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Lewin differentiates between multiple realities. For example, the psychological reality encompasses everything that an individual perceives and believes to be true. Only what is contained within the psychological reality can affect behavior. In contrast, things that may be outside the psychological reality, such as bit... | Wikipedia - Lewin's equation | null | null | null |
Lewin states:"The psychological reality...does not depend upon whether or not the content...exists in a physical or social sense....The existence or nonexistence...of a psychological fact are independent of the existence or nonexistence to which its content refers. "As a result, the only reality that is contained withi... | Wikipedia - Lewin's equation | null | null | null |
Lewin refers to the example of a "child living in a 'magic world.'" Lewin asserts that, for this child, the realities of the 'magic world' are a psychological reality, and thus must be considered as an influence on their subsequent behavior, even though this 'magic world' does not exist within the physical reality. Lik... | Wikipedia - Lewin's equation | null | null | null |
Lewis employed autoradiographic incorporation of 3H-thymidine to calculate mitotic indices and estimate pyknotic indices. This technique can be used to study the tissue kinetics of tumors and has applications in the scanning electron microscope. | Wikipedia - Identification of cell death | null | null | null |
Lewis studied logic under his eventual Ph.D. thesis supervisor, Josiah Royce, and is a principal architect of modern philosophical logic. In 1912, two years after the publication of the first volume of Principia Mathematica, Lewis began publishing articles taking exception to Principia' s pervasive use of material impl... | Wikipedia - C. I. Lewis | null | null | null |
Lewis restated this criticism in his reviews of both editions of Principia Mathematica. Lewis's reputation as a promising young logician was soon assured. | Wikipedia - C. I. Lewis | null | null | null |
Material implication (the rule of inference which claims that stating "P implies Q" is equivalent to stating "Q OR not P") allows a true consequent to follow from a false antecedent (so if P is not true still Q may be true since you only stated what a true P implies, but did not state what is implied if P is untrue). L... | Wikipedia - C. I. Lewis | null | null | null |
Stating strictly that P implies Q is explicitly not stating what the untrue P implies. And therefore if P is not true, Q may be true, but may be false as well.As opposed to material implication, in strict implication the statement is not primitive - it is not defined in positive terms, but rather in the combined terms ... | Wikipedia - C. I. Lewis | null | null | null |
Lewis's strict implication is now a historical curiosity, but the formal modal logic in which he grounded that notion is the ancestor of all modern work on the subject. Lewis' ◊ {\displaystyle \Diamond } notation is still standard, but current practice usually takes its dual, the square notation ◻ {\displaystyle \squar... | Wikipedia - C. I. Lewis | null | null | null |
At the time of its publication, it included the only discussion in English of the logical writings of Charles Sanders Peirce. This book followed Russell's 1900 monograph on Leibnitz, and in later editions he removed a section that seemed similar to it.Lewis went on to devise modal logic which he described in his next b... | Wikipedia - C. I. Lewis | null | null | null |
Lewis's trilemma is an apologetic argument traditionally used to argue for the divinity of Jesus by arguing that the only alternatives were that he was evil or deluded. One version was popularised by University of Oxford literary scholar and writer C. S. Lewis in a BBC radio talk and in his writings. It is sometimes de... | Wikipedia - Christological argument | null | null | null |
It takes the form of a trilemma — a choice among three options, each of which is in some way difficult to accept. This argument is very popular with Christian apologists, although some theologians and biblical scholars do not view Jesus as having claimed to be God. Some argue that he identified himself as a divine agen... | Wikipedia - Christological argument | null | null | null |
Others see him as wanting to direct attention to the divine kingdom he proclaimed.The argument relies on the premise that Jesus was a great moral teacher. The structure of the argument is as follows: Jesus claimed to be God Jesus was a wise moral teacher By the trilemma, Jesus was dishonest, deluded or God No wise mora... | Wikipedia - Christological argument | null | null | null |
Many modern New Testament scholars argue that Jesus did not, in fact, claim to be God. Disputing premise 2: C.S. Lewis expressed the opinion that any mere man who claimed to be God could not, by definition, be a wise moral teacher (and that, conversely, any wise moral teacher would not claim to be God). | Wikipedia - Christological argument | null | null | null |
