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Gladstone Gander

Gladstone Gander is a cartoon character created in 1948 by The Walt Disney Company. He is an anthropomorphic gander (male goose) who possesses exceptionally good luck that grants him anything he desires as well as protecting him from any harm. This is in contrast to his cousin Donald Duck, who is often characterized for having bad luck. Gladstone is also a rival of Donald for the affection of Daisy Duck. Gladstone dresses in a very debonair way, often in a suit, and wearing a bow-tie, fedora, and spats. He has a wavy hairstyle which is depicted either as white or blonde. In the story "Luck of the North" (December 1949), he is described as having a brassy voice.

Origin
Gladstone Gander first appeared in "Wintertime Wager" in Walt Disney's Comics and Stories #88 (January 1948), written and drawn by Carl Barks. In that story he arrives at Donald Duck's house during a freezing cold Christmas Day to remind him of a wager Donald made the previous summer; that he could swim in the Frozenbear Lake during Christmas Day or forfeit his house to Gladstone. Donald eventually loses the wager but Gladstone later on loses a wager of his own, brought to light by Daisy Duck, and thus Donald's house is returned to him.

Barks gradually developed Gladstone's personality and demeanor after his first appearance and used him quite frequently—in 24 stories between 1948 and 1953, the first five years of his existence. In his first three appearances in 1948 ("Wintertime Wager", "Gladstone Returns", "Links Hijinks"), he was portrayed as the mirror image of Donald: an obstinate braggart, perhaps just a little bit more arrogant, but did not yet have his characteristic luck. In his next two appearances, "Rival Beachcombers" and "The Goldilocks Gambit", Gladstone is portrayed as merely lazy and irritable, and also gullible. The breakthrough of his lucky streak occurs in 1949, within the adventure story "Race to the South Seas!" (March of Comics #41). In that story, Donald and his three nephews set sail on a rescue mission from Duckburg to a remote Pacific island on which Scrooge McDuck is believed to have stranded, in an attempt to gain their uncle's favor. For the same reason Gladstone is in hot pursuit as well, but because he was "born lucky" as Donald explains to his nephews, without lifting a finger, is having a much easier time than them.

His and Donald's rivalry over Daisy is established in "Donald's Love Letters" (1949), "Wild About Flowers" (1950), and "Knightly Rivals" (1951), and as potential heirs to Scrooge's fortune in "Some Heir Over the Rainbow" (1953). After that, Barks felt unable to develop the character further, finding him basically unsympathetic, and began using him less frequently. But by then, Gladstone had found a steady place in the Duck universe as one of the main established characters; frequently used by other writers and artists both in the Americas and Europe. He was first used by an artist other than Barks in 1951: "Presents For All" by Del Connell and Bob Moore. He appears as a main character in the Big Little Book series book "Luck of the Ducks" (1969).

Character
Gladstone's good luck defies probability and provides him with anything that would be to his benefit or enjoyment; as well as things he specifically wishes for, which are at times related to the plot of some stories. This could range from finding wallets and other valuables on the sidewalk to pieces of a ripped apart treasure map floating together in river to form it whole again; revealing the treasure’s location. His good luck also protects him from any harm. At times he might not even know that a situation will work out in his favor in a later stage and by then often feeling confused or at times even thinking his luck has abandoned him; just to learn that it has been with him all along.

There have been various explanations for Gladstone's good luck over the years by various writers and artists. In Barks' story "Luck of the North" Gladstone proclaims: "I was born under a lucky star, and everything I do will bring me good fortune [...]." This is read from a horoscope book he owns which also has a map showing his lucky star conniving with the planet Neptunus. In many of the Italian comic books stories, Gladstone is bestowed with his luck because Fortuna, the goddess of fortune, is in love with him. In Don Rosa's story "The Sign of the Triple Distelfink" (1997), he added the fact that Gladstone was born on the day of his mother Daphne's birthday in 1920, under the protection sign of the Triple Distelfink, thus inheriting his mother's luck. In some stories he also uses good luck charms like lucky horseshoes or rabbits foots.

However, for all his luck, Gladstone has no achievements to be proud of and no true ambitions, as he is incapable of long-term planning. This is all because of that he does not have to make the slightest of efforts to get what he wants, as his good luck will just give it to him in the end. He also often does not learn any life lessons from any misfortunes he could experience. This leads him to be extremely lazy; at times even thinking that willfully wishing for something is hard work, and disconnected from the realities of ordinary life. All of this is in stark contrast to his relative Scrooge McDuck, who is also capable of taking advantage of opportunities but works hard to create situations favorable for him; is strongly motivated by his ambitions and takes pride in forming his fortune by his own efforts and experiences. Instead, Gladstone often shows pride in his effortlessness and expresses great anxiety if he would betray those ideals. This is something first explored in more detail in Carl Barks's story "Gladstone's Terrible Secret" (May, 1952). Comic artist and writer Don Rosa has commented about this on the character: "Gladstone is unwilling to make the slightest effort to gain something that his luck cannot give him, and, when things go wrong, he resigns immediately, certain that around the next corner a wallet, dropped by a passer-by, will be waiting for him". Gladstone feels open contempt towards work of any kind, claiming it is beneath him, to the point of hesitating to help others even if it is to his own advantage, claiming that it seems alarmingly like work.

Also because of his good fortune, Gladstone is most often characterized to be very snobbish and a gloat; especially (and in some stories exclusively) toward his cousin Donald to whom he also can be very aggravating. For all of these reasons, he and Donald have formed an intense rivalry with each other. Gladstone's outrageous luck and boastfulness toward his cousin, combined with Donald's own ego and belief he can still best him despite all odds — or as Don Rosa's version of Donald comments, "Donald's eternal tendency towards self-destruction" — have set the stage for many stories featuring the two cousins' confrontations. They have, however, worked together or at least tolerated each other at times, but this is very rare and not without some tension that easily can turn into their ordinary rivalry.

Occasionally, he is a rival to Scrooge McDuck himself who resents his complete reliance on his uncanny good luck. In such stories, often the only way Gladstone can be believably defeated is to have him win by the letter of the original story preset while the heroes later on take the bigger prize. In Carl Barks' story "The Golden Nugget Boat" (Uncle Scrooge #35, September 1961), Gladstone and Scrooge are competing in a gold prospecting contest in which Gladstone finds a gold nugget the nephews fashioned from a gold item Scrooge already owns to stop him from killing himself from overexertion in the contest. Gladstone finds the nugget and returns to win, but Scrooge and his nephews then find a much bigger nugget they are able to fashion into a boat to return to civilization afterward with a monetary worth that is easily more than Gladstone's find. Another instance of this with his rivalry with Donald was in the "Salmon Derby" (Walt Disney's Comics and Stories #167, August 1954), where Gladstone catches the biggest fish and wins a new car but Donald manages to save a wealthy tycoon's daughter and is able to purchase a much bigger car. Another instance was when both Donald and Gladstone were competing for a job as a cameraman for a nature film director because Daisy was the director's assistant, Gladstone got the job but wished he had not because he wound up trudging through a swamp to film giant spiders while Daisy stayed behind in America, with Donald.

In more modern stories where he takes a more protagonistic role, writers most often downplay his unlikability to make him more relatable. For example, showing that he can learn from potential bad experiences, even if they never would become reality, and put more emphasis on his love for his family, even if he at times has a hard time expressing these feelings.

