title: Renin–angiotensin system

text: Locally expressed renin–angiotensin systems have been found in a number of tissues, including the kidneys, adrenal glands, the heart, vasculature and nervous system, and have a variety of functions, including local cardiovascular regulation, in association or independently of the systemic renin–angiotensin system, as well as non-cardiovascular functions. Outside the kidneys, renin is predominantly picked up from the circulation but may be secreted locally in some tissues; its precursor prorenin is highly expressed in tissues and more than half of circulating prorenin is of extrarenal origin, but its physiological role besides serving as precursor to renin is still unclear. Outside the liver, angiotensinogen is picked up from the circulation or expressed locally in some tissues; with renin they form angiotensin I, and locally expressed angiotensin-converting enzyme, chymase or other enzymes can transform it into angiotensin II. This process can be intracellular or interstitial.


title: Renin–angiotensin system

text: In the adrenal glands, it is likely involved in the paracrine regulation of aldosterone secretion; in the heart and vasculature, it may be involved in remodeling or vascular tone; and in the brain, where it is largely independent of the circulatory RAS, it may be involved in local blood pressure regulation. In addition, both the central and peripheral nervous systems can use angiotensin for sympathetic neurotransmission. Other places of expression include the reproductive system, the skin and digestive organs. Medications aimed at the systemic system may affect the expression of those local systems, beneficially or adversely.


title: Renin–angiotensin system

text: In the fetus, the renin–angiotensin system is predominantly a sodium-losing system, as angiotensin II has little or no effect on aldosterone levels. Renin levels are high in the fetus, while angiotensin II levels are significantly lower; this is due to the limited pulmonary blood flow, preventing ACE (found predominantly in the pulmonary circulation) from having its maximum effect.


title: Solenoid

text: A solenoid () is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter, which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it.


title: Solenoid

text: The helical coil of a solenoid does not necessarily need to revolve around a straight-line axis; for example, William Sturgeon's electromagnet of 1824 consisted of a solenoid bent into a horseshoe shape (not unlike an arc spring).


title: Solenoid

text: Solenoids provide magnetic focusing of electrons in vacuums, notably in television camera tubes such as vidicons and image orthicons. Electrons take helical paths within the magnetic field. These solenoids, focus coils, surround nearly the whole length of the tube.


title: Solenoid

text: In engineering, the term "solenoid" refers not only to the electromagnet but to a complete apparatus providing an actuator that converts electrical energy to mechanical energy.


title: Solenoid

text: An infinite solenoid has infinite length but finite diameter. "Continuous" means that the solenoid is not formed by discrete finite-width coils but by many infinitely thin coils with no space between them; in this abstraction, the solenoid is often viewed as a cylindrical sheet of conductive material.


title: Solenoid

text: The magnetic field inside an infinitely long solenoid is homogeneous and its strength neither depends on the distance from the axis nor on the solenoid's cross-sectional area.


title: Solenoid

text: This is a derivation of the magnetic flux density around a solenoid that is long enough so that fringe effects can be ignored. In Figure 1, we immediately know that the flux density vector points in the positive "z" direction inside the solenoid, and in the negative "z" direction outside the solenoid. We confirm this by applying the right hand grip rule for the field around a wire. If we wrap our right hand around a wire with the thumb pointing in the direction of the current, the curl of the fingers shows how the field behaves. Since we are dealing with a long solenoid, all of the components of the magnetic field not pointing upwards cancel out by symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards.


title: Solenoid

text: Now consider the imaginary loop "c" that is located inside the solenoid. By Ampère's law, we know that the line integral of B (the magnetic flux density vector) around this loop is zero, since it encloses no electrical currents (it can be also assumed that the circuital electric field passing through the loop is constant under such conditions: a constant or constantly changing current through the solenoid). We have shown above that the field is pointing upwards inside the solenoid, so the horizontal portions of loop "c" do not contribute anything to the integral. Thus the integral of the up side 1 is equal to the integral of the down side 2. Since we can arbitrarily change the dimensions of the loop and get the same result, the only physical explanation is that the integrands are actually equal, that is, the magnetic field inside the solenoid is radially uniform. Note, though, that nothing prohibits it from varying longitudinally, which in fact, it does.


title: Solenoid

text: A similar argument can be applied to the loop "a" to conclude that the field outside the solenoid is radially uniform or constant. This last result, which holds strictly true only near the center of the solenoid where the field lines are parallel to its length, is important as it shows that the flux density outside is practically zero since the radii of the field outside the solenoid will tend to infinity. An intuitive argument can also be used to show that the flux density outside the solenoid is actually zero. Magnetic field lines only exist as loops, they cannot diverge from or converge to a point like electric field lines can (see Gauss's law for magnetism). The magnetic field lines follow the longitudinal path of the solenoid inside, so they must go in the opposite direction outside of the solenoid so that the lines can form loops. However, the volume outside the solenoid is much greater than the volume inside, so the density of magnetic field lines outside is greatly reduced. Now recall that the field outside is constant. In order for the total number of field lines to be conserved, the field outside must go to zero as the solenoid gets longer. Of course, if the solenoid is constructed as a wire spiral (as often done in practice), then it emanates an outside field the same way as a single wire, due to the current flowing overall down the length of the solenoid.


title: Solenoid

text: where formula_2 is the magnetic flux density, formula_3 is the length of the solenoid, formula_4 is the magnetic constant, formula_5 the number of turns, and formula_6 the current. From this we get


title: Solenoid

text: This equation is valid for a solenoid in free space, which means the permeability of the magnetic path is the same as permeability of free space, μ.


title: Solenoid

text: If the solenoid is immersed in a material with relative permeability μ, then the field is increased by that amount:


title: Solenoid

text: In most solenoids, the solenoid is not immersed in a higher permeability material, but rather some portion of the space around the solenoid has the higher permeability material and some is just air (which behaves much like free space). In that scenario, the full effect of the high permeability material is not seen, but there will be an effective (or apparent) permeability "μ" such that 1 ≤ "μ" ≤ "μ".


title: Solenoid

text: The inclusion of a ferromagnetic core, such as iron, increases the magnitude of the magnetic flux density in the solenoid and raises the effective permeability of the magnetic path. This is expressed by the formula


title: Solenoid

text: where "μ" is the effective or apparent permeability of the core. The effective permeability is a function of the geometric properties of the core and its relative permeability. The terms relative permeability (a property of just the material) and effective permeability (a property of the whole structure) are often confused; they can differ by many orders of magnitude.


title: Solenoid

text: For an open magnetic structure, the relationship between the effective permeability and relative permeability is given as follows:


title: Solenoid

text: A finite solenoid is a solenoid with finite length. Continuous means that the solenoid is not formed by discrete coils but by a sheet of conductive material. We assume the current is uniformly distributed on the surface of the solenoid, with a surface current density "K"; in cylindrical coordinates:


title: Solenoid

text: The magnetic field can be found using the vector potential, which for a finite solenoid with radius "R" and length "l" in cylindrical coordinates formula_12 is


title: Solenoid

text: Here, formula_20, formula_21, and formula_22 are complete elliptic integrals of the first, second, and third kind.


title: Solenoid

text: Inside the solenoid, far away from the ends (formula_27), this tends towards the constant value formula_28.


title: Solenoid

text: For the case in which the radius is much larger than the length of the solenoid (formula_29), the magnetic flux density through the centre of the solenoid (in the "z" direction, parallel to the solenoid's length, where the coil is centered at "z"=0) can be estimated as the flux density of a single circular conductor loop:


title: Solenoid

text: Within the category of finite solenoids, there are those that are sparsely wound with a single pitch, sparsely wound with varying pitches (varied-pitch solenoid), or those with a varying radius for different loops (non-cylindrical solenoids). They are called irregular solenoids. They have found applications in different areas, such as sparsely wound solenoids for wireless power transfer, varied-pitch solenoids for magnetic resonance imaging (MRI), and non-cylindrical solenoids for other medical devices.


title: Solenoid

text: The calculation of the intrinsic inductance and capacitance cannot be done using those for the traditional solenoids, i.e. the tightly wound ones. New calculation methods were proposed for the calculation of intrinsic inductance(codes available at ) and capacitance. (codes available at )


title: Solenoid

text: As shown above, the magnetic flux density formula_2 within the coil is practically constant and given by


title: Solenoid

text: where "μ" is the magnetic constant, formula_5 the number of turns, formula_6 the current and formula_3 the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density formula_2 by the cross-section area formula_37:


title: Solenoid

text: A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.


title: Solenoid

text: This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.


title: Solenoid

text: Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative permeability of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. The presence of a core can be taken into account in the above equations by replacing the magnetic constant "μ" with "μ" or "μμ", where "μ" represents permeability and "μ" relative permeability. Note that since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.


title: Pay television

text: Pay television, also known as subscription television, premium television or, when referring to an individual service, a premium channel, refers to subscription-based television services, usually provided by multichannel television providers, but also increasingly via digital terrestrial, and streaming television. In the United States, subscription television began in the late 1970s and early 1980s in the form of encrypted analog over-the-air broadcast television which could be decrypted with special equipment. The concept rapidly expanded through the multi-channel transition and into the post-network era. Other parts of the world beyond the United States, such as France and Latin America have also offered encrypted analog terrestrial signals available for subscription.