Christopher Hitchens argued that Jesus was not a wise moral teacher by arguing against several of his teachings. For example, of Jesus' teaching "Let him who is without sin cast the first stone", Hitchens wrote: "if only the non-sinners have the right to punish, then how could an imperfect society ever determine how to... | Wikipedia - Christological argument | null | null | null |
Jesus could have believed (as some later philosophers have held) that religion is false but beneficial to society, and that by establishing a new religion (or a reform of Judaism) he was doing a good deed nonetheless. Disputing premise 5: A person can be a wise moral teacher despite being delusional. | Wikipedia - Christological argument | null | null | null |
Granting credence to some, or even most, of someone's claims does not require that we give credence to all of them. Someone can believe Socrates' philosophical claims about justice without also believing Socrates' theological speculations about the Greek gods, or accept Aristotle's views on poetry without also acceptin... | Wikipedia - Christological argument | null | null | null |
Lexell's first work at the Russian Academy of Sciences was to analyse data collected from the observation of the 1769 transit of Venus. He published four papers in "Novi Commentarii Academia Petropolitanae" and ended his work with a monograph on determining the parallax of the Sun, published in 1772.Lexell aided Euler ... | Wikipedia - Anders Johan Lexell | null | null | null |
This comet was later named Lexell's Comet. Lexell also was the first to calculate the orbit of Uranus and to actually prove that it was a planet rather than a comet. He made preliminary calculations while travelling in Europe in 1781 based on Hershel's and Maskelyne's observations. | Wikipedia - Anders Johan Lexell | null | null | null |
Having returned to Russia, he estimated the orbit more precisely based on new observations, but due to the long orbital period it was still not enough data to prove that the orbit was not parabolic. Lexell then found the record of a star observed in 1759 by Christian Mayer in Pisces that was neither in the Flamsteed ca... | Wikipedia - Anders Johan Lexell | null | null | null |
In addition to calculating the parameters of the orbit Lexell also estimated the planet's size more precisely than his contemporaries using Mars that was in the vicinity of the new planet at that time. Lexell also noticed that the orbit of Uranus was being perturbed. He then stated that, based on his data on various co... | Wikipedia - Anders Johan Lexell | null | null | null |
Lexical resources in digital lexicography are often referred to as machine-readable dictionary (MRD), a dictionary stored as machine (computer) data instead of being printed on paper. It is an electronic dictionary and lexical database. The term MRD is often contrasted with NLP dictionary, in the sense that an MRD is t... | Wikipedia - Lexical database | null | null | null |
Li (2019) proposes that there are three generations of construction safety informatics which are relevant to construction safety enhancement: The first generation of construction safety informatics consisted of technologies that relied completely on control by human beings; for example, structural equation modelling re... | Wikipedia - Construction safety | null | null | null |
Li added corrections to certain mathematical works. Examples of this are in Nine Chapters on the Mathematical Art by Liu Hui. He demonstrated that the least common multiple of the numbers {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} was 27720—the answer was flawed in the original text of Nine Chapters. Yet another instance of ... | Wikipedia - Li Chunfeng | null | null | null |
Li gave 22/7 (=3.142857 repeating) instead of 3 as a better approximation of what we know now as pi. He began each annotation with the words “Your servant, Chunfeng, and his collaborators comment respectfully on…” Li did write some mathematical works of his own, little is known about them. | Wikipedia - Li Chunfeng | null | null | null |
They are usually dismissed as unimportant in comparison to his other accomplishments. With Liang Shu and Wang Zhenru, he wrote Shibu Suanjing (十部算经) in 656. These were ten mathematical manuals submitted to the emperor. | Wikipedia - Li Chunfeng | null | null | null |
LiClO4 is highly soluble in organic solvents, even diethyl ether. Such solutions are employed in Diels–Alder reactions, where it is proposed that the Lewis acidic Li+ binds to Lewis basic sites on the dienophile, thereby accelerating the reaction.Lithium perchlorate is also used as a co-catalyst in the coupling of α,β-... | Wikipedia - LiClO4 | null | null | null |
LiDAR (Light Detection and Ranging) is a rapid surveying process that emits and receives laser pluses to acquire 3-D information. By illuminating lights with different wavelengths to the object of interest, LiDAR can be used to create precise topographic maps, with applications in: geology, geomorphology, surveying and... | Wikipedia - Geological structure measurement by LiDAR | null | null | null |
