF) to weakly ferromagnetic (WFM) spin-flop transition at H$_{c}$$\approx$0.15 T.\cite{Cao1998,Kim2009}
In the results reported here, we show that the in-plane spin dynamics of Sr$_{2}$IrO$_{4}$ at high fields (H$>$1.5 T) are well-described by isotropic, two-dimensional S=1/2 Heisenberg model predictions. By contrast, the low-field (H$<$1.5 T) spin dynamics of Sr$_{2}$IrO$_{4}$ exhibit important effects associated with interlayer coupling and in-plane anisotropy that are not accounted for in standard descriptions of the spin-dynamics of Sr$_{2}$IrO$_{4}$. These effects include an anisotropic field-dependence of the spin-dynamics for H$<$1.5 T, and an AF-to-WFM transition that occurs via either discontinuous spin-flop or continuous spin-reorientation transitions for different in-plane field orientations.
\section{Experimental Set up}
\subsection{Sample Preparation}
The single crystals of Sr$_{2}$IrO$_{4}$ (T$_{N}$$\sim$240K) studied were grown from off-stoichiometric quantities of SrCl$_{2}$, SrCO$_{3}$ and IrO$_{2}$ using self-flux techniques. Technical details are described elsewhere.\cite{Cao1998} The structures of Sr$_{2}$IrO$_{4}$ samples were determined using a Nonius Kappa CCD X-ray diffractometer. The data were collected between 90 K and 300 K, and the structures were refined using the SHELX-97 program.\cite{Sheldrick2008} Chemical compositions of the single crystals were determined using energy dispersive X-ray analysis (EDX) (Hitachi/Oxford 3000).
The Eu-doped Sr$_{2}$IrO$_{4}$ sample (T$_{N}$$\sim$200K) was synthesized at Argonne National Laboratory using a Eu-enriched SrCl$_{2}$ flux method. Samples were characterized by DC magnetization using a Quantum Design SQUID magnetometer.\footnote{STM measurements on the Eu-doped sample (A. Satpathy, private communication) indicate that the Eu concentration is $<$1\% and suggests the possible role of O vacancies at the $\sim$1\% level. Both species in principle are electron donors}
The samples were cleaved to create c-axis normal surfaces, as verified using room temperature x-ray diffraction measurements.
\subsection{Raman Measurements}
Raman scattering measurements were performed using the 647.1 nm excitation line from a Kr$^{+}$ laser. The incident laser power was limited to 5 mW and was focused to a $\sim$50 $\mu$m-diameter spot to minimize laser heating of the samples. The scattered light from the samples was collected in a backscattering geometry, dispersed through a triple stage spectrometer, and then detected with a liquid-nitrogen-cooled CCD detector. The incident light polarization was selected with a combination of a polarization rotator and a 1/4-waveplate and the scattered light polarization was analyzed with a linear polarizer. The scattering geometry used for all measurements had both the incident and scattered polarizations oriented in the ab-planes of the crystals. The incident and scattered light polarizations, \textbf{e}$_{i}$ and \textbf{e}$_{s}$, were kept in a (\textbf{e}$_{i}$,\textbf{e}$_{s}$)=(R,x) configuration for all measurements, where R represents right circular polarized light and x represents linear polarized light oriented in the ab-planes of the crystals.
The samples were inserted into a continuous He-flow cryostat, which was horizontally mounted in the open bore of a superconducting magnet. This experimental arrangement allowed Raman scattering measurements under the simultaneous conditions of low temperature (3-290 K) and high magnetic field (0-8 Tesla). Field-dependent Raman measurements were performed after zero-field cooling the samples to T$\sim$3 K in order to avoid inducing the antiferromagnetic (AF) to weakly ferromagnetic (WFM) alignment of the ferromagnetic spin components in adjacent layers, which occurs for very low critical fields (H$_{c}\sim$0.15 T) in Sr$_{2}$IrO$_{4}$.\cite{Cao1998,Kim2009}
\begin{figure}
\includegraphics[width=9cm] {Fig1}
\caption{\label{Fig1} Temperature dependence of the spin-wave spectra of (a) Sr$_{2}$IrO$_{4}$ and (b) Eu-doped Sr$_{2}$IrO$_{4}$. (c) Summary of the temperature dependence of spin-wave energies for Sr$_{2}$IrO$_{4}$ and Eu-doped Sr$_{2}$IrO$_{4}$. The left inset shows the crystal structure of Sr$_{2}$IrO$_{4}$. Because of octahedral rotations, the unit cell of Sr$_{2}$IrO$_{4}$ contains 4 layers of IrO$_{2}$. The right inset shows the in-plane orientations of the J$_{eff}$=1/2 moments. }
\end{figure}
Temperature- and field-dependent Raman scattering measurements were performed on two different Sr$_{2}$IrO$_{4}$ samples and one Eu-doped Sr$_{2}$IrO$_{4}$ sample. The two Sr$_{2}$IrO$_{4}$ samples studied$-$one of which was used to obtain the temperature-dependent data of Fig. \ref{Fig1}(a) and the second of which was used to obtain the field-dependent data shown in Fig. \ref{Fig2}-\ref{Fig4}$-$exhibited slightly different spin-wave energies (on the order of ~1 cm$^{-1}$ or 0.13 meV energy difference). However, the qualitative temperature- and field-dependences of the spin-wave excitation energies were nearly identical in both Sr$_{2}$IrO$_{4}$ samples. In addition to the spin-wave excitations, a temperature- and field-independent peak was observed in many of the spectra near 29 cm$^{-1}$ (peaks denoted with asterisks (*) in the H=0 T spectra of Fig. \ref{Fig2}(a) and 3(a)). This 29 cm$^{-1}$ peak is associated with unfiltered light from the laser and was fit and subtracted from the spectra at other fields so the field-dependences of the spin-wave excitations could be more clearly observed. Because of the very narrow linewidth of the 29 cm$^{-1}$ peak, its subtraction from the spectra did not affect our determination of the spin-wave energies at different magnetic fields. Note that the higher frequency phonon spectra of the samples studied were also measured and the phonon results obtained were similar in most respects to results reported earlier.\cite{Cetin2012} However, the focus of this paper will be on the spin-wave excitation spectra of Sr$_{2}$IrO$_{4}$ and the phonon spectra will not be shown or discussed further here.
\section{Results}
\subsection{Temperature- and Doping-dependent Results}
Figure \ref{Fig1}(a) shows the low frequency (5$-$35 cm$^{-1}$) excitation spectrum of Sr$_{2}$IrO$_{4}$ as a function of temperature for H=0 T. At temperatures near T$_{N}\sim$240 K, the low energy spectrum exhibits a diffusive background, most likely associated with incoherent spin scattering. Below T$_{N}$, this diffusive background develops into a sharp mode that increases in energy with decreasing temperature to a slightly sample-dependent value near $\omega_{2}$$\sim$17-18 cm$^{-1}$ (2.1-2.3 meV) at T=3 K. Additionally, a weak second peak near $\omega_{1}$=9-10 cm$^{-1}$ is observed in the 3 K spectrum; this lower-energy mode is more clearly observed in the second Sr$_{2}$IrO$_{4}$ sample used for the field dependent measurements (see Fig. \ref{Fig2}) and will be discussed in more detail in the field-dependent results section below.
The effects of doping on the low energy magnetic excitation spectrum of Sr$_{2}$IrO$_{4}$ are also shown in Fig. \ref{Fig1}(b), which displays the temperature dependence of the $\sim$18 cm$^{-1}$ spin wave excitation near in Eu-doped Sr$_{2}$IrO$_{4}$. The temperature dependences of the spin-wave energies in Eu-doped Sr$_{2}$IrO$_{4}$ (filled circles) and Sr$_{2}$IrO$_{4}$ (filled triangles) are summarized in Fig. \ref{Fig1}(c). Several slight differences between the spin wave modes in the doped and undoped Sr$_{2}$IrO$_{4}$ samples are observed: the linewidths of the $\sim$18 cm$^{-1}$ spin wave mode are slightly broader in the doped sample compared to the undoped sample ($\Gamma_{doped}/\Gamma_{undoped}\approx$1.25), which is likely associated with greater spin and potential disorder in the doped sample. The doped sample also exhibits a slightly higher value for the spin-wave mode energy at T=3 K, but this difference is consistent with the sample-to-sample variations we noted for the measured spin-wave energies in undoped Sr$_{2}$IrO$_{4}$; consequently, this energy difference is not believed to be significant. Thus, the most noteworthy feature of Fig. \ref{Fig1}(b) is that there is not a substantial influence of slight doping on the \textbf{q}=0 spin-wave energies in Sr$_{2}$IrO$_{4}$. This conclusion is consistent with evidence that electron doping in Sr$_{2}$IrO$_{4}$ causes a subtle unbuckling of the IrO$_{6}$ octahedra and a crossover to metallic behavior, but does not significantly affect the magnetic properties of Sr$_{2}$IrO$_{4}$.\cite{Ge2011}
\begin{figure}
\includegraphics[width=9cm] {Fig2}
\caption{\label{Fig2} Field dependence of spin-wave spectra of (a) Sr$_{2}$IrO$_{4}$ for H$\parallel$b-axis (=[010]) at T=3 K. The peak marked with an asterisk (*) in the H=0 T spectra is an artifact from the laser and this peak has been removed from the spectra at other fields. The spectra have been offset for clarity. Summary of the field-dependence of the spin wave energies for $\omega_{1}$ (filled triangles) and $\omega_{2}$ (open triangles) in Sr$_{2}$IrO$_{4}$ with H$\parallel$b-axis are shown in (b) (expanded view) and (c) (full range).}
\end{figure}
\subsection{Field-dependent Results}
Figure \ref{Fig2}(a) shows the magnetic-field dependence of the spin-wave spectrum of Sr$_{2}$IrO$_{4}$ for a field orientation parallel to the FM moment (i.e., H$\parallel$b-axis=[010]). The field-dependent results for Eu-doped Sr$_{2}$IrO$_{4}$ are similar and will not be shown. Two spin-wave modes, $\omega_{1}$ and $\omega_{2}$, are clearly evident in the H=0 T spectrum at $\omega_{1}$=11 cm$^{-1}$ (1.38 meV) and $\omega_{2}$=18 cm$^{-1}$ (2.25 meV). As a first step towards identifying these modes, note that in the simplest description of Sr$_{2}$IrO$_{4}$ as a two-dimensional canted antiferromagnet$-$which ignores, in particular, interlayer coupling between the antiferromagnetically coupled layers \cite{Thio1988}$-$the two-fold degenerate \textbf{q}=0 spin-wave branch is expected to split into a low-frequency ``ferromagnetic (FM) mode" and a higher frequency ``antiferromagnetic (AF) mode", associated with precession of the spins about the FM and AF axes, respectively.\cite{White1982,Cottam1986,Gozar2004,Bahr2014} However, we can likely rule out assigning either $\omega_{1}$ or $\omega_{2}$ to the FM mode of Sr$_{2}$IrO$_{4}$, because previous ESR measurements have reported that the FM spin-wave mode in Sr$_{2}$IrO$_{4}$ has an H$\approx$0 T value of $\omega_{FM}$=0.32 cm$^{-1}$,\cite{Bahr2014} which is well below the spectral range of our light scattering study. We can also rule out the possibility that the modes at $\omega_{1}$ and $\omega_{2}$ in Fig. \ref{Fig1}(a) are the same spin-wave mode associated with different magnetic domains in Sr$_{2}$IrO$_{4}$. Magnetic domains have been reported in Sr$_{2}$IrO$_{4}$, but likely involve simple 90$^{\circ}$ rotations of the unit cell,\cite{Dhital2013} which cannot account for the significantly different energies (~$\sim$1 meV) of the $\omega_{1}$ and $\omega_{2}$ modes shown in Sr$_{2}$IrO$_{4}$ (see Fig. \ref{Fig2}(a)). Magnetic domains with a different stacking sequence of the layers$-$such as domains already in the WFM phase at H=0 T$-$would cause the same spin-wave mode to have slightly different energies in the different domains. However, the energy difference in this case would probably not be large enough to account for the large ($\sim$ 1 meV) observed energy difference between the $\omega_{1}$ and $\omega_{2}$ spin wave modes in Sr$_{2}$IrO$_{4}$. Further, to our knowledge there have been no reports that domains associated with the WFM phase are present at H=0 T in Sr$_{2}$IrO$_{4}$.
Therefore, the two spin-wave modes $\omega_{1}$ and $\omega_{2}$ in Fig. \ref{Fig2}(a) are most likely associated with the effects of interlayer coupling between antiferromagnetically coupled IrO layers. As discussed by Thio et al. for La$_{2}$CuO$_{4}$,\cite{Thio1990} interlayer coupling between the two inequivalent (antiferromagnetically coupled) layers in Sr$_{2}$IrO$_{4}$ results in a magnetic unit cell that contains four spins and two 2-fold FM and AF magnon branches whose degeneracies at H=0 T are split by interlayer coupling.\cite{Chovan2000} We associate the spin-wave modes $\omega_{1}$ and $\omega_{2}$ in Sr$_{2}$IrO$_{4}$ with the in-phase and out-of-phase combinations of the AF spin-waves on adjacent layers, respectively. This interpretation is supported by the observed reduction from two \textbf{q}=0 AF spin-wave modes in the antiferromagnetic (AF) phase of Sr$_{2}$IrO$_{4}$$-$which has two magnetically inequivalent layers per unit cell in the simplest model description$-$to a single \textbf{q}=0 AF spin-wave mode (see Figs. \ref{Fig2}(a), \ref{Fig2}(b), and \ref{Fig2}(c)) in the weakly ferromagnetic (WFM) phase of Sr$_{2}$IrO$_{4}$, which has only a single layer per unit cell. In particular, in the WFM phase, the out-of-phase AF spin-wave mode $\omega_{2}$ becomes a zone-boundary mode and only the in-phase AF mode $\omega_{1}$ is expected to be present at \textbf{q}=0.
The importance of interlayer coupling on the spin-wave excitation spectrum of Sr$_{2}$IrO$_{4}$ is also supported by the abrupt increase in the in-phase AF spin-wave energy ( $\omega_{1}$=3.4 cm$^{-1}$ or 0.43 meV) (Figs. \ref{Fig2}(b) and \ref{Fig2}(c)) at H$_{c}$, which reflects an increase in the AF spin-wave stiffness through the AF-to-WFM transition. The energy shift of $\omega_{1}$ at H$_{c}$ allows an estimate of the interlayer coupling energy in Sr$_{2}$IrO$_{4}$: Using the measured change in the energy of $\omega_{1}$ at H$_{c}$ (see Fig. \ref{Fig2}(b)) and the result that,\cite{Chovan2000, Benfatto2006b}
\begin{equation}
4JJ_{\perp}=[\omega_{1}^{2}(H_{c}^{+})-\omega_{1}^{2}(H_{c}^{-})]/\sqrt{2}
\label{eq0}
\end{equation}
we find $4JJ_{\perp}$$\sim$59 cm$^{-2}$ in Sr$_{2}$IrO$_{4}$, giving an estimate for the value of the interlayer coupling energy J$_{\perp}$$\sim$0.018 cm$^{-1}$ (2.3 $\mu$eV) (using J$\sim$800 cm$^{-1}$).\cite{Fujiyama2012} This estimate of J$_{\perp}$ is consistent with published reports for Sr$_{2}$IrO$_{4}$, including estimates based upon the measured critical field H$_{c}$ in Sr$_{2}$IrO$_{4}$: J$_{\perp}$=mH$_{c}$/S$^{2}$,\cite{Thio1988, Chovan2000} which gives J$_{\perp}$$\sim$3 $\mu$eV for Sr$_{2}$IrO$_{4}$, using m=0.07$\mu_{B}$ per Ir atom, H$_{c}$=0.15 T, and S=1/2.
\begin{figure}
\includegraphics[width=9cm] {Fig3}
\caption{\label{Fig3} Field dependences of the spin-wave spectra of Sr2IrO4 for (a) H$\parallel$[100], (b) H$\parallel$[110], and (c) H$\parallel$[001] at T=3 K. The peak marked with an asterisk (*) in the H=0 T spectrum is an artifact from the laser and has been removed from the spectra at other fields. The spectra have been offset for clarity. (d) Summary of the field-dependences of the spin wave energies for different applied field orientations for both $\omega_{1}$(filled symbols) and $\omega_{2}$(open symbols). Also shown for comparison are results from ESR measurements \cite{Bahr2014} for H$\parallel$[001] (filled stars). The dashed line is a fit to the data with the functional form $\omega$=$\sqrt{\gamma H}$ using $\gamma$=209.38 cm$^{-2}$T$^{-1}$.}
\end{figure}
The magnetic-field dependences of the AF spin-wave energies $\omega_{1}$ and $\omega_{2}$ of Sr$_{2}$IrO$_{4}$ are shown for different applied field orientations in Fig. \ref{Fig3}. Figure \ref{Fig3}(a) shows the magnetic field dependences of $\omega_{1}$ and $\omega_{2}$ with H roughly parallel to the spin direction, H$\parallel$a-axis=[100], while Fig. \ref{Fig3}(b) shows the magnetic field dependences of $\omega_{1}$ and $\omega_{2}$ with H oriented roughly 45$^{\circ}$ from the a-axis, i.e., H$\parallel$[110]. Note that the H=0.4 T spectrum in Fig. 3 (a) shows three peaks, consisting of a superposition between the two spin-wave modes of the AF phase and the single spin-wave mode in the WFM phase. This superposition is consistent with a coexistence of AF and WFM phases expected near the first-order transition at H$_{c}$.
Also shown in Fig. \ref{Fig3}(c) is the magnetic field dependence of the \textbf{q}=0 spin-wave spectrum in Sr$_{2}$IrO$_{4}$ for the out-of-plane magnetic field orientation, i.e., with H roughly parallel to the c-axis direction, H$\parallel$[001]. The $\omega_{1}$$\sim$8 cm$^{-1}$ (not shown) and $\omega_{2}$$\sim$18 cm$^{-1}$ spin-wave mode energy exhibits a much weaker magnetic field dependence for H$\parallel$c-axis=[001], consistent with previous electron spin resonance (ESR) results (filled stars).\cite{Bahr2014} An additional weak mode develops near $\sim$13 cm $^{-1}$ for H$>$4 T with H$\parallel$c-axis=[001]. This mode may be associated with the presence of a small in-plane field caused by a slight misalignment of the magnetic field in the H$\parallel$c-axis configuration, which can induce an AF-to-WFM transition$-$and a lower value for the spin-wave energy (as discussed above)$-$in parts of the sample.
\begin{figure}[!]
\includegraphics {Fig4}
\caption{\label{Fig4} Summaries of the field dependences of the spin-wave energies of Sr$_{2}$IrO$_{4}$ for different in-plane field orientations, including (a) H$\parallel$a-axis=[100], (b) H$\parallel$[110], and (c) H$\parallel$b-axis=[010] at T=3 K. (Closed symbols= $\omega_{1}$ mode, Open symbols= $\omega_{2}$ mode) The dashed lines are plots of $\omega$=$\sqrt{\gamma H}$ with $\gamma$=209.38 cm$^{-2}$T$^{-1}$ for comparison with the data. The dotted-dashed line is a fit to the data with the functional form of $\omega$=$\sqrt{\Delta^{2}-\alpha H^{2}}$ with $\Delta$=18.9 cm$^{-1}$ and $\alpha$=1014 cm$^{-2}$T$^{-2}$. Also shown for comparison are results from ESR measurements\cite{Bahr2014} for H$\parallel$[110] (filled stars). (d) Schematic illustration of the rotation of the staggered spin components (red arrows) and uniform spin component (black arrows) on adjacent layers for an applied field (green arrow) oriented transverse to easy axis direction of FM component of the spin, H$\parallel$a-axis=[100], illustrating the continuous rotation of the spins on adjacent layers for this applied field orientation. (e) Schematic illustration of the rotation of the staggered spin components (red arrows) and uniform spin component (black arrows) on adjacent layers for an applied field (green arrow) oriented parallel to easy axis direction of FM component of the spin, H$\parallel$ b-axis=[010], illustrating the abrupt flipping of the spins in one layer for this field orientation. (f) Diagram showing the angle $\theta$ of the FM component of the spins (\textbf{m}) and the angle $\phi$ of the applied field (H) relative to the easy axis in-plane orientation of \textbf{m} (i.e., [010]).}
\end{figure}
\section{Discussion}
The central result of this study concerns the magnetic-field-dependences (0$\leq$H$\leq$8 T) of the AF spin-wave mode energies summarized in Fig. \ref{Fig3}(d) for different in-plane magnetic field orientations, H$\parallel$a-axis=[100] (filled squares), H$\parallel$b-axis=[010] (filled triangles), and H$\parallel$[110] (filled circles). Fig. \ref{Fig3}(d) illustrates that there are 2 distinct field regimes for the in-plane spin dynamics in Sr$_{2}$IrO$_{4}$, (A) an isotropic regime for H$\gtrsim$1.5T and (B) an anisotropic regime for H$\lesssim$1.5T.
\subsection{Isotropic regime H$\gtrsim$1.5T }
For H$>$1.5 T, the in-plane spin dynamics are isotropic and the AF spin-wave mode $\omega_{1}$ energy in the WFM phase region is well-described by a square-root field dependence, $\omega_{1}=\sqrt{\gamma H}$, with $\gamma$=209.4 cm$^{-2}$T$^{-1}$ (dashed line). The isotropic square-root field dependence for H$>$1.5 T indicates that the FM components of the spins simply follow the applied field direction in Sr$_{2}$IrO$_{4}$, due to the dominant interaction between the applied field and the weak FM moments induced by the DM interaction.\cite{Cottam1986,Benfatto2006a,Bahr2014}
The spin dynamics above H$>$1.5 T in Sr$_{2}$IrO$_{4}$ are consistent with an isotropic, two-dimensional effective S=1/2 Hamiltonian given by: \cite{Jackeli2009,Bahr2014}
\begin{equation}
H_{12}=J\vec{S_{1}}\cdot\vec{S_{2}}+\Gamma S_{1}^{z}S_{2}^{z}+D(S_{1}^{x}S_{2}^{y}-S_{1}^{y}S_{2}^{x})
\label{eq1}
\end{equation}
where the first term (J) is associated with isotropic antiferromagnetic exchange between the two inequivalent spins, 1 and 2, in the IrO plane, the second term ($\Gamma$) represents symmetric exchange anisotropy that favors collinear c-axis spin order, and the third term (D) represents antisymmetric exchange anisotropy that favors canted in-plane spin order. Bahr \textit{et al}. predict that for $\Gamma$,D $\ll$ J, the AF spin-wave energy associated with the model Hamiltonian in Eq.(\ref{eq1}) should have a field-dependence given by,\cite{Bahr2014}
\begin{equation}
\omega_{1}\approx\sqrt{\Delta^{2}+8Jm_{FM}H}
\label{eq2}
\end{equation}
where J$\sim$100 meV in Sr$_{2}$IrO$_{4}$ and m$_{FM}$ is the FM canting moment. This prediction is consistent with the square-root field dependence we observe for $\omega_{1}$ in Fig. \ref{Fig3}(d). Using our value of $\gamma$=209.4 cm$^{-2}$T$^{-1}$ from the fit to the data in Fig. \ref{Fig3}(d) (dashed line) with $\Delta$$\sim$ 0, we obtain an estimated FM canting moment of m$_{FM}$$\sim$$\gamma$/8J $\sim$ 0.07$\mu_{B}$ in Sr$_{2}$IrO$_{4}$, which is consistent with other estimates (e.g., see ref. 16). Notably, the $\gamma$ value determined from the field-dependence of the AF spin wave in La$_{2}$CuO$_{4}$ ($\gamma_{LCO}$= 22.6 cm$^{-2}T^{-1}$)\cite{Gozar2004} is much smaller than our value for Sr$_{2}$IrO$_{4}$, reflecting the much smaller FM moment associated with spin canting in La$_{2}$CuO$_{4}$ (m$_{FM}$$\sim$0.002$\mu_{B}$).\cite{Thio1988}
\subsection{Anisotropic regime H$\lesssim$1.5T}
Figure \ref{Fig3}(d) shows that the field-dependent spin-wave dynamics for H$<$1.5 T are highly anisotropic in the planes, revealing interaction effects in Sr$_{2}$IrO$_{4}$ that are not accounted for in Eq.(\ref{eq1}). A more detailed view of the anisotropic magnetic field dependence of the AF spin wave energy in Sr$_{2}$IrO$_{4}$ is provided in Fig. \ref{Fig4}, which shows the field-dependences of spin-wave energies $\omega_{1}$ and $\omega_{2}$ in the field range 0$\leq$H$\leq$1.5 T for several in-plane field orientations, including (a) H$\parallel$a-axis=[100], (b) H$\parallel$[110], and (c) H$\parallel$b-axis=[010]. As discussed above, Fig. \ref{Fig4}(c) shows that the in-phase AF spin-wave energy $\omega_{1}$ exhibits an abrupt increase in energy ($\Delta$$\omega_{1}$=3.4 cm$^{-1}$ or 0.43 meV) through the AF-to-WFM spin-flop transition at H$_{c}$$\sim$0.15 T when the applied field is oriented in the direction of the FM (uniform) spin component of the spins, H$\parallel$b-axis=[010]. This behavior indicates that the AF-to-WFM transition in Sr$_{2}$IrO$_{4}$ occurs via a discontinuous spin-flop transition, and results in a discontinuous change in interlayer coupling, when the applied field is oriented along the weak FM component of the spins (see Fig. \ref{Fig4}(e)).
On the other hand, Fig. \ref{Fig4}(a) shows that when the applied field is oriented parallel to the staggered spins, H$\parallel$a-axis=[100], AF spin-wave modes $\omega_{1}$ and $\omega_{2}$ exhibit ``soft mode" behavior: the field-dependence of $\omega_{2}$ with H$\parallel$a-axis=[100] is well described by the functional form $\omega_{2}=\sqrt{\Delta^{2}-\alpha H^{2}}$ (dashed-dotted line) with $\Delta$=18 cm$^{-1}$ and $\alpha$=1014 cm$^{-2}$$T^{-2}$. The soft spin-wave mode behavior shown in Fig. \ref{Fig4}(a) indicates that the AF-to-WFM transition involves a continuous spin reorientation and a gradual crossover when H$\parallel$a-axis=[100] (Figure \ref{Fig4}(d)). The field-dependence of the AF spin wave energy of Sr$_{2}$IrO$_{4}$ for H$\parallel$[110], shown in Fig. \ref{Fig4}(b), exhibits behavior intermediate to that observed for the H$\parallel$[010] and H$\parallel$[100] orientations. Also shown for comparison in Fig. \ref{Fig4}(b) is the field-dependence of the AF spin-wave mode $\omega_{AF}$ determined from electron-spin-resonance (ESR) measurements with H$\parallel$[110] (filled stars),\cite{Bahr2014} showing that there is a good agreement between the AF spin-wave energies measured with Raman scattering and ESR for this H$\parallel$[110] orientation.
The dramatic difference in the nature of the AF-to-WFM transition for different in-plane field orientations (Fig. \ref{Fig4}) reflects the importance of in-plane anisotropy for H$\lesssim$1.5T in Sr$_{2}$IrO$_{4}$. Similar effects of in-plane anisotropy on the spin dynamics of ferrimagnets \cite{Horner1968} and canted antiferromagnets \cite{Pincus1960,Shapiro1974,Fainstein1993} have been observed previously, particularly in iridates,\cite{Wang2014} cuprates \cite{Thio1990,Fainstein1993} and ferrites. \cite{Pincus1960, Levinson1969,Koshizuka1988,Gorodetsky1970,Shapiro1974} The AF spin-wave mode softening observed in Sr$_{2}$IrO$_{4}$ (Fig. \ref{Fig4}(a)) reflects a continuous decrease in the interlayer exchange energy in Sr$_{2}$IrO$_{4}$ with applied field for H$\perp$m$_{FM}$, caused by the continuous field-induced rotation of the FM moments in opposite directions in the antiferromagnetically coupled layers (see Fig. \ref{Fig4}(d)).
An estimate of the in-plane anisotropy field, H$_{A}$, can be obtained from our data by first developing a simple phenomenological description of the interlayer coupling energy between two adjacent layers can be written, E$_{\perp}$$\sim$J$_{\perp}$cos(2$\theta$)=J$_{\perp}$(1-2sin$^{2}$($\theta$)), where $\theta$ is the angle between the FM spin components and their zero-field (easy axis) directions in the each of two layers (see Fig. \ref{Fig4}(f)). The interlayer coupling energy can be written in terms of the applied in-plane field H, using the result that the equilibrium in-plane orientation for the weak FM moment in each layer for a particular field H is given by:\cite{Fainstein1993}
\begin{equation}
(H_{DM}/H_{E})Hsin(\phi-\theta)=(H_{A})sin(2\theta)
\label{eq3}
\end{equation}
where H$_{DM}$ is the Dzyaloshinskii-Moriya field, H$_{E}$ is the exchange field, H$_{A}$ is the in-plane anisotropy field, H is the applied field, $\theta$ is the angle between the FM spin component, m$_{FM}$, and its zero-field (easy axis) orientation, $\phi$ is the angle between the applied field and the easy-axis, and assuming H$_{E}$$\gg$H$_{DM}$$\gg$H$_{A}$$\sim$H. Equation (\ref{eq3}) shows that a field applied perpendicular to the easy-axis orientation of m$_{FM}$ (i.e., H$\perp$m$_{FM}$ or $\phi$=$\pi$/2), which is the field orientation for which we observe soft magnon behavior (see Fig. \ref{Fig4}(a)), induces an in-plane rotation of m$_{FM}$ by an angle $\theta$ that increases continuously with the applied field according to sin($\theta$)=(H$_{DM}$/2H$_{A}$H$_{E}$)H, as schematically depicted in Fig. \ref{Fig4}(d). In the AF phase, the ferromagnetic components in adjacent layers rotate in opposite directions in response to an applied transverse in-plane field. Consequently, the interplane exchange energy for a pair of coupled layers will continuously decrease with field H for H$<$H$_{c}$ according to E$_{\perp}$$\sim$J$_{\perp}$(1-2$\beta^{2}$H$^{2}$), where $\beta$=(H$_{DM}$/2H$_{E}$H$_{A}$). This functional form for E$_{\perp}$ is consistent with the observed field-dependences of the spin wave mode energies near H$\sim$0.4 T for H$\parallel$a-axis=[100] (see Fig. \ref{Fig4}(a)). Note that the interplane exchange energy goes to zero, E$_{\perp}\rightarrow$0, at a critical field given by H$_{c}$=$\sqrt{2}$(H$_{E}$/H$_{DM}$)H$_{A}$. Using our rough measurement of the field at which the AF spin-wave mode energy approaches zero, H$_{c}$$\approx$0.4T, and an estimate of the ratio (H$_{DM}$/H$_{E}$) using tan(2$\xi$)=(H$_{DM}$/H$_{E}$),\cite{Bahr2014} where $\xi$=11$^{\circ}$ is the canting angle, we obtain a value for the in-plane anisotropy field in Sr$_{2}$IrO$_{4}$, H$_{A}$=1/$\sqrt{2}$(H$_{DM}$/H$_{E}$)H$_{c}$$\approx$0.1T. This estimate compares well with the coercive field $\sim$0.15T needed to induce an abrupt ``spin-flip" transition between AF and WFM phases for H$\parallel$b-axis =[010] (see Fig. \ref{Fig4}(c)). Additionally, the minimum value for the spin-wave energy at H=0.4 T (see Fig. \ref{Fig4}(a)), $\Delta$$\sim$8 cm$^{-1}$ (1 meV), offers a good estimate of the spin-gap energy in Sr$_{2}$IrO$_{4}$ without the effects of interlayer coupling.