title: Pay television

text: The term is most synonymous with premium entertainment services focused on films or general entertainment programming such as, in the United States, Cinemax, Epix, HBO, Showtime, and Starz, but such services can also include those devoted to sports, as well as adult entertainment.


title: Pay television

text: In contrast to most other multichannel television broadcasters, which depend on advertising and carriage fees as their sources of revenue, the majority of pay television services rely almost solely on monthly subscription fees paid by individual customers. As a result, pay television outlets are most concerned with offering content that can justify the cost of the service, which helps to attract new subscribers, and retain existing subscribers.


title: Pay television

text: Many pay television services consist of multiple individual channels, referred to as "multiplex" services (in reference to multiplex cinemas), where a main flagship channel is accompanied by secondary services with distinct schedules focusing on specific genres and audiences (such as multiplexes focusing more on "classic" films, or family-oriented programming), time shifting, or brand licensing deals (such as channels focusing specifically on Disney films, or content from U.S. pay television brands if they do not specifically run their own network in a specific market). Typically, these services are bundled together with the main channel at no additional charge, and cannot be purchased separately.


title: Pay television

text: Depending on local regulations, pay television services generally have more lenient content standards because of their relatively narrower distribution, and not being subject to pressure from sponsors to tone down content. As a result, programming is typically aired with limited to no edits for time or, where applicable, mature content such as graphic violence, profanity, nudity, and sexual activity.


title: Pay television

text: As premium television services are commonly devoid of traditional commercial advertising, breaks between programming typically include promotions for upcoming programs, and interstitial segments (such as behind-the-scenes content, interviews, and other feature segments). Some sports-based pay services, however, may feature some commercial advertising, particularly if they simulcast sporting events that are broadcast by advertiser-supported television networks.


title: Pay television

text: In addition, most general interest or movie-based pay services do not adhere to the common top and bottom of the hour scheduling of other cable channels and terrestrial broadcasters. As such, programs often air using either conventional scheduling or have airtimes in five-minute increments (for example, 7:05 a.m. or 4:40 p.m.); since such channels broadcast content without in-program break interruptions, this sometimes leads to extended or abbreviated breaks between programs, depending on when the previous program concludes and when the start time of the next program is. The only universal variation to this is prime time, where the main channel in each pay service's suite usually schedules films to start on the hour.


title: Pay television

text: Films comprise much of the content seen on most pay television services, particularly those with a general entertainment format and those that focus exclusively on films. Services often obtain rights to films through exclusive agreements with film distributors. Films acquired during the original term of license agreements with a distributor may also be broadcast as "sub-runs," in which a service holds rights to film long after the conclusion of a distribution agreement (under this arrangement, the pay service that originally licensed the rights to a particular film title, or one other than that which had held rights, may hold the broadcast rights through a library content deal).


title: Pay television

text: Many general interest premium channels also produce original television series. Due to the aforementioned leniency in content standards, they too can contain content that is more mature than those of other cable channels or television networks. These series also tend to be high-budget and aim for critical success in order to attract subscribers: notable premium series, such as HBO's "Curb Your Enthusiasm", "Game of Thrones", "Sex and the City", and "The Sopranos", and Showtime's "Dexter", "Homeland", and "Weeds", have achieved critical acclaim and have won various television awards. Some premium channels also broadcast television specials, which most commonly consist of concerts and concert films, documentaries, stand-up comedy, and in the past, theatrical plays.


title: Pay television

text: Sports programming is also featured on some premium services; HBO was historically known for its broadcasts of boxing, while Showtime and Epix also carry mixed martial arts events. Some general interest premium channels have aired other professional sporting events in the past: HBO for example, carried games from the National Hockey League (NHL), National Basketball Association (NBA) and American Basketball Association (ABA) in its early years, and from 1975 to 1999 aired the Wimbledon tennis tournament. Specialty pay sports channels also exist—often focusing on international sports considered niche to domestic audiences (such as, in the United States, cricket), and are typically sold at a higher expense than traditional premium services. Out-of-market sports packages in North America are multi-channel pay services carrying professional or collegiate sporting events which are sold in a seasonal package. They are typically the most expensive type of pay services, generally running in the range of $35 to $50 per month.


title: Pay television

text: Some pay services also offer pornographic films; Cinemax was well known for carrying a late-night block of softcore films and series known as "Max After Dark"—a reputation that led to the network often being nicknamed "Skinemax" by viewers. Cinemax phased out this programming in the 2010s, noting the increasing amount of sexual content in other mainstream premium series (such as "Game of Thrones") as having made a specific block redundant, the increased availability of porn via other digital platforms, and an increasing shift towards action series as part of its original programming strategy. Specialized channels dedicated to pornographic films also exist, that carry either softcore adult programs (such as Playboy TV), or more hardcore content (such as The Erotic Network and Hustler TV).


title: Pay television

text: Pay television channels come in different price ranges. Many channels carrying advertising combine this income with a lower subscription fee. These are called "mini-pay" channels (a term also used for smaller scale commercial-free pay television services) and are often sold as a part of a package with numerous similarly priced channels. Usually, however, the regular pricing for premium channels ranges from just under $10 to near $25 per month per suite, with lower prices available via bundling options with cable or satellite providers, or special limited offers which are available during free preview periods or before the launch of a network's prestige series. However, some other channels, such as sports and adult networks may ask for monthly pricing that may go as high as near $50 a month. There are also premium television services which are priced significantly higher than the mini-pay channels, but they compensate for their higher price by carrying little or no advertising and also providing a higher quality program output. As advertising sales are sensitive to the business cycle, some broadcasters try to balance them with more stable income from subscriptions.


title: Pay television

text: Some providers offer services owned by the same company in a single package. For example, American satellite provider DirecTV offers the Encore channels along with the Starz multiplex (both owned by Lionsgate's Starz Inc.) in its "Starz Super Pack"; and The Movie Channel, Flix and SundanceTV (the latter of which continues to be sold in the DirecTV package despite Showtime Networks no longer owning Sundance, that channel is now owned by AMC Networks) along with Showtime in its "Showtime Unlimited" package; Cinemax and its multiplex networks, in turn, are almost always packaged with HBO (both owned by Warner Bros. Discovery).


title: Pay television

text: Though selling premium services that are related by ownership as a package is common, that may not always be the situation: for example, in the United States, Cinemax and Encore are optionally sold separately from or in a single package with their respective parent networks HBO and Starz, depending on the service provider. The Movie Channel and Flix meanwhile, are usually sold together with Showtime (all three channels are owned by Paramount Global); though subscribers are required to purchase Showtime in order to receive Flix, The Movie Channel does not have such a restriction as a few providers optionally sell that service without requiring a Showtime subscription.


title: Pay television

text: Unlike other cable networks, premium services are almost always subscribed to "a la carte", meaning that one can, for example, subscribe to HBO without subscribing to Showtime (in Canada, there are slight modifications, as most providers include U.S. superstations – such as WAPA-TV – with their main premium package by default). However, subscribing to an "individual" service automatically includes access to all of that service's available multiplex channels and, in some cases, access to content via video-on-demand (in the form of a conventional VOD television service, and in some cases, a companion on-demand streaming service as well). Most pay television providers also offer a selection of premium services (for example, the HBO, Showtime and Starz packages) in one bundle at a greatly reduced price than it would cost to purchase each service separately, as an inducement for subscribers to remain with their service provider or for others to induce subscribers into using their service. Similarly, many television providers offer general interest or movie-based premium channels at no additional charge for a trial period, often one to three months, though there have been rare instances of free trials for pay services that last up to one year for newer subscribers to that provider's television service.


title: Pay television

text: Pay television has become popular with cable and satellite television. Pay television services often, at least two to three times per year, provide free previews of their services, in order to court potential subscribers by allowing this wider audience to sample the service for a period of days or weeks; these are typically scheduled to showcase major special event programming, such as the pay cable premiere of a blockbuster feature film, the premiere (either a series or season premiere) of a widely anticipated or critically acclaimed original series or occasionally, a high-profile special (such as a concert).


title: Pay television

text: Subscription services transmitted via analogue terrestrial television have also existed, to varying degrees of success. The most known example of such service in Europe is Canal+ and its scrambled services, which operated in France from 1984 to the 2011 closedown of analogue television, Spain from 1990 to 2005 and Poland from 1995 to 2001. Some U.S. television stations launched pay services (known simply as "subscription television" services) such as SuperTV, Wometco Home Theater, PRISM (which principally operated as a cable service, only being simultaneously carried over-the-air for a short time during the 1980s, and unlike other general-interest pay services accepted outside advertising for broadcast during its sports telecasts), Preview, SelecTV and ONTV in the late 1970s, but those services disappeared as competition from cable television expanded during the 1980s.


