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81,500 | The Society for the Study of Artificial Intelligence and Simulation of Behaviour or SSAISB or AISB is a nonprofit, scientific society devoted to advancing the scientific understanding of the mechanisms underlying thought and intelligent behaviour and their simulation and embodiment in machines. AISB also aims to facili... | Society for the Study of Artificial Intelligence and the Simulation of Behaviour | 0.784566 |
81,501 | WSMR Launch Complex 33 for the captured V-2s was built around this blockhouse.Initial V-2 assembly efforts produced 25 rockets available for launch. The Army assembled an Upper Atmosphere Research Panel of representative from the Air Materiel Command, Naval Research Laboratory (NRL), Army Signal Corps, Ballistic Resear... | V-2 sounding rocket | 0.784565 |
81,502 | A 1946 Naval Research Laboratory launch took the first photographs of the Sun in the ultraviolet spectrum up to an altitude of 88 km (55 mi).The first night flight of a V-2 sounding rocket began at 10:00 pm (MST) 17 December 1946 on an Applied Physics Laboratory flight. This rocket carried several explosive charges tha... | V-2 sounding rocket | 0.784565 |
81,503 | Google PowerMeter was a software project of Google's philanthropic arm, Google.org, to help consumers track their home electricity usage. It was launched on October 5, 2009 and ended on September 16, 2011. The development of the software was part of an effort by Google to invest in renewable energy, electricity grid up... | Google PowerMeter | 0.784564 |
81,504 | The development of the software was part of an effort by Google to invest in renewable energy, electricity grid upgrades, and other measures that would reduce greenhouse gas emissions. It was hoped that this tool would raise the home-owner's awareness of how much energy they use and make users more energy efficient. Po... | Google PowerMeter | 0.784564 |
81,505 | A nested quotation is a quotation that is encapsulated inside another quotation, forming a hierarchy with multiple levels. When focusing on a certain quotation, one must interpret it within its scope. Nested quotation can be used in literature (as in nested narration), speech, and computer science (as in "meta"-stateme... | Nested quotation | 0.784563 |
81,506 | : 141 Nylon was introduced as part of "The world of tomorrow" at the 1939 New York World's Fair and was featured at DuPont's "Wonder World of Chemistry" at the Golden Gate International Exposition in San Francisco in 1939. Actual nylon stockings were not shipped to selected stores in the national market until May 15, 1... | Nylon | 0.784563 |
81,507 | Some of the key ingredients of nylon had to be produced using high pressure chemistry, the main area of expertise of the Ammonia Department. Nylon was considered a "godsend to the Ammonia Department", which had been in financial difficulties. The reactants of nylon soon constituted half of the Ammonia Department's sale... | Nylon | 0.784563 |
81,508 | Researchers at DuPont began developing cellulose-based fibers, culminating in the synthetic fiber rayon. DuPont's experience with rayon was an important precursor to its development and marketing of nylon. : 8, 64, 236 DuPont's invention of nylon spanned an eleven-year period, ranging from the initial research program ... | Nylon | 0.784563 |
81,509 | Integrated topside design is a design approach used by military ship and ship equipment designers to overcome the challenges of effectively operating shipboard antenna systems and equipment susceptible to electromagnetic fields in the high electromagnetic environment of a warship's topside. The approach primarily uses ... | Integrated topside design | 0.784561 |
81,510 | The lack of evidence for evolutionary transitions in the fossil record is best accounted for, first, by the poorness of the geological record, and, second, as a consequence of quantum evolution (which is responsible for "the origin of taxonomic units of relatively high rank, such as families, orders, and classes"). Qua... | Tempo and Mode in Evolution | 0.784559 |
81,511 | Tempo and Mode in Evolution (1944) was George Gaylord Simpson's seminal contribution to the evolutionary synthesis, which integrated the facts of paleontology with those of genetics and natural selection. Simpson argued that the microevolution of population genetics was sufficient in itself to explain the patterns of m... | Tempo and Mode in Evolution | 0.784559 |
