OpenLLM-France/wikimedia huggingface dataset:
# Peano kernel theorem

In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano.

## Statement

Let ${\mathcal {V}}[a,b]$ be the space of all functions $f$ that are differentiable on $(a,b)$ that are of bounded variation on $[a,b]$, and let $L$ be a linear functional on ${\mathcal {V}}[a,b]$. Assume that that $L$ annihilates all polynomials of degree $\leq \nu$, i.e.$Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].$Suppose further that for any bivariate function $g(x,\theta )$ with $g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]$, the following is valid:$L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,$and define the Peano kernel of $L$ as$k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],$using the notation$(x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}$The Peano kernel theorem states that, if $k\in {\mathcal {V}}[a,b]$, then for every function $f$ that is ν+1{\textstyle \nu +1} times continuously differentiable, we have $Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .$

### Bounds

Several bounds on the value of $Lf$ follow from this result:${\begin{aligned}|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{1}\|f^{(\nu +1)}\|_{\infty }\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{\infty }\|f^{(\nu +1)}\|_{1}\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}$
where $\|\cdot \|_{1}$, $\|\cdot \|_{2}$ and $\|\cdot \|_{\infty }$are the taxicab, Euclidean and maximum norms respectively.

## Application

In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all $f\in \mathbb {P} _{\nu }$. The theorem above follows from the Taylor polynomial for $f$ with integral remainder:
    ${\begin{aligned}f(x)=f(a)+&(x-a)f'(a)+{\frac {(x-a)^{2}}{2}}f''(a)+\cdots \\[6pt]&\cdots +{\frac {(x-a)^{\nu }}{\nu !}}f^{(\nu )}(a)+{\frac {1}{\nu !}}\int _{a}^{x}(x-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}$
defining $L(f)$ as the error of the approximation, using the linearity of $L$ together with exactness for $f\in \mathbb {P} _{\nu }$ to annihilate all but the final term on the right-hand side, and using the $(\cdot )_{+}$ notation to remove the $x$-dependence from the integral limits.
# Guardiola odontophylla

Guardiola odontophylla is a rare North American species of plants in the family Asteraceae. It is found only in northern Mexico in the state of Durango.
Guardiola odontophylla is a branching perennial with a purple stem, hairless and appearing whitish because of a waxy coating. Flower heads have both ray flowers and disc flowers.
# Vauvenargues

Vauvenargues may refer to:

## People

* Luc de Clapiers, marquis de Vauvenargues (1715–1747), French writer

## Places

* Château of Vauvenargues, the family home of Luc de Clapiers, in the village of Vauvenargues
* Vauvenargues, Bouches-du-Rhône, a commune in southern France
# Vedano Olona

Vedano Olona is a comune (municipality) in the Province of Varese in the Italian region Lombardy, located about 40 kilometres (25 mi) northwest of Milan and about 7 kilometres (4 mi) southeast of Varese.
Vedano Olona borders the following municipalities: Binago, Castiglione Olona, Lozza, Malnate, Varese, Venegono Superiore.
# Sitapur Junction railway station

Sitapur Junction is a railway junction in the North Eastern Railway zone in Sitapur district, Uttar Pradesh. This junction lies on Aishbagh-Bareilly Section (Vaya Sitapur, Pilibhit) and Roza-Burhwal Line. Its code is STP. It serves Sitapur city. The main line of the Lucknow–Basti–Gorakhpur section is 277 km (172 mi), and also features a branch line which connects to Roza Jn. (Shahjahanpur) from Burhwal Junction with a length of 188 km (117 mi) , and joins to the Lucknow–Moradabad line. The upgraded station now consists of five platforms. Three platforms have been completely renovated to have basic facilities including water and sanitation.
# Sitapur Ki Geeta

Sitapur Ki Geeta is a 1987 Hindi movie which stars Hema Malini, Rajesh Khanna and Pran. The music of the film was composed by the duo Sonik Omi. The movie, set against a rural backdrop, was most successful in the North and the East of India. Lachak Lachak Lachak Jaye Jawani is the most famous song from this film, sung by Amit Kumar and Asha Bhosle. Rajesh Khanna makes special appearance in the film as the presumably intended groom of Hema Malini. The film was dubbed into Bhojpuri and re-released in 2006.