title: Pay television

text: In some countries, the launch of digital terrestrial television has meant that pay television has become increasingly popular in countries with regular antennas. The UK's original DTT service, ONdigital (later rebranded as ITV Digital), provided both free-to-air channels and a selection of pay channels; however, it infamously suffered many problems and went under in 2002. The replacement DTT service, Freeview, does not have any pay services available. Conversely, even as Cord-cutting by pay television subscribers due to price increases resulting from rising carriage fees and as the use of digital multicasting by terrestrial broadcasters has increased since the late 2000s, there have not been any attempts to launch new over-the-air pay services in North America, aside from the short-lived USDTV and, to an extent, the short-lived MovieBeam service (which used digital multicasting to assist subscribers in purchasing and downloading movies). However, as of 2020, a service called Evoca TV has been launched in markets including Boise and Denver, providing an ATSC 3.0 based service, combined with an internet connection, offering 80 channels.


title: Pay television

text: In Australia, Foxtel, Optus Television and TransACT are the major pay television distributors, all of which provide cable services in some metropolitan areas, with Foxtel providing satellite service for all other areas where cable is not available. Austar formerly operated as a satellite pay service, until it merged with Foxtel and SelecTV. The major distributors of pay television in New Zealand are Sky Network Television on satellite and Vodafone on cable.


title: Pay television

text: In the 2010s, over-the-top subscription video on demand (SVOD) services distributed via internet video emerged as a major competitor to traditional pay television, with services such as Amazon Video, Hulu, and Netflix gaining prominence. Similarly to pay television services, their libraries include acquired content (which can not only include films, but acquired television series as well), and a mix of original series, films, and specials. The shift towards SVOD has resulted in increasing competition within the sector, with media conglomerates such as Paramount (Paramount+), Disney (Disney+, and also having acquired majority ownership in Hulu—originally a joint venture of multiple broadcasters), and NBCUniversal (Peacock) having launched their own services to compete, and existing premium services such as HBO (HBO Now, succeeded in 2020 by HBO Max—a revamp of the service adding content from other Warner Bros. Discovery properties and third-parties) launching OTT versions of their services sold on a direct-to-consumer basis to appeal to cord cutters.


title: Pay television

text: Pay-per-view (PPV) services are similar to subscription-based pay television services in that customers must pay to have the broadcast decrypted for viewing, but usually only entail a one-time payment for a single or time-limited viewing. Programs offered via pay-per-view are most often movies or sporting events, but may also include other events, such as concerts and even softcore adult programs. In the U.S., the initial concept and technology for pay-per-view for broadcast television was first developed in the early 1950s, including a crude decrypting of the over-the-air television signal and a decoding box, but never caught on for use at that time. It took another four decades when cable broadcasters started using pay-per-view on a widespread basis.


title: Pay television

text: "Free" variants are free-to-air (FTA) and free-to-view (FTV); however, FTV services are normally encrypted and decryption cards either come as part of an initial subscription to a pay television bouquet – in other words, an offer of pay-TV channels – or can be purchased for a one-time cost. FTA and FTV systems may still have selective access. ABC Australia is one example, as much of its programming content is free-to-air except for National Rugby League (NRL) games, which are encrypted.


title: Anoushka Shankar

text: Anoushka Shankar (born 9 June 1981) is a British-American sitar player, producer, film composer and activist. She was the youngest and first woman to receive a British House of Commons Shield; she has had 7 Grammy Awards nominations and was the first musician of Indian origin to perform live and to serve as presenter at the ceremony. She performs across multiple genres and styles - classical and contemporary, acoustic and electronic.


title: Anoushka Shankar

text: Shankar was born in London and her childhood was divided between London and Delhi. She is the daughter of Sukanya Shankar and Indian sitar maestro Ravi Shankar, who was 61 when she was born. Through her father, she is also the half-sister of American singer Norah Jones (born Geetali Norah Shankar), and Shubhendra "Shubho" Shankar, who died in 1992.


title: Anoushka Shankar

text: As a teenager, Shankar lived in Encinitas, California, and attended San Dieguito High School Academy. A 1999 honours graduate and Homecoming Queen, she decided to pursue a career in music rather than attend college.


title: Anoushka Shankar

text: Shankar began training on the sitar with her father Ravi Shankar at the age of seven. As part of her training, she began accompanying him on the tanpura at his performances from the age of ten, soaking up the music and becoming acclimated to the stage. She gave her first public sitar performance on 27 February 1995 at the age of 13, at Siri Fort in New Delhi as part of her father's 75th birthday celebration concert. For this solo debut, she was accompanied by tabla maestro Zakir Hussain. Her first experience in the recording studio came that same year when Angel Records released a special four-CD box set called "In Celebration", to mark her father's birthday. By the age of fourteen, she was accompanying her father at concerts around the world. At fifteen, she assisted her father on the landmark album "Chants of India", produced by George Harrison. Under both their guidance, she was in charge of notation and eventually of conducting the performers who took part in the recording. After this experience, the heads of Angel Records came to her parents' home to ask to sign her, and Shankar signed her first exclusive recording contract with Angel/EMI when she was sixteen.


title: Anoushka Shankar

text: She released her first album, "Anoushka", in 1998, followed by "Anourag" in 2000. In 1999 Shankar graduated from high school with honours, but decided against university in favour of beginning to tour as a solo artist. Both Shankar and her half-sister Norah Jones were nominated for Grammy awards in 2003 when Anoushka became the youngest nominee in the World Music category for her third album, "Live at Carnegie Hall".


title: Anoushka Shankar

text: Having released three albums of Indian classical music, Shankar took several years away from recording and focused her energy on establishing herself as a solo concert performer outside of her father's ensemble. In that time, she toured worldwide, playing an average of 50–60 concerts per year. 2005 brought the release of her fourth album "RISE", her first self-produced, self-composed, non-classical album, earning her another Grammy nomination in the Best Contemporary World Music category. In February 2006 she became the first Indian to play at the Grammy Awards, playing material from "RISE".


title: Anoushka Shankar

text: Shankar, in collaboration with Indian producer, composer and musician, Karsh Kale, released "Breathing Under Water" on 28 August 2007. It is a mix of classical sitar and electronica beats and melodies. Notable guest vocals included her paternal half-sister Norah Jones, Sting, and her father, who performed a sitar duet with her.


title: Anoushka Shankar

text: In 2011 Shankar signed with record label Deutsche Grammophon as an exclusive artist. This marked the beginning of a prolific recording and creative period for Shankar, during which time she continued to refine the sitar sound and musical ideas she had become known for. She earned a third Grammy Award nomination in 2013 for "Traveller", an exploration of the shared history between flamenco and Indian classical music, which was produced by Javier Limón and featured artists such as Buika, Pepe Habichuela and Duquende. As Shankar had begun to do with "Rise", she created a specially handpicked ensemble of musicians with whom to perform this cross-genre music, and played over a hundred concerts worldwide in support of "Traveller". In 2013 she released a personal album called "Traces of You", which was released several months after the passing of her father Ravi Shankar. Produced by Nitin Sawhney, and featuring her half-sister Norah Jones as the sole vocal performer, "Traces of You" earned Shankar a fourth Grammy nomination in the World Music category. In July 2015 Shankar released "Home", her first purely classical album of Indian Ragas. Self-composed and produced, Home was recorded over a week in October 2014 in Shankar's new, purpose-built home-studio.


title: Anoushka Shankar

text: Following "Home," Shankar released "Land of Gold" (2016) – her fourth album with Deutsche Grammophon; the global refugee crisis and the intense contrast between being able to provide for her newly born second child and the sense of powerlessness to alleviate the injustice and pain happening as the world looked on provided the inspiration for this heartfelt and hopeful album. The release features different types of contributions – from the vocal of MIA and Alev Lenz, to the monologue of Remain the Sea by Veteran British actress Vanessa Redgrave. There is also a strong cinematic influence in the videos, lent by the production of film Director Joe Wright (Pride & Prejudice, Atonement). "Land of Gold remixes" came out in the same year with remixes by Mowgli, Karsh Kale, Matt Robertson, Grain and East London-based collective Shiva Soundsystem.


title: Anoushka Shankar

text: In 2019 came the compilation "Reflections", a 20-year retrospective album that revisits the best of Shankar's cross cultural back catalogue with Deutsche Grammophon, including many of her previous collaborations with fellow artists and producers.


title: Anoushka Shankar

text: The release of "Love Letters" in 2020 marked a different direction for Shankar. The co-produced EP is a compilation of songs written across 2018-19 and released on her new record label, Mercury KX. Several guest musicians feature on the EP, including singer and co-producer Alev Lenz, twin sister vocal duo Ibeyi, singer and cellist Ayanna Witter-Johnson, Indian singer Shilpa Rao, Brooklyn-based mastering engineer Heba Kadry (Björk, Slowdive) and British audio mastering engineer Mandy Parnell (Aphex Twin, The XX).


title: Anoushka Shankar

text: Shankar regularly collaborates with electronic music producer Gold Panda, percussionist Manu Delago and the Britten Sinfonia strings, all of which appeared, including Shankar, at the 2020 BBC Proms at Royal Albert Hall.


