title: National Weather Service

text: The National Weather Service also maintains connections with privately operated mesonets such as the Citizen Weather Observer Program for data collection, in part, through the Meteorological Assimilated Data Ingest System (MADIS). Funding is also provided to the CoCoRaHS volunteer weather observer network through parent agency NOAA.


title: National Weather Service

text: NWS forecasters need frequent, high-quality marine observations to examine conditions for forecast preparation and to verify their forecasts after they are produced. These observations are especially critical to the output of numerical weather models because large bodies of water have a profound impact on the weather. Other users rely on the observations and forecasts for commercial and recreational activities. To help meet these needs, the NWS's National Data Buoy Center (NDBC) in Hancock County, Mississippi operates a network of about 90 buoys and 60 land-based coastal observing systems (C-MAN). The stations measure wind speed, direction, and gust; barometric pressure; and air temperature. In addition, all buoy and some C-MAN stations measure sea surface temperature, and wave height and period. Conductivity and water current are measured at selected stations. All stations report on an hourly basis.


title: National Weather Service

text: Supplemental weather observations are acquired through the United States Voluntary Observing Ship (VOS) program. It is organized for the purpose of obtaining weather and oceanographic observations from transiting ships. An international program under World Meteorological Organization (WMO) marine auspices, the VOS has 49 countries as participants. The United States program is the largest in the world, with nearly 1,000 vessels. Observations are taken by deck officers, coded in a special format known as the "ships synoptic code", and transmitted in real-time to the NWS. They are then distributed on national and international circuits for use by meteorologists in weather forecasting, by oceanographers, ship routing services, fishermen, and many others. The observations are then forwarded for use by the National Climatic Data Center (NCDC) in Asheville, North Carolina.


title: National Weather Service

text: Upper air weather data is essential for weather forecasting and research. The NWS operates 92 radiosonde locations in North America and ten sites in the Caribbean. A small, expendable instrument package is suspended below a wide balloon filled with hydrogen or helium, then released daily at or shortly after 1100 and 2300 UTC, respectively. As the radiosonde rises at about 300 meters/minute (1,000 ft/min), sensors on the radiosonde measure profiles of pressure, temperature, and relative humidity. These sensors are linked to a battery-powered radio transmitter that sends the sensor measurements to a ground receiver. By tracking the position of the radiosonde in flight, information on wind speed and direction aloft is also obtained. The flight can last longer than two hours, and during this time the radiosonde can ascend above and drift more than from the release point. When the balloon has expanded beyond its elastic limit and bursts (about in diameter), a small parachute slows the descent of the radiosonde, minimizing the danger to lives and property. Data obtained during the flights is coded and disseminated, at which point it can be plotted on a Skew-T or Stuve diagram for analysis. In recent years, the National Weather Service has begun incorporating data from AMDAR in its numerical models (however, the raw data is not available to the public).


title: National Weather Service

text: The National Weather Service has developed a multi-tier concept for forecasting or alerting the public to all types of hazardous weather:


title: National Weather Service

text: Short-fused weather warnings and advisories issued by local NWS forecast offices are generally less than in area. Warnings for severe local storms are intended to be issued preceding the arrival of severe weather at a particular locale by one hour or less; the NWS also issues warnings and advisories for various hydrological and non-hydrological events including floods, non-thunderstorm high winds, winter storms, intense heat or cold, fire weather and marine hazards, which vary in timepsan depending on the weather situation (inland and coastal warnings for tropical cyclones are issued by the National Hurricane Center (NHC), a guidance center of the National Oceanic and Atmospheric Administration). The NWS defines a warning as a "hazardous weather or hydrologic event [that] is occurring, is imminent, or has a very high probability of occurring" and an advisory as "[highlighting] special weather conditions that are less serious than a warning [...] for events that may cause significant inconvenience, and if caution is not exercised, [..] could lead to situations that may threaten life and/or property." In earnest, they indicate that hazardous weather conditions are occurring that may pose a risk to life and property, and are intended to direct the general public to take immediate action and heed safety precautions; it also has the side purpose of directing emergency management personnel to be on standby in case the weather situation leads to property damage or casualties. Severe thunderstorm and flood warnings indicate that organized severe thunderstorms or flooding are occurring, whereas tornado warnings are issued if a storm is indicated to be producing an observed tornado or exhibits strong, low-level rotation.


title: National Weather Service

text: The process of issuing a warning or advisory begins with observations of a hydrological or extreme weather event that is either occurring at present (through radar imagery, reports from local television and radio stations, or ground observations by local law enforcement, civil defense officials, media outlets or storm spotters) or is forecast to occur within 12 to 24 hours. If after collaboration a warning or advisory is deemed necessary, the Weather Forecast Office will generate a bulletin product via the Advance Weather Interactive Processing System (AWIPS) and then disseminate the alert through various communication routes accessed by the media and various agencies, on the internet, to NOAA satellites, and on NOAA Weather Radio.


title: National Weather Service

text: The product outlines the alert type, the issuing WFO, the sections of government subdivisions (county, parish or boroughs) covered by the alert, and its time of expiration (based on the local time zone). Some products – particularly for severe thunderstorm, tornado and flood warnings – include a tag requesting Emergency Alert System activation to trigger public alert messages via television, radio stations, NOAA Weather Radio, and smartphone apps and messaging services. For local storm events, the warning or advisory product also outlines a meteorological summary of the most recent storm location or local storm report issued prior to the product's issuance (including the approximate area in statute miles and estimated speed and direction), associated hazards, impacts, municipalities and designated land areas (and, if applicable, highway mile markers) covered by the alert, and boilerplate action messages informing the public of safety precautions they need to take or advising them to be vigilant of any warnings or weather statements that may be issued by their local National Weather Service office. A statement may be issued as a follow-up message to a warning, watch, or emergency, which may update, extend, or cancel the previously issued product or be used as a notification of significant weather for which no type of alert is currently in effect for a given location or is expected to be in effect.


title: National Weather Service

text: In situations where a forecaster indicates a significant threat of extremely severe and life-threatening weather with an ongoing local weather event, enhanced wording may be used to note the heightened threat by a significant local storm event. In April 2012, the NWS introduced the Impact Based Warning system at its Weather Forecast Offices in Wichita and Topeka, Kansas, and Springfield, St. Louis and Kansas City/Pleasant Hill, Missouri; the pilot project – which would expand to 80 Weather Forecast Offices overseen by the Central, Eastern, Southern and Western Region Headquarters by the spring of 2015 – incorporate message tags within the main body of the product describing the source of the hazard report, damage potential, and if applicable, radar indications or physical observations of tornadoes or the possibility of a tornado; hazards are also summarized at the close of the product text (describing estimated maximum hail size and wind gusts, and if applicable, if a storm has the potential to produce a tornado or in the event of a tornado warning, the basis of the warning or its damage threat). The wording "Particularly Dangerous Situation" (PDS), which originated by the Storm Prediction Center for use in tornado watch products during expected high-end severe weather outbreaks, is subjectively issued. It is occasionally issued with tornado warnings, normally if a large tornado capable of producing EF3 to EF5 damage or staying on the ground for long-duration – sometimes uninterrupted – paths has been reported (although a tornado emergency may be issued in such cases if the tornado is expected to track into a densely populated area). PDS warnings for other alerts occur with even less frequency, and their criteria varies depending on the alert type to which the wording is applied.


title: National Weather Service

text: Until September 30, 2007, local offices of the National Weather Service issued warnings for severe thunderstorms, tornadoes, flash flooding and marine hazards using geopolitical boundaries. The implementation of storm-based warnings on October 1, 2007, saw alerts for these meteorological or hydrological threats be delineated by polygonal shapes in map-based weather hazard products, which outline the specified sections of government sub-jurisdictions that the warning covers, based on the projected path of a storm as determined by Doppler radar at the time of the warning's issuance; however, entire counties/parishes may sometimes be included in the warning polygon, especially if they encompass a small geographical area. Warnings can be expanded, contracted (by removing jurisdictions where SPC and NWS forecasters no longer consider there to be a viable threat of severe weather, in which case, the storm-based warning may take on a trapezoidal representation in map-based watch products) or canceled before their set time of expiration by local NWS offices.


title: National Weather Service

text: The NWS also releases Experimental Severe Weather Impact products for use on social media accounts maintained by local forecast offices as well as the Enhanced Data Display (EDD), an experimental pilot project created by the Charleston, West Virginia office's WeatherReady Nation initiative. The product provides a graphical depiction of short-fuse warnings and watches (specifically, tornado and severe thunderstorm watches and warnings, and flash flood warnings), showing a map of the warning area (outlined as a red polygon) and locations (including communities and interstate highways) that will be impacted. For severe thunderstorm, tornado and flash flood warnings, the estimated population count of the warned area and approximate totals of public schools and hospitals within the warning area as well as the maximum forecast intensity of hail size, wind gusts and potential tornadoes; tornado warnings referenced in the impact product also denote whether the warning was issued based on radar indication or ground confirmation.


title: National Weather Service

text: NOAA Weather Radio All Hazards (NWR), promoted as "The Voice of the National Weather Service", is a special radio system that transmits uninterrupted weather watches, warnings and forecasts 24 hours a day directly from a nearby NWS office, with the broadcasts covering across 95–97% of the United States' population. The system – which is owned and operated by the NWS – consists of 1,030 transmitters, covering all 50 states; adjacent coastal waters; Puerto Rico; the U.S. Virgin Islands; and the U.S. Pacific Territories of American Samoa, Guam and the Northern Mariana Islands. NWR requires a scanner or special radio receiver capable of picking up the signal. Individual NWR stations broadcast any one of seven allocated frequencies centered on 162 MHz (known collectively as "weather band") in the marine VHF radio band. In recent years, national emergency response agencies such as the Federal Emergency Management Agency (FEMA) and the Department of Homeland Security have begun to take advantage of NWR's ability to efficiently reach a large portion of the U.S. population. When necessary, the system can also be used (in conjunction with the Emergency Alert System) to broadcast civil, natural and technological emergency and disaster alerts and information, in addition to those related to weather – hence the addition of the phrasing ""All Hazards"" to the name.


