title: Murder of Emani Moss

text: Emani eventually became too weak to move and could not leave her bed, urinate or defecate. Though Moss denied food to Emani, she did take care of and feed her two biological children. On several occasions, she sent Eman pictures of meals she had prepared for her children, and in another instance, she asked her husband to bring home cookie dough so she could bake. At trial, prosecutors pointed out that Emani would have had to smell the baked cookies as she lay in her room starving to death. According to Dr. Staffenberg, the medical examiner who testified at trial, the process of starving to death would have been painful. At first, Emani would have experienced hunger pangs. She then would have become fatigued. She would continue losing energy and weight until she ultimately died.


title: Murder of Emani Moss

text: In the early evening of October 24, Emani suffered what Eman believed was a seizure. Eman testified that when he came home, Moss told him something was wrong with Emani. He then went into the bathroom and found his daughter in the bathtub shaking. Emani was unresponsive, and her eyes were rolling back and forth. Eman moved Emani to her bed, where she stayed for the next couple of days. Eman visited her during this time and tried to feed her but was unsuccessful. Emani died on October 28, 2013. By the time Emani died, she was severely underweight, with Dr. Staffenberg describing her as "more or less skin and bones." She weighed , the weight of the average three-year-old. Additionally, her organs were found at autopsy to be very small.


title: Murder of Emani Moss

text: On October 28, Moss called Eman at work to tell him that Emani was dead. Eman testified at trial that when he came home from work, the family seemed normal, with the children playing and Moss watching TV. He found his daughter lying on a blanket on her bedroom floor. He told his wife they should call the police though she insisted they couldn't because she would lose her children. Moss told Eman they needed to hide Emani's body and "be on our criminal mind." Eman wrapped Emani's body with blankets and moved her to the computer room. The couple kept Emani's body in their apartment for several days, and their lives largely went back to normal. Eman testified at trial that he would go to work and spend time at home with Emani's body grieving.


title: Murder of Emani Moss

text: The Moss couple agreed to cover up Emani's death. The day after Emani's death, Moss went to Anna's Linens and bought new sheets and a new coverall, as the ones Emani had used were covered with excrement and urine. Moss suggested burying Emani and reporting her as a runaway. Eman went to Walmart and bought a galvanized trash can, trash bags, charcoal, and lighter fluid. On Halloween, Moss and Eman decided to put Emani's body in the trash can and burn it. When they tried to place her in the trash can, they found that she was stiff with rigor mortis and used duct tape to compress her body. Eman covered her with a comforter. The Mosses then stuffed Emani's body in the trash bag. In the early morning of November 1, the Moss couple put the trash can containing Emani's body in the back of their car and took their children to find a place to burn it. They found a secluded location to commit the arson and removed the trash can from the vehicle. They added charcoal briquettes to the bottom of the can, doused Emani's body with lighter fluid, and set it on fire. As the couple watched the body burn, they found it would not burn to ash. After about five minutes, they extinguished the fire and took the trash can and Emani's body back to the apartment.


title: Murder of Emani Moss

text: The day after attempting to burn Emani's body, Eman went to work with Emani's body still in the back of his car. He confessed to a friend about the crime. The friend urged him to call the police. Eman thought about it, and after coming home from his second job, he decided to contact the police. At 4:00 am he made the call, saying he was suicidal. Eman informed Moss about his call to the police. Upon learning that the police had been called, Moss placed the trash can with Emani's remains in a grassy area at the apartment and fled in the family vehicle with her children. Upon arrival at the apartment, the police were told by Eman that Emani had drunk some chemicals and died. He claimed that he had panicked and put her body in a trash can outside the apartment and tried to cremate it. Police found the body and did not believe Eman's story, quickly identifying him and his wife as suspects. Eman was arrested almost immediately. Moss dropped her children off at her mother's house and ultimately turned herself in.


title: Murder of Emani Moss

text: Eman ultimately confessed to covering up Emani's murder by reporting her as a runaway and trying to burn her body. In 2015 he pled guilty to felony murder and concealing a death. He was sentenced to life in prison without the possibility of parole. In exchange, he agreed to testify against Moss, who rejected a plea deal that would have allowed her to be sentenced to life in prison. Eman is currently incarcerated at the Smith State Prison in Glennville, Georgia. The case against Moss instead went to trial.


title: Murder of Emani Moss

text: Moss's trial began on April 15, 2019, and was presided over by Judge George Hutchinson. The jury consisted of six men and six women. Moss, who was appointed lawyers through the State Office of the Capital Defender, decided to represent herself, despite Judge Hutchinson's efforts to persuade her otherwise. Instead of representing her, her defense attorneys served as standby attorneys to answer any legal questions. During pre-trial hearings and jury selection, Judge Hutchinson urged Moss to rely upon the standby counsel, but she refused. Moss did not give an opening statement. Nor did she cross-examine any witnesses, or give a closing argument. District Attorney Danny Porter and Assistant District Attorney Lisa Jones called eighteen witnesses, including Emani's aunt, grandmother, and fourth-grade teacher. They also called Eman and Dr. Staffenberg, the medical examiner who performed an autopsy on Emani's body.


title: Murder of Emani Moss

text: On April 29, Moss was convicted on all six counts, including one count of malice murder, two counts of felony murder, two counts of cruelty to children, and one count of concealing a death. The jury deliberated for less than three hours.


title: Murder of Emani Moss

text: During the sentencing phase, Moss declined to address the jury, present mitigating evidence, or have her relatives who had attended the trial testify on her behalf. She also refused to make a closing statement. In the state's closing argument, Porter argued that Moss did not deserve a life sentence, whether it be a life sentence with parole or a life without parole sentence. She should not be given the opportunity to be released, he argued, because she would never change. "She's shown you too much of her capacity for cruelty. There will always be that dark side waiting to come out." He also argued that for Moss, life without parole would not be a worse sentence than death because she did not regret her crimes and would never be bothered by them.


title: Murder of Emani Moss

text: After closing arguments, the jury began deliberating. After the first day, they appeared conflicted and were told by Judge Hutchinson to go home and "sleep on it." They continued deliberations the next day and ultimately agreed on the death penalty as a punishment for Moss. Sentencing occurred on May 1. Judge Hutchinson agreed with the jury's recommendation and sentenced Moss, then age thirty-six, to death by lethal injection. Moss was the first person to be sentenced to death in Georgia in over five years. Judge Hutchinson scheduled her execution for between June 7 and 14, 2019. Porter said of the death sentence, "there's no joy when a jury imposes a death sentence. But this was one of the worst cases I've ever seen. The first time you look at it, it made you sick. The last time you look at it it makes you sick." The execution did not occur during the June 7–14 time frame due to the appeals process, which resulted in an automatic stay of execution being applied to the initial scheduled execution date. She is currently incarcerated at Arrendale State Prison and is Georgia's only female death row inmate. If Moss is eventually executed, she will be the third woman in Georgia to be executed since 1945.


title: Murder of Emani Moss

text: Despite representing herself at trial, Moss accepted legal representation in her appeal. Soon after she was sentenced to death, the Georgia Capital Defender group filed a motion asking for a new trial, arguing, among other things, that Moss was not competent to act as her attorney. According to her attorneys, Moss has "neuropsychological testing data that showed the defendant to have damage to the premotor and prefrontal regions of the brain." A status hearing regarding the motion was held on August 23, 2019.


title: Murder of Emani Moss

text: Eman and Moss, who remain married, lost custody of their two children after Emani's murder. The children were sent to live with foster parents. The Moss couple, along with both Moss and Eman's mothers, tried to gain custody. In 2019, Porter told local news that the children were adopted by their foster parents. The couple's son, who was three years old when Emani was murdered, reportedly did have some memories of the crime. However, according to Porter, at the time of the interview in April 2019, the children were "doing great" and may never know what happened.


title: Murder of Emani Moss

text: Emani's murder led to criticisms of and systemic changes in Georgia's child welfare system. After the murder, an intake case manager, a social services administrator, and a program assistant at the GDFCS were all terminated. Others were reportedly disciplined. The murder led the department to enact reforms, including deeper investigations into allegations of abuse and changing how it assembles maltreatment reports. New case managers and supervisors were hired, reducing caseloads. Agency workers no longer decide whether reports warrant investigations based entirely on information gathered over the telephone. Additionally, no case is assigned a less-serious, lower-priority status until a caseworker meets a child who allegedly has been abused. After Emani's death, along with several other abuse-related deaths, the Child Welfare Reform Council was commissioned by Governor Nathan Deal. In January 2015, the Council released a report with recommendations on ways to improve Georgia's child welfare system.


title: Murder of Emani Moss

text: In 2018, Emani's grandmother Robin filed a lawsuit in the Gwinnett County State Court against the GDFCS, arguing that caseworkers were aware of deteriorating conditions and abuse in the Moss family and could have acted earlier. Robin is also suing the Department of Human Services. She is seeking a jury trial and "reasonable compensatory damages." According to the lawsuit, there were multiple occasions where the department could have investigated the Moss household and intervened on behalf of Emani: "As a result of the negligence of DFCS and its agents, Emani suffered constant abuse and deprivation from 2008 until her untimely death." The lawsuit also states: "As a direct and proximate result of defendant's wrongful conduct, plaintiff is entitled to recover for the wrongful death of Emani, including the full value of the economic and non-economic value of her life had she lived."


title: Bobby Goldsboro

text: Robert Charles Goldsboro (born January 18, 1941) is an American pop and country singer and songwriter. He had a string of pop and country hits in the 1960s and 1970s, including his signature No. 1 hit "Honey", which sold over 1 million copies in the United States, and the UK top-10 single "Summer (The First Time)".


title: Bobby Goldsboro

text: Goldsboro starred in his own television show, "The Bobby Goldsboro Show", from 1973 to 1976. He also created the children's series "The Swamp Critters of Lost Lagoon".


