title: Operator (mathematics)

text: This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphisms between vector spaces.


title: Operator (mathematics)

text: In the finite-dimensional case linear operators can be represented by matrices in the following way. Let formula_4 be a field, and formula_5 and formula_6 be finite-dimensional vector spaces over formula_4. Let us select a basis formula_8 in formula_5 and formula_10 in formula_6. Then let formula_12 be an arbitrary vector in formula_5 (assuming Einstein convention), and formula_14 be a linear operator. Then


title: Operator (mathematics)

text: Then formula_16 is the matrix of the operator formula_17 in fixed bases. formula_18 does not depend on the choice of formula_19, and formula_20 if formula_21. Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from formula_5 to formula_6.


title: Operator (mathematics)

text: The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace.


title: Operator (mathematics)

text: Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as functional analysis (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).


title: Operator (mathematics)

text: The space of sequences of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations.


title: Operator (mathematics)

text: Bounded linear operators over Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.


title: Operator (mathematics)

text: Let "U" and "V" be two vector spaces over the same ordered field (for example, formula_24), and they are equipped with norms. Then a linear operator from "U" to "V" is called bounded if there exists "C" > 0 such that


title: Operator (mathematics)

text: Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of "U" and "V":


title: Operator (mathematics)

text: Any unital normed algebra with this property is called a Banach algebra. It is possible to generalize spectral theory to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics.


title: Operator (mathematics)

text: In geometry, additional structures on vector spaces are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form groups by composition.


title: Operator (mathematics)

text: For example, bijective operators preserving the structure of a vector space are precisely the invertible linear operators. They form the general linear group under composition. They "do not" form a vector space under the addition of operators, e.g. both id and −id are invertible (bijective), but their sum, 0, is not.


title: Operator (mathematics)

text: Operators preserving the Euclidean metric on such a space form the isometry group, and those that fix the origin form a subgroup known as the orthogonal group. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations.


title: Operator (mathematics)

text: Operators are also involved in probability theory, such as expectation, variance, and covariance. Indeed, every covariance is basically a dot product; every variance is a dot product of a vector with itself, and thus is a quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the Pearson correlation coefficient; expected value is basically an integral operator (used to measure weighted shapes in the space).


title: Operator (mathematics)

text: From the point of view of functional analysis, calculus is the study of two linear operators: the differential operator formula_28, and the Volterra operator formula_29.


title: Operator (mathematics)

text: The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively invertible. No information is lost, as there is an inverse transform operator. In the simple case of periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves and cosine waves:


title: Operator (mathematics)

text: The tuple ("a", "a", "b", "a", "b", …) is in fact an element of an infinite-dimensional vector space "ℓ", and thus Fourier series is a linear operator.


title: Operator (mathematics)

text: The "Laplace transform" is another integral operator and is involved in simplifying the process of solving differential equations.


title: Operator (mathematics)

text: As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with tensor calculus as well as vector calculus.


title: Glen or Glenda

text: Glen or Glenda is a 1953 American exploitation film directed, written by and starring Ed Wood (credited in his starring role as "Daniel Davis"), and featuring Wood's then-girlfriend Dolores Fuller and Bela Lugosi. It was produced by George Weiss who also made the exploitation film "Test Tube Babies" that same year.


title: Glen or Glenda

text: The film is a docudrama about cross-dressing and transvestism, and is semi-autobiographical in nature. Wood himself was a cross-dresser, and the film is a plea for tolerance. It was widely considered one of the worst films ever made upon release. However, it has since been reevaluated and has become a cult film due to its low-budget production values, idiosyncratic style, and early cinematic themes of transgender acceptance.


title: Glen or Glenda

text: A narrator, called The Scientist (Bela Lugosi), makes cryptic comments about humanity. He first comments that humanity's constant search for the unknown results in startling things coming to light. The cries of a newborn baby are followed by the sirens of an ambulance. One is a sign that a new life has begun, the other that a life has ended.


title: Glen or Glenda

text: This last comment starts the narrative of the film. The life which has ended is that of a transvestite named Patrick/Patricia, who has committed suicide. A suicide note explains the reasons behind the suicide. Patrick/Patricia had been arrested four times for cross-dressing in public, and had spent time in prison, so he ended his own life and wished to be buried with his women's clothing.


title: Glen or Glenda

text: Inspector Warren is puzzled and wants to know more about cross-dressing, so he seeks the office of Dr. Alton, who narrates for him the story of Glen/Glenda. Glen is shown studying women's clothes in a shop window. A flashback scene reveals that a young Glen started out by asking to wear his sister's dress for a Halloween party.


title: Glen or Glenda

text: The narrative explains that Glen is a transvestite, but not a homosexual. He hides his cross-dressing from his fiancée, Barbara, fearing that she will reject him. She voices her suspicion that there is another woman in his life, unaware that the woman is his feminine alter ego, Glenda. The scene shifts from a speechless Glen to footage of a stampeding herd of bison, while the Scientist calls for Glen to ""Pull the string. Dance to that for which one is created for!"", referring to the narrator pulling the strings of a hapless puppet who is not in control of his own destiny.


title: Glen or Glenda

text: Alton narrates that Glen is torn between the idea of being honest with Barbara before their wedding or waiting until after. The narrative shifts briefly from Glen's story to how society reacts to sex change operations. A conversation between two "average joes", concludes that society should be more "lenient" when it comes to people with tranvestite tendencies. The story returns to Glen, who confides in a transvestite friend of his, John, whose wife left him after catching him wearing her clothes.


title: Glen or Glenda

text: Later, Glen/Glenda is walking the city streets at night. He returns home when the sound of thunder causes him to collapse to the floor; an extended dream sequence begins. Barbara is depicted trapped under a tree. Glenda fails to lift the tree, then is replaced by Glen, who completes the task with ease. The dream then depicts Glen and Barbara getting married, where the best man is a devil. The sequence continues with a series of erotic vignettes containing BDSM, striptease, lesbian, autoerotic, and rape fantasy themes. Throughout these vignettes, the faces of Glen and the Scientist appear, silently reacting to the various images.


title: Glen or Glenda

text: The dream returns to Glen in the midst of a jury of public opinion. A blackboard appears, with messages recording what the Scientist or the mocking voices said in previous scenes. The Devil and the various specters on the jury menacingly approach Glen. Then the Devil departs, Glen turns into Glenda, and the specters retreat. A victorious Glenda sees Barbara and approaches her, but she turns into a mocking Devil. Barbara starts appearing and disappearing, always evading Glenda's embrace. The dream sequence ends.


title: Glen or Glenda

text: Glen/Glenda wakes and stares at his mirror reflection. He decides to tell Barbara the truth. She initially reacts with distress, but ultimately decides to stay with him. She offers him an angora sweater as a sign of acceptance. The scene effectively concludes their story.


title: Glen or Glenda

text: Back in Dr. Alton's office, he starts another narrative. This one concerns another tranvestite, called Alan/Anne. Anne was born Alan, a boy, but her mother wanted a girl and raised her as such. Her father did not care either way. She was an outsider as a child, trying to be one of the girls and consequently rejected by schoolmates of both sexes. As a teenager, she self-identified as a woman. She was conscripted in World War II, maintaining a secret life throughout her military service. She first heard of sex change operations during the War while recovering from combat wounds in a hospital. She eventually did have a sex change operation, enduring the associated pains to fulfill her dreams. The World War II veteran becomes a "lovely young lady". Following a brief epilogue, the film ends.


title: Glen or Glenda

text: Shot in four days, the film was loosely inspired by the sex reassignment surgery of Christine Jorgensen, which made national headlines in the U.S. in 1952. George Weiss, a Hollywood producer of low-budget films, commissioned a movie to exploit the case. Originally Weiss made Jorgensen several offers to appear in the film, but these were turned down. Wood convinced Weiss that his own transvestism made him the perfect director despite his modest resume. Wood was given the job, but instead made a movie about transvestism. Nonetheless, posters for the film still claimed it was based on Jorgensen's case.


title: Glen or Glenda

text: Wood persuaded Lugosi, at the time poor and drug-addicted, to appear in the movie. Lugosi's scenes were shot at the Jack Miles Studios in Los Angeles. He was reportedly paid $5000 for the role, although some stories state the actual amount was only $1000. Lugosi is credited as "The Scientist", a character whose purpose is unclear. He acts as a sort of narrator but gives no narration relevant to the plot; that job is reserved for the film's primary narrator, Timothy Farrell.


title: Glen or Glenda

text: This was the only film Wood directed but did not also produce. Wood played the eponymous character, but under the pseudonym "Daniel Davis". His then-girlfriend, Dolores Fuller, played Glen's girlfriend/fiancée Barbara. Wood later returned to "Glen or Glenda" in his pulp novel "Killer in Drag" (1963). The plot features a transvestite called Glen whose alter-ego is called Glenda. He is executed in the sequel "Death of a Transvestite" (1967) after a struggle for the right to go to the electric chair dressed as Glenda.


title: Glen or Glenda

text: The erotic-themed vignettes were not created by Wood. They were reportedly added by producer George Weiss. He needed extra scenes to add to what he felt was an overly-short film. While not organic parts of the narrative, they seem to tell their own tales of gender dynamics and so fit in with the general themes of the film. The whipping scene suggests a master/slave relationship. That the man is dominant and the woman submissive, seems to reflect male chauvinism. The flirtatious and striptease-themed vignettes were typical of 1950s exploitation films and grindhouse films, as was the rape scene.


