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2014-12-01T14:55:59Z

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2014-11-19T20:43:26Z

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2014-05-19T13:05:02Z

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2014-05-15T09:38:51Z

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2014-03-03T21:42:50Z

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130.88.213.204: /* Endocytosis and exocytosis */

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130.88.168.24: /* Simplification */

{{Cleanup|date=October 2013|reason=Update CEO information}} {{Advert|date=October 2013}} {{Infobox company
|company_name = Weight Watchers International, Inc.
|company_logo =
|company_type = [[Public company|Public]]
|traded_as = {{nyse|WTW}}
|company_slogan = ''Stop Dieting. Start Living''
|foundation = 1963
|location = 11 [[Madison Avenue]] [[New York City|New York, New York]] 10010
|key_people = Jim Chambers, President & CEO
|products = [[Weight loss]], [[Packaged food]]s, [[Exercise]] products
|num_employees = 28,000 ({{As of|2010|lc=on}})
|homepage = [http://www.weightwatchers.com weightwatchers.com] [http://www.weightwatchers.co.uk weightwatchers.co.uk] [http://www.weightwatchers.ca weightwatchers.ca] [http://www.fr.weightwatchers.ca fr.weightwatchers.ca]
}}

[[Image:Weight Watchers Center, Newton Highlands MA.jpg|thumb|right|250px|<center>A former Weight Watchers location in [[Newton Highlands, Massachusetts]]</center>]]

'''Weight Watchers International''' is an international company based in the United States that offers various products and services to assist [[weight loss]] and maintenance. Founded in 1963 by [[Brooklyn]], [[New York]], homemaker [[Jean Nidetch]], it {{As of|2007|alt=now}} operates in about 30 [[countries]] around the world, generally under names that are local translations of “Weight Watchers”. The core philosophy behind Weight Watchers programs is to use a science-driven approach to help participants lose weight by forming helpful habits, eating smarter, getting more exercise and providing support.<ref name="2010splashpage">{{cite web |url=http://www.weightwatchers.com/plan/index.aspx |title=Helping People Lose Weight for 45 Years |accessdate=2010-12-29}}</ref><ref name="2010ceoletter">{{cite |url=http://www.weightwatchers.com/util/art/index_art.aspx?tabnum=1&art_id=105421 |title=Our New Program the Next Level |author=David Kirchhoff |accessdate=2010-12-29}}</ref>

The term ''weight-watcher'', in the same sense, had circulated publicly for several years before the company was formed.<ref>{{cite web|url=http://dbase1.lapl.org/images/menus/fullsize/j/rb03902-02.jpg |title=Menu from Stouffer's restaurant, probably Chicago, dated November 7, 1959: "Weight-Watcher's Luncheon ... $1.20." |date= |accessdate=2012-01-14}}</ref>

== Approach and Philosophy ==
Weight Watchers' core approach is to assist members in losing weight by forming helpful habits, eating smarter, getting more exercise and providing support.<ref name="2010splashpage" /> In contrast to other diets that might provide a list of foods that either must be or should never be eaten, Weight Watchers ' primary plan has no directly comparable requirements and no food is off limits. Participants lose weight by creating a calorie deficit; the PointsPlus system provides a framework for creating that deficit.<ref name="2010ceoletter" /> Accordingly, Weight Watchers is generally compatible with other diet approaches and/or food intake restrictions (e.g., the [[paleolithic diet]] and [[South Beach diet]]), provided participants use the Weight Watchers framework to measure and limit the quantity of food consumed while using the other diet plan to dictate the range of acceptable food choices.

The newest plan has adjusted the "cost" of items while retaining the fundamental "counting, budgeting, and planning approach"<ref name="2010ceoletter" /> that has been the hallmark of Weight Watchers plans since the introduction of points-based counting systems.<ref name="2010ceoletter" />

There are two primary ways individuals can work with Weight Watchers: via in-person meetings and an online-only program.<ref name="2010splashpage" /> Both programs use the same basic materials and computations.<ref name="2010ceoletter" /> For in-person meetings, Weight Watchers encourages members to select a goal weight that results in a [[body mass index]] generally accepted as healthy (18 to 24.9), although a member may also establish a goal weight outside of that range after providing a doctor's note to that effect. In order to join Weight Watchers in the United States, one must weigh at least {{convert|5|lb}} more than the minimum weight for his or her height.{{citation needed|date=December 2010}}

Once a member reaches his or her goal weight, he or she starts a maintenance period. For the following six weeks, the member gradually adjusts his or her food intake until the member no longer loses or gains weight. If, at the end of six weigh-ins during the maintenance period, the member weighs in within {{convert|2|lb}} of his or her goal weight, he or she becomes a "Lifetime" member.<ref name="wwlifetimefaq">{{cite web |url=http://www.weightwatchers.com/help/index.aspx?pageid=1027981 |title=Lifetime Membership FAQ |accessdate=2010-12-29 }}</ref> A Lifetime member may attend meetings free of charge as long as he or she weighs in at least once per calendar month and, at any weigh-in, does not weigh more than {{convert|2|lb}} more than his or her goal weight.<ref name="wwlifetimefaq" /> If the member weighs more than {{convert|2|lb}} over the goal weight at a weigh-in, he or she will be charged the weekly fee for any meeting he or she attends (but not for skipped meetings) until he or she again obtains a weight within the target range.<ref name="wwlifetimefaq" /> As long as a Lifetime member weighs in each month and stays within the target range, there is no fee for attending meetings, and the member may attend for the rest of his or her life. Unlike new members, Lifetime members are never charged a registration fee, even if they regain weight or fail to weigh in every calendar month.<ref name="wwlifetimefaq" />

Members who reached Lifetime status in the early days of Weight Watchers and who received a contract with their Lifetime package never pay more than the weekly meeting fee that was in force at the time they reached Lifetime status; however, in order to receive this special rate, a member must be able to show their contract that states what the fee is. Weight Watchers tried to stop honoring this early promise and was sued. The case ended with a ruling ordering Weight Watchers to honor their contract for any member who can produce it.

Weight Watchers’ ''eTools'' is a Web-based service for members that includes access to support materials and tracking tools. The service is available for an additional fee for members who pay as they go, and is included in the fee for members who opt for the recurring "Monthly Pass" membership plan. In some areas Weight Watchers meetings are operated by a locally franchised organization rather than by Weight Watchers International.<ref name="WWPhila">{{cite web
|url= http://www.wwphl.com
|title= Home Page
|accessdate= 2008-06-07
|author=
|last=
|first=
|authorlink=
|coauthors=
|date=
|year=
|month=
|publisher= Weight Watchers of Philadelphia
|doi=
|archiveurl=
|archivedate=
|quote= This website is operated by Weight Watchers of Philadelphia, Inc.<nowiki> [punctuation sic]</nowiki> a franchise of Weight Watchers International, Inc.
}}</ref>

Additionally, a fully online program is available via the Weight Watchers website.<ref>{{cite web |url= http://www.weightwatchers.com/plan/apr/ways.aspx |accessdate=2010-12-27 |title=Weight Watchers Summary of Plan Options}}</ref> Online and meeting memberships are entirely separate and online-only members are not eligible for "Lifetime" status.<ref name="wwlifetimefaq" /> The support system for Weight Watchers Online is a message board where members can post support and share advice. The message board (but not the other support tools) is available free of charge to the public; in an attempt to limit access to the essential plan information to its members, Weight Watchers prohibits the posting of Points or PointsPlus values for foods on the boards.

In most locations, Weight Watchers holds meetings for members which in some cases may cause [[positive reinforcement]] for participants.
<ref>MOISIO, R., & BERUCHASHVILI, M. (2010). "Questing for Well-Being at Weight Watchers: The Role of the Spiritual-Therapeutic Model in a Support Group". ''journal Of Consumer Research'', 36(5), 857-875.</ref>

== Weight loss plans ==
===In the United States===
[[File:WW Rechenschieber.jpg|thumb|Weight Watcher Slide Ruler]]

The primary, current plan offered in the United States, started in November 2010, is called "PointsPlus". Touted by Weight Watchers as incorporating a decade of science compared to the prior Points-based plans, the focus remains on assisting members in creating a calorie deficit to lose weight using a reformulated calculation approach for computing target daily points (e.g., approximately how many calories per day should be eaten) and the "costs" (PointsPlus values) of food.<ref>{{cite web|url=http://www.webmd.com/diet/features/weight-watchers-diet |title=Weight Watchers Points Plus: Diet Review By Kathleen M. Zelman, MPH, RD, LD WebMD Expert Review |publisher=Webmd.com |date= |accessdate=2013-02-22}}</ref>

The calculation of the daily points targets is based on creating approximately a 1,000 calorie/day deficit ({{cite patent
|country=WO
|number=2010025422
|status= application
}} [0098]-[00109] and accompanying equations 16-19). This in turn means that members are assigned a daily point target in the range of 26-71 PointsPlus to consume each day. Additionally, members are allowed 49 PointsPlus each week, or weekly points, that they can spend how they wish throughout the week. Physical activity earns activity points that, like weekly points, can be used to supplement the daily points allowance.

As compared to prior Points-based plans, the main change is the direct computation of PointsPlus-values from macronutrients (fat, protein, carbohydrates) vs. calories and fat. Both formulas used dietary fiber information when it was available. The formulas are country specific based on the information on nutrition labels in a given country. The patent provides equations for different country labeling techniques.

This change allowed Weight Watchers to change the relative cost of the different macronutrients. Specifically, the relative weight of fat appears to be unchanged vs. the Momentum calculation; however, carbohydrates are "more expensive" and fiber is "less beneficial" and protein is only modestly changed. This increases the PointsPlus cost vs. the Point cost of a number of foods that were specially designed to take advantage of the old formula, e.g., muffins loaded with fiber to stay at 1 Point. The range of calories for one PointsPlus point is approximately in the range of 35-45 calories, in contrast one Point had a broader range of approximately 25-80 calories. However, because the approach for computing the daily points target has been refined ({{cite patent
|country=WO
|number=2010025422
|status= application
}} [0098]-[00104] and accompanying equations 16 (male) and 17 (female)), most people have significantly more PointsPlus available as part of their daily points target than on the prior Momentum plan. For example, under the Momentum plan 18 was the minimum number of points; now 26 (as of 2012, down from 29) is the minimum number of points. Additionally, irrespective of the underlying nutritional information, on the PointsPlus plan, fruits and most vegetables are considered to have a PointsPlus value of 0.

As an alternative to the PointsPlus plan, participants can use the Simply Filling Technique.<ref name="wwsft">{{cite web |url=http://www.weightwatchers.com/util/art/index_art.aspx?tabnum=2&art_id=65271&sc=3002 |title=What's included in the Simply Filling Technique |accessdate=2011-01-03}}</ref> On the Simply Filling Technique, participants are intended to eat from a designated list of foods without the requirement to track.<ref name="wwsft" /> Categories of foods on the list include: most fruits and vegetables, whole grains, non-fat dairy and dairy substitutes, lean proteins,and a handful of other items.<ref name="wwsft" /> For items that a person eats that are ''not'' on the list, a weekly points allowance of PointsPlus points is available to be used and the value of that item must be tracked.<ref name="wwsft" /> Because the plan does not require tracking, participants must be mindful to "[e]at portions that feel right for [them]. Not so much that [they] feel too full, and not too little that [they] still feel hungry."<ref name="wwsft" />

On December 3, 2012, Weight Watchers introduced additional aspects to the program. This was dubbed "Weight Watchers 360". Additions included introducing more interactivity to meetings, more focus on building healthful habits, and fighting [[hedonic hunger]]: "the desire to seek out high-sugar, high-fat foods that bring pleasure." The 360 plan also introduced the ActiveLink physical activity monitor as an optional purchase item.<ref name="Hellmich">{{cite web|url=http://www.usatoday.com/story/news/nation/2012/12/02/weight-watchers-new-program/1732297/|title=New Weight Watchers 360 plan unveiled|last=Hellmich|first=Nancy|date=December 2, 2012|work=[[USA Today]]|accessdate=December 8,2013}}</ref>

====Momentum====
Weight Watchers implemented the Momentum Plan on December 7, 2008. The cornerstone of the plan is the proprietary ''Points'' formula, which allocates each food a value based on its calories, fat, and fiber. Members are allocated a certain number of points each day based on their height, current weight, age, and activity level. Members were also allowed 35 optional Points each week. Finally, members earn additional Points through exercise. These "Activity Points" are calculated based on the member's weight and the duration and intensity of the activity. Activity Points previously had to be consumed on the day that they were earned or they were lost; with the Momentum plan, they can be eaten on any day during the week.

The Momentum Plan encourages members to choose foods by meeting the "Good Health Guidelines." These include eating lean protein and whole grains, meeting target servings of fruits and vegetables and dairy or soy products, taking a multivitamin, exercising, eating healthy oils, drinking adequate liquids, and limiting sugar and alcohol. Additionally, the Momentum Plan encourages members to remain satisfied while dieting by focusing on eating foods identified as ''Filling Foods,'' which include fruits, vegetables, whole grains, dairy and soy products, and lean proteins, but this is not required. The effect of this is that the member is not prevented from eating any specific type of food, but he or she must consume foods only with his or her allotment of points. This stands in marked contrast to approaches such as the [[South Beach Diet|South Beach]] or [[Atkins diet|Atkins]] diets, in which some foods are completely forbidden and others are permitted in as great a quantity as the dieter likes. The member also has the choice to exercise — which will entitle him or her to consume more food — or to eat food of lower Points value if he or she prefers not to exercise.

Many (but not all) of the Filling Foods also have a ''SetPoints'' value that allows members to eat a reasonable portion of the food and track the SetPoints value, rather than having to measure and calculate the exact Points value of the portion eaten. The Momentum Plan also includes an option that allows members to eat from the Filling Foods list without tracking Points values at all. This option, known as the ''Simply Filling Technique,'' is very similar to the former Core Plan. Members following the Simply Filling Technique do not receive a daily Points allowance, but eat to satisfaction from the Filling Foods list. They do, however, receive the 35 weekly Points, and are entitled to eat any Activity Points that they earn. The weekly Points and Activity Points may be used for foods that are not on the Filling Foods list.

Many Weight Watchers proponents enjoy the tracking option of the Momentum Plan precisely because no food is out of bounds as long as it is eaten in moderation.{{citation needed|date=November 2008}} (In the [[United Kingdom|UK]], Weight Watchers advertises under the slogan "Where no food is a sin"; this is a reference to its chief competitor [[Slimming World|Slimming World’s]] system of giving some food "syn" values.) Many others, however, dislike the record-keeping that the plan requires of participants, who must keep track of the Points values of everything they eat; they prefer the Simply Filling option or other plans that place restrictions on types of food rather than quantities.

====TurnAround====
In August 2004, Weight Watchers introduced the TurnAround program, incorporating parts of the Flex and Core food plans, but intended to assist people in developing an overall healthy lifestyle. Aside from encouraging healthy eating and providing member support, the program encouraged participants to follow eight Good Health Guidelines.{{citation needed|date=November 2008}}

====Historical Plans====
Jean Nidetch, the founder of Weight Watchers, wrote a book in the early 1970s ''The Memoir of a Successful Loser The Story of Weight Watchers'' that documented the original Weight Watchers plan.

