title: Abandonware

text: Several websites archive abandonware for download, including old versions of applications which are difficult to find by any other means. Much of this software fits the definition of "software that is no longer current, but is still of interest," but the line separating the use and distribution of abandonware from copyright infringement is blurry, and the term "abandonware" could be used to distribute software without proper notification of the owner.


title: Abandonware

text: The Internet Archive has created an archive of what it describes as "vintage software," as a way to preserve them. The project advocated for an exemption from the United States Digital Millennium Copyright Act to permit them to bypass copy protection, which was approved in 2003 for a period of three years. The exemption was renewed in 2006, and , has been indefinitely extended pending further rulemakings. The Archive does not offer this software for download, as the exemption is solely "for the purpose of preservation or archival reproduction of published digital works by a library or archive." Nevertheless, in 2013 the Internet Archive began to provide antique games as browser-playable emulation via MESS, for instance the Atari 2600 game "E.T. the Extra-Terrestrial". Since 23 December 2014 the Internet Archive presents via a browser based DOSBox emulation thousands of archived DOS/PC games for ""scholarship and research purposes only"".


title: Abandonware

text: Starting around 2006, The Library of Congress began the long-time preservation of video games with the "Game canon" list. In September 2012 the collection had nearly 3,000 games from many platforms and also around 1,500 strategy guides. For instance, the source code of the unreleased PlayStation Portable game was discovered in August 2014 to be preserved at the Library of Congress.


title: Abandonware

text: Since around 2009 the International Center for the History of Electronic Games (ICHEG) has taken a five-pronged approach to video game preservation: original software and hardware, marketing materials and publications, production records, play capture, and finally the source code. In December 2013 the ICHEG received a donation of several SSI video games, for instance Computer Bismarck, including the source code for preservation. In 2014 a collection of Brøderbund games and a "virtually complete" Atari arcade machine source code and asset collection was added.


title: Abandonware

text: In 2010 Computer History Museum began with the preservation of source code of important software, beginning with Apple's MacPaint 1.3. In 2012 the APL programming language followed. Adobe Systems, Inc. donated the Photoshop 1.0.1 source code to the collection in February 2013. The source code is made available to the public under an own non-commercial license. On March 25, 2014, Microsoft followed with the donation of MS-DOS variants as well as Word for Windows 1.1a under their own license. On October 21, 2014, Xerox Alto's source code and other resources followed.


title: Abandonware

text: In 2012 a group of European museums and organizations started the "European Federation of Game Archives, Museums and Preservation Projects" (EFGAMP) to join forces to ""Preserve Gaming Legacy"". Also in Japan video game software archival happens since several years.


title: Abandonware

text: In 2012 the MOMA started with archiving video games and explicitly tries to get the source code of them.


title: Abandonware

text: There are also some cases in which the source code of games was given to a fan community for long-time preservation, e.g. several titles of the "Wing Commander" video game series or Ultima 9 of the Ultima series.


title: Abandonware

text: In response to the missing software support, sometimes the software's user community begins to provide support (bug fixes, compatibility adaptions etc.) even without available source code, internal software documentation and original developer tools. Methods are debugging, reverse engineering of file and data formats, and hacking the binary executables. Often the results are distributed as unofficial patches. Notable examples are "Fallout 2", "" or even Windows 98. For instance in 2012, when the multiplayer game "" became unsupported abandonware as the official multiplayer server and support was shut down, the game community itself took over with a self-developed multiplayer server and client.


title: Abandonware

text: With the new possibility of digital distribution arising in mid-2000, the commercial distribution for many old titles became feasible again as deployment and storage costs dropped significantly. A digital distributor specialized in bringing old games out of abandonware is "GOG.com" (formerly called "Good Old Games") who started in 2008 to search for copyright holders of classic games to release them legally and DRM-free again. For instance, on December 9, 2013 the real-time strategy video game "" was, after ten years of non-availability, re-released by gog.com, also including the source code.


title: Abandonware

text: Proponents of abandonware preservation argue that it is more ethical to make copies of such software than new software that still sells. Those ignorant of copyright law have incorrectly taken this to mean that abandonware is legal to distribute, although no software written since 1964 is old enough for copyright to have expired in the US. Even in cases where the original company no longer exists, the rights usually belong to someone else, though no one may be able to trace actual ownership, including the owners themselves.


title: Abandonware

text: Abandonware advocates also frequently cite historical preservation as a reason for trading abandoned software. Older computer media are fragile and prone to rapid deterioration, necessitating transfer of these materials to more modern, stable media and generation of many copies to ensure the software will not simply disappear. Users of still-functional older computer systems argue for the need of abandonware because re-release of software by copyright holders will most likely target modern systems or incompatible media instead, preventing legal purchase of compatible software.


title: Abandonware

text: Those who oppose these practices argue that distribution denies the copyright holder potential sales, in the form of re-released titles, official emulation, and so on. Likewise, they argue that if people can acquire an old version of a program for free, they may be less likely to purchase a newer version if the old version meets their needs.


title: Abandonware

text: Some game developers showed sympathy for abandonware websites as they preserve their classical game titles.


title: Abandonware

text: In most cases, software classed as abandonware is not in the public domain, as it has never had its original copyright officially revoked and some company or individual may still own rights. While sharing of such software is usually considered copyright infringement, in practice copyright holders rarely enforce their abandonware copyrights for a number of reasons – chiefly among which the software is technologically obsolete and therefore has no commercial value, therefore rendering copyright enforcement a pointless enterprise. By default, this may allow the product to "de facto" lapse into the public domain to such an extent that enforcement becomes impractical.


title: Abandonware

text: Rarely has any abandonware case gone to court, but it is still unlawful to distribute copies of old copyrighted software and games, with or without compensation, in any Berne Convention signatory country.


title: Abandonware

text: Old copyrights are usually left undefended. This can be due to intentional non-enforcement by owners due to software age or obsolescence, but sometimes results from a corporate copyright holder going out of business without explicitly transferring ownership, leaving no one aware of the right to defend the copyright.


title: Abandonware

text: Even if the copyright is not defended, copying of such software is still unlawful in most jurisdictions when a copyright is still in effect. Abandonware changes hands on the assumption that the resources required to enforce copyrights outweigh benefits a copyright holder might realize from selling software licenses. Additionally, abandonware proponents argue that distributing software for which there is no one to defend the copyright is morally acceptable, even where unsupported by current law. Companies that have gone out of business without transferring their copyrights are an example of this; many hardware and software companies that developed older systems are long since out of business and precise documentation of the copyrights may not be readily available.


title: Abandonware

text: Often the availability of abandonware on the Internet is related to the willingness of copyright holders to defend their copyrights. For example, unencumbered games for Colecovision are markedly easier to find on the Internet than unencumbered games for Mattel Intellivision in large part because there is still a company that sells Intellivision games while no such company exists for the Colecovision.


title: Abandonware

text: The Digital Millennium Copyright Act (DMCA) can be a problem for the preservation of old software as it prohibits required techniques. In October 2003, the US Congress passed 4 clauses to the DMCA which allow for reverse engineering software in case of preservation.


title: Abandonware

text: In November 2006 the Library of Congress approved an exemption to the DMCA that permits the cracking of copy protection on software no longer being sold or supported by its copyright holder so that they can be archived and preserved without fear of retribution.


title: Abandonware

text: Currently, US copyright law does not recognize the term or concept of "abandonware" while the general concept "orphan works" is recognized (see Orphan works in the United States). There is a long-held concept of abandonment in trademark law as a direct result of the infinite term of trademark protection. Currently, a copyright can be released into the public domain if the owner clearly does so in writing; however this formal process is not considered abandoning, but rather releasing. Those who do not own a copyright cannot merely claim the copyright abandoned and start using protected works without permission of the copyright holder, who could then seek legal remedy.


title: Abandonware

text: Hosting and distributing copyrighted software without permission is illegal. Copyright holders, sometimes through the Entertainment Software Association, send cease and desist letters, and some sites have shut down or removed infringing software as a result. However, most of the Association's efforts are devoted to new games, due to those titles possessing the greatest value.


title: Abandonware

text: In the EU in 2012 an "Orphan Works Directive" (Directive 2012/28/EU) was constituted and is transferred into the member's laws. While the terminology has ambiguities regarding software and especially video games, some scholars argue that abandonware software video games fall under the definition of "audiovisual works" mentioned there.


title: Abandonware

text: Once the copyright on a piece of software has expired, it automatically falls into public domain. Such software can be legally distributed without restrictions. However, due to the length of copyright terms in most countries, this has yet to happen for most software. All countries that observe the Berne Convention enforce copyright ownership for at least 50 years after publication or the author's death. However, individual countries may choose to enforce copyrights for longer periods. In the United States, copyright durations are determined based on authorship. For most published works, the duration is 70 years after the author's death. However, for anonymous works, works published under a pseudonym or works made for hire, the duration is 120 years after creation, or 95 years after publication, whichever comes first. In France, copyright durations are 70 years after the relevant date (date of author's death or publication) for either class.


title: Abandonware

text: However, because of the length of copyright enforcement in most countries, it is likely that by the time a piece of software defaults to public domain, it will have long become obsolete, irrelevant, or incompatible with any existing hardware. Additionally, due to the relatively short commercial, as well as physical, lifespans of most digital media, it is entirely possible that by the time the copyright expires for a piece of software, it will no longer exist in any form. However, since the largest risk in dealing with abandonware is that of distribution, this may be mitigated somewhat by private users (or organizations such as the Internet Archive) making private copies of such software, which would then be legally redistributable at the time of copyright expiry.


title: Abandonware

text: There are alternatives for companies with a software product which faces the end-of-life instead of abandoning the software in an unsupported state.


title: Abandonware

text: Some user-communities convince companies to voluntarily relinquish copyright on software, putting it into the public domain, or re-license it as free software (freeware). Unlike so-called "abandonware", it is perfectly legal to transfer public domain or freely licensed software.


