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Louisville Bats

The Louisville Bats are a Minor League Baseball team of the International League (IL) and the Triple-A affiliate of the Cincinnati Reds. They are located in Louisville, Kentucky, and are named in dual reference to the winged mammal and baseball bats, such as those manufactured locally under the Louisville Slugger brand. The team plays their home games at Louisville Slugger Field, which opened in 2000. The Bats previously played at Cardinal Stadium from 1982 to 1999.

The club began play as the Louisville Redbirds in the Triple-A American Association (AA) in 1982. Louisville won three AA championships: in 1984, 1985, and 1995 as the top affiliate of the St. Louis Cardinals. They joined the Triple-A International League in 1998 following the dissolution of the AA after the 1997 season. The Redbirds rebranded as the Louisville RiverBats in 1999; this was shortened to Bats in 2002. In conjunction with Major League Baseball's restructuring of Minor League Baseball in 2021, the Bats were shifted to the Triple-A East, but this was renamed the International League in 2022. Their lone IL championship was won in 2001 with Cincinnati.

American Association (1982–1997)
In 1982, the St. Louis Cardinals moved their Triple-A team of the American Association (AA), the Springfield Redbirds, from Springfield, Illinois, to Louisville, Kentucky, where they became the Louisville Redbirds. They played their home games at Cardinal Stadium, which was located at the Kentucky Exposition Center and seated over 30,000 people. The stadium previously hosted the Louisville Colonels from 1957 to 1972.

During their inaugural 1982 season, the Redbirds broke the minor league attendance record by drawing 868,418 people to Cardinal Stadium. On the field, the team narrowly missed the playoffs when they finished in second place, just one-and-a-half games out of first at 73–62. In 1983, the Redbirds became the first minor league team to draw over one million fans in a single season (1,052,438). The team improved to league-best 78–57 and won the Eastern Division title, which qualified them for the postseason. Though Louisville won the semifinals versus the Oklahoma City 89ers, 3–2, they lost the AA championship to the Denver Bears, 4–0, in the finals. Manager Jim Fregosi was selected as the AA Manager of the Year. The franchise was awarded the 1983 John H. Johnson President's Award, recognizing them as the "complete baseball franchise—based on franchise stability, contributions to league stability, contributions to baseball in the community, and promotion of the baseball industry."

Fergosi led the Redbirds to win back-to-back American Association championships in 1984 and 1985. The 1984 squad won the semifinals against the Indianapolis Indians, 4–2, before winning their first league title over the Denver Zephyrs, 4–1. Their second came the next season after winning the 1985 Eastern Division title and defeating Oklahoma City, 4–1, in the only round of playoffs. Fergosi won his second Manager of the Year Award for the 1985 campaign.

The 1987 team made a return to the postseason, but they were eliminated by Indianapolis, 3–2, in the semifinals. Outfielder Lance Johnson was honored with both the AA Most Valuable Player (MVP) Award and Rookie of the Year Award. Louisville regularly finished third or fourth in their four-team division over the next six seasons. A pair of Redbirds won league awards, however, during this period. René Arocha was selected as the 1992 Most Valuable Pitcher, and catcher Todd Zeile was the 1989 Rookie of the Year. The team qualified for a playoff spot in 1994, but they were quickly swept out of the semifinals, 3–0, by Indianapolis. The Redbirds came back in 1995 to sweep Indianapolis in the semis before winning a third AA championship versus the Buffalo Bisons, 3–2, behind manager Joe Pettini.

International League (1998–2020)
Louisville experienced a number of changes around the turn of the millennium. The American Association, of which the Redbirds had been members since 1982, disbanded after the 1997 season. Its teams were absorbed by the two remaining Triple-A leagues—the International League (IL) and Pacific Coast League (PCL), with Louisville joining the IL. Their 16-year relationship with the St. Louis Cardinals also ended after the 1997 season when the Cardinals chose to move their Triple-A affiliation to the Memphis Redbirds, a PCL expansion team. Louisville then became the top farm club of the Milwaukee Brewers in 1998. They started their time in the IL by winning the Western Division title, but in the Governors' Cup playoffs, they were eliminated in the semifinals by the Durham Bulls, 3–0. In 1999, the team rebranded as the Louisville RiverBats. The name was a dual reference to Louisville's location on the Ohio River and baseball bats, such as those manufactured locally under the Louisville Slugger brand. Their new logos incorporated a bat (the animal) swinging a baseball bat. After a second season with the Brewers, Louisville entered into a new affiliation with the Cincinnati Reds in 2000. They also left Cardinal Stadium for the new US$27.8 million Louisville Slugger Field. Located in downtown Louisville, the new ballpark seats 13,131 people with a more intimate setting than their previous facility.

In 2001, manager Dave Miley led the RiverBats to an 84–60 record and the Western Division title. They defeated the Norfolk Tides, 3–2, in the semifinals to advance to the final round against the Scranton/Wilkes-Barre Red Barons. Louisville won the first game of the series, 2–1, before the September 11 terrorist attacks resulted in the cancellation of the rest of the playoffs. The RiverBats were declared the International League champions for 2001.

Louisville dropped "River" from its name to become simply the Louisville Bats in 2002. Outfielder Raúl González won that season's IL MVP Award. The team returned to the playoffs in 2003, with another division title, but they were ousted in the semifinals by Durham, 3–1. In 2007, first baseman Joey Votto was selected for the IL Rookie of the Year Award. The Bats made three consecutive trips to the postseason from 2008 to 2010 with back-to-back-to-back division titles, but they were eliminated by Durham each time in the semifinals. Rick Sweet won the IL Manager of the Year Award in 2008 and 2009, and Justin Lehr won the Most Valuable Pitcher Award in 2009. Typically finishing third or fourth out of four teams in the Western Division, Louisville did not again qualify for the Governors' Cup playoffs after 2010.

Right fielder Aristides Aquino was selected as the 2019 IL Rookie of the Year. The start of the 2020 season was postponed due to the COVID-19 pandemic before being cancelled on June 30.

Triple-A East / International League (2021–present)
Following the 2020 season, Major League Baseball assumed control of Minor League Baseball in a move to increase player salaries, modernize facility standards, and reduce travel. The Cincinnati Reds retained Louisville as their Triple-A affiliate, but the International League disbanded, and the Bats followed the other IL teams into the Triple-A East. Louisville ended the season in sixth place in the Midwestern Division with a 51–68 record. No playoffs were held to determine a league champion; instead, the team with the best regular-season record was declared the winner. However, 10 games that had been postponed from the start of the season were reinserted into the schedule as a postseason tournament called the Triple-A Final Stretch in which all 30 Triple-A clubs competed for the highest winning percentage. Louisville finished the tournament tied for 18th place with a 4–5 record. In 2022, the Triple-A East became known as the International League, the name historically used by the regional circuit prior to the 2021 reorganization. In December 2023, the team was purchased by Diamond Baseball Holdings.

