Note: One can make a non-surjective function into a surjection by "restricting" its codomain to elements of its range. For example, the "new" function, "f"("x"):ℝ → [0,+∞) where "f"("x") = "x" is a surjective function. (This is not the same as the restriction of a function which restricts the domain!)
Example: The exponential function "f"("x") = 10 is not a surjection. The range of "10" is (0,+∞), that is, the set of positive numbers. (This function is an injection.)
Example: The logarithmic function base 10 "f"("x"):<nowiki>(0,+∞)</nowiki>→ℝ defined by "f"("x")=log("x") or "y"=log("x") is a surjection (and an injection). (This is the inverse function of 10.)
Example: The function "f"(("x","y")):ℝ²→ℝ defined by "z"="y" is a surjection. Its graph is a plane in 3-dimensional space. The pre-image of "z" is the line y="z" in the "x"0"y" plane.
Range (mathematics)
In mathematics, the range is a concept which has various meanings in different topics:
Organic reaction
Organic reactions are chemical reactions involving organic compounds. The basic organic chemistry reaction types are addition reactions, elimination reactions, substitution reactions, pericyclic reactions, rearrangement reactions, photochemical reactions and redox reactions. In organic synthesis, organic reactions build new organic molecules. Organic reactions produce many man-made chemicals such as drugs, plastics, food additives, and fabrics.
Shikasta (novel)
Shikasta is a 1979 science fiction novel. The full title is Re: Colonised Planet 5, Shikasta. British Nobel Prize in Literature-winner Doris Lessing wrote it as the first book in her five-book "Canopus in Argos" series. Alfred A. Knopf published it first in the United States in October 1979. Jonathan Cape published the book in the United Kingdom in November 1979. Shikasta is also the name of the fictional planet featured in the novel.
The book has a subtitle, ""Personal, psychological, historical documents relating to visit by Johor (George Sherban) Emissary (Grade 9) 87th of the Period of the Last Days"". 'The novel is the history of the planet Shikasta under the influence of three galactic empires: Canopus, Sirius, and their shared enemy, Puttiora. The book is a series of reports by Canopean visitors to Shikasta. They record what happens on the planet which is very similar to Earth. The novel shows the Earth's prehistory, how things became worse, the "Century of Destruction" (the 20th-century), and the Apocalypse (World War III).
"Shikasta" uses stories and images from the Old Testament and is influenced by spiritual and mystical themes in Sufism. Lessing had become interested in this Islamic belief system in the mid-1960s. The book was a major shift of focus in Lessing's writing, from realism to science fiction. Many of her readers were unhappy with the change. Critics gave the book both good and bad reviews. Some were impressed by the way the book showed a wide view of humanity and the universe. One reviewer called it a brave "and disturbing work from one of the world's great living writers". Others reviewers disliked the novel because it was depressing and humanity had no free will. People's lives were controlled by galactic empires.
The story of Shikasta is retold in the third book of the "Canopus" series, "The Sirian Experiments" (1980). In that book, the story is told from the point of view of Sirius. Shikasta appears again in the fourth book in the series, "The Making of the Representative for Planet 8" (1982). The Zones that are briefly mentioned in "Shikasta" are the subject of the second book in the series, "The Marriages Between Zones Three, Four and Five" (1980).
HOMO/LUMO
HOMO stands for highest occupied molecular orbital. LUMO stands for lowest unoccupied molecular orbital. The energy difference between the HOMO and LUMO is termed the HOMO-LUMO gap. HOMO and LUMO are sometimes referred to as frontier orbitals.
Each molecular orbital has a calculated energy level. Chemists sort the molecular orbitals (MOs) by energy levels. Chemists assume that the electrons will occupy the lowest energy level MOs first. For example, if a molecule has enough electrons to fill 15 MOs, the 15 MOs with the lowest energy levels will be occupied. The 15th MO on the list would be called the "highest occupied molecular orbital" (HOMO) and the 16th MO on the list would be the "lowest unoccupied molecular orbital" (LUMO). The difference in the HOMO's energy level and the LUMO's energy level is called the band gap. The band gap can sometimes serve as a measure of the excitability of the molecule: the smaller the energy, the more easily a molecule's electrons will be excited. For example, this can help predict whether a substance will have luminescence.
A SOMO is a "singly occupied molecular orbital" such as half-filled HOMO of a radical.
Roald Hoffmann
Roald Hoffmann (born July 18, 1937) is a Polish-born American theoretical chemist of Jewish descent. He won the 1981 Nobel Prize in Chemistry. He currently teaches at Cornell University in Ithaca, New York.
He was born in Zloczov, Poland (now Ukraine)
Hoffmann survived The Holocaust and moved to the United States in 1949.
Substituent
In organic chemistry and biochemistry, substituent is the name for a single atom or for a group of atoms that take the position of another atom in a molecule. They replace this atom. This reaction is called a substitution reaction. The atom or group of atoms that gets replaced is called the leaving group.
The terms substituent, side chain, group, branch, or pendant group are used in the same way to describe branches from a parent structure. These terms have different meanings in polymer chemistry. In polymers, side chains extend from a backbone structure, while they are attached to the alpha carbon atoms of the amino acid backbone in proteins.
Organic chemistry has rules for naming compounds with substituent groups. The substituent group is listed first with a suffix to describe how it is attached to the main carbon chain. The suffix -yl is used when naming organic compounds that contain a single bond replacing one hydrogen. -ylidene and -ylidyne are used for double bonds and triple bonds respectively. When naming hydrocarbons that contain a substituent, position numbers may be used. These number will indicate which carbon atom the substituent is attached to. One case where this information is needed is that of isomers.
The phrases most-substituted and least-substituted are often used to describe molecules and predict their products. For example:
In a structural diagram or formula, the letters R or X are used to represent a substituent group. An organic substituent such as methyl, ethyl, aryl can be written as R (or R, R, etc.) This is a generic placeholder. The "R" derived from "radical" or "rest". The R can represent as many atoms as the author wants. The first to use this symbol was Charles Frédéric Gerhardt in 1844.
The symbol X can represent electronegative substituents such as the halides.
Greater rhea
The greater rhea, "Rhea americana", is a flightless bird found in South America. The greater rhea is also known as grey, common rhea, or the American rhea. The species, first described by Carl Linnaeus in 1758, is found in Argentina, Bolivia, Brazil, Paraguay, and Uruguay.
The greater rhea takes its scientific name from the Greek goddess Rhea and the Latin word for "America".
It has a length of and a wingspan of . Weighing , the greater rhea is the largest bird in South America.
Petr Kalandra
Petr Kalandra (March 10, 1950 – September 7, 1995) was a Czech musician, guitarist, harmonica player, vocalist and songwriter. He was member of A.S.P.M., Blues Session and Marsyas. He died in September 1995. In 2006 he was inducted into the Beatová síň slávy (Beat Hall of Fame).
Till Eulenspiegel's Merry Pranks