title: Abandonware

text: Amstrad is an example which supports emulation and free distribution of CPC and ZX Spectrum hardware ROMs and software. Borland released "antique software" as freeware. Smith Engineering permits not-for-profit reproduction and distribution of Vectrex games and documentation.


title: Abandonware

text: Groups that lobby companies to release their software as freeware have met with mixed results. One example is the library of educational titles released by MECC. MECC was sold to Brøderbund, which was sold to The Learning Company. When TLC was contacted about releasing classic MECC titles as freeware, the documentation proving that TLC held the rights to these titles could not be located, and therefore the rights for these titles are "in limbo" and may never be legally released. Lost or unclear copyrights to vintage out-of-print software is not uncommon, as rights to the "No One Lives Forever" series illustrates.


title: Abandonware

text: The problem of missing technical support for a software can be most effectively solved when the source code becomes available. Therefore, several companies decided to release the source code specifically to allow the user communities to provide further technical software support (bug fixes, compatibility adaptions etc.) themselves, e.g. by community patches or source ports to new computing platforms. For instance, in December 2015 Microsoft released the Windows Live Writer source code to allow the community to continue the support.


title: Abandonware

text: Id Software and 3D Realms are early proponents in this practice, releasing the source code for the game engines of some older titles under a free software license (but not the actual game content, such as levels or textures). Also Falcon 4.0's lead designer Kevin Klemmick argued in 2011 that availability of the source code of his software for the community was a good thing:


title: Abandonware

text: The chilling effect of drawing a possible lawsuit can discourage release of source code. Efforts to persuade IBM to release OS/2 as open source software were ignored since some of the code was co-developed by Microsoft.


title: Abandonware

text: Nevertheless, several notable examples of successfully opened commercial software exist, for instance, the web browser Netscape Communicator, which was released by Netscape Communications on March 31, 1998. The development was continued under the umbrella of the Mozilla Foundation and Netscape Communicator became the basis of several browsers, such as Mozilla Firefox.


title: Abandonware

text: Another important example for open sourced general software is the office suite StarOffice which was released by Sun Microsystems in October 2000 as OpenOffice.org and is in continued development as LibreOffice and Apache OpenOffice.


title: Abandonware

text: There are also many examples in the video game domain: Revolution Software released their game "Beneath a Steel Sky" as freeware and gave the engine's source code to the authors of ScummVM to add support for the game. Other examples are "Myth II", "Call to Power II" and Microsoft's "Allegiance" which were released to allow the community to continue the support.


title: Abandonware

text: Museums, whether physical or virtual, provide a legal means for preserving what otherwise is an "orphan work".


title: Abandonware

text: WinWorld is an online museum that contains information about old computers and the software that was used with them. The website also has screenshots and


title: Manna Dey

text: Prabodh Chandra Dey (May 1, 1919 − October 24, 2013), known by his stage name Manna Dey, was an internationally acclaimed and celebrated Indian playback singer, music director, and a musician. As a classical vocalist, he belonged to the Bhendibazaar Gharana and was trained under Ustad Aman Ali Khan. He is considered one of the most versatile and celebrated vocalists of the Hindi film industry, often credited with the success of Indian classical music in Hindi commercial movies. As a musician, Dey is best known for infusing Indian classical music in a pop framework that ushered the golden period in Hindi cinema.


title: Manna Dey

text: In a career spanning over five decades, Dey recorded total 3,047 songs, though most primarily in Bengali and Hindi; Dey also sang in 14 other Indian languages, including Bhojpuri, Punjabi, Assamese, Gujarati, Kannada, Malayalam, and Chhattisgarhi. The mid-50s to 70s were considered the peak of his musical career.


title: Manna Dey

text: The Government of India honored him with the Padma Shri in 1971, the Padma Bhushan in 2005 and the Dadasaheb Phalke Award in 2007.


title: Manna Dey

text: Dey was born in a Bengali family to Mahamaya and Purna Chandra Dey on 1 May 1919 in Calcutta (Now Kolkata). Besides his parents, his youngest paternal uncle, Sangeetacharya Krishna Chandra Dey highly inspired and influenced him. He received his early education at Indu Babur Pathshala, a small pre-primary school. He started doing stage shows in school from 1929. He attended Scottish Church Collegiate School and Scottish Church College. He participated in sports events like wrestling and boxing in his college days, taking training from Gobar Guha. He graduated from Vidyasagar College.


title: Manna Dey

text: Dey began taking music lessons from Krishna Chandra Dey and Ustad Dabir Khan. During this period of learning, he stood first for three consecutive years in three different categories of inter-collegiate singing competitions.


title: Manna Dey

text: In 1942, Dey accompanied Krishna Chandra Dey on a visit to Bombay. There he started working as an assistant music director first under Krishna Chandra Dey in a Bengali film "Chanakya" in 1939 and then under Sachin Dev Burman. Later, he assisted other music composers and then started to work independently. While working independently as a music director for various Hindi movies, Dey continued to take musical lessons in Hindustani classical music from Ustad Aman Ali Khan and Ustad Abdul Rahman Khan. He often works with near about 185 music directors in his singing career in Bollywood.


title: Manna Dey

text: Dey started his career in playback singing with the movie "Tamanna", in 1942. The musical score was by Krishna Chandra Dey and Manna sang a duet named "Jago Aayee Usha Ponchi Boley Jago" with Suraiya which was an instant hit.


title: Manna Dey

text: But it was only in 1943 that he got his first solo break with "Ram Rajya". Incidentally, the producer of the film Vijay Bhatt and its composer Shankar Rao Vyas had approached K C Dey with an offer for playback in the film. When K C Dey refused the offer on the grounds that he would not lend his voice to other actors, they spotted Manna Dey sitting in the corner of the room and offered him the opportunity.


title: Manna Dey

text: Shankar Rao Vyas taught Manna Dey the songs and he chose to sing them in his uncle's distinct style. And thus started the illustrious career with the first song ""Gayi tu gayi Seeta sati"(Ram Rajya", 1943).


title: Manna Dey

text: His songs like "O Prem Diwani Sambhal Ke Chalna" from 1944 film "Kadambari" composed by Anil Biswas, "Dil Churaney Ki Liye from Dur Chaley" (1946) composed by Jafar Khurshid, his duets with Amira Bahee like "E Diniya Jara Suney" from "Kamala" (1946) and duet song "Aaj Bor Aayee" with Meena Kapoor from 1947 film "Chaltey Chaltey" became chartbusters in respective years. Between 1945 and 1947, many Dey-Rajkumari duets like "Hay Gagan Me Badal Tharey" in 1945 for the film "Vikaramaditya", "Aowji Morey" from "Insaaf" (1946), all four duets from the film "Geet Govind" composed by Pandit Indra – "Kit Ho Nando Kumar", "Chorr Sakhi Aaj Laj", "Apney Hi Rang", "Lalit Labang Lata" from "Geet Govind" became popular.


title: Manna Dey

text: He sang for first time songs composed by Sachin Dev Burman, "Upar Gagan Vishal" and "Duniya Ke Logo" in the 1950 movie "Mashal", which became popular and from here his association with S.D.Burman began. Its lyrics were written by Kavi Pradeep. In 1952, Dey sang for a Bengali and a Marathi movie with the same name and storyline, "Amar Bhupali". This established him as a leading playback singer in Bengali films and Marathi films as well by 1953.


title: Manna Dey

text: In the post independence period, after 1947, Dey was regularly used by music composers Anil Biswas, Shankar Rao Vyas, S.K.Pal, S.D.Burman, Khem Chand Prakash, Mohd.Safi from 1947 to 1957. Dey-Anil Biswas combination gave hit numbers from films like "Gajre" (1948), "Hum Bhi Insaan Hai" (1948), "Do Sitaare" (1951), "Hamdard" (1953), "Mahatma Kabir" (1954), "Jasoos" (1957) and "Pardesi" (1957). Though Anil Biswas worked with Dey in very few films, their songs remain famous. He recorded his first duet with Shamshad Begum, who was the Hindi female singer most in demand from 1940 to 1961, "Phoolon Ka Swapna" in the films "Girls School" (1949) composed by S.K.Pal.


title: Manna Dey

text: His first duet with the then upcoming singer Lata Mangeshkar was "Lapat Ke Pot Pahaney Bikral" composed by Vasant Desai for "Narsingh Avtar" (1949), and with Kishore Kumar it was "Subaho Ki Paheli Kiran" from 1951 film "Anadolan" composed by Pannalal Ghosh. His first duet with Geeta Dutt was "Dhonyo Dhonyo He Ayodh Puri" from the film "Ram Vivah" (1949) composed by Shankar Rao Vyas, first duet with Umadevi (Tun Tun) was "Hay Ye Hain" from "Jangal Ka Jaanwar" (1951) composed by Ghantshala. His first duet with the then struggling singer Asha Bhosle was "O Raat Gayee Fir Din Aya" from 1953 film "Bootpolish".


title: Manna Dey

text: Dey established his verstality between 1948 and 1954 by singing not only the classical based film songs but also singing such film songs which were fusion of Indian classical music and pop music and by giving classical music concerts. His experimentation with western music too produced many unforgettable melodies resulting in an increase in singing offers in films from 1955. He began singing ghazals in Hindi films from 1953. He became a music composer in Hindi films when he composed music along with Khemchand Prakash for both "Shri Ganesh Janma" (1951) and "Vishwamitra" (1952).


