title: Chamorro language

text: Current common Chamorro uses only the Days of the Week which are Spanish in origin but are spelled differently. There is currently an effort by Chamorro Language advocates to introduce/re-introduce native terms for the Chamorro Days of the Week.


title: Chamorro language

text: Unfortunately, both major dialects differ in the terminology used. Guamanian advocates support a number-based system derived from Old Chamorro Numerals, whereas the NMI advocates support a more unique system.


title: Chamorro language

text: Before the Spanish-based 12-month calendar became predominant, the Chamoru 13-month lunar calendar was commonly used. The first month in the left column below corresponds with January. In the right column are the Spanish-based months.


title: Chamorro language

text: Chamorro is studied at the University of Guam and in several academic institutions of Guam and the Northern Marianas.


title: Chamorro language

text: Researchers in several countries are studying aspects of Chamorro. In 2009, the Chamorro Linguistics International Network (CHIN) was established in Bremen, Germany. CHiN was founded on the occasion of the Chamorro Day (27 September 2009) which was part of the programme of the Festival of Languages. The foundation ceremony was attended by people from Germany, Guam, the Netherlands, New Zealand, Spain, Switzerland, and the United States of America.


title: Wendy Rogers (politician)

text: Wendy Rogers (born July 24, 1954) is an American far-right Republican Party politician serving as a member of the Arizona State Senate from the 6th legislative district. Elected in November 2020, she assumed office on January 11, 2021. Rogers was an officer in the United States Air Force from 1976 to 1996.


title: Wendy Rogers (politician)

text: An outspoken supporter of Donald Trump, Rogers led a successful primary challenge in 2020 against incumbent Republican state senator Sylvia Allen and defeated Democrat Felicia French in the general election. She had previously made five unsuccessful campaigns for public office.


title: Wendy Rogers (politician)

text: Since her election, Rogers has emerged as a divisive and controversial figure, embracing inflammatory rhetoric including the Great Replacement conspiracy theory and appearing on TruNews. Rogers is a member of Oath Keepers, an anti-government militia whose members took part in the 2021 U.S. Capitol attack.


title: Wendy Rogers (politician)

text: In March 2022, the Republican-controlled Arizona Senate censured her for remarks she made at the America First Political Action Conference, and weeks later opened an ethics inquiry after she suggested the 2022 Buffalo shooting was a false flag operation conducted by the federal government.


title: Wendy Rogers (politician)

text: Rogers was born in Fort Knox, Kentucky, on July 24, 1954. She holds a Bachelor of Social Work from Michigan State University, a Master of Social Work from the University of Alabama, and a Master of Science in national security studies from California State University, San Bernardino.


title: Wendy Rogers (politician)

text: Rogers served in the United States Air Force from 1976 to 1996, retiring with the rank of lieutenant colonel. One of the first 100 female pilots in the Air Force, Rogers earned her flight wings in 1981 and went on to fly the C-141 Starlifter heavy military transport aircraft and C-21 "Learjet" transport. Rogers was later stationed in Europe. She currently owns a home inspection business.


title: Wendy Rogers (politician)

text: Beginning in 2010, Rogers ran unsuccessfully for public office five times. In 2010, Rogers ran for the Arizona Senate in the 17th legislative district, losing to Democrat David Schapira in the general election. In 2012, she ran for the U.S. House of Representatives in Arizona's 9th congressional district, losing the Republican primary to Vernon Parker. In 2014, she ran again for the 9th district and won the Republican nomination, but lost to incumbent Democratic congresswoman Kyrsten Sinema. During that campaign, Rogers used footage of the beheading video of American journalist James Foley by ISIL terrorists in a campaign ad seeking to attack Sinema as weak on national security. Democrats condemned the ad as a "reprehensible" smear tactic, while Rogers's campaign defended it.


title: Wendy Rogers (politician)

text: In 2016, Rogers unsuccessfully sought the Republican nomination for Arizona's 1st congressional district; she was one of a five-person field, along with Pinal County Sheriff Paul Babeu, former state Senator and Arizona Secretary of State Ken Bennett, rancher and businessman Gary Kiehne, and Shawn Redd. During her primary campaign, Rogers was the sole candidate to support Donald Trump's proposal to build a wall on the border with Mexico. She also supported increasing the number of U.S. military personnel deployed to foreign conflicts. Rogers lost, coming in third place behind Babeu, who won the nomination, and Kiehne, the runner-up. In 2018, Rogers ran again and won the Republican nomination for the 1st congressional district, but lost to incumbent Democratic congressman Tom O'Halleran.


title: Wendy Rogers (politician)

text: In 2020, Rogers ran for the Arizona Senate in the 6th legislative district, which encompasses Rim Country and the White Mountains, and extends from Flagstaff to the Arizona–New Mexico border. Rogers unseated longtime Republican incumbent Sylvia Allen in a bitterly contested primary election, and defeated Democratic nominee Felicia French in the general election. During the campaign, Rogers made few public appearances and did not participate in debates; she also avoided taking positions on local political issues, such as forest management, education funding, or Arizona's response to the COVID-19 pandemic. Dark money organizations on both sides spent large sums to support and oppose the two candidates. Rogers raised $1 million in campaign contributions, a record for the district.


title: Wendy Rogers (politician)

text: Rogers is Vice Chairman of the Arizona Senate Judiciary Committee, a member of the Senate's Health and Human Services Committee and Natural Resources, Energy and Water Committee, and a member of the legislature's joint Blockchain and Cryptocurrency Study Committee.


title: Wendy Rogers (politician)

text: Following the 2020 United States presidential election, in which President Donald Trump was defeated by Joe Biden, Rogers promoted the false claim that Trump had won the election nationally and in Arizona. As Arizona's slate of electors met in Phoenix to formally cast the state's electoral votes for Biden, Rogers tweeted "Buy more ammo." Arizona Secretary of State Katie Hobbs condemned Rogers's statement.


title: Wendy Rogers (politician)

text: After a mob of Trump supporters attacked the U.S. Capitol in January 2021 in a failed attempt to halt the counting of the electoral votes and keep Trump in power, Rogers falsely claimed the attack had been conducted by antifa groups. Rogers was one of a number of Republican state legislators in Arizona who either defended and excused the attempted insurrection, or spread disinformation about responsibility for the attack.


title: Wendy Rogers (politician)

text: Rogers strongly supported the 2021 Maricopa County presidential ballot audit initiated by Arizona Senate Republicans to challenge Joe Biden's victory in the county and the state. She promoted the review extensively on social media, which helped her gain a large following. The results had previously been repeatedly recounted, with no discrepancies found. After the audit found no proof of election fraud, and that Biden's margin of victory was actually larger, Rogers began a campaign to audit elections in all 50 states and called for each state to "decertify its electors where it has been shown the elections were certified prematurely and inaccurately". She is one of two Arizona legislators endorsed for reelection by former President Donald Trump.


title: Wendy Rogers (politician)

text: In 2021, during the COVID-19 pandemic in Arizona, Rogers sponsored legislation to declare gun shops "essential businesses" permitted to remain open during emergencies. In February 2021, Rogers sponsored legislation seeking to rename a portion of Arizona State Route 260 as the "Donald J. Trump Highway", which State Senator Martin Quezada, a Democrat from Phoenix, described as "a desperate attempt to really pander to a base of voters even though the state of Arizona rejected Donald Trump". In the legislature, Rogers has introduced anti-abortion measures.


title: Wendy Rogers (politician)

text: In January 2022, Rogers proposed a bill that, if accepted, would make Arizona the first state to accept Bitcoin as legal tender, though doubts were raised about its compliance with the Contract Clause of the United States Constitution. Rogers subsequently introduced legislation to allow state agencies to accept cryptocurrencies in the payment of debts, and to exclude cryptocurrencies from Arizona taxes.


title: Wendy Rogers (politician)

text: Rogers successfully ran for re-election to the Arizona Senate in 7th legislative district as a result of a "last-minute change" by Arizona's Independent Redistricting Commission, which altered the district boundaries in Flagstaff, where Rogers resides. Rogers was endorsed by former president Donald Trump.


title: Wendy Rogers (politician)

text: In January 2022, it was reported that Rogers had raised a "record-breaking" $2.5 million for her re-election campaign. Rogers defeated fellow incumbent Arizona Senator Kelly Townsend in the Republican primary by an 18-point margin, receiving 59 percent of the vote to Townsend's 41 percent. Townsend initially conceded to Rogers, before threatening a lawsuit to overturn the election results.


title: Wendy Rogers (politician)

text: Rogers faced Democrat Kyle Nitschke in the general election, winning a second term with 64% of the vote.


title: Wendy Rogers (politician)

text: While running for Congress in 2018, Rogers ran ads attacking one of her Republican primary opponents, Steve Smith, calling him a "slimy character" and linking his employer, a modeling agency, to "websites linked to sex trafficking". The modeling agency and its owner sued Rogers for defamation in October 2018. In a 4–3 decision issued in February 2022, the Supreme Court ruled that Rogers's ad was protected by the First Amendment. The majority of the court ruled that allowing the case to proceed would "inevitably and intolerably chill political speech" by opening the door to lawsuits by "any third party who might indirectly be identified in a passing reference in a political advertisement"; the dissenting justices criticized the decision, writing that it "effectively weaponizes the First Amendment against innocent bystanders ensnared by often-vitriolic political campaigns, disregards well-established precedent, and is unnecessary for protecting political speech."


