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-gry puzzle

The -gry puzzle is a popular word puzzle that asks for the third English word that ends with the letters -gry other than angry and hungry. Specific wording varies substantially, but the puzzle has no clear answer, as there are no other common English words that end in -gry. Interpretations of the puzzle suggest it is either an answerless hoax; a trick question; a sincere question asking for an obscure word; or a corruption of a more straightforward puzzle, which may have asked for words containing gry (such as gryphon). Of these, countless trick question variants and obscure English words (or nonce words) have been proposed. The lack of a conclusive answer has ensured the enduring popularity of the puzzle, and it has become one of the most frequently asked word puzzles.

The ultimate origin and original form of the puzzle is unknown, but it was popularized in 1975, starting in the New York area, and has remained popular into the 21st century. Various similar puzzles exist, though these have straightforward answers. The most notable is "words ending in -dous", which has been popular since the 1880s.

Answers
Various proposed answers exist, stating that the question is one of the following:
 * A hoax – there is no answer, and its purpose (or effect) is to frustrate.
 * A trick question, with various answers depending on precise wording.
 * A sincere question asking for an obscure word, most often proposed as, , or . This does not apply to wordings that explicitly ask for a very common word.
 * A corruption of a more straightforward word puzzle, namely a word containing the sequence "gry", though not necessarily at the (tail) end, in which case the answer is which is uncommon but in use.

This topic is a source of lively interest, both to lovers of word puzzles and lovers of words. For both groups, much of the appeal lies in the quest, either to trace the origin of the puzzle or compile a complete list of words ending in -gry.

More recently, the word hangry—a portmanteau of 'hungry' and 'angry'—has been used to refer to an irritable state induced by lack of food. Oxford Dictionaries (controlled by, but less restrictive than, the Oxford English Dictionary) added hangry on 27 August 2015,  and the full Oxford English Dictionary added hangry in 2018.

History
There are anecdotal reports of various forms of the puzzle dating to the 1950s or earlier; the ultimate origin is presumably an oral tradition or a lost book of puzzles. However, the first documented evidence is from early 1975 in the New York metropolitan area, and the puzzle rapidly gained popularity in this year. The most likely source is the talk show of Bob Grant, from some program in early or mid March 1975.

Merriam-Webster, publishers of the leading American dictionaries, first heard of this puzzle in a letter dated March 17, 1975, from Patricia Lasker of Brooklyn, New York. Lasker says her plant manager heard the question on an unnamed quiz show. Since that time Merriam-Webster has received about four letters each year asking the question.

The puzzle first appears in print in Anita Richterman's "Problem Line" column in Newsday on April 29, 1975. One "M.Z." from Wantagh, New York states that the problem was asked on a TV quiz program. Richterman states that she asked a learned professor of English for help when she first received the inquiry, and he did not respond for over a month. This agrees with the Merriam-Webster report, suggesting a quiz show in early or mid March 1975.

In Anita Richterman's column on May 9, 1975, several correspondents reported that they had heard the puzzle on the Bob Grant radio talk show on WMCA in New York City. This suggests either that the earlier claims of a (TV) quiz show confused a talk show with a quiz show, or that there was another unspecified quiz show that was then repeated by Grant. The majority of readers gave the answer "gry", an obsolete unit of measure invented by John Locke. It is unclear whether this was the answer given on the Grant show, or what the precise wording had been.

By fall 1975 the puzzle had reached the Delaware Valley, again apparently by radio, by which time the puzzle seems to have mutated to a form in which the missing word is an adjective that describes the state of the world.

The puzzle has had occasional bouts of popularity: after its initial popularity in 1975, it was popular in 1978, then again in 1995–1996.

Reports of earlier versions
The most credible report of an early version was given on Stumpers-L, which reported a trick question formulation from an eight-page pamphlet entitled Things to Think About, probably dating to the 1940s:

Trick versions

 * 1) This version only works when spoken: There are three words in English that end in a "gree". The first two are "angry" and "hungry", and if you've listened closely, you'll agree that I've already told you the third one.
 * 2) *The answer is "agree".
 * 3) There are three words in the English language that end with the letters 'g', 'r', and 'y'. Two are "hungry" and "angry". The third word is something everyone uses every day. Everyone knows what the third word means. What is the third word?
 * 4) *The answer is "energy". The riddle says that the word ends in the letters g-r-y; it says nothing about the order of the letters. Many words end with "-rgy", but energy is something everyone uses every day.
 * 5) There are at least three words in the English language that end in "g" or "y". One of them is "hungry", and another one is "angry". There is a third word, a short one, which you probably say every day. If you are listening carefully to everything I say, you just heard me say it three times. What is it?
 * 6) *The answer is "say". This version depends upon the listener confusing the spoken word "or" and the spoken letter "r".
 * 7) There are three words in the English language that end in "gry". Two words that end in "gry" are "hungry" and "angry". Everyone knows what the third word means, and everyone uses them every day. If you listened very carefully, I have already stated to you what the third word is. The three words that solve this riddle are...?
 * 8) *The answer is the three-word sentence "I am hungry". This version asks for three words that end in "gry", not three words each of which ends in "gry".
 * 9) I know two words that end in "gry". Neither one is angry or hungry. What are they?
 * 10) *The answer is "angry" and "hungry". Since these are words, they are not capable of being angry or hungry.
 * 11) Give me three English words, commonly spoken, ending in g-r-y.
 * 12) *There are many possible answers, such as "Beg for mercy", or "Bring your money".
 * 13) There are three words in the English language that end g-r-y. One is angry and another is hungry. The third word is something that "everyone" uses. If you have listened carefully, I have already told you what it is.
 * 14) *The answer is "every". First word is "fuming" which ends with "g". The word "fuming" is angry (when personified). Second word is "eager" which ends with "r". The word "eager" is hungry (when personified). Third word is "every" which ends with "y". The word "everyone" uses the word "every".

Meta-puzzle versions
The remaining versions are a form of meta-puzzle, in the sense that they make no use of the actual letters "gry" themselves, which therefore are a red herring. The red herring only works because there is another puzzle that does use these letters (even though that puzzle has no good answer).

 * 1) Think of words ending in "gry". "Angry" and "hungry" are two of them. There are only three words in "the English language". What is the third word? Hint: The word is something that everyone uses every day. If you have listened carefully, I have already told you what it is.
 * 2) *The answer is "language", and the logic is as follows: There are only three words in "the English language"; the third word is "language". Since this version requires quotation marks around the phrase, "the English language", the written version gives away the trick.
 * 3) Angry and hungry are two words in the English language that end in "gry". "What" is the third word. The word is something that everyone uses every day. If you have listened carefully, I have already told you what it is.
 * 4) *The answer is "what". But again, the quotation marks spoil the puzzle when it is printed.
 * 5) There are three words in the English language that end with "gry". Two of these are "angry" and "hungry". The third word is a very common word, and you use it often. If you have read what I have told you, you will see that I have given you the third word. What is the third word? Think very carefully.
 * 6) *The answer is "three", the third word in the paragraph. The rest of the paragraph is a red herring.
 * 7) There are three words in the English language that end in "gry". The first "one" is "hungry", the second "one" is "angry", what is the third "one"? If you have read this carefully I have given a clue.
 * 8) *The answer is the word "one", which is the third "one". Again the quotation marks ruin the written puzzle, so this version is usually written without the quotation marks and with the word "one" capitalized.

Similar puzzles
There are numerous similar puzzles, giving letter sequences that rarely occur in words. The most-notable of these is the -dous puzzle of finding words ending in -dous, which was popular in the 1880s. This took various forms, sometimes simply listing all words or all common words, sometimes being posed as a riddle, giving the three common words, tremendous, stupendous, and hazardous, and requesting the rarer fourth, which is jeopardous. This form originated in 1883, with an A.A. of Glasgow writing to George Augustus Henry Sala in his "Echoes of the Week" column in the Illustrated London News. This question has had enduring popularity, even inspiring a contest, though the words have proven less stable: today jeopardous is considered too rare, and the formerly unpopular horrendous has taken its place; this change occurred as early as 1909. At times other words such as hybridous have been accepted. Today hazardous is typically the omitted word, and differs from the others in being a visible compound hazard + -ous. This puzzle has continued in popularity through the end of the 20th century, with recent versions giving it as an alternative to the gry puzzle. There is a Russian puzzle which goes: "There are three words in the Russian language which end in -zo. Two of them are zhelezo "iron" and puzo "belly". What is the third word?" There is quite a handful of other nouns that end in -zo, in the Russian language, but most of them are fairly obscure terms like авизо, abbreviations or proper names like Кензо. Another similar one is words ending in -cion, of which the common words are coercion, scion, and suspicion.

The most similar to the gry puzzle in form is to find three words that contain the letter sequence shion, to which the answer is cushion, fashion, and parishioner; this is typically stated by giving cushion and fashion, and requesting the third word, namely parishioner. This can be modified to finding words ending with -shion, in which case the answer is the obsolete word parishion, which is a synonymous variant of parishioner. This has not been nearly as popular as the gry puzzle.