title: Manna Dey

text: By 1954, Dey became popular among musical circle across film industries of various Indian languages. He became nationally famous after release of "Do Bigha Zamin" (1953) where two of the songs sung by him and composed by Salil Choudhury became hits. Salil Choudhury worked with Dey from 1953 to 1992 in Hindi films and made Dey sing even in Bengali and Malayalam films right from the late 1950s to early 1990s. His association with Shankar–Jaikishan and producer Raj Kapoor had begun while working for "Awara", but their combination became famous while working together for "Boot Polish" in 1954. The trio worked in many films together from 1954 to 1971 whose musical scores were appreciated, irrespective of their box-office fate, like "Shree 420", "Chori Chori", "Mera Naam Joker", "Awaara", and "Kal Aaj Aur Kal". He also sang for Raj Kapoor in Parvarish, Sriman Satyavadi (both with music by Dattaram), Dil Hi To Hai (music by Roshan), and Abdullah (music by R.D. Burman). The Raj Kapoor-Manna Dey combination produced superhits (both music and film, the only exception being "Mera Naam Joker" where the songs were chartbusters but the film was a flop) and was a pair to reckon with. While Mukesh sang the slow pathos for Raj Kapoor, Dey sang the fast peppy ones, the classical numbers, the romantic duets (in case one analyses Dey has sung more than 95% of Raj Kapoor's superhit duets) and the naughty numbers. Dey has the rare distinction of doing playback for both father and son – Raj Kapoor and Randhir Kapoor (In fact Dey later sang for Rishi Kapoor in Zamane ko dikhana hai). C. Ramchandra worked with Dey very first time in "Insaniyaat" in 1955 and then recorded songs with Dey consistently even in the 1960s in films like – "Tallaq" (1959), "Navrang", "Paigham", "Stree" (1961), "Veer Bhimsen" etc.


title: Manna Dey

text: In 1956, he sang with a new batch of singers. He recorded his first duets with Sudha Malhotra "Ghar Ghar Deep Jalao Re" from "Ayodhyapati" in 1956, with Binata Chatterjee in the duet "Tum Mile Mil Gaye Karaar" from "Grand Hotel" (1956) composed by Suresh Talwar, and then his first duet with Sabita Banerjee in the song "Jamine Hamari Jamana" composed by A.R.Quereshi from "Laal-E-Yaman" (1956) and with singer Meena Kapoor in the song "Rim Jhim Jhim Rim Jhim" from Pardesi (1957) composed by Anil Biswas.


title: Manna Dey

text: Dey turned into an independent music composer in Hindi films with "Maha Poojaa" in 1954. He sang eighty-three Hindi songs in three years from 1953 to 1955 and his demand increased such that he sang 45 songs in the year 1956. His career reached in peak form when he recorded 95 Hindi songs in single year 1957 and 64 in 1958. His peak period in Hindi film industry is considered to be from 1953 to 1969 where he recorded 758 Hindi songs of which 631 came between 1957 and 1969. He worked extensively with other music directors like Naushad, K.Dutta, Vasant Pawar & Ram, Vasant Desai, Ravi, S.K.Pal, Avinash Vyas, S. N. Tripathi, Sanmukh Babu, Nissar Bazmi, Husanlal Bhagatram, B.N. Bali, Sushanta Banerjee, O.P. Nayyar, G.Ramanathan, T.G.Lingappa, Nirmal Kumar, Ghulam Mohhammed, Bipin Dutta, Rabin Banerjee, Roshan, Sapan Jagmohan from 1954 to 1968.


title: Manna Dey

text: New age composers like Kalyanji-Anandji started recording songs with Dey from 1958 and Laxmikant Pyarelal from 1964. Rahul Dev Burman made Dey sing the westernised songs – "Aao Twist Karen" and "Pyar Karta Ja" which became chartbusters in 1965. But the composers who gave Dey consistent popular song numbers in commercially successful films from 1955 to 1969 were S.D.Burman, C.Ramachandra, Ravi, Avinash Vyas, Vasant Desai, Anil Biswas, Salil Choudhury and Shankar Jaikishan. The solo songs sung by Dey like "Lapak Jhapak Tu Aa Re" from "Boot Polish" (1954), "O Gori Tori Tu Pyar Ka Sagar Hai" from "Seema" (1955), "Yeh Kahani Hai Diye Aur Toofan Ki" from "Diya Aur Toofan" (1956) composed by Vasant Desai, "Humdum Se Gaye" from "Manzil" (1960), "Aye Mere Pyare Watan" from "Kabuliwala" (1961), "Laga Chunari Mein Daag" from "Dil Hi Toh Hai" (1963), classical songs like "Sur Na Saje" from "Basant Bahar" (1956), "Kaun Aya Mere Mann" from "Dekh Kabira Roya" (1957), "Pucho Na Kaise Maine Rain" from "Meri Surat Teri Aankhen" (1963), "Jhanak Jhanak Tore Baje Payalia" from "Mere Huzoor" (1965); folk based songs like "Kisi Chilman Se" from "Baat Ek Raat Ki" (1962), "Ae Meri Zohra Jabeen" from "Waqt" (1965), "Chalat Musafir Moh Liya" from "Teesri Kasam" (1967), "Aao Aao Sawariya", and duets with Lata like "Masti Bhara Yeh Sama" from "Parvarish" (1958), "Nain Mile Chain Kahan" from "Basant Bahar"(1956), "Kehdoji Kehdo Chupaona Pyar" from "Kismat Ka Khel" (1956), "Tum Gagan Ke Chandrama" from "Sati Savitri" (1964), "Dil Ki Girah" from "Raat Aur Din" (1966), "Chunari Sambhal Gori" from "Baharon Ke Sapne" (1967), were chartbusters in their respective year of release. Credit is also given to Dey for popularising classical based solo and duet songs to the masses like the duet with Lata – "Pritam Daras Dikhao" from "Chacha Zindabad" (1959), became a popular song though it was based on classical Raag Lalit. His rendition of "Kasame Vaade Pyar" from "Upkar" (1967) pictured on Pran and composed by Kalyanji Anandji won accolades for Dey, and was also significant in Pran's career as he began to do positive roles.


title: Manna Dey

text: Dey gave playback for Raj Kapoor in "Shree 420" (Mud Mud Ke Na Dekh), "Chori Chori" (Yeh Raat Bheegi, Jahan Main Jati and Aja Sanam), "Parvarish" (Masti bhara hai sama), "Dil Hi To Hai" (Laga chunari mein daag), "Mera Naam Joker" (Ae bhai zara dekh ke chalo) and "Ek Paon Chal Raha Hai" from "Kal Aaj Aur Kal" (1971). R.D.Burman recorded hit songs with Dey regularly from 1965 with films like "Bhoot Bangla", "Gomti Ke Kinare", "Chandan Ka Palna", "Baharon Ke Sapne", "Padosan", in late 60s. He sang for Balraj Sahni in many films from 1950 till 1969 and was voice of Mehmood and Anoop Kumar in many films from 1960 to 1975.


title: Manna Dey

text: Among the new breed of singers which emerged after 1956, Dey's duets with Suman Kalyanpur were popular making them a celebrated team. Their first duet was "Prem Bada Balwan Jagat" from "Maayaa Nagri" (1957) composed by B.N.Bali and since then have sung around 45 songs together. Their popular duets include "Tum Jo Aao" from "Sakhi Robin" (1962) composed by Robin Bannerjee, "Dil Se Jo Baat" from "Al Hilal" (1958) composed by C.Ramachandra, "Na Jane Kahan" from "Zindagi Aur Khwab" (1963), "Ye Din Hai Khushi Ke" from "Jab Se Tumhe Dekha Hai" (1963) and "Bheegi Hawaon Mein" from "Shriman Sataywadi" (1960)- all 3 composed by Dattaram Wadkar, "Aankh Mein Shokhi" from Reshmi Roomal (1960) composed by Babul, "Aao Hilmil Ke Nacho Re" from "Jaane Anjaane" (1970) composed by Shankar–Jaikishan and "Dil To Dil Hain Phool Bhi" from "Dafaa 302" (1975) composed by Laxmikant Pyarelal.


title: Manna Dey

text: Dey recorded popular duets with Mohd. Rafi, Dey sang 101 Hindi songs along with Rafi which include 58 duets with Rafi such as "Ishq Ishq" (Barsaat Ki Raat) "Tu hai mera prem Devta" (Kalpana), "Mama o mama" (Parvarish), Duniyaan ke liye from "Maan Gaye Ustaad" (1981), "Main Hoon Tera Prem Aur Tu Ho Meri Pran" from "Rahu Ketu" (1979), the song "Hindustan Ki Kasam" from "Hindustan Ki Kasam" (1973), "Hum To Tere Hai Deewane" from "Johar Mehmood in Hong Kong", "Badey Miya Diwane" from "Shagird", "Ye Do Diwane Dil Ke" from "Johar Mehmood in Goa", "Agar Dil Dil Se" from "Shola Aur Shabnam" (1961).


title: Manna Dey

text: With Asha Bhosle, recorded around 160 Hindi songs from 1953 to 1982 though their duets kept releasing till 1986 and their last song sung together was in "Teri Maang Sitaaron Se Bhar Doon" (1982). Popular duets of Asha-Dey include "Ye Hawa Yeh Nadi Ka" from "Ghar Sansar" (1958), "Tu chupi hai kahan" from "Navrang" (1959), "Jodi Hamari Jamegi" from "Aulad ("1968), "Sanjh Dhali Dil Ki Lagi" from "Kala Bazar" (1960), "Aye Kash Chalte Milte" from "Manzil" (1960), "Na Tohh karvan Ki Talaash Hai" from "Barsaat Ki Raat" (1960), "Jane Na Doonga" from "Dadimaa" (1966), "Re man sur me ga" from "Lal Patthar" (1971), "Zindegi hai khel" from "Seeta Aur Geeta" (1972) and "Paisa Daulat" from "Dharkan" (1972). Dey sang around 27 Hindi duets with Geeta Dutt from 1949 with the last being from "Ziddi" in 1964. The popular duets of Geeta Dutt-Dey include both classical songs as well as those which have style of Twist dance, Rock & Roll, cha-cha-cha like "Aan aan milo" from "Devdas", :O Mister Suno Ek Baat" from "Agra Road" (1957), "Karo Na Phere Gali Ke Mere" from Bengali film "Gali Theke Rajpath" (1959) composed by Sudhin Dasgupta, "Naya Naya Chand Hai Jee" "Khuda Ka Banda" (1957) starring actor Chandrashekar and composed by S.N.Tripati. The song "Picnic Me Tick Tick" from "Piya Milan Ki Aas" (1961) composed by S.N.Tripati is noted for Dey yodeling and singing in a style which is known as forte of Kishore. With Lata Mangeshkar, Day recorded around 103 Hindi duets of which songs like "Tere Bina Aag Yeh Chandni" from "Awara", "Yeh raat bhigi bhigi" and "Aaja Sanam Madhur Chandni Mein" from "Chori Chori", "Pyar hua iqrar hua" from "Shree 420", "Woh Chand Muskaye" from "Akhri Dao" (1958), "Ritu aye", "Dil Ki Girah Khol Do" from "Raat Aur Din" (1966), "Aya Abdulla Aya" from "Juaari" (1968), "Soch Ke Ye Gagan Jhume" from the flop film "Jyoti" (1969) and "main buddho lambo lambo" of "Buddha Mil Gaya" (1971) are still extremely popular.


