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Olene Walker

Olene Walker (née Smith; November 15, 1930 – November 28, 2015) was an American politician who served as the 15th governor of Utah from 2003 to 2005, succeeding the governorship after Mike Leavitt's resignation. A member of the Republican Party, Walker was Utah's first female governor and the first female lieutenant governor of Utah.

Early life and education
Walker was born Olene Smith in Ogden, Utah, in 1930 to Thomas Ole Smith and Nina (née Hadley) Smith, the second of their five children. She graduated from Weber High School. Walker received her bachelor's degree from Brigham Young University in 1953, her master's from Stanford University, and her doctorate in education administration from the University of Utah.

Career
Walker's began her career in the Utah House of Representatives, including a term as Majority Whip, during which she helped create Utah’s Rainy Day Fund. She served in the Utah House from 1981 to 1989. She served as the fourth Lieutenant Governor of Utah for the 10 years prior to becoming governor. She founded the Salt Lake Education Foundation and served as its director. She served as director of the Utah Division of Community Development. She has chaired the Commission on Criminal and Juvenile Justice, the Utah State Housing Coordinating Committee, the Governor's Commission on Child Care, and the National Conference of Lieutenant Governors. She also became the first woman governor of Utah. Senator Orrin Hatch said Walker "truly paved the way for women in government in Utah." Also, throughout her political career "her passion across the board" was education.

Governorship
Walker assumed the office of Governor of the State of Utah after previous governor Mike Leavitt was nominated by President George W. Bush to lead the Environmental Protection Agency in 2003. Walker served as governor until the end of the term on January 3, 2005.

During her short term as governor, Walker focused on supporting education in Utah. She created the Read With a Child Early Literacy Initiative, visited classrooms often to read to children, and vetoed a proposal for vouchers for private schools with the reasoning that the proposal would take funding away from public schools. She also signed legislation ending the use of firing squads for execution in Utah and worked to preserve Utah wilderness and to create affordable housing.

In a move that caused a degree of controversy within the state, the Utah Republican Party at its convention on May 8, 2004, elected not to place Walker on the ballot for the party primary (held on June 22, 2004), selecting instead Jon Huntsman, Jr. and Nolan Karras as the two potential Republican party candidates for the office of Governor of the State of Utah, the first time in 48 years that an incumbent Utah governor failed to win a party nomination. Huntsman went on to win the primary election with more than 66% of the vote. These events effectively ruled out any possibility of Walker being on the ballot in the 2004 general election. Convention delegates defended their choice by claiming that many of the delegates were already pledged to other candidates, because Walker had served only six months as Governor before the party convention. She had also waited until two months before the nominating convention to choose to run for election, giving her rivals a head start to build their campaigns. She left office with an 87% approval rating.

Olene S. Walker Institute of Politics & Public Service
In 2012 Walker created the Olene S. Walker Institute of Politics & Public Service to "help foster in Weber State University students and the broader community the ideals of public service and political engagement that motivated her decades-long career in Utah politics." The Walker Institute coordinates internships for Weber State University students, holds public forums and debates on public policy issues, and provides leadership and engaged-citizenship workshops for students and the community at large.

Personal life
Walker was married to Myron Walker; they had seven children. Walker and her husband were Latter-day Saints (Mormons). She and her husband served as International Affairs missionaries for The Church of Jesus Christ of Latter-day Saints in New York City. Starting in spring of 2010 she served as the Primary president in the Bloomington 7th Ward on the south side of St. George, Utah.

Olene Walker died of natural causes on November 28, 2015, thirteen days after her 85th birthday.

Recognition
The State of Utah operates the Olene Walker Housing Loan Fund, which seeks to provide affordable housing throughout the state, renovate rural housing, protect tenants from landlords who seek to exploit them, and in other ways provide livable housing options to low income residents of the state.

In 2003, The Salt Lake Tribune named her Utahn of the Year.

In May 2010 the Utah County Democratic Party gave Walker its first distinguished service award.

Walker was one of the first co-chairs of the Utah Debate Commission.

Olene Walker Elementary School opened in 2020.

title: Olene Walker


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Chi Rho

The Chi Rho (☧, English pronunciation ; also known as chrismon ) is one of the earliest forms of the Christogram, formed by superimposing the first two (capital) letters—chi and rho (ΧΡ)—of the Greek ΧΡΙΣΤΟΣ ( rom: Christos) in such a way that the vertical stroke of the rho intersects the center of the chi.

The Chi-Rho symbol was used by the Roman Emperor Constantine the Great (r. 306–337 AD) as part of a military standard (vexillum). Constantine's standard was known as the Labarum. Early symbols similar to the Chi Rho were the Staurogram and the IX monogram.

In pre-Christian times, the Chi-Rho symbol was also used to mark a particularly valuable or relevant passage in the margin of a page, abbreviating chrēston (good). Some coins of Ptolemy III Euergetes (r. 246–222 BC) were marked with a Chi-Rho.

Although formed of Greek characters, the device (or its separate parts) is frequently found serving as an abbreviation in Latin text, with endings added appropriate to a Latin noun, thus XPo, signifying Christo, "to Christ", the dative form of Christus, or χρ̅icola, signifying Christicola, "Christian", in the Latin lyrics of Sumer is icumen in.

Origin and adoption
According to Lactantius, a Latin historian of North African origins saved from poverty by the Emperor Constantine the Great (r. 306–337), who made him tutor to his son Crispus, Constantine had dreamt of being ordered to put a "heavenly divine symbol" (coeleste signum dei) on the shields of his soldiers. The description of the actual symbol chosen by Emperor Constantine the next morning, as reported by Lactantius, is not very clear: it closely resembles a Tau-Rho or a staurogram, a similar Christian symbol. That very day Constantine's army fought the forces of Maxentius and won the Battle of the Milvian Bridge (312), outside Rome.

Eusebius of Caesarea (died in 339) gave two different accounts of the events. In his church history, written shortly after the battle, when Eusebius had not yet had contact with Constantine, he does not mention any dream or vision, but compares the defeat of Maxentius (drowned in the Tiber) to that of the biblical pharaoh and credits Constantine's victory to divine protection. In a memoir of the Roman emperor that Eusebius wrote after Constantine's death (On the Life of Constantine, c. 337–339), a miraculous appearance is said to have come in Gaul long before the Battle of the Milvian Bridge. In this later version, the Roman emperor had been pondering the misfortunes that befell commanders who invoked the help of many different gods, and decided to seek divine aid in the forthcoming battle from the One God. At noon, Constantine saw a cross of light imposed over the sun. Attached to it, in Greek characters, was the saying "Εν τούτῳ Νίκα!" ("In this, conquer!"). Not only Constantine, but the whole army saw the miracle. That night, Christ appeared to the Roman emperor in a dream and told him to make a replica of the sign he had seen in the sky, which would be a sure defence in battle.

Eusebius wrote in the Vita that Constantine himself had told him this story "and confirmed it with oaths" late in life "when I was deemed worthy of his acquaintance and company." "Indeed", says Eusebius, "had anyone else told this story, it would not have been easy to accept it."

Eusebius also left a description of the labarum, the military standard which incorporated the Chi-Rho sign, used by Emperor Constantine in his later wars against Licinius.

Late antiquity
An early visual representation of the connection between the Crucifixion of Jesus and his resurrection, seen in the 4th century sarcophagus of Domitilla in Rome, the use of a wreath around the Chi-Rho symbolizes the victory of the Resurrection over death.

After Constantine, the Chi-Rho became part of the official imperial insignia. Archaeologists have uncovered evidence demonstrating that the Chi-Rho was emblazoned on the helmets of some Late Roman soldiers. Coins and medallions minted during Emperor Constantine's reign also bore the Chi-Rho. By the year 350, the Chi-Rho began to be used on Christian sarcophagi and frescoes. The usurper Magnentius appears to have been the first to use the Chi-Rho monogram flanked by Alpha and Omega, on the reverse of some coins minted in 353. In Roman Britannia, a tesselated mosaic pavement was uncovered at Hinton St. Mary, Dorset, in 1963. On stylistic grounds, it is dated to the 4th century; its central roundel represents a beardless male head and bust draped in a pallium in front of the Chi-Rho symbol, flanked by pomegranates, symbols of eternal life. Another Romano-British Chi-Rho, in fresco, was found at the site of a villa at Lullingstone (illustrated). The symbol was also found on Late Roman Christian signet rings in Britain.

In 2020, archaeologists discovered in Vindolanda in northern England a 5th-century chalice covered in religious iconography, including the Chi-Rho.

Insular Gospel books
In Insular Gospel books, the beginning of Matthew 1:18, at the end of his account of the genealogy of Christ and introducing his account of the life, so representing the moment of the Incarnation of Christ, was usually marked with a heavily decorated page, where the letters of the first word "Christi" are abbreviated and written in Greek as "XPI", and often almost submerged by decoration. Though the letters are written one after the other and the "X" and "P" not combined in a monogram, these are known as Chi-Rho pages.

Famous examples are in the Book of Kells and Book of Lindisfarne. The "X" was regarded as the crux decussata, a symbol of the cross; this idea is found in the works of Isidore of Seville and other patristic and Early Medieval writers. The Book of Kells has a second Chi-Rho abbreviation on folio 124 in the account of the Crucifixion of Christ, and in some manuscripts the Chi-Rho occurs at the beginning of Matthew rather than mid-text at Matthew 1:18. In some other works like the Carolingian Godescalc Evangelistary, "XPS" in sequential letters, representing "Christus" is given a prominent place.

In Unicode
The Chi Rho symbol has two Unicode codepoints:
 * in the Miscellaneous symbols block and
 * in the Coptic block.

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

title: Chi Rho


text:
Meatloaf

Meatloaf is a dish of ground meat that has been combined with other ingredients and formed into the shape of a loaf, then baked or smoked. The final shape is either hand-formed on a baking tray, or pan-formed by cooking it in a loaf pan. It is usually made with ground beef, although ground lamb, pork, veal, venison, poultry, and seafood are also used, sometimes in combination. Vegetarian adaptations of meatloaf may use imitation meat or pulses.

The cooked meatloaf can be sliced like a loaf of bread to make individual portions. It can easily become dry; therefore, various techniques exist to keep the dish moist, like mixing in bread crumbs and egg, covering it with sauce, wrapping it, or using moisture-enhancing ingredients in the mixture, such as filling it with fatty meats, rich, cheeses, or vegetables.

History
Meatloaf is a traditional German, Scandinavian and Belgian dish, and it is a cousin to the meatball in Dutch cuisine.

American meatloaf has its origins in scrapple, a mixture of ground pork and cornmeal served by German-Americans in Pennsylvania since colonial times. Meatloaf in the contemporary American sense did not appear in cookbooks until the late 19th century.