title: Glen or Glenda

text: The film has deleted scenes. In the theatrical trailer, included in laserdisc and DVD editions, the scene in which Fuller hands over her angora sweater, is a different take than the one in the release version — in the trailer, she tosses it to Wood in a huff, while the release version shows her handing it over more acceptingly. There is also a shot of Wood in drag, mouthing the word "Cut!"


title: Glen or Glenda

text: The second part of the film, titled "Alan or Anne", is much shorter, told largely through stock footage, and was made to meet the distributor's demand for a sex change film. Alan is a pseudo-hermaphrodite who fights in World War II wearing women's underwear. After returning, Alan undergoes surgery to become a woman.


title: Glen or Glenda

text: Domestically, the film was limited in release, having been pre-sold to some theaters (under alternative titles such as "I Led Two Lives", "He or She?" and "I Changed My Sex"). Internationally, the film was also limited, and in France and Belgium, the title was translated as "Louis ou Louise" and in Argentina as "Yo Cambié Mi Sexo" (I changed my sex); the film had a brief screening in the Republic of China. It was re-released to theaters in 1981 by Paramount.


title: Glen or Glenda

text: According to Tim Dirks, the film was one of a wave of "cheap teen movies" released for the drive-in market. They consisted of "exploitative, cheap fare created especially for them [teens] in a newly-established teen/drive-in genre."


title: Glen or Glenda

text: It was denied classification by the British Board of Film Classification upon submission on February 26, 1958.


title: Glen or Glenda

text: In 2009, "Glen or Glenda" became the final film to be restored and colorized by Legend Films, who subsequently released it on DVD.


title: Glen or Glenda

text: In 1980, Wood was posthumously given the accolade of 'Worst Director of All Time' at the Golden Turkey Awards, and a revival of interest in his work followed. This led to "Glen or Glenda" being reissued in 1982. This cut included six minutes of additional footage. One of the restored scenes features Glen rejecting a pass made to him by a man. At this point, the film was reviewed seriously, and reclaimed as a radical work, by Steve Jenkins in the "Monthly Film Bulletin".


title: Glen or Glenda

text: The critic Leonard Maltin names "Glen or Glenda" as "possibly the worst movie ever made". Richard Barrios describes "Glen or Glenda" as "one of the funniest and worst movies ever made".


title: Glen or Glenda

text: In his book "Cult Movies 3", Danny Peary suggests this is actually a radical, if ineptly made, film that presents a far more personal story than is contained in films by more well-respected auteurs.


title: Glen or Glenda

text: In 1994, Tim Burton chronicled the troubled production of "Glen or Glenda" in "Ed Wood". The film includes re-creations of several key scenes, including Lugosi's narration and Glen's plea for his girlfriend's understanding at the end of the film.


title: Work (thermodynamics)

text: In thermodynamics, work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. An exchange of energy is facilitated by a mechanism through which the system can spontaneously exert macroscopic forces on its surroundings, or vice versa. In the surroundings, this mechanical work can lift a weight, for example.


title: Work (thermodynamics)

text: The externally measured forces and external effects may be electromagnetic, gravitational, or mechanical (such as pressure-volume) variables. For thermodynamic work, these externally measured quantities are exactly matched by values of or contributions to changes in macroscopic internal state variables of the system, which always occur in conjugate pairs, for example pressure and volume or magnetic flux density and magnetization.


title: Work (thermodynamics)

text: By an external system that lies in the surroundings, not necessarily a thermodynamic system as strictly defined by the usual thermodynamic state variables, otherwise than by transfer of matter, work can be said to be done on a thermodynamic system. Part of such surroundings-defined work can have a mechanism just as for system-defined thermodynamic work done by the system, while the rest of such surroundings-defined work appears, to the thermodynamic system, not as a negative amount of thermodynamic work done by it, but, rather, as heat transferred to it. The paddle stirring experiments of Joule provide an example, illustrating the concept of isochoric (or constant volume) mechanical work, in this case sometimes called "shaft work". Such work is not thermodynamic work as defined here, because it acts through friction, within, and on the surface of, the thermodynamic system, and does not act through macroscopic forces that the system can spontaneously exert on its surroundings, describable by its state variables. Surroundings-defined work can also be non-mechanical. An example is Joule heating, because it occurs through friction as the electric current passes through the thermodynamic system. When it is done isochorically, and no matter is transferred, such an energy transfer is regarded as a heat transfer into the system of interest.


title: Work (thermodynamics)

text: In the International System of Units (SI), work is measured in joules (symbol: J). The rate at which work is performed is power, measured as joules per second and denoted with the unit watt (W).


title: Work (thermodynamics)

text: Work, i.e. "weight "lifted" through a height", was originally defined in 1824 by Sadi Carnot in his famous paper "Reflections on the Motive Power of Fire", where he used the term "motive power" for work. Specifically, according to Carnot:


title: Work (thermodynamics)

text: We use here motive power to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.


title: Work (thermodynamics)

text: In 1845, the English physicist James Joule wrote a paper "On the mechanical equivalent of heat" for the British Association meeting in Cambridge. In this paper, he reported his best-known experiment, in which the "mechanical power" released through the action of a "weight "falling" through a height" was used to turn a paddle-wheel in an insulated barrel of water.


title: Work (thermodynamics)

text: In this experiment, the motion of the paddle wheel, through agitation and friction, heated the body of water, so as to increase its temperature. Both the temperature change ∆T of the water and the height of the fall ∆h of the weight mg were recorded. Using these values, Joule was able to determine the mechanical equivalent of heat. Joule estimated a mechanical equivalent of heat to be 819 ft•lbf/Btu (4.41 J/cal). The modern day definitions of heat, work, temperature, and energy all have connection to this experiment. In this arrangement of apparatus, it never happens that the process runs in reverse, with the water driving the paddles so as to raise the weight, not even slightly. Mechanical work was done by the apparatus of falling weight, pulley, and paddles, which lay in the surroundings of the water. Their motion scarcely affected the volume of the water. Work that does not change the volume of the water is said to be isochoric; it is irreversible. The energy supplied by the fall of the weight passed into the water as heat.


title: Work (thermodynamics)

text: A pre-supposed guiding principle of thermodynamics is the conservation of energy. The total energy of a system is the sum of its internal energy, of its potential energy as a whole system in an external force field, such as gravity, and of its kinetic energy as a whole system in motion. Thermodynamics has special concern with transfers of energy, from a body of matter, such as, for example a cylinder of steam, to the surroundings of the body, by mechanisms through which the body exerts macroscopic forces on its surroundings so as to lift a weight there; such mechanisms are the ones that are said to mediate "thermodynamic" work.


title: Work (thermodynamics)

text: Besides transfer of energy as work, thermodynamics admits transfer of energy as heat. For a process in a closed (no transfer of matter) thermodynamic system, the first law of thermodynamics relates changes in the internal energy (or other cardinal energy function, depending on the conditions of the transfer) of the system to those two modes of energy transfer, as work, and as heat. Adiabatic work is done without matter transfer and without heat transfer. In principle, in thermodynamics, for a process in a closed system, quantity of heat transferred is defined by the amount of adiabatic work that would be needed to effect the change in the system that is occasioned by the heat transfer. In experimental practice, heat transfer is often estimated calorimetrically, through change of temperature of a known quantity of calorimetric material substance.


title: Work (thermodynamics)

text: Energy can also be transferred to or from a system through transfer of matter. The possibility of such transfer defines the system as an open system, as opposed to a closed system. By definition, such transfer is neither as work nor as heat.


title: Work (thermodynamics)

text: Changes in the potential energy of a body as a whole with respect to forces in its surroundings, and in the kinetic energy of the body moving as a whole with respect to its surroundings, are by definition excluded from the body's cardinal energy (examples are internal energy and enthalpy).


title: Work (thermodynamics)

text: In the surroundings of a thermodynamic system, external to it, all the various mechanical and non-mechanical macroscopic forms of work can be converted into each other with no limitation in principle due to the laws of thermodynamics, so that the energy conversion efficiency can approach 100% in some cases; such conversion is required to be frictionless, and consequently adiabatic. In particular, in principle, all macroscopic forms of work can be converted into the mechanical work of lifting a weight, which was the original form of thermodynamic work considered by Carnot and Joule (see History section above). Some authors have considered this equivalence to the lifting of a weight as a defining characteristic of work. For example, with the apparatus of Joule's experiment in which, through pulleys, a weight descending in the surroundings drives the stirring of a thermodynamic system, the descent of the weight can be diverted by a re-arrangement of pulleys, so that it lifts another weight in the surroundings, instead of stirring the thermodynamic system.


title: Work (thermodynamics)

text: Such conversion may be idealized as nearly frictionless, though it occurs relatively quickly. It usually comes about through devices that are not simple thermodynamic systems (a simple thermodynamic system is a homogeneous body of material substances). For example, the descent of the weight in Joule's stirring experiment reduces the weight's total energy. It is described as loss of gravitational potential energy by the weight, due to change of its macroscopic position in the gravity field, in contrast to, for example, loss of the weight's internal energy due to changes in its entropy, volume, and chemical composition. Though it occurs relatively rapidly, because the energy remains nearly fully available as work in one way or another, such diversion of work in the surroundings may be idealized as nearly reversible, or nearly perfectly efficient.