That original plan was supplanted shortly after the book with materials bearing a 1972-73 copyright: “Weight Watchers(R) Program Handbook for Ladies”.<ref name="ww1972">{{cite web |url=http://b.feli.me/post/4959873786/ww-1972-plan |accessdate=2011-05-04 |title=WW 1972 Plan |publisher=B.feli.me}}</ref> This plan was similar to the original plan.

By 1989, the plan had switched to an exchange-based diet.<ref name="ww1989">{{cite web |url=http://b.feli.me/post/4989735431/ww-1989-plan |accessdate=2011-05-04 |title=WW 1989 Plan |publisher=B.feli.me}}</ref> The document's subtitle and copyright dates make determining the exact date difficult, “The Quick Success(R) Program” with the subtitle “New for 1989” has a copyright date of 1987, 1988.<ref name="ww1989" />

By 1997, the first Points-based program was introduced.

On April 15, 2005, A sputtering low-carbohydrate fad and the end of a licensing deal with Weight Watchers International Inc. had carved a big chunk out of CoolBrands International Inc.'s bottom line.<ref>{{cite web |url=https://www.lexis.com/research/retrieve?_m=e727a207b8d2759b248388682fbaff35&docnum=37&_fmtstr=FULL&_startdoc=1&wchp=dGLzVzB-zSkAz&_md5=832cadb02ee6497747bb68a13cb3df8d |title= Low-carb, low-fat crazes hit CoolBrands; Shares drop on second-quarter loss}}</ref>
On December 20, 2005, CoolBrands International Inc. said it intended to wield the power of three big-name licensed brands to help it recover from the loss of Weight Watchers from its product lineup.
<ref>{{cite web |url=https://www.lexis.com/research/retrieve?_m=e727a207b8d2759b248388682fbaff35&docnum=5&_fmtstr=FULL&_startdoc=1&wchp=dGLzVzB-zSkAz&_md5=832cadb02ee6497747bb68a13cb3df8d |title= CoolBrands plays down its loss of Weight Watchers
}}</ref>

===In the UK===
The original Points Plan (aka free money plan) was developed in the Uk by the Weight Watchers programme development team headed by Sarah Watson and Marian Way who took out the first patent for a calculator to embody the algorithm (based on the summation of calories/70 and saturated fat/4 with the sum rounded to the nearest half) the patent was filed on 1/11/95 <ref name="Sarah Watson, Marian Way">{{cite web|title=UK Patent Application 2 302 605 A|url=http://worldwide.espacenet.com/publicationDetails/originalDocument?FT=D&date=19970122&DB=EPODOC&locale=en_gb&CC=GB&NR=2302605A&KC=A&ND=4}}</ref>
In the UK during November 2010 the Points plan has been replaced with ProPoints. This is similar to the Points plan only the calculation now takes protein, carbohydrates, all fat and fiber into consideration. The following main changes were made from the Points to ProPoints plan:
* Change the formula, so the majority of foods now cost more ProPoints than Points
* Daily ProPoint allowances (give you free money) were increased however you are no longer allowed to carry your daily ProPoints over to the next day
* Most fruit and vegetables are now zero ProPoints
* All participants also have a weekly ProPoints allowance of 49 ProPoints which they can choose to use as they see fit throughout the week.

===In Ireland and Australia===
In Ireland there has recently been a new plan launched called ProPoints Plan.Traditional weight loss plans are based on the calories on a food label. But the new ProPoints system incorporates the latest science that shows there is a more accurate way to assess the impact a food has on weight loss. ProPoints values are calculated using 4 macronutrients: fat, fibre, protein and carbohydrate. There is also a new weekly ProPoints allowance and zero ProPoints values fruit and veg.<ref>[http://www.weightwatchers.ie Ireland]</ref><ref>[http://www.weightwatchers.com.au Australia]</ref> In Australia, Weight Watchers also offers free trials for ProPoints online program to attract customers. <ref>[http://www.topbargains.com.au/forum/weight-watchers-online-free-14-day-trial-new-propoints-program Weight Watchers Online 14 Days FREE Trial]</ref>

==Products==
In addition to Weight Watchers membership plans, other products (such as packaged foods, exercise equipment and DVDs, food preparation and storage tools, cookbooks, etc.) are available for purchase. In 2012, the company partnered with [[Philips]] to create the ActiveLink [[activity tracker]].<ref name="Dolan">{{cite web|url=http://mobihealthnews.com/18687/weight-watchers-taps-philips-for-activity-tracking/|title=Weight Watchers taps Philips for activity tracking|last=Dolan|first=Brian|date=October 11, 2012|work=mobihealthnews|accessdate=December 8, 2013}}</ref><ref name="ActiveLink press release">{{cite web|url=http://www.newscenter.philips.com/us_en/standard/news/press/2012/20121112_weightwatchers_activelink.wpd|title=Weight Watchers Partners with Philips to Launch ActiveLink™|first=November 12, 2012|accessdate=December 8, 2013}}</ref>

==Effectiveness==
Weight Watchers claims that “Research shows that people who attend Weight Watchers meetings lose three times more weight than [those who diet on their own].”<ref name="WWClaim1.4">
{{cite web
|url= http://www.weightwatchers.com/templates/print.aspx?PageId=1064291&PrintFlag=yes&previewDate=6/7/2008
|title= The Benefits of a Supportive Environment
|accessdate= 2008-06-07
|author= The Weight Watchers Research Dept.
|date=
|year= 2008
|month=
|work=
|publisher= Weight Watchers International
|doi=
|archiveurl=
|archivedate=
|quote= Research has found that those who attend Weight Watchers meetings lose three times more weight than those who try to lose weight on their own.}}</ref><ref name="WWClaim1.3">
{{cite web
|url= http://www.weightwatchers.com/international/bahamas/meetings/
|title= Weight Watchers - Bahamas
|accessdate= 2008-06-07
|date=
|year= 2007
|month=
|work=
|publisher= Weight Watchers International
|doi=
|archiveurl=
|archivedate=
|quote= Research shows that people who attend Weight Watchers meetings lose three times more weight than those who diet on their own.
}}
</ref><ref name="WWClaim1.2">
{{cite web
|url= http://www.weightwatchers.com/images/1033/dynamic/GCMSImages/WW_EE_LMV.pdf
|title= Open Enrollment
|accessdate= 2008-06-07
|last= Broughton
|first= Sherri
|date=July 2006
|format= PDF
|work=
|publisher= Weight Watchers International
|page= 1
|doi=
|archiveurl=
|archivedate=
|quote= Research shows that people who attend Weight Watchers meetings lose three times more weight than people dieting on their own<nowiki>[.]</nowiki>
}}
</ref><ref name="WWClaim1.1">
{{cite web
|url= http://www.weightwatchers.com/images/1033/dynamic/GCMSImages/Fact_Sheet_for_employers.pdf
|title= Sell Sheet
|accessdate= 2008-06-07
|author=
|last=
|first=
|authorlink=
|coauthors=
|date=October 2005
|format= PDF
|work=
|publisher= Weight Watchers International
|page= 1
|doi=
|archiveurl=
|archivedate=
|quote= It’s been proven that attending meetings leads to successful weight loss. Research shows that people who attend Weight Watchers meetings lose three times more weight than people dieting on their own.
}}
</ref> Specifically, the results of the study were that the mean weight loss of Weight Watchers participants was 2.3 times more than (3.3 times as much as) the self-help group at one year, and essentially undefined at two years.<ref name="Heshka 2003">{{cite journal
| author = Heshka, Stanley
| authorlink =
| coauthors = Anderson, James W.; Atkinson, Richard L.; Greenway, Frank L.; Hill, James O.; Phinney, Stephen D.; Kolotkin, Ronette L.; Miller-Kovach, Karen; Pi-Sunyer, F. Xavier
| date = 2003-04-09
| title = Weight Loss With Self-help Compared With a Structured Commercial Program: A Randomized Trial
| journal = JAMA: the Journal of the American Medical Association
| volume = 289
| issue = 14
| pages = 1792–1798
| publisher = American Medical Association
| location =
| pmid =12684357
| pmc =
| doi = 10.1001/jama.289.14.1792
| bibcode =
| oclc =
| id =
| url = http://jama.ama-assn.org/cgi/content/full/289/14/1792
| accessdate = 2008-06-06
| laysummary =
| laysource =
| laydate =
| quote = ...[M]ean (SD) weight loss... in the commercial group was greater than in the self-help group at 1 year (-4.3 [6.1] kg vs -1.3 [6.1] kg, respectively...) and at 2 years (-2.9 [6.5] kg vs -0.2 [6.5] kg, respectively...).
}}</ref> Weight Watchers also claims that members who both use Weight Watchers’ Web-based eTools and attend meetings lose half again as much weight as those who only attend meetings, but it fails to cite a study to back up this claim, instead only referring to an unspecified “12 week study comparing people who were instructed to attend Weight Watchers meetings and use eTools to people who were instructed to attend Weight Watchers meetings alone”.<ref name="WWClaim2.1">
{{cite web
|url= http://www.weightwatchers.com/plan/www/etools_01.aspx
|title= eTools, the Internet companion
|accessdate= 2008-06-07
|author=
|date=
|year= 2008
|month=
|work=
|publisher= Weight Watchers International
|doi=
|archiveurl=
|archivedate=
|quote= People who both attend Weight Watchers meetings and use eTools lose over 50% more weight on average than those who attend weekly meetings alone. ...Weight loss data based on 12 week study comparing people who were instructed to attend Weight Watchers meetings and use eTools to people who were instructed to attend Weight Watchers meetings alone.
}}
</ref>

A clinical study involving [[Atkins Nutritional Approach|Atkins]], [[Dean Ornish|Ornish]], Weight Watchers and [[Zone diet|The Zone]] diets, published in 2005, reported that among the Weight Watchers participants the average net weight loss in a one-year period was {{convert|3.0|kg}}.{{citation needed|date=November 2008}} However, the study only included two months of “maximum effort”, letting the participants decide their level of adherence for the following ten months.{{citation needed|date=November 2008}} Weight Watchers was the third most effective diet in terms of weight loss, and those that continued to adhere to any of the diets significantly decreased cardiac risk factors.<ref name="Dansinger 2005">{{cite journal
| author = Dansinger, Michael L.
| authorlink =
| coauthors = Gleason, Joi Augustin; Griffith, John L.; Selker, Harry P.; Schaefer, Ernst J.
| date = 2005-01-05
| title = Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Risk Reduction: A Randomized Trial
| journal = JAMA: the Journal of the American Medical Association
| volume = 293
| issue = 1
| pages = 43–53
| publisher = American Medical Association
| location =
| pmid =15632335
| pmc =
| doi = 10.1001/jama.293.1.43
| bibcode =
| oclc =
| id =
| url = http://jama.ama-assn.org/cgi/content/full/293/1/43
| accessdate = 2008-06-07
| laysummary =
| laysource =
| laydate =
| quote =
}}</ref>

==Spokespersons==
The current spokesperson for Weight Watchers is Grammy/Academy-Award winner singer/actress [[Jennifer Hudson]].<ref>[http://www.weightwatchers.com/jenniferhudson Jennifer's Weight Watchers Story] Retrieved 2010-04-01.</ref><ref>[http://abcnews.go.com/GMA/Entertainment/jennifer-hudson-weight-watchers-spokeswoman/story?id=10253977 "Jennifer Hudson is New Weight Watchers Spokeswoman - ABC News"] Retrieved 2010-04-01.</ref>
The newest spokesperson to sign with Weight Watchers is [[Jessica Simpson]]. Prior spokespersons associated with the product were actresses [[Lynn Redgrave]] and [[Jenny McCarthy]], and Duchess of York [[Sarah, Duchess of York|Sarah Ferguson]].<ref>{{cite journal|last=Hendley|first=Joyce|title=Weight Watchers at Forty: A Celebration|journal=Gastronomica: The Journal of Food and Culture|pages=Abstract|url=http://www.jstor.org/discover/10.1525/gfc.2003.3.1.16?uid=2&uid=4&sid=21101490393911}}</ref>

==Corporate information==
===Corporate history===
From 1978 until 1999, Weight Watchers was owned by the [[H. J. Heinz Company]], which continues to produce packaged foods bearing the Weight Watchers brand (and with point values clearly identified). Weight Watchers was acquired in a [[leveraged buyout]] in 1999 by [[Artal Luxembourg]] and [[initial public offering|went public]] in 2001. Artal continues to own over 50% of the shares of Weight Watchers as of 2010.

===Corporate governance===
The {{As of|2007|alt=current}} members of the [[board of directors]] of Weight Watchers are Philippe Amouyal, John Bard, Raymond Debbane, Marsha Evans, Jonas Fajgenbaum, Linda Huett, Sacha Lainovic, Steven M. Altschuler and Christopher Sobecki.

== Formulas ==
Weight Watchers has not provided official confirmation of the Points or PointsPlus formulas and has aggressively sent cease and desist letters to websites and a number of third party tools that claimed to provide Points, or PointsPlus, calculations.<ref>{{cite web |url=http://www.chillingeffects.org/derivative/notice.cgi?NoticeID=162 |title=Chilling Effects Notice |accessdate=27 December 2010}}</ref> Nonetheless, some companies such as [[ConAgra Foods]] have computed values for their frozen meals, [[Healthy Choice]] brand, without Weight Watchers' consent.

Additionally, as shown in Weight Watchers patent applications, the formulas are adapted in each country based on the information regularly provided on nutrition labels within a country.

For each program there should be three groups of formulas: a formula or chart to compute (daily) target consumption, a formula for computing the cost of foods, and a formula for computing the value of activity under the program to earn extra consumption. These formulas are present in Weight Watchers patents and are presented from those patents here.

=== PointsPlus (US; Nov 2010-) ===
==== Food Values ====
In November 2010, Weight Watchers in the United States announced that its Points system was being replaced by a new PointsPlus system. The old system was based on a calculation based on calories, fat, and fiber content. The new PointsPlus system is based on protein, carbohydrates, fat and fiber:

:<math>\text{PointsPlus} = \max \left\{ \mathrm{round} \left( \frac{(16 \cdot \text{protein}) + (19 \cdot \text{carbohydrates}) + (45 \cdot \text{fat}) - (14 \cdot \text{fiber})}{175}\right) , 0 \right\} </math>

:<math>\text{PointsPlus} =\max \left\{ \mathrm{round} \left( \frac{\text{protein}}{10.9375} + \frac{\text{carbohydrates}}{9.2105} + \frac{\text{fat}}{3.8889} - \frac{\text{fiber}}{12.5} \right), 0 \right\}</math>

where all units are grams. This formula is Equation 10 of <ref name="2010patent">{{cite patent
|country=WO
|number=2010025422
|status= application
}}</ref> with the preferred weightings for the macronutrients entered. Additionally, while not confirmed by Weight Watchers it is an empirical match to the Weight Watchers calculators.

Some foods such as fruits and fresh vegetables are treated has having a PointsPlus value of 0 by Weight Watchers standards<ref name="2010patent" /><ref>{{cite news |publisher=NY Times |date=2010-12-03 |title=Weight Watchers Upends Its Points System |url=http://www.nytimes.com/2010/12/04/nyregion/04watchers.html}}</ref> despite having a non-zero value if entered directly into the Weight Watcher's calculator using the [[Nutrition facts label]] information.