Europe

The Austrian version of meatloaf is called faschierter Braten (literally "minced roast", from faschieren, "to mince", and ein Braten, "a roast"), also known as falscher Hase, or 'fake rabbit'. Most of the time, it is not filled (as it may be in Germany), though the variation im Speckmantel involves wrapping the exterior in ham or bacon before baking. Often, it is served with mashed potatoes (when warm), or with Cumberland sauce (when cold).

The Belgian version of meatloaf is called vleesbrood (meatbread); however, fricandon is also used to refer to it in Dutch. In French, it is called pain de viande. It is usually served warm and can then be served with various sauces, but can also be eaten cold with a loaf of bread.

Rulo Stefani (Руло Стефани), a Bulgarian meatloaf is similar to the Hungarian Stefánia meatloaf, with hard-boiled eggs, and sometimes with chopped carrots and pickled gherkins in the middle.

In the Czech Republic, meatloaf is referred to as sekaná ('chopped'). It is optional to put hard boiled eggs, gherkins, or wienerwurst inside.

Danish meatloaf is called forloren hare ('mock hare') or farsbrød ('ground-meat bread') and is usually made from a mixture of ground pork and beef with strips of bacon or cubed bacon on top. It is served with boiled or mashed potatoes and brown gravy sweetened with red currant jam.

Finnish meatloaf is called lihamureke. It is entirely based on the basic meatball recipe. The only spices used are salt and pepper. It is not customary to stuff lihamureke with anything. The usual side dish is mashed potatoes, and lihamureke is usually served with brown sauce.

In Germany, meatloaf is referred to as Hackbraten (literally "ground roast", from Hackfleisch, "ground meat", and ein Braten, "a roast"), faschierter Braten (literally "minced roast", from faschieren, "to mince", and ein Braten, "a roast"), Wiegebraten, falscher Hase ("false hare" or "faux hare") and Heuchelhase ("mock-hare"). In some regions, it often has boiled eggs inside. Fleischlaib does literally mean "meat-loaf", but is actually another name for Leberkäse (literally "liver-cheese"), which is not a meatloaf.

In Greece, meatloaf is referred to as rolo (Ρολό) and it is usually filled with hard boiled eggs, although several other variations exist.

Stefania meatloaf (Stefánia szelet) or Stefania slices are a type of Hungarian long meatloaf baked in a loaf pan, with 3 hard boiled eggs in the middle, making decorative white and yellow rings in the middle of the slices.

In Italy, meatloaf is called polpettone and can be filled with eggs, ham and cheese, and other ingredients.

The Macedonian Rolat is a similar dish to the chiefly Arab, though also Persian and South-Asian, kofta. Ground beef is rolled and cooked until brown. It can be cooked with vegetables and various sauces.

The Dutch version of meatloaf is called gehaktbrood and can be eaten warm or cold. Slavink is sometimes thought of as a small meatloaf, though it is pan-fried.

In Polish cuisine, the dish called pieczeń rzymska ("Roman roast") or klops is made of ground pork, beef, onions and garlic, with an obligatory hard boiled egg inside.

In Romanian cuisine, there is a meatloaf dish called drob, similar to other minced meat dishes in the region like the Bulgarian Rulo Stefani or the Hungarian Stefánia meatloaf. The major difference is that drob is always made with lamb organs (or a mixture of lamb organs and pork or veal), and the hard boiled eggs in the centre of the drob are optional.

Swedish meatloaf is called köttfärslimpa (literally "minced meat-loaf", from köttfärs, "minced meat", and limpa, "loaf") and is usually made from a mixture of ground pork and beef. It is served with boiled or mashed potatoes, brown sauce gravy, often made from the meat juice that comes from cooking the meatloaf, and lingonberry jam. It is also used thinly sliced as a spread on sandwiches.

In the UK, there are regional pork meatloaf dishes known as haslet, which can be eaten cold or hot.

In British English, haslet or acelet is a pork meatloaf with herbs, originally from Lincolnshire. The word is derived from the Old French hastilles meaning entrails. In Lincolnshire, haslet (pronounced '/ˈhæslɪt/' locally) is typically made from stale white bread, minced pork, sage, salt and black pepper.[1] It is typically served cold with pickles and salad, or as a sandwich filling. In England, it is occasionally sold on a delicatessen counter.

In France, meatloaf is not a dish well-known or appreciated by the locals.

Asia
The Bangladeshi version of meatloaf is called Maṅśer lof.

In Korea, there is a meatloaf dish called Tteok-galbi and Neobiani.

Khuchmal (хучмал) is served with mashed potatoes cooked over the ground meat.

Embutido (not to be confused with the Spanish embutido) is made of well-seasoned ground pork, raisins, minced carrots, sausages, and whole boiled eggs. The meat is molded into a roll with the sausages and hard boiled eggs set in the middle. Another variation of the dish involves wrapping the meatloaf with pork crow or mesentery. It is then wrapped in aluminum foil (historically, banana leaves) and steamed for an hour. The cooked embutido may be stored in freezers. It is usually served fried and sliced for breakfast.

Embutido is sometimes confused with morcón (also not to be confused with Spanish morcón), due to their similarity in appearance. However, morcón is a beef roulade, not a meatloaf.

Hardinera is a Filipino meatloaf made with diced or ground pork topped with sliced hard-boiled eggs, pineapples, carrots, bell peppers, peas, tomatoes, and raisins, among others.

In Turkish cuisine, there is a version of meatloaf called dalyan köfte or rulo köfte; it is typically filled with carrots, peas, and whole boiled eggs.

The Vietnamese meatloaf version is called chả. It is boiled rather than baked or smoked. There are many versions of chả that differ in the ingredients used.

South America
Chilean meatloaf, known as Asado Alemán (German roasted meat) is a staple of southern Chilean cuisine, especially in areas known for having been influenced by the arrival of German immigrants during the 18th and 19th century. The most common recipe nowadays consists of ground beef, carrots, sausages, boiled eggs and breadcrumbs, cooked in the oven and normally served with a side-dish of mashed potatoes or rice.

The Cuban version of meatloaf is called pulpeta. It is made with ground beef and ground ham, and stuffed with hard boiled eggs, and it is cooked on the stovetop. The dish was brought to public attention, albeit mistakenly referred to as a sausage, in the second episode of the third season of The Cosby Show, entitled "Food for Thought". However, due to Cuba's strict laws regarding the purchasing of meat products, especially beef, meatloaf is not a common dish in Cuba.

Jewish cuisine
In Ashkenazi Jewish cuisine, meatloaf is called Klops (קלופס) and can be served cold or hot. It is sometimes filled with whole boiled eggs. The name presumably comes from the German Klops 'meatball'.

Middle East
In Lebanon, kibbeh (ground beef or lamb mixed with Bulgar) can sometimes be formed into a loaf and baked. It is sometimes made from raw meat.

North America
Mexican meatloaf is known as albondigón and is small in size.

The first recorded recipe for the modern American meatloaf dates from the late 1870s. Those preparing the dish were told to chop up whatever meat was on hand, the meat most likely being beef. To that they added salt, pepper, onion, egg and milk-soaked bread. This meatloaf was originally served as a breakfast food. During the Great Depression, cooking meatloaf was a way for families to stretch the food budget by using an inexpensive type of meat and left-over ingredients. Along with spices, it was popular to add cereal grains, bread or saltine crackers to the meatloaf to add bulk and stretch the meat. This tradition of additions still lives on, but with new goals: primarily, producing a lower-fat dish with superior binding and consistency.

Meatloaf recipes in America are typically made with a sauce or relish, often applied before cooking. Many recipes call for a tomato sauce to be poured over the loaf, which forms a crust during baking. A simple brown or onion gravy or a can of cream of mushroom soup can substitute for tomato-based sauce, but the meatloaf is prepared in a similar manner. Barbecue sauce, tomato ketchup, or a mixture of ketchup and prepared mustard may also be used. This style of meatloaf may be topped with a "meatloaf sauce" consisting of ketchup and brown sugar. Another variety of meatloaf, in the same style, is prepared by "frosting" the loaf with mashed potatoes, drizzling a small amount of butter over the top, and then browning it in the oven.

Some recipes are even more imaginative. There are vegetarian meatloaves, vegan meatloaves, even meatloaves made with ahi tuna, French fries, Fritos, Spanish chorizo, mint and pine nuts. Sculpted versions such as feetloaf are also becoming popular, especially on social media.

Meatloaf in America is normally served warm, as part of a main course, but it can also be sliced as a cold cut (and then used in sandwiches). This dish can be considered a typical comfort food in the US and Canada, and so it is served in many diners and restaurants. Indeed, meatloaf is said to have attained iconic comfort food status along with hamburgers, fried chicken and mac and cheese. In a 2007 poll by Good Housekeeping, meatloaf was the seventh-favorite dish of Americans.

In Puerto Rican cuisine, meatloaf is known as albondigón or butifarrón al horno. Puerto Rican style meatloaf is made with ground pork, beef, turkey, adobo, Worcestershire sauce, milk, ketchup, potatoes, red beans, breadcrumbs, parsley, and a hard-boiled egg in the middle.

title: Meatloaf


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Locale (computer software)

In computing, a locale is a set of parameters that defines the user's language, region and any special variant preferences that the user wants to see in their user interface. Usually a locale identifier consists of at least a language code and a country/region code. Locale is an important aspect of i18n.

General locale settings
These settings usually include the following display (output) format settings:

 * Number format setting (LC_NUMERIC, C/C++)
 * Character classification, case conversion settings (LC_CTYPE, C/C++)
 * Date-time format setting (LC_TIME, C/C++)
 * String collation setting (LC_COLLATE, C/C++)
 * Currency format setting (LC_MONETARY, C/C++)
 * Paper size setting (LC_PAPER, ISO 30112)
 * Color setting
 * UI font setting (especially for CJKV language)
 * Location setting (country or region)

The locale settings are about formatting output given a locale. So, the time zone information and daylight saving time are not usually part of the locale settings. Less usual is the input format setting, which is mostly defined on a per application basis.

Programming and markup language support
In these environments,

 * C
 * C++
 * Eiffel
 * Java
 * .NET Framework
 * REBOL
 * Ruby
 * Perl
 * PHP
 * Python
 * XML
 * JSP
 * JavaScript

and other (nowadays) Unicode-based environments, they are defined in a format similar to BCP 47. They are usually defined with just ISO 639 (language) and ISO 3166-1 alpha-2 (2-letter country) codes.

International standards
In standard C and C++, locale is defined in "categories" of (text collation),  (character class),  (currency format),  (number format), and  (time format). The special category can be used to set all locale settings.

There is no standard locale names associated with C and C++ standards besides a "minimal locale" name "C", although the POSIX format is a commonly-used baseline.