\section{Summary}
In the field- and temperature-dependent Raman scattering studies of the angle-dependence of spin excitations of Sr$_{2}$IrO$_{4}$ presented here, we show clear evidence for a magnetic field scale H$\sim$1.5 T above which the in-plane spin dynamics behave in accordance with the predictions of an isotropic, two-dimensional effective S=1/2 Hamiltonian. The field-dependence of spin-wave excitations in this ``high field" regime are isotropic, two-dimensional, and solely governed by the interplay between the applied field and the FM component of the spins associated with the DM interaction. However, dramatic deviations from this isotropic and two-dimensional behavior are clearly observed at lower fields, H$<$1.5 T, manifested, for example, in a highly anisotropic field-dependence of the spin dynamics and interlayer-exchange-split spin-wave modes. Particularly noteworthy is the observation of field-induced magnon soft mode behavior near H$_{c}$ for a field applied transverse to the FM spin components, H$\perp$m$_{FM}$, which reveals a continuous spin rearrangement transition at the antiferromagnetic-to-weakly ferromagnetic transition at H$_{c}$ in Sr$_{2}$IrO$_{4}$. Our results also show that when the in-plane field is aligned perpendicular to the easy-axis direction of the FM moment, the field dependence of the \textbf{q}=0 spin-wave energy evolves according to $\omega$$\sim$$H^{1/2}$ above H$_{c}$, i.e., in a manner consistent with a 2D canted antiferromagnet with no spin gap. These studies highlight the importance of considering in-plane anisotropy and interlayer coupling effects on the low energy spin dynamics when interpreting and calculating the low-field magnetic and dynamical properties of Sr$_{2}$IrO$_{4}$.\\
\textbf{Acknowledgments}$-$Research was supported by the National Science Foundation under Grant NSF DMR 14-64090. Work at the University of Kentucky was supported by the National Science Foundation via Grant Nos. DMR-1265162. Work at Argonne National Laboratory (crystal growth and magnetic characterization) was supported by the U.S. Department of Energy Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. We thank Sam Gleason for useful discussions
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,021 |
Division of Neonatololgy-Departments & Services-For Patients-West China the second hospital
Doctor's Clinic Schedule
For Outpatients
For Inpatients
For Emergency Patients
Division of Neonatololgy
The Division of Neonatololgy of West China Second University Hospital was founded in 1973. It is one of the earliest neonatal medical units in China and the only unit with National Key Clinical Discipline (Neonatology) and National Neonatal Health Care Specialty in Yunnan, Guizhou, Sichuan and Tibet. The overall level of our discipline is leading in the country. It boasts for its "National Fund for Distinguished Young Scholars" and the only "Changjiang Scholars' innovation team of the Ministry of education" in Chinese pediatric community. The division is among the first batch of standardized training bases for neonatal and perinatal medical specialist in China and neonatal specialist nurses in Sichuan Province. It is the only unit with Key Clinical Specialty Program of the Ministry of Health of China in western region, and the therapeutic center for critically ill newborns in Sichuan Province. It is a referral center in Western China for critically ill neonates with most advanced equipment and the strongest medical care technology.
The Division of Neonatology has set up two wards at Huaxi and Jinjiang Campuses respectively. It includes Neonatal Intensive Care Unit (NICU), general neonatal ward and neonatal surgery ward, with 305 beds, including 200 for NICU and 12 for surgery. More than 6,000 patients were admitted in 2021, in which more than 95% patients were critically ill, and more than 70% patients were difficulty and critically ill babies transferred in utero or referred by all levels medical units after birth(是否是这个意思:and more than 70% patients were referred by hospitals of all levels, coming with complicated and critical in utero conditions ). The success rate of rescue has reached 99.7%.
There are 6 doctoral tutors and 15 master tutors in the division. Nearly 30 doctoral students and more than 50 master students have graduated in the past five years. The academic leader, professor MU Dezhi is the first neonatal physician in China who has secrured the grant of the National Funds for Distinguished Young Scientists, is the leader of Changjiang Scholars Innovation Team of the Ministry of Education, has been honored the young and middle-aged experts with outstanding contributions, the first National Famous Doctor with Outstanding Demeanor. He is also the vice chairman of pediatrics branch and perinatal medicine branch of Chinese Medical Association, vice president of pediatric branch and neonatal branch of Chinese Medical Doctor Association. The division has a strong team with powful technical strength. There are 51 physicians, 3 reemployed retired professors, 8 neonatal respiratory therapists, and 227 nurses. In the team, 13 members carry senior professional titles, 5 associalte senior professional titles. Most of the physicians hold doctoral degrees and have been trained overseas.
We have a first-class scientific research platform in China, including the Pediatric Neurodevelopmental Laboratory, the Laboratory of Lung Development and Diseases, and the Laboratory of Perinatal Medicine. From 2017 to 2021, we conducted 22 national scientific research projects, 22 provincial and ministerial projects, and held 48 patents. We have published about 500 journal articles, including more than 140 on SCI journals. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,466 |
Clear day ahead in Delhi
New Delhi, It was a pleasant Monday morning in Delhi with the minimum temperature recorded at 25 degrees Celsius, a notch above the season's average. The Met Office has forecast a clear day ahead.
"Sky will remain clear and the maximum temperature is likely to hover around 36 degrees Celsius," said an official of the India Meteorological Department.
Humidity at 8.30 a.m. was 61 percent.
Sunday's maximum temperature was recorded two notches above the season's average at 36 degrees Celsius and the minimum temperature settled at 25.2 degrees Celsius, slightly above the average.
Source:IANS | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,968 |
Q: Reg~x for the N~number Would be grateful your help.
What would be elegant Regx to check a String whether it holds a number ? any suggestions? thanks.
A: I don't think that an exception (even if it is triggered) is that much more expensive than a regular expression - you'd have to profile it and see if it really makes a difference.
That said, a regular expression that implements the BigDecimal grammar according to the Java Docs is
[+-]?(?:\d+(?:\.\d*)?|\.\d+)(?:[eE][+-]?\d+)?
Explanation:
[+-]? # optional sign
(?: # required significand: either...
\d+ # a number
(?:\.\d*)? # optionally followed by a dot, optionally followed by more digits
| # or...
\.\d+ # just a dot, followed by digits (in this case required)
) # end of significand
(?: # optional exponent
[eE] # required exponent indicator
[+-]? # optional sign
\d+ # required digits
)? # end of exponent
If you want to allow different number formats (and thousands separators) you could first get those values from DecimalFormatSymbols and build your regex with it.
Something like this (I don't know Java, so feel free to correct my Syntax Errors):
// you need java.util.regex.Pattern and java.text.DecimalFormatSymbols
string ds = Pattern.quote(DecimalFormatSymbols.getDecimalSeparator())
string gs = Pattern.quote(DecimalFormatSymbols.getGroupingSeparator())
string ms = Pattern.quote(DecimalFormatSymbols.getMinusSign())
string es = Pattern.quote(DecimalFormatSymbols.getExponentSeparator())
string myre =
"(?xi)\n # verbose, case-insensitive regex" +
"[+" +ms+ "]? # optional sign\n" +
"(?: # required significand: either...\n" +
" (?:\\d{1,3}(?:" +gs+ "\\d{3}|\\d++) # a number with optional thousand separators,\n" +
" (?:" +ds+ "\\d*)? # optionally followed by a dot, optionally followed by more digits\n" +
" | # or...\n" +
ds+ "\\d+ # just a dot, followed by digits (in this case required)\n" +
") # end of significand\n" +
"(?: # optional exponent\n" +
es+ " # required exponent indicator\n" +
" [+" +ms+ "]? # optional sign\n" +
" \\d+ # required digits\n" +
")? # end of exponent"
boolean foundMatch = subjectString.matches(myregex);
You could then replace the locale-dependent bits back into their US counterparts before handing the number over for BigDecimal conversion, if that's not locale-aware.
A: Take a look at the implementation of the Apache's NumberUtils.isNumber. I wouldn't recommend using regular expressions for such validation.
A: Do not use regular expressions, they almost always are not the answer to any problem. You might accept some regex and then later on find out that it does not cover all situations. If you really want to know if a string has a BigDecimal, why not ask the source BigDecimal itself, after all it by definition is always right while using a regular expression cant always be right, you might just get it wrong.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,769 |
This website provides a tool that allows us to create diagrams and flow charts. It is an easy collaboration tool. The basic version is free, and allows us to make 5 diagrams.
Online tool that allows us create videos and presentations. We select images, select music and share it with others. It's free but we need to register.
This website provides a tool that allows us to organizing the tasks and work collaborately. It's free but first we need to register.
This website provides an online tool that allows us to create an online newspaper with all our Twitter and/or Facebook activity. We can read other newspapers in the website.
This webpage contains an interactive book that with cartoon animals in letter shapes. It is designed to help us learn the alphabet letters. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,640 |
{"url":"https:\/\/matbesancon.xyz\/post\/2022-04-29-expression-trees\/","text":"# Pruning the expression tree with recursive value identification\n\nToday was the release of SCIP.jl v0.11, the first release switching to SCIP 8. The major change in this (massive) release was the rewrite of the nonlinear optimization part, using a so-called expression framework. The rewrite of the wrapper had some fairly tedious parts, debugging C shared libraries is quickly a mess with cryptic error messages. But the nonlinear rewrite gave me the opportunity to tweak the way Julia expressions are passed to SCIP in a minor way.\n\n# SCIP expressions\n\nI will not go in depth into the new expression framework and will instead reference these slides but more importantly the SCIP 8 release report\n\nThe key part is that in a nonlinear expression, each operand is defined as an expression handler, and new ones can be introduced by users. Several specialized constraint types or constraint handlers in SCIP terminology were also removed, using the expression framework with a generic nonlinear constraint instead.\n\n# The Julia wrapper initial framework\n\nAs a Lisp-inspired language, (some would even a Lisp dialect), Julia is a homoiconic language: valid Julia code can always be represented and stored in a primitive data structure. In this case, the tree-like structure is Expr with fields head and args:\n\njulia> expr = :(3 + 1\/x)\n:(3 + 1 \/ x)\n\n:call\n\njulia> expr.args\n3-element Vector{Any}:\n:+\n3\n:(1 \/ x)\n\n\nThe SCIP.jl wrapper recursively destructures the Julia expression and builds up corresponding SCIP expressions, a SCIP data structure defined either as a leaf (a simple value or a variable) or as an operand and a number of subexpressions. This is done through a push_expr! function which either:\n\n\u2022 Creates and returns a single variable expression if the expression is a variable\n\u2022 Creates and returns a single value expression if the expression is a constant\n\u2022 If the expression is a function f(arg1, arg2...), calls push_expr! on all arguments, and then creates and returns the SCIP expression corresponding to f.\n\nOne part remains problematic, imagine an expression like 3 * exp(x) + 0.5 * f(4.3), where f is not a primitive supported by SCIP. It should not have to be indeed, because that part of the expression could be evaluated at expression compile-time. But if one is walking down the expression sub-parts, there was no way to know that a given part is a pure value, the expression-constructing procedure would first create a SCIP expression for 4.3 and then try to find a function for f to apply with this expression pointer as argument. This was the use case initially reported in this issue at a time when SCIP did not support trigonometric functions yet.\n\nAnother motivation for solving this issue is on the computational and memory burden. Imagine your expression is now 3 * exp(x) + 0.1 * cos(0.1) + 0.2 * cos(0.2) + ... + 100.0 * cos(100.0). This will require producing 2 * 1000 expressions for a constant, declared, allocated and passed down to SCIP. The solver will then likely preprocess all constant expressions to reduce them down, so it ends up being a lot of work done on one end to undo immediately on the other.\n\n# A lazified expression declaration\n\nMake push_expr! return two values (scip_expr, pure_value), with the second being a Boolean for whether the expression is a pure value or not. At any leaf computing f(arg1, arg2...).\n\nIf the expression of all arguments are pure_value, do not compute the expression and just return a null pointer, pure_value is true for this expression.\n\nIf at least one of the arguments is not a pure_value, we need to compute the actual expression. None of the pure_value arguments were declared as SCIP expressions yet, we create a leaf value expression for them with Meta.eval(arg_i). The non-pure value arguments already have a correct corresponding SCIP expression pointer. pure_value is false for this expression.\n\nNote here that we are traversing some sub-expressions twice, once when walking down the tree and once more hidden with Meta.eval(arg_i) which computes the value for said expression, where we delegate the expression value computation to Julia. An alternative would be to return a triplet from every push_expr! call (expr_pointer, pure_value, val) and evaluate at each pure_value node the value of f(args...), with the value of the arguments already computed. This would however complexity the code in the wrapper with no advantage of the runtime, the expression evaluation is not a bottleneck for expressions that can realistically be tackled by a global optimization solver like SCIP.\n\n##### Mathieu Besan\u00e7on\n###### Researcher in mathematical optimization\n\nMathematical optimization, scientific programming and related.","date":"2023-03-27 05:07:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4434530735015869, \"perplexity\": 1797.9703140088884}, \"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-2023-14\/segments\/1679296946637.95\/warc\/CC-MAIN-20230327025922-20230327055922-00611.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaC4"
} | 4,004 |
The 1961 Brussels Grand Prix was a motor race set to Formula One rules, held on 9 April 1961 at Heysel Park, Belgium. The race was run in three "heats" of 22 laps each and the times were aggregated. The race was won by Australian driver Jack Brabham in a Cooper T53.
Results
Notes
Maurice Trintignant had previously entered as the No. 40 Cooper-Climax car, but withdrew.
Wolfgang von Trips and Mauro Bianchi had also entered, but did not participate.
Jo Schlesser made his debut in this race.
Although performing extremely well in the practice and qualifying sessions, Jack Lewis was not admitted to the race, since he had failed to be on the Belgian organizers' list of fame.
Equipe Nationale Belge had entered two modified Emeryson chassis fitted with the heavy Maserati engine for Olivier Gendebien and Lucien Bianchi, and a standard one with the Coventry Climax engine, for Willy Mairesse.
References
Brussels Grand Prix | {
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\section{Introduction}
Coupled-oscillator models have been widely used in the studies of a variety of rhythmic phenomena in physics, chemistry, and biology~\cite{kuramoto2012chemical, winfree2001geometry,ashwin2016mathematical}.
While many results for coupled oscillators have been obtained for phase models, in which the amplitude variable of oscillators is eliminated and only the phase of oscillators is considered, the effect of amplitude degrees of freedom often leads to intriguing collective dynamics that cannot occur in phase models.
Regarding collective oscillations in coupled oscillators with amplitude degrees of freedom, Daido and Nakanishi performed a pioneering study that revealed the effect of deteriorated inactive oscillators on macroscopic collective oscillation of the whole system, which they called aging transition~\cite{daido2004aging}. In their model, macroscopic synchronized oscillation turns into a quiescent state as the proportion of inactive elements exceeds a certain critical value when coupling strength is sufficiently large.
The aging transition framework of Daido and Nakanishi, originally analyzed for globally coupled oscillators~\cite{daido2004aging, pazo2006universal, tanaka2010phase, daido2011strong, daido2013onset}, has been generalized to various kinds of coupled-oscillator models with different coupling topologies. For example, aging transitions in a one-dimensional ring of locally coupled oscillators~\cite{daido2008aging} and in networks of coupled oscillators with random coupling~\cite{morino2011robustness,tanaka2012dynamical} have been analyzed.
However, compared with global coupling, mathematical treatment of other coupling topologies is difficult and relied mostly upon numerical simulations in most cases. As for coupled oscillators with lattice Laplacian coupling, which is a discretized version of ordinary diffusion in continuous media, spatial arrangement of the active and inactive oscillators decisively affects the aging transition in contrast to the case of global coupling.
The main aim of this paper is providing a theory to deal with heterogeneous spatial arrangement of active and inactive oscillators. We consider the suppression of macroscopic oscillations in mixed populations of active and inactive oscillators (i.e., aging transition) coupled through lattice Laplacian in general dimensions. By introducing a generalized free energy (Lyapunov function) of the system and approximating it with two variables, we show that an effective wavenumber of the system can be introduced, which characterizes the arrangement of the active and inactive oscillators, and can be used for determining whether suppression of macroscopic oscillation occurs or not.
Another aim of this paper is to show how the aging transition framework with the proposed theory provides insights into realistic situations in a cortico-thalamic model of epileptic seizures.
Though the oscillating state and quiescent state were interpreted as alive and dead respectively in the original study~\cite{daido2004aging}, the aging transition framework can also be interpreted in the opposite sense.
That is, we consider whether pathological macroscopic oscillation caused by pathological oscillators can be suppressed when the proportion of the healthy oscillators is increased~\cite{kim2009dynamics}.
Using the proposed theory, we analyze a cortico-thalamic model of epileptic seizures, where active and inactive oscillators correspond to epileptic foci and healthy region, respectively.
\section{Theory}
\subsection{Model}
We consider a mixed population of active and inactive oscillators, where the active group is denoted as S (seizure) and the inactive group is denoted as H (healthy).
The oscillators are placed on a $p$-dimensional hypercubic lattice of $N^{p}$ sites with a lattice constant $d$, where $L=Nd$ is the side length of the hypercube, and periodic boundary conditions are assumed.
The system is described by the following lattice complex Ginzburg-Landau model on a $p$-dimensional lattice ($p$ is a natural number) of lattice constant $d$:
\begin{eqnarray}
\frac{dB}{dt}=-i \Omega B + (a+ib) \left[ \left( \frac{1}{2} \mu+\frac{3}{8} \epsilon \left | B \right |^{2} \right ) B + \frac{1}{2} r_{e}^{2} \Delta B \right ].
\label{CGL_0}
\end{eqnarray}
Here, $B=B(\bm{j}; t)$ is the complex amplitude of the oscillator at lattice coordinate $\bm{j}=(j_1, j_2, \dots, j_p)$ and at time $t$, $\Omega$ is the natural frequency of the oscillator at the onset of oscillation, $a >0$, $b$ and $\epsilon <0$ are real parameters, and $r_{e}$ represents a diffusion constant.
The Hopf bifurcation parameter $\mu = \mu({\bm j})$ determines if the oscillator is active ($\mu>0$) or inactive ($\mu<0$), and $\Delta$ denotes the $p$-dimensional discrete (lattice) Laplacian defined as
\begin{equation}
\Delta B(\bm{j}) = \frac{1}{d^2} \sum_{q=1}^{p}\{ B(j_1, \dots, j_q-1, \dots, j_p) + B(j_1, \dots, j_q+1, \dots, j_p) - 2 B(\bm{j}) \}.
\end{equation}
For simplicity, we restrict our investigation to the case where all oscillators have the same frequency parameter $\Omega$.
Then, by introducing a rotating frame with frequency $\Omega$ and denoting $B=Ae^{i \Omega}$, we obtain
\begin{eqnarray}
\frac{dA}{dt}= (a+ib) \left[
\left( \frac{1}{2} \mu+\frac{3}{8} \epsilon \left | A \right |^{2} \right ) A + \frac{1}{2} r_{e}^{2} \Delta A \right ].
\label{CGL}
\end{eqnarray}
We also assume that the characteristics of the oscillators are binary. That is,
if the oscillator belongs to group S, then its bifurcation parameter satisfies $\mu=\mu_{S}>0$; if the oscillator belongs to group H, $\mu=\mu_{H}<0$. All parameters other than $\mu$ are identical in both groups.
Depending on the parameters and the spatial arrangement of the active and inactive oscillators, the system is in the quiescent state ($A=0$) or in the active state ($|A|>0$).
When the system is in the active state, that is, when the oscillators undergo collective oscillations, we can assume complete synchronization of the oscillators in each group because the frequency parameters of the oscillators are the same and the diffusion constant is real.
That is, we suppose that the oscillation amplitudes of the oscillators are uniform in each group, i.e., $A=A_{S} e^{i \theta (t)}$ in group S and $A=A_{H} e^{i \theta (t)}$ in group H, where $A_S$ and $A_H$ are real amplitudes and the real-valued function $\theta(t)$ represents the phase of the oscillation in the rotating frame. Namely, we assume that the oscillators are approximately binarized into either of the two groups.
This binarization assumption holds strictly when $r_{e}$ is infinitesimal or when the oscillators of group S and H are arranged in the smallest possible checkered pattern.
Though some error may generally be expected, this assumption also works reasonably well for other arrangements and makes the problem analytically tractable.
\subsection{Two-variable free energy}
It is known that Eq.~(\ref{CGL}) has a Lyapunov function, or the generalized free energy (GFE), which monotonously decreases with the evolution of the system~\cite{aranson2002world}.
As we derive below, we can approximate the GFE of Eq.~(\ref{CGL}) by a two-variable function $F(A_{S}, A_{H})$ as
\begin{eqnarray}
\frac{4F}{L^{p}}=-(\alpha_{S} \mu_{S} A_{S}^{2}+\alpha_{H} \mu_{H} A_{H}^{2})+\alpha_{S} \alpha_{H} r_{e}^{2} k_{m}^{2} (A_{S}-A_{H})^2 +
O(|A|^{4}),
\label{FE}
\end{eqnarray}
under the \textit{two-group approximation} that we introduce below, where $O(|A|^4)$ represents $4$th-order terms in $A_{S}$ and $A_{H}$.
Here, $\alpha_{H}(>0)$ and $\alpha_{S}(>0)$ are the proportions of the oscillators in group S and group H, respectively ($\alpha_{H}+\alpha_{S}=1$), and the parameter $k_m$ represents the \textit{effective wavenumber} characterizing the spatial arrangement of group S and H defined below.
The approximate GFE Eq.~(\ref{FE}) can be obtained as follows. The exact GFE of Eq.~(\ref{CGL}) is given by ~\cite{aranson2002world}
\begin{eqnarray}
F &=& d^{p} \sum_{\bm{j}} f(A(\bm{j})), \cr
f(A) &=& \frac{1}{2}\left[-\frac{1}{2}\mu \left | A \right |^{2} -\frac{3}{16} \epsilon \left | A \right |^{4} + \frac{1}{2} r_{e}^{2} \left | \nabla A \right|^{2} \right]
\label{CE_F},
\end{eqnarray}
where $\nabla$ denotes the discrete gradient operator on the $p$-dimensional lattice defined as
\begin{equation}
\nabla A(\bm{j}) = \frac{1}{d} \left( A(j_1+1, j_2, \dots, j_p) - A(\bm{j}),\dots, A(j_1, \dots, j_p+1) - A(\bm{j}) \right).
\label{DGra}
\end{equation}
Using the above GFE, Eq.~(\ref{CGL}) and its complex conjugate can be expressed as
\begin{eqnarray}
\frac{dA}{dt}=-(a+ib) \frac{\partial F}{\partial A^{*}}, \; \; \frac{dA^{*}}{dt}=-(a-ib) \frac{\partial F}{\partial A},
\end{eqnarray}
where $A = A(\bm{j})$ and $A^{*} = A^{*}(\bm{j})$.
Note that $F$ is monotonically decreasing:
\begin{eqnarray}
\begin{aligned}
\frac{dF}{dt}&=d^{p} \sum_{\bm{j}} \left ( \frac{\partial f}{\partial A}\frac{dA}{dt}+\frac{\partial f}{\partial A^{*}}\frac{dA^{*}}{dt} \right ) \\
&=-d^{p} \sum_{\bm{j}} \left ( \frac{\partial f}{\partial A} (a+ib) \frac{\partial f}{\partial A^{*}} +\frac{\partial f}{\partial A^{*}} (a-ib) \frac{\partial f}{\partial A} \right ) \\
&=-2ad^{p}\sum_{\bm{j}} \left | \frac{\partial F}{\partial A} \right |^{2} \leq 0,
\end{aligned}
\end{eqnarray}
where $a>0$ is used.
Using the binarization assumption, the summation of the square amplitude in Eq.~(\ref{CE_F}) can be approximated as
\begin{eqnarray}
d^{p} \sum_{\bm{j}}\mu |A|^{2} &=& d^{p} \sum_{\bm{j}} \mu(\bm{j})|A(\bm{j})|^{2} \nonumber \\
&\simeq&L^{p} \left( \alpha_{S} \mu_{S}A_{S}^{2}+\alpha_{H} \mu_{H} A_{H}^{2} \right ),
\label{CE_Am}
\end{eqnarray}
which we call the two-group approximation.
To rewrite the summation of the discrete Laplacian term in Eq.~(\ref{CE_F}), we use the discrete Fourier transformation (DFT),
\begin{eqnarray}
\begin{aligned}
A(\bm{j})&=\sum_{\bm{k}} A(\bm{k})e^{i \bm{k}\bm{x}}, \\
\mu (\bm{j})&=\sum_{\bm{k}} \mu (\bm{k})e^{i \bm{k}\bm{x}}, \\
\bm{k}\bm{x} &\equiv \sum_{q=1}^{p} k_{q}j_{q}d.
\end{aligned}
\end{eqnarray}
We can then rewrite the summation in Eq.~(\ref{CE_F}) as
\begin{eqnarray}
\begin{aligned}
&d^{p} \sum_{{\bm j}} |\nabla A|^{2} = d^{p} \left| \nabla \sum_{\bm{k}} A(\bm{k})e^{i \bm{k}\bm{x}} \right |^{2} \\
&=d^{p} \sum_{\bm{j}} \left ( \nabla \sum_{\bm{k}} A(\bm{k}) e^{i \bm{k}\bm{x}} \right ) \left ( \nabla \sum_{\bm{k'}} A^{*}(\bm{k'})e^{-i \bm{k'}\bm{x}} \right ) \\
&=d^{p} \sum_{\bm{j}} \left ( \sum_{\bm{k}} i \bm{k}_{\bm{d}} A(\bm{k}) e^{i \bm{k}\bm{x}} \right ) \left ( \sum_{\bm{k'}} (i \bm{k'}_{\bm{d}})^{*} A^{*}(\bm{k'})e^{-i \bm{k'}\bm{x}} \right ) \\
&=\sum_{\bm{k},\bm{k'}}\bm{k}_{d}(\bm{k'}_{d})^{*} A(\bm{k})A^{*}(\bm{k'})d^{p} \sum_{\bm{j}}e^{i(\bm{k}-\bm{k'})\bm{x}} \\
&=\sum_{\bm{k},\bm{k'}}\bm{k}_{d}(\bm{k'}_{d})^{*} A(\bm{k})A^{*}(\bm{k'}) L^{p} \delta_{\bm{k},\bm{ k'}} \\
&= L^{p} \sum_{\bm{k}} |\bm{k}_{d} |^{2} |A(\bm{k})|^{2},
\label{CE_20}
\end{aligned}
\end{eqnarray}
where
\begin{eqnarray}
\bm{k}_{d} \equiv \left ( \frac{2\sin \left( \frac{k_{1}d}{2} \right) }{d} e^{i \frac{k_{1}d}{2}}, \dots, \frac{2\sin \left( \frac{k_{p}d}{2} \right) }{d} e^{i \frac{k_{p}d}{2}} \right).
\end{eqnarray}
This expression for $\bm{k}_d$ can be derived from the definition of the discrete gradient Eq.~(\ref{DGra}) as
\begin{eqnarray}
e^{i[k_{1}(j_{1}+1)+k_{2}j_{2} \dots+k_{p}j_{p}]d}-e^{i[\bm{k}\bm{j}]d}=e^{i\bm{kx}}(e^{ik_{1}d}-1)=e^{i\bm{kx}}e^{i\frac{k_{1}d}{2}}2i\sin\frac{k_{1}d}{2}
\end{eqnarray}
for the first component, and similarly for the other components. Note that $\bm{k}_{d} \to \bm{k} $ when $d \to 0$, and also that the difference between $\bm{k}$ and $\bm{k}_{d}$ is not negligible when $\bm{k}$ is lager than or comparable to $1/d$.
We now introduce an \textit{effective wavenumer} $k_{m}$:
\begin{eqnarray}
k^{2}_{m} \equiv \frac{\sum_{\bm{k}} |\bm{k}_{d}|^{2} |A(\bm{k})|^{2}}{\sum_{\bm{k} \ne 0}|A(\bm{k})|^{2}} \simeq \frac{\sum_{\bm{k}} |\bm{k}_{d}|^{2} |\mu(\bm{k})|^{2}}{\sum_{\bm{k} \ne 0}|\mu(\bm{k})|^{2}},
\label{CE_31}
\end{eqnarray}
which characterizes the spatial arrangement of the active and inactive oscillators. The second expression follows from the approximation that the oscillators are binarized into two classes depending on the sign of $\mu$.
Plugging this definition into Eq.~(\ref{CE_20}) and using Parseval's identity, the summation can be approximated as
\begin{eqnarray}
\begin{aligned}
&d^{p} \sum_{\bm{j}} |\nabla A|^{2} = L^{p} k_{m}^{2}{\sum_{\bm{k} \ne 0}|A(\bm{k})|^{2}} \\
&=\frac{L^{p} k_{m}^{2}}{N^{p}}\sum_{\bm{j}}|A(\bm{j})-\overline{A}|^{2} \\
&=L^{p} k_{m}^{2}\left \{ \alpha_{S}[A_{S}-(\alpha_{S} A_{S} + \alpha_{H} A_{H})]^{2} + \alpha_{H}[A_{H}-(\alpha_{S} A_{S} + \alpha_{H} A_{H})]^{2}\right \} \\
&=L^{p}k_{m}^{2} \left \{ \alpha_{S} [\alpha_{H} A_{S} - \alpha_{H} A_{H}]^2 + \alpha_{H} [\alpha_{S}A_{H}-\alpha_{S}A_{S}]^{2}
\right \} \\
&=L^{p} k_{m}^{2} \alpha_{S} \alpha_{H} (\alpha_{S} + \alpha_{H})(A_{S}-A_{H})^{2} \\
&=L^{p} k_{m}^{2}\alpha_{S} \alpha_{H} (A_{S}-A_{H})^{2},
\end{aligned}
\label{CE_Gr}
\end{eqnarray}
where $\overline{A} = \alpha_S A_S + \alpha_H A_H$ denotes the direct-current component of $A$ in the DFT under the binarization assumption, i.e., the area average of $A$. Substituting Eq.~(\ref{CE_Am}) and Eq.~(\ref{CE_Gr}) into Eq.~(\ref{CE_F}), we obtain Eq.~(\ref{FE}).\\
\subsection{Stability condition}
Using the approximate GFE in Eq.~(\ref{FE}), we can derive the approximate conditions for the linear stability of the stationary quiescent state of the system.
The Hessian matrix of Eq.~(\ref{FE}) at the quiescent state $(A_{S},A_{H})=(0,0)$ is given by
\begin{eqnarray}
\begin{aligned}
M &\equiv \left [
\begin{array}{cc}
\frac{\partial^{2}F}{\partial A_{S} \partial A_{S}} & \frac{\partial^{2}F}{\partial A_{S} \partial A_{H}} \\
\frac{\partial^{2}F}{\partial A_{H} \partial A_{S}} & \frac{\partial^{2}F}{\partial A_{H} \partial A_{H}}
\end{array}
\right ]_{(A_{S},A_{H})=(0,0)} \\
&= \frac{1}{2} L^{p} \left[
\begin{array}{cc}
\alpha_{S}(\alpha_{H} r_{e}^{2} k_{m}^{2}-\mu_{S}) & -\alpha_{S} \alpha_{H} r_{e}^{2} k_{m}^{2} \\
-\alpha_{S} \alpha_{H} r_{e}^{2} k_{m}^{2} & \alpha_{H}(\alpha_{S} r_{e}^{2} k_{m}^{2}-\mu_{H})
\end{array}
\right ].
\end{aligned}
\end{eqnarray}
Thus, the quiescent state is linearly stable if and only if $\mbox{tr}(M)>0$ and $\mbox{det}(M)>0$, which results in the following two inequalities:
\begin{eqnarray}
\alpha_{S} \mu_{S}+\alpha_{H} \mu_{H}<0, \; \; \frac{\mu_{S} \mu_{H}}{\alpha_{S} \mu_{S}+\alpha_{H} \mu_{H}}<r_{e}^{2}k_{m}^{2}.
\label{CE_stable}
\end{eqnarray}
The first inequality indicates that the average value of $\mu$ should be negative for the linear stability of the system. If the average value of $\mu$ is positive, the macroscopic oscillation cannot be suppressed by any $r_{e}$.
This inequality is interpreted as the condition that the inactive group wins over the active group when $r_{e}$ becomes infinitely large.
The second inequality shows that larger diffusion constant $r_{e}$ is required to suppress the macroscopic oscillation when the effective wavelength of the arrangement of active and inactive oscillators is larger (i.e., when the effective wavenumber $k_{m}$ is smaller).
It is remarkable that both inequalities do not depend on $p$ and therefore they are valid in any dimensions.
\section{Application to the cortico-thalamic model}
To illustrate how our approximate theory can provide insights into realistic problems, we consider the cortico-thalamic model proposed by Kim and Robinson~\cite{kim2007compact, kim2009dynamics},
\begin{eqnarray}
\left[(t_e \frac{d}{dt}+1)^2-r_e^2 \Delta \right] \chi= (1+c_1) \chi
+c_2\tilde{\chi}+\epsilon\chi(t)^3,
\label{Compact}
\end{eqnarray}
where the field variable $\chi=\chi(j_{x}, j_{y}; t)$ represents a mean firing rate of the neurons within each local area of the cortex, $\tilde{\chi} \equiv \chi(j_{x}, j_{y}; t-t_{0})$ is a delayed value of $\chi$ (where $t_{0}$ is the time delay), $c_1$ parameterizes the strength of cortico-cortical activities, $c_2$ characterizes cortico-thalamic feedback, and $\epsilon$ controls the nonlinear term that takes into account the characteristics of neuronal firing.
The Kim-Robinson model is categorized into a neural \textit{field} model, which describes how a mesoscopic quantity characterizing neural activity evolves over both \textit{space} and time~\cite{pinotsis2014neural}.
This model undergoes Hopf bifurcation and exhibits collective oscillations, which is interpreted as the onset of an epileptic seizure~\cite{kim2009dynamics}.
Though Eq.~(\ref{Compact}) has a simple expression, it is actually a system with time delay and exhibits a wide variety of oscillatory dynamics. It is known that the model can reproduce many of the typical oscillatory waveforms observed in real electroencephalograms.