Alcohol and sugar alcohols (sorbitol, malitol, etc.) are exceptions to the above formula. See <ref name="2010patent" /> at Equation 12 (paragraph [0090]) and Equation 14 (paragraph [0094]), respectively for specific formulas for those items. As a shorthand, you can enter alcohol grams as fat grams and sugar alcohols as carbohydrates grams in the formula provided. The reason alcohol is weighted more heavily than would otherwise occur is due to two reasons, see <ref name="2010patent" /> at paragraph [0089], (a) "based on the principle that alcohol is metabolized in the same pathway as fat" (treats alcohol as having 9 calories per gram) and (b) "based, in part, on the desirability of discouraging consumption of alcohol, due to the ill-healthy effects associated with this nutrient" (weights alcohol more than other sources of calories). This latter explanation is in spite of evidence linking moderate alcohol consumption to reduced incidence of coronary disease.<ref>{{Cite news|publisher=BBC News|url=http://news.bbc.co.uk/1/hi/health/2638399.stm|title=Frequent tipple cuts heart risk|date=2008-01-09}}</ref> This may have been done because of the inelastic nature of the point system, which is unable to distinguish between occasional and heavy drinking.

===== Food with Alcohol and Sugar Alcohol =====
:<math>\text{PointsPlus} = \max \left\{ \mathrm{round} \left( \frac{(16 \cdot \text{protein}) + (19 \cdot (\text{carbohydrates} - \text{sugar alcohol})) + (45 \cdot \text{fat}) - (14 \cdot \text{fiber}) +(58.05 \cdot \text{alcohol}) + (11.4 \cdot \text{sugar alcohol})}{175} \right) , 0 \right\} </math>

or after simplifying so sugar alcohols which are included in the carbohydrate count on US nutrition labels only appears once:

:<math>\text{PointsPlus} =\max \left\{ \mathrm{round} \left( \frac{\text{protein}}{10.9375} + \frac{\text{carbohydrates}}{9.2105} + \frac{\text{fat}}{3.8889} - \frac{\text{fiber}}{12.5} + \frac{\text{alcohol}}{3.0147} - \frac{\text{sugar alcohol}}{23.0263} \right), 0 \right\}</math>

==== Daily Points Target ====
For 2010 PointsPlus, the formulas in Equations 16-19 of <ref name="2010patent" /> show the basic calculation of daily points targets for adults age 19+:

:<math> TEE_m = 864 - (9.72 \cdot age) + 1.12 \cdot (14.2 \cdot weight + 503 \cdot height)</math>

:<math> TEE_f = 387 - (7.31 \cdot age) + 1.14 \cdot (10.9 \cdot weight + 660.7 \cdot height)</math>

Where <math>TEE_m</math> and <math>TEE_f</math> represent the total energy expenditure (in kCal) for men and women, respectively, on a basic day. Weight is provided in kilograms and height in meters.

The <math>TEE</math> is adjusted for foods treated as having a PointsPlus value of 0 despite having a caloric content:

:<math> ATEE =0.9 \cdot TEE + 200 </math>

Finally, the daily target can be computed:

:<math> Target = \mathrm{round} \left( \frac{\min \left( \max \left( ATEE - 1000, 1000 \right), 2500 \right) }{35} \right) </math>

Note that the patent also suggests that 3000 might be the upper kCal limit as opposed to 2500 kCal and 500 might be the lower limit rather than 1000 kCal. The results of these patent-provided formulas produce a range from 29 to 71 matching the high and low range reported by Weight Watchers members.<ref>{{cite web |url=http://www.weightwatchers.com/community/mbd/post.aspx?page_size=25&rownum=1&threadpage_no=1&sincedate=12/24/2010%2012:00:00%20AM&thread_id=147816280&board_id=360&forum_id=1&thread_name=Maintenance%20Points/Lifetime%20Member?&mod_no=&daterange=2days&viewchange=OPENDATEDESC |title=Forum Posting 1 re New Plan |accessdate=2010-12-30 |date=2010-12-08}}</ref><ref>{{cite web |title=Forum Posting 2 re New Plan |accessdate=2010-12-30 |date=2010-12-04 |url=http://www3.weightwatchers.com/community/mbd/post.aspx?threadpage_no=1&page_size=25&rownum=5&board_name=Guys+on+a+Diet&thread_id=147701320&board_id=15&forum_name=message+boards&post_id=1265677780&forum_id=1&thread_name=Just+got+back+from+my+meeting+and...&mod_no=&sincedate=11/9/2010+12:00:00+AM&viewchange=DATECREATEDDESC&gotolast=true}}</ref>

The above would not account for the 49 weekly points, or 7/day.<ref>
{{cite web
|url=http://www.weightwatchers.com/util/art/index_art.aspx?tabnum=1&art_id=48151
|title=I’ve eaten all my PointsPlusvalues, and it’s still only lunchtime!
|accessdate=2010-12-30}}
</ref> Nor would it account for comments on the Weight Watchers boards discussing the patent and identifying an additional constant offset from the patent by a value of 11 in the middle of the points range, or 7 + 4.<ref>
{{cite web
|url=http://www.weightwatchers.com/community/mbd/post.aspx?threadpage_no=12&page_size=25&rownum=300&board_name=Guys+on+a+Diet&thread_id=147761410&board_id=15&forum_name=message+boards&forum_id=1&thread_name=Deep+thoughts,+by+Jack+Handey+(Steve0mania)+(long)&mod_no=&sincedate=12/2/2010+12:00:00+AM&viewchange=DATECREATEDDESC
|title=Forum Posting 3 Re New Plan
|accessdate=2010-12-30
|date=2010-12-06
}}</ref><ref>{{cite web |url=http://www.weightwatchers.com/community/mbd/post.aspx?page_size=25&rownum=15&threadpage_no=1&sincedate=1/18/2011+12:00:00+AM&thread_id=148946590&board_id=360&forum_id=1&thread_name=Points+Plus+Target+Quiz%3f&mod_no=&daterange=2days&viewchange=OPENDATEDESC |title=Forum Posting Validating Wikipedia Formula |date=2011-01-19 |accessdate=2011-01-19 }}</ref> To accommodate 2012 plan changes and in order to remain in the 26-71 range and account for this constant offset, the target formula would be:

:<math>Target_{mod} = \min \left (\max \left( \mathrm{round} \left( \frac{ \max \left( ATEE - 1000, 1000 \right) }{35} \right) - 7 - 4, 26 \right), 71 \right) </math>

==== Activity Points Under PointsPlus ====
The prior system described in {{US patent|6663564}} at Col 6, lines 14-61, provided one activity point for approximately every 100 Calories expended vs. 1 Point of food having approximately 50 calories. The formulas used under the PointsPlus system must still be identified; however, if the same ratio of exercise to additional consumption is maintained, then each new activity point would correspond to 70 Calories expended vs. 1 PointPlus of food having approximately 35 calories.

=== ProPoints (UK, Nov 2010-) ===
Aside from differences in nutrition labeling resulting in a slightly different calculation of ProPoints from the label, there is no evidence to suggest the ProPoints system and PointsPlus systems are significantly different as of December 2010.

==== Food Values ====
In the UK during November 2010, Weight Watchers announced that its Points system was being replaced by a new ProPoints system. The new PointsPlus system is based on protein, carbohydrates, fat and fiber:

:<math>ProPoints = \max \left\{ \mathrm{round} \left( \frac{(16 \cdot protein) + (19 \cdot carbohydrates) + (45 \cdot fat) + (5 \cdot fiber)}{175}\right), 0 \right\}</math>

or

:<math>ProPoints = \max \left\{ \mathrm{round} \left( \frac{protein}{10.9375} + \frac{carbohydrates}{9.2105} + \frac{fat}{3.8889} + \frac{fiber}{35} \right) , 0 \right\} </math>

where all input units are grams. This formula is based on Equation 10 of <ref name="2010patent" /> and taking into account that the carbohydrate line on nutrition labels in the UK do not count the fiber. The formula also has been empirically compared with a third party ProPoints calculator.<ref>[http://www.wijvallenaf.nl/Weight-Watchers/ProPoints-Calculator.html ProPoints Calculator from 3rd Party Website] Retrieved 2010-12-02.</ref>

=== Points (US; Circa 1998-2010) ===
==== Food Values ====
The formula for calculating the value of a specific serving in points is described in {{US patent|6040531}}:

:<math>p(c,f,r) = \mathrm{round} \left( \frac{c}{50} + \frac{f}{12} - \frac{\min\{r,4\}}{5} \right) </math>

where <math>p</math> is the number of points, <math>c</math> is the amount of [[energy]] in [[Calorie]]s (1 Calorie (Cal) ≡ 1 kilocalorie (kcal) ≈ 4.2 kilojoules (kJ)), <math>f</math> is the amount of [[fat]] in [[gram]]s, and min{<math>r</math>,4} is the lesser of the amount of [[dietary fiber]] in grams or four.<ref>
{{cite web
|url= http://www.alanlight.com/WWPoints.htm
|title= Weight Watchers Points Formula
|accessdate= 2008-07-29
|author=
|last=
|first=
|authorlink=
|coauthors=
|date=October 2005
|work=
|publisher=
|pages=
|doi=
|archiveurl=
|archivedate=
|quote=
}}
</ref> The value is always an [[integer]], with [[Fraction (mathematics)|fractional]] values rounded to the nearest point. An alternative version mentioned in the patent and used on some Web sites rounds values to the nearest half-point.

==== Daily Targets ====
===== Patent Based =====
Various versions of the prior Points-based programs have used different approaches to calculate the daily target points (and then providing 35 weekly points). Keep in mind the new PointsPlus program is different, see [[Weight Watchers#Daily Points Target|Daily Points Target]] (49 weekly points provided on PointsPlus plan).

According to {{US patent|6040531}} at Col. 5, lines 4-16, the points targets were assigned based on weight bands and ranged from 18-35 Points per day.

===== Survey Based =====
The above, patent-provided daily Points targets are not well correlated with the energy requirements of typical males or females. Accordingly, in one of the later incarnations of the Points-based plans (there were several: 1-2-3 Success, Winning Points, Flex, Turnaround, Momentum) Weight Watchers switched to a survey-based approach that provided a range from 18-44 points<ref name="pocketguide2010">{{cite book |title=WeightWatchers Pocket Guide 2010}}</ref> together with 35 weekly points:<ref name="pocketguide2010" />

::10% of your weight in pounds
:::;PLUS
::2 points for females
::8 points for males
:::;PLUS
::4 points if you're between 17-26 years old
::3 points if you're between 27-37 years old
::2 points if you're between 38-47 years old
::1 point if you're between 48-58 years old
::0 points if you're over 58
:::;PLUS
::0 points if you're under 5' 1"
::1 point if you're between 5' 1" and 5' 10"
::2 points if you're over 5' 10"
:::;PLUS
::0 points if you spend most of your day sitting down
::2 point if you spend most of your day standing
::4 points if you are walking most of the time
::6 points if you are doing physically hard work
:::;PLUS
::10 points for mothers who are solely breast feeding
::5 points for supplementing breastfeeding with some solid foods or formula

The sum (bounded below by 18 and at the top by 44) is the daily target.

==== Activity Points ====
According to {{US patent|6663564}} at Col 6, lines 14-61, the formula for computing the additional consumption (extra Points) available for activity is described. Analysis of the formulas indicates<ref>{{cite web |url=http://b.feli.me/post/5248716882/old-weight-watchers-activity-points-formula |title=Old Weight Watchers Activity Points Formula |accessdate=2011-07-06 }}</ref> that 1 point for activity is earned for approximately every 100 Calories of effort. Given that 1 PointsPlus could be approximated as 40 Calories, consuming activity points would still create a calorie deficit.

==== Food Values Outside the United States ====
Outside the United States, [[Nutrition facts label]]s are markedly different from those in the United States; in particular, energy is usually expressed in standard kilojoules (kJ) rather than Calories (kcal), and labels in some countries may not show fiber content. The points formula for some markets is based on energy (in kilojoules) and ''saturated'' (not ''total'') fat. This formula is also described in {{US patent|6040531}} at Col 3, lines 10-33 and Col 7, lines 10-20:

:<math>p = \frac{e}{k_1} + \frac{f}{k_2}</math>

Where <math>p</math> is the number of Points, <math>e</math> is the energy value (in either kilojoules (kJ) or Calories (kcal)), and <math>f</math> is the amount of saturated fat in grams. The constants <math>k_1</math> and <math>k_2</math> are weights used in the formula. A practical implementation of this formula for a kilojoule-based calculation appears to be as follows:

:<math>p = \frac{e}{290} + \frac{f}{4}</math>

or the following for a Calorie-based calculation:

:<math>p = \frac{e}{70} + \frac{f}{4}</math>

The resultant value <math>p</math> is then rounded to the nearest half.

Note, the calorie based-formula is provided directly in Col 7, the kJ formula is a close match to the calorie based formula based on 70 kCal = 292.88 kJ.

====Food Values, Alternative Calculation====
An early version of the points system did not limit the fiber “credit” to four grams. Another variation, which may be explained by rounding, is that the “points finder” slide-rule distributed at Weight Watchers meetings does not reach a value of 1 unless an item has at least {{convert|70|Cal|sigfig=1|lk=on}}, rather than {{convert|50|Cal|sigfig=1|abbr=on}} as the formula indicates. As a result, point boundaries are {{convert|20|Cal|sigfig=1}} or 4.8 grams of fat higher than might be expected.

More recent versions of the program, such as that described in {{US patent|6878885}}, take exercise and physical activity into account to grant additional points in the daily allowance.

=== Calculation Aides ===
Members can either use Weight Watchers' published books with the PointsPlus values (or ProPoints values) for a variety of foods or they can use a Weight Watchers calculator designed to accept the inputs required to compute PointsPlus values. The calculators sell for approximately $12.

Weight Watchers also sells small paperback books that list thousands of foods sold by fast-food and full-service restaurant chains as well as generic restaurant foods, e.g., “Mexican: refried beans, ½ cup” (120 mL), and corresponding point values. This is useful in cases where precise nutritional data are not available. These books are available for purchase at meetings and to online members via the Weightwatchers.com website.

==== Former Calculation Aides ====
Prior to PointsPlus, members were given a cardboard slide rule to assist in calculating point values of foods. Because the new program has more inputs, the slide rule approach is no longer feasible. By reading the nutrition label on any food package and quickly adjusting three slide-rule scales for energy, fat, and fiber, the point value was determined by inspection of a fourth scale. Weight Watchers also sold an electronic calculator that performed the same functions.

==See also==
* [[American Dietetic Association]]

==References==
{{reflist|colwidth=30em}}

==External links==
* [http://www.weightwatchers.com Weight Watchers]
{{Heinz}}

[[Category:Companies listed on the New York Stock Exchange]]
[[Category:Brand name diet products]]
[[Category:Companies established in 1963]]
[[Category:Franchises]]

Elias M. Stein

2014-01-25T15:53:05Z

130.88.141.34: Barack Obama

[[File:Strain gauge.svg|thumb|Typical foil strain gauge. The gauge is far more sensitive to strain in the vertical direction than in the horizontal direction. The markings outside the active area help to align the gauge during installation.]]

A '''strain gauge''' (also '''strain gage''') is a device used to measure [[Deformation (mechanics)#Strain|strain]] on an object. Invented by [[Edward E. Simmons]] and [[Arthur C. Ruge]] in 1938, the most common type of strain gauge consists of an [[Electrical insulation|insulating]] flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable adhesive, such as [[cyanoacrylate]].<ref>[http://www.efunda.com/designstandards/sensors/strain_gages/strain_gage_selection_matl.cfm Strain Gage: Materials]</ref> As the object is deformed, the foil is deformed, causing its [[electrical resistance]] to change. This resistance change, usually measured using a [[Wheatstone bridge]], is related to the strain by the quantity known as the ''[[gauge factor]]''.