Solution techniques
The standard way to solve such puzzles is to use a reverse dictionary, or to perform an exhaustive search through a dictionary, either manually, which is tedious and error-prone, or using computer tools such as grep, which requires an electronic word list. At the origin of the gry puzzle, the standard reverse dictionary in modern English was the "Air Force Reverse Dictionary" (formally the Normal and Reverse Word List, compiled under the direction of A. F. Brown), which did not have additional answers for gry. The most plausible answer at the time was meagry, found in the Oxford English Dictionary. A more elaborate strategy is to list words that have endings similar to gry, such as -gary, and then search a larger dictionary for obsolete variants ending in -gry, for example begry for beggary.

From around 1980 electronic word lists became widely available on Unix systems, and searching for answers to the gry puzzle was an occasional benchmark; this also turned up gryphon in some cases, if match is not required to be at the end. This is now easily done in milliseconds on modern personal computers:

title: -gry puzzle


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Giosuè Carducci

Giosuè Alessandro Giuseppe Carducci (27 July 1835 – 16 February 1907) was an Italian poet, writer, literary critic and teacher. He was noticeably influential, and was regarded as the official national poet of modern Italy. In 1906, he became the first Italian to receive the Nobel Prize in Literature. The Swedish Academy's motivation was that "not only in consideration of his deep learning and critical research, but above all as a tribute to the creative energy, freshness of style, and lyrical force which characterize his poetic masterpieces."

Biography
He was born in Valdicastello (part of Pietrasanta), a small town in the Province of Lucca in the northwest corner of the region of Tuscany. His father, a doctor, was an advocate of the unification of Italy and was involved with the Carbonari. Because of his politics, the family was forced to move several times during Carducci's childhood, eventually settling for a few years in Florence.

From the time he was in school, he was fascinated with the restrained style of Greek and Roman Antiquity, and his mature work reflects a restrained classical style, often using the classical meters of such Latin poets as Horace and Virgil. He translated Book 9 of Homer's Iliad into Italian.

Carducci was awarded a scholarship to study at the prestigious Scuola Normale Superiore di Pisa. After graduating in 1856, he began teaching school. The following year, he published his first collection of poems, Rime. These were difficult years for Carducci: his father died, and his brother committed suicide.

In 1859, he married Elvira Menicucci, and they had four children. He briefly taught Greek at a high school in Pistoia, and then was appointed Professor of Italian Literature at the University of Bologna. Here, one of his students was Giovanni Pascoli, who became an eminent poet himself and later succeeded him at the university.

Carducci was a popular lecturer and a fierce critic of literature and society. In his youth he was an atheist, whose political views were vehemently hostile to the Catholic Church. In the course of his life his views on religion shifted towards a socially oriented theism which he exposed in his famous "Discorso sulla libertà perpetua di San Marino" ("A Speech on San Marino's Perpetual Freedom"), pronounced on September the 30th, 1894 before the authorities and people of that ancient Republic and celebrating "the Universal God of Peoples, Mazzini's and Washington's God".

His anti-clerical revolutionary vehemence was prominently showcased in one famous poem, the deliberately blasphemous and provocative Inno a Satana ("Hymn to Satan"). "Satan" / "Lucifer" was considered by Italian leftists of the time as a metaphor of the rebellious and freethinking spirit. The poem was composed in 1863 as a dinner party toast, published in 1865, then republished in 1869 by Bologna's radical newspaper, Il Popolo, as a provocation timed to coincide with the First Vatican Council, a time when revolutionary fervor directed against the papacy was running high as republicans pressed both politically and militarily for an end to the Vatican's domination over the papal states.

While "Inno a Satana" had quite a revolutionary impact, Carducci's finest poetry came in later years. His collections Rime Nuove (New Rhymes) and Odi Barbare (Barbarian Odes) contain his greatest works.

He was the first Italian to receive the Nobel Prize in Literature, in 1906. He was also appointed senator by the King of Italy (1890). In politics he remained a strong Liberal throughout his life; through the years he progressively evolved from republicanism to a sort of support to monarchy. He was a Freemason of the Grand Orient of Italy. His father Michele, a physician, was also a member of the Italian Carboneria. Although his reputation rests primarily on his poetry, he also produced a large body of prose works. Indeed, his prose writings, including literary criticism, biographies, speeches and essays, fill some 20 volumes. Carducci was also an excellent translator and translated some of Goethe and Heine into Italian.

The Museum of the Risorgimento, Bologna is housed in the Casa Carducci, the house where he died at the age of 71, and contains an exhibit on the author.

Legacy
Carducci confessed his sins and was reconciled to the Catholic Church in 1895. On 11 September 1978, Pope John Paul I mentioned him as a "model" for university professors and teachers of Latin.

Works
It is not always easy to follow the development of Carducci's poetry through the collections he edited. The poet in fact organized his compositions several times and in different ways and gave a definitive arrangement only later in the edition of his Opere published for Zanichelli between 1889 and 1909. The following is a list of poetic works published in one volume, then rearranged into the 20 volumes of his Opere.

 * Rime, San Miniato, 1857.
 * Levia Gravia, 1868.
 * Poesie, Firenze, Barbera, 1871.
 * Primavere elleniche, 1872.
 * Nuove poesie, 1873.
 * Odi barbare, 1877.
 * Juvenilia, 1880.
 * Levia Gravia, 1881.
 * Giambi ed Epodi, 1882.
 * Nuove odi barbare, 1882.
 * Rime nuove, 1887.
 * Terze odi barbare, 1889.
 * Delle Odi barbare. Libri II ordinati e corretti, 1893.
 * Rime e ritmi, 1899.
 * Poesie. MDCCCL-MCM, 1901.

Below are the poetic volumes in the Opere. The volumes, however, do not correspond to the chronological order with which the poet had published his first collections, but refer more than anything else to the distinctions of genres and therefore we find poems of the same period in different collections. The collections follow this order:

 * Juvenilia, in six books, 1850–1860
 * Levia Gravia, in two books, 1861–1871
 * Inno a Satana, 1863
 * Giambi ed Epodi, in two books, 1867–1879
 * Intermezzo, 1874–1887
 * Rime Nuove, in nine books, 1861–1887
 * Odi barbare, in two books, 1873–1889
 * Rime e Ritmi, 1889–1898
 * Della Canzone di Legnano, Part I, 1879

Juvenilia
The first collection of lyrical poems, which Carducci collected and divided in six books under the title Juvenilia (1850–1860), is undoubtedly inspired by the classical tradition of the Amici pedanti group that was constituted at that time for the purpose of fighting the romanticism of the Florentines. In the verses of the collection we can immediately see his imitation of the ancient classics, of the stilnovo style, of Dante and Petrarch and, among the moderns, Vittorio Alfieri, Monti, Foscolo and Leopardi.

But the Carduccian spirit is already visible; his love for the beauty of style, the purity of sentiments and the celebration of liberty, as well as the ability to appreciate all that is genuine, therefore also the language of the common people.

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

title: Giosuè Carducci


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List of glues

There are many adhesive substances that are considered or commonly referred to as glue. Historically, the term only referred to protein colloids prepared from animal flesh. The meaning has been extended to refer to any fluid adhesive.

Solvent-type glues
Solvent adhesives temporarily dissolve the substance they are gluing, and bond by solvent-welding material together with the re-solidified material.

Thermoplastic polymers
Thermoplastic glues including hot-melt adhesives cure reversibly as they cool, like the gelatin and keratin glues listed above. They frequently don't emit volatiles unless overheated.

 * Styrene acrylic copolymer – e.g. "No More Nails"

Thermosetting polymers
Thermosetting glues or thermosets cure irreversibly by polymerization. The polymerization can be triggered by heat or other radiation, or high pressure or a catalyst/hardener may be added.

 * Epoxy resins
 * Epoxy putty
 * Polyvinyls
 * Polyvinyl acetate (PVA) Includes white glue (e.g. Elmer's Glue) and yellow carpenter's glue (Aliphatic resin) (Brands include Titebond and Lepage)
 * Polyvinyl alcohol
 * Polyvinyl chloride (PVC)
 * Polyvinyl chloride emulsion (PVCE) – a water-miscible emulsion that polymerizes as it cures
 * Polyvinylpyrrolidone (component of glue sticks)
 * Silicone resins
 * Silyl modified polymers

By use

 * Postage stamp gum may be a mixture of assorted starch and resin adhesives
 * Library paste is usually starch-based
 * Meat glue is a variety of culinary binding agents

title: List of glues


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Geyser

A geyser  is a spring characterized by an intermittent discharge of water ejected turbulently and accompanied by steam. As a fairly rare phenomenon, the formation of geysers is due to particular hydrogeological conditions that exist only in a few places on Earth.

Generally, geyser field sites are located near active volcanic areas, and the geyser effect is due to the proximity of magma. Surface water works its way down to an average depth of around 2000 m where it contacts hot rocks. The resultant boiling of the pressurized water results in the geyser effect of hot water and steam spraying out of the geyser's surface vent.

A geyser's eruptive activity may change or cease due to ongoing mineral deposition within the geyser plumbing, exchange of functions with nearby hot springs, earthquake influences, and human intervention. Like many other natural phenomena, geysers are not unique to Earth. Jet-like eruptions, often referred to as cryogeysers, have been observed on several of the moons of the outer solar system. Due to the low ambient pressures, these eruptions consist of vapor without liquid; they are made more easily visible by particles of dust and ice carried aloft by the gas. Water vapor jets have been observed near the south pole of Saturn's moon Enceladus, while nitrogen eruptions have been observed on Neptune's moon Triton.

