In mathematics, an isomorphism is a mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic; the word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", μορφή morphe "form" or "shape". The interest in isomorphisms lies in the fact, thus isomorphic structures cannot be distinguished from the point of view of structure only, may be identified. In mathematical jargon, one says. An automorphism is an isomorphism from a structure to itself; when there is only one isomophism between two structures, it is called a canonical isomorphism, one says that the structures are canonically isomorphic. For example, for every prime number p, all fields with p elements are canonically isomorphic; the term isomorphism is used for algebraic structures. In this case, mappings are called homomorphisms, a homomorphism is an isomorphism if and only if it is bijective. In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration.

For example: An isometry is an isomorphism of metric spaces. A homeomorphism is an isomorphism of topological spaces. A diffeomorphism is an isomorphism of spaces equipped with a differential structure differentiable manifolds. A permutation is an automorphism of a set. Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea. In geometry and automorphisms are called transformations, for example rigid transformations, affine transformations, projective transformations. Let R + be the multiplicative group of positive real numbers, let R be the additive group of real numbers; the logarithm function log: R + → R satisfies log ⁡ = log ⁡ x + log ⁡ y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp: R → R + satisfies exp ⁡ = for all x, y ∈ R, so it too is a homomorphism; the identities log ⁡ exp ⁡ x = x and exp ⁡ log ⁡ y = y show that log and exp are inverses of each other.

Since log is a homomorphism that has an inverse, a homomorphism, log is an isomorphism of groups. Because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers; this facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale. Consider the group, the integers from 0 to 5 with addition modulo 6. Consider the group, the ordered pairs where the x coordinates can be 0 or 1, the y coordinates can be 0, 1, or 2, where addition in the x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3; these structures are isomorphic under addition, under the following scheme: ↦ 0 ↦ 1 ↦ 2 ↦ 3 ↦ 4 ↦ 5or in general ↦ mod 6. For example, + =, which translates in the other system as 1 + 3 = 4. Though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are the same. More the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime, per the Chinese remainder theorem.

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S an isomorphism from X to Y is a bijective function ƒ: X → Y such that: S ⁡ ⟺ R ⁡ {\displaystyle \operatorname (f,f(

Chen Wu, courtesy name Zilie, was a military general serving under the warlord Sun Quan in the late Eastern Han dynasty. He served under Sun Ce, Sun Quan's elder brother and predecessor. Chen Wu was from Songzi County, Lujiang Commandery, in present-day Susong County, Anhui, he was seven chi and seven cun tall. When he was about 17 years old, he travelled to Shouchun to meet Sun Ce, a subordinate of the warlord Yuan Shu. Chen Wu accompanied Sun Ce on his conquests in the Jiangdong region in the 190s, was appointed as a Major of Separate Command for his contributions in battle, he followed Sun Ce to attack a minor warlord Liu Xun and helped to recruit many capable men from Lujiang to join Sun Ce's army, with him as their commander. After Sun Ce's death in the year 200, Chen Wu continued serving under Sun Quan, Sun Ce's younger brother and successor, he was put in command of five regiments. Chen Wu was known to a kind and generous person, he won the hearts of many people from his hometown and other places.

He was favoured by Sun Quan, who visited his house on several occasions. Chen Wu was promoted to Lieutenant-General for his achievements. In 215, Chen Wu followed Sun Quan to attack Hefei, a city under the control of a rival warlord Cao Cao, which led to the Battle of Xiaoyao Ford. Sun Quan's forces suffered a disastrous defeat at the hands of Cao Cao's general Zhang Liao, defending Hefei. Chen Wu was killed in action. Sun Quan attended the latter's funeral. Sun Quan had Chen Wu's favourite concubine sacrificed to join Chen in death, he awarded Chen's family 200 taxable households in their estate; the historian Sun Sheng criticised Sun Quan's act of forcing Chen Wu's concubine to join Chen in death, citing an earlier negative example of Duke Mu of Qin and a positive example of Wei Ke. Chen Wu had two sons -- Chen Biao. Chen Xiu had a personality; when he was 19 years old, Sun Quan summoned him and appointed him as a Major of Separate Command and put him in charge of 500 troops. At the time, many newly recruited soldiers were unwilling to serve in the army.