==Physical operation==
A strain gauge takes advantage of the physical property of [[electrical conductance]] and its dependence on the conductor's geometry. When an [[electrical conductor]] is stretched within the limits of its [[elasticity (physics)|elasticity]] such that it does not break or permanently deform, it will become narrower and longer, changes that increase its electrical resistance end-to-end. Conversely, when a conductor is compressed such that it does not buckle, it will broaden and shorten, changes that decrease its electrical resistance end-to-end. From the measured [[electrical resistance]] of the strain gauge, the amount of applied [[stress (physics)|stress]] may be inferred. A typical strain gauge arranges a long, thin conductive strip in a zig-zag pattern of parallel lines such that a small amount of stress in the direction of the orientation of the parallel lines results in a multiplicatively larger [[Strain (materials science)|strain measurement]] over the effective length of the conductor surfaces in the array of conductive lines—and hence a multiplicatively larger change in resistance—than would be observed with a single straight-line conductive wire.

==Gauge factor==
The [[gauge factor]] <math>GF</math> is defined as:
:<math>GF=\frac{\Delta R/R_G}{\epsilon}</math>
where
:<math>\Delta R</math> is the change in resistance caused by strain,
:<math>R_G</math> is the resistance of the undeformed gauge, and
:<math>\epsilon</math> is strain.

For metallic foil gauges, the gauge factor is usually a little over 2.<ref name="Strain Gage: Sensitivity">[http://www.efunda.com/designstandards/sensors/strain_gages/strain_gage_sensitivity.cfm Strain Gage: Sensitivity]</ref> For a single active gauge and three dummy resistors, the output <math>v</math> from the bridge is:
:<math>v=\frac{BV \cdot GF \cdot \epsilon}4</math>
where
:<math>BV</math> is the bridge excitation voltage.

Foil gauges typically have active areas of about 2–10 mm2 in size. With careful installation, the correct gauge, and the correct [[adhesive]], strains up to at least 10% can be measured.

==Gauges in practice==

[[Image:StrainGaugeVisualization.svg|400px|frame|Visualization of the working concept behind the strain gauge on a [[Beam (structure)|beam]] under exaggerated bending.]]

An excitation voltage is applied to input leads of the gauge network, and a voltage reading is taken from the output leads. Typical input voltages are 5 V or 12 V and typical output readings are in millivolts.

Foil strain gauges are used in many situations. Different applications place different requirements on the gauge. In most cases the orientation of the strain gauge is significant.

Gauges attached to a [[load cell]] would normally be expected to remain stable over a period of years, if not decades; while those used to measure response in a dynamic experiment may only need to remain attached to the object for a few days, be energized for less than an hour, and operate for less than a second.

Strain gauges are attached to the substrate with a special glue. The type of glue depends on the required lifetime of the measurement system. For short term measurements (up to some weeks) cyanoacrylic glue is appropriate, for long lasting installation epoxy glue is required. Usually epoxy glue requires high temperature curing (at about 80-100°C). The preparation of the surface where the strain gauge is to be glued is of the utmost importance. The surface must be smoothed (e.g. with very fine sand paper), deoiled with solvents, the solvent traces must then be removed and the strain gauge must be glued immediately after this to avoid oxidation or pollution of the prepared area. If these steps are not followed the strain gauge binding to the surface may be unreliable and unpredictable measurement errors may be generated.

Strain gauge based technology is utilized commonly in the manufacture of [[pressure sensor]]s. The gauges used in pressure sensors themselves are commonly made from silicon, polysilicon, metal film, thick film, and bonded foil.

===Variations in temperature===
Variations in temperature will cause a multitude of effects. The object will change in size by thermal expansion, which will be detected as a strain by the gauge. Resistance of the gauge will change, and resistance of the connecting wires will change.

Most strain gauges are made from a [[constantan]] alloy.<ref>[http://www.vishaypg.com/docs/11055/tn505.pdf Constantan Alloy: Strain Gauge Selection]</ref> Various constantan alloys and Karma alloys have been designed so that the temperature effects on the resistance of the strain gauge itself cancel out the resistance change of the gauge due to the thermal expansion of the object under test. Because different materials have different amounts of thermal expansion, self-temperature compensation (STC) requires selecting a particular alloy matched to the material of the object under test.

Strain gauges that are not self-temperature-compensated (such as isoelastic alloy) can be temperature compensated by use of the dummy gauge technique. A dummy gauge (identical to the active strain gauge) is installed on an unstrained sample of the same material as the test specimen. The sample with the dummy gauge is placed in thermal contact with the test specimen, adjacent to the active gauge. The dummy gauge is wired into a [[Wheatstone bridge]] on an adjacent arm to the active gauge so that the temperature effects on the active and dummy gauges cancel each other.<ref>Shull, Larry C., "Basic Circuits", Hannah, R.L. and Reed, S.E. (Eds.) (1992).''Strain Gage Users' Manual'', p. 122. Society for Experimental Mechanics. ISBN 0-912053-36-4. </ref> ([[Murphy's Law]] was originally coined in response to a set of gauges being incorrectly wired into a Wheatstone bridge.<ref>Spark, N. (2006). ''A History of Murphy's Law''. Periscope Film. ISBN 978-0-9786388-9-4</ref>)

Temperature effects on the lead wires can be cancelled by using a "3-wire bridge" or a "4-wire ohm circuit"<ref>[http://www.omega.com/literature/transactions/volume3/strain2.html The Strain Gage]</ref> (also called a "4-wire [[Kelvin connection]]").

In any case it is a good engineering practice to keep the Wheatstone bridge voltage drive low enough to avoid the self heating of the strain gauge. The self heating of the strain gauge depends on its mechanical characteristic (large strain gauges are less prone to self heating). Low voltage drive levels of the bridge reduce the sensitivity of the overall system.

==Errors and compensation==
Zero Offset - If the impedance of the four gauge arms are not exactly the same after bonding the gauge to the force collector, there will be a zero offset which can be compensated by introducing a parallel resistor to one or more of the gauge arms.

Temperature coefficient of Gauge Factor (TCGF) - This is the change of sensitivity of the device to strain with change in temperature. This is generally compensated for by the introduction of a fixed resistance in the input leg, whereby the effective supplied voltage will increase with temperature, compensating for the decrease in sensitivity with temperature.

Zero Shift with temperature - If the TCGF of each gauge is not the same, there will be a zero shift with temperature. This is also caused by anomalies in the force collector. This is usually compensated for with one or more resistors strategically placed in the compensation network.

Linearity - This is an error whereby the sensitivity changes across the pressure range. This is commonly a function of the force collection thickness selection for the intended pressure and/or the quality of the bonding.

Hysteresis - This is an error of return to zero after pressure excursion.

Repeatability - This error is sometimes tied-in with hysteresis but is across the pressure range.

EMI induced errors - As strain gauges output voltage is in the mV range, even μV if the Wheatstone bridge voltage drive is kept low to avoid self heating of the element, special care must be taken in output signal amplification to avoid amplifying also the superimposed noise. A solution which is frequently adopted is to use "carrier frequency" amplifiers which convert the voltage variation into a frequency variation (as in VCOs) and have a narrow bandwidth thus reducing out of band EMI.

Overloading - If a strain gauge is loaded beyond its design limit (measured in microstrain) its performance degrades and can not be recovered. Normally good engineering practice suggests not to stress strain gauges beyond +/-3000 microstrain.

Humidity - If the wires connecting the strain gauge to the signal conditioner are not protected against humidity (bare wire) a parasitic resistance creates between the wires and the substrate to which the strain gauge is glued, or between the two wires themselves. This resistance introduces an error which is proportional to the resistance of the strain gauge. For this reason low resistance strain gauges (120 ohm) are less prone to this type of error. To avoid this error it is sufficient to protect the strain gauges wires with insulating enamel (e.g., epoxy or polyurethanic type). Strain gauges with unprotected wires may be used only in a dry laboratory environment but not in an industrial one.

In some applications, strain gauges add mass and damping to the vibration profiles of the hardware they are intended to measure. In the turbomachinery industry, one used alternative to strain gauge technology in the measurement of vibrations on rotating hardware is the [[Non-Intrusive Stress Measurement System]], which allows measurement of blade vibrations without any blade or disc-mounted hardware.

==Other gauge types==
For measurements of small strain, [[semiconductor]] strain gauges, so called [[Piezoresistive effect|piezoresistors]], are often preferred over foil gauges. A [[piezoresistor|semiconductor gauge]] usually has a larger gauge factor than a foil gauge. Semiconductor gauges tend to be more expensive, more sensitive to temperature changes, and are more fragile than foil gauges.

In biological measurements, especially [[blood flow]] / tissue swelling, a variant called '''mercury-in-rubber strain gauge''' is used. This kind of strain gauge consists of a small amount of liquid mercury enclosed in a small rubber tube, which is applied around e.g., a toe or leg. Swelling of the body part results in stretching of the tube, making it both longer and thinner, which increases electrical resistance.

[[Fiber optic sensor|Fiber optic sensing]] can be employed to measure strain along an [[optical fiber]]. Measurements can be distributed along the fiber, or taken at predetermined points on the fiber. The 2010 [[America's Cup]] boats [[Alinghi 5]] and [[USA-17]] both employ embedded sensors of this type [http://www.nytimes.com/2010/02/09/science/09sail.html].

Capacitive strain gauges use a [[variable capacitor]] to indicate the level of mechanical deformation.

Vibrating wire strain gauges are used in Geotechnical and Civil Engineering applications. The gauge consists of a vibrating, tensioned wire. The strain is calculated by measuring the resonant frequency of the wire (an increase in tension increases the resonant frequency).

==Mechanical types==
[[Image:Hudson-Athens Strain Gauge.jpg|thumb|left|Mechanical strain gauge used to measure the growth of a crack in a masonry foundation. This one is installed on the [[Hudson-Athens Lighthouse]]]]
Simple mechanical types (such as illustrated to the left) are used in [[civil engineering]] to measure movement of buildings, foundations, and other structures. In the illustrated example, the two halves of the device are rigidly attached to the foundation wall on opposite sides of the crack. The red reference lines are on the transparent half and the grid is on the opaque white half. Both vertical and horizontal movement can be monitored over time. In this picture, the crack can be seen to have widened by approximately 0.3 mm (with no vertical movement) since the gauge was installed.

More sophisticated mechanical types incorporate [[dial indicator]]s and mechanisms to compensate for temperature changes. These types can measure movements as small as 0.002 mm.<ref>[http://www.mastrad.com/demecsg.htm Mastrad Quality and Test Systems web site]</ref>

==See also==
*[[Resistance thermometer#Wiring configurations|Resistance thermometer]]

==References==
{{reflist}}

==External links==
* British Society for Strain Measurement (http://www.bssm.org)
* [http://www.loadcelltheory.com/LoadCellSupportTheoryPDF/FEA_StrainGageDesign.pdf Applying Finite Element Analysis Methods to Strain Gage Design]
* [http://www.hbm.com/fileadmin/mediapool/techarticles/hoffmannbook/Hoffmann-book_EN.pdf An Introduction to Measurements using Strain Gages by Karl Hoffmann]
* [http://www.4fos.com Fiber optic strain gauges - NASA patented technology]

[[Category:Sensors]]
[[Category:Elasticity (physics)]]

London equations

2014-01-20T19:53:00Z

130.88.99.231: /* Formulations */

In [[mathematics]], the '''greedy algorithm for Egyptian fractions''' is a [[greedy algorithm]], first described by [[Fibonacci]], for transforming [[rational number]]s into [[Egyptian fraction]]s. An Egyptian fraction is a representation of an [[irreducible fraction]] as a sum of [[unit fraction]]s, as e.g. 5/6 = 1/2 + 1/3. As the name indicates, these representations have been used as long ago as [[Egyptian mathematics|ancient Egypt]], but the first published systematic method for constructing such expansions is described in the [[Liber Abaci]] ([[#{{harvid|Sigler|2002}}|1202]]) of [[Leonardo of Pisa]] (Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible [[unit fraction]] that can be used in any representation of the remaining fraction.

Fibonacci actually lists several different methods for constructing Egyptian fraction representations ({{harvnb|Sigler|2002}}, chapter II.7). He includes the greedy method as a last resort for situations when several simpler methods fail; see [[Egyptian fraction]] for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by {{harvs|authorlink=James Joseph Sylvester|last=Sylvester|first=J. J.|year=1880|txt}}; see for instance {{harvtxt|Cahen|1891}} and {{harvtxt|Spiess|1907}}. A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to {{harvtxt|Lambert|1770}}.

The expansion produced by this method for a number ''x'' is called the '''greedy Egyptian expansion''', '''Sylvester expansion''', or '''Fibonacci–Sylvester expansion''' of ''x''. However, the term ''Fibonacci expansion'' usually refers, not to this method, but to representation of integers as sums of [[Fibonacci number]]s.

==Algorithm and examples==
Fibonacci's algorithm expands the fraction ''x''/''y'' to be represented, by repeatedly performing the replacement
:<math>\frac{x}{y}=\frac{1}{\lceil y/x\rceil}+\frac{(-y)\bmod x}{y\lceil y/x\rceil}</math>
(simplifying the second term in this replacement as necessary). For instance:
:<math>\frac{7}{15}=\frac{1}{3}+\frac{2}{15}=\frac{1}{3}+\frac{1}{8}+\frac{1}{120}.</math>
in this expansion, the denominator 3 of the first unit fraction is the result of rounding 15/7 up to the next larger integer, and the remaining fraction 2/15 is the result of simplifying (-15 mod 7)/(15×3) = 6/45. The denominator of the second unit fraction, 8, is the result of rounding 15/2 up to the next larger integer, and the remaining fraction 1/120 is what is left from 7/15 after subtracting both 1/3 and 1/8.

As each expansion step reduces the numerator of the remaining fraction to be expanded, this method always terminates with a finite expansion; however, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators. For instance, this method expands
:<math>\frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{873960180913}+\frac{1}{1527612795642093418846225},</math>
while other methods lead to the much better expansion
:<math>\frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.</math>
{{harvtxt|Wagon|1991}} suggests an even more badly-behaved example, 31/311. The greedy method leads to an expansion with ten terms, the last of which has over 500 digits in its denominator; however, 31/311 has a much shorter non-greedy representation, 1/12 + 1/63 + 1/2799 + 1/8708.

==Sylvester's sequence and closest approximation==
[[Sylvester's sequence]] 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator <math>\lfloor y/x\rfloor+1</math> instead of <math>\lceil y/x\rceil</math>. Truncating this sequence to ''k'' terms and forming the corresponding Egyptian fraction, e.g. (for ''k'' = 4)
:<math>\frac12+\frac13+\frac17+\frac1{43}=\frac{1805}{1806}</math>
results in the closest possible underestimate of 1 by any ''k''-term Egyptian fraction ({{harvnb|Curtiss|1922}}; {{harvnb|Soundararajan|2005}}). That is, for example, any Egyptian fraction for a number in the open interval (1805/1806,1) requires at least five terms. {{harvtxt|Curtiss|1922}} describes an application of these closest-approximation results in lower-bounding the number of divisors of a [[perfect number]], while {{harvtxt|Stong|1983}} describes applications in [[group theory]].