Logos and colors

The Louisville Redbirds based their original visual identity on that of their major league parent, the St. Louis Cardinals, from 1982 to 1998. The team's uniforms were of red and white, while their logo featured a cardinal preparing to swing a baseball bat. When they rebranded as the RiverBats in 1999, their color scheme became purple and green, while the logo depicted a bat swinging a baseball bat. When this moniker was shortened to "Bats" in 2002, the palate was switched to purple, black, and gold, and the logo to an abstract bat shape. The Bats' current color scheme of red, navy blue, and white was adopted in 2016. Additionally, the rebranding updated the logo to depict a flying bat in front of a moon, which resembled a baseball, while gripping a baseball bat in its talons.

Radio and television
Since 2019, Jim Kelch has been the play-by-play announcer for Bats radio broadcasts. He previously served in the role from 1989 to 2009 before leaving to work for the Cincinnati Reds broadcast team from 2010 to 2017. Other Louisville broadcasters that went on to broadcast for MLB teams include Joe Buck (1989–1990), Todd Kalas (1991), and Mark Neely (1992–1994).

As of 2019, all Bats home and road games were broadcast on WKRD 790 AM. Live audio broadcasts are also available online through the team's website and the MiLB First Pitch app. Games can be viewed through the MiLB.TV subscription feature of the official website of Minor League Baseball, with audio provided by a radio simulcast.

Awards

The franchise has been awarded these honors by Minor League Baseball.

Seven players, two managers, and two executives have won league awards in recognition for their performance with Louisville.

Retired numbers
Louisville has honored one of its players by retiring their uniform number. Corky Miller's number 8 was retired on August 31, 2014. A catcher from 2001 to 2004 and 2009 to 2014, Miller held the franchise record for games played (548) and doubles (99) at the time of its retirement.

The Bats also retired the number 1 for Louisville native and Baseball Hall of Famer Harold "Pee Wee" Reese in a ceremony on July 28, 2018. Reese's Brooklyn Dodgers teammate Jackie Robinson, whose number 42 was retired throughout professional baseball on April 15, 1997, was honored at the same ceremony.

Managers
Louisville has had 17 managers since their inaugural 1982 season.

title: Louisville Bats


text:
Talk:Bat

How much they eat
It says: "Insectivorous bats may eat over 120 percent of their body weight, while frugivorous bats may eat over twice their weight.".

It stands to reason that a bat must have eaten at least 100% of its body-weight over the course of its lifetime. But this statistic is meaningless without saying what the timescale is. I presume it's per day, but I don't have access to the cited source.

Can someone with access to the source fix it?

MrDemeanour (talk) 14:16, 21 May 2023 (UTC)

Semi-protected edit request on 8 June 2023
Change "Most of the oldest known bat fossils were already very similar to modern microbats, such as Archaeopteropus (32 million years ago)." to "Most of the oldest known bat fossils were already very similar to modern microbats, such as Archaeopteropus (32 million years ago). The oldest known bat fossil is the Icaronycteris gunnelli (52 million years ago). The two sets of fossils were discovered in Wyoming." AAAAAA14 (talk) 13:35, 8 June 2023 (UTC)
 * Red information icon with gradient background.svg Not done: please provide reliable sources that support the change you want to be made. Actualcpscm (talk) 14:36, 8 June 2023 (UTC)
 * done Ktkvtsh (talk) 01:49, 4 September 2023 (UTC)
 * ✅ Ktkvtsh (talk) 01:56, 4 September 2023 (UTC)

Potentially incorrect citation about heterothermy.
The Thermoregulation subheading specifically cites Nowack, J.; Stawski, C.; Geiser, F. (2017) in the first sentence as the source for information about the differential nature of heterothermy between bats. Whilst this citation is useful in the general context it doesn't specifically address this question at all, instead focusing on broadening our concept of the reasons for torpor instead!

A more appropriate source would instead be Stawski, Willis, & Geiser, 2014 which has a table (Table 1) that summarises thermal physiology of bats by family. Skeletorfw (talk) 11:52, 21 August 2023 (UTC)
 * ✅; added the source you suggested. —TechnoSquirrel69 (sigh) 03:52, 8 September 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Bat


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User talk:Rmilson

Administration
Hi Rmilson. Please don't have a go at me on AfD - assume good faith and all that. I'm just following procedure where speedy delete and PROD tags have been removed, which is to send the articles to AfD to allow the community to come to a consensus and the author of the articles to explain any reasoning for keeping them. It isn't a vote, it's a discussion, so assuming any motivation in me where there is none helps no-one. Thanks. ➨ ❝ R E  DVERS ❞ 10:44, 24 April 2006 (UTC)

No worries. I didn't mean for my comment to come across as overly adversarial, but I did want to contest your assertions. I will try for a different choice of wording. Rmilson 10:47, 24 April 2006 (UTC)

draft revision of the intoductory section of the manifold entry.
In mathematics, a manifold is an abstract multi-dimensional space where one cannot meaningfully make a choice of coordinate system. In contrast to Euclidean space, one cannot distinguish between a straight line and a curve, nor define the distance between two points. Another difference is that many manifolds possess a complicated spatial structure when viewed as a whole. There are finitary manifolds that are nonetheless unbounded ; other manifolds possess holes, regions where a circle cannot be continuously shrunk to a point.

The surface of Earth is an example of a manifold; locally it seems to be flat, but viewed as a whole it is round. However, since manifold geometry aims to be a minimalist theory, the surface of a sphere and the surface of an ellipsoid cannot be distinguished as manifolds. A manifold can be constructed by "gluing" separate Euclidean spaces together; for example, a world map can be made by gluing many maps of local regions together, and accounting for the resulting distortions. Alternatively, a manifold can be defined by a system of equations. However, as a geometric entity in and of itself, a manifold exists independently of any ambient space.

title: User talk:Rmilson


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Talk:Hamilton Fish III

Harlem Hellfighters
Harlem Hellfighters: The 369th Infantry Regiment of the New York National Guard 15 Regiment. An all African American unit, the 369th fought in World War I under the French Flag because the United States refused to have African American soldiers in combat. The 369th compiled an astounding war record and were decorated by the French government. But when they returned home to the United States, they were subjected to the racism of the era, and discovered their service in World War I meant nothing to their fellow Americans. —Preceding unsigned comment added by Special:Contributions/ (talk)
 * Please do not comment here on the struck-thru material above: the level of detail is irrelevant to this talk page's article, its use concerns Harlem Hellfighters, and thus discussion belongs at talk:Harlem Hellfighters instead. --Jerzy•t 02:32, 20 May 2006 (UTC)

oldest living American politician
I removed
 * When Fish celebrated his 102nd birthday in 1990, he was the oldest living American politician.