title: Work (thermodynamics)

text: In contrast, the conversion of heat into work in a heat engine can never exceed the Carnot efficiency, as a consequence of the second law of thermodynamics. Such energy conversion, through work done relatively rapidly, in a practical heat engine, by a thermodynamic system on its surroundings, cannot be idealized, not even nearly, as reversible.


title: Work (thermodynamics)

text: Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. Historically, thermodynamics was about how a thermodynamic system could do work on its surroundings.


title: Work (thermodynamics)

text: Work done on, and work done by, a thermodynamic system need to be distinguished, through consideration of their precise mechanisms. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Such work done by compression is thermodynamic work as here defined. But shaft work, stirring, and rubbing are not thermodynamic work as here defined, in that they do not change the volume of the system against its resisting pressure. Work without change of volume is known as isochoric work, for example when an agency, in the surroundings of the system, drives a frictional action on the surface or in the interior of the system.


title: Work (thermodynamics)

text: In a process of transfer of energy from or to a thermodynamic system, the change of internal energy of the system is defined in theory by the amount of adiabatic work that would have been necessary to reach the final from the initial state, such adiabatic work being measurable only through the externally measurable mechanical or deformation variables of the system, that provide full information about the forces exerted by the surroundings on the system during the process. In the case of some of Joule's measurements, the process was so arranged that some heating that occurred outside the system (in the substance of the paddles) by the frictional process also led to heat transfer from the paddles into the system during the process, so that the quantity of work done by the surrounds on the system could be calculated as shaft work, an external mechanical variable.


title: Work (thermodynamics)

text: The amount of energy transferred as work is measured through quantities defined externally to the system of interest, and thus belonging to its surroundings. In an important sign convention, preferred in chemistry, work that adds to the internal energy of the system is counted as positive. On the other hand, for historical reasons, an oft-encountered sign convention, preferred in physics, is to consider work done by the system on its surroundings as positive.


title: Work (thermodynamics)

text: One kind of heat transfer, through direct contact between a closed system and its surroundings, is by the "microscopic" thermal motions of particles and their associated inter-molecular potential energies. Microscopic accounts of such processes are the province of statistical mechanics, not of macroscopic thermodynamics. Another kind of heat transfer is by radiation. Radiative transfer of energy is irreversible in the sense that it occurs only from a hotter to a colder system, never the other way. There are several forms of dissipative transduction of energy that can occur internally within a system at a microscopic level, such as friction including bulk and shear viscosity chemical reaction, unconstrained expansion as in Joule expansion and in diffusion, and phase change.


title: Work (thermodynamics)

text: Thermodynamic work does not account for any energy transferred between systems as heat or through transfer of matter.


title: Work (thermodynamics)

text: For an open system, the first law of thermodynamics admits three forms of energy transfer, as work, as heat, and as energy associated with matter that is transferred. The latter cannot be split uniquely into heat and work components.


title: Work (thermodynamics)

text: One-way convection of internal energy is a form a transport of energy but is not, as sometimes mistakenly supposed (a relic of the caloric theory of heat), transfer of energy as heat, because one-way convection is transfer of matter; nor is it transfer of energy as work. Nevertheless, if the wall between the system and its surroundings is thick and contains fluid, in the presence of a gravitational field, convective circulation within the wall can be considered as indirectly mediating transfer of energy as heat between the system and its surroundings, though the source and destination of the transferred energy are not in direct contact.


title: Work (thermodynamics)

text: For purposes of theoretical calculations about a thermodynamic system, one can imagine fictive idealized thermodynamic "processes" that occur so slowly that they do not incur friction within or on the surface of system; they can then be regarded as virtually reversible. These fictive processes proceed along paths on geometrical surfaces that are described exactly by a characteristic equation of the thermodynamic system. Those geometrical surfaces are the loci of possible states of thermodynamic equilibrium for the system. Really possible thermodynamic processes, occurring at practical rates, even when they occur only by work assessed in the surroundings as adiabatic, without heat transfer, always incur friction within the system, and so are always irreversible. The paths of such really possible processes always depart from those geometrical characteristic surfaces. Even when they occur only by work assessed in the surroundings as adiabatic, without heat transfer, such departures always entail entropy production.


title: Work (thermodynamics)

text: The definition of thermodynamic work is in terms of the changes of the system's extensive deformation (and chemical constitutive and certain other) state variables, such as volume, molar chemical constitution, or electric polarisation. Examples of state variables that are not extensive deformation or other such variables are temperature and entropy , as for example in the expression . Changes of such variables are not actually physically measureable by use of a single simple adiabatic thermodynamic process; they are processes that occur neither by thermodynamic work nor by transfer of matter, and therefore are said occur by heat transfer. The quantity of thermodynamic work is defined as work done by the system on its surroundings. According to the second law of thermodynamics, such work is irreversible. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle. A different cycle would be needed to actually measure the work done by the surroundings on the system. This is a reminder that rubbing the surface of a system appears to the rubbing agent in the surroundings as mechanical, though not thermodynamic, work done on the system, not as heat, but appears to the system as heat transferred to the system, not as thermodynamic work. The production of heat by rubbing is irreversible; historically, it was a piece of evidence for the rejection of the caloric theory of heat as a conserved substance. The irreversible process known as Joule heating also occurs through a change of a non-deformation extensive state variable.


title: Work (thermodynamics)

text: Accordingly, in the opinion of Lavenda, work is not as primitive concept as is heat, which can be measured by calorimetry. This opinion does not negate the now customary thermodynamic definition of heat in terms of adiabatic work.


title: Work (thermodynamics)

text: Known as a thermodynamic operation, the initiating factor of a thermodynamic process is, in many cases, a change in the permeability of a wall between the system and the surroundings. Rubbing is not a change in wall permeability. Kelvin's statement of the second law of thermodynamics uses the notion of an "inanimate material agency"; this notion is sometimes regarded as puzzling. The triggering of a process of rubbing can occur only in the surroundings, not in a thermodynamic system in its own state of internal thermodynamic equilibrium. Such triggering may be described as a thermodynamic operation.


title: Work (thermodynamics)

text: In thermodynamics, the quantity of work done by a closed system on its surroundings is defined by factors strictly confined to the interface of the surroundings with the system and to the surroundings of the system, for example, an extended gravitational field in which the system sits, that is to say, to things external to the system.


title: Work (thermodynamics)

text: A main concern of thermodynamics is the properties of materials. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. Consequently, thermodynamic work is defined in terms of quantities that describe the states of materials, which appear as the usual thermodynamic state variables, such as volume, pressure, temperature, chemical composition, and electric polarization. For example, to measure the pressure inside a system from outside it, the observer needs the system to have a wall that can move by a measurable amount in response to pressure differences between the interior of the system and the surroundings. In this sense, part of the definition of a thermodynamic system is the nature of the walls that confine it.


title: Work (thermodynamics)

text: Several kinds of thermodynamic work are especially important. One simple example is pressure–volume work. The pressure of concern is that exerted by the surroundings on the surface of the system, and the volume of interest is the negative of the increment of volume gained by the system from the surroundings. It is usually arranged that the pressure exerted by the surroundings on the surface of the system is well defined and equal to the pressure exerted by the system on the surroundings. This arrangement for transfer of energy as work can be varied in a particular way that depends on the strictly mechanical nature of pressure–volume work. The variation consists in letting the coupling between the system and surroundings be through a rigid rod that links pistons of different areas for the system and surroundings. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. This cannot be done for the transfer of energy as heat because of its non-mechanical nature.


title: Work (thermodynamics)

text: Another important kind of work is isochoric work, i.e., work that involves no eventual overall change of volume of the system between the initial and the final states of the process. Examples are friction on the surface of the system as in Rumford's experiment; shaft work such as in Joule's experiments; stirring of the system by a magnetic paddle inside it, driven by a moving magnetic field from the surroundings; and vibrational action on the system that leaves its eventual volume unchanged, but involves friction within the system. Isochoric mechanical work for a body in its own state of internal thermodynamic equilibrium is done only by the surroundings on the body, not by the body on the surroundings, so that the sign of isochoric mechanical work with the physics sign convention is always negative.


title: Work (thermodynamics)

text: When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. When mechanical work is done on such an adiabatically enclosed system by the surroundings, it can happen that friction in the surroundings is negligible, for example in the Joule experiment with the falling weight driving paddles that stir the system. Such work is adiabatic for the surroundings, even though it is associated with friction within the system. Such work may or may not be isochoric for the system, depending on the system and its confining walls. If it happens to be isochoric for the system (and does not eventually change other system state variables such as magnetization), it appears as a heat transfer to the system, and does not appear to be adiabatic for the system.


title: Work (thermodynamics)

text: In the early history of thermodynamics, a positive amount of work done "by" the system on the surroundings leads to energy being lost from the system. This historical sign convention has been used in many physics textbooks and is used in the present article.


title: Work (thermodynamics)

text: According to the first law of thermodynamics for a closed system, any net change in the internal energy "U" must be fully accounted for, in terms of heat "Q" entering the system and work "W" done by the system:


title: Work (thermodynamics)

text: An alternate sign convention is to consider the work performed "on" the system by its surroundings as positive. This leads to a change in sign of the work, so that formula_2. This convention has historically been used in chemistry, and has been adopted by most physics textbooks.