title: Manna Dey

text: Kishore Kumar and Dey had recorded only 6 songs together until 1968 and all proved to be hits, some of them being "Ye Duniya hai usaki nyari hai" from Sauraksha composed by Bappi Lahiri, "Ye Dosti humnai todenge" from Sholay, "Tubhi Piya Chikara Hoon" and "Tu Jaam Liye Jaa" from "Bewakoof" (1960), "Babu Samjho Ishaare" from "Chalti Ka Naam Gaadi" (1958) all three composed by S.D. Burman and "Baheta Pani Baheta Jaye" from "Dhaake Ke Malmal" composed by C. Ramachandra, "Joyo Joyo Mere Lal" composed by S.K. Pal in 1952 and their first duet together – "Subaho Ki Paheli Kiran" in 1951 from "Andolan". In 1968, R.D. Burman brought them together for "Ek Chatur Naar" in Padosan. Reportedly the song "Ek Chatur Naar" (a duet by Kishore Kumar and Manna Dey) from "Padosan" (1965) was partly improvised by Kishore Kumar at the time of recording and Dey, determined to show Kishore Kumar how he would sing the duet better (since Kishore had not been trained classically), got into the mood of the song and immortalised "Ek Chatur Naar". Dey recorded around 31 songs with Kishore from 1951 to 1987 and all of them became chartbusters. From 1969, after release of "Aradhana", the highest in demand playback singer was Kishore Kumar, so careers of Mukesh, Rafi and Dey were affected and they got lesser number of songs to sing. His peak period as a playback singer is considered to be from 1953 to 1969 and in the year 1969 it seemed that Dey's career in Hindi films would be over due to resurgence of Kishore Kumar but he had an extended peak period till 1976. The song "Tujhe Suraj Kahun Ya Chanda" from "Ek Phool Do Mali" in 1969, S.D. Burman composition "Mitwa Mitwa Piya Maine Kya Kiya" from "Us Paar" (1974) and his duet with Shailendra Singh from "Bobby" – "Na Maangoon Sona Chandi" were a chartbusters in respective years. His song "Sawan Ki Rimjhim Main" was aired in non-film program section in Vividh Bharati and became popular on the radio.


title: Manna Dey

text: He received a fresh lease of life in his playback singing career from 1971 after Rajesh Khanna allowed music directors to picturise or feature songs sung by Dey in films with Khanna in lead role beginning with Anand, composed by Salil Chowdahry. For Rajesh Khanna, Dey had sung songs "Zindagi kaisi hai paheli" ("Anand", 1971), "Tum bin jeewan kaisa jeewan" and "Bhor Aaye Gaya Andhera" ("Bawarchi", 1972), "Nadiya Chale Re" (song sung with Rajesh and Dey), "Hasne ki chah ne kitna mujhe" ("Avishkaar", 1973) and "Gori tori paijaniya" ("Mehbooba", 1976). Later in a 2012 interview, Dey said, "I loved the way he picturised music. The success of a song depends upon how an actor picturises it. He was the number one in picturising songs. I will be ever indebted to him." From 1970 to 1983, he recorded around 503 songs used in Hindi films. Thereafter he became selective of the kinds of songs he sings and chose to do less work in Hindi.


title: Manna Dey

text: The demand for Kishore-Dey combination only grew after 1972. Their popular songs were "Mere Pyale Mein" from "Aamir Garib" (1974), "Is Ishq Mein Har" from "Mr. Romeo" (1974), "Yeh Dosti" from "Sholay" (1975), "Duniya Me Jeeney" from "Naukri", "Kamal Hain" from "Karz", "Phool Chaahiye Na" from "Pyas" (1982) and their last duet was in 1986 film "Maqqar" – "Tu Hi Mera Sapna". Salil Chowdhry compositions rendered by Dey from films like "Ananad" and "Anadatta" became popular as well. Also the trio of Kishore-Rafi-Dey sang together hits like "Tujhme Ishwar, Allah Tujhme" from "Nanha Farishta" (1969) and "Band Mutthee" from "Chalti Ka Naam Zindagi" (1981). Kishore-Dey together also sang with Asha in many films and famous among them are "Khan Chacha" from "Dil Deewana" (1974), "Aa Kitni Sundar Jagah" from "Chandi Sona", both composed by R.D. Burman and with Lata in "Goyakechu Nanche" from "Manoranjan" (1974), "Logo Ke Juban Me Apna" from "Nehla Pe Dehla", both composed by Pancham.


title: Manna Dey

text: Lata and Dey recorded fusion songs, which used both Indian and western instruments. Several such songs included the duets "Kaanhaa Bole Naa" and "Balma Moraa Aanchaar" for the film "Sangat", composed by Salil Chowdhury in 1975, which became very popular and was picturised on Rakesh Pandey and Kajri. The song "Neela peela hara gulabi" is a famous Holi song sung by Lata with Dey in "Aap Bete" (1976). In 1974, Dey sang along with Lata-Kishore in the song "Goyakechu Nanche" from "Manoranjan" (1974) which became a chartbuster in 1974. Dey continued to have hit Hindi songs to his credit from 1971 to 1986, such as "Dulhan Banungi" from "Who Jo Hasina" (1983), composed by Raamlaxman and sung with Lata, "Aankhon Ka Salaam Lo" with Lata-Rafi in "Samraat", "Dilwaale Dilwaale" from "Kranti" (1981), "Yaari Hai Imaan" from "Zanjeer", "Ye Duniya Hey Usi Ko" sung with Kishore-Usha from "Suraksha" (1979), "Tum Besahara Ho To Kisi" from "Anurodh" (1977), "Jo Likha Gaya Hai" from "Umar Qaid" (1975), among others. "Sab Kho Diya Sab Pa Liya" from "Maqqar" (1986), composed by Rajesh Roshan was the last duet of Lata-Dey for a Hindi film and was a chartbuster. He worked with composers like Laxmikant Pyarelal, Kamlyanji Anadji, Pancham, Bappi Lahiri, Raam Laxman, Sonik Omi, S. Rajeswar Rao, Ravi, Rajesh Roshan extensively from late 70s to 1990 in Hindi films. However, offers for him to sing solo songs in Hindi films reduced since 1976 gradually and rarely songs sung by him was picturised in the hero.


title: Manna Dey

text: He sang "Bohey Nirontaro Ananto Anandadhara" from the album "Rabindra Sudha" in 1961, and then went on to sing 14 songs composed by Rabindranath Tagore until 2002 in Hindi. He sang title songs of Hindi serials like "Hum Hindusthaani" (1986), "Khari Khari" (1984), "Aasmaan Se Oonchaa" (1997) and "Aao Jhoome Gaaye" (2001), all composed by Barma Malik.


title: Manna Dey

text: His first duet with Usha Khanna was "Janemon Janemon Tum Din Raat" from film "Haye Mera Dil" (1968), composed by Kalyanji Anandji. His first song with S. Janaki was "Itna Manta Tu Mera" from "Aashiq C.I.D" (1973) and with P. Susheela was "Joy Ho Gaanga Maiya Ki" from "Gangaa Ki God Mein" (1980). He recorded his first song with Vani Jayram, "Rataiya Baba, Rajania Baba" for the film "Ratnaa Dakoo" in 1972 and their other popular duets include "Mitwa More Man Mitwa" from the film "Parinay" (1974), composed by Jaidev. Dey also sang with singer/composer, Hemant Kumar (Hemanta Mukherjee), in Bengali movies, and also with some other Bengali composers like Nachiketa Ghosh and Sudhin Dasgupta. He sang a duet, "Ke Prothom Kachhe Esechi", with Lata Mangeshkar in the movie "Sankhyabela". He also performed Rabindra Sangeet and recorded over 4000 songs till 2012. In addition to film songs, Dey released several albums of devotional songs of Jagadguru Shree Kripaluji Maharaj.


title: Manna Dey

text: Since 1992, Dey withdrew himself from Hindi film music. But he continued to sing in Bengali movies, bhajans and gazals in different languages and appeared in live performances from 1992 to 2012. His last live performance was in 2012 in Mumbai. His last recorded song in Hindi films was for the film Umar in 2006 composed by Shamir Tandon, "Duniyawaalo Kee Nahee Kuchh Bhee Khabar", which he sang along with Kavita Krishnamoorthy and Sonu Nigam. He was presented the Filmfare Life Time Achievement Award in 2011.


title: Manna Dey

text: He sang duets with singers like Md Rafi, Mukesh, Sandhya Mukherjee, Mahendra Kapoor, Talat Mahmood, Amit Kumar, Shailendra Singh, Krishna Kalle, Shardha Rajan Iyengar, Arati Mukherjee, Chandrani Mukherjee, Anuradha Pudwal, Hemlata (singer), Minu Purushottom, Bhupinder Singh, K.J. Yesudas, P Jayachandran, Suresh Wadkar, Kavita Krishnamoorty, Alka Yagnik, Antara Chowdhury, Preeti Sagar, Dilraj Kaur, Yunus Fazmi, Jaspal Singh, Anwar, Manhar Udhas, Joginder and Mubarak Begum. He worked with more than 102 music directors in Hindi film industry from working with Krishna Chandra Dey in Tamanna in 1942 to music composer Shamir Tandon in 2006.