title: National Weather Service

text: The NOAA Weather Wire Service (NWWS) is a satellite data collection and dissemination system operated by the National Weather Service, which was established in October 2000. Its purpose is to provide state and federal government, commercial users, media and private citizens with timely delivery of meteorological, hydrological, climatological and geophysical information. All products in the NWWS data stream are prioritized, with weather and hydrologic warnings receiving the highest priority (watches are next in priority). NWWS delivers severe weather and storm warnings to users in ten seconds or less from the time of their issuance, making it the fastest delivery system available. Products are broadcast to users via the AMC-4 satellite.


title: National Weather Service

text: The Emergency Managers Weather Information Network (EMWIN) is a system designed to provide the emergency management community with access to a set of NWS warnings, watches, forecasts and other products at no recurring cost. It can receive data via radio, internet, or a dedicated satellite dish, depending on the needs and capabilities of the user.


title: National Weather Service

text: NOAAPORT is a one-way broadcast communication system which provides NOAA environmental data and information in near real-time to NOAA and external users. This broadcast service is implemented by a commercial provider of satellite communications utilizing C band.


title: National Weather Service

text: The agency's online service, Weather.gov, is a data rich website operated by the NWS that serves as a portal to hundreds of thousands of webpages and more than 300 different NWS websites. Through its homepage, users can access local forecasts by entering a place name in the main forecast search bar, view a rapidly updated map of active watches and warnings, and select areas related to graphical forecasts, national maps, radar displays, river and air quality data, satellite images and climate information. Also offered are XML data feeds of active watches and warnings, ASOS observations and digital forecasts for 5x5 kilometer grids. All of NWS local weather forecast offices operate their own region-tailored web pages, which provide access to current products and other information specific to the office's local area of responsibility. Weather.gov superseded the Interactive Weather Information Network (IWIN), the agency's early internet service which provided NWS data from the 1990s through the mid-2000s.


title: National Weather Service

text: Since 1983, the NWS has provided external user access to weather information obtained by or derived from the U.S. Government through a collection of data communication line services called the Family of Services (FOS), which is accessible via dedicated telecommunications access lines in the Washington, D.C., area. All FOS data services are driven by the NWS Telecommunication Gateway computer systems located at NWS headquarters in Silver Spring, Maryland. Users may obtain any of the individual services from NWS for a one-time connection charge and an annual user fee.


title: National Weather Service

text: The WSR-88D Doppler weather radar system, also called NEXRAD, was developed by the National Weather Service during the mid-1980s, and fully deployed throughout the majority of the United States by 1997. There are 158 such radar sites in operation in the U.S., its various territorial possessions and selected overseas locations. This technology, because of its high resolution and ability to detect intra-cloud motions, is now the cornerstone of the agency's severe weather warning operations.


title: National Weather Service

text: National Weather Service meteorologists use an advanced information processing, display and telecommunications system, the Advance Weather Interactive Processing System (AWIPS), to complete their work. These workstations allow them to easily view a multitude of weather and hydrologic information, as well as compose and disseminate products. The NWS Environmental Modeling Center was one of the early users of the ESMF common modeling infrastructure. The Global Forecast System (GFS) is one of the applications that is built on the framework.


title: National Weather Service

text: In 2016, the NWS significantly increased the computational power of its supercomputers, spending $44 million on two new supercomputers from Cray and IBM. This was driven by relatively lower accuracy of NWS' Global Forecast System (GFS) numerical weather prediction model, compared to other global weather models. This was most notable in the GFS model incorrectly predicting Hurricane Sandy turning out to sea until four days before landfall; while the European Centre for Medium-Range Weather Forecasts' model predicted landfall correctly at seven days. The new supercomputers increased computational processing power from 776 teraflops to 5.78 petaflops.


title: National Weather Service

text: Critics such as University of Washington professor Cliff Mass have claimed that NWS forecasts are not as accurate as they could be, and that this has resulted in inaccurate daily weather forecasts and dangerously bad predictions concerning the location and intensity of extreme weather events like blizzards and hurricanes. The British Met Office, the European Centre for Medium-Range Weather Forecasts (ECMWF), and the Northwest Regional Modeling Consortium in Seattle have been cited as producing more accurate predictions in certain circumstances. According to critics, causes include:


title: National Weather Service

text: The Next Generation Global Prediction System project at NWS aims to address some of these criticisms by running a unified high-quality model that takes advantage of more recent research results.


title: National Weather Service

text: While respected as one of the premier weather organizations in the world, the National Weather Service has been perceived by some conservatives as competing unfairly with the private sector. National Weather Service forecasts and data, being works of the federal government, are in the public domain and thus available to anyone for free in accordance with United States law. From time to time, the situation receives official review to ascertain if a leaner, more efficient approach may be had by some degree of privatization.


title: National Weather Service

text: In 1983, the Reagan administration and NOAA administrator John V. Byrne announced a proposal to sell all of the agency's weather satellites at auction with the intent to repurchase the weather data from private contractors that would acquire the satellites. Under the proposal, 30% of NOAA's workforce would be reviewed for potential layoffs, and certain specialty forecasts of agricultural and economic importance would be eliminated. NOAA also proposed outsourcing of weather observation stations, NOAA Weather Radio and computerized surface analysis to private companies. The proposal was met with negative reaction among the public, members of Congress and consumer advocacy groups (including most notably, Ralph Nader), objecting to the possibility of weather information intended for the public domain being sold to private entities that would profit from the sale of the data. The proposal to sell the satellite network failed in a Congressional vote, while other aspects of the proposal to dismantle portions of NOAA's agencies were eventually scuttled.


title: National Weather Service

text: In 2005, Pennsylvania Senator Rick Santorum introduced the National Weather Service Duties Act of 2005, a bill which would have prohibited the NWS from freely distributing weather data. The bill was widely criticized by users of the NWS's services, especially by emergency management officials who rely on the National Weather Service for information during situations such as fires, flooding, or severe weather. Groups such as the Aircraft Owners and Pilots Association condemned the bill's restrictions on weather forecasting as threatening the safety of air traffic, noting that 40% of all aviation accidents are at least partially weather-related. The bill attracted no cosponsors, and died in committee during the 2005 Congressional session.


title: Johnny Isakson

text: John Hardy Isakson (December 28, 1944 – December 19, 2021) was an American businessman and politician who served as a United States senator from Georgia from 2005 to 2019 as a member of the Republican Party. He represented in the United States House of Representatives from 1999 to 2005.


title: Johnny Isakson

text: Born in Atlanta, Georgia, Isakson served in the Georgia Air National Guard (1966–1972) and graduated from the University of Georgia. He opened a real estate branch for Northside Realty and later served 22 years as the company's president. After a failed bid for the Georgia House of Representatives in 1974, he was elected in 1976. He served seven terms, including four as minority leader. Isakson was the Republican candidate for governor of Georgia in 1990, but lost. Two years later, he was elected to the Georgia Senate and served one term. He unsuccessfully ran in the Republican primary in the 1996 U.S. Senate election.


title: Johnny Isakson

text: After 6th District Congressman and Speaker of the House Newt Gingrich resigned, Isakson ran in the February 1999 special election to succeed him, winning by a 40-point margin. He ran for the U.S. Senate in 2004 after Democratic incumbent Zell Miller opted not to run for re-election. With the backing of much of Georgia's Republican establishment, he won both the primary and general elections by wide margins. He became the senior senator from Georgia when Saxby Chambliss retired in 2015. On December 31, 2019, midway through his third Senate term, Isakson resigned from the Senate due to health concerns and was succeeded by fellow Republican Kelly Loeffler who was appointed by Brian Kemp, the Republican Governor of Georgia, to fill the vacant seat.


title: Johnny Isakson

text: Isakson was born on December 28, 1944, in Atlanta, Georgia, the son of Julia ("née" Baker) and Edwin Andrew Isakson, a Greyhound bus driver, who later established an Atlanta real estate firm. His paternal grandparents were of Swedish descent, and his paternal grandfather was born in Östersund. His mother was of mostly British ancestry, and her family has been in the American South since the colonial era.


title: Johnny Isakson

text: Isakson served in the Georgia Air National Guard from 1966 to 1972, leaving service as a staff sergeant. Isakson enrolled at the University of Georgia, where he became a member of the Sigma Alpha Epsilon social fraternity. Shortly after graduating from UGA, he opened the first Cobb County office of Northside Realty, a prominent Atlanta-area real estate firm that his father, Ed, helped to establish. Isakson became company president in 1979, a post he held for 22 years, during which Northside became the biggest independent real estate company in the Southeast and one of the largest in the United States.


title: Johnny Isakson

text: In 1974, Isakson first ran for the Georgia House of Representatives in an eastern Cobb County district and lost. He ran again in 1976 and won. He served seven terms in the House. He won re-election unopposed in 1984 and 1988. In his last four terms (1983–1990), he was the Republican Minority leader. In 1988 and 1996, he was co-chair for U.S. Senator Bob Dole's presidential primary campaigns.


title: Johnny Isakson

text: Isakson was the Republican candidate for Governor of Georgia in 1990. He won the Republican primary with 74% of the vote in a four candidate field. In the general election, he was defeated by Democratic Lieutenant Governor Zell Miller 53%–45%. His campaign was managed by Jay Morgan while Miller's campaign was managed by James Carville. Miller ran on a pledge to start a state lottery and use the revenue for public schools. Isakson proposed a ballot referendum on the lottery.


title: Johnny Isakson

text: In 1996, Isakson ran in the Republican primary for the U.S. Senate seat being vacated by retiring Democratic U.S. Senator Sam Nunn. During his campaign, Isakson expressed his support for abortion rights in a campaign advertisement. Isakson finished second in the primary election with 35% of the vote, but the winner Guy Millner, a millionaire businessman, failed to get a majority of the vote (receiving only 42%). Therefore, per Georgia law, he was forced into a primary runoff election. Millner defeated Isakson in the runoff 53%–47%. Millner lost the general election to Democrat Max Cleland.