There are also signs of carbon dioxide eruptions from the southern polar ice cap of Mars. In the case of Enceladus, the plumes are believed to be driven by internal energy. In the cases of the venting on Mars and Triton, the activity may be a result of solar heating via a solid-state greenhouse effect. In all three cases, there is no evidence of the subsurface hydrological system which differentiates terrestrial geysers from other sorts of venting, such as fumaroles.

Etymology
The term 'geyser' in English dates back to the late 18th century and comes from Geysir, which is a geyser in Iceland. Its name means "one who gushes".

Form and function

Geysers are nonpermanent geological features. Geysers are generally associated with volcanic areas. As the water boils, the resulting pressure forces a superheated column of steam and water to the surface through the geyser's internal plumbing. The formation of geysers specifically requires the combination of three geologic conditions that are usually found in volcanic terrain: intense heat, water, and a waterworks system.

The heat needed for geyser formation comes from magma that needs to be close to the surface of the earth. In order for the heated water to form a geyser, a plumbing system (made of fractures, fissures, porous spaces, and sometimes cavities) is required. This includes a reservoir to hold the water while it is being heated. Geysers are generally aligned along faults.

Eruptions
Geyser activity, like all hot spring activity, is caused by surface water gradually seeping down through the ground until it meets rock heated by magma. In non-eruptive hot springs, the geothermally heated water then rises back toward the surface by convection through porous and fractured rocks, while in geysers, the water instead is explosively forced upwards by the high steam pressure created when water boils below. Geysers also differ from non-eruptive hot springs in their subterranean structure; many consist of a small vent at the surface connected to one or more narrow tubes that lead to underground reservoirs of water and pressure tight rock.

As the geyser fills, the water at the top of the column cools off, but because of the narrowness of the channel, convective cooling of the water in the reservoir is impossible. The cooler water above presses down on the hotter water beneath, not unlike the lid of a pressure cooker, allowing the water in the reservoir to become superheated, i.e. to remain liquid at temperatures well above the standard-pressure boiling point.

Ultimately, the temperatures near the bottom of the geyser rise to a point where boiling begins, forcing steam bubbles to rise to the top of the column. As they burst through the geyser's vent, some water overflows or splashes out, reducing the weight of the column and thus the pressure on the water below. With this release of pressure, the superheated water flashes into steam, boiling violently throughout the column. The resulting froth of expanding steam and hot water then sprays out of the geyser vent.

A key requirement that enables a geyser to erupt is a material called geyserite found in rocks nearby the geyser. Geyserite, mostly silicon dioxide (SiO2), is dissolved from the rocks and is deposited on the walls of the geyser's plumbing system and on the surface. The deposits make the channels carrying the water up to the surface pressure-tight. This allows the pressure to be carried all the way to the top and not leak out into the loose gravel or soil that are normally under the geyser fields.

Eventually the water remaining in the geyser cools back to below the boiling point and the eruption ends; heated groundwater begins seeping back into the reservoir, and the whole cycle begins again. The duration of eruptions and time between successive eruptions vary greatly from geyser to geyser; Strokkur in Iceland erupts for a few seconds every few minutes, while Grand Geyser in the United States erupts for up to 10 minutes every 8–12 hours.

General categorization
There are two types of geysers: fountain geysers which erupt from pools of water, typically in a series of intense, even violent, bursts; and cone geysers which erupt from cones or mounds of siliceous sinter (including geyserite), usually in steady jets that last anywhere from a few seconds to several minutes. Old Faithful, perhaps the best-known geyser at Yellowstone National Park, is an example of a cone geyser. Grand Geyser, the tallest predictable geyser on earth, (although Geysir in Iceland is taller, it is not predictable), also at Yellowstone National Park, is an example of a fountain geyser.

There are many volcanic areas in the world that have hot springs, mud pots and fumaroles, but very few have erupting geysers. The main reason for their rarity is because multiple intense transient forces must occur simultaneously for a geyser to exist. For example, even when other necessary conditions exist, if the rock structure is loose, eruptions will erode the channels and rapidly destroy any nascent geysers.

Geysers are fragile phenomena and if conditions change, they may go dormant or extinct. Many have been destroyed simply by people throwing debris into them while others have ceased to erupt due to dewatering by geothermal power plants. However, the Geysir in Iceland has had periods of activity and dormancy. During its long dormant periods, eruptions were sometimes artificially induced—often on special occasions—by the addition of surfactant soaps to the water.

Biology

The specific colours of geysers derive from the fact that despite the apparently harsh conditions, life is often found in them (and also in other hot habitats) in the form of thermophilic prokaryotes. No known eukaryote can survive over 60 °C.

In the 1960s, when the research of the biology of geysers first appeared, scientists were generally convinced that no life can survive above around 73 °C—the upper limit for the survival of cyanobacteria, as the structure of key cellular proteins and deoxyribonucleic acid (DNA) would be destroyed. The optimal temperature for thermophilic bacteria was placed even lower, around 55 °C.

However, the observations proved that it is actually possible for life to exist at high temperatures and that some bacteria even prefer temperatures higher than the boiling point of water. Dozens of such bacteria are known. Thermophiles prefer temperatures from 50 to 70 °C, whilst hyperthermophiles grow better at temperatures as high as 80 to 110 °C. As they have heat-stable enzymes that retain their activity even at high temperatures, they have been used as a source of thermostable tools, that are important in medicine and biotechnology, for example in manufacturing antibiotics, plastics, detergents (by the use of heat-stable enzymes lipases, pullulanases and proteases), and fermentation products (for example ethanol is produced). Among these, the first discovered and the most important for biotechnology is Thermus aquaticus.

Major geyser fields and their distribution

Geysers are quite rare, requiring a combination of water, heat, and fortuitous plumbing. The combination exists in few places on Earth.

Yellowstone National Park, U.S.
Yellowstone is the largest geyser locale, containing thousands of hot springs, and approximately 300 to 500 geysers. It is home to half of the world's total number of geysers in its nine geyser basins. It is located mostly in Wyoming, USA, with small portions in Montana and Idaho. Yellowstone includes the world's tallest active geyser (Steamboat Geyser in Norris Geyser Basin).

Valley of Geysers, Russia

The Valley of Geysers (Долина гейзеров) located in the Kamchatka Peninsula of Russia is the second largest concentration of geysers in the world. The area was discovered and explored by Tatyana Ustinova in 1941. Approximately 200 geysers exist in the area along with many hot-water springs and perpetual spouters. The area was formed due to a vigorous volcanic activity. The peculiar way of eruptions is an important feature of these geysers. Most of the geysers erupt at angles, and only very few have the geyser cones that exist at many other of the world's geyser fields. On June 3, 2007, a massive mudflow influenced two thirds of the valley. It was then reported that a thermal lake was forming above the valley. Few days later, waters were observed to have receded somewhat, exposing some of the submerged features. Velikan Geyser, one of the field's largest, was not buried in the slide and has recently been observed to be active.

El Tatio, Chile

The name "El Tatio" comes from the Quechua word for oven. El Tatio is located in the high valleys on the Andes surrounded by many active volcanoes in Chile, South America at around 4200 m above mean sea level. The valley is home to approximately 80 geysers at present. It became the largest geyser field in the Southern Hemisphere after the destruction of many of the New Zealand geysers, and is the third largest geyser field in the world. The salient feature of these geysers is that the height of their eruptions is very low, the tallest being only 6 m high, but with steam columns that can be over 20 m high. The average geyser eruption height at El Tatio is about 750 mm.

Taupō Volcanic Zone, New Zealand
The Taupō Volcanic Zone is located on New Zealand's North Island. It is 350 km long by 50 km and lies over a subduction zone in the Earth's crust. Mount Ruapehu marks its southwestern end, while the submarine Whakatāne seamount (85 km beyond Whakaari / White Island) is considered its northeastern limit. Many geysers in this zone were destroyed due to geothermal developments and a hydroelectric reservoir, but several dozen geysers still exist.

In the beginning of the 20th century, the largest geyser ever known, the Waimangu Geyser existed in this zone. It began erupting in 1900 and erupted periodically for four years until a landslide changed the local water table. Eruptions of Waimangu would typically reach 160 m and some superbursts are known to have reached 500 m. Recent scientific work indicates that the Earth's crust below the zone may be as little as 5 km thick. Beneath this lies a film of magma 50 km wide and 160 km long.

Iceland
Due to the high rate of volcanic activity in Iceland, it is home to some of the most famous geysers in the world. There are around 20–29 active geysers in the country as well as numerous formerly active geysers. Icelandic geysers are distributed in the zone stretching from south-west to north-east, along the boundary between the Eurasian Plate and the North American Plate. Most of the Icelandic geysers are comparatively short-lived, it is also characteristic that many geysers here are reactivated or newly created after earthquakes, becoming dormant or extinct after some years or some decades.