## Cast

* Hema Malini as Geeta Singh
* Rajesh Khanna as Ramu (Sp. appearance)
* Pran as Dharam Singh "Dharma"
* Rajan Sippy as Inspector Karan Singh
* Shoma Anand as Pinky Shrivastav
* Raza Murad as Thakur Vikram Singh
* Shakti Kapoor as Thakur Bahadur Singh
* Amjad Khan as Thakur Pratap Singh
* Tej Sapru as Teju
* Chandrashekhar as Bandit who supported Geeta
* Om Shivpuri as IG Shrivastav
* Yunus Parvez as Pratap Singh's Munim
* Jayshree T. as Courtesan
* Chand Usmani as Mrs. Yashoda Singh
* Jankidas as Bread Man

## Soundtrack

The film's music was composed by the duo Sonik Omi and the songs were written by lyricist Varma Malik.
| Song | Singer |
| "Zamana Kaisa Zamana" | Mahendra Kapoor |
| "Yaad Kar, Yaad Kar" | Asha Bhosle |
| "Phenk More Raja Rupaiya" | Asha Bhosle |
| "Lachak Lachak Lachak Jaye Jawani" | Asha Bhosle, Amit Kumar |
| "Hum Kaun Hain, Kyun Aaye Hain" | Mohammed Aziz, Dilraj Kaur |
| "Na Chhatri, Na Chhat, Na Chhappad" | Mohammed Aziz, Dilraj Kaur |
# Veguita, New Mexico

Veguita is a census-designated place in Socorro County, New Mexico, United States. Its population was 232 as of the 2010 census. Veguita has a post office with ZIP code 87062. New Mexico State Road 304 passes through the community.
One community fixture is "Veguita Trading Post" which has a fueling station, and provides commodities. In 2015, the Veguita community welcomed KVNM-LP 101.1 FM, one of two low power FM Radio Stations in Socorro County.
Veguita has two independent Christian churches, Chihuahua Bible Chapel, and Iglesia Evangelica "Camino De Fe", which share facilities on Carlos Martinez Road. There also is the Mission Church of the Catholic Diocese. Veguita is the home of iDea Ministries, which is the owner-operator of KVNM-LP 101.1 FM, and produces "Jazz, Power, & The Glory!".

## Education

Its school district is Belén Consolidated Schools. Belén High School is the district's comprehensive high school.
# Veguitas, Jayuya, Puerto Rico

Veguitas is a barrio in the municipality of Jayuya, Puerto Rico. Its population in 2010 was 3,685.
# Velddriel

Velddriel is a village in the Dutch province of Gelderland. The village is a part of the municipality of Maasdriel, and lies about 8 km north of 's-Hertogenbosch.
It was first mentioned in 996 as Thrile. The veld (field) was later added to distinguish from Kerkdriel (church). In 1840, it was home to 422 people. The St. Martinus Church was built in 1858 and destroyed in 1944. In 1953, a new church was completed.

## Gallery

* Former school
* Former monastery
# Peire Milo

Peire Milo, Milon or Millon was a mid-13th-century Italian troubadour. His cobla esparsa "En amor trop pietat gran", he derives the word amor (love) from suffering and death (mor). His only other songs are cansos:
* "Aissi m'ave com cel qui seignors dos"
* "A vos, merces, voill retrar mon afaire"
* "Per pratz vertz ni per amor"
* "Pos l'us auzels envas l'autre s'atura"
* "Pos que dal cor m'ave, farai chansos"
* "Quant hom troba dos bos combatedo"
* "S'eu anc d'amor sofers ni mal ni pena"
* "Si com lo metge fa crer"
# Peleteria iavana

Peleteria iavana is a species of bristle fly in the family Tachinidae.