In many stories, Gladstone is also considered among the prime candidates for Scrooge McDuck's succession. In "Some Heir Over the Rainbow" (Walt Disney's Comics and Stories #155, August 1953) by Carl Barks, Scrooge gives $1,000 to Donald, Gladstone, and Huey, Dewey and Louie to determine how they use it in order to be the most suitable heir to his fortune. Gladstone thinks of using the money to buy a television set until he finds a raffle ticket, in which he decides to save the money by placing it in a tree. Scrooge is unimpressed that Gladstone did not increase the money at all, but acknowledges that at least he still "had" it. Huey, Dewey, and Louie infuriate Scrooge by investing their money in what Scrooge assumes was a scam. Finally, Scrooge thinks to himself, "I guess my heir will have to be Gladstone Gander! .... What an awful injustice to the world!" Soon afterwards, however, the children's decision proves to be the wisest, and they become Scrooge's heirs instead of Gladstone.

He is a rival of Donald for the love of his girlfriend Daisy Duck. This is often portrayed either by showing Daisy uncertain of which one she likes the most or; the more common version, that she is angry with Donald and goes out with Gladstone instead to make Donald jealous. In modern comics this love-triangle is at times often more downplayed. However, despite having an eternal crush on Daisy Duck, Gladstone has appeared in love with other duck girls in Italian and Danish comic stories. One of those is Linda Paper, who really conquered Gladstone, making him want to give up all his luck, since she is unlucky and absolutely hates lucky people. She appeared in two subsequent comic stories. Another is Feather Mallard, who is as lucky as Gladstone but whenever they are together they are instead experiencing bad luck. A number of recent Italian stories feature Gladstone's protector goddess Fortuna appearing in person. She appears in one noteworthy tale, "Gastone e il debole dalla Fortuna" ("Gladstone and the Struck of Luck"), by Enrico Faccini and Augusto Macchetto, first published on March 24, 1998. In this story, the Goddess takes mortal guise in order to be able to date her favorite gander, and appears to be love-struck with him. Although Gladstone competes for Daisy's attention and affection with Donald and often loses, he reassures himself that at least he still has his extraordinary luck, because in the end he is a hedonist and the only person he really adores is himself.

His exact relation to the Duck Family Tree at its early stages was somewhat uncertain. In Carl Barks' original version of the family tree from the 1950s, Gladstone was the son of Luke the Goose and Daphne Duck who died by overeating at a free-lunch picnic. He was later adopted by Matilda McDuck and Goosetave Gander. Later, Barks is reported to have done away with the adoption, which was never featured in any story. (Of course, no stories denying the event were published.) In a more recent version of the family tree created by Don Rosa, with input from Barks, it was established that Daphne Duck (Donald's paternal aunt) married Goostave Gander and the two were Gladstone's parents. This is consistent with what Gladstone says in "Race to the South Seas": "Scrooge McDuck is my mother's brother's brother-in-law".

Gladstone has a nephew named Shamrock Gladstone who shares his propensity for luck. He also has a con artist cousin (not on Donald's side of the family) named Disraeli.

Animation
Gladstone makes two speaking appearances in the animated series DuckTales, where he was voiced by Rob Paulsen. In these episodes he is not characterized as much of a braggart he is in the comics but rather focusing much more on his care-free nature, and additionally acting a bit more foolishly. In the episode "Dime Enough for Luck", Gladstone is an unwitting stooge for Magica De Spell in one of her attempts to steal Scrooge's Number One Dime. He returns in the episode "Dr. Jekyll & Mr. McDuck", where he accidentally bids on an item that turns out to be valuable. This inspires Scrooge to bid on the next item—a trunk containing Dr. Jekyll's formula—which sets the plot in motion. He also makes non-speaking cameo appearances in the episodes "Sweet Duck of Youth" and "Till Nephews Do Us Part", as well in episode of House of Mouse titled "Goofy For A Day". Gladstone appears in the 2000 video game Donald Duck: Goin' Quackers in his traditional role of Donald's rival for Daisy's affection, and every time a Boss Battle is about to start, Gladstone greets Donald, but always gets hurt, becoming squished by a giant bird, getting knocked off a building by a wrecking ball, being sent crashing to the bottom of a haunted mansion, and even gets sent back to Duckburg inside a pipe, and every time he gets hurt, he says that he has found a nickel. He also appears in the DuckTales reboot, first appearing in the episode "The House of the Lucky Gander!". In this series, he is voiced by Paul F. Tompkins. This series brings his personality far closer to how he acts in the comics, although it is implied that he is genuinely unaware of how arrogant he is.

title: Gladstone Gander


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Gordon Michael Woolvett

Gordon Michael Woolvett (born 1970) is a Canadian actor from Hamilton, Ontario.

Career
Woolvett's most enduring role was as Seamus Zelazny Harper on the television series Andromeda (2000–2005). Prior to Andromeda he starred in another science fiction television series, Deepwater Black. He was also credited as playing Mitch in the "Pariah" episode of the short-lived 1980s science-fiction/action series Captain Power and the Soldiers of the Future. He was in two episodes of Psi Factor: Chronicles of the Paranormal.

Woolvett was also one of the first program jockeys for YTV's The Zone (then called The After-School Zone) and the original main host for a program called Video & Arcade Top 10 which also aired on YTV. He acted in the 1999 made-for-TV film Ultimate Deception with Yasmine Bleeth. He also appeared on the Canadian television series The Guard, which aired from 2008 to 2009 on Global Television Network.

In 1992, Woolvett was nominated for the Gemini Award for Best Performance by a Supporting Actor for his role in Princes in Exile (1990). He directed the documentary Around the World in 80 Anthems, which won Best Documentary at the seventh International Film Festival Manhattan in 2017.

In 2020, Woolvett began teaching acting, improv, scene study, movie making, screenwriting and performance to young actors aged 9–18 on the Outschool homeschooling platform. His first year brought more than 70 five-star reviews from parents and students.

Personal life
Woolvett was born in 1970 in Hamilton, Ontario, Canada. He married Michele Morand on January 15, 2000. His brother is actor Jaimz Woolvett. He has three children.

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

title: Gordon Michael Woolvett


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Gypsum

Gypsum is a soft sulfate mineral composed of calcium sulfate dihydrate, with the chemical formula CaSO4*2H2O. It is widely mined and is used as a fertilizer and as the main constituent in many forms of plaster, drywall and blackboard or sidewalk chalk. Gypsum also crystallizes as translucent crystals of selenite. It forms as an evaporite mineral and as a hydration product of anhydrite. The Mohs scale of mineral hardness defines gypsum as hardness value 2 based on scratch hardness comparison.

Fine-grained white or lightly tinted forms of gypsum known as alabaster have been used for sculpture by many cultures including Ancient Egypt, Mesopotamia, Ancient Rome, the Byzantine Empire, and the Nottingham alabasters of Medieval England.

Etymology and history
The word gypsum is derived from the Greek word γύψος, "plaster". Because the quarries of the Montmartre district of Paris have long furnished burnt gypsum (calcined gypsum) used for various purposes, this dehydrated gypsum became known as plaster of Paris. Upon adding water, after a few dozen minutes, plaster of Paris becomes regular gypsum (dihydrate) again, causing the material to harden or "set" in ways that are useful for casting and construction.