POSIX platforms
On POSIX platforms such as Unix, Linux and others, locale identifiers are defined in a way similar to the BCP 47 definition of language tags, but the locale variant modifier is defined differently, and the character set is optionally included as a part of the identifier. The POSIX or "XPG" format is [language[_territory][.codeset][@modifier]] |undefined. (For example, Australian English using the UTF-8 encoding is en_AU.UTF-8.) Separately, ISO/IEC 15897 describes a different form, language_territory+audience+application,sponsor_version, though it's highly dubious whether it is used at all.

In the next example there is an output of command  for Czech language (cs), Czech Republic (CZ) with explicit UTF-8 encoding:

$ locale LANG=cs_CZ.UTF-8 LC_CTYPE="cs_CZ.UTF-8" LC_NUMERIC="cs_CZ.UTF-8" LC_TIME="cs_CZ.UTF-8" LC_COLLATE="cs_CZ.UTF-8" LC_MONETARY="cs_CZ.UTF-8" LC_MESSAGES="cs_CZ.UTF-8" LC_PAPER="cs_CZ.UTF-8" LC_NAME="cs_CZ.UTF-8" LC_ADDRESS="cs_CZ.UTF-8" LC_TELEPHONE="cs_CZ.UTF-8" LC_MEASUREMENT="cs_CZ.UTF-8" LC_IDENTIFICATION="cs_CZ.UTF-8" LC_ALL=

Specifics for Microsoft platforms
Windows uses specific language and territory strings. The locale identifier (LCID) for unmanaged code on Microsoft Windows is a number such as 1033 for English (United States), or 2057 for English (United Kingdom), or 1041 for Japanese (Japan). These numbers consist of a language code (lower 10 bits) and a culture code (upper bits), and are therefore often written in hexadecimal notation, such as 0x0409, 0x0809 or 0x0411. Microsoft is starting to introduce managed code application programming interfaces (APIs) for .NET that use this format. One of the first to be generally released is a function to mitigate issues with internationalized domain names, but more are in Windows Vista Beta 1.

Starting with Windows Vista, new functions that use BCP 47 locale names have been introduced to replace nearly all LCID-based APIs.

A POSIX-like locale name format of language[_country-region[.code-page]]|undefined is available in the UCRT (Universal C Run Time) of Windows 10 and 11.

title: Locale (computer software)


text: User talk:66.250.190.107

title: User talk:66.250.190.107


text:
Fitness proportionate selection

Fitness proportionate selection, also known as roulette wheel selection, is a genetic operator used in genetic algorithms for selecting potentially useful solutions for recombination.

In fitness proportionate selection, as in all selection methods, the fitness function assigns a fitness to possible solutions or chromosomes. This fitness level is used to associate a probability of selection with each individual chromosome. If $$f_i$$ is the fitness of individual $$i$$ in the population, its probability of being selected is
 * $$p_i = \frac{f_i}{\Sigma_{j=1}^{N} f_j},$$

where $$N$$ is the number of individuals in the population.

This could be imagined similar to a Roulette wheel in a casino. Usually a proportion of the wheel is assigned to each of the possible selections based on their fitness value. This could be achieved by dividing the fitness of a selection by the total fitness of all the selections, thereby normalizing them to 1. Then a random selection is made similar to how the roulette wheel is rotated.

While candidate solutions with a higher fitness will be less likely to be eliminated, there is still a chance that they may be eliminated because their probability of selection is less than 1 (or 100%). Contrast this with a less sophisticated selection algorithm, such as truncation selection, which will eliminate a fixed percentage of the weakest candidates. With fitness proportionate selection there is a chance some weaker solutions may survive the selection process. This is because even though the probability that the weaker solutions will survive is low, it is not zero which means it is still possible they will survive; this is an advantage, because there is a chance that even weak solutions may have some features or characteristics which could prove useful following the recombination process.

The analogy to a roulette wheel can be envisaged by imagining a roulette wheel in which each candidate solution represents a pocket on the wheel; the size of the pockets are proportionate to the probability of selection of the solution. Selecting N chromosomes from the population is equivalent to playing N games on the roulette wheel, as each candidate is drawn independently.

Other selection techniques, such as stochastic universal sampling or tournament selection, are often used in practice. This is because they have less stochastic noise, or are fast, easy to implement and have a constant selection pressure.

The naive implementation is carried out by first generating the cumulative probability distribution (CDF) over the list of individuals using a probability proportional to the fitness of the individual. A uniform random number from the range [0,1) is chosen and the inverse of the CDF for that number gives an individual. This corresponds to the roulette ball falling in the bin of an individual with a probability proportional to its width. The "bin" corresponding to the inverse of the uniform random number can be found most quickly by using a binary search over the elements of the CDF. It takes in the O(log n) time to choose an individual. A faster alternative that generates individuals in O(1) time will be to use the alias method.

In 2011, a very simple algorithm was introduced that is based on "stochastic acceptance". The algorithm randomly selects an individual (say $$i$$) and accepts the selection with probability $$f_i/f_M$$, where $$f_M$$ is the maximum fitness in the population. Certain analysis indicates that the stochastic acceptance version has a considerably better performance than versions based on linear or binary search, especially in applications where fitness values might change during the run. While the behavior of this algorithm is typically fast, some fitness distributions (such as exponential distributions) may require $$O(n)$$ iterations in the worst case. This algorithm also requires more random numbers than binary search.

Pseudocode
For example, if you have a population with fitnesses [1, 2, 3, 4], then the sum is (1 + 2 + 3 + 4 = 10). Therefore, you would want the probabilities or chances to be [1/10, 2/10, 3/10, 4/10] or [0.1, 0.2, 0.3, 0.4]. If you were to visually normalize this between 0.0 and 1.0, it would be grouped like below with [red = 1/10, green = 2/10, blue = 3/10, black = 4/10]:

0.1 ] 0.2 \ 0.3 /  0.4 \ 0.5 | 0.6 /  0.7 \ 0.8 | 0.9 | 1.0 /

Using the above example numbers, this is how to determine the probabilities:

sum_of_fitness = 10 previous_probability = 0.0 [1] = previous_probability + (fitness / sum_of_fitness) = 0.0 + (1 / 10) = 0.1 previous_probability = 0.1 [2] = previous_probability + (fitness / sum_of_fitness) = 0.1 + (2 / 10) = 0.3 previous_probability = 0.3 [3] = previous_probability + (fitness / sum_of_fitness) = 0.3 + (3 / 10) = 0.6 previous_probability = 0.6 [4] = previous_probability + (fitness / sum_of_fitness) = 0.6 + (4 / 10) = 1.0

The last index should always be 1.0 or close to it. Then this is how to randomly select an individual:

random_number # Between 0.0 and 1.0 if random_number < 0.1 select 1 else if random_number < 0.3 # 0.3 − 0.1 = 0.2 probability select 2 else if random_number < 0.6 # 0.6 − 0.3 = 0.3 probability select 3 else if random_number < 1.0 # 1.0 − 0.6 = 0.4 probability select 4 end if

title: Fitness proportionate selection


text:
Hon'inbō Sansa

Hon'inbō Sansa (本因坊 算砂, 1559 – June 13, 1623) was the assumed name of Kanō Yosaburō  (加納 與三郎), one of the strongest Japanese Go players of the Edo period (1603–1867), and founder of the house of Hon'inbō, first among the four great schools of Go in Japan. He was a Buddhist priest of the Nichiren sect, and his original dharma name was Nikkai (日海).

Life and career
Nikkai was born in Kyōto and became a monk at age nine. The name "Hon'inbō", (originally pronounced "Honninbō"), comes from a sub-temple of the Jakkōji temple complex in Kyōto where Nikkai, the first "Hon'inbō", resided.

Among his students were the daimyōs Oda Nobunaga, Toyotomi Hideyoshi, and Tokugawa Ieyasu, the three great "unifiers" of feudal Japan. Nikkai considered the three generals to be "fifth-degree" players (五子, comparable to amateur 4 or 5 dan), but "diplomacy" was likely a factor in Nikkai's even-handed assessment of these imposing figures.

It was Nobunaga who, in 1578, recognized twenty-year-old Nikkai as the first Meijin of Go. In 1582, Nikkai, at the behest of Nobunaga, was involved in a notorious game at the Honnōji Temple against his rival, another Nichiren priest, Kashio Rigen (鹿塩 利玄 b. 1565). (Apparently, there is a theory that "Kashio" and "Rigen" were two different people. See the Japanese version of this article.) The game is traditionally held to have taken place on the eve of the treacherous Incident at Honnōji, (in which Nobunaga was forced to commit seppuku), and is said to have ended in a "triple ko". Hence the notion of triple ko as bad omen. There is a go game record, but typical for the period, it is incomplete. The triple ko may have occurred, somewhat implausibly, in unrecorded remaining plays, or in another game that day, but in the end, there is no evidence that this is more than a good story.

In 1587, Nobunaga's successor, the regent (kampaku) Hideyoshi gave Nikkai an official position, (forerunner to the office of Go-dokoro), granting his temple an annual four koku of rice and setting up a tournament that he attended in person. In 1592, Nikkai was given the rank of Gon-no-Daisoku by the Go-Yōzei Emperor.

In 1603, the Go-Yōzei Emperor recognized sixty-year-old Tokugawa Ieyasu as shōgun, inaugurating the 250-year rule of the Tokugawa bakufu. At the direction of Ieyasu, Nikkai turned priestly duties at Jakkoji over to his younger brother Nichiei and became the first go-dokoro, in which office Nikkai was provided fifty koku of rice and five servants. Also around this time (c. 1605), Nikkai took the name Hon'inbō Sansa (Sansa written 算砂: the characters for "calculating" and "sand").

Sansa was also a strong shogi player, but in the Tokugawa period, go was organised into four "houses" (or "academies"), and shogi into three, which would compete in oshirogo "Castle Go" (and "Castle shogi") tournaments for the title Meijin. (See more at History of shogi § Modern shogi.) The official nature of these arrangements meant that go and shogi masters held different offices. In 1612, eight go and shogi players were given individual state support. This system persisted over two and a half centuries, until the collapse of the Tokugawa government itself in the Meiji Restoration.

The beginning of the oshirogo matches in Edo, seat of the Tokugawa government, meant that at the end of each year, Sansa had to travel east for a month. He is said to have turned down an official residence in Edo's early Nihonbashi district, but was given other accommodations. (It was the unrivaled fourth Hon'inbō, Hon'inbō Dōsaku (1645–1702) who finally moved the school to Edo.) Sansa died in Edo at the age of 65.

Legacy
Honinbō Sansa was succeeded by a youth named Sugimura, who due to his age, had to wait several years after Sansa's death to re-establish the Hon'inbō house as Hon'inbō San'etsu when he turned twenty. However, the title of Meijin was regained by the third head of Hon'inbō, and altogether, seven of the ten generations of historic Meijin were Hon'inbō. In addition, all three traditional Kisei ("Go Sages") were Hon'inbō masters. (Both Meijin and Kisei are now titles administered by the Nihon Ki-in.)