However, Chen Xiu treated his men well and not a single soldier under him deserted. Sun Quan was surprised and pleased, so he promoted Chen Xiu to Colonel. In the late 210s, when Sun Quan granted awards to the descendants of his deceased subjects to honour his subjects for their service, Chen Xiu received the title of a Marquis of a Chief Village in recognition of his father's contributions, he became a commander in the jiefan corps, one of the elite units in Sun Quan's military forces. He died in 229, shortly after Sun Quan proclaimed himself emperor and established the state of Eastern Wu. Chen Biao was born to one of Chen Wu's concubines, he served in Eastern Wu. After Chen Biao's death, his son Chen Ao was commissioned as a Major of Separate Command at the age of 16 and was placed in charge of 400 troops; when Chen Ao died, his military appointment was inherited by his cousin Chen Yan, Chen Xiu's son. Chen Yan had Chen Yong, who became a general and received a marquis title. Although no details were given on how Chen Wu died at the Battle of Xiaoyao Ford in 215, his death was dramatised in chapter 68 of the historical novel Romance of the Three Kingdoms by Luo Guanzhong, which romanticises the historical events before and during the Three Kingdoms period.

In the novel, Chen Wu encountered Pang De in another battle right after the Battle of Xiaoyao Ford. While fighting with Pang De, he was driven into a valley full of thick vegetation and his sleeve was caught in some tree branches, he was killed by Pang De while attempting to free himself. Lists of people of the Three Kingdoms Chen, Shou. Records of the Three Kingdoms. Luo, Guanzhong. Romance of the Three Kingdoms. Pei, Songzhi. Annotations to Records of the Three Kingdoms

The Angolan fruit bat or Angolan rousette is a species of megabat in the family Pteropodidae. It is found in Angola, Cameroon, Central African Republic, Republic of Congo, Democratic Republic of Congo, Nigeria, Sudan, Tanzania and Zambia, its natural habitats are subtropical or tropical moist lowland forest, moist savanna, rocky areas. It was described in 1898 by Portuguese zoologist José Vicente Barbosa du Bocage. Bocage placed it in the now-defunct genus Cynonycteris, with a binomial of C. angolensis. Its species name "angolensis" is Latin for "Angolan," in reference to the fact that the holotype was collected near Pungo Andongo in Angola, its forearm length is 68–90 mm and it weighs 60–87 g. It is frugivorous. Consumed fruits include fruits of various trees, including fig trees, Milicia excelsa, Adenia; the Angolan rousette has a wide range encompassing parts of West and Central Africa. It is found from sea level to 4,000 m above sea level; as of 2017, it is evaluated as a least-concern species by the IUCN.

It meets the criteria for this classification.

Kelvin Barry "Kel" Richards is an Australian author, radio personality and lay Christian. Richards has written a series of crime novels and thrillers for adult readers which includes The Case of the Vanishing Corpse, Death in Egypt and An Outbreak of Darkness. Richards presented ABC NewsRadio's weekend afternoons, which included regular Wordwatch segments until 15 August 2010. Wordwatch was a feature introduced by Richards. Developed as a "filler" program for the radio programs to allow time for changes of people or locations, it tapped into an interest by the listening public and several books have been produced based on the show's research. In November 2003 the thousandth episode of the show was produced. Kel Richards presents Sydney radio station 2CH's Sunday night program "Sunday Night with Kel Richards", he is a lay canon at St Andrew's Cathedral and the author of the Aussie Bible, which has sold over 100,000 copies. This book translates sections of the Bible into Australian vernacular; the Case of the Vanishing Corpse The Case of the Secret Assassin The Second Death The Case of the Damascus Dagger Moonlight Shadows The Case of the Dead Certainty The Third Bloodstain Death in Egypt: a Murder Mystery "The Sinister Student: A 1930s Murder Mystery" The Floating Corpse C.

S Lewis and the Body in the Basement Sherlock Holmes: Tales of Terror: Curse of the Pharaohs Sherlock Holmes: Tales of Terror: The Headless Monk Sherlock Holmes: Tales of Terror: The Vampire Serpent "The Amateur Hangman", More Crimes for a Summer Christmas edited by Stephen Knight The Ballad of the Two Sons illustrated by Meng Aussie Yarns Aussie Pilgrim's Progress: John Bunyan's immortal story An Aussie Christmas Carol: Charles Dickens' immortal tale retold as an Aussie bush yarn "The Lamington Man" with Glen Singleton ABC Classic FM's Word of the Day with Clive Robertson The Story of Australian English Word Map: What words are Used Where in Australia with the Macquarie DictionaryReview Word of the day 2: Wordwatching: a Wonderful Look Into the World of Words Murder Call TV episodes– Deadfall – A Blow to the Heart – House of Spirits – A Stab in the Dark Austlit – Richards, Kel Kel Richards Books