By the center manifold reduction, the lattice complex Ginzburg-Landau model, Eq.~(\ref{CGL}), can be derived from Eq.~(\ref{Compact}) after
spatial discretization (see Ref.~\cite{yamaguchi2011reduction} for details). In the following, we consider Eq.~(\ref{CGL}) with the spatial arrangements of active and inactive oscillators assumed in Ref.~\cite{kim2009dynamics} and analyze the transition to macroscopic oscillations.
In Ref.~\cite{kim2009dynamics}, the authors analyzed focal epilepsy both numerically and analytically using the model Eq.~(\ref{Compact}) by assuming that the parameters $c_1$ and $c_2$ are in the unstable regime (i.e., the field variable $\chi$ undergoes oscillations) in the epileptic foci of the cortex, while $c_1$ and $c_2$ are in the stable regime ($\chi$ tends to vanish) in the remaining tissue.
Their study implied that areas surrounding the epileptic foci play an essential role in the suppression of epileptic seizures. However, the analytical condition derived in Ref.~\cite{kim2009dynamics} was based on simplified assumptions and therefore not general enough.
In Ref.~\cite{kim2009dynamics}, it is assumed that the inactive region (inactive oscillators in our discrete model) never oscillates at all, there is only a single active area and, the oscillation decays at the edge of the boundary. These assumptions lead to the prediction that the active region can be suppressed whenever the diffusion constant $r_e$ is larger than a critical value. This prediction, however, is valid only in special cases.
Consider the limiting case where the proportion of the active area is close to $1$, where the activity of the active area is infinitely large, where the inactive area is vanishingly small, or where the inactivity (resistance to activation) of the inactive area is infinitely small. In these cases, it is obvious that the active region would not be suppressed no matter how large the diffusion constant $r_e$ is. As we show below, not the inactive group but the active group \textit{wins} under sufficiently strong coupling in these cases.
The major difficulty in deriving a better analytical prediction is that Eq.~(\ref{Compact}) has a delay term, which complicates theoretical analysis.
This difficulty can be circumvented by reducing Eq.~(\ref{Compact}) with time delay to the lattice complex Ginzburg-Landau model Eq.~(\ref{CGL}) by the center manifold reduction near the bifurcation point, as we considered in Ref.~\cite{yamaguchi2011reduction}.
In the reduced Eq.~(\ref{CGL}), epileptic foci and areas surrounding them correspond to the regions with $\mu>0$ and $\mu<0$, respectively.
See Ref.~\cite{yamaguchi2011reduction} for further details on Eq.~(\ref{Compact}) and its reduction to Eq.~(\ref{CGL}).
The other problem is that the stability condition for mixed populations of coupled active and inactive oscillators on a two-dimensional lattice is difficult to derive analytically. By using the approximate theory that we developed in the previous section, we can analyze the effect of spatial arrangement of the active and inactive oscillators, extending the result of Ref.~\cite{kim2009dynamics}.
In the following, we illustrate our theory by numerically integrating Eq.~(\ref{Compact}), which is spatially discretized on a 6 $\times$ 6 lattice, for several spatial arrangements of the oscillators.
\subsubsection{Investigated patterns}
Figure~\ref{Patterns_PRE} shows the investigated arrangements of the active and inactive oscillators. The yellow cells represent active oscillators in group S, while the white cells represent inactive oscillators in group H.
We study the following two series of spatial arrangements: K-series (K0, K1, \dots, K9) and V-series (V0, V1, \dots, V9). The pattern identifier is indicated to the left of each pattern. The value below the identifier represents the effective wavenumber multiplied by the side length, $k_m L$, defined by Eq.~(\ref{CE_31}).
In the K-series, $\alpha_{S}$ = 4/36 is fixed and all the patterns satisfy the first inequality of Eq.~(\ref{CE_stable}), while in the V-series, $\alpha_{S}$ varies from 4/36 to 28/36 and V3, V4, V8 and V9 do not satisfy the first inequality of Eq.~(\ref{CE_stable}).
\subsubsection{Results for the K-series}
Using our approximate theory, we first reproduce the typical results presented in Ref.~\cite{kim2009dynamics}. Here, $L$ (= 60 cm) corresponds to the side length of a square of the cortex. There are four active oscillators at the center of the square as in the pattern K0 in Fig.1, which correspond to the focal epileptic area of the cortex. The parameter values used in our calculations are shown Table~\ref{CE_param_Kim}, which are the same as those used in Ref.~\cite{kim2009dynamics}.
In Ref.~\cite{kim2009dynamics}, it is shown that the macroscopic oscillation, which is considered to be a seizure, does not occur if $r_{e}$ is larger than about 2.5 cm in this case.
This suggests that inhibition by the surrounding area of the epileptic focus plays an important role in the suppression of epileptic seizures.\\
\begin{table}[hbtp]
\caption{Parameter values \label{CE_param_Kim}}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|c|}
\hline
Parameter & $c_{1}$ & $c_{2}$ & $t_{e}$ & $t_{0}$ & $\epsilon$ & $\Omega$ & $\mu$ \\ \hline
Group S & -0.1 & -0.5 & 10msec & 80msec & -0.1 & 24rad/sec & 0.116
\\ \hline
Group H & -0.4 & -0.5 & 10msec & 80msec & -0.1 & 24rad/sec & -0.184
\\ \hline
\end{tabular}
\end{center}
\end{table}
Steady states of the system after a sufficiently long transient period are shown in Fig.~\ref{CE_K0} for several values of $r_{e}$. The top row shows peak to peak amplitude of the oscillators.
The waveforms of all oscillators are shown in the bottom row of Fig.~\ref{CE_K0}. When $r_{e}=0$, the oscillators are completely separated into two groups, i.e., oscillating group and quiescent group (the waveform of the quiescent group lies on the horizontal axis). As $r_{e}$ becomes larger, the inactive oscillators start to oscillate, while the active oscillators start to decrease their oscillation amplitudes, and all oscillators stop at the critical value of $r_{e}$. We evaluated the critical value of the diffusion constant $r_{e}^{*}$ from the stability condition given by Eq.~(\ref{CE_stable}) and obtained $r_{e}^{*}=$ 2.51 cm for the pattern K0. We can confirm that the oscillations actually disappear near this predicted point, in good agreement with Ref.~\cite{kim2009dynamics}.
Figure~\ref{CE_K} shows the results for all K-series patterns in Fig.\ref{Patterns_PRE} in grayscale, and Fig.~\ref{CE_K_power_abs} shows the power of the macroscopic oscillation vs. $r_{e}$, where the power is calculated as the areal and temporal mean square of the amplitudes of all oscillators. The oscillations disappear near the critical diffusion constants predicted by the present approximate theory.
\subsubsection{Results for the V-series}
Figures~\ref{CE_V} and \ref{CE_V_p} show the results for the V-series patterns in Fig.~\ref{Patterns_PRE}.
In these cases, the oscillations should remain at any large $r_{e}$ in V3, V4, V8 and V9, because the first inequality of Eq.~(\ref{CE_stable}) is not satisfied in these patterns. The validity of the prediction can be seen in Fig.~\ref{CE_V_p}. As for the other patterns, we can again confirm that the oscillations disappear near the critical diffusion constants predicted by the second inequality of Eq.~(\ref{CE_stable}).
The prediction errors are less than about 30\% of the true values through K-series and V-series. These errors are caused by the fact that the binarization assumption does not hold exactly, especially when the characteristic length scale of the patterns is large. But the error is still not fatal and does not undermine the insight; we can still predict whether the suppression of the macroscopic oscillation occurs or not by the approximate theory.
\section{DISCUSSION AND CONCLUSION}
We have derived an approximate condition for the suppression of macroscopic oscillations in mixed populations of coupled active and inactive oscillators using the two-group approximation for the lattice complex Ginzburg-Landau model, which can be considered an aging transition due to local diffusive coupling. As illustrated by numerical simulations, the approximate condition, Eq.~(\ref{CE_stable}), explains the transitions qualitatively well.
Although quantitative accuracy is not very well and further improvement in the approximation is desirable, we believe that the theory
developed in this paper provides an important step toward understanding of the aging transition in mixed populations coupled oscillators.
In the present study, we focused our attention on the generalized free energy (GFE) and approximated it to analyze the transition in a simple way.
Although the approximate GFE does not have the same information as the original dynamical equation in the sense that we cannot derive the original equation from the approximate GFE, it contains essential information for predicting the stability and helps us derive the stability condition.
Note that, unlike the real Ginzburg-Landau equation, several conditions must be satisfied for the complex Ginzburg-Landau equation to possess a GFE function.
In our analysis, Fourier transform to the wavenumber space was used to extract the effective wave number $k_{m}$ of the spatial arrangement of the mixed oscillator population, which played an important role in determining the stability of the quiescent state in addition to the proportions of the oscillators ($\alpha_{S}, \alpha_{H}$) and the bifurcation parameters ($\mu_{S}, \mu_{H}$).
The following proposition is useful in understanding the meaning of $k_m$ and in calculating its actual value: the binding number $n_b$ between group S and group H is related to the effective wavenumber $k_{m}$ as $n_{b}=N^{p} \alpha_{S}\alpha_{H} (k_{m}d)^{2}$. We can prove this proposition by noting that $|\nabla \mu|^{2}$ takes a non-zero value $|(A_S-A_H)/d|^{2}$ only at the boundaries between the group S and group H under our binarization assumption. Plugging $|\nabla \mu|^{2}=|(A_S-A_H)/d|^{2}$ at the boundaries and otherwise $|\nabla \mu|^{2}=0$ into Eq.~(\ref{CE_Gr}) leads
\begin{equation}
d^{p-2} n_{b}|(A_{S}-A_{H})|^{2} = L^{p} k_{m}^{2}\alpha_{S}\alpha_{H}|(A_{S}-A_{H})|^{2},
\end{equation}
which gives
$n_{b}=N^{p} \alpha_{S}\alpha_{H} (k_{m}d)^{2}$, using $L=Nd$.\\
Finally, though we illustrated the theory for a model of epileptic seizures, we stress that our result is generally applicable to coupled-oscillator models near the Hopf bifurcation, because the complex Ginzburg-Landau equation is a normal form of coupled oscillators near the Hopf bifurcation derived by the center manifold reduction method from the original model ~\cite{yamaguchi2011reduction, yang2017critical}.
\begin{acknowledgements}
The authors would like to thank Yoshiharu Yamamoto, Toru Nakamura, Fumiharu Togo, Akifumi Kishi and Jerome Foo for useful discussion. This work is partly supported by JSPS KAKENHI Grant number 15K01499 and 18K17887 to IY.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,505 |
Q: What are some use-cases for Meteor.isClient? The docs on docs.meteor.com are very lacking for this check. I've seen elsewhere that it is useful for setting up helper functions in a Handlebars (and the new Spacebars?) JS. But where else would a Meteor.isClient check be required?
A: It's useful whenever you have shared code between the client and the server. For example, the default code that comes with any new meteor project puts all of the javascript into a single file. Template definitions won't work on the server, so they need to be wrapped within a Meteor.isClient check. Of course in a larger project, you can easily separate these sections into their respective /client and /server directories. However, you could still have utility functions, or methods defined in a shared directory. In those cases you may again find that some portions of the code only make sense when executed in one of the two environments.
TL;DR
They are critical for small apps where all of the code exists in a single file. Larger apps tend to use them only for things like meteor methods which can have a single definition but work differently depending on the environment.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,498 |
This is a column from Chief Executive Officer, Duncan Hughes-Phillips published by Construction News, 19.08.2016. Right now in the construction industry, the only certainty is uncertainty. Political and market conditions mean it is more important than ever to be diligent in understanding a project's potential risks and negotiating contractual responsibilities.
Josiah Lee, Quantity Surveyor at Base Quantum, explores how rising costs are connected to an increase in the number of projects which end in dispute. As the UK climbs out of recession the total value of construction disputes remains high. Perhaps this is connected to the rising cost of consultants. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,111 |
Q: Using the data of the index column I just added I was just reading about PDO::lastInsertId from the PHP documentation, but after some tests I reallized I absolutelly suck at PHP xD and I should investigate more about classes and static methods and more to even understand where to put that code.
Anyway, I'm not actually trying to know the last Inserted Id, I want to get the Index value of the last inserted row of MySQL table. My Index column is called "id" and is set to auto increment everytime I add a new row, yet, when I delete a row the number doesn't update, so the value in the row called "id" may not allways be the same as the row number.
Here is the code that inserts a new row.
<?php
require('conexion.php');
$name=$_POST['name'];
$email=$_POST['email'];
$pass=$_POST['pass'];
$localidad=$_POST['localidad'];
$saldo=$_POST['saldo'];
$slastupdate=$_POST['slastupdate'];
$query="INSERT INTO usuarios (name, email, pass, localidad, saldo, slastupdate) VALUES ('$name','$email','$pass','$localidad','$saldo','$slastupdate')";
$resultado=$mysqli->query($query);
?>
Update: For now on, I'm using another query to get the last inserted's row Id... In my case using PDO is not necessary since I'm developing a private app, and there's no risk of mysql injections
require_once "../php/new_mysql.php";
$query="INSERT INTO clientes (gira, localidad, nombre, email, proxivisi, saldo) VALUES ('$gira', '$localidad', '$nombre', '$email', '$proxivisi', '$saldo')";
$agregar=$mysql->query($query) or die(mysql_error());
$querygetid=$mysql->query("SELECT id, proxivisi FROM clientes WHERE nombre='$nombre' AND email='$email'");
$getid=$querygetid->fetch_assoc();
$id=$getid['id'];
A: @FranCano use @user to notify us of your reply.
Your first problem is you seem to be using mysqli, not PDO in the above code. mysqli and PDO are different APIs to run MySQL queries and don't share functions. PDO::lastInsertId will give you the auto increment value of the last inserted row.
Start with PDO::__construct() to see how to start a PDO instance and connect to a database. Check PDO::query() and the examples listed to see how PDO works. However, PDO::query() is basic and does not protect you from SQL injections. For your above query you should be used prepared statements since you use user input (POST). Prepared statements are started with PDO::prepare(). MySQLi has prepared statements too but they are slightly different.
PDO is object oriented, you can't just call PDO::query() and PDO::lastInsertId(). You create a PDO instance to connect to the database, then you create a PDOStatement object by using prepare(), then you execute (PDOStatement::execute) that PDOStatement object to run the query. Then you can resort to the lastInsertId() on the original PDO object.
If you don't know OOP or object oriented programming then this will all sound pretty confusing so that is where you need to start your reading.
A quick example is:
$pdo = new PDO(..mysql connection information..);
// Create a PDOStatement object
$stmt = $pdo->prepare("INSERT into test ('field1', 'field2') VALUES (?, ?)");
// Execute the prepared statement safely inserting the post variables into the ?'s. This protects against SQL injection attacks.
if ($stmt->execute(array($_POST['value1'], $_POST['value2']))) {
$id = $pdo->lastInsertId();
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,505 |
\section{Introduction}
\label{s1}
According to the well-known Ostrovsky theorem, there are two equal in
rights ``universes'': the real ``universe'' and the $p$-adic one.
The real ``universe'' is based on the field ${\mathbb R}$ of real numbers,
which is defined as the completion of the field of rational numbers
${\mathbb Q}$ with respect to the usual Euclidean distance between rational
numbers. In its turn, the $p$-adic ``universe'' is based on the field
${\mathbb Q}_p$ of $p$-adic numbers, which is defined as the completion
of ${\mathbb Q}$ with respect to the $p$-adic norm $|\cdot|_p$. This norm
is defined as follows: if an arbitrary rational number $x\ne 0$ is
represented as $x=p^{\gamma}\frac{m}{n}$, where $\gamma=\gamma(x)\in {\mathbb Z}$
and the integers $m$, $n$ are not divisible by $p$, then
\begin{equation}
\label{1}
|x|_p=p^{-\gamma}, \quad x\ne 0, \qquad |0|_p=0.
\end{equation}
The norm $|\cdot|_p$ satisfies the strong triangle inequality
$|x+y|_p\le \max(|x|_p,|y|_p)$ and is {\em non-Archimedean}.
During a few hundred years theoretical physics has been developed
on the basis of real (and later also complex) numbers. However, in the last 20
years the field of $p$-adic numbers ${{\mathbb Q}}_p$ (as well as its
algebraic extensions) has been intensively used in theoretical
and mathematical physics, stochastics, cognitive sciences and
psychology~\cite{Ar-Dr-V},~\cite{Av-Bik-Koz-O},~\cite{Kh1}--~\cite{Kh4},
~\cite{Kh-Koz3},~\cite{Koch3},~\cite{Koz-Os-Av-1},~\cite{Vl-V-Z}--~\cite{V2}
(see also the references therein). However, since a $p$-adics is a young area
there are many unsufficiently studied problems which have been intensively
studied in the real ``universe''. One of them is the $p$-adic wavelet theory.
Nowadays it is difficult to find an engineering area where wavelets
(in the real setting) are not applied. There is a general scheme for the
construction of wavelets in the real setting, which was developed
in the early nineties. This scheme is based on the notion of the
{\em multiresolution analysis} introduced by Y.~Meyer and
S.~Mallat~\cite{Mallat-1},~\cite{Meyer-1}. The $p$-adic wavelet theory
is now in conceptual stage of investigation.
In this theory the situation is as follows.
In 2002, S.~V.~Kozyrev~\cite{Koz0} found a compactly supported
$p$-adic wavelet basis for ${{\mathcal L}}^2({\mathbb Q}_p)$ which is an analog of
the real Haar basis:
\begin{equation}
\label{62.0-1}
\theta_{k;j a}(x)=p^{-j/2}\chi_p\big(p^{-1}k(p^{j}x-a)\big)
\Omega\big(|p^{j}x-a|_p\big), \quad x\in {\mathbb Q}_p,
\end{equation}
$k=1,2,\dots,p-1$, $j\in {\mathbb Z}$, $a\in I_p={\mathbb Q}_p/{\mathbb Z}_p$,
where $\Omega(t)$ is the characteristic
function of the segment $[0,1]\subset{\mathbb R}$,
the function $\chi_p(\xi x)$ is an {\em additive character} of the field ${\mathbb Q}_p$
for every fixed $\xi \in {\mathbb Q}_p$ (see Sec.~\ref{s2}).
Kozyrev's wavelet basis (\ref{62.0-1}) is generated by dilatations and
translations of the wavelet functions:
\begin{equation}
\label{62.0-1-0}
\theta_{k}(x)=\chi_p(p^{-1}kx)\Omega(x|_p), \quad x\in {\mathbb Q}_p, \quad k=1,2,\dots,p-1.
\end{equation}
Multidimensional $p$-adic bases obtained by direct multiplying
out the wavelets (\ref{62.0-1}) were considered in~\cite{Al-Kh-Sh3}.
The Haar wavelet basis (\ref{62.0-1}) was extended to
the ultrametric spaces in~\cite{Kh-Koz1},~\cite{Kh-Koz2},~\cite{Koz2}.
J.~J.~Benedetto and R.~L.~Benedetto~\cite{Ben-Ben},
R.~L.~Benedetto~\cite{Ben1} suggested a method for finding wavelet
bases on the locally compact abelian groups with compact open
subgroups, which includes the $p$-adic setting. They did not develop
the {\em multiresolution analysis {\rm(}MRA{\rm)}}, their method
being based on the {\em theory of wavelet sets}. Their method only
allows the construction of wavelet functions whose
Fourier transforms are the characteristic functions
of some sets (see~\cite[Proposition~5.1.]{Ben-Ben}).
Note that Kozyrev's wavelet basis (\ref{62.0-1}) can be constructed
in the framework of Benedettos' approach~\cite[5.1.]{Ben-Ben}.
The notion of $p$-adic MRA was introduced and a general scheme for
its construction was described in~\cite{S-Sk-1}.
To construct a $p$-adic analog of a classical MRA we need a proper $p$-adic
{\em refinement equation}. In our preprint~\cite{Kh-Sh1}, the following
conjecture was proposed: the equality
\begin{equation}
\label{62.0-3**}
\phi(x)=\sum_{r=0}^{p-1}\phi\Big(\frac{1}{p}x-\frac{r}{p}\Big),
\quad x\in {\mathbb Q}_p,
\end{equation}
can be considered as a {\em refinement equation}. A solution $\phi$ to this
equation ({\it a refinable function}) is the characteristic function of the unit disc
\begin{equation}
\label{62.0-3**-1}
\phi(x)=\Omega\big(|x|_p\big), \quad x\in {\mathbb Q}_p.
\end{equation}
The equation (\ref{62.0-3**}) reflects {\it natural} ``self-similarity'' of
the space ${\mathbb Q}_p$: the unit disc $B_{0}(0)=\{x: |x|_p \le 1\}$ is represented
by a sum of $p$ mutually {\it disjoint} discs
$$
B_{0}(0)=B_{-1}(0)\cup\Big(\cup_{r=1}^{p-1}B_{-1}(r)\Big),
$$
where $B_{-1}(r)=\bigl\{x: |x-r|_p \le p^{-1}\bigr\}$
(see formula (\ref{79}) in Proposition~\ref{pr1-int}).
The equation (\ref{62.0-3**}) is an analog of the {\em refinement equation}
generating the Haar MRA in the real analysis.
Using this idea, the notion of $p$-adic MRA was introduced and a general
scheme for its construction was described in~\cite{S-Sk-1}.
The scheme was realized for construction $2$-adic Haar MRA
with using (\ref{62.0-3**-1}) as the {\em generating refinement equation}.
In contrast to the real setting, the {\em refinable function} $\phi$ generating
the Haar MRA is {\em periodic}, which {\em never holds} for real
refinable functions. Due to this fact, there exist {\em infinity many different}
orthonormal wavelet bases in the same Haar MRA (see~\cite{S-Sk-1}). One of them
coincides with Kozyrev's wavelet basis (\ref{62.0-1}).
From the standpoint of results of the papers~\cite{Kh-Sh-Sk},~\cite{Al-Ev-Sk},
in~\cite{S-Sk-1} all compactly supported wavelet Haar bases were constructed.
It turned out that the above-mentioned $p$-adic wavelets are eigenfunctions of
$p$-adic pseudo-differential operators~\cite{Al-Kh-Sh3}--\cite{Al-Kh-Sh5},
~\cite{Kh-Sh1},~\cite{Kh-Sh2},~\cite{Koz0},~\cite{Koz2} (see also Sec.~\ref{s4}).
Thus the spectral theory of $p$-adic pseudo-differential operators is related
to the wavelet theory. On the other hand, it is well-known that numerous models
connected with $p$-adic differential equations use pseudo-differential operators
(see~\cite{Kh2},~\cite{Koch3},~\cite{Vl-V-Z} and the references therein).
This is closely related to the fact that for the $p$-adic analysis associated with the
mapping ${\mathbb Q}_p \to {\mathbb C}$, the operation of differentiation is {\it not defined\/},
and as a result, many models connected with $p$-adic differential equations
use pseudo-differential operators, in particular, the fractional operator
$D^{\alpha}$ (see the above-mentioned papers and books).
These two facts imply that study of wavelets is important since it gives
a new powerful technique for solving $p$-adic problems.
{\bf Contents of the paper.}
The main goal of this paper is to construct a {\em countable family of
new compactly supported non-Haar $p$-adic wavelet bases} in ${{\mathcal L}}^2({\mathbb Q}_p)$.
Another goal is to study the connections between
{\em wavelet analysis and spectral analysis of $p$-adic pseudo-differential operators}.
In addition, we use our results to solve the Cauchy problems for $p$-adic
pseudo-differential equations.
In Sec.~\ref{s2}, we recall some facts from the theory of $p$-adic
distributions~\cite{G-Gr-P}, \cite{Taib1}--~\cite{Vl-V-Z}. In particular,
in Subsec.~\ref{s2.2}, some facts from the theory of the $p$-adic Lizorkin
spaces of test functions $\Phi({\mathbb Q}_p^n)$ and distributions $\Phi'({\mathbb Q}_p^n)$
are recalled (for details, see~\cite{Al-Kh-Sh3}).
In Sec.~\ref{s3}, {\it non-Haar $p$-adic compactly supported
wavelet bases} are introduced.
In Subsec.~\ref{s3.1}, we construct the non-Haar basis
(\ref{62.1}) which was introduced in the preprint~\cite{Kh-Sh1}
(for the brief review see~\cite{Kh-Sh2}).
In contrast to (\ref{62.0-1}), for the basis (\ref{62.1}) the number
of generating wavelet functions is not minimal, for example,
for $p=2$ we have $2^{m-1}$ wavelet functions (instead
of one as it is for (\ref{62.0-1}) and for classical wavelet
bases in real analysis).
The basis (\ref{62.1}) is the {\em non-Haar} wavelet basis, since it cannot be
constructed in the framework of the $p$-adic Haar MRA (see~\cite{S-Sk-1} and Theorem~\ref{th2}).
According to Remark~\ref{rem1}, Kozyrev's wavelet basis (\ref{62.0-1}) is
a particular case of the basis (\ref{62.1}) for $m=1$.
According to the same remark, our non-Haar wavelet basis (\ref{62.1}) can be
obtained by using the algorithm developed by the Benedettos~\cite{Ben-Ben}.
However, using our approach, we obtained the {\em explicit formulas} (\ref{62.1})
for this basis. Moreover, our technique allows to produce new wavelet bases
(see in Subsec.~\ref{s3.2}).
In Subsec.~\ref{s3.2}, using the proof scheme of~\cite[Theorem~1]{S-Sk-1},
we construct {\em infinitely many new different non-Haar wavelet bases}
(\ref{109-11}), (\ref{101-11}), (\ref{108-11}) which are distinct from the
basis (\ref{62.1}). These new bases cannot be obtained in the framework of
the standard scheme of the MRA~\cite{S-Sk-1}.
Our bases given by formulas (\ref{109-11}),
(\ref{101-11}), (\ref{108-11}) {\it cannot be constructed} by Benedettos'
method~\cite{Ben-Ben}.
For example, it is easy to see that the Fourier transform of our
generating wavelet-functions $\psi_{s}^{(m)[1]}(x)$, $s\in J_{p;m}$ defined by
(\ref{101-11}), (\ref{108-11}) and all their shifts {\em are not characteristic
functions} (see Remark~\ref{rem2}).
In Subsec.~\ref{s3.3}, $n$-dimensional non-Haar wavelet bases (\ref{62.8})
and (\ref{62.8-1}) are introduced as $n$-direct products of the corresponding
one-dimensional non-Haar wavelet bases.
All above wavelets belong to the Lizorkin space of test functions $\Phi({\mathbb Q}_p^n)$.
In Subsec.~\ref{s3.4}, the characterizations of the spaces of Lizorkin test functions and
distributions in terms of wavelets are given (see Lemma~\ref{lem-w-1**} and
Proposition~\ref{pr-w-2**}), which are very useful for solution of $p$-adic
pseudo-differential equations. The assertions of the type of
Lemma~\ref{lem-w-1**} and Proposition~\ref{pr-w-2**} were stated for ultrametric
Lizorkin spaces in~\cite{Al-Koz}.
In Sec.~\ref{s4}, the spectral theory of one class of $p$-adic multidimensional
pseudo-differential operators (\ref{64.3}) (which were introduced in~\cite{Al-Kh-Sh3})
is studied.
In Subsec.~\ref{s4.1},~\ref{s4.2}, we recall some facts on this class of
pseudo-differential operators defined in the Lizorkin space ${{\mathcal D}}'({\mathbb Q}_p^n)$.
Our operators (\ref{64.3}) include the fractional
operator~\cite[\S2]{Taib1},~\cite[III.4.]{Taib3} and the
pseudo-differential operators studied in~\cite{Koch3},~\cite{Z1},~\cite{Z2}.
The Lizorkin spaces are {\it invariant\/} under our
pseudo-differential operators.
In Subsec.~\ref{s4.3}, by Theorems~\ref{th4.1},~\ref{th4.1-1} the
criterion (\ref{64.1***}) for multidimensional $p$-adic
pseudo-differential operators (\ref{64.3}) to have multidimensional
wavelets (\ref{62.8}) and (\ref{62.8-1}) as eigenfunctions is derived.
In particular, the multidimensional wavelets (\ref{62.8}) and (\ref{62.8-1})
are eigenfunctions of the Taibleson fractional operator (see
Corollaries~\ref{cor5}--\ref{cor7}).
In Sec.~\ref{s5}, the results of Sec.~\ref{s3},~\ref{s4} are used to solve
the Cauchy problems for $p$-adic evolutionary pseudo-differential equations.
Note that the Cauchy problem (\ref{76-sl}) was solved in~\cite{Al-Kh-Sh5}
for a particular case. These results give significant advance in the theory
of $p$-adic pseudo-differential equations. Moreover, since many $p$-adic models
use pseudo-differential operators (in particular, fractional operator),
these results can be used in applications.
It easy to see that formulas (\ref{109-11}), (\ref{101-11}), (\ref{108-11})
do not give description of all {\em non-Haar wavelet bases}.
Due to the results of Sec.~\ref{s3}, there arise
two important problems: to construct an analog of MRA scheme and describe all
compactly supported {\em non-Haar wavelet bases}. It is necessary
to verify if all {\em non-Haar wavelet bases} are given by formulas (\ref{109-11}),
(\ref{101-11}), (\ref{108-11})?
Taking into account representation (\ref{62.0-5}), it is natural to suggest that in this
case we must use the {\em refinement type equation}:
$$
\phi(x)=\sum_{b}\phi\Big(\frac{1}{p^m}x-\frac{b}{p^m}\Big),
\quad x\in {\mathbb Q}_p,
$$
instead of the {\em Haar refinement equation} (\ref{62.0-3**}),
where $b=0$ or $b=b_{r}p^{r}+b_{r+1}p^{r+1}+\cdots+b_{m-1}p^{m-1}$,
$r=0,1,\dots,m-1$, $0\le b_j\le p-1$, $b_r\ne 0$.
This equation reflects the geometric fact that
the unit disc $B_{0}=\{x: |x|_p \le 1\}$ is represented by a sum of $p^m$
mutually {\em disjoint} discs $B_{-m}(b)=\{x:|x-b|_p \le p^{-m}\}$.
\section{Preliminary results in $p$-adic analysis}
\label{s2}
\subsection{$p$-Adic functions and distributions.}\label{s2.1}
We shall systematically use the notations and results from~\cite{Vl-V-Z}.
Let ${\mathbb N}$, ${\mathbb Z}$, ${\mathbb C}$ be the sets of positive integers, integers,
complex numbers, respectively.
Any $p$-adic number $x\in{\mathbb Q}_P$, $x\ne 0$, is represented in the {\em canonical form}
\begin{equation}
\label{8.1}
x=p^{\gamma}(x_0 + x_1p + x_2p^2 + \cdots )
\end{equation}
where $\gamma=\gamma(x)\in {\mathbb Z}$, \ $x_k=0,1,\dots,p-1$, $x_0\ne 0$,
$k=0,1,\dots$. The series is convergent in the $p$-adic norm $|\cdot|_p$,
and one has $|x|_p=p^{-\gamma}$.
The {\it fractional part} of a number $x\in {\mathbb Q}_p$ (given by (\ref{8.1}))
is defined as follows
\begin{equation}
\label{8.2**}
\{x\}_p=\left\{
\begin{array}{lll}
0,\quad \text{if} \quad \gamma(x)\ge 0 \quad \text{or} \quad x=0,&& \\
p^{\gamma}(x_0+x_1p+x_2p^2+\cdots+x_{|\gamma|-1}p^{|\gamma|-1}),
\quad \text{if} \quad \gamma(x)<0. && \\
\end{array}
\right.
\end{equation}
The space ${\mathbb Q}_p^n={\mathbb Q}_p\times\cdots\times{\mathbb Q}_p$ consists of points
$x=(x_1,\dots,x_n)$, where $x_j \in {\mathbb Q}_p$, $j=1,2\dots,n$, \ $n\ge 2$.
The $p$-adic norm on ${\mathbb Q}_p^n$ is
\begin{equation}
\label{8}
|x|_p=\max_{1 \le j \le n}|x_j|_p, \quad x\in {\mathbb Q}_p^n,
\end{equation}
where $|x_j|_p$, $x_j\in {\mathbb Q}_p$, is defined by (\ref{1}), $j=1,\dots,n$.
Denote by $B_{\gamma}^n(a)=\{x: |x-a|_p \le p^{\gamma}\}$ the ball
of radius $p^{\gamma}$ with the center at a point $a=(a_1,\dots,a_n)\in {\mathbb Q}_p^n$
and by $S_{\gamma}^n(a)=\{x: |x-a|_p = p^{\gamma}\}
=B_{\gamma}^n(a)\setminus B_{\gamma-1}^n(a)$ its boundary (sphere),
$\gamma \in {\mathbb Z}$. For $a=0$ we set $B_{\gamma}^n(0)=B_{\gamma}^n$ and
$S_{\gamma}^n(0)=S_{\gamma}^n$. For the case $n=1$ we will omit the
upper index $n$.