title: Manna Dey

text: In December 1953, Dey married Sulochana Kumaran. She was originally from Kannur, Kerala. Together they had two daughters – Shuroma Herekar (1956-2016), an U.S. based scientist and Shumita Dev (b. 1958), a Bangalore-based businesswoman. Sulochana died in Bengaluru in January 2012. She had been suffering from cancer for some time. After her death, Dey moved to Kalyan Nagar in Bengaluru after spending more than fifty years in Mumbai.


title: Manna Dey

text: On 8 June 2013, Dey was admitted to the ICU in a Bengaluru hospital after a chest infection gave rise to other complications. His health gradually improved and about a month later doctors took him off the ventilator support. Later, he was released from hospital.


title: Manna Dey

text: He was hospitalized again in the first week of October 2013, and died of a cardiac arrest at 3:45 pm on 24 October at Narayana Hrudayalaya hospital in Bengaluru, aged 94. Musicians, politicians, cricketers and other notable persons issued statements on his death. He was cremated at Bengaluru itself.


title: Manna Dey

text: Dey's Bengali language autobiography, "Jiboner Jalsaghorey", has been published by the renowned Ananda Publishers in the year 2005 which has been translated in English as "Memories Come Alive", in Hindi as "Yaden Jee Uthi" and in Marathi as "Jivanacha Jalasagarat".


title: Manna Dey

text: "Jibaner Jalsaghore", a documentary on Dey's life, was released in 2008. Manna Dey Sangeet Academy is developing a complete archive on Manna Dey. In association with Rabindra Bharati University, Kolkata, the Manna Dey Music Archive has been developed in the Sangeet Bhawan.


title: Manna Dey

text: Song Bissar Gayi Sab Tat Parayi Artist : Manna Dey Music Director : S. Mohinder Lyricist : Shabad Gurbani


title: Duality (optimization)

text: In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality.


title: Duality (optimization)

text: In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality.


title: Duality (optimization)

text: Usually the term "dual problem" refers to the "Lagrangian dual problem" but other dual problems are used – for example, the Wolfe dual problem and the Fenchel dual problem. The Lagrangian dual problem is obtained by forming the Lagrangian of a minimization problem by using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for the primal variable values that minimize the original objective function. This solution gives the primal variables as functions of the Lagrange multipliers, which are called dual variables, so that the new problem is to maximize the objective function with respect to the dual variables under the derived constraints on the dual variables (including at least the nonnegativity constraints).


title: Duality (optimization)

text: In general given two dual pairs of separated locally convex spaces formula_1 and formula_2 and the function formula_3, we can define the primal problem as finding formula_4 such that formula_5


title: Duality (optimization)

text: In other words, if formula_4 exists, formula_7 is the minimum of the function formula_8 and the infimum (greatest lower bound) of the function is attained.


title: Duality (optimization)

text: If there are constraint conditions, these can be built into the function formula_8 by letting formula_10 where formula_11 is a suitable function on formula_12 that has a minimum 0 on the constraints, and for which one can prove that formula_13. The latter condition is trivially, but not always conveniently, satisfied for the characteristic function (i.e. formula_14 for formula_15 satisfying the constraints and formula_16 otherwise). Then extend formula_17 to a perturbation function formula_18 such that formula_19.


title: Duality (optimization)

text: where formula_21 is the convex conjugate in both variables and formula_22 denotes the supremum (least upper bound).


title: Duality (optimization)

text: The duality gap is the difference between the values of any primal solutions and any dual solutions. If formula_23 is the optimal dual value and formula_24 is the optimal primal value, then the duality gap is equal to formula_25. This value is always greater than or equal to 0 (for minimization problems). The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds.


title: Duality (optimization)

text: In computational optimization, another "duality gap" is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem. This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the "convex relaxation" of the primal problem: The convex relaxation is the problem arising replacing a non-convex feasible set with its closed convex hull and with replacing a non-convex function with its convex closure, that is the function that has the epigraph that is the closed convex hull of the original primal objective function.


title: Duality (optimization)

text: Linear programming problems are optimization problems in which the objective function and the constraints are all linear. In the primal problem, the objective function is a linear combination of "n" variables. There are "m" constraints, each of which places an upper bound on a linear combination of the "n" variables. The goal is to maximize the value of the objective function subject to the constraints. A "solution" is a vector (a list) of "n" values that achieves the maximum value for the objective function.


title: Duality (optimization)

text: In the dual problem, the objective function is a linear combination of the "m" values that are the limits in the "m" constraints from the primal problem. There are "n" dual constraints, each of which places a lower bound on a linear combination of "m" dual variables.


title: Duality (optimization)

text: In the linear case, in the primal problem, from each sub-optimal point that satisfies all the constraints, there is a direction or subspace of directions to move that increases the objective function. Moving in any such direction is said to remove slack between the candidate solution and one or more constraints. An "infeasible" value of the candidate solution is one that exceeds one or more of the constraints.


title: Duality (optimization)

text: In the dual problem, the dual vector multiplies the constraints that determine the positions of the constraints in the primal. Varying the dual vector in the dual problem is equivalent to revising the upper bounds in the primal problem. The lowest upper bound is sought. That is, the dual vector is minimized in order to remove slack between the candidate positions of the constraints and the actual optimum. An infeasible value of the dual vector is one that is too low. It sets the candidate positions of one or more of the constraints in a position that excludes the actual optimum.


title: Duality (optimization)

text: In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply.


title: Duality (optimization)

text: To ensure that the global maximum of a non-linear problem can be identified easily, the problem formulation often requires that the functions be convex and have compact lower level sets.


title: Duality (optimization)

text: This is the significance of the Karush–Kuhn–Tucker conditions. They provide necessary conditions for identifying local optima of non-linear programming problems. There are additional conditions (constraint qualifications) that are necessary so that it will be possible to define the direction to an "optimal" solution. An optimal solution is one that is a local optimum, but possibly not a global optimum.


title: Duality (optimization)

text: with the domain formula_27 having non-empty interior, the "Lagrangian function" formula_28 is defined as


title: Duality (optimization)

text: The vectors formula_30 and formula_31 are called the "dual variables" or "Lagrange multiplier vectors" associated with the problem. The "Lagrange dual function" formula_32 is defined as


title: Duality (optimization)

text: The dual function "g" is concave, even when the initial problem is not convex, because it is a point-wise infimum of affine functions. The dual function yields lower bounds on the optimal value formula_24 of the initial problem; for any formula_35 and any formula_31 we have formula_37.


title: Duality (optimization)

text: If a constraint qualification such as Slater's condition holds and the original problem is convex, then we have strong duality, i.e. formula_38.


title: Duality (optimization)

text: where the objective function is the Lagrange dual function. Provided that the functions formula_8 and formula_42 are continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem


title: Duality (optimization)

text: is called the Wolfe dual problem. This problem may be difficult to deal with computationally, because the objective function is not concave in the joint variables formula_44. Also, the equality constraint formula_45 is nonlinear in general, so the Wolfe dual problem is typically a nonconvex optimization problem. In any case, weak duality holds.


title: Duality (optimization)

text: According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem. Von Neumann noted that he was using information from his game theory, and conjectured that two person zero sum matrix game was equivalent to linear programming. Rigorous proofs were first published in 1948 by Albert W. Tucker and his group. (Dantzig's foreword to Nering and Tucker, 1993)


title: Matching (graph theory)

text: In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.


title: Matching (graph theory)

text: Given a graph a matching "M" in "G" is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices.


title: Matching (graph theory)

text: A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated).


title: Matching (graph theory)

text: A maximal matching is a matching "M" of a graph "G" that is not a subset of any other matching. A matching "M" of a graph "G" is maximal if every edge in "G" has a non-empty intersection with at least one edge in "M". The following figure shows examples of maximal matchings (red) in three graphs.


title: Matching (graph theory)

text: A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number formula_1 of a graph is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs.


title: Matching (graph theory)

text: A perfect matching is a matching that matches all vertices of the graph. That is, a matching is perfect if every vertex of the graph is incident to an edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used. In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover. Thus, the size of a maximum matching is no larger than the size of a minimum edge cover: . A graph can only contain a perfect matching when the graph has an even number of vertices.


title: Matching (graph theory)

text: A near-perfect matching is one in which exactly one vertex is unmatched. Clearly, a graph can only contain a near-perfect matching when the graph has an odd number of vertices, and near-perfect matchings are maximum matchings. In the above figure, part (c) shows a near-perfect matching. If every vertex is unmatched by some near-perfect matching, then the graph is called factor-critical.


title: Matching (graph theory)

text: Given a matching "M", an alternating path is a path that begins with an unmatched vertex and whose edges belong alternately to the matching and not to the matching. An augmenting path is an alternating path that starts from and ends on free (unmatched) vertices. Berge's lemma states that a matching "M" is maximum if and only if there is no augmenting path with respect to "M".


title: Matching (graph theory)

text: In any graph without isolated vertices, the sum of the matching number and the edge covering number equals the number of vertices. If there is a perfect matching, then both the matching number and the edge cover number are .


title: Matching (graph theory)

text: If and are two maximal matchings, then and . To see this, observe that each edge in can be adjacent to at most two edges in because is a matching; moreover each edge in is adjacent to an edge in by maximality of , hence


title: Matching (graph theory)

text: In particular, this shows that any maximal matching is a 2-approximation of a maximum matching and also a 2-approximation of a minimum maximal matching. This inequality is tight: for example, if is a path with 3 edges and 4 vertices, the size of a minimum maximal matching is 1 and the size of a maximum matching is 2.