## Distribution

China, Belarus, Czech Republic, Hungary, Latvia, Moldova, Poland, Romania, Slovakia, Ukraine, Andorra, Bulgaria, Croatia, Greece, Italy, Macedonia, Portugal, Serbia, Spain, Turkey, Austria, Belgium, Channel Islands, France, Netherlands, Switzerland, Japan, Kazakhstan, South Korea, Israel, Saudi Arabia, Algeria, Russia, Democratic Republic of the Congo, Ethiopia, Kenya, Madagascar, South Africa, Sudan, Tanzania, Zambia, Zimbabwe, India, Borneo, Java, Sulawesi, Sumatra, Malaysia, Myanmar, Nepal, Philippines, Sri Lanka, Taiwan, Thailand, Australia, Papua New Guinea, Solomon Islands.
# Guidi

Guidi is an Italian surname shared by several notable people:
* Alessandro Pier Guidi (born 1983), racing driver from Italy
* Angela Maria Guidi Cingolani (1896–1991), Italian politician
* Angelo Guidi (1888–1953), Italian racing cyclist
* Antonio Guidi (1927–2013), Italian actor and voice actor
* Antonio Guidi (born 1945), Italian politician
* Carlo Alessandro Guidi (1650–1712), Italian lyric poet
* Chen Guidi (born 1942), Chinese writer from Huaiyuan county, Anhui
* Diego Guidi (born 1981), Argentine retired footballer
* Domenico Guidi (1625–1701), Italian sculptor
* Dominique Guidi (born 1996), French professional footballer
* Donna Rachele Guidi (1890–1979), wife of Italian dictator Benito Mussolini
* Emanuele Guidi (born 1969), Sammarinese archer
* Fabrizio Guidi (born 1972), Italian cyclist
* Federica Guidi (born 1969), Italian businesswoman and the former Minister of Economic Development
* Gianluca Guidi (born 1968), Italian rugby union coach and former player
* Gina Guidi (born 1962), American professional female boxer
* Giovanni Francesco Guidi di Bagno, Italian cardinal, brother of cardinal Nicola and nephew of cardinal Colonna
* Giovanni di ser Giovanni Guidi (1406–1486), Italian painter, brother of Masaccio
* Guidarino Guidi (1922–2003), Italian film actor and director
* Guido Guidi (fl. 20th century), Italian comic book artist and penciller
* Guido Guidi (photographer) (born 1941), Italian photographer
* Guido Buffarini Guidi (1895–1945), Italian politician active during the Second World War
* Ignazio Guidi (1844–1935), Italian orientalist
* Jennifer Guidi (born 1972), American artist
* Juan Héctor Guidi (1930–1973), Argentine footballer
* Leonardo Guidi (born 1974), Italian racing cyclist
* Nicholas Guidi (born 1983), Italian football defender
* Nicola Guidi di Bagno (1583–1663), titular archbishop of Atenia, bishop of Senigallia, and cardinal
* Osvaldo Guidi (1964–2011), Italian cinema, theater and television actor, and a dramaturge and theater director
* Peter Guidi (1949–2018), American jazz musician
* Scipione Guidi (1884–1966), Italian violinist
* Tommaso di Giovanni di Simone Guidi Masaccio (1401-1428), Italian leading painter
* Valentina Guidi Ottobri (born 1988), Italian art curator
* Virgilio Guidi (1891–1984), Italian artist and writer
# Guillaume Herincx

Guillaume Herincx (Willem Herinx; 1621 – 17 August 1678), was a Belgian Franciscan theologian. He became bishop of Ypres.