Here
\begin{equation}
\label{9}
B_{\gamma}^n(a)=B_{\gamma}(a_1)\times\cdots\times B_{\gamma}(a_n),
\end{equation}
where $B_{\gamma}(a_j)=\{x_j: |x_j-a_j|_p \le p^{\gamma}\}$ is a
disc of radius $p^{\gamma}$ with the center at a point $a_j\in {\mathbb Q}_p$,
$j=1,2\dots,n$. Any two balls in ${\mathbb Q}_p^n$ either are
disjoint or one contains the other. Every point of the ball is its
center.
\begin{Proposition}
\label{pr1-int}
{\rm (~\cite[I.3, Examples 1,2.]{Vl-V-Z})}
The disc $B_{\gamma}$ is represented by the sum of $p^{\gamma-\gamma'}$
{\em disjoint} discs $B_{\gamma'}(a)$, $\gamma'<\gamma$:
\begin{equation}
\label{79.0}
B_{\gamma}=B_{\gamma'}\cup\cup_{a}B_{\gamma'}(a),
\end{equation}
where $a=0$ and
$a=a_{-r}p^{-r}+a_{-r+1}p^{-r+1}+\cdots+a_{-\gamma'-1}p^{-\gamma'-1}$
are the centers of the discs $B_{\gamma'}(a)$, \
$0\le a_j\le p-1$, $j=-r,-r+1,\dots,-\gamma'-1$, $a_{-r}\ne 0$, \,
$r=\gamma,\gamma-1,\gamma-2,\dots,\gamma'+1$.
In particular, the disc $B_{0}$ is represented by the sum of $p$
{\em disjoint} discs
\begin{equation}
\label{79}
B_{0}=B_{-1}\cup\cup_{r=1}^{p-1}B_{-1}(r),
\end{equation}
where $B_{-1}(r)=\{x\in S_{0}: x_0=r\}=r+p{\mathbb Z}_p$, $r=1,\dots,p-1$;
$B_{-1}=\{|x|_p\le p^{-1}\}=p{\mathbb Z}_p$; and
$S_{0}=\{|x|_p=1\}=\cup_{r=1}^{p-1}B_{-1}(r)$. Here all the discs
are disjoint.
\end{Proposition}
We call covering (\ref{79.0}), (\ref{79}) the {\it canonical covering} of the disc $B_{0}$.
A complex-valued function $f$ defined on ${\mathbb Q}_p^n$ is called
{\it locally-constant} if for any $x\in {\mathbb Q}_p^n$ there exists
an integer $l(x)\in {\mathbb Z}$ such that
$$
f(x+y)=f(x), \quad y\in B_{l(x)}^n.
$$
Let ${{\mathcal E}}({\mathbb Q}_p^n)$ and ${{\mathcal D}}({\mathbb Q}_p^n)$ be the
linear spaces of locally-constant ${\mathbb C}$-valued functions on ${\mathbb Q}_p^n$
and locally-constant ${\mathbb C}$-valued functions with compact supports
(so-called test functions), respectively; ${{\mathcal D}}({\mathbb Q}_p)$,
${{\mathcal E}}({\mathbb Q}_p)$~\cite[VI.1.,2.]{Vl-V-Z}.
If $\varphi \in {{\mathcal D}}({\mathbb Q}_p^n)$, according to Lemma~1 from~\cite[VI.1.]{Vl-V-Z},
there exists $l\in {\mathbb Z}$, such that
$$
\varphi(x+y)=\varphi(x), \quad y\in B_l^n, \quad x\in {\mathbb Q}_p^n.
$$
The largest of the numbers $l=l(\varphi)$ is called the
{\it parameter of constancy} of the function $\varphi$.
Let us denote by ${{\mathcal D}}^l_N({\mathbb Q}_p^n)$ the finite-dimensional space of
test functions from ${{\mathcal D}}({\mathbb Q}_p^n)$ with supports in the ball $B_N^n$
and with parameters of constancy $\ge l$~\cite[VI.2.]{Vl-V-Z}.
We have ${{\mathcal D}}^l_N({\mathbb Q}_p^n) \subset {{\mathcal D}}^{l'}_{N'}({\mathbb Q}_p^n)$, \
$N\le N'$, \ $l\ge l'$.
\begin{Lemma}
\label{lem-four-1}
{\rm(~\cite[VI.5.,(5.2')]{Vl-V-Z})}
Any function $\varphi \in {{\mathcal D}}^l_N({\mathbb Q}_p^n)$ can be represented as a {finite}
linear combination
$$
\varphi(x)=\sum_{\nu=1}^{p^{n(N-l)}}\varphi(a^{\nu})\Delta_{l}(x-a^{\nu}),
\quad x\in {\mathbb Q}_p^n,
$$
where $\Delta_{l}(x-a^{\nu})=\Omega(p^{-l}|x-a^{\nu}|_p)](x)$ is the characteristic
function of the ball $B_{l}^n(a^{\nu})$, and the points
$a^{\nu}=(a_1^{\nu},\dots a_n^{\nu})\in B_N^n$ do not depend on $\varphi$
and are such that the bolls
$B_{l}^n(a^{\nu})$, $\nu=1,\dots,p^{n(N-l)}$, are disjoint and cover
the ball $B_{N}^n$.
\end{Lemma}
Denote by ${{\mathcal D}}'({\mathbb Q}_p^n)$ the set of all linear functionals on
${{\mathcal D}}({\mathbb Q}_p^n)$~\cite[VI.3.]{Vl-V-Z}.
The Fourier transform of $\varphi\in {{\mathcal D}}({\mathbb Q}_p^n)$ is defined by the
formula
$$
F[\varphi](\xi)=\int_{{\mathbb Q}_p^n}\chi_p(\xi\cdot x)\varphi(x)\,d^nx,
\quad \xi \in {\mathbb Q}_p^n,
$$
where $d^n x=dx_1\cdots dx_n$ is the Haar measure such that
$\int_{|\xi|_p\le 1}\,d^nx=1$;
$\chi_p(\xi\cdot x)=\chi_p(\xi_1x_1)\cdots\chi_p(\xi_nx_n)$;
$\xi\cdot x$ is the scalar product of vectors and
$\chi_p(\xi_jx_j)=e^{2\pi i\{\xi_jx_j\}_p}$
are additive characters, $\{x\}_p$ is the {\it fractional part} (\ref{8.2**})
of a number $x\in {\mathbb Q}_p$.
\begin{Lemma}
\label{lem-four-2}
{\rm(~\cite[Lemma~A.]{Taib1},~\cite[III,(3.2)]{Taib3},~\cite[VII.2.]{Vl-V-Z})}
The Fourier transform is a linear isomorphism ${{\mathcal D}}({\mathbb Q}_p^n)$ into
${{\mathcal D}}({\mathbb Q}_p^n)$. Moreover,
\begin{equation}
\label{12}
\varphi(x) \in {{\mathcal D}}^l_N({\mathbb Q}_p^n) \quad \text{iff} \quad
F\big[\varphi(x)\big](\xi) \in {{\mathcal D}}^{-N}_{-l}({\mathbb Q}_p^n).
\end{equation}
\end{Lemma}
The Fourier transform $F[f]$ of a distribution
$f\in {{\mathcal D}}'({\mathbb Q}_p^n)$ is defined by the relation
$\langle F[f],\varphi\rangle=\langle f,F[\varphi]\rangle$,
for all $\varphi\in {{\mathcal D}}({\mathbb Q}_p^n)$.
Let $A$ be a matrix and $b\in {\mathbb Q}_p^n$. Then for a distribution
$f\in{{\mathcal D}}'({\mathbb Q}_p^n)$ the following relation holds~\cite[VII,(3.3)]{Vl-V-Z}:
\begin{equation}
\label{14}
F[f(Ax+b)](\xi)
=|\det{A}|_p^{-1}\chi_p\big(-A^{-1}b\cdot \xi\big)F[f(x)]\big(A^{-1}\xi\big),
\end{equation}
where $\det{A} \ne 0$.
According to~\cite[IV,(3.1)]{Vl-V-Z},
\begin{equation}
\label{14.1}
F[\Omega(p^{-k}|\cdot|_p)](x)=p^{nk}\Omega(p^k|x|_p), \quad k\in {\mathbb Z}, \qquad x \in {\mathbb Q}_p^n.
\end{equation}
In particular, $F[\Omega(|\xi|_p)](x)=\Omega(|x|_p)$. Here
$\Omega(t)$ is the characteristic function of the segment $[0,1]\subset{\mathbb R}$,
\subsection{$p$-Adic Lizorkin spaces}\label{s2.2}
According to~\cite{Al-Kh-Sh3},~\cite{Al-Kh-Sh4}, the $p$-adic
{\it Lizorkin space of test functions\/} is defined as
$$
\Phi({\mathbb Q}_p^n)=\{\phi: \phi=F[\psi], \, \psi\in \Psi({\mathbb Q}_p^n)\},
$$
where $\Psi({\mathbb Q}_p^n)
=\{\psi(\xi)\in {\mathcal D}({\mathbb Q}_p^n): \psi(0)=0\}$.
The space $\Phi({\mathbb Q}_p^n)$ can be equipped with the
topology of the space ${\mathcal D}({\mathbb Q}_p^n)$ which makes it a
complete space. In view of Lemma~\ref{lem-four-2}, the Lizorkin space
$\Phi({\mathbb Q}_p^n)$ admits the following characterization.
\begin{Lemma}
\label{lem1}
{\rm (~\cite{Al-Kh-Sh3},~\cite{Al-Kh-Sh4})}
{\rm (a)} $\phi\in \Phi({\mathbb Q}_p^n)$ iff $\phi\in {\mathcal D}({\mathbb Q}_p^n)$ and
\begin{equation}
\label{54}
\int_{{\mathbb Q}_p^n}\phi(x)\,d^nx=0.
\end{equation}
{\rm (b)} $\phi \in {{\mathcal D}}^l_N({\mathbb Q}_p^n)\cap\Phi({\mathbb Q}_p^n)$, i.e.,
$\int_{B^n_{N}}\phi(x)\,d^nx=0$,
iff \ $\psi=F^{-1}[\phi]\in {{\mathcal D}}^{-N}_{-l}({\mathbb Q}_p^n)\cap\Psi({\mathbb Q}_p^n)$,
i.e., $\psi(\xi)=0$, $\xi \in B^n_{-N}$.
\end{Lemma}
Suppose that $\Phi'({\mathbb Q}_p^n)$ and $\Psi'({\mathbb Q}_p^n)$ denote the topological
dual of the spaces $\Phi({\mathbb Q}_p^n)$ and $\Psi({\mathbb Q}_p^n)$, respectively.
We call $\Phi'({\mathbb Q}_p^n)$ the space of $p$-adic {\em Lizorkin
distributions}.
The space $\Phi'({\mathbb Q}_p^n)$ can be obtained from ${\mathcal D}'({\mathbb Q}_p^n)$ by
``sifting out'' constants. Thus two distributions in ${\mathcal D}'({\mathbb Q}_p^n)$
differing by a constant are indistinguishable as elements of $\Phi'({\mathbb Q}_p^n)$.
We define the Fourier transform of
$f\in \Phi'({\mathbb Q}_p^n)$ and $g\in \Psi'({\mathbb Q}_p^n)$
respectively by formulas $\langle F[f],\psi\rangle=\langle f,F[\psi]\rangle$,
for all $\psi\in \Psi({\mathbb Q}_p^n)$, and
$\langle F[g],\phi\rangle=\langle g,F[\phi]\rangle$, for all
$\phi\in \Phi({\mathbb Q}_p^n)$.
It is clear that $F[\Phi'({\mathbb Q}_p^n)]=\Psi'({\mathbb Q}_p^n)$
and $F[\Psi'({\mathbb Q}_p^n)]=\Phi'({\mathbb Q}_p^n)$~\cite{Al-Kh-Sh3}.
Recall that in the real setting the Lizorkin spaces were introduced in the
excellent papers by P.~I.~Lizorkin~\cite{Liz1},~\cite{Liz3}.
\section{Non-Haar $p$-adic wavelet bases}
\label{s3}
\subsection{One non-Haar wavelet basis in ${{\mathcal L}}^2({\mathbb Q}_p)$.}\label{s3.1}
It is well known that
${\mathbb Q}_p=B_{0}(0)\cup\cup_{\gamma=1}^{\infty}S_{\gamma}$, where
$S_{\gamma}=\{x\in {\mathbb Q}_p: |x|_p = p^{\gamma}\}$. Due to
(\ref{8.1}), $x\in S_{\gamma}$, $\gamma\ge 1$, if and only if
$x=x_{-\gamma}p^{-\gamma}+x_{-\gamma+1}p^{-\gamma+1}+\cdots+x_{-1}p^{-1}+\xi$,
where $x_{-\gamma}\ne 0$, $\xi \in B_{0}(0)$. Since
$x_{-\gamma}p^{-\gamma}+x_{-\gamma+1}p^{-\gamma+1}
+\cdots+x_{-1}p^{-1}\in I_p$, we have a ``natural'' decomposition of
${\mathbb Q}_p$ into a union of mutually disjoint discs:
$$
{\mathbb Q}_p=\bigcup\limits_{a\in I_p}B_{0}(a).
$$
Therefor,
$$
I_p=\{a=p^{-\gamma}\big(a_{0}+a_{1}p+\cdots+a_{\gamma-1}p^{\gamma-1}\big):
\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{62.0**}
\gamma\in {\mathbb N}; \, a_j=0,1,\dots,p-1; \, j=0,1,\dots,\gamma-1\}
\end{equation}
is a {\em ``natural'' set of shifts} for ${\mathbb Q}_p$, which will
be used in the sequel.
This set $I_p$ can be identified with the factor group ${\mathbb Q}_p/{\mathbb Z}_p$.
Let
$$
J_{p;m}=\{s=p^{-m}\big(s_{0}+s_{1}p+\cdots+s_{m-1}p^{m-1}\big):
\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{62.0*}
\qquad
s_j=0,1,\dots,p-1; \, j=0,1,\dots,m-1; s_0\ne 0\},
\end{equation}
where $m\ge 1$ is a {\it fixed} positive integer.
Let us introduce the set of $(p-1)p^{m-1}$ functions
\begin{equation}
\label{62.1-11}
\theta_{s}^{(m)}(x)=\chi_p(sx)\Omega\big(|x|_p\big), \quad s\in J_{p;m},
\quad x\in {\mathbb Q}_p,
\end{equation}
and the family of functions generated by their dilatations and shifts:
\begin{equation}
\label{62.1}
\theta_{s;\,j a}^{(m)}(x)=p^{-j/2}\chi_p\big(s(p^{j}x-a)\big)
\Omega\big(|p^{j}x-a|_p\big), \quad x\in {\mathbb Q}_p,
\end{equation}
where $s\in J_{p;m}$, $j\in {\mathbb Z}$, $a\in I_p$, $\Omega(t)$ is the
characteristic function of the segment $[0,1]\subset{\mathbb R}$.
\begin{Theorem}
\label{th2}
The functions {\rm(\ref{62.1})} form an orthonormal {\em non-Haar $p$-adic wavelet
basis} in ${{\mathcal L}}^2({\mathbb Q}_p)$.
\end{Theorem}
\begin{proof}
1. Consider the scalar product
$$
\big(\theta_{s';\,j' a'}^{(m)}(x),\theta_{s;\,j a}^{(m)}(x)\big)
=p^{-(j+j')/2}\int_{{\mathbb Q}_p}\chi_p\big(s'(p^{j'}x-a')-s(p^{\gamma}x-a)\big)
$$
\begin{equation}
\label{62.3}
\quad
\times
\Omega\big(|p^{j}x-a|_p\big)\Omega\big(|p^{j'}x-a'|_p\big)\,dx.
\end{equation}
If $j \le j'$, according to formula~\cite[VII.1]{Vl-V-Z},~\cite{Koz0}
\begin{equation}
\label{62.4-1}
\Omega\big(|p^{j}x-a|_p\big)\Omega\big(|p^{j'}x-a'|_p\big)
=\Omega\big(|p^{j}x-a|_p\big)\Omega\big(|p^{j'-j}a-a'|_p\big),
\end{equation}
(\ref{62.3}) can be rewritten as
$$
\big(\theta_{s';\, j' a'}^{(m)}(x),\theta_{s;\,j a}^{(m)}(x)\big)
=p^{-(j+j')/2}\Omega\big(|p^{j'-j}a-a'|_p\big)
\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{62.4}
\times
\int_{{\mathbb Q}_p}
\chi_p\big(s'(p^{j'}x-a')-s(p^{j}x-a)\big)\Omega\big(|p^{j}x-a|_p\big)\,dx.
\end{equation}
Let $j < j'$. Making the change of variables $\xi=p^{j}x-a$
and taking into account (\ref{14.1}), we obtain from (\ref{62.4})
$$
\big(\theta_{s';\, j' a'}^{(m)}(x),\theta_{s;\,j a}^{(m)}(x)\big)
=p^{-(j+j')/2}\chi_p\big(s'(p^{j'-j}a-a')\big)
\qquad\qquad\qquad\qquad
$$
$$
\qquad\qquad
\times
\Omega\big(|p^{j'-j}a-a'|_p\big)
\int_{{\mathbb Q}_p}\chi_p\big((p^{j'-j}s'-s)\xi\big)
\Omega\big(|\xi|_p\big)\,d\xi
$$
$$
=p^{-(j+j')/2}\chi_p\big(s'(p^{j'-j}a-a')\big)
\quad
$$
\begin{equation}
\label{62.5}
\times
\Omega\big(|p^{j'-j}a-a'|_p\big)\Omega\big(|p^{j'-j}s'-s|_p\big).
\end{equation}
Since
$$
\begin{array}{rclrcl}
\displaystyle
p^{j'-j}s'&=&p^{j'-j-m}
\big(s_{0}'+s_{1}'p+\cdots+s_{j-1}'p^{m-1}\big), \smallskip \\
s&=&p^{-m}\big(s_{0}+s_{1}p+\cdots+s_{j-1}p^{m-1}\big), \\
\end{array}
$$
where $s_{0}',s_{0}\ne 0$, $j'-j\le 1$, it is clear that the
fractional part $\{p^{j'-j}s'-s\}_p\ne 0$. Thus
$\Omega\big(|p^{j'-j}s'-s|_p\big)=0$ and
$\big(\theta_{s';\, j' a'}^{(m)}(x),\theta_{s;\, j a}^{(m)}(x)\big)=0$.
Consequently, the scalar product $\big(\theta_{s';\, j' a'}^{(m)}(x),
\theta_{s;\,j a}^{(m)}(x)\big)=0$ can be nonzero
only if $j=j'$. In this case (\ref{62.5}) implies
$$
\big(\theta_{s';\, j a'}^{(m)}(x),\theta_{s;\,j a}^{(m)}(x)\big)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{62.6}
=p^{-j}
\chi_p\big(s'(a-a')\big)\Omega\big(|a-a'|_p\big)\Omega\big(|s'-s|_p\big),
\end{equation}
where $\Omega\big(|a-a'|_p\big)=\delta_{a'a}$,
$\Omega\big(|s'-s|_p\big)=\delta_{s's}$, and $\delta_{s's}$,
$\delta_{a'a}$ are the Kronecker symbols.
Since $\int_{{\mathbb Q}_p}\Omega\big(|p^{j}x-a|_p\big)\,dx=p^{j}$, formulas
(\ref{62.5}), (\ref{62.6}) imply that
\begin{equation}
\label{62.7}
\big(\theta_{s';\, j' a'}^{(m)}(x),\theta_{s;\,j a}^{(m)}(x)\big)
=\delta_{s's}\delta_{j'j}\delta_{a'a}.
\end{equation}
Thus the system of functions (\ref{62.1}) is orthonormal.
To prove the completeness of the system of functions (\ref{62.1}),
we repeat the corresponding proof~\cite{Koz0} almost word for word.
Recall that the system of the characteristic functions of the discs
$B_{k}(0)$ is complete in ${{\mathcal L}}^2({\mathbb Q}_p)$. Consequently, taking into
account that the system of functions
$\{\theta_{s;\,j a}^{(m)}(x): s\in J_{p;m}; j\in {\mathbb Z}, a\in I_p\}$
is invariant under dilatations and shifts, in order to prove that
it is a complete system, it is sufficient to verify the Parseval identity
for the characteristic function $\Omega(|x|_p)$.
If $0\le j$, according to (\ref{62.4-1}), (\ref{14.1}),
$$
\big(\Omega(|x|_p),\theta_{s;\,j a}^{(m)}(x)\big)
=p^{-j/2}\Omega\big(|-a|_p\big)
\int_{{\mathbb Q}_p}\chi_p\big(s(p^{j}x-a)\big)\Omega\big(|x|_p\big)\,dx
$$
$$
\qquad\qquad
=p^{-j/2}\chi_p\big(-sa)\big)
\Omega\big(|sp^{j}|_p\big)\Omega\big(|-a|_p\big)
$$
\begin{equation}
\label{62.7-11}
\qquad\quad
=\left\{
\begin{array}{lll}
0, && a\ne 0, \\
0, && a=0, \quad j \le m-1, \\
p^{-j/2}, && a=0, \quad j \ge m. \\
\end{array}
\right.
\end{equation}
If $0> j$, according to (\ref{62.4-1}), (\ref{14.1}),
$$
\big(\Omega(|x|_p),\theta_{s;\,j a}^{(m)}(x)\big)
=p^{-j/2}\Omega\big(|p^{-j}a|_p\big)
\int_{{\mathbb Q}_p}\chi_p\big(s(p^{j}x-a)\big)
\Omega\big(|p^{j}x-a|_p\big)\,dx
$$
\begin{equation}
\label{62.7-12}
=p^{j/2}\Omega\big(|p^{-j}a|_p\big)
\int_{{\mathbb Q}_p}\chi_p\big(s\xi\big)\Omega\big(|\xi|_p\big)\,d\xi
=p^{j/2}\Omega\big(|p^{-j}a|_p\big)\Omega\big(|s|_p\big)=0.
\end{equation}
Thus,
$$
\sum_{s\in J_{p;m};j\in {\mathbb Z},a\in I_p}
\big|\big(\Omega(|x|_p),\theta_{s;\,j a}^{(m)}(x)\big)\big|^2
=\sum_{j=m}^{\infty}\sum_{s\in J_{p;m}}p^{-j}
\qquad\qquad\qquad\quad
$$
$$
\qquad\qquad
=p^{m-1}(p-1)\frac{p^{-m}}{1-p^{-1}}=1
=\big|\big(\Omega(|x|_p),\Omega(|x|_p)\big|^2.
$$
Thus the system of functions (\ref{62.1}) is an orthonormal basis in
${{\mathcal L}}^2({\mathbb Q}_p)$.
2. Since elements of basis (\ref{62.1}) can be obtained
by dilatations and shifts of the set of $(p-1)p^{m-1}$ functions
(\ref{62.1-11}), it is the $p$-adic wavelet basis.
3. According to~\cite{S-Sk-1}, the Haar wavelet functions are constructed by
the Haar type {\em refinement equation} (\ref{62.0-3**}). In particular, it is easy
to see that Kozyrev's wavelet functions (\ref{62.0-1-0}) can be expressed in terms
of the {\em refinable function} (\ref{62.0-3**-1}) as
\begin{equation}
\label{62.0-1-2}
\theta_{k}(x)=\chi_p(p^{-1}kx)\Omega\big(|x|_p\big)
=\sum_{r=0}^{p-1}h_{k r}\phi\Big(\frac{1}{p}x-\frac{r}{p}\Big),
\quad x\in {\mathbb Q}_p,
\end{equation}
where $h_{k r}=e^{2\pi i\{\frac{kr}{p}\}_p}$, $r=0,1,\dots,p-1$, \
$k=1,2,\dots,p-1$.
In this case the wavelet functions $\theta_{k}(x)=\chi_p(p^{-1}kx)\Omega\big(|x|_p\big)$
takes values in the set $\{e^{2\pi i\frac{kr}{p}}:r=0,1,\dots,p-1\}$ of $p$ elements
on the discs $B_{-1}(r)$, $r=0,1,\dots,p-1$.
Thus any wavelet function $\theta_{k}(x)$ is represented as a linear combination
of the characteristic functions of the disks of the radius of $p^{-1}$.
In contrast to the Kozyrev wavelet basis (\ref{62.0-1}), the number
of generating wavelet functions (\ref{62.1-11}) for the wavelet basis (\ref{62.1})
is not minimal. For example, if $p=2$, then we have $2^{m-1}$
wavelet functions (instead of one as it is for (\ref{62.0-1}) and for classical
wavelet bases in real analysis.
Let $B_{0}=\cup_{b}B_{-m}(a)\cup B_{-m}$ be the {\it canonical covering}
(\ref{79.0}) of the disc $B_{0}$ with $p^{m}$ discs, $m\ge 1$, where
$b=0$ and $b=b_{r}p^{r}+b_{r+1}p^{r+1}+\cdots+b_{m-1}p^{m-1}$
is the center of the discs $B_{-m}$ and $B_{-m}(b)$, respectively,
$0\le b_j\le p-1$, $j=r,r+1,\dots,m-1$, $b_r\ne 0$, \ $r=0,1,2,\dots,m-1$.
For $x\in B_{-m}(b)$, $s\in J_{p;m}$, we have
$x=b+p^{m}\big(y_{0}+y_{1}p+y_{2}p^2+\cdots\big)$,
$s=p^{-m}\big(s_{0}+s_{1}p+\cdots+s_{m-1}p^{m-1}\big)$, $s_0\ne 0$;
$sx=sb+\xi$, $\xi\in {\mathbb Z}_p$;
and $\{sx\}_p=\{sb\}_p=\{p^{r-m}\big(b_{r}+a_{r+1}p+\cdots+b_{m-1}p^{m-r-1}\big)
\big(s_{0}+s_{1}p+\cdots+s_{m-1}p^{m-1}\big)\}_p$, \
$r=0,1,2,\dots,m-1$, (see (\ref{8.2**})).
Thus,
$$
\theta_{s}^{(m)}(x)=\chi_p(sx)\Omega\big(|x|_p\big)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{62.0-2-1*}
\qquad
=\left\{
\begin{array}{lll}
0, && |x|_p\ge p, \\
e^{2\pi i\{sa\}_p}, && x\in B_{-m}(b), \quad
b=\sum_{l=r}^{m-1}b_{l}p^{l}, \\
1, && x\in B_{-m}, \\
\end{array}
\right.
\end{equation}
where $0\le b_j\le p-1$, $j=r,\dots,m-1$, $b_r\ne 0$, $r=0,1,\dots,m-1$;
\, $s=p^{-m}\big(s_{0}+s_{1}p+\cdots+s_{m-1}p^{m-1}\big)$,
$0\le s_j\le p-1$, $j=0,1,\dots,m-1$, $s_0\ne 0$.
Thus the function $\theta_{s}^{(m)}(x)=\chi_p(sx)\Omega\big(|x|_p\big)$
takes values in the set $\{e^{2\pi i\{sb\}_p}:b=\sum_{l=r}^{m-1}b_{l}p^{l},
0\le b_j\le p-1, j=r,\dots,m-1, b_r\ne 0, r=0,1,\dots,m-1\}$
of $p^m$ elements on the discs $B_{-m}(b)$. By using (\ref{62.0-2-1*}),
one can see that in contrast to the Kozyrev wavelet functions (\ref{62.0-1-0}),
any wavelet function $\theta_{s}^{(m)}(x)$ is represented as a
linear combination of the characteristic functions of the disks of the radius
of $p^{-m}$:
\begin{equation}
\label{62.0-5}
\theta_{s}^{(m)}(x)=\chi_p(sx)\Omega\big(|x|_p\big)
=\sum_{b}{\tilde h}_{s b}\phi\Big(\frac{1}{p^m}x-\frac{b}{p^m}\Big),
\end{equation}
$x\in {\mathbb Q}_p$, where ${\tilde h}_{s b}=e^{2\pi i\{sb\}_p}$,
$b=0$ or $b=b_{r}p^{r}+b_{r+1}p^{r+1}+\cdots+b_{m-1}p^{m-1}$,
$r=0,1,\dots,m-1$, $0\le b_j\le p-1$, $b_r\ne 0$; $s\in J_{p;m}$.
According to formulas (\ref{62.0-2-1*}), (\ref{62.0-5}), the wavelet
function (\ref{62.1-11}) {\em cannot be represented} in terms of the
{\em Haar refinable function} $\phi(x)=\Omega\big(|x|_p\big)$ (which is
a solution of the Haar {\it refinement equation} (\ref{62.0-3**})), i.e.,
in the form
$$
\theta_{s}^{(m)}(x)=\sum_{a\in I_p}\beta_a\phi(p^{-1}x-a),
\quad \beta_a\in {\mathbb C}.
$$
Consequently, the wavelet basis (\ref{62.1}) is non-Haar type for $m\ge 2$.
\end{proof}
Making the change of variables $\xi=p^{j}x-a$ and taking into
account (\ref{14.1}), we obtain
$\int{{\mathbb Q}_p}\theta_{s;\,j a}^{(m)}(x)\,dx
=p^{j/2}\int\limits_{{\mathbb Q}_p}\chi_p\big(s\xi\big)\Omega\big(|\xi|_p\big)\,d\xi
=p^{j/2}\Omega\big(|s|_p\big)=0$,
i.e., according to Lemma~\ref{lem1}, the wavelet function
$\theta_{s;\,j a}^{(m)}(x)$ belongs to the Lizorkin space $\Phi({\mathbb Q}_p)$.
\begin{Remark}
\label{rem1} \rm
1. Kozyrev's wavelet basis (\ref{62.0-1}) is a particular case of the wavelet
basis (\ref{62.1}) for $m=1$. Indeed, in this case we have
$\theta_{s;\,j a}^{(1)}(x)\equiv \theta_{k;\,j a}(x)$,
where $s\equiv p^{-1}k$, $k=1,2,\dots,p-1$; \,$j\in {\mathbb Z}$, $a\in I_p$.
2. Our non-Haar wavelet basis (\ref{62.1}) can be constructed in the framework of the
approach developed by J.~J.~Benedetto and R.~L.~Benedetto~\cite{Ben-Ben}
\footnote{Here we follow the referee report of the previous version of our paper.}.
In the notation~\cite{Ben-Ben}, $H^{\perp}$ is the ball $B_0$, and $A^*_1:{\mathbb Q}_p\to{\mathbb Q}_p$
is multiplication by $p^{-1}$, and $W=(A^*_1)^{m-1}H^{\perp}=B_{m-1}$.
In addition, the ``choice coset of representatives'' ${\mathcal D}$ in~\cite{Ben-Ben}
is the set (\ref{62.0**}) of shifts $I_p$, and the set $J_{p;m}$ given by (\ref{62.0*})
is precisely the set ${\mathcal D}\cap\big((A^*_1W)\setminus W\big)$ that appears in
equation~\cite[(4.1)]{Ben-Ben}.
The set $J_{p;m}$ consists of $N=p^{m}-p^{m-1}$ elements, where $N$ is the number
of wavelet generators.
The algorithm of~\cite{Ben-Ben} starts with $N$ sets $\Omega{s,0}$ and $N$
local translation functions $T_s$, one for each $s\in J_{p;m}$. In order to
construct the wavelets (\ref{62.1}) by using the algorithm of~\cite{Ben-Ben},
we set
$$
\Omega_{s,0}=B_0\big(p^{-m}(s_{1}p+\cdots+s_{m-1}p^{m-1})\big)
\qquad\qquad\quad
$$
$$
\qquad
=p^{-m}(s_{1}p+\cdots+s_{m-1}p^{m-1})+B_0\subset B_{m-1},
$$
i.e., remove the term $s_0$ and add the set $B_0=H^{\perp}$. We also define
$$
T_s:B_{m-1}\to B_{m}\setminus B_{m-1} \quad \text{by} \quad T_s(w)=w+s_0p^{-m},
$$
so that $T_s$ maps $\Omega_{s,0}$ to $B_0(s)$ by translation. It is easy to verify
that this data fit the requirements of~\cite{Ben-Ben}: each $\Omega_{s,0}$ is
$(\tau,{\mathcal D})$-congruent to $H^{\perp}=B_0$, the union of all such sets contains
a neighborhood of $0$, each $T_s$ has the form required by formula~\cite[(4.1)]{Ben-Ben},
and for each $s\ne s'$ in $J_{p;m}$, one of the two compatibility
condition~\cite[(4.2) or (4.3) ]{Ben-Ben} holds.
Thus, by using the algorithm of~\cite{Ben-Ben}, one can produce the wavelets (\ref{62.1}).
Moreover, according to (\ref{o-64.7}), any wavelet $\theta_{s}^{(m)}$ is the Fourier transform
of the characteristic function of each disc $B_0(s)$, $s\in J_{p;m}$. Recall that
the algorithm of~\cite{Ben-Ben} only allows the construction of wavelet functions whose
Fourier transforms are the characteristic functions
of some sets (see~\cite[Proposition~5.1.]{Ben-Ben}).