title: Johnny Isakson

text: In November 1998, 6th District U.S. Congressman and Speaker of the House Newt Gingrich faced a revolt in his caucus after the Republicans lost five seats in the midterm elections. Amid the turmoil, Gingrich announced on Friday after the Tuesday elections not only that he would not run for a third term as Speaker, but he would also not take his seat for an eleventh term beginning in January 1999. Isakson ran for the seat in a special election in February. He won the election with 65% of the vote, forty points ahead of the second-place finisher Christina Fawcett Jeffrey.


title: Johnny Isakson

text: During his tenure in the House of Representatives, Isakson served on the Committee on Education and the Workforce, aiding President Bush in passing the No Child Left Behind Act. As a Representative, Isakson sponsored 27 bills. He was a member of the U.S. House Education Committee. In October 2002, Isakson voted in favor of the authorization of force against the country of Iraq.


title: Johnny Isakson

text: In early 2003, conservative Democratic U.S. Senator Zell Miller—who had been appointed to fill out the term of the late Republican Senator Paul Coverdell and elected to the post in his own right in 2000—declared his intention not to run for a full term in the Senate in 2004. Isakson immediately entered the race. He faced 8th District U.S. Congressman Mac Collins and businessman Herman Cain in the primary.


title: Johnny Isakson

text: It was initially thought Isakson would face a difficult primary since many socially conservative Republicans still felt chagrin at Isakson's declared support for abortion rights in 1990. However, he won the Republican primary with 53%, with Cain a distant second and Collins third, averting the need for a runoff. In the general election, he easily defeated the Democratic candidate, 4th District Congresswoman Denise Majette, by 18 points. Isakson's election marked the first time in Georgia's history that both of the state's U.S. Senate seats had been held by Republicans, as Saxby Chambliss had won the other seat by defeating Nunn's successor, Max Cleland, two years earlier.


title: Johnny Isakson

text: In 2010, Isakson was unopposed in the primary. He won re-election with 58.3% of the vote in 2010, defeating State Commissioner of Labor Mike Thurmond.


title: Johnny Isakson

text: In 2010, Isakson apologized for referring to voters as "the unwashed" in off-hand comments, saying he "didn't mean anything derogatory by it."


title: Johnny Isakson

text: Isakson resigned from the Senate for health reasons on December 31, 2019. He is the longest serving Republican senator in the history of Georgia.


title: Johnny Isakson

text: When compared to his Republican peers in the Senate, Isakson was neither more liberal than average nor more conservative than average.


title: Johnny Isakson

text: During his campaign for U.S. senator in 1996, Isakson expressed his support for abortion rights in a campaign advertisement. In 2005, Isakson reportedly identified himself as pro-life with exceptions. In March 2017, Isakson—who was recovering from back surgery—came to the U.S. Capitol in a wheelchair to vote to repeal an Obama administration rule that had made it unlawful for states to bar abortion providers from receiving Title X funding. The Senate vote on the bill was 50–50, and Vice President Mike Pence cast a tie-breaking vote that allowed the bill to pass.


title: Johnny Isakson

text: In July 2019, Isakson was one of eight senators to introduce the Agricultural Trucking Relief Act, a bill that would alter the definition of an agricultural commodity to include both horticultural and aquacultural products and promote a larger consistency in regulation through both federal and state agencies as part of an attempt to ease regulatory burdens on trucking and the agri-community.


title: Johnny Isakson

text: In 2017, Isakson said that while he did support concealed carry nationwide, he did not support campus carry and stated that it is "not the appropriate thing to do."


title: Johnny Isakson

text: In February 2018, in response to the Stoneman Douglas High School shooting, Isakson said, "We have to do everything we can within our powers to make sure it never happens again."


title: Johnny Isakson

text: Isakson voted against the Patient Protection and Affordable Care Act and voted more than 60 times to repeal it.


title: Johnny Isakson

text: In 2019, Isakson voted to support President Donald Trump's national emergency declaration regarding border security.


title: Johnny Isakson

text: Isakson and his wife, Dianne, were married in 1968, and had three children. His wife is a watercolor artist, and served as honorary co-chair for Marietta's Theatre in the Square playhouse in 2007.


title: Johnny Isakson

text: In June 2015, Isakson disclosed that he had been diagnosed with Parkinson's disease, but added that the diagnosis would not affect his 2016 re-election plans. He continued his campaign and was elected in November 2016 to serve a third six-year term in the Senate. On August 28, 2019, however, Isakson announced that he would resign his Senate seat for health reasons on December 31, 2019.


title: Don Lake

text: Donald Lake (born November 26, 1956) is a Canadian actor, writer, and television producer. He is frequently cast by director Christopher Guest, and is also a close friend and the collaborative partner of Bonnie Hunt.


title: Don Lake

text: He had a role in "The Bonnie Hunt Show", for which he received comedic praise. He also had roles in the comedy films "Police Academy," "Hot Shots!", "Dumb & Dumber To", and "". He played more serious roles in "" and "Super Mario Bros.", along with a voice role as Stu Hopps in "Zootopia". He is also known as Dr. Carl Whitehorn on "The Fresh Prince of Bel-Air".


title: Don Lake

text: After graduating from the California Institute of the Arts, he returned to Toronto to join the Second City Touring Company, and later was promoted to The Second City. He also appeared in the Netflix comedy series Space Force.


title: Emerald ash borer

text: The emerald ash borer (Agrilus planipennis), also known by the acronym EAB, is a green buprestid or jewel beetle native to north-eastern Asia that feeds on ash species. Females lay eggs in bark crevices on ash trees, and larvae feed underneath the bark of ash trees to emerge as adults in one to two years. In its native range, it is typically found at low densities and does not cause significant damage to trees native to the area. Outside its native range, it is an invasive species and is highly destructive to ash trees native to Europe and North America. Before it was found in North America, very little was known about emerald ash borer in its native range; this has resulted in much of the research on its biology being focused in North America. Local governments in North America are attempting to control it by monitoring its spread, diversifying tree species, and through the use of insecticides and biological control.


title: Emerald ash borer

text: French priest and naturalist Armand David collected a specimen of the species during one of his trips through Imperial China in the 1860s and 1870s. He found the beetle in Beijing and sent it to France, where the first brief description of "Agrilus planipennis" by the entomologist Léon Fairmaire was published in the "Revue d'Entomologie" in 1888. Unaware of Fairmaire's description, a separate description naming the species as "Agrilus marcopoli" was published in 1930 by Jan Obenberger.


title: Emerald ash borer

text: Adult beetles are typically bright metallic green and about long and wide. Elytra are typically a darker green, but can also have copper hues. Emerald ash borer is the only North American species of "Agrilus" with a bright red upper abdomen when viewed with the wings and elytra spread. The species also has a small spine found at the tip of the abdomen and serrate antennae that begin at the fourth antennal segment. They leave tracks in the trees they damage below the bark that are sometimes visible. Adults beetles of other species can often be misidentified by the public.


title: Emerald ash borer

text: The emerald ash borer life cycle can occur over one or two years depending on the time of year of oviposition, the health of the tree, and temperature.


title: Emerald ash borer

text: After 400–500 accumulated degree-days above , adults begin to emerge from trees in late spring, and peak emergence occurs around 1,000 degree-days. After emergence, adults feed for one week on ash leaves in the canopy before mating, but cause little defoliation in the process. Males hover around trees, locate females by visual cues, and drop directly onto the female to mate. Mating can last 50 minutes, and females may mate with multiple males over their lifespan. A typical female can live around six weeks and lay approximately 40–70 eggs, but females that live longer can lay up to 200 eggs.


title: Emerald ash borer

text: Eggs are deposited between bark crevices, flakes, or cracks and hatch about two weeks later. Eggs are approximately in diameter, and are initially white, but later turn reddish-brown if fertile. After hatching, larvae chew through the bark to the inner phloem, cambium, and outer xylem where they feed and develop. Emerald ash borer has four larval instars. By feeding, larvae create long serpentine galleries. Fully mature fourth-instar larvae are long. In fall, mature fourth-instars excavate chambers about into the sapwood or outer bark where they fold into a J-shape. These J-shaped larvae shorten into prepupae and develop into pupae and adults the following spring. To exit the tree, adults chew holes from their chamber through the bark, which leaves a characteristic D-shaped exit hole. Immature larvae can overwinter in their larval gallery, but can require an additional summer of feeding before overwintering again and emerging as adults the following spring. This two-year life cycle is more common in cool climates, such as European Russia.


title: Emerald ash borer

text: The native range of the emerald ash borer is temperate north-eastern Asia, which includes Russia, Mongolia, northern China, Japan, and Korea.


title: Emerald ash borer

text: The beetle is invasive in North America where it has a core population in Michigan and surrounding states and provinces. Populations are more scattered outside the core area, and the edges of its known distribution range north to Ontario, south to northern Louisiana, west to Colorado, and east to New Brunswick, and in the Pacific Northwest in Oregon. In eastern Europe, a population was found in Moscow in 2003. From 2003 to 2016, this population has spread west towards the European Union at up to per year and is expected to reach central Europe between 2031 and 2036. Although not recorded from the European Union as of 2019, it has already spread to far eastern Ukraine from neighboring Russia.


title: Emerald ash borer

text: In its native range, emerald ash borer is only a nuisance pest on native trees, as population densities typically do not reach levels lethal to healthy trees. In China, it infests native "F. chinensis", "F. mandshurica", and "F. rhynchophylla"; in Japan it also infests "F. japonica" and "F. lanuginosa".


title: Emerald ash borer

text: Emerald ash borer primarily infest and can cause significant damage to ash species including green ash ("Fraxinus pennsylvanica"), black ash ("Fraxinus nigra"), white ash ("Fraxinus americana"), and blue ash ("Fraxinus quadrangulata") in North America. In Europe, "Fraxinus excelsior" is the main ash species colonized, which is moderately resistant to emerald ash borer infestation. Ash susceptibility can vary depending on the attractiveness of chemical volatiles to adults, or the ability of larvae to detoxify phenolic compounds. Emerald ash borer has also been found infesting white fringe tree in North America, which is a non-ash host, but it is unclear whether the trees were healthy when first infested, or were already in decline because of drought. Another non-ash host has also been discovered, "Olea europaea", albeit in a lab setting.