Two most prominent geysers of Iceland are located in Haukadalur. The Great Geysir, which first erupted in the 14th century, gave rise to the word geyser. By 1896, Geysir was almost dormant before an earthquake that year caused eruptions to begin again, occurring several times a day, but in 1916, eruptions all but ceased. Throughout much of the 20th century, eruptions did happen from time to time, usually following earthquakes. Some man-made improvements were made to the spring and eruptions were forced with soap on special occasions. Earthquakes in June 2000 subsequently reawakened the giant for a time but it is not currently erupting regularly. The nearby Strokkur geyser erupts every 5–8 minutes to a height of some 30 m.

Geysers are known to have existed in at least a dozen other areas on the island. Some former geysers have developed historical farms, which benefitted from the use of the hot water since medieval times.

Extinct and dormant geyser fields
There used to be two large geysers fields in Nevada—Beowawe and Steamboat Springs—but they were destroyed by the installation of nearby geothermal power plants. At the plants, geothermal drilling reduced the available heat and lowered the local water table to the point that geyser activity could no longer be sustained.

Many of New Zealand's geysers have been destroyed by humans in the last century. Several New Zealand geysers have also become dormant or extinct by natural means. The main remaining field is Whakarewarewa at Rotorua. Two thirds of the geysers at Orakei Korako were flooded by the construction of the hydroelectric Ohakuri dam in 1961. The Wairakei field was lost to a geothermal power plant in 1958. The Rotomahana field was destroyed by the 1886 eruption of Mount Tarawera.

Misnamed geysers
There are various other types of geysers which are different in nature compared to the normal steam-driven geysers. These geysers differ not only in their style of eruption but also in the cause that makes them erupt.

Artificial geysers
In a number of places where there is geothermal activity, wells have been drilled and fitted with impermeable casements that allow them to erupt like geysers. The vents of such geysers are artificial, but are tapped into natural hydrothermal systems. These so-called artificial geysers, technically known as erupting geothermal wells, are not true geysers. Little Old Faithful Geyser, in Calistoga, California, is an example. The geyser erupts from the casing of a well drilled in the late 19th century. According to Dr. John Rinehart in his book A Guide to Geyser Gazing (1976 p. 49), a man had drilled into the geyser in search for water. He had "simply opened up a dead geyser".

In the case of the Big Mine Run Geyser in Ashland, Pennsylvania, the heat powering the geyser (which erupts from an abandoned mine vent) comes not from geothermal power, but from the long-simmering Centralia mine fire.

Perpetual spouter
This is a natural hot spring that spouts water constantly without stopping for recharge. Some of these are incorrectly called geysers, but because they are not periodic in nature they are not considered true geysers.

Commercialization

Geysers are used for various activities such as electricity generation, heating and tourism. Many geothermal reserves are found all around the world. The geyser fields in Iceland are some of the most commercially viable geyser locations in the world. Since the 1920s hot water directed from the geysers has been used to heat greenhouses and to grow food that otherwise could not have been cultivated in Iceland's inhospitable climate. Steam and hot water from the geysers has also been used for heating homes since 1943 in Iceland. In 1979 the U.S. Department of Energy (DOE) actively promoted development of geothermal energy in the "Geysers-Calistoga Known Geothermal Resource Area" (KGRA) near Calistoga, California through a variety of research programs and the Geothermal Loan Guarantee Program. The Department is obligated by law to assess the potential environmental impacts of geothermal development.

Extraterrestrial geyser-like features
There are many bodies in the Solar System where eruptions which superficially resemble terrestrial geysers have been observed or are believed to occur. Despite being commonly referred to as geysers, they are driven by fundamentally different processes, consist of a wide range of volatiles, and can occur on vastly disparate scales; from the modestly sized Martian carbon dioxide jets to the immense plumes of Enceladus. Generally, there are two broad categories of feature commonly referred to as geysers: sublimation plumes, and cryovolcanic plumes (also referred to as cryogeysers).

Sublimation plumes are jets of sublimated volatiles and dust from shallow sources under icy surfaces. Known examples include the CO2 jets on Mars, and the nitrogen eruptions on Neptune's moon Triton.

On Mars carbon dioxide jets are believed to occur in the southern polar region of Mars during spring, as a layer of dry ice accumulated over winter is warmed by the sun. Although these jets have not yet been directly observed, they leave evidence visible from orbit in the form of dark spots and lighter fans atop the dry ice. These features consist primarily of sand and dust blown out by the outbursts, as well as spider-like patterns of channels created below the ice by the rapid flow of CO2 gas. There are a plethora of theories to explain the eruptions, including heating from sunlight, chemical reactions, or even biological activity.

Triton was found to have active eruptions of nitrogen and dust by Voyager 2 when it flew past the moon in 1989. These plumes were up to 8   km high, where winds would blow them up to 150    km downwind, creating long, dark streaks across the otherwise bright south polar ice cap. There are various theories as to what drives the activity on Triton, such as solar heating through transparent ice, cryovolcanism, or basal heating of nitrogen ice sheets.

Cryovolcanic plumes or cryogeysers generally refer to large-scale eruptions of predominantly water vapour from active cryovolcanic features on certain icy moons. Such plumes occur on Saturn's moon Enceladus and Jupiter's moon Europa.

Plumes of water vapour, together with ice particles and smaller amounts of other components (such as carbon dioxide, nitrogen, ammonia, hydrocarbons and silicates), have been observed erupting from vents associated with the "tiger stripes" in the south polar region of Enceladus by the Cassini orbiter. These plumes are the source of the material in Saturn's E ring. The mechanism which causes these eruptions are generated remains uncertain, as well as to what extent they are physically linked to Enceladus' subsurface ocean, but they are believed to be powered at least in part by tidal heating. Cassini flew through these plumes several times, allowing direct analysis of water from inside another solar system body for the first time.

In December 2013, the Hubble Space Telescope detected water vapor plumes potentially 200   km high above the south polar region of Europa. Re-examination of Galileo data also suggested that it may have flown through a plume during a flyby in 1997. Water was also detected by the Keck Observatory in 2016, announced in a 2019 Nature article speculating the cause to be a cryovolcanic eruption. It is thought that Europa's lineae might be venting this water vapor into space in a similar manner to the "tiger stripes" of Enceladus.

title: Geyser


text:
Gaussian elimination

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the matrix is said to be in reduced row echelon form. This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form. $$\begin{bmatrix} 1 & 3 & 1 & 9 \\ 1 & 1 & -1 & 1 \\ 3 & 11 & 5 & 35 \end{bmatrix}\to \begin{bmatrix} 1 & 3 & 1 & 9 \\ 0 & -2 & -2 & -8 \\ 0 & 2 & 2 & 8 \end{bmatrix}\to \begin{bmatrix} 1 & 3 & 1 & 9 \\ 0 & -2 & -2 & -8 \\ 0 & 0 & 0 & 0 \end{bmatrix}\to \begin{bmatrix} 1 & 0 & -2 & -3 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$
 * Swapping two rows,
 * Multiplying a row by a nonzero number,
 * Adding a multiple of one row to another row.

Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes preferable to stop row operations before the matrix is completely reduced.

Definitions and example of algorithm
The process of row reduction makes use of elementary row operations, and can be divided into two parts. The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions. The second part (sometimes called back substitution) continues to use row operations until the solution is found; in other words, it puts the matrix into reduced row echelon form.

Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix. The elementary row operations may be viewed as the multiplication on the left of the original matrix by elementary matrices. Alternatively, a sequence of elementary operations that reduces a single row may be viewed as multiplication by a Frobenius matrix. Then the first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row echelon matrix.

Row operations
There are three types of elementary row operations which may be performed on the rows of a matrix:
 * 1) Swap the positions of two rows.
 * 2) Multiply a row by a non-zero scalar.
 * 3) Add to one row a scalar multiple of another.

If the matrix is associated to a system of linear equations, then these operations do not change the solution set. Therefore, if one's goal is to solve a system of linear equations, then using these row operations could make the problem easier.

Echelon form
For each row in a matrix, if the row does not consist of only zeros, then the leftmost nonzero entry is called the leading coefficient (or pivot) of that row. So if two leading coefficients are in the same column, then a row operation of type 3 could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above. If this is the case, then matrix is said to be in row echelon form. So the lower left part of the matrix contains only zeros, and all of the zero rows are below the non-zero rows. The word "echelon" is used here because one can roughly think of the rows being ranked by their size, with the largest being at the top and the smallest being at the bottom.

For example, the following matrix is in row echelon form, and its leading coefficients are shown in red: $$\begin{bmatrix} 0 & \color{red}{\mathbf{2}} &                    1   & -1 \\ 0 &                    0   & \color{red}{\mathbf{3}} &  1 \\ 0 &                    0   &                     0   &  0 \end{bmatrix}.$$

It is in echelon form because the zero row is at the bottom, and the leading coefficient of the second row (in the third column), is to the right of the leading coefficient of the first row (in the second column).

A matrix is said to be in reduced row echelon form if furthermore all of the leading coefficients are equal to 1 (which can be achieved by using the elementary row operation of type 2), and in every column containing a leading coefficient, all of the other entries in that column are zero (which can be achieved by using elementary row operations of type 3).