\end{Remark}
\subsection{Countable family of non-Haar wavelet bases in ${{\mathcal L}}^2({\mathbb Q}_p)$.}\label{s3.2}
Now, using the proof scheme of~\cite[Theorem~1]{S-Sk-1}, we construct
infinitely many different non-Haar wavelet bases, which are distinct from the basis (\ref{62.1}).
In what follows, we shall write the $p$-adic number
$a=p^{-\gamma}\big(a_{0}+a_{1}p+\cdots+a_{\gamma-1}p^{\gamma-1}\big)\in I_p$, \
$a_{j}=0,1,\dots,p-1$, \ $j=0,1,\dots,\gamma-1$, in the form $a=\frac{r}{p^{\gamma}}$,
where $r=a_{0}+a_{1}p+\cdots+a_{\gamma-1}p^{\gamma-1}$.
Since the $p$-adic norm is non-Archimedean, it is easy to see that the wavelet
functions (\ref{62.1-11}) have the following property:
\begin{equation}
\label{100-11}
\theta_{s}^{(m)}(x\pm 1)=\chi_p(\pm s)\theta_{s}^{(m)}(x), \quad s\in J_{p;m}.
\end{equation}
\begin{Theorem}
\label{th4-11}
Let $\nu=1,2,\dots$. The functions
\begin{equation}
\label{101-11}
\psi_{s}^{(m)[\nu]}(x)=\sum_{k=0}^{p^\nu-1}\alpha_{s;k}\theta_{s}^{(m)}\Big(x-\frac{k}{p^\nu}\Big),
\qquad s\in J_{p;m},
\end{equation}
are wavelet functions if and only if
\begin{equation}
\label{108-11}
\alpha_{s;k}=p^{-\nu}\sum_{r=0}^{p^\nu-1}\gamma_{s;r}e^{-2i\pi\frac{-s+r}{p}k},
\end{equation}
where $\gamma_{s;k}\in {\mathbb C}$, $|\gamma_{s;k}|=1$, $k=0,1,\dots,p^\nu-1$, \,$s\in J_{p;m}$.
\end{Theorem}
\begin{proof}
Let $\psi_{s}^{(m)[\nu]}(x)$ be defined by (\ref{101-11}), $s\in J_{p;m}$.
According to Theorem~\ref{th2}, $\{\theta_{s}^{(m)}(\cdot-a), s\in J_{p;m}, a\in I_p\}$
is an orthonormal system. Hence, taking into account (\ref{100-11}),
we see that $\psi_{s}^{(m)[\nu]}$ is orthogonal to $\psi_{s}^{[\nu]}(x)(\cdot-a)$
whenever $a\in I_p$, $a\ne \frac{k}{p^\nu}$, $k=0,1,\dots p^\nu-1$; \ $\nu=1,2,\dots$.
Thus the system $\{\psi_{s}^{(m)[\nu]}(x-a), s\in J_{p;m}, a\in I_p\}$ is
orthonormal if and only if the system consisting of the functions
$$
\psi_{s}^{(m)[\nu]}\Big(x-\frac{r}{p^\nu}\Big)
=\chi_p(-s)\alpha_{s;p^\nu-r}\theta_{s}^{(m)}(x)
+\chi_p(-s)\alpha_{s;p^\nu-r+1}\theta_{s}^{(m)}\Big(x-\frac{1}{p^\nu}\Big)
$$
$$
\qquad\qquad\qquad\qquad
+\cdots
+\chi_p(-s)\alpha_{s;p^\nu-1}\theta_{s}^{(m)}\Big(x-\frac{r-1}{p^\nu}\Big)
$$
$$
\qquad\qquad
+\alpha_{s;0}\theta_{s}^{(m)}\Big(x-\frac{r}{p^\nu}\Big)
+\alpha_{s;1}\theta_{s}^{(m)}\Big(x-\frac{r+1}{p^\nu}\Big)
$$
\begin{equation}
\label{104-11}
\qquad\qquad
+\cdots
+\alpha_{s;p^\nu-1-r}\theta_{s}^{(m)}\Big(x-\frac{p^\nu-1}{p^\nu}\Big),
\end{equation}
$r=0,\dots,p^\nu-1$, $s\in J_{p;m}$, is orthonormal, $\nu=1,2,\dots$.
Set
$$
\Xi_{\nu}^{[0]}=\left(\theta_{s}^{(m)},
\theta_{s}^{(m)}\left(\cdot-\frac{1}{p^\nu}\right),\dots,
\theta_{s}^{(m)}\left(\cdot-\frac{p^\nu-1}{p^\nu}\right)\right)^T,
$$
$$
\Xi_{\nu}^{[\nu]}=\left(\psi_{s}^{(m)[\nu]},
\psi_{s}^{(m)[\nu]}\left(\cdot-\frac{1}{p^\nu}\right),\dots,
\psi_{s}^{(m)[\nu]}\left(\cdot-\frac{p^\nu-1}{p^\nu}\right)\right)^T,
$$
where $T$ is the transposition operation.
By (\ref{104-11}), we have $\Xi_{s}^{[\nu]}=D_{s}\Xi_{s}^{[0]}$,
where
$$
D_{s}=
\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad\quad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{105-11}
\left(
\begin{array}{ccccc}
\alpha_{s;0}&\alpha_{s;1}&\ldots&\alpha_{s;p^\nu-2}&\alpha_{s;p^\nu-1} \\
\chi_p(-s)\alpha_{s;p^\nu-1}&\alpha_{s;0}&\ldots&\alpha_{s;p^\nu-3}&\alpha_{s;p^\nu-2} \\
\chi_p(-s)\alpha_{s;p^\nu-2}&\chi_p(-s)\alpha_{s;p^\nu-1}&\ldots&\alpha_{s;p^\nu-4}&\alpha_{s;p^\nu-3} \\
\hdotsfor{5} \\
\chi_p(-s)\alpha_{s;2}&\chi_p(-s)\alpha_{s;3}&\ldots&\alpha_{s;0}&\alpha_{s;1} \\
\chi_p(-s)\alpha_{s;1}&\chi_p(-s)\alpha_{s;2}&\ldots&\chi_p(-s)\alpha_{s;p^\nu-1}&\alpha_{s;0} \\
\end{array}
\right),
\end{equation}
and $s\in J_{p;m}$; \,$\nu=1,2,\dots$.
Due to orthonormality of $\{\psi_{s}^{(m)[\nu]}(x)(\cdot-a), s\in J_{p;m}, a\in I_p\}$,
the coordinates of $\Xi_{s}^{[\nu]}$ form an orthonormal system
if and only if the matrixes $D_{s}$ are unitary, $s\in J_{p;m}$.
Let $u_{s}=(\alpha_{s;0},\alpha_{s;1},\dots,\alpha_{s;p^\nu-1})^{T}$ be a vector
and let
$$
A_{s}=\left(
\begin{array}{cccccc}
0&0&\ldots&0&0&\chi_p(-s) \\
1&0&\ldots&0&0&0 \\
0&1&\ldots&0&0&0 \\
\hdotsfor{6} \\
0&0&\ldots&1&0&0 \\
0&0&\ldots&0&1&0 \\
\end{array}
\right)
$$
be a $p^\nu\times p^\nu$ matrix, $s\in J_{p;m}$.
It is not difficult to see that
$$
A_{s}^ru_{\nu}
=\big(\chi_p(-s)\alpha_{s;p^\nu-r},\chi_p(-s)\alpha_{s;p^\nu-r+1},\dots,
\chi_p(-s)\alpha_{s;p^\nu-1},
$$
$$
\qquad\quad\qquad\qquad
\alpha_{s;0},\alpha_{s;1},\dots,\alpha_{s;p^\nu-r-1}\big)^{T},
$$
where $r=0,1,\dots,p^s-1$, \,$s\in J_{p;m}$; \,$\nu=1,2,\dots$. Thus we have
$$
D_{s}=\big(u_{s},A_{s}u_{s},\dots,A_{s}^{p^\nu-1}u_{s}\big)^T.
$$
Consequently, to describe all unitary matrixes $D_{s}$, we should find all vectors
$u_{s}=(\alpha_{s;0},\alpha_{s;1},\dots,\alpha_{s;p^\nu-1})^{T}$
such that the system of vectors $\{A_{s}^ru_{s}, r=0,\dots,p^\nu-1\}$ is orthonormal,
$s\in J_{p;m}$.
We already have such a vector $u_0=(1,0,\dots,0,0)^{T}$
because the matrix $D_0=\big(u_0,Au_0,\dots,A^{p^\nu-1}u_0\big)^T$
is the identity matrix.
Now we prove that the system
$\{A_{s}^ru_{s}, r=0,\dots,p^\nu-1\}$ is orthonormal if and only if
$u_{s}=B_{s}u_0$, where $B_{s}$ is a unitary matrix such that
$A_{s}B_{s}=B_{s}A_{s}$, \,$s\in J_{p;m}$. Indeed,
if $u_{s}=B_{s}u_0$, $B_{s}$ is a unitary matrix such that $A_{s}B_{s}=B_{s}A_{s}$,
then $A_{s}^ru_{s}=B_{s}A_{s}^ru_0$, \ $r=0,1,\dots,p^\nu-1$, \,$s\in J_{p;m}$.
Since the system $\{A_{s}^ru_0, r=0,1,\dots,p^\nu-1\}$ is orthonormal and the
matrix $B_{s}$ is unitary, the vectors $A_{s}^ru_{s}$, \ $r=0,1,\dots,p^\nu-1$
are also orthonormal, $s\in J_{p;m}$.
Conversely, if the system $A_{s}^ru_{s}$, \ $r=0,1,\dots,p^\nu-1$ is
orthonormal, taking into account that $\{A_{s}^ru_0, r=0,1,\dots,p^\nu-1\}$
is also an orthonormal system, we conclude that
there exists a unitary matrix $B_{s}$ such that $A_{s}^ru_{s}=B_{s}(A_{s}^ru_0)$, \
$r=0,1,\dots,p^\nu-1$. Since $A_{s}^{p^\nu}u_{s}=\chi_p(-s)u_{s}$ and
$A_{s}^{p^\nu}u_0=\chi_p(-s)u_0$, we obtain additionally
$A_{s}^{p^\nu}u_{s}=B_{s}A_{s}^{p^\nu}u_0$. It follows
from the above relations that $(A_{s}B_{s}-B_{s}A_{s})(A_{s}^{r}u_0)=0$, \
$r=0,1,\dots,p^\nu-1$. Since the vectors $A_{s}^ru_0$, \
$r=0,1,\dots,p^\nu-1$ form a basis in the $p^\nu$-dimensional
space, we conclude that $A_{s}B_{s}=B_{s}A_{s}$, \,$s\in J_{p;m}$.
Thus all required unitary matrixes (\ref{105-11}) are given by
$$
D_{s}=\big(B_{s}u_0,B_{s}A_{\nu}u_0,\dots,B_{s}A_{s}^{p^\nu-1}u_0\big)^T,
$$
where $u_0=(1,0,\dots,0,0)^{T}$ and $B_{s}$ is a unitary matrix such that
$A_{s}B_{s}=B_{s}A_{s}$, \,$s\in J_{p;m}$.
It remains to describe all such matrixes $B_{s}$.
It is easy to see that the eigenvalues of $A_{s}$ and
the corresponding normalized eigenvectors are respectively
\begin{equation}
\label{106-11}
\lambda_{s;r}=e^{2i\pi\frac{-s+r}{p^\nu}}
\end{equation}
and
$$
v_{s;r}=\big((v_{s;r})_0,\dots,(v_{s;r})_{p^\nu-1}\big)^T,
$$
where
\begin{equation}
\label{107-11}
(v_{s;r})_l=p^{-\nu/2}e^{-2i\pi\frac{-s+r}{p^\nu}l},
\quad l=0,1,2,\dots,p^\nu-1,
\end{equation}
$r=0,1,\dots,p^\nu-1$, \,$s\in J_{p;m}$.
Hence the matrix $A_{s}$ can be represented as
$A_{s}=C_{s}\widetilde{A}_{s}C_{s}^{-1}$, where
$$
\widetilde{A}_{s}=\left(
\begin{array}{ccccc}
\lambda_{0}&0&\ldots&0 \\
0&\lambda_{1}&\ldots&0 \\
\vdots&\vdots&\ddots&\vdots \\
0&0&\ldots&\lambda_{p^\nu-1} \\
\end{array}
\right)
$$
is a diagonal matrix, $C_{s}=\big(v_{s;0},\dots,v_{s;p^\nu-1}\big)$
is a unitary matrix. It follows that the matrix $B_{s}=C_{s}\widetilde{B}_{s}C_{s}^{-1}$
is unitary if and only if $\widetilde{B}_{s}$ is unitary. On the other hand,
$A_{s}B_{s}=B_{s}A_{s}$ if and only if
$\widetilde{A}_{s}\widetilde{B}_{s}=\widetilde{B}_{s}\widetilde{A}_{s}$.
Moreover, since according to (\ref{106-11}), $\lambda_{s;k}\ne\lambda_{s;l}$
whenever $k\ne l$, all unitary matrixes $\widetilde{B}_{s}$ such that
$\widetilde{A}_{s}\widetilde{B}_{s}=\widetilde{B}_{s}\widetilde{A}_{s}$, are given by
$$
\widetilde{B}_{s}=\left(
\begin{array}{ccccc}
\gamma_{s;0}&0&\ldots&0 \\
0&\gamma_{s;1}&\ldots&0 \\
\vdots&\vdots&\ddots&\vdots \\
0&0&\ldots&\gamma_{s;p^\nu-1} \\
\end{array}
\right),
$$
where $\gamma_{s;k}\in {\mathbb C}$, \,$|\gamma_{s;k}|=1$, $k=0,1,\dots,p^\nu-1$, \,$s\in J_{p;m}$.
Hence all unitary matrixes $B_{s}$ such that $A_{s}B_{s}=B_{s}A_{s}$,
are given by $B_{s}=C\widetilde{B}_{s}C_{s}^{-1}$.
Using (\ref{107-11}), one can calculate
$$
\alpha_{s;k}=(B_{s}u_0)_k=(C_{s}\widetilde{B}_{s}C_{s}^{-1}u_0)_k
\qquad\qquad\qquad
$$
$$
\qquad\qquad
=\sum_{r=0}^{p^\nu-1}\gamma_{s;r}(v_{s;r})_k(\overline{v}_{s;r})_0
=p^{-\nu}\sum_{r=0}^{p^\nu-1}\gamma_{\nu;r}e^{-2i\pi\frac{-s+r}{p^\nu}k},
$$
where $\gamma_{s;k}\in {\mathbb C}$, $|\gamma_{s;k}|=1$, $k=0,1,\dots,p^\nu-1,$, \,$s\in J_{p;m}$.
It remains to prove that
\begin{equation}
\label{109-11}
\{p^{-j/2}\psi_{s}^{(m)[\nu]}(p^{j}x-a), x\in{\mathbb Q}_p: s\in J_{p;m}, \, j\in {\mathbb Z}, \,a\in I_p\}
\end{equation}
is a basis for ${{\mathcal L}}^2({\mathbb Q}_p)$ whenever $\psi_{s}^{(m)[\nu]}$ is defined by (\ref{101-11}),
(\ref{108-11}), \,$\nu=1,2,\dots$. Since according to Theorem~\ref{th2},
$$
\{p^{-j/2}\theta_{s}^{(m)}(p^{j}x-a), x\in{\mathbb Q}_p: s\in J_{p;m},\, j\in {\mathbb Z}, \,a\in I_p\}
$$
is a basis for ${{\mathcal L}}^2({\mathbb Q}_p)$, it suffices to check that any function
$p^{-j/2}\theta_{s}^{(m)}(p^{j}x-c)$, $c\in I_p$, can be
decomposed with respect to the functions $p^{-j/2}\psi_{s}^{(m)[\nu]}(p^{j}x-a)$,
$a\in I_p$; where $s\in J_{p;m}$, $j\in {\mathbb Z}$. Any $c\in I_p$, $c\ne 0$,
can be represented in the form $c=\frac{r}{p^\nu}+b$, where $r=0,1,\dots,p^\nu-1$,
$|b|_p\ge p^{\nu+1}$, $bp^\nu\in I_p$. Taking into account that
$\Xi_{s}^{[0]}=D_{s}^{-1}\Xi_{s}^{[\nu]}$, i.e.,
$$
\theta_{s}^{(m)}\Big(p^{j}x-\frac{r}{p^\nu}\Big)
=\sum_{k=0}^{p^\nu-1}\beta_{s;k}^{(r)}\psi_{s}^{(m)[\nu]}\Big(p^{j}x-\frac{k}{p^\nu}\Big),
\quad r=0,1,\dots,p^\nu-1,
$$
we have
$$
\theta_{s}^{(m)}\Big(p^{j}x-c\Big)=\theta_{s}^{(m)}\Big(p^{j}x-\frac{r}{p^\nu}-b\Big)=
\sum_{k=0}^{p^\nu-1}\beta_{s;k}^{(r)}\psi_{s}^{(m)[\nu]}\Big(p^{j}x-\frac{k}{p^\nu}-b\Big),
$$
and $\frac{k}{p^\nu}+b\in I_p$, $k=0,1,\dots,p^\nu-1$, \,$\nu=1,2,\dots$. Consequently, any function
$f\in {{\mathcal L}}^2({\mathbb Q}_p)$ can be decomposed with respect to the system of functions (\ref{109-11}).
\end{proof}
Thus, we have constructed a countable family of non-Haar wavelet bases
given by formulas (\ref{109-11}), (\ref{101-11}), (\ref{108-11}).
\begin{Remark}
\label{rem2} \rm
Let $\nu=1$. According to (\ref{109-11}), (\ref{101-11}), (\ref{108-11});
(\ref{62.1-11}), (\ref{62.1}), in this case we have
$$
\psi_{s}^{(m)[1]}(x-a)
=\sum_{k=0}^{p-1}\alpha_{s;k}\theta_{s}^{(m)}\Big(x-a-\frac{k}{p}\Big),
\quad s\in J_{p;m}, \quad a\in I_p.
$$
Applying formulas (\ref{o-64.7}), (\ref{14}) to the last relation, we obtain
$$
F[\psi_{s}^{(m)[1]}(x-a)](\xi)
=\chi_p(a\xi)\Omega\big(|\xi+s|_p\big)\sum_{k=0}^{p-1}\alpha_{s;k}\chi_p\Big(\frac{k}{p}\xi\Big),
\quad s\in J_{p;m}, \quad a\in I_p.
$$
Thus the right-hand side of the last relation is not equal to zero only if $\xi=-s+\eta$,
$\eta=\eta_0+\eta_1p+\cdots \in {\mathbb Z}_p$, i.e., $\xi \in B_0(-s)$.
Hence, for $\xi \in B_0(-s)$ we have
$$
F[\psi_{s}^{(m)[1]}(x-a)](\xi)
=\chi_p(-as)\chi_p(a\eta)\sum_{k=0}^{p-1}\alpha_{s;k}\chi_p\Big(\frac{k}{p}(-s+\eta_0)\Big),
\quad \eta\in {\mathbb Z}_p,
$$
where $s\in J_{p;m}$, $a\in I_p,$.
It is easy to verify that for any $s\in J_{p;m}$ and $a\in I_p$ the right-hand
side of last relation {\em is not a characteristic function of any set}.
According to~\cite[Proposition~5.1.]{Ben-Ben}, {\em only} the functions whose Fourier
transforms are the characteristic functions of some sets may be wavelet
functions obtained by Benedettos' method. Consequently, the wavelet basis corresponding
to the generating wavelet functions $\psi_{s}^{(m)[1]}$, $s\in J_{p;m}$,
{\em cannot be constructed} by the method of~\cite{Ben-Ben}.
\end{Remark}
\subsection{Multidimensional non-Haar $p$-adic wavelets.}\label{s3.3}
Since the one- dimensional wavelets (\ref{62.1}) are non-Haar type,
we cannot construct the $n$-dimensional wavelet basis as the tensor products
of the one-dimensional MRAs (see~\cite{S-Sk-1}).
In this case we introduce $n$-dimensional non-Haar
wavelet functions as the $n$-direct product of the one-dimensional
non-Haar wavelets (\ref{62.1}). For $x=(x_1,\dots,x_n)\in {\mathbb Q}_p^n$ and
$j=(j_1,\dots,j_n)\in {\mathbb Z}^n$ we introduce a multi-dilatation
\begin{equation}
\label{62.8-md}
\widehat{p^{j}}x\stackrel{def}{=}(p^{j_1}x_1,\dots,p^{j_n}x_n)
\end{equation}
and define the $n$-direct products of the one-dimensional $p$-adic
wavelets (\ref{62.1}) as
\begin{equation}
\label{62.8}
\Theta_{s;\,j a}^{(m)\times}(x)=p^{-|j|/2}
\chi_p\big(s\cdot(\widehat{p^{j}}x-a)\big)
\Omega\big(|\widehat{p^{j}}x-a|_p\big), \quad x\in {\mathbb Q}_p^n,
\end{equation}
where $j=(j_1,\dots,j_n)\in {\mathbb Z}^n$; $|j|=j_1+\cdots+j_n$;
$a=(a_1,\dots,a_n)\in I_p^n$;
$s=(s_1,\dots,s_n)\in J_{p;m}^n$; $m=(m_1,\dots,m_n)$, $m_l\ge 1$
is a {\it fixed} positive integer, $l=1,2,\dots,n$. Here $I_p^n$, $J_{p;m}^n$
are the $n$- direct products of the corresponding sets (\ref{62.0**}) and
(\ref{62.0*}).
In view of (\ref{9}), Theorem~\ref{th2} implies the following
statement.
\begin{Theorem}
\label{th2.1}
The non-Haar wavelet functions {\rm(\ref{62.8})} form an orthonormal
basis in ${{\mathcal L}}^2({\mathbb Q}_p^n)$.
\end{Theorem}
Using (\ref{9}), it is easy to verify that
$\int_{{\mathbb Q}_p^n}\Theta_{s;\, j a}^{(m)\times}(x)\,d^nx=0$,
i.e., in view of Lemma~\ref{lem1}, the wavelet function
$\Theta_{s;\, j a}^{(m)\times}(x)$, $j\in {\mathbb Z}^n$, $a\in I_p^n$, $s\in J_{p;m}^n$,
belongs to the Lizorkin space $\Phi({\mathbb Q}_p^n)$.
\begin{Corollary}
\label{cor3-1}
The $n$-direct product of one-dimensional Kozyrev's wavelets
{\rm (\ref{62.0-1})}
\begin{equation}
\label{64**}
\Theta_{k;j a}^{\times}(x)
=p^{-|j|/2}\chi_p\big(p^{-1}k\cdot(\widehat{p^{j}}x-a)\big)
\Omega\big(|\widehat{p^{j}}x-a|_p\big), \quad x\in {\mathbb Q}_p^n,
\end{equation}
form an orthonormal complete basis in ${{\mathcal L}}^2({\mathbb Q}_p^n)$, \ $k \in J_p^n$, \
$j\in {\mathbb Z}^n$, $a\in I_p^n$.
\end{Corollary}
The proof follows from Theorem~\ref{th2.1} if we set $m=1$.
\begin{Corollary}
\label{cor2}
The family of functions
\begin{equation}
\label{o-64.8*}
{\widetilde\Theta}_{s;\, j a}^{(m)\times}(\xi)
=F[\Theta_{s;\, j a}^{(m)\times}](\xi)
=p^{|j|/2}\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)
\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big),\quad \xi\in {\mathbb Q}_p^n,
\end{equation}
form an orthonormal complete basis in ${{\mathcal L}}^2({\mathbb Q}_p^n)$, $j\in {\mathbb Z}^n$;
$a\in I_p^n$; $s\in J_{p;m}^n$; $m=(m_1,\dots,m_n)$, $m_l\ge 1$ is
a {\it fixed} positive integer, $l=1,2,\dots,n$.
\end{Corollary}
\begin{proof}
Consider the function $\Theta_{s}^{(m)\times}(x)=\chi_p(s\cdot x)\Omega(|x|_p)$
generated by the direct product of functions (\ref{62.1-11}),
$x\in {\mathbb Q}_p^n$, $s=(s_1,\dots,s_n)\in J_{p;m}^n$, $s_k\in J_{p;m_k}$,
$k=1,2,\dots,n$. Using (\ref{9}), (\ref{14.1}), (\ref{14}), we have
$$
F[\Theta_{s}^{(m)\times}(x)](\xi)
=F\Big[\prod_{k=1}^n\chi_p(x_ks_k)\Omega(|x_k|_p)\Big](\xi)
=\prod_{k=1}^nF\Big[\Omega(|x_k|_p)\Big](\xi_k+s_k|_p)
$$
\begin{equation}
\label{o-64.7}
=\prod_{k=1}^n\Omega\big(|\xi_k+s_{k}|_p\big)
=\Omega\big(|\xi+s|_p\big), \quad \xi\in {\mathbb Q}_p^n.
\end{equation}
Here, in view of (\ref{9}), $\Omega\big(|\xi+s|_p\big)
=\Omega\big(|\xi_1+s_{1}|_p\big)\times\cdots\times
\Omega\big(|\xi_n+s_{n}|_p\big)$.
According to (\ref{62.0*}), $|s_{k}|_p=p^{m_k}$ and
$\Omega\big(|\xi_k+s_{k}|_p\big)\ne 0$ only if
$\xi_k=-s_{k}+\eta_k$, where $\eta_k\in {\mathbb Z}_p$,
$s_k \in J_{p;m_k}$, \ $k=1,2,\dots,n$. This yields $\xi=-s+\eta$,
where $\eta \in {\mathbb Z}_p^n$, $s\in J_{p;m}^n$, and in view
of (\ref{8}), $|\xi|_p=p^{\max\{m_1,\dots,m_n\}}$.
In view of formulas (\ref{62.8}), (\ref{o-64.7}), (\ref{14}), we have
$$
F[\Theta_{s;\, j a}^{(m)\times}(x)](\xi)
=p^{-|j|/2}F[\Theta_{s}^{(m)\times}(\widehat{p^{j}}x-a)](\xi)
\qquad\qquad\qquad\qquad\quad
$$
$$
=p^{|j|/2}\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)
\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big),
$$
i.e., (\ref{o-64.8*}).
The formula (\ref{o-64.8*}), the Parseval formula~\cite[VII,(4.1)]{Vl-V-Z}, and
Theorem~\ref{th2.1} imply the statement.
\end{proof}
Similarly, one can construct $n$-dimensional non-Haar wavelet bases generated
by the one-dimensional non-Haar wavelets (\ref{109-11}) (as the $n$-direct product):
\begin{equation}
\label{62.8-1}
\Psi_{s;\,j a}^{(m)[\nu]\times}(x)=p^{-|j|/2}\psi_{s}^{(m_1)[\nu]}(p^{j_1}x_1-a_1)\cdots
\psi_{s}^{(m_n)[\nu]}(p^{j_n}x_n-a_n),
\end{equation}
where $x\in {\mathbb Q}_p^n$ and $\psi_{s_{j_k}}^{(m_{j_k})[\nu]}$ is defined by (\ref{101-11}),
(\ref{108-11}), $j=(j_1,\dots,j_n)\in {\mathbb Z}^n$; $|j|=j_1+\cdots+j_n$;
$a=(a_1,\dots,a_n)\in I_p^n$;
$s=(s_1,\dots,s_n)\in J_{p;m}^n$; $m=(m_1,\dots,m_n)$, $m_k\ge 1$ is
a {\it fixed} positive integer, $k=1,2,\dots,n$; \,$\nu=1,2,\dots$.
\subsection{$p$-Adic Lizorkin spaces and wavelets.}\label{s3.4}
Now we prove an analog of Lemma~\ref{lem-four-1} for the Lizorkin
test functions from $\Phi({\mathbb Q}_p^n)$.
\begin{Lemma}
\label{lem-w-1**}
Any function $\phi \in \Phi({\mathbb Q}_p^n)$ can be represented in the form of a {\em finite} sum
\begin{equation}
\label{wav-9.4=1}
\phi(x)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}c_{s;\,j, a}
\Theta_{s;\,j a}^{(m)\times}(x), \quad x \in {\mathbb Q}_p^n,
\end{equation}
where $c_{s;\,j, a}$ are constants; $\Theta_{s;\,j a}^{(m)\times}(x)$ are elements
of the non-Haar wavelet basis {\rm(\ref{62.8})}; $s=(s_1,\dots,s_n)\in J_{p;m}^n$;
$j=(j_1,\dots,j_n)\in {\mathbb Z}^n$, $|j|=j_1+\cdots+j_n$; $a=(a_1,\dots,a_n)\in I_p^n$;
$m=(m_1,\dots,m_n)$, $m_l\ge 1$ is a {\it fixed} positive integer, $l=1,2,\dots,n$.
\end{Lemma}
\begin{proof}
Let us calculate ${\mathcal L}^2({\mathbb Q}_p^n)$-scalar product $(\phi(x),\Theta_{s;\,j a}^{(m)\times}(x))$.
Taking into account formula (\ref{o-64.8*}) and using the Parseval-Steklov theorem,
we obtain
$$
c_{s;\,j, a}=\big(\phi(x),\Theta_{s;\,j a}^{(m)\times}(x)\big)
=\big(F[\phi](\xi),F[\Theta_{s;\,j a}^{(m)\times}](\xi)\big)
$$
\begin{equation}
\label{wav-9.4=2}
=\big(\psi(\xi), p^{|j|/2}\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)
\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big)\big),
\end{equation}
where $j\in {\mathbb Z}^n$, $a\in I_p^n$, $s\in J_{p;m}^n$. Here,
according to Lemma~\ref{lem1}, any function
$\phi \in \Phi({\mathbb Q}_p^n)$ belongs to one of the spaces ${{\mathcal D}}^l_N({\mathbb Q}_p^n)$,
$\psi=F^{-1}[\phi]\in \Psi({\mathbb Q}_p^n)\cap {\mathcal D}^{-N}_{-l}({\mathbb Q}_p^n)$, and
${\rm supp}\,\psi\subset B^n_{-l}\setminus B^n_{-N}$.
Let $|s|_p\ne |\widehat{p^{-j}}\xi|_p$. Since $p^{-N}\le|\xi|_p\le p^{-l}$
and
$$
|s+\widehat{p^{-j}}\xi|_p
=\max(|s|_p,|\widehat{p^{-j}}\xi|_p)=\max\big(p^{\max(m_1,\dots,m_n)},\max(p^{j_k}|\xi_k|_p\big),
$$
in view of (\ref{wav-9.4=2}), it is clear that there are finite quantity of indexes
$s=(s_1,\dots,s_n)\in J_{p;m}^n$, $j=(j_1,\dots,j_n)\in {\mathbb Z}^n$ such that $c_{s;\,j, a}\ne 0$.
The case $|s|_p=|\widehat{p^{-j}}\xi|_p=p^{\max(m_1,\dots,m_n)}$ can be considered
in the same way.
Thus equality (\ref{wav-9.4=1}) holds in the sense of ${{\mathcal L}}^2({\mathbb Q}_p^n)$. Consequently,
it holds in the usual sense.
\end{proof}
Using standard results from the book~\cite{Schaefer} or repeating the
reasoning~\cite{Kh-Sh-Sm1}, \cite{Kh-Sh-Sm2} almost word for word,
we obtain the following assertion.
\begin{Proposition}
\label{pr-w-2**}
Any distribution $f \in \Phi'({\mathbb Q}_p^n)$ can be realized in the form of an
{\em infinite} sum of the form
\begin{equation}
\label{wav-9.4=3}
f(x)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}d_{s;\,j, a}
\Theta_{s;\,j a}^{(m)\times}(x), \quad x \in {\mathbb Q}_p^n,
\end{equation}
where $d_{s;\,j, a}$ are constants; $\Theta_{s;\,j a}^{(m)\times}(x)$ are elements
of the non-Haar wavelet basis {\rm(\ref{62.8})}; $s=(s_1,\dots,s_n)\in J_{p;m}^n$;
$j=(j_1,\dots,j_n)\in {\mathbb Z}^n$, $|j|=j_1+\cdots+j_n$; $a=(a_1,\dots,a_n)\in I_p^n$;
$m=(m_1,\dots,m_n)$, $m_l\ge 1$ is a {\it fixed} positive integer, $l=1,2,\dots,n$.
\end{Proposition}
Here any distribution $f \in \Phi'({\mathbb Q}_p^n)$ is associated with the representation
(\ref{wav-9.4=3}), where the coefficients
\begin{equation}
\label{wav-9.4=4}
d_{s;\,j, a}\stackrel{def}{=}\bigl\langle f,\Theta_{s;\,j a}^{(m)\times}\bigr\rangle,
\quad s\in J_{p;m}^n, \quad j\in {\mathbb Z}^n, \quad a\in I_p^n.
