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80,800 | Hence the notion of a localizing subcategory is equivalent to the notion of a hereditary torsion class. If A {\displaystyle {\mathcal {A}}} is a Grothendieck category and C {\displaystyle {\mathcal {C}}} a localizing subcategory, then C {\displaystyle {\mathcal {C}}} and the quotient category A / C {\displaystyle {\mat... | Localizing subcategory | 0.784825 |
80,801 | The theoretical basis for compression is provided by information theory and, more specifically, Shannon's source coding theorem; domain-specific theories include algorithmic information theory for lossless compression and rate–distortion theory for lossy compression. These areas of study were essentially created by Cla... | Source coding | 0.784824 |
80,802 | It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptiv... | Source coding | 0.784824 |
80,803 | Their definition consists of three categories: techniques for processing, the application of statistical and mathematical methods to decision-making, and the simulation of higher-order thinking through computer programs.The term is commonly used as a synonym for computers and computer networks, but it also encompasses ... | Information technologist | 0.784824 |
80,804 | The Modane Underground Laboratory (LSM) (French: Laboratoire Souterrain de Modane; also known as the Fréjus Underground Laboratory) is a subterranean particle physics laboratory located within the Fréjus Road Tunnel near Modane, France. It is jointly operated by the French National Center for Scientific Research and th... | Modane Underground Laboratory | 0.784824 |
80,805 | Other encryption techniques like elliptic curve cryptography and symmetric key encryption are also vulnerable to quantum computing.While quantum computing could be a threat to encryption security in the future, quantum computing as it currently stands is still very limited. Quantum computing currently is not commercial... | Data encryption | 0.784822 |
80,806 | It would take a supercomputer anywhere between weeks to months to factor in this key. However, quantum computing can use quantum algorithms to factor this semiprime number in the same amount of time it takes for normal computers to generate it. This would make all data protected by current public-key encryption vulnera... | Data encryption | 0.784822 |
80,807 | For example, the original encryption key, DES (Data Encryption Standard), was 56 bits, meaning it had 2^56 combination possibilities. With today's computing power, a 56-bit key is no longer secure, being vulnerable to brute force attacks.Quantum computing utilizes properties of quantum mechanics in order to process lar... | Data encryption | 0.784822 |
80,808 | Created in 1978, it is still used today for applications involving digital signatures. Using number theory, the RSA algorithm selects two prime numbers, which help generate both the encryption and decryption keys.A publicly available public-key encryption application called Pretty Good Privacy (PGP) was written in 1991... | Data encryption | 0.784822 |
80,809 | Novaquark formed in 2014 with Ubisoft, Sony, Apple and Aldebaran Robotics former employees. Jean-Christophe Baillie, Novaquark's original CEO, claims to have originated the idea for Dual Universe in 2011.The game was revealed in May 2016 in its pre-alpha state, with the subsequent release of a brief teaser trailer foll... | Dual Universe | 0.784819 |
80,810 | The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then further researched.Algebraic coding theory is basically divided into two major types of codes: Linear block codes Convolutional codesIt analyzes the following three properties ... | Analog encryption | 0.784818 |
80,811 | Linear block codes have the property of linearity, i.e. the sum of any two codewords is also a code word, and they are applied to the source bits in blocks, hence the name linear block codes. There are block codes that are not linear, but it is difficult to prove that a code is a good one without this property.Linear b... | Analog encryption | 0.784818 |
80,812 | Information is encoded analogously in the neural networks of brains, in analog signal processing, and analog electronics. Aspects of analog coding include analog error correction, analog data compression and analog encryption. | Analog encryption | 0.784818 |
80,813 | ISO/IEC 24744 Software Engineering—Metamodel for Development Methodologies, is a power type-based metamodel for software development methodologies. SPEM 2.0 by the Object Management Group. Soft systems methodology - a general method for improving management processes. Method engineering - a general method for improving... | Software development life cycle | 0.784815 |
80,814 | As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). Each student is described by three criteria: level in Phys... | Dominance-based rough set approach | 0.784815 |