The Journal of Theoretical and Computational Chemistry was founded in 2002 and is published by World Scientific. It is an interdisciplinary journal covering developments in theoretical and computational chemistry, as well as their applications to other scientific fields, such as physics and material sciences. Articles are broadly categorized into quantum chemistry, chemical dynamics, statistical mechanics, kinetic theory, chemical biology and nanotechnology, ranging from fundamental theory, theoretical methods and computational algorithms to numerical applications, such as in chemical and nano–bio systems; the journal is abstracted and indexed in: Science Citation Index Expanded ISI Alerting Services Chemistry Citation Index Current Contents/Physical and Earth Sciences InspecAccording to the Journal Citation Reports, the journal has a 2014 impact factor of 0.638, ranking it 127th out of 157 journals in the category "Chemistry Multidisciplinary". Journal Website

Michael "Mike" Whorf is an American radio personality based in Detroit, Michigan. He was an announcer and program host on WJR from 1964 to 2003. Whorf was producer and host of the George Foster Peabody Award-winning documentary/narrative program Kaleidoscope, a combination of storytelling, historic recordings and music on a particular topic. Whorf was born in Brookline and spent his childhood and teen years in Provincetown, Massachusetts, his father was internationally renowned watercolorist John Whorf. Whorf's sisters Carol Whorf Westcott and Nancy Whorf Kelly were Provincetown artists, his brother John was an established applied artist in Hingham. Whorf is nephew of linguist Benjamin Lee Whorf and nephew of actor and television director Richard Whorf. Whorf graduated from Provincetown High School in 1950. Upon graduation, Whorf enlisted in the United States Air Force where he served as a radio announcer and entertainer on the Armed Forces Network, his tour of duty included assignments at air bases in California and Morocco.

After his honorable discharge from the USAF, Whorf worked as an announcer at WOCB in West Yarmouth, Massachusetts and at WCOJ in Coatesville, where he met and married his wife Barbara Ann Brown. He developed the Kaleidoscope predecessor "Tempo" while on the air at WTAG in Worcester, Massachusetts. Whorf worked at WWL in New Orleans, Louisiana before returning to WTAG. Shortly after, he went on to WJR. Throughout the late 1960s and 1970s, Whorf was part of a line-up of radio personalities known throughout the region including J. P. McCarthy, Karl Haas, Jimmy Launce and MLB Hall of Fame broadcaster Ernie Harwell. Whorf's Kaleidoscope topics ranged from religion from the arts to sports; some of Whorf's extended series included oral histories of Native American tribes and a collection of personal interviews with popular American song composers of the 1920s and 1930s entitled "The Bards of Tinpan Alley". Whorf is himself a published composer, his 1979 Christmas song "The Man with a Hundred Names" muses on the many names by which Santa Claus is known worldwide.

In 1970, Whorf formed the company Mike Whorf Inc. which sold to schools and individuals cassette tape copies of Kaleidoscope. In the late 1970s and 1980s, he partnered with nephew and retired United States Marine Corps Captain Charles "Charlie" T. Westcott III on radio dramas and comedies that were featured during the Kaleidoscope hour, including the old-time radio parodies "Big Jim Small" and "Another Man’s Family." Whorf parted from WJR from 1983 to 1984 to serve as program director of classical station WQRS-FM in Detroit. While with WQRS, Whorf created the program "Quest for Excellence," a juried music competition show for young talents broadcast live before a studio audience. Whorf brought that program to CKLW in Windsor and took it with him upon his return to WJR. Whorf's programs were broadcast by WVXU-FM, the National Public Radio affiliate station of Xavier University in Cincinnati, Ohio. In addition to his Peabody-winning work on topics including the life and times of Dr. Martin Luther King Jr. Whorf received multiple broadcasting awards from the Freedoms Foundation at Valley Forge.

In 2008 he was inducted into the Michigan Broadcasting Hall of Fame by the Michigan Association of Broadcasters. "Kaleidoscope" - archive at The Public Radio Exchange. "The Man with a Hundred Names" - used print copies at Amazon. Copyright 1979 Shawnee Press sheet music # EA-7. - contact to get reproduction copyright permission