\end{equation}
And vice versa, taking into account
Lemma~\ref{lem-w-1**} and orthonormality of the wavelet basis (\ref{62.8}), any
infinite sum is associated with the distribution $f \in \Phi'({\mathbb Q}_p^n)$ whose
action on a test function $\phi \in \Phi({\mathbb Q}_p^n)$ is defined as
\begin{equation}
\label{wav-9.4=5}
\langle f,\phi\rangle=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}d_{s;\,j, a}c_{s;\,j, a},
\end{equation}
where the sum is finite.
It is clear that in Lemma~\ref{lem-w-1**} and Proposition~\ref{pr-w-2**}
instead of the basis (\ref{62.1}) or its multidimensional generalization
(\ref{62.8}), one can use the bases (\ref{109-11}), (\ref{101-11}), (\ref{108-11})
and their multidimensional generalizations.
In~\cite{Al-Koz}, the assertions of the type of
Lemma~\ref{lem-w-1**} and Proposition~\ref{pr-w-2**} were stated for ultrametric
Lizorkin spaces.
\section{Spectral theory of $p$-adic pseudo-differential operators}
\label{s4}
\subsection{Pseudo-differential operators in the Lizorkin spaces.}\label{s4.1}
In this subsection we present some facts on pseudo-differential operators which
were introduced in~\cite{Al-Kh-Sh3},~\cite{Al-Kh-Sh4}. Consider a class of
pseudo-differential operators in the Lizorkin space $\Phi({\mathbb Q}_p^n)$
$$
(A\phi)(x)=F^{-1}\big[{\mathcal A}(\cdot)\,F[\phi](\cdot)\big](x)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{64.3}
=\int_{{\mathbb Q}_p^n}\int_{{\mathbb Q}_p^n}\chi_p\big((y-x)\cdot \xi\big)
{\mathcal A}(\xi)\phi(y)\,d^n\xi\,d^ny,
\quad \phi \in \Phi({\mathbb Q}_p^n)
\end{equation}
with symbols ${\mathcal A}\in {\mathcal E}({\mathbb Q}_p^n\setminus \{0\})$.
\begin{Remark}
\label{rem3} \rm
The class of operators (\ref{64.3}) includes the Taibleson fractional
operator with the symbol of the form $|\xi|^{\alpha}_p$ (see (\ref{59**}));
the Kochubei operator with the symbol of the form
${\mathcal A}(\xi)=|f(\xi_1,\dots,\xi_n)|_p^{\alpha}$, $\alpha> 0$, where
$f(\xi_1,\dots,\xi_n)$ is a quadratic form such that
$f(\xi_1,\dots,\xi_n)\ne 0$ when $|\xi_1|_p+\cdots |\xi_n|_p\ne 0$
(see~\cite{Koch3},~\cite{Koch4}); the Zuniga-Galindo operator
with the symbol of the form ${\mathcal A}(\xi)=|f(\xi_1,\dots,\xi_n)|_p^{\alpha}$,
$\alpha> 0$, where $f(\xi_1,\dots,\xi_n)$ is a non-constant polynomial
(see~\cite{Z1},~\cite{Z2}).
\end{Remark}
If we define a conjugate pseudo-differential operator $A^{T}$ as
$$
(A^{T}\phi)(x)=F^{-1}[{\mathcal A}(-\xi)F[\phi](\xi)](x)
=\int_{{\mathbb Q}_p^n}\chi_p(-x\cdot \xi){\mathcal A}(-\xi)F[\phi](\xi)\,d^n\xi
$$
then one can define the operator $A$ in the Lizorkin space of
distributions: for $f \in \Phi'({\mathbb Q}_p^n)$ we have
\begin{equation}
\label{64.4}
\langle Af,\phi\rangle=\langle f,A^{T}\phi\rangle,
\qquad \forall \, \phi \in \Phi({\mathbb Q}_p^n).
\end{equation}
\begin{Lemma}
\label{lem4}
{\rm(~\cite{Al-Kh-Sh3})}
The Lizorkin spaces $\Phi({\mathbb Q}_p^n)$ and $\Phi'({\mathbb Q}_p^n)$ are invariant under the
pseudo-differential operators {\rm(\ref{64.3})}.
\end{Lemma}
\begin{proof}
In view of (\ref{12}) and results of Subsec.~\ref{s2.2}, both functions
$F[\phi](\xi)$ and ${\mathcal A}(\xi)F[\phi](\xi)$ belong to $\Psi({\mathbb Q}_p^n)$,
and, consequently, $(A\phi)(x)\in \Phi({\mathbb Q}_p^n)$, i.e.,
$A(\Phi({\mathbb Q}_p^n))\subset \Phi({\mathbb Q}_p^n)$. Thus the
pseudo-differential operators (\ref{64.3}) are well defined, and the
Lizorkin space $\Phi({\mathbb Q}_p^n)$ is invariant under them. In view of
(\ref{64.4}), the Lizorkin space of distributions $\Phi'({\mathbb Q}_p^n)$
is also invariant under pseudo-differential operator $A$.
\end{proof}
\subsection{The Taibleson fractional operator in the Lizorkin spaces.}\label{s4.2}
In particular, setting ${\mathcal A}(\xi)=|\xi|_p^{\alpha}$, $\xi\in {\mathbb Q}_p^n$, we obtain
the multi-dimensional Taibleson fractional operator. This operator was introduced
in~\cite[\S2]{Taib1},~\cite[III.4.]{Taib3} on the space of distributions
${{\mathcal D}}'({\mathbb Q}_p^n)$ for $\alpha\in {\mathbb C}$, $\alpha\ne -n$. Next, in~\cite{Al-Kh-Sh3},
the Taibleson fractional operator was defined and studied in the Lizorkin space of
distributions $\Phi'({\mathbb Q}_p^n)$ for all $\alpha\in {\mathbb C}$.
According to (\ref{64.3}), (\ref{64.4}),
\begin{equation}
\label{61**}
\big(D^{\alpha}f\big)(x)
=F^{-1}\big[|\cdot|^{\alpha}_pF[f](\cdot)\big](x),
\quad f \in \Phi'({\mathbb Q}_p^n).
\end{equation}
Representation (\ref{61**}) can be rewritten as a convolution
\begin{equation}
\label{59**}
\big(D^{\alpha}f\big)(x)=\kappa_{-\alpha}(x)*f(x)
=\langle \kappa_{-\alpha}(x),f(x-\xi)\rangle,
\quad f\in \Phi'({\mathbb Q}_p^n), \quad \alpha \in {\mathbb C},
\end{equation}
where according to~\cite{Al-Kh-Sh3}, the multidimensional {\it Riesz kernel\/}
is given by the formula
$$
\kappa_{\alpha}(x)=\left\{
\begin{array}{lll}
\frac{|x|_p^{\alpha-n}}{\Gamma_p^{(n)}(\alpha)}, &&
\alpha \ne 0, \, \, n, \\
\delta(x) && \alpha=0, \\
-\frac{1-p^{-n}}{\log p}\log|x|_p && \alpha=n \\
\end{array}
\right.
$$
the function $|x|_p$, \ $x\in {\mathbb Q}_p^n$ is defined by (\ref{8}). Here the
multidimensional homogeneous distribution
$|x|_p^{\alpha-n}\in {{\mathcal D}}'({\mathbb Q}_p^n)$ of degree $\alpha-n$ was defined
in~\cite[(*)]{Taib1},~\cite[III,(4.3)]{Taib3},~\cite[VIII,(4.2)]{Vl-V-Z},
$\Gamma^{(n)}_p(\alpha)$ is the $n$-dimensional $\Gamma$-{\it function\/}
defined in~\cite[Theorem~1.]{Taib1},~\cite[III,Theorem~(4.2)]{Taib3},
~\cite[VIII,(4.4)]{Vl-V-Z}.
According to Lemma~\ref{lem4} and (\ref{64.3}), (\ref{64.4}), the Lizorkin
space $\Phi'({\mathbb Q}_p^n)$ is invariant under the Taibleson fractional operator
$D^{\alpha}$ for all $\alpha \in {\mathbb C}$~\cite{Al-Kh-Sh3}.
\subsection{$p$-Adic wavelets as eigenfunctions of pseudo-differential operators.}\label{s4.3}
As mentioned above in Sec.~\ref{s1}, it is typical that $p$-adic
compactly supported wavelets are eigenfunctions of $p$-adic pseudo-differential
operators. For example, in~\cite{Koz0} S.~V.~Kozyrev proved that wavelets
(\ref{62.0-1}) are eigenfunctions of the one-dimensional fractional operator
(\ref{61**}), (\ref{59**}) for $\alpha>0$:
\begin{equation}
\label{62.2-v}
D^{\alpha}\theta_{k;\,j a}(x)=p^{\alpha(1-j)}\theta_{k;\,j a}(x),
\quad x\in {\mathbb Q}_p,
\end{equation}
where $k=1,2,\dots p-1$, \ $j\in {\mathbb Z}$, $a\in I_p$.
Since wavelet functions (\ref{62.0-1}) belong to the Lizorkin space,
the relation (\ref{62.2-v}) holds for all $\alpha\in {\mathbb C}$.
Now we study the spectral problem for pseudo-differential operators (\ref{64.3})
in connection with the wavelet functions (\ref{62.8}) and (\ref{62.8-1}).
\begin{Theorem}
\label{th4.1}
Let $A$ be a pseudo-differential operator {\rm(\ref{64.3})}
with a symbol ${\mathcal A}(\xi)\in {\mathcal E}({\mathbb Q}_p^n\setminus \{0\})$.
Then the $n$-dimensional non-Haar wavelet function {\rm (\ref{62.8})}
$$
\Theta_{s;\,j a}^{(m)\times}(x)=p^{-|j|/2}
\chi_p\big(s\cdot(\widehat{p^{j}}x-a)\big)
\Omega\big(|\widehat{p^{j}}x-a|_p\big), \quad x\in {\mathbb Q}_p^n,
$$
is an eigenfunction of $A$ if and only if
\begin{equation}
\label{64.1***}
{\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)={\mathcal A}\big(-\widehat{p^{j}}s\big),
\qquad \forall \, \eta \in {\mathbb Z}_p^n,
\end{equation}
where $j=(j_1,\dots,j_n)\in {\mathbb Z}^n$; $a\in I_p^n$;
$s\in J_{p;m}^n$; and $m=(m_1,\dots,m_n)$, $m_j\ge 1$ is
a {\it fixed} positive integer, $j=1,2,\dots,n$.
The corresponding eigenvalue is $\lambda={\mathcal A}\big(-\widehat{p^{j}}s\big)$, i.e.,
$$
A\Theta_{s;\,j a}^{(m)\times}(x)
={\mathcal A}(-\widehat{p^{j}}s)\Theta_{s;\,j a}^{(m)\times}(x).
$$
Here the multi-dilatation is defined by {\rm(\ref{62.8-md})},
$I_p^n$ and $J_{p;m}^n$ are the $n$-direct products of the
corresponding sets {\rm(\ref{62.0**})} and {\rm(\ref{62.0*})}.
\end{Theorem}
\begin{proof}
Let condition (\ref{64.1***}) be satisfied. Then (\ref{64.3})
and the above formula (\ref{o-64.8*}) imply that
$$
A\Theta_{s;\,j a}^{(m)\times}(x)
=F^{-1}\big[{\mathcal A}(\xi)F[\Theta_{s;\,j a}^{(m)\times}](\xi)\big](x)
\qquad\qquad\qquad\qquad\qquad\qquad\quad
$$
\begin{equation}
\label{o-64.9}
=p^{|j|/2}F^{-1}\big[{\mathcal A}(\xi)\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)
\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big)\big](x).
\end{equation}
Making the change of variables $\xi=\widehat{p^{j}}(\eta-s)$
and using (\ref{14.1}), we obtain
$$
A\Theta_{s;\,j a}^{(m)\times}(x)
=p^{-|j|/2}
\int\limits_{{\mathbb Q}_p^n}\chi_p\big(-(\widehat{p^{j}}x-a)\cdot (\eta-s)\big)
{\mathcal A}(\widehat{p^{j}}(\eta-s))\,\Omega(|\eta|_p)\,d^n\eta
$$
$$
\qquad\qquad\quad
=p^{-|j|/2}{\mathcal A}(-\widehat{p^{j}}s)
\chi_p\big(s\cdot(\widehat{p^{j}}x-a)\big)
\int_{B_{0}^n}\chi_p(-(\widehat{p^{j}}x-a)\cdot\eta)\,d^n\eta
$$
$$
={\mathcal A}(-\widehat{p^{j}}s)\Theta_{s;\,j a}^{(m)}(x).
\qquad\qquad\qquad\qquad\qquad\qquad
$$
Consequently,
$A\Theta_{s;\,j a}^{(m)\times}(x)=\lambda\Theta_{s;\,j a}^{(m)\times}(x)$,
where $\lambda={\mathcal A}(-\widehat{p^{j}}s)$.
Conversely, if
$A\Theta_{s;\,j a}^{(m)\times}(x)=\lambda\Theta_{s;\,j a}^{(m)\times}(x)$,
$\lambda\in {\mathbb C}$, taking the Fourier transform of
both left- and right-hand sides of this identity and
using (\ref{64.3}), (\ref{o-64.8*}), (\ref{o-64.9}), we have
\begin{equation}
\label{o-64.9==}
\big({\mathcal A}(\xi)-\lambda\big)\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)
\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big)=0, \quad \xi\in {\mathbb Q}_p^n.
\end{equation}
Now, if $s+\widehat{p^{-j}}\xi=\eta$, $\eta \in {\mathbb Z}_p^n$, then
$\xi=\widehat{p^{j}}(-s+\eta)$. Since
$\chi_p\big(\widehat{p^{-j}}a\cdot\xi\big)\ne0$,
$\Omega\big(|s+\widehat{p^{-j}}\xi|_p\big)\ne0$,
it follows from (\ref{o-64.9==}) that
$\lambda={\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)$ for any
$\eta \in{\mathbb Z}_p^n$. Thus $\lambda={\mathcal A}(-\widehat{p^{j}}s)$, and,
consequently, (\ref{64.1***}) holds.
The proof of the theorem is complete.
\end{proof}
\begin{Corollary}
\label{cor3}
Let $A$ be a pseudo-differential operator {\rm(\ref{64.3})}
with the symbol ${\mathcal A}(\xi)\in {\mathcal E}({\mathbb Q}_p^n\setminus \{0\})$.
Then the $n$-dimensional wavelet function {\rm (\ref{64**})}
$$
\Theta_{k;j a}^{\times}(x)
=p^{-|j|/2}\chi_p\big(p^{-1}k\cdot(\widehat{p^{j}}x-a)\big)
\Omega\big(|\widehat{p^{j}}x-a|_p\big), \quad x\in {\mathbb Q}_p^n,
$$
is an eigenfunction of $A$ if and only if
$$
{\mathcal A}\big(\widehat{p^{j}}(-p^{-1}k+\eta)\big)
={\mathcal A}\big(-\widehat{p^{j-I}}k\big),
\qquad \forall \, \eta \in {\mathbb Z}_p^n,
$$
where $k \in J_p^n$, $j\in {\mathbb Z}^n$, $a\in I_p^n$, $I=(1,\dots,1)$.
The corresponding eigenvalue is $\lambda={\mathcal A}\big(-\widehat{p^{j-I}}j\big)$,
i.e.,
$$
A\Theta_{k;j a}^{\times}(x)={\mathcal A}(-\widehat{p^{j-I}}k)
\Theta_{k;j a}^{\times}(x).
$$
\end{Corollary}
Representation (\ref{101-11}) and Theorem~\ref{th4.1} imply the following statement.
\begin{Theorem}
\label{th4.1-1}
Let $A$ be a pseudo-differential operator {\rm(\ref{64.3})}
with the symbol ${\mathcal A}(\xi)\in {\mathcal E}({\mathbb Q}_p^n\setminus \{0\})$.
Then the $n$-dimensional wavelet function {\rm (\ref{62.8-1})}
$$
\Psi_{s;\,j a}^{(m)[\nu]\times}(x)=p^{-|j|/2}\psi_{s}^{(m_1)[\nu]}(p^{j_1}x_1-a_1)\cdots
\psi_{s}^{(m_n)[\nu]}(p^{j_n}x_n-a_n), \quad x\in {\mathbb Q}_p^n,
$$
is an eigenfunction of $A$ if and only if condition {\rm(\ref{64.1***})} holds,
where $\psi_{s_{j_k}}^{(m_{j_k})[\nu]}$ is defined by {\rm(\ref{101-11})},
{\rm(\ref{108-11})}, $j=(j_1,\dots,j_n)\in {\mathbb Z}^n$; $|j|=j_1+\cdots+j_n$;
$a=(a_1,\dots,a_n)\in I_p^n$;
$s=(s_1,\dots,s_n)\in J_{p;m}^n$; $m=(m_1,\dots,m_n)$, $m_k\ge 1$ is
a {\it fixed} positive integer, $k=1,2,\dots,n$; \,$\nu=1,2,\dots$.
The corresponding eigenvalue is $\lambda={\mathcal A}(-\widehat{p^{j}}s)$,
i.e.,
$$
A\Psi_{s;\,j a}^{(m)[\nu]\times}(x)={\mathcal A}(-\widehat{p^{j}}s)
\Psi_{s;\,j a}^{(m)[\nu]\times}(x).
$$
\end{Theorem}
\subsection{$p$-Adic wavelets as eigenfunctions of the Taibleson fractional operator.}\label{s4.4}
As mentioned above, the Taibleson fractional operator $D^{\alpha}$
has the symbol ${\mathcal A}(\xi)=|\xi|_p^{\alpha}$.
The symbol ${\mathcal A}(\xi)=|\xi|_p^{\alpha}$ satisfies the condition (\ref{64.1***}):
$$
{\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)
=|\widehat{p^{j}}(-s+\eta)|_p^{\alpha}
=\Big(\max_{1\le r\le n}\big(p^{-j_r}|-s_r|_p\big)\Big)^{\alpha}
\qquad
$$
$$
\quad
={\mathcal A}\big(-\widehat{p^{j}}s\big)
=p^{\alpha\max_{1\le r\le n}\{m_r-j_r\}}
$$
for all $\eta \in {\mathbb Z}_p^n$. Consequently, according to Theorem~\ref{th4.1},
we have
\begin{Corollary}
\label{cor5}
The $n$-dimensional non-Haar $p$-adic wavelet {\rm(\ref{62.8})} is an
eigenfunction of the Taibleson fractional operator {\rm (\ref{59**})}:
$$
D^{\alpha}\Theta_{s;\,j a}^{(m)\times}(x)
=p^{\alpha\max_{1\le r\le n}\{m_r-j_r\}}
\Theta_{s;\,j a}^{(m)\times}(x), \quad \alpha \in{\mathbb C}, \qquad x\in {\mathbb Q}_p^n,
$$
$s \in J_{p;m}^n$, $j\in {\mathbb Z}^n$, $a\in I_p^n$.
\end{Corollary}
In particular, in view of Corollary~\ref{cor3}, we have
\begin{Corollary}
\label{cor6}
The $n$-dimensional $p$-adic wavelet {\rm(\ref{64**})} is an
eigenfunction of the Taibleson fractional operator {\rm (\ref{59**})}:
$$
D^{\alpha}\Theta_{k;\, j a}^{\times}(x)
=p^{\alpha(1-\min_{1\le r\le n}j_r)}\Theta_{k;\, j a}^{\times}(x),
\quad \alpha \in{\mathbb C}, \quad x\in {\mathbb Q}_p^n,
$$
$k \in J_p^n$, $j\in {\mathbb Z}^n$, $a\in I_p^n$.
\end{Corollary}
\begin{Corollary}
\label{cor7}
The $n$-dimensional $p$-adic wavelet {\rm(\ref{62.8-1})} is an
eigenfunction of the Taibleson fractional operator {\rm (\ref{59**})}:
$$
D^{\alpha}\Psi_{s;\,j a}^{(m)[\nu]\times}(x)
=p^{\alpha\max_{1\le r\le n}\{m_r-j_r\}}\Psi_{s;\,j a}^{(m)[\nu]\times}(x),
\quad \alpha \in{\mathbb C}, \quad x\in {\mathbb Q}_p^n,
$$
$s \in J_{p;m}^n$, $j\in {\mathbb Z}^n$, $a\in I_p^n$,
\,$\nu=1,2,\dots$.
\end{Corollary}
\section{Application of $p$-adic wavelets to evolutionary pseudo-differential equations}
\label{s5}
\subsection{Linear equations.}\label{s5.1}
{\bf (a)} Let us consider the Cauchy problem for the {\em linear evolutionary
pseudo-differential equation}
\begin{equation}
\label{eq-70}
\left\{
\begin{array}{rclrcl}
\frac{\partial u(x,t)}{\partial t}+A_xu(x,t)&=&0,
&&\text{in} \quad {\mathbb Q}_p^n\times (0, \ \infty), \medskip \\
u(x,t)&=&u^0(x),
&&\text{in} \quad {\mathbb Q}_p^n\times \{t=0\}, \\
\end{array}
\right.
\end{equation}
where $t\in {\mathbb R}$, $u^0\in \Phi'({\mathbb Q}_p^n)$ and
\begin{equation}
\label{eq-70-op}
A_xu(x,t)=F^{-1}\big[{\mathcal A}(\xi)\,F[u(\cdot,t)](\xi)\big](x)
\end{equation}
is a pseudo-differential operator (\ref{64.3}) (with
respect to $x$) with symbols ${\mathcal A}(\xi)\in {\mathcal E}({\mathbb Q}_p^n\setminus \{0\})$,
$u(x,t)$ is the desired distribution such that $u(x,t)\in \Phi'({\mathbb Q}_p^n)$
for any $t\ge 0$.
In particular, we will consider the Cauchy problem
\begin{equation}
\label{eq-70-c}
\left\{
\begin{array}{rclrcl}
\frac{\partial u(x,t)}{\partial t}+D^{\alpha}_xu(x,t)&=&0,
&&\text{in} \quad {\mathbb Q}_p^n\times (0, \ \infty), \medskip \\
u(x,t)&=&u^0(x),
&&\text{in} \quad {\mathbb Q}_p^n\times \{t=0\}, \\
\end{array}
\right.
\end{equation}
where $D^{\alpha}_xu(x,t)=F^{-1}\big[|\xi|_p^{\alpha}\,F[u(\cdot,t)](\xi)\big](x)$
is the Taibleson fractional operator (\ref{61**})
with respect to $x$, \ $\alpha\in{\mathbb C}$.
\begin{Theorem}
\label{eq-th5.1}
The Cauchy problem {\rm (\ref{eq-70})} has a unique solution
\begin{equation}
\label{eq-71}
u(x,t)=F^{-1}\big[F[u^{0}(\cdot)](\xi)e^{-{\mathcal A}(\xi)\,t}\big](x).
\end{equation}
\end{Theorem}
\begin{proof}
Since $u(x,t)$ is a distribution such that $u(x,t)\in \Phi'({\mathbb Q}_p^n)$
for any $t\ge 0$, the relation (\ref{eq-70}) is well-defined. Applying
the Fourier transform to (\ref{eq-70}), we obtain the following equation
$$
\frac{\partial F[u(\cdot,t)](\xi)}{\partial t}+{\mathcal A}(\xi)\,F[u(\cdot,t)](\xi)=0.
$$
Solving this equation, we obtain
$$
F[u(x,t)](\xi)=F[u(x,0)](\xi)e^{-{\mathcal A}(\xi)\,t}.
$$
This implies (\ref{eq-71}).
\end{proof}
\begin{Theorem}
\label{eq-th5}
Let a pseudo-differential operator $A_{x}$ in {\rm (\ref{eq-70})} be
such that its symbol ${\mathcal A}(\xi)$ satisfies the condition {\rm (\ref{64.1***})}:
$$
{\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)={\mathcal A}\big(-\widehat{p^{j}}s\big),
\qquad \forall \, \eta \in {\mathbb Z}_p^n,
$$
for any $j\in {\mathbb Z}^n$, $s\in J_{p;m}^n$.
Then the Cauchy problem {\rm (\ref{eq-70})} has a unique solution
\begin{equation}
\label{eq-75}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle
e^{-{\mathcal A}(-\widehat{p^{j}}s)t}\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
for $t\ge 0$, where $\Theta_{s;\,j a}^{(m)\times}(x)$ are $n$-dimensional
$p$-adic wavelets {\rm (\ref{62.8})}.
\end{Theorem}
\begin{proof}
According to the formula (\ref{wav-9.4=3}) from Proposition~\ref{pr-w-2**},
we will seek a solution of the Cauchy problem (\ref{eq-70}) in the form
of an infinite sum
\begin{equation}
\label{eq-73}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}\Lambda_{s;\,j, a}(t)
\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
where $\Lambda_{s;\,j, a}(t)$ are the desired functions, $s\in J_{p;m}^n$,
$j\in {\mathbb Z}^n$, $a\in I_p^n$.
Substituting (\ref{eq-73}) into equation (\ref{eq-70}), in view of
of Theorem~\ref{th4.1}, we obtain
$$
\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\Big(\frac{d\Lambda_{s;\,j, a}(t)}{dt}
+{\mathcal A}(-\widehat{p^{j}}s)\Lambda_{s;\,j, a}(t)\Big)
\Theta_{s;\,j a}^{(m)\times}(x)=0.
$$
The last equation is understood in the weak sense, i.e.,
\begin{equation}
\label{eq-73-1}
\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\Bigl\langle\Big(\frac{d\Lambda_{s;\,j, a}(t)}{dt}
+{\mathcal A}(-\widehat{p^{j}}s)\Lambda_{s;\,j, a}(t)\Big)
\Theta_{s;\,j a}^{(m)\times}(x),\phi(x)\Bigr\rangle=0,
\end{equation}
for all $\phi \in \Phi({\mathbb Q}_p^n)$.
Since according to Lemma~\ref{lem-w-1**}, any test function $\phi \in \Phi({\mathbb Q}_p^n)$
is represented in the form of a {\em finite} sum (\ref{wav-9.4=1}), the equality
(\ref{eq-73-1}) implies that
$$
\frac{d\Lambda_{s;\,j, a}(t)}{dt}
+{\mathcal A}(-\widehat{p^{j}}s)\Lambda_{s;\,j, a}(t)=0, \quad
\forall \, s\in J_{p;m}^n, \, j\in {\mathbb Z}^n, \, a\in I_p^n,
$$
for all $t\ge 0$.
Solving this differential equation, we obtain
\begin{equation}
\label{eq-74}
\Lambda_{s;\,j, a}(t)
=\Lambda_{s;\,j, a}(0)e^{-{\mathcal A}(-\widehat{p^{j}}s)t},
\quad s\in J_{p;m}^n, \quad j\in {\mathbb Z}^n, \quad a\in I_p^n.
\end{equation}
By substituting (\ref{eq-74}) into (\ref{eq-73}) we find a solution of
the Cauchy problem (\ref{eq-70}) in the form
\begin{equation}
\label{eq-75*}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\Lambda_{s;\,j, a}(0)e^{-{\mathcal A}(-\widehat{p^{j}}s)t}
\Theta_{s;\,j a}^{(m)\times}(x).
\end{equation}
Setting $t=0$, we find
$$
u^{0}(x)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\Lambda_{s;\,j, a}(0)\Theta_{s;\,j a}^{(m)\times}(x),
$$
where $u^0\in \Phi'({\mathbb Q}_p^n)$ and according to (\ref{wav-9.4=3}),
the coefficients $\Lambda_{s;\,j, a}(0)$ are uniquely determined by
(\ref{wav-9.4=4}) as
\begin{equation}
\label{eq-75*-1}
\Lambda_{s;\,j, a}(0)
=\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle,
\quad s\in J_{p;m}^n, \quad j\in {\mathbb Z}^n, \quad a\in I_p^n.
\end{equation}
The relations (\ref{eq-75*}), (\ref{eq-75*-1}) imply (\ref{eq-75}).
In view of (\ref{wav-9.4=5}), the sum (\ref{eq-75}) is finite on
any test function from the Lizorkin space $\Phi({\mathbb Q}_p^n)$.
The theorem is thus proved.
\end{proof}
Theorem~\ref{eq-th5} and Corollary~\ref{cor5} imply the following assertion.
\begin{Corollary}
\label{eq-cor5}
The Cauchy problem {\rm (\ref{eq-70-c})} has a unique solution
\begin{equation}
\label{eq-75-c}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle
e^{-p^{\alpha\max_{1\le r\le n}\{m_r-j_r\}}t}\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
for $t\ge 0$, where $\Theta_{s;\,j a}^{(m)\times}(x)$ are $n$-dimensional
$p$-adic wavelets {\rm (\ref{62.8})}.
\end{Corollary}
Solutions of the Cauchy problems (\ref{eq-70}) and (\ref{eq-70-c})
describe the diffusion processes in the space ${\mathbb Q}_p^n$.
If ${\mathcal A}(-\widehat{p^{j}}s)>0$, according to (\ref{eq-75}), $u(x,t)\to 0$,
as $t\to \infty$. In particular, this fact holds for the solution
of the Cauchy problem (\ref{eq-70-c}).
\begin{Example}
\label{ex1}\rm
Consider the one-dimensional Cauchy problem (\ref{eq-70-c}) for the
initial data
$$
u^{0}(x)=\Omega(|x|_p)=\left\{
\begin{array}{rcl}
1, &&|x|_p\le 1, \\
0, &&|x|_p> 1. \\
\end{array}
\right.
$$
Substituting (\ref{62.1}), (\ref{62.7-11}), (\ref{62.7-12}) into (\ref{eq-75-c}),
we obtain a solution of this Cauchy problem:
$$
u(x,t)=\sum_{s\in J_{p;m}}\sum_{j=m}^{\infty}p^{-j}
e^{-p^{\alpha(m-j)}t}\chi_p\big(sp^{j}x\big)
\Omega\big(|p^{j}x|_p\big).
$$
\end{Example}
{\bf (b)} Now we consider the Cauchy problem
\begin{equation}
\label{eq-70.1}
\left\{
\begin{array}{rclrcl}
i\frac{\partial u(x,t)}{\partial t}-A_xu(x,t)&=&0,
&&\text{in} \quad {\mathbb Q}_p^n\times (0, \ \infty), \medskip \\
u(x,t)&=&u^0(x),
&&\text{in} \quad {\mathbb Q}_p^n\times \{t=0\}, \\
\end{array}
\right.
\end{equation}
where $u^0\in \Phi'({\mathbb Q}_p^n)$ and a pseudo-differential operator operator $A_x$
is given by (\ref{eq-70-op}). In particular, we have the Cauchy problem
\begin{equation}
\label{eq-70.1-c}
\left\{
\begin{array}{rclrcl}
i\frac{\partial u(x,t)}{\partial t}-D^{\alpha}_xu(x,t)&=&0,
&&\text{in} \quad {\mathbb Q}_p^n\times (0, \ \infty), \medskip \\
u(x,t)&=&u^0(x),
&&\text{in} \quad {\mathbb Q}_p^n\times \{t=0\}, \\
\end{array}
\right.
\end{equation}
where $D^{\alpha}_x$ is the Taibleson fractional operator (\ref{61**})
with respect to $x$, \ $\alpha\in{\mathbb C}$.
Using the above results, one can construct a solution of
the Cauchy problems (\ref{eq-70.1}) and (\ref{eq-70.1-c}).
\begin{Theorem}
\label{eq-th6}
Let a pseudo-differential operator $A_{x}$ in {\rm (\ref{eq-70.1})} be
such that its symbol ${\mathcal A}(\xi)$ satisfies the condition {\rm (\ref{64.1***})}:
$$
{\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)={\mathcal A}\big(-\widehat{p^{j}}s\big),
\qquad \forall \, \eta \in {\mathbb Z}_p^n,
$$
for any $j\in {\mathbb Z}^n$, $s\in J_{p;m}^n$.
Then the Cauchy problem {\rm (\ref{eq-70.1})} has a unique solution
\begin{equation}
\label{eq-75.1}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle
e^{-i{\mathcal A}(-\widehat{p^{j}}s)t}\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
for $t\ge 0$, where $\Theta_{s;\,j a}^{(m)\times}(x)$ are $n$-dimensional
$p$-adic wavelets {\rm (\ref{62.8})}.
\end{Theorem}
\begin{Corollary}
\label{eq-cor6}
The Cauchy problem {\rm (\ref{eq-70.1-c})} has a unique solution
\begin{equation}
\label{eq-75.1-c}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle
e^{-ip^{\alpha\max_{1\le r\le n}\{m_r-j_r\}}t}\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
for $t\ge 0$, where $\Theta_{s;\,j a}^{(m)\times}(x)$ are $n$-dimensional
$p$-adic wavelets {\rm (\ref{62.8})}.
\end{Corollary}
\subsection{Semi-linear equations}\label{s5.2}
Consider the Cauchy problem for the semi-linear pseudo-differential equation:
\begin{equation}
\label{76-sl}
\left\{
\begin{array}{rcl}
\frac{\partial u(x,t)}{\partial t}+A_xu(x,t) + u(x,t)|u(x,t)|^{2m}=0,
&&\text{in} \quad {\mathbb Q}_p^n\times (0, \ \infty), \medskip \\
u(x,t)=u^0(x), &&\text{in} \quad {\mathbb Q}_p^n\times \{t=0\}, \\
\end{array}
\right.