When stated clearly enuf, it may belong in the article, but for now:
 * 1) "Politician" does not have a precise definition;
 * 2) Once qualifying as a politician, are you one for life, or do you become a retired politician at some point?
 * 3) In practice, there are too many politicians to keep track of which, e.g., town supervisors and school board members (even the non-retired ones, i expect) are still alive, so this is not verifiable unless it is trying to say, e.g., Federal office-holders, or Federal and statewide, or Federal and state (the third taking in state legislators) -- and it can't be verified without clarifying which restricted group is intended. --Jerzy•t 02:32, 20 May 2006 (UTC)

All-black
I modified the description of the regiment to avoid implying that Fish was black, without bothering to do any research: almost certainly all commissioned officers were white; it might be worth knowing about the non-coms. --Jerzy•t 02:32, 20 May 2006 (UTC)

Trivia
Shared an apartment with the widow of Jose Capablanca -- apparently the two did not speak to each other —Preceding unsigned comment added by Jrm2007 (talk • contribs) 09:41, 19 November 2007 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot (talk) 19:07, 5 January 2008 (UTC)

"Reds"
Is this the Hamilton Fish who was one of the witnesses in Warren Beatty's 1981 movie "Reds"? If so, shouldn't this be mentioned? — Preceding unsigned comment added by Jtyroler (talk • contribs) 20:38, 27 August 2012 (UTC)

Yes, it was that Hamilton Fish, and I agree that it should be mentioned. Lost Jaguar (talk) 18:04, 17 December 2015 (UTC)

 * Done.
 * Billmckern (talk) 19:44, 17 December 2015 (UTC)

Fish and civil rights
The article ought to say something about Fish' lifelong commitment to civil rights for African-Americans, including his championing of allowing African-Americans to serve in combat units in the Second World War. For example: http://articles.latimes.com/1991-01-20/news/mn-630_1_hamilton-fish and http://www.lwfaam.net/ww2/

I believe this would bring a better balance to the article. --Yaush (talk) 04:00, 8 September 2013 (UTC)

 * Done.
 * Billmckern (talk) 20:22, 17 December 2015 (UTC)

Wife Lydia Died
I was looking at the Poughkeepsie Journal and his 4th wife, Lydia died this past week - someone my want to edit this - here is the reference - http://www.legacy.com/obituaries/poughkeepsiejournal/obituary.aspx?pid=173840110 — Preceding unsigned comment added by 107.181.17.194 (talk) 01:49, 20 January 2015 (UTC)

 * Done.
 * Billmckern (talk) 19:45, 17 December 2015 (UTC)

To add to article
To add to this article: the fact that, in 1938, Fish spoke at a German Day rally at Madison Square Garden from a stage decorated with swastikas. 204.11.189.94 (talk) 12:25, 3 November 2022 (UTC)

Coverage in Ultra podcast
To consider adding to this article: Hamilton Fish's pro-Nazi efforts are discussed in Episode 5, 7, and 8 of Rachel Maddow's podccast Ultra (2022).

Links:

 * https://www.msnbc.com/msnbc-podcast/rachel-maddow-presents-ultra/episode-5-shut-it-down-n1300315
 * https://www.msnbc.com/msnbc-podcast/rachel-maddow-presents-ultra/episode-7-rinse-repeat-n1300745
 * https://www.msnbc.com/msnbc-podcast/rachel-maddow-presents-ultra/episode-8-ultra-vires-n1300884

204.11.189.94 (talk) 12:32, 3 November 2022 (UTC)

 * @204.11.189.94 Agreed. Some of this is mentioned in the article, but as currently written it buries the "minor detail" that he was a Nazi agent while serving in Congress. Pstemari (talk) 07:18, 19 November 2022 (UTC)

Specifics? References?
"During his tenure in Congress, accusations of collusion with Nazis came to light. These accusations were later confirmed by the discovery of Nazi documents seized after the end of World War II. Later, his ties to George Sylvester Viereck, a top Nazi spy working in the US to spread pro-Hitler and anti-Semitic propaganda, were confirmed by the discovery of correspondence between Viereck and US Senator Ernest Lundeen."

This needs elaboration in the main article. Lundeen and his correspondence is neither mentioned nor referenced in the main body of the article and Fish is only related to Viereck in the actions of his chief of staff. The latter related to an earlier statement about German support for isolationist Republicans. A passing remark about Fish and other isolationists aping German talking points could be fleshed out better. More evidence is needed to effectively support the specific claim of collusion, so the Nazi documents should be discussed and not just mentioned in passing. 69.117.197.29 (talk) 08:45, 10 July 2023 (UTC)

title: Talk:Hamilton Fish III


text:
Henrico

Henrico may refer to:

Colonial America

 * City of Henrico (Virginia Company), also known as Henrico, a county in the Colony of Virginia
 * Henricus, also called Henrico, a settlement in the county, founded in 1611
 * Henrico Shire, one of the eight Shires of Virginia, established in 1634

United States

 * Henrico, North Carolina, an unincorporated community
 * Henrico County, Virginia

People

 * Henrico Atkins (born 1966), Barbadian former sprinter
 * Henrico Botes (born 1979), Namibian former footballer
 * Henrico Drost (born 1987), Dutch footballer

Other uses

 * USS Henrico (APA-45), an attack transport of World War II and the Korean War
 * Henrico High School, Henrico County, Virginia
 * Henrico Theatre, Henrico County, on the National Register of Historic Places

title: Henrico


text:
Calculus (dental)

In dentistry, calculus or tartar is a form of hardened dental plaque. It is caused by precipitation of minerals from saliva and gingival crevicular fluid (GCF) in plaque on the teeth. This process of precipitation kills the bacterial cells within dental plaque, but the rough and hardened surface that is formed provides an ideal surface for further plaque formation. This leads to calculus buildup, which compromises the health of the gingiva (gums). Calculus can form both along the gumline, where it is referred to as supragingival ("above the gum"), and within the narrow sulcus that exists between the teeth and the gingiva, where it is referred to as subgingival ("below the gum").

Calculus formation is associated with a number of clinical manifestations, including bad breath, receding gums and chronically inflamed gingiva. Brushing and flossing can remove plaque from which calculus forms; however, once formed, calculus is too hard (firmly attached) to be removed with a toothbrush. Calculus buildup can be removed with ultrasonic tools or dental hand instruments (such as a periodontal scaler).

Etymology
The word comes from Latin calculus "small stone", from calx "limestone, lime", probably related to Greek χάλιξ chalix "small stone, pebble, rubble", which many trace to a Proto-Indo-European root for "split, break up". Calculus was a term used for various kinds of stones. This spun off many modern words, including "calculate" (use stones for mathematical purposes), and "calculus", which came to be used, in the 18th century, for accidental or incidental mineral buildups in human and animal bodies, like kidney stones and minerals on teeth.

Tartar, on the other hand, originates in Greek as well (tartaron), but as the term for the white encrustation inside casks (a.k.a. potassium bitartrate, commonly known as cream of tartar). This came to be a term used for calcium phosphate on teeth in the early 19th century.

Chemical composition
Calculus is composed of both inorganic (mineral) and organic (cellular and extracellular matrix) components.

In supra-gingival calculus
The mineral proportion of supragingival calculus ranges from approximately 40–60%, depending on its location in the dentition, and consists primarily of calcium phosphate crystals organized into four principal mineral phases, listed here in order of decreasing ratio of phosphate to calcium:
 * whitlockite, Ca9(Mg,Fe)(PO4)6(PO3OH)
 * hydroxyapatite, Ca5(PO4)3OH
 * octacalcium phosphate, Ca8H2(PO4)6 * 5 H2O
 * and brushite, CaHPO4 * 2 H2O

The organic component is approximately 85% cellular and 15% extracellular matrix. Cell density within dental plaque and calculus is very high, consisting of an estimated 200,000,000 cells per milligram. The cells within calculus are primarily bacterial, but also include at least one species of archaea (Methanobrevibacter oralis) and several species of yeast (e.g., Candida albicans). The organic extracellular matrix in calculus consists primarily of proteins and lipids (fatty acids, triglycerides, glycolipids, and phospholipids), as well as extracellular DNA. Trace amounts of host, dietary, and environmental microdebris are also found within calculus, including salivary proteins, plant DNA, milk proteins, starch granules, textile fibers, and smoke particles.