title: Matching (graph theory)

text: A spectral characterization of the matching number of a graph is given by Hassani Monfared and Mallik as follows: Let formula_4 be a graph on formula_5 vertices, and formula_6 be formula_7 distinct nonzero purely imaginary numbers where formula_8. Then the matching number of formula_4 is formula_7 if and only if (a) there is a real skew-symmetric matrix formula_11 with graph formula_4 and eigenvalues formula_13 and formula_14 zeros, and (b) all real skew-symmetric matrices with graph formula_4 have at most formula_16 nonzero eigenvalues. Note that the (simple) graph of a real symmetric or skew-symmetric matrix formula_11 of order formula_5 has formula_5 vertices and edges given by the nonozero off-diagonal entries of formula_11.


title: Matching (graph theory)

text: A generating function of the number of "k"-edge matchings in a graph is called a matching polynomial. Let "G" be a graph and "m" be the number of "k"-edge matchings. One matching polynomial of "G" is


title: Matching (graph theory)

text: where "n" is the number of vertices in the graph. Each type has its uses; for more information see the article on matching polynomials.


title: Matching (graph theory)

text: A fundamental problem in combinatorial optimization is finding a "maximum matching". This problem has various algorithms for different classes of graphs.


title: Matching (graph theory)

text: In an "unweighted bipartite graph", the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.


title: Matching (graph theory)

text: In a "weighted" "bipartite graph," the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem. The Hungarian algorithm solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path search in the augmenting path algorithm. If the Bellman–Ford algorithm is used for this step, the running time of the Hungarian algorithm becomes formula_23, or the edge cost can be shifted with a potential to achieve formula_24 running time with the Dijkstra algorithm and Fibonacci heap.


title: Matching (graph theory)

text: In a "non-bipartite weighted graph", the problem of maximum weight matching can be solved in time formula_25 using Edmonds' blossom algorithm.


title: Matching (graph theory)

text: A maximal matching can be found with a simple greedy algorithm. A maximum matching is also a maximal matching, and hence it is possible to find a "largest" maximal matching in polynomial time. However, no polynomial-time algorithm is known for finding a minimum maximal matching, that is, a maximal matching that contains the "smallest" possible number of edges.


title: Matching (graph theory)

text: A maximal matching with "k" edges is an edge dominating set with "k" edges. Conversely, if we are given a minimum edge dominating set with "k" edges, we can construct a maximal matching with "k" edges in polynomial time. Therefore, the problem of finding a minimum maximal matching is essentially equal to the problem of finding a minimum edge dominating set. Both of these two optimization problems are known to be NP-hard; the decision versions of these problems are classical examples of NP-complete problems. Both problems can be approximated within factor 2 in polynomial time: simply find an arbitrary maximal matching "M".


title: Matching (graph theory)

text: The number of matchings in a graph is known as the Hosoya index of the graph. It is #P-complete to compute this quantity, even for bipartite graphs. It is also #P-complete to count perfect matchings, even in bipartite graphs, because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings. A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.


title: Matching (graph theory)

text: The number of perfect matchings in a complete graph "K" (with "n" even) is given by the double factorial ("n" − 1)!!. The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers.


title: Matching (graph theory)

text: One of the basic problems in matching theory is to find in a given graph all edges that may be extended to a maximum matching in the graph (such edges are called maximally-matchable edges, or allowed edges). Algorithms for this problem include:


title: Matching (graph theory)

text: The problem of developing an online algorithm for matching was first considered by Richard M. Karp, Umesh Vazirani, and Vijay Vazirani in 1990.


title: Matching (graph theory)

text: In the online setting, nodes on one side of the bipartite graph arrive one at a time and must either be immediately matched to the other side of the graph or discarded. This is a natural generalization of the secretary problem and has applications to online ad auctions. The best online algorithm, for the unweighted maximization case with a random arrival model, attains a competitive ratio of .


title: Matching (graph theory)

text: Kőnig's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs.


title: Matching (graph theory)

text: Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.


title: Icon-class cruise ship

text: The "Icon" class (formally Project Icon) is a planned class of cruise ships ordered by Royal Caribbean International and to be built by Meyer Turku in Turku, Finland. Royal Caribbean plans to have 3 "Icon"-class ships by 2030.


title: Icon-class cruise ship

text: On 10 October 2016, Royal Caribbean and Meyer Turku announced an order to build two ships under the project name "Icon". The first two ships are planned to be delivered in the second quarters of 2022 and 2024. The ships will be classified by DNV GL.


title: Icon-class cruise ship

text: Royal Caribbean applied to register a trademark for "Icon of the Seas" in 2016, which was at the time suggested as an indication of the name of the first ship.


title: Icon-class cruise ship

text: On 2 July 2019, Royal Caribbean announced an order for a third ship in the "Icon" class. The third ship is planned to be delivered in 2025, one year after the second "Icon" ship.


title: Icon-class cruise ship

text: The ships will employ fuel cell technology, to be supplied by ABB Group, and be powered by liquefied natural gas, with a gross tonnage of 200,000 GT. Ships will contain other alternative energy features, like the use of fuel cells to produce electricity and fresh water. It will have a capacity of 5,600 berths.


title: Icon-class cruise ship

text: In 2020, the director of projects and facilities at Nassau Cruise Port said that the specifications for the Icon class indicate it would be larger than the . Later, in May 2022, Royal Caribbean confirmed that "Icon of the Seas" would be bigger than the "Oasis" class.


title: Shadaab Khan

text: Shadaab Khan (born 20 September 1973) is an Indian Hindi film actor, writer, and film director. He is the son of actor Amjad Khan.


title: Shadaab Khan

text: Shadaab Khan was born on 20 September 1973 in Bombay, Maharashtra, India (now Mumbai) into a Muslim family to parents Amjad Khan and Shaila Khan. Shadaab is of Pashtun descent on his father's side, traced back to Peshawar, North-West Frontier Province (now in Khyber Pakhtunkhwa, Pakistan), where his grandfather Zakaria Khan, better known as Jayant was born and raised. He has two siblings Seemaab Khan and Ahlam Khan. He is married to Rumana Achwa since 2005.


title: Shadaab Khan

text: Shadaab Khan has written two books: "Shanti Memorial" in 2013 and "Murder in Bollywood", a crime novel, in 2015.


title: Donald Moffat

text: Donald Moffat (26 December 1930 – 20 December 2018) was a British–American actor with a decades-long career in film and stage in the United States. He began his acting career on- and off-Broadway, which included appearances in "The Wild Duck" and "Right You Are If You Think You Are", earning a Tony Award nomination for both, as well as "Painting Churches", for which he received an Obie Award. Moffat also appeared in several feature films, including "The Thing" and "The Right Stuff", along with his guest appearances in the television series "Dr. Quinn, Medicine Woman" and "The West Wing".


title: Donald Moffat

text: Moffat was born in Plymouth, Devon, the only child of Kathleen Mary (née Smith) and Walter George Moffat, an insurance agent. His father was Scottish. His parents ran a boarding house in Totnes. Completing his studies at the local King Edward VI School and national service in the Army from 1949 to 1951, Moffat trained at the Royal Academy of Dramatic Art in London.


title: Donald Moffat

text: Moffat began his career as a stage actor in London and New York City. His first work was at the Old Vic Theatre Company in London.


title: Donald Moffat

text: After moving to the United States, Moffat worked as a bartender and a lumberjack in Oregon, his wife's home state. "After six months," he said, "I realized that I was an actor and I would always be an actor. And an actor must act. So I started acting again." His first acting job in the United States was in Princeton, New Jersey. He worked as a carpenter, and his wife did ironing in order to supplement his $25 per week pay.


title: Donald Moffat

text: He joined APA (The Association of Producing Artists), a repertory company on Broadway, and was nominated for a Tony for Best Actor in a Play in 1967 for his roles in revivals of Henrik Ibsen's "The Wild Duck" and Pirandello's "Right You Are If You Think You Are".


title: Donald Moffat

text: He was nominated for Drama Desk Awards for Outstanding Actor in a Play for his work in "Play Memory" (1984) and for Outstanding Featured Actor in the revival of Eugene O'Neill's "The Iceman Cometh" (1986) with Jason Robards. He won an Obie for "Painting Churches". In 1998, he was nominated for a Gemini Award for his performance as attorney Joe Ruah in the CBC miniseries "The Sleep Room". He also appeared in many Broadway and Off-Broadway plays, including John Guare's "A Few Stout Individuals" (as Ulysses S. Grant), "The Heiress", "The Cherry Orchard", "Much Ado About Nothing", "The School for Scandal", "The Affair" and "Hamlet".


title: Donald Moffat

text: Among Moffat's best-known film roles are as Lyndon B. Johnson in "The Right Stuff" (1983), the corrupt U.S. president in "Clear and Present Danger", and as Garry, the station commander in "The Thing".


title: Donald Moffat

text: Moffat played Enos in the CBS western miniseries "The Chisholms", Lars Lundstrom in the ABC drama "The New Land". and Rem in the CBS science-fiction series "Logan's Run". He also appeared in "Columbo", "The West Wing", "Dr. Quinn, Medicine Woman" and "Tales of the City", in which his performance as dying executive Edgar Halcyon earned him many new fans. One of his final roles was as Baseball Commissioner Ford Frick in the HBO movie, "61*". Moffat's last role was as a judge in an episode of "" in 2005.


title: Donald Moffat

text: Moffat married actress Anne Murray in 1954; they had a daughter, Wendy, and a son, Gabriel, before divorcing in 1968. He later married actress Gwen Arner.


title: Donald Moffat

text: Moffat died on 20 December 2018 in Sleepy Hollow, New York due to complications from a stroke at the age of 87.


title: Ukiyo

text: The famous Japanese woodblock prints known as , or "pictures of the floating world", had their origins in these districts, and often depicted scenes of the floating world itself such as geisha, kabuki actors, sumo wrestlers, samurai, merchants, and prostitutes.


title: Ukiyo

text: The term in medieval Japan was associated with Buddhism and meant "this transient, unreliable world". When written as meaning "the floating world", is also an ironic, homophonous allusion to the earlier Buddhist term , referring to the earthly plane of death and rebirth from which Buddhists sought release.