81,512 | The Centre for Alternative Technology (CAT) (Welsh: Canolfan y Dechnoleg Amgen) is an eco-centre in Montgomeryshire, Powys, Wales dedicated to demonstrating and teaching sustainable development. CAT, despite its name, no longer concentrates its efforts exclusively on alternative technology, but provides information on ... | Centre for Alternative Technology | 0.784559 |
81,513 | In 1975 a permanent exhibition opened in order to generate wider interest. The 40-acre (160,000 m2) site with 7 acres (28,000 m2) of interactive displays is the largest tourist attraction in the area. It is open all-year except Christmas, seven days a week. It is a registered charity.The facilities and exhibits include... | Centre for Alternative Technology | 0.784558 |
81,514 | The centre offers a range of courses which are between a day and a week in length. Some of these courses are offered to the general public while others are accredited courses for professional installers. CAT is involved with school education through training teachers, producing materials and offering special tours and ... | Centre for Alternative Technology | 0.784558 |
81,515 | In the Langmuir-Blodgett method, the nanoparticles are injected at air-water interphase in a special Langmuir-Blodgett Trough. The floating particles are compressed closer to each other with motorized barriers which allow to control the packing density of the particles. After compressing the particles to the desired pa... | Nanoparticle deposition | 0.784556 |
81,516 | In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares.Its square root is called regression standard error, standard error of the re... | Reduced chi-squared statistic | 0.784556 |
81,517 | The libpf library includes functions for the generation and manipulation of hierarchical scene graphs, scene processing (simulation, intersection, culling, and drawing tasks), level-of-detail management, asynchronous database paging, dynamic coordinate systems, environment models, light points, and so on. This library ... | OpenGL Performer | 0.784556 |
81,518 | In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an e... | Hyperbolic tetrahedral-octahedral honeycomb | 0.784556 |
81,519 | Integrative Psychological and Behavioral Science is a quarterly peer-reviewed scientific journal covering the integration of psychology and biology through epigenesis. It was established in 1965 by W. H. Gantt and others as Conditional Reflex; the first issue was published in 1966. In 1974 it was renamed The Pavlovian ... | Integrative Psychological and Behavioral Science | 0.784555 |
81,520 | Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space; as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Smale conjectured that if M {\displaystyle M} is an oriented smooth ... | Diffeomorphism | 0.784554 |
81,521 | The Lie algebra of the diffeomorphism group of M {\displaystyle M} consists of all vector fields on M {\displaystyle M} equipped with the Lie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate x {\displaystyle x} at each point in space: x μ ↦ x μ + ε h μ ( x ) {\display... | Diffeomorphism | 0.784554 |
81,522 | variety (botany) – vertebrate paleontology (a.k.a. vertebrate paleobiology or paleozoology) – viral evolution – The Voyage of the Beagle – vestigiality | Index of evolutionary biology articles | 0.784553 |
81,523 | selection – selective breeding – selfish DNA – The Selfish Gene – sexual selection – signalling theory – sociobiology – social effects of evolutionary theory – species – speciation – species flock – sperm competition – stabilizing selection – strain (biology) – subspecies – survival of the fittest – symbiogenesis – sys... | Index of evolutionary biology articles | 0.784553 |
81,524 | race (biology) – Red Queen hypothesis – recapitulation theory – recent African origin of modern humans – recombination – Bernhard Rensch – reinforcement (speciation) – reproductive coevolution in Ficus – r/K selection theory | Index of evolutionary biology articles | 0.784553 |
81,525 | paleoanthropology – paleobiology – paleobotany – paleontology – paleozoology (of vertebrates – of invertebrates) – parallel evolution – paraphyletic – particulate inheritance – peppered moth – peppered moth evolution – peripatric speciation – phenotype – phylogenetics – phylogeny – phylogenetic tree – Pikaia – Plant ev... | Index of evolutionary biology articles | 0.784553 |