title: Emerald ash borer

text: Adults prefer to lay eggs on open grown or stressed ash but readily lay eggs on healthy trees amongst other tree species. Ashes that grow in pure stands, whether naturally occurring or in landscaping, are more prone to attack than isolated trees or ones located in mixed forest stands. Ashes used in landscaping also tend to be subjected to higher amounts of environmental stresses including compacted soil, lack of moisture, heating effects from urban islands, road salt, and pollution, which may also reduce their resistance to the borer. Furthermore, most ashes used in landscaping were produced from a handful of cultivars, resulting in low genetic diversity. Young trees with bark between to are preferred. Both males and females use leaf volatiles and sesquiterpenes in the bark to locate hosts. Damage occurs in infested trees by larval feeding. The serpentine feeding galleries of the larvae disrupt the flow of nutrients and water, effectively girdling, thus killing the tree, as it is no longer able to transport sufficient water and nutrients to the leaves to survive. Girdled ashes will often attempt to regenerate through stump sprouting, and there is evidence that stressed trees may also generate higher than normal seed crops as an emergency measure.


title: Emerald ash borer

text: Outside its native range, emerald ash borer is an invasive species that is highly destructive to ash trees in its introduced range. Before emerald ash borer was found in North America, very little was known about the insect in its native range aside from a short description of life-history traits and taxonomic descriptions, which resulted in focused research on its biology in North America. The insect was first identified in Canton, Michigan (near Detroit), in 2002, but it may have been in the U.S. since the late 1980s. It is suspected that it was introduced from overseas in shipping materials such as packing crates.


title: Emerald ash borer

text: Without factors that would normally suppress emerald ash borer populations in its native range (e.g., resistant trees, predators, and parasitoid wasps), populations can quickly rise to damaging levels. After initial infestation, all ash trees are expected to die in an area within 10 years without control measures. Every North American ash species has susceptibility to emerald ash borer, as North American species planted in China also have high mortality from infestations, but some Asian ash species are resistant, including "F. baroniana, F. chinensis, F. floribunda, F. mandshurica", and "F. platypoda".


title: Emerald ash borer

text: Green ash and black ash trees are preferred by emerald ash borer. White ash is also killed rapidly but usually only after all green and black ash trees are eliminated. Blue ash is known to exhibit a higher degree of resistance to emerald ash borer, which is believed to be caused by the high tannin content in the leaves making the foliage unpalatable to the insect. While most Asian ashes have evolved this defense, it is absent from American species other than blue ash. Researchers have examined populations of so-called "lingering ash", trees that survived ash borer attack with little or no damage, as a means of grafting or breeding new, resistant stock. Many of these lingering ashes were found to have unusual phenotypes that may result in increased resistance. Aside from their higher tannin content, Asian ashes also employ natural defenses to repel, trap, and kill emerald ash borer larvae. Although studies of American ashes have suggested that they are capable of mustering similar defensive mechanisms, the trees do not appear to recognize when they are under attack. Many of the specialized predators and parasitoids that suppressed emerald ash borer in Asia were not present in North America. Predators and parasitoids native to North America do not sufficiently suppress emerald ash borer, so populations continue to grow. Birds such as woodpeckers feed on emerald ash borer larva, although the adult beetles have not been used by any American fauna as food. Emerald ash borer populations can spread between per year. It primarily spreads through flight or by transportation of ash bark containing products such as firewood or nursery stock, which allows it to reach new areas and create satellite populations outside of the main infestation.


title: Emerald ash borer

text: Other factors can limit spread. Winter temperatures of approximately limit range expansion., and overwintering emerald ash borer survive down to average temperatures of because of antifreeze chemicals in the body and insulation provided by tree bark. Larvae can also survive high heat up to . Conversely, much like ashes grown in the nursery trade, the population of emerald ash borer in North America is believed to have originated from a single group of insects from central China and also exhibits low genetic diversity.


title: Emerald ash borer

text: North American predators and parasitoids can occasionally cause high emerald ash borer mortality, but generally offer only limited control. Mortality from native woodpeckers is variable. Parasitism by parasitoids such as "Atanycolus cappaerti" can be high, but overall such control is generally low.


title: Emerald ash borer

text: The United States Department of Agriculture's Animal and Plant Health Inspection Service published a rule on December 14, 2020—to take effect one month later, January 14, 2021—ending all EAB quarantine activities in the United States due to ineffectiveness so far. Other means will be used instead, especially biological controls (see §Biological control below).


title: Emerald ash borer

text: Emerald ash borer threatens the entire North American genus "Fraxinus". It has killed tens of millions of ash trees so far and threatens to kill most of the 8.7 billion ash trees throughout North America. Emerald ash borer kills young trees several years before reaching their seeding age of 10 years. In both North America and Europe, the loss of ash from an ecosystem can result in increased numbers of invasive plants, changes in soil nutrients, and effects on species that feed on ash.


title: Emerald ash borer

text: Damage and efforts to control the spread of emerald ash borer have affected businesses that sell ash trees or wood products, property owners, and local or state governments. Quarantines can limit the transport of ash trees and products, but economic impacts are especially high for urban and residential areas because of treatment or removal costs and decreased land value from dying trees. Costs for managing these trees can fall upon homeowners or local municipalities. For municipalities, removing large numbers of dead or infested trees at once is costly, so slowing down the rate at which trees die through removing known infested trees and treating trees with insecticides can allow local governments more time to plan, remove, and replace trees that would eventually die. This strategy saves money as it would cost $10.7 billion in urban areas of 25 states over 10 years, while removing and replacing all ash trees in these same areas at once would cost $25 billion (with another estimate putting the removal alone at $20–60 billion). Some urban areas such as Minneapolis have large amounts of ash with slightly more than 20% of their urban forest as ash.


title: Emerald ash borer

text: In areas where emerald ash borer has not yet been detected, surveys are used to monitor for new infestations. Visual surveys are used to find ash trees displaying emerald ash borer damage, and traps with colors attractive to emerald ash borer, such as purple or green, are hung in trees as part of a monitoring program. These traps can also have volatile pheromones applied to them that attract primarily males.


title: Emerald ash borer

text: Sometimes trees are girdled to act as trap trees to monitor for emerald ash borer. The stressed tree attracts egg-laying females in the spring, and trees can be debarked in the fall to search for larvae. If detected, an area is often placed under a quarantine to prevent infested wood material from causing new infestations. Further control measures are then taken within the area to slow population growth by reducing beetle numbers, preventing them from reaching reproductive maturity and dispersing, and reducing the abundance of ash trees.


title: Emerald ash borer

text: Government agencies in both the U.S. and Canada have utilized a native species of parasitoid wasp, "Cerceris fumipennis", as a means of detecting areas to which emerald ash borer has spread. The females of these wasps hunt other jewel beetles and emerald ash borer if it is present. The wasps stun the beetles and carry them back to their burrows in the ground where they are stored until the wasps’ eggs hatch and the wasp larvae feed on the beetles. Volunteers catch the wasps as they return to their burrows carrying the beetles to determine whether emerald ash borer is present. This methodology is known as biological surveillance, as opposed to biological control, because it does not appear that the wasps have a significant negative impact on emerald ash borer populations.


title: Emerald ash borer

text: In areas where emerald ash borer is non-native and invasive, quarantines, infested tree removal, insecticides, and biological control are used to reduce damage to ash trees.


title: Emerald ash borer

text: Once an infestation is detected, quarantines are typically imposed by state, or previously, national government agencies disallowing transport of ash firewood or live plants outside of these areas without permits indicating the material has been inspected or treated (i.e., heat treatment or wood chipping) to ensure no live emerald ash borer are present in the bark and phloem. In urban areas, trees are often removed once an infestation is found to reduce emerald ash borer population densities and the likelihood of further spread. Urban ash are typically replaced with non-ash species such as maple, oak, or linden to limit food sources. In rural areas, trees can be harvested for lumber or firewood to reduce ash stand density, but quarantines may apply for this material, especially in areas where the material could be infested.


title: Emerald ash borer

text: Kentucky Extension specialists suggest selecting uncommon species to replace removed ashes in the landscape. Previous generations created monocultures by planting ash trees in an overabundance, a factor in the extent of the devastation caused by the emerald ash borer. Favoring instead a diversity in species helps keep urban forests healthy. University of Kentucky scientists suggest choosing monotypic species such as the pawpaw, yellowwood, Franklin tree, Kentucky coffeetree, Osage orange, sourwood, and bald cypress.


title: Emerald ash borer

text: Insecticides with active ingredients such as azadirachtin, imidacloprid, emamectin benzoate, and dinotefuran are currently used. Dinotefuran and imidacloprid are systemic (i.e., incorporated into the tree) and remain effective for one to three years depending on the product. Insecticides are typically only considered a viable option in urban areas with high value trees near an infestation. Ash trees are primarily treated by direct injection into the tree or soil drench. Some insecticides cannot be applied by homeowners and must be applied by licensed applicators. Damage from emerald ash borer can continue to increase over time even with insecticide applications. Insecticide treatments are not feasible for large forested areas outside of urban areas.


title: Emerald ash borer

text: The native range of emerald ash borer in Asia was surveyed for parasitoid species that parasitize emerald ash borer and do not attack other insect species in the hope they would suppress populations when released in North America. Three species imported from China were approved for release by the USDA in 2007 and in Canada in 2013: "Spathius agrili", "Tetrastichus planipennisi", and "Oobius agrili", while "Spathius galinae" was approved for release in 2015. Excluding "Spathius galinae", which has only recently been released, the other three species have been documented parasitizing emerald ash borer larvae one year after release, indicating that they survived the winter, but establishment varied among species and locations. "Tetrastichus planipennisi" and "Oobius agrili" established and have had increasing populations in Michigan since 2008; "Spathius agrili" has had lower establishment success in North America, which could be caused by a lack of available emerald ash borer larvae at the time of adult emergence in spring, limited cold tolerance, and better suitability to regions of North America below the 40th parallel.