title: Wendy Rogers (politician)

text: In January 2021, Michael Polloni, a former legislative aide to Rogers, filed a complaint with the Arizona State Senate Ethics Committee accusing Rogers of abusive workplace conduct. In his complaint, Polloni claimed that Rogers subjected him to verbal abuse, removed and damaged his belongings, demanded that he perform campaign work on government time, and demanded that he work while on sick leave for COVID-19. An investigation by a Senate attorney found that Rogers cursed at Polloni during a heated argument, but found "little evidence" to corroborate other claims. In March 2021, the Republican-led Ethics Committee dismissed the complaint, finding no clear and convincing evidence of an ethics violation. Both Democrats on the Committee disagreed with the decision. Polloni subsequently filed a $500,000 notice of claim (a precursor to a lawsuit) against the state, alleging wrongful termination and harassment. In December 2021, Polloni filed suit in Maricopa County Superior Court, alleging wrongful termination, assault and emotional distress.


title: Wendy Rogers (politician)

text: Following the May 2022 mass shooting in Buffalo, New York, Rogers suggested on a social media site that the shooting was a false flag operation perpetrated by federal agents. The Republican-controlled Arizona Senate voted two days later by a margin of 24–3 to open an ethics investigation of Rogers over the social media comment. Rogers was one of the three Republicans who voted against the investigation.


title: Wendy Rogers (politician)

text: On March 1, 2022, Republican Arizona Senate Majority Leader Rick Gray moved to censure Rogers for "conduct unbecoming of a senator, including publicly issuing and promoting social media and video messages encouraging violence". The motion was brought in response to Rogers's February 25 speech to the America First Political Action Conference, in which she called for the public execution of political "traitors". On the Senate floor, Rogers refused to apologize or "back down" and accused Senate Republican leadership of "colluding with the Democrats". Rogers had previously threatened to "destroy the careers" of any Republican Senators who voted to censure her. The motion passed 24–3, with Rogers and two other senators voting no. Rogers's censure was the first such successful motion in three decades.


title: Wendy Rogers (politician)

text: In 2018, Rogers claimed to be a "charter member" of the Oath Keepers, a militia group known to promote conspiracy theories and violent, extremist rhetoric, including talk of a new civil war. During her 2020 campaign for the Arizona Senate, she promoted her membership in the group. After the January 6, 2021 attack on the U.S. Capitol, which resulted in criminal indictments for 26 members of the Oath Keepers, Rogers stated on Twitter: "I am a member of the Oathkeepers and I really like their dedication to our Constitution and to our country," and included a photo of her speaking to the Cottonwood chapter of the organization.


title: Wendy Rogers (politician)

text: In December 2020, as a State Senator-elect, Rogers praised Confederate general Robert E. Lee as a "great patriot and a great leader". In June 2021, Rogers appeared on the streaming channel TruNews on a show hosted by Christian nationalist commentator and politician, Lauren Witzke. TruNews, along with its founder Rick Wiles, is known for its promotion of antisemitic conspiracy theories, including a claim Trump's impeachment was orchestrated by "seditious Jews" and that Americans are "oppressed by Jewish tyrants". In July 2021, Rogers claimed that "Americans who love this country" are "being replaced [...] using mass immigration, education, big tech, big corporations" and "other strategies", echoing the Great Replacement conspiracy theory.


title: Wendy Rogers (politician)

text: In June 2022, QAnon conspiracy theorist Ron Watkins filed a formal ethics complaint against Rogers with the Arizona Senate. In support of his complaint, Watkins cited Rogers' request on the instant messaging service Telegram for the Groypers to "hit" Watkins after he alleged that Rogers had cut a "backroom deal" to prevent election equipment from being examined.


title: Agarose gel electrophoresis

text: Agarose gel electrophoresis is a method of gel electrophoresis used in biochemistry, molecular biology, genetics, and clinical chemistry to separate a mixed population of macromolecules such as DNA or proteins in a matrix of agarose, one of the two main components of agar. The proteins may be separated by charge and/or size (isoelectric focusing agarose electrophoresis is essentially size independent), and the DNA and RNA fragments by length. Biomolecules are separated by applying an electric field to move the charged molecules through an agarose matrix, and the biomolecules are separated by size in the agarose gel matrix.


title: Agarose gel electrophoresis

text: Agarose gel is easy to cast, has relatively fewer charged groups, and is particularly suitable for separating DNA of size range most often encountered in laboratories, which accounts for the popularity of its use. The separated DNA may be viewed with stain, most commonly under UV light, and the DNA fragments can be extracted from the gel with relative ease. Most agarose gels used are between 0.7–2% dissolved in a suitable electrophoresis buffer.


title: Agarose gel electrophoresis

text: Agarose gel is a three-dimensional matrix formed of helical agarose molecules in supercoiled bundles that are aggregated into three-dimensional structures with channels and pores through which biomolecules can pass. The 3-D structure is held together with hydrogen bonds and can therefore be disrupted by heating back to a liquid state. The melting temperature is different from the gelling temperature, depending on the sources, agarose gel has a gelling temperature of 35–42 °C and a melting temperature of 85–95 °C. Low-melting and low-gelling agaroses made through chemical modifications are also available.


title: Agarose gel electrophoresis

text: Agarose gel has large pore size and good gel strength, making it suitable as an anticonvection medium for the electrophoresis of DNA and large protein molecules. The pore size of a 1% gel has been estimated from 100 nm to 200–500 nm, and its gel strength allows gels as dilute as 0.15% to form a slab for gel electrophoresis. Low-concentration gels (0.1–0.2%) however are fragile and therefore hard to handle. Agarose gel has lower resolving power than polyacrylamide gel for DNA but has a greater range of separation, and is therefore used for DNA fragments of usually 50–20,000 bp in size. The limit of resolution for standard agarose gel electrophoresis is around 750 kb, but resolution of over 6 Mb is possible with pulsed field gel electrophoresis (PFGE). It can also be used to separate large proteins, and it is the preferred matrix for the gel electrophoresis of particles with effective radii larger than 5–10 nm. A 0.9% agarose gel has pores large enough for the entry of bacteriophage T4.


title: Agarose gel electrophoresis

text: The agarose polymer contains charged groups, in particular pyruvate and sulphate. These negatively charged groups create a flow of water in the opposite direction to the movement of DNA in a process called electroendosmosis (EEO), and can therefore retard the movement of DNA and cause blurring of bands. Higher concentration gels would have higher electroendosmotic flow. Low EEO agarose is therefore generally preferred for use in agarose gel electrophoresis of nucleic acids, but high EEO agarose may be used for other purposes. The lower sulphate content of low EEO agarose, particularly low-melting point (LMP) agarose, is also beneficial in cases where the DNA extracted from gel is to be used for further manipulation as the presence of contaminating sulphates may affect some subsequent procedures, such as ligation and PCR. Zero EEO agaroses however are undesirable for some applications as they may be made by adding positively charged groups and such groups can affect subsequent enzyme reactions. Electroendosmosis is a reason agarose is used in preference to agar as the agaropectin component in agar contains a significant amount of negatively charged sulphate and carboxyl groups. The removal of agaropectin in agarose substantially reduces the EEO, as well as reducing the non-specific adsorption of biomolecules to the gel matrix. However, for some applications such as the electrophoresis of serum proteins, a high EEO may be desirable, and agaropectin may be added in the gel used.


title: Agarose gel electrophoresis

text: A number of factors can affect the migration of nucleic acids: the dimension of the gel pores (gel concentration), size of DNA being electrophoresed, the voltage used, the ionic strength of the buffer, and the concentration of intercalating dye such as ethidium bromide if used during electrophoresis.


title: Agarose gel electrophoresis

text: Smaller molecules travel faster than larger molecules in gel, and double-stranded DNA moves at a rate that is inversely proportional to the logarithm of the number of base pairs. This relationship however breaks down with very large DNA fragments, and separation of very large DNA fragments requires the use of pulsed field gel electrophoresis (PFGE), which applies alternating current from different directions and the large DNA fragments are separated as they reorient themselves with the changing field.


title: Agarose gel electrophoresis

text: For standard agarose gel electrophoresis, larger molecules are resolved better using a low concentration gel while smaller molecules separate better at high concentration gel. Higher concentration gels, however, require longer run times (sometimes days).


title: Agarose gel electrophoresis

text: The movement of the DNA may be affected by the conformation of the DNA molecule, for example, supercoiled DNA usually moves faster than relaxed DNA because it is tightly coiled and hence more compact. In a normal plasmid DNA preparation, multiple forms of DNA may be present. Gel electrophoresis of the plasmids would normally show the negatively supercoiled form as the main band, while nicked DNA (open circular form) and the relaxed closed circular form appears as minor bands. The rate at which the various forms move however can change using different electrophoresis conditions, and the mobility of larger circular DNA may be more strongly affected than linear DNA by the pore size of the gel.


title: Agarose gel electrophoresis

text: Ethidium bromide which intercalates into circular DNA can change the charge, length, as well as the superhelicity of the DNA molecule, therefore its presence in gel during electrophoresis can affect its movement. For example, the positive charge of ethidium bromide can reduce the DNA movement by 15%. Agarose gel electrophoresis can be used to resolve circular DNA with different supercoiling topology.


title: Agarose gel electrophoresis

text: DNA damage due to increased cross-linking will also reduce electrophoretic DNA migration in a dose-dependent way.


title: Agarose gel electrophoresis

text: The rate of migration of the DNA is proportional to the voltage applied, i.e. the higher the voltage, the faster the DNA moves. The resolution of large DNA fragments however is lower at high voltage. The mobility of DNA may also change in an unsteady field – in a field that is periodically reversed, the mobility of DNA of a particular size may drop significantly at a particular cycling frequency. This phenomenon can result in band inversion in field inversion gel electrophoresis (FIGE), whereby larger DNA fragments move faster than smaller ones.