title: Bobby Goldsboro

text: Goldsboro was born in Marianna, Florida. During his first year of life, his family moved north from Marianna to Dothan, Alabama. He learned to play the ukelele when he was around 12 years old before learning to play the guitar. He was interested in becoming a professional baseball player before turning his interest to music.


title: Bobby Goldsboro

text: Goldsboro attended Dothan High School. In his senior year in high school he started playing in a band called The Webbs. After graduating from high school in 1959, he enrolled at Auburn University, majoring in business administration. Goldsboro left college after his second year to pursue a musical career, playing guitar in the backup band for Roy Orbison.


title: Bobby Goldsboro

text: In high school Goldsboro joined Spider and The Webs which featured Amos "Mugsby" on bass, Dave Robinson on drums, "Flying Fingers" John Rainey on lead guitar, and "Rockin" Steve Murphree on rhythm guitar. He performed semi-professionally as part of The Webs while at college. At the end of his second year in college, the manager of The Webs Buddy Buie arranged for the band to play for Roy Orbison who was without a backing band in a show he organized. The arrangement worked well, and The Webs then became the permanent backing band for Orbison for two-and-a-half years until 1964, touring with Orbison in the US and Europe.


title: Bobby Goldsboro

text: When they were not touring with Orbison, The Webs played in local gigs, recorded songs written by Goldsboro and had some local success. They came to the attention of an independent producer Jack Gold, who was interested in working with Goldsboro. He gave Goldsboro a song titled "Molly" for him to record, and the song managed to chart on "Billboard" Hot 100 in early 1963.


title: Bobby Goldsboro

text: Goldsboro continued performing with The Webs and Orbison until his solo career kicked off with the top 10 hit "See the Funny Little Clown". The single, written by Goldsboro, reached No. 9 on the U.S. national charts in early 1964. It sold over 1 million copies and was awarded a gold disc. In the next few years, he achieved a few Top 40 hits, including "Little Things", "Voodoo Woman", and "It's Too Late". However, his attempt at more serious songs was derailed when "Broomstick Cowboy" was banned from ABC's "American Bandstand". Although Goldsboro was not a prolific performer of soul music, "Too Many People" and "It's Too Late" were huge hits on the Northern soul scene in Great Britain and were played extensively.


title: Bobby Goldsboro

text: In 1968, Goldsboro released the biggest hit of his career, "Honey", a tearjerker about the death of a man's wife. The song, written by Bobby Russell, was recorded in one take. It became the largest-selling record in the world for 1968 and topped the Hot 100 for five weeks, reached number two in the UK Singles Chart in 1968 and 1975, and was a number one single in Australia. It also became his first country hit.


title: Bobby Goldsboro

text: One of Goldsboro's compositions, "With Pen in Hand", was recorded by several artists, including a Grammy-nominated pop version by Vikki Carr that reached the Hot 100's top 40 in 1969; Johnny Darrell had taken the song to No. 3 on the U.S. country chart a year earlier. In 1970, Della Reese included a cover on her album "Black Is Beautiful". Goldsboro's "The Cowboy and the Lady" became a top 10 country hit as "The Cowgirl and the Dandy" for Brenda Lee in 1980; Dolly Parton covered it in 1977, and John Denver had a hit with the song in 1981.


title: Bobby Goldsboro

text: Goldsboro reached No. 11 on "Billboard"'s national charts with the Mac Davis-penned song "Watching Scotty Grow". "Summer (The First Time)", a 1973 reminiscence about a 17-year-old boy's first sexual experience (with a 31-year-old woman), was a Top 25 hit in the U.S. and reached number 9 in the UK. Using a repeating piano riff, 12-string guitar, and an orchestral string arrangement, the song was suggestive enough to spark some controversy. The follow-up "Hello Summertime" was written by Roger Cook and Roger Greenaway and hit No. 14 in the UK in September 1974.


title: Bobby Goldsboro

text: The success of "Honey" led to numerous television appearances for Goldsboro, and he became a regular on "The Mike Douglas Show", co-hosting the show for a few times. Goldsboro was approached to host his own syndicated half-hour television series "The Bobby Goldsboro Show". The show, which ran from 1973 to 1976, was one of the more successful syndicated shows at that time.


title: Bobby Goldsboro

text: In the 1990s, he composed the music for the sitcom "Evening Shade". In 1995, he created the children's television series "The Swamp Critters of Lost Lagoon". Goldsboro voiced all the characters, wrote all the scripts, and played all the musical instruments in the show.


title: Bobby Goldsboro

text: Goldsboro was married to Mary Alice Watson, whom he met while he was in college; they had three children. They divorced in 1982 after a bitter court case where Goldsboro's daughter made accusations against him that he rejected as false. He married Dianne J. Roberts in 1985. Goldsboro is an accomplished oil painter.


title: Foundry

text: A foundry is a factory that produces metal castings. Metals are cast into shapes by melting them into a liquid, pouring the metal into a mold, and removing the mold material after the metal has solidified as it cools. The most common metals processed are aluminum and cast iron. However, other metals, such as bronze, brass, steel, magnesium, and zinc, are also used to produce castings in foundries. In this process, parts of desired shapes and sizes can be formed.


title: Foundry

text: Foundries are one of the largest contributors to the manufacturing recycling movement, melting and recasting millions of tons of scrap metal every year to create new durable goods. Moreover, many foundries use sand in their molding process. These foundries often use, recondition, and reuse sand, which is another form of recycling.


title: Foundry

text: In metalworking, casting involves pouring liquid metal into a mold, which contains a hollow cavity of the desired shape, and then allowing it to cool and solidify. The solidified part is also known as a casting, which is ejected or broken out of the mold to complete the process. Casting is most often used for making complex shapes that would be difficult or uneconomical to make by other methods.


title: Foundry

text: Melting is performed in a furnace. Virgin material, external scrap, internal scrap, and alloying elements are used to charge the furnace. Virgin material refers to commercially pure forms of the primary metal used to form a particular alloy. Alloying elements are either pure forms of an alloying element, like electrolytic nickel, or alloys of limited composition, such as ferroalloys or master alloys. External scrap is material from other forming processes such as punching, forging, or machining. Internal scrap consists of gates, risers, defective castings, and other extraneous metal oddments produced within the facility.


title: Foundry

text: The process includes melting the charge, refining the melt, adjusting the melt chemistry and tapping into a transport vessel. Refining is done to remove harmful gases and elements from the molten metal to avoid casting defects. Material is added during the melting process to bring the final chemistry within a specific range specified by industry and/or internal standards. Certain fluxes may be used to separate the metal from slag and/or dross and degassers are used to remove dissolved gas from metals that readily dissolve in gasses. During the tap, final chemistry adjustments are made.


title: Foundry

text: Several specialised furnaces are used to heat the metal. Furnaces are refractory-lined vessels that contain the material to be melted and provide the energy to melt it. Modern furnace types include electric arc furnaces (EAF), induction furnaces, cupolas, reverberatory, and crucible furnaces. Furnace choice is dependent on the alloy system quantities produced. For ferrous materials EAFs, cupolas, and induction furnaces are commonly used. Reverberatory and crucible furnaces are common for producing aluminium, bronze, and brass castings.


title: Foundry

text: Furnace design is a complex process, and the design can be optimized based on multiple factors. Furnaces in foundries can be any size, ranging from small ones used to melt precious metals to furnaces weighing several tons, designed to melt hundreds of pounds of scrap at one time. They are designed according to the type of metals that are to be melted. Furnaces must also be designed based on the fuel being used to produce the desired temperature. For low temperature melting point alloys, such as zinc or tin, melting furnaces may reach around . Electricity, propane, or natural gas are usually used to achieve these temperatures. For high melting point alloys such as steel or nickel-based alloys, the furnace must be designed for temperatures over . The fuel used to reach these high temperatures can be electricity (as employed in electric arc furnaces) or coke.


title: Foundry

text: The majority of foundries specialize in a particular metal and have furnaces dedicated to these metals. For example, an iron foundry (for cast iron) may use a cupola, induction furnace, or EAF, while a steel foundry will use an EAF or induction furnace. Bronze or brass foundries use crucible furnaces or induction furnaces. Most aluminium foundries use either electric resistance or gas heated crucible furnaces or reverberatory furnaces.


title: Foundry

text: Degassing is a process that may be required to reduce the amount of hydrogen present in a batch of molten metal. Gases can form in metal castings in one of two ways:


title: Foundry

text: Hydrogen is a common contaminant for most cast metals. It forms as a result of material reactions or from water vapor or machine lubricants. If the hydrogen concentration in the melt is too high, the resulting casting will be porous; the hydrogen will exit the molten solution, leaving minuscule air pockets, as the metal cools and solidifies. Porosity often seriously deteriorates the mechanical properties of the metal.


title: Foundry

text: An efficient way of removing hydrogen from the melt is to bubble a dry, insoluble gas through the melt by purging or agitation. When the bubbles go up in the melt, they catch the dissolved hydrogen and bring it to the surface. Chlorine, nitrogen, helium and argon are often used to degas non-ferrous metals. Carbon monoxide is typically used for iron and steel.


title: Foundry

text: There are various types of equipment that can measure the presence of hydrogen. Alternatively, the presence of hydrogen can be measured by determining the density of a metal sample.


title: Foundry

text: In cases where porosity still remains present after the degassing process, porosity sealing can be accomplished through a process called metal impregnating.


title: Foundry

text: In the casting process, a pattern is made in the shape of the desired part. Simple designs can be made in a single piece or solid pattern. More complex designs are made in two parts, called split patterns. A split pattern has a top or upper section, called a cope, and a bottom or lower section called a drag. Both solid and split patterns can have cores inserted to complete the final part shape. Cores are used to create hollow areas in the mold that would otherwise be impossible to achieve. Where the cope and drag separates is called the parting line.