title: Emerald ash borer

text: The USDA is also assessing the application of "Beauveria bassiana", an insect fungal pathogen, for controlling emerald ash borer in conjunction with parasitoid wasps.


title: Multivibrator

text: A multivibrator is an electronic circuit used to implement a variety of simple two-state devices such as relaxation oscillators, timers, and flip-flops. The first multivibrator circuit, the astable multivibrator oscillator, was invented by Henri Abraham and Eugene Bloch during World War I. It consisted of two vacuum tube amplifiers cross-coupled by a resistor-capacitor network. They called their circuit a "multivibrator" because its output waveform was rich in harmonics. A variety of active devices can be used to implement multivibrators that produce similar harmonic-rich wave forms; these include transistors, neon lamps, tunnel diodes and others. Although cross-coupled devices are a common form, single-element multivibrator oscillators are also common.


title: Multivibrator

text: Multivibrators find applications in a variety of systems where square waves or timed intervals are required. For example, before the advent of low-cost integrated circuits, chains of multivibrators found use as frequency dividers. A free-running multivibrator with a frequency of one-half to one-tenth of the reference frequency would accurately lock to the reference frequency. This technique was used in early electronic organs, to keep notes of different octaves accurately in tune. Other applications included early television systems, where the various line and frame frequencies were kept synchronized by pulses included in the video signal.


title: Multivibrator

text: The first multivibrator circuit, the classic astable multivibrator oscillator (also called a "plate-coupled multivibrator") was first described by Henri Abraham and Eugene Bloch in "Publication 27" of the French "Ministère de la Guerre", and in "Annales de Physique 12, 252 (1919)". Since it produced a square wave, in contrast to the sine wave generated by most other oscillator circuits of the time, its output contained many harmonics above the fundamental frequency, which could be used for calibrating high frequency radio circuits. For this reason Abraham and Bloch called it a "multivibrateur". It is a predecessor of the Eccles-Jordan trigger which was derived from the circuit a year later.


title: Multivibrator

text: An astable multivibrator consists of two amplifying stages connected in a positive feedback loop by two capacitive-resistive coupling networks. The amplifying elements may be junction or field-effect transistors, vacuum tubes, operational amplifiers, or other types of amplifier. Figure 1, below right, shows bipolar junction transistors.


title: Multivibrator

text: The circuit is usually drawn in a symmetric form as a cross-coupled pair. The two output terminals can be defined at the active devices and have complementary states. One has high voltage while the other has low voltage, except during the brief transitions from one state to the other.


title: Multivibrator

text: The circuit has two astable (unstable) states that change alternatively with maximum transition rate because of the "accelerating" positive feedback. It is implemented by the coupling capacitors that instantly transfer voltage changes because the voltage across a capacitor cannot suddenly change. In each state, one transistor is switched on and the other is switched off. Accordingly, one fully charged capacitor discharges (reverse charges) slowly thus converting the time into an exponentially changing voltage. At the same time, the other empty capacitor quickly charges thus restoring its charge (the first capacitor acts as a time-setting capacitor and the second prepares to play this role in the next state). The circuit operation is based on the fact that the forward-biased base-emitter junction of the switched-on bipolar transistor can provide a path for the capacitor restoration.


title: Multivibrator

text: In the beginning, the capacitor C1 is fully charged (in the previous State 2) to the power supply voltage "V" with the polarity shown in Figure 1. Q1 is "on" and connects the left-hand positive plate of C1 to ground. As its right-hand negative plate is connected to Q2 base, a maximum negative voltage (-"V") is applied to Q2 base that keeps Q2 firmly "off". C1 begins discharging (reverse charging) via the high-value base resistor R2, so that the voltage of its right-hand plate (and at the base of Q2) is rising from below ground (-"V") toward +"V". As Q2 base-emitter junction is reverse-biased, it does not conduct, so all the current from R2 goes into C1. Simultaneously, C2 that is fully discharged and even slightly charged to 0.6 V (in the previous State 2) quickly charges via the low-value collector resistor R4 and Q1 forward-biased base-emitter junction (because R4 is less than R2, C2 charges faster than C1). Thus C2 restores its charge and prepares for the next State C2 when it will act as a time-setting capacitor. Q1 is firmly saturated in the beginning by the "forcing" C2 charging current added to R3 current. In the end, only R3 provides the needed input base current. The resistance R3 is chosen small enough to keep Q1 (not deeply) saturated after C2 is fully charged.


title: Multivibrator

text: When the voltage of C1 right-hand plate (Q2 base voltage) becomes positive and reaches 0.6 V, Q2 base-emitter junction begins diverting a part of R2 charging current. Q2 begins conducting and this starts the avalanche-like positive feedback process as follows. Q2 collector voltage begins falling; this change transfers through the fully charged C2 to Q1 base and Q1 begins cutting off. Its collector voltage begins rising; this change transfers back through the almost empty C1 to Q2 base and makes Q2 conduct more thus sustaining the initial input impact on Q2 base. Thus the initial input change circulates along the feedback loop and grows in an avalanche-like manner until finally Q1 switches off and Q2 switches on. The forward-biased Q2 base-emitter junction fixes the voltage of C1 right-hand plate at 0.6 V and does not allow it to continue rising toward +"V".


title: Multivibrator

text: Now, the capacitor C2 is fully charged (in the previous State 1) to the power supply voltage "V" with the polarity shown in Figure 1. Q2 is "on" and connects the right-hand positive plate of C2 to ground. As its left-hand negative plate is connected to Q1 base, a maximum negative voltage (-"V") is applied to Q1 base that keeps Q1 firmly "off". C2 begins discharging (reverse charging) via the high-value base resistor R3, so that the voltage of its left-hand plate (and at the base of Q1) is rising from below ground (-"V") toward +"V". Simultaneously, C1 that is fully discharged and even slightly charged to 0.6 V (in the previous State 1) quickly charges via the low-value collector resistor R1 and Q2 forward-biased base-emitter junction (because R1 is less than R3, C1 charges faster than C2). Thus C1 restores its charge and prepares for the next State 1 when it will act again as a time-setting capacitor...and so on... (the next explanations are a mirror copy of the second part of State 1).


title: Multivibrator

text: The duration of state 1 (low output) will be related to the time constant "R""C" as it depends on the charging of C1, and the duration of state 2 (high output) will be related to the time constant "R""C" as it depends on the charging of C2. Because they do not need to be the same, an asymmetric duty cycle is easily achieved.


title: Multivibrator

text: Looking at C2, just before Q2 turns on, the left terminal of C2 is at the base-emitter voltage of Q1 (V) and the right terminal is at "V" (""V"" is used here instead of "+"V"" to ease notation). The voltage across C2 is "V" minus "V" . The moment after Q2 turns on, the right terminal of C2 is now at 0 V which drives the left terminal of C2 to 0 V minus ("V" - "V") or "V" - "V". From this instant in time, the left terminal of C2 must be charged back up to V. How long this takes is half our multivibrator switching time (the other half comes from C1). In the charging capacitor equation above, substituting:


title: Multivibrator

text: For this circuit to work, V»V (for example: V=5 V, V=0.6 V), therefore the equation can be simplified to:


title: Multivibrator

text: The output voltage has a shape that approximates a square waveform. It is considered below for the transistor Q1.


title: Multivibrator

text: During State 1, Q2 base-emitter junction is reverse-biased and capacitor C1 is "unhooked" from ground. The output voltage of the switched-on transistor Q1 changes rapidly from high to low since this low-resistive output is loaded by a high impedance load (the series connected capacitor C1 and the high-resistive base resistor R2).


title: Multivibrator

text: During State 2, Q2 base-emitter junction is forward-biased and capacitor C1 is "hooked" to ground. The output voltage of the switched-off transistor Q1 changes exponentially from low to high since this relatively high resistive output is loaded by a low impedance load (capacitor C1). This is the output voltage of RC integrating circuit.


title: Multivibrator

text: To approach the needed square waveform, the collector resistors have to be low in resistance. The base resistors have to be low enough to make the transistors saturate in the end of the restoration (R < β.R).


title: Multivibrator

text: When the circuit is first powered up, neither transistor will be switched on. However, this means that at this stage they will both have high base voltages and therefore a tendency to switch on, and inevitable slight asymmetries will mean that one of the transistors is first to switch on. This will quickly put the circuit into one of the above states, and oscillation will ensue. In practice, oscillation always occurs for practical values of "R" and "C".


title: Multivibrator

text: However, if the circuit is temporarily held with both bases high, for longer than it takes for both capacitors to charge fully, then the circuit will remain in this stable state, with both bases at 0.60 V, both collectors at 0 V, and both capacitors charged backwards to −0.60 V. This can occur at startup without external intervention, if "R" and "C" are both very small.


title: Multivibrator

text: An astable multivibrator can be synchronized to an external chain of pulses. A single pair of active devices can be used to divide a reference by a large ratio, however, the stability of the technique is poor owing to the variability of the power supply and the circuit elements. A division ratio of 10, for example, is easy to obtain but not dependable. Chains of bistable flip-flops provide more predictable division, at the cost of more active elements.


title: Multivibrator

text: While not fundamental to circuit operation, diodes connected in series with the base or emitter of the transistors are required to prevent the base-emitter junction being driven into reverse breakdown when the supply voltage is in excess of the "V" breakdown voltage, typically around 5-10 volts for general purpose silicon transistors. In the monostable configuration, only one of the transistors requires protection.