The school established by Hon'inbō Sansa, along with the Inoue and Yasui schools, lasted into the 20th century, but struggled without government backing. In 1936, the last Hon'inbō master, the controversial Hon'inbō Shūsai, turned selection of the Hon'inbō title over to the Nihon Ki-in. (See Honinbō Tournament.)

In popular culture
He has been featured in the Japanese manga Hikaru no Go.

title: Hon'inbō Sansa


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Research Triangle Park

Research Triangle Park (RTP) is the largest research park in the United States, occupying 7000 acre in North Carolina and hosting more than 300 companies and 65,000 workers. It is owned and managed by the Research Triangle Foundation, a private non-profit organization.

North Carolina's Research Triangle region is named for the facility, which sits within the geographic triangle formed by three nearby research universities: North Carolina State University, Duke University, and University of North Carolina at Chapel Hill.

The park is bounded by the cities of Raleigh, Durham, and Chapel Hill and the communities of Morrisville and Cary. Most of its territory lies within Durham County, with about one-quarter in Wake County.

Overview
Research Triangle Park is one of the most prominent high-tech research and development parks in the United States. It was created in 1959 by state and local governments, nearby universities, and local business interests. Karl Robbins bought the land where the park is now built. The park covers 7000 acre and is situated in a pine forest with 22500000 sqft of built space. The park is traversed by Interstate 40, the Durham Freeway, and NC 540.

The park is home to more than 300 companies that employ 55,000 workers and an additional 10,000 contractors.

IBM has long been among the largest corporate tenants at RTP, with a four-building complex totaling 774,000 square feet. The park hosts one of GlaxoSmithKline's largest R&D centers with about 5,000 employees. Cisco Systems' campus in the park, with about 5,000 employees, is its second-largest location, after its Silicon Valley corporate headquarters. The National Institutes of Health has its National Institute of Environmental Health Sciences in the park and the city of Durham.

In August 2017, Scott Levitan was named the foundation's new president and CEO, making him the 9th leader since the foundation was established.

History
After World War II, North Carolina's economy was hurt as its agriculture, textile, and furniture industries lost market share. Academics at UNC, NCSU and Duke proposed a park to allow the universities to do research together, harness the area's strengths, and keep graduates in the state.

The "Research Triangle" name is first known to have been used by Romeo Guest in 1953. Guest began advocating the project after unsuccessfully attempting to bring pharmaceutical giant Merck to Aberdeen, North Carolina in the early 1940s. The project was officially greenlit by Governor Luther Hodges in 1956, but the three universities were initially wary of the project.

Although RTP was initially envisioned as a for-profit business, its strategy shifted at the direction of Archibald "Archie" Davis, a former Wachovia Bank chairman, state senator, and president of the United States Chamber of Commerce, who shifted to a non-profit model to purchase the RTP site, and successfully raised $1.425 million in donations from North Carolinians.

The surge in RTP's growth began in 1965, when IBM and the Department of Health, Education, and Welfare announced new facilities in the park. The latter move was attributed to the efforts of Governor Terry Sanford, an ally of President John F. Kennedy. From then until the late 2000s, the park averaged six new companies and 1,800 new employees per year.

IBM remained the largest employer at RTP for decades, with 11,000 workers at the site as of 2013. The RTP facility handled products including the IBM 1050 terminal, Selectric typewriters, PC and accounting operations, and BladeCenter servers.

In 1976, the three major area universities formed a consortium, the Triangle Universities Center for Advanced Studies (TUCASI), to develop joint research facilities and think tanks at RTP. This led to the foundation of the National Humanities Center, Microelectronics Center of North Carolina, the North Carolina Biotechnology Center, and the National Institute for Statistical Sciences.

In their article "The Growth of Research Triangle Park", Link and Scott posit that entrepreneurial culture and leadership contributed the most to its success as a cluster. Archie Davis promoted a culture of innovation and entrepreneurship by locating the park near universities, actively recruiting organizations (like the American Academy of Arts and Sciences), and used his vision to raise funding for the park.

Davis strongly believed that profits could not be the only driver for creating the park and that the betterment of the community should be the key goal. "The love of this state … was the motivation for the Research Triangle idea," he said. "Research Triangle is a manifestation of what North Carolina is all about." Research Triangle Park remains a nonprofit.

Local government
The park is an unincorporated area, and state law prohibits municipalities from annexing areas within the park. Some local government functions are served by the Durham-Wake Counties Research and Production Service District, a special tax district created in 1986 that is conterminous with the park, wherein the property tax rate is limited to 10 cents per $100 valuation. The park has special zoning as a Research Applications District in the Wake County portion, and a Scientific Research Park in the Durham County portion. As of October 2012, both zoning areas are in the process of being revised to allow higher density development. The zoning changes are coupled with legislative changes allowing for Urban Research Service Districts (URSD) within the Park, which can include a mix of retail and residential usages. The URSDs have the power to levy taxes at the same rate as a neighboring city.

The Research Triangle Foundation operates several subsidiaries within the park. These include: the Frontier RTP startup campus, Boxyard RTP, and Hub RTP. Frontier RTP first opened as a free coworking space in a single building in January 2015. Since its inception, the Foundation has expanded the Frontier RTP concept to three additional buildings, creating an affordable campus for growing tech, life science and nonprofit organizations; as of 2021, 100 of the Park's 300 companies are housed in the Frontier campus.

Redevelopment
On October 1, 2015, former President and CEO of the Research Triangle Foundation, Bob Geolas, announced RTP's plans for a $50,000,000 redevelopment involving the formation of "Park Center." $20,000,000 will be allocated from Durham County, $10,000,000 from the Durham-Wake Counties Research and Production Service District, and $20,000,000 as a result of land purchases and site work provided by the Research Triangle Foundation of North Carolina. The redevelopment plans also include exploring partnerships with regional transit groups. The hope of the Research Triangle Foundation is to broaden public transportation to and from the area.

Hub RTP
In 2019, the Research Triangle Foundation announced a re-brand of the site to "Hub RTP." The 100-acre site broke ground in September 2020 and will include 125,000 SF of office over retail, 1200 residential apartments, 16 acres of green space, at least one hotel, and 1M SF of the first high rise office towers in RTP.

Boxyard RTP
In March 2019, RTF announced plans to construct Boxyard RTP, an 15,000-square-foot shipping container complex of retail, dining, and other amenities. The $9M project, which is set on 12 acres of the Frontier RTP campus, delayed its launch for a year because of the impacts of the pandemic and soft-launched in June 2021 with a few of the planned tenants opening for business in the days following. The complex held its grand opening on November 18, 2021, with several anchor tenants open for business. Initial tenants include restaurants, a brewery, a cocktail bar, a coffee shop, a beauty shop, a CBD vendor, and an escape room. The complex also features a dog park named the Barkyard RTP, pop-up yoga classes, live music, and both standalone and roof-mounted heaters in the outdoor pavilion. Boxyard RTP is inspired by a similar development called The Boxyard Tulsa, located in downtown Tulsa's East Village district.

title: Research Triangle Park


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Genetic operator

A genetic operator is an operator used in genetic algorithms to guide the algorithm towards a solution to a given problem. There are three main types of operators (mutation, crossover and selection), which must work in conjunction with one another in order for the algorithm to be successful. Genetic operators are used to create and maintain genetic diversity (mutation operator), combine existing solutions (also known as chromosomes) into new solutions (crossover) and select between solutions (selection). In his book discussing the use of genetic programming for the optimization of complex problems, computer scientist John Koza has also identified an 'inversion' or 'permutation' operator; however, the effectiveness of this operator has never been conclusively demonstrated and this operator is rarely discussed.

Mutation (or mutation-like) operators are said to be unary operators, as they only operate on one chromosome at a time. In contrast, crossover operators are said to be binary operators, as they operate on two chromosomes at a time, combining two existing chromosomes into one new chromosome.

Operators
Genetic variation is a necessity for the process of evolution. Genetic operators used in genetic algorithms are analogous to those in the natural world: survival of the fittest, or selection; reproduction (crossover, also called recombination); and mutation.

Selection
Selection operators give preference to better solutions (chromosomes), allowing them to pass on their 'genes' to the next generation of the algorithm. The best solutions are determined using some form of objective function (also known as a 'fitness function' in genetic algorithms), before being passed to the crossover operator. Different methods for choosing the best solutions exist, for example, fitness proportionate selection and tournament selection; different methods may choose different solutions as being 'best'. The selection operator may also simply pass the best solutions from the current generation directly to the next generation without being mutated; this is known as elitism or elitist selection.

Crossover
Crossover is the process of taking more than one parent solutions (chromosomes) and producing a child solution from them. By recombining portions of good solutions, the genetic algorithm is more likely to create a better solution. As with selection, there are a number of different methods for combining the parent solutions, including the edge recombination operator (ERO) and the 'cut and splice crossover' and 'uniform crossover' methods. The crossover method is often chosen to closely match the chromosome's representation of the solution; this may become particularly important when variables are grouped together as building blocks, which might be disrupted by a non-respectful crossover operator. Similarly, crossover methods may be particularly suited to certain problems; the ERO is generally considered a good option for solving the travelling salesman problem.

Mutation
The mutation operator encourages genetic diversity amongst solutions and attempts to prevent the genetic algorithm converging to a local minimum by stopping the solutions becoming too close to one another. In mutating the current pool of solutions, a given solution may change entirely from the previous solution. By mutating the solutions, a genetic algorithm can reach an improved solution solely through the mutation operator. Again, different methods of mutation may be used; these range from a simple bit mutation (flipping random bits in a binary string chromosome with some low probability) to more complex mutation methods, which may replace genes in the solution with random values chosen from the uniform distribution or the Gaussian distribution. As with the crossover operator, the mutation method is usually chosen to match the representation of the solution within the chromosome.

Combining operators
While each operator acts to improve the solutions produced by the genetic algorithm working individually, the operators must work in conjunction with each other for the algorithm to be successful in finding a good solution. Using the selection operator on its own will tend to fill the solution population with copies of the best solution from the population. If the selection and crossover operators are used without the mutation operator, the algorithm will tend to converge to a local minimum, that is, a good but sub-optimal solution to the problem. Using the mutation operator on its own leads to a random walk through the search space. Only by using all three operators together can the genetic algorithm become a noise-tolerant hill-climbing algorithm, yielding good solutions to the problem.

title: Genetic operator


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Double-barreled cannon

The double-barreled cannon is an American Civil War-era experimental weapon and is now a modern landmark located in Athens, Georgia. While originally built for warfare, the cannon never saw battle. It is part of the Downtown Athens Historic District, in the National Register of Historic Places.