title: Agarose gel electrophoresis

text: The negative charge of its phosphate backbone moves the DNA towards the positively charged anode during electrophoresis. However, the migration of DNA molecules in solution, in the absence of a gel matrix, is independent of molecular weight during electrophoresis. The gel matrix is therefore responsible for the separation of DNA by size during electrophoresis, and a number of models exist to explain the mechanism of separation of biomolecules in gel matrix. A widely accepted one is the Ogston model which treats the polymer matrix as a sieve. A globular protein or a random coil DNA moves through the interconnected pores, and the movement of larger molecules is more likely to be impeded and slowed down by collisions with the gel matrix, and the molecules of different sizes can therefore be separated in this sieving process.


title: Agarose gel electrophoresis

text: The Ogston model however breaks down for large molecules whereby the pores are significantly smaller than size of the molecule. For DNA molecules of size greater than 1 kb, a reptation model (or its variants) is most commonly used. This model assumes that the DNA can crawl in a "snake-like" fashion (hence "reptation") through the pores as an elongated molecule. A biased reptation model applies at higher electric field strength, whereby the leading end of the molecule become strongly biased in the forward direction and pulls the rest of the molecule along. Real-time fluorescence microscopy of stained molecules, however, showed more subtle dynamics during electrophoresis, with the DNA showing considerable elasticity as it alternately stretching in the direction of the applied field and then contracting into a ball, or becoming hooked into a U-shape when it gets caught on the polymer fibres.


title: Agarose gel electrophoresis

text: The details of an agarose gel electrophoresis experiment may vary depending on methods, but most follow a general procedure.


title: Agarose gel electrophoresis

text: The gel is prepared by dissolving the agarose powder in an appropriate buffer, such as TAE or TBE, to be used in electrophoresis. The agarose is dispersed in the buffer before heating it to near-boiling point, but avoid boiling. The melted agarose is allowed to cool sufficiently before pouring the solution into a cast as the cast may warp or crack if the agarose solution is too hot. A comb is placed in the cast to create wells for loading sample, and the gel should be completely set before use.


title: Agarose gel electrophoresis

text: The concentration of gel affects the resolution of DNA separation. The agarose gel is composed of microscopic pores through which the molecules travel, and there is an inverse relationship between the pore size of the agarose gel and the concentration – pore size decreases as the density of agarose fibers increases. High gel concentration improves separation of smaller DNA molecules, while lowering gel concentration permits large DNA molecules to be separated. The process allows fragments ranging from 50 base pairs to several mega bases to be separated depending on the gel concentration used. The concentration is measured in weight of agarose over volume of buffer used (g/ml). For a standard agarose gel electrophoresis, a 0.8% gel gives good separation or resolution of large 5–10kb DNA fragments, while 2% gel gives good resolution for small 0.2–1kb fragments. 1% gels is often used for a standard electrophoresis. High percentage gels are often brittle and may not set evenly, while low percentage gels (0.1-0.2%) are fragile and not easy to handle. Low-melting-point (LMP) agarose gels are also more fragile than normal agarose gel. Low-melting point agarose may be used on its own or simultaneously with standard agarose for the separation and isolation of DNA. PFGE and FIGE are often done with high percentage agarose gels.


title: Agarose gel electrophoresis

text: Once the gel has set, the comb is removed, leaving wells where DNA samples can be loaded. Loading buffer is mixed with the DNA sample before the mixture is loaded into the wells. The loading buffer contains a dense compound, which may be glycerol, sucrose, or Ficoll, that raises the density of the sample so that the DNA sample may sink to the bottom of the well. If the DNA sample contains residual ethanol after its preparation, it may float out of the well. The loading buffer also includes colored dyes such as xylene cyanol and bromophenol blue used to monitor the progress of the electrophoresis. The DNA samples are loaded using a pipette.


title: Agarose gel electrophoresis

text: Agarose gel electrophoresis is most commonly done horizontally in a subaquaeous mode whereby the slab gel is completely submerged in buffer during electrophoresis. It is also possible, but less common, to perform the electrophoresis vertically, as well as horizontally with the gel raised on agarose legs using an appropriate apparatus. The buffer used in the gel is the same as the running buffer in the electrophoresis tank, which is why electrophoresis in the subaquaeous mode is possible with agarose gel.


title: Agarose gel electrophoresis

text: For optimal resolution of DNA greater than 2kb in size in standard gel electrophoresis, 5 to 8 V/cm is recommended (the distance in cm refers to the distance between electrodes, therefore this recommended voltage would be 5 to 8 multiplied by the distance between the electrodes in cm). Voltage may also be limited by the fact that it heats the gel and may cause the gel to melt if it is run at high voltage for a prolonged period, especially if the gel used is LMP agarose gel. Too high a voltage may also reduce resolution, as well as causing band streaking for large DNA molecules. Too low a voltage may lead to broadening of band for small DNA fragments due to dispersion and diffusion.


title: Agarose gel electrophoresis

text: Since DNA is not visible in natural light, the progress of the electrophoresis is monitored using colored dyes. Xylene cyanol (light blue color) comigrates large DNA fragments, while Bromophenol blue (dark blue) comigrates with the smaller fragments. Less commonly used dyes include Cresol Red and Orange G which migrate ahead of bromophenol blue. A DNA marker is also run together for the estimation of the molecular weight of the DNA fragments. Note however that the size of a circular DNA like plasmids cannot be accurately gauged using standard markers unless it has been linearized by restriction digest, alternatively a supercoiled DNA marker may be used.


title: Agarose gel electrophoresis

text: DNA as well as RNA are normally visualized by staining with ethidium bromide, which intercalates into the major grooves of the DNA and fluoresces under UV light. The intercalation depends on the concentration of DNA and thus, a band with high intensity will indicate a higher amount of DNA compared to a band of less intensity. The ethidium bromide may be added to the agarose solution before it gels, or the DNA gel may be stained later after electrophoresis. Destaining of the gel is not necessary but may produce better images. Other methods of staining are available; examples are SYBR Green, GelRed, methylene blue, brilliant cresyl blue, Nile blue sulphate, and crystal violet. SYBR Green, GelRed and other similar commercial products are sold as safer alternatives to ethidium bromide as it has been shown to be mutagenic in Ames test, although the carcinogenicity of ethidium bromide has not actually been established. SYBR Green requires the use of a blue-light transilluminator. DNA stained with crystal violet can be viewed under natural light without the use of a UV transilluminator which is an advantage, however it may not produce a strong band.


title: Agarose gel electrophoresis

text: When stained with ethidium bromide, the gel is viewed with an ultraviolet (UV) transilluminator. The UV light excites the electrons within the aromatic ring of ethidium bromide, and once they return to the ground state, light is released, making the DNA and ethidium bromide complex fluoresce. Standard transilluminators use wavelengths of 302/312-nm (UV-B), however exposure of DNA to UV radiation for as little as 45 seconds can produce damage to DNA and affect subsequent procedures, for example reducing the efficiency of transformation, "in vitro" transcription, and PCR. Exposure of DNA to UV radiation therefore should be limited. Using a higher wavelength of 365 nm (UV-A range) causes less damage to the DNA but also produces much weaker fluorescence with ethidium bromide. Where multiple wavelengths can be selected in the transilluminator, shorter wavelength can be used to capture images, while longer wavelength should be used if it is necessary to work on the gel for any extended period of time.


title: Agarose gel electrophoresis

text: The transilluminator apparatus may also contain image capture devices, such as a digital or polaroid camera, that allow an image of the gel to be taken or printed.


title: Agarose gel electrophoresis

text: The separated DNA bands are often used for further procedures, and a DNA band may be cut out of the gel as a slice, dissolved and purified. Contaminants however may affect some downstream procedures such as PCR, and low melting point agarose may be preferred in some cases as it contains fewer of the sulphates that can affect some enzymatic reactions. The gels may also be used for blotting techniques.


title: Agarose gel electrophoresis

text: In general, the ideal buffer should have good conductivity, produce less heat and have a long life. There are a number of buffers used for agarose electrophoresis; common ones for nucleic acids include Tris/Acetate/EDTA (TAE) and Tris/Borate/EDTA (TBE). The buffers used contain EDTA to inactivate many nucleases which require divalent cation for their function. The borate in TBE buffer can be problematic as borate can polymerize, and/or interact with cis diols such as those found in RNA. TAE has the lowest buffering capacity, but it provides the best resolution for larger DNA. This means a lower voltage and more time, but a better product.


title: Agarose gel electrophoresis

text: Many other buffers have been proposed, e.g. lithium borate (LB), iso electric histidine, pK matched goods buffers, etc.; in most cases the purported rationale is lower current (less heat) and or matched ion mobilities, which leads to longer buffer life. Tris-phosphate buffer has high buffering capacity but cannot be used if DNA extracted is to be used in phosphate sensitive reaction. LB is relatively new and is ineffective in resolving fragments larger than 5 kbp; However, with its low conductivity, a much higher voltage could be used (up to 35 V/cm), which means a shorter analysis time for routine electrophoresis. As low as one base pair size difference could be resolved in 3% agarose gel with an extremely low conductivity medium (1 mM lithium borate).


title: Agarose gel electrophoresis

text: Other buffering system may be used in specific applications, for example, barbituric acid-sodium barbiturate or Tris-barbiturate buffers may be used for in agarose gel electrophoresis of proteins, for example in the detection of abnormal distribution of proteins.


title: Agarose gel electrophoresis

text: Agarose gels are easily cast and handled compared to other matrices and nucleic acids are not chemically altered during electrophoresis. Samples are also easily recovered. After the experiment is finished, the resulting gel can be stored in a plastic bag in a refrigerator.


title: Agarose gel electrophoresis

text: Electrophoresis is performed in buffer solutions to reduce pH changes due to the electric field, which is important because the charge of DNA and RNA depends on pH, but running for too long can exhaust the buffering capacity of the solution. Further, different preparations of genetic material may not migrate consistently with each other, for morphological or other reasons.


title: Agarose gel electrophoresis

text: In the mid- to late 1960s, agarose and related gels were first found to be effective matrices for DNA and RNA electrophoresis.