title: Multivibrator

text: Assume all the capacitors to be discharged at first. The output of the op-amp V at node c is +V initially. At node a, a voltage of +β V is formed due to voltage division where formula_12. The current that flows from nodes c and b to ground charges the capacitor C towards +V. During this charging period, the voltage at b becomes greater than +β Vat some point. The voltage at inverting terminal will be greater than the voltage at the non-inverting terminal of the op-amp. This is a comparator circuit and hence, the output becomes -V. The voltage at node a becomes -βV due to voltage division. Now the capacitor discharges towards -V. At some point, the voltage at b becomes less than -β V. The voltage at the non-inverting terminal will be greater than the voltage at the inverting terminal of the op-amp. So, the output of the op-amp is +V. This repeats and forms a free-running oscillator or an astable multivibrator.


title: Multivibrator

text: If V is the voltage across the capacitor and from the graph, the time period of the wave formed at capacitor and the output would match, then the time period could be calculated in this way:formula_13


title: Multivibrator

text: We are taking values of R, C and β such that we get a symmetrical square wave. Thus, we get "T1" = "T2" and total time period "T" = "T1" + "T2". So, the time period of the square wave generated at the output is:


title: Multivibrator

text: In the monostable multivibrator, one resistive-capacitive network (C-R in Figure 1) is replaced by a resistive network (just a resistor). The circuit can be thought as a 1/2 astable multivibrator. Q2 collector voltage is the output of the circuit (in contrast to the astable circuit, it has a perfect square waveform since the output is not loaded by the capacitor).


title: Multivibrator

text: When triggered by an input pulse, a monostable multivibrator will switch to its unstable position for a period of time, and then return to its stable state. The time period monostable multivibrator remains in unstable state is given by "t" = ln(2)"R""C". If repeated application of the input pulse maintains the circuit in the unstable state, it is called a "retriggerable" monostable. If further trigger pulses do not affect the period, the circuit is a "non-retriggerable" multivibrator.


title: Multivibrator

text: For the circuit in Figure 2, in the stable state Q1 is turned off and Q2 is turned on. It is triggered by zero or negative input signal applied to Q2 base (with the same success it can be triggered by applying a positive input signal through a resistor to Q1 base). As a result, the circuit goes in State 1 described above. After elapsing the time, it returns to its stable initial state.


title: Multivibrator

text: The circuit is useful for generating single output pulse of adjustable time duration in response to a triggering signal. The width of the output pulse


title: Multivibrator

text: depends only on external components connected to the op-amp. A diode D1 clamps the capacitor voltage to 0.7 V when the output is at +Vsat. Let us assume that in the stable


title: Multivibrator

text: state the output Vo = +Vsat. The diode D1 clamps the capacitor to 0.7 V. The voltage at the non-inverting terminal through the potential divider will be + βVsat. Now a negative trigger of magnitude V1 is applied to the non-inverting terminal so that the effective signal at this terminal is less than 0.7 V. Then the output voltage switches from +Vsat to -Vsat. The diode will now get reverse biased and the capacitor starts charging exponentially to -Vsat through R. The voltage at the non-inverting terminal through the potential divider will be - βVsat. After some time the capacitor charges to a voltage more than - βVsat. The voltage on the non-inverting input is now greater than on the inverting input and the output of the op-amp switches again to +Vsat. The capacitor discharges through resistor R and charges again to 0.7 V.


title: Multivibrator

text: In the bistable multivibrator, both resistive-capacitive networks (C-R and C-R in Figure 1) are replaced by resistive networks (just resistors or direct coupling).


title: Multivibrator

text: This latch circuit is similar to an astable multivibrator, except that there is no charge or discharge time, due to the absence of capacitors. Hence, when the circuit is switched on, if Q1 is on, its collector is at 0 V. As a result, Q2 gets switched off. This results in more than half +"V" volts being applied to R4 causing current into the base of Q1, thus keeping it on. Thus, the circuit remains stable in a single state continuously. Similarly, Q2 remains on continuously, if it happens to get switched on first.


title: Multivibrator

text: Switching of state can be done via Set and Reset terminals connected to the bases. For example, if Q2 is on and Set is grounded momentarily, this switches Q2 off, and makes Q1 on. Thus, Set is used to "set" Q1 on, and Reset is used to "reset" it to off state.


title: Janice Wu

text: Wu Qian (, born ), also known as Janice Wu, is a Chinese actress. She rose to fame for portraying the younger counterpart of Zhao Mosheng in the 2015 hit romance drama "My Sunshine", and gained attention for her roles in the television series "My Amazing Boyfriend" (2016), "Fighter of the Destiny" (2017), " The Brightest Star in the Sky" (2019), "Le Coup de Foudre" (2019) and "Skate into Love" (2020).


title: Janice Wu

text: Wu made her acting debut in 2014 with a minor role in "Heroes of Sui and Tang Dynasties 3 & 4". She became known for her role in the wuxia drama "The Deer and the Cauldron", playing Mu Jianping.


title: Janice Wu

text: In 2015, Wu starred as the younger counterpart of Zhao Mosheng in the hit romance drama "My Sunshine", gaining popularity for her portrayal. The following year, she played her first lead role in the romance comedy series "My Amazing Boyfriend" (2016). The series was a success and gained more than 2 billion views by the end of its run.


title: Janice Wu

text: In 2017, Wu starred in the hit fantasy drama "Fighter of the Destiny". The role of Luo Luo endeared her to the audience and contributed to her rise in popularity. The same year, Wu reunited with "My Sunshine" co-star Luo Yunxi in the fantasy romance drama "A Life Time Love".


title: Janice Wu

text: In 2018, Wu had a guest role in the palace drama "Ruyi's Royal Love in the Palace", where she played an antagonistic character for the first time. The same year, she starred in the fantasy action drama "Oriental Odyssey" alongside Zheng Yecheng.


title: Janice Wu

text: In 2019, Wu starred alongside Huang Zitao in the music romance drama "The Brightest Star in the Sky". The same year, she starred in the romance web series "Le Coup de Foudre".


title: Janice Wu

text: In 2021, it was confirmed that she and her Le Coup de Foudre co-star Zhang Yujian are indeed married and have a daughter together. In 2022, they divorced.


title: Bagdogra Airport

text: Bagdogra Airport is a customs airport located in the western part of Siliguri, in Bagdogra in northern West Bengal, India.


title: Bagdogra Airport

text: It is operated as a civil enclave at AFS Bagdogra of the Indian Air Force. It is the gateway airport to the hill stations of Darjeeling, Gangtok, Kurseong, Kalimpong, Mirik and other parts of the North Bengal region. As a major transport hub in the region, the airport sees thousands of tourists annually. The Government of India conferred limited international airport status to the airport in 2002 with limited international operations.


title: Bagdogra Airport

text: Bagdogra Airport experiences humid sub-tropical climate (Köppen climate classification Cwa) - hot humid summers and cooler winter. Significant precipitation in all seasons but drier in the winter. Air traffic at Bagdogra crossed 1 million for the first time growing at 43.6% percent in 2014–15. In 2019–20, the airport served 3.2 million passengers which was an increase of 11.2% from the previous year, making it the 17th-busiest airport in India. It is one of the few airports in India with zero sales tax on aviation turbine fuel.


title: Bagdogra Airport

text: The airbase is home to the IAF No. 20 Wing, as also to the Mikoyan-Gurevich MiG-21 (Mig-21) FL fighter aircraft of the No. 8 Squadron and a Helicopter Unit. Along with the airbase at Hasimara, Alipurduar district; it is responsible for combat air operations over a large area including North Bengal, Sikkim, and if needed, Bhutan. The base caters to all military air traffic for the Indian Army's XXXIII Corps based nearby in Sukna.


title: Bagdogra Airport

text: The state government had handed over of land to the Ministry of Civil Aviation for developing infrastructure for night landing in 2010. AAI also expanded the apron at the same time, enabling the parking of 5 narrow body aircraft simultaneously. The IAF, which maintains the ATC and runway, gave permission for night landings by civilian aircraft allowing flights past 6 pm in 2013. The new terminal building, which will start its construction by December 2022, will be completed by June 2025 with 16 parking bays, 10 aerobridges and a taxi way. It will have a floor area of 1 lakh sq.m at a cost of Rs 1884 crore, and it will be capable of handling 12.5 million passengers per year. After the expansion of the airport, more international flights can land here, and will turn this customs airport into an international one.


title: Bagdogra Airport

text: For its growing traffic over the years, in 2022, From 11 to 25 April, the airport was shut down for 14 days to refurbish its sole runway.


title: Songs in A Minor

text: Songs in A Minor is the debut studio album by American singer and songwriter Alicia Keys, released on June 5, 2001, by J Records.


title: Songs in A Minor

text: Keys began writing songs for the album in 1995 at age 14 and recording the album in 1998 for Columbia Records, but after they rejected it, she signed a recording contract with Clive Davis's Arista Records and eventually J Records. An accomplished, classically trained pianist, Keys wrote, arranged and produced the majority of the album herself. It is a neo soul album with elements of R&B, soul, jazz, hip hop, blues, classical, and gospel music. Lyrically, the songs explore the complexities and various stages of personal relationships. Despite the album's title, only two songs, "Jane Doe" and "Mr. Man," are actually in the key of A minor.


title: Songs in A Minor

text: "Songs in A Minor" debuted at number one on the US "Billboard" 200, selling 236,000 copies in its first week. The album has since sold over 6.2 million copies in the United States and 12 million copies worldwide. It was also an immediate critical success and has since been regarded as a classic. The album earned Keys several accolades, including five Grammy Awards at the 44th Annual Grammy Awards. To promote the album, Keys embarked on her first headlining concert tour, entitled Songs in A Minor Tour.


title: Songs in A Minor

text: In 2022, the album was selected by the Library of Congress for preservation in the National Recording Registry.