==Maximum-length expansions and congruence conditions==
Any fraction ''x''/''y'' requires at most ''x'' terms in its greedy expansion. {{harvtxt|Mays|1987}} and {{harvtxt|Freitag|Phillips|1999}} examine the conditions under which ''x''/''y'' leads to an expansion with exactly ''x'' terms; these can be described in terms of congruence conditions on ''y''.

* Every fraction 1/''y'' requires one term in its expansion; the simplest such fraction is 1/1.

* Every fraction 2/''y'' for odd ''y'' > 1 requires two terms in its expansion; the simplest such fraction is 2/3.

* A fraction 3/''y'' requires three terms in its expansion if and only if ''y'' ≡ 1 (mod 6), for then -''y'' mod ''x'' = 2 and y(y+2)/3 is odd, so the fraction remaining after a single step of the greedy expansion,
::<math>\frac{(-y)\bmod x}{y\lceil y/x\rceil} = \frac2{y(y+2)/3}</math>
:is in simplest terms. The simplest fraction 3/''y'' with a three-term expansion is 3/7.

* A fraction 4/''y'' requires four terms in its expansion if and only if ''y'' ≡ 1 or 17 (mod 24), for then the numerator -''y'' mod ''x'' of the remaining fraction is 3 and the denominator is 1 (mod 6). The simplest fraction 4/''y'' with a four-term expansion is 4/17. The [[Erdős–Straus conjecture]] states that all fractions 4/''y'' have an expansion with three or fewer terms, but when ''y'' ≡ 1 or 17 (mod 24) such expansions must be found by methods other than the greedy algorithm.

More generally the sequence of fractions ''x''/''y'' that have ''x''-term expansions and that have the smallest possible denominator ''y'' for each ''x'' is
:<math>1, \frac{2}{3}, \frac{3}{7}, \frac{4}{17}, \frac{5}{31}, \frac{6}{109}, \frac{7}{253}, \frac{8}{97}, \frac{9}{271}, \dots</math> {{OEIS|id=A048860}}.

==Approximation of polynomial roots==
{{harvtxt|Stratemeyer|1930}} and {{harvtxt|Salzer|1947}} describe a method of finding an accurate [[Root-finding algorithm|approximation for the roots of a polynomial]] based on the greedy method. Their algorithm computes the greedy expansion of a root; at each step in this expansion it maintains an auxiliary polynomial that has as its root the remaining fraction to be expanded. Consider as an example applying this method to find the greedy expansion of the [[golden ratio]], one of the two solutions of the polynomial equation ''P''0(''x'') = ''x''2 - x - 1 = 0. The algorithm of Stratemeyer and Salzer performs the following sequence of steps:

#Since ''P''0(''x'') < 0 for ''x'' = 1, and ''P''0(''x'') > 0 for all ''x'' ≥ 2, there must be a root of ''P''0(''x'') between 1 and 2. That is, the first term of the greedy expansion of the golden ratio is 1/1. If ''x''1 is the remaining fraction after the first step of the greedy expansion, it satisfies the equation ''P''0(''x''1 + 1) = 0, which can be expanded as ''P''1(''x''1) = ''x''12 + ''x''1 - 1 = 0.
#Since ''P''1(''x'') < 0 for ''x'' = 1/2, and ''P''1(''x'') > 0 for all ''x'' > 1, the root of ''P''1 lies between 1/2 and 1, and the first term in its greedy expansion (the second term in the greedy expansion for the golden ratio) is 1/2. If ''x''2 is the remaining fraction after this step of the greedy expansion, it satisfies the equation ''P''1(''x''2 + 1/2) = 0, which can be expanded as ''P''2(''x''2) = 4''x''22 + 8''x''2 - 1 = 0.
#Since ''P''2(''x'') < 0 for ''x'' = 1/9, and ''P''2(''x'') > 0 for all ''x'' > 1/8, the next term in the greedy expansion is 1/9. If ''x''3 is the remaining fraction after this step of the greedy expansion, it satisfies the equation ''P''2(''x''3 + 1/9) = 0, which can again be expanded as a polynomial equation with integer coefficients, ''P''3(''x''3) = 324''x''32 + 720''x''3 - 5 = 0.

Continuing this approximation process eventually produces the greedy expansion for the golden ratio,

:<math>\varphi = \frac11+\frac12+\frac19+\frac1{145}+\frac1{37986}+\cdots</math> {{OEIS|id=A117116}}.

==Other integer sequences==
The length, minimum denominator, and maximum denominator of the greedy expansion for all fractions with small numerators and denominators can be found in the [[On-Line Encyclopedia of Integer Sequences]] as sequences {{OEIS2C|A050205}}, {{OEIS2C|A050206}}, and {{OEIS2C|A050210}}, respectively. In addition, the greedy expansion of any [[irrational number]] leads to an infinite increasing sequence of integers, and the OEIS contains [http://oeis.org/search?q=greedy-Egyptian-fraction-expansion expansions of several well known constants]. Some [http://oeis.org/search?q=Egyptian-fraction-for additional entries in the OEIS], though not labeled as being produced by the greedy algorithm, appear to be of the same type.

==Related expansions==
In general, if one wants an Egyptian fraction expansion in which the denominators are constrained in some way, it is possible to define a greedy algorithm in which at each step one chooses the expansion
:<math>\frac{x}{y}=\frac{1}{d}+\frac{xd-y}{yd},</math>
where ''d'' is chosen, among all possible values satisfying the constraints, as small as possible such that ''xd'' > ''y'' and such that ''d'' is distinct from all previously chosen denominators. For instance, the [[Engel expansion]] can be viewed as an algorithm of this type in which each successive denominator must be a multiple of the previous one. However, it may be difficult to determine whether an algorithm of this type can always succeed in finding a finite expansion. In particular, the [[odd greedy expansion]] of a fraction ''x''/''y'' is formed by a greedy algorithm of this type in which all denominators are constrained to be odd numbers; it is known that, whenever ''y'' is odd, there is a finite Egyptian fraction expansion in which all denominators are odd, but it is not known whether the odd greedy expansion is always finite.

== References ==
{{refbegin|colwidth=30em}}
*{{citation
| last = Cahen | first = E.
| journal = Nouvelles Annales des Mathématiques | series = Ser. 3
| pages = 508–514
| title = Note sur un développement des quantités numériques, qui presente quelque analogie avec celui en fractions continues
| volume = 10
| year = 1891}}.
*{{citation
| last = Curtiss | first = D. R. | authorlink = David Raymond Curtiss
| doi = 10.2307/2299023
| issue = 10
| journal = [[American Mathematical Monthly]]
| jstor = 2299023
| pages = 380–387
| title = On Kellogg's diophantine problem
| volume = 29
| year = 1922}}.
*{{citation
| last1 = Freitag | first1 = H. T. | author1-link = Herta Freitag
| last2 = Phillips | first2 = G. M.
| contribution = Sylvester's algorithm and Fibonacci numbers
| location = Dordrecht
| mr = 1737669
| pages = 155–163
| publisher = Kluwer Acad. Publ.
| title = Applications of Fibonacci numbers, Vol. 8 (Rochester, NY, 1998)
| year = 1999}}.
*{{citation
| last = Lambert | first = J. H. | author-link = Johann Heinrich Lambert
| location = Berlin
| pages = 99–104
| publisher = Zweyter Theil
| title = Beyträge zum Gebrauche der Mathematik und deren Anwendung
| year = 1770}}.
*{{citation
| last = Mays | first = Michael
| journal = Journal of Combinatorial Mathematics and Combinatorial Computing
| mr = 0888838
| pages = 141–148
| title = A worst case of the Fibonacci–Sylvester expansion
| volume = 1
| year = 1987}}.
*{{citation
| last = Salzer | first = H. E.
| doi = 10.2307/2305906
| issue = 3
| journal = [[American Mathematical Monthly]]
| jstor = 2305906
| mr = 0020339
| pages = 135–142
| title = The approximation of numbers as sums of reciprocals
| volume = 54
| year = 1947}}.
*{{citation
| last = Salzer | first = H. E.
| doi = 10.2307/2304960
| issue = 6
| journal = [[American Mathematical Monthly]]
| jstor = 2304960
| mr = 0025512
| pages = 350–356
| title = Further remarks on the approximation of numbers as sums of reciprocals
| volume = 55
| year = 1948}}.
*{{citation
| last = Sigler | first = Laurence E. (trans.)
| isbn = 0-387-95419-8
| publisher = Springer-Verlag
| title = Fibonacci's Liber Abaci
| year = 2002}}.
*{{citation
| last = Soundararajan | first = K.
| arxiv = math.CA/0502247
| title = Approximating 1 from below using ''n'' Egyptian fractions
| year = 2005}}.
*{{citation
| last = Spiess | first = O.
| journal = Archiv der Mathematik und Physik, Ser. 3
| pages = 124–134
| title = Über eine Klasse unendlicher Reihen
| volume = 12
| year = 1907}}.
*{{citation
| last = Stong | first = R. E. | authorlink = Robert Evert Stong
| doi = 10.1007/BF01455950
| issue = 4
| journal = Mathematische Annalen
| mr = 0721884
| pages = 501–512
| title = Pseudofree actions and the greedy algorithm
| volume = 265
| year = 1983}}.
*{{citation
| last = Stratemeyer | first = G.
| doi = 10.1007/BF01246446
| journal = Mathematische Zeitschrift
| pages = 767–768
| title = Stammbruchentwickelungen für die Quadratwurzel aus einer rationalen Zahl
| volume = 31
| year = 1930}}.}
*{{citation
| last = Sylvester | first = J. J. | author-link = J. J. Sylvester
| doi = 10.2307/2369261
| issue = 4
| journal = [[American Journal of Mathematics]]
| jstor = 2369261
| pages = 332–335
| title = On a point in the theory of vulgar fractions
| volume = 3
| year = 1880}}.
*{{citation
| last = Wagon | first = S. | authorlink = Stan Wagon
| pages = 271–277
| publisher = W. H. Freeman
| title = Mathematica in Action
| year = 1991}}.
{{refend}}

[[Category:Number theory]]
[[Category:Integer sequences]]
[[Category:Egyptian fractions]]

Jensen's inequality

2014-01-16T10:29:14Z

130.88.73.77: /* Finite form */

A '''chord''' of a [[circle]] is a [[geometry|geometric]] [[line segment]] whose endpoints both lie on the circle. A [[secant line]], or just ''secant'', is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an [[Ellipse#Chords|ellipse]]. A chord that passes through a circle's center point is the circle's diameter.
[[Image:Chord in mathematics.svg|right|thumb|200px|The red segment ''BX'' is a '''chord''' 
(as is the diameter line ''AB'').]]

== Chords of a circle ==
{{Further2|[[Circle#Chord|Chord properties]]}}
Among properties of chords of a [[circle]] are the following:
# Chords are equidistant from the center only if their lengths are equal.
# A chord that passes through the center of a circle is called a diameter, and is the longest chord.
# If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD ([[power of a point theorem]]).

The area that a circular chord "cuts off" is called a [[circular segment]].

Chord is from the Latin ''chorda'' meaning ''bowstring''.

==Chords of an ellipse==

The midpoints of a set of parallel chords of an ellipse are [[Collinearity|collinear]].<ref>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.</ref>{{rp|p.147}}

== Chords in trigonometry ==
[[Image:TrigonometricChord.svg|left|200px]]
Chords were used extensively in the early development of [[trigonometry]]. The first known trigonometric table, compiled by [[Hipparchus]], [[table of chords|tabulated the value of the chord function]] for every 7.5 [[Degree (angle)|degree]]s. In the second century AD, [[Ptolemy]] of Alexandria compiled a more extensive table of chords in [[Almagest|his book on astronomy]], giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.

The chord function is defined geometrically as in the picture to the left. The chord of an [[angle]] is the [[length]] of the chord between two points on a unit circle separated by that angle. The chord function can be related to the modern [[sine]] function, by taking one of the points to be (1,0), and the other point to be (cos <math>\theta</math>, sin <math>\theta</math>), and then using the [[Pythagorean theorem]] to calculate the chord length:

: <math> \mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right). </math>

The last step uses the [[Trigonometric_identity#Double-angle.2C_triple-angle.2C_and_half-angle_formulae|half-angle formula]]. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones:

{| class="wikitable"
!Name!!Sine-based!!Chord-based
|-
|Pythagorean
|<math>\sin^2 \theta + \cos^2 \theta = 1 \, </math>
|<math>\mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \, </math>
|-
|Half-angle
|<math>\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \, </math>
|<math>\mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \,</math>
|-
| '''[[Apothem]] (a)'''
|<math>c=2 \sqrt{r^2- a^2}</math>
|<math>c=\sqrt{D ^2-4 a^2}</math>
|-
| '''Angle (θ)'''
|<math>c=2 r \sin \left(\frac{\theta }{2}\right)</math>
|<math>c=D \sin \left(\frac{\theta }{2}\right)</math>
|}

==See also==
*[[Circular segment]]
*[[Circle graph]]

==References==

{{reflist}}

== External links ==
{{Commons category|Chord (geometry)}}
* [http://aleph0.clarku.edu/~djoyce/ma105/trighist.html History of Trigonometry Outline]
* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html Trigonometric functions], focusing on history
*[http://www.mathopenref.com/chord.html Chord (of a circle)] With interactive animation

[[Category:Curves]]
[[Category:Trigonometry]]

[[ja:円 (数学)#円の性質]]

Systematic risk

2013-12-20T10:28:36Z

130.88.141.34: sp

In [[mathematics]], particularly the study of [[Lie groups]], a '''Dunkl operator''' is a certain kind of [[mathematical operator]], involving [[differential operator]]s but also [[Reflection (mathematics)|reflection]]s in an underlying space.

Formally, let ''G'' be a [[Coxeter group]] with reduced root system ''R'' and ''k''''v'' a multiplicity function on ''R'' (so ''k''''u'' = ''k''''v'' whenever the reflections σ''u'' and σ''v'' corresponding to the roots ''u'' and ''v'' are conjugate in ''G''). Then, the '''Dunkl operator''' is defined by:

:<math>T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i</math>

where <math>v_i </math> is the ''i''-th component of ''v'', 1 ≤ ''i'' ≤ ''N'', ''x'' in ''R''''N'', and ''f'' a smooth function on ''R''''N''.

Dunkl operators were introduced by {{harvs|txt|authorlink=Charles F. Dunkl|last=Dunkl|year=1989}}. One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy <math>T_i (T_j f(x)) = T_j (T_i f(x))</math> just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.

==References==

*{{Citation | last1=Dunkl | first1=Charles F. | title=Differential-difference operators associated to reflection groups | doi=10.2307/2001022 | mr=951883 | year=1989 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=311 | issue=1 | pages=167–183}}

[[Category:Lie groups]]

Hume-Rothery rules

2013-12-09T15:54:23Z

130.88.205.16: /* Substitutional Solid Solution Rules */

{{enzyme
| Name = chloroplast protein-transporting ATPase
| EC_number = 3.6.3.52
| CAS_number =
| IUBMB_EC_number = 3/6/3/52
| GO_code = 0016464
| image =
| width =
| caption =
}}
In [[enzymology]], a '''chloroplast protein-transporting ATPase''' ({{EC number|3.6.3.52}}) is an [[enzyme]] that [[catalysis|catalyzes]] the [[chemical reaction]]

:ATP + H2O <math>\rightleftharpoons</math> ADP + phosphate

Thus, the two [[substrate (biochemistry)|substrates]] of this enzyme are [[adenosine triphosphate|ATP]] and [[water|H2O]], whereas its two [[product (chemistry)|products]] are [[adenosine diphosphate|ADP]] and [[phosphate]].