81,526 | natural selection – natural genetic engineering – nature versus nurture – Neo-Darwinism – neutral theory of molecular evolution – Baron Franz Nopcsa – "Nothing in Biology Makes Sense Except in the Light of Evolution" | Index of evolutionary biology articles | 0.784553 |
81,527 | macroevolution – macromutation – The Major Transitions in Evolution – maladaptation – The Malay Archipelago – mass extinctions – mating systems – John Maynard Smith – Ernst Mayr – Gregor Mendel – memetics – Mendelian inheritance – Mesozoic–Cenozoic radiation – microevolution – micropaleontology (a.k.a. micropaleobiolog... | Index of evolutionary biology articles | 0.784553 |
81,528 | inclusive fitness – insect evolution – Invertebrate paleontology (a.k.a. invertebrate paleobiology or paleozoology) | Index of evolutionary biology articles | 0.784553 |
81,529 | This is a list of topics in evolutionary biology. | Index of evolutionary biology articles | 0.784553 |
81,530 | E. coli long-term evolution experiment – ecological genetics – ecological selection – ecological speciation – Endless Forms Most Beautiful – endosymbiosis – error threshold (evolution) – evidence of common descent – evolution – evolutionary arms race – evolutionary capacitance Evolution: of ageing – of the brain – of c... | Index of evolutionary biology articles | 0.784553 |
81,531 | Cambrian explosion – camouflage – Sean B. Carroll – catagenesis – gene-centered view of evolution – cephalization – Sergei Chetverikov – chronobiology – chronospecies – clade – cladistics – climatic adaptation – coalescent theory – co-evolution – co-operation – coefficient of relationship – common descent – convergent ... | Index of evolutionary biology articles | 0.784553 |
81,532 | J. B. S. Haldane – W. D. Hamilton – Hardy–Weinberg principle – heredity – hierarchy of life – history of evolutionary thought – history of speciation – homologous chromosomes – homology (biology) – horizontal gene transfer – human evolution – human evolutionary genetics – human vestigiality – Julian Huxley – Thomas Hen... | Index of evolutionary biology articles | 0.784553 |
81,533 | Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-037941-8. | Trivalent chromium | 0.784553 |
81,534 | Chromium also has two metastable nuclear isomers.53Cr is the radiogenic decay product of 53Mn (half-life 3.74 million years). Chromium isotopes are typically collocated (and compounded) with manganese isotopes. This circumstance is useful in isotope geology. | Trivalent chromium | 0.784553 |
81,535 | The opposite is true in the algebraic and holomorphic settings. Here the space of global sections is often finite dimensional, but there may not be any non-vanishing global sections at a given point. (As in the case when this procedure constructs a Lefschetz pencil.) In fact, it is possible for a bundle to have no non-... | Determinant bundle | 0.784553 |
81,536 | Every locally normal T1 space is locally regular and locally Hausdorff. A locally compact Hausdorff space is always locally normal. A normal space is always locally normal. A T1 space need not be locally normal as the set of all real numbers endowed with the cofinite topology shows. | Locally normal space | 0.78455 |
81,537 | A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology.Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology). Note however, that if a space ... | Locally normal space | 0.78455 |
81,538 | In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology. | Locally normal space | 0.78455 |
81,539 | Software packages have been designed for 1D/2D/3D basin modelling purposes to simulate the burial and thermal history of a basin as well as petroleum migration modelling. Basin Modeling software programs: Permedia (Halliburton), PBM-Pars Basin Modeler(Research Institute of Petroleum Industry), PetroMod (Schlumberger), ... | Basin modelling | 0.784548 |
81,540 | The Applied Catalysis Award is awarded by the Royal Society of Chemistry to individuals for "creativity and excellence in novel approaches or use of catalysis in industry." The award was established in 2008. The winner of the award is chosen by the Industry & Technology Division Awards Committee, and receives £2000, a ... | Applied Catalysis Award | 0.784547 |
81,541 | John Leonard; B. Lygo; Garry Procter (2 June 1994). Advanced practical organic chemistry. pp. 198–200. ISBN 9780748740710. | Perkin triangle | 0.784546 |