title: Ukiyo

text: In its modern usage, the term is used to refer to a state of mind emphasising living in the moment, detached from the difficulties of life.


title: Visage (band)

text: Visage were a British synthpop band, formed in London in 1978. The band became closely linked to the burgeoning New Romantic fashion movement of the early 1980s, and are best known for their hit "Fade to Grey" which was released in late 1980. In the UK, the band achieved two Top 20 albums ("Visage" and "The Anvil") and five Top 30 singles before the commercial failure of their third album ("Beat Boy") led to their break-up in 1985.


title: Visage (band)

text: The band has seen various line-up changes over the years, all fronted by vocalist Steve Strange, who resurrected the band name in the 2000s. In 2013, the most recent line-up of the band released "Hearts and Knives", the first new Visage album in 29 years. The band's fifth and final album, "Demons to Diamonds", was released in 2015, nine months after Strange had died following a heart attack.


title: Visage (band)

text: Founding members Midge Ure and Rusty Egan started working on Visage to produce music to play at the clubs Egan was DJing at. Egan alongside Steve Strange was hosting David Bowie and Roxy Music club nights at "Billy's" nightclub in London's Soho district at the time and Egan was eager to find new music to play, ultimately opting to create music himself with Ure. Steve Strange had briefly been in the punk/new wave bands The Moors Murderers and The Photons, and Egan was working with Midge Ure in the band The Rich Kids. Ure and Egan recorded a demo which included the original track "The Dancer" of which a version appeared on the debut Visage album and a cover of the Zager and Evans hit "In the Year 2525". Strange was then brought into the band to provide the face and voice of Visage with the line-up being completed with the addition of Ultravox keyboardist Billy Currie and three-fifths of the post-punk band Magazine – guitarist John McGeoch, keyboardist Dave Formula and bassist Barry Adamson (who left the band after playing on its debut single, but returned as a session musician). Producer Martin Rushent had heard some of the band's material at "Billy's" nightclub and financed further recordings with a view to signing the band to his new Genetic Records label. Visage recorded their first album at Rushent's home studio in Berkshire, but Rushent's label collapsed, and the band instead signed to Radar Records, a new independent label run by Rushent's former colleague Martin Davis (the pair had worked together at United Artists Records). Visage released their first single "Tar" on Radar in September 1979, though the single failed to chart. By this time, however, Strange and Egan had relocated their themed club nights to the "Blitz" club in Covent Garden and the New Romantic movement had begun in earnest. In mid-1980, David Bowie himself visited the club and asked Strange and three other regulars to appear in the video for his single "Ashes to Ashes", which helped to propel the New Romantic movement into the mainstream.


title: Visage (band)

text: Although Visage's self-titled debut album had been completed for several months, it was not released until November 1980 when the band was now signed to the major label Polydor Records. The band's second single, "Fade to Grey", was released at the same time. The single became a hit in early 1981, making the top ten in the UK Singles Chart and several other countries, and reaching no. 1 in Germany and Switzerland. The album also became a Top 20 hit in the UK and was certified Silver by the British Phonographic Industry.


title: Visage (band)

text: After further hits with the singles "Mind of a Toy" and the title track "Visage", Strange struggled to reunite the band's members again to record a second album due to their commitments with their respective bands; Ure had now joined Currie in Ultravox, Formula and Adamson with Magazine and McGeoch had joined Siouxsie and the Banshees. Nonetheless, in the autumn of 1981, Visage went into the studio and recorded "The Anvil" as a five-piece band without McGeoch and only limited guest work from Adamson. The album, which was named after an infamous gay nightclub in New York City, was released in March 1982 and became the band's only Top 10 hit in the UK Albums Chart, producing two top-twenty singles with "The Damned Don't Cry" and "Night Train". Like their first album, "The Anvil" earned a Silver disc in the UK. Following this, Ure quit from Visage to concentrate on his work with Ultravox, who were now even more successful than Visage. Creative differences with Strange and Egan were also cited as reasons for his departure at the time.


title: Visage (band)

text: Visage were now without Ure, McGeoch and Adamson, who went on to join Nick Cave and the Bad Seeds. Bassist Steve Barnacle was recruited to replace Adamson, and the remaining lineup recorded the stand-alone single "Pleasure Boys", which was released in October 1982. The single failed to prolong their string of hits and peaked just outside the UK Top 40.


title: Visage (band)

text: Although still recording, Visage then took a two-year hiatus from releasing any new material due to contractual difficulties with their management company. Polydor issued a "best of" compilation, "Fade to Grey – The Singles Collection", which included all of the singles released to date and the previously unreleased "In the Year 2525". Although the album was certified Gold in the UK for pre-release sales to stores, it only peaked at No. 38 in the UK Albums Chart after its release in November 1983.


title: Visage (band)

text: In 1984, with their contractual problems resolved, Visage returned with their third album, "Beat Boy". Released in October 1984, the album was a critical and commercial failure, peaking at No.  79 in the UK. Two singles from the album, "Love Glove" and "Beat Boy", also failed to make the UK top 40. By this time, Billy Currie and Dave Formula had also left the band (though they received a "special thanks" credit on the album sleeve for their input), leaving only Strange and Egan from the original line-up along with Steve Barnacle and new recruits Gary Barnacle (Steve Barnacle's brother) and Andy Barnett who also was a member of FM and ASAP. A decision to make Visage a live band instead of a strictly studio-based project also failed to meet with success and the band split in 1985. Their final release was a "Visage" video compilation of the band's promotional videos and other footage, including Strange's 1983 trip to North Africa. The compilation does not, however, include the original video for the "Love Glove" single which was filmed at a late-night Dockland location in London in 1984.


title: Visage (band)

text: Following the demise of Visage, Strange and Barnacle then formed the short-lived band Strange Cruise, though this too proved unsuccessful. Visage returned to the charts once more when a Bassheads remix of "Fade to Grey" was a UK Top 40 hit in 1993.


title: Visage (band)

text: Steve Strange reappeared on the music scene in 2002, after several years of battling a heroin addiction and other personal problems. He performed several Visage songs on the Here and Now Xmas Tour – a revival of 1980s pop acts. Some time after the performance, Strange decided to create a "Mark II" of Visage with people from several electronic bands and projects: Steven Young, Sandrine Gouriou and Rosie Harris from Seize and Ross Tregenza from Jetstream Lovers/Goteki. After the announcement of the formation of the new line-up and several television appearances, plans for reworking old material and releasing a new record made slow progress. An updated version of "Fade to Grey" was produced in 2005. In 2006, Strange also collaborated with the electronic duo Punx Soundcheck and provided vocals on the track "In the Dark", which was included on the duo's debut double album "When Machines Ruled the World". The first Visage mk II track was released in 2007, entitled "Diary of a Madman". Written by Strange with Visage mk II member Ross Tregenza, the track was co-produced by original Visage member Dave Formula. This song was made available for download from their official website in return for a donation to the charity Children in Need. However, no further new material surfaced from this line-up.


title: Visage (band)

text: The long-since deleted Visage VHS video collection was repackaged for release on DVD in Summer 2006, though it was mistakenly titled "Visage Live".


title: Visage (band)

text: In 2008, Strange (and Visage II keyboardist Sandrine Gouriou) made an appearance in the BBC drama series "Ashes to Ashes" which is set in 1981. In it, they performed the song "Fade to Grey" in a scene set in the "Blitz" nightclub.


title: Visage (band)

text: In 2009, Strange and Egan appeared in Living TV's "Pop Goes the Band", a series in which pop stars from the 1980s are given a complete makeover in return for a one-off performance. The Visage episode aired on 16 March 2009, and was the first time that the two men had spoken in over 20 years. The episode focused (like others in the series) more on getting them fit in the gym than on the current state of their relationship, though they appeared to get on well enough. At the culmination of the episode, they performed "Fade to Grey".


title: Visage (band)

text: In 2010, new remixes of "Fade to Grey" were produced by club DJs Michael Gray and Lee Mortimer to celebrate the 30th anniversary of the song. The remixes were included on a new compilation album, "The Face – The Very Best of Visage", which was released in March 2010.


title: Visage (band)

text: On 8 January 2013, Strange appeared as a guest on the "Channel 4 News" programme to discuss the forthcoming David Bowie album "The Next Day". During the interview he mentioned that a new Visage album was also due for release in Spring 2013.


title: Visage (band)

text: Also on 8 January 2013, Visage launched their new website, Twitter, Facebook and Soundcloud accounts and announced their new line-up, consisting of Steve Strange, former Visage bassist Steve Barnacle, former Ultravox guitarist Robin Simon, and Lauren Duvall on vocals. A single from the album, "Shameless Fashion", was made available as a free download from the band's Facebook page, and a physical CD-single was released on 6 May 2013. The band's new album, "Hearts and Knives", was released on 20 May 2013. A second single from the album, "Dreamer I Know", was released in July 2013, and a third single, "Never Enough", was released in December. Throughout the second half of 2013, the band also embarked on a series of live dates in the UK and Europe. "Hidden Sign" was the fourth single to be taken from "Hearts and Knives", released in May 2014 and "She's Electric (Coming Around)" was released as the fifth and final single in August 2014.


title: Visage (band)

text: In December 2014, Visage released "Orchestral", a mostly live album containing twelve Visage songs remade with a symphony orchestra. A single of the orchestral version of "Fade To Grey" was released ahead of the album in November 2014.


title: Visage (band)

text: On 12 February 2015, frontman Steve Strange died of a heart attack while on holiday in Egypt. While Strange's death ultimately meant the end of Visage, the band opted to complete the album they had already been working on with Strange prior to his death. On 2 September 2015, an organisation known as The Steve Strange Collective was announced. Run by Strange's friends and relatives, they oversaw the release of the final Visage album, "Demons to Diamonds", which was released on 6 November 2015.