Gypsum was known in Old English as spærstān, "spear stone", referring to its crystalline projections. Thus, the word spar in mineralogy, by comparison to gypsum, refers to any non-ore mineral or crystal that forms in spearlike projections. In the mid-18th century, the German clergyman and agriculturalist Johann Friderich Mayer investigated and publicized gypsum's use as a fertilizer. See:
 * From p. 544: " … er bewirtschaftete nebenbei ein Pfarrgüttchen, … für die Düngung der Felder mit dem in den nahen Waldenburger Bergen gefundenen Gips einsetzte." ( … he also managed a small parson's estate, on which he repeatedly conducted agricultural experiments.  In 1768, he first published the fruits of his experiences during this time as "Instruction about Gypsum", in which he espoused the fertilizing of fields with the gypsum that was found in the nearby Waldenburg mountains.)
 * From p. 60: "Schon seit undenklichen Zeiten … ein Gewinn zu erhalten seyn wird."  (Since times immemorial, in our vicinity, in the ministry of Niedeck [a village southeast of Göttingen], one has already made this use of gypsum; but Mr. Mayer has the merit to have made it generally known.  In the History of Farming in Kupferzell, he had depicted a crushing mill (p. 74), in order to pulverize gypsum, from which a profit has been obtained, albeit with difficulty.)
 * Gypsum may act as a source of sulfur for plant growth, and in the early 19th century, it was regarded as an almost miraculous fertilizer. American farmers were so anxious to acquire it that a lively smuggling trade with Nova Scotia evolved, resulting in the so-called "Plaster War" of 1820.
 * Gypsum may act as a source of sulfur for plant growth, and in the early 19th century, it was regarded as an almost miraculous fertilizer. American farmers were so anxious to acquire it that a lively smuggling trade with Nova Scotia evolved, resulting in the so-called "Plaster War" of 1820.

Physical properties
Gypsum is moderately water-soluble (~2.0–2.5 g/L at 25 °C) and, in contrast to most other salts, it exhibits retrograde solubility, becoming less soluble at higher temperatures. When gypsum is heated in air it loses water and converts first to calcium sulfate hemihydrate (bassanite, often simply called "plaster") and, if heated further, to anhydrous calcium sulfate (anhydrite). As with anhydrite, the solubility of gypsum in saline solutions and in brines is also strongly dependent on NaCl (common table salt) concentration.

The structure of gypsum consists of layers of calcium (Ca2+) and sulfate (SO4(2-)) ions tightly bound together. These layers are bonded by sheets of anion water molecules via weaker hydrogen bonding, which gives the crystal perfect cleavage along the sheets (in the {010} plane).

Crystal varieties
Gypsum occurs in nature as flattened and often twinned crystals, and transparent, cleavable masses called selenite. Selenite contains no significant selenium; rather, both substances were named for the ancient Greek word for the Moon.

Selenite may also occur in a silky, fibrous form, in which case it is commonly called "satin spar". Finally, it may also be granular or quite compact. In hand-sized samples, it can be anywhere from transparent to opaque. A very fine-grained white or lightly tinted variety of gypsum, called alabaster, is prized for ornamental work of various sorts. In arid areas, gypsum can occur in a flower-like form, typically opaque, with embedded sand grains called desert rose. It also forms some of the largest crystals found in nature, up to 12 m long, in the form of selenite.

Occurrence
Gypsum is a common mineral, with thick and extensive evaporite beds in association with sedimentary rocks. Deposits are known to occur in strata from as far back as the Archaean eon. Gypsum is deposited from lake and sea water, as well as in hot springs, from volcanic vapors, and sulfate solutions in veins. Hydrothermal anhydrite in veins is commonly hydrated to gypsum by groundwater in near-surface exposures. It is often associated with the minerals halite and sulfur. Gypsum is the most common sulfate mineral. Pure gypsum is white, but other substances found as impurities may give a wide range of colors to local deposits.

Because gypsum dissolves over time in water, gypsum is rarely found in the form of sand. However, the unique conditions of the White Sands National Park in the US state of New Mexico have created a 710 km2 expanse of white gypsum sand, enough to supply the US construction industry with drywall for 1,000 years. Commercial exploitation of the area, strongly opposed by area residents, was permanently prevented in 1933 when President Herbert Hoover declared the gypsum dunes a protected national monument.

Gypsum is also formed as a by-product of sulfide oxidation, amongst others by pyrite oxidation, when the sulfuric acid generated reacts with calcium carbonate. Its presence indicates oxidizing conditions. Under reducing conditions, the sulfates it contains can be reduced back to sulfide by sulfate-reducing bacteria. This can lead to accumulation of elemental sulfur in oil-bearing formations, such as salt domes, where it can be mined using the Frasch process Electric power stations burning coal with flue gas desulfurization produce large quantities of gypsum as a byproduct from the scrubbers.

Orbital pictures from the Mars Reconnaissance Orbiter (MRO) have indicated the existence of gypsum dunes in the northern polar region of Mars, which were later confirmed at ground level by the Mars Exploration Rover (MER) Opportunity.

Mining
Commercial quantities of gypsum are found in the cities of Araripina and Grajaú in Brazil; in Pakistan, Jamaica, Iran (world's second largest producer), Thailand, Spain (the main producer in Europe), Germany, Italy, England, Ireland, Canada and the United States. Large open pit quarries are located in many places including Fort Dodge, Iowa, which sits on one of the largest deposits of gypsum in the world, and Plaster City, California, United States, and East Kutai, Kalimantan, Indonesia. Several small mines also exist in places such as Kalannie in Western Australia, where gypsum is sold to private buyers for additions of calcium and sulfur as well as reduction of aluminum toxicities on soil for agricultural purposes.

Crystals of gypsum up to 11 m long have been found in the caves of the Naica Mine of Chihuahua, Mexico. The crystals thrived in the cave's extremely rare and stable natural environment. Temperatures stayed at 58 C, and the cave was filled with mineral-rich water that drove the crystals' growth. The largest of those crystals weighs 55 t and is around 500,000 years old.

Synthesis
Synthetic gypsum is produced as a waste product or by-product in a range of industrial processes.

Desulfurization
Flue gas desulfurization gypsum (FGDG) is recovered at some coal-fired power plants. The main contaminants are Mg, K, Cl, F, B, Al, Fe, Si, and Se. They come both from the limestone used in desulfurization and from the coal burned. This product is pure enough to replace natural gypsum in a wide variety of fields including drywalls, water treatment, and cement set retarder. Improvements in flue gas desulfurization have greatly reduced the amount of toxic elements present.

Desalination
Gypsum precipitates onto brackish water membranes, a phenomenon known as mineral salt scaling, such as during brackish water desalination of water with high concentrations of calcium and sulfate. Scaling decreases membrane life and productivity. This is one of the main obstacles in brackish water membrane desalination processes, such as reverse osmosis or nanofiltration. Other forms of scaling, such as calcite scaling, depending on the water source, can also be important considerations in distillation, as well as in heat exchangers, where either the salt solubility or concentration can change rapidly.

A new study has suggested that the formation of gypsum starts as tiny crystals of a mineral called bassanite (2CaSO4·H2O). This process occurs via a three-stage pathway:
 * 1) homogeneous nucleation of nanocrystalline bassanite;
 * 2) self-assembly of bassanite into aggregates, and
 * 3) transformation of bassanite into gypsum.

Refinery waste
The production of phosphate fertilizers requires breaking down calcium-containing phosphate rock with acid, producing calcium sulfate waste known as phosphogypsum (PG). This form of gypsum is contaminated by impurities found in the rock, namely fluoride, silica, radioactive elements such as radium, and heavy metal elements such as cadmium. Similarly, production of titanium dioxide produces titanium gypsum (TG) due to neutralization of excess acid with lime. The product is contaminated with silica, fluorides, organic matters, and alkalis.

Impurities in refinery gypsum waste have, in many cases, prevented them from being used as normal gypsum in fields such as construction. As a result, waste gypsum is stored in stacks indefinitely, with significant risk of leaching their contaminants into water and soil. To reduce the accumulation and ultimately clear out these stacks, research is underway to find more applications for such waste products.