This enzyme belongs to the family of [[hydrolase]]s, specifically those acting on acid anhydrides to catalyse transmembrane movement of substances. The systematic name of this enzyme class is '''ATP phosphohydrolase (chloroplast protein-importing)'''.

==References==
{{reflist|1}}
* {{cite journal | author = Cline K, Ettinger WF, Theg SM | year = 1992 | title = Protein-specific energy requirements for protein transport across or into thylakoid membranes. Two lumenal proteins are transported in the absence of ATP | journal = J. Biol. Chem. | volume = 267 | pages = 2688–96 | pmid = 1733965 | issue = 4 }}
* {{cite journal | author = Nakai M, Goto A, Nohara T, Sugita D, Endo T | year = 1994 | title = Identification of the SecA protein homolog in pea chloroplasts and its possible involvement in thylakoidal protein transport | journal = J. Biol. Chem. | volume = 269 | pages = 31338–41 | pmid = 7989297 | issue = 50 }}
* {{cite journal | author = Scott SV, Theg SM | year = 1996 | title = A new chloroplast protein import intermediate reveals distinct translocation machineries in the two envelope membranes: energetics and mechanistic implications | journal = J. Cell. Biol. | volume = 132 | pages = 63–75 | pmid = 8567731 | doi = 10.1083/jcb.132.1.63 | issue = 1–2 | pmc = 2120703 }}

[[Category:EC 3.6.3]]
[[Category:Enzymes of unknown structure]]

{{hydrolase-stub}}

Hyperelastic material

2013-10-23T10:16:26Z

130.88.123.93: /* Consistency conditions for isotropic hyperelastic models */

{{Lowercase|title=d'Alembert's Solution of the wave equation}}
{{Refimprove|date=September 2009}}
{{Textbook|date=June 2010}}
In [[mathematics]], and specifically [[partial differential equations]], '''d´Alembert's formula''' is the general solution to the one-dimensional [[wave equation]]:
:<math>u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),</math>
for <math>-\infty < x<\infty,\,\, t>0</math>. It is named after the mathematician [[Jean le Rond d'Alembert]].<ref>D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA214#v=onepage&q&f=false "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219. See also: D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA220#v=onepage&q&f=false "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"] (Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 220-249. See also: D'Alembert (1750) [http://books.google.com/books?id=m5UDAAAAMAAJ&pg=PA355#v=onepage&q&f=false "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"] ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 6, pages 355-360.</ref>

The [[method of characteristics|characteristics]] of the PDE are <math>x\pm ct=\mathrm{const}\,</math>, so use the change of variables <math>\mu=x+ct, \eta=x-ct\,</math> to transform the PDE to <math>u_{\mu\eta}=0\,</math>. The general solution of this PDE is <math>u(\mu,\eta) = F(\mu) + G(\eta)\,</math> where <math>F\,</math> and <math>G\,</math> are <math>C^1\,</math> functions. Back in <math>x,t\,</math> coordinates,

:<math>u(x,t)=F(x+ct)+G(x-ct)\,</math>
:<math>u\,</math> is <math>C^2\,</math> if <math>F\,</math> and <math>G\,</math> are <math>C^2\,</math>.

This solution <math>u\,</math> can be interpreted as two waves with constant velocity <math>c\,</math> moving in opposite directions along the x-axis.

Now consider this solution with the [[Cauchy data]] <math>u(x,0)=g(x), u_t(x,0)=h(x)\,</math>.

Using <math>u(x,0)=g(x)\,</math> we get <math>F(x)+G(x)=g(x)\,</math>.

Using <math>u_t(x,0)=h(x)\,</math> we get <math>cF'(x)-cG'(x)=h(x)\,</math>.

Integrate the last equation to get

:<math>cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1.\,</math>

Now solve this system of equations to get

:<math>F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right)\,</math>

:<math>G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right).\,</math>

Now, using

:<math>u(x,t) = F(x+ct)+G(x-ct)\,</math>

d´Alembert's formula becomes:

:<math>u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi.</math>

==See also==
*[[D'Alembert operator]]

== Notes ==

{{reflist}}

==External links==
*[http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com

[[Category:Partial differential equations]]

Superconducting coherence length

2013-10-18T12:31:13Z

130.88.75.149:

In [[mathematics]], '''SO(5)''', also denoted '''SO5(R)''' or '''SO(5,R)''', is the [[special orthogonal group]] of degree 5 over the field '''R''' of [[real number]]s, i.e. ([[isomorphic]] to) the group of [[orthogonal matrix|orthogonal]] 5×5 [[Matrix (mathematics)|matrices]] of [[determinant]] 1.

==Geometric interpretation==

SO(5) is a subgroup of the [[Euclidean group#Direct and indirect isometries|direct Euclidean group ''E''+(5)]], the group of ''direct'' [[isometry|isometries]], i.e., isometries preserving [[orientation (mathematics)|orientation]], of '''R'''5, consisting of elements which leave the origin fixed.

More precisely, we have:
:''SO''(''5'') <math>\cong</math> ''E''+(5) ''/ T''
where ''T'' is the [[translation (geometry)|translational]] group of '''R'''5.

==Lie group==

SO(5) is a [[simple Lie group]] of dimension 10.

==See also==

*[[Orthogonal matrix]]
*[[Orthogonal group]]
*[[Rotation group SO(3)]]
*[[List of simple Lie groups]]

[[Category:Lie groups]]

{{geometry-stub}}

Pitchfork bifurcation

2013-10-08T21:47:23Z

130.88.166.46: /* References */

In [[aerodynamics]], the '''zero-lift drag coefficient''' <math>C_{D,0}</math> is a dimensionless parameter which relates an aircraft's zero-lift [[drag (physics)|drag]] [[force]] to its size, speed, and flying altitude.

Mathematically, zero-lift [[drag coefficient]] is defined as <math>C_{D,0} = C_D - C_{D,i}</math>, where <math>C_D</math> is the total drag coefficient for a given power, speed, and altitude, and <math>C_{D,i}</math> is the [[lift-induced drag]] coefficient at the same conditions. Thus, zero-lift drag coefficient is reflective of [[parasitic drag]] which makes it very useful in understanding how "clean" or streamlined an aircraft's aerodynamics are. For example, a [[Sopwith Camel]] biplane of [[World War I]] which had many wires and bracing struts as well as fixed landing gear, had a zero-lift drag coefficient of approximately 0.0378. Compare a <math>C_{D,0}</math> value of 0.0161 for the streamlined [[P-51 Mustang]] of [[World War II]]<ref name="Loftin">{{cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468|url=http://www.hq.nasa.gov/pao/History/SP-468/cover.htm|accessdate=2006-04-22}}</ref> which compares very favorably even with the best modern aircraft.

The zero-lift drag coefficient can be more easily conceptualized as the '''drag area''' ('''<math>f</math>''') which is simply the product of zero-lift drag coefficient and aircraft's wing area (<math>C_{D,0} \times S</math> where <math>S</math> is the wing area). Parasitic drag experienced by an aircraft with a given drag area is approximately equal to the drag of a flat square disk with the same area which is held perpendicular to the direction of flight. The Sopwith Camel has a drag area of {{convert|8.73|sqft|m2|abbr=on}}, compared to {{convert|3.80|sqft|m2|abbr=on}} for the P-51. Both aircraft have a similar wing area, again reflecting the Mustang's superior aerodynamics in spite of much larger size.<ref name="Loftin" /> In another comparison with the Camel, a very large but streamlined aircraft such as the [[Lockheed Constellation]] has a considerably smaller zero-lift drag coefficient (0.0211 vs. 0.0378) in spite of having a much larger drag area (34.82 ft² vs. 8.73 ft²).

Furthermore, an aircraft's maximum speed is proportional to the [[cube root]] of the ratio of power to drag area, that is:

:<math>V_{max}\ \propto\ \sqrt[3]{power/f}</math>.<ref name="Loftin"/>

==Estimating zero-lift drag<ref name="Loftin"/>==
As noted earlier, <math>C_{D,0} = C_D - C_{D,i}</math>.

The total drag coefficient can be estimated as:

:<math>C_D = \frac{550 \eta P}{\frac{1}{2} \rho_0 [\sigma S (1.47V)^3]}</math>,

where <math>\eta</math> is the [[propulsive efficiency]], P is engine power in [[horsepower]], <math>\rho_0</math> sea-level air density in [[slug (mass)|slugs]]/cubic foot, <math>\sigma</math> is the atmospheric density ratio for an altitude other than sea level, S is the aircraft's wing area in square feet, and V is the aircraft's speed in miles per hour. Substituting 0.002378 for <math>\rho_0</math>, the equation is simplified to:

:<math>C_D = 1.456 \times 10^5 (\frac{\eta P}{\sigma S V^3})</math>.

The induced drag coefficient can be estimated as:

:<math>C_{D,i} = \frac{C_L^2}{\pi A \epsilon}</math>,

where <math>C_L</math> is the [[lift coefficient]], ''A'' is the [[aspect ratio (wing)|aspect ratio]], and <math>\epsilon</math> is the aircraft's efficiency factor.

Substituting for <math>C_L</math> gives:

:<math>C_{D,i}=\frac{4.822 \times 10^4}{A \epsilon \sigma^2 V^4} (W/S)^2</math>,

where W/S is the [[wing loading]] in lb/ft².

==References==

{{reflist}}

[[Category:Aerodynamics]]

Building (mathematics)

2013-07-11T14:26:19Z

130.88.16.223: Undid revision 563818551 by 130.88.16.205 (talk)

A '''list comprehension''' is a [[Syntax of programming languages|syntactic]] construct available in some [[programming language]]s for creating a list based on existing [[list (computing)|lists]]. It follows the form of the mathematical ''[[set-builder notation]]'' (''set comprehension'') as distinct from the use of [[Map (higher-order function)|map]] and [[Filter (higher-order function)|filter]] functions.

==Overview==
Consider the following example in [[set-builder notation]].

:<math>S=\{\,2\cdot x\mid x \in \mathbb{N},\ x^2>3\,\}</math>

This can be read, "<math>S</math> is the set of all numbers "2 times <math>x</math>" where <math>x</math> is an item in the set of [[natural number]]s (<math>\mathbb{N}</math>), for which <math>x</math> squared is greater than <math>3</math>."

In this annotated version of the example:
:<math>S=\{\,\underbrace{2\cdot x}_{\color{Violet}\text{output expression}}\mid \underbrace{x}_{\color{Violet}\text{variable}} \in \underbrace{\mathbb{N}}_{\color{Violet}\text{input set}},\ \underbrace{x^2>3}_{\color{Violet}\text{predicate}}\,\}</math>
* <math>x</math> is the variable representing members of an input set.
* <math>\mathbb{N}</math> represents the input set, which in this example is the set of natural numbers
* <math>x^2>3</math> is a [[Predicate (logic)|predicate]] expression acting as a filter on members of the input set.
* <math>2\cdot x</math> is an output expression producing members of the new set from members of the input set that satisfy the predicate expression.
* <math>\{\}</math> brackets contain the expression
* <math>\mid</math> <math>,</math> the vertical bar and the comma are separators.

A list comprehension has the same syntactic components to represent generation of a list in order from an input [[List (computing)|list]] or [[iterator]]:
* A variable representing members of an input list.
* An input list (or iterator).
* An optional predicate expression.
* And an output expression producing members of the output list from members of the input iterable that satisfy the predicate.
The order of generation of members of the output list is based on the order of items in the input.

In [[Haskell (programming language)|Haskell's]] list comprehension syntax, this set-builder construct would be written similarly, as:
<source lang="haskell">
s = [ 2*x | x <- [0..], x^2 > 3 ]
</source>

Here, the list <code>[0..]</code> represents <math>\mathbb{N}</math>, <code>x^2>3</code> represents the predicate, and <code>2*x</code> represents the output expression.

List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may [[Generator (computer science)|generate]] the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.

==History==

The [[SETL programming language]] (later 1960s) had a set formation construct, and the [[computer algebra system]] [[Axiom (computer algebra system)|AXIOM]] (1973) has a similar construct that processes [[stream (computing)|stream]]s,
but the first use of the term "comprehension" for such constructs was in Rod Burstall and John Darlington's description of their functional programming language [[NPL programming language|NPL]] from 1977.

[[Smalltalk]] block context messages which constitute list comprehensions have been in that language since at least Smalltalk-80.

Burstall and Darlington's work with NPL influenced many functional programming languages during the 1980s, but not all included list comprehensions. An exception was the influential pure lazy functional programming language [[Miranda programming language|Miranda]], which was released in 1985. The subsequently developed standard pure lazy functional language [[Haskell programming language|Haskell]] includes many of Miranda's features, including list comprehensions.

Comprehensions were proposed as a query notation for databases<ref>[http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=135271 Comprehensions, a query notation for DBPLs]</ref> and were implemented in the ''Kleisli'' database query language.<ref>[http://portal.acm.org/citation.cfm?id=351241&dl=ACM&coll=portal The functional guts of the Kleisli query system]</ref>

== Examples in different programming languages ==

{{main|Comparison of programming languages (list comprehension)}}

The following provides a few examples of specific syntax used in programming languages.

Although the original example denotes an infinite list, few languages can express that, so in some of those cases we show how to take a subset of <math>\{0, 1, ..., 100\}</math> rather than a subset of <math>\mathbb{N}</math>.

=== B-Prolog ===
{{further2|[[BProlog]]}}

<source lang="prolog">
L @= [Y : X in 1..100, [Y], (X*X>3, Y is 2*X)]
</source>

A list of the form <code>[T : E1 in D1, ..., En in Dn, LocalVars, Goal]</code> is interpreted as a list comprehension in calls to <code>@=/2</code> and constraints. A list comprehension is translated into a foreach construct with an accumulator.

=== Ceylon ===
{{further2|[[Ceylon Project]]}}

<pre>
{ for (x in 0..100) if ( x**2 > 3) x * 2 }
</pre>

=== Clojure ===
{{further2|[[Clojure]]}}

Clojure generates infinite lazy sequences (similar to Haskell's lazy lists or Python's generators). Use '''take''' to get the first N results from the infinite sequence.

<source lang="lisp">
(take 20 (for [x (range) :when (> (* x x) 3)] (* 2 x)))
;; ⇒ (4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42)
</source>

=== CoffeeScript ===
{{further2|[[CoffeeScript]]}}

CoffeeScript brings pretty list comprehensions to JavaScript.

<source lang="CoffeeScript">
(x * 2 for x in [0..20] when x*x > 3)
</source>

=== Common Lisp ===
{{further2|[[Common Lisp]]}}

List comprehensions can be expressed with the <code>loop</code> macro's <code>collect</code> keyword. Conditionals are expressed with <code>if</code>, as follows:

<source lang="lisp">
(loop for x from 0 to 100 if (> (* x x) 3) collect (* 2 x))
</source>

An infinite lazy sequence can be created in a variety of ways, such as the [[CLOS]] object system or a [[Generator (computer science)|yield]] macro.

=== Elixir ===
{{further2|[[Elixir (programming language)|Elixir]]}}

The same example in Elixir:
<source lang="erlang">
S = lc x inlist Enum.to_list(0..100), x*x > 3, do: x*2
</source>

=== Erlang ===
{{further2|[[Erlang (programming language)|Erlang]]}}

The same example in Erlang:
<source lang="erlang">
S = [2*X || X <- lists:seq(0,100), X*X > 3].</source>

=== F# ===
{{further2|[[Fsharp|F#]]}}

The F# generator comprehension has the list comprehension syntax elements.
Generator comprehensions can be used to generate Lists, Sequences (lazy lists) and Arrays (not discussed here).