## Life

Herincx was born at Helmond, North Brabant. After receiving his preliminary education at 's-Hertogenbosch he entered the University of Louvain, where he devoted himself to the study of the ancient classics and obtained the degree of doctor of philosophy. After completing his university course, he resolved to embrace the religious state and entered the Franciscan Order.
In 1653 he was appointed lecturer in theology at Louvain. After fifteen years spent in teaching theology, Father Herincx was honoured with the title of lector jubilate, equivalent to the university degree of doctor of divinity.
He was twice elected minister provincial, then definitor general, and finally commissary general for the northern countries of Europe. On 28 April 1677, whilst making a canonical visitation in England, he received word at Newport that Charles II had nominated him Bishop of Ypres.
He was consecrated on 24 October in the same year, in the Franciscan church, Brussels. He left immediately for his diocese but ruled it for less than a year; he died while making his first diocesan visitation.
The epitaph on his tombstone in the cathedral of Ypres says: "Ob virtutem et omnimodam eruditionem ad has infulas assumptus". Letters found in his room after his death show that his promotion to the cardinalate had been determined on by the pope.

## Works

His superiors, who had observed his talent and success in teaching, ordered him (1658) to draw up a course of theology for use in the Franciscan schools, and the first volume of his work was published in 1660. The style is concise and clear. In the preface to his Summa Theologica, he writes:
   The teaching of theology does not consist alone in the search after truth, but it behooves us to make use of the truth for our own sanctification and for the sanctification of others, and above all for kindling and nourishing in ourselves and in others the love of God.
According to the constitutions of his order, Father Herincx propounds the doctrine of Duns Scotus, but he does not neglect the teachings of Bonaventure or Thomas Aquinas.
Father Herincx was a Probabilist, and his tractate "De conscientia" is a masterpiece. He shows that the system of Probabilism is not altogether new, and he draws his proofs from Aquinas, Bonaventune, St. Antonine, and Scotus, although the Subtle Doctor is not so explicit on the matter as the other ancient writers. According to Herincx, the tempest that arose in the seventeenth century against Probabilism had its origin in Jansenism, for Rigorism was unknown among the theologians of the Middle Ages.
The decrees of Pope Alexander VII, issued in 1665 and 1666, after the publication of Herincx's work, called for some modifications in the latter, and Father Guillaume van Goorlaeken, lector jubilate, was commissioned to bring out a new edition.
His "Summa Theologica Scholastica et Moralis" was published at Antwerp, 1660–63; 2nd ed., 1680; 3rd, 1702-04. His Centiloqium Theologicum was apparently influenced by Theodore Smising.
# Skyways

Skyways may refer to:
* Skyway, walk bridge
* Skyways (British airline), defunct British airline also known as Skyways Limited
* Skyways (Swedish airline), defunct Swedish airline also known as Skyways Express
* Skyways (TV series), Australian TV serial broadcast 1979-1981
* Skyways Coach-Air Limited
# Guido Majno

Guido Majno was an Italian-born American physician who served as a chairman of Department of Pathology at the University of Massachusetts Medical School, Worcester.
Majno was born and raised in Milan, Italy. In 1947, he completed his doctorate degree from the University of Milan School of Medicine in medicine and surgery. After receiving his education, he started his career with University of Geneva’s Institute of Pathology.
In 1952, he moved to the United States and started working at Tufts University School of Medicine.
In 1953, he joined Harvard Medical School as an instructor.
In 1975, he published his book The Healing Hand: Man and Wound in the Ancient World which was reviewed by notable medical journals.
In 1999, he received Humanism in Medicine Award.
In 2010, he died at the age of 88.
A June 4, 1982 article on Majno in JAMA (Journal of the American Medical Association) provides more extensive background and a description of Dr. Majno's research for The Healing Hand: Man and Wound in the Ancient World.

## Books

* The Healing Hand: Man and Wound in the Ancient World (1975)
* Cells, Tissues, and Disease — Principles of General Pathology
# Penntown, Indiana

Penntown is an unincorporated community in Adams Township, Ripley County, in the U.S. state of Indiana.