Occupational safety
People can be exposed to gypsum in the workplace by breathing it in, skin contact, and eye contact. Calcium sulfate per se is nontoxic and is even approved as a food additive, but as powdered gypsum, it can irritate skin and mucous membranes.

United States
The Occupational Safety and Health Administration (OSHA) has set the legal limit (permissible exposure limit) for gypsum exposure in the workplace as TWA 15 mg/m3 for total exposure and TWA 5 mg/m3 for respiratory exposure over an eight-hour workday. The National Institute for Occupational Safety and Health (NIOSH) has set a recommended exposure limit (REL) of TWA 10 mg/m3 for total exposure and TWA 5 mg/m3 for respiratory exposure over an eight-hour workday.

Uses
Gypsum is used in a wide variety of applications:

Construction industry

 * Gypsum board is primarily used as a finish for walls and ceilings, and is known in construction as plasterboard, "sheetrock", or drywall. Gypsum provides a degree of fire-resistance to these materials and glass fibers are added to their composition to accentuate this effect. Gypsum has little heat conductivity, giving its plaster some insulative properties.
 * Gypsum blocks are used like concrete blocks in building construction.
 * Gypsum mortar is an ancient mortar used in building construction.
 * A component of Portland cement used to prevent flash setting (too rapid hardening) of concrete.
 * A wood substitute in the ancient world: For example, when wood became scarce due to deforestation on Bronze Age Crete, gypsum was employed in building construction at locations where wood was previously used.

Agriculture

 * Fertilizer: In the late 18th and early 19th centuries, Nova Scotia gypsum, often referred to as plaster, was a highly sought fertilizer for wheat fields in the United States. Gypsum provides two of the secondary plant macronutrients, calcium and sulfur. Unlike limestone, it generally does not affect soil pH.
 * Reclamation of saline soils, regardless of pH. When gypsum is added to sodic (saline) and acidic soil, the highly soluble form of boron (sodium metaborate) is converted to the less soluble calcium metaborate. Exchangeable sodium percentage is also reduced by gypsum application. The Zuiderzee Works uses gypsum for the recovered land.
 * Other soil conditioner uses: Gypsum reduces aluminium and boron toxicity in acidic soils. It also improves soil structure, improving water absorption and aeration.
 * Soil water potential monitoring: a gypsum block can be inserted into soil, its electrical resistance measured to derive soil moisture.

Modeling, sculpture and art

 * Plaster for casting moulds and modeling.
 * As alabaster, a material for sculpture, it was used especially in the ancient world before steel was developed, when its relative softness made it much easier to carve. During the Middle Ages and Renaissance, it was preferred even to marble.
 * In the medieval period, scribes and illuminators used it as an ingredient in gesso, which was applied to illuminated letters and gilded with gold in illuminated manuscripts.

Food and drink

 * A tofu (soy bean curd) coagulant, making it ultimately a significant source of dietary calcium.
 * Adding hardness to water used for brewing.
 * Used in baking as a dough conditioner, reducing stickiness, and as a baked-goods source of dietary calcium. The primary component of mineral yeast food.
 * Used in mushroom cultivation to stop grains from clumping together.

Medicine and cosmetics

 * Plaster for surgical splints.
 * Impression plasters in dentistry.

Other

 * An alternative to iron oxide in some thermite mixes.
 * Tests have shown that gypsum can be used to remove pollutants such as lead or arsenic from contaminated waters.

title: Gypsum


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Growth factor

A growth factor is a naturally occurring substance capable of stimulating cell proliferation, wound healing, and occasionally cellular differentiation. Usually it is a secreted protein or a steroid hormone. Growth factors are important for regulating a variety of cellular processes.

Growth factors typically act as signaling molecules between cells. Examples are cytokines and hormones that bind to specific receptors on the surface of their target cells.

They often promote cell differentiation and maturation, which varies between growth factors. For example, epidermal growth factor (EGF) enhances osteogenic differentiation (osteogenesis or bone formation), while fibroblast growth factors and vascular endothelial growth factors stimulate blood vessel differentiation (angiogenesis).

Comparison to cytokines
Growth factor is sometimes used interchangeably among scientists with the term cytokine. Historically, cytokines were associated with hematopoietic (blood and lymph forming) cells and immune system cells (e.g., lymphocytes and tissue cells from spleen, thymus, and lymph nodes). For the circulatory system and bone marrow in which cells can occur in a liquid suspension and not bound up in solid tissue, it makes sense for them to communicate by soluble, circulating protein molecules. However, as different lines of research converged, it became clear that some of the same signaling proteins which the hematopoietic and immune systems use were also being used by all sorts of other cells and tissues, during development and in the mature organism.

While growth factor implies a positive effect on cell proliferation, cytokine is a neutral term with respect to whether a molecule affects proliferation. While some cytokines can be growth factors, such as G-CSF and GM-CSF, others have an inhibitory effect on cell growth or cell proliferation. Some cytokines, such as Fas ligand, are used as "death" signals; they cause target cells to undergo programmed cell death or apoptosis.

The nerve growth factor (NGF) was first discovered by Rita Levi-Montalcini, which won her a Nobel Prize in Physiology or Medicine.

List of classes
Individual growth factor proteins tend to occur as members of larger families of structurally and evolutionarily related proteins. There are many families, some of which are listed below:
 * Adrenomedullin (AM)
 * Angiopoietin (Ang)
 * Autocrine motility factor
 * Bone morphogenetic proteins (BMPs)
 * Ciliary neurotrophic factor family
 * Ciliary neurotrophic factor (CNTF)
 * Leukemia inhibitory factor (LIF)
 * Interleukin-6 (IL-6)
 * Colony-stimulating factors
 * Macrophage colony-stimulating factor (M-CSF)
 * Granulocyte colony-stimulating factor (G-CSF)
 * Granulocyte macrophage colony-stimulating factor (GM-CSF)
 * Epidermal growth factor (EGF)
 * Ephrins
 * Ephrin A1
 * Ephrin A2
 * Ephrin A3
 * Ephrin A4
 * Ephrin A5
 * Ephrin B1
 * Ephrin B2
 * Ephrin B3
 * Erythropoietin (EPO)
 * Fibroblast growth factor (FGF)
 * Fibroblast growth factor 1(FGF1)
 * Fibroblast growth factor 2(FGF2)
 * Fibroblast growth factor 3(FGF3)
 * Fibroblast growth factor 4(FGF4)
 * Fibroblast growth factor 5(FGF5)
 * Fibroblast growth factor 6(FGF6)
 * Fibroblast growth factor 7(FGF7)
 * Fibroblast growth factor 8(FGF8)
 * Fibroblast growth factor 9(FGF9)
 * Fibroblast growth factor 10(FGF10)
 * Fibroblast growth factor 11(FGF11)
 * Fibroblast growth factor 12(FGF12)
 * Fibroblast growth factor 13(FGF13)
 * Fibroblast growth factor 14(FGF14)
 * Fibroblast growth factor 15(FGF15)
 * Fibroblast growth factor 16(FGF16)
 * Fibroblast growth factor 17(FGF17)
 * Fibroblast growth factor 18(FGF18)
 * Fibroblast growth factor 19(FGF19)
 * Fibroblast growth factor 20(FGF20)
 * Fibroblast growth factor 21(FGF21)
 * Fibroblast growth factor 22(FGF22)
 * Fibroblast growth factor 23(FGF23)
 * Foetal Bovine Somatotrophin (FBS)
 * GDNF family of ligands
 * Glial cell line-derived neurotrophic factor (GDNF)
 * Neurturin
 * Persephin
 * Artemin
 * Growth differentiation factor-9 (GDF9)
 * Hepatocyte growth factor (HGF)
 * Hepatoma-derived growth factor (HDGF)
 * Insulin
 * Insulin-like growth factors
 * Insulin-like growth factor-1 (IGF-1)
 * Insulin-like growth factor-2 (IGF-2)
 * Interleukins
 * IL-1- Cofactor for IL-3 and IL-6. Activates T cells.
 * IL-2 – T-cell growth factor. Stimulates IL-1 synthesis. Activates B-cells and NK cells.
 * IL-3 – Stimulates production of all non-lymphoid cells.
 * IL-4 – Growth factor for activated B cells, resting T cells, and mast cells.
 * IL-5 – Induces differentiation of activated B cells and eosinophils.
 * IL-6 – Stimulates Ig synthesis. Growth factor for plasma cells.
 * IL-7 – Growth factor for pre-B cells.
 * Keratinocyte growth factor (KGF)
 * Migration-stimulating factor (MSF)
 * Macrophage-stimulating protein (MSP), also known as hepatocyte growth factor-like protein (HGFLP)
 * Myostatin (GDF-8)
 * Neuregulins
 * Neuregulin 1 (NRG1)
 * Neuregulin 2 (NRG2)
 * Neuregulin 3 (NRG3)
 * Neuregulin 4 (NRG4)
 * Neurotrophins
 * Brain-derived neurotrophic factor (BDNF)
 * Nerve growth factor (NGF)
 * Neurotrophin-3 (NT-3)
 * Neurotrophin-4 (NT-4)
 * Placental growth factor (PGF)
 * Platelet-derived growth factor (PDGF)
 * Renalase (RNLS) – Anti-apoptotic survival factor
 * T-cell growth factor (TCGF)
 * Thrombopoietin (TPO)
 * Transforming growth factors
 * Transforming growth factor alpha (TGF-α)
 * Transforming growth factor beta (TGF-β)
 * Tumor necrosis factor-alpha (TNF-α)
 * Vascular endothelial growth factor (VEGF)