title: Singapore Armed Forces ranks

text: The military ranks of Singapore are the military insignia used by the Singapore Armed Forces (SAF). The SAF has five rank schemes for active and reservist personnel, with a sixth for the auxiliaries of the SAF Volunteer Corps. The SAF has a unique rank structure as an integrated force, ranks are the same in the Singapore Army, the Republic of Singapore Navy (RSN), the Republic of Singapore Air Force (RSAF), and the Digital and Intelligence Service (DIS).


title: Singapore Armed Forces ranks

text: Like many Commonwealth countries, the SAF draws its heritage from the British military, however Singapore no longer uses British-style rank insignia across its armed services and also does not use the corresponding Royal Navy and Royal Air Force rank titles for its navy and air force. The three branches of the SAF was unified in 1972, but prior to 1982, the three component services maintained their own distinct sets of uniform insignia, particularly for officer-level ranks. In 1982, the SAF implemented identical rank insignia across all three services along with a new service dress uniform. Rank titles are also largely identical, with only two exceptions in the Republic of Singapore Navy, where general officers have admiral ranks while officer cadets are referred to as midshipmen.


title: Singapore Armed Forces ranks

text: The ranks of uniformed personnel are outlined in the "Singapore Armed Forces (Ranks of Servicemen) Regulations". While the legislation provides for the ranks of general and admiral, there have not been any holders of these ranks , and their rank insignia are not known. As below is a list of ranks prescribed by legislation, arranged in order of increasing seniority.


title: Singapore Armed Forces ranks

text: Enlistees make up the bulk of manpower in the SAF and comprise the majority of full-time national servicemen. All new soldiers in the SAF, also known as enlistees, must first undergo Basic Military Training (BMT), during which they hold the most basic rank of recruit (REC) until they successfully complete their training. Once recruits graduate from their BMT they attain the rank of private (PTE). Selected recruits may also be chosen for further leadership training at the Specialist Cadet School or Officer Cadet School after graduating from BMT to become future specialists or officers respectively. Conscripts who do not complete BMT will continue to hold the rank of recruit and may attain the rank of private only at the end of their 2 years of active service.


title: Singapore Armed Forces ranks

text: Recruits and privates of all armed services do not wear any rank insignia. Privates can be further promoted to the ranks of private (first class) (PFC), lance corporal (LCP), corporal (CPL), and corporal (first class) (CFC), if they perform well.


title: Singapore Armed Forces ranks

text: Specialists serve as the junior commanders in the Singapore Armed Forces. They are specifically trained in a variety of equipment and skills and are considered "subject matter experts". They take responsibility for leading and training small units and work together with commissioned officers. The specialist corps (as well as the warrant officer corps) were introduced in 1992 to replace the previous non-commissioned officer corps.


title: Singapore Armed Forces ranks

text: Potential specialists are selected while training as recruits during BMT and, following graduation, undergo further training at the Specialist Cadet School (SCS). Since 22 December 2008, all specialist trainees bear the rank of specialist cadet (SCT) while undergoing specialist training. This rank is considered equivalent to a private, and is worn until they graduate from SCS, upon which they attain the rank of third sergeant (3SG). Servicemen may subsequently be promoted to second sergeant (2SG) and first sergeant (1SG). Regular personnel may progress further to the staff-level ranks of staff sergeant (SSG) or master sergeant (MSG). The master sergeant is the pinnacle rank of the specialist corps.


title: Singapore Armed Forces ranks

text: Specialists in general are addressed as "Sergeant", while staff sergeants are addressed as "Staff" and master sergeants as "Master".


title: Singapore Armed Forces ranks

text: The rank insignia of specialists consists of three downward-pointing chevrons for the lowest rank of third sergeant, with additional upward-pointing chevrons depending on their seniority. In addition, staff-level ranks contain a coat of arms of Singapore between the upward-pointing and downward-pointing chevrons.


title: Singapore Armed Forces ranks

text: Warrant officers serve as senior mentors and disciplinarians in units as well as training institutes. Warrant officers are appointed by the Armed Forces Council (AFC), and may be given command responsibility of units and serve as disciplinary or investigating officers for military offences. In addition, they may only be charged for military offences by superior commanders. Like officers, they may also carry ceremonial swords on parades, and wear their ranks on their shoulder epaulettes.


title: Singapore Armed Forces ranks

text: Warrant officers are typically promoted from the ranks of specialists and generally have more than ten years of service, although outstanding specialists can attain the rank of third warrant officer as soon as seven years into service. Senior specialists are trained at the SAFWOS Leadership School (SAFWOS) before becoming warrant officers, and selected operationally-ready national servicemen may also be selected for warrant officer rank.


title: Singapore Armed Forces ranks

text: While technically ranking below commissioned officers, warrant officers are addressed by junior commissioned officers as "" for males or "" for females, in respect of their experience and knowledge.


title: Singapore Armed Forces ranks

text: The rank insignia of a third warrant officer (3WO) consists of a coat of arms of Singapore with an arc below and a thin upward-pointing chevron, while that of a second warrant officer (2WO) through to senior warrant officer (SWO) have a thicker chevron which vary in number according to the rank. Finally, the rank of chief warrant officer (CWO) is distinguished from the rank of senior warrant officer with the addition of a laurel.


title: Singapore Armed Forces ranks

text: Officers exercise leadership and command authority in the SAF. Prospective officers are selected from trainees in BMT and SCS, and undergo training at the Officer Cadet School (OCS). While training to become an officer, they hold the rank of officer cadet or midshipman, which are considered equivalent in rank to a private. Prospective officers are required to hold A-Level, International Baccalaureate, polytechnic diploma or equivalent tertiary qualifications, and career officers are encouraged to obtain higher studies in universities. They are appointed by the President of Singapore.


title: Singapore Armed Forces ranks

text: Junior officers are in charge of sub-units, and begin at the rank of second lieutenant (2LT), and may be promoted to the ranks of lieutenant (LTA). Regular and reservist personnel may be further promoted to the rank of captain (CPT) after attending courses at the SAF Advanced Schools. The rank insignia of junior officers are depicted by bars.


title: Singapore Armed Forces ranks

text: Senior officers are typically placed in charge of units or higher branches and responsible for making command decisions. They begin duties at the rank of major (MAJ), and may progress to lieutenant colonel (LTC), senior lieutenant colonel (SLTC), or colonel (COL). The rank insignia for senior officers uses the coat of arms of Singapore, also informally known as "crabs".


title: Singapore Armed Forces ranks

text: The general officers are the highest-ranking officers in the SAF, and are responsible for broad strategy and policies which affect the entire armed forces. General officers of the Army and Air Force use general-based ranks, while those of the Navy use admiral-based ranks. Their rank insignia consists of stars, ranging from one for brigadier generals (BG) and rear admiral (one-star) (RADM(1)), two for major generals (MG) and rear admiral (two-star) (RADM(2)), and three for lieutenant generals (LG) and vice admirals (VADM). Finally, the ranks of general and admiral are also provided for in the Singapore Armed Forces (Ranks of Servicemen) Regulations.


title: Singapore Armed Forces ranks

text: Military experts are soldiers that serve in specific military domains under the Military Domain Experts Scheme (MDES), which is a separate rank scheme that was introduced in April 2010. The rank structure runs in parallel with the conventional ranks of enlistees, specialists, warrant officers, and commissioned officers, and allows for continual advancement from ME1 to higher ranks all the way to ME9 based upon expertise and performance levels.


title: Singapore Armed Forces ranks

text: Military experts of Military Expert 3 (ME3) rank are considered equivalent to warrant officers; while those ranking Military Expert 4 (ME4) or higher (collectively referred to as "senior military experts") are considered equivalent to commissioned officers and given the same legal authority. The rank titles do not have unique names, but are instead suffixed by a cardinal number from 1 to 9 (for example, Military Expert 3 or ME3), with 1 indicating the most junior rank and 9 the most senior. Each of the ranks may be further qualified by another suffix: this may be a letter to denote military experts in different stages training, such as T for trainees (for example, Military Expert 4 Trainee or ME4T) or A for apprentices (for example, Military Expert 4 Apprentice or ME4A); or a number to distinguish paygrades for fully qualified military experts (for example, Military Expert 4–2 or ME4-2). Each of the rank has two or three paygrades except for the ranks of ME7, ME8, and ME9, where there is only one paygrade and no suffix is used.


title: Singapore Armed Forces ranks

text: Military expert trainees are considered equivalent to the rank of private, while military expert apprentices are considered equivalent to the rank of Military Expert 1.


title: Singapore Armed Forces ranks

text: The SAF Volunteer Corps (SAFVC) was established in October 2014 to allow female Singapore citizens, first generation permanent residents and naturalised citizens, all of whom would not ordinarily have any national service obligation, to serve as volunteers in the SAF. The SAFVC Volunteers (SV) have a unique rank structure that does not correlate with the other ranks in the SAF.


title: Singapore Armed Forces ranks

text: The SAFVC ranks comprise five tiers from SAFVC Volunteer (Trainee), abbreviated as SV (Trainee), to SAFVC Volunteer 4 (SV4), enumerated by winged chevrons.


title: Singapore Armed Forces ranks

text: Where the rank insignia is worn depends on a number of factors, including the uniform type, rank scheme, gender, and service of the serviceman.


title: Singapore Armed Forces ranks

text: When wearing the No.4 uniform (combat uniform), all servicemen wear their rank insignia on a chest strap. Otherwise, when wearing other uniforms, the rank insignia of enlistees and specialists are generally worn on the sleeves, while that of warrant officers, officers, and military experts are worn on shoulder epaulettes. In addition, female warrant officers, officers, and military experts wear their rank insignia on the collars when wearing the No.3 or No.5 uniforms (service dress uniforms), except for those of Army, who may wear their insignia on the shoulders if they are wearing a suit with their No.5 uniform.