Generators are of the form <code>[for x in collection do ... yield expr]</code> for lists and <code>seq {for x in collection do ... yield expr}</code> for sequences.
For example:

<source lang="fsharp">
> seq { for x in 0..100 do
if x*x > 3 then yield 2*x } ;;
val it : seq<int> = seq [4; 6; 8; 10; ...]</source>

=== Falcon ===
{{further2|[[Falcon (programming language)|Falcon]]}}

The "comp" generic method family provides wide support for comprehension. For example, the "mfcomp" method can be applied to an array:
<syntaxhighlight lang=falcon>
s = [].mfcomp( { i => if i*i > 3: return 2*i; return oob(1)}, [1:101] )
</syntaxhighlight>
Falcon can also use functional generators to provide input lists. For example, the following code uses a continuation to create a set of pairs.
<syntaxhighlight lang=falcon>
gen = Continuation( function( max, c )
i = 0
while i < max: c(++i)
return oob(0)
end )
data = [10,11,12]
s = Set().mfcomp( {x, y => x+y}, .[gen 3], data )
</syntaxhighlight>
Method "comp" was introduced in version 0.9.6, and methods "mcomp" and "mfcomp" in version 0.9.6.2.

=== Groovy ===
{{further2|[[Groovy (programming language)|Groovy]]}}

Groovy supports list comprehension style expressions for any kind of Java Collection including lists, sets, and maps.
<source lang="groovy">
s = (1..100).grep { it ** 2 > 3 }.collect { it * 2 }
</source>

The "it" variable is shorthand for the implicit parameter to a closure. The above is equivalent to:
<source lang="groovy">
s = (1..100).grep { x -> x ** 2 > 3 }.collect { x -> x * 2 }
</source>

=== Haskell ===
{{further2|[[Haskell (programming language)|Haskell]]}}

Please refer to the main example in the [[#Overview|overview]].
<source lang="haskell">
s = [ 2*x | x <- [0..], x^2 > 3 ]
</source>

Here, the list <code>[0..]</code> generates natural numbers one by one which get bound to variable <code>x</code>, <code>x^2>3</code> represents the predicate that either accepts or rejects a given variable's value, and <code>2*x</code> represents the result expression. There might be several generators and test predicates in one list compehension expression in Haskell, in effect defining nested loops, e.g.:
<source lang="haskell">
s = [ 2*x*y | x <- [0..], x^2 > 3, y <- [1,3..x], y^2 < 100-x^2]
-- for each x from 0 by 1:
-- if x^2 > 3:
-- for each y from 1 by 2 upto x:
-- if y^2 < 100 - x^2:
-- produce 2*x*y
</source>
The above expression becomes unproductive (''"stuck"'') at some point, when new '''''x'''''s keep being generated only to be rejected later on. This is so because any test can only reject a value it is given, not any future ones (there is no ''cut'' mechanism here, in [[Prolog]] terms - a generator in general might produce its values unordered, like e.g. the above expression itself). This can be dealt with using bounded list generators always or by enclosing a generator inside a <code>'''take'''</code> or <code>'''takeWhile'''</code> call, limiting the amount of generated values.

=== Haxe ===
{{further2|[[Haxe]]}}

Haxe 3 released with array and map comprehension.<ref>http://haxe.org/manual/haxe3/features#array-comprehension</ref>
<source lang="haxe">
var s = [for(x in [0, 1, 2, 3, 4, 5, 6, 7]) if(x * x < 5) x];
</source>
However, Haxe 2's syntax required use of Lambda:
<source lang="haxe">
var a = [0, 1, 2, 3, 4, 5, 6, 7];
var s = Lambda.array(Lambda.filter(a, function(x) return x * x < 5));
</source>

=== JavaScript 1.7 ===

JavaScript 1.7 has array comprehensions. The [[JavaScript engine]] of the popular [[Firefox browser]] from [[Mozilla Foundation]]---[[SpiderMonkey (JavaScript engine)|SpiderMonkey]]---supports them,<ref>https://developer.mozilla.org/en-US/docs/JavaScript/Guide/Predefined_Core_Objects#Array_comprehensions</ref> for example,
<source lang="Javascript">
js> [2*x for each (x in [0,1,2,3,4,5,6,7]) if (x*x<5)]
[0, 2, 4]
</source>

The sequence of integers can be obtained by prototyping the Number object,
<source lang="Javascript">
Number.prototype.__iterator__=function(){for (var i=0; i<this; i++) yield i}
var s = [2*x for (x in 100) if (x*x<5)]
</source>
Or introducing a range function,
<source lang="Javascript">
var range = function(start,end){for (var i=start; i<=end; i++) yield i}
var s = [2*x for (x in range(0,100)) if (x*x<5)]
</source>

=== Mathematica ===
{{further2|[[Mathematica]]}}

The Cases command with a <tt>RuleDelayed</tt> in the second argument provides a list comprehension mechanism:

<code>
s = Cases[Range[0,100], i_ /; i^2 > 3 :> 2i]</code>
Alternatively

<code>
Table[If[i^2 > 3, 2i, Unevaluated[]], {i, 0, 100}]
</code>

<code>
Do[If[i^2 > 3, Sow[2i]], {i, 0, 100}] // Reap
</code>

=== OCaml ===
{{further2|[[OCaml]]}}

OCaml Batteries Included has uniform comprehension syntax for lists, arrays, enumerations (like streams), lazy lists (like lists but evaluated on-demand), sets, hashtables, etc.

Comprehension are of the form <code>[? expression | x <- enumeration ; condition; condition ; ... ?]</code>.

For instance,
<source lang="ocaml">
# [? 2 * x | x <- 0 -- max_int ; x * x > 3 ?];;
- : int Enum.t = <abstr>
</source>
or, to compute a list,
<source lang="ocaml">
# [? List: 2 * x | x <- 0 -- 5 ; x * x > 3 ?];;
- : int list = [4; 6; 8; 10]
</source>
or, to compute a set,
<source lang="ocaml">
# [? PSet: 2 * x | x <- 0 -- 5 ; x * x > 3 ?];;
- : int PSet.t = <abstr>
</source>

etc.

=== Octave ===
[[GNU Octave]] can do list (vector) comprehensions in the form <code>(vector expression)(vector condition)</code>.

For example,
<source lang="octave">
octave:1> x=0:100; s=(2*x)(x.**2<5)
s =
0 2 4</source>

=== Perl 5 ===
{{further2|[[Perl 5]]}}

Perl 5 provides an alternate approach

<source lang="perl">
my @s = map { $_ * 2 } grep { $_ * $_ < 5 } (0..7);
</source>

=== Perl 6 ===
{{further2|[[Perl 6]]}}

[[Perl 6]] provides more than one way to implement list comprehensions.

<source lang="perl6">
my @s = ($_ * 2 if $_ ** 2 > 3 for ^100);
</source>

Or, using [[Perl 6#Gather|gather]]:

<source lang="perl6">
my @s = gather { for ^100 { take 2 * $_ if $_ ** 2 > 3 } };
</source>

=== Picat ===

<source lang="text">
[2*X : X in 1..100, X*X>3]
</source>

A list comprehension in [http://picat-lang.org Picat] takes the form <code>[T : E1 in D1, Cond1, ..., En in Dn, Condn]</code>. A list comprehension is compiled into a foreach loop, which is further compiled into a tail-recursive predicate.

=== PowerShell ===
{{further2|[[PowerShell]]}}

<source lang="PowerShell">
0..100 | Where {$_ * $_ -gt 3} | ForEach {$_ * 2}
</source>

=== Pure ===
{{further2|[[Pure (programming language)|Pure]]}}

The same example in Pure:
s = [2*n | n=1..100; n*n > 3];

=== Python ===
{{further2|[[Python syntax and semantics#Generators|Python syntax and semantics: Generators]]}}

The [[Python (programming language)|Python programming language]] has a corresponding syntax for expressing list comprehensions.
The near-equivalent in Python to the example above is as follows:

<source lang="python">
S = [2 * x for x in range(101) if x ** 2 > 3]
</source>

List comprehensions were introduced in Python version 2.0.<ref>{{cite web | url = http://www.python.org/dev/peps/pep-0202/ | title = PEP 202 – List Comprehensions | work = python.org | date = 2 October 2008 | first = Barry | last = Warsaw}}</ref>

=== Racket ===
{{further2|[[Racket (programming language)|Racket]]}}

Racket provides functional versions of for-loops, which are essentially list comprehension syntax:
<source lang="scheme">
(for/list ([x (in-range 100)] #:when (> (* x x) 3))
(* 2 x))
</source>
The imperative <code>for</code> can also be used, combined with Racket's generator library to produce an infinite generator:
<source lang="scheme">
(require racket/generator)
(generator ()
(for ([x (in-naturals)] #:when (> (* x x) 3))
(yield (* 2 x))))
</source>

=== Ruby ===
{{further2|[[Ruby (programming language)|Ruby]]}}

In the Ruby language you can use multiple ways to simulate this function, for example:
<source lang="Ruby">
(1..100).select{|x| x ** 2 > 3 }.collect{|x| 2 * x}
</source>
Or you can define your own method:

<source lang="Ruby">
module Enumerable
def comprehend(&block)
return self if block.nil?
collect(&block).compact
end
end

(1..100).comprehend {|x| 2 * x if x ** 2 > 3}
</source>

=== Rust ===
{{further2|[[Rust (programming language)|Rust]]}}

The Rust language currently doesn't have a built-in support for list comprehensions but the functionality can be implemented as a macro:
<syntaxhighlight lang="text" line highlight="8">
macro_rules! compr(
($o:expr : $v:ident <- $i:expr $(,$p:expr)+) =>
($i.$(filtered(|&$v| { $p })).*.map(|&$v| { $o }));
($o:expr : $v:ident <- $i:expr) =>
($i.map(|&$v| { $o }))
)

let s = compr!(x * 2 : x <- [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], x > 2);
</syntaxhighlight>

=== Scala ===
{{further2|[[Scala (programming language)|Scala]]}}

Using a for-expression:

<source lang="scala">
val s = for (x <- Stream.from(0) if x*x > 3) yield 2*x
</source>

=== Scheme ===
{{further2|[[Scheme (programming language)|Scheme]]}}
Although there is no standard list comprehension syntax in [[R5RS]], many implementations provide an extension for this. For example, in [[Chicken (Scheme implementation)|Chicken Scheme]]:
<source lang="scheme">
(require-extension list-of)
(list-of (* 2 x) (x range 0 101) (> (* x x) 3))
</source>
There is also a portable library SRFI/42 "Eager Comprehensions", which in particular includes list comprehensions:
<source lang="scheme">
(require srfi/42) ; example import into Racket Scheme
(list-ec (: x 101) (if (> (* x x) 3)) (* 2 x))
</source>

=== SETL ===
{{further2|[[SETL]]}}
<source lang="html4strict">
s := {2*x : x in {0..100} | x**2 > 3 };
</source>

=== Smalltalk ===
{{further2|[[Smalltalk]]}}
<source lang="smalltalk">
((1 to: 100) select: [:x|x*x>3]) collect: [:x|2*x]
</source>

=== SuperCollider ===
{{further2|[[SuperCollider]]}}
In SuperCollider list comprehensions are implemented as Routines, whose results can be collected with the message 'all'. A shortcut syntax is provided for defining list comprehensions, which internally translates to a routine.
<source lang="text">
all {: x * 2, x <- (1..100), x ** 2 > 3 }
</source>

=== Visual Prolog ===

{{further2|[[Visual Prolog]]}}

<source lang="VisualProlog">S = [ 2*X || X = std::fromTo(1, 100), X^2 > 3 ]</source>

==Similar constructs==

===Monad comprehension===
In Haskell, a [[monads in functional programming#do-notation|monad comprehension]] is a generalization of the list comprehension to other [[monads in functional programming]].

===Set comprehension===
Version 3.x and 2.7 of the Python language introduces syntax for [[set (computer science)|set]] comprehensions. Similar in form to list comprehensions, set comprehensions generate Python sets instead of lists.
<source lang="python">
>>> s = {v for v in 'ABCDABCD' if v not in 'CB'}
>>> print(s)
{'A', 'D'}
>>> type(s)
<class 'set'>
>>>
</source>

Racket set comprehensions generate Racket sets instead of lists.
<source lang="scheme">
(for/set ([v "ABCDABCD"] #:unless (member v (string->list "CB")))
v))
</source>

===Dictionary comprehension===
Version 3.x and 2.7 of the Python language introduced a new syntax for [[associative array|dictionary]] comprehensions, similar in form to list comprehensions but which generate Python [http://docs.python.org/library/stdtypes.html#dict dicts] instead of lists.
<source lang="python">
>>> s = {key: val for key, val in enumerate('ABCD') if val not in 'CB'}
>>> s
{0: 'A', 3: 'D'}
>>>
</source>

Racket hash table comprehensions generate Racket hash tables (one implementation of the Racket dictionary type).
<source lang="scheme">
(for/hash ([(val key) (in-indexed "ABCD")]
#:unless (member val (string->list "CB")))
(values key val))
</source>

===Parallel list comprehension===
The [[Glasgow Haskell Compiler]] has an extension called '''parallel list comprehension''' (also known as '''zip-comprehension''') that permits multiple independent branches of qualifiers within the list comprehension syntax.
Whereas qualifiers separated by commas are dependent ("nested"), qualifier branches separated by pipes are evaluated in parallel (this does not refer to any form of multithreadedness: it merely means that the branches are [[Map (higher-order function)|zipped]]).
<source lang="haskell">
-- regular list comprehension
a = [(x,y) | x <- [1..5], y <- [3..5]]
-- [(1,3),(1,4),(1,5),(2,3),(2,4) ...

-- zipped list comprehension
b = [(x,y) | (x,y) <- zip [1..5] [3..5]]
-- [(1,3),(2,4),(3,5)]

-- parallel list comprehension
c = [(x,y) | x <- [1..5] | y <- [3..5]]
-- [(1,3),(2,4),(3,5)]
</source>

Racket's comprehensions standard library contains parallel and nested versions of its comprehensions, distinguished by "for" vs "for*" in the name. For example, the vector comprehensions "for/vector" and "for*/vector" create vectors by parallel versus nested iteration over sequences. The following is Racket code for the Haskell list comprehension examples.
<source lang="scheme">
> (for*/list ([x (in-range 1 6)] [y (in-range 3 6)]) (list x y))
'((1 3) (1 4) (1 5) (2 3) (2 4) (2 5) (3 3) (3 4) (3 5) (4 3) (4 4) (4 5) (5 3) (5 4) (5 5))
> (for/list ([x (in-range 1 6)] [y (in-range 3 6)]) (list x y))
'((1 3) (2 4) (3 5))
</source>

In Python we could do as follows:
<source lang="python">
# regular list comprehension
>>> a = [(x, y) for x in range(1, 6) for y in range(3, 6)]
[(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), ...

# parallel/zipped list comprehension
>>> b = [x for x in zip(range(1, 6), range(3, 6))]
[(1, 3), (2, 4), (3, 5)]
</source>

=== XQuery and XPath ===
Like the original NPL use, these are fundamentally database access languages.