Example of the algorithm
Suppose the goal is to find and describe the set of solutions to the following system of linear equations: $$ \begin{alignat}{4} 2x &{}+{}& y &{}-{}& z &{}={}&   8 & \qquad (L_1) \\ -3x &{}-{}& y &{}+{}& 2z &{}={}& -11 & \qquad (L_2) \\ -2x &{}+{}& y &{}+{}& 2z &{}={}& -3 & \qquad (L_3) \end{alignat} $$

The table below is the row reduction process applied simultaneously to the system of equations and its associated augmented matrix. In practice, one does not usually deal with the systems in terms of equations, but instead makes use of the augmented matrix, which is more suitable for computer manipulations. The row reduction procedure may be summarized as follows: eliminate $x$ from all equations below $L_{1}$, and then eliminate $y$ from all equations below $L_{2}$. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.

 * {| style="background-color:white;" class="wikitable"

! System of equations !! Row operations !! Augmented matrix \begin{alignat}{4} 2x &{}+{}& y &{}-{}& z &{}={}&   8 & \\ -3x &{}-{}& y &{}+{}& 2z &{}={}& -11 & \\ -2x &{}+{}& y &{}+{}& 2z &{}={}& -3 & \end{alignat} $$ \left[\begin{array}{rrr|r} 2 & 1 & -1 &   8 \\ -3 & -1 &  2 & -11 \\ -2 &  1 &  2 &  -3 \end{array}\right] $$ \begin{alignat}{4} 2x &{}+{}&         y &{}-{}&          z &{}={}& 8 & \\ &    & \tfrac12 y &{}+{}& \tfrac12 z &{}={}& 1 & \\ &    &         2y &{}+{}&          z &{}={}& 5 & \end{alignat} $$ \begin{align} L_2 + \tfrac32 L_1 &\to L_2 \\ L_3 +         L_1 &\to L_3 \end{align} $$ \left[\begin{array}{rrr|r} 2 &     1  &     -1  & 8 \\ 0 & \frac12 & \frac12 & 1 \\ 0 &     2  &      1  & 5 \end{array}\right] $$ \begin{alignat}{4} 2x &{}+{}&         y &{}-{}&          z &{}={}& 8 & \\ &    & \tfrac12 y &{}+{}& \tfrac12 z &{}={}& 1 & \\ &    &            &     &         -z &{}={}& 1 & \end{alignat} $$ L_3 + -4 L_2 \to L_3$$ \left[\begin{array}{rrr|r} 2 &     1  &     -1  & 8 \\ 0 & \frac12 & \frac12 & 1 \\ 0 &     0  &     -1  & 1 \end{array}\right] $$ \begin{alignat}{4} 2x &{}+{}&         y &     &   &{}={}       7  & \\ &    & \tfrac12 y &     &   &{}={} \tfrac32 & \\ &    &            &{}-{}& z &{}={}       1  & \end{alignat} $$ \begin{align} L_1 -         L_3 &\to L_1\\ L_2 + \tfrac12 L_3 &\to L_2 \end{align} $$ \left[\begin{array}{rrr|r} 2 &     1  &  0 &      7  \\ 0 & \frac12 &  0 & \frac32 \\ 0 &     0  & -1 &      1 \end{array}\right] $$ \begin{alignat}{4} 2x &{}+{}& y &\quad&  &{}={}&  7 & \\ &    & y &\quad&   &{}={}&  3 & \\ &    &   &\quad& z &{}={}& -1 & \end{alignat} $$ \begin{align} 2 L_2 &\to L_2 \\ -L_3 &\to L_3 \end{align} $$ \left[\begin{array}{rrr|r} 2 & 1 & 0 & 7 \\ 0 & 1 & 0 &  3 \\ 0 & 0 & 1 & -1 \end{array}\right] $$ \begin{alignat}{4} x &\quad&  &\quad&   &{}={}&  2 & \\ &\quad& y &\quad&  &{}={}&  3 & \\ &\quad&  &\quad& z &{}={}& -1 & \end{alignat} $$ \begin{align} L_1 - L_2 &\to L_1 \\ \tfrac12 L_1      &\to L_1 \end{align} $$ \left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 &  3 \\ 0 & 0 & 1 & -1 \end{array}\right] $$
 * - align="center"
 * - align="center"
 * - align="center"
 * colspan=3 align="center"| The matrix is now in echelon form (also called triangular form)
 * - align="center"
 * - align="center"
 * - align="center"
 * - align="center"
 * }

The second column describes which row operations have just been performed. So for the first step, the $x$ is eliminated from $L_{2}$ by adding $3⁄2L_{1}$ to $L_{2}$. Next, $x$ is eliminated from $L_{3}$ by adding $L_{1}$ to $L_{3}$. These row operations are labelled in the table as $$\begin{align} L_2 + \tfrac32 L_1 &\to L_2, \\ L_3 +         L_1 &\to L_3. \end{align}$$

Once $y$ is also eliminated from the third row, the result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. From a computational point of view, it is faster to solve the variables in reverse order, a process known as back-substitution. One sees the solution is $z = −1$, $y = 3$, and $x = 2$. So there is a unique solution to the original system of equations.

Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish it from stopping after reaching echelon form.

History
The method of Gaussian elimination appears – albeit without proof – in the Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC. It was commented on by Liu Hui in the 3rd century.

The method in Europe stems from the notes of Isaac Newton. In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton had left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject.

Some authors use the term Gaussian elimination to refer only to the procedure until the matrix is in echelon form, and use the term Gauss–Jordan elimination to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by Wilhelm Jordan in 1888. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss–Jordan elimination independently.

Applications
Historically, the first application of the row reduction method is for solving systems of linear equations. Below are some other important applications of the algorithm.

Computing determinants
To explain how Gaussian elimination allows the computation of the determinant of a square matrix, we have to recall how the elementary row operations change the determinant:
 * Swapping two rows multiplies the determinant by −1
 * Multiplying a row by a nonzero scalar multiplies the determinant by the same scalar
 * Adding to one row a scalar multiple of another does not change the determinant.

If Gaussian elimination applied to a square matrix $A$ produces a row echelon matrix $B$, let $d$ be the product of the scalars by which the determinant has been multiplied, using the above rules. Then the determinant of $A$ is the quotient by $d$ of the product of the elements of the diagonal of $B$: $$\det(A) = \frac{\prod\operatorname{diag}(B)}{d}.$$

Computationally, for an $n × n$ matrix, this method needs only $O(n^{3})$ arithmetic operations, while using Leibniz formula for determinants requires $O(n!)$ operations (number of summands in the formula), and recursive Laplace expansion requires $O(2^{n})$ operations (number of sub-determinants to compute, if none is computed twice). Even on the fastest computers, these two methods are impractical or almost impracticable for $n$ above 20.

Finding the inverse of a matrix
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If $A$ is an $n × n$ square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the $n × n$ identity matrix is augmented to the right of $A$, forming an $n × 2n$ block matrix $[A | I]$. Now through application of elementary row operations, find the reduced echelon form of this $n × 2n$ matrix. The matrix $A$ is invertible if and only if the left block can be reduced to the identity matrix $I$; in this case the right block of the final matrix is $A^{−1}$. If the algorithm is unable to reduce the left block to $I$, then $A$ is not invertible.

For example, consider the following matrix: $$A = \begin{bmatrix} 2 & -1 & 0 \\ -1 &  2 & -1 \\  0 & -1 &  2 \end{bmatrix}. $$

To find the inverse of this matrix, one takes the following matrix augmented by the identity and row-reduces it as a 3 × 6 matrix: $$[ A | I ] = \left[\begin{array}{ccc|ccc} 2 & -1 & 0 & 1 & 0 & 0 \\  -1 &  2 & -1 & 0 & 1 & 0 \\   0 & -1 &  2 & 0 & 0 & 1 \end{array}\right]. $$

By performing row operations, one can check that the reduced row echelon form of this augmented matrix is $$[ I | B ] = \left[\begin{array}{rrr|rrr} 1 & 0 & 0 & \frac34 & \frac12 & \frac14 \\ 0 & 1 & 0 & \frac12 &     1  & \frac12 \\ 0 & 0 & 1 & \frac14 & \frac12 & \frac34 \end{array}\right]. $$

One can think of each row operation as the left product by an elementary matrix. Denoting by $B$ the product of these elementary matrices, we showed, on the left, that $BA = I$, and therefore, $B = A^{−1}$. On the right, we kept a record of $BI = B$, which we know is the inverse desired. This procedure for finding the inverse works for square matrices of any size.

Computing ranks and bases
The Gaussian elimination algorithm can be applied to any $m × n$ matrix $A$. In this way, for example, some 6 × 9 matrices can be transformed to a matrix that has a row echelon form like $$ T= \begin{bmatrix} a & * & * & *& * & * & * & * & * \\ 0 & 0 & b & * & * & * & * & * & * \\ 0 & 0 & 0 & c & * & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & d & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}, $$ where the stars are arbitrary entries, and $a, b, c, d, e$ are nonzero entries. This echelon matrix $T$ contains a wealth of information about $A$: the rank of $A$ is 5, since there are 5 nonzero rows in $T$; the vector space spanned by the columns of $A$ has a basis consisting of its columns 1, 3, 4, 7 and 9 (the columns with $a, b, c, d, e$ in $T$), and the stars show how the other columns of $A$ can be written as linear combinations of the basis columns.

All of this applies also to the reduced row echelon form, which is a particular row echelon format.