80,815 | Consider the following problem of high school students’ evaluations: Each object (student) is described by three criteria q 1 , q 2 , q 3 {\displaystyle q_{1},q_{2},q_{3}\,\!} , related to the levels in Mathematics, Physics and Literature, respectively. According to the decision attribute, the students are divided into... | Dominance-based rough set approach | 0.784814 |
80,816 | The Morris water navigation task, also known as the Morris water maze (not to be confused with water maze), is a behavioral procedure mostly used with rodents. It is widely used in behavioral neuroscience to study spatial learning and memory. It enables learning, memory, and spatial working to be studied with great acc... | Morris water navigation task | 0.784813 |
80,817 | A 2015 article by Kotaku noted that "their games, wonderful in many ways as they may be, have been accompanied by an undercurrent of fan anger" over widespread bugs and glitches. The article concluded that Telltale's support forums "paint a portrait of a publisher that is constantly releasing buggy and even outright br... | Telltale Tool | 0.784813 |
80,818 | Until 2016, Telltale Tool did not have a physics engine, meaning that if a scene required an object to fall, this had to be animated by hand, taking time from other more productive activities. This also prevented them from using elements like dynamic lighting, and required them to develop lighting models using 3D model... | Telltale Tool | 0.784813 |
80,819 | Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry (2nd revised ed.). Springer-Verlag. | Quadtree | 0.784811 |
80,820 | Polyferrocenes are polymers containing ferrocene units. Ferrocene offers many advantages over pure hydrocarbons when used as a building block of macromolecular chemistry. The variety of possible substitutions at the ferrocene parent body results in a multitude of accessible polymers with interesting electronic and phot... | Polyferrocenes | 0.78481 |
80,821 | Given an integral domain D, let Q = ∑ i = 0 n a i x i {\displaystyle Q=\sum _{i=0}^{n}a_{i}x^{i}} be an element of D, the polynomial ring with coefficients in D. Suppose there exists a prime ideal p of D such that ai ∈ p for each i ≠ n, an ∉ p, and a0 ∉ p2, where p2 is the ideal product of p with itself.Then Q cannot b... | Eisenstein criterion | 0.784809 |
80,822 | In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials... | Eisenstein criterion | 0.784809 |
80,823 | The Journal of Time Series Analysis is a bimonthly peer-reviewed academic journal covering mathematical statistics as it relates to the analysis of time series data. It was established in 1980 and is published by John Wiley & Sons. The editor-in-chief is Robert Taylor (University of Essex). According to the Journal Cit... | Journal of Time Series Analysis | 0.784806 |
80,824 | ( n + 1 ) ! {\displaystyle \dim(TL_{n}(\delta ))={\frac {(2n)!}{n!(n+1)!}}} The Temperley–Lieb algebra T L n ( δ ) {\displaystyle TL_{n}(\delta )} is a subalgebra of the Brauer algebra B n ( δ ) {\displaystyle {\mathfrak {B}}_{n}(\delta )} , and therefore also of the partition algebra P n ( δ ) {\displaystyle P_{n}(\de... | Temperley-Lieb algebra | 0.784806 |
80,825 | Erythrocyte deformability refers to the ability of erythrocytes (red blood cells, RBC) to change shape under a given level of applied stress, without hemolysing (rupturing). This is an important property because erythrocytes must change their shape extensively under the influence of mechanical forces in fluid flow or w... | Erythrocyte deformability | 0.784804 |
80,826 | Shape change of erythrocytes under applied forces (i.e., shear forces in blood flow) is reversible and the biconcave-discoid shape, which is normal for most mammals, is maintained after the removal of the deforming forces. In other words, erythrocytes behave like elastic bodies, while they also resist to shape change u... | Erythrocyte deformability | 0.784804 |
80,827 | In mathematics, the Denjoy–Luzin–Saks theorem states that a function of generalized bounded variation in the restricted sense has a derivative almost everywhere, and gives further conditions of the set of values of the function where the derivative does not exist. N. N. Luzin and A. Denjoy proved a weaker form of the t... | Denjoy–Luzin–Saks theorem | 0.784803 |
80,828 | In that case the discount factor, and thus the present value, of a cash flow at time T is given by the integral of the continuously compounded rate r(t): PV = FV ⋅ exp ( − ∫ 0 T r ( t ) d t ) {\displaystyle {\text{PV}}={\text{FV}}\cdot \exp \left(-\int _{0}^{T}r(t)\,dt\right)} Indeed, a key reason for using continuou... | Time-value of money | 0.784803 |
80,829 | Ordinary and partial differential equations (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics. While time value of money can be understood without using the framework of differential equations, the added sophi... | Time-value of money | 0.784803 |