LiDAR data can be collected on ground-based, airborne and mobile platforms. For example, Airborne LiDAR Scanning (ALS), Unmanned Aerial Vehicles (UAV), Terrestrial Laser Scanning (TLS) and Handheld Laser Scanning (HLS). The tables below will compare the mentioned data collection platforms: 1) The data collection method... | Wikipedia - Geological structure measurement by LiDAR | null | null | null |
LiDAR data is mainly stored in a point cloud format(.las). The captured point cloud data store X, Y Z geometric data. Each data point is obtained from a single laser scan and represents a local geo-referenced spatial datum. It can represent realistic and three-dimensional rock faces in a remote and inaccessible natural... | Wikipedia - Geological structure measurement by LiDAR | null | null | null |
LiDAR system emits pulsed and continuous-wave lasers to acquire 3-D information. The laser scanner is the main component of LiDAR. Lasers with a wavelength of 550-600 nm are used on a ground-based system (handheld laser scanning and terrestrial laser scanning), whereas airborne systems use lasers with 1000-1600 nm wave... | Wikipedia - Geological structure measurement by LiDAR | null | null | null |
A laser is reflected whenever it reaches any surfaces. The full-waveform return is able to penetrate down into canopies and return vegetation information at different heights. Discrete return can only return the superficial materials. Thus, airborne LiDAR is often used for forestry studies. | Wikipedia - Geological structure measurement by LiDAR | null | null | null |
Liability for payment of an award of damages is established when the claimant proves, on the balance of probabilities, that a defendant's wrongful act caused a tangible, harm, loss or injury to the plaintiff. Once that threshold is met, the plaintiff is entitled to some amount of recovery for that loss or injury. No re... | Wikipedia - Compensatory damages | null | null | null |
The amount of damages a plaintiff would recover is usually measured on a "loss of bargain" basis, also known as expectation loss, or "economic loss". This concept reflects the difference between "the value of what has been received and its value as represented".Damages are usually assessed at the date of the wrongful a... | Wikipedia - Compensatory damages | null | null | null |
As the thresholds were not met, the loan notes were not redeemable, but at the date of the advisors' breach of contract this could not be known, only the loan notes' face value could be known. The conclusion was that in this case valuation could not be done until after the profit performance became known. In his judgem... | Wikipedia - Compensatory damages | null | null | null |
Liao is well-known for pioneering the idea of human engineering as a potential solution to accelerating climate change. In the paper, Human Engineering and Climate Change, he assesses proposed behavioral, market, and geoengineering solutions to climate change, arguing that they fall short because of insufficient motiva... | Wikipedia - S. Matthew Liao | null | null | null |
Similarly, he proposes that we research cognitive enhancement, as intelligence is linked to lower birth rates, and moral enhancement, as a way to amplify our altruism and empathy. Liao argues that human engineering is less risky that global-scale geoengineering projects and would increase the efficacy of behavioral and... | Wikipedia - S. Matthew Liao | null | null | null |
Liao works at the forefront of the emerging field of AI ethics, having recently completed the volume Ethics of Artificial Intelligence, which brings together perspectives from some of the most prominent philosophers and AI researchers on AI and morality. In 2016, he co-organized an international conference focusing on ... | Wikipedia - S. Matthew Liao | null | null | null |
On this account, algorithms that operate inside of the body tend to disrupt organismic continuity and infringe on bodily integrity, both of which warrant additional regulatory scrutiny. One issue in AI ethics that Liao is particularly interested in the moral status and rights of Artificial General Intelligences. He arg... | Wikipedia - S. Matthew Liao | null | null | null |
Libersky and Petschek extended SPH to Solid Mechanics. The main advantage of SPH in this application is the possibility of dealing with larger local distortion than grid-based methods. This feature has been exploited in many applications in Solid Mechanics: metal forming, impact, crack growth, fracture, fragmentation, ... | Wikipedia - Smoothed-particle hydrodynamics | null | null | null |
However, classical SPH formulations suffer from tensile instabilities and lack of consistency. Over the past years, different corrections have been introduced to improve the accuracy of the SPH solution, leading to the RKPM by Liu et al. Randles and Libersky and Johnson and Beissel tried to solve the consistency proble... | Wikipedia - Smoothed-particle hydrodynamics | null | null | null |