title: Foundry

text: When making a pattern it is best to taper the edges so that the pattern can be removed without breaking the mold. This is called draft. The opposite of draft is an undercut where there is part of the pattern under the mold material, making it impossible to remove the pattern without damaging the mold.


title: Foundry

text: The pattern is made of wax, wood, plastic, or metal. The molds are constructed by several different processes dependent upon the type of foundry, metal to be poured, quantity of parts to be produced, size of the casting, and complexity of the casting. These mold processes include:


title: Foundry

text: In a foundry, molten metal is poured into molds. Pouring can be accomplished with gravity, or it may be assisted with a vacuum or pressurized gas. Many modern foundries use robots or automatic pouring machines to pour molten metal. Traditionally, molds were poured by hand using ladles.


title: Foundry

text: The solidified metal component is then removed from its mold. Where the mold is sand based, this can be done by shaking or tumbling. This frees the casting from the sand, which is still attached to the metal runners and gates — which are the channels through which the molten metal traveled to reach the component itself.


title: Foundry

text: Degating is the removal of the heads, runners, gates, and risers from the casting. Runners, gates, and risers may be removed using cutting torches, bandsaws, or ceramic cutoff blades. For some metal types, and with some gating system designs, the sprue, runners, and gates can be removed by breaking them away from the casting with a sledge hammer or specially designed knockout machinery. Risers must usually be removed using a cutting method (see above) but some newer methods of riser removal use knockoff machinery with special designs incorporated into the riser neck geometry that allow the riser to break off at the right place.


title: Foundry

text: The gating system required to produce castings in a mold yields leftover metal — including heads, risers, and sprue (sometimes collectively called sprue) — that can exceed 50% of the metal required to pour a full mold. Since this metal must be remelted as salvage, the yield of a particular gating configuration becomes an important economic consideration when designing various gating schemes, to minimize the cost of excess sprue, and thus overall melting costs.


title: Foundry

text: Heat treating is a group of industrial and metalworking processes used to alter the physical, and sometimes chemical, properties of a material. The most common application is metallurgical. Heat treatments are also used in the manufacture of many other materials, such as glass. Heat treatment involves the use of heating or chilling, normally to extreme temperatures, to achieve a desired result such as hardening or softening of a material. Heat treatment techniques include annealing, case-hardening, precipitation strengthening, tempering, and quenching. Although the term "heat treatment" applies only to processes where the heating and cooling are done for the specific purpose of altering properties intentionally, heating and cooling often occur incidentally during other manufacturing processes such as hot forming or welding.


title: Foundry

text: After degating and heat treating, sand or other molding media may remain adhered to the casting. To remove any mold remnants, the surface is cleaned using a blasting process. This means a granular media will be propelled against the surface of the casting to mechanically knock away the adhering sand. The media may be blown with compressed air, or may be hurled using a shot wheel. The cleaning media strikes the casting surface at high velocity to dislodge the mold remnants (for example, sand, slag) from the casting surface. Numerous materials may be used to clean cast surfaces, including steel, iron, other metal alloys, aluminium oxides, glass beads, walnut shells, baking powder, and many others. The blasting media is selected to develop the color and reflectance of the cast surface. Terms used to describe this process include cleaning, bead blasting, and sand blasting. Shot peening may be used to further work-harden and finish the surface.


title: Foundry

text: The final step in the process of casting usually involves grinding, sanding, or machining the component in order to achieve the desired dimensional accuracies, physical shape, and surface finish.


title: Foundry

text: Removing the remaining gate material, called a gate stub, is usually done using a grinder or sander. These processes are used because their material removal rates are slow enough to control the amount of material being removed. These steps are done prior to any final machining.


title: Foundry

text: After grinding, any surfaces that require tight dimensional control are machined. Many castings are machined in CNC milling centers. The reason for this is that these processes have better dimensional capability and repeatability than many casting processes. However, it is not uncommon today for castings to be used without machining.


title: Foundry

text: A few foundries provide other services before shipping cast products to their customers. It is common to paint castings to prevent corrosion and improve visual appeal. Some foundries assemble castings into complete machines or sub-assemblies. Other foundries weld multiple castings or wrought metals together to form a finished product.


title: Foundry

text: More and more, finishing processes are being performed by robotic machines, which eliminate the need for a human to physically grind or break parting lines, gating material, or feeders. Machines can reduce risk of injury to workers and lower costs for consumables — while also increasing productivity. They also limit the potential for human error and increase repeatability in the quality of grinding.


title: Aahana Kumra

text: Aahana Kumra is an Indian feature film, television, and theatre actress. Kumra is known for her small screen debut in Sony Entertainment Television's "Yudh" with Amitabh Bachchan and also for her lead role in the TV series "Agent Raghav - Crime Branch" with Sharad Kelkar, followed by leading roles in a number of feature films.


title: Aahana Kumra

text: Kumra was born and brought up in Lucknow. She had an inclination towards acting since childhood when she began theatre roles.


title: Aahana Kumra

text: In 2013, Kumra was cast in the television series "Yudh" - a 20-episode show starring Amitabh Bachchan in which Kumra plays the pivotal role of Bachchan's daughter. Later, she starred in "Agent Raghav – Crime Branch", a biweekly television crime thriller show, as the female lead Agent Trisha Dewan opposite Sharad Kelkar. Kumra won Best Actress Award at the Third Moida International Awards for her short film "Siberia". She made her Hindi feature film debut in "Sona Spa" in 2013, and her Tulu feature film debut in 2015 in "Kudla Cafe". She hosted the Pro Kabbadi 2016 series. In 2017, Kumra had a leading role in the black comedy film "Lipstick Under My Burkha" as Leela. The following year she portrayed the role of Priyanka Gandhi in the political biography film "The Accidental Prime Minister".


title: 2021–22 Ashes series

text: The 2021–22 Ashes series, named the Vodafone Men's Ashes Series for sponsorship reasons, was a series of five Test cricket matches that were contested between England and Australia for The Ashes. The series was played at five venues across Australia from 8 December 2021 and was scheduled to finish on 18 January 2022. Australia were the defending holders of the Ashes going into the series, having won in 2017–18 with the 2019 Ashes series ending in a draw. The series was part of the 2021–2023 ICC World Test Championship.


title: 2021–22 Ashes series

text: Australia retained the Ashes by winning the first three Test matches. The fourth Test ended in a draw, with Australia winning the fifth Test by 146 runs to win the series 4–0.


title: 2021–22 Ashes series

text: On 25 October 2021, Ben Stokes was added to England's squad. On 7 December 2021, James Anderson was left out of the first Test as a precaution, despite being declared fit to play. On 7 January 2022, Sam Billings was added to the squad as injury cover. Jos Buttler was ruled out of England's squad for the fifth Test, after breaking a finger during the fourth Test.


title: 2021–22 Ashes series

text: On 17 November 2021, Cricket Australia named a 15-member squad for the first two Test matches. On 19 November 2021, Tim Paine resigned from the captaincy role and, on 26 November 2021, was replaced by Pat Cummins as captain with Steve Smith as Cummins' deputy. Paine then withdrew from the squad and announced he was taking an indefinite leave of absence from cricket. On 2 December 2021, Alex Carey was added to Australia's squad as Paine's replacement and a wicket-keeper, receiving his maiden Test call-up in the process. Ahead of the second Test, Josh Hazlewood was ruled out of Australia's squad due to a side strain picked up during the first Test of the series. Pat Cummins was ruled out of the second Test after being named as a close contact of a positive COVID-19 case; Steve Smith was made captain, and Michael Neser made his Test debut for Australia. During the fifth day of the second Test, Cricket Australia named an unchanged squad for the rest of the series. Ahead of the third Test, Australia also added Scott Boland to their squad as a cover for their fast bowlers.


title: 2021–22 Ashes series

text: The 2021–22 Ashes series took place against the backdrop of the ongoing COVID-19 pandemic, which threatened to derail the series numerous times. As far back as June 2021, concerns had been raised that key England players could pull out of the tour if their families were not able to accompany them due to the Australian government's restrictions on international travel.


title: 2021–22 Ashes series

text: On the morning of the first day of the Second Test, Australian captain Pat Cummins was ruled out of the match after he was deemed to be a close contact of someone with COVID-19. In mid-December, a number of people at the match at Adelaide Oval tested positive, including members of the media.


title: 2021–22 Ashes series

text: On 31 December 2021, Travis Head was ruled out of the fourth test due to contracting COVID-19. Mitchell Marsh, Nic Maddinson and Josh Inglis were added to the squad as additional cover. Several key staff and officials, including ICC match referee David Boon and England coach Chris Silverwood were also forced to miss the fourth test; Boon had tested positive for COVID-19 while Silverwood was forced to isolate with his family following an outbreak of the virus in the England camp.


title: 2021–22 Ashes series

text: In May 2021, Cricket Australia (CA) announced that The Gabba, the Adelaide Oval, the Melbourne Cricket Ground (MCG), the Sydney Cricket Ground and the Perth Stadium would host the Test matches. However, in late November 2021, doubts were raised about the final Test taking place in Perth due to COVID-19 restrictions. Mark McGowan, the premier of Western Australia, insisted on a 14-day quarantine period for everyone travelling into the state. On 6 December 2021, CA confirmed the fifth Test would not be held in Perth, and that the relocated Test would be played as a day/night match. On 11 December 2021, CA confirmed that the venue for the fifth Test match would be the Bellerive Oval, in Hobart. This was the first ever Ashes Test match to be played in Hobart.