## History

Penntown was originally called Pennsylvaniaburg, and under the latter name was laid out in 1837. The community was named after Pennsylvania, the native state of a share of the early settlers.
# Fjell (disambiguation)

A fjell or fell is a high and barren landscape feature in Fennoscandia, the Isle of Man, parts of Northern England, and Scotland.
Fjell may also refer to:

## People

* Anders Fjell (born 1974), a Norwegian psychologist, neuroscientist and professor
* Jan-Erik Fjell (born 1982), a Norwegian novelist
* Kai Fjell (1907-1989), a Norwegian painter, printmaker and scenographer
* Olav Fjell (born 1951), a Norwegian businessperson

## Places

* Fjell, a former municipality in the old Hordaland county, Norway
* Fjell (village), a village in Øygarden municipality in Vestland county, Norway
* Fjell Church, a church in Øygarden municipality in Vestland county, Norway
* Fjell Church (Drammen), a church in Drammen municipality in Buskerud county, Norway
* Fjell Fortress, a fortress on the island of Sotra in Vestland county, Norway
* Fjell, Innlandet, a village in Nord-Odal municipality in Innlandet county, Norway
* Fjell skole, a school in Drammen municipality in Buskerud county, Norway
# Fish Ponds

Fish Ponds is one of the locations where the Mojave River rose to run on the surface of its course through the Mojave Desert. The site is located in the river bed in Nebo Center, in the eastern part of Barstow, in San Bernardino County, California. Fish Ponds was named for the Mohave chub that were to be found in the ponds in that location.$^{:218}$$^{:105}$
# Skryje (Rakovník District)

Skryje is a municipality and village in Rakovník District in the Central Bohemian Region of the Czech Republic. It has about 200 inhabitants. The historic centre of the village is well preserved and is protected by law as a village monument zone.

## Etymology

The word skryj denoted a person who was hiding or who was hiding something. The name Skryje denoted the village where such people lived.

## Geography

Skryje is located about 15 kilometres (9 mi) south of Rakovník and 40 km (25 mi) west of Prague. The northern part of the municipal territory lies in the Plasy Uplands, the southern part lies in the Křivoklát Highlands. The highest point is the hill Vlastec at 612 m (2,008 ft) above sea level. The municipality is situated on the right bank of the Berounka River. The entire municipality is located within the Křivoklátsko Protected Landscape Area.

## History

The first written mention of Skryje is from 1239, when it was owned by the Kladruby Abbey. From 1375 to 1601, the village was owned by various lower nobility. From 1601 until the abolition of manorialism, Skryje was part of the Křivoklát estate.

## Demographics

| Year | Pop. | ±% |
| 1869 | 597 | — |
| 1880 | 592 | −0.8% |
| 1890 | 546 | −7.8% |
| 1900 | 498 | −8.8% |
| 1910 | 510 | +2.4% |
| Year | Pop. | ±% |
| 1921 | 476 | −6.7% |
| 1930 | 434 | −8.8% |
| 1950 | 295 | −32.0% |
| 1961 | 265 | −10.2% |
| 1970 | 199 | −24.9% |
| Year | Pop. | ±% |
| 1980 | 204 | +2.5% |
| 1991 | 163 | −20.1% |
| 2001 | 144 | −11.7% |
| 2011 | 151 | +4.9% |
| 2021 | 153 | +1.3% |
Source: Censuses :

## Transport

There are no railways or major roads passing through the municipality.

## Sights

Skryje is known for paleontological and geological research. The first mention of the collection of local fossils comes from 1832. The village became famous when French geologist Joachim Barrande discovered rich fossil deposits of trilobites in the close surroundings of the village, when he was surveying the terrain for a horse-pulled railway.
Today there is the Monument to Joachim Barrand, which is a small museum with paleontological and geological exposition. Part of the museum is dedicated to a country life at the turn of 19th and 20th century in Skryje and its surroundings.
There is also an educational trail dedicated to fossil sites and creatures whose fossils have been found here. It is still possible to find fossils in several places.
The Church of Saint Michael the Archangel was first mentioned in 1350, already as a parish church. The current Baroque building dates from 1713.
# Flag of Białogard

The flag, which serves as the symbol of the city of Białogard, West Pomeranian Voivodeship, in western Poland, is divided into three horizontal stripes, including a blue stripe in the middle, and two white stripes, twice its size, placed on top and bottom, and with two light red rhombuses, each placed in the left corner of white stripe. The flag was established in 2004.