In sub-gingival calculus
Sub-gingival calculus is composed almost entirely of two components: fossilized anaerobic bacteria whose biological composition has been replaced by calcium phosphate salts, and calcium phosphate salts that have joined the fossilized bacteria in calculus formations.

The following minerals are detectable in calculus by X-ray diffraction:
 * brushite (CaHPO4 * 2 H2O)
 * octacalcium phosphate (Ca8H2(PO4)6 * 5 H2O)
 * magnesium-containing whitlockite (Ca9(Mg,Fe)(PO4)6(PO3OH))
 * carbonate-containing hydroxyapatite (approximately Ca5(PO4)3OH but containing some carbonate).

Calculus formation
Dental calculus typically forms in incremental layers that are easily visible using both electron microscopy and light microscopy. These layers form during periodic calcification events of the dental plaque, but the timing and triggers of these events are not well understood. The formation of calculus varies widely among individuals and at different locations within the mouth. Many variables have been identified that influence the formation of dental calculus, including age, sex, ethnic background, diet, location in the oral cavity, oral hygiene, bacterial plaque composition, host genetics, access to professional dental care, physical disabilities, systemic diseases, tobacco use, and drugs and medications.

Supragingival calculus formation is most abundant on the buccal (cheek) surfaces of the maxillary (upper jaw) molars and on the lingual (tongue) surfaces of the mandibular (lower jaw) incisors. These areas experience high salivary flow because of their proximity to the parotid and sublingual salivary glands.

Subgingival calculus forms below the gumline and is typically darkened in color by the presence of black-pigmented bacteria, whose cells are coated in a layer of iron obtained from heme during gingival bleeding. The reason fossilized bacteria are initially attracted to one part of the subgingival tooth surface over another is not fully understood. However, once the first layer is attached, more calculus components are naturally attracted to the same places due to electrical charge. This is because the calcium phosphate salts contained in them exist as electrically unstable ions (unlike calcium phosphate, the primary component of teeth). The fossilized bacteria pile up rather haphazardly, while free-floating ionic components (calcium phosphate salts) fill in the gaps.

The resultant hardened structure can be compared to concrete; with the fossilized bacteria playing the role of aggregate, and the smaller calcium phosphate salts being the cement. The "hardened" calculus formations are at the heart of periodontal disease and treatment.

Clinical significance
Plaque accumulation causes the gingiva to become irritated and inflamed, and this is referred to as gingivitis. When the gingiva become so irritated that there is a loss of the connective tissue fibers that attach the gums to the teeth and bone that surrounds the tooth, this is known as periodontitis. Dental plaque is not the sole cause of periodontitis; however it is many times referred to as a primary aetiology. Plaque that remains in the oral cavity long enough will eventually calcify and become calculus. Calculus is detrimental to gingival health because it serves as a trap for increased plaque formation and retention; thus, calculus, along with other factors that cause a localized build-up of plaque, is referred to as a secondary aetiology of periodontitis.

When plaque is supragingival, the bacterial content contains a great proportion of aerobic bacteria and yeast, or those bacteria which utilize and can survive in an environment containing oxygen. Subgingival plaque contains a higher proportion of anaerobic bacteria, or those bacteria which cannot exist in an environment containing oxygen. Several anaerobic plaque bacteria, such as Porphyromonas gingivalis, secrete antigenic proteins that trigger a strong inflammatory response in the periodontium, the specialized tissues that surround and support the teeth. Prolonged inflammation of the periodontium leads to bone loss and weakening of the gingival fibers that attach the teeth to the gums, two major hallmarks of periodontitis. Supragingival calculus formation is nearly ubiquitous in humans,  but to differing degrees. Almost all individuals with periodontitis exhibit considerable subgingival calculus deposits. Dental plaque bacteria have been linked to cardiovascular disease and mothers giving birth to pre-term low weight infants, but there is no conclusive evidence yet that periodontitis is a significant risk factor for either of these two conditions.

Prevention
Toothpaste with pyrophosphates or zinc citrate has been shown to produce a statistically significant reduction in plaque accumulation, but the effect of zinc citrate is so modest that its clinical importance is questionable. Some calculus may form even without plaque deposits, by direct mineralisation of the pellicle.

Calculus in other animals
Calculus formation in other animals is less well studied than in humans, but it is known to form in a wide range of species. Domestic pets, such as dogs and cats, frequently accumulate large calculus deposits. Animals with highly abrasive diets, such as ruminants and equids, rarely form thick deposits and instead tend to form thin calculus deposits that often have a metallic or opalescent sheen. In animals, calculus should not be confused with crown cementum, a layer of calcified dental tissue that encases the tooth root underneath the gingival margin and is gradually lost through periodontal disease.

Archaeological significance
Dental calculus has been shown to contain well preserved microparticles, DNA and protein in archaeological samples. The information these molecules contain can reveal information about the oral microbiome of the host and the presence of pathogens. It is also possible to identify dietary sources as well as study dietary shifts and occasionally evidence of craft activities.

Removal of calculus after formation
Plaque and calculus deposits are a major etiological factor in the development and progression of oral disease. An important part of the scope of practice of a dental hygienist is the removal of plaque and calculus deposits. This is achieved through the use of specifically designed instruments for debridement of tooth surfaces. Treatment with these types of instruments is necessary as calculus deposits cannot be removed by brushing or flossing alone. To effectively manage disease or maintain oral health, thorough removal of calculus deposits should be completed at frequent intervals. The recommended frequency of dental hygiene treatment can be made by a registered professional, and is dependent on individual patient needs. Factors that are taken into consideration include an individual's overall health status, tobacco use, amount of calculus present, and adherence to a professionally recommended home care routine.

Hand instruments are specially designed tools used by dental professionals to remove plaque and calculus deposits that have formed on the teeth. These tools include scalers, curettes, jaquettes, hoes, files and chisels. Each type of tool is designed to be used in specific areas of the mouth. Some commonly used instruments include sickle scalers which are designed with a pointed tip and are mainly used supragingivally. Curettes are mainly used to remove subgingival calculus, smooth root surfaces and to clean out periodontal pockets. Curettes can be divided into two subgroups: universals and area specific instruments. Universal curettes can be used in multiple areas, while area specific instruments are designed for select tooth surfaces. Gracey curettes are a popular type of area specific curettes. Due to their design, area specific curettes allow for better adaptation to the root surface and can be slightly more effective than universals. Hoes, chisels, and files are less widely used than scalers and curettes. These are beneficial when removing large amounts of calculus or tenacious calculus that cannot be removed with a curette or scaler alone. Chisels and hoes are used to remove bands of calculus, whereas files are used to crush burnished or tenacious calculus.