title: Subcutaneous tissue

text: The subcutaneous tissue (), also called the hypodermis, hypoderm (), subcutis, superficial fascia, is the lowermost layer of the integumentary system in vertebrates. The types of cells found in the layer are fibroblasts, adipose cells, and macrophages. The subcutaneous tissue is derived from the mesoderm, but unlike the dermis, it is not derived from the mesoderm's dermatome region. It consists primarily of loose connective tissue, and contains larger blood vessels and nerves than those found in the dermis. It is a major site of fat storage in the body.


title: Subcutaneous tissue

text: In arthropods, a hypodermis can refer to an epidermal layer of cells that secretes the chitinous cuticle. The term also refers to a layer of cells lying immediately below the epidermis of plants.


title: Subcutaneous tissue

text: In some animals, such as whales and hibernating mammals, the hypodermis forms an important insulating layer and/or food store.


title: Subcutaneous tissue

text: In some plants, the hypodermis is a layer of cells immediately below the epidermis of leaves. It is often mechanically strengthened, for example, in pine leaves, forming an extra protective layer or a water storage tissue.


title: Subcutaneous tissue

text: Subcutaneous fat is the layer of subcutaneous tissue that is most widely distributed. It is composed of adipocytes, which are grouped together in lobules separated by connective tissue. The number of adipocytes varies among different areas of the body, while their size varies according to the body's nutritional state. It acts as padding and as an energy reserve, as well as providing some minor thermoregulation via insulation. Subcutaneous fat is found just beneath the skin, as opposed to visceral fat, which is found in the peritoneal cavity, and can be measured using body fat calipers to give a rough estimate of total body adiposity.


title: Subcutaneous tissue

text: Injection into the subcutaneous tissue is a route of administration used for drugs such as insulin: because it is highly vascular, the tissue absorbs drugs quickly. Subcutaneous injection is believed to be the most effective manner to administer some drugs, such as human growth hormones. Just as the subcutaneous tissue can store fat, it can also provide good storage space for drugs that need to be released gradually because there is limited blood flow. "Skin popping" is a slang term that includes this method of administration, and is usually used in association with recreational drugs.


title: Sepal

text: A sepal () is a part of the flower of angiosperms (flowering plants). Usually green, sepals typically function as protection for the flower in bud, and often as support for the petals when in bloom. The term "sepalum" was coined by Noël Martin Joseph de Necker in 1790, and derived .


title: Sepal

text: Collectively the sepals are called the calyx (plural calyces), the outermost whorl of parts that form a flower. The word "calyx" was adopted from the Latin , not to be confused with 'cup, goblet'. "Calyx" is derived from Greek 'bud, calyx, husk, wrapping' ( Sanskrit 'bud'), while is derived from Greek 'cup, goblet', and the words have been used interchangeably in botanical Latin.


title: Sepal

text: After flowering, most plants have no more use for the calyx which withers or becomes vestigial. Some plants retain a thorny calyx, either dried or live, as protection for the fruit or seeds. Examples include species of "Acaena", some of the Solanaceae (for example the Tomatillo, "Physalis philadelphica"), and the water caltrop, "Trapa natans". In some species the calyx not only persists after flowering but instead of withering, begins to grow until it forms a bladder-like enclosure around the fruit. This is an effective protection against some kinds of birds and insects, for example in "Hibiscus trionum" and the Cape gooseberry. In other species, the calyx grows into an accessory fruit.


title: Sepal

text: Morphologically, both sepals and petals are modified leaves. The calyx (the sepals) and the corolla (the petals) are the outer sterile whorls of the flower, which together form what is known as the "perianth". Similarly to ordinary leaves, sepals are capable of performing photosynthesis. However, photosynthesis in sepals occurs at a slower rate than in ordinary leaves due to sepals having a lower stomatal density which limits the spaces for gas exchange.


title: Sepal

text: The term "tepal" is usually applied when the parts of the perianth are difficult to distinguish, e.g. the petals and sepals share the same color or the petals are absent and the sepals are colorful. When the undifferentiated tepals resemble petals, they are referred to as "petaloid", as in petaloid monocots, orders of monocots with brightly colored tepals. Since they include Liliales, an alternative name is lilioid monocots. Examples of plants in which the term tepal is appropriate include genera such as "Aloe" and "Tulipa". In contrast, genera such as "Rosa" and "Phaseolus" have well-distinguished sepals and petals.


title: Sepal

text: The number of sepals in a flower is its merosity. Flower merosity is indicative of a plant's classification. The merosity of a eudicot flower is typically four or five. The merosity of a monocot or palaeodicot flower is three, or a multiple of three.


title: Sepal

text: The development and form of the sepals vary considerably among flowering plants. They may be free (polysepalous) or fused together (gamosepalous). Often, the sepals are much reduced, appearing somewhat awn-like, or as scales, teeth, or ridges. Most often such structures protrude until the fruit is mature and falls off.


title: Sepal

text: In some flowers, the sepals are fused towards the base, forming a calyx tube (as in the family Lythraceae, and Fabaceae). In other flowers (e.g., Rosaceae, Myrtaceae) a hypanthium includes the bases of sepals, petals, and the attachment points of the stamens.


title: Sepal

text: Mechanical cues may be responsible for sepal growth and there is a strong evidence suggesting that microtubules are present and determine the tensile strength and direction of growth at a molecular level.


title: Stochastic differential equation

text: A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes.


title: Stochastic differential equation

text: Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force.


title: Stochastic differential equation

text: The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician Kiyosi Itô, who introduced the concept of stochastic integral and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist Stratonovich, leading to a calculus similar to ordinary calculus.


title: Stochastic differential equation

text: The most common form of SDEs in the literature is an ordinary differential equation with the right hand side perturbed by a term dependent on a white noise variable. In most cases, SDEs are understood as continuous time limit of the corresponding stochastic difference equations. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral. Such a mathematical definition was first proposed by Kiyosi Itô in the 1940s, leading to what is known today as the Itô calculus.


title: Stochastic differential equation

text: The Itô integral and Stratonovich integral are related, but different, objects and the choice between them depends on the application considered. The Itô calculus is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time.


title: Stochastic differential equation

text: The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion on manifolds.


title: Stochastic differential equation

text: An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the Smoluchowski equation or the Fokker–Planck equation, an equation describing the time evolution of probability distribution functions. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of stochastic evolution operator.


title: Stochastic differential equation

text: In physical science, there is an ambiguity in the usage of the term "Langevin SDEs". While Langevin SDEs can be of a more general form, this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure, leading to a N=2 supersymmetric model closely related to supersymmetric quantum mechanics. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e., (overdamped) Langevin SDEs are never chaotic.


title: Stochastic differential equation

text: Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Itô stochastic calculus and the Stratonovich stochastic calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003) and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down.


title: Stochastic differential equation

text: Numerical methods for solving stochastic differential equations include the Euler–Maruyama method, Milstein method and Runge–Kutta method (SDE).


title: Stochastic differential equation

text: In physics, SDEs have widest applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the dynamical systems theory to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.


title: Stochastic differential equation

text: There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:


title: Stochastic differential equation

text: where formula_2 is the position in the system in its phase (or state) space, formula_3, assumed to be a differentiable manifold, the formula_4 is a flow vector field representing deterministic law of evolution, and formula_5 is a set of vector fields that define the coupling of the system to Gaussian white noise, formula_6. If formula_7 is a linear space and formula_8 are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case in which formula_9.


title: Stochastic differential equation

text: For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition. Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a stochastic difference equation. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a uniquely defined mathematical object that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.


title: Stochastic differential equation

text: In physics, the main method of solution is to find the probability distribution function as a function of time using the equivalent Fokker–Planck equation (FPE). The Fokker–Planck equation is a deterministic partial differential equation. It tells how the probability distribution function evolves in time similarly to how the Schrödinger equation gives the time evolution of the quantum wave function or the diffusion equation gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by Monte Carlo simulation. Other techniques include the path integration that draws on the analogy between statistical physics and quantum mechanics (for example, the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables) or by writing down ordinary differential equations for the statistical moments of the probability distribution function.


title: Stochastic differential equation

text: The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different. It is also the notation used in publications on numerical methods for solving stochastic differential equations. This notation makes the exotic nature of the random function of time formula_10 in the physics formulation more explicit. In strict mathematical terms, formula_10 cannot be chosen as an ordinary function, but only as a generalized function. The mathematical formulation treats this complication with less ambiguity than the physics formulation.


title: Stochastic differential equation

text: This equation should be interpreted as an informal way of expressing the corresponding integral equation


title: Stochastic differential equation

text: The equation above characterizes the behavior of the continuous time stochastic process "X" as the sum of an ordinary Lebesgue integral and an Itô integral. A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length "δ" the stochastic process "X" changes its value by an amount that is normally distributed with expectation "μ"("X", "t") "δ" and variance "σ"("X", "t") "δ" and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function "μ" is referred to as the drift coefficient, while "σ" is called the diffusion coefficient. The stochastic process "X" is called a diffusion process, and satisfies the Markov property.


title: Stochastic differential equation

text: The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution. Both require the existence of a process "X" that solves the integral equation version of the SDE. The difference between the two lies in the underlying probability space (formula_15). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space.


title: Stochastic differential equation

text: which is the equation for the dynamics of the price of a stock in the Black–Scholes options pricing model of financial mathematics.


title: Stochastic differential equation

text: There are also more general stochastic differential equations where the coefficients "μ" and "σ" depend not only on the present value of the process "X", but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process, "X", is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values of "X", the defining equation is called a stochastic delay differential equation.


title: Stochastic differential equation

text: As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in "n"-dimensional Euclidean space R and driven by an "m"-dimensional Brownian motion "B"; the proof may be found in Øksendal (2003, §5.2).


title: Stochastic differential equation

text: Let "Z" be a random variable that is independent of the "σ"-algebra generated by "B", "s" ≥ 0, and with finite second moment:


title: Stochastic differential equation

text: has a P-almost surely unique "t"-continuous solution ("t", "ω") ↦ "X"("ω") such that "X" is adapted to the filtration "F" generated by "Z" and "B", "s" ≤ "t", and