Concept
This concept dates from 1642 and Florentine gun maker Antonio Petrini. He cast the first cannon with the intention of simultaneously firing two balls linked by a chain from side-by-side barrels, which were meant to scythe down enemy soldiers like standing wheat when it reached them. For the cannon to work, the powder behind each round shot had to ignite at the same instant, which rarely happened.

In 1862, Georgia dentist, builder, and mechanic John Gilleland raised money from a coterie of Confederate citizens in Athens, Georgia to build the chain-shot gun for a cost of $350. Cast in one piece, the gun featured side-by-side bores, each a little over 3 inches in diameter and splayed slightly outward so the shots would diverge and stretch the chain taut. The two barrels have a divergence of 3 degrees, and the cannon was designed to shoot simultaneously two cannonballs connected with a chain to "mow down the enemy somewhat as a scythe cuts wheat". During tests, the Gilleland cannon effectively mowed down trees, tore up a cornfield, knocked down a chimney, and killed a cow. These experiments took place along Newton Bridge Road northwest of downtown Athens. None of the previously mentioned items were anywhere near the gun's intended target.

A treatise that describes Antonio Petrini's cannon survives in the Royal Armories of the Tower of London, while Gilleland's gun sits on the lawn of the Athens, Georgia, city hall.

Execution
Gilleland's invention was a failure. When it was first tested on 22 April 1862, it was aimed at a target of two upright poles. Uneven combustion of the powder and casting imperfections in the barrels gave the connected balls a spinning movement in an off-center direction, with witnesses reporting that on its first firing it "plowed up about an acre of ground, tore up a cornfield, mowed down saplings, and then the chain broke, the two balls going in different directions".

On its second firing, the chain shot across the horizon and into a thicket of pine. "[The] thicket of young pines at which it was aimed looked as if a narrow cyclone or a giant mowing machine had passed through," reported another witness.

On its third firing, the chain snapped immediately and one ball tore into a nearby cabin, knocking down its chimney; the other spun off erratically and struck a nearby cow, killing it instantly. Gilleland considered the test-firings a success.

Civil War use
Gilleland tried to promote his invention to the Confederate States Army's arsenal in Augusta, Georgia, where it was found unfit for its purpose. He continued to try to promote his invention to other military leaders around Augusta, but failed to interest anyone. Finally his contraption was used as a signal gun in Athens to warn the citizens about an invasion by the Union Army.

On 27 July 1864, the cannon was fired after a report was heard of several thousand Union soldiers approaching nearby Monroe, Georgia. However, this report turned out to be false. The cannon disappeared in 1891 and was found again twelve years later.

Modern use
Currently, the cannon is on display on the front lawn of the City Hall of Athens, Georgia, where it is a contributing property of the Downtown Athens Historic District. A local landmark and public curiosity, the cannon is one of the most popular and well-known attractions in the city. It is still pointing northward in a symbolic gesture of defiance against the North it was built to fight.

Other multi-barreled cannons
While the cannon preserved at Athens is the most famous double-barreled cannon, it is not the only one. Another notable example was called Elizabeth-Henry, named after Charles I's youngest children. It was used by the Cavaliers during the English Civil War, and fired 2-ounce charges. It could also fire grapeshot. The barrels were wrapped in leather to prevent rusting.

Multi-barreled cannons have also been built in India, and an example with six barrels, similar to the ribauldequins of the Middle Ages, has been preserved in the Nehru Scientific Center.

The French army fielded a three barrelled canon during the War of the Spanish Succession.

In Poland, cannons with as many as seven barrels were in use in the 16th century. These were similar to the volley guns and organ battery of the 19th century, but in a larger caliber.

However, these other cannons did not attempt the Athens gun's idea of firing a connected round simultaneously from multiple barrels.

In 2012 a slightly modernized version was built, at customer request, by the Gunsmoke gun shop on the reality television series American Guns (Season 2, Ep 8). By using a more modern ignition system than Civil War-era the Gunsmoke crew were able to achieve some success in firing the barrels properly. The gun used in the episode was actually built by Sprik's Cannon Works and sold to Gunsmoke for their TV show. The barrel was to be welded together by Sprik's Cannon Works, who also did the ignition system for a more accurate discharge of both barrels at the same time.

title: Double-barreled cannon


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Talk:Semmering railway

Someone with an active interest and good English should comb this article a bit. Too late for me now, gotta sleep! :) --Gutza 00:19, 19 Nov 2003 (UTC)

Map
Can some one please insert this in the infobox rail line of this article under Map? Peter Horn 22:50, 15 May 2009 (UTC)

 * And add "map caption" to Template:Infobox rail line? Peter Horn 23:53, 15 May 2009 (UTC)
 * Never mind, I managed to get the map in place myself and I noticed that "map caption" is not needed in this template. Peter Horn 20:47, 16 May 2009 (UTC)

External links modified
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title: Talk:Semmering railway


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Hetaira

A hetaira, (: hetairai, ), also hetaera,  (: hetaerae, ) (ἑταίρα, 'companion'; : ἑταῖραι; hetaera; : hetaerae), was a type of courtesan or prostitute in ancient Greece, who served as an artist, entertainer and conversationalist in addition to providing sexual service. Custom excluded the wives and daughters of Athenian citizens from the symposium, but this prohibition did not extend to hetairai, who were often foreign born and could be highly educated. Other female entertainers made appearances in the otherwise male domain, but hetairai joined the male guests in their sexual joking, sometimes evidencing a wide knowledge of literature in their contributions.

Summary
Traditionally, historians of ancient Greece have distinguished between hetairai and pornai, another class of prostitute in ancient Greece. In contrast to pornai, who provided sex for numerous clients in brothels or on the street, hetairai were thought to have had only a few men as clients at any one time, to have had long-term relationships with them, and to have provided companionship and intellectual stimulation as well as sex. For instance, Charles Seltman wrote in 1953 that "hetaeras were certainly in a very different class, often highly educated women".

More recently, however, historians have questioned the extent to which there was really a distinction between hetairai and pornai. The second edition of the Oxford Classical Dictionary, for instance, held that hetaira was a euphemism for any kind of prostitute. This position is supported by Konstantinos Kapparis, who holds that Apollodorus' famous tripartite division of the types of women in the speech Against Neaera ("We have courtesans for pleasure, concubines for the daily tending of the body, and wives in order to beget legitimate children and have a trustworthy guardian of what is at home." ) classes all prostitutes together, under the term hetairai.

A third position, advanced by Rebecca Futo Kennedy, suggests that hetairai "were not prostitutes or even courtesans". Instead, she argues, hetairai were "elite women ... who participated in sympotic and luxury culture", just as hetairoithe masculine form of the wordwas used to refer to groups of elite men at symposia.

Even when the term hetaira was used to refer to a specific class of prostitute, though, scholars disagree on what precisely the line of demarcation was. Kurke emphasises that hetairai veiled the fact that they were selling sex through the language of gift-exchange, while pornai explicitly commodified sex. Leslie Kurke claims that both hetairai and pornai could be slaves or free, and might or might not work for a pimp. Kapparis says that hetairai were high-class prostitutes, and cites Dover as pointing to the long-term nature of hetairai's relationships with individual men. Miner disagrees with Kurke, claiming that hetairai were always free, not slaves.

Along with sexual services, women described as hetairai rather than pornai seem to have often been educated, and have provided companionship. According to Kurke, the concept of hetairism was a product of the symposium, where hetairai were permitted as sexually available companions of the male party-goers. In Athenaeus' Deipnosophistai, hetairai are described as providing "flattering and skillful conversation": something which is, elsewhere in classical literature, seen as a significant part of the hetaira's role. Particularly, "witty" and "refined" (ἀστείᾱ) were seen as attributes which distinguished hetairai from common pornai. Hetairai are likely to have been musically educated, too.

Free hetairai could become very wealthy, and control their own finances. However, their careers could be short, and if they did not earn enough to support themselves, they might have been forced to resort to working in brothels, or working as pimps, in order to ensure a continued income as they got older.

title: Hetaira


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Baptist Union of Wales

The Baptist Union of Wales (Undeb Bedyddwyr Cymru) is a Baptist Christian association of churches in Wales. It is affiliated with the Free Church Council, Cytûn (Churches Together in Wales), the European Baptist Federation, and the Baptist World Alliance. The headquarters is in Carmarthen.

History
The General Baptist minister Hugh Evans was one of the first Baptists to preach in Wales around 1646, in the parishes of Llan-hir, Cefnllys, Nantmel, and Llanddewi Ystradenny, as well as in districts across the upper Wye Valley in Breconshire. In 1649 John Myles (1621–1683) and Thomas Proud led in the formation of a congregation at Ilston, before Myles emigrated to Swansea, Massachusetts, in 1663. Myles and Proud were connected to the Particular Baptists in London. In 1650 three Baptist congregations held the first general meeting of Baptists in Wales. The national union was organized in 1866.

Membership
According to a census published by the association in 2023, it claimed 8,105 members and 315 churches.

title: Baptist Union of Wales


text:
Talk:Hetaira

A note
The link to courtisanerie seems to be misdirected in the context of this article. That article seems to discuss the activities of courtiers, not courtesans. Perhaps a link to courtesan instead? -- April 00:25, 19 Nov 2003 (UTC)

 * The former has been redirected to the latter. Ellsworth 20:50, 8 Mar 2005 (UTC)

"Physical talents"
What does that term mean? Is it a euphemism for 'sexual talents'? If so, I think we should say the latter. Anchoress · Weigh Anchor · Catacomb 19:28, 10 March 2008 (UTC)

 * I just came to say the same thing. Dance and music are largely physical; certainly they are also mental, but what is not? -Pgan002 (talk) 02:10, 29 October 2008 (UTC)

NSFW pictures?
Just a question about NSFW pictures - the illustrative picture on this page appears to be close to the line, in that I would be uncomfortable having that pop up on my screen at work. Is there a policy on this and where does this picture fall? —Preceding unsigned comment added by 207.112.94.152 (talk) 00:35, 11 March 2008 (UTC)
 * Wikipedia is not censored, so the only policy is to use illustrative freely-licensed pictures exactly where they are the most illustrative. Anchoress · Weigh Anchor · Catacomb 00:46, 11 March 2008 (UTC)

public exclusion of wives
Should mention that one of the main foundations of the whole hetaerism system (as opposed to simple prostitution) was that legitimate wives were often forbidden from participating in many public events, including the banquets or "symposia" which often made up such a large part of ancient Greek social life (such exclusion of "virtuous" women seems to have been especially strict in Athens), which made it necessary to have "unvirtuous" women available if men wanted to have female companionship at such events... AnonMoos (talk) 09:46, 24 November 2008 (UTC)

Modern parallel?
By the way, a somewhat modern parallel would seem to be Japanese geisha and bar "hostesses" -- just like the socially-active and moderately wealthy citizens of ancient Athens (who spent many hours at convivial feasts or "symposia"), modern Japanese salarymen spend many hours socializing after work with their fellow employees, and in both cases (as a matter of firmly-established custom) their wives can't accompany them to these social occasions, so that they need other female companionship there... AnonMoos (talk) 13:25, 7 December 2008 (UTC)
 * No we should not make the claim that Japanese barpersons are prostitutes without sufficient reliable sources to say that is what they are. ~ R.T.G 13:53, 2 April 2020 (UTC)

Etymology of Hetaera
Why does this article refer to these women as “hetaerae” instead of “hetairai”? The former spelling apparently comes from Medieval Latin, while the latter spelling comes directly from the Ancient Greek. See “Hetaera” Entry in Online Etymology Dictionary as a reference. The “hetairai” spelling might be more appropriate, especially since this article deals primarily with Greek historical figures while referencing Greek primary sources. 67.174.12.194 (talk) 09:34, 29 November 2013 (UTC)
 * Google ngrams shows that 'hetaira' is more common than 'hetaera', being used between 2 and 5 times as often. --ABehrens (talk) 19:33, 1 February 2020 (UTC)

title: Talk:Hetaira


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Worshipful Company of Woolmen

The Worshipful Company of Woolmen is one of the livery companies in the City of London. It is known to have existed in 1180, making it one of the older livery companies of the city. It was officially incorporated in 1522. The company's original members were concerned with the winding and selling of wool; presently, a connection is retained by the company's support of the field of wool production and related fields. However, the company is now primarily a charitable institution.