80,830 | The word ‘bionic architecture’ is derived from the Greek word ‘bios’ (life) as well as the English word ‘technics’ (to study). The term was originally used to describe the scientific trend of ‘transferring technologies into life-forms’. The term ‘bionic’ was first used in 1958 by U.S army colonel, Jack E. Steele and So... | Bionic architecture | 0.784803 |
80,831 | The built environment contributes to a majority of waste, material production, energy use and fossil fuel emissions. Thus, there is a responsibility to develop a more efficient and ecologically friendly construction design that still allows for daily activities in society to take place. This is achieved through the use... | Bionic architecture | 0.784803 |
80,832 | Rudin's conjecture is a mathematical hypothesis (in additive combinatorics and elementary number theory) concerning an upper bound for the number of squares in finite arithmetic progressions. The conjecture, which has applications in the theory of trigonometric series, was first stated by Walter Rudin in his 1960 paper... | Rudin's conjecture | 0.784803 |
80,833 | In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a... | Jacobson's conjecture | 0.784802 |
80,834 | Jacobson's conjecture has been verified for particular types of Noetherian rings: Commutative Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the Krull intersection theorem. Fully bounded Noetherian rings Noetherian rings with Krull dimension 1 Noetherian rings satisfying the second layer c... | Jacobson's conjecture | 0.784802 |
80,835 | In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where r {\dis... | Bloch wave | 0.7848 |
80,836 | Bloch's theorem is as follows: For electrons in a perfect crystal, there is a basis of wave functions with the following two properties: each of these wave functions is an energy eigenstate, each of these wave functions is a Bloch state, meaning that this wave function ψ {\displaystyle \psi } can be written in the form... | Bloch wave | 0.7848 |
80,837 | Let T ^ n 1 , n 2 , n 3 {\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}} denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3 (as above, nj are integers). The following fact is helpful for the proof of Bloch's theorem: Finally, we are ready for the main proof of Bloch's theorem w... | Bloch wave | 0.7848 |
80,838 | All observed red dwarfs contain "metals", which in astronomy are elements heavier than hydrogen and helium. The Big Bang model predicts that the first generation of stars should have only hydrogen, helium, and trace amounts of lithium, and hence would be of low metallicity. With their extreme lifespans, any red dwarfs ... | Red dwarf stars | 0.7848 |
80,839 | In organic chemistry, an alkyne is an unsaturated hydrocarbon containing at least one carbon—carbon triple bond. The simplest acyclic alkynes with only one triple bond and no other functional groups form a homologous series with the general chemical formula CnH2n−2. Alkynes are traditionally known as acetylenes, althou... | C2R2 | 0.784799 |
80,840 | They are mainly used for large family cars and executive cars, as well as large MPVs, although the 2.0 L is also used for some hot hatch models. The DW started with an SOHC 2-valve design between 2.0 and 2.0 L (1,968 and 1,997 cc), later receiving DOHC and four valves per cylinder upon the introduction of the 2.2 L in ... | PSA EW/DW engine | 0.784799 |
80,841 | The DW10 was used as the basis for the new family of Duratorq Diesel engines co-developed with Ford, and it is used in the Focus, Kuga, Mondeo, C-Max and Volvo C30/S40/V50, besides various Citroën and Peugeot passenger models. The DOHC 16-valve powerplants were mated to a second generation common rail injection system ... | PSA EW/DW engine | 0.784799 |
80,842 | In particular, Alexei Abrikosov showed that Ginzburg–Landau theory predicts the division of superconductors into the two categories now referred to as Type I and Type II. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize in Physics for their work (Landau having died in 1968). | History of superconductivity | 0.784797 |
80,843 | Abrikosov, "On the magnetic properties of superconductors of the second group," Soviet Physics JETP 5, 1174 (1957) Little, W. A.; Parks, R. D. (1962-07-01). "Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder". | History of superconductivity | 0.784797 |
80,844 | In 1986, J. Georg Bednorz and K. Alex Mueller discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987) and was the first of the high-temperature superconductors. It was shortly found (by Ching-Wu Chu) that replacing the lant... | History of superconductivity | 0.784797 |
80,845 | For his research, he was awarded the Nobel Prize in Physics in 1913. Onnes conducted an experiment, in 1912, on the usability of superconductivity. Onnes introduced an electric current into a superconductive ring and removed the battery that generated it. | History of superconductivity | 0.784797 |
80,846 | The complete microscopic theory of superconductivity was finally proposed in 1957 by John Bardeen, Leon N. Cooper, and Robert Schrieffer. This BCS theory explained the superconducting current as a superfluid of Cooper pairs, pairs of electrons interacting through the exchange of phonons. For this work, the authors were... | History of superconductivity | 0.784797 |