title: 2021–22 Ashes series

text: Rory Burns was bowled for a golden duck by Mitchell Starc off the first ball of the series. Dawid Malan was caught behind by Alex Carey from a delivery by Josh Hazlewood. Captain Joe Root followed with a duck after getting caught at first slip by David Warner from a delivery by Hazlewood. Ben Stokes was only able to score five runs before being caught out off a delivery by Pat Cummins. Ultimately, England was dismissed for 147 in their first innings. Although England was all out before tea, Australia were unable to begin their first innings due to weather.


title: 2021–22 Ashes series

text: Marcus Harris was only able to score three runs before being dismissed by Ollie Robinson. Marnus Labuschagne, Steve Smith, David Warner, and Cameron Green were dismissed in quick succession, going four for 29. England were unable to capitalise on three close opportunities to dismiss David Warner on 17, 48, and 60. Warner was able to score 94 runs before being caught out off a delivery by Robinson. Green was bowled out by Robinson in one ball. Australia ended the day 7/343. Travis Head was not out, having scored 112 runs with twelve fours and two sixes.


title: 2021–22 Ashes series

text: Australia were all out before lunch, with Travis Head bowled out off a yorker from Mark Wood. Australia's run total for the innings was 425, with Head contributing 152. Haseeb Hameed and Rory Burns were the first two batsmen for England's second innings. Burns was called out lbw with zero runs, but the call was overturned by DRS. Burns was later dismissed for 13 and Hameed soon followed, being dismissed for 27. The next two batsmen, Dawid Malan and Joe Root, ended the day not out with 80 and 86 runs, respectively. As a result, England ended the day trailing Australia by 58 runs.


title: 2021–22 Ashes series

text: Despite strong performances in day three, Malan and Root were only able to add two and three runs on day four before falling. Ollie Pope was the next to fall after being caught at the slip. The global broadcast from The Gabba was then cut off for approximately 30 minutes while the Test continued. After the broadcast was able to return, Stokes edged a ball from Pat Cummins and was caught out in the gully, leaving England 6/266. The next four batsmen, Jos Buttler, Chris Woakes, Ollie Robinson, and Mark Wood, were dismissed in quick succession, ending the innings. In total, England scored 297 with eight wickets and only 77 runs in day four.


title: 2021–22 Ashes series

text: With a target of 20 runs, Australia opened their second innings with Alex Carey and Marcus Harris. Carey was dismissed, but Harris was able to reach the target score before Australia's third batsman faced a ball.


title: 2021–22 Ashes series

text: Despite testing negative to COVID-19, Pat Cummins was forced to withdraw from the match and isolate in accordance with the South Australian government's public health regulations. Steve Smith was appointed Australian captain for the first time since his comeback from the Sandpapergate incident, while Michael Neser was named as Cummins' replacement and hastily awarded his baggy green.


title: 2021–22 Ashes series

text: Smith won the toss and decided to bat first. Marcus Harris and David Warner opened the batting for the home team. However, Harris was out soon, caught by England wicketkeeper Jos Buttler off the bowling of Stuart Broad. Marnus Labuschagne and Warner stabilized the innings putting on a partnership of 172 runs for the second wicket before Warner was out for 95, falling five short of a century, caught by Broad off the bowling of Ben Stokes. Australia ended the day at 225 for two, with Labuschagne remaining not out at 95 (having been dropped twice by Buttler on 21 and 95) and Smith on 18 not out.


title: 2021–22 Ashes series

text: Early in the day, Labuschagne reached his sixth test hundred with a controlled edge past second slip to the boundary. Having come off 287 balls, it was the slowest Test century by an Australian since Jason Gillespie's effort against Bangladesh in 2006. He was dismissed LBW shortly afterward on 103 by Ollie Robinson, offering no shot to a delivery. Australia reviewed the decision, however, the decision by the on-field umpire stayed. Travis Head joined the captain Smith at the wicket and the duo put on fifty runs for the fourth wicket before Head was bowled off a full toss from the bowling of English captain Joe Root. Cameron Green came to the crease but was back to the pavilion adding only two runs. Australia were at 294 for five. Wicketkeeper Alex Carey joined Smith to put on 91 runs for the sixth wicket before Smith was out leg before to James Anderson for 93. Australia reviewed the decision but the on-field umpire's decision stayed as an 'umpire's call'. Carey brought up his fifty before being out for 51 caught at cover by Haseeb Hameed off the bowling of Anderson. Australia brought up their 450 before declaring for 473 at the loss of 9 wickets.


title: 2021–22 Ashes series

text: England opened their innings with Hameed and Rory Burns, both of whom were out for single digit scores bringing England to a total of 12 runs for the loss of two wickets. Hameed was out for 6 runs caught at mid-on by Mitchell Starc off the bowling of Michael Neser, while Burns was out for 4 runs caught by Smith at second slip off the bowling of Starc. England ended the day at 17 for two with Dawid Malan remaining not out at one and Joe Root remaining not out at 5.


title: 2021–22 Ashes series

text: Malan and Root resumed from their overnight positions and carried their bats through the first session with England getting to lunch at 140 runs for the loss of no additional wickets. Malan was unbeaten at 58 while Root was unbeaten at 40. Root was the first to be dismissed for 62 runs out caught by Smith at first slip off the bowling of Green. England were 150 for the loss of three wickets. Malan followed soon with the team adding 7 runs before Malan was out for 80 runs caught by Smith again at first slip off the bowling of Starc. England lost their next six wickets within 79 runs folding for a total of 236 runs. For the Australians Starc took four wickets conceding 37 runs, while spinner Nathan Lyon took three wickets conceding 58 runs.


title: 2021–22 Ashes series

text: Australia chose not to enforce the follow-on and ended the day 45 for one, with Warner run-out in a mixup with batsman Harris, with a combination of Broad and Buttler dislodging the stumps. Harris remained unbeaten at 21 while Neser was not out having scored two runs.


title: 2021–22 Ashes series

text: England took the field without Root, who was being treated for a lower abdominal injury sustained in the warm up session. Stokes took over as captain during Root's brief absence. Despite this setback, England started the day well, dismissing Neser, Harris and then the captain Smith in quick succession to have Australia 55 for four. But then Robinson, who had taken two of the day's wickets, switched from seam bowling to off-spin, presumably to lift the over rate. Root returned soon afterwards to the field, but by that time Australia had steadied, Head having combined with Labuschagne and helping Australia end the first session 134 for four, leading by 371 runs.


title: 2021–22 Ashes series

text: Head was first to depart when play resumed, caught by Stokes at deep square leg off Robinson; his 51 had come off only 54 deliveries. Labuschagne also reached a half-century to back up his mighty first-innings effort before departing, also for 51, caught by Stokes (this time fielding at deep mid wicket) off the part-time leg spin of Malan, his first test wicket. Although wickets continued to fall at regular intervals, the remaining batters continued to add runs and when Smith declared at 230 for nine after Richardson had been dismissed, England had been set a winning target of 468 in the remaining four sessions.


title: 2021–22 Ashes series

text: England resumed their second innings chasing a target of 468 runs and were immediately down a wicket when Hameed was out for a duck caught by the wicketkeeper Carey off the bowling of Jhye Richardson. Malan joined Burns and the duo put on 44 runs before Malan was out for 20 runs leg before to Neser. Burns and Root followed soon after with England ending the day at 82 runs for the loss of four wickets.


title: 2021–22 Ashes series

text: Chasing a stiff target, England were at 142 runs for six wickets at dinner with Buttler at 16 and Chris Woakes at 28. The duo added 61 runs for the seventh wicket before Woakes was out for 44 runs out, bowled Richardson. Buttler, having faced 207 deliveries, trod on his stumps trying to work a cut behind square; his was the ninth wicket and England were all out for 192. Australia won the game by 275 runs, going 2-0 up in the five-match series. Labuschagne was named player of the match for his century in the first innings and half century in the second innings, while Richardson was the pick of the bowlers, taking five wickets for 42 runs.


title: 2021–22 Ashes series

text: England made four changes to its team, with Jonny Bairstow, Zak Crawley, Jack Leach and Mark Wood replacing Stuart Broad, Rory Burns, Ollie Pope and Chris Woakes. Australia also made some changes, with Jhye Richardson and Michael Neser making way for Pat Cummins returning as captain, and debutant Scott Boland, a local boy who became only the fourth Indigenous Australian cricketer to play at Test level. After the start of play was delayed by rain, England's batting woes at the top of the order continued, with Cummins having dismissed openers Hameed and Crawley with only 13 runs on the board. It was yet again up to Root and Malan to rescue the innings, and the pair added 48 runs before Malan lost his wicket on the last ball before lunch, caught in the slips by Warner off Cummins for 14. England's score was 61 for three.


title: 2021–22 Ashes series

text: Warner and Harris gave Australia its best start of the series, combining for an opening stand of 57 in 14 overs before Warner fell for a brisk 38 (off 42 balls) when a thick edge off Anderson was caught by Crawley at gully. Lyon was sent in as nightwatchman for the remaining two overs as Australia ended the day's play at 61 for one.


title: 2021–22 Ashes series

text: The start of the second day was delayed by half an hour after it was revealed that four non-playing members of the England squad – two support staff and two family members – had tested positive to COVID-19. Australia increased their run total to 267 before being bowled out. Anderson claimed three more wickets to bring his total to 4/33 in the first innings.


title: 2021–22 Ashes series

text: England once again found themselves in trouble early in the innings. Mitchell Starc dismissed Zak Crawley and Dawid Malan on consecutive deliveries in the fifth over. In the eleventh over, Scott Boland dismissed Haseeb Hameed and Jack Leach in three balls. Day two ended with England 4/31 and trailing Australia by 51 runs.


title: 2021–22 Ashes series

text: In the fifth over of the day, Starc bowled Ben Stokes out with England at 5/46. Boland dismissed the next four batsmen (Jonny Bairstow, Joe Root, Mark Wood, and Ollie Robinson) with ducks from Wood and Robinson. Cameron Green bowled James Anderson out to end the Test with Australia winning by an innings and 14 runs. England's score of 68 runs was their lowest innings score on Australian soil since 1904; their previous lowest had also come at the MCG.