In platelets
The alpha granules in blood platelets contain growth factors PDGF, IGF-1, EGF, and TGF-β which begin healing of wounds by attracting and activating macrophages, fibroblasts, and endothelial cells.

Uses in medicine
For the last two decades, growth factors have been increasingly used in the treatment of hematologic and oncologic diseases and cardiovascular diseases  such as:

 * skin wound healing and regeneration of other tissues such as bone (PDGF-BB)
 * neutropenia
 * myelodysplastic syndrome (MDS)
 * leukemias
 * aplastic anaemia
 * bone marrow transplantation
 * angiogenesis for cardiovascular diseases

title: Growth factor


text:
Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. He also wrote an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, while others are in Italian.

Biography
Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. He attended the Liceo classico Cavour in Turin, and enrolled at the University of Turin in 1876, graduating in 1880 with high honours, after which the University employed him to assist first Enrico D'Ovidio, and then Angelo Genocchi, the Chair of calculus. Due to Genocchi's poor health, Peano took over the teaching of calculus course within two years. His first major work, a textbook on calculus entitled Calcolo differenziale, e principii di calcolo integrale, was published in 1884 and was credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic. Here the modern symbols for the union and intersection of sets appeared for the first time.

In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, known for painting the Refugium Peccatorum Madonna. In 1886, he began teaching concurrently at the Royal Military Academy, and was promoted to Professor First Class in 1889. In that year he published the Peano axioms, a formal foundation for the collection of natural numbers. The next year, the University of Turin also granted him his full professorship. The Peano curve was published in 1890 as the first example of a space-filling curve which demonstrated that the unit interval and the unit square have the same cardinality. Today it is understood to be an early example of what is known as a fractal.

In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891. In 1891 Peano started the Formulario Project. It was to be an "Encyclopedia of Mathematics", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1897, the first International Congress of Mathematicians was held in Zürich. Peano was a key participant, presenting a paper on mathematical logic. He also started to become increasingly occupied with Formulario to the detriment of his other work.

In 1898 he presented a note to the Academy about binary numeration and its ability to be used to represent the sounds of languages. He also became so frustrated with publishing delays (due to his demand that formulae be printed on one line) that he purchased a printing press.

Paris was the venue for the Second International Congress of Mathematicians in 1900. The conference was preceded by the First International Conference of Philosophy where Peano was a member of the patronage committee. He presented a paper which posed the question of correctly formed definitions in mathematics, i.e. "how do you define a definition?". This became one of Peano's main philosophical interests for the rest of his life. At the conference, Peano met Bertrand Russell and gave him a copy of Formulario. Russell was struck by Peano's innovative logical symbols and after the conference, he retired to the country "to study quietly every word written by him or his disciples".

Peano's students Mario Pieri and Alessandro Padoa also had papers presented at the philosophy congress. For the mathematical congress, Peano did not speak, but Padoa's memorable presentation has been frequently recalled. A resolution calling for the formation of an "international auxiliary language" to facilitate the spread of mathematical (and commercial) ideas, was proposed; Peano fully supported it.

By 1901, Peano was at the peak of his mathematical career. He had made advances in the areas of analysis, foundations and logic, made many contributions to the teaching of calculus and also contributed to the fields of differential equations and vector analysis. Peano played a key role in the axiomatization of mathematics and was a leading pioneer in the development of mathematical logic. Peano had by this stage become heavily involved with the Formulario project and his teaching began to suffer. In fact, he became so determined to teach his new mathematical symbols that the calculus in his course was neglected. As a result, he was dismissed from the Royal Military Academy but retained his post at Turin University.

In 1903 Peano announced his work on an international auxiliary language called Latino sine flexione ("Latin without inflexion," later called Interlingua, and the precursor of the Interlingua of the IALA). This was an important project for him (along with finding contributors for 'Formulario'). The idea was to use Latin vocabulary, since this was widely known, but simplify the grammar as much as possible and remove all irregular and anomalous forms to make it easier to learn. On 3 January 1908, he read a paper to the Academia delle Scienze di Torino in which he started speaking in Latin and, as he described each simplification, introduced it into his speech so that by the end he was talking in his new language.

The year 1908 was important for Peano. The fifth and final edition of the Formulario project, titled Formulario mathematico, was published. It contained 4200 formulae and theorems, all completely stated and most of them proved. The book received little attention since much of the content was dated by this time. However, it remains a significant contribution to mathematical literature. The comments and examples were written in Latino sine flexione.

Also in 1908, Peano took over the chair of higher analysis at Turin (this appointment was to last for only two years). He was elected the director of Academia pro Interlingua. Having previously created Idiom Neutral, the Academy effectively chose to abandon it in favour of Peano's Latino sine flexione.

After his mother died in 1910, Peano divided his time between teaching, working on texts aimed for secondary schooling including a dictionary of mathematics, and developing and promoting his and other auxiliary languages, becoming a revered member of the international auxiliary language movement. He used his membership of the Accademia dei Lincei to present papers written by friends and colleagues who were not members (the Accademia recorded and published all presented papers given in sessions).

During the years 1913–1918, Peano published several papers that dealt with the remainder term for various numerical quadrature formulas, and introduced the Peano kernel.

In 1925 Peano switched Chairs unofficially from Infinitesimal Calculus to Complementary Mathematics, a field which better suited his current style of mathematics. This move became official in 1931. Giuseppe Peano continued teaching at Turin University until the day before he died when he suffered a fatal heart attack.