The company ranks forty-third in the order of precedence of the livery companies. Its motto is Lana Spes Nostra, Latin for "Wool Is Our Hope".

title: Worshipful Company of Woolmen


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Magadha

Magadha also called the Kingdom of Magadha or the Magadha Empire, was a kingdom and empire, and one of the sixteen Mahajanapadas of the Second Urbanization, based in southern Bihar in the eastern Ganges Plain, in Ancient India. Magadha was ruled by the Brihadratha dynasty (1700-682 BCE), the Pradyota dynasty (682-544 BCE), the Haryanka dynasty (544–413 BCE), the Shaishunaga dynasty (413–345 BCE), the Nanda dynasty (345–322 BCE), the Mauryan dynasty (322–184 BCE), the Shunga dynasty (184–73 BCE), Kanva dynasty (73–28 BCE) and Gupta Empire (240-550 CE). Kanva dynasty lost much of its territory after being defeated by the Satavahanas of Deccan in 28 BCE and was reduced to a small principality around Pataliputra. However, with the rule of Gupta Empire (240-550 CE), The Gupta Empire regained the Glory of Magadh. Under the Mauryas, Magadha became a pan-Indian empire, covering large swaths of the Indian subcontinent and Afghanistan. The Magadh under the Gupta Empire emerged as the most prosperous Kingdom in the History of Ancient India.

Magadha played an important role in the development of Jainism and Buddhism. It was the core of four of northern India's greatest empires, the Nanda Empire (c. 345), Maurya Empire (c. 322–185 BCE), Shunga Empire (c. 185–78 BCE) and Gupta Empire (c. 240–550 CE). The Pala Empire also ruled over Magadha and maintained a royal camp in Pataliputra.

The Pithipatis of Bodh Gaya referred to themselves as Magadhādipati and ruled in parts of Magadha until the 13th century.

Geography

The territory of the Magadha kingdom proper before its expansion was bounded to the north, west, and east respectively by the Gaṅgā, Son, and Campā rivers, and the eastern spurs of the Vindhya mountains formed its southern border. The territory of the initial Magadha kingdom thus corresponded to the modern-day Patna and Gaya districts of the Indian state of Bihar.

The region of Greater Magadha also included neighbouring regions in the eastern Gangetic plains and had a distinct culture and belief. Much of the Second Urbanisation took place here from (c. 500 BCE) onwards and it was here that Jainism and Buddhism arose.

History

Some scholars have identified the Kīkaṭa tribe—mentioned in the Rigveda (3.53.14) with their ruler Pramaganda—as the forefathers of Magadhas because Kikata is used as synonym for Magadha in the later texts; Like the Magadhas in the Atharvaveda, the Rigveda speaks of the Kikatas as a hostile tribe, living on the borders of Brahmanical India, who did not perform Vedic rituals.

The earliest reference to the Magadha people occurs in the Atharvaveda, where they are found listed along with the Angas, Gandharis and Mujavats. The core of the kingdom was the area of Bihar south of the Ganges; its first capital was Rajagriha (modern day Rajgir), then Pataliputra (modern Patna). Rajagriha was initially known as 'Girivrijja' and later came to be known as so during the reign of Ajatashatru. Magadha expanded to include most of Bihar and Bengal with the conquest of Vajjika League and Anga, respectively. The kingdom of Magadha eventually came to encompass Bihar, Jharkhand, Orissa, West Bengal, eastern Uttar Pradesh, and the areas that are today the nations of Bangladesh and Nepal.

The ancient kingdom of Magadha is heavily mentioned in Jain and Buddhist texts. It is also mentioned in the Ramayana, the Mahabharata and the Puranas.

There is little certain information available on the early rulers of Magadha. The most important sources are the Buddhist Pāli Canon, the Jain Agamas and the Hindu Puranas. Based on these sources, it appears that Magadha was ruled by the Haryanka dynasty for some 200 years, c. 543 to 413 BCE.

Gautama Buddha, the founder of Buddhism, lived much of his life in the kingdom of Magadha. He attained enlightenment in Bodh Gaya, gave his first sermon in Sarnath and the first Buddhist council was held in Rajgriha.

The Hindu Mahabharata calls Brihadratha the first ruler of Magadha. Ripunjaya, last king of Brihadratha dynasty, was killed by his minister Pulika, who established his son Pradyota as the new king. Pradyota dynasty was succeeded by Haryanka dynasty founded by Bimbisara. Bimbisara led an active and expansive policy, conquering the Kingdom of Anga in what is now West Bengal. King Bimbisara was killed by his son, Ajatashatru. Pasenadi, king of neighbouring Kosala and brother-in-law of Bimbisara, promptly reconquered the Kashi province.

Accounts differ slightly as to the cause of King Ajatashatru's war with the Licchavi, a powerful tribe north of the river Ganges. It appears that Ajatashatru sent a minister to the area who worked for three years to undermine the unity of the Licchavis. To launch his attack across the Ganges River, Ajatashatru built a fort at the town of Pataliputra. Torn by disagreements, the Licchavis fought with Ajatashatru. It took fifteen years for Ajatashatru to defeat them. Jain texts tell how Ajatashatru used two new weapons: a catapult, and a covered chariot with swinging mace that has been compared to a modern tank. Pataliputra began to grow as a centre of commerce and became the capital of Magadha after Ajatashatru's death.

The Haryanka dynasty was overthrown by the Shishunaga dynasty. The last Shishunaga ruler, Mahanandin, was assassinated by Mahapadma Nanda in 345 BCE, the first of the so-called "Nine Nandas", i. e. Mahapadma and his eight sons, last being Dhana Nanda.

In 326 BCE, the army of Alexander approached the western boundaries of Magadha. The army, exhausted and frightened at the prospect of facing another giant Indian army at the Ganges, mutinied at the Hyphasis (the modern Beas River) and refused to march further east. Alexander, after the meeting with his officer Coenus, was persuaded that it was better to return and turned south, conquering his way down the Indus to the Ocean.

Around 321 BCE, the Nanda Dynasty ended with the defeat of Dhana Nanda at the hands of Chandragupta Maurya who became the first king of the Mauryan Empire with the help of his mentor Chanakya. The Empire later extended over most of India under King Ashoka The Great, who was at first known as 'Ashoka the Cruel' but later became a disciple of Buddhism and became known as 'Dharma Ashoka'. Later, the Mauryan Empire ended, as did the Shunga and Khārabēḷa empires, to be replaced by the Gupta Empire. The capital of the Gupta Empire remained Pataliputra in Magadha.

During the Pala-period in Magadha from the 11th to 13th century CE, a local Buddhist dynasty known as the Pithipatis of Bodh Gaya ruled as tributaries to Pala Empire.

Buddhism and Jainism
Several Śramaṇic movements had existed before the 6th century BCE, and these influenced both the āstika and nāstika traditions of Indian philosophy. The Śramaṇa movement gave rise to diverse range of heterodox beliefs, ranging from accepting or denying the concept of soul, atomism, antinomian ethics, materialism, atheism, agnosticism, fatalism to free will, idealization of extreme asceticism to that of family life, strict ahimsa (non-violence) and vegetarianism to the permissibility of violence and meat-eating. Magadha kingdom was the nerve centre of this revolution.

Jainism was revived and re-established after Mahavira, the last and the 24th Tirthankara, who synthesised and revived the philosophies and promulgations of the ancient Śramaṇic traditions laid down by the first Jain tirthankara Rishabhanatha millions of years ago. Buddha founded Buddhism which received royal patronage in the kingdom.

According to Indologist Johannes Bronkhorst, the culture of Magadha was in fundamental ways different from the Vedic kingdoms of the Indo-Aryans. According to Bronkhorst, the śramana culture arose in "Greater Magadha," which was Indo-Aryan, but not Vedic. In this culture, Kshatriyas were placed higher than Brahmins, and it rejected Vedic authority and rituals. He argues for a cultural area termed "Greater Magadha", defined as roughly the geographical area in which the Buddha and Mahavira lived and taught.

With regard to the Buddha, this area stretched by and large from Śrāvastī, the capital of Kosala, in the north-west to Rājagṛha, the capital of Magadha, in the south-east". According to Bronkhorst "there was indeed a culture of Greater Magadha which remained recognizably distinct from Vedic culture until the time of the grammarian Patañjali (ca. 150 BCE) and beyond". The Buddhologist Alexander Wynne writes that there is an "overwhelming amount of evidence" to suggest that this rival culture to the Vedic Aryans dominated the eastern Gangetic plain during the early Buddhist period. Orthodox Vedic Brahmins were, therefore, a minority in Magadha during this early period.

The Magadhan religions are termed the sramana traditions and include Jainism, Buddhism and Ājīvika. Buddhism and Jainism were the religions promoted by the early Magadhan kings, such as Srenika, Bimbisara and Ajatashatru, and the Nanda Dynasty (345–321 BCE) that followed was mostly Jain. These Sramana religions did not worship the Vedic deities, practised some form of asceticism and meditation (jhana) and tended to construct round burial mounds (called stupas in Buddhism). These religions also sought some type of liberation from the cyclic rounds of rebirth and karmic retribution through spiritual knowledge.