title: 2021–22 Ashes series

text: Boland earned the Mullagh Medal after taking six wickets for seven runs during England's second innings.


title: 2021–22 Ashes series

text: Abbreviations: "H / S" – Highest score in an innings; "S / R" – Strike rate: runs per 100 balls faced


title: The Highlights

text: The Highlights is the second greatest hits album (and first released globally) by Canadian singer the Weeknd. It was released on February 5, 2021, and follows the release of his fourth studio album "After Hours" (2020) and his first greatest hits album "The Weeknd in Japan" (2018). It was released in anticipation of his performance at the Super Bowl LV halftime show. The track list is composed of songs from his three "Billboard" 200 number one studio albums: "Beauty Behind the Madness" (2015), "Starboy" (2016), and "After Hours" (2020), his debut mixtape "House of Balloons" (2011), his EP "My Dear Melancholy" (2018), and his two co-lead collaborations "Love Me Harder" and "Pray for Me", with Ariana Grande and Kendrick Lamar respectively, from the albums "My Everything" (2014) and "" (2018).


title: The Highlights

text: The album's cover continues the usage of the red suit that The Weeknd donned throughout the promotional material for his fourth studio album "After Hours" (2020). The particular suit worn in the cover art is the same one that he wore in the music video for the song "Save Your Tears", which served as the fourth single from the aforementioned album.


title: The Highlights

text: "The Highlights" debuted at number two on the US "Billboard" 200 with 89,000 album-equivalent units, including 10,000 pure album sales. It is the Weeknd's highest charting compilation album, and marks the biggest first week debut for a greatest hits set since Blake Shelton's "" (2019), which peaked at number two on the chart dated December 28, 2019.


title: Flak tower

text: Flak towers () were large, above-ground, anti-aircraft gun blockhouse towers constructed by Nazi Germany. There were 8 flak tower complexes in the cities of Berlin (three), Hamburg (two), and Vienna (three) from 1940 onwards. Other cities that used flak towers included Stuttgart and Frankfurt. Smaller single-purpose flak towers were built at key outlying German strongpoints, such as at Angers in France, and Helgoland in Germany.


title: Flak tower

text: The towers were operated by the Luftwaffe to defend against Allied strategic air raids against these cities during World War II. They also served as air-raid shelters for tens of thousands of local civilians.


title: Flak tower

text: After the RAF's raid on Berlin in 1940, Adolf Hitler ordered the construction of three massive flak towers to defend the capital from air attack. Each tower had a radar installation with a radar dish which could be retracted behind a thick concrete and steel dome for protection.


title: Flak tower

text: Hitler was interested in the design of the towers, and even made some sketches. They were constructed in six months. The priority of the project was such that the German national rail schedule was altered to facilitate the shipment of concrete, steel and timber to the construction sites.


title: Flak tower

text: With concrete walls up to thick, their designers considered the towers to be invulnerable to attack by the standard ordnance carried by RAF heavy bombers at the time of their construction.


title: Flak tower

text: The towers were able to sustain a rate of fire of per minute from their multi-level guns (albeit mostly smaller-caliber shells, such as the 2cm FlaK 30), with a range of up to in a field of fire. However, only the FlaK 40 guns had effective range to defend against the RAF and USAAF heavy bombers. The three flak towers around the outskirts of Berlin created a triangle of anti-aircraft fire that covered the centre of Berlin.


title: Flak tower

text: The flak towers had also been designed with the idea of using the above-ground bunkers as a civilian shelter, with room for and a hospital ward inside. During the Battle of Berlin, occupants formed their own communities, with up to taking refuge in one tower during the battle. These towers, much like the keeps of medieval castles, were some of the safest places in a fought-over city and so the flak towers were some of the last places to surrender to the Red Army, eventually being forced to capitulate as supplies dwindled.


title: Flak tower

text: The Soviets, in their assault on Berlin, found it difficult to inflict significant damage on the flak towers, even with some of the largest Soviet guns, such as the 203 mm M1931 howitzers.


title: Flak tower

text: After the war, the demolition of the towers was often considered not feasible and many remain to this day, with some having been converted for alternative use.


title: Flak tower

text: The G-Towers were square and tall, usually armed with eight (four twin) 12.8 cm FlaK 40 and numerous 37 mm Flak and 32 (eight quadruple) 20mm Flakvierling guns.


title: Flak tower

text: G-Towers were , usually armed with eight (four twin) 128 mm guns and sixteen (four quadruple) 20 mm guns.


title: Flak tower

text: The G-Towers were , usually armed with eight (four twin) 128 mm guns and thirty-two (eight quadruple) 20 mm guns.


title: Flak tower

text: The evaluation of even larger Battery Towers was commissioned by Adolf Hitler. These would have been three times the size and firepower of flak towers.


title: Flak tower

text: The tower built near the Berlin Zoo was the first generation type and covered the government district. It was also used as a repository for artefacts from the Berlin museum. The occupants surrendered to Soviets on 30 April 1945.


title: Flak tower

text: In 1947 the British blew up the G-Tower on the second attempt with several tons of explosives. The L-Tower was demolished first in July.


title: Flak tower

text: Both towers were covered over and now appear to be natural hills in Volkspark Friedrichshain. The G-Tower, known as "Mont Klamott" (Rubble Mountain) in Berlin, was the inspiration for songs by singer-songwriter Wolf Biermann and the rock band Silly.


title: Flak tower

text: The G-Tower was partially demolished after the war; one side remains visible. The interior can be visited. . The L-Tower was partially demolished after the war; some walls remain visible.


title: Flak tower

text: This tower, containing six levels below the rooftop, includes in its design, as part of its air-raid shelter, two identical spaces for protection against gas attacks, one on the first floor (above ground level) and the other on the second floor. Both in Tower 1, they are about 300 sq. m. (3,230 sq. ft.) in area, and have six windows (openings in the wall).


title: Flak tower

text: The L-Tower was demolished after the war and replaced by a building owned by T-Mobile. . The G-Tower was transformed into a nightclub with a music school and music shops. In 2019 the NH Hotel Group announced plans to turn it into a luxury hotel with a forest on top of it, with construction to take place in 2021 and opening in 2022. After the reconstruction the height was to increase to 58m with some additional floors. There were to be thirteen stairs.


title: Theme from New York, New York

text: "Theme from "New York, New York"", often abbreviated to just "New York, New York", is the theme song from the Martin Scorsese film "New York, New York" (1977), composed by John Kander, with lyrics by Fred Ebb. It was written for and performed in the film by Liza Minnelli. It remains one of the best-known songs about New York City. In 2004 it ranked #31 on AFI's 100 Years...100 Songs survey of top tunes in American cinema.


title: Theme from New York, New York

text: Composer John Kander and Lyricist Fred Ebb stated on the A&E "Biography" episode about Liza Minnelli, that they attribute the song's success to actor Robert De Niro, who rejected their original theme for the film because he thought it was "too weak". The song did not become a popular hit until it was picked up in concert by Frank Sinatra during his performances at Radio City Music Hall in October 1978.


title: Theme from New York, New York

text: In 1979, "Theme from "New York, New York"" was recorded by Frank Sinatra for his album "" (1980), and became closely associated with him as one of his signature songs. Don Costa received a Grammy nomination for the energetic orchestration. Sinatra occasionally performed the song live with Minnelli as a duet. Sinatra recorded it a second time for his 1993 album "Duets", with Tony Bennett.


title: Theme from New York, New York

text: Sinatra's recording peaked at #32 in June 1980, becoming his final Top 40 hit. It was also an Adult Contemporary hit, reaching #10 in the US and #2 in Canada. The song made a minor showing in the UK (#59); however, it recharted several years later and reached #4 in 1986. The song was nominated for a Grammy Award for Best Pop Vocal Performance, Male and Sinatra made two more studio recordings of the song in 1981 (for his NBC TV special "The Man and His Music") and 1993 (for Capitol Records). From the latter, an electronic duet with Tony Bennett was produced for Sinatra's "Duets" album.


title: Theme from New York, New York

text: The lyrics of the Sinatra versions differ slightly from Ebb's original lyrics. Notably, the phrase "A-number-one", which does not appear at all in the original lyrics, is sung twice at the song's rallentando climax. (Ebb has said he "didn't even like" Sinatra's use of "A-number-one".) The phrase is both the first and fourth on a list of three superlative titles the singer strives to achieve — "A-number-one, top of the list, king of the hill, A-number-one" — where Ebb's original lyrics (performed by Minnelli) were "king of the hill, head of the list, cream of the crop, and the top of the heap."


title: Theme from New York, New York

text: Minnelli's original recording of the song (also used in the Tony Bennett version in "Duets") uses the following closing line:


title: Theme from New York, New York

text: Despite Sinatra's version becoming more familiar, original singer Minnelli had two of the tune's most memorable live performances – during the July 4, 1986 ceremony marking the rededication of the Statue of Liberty after extensive renovations, and in the middle of the seventh inning at Shea Stadium during a New York Mets game, the first pro sports event in the metro area after the September 11, 2001 attacks. She also sang it at Los Angeles Memorial Coliseum during the 1984 Summer Olympics opening ceremony, accompanied by 24 pianos and strobe lights.


title: Theme from New York, New York

text: The song has been embraced as a celebration of New York City, and is often heard at New York City social events, such as weddings and bar mitzvahs. Many sports teams in New York City have played this song in their arenas/stadiums, but the New York Yankees are the most prominent example. It has been played over the loudspeakers at both the original and current Yankee Stadiums at the end of every Yankee home game since July 1980. Originally, Sinatra's version was played after a Yankees win, and the Minnelli version after a loss. However, due to a complaint from Minnelli, the Sinatra version is now heard regardless of the game's outcome.