title: Singapore Armed Forces ranks

text: Military ranks are sometimes suffixed with an abbreviation to denote certain circumstances under which the personnel holds the rank.


title: Root of unity

text: In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.


title: Root of unity

text: Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field.


title: Root of unity

text: Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if is even, which are complex with a zero imaginary part), and in this case, the th roots of unity are


title: Root of unity

text: However, the defining equation of roots of unity is meaningful over any field (and even over any ring) , and this allows considering roots of unity in . Whichever is the field , the roots of unity in are either complex numbers, if the characteristic of is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo "n" and Finite field for further details.


title: Root of unity

text: In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers.


title: Root of unity

text: Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of modular integers, see Root of unity modulo "n".


title: Root of unity

text: Every th root of unity is a primitive th root of unity for some , which is the smallest positive integer such that .


title: Root of unity

text: This is also true for negative exponents. In particular, the reciprocal of an th root of unity is its complex conjugate, and is also an th root of unity:


title: Root of unity

text: If is an th root of unity and then . Indeed, by the definition of congruence modulo "n", for some integer , and hence


title: Root of unity

text: Let be a primitive th root of unity. Then the powers , , ..., , are th roots of unity and are all distinct. (If where , then , which would imply that would not be primitive.) This implies that , , ..., , are all of the th roots of unity, since an th-degree polynomial equation over a field (in this case the field of complex numbers) has at most solutions.


title: Root of unity

text: From the preceding, it follows that, if is a primitive th root of unity, then formula_7 if and only if formula_8


title: Root of unity

text: If is not primitive then formula_9 implies formula_10 but the converse may be false, as shown by the following example. If , a non-primitive th root of unity is , and one has formula_11, although formula_12


title: Root of unity

text: where formula_14 is the greatest common divisor of and . This results from the fact that is the smallest multiple of that is also a multiple of . In other words, is the least common multiple of and . Thus


title: Root of unity

text: Thus, if and are coprime, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity (where is Euler's totient function). This implies that if is a prime number, all the roots except are primitive.


title: Root of unity

text: In other words, if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the :


title: Root of unity

text: The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if and , then , and , where is the least common multiple of and .


title: Root of unity

text: Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.


title: Root of unity

text: For an integer "n", the product and the multiplicative inverse of two th roots of unity are also th roots of unity. Therefore, the th roots of unity form an abelian group under multiplication.


title: Root of unity

text: Given a primitive th root of unity , the other th roots are powers of . This means that the group of the th roots of unity is a cyclic group. It is worth remarking that the term of "cyclic group" originated from the fact that this group is a subgroup of the circle group.


title: Root of unity

text: Let formula_18 be the field extension of the rational numbers generated over formula_19 by a primitive th root of unity . As every th root of unity is a power of , the field formula_18 contains all th roots of unity, and formula_18 is a Galois extension of formula_22


title: Root of unity

text: If is an integer, is a primitive th root of unity if and only if and are coprime. In this case, the map


title: Root of unity

text: induces an automorphism of formula_18, which maps every th root of unity to its th power. Every automorphism of formula_18 is obtained in this way, and these automorphisms form the Galois group of formula_18 over the field of the rationals.


title: Root of unity

text: The rules of exponentiation imply that the composition of two such automorphisms is obtained by multiplying the exponents. It follows that the map


title: Root of unity

text: defines a group isomorphism between the units of the ring of integers modulo and the Galois group of formula_28


title: Root of unity

text: This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.


title: Root of unity

text: This formula shows that in the complex plane the th roots of unity are at the vertices of a regular -sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for and on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).


title: Root of unity

text: It follows from the discussion in the previous section that this is a primitive th-root if and only if the fraction is in lowest terms; that is, that and are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as formula_35, is called a trigonometric number.


title: Root of unity

text: The th roots of unity are, by definition, the roots of the polynomial , and are thus algebraic numbers. As this polynomial is not irreducible (except for ), the primitive th roots of unity are roots of an irreducible polynomial of lower degree, called the th cyclotomic polynomial, and often denoted . The degree of is given by Euler's totient function, which counts (among other things) the number of primitive th roots of unity. The roots of are exactly the primitive th roots of unity.


title: Root of unity

text: Galois theory can be used to show that cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form formula_36 is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer , there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ( possible values for a th root). (For more details see , below.)


title: Root of unity

text: Gauss proved that a primitive th root of unity can be expressed using only square roots, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the regular -gon. This is the case if and only if is either a power of two or the product of a power of two and Fermat primes that are all different.


title: Root of unity

text: If is a primitive th root of unity, the same is true for , and formula_37 is twice the real part of . In other words, is a reciprocal polynomial, the polynomial formula_38 that has as a root may be deduced from by the standard manipulation on reciprocal polynomials, and the primitive th roots of unity may be deduced from the roots of formula_38 by solving the quadratic equation formula_40 That is, the real part of the primitive root is formula_41 and its imaginary part is formula_42


title: Root of unity

text: The polynomial formula_38 is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular -gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the casus irreducibilis, that is, every expression of the roots in terms of radicals involves "nonreal radicals".


title: Root of unity

text: for are all -periodic (because ). Furthermore, the set } of these sequences is a basis of the linear space of all -periodic sequences. This means that "any" -periodic sequence of complex numbers


title: Root of unity

text: This is a form of Fourier analysis. If is a (discrete) time variable, then is a frequency and is a complex amplitude.


title: Root of unity

text: This is an immediate consequence of Vieta's formulas. In fact, the th roots of unity being the roots of the polynomial , their sum is the coefficient of degree , which is either 1 or 0 according whether or .


title: Root of unity

text: Alternatively, for there is nothing to prove, and for there exists a root – since the set of all the th roots of unity is a group, , so the sum satisfies , whence .


title: Root of unity

text: In the section Elementary properties, it was shown that if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the :


title: Root of unity

text: This is the special case of Ramanujan's sum , defined as the sum of the th powers of the primitive th roots of unity:


title: Root of unity

text: defines a discrete Fourier transform. Computing the inverse transformation using Gaussian elimination requires operations. However, it follows from the orthogonality that is unitary. That is,


title: Root of unity

text: and thus the inverse of is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation). The straightforward application of or its inverse to a given vector requires operations. The fast Fourier transform algorithms reduces the number of operations further to .


title: Root of unity

text: are precisely the th roots of unity, each with multiplicity 1. The th "cyclotomic polynomial" is defined by the fact that its zeros are precisely the "primitive" th roots of unity, each with multiplicity 1.


title: Root of unity

text: where are the primitive th roots of unity, and is Euler's totient function. The polynomial has integer coefficients and is an irreducible polynomial over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime , which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial


title: Root of unity

text: Every th root of unity is a primitive th root of unity for exactly one positive divisor of . This implies that


title: Root of unity

text: Substituting any positive integer ≥ 2 for , this sum becomes a base repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.


title: Root of unity

text: Note that, contrary to first appearances, "not" all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is . It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on as on how many odd prime factors appear in . More precisely, it can be shown that if has 1 or 2 odd prime factors (for example, ) then the th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is . This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if formula_70 where formula_71 are odd primes, formula_72 and "t" is odd, then occurs as a coefficient in the th cyclotomic polynomial.


title: Root of unity

text: Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if is prime, then if and only .


title: Root of unity

text: Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for th roots of unity with the additional property that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive th root of unity. This was already shown by Gauss in 1797. Efficient algorithms exist for calculating such expressions.


title: Root of unity

text: The th roots of unity form under multiplication a cyclic group of order , and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive th root of unity.


title: Root of unity

text: The th roots of unity form an irreducible representation of any cyclic group of order . The orthogonality relationship also follows from group-theoretic principles as described in Character group.


title: Root of unity

text: The roots of unity appear as entries of the eigenvectors of any circulant matrix; that is, matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem. In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.


title: Root of unity

text: By adjoining a primitive th root of unity to formula_73 one obtains the th cyclotomic field formula_74This field contains all th roots of unity and is the splitting field of the th cyclotomic polynomial over formula_22 The field extension formula_76 has degree φ("n") and its Galois group is naturally isomorphic to the multiplicative group of units of the ring formula_77


title: Root of unity

text: As the Galois group of formula_76 is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every "n"th root of unity may be expressed in term of "k"-roots, with various "k" not exceeding φ("n"). In these cases Galois theory can be written out explicitly in terms of Gaussian periods: this theory from the "Disquisitiones Arithmeticae" of Gauss was published many years before Galois.


title: Root of unity

text: Conversely, "every" abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the "Kronecker–Weber theorem" on the grounds that Weber completed the proof.


title: Root of unity

text: For four other values of , the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an th root of unity) is a quadratic integer.


title: Root of unity

text: For , none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z[] (). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.


title: Wichita Massacre

text: The Wichita Massacre, also known as the Wichita Horror, was a week-long series of random brutal crimes perpetrated by brothers Reginald and Jonathan Carr in the city of Wichita, Kansas between December 8 and 15, 2000. Five people were killed, and two people, a man and a woman, were severely wounded. The brothers were arrested and convicted of multiple counts of murder, kidnapping, robbery, and rape. They were both sentenced to death in October 2002. Their vicious crimes created panic in the Wichita area resulting in an increase in the sales of guns, locks, and home security systems.


title: Wichita Massacre

text: The case has received significant attention because the killers' death sentences have been subject to various rulings related to the use of executions in Kansas. In 2004, the Kansas Supreme Court overturned the state's death penalty law, but the Kansas Attorney General appealed to the US Supreme Court, which upheld the constitutionality of the death penalty in Kansas. On July 25, 2014, the Kansas Supreme Court again overturned the Carrs' death sentences on a legal technicality relating to their original trial judge not giving each brother a separate penalty proceeding. After an appeal by the state's attorney general to the US Supreme Court, it overturned the decision of the Kansas Supreme Court in January 2016 and reinstated the death sentences. The Carr brothers are incarcerated on death row at El Dorado Correctional Facility.