\end{equation}
where pseudo-differential operator $A_x$ is given by (\ref{eq-70-op}), $m\in {\mathbb N}$,
$u(x,t)$ is the desired distribution such that $u(x,t)\in \Phi'({\mathbb Q}_p^n)$
for any $t\ge 0$.
According to Proposition~\ref{pr-w-2**}, a distribution $u(x,t)$
can be realized as an infinite sum of the form
\begin{equation}
\label{77-sl}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}\Lambda_{s;\,j, a}(t)
\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
where $\Lambda_{s;\,j, a}(t)$ are the desired functions,
$\Theta_{s;\,j a}^{(m)\times}(x)$ are elements of the wavelet basis (\ref{62.8}).
We will solve the Cauchy problem in a particular class of distributions $u(x,t)$
such that in representation (\ref{77-sl})
\begin{equation}
\label{82.0-1-sl}
\widehat{p^{j'-j}}a-a' \notin {\mathbb Z}_p^n, \quad \text{if}\quad j_k<j_k', \quad
k=1,\dots,n.
\end{equation}
In view of (\ref{62.4-1}), in this case all sets
$\{x\in {\mathbb Q}_p^n:|\widehat{p^{j}}x-a|_p\le 1\}$,
$\{x\in {\mathbb Q}_p^n:|\widehat{p^{j'}}x-a'|_p\le 1\}$
are disjoint.
\begin{Theorem}
\label{th4.2-sl}
Let a pseudo-differential operator $A_{x}$ in {\rm (\ref{76-sl})} be
such that its symbol ${\mathcal A}(\xi)$ satisfies the condition {\rm (\ref{64.1***})}:
$$
{\mathcal A}\big(\widehat{p^{j}}(-s+\eta)\big)={\mathcal A}\big(-\widehat{p^{j}}s\big),
\qquad \forall \, \eta \in {\mathbb Z}_p^n,
$$
for any $j\in {\mathbb Z}^n$, $s\in J_{p;m}^n$.
Then in the above-mentioned class of distributions {\rm(\ref{77-sl})}, {\rm(\ref{82.0-1-sl})}
the Cauchy problem {\rm(\ref{76-sl})} has a unique solution
$$
u(x,t)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{82.0-sl}
\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\frac{\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\rangle\big({\mathcal A}(-\widehat{p^{j}}s)\big)^{1/2m}
e^{-{\mathcal A}(-\widehat{p^{j}}s)t}\,\Theta_{s;\,j a}^{(m)\times}(x)}
{\big({\mathcal A}(-\widehat{p^{j}}s)+\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\rangle^{2m}p^{-m|j|}
\big(1-e^{-2m{\mathcal A}(-\widehat{p^{j}}s)t}\big)\big)^{1/2m}}
\end{equation}
for $t\ge 0$, where $\Theta_{s;\,j a}^{(m)\times}(x)$ are $n$-dimensional
$p$-adic wavelets {\rm (\ref{62.8})}. Moreover, this formula is applicable
for the case ${\mathcal A}\equiv 0$.
\end{Theorem}
\begin{proof}
Since $|\chi_p\big(p^{-1}k\cdot(p^{j}x-a)\big)|=1$, taking into account formulas
(\ref{77-sl}), (\ref{82.0-1-sl}), (\ref{62.8}), we obtain
$$
|u(x,t)|^{2}=\sum_{j\in {\mathbb Z},k\in J_{p \, 0}^n,a\in I_p^n}
\Lambda_{s;\,j, a}^2(t)p^{-|j|}\Omega\big(|\widehat{p^{j}}x-a|_p\big)
$$
and
\begin{equation}
\label{77.3-sl}
u(x,t)|u(x,t)|^{2m}=\sum_{j\in {\mathbb Z},k\in J_{p \, 0}^n,a\in I_p^n}
\Lambda_{s;\,j, a}^{2m+1}(t)p^{-m|j|}\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
where the indexes in the above sums satisfy the condition (\ref{82.0-1-sl}).
Substituting (\ref{77.3-sl}) and (\ref{77-sl}) into (\ref{76-sl}),
in view of of Theorem~\ref{th4.1}, we find that
$$
\sum_{j\in {\mathbb Z},k\in J_{p \, 0}^n,a\in I_p^n}
\Big(\frac{d\Lambda_{s;\,j, a}(t)}{dt}
+{\mathcal A}\big(-\widehat{p^{j}}s\big)\Lambda_{s;\,j, a}(t)
\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{78-sl}
\qquad\qquad
+p^{-m|j|}\Lambda_{s;\,j, a}^{2m+1}(t)\Big)\Theta_{s;\,j a}^{(m)\times}(x)=0,
\end{equation}
where the last equation is understood in the weak sense.
Since, according to Lemma~\ref{lem-w-1**}, any test function $\phi \in \Phi({\mathbb Q}_p^n)$
is represented in the form of a {\em finite} sum (\ref{wav-9.4=1}), the equality
(\ref{78-sl}) implies that
$$
\frac{d\Lambda_{s;\,j, a}(t)}{dt}
+{\mathcal A}\big(-\widehat{p^{j}}s\big)\Lambda_{s;\,j, a}(t)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\label{78.1-sl}
+p^{-m|j|}\Lambda_{s;\,j, a}^{2m+1}(t)=0, \quad
\forall \, s\in J_{p;m}^n, \, j\in {\mathbb Z}^n, \, a\in I_p^n,
\end{equation}
for all $t\ge 0$.
Integrating (\ref{78.1-sl}), we obtain
$$
\frac{\Lambda_{s;\,j, a}^{2m}(t)}
{{\mathcal A}(-\widehat{p^{j}}s)+p^{-m|j|}\Lambda_{s;\,j, a}^{2m}(t)}
=E_{s;\,j, a}e^{-2m{\mathcal A}(-\widehat{p^{j}}s)t},
$$
i.e.,
\begin{equation}
\label{81.0-sl}
\Lambda_{s;\,j, a}(t)=\frac{E_{s;\,j, a}^{1/2m}\big({\mathcal A}(-\widehat{p^{j}}s)\big)^{1/2m}
e^{-{\mathcal A}(-\widehat{p^{j}}s)t}}
{\big(1-E_{s;\,j, a}p^{-m|j|}e^{-2m{\mathcal A}(-\widehat{p^{j}}s)t}\big)^{1/2m}},
\end{equation}
where $E_{s;\,j, a}$ is a constant, $s\in J_{p;m}^n$, $j\in {\mathbb Z}^n$, $a\in I_p^n$.
Substituting (\ref{81.0-sl}) into (\ref{77-sl}), we find
a solution of the problem (\ref{76-sl})
\begin{equation}
\label{81.0-2-sl}
u(x,t)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\frac{E_{s;\,j, a}^{1/2m}\big({\mathcal A}(-\widehat{p^{j}}s)\big)^{1/2m}
e^{-{\mathcal A}(-\widehat{p^{j}}s)t}}
{\big(1-E_{s;\,j, a}p^{-m|j|}e^{-2m{\mathcal A}(-\widehat{p^{j}}s)t}\big)^{1/2m}}
\Theta_{s;\,j a}^{(m)\times}(x),
\end{equation}
$x\in {\mathbb Q}_p^n$, $t\ge 0$. Setting in (\ref{81.0-2-sl}) $t=0$,
we obtain that
$$
u^{0}(x)=\sum_{s\in J_{p;m}^n,j\in {\mathbb Z}^n,a\in I_p^n}
\bigg(\frac{E_{s;\,j, a}{\mathcal A}(-\widehat{p^{j}}s)}
{1-E_{s;\,j, a}p^{-m|j|}}\bigg)^{1/2m}\Theta_{s;\,j a}^{(m)\times}(x),
$$
where $u^0\in \Phi'({\mathbb Q}_p^n)$. Hence, according to (\ref{wav-9.4=3}),
the coefficients $E_{s;\,j, a}$ are uniquely determined by
(\ref{wav-9.4=4}) as
$$
\bigg(\frac{E_{s;\,j, a}{\mathcal A}(-\widehat{p^{j}}s)}
{1-E_{s;\,j, a}p^{-m|j|}}\bigg)^{1/2m}
=\bigl\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\bigr\rangle.
$$
The last equation implies that
$$
E_{s;\,j, a}=\frac{\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\rangle^{2m}}
{{\mathcal A}(-\widehat{p^{j}}s)+p^{-m|j|}\langle u^{0}(x),\Theta_{s;\,j a}^{(m)\times}\rangle^{2m}}
$$
Substituting $E_{s;\,j, a}$ into (\ref{81.0-2-sl}), we obtain (\ref{82.0-sl}).
In view of (\ref{wav-9.4=5}), the sum (\ref{82.0-sl}) is finite on any test function
from the Lizorkin space $\Phi({\mathbb Q}_p^n)$.
Now by passing to the limit as ${\mathcal A}\to0$ in formula (\ref{82.0-sl}),
one can easily see that this formula (\ref{82.0-sl}) is applicable
for the case ${\mathcal A}\equiv 0$.
The theorem is thus proved.
\end{proof}
\section*{Acknowledgments}
The authors are greatly indebted to M.~A.~Skopina for fruitful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,846 |
Let our team help you deliver on the dream you have for your home, faster and more affordably. Call G.M. Roth Design Remodeling today at (866) 221-9641 or click here to schedule a FREE in-home consultation and estimate. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,085 |
Q: postgresql recommended data type for money I am building an app for estimate and invoicing. This app will be able to export or integrate with accounting software.
For PostgreSQL which data type would you recommend? and if NUMERIC, what do you recommend for precision & scale settings?
A: numeric(12,2) it is then.
Thanks everyone. New to stackoverflow but loving it so far.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,626 |
import { IBoleto } from './interfaces';
export declare const CreateBoleto: (options: any) => IBoleto;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,435 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using Xamarin.Auth;
using Xamarin.Forms;
namespace AccountStoreTest
{
public partial class MainPage : ContentPage
{
public MainPage()
{
InitializeComponent();
SaveCredentials("moljac", "test");
return;
}
void OnSaveClicked(object sender, EventArgs args)
{
SaveCredentials(UserName, Password);
return;
}
void OnLoadClicked(object sender, EventArgs args)
{
(string username, string password) data = LoadCredentials(AppName);
UserName = UserName;
Password = Password;
return;
}
void OnClearClicked(object sender, EventArgs args)
{
}
public string UserName
{
get;
set;
} = "user";
public string Password
{
get;
set;
} = "password";
public string AppName
{
get;
set;
} = "AccountStoreTest";
public void SaveCredentials(string userName, string password)
{
if (!string.IsNullOrWhiteSpace(userName) && !string.IsNullOrWhiteSpace(password))
{
Account account = new Account
{
Username = userName
};
account.Properties.Add("Password", password);
AccountStore.Create().Save(account, AppName);
}
return;
}
public (string username, string password) LoadCredentials(string appname)
{
var account = AccountStore.Create().FindAccountsForService(appname).FirstOrDefault();
string u = (account != null) ? account.Username : null;
string p = (account != null) ? account.Properties["Password"] : null;
return (username: u, password: p);
}
public void DeleteCredentials()
{
var account = AccountStore.Create().FindAccountsForService(AppName).FirstOrDefault();
if (account != null)
{
AccountStore.Create().Delete(account, AppName);
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,204 |
Semi;Colon
E. A. Farro
It Becomes a Question
A conversation with Jericho Brown
"Your interview is not happening, the interviewer is sick," the volunteer at author check-in told Jericho Brown. His smile fell, he paused. It was the inaugural Wordplay event hosted by the Loft Literary Center to celebrate readers, writers, and great books in Minneapolis, Minnesota. Jericho is the recipient of a Whiting Writer's Award, and fellowships from the Guggenheim Foundation, Harvard, and the National Endowment for the Arts. He is author of three books of poetry. The most recent, The Tradition, published by Copper Canyon Press, came out in April 2019.
I had seen him that morning with an entourage of cameras in and around the shiny royal-blue Guthrie Theatre. On the four-story-high escalator, he knelt down on one leg with the fluidity of a dancer. On a couch in front of long windows with views of the Mississippi River, he rested chin to fist. Between poses, he talked to the crew, threw his head back to laugh. Just watching him made me smile. Out on the Endless Bridge that extends 178' from the face of the building and overlooks the former mills the city built itself around, Jericho stood face upward, eyes closed, shoulder length braids swaying in a breeze coming off the river. He was beautiful. The photographer was close enough to where I was drinking coffee that I heard him sigh with what sounded like satisfaction at capturing this image on film.
After hearing that his interview was cancelled, Jericho looked up and asked, "Is there someone else who can interview me?" He was both joking and earnest.
Without thinking I stepped out from the group of volunteers in matching conference t-shirts and fanny packs. "I can."
He smiled.
Three months before I had finished an intense job at the Minnesota State Capitol. Instead of getting the next big job, I was writing. When I found creative flow, I know I was where I needed to be, but the battle to get there was exhausting. I wrote with one hand while the other held back a giant wave of demons. I volunteered for the interview because I wanted to know what made this poet glow.
Jericho's presence was both quiet and loud. He paused speaking on stage and before answering my questions. "Trying to tell the truth means that I have to take a second and search myself, make sure I'm saying what is really the case for me," he told me.
When he walked into the house that served as the green room for the authors, he didn't hesitate before approaching people. He was not taking up space, but opening up new space none of us had realized were there, spaces we felt ourselves pulled into as sparks of smiles and jokes took off into full-blown conversations between acquaintances and people meeting for the first time.
I asked what drove him to be so friendly, so welcoming. He told me his parents taught him as a young child to walk into a room and talk to every adult, to make people "feel that you are a part of them when they saw you." Making people feel seen may have made him charming, but that wasn't why his parents taught him this behavior. "My mom and dad really believed that it was sinful to greet people without a smile. You greet people with a smile, that is the right thing to do."
Up on the roof deck, a chill moved over us and then lifted as the clouds shifted back and forth across the sun. We sat on wood benches that ran the perimeter of the deck, and pulled layers on over our long sleeves. Jericho sat close, leaned close, looked into my eyes. After answering a question he would say, Ya' Know? Do you follow? Do you understand what I mean? He paused, made sure I was with him.
Jericho pulled a KIND Bar out of his gift bag. Voices from the closest stage floated up from a National Book Awards panel. He told me he loved eating junk: Doritos, Lay's STAX sour cream and onion, white cheddar Smartfood popcorn. He followed this up by saying, "I go through these periods where I only eat very healthy and I try to do it for a twelve week period. I'll only eat some protein and something green. I'm a big kale fan. And then for carbohydrates, I'll have sweet potato or brown rice or quinoa."
"When did you start eating quinoa?" I asked.
"When I moved to Atlanta." Jericho moved from San Diego to Atlanta in 2012 where he is the Director of the Creative Writing Program at Emory University.
During our interview, other writers came up to greet Jericho. They paused before leaving, one offered to read him a poem later.
By this time we had known each other for less than an hour and I felt like we had everything in common. We both worked for elected officials! I only recently discovered quinoa too! We both liked plants! That this interview was happening was obviously because we were both the kind of people who said Yes!
This sense of being kindred spirits didn't come from our chemistry, nor was it something about me in particular. It was him. It was his particular magic, a special ability to connect with others. Poets sometimes get stereotyped as quiet and withdrawn. Jericho is not just a poet on the page; he is also a performer. When he is on stage, he feels like he could, "do it forever."
But his laughter and smiles were not performance. They were the laughter of someone who knows the sadness and joy of their own heart.
I asked where he gets his energy from, Jericho told me, "I like a lot of old school music. I like Motown from the sixties and seventies. I like to hear women, Black women in particular, singing, hollering." He went on, "I exercise a lot. I do burpees. I do weight training. I think it's a good idea because it gets me out of my head. You know when you are out of breath and you gotta do ten more reps?" Jericho has put out burpee challenges on Twitter where he has over 15,000 followers, pushing, encouraging, and asking people to let him know how it goes.
The Tradition came to Jericho quickly. He wrote most of the book between Thanksgiving of 2017 and Martin Luther King Jr. Day of 2018. "It was chasing me," he said. "I couldn't stop writing. I was actually scared I was going to die because I was writing so much."
"I invented a new form, I cut up all the lines I had left over from poems failed going back as far as 2004. I was splicing things together, making fragments work in ways I had never made them work before." Jericho's new form is called the duplex. It was like listening to a scientist explained the set-up of an experiment when he explained the structure of the duplex. Put simply, it is a combination of the sonnet, the ghazal, and the blues.
The book is about tradition in our culture, exploring what it means to live in a country where we normalize rape culture, mass shootings, and police violence. It's about tradition in our families, like planting gardens. "This is the stuff men in my family have cared about for generations just for the sake of beauty." Jericho recently bought a house and found himself planting begonias and creeping myrtle. The book is about tradition in our daily lives, "If people see me as someone who falls in love, it might keep them from shooting me."
Jericho juxtaposes images of black men and flowers in his poem The Tradition. The first line names flowers. Aster. Nasturtium, Delphinium. And the last line names black men killed by police. John Crawford. Eric Garner. Mike Brown. In his headshot on the back of the book, Jericho wears a bright yellow shirt that matches a garden of yellow daffodils behind him. In his iconic social media headshot, he wears a wreath of flowers, mirroring the image of the African American child painted on the cover of his newest book. By juxtaposing flowers and black men, he asks a question of the reader about what images come to mind when they think of a flower, a black child, a black man. It's a question that creates an opportunity for examination, opens the possibility for change.
"Can poetry change the world?" I asked.
"Yeah!" He exclaimed, then went on in a quieter voice, "but, I also think its a powerful force for those who let it have power on them. You have to be exposed to it; you have to be in the position to know it. That is part of what we are doing as ambassadors of poetry. We poets are putting ourselves in a position so that if you might be interested, here we are."
Jericho doesn't wait for inspiration to strike; he prepares himself for it by consuming art and writing daily. He writes for two hours first thing every morning. "I set appointments with myself, and I honor them. I feel exhilarated by that fact."
"You get to create your life," he told me. "As soon as you realize it, it becomes a question. I get to create my life? And then you say to yourself, yes!" He laughed before saying, "You realize you are looking into a void."
He went on, "It's the same thing that happens when you make a poem. You are looking at a blank sheet and suddenly you have made a thing that literally changes thoughts and emotions simply by typing. Suddenly you have a trigger." A question, like a poem, is an act of change.
His daily writing practice starts with meditation and prayer. He reads modern spiritual writers like Ernest Holmes, Marianne Williamson, and Michael Bernard Beckwith. "The spiritual part has always been there, because that is how I was raised. But, I started taking responsibility for my own spirituality at a certain point," he explained, "and when I did, my poems got to be a lot better." His spirituality helps him "put myself in a place of faith and trust," this creates the space and safety for play.
Jericho's daily appointment has many forms. "Writing sometimes means revision, sometimes means drafting, sometimes means putting up things that aren't working. Sometimes writing doesn't go so well. It doesn't mean any of that. It just means reading stuff or wishing you had something. But there are two hours I dedicate to it everyday."
I pictured him like a train going up a mountain at a steady pace then reaching the top and coming down the other side faster and faster as he neared the end of his last book.
"When a book comes out, I run behind it pretty hard for a year, which is why I am here. I say yes to everything. I try to drum up whatever I need to send the book out into the world so people know it exists."
Jericho's goal was to sell at least 4000 copies of The Tradition in the next three months. I asked, "Who do you want those 4000 people to be? Who do you want to buy the book?" He started listing states he had never been to.
THIS IS A PUBLIC SERVICE ANNOUNCEMENT – if you are in North Dakota, South Dakota, Montana, Hawaii, Alaska, or Idaho, Jericho has you on his mind. He has never been to your state, and he wants to come. He will walk in, look you in the eye, and shake your hand. And if you meet him again, a year later, in different clothing and in a different place, he will remember you. He has timed how long it takes to read each poem; he is ready for whatever venue you have. The room where his parents taught him to walk in and greet everyone, it has expanded to be the entire country.
Later that afternoon Jericho spoke on a low stage with a backdrop of exposed brick. The panel was on Art & The Body with T. Fleischmann, moderated by Lisa Marie Brimmer. Jericho's preparation was evident in his clothing; his thin peach sweater matched the color of the sky on the cover of The Tradition, and his grey-blue pants matched the color of the ocean.
The panel started with an acknowledgment that we were on stolen Indigenous land. This is something said more and more often at events in Minnesota. The statement was followed by a beat to consider—stolen lands, Indigenous—and then the schedule continued. In my own head, I heard Jericho's voice, It becomes a question. We create our own lives.
At the end of the Q&A, an older African American woman asked Jericho to read his poem about cuddling. It's called Stand. "Cuddling is my favorite thing to do," Jericho said with an open-mouth smile, and we all smiled back. At each line break, I was uncertain of what would come next: violence or beauty?
When he was done, he didn't look up to see the crowd holding its breath and considering how bodies making love fit into our landscapes of crisis. He didn't smile or laugh. The poem had taken him elsewhere, as it had taken each of us.
I had asked him earlier if poetry is a political tool, and he didn't pause before shaking his head no. "I think every poem, I think every book, is for a single heart. When someone is reading it, it is your heart to their heart. It's not your heart to a mass of people."
I biked home along the beautiful Ȟaȟáwakpa Misi-ziibi Mississippi River. Cars zoomed by me on one side. On the other, trees blushed green with spring leaves along the steep riverbank. I cast questions into the void: Is it a responsibility to greet everyone when I walk into a room? What would it mean to have a culture of connection? Every room of people an opportunity to change myself, every person a poem?
E. A. Farro is a scientist and artist who spent the last seven years working in politics. She is the founder of The Nature Library, a literary art instillation in Saint Paul, Minnesota. Her publications have appeared in Rumpus, The Kenyon Review, and The Normal School, among others. She is a recipient of a 2010 Loft Literary Center Mentor Award and a 2019 Minnesota State Art Board grant.
EA FarroEssayIt Becomes a QuestionJericho Brown
Sydney Sargis
Marc Frazier
Sharon Goldberg
About Punctuate
Punctuate is a nonfiction magazine of the Department of English and Creative Writing at Columbia College Chicago. Published year round online.
Semi;Colon (53)
Riley McFarlane: Lia Purpura Taught Me to Dissect the Scene January 21, 2020
Student Editors of Punctuate Present Points of View & Reviews of F(r)iction: A Literary Journal with a Difference January 4, 2020
Tracie Taylor: My girlfriend burnt a piece of toast while I'm writing this and I can't wait to write a poem about it (praise for Olivia Gatwood) December 17, 2019
Anorexia anthology art Best American Essays blog Book Reviews Columbia College Chicago comic writer Coyote Jaws Death Eating Disorder Elizabeth Kadetsky Esmé Weijun Wang Essay Essays Graywolf Press Gretchen Lida Illness Interview Kelsey Hoff Kelsey Hoff Writes Ladies Lazarus Laura Manardo Lyric Essay memoir Mental Illness multi-genre writing narrative medicine nonfiction Penn State Piper J. Daniels poetry Prose Randon Billings Noble semicolon Spoon Collecting Spoons Sylvia Plath Tarpaulin Sky Press Travel University of Nebraska Press Wood Woodworking Wuthering Heights yoga
© 2018 Columbia College Chicago
|Department of English and Creative Writing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,598 |
Cliburn is a parish and small village 7 m. N.W. of Appleby. The church and Cliburn Hall are the principal monuments.
b(1). Parish Church of St. Cuthbert stands at the S. end of the village. The walls are of sandstone rubble with dressings of the same material and some tufa; the roofs are covered with slates and lead. The church, consisting of Chancel and Nave, was built about the middle of the 12th century, and at a later period a S. porch was added. The building was drastically restored in 1886–7 when the South Aisle and Chapel were added and the South Porch re-built.
Architectural Description—The Chancel (12¼ ft. by 13½ ft.) has a 13th-century lancet-window in the E. wall; below and S. of it are traces of an earlier window or opening. In the N. wall is a 12th-century window of one round-headed light. In the S. wall is a modern arch. The late 12th or early 13th-century chancel-arch has jambs and round arch of one chamfered order.
The Nave (29½ ft. by 17 ft.) has three modern windows in the N. wall, a modern S. arcade and a modern window in the W. wall. The bell-cote on the W. gable is also modern.
The South Aisle is modern, but in the S. wall is a re-set 12th-century doorway with plain jambs, corbelled lintel with a small figure carved at each end and a round head with cheveron-ornament enclosing a plain tympanum. The re-built S. porch incorporates old material.
Fittings—Chest: In nave—of hutch-type, with the initials and date T.H. 1696 incised on front. Churchyard Cross: S. of chancel—square tapering base, mediæval, shaft modern. Coffin-lids: In porch— (1) fragment with base of cross-shaft; (2) fragment with branched shaft of cross and stepped base; probably 13th-century. Consecration Cross: On W. jamb of S. doorway—small incised cross. Font: octagonal bowl (Plate 43) with moulded underside, shaft with zig-zag cutting, moulded necking and chamfered base, late 12th-century, bowl later or re-cut. Plate: includes Elizabethan cup (Plate 54) with band of ornament round bowl, a 17th-century pewter flagon and paten and an olive-wood cross inlaid with ebony and mother of pearl, said to have come from Vallombrosa. Sundial: On E. jamb of S. doorway, scratch-dial. Miscellanea: Incorporated in porch—two inscribed Roman stones (Plates 3, 4) (p. xlii) found during the restoration.
b(2). Cliburn Hall, house and outbuildings, 150 yards E.N.E. of the church. The House is of three storeys; the walls are of local rubble and ashlar and the roofs are slate-covered. The house was built or re-built by Richard Cliburn in 1567, but the thickness of some of the walls may indicate that portions of an earlier building were incorporated. At the N.W. end is a rather later extension with a two-storeyed wing extending towards the S.W. The house was re-roofed in the 19th century when the former parapet was removed; other alterations have been made in recent years. The S.W. front has a projecting two-storeyed wing, probably a porch-wing; the upper storey projects slightly on continuous corbelling; the former mullioned and transomed windows have been replaced by modern work in recent years. The former doorway in the N.W. return-wall is blocked. The main block has a doorway at the first-floor level, approached by steps; it has a square head and above it is a re-set panel inscribed "Rychard Cleburn thus they me cawl Wch in my tyme hath bealded ys hall, the yeare of owre lorde God who lyst for to neam (?) 1567— R.D. Mayson"; above the panel are the initials R.C. and a cartouche of the arms of Cliburn quartering Kirkbride. Farther S.E. are two windows each of four transomed and elliptical-headed lights with a moulded label; two other windows have been destroyed by a rebuilding round and above the doorway. The ground floor, approached by steps, has an original window with an iron grate. Both the S.E. end and the back (Plate 18) have a series of original transomed windows similar to those on the S.W. front, and the ground floor has a window with a grate. In the addition, at the first-floor level, is a doorway with a modern head, opening on to a solid stone platform approached by steps and giving access to a well enclosed in a square block of masonry. Inside the building, the ground floor both of the main block and the extension has elliptical barrel-vaults of rubble, groined back over the windows; two doorways have triangular arches in square heads and the large S.E. fireplace has an arched head. On the first floor, the middle beam of the main room or hall rests on shaped corbels; the blocked fireplace has moulded jambs and flat three-centred head; the original doorway, N.E. of the fireplace, has the scratched name John Lowther; another doorway has the names James Lowther, William Lowther and the figures 68. In the N.W. wall of the former hall are two more 16th-century doorways, one with a triangular and one with a three-centred head. The second floor has an original fireplace with a triangular arch in a square head.
The Outbuildings, extending S.W. from the S.E. end of the house, are probably of late 16th-century date and retain a series of original doorways and remains of an original window at the first-floor level.
d(3). Winderwath, house and outbuilding in a detached part of the parish, about 3 m. N.N.E. of the church. The House is of two storeys; the walls are of rubble and the roofs are slate-covered. There are remains of a mediæval building incorporated in the present structure, which was, however, largely re-built and remodelled late in the 17th century. The house was again altered c. 1860, when a S.E. wing was added; this has been extended in recent years. The exterior has no ancient features except a 17th-century doorway on the W. front; it has a moulded architrave and panelled and enriched pilasters, supporting consolebrackets, enriched frieze and cornice; above the doorway is a re-set shield-of-arms of Clifford impaling Vipont. Re-set in a gable of the N. wing is a grotesque head-corbel, a stone carved with a cinquefoil and a shield bearing a cross with a cinquefoil in the quarter. Inside the building, the dining-room has a 17th-century fireplace with rusticated jambs and head, enriched Ionic side-pilasters, frieze with blank shields and a wreath and a moulded cornice. In the main E. wall is a 15th-century window of one trefoiled light with a moulded label. In the N. wall of the kitchen is a mediæval doorway with a two-centred head and E. of it is a late 17th-century fireplace; it has a corbelled lintel enriched with mouldings and carved blocks.
The Stable and barn, W. of the house, is a two-storeyed building of the 17th century, retaining an original doorway and window. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,310 |
{"url":"https:\/\/byjus.com\/question-answer\/the-value-of-x-which-satisfies-the-equation-ax-b-0-a-neq-0-is-58\/","text":"Question\n\n# The value of x\u00a0which satisfies the equation ax+b=0,a\u22600 is known as solution of the equation. Solution of the equationDegree of the equation\n\nSolution\n\n## The correct option is B Degree of the equationA\u00a0solution\u00a0to an\u00a0equation\u00a0is a value of the variable, when substituted, makes the\u00a0equation\u00a0true. The value of x which satisfies the equation ax+b=0,a\u22600 is known as solution of the equation.\n\nSuggest corrections","date":"2021-11-27 06:38:57","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.951740562915802, \"perplexity\": 860.1560499400057}, \"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\/1637964358118.13\/warc\/CC-MAIN-20211127043716-20211127073716-00616.warc.gz\"}"} | null | null |
Hayley Kiyoko Quotes
Enjoy the top 13 famous quotes, sayings and quotations by Hayley Kiyoko.
"The music industry is such a different world from the acting world. Everything is really last minute, but I love the challenge, and I love owning my material and being able to put out what I want."
— Hayley Kiyoko —
"I really never imagined that I could ever even direct anything, so 'Girls Like Girls' was co-directed."
— Hayley Kiyoko
"For me, I think 'Jem' fans were expecting a remake of the cartoon, and the movie really is inspired by the cartoon based in a 2015, modern-day setting. It is going to be very different, but it's also going to be very familiar as well."
"I just love storytelling. I write music to tell stories. So when I'm done writing a song, I take it and go, 'Okay. How can I interpret these lyrics differently?' I love taking lyrics that were so close to me at a certain point in my life and then revamping them. I always want to take things to the next level."
I just want to make art that connects with people and moves them on an emotional level. Any time I can put out music and place a story behind it and have people watch it and go, 'Wow, I was affected by that,' to me, feels like I've done my job.
"I loved the idea of how all these guys always are stealing other guys' girls and I was like, 'There's no female anthem for a girl stealing another guy's girl,' and that is the coolest thing ever."
"I've always wanted my lyrics to say something meaningful and, you know, you always want to tell a message with your art. So yes, as I continue to write music, I will write about things that are real and things that I feel aren't written about a lot."
"It's tough as an artist to have such a specific vision for your video in mind when you write a song. Reaching out to directors is like going on blind dates and trying to find someone who sees the exact vision behind your music, which can be really difficult."
"My friends call my style 'old man chic.' I wear loafers and stylish sweatpants. I love to stay comfortable, so I definitely funk it up, but I'm always comfortable. I wear lots of hats and feathers, and I kind of have a little obsession with Native American jewelry."
"My hair has been this chapter thing for me. In 'Jem,' I have blue hair. 'Insidious,' it's pink. In 'CSI,' I have blonde. I love changing my hair. It's just hair and it grows all the time."
"My mom choreographed the top Olympians; she's really the queen of ice in her world, so I kind of get my directorial bug from her because she's really good at telling people what to do!"
"On a very small scale, I kind of understand why the directors that I work with do certain things. I don't consider myself an incredible director. I'm not ready to do movies by any means. But, I feel like I can be a better actress now that I've been on the other side and kind of understand the process and more of the technical aspects of it."
When I'm on set, I'm on set, and I focus and get the work done. Then when I'm done, I kind of have this button that I switch. I'm constantly switching this button and putting on different masks, and that kind of keeps me organized.
Hayley Kiyoko Quotes Pictures
Want to see more pictures of Hayley Kiyoko quotes? Click on image of Hayley Kiyoko quotes to view full size.
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Stacking The Deck Quotes | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,598 |
Creative Link offers high quality, cost efficient and innovative design solutions on budgets to suit you. We see every project as a chance to produce something of value and lasting quality, effectively communicating your message whatever the medium.
Meeting and discussing projects with our clients helps us build lasting relationships, as well as understand their visions, plans and goals for the future. We provide a fresh pair of eyes outwith the organisation which can be very beneficial. Our tailored approach has proven to be extremely successful when meeting our clients needs and requirements.