title: Theme from New York, New York

text: Liza Minnelli performed it live on September 21, 2001 at Shea Stadium during the seventh inning stretch which was the first game in New York after the attacks on the World Trade Center. The Mets beat the Braves with a dramatic home run by Mike Piazza. It is known as the “9/11 game”.


title: Theme from New York, New York

text: From the 2005 season until 2020, at the Richmond County Bank Ballpark following Staten Island Yankees games, the Sinatra version was heard regardless of the game's outcome, and was formerly done at Shea Stadium at the end of New York Mets games after the September 11, 2001 attack. Previously, Mets fans felt it was a "Yankee song", and began booing it when it was played. It actually first had snippets of the song played after World Series home runs by Ray Knight and Darryl Strawberry during Game 7 of the 1986 World Series. The song is also sometimes played at New York Knicks games. The Sinatra version is played at the end of every New York Rangers game at Madison Square Garden. It was played at the opening faceoff of Game 7 of the 1994 Stanley Cup Finals at the Garden. The song has also been the post parade song for the Belmont Stakes from 1997 to 2009, and since 2011. Sinatra's version of the song has been played at the end of all four Super Bowls that the New York Giants have won to date, as well as before kickoff of Super Bowl XLVIII, while Minnelli's version was heard after the Giants' Super Bowl XXXV loss.


title: Theme from New York, New York

text: The song was the musical basis for Jimmy Picker's 1983 three-minute animated short, "Sundae in New York", which won the Oscar for Best Short Film (Animated) that year, with a likeness of then-mayor Ed Koch somewhat stumbling through the song, with clay caricatures of New York-based celebrities (including Alfred E. Neuman) and finishing the song with "Basically, I think New York is very therapeutic. Hey, an apple a day is... uh... great for one's constitution!" and burying his face in a big banana split with "THE END" written on his bald head. (Koch used the same "rallentando" climax Sinatra used, albeit with one big difference: "A-number one, top of the list, king of the hill..." followed by his impression of Groucho Marx completing, "...and incidentally a "heckuva" nice guy!")


title: Theme from New York, New York

text: An instrumental version of the song is used as the main theme music for NBC's broadcasts of the Macy's Thanksgiving Day Parade. The Frank Sinatra version is also played during the annual Times Square New Year's Day celebrations, immediately after the first verse of "Auld Lang Syne" and the ball drop that signifies the new year.


title: Theme from New York, New York

text: Queen covered the song for the 1986 fantasy film "Highlander". Unlike the other songs recorded for the film, it has never appeared on a Queen album.


title: Theme from New York, New York

text: The song is performed during the end credits "The City of New York vs. Homer Simpson", the first episode of the ninth season of "The Simpsons".


title: Theme from New York, New York

text: In "Arrested Development" episode 8 of season 2, which originally aired in January 2005, Tobias, played by David Cross, starts singing the song in his club. Lucille 2, played by Liza Minnelli, is in the audience and remarks "Everyone thinks they're Frank Sinatra."


title: Theme from New York, New York

text: In DreamWorks Animation's "Madagascar" (2005), the song is introduced in Central Park Zoo, and Marty later sings the song in the midst of Alex the Lion's delirium.


title: Theme from New York, New York

text: In the series premiere of the popular CBS crime-drama series "Blue Bloods", back in 2010, the song is playing while Jamie is walking out in front of his family while about to graduate from the police academy and when they throw their hats.


title: Theme from New York, New York

text: In the 2011 film "Shame", Sissy Sullivan, played by Carey Mulligan, sings a slower version of this song at a bar.


title: Theme from New York, New York

text: The song was also featured in the James Bond film Spectre, during the car chase in Rome, much to Bond's annoyance.


title: Theme from New York, New York

text: In 2016, the American filmmaker and YouTube personality Casey Neistat published a viral video titled "Snowboarding with the NYPD" which was set to the song.


title: Theme from New York, New York

text: The song was played thoroughly throughout New York City during the 2020 Coronavirus pandemic in the spring of 2020 by residents expressing pride and solidarity.


title: Theme from New York, New York

text: It is also commonly played at the AESSEAL New York Stadium, home of football team Rotherham United following a win. The stadium is named this due to the land it is built on being an old steel works factory which built the red fire hydrants that characterise New York, USA.


title: Fire alarm system

text: A fire alarm system warns people when smoke, fire, carbon monoxide or other fire-related or general notification emergency|emergencies are detected. These alarms may be activated automatically from smoke detectors and heat detectors or may also be activated via manual fire alarm activation devices such as manual call points or pull stations. Alarms can be either motorized bells or wall mountable sounders or horns. They can also be speaker strobes which sound an alarm, followed by a voice evacuation message which warns people inside the building not to use the elevators. Fire alarm sounders can be set to certain frequencies and different tones including low, medium, and high, depending on the country and manufacturer of the device. Most fire alarm systems in Europe sound like a siren with alternating frequencies. Fire alarm electronic devices are known as horns in the United States and Canada and can be either continuous or set to different codes. Fire alarm warning devices can also be set to different volume levels.


title: Fire alarm system

text: After the fire protection are established—usually by referencing the minimum levels of protection mandated by the appropriate model building code, insurance agencies, and other authorities—the fire alarm designer undertakes to detail specific components, arrangements, and interfaces necessary to accomplish these goals. Equipment specifically manufactured for these purposes is selected and standardized installation methods are anticipated during the design.


title: Fire alarm system

text: There are national codes in each European country for planning, design, installation, commissioning, use and maintenance of fire detection system with additional requirements that are mentioned on TS 54 -14


title: Fire alarm system

text: As per NFPA 72, 18.4.2 (2010 Edition)Temporal Code 3 is the standard audible notification in a modern system. It consists of a repeated three-pulse cycle (0.5 s on, 0.5 s off, 0.5 s on, 0.5 s off, 0.5 s on, 1.5 s off). Voice Evacuation is the second most common audible in a modern system. Legacy systems, typically found in older schools and buildings, have used continuous tones alongside other audible schemas.


title: Fire alarm system

text: Mass notification systems often extend the notification appliances of a standard fire alarm system to include PC based workstations, text-based digital signage, and a variety of remote notification options including email, text message, RSS feed , or IVR-based telephone text-to-speech messaging.


title: Fire alarm system

text: Fire alarm systems in nondomestic premises are generally designed and installed in accordance with the guidance given in BS 5839 Part 1. There are many types of fire alarm systems each suited to different building types and applications. A fire alarm system can vary dramatically in both price and complexity, from a single panel with a detector and sounder in a small commercial property to an addressable fire alarm system in a multioccupancy building.


title: Fire alarm system

text: An important consideration when designing fire alarms is that of individual zones. The following recommendations are found in BS 5839 Part 1:


title: Fire alarm system

text: Also, the NFPA recommends placing a list for reference near the FACP showing the devices contained in each zone.


title: Dakshineswar Kali Temple

text: Dakshineswar Kali Temple is a Hindu navaratna temple in Dakshineswar, Kolkata, West Bengal, India. Situated on the eastern bank of the Hooghly River, the presiding deity of the temple is Bhavatarini, a form of Parashakti Adya Kali, otherwise known as Adishakti Kalika. The temple was built in 1855 by Rani Rashmoni, a Zamindar, philanthropist and a devotee of Kali. The temple is known for its association with Ramakrishna and Ma Sarada Devi, mystics of 19th century Bengal.


title: Dakshineswar Kali Temple

text: The main temple was inspired by Navaratna style Radhakanta temple, built by Babu Ramnath Mondal of Tollygunge. The temple compound, apart from the nine-spired main temple, contains a large courtyard surrounding the temple, with rooms along the boundary walls. There are twelve shrines dedicated to Shiva—Kali's consort—along the riverfront, a temple to Radha-Krishna, a bathing ghat on the river, a shrine dedicated to Rani Rashmoni. 'Nahabat', the chamber in the northwestern corner just beyond the last of the Shiva temples, is where Ramakrishna and Maa Sarada spent a considerable part of their lives.


title: Dakshineswar Kali Temple

text: The Dakshineswar Kali Temple was founded around the middle of the 19th century by Rani Rashmoni. Rani Rashmoni was a Mahishya by caste and was well known for her philanthropic activities. In the year 1847, Rashmoni prepared to go upon a long pilgrimage to the sacred Hindu city of Kashi to express her devotions to the Divine Mother. Rani was to travel in twenty-four boats, carrying relatives, servants, and supplies. According to traditional accounts, the night before the pilgrimage began, Rashmoni had a vision of the Divine Mother goddess Kali in a dream and reportedly said,


title: Dakshineswar Kali Temple

text: Profoundly affected by the dream, Rani immediately looked for and purchased a 30,000-acre plot in the village of Dakshineswar. The large temple complex was built between 1847 and 1855. The plot was bought from an Englishman, Jake Hastie, and was then popularly known as "Saheban Bagicha". Partly old Muslim burial ground shaped like a tortoise, considered befitting for the worship of Shakti according to Tantra traditions, it took eight years and nine hundred thousand rupees to complete the construction. The idol of Goddess Kali was installed on the "Snana Yatra" day on 31 May 1855 amid festivities at the temple formerly known as "Sri Sri Jagadishwari Kali", with Ramkumar Chhattopadhyay as the head priest. Soon his younger brother Gadai or Gadadhar (later known as Ramakrishna) moved in and so did his nephew Hriday to assist him. On 31 May 1855 more than 1 lakh (one hundred thousand) Brahmins were invited from different parts of the country to grace the auspicious occasion. The next year, Ramkumar Chattopadhyay died, and the position was given to Ramakrishna. His wife, Sarada Devi, stayed in the south side of the "Nahabat" (music room) in a small room on the ground floor, which is now a shrine dedicated to her. Ramakrishnana was responsible for bringing much in the way of both fame and pilgrims to the temple.