title: Songs in A Minor

text: Keys began writing the songs that would constitute "Songs in A Minor" at age 14, "Butterflyz" being her first composition for the album. Keys had been accepted to Columbia University, which she attended after graduating from the Professional Performing Arts School at age 16. She dropped out after four weeks to pursue her music career full time. She signed a demo deal with Jermaine Dupri and his So So Def label. Keys co-wrote and recorded a song titled "Dah Dee Dah (Sexy Thing)", which appeared on the soundtrack to the 1997 film "Men in Black". She also contributed to the So So Def Christmas recordings. Keys began producing and recording the album in 1998. She completed it that same year, but it was rejected by Columbia Records. Keys explained that the producers she was required to work with by the label would tell her to "just get in the booth and sing", which frustrated her. Her record contract with Columbia ended after a dispute with the label. Keys then performed for Clive Davis, who sensed a "special, unique" artist; he bought Keys' contract from Columbia and signed her to Arista Records, which later disbanded.


title: Songs in A Minor

text: Following Davis to his newly formed J Records label, Keys rented an apartment and struggled to create an album. She began writing the song "Troubles" and came to a realization: "That's when the album started comin' together. Finally, I knew how to structure my feelings into something that made sense, something that can translate to people. That was a changing point. My confidence was up, way up." Keys learned how to produce by asking questions to the producers and engineers; she wrote, arranged and produced a majority of the album. She recorded the songs "Rock wit U" and "Rear View Mirror", which were featured on the soundtracks to the films "Shaft" (2000) and "Dr. Dolittle 2" (2001), respectively. One of the final songs Keys recorded was "Fallin". A total of 32 songs were recorded for the album. Originally titled "Soul Stories in A Minor", the title of the album was changed over concerns that it would limit exposure only to black radio stations.


title: Songs in A Minor

text: "Songs in A Minor" is a neo soul album with classical piano references and arpeggios. Keys incorporates classical piano with R&B, soul and jazz into the album's music. With influences of classical piano, classic soul and East Coast hip hop, Keys described the album as a "fusion of my classical training, meshed with what I grew up listening to [...] things I've been exposed to and drawn from and my life experiences". Jane Stevenson of Jam! described the music as "old-school urban sounds and attitude set against a backdrop of classical piano and sweet, warm vocals". "USA Today"s Steve Jones wrote that Keys "taps into the blues, soul, jazz and even classical music to propel haunting melodies and hard-driving funk". John Mulvey of Yahoo! Music called the album "a gorgeous and ambitious melding of classic soul structures and values to hyper-modern production technique".


title: Songs in A Minor

text: The album's opening track, "Piano & I", begins with a rendition of Ludwig van Beethoven's "Moonlight Sonata", combined with a hip hop beat. The introduction is followed by "Girlfriend", which was produced by Jermaine Dupri. Commended for its "crisp production", the song samples Ol' Dirty Bastard's "Brooklyn Zoo". Keys' cover of Prince's 1982 ballad "How Come U Don't Call Me Anymore?" (retitled "How Come You Don't Call Me") was inspired by a long-term relationship with a partner. The music critic for "PopMatters" felt that the song was credible, but fell short from the original and Stephanie Mills's 1980s cover. "Fallin", the gospel-driven lead single and often considered Keys' signature song, contains a sample of James Brown's "It's a Man's Man's Man's World". The song earned Keys comparisons to Aretha Franklin.


title: Songs in A Minor

text: "A Woman's Worth", the second single released from the album, is a "gospel-tinged" song that recommends that men show respect to their female partners. "Jane Doe" is a funk-driven song, with backing vocals provided by Kandi Burruss. "The Life", which elicits Curtis Mayfield's "Gimmie Your Love", describes Keys' "philosophy of life and struggle". The song was compared to the work of the English band Sade. "Mr. Man" contains elements of Latin American music and was described as a "sexy and soulful duet", in which Jimmy Cozier "adds his spice". The album ends with the hidden track "Lovin' U", which Christian Ward of "NME" compared to works of the musical group the Supremes.


title: Songs in A Minor

text: In advance of "Songs in A Minor", "Girlfriend" was serviced to urban contemporary radio as a promotional single in early 2001 to "introduce" Keys to the general public. In order to promote her, music executive Clive Davis booked Keys to "The Tonight Show with Jay Leno". Davis also wrote a letter to Oprah Winfrey, asking her to allow Keys, along with Jill Scott and India Arie, to perform on her show. The singers performed on "The Oprah Winfrey Show", where Keys "wowed" the audience. This led to the album's pre-orders to double that night. From August to October 2001, Keys toured alongside recording artist Maxwell in promotion of the album. Soon after, she embarked on her Songs in A Minor Tour. A concert at KeyArena in Seattle was partly recorded and the live performances were included on the bonus disc of a European reissue of the album, titled "Songs in A Minor: Remixed & Unplugged", which was released on October 28, 2002; the bonus disc also includes remixes to the songs from "Songs in A Minor". The bonus disc was made available on its own exclusively in Japan, under the title "Remixed & Unplugged in A Minor", on February 26, 2003.


title: Songs in A Minor

text: On June 28, 2011, "Songs in A Minor" was re-released in two editions to commemorate its tenth anniversary of release. Both editions feature previously unreleased material and a documentary chronicling the making of "Songs in A Minor". At the BET Awards 2011 on June 26, Keys performed a medley of songs, including "Typewriter", "A Woman's Worth" with Bruno Mars and "Maybach Music" with Rick Ross and "Fallin. On June 28, Keys performed "Fallin, "Butterflyz" and "Empire State of Mind (Part II) Broken Down" on "Good Morning America". BET aired "The Story So Far... Alicia Keys" special, highlighting Keys' ten-year career through her BET moments, on June 28. On June 30, Keys performed "Songs in A Minor" in its entirety and told stories of its recording in a show titled at the Beacon Theatre in New York City. In an interview for MTV, Keys called its tenth anniversary "incredibly surreal for me" and said of the album in retrospect: "This album is possibly the most precious to me as your first album only happens once, and so "Songs in A Minor" will always hold a special place in my life that's filled with amazing memories. I'm so proud the songs are still being enjoyed, and I'm crazy excited to share songs never heard before." To celebrate the 20th anniversary of the release of "Songs in A Minor", the album was yet again re-released on June 4, 2021 with four bonus tracks, including the previously unreleased "Foolish Heart" and "Crazy (Mi Corazon)".


title: Songs in A Minor

text: Davis sent the music video of the album's lead single "Fallin'" to MTV; "half the women had tears down their face" when the video finished playing. The song was serviced to rhythmic contemporary, urban adult contemporary and urban contemporary radio stations in April 2001. "Fallin'" peaked at number one on the "Billboard" Hot 100 and Hot R&B/Hip-Hop Songs charts, remaining atop the charts for six and four weeks, respectively. It became the most played song in the United States at the time and was certified gold by the Recording Industry Association of America (RIAA). Internationally, "Fallin'" peaked within the top ten in almost all countries it charted in, topping the charts in Flanders, the Netherlands and New Zealand, as well as the UK R&B Singles chart.


title: Songs in A Minor

text: "A Woman's Worth" was released as the album's second single on October 2, 2001. It peaked at number seven on the "Billboard" Hot 100. The single was certified gold by the RIAA. Internationally, the song reached number one in Croatia and top ten in Hungary and New Zealand. The single's accompanying music video was directed by Chris Robinson, who directed the video for "Fallin'". Its plot continues from the video for "Fallin'", which revolves around Keys' travel to her imprisoned boyfriend, and picks up where it left, depicting his release from prison and tries to acclimate to society.


title: Songs in A Minor

text: The third single "How Come You Don't Call Me" was released on March 11, 2002. It peaked at number 59 on the "Billboard" Hot 100, failing to duplicate the commercial success of its predecessors. Elsewhere, the song peaked within the top ten in Hungary and within the top 40 in Australia, Ireland, Scotland and the United Kingdom.


title: Songs in A Minor

text: "Girlfriend" peaked at number 82 on the Hot R&B/Hip-Hop Songs chart upon its 2001 release as a promotional single. It was released as the album's fourth and final single outside the United States on November 25, 2002. The song reached top 20 in Australia and the Netherlands, while peaking at number 24 in the UK.


title: Songs in A Minor

text: "Songs in A Minor" received positive reviews from critics. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream critics, it received an average score of 78, based on 10 reviews.


title: Songs in A Minor

text: Reviewing the album in "NME", Sam Faulkner described the balance between contemporary music and retrospective as "an act of pure genius". "Q" magazine hailed it as "a prime candidate to head up the nu-soul revolution ... with a voice that challenges Mary J. Blige's". Steve Jones of "USA Today" said that "Keys already has a musical, artistic and thematic maturity that many more experienced artists never achieve". "The Washington Post"s Richard Harrington wrote favorably of Keys' musical influences on the album and expressed that she has "vocal maturity and writing instincts beyond her years". "PopMatters" critic Mark Anthony Neal praised Keys' performance on the album and called it "a distinct and oft-times brilliant debut from an artist who clearly has a fine sense of her creative talents". Robert Christgau, writing in "The Village Voice", said that the "grace and grit" of the first half warrant the "auspicious debut" label and that, after some "bores that threaten to sink the project midway through," Keys sustains the album with the songs at the end.


title: Songs in A Minor

text: Keys' vocal performance was lauded; Sal Cinquemani from "Slant Magazine" declared that Keys' displayed a "powerful range, proving she can belt along with the best of them". "Uncut" called the album "frequently stunning" and said that Keys sings like "a young Aretha Franklin". However, some found her lyrics to be sub-par to her singing and musical ability. "The New Zealand Herald"s Russell Baillie stated that Keys "might indicate abundant talent aligned to neatly reverential vintage soul style", but expressed that the songs "don't add up to anything particularly memorable". "Entertainment Weekly"s Beth Johnson called the second half of the album slacked with "sad sack teen themes", but called it a promising album. "Rolling Stone"s Barry Walters perceived her singing as more mature than her songwriting, but commended Keys for her "commanding presence" on the album. "Los Angeles Times" writer Robert Hilburn said that it "makes a convincing case that's she's going far—in both a commercial and creative sense".