## Design

The flag of the town of Białogard is divided into three horizontal stripes, which are: a blue stripe in the middle, with a height of 1/5 of the height of the flag, and two white stripes, one on the top, and one on the bottom, both with the height of 2/5 of the height of the flag.
In both of the white fields, in their left corners, is placed a light red symmetrical rhombus. The length of its longer horizontal diagonal line is equal to 2/8 of the height of the flag, and the length of its shorter vertical diagonal line is equal 1/5 of the height of the flag. The top and bottom angles of the rhombus have 120 decrees each, and the left and right angles have 60 degrees each. Each rhombus is placed in the middle of the height of the white field, and at a distance from the left edge of the flag, equal to 1/5 of the height of the flag. Around the edges of the flag is a thin blue border. The flag proportions have the aspect ratio of its height to its width equal to 5:8.
The blue stripe on the flag symbolizes the location of the town of Białogard on the Parsęta river, while two red rhombuses symbolize the historical association of the town with the Hanseatic League. Colours blue, red, and white are also colours of the coat of arms of Białogard.

## History

The flag was established by the Białogard City Council on 3 March 2004.
# Guinardia

Guinardia is a genus of diatoms belonging to the family Rhizosoleniaceae.
The genus was first described by H. Peragallo in 1892.
The genus has cosmopolitan distribution.
Species:
* Guinardia delicatula
* Guinardia flaccida
* Guinardia pungens
* Guinardia striata
# Guinas Constituency

Guinas Constituency is an electoral constituency in the Oshikoto Region of Namibia. It had 8,435 inhabitants in 2004 and 4,210 registered voters in 2020. Guinas consists of the rural area surrounding the mining town of Tsumeb. Larger settlements in the Guinas constituency are Oshivelo and Tsintsabis.
The constituency is named after Lake Guinas.

## Politics

Guinas constituency is traditionally a stronghold of the South West Africa People's Organization (SWAPO) party.

### Regional elections

In the 2004 regional election SWAPO candidate Isack Gomachab received 2,030 of the 3,094 votes cast. In the 2015 local and regional elections the SWAPO candidate won uncontested and became councillor after no opposition party nominated a candidate.
The SWAPO candidate also won the 2020 regional election by a landslide. Elias Marthinu received 1,268 votes, well ahead of the only opposition candidate Pinehas Heita of the Independent Patriots for Change (IPC), a party formed in August 2020, with 95 votes.

### National elections

In the Namibian general election, 2009, Hifikepunye Pohamba received approximately 77% of the votes in Guinas. Hidipo Hamutenya finished in second place with approximately 10% of the vote.
# Guinara Zharafutdinova

Guinara Zharafutdinova is a Soviet sprint canoer who competed in the mid to late 1980s. She won two medals in the K-4 500 m event at the ICF Canoe Sprint World Championships with a silver in 1985 and a bronze in 1987.
# USS Narcissus

USS Narcissus may refer to the following ships of the United States Navy:
* USS Narcissus (1863), a screw steamer which served in the Union, during the American Civil War.
* USS Narcissus (WAGL-238), a U.S. Coast Guard ship that was transferred to the U.S. Navy; served as a large inland buoy tender until the end of World War II; returned to the Coast Guard.
# New York City (T. Rex song)

"New York City" is a 1975 single by the British glam rock band T. Rex. The track and its B-side "Chrome Sitar", are taken from the 1976 album Futuristic Dragon. The song consists of a couple of lines of lyric, "Did you ever see a woman coming out of New York City/With a frog in her hand?" and "I did don't you know/And don't it show?" repeated over a bass line.
The single was released on 27 June 1975, and was in the UK Singles Chart for a total of eight weeks, peaking at No. 15.
In some countries, the B-side was "Solid Baby" taken from the previous album Bolan's Zip Gun.