title: Work (thermodynamics)

text: This equation reflects the fact that the heat transferred and the work done are "not" properties of the state of the system. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. This can be summarized by saying that heat and work are not state functions of the system. This is in contrast to classical mechanics, where net work exerted by a particle is a state function.


title: Work (thermodynamics)

text: Pressure–volume work (or "PV" work) occurs when the volume of a system changes. "PV" work is often measured in units of litre-atmospheres where . However, the litre-atmosphere is not a recognized unit in the SI system of units, which measures "P" in Pascal (Pa), "V" in m, and "PV" in Joule (J), where 1 J = 1 Pa·m. "PV" work is an important topic in chemical thermodynamics.


title: Work (thermodynamics)

text: For a process in a closed system, occurring slowly enough for accurate definition of the pressure on the inside of the system's wall that moves and transmits force to the surroundings, described as "quasi-static", work is represented by the following equation between differentials:


title: Work (thermodynamics)

text: P–V work is path-dependent and is, therefore, a thermodynamic process function. In general, the term formula_12 is not an exact differential. The statement that a process is quasi-static gives important information about the process but does not determine the P–V path uniquely, because the path can include several slow goings backwards and forward in volume, slowly enough to exclude friction within the system occasioned by departure from the quasi-static requirement. An adiabatic wall is one that does not permit passage of energy by conduction or radiation.


title: Work (thermodynamics)

text: For a quasi-static adiabatic process, the change in internal energy is equal to minus the integral amount of work done by the system, and depends only on the initial and final states of the process and is one and the same for every intermediate path.


title: Work (thermodynamics)

text: If the process path is other than quasi-static and adiabatic, there are indefinitely many different paths, with significantly different work amounts, between the initial and final states.


title: Work (thermodynamics)

text: In another notation, is written (with a line through the d). This notation indicates that is not an exact one-form. The line-through is merely a flag to warn us there is actually no function (0-form) which is the potential of . If there were, indeed, this function , we should be able to just use Stokes Theorem to evaluate this putative function, the potential of , at the boundary of the path, that is, the initial and final points, and therefore the work would be a state function. This impossibility is consistent with the fact that it does not make sense to refer to "the work on a point" in the PV diagram; work presupposes a path.


title: Work (thermodynamics)

text: There are several ways of doing mechanical work, each in some way related to a force acting through a distance. In basic mechanics, the work done by a constant force "F" on a body displaced a distance s in the direction of the force is given by


title: Work (thermodynamics)

text: If the force is not constant, the work done is obtained by integrating the differential amount of work,


title: Work (thermodynamics)

text: Energy transmission with a rotating shaft is very common in engineering practice. Often the torque "T" applied to the shaft is constant which means that the force "F" applied is constant. For a specified constant torque, the work done during "n" revolutions is determined as follows: A force "F" acting through a moment arm "r" generates a torque "T"


title: Work (thermodynamics)

text: When a force is applied on a spring, and the length of the spring changes by a differential amount "dx", the work done is


title: Work (thermodynamics)

text: where "K" is the spring constant and has the unit of N/m. The displacement "x" is measured from the undisturbed position of the spring (that is, when ). Substituting the two equations


title: Work (thermodynamics)

text: where "x" and "x" are the initial and the final displacement of the spring respectively, measured from the undisturbed position of the spring.


title: Work (thermodynamics)

text: Solids are often modeled as linear springs because under the action of a force they contract or elongate, and when the force is lifted, they return to their original lengths, like a spring. This is true as long as the force is in the elastic range, that is, not large enough to cause permanent or plastic deformation. Therefore, the equations given for a linear spring can also be used for elastic solid bars. Alternately, we can determine the work associated with the expansion or contraction of an elastic solid bar by replacing the pressure "P" by its counterpart in solids, normal stress in the work expansion


title: Work (thermodynamics)

text: Consider a liquid film such as a soap film suspended on a wire frame. Some force is required to stretch this film by the movable portion of the wire frame. This force is used to overcome the microscopic forces between molecules at the liquid-air interface. These microscopic forces are perpendicular to any line in the surface and the force generated by these forces per unit length is called the surface tension "σ" whose unit is N/m. Therefore, the work associated with the stretching of a film is called surface tension work, and is determined from


title: Work (thermodynamics)

text: where is the change in the surface area of the film. The factor 2 is due to the fact that the film has two surfaces in contact with air. The force acting on the moveable wire as a result of surface tension effects is , where "σ" is the surface tension force per unit length.


title: Work (thermodynamics)

text: The amount of useful work which may be extracted from a thermodynamic system is determined by the second law of thermodynamics. Under many practical situations this can be represented by the thermodynamic availability, or Exergy, function. Two important cases are: in thermodynamic systems where the temperature and volume are held constant, the measure of useful work attainable is the Helmholtz free energy function; and in systems where the temperature and pressure are held constant, the measure of useful work attainable is the Gibbs free energy.


title: Work (thermodynamics)

text: Non-mechanical work in thermodynamics is work caused by external force fields that a system is exposed to. The action of such forces can be initiated by events in the surroundings of the system, or by thermodynamic operations on the shielding walls of the system.


title: Work (thermodynamics)

text: The non-mechanical work of force fields can have either positive or negative sign, work being done by the system on the surroundings, or "vice versa". Work done by force fields can be done indefinitely slowly, so as to approach the fictive reversible quasi-static ideal, in which entropy is not created in the system by the process.


title: Work (thermodynamics)

text: In thermodynamics, non-mechanical work is to be contrasted with mechanical work that is done by forces in immediate contact between the system and its surroundings. If the putative 'work' of a process cannot be defined as either long-range work or else as contact work, then sometimes it cannot be described by the thermodynamic formalism as work at all. Nevertheless, the thermodynamic formalism allows that energy can be transferred between an open system and its surroundings by processes for which work is not defined. An example is when the wall between the system and its surrounds is not considered as idealized and vanishingly thin, so that processes can occur within the wall, such as friction affecting the transfer of matter across the wall; in this case, the forces of transfer are neither strictly long-range nor strictly due to contact between the system and its surrounds; the transfer of energy can then be considered as by convection, and assessed in sum just as transfer of internal energy. This is conceptually different from transfer of energy as heat through a thick fluid-filled wall in the presence of a gravitational field, between a closed system and its surroundings; in this case there may convective circulation within the wall but the process may still be considered as transfer of energy as heat between the system and its surroundings; if the whole wall is moved by the application of force from the surroundings, without change of volume of the wall, so as to change the volume of the system, then it is also at the same time transferring energy as work. A chemical reaction within a system can lead to electrical long-range forces and to electric current flow, which transfer energy as work between system and surroundings, though the system's chemical reactions themselves (except for the special limiting case in which in they are driven through devices in the surroundings so as to occur along a line of thermodynamic equilibrium) are always irreversible and do not directly interact with the surroundings of the system.


title: Work (thermodynamics)

text: Non-mechanical work contrasts with pressure–volume work. Pressure–volume work is one of the two mainly considered kinds of mechanical contact work. A force acts on the interfacing wall between system and surroundings. The force is that due to the pressure exerted on the interfacing wall by the material inside the system; that pressure is an internal state variable of the system, but is properly measured by external devices at the wall. The work is due to change of system volume by expansion or contraction of the system. If the system expands, in the present article it is said to do positive work on the surroundings. If the system contracts, in the present article it is said to do negative work on the surroundings. Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. It is accurately described by changes in state variables of the system, such as the time courses of changes in the pressure and volume of the system. The volume of the system is classified as a "deformation variable", and is properly measured externally to the system, in the surroundings. Pressure–volume work can have either positive or negative sign. Pressure–volume work, performed slowly enough, can be made to approach the fictive reversible quasi-static ideal.


title: Work (thermodynamics)

text: Non-mechanical work also contrasts with shaft work. Shaft work is the other of the two mainly considered kinds of mechanical contact work. It transfers energy by rotation, but it does not eventually change the shape or volume of the system. Because it does not change the volume of the system it is not measured as pressure–volume work, and it is called isochoric work. Considered solely in terms of the eventual difference between initial and final shapes and volumes of the system, shaft work does not make a change. During the process of shaft work, for example the rotation of a paddle, the shape of the system changes cyclically, but this does not make an eventual change in the shape or volume of the system. Shaft work is a kind of contact work, because it occurs through direct material contact with the surrounding matter at the boundary of the system. A system that is initially in a state of thermodynamic equilibrium cannot initiate any change in its internal energy. In particular, it cannot initiate shaft work. This explains the curious use of the phrase "inanimate material agency" by Kelvin in one of his statements of the second law of thermodynamics. Thermodynamic operations or changes in the surroundings are considered to be able to create elaborate changes such as indefinitely prolonged, varied, or ceased rotation of a driving shaft, while a system that starts in a state of thermodynamic equilibrium is inanimate and cannot spontaneously do that. Thus the sign of shaft work is always negative, work being done on the system by the surroundings. Shaft work can hardly be done indefinitely slowly; consequently it always produces entropy within the system, because it relies on friction or viscosity within the system for its transfer. The foregoing comments about shaft work apply only when one ignores that the system can store angular momentum and its related energy.