Computational efficiency
The number of arithmetic operations required to perform row reduction is one way of measuring the algorithm's computational efficiency. For example, to solve a system of $n$ equations for $n$ unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires $n(n + 1)/2$ divisions, $(2n^{3} + 3n^{2} − 5n)/6$ multiplications, and $(2n^{3} + 3n^{2} − 5n)/6$ subtractions, for a total of approximately $2n^{3}/3$ operations. Thus it has a arithmetic complexity (time complexity, where each arithmetic operation take a unit of time, independently of the size of the inputs) of $O(n^{3})$.

This complexity is a good measure of the time needed for the whole computation when the time for each arithmetic operation is approximately constant. This is the case when the coefficients are represented by floating-point numbers or when they belong to a finite field. If the coefficients are integers or rational numbers exactly represented, the intermediate entries can grow exponentially large, so the bit complexity is exponential. However, Bareiss' algorithm is a variant of Gaussian elimination that avoids this exponential growth of the intermediate entries; with the same arithmetic complexity of $O(n^{3})$, it has a bit complexity of $O(n^{5})$, and has therefore a strongly-polynomial time complexity.

Gaussian elimination and its variants can be used on computers for systems with thousands of equations and unknowns. However, the cost becomes prohibitive for systems with millions of equations. These large systems are generally solved using iterative methods. Specific methods exist for systems whose coefficients follow a regular pattern (see system of linear equations).

Bareiss algorithm
The first strongly-polynomial time algorithm for Gaussian elimination was published by Jack Edmonds in 1967. Independently, and almost simultaneously, Erwin Bareiss discovered another algorithm, based on the following remark, which applies to a division-free variant of Gaussian elimination.

In standard Gaussian elimination, one subtracts from each row $$R_i$$ below the pivot row $$R_k$$ a multiple of $$R_k$$ by $$r_{i,k}/r_{k,k},$$ where $$r_{i,k}$$ and $$r_{k,k}$$ are the entries in the pivot column of $$R_i$$ and $$R_k,$$ respectively.

Bareiss variant consists, instead, of replacing $$R_i$$ with $\frac{r_{k,k}R_i-r_{i,k}R_k}{r_{k-1,k-1}}.$ This produces a row echelon form that has the same zero entries as with the standard Gaussian elimination.

Bareiss' main remark is that each matrix entry generated by this variant is the determinant of a submatrix of the original matrix.

In particular, if one starts with integer entries, the divisions occurring in the algorithm are exact divisions resulting in integers. So, all intermediate entries and final entries are integers. Moreover, Hadamard inequality provides an upper bound on the absolute values of the intermediate and final entries, and thus a bit complexity of $$\tilde O(n^5),$$ using soft O notation.

Moreover, as an upper bound on the size of final entries is known, a complexity $$\tilde O(n^4)$$ can be obtained with modular computation followed either by Chinese remaindering or Hensel lifting.

As a corollary, the following problems can be solved in strongly polynomial time with the same bit complexity:

 * Testing whether m given rational vectors are linearly independent
 * Computing the determinant of a rational matrix
 * Computing a solution of a rational equation system Ax = b
 * Computing the inverse matrix of a nonsingular rational matrix
 * Computing the rank of a rational matrix

Numeric instability
One possible problem is numerical instability, caused by the possibility of dividing by very small numbers. If, for example, the leading coefficient of one of the rows is very close to zero, then to row-reduce the matrix, one would need to divide by that number. This means that any error which existed for the number that was close to zero would be amplified. Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable, when using partial pivoting, even though there are examples of stable matrices for which it is unstable.

Generalizations
Gaussian elimination can be performed over any field, not just the real numbers.

Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination, manifesting as the choice to work from left to right when selecting pivot positions.

Computing the rank of a tensor of order greater than 2 is NP-hard. Therefore, if $P ≠ NP$, there cannot be a polynomial time analog of Gaussian elimination for higher-order tensors (matrices are array representations of order-2 tensors).

Pseudocode
As explained above, Gaussian elimination transforms a given $m × n$ matrix $A$ into a matrix in row-echelon form.

In the following pseudocode,  denotes the entry of the matrix $A$ in row $i$ and column $j$ with the indices starting from 1. The transformation is performed in place, meaning that the original matrix is lost for being eventually replaced by its row-echelon form.

h := 1 /* Initialization of the pivot row */ k := 1 /* Initialization of the pivot column */ while h ≤ m and k ≤ n    /* Find the k-th pivot: */ i_max := argmax (i = h ... m, abs(A[i, k])) if A[i_max, k] = 0 /* No pivot in this column, pass to next column */ k := k + 1 else swap rows(h, i_max) /* Do for all rows below pivot: */ for i = h + 1 ... m:            f := A[i, k] / A[h, k]             /* Fill with zeros the lower part of pivot column: */ A[i, k] := 0 /* Do for all remaining elements in current row: */ for j = k + 1 ... n:                A[i, j] := A[i, j] - A[h, j] * f         /* Increase pivot row and column */ h := h + 1 k := k + 1

This algorithm differs slightly from the one discussed earlier, by choosing a pivot with largest absolute value. Such a partial pivoting may be required if, at the pivot place, the entry of the matrix is zero. In any case, choosing the largest possible absolute value of the pivot improves the numerical stability of the algorithm, when floating point is used for representing numbers.

Upon completion of this procedure the matrix will be in row echelon form and the corresponding system may be solved by back substitution.

title: Gaussian elimination


text:
Guantanamo Bay Naval Base

Guantanamo Bay Naval Base (Base Naval de la Bahía de Guantánamo), officially known as Naval Station Guantanamo Bay or NSGB, (also called GTMO, pronounced Gitmo  as jargon  by members of the U.S. military ) is a United States military base located on 45 sqmi of land and water on the shore of Guantánamo Bay at the southeastern end of Cuba. It has been leased to the United States with no end date since 1903 as a coaling station and naval base, making it the oldest overseas U.S. naval base. The lease was $2,000 in gold per year until 1934, when the payment was set to match the value of gold in dollars; in 1974, the yearly lease was set to $4,085.

Since taking power in 1959, the Cuban communist government has consistently protested against the U.S. presence on Cuban soil, arguing that the base "was imposed on Cuba by force" and is "illegal under international law." Since 2002, the naval base has contained a military prison, for alleged unlawful combatants captured in Afghanistan, Iraq, and other places during the War on Terror. Cases of alleged torture of prisoners by the U.S. military, and their denial of protection under the Geneva Conventions, have been criticized.

The 1903 lease has no fixed expiration date; as such, it can only be ended if the US Navy decided to abandon the area or both countries agreed mutually to end the lease.

Geography
The Naval Base occupies three main geographical sections: Leeward Point, Windward Point, and Guantánamo Bay. Guantánamo Bay physically divides the Naval Station into sections. The bay extends past the boundaries of the base into Cuba, where the bay is then referred to as Bahía de Guantánamo. Guantánamo Bay contains several cays, which are identified as Hospital Cay, Medico Cay, North Toro Cay, and South Toro Cay.

Leeward Point of the Naval Station is the site of the active airfield. Major geographical features on Leeward Point include Mohomilla Bay and the Guantánamo River. Three beaches exist on the Leeward side. Two are available for use by base residents, while the third, Hicacal Beach, is closed. Nearby is the US Naval Cemetery at Guantanamo Bay, also known as Cuzco Wells Cemetery. The cemetery holds over 350 burials including that of Captain Henry Smith who died as a result of injuries received during the sinking of his ship the SS Delmundo by the German submarine U-600 on 13 August 1942.

Windward Point contains most of the activities at the Naval Station. There are nine beaches available to base personnel. The highest point on the base is John Paul Jones Hill (named after the Revolutionary War hero of the same name) at a total of 495 ft. The geography of Windward Point is such that there are many coves and peninsulas along the bay shoreline providing ideal areas for mooring ships. According to Stephen Benz, Stephen Crane described the area as "craggy" and "cut with ravines." He also described the coastline as covered in ridges and "chalky cliffs".

Spanish era
The area surrounding Guantanamo Bay was originally inhabited by the Taíno people. On 30 April 1494, Christopher Columbus, on his second voyage, arrived and spent the night. The place where Columbus landed is now known as Fisherman's Point. Columbus declared the bay Puerto Grande. The bay and surrounding areas briefly came under British control during the War of Jenkins' Ear. Prior to the outbreak of the conflict, the bay was referred to as Walthenham Harbor by British cartographers. The British expeditionary force renamed the bay Cumberland Bay. They eventually retreated from the area after an attempt to march to Santiago de Cuba was repulsed by Spanish troops.

Spanish–American War
During the Spanish–American War, the U.S. fleet attacking Santiago secured Guantánamo's harbor for protection during the hurricane season of 1898. The Marines landed at Guantanamo Bay with naval support, and American and Cuban forces routed the defending Spanish troops. There is a monument on McCalla Hill to one Navy officer and five Marines who died in battle at Guantanamo Bay.

The war ended with the Treaty of Paris of 1898, in which Spain formally relinquished control of Cuba. Although the war was over, the United States maintained a strong military presence on the island. In 1901 the United States government passed the Platt Amendment as part of an Army Appropriations Bill. Section VII of this amendment read

"That to enable the United States to maintain the independence of Cuba, and to protect the people thereof, as well as for its own defense, the government of Cuba will sell or lease to the United States lands necessary for coaling or naval stations at certain specified points to be agreed upon with the President of the United States."