Ultrasonic scalers, also known as power scalers, are effective in removing calculus, stain, and plaque. These scalers are also useful for root planing, curettage, and surgical debridement. Not only is tenacious calculus and stain removed more effectively with ultrasonic scalers than with hand instrumentation alone, it is evident that the most satisfactory clinical results are when ultrasonics are used in adjunct to hand instrumentation. There are two types of ultrasonic scalers; piezoelectric and magnetostrictive. Oscillating material in both of these handpieces cause the tip of the scaler to vibrate at high speeds, between 18,000 and 50,000 Hz. The tip of each scaler uses a different vibration pattern for removal of calculus. The magnetostrictive power scaler vibration is elliptical, activating all sides of the tip, whereas the piezoelectric vibration is linear and is more active on the two sides of the tip.

Special tips for ultrasonic scalers are designed to address different areas of the mouth and varying amounts of calculus buildup. Larger tips are used for heavy subgingival or supragingival calculus deposits, whereas thinner tips are designed more for definitive subgingival debridement. As the high frequency vibrations loosen calculus and plaque, heat is generated at the tip. A water spray is directed towards the end of the tip to cool it as well as irrigate the gingiva during debridement. Only the first 1–2 mm of the tip on the ultrasonic scaler is most effective for removal, and therefore needs to come into direct contact with the calculus to fracture the deposits. Small adaptations are needed in order to keep the tip of the scaler touching the surface of the tooth, while overlapping oblique, horizontal, or vertical strokes are used for adequate calculus removal.

Current research on potentially more effective methods of subgingival calculus removal focuses on the use of near-ultraviolet and near-infrared lasers, such as Er,Cr:YSGG lasers. The use of lasers in periodontal therapy offers a unique clinical advantage over conventional hand instrumentation, as the thin and flexible fibers can deliver laser energy into periodontal pockets that are otherwise difficult to access. Near-infrared lasers, such as the Er,CR:YSGG laser, have been proposed as an effective adjunct for calculus removal as the emission wavelength is highly absorbed by water, a large component of calculus deposits. An optimal output power setting of 1.0-W with the near-infrared Er,Cr:YSGG laser has been shown to be effective for root scaling. Near-ultraviolet lasers have also shown promise as they allow the dental professional to remove calculus deposits quickly, without removing underlying healthy tooth structure, which often occurs during hand instrumentation. Additionally, near-ultraviolet lasers are effective at various irradiation angles for calculus removal. Discrepancies in the efficiency of removal are due to the physical and optical properties of the calculus deposits, not to the angle of laser use. Dental hygienists must receive additional theoretical and clinical training on the use of lasers, where legislation permits.

title: Calculus (dental)


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Talk:Riemann integral

Animated image
Hello. Riemann integral is a very nice article. I wonder about the animated image. I agree that it's useful for viewing in a browser, but I don't think it would be as useful on a printed page. I don't see a good reason to rule out printed copies; this article could easily be something a student would print out for reference, for example. Can we convey the same information by displaying each frame of the animation separately? -- Sequences of graphs are going to be pretty common in the math articles, and so I think it's important to set a good precedent here. I look forward to your comments. Wile E. Heresiarch 16:38, 21 Jun 2004 (UTC)

 * Personally I think the animated image is great. It really gets accross the idea well and takes advantage of what the web format can offer.  I do see your point that in print it would not work.  So the answer would not be to remove the animation from the web version, but to make sure it translated to a series of images perhaps in a future print version. - Taxman 20:38, Jun 21, 2004 (UTC)

Older discussion
Maybe needs paragraph about Stieltjes generalization. Or perhaps link to separate article about same.
 * Done

Do you mean totally bounded functions and all that stuff? Loisel 07:30 30 Jun 2003 (UTC)

Image
Isn't it far TOO BIG?

Merge?
I think the Riemann sum and the Riemann integral have too much in common. I suggest merging information from them into Riemann integral, and make Riemann sum a redirect. —Preceding unsigned comment added by Igny (talk • contribs)

 * I would not be opposed to it. But I think you would need to also write your proposal on talk:Riemann sum and maybe even contact User:Icedemon who had put a lot of effort into that one. Oleg Alexandrov (talk) 18:19, 5 December 2005 (UTC)

 * I was not aware that a Riemann integral site existed. I agree that hte two should be merged, perhaps entering the work I did on Right/Left/Middle/Trapezoidal into the Examples section of the Riemann Integral page. Icedemon 03:29, 7 December 2005 (UTC)
 * Well, in this case I would not be for the move. The article Riemann integral is rather well-written and self-contained, while the article Riemann sum has a lot of wording which I find reduntant. All those sections on left Riemann sum, right Riemann sum, middle Riemann sum could be described just in a couple of sentences instead of several screenfuls of text. In fact, I already did that, see the ==Defintion== section in Riemann sum. So I would agree to move from Riemann sum to Riemann integral just the picture and a paragraph or two; most of the work there is reduntant. Otherwise, I would suggest leaving thing the way they are. Oleg Alexandrov (talk) 04:26, 7 December 2005 (UTC)
 * I agree that Left, Right, Middle, and Trapezoidal sums can be explained in a few sentences, but not to anybody that is not a mathematician. Please remember that the goal of Wikipedia is not only to be succint, but also to explain the subject in a manner that is understandable by everybody. I have attempted to do so in Riemann sum, and you still refer to my work as "redundant". Redundancy is necessary, in this case, in order to properly explain the material to anybody that does not have a degree in mathematics. As for hte merge, I maintain that either the Riemann Sum Left,Middle,Right, and Trapezoidal be moved in their entirety to Riemann integral or the pages remain separate. Icedemon 00:32, 9 December 2005 (UTC)
 * That's what I am saing too. The material at Riemann sum is way too detailed to be included in Riemann integral. So then the two articles should be separate. Oleg Alexandrov (talk) 01:29, 9 December 2005 (UTC)

LaTeX formatting
Who tagged this page as needing LaTeX formatting, and what do you need done? –Ryan McDaniel 15:02, 15 March 2006 (UTC)
 * You can tell who put the tag from the article history. The work to be done is converting html formulas to TeX formulas, which I think is not urgent at all, or maybe not even necessary. I'd suggest the template be removed from this talk page. Oleg Alexandrov (talk) 02:29, 16 March 2006 (UTC)

direct proof of f *g is Riemann integrable?
Given that both f and g are real functions, Riemann integrable on [a,b], then how to prove DIRECTLY that f * g (f times g) is also Riemann - integrable on [a,b]? —The preceding unsigned comment was added by 142.104.2.101 (talk) 18:50, 16 March 2007 (UTC).

Original definition
This article is really well-written and interesting. Does this article give the original definition of Riemann?

In most textbooks, e.g. Cohn: Measure theory, the equivalent definition by Darboux is given and they still call it the Riemann integral.

The Darboux definition is easier for a student to understand. Perhaps the main article should give the easier definition by Darboux, and then another article should give the original definition by Riemann. Pierreback 00:56, 25 March 2007 (UTC)

Link to Java program
I included the following link:

[] Interactive computer program about the Darboux integral (in German)

I think it is a really nice java program which gives a feeling about the definition. Unfortunately it is in German and has now been removed (by another user) from the article. In my opinion it is good enough to be included. What is the general opinion? 83.253.10.236 23:55, 25 March 2007 (UTC)

Merger proposal
Rectangle method and Riemann sum should be merged into Riemann integral. Rectangle method is about the method of finding a Riemann sum; it is not a long article and could be its own section in Riemann integral. Riemann sum is the same thing as Riemann integral, with a simpler name. (See above for earlier discussion). Mazin07C₪T 15:49, 7 April 2007 (UTC)
 * I don't think that all those facts about left Riemann sum, right sum, etc, and all those pictures from Riemann sum belong at Riemann integral. That's too much detail. In Riemann integration all that matters is that the Riemann sum converges regardless of how the points inside the intervals are chosen.