title: Work (thermodynamics)

text: Gravitational work is defined by the force on a body measured in a gravitational field. It may cause a generalized displacement in the form of change of the spatial distribution of the matter within the system. The system gains internal energy (or other relevant cardinal quantity of energy, such as enthalpy) through internal friction. As seen by the surroundings, such frictional work appears as mechanical work done on the system, but as seen by the system, it appears as transfer of energy as heat. When the system is in its own state of internal thermodynamic equilibrium, its temperature is uniform throughout. If the volume and other extensive state variables, apart from entropy, are held constant over the process, then the transferred heat must appear as increased temperature and entropy; in a uniform gravitational field, the pressure of the system will be greater at the bottom than at the top.


title: Work (thermodynamics)

text: By definition, the relevant cardinal energy function is distinct from the gravitational potential energy of the system as a whole; the latter may also change as a result of gravitational work done by the surroundings on the system. The gravitational potential energy of the system is a component of its total energy, alongside its other components, namely its cardinal thermodynamic (e.g. internal) energy and its kinetic energy as a whole system in motion.


title: Wyc Grousbeck

text: Wycliffe "Wyc" K. Grousbeck (born June 13, 1961) is an American entrepreneur who is the majority owner and Governor of the National Basketball Association's Boston Celtics.


title: Wyc Grousbeck

text: Grousbeck was born in Worcester, Massachusetts and graduated from Noble and Greenough School in Dedham, Massachusetts. He attended Princeton University, B.A. in history in 1983, and rowed on the 1983 undefeated lightweight crew team that claimed the Ivy League and National rowing championship. He received a J.D. degree from the University of Michigan in 1986 and an M.B.A. in 1992 (Miller Scholar) from the Stanford Graduate School of Business.


title: Wyc Grousbeck

text: After spending seven years as a partner at the venture capital firm, Highland Capital Partners, Grousbeck founded and led the group Boston Basketball Partners L.L.C. that bought the Boston Celtics for $360 million in 2002. In 2008, the Celtics won their seventeenth NBA Championship.


title: Wyc Grousbeck

text: While continuing his duties with the Celtics, in 2010 Grousbeck became Chairman of the Massachusetts Eye and Ear Infirmary, the world's largest research and clinical hospital specializing in blindness and deafness research. He instituted and led a capital campaign that raised $250 million for Mass Eye and Ear, and led them to join the world class Mass General Brigham medical system, as a first-tier hospital. In 2013, Grousbeck co-founded Causeway Media Partners, LP, a growth equity partnership managing over $330 million, investing in sports technology and media companies such as Omaze, Inc., Zwift, FloSports, Freeletics, Formula E Racing, and SeatGeek.


title: Wyc Grousbeck

text: Wyc married Emilia Fazzalari in January 2017. Wyc and Emilia joined with fellow NBA owners Michael Jordan, Jeanie Buss and Wes Edens to found Cincoro Tequila, which launched to acclaim in September, 2019. Wyc sits on the Executive, Labor, and Media Committees of the NBA, is Board Chair of the Boston Celtics, and Mass Eye and Ear, and is a board member of the Boston Celtics, the Shamrock Foundation, Giving Grousbeck Fazzalari, Omaze, Inc., Cinco Spirits Group, Mass General Brigham, and NBC Sports Boston.


title: Wyc Grousbeck

text: In 2022, Princeton, in an expansion effort to support additional/diverse students, will open a new Dormitory, Grousbeck Hall, based on gift by Wyc and his family; https://alumni.princeton.edu/stories/wyc-grousbeck-dormitory and https://www.princeton.edu/news/2021/05/18/wyc-grousbeck-and-emilia-fazzalari-support-undergraduate-student-expansion-new


title: Bilinear form

text: In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called "vectors") over a field "K" (the elements of which are called "scalars"). In other words, a bilinear form is a function that is linear in each argument separately:


title: Bilinear form

text: The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.


title: Bilinear form

text: When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.


title: Bilinear form

text: If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then:


title: Bilinear form

text: A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if is another basis of , then


title: Bilinear form

text: where the formula_4 form an invertible matrix . Then, the matrix of the bilinear form on the new basis is .


title: Bilinear form

text: where the dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).


title: Bilinear form

text: For a finite-dimensional vector space , if either of or is an isomorphism, then both are, and the bilinear form is said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:


title: Bilinear form

text: The corresponding notion for a module over a commutative ring is that a bilinear form is if is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from to is multiplication by 2.


title: Bilinear form

text: If is finite-dimensional then one can identify with its double dual . One can then show that is the transpose of the linear map (if is infinite-dimensional then is the transpose of restricted to the image of in ). Given one can define the "transpose" of to be the bilinear form given by


title: Bilinear form

text: The left radical and right radical of the form are the kernels of and respectively; they are the vectors orthogonal to the whole space on the left and on the right.


title: Bilinear form

text: If is finite-dimensional then the rank of is equal to the rank of . If this number is equal to then and are linear isomorphisms from to . In this case is nondegenerate. By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the "definition" of nondegeneracy:


title: Bilinear form

text: If is finite-dimensional then, relative to some basis for , a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example over the integers.


title: Bilinear form

text: If the characteristic of is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.


title: Bilinear form

text: A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when ).


title: Bilinear form

text: A bilinear form is symmetric if and only if the maps are equal, and skew-symmetric if and only if they are negatives of one another. If then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows


title: Bilinear form

text: When , the quadratic form "Q" is determined by the symmetric part of the bilinear form "B" and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.


title: Bilinear form

text: A bilinear form is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the "kernel" or the "radical" of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector , with matrix representation , is in the radical of a bilinear form with matrix representation , if and only if . The radical is always a subspace of . It is trivial if and only if the matrix is nonsingular, and thus if and only if the bilinear form is nondegenerate.


title: Bilinear form

text: For a non-degenerate form on a finite-dimensional space, the map is bijective, and the dimension of is .


title: Bilinear form

text: Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field


title: Bilinear form

text: Here we still have induced linear mappings from to , and from to . It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, "B" is said to be a perfect pairing.


title: Bilinear form

text: In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance via is nondegenerate, but induces multiplication by 2 on the map .


title: Bilinear form

text: Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product". To define them he uses diagonal matrices "A" having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field , the instances with real numbers , complex numbers , and quaternions are spelled out. The bilinear form


title: Bilinear form

text: is called the real symmetric case and labeled , where . Then he articulates the connection to traditional terminology:


title: Bilinear form

text: By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on and linear maps . If is a bilinear form on the corresponding linear map is given by


title: Bilinear form

text: In the other direction, if is a linear map the corresponding bilinear form is given by composing "F" with the bilinear map that sends to .


title: Bilinear form

text: The set of all linear maps is the dual space of , so bilinear forms may be thought of as elements of which (when is finite-dimensional) is canonically isomorphic to .


title: Bilinear form

text: Likewise, symmetric bilinear forms may be thought of as elements of (the second symmetric power of ), and alternating bilinear forms as elements of (the second exterior power of ).


title: Bilinear form

text: Definition: A bilinear form on a normed vector space is bounded, if there is a constant such that for all ,


title: Bilinear form

text: Definition: A bilinear form on a normed vector space is elliptic, or coercive, if there is a constant such that for all ,


title: Mentor (A'lars)

text: Mentor (A'lars) is a fictional character appearing in American comic books published by Marvel Comics. The character first appeared in "Iron Man" #55 (February 1973), and was created by Jim Starlin.


title: Mentor (A'lars)

text: Mentor is the leader of the Titanian colony where he is the father of Thanos and Starfox, the brother of Zuras, and the nephew of Uranos. His back-story was based on Greek mythology. Some years later, he was retconned to be a member of the Eternals, a race in the Marvel Universe created by Jack Kirby, separately based on Greek mythology.


title: Mentor (A'lars)

text: A'lars was the second son of Kronos and Daina, two of the first generation of The Eternals, and was born eons ago in the city of Titanos. A'lars and his brother Zuras were raised in the aftermath of the great civil war which divided the first colony of Eternals. Their father Kronos, who was on the winning side of the civil war, toiled away in his laboratory attempting to discover the secrets of the self-regenerating cosmic life force that he felt his race was capable of attaining. A force field cylinder containing cosmic energy exploded, disintegrating Kronos' body and imparted immortality to all nearby Eternals. A'lars then became a scientist like his father.


title: Mentor (A'lars)

text: A'lars and Zuras, both adults at the time of Kronos's death, held a general assembly to determine who would succeed him. The assembly chose Zuras, so A'lars chose to leave the colony to avoid the sort of fraternal rivalry that led Kronos and his brother Uranos to the first civil war. He settled on Saturn's moon Titan, taking over as leader of the underground colony that his uncle Uranos had built centuries earlier.


title: Mentor (A'lars)

text: Upon arriving on Titan, A'lars discovered the Uranian Eternal Sui-San, the sole survivor of the civil war that destroyed the Eternals of Titan previously. Sui-San bore two children for Mentor, Thanos and Eros (later known as Starfox). Sui-San died in Thanos' first major attack against Titan.


title: Mentor (A'lars)

text: Mentor implored his father Kronos to aid Titan against Thanos, causing Kronos to create Drax the Destroyer.


title: Mentor (A'lars)

text: Mentor later met Rick Jones and Captain Mar-Vell. With his son Eros, he informed Jones and Mar-Vell of Thanos's degradation of Titan. Mentor enlisted Mar-Vell's aid against Thanos. Mentor was imprisoned by Thanos, but freed by Captain Mar-Vell. Mentor joined in the battle against Thanos. Later, Mentor joined in the attempt to cure Captain Mar-Vell of cancer, and attended Mar-Vell's last hours.