title: Wichita Massacre

text: The Carr brothers were from Dodge City. Both 22-year-old Reginald and 20-year-old Jonathan had lengthy criminal records.


title: Wichita Massacre

text: On December 8, 2000, having recently arrived in Wichita, the brothers robbed and wounded 23-year-old Andrew Schreiber, an assistant baseball coach. Three days later, on December 11, they shot 55-year-old cellist and librarian Ann Walenta as she tried to escape from them in her car. She died three days later in hospital from her wounds.


title: Wichita Massacre

text: On December 14, the brothers broke into a house at 12727 E Birchwood Drive in Wichita. Inside the property, which they had chosen at random, were Brad Heyka, Heather Muller, Aaron Sander, Jason Befort, and his girlfriend, a young woman identified as "Holly G.". Befort was a local high school teacher; Heyka was a director of finance with a local financial services company; Muller was a local preschool teacher; Sander a former financial analyst who had been studying to become a priest; Holly G. was a teacher.


title: Wichita Massacre

text: First, the Carrs searched the house for valuables. Befort had intended to propose to Holly, and she found this out when the Carrs discovered an engagement ring hidden in a popcorn box. After burgling the house, the Carrs forced their victims to strip naked and bound them. The brothers then repeatedly raped the two women and forced the men to engage in sexual acts with the women and the women with each other.


title: Wichita Massacre

text: After taking the victims in Befort's truck to ATMs to empty their bank accounts, they drove them to the closed Stryker Soccer Complex on the outskirts of Wichita, where they shot all five execution-style in the back of their heads. The Carrs then drove Befort's truck over their bodies and left them for dead. Holly G. survived because her plastic barrette deflected the bullet to the side of her head. She walked naked for more than a mile in freezing weather to seek first aid and shelter at a house. Before getting medical treatment, she reported the incident and descriptions of her attackers to the couple who took her in before the police arrived.


title: Wichita Massacre

text: After the killings, the Carrs returned to the house to ransack it for more valuables; while there, they used a golf club to beat Holly's pet dog, Nikki, to death.


title: Wichita Massacre

text: Police captured both Carr brothers the next day. Both Schreiber and the dying Walenta identified Reginald. The District Attorney said that they believed their motive was robbery based on evidence.


title: Wichita Massacre

text: With the help of Holly's testimony at the trial, both brothers were convicted of nearly all 113 counts against them, including kidnapping, robbery, rape, four counts of capital murder, and one count of first-degree murder. Reginald Carr was convicted of 50 counts and Jonathan Carr of 43. They were each sentenced to death for the capital murders and life in prison, with decades to serve before being eligible for parole. Their cases were appealed.


title: Wichita Massacre

text: There has been continuing attention for the Carr brothers' case because of various rulings about the Kansas death penalty law and decisions by its high court on such cases. In 2004, the Kansas Supreme Court overturned the state's death penalty law, but the state attorney general appealed this ruling to the U.S. Supreme Court. It upheld the constitutionality of the state's death penalty law, which returned the Carrs and other condemned killers to death row.


title: Wichita Massacre

text: On July 25, 2014, the Kansas Supreme Court announced it had overturned the Carr death sentences on appeal. The six justice-majority said they did so because the trial judge failed to adequately separate the penalty proceedings for each defendant. According to a release from the Kansas Supreme Court public information officer, the court unanimously reversed three of each defendant's four capital convictions because jury instructions on sex-crime-based capital murder were "fatally erroneous and three of the multiple-homicide capital murder charges duplicated the first."


title: Wichita Massacre

text: The high court upheld most of the convictions against each of the brothers despite other purported lower-court errors. The court ruled that the brothers were entitled to separate sentencing trials, as "differentiation in the moral culpability of two defendants" can cause a jury "to show mercy to one while refusing to show mercy to the other." Even if the death penalties were not upheld, each of the Carrs was already sentenced to serve at least "70 to 80 years" in prison before being eligible for parole.


title: Wichita Massacre

text: The Kansas Attorney General appealed the high court's ruling to the US Supreme Court, which in March 2015 agreed to hear the Carr brothers' sentencing case, together with another death-penalty case from the state. In January 2016, the United States Supreme Court (in an 8–1 ruling) reinstated the death sentences, overturning the Kansas Supreme Court, deciding that neither the jury instructions which were challenged by the Carrs' legal counsel, nor the combined sentencing proceedings, violated the Constitution. The Kansas Supreme Court affirmed the death penalty for the brothers on January 21, 2022.


title: Wichita Massacre

text: The victims were white and the Carr brothers are black, but the District Attorney held that there was no "prima facie" evidence of a racial crime. Based on the robberies, Sedgwick County District Attorney Nola Foulston decided against treating these incidents as hate crimes. Conservative media commentators David Horowitz, Michelle Malkin, and Thomas Sowell said the crimes were downplayed; they felt the national media suppressed the stories due to political correctness. Sowell, an African-American conservative, has said the media has a double standard regarding interracial offenses, tending to play up "vicious crimes by whites against blacks" but play down "equally vicious crimes by blacks against whites".


title: Wichita Massacre

text: Years later, "The Wichita Eagle" commented that the deaths of four young black people who were murdered by a young black man eight days before the "Wichita Massacre" in 2000 received less general media coverage than the murders committed by the Carr brothers. In this case, Cornelius Oliver, 19, killed his girlfriend, Raeshawnda Wheaton, 18, at her house, as well as her roommate Dessa Ford, and friends Jermaine Levy and Quincy Williams, who were visiting the women. Oliver shot the men in the back of the head where they sat on a couch. The couple had been known to have a "volatile, violent relationship". The police arrested Oliver that day; he still had blood on his shoes.


title: Wichita Massacre

text: Some members of the black community questioned why the murders of the four young black people was superseded by attention given to the Carr brothers' killing of four young white people. A relative of Wheaton asked, "How could one be any worse than the other, if the results [multiple deaths] were the same?" The deaths of Wheaton and her friends were characterized by Oliver having had a personal relationship with at least one of his victims, unlike the Carrs who chose their victims at random. In addition, the Carrs committed rapes, as well as other assaults and abuses, to the victims. There was no prolonged torture or sexual component in the Wheaton case. Oliver was convicted of the four murders and is serving a life sentence in prison, with no possibility of parole before 2140.


title: Scooby-Doo! and the Witch's Ghost

text: Scooby-Doo! and the Witch's Ghost is a 1999 American direct-to-video animated supernatural horror comedy film, and the second of the direct-to-video films based upon Hanna-Barbera's "Scooby-Doo" Saturday morning cartoons. It was produced by Hanna-Barbera Cartoons. The film was released on VHS on October 5, 1999, then on DVD on March 6, 2001.


title: Scooby-Doo! and the Witch's Ghost

text: The plot involves Mystery Inc. travelling to a New England town called Oakhaven (a fictional suburb of Salem, Massachusetts) after being invited by horror writer Ben Ravencroft. Like a number of direct-to-video "Scooby-Doo" animated films released in the late-1990s and early-2000s, "The Witch's Ghost" features real supernatural elements instead of the traditionally fabricated ones the franchise is associated with, giving the film a darker tone. The film has been adapted into a book.


title: Scooby-Doo! and the Witch's Ghost

text: It is the second of the first four "Scooby-Doo" direct-to-video films to be animated overseas by Japanese animation studio Mook Animation. The film marks the first time voice actor and radio-personality Scott Innes voiced Shaggy, as Billy West (who voiced Shaggy in "Scooby-Doo on Zombie Island") needed time for his voice work on "Futurama". This was also the final film starring Mary Kay Bergman that was released during her lifetime.


title: Scooby-Doo! and the Witch's Ghost

text: After Ben Ravencroft, a famous horror writer of whom Velma Dinkley is a huge fan, assists her and Mystery Inc. in solving a case at a museum, he invites them to his hometown, Oakhaven, Massachusetts. When they arrive, they find the town converted into a tourist attraction by Mayor Corey, with 17th-century replicas based on the ghost of Sarah Ravencroft, an ancestor of Ben's who was persecuted as a witch and executed by the Puritan townspeople in 1657. Ben disputes this, claiming that Sarah was a Wiccan who used herbal remedies to heal the less fortunate, and he has spent years searching for her medical journal to prove her innocence.


title: Scooby-Doo! and the Witch's Ghost

text: Scooby-Doo and Shaggy Rogers are chased by the ghost of a witch. The gang is drawn to an all-female gothic rock band, The Hex Girls, led by Sally "Thorn" McKnight, during one of their rehearsals. Fred Jones and Daphne Blake follow Thorn and discover her performing a ritual and are convinced the Hex Girls are witches.


title: Scooby-Doo! and the Witch's Ghost

text: The ghost witch is captured by Velma and revealed to be Mr. McKnight, Thorn's father and Oakhaven's pharmacist, and the townspeople were involved. Thorn explains the "ritual" Fred and Daphne witnessed was an herbal remedy made for soothing her vocal cords and that she is actually 1/16th Wiccan. Corey and Mr. McKnight apologize to Ben for using his ancestor in their publicity stunt, explaining that they created the ghost witch to boost the town's failing tourist economy and that they found inspiration from digging up the head marker for Sarah's grave, though her body was not found. It is revealed that a shoe buckle Scooby had found earlier was actually the lock from Sarah's journal.


title: Scooby-Doo! and the Witch's Ghost

text: Returning to where the lock had been found, Scooby digs and finds a box containing the buried journal, which is actually a spell book. Ben reveals that Sarah was in fact a witch, who wielded her witchcraft against the townspeople before the Wiccans used their nature-based powers to imprison her within her own spell book; his ancestry, therefore, makes him a warlock. He engineered the mystery at the museum just so he could meet with Mystery Inc., knowing they would be able help him find the book; the town's fake ghost wasn't part of his plan, though he was able to use it to his advantage. Awakening his warlock powers, Ben releases Sarah but discovers that she has no loyalty to him, and her ambitions are to destroy the world to avenge her imprisonment rather than rule it alongside him.