title: Dakshineswar Kali Temple

text: Rani Rashmoni lived for only five years and nine months after the inauguration of the temple. She fell seriously ill in 1861. Realizing that her death was near, she decided to hand over the property she had purchased in Dinajpur (now in Bangladesh) as a legacy for the maintenance of the temple to the temple trust. She accomplished her task on 18 February 1861 and died on the next day. After her death, her sons-in-law took to celebrating Durga Puja in their respective premises.


title: Dakshineswar Kali Temple

text: Built in the navaratna or nine spires style of Bengal architecture, the three-storeyed south-facing temple has nine spires distributed in the upper two storeys, and stands on a high platform with a flight of stairs, overall it measures square and rises over high.


title: Dakshineswar Kali Temple

text: The "garbha griha" (sanctum sanctorum) houses an idol of goddess Kali, known as "Bhavatarini", standing on the chest of a supine Shiva, and the two idols are placed on a thousand-petaled lotus throne made of silver.


title: Dakshineswar Kali Temple

text: Close to the main temple are the row of twelve identical Shiva temples built facing the east in the typical "aat-chala" Bengal architecture, they are built on either side of the ghat on the Hooghly river. To the North east of the Temple Complex is the Vishnu Temple or the Radha Kanta Temple. A flight of steps lead to the columned verandah and into the temple where a silver throne rests with a idol of Lord Krishna and a idol of Radha.


title: Invincible-class aircraft carrier

text: The "Invincible" class was a class of light aircraft carrier operated by the Royal Navy. Three ships were constructed: , and . The vessels were built as aviation-capable anti-submarine warfare (ASW) platforms to counter the Cold War North Atlantic Soviet submarine threat, and initially embarked Sea Harrier aircraft and Sea King HAS.1 anti-submarine helicopters. With cancellation of the aircraft carriers renewal programme in the 1960s, the three ships became the replacements for "Ark Royal" and "Eagle" fleet carriers and the light fleet carriers, and the Royal Navy's sole class of aircraft carrier.


title: Invincible-class aircraft carrier

text: The three vessels saw active service in a number of locations, including the South Atlantic during the Falklands War, the Adriatic during the Bosnian War, and in the Middle East for the 2003 Invasion of Iraq.


title: Invincible-class aircraft carrier

text: "Invincible" was decommissioned in 2005 and put in reserve in a low state of readiness. She was sold to a Turkish scrapyard in February 2011, and left Portsmouth under tow on 24 March 2011. Pursuant to the Strategic Defence and Security Review, 2010, "Ark Royal" followed, decommissioning on 13 March 2011. This left "Illustrious" as the sole remaining ship, serving as a helicopter carrier from 2011 to 2014 when it was decommissioned as well. The Royal Navy was without an aircraft carrier for the first time in nearly a century, until the commissioning of the first of two aircraft carriers in December 2017.


title: Invincible-class aircraft carrier

text: The "Invincible" class has its origins in a sketch design for a 6,000-ton, guided-missile armed, helicopter carrying escort cruiser intended as a complement to the much larger CVA-01-class fleet aircraft carrier. The cancellation of CVA-01 in 1966 meant that the smaller cruiser would now have to provide the anti-submarine warfare (ASW) taskforce with command and control facilities. Two new designs were prepared for this requirement; a 12,500-ton cruiser with missiles forward, six Westland Sea King helicopters and a flight deck aft, somewhat similar to of the Italian Navy and a larger 17,500-ton vessel with a "through-deck", nine Sea Kings and missiles right forward. By 1970, the "through-deck" design had advanced into a Naval Staff Requirement for an 18,750-ton Through-Deck Command Cruiser (TDCC).


title: Invincible-class aircraft carrier

text: In February 1963, the Hawker P.1127 VTOL (vertical take-off and landing) aircraft had landed and taken-off from the carrier and the subsequent Hawker-Siddeley Kestrel had undergone trials from the "Commando carrier" (an aircraft carrier operating helicopters) . It was therefore perfectly possible that the new "cruisers" could be used to operate VTOL aircraft. The new ships were called "through-deck cruisers" and not "aircraft carrier". This was in part because CVA-01's cancellation was so recent, but also because the ships were intended to serve in traditional cruiser roles of CI and anti-submarine warfare, and were constructed like cruisers. The "aircraft carrier" name did not officially appear in association with the ships until the 1980 Defence Estimates referred to the "Invincible"s as such.


title: Invincible-class aircraft carrier

text: Economic problems in the UK in the early 1970s delayed progress on the new ships, but the design continued to evolve. The order for the first ship was given to Vickers (Shipbuilding) on 17 April 1973. By now, the design was for a 19,000-ton "CAH" (helicopter carrying heavy cruiser, styled after the US Navy hull classification symbols) with up to fourteen aircraft and a Sea Dart missile launcher on the bows.


title: Invincible-class aircraft carrier

text: The government decided that the carrier needed fixed-wing aircraft to defend against Soviet reconnaissance aircraft. In May 1975, it authorised the maritime version of the Hawker Siddeley Harrier, which was successfully developed into the Sea Harrier. This meant that the design was reworked again to include a small complement of these VTOL aircraft. The comparatively short flight deck made even STOVL rolling take-offs marginal for launching fully laden Harriers. The development of the ski-jump made it possible to launch in all conditions by propelling the aircraft upwards at the end of its take-off roll. "Invincible" and "Illustrious" were equipped with 7° ski-jumps, and "Ark Royal" carried a 12° ski-jump. The class also had, since 1976, a secondary role as a helicopter carrier, or LPH, in the reinforcement of NATO's Northern flank in Norway. In 1998, , with a hull form based on that of the "Invincible" class, was commissioned specifically for this role.


title: Invincible-class aircraft carrier

text: After the 1982 Falklands War, CIWS guns were added to the design. "Illustrious" had them fitted at the last minute before commissioning, "Ark Royal" had them added as a normal part of the building process, and "Invincible" had them fitted during her first overhaul after the Falklands. Initially, "Invincible" and "Illustrious" were fitted with two Vulcan Phalanx units; these were replaced with three Goalkeeper systems. "Ark Royal" retained the three Phalanx CIWS systems she was fitted with when built (she could be easily distinguished from her sisters by the Phalanx's distinctive white "R2-D2" radome). Electronic countermeasures were provided by a Thales jamming system and ECM system. Seagnat launchers were provided for chaff or flare decoys. As part of upgrades during the mid-1990s, all three ships had the Sea Dart removed, with the forecastle filled in to increase the size of the flight deck.


title: Invincible-class aircraft carrier

text: In the mid-1970s, the Shah of Iran expressed interest in acquiring three "Invincible"-class ships and a fleet of twenty-five Sea Harriers to provide fleet defence. When the Iranian Navy could not commit to providing sufficient personnel for manning the vessels, the ship order was cancelled in 1976. A later proposal to buy four "Harrier-type" vessels was also discarded, as were later negotiations to buy the Sea Harrier.


title: Invincible-class aircraft carrier

text: The 1981 Defence White Paper and its planned reduction in the size of the carrier fleet saw "Invincible" marked as surplus to requirements, and the ship was offered for sale to the Royal Australian Navy in July 1981 as a replacement for the ageing aircraft carrier . The class had previously been considered and discarded as a potential replacement for the Australian ship, but the low GB£175 million (A$285 million) offer price and the already-constructed state of the vessel prompted the Australian government to announce in February 1982 their intention to accept the British offer. In Australian service, the ship would have been named HMAS "Australia", and would operate as a helicopter carrier until a later decision on the acquisition of Sea Harriers was made. "Invincible"s service during the Falklands War showed that the White Paper's suggested reductions were flawed and both nations withdrew from the deal in July 1982.


title: Invincible-class aircraft carrier

text: Prior to 1982, "Invincible"s air group consisted purely of Sea King HAS.5 anti-submarine helicopters and Sea Harrier FRS.1 aircraft. Typically, nine Sea Kings, and four or five Sea Harriers were embarked. This was due to the fact that the originally envisioned mission for the ships was to provide the heart of ASW hunter-killer groups in the North Atlantic during a war against the Soviet Union. In that context, the main weapon of the carrier would not be its fighter aircraft, but its ASW helicopters. The fighters were on board to shoot down the occasional Soviet maritime patrol aircraft nosing around the ship and its escorts.


title: Invincible-class aircraft carrier

text: The Falklands War changed that posture, since it proved that Britain needed to retain the capability to use carrier air power in its traditional role of power projection, both over land, and against enemy fleets. The Falklands War saw "Invincible", and the larger and older filled to capacity with both the Sea Harrier and the Royal Air Force Harrier GR3 ground attack variant of the aircraft, along with ASW helicopters. The RAF Harriers proved to be a temporary aberration at the time, but a permanent addition to the usual air group was made due to lessons learned during the war: the Sea King AEW2A (airborne early warning) version. "Illustrious" carried the first examples of the type when it was rushed south in the aftermath of the Falklands War to relieve "Invincible" of its guard duty around the islands.


title: Invincible-class aircraft carrier

text: In the aftermath of the Falklands, the typical air group was three AEW Sea Kings, nine ASW Sea Kings and eight or nine Sea Harriers. Analysis of the Sea Harrier's performance during the war lead to the requirement for an upgrade, approval for which was granted in 1984. The Sea Harrier FA2 entered service in 1993 and deployed on "Invincible" to Bosnia in 1994. The FA2 featured the Blue Vixen radar which was described as one of the most advanced pulse Doppler radars in the world. The FA2 carried the AIM-120 AMRAAM. The final new build Sea Harrier FA2 was delivered on 18 January 1999. Other improvements were made to the class during the 1980s and early 1990s, in particular to increase the ski-jump exit angle on "Invincible" and "Illustrious" to 12° to match "Ark Royal".