title: Mentor (A'lars)

text: Thanos escapes captivity and reclaims his Black Order forces from Corvus Glaive. After retaking command of his Black Quadrant outpost, Thanos discovers that he is dying. Thanos tries to force Mentor to find a cure for his malady, but kills him when he is unable to.


title: Mentor (A'lars)

text: As a member of the Eternals, Mentor has superhuman stamina and durability, and a gifted intellect. He possesses various superhuman powers, including the ability to project cosmic energy as waves of concussive force and the ability to manipulate matter on the molecular level. Mentor has displayed telepathic abilities, the limits of which are as yet unknown.


title: Mentor (A'lars)

text: Mentor has knowledge of science and technology far in advance of present-day Earth. As ruler of Titan, Mentor has access to and full use of a wide range of advanced technology located on that artificial moon.


title: Mentor (A'lars)

text: Mentor appears in the "Silver Surfer" episode "Learning Curve", voiced by Cedric Smith. He is Thanos' brother, rather than Starfox, due to a typographical error in the script.


title: McGruff the Crime Dog

text: McGruff the Crime Dog is an anthropomorphic animated bloodhound created by Dancer Fitzgerald Sample advertising executive Jack Keil (who also voiced the character) through the Ad Council and later the National Crime Prevention Council to increase crime awareness and personal safety in the United States. McGruff costumes are used by police outreach efforts, often with children. McGruff was created in 1979 and debuted in 1980 with a series of public service announcements educating citizens on personal security measures, such as locking doors and putting lights on timers, in order to reduce crime. His name was selected as part of a nationwide contest in July 1980.


title: McGruff the Crime Dog

text: McGruff proved to be a successful campaign with over $100 million in free air time donated in the first year reaching over 50% of adults. McGruff campaigns continued over the years to cover topics such as child abduction, robbery, anti-drug messages, and anti-bullying campaigns. From 1982 to 2012, a number of municipalities participated in the McGruff house program which offered temporary haven to children fearing immediate harm. McGruff has continued to be well-recognized, with nine out of ten people recognizing him in a 2021 survey. This is thanks partly to recent campaigns against cyber-bullying, stopping online fakes, and elder-crime.


title: McGruff the Crime Dog

text: The decades prior to McGruff's creation saw an increase in U.S. public concern over crime. In the 1960s, a number of riots broke out across the U.S. and numerous public figures were assassinated, including President Kennedy, Martin Luther King Jr., and Malcolm X. Accepting the Republican nomination for president, Barry Goldwater positioned crime as one of the biggest issues facing the nation. While Goldwater lost to Lyndon Johnson, the issue of crime did not stop there. In July 1965, President Johnson formed the President's Commission on Law Enforcement and Administration of Justice to "probe ... fully and deeply into the problems of crime in our nation."


title: McGruff the Crime Dog

text: After two years and $2.5 million, the Commission delivered its report, "The Challenge of Crime in a Free Society", in February 1967 which influenced the Crime Control and Safe Streets Act of 1968. The act gave $300 million to local police forces for more personnel and equipment. With the election of Richard Nixon in 1968, the attempts to control rising crime rates shifted from a social approach—the "War on Poverty"—to a tough on crime approach—the "War on Crime". Despite Nixon's attempts, crime continued to rise from 363.5 crimes per 100,000 people in 1970 to 549.5 in 1979. The Carter administration took the focus away from crime and onto nuclear arms control and human rights. During Carter's presidency, crime continued to be a concern with the "kill for thrill murders" of 1979, when two men killed four people over eight days in Western Pennsylvania.


title: McGruff the Crime Dog

text: The Ad Council was first approached by the Department of Justice in 1977 to create a public campaign to engage the public in reducing crime. The FBI director recommended a campaign playing on fears to convince citizens to take personal safety steps, but the Ad Council rejected their proposal believing it would largely be ignored by an already frightened public. However, the Ad Council was still interested in a crime prevention campaign. Leo Perlis, a member of the Ad Council's Public Policy Committee, heard the proposal and liked the idea. He met with FBI Director Clarence M. Kelley, the head of the National Council on Crime and Delinquency, and a board member of the National Sheriffs' Association to form a coalition to direct the ad campaign.


title: McGruff the Crime Dog

text: The Ad Council gave the creative responsibilities to Dancer Fitzgerald Sample who they had previously worked with on the Keep America Beautiful campaign. On February 8, 1979, the Ad Council's board of directors held a meeting where they and public officials met to listen to data Dancer Fitzgerald Sample had compiled. Dancer Fitzgerald Sample had conducted focus groups in a number of cities to determine public perceptions on crime. The focus groups found that the public believed police should be the ones to prevent crime, but that they were unwilling to pay more in taxes to support more officers. They recommended a campaign which would "emphasize that individual actions can reduce crime" and "offer easily accessible opportunities for people to participate."


title: McGruff the Crime Dog

text: The task was given to Jack Keil, executive vice president and creative director of Dancer Fitzgerald Sample. Keil, thinking of Smokey Bear, came up with the idea of an animal mascot. After coming up with the slogan—"Take a bite out of crime"—he settled upon the idea of a dog. His first version was "a Snoopy look-alike wearing a Keystone Cop hat." His creative team however did not believe the dog would be taken seriously. In response, he gave the team a day to come up with a new version.


title: McGruff the Crime Dog

text: Five teams of two—a copywriter and an art director—produced proposals. They rejected proposals included a bulldog version of J. Edgar Hoover, a golden retriever, an "aggressive-looking deputy dog", and a "mongrel who became a wonder dog". The proposal Keil selected, which would go on to become McGruff, was a talking dog in a trench coat produced by Sherry Nemmers and Ray Krivascy who "was tired...he had seen the world, and he had epitomized all the detectives we had seen from Raymond Chandler to Dashiell Hammett and even Columbo."


title: McGruff the Crime Dog

text: While lauded by Keil, the U.S. Department of Justice was less enthused by the idea of a talking dog as the spokesman for crime prevention. By 1979, the Law Enforcement Assistance Administration (LEAA), created by President Johnson, was being criticized for its wastefulness prompting President Carter to shut down the program. The public safety outreach, part of the LEAA, was one of the few programs saved by Robert Diegelman who was tasked with dismantling the LEAA. Diegelman saw value in the public outreach effort and so sent monthly reports to his superiors in order to assuage their concerns. Despite this, Attorney General Benjamin Civiletti criticised the effort saying: "Why has the LEAA gotten into a campaign that is spending good money on a talking dog?" Civiletti ordered the campaign be shut down, but the advertisements had already been distributed to the media and were set to run.


title: McGruff the Crime Dog

text: In November 1979, the dog was introduced at a press Conference in New York City with his slogan "Take a bite out of crime." Eight months later, in July 1980, a nationwide contest to name the dog was concluded. "McGruff the Crime Dog" was selected as the winner with "Shurlocked Homes" as the runner-up. The winning name was submitted by Officer John Isbell of the New Orleans Police Department.


title: McGruff the Crime Dog

text: McGruff was the first Ad Council campaign to be independently evaluated. Garrett O'Keefe of the University of Denver was given a grant of $900,000 by the Justice Department's National Institute of Justice in order to evaluate the campaign. O'Keefe found that "media response to the campaign was excellent. More than $100 million of [ad] time and space had been donated by mid-1981, making McGruff one of the most popular Ad Council campaigns." As a result of the advertisements, over 1 million free booklets had been distributed, and another 250,000 were purchased from the Government Printing Office. The Army printed 300,000 booklets for their own programs as well.


title: McGruff the Crime Dog

text: By the end of 1981, over 50% of Americans had seen at least one McGruff advertisement with one third reporting they had seen the advertisements more than ten times. The dominant medium of exposure was television advertisements, comprising 78% of views, followed by posters and billboards at 14%, and newspapers at 8%. While the demographics of exposure were notably diverse, there were some trends in who saw the ads more often than others. The ads were found to reach demographics prone to crime—men, youth, people with less stable residences, and those living in lower-working-class neighborhoods—slightly more often than those populations less prone to crime. Of those who had seen the advertisements, 88% were able to articulate what they were "trying to get across" with 28% pointing out the advertisements' goals of getting citizens to participate in crime prevention programs and reporting crime to the police.


title: McGruff the Crime Dog

text: O'Keefe also asked some questions related to public perception of McGruff. He found that only 3% disliked McGruff, most calling him "too cutesy", while 57% liked him for being "attention-getting, clever, different, or appealing to all ages." 36% of respondents were neutral to McGruff. 8% said that they were annoyed by the commercials while 59% said that they were "pleased" by them.


title: McGruff the Crime Dog

text: In order to assess the impact of the McGruff advertisements, O'Keefe surveyed adults in 1979 and in 1981, a year before and a year after the premiere of the first McGruff advertisement. Of the forty personal security measures that McGruff advertisements recommended, only seven were explicitly mentioned in TV advertisements: locking doors, leaving outdoor lights on, putting indoor lights on timers, asking neighbors to watch your house, watching the neighborhood, reporting suspicious activity, and forming community groups to prevent crime. Of those seven, six saw a significant increase in usage by the public after seeing McGruff advertisements. The only activity not to see an increase was locking doors, despite the first McGruff spot specifically advocating this. O'Keefe hypothesizes that this is due to a plateau effect, as 75% of respondents in 1979 already reported locking their doors; the only personal security measure not mentioned in a television advertisement to see a significant increase was getting a dog.


title: McGruff the Crime Dog

text: McGruff debuted in 1980 with television, newspaper, billboard, and radio advertisements. The Ad Council and the National Crime Prevention Council still use McGruff in national campaigns to raise awareness about crime and crime prevention strategies. About 1500 law enforcement agencies use McGruff costumes as part of their outreach efforts in communities. McGruff advertisements feature a "fulfillment strategy", a means of contact for more information. Early advertisements contained PO boxes that could be written to for more information, but now contain phone numbers and websites.


title: McGruff the Crime Dog

text: McGruff was well received in the 1980s, and current campaigns are similarly recognizable. In a survey done by Harris Interactive for the National Crime Prevention Council, McGruff was known by 9 in 10 adults, teens, and children once being prompted; about 3 in 4 adults, 8 in 10 teens, and 8 in 10 children recognized McGruff without being prompted. Respondents were asked how likely they were to take McGruff's advice. Children were found to be very receptive, with 8 in 10 responding they were likely to take his advice. 7 in 10 teens and 6 in 10 adults gave similar responses.


title: McGruff the Crime Dog

text: The first McGruff campaign featured three television and radio advertisements as well as billboards and posters. The campaign focused on raising awareness of the ability for citizens to help prevent crime through personal security steps, community awareness, and reporting crimes in progress.


title: McGruff the Crime Dog

text: The first television advertisement, "Stop a Crime", debuted in February 1980. The full 60-second advertisement features McGruff (voiced by Jack Keil) entering an unlocked house and telling the viewer, "All crime needs is a chance. Don't give it a chance" before giving tips on preventive measures. These measures included locking doors, turning on exterior lights, securing windows, asking neighbors to watch the house during long absences, and putting lights on timers.


title: McGruff the Crime Dog

text: The advertisement was followed by two more which focused on community crime prevention tactics: "The Gilstraps" and "Mimi Marth". In "The Gilstraps", McGruff is backgrounded by men loading furniture into a moving truck. McGruff points out that these are actually thieves stealing from the home of the titular Gilstraps. The camera cuts to the Gilstraps' neighbors who, knowing the Gilstraps are out of town, call the police.


title: McGruff the Crime Dog

text: In order to show the effectiveness of community watch, McGruff creator Sherry Nemmers selected actual Hartford resident Mimi Marth for the advertisement which now bears her name. "Mimi Marth" shows Marth and another watch member, Albert Bell, responding to crimes in progress by reporting them to police on their radios. McGruff tells the viewer that "There's 126 of them, regular people like you and me, working against crime."


title: McGruff the Crime Dog

text: In addition to advertising and media campaigns, a costume was created for in-person appearances. Approximately 1,500 state and local law enforcement agencies use officers wearing a McGruff costume to educate children and others about crime prevention.


title: McGruff the Crime Dog

text: The National Crime Prevention Council hired their first president and CEO, Jack Calhoun. Calhoun wanted to address the roots of crime saying, "At some point, I have to step out from my locked house and barred windows." Current plans are to introduce McGruff to a new generation and to have the Crime Dog become a watchdog to combat new criminal and 21st Century online crime. "A greater investment in community interventions will help take a bite out of violent crime," said Paul DelPonte, current head of the National Crime Prevention Council. "Strategies that increase public engagement in public safety are proven crime stoppers." DelPonte also has urged public health officials to use McGruff and crime prevention in health prevention programs.


title: McGruff the Crime Dog

text: NCPC and McGruff are currently working with the U.S. Patent and Trademark Office to combat the sale of fake products online. The campaign, "You're Smart. Buy Smart.", premiered the first television ads featuring McGruff in 3D animnation in August 2022. The ads, fimed on the lot of Jim Henson Studios in Hollywood, are also in Spanish. Among the largest campaigns in the Crime Dog's lengthy career, it includes a partnership with NASCAR and featured a new car with driver Joey Gase.


title: McGruff the Crime Dog

text: In December 2021, the animated television show "Family Guy" featured McGruff and the National Crime Prevention Council on preventing kidnapping. The re-emergence of McGruff generated an outpouring of fan support.


title: McGruff the Crime Dog

text: In April 2022, the National Crime Prevention Council announced a partnership with McGruff the Crime Dog in creating Fentanyl Prevention Awareness Day scheduled for August 21, 2022.


title: McGruff the Crime Dog

text: The McGruff House program was a program that designated temporary safe havens for children in emergency situations. The program was first created in Utah in 1982 in response to the abduction and murder of five children by Arthur Gary Bishop. Owners of houses and apartments, after clearing a background check, would display a sign in their window with the image of McGruff. Children would be educated at school and community events to go to these houses when they felt threatened or in need of help.


title: McGruff the Crime Dog

text: The program operated under the motto "we'll call for help" and emphasized its use as a temporary haven. Volunteers were trained to call the appropriate authorities in emergency situations and would provide emotional support to children in danger. The program made clear that volunteers were for emergency situations, and even in such situations, volunteers would not act as escorts or provide first aid "except in extreme emergency situations and then only if qualified."


title: McGruff the Crime Dog

text: The first McGruff truck was established in Utah in 1986. A utility company asked that its trucks be designated as "rolling McGruff Houses" and were approved. The program was extended to other companies and municipalities, and in 2006 there were over 170 participants. In February 2012, the McGruff House program was ended after nearly 30 years. The program was ended because, with the advent and growing prominence of cell phones, the need for McGruff Houses declined combined with tightening budgets.


title: McGruff the Crime Dog

text: In 2018, the Martin Agency brought back McGruff as part of GEICO Insurance's 'count on GEICO' campaign. The TV spot has an animated McGruff attempting to share his investigation evidence with several human colleagues. They respond by not taking him seriously and treating him like an actual dog with 'baby talk', leading to McGruff's throwing his paperwork in the air and storming out of the scene.


title: My Absolute Boyfriend

text: My Absolute Boyfriend () is a 2019 South Korean television drama series based on the manga "Absolute Boyfriend" by Yuu Watase and starring Yeo Jin-goo, Bang Min-ah, Hong Jong-hyun, Hong Seo-young and Choi Sung-won. It aired on SBS TV on Wednesdays and Thursdays at 22:00 KST from May 15 to July 11, 2019. It is the third television drama based on the manga, following previous adapations in Japan and Taiwan.


title: My Absolute Boyfriend

text: It is titled just Absolute Boyfriend on Viki in English-speaking countries, but the official English title in South Korea and on Netflix in Asia is "My Absolute Boyfriend".


title: My Absolute Boyfriend

text: Um Da-da (Bang Min-ah) is a special effects makeup artist. She is secretly dating the popular actor Ma Wang-joon (Hong Jong-hyun), who is on the rise. As Wang-joon gets busy because of work and tired of their relationship, their seven-year-long relationship ends. And that is when Zero Nine (Yeong-gu) (Yeo Jin-goo), a humanoid robot who is programmed to be in love with his one and only girlfriend, appears in her life. An accidental kiss activates Yeong-gu and he takes Da-da as his girlfriend. As Da-da and Yeong-gu get closer and fall for each other, a mystery man tries to harm Da-da by threatening her to break up with Wang-joon, who secretly wants her back. Meanwhile, Diana (Hong Seo-young), the original owner of Zero Nine, comes after them to take Yeong-gu back. In the end, "My Absolute Boyfriend" is about the love with a robot who is humane and unstinting. Throughout the series, Yeong-gu shows what true love is as he learns about love and becomes the absolute boyfriend.


title: Chuncheon

text: Chuncheon (; ; formerly romanized as Chunchŏn; literally "spring river") is the capital of Gangwon Province in South Korea. The city lies in the north of the county, located in a basin formed by the Soyang River and Han River. There are some large lakes around the city, most notably Soyang Lake and Uiam Lake (or Uiam Dam). The area is renowned for its small river islands, such as Sangjungdo, Ha-Jungdo, Bungeodo, and Wido.


title: Chuncheon

text: It is a popular destination among east Asian tourists as it was featured in the popular Korean drama "Winter Sonata" (겨울연가). It is where the resort island of Namiseom is located.


title: Chuncheon

text: The area now occupied by the city was first settled several thousands of years ago, in prehistoric times, as demonstrated by stone-age archaeological evidence in the collections of Chuncheon National Museum and Hallym University Museum. In 637 AD the city was called "Usooju". In 757 AD it was renamed "Saku" and again in 940 AD as "Chunju" () before receiving its current name in 1413. In 1896, Chuncheon became the capital city of Gangwon province. The city was largely destroyed during the Korean War in the Battle of Chuncheon.


title: Chuncheon

text: Uiam Dam on the Bukhan River was completed in 1967. In 1995 Chuncheon city was merged with the government of the surrounding Chuncheon county.


title: Chuncheon

text: In the spring, the Chuncheon International Mime Festival is held, as well as the Spring Season Art Festival. In the summer, the Chuncheon Puppet Festival takes place. There is also a Makguksu festival; a cold noodle dish that originated in Chuncheon. Also celebrated in the summer is the International Animation Festival. The city has had its own annual marathon race since 1946; the Chuncheon Marathon is held every October. In winter, snow and ice festivals are organized. Currently, MAC Architects-Consultants Group Ltd. () is building the Design and Arts Arcadia of Myungseung in Chuncheon.


title: Chuncheon

text: According to the legend surrounding Cheongpyeongsa temple in Chuncheon-si, a man loved a princess so much that he became a snake and would not leave her alone. When she begged leave to collect rice from the temple, the snake let her go but then went looking for her, only to be struck dead by lightning. The princess then buried him at the temple.


title: Chuncheon

text: The city was historically known as a chicken farming region, and is famous for "dak galbi", which translates as chicken ribs. It consists of chicken, sliced cabbage, sweet potatoes, "tteok" (rice cakes), and scallions, mixed with a spicy sauce heavy on gochujang. It is usually cooked in a big iron pan in the center of the table, but the original method is over charcoal. Despite the name, the dish is traditionally made from leg meat, not ribs. There is even a street in the city dedicated to restaurants serving the dish, with some 25 restaurants populating the area.


title: Chuncheon

text: Gim Yujeong munhakje is a literature festival that takes place in April every year in memorial of the novelist Gim Yujeong. Uiam jae is another festival in April in which people worship Nongae who sacrificed herself for the country. In May, the Chuncheon International Mime Festival gathers troupes from places including Denmark, Taiwan, Germany etc., to give theatre and dance performance. Chuncheon art festival in July provides performers with opportunities to show off their talents and skills. Chuncheon International Early Music Festival also takes place in July with a wide range of classical Music of Korea. Chuncheon Puppet Festival showcases traditional puppet shows every August while traditional theatre plays are performed on Chuncheon International Theatre Festival in September Chuncheon International AniTown Festival in September is a free event that enables participants to learn more about animation production.


title: Chuncheon

text: In August, visitors can enjoy traditional Chuncheon cuisines like Dak-galbi (spicy stir-fried chicken) and Mak-guksu (buckwheat noodles) during the Chuncheon Dakgalbi & Makguksu Festival. The Soyang festival, which takes place in September, promotes the cultural history of the Soyang Dam.


title: Chuncheon

text: There are three marathons in Chuncheon every year. The National inlineskate Chuncheon Marathon takes place in April while The Chosun Ilbo Chuncheon Marathon and the Lakeside Chuncheon Marathon Festival are both held in October. People participates in Chuncheon Open International Taekwondo Championships in June to compete for championship in the Korean martial arts Taekwondo. Gangchon national MTB Challenge Competition September is a Mountain bike racing festival.


title: Chuncheon

text: Chuncheon is the market center for agricultural produce from the surrounding area. The main products are rice and soybeans. Since the 1960s light industry has become dominant in the city. After the huge success of the TV drama "Winter Sonata", the city has also become a major "Korean Wave" ("Han-ryu", ) tourist destination, attracting visitors from around East Asia.


title: Chuncheon

text: Legoland Korea is also under construction in Chuncheon. The first Legoland in Northeast Asia, it will be built in a 1.29 square kilometer parcel of land in Jungdo island and is expected to draw over a million tourists when it opens. It will be the largest Legoland in the world.


title: Chuncheon

text: There are hydroelectric powerplants in the area around Chuncheon. The Soyang Dam is the largest sand gravel dam in East Asia.


title: Chuncheon

text: As of June 2014, there are two train stations in the Chuncheon suburban area: the terminus of the Gyeongchun Line, Chuncheon station, and the busier Namchuncheon station. Gimyujeong, Baegyang-ri, Gulbongsan, and Gangchon stations, all located in rural Chuncheon, attract tourists. Gangchon is one of the most favorite recreational destinations for university students. The Gyeongchun Line is a double track rapid transit commuter train that connects to Yongsan, Cheongnyangni and Sangbong stations in Seoul. On 21 December 2010, a new realignment of the Gyeongchun Line was opened, and several new or transferable stations have since been opened to replace the existing stations outside Chuncheon, such as Cheonmasan, Sinnae, and Byeollae.


title: Chuncheon

text: There is the Gyeongchun Line subway in Chuncheon. If you use the Gyeongchun Line, you can get to Chuncheon in an hour and a half from Seoul.


title: Chuncheon

text: Chuncheon is the northern terminus of the publicly funded Jungang Expressway, and the main destination of the privately run Seoul-Chuncheon Expressway, as also known as Gyeongchun Expressway that connects to Seoul and Hongcheon-east. As of 2017, the expressway has been extended to east coast city of Yangyang, thus renamed to Seoul-Yangyang Expressway. Chuncheon Bus Terminal connects to all major cities in mainland South Korea.


title: Chuncheon

text: On National roads-wise, Chuncheon is also the main strategic destination that connects between Changwon to Cheolwon of Route (Number) 5, Incheon to Goseong of 46, and Cheolwon to Yangyang of 56.


title: Chuncheon

text: There are also river transport facilities in Soyangho (Lake Soyang), which connects to Cheongpyeongsa (청평사, Buddhist) temple.


title: Chuncheon

text: Chuncheon has a monsoon-influenced humid continental climate (Köppen: "Dwa") with cold, dry winters and hot, rainy summers. The lowest measured temperature was (6 February 1969), and the highest was (1 August 2018). The annual precipitation is an average of 1347.3 mm.


title: Chuncheon

text: High1 (Korean: 하이원) is an ice hockey team based in Chuncheon. The club was a member of the Asia League Ice Hockey between 2005 and 2019.


title: Chuncheon

text: Chuncheon's association football club, Chuncheon FC, is a member of the K4 League, a semi-professional football competition and the fourth tier of the South Korean football league system. The club was founded on 27 February 2010 with its home at the Chuncheon Stadium.


title: Chuncheon

text: Chuncheon-si protested against the enactment of "Day of Takeshima" ordinance of Shimane Prefecture , which expressed dispute over the Dokdo Islands, and announced suspension of its sisterhood relationship with Kakamigahara in March 2005.


title: Empress Lü

text: Empress (Dowager) Lü Zhi (241–18 August 180 BC), commonly known as Empress Lü () and formally Empress Gao of Han (), was the empress consort of Gaozu, the founding emperor of the Han dynasty. They had two known children, Liu Ying (later Emperor Hui of Han) and Princess Yuan of Lu. Lü was the first woman to assume the title Empress of China and paramount power. After Gaozu's death, she was honoured as Empress Dowager and Regent during the short reigns of Emperor Hui and his successors Emperor Qianshao of Han and Liu Hong (Emperor Houshao).


title: Empress Lü

text: She played a role in the rise and establishment of her husband, Emperor Gaozu, and his established dynasty, and in some of the laws and customs laid down by Gaozu. Empress Lü even in the absence of her husband from the capital killed two prominent generals who played an important role in Gaozu's rise to power, namely Han Xin and Peng Yue, as a lesson for the aristocracy and other generals. In 195, with the death of Gaozu, Empress Lü became as the widow of the late emperor and mother of the new emperor, Empress Dowager (皇太后, Huángtàihòu), and assumed a leadership role in her son's administration. Less than a year after Emperor Hui's accession to the throne in 194 BC, Lü had Concubine Qi (one of the late Emperor Gaozu's consorts), whom she deeply hated, put to death in a cruel manner. She also had Concubine Qi's son Liu Ruyi poisoned to death. Emperor Hui was shocked by his mother's cruelty and fell sick for a year, and thereafter no longer became involved in state affairs, and gave more power to his mother. As a result, Empress dowager Lü held the court, listened to the government, spoke on behalf of the emperor, and did everything (臨朝聽政制, linchao tingzheng zhe). With the untimely death of his 22-year-old son, Emperor Hui, Empress dowager Lü, his two infant sons, respectively; announced Emperor Qianshao and Emperor Houshao and gained more power than ever before, and these two young emperors had no legitimacy as emperors in history, and the history of this 8-year period is considered and recognized as the reign of Empress dowager Lü. Lü dominated the political scene for 15 years until her death in 180 BC and actually ruled as Empress regnant of Han dynasty. She is often depicted as the first woman to have ruled China; while four women are noted as politically active before her - Fu Hao, Yi Jiang, Lady Nanzi and Queen Dowager Xuan - Lü was the perhaps first woman to have ruled over united China.


title: Empress Lü

text: Lü Zhi was born in Shanfu County (單父; present-day Shan County, Shandong) during the late Qin Dynasty. Her courtesy name was Exu (). To flee from enemies, her father Lü Wen () brought their family to Pei County, settled there, and became a close friend of the county magistrate. Many influential men in town came to visit Lü Wen. Xiao He, then an assistant of the magistrate, was in charge of the seating arrangement and collection of gifts from guests at a banquet in Lü Wen's house, and he announced "Those who do not offer more than 1,000 coins in gifts shall be seated outside the hall." Liu Bang (later Emperor Gaozu of Han), then a minor patrol officer (), went there bringing a single cent and said, "I offer 10,000 coins." Lü Wen saw Liu Bang and was impressed with him on first sight, that he immediately stood up and welcomed Liu into the hall to sit beside him. Xiao He told Lü Wen that Liu Bang was not serious, but Liu ignored him and chatted with Lü. Lü Wen said, "I used to predict fortunes for many people but I've never seen someone so exceptional like you before." Lü Wen then offered his daughter Lü Zhi's hand in marriage to Liu Bang and they were wed. Lü Zhi bore Liu Bang a daughter (later Princess Yuan of Lu) and a son Liu Ying (later Emperor Hui of Han).


title: Empress Lü

text: Liu Bang later participated in the rebellion against the Qin Dynasty under the insurgent Chu kingdom, nominally ruled by King Huai II. Lü Zhi and her two children remained with her father and family for most of the time during this period.