After initial resistance by the Cuban Constitutional Convention, the Platt Amendment was incorporated into the Constitution of the Republic of Cuba in 1901. The Constitution took effect in 1902, and land for a naval base at Guantanamo Bay was granted to the United States the following year.

Permanent lease
The 1903 lease agreement, which has no fixed expiration date, was executed in two parts. The first, signed in February, consisted of the following provisions:
 * 1) Agreement – The United States of America and the Republic of Cuba, desiring to maintain the Independence of Cuba, will enter into a lease for lands necessary for US Naval Stations.
 * 2) Article 1 – Describes the boundaries of the areas being leased, Guantanamo Bay and Bahia Honda.
 * 3) Article 2 – The U.S. may occupy, use, and modify the properties to fit the needs of a coaling and naval station, only. Vessels in the Cuban trade shall have free passage.
 * 4) Article 3 – Cuba retains ultimate sovereignty, but during the occupation, the U.S. exercises sole jurisdiction over the areas described in Article 1. Under conditions to be agreed on, the U.S. has the right to acquire, by purchase or eminent domain, any land included therein.

The second part, signed five months later in July 1903, consisted of the following provisions: SIGNED Theodore Roosevelt and Jose M Garcia Montes.
 * 1) Article 1 – Payment is $2000 gold coin, annually. All private lands within the boundaries shall be acquired by Cuba. The U.S. will advance rental payments to Cuba to facilitate those purchases.
 * 2) Article 2 – The U.S. shall pay for a survey of the sites and mark the boundaries with fences.
 * 3) Article 3 – There will be no commercial or other enterprise within the leased areas.
 * 4) Article 4 – Mutual extradition
 * 5) Article 5 – Not ports of entry.
 * 6) Article 6 – Ships shall be subject to Cuban port police. The U.S. will not obstruct entry or departure into the bay.
 * 7) Article 7 – This proposal is open for seven months.

In 1934, the United States unilaterally changed the payment from gold coin to U.S. dollars per the Gold Reserve Act. The lease amount was set at US$3,386.25, based on the price of gold at the time. In 1973, the U.S. adjusted the lease amount to $3,676.50, and in 1974 to $4,085, based on further increases to the price of gold in USD. Payments have been sent annually, but only one lease payment has been accepted since the Cuban Revolution and Fidel Castro claimed that this check was deposited due to confusion in 1959. The Cuban government has not deposited any other lease check since that time.

Before and during World War II
The Naval Base became a prime area for winter training for the Atlantic fleet. Due to the large increase in population during the training months, the Naval Base quickly established facilities to support everyday functions. Construction of the Guantanamo Bay fleet range system began as early as 1905 under Captain Rogers. Four primary ranges were built during its construction: Range Alpha, Range Beta, Range Charlie, and Range Delta. Due to the size of the base, sections were shut down in order for range operations to proceed. Ironically, the concrete range system's size and cost led to its downfall. Strong evidence suggests the fleet ranges were used throughout World War I and the beginning of World War II.

During World War II, the base was set up to use a nondescript number for postal operations. The base used the Fleet Post Office, Atlantic, in New York City, with the address: 115 FPO NY. The base was also an important intermediate distribution point for merchant shipping convoys from New York City and Key West, Florida, to the Panama Canal and the islands of Puerto Rico, Jamaica, and Trinidad and Tobago.

Cold War and beyond
From 1953 to 1959, thousands of Cubans commuted daily from outside the base to jobs within it. In mid-1958, vehicular traffic was stopped; workers were required to walk through the base's several gates. Public Works Center buses were pressed into service almost overnight to carry the tides of workers to and from the gate.

The "Cactus Curtain" is a term describing the line separating the naval base from Cuban-controlled territory, an allusion to Europe's Iron Curtain, the Bamboo Curtain in East Asia, and the similar Ice Curtain in the Bering Strait. After the Cuban Revolution, some Cubans sought refuge on the Guantanamo Bay Naval Base. In late 1961, Cuban troops planted an 8 mi long, 10 ft wide barrier of Opuntia (prickly pear) cactus along the northeastern section of the 17 mi fence surrounding the base in order to prevent checkpoint evasion when moving between the base and Cuba proper.

The curtain forms part of a "no man's land" that encircles the base. This area is complete with perimeter patrols, outposts featuring sandbags, and watchtowers, and has been complemented with barbed wire fences, minefields, and cacti. Apart from the cacti, both U.S. and Cuban troops erected, maintained, and otherwise manned these defenses, primarily to prevent checkpoint evasion and possible invasion from the other side.

U.S. and Cuban troops placed some 55,000 anti-personnel and anti-tank land mines across the "no man's land" around the perimeter of the naval base, creating the second-largest minefield in the world, and the largest in the Western Hemisphere. Initially, the mines were laid down by US troops, who also laid down signs stating that the landmines were "precautions" and should not be viewed as "aggressive". In response, Cuban troops also laid down their own mines, with both sides completing their minefields in 1961. Between 1961 and 1965, landmine explosions resulted in the deaths of at least 10 people, including as the result of engineering accidents and late-night partygoers. On 16 May 1996, U.S. President Bill Clinton ordered the demining of the American field. They have since been replaced with motion and sound sensors to detect intruders on the base. The Cuban government has not removed its corresponding minefield outside the perimeter.

During the Cuban Missile Crisis in 1962, the families of military personnel were evacuated from the base. Notified of the evacuation on 22 October, evacuees were told to pack one suitcase per family member, to bring evacuation and immunization cards, to tie pets in the yard, to leave the keys to the house on the dining table, and to wait in front of the house for buses. Dependents travelled to the airfield for flights to the United States, or to ports for passage aboard evacuation ships. After the crisis was resolved, family members were allowed to return to the base in December 1962.

From 1939, the base's water was supplied by pipelines that drew water from the Yateras River about 4.5 mi northeast of the base. The U.S. government paid a fee for this; in 1964, it was about $14,000 a month for about 2.5 e6U.S.gal per day. In 1964, the Cuban government stopped the flow. The base had about 14 e6U.S.gal of water in storage, and strict water conservation was put into effect immediately. The U.S. first imported water from Jamaica by barge, then relocated a desalination plant from San Diego (Point Loma). When the Cuban government accused the United States of "stealing water", base commander John D. Bulkeley ordered that the pipelines be cut and a section removed. A 38 in length of the 14 in diameter pipe and a 20 in length of the 10 in diameter pipe were lifted from the ground and the openings sealed.

During the 1960s and 1970s, the base had problems with alcohol and racial tension. Harassment and strip searches also became a regular occurrence for Cuban workers on the base. Several old guns from the USS Monongahela (1862) have been salvaged and placed around the base. The old warship served as a storeship at Guantanamo until destroyed by fire on 17 March 1908. A 4 in gun was salvaged from her wreck and put on display at the Naval Station. Since the gun was deformed by the heat from the fire, it was nicknamed "Old Droopy". A similar gun, possibly also salvaged from the Monongahela, is on display near the Bay View Club on the Naval Station.

By 2006, only two elderly Cubans, Luis Delarosa and Harry Henry, still crossed the base's North East Gate daily to work on the base, because the Cuban government prohibited new recruitment since 1959. They both retired at the end of 2012. At the United Nations Human Rights Council in 2013, Cuba's Foreign Minister demanded the U.S. return the base.

Current operations

The military facility has over 8,500 U.S. sailors and Marines stationed there. It is the only military base the U.S. maintains in a socialist country.

In 2005, the U.S. Navy completed a $12 million wind-power project at the base, erecting four 950 kilowatt, 275 ft wind turbines, reducing the reliance on diesel fuel to power the existing diesel generators (the base's primary electricity generation). In 2006, the wind turbines reduced diesel fuel consumption by 650000 USgal annually.

Units and commands
Access to the Naval Station is very limited and must be pre-approved through the appropriate local chain of command with the Commander of the station as the final approval. Since berthing facilities are limited, visitors must be sponsored indicating that they have an approved residence for the duration of the visit.

 * Resident units
 * Headquarters, Naval Station Guantanamo Bay
 * Customer Service Desk (CSD)
 * Joint Task Force Guantanamo
 * Headquarters, JTF Guantanamo
 * Joint Detention Group
 * Joint Intelligence Group
 * Joint Medical Group
 * U.S. Coast Guard Maritime Security Detachment Guantanamo Bay
 * AFN Guantanamo Bay radio and television unit
 * Marine Corps Security Force Company
 * Naval Computer and Telecommunications Area Master Station Atlantic Detachment Guantanamo Bay
 * Naval Hospital Guantanamo Bay
 * Navy Supply
 * Navy Security Forces
 * SEABEE Detachment
 * U.S. Coast Guard Aviation Detachment Guantanamo Bay

 * Assigned units
 * Fleet Composite Squadron Ten (VC-10) (1965–1993)
 * U.S. Marine Corps Ground Defense Force (GDF) (1971–2000 [Redesignated as Marine Corps Security Forces Company on 1 Sep 2000])
 * Naval Security Group Activity (Company L) (1966–2001)
 * Shore Intermediate Maintenance Activity (SIMA) (1903–1995)
 * Fleet Training Group (FTG) (1943–1995)

 * Homeported watercraft
 * YC 1639 (open lighter)
 * Leeward (YFB-92) (ferry boat)
 * Windward (YFB-93) (ferry boat)
 * YON 258 (non-self propelled fuel oil barge)
 * USS Wanamassa (YTB-820) (large harbor tug)
 * LCU 1671 and MK-8: landing craft used as an alternate ferry for transportation to areas inaccessible by the primary ferry and for moving hazardous cargo.
 * GTMO-5, GTMO-6 and GTMO-7 (50-ft. utility boats): used for personnel transportation during off-ferry hours.