title: Anoushka Shankar

text: Throughout her career, Shankar has made many guest appearances on recordings by other artists, among them Sting, Lenny Kravitz, Thievery Corporation and Nitin Sawhney.


title: Anoushka Shankar

text: In 2010 she collaborated with Jazz legend Herbie Hancock, on his album "The Imagine Project", alongside Pink, India.Arie, Jeff Beck, John Legend, Chaka Khan, The Chieftains, Wayne Shorter and Dave Matthews. The project was also filmed for a documentary accompanying the release and showing him working with the artists who collaborated on the project.


title: Anoushka Shankar

text: In the same year she featured on the track "Rebirth by Indian" fusion group MIDIval Punditz. In 2012 the album "Area 52" by Rodrigo & Gabriela's album featured a guest appearance from Shankar's sitar improvisation. She performed the "Raga Piloo" with violinist Patricia Kopatchinskaja during a concert in Konzerthaus Berlin, Germany in April 2016 - the song was originally composed, performed and recorded by Ravi Shankar as a duet with Yehudi Menuhin on the album "West Meets East, Volume 2" in 1968.


title: Anoushka Shankar

text: More recently, she has played on the track "Ama La" on the Dalai Lama's first music album, "Inner World" that was released in July 2020 on his 85th birthday.


title: Anoushka Shankar

text: Shortly after she appeared on "Peradam," the third and final instalment of album trilogy "Perfect Vision" by Patti Smith and Soundwalk Collective alongside Charlotte Gainsbourg, and Tenzin Choegyal; she also collaborated with Grammy-Nominated Deva Premal on the song "Prabhujee", released in November 2020.


title: Anoushka Shankar

text: On 13 November 2020, Shankar was featured on "Stop Crying Your Heart Out" as part of the BBC Radio 2's Allstars' Children in Need charity single. The single debuted at number 7 on the Official UK Singles Chart and number 1 on both the Official UK Singles Sales Chart and the Official UK Singles Download Chart.


title: Anoushka Shankar

text: Anoushka has always continued to tour and perform as a classical sitarist, both within purely Indian classical ensemble but also as a soloist championing her father's compositions with the world's leading orchestras including the London Symphony Orchestra, New York Philharmonic, Berliner Philharmoniker, MDR Sinfonieorchester, Metropole Orkest and Lucerne Symphony.


title: Anoushka Shankar

text: Shankar is the sole performer of Ravi Shankar's First and Second Concertos for Sitar and Orchestra since the composer’s death, performing multiple times under the leadership of esteemed conductors such as legendary Zubin Mehta, Jules Buckley, Kristjan Järvi, and Jakob Hrusa. In January 2009 she was the sitar soloist alongside the Orpheus Chamber Orchestra premiering her father's Third Concerto for Sitar and Orchestra, and in July 2010 she premiered Ravi Shankar's first symphony for sitar and orchestra with the London Philharmonic Orchestra at London's Royal Festival Hall.


title: Anoushka Shankar

text: She has also performed in duets with artists such as violinist Joshua Bell, in a sitar-cello duet with Mstislav Rostropovich, and with flutist Jean-Pierre Rampal, playing both sitar and piano.


title: Anoushka Shankar

text: Shankar took her first steps into scoring with a Bengali lullaby composed for Director Joe Wright's "Anna Karenina", starring Keira Knightley, Jude Law and Tannishtha Chatterjee.


title: Anoushka Shankar

text: Shankar's progression into composition for films led her to score the British Film Institute's restoration of rare Indian silent "Shiraz: A Romance of India" (Original 1928, scored in 2017). She also performed the composition live at screenings of the movie that was premiered as 2017's London Film Festival Archive Gala and then appeared at the global film festival We Are One on 2 June 2020.


title: Anoushka Shankar

text: In the same year (2017) she co-wrote and performed on the end-title song “Gain the Ocean" for the Judi Dench-starred British-American biographical drama "Victoria & Abdul" directed by Stephen Frears.


title: Anoushka Shankar

text: More recently she co-composed the score to Mira Nair's ("Monsoon Wedding", "Vanity Fair", "The Namesake") BBC 6-part series "A Suitable Boy" (2020); based on Vikram Seth's classic novel of the same title, the show developed by Andrew Davies ("Bridget Jones's Diary", "War and Peace", "Les Misérables") stars Tabu, Tanya Maniktala and Ishaan Khatter.


title: Anoushka Shankar

text: Shankar has also ventured into acting ("Dance Like a Man", 2004) and writing. She wrote a biography of her father, "", in 2002 and has contributed to various books. As a columnist she wrote monthly columns for India's "First City Magazine" for three years, and spent one year as a weekly columnist for India's second largest newspaper, the "Hindustan Times".


title: Anoushka Shankar

text: She narrated "Stolen Innocence: India's Untold Story of Human Trafficking", a 2017 documentary by filmmakers Chris Davis, Casey Allred and Lindsay Daniels, telling the true stories of young women who escaped from sex slavery in India and Nepal.


title: Anoushka Shankar

text: Over the span of her musical career, Shankar has frequently been invited to perform for benefit concerts around the world.


title: Anoushka Shankar

text: In 2000 she shared the stage with Madonna and Bryan Adams st the Summer's Tibetan Peace Garden Benefit concert organised in London by Sting's wife Trudie Styler; on 29th November 2002, Shankar was the featured performer of the "Indian" half of the Concert for George, a posthumous tribute to the life and music of George Harrison, held at the Royal Albert Hall in London. She opened the show by playing a solo sitar instrumental titled "Your Eyes". Also on the sitar, she performed George Harrison's "The Inner Light" with Jeff Lynne. Lastly, she conducted a new composition, "Arpan", written by her father. The composition featured Eric Clapton playing acoustic guitar, and a full orchestra of Indian and Western musicians. The concert was modelled after Ravi Shankar's benefit concert with Harrison, the 1971 Concert for Bangladesh.


title: Anoushka Shankar

text: Also in 2002 she performed alongside Patti Labelle, Elton John, Nina Simone and others for Rock for the Rainforest, the benefit concert organized by Sting and Trudie Styler at Carnegie Hall.


title: Anoushka Shankar

text: Shankar was invited by Richard Gere and Philip Glass to perform in a concert at Avery Fisher Hall in 2003 in aid of the Healing the Divide: A Concert for Peace and Reconciliation.


title: Anoushka Shankar

text: Shankar and Jethro Tull postponed a concert scheduled for 29 November 2008 in Mumbai after the 2008 Mumbai attacks. They reorganised the performance as A Billion Hands Concert, a benefit performance for victims of the attacks, and held it on 5 December 2008. Shankar commented on this decision stating that: "As a musician, this is how I speak, how I express the anger within me [...] our entire tour has been changed by these events and even though the structure of the concert may remain the same, emotionally perhaps we are saying a lot more."


title: Anoushka Shankar

text: In 2016 Shankar helped gather names and signed Letter to the Guardian for a call for action to help Syrian Refugees.


title: Anoushka Shankar

text: In February 2018 she read a poem as part of Letters Live for Help Refugees, alongside other performers such as Gemma Arterton, Andrew Scott, Jade Anouka, Cara Theobaold, Florence Welch, Chiwetel Ejiofor, Joely Richardso among many others.


title: Anoushka Shankar

text: In July 2018 she embarked on a short US tour of "Land of Gold" to help raise funds for the nonprofit organization Help Refugees.


title: Anoushka Shankar

text: Shankar is an activist working with multiple causes and charitable organisations, in particular supporting women and refugees; she is also an advocate for animal rights.


title: Anoushka Shankar

text: In 2013, responding to the horrific gang-rape of a young girl in Delhi, whom the Indian media referred to as Nirbhaya, Shankar threw her weight behind an online campaign One Billion Rising on Change.org, demanding an end to crime against women. As part of the campaign, she released a video in which she revealed she had been sexually abused for many years as a child.


title: Anoushka Shankar

text: In 2018 she joined musicians, actors and artists such as Patrick Stewart, Peter Capaldi, Vivienne Westwood, Anish Kapoor, The Kaiser Chiefs and many others in a call to urge MPs to attend the Refugees Family Reunion Bill and ensure refugee families torn apart by war and conflict are reunited.


title: Anoushka Shankar

text: Shankar is one of the faces of the Help Refugees campaign to raise funds and awareness for the refugee crisis worldwide.


title: Anoushka Shankar

text: Shankar has joined the charity, founded by Annie Lennox in 2008, working to achieve equality for women and girls across the world.


title: Anoushka Shankar

text: In 2020 Shankar was announced as the inaugural President of the F-List: a UK database created to help bridge the gender-gap in music.


title: Anoushka Shankar

text: In 2020 Shankar designed a pin for the "Pin Your Thanks" initiative that supports the National Health Service (NHS) charities during the COVID-19 pandemic in the United Kingdom.


title: Anoushka Shankar

text: In 2007 the UN World Food Programme appointed Shankar as spokesperson to raise awareness about the issues of hunger and malnutrition, especially among children, in India.


title: Anoushka Shankar

text: Shankar and her father, also a supporter of animal rights, appeared in a thirty-second public-service announcement against animal suffering for People for the Ethical Treatment of Animals (PETA).