81,542 | Stellar parallax is so small that it was unobservable until the 19th century, and its apparent absence was used as a scientific argument against heliocentrism during the early modern age. It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons,... | Stellar parallax | 0.784545 |
81,543 | In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier ... | Microlocalization functor | 0.784544 |
81,544 | The degree of A is defined to be the square root of its dimension. As a result, the isomorphism classes of CSAs over K form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse of an algebra A is given by its opposite algebra Aop (the opposite ring wi... | Brauer equivalence | 0.784543 |
81,545 | We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n, D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m, D) with M(n, D) for all positive integers m and n, we get the B... | Brauer equivalence | 0.784543 |
81,546 | A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A such that A is a simple ring and the center of A is equal to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over the... | Brauer equivalence | 0.784543 |
81,547 | In 2021, Nicholas Seaton of the University of Wollongong defined the Brauer Group for Fell Algebras, a significant breakthrough in the field. | Brauer equivalence | 0.784543 |
81,548 | For any central simple algebra A over a field K, the period of A divides the index of A, and the two numbers have the same prime factors. The period-index problem is to bound the index in terms of the period, for fields K of interest. For example, if A is a central simple algebra over a local field or global field, the... | Brauer equivalence | 0.784543 |
81,549 | For an arbitrary field K, the Brauer group can be expressed in terms of Galois cohomology as follows: Br ( K ) ≅ H 2 ( K , G m ) , {\displaystyle {\textrm {Br}}(K)\cong H^{2}(K,G_{m}),} where Gm denotes the multiplicative group, viewed as an algebraic group over K. More concretely, the cohomology group indicated means ... | Brauer equivalence | 0.784543 |
81,550 | Then the subgroup of the Brauer group of K killed by n is generated by cyclic algebras of degree n. Equivalently, any division algebra of period dividing n is Brauer equivalent to a tensor product of cyclic algebras of degree n. Even for a prime number p, there are examples showing that a division algebra of period p n... | Brauer equivalence | 0.784543 |
81,551 | (When n is not invertible in K or K does not have a primitive nth root of unity, a similar construction gives the cyclic algebra (χ, a) associated to a cyclic Z/n-extension χ of K and a nonzero element a of K.) The Merkurjev–Suslin theorem in algebraic K-theory has a strong consequence about the Brauer group. Namely, f... | Brauer equivalence | 0.784543 |
81,552 | For a positive integer n, let K be a field in which n is invertible such that K contains a primitive nth root of unity ζ. For nonzero elements a and b of K, the associated cyclic algebra is the central simple algebra of degree n over K defined by ( a , b ) ζ = K ⟨ u , v ⟩ / ( u n = a , v n = b , u v = ζ v u ) . {\displ... | Brauer equivalence | 0.784543 |
81,553 | Another important interpretation of the Brauer group of a field K is that it classifies the projective varieties over K that become isomorphic to projective space over an algebraic closure of K. Such a variety is called a Severi–Brauer variety, and there is a one-to-one correspondence between the isomorphism classes of... | Brauer equivalence | 0.784543 |
81,554 | TET2 can be recruited by WT1 to its target genes and activates WT1-target genes by converting 5mC into 5hmC residues at the genes’ promoters, representing an important feature of a new regulatory WIT pathway linked to the development of AML.The serine protease HtrA2 binds to WT1 and it cleaves WT1 at multiple sites fol... | Wilms tumor protein | 0.784543 |
81,555 | In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative... | Bose–Mesner algebra | 0.784542 |
81,556 | This means that A {\displaystyle {\mathcal {A}}} is semi-simple and has a unique basis of primitive idempotents J 0 , … , J n {\displaystyle J_{0},\ldots ,J_{n}} . These are complex n × n matrices satisfying J i 2 = J i , i = 0 , … , n , ( 3 ) {\displaystyle J_{i}^{2}=J_{i},i=0,\ldots ,n,\qquad (3)} J i J k = 0 , i ≠ k... | Bose–Mesner algebra | 0.784542 |
81,557 | This associative commutative algebra A {\displaystyle {\mathcal {A}}} is called the Bose–Mesner algebra of the association scheme. Since the matrices in A {\displaystyle {\mathcal {A}}} are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix S {\displaystyle... | Bose–Mesner algebra | 0.784542 |
81,558 | In this representation, there is a loop at each vertex and all the loops receive the same 0th color. The association scheme can also be represented algebraically. | Bose–Mesner algebra | 0.784542 |
81,559 | The ABC was built and tested over the next two years. A January 15, 1941, story in the Des Moines Register announced the ABC as "an electrical computing machine" with more than 300 vacuum tubes that would "compute complicated algebraic equations" (but gave no precise technical description of the computer). The system w... | Atanasoff–Berry computer | 0.784539 |
81,560 | $5,000 of further funding (equivalent to $105,000 in 2022) to complete the machine came from the nonprofit Research Corporation of New York City.The ABC was built by Atanasoff and Berry in the basement of the physics building at Iowa State College during 1939–1942. The initial funds were released in September, and the ... | Atanasoff–Berry computer | 0.784539 |
81,561 | This process would be repeated manually for each of the equations, which would result in a system of equations with one fewer variable. Then the whole process would be repeated to eliminate another variable. George W. Snedecor, the head of Iowa State's Statistics Department, was very likely the first user of an electro... | Atanasoff–Berry computer | 0.784539 |
81,562 | The ABC was designed for a specific purpose – the solution of systems of simultaneous linear equations. It could handle systems with up to 29 equations, a difficult problem for the time. Problems of this scale were becoming common in physics, the department in which John Atanasoff worked. The machine could be fed two l... | Atanasoff–Berry computer | 0.784539 |
81,563 | Conceived in 1937, the machine was built by Iowa State College mathematics and physics professor John Vincent Atanasoff with the help of graduate student Clifford Berry. It was designed only to solve systems of linear equations and was successfully tested in 1942. However, its intermediate result storage mechanism, a p... | Atanasoff–Berry computer | 0.784539 |
81,564 | Understanding the differences between the R* theory and its major alternative the CSR triangle theory is a major goal in community ecology for many years. Unlike the R* theory, the CSR theory predicts that competitive ability is determined by relative growth rate and other size related traits. While some experiments su... | R* rule | 0.784538 |
81,565 | The R* rule (also called the resource-ratio hypothesis) is a hypothesis in community ecology that attempts to predict which species will become dominant as the result of competition for resources. The hypothesis was formulated by American ecologist David Tilman. It predicts that if multiple species are competing for a ... | R* rule | 0.784538 |
81,566 | In medicine, common types of salines include: Ringer's lactate solution Acetated Ringer's solution Intravenous sugar solutions 5% dextrose in normal saline (D5NS) 10% dextrose in normal saline (D10NS) 5% dextrose in half-normal saline (D5HNS) 10% dextrose in half-normal saline (D10HNS)And in cell biology, in addition t... | 3% normal saline | 0.784537 |
81,567 | The chemistry of ununennium is predicted to be similar to that of the alkali metals, but it would probably behave more like potassium or rubidium than caesium or francium. This is due to relativistic effects, as in their absence periodic trends would predict ununennium to be even more reactive than caesium and francium... | Element 119 | 0.784536 |
81,568 | In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets. In the calculus of variations, where one is typically interested in infinite-dimensional function spac... | Palais–Smale compactness condition | 0.784535 |
81,569 | The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional... | Palais–Smale compactness condition | 0.784535 |
81,570 | The International Society for Magnetic Resonance in Medicine is a "multi-disciplinary nonprofit association that promotes innovation, development, and application of magnetic resonance techniques in medicine and biology throughout the world".The Society has over 9000 members and is based in Concord, California, USA. | International Society for Magnetic Resonance in Medicine | 0.784534 |
81,571 | However, this is only one of many estimates published by the UN; in 2009, UN population projections for 2050 ranged between around 8 billion and 10.5 billion. An alternative scenario is given by the statistician Jorgen Randers, who argues that traditional projections insufficiently take into account the downward impact... | Human population | 0.784533 |
81,572 | Given a group G with a subgroup H, there are two fundamental functors between the categories of representations of G and H (over a fixed field): one is a forgetful functor called the restriction functor Res H G: Rep ( G ) ⟶ Rep ( H ) π ⟼ π | H {\displaystyle {\begin{aligned}\operatorname {Res} _{H}^{G}:\operatornam... | Category of representations | 0.784532 |
81,573 | Depending on the types of the representations one wants to consider, it is typical to use slightly different definitions. For a finite group G and a field F, the category of representations of G over F has objects: pairs (V, f) of vector spaces V over F and representations f of G on that vector space morphisms: equivar... | Category of representations | 0.784532 |
81,574 | In representation theory, the category of representations of some algebraic structure A has the representations of A as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an obje... | Category of representations | 0.784532 |
81,575 | The category of linear representations of a group has a monoidal structure given by the tensor product of representations, which is an important ingredient in Tannaka-Krein duality (see below). Maschke's theorem states that when the characteristic of F doesn't divide the order of G, the category of representations of G... | Category of representations | 0.784532 |
81,576 | Tannaka–Krein duality concerns the interaction of a compact topological group and its category of linear representations. Tannaka's theorem describes the converse passage from the category of finite-dimensional representations of a group G back to the group G, allowing one to recover the group from its category of repr... | Category of representations | 0.784532 |
81,577 | The Haber–Weiss reaction generates •OH (hydroxyl radicals) from H2O2 (hydrogen peroxide) and superoxide (•O2−) catalyzed by iron ions. It was first proposed by Fritz Haber and his student Joseph Joshua Weiss in 1932.This reaction has long been studied and revived in different contexts, including organic chemistry, free... | Haber–Weiss reaction | 0.784532 |
81,578 | LAH crystallizes in the monoclinic space group P21/c. The unit cell has the dimensions: a = 4.82, b = 7.81, and c = 7.92 Å, α = γ = 90° and β = 112°. In the structure, Li+ cations are surrounded by five − anions, which have tetrahedral molecular geometry. The Li+ cations are bonded to one hydrogen atom from each of the... | Lithium aluminium hydride | 0.784531 |
81,579 | In mathematics, a tangent Lie group is a Lie group whose underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian subgroup with underlying space the Lie algebra of G, and G itself. | Tangent Lie group | 0.78453 |
81,580 | Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functio... | Abel's functional equation | 0.784529 |
81,581 | In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates sever... | Abel's functional equation | 0.784529 |
81,582 | In 1991, two self-professed pranksters, Doug Bower and Dave Chorley, made headlines by claiming that it was they who started the phenomenon in 1978 with the use of simple tools consisting of a plank of wood, rope, and a baseball cap fitted with a loop of wire to help them walk in straight lines. To prove their case the... | Crop circle | 0.784528 |
81,583 | Some paranormal advocates think that crop circles are caused by ball lighting and that the patterns are so complex that they have to be controlled by some entity. Some proposed entities are: Gaia asking to stop global warming and human pollution, God, supernatural beings (for example Indian devas), the collective minds... | Crop circle | 0.784528 |
81,584 | These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as... | Infinitesimal analysis | 0.784526 |
81,585 | Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is a von Neumann ordinal). However, smooth infinitesimal... | Infinitesimal analysis | 0.784526 |
81,586 | This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT (ε ≠ 0); yet it is provably false that all infinitesi... | Infinitesimal analysis | 0.784526 |
81,587 | Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of ... | Infinitesimal analysis | 0.784526 |
81,588 | The cross-product is given by the matrix commutator (up to a factor of 2 i {\displaystyle 2i} ) In fact, the existence of a norm follows from the fact that R 3 {\displaystyle \mathbb {R} ^{3}} is a Lie algebra: see Killing form. This cross-product can be used to prove the orientation-preserving property of the map abov... | Pauli spin matrix | 0.784526 |
81,589 | The real linear span of {I, iσ1, iσ2, iσ3} is isomorphic to the real algebra of quaternions, H {\displaystyle \mathbb {H} } , represented by the span of the basis vectors { 1 , i , j , k } . {\displaystyle \left\{\;\mathbf {1} ,\,\mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} \;\right\}.} The isomorphism from H {\displaysty... | Pauli spin matrix | 0.784526 |
81,590 | In coordination chemistry, a macrocyclic ligand is a macrocyclic ring having at least nine atoms (including all hetero atoms) and three or more donor sites that serve as ligands that can bind to a central metal ion. Crown ethers and porphyrins are prominent examples. Macrocyclic ligands exhibit high affinity for metal ... | Macrocyclic ligand | 0.784526 |
81,591 | Porphyrins and phthalocyanines have long been recognized as potent ligands in coordination chemistry as illustrated by numerous transition metal porphyrin complexes and the commercialization of copper phthalocyanine pigments. In the 1960s the synthesis of macrocylic ligands received much attention. One early contributi... | Macrocyclic ligand | 0.784526 |
81,592 | Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real roots. Given the existence of the imaginary scalar, a number of expressions can be written and given proper names. All of these were part of Hamilton's original quaternion calculus. In symbols: q + q ′ − 1... | Classical Hamiltonian quaternions | 0.784526 |
81,593 | . In Hamilton's calculus the product is not commutative, i.e., the order of the variables is of great importance. If the order of q and β were to be reversed the result would not in general be α. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, formerly an act of version an... | Classical Hamiltonian quaternions | 0.784526 |
81,594 | Subtraction is a type of analysis called ordinal analysis ...let space be now regarded as the field of progression which is to be studied, and POINTS as states of that progression. ...I am led to regard the word "Minus," or the mark −, in geometry, as the sign or characteristic of analysis of one geometric position (in... | Classical Hamiltonian quaternions | 0.784526 |
81,595 | Phthalocyanines were the first macrocycles synthesized by template reaction. Featuring planar, dianionic, 18-membered rings with four nitrogenous ligands, phthalocyanines resemble porphyrins. The size of the metal cation used as the template has proved to be of importance in directing the synthetic pathway for the Schi... | Macrocyclic ligand | 0.784526 |
81,596 | The purpose is to have interchangeable and high quality format that can be used as a main archive format for this kind of data.The XDF project and related development have been halted (2002,2006). The existing information have been archived to the UMD Astronomy Information and Knowledge Group site as a reference. == Re... | Extensible Data Format | 0.784526 |
81,597 | Tetraphenylborates are often studied in organometallic chemistry because of their good solubility in nonpolar solvents and their crystallinity. For example, the homoleptic trimethylphosphite complexes {M5}2+ (Ni, Pd, and Pt) have been prepared as their tetraphenylborate salts. Similarly, sodium tetraphenylborate has be... | Sodium tetraphenylborate | 0.784525 |
81,598 | Sodium tetraphenylborate is the organic compound with the formula NaB(C6H5)4. It is a salt, wherein the anion consists of four phenyl rings bonded to boron. This white crystalline solid is used to prepare other tetraphenylborate salts, which are often highly soluble in organic solvents. The compound is used in inorgani... | Sodium tetraphenylborate | 0.784525 |
81,599 | There are numerous partial results.Claire Voisin was awarded the Ruth Lyttle Satter Prize in Mathematics for her solution of the generic case of Green's conjecture in two papers. The case of Green's conjecture for generic curves had attracted a huge amount of effort by algebraic geometers over twenty years before final... | Green's conjecture | 0.784525 |
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