 * So if a merger is done, I think very little from Riemann sum should make it in Riemann integral.

 * My preferred solution would be to keep things the way they are, and refer to the Riemann sum article in the "Riemann sum" section on Riemann integral, for people who want extended knowledge of the sums. Oleg Alexandrov (talk) 16:13, 7 April 2007 (UTC)

 * I'd agree that Riemann sum goes into too much detail for this article, and there is even more to say about Riemann sums if you want to. It does make sense to merge Rectangle method and Riemann sum, but I'd keep them separate from Riemann integral. -- Jitse Niesen (talk) 04:16, 11 April 2007 (UTC)

 * Rectangle method would be much better merged with midpoint method than Riemann integral. 69.231.241.75 06:12, 29 June 2007 (UTC)

Remember the Reader
I looked at the article on Riemman sum, and it's done up in advanced formal notation. While there are, of course similarities, Rectangle and Riemman are in in completely different leagues; not to mention the significant historical and pedagogical differences. To give an other example, playdoh and Riemman's contributions to differential geometry are related, and deformable solids such as playdoh (or some unspecified dough) are often used to convey the idea of smooth continuous deformation; however, it really just wouldn't be appropriate to lump (no pun intended) playdoh and non-euclidian geometry together in the same article; instead you have links pointing to each. And likewise, the Riemman integral could point to and be pointed to by the Rectangle article.

A merger could hold back Riemman (properly in the realm of advanced mathematics), and would certainly add luggage to the Rectangle article (a famous person, familiar with non-euclidean geometry once said: 'as simple as possible, just not too simple' [paraphrased from the original german])--ElectronicsEnthusiast 00:28, 22 July 2007 (UTC)

Unencylopedic tone and scope
This piece is very nicely written, but it has a didactic quality which is not appropriate for an encylcopedia article. It reads like a set of lecture notes, often saying "we see" and giving first provisional definitions and then after some discussion fixing them up to give the correct definition.

The article is also sorely lacking in historical information. References to Riemann's original papers are required. As another example, the claim that the Riemann integral is the first rigorous definition of an integral needs to be supported and explained by a comparison to the concept of integral used in the 200 years between the founding of calculus and the Riemann integral. Plclark 04:07, 30 September 2007 (UTC)Plclark

 * If you don't like it, be bold and fix it: Write out all the "we see"s and find references to Riemann's papers. I agree that the article sounds didactic, but when I wrote it (I'm responsible for most of the text) I figured this was the most comprehensible way to describe the Riemann integral to non-specialists.

 * I don't think the claim that the Riemann integral is the first rigorous definition is correct, though I did hear it made somewhere; what is more likely true is that it is the first definition of an integral that is independent of the notion of the derivative. My understanding is that before Riemann, the Fundamental Theorem of Calculus was taken as the definition of a definite integral.  However, I'm no historian and I could be wrong (even completely wrong) about this.  141.211.62.20 00:43, 4 October 2007 (UTC)

Continuity
I want to point out that the discussion on continuity requirements for Riemann integration refer to measure theory. This seems unfortunate as one reason for using Riemann's approach is to avoid measure theory! Is there a concise and precise way to describe the requirements that does not use measure theory? Acolombi 17:07, 15 October 2007 (UTC)

 * Yes and no. The condition, precisely stated, is that the set of discontinuities has measure zero.  But the concept of measure zero (with respect to Lebesgue measure) can be defined independently of the whole framework of measure theory: A measure zero set is simply one which can be covered by a set of balls B(xi, εi) such that the sum of all εi (2εi if you're pedantic) can be made arbitrarily small.  Perhaps that should be added to the article on almost everywhere.  141.211.62.20 01:11, 18 October 2007 (UTC)

What the damn?
what the damn does half this stuff mean? I am currently doing senior mathematics in high school and I have no idea what this means can someone explain it in layman's terms if need be? what i know is 'elongated "s" (integral sign) of a (lower limit) and b (upper limit) of f(x) dx is the integral. Can someone please explain it to me without the x sub i's and weighted sums and stuff like that what i mean is is in the form of ....(b-a/2 + f(a) + (a+b/6),etc...) something like that so that I can understand what the **** they are talking about. —Preceding unsigned comment added by 61.68.208.134 (talk) 08:32, 29 April 2009 (UTC)

 * You may find Darboux integral a little more approachable. In general, however, I would advise you to study hard and think carefully about what the article is saying. The subject is intrinsically difficult; there is no royal road. You will understand integrals better the more you think about them. Ozob (talk) 17:02, 29 April 2009 (UTC)

 * If you're in high school, you've probably gotten the same relatively shallow mathematics education that almost everyone else gets in high school. The topic of this article is a fair bit deeper than that and if you want to understand it all you'll probably have to get some more background. In other words it's not likely that someone can explain all that much more simply than the article already does. Perhaps try reading the article again and focus on the overview parts, then dig into the notation and specifics as you gain experience. The topic of this article is commonly covered in textbooks with titles like Advanced Calculus. Try picking up a few used or new texts like that until you find one that seems the most approachable, and consider getting some books that lead up to it. Most likely you will need to become comfortable with the x sub i notation (known as subscript notation). Also try looking for open textbooks online. There are many good sources of free math texts, though most are at a high level. If you look enough you can find good ones. - Taxman Talk 17:10, 3 May 2010 (UTC)

Cauchy's definition of the integral
I've been reading a little about different definitions of the integral, and a couple of books mention the "Cauchy Integral" which was formulated before the Riemann Integral and is in fact a special case of the latter where the "tag" of each interval in the partition is chosen to be the left endpoint of the interval. I notice that Wikipedia (and seemingly most other online sources from a quick google) doesn't mention it at all, and Riemann Integral even goes as far to say "the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval." when it seems that Riemann merely generalized Cauchy's integral. Cauchy integral is also a redirect to Cauchy's integral theorem.

I feel like it should be included for completeness, at least from a historical perspective if nothing else. I'm no expert by any means, but I might have a go at making an article. It seems fairly odd that it is not referred to on WP at all so I'm not sure whether to just plow ahead and make an article (and change the Integral and Riemann Integral accordingly). It seems it would be fairly straightforward to include since the definitions are so similar to the Riemann.

So basically I'm just wondering if anyone objects to including this integral in WP, or knows anything about it. (also posting this at Talk:Integral) slimeknight (talk) —Preceding undated comment added 02:38, 25 November 2009 (UTC).