title: Anoushka Shankar

text: Shankar is the daughter of Pandit Ravi Shankar and Sukanya Rajan, and the half-sister of Norah Jones. She grew up in the US, the UK, and India. In 2010, she married British director Joe Wright. They have two sons (born 2011 and 2015). They separated in December 2017 and finalised their divorce in September 2019. She lives in London with her sons.


title: Continental Army

text: The Continental Army was the army of the United Colonies (the Thirteen Colonies) in the Revolutionary-era United States. It was formed by the Second Continental Congress after the outbreak of the American Revolutionary War, and was established by a resolution of Congress on June 14, 1775. The Continental Army was created to coordinate military efforts of the Colonies in their war for independence against the British, who sought to keep their American lands under control. General George Washington was the commander-in-chief of the army throughout the war.


title: Continental Army

text: The Continental Army was supplemented by local militias and volunteer troops that were either loyal to individual states or otherwise independent. Most of the Continental Army was disbanded in 1783 after the Treaty of Paris formally ended the fighting. The 1st and 2nd Regiments of the Army went on to form what was to become the Legion of the United States in 1792. This became the foundation of what is now the United States Army.


title: Continental Army

text: The Continental Army consisted of soldiers from all 13 colonies and, after 1776, from all 13 states. When the American Revolutionary War began (at the Battles of Lexington and Concord on April 19, 1775) the colonial revolutionaries did not have a standing army. Previously, each colony had relied upon the militia (which was made up of part-time citizen-soldiers) for local defense, or the raising of temporary provincial troops during such crises as the French and Indian War of 1754–1763. As tensions with Great Britain increased in the years leading to the war, colonists began to reform their militias in preparation for the perceived potential conflict. Training of militiamen increased after the passage of the Intolerable Acts in 1774. Colonists such as Richard Henry Lee proposed forming a national militia force, but the First Continental Congress rejected the idea.


title: Continental Army

text: On April 23, 1775, the Massachusetts Provincial Congress authorized the raising of a colonial army consisting of 26 company regiments. New Hampshire, Rhode Island, and Connecticut soon raised similar but smaller forces. On June 14, 1775, the Second Continental Congress decided to proceed with the establishment of a Continental Army for purposes of common defense, adopting the forces already in place outside Boston (22,000 troops) and New York (5,000). It also raised the first ten companies of Continental troops on a one-year enlistment, riflemen from Pennsylvania, Maryland, Delaware, and Virginia to be used as light infantry, who became the 1st Continental Regiment in 1776. On June 15, 1775, Congress elected by unanimous vote George Washington as Commander-in-Chief, who accepted and served throughout the war without any compensation except for reimbursement of expenses. Supporting Washington as commander in chief were four major-generals (Artemas Ward, Charles Lee, Philip Schuyler, and Israel Putnam) and eight brigadier-generals (Seth Pomeroy, Richard Montgomery, David Wooster, William Heath, Joseph Spencer, John Thomas, John Sullivan, and Nathanael Greene) As the Continental Congress increasingly adopted the responsibilities and posture of a legislature for a sovereign state, the role of the Continental Army became the subject of considerable debate. Some Americans had a general aversion to maintaining a standing army; but on the other hand, the requirements of the war against the British required the discipline and organization of a modern military. As a result, the army went through several distinct phases, characterized by official dissolution and reorganization of units.


title: Continental Army

text: The Continental Army was a dirty, unorganized rabble. They did not have the discipline typically expected of an army. When they first assembled, the count of how many soldiers George Washington had was delayed a little over a week. Instead of obeying their commanders and officers without question, each unit was a community that had democratically chosen its leaders. The regiments, coming from different states, were uneven in numbers. Logically, they should be evened, which would mean moving soldiers around. In the spirit of American republicanism, if George Washington separated the soldiers from the officers they had chosen they did not believe they should have to serve. George Washington had to give in to the soldiers and negotiate with them. He needed them to have an army.


title: Continental Army

text: Soldiers in the Continental Army were volunteers; they agreed to serve in the army and standard enlistment periods lasted from one to three years. Early in the war, the enlistment periods were short, as the Continental Congress feared the possibility of the Continental Army evolving into a permanent army. The army never numbered more than 48,000 men overall and 13,000 troops in one area. The turnover proved a constant problem, particularly in the winter of 1776–77, and longer enlistments were approved. As the new country (not yet fully independent) had no money, the government agreed to give grants to the soldiers which they could exchange for money.


title: Continental Army

text: The officers of both the Continental Army and the state militias were typically yeoman farmers with a sense of honor and status and an ideological commitment to oppose the policies of the British Crown. The enlisted men were very different. They came from the working class or minority groups (Irish, German, African American). They were motivated to volunteer by specific contracts that promised bounty money; regular pay at good wages; food, clothing, and medical care; companionship; and the promise of land ownership after the war. They were unruly and would mutiny if the contractual terms were not met. By 1780–1781, threats of mutiny and actual mutinies were becoming serious. Upwards of a fourth of Washington's army were of Irish origin, many being recent arrivals and in need of work.


title: Continental Army

text: The Continental Army was racially integrated, a condition the United States Army would not see again until the 1950s. During the Revolution African-American slaves were promised freedom in exchange for military service by both the Continental and British armies. Approximately 6,600 people of color (including African American, indigenous, and multiracial men) served with the colonial forces, and made up one-fifth of the Northern Continental Army.


title: Continental Army

text: In addition to the Continental Army regulars, state militia units were assigned for short-term service and fought in campaigns throughout the war. Sometimes the militia units operated independently of the Continental Army, but often local militias were called out to support and augment the Continental Army regulars during campaigns. The militia troops developed a reputation for being prone to premature retreats, a fact that General Daniel Morgan integrated into his strategy at the Battle of Cowpens and used to fool the British in 1781.


title: Continental Army

text: The financial responsibility for providing pay, food, shelter, clothing, arms, and other equipment to specific units was assigned to states as part of the establishment of these units. States differed in how well they lived up to these obligations. There were constant funding issues and morale problems as the war continued. This led to the army offering low pay, often rotten food, hard work, cold, heat, poor clothing and shelter, harsh discipline, and a high chance of becoming a casualty.


title: Continental Army

text: Keeping the continentals clothed was a difficult task and to do this Washington appointed James Mease, a merchant from Philadelphia. Mease worked closely with state-appointed agents to purchase clothing and things such as cow hides to make clothing and shoes for soldiers. Mease eventually resigned in 1777 and had compromised much of the organization of the Clothing Department. After this, on many accounts, the soldiers of the Continental Army were often poorly clothed, had few blankets, and often did not even have shoes. The problems with clothing and shoes for soldiers were often not the result of not having enough but of organization and lack of transportation. To reorganize the Board of War was appointed to sort out the clothing supply chain. During this time they sought out the help of France and for the remainder of the war, clothing was coming from over-sea procurement.


title: Continental Army

text: At the time of the Siege of Boston, the Continental Army at Cambridge, Massachusetts, in June 1775, is estimated to have numbered from 14 to 16,000 men from New England (though the actual number may have been as low as 11,000 because of desertions). Until Washington's arrival, it remained under the command of Artemas Ward. The British force in Boston was increasing by fresh arrivals. It numbered then about 10,000 men. The British controlled Boston, and defended it with their fleet, but were outnumbered and did not attempt to challenge the American control of New England. Washington selected young Henry Knox, a self-educated strategist, to take charge of the artillery from an abandoned British fort in upstate New York, and dragged across the snow to and placed them in the hills surrounding Boston in March 1776. The British situation was untenable. They negotiated an uneventful abandonment of the city and relocated their forces to Halifax in Canada. Washington relocated his army to New York. For the next five years, the main bodies of the Continental and British armies campaigned against one another in New York, New Jersey, and Pennsylvania. These campaigns included the notable battles of Trenton, Princeton, Brandywine, Germantown, and Morristown, among many others.


title: Continental Army

text: The army increased its effectiveness and success rate through a series of trials and errors, often at a great human cost. General Washington and other distinguished officers were instrumental leaders in preserving unity, learning and adapting, and ensuring discipline throughout the eight years of war. In the winter of 1777–1778, with the addition of Baron von Steuben, a Prussian expert, the training and discipline of the Continental Army was dramatically upgraded to modern European standards. This was during the infamous winter at Valley Forge. Washington always viewed the Army as a temporary measure and strove to maintain civilian control of the military, as did the Continental Congress, though there were minor disagreements about how this was to be carried out.


title: Continental Army

text: Throughout its existence, the Army was troubled by poor logistics, inadequate training, short-term enlistments, interstate rivalries, and Congress's inability to compel the states to provide food, money, or supplies. In the beginning, soldiers enlisted for a year, largely motivated by patriotism; but as the war dragged on, bounties and other incentives became more commonplace. Major and minor mutinies—56 in all—diminished the reliability of two of the main units late in the war.