80,847 | Superconductivity is the phenomenon of certain materials exhibiting zero electrical resistance and the expulsion of magnetic fields below a characteristic temperature. The history of superconductivity began with Dutch physicist Heike Kamerlingh Onnes's discovery of superconductivity in mercury in 1911. Since then, many... | History of superconductivity | 0.784797 |
80,848 | This phenomenon, now called the Josephson effect, is exploited by superconducting devices such as SQUIDs. It is used in the most accurate available measurements of the magnetic flux quantum h/2e, and thus (coupled with the quantum Hall resistivity) for Planck's constant h. Josephson was awarded the Nobel Prize in Physi... | History of superconductivity | 0.784797 |
80,849 | Physics Letters. Elsevier BV. 1 (7): 251–253. | History of superconductivity | 0.784797 |
80,850 | We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. Sen's method of informational broadening allows social choice theory to escape the objections of Robbins, which looked as though they would permanently harm social choice theory. Additionally, si... | Voting theory | 0.784794 |
80,851 | CA-groups were important in the context of the classification of finite simple groups. Michio Suzuki showed that every finite, simple, non-abelian, CA-group is of even order. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to ... | CA-group | 0.784794 |
80,852 | Locally finite CA-groups were classified by several mathematicians from 1925 to 1998. First, finite CA-groups were shown to be simple or solvable in (Weisner 1925). Then in the Brauer–Suzuki–Wall theorem (Brauer, Suzuki & Wall 1958), finite CA-groups of even order were shown to be Frobenius groups, abelian groups, or t... | CA-group | 0.784794 |
80,853 | In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theor... | CA-group | 0.784794 |
80,854 | The Whetstone Compiler was built at the Atomic Power Division of the English Electric Company in Whetstone, Leicestershire, England, hence its name. Dr. B.A. Wichman at NPL produced a set of 42 simple ALGOL 60 statements, which in a suitable combination matched the execution statistics. | Whetstone (benchmark) | 0.784793 |
80,855 | The Whetstone benchmark is a synthetic benchmark for evaluating the performance of computers. It was first written in Algol 60 in 1972 at the Technical Support Unit of the Department of Trade and Industry (later part of the Central Computer and Telecommunications Agency) in the United Kingdom. It was derived from stati... | Whetstone (benchmark) | 0.784793 |
80,856 | This method is known as the Caldeira–Leggett model, or harmonic bath model. To proceed and obtain explicit solutions, the path integral formulation description of quantum mechanics is typically employed. A large part of the power behind this method is the fact that harmonic oscillators are relatively well-understood co... | Open quantum system | 0.784791 |
80,857 | (2008). Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Amsterdam, Boston, London, New York: Elsevier Science. | Open quantum system | 0.784791 |
80,858 | (2012). "On the Origin of Nuclear Clustering". Progress of Theoretical Physics Supplement. | Open quantum system | 0.784791 |
80,859 | In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the syst... | Open quantum system | 0.784791 |
80,860 | In molecular biology, the group I pyridoxal-dependent decarboxylases, also known as glycine cleavage system P-proteins, are a family of enzymes consisting of glycine cleavage system P-proteins (glycine dehydrogenase (decarboxylating)) EC 1.4.4.2 from bacterial, mammalian and plant sources. The P protein is part of the ... | Group I pyridoxal-dependent decarboxylases | 0.784788 |
80,861 | The 1964 TV series My Living Doll features a robot, portrayed by Julie Newmar, who is similarly described. The film Her (2013) depicts an Artificial Intelligence assistant called Samantha, whom the protagonist, Theodore, falls in love with until her intelligence surpasses human comprehension and she leaves to fulfil he... | Gynoid | 0.784787 |
80,862 | Terrain types: Simple terrain with no more than 10 degree slopes. Building effects: Building downwash algorithms are included. Meteorological data: The model includes a preprocessor to produce the meteorological parameters needed to characterize the atmospheric turbulence as well as to produce wind speed and direction ... | DISPERSION21 | 0.784786 |
80,863 | It is for instance possible to construct counterexamples where the domain is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the maximum norm and R m {\displaystyle \mathbb {R} ^{m}} carries the Euclidean norm. More generally, the theorem fails for R m {\displaystyle \mathbb {R} ^{m}} equipped with any ℓ p {\disp... | Kirszbraun theorem | 0.784786 |