title: Invincible-class aircraft carrier

text: In later years, three other changes were made. One was the removal of the Sea Dart system, creating an increased deck park for aircraft. The Sea Dart magazines were converted to increase air-to-surface weapons stowage, and new aircrew briefing facilities created under the extended flight deck, both to support the embarkation of RAF Harrier GR7s as a routine part of the air group. The ships were all fitted to handle Merlin helicopters as the Merlin HM1 replaced the Sea King HAS6 in the carrier-borne ASW role. Following the integration of the Harrier GR7, typical deployments included seven or eight of those aircraft, pushing the Merlin onto the carrier's accompanying "Fort"-class auxiliaries.


title: Invincible-class aircraft carrier

text: The last wartime deployments of the class saw them in their secondary LPH role, as it was officially judged that Sea Harriers could provide no useful role in the missions. During those deployments, the class embarked RAF Chinook helicopters, in lieu of their fixed-wing complement.


title: Invincible-class aircraft carrier

text: "Illustrious" underwent a 16-month £40 million refit at Rosyth Dockyard during 2010 and 2011 in preparation for her new role as a helicopter carrier during the refit of .


title: Invincible-class aircraft carrier

text: The Sea Harrier was officially retired on 1 April 2006. The principal weapon of the "Invincible"-class carriers then became the Harrier GR9 flown by two Fleet Air Arm and two RAF squadrons until they were retired in 2010.


title: Invincible-class aircraft carrier

text: "Invincible" was decommissioned in July 2005, and was mothballed until September 2010. On 24 March 2011 "Invincible" left Portsmouth under tow for scrapping at Leyal Ship Recycling, Turkey.


title: Invincible-class aircraft carrier

text: "Ark Royal" took over as the flagship, was planned to be decommissioned in 2016, but retired in 2010 following the Strategic Defence and Security Review.


title: Invincible-class aircraft carrier

text: "Illustrious" remained the only one of the class in service, but was also retired in 2014. After being laid up it left Portsmouth under tow to the shipbreakers in Turkey on 7 December 2016.


title: Invincible-class aircraft carrier

text: Two larger s replaced the Invincible class, with the first, HMS "Queen Elizabeth", commissioned in late 2017. They displace around 65,000 tonnes each – more than three times the displacement of the "Invincible" class.


title: Differential form

text: In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.


title: Differential form

text: For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of :


title: Differential form

text: The symbol denotes the exterior product, sometimes called the "wedge product", of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials formula_3


title: Differential form

text: The differential forms form an alternating algebra. This implies that formula_4 and formula_5 This alternating property reflects the orientation of the domain of integration.


title: Differential form

text: The exterior derivative is an operation on differential forms that, given a -form formula_6, produces a -form formula_7 This operation extends the differential of a function (a function can be considered as a -form, and its differential is formula_8) This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.


title: Differential form

text: Differential -forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and -forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.


title: Differential form

text: Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, "Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics)".


title: Differential form

text: A differential -form can be integrated over an oriented manifold of dimension . A differential -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential -form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.


title: Differential form

text: Integration of differential forms is well-defined only on oriented manifolds. An example of a 1-dimensional manifold is an interval , and intervals can be given an orientation: they are positively oriented if , and negatively oriented otherwise. If then the integral of the differential -form over the interval (with its natural positive orientation) is


title: Differential form

text: which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:


title: Differential form

text: This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (), the increment is negative in the direction of integration.


title: Differential form

text: More generally, an -form is an oriented density that can be integrated over an -dimensional oriented manifold. (For example, a -form can be integrated over an oriented curve, a -form can be integrated over an oriented surface, etc.) If is an oriented -dimensional manifold, and is the same manifold with opposite orientation and is an -form, then one has:


title: Differential form

text: These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function with respect to a measure and integrates over a subset , without any notion of orientation; one writes formula_12 to indicate integration over a subset . This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details.


title: Differential form

text: Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra. The differentials of a set of coordinates, , ..., can be used as a basis for all -forms. Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general -form is a linear combination of these differentials at every point on the manifold:


title: Differential form

text: where the are functions of all the coordinates. A differential -form is integrated along an oriented curve as a line integral.


title: Differential form

text: The expressions , where can be used as a basis at every point on the manifold for all -forms. This may be thought of as an infinitesimal oriented square parallel to the –-plane. A general -form is a linear combination of these at every point on the manifold: and it is integrated just like a surface integral.


title: Differential form

text: A fundamental operation defined on differential forms is the exterior product (the symbol is the wedge ). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,


title: Differential form

text: because the square whose first side is and second side is is to be regarded as having the opposite orientation as the square whose first side is and whose second side is . This is why we only need to sum over expressions , with ; for example: . The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that , in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, if any two of the indices , ..., are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.


title: Differential form

text: A common notation for the wedge product of elementary -forms is so called multi-index notation: in an -dimensional context, for we define Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length , in a space of dimension , denoted Then locally (wherever the coordinates apply), formula_15 spans the space of differential -forms in a manifold of dimension , when viewed as a module over the ring of smooth functions on . By calculating the size of formula_16 combinatorially, the module of -forms on an -dimensional manifold, and in general space of -covectors on an -dimensional vector space, is  choose : This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.


title: Differential form

text: In addition to the exterior product, there is also the exterior derivative operator . The exterior derivative of a differential form is a generalization of the differential of a function, in the sense that the exterior derivative of is exactly the differential of . When generalized to higher forms, if is a simple -form, then its exterior derivative is a -form defined by taking the differential of the coefficient functions:


title: Differential form

text: with extension to general -forms through linearity: if a_I \, dx^I \in \Omega^k(M)</math>,}} then its exterior derivative is


title: Differential form

text: In , with the Hodge star operator, the exterior derivative corresponds to gradient, curl, and divergence, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.


title: Differential form

text: The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in differential geometry, differential topology, and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an antiderivation of degree 1 on the exterior algebra of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on manifolds. It also allows for a natural generalization of the fundamental theorem of calculus, called the (generalized) Stokes' theorem, which is a central result in the theory of integration on manifolds.


title: Differential form

text: Let be an open set in . A differential -form ("zero-form") is defined to be a smooth function on – the set of which is denoted . If is any vector in , then has a directional derivative , which is another function on whose value at a point is the rate of change (at ) of in the direction:


title: Differential form

text: In particular, if is the th coordinate vector then is the partial derivative of with respect to the th coordinate vector, i.e., , where , , ..., are the coordinate vectors in . By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates , , ..., are introduced, then


title: Differential form

text: for any vectors , and any real number . At each point "p", this linear map from to is denoted and called the derivative or differential of at . Thus . Extended over the whole set, the object can be viewed as a function that takes a vector field on , and returns a real-valued function whose value at each point is the derivative along the vector field of the function . Note that at each , the differential is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential -form.


title: Differential form

text: Since any vector is a linear combination of its components, is uniquely determined by for each and each , which are just the partial derivatives of on . Thus provides a way of encoding the partial derivatives of . It can be decoded by noticing that the coordinates , , ..., are themselves functions on , and so define differential -forms , , ..., . Let . Since , the Kronecker delta function, it follows that


title: Differential form

text: The meaning of this expression is given by evaluating both sides at an arbitrary point : on the right hand side, the sum is defined "pointwise", so that


title: Differential form

text: Applying both sides to , the result on each side is the th partial derivative of at . Since and were arbitrary, this proves the formula .


title: Differential form

text: for each . Any differential -form arises this way, and by using it follows that any differential -form on may be expressed in coordinates as


title: Differential form

text: The second idea leading to differential forms arises from the following question: given a differential -form on , when does there exist a function on such that ? The above expansion reduces this question to the search for a function whose partial derivatives are equal to given functions . For , such a function does not always exist: any smooth function satisfies


title: Differential form

text: The skew-symmetry of the left hand side in and suggests introducing an antisymmetric product on differential -forms, the exterior product, so that these equations can be combined into a single condition


title: Differential form

text: This is an example of a differential -form. This -form is called the exterior derivative of . It is given by


title: Differential form

text: Differential -forms, -forms, and -forms are special cases of differential forms. For each , there is a space of differential -forms, which can be expressed in terms of the coordinates as


title: Differential form

text: for a collection of functions . Antisymmetry, which was already present for -forms, makes it possible to restrict the sum to those sets of indices for which .


title: Differential form

text: Differential forms can be multiplied together using the exterior product, and for any differential -form , there is a differential -form called the exterior derivative of .


title: Differential form

text: Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any smooth manifold . One way to do this is cover with coordinate charts and define a differential -form on to be a family of differential -forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.


title: Differential form

text: Let be a smooth manifold. A smooth differential form of degree is a smooth section of the th exterior power of the cotangent bundle of . The set of all differential -forms on a manifold is a vector space, often denoted .


title: Differential form

text: The definition of a differential form may be restated as follows. At any point , a -form defines an element


title: Differential form

text: where is the tangent space to at and is its dual space. This space is to the fiber at of the dual bundle of the th exterior power of the tangent bundle of . That is, is also a linear functional formula_32, i.e. the dual of the th exterior power is isomorphic to the th exterior power of the dual:


title: Differential form

text: Consequently, a differential -form may be evaluated against any -tuple of tangent vectors to the same point of . For example, a differential -form assigns to each point a linear functional on . In the presence of an inner product on (induced by a Riemannian metric on ), may be represented as the inner product with a tangent vector . Differential -forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics.


title: Differential form

text: The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping


title: Differential form

text: where is the symmetric group on elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding


title: Differential form

text: This map exhibits as a totally antisymmetric covariant tensor field of rank . The differential forms on are in one-to-one correspondence with such tensor fields.


title: Differential form

text: As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.