title: Continental Army

text: The French played a decisive role in 1781 as Washington's Army was augmented by a French expeditionary force (under General Rochambeau) and a squadron of the French navy (under the Comte de Barras). By disguising his movements, Washington moved the combined forces South to Virginia without the British commanders in New York realizing it. This resulted in the capture of the main British invasion force in the South at Siege of Yorktown. The Americans and their allies had won the land war in North America and independence was assured. Before the peace treaty went into effect in 1783, the British partly recovered by defeating the French fleet at the Battle of the Saintes.


title: Continental Army

text: A small residual force remained at West Point and some frontier outposts until Congress created the United States Army by their resolution of June 3, 1784.


title: Continental Army

text: Planning for the transition to a peacetime force had begun in April 1783 at the request of a congressional committee chaired by Alexander Hamilton. The commander-in-chief discussed the problem with key officers before submitting the army's official views on May 2. Significantly, there was a broad consensus on the basic framework among the officers. Washington's proposal called for four components: a small regular army, a uniformly trained and organized militia, a system of arsenals, and a military academy to train the army's artillery and engineer officers. He wanted four infantry regiments, each assigned to a specific sector of the frontier, plus an artillery regiment. His proposed regimental organizations followed Continental Army patterns but had a provision for increased strength in the event of war. Washington expected the militia primarily to provide security for the country at the start of a war until the regular army could expand—the same role it had carried out in 1775 and 1776. Steuben and Duportail submitted their own proposals to Congress for consideration.


title: Continental Army

text: Although Congress declined on May 12 to make a decision on the peace establishment, it did address the need for some troops to remain on duty until the British evacuated New York City and several frontier posts. The delegates told Washington to use men enlisted for fixed terms as temporary garrisons. A detachment of those men from West Point reoccupied New York without incident on November 25. When Steuben's effort in July to negotiate a transfer of frontier forts with Major General Frederick Haldimand collapsed, however, the British maintained control over them, as they would into the 1790s. That failure and the realization that most of the remaining infantrymen's enlistments were due to expire by June 1784 led Washington to order Knox, his choice as the commander of the peacetime army, to discharge all but 500 infantry and 100 artillerymen before winter set in. The former regrouped as Jackson's Continental Regiment under Colonel Henry Jackson of Massachusetts. The single artillery company, New Yorkers under John Doughty, came from remnants of the second Continental Artillery Regiment.


title: Continental Army

text: Congress issued a proclamation on October 18, 1783, which approved Washington's reductions. On November 2, Washington, then at Rockingham near Rocky Hill, New Jersey, released his "Farewell Orders issued to the Armies of the United States of America" to the Philadelphia newspapers for nationwide distribution to the furloughed men. In the message he thanked the officers and men for their assistance and reminded them that "the singular interpositions of Providence in our feeble condition were such, as could scarcely escape the attention of the most unobserving; while the unparalleled perseverance of the Armies of the United States, through almost every possible suffering and discouragement for the space of eight long years, was little short of a standing Miracle."


title: Continental Army

text: Washington believed that the blending of persons from every colony into "one patriotic band of Brothers" had been a major accomplishment, and he urged the veterans to continue this devotion in civilian life.


title: Continental Army

text: Washington said farewell to his remaining officers on December 4 at Fraunces Tavern in New York City. On December 23 he appeared in Congress, then sitting at Annapolis, and returned his commission as commander-in-chief: "Having now finished the work assigned me, I retire from the great theatre of Action; and bidding an Affectionate farewell to this August body under whose orders I have so long acted, I here offer my Commission, and take my leave of all the employments of public life." Congress ended the War of American Independence on January 14, 1784, by ratifying the definitive peace treaty that had been signed in Paris on September 3.


title: Continental Army

text: Congress again rejected Washington's concept for a peacetime force in October 1783. When moderate delegates then offered an alternative in April 1784 which scaled the projected army down to 900 men in one artillery and three infantry battalions, Congress rejected it as well, in part because New York feared that men retained from Massachusetts might take sides in a land dispute between the two states. Another proposal to retain 350 men and raise 700 new recruits also failed. On June 2 Congress ordered the discharge of all remaining men except twenty-five caretakers at Fort Pitt and fifty-five at West Point. The next day it created a peace establishment acceptable to all interests.


title: Continental Army

text: The plan required four states to raise 700 men for one year's service. Congress instructed the Secretary of War to form the troops into eight infantry and two artillery companies. Pennsylvania, with a quota of 260 men, had the power to nominate a lieutenant colonel, who would be the senior officer. New York and Connecticut each were to raise 165 men and nominate a major; the remaining 110 men came from New Jersey. The economy was the watchword of this proposal, for each major served as a company commander, and line officers performed all staff duties except those of chaplain, surgeon, and surgeon's mate. Under Josiah Harmar, the First American Regiment slowly organized and achieved permanent status as an infantry regiment of the new Regular Army. The lineage of the First American Regiment is carried on by the 3rd United States Infantry Regiment (The Old Guard).


title: Continental Army

text: However, the United States military realized it needed a well-trained standing army following St. Clair's Defeat on November 4, 1791, when a force led by General Arthur St. Clair was almost entirely wiped out by the Western Confederacy near Fort Recovery, Ohio. The plans, which were supported by U.S. President George Washington and Henry Knox, Secretary of War, led to the disbandment of the Continental Army and the creation of the Legion of the United States. The command would be based on the 18th-century military works of Henry Bouquet, a professional Swiss soldier who served as a colonel in the British Army, and French Marshal Maurice de Saxe. In 1792, Anthony Wayne, a renowned hero of the American Revolutionary War, was encouraged to leave retirement and return to active service as Commander-in-Chief of the Legion with the rank of major general.


title: Continental Army

text: The legion was recruited and raised in Pittsburgh, Pennsylvania. It was formed into four sub-regions. These were created from elements of the 1st and 2nd Regiments from the Continental Army. These units then became the First and Second Sub-Legions. The Third and Fourth Sub-Legions were raised from further recruits. From June 1792 to November 1792, the Legion remained cantoned at Fort LaFayette in Pittsburgh.


title: Continental Army

text: Throughout the winter of 1792–1793, existing troops along with new recruits were drilled in military skills, tactics, and discipline at Legionville on the banks of the Ohio River near present-day Baden, Pennsylvania. The following Spring the newly named Legion of the United States left Legionville for the Northwest Indian War, a struggle between American Indian tribes affiliated with the Western Confederacy in the area south of the Ohio River. The overwhelmingly successful campaign was concluded with the decisive victory at Fallen Timbers on August 20, 1794, Maj. Gen. Anthony Wayne applied the techniques of wilderness operations perfected by Sullivan's 1779 expedition against the Iroquois. The training the troops received at Legionville was also seen as instrumental to this overwhelming victory.


title: Continental Army

text: Nevertheless, Steuben's Blue Book remained the official manual for the legion, as well as for the militia of most states, until Winfield Scott in 1835.


title: Continental Army

text: In 1796, the United States Army was raised following the discontinuation of the legion of the United States. This preceded the graduation of the first cadets from United States Military Academy at West Point, New York, which was established in 1802.


title: Continental Army

text: During the American Revolutionary War, the Continental Army initially wore ribbons, cockades and epaulettes of various colors as an "ad hoc" form of rank insignia, as General George Washington wrote in 1775:


title: Continental Army

text: Later on in the war, the Continental Army established its own uniform with a black and white cockade among all ranks. Infantry officers had silver and other branches gold insignia:


title: Continental Army

text: This choice of units/ranks was introduced by Captain John Hancock. The suggested titles were then released for the Continental Army in 1779.


title: Khushbu

text: Khushbu Sundar (born Nakhat Khan; 29 September 1970) is an Indian actress, politician, film producer and television presenter. She is known for her works predominantly in Tamil, besides Telugu, Hindi, Malayalam and Kannada films. She is a recipient of three Tamil Nadu State Film Awards, one Kerala state Film Special Mention and one Kalaimamani by Tamil Nadu government.


title: Khushbu

text: Khushbu was born as Nakhat Khan on 29 September 1970 into a Muslim family in Mumbai, Maharashtra, India. Her parents gave her the stage name Khushbu when she started her career as a child actress. She has been living in Chennai for 36 years.


title: Khushbu

text: Khushbu started her career as a child artist in the Hindi film "The Burning Train" (1980) in the song "Teri Hai Zameen Tera Aasman". Between 1980 and 1985 she performed as a child actor in the Hindi films "Naseeb", "Laawaris", "Kaalia", "Dard Ka Rishta", and "Bemisal".


title: Khushbu

text: Her Pari/Fairy song "Main Pariyon Ki Shehzaadi" from "Dard Ka Rishta", a 1982 Hindi movie, was very successful and is still a highly popular song at annual Day School performances, children's parties and Pari/Fairy themed parties in India.


title: Khushbu

text: She made her adult role acting debut by playing a supporting role of Anil Kapoor's younger sister in much-acclaimed film "Meri Jung" in the year 1985. She danced with Javed Jaffrey in the famous song "Bol Baby Bol, Rock'n Roll" in this film. She played her first lead role in the film "Jaanoo" in the very same year and was cast opposite veteran actor Jackie Shroff in this film. She also starred in "Tan-Badan" (1986), opposite Govinda. Khushbu played a significant supporting role in "Deewana Mujh Sa Nahin" (1990), which also starred Aamir Khan and Madhuri Dixit.