Besides servicemembers, the base houses a large number of civilian contractors working for the military. Many of these contractors are migrant workers from Jamaica and the Philippines, and are thought to constitute up to 40% of the base's population.
 * Civilian contractors

Major contractors working at NSGB have included the following:
 * KBR
 * Schuyler Line Navigation Company (SLNC)
 * Satellite Communication Systems Incorporated
 * Centerra
 * EMCOR
 * Islands Mechanical Contractor
 * Munilla Construction Management
 * RQ Construction
 * MCM Construction
 * J&J Worldwide Services

Cargo shipping
Ocean transportation is provided by Schuyler Line Navigation Company, a U.S. Flag Ocean Carrier. Schuyler Line operates under government contract to supply sustainment and building supplies to the base.

Airfields
There are two airfields within the base, Leeward Point Field and McCalla Field. Leeward Point Field is the active military airfield, with the ICAO code MUGM and IATA code NBW. McCalla Field was designated as the auxiliary landing field in 1970s, but was no longer a viable airfield by the 1990s.

Leeward Point Field
Leeward Point Field was constructed with a 6,000-foot main runway in 1943; the runway was extended in 1953 to 8,000 feet to accommodate jet aircraft.

Leeward Point Field has a single active runway, 10/28, measuring 8000 ft. The former runway, 9/27 was 8500 ft. Currently, Leeward Point Field operates several aircraft and helicopters supporting base operations. Leeward Point Field was home to Fleet Composite Squadron 10 (VC-10) until the unit was phased out in 1993. VC-10 was one of the last active-duty squadrons flying the Douglas A-4 Skyhawk.

McCalla Field
McCalla Field was established in 1931 and remained operational until 1970. The airfield was named for Bowman H. McCalla, who was a United States Navy admiral in charge of the Battle of Guantánamo Bay. The current field was expanded in 1941 when the original grass runway was replaced. Naval Air Station Guantanamo Bay was officially established 1 February 1941. Aircraft routinely operating out of McCalla included JRF-5, N3N, J2F, C-1 Trader, and dirigibles.

The airfield was deactivated in the 1970s and was used to house Cuban and Haitian refugees beginning in the 1990s. Sometime between 1996 and 2001, the refugee camps were dismantled and the area became a collection of abandoned buildings. McCalla Field is now listed as a closed airfield. The USN aviation assets are now located across the bay at Leeward Point Field, about 1.5 miles to the west.

After the events of September 11, the area was reevaluated as a possible location for a detention facility. Camp Justice is now located on the grounds of the former airfield.

The area formerly consisted of three runways, all now closed: 1/19 at 4500 ft, 14/32 at 2210 ft, and 10/28 at 1850 ft.

The former airfield resides at an elevation of 60 ft above mean sea level.

Detention camp

Beginning in the last quarter of the 20th century, the base was used to house Cuban and Haitian refugees intercepted on the high seas. In the early 1990s, it held refugees who fled Haiti after military forces overthrew president Jean-Bertrand Aristide. These refugees were held in a detainment area called Camp Bulkeley until United States district court Judge Sterling Johnson, Jr. declared the camp unconstitutional on 8 June 1993. This decision was later vacated. The last Haitian migrants departed Guantanamo on 1 November 1995.

Beginning in 2002, following the U.S. invasion of Afghanistan, started in response to the September 11 attacks, a small portion of the base was used to detain several hundred individuals with ties or suspected ties to Islamic terrorist groups such as al-Qaeda and the Taliban. These detainees were not afforded POW status or facing formal charges but instead designated enemy combatants; assigned to confinement in Camp Delta, Camp Echo, Camp Iguana, and the now-closed Camp X-Ray. In litigation regarding the availability of fundamental rights to those imprisoned at the base, the U.S. Supreme Court has recognized that the detainees "have been imprisoned in territory over which the United States exercises exclusive jurisdiction and control." Therefore, the detainees have the fundamental right to due process of law under the Fifth Amendment. A district court has since held that the "Geneva Conventions applied to the Taliban detainees, but not to members of Al-Qaeda terrorist organization."

On 10 June 2006, the Department of Defense reported that three Guantanamo Bay detainees took their own lives. The military reported the men hanged themselves with nooses made of sheets and clothes. A study published by Seton Hall Law's Center for Policy and Research, while making no conclusions regarding what actually transpired, asserts that the military investigation failed to address significant issues detailed in that report.

On 6 September 2006, President George W. Bush announced that alleged or non-alleged combatants held by the CIA would be transferred to the custody of Department of Defense, and held at Guantanamo Prison. Of approximately 500 prisoners in Guantanamo Bay, only 10 have been tried by the Guantanamo military commission, but all cases have been stayed pending the adjustments being made to comply with the U.S. Supreme Court decision in Hamdi v. Rumsfeld.

In January 2009, President Obama signed an executive order directing the closing of the Guantánamo detention camp within a year. This plan was thwarted for the time being on 20 May 2009, when the United States Senate voted to keep the prison at Guantanamo Bay open for the foreseeable future and forbid the transfer of any detainees to facilities in the United States. Senator Daniel Inouye, a Democrat from Hawaii and chairman of the appropriations committee, said he favored keeping Guantanamo open until Obama produced a "coherent plan for closing the prison." Consequently, Obama decided to postpone difficult decisions on the details for at least six months. On 7 March 2011, Obama issued a new executive order permitting indefinite detention of Guantánamo detainees. This decision was codified into federal law by provision added to the National Defense Authorization Act for Fiscal Year 2012.

In early February 2021, the administration of U.S. President Joe Biden declared his intention to shut down the facility before he leaves office. In July 2021, an additional detainee was released.

In June 2022, an Afghan prisoner held at Guantanamo Bay for about 15 years without trial was released.

In February 2023, three prisoners were released from Guantanamo Bay. Majid Khan, who pled guilty before a Military Commission in 2012, had completed his sentence of 10 years, and was released to Belize. Brothers Abdul Al-Rahim Ghulam Rabbani and Mohammed Ahmad Ghulam Rabbani, whose detentions were deemed no longer necessary in order to protect the United States from a national security threat, were released and repatriated to the government of Pakistan.

In March 2023, Ghassan al-Sharbi was released to his home country of Saudi Arabia from Guantanamo Bay, after being deemed no longer necessary to detain in order to protect the United States from a national security threat.

In June 2023, United Nations Special Rapporteur on Human Rights and Counter-Terrorism Fionnuala Ní Aoláin released her final report on the detention center. The report concludes that prisoners endure "ongoing cruel, inhuman, and degrading treatment" and that the detention center should be closed.

Represented businesses
Despite the prohibition on the establishment of "commercial or other enterprises" as stated in Article 3 of the second part of the lease, several recognized American food outlets have been opened at the military base. Most of the restaurants on the installation are franchises which are owned and operated by the Department of the Navy. Proceeds from these restaurants are used to support Morale, Welfare and Recreation (MWR) activities for service personnel and their families. These restaurants are located inside the base; as such, they are not accessible to Cubans.

A Baskin-Robbins ice cream stand, which opened in the 1980s, was one of the first business franchises allowed on the base. In early 1986, the base added the first and only McDonald's restaurant within Cuba. A Subway restaurant was opened in 1987 and closed in 1994 when the Navy Department refused to renew the lease. A Hooters restaurant was opened in 2002. In 2004, a combined KFC and A&W restaurant was opened at the bowling alley and a Pizza Hut Express was added to the Windjammer Restaurant. There is also a cafe that sells Starbucks coffee, and there is a combined KFC and Taco Bell restaurant.

Education and schooling
Department of Defense Education Activity (DoDEA) provides for the education of dependent personnel with two schools. Both schools are named for Rear Admiral William Thomas Sampson. W.T. Sampson Elementary School serves grades K–5 and W. T. Sampson High School serves grades 6–12. The Villamar Child Development Center provides child care for dependents from six weeks to five years old. MWR operates a Youth Center that provides activities for dependents.

Some former students of Guantánamo have shared stories of their experiences with the Guantánamo Public Memory Project. The 2013 documentary Guantanamo Circus directed by Christina Linhardt and Michael Rose reveals a glimpse of day-to-day life on GTMO as seen through the eyes of circus performers who visit the base. It is used as a reference by the Guantánamo Public Memory Project.

Climate
The bay has an annual rainfall of about 24 in. The amount of rainfall has resulted in the base being classified as a semi-arid desert environment. The annual average high temperature on the base is 31.2 °C, the annual average low is 22.5 °C.

Notable people
Notable people born at the naval base include actor Peter Bergman and American guitarist Isaac Guillory.

title: Guantanamo Bay Naval Base