80,864 | In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and f: U → H 2 {\displaystyle f:U\rightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map F: H 1 → H 2 {\di... | Kirszbraun theorem | 0.784786 |
80,865 | The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem. | Kirszbraun theorem | 0.784786 |
80,866 | Mark Fox, Optical properties of solids, Oxford, New York, 2001. Hartmut Haug, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientific, 2004. https://web.archive.org/web/20100728030241/http://www.rle.mit.edu/sclaser/6.973%20lecture%20notes/Lecture%2013c.pdf Shun Lien Chuang, Physics o... | Quantum-confined Stark effect | 0.784784 |
80,867 | The LL chondrites are a group of stony meteorites, the least abundant group of the ordinary chondrites, accounting for about 10–11% of observed ordinary-chondrite falls and 8–9% of all meteorite falls (see meteorite fall statistics). The ordinary chondrites are thought to have originated from three parent asteroids, wi... | LL chondrite | 0.784784 |
80,868 | Let A be a Grothendieck category (an AB5 category with a generator), G a generator of A and R be the ring of endomorphisms of G; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(G,X). Then the Gabriel–Popescu theorem states that S is full and faithful and has an exact l... | Gabriel-Popescu theorem | 0.784783 |
80,869 | In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of th... | Gabriel-Popescu theorem | 0.784783 |
80,870 | : 705–707 The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being ... | Relativistic quantum theory | 0.784783 |
80,871 | Manipulation now yields M X = N X Λ {\displaystyle MX=NX\Lambda } where X {\displaystyle X} is a matrix having as columns the generalized eigenvectors and Λ {\displaystyle \Lambda } is a diagonal matrix of the generalized eigenvalues. Now premultiplication with X T {\displaystyle X^{\textsf {T}}} gives the final result... | Positive definite matrix | 0.784782 |
80,872 | : 19 The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields,: 52 vector fields (e.g. the electromagnetic field), and even strings. However, creation and annihilation operators are only well defined in the simplest theories ... | Relativistic quantum theory | 0.784782 |
80,873 | Using first-order perturbation theory, he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy s... | Relativistic quantum theory | 0.784782 |
80,874 | For any morphism f in E {\displaystyle \mathbf {E} } there is an associated "pullback functor" f ∗ := − ↦ f × − → f {\displaystyle f^{*}:=-\mapsto f\times -\rightarrow f} which is key in the proof of the theorem. For any other morphism g in E {\displaystyle \mathbf {E} } which shares the same codomain as f, their produ... | Fundamental theorem of topos theory | 0.784781 |
80,875 | In mathematics, The fundamental theorem of topos theory states that the slice E / X {\displaystyle \mathbf {E} /X} of a topos E {\displaystyle \mathbf {E} } over any one of its objects X {\displaystyle X} is itself a topos. Moreover, if there is a morphism f: A → B {\displaystyle f:A\rightarrow B} in E {\displaystyle \... | Fundamental theorem of topos theory | 0.784781 |
80,876 | Consider a pair of ground formulas ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } whose extensions {\displaystyle } and {\displaystyle } (where the underscore here denotes the null context) are objects of the base topos. Then ϕ {\displaystyle \phi } implies ψ {\displaystyle \psi } if and only if there is a monic... | Fundamental theorem of topos theory | 0.784781 |
80,877 | But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane H2 by a certain cocompact group G that acts freely on H2 by isometries. This gives the Klein quartic a Riemannian m... | Klein quartic | 0.78478 |
80,878 | Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties. Originally, the "Klein... | Klein quartic | 0.78478 |
80,879 | In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 automorphisms if orientation may be reversed. As such, the Klein quartic... | Klein quartic | 0.78478 |
80,880 | Namely, Γ(I) is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators i,j and relations i 2 = j 2 = η , i j = − j i . {\displaystyle i^{2}=j^{2}=\eta ,\qquad ij=-ji.} One chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathc... | Klein quartic | 0.78478 |
80,881 | The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal I = ⟨ η − 2 ⟩ {\displaystyle I=\langle \eta -2\rangle } in the ring of algebraic integers Z(η) of the field Q(η) where... | Klein quartic | 0.78478 |
80,882 | The Klein quartic is related to various other Riemann surfaces. Geometrically, it is the smallest Hurwitz surface (lowest genus); the next is the Macbeath surface (genus 7), and the following is the First Hurwitz triplet (3 surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being ... | Klein quartic | 0.78478 |
80,883 | The Klein quartic can be viewed as a projective algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates on P2(C): x 3 y + y 3 z + z 3 x = 0. {\displaystyle x^{3}y+y^{3}z+z^{3}x=0.} The locus of this equation in P2(C) is the original Riemannian surface that Klein... | Klein quartic | 0.78478 |