title: Khushbu

text: Khushbu was introduced to South Indian screens through the Telugu film "Kaliyuga Pandavulu" (1986), opposite Venkatesh.


title: Khushbu

text: She then moved her base to Chennai and started focusing on Tamil and other South Indian film industries.


title: Khushbu

text: Prabhu and Khushbu rocked as the most loved pair in Tamil cinema during the nineties and almost all their films together were blockbuster hits including "Dharmathin Thalaivan" (1988), "Vettri Vizhaa" (1989), "My Dear Marthandan" (1990), "Chinna Thambi" (1991), "Pandithurai" (1992), "Uthama Raasa" (1993), "Maravan" (1993) and "Chinna Vathiyar" (1995).


title: Khushbu

text: During her peak, Khushbu was the leading actress of Tamil cinema. Among regional South Indian films, she acted in over 100 Tamil films and became the most sought-after actresses of Tamil film industry. She in her illustrious acting career has worked alongside all the superstars of South Indian film industry. After sparkling through the '90s, the number of films reduced in the 2000s. Khushbu forayed into television, plunged into politics and had to take care of her children.


title: Khushbu

text: During her peak, Khushbu was the leading actress of Tamil cinema. She became the first and only Indian actress for whom her fans built a dedicated temple.


title: Khushbu

text: Khushbu and sprinter Usain Bolt are honorary members of Richmond Football Club, based in Melbourne, Australia. Khushbu was made the number-one ticket holder of Richmond Football Club in 2016, and was the number-one female ticket holder in 2017. She is the first Indian celebrity, and the only Indian female to be bestowed with such an honour in the AFL.


title: Khushbu

text: Khushbu was introduced to Australian Rules Football by former Ministerial Adviser Nitin Gupta. She had agreed to help promote AFL Football in India after taking a guided tour of the Richmond Football Club. This offer was a goodwill gesture to further strengthen Australia–India relations.


title: Khushbu

text: During her peak in Tamil cinema, many dishes in Tamil Nadu were named after Khushbu. Kushboo Idli, a rice cake, was the most popular food item named after the actress. Kushboo Jhumki, Kushboo Sarees, Kushboo Sharbet, Kushboo Coffee, Kushboo Cocktails and various other food items were named after her. Many items still retain those names.


title: Khushbu

text: Khushbu has offered to help in the promotion of Little India cultural precincts in Melbourne by becoming its Brand Ambassador.


title: Khushbu

text: Khushbu joined the DMK on 14 May 2010. She was welcomed by DMK Leader Karunanidhi at the party headquarters in Chennai. Khushbu quit DMK on 16 June 2014.


title: Khushbu

text: Khushbu joined the Indian National Congress on 26 November 2014 after meeting with then Congress President Mrs. Sonia Gandhi, and then Vice-president Mr. Rahul Gandhi. She was the National Spokesperson of the INC. Kushbu was very critical of Narendra Modi's policies and has often taken to Twitter to criticise the ruling dispensation.


title: Khushbu

text: She resigned from Congress on 12 October 2020, and joined the Bharatiya Janata Party same day ahead of the Tamil Nadu Assembly elections in 2021. The congress secretary of Media said that Khushbu was being pressurized by her husband Sundar C to join the BJP. She claimed that her previous party "does not want an intelligent woman" and there is no freedom to speak the truth within the party and called the Congress as "mentally retarded". The National Platform for the Rights of the Disabled (NPRD) filed 30 complaints at different police stations in the state over her statements for allegedly making derogatory remarks against people with disabilities.


title: Khushbu

text: The Bharatiya Janata Party has fielded Khushbu for the 2021 Tamil Nadu Legislative Assembly election from Thousand Lights constituency in Chennai. She lost the election by a margin of 32200 votes to Ezhilan N of Dravida Munnetra Kazhagam.


title: Khushbu

text: Kushbu created a controversy in December 2012 when she wore a saree which had images of Hindu gods of Ram, Krishna and Hanuman. The Hindu Makkal Katchi demanded her for an apology and threatened to start agitations. She responded by saying that "I am not going to answer every Tom, Dick and Harry. Why should I? There is no need at all. Why are they worried about what a woman sports. Don't they have any other worthwhile work?".


title: Khushbu

text: The Hindu Munnani and the Hindu Makkal Katchi filed several cases against Khushbu, accusing the actress that she had disrespected Hindu gods by sitting cross-legged with her slippers in front of the idols of Goddesses Lakshmi, Sarswathy and Parvathy during 'pooja' for the muhurat at Chennai on 22 November 2007.


title: Khushbu

text: Khushbu has been a very vocal supporter of the bull-taming sport jallikattu, and has voiced her opinion at various national and international platforms. She even visited a cattle station in Warrnambool, Australia to get a better understanding of how cattle is treated and raised on farms outside India.


title: Khushbu

text: In 2005, she said in an interview it was fine for girls to indulge in pre-marital sex if they safeguard themselves and take precautions to prevent pregnancy and sexually transmitted diseases. Later, she justified her statement by saying no educated man could expect his partner to be a virgin. Dalit Panthers of India stormed the office of the South India Film Artistes' Association in Chennai demanding an apology from her. The Pattali Makkal Katchi said it will protest outside her house. 22 complaints that she was "defaming Tamil womanhood and chastity" were brought against her, but in 2010 the Supreme Court dismissed all cases.


title: Khushbu

text: In August 2017, Khushbu accused Mitu Bhowmick Lange, the tender provider and director of Indian Film Festival of Melbourne (IFFM) of discriminating against non-Hindi films and film stars.


title: Khushbu

text: Khushbu has been very vocal about getting more recognition for South Indian cinema, and the non-Hindi cinema from India on global platforms and film festivals. She has always maintained that Indian cinema is not just about Bollywood movies.


title: Khushbu

text: In January 2006, in their first Indian edition, "Maxim" magazine featured a full-page composite photograph with her head pasted onto the body of a model wearing a bikini. Khushbu filed two complaints: defamation and the indecent representation of women against the editor and four others involved with the magazine. The proceedings were later stayed by the Madras High Court in December 2007 based on a plea filed by one of those charged.


title: Khushbu

text: She dated actor Prabhu while filming "Chinna Thambi" (1991). They had been in a live-in relationship for four and half years, they got married on September 12, 1993 at the house in Poes Garden that they had bought. Sivaji Ganesan, father of Prabhu, strictly opposed their relationship and finally Khushbu and Prabhu decided to break up their relationship and ended it within four months.


title: Khushbu

text: In 2000, she married film actor, director and producer Sundar C. She has been using the married name Khushbu Sundar since then. They have two daughters, Avantika and Anandita, after whom they named their production house, Avni Cinemax.


title: Khushbu

text: Khushbu added her husband's name Sundar to her name after marriage. In 2018, she was accused by BJP and the Hindutva of hiding her name Nakat Khan.


title: Heat equation

text: In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.


title: Heat equation

text: As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.


title: Heat equation

text: The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.


title: Heat equation

text: In mathematics, if given an open subset of and a subinterval of , one says that a function is a solution of the heat equation if


title: Heat equation

text: where denotes a general point of the domain. It is typical to refer to as "time" and as "spatial variables," even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as . For any given value of , the right-hand side of the equation is the Laplacian of the function . As such, the heat equation is often written more compactly as


title: Heat equation

text: In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function of three spatial variables and time variable . One then says that is a solution of the heat equation if


title: Heat equation

text: in which is a positive coefficient called the thermal diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with being the temperature at the point and time . If the medium is not homogeneous and isotropic, then would not be a fixed coefficient, and would instead depend on ; the equation would also have a slightly different form. In the physics and engineering literature, it is common to use to denote the Laplacian, rather than .


title: Heat equation

text: In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that formula_3 is used to denote , so the equation can be written


title: Heat equation

text: Note also that the ability to use either or to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is "translationally and rotationally invariant." In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.


title: Heat equation

text: The "diffusivity constant" is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let be a function with


title: Heat equation

text: Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of and solutions of the heat equation with . As such, for the sake of mathematical analysis, it is often sufficient to only consider the case .


title: Heat equation

text: Since formula_6 there is another option to define a formula_7 satisfying formula_8 as in () above by setting formula_9. Note that the two possible means of defining the new function formula_7 discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.


title: Heat equation

text: Informally, the Laplacian operator gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if is the temperature, tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.


title: Heat equation

text: By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.


title: Heat equation

text: By the combination of these observations, the heat equation says the rate formula_3 at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient in the equation takes into account the thermal conductivity, specific heat, and density of the material.


title: Heat equation

text: The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed , one has


title: Heat equation

text: where is the single-variable function denoting the "average value" of over the surface of the sphere of radius centered at ; it can be defined by


title: Heat equation

text: in which denotes the surface area of the unit ball in -dimensional Euclidean space. This formalizes the above statement that the value of at a point measures the difference between the value of and the value of at points nearby to , in the sense that the latter is encoded by the values of for small positive values of .