Religious sites in Magadha
Among the Buddhist sites currently found in the Magadha region include two UNESCO World Heritage Sites such as the Mahabodhi temple at Bodh Gaya and the Nalanda monastery. The Mahabodhi temple is one of the most important places of pilgrimage in the Buddhist world and is said to mark the site where the Buddha attained enlightenment.

Language
Beginning in the Theravada commentaries, the Pali language has been identified with Magahi, the language of the kingdom of Magadha, and this was taken to also be the language that the Buddha used during his life. In the 19th century, the British Orientalist Robert Caesar Childers argued that the true or geographical name of the Pali language was Magadhi Prakrit, and that because pāḷi means "line, row, series", the early Buddhists extended the meaning of the term to mean "a series of books", so pāḷibhāsā means "language of the texts". Nonetheless, Pali does retain some eastern features that have been referred to as Māgadhisms.

Magadhi Prakrit was one of the three dramatic prakrits to emerge following the decline of Sanskrit. It was spoken in Magadha and neighbouring regions and later evolved into modern eastern Indo-Aryan languages like Magahi, Maithili and Bhojpuri.

Dynasties and rulers
The history of Magadha region is very vast, it can be divided into many periods as:

 * Vedic Magadha Kingdom :- Kikata Kingdom, Brihadratha dynasty
 * Magadha Mahajanapada :- Pradyota dynasty, Haryanka dynasty, Shaishunaga dynasty
 * Magadha Empire :- Nanda Empire, Maurya Empire, Shunga Empire, Kanva dynasty
 * Classical Magadha :- Gupta Empire, Later Gupta dynasty
 * Medieval Magadha :- Pala Empire, Pithipatis of Bodh Gaya

There is much uncertainty about the succession of kings and the precise chronology of Magadha prior to Mahapadma Nanda; the accounts of various ancient texts (all of which were written many centuries later than the era in question) contradict each other on many points.

Two notable rulers of Magadha were Bimbisara (also known as Shrenika) and his son Ajatashatru (also known as Kunika), who are mentioned in Buddhist and Jain literature as contemporaries of the Buddha and Mahavira. Later, the throne of Magadha was usurped by Mahapadma Nanda, the founder of the Nanda Dynasty (c. 345), which conquered much of north India. The Nanda dynasty was overthrown by Chandragupta Maurya, the founder of the Maurya Empire (c. 322–185 BCE).

Furthermore, there is a "Long Chronology" and a contrasting "Short Chronology" preferred by some scholars, an issue that is inextricably linked to the uncertain chronology of the Buddha and Mahavira. According to historian K. T. S. Sarao, a proponent of the Short Chronology wherein the Buddha's lifespan was c.477–397 BCE, it can be estimated that Bimbisara was reigning c.457–405 BCE, and Ajatashatru was reigning c.405–373 BCE. According to historian John Keay, a proponent of the "Long Chronology," Bimbisara must have been reigning in the late 5th century BCE, and Ajatashatru in the early 4th century BCE. Keay states that there is great uncertainty about the royal succession after Ajatashatru's death, probably because there was a period of "court intrigues and murders," during which "evidently the throne changed hands frequently, perhaps with more than one incumbent claiming to occupy it at the same time" until Mahapadma Nanda was able to secure the throne.

List of rulers
The following "Long Chronology" is according to the Buddhist Mahavamsa:

 * Haryanka dynasty (c. 544 – 413 BCE)

 * Shishunaga dynasty (c. 413 – 345 BCE)

 * Nanda Empire (c. 345)

Other lists
The Hindu Literature mostly Puranas give a different sequence:
 * Puranic list

 * Shishunaga dynasty (360 years)
 * Shishunaga (reigned for 40 years)
 * Kakavarna (36 years)
 * Kshemadharman (20 years)
 * Kshatraujas (29 years)
 * Bimbisara (28 years)
 * Ajatashatru (25 years)
 * Darbhaka or Darshaka or Harshaka (25 years)
 * Udayin (33 years)
 * Nandivardhana (42 years)
 * Mahanandin (43 years)
 * Nanda dynasty (100 years)

A shorter list appears in the Jain tradition, which simply lists Shrenika (Bimbisara), Kunika (Ajatashatru), Udayin, followed by the Nanda dynasty.
 * List by Jain literature

Historical figures from Magadha

Important people from the region of Magadha include:

 * Śāriputra – born to a wealthy Brahmin in a village located near Rājagaha in Magadha. He is considered the first of the Buddha's two chief male disciples, together with Maudgalyāyana.
 * Maudgalyāyana – born in the village of Kolita in Magadha. He was one of the Buddha's two main disciples. In his youth, he was a spiritual wanderer before meeting the Buddha.
 * Mahavira – the 24th Tirthankara of Jainism. Born into a royal kshatriya family in what is now Vaishali district of Bihar. He abandoned all worldly possessions at the age of 30 and became an ascetic. He is considered a slightly older contemporary of the Buddha.
 * Maitripada – an 11th-century Indian Buddhist Mahasiddha associated with the Mahāmudrā transmission. Born in the village of Jhatakarani in Magadha. Also associated with the monasteries of Nalanda and Vikramashila.
 * Dhyānabhadra - 13th/14th century monk of Nalanda born to a minor chief in Magadha and later travelled across South and East Asia.
 * Subhūticandra - 11/12th-century Indian Buddhist monk associated with Nalanda and Vikramashila who belonged to Magadha.

title: Magadha


text: User talk:207.189.98.44

title: User talk:207.189.98.44


text:
Truncation selection

In animal and plant breeding, truncation selection is a standard method in selective breeding in selecting animals to be bred for the next generation. Animals are ranked by their phenotypic value on some trait such as milk production, and the top percentage is reproduced. The effects of truncation selection for a continuous trait can be modeled by the standard breeder's equation by using heritability and truncated normal distributions. On a binary trait, it can be modeled easily using the liability threshold model. It is considered an easy and efficient method of breeding.

Computer science
In computer science, truncation selection is a selection method used in genetic algorithms to select potential candidate solutions for recombination modeled after the breeding method.

In truncation selection the candidate solutions are ordered by fitness, and some proportion, p, (e.g. p = 1/2, 1/3, etc.) of the fittest individuals are selected and reproduced 1/p times. Truncation selection is less sophisticated than many other selection methods, and is not often used in practice. It is used in Muhlenbein's Breeder Genetic Algorithm.

title: Truncation selection


text:
Worshipful Company of Weavers

The Worshipful Company of Weavers is the most ancient of the Livery Companies in the City of London. It existed in the year 1130, and was perhaps formed earlier. The company received a Royal Charter in 1155. At present, the Company retains a connection to textiles through its contributions to the textile industry. It has, however, like most other Livery Companies, evolved into a charitable institution rather than remaining a trade association.

The Company ranks forty-second in the order of precedence of the Livery Companies. Its motto is Weave Truth With Trust.

Members of the Livery elect annually an Upper Bailiff who bears an ancient title unique to the Weavers' Company, and a Renter Bailiff. Previous Upper Bailiffs have included:
 * George Kemp, 1st Baron Rochdale 1926–1927 and 1939–1940
 * George Hayter Chubb, 1st Baron Hayter 1931–32
 * Sir Henry Birchenough 1934–35
 * John Garbutt

title: Worshipful Company of Weavers


text:
Talk:John F. Kennedy International Airport

Aeroflot suspension
Aeroflot service shouldnt be removed as their is no indication that service wont return DisneyAviationRollerCoasterEnthusiast (talk) 18:54, 2 March 2022 (UTC)

In agreement here, similar to COVID suspensions there’s nothing telling us the routes won’t come back when the Russian situation is resolved. I’d say just leaving the route with ‘suspended’ is fine (VenFlyer98 (talk) 23:57, 2 March 2022 (UTC))

 * Russian Airplanes are banned from entering United States Airspace therefore Aeroflot cannot fly into JFK 108.50.206.142 (talk) 22:20, 9 March 2022 (UTC)

 * Sure, they’re banned right now but there’s nothing telling us that the routes won’t come back eventually. No need to remove them when we can just list them as suspended. (VenFlyer98 (talk) 05:28, 10 March 2022 (UTC))

Safe to say any mentioned Aeroflot return now borders on WP:CRYSTALBALL. Sure they may 'come back' someday, but without a firm date due to the political/war situation out there, it's very much breaches CRYSTALBALL (and trumps any WikiProject Guideline/rules) at this point. Coastie43 (talk) 09:55, 6 August 2023 (UTC)

 * I agree. This is clearly a situation where the return of service would require a significant change to come back any time soon. They're more than just suspended, they're banned. If the chart is supposed to represent the current routes at the airport, including routes that haven't flown for over a year and would require a major change in real-world geopolitical events to begin again doesn't fulfill that purpose. There's no reason for keeping them here without engaging in CRYSTALBALL speculation. oknazevad (talk) 00:59, 7 August 2023 (UTC)

Major robberies section
Like to propose adding that the robberies/ heists covered under this section inspired the 1990 film Goodfellas. 173.59.118.212 (talk) 20:46, 5 March 2024 (UTC)


text:
Talk:Pontryagin duality

Older comments
It would be nice to add the dual of the set of p-adic numbers in the section Examples [march, 10, 2009].
 * Done. Ringspectrum (talk) 08:42, 10 March 2009 (UTC)

Just a note here with a copy of a discussion on User talk:Charles Matthews outlining plans for three pages: Pontryagin duality, dual group and Harmonic analysis. This is just in case anyone else wants to do anything to these pages.