Milestones and honours received

 * 1881: Published first paper.
 * 1884: Calcolo Differenziale e Principii di Calcolo Integrale.
 * 1887: Applicazioni Geometriche del Calcolo Infinitesimale.
 * 1889: Appointed Professor First Class at the Royal Military Academy.
 * 1889: Arithmetices principia: nova methodo exposita.
 * 1890: Appointed Extraordinary Professor of infinitesimal calculus at the University of Turin.
 * 1891: Made a member of the Academy of Science, Torino.
 * 1893: Lezioni di Analisi Infinitesimale, 2 vols.
 * 1895: Promoted to Ordinary Professor.
 * 1901: Made Knight of the Order of Saints Maurizio and Lazzaro.
 * 1903: Announces Latino sine flexione.
 * 1905: Made Knight of the Order of the Crown of Italy. Elected a corresponding member of the Accademia dei Lincei in Rome, the highest Italian honour for scientists.
 * 1908: Fifth and final edition of the Formulario mathematico.
 * 1917: Made an Officer of the Crown of Italy.
 * 1921: Promoted to Commendatore of the Crown of Italy.

title: Giuseppe Peano


text:
Gin and tonic

A gin and tonic is a highball cocktail made with gin and tonic water poured over a large amount of ice. The ratio of gin to tonic varies according to taste, strength of the gin, other drink mixers being added, etc., with most recipes calling for a ratio between 1:1 and 1:3. It is usually garnished with a slice or wedge of lime. To preserve effervescence, the tonic can be poured down a bar spoon. The ice cools the gin, dulling the effect of the alcohol in the mouth and making the drink more pleasant and refreshing to taste.

It is commonly referred to as a G and T in the UK, US, Canada, Australia, New Zealand and Ireland. In some parts of the world (e.g., in Germany, Italy, France, Japan, the Netherlands, Spain, Turkey), it is called a gin tonic (ジン・トニック). It is also referred to as ginto in the Netherlands, and as GT in the Nordics.

Garnish and serving
Gin and tonic is traditionally garnished with a slice or wedge of lime, often slightly squeezed into the drink before being placed in the glass. In most parts of the world, lime remains the only usual garnish; however, lemon is often used as an alternative fruit. In the United Kingdom, the use of both lemon and lime together is known as an "Evans". Although the origins of the use of lemons are unknown, their use dates back at least as far as the late 1930s. The use of lemon or lime is a debated issue – some leading brands, such as Gordon's, Tanqueray, and Bombay Sapphire, recommend the use of lime in their gin.

The use of a balloon glass for serving gin has become popular, possibly through promotion by the Bombay Sapphire gin brand. The use of such a glass, with plenty of ice and a garnish tailored to the flavours of the gin, is sometimes said to allow the aromas of the drink to gather at its opening for the drinker to more easily appreciate.

The use of assorted fruits, herbs, and vegetables, as garnish (reflecting the botanicals of the individual gin), is increasingly popular. Besides the classic lime wheel or wedge, alternative garnishes can include orange peel, star anise, thyme-elderflower, a slice of ginger, pink grapefruit, rosemary, cucumber, mint, black peppercorns, strawberry and basil, strawberry syrup, or chillies. Fruits such as kumquats or other citrus or cucumber can be included.

History
The cocktail was created by officers of the Presidency armies, the military force of the East India Company which operated on the Indian subcontinent. On the Indian subcontinent and in other tropical regions, malaria was a persistent problem for Europeans, and in the 18th century, Scottish doctor George Cleghorn studied how quinine, a traditional cure for malaria, could be used to prevent the disease. The quinine was drunk in tonic water but the bitter taste was unpleasant. Presidency armies officers in India in the early 19th century took to adding a mixture of water, sugar, lime and gin to the quinine in order to make the drink more palatable, and thus the gin and tonic cocktail was born. The officers were already given a gin ration as part of their rations, and the sweet concoction made sense. Since it is no longer used as an antimalarial, tonic water today contains much less quinine, is usually sweetened, and is consequently much less bitter.

A 2004 study found that after 12 hours, "considerable quantities (500 to 1,000 ml) of tonic water may, for a short period of time, lead to quinine plasma levels at the lower limit of therapeutic efficacy and may, in fact, cause transitory suppression of parasites". This method of consumption of quinine was impractical for malaria prophylaxis, as the amount of drug needed "cannot be maintained with even large amounts of tonic". The authors concluded that it is not an effective form of treatment for malaria.

Variations
Mixers can include lime juice, lemon juice, orange juice and spiced simple syrup, grenadine, tea, etc.

A gin and tonic can also be mixed with a sorbet.

Some gin-and-tonic inspired drinks also have champagne (e.g., the Parisian), vermouth and Campari (e.g. the Negroni Sbagliato), vermouth and bitters (e.g., the Posh G&T), super smokey whiskey (e.g., the Ol' Smokey), peach liqueur and grapefruit bitters (e.g., the Tonic Delight), mint bitters, and chocolate liqueur (e.g., the Guilty Pleasure), etc.

In popular culture
The trans-galactic nature of the gin and tonic is imagined in Douglas Adams' novel The Restaurant at the End of the Universe, which describes how "85% of all known worlds in the Galaxy, be they primitive or highly advanced, have invented a drink called jynnan tonnyx, or gee-N'N-T'N-ix, or jinond-o-nicks, or any one of a thousand or more variations on the same phonetic theme. The drinks themselves are not the same, and vary between the Sivolvian 'chinanto/mnigs' which is ordinary water served at slightly above room temperature, and the Gagrakackan 'tzjin-anthony-ks' which kills cows at a hundred paces; and in fact the one common factor between all of them, beyond the fact that the names sound the same, is that they were all invented and named before the worlds concerned made contact with any other worlds."

James Bond specifies a recipe for a gin and tonic while in Kingston, Jamaica, in the book Dr. No. Unusually, it involves the juice of a whole lime.

In the BBC1 and Amazon Prime television series Fleabag, Fleabag and the Priest enjoy canned G&Ts from Marks and Spencer. The store reported a 24% increase in sales after the episodes aired.

Founded in 2010, International Gin & Tonic Day is celebrated worldwide on 19 October.

title: Gin and tonic


text:
Gimlet (tool)

A gimlet is a hand tool for drilling small holes, mainly in wood, without splitting. It was defined in Joseph Gwilt's Architecture (1859) as "a piece of steel of a semi-cylindrical form, hollow on one side, having a cross handle at one end and a worm or screw at the other".

A gimlet is always a small tool. A similar tool of larger size is called an auger. The cutting action of the gimlet is slightly different from an auger and the initial hole it makes is smaller; the cutting edges pare away the wood, which is moved out by the spiral sides, falling out through the entry hole. This also pulls the gimlet farther into the hole as it is turned; unlike a bradawl, pressure is not required once the tip has been drawn in.

The name gimlet comes from the Old French guinbelet, guimbelet, later guibelet, probably a diminutive of the Anglo-French wimble, a variation of "guimble", from the Middle Low German wiemel (cf. the Scandinavian wammie, 'to bore or twist'). Modern French uses the term vrille, also the French for "tendril".

Use as a metaphor
The term is also used figuratively to describe something as sharp or piercing, and also to describe the twisting, boring motion of using a gimlet. For example, the gimlet cocktail may be named after the tool. The term gimlet-eyed can mean sharp-eyed or squint-eyed; one example of this use is Major General Smedley Darlington Butler, who was known as "Old Gimlet Eye".

title: Gimlet (tool)


text:
Geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the $n$th root of the product of $n$ numbers, i.e., for a set of numbers $a_{1}, a_{2}, ..., a_{n}$, the geometric mean is defined as

 * $$\left(\prod_{i=1}^n a_i\right)^\frac{1}{n} = \sqrt[n]{a_1 a_2 \cdots a_n}$$

or, equivalently, as the arithmetic mean in logscale:

 * $$\exp{\left( {\frac{1}{n}\sum\limits_{i=1}^{n}\ln a_i} \right)}$$

Most commonly the numbers are restricted to being non-negative, to avoid complications related to negative numbers not having real roots, and frequently they are restricted to being positive, to enable the use of logarithms. In any case, the geometric mean is equal to zero for any data set where one or more values is equal to zero. The geometric mean can be an unreliable measure of central tendency for a dataset where one or more values are extremely close to zero in comparison to the other members of the dataset.

The geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, $$\sqrt{2 \cdot 8} = 4$$. The geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, $$\sqrt[3]{4 \cdot 1 \cdot 1/32} = 1/2$$.

The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time. It also applies to benchmarking, where it is particularly useful for computing means of speedup ratios: since the mean of 0.5x (half as fast) and 2x (twice as fast) will be 1 (i.e., no speedup overall).

Suppose for example a person invests 1000 dollars in shares and achieves annual returns of +10%, -12%, +90%, -30% and +25% over 5 consecutive years to give a final investment value of 1,609 dollars. The arithmetic mean of the annual percent changes is 16.6%. However, this value is unrepresentative. If the initial investment grew by 16.6% per annum, it would be worth 2155 dollars after 5 years. In fact, to find the average percentage growth it is necessary compute the geometric mean of the successive annual growth ratios (1.1, 0.88, 1.9, 0.7, 1.25). This gives a value of 1.0998 which corresponds to an annual average growth of 9.98%. It can be readily verified that an investment of 1000 dollars which grows by 9.98% over five years would achieve a final investment value of 1,609 dollars. In this case, the geometric mean is appropriate because investment growth is multiplicative rather than additive.

The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, $$a$$ and $$b$$, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths $$a$$ and $$b$$. Similarly, the geometric mean of three numbers, $$a$$, $$b$$, and $$c$$, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers.

The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)

Calculation
The geometric mean of a data set $\left\{a_1, a_2,\, \ldots,\, a_n\right\}$ is given by:
 * $$\left(\prod_{i=1}^n a_i \right)^\frac{1}{n} = \sqrt[n]{a_1 a_2 \cdots a_n}.$$

The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession (the number of values is represented by "n") to give a total product of the set, and then the nth root of the total product is taken to give the geometric mean of the original set. For example, in a set of four numbers $\{1, 2, 3, 4\}$, the product of $1 \times 2 \times 3 \times 4$ is $24$ , and the geometric mean is the fourth root of 24, or ~ 2.213. The exponent $\frac{1}{n}$ on the left side is equivalent to the taking nth root. For example, $24^\frac{1}{4} = \sqrt[4]{24}$.

Iterative means
The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, an intersection of the two which always lies in between.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences ($a_n$ ) and ($h_n$ ) are defined:
 * $$a_{n+1} = \frac{a_n + h_n}{2}, \quad a_0 = x$$

and
 * $$h_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{h_n}}, \quad h_0 = y$$

where $h_{n+1}$ is the harmonic mean of the previous values of the two sequences, then $a_n$  and $h_n$  will converge to the geometric mean of $x$  and $y$. The sequences converge to a common limit, and the geometric mean is preserved:
 * $$\sqrt{a_i h_i} =

\sqrt{\frac{a_i + h_i}{\frac{a_i + h_i}{h_i a_i}}} = \sqrt{\frac{a_i + h_i}{\frac{1}{a_i} + \frac{1}{h_i}}} = \sqrt{a_{i+1} h_{i+1}} $$

Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

Relationship with logarithms
The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms. By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:

When $$a_1, a_2, \dots, a_n > 0$$
 * $$\left( \prod_{i=1}^n a_i \right)^\frac{1}{n} = \exp\left[\frac{1}{n} \sum_{i=1}^n \ln a_i\right];$$
 * As:
 * $$\begin{align}

\left ( \prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}&= \sqrt[n]{a_1a_2\cdots a_n}\\ &=e ^{\ln( a_1a_2\cdots a_n)^{1/n}}\\ &=e ^{\frac{1}{n}\left (\ln a_1+\ln a_2+\cdots +\ln a_n \right )}\\ &=e ^\\ \text{geometric mean(}a\text{)}&=e^{\text{arithmetic mean(ln(}a\text{))}} \end{align}$$
 * alternatively, use any positive real number base, for both the logarithms and the number you are raising to the power of the arithmetic mean of the individual logarithms at that same base.

additionally, if negative values of the $$a_i$$ are allowed,
 * $$\left( \prod_{i=1}^n a_i \right)^\frac{1}{n} = \left(\left(-1\right)^m\right)^\frac{1}{n} \exp\left[\frac{1}{n}\sum_{i=1}^n \ln \left|a_i\right| \right],$$

where $m$ is the number of negative numbers. Please take note that this formula can produce POSITIVE geometric mean value for a set of ALL NEGATIVE numbers, so this formula is not reliable. A single negative value, eg -1, on a set of n numbers, where n is an odd number, will negate the calculated geometric mean regardless of how huge the range of positive values are, which does not make sense. It does not make sense that the formula's usefulness depends on the number of input values, ie n and m, and will not work when m is odd and n is even.

This is sometimes called the log-average (not to be confused with the logarithmic average). It is simply computing the arithmetic mean of the logarithm-transformed values of $$a_i$$ (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with $$f(x) = \log x$$. For example, the geometric mean of 2 and 8 can be calculated as the following, where $$b$$ is any base of a logarithm (commonly 2, $e$ or 10):
 * $$b^{\frac{1}{2}\left[\log_b (2) + \log_b (8)\right]} = 4$$

Related to the above, it can be seen that for a given sample of points $$a_1, \ldots, a_n$$, the geometric mean is the minimizer of
 * $$f(a) = \sum_{i=1}^n (\log(a_i) - \log(a))^2 = \sum_{i=1}^n (\log(a_i/a) )^2$$,

whereas the arithmetic mean is the minimizer of
 * $$f(a) = \sum_{i=1}^n (a_i - a)^2$$.

Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).

The log form of the geometric mean is generally the preferred alternative for implementation in computer languages because calculating the product of many numbers can lead to an arithmetic overflow or arithmetic underflow. This is less likely to occur with the sum of the logarithms for each number.

Comparison to arithmetic mean
The geometric mean of a non-empty data set of (positive) numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.

Average growth rate
In many cases the geometric mean is the best measure to determine the average growth rate of some quantity. (For example, if in one year sales increases by 80% and the next year by 25%, the end result is the same as that of a constant growth rate of 50%, since the geometric mean of 1.80 and 1.25 is 1.50.) In order to determine the average growth rate, it is not necessary to take the product of the measured growth rates at every step. Let the quantity be given as the sequence $$a_0, a_1,..., a_n$$, where $$n$$ is the number of steps from the initial to final state. The growth rate between successive measurements $$a_k$$ and $$a_{k+1}$$ is $$a_{k+1}/a_k$$. The geometric mean of these growth rates is then just:
 * $$\left( \frac{a_1}{a_0} \frac{a_2}{a_1} \cdots \frac{a_n}{a_{n-1}} \right)^\frac{1}{n} = \left(\frac{a_n}{a_0}\right)^\frac{1}{n}.$$

Application to normalized values
The fundamental property of the geometric mean, which does not hold for any other mean, is that for two sequences $$X$$ and $$Y$$ of equal length,

 * $$\operatorname{GM}\left(\frac{X_i}{Y_i}\right) = \frac{\operatorname{GM}(X_i)}{\operatorname{GM}(Y_i)}$$

This makes the geometric mean the only correct mean when averaging normalized results; that is, results that are presented as ratios to reference values. This is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources (for example, life expectancy, education years, and infant mortality). In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of execution time of computer programs:

Table 1

The arithmetic and geometric means "agree" that computer C is the fastest. However, by presenting appropriately normalized values and using the arithmetic mean, we can show either of the other two computers to be the fastest. Normalizing by A's result gives A as the fastest computer according to the arithmetic mean:

Table 2

while normalizing by B's result gives B as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

Table 3

and normalizing by C's result gives C as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

Table 4

In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.

However, this reasoning has been questioned. Giving consistent results is not always equal to giving the correct results. In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time (using the arithmetic mean), and then normalize that result to one of the computers. The three tables above just give a different weight to each of the programs, explaining the inconsistent results of the arithmetic and harmonic means (Table 4 gives equal weight to both programs, the Table 2 gives a weight of 1/1000 to the second program, and the Table 3 gives a weight of 1/100 to the second program and 1/10 to the first one). The use of the geometric mean for aggregating performance numbers should be avoided if possible, because multiplying execution times has no physical meaning, in contrast to adding times as in the arithmetic mean. Metrics that are inversely proportional to time (speedup, IPC) should be averaged using the harmonic mean.

The geometric mean can be derived from the generalized mean as its limit as $$p$$ goes to zero. Similarly, this is possible for the weighted geometric mean.

Geometric mean of a continuous function
If $$f:[a,b]\to(0, \infty)$$ is a positive continuous real-valued function, its geometric mean over this interval is

 * $$\text{GM}[f] = \exp\left(\frac{1}{b-a}\int_a^b\ln f(x)dx\right)$$

For instance, taking the identity function $$f(x) = x$$ over the unit interval shows that the geometric mean of the positive numbers between 0 and 1 is equal to $$\frac{1}{e}$$.

Proportional growth
The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.6666% and 42.8571% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5079% (80% + 16.6666% + 42.8571%, that sum then divided by 3). However, if we start with 100 oranges and let it grow 46.5079% each year, the result is 314 oranges, not 300, so the linear average over-states the year-on-year growth.

Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.166666 and 1.428571, i.e. $$\sqrt[3]{1.80 \times 1.166666 \times 1.428571} \approx 1.442249$$; thus the "average" growth per year is 44.2249%. If we start with 100 oranges and let the number grow with 44.2249% each year, the result is 300 oranges.

Financial
The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean. It is also used in the CPI calculation and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.

This has the effect of understating movements in the index compared to using the arithmetic mean.

Applications in the social sciences
Although the geometric mean has been relatively rare in computing social statistics, starting from 2010 the United Nations Human Development Index did switch to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared:
 * The geometric mean decreases the level of substitutability between dimensions [being compared] and at the same time ensures that a 1 percent decline in say life expectancy at birth has the same impact on the HDI as a 1 percent decline in education or income. Thus, as a basis for comparisons of achievements, this method is also more respectful of the intrinsic differences across the dimensions than a simple average.

Not all values used to compute the HDI (Human Development Index) are normalized; some of them instead have the form $$\left(X - X_\text{min}\right) / \left(X_\text{norm} - X_\text{min}\right)$$. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

The equally distributed welfare equivalent income associated with an Atkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is an Lp norm divided by the number of elements, with p equal to one minus the inequality aversion parameter.

Geometry
In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as the geometric mean theorem.

In an ellipse, the semi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from a focus; it is also the geometric mean of the semi-major axis and the semi-latus rectum. The semi-major axis of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. Another way to think about it is as follows:

Consider a circle with radius $$r$$. Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengths $$a$$ and $$b$$. Since the area of the circle and the ellipse stays the same, we have:

$$ \begin{align} \pi r^2 &= \pi a b \\

r^2 &= a b \\ r &= \sqrt{a b} \end{align} $$

The radius of the circle is the geometric mean of the semi-major and the semi-minor axes of the ellipse formed by deforming the circle.

Distance to the horizon of a sphere (ignoring the effect of atmospheric refraction when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

The geometric mean is used in both in the approximation of squaring the circle by S.A. Ramanujan and in the construction of the heptadecagon with "mean proportionals".

Aspect ratios
The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.

In the choice of 16:9 aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is $\sqrt{2.35 \times \frac{4}{3}} \approx 1.7701$, and thus $16:9 = 1.77\overline{7}$ ... was chosen. This was discovered empirically by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios. When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1. The value found by Powers is exactly the geometric mean of the extreme aspect ratios, 4:3(1.33:1) and CinemaScope(2.35:1), which is coincidentally close to $16:9$ ($1.77\overline{7}:1$ ). The intermediate ratios have no effect on the result, only the two extreme ratios.

Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the 14:9 ($1.55\overline{5}$ ...) aspect ratio, which is likewise used as a compromise between these ratios. In this case 14:9 is exactly the arithmetic mean of $16:9$ and $4:3 = 12:9$, since 14 is the average of 16 and 12, while the precise geometric mean is $\sqrt{\frac{16}{9}\times\frac{4}{3}} \approx 1.5396 \approx 13.8:9,$  but the two different means, arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).

Paper formats
The geometric mean is also used to calculate B and C series paper formats. The $$B_n$$ format has an area which is the geometric mean of the areas of $$A_n$$ and $$A_{n-1}$$. For example, the area of a B1 paper is $$\frac{\sqrt{2}}{2}\mathrm m^2$$, because it is the geometric mean of the areas of an A0 ($$1\mathrm m^2$$) and an A1 ($$\frac{1}{2}\mathrm m^2$$) paper ($$\sqrt{1\mathrm m^2 \cdot \frac{1}{2}\mathrm m^2}=\sqrt{\frac{1}{2}\mathrm m^4}=\frac{1}{\sqrt 2}\mathrm m^2= \frac{\sqrt 2}{2}\mathrm m^2$$).

The same principle applies with the C series, whose area is the geometric mean of the A and B series. For example, the C4 format has an area which is the geometric mean of the areas of A4 and B4.

An advantage that comes from this relationship is that an A4 paper fits inside a C4 envelope, and both fit inside a B4 envelope.

Other applications

 * Spectral flatness: in signal processing, spectral flatness, a measure of how flat or spiky a spectrum is, is defined as the ratio of the geometric mean of the power spectrum to its arithmetic mean.
 * Anti-reflective coatings: In optical coatings, where reflection needs to be minimised between two media of refractive indices n0 and n2, the optimum refractive index n1 of the anti-reflective coating is given by the geometric mean: $$n_1 = \sqrt{n_0 n_2}$$.
 * Subtractive color mixing: The spectral reflectance curve for paint mixtures (of equal tinting strength, opacity and dilution) is approximately the geometric mean of the paints' individual reflectance curves computed at each wavelength of their spectra.
 * Image processing: The geometric mean filter is used as a noise filter in image processing.
 * Labor compensation: The geometric mean of a subsistence wage and market value of the labor using capital of employer was suggested as the natural wage by Johann von Thünen in 1875.

title: Geometric mean


text:
Gent

Gent is a shortened form of the word gentleman. It may also refer to:

 * Ghent (Dutch: Gent), a Belgian city
 * K.A.A. Gent, a football club from Ghent
 * K.R.C. Gent, a football club from Ghent
 * Gent RFC, a rugby club in Ghent
 * .gent, a GeoTLD (top-level domain) for the city of Ghent
 * Gent (hyperelastic model), rubber elasticity model
 * Gent Cakaj (born 1990), Albanian politician
 * Gent Strazimiri (born 1972), Albanian politician and former Deputy Minister of Interior Affairs
 * Gent (surname)
 * Gent (magazine), a defunct pornographic magazine
 * Honeywell Gent, a brand of fire alarm systems previously known as Gents' of Leicester

title: Gent