 * If Cauchy really did have a working rigorous definition of an integral, that would be news to me. I don't know where I first heard the claim that it was Riemann who first formulated a correct definition of an integral—it may have been this very article. If you have a reference that says otherwise and explains what Cauchy did, then adding it would be great! Ozob (talk) 20:59, 25 November 2009 (UTC)

 * Slimeknight's comment above appears to be quite correct. When I read the statement "the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval", my immediate reaction was that this is dead wrong. I have seen this integral referred to as the Cauchy-Riemann integral. The catch is that Cauchy appears to have applied this definition only to continuous functions. Riemann seems to deserve credit for having made the first examination of Cauchy's definition applied to all functions (or, at least, all bounded ones) and to have asked the question "What functions are integrable?". You can read all about it here: Look at Chapter 1, Section 1.2 (Catgod119 (talk) 00:10, 18 November 2016 (UTC))

Riemann integral "unworkable"?
The final text about improper integrals, from "This is also unacceptable..." seem to me a little harsh and misleading. First, I think that the final alternative of a improper integral definition (over the entire real line), with the double limit, has some undesirable properties but it is nonetheless "workable", and is the commonly accepted definition of that improper integral in the Riemmann integral context. This should be stressed. Second, the example of a uniformly convergent function that -in that definition- does not conmute with the integral is interesting, but misleading: it seems to imply that the example can be dealt by the Lebesgue integral, but (AFAIK) it cannot, nor could it: the integral of the succession fuctions are obviously 1, the function to which they converge is identically zero, so no integral (Riemman, Lebesgue or whatever) would permit us to interchange limits and integral in that case. Finally, to state that "This makes the Riemann integral unworkable in applications." is way too much. Riemman integral is just less useful for some theoretical work, that's all. —Preceding unsigned comment added by Leonbloy

Riemann integral equivalence to Darboux Integral
I have a question. Is the Riemann Integral really equivalent to the Darboux Integral?. For example, consider the function f(x)=sin(x)/x. I want to know if it is integrable between -1 and 1. The problem arises at x=0. Now, this function is clearly Darboux integrable, since you solve the problem of the non-existance of f(0) with the supremum operator. But I don't think it is Riemann integrable, since you can not evaluate f(0), and the Riemann definition clearly states that the sum must exist fon ANY election of evaluation numbers. If you defined f(0) as 1, you would not be solving the problem, because you would be integrating a different function,not f(x), since you are changing the domain of the function. I think it would be better to say that the Riemann and Darboux integral are equivalent iff f(x) is well-defined for every single point of the interval. Please correct me if I am wrong.I will deeply appreciate any replies. (DanteEspinoza1989 (talk) 06:20, 1 May 2010 (UTC))

 * I think the problem is that your f(x) is not integrable between −1 and 1. It's not defined everywhere in that interval, so it can't be integrable. After all, you can't integrate something that's not a function, and since f doesn't assign a value to zero, it's not a function.

 * A perhaps confusing aspect of this is that we can still integrate partially defined "functions" like f. We can change the value of a function at any single point (or more generally, on a set of measure zero) without changing the integral. What this means is that we can set f(0) = 1 and integrate, and we get the same thing as if we set f(0) = -999 and integrate. Or you could set f(0) to 37, or to 245.20485254, or whatever. So in a sense f does have an integral: To compute it, you first choose a value for f(0), then you take its (Riemann or Darboux or Lebesgue or ...) integral. That doesn't depend on your choice of f(0), so you can call that the integral of f.

 * You might appreciate the article Lp space (which is unfortunately a bit technical). We're considering p = 1. Ozob (talk) 20:06, 2 May 2010 (UTC)

Not all Riemann Integrals are Lebesgue Integrable
A function f is Lebesgue Integrable over a region if and only the function of the absolute value of f is Lebesgue Integrable over the same region. It is possible to construct a function f that is Riemann Integrable over a region but abs(f) is not Riemann Integrable over the same region. Provided that abs(f) were not Lebesgue Integrable over that region, neither would f be. I remember this as an exercise in my Measure Theory course in grad school, to provide such an f, but I don't remember how to provide an example.--Moly 12:41, 10 June 2010 (UTC) —Preceding unsigned comment added by Moly (talk • contribs)

 * This is not true. It may be true if you consider improper Riemann integrals, because improper integrals do funny things.  But choose a and suppose that for all ε > 0 there exists a δ > 0 such that |x - a| < δ implies |f(x) − f(a)| < ε; then also ||f(x)| − |f(a)|| < ε, so |f| is continuous at a.  I.e., the set where |f| is continuous contains the set where f is continuous, and hence the set of discontinuities has measure zero. Ozob (talk) 21:46, 10 June 2010 (UTC)
 * And even if you consider improper Riemann integrals, you should as well consider improper Lebesgue integrals, and again the latter include the former.—Emil J. 12:16, 11 June 2010 (UTC)

Cleanup of this Article
It looks as if this article needs some cleanup, as to increase clarity. I did so with the "Partitions of an Interval" section, and I would encourage others to do so too. Try to reduce redundancy of the same equations, and do not use excessive latex in the middle of the paragraph. Fraqtive42 (talk) 00:16, 22 July 2011 (UTC)

Counter example
I think it would be interesting to show as well what kind of integral is not a Riemann integral. — Preceding unsigned comment added by 189.115.235.6 (talk) 18:31, 4 October 2011 (UTC)

 * The article already gives an example of a non-Riemann-integrable function, and it links to Lebesgue integral, an example of another integration theory. Ozob (talk) 21:52, 4 October 2011 (UTC)

Definition
In the definition part of the article the following equation is found:

 * $$\left|\sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i) - s\right| < \varepsilon.$$

Shouldn't it be made clear that "s" (the value of the integral) is not part of the summation (by using parenthesis)? — Preceding unsigned comment added by 178.199.110.247 (talk) 16:27, 19 October 2011 (UTC)

Wrong proof at "Integrability section"
It was noticed by User:186.48.9.230 that the given proof at Integrability of the Vitali-Lebesgue criterion is wrong. The discontinuity points of the function may be dense, so that there is no chance to cut a given (finite) partition of the segment into those that contain discontinuities of the function and the others. What shoud be done is to consider the closed set with measure zero where the oscillation of the function is larger than some ε. I don't have time to do it now, can someone do it? Bdmy (talk) 10:05, 31 May 2013 (UTC) Bdmy (talk) 10:10, 31 May 2013 (UTC)
 * I made the needed corrections - actually, the whole proof had to be rewritten. This was tiring... Dan Gluck (talk) 20:14, 23 October 2015 (UTC)

Generalizations section
I think the assertion of an improper integral not being easily extended is not entirely correct, or at least not the best way to explain it. My textbook defines $$\int_{-\infty}^{\infty} f(x) dx = \int_{-\infty}^a f(x) dx + \int_a^{\infty} f(x) dx$$ and further states that the integral on the left exists if and only if both integrals on the right exist, where a is any arbitrary number. Since neither of those two exist in the example given, the integral over the entire real line of that function does not exist.

I alternatively viewed the problem as a multivariate limit $$\lim_{(a,b)\to(\infty,\infty)} \int_{-a}^b f(x) dx $$. Then the notation $$\lim_{c\to\infty} \int_{-c}^c f(x) dx $$ is really $$\lim_{(c,c)\to(\infty,\infty)} \int_{-c}^c f(x) dx $$, i.e. only one path (the graph of a=b) along which a and b can approach infinity. If the first limit where the path is arbitrary exists, then it is unique, so the last one must also exist too (in other words, if that bivariate limit exists, then we automatically have the Cauchy principal value as the value of the integral, but the converse is false). Likewise, the nested univariate limits simply consider a path consisting of line segments parallel to a=0 and/or b=0, which again can't prove that the bivariate limit along any path exists. Therefore, the example given in this section is overdoing it: the bivariate limit does not exist, so all the talk of the integral of that function over the entire real line is meaningless.