title: Scooby-Doo! and the Witch's Ghost

text: Disillusioned, Ben attempts to reimprison Sara, but she tells him that only a Wiccan can defeat her, and traps him in a magical sphere. The gang launches an attempt to get the book while Sarah turns pumpkins and trees into monsters and enlarges a turkey in order to stop them. Daphne and Velma free the Hex Girls and the latter convinces Thorn to use her inherited Wiccan power to reimprison Sarah. The plan works, sucking her back into the book and turning the monsters she created, except for the turkey, back to normal. Refusing to return to her imprisonment alone, Sarah drags Ben into the book with her.


title: Scooby-Doo! and the Witch's Ghost

text: A burning branch then falls onto the book and incinerates it, ensuring that the Ravencrofts can never return. The gang and townsfolk celebrate their deliverance with a concert from the Hex Girls with the gang and the still giant turkey joining in on the performance.


title: Scooby-Doo! and the Witch's Ghost

text: After the success of "Scooby-Doo on Zombie Island", which received better sales than Warner Bros. had expected, the team were tasked with creating a second "Scooby-Doo" direct-to-video film. Its predecessor was considered a one-off experiment and, as such, the crew producing it worked with little oversight from executives. For "Witch's Ghost", this creative freedom was scaled back considerably. Warner Bros. suggested screenwriters Rick Copp and David A. Goodman, which insulted the team that had produced the first film in total autonomy. In addition, the studio requested the filmmakers "tone down" their content, as they feared "Zombie Island" had proved too scary for its intended audience.


title: Scooby-Doo! and the Witch's Ghost

text: Copp and Goodman's script concluded with the revelation that the townspeople were using the witch as a publicity stunt. The original team found this unsatisfactory and Glenn Leopold re-wrote the last third of the film, introducing the concept that the ghost is real.


title: Scooby-Doo! and the Witch's Ghost

text: To coincide with the release of Scooby-Doo on Zombie Island, Warner Bros. decided to release the album . It went on to peak at number 5 on Billboard's Kid Albums chart and stayed in the top 25 for over 26-weeks.


title: Scooby-Doo! and the Witch's Ghost

text: Kid Rhino partnered with Warner Home Video and Cartoon Network to release the soundtrack for the film. According to Rhino VP Carol Lee, "We [worked] closely with Warner Home Video so that we're part of everything they do." She added the soundtrack to the film was, "treated like that of a theatrical release. We created a Music Video which appeared on the home video."


title: Scooby-Doo! and the Witch's Ghost

text: On September 14, 1999, the soundtrack was released on CD and Audio Cassette, featuring songs by The Hex Girls, and Billy Ray Cyrus performing "Scooby-Doo, Where Are You?".


title: Scooby-Doo! and the Witch's Ghost

text: "Scooby-Doo and the Witch's Ghost" was released on VHS on October 5, 1999, then on DVD on March 6, 2001.


title: Scooby-Doo! and the Witch's Ghost

text: The film earned a 50% approval rating on Rotten Tomatoes. David Parkinson of "Radio Times", gave the film two out of five stars, saying, "This full-length cartoon featuring the ghost-hunting teenage detectives is something of a mixed bag." Joe Neumaier from Entertainment Weekly said, "Though slyly written, it doesn't have the punch of last year's Scooby-Doo on Zombie Island – but it's still scarier than The Blair Witch Project."


title: Scooby-Doo! and the Witch's Ghost

text: The film was criticized by religious groups upon its release, who claimed The Hex Girls were "of the Devil, luring young girls into Wicca witchcraft."


title: Cesare Maldini

text: Cesare Maldini (; 5 February 1932 – 3 April 2016) was an Italian professional football manager and player who played as a defender.


title: Cesare Maldini

text: Father to Paolo Maldini and grandfather to Daniel Maldini, Cesare began his career with Italian side Triestina, before transferring to AC Milan in 1954, whom he captained to win four Serie A league titles and one European Cup during his twelve seasons with the club. He retired in 1967, after a season with Torino. Internationally, he played for Italy, earning 14 caps and participating in the 1962 World Cup. He served as team captain for both Milan and Italy.


title: Cesare Maldini

text: As a manager, he also coached his former club Milan on two occasions, as well as Italian sides Foggia, Ternana and Parma. He had a successful career in charge of the Italy under-21 side, winning the European Under-21 Championship a record three consecutive times; he later also coached the Italy senior team at the 1998 FIFA World Cup, and the Paraguay national football team at the 2002 FIFA World Cup.


title: Cesare Maldini

text: The son of Albino Maldini, a sailor, and Maria Vodeb, Cesare Maldini was born in Trieste, Friuli-Venezia Giulia, Italy. His family was of Slovenian descent. He married Maria Luisa (Marisa; née Mazzucchelli, 1939–2016) in 1962; together they had six children: three sons and three daughters. One of his sons, Paolo, also had a successful football career as a defender with Milan, and also once held the record for the most caps for the Italy national team (now third behind Gianluigi Buffon and Fabio Cannavaro). The younger Maldini captained Milan to the UEFA Champions League title in 2003 and 2007, and won the trophy five times in total. Maldini's grandsons Christian and Daniel played football in the AC Milan youth teams.


title: Cesare Maldini

text: Maldini began his playing career with local side Triestina, in 1952, and made his Serie A debut in his first season with the club, on 24 May 1953, in a 0–0 away draw against Palermo. After two seasons with Triestina, Maldini transferred to AC Milan in 1954, where he went on to achieve notable successes both domestically and internationally in the team's starting line-up, also becoming an important figure at the club. He made his debut with the club on 19 September 1954, in a 4–0 league win over his former side, featuring in a Milan team which included several important players at the time, such as Lorenzo Buffon, Francesco Zagatti, Nils Liedholm, Gunnar Nordahl, and Juan Alberto Schiaffino; he immediately broke into the first team and won his first league title in his debut season with the squad. In total, he made 347 appearances for Milan in Serie A, scoring 3 goals, and made 412 appearances for the club in all competitions. Maldini won four league titles with Milan, and also later became the team's captain in 1961, a role which he held for several years, until he left the club, and was succeeded by Gianni Rivera. During his time with Milan, he also won a Coppa Latina, and went on to capture the club's first ever European Cup as team captain in 1963, as Milan defeated Benfica 2–1 at Wembley Stadium on 22 May, with two goals from José Altafini; as a result, Milan became the first Italian side ever to win the trophy, while Maldini became the first Italian captain to lift the cup. He made his final appearance for Milan on 22 May 1966, in a 6–1 home win over Catania, in Serie A. In 1966 he moved to Torino for a season, before retiring in 1967; he made his final Serie A appearance on 28 May, in a 2–1 away defeat to Napoli.


title: Cesare Maldini

text: At international level Maldini earned 14 caps for the Italy national football team between 1960 and 1963, also serving as the national team's captain between 1962 and 1963, but was less successful than with Milan. He made his debut on 6 January 1960, in a 3–0 win over Switzerland in the 1955–60 Central European International Cup, and later took part in the 1962 FIFA World Cup with Italy, making two appearances in the competition. The team suffered a disappointing and controversial first-round elimination, although he was named to the team of the tournament for his performances. He made his final appearance for Italy in a European championship qualifier in Moscow, in 1963, as Italy suffered a 2–0 away defeat to the Soviet Union.


title: Cesare Maldini

text: A commanding, elegant, and respected defender, with good physical attributes, excellent technique, passing range, and an ability to set the tempo of his team's play, Maldini was known for being strong in the air, and for his outstanding ability to read the game, time his challenges, and anticipate opponents; he is considered one of the greatest defenders of his generation, and one of Italy's best ever defenders. He was also highly regarded for his leadership, and consistency, as well as his discipline and class, both on and off the pitch; throughout his career, he only received five yellow cards and was sent off only once. A tactically versatile player, although he was usually used as a man-marking centre-back, or as a sweeper, he was also capable of functioning as a full-back on either side of the pitch, but was usually fielded on the right flank when played in this position; he was even used as a defensive midfielder on occasion, due to his tendency to advance into midfield when in possession of the ball and start attacking plays from the back. Despite his reputation as a consistently world-class defender, in his early career Maldini was at times also known for being overly confident in possession, and for having a penchant to take risks when carrying or playing the ball out of the defence, due to his passing accuracy and ability on the ball; this tendency occasionally led to sporadic and atypical defensive errors, which were later humorously dubbed "Maldinate" by the Italian media.


title: Cesare Maldini

text: After retiring from playing Maldini became a coach, starting his career as an assistant manager for Nereo Rocco at Milan in 1970, and later also coached the team between 1972 and 1974, while Rocco served as the club's technical director. Maldini's first official match as Milan's head coach came on 6 September 1972, in a 4–1 away win over FA Red Boys Differdange in the Cup Winners' Cup; Maldini won a Coppa Italia and a Cup Winners' Cup double in 1973, and also narrowly missed out on the league title that season. However, Milan also suffered a 6–1 defeat on aggregate to Ajax in the European Super Cup; this was the worst defeat for an Italian team in an UEFA competition final. He was sacked by the club in 1974, following a disappointing 2–1 defeat to Verona on 6 April, and was replaced by Giovanni Trapattoni for the final few games of the season. He later went on to coach Foggia (1974–1976), Ternana (1976–77), and Serie C1 side Parma (1978–80), helping the team to obtain Serie B promotion during his tenure with the club, after managing a second-place finish in the league during the 1978–79 Serie C1 season.