Many other recent studies can be found in the literature devoted to improve the convergence of the SPH method. Recent improvements in understanding the convergence and stability of SPH have allowed for more widespread applications in Solid Mechanics. Other examples of applications and developments of the method include... | Wikipedia - Smoothed-particle hydrodynamics | null | null | null |
SPH-based method SPAM (Smoothed Particle Applied Mechanics) for impact fracture in solids by William G. Hoover. Modified SPH (SPH/MLSPH) for fracture and fragmentation. Taylor-SPH (TSPH) for shock wave propagation in solids. Generalized coordinate SPH (GSPH) allocates particles inhomogeneously in the Cartesian coordina... | Wikipedia - Smoothed-particle hydrodynamics | null | null | null |
Libertarian socialism is a political philosophy that promotes a non-hierarchical, non-bureaucratic society without private property in the means of production. Libertarian socialists believe in converting present-day private productive property into common or public goods. Libertarian socialism is opposed to coercive f... | Wikipedia - Decentralized information technology | null | null | null |
The term libertarian socialism is used by some socialists to differentiate their philosophy from state socialism, and by some as a synonym for left anarchism.Accordingly, libertarian socialists believe that "the exercise of power in any institutionalized form – whether economic, political, religious, or sexual – brutal... | Wikipedia - Decentralized information technology | null | null | null |
Adherents propose achieving this through decentralization of political and economic power, usually involving the socialization of most large-scale private property and enterprise (while retaining respect for personal property). Libertarian socialism tends to deny the legitimacy of most forms of economically significant... | Wikipedia - Decentralized information technology | null | null | null |
In Germany its foremost spokesman during the Peasant Wars was Thomas Muenzer; in England, Gerrard Winstanley, a leading participant in the Digger movement. The concepts held by Muenzer and Winstanley were superbly attuned to the needs of their time – a historical period when the majority of the population lived in the ... | Wikipedia - Decentralized information technology | null | null | null |
What is of real importance is that they spoke to their time; their anarchist concepts followed naturally from the rural society that furnished the bands of the peasant armies in Germany and the New Model in England." The term "anarchist" first entered the English language in 1642, during the English Civil War, as a ter... | Wikipedia - Decentralized information technology | null | null | null |
By the turn of the 19th century, the English word "anarchism" had lost its initial negative connotation.For Pierre-Joseph Proudhon, mutualism involved creating "industrial democracy", a system where workplaces would be "handed over to democratically organised workers' associations . . | Wikipedia - Decentralized information technology | null | null | null |
. We want these associations to be models for agriculture, industry and trade, the pioneering core of that vast federation of companies and societies woven into the common cloth of the democratic social Republic." He urged "workers to form themselves into democratic societies, with equal conditions for all members, on ... | Wikipedia - Decentralized information technology | null | null | null |
This would result in "Capitalistic and proprietary exploitation, stopped everywhere, the wage system abolished, equal and just exchange guaranteed." Workers would no longer sell their labour to a capitalist but rather work for themselves in co-operatives. Anarcho-communism calls for a confederal form in relationships o... | Wikipedia - Decentralized information technology | null | null | null |
Peter Kropotkin thus suggested that "Representative government has accomplished its historical mission; it has given a mortal blow to court-rule; and by its debates it has awakened public interest in public questions. But to see in it the government of the future socialist society is to commit a gross error. Each econo... | Wikipedia - Decentralized information technology | null | null | null |
Life already shows in which direction the change will be made. Not in increasing the powers of the State, but in resorting to free organization and free federation in all those branches which are now considered as attributes of the State." When the First Spanish Republic was established in 1873 after the abdication of ... | Wikipedia - Decentralized information technology | null | null | null |
His acquaintance with Proudhon enabled Pi to warm relations between the Republicans and the socialists in Spain. Pi i Margall became the principal translator of Proudhon's works into Spanish and later briefly became president of Spain in 1873 while being the leader of the Democratic Republican Federal Party. According ... | Wikipedia - Decentralized information technology | null | null | null |
According to the Encyclopædia Britannica "During the Spanish revolution of 1873, Pi y Margall attempted to establish a decentralized, cantonalist political system on Proudhonian lines. "To date, the best-known examples of an anarchist communist society (i.e., established around the ideas as they exist today and achievi... | Wikipedia - Decentralized information technology | null | null | null |
During the Russian Revolution, anarchists such as Nestor Makhno worked to create and defend – through the Revolutionary Insurgent Army of Ukraine – anarchist communism in Ukraine from 1919 before being conquered by the Bolsheviks in 1921. Several libertarian socialists, notably Noam Chomsky among others, believe that a... | Wikipedia - Decentralized information technology | null | null | null |
Libido is governed primarily by activity in the mesolimbic dopamine pathway (ventral tegmental area and nucleus accumbens). Consequently, dopamine and related trace amines (primarily phenethylamine) that modulate dopamine neurotransmission play a critical role in regulating libido.Other neurotransmitters, neuropeptides... | Wikipedia - Libido | null | null | null |
Libra is the cardinal modality of the three air signs, the others being Gemini and Aquarius. The main connotations of planets under this sign are sociable, rational, and detached when dealing with situations. Libra is symbolized by the scales and is associated with the Roman deity Iustitia. According to the writer Mani... | Wikipedia - Libra (astrology) | null | null | null |
Everything was balanced under this righteous sign. Manilius once said that Libra was the sign "in which the seasons are balanced". | Wikipedia - Libra (astrology) | null | null | null |
Both the hours of the day and the hours of the night match each other. Thus why the Romans put so much trust in the "balanced sign". Going back to ancient Greek times, Libra the constellation between Virgo and Scorpio used to be ruled over by the constellation of Scorpio. They called the area the Latin word "chelae", w... | Wikipedia - Libra (astrology) | null | null | null |
Libraries known to support Smile include: Cheshire (Clojure) is a data encoding library that supports Smile as binary alternative to JSON Jackson (Java) with Binary dataformat module supporting Smile, Avro, CBOR, Ion and Protocol Buffers libsmile (C/C++; wrappers for Ruby, Perl) go-smile for decoding Smile data in Gola... | Wikipedia - Smile (data interchange format) | null | null | null |
Libraries unify access to databases by providing a single low-level programming interface to the application developer. Their advantages are most often speed and flexibility because they are not tied to a specific query language (subset) and only have to implement a thin layer to reach their goal. As all SQL dialects a... | Wikipedia - Database abstraction layer | null | null | null |
Popular use for database abstraction layers are among object-oriented programming languages, which are similar to API-level abstraction layers. In an object-oriented language like C++ or Java, a database can be represented through an object, whose methods and members (or the equivalent thereof in other programming lang... | Wikipedia - Database abstraction layer | null | null | null |
Library of Congress Subject Headings (LCSH) Library of Congress Name Authority File (LCNAF) Library of Congress Classification—because LC Classification uses a different MARC format than LC Authorities, mapping LC Classification to MADS/RDF was more difficult than mapping LCSH or LCNAF. Library of Congress Thesaurus fo... | Wikipedia - Library of Congress Linked Data Service | null | null | null |
LibreOffice Base (an OpenOffice.org fork) | Wikipedia - OpenDocument software | null | null | null |
Lichens are pioneer species, among the first living things to grow on bare rock or areas denuded of life by a disaster. Lichens may have to compete with plants for access to sunlight, but because of their small size and slow growth, they thrive in places where higher plants have difficulty growing. Lichens are often th... | Wikipedia - Lichen | null | null | null |
Like some mosses, liverworts, ferns and a few resurrection plants, upon desiccation, lichens enter a metabolic suspension or stasis (known as cryptobiosis) in which the cells of the lichen symbionts are dehydrated to a degree that halts most biochemical activity. In this cryptobiotic state, lichens can survive wider ex... | Wikipedia - Lichen | null | null | null |
Many lichens also grow as epiphytes (epi- on the surface, phyte- plant) on plants, particularly on the trunks and branches of trees. When growing on plants, lichens are not parasites; they do not consume any part of the plant nor poison it. Lichens produce allelopathic chemicals that inhibit the growth of mosses. | Wikipedia - Lichen | null | null | null |
Some ground-dwelling lichens, such as members of the subgenus Cladina (reindeer lichens), produce allelopathic chemicals that leach into the soil and inhibit the germination of seeds, spruce and other plants. Stability (that is, longevity) of their substrate is a major factor of lichen habitats. Most lichens grow on st... | Wikipedia - Lichen | null | null | null |
In these latter cases, lichens are often an important part of soil stabilization; indeed, in some desert ecosystems, vascular (higher) plant seeds cannot become established except in places where lichen crusts stabilize the sand and help retain water. Lichens may be eaten by some animals, such as reindeer, living in ar... | Wikipedia - Lichen | null | null | null |
These include common footman and marbled beauty. They are very low in protein and high in carbohydrates, making them unsuitable for some animals. The Northern flying squirrel uses it for nesting, food and winter water. | Wikipedia - Lichen | null | null | null |
Lidar Radar Stereo photogrammetry from aerial surveys Structure from motion / Multi-view stereo applied to aerial photography Block adjustment from optical satellite imagery Interferometry from radar data Real Time Kinematic GPS Topographic maps Theodolite or total station Doppler radar Focus variation Inertial surveys... | Wikipedia - Digital Terrain Model | null | null | null |
Lidar has been widely used in rock mechanics for rock mass characterization and slope change detection. Some important geomechanical properties from the rock mass can be extracted from the 3-D point clouds obtained by means of the lidar. Some of these properties are: Discontinuity orientation Discontinuity spacing and ... | Wikipedia - Laser radar | null | null | null |
Lidar technology is being used in robotics for the perception of the environment as well as object classification. The ability of lidar technology to provide three-dimensional elevation maps of the terrain, high precision distance to the ground, and approach velocity can enable safe landing of robotic and crewed vehicl... | Wikipedia - Airborne lidar bathymetry | null | null | null |
Liden applied the HBV model to estimate the riverine transport of three different substances, nitrogen, phosphorus and suspended sediment in four different countries: Sweden, Estonia, Bolivia and Zimbabwe. The relation between internal hydrological model variables and nutrient transport was assessed. A model for nitrog... | Wikipedia - Hydrologic transport model | null | null | null |
It was shown that riverine total nitrogen could be well simulated in the Nordic climate and riverine suspended sediment load could be estimated fairly well in tropical and semi-arid climates. The HBV model for material transport generally estimated material transport loads well. The main conclusion of the study was tha... | Wikipedia - Hydrologic transport model | null | null | null |
In a different work, Castanedo et al. applied an evolutionary algorithm to automated watershed model calibration. The United States EPA developed the DSSAM Model to analyze water quality impacts from land use and wastewater management decisions in the Truckee River basin, an area which include the cities of Reno and Sp... | Wikipedia - Hydrologic transport model | null | null | null |
It is based on a pollutant loading metric called "Total Maximum Daily Load" (TMDL). The success of this model contributed to the EPA's commitment to the use of the underlying TMDL protocol in EPA's national policy for management of many river systems in the United States.The DSSAM Model is constructed to allow dynamic ... | Wikipedia - Hydrologic transport model | null | null | null |
Lie algebra extensions are formalized in terms of short exact sequences. A short exact sequence is an exact sequence of length three, such that i is a monomorphism, s is an epimorphism, and ker s = im i. From these properties of exact sequences, it follows that (the image of) h is an ideal in e. Moreover, g ≅ e / Im ... | Wikipedia - Lie algebra extension | null | null | null |
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups. | Wikipedia - Lie subalgebra | null | null | null |
Lie algebras can be generated by a generating set of infinitesimal generators as defined above. To every Lie group, one can associate a Lie algebra. Roughly, a Lie algebra g {\displaystyle {\mathfrak {g}}} is an algebra constituted by a vector space equipped with Lie bracket as additional operation. The base field of a... | Wikipedia - Lie point symmetry | null | null | null |
Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps. A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie g... | Wikipedia - Manifold | null | null | null |
Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n × n {\displaystyle n\times n} matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n 2 {\displaystyle n^{2}} -dimensional disconnected manifold. The... | Wikipedia - Manifold | null | null | null |
Lie sphere geometry in n-dimensions is obtained by replacing R3,2 (corresponding to the Lie quadric in n = 2 dimensions) by Rn + 1, 2. This is Rn + 3 equipped with the symmetric bilinear form ( x 0 , x 1 , … x n , x n + 1 , x n + 2 ) ⋅ ( y 0 , y 1 , … y n , y n + 1 , y n + 2 ) {\displaystyle (x_{0},x_{1},\ldots x_{n},x... | Wikipedia - Lie sphere transformation | null | null | null |
The quadric parameterizes oriented (n – 1)-spheres in n-dimensional space, including hyperplanes and point spheres as limiting cases. Note that Qn is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius). The incidence relation carries over without change: the spheres corresponding to p... | Wikipedia - Lie sphere transformation | null | null | null |
The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles. The space of contact elements is a (2n – 1)-dimensional contact manifold Z 2n – 1: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in n-dimens... | Wikipedia - Lie sphere transformation | null | null | null |
This identification is not invariant under Lie transformations: in Lie invariant terms, Z 2n – 1 is the space of (projective) lines on the Lie quadric. Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z 2n – 1 determined by its oriented tangent spaces. There is no longer a preferred Lie c... | Wikipedia - Lie sphere transformation | null | null | null |
Lie sphere geometry provides a natural description of Dupin cyclides. These are characterized as the common envelope of two one parameter families of spheres S(s) and T(t), where S and T are maps from intervals into the Lie quadric. In order for a common envelope to exist, S(s) and T(t) must be incident for all s and t... | Wikipedia - Lie sphere geometry | null | null | null |
This map is Legendrian if and only if the derivatives of S (or T) are orthogonal to T (or S), i.e., if and only if there is an orthogonal decomposition of R4,2 into a direct sum of 3-dimensional subspaces σ and τ of signature (2,1), such that S takes values in σ and T takes values in τ. Conversely such a decomposition ... | Wikipedia - Lie sphere geometry | null | null | null |
This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional Euclidean space embeds into the Lie quadric Q3 as the set of point spheres apart from the ideal point at infinity. Explicitly, th... | Wikipedia - Lie sphere geometry | null | null | null |
Lie superalgebra supergroup (physics) supercharge supermultiplet supergravity | Wikipedia - List of mathematical topics in quantum theory | null | null | null |
Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a Z2-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras. | Wikipedia - Representation space | null | null | null |
Lie's Theorem states that if V {\displaystyle V} is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and g {\displaystyle {\mathfrak {g}}} is a solvable Lie algebra, and if π {\displaystyle \pi } is a representation of g {\displaystyle {\mathfrak {g}}} over V {\displaystyle V... | Wikipedia - Derived Lie algebra | null | null | null |
If a , b ⊂ g {\displaystyle {\mathfrak {a}},{\mathfrak {b}}\subset {\mathfrak {g}}} are solvable ideals, then so is a + b {\displaystyle {\mathfrak {a}}+{\mathfrak {b}}} . Consequently, if g {\displaystyle {\mathfrak {g}}} is finite-dimensional, then there is a unique solvable ideal r ⊂ g {\displaystyle {\mathfrak {r}}... | Wikipedia - Derived Lie algebra | null | null | null |
Lieb is famous for many groundbreaking results in statistical mechanics concerning, in particular, soluble systems. His numerous works have been collected in the Selecta ″Statistical mechanics″ and ″Condensed matter physics and exactly soluble models″, as well as in a book with Daniel Mattis. They treat (among other th... | Wikipedia - Elliott H. Lieb | null | null | null |
In 1968, together with Fa-Yueh Wu, he gave the exact solution of the one-dimensional Hubbard model. In 1971 he and Neville Temperley introduced the Temperley-Lieb algebra in order to build certain transfer matrices. | Wikipedia - Elliott H. Lieb | null | null | null |
This algebra also has links with knot theory and the braid group, quantum groups and subfactors of von Neumann algebras. Together with Derek W. Robinson in 1972 he derived bounds on the propagation speed of information in non relativistic spin systems with local interactions. They have become known as Lieb-Robinson bou... | Wikipedia - Elliott H. Lieb | null | null | null |
They can be used to prove the exponential decay of correlations in spin systems or to make assertions about the gap above the ground state in higher-dimensional spin systems (generalized Lieb-Schultz-Mattis theorems). In 1972 he and Mary Beth Ruskai proved the strong subadditivity of quantum entropy, a theorem that is ... | Wikipedia - Elliott H. Lieb | null | null | null |
Life Cycle Analysis (LCA) is a tool used to understand the how a product impacts the environment at each stage of its life cycle, from raw input to the end of the products' life cycle. Life Cycle Cost (LCC) is an economic metric that "identifies the minimum cost for each life cycle stage which would be presented in the... | Wikipedia - Ecological Design | null | null | null |
However, switching material may increase environmental effects later in a products life time; LCA takes into account the whole life cycle of a product and can alert designers to the many impacts of a product, which is why LCA is important. Some of the factors that LCA takes into account are the costs and emissions of: ... | Wikipedia - Ecological Design | null | null | null |
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