This makes the comprehension concept more important, because it is computationally infeasible to retrieve the entire list and operate on it (the initial 'entire list' may be an entire XML database).

In XPath, the expression:

/library/book//paragraph[@style='first-in-chapter']

is conceptually evaluated as a series of "steps" where each step produces a list and the next step applies a filter function to each element in the previous step's output.<ref>{{cite web | url = http://www.w3.org/TR/xpath#section-Location-Steps | title = 2.1 Location Steps | work = XML Path Language (XPath) | date = 16 November 1999 | publisher = [[W3C]]}}</ref>

In XQuery, full XPath is available, but [[FLWOR]] statements are also used, which is a more powerful comprehension construct.<ref>{{cite web | url = http://www.w3schools.com/XQuery/xquery_flwor.asp | title = XQuery FLWOR Expressions | work = [[W3Schools]]}}</ref>
<source lang="sql">
for $b in //book
where $b[@pages < 400]
order by $b//title
return
<shortBook>
<title>{$b//title}</title>
<firstPara>{($book//paragraph)[1]}</firstPara>
</shortBook>
</source>
Here the XPath //book is evaluated to create a sequence (aka list); the where clause is a functional "filter", the order by sorts the result, and the <shortBook>...</shortBook> XML snippet is actually an anonymous function that builds/transforms XML for each element in the sequence using the 'map' approach found in other functional languages.

So, in another functional language the above FLWOR statement may be implemented like this:

map(
newXML(shortBook, newXML(title, $1.title), newXML(firstPara, $1...))
filter(
lt($1.pages, 400),
xpath(//book)
)
)

=== LINQ in C# ===

[[C Sharp (programming language)|C#]] 3.0 has a group of related features called [[LINQ]], which defines a set of query operators for manipulating object enumerations.

<source lang="csharp">
var s = Enumerable.Range(0, 100).Where(x => x*x > 3).Select(x => x*2);
</source>

It also offers an alternative comprehension syntax, reminiscent of SQL:

<source lang="csharp">
var s = from x in Enumerable.Range(0, 100) where x*x > 3 select x*2;
</source>

LINQ provides a capability over typical List Comprehension implementations. When the root object of the comprehension implements the IQueryable interface, rather than just executing the chained methods of the comprehension, the entire sequence of commands are converted into an Abstract Syntax Tree (AST) object, which is passed to the IQueryable object to interpret and execute.

This allows, amongst other things, for the IQueryable to
- rewrite an incompatible or inefficient comprehension
- translate the AST into another query language (e.g. SQL) for execution

=== C++ ===
C++ does not have any language features directly supporting list comprehensions. List comprehensions can be constructed using the [[erase-remove idiom]] to select elements in a container and the STL algorithm for_each to transform them.
<source lang="cpp">
#include <algorithm>
#include <list>

using namespace std;

template<class C, class P, class T>
C&& comprehend(C&& source, const P& predicate, const T& transformation)
{
// initialize destination
C d = forward(source);

// filter elements
d.erase(remove_if(begin(d), end(d), predicate), end(d));

// apply transformation
for_each(begin(d), end(d), transformation);

return d;
}

int main()
{
list<int> range(10);
// range is a list of 10 elements, all zero
iota(begin(range), end(range), 1);
// range now contains 1,2,...,10

list<int> result = comprehend(
range,
[](int x){return x*x <= 3;},
[](int &x){x *= 2;});
// result now contains 4,6,...,20
}
</source>
There is some effort in providing C++ with list-comprehension constructs/syntax similar to the set builder notation.
* In [[Boost C++ Libraries|Boost]].Range [http://www.boost.org/libs/range] library there is a notion of adaptors [http://www.boost.org/libs/range/doc/html/range/reference/adaptors.html] that can be applied by to any range and do filtering, transformation etc. With this library, the original Haskell example would look like (using Boost.Lambda [http://www.boost.org/libs/lambda] for anonymous filtering and transforming functions):
<source lang="cpp">
counting_range(1,10) | filtered( _1*_1 > 3 ) | transformed(ret<int>( _1*2 ))
</source>
Full example is here: http://codepad.org/y4bpgLJu

* This<ref>{{cite web | url = http://mfoliveira.org/blog/2011/01/04/simple-list-comprehension-in-cp-using-preprocessor-macros/ | title = Single-variable List Comprehension in C++ using Preprocessor Macros}}</ref> implementation uses a macro and overloads the << operator. It evaluates any expression valid inside an 'if', and any variable name may be chosen. It's not [[threadsafe]], however. Usage example:
<source lang="cpp">
list<int> N;
list<double> S;

for (int i = 0; i < 10; i++)
N.push_back(i);

S << list_comprehension(3.1415 * x, x, N, x*x > 3)
</source>

* This<ref>{{cite web | url = http://www.tedunangst.com/listcc.html | title = C++ list comprehensions}}</ref> implementation provides head/tail slicing using classes and operator overloading, and the | operator for filtering lists (using functions). Usage example:
<source lang="cpp">
bool even(int x) { return x % 2 == 0; }
bool x2(int &x) { x *= 2; return true; }

list<int> l, t;
int x, y;

for (int i = 0; i < 10; i++)
l.push_back(i);

(x, t) = l | x2;
(t, y) = t;

t = l < 9;
t = t < 7 | even | x2;
</source>

==See also==
*The [[Select (SQL)|SELECT]] statement together with its FROM and WHERE clauses in [[SQL#Queries|SQL]]
*[[Programming language]]
*[[Mathematical notation]]
*[[Monads in functional programming]] for monads and monadic notation in general
*For other programming language constructs used to process sequences:
**[[Generator (computer science)]]
**[[Map (higher-order function)]]
*For other programming language constructs copied from the mathematical notation:
**[[Guard (computing)]]
**[[Pattern matching]]
**[[Operator (programming)]]

==Notes and references==
{{reflist}}
*[http://ftp.sunet.se/foldoc/foldoc.cgi?list+comprehension List Comprehension] in The Free On-line Dictionary of Computing, Editor Denis Howe.
*{{cite conference | first = Phil | last = Trinder | url = http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=135271 | title = Comprehensions, a query notation for DBPLs | booktitle = Proceedings of the third international workshop on Database programming languages: bulk types & persistent data, Nafplion, Greece | pages = 55–68 | year = 1992}}
*{{cite conference | first = Philip | last = Wadler |url = http://citeseer.ist.psu.edu/wadler92comprehending.html | title = Comprehending Monads | booktitle = Proceedings of the 1990 ACM Conference on LISP and Functional Programming, Nice | year = 1990}}
*{{cite conference | first = Limsoon | last = Wong | url = http://portal.acm.org/citation.cfm?id=351241&dl=ACM&coll=portal | title = The Functional Guts of the Kleisli Query System | conference = International Conference on Functional Programming | booktitle = Proceedings of the fifth ACM SIGPLAN international conference on Functional programming | pages = 1–10 | year = 2000}}

===Haskell===
*The Haskell 98 Report, chapter [http://haskell.org/onlinereport/exps.html#list-comprehensions 3.11 List Comprehensions].
*The Glorious Glasgow Haskell Compilation System User's Guide, chapter [http://www.haskell.org/ghc/docs/latest/html/users_guide/syntax-extns.html#parallel-list-comprehensions 7.3.4 Parallel List Comprehensions].
*The Hugs 98 User's Guide, chapter [http://cvs.haskell.org/Hugs/pages/users_guide/hugs-ghc.html#ZIP-COMPREHENSION 5.1.2 Parallel list comprehensions (a.k.a. zip-comprehensions)].

===OCaml===
* [http://batteries.forge.ocamlcore.org/ OCaml Batteries Included]
* [http://batteries.forge.ocamlcore.org/doc.preview:batteries-alpha3/html/extensions.html Language extensions introduced in OCaml Batteries Included]

===Python===
* Python Reference Manual, chapter [http://www.python.org/doc/2.4.1/ref/lists.html 5.2.4 List displays].
* Python Enhancement Proposal [http://www.python.org/peps/pep-0202.html PEP 202: List Comprehensions].
* Python Reference Manual, chapter [http://www.python.org/doc/2.4.1/ref/genexpr.html 5.2.5 Generator expressions].
* Python Enhancement Proposal [http://python.org/peps/pep-0289.html PEP 289: Generator Expressions].

===Common Lisp===
* [http://rali.iro.umontreal.ca/Publications/urls/LapalmeLispComp.pdf Implementation of a Lisp comprehension macro] by Guy Lapalme

===Clojure===
* [http://clojure.org/api#for Clojure API documentation - ''for'' macro]

===Axiom===
* [http://page.axiom-developer.org/zope/mathaction/Streams Axiom stream examples]

==External links==
* SQL-like set operations with list comprehension one-liners in the [http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/159974 Python Cookbook]
* [http://lambda-the-ultimate.org/classic/message11326.html Discussion on list comprehensions in Scheme and related constructs]
* [http://langexplr.blogspot.com/2007/02/list-comprehensions-across-languages_18.html List Comprehensions across languages]

[[Category:Programming constructs]]
[[Category:Articles with example code]]
[[Category:Articles with example Haskell code]]
[[Category:Articles with example Python code]]
[[Category:Articles with example Racket code]]

Suspension (dynamical systems)

2013-01-09T12:04:30Z

130.88.141.34: more specific

In [[statistics]], '''Cochran's C test''',<ref>W.G. Cochran, The distribution of the largest of a set of estimated variances as a fraction of their total, Annals of Human Genetics (London) 11(1), 47–52 (January 1941).</ref> named after [[William Gemmell Cochran|William G. Cochran]], is a [[one-tailed test|one-sided]] upper limit variance [[outlier]] test. The C test is used to decide if a single [[Estimation theory|estimate]] of a [[variance]] (or a [[standard deviation]]) is [[statistical significance|significantly]] larger than a group of variances (or standard deviations) with which the single estimate is supposed to be comparable. The C test is discussed in many text books <ref>D.L. Massart, B.G.M. Vandeginste, L.M.C. Buydens, S. de Jong, P.J. Lewi, J. Smeyers-Verbeke, ''Handbook of Chemometrics and Qualimetrics'': Part A, Elsevier, Amsterdam, The Netherlands, 1997 ISBN 0-444-89724-0.</ref><ref name="QA">P. Konieczka, J. Namieśnik, Quality Assurance and Quality Control in the Analytical Chemical Laboratory – A Practical Approach, CRC Press, Boca Raton, Florida, 2009; ISBN 978-1-4200-8270-8.</ref><ref>J.K. Taylor, Quality Assurance of Chemical Measurements, 4th printing, Lewis Publishers, Chelsea, Michigan, 1988; ISBN 0-87371-097-5.</ref> and has been recommended by [[IUPAC]] <ref>[[William Horwitz|W. Horwitz]], Harmonized protocol for the design and interpretation of collaborative studies, Trends in Analytical Chemistry 7(4), 118–120 (April 1988).</ref> and [[International Organization for Standardization|ISO]].<ref name="ISO">[[International Organization for Standardization|ISO]] Standard 5725–2:1994, “[[Accuracy]] (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of [[repeatability]] and [[reproducibility]] of a standard measurement method”, International Organization for Standardization, Geneva, Switzerland, 1994;
http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=11834</ref> Cochran's C test should not be confused with [[Cochran's Q test|Cochran's Q]] test, which applies to the [[analysis]] of two-way [[randomized block design]]s.

The C test assumes a balanced design, i.e. the considered full [[data set]] should consist of individual data series that all have equal size. The C test further assumes that each individual data series is [[normally distributed]]. Although primarily an outlier test, the C test is also in use as a simple alternative for regular [[homoscedasticity]] tests such as [[Bartlett's test|Bartlett's]] test, [[Levene's test|Levene's]] test and the [[Brown–Forsythe test]] to check a [[statistical data]] set for [[homogeneity of variance]]s. An even simpler way to check homoscedasticity is provided by [[Hartley's test|Hartley's Fmax test]],<ref name="QA" /> but Hartley's Fmax test has the disadvantage that it only accounts for the minimum and the maximum of the variance range, while the C test accounts for all variances within the range.

==Description==

The C test detects one exceptionally large variance value at a time. The corresponding data series is then omitted from the full data set. According to ISO standard 5725 <ref name="ISO" /> the C test may be [[iterate]]d until no further exceptionally large variance values are detected, but such practice may lead to excessive rejections if the underlying data series are not normally distributed.
The C test evaluates the [[ratio]]:

::<math>C_j = \frac{S_j^2}{\displaystyle \sum_{i=1}^N S_i^2}</math>

where:
:''Cj'' = Cochran's C statistic for data series ''j''
:''Sj'' = standard deviation of data series ''j''
:''N'' = number of data series that remain in the data set; ''N'' is decreased in steps of 1 upon each iteration of the C test
:''Si'' = standard deviation of data series i (1 ≤ ''i'' ≤ ''N'')

The C test tests the [[null hypothesis]] (H0) against the [[alternative hypothesis]] (Ha):

:H0: All variances are equal.
:Ha: At least one variance value is significantly larger than the other variance values.

==Critical values==

The sample variance of data series ''j'' is considered an outlier at [[significance level]] ''α'' if ''Cj'' exceeds the upper limit [[critical value]] CUL. CUL depends on the desired significance level ''α'', the number of considered data series ''N'', and the number of data points (''n'') per data series. Selections of values for CUL have been tabulated at significance levels α = 0.01,<ref name="ISO" /><ref name="Moore">R. Moore, Mathematics Department, Macquarie University, Sydney, Australia, 1999: http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran.</ref><ref name="tabulated">R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, ''Analytica Chimica Acta'' 659, 68–84 (2010); {{doi|10.1016/j.aca.2009.11.032}}</ref> α = 0.025,<ref name="tabulated" /> and α = 0.05.<ref name="ISO" /><ref name="Moore" /><ref name="tabulated">R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, Analytica Chimica Acta 659, 68–84 (2010); {{doi|10.1016/j.aca.2009.11.032}}</ref> ''C''UL can also be calculated from:<ref name="tabulated" /><ref name="exist" />
:<math>C_\text {UL}(\alpha,n,N) = \left [ 1+ \frac{N-1}{F_\text {c}(\alpha/N,(n-1),(N-1)(n-1))} \right ]^{-1} .</math>

Here:
:''C''UL = upper limit critical value for one-sided test on a balanced design
:''α'' = significance level
:''n'' = number of data points per data series
:''F''c = critical value of [[F-test of equality of variances|Fisher's F]] ratio; ''F''c can be obtained from tables of the [[F distribution]]<ref name="values">Table of critical values of the F-distribution:[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm NIST]</ref> or using computer software for this function.

==Generalization==

The C test can be generalized to include unbalanced designs, one-sided lower limit tests and [[two-tail|two-sided]] tests at any significance level ''α'', for any number of data series ''N'', and for any number of individual data points ''nj'' in data series ''j''.<ref name="tabulated" /><ref name="exist">R.U.E. 't Lam, Variance Outlier Test, blog: http://rtlam.blogspot.com/</ref>

==See also==
*[[Bartlett's test]]
*[[Levene's test]]
*[[Brown–Forsythe test]]
*[[Hartley's test]]
*[[F-test of equality of variances]]

==External links==
*[http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran Critical C values] <ref name="Moore" />
*[http://rtlam.blogspot.com/ Generalized Variance Outlier Test] <ref name="exist" />
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Critical F values] <ref name="values" />

==References==
{{Reflist}}

{{Statistics}}

[[Category:Statistical tests]]

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