title: Songs in A Minor

text: In a retrospective review, AllMusic's Stephen Thomas Erlewine perceived the album's music as "rich enough to compensate for some thinness in the writing" and called it "a startling assured, successful debut that deserved its immediate acclaim and is already aging nicely". Barry Walters wrote in a later article for "Rolling Stone", "the album has aged well – excepting a drum-machine beat or two, it feels timeless." In the "Encyclopedia of Popular Music" (2011), Colin Larkin said Keys had fused urban R&B, hip hop, and blues on what he called "a minor classic of modern soul". "Songs in A Minor" is regarded as an influential and distinctive album of its era.


title: Songs in A Minor

text: "Songs in A Minor" led Keys to win five awards at the 44th Annual Grammy Awards: Song of the Year, Best Female R&B Vocal Performance, and Best R&B Song for "Fallin, Best New Artist, and Best R&B Album; "Fallin was also nominated for Record of the Year. Keys became the second female solo artist to win five Grammy Awards in a single night, following Lauryn Hill at the 41st Annual Grammy Awards (1999). The album also won a NAACP Image Award for Outstanding Album. Keys was also named Best New Artist at the 2002 World Music Awards. "Fallin was ranked at number 37 on VH1's 100 Greatest Songs of the Past 25 Years in 2003 and was ranked the 413th greatest song of all time by "Blender" magazine. The album was ranked at number two on the "Rolling Stone" magazine's Top 10 of 2001, number 18 on "The Village Voice"s 2001 Pazz & Jop list, number 27 on "Mojo" magazine's Best 40 Albums of 2001 and was also named one of "Q" magazine's 100 Greatest Albums Ever. "Q" also listed the album as one of the best 50 albums of 2001. In 2009, "Rolling Stone" named it the 95th best album of the past decade, while "Fallin ranked at number 62 on the magazine's "100 Best Songs of the Decade" list. In 2012, "Entertainment Weekly" ranked "Songs in A Minor" the 57th best album of all time.


title: Songs in A Minor

text: "Songs in A Minor" debuted at number one on the US "Billboard" 200, selling 236,000 copies in its first week. Through word of mouth and promotion, the album remained at number one on the chart, selling 450,000 copies in its second week. The album spent a total of three non-consecutive weeks at number one, and became one of the best-selling albums of 2001. As of June 2014, the album had sold 6,348,000 copies in the United States. "Billboard" ranked the album at number 32 on the "Billboard" 200 decade-end chart for the 2000s and at number 12 on the Top R&B/Hip-Hop Albums decade-end chart. In 2001 the album was the 7th best-selling album globally, selling 6.7 million copies. By March 2008, "Songs in A Minor" had sold over 12 million copies worldwide. On August 19, 2020, the album was certified septuple platinum by the Recording Industry Association of America (RIAA) for combined sales and album-equivalent units of seven million units in the United States.


title: Laurent series

text: In mathematics, the Laurent series of a complex function formula_1 is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.


title: Laurent series

text: where formula_5 and formula_3 are constants, with formula_5 defined by a line integral that generalizes Cauchy's integral formula:


title: Laurent series

text: The path of integration formula_9 is counterclockwise around a Jordan curve enclosing formula_3 and lying in an annulus formula_11 in which formula_1 is holomorphic (analytic). The expansion for formula_1 will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled formula_9. If we take formula_9 to be a circle formula_16, where formula_17, this just amounts


title: Laurent series

text: to computing the complex Fourier coefficients of the restriction of formula_18 to formula_9. The fact that these integrals are unchanged by a deformation of the contour formula_9 is an immediate consequence of Green's theorem.


title: Laurent series

text: One may also obtain the Laurent series for a complex function formula_1 at formula_22. However, this is the same as when formula_23 (see the example below).


title: Laurent series

text: In practice, the above integral formula may not offer the most practical method for computing the coefficients


title: Laurent series

text: formula_5 for a given function formula_1; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever


title: Laurent series

text: Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.


title: Laurent series

text: Consider for instance the function formula_28 with formula_29. As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at formula_30. By replacing formula_31 with formula_32 in the power series for the exponential function, we obtain its Laurent series which converges and is equal to formula_33 for all complex numbers formula_31 except at the singularity formula_30. The graph opposite shows formula_36 in black and its Laurent approximations


title: Laurent series

text: for formula_38 = 1, 2, 3, 4, 5, 6, 7 and 50. As formula_39, the approximation becomes exact for all (complex) numbers formula_31 except at the singularity formula_41.


title: Laurent series

text: More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc.


title: Laurent series

text: is a given Laurent series with complex coefficients formula_5 and a complex center formula_3. Then there exists a unique inner radius formula_45 and outer radius formula_46 such that:


title: Laurent series

text: It is possible that formula_45 may be zero or formula_46 may be infinite; at the other extreme, it's not necessarily true that formula_45 is less than formula_46.


title: Laurent series

text: Conversely, if we start with an annulus of the form formula_47 and a holomorphic function formula_1 defined on formula_11, then there always exists a unique Laurent series with center formula_3 which converges (at least) on formula_11 and represents the function formula_1.


title: Laurent series

text: This function has singularities at formula_64 and formula_65, where the denominator of the expression is zero and the expression is therefore undefined.


title: Laurent series

text: A Taylor series about formula_66 (which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at 1.


title: Laurent series

text: The case formula_80; i.e., a holomorphic function formula_1 which may be undefined at a single point formula_3, is especially important. The coefficient formula_83 of the Laurent expansion of such a function is called the residue of formula_1 at the singularity formula_3; it plays a prominent role in the residue theorem. For an example of this, consider


title: Laurent series

text: To determine the Laurent expansion about formula_88, we use our knowledge of the Taylor series of the exponential function:


title: Laurent series

text: Multiply both sides by formula_95, where k is an arbitrary integer, and integrate on a path γ inside the annulus,


title: Laurent series

text: The series converges uniformly on formula_97, where "ε" is a positive number small enough for "γ" to be contained in the constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity


title: Laurent series

text: A Laurent polynomial is a Laurent series in which only finitely many coefficients are non-zero. Laurent polynomials differ from ordinary polynomials in that they may have terms of negative degree.


title: Laurent series

text: If the principal part of formula_18 is a finite sum, then formula_18 has a pole at formula_3 of order equal to (negative) the degree of the highest term; on the other hand, if formula_18 has an essential singularity at formula_3, the principal part is an infinite sum (meaning it has infinitely many non-zero terms).


title: Laurent series

text: If the inner radius of convergence of the Laurent series for formula_18 is 0, then formula_18 has an essential singularity at formula_3 if and only if the principal part is an infinite sum, and has a pole otherwise.


title: Laurent series

text: If the inner radius of convergence is positive, formula_18 may have infinitely many negative terms but still be regular at formula_3, as in the example above, in which case it is represented by a "different" Laurent series in a disk about formula_3.


title: Laurent series

text: Laurent series with only finitely many negative terms are well-behaved—they are a power series divided by formula_112, and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.


title: Laurent series

text: Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences).


title: Laurent series

text: Two Laurent series with only "finitely" many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at formula_3, and inner radius of convergence 0, so they both converge on an overlapping annulus.


title: Laurent series

text: Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms.


title: Laurent series

text: Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.


title: Laurent series

text: Also, for a field formula_114, by the sum and multiplication defined above, formal Laurent series would form a field formula_115 which is also the field of fractions of the ring formula_116 of formal power series.


title: Hair conditioner

text: Hair conditioner is a hair care cosmetic product used to improve the feel, texture, appearance, and manageability of hair. Its main purpose is to reduce friction between strands of hair to allow smoother brushing or combing, which might otherwise cause damage to the scalp. Various other benefits are often advertised, such as hair repair, strengthening, or a reduction in split ends.


title: Hair conditioner

text: Conditioners are available in a wide range of forms including viscous liquids, gels, and creams, as well as thinner lotions and sprays. Hair conditioner is usually used after the hair has been washed with shampoo. It is applied and worked into the hair, and may either be rinsed out a short time later or left in.


title: Hair conditioner

text: For centuries, natural oils have been used to condition human hair. A conditioner popular with men in the late Victorian era was Macassar oil, but this product was quite greasy and necessitated the pinning of a small cloth, known as an antimacassar, to the headrests of chairs and sofas to preserve the upholstery from being damaged by the oil.


title: Hair conditioner

text: Modern hair conditioner was created at the turn of the 20th century when the Edouard Pinaud company presented a product he called "Brilliantine" at the 1900 Exposition Universelle in Paris. His product was intended to soften men's hair, including beards and moustaches. Since the invention of Pinaud's early products, modern science has advanced the hair conditioner industry to include those made with silicone, fatty alcohols, and quaternary ammonium compounds. These chemical products have the benefits of hair conditioner without feeling greasy or heavy.


title: Hair conditioner

text: The outermost layer of a hair follicle is called the cuticle and is composed largely of keratin. This is rich in cysteine groups which are mildly acidic. When the hair is washed these groups can deprotonate, giving the hair a negative charge.


title: Hair conditioner

text: The ingredients in conditioner, especially positively charged quaternary ammonium species, such as behentrimonium chloride or polymers that are known as Polyquaternium-XX, where XX is an arbitrary number, can then become attached to the hair via electrostatic interactions. Once attached these compounds have several effects. Their long hydrocarbon backbone helps to lubricate the surface of each hair follicle, reducing the sensation of roughness and assisting combing. The surface coating of cationic groups means that hairs are repelled from each other electrostatically, which reduces clumping. The compounds can also act as antistatic agents, which helps to reduce frizzing.


title: Hair conditioner

text: Conditioners are frequently acidic, as low pH protonates the keratin's amino acids. The hydrogen ions give the hair a positive charge and create more hydrogen bonds among the keratin scales, giving the hair a more compact structure. Organic acids such as citric acid are usually used to maintain acidity.


title: Body Snatchers (1993 film)

text: Body Snatchers is a 1993 American science fiction horror film directed by Abel Ferrara and starring Gabrielle Anwar, Billy Wirth, Terry Kinney, Meg Tilly, Christine Elise, R. Lee Ermey, and Forest Whitaker. It is loosely based on the 1955 novel "The Body Snatchers" by Jack Finney, with a screenplay by Nicholas St. John, Stuart Gordon, and Dennis Paoli.