Basically, I'm asserting that this problem with Riemann integrals over the entire real line is not an issue with interchanging limits, "shifting the function", or anything along those lines. It's just a direct consequence of the definition of a bivariate limit. I personally would find this to be a much simpler explanation.--Jasper Deng (talk) 07:18, 7 April 2014 (UTC)

 * Philosophically, if you think of an integral as computing a signed area, then you should simply be able to compute the positive area, compute the negative area, and subtract to get the result. When you can't do this, then either you need to worry about symmetry (as with the Cauchy principal value or singular integrals) or your integral is too weak (as with the Riemann integral).  For a well-behaved integral the question of taking a limit of the domain should never come up; it should either be unnecessary or should be obviously and always true.  What I think the current text is trying to get at is that the problem with improper Riemann integrals is embedded in their very definition as limits.  We can't always avoid taking limits of the domain when working with integrals, but usually their presence indicates a problem, and they should never play a fundamental role in the theory.  This is the case for the Lebesgue integral, for example: The Lebesgue integral is defined as the positive area minus the negative area, each of those areas is defined by a limiting process with respect to the values of the function, and taking a limit of the domain is unnecessary; and if for some reason we do need to take a limit of the domain, then various theorems (dominated convergence, monotone convergence, etc.) make this easy (at least in principle).

 * I'm not sure that the article is expressing this well. Yes, it's true that the bivariate limit you described does not exist, and that its failure to exist underlies everything else that the article is saying.  But it's also true that a definition is justified by its properties, and the article shows why the definition as an improper integral has the wrong properties.  We really want a definition of the integral which takes the right value for, say, compactly supported continuous functions, and which is continuous with respect to an appropriate topology on that function space (and therefore defines and extends to L1).  That might make the article better than details of limits.  Ozob (talk) 14:26, 7 April 2014 (UTC)
 * Well, I think the explanation of those issues could be made more concise using something like "The Riemann integral of a function over the entire real line can be considered to be the bivariate limit $$\lim_{(a,b)\to(\infty,\infty)} \int_{-a}^b f(x) dx $$. However, this creates several problems. For example, in the case of $$f(x)=1$$, this limit does not exist, meaning the Cauchy principal value is not translation-independent." etc. We could elaborate on how this makes defining improper integrals as limits problematic as general, as you say, and thus emphasize this limitation of the Riemann integral. This would replace all of those "tries" at limit definitions that don't work as aformentioned.--Jasper Deng (talk) 07:19, 8 April 2014 (UTC)
 * OK, I rewrote that section. What do you think? Ozob (talk) 15:00, 8 April 2014 (UTC)
 * Seems pretty good. I made one minor addition.--Jasper Deng (talk) 17:14, 8 April 2014 (UTC)

Links [8] and [9] at "Notes" aren't in the article
The links [8] and [9] don't make a appeareance anywhere on the article, and in fact when i Click them, nothing occurs. I think someone deleted the links in the article but they remained in the "Notes" section. — Preceding unsigned comment added by Santropedro (talk • contribs) 20:43, 11 July 2015 (UTC)

 * These notes appear in the proof between notes [7] and [10], which is displayed only when you click on "show". D.Lazard (talk) 02:45, 12 July 2015 (UTC)

Assessment comment
Substituted at 02:33, 5 May 2016 (UTC)

Lebesgue's criterion
In the section "Integrability", it is stated A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). Later, in the section on "Generalisations", it is stated Moreover, a function ƒ defined on a bounded interval is Riemann-integrable if and only if it is bounded and the set of points where f is discontinuous has Lebesgue measure zero.

These two statements are close but not identical; in particular, the second statement says something about unbounded functions on a bounded interval (namely, it claims they are not Riemann-integrable), whereas the first statement does not. I believe that the second statement is correct (at least if the 'bounded interval' is replaced with a compact interval); and if this is so, perhaps the first statement should be amended to give this stronger claim. 130.88.123.107 (talk) 14:56, 24 February 2017 (UTC)

ℛ...?
This page is the destination of the redirect on ℛ. But the ℛ character is never mentioned. Is the redirect correct? Railwayfan2005 (talk) 22:17, 15 October 2018 (UTC)
 * I have nominate it for deletion (Redirects for discussion/Log/2018 October 16). D.Lazard (talk) 09:27, 16 October 2018 (UTC)

Lebesgue-Vitali Theorem
The proof of the Lebesgue-Vitali characterization theorem could maybe easier to read if broken as follows:

 * A bounded $$f:[a,b]\to\mathbb{R}$$ is Riemann integrable if and only if its upper and lower Darboux integrals coincide;
 * For any bounded $$f:[a,b]\to\mathbb{R}$$, the difference between the  upper and lower Darboux integrals  of $$f$$ is equal to the upper Darboux integral of the point oscillation of $$f$$:
 * $$\overline{\int_{a}^{b}} f(x)dx- \underline{\int_{a}^{b}} f(x)dx=\overline{\int_{a}^{b}} \omega_f(x)dx. $$

 * The point oscillation is a bounded, upper semicontinuous, non-negative function.
 * For any bounded, upper semicontinuous non-negative function $$g:[a,b]\to\mathbb{R}$$ one has
 * $$\overline{\int_{a}^{b}} g(x)dx=0$$ if and only if $$g(x)=0$$ a.e.

 * The point oscillation of $$f$$ vanishes exactly at the continuity points of $$f$$.

Therefore, a bounded function on $$[a,b]$$ is Riemann integrable if and only if it is continuous at almost any point $$x\in[a,b]$$. pm a 21:46, 24 March 2021 (UTC)

Riemann Darboux Equivalence
In Definition subsection Riemann integral, it's stated that the Darboux Integral article proves equivalence to the second definition given. I'm not convinced this is the case. The Darboux article proves that Darboux integrable implies Definition 2 and that Definition 1 implies Darboux Integrable. But the second proof (under "Details of finding the tags" on the Darboux page) doesn't begin with a tagged partition that is then refined; it just shows that arbitrarily fine partitions can be tagged to be arbitrarily close to the lower Darboux sum.

Maybe there's a subtle reason that I'm missing for why the second proof works when starting from Definition 2. But to me, beginning with that definition gives you that for some $$\varepsilon>0$$ there is a tagged partition whose refinements are that close to some value, and we want to show that some of those refinements can be made close to the upper and lower Darboux integrals, but we have to deal first with the tags we already have and must keep, which won't necessarily be close to their interval's supremum and infimum and so we would need to enclose them in within a sufficiently small interval, and the rest of the intervals could be assigned tags close to their infimum.

If I'm right and the Darboux article's proof is insufficient, one would need to be added to this section to satisfactorily prove the two definitions' equivalence, at which point it might make sense to make the equivalence arguments a separate section. The page Characterizations of the exponential function would be a good template. I don't think that I have the math or Wikipedia editing experience to make those changes though. Lnkov1 (talk) 00:42, 11 November 2022 (UTC)

title: Talk:Riemann integral