Based in Haddington, East Lothian our client base covers a variety of businesses, both large and small. Please find below a few examples of the servcies we provide, or get in touch to see how we can help you. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,528 |
HomeTinton Falls - Eatontown SunTinton Falls - Eatontown SportsDuarte set for busy weekend of turf sprints at Monmouth Park
Duarte set for busy weekend of turf sprints at Monmouth Park
OCEANPORT – With turf sprints dominating the three-day racing weekend at Monmouth Park, trainer Jorge Duarte Jr. is prepared for all of them. And if it goes well, he could hit a personal milestone by notching his 100th career win.
Duarte, three wins shy of 100, could have a starter in the three turf sprints of note at Monmouth Park starting on Friday, when Networking, a full brother to Grade 1 winner Force the Pass, debuts in 2-year-old Maiden Special Weight company at five furlongs on the grass in the first race on the card, according to a press release.
He will then send out Mac The Pee H Dee, 2 for 3 lifetime, in the $100,000 Blue Sparkler Stakes for 3-year-old fillies at five-and-a-half furlongs on the turf on July 16.
On July 17, he is considering a quick turnaround for Grooms All Bizness in the $100,000 My Frenchman Stakes for 3-year-olds at five-and-a-half furlongs on the turf.
Grooms All Bizness last raced on July 4.
"Hopefully it all goes well this weekend," said Duarte, the private trainer for nearby Colts Neck Stables, last year's leading owner at Monmouth Park. "If I get that 100th I hope it's at Monmouth Park."
All three horses were bred and are owned by Colts Neck Stable.
Networking, a son of Speightstown-Social Queen by Dynaformer, will face seven other 2-year-olds (not counting the Wesley Ward-bred, owned and trained Insanity It Seems, who is on the also eligible list).
Of the eight in the main body of the race, six are first-time starters, including one from Ward (Rylan Jayne) and one from Graham Motion (Hendrickson).
"He got ready fairly quickly," Duarte said of Networking. "He probably wants to go a mile, but the race is here, it's close, and we want to get him started. We have breezed him on the turf and he seems to handle it well. He's not the biggest horse out there, but he looks to have potential."
Mac The Pee H Dee has won two straight races since faltering in her debut at a mile-and-a-sixteenth on the turf at Gulfstream on Feb. 27.
She beat 11 fillies to break her maiden at Aqueduct at six furlongs on the grass on April 9, then was in a dead heat for a win in allowance company at Belmont Park on May 19 in a scheduled grass sprint that was switched to six furlongs on the main track.
Mac The Pee H Dee is a Kentucky-bred daughter of Fed Biz-American Girl by High Chaparral. She is expected to face seven other 3-year-old fillies for her stakes debut in the Blue Sparkler.
"She has been solid for her three starts, though she seems to prefer shorter," said Duarte. "She's a sprinter and I think her best game is on the turf, but she has shown she can handle the dirt. She's very competitive and has a lot of speed. She's fresh and going against 3-year-old fillies again, so that's a plus.
"I tried her two turns at Gulfstream because pedigree-wise I had to try two turns. I'm looking forward to running her 10 minutes from where she is stabled."
Grooms All Bizness, a Kentucky-bred son of Fed Biz-Bride to Be by Candy Ride, was fifth in his most recent start against allowance company at Monmouth Park on July 4. He shows a win and a second from four career starts, all of them at Monmouth Park.
"He ran against older horses his last start and really didn't get a chance to race his way," said Duarte. "He's a closing sprinter and the jockey sent him a little too early and he was flat."
First race post time for Friday's eight-race card is 2 p.m. On Saturday and Sunday, first race post time is 12:15 p.m.
Blue Sparkler Stakes
My Frenchman Stakes
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Mustangs continue to battle following tough stretch on the court
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Lions skate past Monroe Township; gear up for conference tournament | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,156 |
Q: Is there a command to refresh the applications listed on the Dash? Sometimes, after installing an application (Codeweavers Crossover to be more specific on this case) it fails to show on the Dash until you logout and back on. That happens with Unity and GNOME Shell, remember that with GNOME Fallback it is automatically added as a menu.
Is there a command that allows one to force a refresh on the cached application launchers in the Unity Dash or the GNOME Shell desktop environments?
A: To restart the GNOME Shell (no logout):
Press ALT + F2 and type r or restart and press enter
To restart Unity (no logout):
Press ALT + F2 and type setsid unity or unity --replace
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,884 |
Arrive est une ancienne commune française du département des Pyrénées-Atlantiques. Le , la commune fusionne avec Saint-Gladie et Munein pour former la nouvelle commune de Saint-Gladie-Arrive-Munein.
Géographie
Le village est situé au sud de Sauveterre-de-Béarn.
Toponymie
Le toponyme Arrive apparaît sous les formes
Arive (1385, censier de Béarn),
Arribe, Aribe, Arriba et Ribbe (respectivement 1538, 1546 et 1548 pour les deux dernières formes, réformation de Béarn) et
Arrive sur la carte de Cassini (fin ).
Michel Grosclaude indique que le toponyme est d'origine gasconne, arriba (du latin ripam) désignant une « terre en bordure d'un cours d'eau ».
Son nom béarnais est Arriva.
Histoire
En 1385, Arrive dépendait du bailliage de Sauveterre.
Démographie
En 1385, Arrive comptait 9 feux.
Notes et références
Pour approfondir
Article connexe
Anciennes communes des Pyrénées-Atlantiques
Ancienne commune dans les Pyrénées-Atlantiques
Ancienne commune du Béarn | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,297 |
"A Mediterranean diet can help you live longer," reported The Guardian and other newspapers today. They said a Mediterranean diet was one high in vegetables (excluding potatoes), fruits, nuts, beans, grains and fish. It was also low in saturated fat, meat and dairy products and contained a moderate amount of alcohol.
Mediterranean diet, exercise are key to long life.
Mediterranean diet cuts mortality rates, says study.
A med-style diet is key to living longer.
How the Mediterranean diet could help you live longer.
Mitrou PN, Kipnis V, Thiébaut ACM, et al. Mediterranean Dietary Pattern and Prediction of All-Cause Mortality in a US Population: Results From the NIH-AARP Diet and Health Study.Arch Intern Med 207; 167:2461-2468. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,593 |
The presentation will describe the Intelligent Cargo Concept as it has been implemented in the Euridice project. First will be presented the challenges in the logistics industry that led to the project, then will be presented the approach that Euridice used to address such challenges and to approach the market. The Intelligent Cargo capabilities will be described: capable of autonomous decisions (1), capable to start processes (2), capable to monitor and register its status (3), capable to grant access to services (4), capable to detect its context (5), capable to identify itself (6). Finally the architectural approach and a description of the platform and pilot application implemented with special attention to the design decisions tha have been taken. the presentation will end with the results achieved so far and the open issues. | {
"redpajama_set_name": "RedPajamaC4"
} | 376 |
picturesinpowell
I. A Question of Upbringing
II. A Buyer's Market
III. The Acceptance World
IV. At Lady Molly's
IX. The Military Philosophers
Pictures in Powell
V. Casanova's Chinese Restaurant
VI. The Kindly Ones
VII. The Valley of Bones
VIII. The Soldier's Art
X. Books Do Furnish a Room
XI. Temporary Kings
XII. Hearing Secret Harmonies
← Rainy Day at Marrakesh
Barnby's Murals Destroyed →
Egyptian Deities at General Liddament's Mess
Posted on July 15, 2015 by picturesinpowell
Nick attends the mess of General Liddament, a severe presence who is flanked at table by two colonels: "Here was Pharaoh, carved in the niche of a shrine between two tutelary deities who shielded him from human approach. All was manifest. Colonel Hogbourne-Johnson and Colonel Pedlar were animal-headed gods of Ancient Egypt. Colonel Hogbourne-Johnson was, of course, Horus, one of those sculptured representations in which the Lord of the Morning Sun resembles an owl rather than a falcon; a bad-tempered owl at that. Colonel Pedlar's dogs's muzzle, on the other hand, was a milder than normal version of the Jackal-faced Anubis, whose dominion over Tombs and the Dead did indeed fall within A&Q's province." [SA 37/35]
As he did in A Question of Upbringing (see QU 214/221, Le Bas' Appearance), Powell here evokes impressions of Ancient Egyptian imagery more than references to particular works. Below is a sculpture of the Pharaoh Ramses II "shielded from human approach" by the gods Ptah (left) and Sekhmet (right).
Reliefs of Ramses II, Ptah and Sekhmet The Egyptian Museum, Cairo photo by Daniel Mayer, cropped by AnnekeBart from Wikimedia Commons by GNU Free Documentation License
Colonel Hogbourne-Johnson as the god Horus was easy to envision, though not as an owl, as Nick has it, but as a falcon, bad-tempered to be sure. Here is Horus as he is depicted at the Temple at Edfu from the first couple of centuries B.C.
Temple of Horus, Edfu
granite, 237 B.C. or earlier
photo in public domain from Wikimedia Commons
The identities of Egyptian deities is not constant over the long evolution of the Egyptian pantheon, but Horus is generally known as the son of Isis and the rival of Set, slayer of Osiris, Horus' father, or sometimes brother. Horus is also identified with the sun and the moon, and Pharaoh himself came to be identified with Horus while alive, then as Osiris after death. The Horus at Edfu, judging from his fierce expression, might easily be a colonel who aspires to become a general as soon as possible.
white marble, height 62″
1st-2nd century AD
From Anzio, Villa Pamphili
Vatican Museum, Rome
photo from Wikimedia Commons by Creative Commons GNU license
Anubis, shown above, was the jackal-headed god who was associated, as Nick suggests, with mummification of the dead and supervision of souls into the afterlife. Colonel Pedlar as a mild-faced Anubis came readily to mind when we saw this Anubis in the Vatican collection. It is from the Roman period late in Egypt's history and is actually a blend of Anubis and the Greek god Hermes. We liked how this more doggy Anubis sports military braids and medals and looks as if he would be right at home at General Liddament's mess.
1 Response to Egyptian Deities at General Liddament's Mess
kcm76 says:
Along with the opening of A Question of Upbringing, that section from The Soldier's Art is one of my favourite passages in the whole of Dance. Not just for the imagery, which is superb, but also for the tightness of the prose.
Leave a Reply to kcm76 Cancel reply
The Modigliani reappears
Pop Art Armchairs
The Duport Collection
Apollyon, the Fiend Illustrated
The Whispering Knights
Tony Marshall on The Duport Collection
At the Isbister Memo… on A Pictograph of Widmerpoo…
vgreig007 on Aubrey Beardsley
Tokenhouse III: Plei… on Gauguin and Rimbaud
Virginia on Trajan's Column
Anthony Powell — The Artist as a Young Man
Powell is known as a novelist and book critic, but he probably began drawing before he knew how to write. In his autobiography, he relates that by the time he was six, his drawings, including a Mephistopheles, were shown to a visitor to his family. The term Post Impressionism (then recently introduced by Roger Fry) was bantered as the pictures were critiqued. He began at Eton in 1919 and took Extra Drawing from the drawing master Sidney Evans, who first told him of Picasso and Matisse. At Eton he drew for an art magazine, The Eton Candle (1922) , and at Oxford, which he attended from 1923 to 1026, his drawings appeared in another magazine, The Cherwell. His drawing Colonel Caesar Cannonbrains of the Black Hussars (1922) is reproduced in To Keep the Ball Rolling (p56). | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,902 |
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<h1 id="module-isortio">Module isort.io</h1>
<p>Defines any IO utilities used by isort</p>
<p>None</p>
<details class="example"><summary>View Source</summary><div class="codehilite"><pre><span></span><code><span class="sd">"""Defines any IO utilities used by isort"""</span>
<span class="kn">import</span> <span class="nn">re</span>
<span class="kn">import</span> <span class="nn">tokenize</span>
<span class="kn">from</span> <span class="nn">contextlib</span> <span class="kn">import</span> <span class="n">contextmanager</span>
<span class="kn">from</span> <span class="nn">io</span> <span class="kn">import</span> <span class="n">BytesIO</span><span class="p">,</span> <span class="n">StringIO</span><span class="p">,</span> <span class="n">TextIOWrapper</span>
<span class="kn">from</span> <span class="nn">pathlib</span> <span class="kn">import</span> <span class="n">Path</span>
<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Any</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">Iterator</span><span class="p">,</span> <span class="n">TextIO</span><span class="p">,</span> <span class="n">Union</span>
<span class="kn">from</span> <span class="nn">isort._future</span> <span class="kn">import</span> <span class="n">dataclass</span>
<span class="kn">from</span> <span class="nn">isort.exceptions</span> <span class="kn">import</span> <span class="n">UnsupportedEncoding</span>
<span class="n">_ENCODING_PATTERN</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="sa">br</span><span class="s2">"^[ \t\f]*#.*?coding[:=][ \t]*([-_.a-zA-Z0-9]+)"</span><span class="p">)</span>
<span class="nd">@dataclass</span><span class="p">(</span><span class="n">frozen</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">class</span> <span class="nc">File</span><span class="p">:</span>
<span class="n">stream</span><span class="p">:</span> <span class="n">TextIO</span>
<span class="n">path</span><span class="p">:</span> <span class="n">Path</span>
<span class="n">encoding</span><span class="p">:</span> <span class="nb">str</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Path</span><span class="p">],</span> <span class="n">readline</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="nb">bytes</span><span class="p">])</span> <span class="o">-></span> <span class="nb">str</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="n">tokenize</span><span class="o">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">readline</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">UnsupportedEncoding</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">from_contents</span><span class="p">(</span><span class="n">contents</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">filename</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-></span> <span class="s2">"File"</span><span class="p">:</span>
<span class="n">encoding</span> <span class="o">=</span> <span class="n">File</span><span class="o">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">BytesIO</span><span class="p">(</span><span class="n">contents</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="s2">"utf-8"</span><span class="p">))</span><span class="o">.</span><span class="n">readline</span><span class="p">)</span>
<span class="k">return</span> <span class="n">File</span><span class="p">(</span> <span class="c1"># type: ignore</span>
<span class="n">stream</span><span class="o">=</span><span class="n">StringIO</span><span class="p">(</span><span class="n">contents</span><span class="p">),</span> <span class="n">path</span><span class="o">=</span><span class="n">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span><span class="o">.</span><span class="n">resolve</span><span class="p">(),</span> <span class="n">encoding</span><span class="o">=</span><span class="n">encoding</span>
<span class="p">)</span>
<span class="nd">@property</span>
<span class="k">def</span> <span class="nf">extension</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-></span> <span class="nb">str</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">suffix</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s2">"."</span><span class="p">)</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">_open</span><span class="p">(</span><span class="n">filename</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Path</span><span class="p">])</span> <span class="o">-></span> <span class="n">TextIOWrapper</span><span class="p">:</span>
<span class="sd">"""Open a file in read only mode using the encoding detected by</span>
<span class="sd"> detect_encoding().</span>
<span class="sd"> """</span>
<span class="n">buffer</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s2">"rb"</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">encoding</span> <span class="o">=</span> <span class="n">File</span><span class="o">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">buffer</span><span class="o">.</span><span class="n">readline</span><span class="p">)</span>
<span class="n">buffer</span><span class="o">.</span><span class="n">seek</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">text</span> <span class="o">=</span> <span class="n">TextIOWrapper</span><span class="p">(</span><span class="n">buffer</span><span class="p">,</span> <span class="n">encoding</span><span class="p">,</span> <span class="n">line_buffering</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">newline</span><span class="o">=</span><span class="s2">""</span><span class="p">)</span>
<span class="n">text</span><span class="o">.</span><span class="n">mode</span> <span class="o">=</span> <span class="s2">"r"</span> <span class="c1"># type: ignore</span>
<span class="k">return</span> <span class="n">text</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="n">buffer</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
<span class="k">raise</span>
<span class="nd">@staticmethod</span>
<span class="nd">@contextmanager</span>
<span class="k">def</span> <span class="nf">read</span><span class="p">(</span><span class="n">filename</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Path</span><span class="p">])</span> <span class="o">-></span> <span class="n">Iterator</span><span class="p">[</span><span class="s2">"File"</span><span class="p">]:</span>
<span class="n">file_path</span> <span class="o">=</span> <span class="n">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span><span class="o">.</span><span class="n">resolve</span><span class="p">()</span>
<span class="n">stream</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">stream</span> <span class="o">=</span> <span class="n">File</span><span class="o">.</span><span class="n">_open</span><span class="p">(</span><span class="n">file_path</span><span class="p">)</span>
<span class="k">yield</span> <span class="n">File</span><span class="p">(</span><span class="n">stream</span><span class="o">=</span><span class="n">stream</span><span class="p">,</span> <span class="n">path</span><span class="o">=</span><span class="n">file_path</span><span class="p">,</span> <span class="n">encoding</span><span class="o">=</span><span class="n">stream</span><span class="o">.</span><span class="n">encoding</span><span class="p">)</span> <span class="c1"># type: ignore</span>
<span class="k">finally</span><span class="p">:</span>
<span class="k">if</span> <span class="n">stream</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">stream</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
<span class="k">class</span> <span class="nc">_EmptyIO</span><span class="p">(</span><span class="n">StringIO</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">write</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">:</span> <span class="n">Any</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">:</span> <span class="n">Any</span><span class="p">)</span> <span class="o">-></span> <span class="kc">None</span><span class="p">:</span> <span class="c1"># type: ignore # skipcq: PTC-W0049</span>
<span class="k">pass</span>
<span class="n">Empty</span> <span class="o">=</span> <span class="n">_EmptyIO</span><span class="p">()</span>
</code></pre></div>
</details>
<h2 id="variables">Variables</h2>
<div class="codehilite"><pre><span></span><code><span class="n">Empty</span>
</code></pre></div>
<h2 id="classes">Classes</h2>
<h3 id="file">File</h3>
<div class="codehilite"><pre><span></span><code><span class="k">class</span> <span class="nc">File</span><span class="p">(</span>
<span class="n">stream</span><span class="p">:</span> <span class="o"><</span><span class="k">class</span> <span class="err">'</span><span class="nc">TextIO</span><span class="s1">'>,</span>
<span class="n">path</span><span class="p">:</span> <span class="n">pathlib</span><span class="o">.</span><span class="n">Path</span><span class="p">,</span>
<span class="n">encoding</span><span class="p">:</span> <span class="nb">str</span>
<span class="p">)</span>
</code></pre></div>
<details class="example"><summary>View Source</summary><div class="codehilite"><pre><span></span><code><span class="nv">@dataclass</span><span class="p">(</span><span class="n">frozen</span><span class="o">=</span><span class="k">True</span><span class="p">)</span><span class="w"></span>
<span class="k">class</span><span class="w"> </span><span class="k">File</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="nl">stream</span><span class="p">:</span><span class="w"> </span><span class="n">TextIO</span><span class="w"></span>
<span class="w"> </span><span class="k">path</span><span class="err">:</span><span class="w"> </span><span class="k">Path</span><span class="w"></span>
<span class="w"> </span><span class="nl">encoding</span><span class="p">:</span><span class="w"> </span><span class="nf">str</span><span class="w"></span>
<span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">detect_encoding</span><span class="p">(</span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="ow">Union</span><span class="o">[</span><span class="n">str, Path</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="nl">readline</span><span class="p">:</span><span class="w"> </span><span class="n">Callable</span><span class="o">[</span><span class="n">[</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">bytes</span><span class="err">]</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="nf">str</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="k">try</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">tokenize</span><span class="p">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">readline</span><span class="p">)</span><span class="o">[</span><span class="n">0</span><span class="o">]</span><span class="w"></span>
<span class="w"> </span><span class="ow">except</span><span class="w"> </span><span class="k">Exception</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">raise</span><span class="w"> </span><span class="n">UnsupportedEncoding</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">from_contents</span><span class="p">(</span><span class="nl">contents</span><span class="p">:</span><span class="w"> </span><span class="nf">str</span><span class="p">,</span><span class="w"> </span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="nf">str</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="ss">"File"</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">encoding</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">File</span><span class="p">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span><span class="w"> </span><span class="n">BytesIO</span><span class="p">(</span><span class="n">contents</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="ss">"utf-8"</span><span class="p">)).</span><span class="n">readline</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">File</span><span class="p">(</span><span class="w"> </span><span class="err">#</span><span class="w"> </span><span class="nl">type</span><span class="p">:</span><span class="w"> </span><span class="k">ignore</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="o">=</span><span class="n">StringIO</span><span class="p">(</span><span class="n">contents</span><span class="p">),</span><span class="w"> </span><span class="k">path</span><span class="o">=</span><span class="k">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">).</span><span class="n">resolve</span><span class="p">(),</span><span class="w"> </span><span class="n">encoding</span><span class="o">=</span><span class="n">encoding</span><span class="w"></span>
<span class="w"> </span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="nv">@property</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">extension</span><span class="p">(</span><span class="n">self</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="nf">str</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="k">path</span><span class="p">.</span><span class="n">suffix</span><span class="p">.</span><span class="n">lstrip</span><span class="p">(</span><span class="ss">"."</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">_open</span><span class="p">(</span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="ow">Union</span><span class="o">[</span><span class="n">str, Path</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="nl">TextIOWrapper</span><span class="p">:</span><span class="w"></span>
<span class="w"> </span><span class="ss">"""Open a file in read only mode using the encoding detected by</span>
<span class="ss"> detect_encoding().</span>
<span class="ss"> """</span><span class="w"></span>
<span class="w"> </span><span class="n">buffer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span><span class="w"> </span><span class="ss">"rb"</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="k">try</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">encoding</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">File</span><span class="p">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span><span class="w"> </span><span class="n">buffer</span><span class="p">.</span><span class="n">readline</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">buffer</span><span class="p">.</span><span class="n">seek</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="nc">text</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TextIOWrapper</span><span class="p">(</span><span class="n">buffer</span><span class="p">,</span><span class="w"> </span><span class="n">encoding</span><span class="p">,</span><span class="w"> </span><span class="n">line_buffering</span><span class="o">=</span><span class="k">True</span><span class="p">,</span><span class="w"> </span><span class="n">newline</span><span class="o">=</span><span class="ss">""</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="nc">text</span><span class="p">.</span><span class="n">mode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ss">"r"</span><span class="w"> </span><span class="err">#</span><span class="w"> </span><span class="nl">type</span><span class="p">:</span><span class="w"> </span><span class="k">ignore</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nc">text</span><span class="w"></span>
<span class="w"> </span><span class="ow">except</span><span class="w"> </span><span class="k">Exception</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">buffer</span><span class="p">.</span><span class="k">close</span><span class="p">()</span><span class="w"></span>
<span class="w"> </span><span class="n">raise</span><span class="w"></span>
<span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="nv">@contextmanager</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="k">read</span><span class="p">(</span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="ow">Union</span><span class="o">[</span><span class="n">str, Path</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="n">Iterator</span><span class="o">[</span><span class="n">"File"</span><span class="o">]</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">file_path</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">).</span><span class="n">resolve</span><span class="p">()</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">None</span><span class="w"></span>
<span class="w"> </span><span class="k">try</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">File</span><span class="p">.</span><span class="n">_open</span><span class="p">(</span><span class="n">file_path</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">yield</span><span class="w"> </span><span class="k">File</span><span class="p">(</span><span class="n">stream</span><span class="o">=</span><span class="n">stream</span><span class="p">,</span><span class="w"> </span><span class="k">path</span><span class="o">=</span><span class="n">file_path</span><span class="p">,</span><span class="w"> </span><span class="n">encoding</span><span class="o">=</span><span class="n">stream</span><span class="p">.</span><span class="n">encoding</span><span class="p">)</span><span class="w"> </span><span class="err">#</span><span class="w"> </span><span class="nl">type</span><span class="p">:</span><span class="w"> </span><span class="k">ignore</span><span class="w"></span>
<span class="w"> </span><span class="nl">finally</span><span class="p">:</span><span class="w"></span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="ow">not</span><span class="w"> </span><span class="k">None</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="p">.</span><span class="k">close</span><span class="p">()</span><span class="w"></span>
</code></pre></div>
</details>
<hr />
<h4 id="static-methods">Static methods</h4>
<h4 id="detect_encoding">detect_encoding</h4>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">detect_encoding</span><span class="p">(</span>
<span class="n">filename</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">pathlib</span><span class="o">.</span><span class="n">Path</span><span class="p">],</span>
<span class="n">readline</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="nb">bytes</span><span class="p">]</span>
<span class="p">)</span> <span class="o">-></span> <span class="nb">str</span>
</code></pre></div>
<details class="example"><summary>View Source</summary><div class="codehilite"><pre><span></span><code><span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">detect_encoding</span><span class="p">(</span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="ow">Union</span><span class="o">[</span><span class="n">str, Path</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="nl">readline</span><span class="p">:</span><span class="w"> </span><span class="n">Callable</span><span class="o">[</span><span class="n">[</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">bytes</span><span class="err">]</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="nf">str</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="k">try</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">tokenize</span><span class="p">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">readline</span><span class="p">)</span><span class="o">[</span><span class="n">0</span><span class="o">]</span><span class="w"></span>
<span class="w"> </span><span class="ow">except</span><span class="w"> </span><span class="k">Exception</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">raise</span><span class="w"> </span><span class="n">UnsupportedEncoding</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span><span class="w"></span>
</code></pre></div>
</details>
<h4 id="from_contents">from_contents</h4>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">from_contents</span><span class="p">(</span>
<span class="n">contents</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span>
<span class="n">filename</span><span class="p">:</span> <span class="nb">str</span>
<span class="p">)</span> <span class="o">-></span> <span class="s1">'File'</span>
</code></pre></div>
<details class="example"><summary>View Source</summary><div class="codehilite"><pre><span></span><code><span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="n">from_contents</span><span class="p">(</span><span class="nl">contents</span><span class="p">:</span><span class="w"> </span><span class="nf">str</span><span class="p">,</span><span class="w"> </span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="nf">str</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="ss">"File"</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">encoding</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">File</span><span class="p">.</span><span class="n">detect_encoding</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span><span class="w"> </span><span class="n">BytesIO</span><span class="p">(</span><span class="n">contents</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="ss">"utf-8"</span><span class="p">)).</span><span class="n">readline</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">File</span><span class="p">(</span><span class="w"> </span><span class="err">#</span><span class="w"> </span><span class="nl">type</span><span class="p">:</span><span class="w"> </span><span class="k">ignore</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="o">=</span><span class="n">StringIO</span><span class="p">(</span><span class="n">contents</span><span class="p">),</span><span class="w"> </span><span class="k">path</span><span class="o">=</span><span class="k">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">).</span><span class="n">resolve</span><span class="p">(),</span><span class="w"> </span><span class="n">encoding</span><span class="o">=</span><span class="n">encoding</span><span class="w"></span>
<span class="w"> </span><span class="p">)</span><span class="w"></span>
</code></pre></div>
</details>
<h4 id="read">read</h4>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">read</span><span class="p">(</span>
<span class="n">filename</span><span class="p">:</span> <span class="n">Union</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">pathlib</span><span class="o">.</span><span class="n">Path</span><span class="p">]</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Iterator</span><span class="p">[</span><span class="n">ForwardRef</span><span class="p">(</span><span class="s1">'File'</span><span class="p">)]</span>
</code></pre></div>
<details class="example"><summary>View Source</summary><div class="codehilite"><pre><span></span><code><span class="w"> </span><span class="nv">@staticmethod</span><span class="w"></span>
<span class="w"> </span><span class="nv">@contextmanager</span><span class="w"></span>
<span class="w"> </span><span class="n">def</span><span class="w"> </span><span class="k">read</span><span class="p">(</span><span class="nl">filename</span><span class="p">:</span><span class="w"> </span><span class="ow">Union</span><span class="o">[</span><span class="n">str, Path</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="n">Iterator</span><span class="o">[</span><span class="n">"File"</span><span class="o">]</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">file_path</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">Path</span><span class="p">(</span><span class="n">filename</span><span class="p">).</span><span class="n">resolve</span><span class="p">()</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">None</span><span class="w"></span>
<span class="w"> </span><span class="k">try</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">File</span><span class="p">.</span><span class="n">_open</span><span class="p">(</span><span class="n">file_path</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">yield</span><span class="w"> </span><span class="k">File</span><span class="p">(</span><span class="n">stream</span><span class="o">=</span><span class="n">stream</span><span class="p">,</span><span class="w"> </span><span class="k">path</span><span class="o">=</span><span class="n">file_path</span><span class="p">,</span><span class="w"> </span><span class="n">encoding</span><span class="o">=</span><span class="n">stream</span><span class="p">.</span><span class="n">encoding</span><span class="p">)</span><span class="w"> </span><span class="err">#</span><span class="w"> </span><span class="nl">type</span><span class="p">:</span><span class="w"> </span><span class="k">ignore</span><span class="w"></span>
<span class="w"> </span><span class="nl">finally</span><span class="p">:</span><span class="w"></span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">stream</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="ow">not</span><span class="w"> </span><span class="k">None</span><span class="err">:</span><span class="w"></span>
<span class="w"> </span><span class="n">stream</span><span class="p">.</span><span class="k">close</span><span class="p">()</span><span class="w"></span>
</code></pre></div>
</details>
<h4 id="instance-variables">Instance variables</h4>
<div class="codehilite"><pre><span></span><code><span class="n">extension</span>
</code></pre></div>
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{{Taxobox
| name = Bacteriophage AP205
| virus_group = iv
| realm = Riboviria
| regnum = Orthornavirae
| phylum = Lenarviricota
| classis = Leviviricetes
| ordo = Norzivirales
| familia = Duinviridae
| genus = Apeevirus
| species = Bacteriophage AP205}}Bacteriophage AP205 is a bacteriophage that infects Acinetobacter bacteria. Contains a genome linear of positive single-stranded RNA. The bacteriophage belongs to the genus Apeevirus of the Duinviridae'' family and is the type species of the family.
References
Bacteriophages | {
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You are here: Home News A paper-based sensor for monitoring cadmium contamination in water
A paper-based sensor for monitoring cadmium contamination in water
ICREA Research Professor Arben Merkoci and colleagues report a paper-based sensor for detection of cadmium (Cd2+) in water at record levels of sensitivity, in latest issue of Analytical Chemistry.
The sensor is cheap, portable and easy-to-use, making it a potentially ideal replacement for the costly, large time-consuming and complicated equipment traditionally employed for metal detection in water, and enabling water analysis in remote locations with limited resources
Reporting in the latest issue of Analytical Chemistry a team of researchers from ICN, the Universitat Autonoma de Barcelona and Tulane University (New Orleans, USA) have described a new, portable, low-cost and easy to use sensor for detecting the heavy metal cadmium (as Cd2+) in drinking water ("All-integrated and highly sensitive paper-based device with sample treatment platform for Cd2+ immunodetection in drinking waters").
Water pollution continues to cause health and environmental problems globally, meaning that much of the world's population does not have regular access to clean drinking water. Unfortunately, water analysis usually involves bulky and complex equipment, a dedicated laboratory and trained technicians, which are out of reach for most developing nations and which make field detection nearly impossible, especially in remote areas. Thus, there is a pressing need for affordable, easy to use devices for on-site water testing.
Now, ICREA Research Professor and ICN Group Leader Arben Merkoci, doctoral student Adaris López Marzo and colleagues have created a paper-based sensor that employs an immunoassay and gold nanoparticles to detect Cd2+, a common pollutant whose presence in drinking water has been linked to various health problems, including kidney and liver malfunction, and cardiovascular diseases.
The functionalised paper strip, which measures about 6 x 0.7 cm (l x w), changes from pink to white in the presence of Cd2+. The intensity of the colour band corresponds to the concentration of Cd2+, which Merkoci and colleagues quantified by scanning the band with a RapidScan® reader.
The team then validated the device using real water samples, for which it offered the highest sensitivity ever reported for detection of metals using paper-based sensors: a Limit of Detection (LOD) of 0.1 ppb and a Limit of Quantification (LOQ) of 0.4 ppb. This LOD is 50 times lower than the legally permissible limit for Cd2+ in drinking water. They also tested their device by testing recovery using spiked samples (achieving a recovery rate of nearly 100%) and by comparing their results to those obtained using standard methods, such as inductively-coupled plasma emission spectroscopy (ICPES).
The Cd2+ in water is selectively detected through a two-step immunochemical reaction whereby the free ionic species conjugates to free EDTA on the first of two reaction pads in the sensor (this first pad also contains free ovalbumin, which is used to capture other metal ion species to mask background signals). The Cd-EDTA complex then competes with pre-formed complex of Cd-EDTA-BSA-gold nanoparticles for binding to same sites in the monoclonal antibody 2A8G15.
"We are confident that our new sensor could be particularly useful for routine screening of drinking water samples or for testing industrial process streams, especially in developing countries, where heavy-metal concentrations tend to be high," affirms Prof Merkoci, "and that this technology could easily be extended to other analytes of interest."
To access the article "All-integrated and highly sensitive paper-based device with sample treatment platform for Cd2+ immunodetection in drinking waters", click here. | {
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