80,884 | The Klein quartic can be obtained as the quotient of the hyperbolic plane by the action of a Fuchsian group. The fundamental domain is a regular 14-gon, which has area 8 π {\displaystyle 8\pi } by the Gauss-Bonnet theorem. This can be seen in the adjoining figure, which also includes the 336 (2,3,7) triangles that tess... | Klein quartic | 0.78478 |
80,885 | A quotient of any tiling in the (2,3,7) family can be used (and will have the same automorphism group); of these, the two regular tilings are the tiling by 24 regular hyperbolic heptagons, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56 equilateral triangles, each of degree 7 (meeting at 24 vertice... | Klein quartic | 0.78478 |
80,886 | It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically t... | Klein quartic | 0.78478 |
80,887 | The following outline is provided as an overview of and topical guide to ecology: Ecology – scientific study of the distribution and abundance of living organisms and how the distribution and abundance are affected by interactions between the organisms and their environment. The environment of an organism includes both... | Outline of ecology | 0.78478 |
80,888 | Human ecology – Study of the relationship between humans and their natural, social, and built environments – Animal ecology – Scientific study of the relationships between living animals and their environment – Insect ecology – The study of how insects interact with the surrounding environment – Microbial ecology – Stu... | Outline of ecology | 0.78478 |
80,889 | History of ecology – Aspect of history covering the study of ecology History of human ecology | Outline of ecology | 0.78478 |
80,890 | Global ecology – Global sum of all ecosystems on Earth – Macroecology – Study of relationships between organisms and their environment at large spatial scales – Microecology – Microbial ecology or ecology of a microhabitat. | Outline of ecology | 0.78478 |
80,891 | Field ecology – Collection of information outside a laboratory, library or workplace setting – Quantitative ecology – application of advanced mathematical and statistical tools to perform ecological researchPages displaying wikidata descriptions as a fallback – Theoretical ecology – scientific discipline devoted to the... | Outline of ecology | 0.78478 |
80,892 | Ecology is a broad discipline comprising many subdisciplines. The field of ecology can be subdivided according to several classification schemes: | Outline of ecology | 0.78478 |
80,893 | Ecology has also inspired (and lent its name to) other non-biological disciplines such as: Media ecology Industrial ecology Information ecology | Outline of ecology | 0.78478 |
80,894 | Ecology can also be classified on the basis of: the primary kinds of organism under study, e.g. animal ecology, plant ecology, insect ecology; the biomes principally studied, e.g. forest ecology, grassland ecology, desert ecology, benthic ecology, marine ecology, urban ecology; the geographic or climatic area, e.g. arc... | Outline of ecology | 0.78478 |
80,895 | Benthic ecology – The study of the interaction of sea-floor organisms with each other and with the environment – Desert ecology – The study of interactions between both biotic and abiotic components of desert environments – Forest ecology – Study of interactions between the biota and environment in forets – Grassland e... | Outline of ecology | 0.78478 |
80,896 | Behavioral ecology – Study of the evolutionary basis for animal behavior due to ecological pressures – Chemical ecology – Study of chemically-mediated interactiond between living organisms – which deals with the ecological role of biological chemicals used in a wide range of areas including defense against predators an... | Outline of ecology | 0.78478 |
80,897 | Arranged from lowest to highest level of organisation: Autecology – ecological study of a single individual, population, or speciesPages displaying wikidata descriptions as a fallback – the study of individual organisms of a single species in relation to their environment; Synecology – Associated populations of species... | Outline of ecology | 0.78478 |
80,898 | Arctic ecology – Study of the relationships between biotic and abiotic factors in the arctic – Polar ecology – Relationship between plants and animals and a polar environment – Tropical ecology – Study of the relationships between the biotic and abiotic components of the tropics. | Outline of ecology | 0.78478 |
80,899 | Agroecology – Study of ecological processes in agriculture – Applied ecology – sub-field within ecology that considers the application of the science of ecology to real-world (usually management) questionsPages displaying wikidata descriptions as a fallback – the practice of employing ecological principles and understa... | Outline of ecology | 0.78478 |
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