title: Cesare Maldini

text: Between 1980 and 1986, he was one of Enzo Bearzot's assistants for the Italy national team, and even served as his assistant manager during the 1982 FIFA World Cup in Spain, which Italy won.


title: Cesare Maldini

text: Maldini took charge of the Italian Under-21 side in 1986, and coached the team for ten years, winning the European Under-21 Football Championship tournament a record three consecutive times between 1992 and 1996. As Italy's Under-21 coach, Maldini was also known as a mentor for many of the team's players, who later stated that he played a key role in their development; indeed, former protégés Fabio Cannavaro, Gianluigi Buffon and Francesco Totti from Maldini's 1996 Under-21 European Championship-winning squad went on to win the 2006 FIFA World Cup. He also coached the Italian Olympic teams that took part at the 1992 and 1996 Summer Olympics.


title: Cesare Maldini

text: Following his successes with the Under-21 side, Maldini took charge of the senior team in December 1996, replacing Arrigo Sacchi. After taking over as the national team's manager, he helped Italy to a crucial 1–0 away victory over England at Wembley Stadium on 12 February 1997, and successfully aided the team to qualify for the 1998 FIFA World Cup undefeated through the play-offs against Russia. Under Maldini, Italy were one of the four teams who also took part in the 1997 Tournoi de France friendly tournament, against his wishes, in which they finished in last place. Despite initially struggling in qualification, the Italian media and fans had great expectations of the 1998 side, which included a strong defence, and several prolific attacking players, such as Christian Vieri, Alessandro Del Piero and Filippo Inzaghi, among others, in their prime. Cesare Maldini's son, Paolo, was captain of the team.


title: Cesare Maldini

text: Italy were drawn in Group B of the tournament with Chile, Cameroon and Austria. They won their group, drawing 2–2 with Chile in their opening game, and then beating Cameroon 3–0 and Austria 2–1. Advancing to the Round of 16, Italy then beat Norway 1–0 to secure a spot in the quarter-finals against tournament hosts France. After a goalless draw, Italy were eventually eliminated by the eventual champions on penalties.


title: Cesare Maldini

text: Although Italy did not lose a match at the World Cup, Maldini resigned after the tournament due to heavy criticism in the Italian media over his allegedly ultra-defensive and "old-fashioned" catenaccio-inspired tactics, which included the use of a traditional man-marking defensive back-line, as well as a sweeper. He was also condemned for leaving Gianfranco Zola out of the squad, and for his reluctance to field creative forwards Roberto Baggio and Alessandro Del Piero alongside each other in the front-line, opting instead to have one player come on for the other in the second half; these controversial substitutions were compared to those made between Sandro Mazzola and Gianni Rivera by Ferruccio Valcareggi, the Italian manager at the 1970 World Cup.


title: Cesare Maldini

text: After serving as a head scout for his former team Milan from February 1999, Maldini briefly returned to coach the Milan first team in March 2001, serving as an interim manager for the club (whose captain was his son, Paolo) alongside youth coach Mauro Tassotti, following Alberto Zaccheroni's sacking, and led the squad for their final games of the season. Although the club endured a disappointing season, finishing in sixth place and failing to qualify for the Champions League, Maldini did lead the team to a memorable 6–0 victory over cross-city rivals Inter in the Milan Derby on 11 May; the result was Inter's worst-ever home defeat in Serie A, and Milan's biggest league win in the Derby. After helping the club to qualify for the UEFA Cup, Maldini was replaced by Fatih Terim the following season, although he initially remained with the club as an advisor to the coach.


title: Cesare Maldini

text: In January 2002, Maldini became coach of Paraguay. His appointment caused some controversy as domestic managers were overlooked (prompting the managers union to try to unsuccessfully expel him for immigration breaches), and because he spoke little Spanish. Maldini nonetheless had the support of star goalkeeper José Luis Chilavert and several other senior players. He took over the team which had already qualified for the 2002 World Cup hosted by South Korea and Japan; at the age of 70, he became, what was at the time, the oldest coach to ever be in charge of a national side at a World Cup tournament (his son Paolo captained Italy in the same tournament); this record was later broken by Greece's coach Otto Rehhagel in the 2010 World Cup.


title: Cesare Maldini

text: Despite missing Chilavert for the first game due to suspension, Paraguay managed to advance from Group B to the Round of 16, after a 2–2 draw with South Africa, a 3–1 defeat from Spain, and a 3–1 victory over Slovenia. In the Round of 16, Paraguay were eliminated by the finalist Germany, losing 1–0 thanks to a 89th-minute goal by Oliver Neuville.


title: Cesare Maldini

text: From a tactical standpoint, Maldini was known for his cautious, traditional, and conservative approach as a manager, and he used a solid, defensive-minded "catenaccio"-inspired system as the head coach of the Italian national side, a system which had been widely used by his mentor, Nereo Rocco; as he preferred to build his team on a very strong back-line, he often fielded a 4–4–2 formation, but which usually employed a sweeper (or "libero," in Italian) and three-man-marking centre-backs (or "stopper," in Italian) in front of the goalkeeper, although at times, he would also use a more common flat four-man defensive line and a zonal marking system, with two centre-backs, a left-back, and a right-back. His teams were often known for being very compact defensively and for their tight and physical marking of opponents and would often play on the counter-attack. In attack, he usually paired two forwards who would complement each other alongside one another, namely a traditionally large, physical, and prolific centre-forward – such as Christian Vieri – with a smaller, faster, creative and more technical second striker – such as Roberto Baggio or Alessandro Del Piero; on rare occasions, he also employed an attacking trident.


title: Cesare Maldini

text: After the 2002 World Cup, Maldini returned to AC Milan as a talent scout for the "Rossoneri". He also later worked as a sports analyst for several radio stations and sports channels, such as beIN SPORTS, and also for Al Jazeera in 2012, with Alessandro Altobelli.


title: Cesare Maldini

text: On 3 April 2016, at the age of 84, Cesare died. In his honour, a minute of silence was held before every league game in Italy that weekend, while Milan players wore black armbands in their match against Atalanta. Maldini's grandson Christian also captained the youth team in a friendly match in his honor Maldini's funeral service was held on 5 April, at the Basilica of Sant'Ambrogio, in Milan, and was attended by several important footballing figures. His wife died later that year, on 28 July.


title: House of Representatives (Nepal)

text: The House of Representatives, or Pratinidhi Sabha (; ) is the one of the houses of the Federal Parliament of Nepal, with the other house being the National Assembly. Members of the House of Representatives are elected through a parallel voting system. They hold their seats for five years or until the body is dissolved by the President on the advice of the council of ministers. The house meets in the International Convention Centre in Kathmandu.


title: House of Representatives (Nepal)

text: The House has 275 members; 165 elected from single-member constituencies by first-past-the-post voting and 110 elected through proportional electoral system where voters vote for political parties, considering the whole country as a single election constituency. The House of Representatives, unless dissolved, continues to operate for five years from the date appointed for its first meeting. However, in a state of emergency, the term of the House of Representatives may be extended, not exceeding one year in accordance with federal law.


title: House of Representatives (Nepal)

text: The current House of Representatives was elected in 2017 and its first meeting was held on 5 March 2018. On 20 December 2020, it was dissolved by a cabinet decision. However, the Supreme Court reinstated the House on 22 February 2021. On 22 May 2021, it was again dissolved by a cabinet decision followed by approval of President on mid-night with elections set to be held on 12 and 19 November in two phases. The Supreme Court, by the order of certiorari, has reinstated the House for the second time in a landmark decision on 12 July 2021.


title: House of Representatives (Nepal)

text: The 1959 constitution of the Kingdom of Nepal, proclaimed on 12 February 1959, first mentions the Pratinidhi Sabha first as follows: "There shall be a Parliament which shall consist of His Majesty and two Houses, to be known respectively as the Senate (Maha Sabha) and the House of Representatives (Pratinidhi Sabha)" "(Article No. 18, Constitution of the Kingdom of Nepal, 1959)."


title: House of Representatives (Nepal)

text: The 1959 constitution was abrogated on 16 December 1962 when the new Constitution of the Kingdom of Nepal, 1962 was proclaimed and the parliament of the kingdom became unicameral.


title: House of Representatives (Nepal)

text: The House of Representatives was first provided for by the "Constitution of the Kingdom of Nepal 1990", which replaced the former panchayat system of parliament with a bicameral parliament. It consisted of 205 members directly elected from single member constituencies. It had five-year terms, but it could be dissolved by the King on the advice of the Prime Minister before the end of its term.


title: House of Representatives (Nepal)

text: In May 2002, the House of Representatives was dissolved by King Gyanendra on advice of the then prime minister, Sher Bahadur Deuba, in order to hold new elections. Elections could not take place due to the ongoing civil war, which eventually led King Gyanendra to stage a royal coup. Following the democracy movement of 2006, the King reinstated the earlier legislature. On 15 January 2007, the House of Representatives was transformed into an Interim legislature. This consisted of members appointed under an agreement between the Seven Party Alliance and the Communist Party of Nepal (Maoist) (known by the name Communist Party of Nepal (Maoist Centre), since 2009).


title: House of Representatives (Nepal)

text: The Constitution of Nepal was drafted by the 2nd Constituent Assembly and the provision for a bicameral legislature was re-adopted. The House of Representatives became the lower house of the Federal Parliament of Nepal and its first election was held in 2017.


title: House of Representatives (Nepal)

text: The composition and powers of the house are established by Parts 8 and 9 of the Constitution of Nepal. The qualifications for becoming a member of the House are laid out in Article 87 of the Constitution and House of Representatives Election Act, 2017. Members must be :


title: House of Representatives (Nepal)

text: In addition to this, no member can be a member of both the House of Representatives and the National Assembly.


title: House of Representatives (Nepal)

text: The seat of a member of House of Representatives may be considered vacant in the following circumstances: