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97,500 | Mobile positioning data is also touted to be suitable for studying tourism in less visited natural areas where it is difficult to use other methods such as accommodation statistics or questionnaires.The timeliness of statistics (up to near real-time) and the possibility of producing unconfirmed quick indicators are fur... | Mobile positioning data | 0.777701 |
97,501 | For statistical purposes, mobile positioning data provides new possibilities in terms of the quality of the data. Statistics based on MPD can be compiled automatically, in some cases almost in real time, and it requires less manual labour. The job of analysing and interpreting the resulting statistical indicators is le... | Mobile positioning data | 0.777701 |
97,502 | Since the structure sheaf of a scheme naturally carries only the discrete topology, the notion of formally étale for schemes is analogous to formally étale for the discrete topology for rings. That is, a morphism of schemes f: X → Y is formally étale if for every affine Y-scheme Z, every nilpotent sheaf of ideals J on ... | Formally etale | 0.777699 |
97,503 | Localizations are formally étale. Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale. | Formally etale | 0.777699 |
97,504 | Let A be a topological ring, and let B be a topological A-algebra. Then B is formally étale if for all discrete A-algebras C, all nilpotent ideals J of C, and all continuous A-homomorphisms u: B → C/J, there exists a unique continuous A-algebra map v: B → C such that u = pv, where p: C → C/J is the canonical projection... | Formally etale | 0.777699 |
97,505 | In commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism. | Formally etale | 0.777699 |
97,506 | In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i {\displaystyle i} in time, the times at which the state of a dynamical system returns to the previous state at i {\displaystyle i} , i.e., when the phase space trajectory visits roughly the same area in the phase spa... | Recurrence plot | 0.777698 |
97,507 | If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem): x → ( i ) = ( u ( i ) , u ( i + τ ) , … , u ( i + τ ( m − 1 ) ) , {\displaystyle {\vec {x}}(i)=(u(i),u(i+\tau ),\ldots ,u(i+\tau (m-1)),} where u ( i ) {\displaystyle u(i)} is the time series,... | Recurrence plot | 0.777698 |
97,508 | Research on face recognition to reliably locate a face in an image that contains other objects gained traction in the early 1990s with the principal component analysis (PCA). The PCA method of face detection is also known as Eigenface and was developed by Matthew Turk and Alex Pentland. Turk and Pentland combined the c... | Facial recognition software | 0.777696 |
97,509 | However, these methods are used to prevent infrared facial recognition and would not work on AI facial recognition of plain images. Some projects use adversarial machine learning to come up with new printed patterns that confuse existing face recognition software.One method that may work to protect from facial recognit... | Facial recognition software | 0.777696 |
97,510 | Systematic Entomology is a scientific journal covering the field of systematic entomology, published by the Royal Entomological Society of London. Having begun in 1932 as Proceedings of the Royal Entomological Society of London, Series B: Taxonomy, the title was changed to Journal of Entomology, Series B: Taxonomy in 1... | Systematic Entomology | 0.777695 |
97,511 | The H100 was interesting in that it could run either the CP/M operating system, or their OEM version of MS-DOS named Z-DOS, which were the two leading business PC operating systems at the time. Although the machine had to be rebooted to change modes, the competing operating systems could read each other's disks. In 198... | Heathkit OC-1401 | 0.777694 |
97,512 | Heath developed a business relationship with electronics correspondence schools (e.g., NRI and Bell & Howell), and supplied electronic kits to be assembled as part of their courses, with the schools basing their texts and lessons around the kits. In the 1960s, Heathkit marketed a line of its electronic instruments whic... | Heathkit OC-1401 | 0.777694 |
97,513 | The assignment of the HSQC spectrum requires other experiments, ideally using triple resonance experiments with 15N and 13C-labelled proteins, that provide sequential connectivities between residues so that the resonances can be linked to particular residues and sequentially assigned. The assignment of the spectrum is ... | Heteronuclear single quantum correlation | 0.777689 |
97,514 | Economists use the extreme value theorem to guarantee that a solution to the utility maximization problem exists. That is, since the budget constraint is both bounded and closed, a solution to the utility maximization problem exists. Economists call the solution to the utility maximization problem a Walrasian demand fu... | Microeconomic theory | 0.777689 |
97,515 | The utility maximization problem serves not only as the mathematical foundation of consumer theory but as a metaphysical explanation of it as well. That is, the utility maximization problem is used by economists to not only explain what or how individuals make choices but why individuals make choices as well. The utili... | Microeconomic theory | 0.777689 |
97,516 | The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that ... | Laplacian operator | 0.777688 |
97,517 | The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic. In Minkowski space the Laplace–Beltrami operator becomes the D'Alembert operator ◻ {\displaystyle \Box } or D'Alembertian: It is the generalization of the Laplace operator in the sense tha... | Laplacian operator | 0.777688 |
97,518 | Since the CPU performs many of geometry calculations, using a faster CPU will increase the speed of a geometry-limited application. The O2's graphics is known to have slower rasterization speed than the Indigo2's Maximum IMPACT graphics boards, though the Maximum IMPACT graphics is limited to 4 MB of texture memory, wh... | SGI O2 | 0.777687 |
97,519 | There is another commuting algebra which is 'universal', underlying the Gaudin algebra for any choice of sites and weights, called the Feigin–Frenkel center. See here. Then eigenvectors of the Gaudin algebra define linear functionals on the algebra. If X {\displaystyle X} is an element of the Gaudin algebra G {\display... | Gaudin model | 0.777687 |
97,520 | When g {\displaystyle {\mathfrak {g}}} has rank greater than 1, the commuting algebra spanned by the Gaudin Hamiltonians and the identity can be expanded to a larger commuting algebra, known as the Gaudin algebra. Similarly to the Harish-Chandra isomorphism, these commuting elements have associated degrees, and in part... | Gaudin model | 0.777687 |
97,521 | There is an ODE/IM isomorphism between the Gaudin algebra (or the universal Feigin–Frenkel center), which are the 'integrals of motion' for the theory, and opers, which are ordinary differential operators, in this case on P 1 {\displaystyle \mathbb {P} ^{1}} . | Gaudin model | 0.777687 |
97,522 | There exist generalizations arising from weakening the restriction on g {\displaystyle {\mathfrak {g}}} being a strictly semi-simple Lie algebra. For example, when g {\displaystyle {\mathfrak {g}}} is allowed to be an affine Lie algebra, the model is called an affine Gaudin model. A different way to generalize is to pi... | Gaudin model | 0.777687 |
97,523 | The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be s l 2 {\displaystyle {\mathfrak {sl}}_{2}} , the two-dimensional special linear group. | Gaudin model | 0.777687 |
97,524 | Analogues of the Bethe ansatz equation can be derived for Lie algebras of higher rank. However, these are much more difficult to derive and solve than the s l 2 {\displaystyle {\mathfrak {sl}}_{2}} case. Furthermore, for g {\displaystyle {\mathfrak {g}}} of rank greater than 1, that is, all others besides s l 2 {\displ... | Gaudin model | 0.777687 |
97,525 | A solution to a Gaudin model often means determining the spectrum of the Gaudin Hamiltonian or Gaudin Hamiltonians. There are several methods of solution, including Algebraic Bethe ansatz, used by Gaudin Separation of variables, used by Sklyanin Correlation functions/opers, using a method described by Feigin, Frenkel a... | Gaudin model | 0.777687 |
97,526 | It depends on a complex parameter u {\displaystyle u} , and also on the quadratic Casimir, which is an element of the universal enveloping algebra U ( g ) {\displaystyle U({\mathfrak {g}})} , defined as This acts on representations V ( λ ) {\displaystyle V_{({\boldsymbol {\lambda }})}} by multiplying by a number depend... | Gaudin model | 0.777687 |
97,527 | Let g {\displaystyle {\mathfrak {g}}} be a semi-simple Lie algebra of finite dimension d {\displaystyle d} . Let N {\displaystyle N} be a positive integer. On the complex plane C {\displaystyle \mathbb {C} } , choose N {\displaystyle N} different points, z i {\displaystyle z_{i}} . Denote by V λ {\displaystyle V_{\lamb... | Gaudin model | 0.777687 |
97,528 | In physics, the Gaudin model, sometimes known as the quantum Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also examples of quantum spin chains. | Gaudin model | 0.777687 |
97,529 | Gross works on complex geometry, algebraic geometry, and mirror symmetry. Gross and Bernd Siebert jointly developed a program (known as the Gross–Siebert Program) for studying mirror symmetry within algebraic geometry. The Gross–Siebert program builds on an earlier, differential-geometric, proposal of Strominger, Yau, ... | Mark Gross (mathematician) | 0.777686 |
97,530 | Mark William Gross (born 30 November 1965) is an American mathematician, specializing in differential geometry, algebraic geometry, and mirror symmetry. | Mark Gross (mathematician) | 0.777686 |
97,531 | In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the Ck- case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simpler proof of the second Nash embedding theorem was obtained b... | Nash embedding theorems | 0.777681 |
97,532 | The C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by Greene & Jacobowitz (1971). (A local version of this result was proved by Élie Cartan and Maurice Janet in the 1920s.) | Nash embedding theorems | 0.777681 |
97,533 | The first theorem is for continuously differentiable (C1) embeddings and the second for embeddings that are analytic or smooth of class Ck, 3 ≤ k ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem... | Nash embedding theorems | 0.777681 |
97,534 | The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isom... | Nash embedding theorems | 0.777681 |
97,535 | The Subaru Telescope is a Ritchey-Chretien reflecting telescope. Instruments can be mounted at a Cassegrain focus below the primary mirror; at either of two Nasmyth focal points in enclosures on the sides of the telescope mount, to which light can be directed with a tertiary mirror; or at the prime focus in lieu of a s... | Subaru Telescope | 0.777679 |
97,536 | In molecular biology, a CCAAT box (also sometimes abbreviated a CAAT box or CAT box) is a distinct pattern of nucleotides with GGCCAATCT consensus sequence that occur upstream by 60–100 bases to the initial transcription site. The CAAT box signals the binding site for the RNA transcription factor, and is typically acco... | CAAT box | 0.777677 |
97,537 | The boxes were used in the United Kingdom by engineers from then General Post Office, who were responsible for UK telecommunications to trace electrical faults, i.e. to determine where a break occurred in a cable which could be several miles in length. It works on the principle of Wheatstone bridge to identify the resi... | Post office box (electricity) | 0.777677 |
97,538 | Since each of the two above transformation steps preserved this quantity, we have shown V − E + F = 1 {\displaystyle \ V-E+F=1\ } for the deformed, planar object thus demonstrating V − E + F = 2 {\displaystyle \ V-E+F=2\ } for the polyhedron. This proves the theorem. For additional proofs, see Eppstein (2013). Multiple... | Euler–Poincaré characteristic | 0.777677 |
97,539 | Each new diagonal adds one edge and one face and does not change the number of vertices, so it does not change the quantity V − E + F . {\displaystyle \ V-E+F~.} (The assumption that all faces are disks is needed here, to show via the Jordan curve theorem that this operation increases the number of faces by one.) | Euler–Poincaré characteristic | 0.777677 |
97,540 | For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold. The Euler class, in turn, relates to all other characteristic classes of vector bundles. For closed Riemannian manifolds, the Euler charact... | Euler–Poincaré characteristic | 0.777677 |
97,541 | It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and... | Euler–Poincaré characteristic | 0.777677 |
97,542 | In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by ... | Euler–Poincaré characteristic | 0.777677 |
97,543 | homest is a GPL C/C++ library for robust, non-linear (based on the Levenberg–Marquardt algorithm) homography estimation from matched point pairs (Manolis Lourakis). OpenCV is a complete (open and free) computer vision software library that has many routines related to homography estimation (cvFindHomography) and re-pro... | Homography (computer vision) | 0.777676 |
97,544 | In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image rectification, image registration, or camera motion—rotation and translation—between two images. Once camera resectioning... | Homography (computer vision) | 0.777676 |
97,545 | In coordination chemistry, a scorpionate ligand is a tridentate (three-donor-site) ligand that binds to a central atom in a fac manner. The most popular class of scorpionates are the hydrotris(pyrazolyl)borates or Tp ligands. These were also the first to become popular. These ligands first appeared in journals in 1966 ... | Scorpionate ligand | 0.777675 |
97,546 | A range of tripodal phosphines such as HC(CH2PR2)3, N(CH2CH2PPh2)3 and P(CH2CH2PMe2)3 have been reviewed. The tetra amine (tris(2-aminoethyl)amine) can be reacted with salicylaldehyde to form a ligand which can bind with three oxygens and three nitrogens to a metal. Trispyrazolylmethane (Tpm) is another class of scorpi... | Scorpionate ligand | 0.777675 |
97,547 | By replacing the nitrogen donor of a Tp ligand atoms with sulfur atoms, a class of ligands known as Tm can be made. These are related to the thioureas.¹; Several research groups including Anthony F. Hill's group have been working on this ligand class. To form NaTm {Na+ HB(mt)3−), Methimazole and sodium borohydride are ... | Scorpionate ligand | 0.777675 |
97,548 | The tricarbonyl manganese complex of trithia-9-crown-3 has the three sulfur atoms binding to the metal atom in the same place as the Cp ligand and using the same sort of orbitals for the bonding. While the geometry of the Tp ligands do not allow the formation of simple borane complexes with the metals, the geometry of ... | Scorpionate ligand | 0.777675 |
97,549 | Since the work by Wilkinson and others on ferrocene a vast amount of work has been done on cyclopentadienyl complexes. It was soon understood by many organometallic chemists that a Cp ligand is isolobal to Tp. As many insights into chemistry can be obtained by the study of a series of closely related compounds (where o... | Scorpionate ligand | 0.777675 |
97,550 | Like the other two exoplanets, HD 40307 b and HD 40307 d, orbiting the star HD 40307, HD 40307 c was discovered by measuring variations in the radial velocity of HD 40307 caused by the star's orbit around the center of mass of the planetary system. These measurements were made by the High Accuracy Radial Velocity Plane... | HD 40307 c | 0.777674 |
97,551 | One chemical process has been demonstrated to pilot plant stage but not used for production. The French CHEMEX process exploited a very slight difference in the two isotopes' propensity to change valency in oxidation/reduction, using immiscible aqueous and organic phases. An ion-exchange process was developed by the As... | Uranium enrichment | 0.777673 |
97,552 | The foundation is classified as NTEE T99—Other Philanthropy, Voluntarism, and Grantmaking Foundations N.E.C. Since 1998 the Ellison Medical Foundation has spent hundreds of millions of dollars funding fundamental research on the biology of ageing. $40 million per year was given to 25 Senior Scholars and 25 New Scholars... | Ellison Medical Foundation | 0.777672 |
97,553 | The International Organization of Biological Control (IOBC) describes integrated farming according to the UNI 11233-2009 European standard as a farming system where high quality organic food, feed, fibre and renewable energy are produced by using resources such as soil, water, air and nature as well as regulating facto... | Integrated farming | 0.777672 |
97,554 | Lexical entrainment has applications in natural language processing in computers, as well as human–human interaction. Currently, the adaptability of computers to modify their referencing to the terms of their human interlocutor is limited, so the entrainment adaptation falls to the human operator; this phenomenon is re... | Lexical entrainment | 0.77767 |
97,555 | He predicted that intrinsic to a material, which manifests negative values of effective permittivity and permeability, are several types of reversed physics phenomena. Hence, there was then a critical need for a higher-dimensional LHMs to confirm Veselago's theory, as expected. The confirmation would include reversal o... | Negative index metamaterials | 0.777669 |
97,556 | Vertex shaders are the most established and common kind of 3D shader and are run once for each vertex given to the graphics processor. The purpose is to transform each vertex's 3D position in virtual space to the 2D coordinate at which it appears on the screen (as well as a depth value for the Z-buffer). Vertex shaders... | Tessellation Control Shader | 0.777668 |
97,557 | Circa 2017, the AMD Vega microarchitecture added support for a new shader stage—primitive shaders—somewhat akin to compute shaders with access to the data necessary to process geometry. Similarly, Nvidia introduced mesh and task shaders with its Turing microarchitecture in 2018 which provide similar functionality and l... | Tessellation Control Shader | 0.777668 |
97,558 | 2D shaders act on digital images, also called textures in the field of computer graphics. They modify attributes of pixels. 2D shaders may take part in rendering 3D geometry. Currently the only type of 2D shader is a pixel shader. | Tessellation Control Shader | 0.777668 |
97,559 | The field has also seen a recent surge in interest and modification due to the results of molecular phylogenetics. George Gaylord Simpson's classic "Principles of Classification and a Classification of Mammals" (Simpson, 1945) taxonomy text laid out a systematics of mammal origins and relationships that was universally... | Mammal taxonomy | 0.777668 |
97,560 | Optical gain is the most important requirement for the realization of a semiconductor laser because it describes the optical amplification in the semiconductor material. This optical gain is due to stimulated emission associated with light emission created by recombination of electrons and holes. While in other laser m... | Gain spectroscopy of semiconductors | 0.777668 |
97,561 | Reinecke's salt is a chemical compound with the formula NH4·H2O. The dark-red crystalline compound is soluble in boiling water, acetone, and ethanol. The chromium atom is surrounded by six nitrogen atoms in an octahedral geometry. The NH3 ligands are mutually trans and the Cr–NCS groups are linear. The salt crystallize... | Reinecke's salt | 0.777667 |
97,562 | Polymers come in a wide variety of compositions, and this diversity of chemistry results in wide-ranging embrittlement mechanisms. The most common sources of polymer embrittlement include oxygen in the air, water in liquid or vapor form, ultraviolet radiation from the sun, acids, and organic solvents.One of the ways th... | Embrittlement | 0.777667 |
97,563 | Two models are used for the calculations in AFAM: the cantilever dynamics model and the contact mechanics model. Using these two models the elastic properties of the materials can be determined. All the calculations are done using LabView software. The frequency of the eigen modes of the cantilever depends, amongst oth... | Atomic force acoustic microscopy | 0.777664 |
97,564 | It first cleans the data by filtering out any junk or missing values. Then, it utilizes machine learning algorithms to find patterns across sets of users and organize them on a broad scale. | Data Management Platform | 0.777663 |
97,565 | DMPs are used for profiling, analyzing, and targeting online customers in digital marketing. They work in the following areas: Ad targeting - creating audience segments and targeting specific users with personalized ad campaigns, such as displaying advertisements for cars to users who demonstrate interest in buying a n... | Data Management Platform | 0.777663 |
97,566 | A data management platform (DMP) is a software platform used for collecting and managing data. They allow businesses to identify audience segments, which can be used to target specific users and contexts in online advertising campaigns. DMPs may use big data and artificial intelligence algorithms to process and analyze... | Data Management Platform | 0.777663 |
97,567 | Nowadays, data management has transferred over from local storage to the cloud. In the late 1990s and early 2000s, Salesforce and Amazon popularized the concept of internet-based services, which appealed to customers as it reduced in-house maintenance costs and increased flexibility in changing the needs of a business.... | Data Management Platform | 0.777663 |
97,568 | Piecewise linear models such as Markov chains, continuous-time Markov chains, the M/G/1 queue, the GI/G/1 queue and the fluid queue can be encapsulated as PDMPs with simple differential equations. | Piecewise deterministic Markov processes | 0.777663 |
97,569 | In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X {\displaystyle X} follow an ordinary normal distribution, N ( 0 , σ 2 ) {\displaystyle N(0,\sigma ^{2})} . Then, Y = | X | {\displaystyle Y=|X|} follows a half-normal distribution. Thus, the hal... | Half-normal distribution | 0.777663 |
97,570 | The platform underlying the Industrial Revolution was factories. What can be seen as a fourth revolution is centered on an intelligent computing platform. The Internet of Things, Artificial intelligence, and virtual reality are realizations of such advancements. | Intelligent information society | 0.777662 |
97,571 | Intelligent information society is a hypothetical social transformation based around communication technology infrastructure and artificial intelligence. In the Fourth Industrial Revolution, the convergence of artificial intelligence, robot technology, big data and software disrupts fields such as labor, welfare, emplo... | Intelligent information society | 0.777662 |
97,572 | Computational chemistry is simplest in gas-phase chemistry, in which interactions between molecules may be ignored as negligible. Multiple authors have researched dubnium pentachloride; calculations show it to be consistent with the periodic laws by exhibiting the properties of a compound of a group 5 element. For exam... | Dubnium | 0.777662 |
97,573 | According to the periodic law, dubnium should belong to group 5, with vanadium, niobium, and tantalum. Several studies have investigated the properties of element 105 and found that they generally agreed with the predictions of the periodic law. Significant deviations may nevertheless occur, due to relativistic effects... | Dubnium | 0.777662 |
97,574 | For element 105, they proposed joliotium (Jl) after the French physicist Frédéric Joliot-Curie, a contributor to the development of nuclear physics and chemistry; this name was originally proposed by the Soviet team for element 102, which by then had long been called nobelium. This recommendation was criticized by the ... | Dubnium | 0.777662 |
97,575 | Experimental results of the chemistry of dubnium date back to 1974 and 1976. JINR researchers used a thermochromatographic system and concluded that the volatility of dubnium bromide was less than that of niobium bromide and about the same as that of hafnium bromide. It is not certain that the detected fission products... | Dubnium | 0.777662 |
97,576 | The long-standing dispute was resolved in 1993 by an official investigation of the discovery claims by the Transfermium Working Group, formed by the International Union of Pure and Applied Chemistry and the International Union of Pure and Applied Physics, resulting in credit for the discovery being officially shared be... | Dubnium | 0.777662 |
97,577 | In 1985, the International Union of Pure and Applied Chemistry (IUPAC) and the International Union of Pure and Applied Physics (IUPAP) formed a Transfermium Working Group (TWG) to assess discoveries and establish final names for the controversial elements. The party held meetings with delegates from the three competing... | Dubnium | 0.777662 |
97,578 | When LBL first announced their synthesis of element 105, they proposed that the new element be named hahnium (Ha) after the German chemist Otto Hahn, the "father of nuclear chemistry", thus creating an element naming controversy.In the early 1970s, both teams reported synthesis of the next element, element 106, but did... | Dubnium | 0.777662 |
97,579 | In molecular biology, entericidins are bacterial antidote/toxin peptides. The entericidin locus is activated in the stationary phase of growth under high osmolarity conditions by rho-S and simultaneously repressed by the osmoregulatory EnvZ/OmpR signal transduction pathway. The entericidin locus encodes tandem paralogo... | Entericidin | 0.777661 |
97,580 | Oolitic aragonite sand is composed of the calcium carbonate mineral, aragonite, with an egg-like shape ("oolitic" from the Ancient Greek word ᾠόν for "egg") and sand grain size. This sand type forms in tropical waters through precipitation, sedimentation, and microbial activity, and is indicative of high energy environ... | Oolitic aragonite sand | 0.777659 |
97,581 | Never covered by ice during the last ice age, the area lacks the characteristic glacial deposits known as drift. Its landscape is characterized by steep hills, forested ridges, deeply carved river valleys, and karst geology with spring-fed waterfalls and cold-water trout streams. Ecologically, the Driftless Area's flor... | Driftless Area | 0.777659 |
97,582 | If it were, then the center of H has to contain C, but the center of H is R. Therefore, the only finite-dimensional division algebra over C is C itself. This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are R, C, H, and the (non-associative) algebra O. Pontry... | Frobenius theorem (real division algebras) | 0.777657 |
97,583 | The fact that D is generated by e1, ..., en subject to the above relations means that D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are division algebras are Cℓ0, Cℓ1 and Cℓ2. As a consequence, the only commutative division algebras are R and C. Also note that H is not ... | Frobenius theorem (real division algebras) | 0.777657 |
97,584 | The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra. | Frobenius theorem (real division algebras) | 0.777657 |
97,585 | So V is the subset of all a with tr(a) = 0. In particular, it is a vector subspace. The rank–nullity theorem then implies that V has dimension n - 1 since it is the kernel of tr: D → R {\displaystyle \operatorname {tr} :D\to \mathbf {R} } . Since R and V are disjoint (i.e. they satisfy R ∩ V = { 0 } {\displaystyle \mat... | Frobenius theorem (real division algebras) | 0.777657 |
97,586 | Since zj ∈ C\R, the polynomials Q(zj; x) are all irreducible over R. By the Cayley–Hamilton theorem, p(a) = 0 and because D is a division algebra, it follows that either a − ti = 0 for some i or that Q(zj; a) = 0 for some j. The first case implies that a is real. In the second case, it follows that Q(zj; x) is the mini... | Frobenius theorem (real division algebras) | 0.777657 |
97,587 | The key to the argument is the following Claim. The set V of all elements a of D such that a2 ≤ 0 is a vector subspace of D of dimension n - 1. Moreover D = R ⊕ V as R-vector spaces, which implies that V generates D as an algebra.Proof of Claim: Let m be the dimension of D as an R-vector space, and pick a in D with cha... | Frobenius theorem (real division algebras) | 0.777657 |
97,588 | Assume that n > 2. Let u = e1e2en. It is easy to see that u2 = 1 (this only works if n > 2). If D were a division algebra, 0 = u2 − 1 = (u − 1)(u + 1) implies u = ±1, which in turn means: en = ∓e1e2 and so e1, ..., en−1 generate D. This contradicts the minimality of W. | Frobenius theorem (real division algebras) | 0.777657 |
97,589 | Hence it is isomorphic to C. If n = 2, it has been shown above that D is generated by 1, e1, e2 subject to the relations e 1 2 = e 2 2 = − 1 , e 1 e 2 = − e 2 e 1 , ( e 1 e 2 ) ( e 1 e 2 ) = − 1. {\displaystyle e_{1}^{2}=e_{2}^{2}=-1,\quad e_{1}e_{2}=-e_{2}e_{1},\quad (e_{1}e_{2})(e_{1}e_{2})=-1.} These are precisely t... | Frobenius theorem (real division algebras) | 0.777657 |
97,590 | For a, b in V define B(a, b) = (−ab − ba)/2. Because of the identity (a + b)2 − a2 − b2 = ab + ba, it follows that B(a, b) is real. Furthermore, since a2 ≤ 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive definite symmetric bilinear form, in other words, an inner product on V. Let W be a subspace of V that gener... | Frobenius theorem (real division algebras) | 0.777657 |
97,591 | Let D be the division algebra in question. Let n be the dimension of D. We identify the real multiples of 1 with R. When we write a ≤ 0 for an element a of D, we imply that a is contained in R. We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplicati... | Frobenius theorem (real division algebras) | 0.777657 |
97,592 | In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C ... | Frobenius theorem (real division algebras) | 0.777657 |
97,593 | With the same table, the query SELECT * FROM T WHERE C1 = 1 will result in all the elements of all the rows where the value of column C1 is '1' being shown – in relational algebra terms, a selection will be performed, because of the WHERE clause. This is also known as a Horizontal Partition, restricting rows output by ... | SQL query | 0.777657 |
97,594 | Given a table T, the query SELECT * FROM T will result in all the elements of all the rows of the table being shown. With the same table, the query SELECT C1 FROM T will result in the elements from the column C1 of all the rows of the table being shown. This is similar to a projection in relational algebra, except that... | SQL query | 0.777657 |
97,595 | Some non-standard implementations of SELECT can have persistent effects, such as the SELECT INTO syntax provided in some databases.Queries allow the user to describe desired data, leaving the database management system (DBMS) to carry out planning, optimizing, and performing the physical operations necessary to produce... | SQL query | 0.777657 |
97,596 | In topology, a branch of mathematics, a lamination is a: "topological space partitioned into subsets" decoration (a structure or property at a point) of a manifold in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel.A lamination of a surface is a ... | Lamination (topology) | 0.777656 |
97,597 | Sirius is mainly used to design complex systems (industrial systems or IT applications). The first use case was Capella, a Systems Engineering workbench contributed to the Eclipse Working Group PolarSys in 2014 by Thales. The Sirius gallery on the project website lists some of the projects using Sirius. | Eclipse Sirius | 0.777654 |
97,598 | Rosen earned a bachelor's degree from Brandeis University in 1959 and a PhD from Princeton University in 1963 under John Coleman Moore with thesis Representations of twisted group rings. He is a mathematics professor at Brown University. Rosen is known for his textbooks, especially for the book with co-author Kenneth I... | Michael Rosen (mathematician) | 0.777653 |
97,599 | Michael Ira Rosen (born March 7, 1938) is an American mathematician who works on algebraic number theory, arithmetic theory of function fields, and arithmetic algebraic geometry. | Michael Rosen (mathematician) | 0.777653 |
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