Hi Charles,  Pontryagin duality looks good, but I wondered why you created this instead of adding to and/or rewriting dual group. Both are about the same thing at different levels of sophistication. I threw dual group together the other day to expand an ugly stub -- essentially the first few lines describing characters. I have no problem with dual group being improved -- such is the way of the wiki. It suffers from the its history and while trying to be consistent with related pages. The best solution may be a merged page under Pontryagin duality (a more accurate title) with dual group redirecting.  If a merge was to happen, dual group suffers from not having any reasons for wanting duality (which yours covers from a couple of angles) and Pontryagin duality suffers from not actually saying what the dual group is (in terms that an undergrad maths student can handle). Your account of the development of the theory is also better. Maybe a solution is to import and tidy up some of the material on the concrete construction of the dual group (characters, topology) from dual group and make dual group a redirect? Possibly keep the classical examples as well, as they tie it to the Fourier transform/series articles. My attempt at referencing was deliberately lazy, so could be improved.  I have also since noticed some overlap with my version of dual group with the middle chunk of Harmonic analysis. I think it doesn't really belong there, as harmonic analysis on LCA groups is only a part of HA, whereas it reads as though it is the most important idea. My feeling is that HA (also served by a redirect from Fourier analysis) should really be a high-level page with a general flavour of what harmonic analysis is about, some history and many links to the disparate fields that come under or are related to HA - from the abstract to the applications (classical things like PDE solutions through to wavelets and whatever else). Do you know of anyone who has an interest in (or oversees/coordinates) this higher-level organisation of WP? The WikiProject Mathematics page only deals with things at the article level.  Cheers, AndrewKepert 01:21, 18 Nov 2003 (UTC)  Generally, I agree with your assessment. I wrote the PD page quickly once I'd actually found dual group (I've thought subsequently of adding a bit more, on discrete and compact R-modules, and End(G) = opposite ring of End (G^)). It could indeed be good to merge the two under PD as main title, putting the dual group material first in the article (and moving across from HA anything that is really about LCA in general as you suggest).  There is no 'they' who sorts out policy-level stuff. We're on our own here ...  Charles Matthews 10:12, 18 Nov 2003 (UTC)  Thanks Charles - yes I know Wiki is anarchic. I also thought that informing somebody(s) who might consider the higher-level organisation as their patch might bring in a few tips and avoid subsequent angst when somebody reverted the lot!  I will go ahead with the merge as discussed (not today - too busy) and strip the corresponding section out of Harmonic analysis. I'll also tidy up notation - I prefer &gamma ∈ Γ = G^ to φ ∈ G' myself, but will change additive notation in the group to multiplicative. I think the merge will work well, given the complementary POV of the two pages. Then I will start thinking about how HA should be structured and fleshing it out. Hopefully whatever I do will offend sufficiently many people, thus attracting interest and contributions.  Cheers AndrewKepert 01:00, 19 Nov 2003 (UTC)

I've done the big move from dual group, and added some headers. This leaves detailed editing of the page.

Charles Matthews 08:47, 19 Nov 2003 (UTC)

Now moved the LCA stuff from harmonic analysis, and spliced it in. Leaves the page structure OK, I think - but needing a thorough edit on notation, excess links and so on.

Charles Matthews 09:30, 19 Nov 2003 (UTC)

What is it good for?
Can someone give some examples that show where Pontryagin duality can be used to prove something or understand something better? —The preceding unsigned comment was added by 217.189.228.213 (talk) 08:49, 3 May 2007 (UTC).

Dual of finite-dimensional vector space
It is analogous to the dual vector space of a finite-dimensional vector space: a vector space V and its dual vector space V * are not naturally isomorphic

This is unclear, but seems to say that a finite-dimensional vector space is not isomorphic to its dual. That's not right, is it? 72.75.67.226 (talk) 01:53, 3 October 2009 (UTC)

It says they are not naturally isomorphic (a term from category theory), but they do become isomorphic once we pick a basis. See for example Natural_transformation. Terminus0 (talk) 05:50, 12 November 2009 (UTC)

Topological Groups
I think one should require the the topological group is also Hausdorff space. —Preceding unsigned comment added by 132.64.102.21 (talk) 13:12, 24 November 2009 (UTC)

What is it?
The article's first para gives a suitably gentle description of where Pontryagin duality might be useful, how, and why. But I don't anywhere find anything approaching a definition. The overall feeling I get from this article is similar to the effect I'd expect if the article on lion consisted of this text:

 * "Lions are native to Africa. The males have manes. Their complex behavior, including apex predation, gives them a central role in the culture of many indigenous social groups. See also: circus."

(Ummm, OK, but shouldn't we mention, for instance, that lion is a species (Panthera leo) of large cat?)

Can somebody please mend this shortcoming. I'd do it myself if I knew enough about the subject matter.—PaulTanenbaum (talk) 16:18, 1 March 2010 (UTC)

About non-commutative theories
Excuse me, this propositions is a little bit vague to me:

Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^ of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group.

Is it possible to reformulate it is such a way that it could be presented as a theorem?

And if you don't mind, I would insert here a little review of non-commutative generalizations.Eozhik (talk) 08:39, 10 November 2012 (UTC)

 * Seems to be all that statement says is, for non-abelian groups, one cannot multiply irreducible unitary representations to get another one unlike the characters in the abelian case. That said, clarification/elaboration is always good. The simplest nonabelian case is listed in See Also:Peter-Weyl theorem. Maybe the main body of the article can say something about that. Mct mht (talk) 10:12, 10 November 2012 (UTC)

 * But if we do not see how to multiply representations, I suppose, this doesn't mean that such a multiplication does'n t exist... Perhaps one can define an operation of multiplication such that the space of all irreducible unitary representrations becomes a group, is it possible? Eozhik (talk) 14:02, 10 November 2012 (UTC)

 * In a very naive sense, and assuming choice, yes. The axiom of choice implies that every nonempty set carries some group structure, so we can pick one essentially at random for each of our sets of representations. Of course, this will not have any nice properties; the interesting question is whether there is a functor between appropriate categories that we can define. Honestly, the statement above confuses me, too: what isomorphism classes are we taking? where did the notion of dimension come in? I suspect that there is not such a functor; perhaps one could prove this by taking a small (i.e. finite) nonabelian group and doing some casework. In particular, functoriality (which is really what we want) would require our multiplication on characters to satisfy (χf)•(ρf) = (χ•ρ)f where the dot is our supposed multiplication and juxtaposition is computation, for every homomorphism f : A -> A, A being the nonabelian group under consideration. Cyrapas (talk) 01:47, 21 January 2014 (UTC)

 * I interpreted that sentence to mean that in a more abstract treatment of G^, we would find that its elements are precisely the 1-dimensional representations (up to isomorphism) and the group operation is the tensor product. This has a chance of being invertible for one-dimensional representations, but since dimensions multiply under tensoring, it has no chance of being invertible in general, so we would generally just have a monoid. Still, the classical definition of the Brauer group for CSA's naively has the same issue--tensoring only gives a monoid structure--which gets resolved by weakening the notion of equivalence slightly. The quoted text is vague about why one should never expect such a theory to exist, and it's also vague about its own abstract underpinnings. I like it, I just wish it were expanded, clarified, and referenced. (I know nothing about this subject.) 50.132.4.93 (talk) 22:20, 17 November 2015 (UTC)

 * I must say that I don't like this vagueness. It seems to me it would be more honest to write that in the noncommutative case the classical construction stops working for various reasons, in particular, because characters stop separating points, irreducible representations cease to be one-dimensional, it is not clear how to introduce multiplication on the dual object if it is supposed to be the set of irreducible unitary representatioons, and it is even not clear whether the set of irrducuble unitary representations is indeed a good choice for the status of the dual object for the group. So the task of constructing duality in this case requires complete rethinking. Would anybody mind if I write something like this? Eozhik (talk) 14:54, 1 March 2019 (UTC)

 * I'll make some edits this week-end, if there will be questions, we can roll back the text and discuss. Eozhik (talk) 06:04, 2 March 2019 (UTC)

Invertibility of canonical map
In the section about the duality theorem and the canonical map, the canonical map is defined, but it is not shown to be an isomorphism. This, to me, seems a rather important step, since in other contexts (e.g. infinite-dimensional vector spaces) analogous canonical maps can fail to be isomorphisms. Presumably this proof uses the fact that the groups are "small" because they are locally compact. Does anyone know it? Cyrapas (talk) 01:52, 21 January 2014 (UTC)

 * Not sure what you want in a Wiki article--a sketch of a proof? The explicit map was probably only included because it's so easy to define. Any textbook presentation of this material will include a proof that it's in fact an isomorphism. Of course "locally compact" is standing in for "finite-dimensional" here. 50.132.4.93 (talk) 22:57, 17 November 2015 (UTC)

Meaning of picture?
What are those patterns of colored hyperbolic tiles supposed to represent? 166.170.23.122 (talk) 04:57, 15 January 2016 (UTC)Collin237
 * Does Commons:File:2-adic_integers_with_dual_colorings.svg answer your question? Melchoir (talk) 04:59, 15 January 2016 (UTC)

Suitability of Picture

I am considering removing the picture at the top of the page, though I thought it prudent to begin a discussion first, giving those who may disagree the opportunity to articulate their reasons; mine are straight forward enough: It is too technical to be understood by a lay audience. I believe for those who have a working familiarity with the subject, this fact may not be immediately apparent. However, a failure to understand is obviously not sufficient reason to remove information (image or otherwise) from a page. It would perhaps be best for the editor who added the picture to remove it themselves, however, if I remain convinced in another few months, I will probably cut to the chase and remove it myself. Or not? Please chime in. Ciao.

Phrase about Plancherel measure
I must say that I don't understand the meaning of this phrase in the text: "but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on $\widehat{G}$." I would like to ask its author to clarify this. The duality theories for non commutative groups are constructed. What is the reason to mention the Plancherel measure in this connection? Eozhik (talk) 10:52, 24 February 2019 (UTC)

 * If there are no objections, I would like to move this subsection, Dualities for non-commutative topological groups, to the end, and to widen it. Eozhik (talk) 08:07, 1 March 2019 (UTC)

 * And to tell the truth, I want to delete this phrase about the Plancherel measure. It is too vague, in my opinion. At least for some groups the theorem remains true as this is written in A.A.Kirillov's book. Eozhik (talk) 14:18, 1 March 2019 (UTC)

Pairing???
The section Pontrjagin duality and the Fourier transform has a subsection titled Fourier transform and Fourier inversion formula for L1-functions.

In that subsection, this sentence appears: "''As an example, suppose $$G=\R^n$$, so we can think about $$\widehat{G}$$ as $$\R^n$$ by the pairing $$(\mathbf{v}, \mathbf{w}) \mapsto e^{i \mathbf{v}\cdot \mathbf{w}}.$$

But this is the first use of the word "pairing" in the article, so the reader does not know why or how one is supposed to think about $$\widehat{G}$$ by this pairing.

It would be very helpful if the article elaborated on how one is supposed to think of $$\widehat{G}$$ by this pairing, and why.50.205.142.50 (talk) 18:40, 25 March 2020 (UTC)

Good introduction, but not appropriate for the introduction
The introduction reads as follows:

"''In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as $$\R$$, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.

"The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940."c

This is all certainly worthwhile information to include in the article.

But not in the introduction.

The introduction should inform the reader of what Pontrjagin duality is, at least roughly. That is much more important to include in the introduction than issues of how Pontrjagin duality fits with the rest of mathematics.

I hope someone knowledgeable about this subject can write an appropriate introduction. And perhaps move the current introduction to somewhere else in the article. 2601:200:C000:1A0:184C:28DD:BFD5:B103 (talk) 19:51, 17 April 2021 (UTC)
 * I agree with your remark. Many articles of mathematics have a similar issue because many editors forget that Wikipedia is not a text book, and thus that, generally, readers do not come to an article for learning why the subject is important, but for learning more on a subject that they have encountered elsewhere.
 * I have rewritten the lead for fixing the issue. Be free to improve my version further. D.Lazard (talk) 11:52, 18 April 2021 (UTC)

title: Talk:Pontryagin duality