title: Heat equation

text: Following this observation, one may interpret the heat equation as imposing an "infinitesimal averaging" of a function. Given a solution of the heat equation, the value of for a small positive value of may be approximated as times the average value of the function over a sphere of very small radius centered at .


title: Heat equation

text: The heat equation implies that peaks (local maxima) of formula_14 will be gradually eroded down, while depressions (local minima) will be filled in. The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings. In particular, if the values in a neighborhood are very close to a linear function formula_15, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative formula_3 will be zero).


title: Heat equation

text: A more subtle consequence is the maximum principle, that says that the maximum value of formula_14 in any region formula_18 of the medium will not exceed the maximum value that previously occurred in formula_18, unless it is on the boundary of formula_18. That is, the maximum temperature in a region formula_18 can increase only if heat comes in from outside formula_18. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below).


title: Heat equation

text: Another interesting property is that even if formula_14 initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures formula_24 and formula_25, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where formula_14 will gradually vary between formula_24 and formula_25.


title: Heat equation

text: If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a diffusion wave. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.


title: Heat equation

text: For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy .


title: Heat equation

text: By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it:


title: Heat equation

text: where formula_30 is the thermal conductivity of the material, formula_31 is the temperature, and formula_32 is a vector field that represents the magnitude and direction of the heat flow at the point formula_33 of space and time formula_34.


title: Heat equation

text: If the medium is a thin rod of uniform section and material, the position is a single coordinate formula_35, the heat flow towards increasing formula_35 is a scalar field formula_37, and the gradient is an ordinary derivative with respect to the formula_35. The equation becomes


title: Heat equation

text: Let formula_40 be the internal heat energy per unit volume of the bar at each point and time. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, formula_41, is proportional to the rate of change of its temperature, formula_42. That is,


title: Heat equation

text: where formula_44 is the specific heat capacity (at constant pressure, in case of a gas) and formula_45 is the density (mass per unit volume) of the material. This derivation assumes that the material has constant mass density and heat capacity through space as well as time.


title: Heat equation

text: Applying the law of conservation of energy to a small element of the medium centered at formula_35, one concludes that the rate at which heat accumulates at a given point formula_35 is equal to the derivative of the heat flow at that point, negated. That is,


title: Heat equation

text: An additional term may be introduced into the equation to account for radiative loss of heat. According to the Stefan–Boltzmann law, this term is formula_51, where formula_52 is the temperature of the surroundings, and formula_53 is a coefficient that depends on physical properties of the material. The rate of change in internal energy becomes


title: Heat equation

text: Note that the state equation, given by the first law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer or radiation). This form is more general and particularly useful to recognize which property (e.g. "c" or "formula_45") influences which term.


title: Heat equation

text: In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is


title: Heat equation

text: If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for "u". To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions or a sign condition (nonnegative solutions are unique by a result of David Widder).


title: Heat equation

text: Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable equilibrium. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods.


title: Heat equation

text: Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as


title: Heat equation

text: where the Laplace operator, Δ or ∇, the divergence of the gradient, is taken in the spatial variables.


title: Heat equation

text: The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some phenomena arising in finance, like the Black–Scholes or the Ornstein-Uhlenbeck processes. The equation, and various non-linear analogues, has also been used in image analysis.


title: Heat equation

text: The heat equation is, technically, in violation of special relativity, because its solutions involve instantaneous propagation of a disturbance. The part of the disturbance outside the forward light cone can usually be safely neglected, but if it is necessary to develop a reasonable speed for the transmission of heat, a hyperbolic problem should be considered instead – like a partial differential equation involving a second-order time derivative. Some models of nonlinear heat conduction (which are also parabolic equations) have solutions with finite heat transmission speed.


title: Heat equation

text: The function "u" above represents temperature of a body. Alternatively, it is sometimes convenient to change units and represent "u" as the heat density of a medium. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units.


title: Heat equation

text: Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function "q" varying in space and time. Then the heat per unit volume "u" satisfies an equation


title: Heat equation

text: For example, a tungsten light bulb filament generates heat, so it would have a positive nonzero value for "q" when turned on. While the light is turned off, the value of "q" for the tungsten filament would be zero.


title: Heat equation

text: The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise "Théorie analytique de la chaleur", published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is


title: Heat equation

text: Let us attempt to find a solution of that is not identically zero satisfying the boundary conditions but with the following property: "u" is a product in which the dependence of "u" on "x", "t" is separated, that is:


title: Heat equation

text: Since the right hand side depends only on "x" and the left hand side only on "t", both sides are equal to some constant value −λ. Thus:


title: Heat equation

text: In general, the sum of solutions to that satisfy the boundary conditions also satisfies and . We can show that the solution to , and is given by


title: Heat equation

text: The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator "u" with the zero boundary conditions can be represented in terms of its eigenfunctions. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.


title: Heat equation

text: Moreover, any eigenfunction "f" of Δ with the boundary conditions "f"(0) = "f"("L") = 0 is of the form "e" for some "n" ≥ 1. The functions "e" for "n" ≥ 1 form an orthonormal sequence with respect to a certain inner product on the space of real-valued functions on [0, "L"]. This means


title: Heat equation

text: Finally, the sequence {"e"} spans a dense linear subspace of "L"((0, "L")). This shows that in effect we have diagonalized the operator Δ.


title: Heat equation

text: In general, the study of heat conduction is based on several principles. Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.


title: Heat equation

text: A fundamental solution, also called a "heat kernel", is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains; see, for instance, for an introductory treatment.


title: Heat equation

text: In one variable, the Green's function is a solution of the initial value problem (by Duhamel's principle, equivalent to the definition of Green's function as one with a delta function as solution to the first equation)


title: Heat equation

text: where "formula_96" is the Dirac delta function. The solution to this problem is the fundamental solution (heat kernel)


title: Heat equation

text: One can obtain the general solution of the one variable heat equation with initial condition "u"("x", 0) = "g"("x") for −∞ < "x" < ∞ and 0 < "t" < ∞ by applying a convolution:


title: Heat equation

text: The "n"-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,


title: Heat equation

text: The general solution of the heat equation on R is then obtained by a convolution, so that to solve the initial value problem with "u"(x, 0) = "g"(x), one has


title: Heat equation

text: with either Dirichlet or Neumann boundary data. A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms (Cole, 2011).


title: Heat equation

text: A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere. In some of these, the spatial domain is (−∞,∞). In others, it is the semi-infinite interval (0,∞) with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation


title: Heat equation

text: "Comment". This solution is the convolution with respect to the variable "x" of the fundamental solution


title: Heat equation

text: Therefore, according to the general properties of the convolution with respect to differentiation, "u" = "g" ∗ Φ is a solution of the same heat equation, for


title: Heat equation

text: so that, by general facts about approximation to the identity, Φ(⋅, "t") ∗ "g" → "g" as "t" → 0 in various senses, according to the specific "g". For instance, if "g" is assumed bounded and continuous on R then converges uniformly to "g" as "t" → 0, meaning that "u"("x", "t") is continuous on with


title: Heat equation

text: "Comment." This solution is obtained from the preceding formula as applied to the data "g"("x") suitably extended to R, so as to be an odd function, that is, letting "g"(−"x") := −"g"("x") for all "x". Correspondingly, the solution of the initial value problem on (−∞,∞) is an odd function with respect to the variable "x" for all values of "t", and in particular it satisfies the homogeneous Dirichlet boundary conditions "u"(0, "t") = 0.


title: Heat equation

text: "Comment." This solution is obtained from the first solution formula as applied to the data "g"("x") suitably extended to R so as to be an even function, that is, letting "g"(−"x") := "g"("x") for all "x". Correspondingly, the solution of the initial value problem on R is an even function with respect to the variable "x" for all values of "t" > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditions "u"(0, "t") = 0. The Green's function number of this solution is X20.


title: Heat equation

text: the function "ψ"("x, t") is also a solution of the same heat equation, and so is "u" := "ψ" ∗ "h", thanks to general properties of the convolution with respect to differentiation. Moreover,


title: Heat equation

text: so that, by general facts about approximation to the identity, "ψ"("x", ⋅) ∗ "h" → "h" as "x" → 0 in various senses, according to the specific "h". For instance, if "h" is assumed continuous on R with support in [0, ∞) then "ψ"("x", ⋅) ∗ "h" converges uniformly on compacta to "h" as "x" → 0, meaning that "u"("x, t") is continuous on with


title: Heat equation

text: "Comment". This solution is the convolution in R, that is with respect to both the variables "x" and "t", of the fundamental solution


title: Heat equation

text: and the function "f"("x, t"), both meant as defined on the whole R and identically 0 for all "t" → 0. One verifies that


title: Heat equation

text: "Comment". This solution is obtained from the preceding formula as applied to the data "f"("x", "t") suitably extended to R × [0,∞), so as to be an odd function of the variable "x", that is, letting "f"(−"x", "t") := −"f"("x", "t") for all "x" and "t". Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an odd function with respect to the variable "x" for all values of "t", and in particular it satisfies the homogeneous Dirichlet boundary conditions "u"(0, "t") = 0.